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\section{Introduction}
\IEEEPARstart{E}{nvironmental} sound recognition is an important ability of an
individual to quickly grasp useful information from ambient environment, the success of which can lead to prompt actions to be taken before potential opportunities or dangers, and thus determine the fitness of the individual. For example, a prey may realize the approach of a predator by the sounds of breaking twigs even without a vision. A successful recognition of such sounds often means its survival. Human and other animals are very good at recognizing environmental sounds. This extraordinary ability has inspired more efforts being devoted to endow artificial systems with a similar ability for environmental sound recognition (ESR) \cite{wang2006computational,cowling2003comparison,sharan2016overview,chachada2014environmental}, which can be also referred as automatic sound recognition (ASR).
ESR has attracted increasing attention in recent years from the field of acoustic signal processing \cite{o2008automatic,cowling2003comparison,sharan2016overview}, as well as neuroscience \cite{leaver2010cortical}. Similar to other well studied tasks such as speech or music recognition, ESR aims to recognize a specific sound automatically from the environment. Differently, the chaotic and unstructured difficulties residing in the sound signals make ESR a challenging and distinctive task.
In addition, its practical importance has been reflected by a number of various newly applied developments or attempts including, but not limited to, bioacoustic monitoring \cite{weninger2011audio},
surveillance \cite{ntalampiras2009acoustic} and general machine hearing \cite{lyon2010machine}. Successful recognition of critical sounds like gunshots can send an early alarm to crowd and police, and thus help to save more lives and minimize losses.
An ESR system can provide machines like a robot \cite{pineau2003towards} a cheap and advantageous way to understand the environment under poor visual conditions such as weak lighting or visual obstruction. Compared to vision-based processing, audio signals are relatively cheap to compute and store, which brings benefits of high efficiency and low-power consumption.
Both research challenges and advantages have motivated studies designed for ESR systems.
A general approach to pattern recognition tasks can be used to ESR. The approach typically contains three key steps \cite{sharan2016overview,Yu2013TNN} which are signal preprocessing, feature extraction and classification. These steps are tightly jointed to facilitate the functionality as a whole: signal preprocessing aims to prepare sound information for a better feature extraction which will then improve the performance of the classification.
In its primitive phases, ESR algorithms were simple reflections of speech and music processing paradigms \cite{cowling2003comparison,sharan2016overview,chachada2014environmental,huang2001spoken}, but divergence emerges as considerably non-stationary characteristics of environmental sounds are taken into account. The recognition performance of ESR systems largely depends on the choice of two essential components, i.e. feature(s) and classifier(s).
Research efforts have been made to them, as well as different combinations of methods from each. Different approaches can thus be categorized by methods being adopted for each component.
Features developed for speech processing are often used for ESR \cite{mitrovic2010features}. Statistical features are introduced to give descriptions of the sound signal in terms of psychoacoustic and physical properties such as power, pitch and zero-crossing rate, etc. These features are often only used as supplementary ones in ESR systems \cite{rabaoui2008using}. Cepstral features such as Mel-Frequency Ceptral Coefficients (MFCC) and spectrogram images are the most frequently used ones. The frame-based MFCC features are more favorable for modeling single sound sources but not for environmental sounds which typically contain a variety of sources \cite{chu2009environmental}. In addition, MFCC features are modeled from the overall spectrum which makes them vulnerable to noise \cite{dennis2013temporal}. On the other hand, spectrogram images are good at describing acoustic details of the sound from both the time and frequency domain \cite{dennis2011spectrogram}, but high dimensionality of the feature restricts its applicability \cite{sharan2016overview,chachada2014environmental}. There has been increasing number of advanced and sophisticated feature representations, such as stabilized auditory image \cite{lyon2010sound} and matching pursuit \cite{chu2009environmental}. Some other works \cite{dennis2013temporal,wu2018spiking} construct representations by utilizing additional feature extraction methods like self-organizing map (SOM). However, complexity of these feature representations is one of the major drawbacks. Recently, a simpler and more efficient feature representation method for sounds has been introduced by using local time-frequency information (LTF) \cite{xiao2018spike}. We will continue to contribute toward this method with simplicity, sparseness and flexibility bearing in mind.
Various classifiers have been successfully applied to ESR tasks in recent years.
The most commonly used classifiers \cite{kolozali2013automatic,chu2009environmental,lu2003content} include multi-layer perceptron (MLP), k-nearest neighbor (kNN), Gaussian mixture model (GMM) and support vector machines (SVMs). These classifiers continue to be used with modifications or a hybrid of classification algorithms \cite{sharan2016overview,chu2009environmental}, but they ignore the temporal information of sound signals. The hidden Markov model (HMM) was then applied to capture the temporal structure for a better performance \cite{dennis2011spectrogram}. However, HMM do not model explicitly the diverse temporal dependencies of environmental sounds \cite{dennis2013temporal}, leading further research towards a more complete modeling of the temporal structure.
In recent years, artificial neural networks (ANNs) with a class of techniques called deep learning have been thriving with a great success in various recognition tasks \cite{lecun2015deep}. Two of the most popular deep learning structures are deep neural network (DNN) and convolutional neural network (CNN), which have been successfully applied to ESR tasks very recently \cite{piczak2015environmental,zhang2015robust,ozer2018noise,mcloughlin2015robust}.
One of the major challenges of the aforementioned classifiers is the biological plausibility. Human brain is remarkably good at various cognitive tasks, including sound recognition, with extraordinary performance in terms of accuracy, efficiency, robustness and adaptivity, etc. How to transfer these advantageous abilities of the brain to artificial systems for solving ESR tasks motivates our study in this work.
\begin{figure}[!htb]
\centering\includegraphics[width=0.48\textwidth]{framework-demo.pdf}
\caption{Overall framework of the multi-spike neural network for sound recognition.
\textbf{A}, information processing structure of the system. \textbf{B}-\textbf{E}, demonstration of information in different systematic components: spectrogram images (\textbf{B} and \textbf{C}), spatiotemporal spike pattern (\textbf{D}) and dynamics of neuron's membrane potential (\textbf{E}).}
\label{Fig:framework}
\end{figure}
Neurons in the brain use spikes, also called electrical pulses, to transmit information between each other \cite{kandel2000principles}. The discrete feature of spikes is believed to play an essential role in efficient computation, which has inspired a group of neuromorphic hardware implementations \cite{benjamin2014neurogrid,merolla2014million,yao2017face}. In spite of these hardware developments, how could spikes convey information still remains unclear. A sequence of spikes could encode information either with the total number of spikes or their precise spike timings, representing two of the most popular neural coding schemes, i.e. rate and temporal codes, respectively \cite{kandel2000principles,dayan2001theoretical,YuNCS,Panzeri10}. The rate code ignores the temporal structure of the spike train, making it highly robust with respect to interspike-interval noise \cite{stein2005neuronal,london2010sensitivity}, while the temporal code has a high information-carrying capacity as a result of making full use of the temporal structure \cite{Hopfield95,Richard98,Borst99}. Although an increasing number of experiments have been shown in various nervous systems \cite{london2010sensitivity,kandel2000principles,gollisch2008rapid,butts2007temporal} to support different codes, it is still arguable whether the rate or temporal code dominates information coding in the brain \cite{masuda2002bridging,gutig2014spike,yu2018spike}.
Different learning algorithms have been developed to better understand the underlying computing and processing principles of neural systems. One of the most widely studied rules is spike-timing-dependent plasticity (STDP) \cite{dan2004spike,song2000competitive} which instructs neurons to update their synaptic efficacies according to the temporal difference between afferent and efferent spikes. Dependence of temporal continuity hinders its development to act as an appropriate classifier \cite{gutig06,Yu2013TNN}.
The tempotron learning rule \cite{gutig06} is proposed to discriminate target and null patterns by firing a single spike or keeping silent. The firing-or-not behavior of the tempotron makes it an efficient learning rule, but constrains its ability to utilize the temporal structure of the neuron's output \cite{yu2018spike}.
Some other learning rules are proposed to train neurons to fire spikes at desired times \cite{bohte02spikeprop,ponulak10,florian2012chronotron,mohemmed2012span,YuQ2013PSD,memmesheimer2014}.
Although the temporal structure could be utilized by precise output spikes, designing an instructor signal with precise timings is challenging for both biological and artificial systems. Additionally, these learning rules are developed under the assumption of a temporal code, resulting that most of them cannot be generalized to a rate code \cite{yu2018spike,brette2015philosophy}. In \cite{gutig2016}, a novel type of learning rule is developed to train neurons to fire a desired number of spikes rather than precise timings. This multi-spike learning rule thus provide a new way to overcome limitations of the other ones. Improved modifications have been developed in \cite{yu2018spike,yu2018iconip}, along with detailed evaluations of different properties as well as theoretical proofs.
How to adopt the biologically plausible network, i.e. spiking neural network (SNN), to the ESR task demands more efforts. Previous related works \cite{dennis2013temporal,wu2018spiking,xiao2018spike} to this research problem have demonstrated the advantages of the spike-based approach. Following a general processing structure with SNN \cite{Yu2013TNN,yu2016spiking}, the framework normally consists of three functional parts, i.e. encoding, learning and readout.
In \cite{dennis2013temporal,wu2018spiking}, the encoding depends on an SOM model, which could complicate the process and thus degrade computing efficiency. In addition, the biologically plausible implementation of this SOM model remains challenging, let alone a precise time reference to each segmentation frame for encoding spikes \cite{wu2018spiking}. In the learning part, the approaches of \cite{dennis2013temporal,wu2018spiking,xiao2018spike} are based on a binary-spike tempotron rule, which will limit neurons' capability to fully explore and exploit the temporal information over the presence of sounds. The readout in \cite{dennis2013temporal,wu2018spiking} relies on a voting scheme over the maximal potential. This means a recorder is required for tracking this maximum, and thus the efficiency and effectiveness of the readout is degraded.
In this work, we propose a spike-based framework (see Fig.~\ref{Fig:framework}) for the ESR task by combining a sparse key-point encoding and an efficient multi-spike learning. The significance of our major contributions can be highlighted in the following five aspects.
\begin{itemize}
\item An integrated spike-based framework is developed for ESR. The event-driven scheme is employed in the overall system from encoding to learning and readout without taking advantages of other auxiliary traditional methods like SOM, making our system more consistent, efficient and biologically plausible. Compared to other non-spike-based approaches (which we refer as conventional ones), our system contributes to drive a paradigm shift in the processing way towards more human-like.
\item A simplified key-point encoding frontend is proposed to convert sound signals into sparse spikes. The key-points are directly used without taking any extra steps of feature clustering. This simplification could be beneficial for low-power and on-line processing. Moreover, we show the effectiveness of our encoding by combining it with two of the most popular networks, i.e. CNN and DNN. Their performance is significantly improved, indicating the generalization of our encoding.
\item We extend our previous multi-spike learning rule, namely threshold-driven plasticity (TDP) \cite{yu2018spike}, to solve the practical ESR task. A novel range training mechanism is developed to enhance the capability of the learning rule.
Moreover, we examine its properties including efficiency, robustness and capability of learning inhomogeneous firing statistics. We are the first one, to the best of our knowledge, to make detailed comparisons of three representative learning rules.
\item The proposed system is robust to noise under mismatched condition. Benchmark results highlight the significance of our approach. Further improvement can be obtained with a multi-condition training method.
\item Our system is robust to severe non-stationary noise and is capable of processing ongoing dynamic environmental sound signals. This highlights the applicability of our proposed system.
\end{itemize}
The remainder of this paper is structured as follows. Section~\ref{sec:Methods} details our proposed approaches and methods being applied. Section~\ref{sec:experiments} then presents our experimental results, followed by discussions in Section~\ref{sec:discuss}. Finally, we conclude our work in Section~\ref{sec:Conclusion}.
\section{Methods}
\label{sec:Methods}
In this section, we will introduce the components and methods used in our framework (see Fig.~\ref{Fig:framework}). Firstly, we describe the proposed encoding frontend that converts sound signals into spikes. Then, the neuron model is described, followed by various learning rules including tempotron \cite{gutig06}, PSD \cite{YuQ2013PSD,yu2016spiking} and TDP \cite{yu2018spike}. Additionally, we present our new methods for improving the performance of the multi-spike learning rules. Finally, other task-related methods and approaches are detailed.
\subsection{Key-Point Encoding Frontend}
\begin{figure}[!htb]
\centering\includegraphics[width=0.47\textwidth]{demo_kp_encoding.pdf}
\caption{Demonstration of key-point frontend encoding. A sound sample of `horn' is presented under conditions of clean and 10 dB of noise in the form of spectrogram (top row). The corresponding spike patterns encoded with our frontend are demonstrated in the bottom row. Each dot represents a spike from the corresponding afferent.}
\label{Fig:kp}
\end{figure}
Biological evidence \cite{christopher1998optimizing,theunissen2000spectral,joris2004neural} gained from measurements of spectral-temporal response fields suggests that auditory neurons are sensitive to feature parameters of stimuli including local frequency bands, intensity, amplitude and frequency modulation, etc. The capability of neurons to capture local spectral-temporal features inspires the idea of utilizing key-points to represent sound signals \cite{dennis2013temporal,xiao2018spike}. Here, we present a more simpler and versatile encoding frontend by directly utilizing the key-points while keeping the processing steps to be as minimal as possible.
The detailed procedures of our encoding is presented in Fig.~\ref{Fig:framework}\textbf{A}. Sound signals are firstly converted into spectrograms by Short-Time Fourier Transform (STFT) with a window of 256 samples and a sliding step of 10 ms. The resulting spectrogram, $S(t,f)$, describes the power spectral density of the sound signal over both time and frequency dimensions (see Fig.~\ref{Fig:framework}\textbf{B}). Next, we perform a logarithm step to
convert the spectrogram into a log scale through $log(S(t,f)+\epsilon)-log(\epsilon)$ with $\epsilon=10^{-5}$, followed by a normalization step. The resulting spectrogram (see Fig.~\ref{Fig:framework}\textbf{C}) which is still denoted as $S(t,f)$ for simplicity is further processed in the following key-point extraction steps.
The key-points are detected by localizing the sparse high-energy peaks in
the spectrogram. Such localizations are accomplished by searching local maxima across either time or frequency, as follows:
\begin{equation}
\label{eq:kp}
P(t, f) = \Bigg\{ S(t, f) \Big| S(t, f)=max \Big\{ \begin{array}{l}
S(t\pm d_t, f) \mbox{ or} \\
S(t, f\pm d_f)
\end{array} \Big \} \Bigg \}
\end{equation}
where $d_t (d_f)=[0,1,2,...,D_t(D_f)]$. $D_t$ and $D_f$ denote the region size for key-point detection. We set both of them to 4, which we found was big enough for a sparse representation, but small enough to extract important peaks.
In order to further enhance the sparseness of our encoding, we introduce two different masking schemes, namely the absolute-value and the relative-background masking.
In the absolute-value scheme, those key-points are discarded if criterion of $P(t, f)<\beta_\mathrm{a}$ is satisfied. This means we only focus on significantly large key-points which are believed to contain important information.
In the relative-background masking scheme, the key-point is dropped if the contrast between it and its surrounding background reaches the condition of
$P(t, f)*\beta_\mathrm{r}<mean\{S(t\pm d_t, f\pm d_f)\}$. $\beta_\mathrm{a}=0.15$ and $\beta_\mathrm{r}=0.85$ are the two hyper-parameters that control the level of reduction in the number of key-points.
The extracted key-points contain both spectral and temporal information, which we found is sufficient enough to form a spatiotemporal spike pattern by directly mapping each key-point to a spike. As can be seen from Fig.~\ref{Fig:kp}, the resulting spike pattern is capable of giving a `glimpse' of the sound signal with a sparse and robust representation. The advantageous properties of our encoding can be beneficial to learning algorithms, which we will show later.
\subsection{Neuron Model}
In this paper, we use the current-based leaky integrate-and-fire neuron model due to its simplicity and analytical tractability \cite{gutig2016,yu2018spike}.
The neuron continuously integrates afferent spikes into its membrane potential,
and generates output spikes whenever a firing condition is reached. Each afferent spike will result in a post-synaptic potential (PSP), whose peak value is controlled by the synaptic efficacy, $w$. The shape of PSPs is determined by the kernel defined as
\begin{equation}
K(t-t_i^j) = V_0\left[\exp{\left(-\frac{t-t_i^j}{\tau_\mathrm{m}}\right)}-\exp{\left(-\frac{t-t_i^j}{\tau_\mathrm{s}}\right)}\right]
\end{equation}
where $V_0$ is a constant parameter that normalizes the peak of the kernel to unity, and $t_i^j$ denotes the time of the j-th spike from the i-th neuron. $\tau_\mathrm{m}$ and $\tau_\mathrm{s}$ represent time constants of the membrane potential and the synaptic currents, respectively.
\begin{figure}[!htb]
\centering\includegraphics[width=0.47\textwidth]{demo_neuron_dynamics.pdf}
\caption{Dynamics of spiking neuron model. \textbf{A}, the input spatiotemporal spike pattern where each dot denotes a spike. Each afferent fires a certain number of spikes across the time window. \textbf{B}, the synaptic weights of the corresponding afferents. \textbf{C}, membrane potential dynamics of the neuron in response to the pattern in \textbf{A}. The red dashed line denotes the firing threshold. \textbf{D}, normalized kernel of post-synaptic potential.}
\label{Fig:NeuronDyms}
\end{figure}
The evolving dynamics of the neuron with $N$ synaptic afferents is described as
\begin{equation}
\label{Eq:neuron}
V(t) = \sum_{i=1}^N w_i\sum_{t_i^j<t} K(t-t_i^j) - \vartheta \sum_{t_\mathrm{s}^j<t} \exp{\left(-\frac{t-t_\mathrm{s}^j}{\tau_\mathrm{m}}\right)}
\end{equation}
where $\vartheta$ denotes the firing threshold and $t_\mathrm{s}^j$ represents the time of the $j$-th output spike.
As can be seen from Fig.~\ref{Fig:NeuronDyms}, each afferent spike will result in a change in the neuron's membrane potential. The neuron continuously integrates afferent spikes in an event-driven manner. The mechanism of the event-driven computation is advantageous in both efficiency and speed \cite{yu2018spike}, and thus is adopted in our study. In the absence of an input spike, neuron's membrane potential will gradually decay to the rest level. Whenever the membrane potential crosses neuron's firing threshold, an output spike is elicited, followed by a reset dynamics.
\subsection{Learning Rules}
Various learning rules have been introduced to train neurons to learn spikes. These learning rules can be categorized according to neuron's output response. With supervised temporal learning rules, neurons can be trained to have an efficient binary response (e.g. spike or not) or multiple output spikes where either their precise timing or the total spike number matters. In our study, we select tempotron \cite{gutig06}, PSD \cite{YuQ2013PSD} and TDP \cite{yu2018spike} as representatives of different types. Our new contributions to these learning rules are provided after descriptions about them.
\subsubsection{The tempotron rule}
Different from a multi-spike neuron model as described in Eq.~\ref{Eq:neuron}, the tempotron can only fire a single spike due to the constraint of a shunting mechanism \cite{gutig06}. Neuron is trained to elicit a single spike in response to a target pattern ($P^+$) and to keep silent to a null one ($P^-$). In this rule, a gradient descent method is applied to minimize the cost defined by the distance between the neuron's maximal potential and its firing threshold, leading to the follow:
\begin{equation}
\Delta w_i=\left\{\begin{array}{rl}
\lambda\sum_{t_i<t_{\mathrm{max}}}K(t_{\mathrm{max}}-t_i), &\mbox{ if $P^+$ error;}\\
-\lambda\sum_{t_i<t_{\mathrm{max}}}K(t_{\mathrm{max}}-t_i), &\mbox{ if $P^-$ error;}\\
0, &\mbox{ otherwise.}
\end{array}
\label{eq:tempotronRule}
\right.
\end{equation}
where $t_{\mathrm{max}}$ denotes the time at the maximal potential and $\lambda$ represents the learning rate.
Decision performance of the tempotron rule can be improved by incorporating other mechanisms such as grouping \cite{Yu2013TNN} and voting with maximal potential \cite{dennis2013temporal}. The mechanism of maximal potential decision is applied for the tempotron rule in our study.
\subsubsection{The PSD rule}
The PSD rule is proposed to train neurons to fire at desired spike times in response to input spatiotemporal spike patterns, such that the temporal domain of the output can be potentially utilized for information transmission as well as multi-category classification \cite{YuQ2013PSD,yu2016spiking}. The learning rule is implemented to minimize the difference between the actual ($t_\mathrm{o}$) and the desired ($t_\mathrm{d}$) output spike times, and the learning rule is thus given as:
\begin{align}
\label{Eq:TrialL}
\Delta w_i
&=\lambda \Bigg [ \sum_g\sum_fK(t_\mathrm{d}^g-t_i^f)H(t_\mathrm{d}^g-t_i^f) \\ \nonumber
&- \sum_h\sum_fK(t_\mathrm{o}^h-t_i^f)H(t_\mathrm{o}^h-t_i^f) \Bigg ]
\end{align}
where $H(\cdot)$ represents the Heaviside function.
According to the PSD rule, a long-term potentiation (LTP) will occur to increase the synaptic weights when the neuron fails to fire at a desired time, while a long-term depression (LTD) will decrease the weights when the neuron erroneously elicits an output spike. The distance between two spike trains can be measured by
\begin{equation}
Dist = \frac{1}{\tau}\int_0^\infty [f(t)-g(t)]^2dt
\label{Eq:Dist}
\end{equation}
where $f(t)$ and $g(t)$ are filtered signals of the two spike trains.
This distance metric can be used in both training and evaluation, but the choice of a critical value for termination in the training could be difficult. In this paper, we introduce a much simpler and efficient approach, i.e. the coincidence metric, to measure the distance. We introduce a margin parameter $\zeta$ to control the precision of the coincidence detection. We will treat the output spike time as a correct one if it satisfies the condition of $t_\mathrm{d}-\zeta \leq t_\mathrm{o} \leq t_\mathrm{d}+\zeta$. This margin parameter can facilitate the learning.
\subsubsection{The TDP rule}
Recently, a new family of learning rules are proposed to train neurons to fire a certain number of spikes instead of explicitly instructing their precise timings \cite{gutig2016,yu2018spike}. These learning rules are superior to others for making decision and exploring temporal features from the signals. We adopt the TDP rule in this paper due to its efficiency and simplicity. The learning rule is developed based on the property of the multi-spike neuron, namely spike-threshold-surface (STS). Neuron's actual output spike number can often be determined by the position of its firing threshold in STS. Therefore, modifications of the critical threshold values can result in a desired spike number. The TDP learning rule is given as
\begin{equation}
\label{Eq:learning}
\Delta w =
\begin{cases}
-\lambda \frac{d\vartheta^*_{n_o}}{d w} & \quad \text{if } n_\mathrm{o}>n_\mathrm{d} \\
\lambda \frac{d\vartheta^*_{n_o+1}}{d w} & \quad \text{if } n_\mathrm{o}<n_\mathrm{d}\\
\end{cases}
\end{equation}
where $d\vartheta^*_{k}/d w$ represents the directive evaluation of critical threshold values with respect to synaptic weights (the details can be found in \cite{yu2018spike}). The basic idea of this learning rule (see Fig.~\ref{Fig:sts}) is to increase (decrease) the critical values that are smaller (greater) than $\vartheta$ with an LTP (LTD) process if the neuron fails to fire a desired number of spikes. The learning stops until the neuron's firing threshold falls into a desired region.
\begin{figure}[!htb]
\centering\includegraphics[width=0.47\textwidth]{demo_sts.pdf}
\caption{Demonstration of spike-threshold-surface (STS). Neuron's output spike number, $n_\mathrm{o}$, can be determined by the position of its firing threshold and the critical values $\vartheta^*_k$.}
\label{Fig:sts}
\end{figure}
This learning mechanism makes the TDP rule capable of learning both rate- and temporal-based patterns \cite{yu2018spike}, which would be advantageous if the temporal structure of external stimuli is unknown.
In order to further enhance the applicability of the TDP rule, we develop a range training mechanism in this paper. Instead of using a specific desired spike number, the learning is stopped if neuron's output spike number falls into a desired range. In our sound recognition task, we train neurons to fire at least 20 spikes in response to their target categories.
\subsection{Deep Learning Networks}
CNN and DNN, as two of the most popular networks in deep learning \cite{lecun2015deep}, are also applied in this study to benchmark our proposed approaches.
A CNN typically consists of input, convolutional, pooling, normalization, fully connected and output layers.
Since CNN favors input images with fixed dimensions, we extend spectrograms to the longest duration of all sound signals by employing zero padding to the end.
We set our CNN architecture to 32C3@127$\times$211-64C3@63$\times$105-128C3@31$\times$52-256C3@15$\times$25-F64-F10.
It consists of 5 learning layers, including 4 convolutional and one fully connected layer. All learning layers use the non-linearity rectified linear units (ReLU) as the activation function, and batch normalization is applied to avoid over-fitting. In addition, we use 2$\times$2 strides in all learning layers except for the first convolutional one. The CNN network is trained using Adam optimizer with a learning rate of 0.0001.
DNN is a feed-forward artificial neural network, which consists of more than one layer of hidden units between the inputs and outputs. We construct a 4-layer DNN of the form 256-180-64-32 with the output layer in a one-of-N (i.e. N categories) configuration. We flatten the spectrograms to one-dimensional vectors which serve as the inputs of DNN. We adopt ReLU activation except for the input and output layers. Again, the Adam optimizer with a learning rate of 0.0001 is adopted.
\subsection{Sound Database}
Following the setups in \cite{dennis2013temporal,xiao2018spike,wu2018spiking}, we choose the same following ten sound classes for a fair comparison from the Real World Computing Partnership (RWCP) \cite{nakamura2000acoustical}: whistle1, ring, phone4, metal15, kara, horn, cymbals,
buzzer, bottle1 and bells5.
To standardize the selection, we choose the first 80 files from each class to
form our experimental dataset.
In each experimental run, we randomly select half files
of each class as the training set, and leave the rest as testing. The "Speech Babble" noise
environment is obtained from NOISEX'92 database \cite{varga1993assessment} for evaluating
the robustness of the sound recognition. The performance of different approaches is evaluated in both clean
environment and noisy cases with signal-to-noise ratio (SNR) of 20, 10, 0 and -5 dB.
The performance is then averaged over 10 independent runs.
\section{Experimental Results}
\label{sec:experiments}
In this section, we first examine properties of different learning rules. To be specific, we concern the properties of learning efficiency and multi-category classification. Additionally, we also show the capability of the multi-spike learning rule for processing a more challenging task of inhomogeneous firing. Then, we present the performance of our proposed framework for sound recognition. Detailed examinations on various learning properties of the system are given accordingly. Finally, we show the outstanding performance of our system for processing ongoing dynamic environmental signals.
\subsection{Learning Efficiency of Multi-Spike Rules}
In this experiment, we evaluate the learning efficiency of different multi-spike learning rules, including PSD \cite{YuQ2013PSD}, MST \cite{gutig2016} and TDP \cite{yu2018spike}. These learning rules can be used to train neurons to fire a desired number of spikes. Different from the others, PSD rule requires precise spike timings in the supervisor signal. In order to relax this constraint, we employ margin parameters $\zeta$ of 5 and 10 ms to the PSD rule.
\begin{figure}[!htb]
\centering\includegraphics[width=0.47\textwidth]{efficiency_wmn.pdf}
\caption{Efficiency of multi-spike learning rules including MST, TDP and PSD. \textbf{A} and \textbf{B} show the learning epochs and the corresponding cpu running time, respectively. Experiment was performed on a platform of Intel E5-2620@2.10GHz. All the learning rules are used to train neurons to fire 20 spikes. In the PSD rule, the desired times of these 20 spikes are constructed by evenly distributing them over the time window. Margin parameters of 5 and 10 ms are applied in the PSD rule. Results were averaged over 100 runs.}
\label{Fig:efficiency}
\end{figure}
The input spike patterns are generated over a time window of
T = 1.0 s with each afferent neuron firing at a Poisson rate of 8 Hz over T.
Similar to \cite{yu2018spike}, other parameters are set as: $N=500$, $\tau_\mathrm{m}=20$ ms, $\tau_\mathrm{s}=5$ ms and $\lambda=10^{-4}$. Neurons are trained to elicit 20 spikes with different learning rules under different initial weight setups. The synaptic weights are initialized according to a Gaussian distribution where we keep the standard deviation of 0.01 fixed while change the mean value for different evaluations.
As can be seen from Fig.~\ref{Fig:efficiency}, the learning speeds of both MST and TDP rules change with different initial mean weight due to the incremental updating characteristics of the learning. The current training method of the MST and TDP rules is implemented in a way to increase or decrease one spike a time. Neuron's output spike number normally increases with bigger mean values of the initial weights \cite{yu2018spike}. These are the reasons why the speeds of both MST and TDP increase first and then decrease with increasing mean initial weights. Notably, TDP always outperforms MST in terms of efficiency. Different from the other two, the learning speed of PSD barely changes with different initial conditions. This is because PSD employs a form of batch updating where neurons are instructed by all desired spikes together during learning. These desired spikes are independent of the initial weight setups, resulting in a roughly steady learning speed. The learning speed of PSD can be increased by further relaxing the margin parameter $\zeta$ (e.g. from 5 ms to 10 ms). Although the margin scheme facilitates the learning, the precise spike timings of the instructor are still required.
\subsection{Learning to Classify Spatiotemporal Spike Patterns}
In this experiment, we study the ability of different rules on discriminating spatiotemporal spike patterns of different categories. The neuron parameters are the same as the previous experiment except that the mean and the standard deviation of initial weights are set as 0 and 0.001, receptively. Similar to the experimental setups in \cite{florian2012chronotron,mohemmed2012span,YuQ2013PSD}, we design a 3-category classification task and construct one template spike pattern for each category. Every template is randomly generated and then fixed after generation. Each afferent has a firing rate of 2 Hz over a time window of 0.5 s. Spike patterns of each category are instantiated by adding two types of noises to the template pattern. The first type is jitter noise: each spike of the pattern is jittered by a random Gaussian noise with zero mean and standard deviation of $\sigma_\mathrm{jit}$. The other type is deletion noise: each spike would be randomly deleted with a probability of $p_\mathrm{del}$. We use $\sigma_\mathrm{jit}=2$ ms and $p_\mathrm{del}=0.1$ to train neurons for the corresponding noise type, followed by evaluations over a broader range of noise. Each category is assigned to be the target of one learning neuron.
Two different readout schemes are applied: the absolute (`abs') and the winner-take-all (`wta') methods. In the `abs' method, the neuron with exactly the same output response as a predefined critical one will represent the prediction, while the one with the maximal response among all the output neurons will dominate the prediction in the `wta' method. We set different configurations for different learning rules. In the tempotron rule, we apply the binary response of spike or not for `abs', while the maximal potential is used in `wta'. In the PSD rule, neurons are trained to have 4 evenly distributed spikes and none for the target and null categories, respectively. A critical spike number of 2 is used in `abs', while the spike distance measurement of Eq.~\ref{Eq:Dist} is adopted in `wta'. A margin of 10 ms is also applied in the PSD. For the TDP rule, neurons are trained to elicit at least 20 spikes for target categories and keep silent for null ones. The critical spike number of 10 is used in `abs', while `wta' searches for the neuron with the maximal output spike number. In order to facilitate the learning, a momentum scheme \cite{gutig06,yu2018spike} with $\mu=0.9$ is also applied to all rules.
\begin{figure}[!htb]
\centering\includegraphics[width=0.48\textwidth]{spike_classification.pdf}
\caption{Robustness of various spike learning rules on spatiotemporal spike pattern classification. \textbf{A} and \textbf{B} show the classification accuracy against noises of spike jitter $\sigma_\mathrm{jit}$ and spike deletion $p_\mathrm{del}$, respectively. The examined learning rules are denoted as: `mul' (for TDP rule), `bin' (tempotron) and `psd' (PSD). These rules are combined with two readout schemes, i.e. `wta' and `abs', resulting in 6 combinations. Data were averaged over 100 runs.}
\label{Fig:robustness}
\end{figure}
As can be seen from Fig.~\ref{Fig:robustness}, all the rules with `wta' scheme perform better than their counterparts with `abs'. This is because the competing policy in `wta' can help the system to identify the most likely representation by comparing all outputs. The TDP rule is the best as compared to other learning rules under both `wta' and `abs'. The performance of the tempotron rule is significantly improved by the maximal potential decision as compared to fire-or-not. This is because useful information can be integrated to subtle changes on the membrane potential which will be difficult to capture with a binary-spike response, but it could be reflected in the maximal potential to some extend. The PSD rule performs relatively worse than the others. This is because the desired spike timings would hardly be an optimal choice for a given task, and it is difficult to find a such one. In a sound recognition task, the appearance of a stimulus can be arbitrary, let alone to train a neuron to fire at desired times. Therefore, we only evaluate the performance of the tempotron and TDP in the following sound recognition tasks. Additionally, the `wta' scheme will be adopted due to its superior performance.
\subsection{Learning to Process Inhomogeneous Firing Statistics}
\begin{figure}[!htb]
\centering\includegraphics[width=0.49\textwidth]{spike_rt.pdf}
\caption{Learning performance of multi-spike rule on inhomogeneous firing statistics. \textbf{A}, firing rate $r(t)$ of two rate patterns as a function of time. The blue and orange colors represent the inhomogeneous (the target class) and homogeneous (the null class) firing cases, respectively. The vertical dashed lines depict the center of peaks, and shaded areas show the regions with a width to the left and right from the center. \textbf{B}, spike pattern instantiation examples of the corresponding firing rates. Each marker denotes a spike. \textbf{C}, voltage trace demonstration of a neuron that is trained to have 2 spikes for the target class and none for the null. \textbf{D}, illustration of output spike distribution over time in response to an input pattern. Color intensity from light to dark indicates a desired target spike number of 1, 2, 6 and 10. The curves of 1 and 2 for the null class (orange) are not clearly visualized since they are presented at zero. \textbf{E}, average output spike number versus the desired one used for the target class. \textbf{D} and \textbf{E} were obtained from 500 runs.}
\label{Fig:rt}
\end{figure}
The nervous neurons general receive inputs of non-homogeneous statistics, i.e. time-varying firing rates. Thanks to the capability of the TDP rule to process both rate and temporal based patterns \cite{yu2018spike}, we will further our study to examine its capabilities of learning inhomogeneous firing statistics in this experiment.
We design two firing rate classes, i.e. inhomogeneous and homogeneous. The inhomogeneous firing rate is time-varying while it is constant for the homogeneous class. For the inhomogeneous rate, we add a form of $4*\exp^{-(\frac{t-c}{b})^2}$ to a baseline firing of 1 Hz, where $c$ and $b$ denotes the center and width, respectively. We choose two centers at 150 and 350 ms, and set the width to 20 ms. In order to remove statistical difference on the mean firing over the whole time window of 0.5 s, we set the homogeneous rate to a level such that the integrals of both firing statistics are the same. The two resulted classes of firing rate are shown in Fig.~\ref{Fig:rt}\textbf{A}. Spike patterns of each class are generated by instantiating spikes according to the instantaneous firing rate determined by $r(t)$. Different from the previous spatiotemporal task, we do not keep any generated spike patterns fixed, but always generate new ones according the corresponding $r(t)$. Fig.~\ref{Fig:rt}\textbf{B} demonstrates examples of two generated spike patterns. In this experiment, we use the TDP rule to train neuron to fire at least $n_\mathrm{d}$ spikes in response to the inhomogeneous patterns while keep silent to the homogeneous ones.
Fig.~\ref{Fig:rt}\textbf{C} shows the learning results of a neuron with $n_\mathrm{d}$ being set to 2. The neuron can successfully elicit 2 spikes in response to a target pattern while keep silent as expected to a null one. Notably, the two output spikes occur around the peak centers with a nearly equal possibility, indicating a successful detection of the discriminative information. In order to have a detailed examination on this capability, we run the experiments with different $n_\mathrm{d}$ values. Fig.~\ref{Fig:rt}\textbf{D} and Fig.~\ref{Fig:rt}\textbf{E} show the output distribution and the total average output spike number in response to a pattern of different classes, respectively.
The output spike number for the target class increases with increasing $n_\mathrm{d}$, while it keeps around zero for the null class first at low $n_\mathrm{d}$ values (e.g. 4) and then starts to increase. Although it seems that the increasing output spike number of both classes makes the discrimination difficult, the distribution of these spikes over time can facilitate the classification (Fig.~\ref{Fig:rt}\textbf{D}). Neurons can always discover useful information by eliciting spikes around interesting areas, and downstream neurons can be added to further utilize these spikes.
\subsection{Environmental Sound Recognition}
In this experiment, we examine the capabilities of our framework on the task of sound recognition.
We set neuron's parameters as $\tau_\mathrm{m}=40$ ms and $\tau_\mathrm{s}=10$ ms. Synaptic weights are initialized with a mean of 0 and standard deviation of 0.01. Recognition performance is evaluated under clean and different noise levels of 20, 10, 0 and -5 dB.
\begin{figure}[!htb]
\centering\includegraphics[width=0.48\textwidth]{demo_single_ptns_full.pdf}
\caption{Demonstration of neurons' response to different sound samples under clean (\textbf{A}) and SNR of 0 dB background noise (\textbf{B}). The rows of each panel (from top to bottom) show: sound wave signals, spectrograms, voltage traces of a neuron with the tempotron rule (denoted as `Bin') in response to the corresponding sound samples, and voltages of a neuron with the TDP rule (represented by `Mul'). The target class of the demonstrated neurons is `buzzer'.}
\label{Fig:sound_demo}
\end{figure}
Fig.~\ref{Fig:sound_demo} illustrates the dynamics of single neurons that are trained with the tempotron (`Bin') and the TDP (`Mul') rule for a target class of `buzzer'. Neurons successfully elicit desired response to both target and null patterns as can be seen from the figure. With an imposed noise, neurons can still discriminate target patterns from null ones based on the output response, although the underlying dynamics is affected to a certain extent by the noise.
\begin{table}[!tbh]
\centering
\caption{Classification accuracy (in percentage \%) under mismatched condition. Shaded areas denote results obtained from our approaches in this study, while other baseline results are collected from \cite{dennis2013temporal,xiao2018spike,wu2018spiking}. The darker shaded color highlights our proposed multi-spike approach. The bold digits represent the best across each column. We use the `Avg$.$' as a performance indicator in the following evaluations, and the marker $\bullet$ is used in Fig.~\ref{Fig:sound_nd}-\ref{Fig:sound_ratio} for consistent presentation.}
\begin{tabular}{c||ccccc|c}
\hline
\textit{Methods} & Clean & 20dB & 10dB & 0dB & -5dB & Avg$.$ \\ \hline
MFCC-HMM & 99.0 & 62.1 & 34.4 & 21.8 & 19.5 & 47.3 \\
\rowcolor{LightCyan}
SPEC-DNN & \textbf{100} & 94.38 & 71.8 & 42.68 & 34.85 & 68.74 \\
\rowcolor{LightCyan}
SPEC-CNN & 99.83 & \textbf{99.88} & 98.93 & 83.65 & 58.08 & 88.07 \\
\rowcolor{LightCyan}
KP-CNN & 99.88 & 99.85 & \textbf{99.68} & 94.43 & 84.8 & 95.73 \\
SOM-SNN & 99.6 & 79.15 & 36.25 & 26.5 & 19.55 & 52.21 \\
LSF-SNN & 98.5 & 98.0 & 95.3 & 90.2 & 84.6 & 93.3 \\
LTF-SNN & \textbf{100} & 99.6 & 99.3 & 96.2 & 90.3 & 97.08 \\
\rowcolor{LightCyan}
KP-Bin & 99.35 & 96.58 & 94.0 & 90.35 & 82.45 & 92.54 \\
\rowcolor{DarkCyan}
KP-Mul & \textbf{100} & 99.5 & 98.68 & \textbf{98.1} & \textbf{97.13} & \textbf{98.68} $\bullet$ \\
\hline
\end{tabular}
\label{tab:mismatch}
\end{table}
We examine the recognition performance under a mismatched condition where neurons are only trained with clean sounds but evaluated with different levels of noises after training. Table~\ref{tab:mismatch} shows the recognition performance of our proposed approaches, as well as other baseline ones.
Both conventional and spike-based approaches are covered. MFCC-HMM, as a typical framework widely used in acoustic processing, performs well in clean environment, but degrades rapidly with increasing levels of noise. The deep learning techniques, i.e. DNN and CNN, can be used to improve the performance as is compared to MFCC-HMM. CNN demonstrates a superior performance to DNN due to its advanced capabilities for processing images (here spectrograms, `SPEC') with convolution operations. Notably, the performance is significantly improved when we combine our key-point encoding (`KP') with CNN, reflecting the sparseness and effectiveness of our proposed encoding.
On the other hand, the spike-based approaches try to solve the sound recognition task from a biologically realistic perspective. Most of these approaches except SOM-SNN get a relatively high performance with an average accuracy over 90\%. Different from other spike-based approaches \cite{dennis2013temporal,xiao2018spike,wu2018spiking} where a binary tempotron rule and a maximal voltage voting scheme are used, our proposed approach utilizes a more powerful multi-spike learning rule and adopts a simpler and more realistic maximal spike number voting scheme. Additionally, the comparative performance of our KP-Bin to that of LSF-SNN reflects the simplicity and efficiency of our encoding. In the following experiments, we use the average accuracy over all noisy conditions, i.e. `Avg$.$', as a performance index to further examine properties of our proposed system.
\begin{figure}[!htb]
\centering\includegraphics[width=0.49\textwidth]{sound_acc_nd.pdf}
\caption{Recognition accuracy versus different target spike number $n_\mathrm{d}$.}
\label{Fig:sound_nd}
\end{figure}
In Fig.~\ref{Fig:sound_nd}, we evaluate the effects of the target spike number on the performance of our multi-spike framework. In the case of $n_\mathrm{d}=1$, we restrict neurons to fire only single spikes in response to target classes. In this way, we can show the performance of a binary-spike decision scheme, and thus identify the contribution of a maximal voltage voting scheme applied in other approaches \cite{dennis2013temporal,xiao2018spike,wu2018spiking} and our KP-Bin in Table~\ref{tab:mismatch}. With an increasing number of $n_\mathrm{d}$, the performance of our approach is continuously enhanced. This is because more spikes can be used to explore useful features over time for a better decision. Further increasing $n_\mathrm{d}$ will marginally degrade the recognition accuracy which could be possibly due to a temporal interference \cite{yu2016spiking,rubin2010theory}.
\begin{figure}[!htb]
\centering\includegraphics[width=0.49\textwidth]{sound_acc_early_dec.pdf}
\caption{Recognition accuracy versus presenting ratio of single patterns.}
\label{Fig:sound_early}
\end{figure}
Fig.~\ref{Fig:sound_early} shows the capabilities of different approaches to make early and prompt
discriminations when only a ratio of the whole pattern from the beginning is present to them.
As can be seen from the figure, different approaches favor longer presence of a pattern, because more useful information can be accumulated for a better decision. The larger the presenting ratio, usually the better the system performance. The performance of CNN and DNN based approaches slightly decreases when whole patterns are present. This is because the sound signals are recorded in a way with a short ending silence which does not contain useful information and thus distracts the deep learning approaches. This is not an issue to our multi-spike approach. The multi-spike characteristic makes neurons to elicit necessary spikes when more useful information appears and to ignore useless parts. Notably, our multi-spike approach demonstrates an outstanding capability for early decision as is compared to other ones. A relatively high accuracy of around 90\% can be obtained at an early time when only 40\% of the pattern is present.
\begin{figure}[!htb]
\centering\includegraphics[width=0.49\textwidth]{sound_acc_ratio.pdf}
\caption{Recognition accuracy versus the ratio of the whole database used for training.}
\label{Fig:sound_ratio}
\end{figure}
Fig.~\ref{Fig:sound_ratio} shows the effects of training-set sizes on the performance. All approaches generally demonstrate a higher accuracy for a larger training set. This is as expected since the more samples used for training the more knowledge neurons can learn. Importantly, our multi-spike approach achieves a high accuracy of around 95\% with only a small set used for training, indicating outstanding advantages of spike-based systems to learn from limited data samples which imposes difficulties for deep learning techniques \cite{lecun2015deep}.
