prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What kind of ultracold neutral plasmas does this study focus on?

Read the following text and answer briefly.

Context 1: \section{Introduction}

Ultracold neutral plasmas studied in the laboratory offer access to a regime of plasma physics that scales to describe thermodynamic aspects of important high-energy-density systems, including strongly coupled astrophysical plasmas \cite{VanHorn,Burrows}, as well as terrestrial sources of neutrons \cite{Hinton,Ichimaru_fusion,Atzeni,Boozer} and x-ray radiation \cite{Rousse,Esarey}   Yet, under certain conditions, low-temperature laboratory plasmas evolve with dynamics that are governed by the quantum mechanical properties of their constituent particles, and in some cases by coherence with an external electromagnetic field    

The relevance of ultracold plasmas to such a broad scope of problems in classical and quantum many-body physics has given rise to a great deal of experimental and theoretical research on these systems since their discovery in the late 90s   A series of reviews affords a good overview of progress in the last twenty years \cite{Gallagher,Killian_Science,PhysRept,Lyon}   Here, we focus on the subset of ultracold neutral plasmas that form via kinetic rate processes from state-selected Rydberg gases, and emphasize in particular the distinctive dynamics found in the evolution of molecular ultracold plasmas   

While molecular beam investigations of threshold photoionization spectroscopy had uncovered relevant effects a few years earlier \cite{Scherzer,Alt}, the field of ultracold plasma physics began in earnest with the 1999 experiment of Rolston and coworkers on metastable xenon atoms cooled in a magneto optical trap (MOT) \cite{Killian}   

This work and many subsequent efforts tuned the photoionization energy as a means to form a plasma of very low electron temperature built on a strongly coupled cloud of ultracold ions   Experiment and theory soon established that fast processes associated with disorder-induced heating and longer-time electron-ion collisional rate processes act to elevate the ion temperatures to around one degree Kelvin, and constrain the effective initial electron temperature to a range above 30 K \cite{Kuzmin,Hanson,Laha}   

This apparent limit on the thermal energy of the electrons can be more universally expressed for an expanding plasma by saying that the electron correlation parameter, $\Gamma_e$, does not exceed 0 25, where, 
\begin{equation}
\Gamma_e = \frac{e^2}{4\pi \epsilon_0 a_{ws}}\frac{1}{k_B T_e}
\label{eqn:gamma_e}
\end{equation}
defines the ratio of the average unscreened electron-electron potential energy to the electron kinetic energy   $a_{ws}$ is the Wigner-Seitz radius, related to the electron density by, $\rho_e = 1/(\frac{4}{3} \pi a_{ws}^3)$   These plasmas of weakly coupled electrons and strongly coupled ions have provided an important testing ground for ion transport theory and the study of electron-ion collision physics \cite{Strickler} 

Soon after the initial reports of ultracold plasmas formed by direct photoionization, a parallel effort began with emphasis on the plasma that forms spontaneously by Penning ionization and electron-impact avalanche in a dense ultracold Rydberg gas \cite{Mourachko}   This process affords less apparent control of the initial electron temperature   But, pulsed field-ionization measurements soon established that the photoionized plasma and that formed by the avalanche of a Rydberg gas both evolve to quasi-equilibria of electrons, ions and high-Rydberg neutrals \cite{Rolston_expand,Gallagher}   

Early efforts to understand plasmas formed by Rydberg gas avalanche paid particular attention to the process of initiation   Evolution to plasma in effusive atomic beams was long known for high-Rydberg gases of caesium and well explained by coupled rate equations \cite{Vitrant}   But, low densities and ultracold velocity distributions were thought to exclude Rydberg-Rydberg collisional mechanisms in a MOT   

In work on ultracold Rydberg gases of Rb and Cs, Gallagher, Pillet and coworkers describe the initial growth of electron signal by a model that includes ionization by blackbody radiation and collisions with a background of uncooled Rydberg atoms \cite{Mourachko,Gallagher,Li,Comparat,Tanner}  This picture was subsequently refined to include many-body excitation and autoionization, as well as attractive dipole-dipole interactions \cite{Viteau,Pillet}, later confirmed by experiments at Rice \cite{Mcquillen}   

The Orsay group also studied the effect of adding Rydberg atoms to an established ultracold plasma   They found that electron collisions in this environment completely ionize added atoms, even when selected to have deep binding energies \cite{Vanhaecke}   They concluded from estimates of electron trapping efficiency that the addition of Rydberg atoms does not significantly alter the electron temperature of the plasma   

Tuning pair distributions by varying the wavelength of the excitation laser, Weidem\"uller and coworkers confirmed the mechanical effects of van der Waals interactions on the rates of Penning ionization in ultracold $^{87}$Rb Rydberg gases \cite{Amthor_mech}   They recognized blackbody radiation as a possible means of final-state redistribution, and extended this mechanical picture to include long-range repulsive interactions \cite{Amthor_model}   This group later studied the effects of spatial correlations in the spontaneous avalanche of Rydberg gases in a regime of strong blockade, suggesting a persistence of initial spatial correlations \cite{RobertdeSaintVincent}   

Robicheaux and coworkers have recently investigated the question of prompt many-body ionization from the point of view of Monte Carlo classical trajectory calculations \cite{Goforth}   For atoms on a regular or random grid driven classically by an electromagnetic field, they find that many-body excitation enhances prompt ionization by about twenty percent for densities greater than $5 6 \times 10^{-3}/(n_0^2 a_0)^3$, where $n_0$ is the principal quantum number of the Rydberg gas and $a_0$ is the Bohr radius   They observed that density fluctuations (sampled from the distribution of nearest neighbour distances) have a greater effect, and point to the possible additional influence of secondary electron-Rydberg collisions and the Penning production of fast atoms not considered by the model, but already observed by Raithel and coworkers \cite{Knuffman}   

The Raithel group also found direct evidence for electron collisional $\ell$-mixing in a Rb MOT \cite{Dutta}, and used selective field ionization to monitor evolution to plasma on a microsecond timescale in ultracold $^{85}$Rb $65d$ Rydberg gases with densities as low as $10^8$ cm$^{-3}$ \cite{WalzFlannigan}   Research by our group at UBC has observed very much the same dynamics in the relaxation of Xe Rydberg gases of similar density prepared in a molecular beam \cite{Hung2014}   In both cases, the time evolution to avalanche is well-described by coupled rate equations (see below), assuming an initializing density of Penning electrons determined by Robicheaux's criterion \cite{Robicheaux05}, applied to an Erlang distribution of Rydberg-Rydberg nearest neighbours   

Theoretical investigations of ultracold plasma physics have focused for the most part on the long- and short-time dynamics of plasmas formed by direct photoionization \cite{PhysRept,Lyon}   In addition to studies mentioned above, key insights on the evolution dynamics of Rydberg gases have been provided by studies of Pohl and coworkers exploring the effects of ion correlations and recombination-reionization on the hydrodynamics of plasma expansion \cite{Pohl:2003,PPR}   Further research has drawn upon molecular dynamics (MD) simulations to reformulate rate coefficients for the transitions driven by electron impact between highly excited Rydberg states \cite{PVS}, and describe an effect of strong coupling as it suppresses three-body recombination \cite{Bannasch:2011}   MD simulations confirm the accuracy of coupled rate equation descriptions for systems with $\Gamma$ as large as 0 3   Newer calculations suggest a strong connection between the order created by dipole blockade in Rydberg gases and the most favourable correlated distribution of ions in a corresponding strongly coupled ultracold plasma \cite{Bannasch:2013}   

Tate and coworkers have studied ultracold plasma avalanche and expansion theoretically as well as experimentally   Modelling observed expansion rates, they recently found that $^{85}$Rb atoms in a MOT form plasmas with effective initial electron temperatures determined by initial Rydberg density and the selected initial binding energy, to the extent that these parameters determine the fraction of the excited atoms that ionize by electron impact in the avalanche to plasma \cite{Forest}   This group also returned to the question of added Rydberg atoms, and managed to identify a crossover in $n_0$, depending on the initial electron temperature, that determines whether added Rydberg atoms of a particular initial binding energy act to heat or cool the electron temperature \cite{Crockett}    

Our group has focused on the plasma that evolves from a Rydberg gas under the low-temperature conditions of a skimmed, seeded supersonic molecular beam   In work on nitric oxide starting in 2008 \cite{Morrison2008,Plasma_expan,Morrison_shock,PCCP}, we established an initial kinetics of electron impact avalanche ionization that conforms with coupled rate equation models \cite{Saquet2011,Saquet2012,Scaling,haenelCP} and agrees at early times with the properties of ultracold plasmas that evolve from ultracold atoms in a MOT   We have also observed unique properties of the NO ultracold plasma owing to the fact that its Rydberg states dissociate \cite{Haenel2017}, and identified relaxation pathways that may give rise to quantum effects \cite{SousMBL,SousNJP}   The remainder of this review focuses on the nitric oxide ultracold plasma and the unique characteristics conferred by its evolution from a Rydberg gas in a laser-crossed molecular beam   


\section{Avalanche to strong coupling in a molecular Rydberg gas}

\subsection{The molecular beam ultracold plasma compared with a MOT}

When formed with sufficient density, a Rydberg gas of principal quantum number $n_0>30$ undergoes a spontaneous avalanche to form an ultracold plasma \cite{Li,Morrison2008,RobertdeSaintVincent}   Collisional rate processes combine with ambipolar hydrodynamics to govern the properties of the evolving plasma   For a molecular Rydberg gas, neutral fragmentation, occurs in concert with electron-impact ionization, three-body recombination and electron-Rydberg inelastic scattering   Neutral dissociation combined with radial expansion in a shaped distribution of charged particles, can give rise to striking effects of self-assembly and spatial correlation \cite{Schulz-Weiling2016,Haenel2017}    

The formation of a molecular ultracold plasma requires the conditions of local temperature and density afforded by a high mach-number skimmed supersonic molecular beam   Such a beam propagates at high velocity in the laboratory, with exceedingly well-defined hydrodynamic properties, including a propagation-distance-dependent density and sub-Kelvin temperature in the moving frame \cite{MSW_tutorial}   The low-temperature gas in a supersonic molecular beam differs in three important ways from the atomic gas laser-cooled in a magneto-optical trap (MOT) 

The milli-Kelvin temperature of the gas of ground-state NO molecules entrained in a beam substantially exceeds the sub-100 micro-Kelvin temperature of laser-cooled atoms in a MOT   However, the evolution to plasma tends to erase this distinction, and the two further characteristics that distinguish a beam offer important advantages for ultracold plasma physics:  Charged-particle densities in a molecular beam can exceed those attainable in a MOT by orders of magnitude   A great many different chemical substances can be seeded in a free-jet expansion, and the possibility this affords to form other molecular ultracold plasmas, introduces interesting and potentially important new degrees of freedom governing the dynamics of their evolution 


\subsection{Supersonic molecular beam temperature and particle density}

Seeded in a skimmed supersonic molecular beam, nitric oxide forms different phase-space distributions in the longitudinal (propagation) and transverse coordinate dimensions   As it propagates in $z$, the NO molecules reach a terminal laboratory velocity, $u_{\parallel}$, of about 1400 ${\rm ms^{-1}}$, which varies with the precise seeding ratio   

The distribution of $v_{\parallel}$, narrows to define a local temperature, $T_{\parallel}$, of approximately 0 5 K   The beam forms a Gaussian spatial distribution in the transverse coordinates, $x$ and $y$   In this plane, the local velocity, $v_{\perp}(r)$ is defined for any radial distance almost entirely by the divergence velocity of the beam, $u_{\perp}(r)$   Phase-space sorting cools the temperature in the transverse coordinates, $T_{\perp}$ to a value as low as $\sim 5$ mK \cite{MSW_tutorial}   

The stagnation pressure and seeding ratio determine the local density distribution as a function of $z$   For example, expanding from a stagnation pressure of 500 kPa with a 1:10 seeding ratio, a molecular beam propagates 2 5 cm to a skimmer and then 7 5 cm to a point of laser interaction, where it contains NO at a peak density of $1 6 \times 10^{14}$ cm$^{-3}$   

Here, crossing the molecular beam with a laser beam tuned to the transition sequence, ${\rm X} ~^2 \Pi_{1/2} ~N'' = 1 \xrightarrow{\omega_1} {\rm A} ~^2\Sigma^+ ~N'=0  \xrightarrow{\omega_2} n_0 f(2)$ forms a Gaussian ellipsoidal volume of Rydberg gas in a single selected principal quantum number, $n_0$, orbital angular momentum, $\ell = 3$, NO$^+$ core rotational quantum number, $N^+ = 2$ and total angular momentum neglecting spin, $N=1$   

A typical $\omega_1$ pulse energy of 2 $\mu$J and a Gaussian width of 0 2 mm serves to drive the first step of this sequence in a regime of linear absorption   Overlapping this volume by an $\omega_2$ pulse with sufficient fluence to saturate the second step forms a Rydberg gas ellipsoid with a nominal peak density of $5 \times 10^{12}$ cm$^{-3}$  \cite{Morrison2008,MSW_tutorial}   Fluctuations in the pulse energy and longitudinal mode of $\omega_1$ cause the real density to vary   For certain experiments, we find it convenient to saturate the $\omega_1$ transition, and vary the density of Rydberg gas by delaying $\omega_2$   An $\omega_1$-$\omega_2$ delay, $\Delta t$, reduces the Rydberg gas density by a precise factor, $e^{-\Delta t/\tau}$, where $\tau$ is the 200 ns radiative lifetime of NO ${\rm A} ~^2\Sigma^+ ~N'=0$ \cite{Carter,Hancock} 


\subsection{Penning ionization}

The density distribution of a Rydberg gas defines a local mean nearest neighbour distance, or Wigner-Seitz radius of $ a_{ws} =  \left(3/4 \pi \rho \right)^{1/3} $, where $\rho$ refers to the local Rydberg gas density   For example, a Rydberg gas with a density of $ \rho_0=0 5 \times 10^{12}$ cm$^{-3} $ forms an Erlang distribution \cite{Torquato 1990} of nearest neighbour separations with a mean value of $ 2 a_{ws}=1 6$  $\mu$m   