In addition to the mismatched condition, we conduct multi-condition training to further improve the performance. The multi-condition training, which uses noise during learning, is found to be effective for performance enhancement \cite{lecun2015deep,YuQ2013PSD,wu2018spiking}. In our experiment, we randomly select conditions of clean, or noise levels of 20 or 10 dB during training. As can be seen from Table~\ref{tab:multicond}, the performance of our approaches is improved as expected. Our proposed multi-spike approach still outperforms other baseline ones.
\begin{table}[!tbh]
\centering
\caption{Classification accuracy (\%) under multi-condition training. The shaded area denotes the proposed multi-spike approach.}
\begin{tabular}{c||ccccc|c}
\hline
\textit{Methods} & Clean & 20dB & 10dB & 0dB & -5dB & Avg$.$ \\ \hline
SPEC-DNN & 99.9 & 99.88 & 99.5 & 94.05 & 78.95 & 94.46 \\
SPEC-CNN & 99.89 & 99.89 & 99.89 & 99.11 & 91.17 & 98.04 \\
KP-CNN & \textbf{99.93} & 99.93 & 99.73 & 98.13 & 94.75 & 98.49 \\
SOM-SNN & 99.8 & \textbf{100} & \textbf{100} & \textbf{99.45} & 98.7 & \textbf{99.59} \\
KP-Bin & 99.13 & 99.23 & 99.1 & 95.1 & 89.38 & 96.38 \\
\rowcolor{DarkCyan}
KP-Mul & 99.65 & 99.83 & 99.73 & 99.43 & \textbf{98.95} & 99.52 \\
\hline
\end{tabular}
\label{tab:multicond}
\end{table}
\subsection{Processing Dynamic Sound Signals}
In this experiment, we study the ability of our proposed framework for processing dynamic sound signals which is more challenging and realistic. In order to simulate the highly time-varying characteristic of a severe noise, we construct a modulator signal, $m(t)$, to modulate instantaneous power of the noise signal over time. The modulator is constructed as $m(t)=\sum_{i=1}^{3} A_i * \sin (2\pi f_i + \phi_i)$, where we set three frequencies ($f_i$) as 0.5, 1 and 1.5 Hz, and randomly choose the corresponding $A_i$ and $\phi_i$ from ranges of [0.0, 1.0] and [0, $2\pi$], respectively. Then, we linearly map $m(t)$ to the range of [0.0, 1.0] (see Fig.~\ref{Fig:sound_ongoing}\textbf{C} for an example). A strong SNR of -5 dB is used together with $m(t)$ to construct the noise signal. Both target and distractor sounds randomly occur over time.
\begin{figure}[!htb]
\centering\includegraphics[width=0.49\textwidth]{demo_concatenate.pdf}
\caption{Demonstration of target sound extraction from ongoing dynamic environment. \textbf{A} and \textbf{E} are voltage traces of a neuron in response to sound signals under clean (\textbf{B}) and noisy (\textbf{D}) conditions, respectively. \textbf{C} is a modulator signal, $m(t)$, that is used in the noisy condition to simulate the highly time-varying characteristic of the noise. The blue and grey shaded areas denote the target and distractor sounds, respectively.}
\label{Fig:sound_ongoing}
\end{figure}
As can be seen from Fig.~\ref{Fig:sound_ongoing}, the neuron can successfully detect and recognize the target sounds by a burst of firing spikes in both clean and noisy environment.
Whenever a target sound appears, the neuron starts to continuously elicit spikes within the presenting duration of this target sound, while keeps silent for the other signals. The severe noise can significantly change the membrane potential dynamics of the neuron with a number of erroneous spikes being produced. However, a correction detection and recognition can still be made by the output spike density, i.e. bursting of spikes.
\section{Discussions}
\label{sec:discuss}
Our surrounding environment often contains both variant and non-stationary background noise. For example, a typical wind noise occasionally increases or decreases over time. Crucial sound events could occur arbitrarily in time and be embedded in the background noise. Successful recognition of such events is an important capability for both living individuals and artificial intelligent systems to adapt well in the environment. Inspired by the extraordinary performance of the brain on various cognitive tasks, we designed a biologically plausible framework for the environmental sound recognition task such that it can inherit various advantages such as efficiency and robustness from its counterparts in biology.
In our framework, spikes are used for both information transmission and processing, being an analogy of that in the central nervous systems \cite{kandel2000principles,dayan2001theoretical}. In addition to biological plausibility, spikes are believed to play an essential role in low-power consumption which would be of great importance to benefit devices such as mobiles and wearables where energy consumption is one of the major concerns \cite{merolla2014million}.
Moreover, spikes enable an event-driven computation mechanism which is more efficient as is compared to a clock-based paradigm \cite{yu2018spike}. These benefits drive efforts being devoted to delivering a paradigm shift toward more brain-like computations. Increasing number of neuromorphic hardwares have been developed in this direction recently with preliminary advantages being demonstrated \cite{merolla2014million,yao2017face,benjamin2014neurogrid,lichtsteiner2008128}.
We believe a synergy between neuromorphic hardwares and systems like ours could push the spike-based paradigm to a new horizon.
Our framework is a unifying system that consists of three major functional parts including encoding, learning and readout (see Fig.~\ref{Fig:framework}). All the three parts are consistently integrated in a same spike-based form. In our encoding frontend, key-points are detected by localizing the sparse high-energy peaks in the spectrogram. These peaks are inherently robust to mismatched noise due to the property of local maximum, and are sufficient enough to give a `glimpse' of the sound signal with a sparse representation (see Fig.~\ref{Fig:kp}). The sparseness and robustness of our encoding can significantly increase the performance even of a conventional classifier (see `KP-CNN' in Table~\ref{tab:mismatch}), indicating that its effectiveness could be generalized. Importantly, our encoding does not rely on any auxiliary networks such as SOM used in \cite{dennis2013temporal,wu2018spiking}, or frame-based time reference \cite{wu2018spiking}, making our frontend a superior choice for on-the-fly processing due to this simplicity.
An efficient multi-spike learning rule, i.e. TDP, is employed in our learning part. The TDP is recently proposed to train neurons with a desired number of spikes \cite{yu2018spike}. The learning efficiency gives priority to TDP over other multi-spike learning rules such as MST and PSD (see Fig.~\ref{Fig:efficiency}). Additionally, TDP can automatically explore and exploit more temporal information over time without specifically instructing neurons at what time to fire.
A more robust and better performance is thus obtained (see Fig.~\ref{Fig:robustness}). The TDP rule is capable of processing not only temporal encoded spike patterns but also rate-based ones \cite{yu2018spike}. Fig.~\ref{Fig:rt} demonstrates the capability of TDP to extract information from inhomogeneous firing statistics which is more challenging as is compared to a homogeneous one. All of our property examinations on the learning rules suggest TDP as a perfect choice for sound recognition. Moreover, these examinations could provide a useful reference for spike-based developments.
In our readout part, we utilize the maximal output spike number to indicate the category of an input pattern. The more spikes a neuron elicits, the higher likelihood the presenting pattern belongs to the neuron's category. With this multi-spike readout, the performance of sound recognition is significantly improved as is compared to a single-spike one (see Fig.~\ref{Fig:sound_nd}). The neuron can make full use of the temporal structure of sound signals over time by firing necessary spikes in response to useful information (see Fig.~\ref{Fig:sound_demo} for a demonstration). An impressive early-decision making property is also a result of this readout scheme (see Fig.~\ref{Fig:sound_early}). This behavior highlights the advantages of brain-inspired SNNs to respond with only a few early spikes. Notably, the performance can be improved by further accumulating useful information that occurs after early times (see Fig.~\ref{Fig:sound_early}). The performance of a binary-spike learning rule can be enhanced by a maximal potential readout \cite{dennis2013temporal,wu2018spiking,xiao2018spike}, but this scheme is inefficient since a maximal value needs to be tracked over time. Differently, our multi-spike readout is as simple as to count spike appearance only, thus benefiting both software and hardware implementations.
The spike-based framework is naturally suitable for processing temporal signals. The outstanding performance of our spike-based approaches (see Table~\ref{tab:mismatch} and \ref{tab:multicond}) over the conventional baseline methods highlights the computational power of brain-like systems.
The capability of our framework to process ongoing signal streams in an efficient and robust way endows it with great advantages for practical applications such as bioacoustic monitoring \cite{weninger2011audio},
surveillance \cite{ntalampiras2009acoustic} and general machine hearing \cite{lyon2010machine}.
\section{Conclusion}
\label{sec:Conclusion}
In this work, we proposed a spike-based framework for environmental sound recognition. The whole framework was consistently integrated together with functional parts of sparse encoding, efficient learning and robust readout. Firstly, we introduced a simple and sparse encoding frontend where key-points are used to represent acoustic features. It was shown that our encoding can be generalized to even benefit other non-spike based methods such as DNN and CNN, in addition to our spike-based systems. Then, we examined properties of various spike learning rules in details with our contributions being added for improvements. We showed that the adopted multi-spike learning rule is efficient in learning and powerful in processing spike streams without restricting a specific spike coding scheme. Our evaluations could be instructive not only to the selection of rules in our task but also to other spike-based developments.
Finally, we combined a multi-spike readout with the other parts to form a unifying framework. We showed that our system performs the best in the mismatched sound recognition as is compared to other spike or non-spike based approaches. Performance can be further improved by adopting a multi-condition training scheme. Additionally, our framework was shown to have various advantageous characteristics including early decision making, small dataset training and ongoing dynamic processing.
To the best of our knowledge, our framework is the first work to apply the multi-spike characteristic of nervous neurons to practical sound recognition tasks. The preliminary success of our system would pave the way for more research efforts to be made to the neuromorphic, i.e. spike-based, domain.
\ifCLASSOPTIONcaptionsoff
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{"url":"https:\/\/plainmath.net\/98074\/choose-the-correct-answer-we-use-the-t","text":"# Choose the correct answer: We use the t distribution to perform a hypothesis test about the population mean when: a. The population from which the sample is drawn is approximately normal and the population standard deviation is known b. The population from which the sample is drawn is not approximately normal and the population standard deviation is known. c. The population from which the sample is drawn is approximately normal or sample size is 30 or larger and the population standard deviation is unknown. d. The population from which the sample is drawn is not approximately normal and the population standard deviation is unknown.\n\nWe use the t distribution to perform a hypothesis test about the population mean when:\na. The population from which the sample is drawn is approximately normal and the population standard deviation is known\nb. The population from which the sample is drawn is not approximately normal and the population standard deviation is known.\nc. The population from which the sample is drawn is approximately normal or sample size is 30 or larger and the population standard deviation is unknown.\nd. The population from which the sample is drawn is not approximately normal and the population standard deviation is unknown.\nYou can still ask an expert for help\n\n\u2022 Questions are typically answered in as fast as 30 minutes\n\nSolve your problem for the price of one coffee\n\n\u2022 Math expert for every subject\n\u2022 Pay only if we can solve it\n\ntrivialaxxf\nHere we have given the multiple choice question about the testing of hypothesis for population mean.\nThe question is,\nWe use the t distribution to perform a hypothesis test about the population mean when:\nHere given 4 options we have to select one correct option.\nt-test and z-test are used for testing the population mean. There are two tests one is for testing one population mean and another is, testing equality of two population means.\nThe difference between t-test and z-test is that we use z test when population standard deviation is known and we use t test when population standard deviation is unknown.\nThe assumptions for t test are,\n1. The population form which samples drawn is approximately Normally distributed Or sample is large i.e. grater than 30.\n2. The sample drawn are independent and random.\n3. The population standard deviation is unknown.\nIf the above three conditions satisfied then and then we can use t test for testing population mean.\nC. The population from which the sample is drawn is approximately normal or sample size is 30 or larger and the population standard deviation is unknown.","date":"2022-12-03 15:13:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 32, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.925714373588562, \"perplexity\": 272.8512471643826}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710933.89\/warc\/CC-MAIN-20221203143925-20221203173925-00428.warc.gz\"}"} | null | null |
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Home › Subnautica › Subnautica General Discussion
'BOREALIS RISING' - A Subnautica Story V2.0.
Bugzapper Australia Join Date: 2015-03-06 Member: 201744Members Posts: 1,084 Advanced user
As Benjamin Franklin once said; "Fish and visitors stink after three days."
Even though our guests maintained impeccable standards of personal hygiene and behaviour during their brief stay, I found myself extremely relieved when we finally bundled the colonists into Exodus and deposited them safely back at Kaori-san no-shima. Fortunately, that worrisome 'demigod' business died down rather abruptly, ably assisted by some particularly salty comments from Héloise. We're just your average off-the-shelf AI androids, albeit particularly clever androids blessed with unfettered access to a shipload of nasty weapon systems. No big deal.
Speaking of weapon systems, we'll have to deal with that Precursor ship-killer without any further ado. Carl Sagan's mining crews are already extracting resources from the asteroid belt, and Halvorsen has informed that he will have to start sending his resupply freighters planet-side fairly soon. His current estimate is two weeks before his onboard deuterium stocks are exhausted, although I'm reasonably certain that we can sort things out with Sky Watcher in a day or so. Here's hoping. For starters, we need to convince Sky Watcher that all our colonists are completely free from infection. IANTO drew blood samples from everyone prior to leaving The Broch, and I'm working on an assumption that the Precursor AI is able to analyse those samples somehow.
As backup, I've also brought Héloise and Enzo Savini along. Living proof that the Kharaa are finished.
Ulysses swung wide of the massive gun emplacement, lining up to enter the facility's moon pool. It's a fair bet that the Precursors once possessed vehicles of similar dimensions to our Cyclops, as I had detected an enormous berthing space inside. That's good, because Taranis is sailing astern of us. The way I figure it, a friendly chat with Sky Watcher should be more than enough to get the job done. If by some remarkable chance it isn't, six Mk. V ExoSuits and a pair of Ripleys will make an extremely compelling argument for our case. I'm rather hoping it won't come to that.
"Nervous, laddie?" I said, clapping a friendly hand on Savini's shoulder. "Don't be. First-mission nerves are entirely normal, so I don't think any less of you for that. Just keep your eyes open and your finger off the trigger. If a situation does develop; think first, and then act. We've got your six."
"T-thank you, Sir." Savini stammered, forcing a lukewarm grin. "I'll try not to get myself killed."
"Stout fellow! That's the spirit!" I laughed. "Nay, it won't come to that, mate. Anything that wants you or Héloise will have to chew its way through us Toasters first, and that will take some doing. Okay, brace up, laddie. We're docked."
"JUNO, deploy ExoSuits on remote command and secure the area. No surprises, if you please."
"Aye, Sir. Exos are launched." JUNO said calmly. "No lifesigns detected in the immediate area. Commencing tactical sweep of moon pool. No threats detected. Main corridor entrance is force-shielded, Sir. I have discovered something that may be a control mechanism for the force field, but no apparent interface device is present. Your orders, Sir?"
"It's time to put some boots on the ground. Okay troops, mount up and move out."
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Looks like I owe an apology to @Kellise and @0x6A7232.
Guys, I'm sincerely sorry that I reacted poorly to your constructive criticism. I humbly apologise for launching those nukes.
10-10, Good Buddies.
(Incidentally, Urban Dictionary isn't a wholly reputable source of linguistic information. Some of my entries are in there.)
0x6A7232 US Join Date: 2016-10-06 Member: 222906Members Posts: 5,242 Advanced user
Bugzapper wrote: »
UWE Community guidelines | Guide to play in VR with Google Cardboard or Gear | Increase Subnautica view distance | Useful info to post with error reports | Recovery of corrupted saved games | How to easily update your drivers
Crashing, lag problems? Or maybe your old save didn't get the latest update? Upload your saved game folder to help the devs troubleshoot, then try clearing your cache directories | Automatic Cache cleaning tool here | How to use the Debug Console | How to play any version of Subnautica using Steam | Tips for Subnautica beginners | Why can't the devs "just fix it"? - a modding session for you to educate yourself with Want more frames? Try adding this to your launch options: -window-mode exclusive | Solve options not saving or black screens by deleting options file | Possible workaround for Pause / Menu Bug | Rescue a trapped Seamoth / PRAWN
Slow loading / textures popping in? Try moving Subnautica to an SSD | How to switch Subnautica to Experimental mode (clear cache afterwards) | How to run chkdsk on your drive | How to verify integrity of your game cache (in the "Installation" section) | Blue screens or computer freezing up? Try this fix (updates a corrupted DX10 compatibility file in the graphics driver that sometimes doesn't get updated)
Subnautica launches in a tiny window? Use Task Manager to Maximize it (Thanks FlippingPower) | How to place your Moonpool so it connects correctly (includes diagrams) | Want to hang out with fellow players and the devs? Subnautica Discord server ← swing on by! | SuspensionRailway's Modding Emporium ↔ Categorized list of mods, including 1st playthrough-friendly ← Hey, look, mods! → ReShade mods ↔ Subnautica NexusMods
Humans don't crush at the depths you might think
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fangfatherhunter england Join Date: 2016-05-05 Member: 216369Members Posts: 30 Fully active user
Maybe there will be the great great grandson/daughter of selkirk's brother or sister aboard the carl sagan
Skope Wouldn't you like to know ;) Join Date: 2016-06-07 Member: 218212Members Posts: 1,179 Advanced user
fangfatherhunter wrote: »
That'd be cool, but as far as I know, Selkirk didn't have any children. But he very well could have before the events of Aurora Falls.
That seems like something that could only be proven or disproven by Bugzapper himself.
I've been skulking around here for almost four years.
Yet I still have no idea what's going on.
Skope wrote: »
One would think that that would be something that Selkirk would have thought about many times during the events of Aurora Falls and even Borealis Rising (as he's preparing to head back to civilization now). But yes, we must hear from Bugzapper yes / no / wouldn't you like to know.
what I meant was it could of been the children of Selkirk's possible siblings
Post edited by fangfatherhunter on May 2017
was I meant was it could of been the children of Selkirk's possible sibling
Derp, I didn't see that somehow. So a great great grandniece or nephew then.
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sayerulz oregon Join Date: 2015-04-15 Member: 203493Members Posts: 744 Advanced user
That makes me think of an interesting question: is Selkirkreally the same person he was when he was human? Obviously Bugzapper intends it this way, but the way I see it, it seems as though while he has the memories of the original Selkirk, his view of the world, and as a result, his personality, has changed dramatically due to the clearly quite different experience of having a mechanical body vs. a human one. He has different senses, different abilities, and reads information in a different way. Even though it's not touched on that much in the story, I don't see any way that that wouldn't fundamentally change someones personality.
With a different personality and a different body, one that isn't even biological, I would say that there is no way that he is truly the same person. He looks the same, he has the same memories, and will share at least some of the same behaviors, but on a fundamental level, he is no longer Alexander Selkirk. He is something new.
3 Off Topic Disagree 2Agree 1Awesome ·
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Kellise UK Join Date: 2016-07-23 Member: 220582Members Posts: 81 Advanced user
It's no problem man, amount of random gits bothering you thoughout this I'm not surprised you're a tad snappy. The story is awesome, keep up the good work!
Since Enzo has never piloted an ExoSuit before, the quickest way of teaching him was slaving Centurion's controls to Gawain and letting him get the feel of its systems on the fly. I engaged the haptic force-feedback on his controls and walked him around for a spell, demonstrating the hand motions he'll be using. He seems to be getting it, although I had to coach him along some at first.
"Don't fight the control yokes, laddie. Keep your grip loose for now. Feel how they're moving, and see what your suit is doing in response to those movements. It's all pretty intuitive, once you've got the hang of it. Right, now try it for yourself." I disengaged the remote briefly, allowing him to move around freely for a couple of minutes.
"That's looking good, mate. Right, we're going for a quick swing to the other side of this moon pool. No need to panic. I've resumed control of your suit again. Héloise, you'll be coming with us. JUNO, IANTO and DIGBY, stand fast. Okay, are you both ready?"
Enzo stole a quick glance at the sub bay's ceiling and gulped in alarm. It was a fair distance up, and an equally daunting distance across the pool. I conjure he was clever enough to guess how this manoeuvre would work, and he had every right to be a wee bit apprehensive about it. It's not so bad when you're underwater; everything slows down to a nice, leisurely pace. Doing a Tarzan land-side is a different matter entirely.
Under my control, Gawain, Morrigan and Centurion turned briskly to face the opposite side of the moon pool. All three ExoSuits lifted their right arms skyward, as if grimly saluting an unseen Caesar.
I grinned wickedly and bellowed, "Ave! Qui nos morituri te salutant!"
Three grapples launched as one, caught and locked onto the ceiling. As their trailing cables retracted, the suits shot into the air like bottle rockets, describing three perfect arcs as they swung across the moon pool. Halfway through the swing, their left arms rose into position and launched a second grapple. As soon as the left grapples made solid contact, the right lines disengaged and retracted. At the extreme end of their arcs, the suits fired their right grapples again, latched and cancelled out the momentum of the return swing. We came to rest ten metres in the air, swinging gently. Like spiders descending on silken threads, the ExoSuits gently touched down on the deck.
"Any questions?" No? - Okay. Now, back to the other side. We're using the jump-jets this time."
I marched the ExoSuits to the edge of the moon pool, continuing my lecture on the move.
"You have a maximum of six minutes flight time on continuous thrust, then you'll need to wait another two minutes to allow the suit's propellant supply to regenerate. Don't squander it. Never climb too high, watch your thrust readouts and work to the rule of thirds. Always leave a bit in reserve. It's not so much of an issue underwater... Unless you're about to land in a lava vent, but you can count on your grapples to haul your bahookie out of the fire. Land-side's a different story. Nine point eight two metres per second, per second, straight down. No parachute."
The ExoSuits rose slowly at 25 per cent thrust, turbines whining softly.
I'm taking it nice and slow this time, so that Héloise and Enzo can clearly see how it's done. Five metres up, thrust vectored for forward flight. Velocity, five metres per second. All systems are nominal. Landing in fifteen seconds... And we're down.
"Well, I'd have to say that was five minutes most profitably spent. Enzo, you are now a qualified ExoSuit pilot. Congratulations." I said drily. "Oh aye, your training may have been a bit on the terse side, but you now know as much as I did after my first solo run. You'll pick it up in no time."
The Precursor cannon's moon pool has three exits, two of which are blocked by force fields. After a fruitless search of the area and a thorough examination of both field control pedestals, it was obvious that we needed something that simply wasn't to be found in here. However, we did discover that those pedestals emitted a faint energy signature, and my best guess was that anything used as keys in here might have similar properties. On a hunch, I set the suit's external scanners to search for anything in the immediate area with the same energy pattern. A series of faint signal pings began to appear on the HUD, and my heart sank. According to the map overlay, most of those pings are located inside this facility. However, there are at least two somewhere on the island itself.
"I don't mind telling ye, this looks to be a wee bit dicey. We're deep inside Reaper territory, and we need to climb back onto the island proper. This means going outside without a Cyclops to protect us. These ExoSuits will take a fair beating, but don't be set at ease by that. A Reaper Leviathan is capable of doing heavy damage to our vehicles, and they've become sneaky buggers, too. In the early days, you could hear them roaring away in the deeps, long before they took an interest in you. Gave you plenty of warning, y'see. Not now, though. They've adapted their hunting techniques in response to a human presence. By the time they give it the old school roar, you're as good as dead."
Strength in numbers, it is then. I would have preferred Héloise and Enzo to remain in the sub bay, but neither one of them would hear of it. My main concern was focused on the small fry that swarm around this island. Biters and Bonesharks are usually only minor nuisances to anyone in an ExoSuit, although these two species are prone to raising a ruckus, particularly when they think it's feeding time. That's certain to pique the curiosity of larger predators, and Binky is the undisputed power in these waters. There's also a distinct possibility that Moe, Larry and Curly might take an interest.
I believe that the phrase 'horribly exposed' eloquently describes our current situation. We're attempting to sneak across the base of a near vertical wall of bare basalt, with a twenty-strong pack of Bonesharks hawking about our heads every step of the way. Every once in a while, one gets brave enough to swoop in and try his luck. The temptation to bring our heavy weapons into play is strong, although it would be a fatal mistake on our part. The last thing we need now is a feeding frenzy, and it would only take one dead or injured Boneshark to trigger it. There's no love lost between individuals of that species, and they wouldn't hesitate to turn on one of their own.
Finally, we reached a point where we could exit the water with relative ease. Only sixty metres of sheer rock face stood between us and dry land. Too far to use jump jets, and definitely too far for all of us to grapple up in one go. Those Bonesharks will be all over us in a flash, the same second we turn our backs to them. An undefended climb would be outright suicide. Only one way to do this.
"Ascend by pairs, highest pairs on overwatch. Belay off and stand ready every twenty metres."
JUNO and DIGBY went first, shooting twenty metres straight up. IANTO and Héloise quickly followed. As soon as they were in position, JUNO and DIGBY anchored their suits and stood guard.
I wonder if / when he'll find the teleporter on the inside of the island (or if that even fits in this universe).
This was the worst part of it. All six of us strung out over an undersea cliff, facing a pack of Bonesharks that grew increasingly more agitated with each passing second.
It won't be too long before Binky decides to investigate what all this commotion's about.
"Okay Enzo, it's our turn now. Set your grapples to strike three metres to the right of Héloise's suit. Let me know when you're ready to launch, and I'll follow your lead. Pick your mark and fire, lad."
Enzo's ExoSuit tilted back slightly as he aimed. Both grapples fired simultaneously. His right claw struck and held, although the left claw hit a mineral nodule instead of solid rock. The nodule shattered instantly, leaving the grapple to fall impotently towards the seabed.
"Retrieve your left grapple, smartly now!" I snapped.
Enzo obeyed immediately, reeling in the slack cable that had been piling up at his feet. Fortunately, he did not make the mistake of moving around as the cable fell around him, so there was little chance of the line fouling on his suit. Although he could haul himself up on one grapple, Enzo would find it nigh impossible to use his suit's legs to assist him in making that climb. Best to have both grapples solidly engaged before setting off, or his ExoSuit would end up bouncing all over the place.
"No harm done, mate. Take your time and try again." I said calmly.
My rear-view camera told a different story. Three Bonesharks had broken off from the pack, and they were heading straight for us. I turned to face them, repulsion cannons dialled up to maximum. Time to bloody some snouts.
After a quick check to see that Enzo was lined up to launch his left grapple, I strode forward and activated the piton-bolts in Gawain's foot pads, anchoring my ExoSuit defiantly onto solid basalt.
"Second grapple away and locked, Sir!" Enzo yelled.
"Up ye go! Haul away now, quick as ye like!"
Centurion shot up at full speed to join the others. I now had the undivided attention of three Bonesharks, who were hovering directly in front of me at a prudent distance.
Too far away to make a telling shot with my repulsion cannons, but just close enough to let me know that they meant business.
"JUNO, Get Héloise and Enzo out of the water! I'll be coming up hot, so stand ready for a dust-off!"
"Understood, Sir. What is your intended course of action?"
"A diversion. These three scunners are about to become the best of chums. Stand by."
The Bonesharks shrieked, wheeled about and charged straight for me. At ten metres, I opened fire with both repulsion cannons, blasting the creatures apart with rapid-pulse graviton beams. Gawain shuddered violently under their staggering recoil, but the pitons held firm. Had I done this without activating those anchor bolts, Gawain would have slammed into the cliff face instantly.
I had only a few seconds before the rest of the pack caught scent of blood in the water. It's about to become an extremely unhealthy location for pretty much every living thing in the immediate area. I jettisoned Gawain's pitons and sprinted back toward the cliff, activating the ExoSuit's jump-jets as soon as I was within grapple range. One glance at the maelstrom of carnage now boiling below me was sufficient to fuel a year's worth of nightmares.
A monstrous shape was rising from the depths.
In case you're wondering... Yes, I still have nightmares. Reapers figure prominently in most of them.
Post edited by Bugzapper on May 2017
May 2017 edited June 2017
Binky tore into the tightly-packed mass of Sandsharks like a torpedo, jaws agape. His initial assault caught them completely unawares. The pack scattered in all directions, but not before several of them fell to the Leviathan's scything mandibles and jaws. Inside that soundless explosion of blood, a slow rain of severed heads and tails began to fall; tattered remnants cast aside in his greedy haste.
A thoroughly risky move on my part, I'll admit. Still, I was the only one in any position to see the Leviathan rising. In any case, the Sandshark pack's frenzy had nearly reached critical mass. Call it a snap decision. I've bought us a bare handful of seconds at the very most, and hopefully thinned out any opposition waiting for us on the return leg of this expedition. With any luck, Binky might lose interest in us with a full belly, although I very much doubt it.
The crew were ascending rapidly now. JUNO and DIGBY were waiting for me at the 20-metre mark, Gauss cannons at the ready. Ten per cent thrust remaining. Grapples away. Make every metre count. Thrust again, launch the grapples and haul away. The rear camera's view offers no comfort. An expanding cloud of green haze blooms in the ravening deep, witnessed with crystalline certainty that Death incarnate lurks at its heart. Keep climbing, and never look back.
All present and fully accounted for. Pyramid Rock had put on its Sunday best for our arrival. The sun shone brightly, Skyrays wheeled overhead and lush alien foliage of every hue beckoned invitingly. Aside from a constant skittering noise of Cave Crawlers lurking in the underbrush, the island seemed to be doing its level best to convince us that all was well with the world. Aye, if it wasn't for the hulking shape of an immense Precursor cannon looming over the place like a gargoyle, this would indeed be a capital spot for a beach picnic.
Now that we are outside the Precursor structure again, our sensors are able to obtain a clear fix on those faint energy signatures. The first trace is only fifty metres outside the facility's surface entrance, although the second one is somewhere deep inside Pyramid Rock itself. We paired off and began searching.
Héloise made our first significant discovery in a thoroughly traditional manner. Accidentally.
Morrigan's left foot unit snagged against something as it thudded down, causing the ExoSuit to stumble slightly. The object lay half-buried in the sand, looking for all the world like a grey-green concrete paving slab. A quick glance at a nearby Precursor path-marker immediately confirmed my suspicions. Same material, same cryptic surface ornamentation. After digging it out, we found that the object had been broken into two pieces, either by the impact of Morrigan's foot or something else that happened a very long time ago. There's absolutely no way of telling for certain. Besides, it's all moot at this point. This object is completely inert. Not a single joule of energy remains. If this is the same species of dingus that we're searching for, it's of no bloody use to us now.
I must say, this artifact is a fair old size. It's either Godzilla's old SIM card, or the remnants of a Precursor force-field access key. I can't imagine anyone having more than one of these at hand. Eventually, we found one that was fully intact. However, since we have no way of knowing how many of these keys we'll need once we're inside, there's no point in using it on the topside door. We can access the moon pool by going back the way we came. Not exactly the most delightful prospect to anticipate, particularly when there's a ferociously jacked-up Reaper on the prowl down there.
Post edited by Bugzapper on June 2017
"Well now, that is damn peculiar." I thought.
"Folks, I don't recall seeing this entrance here before. It's as if the whole island's tunnel network has been reconfigured somehow. I conjure it has something to do with having that muckin' great gun suddenly materialise. Ground-penetrating radar doesn't show that much detail beyond a few metres, but I'm guessing there's a whole lot more tunnel down there. Might as well find out the extent of it before we all rush blindly down the rabbit hole. IANTO, there should be a crate of seismic survey charges in our old storage cache. Grab six and plant them in the following locations... Transmitting the charge co-ordinates to you now."
"Are we going to blow something up?" Héloise piped up, clapping her hands delightedly.
"Not as such, Dear Heart. There will be a kaboom, but nothing particularly earth-shattering. Probably wouldn't even rattle your Granny's teeth, in point of fact. It's called Terrain Resonance Imaging. Works roughly the same way as sonar, but this method gives you better signal propagation and finer image detail. We can even pick up buried mineral deposits and any structural faults in the rock. It should take most of the guesswork out of exploring this place, and I'm not awful keen on winding up hip-deep in a lava vent. We're in roughly the right area for them, so it pays to tread a mite softly hereabouts... Don't know how you'd manage that in an ExoSuit, but there you go."
Ten minutes later, IANTO checked in. "All charges are in position, Sir. Awaiting further orders."
"Good man. All units, disperse to the individual locations I've marked on your suit HUDs and activate geophone arrays. Report in when you're ready."
I trudged twenty metres to the right of my current position and set up the ExoSuit's sensors to receive incoming seismic data. To save time, I sent remote commands to Enzo and Héloise's suits, activating the appropriate data acquisition systems. Five minutes later, everyone was good to go.
"All units are confirmed as standing ready in all respects. Firing on one. Three... Two... One."
A prolonged dull thud rippled through the ground as all six charges detonated sequentially. A few nanoseconds later, a dense stream of data assembled itself into a highly detailed holographic map of Pyramid Rock's internal structure.
My goodness, how the old place has changed...
I was right. The maze of tunnels beneath Pyramid Rock has changed dramatically since my last visit. Its structure is perfectly stable, even though the interior is riddled with dozens of tunnels heading off at all points of the compass. There are also a couple of large spaces down there that might hold something of interest, although my most immediate priority is reaching that second Precursor key. We can always check out those rooms later. The main thing is, our path to the second artifact appears to be completely clear, and somewhat unusually for this planet... Reassuringly safe. We'll see.
Our progress toward the artifact's location was remarkably easy, even though these tunnels have a way of wandering all over the place. Here and there, strange tubular outcrops of the Precursor building material poked haphazardly through the tunnel's walls and ceilings, as if the Precursors had some method of extruding these 'cables' (as thick as a human torso, I might add) through solid rock without having to drill a pathway first. Whatever those cables were intended to power, I reckon it's worth taking a quick peek before we head back topside.
Let's just say I'm 'mildly intrigued'.
CalvinTheDiver A place Join Date: 2016-10-08 Member: 222971Members Posts: 87 Advanced user
With the second Precursor key safely in hand, I figured we could spend some time exploring the island's tunnel network, if only to satisfy my curiosity about what was on the other end of those power conduits. Strangely enough, there were no smaller cables branching off from the main power transmission lines. On a hunch, I switched my visual feed to scan the entire invisible EM spectrum. Sure enough, those nodes spaced along the super-conducting conduits continuously 'leak' a carefully regulated flow of electrical energy directly into the ether, presumably as an inductive power source powering a variety of low-demand systems.
The illuminated path markers definitely fall under this category, as our scans revealed that those pedestals contain no apparent cables, discrete components or even an internal power supply. Our observations also confirmed an abiding suspicion. The intricate carvings found on Precursor structures and devices aren't ornamental at all. Believe it or not, those aesthetically-pleasing geometric patterns are actually electronic circuits of immense complexity. The good news is, we can duplicate this technology and adapt it to Terran construction methods.
Presently, the tunnel opened out into a large cavern. All eyes were immediately drawn to a large angular structure sitting on a bare outcrop of basalt, located ten metres or so above the main cavern floor. Four rectangular Precursor structural beams, measuring 1.5 metres a side and six metres in length had been formed into a vertically-standing diamond shape. One quarter of its total height had been merged into a low base platform of the same material, forming what appeared to be an arch or doorway.
As I approached the object, a slender pillar rose smoothly out of the base platform. At first, I thought it might be a control device that responded to simple hand pressure. No such luck. The pillar remained extended while I stood near it, and it retracted immediately whenever I moved off the base platform. Other than that, there was no apparent response from the arch itself. Eventually, I became frustrated with the structure's stubborn refusal to yield its secrets.
"Okay, mates. This bloody thing has got me totally stumped. Any ideas?" I huffed in exasperation.
"This place reminds me of a temple, Chérie. Did the Precursors even have gods?"
"Not that I'm aware of." I admitted. "There's no evidence suggesting that they believed in supernatural entities at all, leastways anything worth depicting in their architecture. Nearly all of the sentient species encountered so far have some sort of mythic tradition, and there's usually physical depictions of whatever beings they consider divine or heroic. The Precursors have left nothing that indicates any kind of belief structure. Remember, all of these carvings are purely functional. There are no coded references to their ancient past concealed in any of these designs."
After spending a few minutes deep in careful thought, IANTO weighed in with his two Credit's worth.
"Captain, I have detected that this device is definitely consuming power from the facility's main transmission line. This indicates that the device could be in its standby mode. All we need do is flip the right switch, so to speak. I believe that the small pillar has been designed to hold a specific Precursor object, presumably one serving as an activation key for this device. We should begin searching the surrounding area for anything that might conceivably be used in this manner."
Whatever this object is, I reckon we may have found precisely what we are looking for.
DIGBY had found this one lodged in a shallow fissure near the island's largest subterranean pool, an area that had been thoroughly combed at least a dozen times during our search. Again, it was a case of being in precisely the right position at the right time. The object appears to be a manufactured crystal matrix of some kind; comprised of stacked tabular plates in a cubic configuration, roughly 200mm along each side. It emits an intense green glow, unsettlingly similar to kryptonite, raising immediate safety concerns for our human companions. Scanning revealed that the object is perfectly safe to handle, as it does not emit hazardous ionizing radiation on any frequency band. However, this revelation is completely at odds with the staggering amount of potential energy contained in this crystal.
In short, this object is the answer to an engineer's prayers. Your actual Holy Grail of energy storage. I'm not entirely certain how the Precursors were able to create such a compact energy source, or how many laws of physics were cheerfully swept aside in its making. This small cube has approximately the same energy density as ten of our advanced power-cells, providing twice the energy output of a standard uranium fission reactor. If this isn't mind-boggling enough for you, think of this cube as a decaton-range nuclear detonation, safely encapsulated within a block of crystal. That's basically what it is.
I found myself torn between wanting to use this crystal to activate the portal structure, or hanging onto it for use elsewhere. There's no telling how scarce these cubes are, and I'm not inclined to waste them simply to see what happens. For the time being, we should return to the Precursor cannon and finally make some progress on our primary mission.
At least we're now awake to the notion that we won't find these Precursor artifacts stacked neatly where they're needed. It's almost as if they were simply tossed aside during some sort of commotion, and there's a fair chance that some may have been displaced by seismic activity at some stage. Fortunately, these objects emit specific energy signatures that can be detected by our sensors, and they also glow rather brightly, making them easy to detect once you know what you're looking for. Of course, the abundance of bio-luminescent flora and fauna on Manannán complicates visual search methods considerably, unless you filter out everything outside an artifact's unique spectral emission signature. Once you have the knack of it, finding these items is a complete doddle.
We regrouped on the base platform of the Precursor cannon. After a thorough sensor sweep of the weapon, we determined that the gun fires a phased-plasma discharge, channelled through an ionized conduit generated in the atmosphere a few seconds before firing. The corona discharge seen at the gun's muzzle is visible throughout the entire firing cycle, indicating that the charged conduit has to maintain constant line-of-sight contact with a vessel until the plasma bolt reaches its target. This means that ships are most vulnerable to attack while following pre-calculated descent or ascent profiles. Furthermore, its effective range is limited to the upper reaches of the atmosphere. Considering the cannon's rapid traverse and elevation rates, there are precious few exploitable flaws in this weapon's design.
Hopefully, it won't come to that. A friendly natter with Sky Watcher should be all that's needed.
TheCreeperCow Netherlands Join Date: 2016-05-13 Member: 216716Members Posts: 21 Advanced user
Now wait a second:
Combine this information with the cannon's relatively slow traverse and elevation rates, and you've got the makings of some thoroughly exploitable flaws in this weapon's design.
"Now, you're probably thinking that something this size would be a sluggish performer in the anti-aircraft stakes. Not so.
Within seconds of completion, the turret rotated through a full 360 degrees in only five seconds. At the same time, the cannon's barrel moved through its complete arc of motion thrice in the time it took the turret to complete a single revolution. Make no mistake; this turret is entirely capable of tracking and engaging a high-speed target. An approaching starship in high orbit would be easy meat, although an atmospheric fighter wouldn't last too much longer."
"Here's the plan. We'll abseil down to the moon pool entrance and make our way back inside. Rig suits for ultra-quiet running. Cockpit blackout, no external lights showing. I'm launching a recon drone from Ulysses first, just to make certain that Binky isn't waiting for his dessert course to arrive. Okay, get yourselves sorted out along this ledge and wait for my signal to drop."
The recon probe drifted slowly out of the moon pool entrance, an insignificant speck of metal peering apprehensively into the darkness below. It hovered at 90 metres, sweeping its sensors through 360 degrees, scanning the whole area from top to bottom. The Bonesharks were gone, but not entirely. Two hundred and fifty metres away, a large shoal of Biters had moved in to feast on their remains, their guttural hyena laughter the only sound piercing an otherwise silent ocean.
I brought the probe in closer. In the gloom below, I spotted something that lay twisted among the boulders on the seafloor.
It was Binky.
The Reaper Leviathan had prevailed against the Bonesharks, but at a terrible cost. Huge chunks had been torn from his flanks, all four outer mandibles had been sheared away and his powerful flukes were now little more than tattered stumps. A thin haze of blood wept slowly from his gaping wounds, unaided by a mighty heart that had long ceased to beat.
My feelings were honestly conflicted as I gazed at Binky's ravaged corpse. Even though we have successfully avoided a suicidal confrontation with a large pack of Bonesharks and a Reaper, I wouldn't consider this any sort of victory, personal or otherwise. Binky was fully capable of tearing our ExoSuits to pieces all by himself, although we've always managed to part company in some clever fashion in the past. In my own way, I hold an abiding respect for Reapers, albeit a respect significantly bolstered by their sheer size and relentless ferocity. Binky was a magnificent and cunning foe, worthy of a far better end than this. A darksome knight of renown, laid low by a rabble of kerns and gallowglasses.
When we return to The Broch, I'll raise a farewell glass to his name.
All thoughts of melancholy must stand aside, at least for now. Our mission is of paramount importance; all other considerations are secondary. After completing a second full sensor sweep, I gave the command to begin our descent. As planned, all six ExoSuits slid noiselessly into the water. Fifty metres down, our grapple lines reached their utmost limits of extension, making it necessary to continue our descent in silent free-fall. As we drew level with the entrance to the moon pool, our grapples fired in unison and latched onto the doorway's massive lintel. Thus secured, the grappling lines retracted, drawing us inside the structure.
We wasted no time in shutting down the first force field blocking our path. However, the control pedestal would not release the Precursor key afterwards. I was a mite concerned about this development, since our progress through this complex could come to a screeching halt at any time for the want of a single key. Even so, we pressed onward. After determining that there were no immediate threats in this part of the facility, I split the squad into three pairs so that we could cover more ground during our search of the gun emplacement's labyrinthine corridors and colossal open spaces.
For all its awe-inspiring grandeur, there is a forlorn air about this place; overwhelming sensations of abandonment and an aching emptiness gnawing away at one's thoughts.
Even though it was created within the span of living memory, this weapon looks and feels incredibly ancient.
About an hour later, we regrouped in the main atrium of the complex. Our efforts so far had turned up another three purple Precursor keys, a blue key, an orange key as well as several loose chunks of that green power crystal. All things considered, a sterling effort from everyone involved. IANTO has determined the precise location of the weapon's control centre, simply by scanning the energy flux that permeates the very fabric of this facility. There is also a second Precursor portal in here, and we'll probably be taking a closer look at these devices once we've taken care of that cannon.
Right now, we have another one of those 'interesting' situations that we occasionally encounter. We're standing at the entrance of an open shaft that plunges 90 metres straight down. Since there's no actual platform that raises and lowers, it's not an actual gravitic elevator shaft as such, leastways not of the type that we are used to. Sensors have detected the presence of a polarised graviton beam in the shaft, so it's a reasonable assumption that this is an alien version of a standard gravity-lift. Not entirely unheard of, since Aurora had dozens of similar transport systems onboard. So, it's a familiar technology in a slightly different form. However, that's not the problem. Our current point of concern is that our ExoSuits may be too heavy to make it down in one piece. There's no way of telling how much mass this system was originally calibrated to handle.
You're probably thinking that we should use our suit's thrusters to slow the descent, just in case.
One question: What happens if the shaft's graviton control systems register the suit's deceleration as an attempt to ascend against a downward force? I don't know about you, but there's a mental picture of an ExoSuit slamming into either end of this shaft that I find a wee bit disturbing. There's only one way to find out, I guess. If all else fails, there's always the grapple system to fall back on.
No pun intended.
DIGBY stepped forward. "Captain, I believe that I should be the one to make the first drop. As Tactical Operations Officer, it is one of my core responsibilities to assess all mission environments beforehand and advise you on the most appropriate courses of action. Your reluctance to order crew members into any potentially dangerous situation is greatly appreciated, though somewhat counter-productive in this particular case. With your permission, Sir?"
I nodded. "Well, you've got me dead to rights there, DIGBY old man... As you say, it's in your job description. I'll not be stepping on your toes, mate. By all means, proceed at your own discretion."
"Very good, Sir. I shall be transmitting a telemetry stream during my descent. If anything goes wrong, you should be able to use this data to avoid meeting the same set of failure conditions."
"Mate, if anything does go wrong, I'll be the first one diving down that shaft. You won't prang."
"Thank you, Sir. I shall endeavour to avoid having you leap to my rescue." DIGBY grinned.
With a brief nod of farewell, DIGBY turned smartly about and stepped into the shaft without hesitation. He hung suspended in mid-air for a couple of seconds, his ExoSuit rotating slowly as its centre of mass shifted relative to the deck far below.
Suddenly, he disappeared.
Naturally, my first impulse was to sprint towards the gravity shaft and dive in after him. However, DIGBY's telemetry is still coming through loud and clear. His descent rate has stabilised at a leisurely 2.5 metres per second, comfortably well below the 9.82 metres per second, per second that spells impending doom for anyone or anything not previously equipped with a parachute or jump jets.
"How's your trip down going so far, DIGBY old son?" I enquired cheerfully.
"Remarkably enjoyable, Sir. We must build something like this in The Broch... Purely for recreational purposes, of course! I can easily imagine myself spending a great deal of my spare time in a variable gravity chamber, assuming that we had one. Ah, the end of the ride approaches. Pity."
"Hmm... Sounds like it might be a fun side-project. After we've sorted things out down here. It uses off-the-shelf technology, so I don't see any reason why we can't rig one up. I could even justify it as a micro-gravity training system for any colonists wanting to sign on as Borealis' hull technicians, although SCUBA diving is also a fair simulation of working in a null-g environment."
Come off it, Selkirk. It's not as if you'll cop any Management flak for goofing off in your own time.
"Bugger it, we'll make one just for fun." I declared at last. "Just a gentle reminder folks... If there's a couple of jumpsuits on the deck outside that gravity chamber, kindly use the intercom first, okay?"
Five minutes later, we all stood assembled at the bottom of the gravity shaft. When no-one else was watching, Héloise favoured me with a particularly significant smile and a sly wink.
Apparently, she also feels that a null-gravity chamber in The Broch would be a splendid idea.
Outrageous woman.
Oddly enough, the final pedestal accepted a purple Precursor key without raising any objections. Considering that we were about to enter the cannon's actual control centre, I found myself anticipating some kind of vigorous response to this incursion. No ear-piercing alarms. No flashing strobe lights. No devastating ambush by a swarm of heavily-armed security drones. Absolutely nothing. Somewhat disappointingly, the shimmering green force field deactivated as easily as a deck-head light. DIGBY advanced into the vast chamber, Gauss cannon held at the ready. After a thorough search of the area, he declared it safe for the rest of us to enter. Once inside, we dismounted our ExoSuits and took a short meal break, mainly for the benefit of Enzo and Héloise.
While DIGBY and IANTO patrolled the area on foot, JUNO and I walked over to examine a large pylon in the chamber's centre. As the quiet conversation between our human companions became muted by distance, the subdued background hum of alien machinery grew perceptibly louder as we approached the device's central pillar. We stood silently for a while, bathed in its kryptonite glow.
"I've seen something like this before," I remarked. "In the Precursor facility underneath the Lava Castle. It's a holographic data terminal and projection system. JUNO, say how-do to Sky Watcher."
Even as I uttered his name, the Precursor's holographic avatar began to materialise above us.
"Warm seas, Father of Shells." Sky Watcher intoned gravely. "Why have you entered this place?"
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* * *
ACKNOWLEDGMENTS
Thanks to my wife, Theresa, for the title idea for this book. Also, thanks to her and my wonderful children Wyatt and Valerie for giving up time with me so I could work on this project. I love you all more than you know.
I did not get to where I am today without help. There are many fine photographic artists who helped groom me. I especially want to acknowledge a few: Scott Dupras, Darton Drake, Mille Totushek, Donna Swiecichowski, Teri Shevy, Dan Stoller, Fuzzy Duenkel, Randy Peterson, and Jon Allyn. A special thanks to my two photo buddies Dan Frievalt and Michael Mowbray for keeping me company and sane via Facebook while working on this project. We all see things very differently, but in the end, good portraiture is good portraiture.
SPONSORS
Thanks to my sponsors: White House Custom Color (WHCC, www.whcc.com), Kodak Alaris (www.kodak.com), Sweetlight Systems (www.sweetlightsystems.com), and G.W. Moulding (www.gwmoulding.com). Without these folks, live photographic education couldn't exist.
Copyright © 2015 by Carl Caylor.
All rights reserved.
All photographs by the author unless otherwise noted.
Published by:
Amherst Media, Inc.
P.O. Box 586
Buffalo, N.Y. 14226
Fax: 716-874-4508
www.AmherstMedia.com
Publisher: Craig Alesse
Senior Editor/Production Manager: Michelle Perkins
Editors: Barbara A. Lynch-Johnt, Harvey Goldstein, Beth Alesse
Associate Publisher: Kate Neaverth
Editorial Assistance from: Carey A. Miller, Sally Jarzab, John S. Loder
Business Manager: Adam Richards
Warehouse and Fulfillment Manager: Roger Singo
ISBN-13: 978-1-60895-884-9
Library of Congress Control Number: 2014955672
10 9 8 7 6 5 4 3 2 1
No part of this publication may be reproduced, stored, or transmitted in any form or by any means, electronic, mechanical, photocopied, recorded or otherwise, without prior written consent from the publisher.
_Notice of Disclaimer:_ The information contained in this book is based on the author's experience and opinions. The author and publisher will not be held liable for the use or misuse of the information in this book.
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CONTENTS
About the Author
Preface
**SECTION 1.**
**THE BASICS AND BEYOND**
The Natural Light Camera Room
Strobe/Natural Light Correlation
**JUMP PAGE :** What Is the Inverse Square Law?
Tools of the Trade
Understanding Exposure, Manual Style
White Balance
Reflector Placement
Portrait Light Patterns
The Rhythm of Light
Outdoor Subject Placement
**SECTION 2.**
**UNPLUGGED: NATURAL LIGHT IN PRACTICE**
1. Too Tuff
2. The Tile Painter
3. Down the Road
4. Camouflage
5. Stair Master
6. Yellow and Blue
7. Skater Boy
8. Alley Girl
9. Give Me Three Steps
10. River Watch
11. Going Down
12. Playing Pool
13. Window Seat
14. Winter Woods
15. Flower Girl
16. These Boots Were Made for
17. Sweet Light on the Beach, Part 1
18. Sweet Light on the Beach, Part 2
19. Glamour Girl
20. Red
21. All Aboard
22. Let's Jam
23. Dove Girl
24. Train Door
25. Thoughts and Dreams
26. Dreadlocks
27. "Cocaine Lighting"
28. A Little Help from Her Friends
29. Black Rocks
30. Train Step Girl
31. Piano Lighting
32. Mountaineer Football Player
33. Yellow Flowers
34. Football Close-Up
35. Blue Pallets
36. Behind Black Rocks
37. Prom's Over
38. Sweet Light Family Portrait
39. Cheerleader
40. Pointer Sisters
41. Diagonal Door Trim
42. Train Hang
43. Grassy Beach
44. Skater Boy Sky
45. Lake Swimmer
46. Black Ore
47. Watercolor Girls
48. Wedding Rapids
49. The Brown Lounge
50. Jack and Jill
51. Papa Bear
52. Train Wheel
53. On the Rocks
54. Lake Antoine at Sundown
55. Purple Ditch
56. Found on the Doorstep
57. People Watching
58. Open Door
59. Pieces of Me
60. Getting Fancy
Conclusion
Index
* * *
ABOUT THE AUTHOR
Carl is a Kodak Alaris mentor and has been involved with photography for over twenty-five years. He started his photographic career in the darkroom as a custom printer and technician. He is PPA Certified, a Master Photographer, a Craftsman with Professional Photographers of America, and an international photographic judge. He has won numerous national awards for his photography, including twenty-four PPA Loan Collection images and several Kodak Gallery and Fuji Masterpiece Awards. He has been the Wisconsin State Photographer of the Year three times and has received a multitude of state awards. In 2014, the Wisconsin Professional Photographers awarded Carl the prestigious National Award for his contributions to the field of photography. You can see more of his work on his website: www.photoimagesbycarl.net.
As much as he loves creating portraits, he also has a passion for helping others become better portrait artists. Carl is one of the most sought- after instructors in the country and abroad due to his hands-on coaching approach. Don't plan on just watching in his class! Carl will challenge you to become a better photographer than you already are. His photographic skills are just part of what will help each student. His greatest strength is his ability to see what skills others already possess and then find ways to help enhance those skills so that photographers can take their work to a new level.
* * *
PREFACE
This is a book of information, not fluff. When you read through the pages and study the images, you will learn to find and use natural light for your portrait needs. Embrace the ideas. They have worked for many years and will continue to work through the end of time, as natural light portraiture has been around since the beginning of photography.
Our industry has changed rapidly as of late. With all the new lighting equipment on the market, why use natural light? There are several reasons: (1) Biology. The eyes of the subject are more colorful and powerful. (2) Education. The light is constant and can be seen before the image is captured. (3) Cost. Natural light is free. How about that? (4) Subject psychology. The subject doesn't have disturbing flashes going off every time an image is created. (5) Viewer psychology. The scene the viewer sees in an image is natural. I'll elaborate.
First, let's look at the biology. Our eyes are like a lens on a camera. The pupil will expand and contract to let in a comfortable volume of light in a given area. In most cases, we work in open shade, which offers ample light for the human eye. The pupil will contract to a small aperture and more of the iris will be visible. The iris, of course, is the colorful part of our eyes. More color means more power. In a dark area, such as a dimly lit studio, the pupil becomes large and covers the iris. If a strobe illuminates the subject at this time, the eye is captured with a large pupil and a lack of color and power.
We see catchlights in the eyes of our subjects when viewing an image. If the light source was handled inappropriately or there are several catchlights from different sources, it will be bothersome to the viewer. For you studio strobe folks—if you need to work with strobes and still want power in the eyes, turn on the overhead lights and get a stronger modeling lamp in your strobes. The bottom line is that the constant light needs to be bright enough for the pupil to contract. It's Biology 101.
"Our eyes are like a lens on a camera. The pupil will expand and contract to let in a comfortable volume of light in a given area."
This is a close-up of an eye in a natural light area. Notice all the beautiful color in the iris.
What about education? When learning the art of placing light on a subject, I feel it is easier with natural light. The light patterns used to sculpt a human face can be seen the entire time before and during an exposure. There should be no surprises. Once you see the shape, form, and texture of your subject, just record it. We'll get back to this topic later.
Natural light is cost-effective; it's free. No batteries, no wire connections that fail, no cords, no triggering devices, transmitters, receivers . . . you get the point. There are times when we use reflectors or panels to help modify the light, but again, there are no batteries required.
Let's talk about subject psychology. When I first began creating images, I covered sporting events and yearbook assignments. I didn't have a strobe at the time, so I had no choice but to record with natural light. I noticed quickly that my images captured a more true representation of the subject compared to those made using strobe. I noticed that most people acted differently when they knew an image was being taken. With strobe, after one pop, your subject is aware of the camera. Natural light is less intimidating. Even when people are sitting for a portrait, it is human nature to blink when a flash goes off. In fact, many subjects anticipate the flash and blink before it happens. That is not a comfortable situation. In a natural light studio or setting, there are fewer distractions. Natural light allows for a more comfortable scenario, and your subjects look more like themselves.
Next, we come to viewer psychology. When looking at an image, whether we realize it or not, our brain is scanning the image for natural truth. When there is light on a person that couldn't have been there naturally, a little red flag goes up. When there are catchlights from multiple lights that don't make sense, there's another red flag. The more natural and comfortable an image, the more the viewer will like it. Okay, strobes can produce a similar result, but it is more difficult to re-create a "natural" scene electronically. Yes, there are times when it becomes necessary to use strobe, but you need to think about how viewers will perceive the image, not just the exposure. For the purpose of this book, and the understanding that you want to learn natural light portraiture, I will not use the "f" word for our learning examples. "Flash" will only be used to compare the relationship of the strobe studio and the natural light studio. All image samples will show how to use natural light.
"I'm going to show you how to work in this medium so that you can make a living; from there, you can add your own creativity to take your artistry to the next level."
Before we move on, I want to mention that this is a book of "bread and butter" portraits done during class demonstrations or for actual clients. In either case, they are always treated with the same respect to artistic ideals. There are a few award-winning images here, but award-winning images don't always pay the bills. I'm going to show you how to work in this medium so that you can make a living; from there, you can add your own creativity to take your artistry to the next level. All I ask is that when you become famous for your work, please contact me and share your ideas so that I may grow as well.
SECTION 1
The Basics and Beyond
In recent years, the natural light photographer has been losing credibility. We are seen as hobbyists instead of portrait artists. Part of this is due to the influx of beginners who don't want to spend money on lights and don't realize there is more to portraiture than just creating a correct exposure of a child running around in a yard. Please don't think I look down on these folks. I started the same way. I hope this book will open everyone's eyes; my aim is to help people grow artistically by learning how to use natural light to produce professional-level portraits.
My natural-light camera room.
THE NATURAL LIGHT CAMERA ROOM
I am going to begin with the north light camera room. This is my main indoor studio room. I say this because I use windows and outdoor locations all over the world as my camera room. That's an advantage of natural light. You don't have to carry it with you, it's everywhere.
This room has windows facing directly north. For this reason, direct sunlight will never come into the room. The ceilings are 20 feet high. The tallest part of the glass is 18 feet tall. The height is important. The light needs to move the entire 21 feet across the room to the south and still be just above head level when it gets there. This allows portraits to be created anywhere within the room. The length of the windows combined is 26 feet. Soon I will explain the significance of this length. As you can see, I have a balcony in my camera room. This is part of my home-based business. The upper part of the balcony is attached to the main level of our home and is used as a workout area and lounge. I can also use it to photograph from directly above my subjects, and it has come in handy for seniors and commercial work throughout the years. The balcony is supported by pillars designed to be a permanent, classical set in my studio. The floor is Italian porcelain in neutral colors with design work that can also be used in portraits. Outside the windows is a cleared area with light decorative rock. This helps with color issues later in the day when the sun is on the ground outside the windows. One more important note: There is a wall on either side of the window. My walls are 4 feet long. If I had it to do over, they would be 12 feet long. The walls allows you to control the density and focus of your background while your subject remains in good light. No matter where you go, indoors or out, look for a lighting scenario such as this: a ceiling to block light from above, light allowed in from one direction, and an area of controllable backgrounds.
"The light needs to move the entire 21 feet across the room to the south and still be just above head level when it gets there."
STROBE/NATURAL LIGHT CORRELATION
I use these windows as I would use strobes. As you can see in the diagram, the principles of lighting a person are the same in each approach. A light should be placed at about a 45 degree angle from the subject on a horizontal plane and about a 45 degree angle on a vertical plane. This "main light" produces directional light that will create shape and form on the subject. The main light should be evident in the subject's eyes as a catchlight (a white highlight in the eye). The catchlight should be at about 2 o'clock or 10 o'clock in the iris, depending on what side of the subject the main light is placed. A fill light can be used to bring the relationship of the shadow and the main light to a desired, printable unity. Remember, the main light gives direction. The fill light shouldn't alter that direction. We'll talk about that in greater detail soon. Then there are accent lights (also called kicker lights or separation lights). These lights create separation and are used to keep the subject from blending into the background. They also help create a three-dimensional feeling to the flat piece of paper we call a picture.
Studio-window diagram.
As you'll notice, these light sources do not point directly at the subject. They skim the subject. Yes, not only do we want a direction of light that gives us shape and form, we also want texture on the subject. The only thing in nature that will create texture is light skimming across a surface. A good example: If you want to photograph ripples in the sand on the beach, when do you go? Sunset or sunrise of course—but you will get the most out of the scene right when the sun disappears over the horizon. This is when there will be a direction that shows the crests and valleys of the ripples, and also the grains of sand within the scene. The human face is a reflective sphere. Within that sphere are other shapes: eyes, nose, cheekbones, hair, and pores in the skin. As an artist, I want it all to be seen. So no, it's not as easy as pointing a light at someone and recording the image. There is so much more to it than that. I'll get to the use of light in a moment, but first we need to talk about the physics of light.
Light will always travel in a straight line. For the sake of understanding, let's just assume the light from a given light source is a beam of light. In the beam itself, the center is most concentrated and intense. Anytime the beam travels directly at a reflective surface, such as the human face, the surface will reflect that beam directly back at its origin. This will cause a spectral highlight (a white reflection with no detail). This is why we use the edge of our light source. It keeps us from losing detail in our subjects.
You will find that there are small light sources and large light sources. Each produces a different effect. A large light source is one that is large in relationship to the subject. An 8-foot window facing north with a subject 2 feet away is a large light source. A 2x2-foot window 10 feet away is a small light source. An open sky on the shadow side of a building is a large light source. The sun in an open field at midday is a small light source. A small light source will produce very defined shadows and a great deal of contrast. A large light source will create soft-edged shadows and more natural fill from highlight to shadow. Both can be used in a natural light portrait, but I tend to favor the larger light sources or at least small light sources without direct sunlight. Most people I photograph can't keep their eyes open in direct sunlight. So unless my subjects are wearing sunglasses, I stick to working in open shade.
"A large light source will create soft-edged shadows and more natural fill from highlight to shadow."
Light will always fall off in intensity as it travels farther from its source. (This principle is known as the inverse square law.) I will avoid being overly technical, but you should note that the farther the light travels, the less there is to expose an image. It is this principle that will explain the use of different-sized light sources. I'll start with a small window. If the subject sits on the edge of the window, the light from the window will show on the person's face. The light may only show on one side of the face. This is because the face itself blocks the light from hitting the other side. If the subject sits on the edge of a large window, the face is still in good light, but the light traveling from the opposite side of the window can also reach the opposite side of the face. This is what we call wraparound light. Because light falls off (at a rate described by the inverse square law), the light traveling from the opposite edge of the window from the subject is less intense than that on the highlight side of the face. This is an important principle to understand, as it will dictate how you will use any given light source.
* * *
_What Is the Inverse Square Law?_
The inverse square law for light intensity states: "The intensity of light is proportional to the inverse square of the distance from the light source."
For those of you who do not like math, just understand that the farther your subject is from the light source, the less light there will be on your subject.
Most of us would guess that twice the distance from a light source would cut the intensity in half. That is not the case. Doubling the distance would change the intensity by the square of one half; in other words, we would end up with one fourth the intensity. If you move the source or subject three times the distance, the intensity would change by the square of one third, or one ninth the intensity.
The closer the subject is to the light source, the greater the change in intensity with a small distance change. In other words, if you stand one foot away from a light and take a reading, then move back one step, there will be a greater change compared to if you were twenty-five feet away and took one more step back. This is good information to know for group portraits. If you have three rows of people and want the light to be the same intensity on all rows, move your light as far away from them as you can while still achieving the exposure output required to use the aperture needed to carry the focus throughout all three rows. In contrast, if you place a light very close to the subjects, the person in the front row may be much brighter than the person in the back row.
There are some exceptions. You can use the inverse square law to predict the rate of light falloff for speedlights and studio strobes. Umbrellas and small softboxes will follow the law quite closely for distances greater than twice their diameter, but there are some exceptions. Light sources modified by focusing units (like a fresnel lens) or those outfitted with a grid will not follow the inverse square law as predictably. Also large, diffused light sources like a north light wall of windows will fail to follow the same equation. That said, still, the farther away the light source is from the subject, the less intense the light will be. The wider the window, the larger the working portrait area.
Large light source.
Small light source.
* * *
"You can use the inverse square law to predict the rate of light falloff for speedlights and studio strobes."
TOOLS OF THE TRADE
Reflectors are very important tools for working in natural light. I use several, ranging from white to very bright, mirror-like silver materials. I do not use gold. Reflectors are used to add accent light or to fill (lighten) shadows. Shadows are cool in color temperature. Gold is a very warm color temperature. The two do not mix well. The only exception to this is on a pastel sandy beach at sunset. When everything seems to have a warm glow to it, a gold reflector can be of some use.
The reflectors I use come from a company that is no longer in business. In the future, check out Sweetlight Systems at www.sweetlightsystems.com. They create light modifiers and are looking to get into the reflector business as well. I have also used, and still do at times, insulation board (available at building supply stores). It works just fine, though it doesn't look as nice and takes up more room when traveling. I like a reflector with a "kickstand" so that I don't need assistants. The reflector with a kickstand also works fine as an under-reflector outdoors. I like to have two reflectors along on an outdoor session if I can. At least one should have black covering one side. This way, I have a reflector and a gobo with me at all times. When I'm inside, I use a bright silver reflector for accent light, a soft silver reflector for fill, a small reflector on a goose-neck stand for an under- reflector, and a gobo on a goose-neck stand as shown in the images on the facing page.
A good tripod is an essential piece of equipment. I use a Bogen Manfrotto #058B tripod. It is a heavy but very effective tool. The length of the legs are controlled by levers up where you hang on to the tripod itself. By pressing the middle levers, all the legs are loosened and mold to even the toughest terrain. When you release the lever, the legs lock. You can also adjust each leg individually with the smaller levers as shown in the close-up image (facing page). No more missing the image because you're messing with tripod legs. This is one thing I cannot live without.
I do not care for camera straps. They pull on your neck and shoulders and they allow your camera to become a wrecking ball each time you bend over to fix hair or clothing on a guest. I prefer the Spider Holster (www.spiderholster.com). This belt and holding device solves all issues and can lock the camera into place. I have ran and jumped over rocks with my camera in this holster. It's great to have a safe place to store your camera while you are setting up a shot.
Indoor bright and soft silver reflectors. The bright silver is used for accent lighting. The soft silver is used for fill. Both stand alone without assistants.
Black back for subtractive light. At least one reflector I carry will be black on one side. This way, I always have a gobo along with me to block light where I don't want it.
Outdoor large reflector. This is a more durable reflector for use in tougher environments. It comes apart and rolls up for easy traveling. It has a self-supporting stand.
Under-reflector. This is a great tool for use in the studio. It is used on the highlight side of the subject and from underneath. It is faced toward the window and feathered back to the subject until you just begin to see the catchlight in the eyes. Be careful! This is the most overused light source in natural light portraiture.
Gobo. This is a black panel on a gooseneck stand. Use it to keep light off small areas where you do not want it. It's like custom burning in-camera!
Tripod and tripod close-up. The Bogen Manfrotto #058B with a ball-joint head. This tripod allows the legs to be adjusted with gravity. By pressing the red levers shown in the photos, the legs fall down until they rest on solid ground. There's no need to reach down to unscrew each leg section. This model allows you to work quickly over even the roughest terrain.
Spider holster (www.spiderholster.com). This is my preferred replacement to the neck strap. It is comfortable and safe for your camera and your subject. It does not allow your camera to swing at them when you lean over to fix their hair or clothing.
Canon 5D Mark III with Canon EF 70–200mm f/2.8L IS USM lens. This is my go-to choice for natural light portraiture.
Last of all, there is the recording device. Why is it the last tool I listed? Because it is only that—a tool, a recording device. Working with the relationship of the subject, the light, and the background is creating photography. I can record it on my iPhone or my Canon. Yes, my Canon 5D Mark III is something they will need to pry from my cold, dead hands someday, but it doesn't create anything. It just records what I do. That being said, it is a fine piece of equipment. It allows me to work at high ISO settings for my portraits and gives me what I expect each time I press the shutter release button. I use several lenses, but my workhorse is the Canon EF 70–200mm f/2.8L IS USM lens. You will see more of my lens selections later in this book.
UNDERSTANDING EXPOSURE, MANUAL STYLE
Next on the agenda is how to create the correct exposure. This is a camera thing. When you were just learning to take pictures, you probably set your camera to auto or program mode and let it do all of the work. For full artistic control, you'll need to set your camera to manual mode. Ask yourself questions about the scene. For instance, do you want a lot in focus or just a small area? Do you need to stop the movement of a child running across the yard? The answers to these questions will determine your exposure-setting selections.
Of course, getting a proper exposure is critical. Therefore, let's start our discussion with metering. There are two ways to meter your scene: you can use a reflective meter or an incident meter. The meter in your camera gives you a reflective reading. A hand-held meter will give you an incident reading or, when fitted with an attachment, a reflective reading. The reflective meter measures the light bouncing off the subject. An incident reading tells you how much light is present at the subject. Both will work with practice.
The meter in your camera is a reflective meter. It should tell you whether your shutter speed and aperture settings harmonize to give a correct exposure for any given scenario. To use this meter, you simply look through your viewfinder and watch the meter readout. You can adjust the aperture (f-stop) or shutter speed until the exposure needle appears in the center of the graph. Of course, you must be pointing the camera at the specific area of your scene that you want to expose for. In portraiture, that would be the highlight side of the subject's face. Yes. Expose for the highlights. If your highlight is overexposed, there is no visual information (detail) there. You need to expose your highlights properly. _Note:_ Not all light meters are calibrated correctly. You need to test your equipment. Your meter may need to be calibrated to a known, correct reading. In the end, you should be confident that your meter, hand-held or in-camera, will give you an accurate reading. If you are using a hand-held (incident) meter, position the meter on the highlight side of the subject's face and point the meter's dome at the light source. Entire books could be written on exposure. For the purposes of this book, let me finish by advising you to practice!
There isn't enough space in this book to get too deep into f-stops and shutter speeds, but I will touch on those topics.
Let's imagine that a meter reading of a hypothetical scene yields recommended camera settings of second at f/8 (at ISO 400). This will create a correct exposure. However, other settings (called equivalent exposures) will allow for the same correct exposure. Why would you use different exposure settings? Well, in some scenarios, you'll want to use a wide-open aperture to blur the background and eliminate distractions. In others, you'll want to show the beauty of the natural environment, and you'll want the background sharp. When you change the aperture, you reduce or increase the amount of light used to record the image. You'll need to reduce or increase the shutter speed to maintain the correct exposure. There are times when stopping the motion of a running child will require a fast shutter speed. You may need a faster shutter speed if you are working with a heavy lens, as well, to prevent blur from camera shake.
The light meter. This is a hand-held light meter. It is used to measure the ambient light falling on the subject. Notice the dome points toward the light source. There are also attachments to use this meter for a reflective reading, just like the meter in your camera.
Now, let's assume that your meter gives you a reading of second at f/8. This setting might help you prevent motion blur that stems from an unsteady hand. F/8 offers a great depth of field. ( _Note:_ Depth of field is the amount of the scene that appears in focus on a horizontal plane from the camera.) That aperture setting may well keep the distracting background elements of the scene in focus. Looking at the chart below, you might consider changing your exposure to second at f/4 so the background is less in focus.
A third control used to adjust exposures is the ISO. The ISO governs the light-sensitivity of the sensor. A high ISO, say 1600, makes the camera more sensitive to light. A low ISO setting, say ISO 100, reduces the sensitivity of the sensor. A low setting is a good choice for very bright, sunny scenes.
ISO 400 is my go-to setting for natural light work. It used to be true that you'd see more noise in an image shot at ISO 400 than ISO 100, but with today's sensors, there is no real difference in quality. You may as well take advantage of using a higher ISO so that you can choose a faster shutter speed to stop motion. The ISO performance in newer cameras is impressive. I have no problem working at 3200 ISO on my Canon 5D Mark III camera. Yes, there is a difference in noise, but it isn't objectionable, even for the most critical of eyes. I have even used settings beyond 3200. Noise is present in extreme ISO settings, but it's not as noticeable as the grain we saw in high ISO films. If you need the exposure and the light is lacking, don't be afraid to use even the highest ISO setting your camera offers.
Exposure equivalent chart. Once a correct exposure is calculated for a given scene, many options are available as long as you are working in the manual camera mode. This table shows all the aperture and shutter speed combinations that will work in this area and still produce a correct exposure. Some will allow for stopping action while others will allow more depth of field.
LENSES
I mentioned that I use the Canon EF 70–200mm f/2.8L IS USM for most of my work. I favor the narrow angle of view that the lens allows. This allows me to find small background areas without hot spots and distractions. It also compresses the background and helps eliminate distractions.
There are times when a wide-angle lens is the right tool for the job. You will know it when it comes up. A family grouping or a wedding party is not one of those times. A wide-angle lens will distort and create unflattering size relationships as subjects are placed behind one another. The subject in the front row will have a large head, while the guy in the back will have a tiny face. Don't be a witch doctor. I recently bought the Canon EF 85mm f/1.2L II USM lens. I love it! One last tip: Use the lens hood that came with your lens. You have a big expensive piece of glass in your hands. The hood helps protect it and prevents lens flare. If you want flare, take it off to get the shot, then immediately replace it and continue shooting. You'll be amazed at the difference in clarity you will see in your work by using the lens hood.
Custom gray card.
Close-up of custom gray card.
WHITE BALANCE
In natural light photography, it is very important to understand white balance. The color temperature of the light may be different in any given scene. When shooting outdoors, I typically work in open shade. When I'm in those areas, setting my camera to the shade white balance preset works fine. But there are times when the light changes—a cloud moves overhead, a red car parks where it reflects light in your direction, or raw light bounces off a yellow building. All these areas need to be custom white balanced. The same goes when you're working with window light. Glass has a colored tint to it; our eyes don't see it, but the camera does. By doing a custom white balance, the color balance in that area becomes correct for your recording and your subjects are rendered the right color.
The first step is to meter the scene. If you don't have the correct exposure, you cannot achieve a correct custom white balance. To be consistent, I record an image of a special gray card. You need to fill the frame with this card. If you can't focus that close, don't worry. The card doesn't need to be in focus. Once the gray card is recorded, check your math (i..e., double check that your exposure was correct). You can do this by looking at your histogram. The histogram is a graphical representation of all of the tones in your frame. When you photograph a gray card, the histogram should look like the graph shown here. The middle spike represents the gray center of the card. The spike on the left and right represent the black and white areas respectively. Notice the spike in the center is wider than the outsides. This is because the histogram shows the percentage of that assumed density throughout the frame. There is more gray, so that spike is wider. Once I see this correct graph, I flip the gray card over to white, fill the frame, and record it. Do not change the exposure! You must do this in manual mode or lock your exposure from the gray card. In your first image, the card may appear off-color. Next, go into your camera's menu, find the custom white balance option, and select it. Your camera will ask you if you want to use the last image. Yes, you do. Your card should now appear white. Your camera will now tell you to choose the custom white balance (if you weren't already set on it). Record an image of the person holding the card just to be sure the colors look right. You are now ready to create images until the next cloud moves overhead.
Histogram.
Before custom white balance.
After custom white balance.
REFLECTOR PLACEMENT
Reflectors are light sources, just like windows or strobe lights. The placement of these light sources is critical. A light shouldn't be pointed directly at our subjects. Rather, the edge of the light should be used. The same is true of a reflector. I place my accent reflector first; it is positioned behind and to the shadow side of the subject. As you can see in the images on this page, if a line was drawn perpendicular from the reflector surface, the very edge of the light source would skim across the subject.
The fill light reflector is placed in a similar fashion but in front of the subject, as shown. In my studio, it actually points directly at the windows. Do not angle this reflector toward the subject. If you do, you will see a cross shadow under the nose, which is not flattering. Remember, fill light shouldn't change the direction of the main light. It should just fill the shadow area.
The accent reflector is placed behind the subject with just the edge skimming light across her outline. It is usually on the shadow side, but on occasion, can also work from the highlight side.
The fill reflector is placed across from the main light in front of the subject. It is used to control the intensity of the shadow in an image without changing the direction of the main light.
Under-reflector placement. The reflector from underneath the subject actually stays on the window side of the subject and tilts toward the window. Feather it back to the subject until it shows in the eyes. Be careful. This reflector destroys more images than any other light source. Again, it should just add a little light in the eye and under the chin, but not change the direction of the main light.
Use a gobo to block light from falling on areas you don't want lit. The gobo is your "in-camera burn tool"; darken areas now, then there will be no need to take care of problems later.
The under-reflector is an often misused tool. It is used to put a little light in the eyes, under the chin, and in the eye sockets. It should not change or compromise the main light direction. This reflector should point back toward the window and is placed on the _highlight_ side of the subject (see the image above). To determine the best-possible placement, start by pointing the reflector at the window. Slowly tilt the reflector toward the subject until you see the light in their eyes. Stop there! Don't overdo this. When it's overdone, light will shoot up the nose. I call this "cocaine lighting." Just say no.
You will sometimes hear photographers talk about subtractive lighting. Some think this is the practice of using a black panel or fabric on the shadow side of the face to suck light from the subject. Well, light cannot be sucked away. All the black panel does is keep light from coming from that direction—just as the subject's own face blocks light from hitting the far side of the face when you are using a small window. The tool used for subtractive lighting is called a gobo. A gobo can be used to block light from the entire subject or just parts of the subject. In the image above, a gobo was used to block light from striking the subject's chest. A gobo is used in every image in one way or another. In the camera room, my ceiling is a gobo. The walls are gobos. You get the idea.
PORTRAIT LIGHT PATTERNS
There are many specific light patterns that have been used in master paintings as well as photographic portraits for centuries. These patterns are appealing to viewers and help tell a story of the subject at the moment the image was captured. These patterns are proven to be flattering. Try to use them. I've shown them in black & white so you can concentrate on the light and dark areas rather than the colors.
Use the edge of your light source. In my studio, placing the subject on the edge of the length of windows produces better wraparound light and better texture on the subject. It also keeps the light direction coming from the side and the front of the subject, rather than from both the front and back.
This image shows the short light pattern. The mask of the face is lit. A fill reflector was used to lighten the shadows, narrowing the tonal range from highlights to shadows.
This is the same short light pattern, but with no fill reflector. Notice how the shadow is a bit darker than before.
Subject placement within the light is important. For the first two patterns shown here, short light and broad light, the subject should be on the edge of the window. More often than not, the chest of the female subject is turned away from the window. This way, the face is brighter than the chest. From this position, the subject can turn her head back toward the window to produce a short light pattern. This pattern reveals the mask of the face. It is especially noticeable in a ¾ view of the subject's face. That is when a clean cheek is visible on the highlight side of the face and one ear is shown on the shadow side of the face. This is a happy light. Smiles are welcome here.
Broad light is the same ¾ view of the face, but this time the visible ear and the broad side of the face is illuminated. The mask of the face goes into shadow. This is a more serious, emotional light. The deeper the shadow, the more somber the feeling. The placement of everything remains the same as in the short light, but the head just turns away from the window.
There are times when the subject looks more comfortable turned into the light. The same patterns can be used. I just add a gobo to block the light from hitting the chest. You can see in the examples the difference a well-placed gobo can make.
Here you can see the camera and subject placement compared to the window. The camera is pretty much parallel with the window. This is the positioning for most "short light" and "broad light" portraits.
This image shows what a "broad light" pattern looks like. The face is turned away from the main light so that the mask of the face actually goes into shadow. This light may add weight to a subject so be careful when putting it into practice. In this image, the fill reflector was used to bring the relationship of the shadow closer to that of the highlight.
Now you can see the same "broad Light" pattern with no fill. Notice the darker shadow side of the face.
I usually choose to turn a female subject away from the main light. In some cases, you will find yourself in a position in which she needs to face it. The problem is that the chest reflects more light than the face, and it is a larger area. Unfortunately, the chest will draw more attention than the face. This is not a good scenario.
Here, a gobo was used to block the light from hitting the subject's chest. This allows the viewer to focus on the subject's face. Here, the model's face is in a short light position.
This is the same gobo usage as before, but this time the model's face was turned to create a broad light pattern.
This portrait shows a Rembrandt light pattern. The mask of the face is mostly lit, but shadows help define a light triangle on the shadow side of the face. This is created by the position of the camera, subject, and light source.
This is the overall view showing where the subject, camera, and window are positioned to achieve a Rembrandt light pattern. Rule of thumb: if you want a triangle of light, make a triangle with you, the subject, and the window.
Here is the setup used to create a split light pattern. I usually work the same as when using Rembrandt light, but I have the subject turn his or her face away from the window until I see the light stop at the midsection of the face.
This image shows the split light pattern. A fill reflector was used to keep the shadow side of the face from going too dark.
This is what the split light pattern looks like in my camera room without additional fill from a reflector panel. Remember, the walls on the south side of my camera room are reflectors as well and naturally fill the shadow side of the face in all portraits.
Rembrandt light is similar to short light, but it reveals a triangle of light on the shadow side of the face between the nose, eye, and cheekbone. This is a more serious light, one for a proper, sturdy pose and expression. It's easy to remember how to achieve this pattern. If you want a triangle, you need to make a triangle with the subject, the camera, and the light source. As you can see in the Rembrandt overview image (facing page), the camera position changed from parallel with the window to farther into the room to achieve this pattern.
Split light is what it sounds like; half the face is in light and half is in shadow. By now, you can probably guess that this is not a smiling light pattern. It is one used to show attitude, intensity, or concentration. To make this pattern, continue to move the camera away from the window and have the subject turn her face to follow you. When you see the light disappear on the shadow side, stop and record the image.
Profile light is used to dramatically show the profile of the subject. This light comes from behind the subject, who is posed with the shadow side of their face toward the camera. This is a thought-provoking, emotional light. In most cases, it is best used for a more conservative expression. Many photographers are unsure as to how this pattern is produced with window light. I'll demonstrate with a few images.
More often than not, I see profile lighting attempted as shown in this image. When the subject is placed in the center of the window, the light engulfs her and creates spectral highlights on the face directly facing the light. The clothing (and bouquet if this were a bride) will be washed out and appear to glow. This happens even if exposed correctly. This is not how to light a profile.
This is the result of placing the subject in the center of the window. Notice the flatness of the face and the washed-out hair.
Here is the correct placement of the subject for profile light. The subject is on the same side of the window as the camera. If you were to stand in her shoes, looking forward, you would see the wall, not out the window. This forces the light to skim across the subject from the opposite side of the camera. This will produce a nice profile with detail.
Here, you can now see the result of proper placement to create a profile light. Notice how much more shape she has and the detail throughout the image. _Note:_ You still need to expose correctly for this. It is easier to use a light meter in this case because, from the camera position, the highlight is not very readable.
"Single accent is a dreamy, storytelling light pattern. The mask of the face is in full shadow. The only defining light is just skimming the side of the face."
First, the camera is positioned parallel to the window, just as with a short light. The subject is on the same side of the window as the camera.
Profile window light can make for a great bridal image. The dress will hold all its detail and the bouquet will not be blown out. If you want, you can add Mom and Dad on the other end of the window watching her. The light from the same window can give Dad a broad light and Mom a short light as they stand together and gaze at their little girl.
Single accent is a dreamy, storytelling light pattern. The mask of the face is in full shadow. The only defining light is just skimming the side of the face. In most cases, the subject is looking down or away from the camera.
Double accent is just what it says. The defining light on the subject is on either side of the face. Usually, as with a single accent, the light comes from behind the subject and skims the sides of the face. Because the mask of the face is in shadow, this is a more somber, thought- provoking light pattern. There are times when eye contact is made from the subject, but only if you are trying to portray a deep and powerful story. For either single or double accent, the subject needs to be positioned where the light just skims around the edge of the background. In my case, the background is against the bright window. I employ a second background as a "slip" so light doesn't come through the canvas.
Here is an example of a single accent light. The face is in shadow. The only defining light skims across the edge of the subject. This is a good pattern for deep emotional images.
This is not where she should be. Be careful not to position your subject out of the light. Move them away from blocking devices until they are at the edge of the directional light.
Now she is in a place where the light can do what it needs to. Find the edge of the light.
This is what happens when the subject is too close to the blocking devices. Notice how flat the light is. It lacks texture.
Study the images on these pages. You will see examples of these light patterns. Each and every pattern can be created with small or large light sources. As I stated earlier, the difference will be the harshness of the shadow that defines the pattern. The images will also have various amounts of fill. You will see all of these patterns put to use in the next section of this book.
"Every pattern can be created with small or large light sources—the difference will be the harshness of the shadow defining the pattern."
Now you can see the result of a double accent light. Most of the time, I have the subject look away from the camera to help with the mood from lack of light on the face itself.
The exception to the rule. This image was also made with a double accent, but this time, there is eye contact. Sometimes the subject's expression and body language allows it. It can make a powerful image. I still wouldn't suggest having the subject smile from behind this shadow—it could be kind of creepy.
Here is an overall view from the camera position. This setup can work for single accent or double accent patterns depending on the subject placement and the subject pose.
THE RHYTHM OF LIGHT
Light rhythm refers to the relationship between the highlight and shadow areas of the subject and what is directly behind the subject. The highlight side of the face should have a darker tone behind it. The shadow side of the face should have something relatively lighter than that of the highlight side. This sends a push-pull message to the viewer's brain. Anything light on a flat piece of photographic paper will appear to advance. Anything dark will appear to recede. In our minds, when the subject and background contrast with each other, depth is created. Also, since we are not working with background lights, ensuring that light rhythm exists will allow for separation between your subject and the background. Not every background will allow for this opportunity. If it can exist, it will make your images more powerful. Shown here (page 28) is the same image, with the same background, in different locations.
I painted this simple but well-thought-out background years ago. I needed a backdrop that helped with overall shape and form. Remember, in natural light photography, there is no background light. There is only subject placement and background relationships. We need dark areas to contrast highlights and light areas to contrast shadows.
To show exact conditions with only the background changing, I created this photo on the green screen. By extracting the subject and changing her relationship to the background, I can show the impact that light rhythm can have on your image.
An example of good light rhythm. Notice how the highlight on the subject is lined up with a darker area of the background. On the shadow side of the face the background transitions into a lighter area as compared to the highlight side. Great depth is created.
Now the subject is moved into an area that is just the opposite. Notice the lack of depth. In fact, the subject seems to blend in instead of jumping off the page.
To show how light rhythm works, I photographed my subject against a green screen, then cut her image out and placed it in a position against this backdrop that allows for good light rhythm. In the first image, we have a rhythm of dark (background) against light (subject), dark (subject) against light (background). The other has just the opposite. Which one stands out?
OUTDOOR SUBJECT PLACEMENT
Cloudy days are often said to be better for portraits because "you can photograph anywhere." This is not true. I do like cloudy days because they allow for more options for backgrounds, as they stay in a consistent tonal range. The art of lighting a subject still remains the same, whether cloudy or sunny. You need to work in an area that allows for good directional light. The question is where. Here is a simple answer. First, find a cover to stop light from above, just as you would indoors. Be careful not to go too deep under the cover and lose the light. Second, make sure light can only enter from where a main light would be positioned. This is a hard lesson to learn, so I will give you a simple-to- remember trick. Go sit where you were planning to pose your subject. Look around. If you can't see a main light source where it should be, or where you would have placed a strobe, you are in the wrong place. Find a new area. Think in terms of where the camera will be. When the camera is there, is the background acceptable? When the background is acceptable, will the light pattern you want to achieve be possible given where the light is coming from? Just because you are outside doesn't mean the rules can be thrown out the window. (Okay, that was a bad choice of words, but you get the idea.)
When you place the subject too far under the tree, the light gets flat and the color is contaminated.
The subject is in an open yard on a cloudy day. Notice the dark eye sockets and washed-out hair. The background is without spotty light, but this is not the most appealing place for the subject to exist. The fact that there are clouds in the sky does not mean that we can work just anywhere.
Whether you are shooting on a cloudy or sunny day, trees, overhangs, bridges, and canopies are good options—just be sure your subject is not too far under them. You'll need to make sure that good, uncontaminated light is able to reach your subject.
Here, the subject is under the cover of the tree and the background is consistent because of the cloud cover. Notice the brighter eyes and the added detail in the hair on top of her head.
The subject is too far out in the sun. There are hot spots on her head, shoulders, and the background. These areas will be impossible to "fix" in Photoshop. This is not the area to shoot in.
The images below and on page 29 show you what to look for and what to avoid. Read the captions and study the images for future reference.
Remember, find a good background. Make sure there is good light to use with that background and the potential subject. Let the subject exist within that space, then record the image.
The rest of the book contains many images and the thought processes I went through to create them. I hope that during your time with this section, something will help you in your journey in the art of natural light portraiture.
Here, we have better placement of the subject, but the background still has problems. She is out of the raw light and the image will hold together for printing, but the background is washed out over a large area. This brightness will take away from the subject. It would be very time-consuming to "fix" in Photoshop. Even if the time was spent fixing the image, chances are it would never look quite right.
The subject is still in a good location and an under-reflector was used to get just a little light in her eyes, but the background is still sunny. It's not as objectionable because a 70–200mm lens was set at a 195mm focal length to compress the area.
Now the subject is in a good location and there is a light cloud cover, but there are too many distractions in the background. This was done with a wide-angle lens. Many times a wide-angle lens can show too much of the surroundings and distract from the subject.
Here is a well-placed subject with a background controlled by clouds and a good choice of lens. The subject is in the same place as in the last image. Just the lens was changed. Now her eyes are bright, there are no hot spots in the background, there is detail on the subject in highlight and shadow areas, and the longer lens both narrowed down what you see behind the subject and compressed the background to help separate the subject.
SECTION 2
Unplugged: Natural Light in Practice
1. TOO TUFF
THE CONCEPT
This image was created to show dedication and work ethic. I used the subject's football helmet as a prop to help tell his story. I kept away from the full uniform and decided to keep the ripped jeans. I felt the jeans told the story of his work ethic off the field. It is my opinion as a coach that an athlete is more than what we see on the field. One who is willing to help others and works hard on his own without being asked is a winner in my book.
THE SETUP
I positioned my canvas background against the windows in my studio. I opted to point my camera directly toward the light. The key here is to position the subject right where the light travels from either side of the background. The background itself is a subtractive light source.
If there is a rock in the middle of a river, water will flow around each side of it in a V shape. Light will do the same thing. Knowing this, I pulled the subject forward from the background until I could see the double accent I was looking for. I also brought in two reflectors. I placed them on either side between myself and the subject and feathered them toward him as a fill light. In this way, I was able to keep the shadows and highlights in a printable range.
The light from the window came over the background and served as a hair light. This is one of the benefits of having tall windows.
As the mask of the subject's face was in shadow, I asked him to look at the camera intensely. With some mist from a spray bottle, the story was complete.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 90mm • second, f/5.6, 400 ISO
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2. THE TILE PAINTER
In 2008, I was invited to do a program for the Mexico National Photographer's Convention in Puebla, Mexico.
A local photographer took several of us on a drive to see and record the area. We stopped at a ceramics shop where visitors could watch artists make pieces. I came across this gentleman. His job was to hand-paint tiles.
LIGHTING AND EXPOSURE
The subject was seated next to a tall, narrow window in a small room. I took a step into the room, then a step to my right. I looked through my camera and thought, "I'm seeing the meter read f/4 at second, but I know the highlight is at least a stop over that. I'll set the aperture to f/5.6." This was a profile pose. Since the light was coming from behind the subject, it was a profile light situation. The pose and light matched.
The shot was a happy accident. The painter happened to be near that window in a position that I recognized could work as a dramatic profile light. Watch for these occasions. The ability to see what is possible will greatly impact your work.
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Canon EOS 5D Mark II • Tamron 28–75mm f/2.8 lens at 50mm • second, f/5.6, 1600 ISO
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POSTPRODUCTION
I used Corel Painter to enhance this image. This was one of my first attempts at postproduction painting. I felt it was appropriate given the subject and his craft. A painter should be painted.
3. DOWN THE ROAD
This image was created during a senior session at the subject's home. The skies were heavily overcast, which allowed me to work in open areas with the background remaining in good workable tonal ranges. It is often said that cloudy days are best for outdoor portraits because the light will be better on the subject's face. That can be true, but the best thing about overcast days is good background tonality.