A semi-classical model \cite{Robicheaux05} suggests that 90 percent of Rydberg molecule pairs separated by a critical distance, $ r_c = 1 8 \cdot 2 n_0^2 a_0 $ or less undergo Penning ionization within 800 Rydberg periods   We can integrate the Erlang distribution from $ r=0 $ to the critical distance $r = r_c$ for a Rydberg gas of given $n_0$, to define the local density of Penning electrons ($ \rho_e$ at $t=0$) produced by this prompt interaction, for any given initial local density, $\rho_0$ by the expression:
\begin{equation}
\rho_e(\rho_0,n_0) = \frac{0 9}{2} \cdot 4 \pi \rho_0 ^2\int_0^{r_{c}} r^2 \mathrm{e}^{-\frac{4\pi}{3}\rho_0 r^3}\mathrm{d}r \quad 
\label{eqn:Erlang}
\end{equation}

Evaluating this definite integral yields an equation in closed form that predicts the Penning electron density for any particular initial Rydberg density and principal quantum number 
\begin{equation}
\rho_e(\rho_0,n_0) =\frac{0 9 \rho_0}{2}(1-\mathrm{e}^{-\frac{4\pi}{3}\rho_0 r_c^3}) \quad 
\label{Eq:PenDens}
\end{equation}
\begin{figure}[h!]
\centering
\includegraphics[scale=0 33]{Penning_Latice pdf}
\caption{Distributions of ion-ion nearest neighbours following Penning ionization and electron-impact avalanche simulated for a predissociating molecular Rydberg gas of initial principal quantum number, $n_0$, from 30 to 80, and density of 10$^{12}$ cm$^{-3}$   Dashed lines mark corresponding values of $a_{ws}$  Calculated by counting ion distances after relaxation to plasma in 10$^6$-particle stochastic simulations  Integrated areas proportional to populations surviving neutral dissociation }
\label{fig:PL}
\end{figure}

Prompt Penning ionization acts on the portion of the initial nearest-neighbour distribution in the Rydberg gas that lies within $r_c$   When a molecule ionizes, its collision partner relaxes to a lower principal quantum number, $n'<n_0/\sqrt{2}$   This close-coupled interaction disrupts the separability of Rydberg orbital configurations in the Penning partner   This causes mixing with core penetrating states that are strongly dissociative   Penning partners are thus very likely to dissociate, leaving a spatially isolated distribution of ions   We refer to the spatial correlation that results as a Penning lattice \cite{Sadeghi:2014}   The extent of this effect varies depending on the local density and the selected initial principal quantum number   Figure \ref{fig:PL} shows the degree to which Rydberg gases with initial principal quantum numbers from 30 to 80 form a Penning lattice for an initial density of $1 \times 10^{12} ~{\rm cm}^{-3}$   

\subsection{Spontaneous electron-impact avalanche}

The electrons produced by prompt Penning ionization start an electron impact avalanche   The kinetics of this process are well described by a set of coupled rate equations that account for state-to-state electron-Rydberg inelastic scattering, electron-impact ionization and three-body ion-electron recombination \cite{PPR,Saquet2011,Saquet2012,Scaling} using detailed rate coefficients,  $k_{ij}$, $k_{i,ion}$ and $k_{i,tbr}$ validated by MD simulations \cite{PVS}   
\begin{eqnarray}
-\frac{d\rho_i}{dt}&=&\sum_{j}{k_{ij}\rho_e\rho_i}-\sum_{j}{k_{ji}\rho_e\rho_j} \nonumber\\
&& +k_{i,ion}\rho_e\rho_i-k_{i,tbr}\rho^3_e 
  \label{level_i}
\end{eqnarray}
\noindent and,
\begin{equation}
\frac{d\rho_e}{dt}=\sum_{i}{k_{i,ion}\rho_e^2}-\sum_{i}{k_{i,tbr}\rho^3_e}
  \label{electron}
\end{equation}

The relaxation of Rydberg molecules balances with collisional ionization to determine an evolving temperature of avalanche electrons to conserve total energy per unit volume  
\begin{equation}
E_{tot}=\frac{3}{2}k_BT_e(t)\rho_e(t)-R\sum_i{\frac{\rho_i(t)}{n_i^2}},
  \label{energy}
\end{equation}
Here, for simplicity, we neglect the longer-time effects of Rydberg predissociation and electron-ion dissociative recombination \cite{Saquet2012} 

Such calculations show that the conversion from Rydberg gas to plasma occurs on a timescale determined largely by the local Penning electron density, or Penning fraction, $P_f = \rho_e/\rho_0$, which depends on the local density of Rydberg molecules and their initial principal quantum number   

Avalanche times predicted by coupled rate equation calculations range widely   For example, in a model developed for experiments on xenon, simulations predict that a Rydberg gas with $n_0 = 42$ at a density of $8 8 \times 10^8 ~{\rm cm}^{-3}$ ($P_f = 6 \times 10^{-5}$) avalanches with a half time of  40 $\mu$s \cite{Hung2014}   At an opposite extreme, rate equations estimate that a Rydberg gas of NO with $n_0=60$ at a density of $1 \times 10^{12} ~{\rm cm}^{-3}$ ($P_f = 0 3$) rises to plasma in about 2 ns \cite{Saquet2012}   

\begin{figure}[h!]
\centering
\includegraphics[width=  49 \textwidth]{SFI_n=49 pdf}
   \caption{Contour plots showing SFI signal as a function the applied field for an $nf(2)$ Rydberg gas with an initial principal quantum number, $n_0=49$   Each frame represents 4,000 SFI traces, sorted by initial Rydberg gas density   Ramp field beginning at 0 and 150 ns (top, left to right), and  300 and 450 ns (bottom) after the $\omega_2$ laser pulse   The two bars of signal most evident at early ramp field delay times represent the field ionization of the $49f(2)$ Rydberg state respectively to NO$^+$ X $^1\Sigma^+$ cation rotational states, $N^+=0$ and 2   The signal waveform extracted near zero applied field represents the growing population of plasma electrons   
   }
\label{fig:SFI}
\end{figure}

Selective field ionization (SFI) probes the spectrum of binding energies in a Rydberg gas   Applied as a function of time after photoexcitation, SFI maps the evolution from a state of selected initial principal quantum number, $n_0$, to plasma \cite{Haenel2017}   Figure \ref{fig:SFI} shows SFI spectra taken at a sequence of delays after the formation of $49f(2)$ Rydberg gases of varying density     

Here, we can see that a $49f(2)$ Rydberg gas with an estimated initial density $\rho_0 = 3 \times 10^{11} ~{\rm cm}^{-3}$ relaxes to plasma on a timescale of about 500 ns   Observations such as these agree well with the predictions of coupled rate-equation calculations   We can understand this variation in relaxation dynamics with $\rho_0$ and $n_0$ quite simply in terms of the corresponding density of prompt Penning electrons these conditions afford to initiate the avalanche to plasma   

Figure \ref{fig:scaled_rise} illustrates this, showing how rise times predicted by coupled rate-equation simulations for a large range of initial densities and principal quantum number match when plotted as a function of time scaled by the ultimate plasma frequency and fraction of prompt Penning electrons   The dashed line gives an approximate account of the scaled rate of avalanche under all conditions of Rydberg gas density and initial principal quantum number in terms of the simple sigmoidal function:

\begin{equation}
\frac{\rho_e}{\rho_0} = \frac{a}{b+e^{-c\tau}},
  \label{scaledEq1}
\end{equation}
where,
\begin{equation}
\tau = t \omega_e P_f^{3/4},
  \label{scaledEq2}
\end{equation}
in which $\omega_e$ is the plasma frequency after avalanche, $P_f$ is the fraction of prompt Penning electrons, and $a = 0 00062$,  $b =   0 00082$ and $c =     0 075$ are empirical coefficients   

\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{sim_analytical_density pdf}
   \caption{Rise in fractional electron density as a function of time scaled by the plasma frequency, $\omega_e$ and fraction, $\rho_e(t=0)/\rho_0 = P_f$, of prompt Penning electrons   Simulation results shown for $n_0 = 30$, 50 and 70 with initial densities, $\rho_0 = 10^9,~10^{10},~10^{11},~{\rm and}~10^{12}~{\rm cm}^{-3}$   
   }
\label{fig:scaled_rise}
\end{figure}


\subsection{Evolution to plasma in a Rydberg gas Gaussian ellipsoid}

As outlined above, the local density and principal quantum number together determine the rate at which a Rydberg gas avalanches to plasma   Our experiment crosses a 2 mm wide cylindrically Gaussian molecular beam with a 1 mm diameter TEM$_{00}$ $\omega_1$ laser beam to produce a Gaussian ellipsoidal distribution of molecules excited to the A $^2\Sigma^+$ $v=0, ~N'=0$ intermediate state   A larger diameter $\omega_2$ pulse then drives a second step that forms a Rydberg gas in a single $n_0f(2)$ state with the spatial distribution of the intermediate state   

We model this shaped Rydberg gas as a system of 100 concentric ellipsoidal shells of varying density \cite{haenelCP}   Coupled rate equations within each shell describe the avalanche to plasma   This rate process proceeds from shell to shell with successively longer induction periods, determined by the local density as detailed above   The rising conversion of Rydberg molecules to ions plus neutral dissociation products conserves the particle number in each shell   We assume that local space charge confines electrons to shells, conserving quasi-neutrality   Electrons exchange kinetic energy at the boundaries of each shell, which determines a single plasma electron temperature   

\begin{figure}[h!]
\centering
\includegraphics[width=  5 \textwidth]{shell_model_100}
   \caption{(top frame) Cross-sectional contour diagram in the $x,y$ plane for $z=0$ describing the distribution of ion plus electron density over 100 shells of Gaussian ellipsoid with initial dimensions, $\sigma_x= 0 75$ mm and $\sigma_y= \sigma_z = 0 42$ mm and an initial $n_0 = 50$ Rydberg gas density, $\rho_0 = 2 \times 10^{11}$ cm$^{3}$ after an evolution time of 100 ns   (bottom frame) Curves describing the (dashed) ascending ion and (solid) descending Rydberg gas densities of each shell as functions of evolution time, for $t=20$, 40, 60, 80 and 100 ns   
   }
\label{fig:shell}
\end{figure}

The upper frame of Figure \ref{fig:shell} shows contours of NO$^+$ ion density after 100 ns obtained from a shell-model coupled rate-equation simulation of the avalanche of a Gaussian ellipsoidal Rydberg gas of nitric oxide with a selected initial state, $50f(2)$ and a density of $2 \times 10^{11}$ cm$^{-3}$   Here, we simulate a relaxation that includes channels of predissociation at every Rydberg level and redistributes the energy released to electrons, which determines a uniform rising electron temperature for all shells   

For comparison, the lower frame plots curves describing the ion density of each shell as a function of time from 20 to 100 ns, as determined by applying Eq \ref{scaledEq1} for the local conditions of initial Rydberg gas density   This numerical approximation contains no provision for predissociation   Coupled rate-equation simulations for uniform volumes show that predissociation depresses yield to some degree, but has less effect on the avalanche kinetics \cite{Saquet2012}   Therefore, we can expect sets of numerically estimated shell densities, scaled to agree with the simulated ion density at the elapsed time of 100 ns to provide a reasonable account of the earlier NO$^+$ density profiles as a function of time    


\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{Shell_pop}
   \caption{Global population fractions of particles as they evolve in the avalanche of a shell-model ellipsoidal Rydberg gas with the initial principal quantum number and density distribution of Figure \ref{fig:shell}
   }
\label{fig:shell_yields}
\end{figure}

For each time step, the difference, $\rho_0 - \rho_e$ defines the neutral population of each shell   We assign a fraction of this population to surviving Rydberg molecules, such that the total population of NO$^*$ as a function of time agrees with the prediction of the shell-model simulation, as shown in Figure \ref{fig:shell_yields}   We consider the balance of this neutral population to reflect NO$^*$ molecules that have dissociated to form N($^4$S) $+$ O($^3$P)   Figure \ref{fig:shell} plots these surviving Rydberg densities as a function of radial distance for each evolution time   At the initial density of this simulation, note at each time step that a higher density of Rydberg molecules encloses the tail of the ion density distribution in $x$    

\subsection{Plasma expansion and NO$^+$ - NO$^*$ charge exchange as an avenue of quench}

We regard the ions as initially stationary   The release of electrons creates a radial electric potential gradient, which gives rise to a force, $-e\nabla \phi_{k,j}(t)$, that accelerates the ions in shell $j$ in direction $k$ according to \cite{Sadeghi 2012}:

\begin{align}
\frac{-e}{m'}\nabla \phi_{k,j}(t) = & \frac{\partial u_{k,j}(t)}{\partial t} \notag  \\
= & \frac{k_BT_e(t)}{m'\rho_j(t)} \frac{\rho_{j+1}(t) - \rho_j(t)}{r_{k,j+1}(t) - r_{k,j}(t)},
  \label{dr_dt}
\end{align}

\noindent where $\rho_j(t)$ represents the density of ions in shell $j$   

The instantaneous velocity, $u_{k,j}(t)$ determines the change in the radial coordinates of each shell, $r_{k,j}(t)$, 
\begin{equation}
\frac{\partial r_{k,j}(t)}{\partial t}=u_{k,j}(t) = \gamma_{k,j}(t) r_{k,j}(t),
  \label{dr_dt}
\end{equation}
\noindent which in turn determines shell volume and thus its density, $ \rho_j(t)$    
The electron temperature supplies the thermal energy that drives this ambipolar expansion   Ions accelerate and $T_e$ falls according to: 

\begin{equation}
\frac{3k_B}{2}\frac{\partial T_e(t)}{\partial t}= -\frac{m'}{\sum_{j}{N_j}}\sum_{k,j}{N_j u_{k,j}(t)\frac{\partial u_{k,j}(t)}{\partial t}},
  \label{dr_dt}
\end{equation}
\noindent where we define an effective ion mass, $m'$, that recognizes the redistribution of the electron expansion force over all the NO$^+$ charge centres by resonant ion-Rydberg charge exchange, which occurs with a very large cross section \cite{PPR}   
\begin{equation}
m' =\left (1+ \frac{\rho^*_{j}(t)}{ \rho_j(t)}\right) m ,
  \label{dr_dt}
\end{equation}
\noindent in which $\rho^*_{j}(t)$ represents the instantaneous Rydberg density in shell $j$ 