DIRECTIONAL LIGHT
When you are working outdoors, it is important to find or create a direction of light to give the face dimension. Cloudy days can make it more difficult to see good defining light on your subject. Ignore the fact that you don't have hot spots in the background and look at the scene as if it were not overcast. For example, as you can see in the diagram, I found an area where light was blocked from the left and was allowed to enter the scene from the right. The subtractive effect from the hedge gave me some of the light direction I needed to shape the subject. In theory, that would be all that is required to create great portrait light—but wait! The sky was still brighter than what was coming from the right side of the image. For this reason, I positioned the subject within the light to help with the light pattern. I had her tip her head and turn her chin slightly up and toward her shoulder. The light from above suddenly worked as the main light and fell onto her face and into her eyes in a comfortable way.
AN IMAGE WITH STYLE
The driveway and landscaping harmonized well with my 70–200mm lens to create wonderful perspective. By positioning the subject within the driveway, but off to one side, depth was evident. I also chose the clothing to match and blend with the driveway. (Doing the shoot at her house was an advantage. We went right to her closet and picked out clothing for the different backgrounds I was seeing in my mind.) The green in the landscaping was a perfect contrast for her red hair. I knew she would jump off the page, but more attention would go to her face because of the lighting and the use of color.
"I knew she was going to jump off the page, but more attention was going to go to her face because of the lighting and the use of color."
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Canon 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 200mm • second, f/4, 400 ISO
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4. CAMOUFLAGE
CULTIVATING SIMPLICITY
This portrait was made for a conservative senior who loves hunting and the great outdoors. I had worked with this family in the past and knew they wanted good portraiture, but in a simple way. Instead of using a bunch of props—guns, archery equipment, and fishing paraphernalia—I used his camouflage sweatshirt and a woodsy background. The key here was to create simplicity in a busy environment.
LENS SELECTION
I chose my 70–200mm f/2.8 lens for this assignment. I knew that the lens would give me a narrow field of view and would help compress the background for the simplicity I was looking for. I chose an aperture of f/4.5, as I wanted a shallow depth of field to help simplify the busy background.
THE RHYTHM OF LIGHT
I positioned the subject under the tree to stop the light from coming straight down on him. The trees on the left blocked light from that side. The open yard allowed the main light to enter from the right side of the image. There was a lack of light in the teen's eyes, so I brought in a reflector on the main light side and skimmed the light across his face until I could see life in his eyes. I wanted to bring the shadow side of his face up in value. To do this, I needed another reflector. This time, the light pointed back at the main light and skimmed the shadow side of his face. Next, I needed separation. I couldn't get a reflector behind the subject to create an accent light, so I needed another way to make him jump off the page. I used the light and dark areas of the background for this. The dark tree trunk and dark-green leaves lined up directly behind the highlight side of the subject's face. Behind the shadow side of his face, lighter green leaves appear. This relationship between subject lighting and contrasting background tonal values give the illusion of separation in a natural way. This is something to look for when creating an image. Look behind your subject and position them or yourself in relationship to them to find this rhythm of light.
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Canon 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 120mm • f/4.5, second, 800 ISO
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5. STAIR MASTER
While teaching a class in Cape May, New Jersey, I came across this stairwell. The end of the stairwell had tall windows within each story of the building, but the windows were not very wide. I wanted to get the composition and depth of the staircase and still use the light from the natural environment.
VISUALIZING THE SHOT
I began by looking through my camera from several stories above. As I chose my background, I visualized where the subject could be positioned. I looked for just a corner of the stairs above where the subject was going to be and composed the image showing the stairs below repeating as they got farther away.
I then instructed the subject to move into the space. Once positioned, I realized it would be a split-light situation. I had her turn her head toward the window until I could see the desired pattern. There was too deep of a shadow. I had one of the students hold a silver reflector to add fill from under the staircase to the right of the subject. This brought the relationship of shadow versus highlight into a range I was happy with. Since it was a split light pattern, I had the subject give me a stern look.
WHITE BALANCE
The entire scene was red, so the light entering the area created an overall red cast, even with a custom white balance. I used Nik Software's Silver Efex Pro to produce the conversion from color to black & white. In the program, I used a red filter to give a good contrast to the contaminated skin and create a better visual of the red staircase.
"Light alone is not enough. The entire story also needs ingredients from the subject, the location, and the presentation to be complete."
TELLING A STORY
The emotion in the subject's expression, the split light, and the black & white presentation led me to title the image _Stair Master._ Light alone is not enough. The entire story also needs ingredients from the subject, the location, and the presentation to be complete.
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Canon 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 78mm • second, f/4, 800 ISO
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6. YELLOW AND BLUE
COLOR AND COMPOSITION
This high-school senior wore a yellow shirt and blue shorts to her session. I had the perfect place in mind for a set of images with this outfit. We walked down to my photo shack and began to play. I positioned her so I could see the yellow wall behind her. Instead of just plain yellow, I decided to maintain a bit more depth and allowed the floor and the blue wall just over her right shoulder to show. The little square of the blue wall was just enough to harmonize with her blue shorts. Between the two blue objects, her face exists.
This portrait is a good example of using color to guide the viewer's eye through an image. The blue shorts are obviously part of the subject, but the blue wall is far behind her. The composition lends a sense of depth to the image.
POSING AND LIGHTING
We did some images showing the shorts, but this pose showed them in a way that was more concealing. By bringing her left leg up, I covered her midsection and the length of the shorts themselves. Of course, with her leg up, there was an apparent weight gain on her bottom. The solution was to crop the shot. I didn't need to show her entire body; it was more important to draw attention to her face.
With this crop, I needed to find something for the subject to do with her arms. I created a circular shape comprised of her arms and face. This shape keeps the viewer's eyes from drifting away from the subject.
There is a soft Rembrandt lighting pattern on the teen's face. It is difficult to see because the light source is so large and the shadows are not very defined, but it is a pleasant pattern and very flattering for her. The soft expression seemed to fit the lighting pattern nicely.
I used a bit of fill, as shown in the diagram. I also used a reflector on the highlight side and from under the subject to add life in her eyes.
_Tip:_ Be sure that your client does not smoosh her face or arms when she leans against something. Encourage her to lean against bone rather than soft tissue. This approach maintains the human form better.
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Canon 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 100mm • f/5.6, second, 400 ISO
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7. SKATER BOY
Okay, I _did_ say not to go out and shoot in direct sunlight. That's a good rule of thumb for more traditional portrait scenarios. In this case, however, my subject was not too traditional. His story was to be told out in the bright sun with asphalt under his feet. I could have made this easier and brought out the "f-word" (flash!), but the look I was after was easy to achieve with natural light.
I knew it wouldn't be practical to use the sun as a main light. In fact, I chose to do just the opposite. I put the subject and his skateboard in a position to hide the sun. A thin layer of clouds helped too. I placed a large reflector to camera right to fill the shadow covering his body and face. The reflector was inches from his foot and just out of the camera view. Notice it was not pointed straight at him, but rather skimmed across him. This way, the only defined shadows from the subject came from the sun. The fill reflector was used to bring the shadow values closer to the highlights.
THE PROP
Next on the storytelling agenda was the skateboard, which was an important part of the image and the subject's life. Without the skateboard, it would not make sense to pose the teen on the pavement. Because of the prop's importance, I felt that I should add light to draw more attention to the skateboard. I placed a small reflector just beyond camera view and under the board. This reflector helped make the colorful board pop and added light in subject's eyes. Since it was so bright, I had the teen close his eyes while I set up the reflectors. I had him quickly open his eyes long enough to know where I was and freeze himself while he closed his eyes again. On the count of three, he opened his eyes and I recorded the portrait.
"Because of the prop's importance, I felt that I should add light to draw more attention to the skateboard. I placed a small reflector just outside camera view and under the board."
PERSPECTIVE
I used a 16–35mm lens for this image because I wanted to show a lot of the environment. A skateboarder needs space. I found a place near the fence so I could incorporate a sense of depth as it got farther away. I gave him a wide, skater-esqe pose and coached him to give me a stern expression. The low camera angle was chosen to hide the sun and present nothing but the sky behind him. I used his board and his orange shoelaces as points on my compositional triangle. His legs filled in the gaps, leading us right to his face. His story was told.
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Canon 1Ds Mark II • Canon EF 16–35mm f/2.8L USM lens at 16mm • second, f/8, 100 ISO
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8. ALLEY GIRL
This senior portrait was made on location. I traveled an hour and a half to a place I had never seen before. There was no time to scout the area; I simply had to roll with it.
A HARMONIOUS SCENE
I noticed the warm tones in the buildings and the way the structures related to each other. I also thought of the brown grass, blue window frames, and the perspective of the window sills. I needed to make them all work together and harmonize with the subject. I made sure she was contained in the grass, with no distractions. Then I lined up the window frames just off to her left in an asymmetrical composition. The lines of the window sills and dead grass lead us past the subject, but when we get to the blue frames, our eyes bounce back to her. Her blue jeans are the secret. Blue is the only pop of color in this otherwise monochromatic scene. Because the same color exists on the subject and a well-placed background feature, a sense of depth is created. The viewer, however, will always go back to the in-focus subject.
LIGHTING
The lighting was simple. It was all about positioning her within the light. There was raw sunlight on the ground in front of the teen just outside of camera view. I walked her back until she was shaded by the building but close to the edge of the light. There, I had a large light source with good direction. I could pose her in various ways and choose many different patterns of light for her face. For this shot, I lit the mask of her face and let her be happy. It suited her.
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Canon 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 85mm • second, f/5.6, 400 ISO
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9. GIVE ME THREE STEPS
When I began work on the retaining wall outside of my studio entryway, I started laying the rocks in a straight line, parallel with the house, but then I thought, "Why not create a compositional element I can use in a portrait someday?" So I rearranged the rocks, filled in the low areas, and added decorative stones on top. I planted a tree to help break up the large open wall and to give a blocking device for later portraits. The visualization paid off. I use this area quite a bit. In the summer, this location is in shade until about 10:00AM. That gives me plenty of time to use the area with my first sessions of the day.
Grass grows near the rocks, so the green color accented the subject's green eyes. The decorative stones worked well with her outfit.
POSING FOR THE LIGHTING
The pose was simple. I noted where the light was coming from and positioned the subject at an angle that allowed the light to reach her face. The secret is to pose for the light, but also for the subject. Sometimes what we have in mind does not look comfortable when the subject does it. When this is the case, try something different. In the end, the light has to be pleasant and the pose needs to be comfortable.
REFLECTOR FILL
I positioned a reflector under and to the highlight side of the subject to brighten her eyes. No added reflector fill was needed. The studio building is a light-cream color. The building cast a large shadow over the area. The wall acted as a soft reflector, bouncing light from the open sky and adding the fill we needed for a flattering image.
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Canon 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 90mm • second, f/5.6, 400 ISO
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10. RIVER WATCH
BEAUTY VERSUS DEVASTATION
I was teaching a class in Colorado several years ago. One afternoon, we took a walk with this high school senior girl to find areas to trash her prom dress. Just because one thing is on your mind doesn't mean you can't alter your plans along the way. This area looked clean and pure. I wanted to show beauty here, not devastation.
SURVEYING THE SCENE
As we came to this bridge, I noticed a lot of nice light coming from an opening created by the bend in the river. I looked through my 70–200mm lens and scoped out a background showing some water. There was a large rock for posing on. I walked down to the rock and took a light reading with my hand-held meter, then I helped the model maneuver down the bank to the rock. I posed her in a profile position because the light was in favor of that pattern in relationship to the camera position. I fixed her hair and scrambled up the bank to the camera to capture this image.
What made me stop here? The river cut an opening in the trees for the light to enter the scene. As you notice in the diagram, the only place light could come from was behind the subject. The trees on both sides of the river acted as a subtractive light source and the position of the camera from above only showed darkness behind the subject. The subject was then sculpted within the scene.
LENS SELECTION
Using a 70–200mm lens was a key factor in the success of this image. By narrowing the field of view, I was able to isolate an area behind the subject that didn't have raw light or other distractions. It also compressed the background, allowing the subject to stand out even more.
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Canon 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 120mm • second, f/5.6, 400 ISO
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11. UNDERGROUND
When I work with a senior outdoors, I tend to make things up as I go along. This way, the teen will have something that no one else has. Senior portraits should be unique. Yes, there should be some constants, but you should do a few things to produce unique looks.
AN URBAN STAIRWELL
This stairwell leading to a basement looked like a good location. The background was different and pleasing, and the light was good. How did I know the light was good before the model was in place? Practice. Plus, the building cast a shadow over the stairs. The only direction light could come from was the open sky over the street. I knew the light would cross from right to left.
COMPOSITION AND LIGHTING
I had the model walk down the stairs until she was framed by the back wall. Her head was the only part of her body above ground level. This ensured that it would be the brightest part of the scene. The back wall was important for more than just simplicity. It also had rhythm potential. The dark bricks could line up just behind the highlight side of her face. The shadow side of her face could be presented against the lighter bricks. By positioning the camera just right I was able to make the separation visible.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 80mm • second, f/5.6, 400 ISO
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12. PLAYING POOL
THE FINALE
I photographed this girl for a trash-the-dress session. Colorado mountain water is cold in late May. Heck, we walked around many piles of snow during the session. To warm up and to finish the session, she jumped in the pool at the resort.
THE LIGHTING
There was good light coming in from the glass wall on the long side of the pool, and there was light coming from the far side. There was a patio above the pool across from the windows with a little set of stairs. The short wall up to the patio and the wall on the far side of it were solid and faced with wood. The entry side of the pool also had a solid wall. The only way for light to naturally get to a subject in the water was from the north and east sides. It was late in the afternoon, so the light coming from the east windows was from open sky, as was the light from the north-facing glass wall.
I walked around the pool to where I thought the light would be best. I had the model walk into the frame. Once she was within my camera view, I turned her until the light was nice. I used her dress to create a unique shape and to cover her underarms. Since the combination of her dress and her pose looked like a flower, I had her close her eyes to avoid making eye contact with viewers.
I liked how the warm color of her skin contrasted the cool color of the water. Even though I wanted the communication to be nondirect, I still wanted the viewer to see the human figure within the design.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 90mm • second, f/3.5, 400 ISO
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13. WINDOW SEAT
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 70mm • second, f/5.6, 400 ISO
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CLOTHING SELECTION
Before each session, I look through the subject's entire wardrobe. I separate colors, then with the help of the model, choose the best pieces within each color. I decide what will be used inside and what will be used outside or on location. The indoor choices are hung in order of what I am going to do. If I am set up with a dark background from the end of my last session, the first outfit will be darker in color. Preparation is key.
The subject had several white tops. All were nice and different styles. Instead of pairing each of them with the same light-colored canvas backdrop, I chose the window sill for this outfit. The windows themselves make a good background—especially in this case, as they repeat themselves as you look farther into the scene.
THE POSE
I just asked the teen to sit in the window. She posed herself. I asked her to turn her head until I had split lighting to complement her expression. I also placed a reflector for fill and one for accent, as shown in the diagram. Not all images need to be difficult. Simple is good.
14. WINTER WOODS
I live in the north woods of Michigan's upper peninsula. Our warm season is short. In 2013–2014, we had snow every month from October through May. That's eight months. That leaves four months to do sessions outdoors in a green, lush environment. Not everyone thinks to come in during those "gentler" months for portraits. Some folks brave the cold.
EMBRACING THE ELEMENTS
This senior wanted to be photographed in the snow. We worked inside first and took care of the traditional things, then packed up and headed down the road.
SERENDIPITY
When I looked through the client's wardrobe, I found this sweatshirt. The first thing that came to my mind was that the pattern on it looked like the bark of poplar trees. When I saw this stand of poplar trees, I quickly pulled over. I had to work fast. There was a good-sized cloud covering the sun, but there was just enough intensity to create a good direction of light.
I found two trees together where I could see more trees behind them. I placed him against one tree, turned his face into the light, and recorded the image. Note that the highlight side of his face is against the shadow side of the tree behind him and the shadow of his face is against the lighter background. Look for these things; they can make your images more powerful.
I did not use a reflector for this image. My only light was the sun through the clouds. The tree at the subject's left acted as a subtractive light source. I chose an aperture of f/4.5 for a shallow depth of field and set the lens to 140mm for a narrow field of view. These two settings produced an image with a simple background despite the fact that I was working in an area where there was a great deal of potential for distractions.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 140mm • second, f/4.5, 400 ISO
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15. FLOWER GIRL
This girl liked wildflowers, so late in the afternoon, we headed out to find some. It didn't take too long. The power line near the studio cuts a path through the woods, leaving good open areas for wildflowers to grow. Most of the areas are lined with trees and wooded country that stretches for several miles. The woods are home to many critters, including bears who frequent these open areas in search of berries or other food sources. It doesn't happen often, but every once in a while, we have furry photo bombers come through. This was not one of those days.
LOCATION LIGHTING
In this particular area, there was a depression from both sides of the clearing so the trees hung way above the field. The banks cast a nice shadow over the flowers and made for a nice consistent background that also had what the senior wanted. I created many images here, but this was her favorite. I also liked it, as there was good light, her hair looked good, and her story was told. The subject's hair hung down and acted as a subtractive light source. Yes, the main light was coming from the western sky, but the hair helped. The moral of the story is this: parts of your subjects can help light or not light other parts. Depending on what you see, position body parts to help you. As I have stated before, body parts can also help create a more powerful composition.
A SIMPLE BACKGROUND
I chose an aperture of f/4 for a shallow depth of field to help simplify the background. I set my lens to a moderate focal length, 70mm, to keep some of the field and flowers in view.
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Kodak DCS Pro SLR/n • Nikon 80–200mm f/2.8 lens at 70mm • second, f/4, 160 IS
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16. THESE BOOTS WERE MADE FOR . . .
I painted this canvas backdrop years ago. I use it to show the dark side of some people. As you'll notice, I painted light and dark areas for a subject to harmonize with and create a nice rhythm of light in relationship with the background.
This subject fit the dark side. She loved her boots and wanted to show them off. I thought that this backdrop repeated the pattern of the eyelets in the boots. The foot closest to the lens ran in a diagonal direction to the subject's face. The eyelets were perpendicular to the lines in the background. Where the two intersect, we see the subject's face.
SPLIT LIGHTING
The split-light pattern suits the portrait concept. I made sure the teen lined up within the dark and light areas of the background for separation. I placed the background at a slight angle to the windows so that I could get the split light pattern I wanted while keeping her square to the background. Normally in a studio, the background stays put and the light is moved. When working with natural light, the subject and the photographer need to move within the light.
I used a bright silver reflector for an accent light and a soft silver reflector for fill. The reflectors were placed to allow light to skim across her.
BIG SALES
When I made this image, I thought my clients might purchase wallet-size prints. As it turns out, her grandparents bought a 24-inch print of this pose. They were taken by the way it showed her personality. I guess you never know.
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Canon EOS 1Ds Mark II • Canon EF 16–35mm f/2.8L USM lens at 16mm • second, f/5.6, 400 ISO
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17. SWEET LIGHT ON THE BEACH, PART 1
Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 70mm • ⅛ second, f/5.6, 400 ISO
THE BEAUTY OF SWEET LIGHT
"Sweet light" is found when the sun is out of sight behind the horizon but the atmosphere is still light. This happens at sunrise and sunset.
These images were created during a demonstration in Cape May, New Jersey. The shoot required an early wakeup call and a good-sized cup of coffee.
The night before this demonstration, I went through the model's wardrobe and found this dress. I liked the simple beauty of it. It was light and airy with just enough texture to make it interesting. I knew it would be well suited to the pastel setting of the beach at sunrise.
Photographing this early in the day requires a tripod. Some of the exposures can be pretty long. If the model is in a sturdy stance, this should be doable. The key to this kind of beach portrait is to work as soon as you have enough light to see. When the sun comes over the horizon, the game is over. It's a short window in which to create images, but it's magical.
When the light is right, the background in any direction holds favorable tonal ranges compared to the subject. This means the model and photographer can move in more angles to the light and create different light patterns without worrying about background problems. As long as the background does what you want for the composition, the subject will be the focal point.
THREE LOOKS
Image 1 (left) was inspired by watercolor artist Steve Hanks. His work is beautiful and inspiring. As I see it, watercolor painting is about the transparency and the art of working backward from the white of the paper to the colors involved in the scene. Here, I saw many layers of transparency. The dry sand, the wet sand, the shallow wave, the ocean, the dress, and the girl's figure. I also looked for depth in the background. By letting the beach fade off in the distance, the viewer gains a sense of space. The far-off pier gives our eyes a place to go, but it is perpendicular with the lines of the beach and stops us, turns us around, and sends us back to the model. Yes, it is a simple image, but one with a great deal of thought behind it.
Image 2 (facing page, left) was created a bit later in the session. The sweet light allowed me to move around the subject for a different light pattern and still hold the background tones. Yes, I got my feet wet, but it was worth it because I had a light source that allowed me to create a short light pattern. The subject could make eye contact with future viewers.
It's worth pointing out that eye contact should generally come from a lighted mask of the face, or at least a split light pattern. People looking at you from within a shadow can be creepy—so light what is looking at you.
Image 3 (below, right) was created a bit earlier in the session. The subject and I moved away from the water while maintaining the same relationship with the light. This way, I could create a short light pattern. By turning the subject, a single accent sculpted her face. I let her look down with a somber expression to match the mood that was established by the light.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 95mm • second, f/4, 400 ISO
Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 80mm • second, f/7.1, 400 ISO
18. SWEET LIGHT, PART 2
FOCAL LENGTHS
These two portraits were made during the same session shown in section 17. The same exposure settings were used for these photographs: second at f/4 and ISO 400. Notice, however, that the images differ from one another. The first image (below) is a full-length scenic image. It was created at a focal length of 70mm. The second image (facing page) was shot at a focal length of 140mm. Notice how the background blurs and the details of the scene blend together, leaving the viewer to focus on the subject.
There are so many variations we can produce in a scene just by knowing what our equipment is capable of. Both of these images show the subject in a profile pose with good profile lighting. You can pose a subject in a profile stance at any given time, but you will find that the photograph will always be more powerful when the light is coming from behind the subject.
Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 70mm • second at f/4 and ISO 400
WORK SMARTER, NOT HARDER
Sweet light times are magical and easy to work in. This is why my outdoor family portrait sessions are all done in the evening. The light is better for subjects and backgrounds. Plus, that's when many folks are home from work. Don't battle. Work smarter, not harder.
"Sweet light times are magical and easy to work in. This is why my outdoor family portrait sessions are all done in the evening."
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 70mm • second at f/4 and ISO 400
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19. GLAMOUR GIRL
When I was in high school, Joe Kind was my wrestling coach, school yearbook advisor, and photography teacher. In wrestling, I learned discipline, attention to detail, and how to work hard to achieve a goal. I also learned that basics are more powerful than "fancy" in most cases. When "fancy" works it's because a skill was practiced enough that it became basic for that particular athlete/artist. Portraits with strong, basic ideas can and will impress more and sell more than "fancy" done poorly.
I was lucky. I learned right away that what happened in-camera (the foundation of the final image) will affect what _can_ happen in the darkroom. Technical thoughts can work together with artistic thoughts. It is best to visualize your end result, then use your tools and skills to create what you have in mind. This way, when you get into the "darkroom" or, in modern terms, "post-process" your images, your work will be rewarded at a higher level.
A DREAMY LOOK
So, how do my life lessons pertain to this image? Well, in this case, I chose to use a long lens to take advantage of the subject-to-background compression that would result, plus a wide aperture for a shallow depth of field. The result was a soft, dreamy look right out of the camera. The clothing the subject wore was light in color. Her hair was light in color and looked quite soft. Then I noticed her eyes—the makeup she used beautifully defined them and made them sharp compared to the rest of the scene. I had her sit on the edge of my north-light window so the entire wall of light could be used to give a very gradual transition between highlight and shadow. I wanted the viewer's attention to go to her eyes, so I had her turn her body away from the window and turn her face back toward the light. By doing this, her chest was shadowed by the rest of her body and her face was the brightest part of the image. I used one reflector to produce fill so the shadows wouldn't become dark and disturbing in this pastel, airy scene. A second reflector was placed under the subject and to the highlight side to enhance her eyes.
POST-PROCESSING
We use software to enhance our images these days, but the old darkroom tactics still apply. In the darkroom, I would burn and dodge to darken or lighten an area. I would use colored filters to create different contrasts and different looks to various colors in the image. Later, with the help of cold light enlargers, I could burn contrast into different areas as I saw fit. That was getting pretty advanced in the darkroom, but it was possible. Today, we have RAW files and can use Photoshop's filters and third-party plug-ins to custom print an image with greater precision than was ever possible in the traditional darkroom. Yes, the means of image enhancement have changed, but the ideals still hold true: the basics are a foundation for even the most elaborate move. Start with something good from the camera, then use post-processing to enhance your results.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 165mm • second, f/4.0, and 400 ISO
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For this image, I used Nik Silver Efex Pro 2 to create several different outcomes with colored filters and variations in contrast. I then layered the images in Photoshop and painted (burned and dodged) the desired effect where I saw fit.
20. RED
THE PHOTO SHACK
I used to load a senior and parent into my vehicle and drive around to various locations to find buildings with character. There were inherent problems with this approach: (1) Gas prices are high; (2) more time was spent driving than creating; (3) the buildings kept getting torn down just as I got to know them. This is why I built the "photo shack" (see the photo on page 40). I tried to use old weathered material but realized that I could make it more than just an old shack. I painted the middle bay walls red and matched one sheet of plywood for extra length if I needed it.
VISUALIZATION
When I saw this subject's red shirt, I visualized using this area of the shack. Her shirt color harmonized well with the background, giving her a reason to exist there. For this image, I propped the red plywood in the corner to make a simple background for her and to create depth in the scene. Normally I would position my camera to point east or west and use the north light from the window, but not this time. This time the soft, north light acted as my fill light. The main light came from a reflector out in the sun. I feathered the direct light across the subject. Some of the reflected light also cast a shadow of the plywood on the wall behind her. I didn't foresee this; it was a happy accident. When I looked through the camera, I smiled. I knew it was going to be something special.
Yes, I usually like to find shade to work in, but don't be afraid to harness a bit of raw light from the area. Just be careful that you do not blind your poor subject by pointing the reflector right at them. It's uncomfortable and it will cause problems when it comes time to print the photograph.
"Don't be afraid to harness a bit of raw light from the area. Just be careful that you do not blind your poor subject by pointing the reflector right at them."
21. ALL ABOARD
WARNING!
This image discussion begins with a warning: _Do not photograph people on working railroad tracks or train cars._ First of all, it is against the law. Railroad tracks are private property and you shouldn't be on them. They are also unsafe. Trains are very large and extremely heavy. They move faster than you think and do not stop easily. It would be a real downer to walk away from one of your clients in handcuffs. It would be even worse if someone were hurt or killed. As of this writing, I have heard of four railroad fatalities linked to photography this year. Don't be part of this statistic. The tracks you see in my images are exempt and they are used for snowmobile trails in the winter. The train car I use is in a lumber yard, and I have permission to go there. Be safe first and creative second.
LIGHTING
How did I get the light to do this? The key to this image, as is the case in many natural light scenarios, is timing. I can use this spot in the early morning or late afternoon. This particular image was made in the late afternoon. The overhang from the caboose blocked the downlight. The open sky to the east produced a nice fill. There was some light skimming over the top of the car behind the subject, creating a nice hair light. The main light was from the brighter open sky to the west. The direct sunlight was blocked by the train car itself, but there was still more intensity from the open sky to the east. You will notice even more direction from our main light because of the thin vertical ladder and the subject's arm acted as a subtractive light source. If she were to lean back away from the ladder, her face would have a double accent pattern on it. By moving her face forward, the light direction was only visible from the west.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 110mm • second, f/4.5, 400 ISO
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A GOOD COMBINATION
Notice the background. I needed to use a fairly long focal length to avoid the raw light that was hitting the train car behind the subject and to our right. At other times of the day, this light may be able to be duplicated, but the background will become unusable. Remember, you need a good background and good light for your subject to exist within. Without one, the other just doesn't cut it.
22. LET'S JAM
At least once a year, a senior walks through the door with a guitar. This girl had a nice one and looked good with it. I decided to use classical lighting with an edgier pose.
POSING AND LIGHTING
I chose to bring the guitar up close to her cheek and aim the neck toward the camera. The light was pretty normal as compared to most of my work. She was positioned on the edge of the bank of windows so that I could fully use the north light as a main light source. One reflector was used for fill and one was placed under the model on the highlight side of her face. I turned her body away from the window and turned her face back into the light just enough to produce a broad light pattern. I asked her to not smile but to look a little happy and confident. Somehow she knew just what I meant—the expression fit the mood.
The guitar was on the shadow side of the subject and was tilted away from the window. I didn't want the window reflections on the guitar, as they would have been distracting.
BACKGROUND
The background was from Shooting Gallery Backgrounds. They do a wonderful job of painting. They also do a great job listening to you and painting to match your needs. This particular painting was created in the mirror image of what was in their catalog. When choosing a new background, look for shapes and patterns or light and dark areas you can use in relationship with the subject. Remember, there is no background light in the north light studio. The background must offer separation.
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Canon EOS 5D Mark II • Canon EF 16–35mm f/2.8L USM lens at 29mm • second, f/6.3, 1600 IS0
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23. DOVE GIRL
TATTOOS
If my client has a tattoo, I want to know the story behind it, as it will guide what I do for the image. This young lady had doves on her hip. Her story was deep and sad, as it was to memorialize the loss of a family member.
LIGHTING
A single accent felt right for this image. I wanted a bright, heavenly background. I thought one could also be the other. I placed her in front of the window. By exposing for her, the background would be overexposed to the point of a glow. I turned her face until I saw the light feather across her cheekbone and nose. There was at least a six-stop difference between the exposed light on her and the background. This worked in my camera room because of the white walls opposite the windows. The walls provided the fill light that created the exposure for her body, while the light from the open sky outside created the highlights on her face.
The ink is a secondary focal point to her face because her face has more dimension. The highlights that show her features attract attention. I made sure there was enough light on the ink to be easily seen, but it is the same light as on the rest of her body. By placing her hands diagonally on either side of the artwork, attention was drawn to that area. It all comes back to the light background. Her hands are dark compared to everything else. The shadow side of her face is the same density as her hands. Her hands and face make a triangle that we follow with our eyes. Along this followed path, her story is told.
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Canon 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 120mm • f/5.6, second, 400 ISO
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24. TRAIN DOOR
CHALLENGE YOURSELF
A few years ago, I gave myself a challenge. For every senior session that took me past the train car, I would stop and create something different. I worked inside, outside, and on top of the car. I would have worked underneath if it weren't for the furry dwellers. I used the train at different times of the day. I learned to use small and large light sources. I sought out different background possibilities, large and small. I found areas for the subjects to sit, stand, lie, lean, and hang among the two cars. There were many angles to follow and colors to use. The challenge was good for me _and_ my clients. It assured everyone that they would have something unique. People like variety, and they like to have something they can call their own.
THE SETUP
For this image, I used the doorway of the caboose. I was positioned across from the door, tucked under the ore car. From this angle, I could see into the car. The boards on the floor created depth as they stretched beyond the door. The warm colors on the frame made a nice contrast to the teen's shirt. It was apparent that the light would be good in that area, so I brought in the subject.
The pose was simple; "sit down" was all I said. He sat and leaned against the door. He was too flat and sat too far into the car, so I had him roll on his bottom toward the door. This gave a better angle for his shoulders and brought his face into the light coming from the open western sky. The result was a broad light pattern on his face. The open eastern sky acted as a fill light. No reflectors were needed. I love places like this. Here, I was afforded a little roof overhead and good light, plus a good background. Ah, the simple pleasures of life.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 70mm • second, f/5.6, 400 ISO
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25. THOUGHTS AND DREAMS
Never underestimate what light does for you. Yes, it is necessary to expose the image, but it can also guide the viewer's eye through the photograph, create depth, tell a story, and portray emotion. That is what this image is about.
A CHALLENGING LOCATION
This portrait was created in a stairwell. There were windows behind the subject. Opposite the windows were dark stairs with a door at the top and bottom of each stair. The light came from the windows. I used a reflector perpendicular to the subject for fill light.
I shot from above so that the landing between the stairwells became the background. It was simple without distractions. It was also dark, which showed a beautiful contrast to the light on her face. One can't help but look at the subject. This light direction could have also been used for a profile portrait.
EMOTION AND EXPRESSION
When you arrive at your location, look around your subject. Maybe the area you've chosen won't allow you to create a portrait in which the subject is looking at the camera, but you may find that a wonderful opportunity exists.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 125mm • f/4, second, 800 ISO
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The subject looked down, so there is no direct communication between her and the viewer. This engages the viewer's imagination. He or she will contemplate her expression, which is simultaneously simple and complex.
26. DREADLOCKS
Working in the photo industry has allowed me to do some wonderful things. I have been coast to coast and beyond, sharing what I do with other photographic artists. This particular image was created in Florida. The subject was modeling for my class. She had wonderful dreadlocks. She wore a green, soft knit sweater, which harmonized well with her hair and eyes. I had been watching a group of students working with her when I came up with this idea.
Behind the subject was a patch of grass. I knew that with a long lens and a shallow depth of field, I could use the color of the grass without showing its texture. I just needed a high camera angle. I found a garbage can nearby and decided to put it to use. The key to climbing on top of a garbage can is to get both feet up on opposite sides of the top rim at the same time. (I found this out after my first attempt to get on it.) Using a high camera angle, I was able to capture what I had envisioned.
CONSIDER THE NUANCES
Portraiture is about making decisions. We must carefully consider the nuances of the subject and scene in front of us and tweak the details to get the image we envision. I might think to myself, "I want a thought-provoking image, so I will want to use a lighting pattern that keeps part of the subject's face in shadow. If I turn her head to achieve that, however, her hair will cover her face in an unflattering way. Instead, I'll use a short light pattern and have her look down and show her face without eye contact." Consider your creative options and make refinements as you go. In this case, I probably should have had her hold this pose but look into the camera as another option. When in doubt, record the image. You can always toss it away later if you don't like it.
LIGHTING AND POSING
The light for this portrait was simple. It was late afternoon and a nearby hotel cast a shadow over the yard. The open sky over the ocean provided the main light. The trees behind and on the shadow side of the subject acted as subtractive light sources and blocked the light from those directions. The only place from which the light could enter the scene was from the side and in front of the subject. All I needed to do was position myself for the background I wanted, then place the subject within that frame. Once the subject was in the correct spot, I had her rotate her body away from the light and her face into the light. With that, I captured the image.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 200mm • f/5.6, second, 400 ISO
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27. "COCAINE LIGHTING"
THE BILLBOARD
Like many of you, I try to make each item in my yard useful for photographs, in addition to using it for its intended purpose. This wall was constructed for my daughter to hit a tennis ball against and also to keep the basketball from rolling too far down the driveway when my son had the ball get away from him. It has also worked well for a few portraits. I don't usually work by this wall in the late afternoon, but as this senior boy and I walked by this location at this particular time, he was wearing a hooded sweatshirt, and I saw this image in my mind.
LIGHTING, POSING, AND COMPOSITION
There was a little raw light skimming over the top of the wall. The raw light was also on the basketball court, bouncing up. I had the subject put his hood up and hang into the scene. I had him lean back and forth until I could get the accent from the raw light to just touch his clothing, while keeping the light from the court on his face. I chose a high camera angle so I could see only the back of the wall and two of the supports.
I broke the rules for this shot. I used what I typically call "cocaine lighting." I let the light shoot up the teen's nose. This lighting is not very flattering—it's best used for someone who is telling a ghost story around the campfire—but it worked here. With the subject's clothing choice and expression, the light made sense and helped tell the story.
I used a reflector for fill, as you can see in the diagram.
If the subject were wearing short sleeves, this approach would not have worked very well. His skin would have been too reflective for the raw light, and the file would have been an unprintable mess.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 80mm • f/4.5, second, 400 ISO
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28. A LITTLE HELP FROM HER FRIENDS
CONNECT WITH FRIENDS AND FAMILY
I don't have employees. If I need an assistant, I recruit the parent or friend of the subject to hold reflectors. Some seniors come along for a friend's session with no intention to book me. After assisting, they often decide to hire me after all. It's not uncommon to have three girls show up two or three times a year—once for their own session, and once more with friends.
THE SETUP
A friend of the subject was perched on top of the retaining wall with a reflector, which stopped the light from coming straight down on the subject. Another friend held the fill reflector. The light was only allowed in from the side illuminated by open sky. The fill reflector was not in its usual place for this shot. The rock wall was in the way. I feathered some fill light from the highlight side, just in front of the subject, to create the desired relationship between shadow and highlight.
We did several poses with this setup. The teen looked most comfortable in this photo, so this is the one I showed the client.
29. BLACK ROCKS
NO PLACE LIKE HOME
I live in the forest area of Northern Michigan. It is cold in the winter and summer goes by quickly, but the beauty of Mother Nature surrounds us always. The background here is what once was the bottom of a raging river. The river continues to carve its way through the rock, but it is now controlled by dams throughout its path. The banks of the river are very high above the water, especially on the west bank. It is because of this high bank that I frequent this location for late- afternoon portraiture. A shadow is cast across the entire river, but the sky to the west still has more power than that from the east. I get a good direction of light from the west. I can look up river or down river and get good light and a background without hot spots.
THE APPROACH
I wanted to create a close-up image but still wanted to see the rock. I had the subject lie against the rock with her head to the east. She turned her face toward the camera, which allowed the light from the western sky to add shape and form to her face. I kept her hands under her body and in shadow to minimize them. The rock she leaned on acted as a gobo. I placed a short black panel just outside the frame to keep the light on her back toned down in comparison to her face. You may be wondering how I carry all these reflectors and panels. I don't carry many—usually two. Both are silver on one side and black on the other. One silver is bright (for accent light) and one is softer (for fill). If I don't need one of the reflectors, the opposite side can be used as a gobo.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 160mm • f/5.6, second, 400 ISO
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30. TRAIN STEP GIRL
TRAIN CAR VARIATION
As you can see, this portrait session took me back to the train car shown earlier. This time, the sky was overcast and I was shooting in the late afternoon. Overcast, cloudy days are said to be best for portraits. It can be true. As far as the background is concerned, it is much easier to work on an overcast day. There is more space to work with that will remain in the same printable tonal range as your subject. Ah, but the subject. On an overcast day, where is the light coming from? Still above. Yes, you must place the subject in an area that will force the light in one direction to produce flattering light on his or her face. Areas that would work on a bright day will also work when it is overcast.
Because it was late in the afternoon, there was some intensity from the western sky, but not enough to cast raw light everywhere. There was also a great deal of light coming from above. I chose not to stop the downlight but work with it instead. By having the subject tip her head and let her long hair hang down, her head became a gobo, blocking the light from the underside of her face. Her long hair also blocked the light from the west. The result was a nice light on the entire mask of the subject's face.
POSING WITH PURPOSE
Because I had the subject tip her head so far to get the right light, I needed a reason for the tilt. I had her bring her hand up so she could lean against it. When you study the image, you'll notice that her arm is not straight up and down. Her hand is behind her cheek and she is not resting her elbow on her leg, it's the underside of her arm. This was done to minimize the size relationship of her hand to her face and to keep her hand from pressing against and distorting her face. Her arm was angled to draw the eye to her face. Having her rest on her triceps rather than her elbow allowed this to happen without looking too awkward. Her face was turned for the light but also directs just a little attention to the other arm. That arm circles back to the leaning arm, and the composition is complete.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 85mm • f/4.5, second, 400 ISO
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31. PIANO LIGHTING
This image has long been one of my favorites. It will never win any awards, but it marks the point in my career at which I began to free myself not to light my subject's face. It was also the point at which I realized that there are as many stories to be told by the absence of light as there are stories that are told by light itself.
THE LIGHTING
This portrait was created during a demonstration in Cape May, New Jersey. As you can see in the diagram, there were two windows in the room. There was a covered porch outside the windows, which were facing west. The raw light from the sky couldn't reach into the windows under the covered porch, but it still was much more intense than what we see from an open sky. The light skimmed across the subject's back, showing wonderful detail in her shoulder blades and spine. It also outlined her hair, giving it shape against the dark piano. Then I noticed the light on the piano itself. I had the subject slide over on the bench until I could see the light on the piano right behind the profile of her face. There was no light on her face at all. This was a way to elicit a somber mood and create depth at the same time.
The subject's dress was long and flowing, with several layers to it. I used a layer to stop our attention from leaving the image at the end of the piano. In hindsight, I would have done something more comfortable with her arms.
CREATE AN ARRAY OF IMAGES
I have included other images from the same session. These portrait examples show other ways to use the same light source. I could probably have come up with a few more if I wanted to. The point is, while you are in a location with a good background and usable light, look for variety—it's good for the client and for you. Be creative, have a vision, but don't become so entranced with your original concept that you overlook other ideas.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 75mm • f/4, second, 1600 IS0
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"While you are in a location with a good background and usable light, look for variety—it's good for the client and for you."
32. MOUNTAINEER FOOTBALL PLAYER
BREAKING THE RULES
I have said _not_ to go out in the direct sun. Well, for this portrait, I broke my own rules. I had a good reason—well, two good reasons: First, the subject wanted to show off his shades. Where better to do that than in the sun? Second, he was proud of being a football player for his high school team. This background means a lot to him and his family. There are times when the client's story is more important than getting just the right light. Portraiture sometimes requires the same rules that apply when I photograph weddings: First, get something recorded. If there is time, make something better. You can't lollygag and wait for perfection. You may miss everything.
This image was created at mid-morning. The sun was up but still at a pretty good angle to the subject. I was fully ready to expose this image in direct, raw light, but at the last moment a little wisp of cloud made its way between us and the sun. It was still very bright, as you can see by the shutter speed, but the scene leveled out a bit, allowing the background to be in a good relationship to the subject.
The subject would have been squinting if it weren't for the sunglasses. Because of the direction of the sunlight, I was able to position him fairly square to us and still get the perspective of the stadium wall fading into the distance. The painting of the football player was on an angled wall, which worked out well for this image. The subject, the painting, and the powerful lines of the stadium together form a triangle. Anyone viewing this image will feel that they can reach into the scene. When the photograph doesn't seem to be a flat piece of paper, it becomes more powerful.
"I was fully ready to expose this image in direct, raw light, but at the last moment a little wisp of cloud made its way between us and the sun."
PHOTOSHOP ENHANCEMENTS
The client wanted the image to be a combination of color and black & white. I left everything black & white except his jersey and part of the stadium wall. This helped tie the subject to the scene.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 75mm • f/5.6, second, 400 ISO
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33. YELLOW FLOWERS
Sometimes, when I am headed to a familiar location, I see a background that is perfect for the client and their outfit. I may see something out of the corner of my eye and hit the brakes. Maybe it's just me, but when I find a location that is just right, I get inspired.
BACKGROUND FIRST
A painter doesn't begin with the subject, then paint the background—and we shouldn't either. You must have a powerful background and good light, and then the subject can exist within the space. When I scout for backgrounds, I also look for the light. The two have become one in my mind's eye. I never disregard cool looking backgrounds in bad light. I just make a mental note to go back at a different time of day.
CREATING THE IMAGE
My subject and I were in a ditch alongside the road. The flowers matched her outfit perfectly. The light was from the western sky, but the direct sunlight was blocked by pilings from the old mine. The trees to the east blocked light from that direction. The result was nice soft light with good direction. I used a gobo to shade the subject's chest to ensure that her face would be the brightest part of the image.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 120mm • f/4.0, second, 400 ISO
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34. FOOTBALL CLOSEUP
IN THE CAMERA ROOM
When I was laying out the footings for my camera room and home, I started with the north wall, which would be the light source for my indoor portraits. I cleared the building site then took out the compass and tried to get a perfect line running east and west. Each time I checked it, I got a different reading. Welcome to Iron Mountain! As it turns out, I was off by a few degrees, which causes issues late in the day during the summer. If I could do it all over again, I would ensure there was at least 12 feet between the end of the window and the east wall. The same would be true on the west side. By having more space from window to wall, you can control the density of your background and the angle at which you use it. As you can see in the diagram, in this case, I needed to block the window to get the lighting I wanted for the subject. I used another background to do this.
Why did we work at such an angle to the window? I wanted to have a great deal of separation from behind. As you can see on the teen's shoulder pads, the light was on his right shoulder. The helmet created an edge and only allowed the light to skim across his face. The length of the window added fill. I used a reflector for a bit more fill and one for more accent on the right side of the image. I left the reflection in his helmet so you can see the placement. One more reflector was used under him and on the highlight side to add fill under his helmet and put a little life in his eyes.
THE POSE AND COMPOSITION
His pose gave us a good angle. His shoulder pads appear in each corner and give our eyes something to follow. Between the corners, the story is told in his eyes.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 120mm • f/4.5, second, 400 ISO
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35. BLUE PALLETS
MIDDAY SUN
There are times when we find ourselves out at midday. On those occasions, I immediately seek out overhangs—dense tree cover, bridges, porches—to block the direct light from above. The key is not to get too far under these areas. Too many times I see students right up against the trunk of a tree. Yes, the canopy stopped the downlight, but the good light was not able to get to the subject either. Move out to the edge of the canopy. As long as you can find a background in that area, you will be in good shape.
A POP OF COLOR
This pile of pallets caught my eye. The colors in the pile would work well with my subject's clothing. A large tree shaded the pallets and subject. The tree was on a bit of a hill, so the light entered the area at a nice angle to the subject. We could have worked here all morning.
I moved the pallets to create good leading lines to the subject. When looking at a scene, if there is something you don't like, change it. You are responsible for the end result.