The initial avalanche in the high-density core of the ellipsoid leaves few Rydberg molecules, so this term has little initial effect   Rydberg molecules predominate in the lower-density wings   There, momentum sharing by charge exchange assumes a greater importance   

We see this most directly in the $\omega_2$ absorption spectrum of transitions to states in the $n_0 f(2)$ Rydberg series, detected as the long-lived signal that survives a flight time of 400 $\mu$s to reach the imaging detector   The balance between the rising density of ions and the falling density of Rydberg molecules depends on the initial density of electrons produced by prompt Penning ionization   As clear from Eq \ref{Eq:PenDens}, this Penning fraction depends sensitively on the principal quantum number, and for all principal quantum numbers, on the initial Rydberg gas density   

\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{w2_spectra}
   \caption{Double-resonant spectra of nitric oxide Rydberg states in the $nf$ series converging to NO$^+$ $v=0$, $N^+=2$ (designated, $nf(2)$), derived from the late-peak signal obtained after a flight time of 400 $\mu$s by scanning $\omega_2$ for $\omega_1$ tuned to NO A $^2\Sigma^+$ $v=0$, $N'=0$ for initial $nf(2)$ densities from top to bottom of 0 07, 0 10, 0 13, 0 19, 0 27, 0 30, 0 32 and $3 \times 10^{12}$ cm$^{3}$   
  }
\label{fig:w2_spectra}
\end{figure}

Figure \ref{fig:w2_spectra} shows a series of $\omega_2$ late-signal excitation spectra for a set of initial densities   Here, we see a clear consequence of the higher-order dependence of Penning fraction - and thus the NO$^+$ ion - NO$^*$ Rydberg molecule balance - on $n_0$, the $\omega_2$-selected Rydberg gas initial principal quantum number   This Penning-regulated NO$^+$ ion - NO$^*$ Rydberg molecule balance appears necessary as a critical factor in achieving the long ultracold plasma lifetime required to produce this signal   We are progressing in theoretical work that explains the stability apparently conferred by this balance   


\subsection{Bifurcation and arrested relaxation}

Ambipolar expansion quenches electron kinetic energy as the initially formed plasma expands   Core ions follow electrons into the wings of the Rydberg gas   There, recurring charge exchange between NO$^+$ ions and NO$^*$ Rydberg molecules redistributes the ambipolar force of the expanding electron gas, equalizing ion and Rydberg velocities   This momentum matching effectively channels electron energy through ion motion into the overall $\pm x$ motion of gas volumes in the laboratory   The internal kinetic energy of the plasma, which at this point is defined almost entirely by the ion-Rydberg relative motion, falls   Spatial correlation develops, and over a period of 500 ns, the system forms the plasma/high-Rydberg quasi-equilibrium dramatically evidenced by the SFI results in Figure \ref{fig:SFI}   

\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{Bifurcation pdf}
   \caption{$x,y$ detector images of ultracold plasma volumes produced by 2:1 aspect ratio ellipsoidal Rydberg gases with selected initial state, $40f(2)$ after a flight time of 402 $\mu$s over a distance of 575 mm   Lower frame displays the distribution in $x$ of the charge integrated in $y$ and $z$   Both images represent the unadjusted raw signal acquired in each case after 250 shots   
   }
\label{fig:bifurcation}
\end{figure}

In the wings, momentum redistribution owing to cycles of ion-Rydberg charge transfer retards radial expansion \cite{Pohl2003,PPR}   By redirecting electron energy from ambipolar acceleration to $\pm x$ plasma motion, NO$^+$ to NO$^*$ charge exchange dissipates electron thermal energy   This redistribution of energy released in the avalanche of the Rydberg gas to plasma, causes the ellipsoidal Rydberg gas to bifurcate \cite{Schulz-Weiling2016,Haenel2017}, forming very long-lived, separating charged-particle distributions   We capture the electron signal from these recoiling volumes on an imaging detector as pictured in Figure \ref{fig:bifurcation}   Here, momentum matching preserves density and enables ions and Rydberg molecules to relax to positions that minimize potential energy, building spatial correlation   

The semi-classical description of avalanche and relaxation outlined above forms an important point of reference from which to interpret our experimental observations   The laser crossed molecular beam illumination geometry creates a Rydberg gas with a distinctively shaped high-density spatial distribution   This initial condition has an evident effect on the evolution dynamics   We have developed semi-classical models that explicitly consider the coupled rate and hydrodynamic processes governing the evolution from Rydberg gas to plasma using a realistic, ellipsoidal representation of the ion/electron and Rydberg densities \cite{haenelCP}   No combination of initial conditions can produce a simulation that conforms classically with the state of arrested relaxation we observe experimentally  


\subsection{A molecular ultracold plasma state of arrested relaxation}

Thus, we find that spontaneous avalanche to plasma splits the core of an ellipsoidal Rydberg gas of nitric oxide  As ambipolar expansion quenches the electron temperature of this core plasma, long-range, resonant charge transfer from ballistic ions to frozen Rydberg molecules in the wings of the ellipsoid quenches the ion-Rydberg molecule relative velocity distribution  This sequence of steps gives rise to a remarkable mechanics of self-assembly, in which the kinetic energy of initially formed hot electrons and ions drives an observed separation of plasma volumes  These dynamics redistribute ion momentum, efficiently channeling electron energy into a reservoir of mass-transport  This starts a process that evidently anneals separating volumes to a state of cold, correlated ions, electrons and Rydberg molecules  

We have devised a three-dimensional spin model to describe this arrested state of the ultracold plasma in terms of two, three and four-level dipole-dipole energy transfer interactions (spin flip-flops), together with Ising interactions that arise from the concerted pairwise coupling of resonant pairs of dipoles \cite{SousMBL,SousNJP}   

The Hamiltonian includes the effects of onsite disorder owing to the broad spectrum of states populated in the ensemble and the unique electrostatic environment of every dipole   Extending ideas developed for simpler systems \cite{Burin1,Sondhi}, one can make a case for slow dynamics, including an arrest in the relaxation of NO Rydberg molecules to predissociating states of lower principal quantum number 

Systems of higher dimension ought to thermalize by energy transfer that spreads from rare but inevitable ergodic volumes (Griffiths regions) \cite{Sarang2, Roeck_griffith, RareRegions_rev, Thermal_inclusions}   However, a feature in the self-assembly of the molecular ultracold plasma may preclude destabilization by rare thermal domains:  Whenever the quenched plasma develops a delocalizing Griffiths region, the local predissociation of relaxing NO molecules promptly proceeds to deplete that region to a void of no consequence 

In summary, the classical dynamics of avalanche and bifurcation appear to create a quenched condition of low temperature and high disorder in which dipole-dipole interactions drive self-assembly to a localized state purified by the predissociation of thermal regions   We suggest that this state of the quenched ultracold plasma offers an experimental platform for studying quantum many-body physics of disordered systems  \\

\section{Acknowledgments}
This work was supported by the US Air Force Office of Scientific Research (Grant No  FA9550-17-1-0343), together with the Natural Sciences and Engineering research Council of Canada (NSERC), the Canada Foundation for Innovation (CFI) and the British Columbia Knowledge Development Fund (BCKDF)  