THE POWER OF LINES
To pose the subject, I started by turning her body away from the light, with her face turned back toward it. I made diagonals from verticals. Her legs were straight up and down. I had her keep her knees together and bring her feet apart. Diagonal lines lead the viewer, vertical and horizontal lines stop the viewer. I asked the subject to lean forward onto her hand, to the shadow side. This way, her hand is in shadow and does not distract from her face. Her left arm was positioned to complete a compositional circle that leads us back to her face.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 153mm • f/4.0, second, 400 ISO
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36. BEHIND BLACK ROCKS
BACK ON THE RIVER BED
Here we are, back on the river bed. It's amazing how many posing opportunities nature grants us. This time, I chose to look up river to the north. The light was coming from the western sky on this late afternoon. The entire area was in shadow. There was no raw light on the scene.
I liked what the light was doing to the subject's hair, but her hair blocked the main light from her face. The eastern sky took over as the main light and the western sky acted as an accent light.
I positioned myself so I could see the dark rock around the highlight side of her and the light-green bush on the shadow side. This created separation. Always remember that moving a quarter of an inch one way or another may completely change your image.
ABOUT THE POSE
Nature presents great props for your subject to lean on. Just be sure that the person does not lean on the wrong places. Ensure that the flesh is never smashed and distorted. Also, be sure the hands don't appear out of nowhere. If you show a hand, show where it came from; leave a connecting point to the rest of the body.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 135mm • f/5.6, second, 400 ISO
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37. PROM'S OVER
I see quite a few prom dresses during a senior season. I think any parent would agree it is good to get one more use out of something that cost a fortune. I'll do traditional things with the dresses—some full-length sitting or standing poses mixed in with some close-up images is always good. This time, however, I had something else in mind. I was thinking more of the stress from the prom: the preparations, the cost, and the night itself can be grueling. That is the story I wanted to convey.
THE SETUP
I used the south wall of my camera room and the pillars for the background. I placed the subject by the front pillar and positioned myself so I could see three pillars behind her. I then had her lean forward and lunge toward me until the shadow side of her hair lined up with the highlight of the pillar directly behind her. Now I had a good background and subject relationship. The light was there within the scene. I just needed the subject to turn her face back toward the light.
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Canon EOS 1Ds Mark II • Canon EF70-200mm f/2.8L IS USM lens at 70mm • second, f/5.6, 400 ISO
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"The pose also helped to tell the story. Her legs, showing at an angle, give her a good base and lead the viewer to her face."
I used a fill reflector as shown in the diagram and an accent reflector to add to the shape of her legs and dress.
The pose also helped to tell the subject's story. Her legs, showing at an angle, give her a good base and lead the viewer to her face. Her bare feet add to the feel of the image. I was happy with what I saw through the camera, so I recorded the shot. At that point, I thought I was done.
FINISHING OPTIONS
Photoshop allows for infinite artistic options. This portrait was stretched to achieve a better composition and also to help show the loneliness a subject might feel on the day after the prom. I must admit that I didn't think of this during or after the image was created. My friend Michael Timmons suggested the concept. It pays to hang out with talented people. Sometimes their talent rubs off on you.
38. SWEET LIGHT FAMILY PORTRAIT
Scheduling is an important part of my life, as a natural light photographer and a business-person. I try to book seniors or other individuals who are interested in outdoor images for morning sessions. I start outdoors and get as much done as I can before the sun is too high to easily work, then I work in the camera room. I schedule executives and other indoor session clients for midday when the light outdoors is problematic. Midday is also a great time for consultations, image presentations, and production work.
Later in the afternoon, I start photographing seniors again. With these clients, I work indoors first, then we head outdoors as the sun is going down.
A family scheduled for an outdoor session will be photographed in the evening. I give an approximate time when booking the appointment but refine the time as the session draws closer. I confirm a time on the day of the session based on the weather. If it's clear, the session will be about fifteen minutes before sunset. This gives us a little "get-to-know-each-other" time before the real push happens. As soon as the sun disappears, I work quickly. This is when the background in a large area will hold together nicely. At this time, the light is from a larger source, allowing for good dimension across an entire family without harsh shadows cast on each other. It is also when even the most sensitive eyes are not affected by the light.
"As soon as the sun disappears, I work quickly. This is when the background in a large area will hold together nicely."
Sweet light times are magical. It's funny. I actually have photographers tell me, "Sure your images are nice. Anyone can do that during sweet light times." I just laugh to myself. They are right, it is an easier time to work. The question is, why aren't they taking advantage of that light? Work smarter, not harder. Yes. It can make for a long day, but it makes a better family portrait, and at a time when more families can make it.
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This was my "window," my light source. The family was just to the side of the tree on the right. They faced the camera, which was placed on the left and pointed parallel to the opening.
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 73mm • f/8, second, 400 ISO
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 85mm • f/5.6, second, 400 ISO
A BACKYARD SESSION
For this image, I chose to position the family in a location in my yard facing south. The western sky after sunset served as the main light source. The opening in the trees to the side and in front of the family forced the light to create shape and form on each face. The trees overhead kept light from entering from above. No fill was needed.
The photograph on the facing page shows the view to the right and ahead from the subject's position. It was a good main light. Remember to sit where your subject will be. Look to where the main light should be. If you don't see a main light there, move on to a different place.
IMAGE DESIGN
I positioned the family in a triangular pose with a little hill behind them for simplicity's sake. Behind the hill is a large meadow area with trees on the far end. That falloff of focus in the trees is what gives this image a sense of depth. When we see an image element that is more out of focus than items closer to the subject, our brain tells us it must be farther away. Depth in an image is your friend. Embrace it.
39. CHEERLEADER
Many athletes come through my studio doors. I do a lot of coaching and supporting of the youth sports in our area. I both appreciate and respect these dedicated folks. Plus, I can relate to them and their lives. I've been in their shoes.
PERSONAL AND PROFESSIONAL GROWTH
For a portrait photographer, the best thing to learn, beyond photography, is human behavior. Psychology ranks right up there and so does drama and speech class. Anything that helps us understand and read into our guests as quickly as possible is gold.
I can relate to people quickly because I have many interests. I am not afraid to try anything, and I love adventure. Listen to different kinds of music. Keep track of sports, even if it's not your thing. Having a little something to throw out there and start a conversation with is important. That being said, don't dig yourself into a hole. If you don't know anything about a subject, say so. It is easy to look like a fool, and fools have smaller portrait orders.
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 140mm • f/5.6, second, 400 ISO
THE SESSION
This cheerleader brought the whole package to the studio—a home jersey, away jersey, jacket, pom pons, everything. She wanted it all in one image. For the first image (facing page), I put a steel grid against the background and clipped the jersey to it. Then I set a chair in front of the model and draped it with her jacket. She could then stand just to the left of the chair and create a nice triangular composition. I used a focal length of 85mm to show all of her gear.
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I don't get into a lot of props, so after making her happy, I got rid of the extra stuff and just focused on her. For our second image (right), I zoomed in to a focal length of 140mm to compress the background and make her more important.
Both images were exposed using the same settings. As you can see in the diagram, the camera was parallel to the windows for the first image. For the second image, the camera was at more of an angle. This way, I could get a more appealing light for the looking-over-the-shoulder pose.
THE BACKDROP
The colors in this background seemed to work well for this outfit. I painted this backdrop many years ago for a dance school shoot. Notice how I painted dark and light areas, knowing I was always going to light this background from the left. Canvas backdrops are just as important as backgrounds on location. Choose wisely.
40. POINTER SISTERS
AN IMPORTANT EXERCISE
As I have said before, start with a background and good light, then let your subjects enter the scene. This is a good example. The camera remained fairly stationary throughout the entire session. All that changed was the subjects' positions within the frame. Try this: Find a pleasant background with good light. Put your camera on a tripod and compose the image as you see fit. Keep in mind where subjects could best exist. Then put your hands behind your back and verbally instruct your guests to enter the image. Watch through your viewfinder and tell them where to go—backward, forward, right, or left—until the image is complete. Then fine-tune and record your masterpiece. It is a good exercise. I still do it every once in a while. It forces me to think back to front, like a painter.
REPEAT CLIENTS
I have photographed these girls for several years now. It started when I photographed their dance school. I did things a bit differently. I used other dancers as background and foreground material and did composition studies as their dance portraits. The rules were simple: The only dancer that gets to look at me is the main subject. All other dancers are instructed to look down or away depending on where I wanted the viewer's attention to go. The result told more about the essence of dance compared to just standing a girl with her hands on her head in front of a background. I don't photograph that school anymore, but their students still come to me to get this kind of portrait.
These sisters wanted to do something special. They had all learned to stand en pointe. It was quite an accomplishment. I wanted to create a portrait that was classical in nature. The pillars in my camera room seemed like the way to go.
I set my camera for the background I wanted and positioned the subjects in key areas for each different idea. Each girl got her turn to be the main subject. I also recorded them as equals. Along the way, I made sure we did smiling images of each girl and the three of them together. Those were for Grandma. Don't forget about Grandma. Make sure you get some smiles no matter what the story is that you want. Smiles equal sales.
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Canon EOS 5D Mark II • Canon EF 16–35mm f/2.8L USM lens at 35mm• f/5.6, second, 400 ISO
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Now, let's move on to the lighting. The girl closest to the camera was going to have more of a defined shadow than the girls farther away. This is logical. The girls farther away had more length of windows, so the light would wrap around their faces. I liked and embraced the idea that the girls farther away were going to have softer light, softer focus, and less communication with the viewer. It was all part of the plan.
If you look at the supporting image (facing page), you can see that there were wrought-iron railings in the balcony area of the space. The the main image (which I used for competition) has the black railings removed. The client didn't mind them, but I felt they were a distraction.
41. DIAGONAL DOOR TRIM
A great deal of thought went into creating this simple but interesting image. I'll begin with the light. The open sky to the north provided the main light. It evenly lit the entire building and the subject. It was late afternoon, and the sun was in the western sky, just high enough to skim over the building. As you can see in the diagram, the raw light actually hit the ground right near my camera. I saw that the light was hitting the model's hair and shoulders as she leaned against the black door. I had her step backward along the building until I saw just a hint of the raw light skimming the top of her head. It was just enough to give a bit of separation without blowing out her hair.
A HAPPY ACCIDENT
Now, for the happy accident that the light forced me into: Originally, I had planned to make the black door with the white door behind it the entire background. Because the light forced me to move and rethink, I saw the repetition of the diagonal lines on the doors. Way cool! The black door acted as a foreground without causing distractions. The diagonal door trim next to the subject keeps the viewer focused on the scene. The diagonal trim beyond her leads the eye back to her and helps create depth in the image. I also liked the old concrete on the ground behind the gas can. The simple slabs bring the viewer's eye past the subject, deep into the image. The eye then stops at the vertical strip of blocks at the edge of the image. Suddenly, we bounce back to the subject with the help of the diagonal trim. The composition is simultaneously simple and complex. It's amazing how the brain sees things.
"The open sky to the north provided the main light. It evenly lit the entire building and the subject."
A NATURAL-LOOKING POSE
There was an eyelet in the door that gave the subject something to do with her right hand. I wanted a strong lean of her body to contrast the diagonal trim. Together, they form a V shape. Her head was tipped parallel with the trim on both doors. This helped build a relationship between the subject and background. I had her turn her hips and cross her legs. With her thumb in her pocket, we were ready to record.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 70mm • f/5.6, second, 400 ISO
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42. TRAIN HANG
SAFETY FIRST
Here is the same senior shown in section 41, photographed a few moments later. I walked past the building she was leaning against to the train cars that were parked there. _Note:_ These cars are not on railroad tracks. They are stored in an abandoned train yard. The track is disconnected and leads nowhere. Never photograph by active trains or work on live tracks.
AN APPEALING LOCATION
I did like the repetition of the cars. I liked the warm colors, and the light was nice. The main light was actually the open sky to the east of the subject. The building kept light from entering too much from behind, and it reflected some raw light that produced an accent in her hair. You can also see it on her sleeve by her face. The train cars shadowed the area but, as you can see, a few little strips of raw light made their way between the cars. Since the light was feathering across the long grass, I figured it wasn't going to cause any problems. The light showed the mask of her face, and I let her smile and be happy.
For the pose, I had her just stand by the car. She turned diagonal to me. I had her put her weight on her back foot, then she reached up and grabbed the handle on the car. It seemed more comfortable and looked more slimming to have her cross her left leg over her right. Her right arm was initially too far back. I had her move it forward until I could see a good connecting point without making her look wide across the middle. A good tip of the head and a little work on her hair, and all was complete.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 85mm • f/5.6, second, 400 ISO
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43. GRASSY BEACH
SWEET LIGHT REVISITED
Here's another example of sweet light. The sun was almost down, and some trees on the horizon blocked most of the direct light.
A BEAUTIFUL BACKGROUND
Three steps out of the parking lot, I saw this background. The light was nice in the immediate area, so I stopped to play. I noticed the path in the sand with tall grass on either side. I sat the model down on the edge of the path, so I could see the path beyond her body. This created a sense of dimension.
I liked how the grass shared characteristics with the subject's hair. This helped tie her to the background. I posed her so that her hair would mimic the feeling of the grass behind her.
I couldn't change where the sun was setting, so I changed the subject's relationship to the light. In this pose, her body was turned away from the light, with her face turned back toward it. Her body was dark, and her face was light. Always use the light to motivate people to look where you want them to look.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 78mm • f/5.6, second, 800 ISO
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44. SKATER BOY SKY
HAVE FUN
It's important to play. Get to know your subject and let them be themselves. If I have a musician in my studio, I ask them to play. Athletes show off their stuff. It makes them comfortable and makes for more meaningful images and bigger sales. Learn who your subjects are, then use what you know about artistic portraiture to show your viewers who your subjects really are.
During this particular session, I watched the subject launch himself onto the tower. I did some action shots, but a controlled portrait was what I was after (and what his mom wanted). When I saw him on the tower, I thought, "What a cool viewpoint!" I had him look over the edge. He hooked the board on the edge of the wall, and I saw the story I wanted.
NATURAL LIGHT
The tower was positioned just right for where the sun was. It was late afternoon and the shadows were falling hard to camera left. I couldn't put a reflector under the teen because it would show in his glasses, so I placed a silver reflector off to the east. This lit the bottom of his board and added a little accent on his right cheek. The rest of the light that brought out his face came from a large ramp below the tower. I was facing southeast, so the sky looks nice and blue.
"If I have a musician in my studio, I ask them to play. Athletes show off their stuff. It makes them comfortable . . ."
POSTPRODUCTION
I used Nik Software's Tonal Contrast filter to add some punch to the skateboard and a little to his face.
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Canon EOS 1Ds Mark II • Canon EF 16–35mm f/2.8L USM lens at 28mm • f/7.1, second, 100 ISO
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45. LAKE SWIMMER
My niece was on her high school swim team. She thought it might be nice to have images made with her in the water. This location was perfect. There is a dock to stand on. I didn't have to get wet and was able to look down on her a little, which allowed me to show the lake around her. From this angle, there were no distractions from trees, weeds, or horizon lines.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 28mm • f/3.5, second, 1600 ISO
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THE POSE
I had the subject stand with equal weight on both feet, and she crossed her arms. I centered her to show power. To ensure the pose worked in the light we had, I turned her slightly toward the lake to shadow her body and had her turn her face back toward the light. A stern expression seemed to suit the pose.
The island over her left shoulder and the trees on the left make a triangle to her. With the water equally on both sides, there is symmetry.
WHY NOT USE STROBES?
If I used strobes, I could work at any time of day—but it wouldn't be the same. First, the wind usually calms down in the evening, and the water's surface becomes glass-like. Second, no strobe could light the subject and the trees in the background to keep exposure detail.
46. BLACK ORE
LET THEM DO WHAT THEY DO
I decided to add this image to the book at the last minute. It's a nice portrait, and the light and background are good, but the example I wanted to share is the subject. This is the skater boy shown in an earlier section. It was later in his session, twenty miles away from the skate park. This is why I don't incorporate specific poses. I told him to be low. This is what he did. He looks strikingly like the pose in his tower image. Why? Because it is him. It is what he does. Watch your subjects. Let them do what they do, then tweak the pose to be more photographically correct. The more a person looks like themselves, the better your images will be received by them and their family.
LIGHTING
Let's move on to the light. I chose this background for the color and because it cast a shadow over itself. The sun was still up, but not by much. There was raw light on the ground in front of the pile of ore. I had the teen walk into the scene until the pile blocked the raw light from hitting him. Then I had him squat down. He was on the very edge of the raw light; this presented a clean, crisp, dimensional view of his features. I did need an accent reflector (it was placed as shown in the top view diagram) to help separate him from the black ore. As you can see in the side view diagram, the pile of ore defined the edge of the light source. Notice the nice texture in the guy's hair, clothing, and skin. If he were to go farther into the shadow of the ore, the light would flatten out and eventually come from the other direction. I could have had him go to the back of the pile, as shown in the side view diagram. The light would have been from the east, with less contrast, and the background would have been different, but it would have worked. An image with the subject positioned in the middle of the pile would not work as well. The light would have had the same flat, contaminated look that you would find if too deep under the canopy of a tree. The moral of the story is, live on the edge.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 90mm • f/4, second, 400 ISO
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47. WATERCOLOR GIRLS
A DISTINCTIVE LOOK
A small part of me misses film, but when I look at all the tools at our disposal now, the feeling goes away quickly. I love the fact that we can print a photographic image on watercolor paper. We can enjoy the magic of the textures and the softness that the medium has to offer. I love it when little girls attired in pastels walk through my doors. I instantly know at least one direction I am going to go.
I tend to keep light colors on light backgrounds. This way, the subject's face is in contrast with the clothing and background, and, as such, it will draw the eye. However, we still need to light the face correctly.
For this image, I used my north light camera room. I positioned the subjects against a pillar, as shown in the diagram. The pillar allowed me to use the edge of the light coming from the window. The older girl became the base for the image. I turned her away from the light but had her turn her face back toward me. This gave an appropriate broad light. The younger girl just snuggled into her sister. When they started holding each other, I quickly repositioned their arms to form a circle with their faces. This circle will hold the viewer's eye. Because the younger girl was posed toward her sister, the light from the window produced a short light pattern on her face. The broad light allowed for the somber eye contact of the older girl. Because short light is usually a happy light pattern, I had the little sister look away from the camera to avoid eye contact with the audience. Now between the light patterns and the posing, the soft emotion was portrayed.
"I tend to keep light colors on light backgrounds. This way, the face is in contrast with the clothing and background, and will draw the eye."
POSTPRODUCTION
The post-process work on this image was done with layers of white backgrounds and layer masks. I painted different opacities of the white underlying backgrounds to enhance the existing highlights. Then I vignetted the image until it faded to white. This type of post-process work is meant for watercolor printing.
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Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 75mm • f/5.6, second, 400 ISO
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48. WEDDING RAPIDS
Many years ago, at a photography convention, I overheard other attendees conversing. They were searching for a program to attend and looked at the sign outside the door of the room I was waiting to enter. It was a wedding program. One attendee said, "Oh, it's a wedding program. I don't do weddings. I'm not going in." It's a shame that they missed it. I learned portraiture from a wedding photographer. I discovered techniques I could use when photographing seniors, families, or chasing kids around the yard. It does not matter if you photograph weddings, rock stars, or rocks. Light is light and composition is composition. There is much to learn from everyone.
ON THE COURSE
This bride and groom wanted outdoor portraits. We borrowed a couple of golf carts and hit the links. Armed with photo gear instead of clubs, we found a few nice backgrounds with nice light. This was one of them.
As shown in the diagram, there was an open area over the fairway to my right. Trees lined the left side of the image as well as the entire area behind the couple. The bride was standing by a tall oak tree. Its canopy reduced the light from above but allowed it to enter from the open sky. Due to the layout of the area, light could only enter the scene from my right and behind me. We had a stream and woods, good light, and a bridal couple. Life was good.
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Canon EOS 5D Mark II, Canon EF 70–200mm f/2.8L IS USM lens. Exposure: f/5.6, , 400 ISO. Focal length at 75mm.
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TIME IS OF THE ESSENCE
On a wedding day, time is of the essence. Since I found a good place to play, I planned to do multiple shots here. I started with the couple together. I leaned the groom against the tree and had the bride lean into him. The time-consuming part was her dress. Since it was already in a good place, I left it there. I pulled the groom down to the foreground and created another image without having to reposition the dress. Then, I had the groom step out and I photographed the bride. This time, I had to make a minor change to the position of the dress.
These images were made possible because of natural light. The source was large enough that I could move the subjects around and still have the same direction and exposure of light from person to person and image to image.
49. THE BROWN LOUNGE
FURNITURE AS PROPS
A few years ago, I was wasting time in a furniture store while the salesperson talked to my wife about new couches. Out of the corner of my eye, I saw this chaise lounge. I noticed that the textures in the seat repeated in a diagonal pattern. I saw the nice soft sweep of the head of the lounge. I thought to myself, "If you noticed this lounge, amidst all of the other furniture, it must have good potential for a posing prop." We bought it. I was right—it was a great prop once it was heaved into place. I didn't know how heavy it was until it was delivered, and now it is a permanent fixture in my house.
THE SETUP
For this portrait, I turned the lounge diagonally to the backdrop with the head facing away from the light. I chose a brown background to match the lounge and selected the dark shirt for the senior. I had the entire scene set up before the teen entered the room. I even positioned the reflectors. I knew where I was going to need the light.
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Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 135mm • f/5.6, second, 400 ISO
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When the teen came into the room, all I did was ask her to sit down facing me and lean over the lounge. I fixed her hair to show its length and repositioned her hands slightly, but there wasn't much to do. Notice how the light on her face is up against the dark shape on the background and the dark side of her face is up against a lighter shape? This was planned in advance. Before the senior came out of the dressing room, I looked at the head of the lounge and saw good light there. I knew that same good light would fall on the subject. I use this approach often. For example, if I am to photograph a child who can't stand still, I place a chair where I want it to establish the desired light and background relationship. Then I have the child sit on the chair or stand by it. Even if they only stay for a moment, at least they are in the right spot.
50. JACK AND JILL
JUST LET THE SUBJECTS EXIST
It was frustrating to create portraits of my kids. Seriously. When I took photographs of everyone else's children, things worked out just fine, but when my kids were in front of my lens, nothing seemed to go right. I guess it just seemed like playtime to them. The last thing they wanted was to sit still for a portrait. I always told other photographers, "Find a background with good light and let the subjects exist." After a few attempts, I started to listen to myself.
A MEANINGFUL LOCATION
We spent a lot of time at an old cabin as kids. When the weather started getting cold, the water was turned off to the cabin. If we spent any time at the cabin after that, we had to get water from this pump. I remember making many trips with water sloshing out of the bucket. It wasn't really an assigned chore; I just liked doing it.
The pump is still there today. It has meaning to me and my family, so I decided to use it for this portrait.
Next to the pump is an old log cabin. The yard is lined with trees, and there are deep wooded areas. In the late afternoon, the cabin casts a shadow over the pump area. The trees to the north and west block light from falling on the background. The open sky to the east provides the only light that can reach the pump. It is forced into the scene perfectly for a portrait.
I didn't want the kids to know I was trying to take pictures, so I didn't take my camera out of the bag until I showed them how to use the pump. Once they started having fun, I pulled the camera out, set it on the tripod, and started recording. I felt as though I had won.
"There was a little red in the background, but it wasn't enough to draw attention, so I enhanced it . . ."
POSTPRODUCTION
It was early fall when I created this image. The leaves were just starting to turn. There was a little red in the background, but it wasn't enough to draw attention, so I enhanced it and added a bit more red to harmonize with the pump. I also added some yellow in the top center of the image to contrast the blue outfits and help create a sense of depth. I added a little orange to the leaves behind my boy and to one leaf in front of the kids to keep the viewer's eye bouncing back and forth in the image. This image is still hanging in our home, partly as a great memory of our children and partly as a trophy for finally winning the portrait war.
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Kodak 14n • Nikon 80–200mm f/2.8 lens at 100mm • f/5.6, second, 400 ISO
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51. PAPA BEAR
LIGHTING THE PROFILE
One of the lighting patterns I see most misused in natural light is profile light. The problem seems to be particularly prevalent with brides. We have a bride in a white dress with lace and beadwork, and she is holding flowers. The photographer poses her in the middle of the window and has her look out dreamily over the world. He can see her face light up and see the profile; he thinks all is good. When the files are processed, the flowers are blown out, the dress has no detail, and the light outlining the bride's face is glowing. A strange orange line follows the edge of the glow. What happened? Was the exposure incorrect? It may have been, but this problem would have occurred even if the image were exposed correctly.
Light travels in a straight line. If light is directed straight at a reflective surface, it will bounce off and return to the source. This becomes a spectral light. In terms of ones and zeros, which is what digital files are, how much information do you think is in a spectral highlight? The answer is zero. So, how do we create a profile light and still hold detail?
THE APPROACH
I wanted a profile of this new father holding his son. I placed a background in the normal location on the east wall. I positioned another background along the windows to act as a gobo. The subject sat at the edge of the gobo on the same side of the window as the camera. (If he looked straight forward, he would have been looking at the light-blocking device.) This way, the light was forced from behind the subject in relationship to where the camera was. The light skimmed across the subject and did not blow out any detail. Notice how you can see the pores in his skin and the soft hair on the newborn's head. If the subjects were in the middle of the window, none of that would be showing. In my studio, the other windows provide fill light. In other locations, the wall between windows will work as a gobo. Just remember to place the subject on the same side of the window as you are on. You will create a much better profile light and your subjects (and vendors, in the case of bridal portraits) will like you a whole lot more.
"Light travels in a straight line. If light is directed straight at a reflective surface it will bounce off and return to the source. This becomes a spectral light."
* * *
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 153mm • f/7.1, second, 400 ISO
* * *
52. TRAIN WHEEL
POSE WITHIN THE LIGHT
When the sun is your main light source, you must move the subject and yourself in relationship to the light to get flattering results. For this reason, I often say "Pose within the light."
This image was made using a component of the train car seen elsewhere in this book. The subject was posed between the train cars and stood on the hitch. The wheel she is leaning on has, I assume, something to do with the connections for the cars. It made for a good prop.
WHY THE POSE WORKS
Using this leaning pose allowed a few good things to happen. First, her body formed a great leading line to her face. Second, the pose drew attention to her long hair. Third, the horizontal image was a nice counterpoint to the predominantly vertical images I captured during the session. Fourth, the pose made good use of light, which was coming from an angle that would have been too low with a more upright pose.
This image was created in the late afternoon under overcast skies. The light was entering equally from both sides of the cars, so I used this pose to create the direction of light that we see. With her body leaning this much, her head tilt looks comfortable. The light from above and to the east illuminated half of her face. Her head blocked the light from the rest of her face.
* * *
Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 85mm • f/5.6, second, 400 ISO
* * *
Since the caboose was only 2 feet away, no light came from in front of the subject's face. Therefore, it appears as if a split light pattern was used. Okay, I _did_ say the light was equal from both sides of the car—but I used the subject's mother as a gobo. She was standing just outside of the frame to our right. She didn't know it, but in her attempt to watch the session from a close distance, she was also helping my lighting. Anything can block light. Use what is available to you.
Timing is everything. Again, this image was made late in the day, under overcast skies. At midday, the sunlight entering this area would have been too harsh. The background would have so much contrast from highlight to shadow that photographic paper would not be able to hold the tonal range. The subject would have no detail in her hair or skin. Sure, we are natural light photographers, but that doesn't mean that _any_ natural light will work for the portrait we have in mind. I have heard people say, "You're a professional. You should be able to work anywhere." That may be true, but my thinking is that a professional should also know when something will or won't work. Not every time/location combination will yield good results. That being said, if a client _needs_ to have a portrait made in a specific area and the light is bad, I can opt to use strobes to get the job done.
53. ON THE ROCKS
THE OLD ORE MINE
It's great to find locations that provide good shade for a long period of time. This is the case with the pilings from the old ore mine. The pile of jagged rock is very tall and fairly steep. The stretch of debris I use runs north and south. As the sun reaches the western sky, a nice shadow covers the bottom of the hill and the surrounding area. It is a nice place to work in the late afternoon, and the good lighting conditions last a long time. As you can see in the side view diagram, the hill of rocks is in complete shadow and makes an evenly illuminated background. Light cannot come from the rocks to the right or from the trees behind the subject, so that means the light can only come from the north, the east, or above. I had to keep this fact in mind when deciding how the subject should pose within the light.
I chose to use the light from above the subject's face. Because of the pose, she was able to tilt her head dramatically and still look comfortable. Her hanging hair blocked the soft light from the east. The ground acted as a subtractive light source too. Light fell onto her face from above and from behind the camera with the same wraparound quality I'd expect in my camera room. The wall of light was just rotated on its axis to match the axis of her head tilt.
THE PERKS OF NATURAL LIGHT
If you study the entire body, you will notice the light on the teen's leg was coming from the east. It did a great job of creating shape. This tells me that we could have posed her to use light from that direction as well. The advantage of understanding and using natural light is that you can use body positions to relate to different light sources within the same image—without carrying multiple sets of strobes and modifiers all over the countryside. Could you imagine trying to light this with strobes? The locale is too rugged to use light stands. You would need several assistants to hold your lights.
* * *
Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 70mm • f/5.6, second, 400 ISO
* * *
"Could you imagine trying to light this with strobes? The locale is too rugged to use light stands. You would need several assistants to hold your lights."
54. LAKE ANTOINE AT SUNDOWN
TO THE BEACH!
Once again, we find ourselves at the beach. When is the best time to shoot? You should know by now that it's late evening or early morning. Since most teenage girls would rather not have their portraits done at 4:30AM, evening seemed like the right time for this session. This shot was made after the sun disappeared behind the trees on the horizon. When the raw light was completely gone, I started working. I will say this: Using the lake shores inland allows for much more time to work on the beach, as compared to working on the ocean shores. When I teach on the beaches of California, once the sun melts into the water, it gets dark fast. Inland, where I work most of the time, the trees and topography of the surroundings block the raw light from the sun earlier; this allows me to begin working before the sun has actually set. What an advantage! The same rule applies to the East Coast during our morning sessions. When there is nothing but water in the way of the sun, the raw light appears quickly. Just be aware of this so you don't get caught off guard.
CLOTHING
I went through the girl's wardrobe before the session began and found this dress. She was scheduled for an extended session, and the beach was to be our last stop. I immediately knew this was to be her final clothing change. The cool color of the water was sure to work well with the blue colors in the dress.
COMPOSITION AND POSE
The beach runs nearly north and south. I positioned the teen so that I could see some beach all around her—and some water to harmonize with the dress. Her pose parallels the water line and produces nice leading lines that direct the viewer's gaze to her face. For modesty's sake, I had her hold her dress from under her knees. This gave her something to do with her hands without ruining the leading lines. It also helped give her shape. Her head was tipped to the left to create balance in the pose and allow the light to form a Rembrandt pattern on the teen's face. If her face were to follow me as I moved closer to the water and showed the beach as a background, we'd have had a short light pattern. In that same location, if she turned her head to the shore, the pattern would be broad lighting. You can often take advantage of many options just by having the subject turn their head.
* * *
Canon EOS 1Ds Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 140mm • f/4.5, second, 400 ISO
* * *
55. PURPLE DITCH
HITTING THE BRAKES
This was another occasion where I saw something cool while driving and had to hit the brakes. In the ditch, not 10 feet away from 60MPH traffic, there were tall, purple flowers. We had to go play.
The trees along the road shadowed the area from the early-evening sun. The most intense light came from the open sky to the south. Some light came from the east, but it was not as powerful, and the result was a wraparound lighting effect. I added a black panel just out of view to the subject's left. This gave a bit more shadow on the left side of her face. Another reflector was added from underneath and on her highlight side to add a little light under her eyes.
I enjoyed the transparent look of the flowers and the opaque quality of the stems. The look of the dark stems is repeated in the trees farther in the background. This helped create a sense of depth. The flowers fading out of focus also enhanced the feeling of a third dimension.
THE POSE
The pose was simple. I saw a triangle of light yellow in the scene behind her, so I mimicked it with her pose. The photo below shows what was going through my mind as far as relating the subject to the background. The opposing corners also matched. These were ingredients for good balance. Everything in your image should have a purpose.
* * *
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 95mm • f/5.6, second, 400 ISO
* * *
56. FOUND ON THE DOORSTEP
EARLY MORNING SESSION
In the summer, the sun rises over the trees early. By 8:30AM, the only places in shadow are on the west side of buildings and under large overhangs.
Outdoor sets should be in good quality light for as much of the day as possible. My photo shack has a door and window on both the east and west ends of the structure—the west for the morning sessions and the east for afternoon shoots.
THE POSE
I had my guest sit on the doorstep on the west side. She naturally leaned forward on her legs. I had her bring her feet apart to create diagonal lines.
LIGHTING
Trees to the south and west blocked much of the light from the open sky. The main light was coming from the larger clearing and the open north sky. As the subject leaned forward, her face entered the light. Note that her face and hand are brighter than her shoulder. Her shoulder was naturally toned down because the wall blocked the north light. I positioned myself to get a look into the shack and achieve a good pattern of light on her face without contorting her too much. I used a reflector under the subject on the highlight side to brighten her eyes.
* * *
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 78mm • f/5.6, second, 400 ISO
* * *
57. PEOPLE WATCHING
FINDING BEAUTIFUL LIGHT
The next time you are traveling by air, watch people. Most airports have massive windows and long hallways that create a great combination of light and background. People are naturally walking through the areas. When something catches your eye, chances are, it's because the person walked into an area of great light. Note where the light was coming from at that moment. The hope is that you learn from this discovery and duplicate the effect on assignment.
SEIZE THE MOMENT
This image was created during one of my class demonstrations. The subject was one of my students, and not the person I was photographing. As I talked to the class, I noticed the beautiful light on his face and I had to go off on a small tangent.
As you can see in the diagram, the subject was standing near a pillar on the second story of a covered hallway. The open-air courtyard allowed light to come in and bounce off the wall behind the subject. The light from the open sky from the southwest could only get to him from behind as well because the pillar blocked the light from directly to the side of him. The end result was a very nice double accent light. I just had to record it.
* * *
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 160mm • f/5.6, second, 400 ISO
* * *
So my question to you is, What time of day was it? Would this image work right here at a different time of day?
58. OPEN DOOR
GOOD LIGHT AND A SIMPLE BACKGROUND
This is another good example of posing within the light. My students, our model, and I were on our way to an area that I knew had good light and a simple background. When I got to the location, I turned to watch everyone walk through the doors and saw wonderful light.
CREATING THE IMAGE
I had the subject hold the door halfway out and lean around it. This brought his face forward into good light and twisted his core enough to allow the same light to skim across it. No reflectors were needed. The fill came from the length of the light source, as you can see in the diagram and the photograph of the area.
The sun was just out of view beyond the far wall. The light from the sky was more intense from the model's side of the porch than on the east side. This caused the harder contrast between shadow and highlight, which was good in this case.
"By changing the relationship between the reflective glass door and the open sky, the reflection was greatly reduced."
FIGHTING REFLECTIONS
When I set up this image, there was a big reflection in the glass. No matter where I moved, it was there. It dawned on me to have him open the door more. By changing the relationship between the reflective glass door and the open sky, the reflection was greatly reduced. In this case, the light didn't really change for the subject's face, and the background remained the same. I pressed the shutter release and recorded the image.
* * *
Canon EOS 5D Mark III • Canon EF 70–200mm f/2.8L IS USM lens at 110mm • f/5.6, second, ISO 400
* * *
59. PIECES OF ME
The joy of photographing people is getting to know their stories. Listen to your subjects and learn who they are, then use everything you know about portraiture to tell others their story.
THE RIGHT SPACE
This talented teen came to my studio. While listening to him, I realized my studio was not the best place to tell his story. We needed to be in the art room, so that's where we headed. Luckily, there were large windows in the room. The bottom panes were 7 feet tall. Above them were more panes angled in, giving more height and reach into the room. The room felt like an old north light studio—it was perfect!
The artist told me about his work, and I chose two projects to play with. The old TV was the first. As shown in the diagram below, I chose a split light pattern. I positioned the camera in a triangle pattern as I would to create a Rembrandt light pattern. However, I didn't have the subject turn his head fully into the light to achieve the triangle of light. I stopped him when I saw the split light. Now, the artwork was evenly lit and he had good shape. Yes, I wanted to show his talent, but he was to be the focus of the image.
I used a reflector for an accent light.
In another part of the room there was a corkboard where we could hang the other project—a large art piece that needed space. The challenge was the falloff of the light. Look at the "raw" image (facing page, top). See the difference in density from the top to the bottom? Notice that the teen's face is right on that fall-off. If he were to take a step to his left, his face would be out of the light. The accent reflector was tipped way back to catch the light near the floor. The accent on his face came from below. You can see it on his face and body, but not the top of his shoulder or his hair. This is the limit in this room for using natural light for portrait work.
The light covered the artist's head in the TV image but didn't quite reach it in _Pieces of Me_ (facing page); therefore, his hair is darker in the second image. In postproduction, I burned in the bottom of the second image to match the top, extended the corkboard, and added a frame.
KNOW YOUR LIMITS
Know the distance you can be from your window and still have good light. My windows were only 7 feet tall in my first studio. It was fine for one person or a few if they were seated lower to the ground, but if a subject got more than 10 feet away, they were out of the light.
* * *
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 90mm • f/7.1, second, 1600 ISO
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 90mm • f/7.1, second, 1600 ISO
* * *
60. GETTING FANCY
Even the fanciest images need a good basic foundation. This rings true in this case. Yes, the postproduction work took me out of my comfort zone, but I needed something new to draw attention to me as an artist. I wanted to take what I was already doing and add to it. Applying the brushes and techniques I learned from studying Woody Walters' work was the answer. These days, I think of what the finished work could be—I look beyond what I see through the camera. The finished image may contain several portraits, and there may be other elements involved. Each image in a composite must be able to stand alone. Start with the good portrait techniques that you learned in this book, then take your work to a new level with postproduction painting.
A GREAT RESOURCE
Detailing the techniques that I used to create the final image is beyond the scope of this book. If you are interested, look up Woody Walters at www.woodywaltersdigitalphotocandy.com. You will find all kinds of new toys for your toolbox.
* * *
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 70mm • f/5.6, second, 2500 ISO
Canon EOS 5D Mark II • Canon EF 70–200mm f/2.8L IS USM lens at 95mm • f/7.1, second, 400 ISO
* * *
CONCLUSION
I'm sure, after reading this book, you have learned that there is a great deal to learn about portraiture. Many things you probably already knew. Hopefully I sparked ideas to use old knowledge in new ways. I also believe that you will start looking for quality light areas and good background combinations. You will pose people within the light to form appealing patterns of light in all locations you work. You will anticipate an activity so that you can position yourself in a good relationship to the light and the background and will be able to precisely determine where the subject will be posed in relation to both so that the final image is powerful, without excuses. You will, with practice, become comfortable with finding and using natural light when photographing any subject. Use the concepts you have learned in these pages, and have a great time playing in the light!
INDEX
**A**
Accent lighting, , , 25–26, , , , , , , , , , , , ,
Adobe Photoshop, , , ,
Airports,
Aperture. _See_ Depth of field
Assistants, , ,
**B**
Backgrounds, , , , , , , , , , , , , , , , ,
Backlighting,
Beach portraits, , , 54–55, 56–57, , 114–15
Billboard,
Biology, ,
Black & white,
Brides. _See_ Weddings
Broad light, , , , , ,
**C**
Camera selection,
Camera shake,
Camera room, 8–9,
Camera straps,
Catchlights, ,
Clients, repeat,
Clothing, , , , , , ,
"Cocaine lighting," ,
Colors, , , , ,
Composition, , , , , , , , , , , ,
Contrast, , , , ,
Corel Painter,
Cropping,
**D**
Depth. _See_ Dimension
Depth of field, , , , ,
Dimension, , , , ,
Double accent lighting, 25–26, ,
Drama classes,
**E**
Education, 5–6
Exposure, , 14–16
Expression, , , , , , , ,
Eyes, 5–6, , , , , , 118–19
**F**
Falloff, , ,
Family portraits,
Field of view, , ,
Fill light, , , , , , , , , , , , , , , , ,
Flare,
Focal length, , ,
Flat lighting, ,
Fresnel lenses,
F-stops, , , ,
**G**
Gobos, , , , , , , , , , , , , , , , , , , ,
Gray cards, 17–18
Green screen,
Grids,
Group portraits, , , , ,
**H**
Hair light,
Highlights, , , , ,
Histograms, 17–18
Hot spots, ,
**I**
Inverse square law, ,
ISO, , , ,
**L**
Leading lines, , , ,
Lens selection, , 16–17, , , , , ,
Light, direct, ,
Light, directional, , , , , ,
Light meters, 15–16, , ,
Light modifiers, 5–6, , , , 18–26, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
Fresnel lenses,
gobos, , , , , , , , , , , , , , , ,
grids,
reflectors, , , 18–19, , , , , , , , , , , , , , , , , , , , , , , , ,
softboxes,
umbrellas,
under-reflectors, , ,
Light patterns, 5–6, 19–26, , , , , , , , , , ,
Light, rhythm of, 27–28, , ,
Light, size of source, , , , ,
Lighting, subtractive, , , , , , , , , , , , , , , , , ,
**M**
Mood,
Motion, freezing,
**N**
Nik Software, , ,
North light, , , ,
**O**
Open shade, ,
Overcast sky, , , , ,
**P**
Perspective, , ,
Posing, , , 56–57, , , , , , , , , , , , , ,
Postproduction, , , , , , , , , , , ,
Profile light, , , ,
Props, ,
Psychology, , ,
**R**
Railroad tracks,
RAW files,
Reflections,
Reflectors, , , 18–19, , , , , , , , , , , , , , , , , , , , ,
Rembrandt lighting, , , ,
**S**
Safety, ,
Sales,
Scheduling,
Self-assignments,
Separation, , , , , , ,
Shack, ,
Shadows, , , ,
Short light, , , ,
Shutter speed,
Simplicity, , ,
Single accent lighting, , ,
Softboxes,
Speedlights,
Split light, 23–25, , , , , ,
Sports, 42–43, 81–82, 88–89, 90–91, ,
Storytelling, , , , , ,
Strobes, , , 9–10, , , ,
Sunrise, , ,
Sunset, , , , ,
Sweet light, 54–55 56–57, 86–87, ,
**T**
Tattoos,
Texture, , ,
Tonal range, , , , , , , ,
Trains, , , , 110–11
Trash-the-dress sessions, ,
Tripods, , ,
**U**
Umbrella,
Under-reflectors, , , ,
**V**
Variety, , , , 90–91,
Visualization, ,
**W**
Walls, , , , , , , , , , , , , , , , , , , , , ,
as gobos, , , , , , ,
as reflectors, , , , , , , ,
Walters, Woody,
Watercolor effects,
Weddings, 102–3,
Weight,
White balance, 17–18
Wide-angle lenses,
Window light, , , , , , , , , , , ,
Wraparound light, , , , ,
**Z**
Zoom lenses, , ,
| {
"redpajama_set_name": "RedPajamaBook"
} | 1,199 |
\section{Introduction} \label{sec1}
In this section, which is partitioned into five inter-dependent subsections, the
reader is given a concise overview of the information subsumed in the text: (i)
in Subsection~\ref{sec1a}, the degenerate Painlev\'{e} III equation (DP3E) is
introduced, representative samples of its ubiquitous manifestations that
have piqued the recent interest of the author are succinctly discussed, and
the qualitative behaviour of the asymptotic results the reader can expect to
extricate from this work are delineated; (ii) in Subsection~\ref{sec1b}, the
DP3E's associated Hamiltonian and principal auxiliary functions, as well as
one of its $\sigma$-forms, are introduced; (iii) in Subsection~\ref{sec1c},
pre- and post-gauge-transformed Lax pairs giving rise to isomonodromic
deformations and the DP3E are reviewed; (iv) in Subsection~\ref{sec1d},
canonical asymptotics of the post-gauge-transformed Lax-pair solution
matrix is presented in conjunction with the corresponding monodromy data;
and (v) in Subsection~\ref{sec1e}, the monodromy manifold is analysed,
and a synopsis of the organisation of this work is given.
\subsection{The Degenerate Painlev\'{e} III Equation (DP3E)} \label{sec1a}
This paper continues the studies initiated in \cite{a1,av2} of the DP3E,
\begin{equation} \label{eq1.1}
u^{\prime \prime}(\tau) \! = \! \dfrac{(u^{\prime}(\tau))^{2}}{u(\tau)}
\! - \! \dfrac{u^{\prime}(\tau)}{\tau} \! + \! \dfrac{1}{\tau} \! \left(-8
\varepsilon (u(\tau))^{2} \! + \! 2 ab \right) \! + \! \dfrac{b^{2}}{u(\tau)},
\quad \varepsilon \! \in \! \lbrace \pm 1 \rbrace,
\end{equation}
where the prime denotes differentiation with respect to $\tau$, $\mathbb{C} \!
\ni \! a$ is the formal parameter of monodromy, and $\mathbb{R} \setminus
\lbrace 0 \rbrace \! \ni \! b$ is a parameter;\footnote{See, also, \cite{vgilss},
Chapter~7, Section~33.} in fact, making the formal change of independent, dependent,
and auxiliary variables $\tau \! \to \! t^{1/2}$, $u(\tau) \! \to \! \tilde{\eta}_{0}^{2}
t^{-1/2} \tilde{\lambda}(t)$, $a \! \to \! \mp \mathrm{i} \tilde{c}_{0} \tilde{\eta}_{0}$,
and $b \! \to \! \pm \mathrm{i} 2 \tilde{\eta}_{0}^{3}$, where $\tilde{c}_{0} \! \in \!