Context 2: Probably one of the most frustrating things about building experimental aircraft, especially when starting with a minimum of pre-fabricated parts, is to start building and ending up with an unexpected result  Every builder starts a new project by wanting it to go "perfectly " So when things aren't going well, especially at the beginning, the frustration can lead to an unfinished airplane 
This is the first article in a series dedicated to helping builders of the Rand Robinson KR series planes build a straight and true fuselage -- the first part of the construction process  Borrowing from modern boatbuliding techniques, focus will be on the KR-2S, but the principles apply to the entire lineup of KR-1 & KR-2 series planes 
While building the KR-2(s) a common surprise is encountered by builders when the completed fuselage sides are laid into position to form the fuselage box section  With many hours spent building the sides flat, finding the once straight longerons that now bow up from the building surface, form a most dissatisfying "banana" shape  Especially when using the preformed fiberglass parts, this curve in the top longeron is not acceptable  The builder is left wondering what went wrong and no amount of clamping or brute force forming will solve the problem to any degree of satisfaction  The problem is not the builder's fault  The solution starts by understanding the three dimensional relationship of the assembled parts being built 
First understand that the plans show the finished form of the plane  They show the "projected" form as you would expect to see it if viewing an actual plane from the top, ends and from the side  Since the sides are sloped (flared) outward, looking from the side, the distances given by measuring the profile drawing are "foreshortened" and don't give the proper shape for building the fuselage with a flat top longeron  What needs to be done is to "develop" the "true" distances and shape of the flat panel so that when it is curved into position, the longerons lay flat 
Second, understand that the dimensions called for in the plans put a twist in the sides that tends to work the panel in two directions of curvature  This twist makes the panel "undevelopable" meaning that that shape cannot be unrolled into an equivalent flat shape  This is important when laying out the side and bottom panels onto flat plywood  To illustrate this, try forming a piece of paper around a soda can  The paper can be formed flat around the can either straight or at a diagonal to it's length  It has only one direction of curvature and is by definition "developable"  Now try to form the same piece of paper around a baseball  It won't lie flat on the surface without some deformation (folding, wrinkling or tearing) of the paper  The ball has curvature in more that one direction and is a "compounded" shape  Paper (or plywood) can only be readily formed in developable shapes as opposed to aluminum or other metal which can accept in plane deformation  A developable surface is needed to lay out a curved surface when the materials used can't be deformed with any degree of in-plane strain 
Initially, the fuselage sides are laid out flat with reference to the top longeron measured to a straight chalk line  The bowing problem starts when the side panels are bent and sloped to form the fuselage box section  If the sides were not sloped (tumbled home), the section formed would be cylindrical and the longerons would lie flat  Since the sides are tumbled home, the section formed is now conical  When a conical shape is cut with a plane (building surface) not perpendicular to it's axis, the shape formed is elliptical -- exactly what happens with the top longeron  When it's built flat, bent to form a cylindrical section, and sloped to form a conical section, it takes on an elliptical shape firewall to tailstock 
This method borrows heavily from proven techniques used in the marine trades  It should be stressed at this point that although the layout procedure is not complicated, it is important to take your time  If the layout is not going well initially, start over! Better to erase layout errors now than to have them built it and cause surprises later 
Layout to ensure a fair and true fuselage starts by drawing a reference line (baseline) on the building surface  Refer to figures 2 & 3 and use a wire guide to draw a very straight baseline  About 500 lbs  Of tension should be adequate  One could use a chalk line, but we're talking airplanes here, not house framing 
The main layout difference is that the baseline isn't used as a reference for the top longeron  The baseline references the mid point of the firewall for the developed (and true dimensioned) side panel  Although the baseline will still be the reference, the top and bottom longerons will be laid separately 
Layout differences don't end there  Each of the stations (vertical members) will be laid out with a calculated separation so that when the panels are formed into position, they land on the spacing called for in the plans  Another major difference is that the bottom & side panels are applied after forming the fuselage box section  This is mainly to obtain the ability to "fair" the side and bottom surfaces and insure a straight and true shape 
Refer to figure 1 for the layout of the new developed side panel  The firewall (station a) is layed out perpendicular to the baseline  Longitudinal (station) measurements are given along the length of the baseline from the firewall  Vertical dimensions are given to reference the angle and breadths of the station at the baseline 
Notice that the top longeron is bowed outward and that the stations are spaced slightly greater than called out in the plans  When the panels are formed into the box frame section ,they will work into the dimensions specified in the plans 
Strike a centerline, longer than is needed on the building surface using a wire guide  Draw off the firewall line perpendicular to the centerline at one end 
Using the distances listed in the balloons, mark them off on the centerline  Distances are measured to the nearest sixteenth of an inch  Take time to mark them off carefully  Don't mark off the distances in a cumulative fashion  Use the firewall as a common reference 
Using the angles listed at each station, mark off a station line longer than is needed  The angles are measured to the nearest hundredth of a degree  Take time to mark them off carefully 
At each station, start by marking off each short (bottom longeron) line distance from the centerline  Use your set of trammels or beam compass for doing this  Mark the intersection of the short line with the station line 
At each station, mark off each long (top longeron) line distance from the intersection of the short line distance and the station line  Again the trammels or beam compass is best for completing this step  Mark the intersection of the long line distance with the station line 
Using the longeron as a batten, trace out the inside and outside curves of the longeron  After the batten is secure, in between each station, fasten a keeper block inside and outside to preserve the shape of the longeron taking care to avoid potential future interference with the diagonal members to be installed later  The fairing blocks can be removed or left in place if they won't interfere with building  The vertical station members and their diagonals can now be measured and positioned  Remember to refer to the plans for the material thickness direction 
After vertical and diagonal members are cut and fitted, take time to draw their outlines on the building surface to cut down on time and confusion when laying out the opposite side 
Finishing the side panel is accomplished in a manner similar to that called for in the handbook with the exception that the side and bottom skin panels will be attached later 
The next article in the series will discuss jigging and building techniques to ensure alignment and straightness of the flat built side panels  Also covered will be building a "strongback" jig to assure alignment of the side panels when they are formed into their final shape 
Part 3 in the series will cover assembly of the side panels using the jigs  Some joint details will be discussed that will ensure a stronger and more fair fuselage assembly  Also covered will be the layout & attachment of the side and bottom ply skins 
U S  Mail: Densmore Associates, inc 
ANSI "D" size, computer generated plots of all the layout drawings in this series are available from the author for $30 plus postage & handling  Full (true size) scale plots may be made available depending on demand 
"Scarfing" is the practice of splicing plywood so that short pieces of plywood can be used to span long distances  On the KR, it is required on both the fuselage skins and spar webs  The angle of the splice should be 10 to 12 degrees to maintain strength across the joint  Also, joints should coincide with structural members, such as spar webs or fuselage truss members 
This scarfer is made by mating a regular plunge router (this one costs about $50) to a table saw  Obviously, you really only need a table saw to cut the chamfer, but it does make a nice heavy table for scarfing  You could just as easily use a large work table as the base First, set the table saw for a 5 5 degree cut (for a 1:12 joint, or 6 5 degree cut for a 10:1 joint), and run a 1 x 6 through on edge to chamfer a corner on the board  Then drill the board for three router mounting holes (two are countersunk) and connect the assembly to the table saw with two 1/4 inch bolts  Use a long (2-3 inch) straight cutting bit to do the cutting  Adjust the bit so it doesn't interfere with your table top, and go to town  Keep pressure on the plywood to ensure contact with the table while you're scarfing  Make sure you feed your material from the same end as you would if you were sawing, or the router will take your plywood away from you and put a big dent in your garage door 
In the late 60's Ken Rand and Stuart Robinson were working as flight system engineers for Douglas Avionics  Ken was working as an electrical engineer, having previously worked for Sperry as an autopilots project engineer, while Stu's degree was in aeronautical engineering from Northrop University  They were two of the guys at the end of the DC-8,9, and 10 assembly lines responsible for correcting some of the nits and picks in various systems before delivery to the customer 
They both wanted to build a fast, inexpensive airplane which was also economical to maintain  Several designs were considered, and plans were bought first for the Jeanie's Teenie and then the Taylor Monoplane  The Monoplane was more to their liking, but would require some modification to fit their needs  A cooperative redesign effort ensued, with virtually no dimensions left untouched  Only the basic fuselage structure, airfoil, and powerplant were retained  The tail shape was Stu's, and came directly from the big DC-8s parked on the ramp outside his office window  The landing gear was designed by Ken, after seeing the gear on a Dewey Bird at Santa Paula airport 
Ken was killed in his KR2 a short time later while flying over Cajon Pass in what was apparently a bad weather / low fuel accident  Ken's wife Jeanette became owner of RR overnight, and stepped up to keep the plans and parts coming  Much of the engineering needs are handled by Bill Marcy of Denver, who's been helping out since early '79 
To date, almost 6000 KR1, 9200 KR2, and 760 KR2S plan sets have been sold  1200 KR2s are estimated to be flying, with 5 KR2Ss now in the air  Much of the development work done on KR's is now done by the builders themselves  KR builders tend to be innovative, which leads to some interesting modifications  Some of the mods that work eventually creep into the plans  The KR2S is a case in point  Many builders who'd heard of the pitch sensitivity and tight cabin of the KR2 began to build an enlarged version, with the length determined by the most commonly available longeron material  The result is a KR2 that is stretched 2" between firewall and main spar, and 14" behind the main spar  Higher gross weights dictated more wing area, with the new standard becoming the Diehl wing skin  Those who plan to carry passengers commonly stretch the cabin width a few inches, although 1 5 inches is the limit if you still want to use RR's premolded parts 
Mike Stearns addresses the KR Forum crowd 
This year's KR Forum featured guest speakers Mike Stearns, Steve Trentman, and Bill Marcey  Mike Stearns spoke on several topics, including the many sources for KR and homebuilding information available on the Internet  He also mentioned KRNet, the list server devoted entirely to KR aircraft, as well as several notable World Wide Web home pages  He also brought a sample of the new Rand Robinson wing skins with him, and discussed their high temperature core prepreg construction  His KR2S will receive the first set, which is currently being installed at Hinson Composites 
Steve Trentman spoke on his turbine installation  It uses a turbine engine which saw duty as an A7 attack jet starter engine  Total weight is about 85 pounds, while putting out around 90 horsepower  There is a small stockpile of these engines available from government surplus  sources  This engine can only be throttled back to 52% power, which leads to some pretty interesting landings  One inflight failure has been logged so far, with very little damage to the aircraft  More on this exciting development in next month's issue of KROnline 
Les Palmer's KR2 N202LP won Best KR2, Best Engine Installation, and People's Choice awards at the 1995 KR Gathering at Columbia, TN  After researching the KR series, and reading Neil Bingham's "A Critical Analysis of the KR2" (Jan 88 Sport Aviation), Les decided to build his as a single seater, stretched 24" in the tail, while maintaining a stock width firewall  His fuselage is made from Douglas fir, which weighs in at 4 lbs heavier than if constructed from spruce  It is skinned with 1/8" birch plywood  Spars are covered with plywoood on both fore and aft sides, ala KR2S  Diehl wing skins provide the lift  Horizontal stabilizer and elevator were stretched 7" longer on each side, while the vertical stabilizer and rudder were stretched 8" taller    The fuselage to cowling junction was made more graceful by adding 1 5 inches to the height of the firewall end of the fuselage sides 
Les's canopy is a Dragonfly, using a four linkage system to swing forward when opening  The canopy frame fits snugly into a recess in the foward deck, providing an excellent wind and water seal  The fiberglass work is exemplary 
Seating is luxurious for one 
The cowling is also a work of art, and uses NACA ducts for efficiency  Female molds were made for all the fiberglass parts on Les's plane, so he could proabably be persuaded to make more, if demand dictates  Les also machines a multitude of KR aluminum and steel parts which he now offers for sale 
The firewall was reinforced with aluminum brackets and angles bolted between the longerons in anticipation of the 200 lb Subaru EA-81 engine installation  His 100 HP Asian version is outfitted with an American Holley 5200 caburetor and manifold  It uses a PSRU of Les's own design, featuring two spur gears with a 1 69:1 reduction ratio and a toothed belt  Other than tapping the crank for larger bolts to mount the redrive, no other engine modifications were required  Also, this is probably the only air conditioned KR2 on the planet  The prop is a 60/63 Hegy 
Originally built as a taildragger, the fixed gear is made from 4130 steel tubing  Custom cast 6 00x6 aluminum wheels and steel rotors are mated with 6" Cleveland calipers for braking  An early taxi test accident damaged the main gear, and prompted Les to change to tricycle gear  Again, he designed his own fiberglass main gear, and uses a Diehl nose wheel fork with a 4130 strut and 6" wheel up front
Context 3: Paper Info

Title: Efficient nonparametric estimation of Toeplitz covariance matrices
Publish Date: March 20, 2023
Author List: Karolina Klockmann (from Department of Statistics and Operations Research, Universität Wien), Tatyana Krivobokova (from Department of Statistics and Operations Research, Universität Wien)

Figure

Figure 1: Spectral density functions (first row) and autocovariance functions (second row) for examples 1, 2, 3 
Figure 2: Distance between the first atom and the first center of mass of aquaporin (left) and the opening diameter y t over time t (right) 
black line in the left plot) confirms that the covariance matrix estimated with our VST-DCT method almost completely decorrelates the channel diameter Y on the training data set Next, we estimated the regression coefficients β with the usual PLS algorithm, ignoring the dependence in the data Finally, we estimated β with PLS that takes into account dependence using our covariance estimator Σ Based on these regression coefficient estimators, the prediction on the test set was calculated The plot on the right side of Figure 2 shows the Pearson correlation between the true channel diameter on the test set and the prediction on the same test set based on raw (grey) and decorrelated data (black) 
Figure 3: On the left, the auto-correlation function of Y (grey) and of Σ−1/2 Y (black), where Σ is estimated with the VST-DCT method; On the right, correlation between the true values on the test data set and prediction based on partial least squares (in grey) and corrected partial least squares (black) 
Uniform distributionThe observations follow a uniform distribution with covariance matrices Σ 1 , Σ 2 , Σ 3 of examples 1, 2, 3, i e , Y i = Σ 1/2 j X i , j = 1, 3, with X 1 ,    the parameter innov of the R function arima sim is used to pass the innovations X 1 ,    , X n i i d Table4, 5 and 6 show respectively the results for (A) p = 5000, n = 1, (B) p = 1000, n = 50 and (C) p = 5000, n = 10 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(B) p = 1000, n = 50: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 

abstract

A new nonparametric estimator for Toeplitz covariance matrices is proposed  This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression  The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data-driven and computationally very fast 
Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices  These results are readily extended to estimation of inverses of Toeplitz covariance matrices  Also, an alternative version of the Whittle likelihood for the spectral density based on the Discrete Cosine Transform (DCT) is proposed 
The method is implemented in the R package vstdct that accompanies the paper 

Introduction

Estimation of covariance and precision matrices is a fundamental problem in statistical data analysis with countless applications in the natural and social sciences  Covariance matrices with a Toeplitz structure arise in the study of stationary stochastic n = 1, to the best of our knowledge, there is no fully data-driven approach for selecting the banding/tapering/thresholding parameter 
suggested first to split the time series into non-overlapping subseries and then apply the cross-validation criterion of   However, it turns out that the right choice of the subseries length is crucial for this approach, but there is no data-based method available for this  In this work, an alternative way to estimate a Toeplitz covariance matrix and its inverse is chosen 
Our approach exploits the one-to-one correspondence between Toeplitz covariance matrices and their spectral densities  First, the given data are transformed into approximate Gaussian random variables whose mean equals to the logarithm of the spectral density  Then, the log-spectral density is estimated by a periodic smoothing spline with a data-driven smoothing parameter 
Finally, the resulting spectral density estimator is transformed into an estimator for Σ or its inverse  It is shown that this procedure leads to an estimator that is fully data-driven, automatically positive definite and achieves the minimax optimal convergence rate under the spectral norm over a large class of Toeplitz covariance matrices 
In particular, this class includes Toeplitz covariance matrices that correspond to long-memory processes with bounded spectral densities  Moreover, the computation is very efficient, does not require iterative or resampling schemes and allows to apply any inference and adaptive estimation procedures developed in the context of nonparametric Gaussian regression 
Estimation of the spectral density from a stationary time series is a research topic with a long history  Earlier nonparametric methods are based on smoothing of the (log-)periodogram, which itself is not a consistent estimator   Another line of nonparametric methods for estimating the spectral density is based on the Whittle likelihood, which is an ap-proximation to the exact likelihood of the time series in the frequency domain 
For example, estimated the spectral density from a penalized Whittle likelihood, while used polynomial splines to estimate the log-spectral density function maximizing the Whittle likelihood  Recently, Bayesian methods for spectral density estimation have been proposed (see , but these may become very computationally intensive in large samples due to posterior sampling 
The minimax optimal convergence rate for nonparametric estimators of Hölder continuous spectral densities from Gaussian stationary time series was obtained by under the L p , 1 ≤ p ≤ ∞, norm  Only few works on spectral density estimation show the optimality of the corresponding estimators  In particular, and derived convergence rates of their estimators for the log-spectral density under the L 2 norm, while neglecting the Whittle likelihood approximation error 
In general, most works on spectral density estimation do not exploit further the close connection to the corresponding Toeplitz covariance matrix estimation  In particular, an upper bound for the L ∞ risk of a spectral density estimator automatically provides an upper bound for the risk of the corresponding Toeplitz covariance matrix estimator under the spectral norm 
This fact is used to establish the minimax optimality of our nonparametric estimator for the Toeplitz covariance matrices  The main contribution of this work is to show that our proposed spectral density estimator is not only numerically very efficient, but also achieves the minimax optimal rate in the L ∞ norm, which in turn ensures the minimax optimality of the corresponding Toeplitz covariance matrix estimator 
The paper is structured as follows  In Section 2, the model is introduced and ap-proximate diagonalization of Toeplitz covariance matrices with the discrete cosine transform is discussed  Moreover, an alternative version of the Whittle's likelihood is proposed  In Section 3, new estimators for the Toeplitz covariance matrix and the precision matrix are derived, while in Section 4 their theoretical properties are presented 
Section 5 contains simulation results, Section 6 presents a real data example, and Section 7 closes the paper with a discussion  The proofs are given in the appendix to the paper 

Set up and diagonalization of Toeplitz matrices

Let Y 1 ,       , Y n i i d  ∼ N p (0 p , Σ), where Σ is a (p × p)-dimensional positive definite covariance matrix with a Toeplitz structure, that is, Σ = {σ |i−j| } p i,j=1 0  The sample size n may tend to infinity or to be a constant  The case n = 1 corresponds to a single observation of a stationary time series, and in this case the data are simply denoted by Y ∼N p (0 p , Σ) 
The dimension p is assumed to grow  The spectral density function f , corresponding to a Toeplitz covariance matrix Σ, is given by so that for f ∈ L 2 (−π, π) the inverse Fourier transform implies Hence, Σ is completely characterized by f , and the non-negativity of the spectral density function implies the positive definiteness of the covariance matrix 
Moreover, the decay of the autocovariance σ k is directly connected to the smoothness of f   Finally, the convergence rate of a Toeplitz covariance estimator and that of the corresponding spectral density estimator are directly related via Σ ≤ f ∞ := sup x∈ |f (x)|, where • denotes the spectral norm (see  
As in , we introduce a class of positive definite Toeplitz covariance matrices with Hölder continuous spectral densities  For β = γ + α > 0, where The optimal convergence rate for estimating Toeplitz covariance matrices over P β (M 0 , M 1 ) depends crucially on β  It is well known that the k-th Fourier coefficient of a function whose γ-th derivative is α-Hölder continuous decays at least with order O(k −β ) (see  
Hence, β determines the decay rate of the autocovariances σ k , which are the Fourier coefficients of the spectral density f , as k → ∞  In particular, this implies that for β ∈ (0, 1], the class P β (M 0 , M 1 ) includes Toeplitz covariance matrices corresponding to long-memory processes with bounded spectral densities, since the sequence of corresponding autocovariances is not summable 
A connection between Toeplitz covariance matrices and their spectral densities is further exploited in the following lemma  Lemma 1  Let Σ ∈ P β (M 0 , M 1 ) and let x j = (j − 1)/(p − 1), j = 1,    , p, then where δ i,j is the Kroneker delta, O(•) terms are uniform over i, j = 1,       , p and divided by √ 2 when i, j ∈ {1, p} is the Discrete Cosine Transform I (DCT-I) matrix 
The proof can be found in Appendix A 1  This result shows that the DCT-I matrix approximately diagonalizes Toeplitz covariance matrices and that the diagonalization error depends to some extent on the smoothness of the corresponding spectral density  In the spectral density literature the discrete Fourier transform (DFT) matrix
, where i is the imaginary unit, is typically employed to approximately diagonalize Toeplitz covariance matrices  Using the fact that introduced an approximation for the likelihood of a single Gaussian stationary time series (case n = 1), the so-called Whittle likelihood (1) The quantity , where F j denotes the j-th column of F , is known as the periodogram at the j-th Fourier frequency 
Note that due to periodogram symmetry, only p/2 data points I 1 ,    , I p/2 are available for estimating the mean f (2πj/p), j = 1,       , p/2 , where x denotes the largest integer strictly smaller than x  The Whittle likelihood has become a popular tool for parameter estimation of stationary time series, e g , for nonparametric and parametric spectral density estimation or for estimation of the Hurst exponent, see e g , ;  
Lemma 1 yields the following alternative version of the Whittle likelihood where W j = (D t j Y ) 2   Note that this likelihood approximation is based on twice as many data points W j as the standard Whittle likelihood  Thus, it allows for a more efficient use of the data Y to estimate the parameter of interest, such as the spectral density or the Hurst parameter 
Equations ( ) or (2) invite for the estimation of f by maximizing the (penalized) likelihood over certain linear spaces (e g , spline spaces), as suggested e g , in or   However, such an approach requires well-designed numerical methods to solve the corresponding optimization problem, since the spectral density in the second term of (1) or ( ) is in the denominator, which does not allow to obtain a closed-form expression for the estimator and often leads to numerical instabilities 
Also, the choice of the smoothing parameter becomes challenging  Therefore, we suggest an alternative approach that allows the spectral density to be estimated as a mean in an approximate Gaussian regression  Such estimators have a closed-form expression, do not require an iterative optimization algorithm and a smoothing parameter can be easily obtained with any conventional criterion 
First Hence, for W j = (D t j Y ) 2 , j = 1,       , p it follows with Lemma 1 that where Γ(a, b) denotes a gamma distribution with a shape parameter a and a scale parameter b  Note that the random variables W 1 ,       , W p are only asymptotically independent  Obviously, E(W j ) = f (πx j ) + O(1), j = 1,      
, p  To estimate f from W 1 ,       , W p , one could use a generalized nonparametric regression framework with a gamma distributed response, see e g , the classical monograph by   However, this approach requires an iterative procedure for estimation, e g , a Newton-Raphson algorithm, with a suitable choice for the smoothing parameter at each iteration step 
Deriving the L ∞ rate for the resulting estimator is also not a trivial task  Instead, we suggest to employ a variance stabilizing transform of that converts the gamma regression into an approximate Gaussian regression  In the next section we present the methodology in more detail for a general setting with n ≥ 1 

Methodology

For Y i ∼ N p (0 p , Σ), i = 1,       , n, it was shown in the previous section that with Lemma 1 the data can be transformed into gamma distributed random variables     , n, j = 1,       , p, where for each fixed i the random variable W i,j has the same distribution as W j given in (3)  Now the approach of Cai et al  ( ) is adapted to the setting n ≥ 1 
First, the transformed data points W i,j are binned, that is, fewer new variables     , T   Note that the number of observations in a bin is m = np/T   In Theorem 1 in Section 4, we show that setting T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) leads to the minimax optimal rate for the spectral density estimator 
To simplify the notation, m is handled as an integer (otherwise, one can discard several observations in the last bin)  Next, applying the variance stabilizing transform (VST) ∼ where H(y) = {φ(m/2) + log (2y/m)} / √ 2 and φ is the digamma function (see   Now, the scaled and shifted log-spectral density H(f ) can be estimated with a periodic smoothing spline
where h > 0 denotes a smoothing parameter, q ∈ N is the penalty order and S per (2q − 1) a space of periodic splines of degree 2q − 1  The smoothing parameter h can be chosen either with generalized cross-validation (GCV) as derived in or with the restricted maximum likelihood, see   Once an estimator H(f ) is obtained, application of the inverse transform function H −1 (y) = m exp √ 2y − φ (m/2) /2 yields the spectral density estimator f = H −1 H(f )  
Finally, using the inverse Fourier transform leads to the fol- The precision matrix Ω is estimated by the inverse Fourier transform of the reciprocal of the spectral density estimator, i e , Ω = (ω |i−j| ) p i,j=1 with ωk = The estimation procedure for Σ and Ω can be summarised as follows  1  Data Transformation:
where D is the (p × p)-dimensional DCT-I matrix as given in Lemma 1 and D j is its j-th column  2  Binning: Set T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and calculate W i,j , k = 1,       , T 

VST:

where k are asymptotically i i d  Gaussian variables  Inverse VST: Estimate the spectral density f with f = H −1 H(f ) , where Note that Σ and Ω are positive definite matrices by construction, since their spectral density functions f and f −1 are non-negative, respectively  Unlike the banding and tapering estimators, the autocovariance estimators σk are controlled by a single smoothing parameter h, which can be estimated fully data-driven with several available automatic methods, which are numerically efficient and well-studied 
In addition, one can also use methods for adaptive mean estimation, see e g , , which in turn leads to adaptive Toeplitz covariance matrix estimation  All inferential procedures developed in the Gaussian regression context can also be adopted accordingly 

Theoretical Properties

In this section, we study the asymptotic properties of the estimators f , Σ and Ω  The results are established under the asymptotic scenario where p → ∞ and p/n → c ∈ (0, ∞], that is, the dimension p grows, while the sample size n either remains fixed or also grows but not faster than p  This corresponds to the asymptotic scenario when the sample covariance matrix is inconsistent 
Let f be the spectral density estimator defined in Section 3, i e , f = m exp{ √ 2 H(f ) − φ(m/2)}/2, where H(f ) is given in (4), m = np/T and φ is the digamma function  Furthermore, let Σ be the Toeplitz covariance matrix estimator and Ω the corresponding precision matrix defined in equations ( ) and (6), respectively 
The following theorem shows that both Σ and Ω attain the minimax optimal rate of convergence over the class and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and q = max{1, γ}, the spectral density estimator f , the corresponding covariance matrix estimator Σ and the precision matrix estimator Ω satisfy sup
For h {log(np)/(np)}   The proof of Theorem 1 can be found in the Appendix A 3 and is the main result of our work  The most important part of this proof is the derivation of the convergence rate for the spectral density estimator f under the L ∞ norm  In the original work, established an L 2 rate for a wavelet nonparametric mean estimator in a gamma regression where the data are assumed to be independent 
In our work, the spectral density estimator f is based on the gamma distributed data W i,1 ,       , W i,p , which are only asymptotically independent  Moreover, the mean of these data is not exactly f (πx 1 ),       , f (πx p ), but is corrupted by the diagonalization error given in Lemma 1  This error adds to the error that arises via binning and VST and that describes the deviation from a Gaussian distribution, as derived in  
Finally, we need to obtain an L ∞ rather than an L 2 rate for our spectral density estimator  Overall, the proof requires different tools than those used in   To get the L ∞ rate for f , we first derive that for the periodic smoothing spline estimator H(f ) of the log-spectral density  To do so, we use a closed-form expression of its effective kernel obtained in , thereby carefully treating various (dependent) errors that describe deviations from a Gaussian nonparametric regression with independent errors and mean f (πx i ) 
Note also that although the periodic smoothing spline estimator is obtained on T binned points, the rate is given in terms of the vector dimension p  Then, using the Cauchy-Schwarz inequality and a mean value argument, this rate is translated into the L ∞ rate for the spectral density estimator f   To obtain the rate for the Toeplitz covariance matrix estimator is enough to note that

Simulation Study

In this section, we compare the performance of the proposed Toeplitz covariance estimator, denoted as VST-DCT, with the tapering estimator of and with the sample covariance matrix  We consider Gaussian vectors Y 1 ,    , (3) such that the corresponding spectral density is Lipschitz continuous but not differentiable: f (x) = 1 44{| sin(x + 0 5π)| 1 7 + 0 45} 
In particular, var(Y i ) = 1 44 in all three examples  Figure shows the spectral densities and the corresponding autocorrelation functions for the three examples  A Monte Carlo simulation with 100 iterations is performed using R (version 4 1 2, seed 42)  For our VST-DCT estimator, we use a cubic periodic spline, i e , q = 2 is set in (4) 
The binning parameters are set to T = 500 bins with m = 10 points for (A) and T = 500 bins with m = 100 points for both (B) and (C)  To select the regularisation parameter for our estimator, we implemented the restricted maximum likelihood (ML) method, generalized cross validation (GCV) and the corresponding oracle versions, i e , as if Σ were known The tapering parameter
where T ap k (Σ ν 2 ) is the tapering estimator of with parameter k  If n = 1, that is, under scenario (A), suggest to split the time series Y into l non-overlapping subseries of length p/l and then proceed as before to select the tuning parameter k  To the best of our knowledge, there is no data-driven method for selecting this parameter l 
Using the true covariance matrix Σ, we selected l = 30 subseries for the example 1 and l = 15 subseries for the exam-ples 2 and 3  The parameter k can then be chosen by cross-validation as above  We employ this approach under scenario (A) instead of an unavailable fully data-driven criterion and name it semi-oracle 
Finally, for all three scenarios (A), (B) and (C), the oracle tapering parameter is computed using grid search for each Monte Carlo sample as kor = arg min k=2,3,   ,p/2 T ap k ( Σ) − Σ , where Σ is the sample covariance matrix  To speed up the computation, one can replace the spectral norm by the 1 norm, as suggested by  
In Tables , the errors of the Toeplitz covariance estimators with respect to the spectral norm and the computation time for one Monte Carlo iteration are given for scenarios (A), (B) and (C), respectively  To illustrate the goodness-of-fit of the spectral density, the L 2 norm f − f 2 is also computed 
The results show that the tapering and VST-DCT estimator perform overall similar in terms of the spectral norm risk  This is not surprising as both estimators are proved to be rate-optimal  Moreover, both the tapering and VST-DCT estimators are clearly superior to the inconsistent sample Toeplitz covariance matrix 
A closer look at the numbers shows that the VST-DCT method has better constants, i e , VST-DCT estimators have somewhat smaller errors in the spectral norm than the tapering estimators across all examples, but especially under scenario (C)  The oracle estimators show similar behaviour, but are slightly less variable compared to the data-driven estimators 
In general, both the tapering and VST-DCT estimators perform best for example 1, second best for example 3 and worst for example 2, which traces back to functions complexity  In terms of computational time, both methods are similarly fast for scenarios (A) and (B)  For scenario (C), the tapering method is much slower due to the multiple high-dimensional matrix multiplications in the cross-validation method 
It is expected that for larger p the tapering estimator is much more computationally intensive than the corresponding VST-DCT estimator  (1) polynomial σ ( (  To test how robust our approach is to deviations from the Gaussian assumption, we simulated the data from gamma and uniform distributions and conducted a simulation study for the same scenarios and examples 
The results are very similar to those of the Gaussian distribution, see supplementary materials for the details 

Application to Protein Dynamics

We revisit the data analysis of protein dynamics performed in Krivobokova et al  (2012) and   We consider data generated by the molecular dynamics (MD) simulations for the yeast aquaporin (Aqy1) -the gated water channel of the yeast Pichi pastoris  MD simulations are an established tool for studying biological systems at the atomic level on timescales of nano-to microseconds 
The data are given as Euclidean coordinates of all 783 atoms of Aqy1 observed in a 100 nanosecond time frame, split into 20 000 equidistant observations  Additionally, the diameter of the channel y t at time t is given, measured by the distance between two centers of mass of certain residues of the protein 
The aim of the analysis is to identify the collective motions of the atoms responsible for the channel opening  In order to model the response variable y t , which is a distance, based on the motions of the protein atoms, we chose to represent the protein structure by distances between atoms and certain fixed base points instead of Euclidean coordinates 
That is, we calculated where A t,i ∈ R 3 , i = 1,       , 783 denotes the i-th atom of the protein at time t, B j ∈ R 3 , j = 1, 2, 3, 4, is the j-th base point and d(•, •) is the Euclidean distance  Figure shows the diameter y t and the distance between the first atom and the first center of mass  It can therefore be concluded that a linear model Y = Xβ + holds, where
  This linear model has two specific features which are intrinsic to the problem: first, the observations are not independent over time and second, X t is high-dimensional at each t and only few columns of X are relevant for Y   have shown that the partial least squares (PLS) algorithm performs exceptionally well on this type of data, leading to a small-dimensional and robust representation of proteins, which is able to identify the atomic dynamics relevant for Y  
Singer et al  ( ) studied the convergence rates of the PLS algorithm for dependent observations and showed that decorrelating the data before running the PLS algorithm improves its performance  Since Y is a linear combination of columns of X, it can be assumed that Y and all columns of X have the same correlation structure 
Hence, it is sufficient to estimate Σ = cov(Y ) to decorrelate the data for the PLS algorithm, i e , Σ −1/2 Y = Σ −1/2 Xβ + Σ −1/2 results in a standard linear regression with independent errors  Our goal now is to estimate Σ and compare the performance of the PLS algorithm on original and decorrelated data 
For this purpose, we divided the data set into a training and a test set (each with p = 10 000 observations)  First, we tested whether the data are stationary  The augmented Dickey-Fuller test confirmed stationarity for Y with a p-value< 0 01  The Hurst exponent of Y is 0 85, indicating moderate long-range dependence supported by a rather slow decay of the sample autocovariances (see grey line in the left plot of Figure ) 
Therefore, we set q = 1 for the VST-DCT estimator to match the low smoothness of the corresponding spectral density  Moreover, the smoothing parameter is selected with the restricted maximum likelihood method and T = 550 bins are used  Obviously, the performance of the PLS algorithm on the decorrelated data is significantly better for the small number of components 
In particular, with just one PLS component, the correlation between the true opening diameter on the test set and its prediction that takes into account the dependence in the data is already 0 54, while it is close to zero for PLS that ignores the dependence in the data  showed that the estimator of β based on one PLS component is exactly the ensemble-weighted maximally correlated mode (ewMCM), which is defined as the collective mode of atoms that has the highest probability to achieve a specific alteration of the response Y  
Therefore, an accurate estimator of this quantity is crucial for the interpretation of the results and can only be achieved if the dependence in the data is taken into account  Estimating Σ with a tapered covariance estimator has two practical problems  First, since we only have a single realization of a time series Y (n = 1), there is no datadriven method for selecting the tapering parameter 
Second, the tapering estimator turned out not to be positive definite for the data at hand  To solve the second problem, we truncated the corresponding spectral density estimator ftap to a small positive value, i e , f + tap = max{ ftap , 1/ log(p)} (see   To select the tapering parameter with cross-validation, we experimented with different subseries lengths and found that the tapering estimator is very sensitive to this choice 
For example, estimating the tapered covariance matrix based on subseries of length 8/15/30 yields a correlation of 0 42/0 53/0 34 between the true diameter and the first PLS component, respectively  Altogether, our proposed estimator is fully data-driven, fast even for large sample sizes, automatically positive definite and can handle certain long-memory processes 
In contrast, the tapering estimator is not data-driven and must be manipulated to become positive definite  Our method is implemented in the R package vstdct 