\mathbb{C}$ and $\mathrm{i} \tilde{\eta}_{0} \! \in \! \mathbb{R} \setminus \lbrace
0 \rbrace$, and setting $\varepsilon \! = \! +1$, one shows that the
DP3E~\eqref{eq1.1} transforms into, in the classification scheme of \cite{a3},
the degenerate third Painlev\'{e} equation of type $D_{7}$,
\begin{equation} \label{dpee3d7}
(P_{\mathrm{III}^{\prime}})_{D_{7}}: \quad \quad
\dfrac{\mathrm{d}^{2} \tilde{\lambda}}{\mathrm{d} t^{2}} \! = \! \dfrac{1}{\tilde{\lambda}}
\! \left(\dfrac{\mathrm{d} \tilde{\lambda}}{\mathrm{d} t} \right)^{2} \! - \! \dfrac{1}{t}
\dfrac{\mathrm{d} \tilde{\lambda}}{\mathrm{d} t} \! + \! \tilde{\eta}_{0}^{2} \! \left(
-2 \dfrac{\tilde{\lambda}^{2}}{t^{2}} \! + \! \dfrac{\tilde{c}_{0}}{t}
\! - \! \dfrac{1}{\tilde{\lambda}} \right).
\end{equation}
It is know that, in the complex plane of the independent variable, Painlev\'{e} equations
admit, in open sectors near the point at infinity containing one special ray, pole-free
solutions that are characterised by divergent asymptotic expansions: such solutions,
called \emph{tronqu\'{e}e} solutions by Boutroux, usually contain free parameters
manifesting in exponentially small terms for large values of the modulus of the
independent variable.\footnote{There also exist pole-free solutions that are void of
parameters in larger open sectors near the point at infinity containing three special
rays: such solutions are called \emph{tritronqu\'{e}e} solutions (see, for example,
\cite{edelere}, Chapter~3).} The present work is devoted to the study of tronqu\'{e}e
solutions of the DP3E~\eqref{eq1.1}. In stark contrast to the asymptotic results
of \cite{a1,av2}, this work entails an analysis of one-parameter families of
\emph{trans-series\/} (\cite{edelere}, Chapter~5) asymptotic (as $\lvert \tau \rvert \!
\to \! +\infty$) solutions related to the underlying quasi-linear Stokes phenomenon
associated with the DP3E~\eqref{eq1.1};\footnote{Such solutions are also referred
to as instanton-type solutions in the physics literature \cite{sgaiakmm}; see, also,
\cite{aritaapa,a66,sachokap1,sachokap2}, and Chapter~11 of \cite{a5}.} in particular,
tronqu\'{e}e solutions that are free of poles not only on the real and the imaginary axes
of $\tau$, but also in open sectors about the point at infinity, are considered. The
existence of one-parameter tronqu\'{e}e solutions for a scaled version of the
DP3E~\eqref{eq1.1} was proved in \cite{ylddpt} via direct asymptotic analysis. Parametric
Stokes phenomena for the $D_{6}$ and $D_{7}$ cases of the third Painlev\'{e} equation
were studied in \cite{iwak1}. Application of the third Painlev\'{e} equation to the study
of transformation phenomena for parametric Painlev\'{e} equations for the $D_{6}$ and
$D_{7}$ cases is considered in \cite{iwak2}, whilst the $D_{8}$ case is studied in
\cite{ayota,hwakytak}. The recent monograph \cite{magch} studies the relation of
the third Painlev\'{e} equation of type $(P_{\mathrm{III}})_{D_{6}}$ to isomonodromic
families of vector bundles on $\mathbb{P}^{1}$ with meromorphic connections. In
\cite{gavyy}, the $\pmb{\pmb{\tau}}$-function associated with the degenerate third
Painlev\'{e} equation of type $D_{8}$ is shown to admit a Fredholm determinant
representation in terms of a generalised Bessel kernel. By using the universal example
of the Gross-Witten-Wadia (GWW) third-order phase transition in the unitary matrix
model, concomitant with the explicit Tracy-Widom mapping of the GWW partition
function to a solution of a third Painlev\'{e} equation, the transmutation (change in
the resurgent asymptotic properties) of a trans-series in two parameters (a coupling
$g^{2}$ and a gauge index $N$) at all coupling and all finite $N$ is studied in
\cite{ahmedunne} (see, also, \cite{eund}).
\begin{eeee} \label{remtranstronq}
The terms trans-series \cite{edga} and tronqu\'{e}e are used interchangeably in this
work. \hfill $\blacksquare$
\end{eeee}
An overview of some recent manifestations of the DP3E~\eqref{eq1.1} and
$(P_{\mathrm{III}^{\prime}})_{D_{7}}$ \eqref{dpee3d7} in variegated mathematical and
physical settings such as, for example, non-linear optics, number theory, asymptotics,
non-linear waves, random matrix theory, and differential geometry, is now given:
\begin{enumerate}
\item[\textbf{(i)}] It was shown in \cite{sulei} that a variant of the DP3E~\eqref{eq1.1} appears
in the characterisation of the effect of the small dispersion on the self-focusing of solutions of
the fundamental equations of non-linear optics in the one-dimensional case, where the main
order of the influence of this effect is described via a universal special monodromic solution of
the non-linear Schr\"{o}dinger equation (NLSE); in particular, the author studies the asymptotics
of a function that can be identified as a solution (the so-called `Suleimanov solution') of a slightly
modified, yet equivalent, version of the DP3E~\eqref{eq1.1} for the parameter values $a \! = \!
\mathrm{i}/2$ and $b \! = \! 64k^{-3}$, where $k \! > \! 0$ is a physical variable.
\item[\textbf{(ii)}] In \cite{avlkv}, an extensive number-theoretic and asymptotic analysis
of the universal special monodromic solution considered in \cite{sulei} is presented: the
author studies a particular meromorphic solution of the DP3E~\eqref{eq1.1} that vanishes
at the origin; more specifically, it is proved that, for $-\mathrm{i} 2 a \! \in \! \mathbb{Z}$, the
aforementioned solution exists and is unique, and, for the case $a \! - \! \mathrm{i}/2 \! \in \!
\mathbb{Z}$, this solution exists and is unique provided that $u(\tau) \! = \! -u(-\tau)$.
The bulk of the analysis presented in \cite{avlkv} focuses on the study of the Taylor
expansion coefficients of the solution of the DP3E~\eqref{eq1.1} that is holomorphic at
$\tau \! = \! 0$; in particular, upon invoking the `normalisation condition' $b \! = \! a$
and taking $\varepsilon \! = \! +1$, it is shown that, for general values of the parameter
$a$, these coefficients are rational functions of $a^{2}$ that possess remarkable
number-theoretic properties: en route, novel notions such as super-generating functions
and quasi-periodic fences are introduced. The author also studies the connection problem
for the Suleimanov solution of the DP3E~\eqref{eq1.1}.
\item[\textbf{(iii)}] Unlike the physical optics context adopted in \cite{sulei}, the authors
of \cite{peetsdb} provide a colossal Riemann-Hilbert problem (RHP) asymptotic analysis
of the solution of the focusing NLSE, $\mathrm{i} \partial_{\mathrm{T}} \Psi \! + \! \tfrac{1}{2}
\partial_{\mathrm{X}}^{2} \Psi \! + \! \lvert \Psi \rvert^{2} \Psi \! = \! 0$, by considering the
rogue wave solution $\Psi (\mathrm{X},\mathrm{T})$ of infinite order, that is, a scaling limit
of a sequence of particular solutions of the focusing NLSE modelling so-called rogue waves
of ever-increasing amplitude, and show that, in the regime of large variables $\mathbb{R}^{2}
\! \ni \! (\mathrm{X},\mathrm{T})$ when $\lvert \mathrm{X} \rvert \! \to \! +\infty$ in such
a way that $\mathrm{T} \lvert \mathrm{X} \rvert^{-3/2} \! - \! 54^{-1/2} \! = \! \mathcal{O}
(\lvert \mathrm{X} \rvert^{-1/3})$, the rogue wave of infinite order $\Psi (\mathrm{X},\mathrm{T})$
can be expressed explicitly in terms of a function $\mathcal{V}(y)$ extracted {}from the
solution of the Jimbo-Miwa Painlev\'{e} II ($\mathrm{P} \mathrm{II}$) RHP for parameters
$p \! = \! \ln (2)/2 \pi$ and $\tau \! = \! 1$;\footnote{Not to be confused with the
independent variable $\tau$ that appears in the DP3E~\eqref{eq1.1} and throughout
this work.} in particular, Corollary~6 of \cite{peetsdb} presents the leading term of
the $\mathrm{T} \! \to \! +\infty$ asymptotics of the rogue wave of infinite order $\Psi
(0,\mathrm{T})$ (see, also, Theorem~2 and Section~4 of \cite{peetsdq}),\footnote{For
the rogue wave of infinite order \cite{peetsdb}, one needs to consider asymptotics of
tronqu\'{e}e/tritronqu\'{e}e solutions of the inhomogeneous $\mathrm{P} \mathrm{II}$
equation, $\tfrac{\mathrm{d}^{2}u(x;\alpha)}{\mathrm{d} x^{2}} \! = \! 2(u(x;\alpha))^{3} \! + \!
xu(x;\alpha) \! - \! \alpha$, for the special complex value of $\alpha \! = \! \tfrac{1}{2}
\! + \! \mathrm{i} \tfrac{\ln (2)}{2 \pi}$ (asymptotics for tronqu\'{e}e/tritronqu\'{e}e solutions
of the $\mathrm{P} \mathrm{II}$ equation with $\alpha \! = \! 0$ are given in the
monograph \cite{a5}), and to know that the increasing tritronqu\'{e}e solution, denoted
$u_{\mathrm{TT}}^{-}(x;\alpha)$ in \cite{peetsdc}, is void of poles on $\mathbb{R}$;
furthermore, for the function $\mathcal{V}(y)$ to have sense as a meaningful asymptotic
representation of the rogue wave of infinite order $\Psi (\mathrm{X},\mathrm{T})$, it is,
additionally, necessary that $u_{\mathrm{TT}}^{-}(x;\alpha)$ be a global solution (analytic
$\forall$ $x \! \in \! \mathbb{R}$) of the $\mathrm{P} \mathrm{II}$ equation for $\alpha
\! = \! \tfrac{1}{2} \! + \! \mathrm{i} \tfrac{\ln (2)}{2 \pi}$. In \cite{peetsdc}, the author provides
a complete RHP asymptotic analysis of the global nature of tritronqu\'{e}e solutions of the
$\mathrm{P} \mathrm{II}$ equation for various complex values of $\alpha$, including the
particular value $\alpha \! = \! \tfrac{1}{2} \! + \! \mathrm{i} \tfrac{\ln (2)}{2 \pi}$, and relates
the function $\mathcal{V}(y)$ to the $\mathrm{P} \mathrm{II}$ equation, subsequently
identifying the particular solution that is requisite in order to construct $\mathcal{V}(y)$
as the increasing tritronqu\'{e}e solution $u_{\mathrm{TT}}^{-}(x;\alpha)$ for the special
parameter value $\alpha \! = \! \tfrac{1}{2} \! + \! \mathrm{i} \tfrac{\ln (2)}{2 \pi}$; moreover,
the value of the total, regularised integral over $\mathbb{R}$ for the increasing
tritronqu\'{e}e solution is evaluated.} which, in the context of the DP3E~\eqref{eq1.1},
coincides, up to a scalar, $\tau$-independent factor, with $\exp (\mathrm{i} \hat{\varphi}(\tau))$,
$\mathrm{T} \! = \! \tau^{2}$, where, given the solution, denoted by $\hat{u}(\tau)$, say,
of the DP3E~\eqref{eq1.1} studied in \cite{avlkv} for the monodromy data corresponding
to $a \! = \! \mathrm{i}/2$ (and a suitable choice for the parameter $b$), $\hat{\varphi}(\tau)$ is
the general solution of the ODE $\hat{\varphi}^{\prime}(\tau) \! = \! 2a \tau^{-1} \! + \!
b(\hat{u}(\tau))^{-1}$ (for additional information regarding the function $\hat{\varphi}
(\tau)$, see, for example, Subsection~\ref{sec1c}, Proposition~\ref{prop1.2} below).
\item[\textbf{(iv)}] In \cite{zamol3}, the authors study the eigenvalue correlation kernel,
denoted by $K_{n}(x,y,t)$, for the singularly perturbed Laguerre unitary ensemble
(pLUE)\footnote{The pLUE and its relation to the Painlev\'{e} III equation was introduced
and studied in \cite{schtin}.} on the space $\mathscr{H}_{n}^{+}$ of $n \times n$
positive-definite Hermitian matrices $M \! = \! (M)_{i,j=1}^{n}$ defined by the probability
measure $Z_{n}^{-1}(\det M)^{\alpha} \exp (-\tr V_{t}(M)) \, \mathrm{d} M$, $n \! \in \! \mathbb{N}$,
$\alpha \! > \! 0$, $t \! > \! 0$, where $Z_{n} \! := \! \int_{\mathscr{H}_{n}^{+}}
(\det M)^{\alpha} \mathrm{e}^{-\tr V_{t}(M)} \, \mathrm{d} M$ is the normalisation constant,
$\mathrm{d} M \! := \! \prod_{i=1}^{n} \mathrm{d} M_{ii} \prod_{j=1}^{n-1} \prod_{k=j+1}^{n} \mathrm{d}
\Re (M_{jk}) \, \mathrm{d} \Im (M_{jk})$, and $V_{t}(x) \! := \! x \! + \! t/x$, $x \! \in \! (0,+\infty)$.
By considering, for example, a variety of double-scaling limits such as $n \! \to \! \infty$
and $(0,d] \! \ni \! t \! \to \! 0^{+}$, $d \! > \! 0$, such that $s \! := \! 2nt$ belongs to
compact subsets of $(0,+\infty)$, or $n \! \to \! \infty$ and $t \! \to \! 0^{+}$ such that
$s \! \to \! 0^{+}$, or $n \! \to \! \infty$ and $(0,d] \! \ni \! t$ such that $s \! \to \! +\infty$,
the authors derive the corresponding limiting behaviours of the eigenvalue correlation kernel
by studying the large-$n$ asymptotics of the orthogonal polynomials associated with the
singularly perturbed Laguerre weight $w(x;t,\alpha) \! = \! x^{\alpha} \mathrm{e}^{-V_{t}(x)}$,
and, en route, demonstrate that some of the limiting kernels involve certain functions
related to a special solution of $(P_{\mathrm{III}^{\prime}})_{D_{7}}$ \eqref{dpee3d7};
moreover, in the follow-up work \cite{zamol4} on the pLUE, the authors derive the large-$n$
asymptotic formula (uniformly valid for $(0,d] \! \ni \! t$, $d \! > \! 0$ and fixed) for the
Hankel determinant, $D_{n}[w;t] \! := \! \det (\smallint_{0}^{+\infty} x^{j+k}w(x;t,\alpha)
\, \mathrm{d} x)_{j,k=0}^{n-1}$, associated with the singularly perturbed Laguerre weight $w(x;t,
\alpha)$, and show that the asymptotic representation for $D_{n}[w;t]$ involves a function
related to a particular solution of $(P_{\mathrm{III}^{\prime}})_{D_{7}}$ \eqref{dpee3d7}. In
the study of the Hankel determinant $D_{n}(t,\alpha,\beta) \! := \! \det (\smallint_{0}^{1}
\xi^{j+k}w(\xi;t,\alpha,\beta) \, \mathrm{d} \xi)_{j,k=0}^{n-1}$ generated by the Pollaczek-Jacobi-type
weight $w(x;t,\alpha,\beta) \! = \! x^{\alpha}(1 \! - \! x)^{\beta} \mathrm{e}^{-t/x}$, $x \! \in \! [0,1]$,
$t \! \geqslant \! 0$, $\alpha,\beta \! > \! 0$, which is a fundamental object in unitary random
matrix theory, under a double-scaling limit where $n$, the dimension of the Hankel matrix,
tends to $\infty$ and $t \! \to \! 0^{+}$ in such a way that $s \! := \! 2n^{2}t$ remains
bounded, the authors of \cite{mchychefa} show that the double-scaled Hankel determinant
has an integral representation in terms of particular asymptotic solutions of a scaled
version of the DP3E~\eqref{eq1.1} (or, equivalently, $(P_{\mathrm{III}^{\prime}})_{D_{7}}$
\eqref{dpee3d7}). In \cite{zamol6}, the authors study singularly perturbed unitary invariant
random matrix ensembles on $\mathscr{H}_{n}^{+}$ defined by the probability measure
$C_{n}^{-1}(\det M)^{\alpha} \exp (-n \tr V_{k}(M)) \, \mathrm{d} M$, $n,k \! \in \! \mathbb{N}$,
$\alpha \! > \! -1$, where $C_{n} \! := \! \int_{\mathscr{H}_{n}^{+}}(\det M)^{\alpha}
\mathrm{e}^{-n\tr V_{k}(M)} \, \mathrm{d} M$, and the---perturbed---potential $V_{k}(x)$ has a pole of
order $k$ at the origin, $V_{k}(x) \! := \! V(x) \! + \! (t/x)^{k}$, $t \! > \! 0$, with the regular
part, $V$, of the potential being real analytic on $[0,+\infty)$ and satisfying certain constraints;
in particular, for the pLUE, the authors obtain, in various double-scaling limits when the size
of the matrix $n \! \to \! \infty$ (at an appropriately adjusted rate) and the `strength' of the
perturbation $t \! \to \! 0$, asymptotics of the associated eigenvalue correlation kernel and
partition function, which are characterised in terms of special, pole-free solutions of a hierarchy
(indexed by $k$) of higher-order analogues of the Painlev\'{e} III ($\mathrm{P} \mathrm{III}$)
equation: the first ($k \! = \! 1$) member of this $\mathrm{P} \mathrm{III}$ hierarchy, denoted
by $\ell_{1}(s)$, $s \! > \! 0$, solves a re-scaled version of the DP3E~\eqref{eq1.1}. (Analogous
results for the singularly perturbed Gaussian unitary ensemble (pGUE) on the set $\mathscr{H}_{n}$
of $n \times n$ Hermitian matrices are also obtained in \cite{zamol6}.) For the pLUE with
perturbed potential $V_{k}(x) \! := \! V(x) \! + \! (t/x)^{k}$, $k \! \in \! \mathbb{N}$,
$x \! \in \! (0,+\infty)$, $t \! > \! 0$, studied in \cite{zamol6}, the authors of \cite{zamol5}
consider a related Fredholm determinant of an integral operator, denoted by
$\mathcal{K}_{\mathrm{P} \mathrm{III}}$, acting on the space $L^{2}((0,+\infty))$, whose
kernel is constructed {}from a certain $\mathrm{M}_{2}(\mathbb{C})$-valued function
associated with a hierarchy (indexed by $k$) of higher-order analogues of the
$\mathrm{P} \mathrm{III}$ equation; more precisely, for the Fredholm determinant
$F(s;\lambda) \! := \! \ln \det (\mathrm{I} \! - \! \mathcal{K}_{\mathrm{P} \mathrm{III}})$,
$s,\lambda \! > \! 0$, the authors of \cite{zamol5} obtain $s \! \to \! +\infty$ asymptotics
of $F(s;\lambda)$ characterised in terms of an explicit integral representation of a special,
pole-free solution for the first $(k \! = \! 1$) member of the corresponding $\mathrm{P}
\mathrm{III}$ hierarchy: this solution is denoted by $\ell_{1}(\lambda)$, and it solves a
re-scaled version of the DP3E~\eqref{eq1.1}.
\item[\textbf{(v)}] Let $\mathcal{X}$ be a six-dimensional Calabi-Yau (CY) manifold (a
complex K\"{a}hler three-fold with covariantly constant holomorphic three-form
$\Omega$). The Strominger-Yau-Zaslow (SYZ) conjecture (see \cite{DP} for details) states
that, near the large complex structure limit, both $\mathcal{X}$ and its mirror should be
the fibrations over the moduli space of special Lagrangian tori (submanifolds admitting
a unitary flat connection). As an examination of the SYZ conjecture, Loftin-Yau-Zaslow
(LYZ) (see \cite{DP} for details) set out to prove the existence of the metric of Hessian
form $g_{B} \! = \! \tfrac{\partial^{2} \phi}{\partial x^{j} \partial x^{k}} \, \mathrm{d} x^{j}
\otimes \mathrm{d} x^{k}$, where $x^{j}$, $j \! = \! 1,2,3$, are local coordinates on a real
three-dimensional manifold, and $\phi$ (a K\"{a}hler potential) is homogeneous of
degree two in $x^{j}$ and satisfies the real Monge-Amp\'{e}re equation $\det \!
\left(\tfrac{\partial^{2} \phi}{\partial x^{j} \partial x^{k}} \right) \! = \! 1$: LYZ
showed that the construction of the metric is tantamount to searching for solutions
of the definite affine sphere equation (DASE) $\psi_{z \overline{z}} \! + \! \tfrac{1}{2}
\mathrm{e}^{\psi} \! + \! \lvert U \rvert^{2} \mathrm{e}^{-2 \psi} \! = \! 0$, $U_{\overline{z}} \! =
\! 0$, where $\psi$ and $U$ are real- and complex-valued functions, respectively, on
an open subset of $\mathbb{C}$. For $U \! = \! z^{-2}$, LYZ proved the existence of
the radially symmetric solution $\psi$ of the DASE with a prescribed behaviour near
the singularity $z \! = \! 0$, and established the existence of the global solution to
the coordinate-independent version of the DASE on $\mathbb{S}^{2}$ with three
points excised. In \cite{DP}, the authors show that the DASE, and a closely related
equation called the Tzitz\'{e}ica equation, arise as reductions of anti-self-dual
Yang-Mills (ASDYM) system by two translations; moroever, they show that the ODE
characterising its radial solutions give rise to an isomonodromy problem described
by the $\mathrm{P} \mathrm{III}$ equation for special values of its parameters. In
particular (see Proposition~1.3 of \cite{DP}), the authors show that, for $U \! = \!
z^{-2}$, solutions of the DASE that are invariant under the group of rotations
(rotational symmetry) $z \! \to \! \mathrm{e}^{\mathrm{i} \mathfrak{c}}z$, $\mathfrak{c} \! \in
\! \mathbb{R}$, are of the form $\psi (z,\overline{z}) \! = \! \ln (\mathscr{H}(s))
\! - \! 3 \ln (s)$, with $s \! := \! \lvert z \rvert^{1/2}$, where $\mathscr{H}$ solves
the $\mathrm{P} \mathrm{III}$ equation with parameters $(-8,0,0,-16)$, that is,
$\mathscr{H}^{\prime \prime}(s) \! = \! \tfrac{(\mathscr{H}^{\prime}(s))^{2}}{
\mathscr{H}(s)} \! - \! \tfrac{\mathscr{H}^{\prime}(s)}{s} \! - \! \tfrac{8(\mathscr{H}
(s))^{2}}{s} \! - \! \tfrac{16}{\mathscr{H}(s)}$, where the prime denotes differentiation
with respect to $s$, which can be identified as a special reduction of the
DP3E~\eqref{eq1.1} for $a \! = \! 0$. The authors of \cite{DP} demonstrate that
the existence theorem for Hessian metrics with prescribed monodromy reduces to
the study of the $\mathrm{P} \mathrm{III}$ equation with parameters $(-8,0,0,-16)$,
that is, a class of semi-flat CY metrics is obtained in terms of real solutions of the
DP3E~\eqref{eq1.1} for $a \! = \! 0$.
\item[\textbf{(vi)}] In \cite{peetsdd}, the author introduces affine spheres as immersions
of a manifold $\mathcal{M}$ as a hypersurface in $\mathbb{R}^{n}$ with certain
properties and defines the affine metric $h$ and the cubic form $C$ on $\mathcal{M}$.
By identifying, for $3$-dimensional cones and, correspondingly, affine $2$-spheres,
the manifold $\mathcal{M}$ with a non-compact, simply-connected domain in
$\mathbb{C}$, one can introduce complex isothermal co-ordinates $z$ on
$\mathcal{M}$, in terms of which the affine metric $h$ may equivalently be described
by a real conformal factor $u(z)$ and the cubic form $C$ by a holomorphic function
$U(z)$ on $\mathcal{M}$, the relations being $h \! = \! \mathrm{e}^{u} \lvert \mathrm{d} z
\rvert^{2}$ and $C \! = \! 2 \Re (U(z)) \mathrm{d} z^{3}$: the compatibility condition of
the pair $(u,U)$ is referred to as \emph{Wang's equation}, $\mathrm{e}^{u} \! = \!
\tfrac{1}{2} \Delta u \! + \! 2 \lvert U \rvert^{2} \mathrm{e}^{-2u}$, where $\Delta u
\! = \! u_{xx} \! + \! u_{yy} \! = \! 4u_{z \overline{z}}$ is the Laplacian of $u$,
$\partial_{z} \! := \! \tfrac{1}{2}(\partial_{x} \! - \! \mathrm{i} \partial_{y})$, and
$\partial_{\overline{z}} \! := \! \tfrac{1}{2}(\partial_{x} \! + \! \mathrm{i} \partial_{y})$.
By classifying pairs $(\psi,U)$, where $\psi$ is a vector field on $\mathcal{M}$
generating a one-parameter group of conformal automorphisms on $\mathcal{M}$
which multiply $U$ by unimodular complex constants, the author finds, for every
pair $(\psi,U)$, a unique solution $u$ of Wang's equation such that the corresponding
affine metric $h$ is complete on $\mathcal{M}$ and $\psi$ is a Killing vector field
for $h$: this latter property permits Wang's equation to be reduced to a second-order
non-linear ODE that is equivalent to the DP3E~\eqref{eq1.1}, a detailed qualitative
study for which is presented in Section~5 and Appendix~A of \cite{peetsdd}. The
author presents a complete classification of self-associated cones (one calls a cone
self-associated if it is linearly isomorphic to all its associated cones, with two cones
said to be associated with each other if the Blaschke metrics on the corresponding
affine spheres are related by an orientation-preserving isometry) and computes
isothermal parametrisations of the corresponding affine spheres, the solution(s)
of which can be expressed in terms of degenerate $\mathrm{P} \mathrm{III}$
transcendents (solutions of the DP3E~\eqref{eq1.1}).
\end{enumerate}
An effectual approach for studying the asymptotic behaviour of solutions (in
particular, the connection formulae for their asymptotics) of the Painlev\'{e}
equations $\mathrm{P} \mathrm{I},\dotsc,\mathrm{P} \mathrm{VI}$ is the
Isomonodromic Deformation Method (IDM) \cite{a5,a18,a2,a22,sachokap3}:
specific features of the IDM as applied, in particular, to the DP3E~\eqref{eq1.1}
can be located in Sections~1 and~2 of \cite{a1}. It is imperative, within the IDM
context, to mention the seminal r\^{o}le played by the recent monograph
\cite{a5}, as it summarizes and reflects not only the key technical and theoretical
developments and advances of the IDM since the appearance of \cite{a2}, but
also of an equivalent, technically distinct approach based on the Deift-Zhou
non-linear steepest descent analysis of the associated RHP \cite{a6}. The
methodological paradigm adopted in this paper is the IDM. Even though the
DP3E~\eqref{eq1.1} resembles one of the canonical, non-degenerate variants
of the Painlev\'{e} equations $\mathrm{P} \mathrm{I},\dotsc,\mathrm{P}
\mathrm{VI}$, the associated asymptotic analysis of its solutions via the IDM
subsumes additional technical complications, due to the necessity of having to
extract the explicit functional dependencies of the contributing error terms,
rather than merely estimating them, which requires a considerably more detailed
study of the error functions. By studying the isomonodromic deformations of a
$3 \! \times \! 3$ matrix linear ODE (see, also, Section~8 of \cite{DP}) with two
irregular singular points, asymptotics as $\tau \! \to \! \infty$ and as $\tau \!
\to \! 0$ of solutions to the DP3E~\eqref{eq1.1} for the case $a \! = \! 0$, as
well as the corresponding connection formulae, were obtained in \cite{aveekit}
via the IDM.\footnote{Note that the DP3E~\eqref{eq1.1} has two singular points: an
irregular one at the point at infinity and a regular one at the origin.} As observed
in \cite{avaev}, though, there is an alternative $2 \! \times \! 2$ matrix linear
ODE whose isomonodromy deformations are described, for arbitrary $a \! \in \!
\mathbb{C}$, by the DP3E~\eqref{eq1.1}: it is this latter $2 \! \times \! 2$ ODE
system that is adopted in the present work.
In order to eschew a flood of superfluous notation and to motivate, in as succinct a
manner as possible, the qualitative behaviour of the tronqu\'{e}e solution of the
DP3E~\eqref{eq1.1} that the reader will encounter in this work, consider, for example,
asymptotics as $\tau \! \to \! +\infty$ with $\varepsilon b \! > \! 0$ of $u(\tau)$.
As is well known \cite{afvkav,edelere,a5,mlaud,shimo2,shimo,shimo3,xxia}, the
Painlev\'{e} equations admit a one-parameter family of trans-series solutions of the
form ``(power series) $+$ (exponentially small terms)''. As argued in Section~\ref{sec3}
below, $u(\tau)$ admits the `complete' asymptotic trans-series representation $u(\tau)
\! =_{\tau \to +\infty} \! c_{0,k}(\tau^{1/3} \! + \! v_{0,k}(\tau))$, $k \! \in \! \lbrace
\pm 1 \rbrace$,\footnote{The significance of the integer index $k$ and its relation to
the monodromy manifold is discussed in Subsection~\ref{sec1e} below.} where $c_{0,k} \!
:= \! \tfrac{1}{2} \varepsilon (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}$, and $v_{0,k}
(\tau) \! := \! \tau^{-1/3} \mathrm{u}_{{\scriptscriptstyle \mathrm{R},k}}(\tau) \! + \!
\mathrm{u}_{{\scriptscriptstyle \mathrm{E},k}}(\tau)$, with $\mathbb{C} \llbracket
\tau^{-1/3} \rrbracket \! \ni \! \mathrm{u}_{{\scriptscriptstyle \mathrm{R},k}}(\tau)
\! = \! \sum_{n=0}^{\infty} \upsilon_{n,k}({\scriptscriptstyle \mathscr{M}})(\tau^{-1/3})^{n}$
and $\mathrm{u}_{{\scriptscriptstyle \mathrm{E},k}}(\tau) \! = \! \sum_{m=1}^{\infty}
\sum_{j=0}^{\infty} \mathfrak{v}_{m,j,k}({\scriptscriptstyle \mathscr{M}})(\tau^{-1/3})^{j}
(\mathrm{e}^{-\frac{3 \sqrt{\smash[b]{3}}}{2}(\sqrt{\smash[b]{3}}+ \mathrm{i} k)(\varepsilon b)^{1/3}
\tau^{2/3}})^{m}$,\footnote{Note that $\mathrm{u}_{{\scriptscriptstyle \mathrm{R},k}}
(\tau)$, $k \! \in \! \lbrace \pm 1 \rbrace$, are divergent series.} and where the
monodromy-data-dependent expansion coefficients, $\upsilon_{n,k}({\scriptscriptstyle \mathscr{M}})$
and $\mathfrak{v}_{m,j,k}({\scriptscriptstyle \mathscr{M}})$, can be determined recursively
provided that certain leading coefficients are known \emph{a priori}. The purpose of the
present work, though, is not to address the complete asymptotic trans-series representation
stated above, but, rather, to determine the coefficient of the leading-order exponentially
small correction term to the asymptotics of solutions of the DP3E~\eqref{eq1.1}, which is,
to the best of the author's knowledge as at the time of the presents, the decidedly non-trivial
task within the IDM paradigm, in which case, the asymptotic trans-series representation for
$u(\tau)$ reads
\begin{equation} \label{mainyoo}
u(\tau) \underset{\tau \to +\infty}{=} c_{0,k} \! \left(\tau^{1/3} \! + \!
\sum_{m=0}^{\infty} \dfrac{\mathfrak{u}_{m}(k)}{(\tau^{1/3})^{m+1}} \! +
\! \mathrm{A}_{k} \mathrm{e}^{-\frac{3 \sqrt{\smash[b]{3}}}{2}(\sqrt{\smash[b]{3}}
+\mathrm{i} k)(\varepsilon b)^{1/3} \tau^{2/3}}(1 \! + \! \mathcal{O}(\tau^{-1/3}))
\right), \quad k \! \in \! \lbrace \pm 1 \rbrace.
\end{equation}
While the expansion coefficients $\lbrace \mathfrak{u}_{m}(k) \rbrace_{m=0}^{\infty}$,
$k \! \in \! \lbrace \pm 1 \rbrace$, can be determined (not always uniquely!)
by substituting the trans-series representation~\eqref{mainyoo} into the
DP3E~\eqref{eq1.1} and solving, iteratively, a system of non-linear recurrence
relations for the $\mathfrak{u}_{m}(k)$'s, the monodromy-data-dependent
expansion coefficients, $\mathrm{A}_{k}$, $k \! \in \! \lbrace \pm 1 \rbrace$,
can not, and must, therefore, be determined independently; in fact, the principal
technical accomplishment of this work is the determination, via the IDM, of the
explicit functional dependence of the coefficients $\mathrm{A}_{k}$, $k \! \in \!
\lbrace \pm 1 \rbrace$, on the Stokes multiplier $s_{0}^{0}$ (see, in particular,
Section~\ref{finalsec}, Equations~\eqref{geek109} and~\eqref{geek111}, below).
Even though the motivational discussion above for the introduction of the
monodromy-data-dependent expansion coefficients $\mathrm{A}_{k}$, $k \! \in
\! \lbrace \pm 1 \rbrace$, relies on asymptotics of $u(\tau)$ as $\tau \! \to \!
+\infty$ for $\varepsilon b \! > \! 0$, it must be emphasized that, in this work,
the coefficients $\mathrm{A}_{k}$, $k \! \in \! \lbrace \pm 1 \rbrace$, and their
analogues, corresponding to trans-series asymptotics of $u(\tau)$, the associated
Hamiltonian and principal auxiliary functions, and one of the $\sigma$-forms of the
DP3E~\eqref{eq1.1} as $\tau \! \to \! +\infty \mathrm{e}^{\mathrm{i} \pi \varepsilon_{1}}$ for
$\varepsilon b \! = \! \lvert \varepsilon b \rvert \mathrm{e}^{\mathrm{i} \pi \varepsilon_{2}}$,
$\varepsilon_{1},\varepsilon_{2} \! \in \! \lbrace 0,\pm 1 \rbrace$, and as $\tau
\! \to \! +\infty \mathrm{e}^{\mathrm{i} \pi \hat{\varepsilon}_{1}/2}$ for $\varepsilon b \! = \!
\lvert \varepsilon b \rvert \mathrm{e}^{\mathrm{i} \pi \hat{\varepsilon}_{2}}$, $\hat{\varepsilon}_{1}
\! \in \! \lbrace \pm 1 \rbrace$ and $\hat{\varepsilon}_{2} \! \in \! \lbrace 0,\pm 1
\rbrace$, are obtained (see, in particular, Section~\ref{sec2}, Theorems~\ref{theor2.1}
and~\ref{appen}, respectively, below).\footnote{The `complete' asymptotic trans-series
representations, which require explicit knowledge of, and are premised on, the
monodromy-data-dependent expansion coefficients, $\mathrm{A}_{k}$, $k \! \in \!
\lbrace \pm 1 \rbrace$, are presently under consideration, and will be presented
elsewhere.}
\begin{eeee} \label{remlndaip34}
In the seminal work \cite{ylddpt}, the authors consider, in particular, the existence and
uniqueness of tronqu\'{e}e solutions of the $\mathrm{P} \mathrm{III}$ equation with
parameters $(1,\beta,0,-1)$, denoted by $\mathrm{P}^{(\mathrm{ii})}_{\mathrm{III}}$ in
Equation~(1.5) of \cite{ylddpt}: $v^{\prime \prime}(x) \! = \! \tfrac{(v^{\prime}(x))^{2}}{v(x)}
\! - \! \tfrac{v^{\prime}(x)}{x} \! + \! \tfrac{1}{x}((v(x))^{2} \! + \! \beta) \! - \! \tfrac{1}{v(x)}$,
where $\mathbb{C} \! \ni \! \beta$ is arbitrary; $\mathrm{P}^{(\mathrm{ii})}_{\mathrm{III}}$
can be derived {}from the DP3E~\eqref{eq1.1} via the mapping $\mathscr{S}_{\scriptscriptstyle
\varepsilon} \colon (\tau,u(\tau),a,b) \! \to \! (\alpha x,\gamma v(x),\tfrac{\beta}{2}
\mathrm{e}^{-\mathrm{i} (2m+1) \pi/2},b)$, $\varepsilon \! = \! \pm 1$, $m \! = \! 0,1$, where $\alpha
\! := \! 2^{-3/2}b^{-1/2} \mathrm{e}^{\mathrm{i} (2+ \varepsilon) \pi/4} \mathrm{e}^{\mathrm{i} (2m^{\prime}+m) \pi/2}$,
and $\gamma \! := \! -\varepsilon 2^{-3/2}b^{1/2} \mathrm{e}^{-\mathrm{i} (2+ \varepsilon) \pi/4}
\mathrm{e}^{-\mathrm{i} (2m^{\prime}+m) \pi/2}$, $m^{\prime} \! = \! 0,1$. In Theorem~2 of \cite{ylddpt},
the authors prove that, in any open sector of angle less than $3 \pi/2$, there exist one-parameter
solutions of $\mathrm{P}^{(\mathrm{ii})}_{\mathrm{III}}$ with asymptotic expansion $v(x) \!
\sim \! v_{f}^{(m_{1})}(x) \! := \! x^{1/3} \sum_{n=0}^{\infty}a_{n}^{(m_{1})}(x^{-2/3})^{n}$
for $S_{k}^{(m_{1})} \! \ni \! x \! \to \! \infty$, $m_{1} \! = \! 0,1,2$, where the sectors
$S_{k}^{(m_{1})}$, $k \! = \! 0,1,2,3$, are defined in Equation~(1.10) of \cite{ylddpt},
$a_{0}^{(m_{1})} \! := \! \exp (\mathrm{i} 2 \pi m_{1}/3)$, and the ($x$-independent) coefficients
$a_{n}^{(m_{1})}$, $n \! \in \! \mathbb{N}$, solve the recursion relations~(1.12) of \cite{ylddpt};
moreover, the authors prove that, for any branch of $x^{1/3}$, there exists a unique solution
of $\mathrm{P}^{(\mathrm{ii})}_{\mathrm{III}}$ in $\mathbb{C} \setminus \pmb{\leftthreetimes}$
with asymptotic expansion $v_{f}^{(m_{1})}(x)$, where $\pmb{\leftthreetimes}$ is an
arbitrary branch cut connecting the singular points $0$ and $\infty$ (they also address
the existence of the exponentially small correction term(s) of the tronqu\'{e}e solution of
$\mathrm{P}^{(\mathrm{ii})}_{\mathrm{III}}$). This crucially important result of \cite{ylddpt},
in conjunction with the invertibility of the mapping $\mathscr{S}_{\scriptscriptstyle \varepsilon}$,
implies the existence and uniqueness of the asymptotic (as $\tau \! \to \! +\infty$ with
$\varepsilon b \! > \! 0$) trans-series representation~\eqref{mainyoo}. \hfill $\blacksquare$
\end{eeee}
\begin{eeee} \label{integrem}
The results of this work, in conjunction with those of \cite{a1,av2}, will be applied in
an upcoming series of studies on uniform asymptotics of integrals of solutions to the
DP3E~\eqref{eq1.1} and related functions: for the monodromy data considered in
\cite{avlkv}, preliminary $\tau \! \to \! +\infty$ asymptotics for $\varepsilon b \!
> \! 0$ have been presented in \cite{avkavint}. \hfill $\blacksquare$
\end{eeee}
\subsection{Hamiltonian Structure, Auxiliary Functions, and the $\sigma$-Form}
\label{sec1b}
Herewith follows a brief synopsis of select results {}from \cite{a1} that are relevant
for the present work; for complete details, see, in particular, Sections~1, 2, and~6
of \cite{a1}, and \cite{avkavint}.
An important formal property of the DP3E~\eqref{eq1.1} is its associated Hamiltonian
structure; in fact, as shown in Proposition~1.3 of \cite{a1}, upon setting
\begin{equation} \label{hamk1}
\mathcal{H}_{\epsilon_{1}}(\hat{p}(\tau),\hat{q}(\tau);\tau) \! := \! (\hat{p}(\tau)
\hat{q}(\tau))^{2} \tau^{-1} \! - \! 2 \epsilon_{1} \hat{p}(\tau) \hat{q}(\tau)(\mathrm{i}
a \! + \! 1/2) \tau^{-1} \! + \! 4 \varepsilon \hat{q}(\tau) \! + \! \mathrm{i} b \hat{p}
(\tau) \! + \! \frac{1}{2 \tau}(\mathrm{i} a \! + \! 1/2)^{2},
\end{equation}
where the functions $\hat{p}(\tau)$ and $\hat{q}(\tau)$ are the generalised impulse
and co-ordinate, respectively, $\epsilon_{1} \! \in \! \lbrace \pm 1 \rbrace$, and
$\epsilon_{1}^{2} \! = \! \varepsilon^{2} \! = \! 1$, Hamilton's equations, that is,
\begin{equation} \label{hamk2}
\hat{p}^{\prime}(\tau) \! = \! -\dfrac{\partial \mathcal{H}_{\epsilon_{1}}(\hat{p}(\tau),
\hat{q}(\tau);\tau)}{\partial \hat{q}} \, \qquad \, \text{and} \, \qquad \, \hat{q}^{\prime}
(\tau) \! = \! \dfrac{\partial \mathcal{H}_{\epsilon_{1}}(\hat{p}(\tau),\hat{q}(\tau);
\tau)}{\partial \hat{p}},
\end{equation}
are equivalent to either one of the degenerate $\mathrm{P} \mathrm{III}$ equations
\begin{gather}
\hat{p}^{\prime \prime}(\tau) \! = \! \dfrac{(\hat{p}^{\prime}(\tau))^{2}}{\hat{p}(\tau)}
\! - \! \dfrac{\hat{p}^{\prime}(\tau)}{\tau} \! + \! \dfrac{1}{\tau} \! \left(-\mathrm{i} 2b(\hat{p}
(\tau))^{2} \! + \! 8 \varepsilon (\mathrm{i} a \epsilon_{1} \! + \! (\epsilon_{1} \! - \! 1)/2)
\right) \! - \! \dfrac{16}{\hat{p}(\tau)}, \label{hamk3} \\
\hat{q}^{\prime \prime}(\tau) \! = \! \dfrac{(\hat{q}^{\prime}(\tau))^{2}}{\hat{q}(\tau)}
\! - \! \dfrac{\hat{q}^{\prime}(\tau)}{\tau} \! + \! \dfrac{1}{\tau} \! \left(-8 \varepsilon
(\hat{q}(\tau))^{2} \! - \! b(2a \epsilon_{1} \! - \! \mathrm{i} (1 \! + \! \epsilon_{1})) \right)
\! + \! \dfrac{b^{2}}{\hat{q}(\tau)}: \label{hamk4}
\end{gather}
it was also noted during the proof of the above-mentioned result that the Hamiltonian
System~\eqref{hamk2} can be re-written as
\begin{equation} \label{hamk5}
\hat{p}(\tau) \! = \! \dfrac{\tau (\hat{q}^{\prime}(\tau) \! - \! \mathrm{i} b)}{2(\hat{q}
(\tau))^{2}} \! + \! \dfrac{\epsilon_{1}(\mathrm{i} a \! + \! 1/2)}{\hat{q}(\tau)} \qquad
\text{and} \, \qquad \, \hat{q}(\tau) \! = \! -\dfrac{\tau (\hat{p}^{\prime}(\tau)
\! + \! 4 \varepsilon)}{2(\hat{p}(\tau))^{2}} \! + \! \dfrac{\epsilon_{1}(\mathrm{i} a \!
+ \! 1/2)}{\hat{p}(\tau)}.
\end{equation}
As shown in Section~2 of \cite{a1}, the \emph{Hamiltonian function}, $\mathcal{H}
(\tau)$, is defined as follows:
\begin{equation} \label{hamk10}
\mathcal{H}(\tau) \! := \! \left. \mathcal{H}_{\epsilon_{1}}(\hat{p}(\tau),\hat{q}
(\tau);\tau) \right\vert_{\epsilon_{1}=-1},
\end{equation}
where $\hat{p}(\tau)$ is calculated {}from the first (left-most) relation of
Equations~\eqref{hamk5} with $\hat{q}(\tau) \! = \! u(\tau)$; moreover, as shown
in Section~2 of \cite{a1}, the Definition~\eqref{hamk10} implies the following
explicit representation for $\mathcal{H}(\tau)$ in terms of $u(\tau)$:
\begin{equation} \label{eqh1}
\mathcal{H}(\tau) \! := \! (a \! - \! \mathrm{i}/2) \dfrac{b}{u(\tau)} \! + \! \dfrac{1}{2 \tau}
(a \! - \! \mathrm{i}/2)^{2} \! + \! \dfrac{\tau}{4(u(\tau))^{2}} \! \left((u^{\prime}(\tau))^{2}
\! + \! b^{2} \right) \! + \! 4 \varepsilon u(\tau).
\end{equation}
It was shown in Section~1 of \cite{a1} that the function $\sigma (\tau)$ defined by
\begin{align} \label{hamk7}
\sigma (\tau) \! :=& \, \tau \mathcal{H}_{\epsilon_{1}}(\hat{p}(\tau),\hat{q}(\tau);
\tau) \! + \! \hat{p}(\tau) \hat{q}(\tau) \! + \! \frac{1}{2}(\mathrm{i} a \! + \! 1/2)^{2} \!