Discussion

In this paper, we proposed a simple, fast, fully data-driven, automatically positive definite and minimax optimal estimator of Toeplitz covariance matrices from a large class that also includes covariance matrices of certain long-memory processes  Our estimator is derived under the assumption that the data are Gaussian 
However, simulations show that the suggested approach yields robust estimators even when the data are not normally distributed  In the context of spectral density estima- , for mixing processes (see Theorem 5 3 of Rosenblatt, 2012), as well as for non-linear processes (see   Since DFT and DCT matrices are closely related, we expect that equation (3) also holds asymptotically for these non-Gaussian time series, but consider a rigorous analysis to be beyond the scope of this paper 
In fact, our numerical experiments have even shown that if the spectral density is estimated from W j = f (πx j ) + j , that is, as if W j were Gaussian instead of gamma distributed, then the resulting spectral density estimator has almost the same L ∞ risk (and hence the corresponding covariance matrix has almost the same spectral norm) 
Of course, such an estimator would lead to a wrong inference about f (πx j ), since the growing variance of W j would be ignored  Since our approach translates Toeplitz covariance matrix estimation into a mean estimation in an approximate Gaussian nonparametric regression, all approaches developed in the context of Gaussian nonparametric regression, such as (locally)
adaptive estimation, as well as the corresponding (simultaneous) inference, can be directly applied  Bayesian tools for adaptive estimation and inference in Gaussian nonparametric regression as proposed in can also be employed 

Appendix

Throughout the appendix, we denote by c, c 1 , C, C 1 ,       etc  generic constants, that are independent of n and p  To simplify the notation, the constants are sometimes skipped and we write for less than or equal to up to constants  We embed the p-dimensional Toeplitz matrix Σ = toep(σ 0 ,       , σ p−1 ) in a (2p − 2)dimensional circulant matrix Σ = toep(σ 0 ,      
, σ p−1 , σ p−2 ,       , σ 1 )  Then, Σ = with the conjugate transpose U * , and Λ is a diagonal matrix with the k-th diagonal value for k = 1,    , p given by Furthermore, Σ = V * ΛV , where V ∈ C (2p−2)×p contains the first p columns of U   In particular, b(j, r) = b(j, 2p−r) and c(j, r) = −c(j, 2p−r) for r = p+1,      
, 2p−2  Together, we have (A 1) Some calculations show that for r = 1,       p Using the Taylor expansion of cot(x) for 0 < |x| < π one obtains for r = 1,       p where the O term does not depend on j and the hidden constant does not depend on r, p  If i = j, equations (A 1) -(A 3) imply where the O terms do not depend on j 
Since the complex exponential function is Lipschitz continuous with constant L = 1, it holds λ r = λ j + L r,j |r − j|p −1 where −1 ≤ L r,j ≤ 1 is a constant depending on r, j  Then, , it is sufficient to consider j = 1,    , p − 1  We begin with first sum  For a shorter notation, we use k := r − 1 and l := j − 1 in the following 
Then, summing the squares of the first term in (A 4) for l = 0,    , p−2 on sums of reciprocal powers  If p is even, then the residual terms are given by where φ and φ (1) denote the digamma function and its derivative  If p is odd, similar remainder terms can be derived  To see that R i (l, p) = O(p −1 ) for i = 1, 2, 3 and uniformly in l we use that asymptotically φ(x)∼ log(x)−1/(2x) and
The mixed term are both of the order p −1   Furthermore, since the harmonic sum diverges at a rate of log(p)  Finally, λ j = f (x j )+O{log(p)p −β } by the uniform approximation properties of the discrete Fourier series for Hölder continuous functions (see   All together, we have shown that (DΣD) j,j = where the O terms are uniform over j = 1,    , p 
Case i = j and |i − j| is even In this case, (DΣD) i,j = a i a j uniformly in i, j  To show that a i a j 2p−2 r=1 λ r c(i, r)c(j, r) = O(p −1 ), we proceed similarly as before  Setting k=r−1, l=j−1, m=i−1 and using that l =m and |l−m| is even, one obtains where for even p the residual terms are given by If p is odd, analogous residual terms can be derived 
Using similar techniques as before, one can show that the two residual terms and the remaining mixed and square terms vanish at a rate of the order O(p −1 ) and uniformly in i, j  Case i = j and |i − j| is odd |r − i| and |r − j| are either odd and even, or even and odd  Without loss of generality, assume that |r − i| is even 
Then, (DΣD) i,j = a i a j 2p−2 r=1 λ r b(i, r)c(j, r)  Since b(i, •) is an even function, c(j, •) is an odd function and λ r = λ 2p−r , it follows (DΣD) i,j = 0  The structure of the proof is as follows  First, we derive the L ∞ rate of the periodic smoothing spline estimator H(f )  Then, using the Cauchy-Schwarz inequality and a mean value argument, the convergence rate of the spectral density estimator f is
∞ the first claim of the theorem follows  Finally, we prove the second statement on the precision matrices  For the sake of clarity, some technical lemmas used in the proof are listed separately in A 4  hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator H(f ) described in Section 3 with q = max{1, γ} satisfies
Proof : Application of the triangle inequality yields a bias-variance decomposition Set T = 2T − 2 and x k = (k − 1)/ T for k = 1,    , T   Using Lemma 4, we can write where Mirroring and renumerating ζ k , η k , k is similar as for Y * k , k = 1,    , T   Using the above representation, one can write First we reduce the supremum to a maximum over a finite number of points 
If q > 1, then W (•, x k ) is Lipschitz continuous with constant L > 0  In this case, it holds almost surely that sup ) is a piecewise linear function with knots at x j = j/ T   The factor (ζ k + ξ k ) can be considered as stochastic weights that do not affect the piecewise linear property  Thus, the supremum is attained at one of the knots x j = j/ T , j = 1,    , T , and (A 7) is also valid for q = 1 
Again with (a + b) 2 ≤ 2a 2 + 2b 2 we obtain We start with bounding   This requires a bound on 1 • ψ 2 denotes the sub-Gaussian norm  In case of a Gaussian random variable the norm equals to the variance  Thus with Lemma 2 and Lemma 4, we obtain Lemma 1 6 of ) then yields Recall that T = p υ for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
Using the inequality log(x) ≤ x a /a one can find constants x υ , C υ > 0 depending on υ but not on n, p such that log(2 T ) log(p) Next, we derive a bound for the second term The exponential decay property of the kernel K stated in Lemma 2 yields The first term in (A 9) can be bounded again with Lemma 1 6 of  
We use the fact that for not necessarily independent random variables X 1 ,    , X N R and R > 0 are constants  This is a consequence of Lemma 1 of which yields , it follows that N i=1 a i X i has a subGaussain distribution and the subGaussian norm is bounded by 2R( N i=1 a 2 i ) 1/2   See for further details on the subgaussian distribution 
T h   For the second inequality Lemma 2(ii) is used  Applying Lemma 1 6 of then yields To bound the second term in (A 9), we use the moment bounds for ξ k derived in Lemma 4  Then, for all integers > 1 Combining the error bounds (A 10) and (A 11) and choosing R=m −1/2 gives By assumption T = p υ and m = np (1−υ) for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
If is an integer such that ≥ 1/(1 − υ), then where we used log(x) ≤ x a /a with a = 1/(4 )  Consider 1/2 < β ≤ 1 and let 0 < χ < 1 be a constant  Applying log(x) ≤ x a /a twice with a = χ/(2 ) yields For any fixed υ∈((4 − 2 min{1, β})/3, 1) one can find an integer which is independent of n, p such that the right side of (A 12) holds 
Since p/n → c ∈ (0, ∞] and thus n/p = O(1) and p −1 = O(n −1 ), it follows for satisfying (A 12) that In total, choosing an integer Using the representation in Lemma 4 once more gives for each x ∈ [0, 1] The bounds on k in Lemma 4 imply Consider the case that β ≥ 1  In particular, q = γ and f (q) is α-Hölder continuous 
Since f is a periodic function with f (x) ∈ [δ, M 0 ] and H(y) ∝ φ(m/2)+ log (2y/m), it follows that {H(f )} (q) is also α-Hölder continuous  Extending g := H(f ) to the entire real line, we get Expanding g(t) in a Taylor series around x and using that h −1 K h is a kernel of order 2q, see Lemma 2(iii), it follows that for any x ∈ [0, 1]
where ξ x,t is a point between x and t  Using the fact that the kernel K h decays exponentially and that g (q) is α-Hölder continuous on [δ, M 0 ] with some constant L, the logarithm is Lipschitz continuous on a compact interval, it follows g = H(f ) is β-Hölder continuous  Expanding g to the entire line and using Lemma 2(iii) with
In a similar way as before, one obtains Note that T −β =o(h β ) as β > 1/2, T h → ∞ and h → 0 by assumption  Since the derived bounds are uniform for x ∈ [0, 1] it holds Putting the bounds A 13 and A 14 together gives If h > 0 such that h → 0 and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator f described in Section 3 with q = max{1, γ} satisfies
Proof : By the mean value theorem, it holds for some function g between H(f ) and To show that the second term on the right hand side of (A 15) is negligible we use the moment generating function of H(f ) ∞   In the next paragraph, we derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞, where λ > 0 may depend on n, p or not 
By the exponential decay property of the kernel K stated in Lemma 2 holds First, H(f ) ∞ is bounded with the maximum over a finite number of points  Calculating the derivative of s : Since δ δx s(x) > 0 almost surely for x = x k , the extrema occur at x k , k = 1,    , T   Thus, for λ > 0 the moment generating function of H(f ) ∞ is bounded by
Let M j = ( T h) −1 T k=1 γ h (x j , x k ), which by Lemma 2 is bounded uniformly in j by some global constant M > 0  By the convexity of the exponential function we obtain √ 2 and by assumption 0 ≤ δ ≤ f ≤ M 0   Using Lemma 3, Q k can be written as a sum of m = np/T independent gamma random variables, i e 
The moment generating function of | log(X)| when X follows a Γ(a, b)-distribution is given by where Γ(a) is the gamma function and γ(a, b) is the lower incomplete gamma function  In particular, To derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞ we first establish the asymptotic order of the ratio Γ(a + t)/Γ(a) for a → ∞ 
We distinguish the two cases where t is independent of a and where t linearly depends on a  Thus, for 0 < t < a and t independent of a, equation (A 17) implies for a → ∞ that Γ(a + t)/Γ(a) = O(a t )  Similarly, it can be seen that Γ(a − t)/Γ(a) = O(a −t )  If 0 < t < a and t linearly depends on a, i e  t = ca for some constant c ∈ (0, 1), then we get Γ(a ± t)/Γ(a) = O(a ±t exp{a}) for a → ∞ 
Hence, for a fixed λ not depending on n, p and such that 0 < λ < m/( √ 2M j ) we get for sufficiently large n, p If λ = cm such that 0 < λ < m/( √ 2M j ), then for sufficiently large n, p b∈{cδ/m,cM 0 /m} (bm/2) for some constant L > 1  Set K = min j=1,   , T 1/( √ 2M j ) which is a constant independent of n, p  Altogether, we showed that for 0 < λ < Km and n, p → ∞
Bounding the right hand side of (A 15) for some constants c 0 , c 1 > 0 and n, p → ∞ Since g lies between H(f ) and H(f ), and f almost surely pointwise  Thus, for C > f ∞ = M 0 it holds where c 1 := H(C − M 0 )  Applying Markov inequality for t = cm with c ∈ (0, K) and C = 2L 4/c + M 0 where c, K, L are the constants in gives
Together with Proposition 1 follows Using the fact that the spectral norm of a Toeplitz matrix is upper bounded by the sup norm of its spectral density we get sup According to the mean value theorem, for a function g between H(f ) and H(f ), it holds that some constant c 1 > 0 not depending on n, p  Chosing the same constant C as in section A 3 2 it follows
Noting that 1/f ∞ ≤ 1/δ and 2/m exp {φ(m/2)} ∈ [0 25, 1] for m ≥ 1, (A 18) implies for some constants c 2 , c 3 > 0 and n, p → ∞ Since the derived bounds hold for each Σ(f ) ∈ F β , we get all together sup This section states some technical lemmata needed for the proof of Theorem 1  The proofs can be found in the supplementary material 
The first lemma lists some properties of the kernel K h and its extension K h on the real line  The proof is based on   Lemma 2  Let h > 0 be the bandwith parameter depending on N   (i) There are constants 0 < C < ∞ and 0 < γ < 1 such that for all for x, t ∈ [0, 1] Lemma 3 states that the sum of the correlated gamma random variables in each bin can be rewritten as a sum of independent gamma random variables 
for i = 1,    , n and j = (k − 1)m + 1,    , km, and x j = (j − 1)/(2p − 2)  Finally, Lemma 4 gives explicit bounds for the stochastic and deterministic errors of the variance stabilizing transform  Thus, it quantifies the difference to an exact Gaussian regression setting  This result is a generalization of Theorem 1 of Cai et al 
(2010) adapted to our setting with n ≥ 1 observations and correlated observations  √ 2 can be written as where for the proof of the first statement  Furthermore, for x, t ∈ [0, 1] holds In particular, for some constants C 1 , C 2 > 0 depending on γ ∈ (0, 1) but not on h and x, it holds h (iii) See Lemma 15 of with p = 2q − 1 
It is sufficient to show the statement for n = 1 by independence of the Y i   Then, the number of points per bin is m = p/T   For simplicity, the index i is skipped in the following  First, we write Q k as a matrix-vector product and refactor it so that it corresponds to a sum of independent scaled χ 2 random variables 
In the second step, we calculate the scaling factors  Let E (km) be a diagonal matrix with ones on the (k − 1)m + 1,    , km-th entries and otherwise zero diagonal elements  Then, By Theorem 1 of for the gamma distribution it follows where Wi,j iid  ∼ Γ(1/2, 2 f (x * k )) and such that Cov( Wi,j , Wi,h ) = Cov(W i,j , W i,h ) for j = (k − 1)p/T + 1,    , kp/T and h ∈ {1,    , p} \ {(k − 1)p/T + 1,    , kp/T } 
Let θ be the maximum difference of the observations' means in each bin  Then, θ = max are defined via quantile coupling, it holds Z k = Φ −1 {F Q( Qk )} (see   Furthermore, define the uniform random variables Let ρ = Cov(Z k , Z l )  Then, the identity implies F Z,Z (x, y) − Φ(x)Φ(y) ≥ 0 for all x, y ∈ R ⇐⇒ ρ ≥ 0, (see  
Since Cov( Qk , Ql ) ≥ 0 and the ratio of two densities is non-negative, x = − 2/m, it follows that f Q(x) is monotone decreasing for x ≥ − 2/m  Furthermore, F Q(− m/2) ≤ 0 5 for all m ∈ N as f Q(x) is right-skewed  In particular, − m/2 ≤ F −1 Q (1/2) for all m ∈ N  Finally, since f Q(− 2/m) → φ(0) for m → ∞ there is a constant c > 0 not depending on m such that
The simulation study in Section 5 is performed in the same way, but with the uniform and the gamma distribution instead of the Gaussian distribution 



ground_truth: A subset that form via kinetic rate processes from state-selected Rydberg gases.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the purpose of the baseline in the layout procedure?

Read the following text and answer briefly.

Context 2: 
Early tests revealed cooling problems, which prompted a radiator move from the firewall to a lower cowling location 
The first flight was almost a disaster, as test pilot Randy Smith lost power right after takeoff  He managed a 180 with a safe downwind landing with only minor nosewheel pant damage  The culprit proved to be a spark plug with too much reach, which was quickly remedied  Subsequent flights have shown water temp to be about 210 degrees, oil temp is 220-230, and airspeed is about 180 mph 
Shopping for the Partially Built KR 
This story starts about twenty years ago when I first started looking at the KR-2 as the plane I'd like to build  The only problem at that time was a lack of money, lack of knowledge, and a lack of job stability  I liked the design, except for the low ground clearance of the retractable gear and that a KR was going to be a tight fit for me to fly 
Over the past twenty years I've owned a number of planes, but still always wanted to build my own  I needed one that would fit me, my budget requirements, and have the speed and performance that I wanted  When "KITPLANES" published the article featuring Roy Marsh's new KR-2S, it was the first I had heard of any major modifications or improvements to the same old KR design  I believe that article and Roy Marsh's workmanship have probably been the greatest boon to Rand Robinson (RR) in the last twenty years  It certainly caught my eye! Here was the same design I had decided I wanted to build twenty years ago, with all of the improvements I wanted  It was sitting on fixed gear with some reasonable ground clearance  It had the capability to be built large enough to accommodate me  It has enough prefab parts available that it didn't have to be 100% scratch built if I decided to hurry the project along  And it had the speed I wanted  I knew that Roy's published speeds were probably not realistic expectations for the average KR, but after knocking around for the last three years in my Champ, anything over 90 mph seems pretty fast to me 
After purchasing the info kit and the sales video from Rand Robinson, the next step after deciding for sure to build this plane was to order the KR-2 plans and the KR-2S addendum  I finally got my plans and was putting together my first order to start the plane, when my partner in the Champ pointed out that there was a partially completed KR-2S for sale in Trade-a-plane  My initial answer was "No, I don't even want to look at it  I want to build my own from scratch " My partner insisted that for the advertised price and the fact that it wasn't too far away, I ought to at least give the guy a call and investigate it  "No, I don't think I want to buy someone else's problems," I persisted  That night I went home and crunched up some numbers on the calculator and finally came to the conclusion that for the sake of my budget for the next several years, I really should give this guy a call 
Three days later, I flew to his place about 400 miles away to take a look at his project  At this point I should probably mention that I consider myself to be fairly knowledgeable about airplane construction, although the vast majority of my experience is with tube and fabric  The rest of this article deals with what I looked for and more importantly what I missed and have had to repair in the last year since I purchased the project 
When we went to the seller's house, I found that the left wing was built using the Dan Diehl wing skins and the right wing skins were leaning against the wall inside the house  Also the canopy was in the house with the canopy covered with paper and tape  I wanted to inspect the fuselage first, so off we went to the shop 
There I found a fuselage sitting on it's gear painted in primer gray  The first step was to inspect the quality of workmanship of what could be seen as it sat  The interior of the fuselage looked as if it had been built with a great deal of care  The fit and finish of all of the interior wood was very nice  Even the gussets looked like they had been painstakingly perfectly fitted  The glass work on the turtle back also looked very precise and clean  It was evenly faired into the vertical and horizontal stabs  The tail also appeared to be well built with the exception of a depression directly over the front and rear spars in the horizontal stabs  He explained that when he moved recently, that he had shot the plane with gray primer to protect it from the weather since he wouldn't have ready access to a shop to put it in right away  It ended up sitting out in the hot south Texas summer sun for a few weeks before he got a shop rented to work in  That caused the glass (or possibly the foam inside the horizontal stab) to swell, except that it held onto the spar, so it was slightly ballooned in front of and behind the spars  His recommendation was to fill it back smooth with micro 
I also found a small linear crack in the lower left wing spar cap on the left wing stub  It appeared to be from over tightening the rear spar wing attach fitting bolts  His explanation was that the crack wasn't important because the rear spars only job is to keep the wings from folding back  I also noticed that the holes for attaching the outer wing to the wing stub were badly rounded out on the rear spar  He explained that the Diehl wing skins require the rear spar to be swept slightly more forward than the stock wings  This won't allow you to use the rear spar attach fittings from RR and that I would need to fabricate a new set of rear spar attach fittings 
I also found that the aileron bellcranks were not built or installed as per plans, but found that they looked professional  I couldn't check for function since the right bellcrank and sheeve wasn't installed, the left wing also wasn't installed, and the right wing didn't exist yet 
Next we pulled the inspection panels off of the fuselage and tail and looked at everything I could see with a good flashlight  I didn't find anything else that might be questionable about the fuselage except for a cracked elevator trim tab that was damaged when it fell off it's hanging place on the wall 
Next we spent some time going over his builders log and builders photo album  I still hadn't seen anything that would dissuade me from buying this project 
At this point it was starting to get late and my ride down needed to get airborne for the flight home  I needed to make a decision about whether I wanted this project or not, but I hadn't inspected the wings and canopy yet  I took a cursory look at the left wing and saw lots on micro built up on it and some bubbles in the leading edge, but nothing that looked seriously wrong to my amateur eye  The right wing was only a set of spars in the shop and the Diehl wing skins in the house, so there wasn't much to look at there  The canopy was wrapped in paper and tape, so there wasn't much to look at there either  I decided that even if there were serious problems in the wing that was built, I would be money ahead to go ahead and buy the project  For the advertised price, I could build a new set of wings and still be way ahead financially  We negotiated a final price, shook hands, took my ride to the airport, and started off in search of a U-haul to haul the project home 
Now, at this point, some of you are thinking about what I surely must have forgotten to inspect and why didn't I take a local A & P or EAA member along for the ride  First of all, I don't know any mechanics locally that have any experience with glass and our EAA chapter of which I am VP is woefully lacking in fiberglass knowledge  Secondly, as you will see, I missed plenty  Some by ignorance, some by just not looking close enough 
Now for a list of the problems that I found over the last year and a few of the fixes that I came up with 
I found that the lower set of rear spar attach fittings on the left rear spar were installed backwards with the longer spaced hole towards the fuselage  Since this is the same place that also had the cracked spar cap, it required a major change  Also in the same area he had drilled through the rear spar with a hole saw to create a place for the aileron cable to pass through and managed to cut out the second from the outside vertical brace in the spar  Then he chose to install the aileron bellcranks in front of the rear spar, and cut another hole through the rear spar for the aileron push rod  He also managed to cut out the outside vertical brace in the spar  Since the holes were already drilled through the spar, the choices were to either cut out that section of spar cap and scarf a new piece in, cut the whole rear spar carrythrough out of the fuselage including ruining the left lower wing skin, or do something else creative to reinforce the spar cap and install a custom built set of attach fittings 
I also found that after I built and installed the right side wing stub ribs and skin that the aileron bellcrank setup would not work as installed  The cable that crosses between the two bellcranks had a sharp uphill from the sheeve to the bellcrank in the last 12 inches on either side  This combined with the radius that the bellcranks turn caused the cross cable to pull up tight when the ailerons were pushed to either end of their travel, but allowed the cables to go very slack when the ailerons were centered  Also the Aileron pushrods needed to pass directly through the lower set of rear wing attach fittings to attach to the aileron  This whole rear spar and aileron bellcrank setup was going to either have to be redesigned or cut out and built to plans  The bottom line is that the problems I observed when I inspected this part were much more serious than expected when I had to fix it 
I decided that I had to remove the rear fittings from the left wing to be replaced with the new set that my neighborhood machinist was cutting out for me  When I put the wing on the work bench to start removing the rear fittings, I thought I had better take a closer look at the bubbles in the leading edge  I found that as I pushed on the leading edge, it delaminated between the glass lay-up on top and the upper and lower wing skin edges that were floxed together underneath  I concluded that that area had to come apart and took a belt sander to the leading edge  What I found was that the leading edge had been floxed together and glassed over, but the mold release had never been scrubbed off the leading edge of the wing  It peeled apart for rebuild quite easily 
When I got back to removing the rear spar attach fittings, I noticed that the woodwork inside the wing looked awfully dull  The reason was that the wing had been closed up without varnishing any of the woodwork  This was rectified with a small hole saw, a number of extensions and a modified undercoating sprayer 
I also found that the aluminum drain fitting in the bottom of the left wing tank had been glassed into place upside down  The tapered pipe threads were tapered the wrong way to install the draincock into the tank  Retapping the fitting the right direction seemed to be a good fix for that problem 
When I finally got around to attaching the wing to the fuselage, I found that the front spar attach fittings were badly misaligned  Although they could be forced into alignment, I didn't think I needed that kind of preload on the main spar fittings  This problem was fixed by calling on my local neighborhood machinist to build me an aligning fixture and reaming the attach holes to the next larger size and ordering the new sized bolts 
On the fuselage I found that although it had new Cleveland wheels and brakes on it, one of the brakes had a severe wobble to it  I must complement the manufacturers for taking care of that problem  One call to the Cleveland factory and they shipped me a new set of wheels and brakes even though the receipt for this set was over four years old and in the original builders name  Their only concern was that this set had never been placed in service yet 
I chose to sand the load of micro off the left wing to see what it was covering  When I got down to the glass, I found that there was no glass for the aft inch and a half of the underside of the wing in front of the aileron hinge  With the Diehl wing skins, you build the wings, then cut the ailerons out of trailing edge of the wing  He had mismeasured and cut too much material off the bottom side of the trailing edge in front of the aileron  It was filled by floxing a piece of spruce into the gap to fill the space between the back edge of the fiberglass and the aileron mount  I chose to wrap the trailing edge of that wing, and the other wing to match with a couple of lay-ups of glass 
When I sanded the primer off the aforementioned damaged trim tab, I found that the hinge was floxed to the leading edge of the foam insides of the tab, but not the glass  I also chose to wrap the front of the trim tab with a lay-up of glass 
I decided to pull the paper off the canopy and take a look at it before I'm ready to bolt it on and fly  The original builder had blown his own canopy and after some of the previous problems, I was beginning to have some concerns about not having looked it over closely enough  The canopy turned out to have been blow a little too large  It ended up with a little larger bubble for headroom, which I didn't object to  However, it had more headroom on the right side than the left  Yes, it was just a little bit lopsided  The main problem was that the canopy is stretched thin enough that it can be easily pushed in with one hand when the weather is warm   My fear was that this is just thin enough that it may decide to lay on my head or in my lap when flying on a warm day  It will have to be replaced 
I'm sure that many that are reading this could see several of the potential problems before I mentioned them, but some others may not have and I'm sure that there could have been many other problems that didn't but could have existed on this project  This is also not intended to be critical of the gentleman that started this project as many parts of it, especially the wood work are better than I could have done and much of his work is outstanding  I prefer to think that I'll end up with a better plane with his woodwork combined with my glasswork  This article is intended to feature some of the problems that you may run into in buying someone else's project 
The final question is, knowing what I have found over the past year, would I have still purchased this project  The answer is yes, but primarily because the price was right in that I am still money and work ahead of where I would be if I had started the project from scratch  There are a few things that I would have done differently, but nothing that I can't live with  Although I won't be able to say that I built it all from scratch, I have built and rebuild enough of the plane that I should have no problem qualifying under the 51% rule 
You can send comments directly to the author via e-mail at "jscott@LANL GOV" 
Here is an brief explanation of how I built my turtledecks  The jig was constructed from scrap plywood and a few 1x4s that I ripped into stringers  I made two temporary bulkheads from the plywood, one for each end  Remember the forward bulkhead needs to be shaped in a way that will closely match the aft end of your canopy frame  Make an aft bulkhead by placing a straight edge at the top of your forward bulkhead and the trailing edge of your horizontal stabilizer  This will give you an idea of how tall your aft bulkhead needs to be  As far as location, I placed my aft bulkhead just forward of the lower/front of my vertical fin  I constructed the jig on the fuselage, it is glued together with automotive bondo 
After the bulkheads were bondoed to the fuselage I used the stringers that I ripped from the 1x4s and bondoed them to the bulkheads  This gave me a male form to cover with thin plastic or posterboard  I stapled two layers of posterboard to the jig(thin plastic would work better)  The posterboard wraps down two inches onto the fuselage  After I was satisfied with the way it looked, I then covered the entire thing with duct tape (fiberglass will not stick to duct tape) On top of this I wetout one layer of tri-ply cloth (22oz) that I had left over from an earlier project, and one layer of 8oz  bid  Remember to mask off your fuselage so you don't get epoxy on it  If you are not familiar with composite lay-ups, you should plan on razor cutting your lay-ups 4 to 6 hours after wetout while the lay-up is still soft enough to cut with a razorblade 
After the lay-up cured (2 or 3 days) it was removed from the jig, and the jig was removed from the fuselage and discarded  (be careful, the bondo sticks very well to the spruce, you could splinter your wood during removal) I now have a fiberglass skin that tends to hold the shape of the jig but is still flexible enough to work with  I made two bulkheads out of 1/4 last-a-foam (AS&S) using the plywood formers from the jig as a guide  I covered these foam bulkheads with one 8oz layer of glass on each side, with a glass to glass edge on the bottom  After cure these bulkheads were bondoed into place (to the fuselage)and the fiberglass skin was pulled down tight and floxed to the bulkheads  When the flox cured the bondo joints were broken, again being careful not to harm the wood  The turtledeck was removed from the fuselage and 2 inch tapes added to the bulkheads inside and out 
At this point the turtledeck looked great and only weighed about 5lbs  but I noticed you could deform the skin by pushing hard on the outside  So I flipped the turtledeck over and from 1/4 inch last-a-foam, I cut two inch wide strips that would run the entire length, forward and aft inside the turtledeck  In effect these would act as composite stringers, I made enough of these two inch wide strips to make up three stringers  One down the center (sort of a backbone) and one on each side of the "backbone" half the distance to the edge of the turtledeck  I sanded the edge of the foam so that when covered with a layer of bid @ 45degrees there would be a nice transition from the turtledeck skin up onto the foam and then back onto the turtledeck I scuff sanded and glued the foam stringers in with micro  I covered the foam stringers with one layer of 8oz bid @ 45degrees 
You can also send me email at: mikemims@pacbell net if you have any questions or want to share your ideas 
KROnline is an online KR Newsletter devoted to sharing KR information with other builders and pilots in a timely manner  The first issue (September 96) is now available as a zipped MicroSoft Word file at http://members aol com/bshadr or as an html document at kronline9 html  If you'd like to submit articles or photos, email Randy Stein at BSHADR@aol com ------------------------------------------------------------ Don't bother to email Randy though  KROnline has been retired since the KR Newsletter has improved 