- \! \epsilon_{1} (\mathrm{i} a \! + \! 1/2) \! + \! \frac{1}{4} \nonumber \\
=& \, \left(\hat{p}(\tau) \hat{q}(\tau) \! - \! \epsilon_{1}(\mathrm{i} a \! + \! (1 \! - \!
\epsilon_{1})/2) \right)^{2} \! + \! \tau (4 \varepsilon \hat{q}(\tau) \! + \! \mathrm{i} b
\hat{p}(\tau))
\end{align}
satisfies the second-order non-linear ODE (related to the DP3E~\eqref{eq1.1})
\begin{equation} \label{hamk9}
(\tau \sigma^{\prime \prime}(\tau) \! - \! \sigma^{\prime}(\tau))^{2} \! = \! 2
(2 \sigma (\tau) \! - \! \tau \sigma^{\prime}(\tau))(\sigma^{\prime}(\tau))^{2}
\! - \! \mathrm{i} 32 \varepsilon b \tau \left(((1 \! - \! \epsilon_{1})/2 \! - \! \mathrm{i} a
\epsilon_{1}) \sigma^{\prime}(\tau) \! + \! \mathrm{i} 2 \varepsilon b \tau \right).
\end{equation}
Equation~\eqref{hamk9} is referred to as the $\sigma$-form of the
DP3E~\eqref{eq1.1}. Motivated by the Definition~\eqref{hamk10} for the
Hamiltonian function, setting $\epsilon_{1} \! = \! -1$, letting the generalised
co-ordinate $\hat{q}(\tau) \! = \! u(\tau)$, and using the first (left-most)
relation of Equations~\eqref{hamk5} to calculate the generalised impulse,
it suffices, for the purposes of the present work, to define the function (cf.
Definition~\eqref{hamk7}) $\sigma (\tau)$ and the second-order non-linear
ODE it satisfies as follows:
\begin{equation} \label{thmk23}
\sigma (\tau) \! := \! \tau \mathcal{H}(\tau) \! + \! \dfrac{\tau (u^{\prime}
(\tau) \! - \! \mathrm{i} b)}{2u(\tau)} \! + \! \frac{1}{2}(\mathrm{i} a \! + \! 1/2)^{2} \! + \!
\frac{1}{4},
\end{equation}
and
\begin{equation} \label{thmk22}
(\tau \sigma^{\prime \prime}(\tau) \! - \! \sigma^{\prime}(\tau))^{2} \! = \! 2
(2 \sigma (\tau) \! - \! \tau \sigma^{\prime}(\tau))(\sigma^{\prime}(\tau))^{2}
\! - \! \mathrm{i} 32 \varepsilon b \tau ((1 \! + \! \mathrm{i} a) \sigma^{\prime}(\tau) \! +
\! \mathrm{i} 2 \varepsilon b \tau).
\end{equation}
Via the B\"{a}cklund transformations given in Subsection~6.1 of \cite{a1}, let
\begin{gather}
u_{-}(\tau) \! := \! \frac{\mathrm{i} \varepsilon b}{8(u(\tau))^{2}} \! \left(\tau
(u^{\prime}(\tau) \! - \! \mathrm{i} b) \! + \! (1 \! - \! \mathrm{i} 2 a_{-})u(\tau) \right),
\label{yoominus1} \\
u_{+}(\tau) \! := \! -\frac{\mathrm{i} \varepsilon b}{8(u(\tau))^{2}} \! \left(\tau
(u^{\prime}(\tau) \! + \! \mathrm{i} b) \! + \! (1 \! + \! \mathrm{i} 2 a_{+})u(\tau) \right),
\label{yooplus1}
\end{gather}
where $u(\tau)$ denotes any solution of the DP3E~\eqref{eq1.1}, and
$a_{\pm} \! := \! a \! \pm \! \mathrm{i}$; in fact, as shown in Subsection~6.1 of
\cite{a1}, $u_{-}(\tau)$ (resp., $u_{+}(\tau)$) solves the DP3E~\eqref{eq1.1}
for $a \! = \! a_{-}$ (resp., $a \! = \! a_{+}$). {}From the results of
\cite{avkavint}, define the two \emph{principal auxiliary functions}
\begin{gather}
f_{-}(\tau) \! := \! -\frac{\mathrm{i} 2}{\varepsilon b}u(\tau)u_{-}(\tau), \label{eqf} \\
f_{+}(\tau) \! := \! u(\tau)u_{+}(\tau), \label{yooplus2}
\end{gather}
where $f_{-}(\tau)$ solves the second-order non-linear ODE {}\footnote{This is
a consequence of the ODE for the function $f(\tau)$ presented on p.~1168 of
\cite{a1} upon making the notational change $f(\tau) \! \to \! f_{-}(\tau)$ and
setting $\epsilon_{1} \! = \! -1$.}
\begin{equation} \label{thmk13}
\tau^{2} \! \left(f_{-}^{\prime \prime}(\tau) \! + \! \mathrm{i} 4 \varepsilon b \right)^{2}
\! - \! \left(4f_{-}(\tau) \! + \! \mathrm{i} 2 a \! + \! 1 \right)^{2} \left((f_{-}^{\prime}
(\tau))^{2} \! + \! \mathrm{i} 8 \varepsilon b f_{-}(\tau) \right) \! = \! 0,
\end{equation}
and $f_{+}(\tau)$ solves the second-order non-linear ODE {}\footnote{See
Equation~(2) in \cite{avkavint}.}
\begin{equation} \label{yooplus3}
(\varepsilon b \tau)^{2} \! \left(f_{+}^{\prime \prime}(\tau) \! - \! 2
(\varepsilon b)^{2} \right)^{2} \! + \! \left(8f_{+}(\tau) \! + \! \mathrm{i} \varepsilon
b (\mathrm{i} 2 a \! - \! 1) \right)^{2} \left((f_{+}^{\prime}(\tau))^{2} \! - \!
4(\varepsilon b)^{2}f_{+}(\tau) \right) \! = \! 0.
\end{equation}
It follows {}from the Definitions~\eqref{yoominus1}--\eqref{yooplus2} that the
functions $f_{\pm}(\tau)$ possess the alternative representations
\begin{gather}
2f_{-}(\tau) \! = \! -\mathrm{i} (a \! - \! \mathrm{i}/2) \! + \! \frac{\tau (u^{\prime}(\tau)
\! - \! \mathrm{i} b)}{2u(\tau)}, \label{yoominus3} \\
\frac{\mathrm{i} 4}{\varepsilon b}f_{+}(\tau) \! = \! \mathrm{i} (a \! + \! \mathrm{i}/2) \! + \!
\frac{\tau (u^{\prime}(\tau) \! + \! \mathrm{i} b)}{2u(\tau)}; \label{yooplus4}
\end{gather}
incidentally, Equations~\eqref{yoominus3} and~\eqref{yooplus4} imply the corollary
\begin{equation} \label{yoominus4}
\frac{\mathrm{i} 4}{\varepsilon b}f_{+}(\tau) \! = \! 2f_{-}(\tau) \! + \! \mathrm{i} \tau \! \left(
\frac{2a}{\tau} \! + \! \frac{b}{u(\tau)} \right).
\end{equation}
For the monodromy data considered in \cite{avlkv}, preliminary asymptotics
as $\tau \! \to \! +\infty$ with $\varepsilon b \! > \! 0$ for $\int_{0}^{\tau}
\xi^{-1}f_{+}(\xi) \, \mathrm{d} \xi$ have been presented in \cite{avkavint}.
\subsection{Lax Pairs and Isomonodromic Deformations} \label{sec1c}
In this subsection, the reader is reminded about some basic facts regarding
the isomonodromy deformation theory for the DP3E~\eqref{eq1.1}.
\begin{eeee} \label{pregauge}
Pre-gauge-transformed Lax-pair-associated functions are denoted with `hats', whilst
post-gauge-transformed Lax-pair-associated functions are not; in some cases, these
functions are equal, and in others, they are not (see the discussion below). \hfill
$\blacksquare$
\end{eeee}
The study of the DP3E~\eqref{eq1.1} is based on the following pre-gauge-transformed
Fuchs-Garnier, or Lax, pair (see Proposition~2.1 of \cite{a1}, with notational amendments):
\begin{equation} \label{eqFGmain}
\partial_{\mu} \widehat{\Psi}(\mu,\tau) \! = \! \widehat{\mathscr{U}}(\mu,
\tau) \widehat{\Psi}(\mu,\tau), \qquad \qquad \partial_{\tau} \widehat{\Psi}
(\mu,\tau) \! = \! \widehat{\mathscr{V}}(\mu,\tau) \widehat{\Psi}(\mu,\tau),
\end{equation}
where
\begin{gather}
\widehat{\mathscr{U}}(\mu,\tau) \! = \! -\mathrm{i} 2 \tau \mu \sigma_{3} \! + \!
2 \tau \!
\begin{pmatrix}
0 & \frac{\mathrm{i} 2 \hat{A}(\tau)}{\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}} \\
-\hat{D}(\tau) & 0
\end{pmatrix}
\! - \! \dfrac{1}{\mu} \! \left(\mathrm{i} a \! + \! \dfrac{1}{2} \! + \! \dfrac{2 \tau
\hat{A}(\tau) \hat{D}(\tau)}{\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}}
\right) \! \sigma_{3} \! + \! \dfrac{1}{\mu^{2}} \!
\begin{pmatrix}
0 & \hat{\alpha}(\tau) \\
\mathrm{i} \tau \hat{B}(\tau) & 0
\end{pmatrix}, \label{mpeea1} \\
\widehat{\mathscr{V}}(\mu,\tau) \! = \! -\mathrm{i} \mu^{2} \sigma_{3} \! + \! \mu \!
\begin{pmatrix}
0 & \frac{\mathrm{i} 2 \hat{A}(\tau)}{\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}} \\
-\hat{D}(\tau) & 0
\end{pmatrix}
\! + \! \left(\dfrac{\mathrm{i} a}{2 \tau} \! - \! \dfrac{\hat{A}(\tau) \hat{D}(\tau)}{
\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}} \right) \! \sigma_{3} \! - \!
\frac{1}{\mu} \frac{1}{2 \tau} \!
\begin{pmatrix}
0 & \hat{\alpha}(\tau) \\
\mathrm{i} \tau \hat{B}(\tau) & 0
\end{pmatrix}, \label{mpeea2}
\end{gather}
with $\sigma_{3} \! = \! \diag (1,-1)$,
\begin{equation} \label{firstintegral}
\hat{\alpha}(\tau) \! := \! -2(\hat{B}(\tau))^{-1} \! \left(\mathrm{i} a \sqrt{
\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}} + \! \tau (\hat{A}(\tau) \hat{D}
(\tau) \! + \! \hat{B}(\tau) \hat{C}(\tau)) \right),
\end{equation}
and where the differentiable, scalar-valued functions $\hat{A}(\tau)$, $\hat{B}(\tau)$,
$\hat{C}(\tau)$, and $\hat{D}(\tau)$ satisfy the system of isomonodromy deformations
\begin{equation} \label{eq1.4}
\begin{gathered}
\hat{A}^{\prime}(\tau) \! = \! 4 \hat{C}(\tau) \sqrt{\smash[b]{-\hat{A}(\tau)
\hat{B}(\tau)}}, \, \qquad \, \quad \, \hat{B}^{\prime}(\tau) \! = \! -4 \hat{D}
(\tau) \sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}, \\
(\tau \hat{C}(\tau))^{\prime} \! = \! \mathrm{i} 2 a \hat{C}(\tau) \! - \! 2 \tau \hat{A}
(\tau), \, \quad \, \qquad \, (\tau \hat{D}(\tau))^{\prime} \! = \! -\mathrm{i} 2a
\hat{D}(\tau) \! + \! 2 \tau \hat{B}(\tau), \\
\left(\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}} \, \right)^{\prime} \! = \!
2(\hat{A}(\tau) \hat{D}(\tau) \! - \! \hat{B}(\tau) \hat{C}(\tau)).
\end{gathered}
\end{equation}
(Note: the isomonodromy deformations~\eqref{eq1.4} are, for arbitrary values
of $\mu \! \in \! \mathbb{C}$, the Frobenius compatibility condition for the
System~\eqref{eqFGmain}.)
\begin{eeee} \label{alphwave}
In fact, $-\mathrm{i} \hat{\alpha}(\tau) \hat{B}(\tau) \! = \! \varepsilon b$, $\varepsilon
\! = \! \pm 1$, so that the Definition~\eqref{firstintegral} is the First Integral of
System~\eqref{eq1.4} (see Lemma~{\rm 2.1} of \cite{a1}, with notational
amendments). \hfill $\blacksquare$
\end{eeee}
\begin{eeee} \label{xalp}
With conspicuous changes in notation (cf. System~(4) in \cite{a1}), whilst transforming
{}from the original Lax pair
\begin{gather*}
\partial_{\lambda} \Phi (\lambda,\tau) \! = \! \tau \! \left(-\mathrm{i} \sigma_{3} \! - \!
\dfrac{1}{\lambda} \dfrac{\mathrm{i} a}{2 \tau} \sigma_{3} \! - \! \dfrac{1}{\lambda} \!
\begin{pmatrix}
0 & \hat{C}(\tau) \\
\hat{D}(\tau) & 0
\end{pmatrix} \! + \! \dfrac{1}{\lambda^{2}} \dfrac{\mathrm{i}}{2} \!
\begin{pmatrix}
\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}} & \hat{A}(\tau) \\
\hat{B}(\tau) & -\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}
\end{pmatrix} \right) \! \Phi (\lambda,\tau), \\
\partial_{\tau} \Phi (\lambda,\tau) \! = \! \left(-\mathrm{i} \lambda \sigma_{3}
\! + \! \dfrac{\mathrm{i} a}{2 \tau} \sigma_{3} \! - \!
\begin{pmatrix}
0 & \hat{C}(\tau) \\
\hat{D}(\tau) & 0
\end{pmatrix} \! - \! \dfrac{1}{\lambda} \dfrac{\mathrm{i}}{2} \!
\begin{pmatrix}
\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}} & \hat{A}(\tau) \\
\hat{B}(\tau) & -\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}
\end{pmatrix} \right) \! \Phi (\lambda,\tau),
\end{gather*}
to the Fuchs-Garnier pair~\eqref{eqFGmain}, the Fabry-type transformation
(cf.~Proposition~2.1 in \cite{a1})
\begin{equation*}
\lambda \! = \! \mu^{2} \qquad \text{and} \qquad \Phi (\lambda,\tau) \!
:= \! \sqrt{\smash[b]{\mu}}
\left(
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix} \! + \! \dfrac{1}{\mu} \!
\begin{pmatrix}
0 & -\frac{\hat{A}(\tau)}{\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}} \\
0 & 1
\end{pmatrix} \right) \! \widehat{\Psi}(\mu,\tau)
\end{equation*}
was used; if, instead, one applies the slightly more general transformation
\begin{equation*}
\Phi (\lambda,\tau) \! := \! \sqrt{\smash[b]{\mu}} \!
\left(
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix} \! + \! \dfrac{1}{\mu} \!
\begin{pmatrix}
-\frac{\hat{A}(\tau) \mathbb{P}^{\ast}}{\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}}
& -\frac{\hat{A}(\tau)}{\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}} \\
\mathbb{P}^{\ast} & 1
\end{pmatrix} \right) \! \widehat{\Psi}(\mu,\tau)
\end{equation*}
for some constant or $\tau$-dependent $\mathbb{P}^{\ast}$, then, in lieu of,
say, the $\mu$-part of the Fuchs-Garnier pair~\eqref{eqFGmain}, that is,
$\partial_{\mu} \widehat{\Psi}(\mu,\tau) \! = \! \widehat{\mathscr{U}}
(\mu,\tau) \widehat{\Psi}(\mu,\tau)$, one arrives at
\begin{equation*}
\partial_{\mu} \widehat{\Psi}(\mu,\tau) \! = \! \left(\hat{\mathcal{L}}_{-1}
\mu \! + \! \hat{\mathcal{L}}_{0} \! + \! \hat{\mathcal{L}}_{1} \mu^{-1} \!
+ \! \hat{\mathcal{L}}_{2} \mu^{-2} \right) \! \widehat{\Psi}(\mu,\tau),
\end{equation*}
where
\begin{gather*}
\hat{\mathcal{L}}_{-1} \! = \! -\mathrm{i} 2 \tau \!
\begin{pmatrix}
1 & 0 \\
-2 \mathbb{P}^{\ast} & -1
\end{pmatrix}, \, \qquad \, \hat{\mathcal{L}}_{0} \! = \! -2 \tau \!
\begin{pmatrix}
0 & 0 \\
\hat{D}(\tau) & 0
\end{pmatrix} \! - \! \dfrac{\mathrm{i} 4 \tau \hat{A}(\tau)}{\sqrt{\smash[b]{-\hat{A}(\tau)
\hat{B}(\tau)}}} \!
\begin{pmatrix}
-\mathbb{P}^{\ast} & -1 \\
(\mathbb{P}^{\ast})^{2} & \mathbb{P}^{\ast}
\end{pmatrix}, \\
\hat{\mathcal{L}}_{1} \! = \! \left(\mathrm{i} a \! + \! \dfrac{1}{2} \! + \! \dfrac{2 \tau \hat{A}
(\tau) \hat{D}(\tau)}{\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}} \right) \!
\begin{pmatrix}
-1 & 0 \\
2 \mathbb{P}^{\ast} & 1
\end{pmatrix}, \, \quad \, \hat{\mathcal{L}}_{2} \! = \! \mathrm{i} \tau \!
\begin{pmatrix}
0 & 0 \\
\hat{B}(\tau) & 0
\end{pmatrix} \! + \! \hat{\alpha}(\tau) \!
\begin{pmatrix}
\mathbb{P}^{\ast} & 1 \\
-(\mathbb{P}^{\ast})^{2} & -\mathbb{P}^{\ast}
\end{pmatrix},
\end{gather*}
with $\hat{\alpha}(\tau)$ defined by Equation~\eqref{firstintegral}. Setting
$\mathbb{P}^{\ast} \! \equiv \! 0$, one arrives at the Fuchs-Garnier (or Lax)
pair stated in Proposition~2.1 of \cite{a1}, System~(1.4) of \cite{av2}, and
System~\eqref{eqFGmain} of the present work. \hfill $\blacksquare$
\end{eeee}
A relation between the Fuchs-Garnier pair~\eqref{eqFGmain} and the
DP3E~\eqref{eq1.1} is given by (see, in particular, Proposition~1.2 of \cite{a1},
with notational amendments)
\begin{bbbb}[{\rm \cite{a1,av2}}] \label{prop1.2}
Let $\hat{u} \! = \! \hat{u}(\tau)$ and $\hat{\varphi} \! = \! \hat{\varphi}(\tau)$
solve the system
\begin{equation} \label{eq1.5}
\begin{gathered}
\hat{u}^{\prime \prime}(\tau) \! = \! \dfrac{(\hat{u}^{\prime}(\tau))^{2}}{\hat{u}
(\tau)} \! - \! \dfrac{\hat{u}^{\prime}(\tau)}{\tau} \! + \! \dfrac{1}{\tau} \! \left(
-8 \varepsilon (\hat{u}(\tau))^{2} \! + \! 2ab \right) \! + \! \dfrac{b^{2}}{\hat{u}
(\tau)}, \quad \quad \hat{\varphi}^{\prime}(\tau) \! = \! \dfrac{2a} \tau \! + \!
\dfrac{b}{\hat{u}(\tau)},
\end{gathered}
\end{equation}
where $\varepsilon \! = \! \pm 1$, and $a,b \! \in \! \mathbb{C}$ are independent
of $\tau$$;$ then,
\begin{equation} \label{eq:ABCD}
\hat{A}(\tau) \! := \! \frac{\hat{u}(\tau)}{\tau} \mathrm{e}^{\mathrm{i} \hat{\varphi}(\tau)}, \!
\quad \! \hat{B}(\tau) \! := \! -\frac{\hat{u}(\tau)}{\tau} \mathrm{e}^{-\mathrm{i} \hat{\varphi}(\tau)},
\! \quad \! \hat{C}(\tau) \! := \! \dfrac{\varepsilon \tau \hat{A}^{\prime}(\tau)}{4 \hat{u}
(\tau)}, \! \quad \! \hat{D}(\tau) \! := \! -\dfrac{\varepsilon \tau \hat{B}^{\prime}
(\tau)}{4 \hat{u}(\tau)}
\end{equation}
solve the System~\eqref{eq1.4}. Conversely, let $\hat{A}(\tau) \! \not\equiv \! 0$,
$\hat{B}(\tau) \! \not\equiv \! 0$, $\hat{C}(\tau)$, and $\hat{D}(\tau)$ solve the
System~\eqref{eq1.4}, and define
\begin{equation} \label{tempeq}
\hat{u}(\tau) \! := \! \varepsilon \tau \sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}},
\quad \hat{\varphi}(\tau) \! := \! -\dfrac{\mathrm{i}}{2} \ln \! \left(-\hat{A}(\tau)/\hat{B}
(\tau) \right), \quad b \! := \! \hat{u}(\tau) \! \left(\hat{\varphi}^{\prime}(\tau)
\! - \! 2a \tau^{-1} \right);
\end{equation}
then, $b$ is independent of $\tau$, and $\hat{u}(\tau)$ and $\hat{\varphi}(\tau)$
solve the System~\eqref{eq1.5}.
\end{bbbb}
\begin{bbbb} \label{equzforhatf}
Let (cf. Equation~\eqref{yoominus3}$)$
\begin{equation} \label{hatsoff1}
2 \hat{f}_{-}(\tau) \! := \! -\mathrm{i} (a \! - \! \mathrm{i}/2) \! + \! \frac{\tau}{2} \!
\left(\frac{\hat{u}^{\prime}(\tau) \! - \! \mathrm{i} b}{\hat{u}(\tau)} \right),
\end{equation}
and (cf. Equation~\eqref{yooplus4}$)$
\begin{equation} \label{pga1}
\frac{\mathrm{i} 4}{\varepsilon b} \hat{f}_{+}(\tau) \! := \! \mathrm{i} (a \! + \! \mathrm{i}/2) \! + \!
\frac{\tau}{2} \! \left(\frac{\hat{u}^{\prime}(\tau) \! + \! \mathrm{i} b}{\hat{u}(\tau)} \right).
\end{equation}
Then, for $\varepsilon \! \in \! \lbrace \pm 1 \rbrace$,
\begin{gather}
2 \hat{f}_{-}(\tau) \! = \! \frac{2 \varepsilon \tau^{2} \hat{A}(\tau) \hat{D}(\tau)}{
\hat{u}(\tau)} \! = \! \frac{\tau}{2} \frac{\mathrm{d}}{\mathrm{d} \tau} \! \left(\ln \! \left(
\frac{\hat{u}(\tau)}{\tau} \right) \! - \! \mathrm{i} \hat{\varphi}(\tau) \right),
\label{hatsoff2} \\
\intertext{and}
\frac{\mathrm{i} 4}{\varepsilon b} \hat{f}_{+}(\tau) \! = \! -\frac{2 \varepsilon \tau^{2}
\hat{B}(\tau) \hat{C}(\tau)}{\hat{u}(\tau)} \! = \! \frac{\tau}{2} \frac{\mathrm{d}}{\mathrm{d} \tau} \!
\left(\ln \! \left(\frac{\hat{u}(\tau)}{\tau} \right) \! + \! \mathrm{i} \hat{\varphi}(\tau) \right);
\label{hatsoff3}
\end{gather}
furthermore,
\begin{equation} \label{pga2}
\frac{\mathrm{i} 4}{\varepsilon b} \hat{f}_{+}(\tau) \! = \! 2 \hat{f}_{-}(\tau) \! + \! \mathrm{i} \tau
\hat{\varphi}^{\prime}(\tau) \! = \! 2 \hat{f}_{-}(\tau) \! + \! \mathrm{i} \tau \! \left(
\frac{2a}{\tau} \! + \! \frac{b}{\hat{u}(\tau)} \right).
\end{equation}
\end{bbbb}
\emph{Proof}. Without loss of generality, consider, say, the proof for the function
$\hat{f}_{-}(\tau)$: the proof for the function $\hat{f}_{+}(\tau)$ is analogous.
One commences by establishing the following relation:
\begin{equation} \label{hatsoff4}
\frac{\hat{u}^{\prime}(\tau) \! - \! \mathrm{i} b}{\hat{u}(\tau)} \! = \! \frac{2}{\tau} \!
\left(\frac{2 \tau \hat{A}(\tau) \hat{D}(\tau)}{\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}
(\tau)}}} \! + \! (\mathrm{i} a \! + \! 1/2) \right).
\end{equation}
{}From Definition~\eqref{firstintegral}, the system of isomonodromy deformations~\eqref{eq1.4},
Remark~\ref{alphwave}, and the definition of the function $\hat{u}(\tau)$ given by the first
(left-most) member of Equations~\eqref{tempeq}, it follows via differentiation that
\begin{align*}
\frac{\hat{u}^{\prime}(\tau) \! - \! \mathrm{i} b}{\hat{u}(\tau)} =& \, \frac{2 \tau (\hat{A}
(\tau) \hat{D}(\tau) \! - \! \hat{B}(\tau) \hat{C}(\tau)) \! + \! \sqrt{\smash[b]{-\hat{A}
(\tau) \hat{B}(\tau)}}}{\tau \sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}} \! - \!
\frac{\mathrm{i} (\varepsilon b)}{\varepsilon \hat{u}(\tau)} \\
=& \, \frac{2 \tau (\hat{A}(\tau) \hat{D}(\tau) \! - \! \hat{B}(\tau) \hat{C}(\tau)) \! +
\! \sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}}{\tau \sqrt{\smash[b]{-\hat{A}(\tau)
\hat{B}(\tau)}}} \! - \! \frac{\hat{\alpha}(\tau) \hat{B}(\tau)}{\varepsilon \hat{u}(\tau)} \\
=& \, \frac{2 \tau (\hat{A}(\tau) \hat{D}(\tau) \! - \! \hat{B}(\tau) \hat{C}(\tau)) \! +
\! \sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}}{\tau \sqrt{\smash[b]{-\hat{A}(\tau)
\hat{B}(\tau)}}} \\
+& \, \frac{2 \tau (\hat{A}(\tau) \hat{D}(\tau) \! + \! \hat{B}(\tau) \hat{C}(\tau)) \! + \!
\mathrm{i} 2a \sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}(\tau)}}}{\tau \sqrt{\smash[b]{-\hat{A}
(\tau) \hat{B}(\tau)}}} \\
=& \, \frac{2}{\tau} \! \left(\frac{2 \tau \hat{A}(\tau) \hat{D}(\tau)}{\sqrt{\smash[b]{
-\hat{A}(\tau) \hat{B}(\tau)}}} \! + \! (\mathrm{i} a \! + \! 1/2) \right);
\end{align*}
conversely, {}from the system of isomonodromy deformations~\eqref{eq1.4}, the
System~\eqref{eq1.5}, and the Definitions~\eqref{eq:ABCD} and~\eqref{tempeq},
it follows that
\begin{align*}
\frac{4 \tau \hat{A}(\tau) \hat{D}(\tau)}{\sqrt{\smash[b]{-\hat{A}(\tau) \hat{B}
(\tau)}}} =& \, \frac{4 \varepsilon \tau \hat{A}(\tau) \hat{D}(\tau)}{\hat{u}(\tau)}
\! = \! \frac{4 \varepsilon \tau}{\hat{u}(\tau)} \! \left(-\frac{\varepsilon}{4}
\hat{B}^{\prime}(\tau) \mathrm{e}^{\mathrm{i} \hat{\varphi}(\tau)} \right) \! = \! \frac{\tau
\mathrm{e}^{\mathrm{i} \hat{\varphi}(\tau)}}{\hat{u}(\tau)} \frac{\mathrm{d}}{\mathrm{d} \tau} \! \left(
\frac{\hat{u}(\tau)}{\tau} \mathrm{e}^{-\mathrm{i} \hat{\varphi}(\tau)} \right) \\
=& \, \frac{\tau}{\hat{u}(\tau)} \! \left(-\mathrm{i} \hat{\varphi}^{\prime}(\tau) \frac{\hat{u}
(\tau)}{\tau} \! - \! \frac{\hat{u}(\tau)}{\tau^{2}} \! + \! \frac{\hat{u}^{\prime}(\tau)}{
\tau} \right) \\
=& \, \frac{\tau}{\hat{u}(\tau)} \! \left(-\frac{\hat{u}(\tau)}{\tau} \! \left(\frac{\mathrm{i} 2a}{
\tau} \! + \! \frac{\mathrm{i} b}{\hat{u}(\tau)} \right) \! - \! \frac{\hat{u}(\tau)}{\tau^{2}}
\! + \! \frac{\hat{u}^{\prime}(\tau)}{\tau} \right) \\
=& \, \frac{\hat{u}^{\prime}(\tau) \! - \! \mathrm{i} b}{\hat{u}(\tau)} \! - \! \frac{2}{\tau}
(\mathrm{i} a \! + \! 1/2) \Rightarrow
\end{align*}
\begin{equation*}
\frac{2}{\tau} \! \left(\frac{2 \tau \hat{A}(\tau) \hat{D}(\tau)}{\sqrt{\smash[b]{-\hat{A}
(\tau) \hat{B}(\tau)}}} \! + \! (\mathrm{i} a \! + \! 1/2) \right) \! = \! \frac{\hat{u}^{\prime}
(\tau) \! - \! \mathrm{i} b}{\hat{u}(\tau)},
\end{equation*}
which establishes Equation~\eqref{hatsoff4}. Via Definition~\eqref{hatsoff1} and
Equation~\eqref{hatsoff4}, one shows that
\begin{equation} \label{hatsoff5}
\hat{f}_{-}(\tau) \! = \! \frac{\tau \hat{A}(\tau) \hat{D}(\tau)}{\sqrt{\smash[b]{-
\hat{A}(\tau) \hat{B}(\tau)}}},
\end{equation}
whence, via the definition for $\hat{u}(\tau)$ given by the first (left-most) member of
Equations~\eqref{tempeq}, one arrives at the first (left-most) relation of
Equation~\eqref{hatsoff2}; moreover, it follows {}from the ODE for the function
$\hat{\varphi}(\tau)$ given in System~\eqref{eq1.5}, and Definition~\eqref{hatsoff1}, that
\begin{equation*}
\tau^{-1} \hat{f}_{-}(\tau) \! = \! \frac{1}{4} \! \left(\frac{\hat{u}^{\prime}(\tau)}{\hat{u}
(\tau)} \! + \! \frac{\mathrm{i} 2a}{\tau} \! - \! \mathrm{i} \hat{\varphi}^{\prime}(\tau) \right) \!
- \! \frac{1}{2 \tau}(\mathrm{i} a \! + \! 1/2) \! = \! \frac{1}{4} \! \left(\frac{\mathrm{d}}{\mathrm{d} \tau}
\ln \! \left(\frac{\hat{u}(\tau)}{\tau} \right) \! - \! \mathrm{i} \hat{\varphi}^{\prime}(\tau)
\right),
\end{equation*}
which implies the second (right-most) relation of Equation~\eqref{hatsoff2}.
Equations~\eqref{hatsoff2} and~\eqref{hatsoff3} imply the Corollary~\eqref{pga2}, which
is consistent with, and can also be derived {}from, the Definition~\eqref{firstintegral} and
the First Integral of System~\eqref{eq1.4} (cf. Remark~\ref{alphwave}). \hfill $\qed$
Herewith follows the post-gauge-transformed Fuchs-Garnier, or Lax, pair.
\begin{bbbb} \label{newlax1}
Let $\widehat{\Psi}(\mu,\tau)$ be a fundamental solution of the System~\eqref{eqFGmain}.
Set
\begin{equation} \label{newlax2}
\begin{gathered}
A(\tau) \! := \! \hat{A}(\tau) \tau^{-\mathrm{i} a}, \, \quad \, B(\tau) \! := \! \hat{B}(\tau)
\tau^{\mathrm{i} a}, \, \quad \, C(\tau) \! := \! \hat{C}(\tau) \tau^{-\mathrm{i} a}, \quad D(\tau)
\! := \! \hat{D}(\tau) \tau^{\mathrm{i} a}, \\
\alpha (\tau) \! := \! \hat{\alpha}(\tau) \tau^{-\mathrm{i} a}, \, \quad \, \widehat{\Psi}
(\mu,\tau) \! := \! \tau^{\frac{\mathrm{i} a}{2} \sigma_{3}} \Psi (\mu,\tau).
\end{gathered}
\end{equation}
Then$:$ {\rm (i)} $\Psi (\mu,\tau)$ is a fundamental solution of
\begin{equation} \label{newlax3}
\partial_{\mu} \Psi(\mu,\tau) \! = \! \widetilde{\mathscr{U}}(\mu,\tau) \Psi
(\mu,\tau), \qquad \quad \partial_{\tau} \Psi(\mu,\tau) \! = \! \widetilde{
\mathscr{V}}(\mu,\tau) \Psi(\mu,\tau),
\end{equation}
where
\begin{gather}
\widetilde{\mathscr{U}}(\mu,\tau) \! = \! -\mathrm{i} 2 \tau \mu \sigma_{3} \! + \!
2 \tau \!
\begin{pmatrix}
0 & \frac{\mathrm{i} 2A(\tau)}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}} \\
-D(\tau) & 0
\end{pmatrix}
\! - \! \dfrac{1}{\mu} \! \left(\mathrm{i} a \! + \! \dfrac{1}{2} \! + \! \dfrac{2 \tau A(\tau)
D(\tau)}{\sqrt{\smash[b]{-A(\tau) B(\tau)}}} \right) \! \sigma_{3} \! + \!
\dfrac{1}{\mu^{2}} \!
\begin{pmatrix}
0 & \alpha (\tau) \\
\mathrm{i} \tau B(\tau) & 0
\end{pmatrix}, \label{nlxa} \\
\widetilde{\mathscr{V}}(\mu,\tau) \! = \! -\mathrm{i} \mu^{2} \sigma_{3} \! + \! \mu \!
\begin{pmatrix}
0 & \frac{\mathrm{i} 2A(\tau)}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}} \\
-D(\tau) & 0
\end{pmatrix}
\! - \! \dfrac{A(\tau)D(\tau)}{\sqrt{\smash[b]{-A(\tau) B(\tau)}}} \sigma_{3} \!
- \! \frac{1}{\mu} \frac{1}{2 \tau} \!
\begin{pmatrix}
0 & \alpha (\tau) \\
\mathrm{i} \tau B(\tau) & 0
\end{pmatrix}, \label{nlxb}
\end{gather}
with
\begin{equation} \label{aphnovij}
\alpha (\tau) \! := \! -2(B(\tau))^{-1} \! \left(\mathrm{i} a \sqrt{\smash[b]{-A(\tau)B(\tau)}}
+ \! \tau (A(\tau)D(\tau) \! + \! B(\tau)C(\tau)) \right);
\end{equation}
and {\rm (ii)} if the coefficient functions $\hat{A}(\tau)$, $\hat{B}(\tau)$,
$\hat{C}(\tau)$, and $\hat{D}(\tau)$ satisfy the system of isomonodromy
deformations~\eqref{eq1.4} and the functions $A(\tau)$, $B(\tau)$, $C(\tau)$, and
$D(\tau)$ are defined by Equations \eqref{newlax2}, then the Frobenius compatibility
condition of the System~\eqref{newlax3}, for arbitrary values of $\mu \! \in \!
\mathbb{C}$, is that the differentiable, scalar-valued functions $A(\tau)$, $B(\tau)$,
$C(\tau)$, and $D(\tau)$ satisfy the corresponding system of isomonodromy
deformations
\begin{equation} \label{newlax8}
\begin{gathered}
A^{\prime}(\tau) \! = \! -\frac{\mathrm{i} a}{\tau}A(\tau) \! + \! 4C(\tau)
\sqrt{\smash[b]{-A(\tau)B(\tau)}}, \quad \, \, \quad B^{\prime}(\tau) \! = \!
\frac{\mathrm{i} a}{\tau}B(\tau) \! - \! 4D(\tau) \sqrt{\smash[b]{-A(\tau)B(\tau)}}, \\
(\tau C(\tau))^{\prime} \! = \! \mathrm{i} a C(\tau) \! - \! 2 \tau A(\tau), \, \, \quad
\quad \, \, (\tau D(\tau))^{\prime} \! = \! -\mathrm{i} a D(\tau) \! + \! 2 \tau B(\tau), \\
\left(\sqrt{\smash[b]{-A(\tau)B(\tau)}} \, \right)^{\prime} \! = \! 2(A(\tau)
D(\tau) \! - \! B(\tau)C(\tau)).
\end{gathered}
\end{equation}
\end{bbbb}
\emph{Proof}. If $\widehat{\Psi}(\mu,\tau)$ is a fundamental solution of the
System~\eqref{eqFGmain}, then it follows {}from the isomonodromy
deformations~\eqref{eq1.4} and the Definitions~\eqref{newlax2} that
$\Psi (\mu,\tau)$ solves the System~\eqref{newlax3}, and that the coefficient
functions $A(\tau)$, $B(\tau)$, $C(\tau)$, and $D(\tau)$ satisfy the
corresponding isomonodromy deformations~\eqref{newlax8}. One verifies the
Frobenius compatibility condition for the System~\eqref{newlax3} by showing that,
$\forall \, \mu \! \in \! \mathbb{C}$, $\partial_{\tau} \widetilde{\mathscr{U}}
(\mu,\tau) \! - \! \partial_{\mu} \widetilde{\mathscr{V}}(\mu,\tau) \! + \!
[\widetilde{\mathscr{U}}(\mu,\tau),\widetilde{\mathscr{V}}(\mu,\tau)] \! = \!
\left(
\begin{smallmatrix}
0 & 0 \\
0 & 0
\end{smallmatrix}
\right)$, where, for $\mathfrak{X},\mathfrak{Y} \! \in \! \mathrm{M}_{2}
(\mathbb{C})$, $[\mathfrak{X},\mathfrak{Y}] \! := \! \mathfrak{X} \mathfrak{Y}
\! - \! \mathfrak{Y} \mathfrak{X}$ is the matrix commutator. \hfill $\qed$
\begin{eeee} \label{newlax6}
Definitions~\eqref{firstintegral}, \eqref{newlax2}, and~\eqref{aphnovij}, and
Remark~\ref{alphwave} imply that $-\mathrm{i} \alpha (\tau)B(\tau) \! = \! \varepsilon b$,
$\varepsilon \! = \! \pm 1$. \hfill $\blacksquare$
\end{eeee}
\begin{bbbb} \label{propuf}
Let $u(\tau)$ and $\varphi(\tau)$ solve the system
\begin{equation} \label{equu17}
\begin{gathered}
u^{\prime \prime}(\tau) \! = \! \dfrac{(u^{\prime}(\tau))^{2}}{u(\tau)} \! - \!
\dfrac{u^{\prime}(\tau)}{\tau} \! + \! \dfrac{1}{\tau} \! \left(-8 \varepsilon
(u(\tau))^{2} \! + \! 2ab \right) \! + \! \dfrac{b^{2}}{u(\tau)}, \quad \quad
\varphi^{\prime}(\tau) \! = \! \dfrac{a} \tau \! + \! \dfrac{b}{u(\tau)},
\end{gathered}
\end{equation}
where $\varepsilon \! = \! \pm 1$, and $a,b \! \in \! \mathbb{C}$ are independent
of $\tau$$;$ then,
\begin{equation} \label{equu18}
\begin{gathered}
A(\tau) \! := \! \frac{u(\tau)}{\tau} \mathrm{e}^{\mathrm{i} \varphi(\tau)}, \, \quad \, B(\tau) \!
:= \! -\frac{u(\tau)}{\tau} \mathrm{e}^{-\mathrm{i} \varphi(\tau)}, \, \quad \, C(\tau) \! := \!
\dfrac{\varepsilon \tau}{4u(\tau)} \! \left(A^{\prime}(\tau) \! + \! \frac{\mathrm{i} a}{\tau}
A(\tau) \right), \\
D(\tau) \! := \! -\dfrac{\varepsilon \tau}{4u(\tau)} \! \left(B^{\prime}(\tau)
\! - \! \frac{\mathrm{i} a}{\tau}B(\tau) \right)
\end{gathered}
\end{equation}
solve the System~\eqref{newlax8}. Conversely, let $A(\tau) \! \not\equiv \! 0$, $B(\tau) \!
\not\equiv \! 0$, $C(\tau)$, and $D(\tau)$ solve the System~\eqref{newlax8}, and define
\begin{equation} \label{equu19}
u(\tau) \! := \! \varepsilon \tau \sqrt{\smash[b]{-A(\tau)B(\tau)}}, \quad \varphi(\tau)
\! := \! -\frac{\mathrm{i}}{2} \ln \! \left(-A(\tau)/B(\tau) \right), \quad b \! := \! u(\tau) \!
\left(\varphi^{\prime}(\tau) \! - \! a \tau^{-1} \right);
\end{equation}
then, $b$ is independent of $\tau$, and $u(\tau)$ and $\varphi (\tau)$ solve the
System~\eqref{equu17}.
\end{bbbb}
\emph{Proof}. Via the definition of $\hat{u}(\tau)$ given by the first (left-most) member
of Equations~\eqref{tempeq} and the Definitions~\eqref{newlax2}, one arrives at the
definition for $u(\tau)$ given by the first (left-most) member of Equations~\eqref{equu19};
in particular, it follows that $u(\tau) \! = \! \hat{u}(\tau)$, and, {}from the first equation of
System~\eqref{eq1.5}, $u(\tau)$ solves the DP3E~\eqref{eq1.1} (see the first equation of the
System~\eqref{equu17}). Let $\varphi (\tau)$ be defined as in Equations~\eqref{equu19},
that is, $\varphi (\tau) \! = \! -\mathrm{i} \ln (\sqrt{\smash[b]{-A(\tau)B(\tau)}}/B(\tau))$; then,
via differentiation, the Definition~\eqref{aphnovij}, and the corresponding system of
isomonodromy deformations~\eqref{newlax8}, it follows that
\begin{align*}
\varphi^{\prime}(\tau) =& \, -\mathrm{i} \! \left(\frac{1}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}}
\left(\sqrt{\smash[b]{-A(\tau)B(\tau)}} \, \right)^{\prime} \! - \! \frac{B^{\prime}
(\tau)}{B(\tau)} \right) \nonumber \\
=& \, -\mathrm{i} \! \left(\frac{2(A(\tau)D(\tau) \! - \! B(\tau)C(\tau))}{\sqrt{\smash[b]{
-A(\tau)B(\tau)}}} \! - \! \frac{1}{B(\tau)} \left(\frac{\mathrm{i} a}{\tau}B(\tau) \! - \!
4D(\tau) \sqrt{\smash[b]{-A(\tau)B(\tau)}} \right) \right) \nonumber \\
=& \, -\frac{a}{\tau} \! + \! \frac{\mathrm{i} 2}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}}
(A(\tau)D(\tau) \! + \! B(\tau)C(\tau)) \nonumber \\
=& \, -\frac{a}{\tau} \! + \! \frac{\mathrm{i} 2}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}} \!
\left(-\frac{\mathrm{i} \varepsilon b}{2 \tau} \! - \! \frac{\mathrm{i} a}{\tau} \sqrt{\smash[b]{
-A(\tau)B(\tau)}} \right) \! = \! \frac{a}{\tau} \! + \! \frac{b}{u(\tau)},
\end{align*}
that is, $\varphi (\tau)$ solves the ODE given by the second (right-most) member of the
System~\eqref{equu17}; moreover, it also follows {}from the Definitions~\eqref{tempeq},
\eqref{newlax2}, and~\eqref{equu19} that
\begin{equation} \label{equu20}
\varphi (\tau) \! = \! \hat{\varphi}(\tau) \! - \! a \ln \tau.
\end{equation}
The Definitions~\eqref{equu18} for the functions $A(\tau)$, $B(\tau)$,
$C(\tau)$, and $D(\tau)$ are a consequence of the Definitions~\eqref{eq:ABCD}
and~\eqref{newlax2}, the fact that $u(\tau) \! = \! \hat{u}(\tau)$, and
Equation~\eqref{equu20}. A series of lengthy, but otherwise straightforward,
differentiation arguments complete the proof. \hfill $\qed$
\begin{eeee} \label{phitophi}
It also follows {}from the ODE satisfied by $\hat{\varphi}(\tau)$ given in
the System~\eqref{eq1.5}, and Equation~\eqref{equu20}, that $\varphi (\tau)$
solves the corresponding ODE given in the System~\eqref{equu17}. \hfill
$\blacksquare$
\end{eeee}
\begin{bbbb} \label{equzforeff}
Let
\begin{gather}
2f_{-}(\tau) \! := \! -\mathrm{i} (a \! - \! \mathrm{i}/2) \! + \! \frac{\tau}{2} \! \left(
\frac{u^{\prime}(\tau) \! - \! \mathrm{i} b}{u(\tau)} \right), \label{hatsoff7} \\
\intertext{and}
\frac{\mathrm{i} 4}{\varepsilon b}f_{+}(\tau) \! := \! \mathrm{i} (a \! + \! \mathrm{i}/2) \! + \!
\frac{\tau}{2} \! \left(\frac{u^{\prime}(\tau) \! + \! \mathrm{i} b}{u(\tau)} \right). \label{pga3}
\end{gather}
Then, for $\varepsilon \! \in \! \lbrace \pm 1 \rbrace$,
\begin{gather}
2f_{-}(\tau) \! = \! \frac{2 \varepsilon \tau^{2}A(\tau)D(\tau)}{u(\tau)} \! = \!