ground_truth: The baseline is used as a reference for the mid point of the firewall for the developed side panel.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the advantage of decorrelating the data before running the PLS algorithm?

Read the following text and answer briefly.

Context 4: 1 & 185 \\
& DSA & 1470 & 33246 \\
& FPSA & 0 438 & 8 \\
& NFPA & 0 484 & 8 \\ \hline  
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with SRK}
\label{SRKresults2} 
\end{table}

The results of all solvers are shown in \cref{SRKresults1,SRKresults2} 
We see that NFPA and FPSA tremendously outperform GMRES and DSA in runtime for all cases 
FPSA is a simpler method than NFPA, requiring less calculations per iteration; therefore, it is expected that it outperforms NFPA in runtime 
We see a reduction in runtime and iterations for FPSA and NFPA as the FP limit is approached, with DSA and GMRES requiring many more iterations by comparison as $\eta$ approaches 0 

An advantage that NFPA offers is that the angular moments of the flux in the LO equation will remain consistent with those of the transport equation even as a problem becomes less forward-peaked 
On the other hand, the moments found using only the FP equation and source iteration lose accuracy 
To illustrate this, Problem 1 was tested using different Screened Rutherford Kernels with increasing $\eta$ parameters 
The percent errors (relative to the transport solution) for the scalar flux obtained with the LO equation and with the standard FP equation at the center of the slab are shown in \cref{momcomp} 
It can be seen that the percent relative errors in the scalar flux of the FP solution is orders of magnitude larger than the error produced using the LO equation 
The same trend can be seen when using the exponential and Henyey-Greenstein kernels  

\begin{figure}[H]
\begin{center}
  \includegraphics[scale=0 15,angle=0]{relerrorlog jpg}
  \caption{Log Scale of $\%$ Relative Error vs $\eta$ for Problem 1 at the Center of the Slab with SRK}
  \label{momcomp}
\end{center}
\end{figure}

\subsubsection{EK: Exponential Kernel}

The exponential kernel \cite{pomraning2, JapanFPSA} is a fictitious kernel made for problems that have a valid Fokker-Planck limit \cite{pomraning1} 
The zero$^{\text{th}}$ moment, $\sigma^{EK}_{s,0}$, is chosen arbitrarily; we define $\sigma^{EK}_{s,0}$ as the same zero$^{\text{th}}$ moment from the SRK 
The $\Delta$ parameter determines the kernel: the first and second moments are given by 
\begin{subequations}
\begin{align}
\sigma^{EK}_{s,1} &= \sigma^{EK}_{s,0} (1-\Delta),\\
\sigma^{EK}_{s,2} &= \sigma^{EK}_{s,0} (1-3\Delta+3\Delta^2),
\end{align}
and the relationship for $l\geq 3$ is
\begin{equation}
\sigma^{EK}_{s,l} = \sigma^{EK}_{s,l-2} - \Delta(2l+1) \sigma^{EK}_{s,l-1} 
\end{equation}
\end{subequations}
As $\Delta$ is reduced, the scattering kernel becomes more forward-peaked 

The EK has a valid FP limit as $\Delta$ approaches 0 \cite{patelFBR} 
Three different values of $\Delta$ were used to generate the scattering kernels shown in \cref{EXP} 
The generated scattering kernels are shown in \cref{EXP} 
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2 
\Cref{EK_plots} shows the solutions for EK with $\Delta = 10^{-7}$ 
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0 1,angle=0]{EXP jpg}
  \caption{Exponential Kernels}
  \label{EXP}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{dta7_iso jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{dta7_beam jpg} }}
    \caption{Results for EK Problems with $\Delta = 10^{-7}$}
    \label{EK_plots}
\end{figure}

The runtimes and iterations for GMRES, DSA, FPSA, and NFPA are shown in \cref{Expresults1,Expresults2} 
We see a similar trend with the EK as seen with SRK 
Smaller $\Delta$ values lead to a reduction in runtime and iterations for NFPA and FPSA, which greatly outperform DSA and GMRES in both categories 

\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\Delta = 10^{-5}$} & GMRES & 196 & 142 \\
& DSA & 3110 & 70140 \\
& FPSA & 0 514 & 11 \\ 
& NFPA & 0 630 & 11 \\\hline 
\multirow{4}{*}{$\Delta = 10^{-6}$} & GMRES & 156 & 132 \\
& DSA & 3120 & 70758 \\
& FPSA & 0 388 & 7 \\ 
& NFPA & 0 393 & 7 \\ \hline 
\multirow{4}{*}{$\Delta = 10^{-7}$} & GMRES & 81 & 127 \\
& DSA & 3120 & 70851  \\
& FPSA & 0 292 & 6 \\ 
& NFPA & 0 318 & 6 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with EK}
\label{Expresults1} 
\end{table}
\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\Delta = 10^{-5}$} & GMRES & 110 & 73 \\
& DSA & 1455 & 33033 \\
& FPSA & 0 492 & 10 \\ 
& NFPA & 0 613 & 10 \\ \hline 
\multirow{4}{*}{$\Delta = 10^{-6}$} & GMRES & 82 7 & 79 \\
& DSA & 1470 & 33309 \\
& FPSA & 0 358 & 7 \\ 
& NFPA & 0 431 & 7 \\ \hline 
\multirow{4}{*}{$\Delta = 10^{-7}$} & GMRES & 56 8 & 90 \\
& DSA & 1470 & 33339 \\
& FPSA & 0 273 & 5 \\ 
& NFPA & 0 319 & 5 \\ \hline  
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with EK}
\label{Expresults2} 
\end{table}

\subsubsection{HGK: Henyey-Greenstein Kernel}

The Henyey-Greenstein Kernel \cite{HGK,JapanFPSA} is most commonly used in light transport in clouds 
It relies on the anisotropy factor $g$, such that
\begin{equation}
\sigma^{HGK}_{s,l} = \sigma_s g^l 
\end{equation}
As $g$ goes from zero to unity, the scattering shifts from isotropic to highly anisotropic 
\begin{figure}[H]
\begin{center}
  \includegraphics[scale=0 1,angle=0]{HGK jpg}
  \caption{Henyey-Greenstein Kernels}
  \label{HGK}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{g099_iso jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{g099_beam jpg} }}
    \caption{Results for HGK Problems with $g = 0 99$}
    \label{HGK_plots}
\end{figure}


The HGK does not have a valid FP limit \cite{patelFBR} 
The three kernels tested are shown in \cref{HGK} 
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2 
\Cref{HGK_plots} shows the solutions for HGK with $g = 0 99$ 
The results of each solver are shown in \cref{HGKresults1,HGKresults2}  
\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$g=0 9$} & GMRES & 9 88 & 76 \\
& DSA & 24 5 & 554 \\
& FPSA & 1 50 & 32 \\ 
& NFPA & 1 39 & 27 \\ \hline 
\multirow{4}{*}{$g=0 95$} & GMRES & 12 2 & 131 \\
& DSA & 47 7 & 1083 \\
& FPSA & 1 75 & 38 \\ 
& NFPA & 1 83 & 35 \\ \hline 
\multirow{4}{*}{$g=0 99$} & GMRES & 40 0 & 27 \\
& DSA & 243 & 5530  \\
& FPSA & 3 38 & 74 \\ 
& NFPA & 3 93 & 73 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with HGK}
\label{HGKresults1} 
\end{table}
\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$g=0 9$} & GMRES & 24 3 & 135 \\
& DSA & 14 8 & 336  \\
& FPSA & 1 15 & 23 \\ 
& NFPA & 1 35 & 24 \\ \hline 
\multirow{4}{*}{$g=0 95$} & GMRES & 31 3 & 107 \\
& DSA & 29 7 & 675 \\
& FPSA & 1 56 & 32 \\ 
& NFPA & 1 90 & 33 \\ \hline 
\multirow{4}{*}{$g=0 99$} & GMRES & 41 4 & 126 \\
& DSA & 146 & 3345 \\
& FPSA & 3 31 & 67 \\ 
& NFPA & 3 99 & 67 \\ \hline  
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with HGK}
\label{HGKresults2} 
\end{table}

Here we see that NFPA and FPSA do not perform as well compared to their results for the SRK and EK 
Contrary to what happened in those cases, both solvers require more time and iterations as the problem becomes more anisotropic 
This is somewhat expected, due to HGK not having a valid Fokker-Planck limit 
However, both NFPA and FPSA continue to greatly outperform GMRES and DSA 
Moreover, NFPA outperforms FPSA in iteration count for problem 1 


\section{Discussion}\label{sec4}

This paper introduced the Nonlinear Fokker-Planck Acceleration technique for steady-state, monoenergetic transport in homogeneous slab geometry 
To our knowledge, this is the first nonlinear HOLO method that accelerates \textit{all $L$ moments} of the angular flux 
Upon convergence, the LO and HO models are consistent; in other words, the (lower-order) modified Fokker-Planck equation \textit{preserves the same angular moments} of the flux obtained with the (higher-order) transport equation 

NFPA was tested on a homogeneous medium with an isotropic internal source with vacuum boundaries, and in a homogeneous medium with no internal source and an incoming beam boundary 
For both problems, three different scattering kernels were used 
The runtime and iterations of NFPA and FPSA were shown to be similar 
They both vastly outperformed DSA and GMRES for all cases by orders of magnitude 
However, NFPA has the feature of preserving the angular moments of the flux in both the HO and LO equations, which offers the advantage of integrating the LO model into multiphysics models  

In the future, we intend to test NFPA capabilities for a variety of multiphysics problems and analyze its performance 
To apply NFPA to more realistic problems, it needs to be extended to include time and energy dependence  
Additionally, the method needs to be adapted to address geometries with higher-order spatial dimensions 
Finally, for the NFPA method to become mathematically ``complete", a full convergence examination using Fourier analysis must be performed 
However, this is beyond the scope of this paper and must be left for future work 

\section*{Acknowledgements}

The authors acknowledge support under award number NRC-HQ-84-15-G-0024 from the Nuclear Regulatory Commission 
The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the view of the U S  Nuclear Regulatory Commission 

J ~K  Patel would like to thank Dr ~James Warsa for his wonderful transport class at UNM, as well as his synthetic acceleration codes 
The authors would also like to thank Dr ~Anil Prinja for discussions involving Fokker-Planck acceleration 







ground_truth: Decorrelating the data before running the PLS algorithm improves the performance of the algorithm.