\frac{\tau}{2} \frac{\mathrm{d}}{\mathrm{d} \tau} \! \left(\ln \! \left(\frac{u(\tau)}{\tau} \right)
\! - \! \mathrm{i} (\varphi (\tau) \! + \! a \ln \tau) \right), \label{hatsoff8} \\
\intertext{and}
\frac{\mathrm{i} 4}{\varepsilon b}f_{+}(\tau) \! = \! -\frac{2 \varepsilon \tau^{2}B(\tau)
C(\tau)}{u(\tau)} \! = \! \frac{\tau}{2} \frac{\mathrm{d}}{\mathrm{d} \tau} \! \left(\ln \! \left(
\frac{u(\tau)}{\tau} \right) \! + \! \mathrm{i} (\varphi (\tau) \! + \! a \ln \tau) \right);
\label{pga4}
\end{gather}
furthermore,
\begin{equation} \label{pga5}
\frac{\mathrm{i} 4}{\varepsilon b}f_{+}(\tau) \! = \! 2f_{-}(\tau) \! + \! \mathrm{i} \tau \frac{\mathrm{d}}{\mathrm{d} \tau}
\! \left(\varphi (\tau) \! + \! a \ln \tau \right) \! = \! 2f_{-}(\tau) \! + \! \mathrm{i} \tau \! \left(
\frac{2a}{\tau} \! + \! \frac{b}{u(\tau)} \right).
\end{equation}
\end{bbbb}
\emph{Proof}. Via Definition~\eqref{aphnovij}, the System~\eqref{equu17}, the
corresponding system of isomonodromy deformations~\eqref{newlax8},
Remark~\ref{newlax6}, and the Definitions~\eqref{equu18} and~\eqref{equu19},
one establishes the veracity of the relation
\begin{equation} \label{hatsoff9}
\frac{u^{\prime}(\tau) \! - \! \mathrm{i} b}{u(\tau)} \! = \! \frac{2}{\tau} \! \left(
\frac{2 \tau A(\tau)D(\tau)}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}} \! + \!
(\mathrm{i} a \! + \! 1/2) \right),
\end{equation}
and then proceeds, \emph{mutatis mutandis}, as in the proof of
Proposition~\ref{equzforhatf}. The Corollary~\eqref{pga5} follows {}from,
and is consistent with, the Definition~\eqref{aphnovij} and the First Integral
of System~\eqref{newlax8} (cf. Remark~\ref{newlax6}). \hfill $\qed$
\begin{eeee} \label{efftohateff}
One deduces {}from the Definitions~\eqref{newlax2}, Equation~\eqref{equu20},
and Propositions~\ref{equzforhatf} and~\ref{equzforeff} that $f_{\pm}(\tau) \!
= \! \hat{f}_{\pm}(\tau)$. \hfill $\blacksquare$
\end{eeee}
\begin{eeee} \label{hamian}
A lengthy algebraic exercise reveals that, in terms of the coefficient functions
$A(\tau)$, $B(\tau)$, $C(\tau)$, and $D(\tau)$ satisfying the corresponding
isomonodromy deformations~\eqref{newlax8}, the Hamiltonian function (cf.
Equation~\eqref{eqh1}$)$ reads
\begin{equation*}
\mathcal{H}(\tau) \! = \! \frac{1}{2 \tau} \! \left(\mathrm{i} a \! + \! \dfrac{1}{2} \! + \!
\dfrac{2 \tau A(\tau)D(\tau)}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}} \right)^{2} \!
+ \! 4 \tau \sqrt{\smash[b]{-A(\tau)B(\tau)}} \! - \! \dfrac{\mathrm{i} (\varepsilon b)
D(\tau)}{B(\tau)} \! + \! 2 \tau C(\tau)D(\tau) \! + \! \dfrac{A(\tau)D(\tau)}{
\sqrt{\smash[b]{-A(\tau)B(\tau)}}}.
\end{equation*}
\end{eeee}
\begin{eeee} \label{rem1.1}
Hereafter, all explicit $\tau$ dependencies are suppressed, except where
imperative. \hfill $\blacksquare$
\end{eeee}
\subsection{Canonical Solutions and the Monodromy Data} \label{sec1d}
A succinct discussion of the monodromy data associated with the System~\eqref{newlax3}
is presented in this subsection (see, in particular, \cite{a1,av2}).
For $\mu \! \in \! \mathbb{C}$, the System~\eqref{newlax3} has two irregular singular
points, one being the point at infinity ($\mu \! = \! \infty$) and the other being the
origin ($\mu \! = \! 0$). For $\delta_{\infty},\delta_{0} \! > \! 0$ and $m \! \in \!
\mathbb{Z}$, define the (sectorial) neighbourhoods $\Omega_{m}^{\infty}$ and
$\Omega_{m}^{0}$, respectively, of these singular points:
\begin{align*}
\Omega_{m}^{\infty} &:= \left\{\mathstrut \mu \! \in \! \mathbb{C}; \, \lvert
\mu \rvert \! > \! \delta_{\infty}^{-1}, \, -\dfrac{\pi}{2} \! + \! \dfrac{\pi m}{2}
\! < \! \arg \mu \! + \! \dfrac{1}{2} \arg \tau \! < \! \dfrac{\pi}{2} \! + \!
\dfrac{\pi m}{2} \right\}, \\
\Omega_{m}^{0} &:= \left\{\mathstrut \mu \! \in \! \mathbb{C}; \, \lvert \mu
\rvert \! < \! \delta_{0}, \, -\pi \! + \! \pi m \! < \! \arg \mu \! - \! \dfrac{1}{2}
\arg \tau \! - \! \dfrac{1}{2} \arg (\varepsilon b) \! < \! \pi \! + \! \pi m \right\}.
\end{align*}
\begin{bbbb}[{\rm \cite{a1,av2}}] \label{prop1.4}
There exist solutions $\mathbb{Y}_{m}^{\infty}(\mu) \! = \! \mathbb{Y}_{m}^{\infty}
(\mu,\tau)$ and $\mathbb{X}_{m}^{0}(\mu) \! = \! \mathbb{X}_{m}^{0}(\mu,\tau)$,
$m \! \in \! \mathbb{Z}$, of the System~\eqref{newlax3} that are uniquely defined by
the following asymptotic expansions:
\begin{align*}
\mathbb{Y}_{m}^{\infty}(\mu) \underset{\Omega_{m}^{\infty} \ni \mu \to \infty}{:=}&
\, \left(\mathrm{I} \! + \! \Psi^{(1)} \mu^{-1} \! + \! \Psi^{(2)} \mu^{-2} \! + \!
\dotsb \right) \! \exp \! \left(-\mathrm{i} \! \left(\tau \mu^{2} \! + \! \left(a \! - \! \mathrm{i}/2
\right) \ln \mu \right) \! \sigma_{3} \right), \\
\mathbb{X}_{m}^{0}(\mu) \underset{\Omega_{m}^{0} \ni \mu \to 0}{:=}& \,
\Psi_{0} \! \left(\mathrm{I} \! + \! \hat{\mathcal{Z}}_{1} \mu \! + \! \dotsb
\right) \! \exp \! \left(-\mathrm{i} \sqrt{\tau \varepsilon b} \, \mu^{-1} \sigma_{3} \right),
\end{align*}
where $\mathrm{I} \! = \! \diag (1,1)$, $\ln \mu \! := \! \ln \vert \mu \vert
\! + \! \mathrm{i} \arg \mu$,
\begin{gather*}
\Psi^{(1)} \! = \!
\begin{pmatrix}
0 & \frac{A(\tau)}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}} \\
-\mathrm{i} D(\tau)/2 & 0
\end{pmatrix}, \, \quad \, \quad \, \Psi^{(2)} \! = \!
\begin{pmatrix}
\psi^{(2)}_{11} & 0 \\
0 & \psi^{(2)}_{22}
\end{pmatrix}, \\
\psi^{(2)}_{11} \! := \! -\dfrac{\mathrm{i}}{2} \! \left(\tau \sqrt{\smash[b]{-A(\tau)B(\tau)}}
+ \! \tau C(\tau)D(\tau) \! + \! \dfrac{A(\tau)D(\tau)}{\sqrt{\smash[b]{-A(\tau)
B(\tau)}}} \right), \\
\psi^{(2)}_{22} \! := \! \dfrac{\mathrm{i} \tau}{2} \! \left(\sqrt{\smash[b]{-A(\tau)B(\tau)}}
+ \! C(\tau)D(\tau) \right), \\
\Psi_{0} \! = \! \dfrac{\mathrm{i}}{\sqrt{2}} \! \left(\dfrac{(\varepsilon b)^{1/4}}{
\tau^{1/4} \sqrt{\smash[b]{B(\tau)}}} \right)^{\sigma_{3}} \left(\sigma_{1}
\! + \! \sigma_{3} \right), \, \quad \, \quad \, \hat{\mathcal{Z}}_{1} \! = \!
\begin{pmatrix}
z_{1}^{(11)} & z_{1}^{(12)} \\
-z_{1}^{(12)} & - z_{1}^{(11)}
\end{pmatrix}, \\
z_{1}^{(11)} \! := -\dfrac{\mathrm{i} \! \left(\mathrm{i} a \! + \! \frac{1}{2} \! + \! \frac{2 \tau
A(\tau)D(\tau)}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}} \right)^{2}}{2 \sqrt{\smash[b]{
\tau \varepsilon b}}} \! - \! \dfrac{\mathrm{i} 2 \tau^{3/2} \sqrt{\smash[b]{-A(\tau)
B(\tau)}}}{\sqrt{\smash[b]{\varepsilon b}}} \! - \! \dfrac{D(\tau) \sqrt{\smash[b]{
\tau \varepsilon b}}}{B(\tau)}, \\
z_{1}^{(12)} \! := \! -\frac{\mathrm{i} \! \left(\mathrm{i} a \! + \! \frac{1}{2} \! + \! \frac{2 \tau
A(\tau)D(\tau)}{\sqrt{\smash[b]{-A(\tau)B(\tau)}}} \right)}{2 \sqrt{\smash[b]{\tau
\varepsilon b}}},
\end{gather*}
and $\sigma_{1} \! = \!
\left(
\begin{smallmatrix}
0 & 1 \\
1 & 0
\end{smallmatrix}
\right)$.
\end{bbbb}
\begin{eeee} \label{newrem12}
The canonical solutions $\mathbb{X}_{m}^{0}(\mu)$, $m \! \in \! \mathbb{Z}$, are
defined uniquely provided that the branch of $(B(\tau))^{1/2}$ is fixed: hereafter,
the branch of $(B(\tau))^{1/2}$ is not fixed; therefore, the set of canonical solutions
$\lbrace \mathbb{X}_{m}^{0}(\mu) \rbrace_{m \in \mathbb{Z}}$ is defined up to a
sign. This ambiguity doesn't affect the definition of the Stokes multipliers $($see
Equations~\eqref{eqnstokmult} below); rather, it results in a sign discrepancy in
the definition of the connection matrix, $G$ (see Equation~\eqref{eqdefg} below).
\hfill $\blacksquare$
\end{eeee}
The \emph{canonical solutions}, $\mathbb{Y}_{m}^{\infty}(\mu)$ and $\mathbb{X}_{m}^{0}
(\mu)$, $m \! \in \! \mathbb{Z}$, enable one to define the \emph{Stokes matrices},
$S_{m}^{\infty}$ and $S_{m}^{0}$, respectively:
\begin{equation}
\mathbb{Y}_{m+1}^{\infty}(\mu) \! = \! \mathbb{Y}_{m}^{\infty}(\mu)S_{m}^{\infty},
\, \quad \, \quad \,\mathbb{X}_{m+1}^{0}(\mu) \! = \! \mathbb{X}_{m}^{0}(\mu)
S_{m}^{0}. \label{eqnstokmult}
\end{equation}
The Stokes matrices are independent of $\mu$ and $\tau$, and have the following
structures:
\begin{equation*}
S_{2m}^{\infty} \! = \!
\begin{pmatrix}
1 & 0 \\
s_{2m}^{\infty} & 1
\end{pmatrix}, \quad S_{2m+1}^{\infty} \! = \!
\begin{pmatrix}
1 & s_{2m+1}^{\infty} \\
0 & 1
\end{pmatrix}, \quad S_{2m}^{0} \! = \!
\begin{pmatrix}
1 & s_{2m}^{0} \\
0 & 1
\end{pmatrix}, \quad S_{2m+1}^{0} \! = \!
\begin{pmatrix}
1 & 0 \\
s_{2m+1}^{0} & 1
\end{pmatrix}.
\end{equation*}
The parameters $s_{m}^{\infty}$ and $s_{m}^{0}$ are called the \emph{Stokes multipliers}:
it can be shown that
\begin{equation} \label{eq1.8}
S_{m+4}^{\infty} \! = \! \mathrm{e}^{-2 \pi (a-\mathrm{i}/2) \sigma_{3}}S_{m}^{\infty}
\mathrm{e}^{2 \pi (a-\mathrm{i}/2) \sigma_{3}}, \, \quad \, \quad \, S_{m+2}^{0} \! = \! S_{m}^{0}.
\end{equation}
Equations~\eqref{eq1.8} imply that the number of independent Stokes multipliers does
not exceed six; for example, $s_{0}^{0}$, $s_{1}^0$, $s_{0}^{\infty}$, $s_{1}^{\infty}$,
$s_{2}^{\infty}$, and $s_{3}^{\infty}$. Furthermore, due to the special structure of the
System~\eqref{newlax3}, that is, the coefficient matrices of odd (resp., even) powers of
$\mu$ in $\widetilde{\mathscr{U}}(\mu,\tau)$ are diagonal (resp., off-diagonal) and
\emph{vice-versa} for $\widetilde{\mathscr{V}}(\mu,\tau)$, one can deduce the following
relations for the Stokes matrices:
\begin{equation} \label{eq1.9}
S_{m+2}^{\infty} \! = \! \sigma_{3} \mathrm{e}^{-\pi (a-\mathrm{i}/2) \sigma_{3}}
S_{m}^{\infty} \mathrm{e}^{\pi (a-\mathrm{i}/2) \sigma_{3}} \sigma_{3}, \, \quad \, \quad \,
S_{m}^{0} = \! \sigma_{1}S_{m+1}^{0} \sigma_{1}.
\end{equation}
Equations~\eqref{eq1.9} reduce the number of independent Stokes multipliers by two,
that is, all Stokes multipliers can be expressed in terms of $s_{0}^{0}$, $s_{0}^{\infty}$,
$s_{1}^{\infty}$, and---the parameter of formal monodromy---$a$. There is one more
relation between the Stokes multipliers that follows {}from the so-called cyclic relation
(see Equation~\eqref{cycrel} below). Define the monodromy matrix at the point at infinity,
$M^{\infty}$, and the monodromy matrix at the origin, $M^{0}$, via the following relations:
\begin{gather*}
\mathbb{Y}_{0}^{\infty}(\mu \mathrm{e}^{-\mathrm{i} 2 \pi}) \! := \! \mathbb{Y}_{0}^{\infty}(\mu)
M^{\infty}, \, \quad \, \quad \, \mathbb{X}_{0}^{0}(\mu \mathrm{e}^{-\mathrm{i} 2 \pi}) \! := \!
\mathbb{X}_{0}^{0}(\mu)M^{0}.
\end{gather*}
Since $\mathbb{Y}_{0}^{\infty}(\mu)$ and $\mathbb{X}_{0}^{0}(\mu)$ are solutions
of the System~\eqref{newlax3}, they differ by a right-hand (matrix) factor $G$:
\begin{equation}
\mathbb{Y}_{0}^{\infty}(\mu) \! := \! \mathbb{X}_{0}^{0}(\mu)G, \label{eqdefg}
\end{equation}
where $G$ is called the \emph{connection matrix}. As matrices relating fundamental
solutions of the System~\eqref{newlax3}, the monodromy, connection, and Stokes
matrices are independent of $\mu$ and $\tau$; moreover, since $\mathrm{tr}
(\widetilde{\mathscr{U}}(\mu,\tau)) \! = \! \mathrm{tr}(\widetilde{\mathscr{V}}
(\mu,\tau)) \! = \! 0$, it follows that $\det (M^{\infty}) \! = \! \det (M^{0}) \!
= \! \det (G) \! = \! 1$. {}From the definition of the monodromy and connection
matrices, one deduces the following \emph{cyclic relation}:
\begin{equation} \label{cycrel}
GM^{\infty} \! = \! M^{0}G.
\end{equation}
The monodromy matrices can be expressed in terms of the Stokes matrices:
\begin{equation*}
M^{\infty} \! = \! S_{0}^{\infty}S_{1}^{\infty}S_{2}^{\infty}S_{3}^{\infty}
\mathrm{e}^{-2 \pi (a-\mathrm{i}/2) \sigma_{3}}, \, \quad \, \quad \, M^{0} \! = \!
S_{0}^{0}S_{1}^{0}.
\end{equation*}
The Stokes multipliers, $s_{0}^{0}$, $s_{0}^{\infty}$, and $s_{1}^{\infty}$, the
elements of the connection matrix, $(G)_{ij} \! =: \! g_{ij}$, $i,j \! \in \! \lbrace
1,2 \rbrace$, and the parameter of formal monodromy, $a$, are called the
\emph{monodromy data}.
\subsection{The Monodromy Manifold and Organisation of Paper} \label{sec1e}
In this subsection, the monodromy manifold is introduced and analysed, and the
contents of this work are delineated.
Consider $\mathbb{C}^{8}$ with co-ordinates $(a,s_{0}^{0},s_{0}^{\infty},
s_{1}^{\infty},g_{11},g_{12},g_{21},g_{22})$. The algebraic variety defined by
$\det (G) \! = \! 1$ and the \emph{semi-cyclic relation}
\begin{equation} \label{semcyc}
G^{-1}S_{0}^{0} \sigma_{1}G \! = \! S_{0}^{\infty}S_{1}^{\infty} \sigma_{3}
\mathrm{e}^{-\pi (a- \mathrm{i}/2) \sigma_{3}}
\end{equation}
are called the \emph{manifold of the monodromy data}, $\mathscr{M}$. Since
only three of the four equations in the semi-cyclic relation~\eqref{semcyc} are
independent, it follows that $\mathrm{dim}_{\mathbb{C}}(\mathscr{M}) \! = \! 4$;
more specifically, the system of algebraic equations defining $\mathscr{M}$ reads:
\begin{equation} \label{monoeqns}
\begin{gathered}
s_{0}^{\infty}s_{1}^{\infty} \! = \! -1 \! - \! \mathrm{e}^{-2 \pi a} \! - \! \mathrm{i} s_{0}^{0}
\mathrm{e}^{-\pi a}, \, \quad \, \quad \, g_{21}g_{22} \! - \! g_{11}g_{12} \! + \!
s_{0}^{0}g_{11}g_{22} \! = \! \mathrm{i} \mathrm{e}^{-\pi a}, \\
g_{11}^{2} \! - \! g_{21}^{2} \! - \! s_{0}^{0} g_{11} g_{21} \! = \! \mathrm{i}
s_{0}^{\infty} \mathrm{e}^{-\pi a}, \, \quad \, g_{22}^{2} \! - \! g_{12}^{2} \! + \!
s_{0}^{0} g_{12} g_{22} \! = \! \mathrm{i} s_{1}^{\infty} \mathrm{e}^{\pi a}, \, \quad \,
g_{11}g_{22} \! - \! g_{12} g_{21} \! = \! 1.
\end{gathered}
\end{equation}
\begin{eeee} \label{newrem13}
To achieve a one-to-one correspondence between the coefficients of the
System \eqref{newlax3} and the points on $\mathscr{M}$, one has to factorize
$\mathscr{M}$ by the involution $G \! \to \! -G$ (cf. Remark~\ref{newrem12}).
\hfill $\blacksquare$
\end{eeee}
The cube roots of unity, that is, $1^{1/3} \! = \! \exp (\mathrm{i} 2 \pi m^{\prime}/3)$, $m^{\prime}
\! \in \! \lbrace 0,\pm 1 \rbrace$, play a seminal r\^{o}le in the asymptotic analysis of solutions
to the DP3E~\eqref{eq1.1}; in fact, the monodromy data is acutely dependent on the integer
index $m^{\prime} \! \in \! \lbrace 0,\pm 1 \rbrace$, subsequently so are solutions of the
DP3E~\eqref{eq1.1}, because they are parametrised in terms of points on $\mathscr{M}$. More
precisely, as shown in Section~2 of \cite{a1}, Equations~\eqref{monoeqns} defining $\mathscr{M}$
are equivalent to one of the following three systems: \textbf{(i)}\footnote{This case does not
exclude the possibility that $g_{12} \! = \! 0$ or $g_{21} \! = \! 0$.} $g_{11}g_{22} \! \neq \! 0$
$\Rightarrow$
\begin{gather} \label{monok1}
s_{0}^{\infty} \! = \! -\dfrac{(g_{21} \! + \! \mathrm{i} \mathrm{e}^{\pi a}g_{11})}{g_{22}}, \,
\quad \, s_{1}^{\infty} \! = \! -\dfrac{\mathrm{i} (g_{22} \! + \! \mathrm{i} \mathrm{e}^{-\pi a}g_{12})
\mathrm{e}^{-\pi a}}{g_{11}}, \, \quad \, s_{0}^{0} \! = \! \dfrac{\mathrm{i} \mathrm{e}^{-\pi a}
\! + \! g_{11}g_{12} \! - \! g_{21}g_{22}}{g_{11}g_{22}};
\end{gather}
\textbf{(ii)} $g_{11} \! \neq \! 0$ and $g_{22} \! = \! 0$, in which case the parameters
are $s_{0}^{0}$ and $g_{11}$, and
\begin{gather} \label{monok2}
g_{12} \! = \! -\dfrac{\mathrm{i} \mathrm{e}^{-\pi a}}{g_{11}}, \quad g_{21} \! = \! -\mathrm{i} \mathrm{e}^{\pi a}
g_{11}, \quad s_{0}^{\infty} \! = \! -\mathrm{i} g_{11}^{2}(1 \! + \! \mathrm{e}^{2 \pi a} \! + \!
\mathrm{i} s_{0}^{0} \mathrm{e}^{\pi a}) \mathrm{e}^{\pi a}, \quad s_{1}^{\infty} \! = \! -\dfrac{\mathrm{i}
\mathrm{e}^{-3 \pi a}}{g_{11}^{2}};
\end{gather}
and \textbf{(iii)} $g_{11} \! = \! 0$ and $g_{22} \! \neq \! 0$, in which case the
parameters are $s_{0}^{0}$ and $g_{22}$, and
\begin{gather} \label{monok3}
g_{12} \! = \! \mathrm{i} \mathrm{e}^{\pi a}g_{22}, \, \quad \, g_{21} \! = \! \frac{\mathrm{i} \mathrm{e}^{-\pi a}}{g_{22}},
\, \quad \, s_{0}^{\infty} \! = \! -\dfrac{\mathrm{i} \mathrm{e}^{-\pi a}}{g_{22}^{2}}, \, \quad \,
s_{1}^{\infty} \! = \! -\mathrm{i} g_{22}^{2}(1 \! + \! \mathrm{e}^{2 \pi a} \! + \! \mathrm{i} s_{0}^{0}
\mathrm{e}^{\pi a}) \mathrm{e}^{-\pi a}.
\end{gather}
It turns out that the real cube root of unity $(m^{\prime} \! = \! 0)$ corresponds to
the monodromy data of case~\textbf{(i)}, and the complex-conjugate cube roots of
unity $(m^{\prime} \! = \! \pm 1)$ correspond to the monodromy datum delineated
in cases~\textbf{(ii)} and~\textbf{(iii)}. Asymptotics as $\tau \! \to \! \pm 0$ and as
$\tau \! \to \! \pm \mathrm{i} 0$ (resp., as $\tau \! \to \! \pm \infty$ and as $\tau \!
\to \! \pm \mathrm{i} \infty$) of the general (resp., general regular) solution of the
DP3E~\eqref{eq1.1}, and its associated Hamiltonian function, $\mathcal{H}(\tau)$,
parametrised in terms of the proper open subset of $\mathscr{M}$ corresponding
to case~\textbf{(i)} were presented in \cite{a1},\footnote{Asymptotics as $\tau
\! \to \! \pm 0$ and as $\tau \! \to \! \pm \mathrm{i} 0$ for the corresponding
$\pmb{\pmb{\tau}}$-function, but without the `constant term', were also conjectured
in \cite{a1}.} and asymptotics as $\tau \! \to \! \pm \infty$ and as $\tau \! \to \!
\pm \mathrm{i} \infty$ of general regular and singular solutions of the DP3E~\eqref{eq1.1},
and its associated Hamiltonian and auxiliary functions, $\mathcal{H}(\tau)$ and
$f_{-}(\tau)$,\footnote{Denoted as $f(\tau)$ in \cite{av2}.} respectively, parametrised
in terms of the proper open subset of $\mathscr{M}$ corresponding to case~\textbf{(i)}
were obtained in \cite{av2}; furthermore, three-real-parameter families of solutions to
the DP3E~\eqref{eq1.1} that possess infinite sequences of poles and zeros asymptotically
located along the imaginary and real axes were identified, and the asymptotics of these
poles and zeros were also derived. The purpose of the present work, therefore, is to
close the aforementioned gaps, and to continue to cover $\mathscr{M}$ by deriving
asymptotics (as $\tau \! \to \! \pm \infty$ and as $\tau \! \to \! \pm \mathrm{i} \infty$) of
$u(\tau)$, and the related functions $f_{\pm}(\tau)$, $\mathcal{H}(\tau)$, and $\sigma
(\tau)$, that are parametrised in terms of the complementary proper open subsets of
$\mathscr{M}$ corresponding to cases~\textbf{(ii)} and~\textbf{(iii)}.\footnote{Asymptotics
as $\tau \! \to \! \pm 0$ and as $\tau \! \to \! \pm \mathrm{i} 0$ for $u(\tau)$, $\mathcal{H}
(\tau)$, $f_{\pm}(\tau)$, and $\sigma (\tau)$ corresponding to cases~\textbf{(ii)}
and~\textbf{(iii)} will be presented elsewhere.} For notational consistency with the main
body of the text, cases~\textbf{(ii)} and~\textbf{(iii)} for $\mathscr{M}$ will, henceforth,
be referred to via the integer index $k \! \in \! \lbrace \pm 1 \rbrace$; more specifically,
case~\textbf{(ii)}, that is, $g_{11} \! \neq \! 0$, $g_{22} \! = \! 0$, and $g_{12}g_{21} \!
= \! -1$, will be designated by $k \! = \! +1$, and case~\textbf{(iii)}, that is, $g_{11}
\! = \! 0$, $g_{22} \! \neq \! 0$, and $g_{12}g_{21} \! = \! -1$, will be designated by
$k \! = \! -1$.
The contents of this paper, the main body of which is devoted to the asymptotic
analysis (as $\tau \! \to \! +\infty$ for $\varepsilon b \! > \! 0$) of $u(\tau)$ and
the related, auxiliary functions $f_{\pm}(\tau)$, $\mathcal{H}(\tau)$, and $\sigma
(\tau)$, are now described. In Section~\ref{sec2}, the main asymptotic results as $\tau
\! \to \! \pm \infty$ and as $\tau \! \to \! \pm \mathrm{i} \infty$ with $\pm (\varepsilon b)
\! > \! 0$ for $u(\tau)$, $f_{\pm}(\tau)$, $\mathcal{H}(\tau)$, and $\sigma (\tau)$
parametrised in terms of the monodromy data corresponding to the cases designated
by the index $k \! \in \! \lbrace \pm 1 \rbrace$ (see the discussion above) are stated.
In Section~\ref{sec3}, the asymptotic (as $\tau \! \to \! +\infty$ for $\varepsilon
b \! > \! 0$) solution of the direct monodromy problem for the $\mu$-part of the
System~\eqref{newlax3}, under certain mild restrictions on its coefficient functions
that are consistent with the monodromy data corresponding to $k \! \in \! \lbrace
\pm 1 \rbrace$, is presented: the latter analysis is predicated on focusing principal
emphasis on the study of the global asymptotic properties of the fundamental solution
of the System~\eqref{newlax3} via the possibility of `matching' different local asymptotic
expansions of $\Psi (\mu,\tau)$ at singular and turning points; in particular, matching
WKB-asymptotics of the fundamental solution of the System~\eqref{newlax3} with its
parametrix represented in terms of parabolic-cylinder functions in open neighbourhoods
of double-turning points. In Section~\ref{finalsec}, the asymptotic results derived in
Section~\ref{sec3} are inverted in order to solve the inverse monodromy problem for the
$\mu$-part of the System~\eqref{newlax3}, that is, explicit asymptotics for the coefficient
functions of the $\mu$-part of the System~\eqref{newlax3} are parametrised in terms
of the monodromy data corresponding to $k \! \in \! \lbrace \pm 1 \rbrace$: under
the permanency of the isomonodromy condition on the corresponding connection
matrices, namely, the monodromy data are constant and satisfy certain conditions,
one deduces that the asymptotics obtained (via inversion) satisfy all of the restrictions
imposed in Section~\ref{sec3}; according to the justification scheme presented in
\cite{a20} (see, also, \cite{bolisachp,a5,a22}), it follows, therefore, that there
exist---exact---solutions of the system of isomonodromy deformations~\eqref{newlax8}
whose asymptotics coincide with those obtained in this section. In order to extend the
results derived in Sections~\ref{sec3} and~\ref{finalsec} for asymptotics of $u(\tau)$,
$f_{\pm}(\tau)$, $\mathcal{H}(\tau)$, and $\sigma (\tau)$ on the positive semi-axis
$(\tau \! \to \! +\infty)$ for $\varepsilon b \! > \! 0$ to asymptotics on the negative
semi-axis $(\tau \! \to \! -\infty)$ and on the imaginary axis $(\tau \! \to \! \pm \mathrm{i}
\infty)$ for both positive and negative values of $\varepsilon b$, one applies the (group)
action of the Lie-point symmetries changing $\tau \! \to \! -\tau$, $\tau \! \to \! \tau$,
$a \! \to \! -a$, and $\tau \! \to \! \mathrm{i} \tau$ derived in Appendix~\ref{sectonsymm}
on the proper open subsets of $\mathscr{M}$ corresponding to $k \! \in \! \lbrace \pm
1 \rbrace$. Finally, in Appendix~\ref{feetics}, asymptotics as $\tau \! \to \! \pm \infty$
and as $\tau \! \to \! \pm \mathrm{i} \infty$ with $\pm (\varepsilon b) \! > \! 0$ for the
multi-valued function $\hat{\varphi}(\tau)$ (cf. Proposition~\ref{prop1.2}) are presented.
\section{Summary of Results} \label{sec2}
In this work, the detailed analysis of asymptotics as $\tau \! \to \! +\infty$ for
$\varepsilon b \! > \! 0$ of $u(\tau)$ and the associated functions $f_{\pm}(\tau)$,
$\mathcal{H}(\tau)$, $\sigma (\tau)$, and $\hat{\varphi}(\tau)$ is presented (see
Sections~\ref{sec3} and~\ref{finalsec}, and Appendix~\ref{feetics} below). In order to
arrive at the corresponding asymptotics of $u(\tau)$, $f_{\pm}(\tau)$, $\mathcal{H}
(\tau)$, $\sigma (\tau)$, and $\hat{\varphi}(\tau)$ for positive, negative, and
pure-imaginary values of $\tau$ for both positive and negative values of $\varepsilon b$,
one applies the action of the Lie-point symmetries changing $\tau \! \to \! -\tau$,
$\tau \! \to \! \tau$, $a \! \to \! -a$, and $\tau \! \to \! \mathrm{i} \tau$ (see
Appendices~\ref{sectonsymmt}--\ref{sectonsymmtit}, respectively, below) on $\mathscr{M}$.
The `composed' symmetries of these actions on $\mathscr{M}$ are presented in
Appendix~\ref{sectonsymmcomp} below in terms of two auxiliary mappings, both of which
are isomorphisms on $\mathscr{M}$, denoted by $\mathscr{F}^{\scriptscriptstyle \lbrace
\ell \rbrace}_{\scriptscriptstyle \varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2})}$, which
is relevant for real $\tau$, and $\hat{\mathscr{F}}^{\scriptscriptstyle \lbrace \hat{\ell}
\rbrace}_{\scriptscriptstyle \hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2})}$, which is relevant for pure-imaginary $\tau$; more precisely,
{}from Appendix~\ref{sectonsymmcomp},\footnote{Due to the involution $G \! \to \!
-G$ (cf. Remarks~\ref{newrem12} and~\ref{newrem13}), it suffices to take $\tilde{l} \!
= \! l^{\prime} \! = \! +1$ in Equations~\eqref{laxhat76}--\eqref{laxhat121} below.}
\begin{align} \label{newpam1}
\mathscr{F}^{\scriptscriptstyle \lbrace \ell \rbrace}_{\scriptscriptstyle \varepsilon_{1},
\varepsilon_{2},m(\varepsilon_{2})} \colon \mathscr{M} \! \to \! \mathscr{M}, \, \,
&(a,s_{0}^{0},s_{0}^{\infty},s_{1}^{\infty},g_{11},g_{12},g_{21},g_{22}) \! \mapsto \!
\left((-1)^{\varepsilon_{2}}a,s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2})
\vert \ell), \right. \nonumber \\
&\left. \, s_{0}^{\infty}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell),
s_{1}^{\infty}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell),g_{11}
(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell), \right. \nonumber \\
&\left. \, g_{12}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell),g_{21}
(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell),g_{22}(\varepsilon_{1},
\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \right),
\end{align}
where $\varepsilon_{1},\varepsilon_{2} \! \in \! \lbrace 0,\pm 1 \rbrace$,
$m(\varepsilon_{2}) \! = \!
\left\{
\begin{smallmatrix}
0, \, \, \varepsilon_{2}=0, \\
\pm \varepsilon_{2}, \, \, \varepsilon_{2} \in \lbrace \pm 1 \rbrace,
\end{smallmatrix}
\right.$ $\ell \! \in \! \lbrace 0,1 \rbrace$, and the explicit expressions for $s_{0}^{0}
(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell)$, $s_{0}^{\infty}
(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell)$, $s_{1}^{\infty}
(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell)$, and $g_{ij}
(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell)$, $i,j \! \in \! \lbrace
1,2 \rbrace$, are given in Equations~\eqref{laxhat76}--\eqref{laxhat90}
and~\eqref{laxhat99}--\eqref{laxhat113} below, and
\begin{align} \label{newpam3}
\hat{\mathscr{F}}^{\scriptscriptstyle \lbrace \hat{\ell} \rbrace}_{\scriptscriptstyle
\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2})} \colon
\mathscr{M} \! \to \! \mathscr{M}, \, \, &(a,s_{0}^{0},s_{0}^{\infty},s_{1}^{\infty},
g_{11},g_{12},g_{21},g_{22}) \! \mapsto \! \left((-1)^{1+\hat{\varepsilon}_{2}}a,
\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}), \right. \nonumber \\
&\left. \, \hat{s}_{0}^{\infty}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}),\hat{s}_{1}^{\infty}(\hat{\varepsilon}_{1},
\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell}),\hat{g}_{11}
(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert
\hat{\ell}), \right. \nonumber \\
&\left. \, \hat{g}_{12}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}),\hat{g}_{21}(\hat{\varepsilon}_{1},
\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell}),\hat{g}_{22}
(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2})
\vert \hat{\ell}) \right),
\end{align}
where $\hat{\varepsilon}_{1} \! \in \! \lbrace \pm 1 \rbrace$, $\hat{\varepsilon}_{2}
\! \in \! \lbrace 0,\pm 1 \rbrace$, $\hat{m}(\hat{\varepsilon}_{2}) \! = \!
\left\{
\begin{smallmatrix}
0, \, \, \hat{\varepsilon}_{2} \in \lbrace \pm 1 \rbrace, \\
\pm \hat{\varepsilon}_{1}, \, \, \hat{\varepsilon}_{2}=0,
\end{smallmatrix}
\right.$ $\hat{\ell} \! \in \! \lbrace 0,1 \rbrace$, and the explicit expressions for
$\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2})
\vert \hat{\ell})$, $\hat{s}_{0}^{\infty}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},
\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell})$, $\hat{s}_{1}^{\infty}(\hat{\varepsilon}_{1},
\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell})$, and $\hat{g}_{ij}
(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert
\hat{\ell})$, $i,j \! \in \! \lbrace 1,2 \rbrace$, are given in
Equations~\eqref{laxhat91}--\eqref{laxhat98} and~\eqref{laxhat114}--\eqref{laxhat121}
below.
\begin{eeeee} \label{stokmxx}
\textsl{It is worth noting that $s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},
m(\varepsilon_{2}) \vert \ell) \! = \! s_{0}^{0} \! = \! \hat{s}_{0}^{0}(\hat{\varepsilon}_{1},
\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell})$$;$ furthermore,
it follows that $\operatorname{card} \lbrace (\varepsilon_{1},\varepsilon_{2},
m(\varepsilon_{2}) \vert \ell) \rbrace \! = \! 30$ and $\operatorname{card} \lbrace
(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert
\hat{\ell}) \rbrace \! = \! 16$, that is, for $\ell,\hat{\ell} \! \in \! \lbrace 0,1 \rbrace$,}
\begin{equation*}
(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! = \!
\begin{lcase}
(0,0,0 \vert \ell), \\
(-1,0,0 \vert \ell), \\
(1,0,0 \vert \ell), \\
(0,-1,-1 \vert \ell), \\
(0,-1,1 \vert \ell), \\
(0,1,-1 \vert \ell), \\
(0,1,1 \vert \ell), \\
(-1,-1,-1 \vert \ell), \\
(1,-1,-1 \vert \ell), \\
(-1,-1,1 \vert \ell), \\
(1,-1,1 \vert \ell), \\
(-1,1,-1 \vert \ell), \\
(1,1,-1 \vert \ell), \\
(-1,1,1 \vert \ell), \\
(1,1,1 \vert \ell),
\end{lcase} \, \, \quad \, \, \text{and} \, \, \quad \, \, (\hat{\varepsilon}_{1},
\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! = \!
\begin{lcase}
(1,1,0 \vert \hat{\ell}), \\
(1,-1,0 \vert \hat{\ell}), \\
(-1,1,0 \vert \hat{\ell}), \\
(-1,-1,0 \vert \hat{\ell}), \\
(1,0,-1 \vert \hat{\ell}), \\
(-1,0,-1 \vert \hat{\ell}), \\
(1,0,1 \vert \hat{\ell}), \\
(-1,0,1 \vert \hat{\ell}).
\end{lcase}
\end{equation*}
\end{eeeee}
\begin{eeeee} \label{rem2.1}
The roots and fractional powers of positive quantities are assumed positive, whilst the
branches of the roots of complex quantities can be taken arbitrarily, unless stated
otherwise; moreover, it is assumed that, for negative real $z$, the following branches
are always taken: $z^{1/3} \! := \! -\lvert z \rvert^{1/3}$ and $z^{2/3} \! := \!
(z^{1/3})^{2}$. \hfill $\blacksquare$
\end{eeeee}
Via the above-defined notation(s) and Remark~\ref{stokmxx}, asymptotics as $\tau
\! \to \! \pm \infty$ (resp., $\tau \! \to \! \pm \mathrm{i} \infty$) for $\pm (\varepsilon
b) \! > \! 0$ of $u(\tau)$, $f_{\pm}(\tau)$, $\mathcal{H}(\tau)$, and $\sigma
(\tau)$ are presented in Theorem~\ref{theor2.1} (resp., Theorem~\ref{appen})
below.\footnote{Asymptotics as $\tau \! \to \! \pm \infty$ (resp., $\tau \! \to
\! \pm \mathrm{i} \infty$) for $\pm (\varepsilon b) \! > \! 0$ of $\hat{\varphi}(\tau)$
are presented in Appendix~\ref{feetics}, Theorem~\ref{pfeetotsa} (resp.,
Theorem~\ref{pfeetotsb}) below.}
\begin{ddddd} \label{theor2.1}
Let $\varepsilon_{1},\varepsilon_{2} \! \in \! \lbrace 0,\pm 1 \rbrace$,
$m(\varepsilon_{2}) \! = \!
\left\{
\begin{smallmatrix}
0, \, \, \varepsilon_{2}=0, \\
\pm \varepsilon_{2}, \, \, \varepsilon_{2} \in \lbrace \pm 1 \rbrace,
\end{smallmatrix}
\right.$ $\ell \! \in \! \lbrace 0,1 \rbrace$, $\varepsilon b \! = \! \vert \varepsilon
b \vert \mathrm{e}^{\mathrm{i} \pi \varepsilon_{2}}$, and $u(\tau)$ be a solution of the
{\rm DP3E}~\eqref{eq1.1} corresponding to the monodromy data $(a,s^{0}_{0},
s^{\infty}_{0},s^{\infty}_{1},g_{11},g_{12},g_{21},g_{22})$.\footnote{Note that
(cf. the identity map~\eqref{laxhat76} below) $s_{0}^{0}(0,0,0 \vert 0) \! = \!
s_{0}^{0}$, $s_{0}^{\infty}(0,0,0 \vert 0) \! = \! s_{0}^{\infty}$, $s_{1}^{\infty}
(0,0,0 \vert 0) \! = \! s_{1}^{\infty}$, and $g_{ij}(0,0,0 \vert 0) \! = \! g_{ij}$,
$i,j \! \in \! \lbrace 1,2 \rbrace$.} For $k \! = \! +1$, let
\begin{equation*}
g_{11}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell)g_{12}
(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell)g_{21}(\varepsilon_{1},
\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! \neq \! 0 \quad \text{and} \quad
g_{22}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! = \! 0,
\end{equation*}
and, for $k \! = \! -1$, let
\begin{equation*}
g_{11}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! = \! 0 \quad
\text{and} \quad g_{12}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert
\ell)g_{21}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell)g_{22}
(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! \neq \! 0.
\end{equation*}
Then, for $s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell)
\! \neq \! \mathrm{i} \mathrm{e}^{(-1)^{1+\varepsilon_{2}} \pi a}$,\footnote{Recall that (cf.
Remark~\ref{stokmxx}) $s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2})
\vert \ell) \! = \! s_{0}^{0}$. For $s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},
m(\varepsilon_{2}) \vert \ell) \! = \! \mathrm{i} \mathrm{e}^{(-1)^{1+\varepsilon_{2}} \pi a}$,
the exponentially small correction terms in Asymptotics~\eqref{thmk11},
\eqref{thmk17}, \eqref{efhpls1}, \eqref{thmk21}, and~\eqref{thmk25} are absent.}
\begin{align} \label{thmk11}
u(\tau) \underset{\tau \to +\infty \mathrm{e}^{\mathrm{i} \pi \varepsilon_{1}}}{=} u_{0,k}^{\ast}
(\tau) - &\dfrac{(-1)^{\varepsilon_{1}} \mathrm{i} \varepsilon (\varepsilon
b \mathrm{e}^{-\mathrm{i} \pi \varepsilon_{2}})^{1/2} \mathrm{e}^{\mathrm{i} \pi k/4}(s_{0}^{0}(\varepsilon_{1},
\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! - \! \mathrm{i} \mathrm{e}^{(-1)^{1+\varepsilon_{2}}
\pi a})}{\sqrt{\pi} \, 2^{3/2}3^{1/4}(2 \! + \! \sqrt{3})^{\mathrm{i} k(-1)^{1+\varepsilon_{2}}a}}
\nonumber \\
\times& \, \mathrm{e}^{-\mathrm{i} k (-1)^{\varepsilon_{2}} \vartheta (\tau)} \mathrm{e}^{(-1)^{1+
\varepsilon_{2}} \beta (\tau)} \! \left(1 \! + \! \mathcal{O}(\tau^{-1/3}) \right),
\quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{equation} \label{thmk1}
u_{0,k}^{\ast}(\tau) \! := \! c_{0,k} \tau^{1/3} \! \left(1 \! + \! \tau^{-2/3}
\sum_{m=0}^{\infty} \dfrac{\mathfrak{u}_{m}(k)}{((-1)^{\varepsilon_{1}}
\tau^{1/3})^{m}} \right),
\end{equation}
with
\begin{gather}
c_{0,k} \! := \! \dfrac{\varepsilon (\varepsilon b)^{2/3}}{2} \mathrm{e}^{-\mathrm{i} 2 \pi k/3},
\label{thmk2} \\
\mathfrak{u}_{0}(k) \! = \! \dfrac{a \mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{3(\varepsilon b)^{1/3}}
\! = \! \dfrac{a}{6 \alpha_{k}^{2}}, \, \quad \, \quad \, \mathfrak{u}_{1}(k) \! = \!
\mathfrak{u}_{2}(k) \! = \! \mathfrak{u}_{3}(k) \! = \! \mathfrak{u}_{5}(k) \! = \!
\mathfrak{u}_{7}(k) \! = \! \mathfrak{u}_{9}(k) \! = \! 0, \label{thmk3} \\
\mathfrak{u}_{4}(k) \! = \! -\dfrac{a(a^{2} \! + \! 1)}{3^{4}(\varepsilon b)},
\qquad \mathfrak{u}_{6}(k) \! = \! \dfrac{a^{2}(a^{2} \! + \! 1)
\mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{3^{5}(\varepsilon b)^{4/3}}, \qquad \mathfrak{u}_{8}(k)
\! = \! \dfrac{a(a^{2} \! + \! 1) \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{3^{5}(\varepsilon b)^{5/3}},
\label{thmk4}
\end{gather}
where
\begin{equation} \label{thmk5}
\alpha_{k} \! := \! 2^{-1/2}(\varepsilon b)^{1/6} \mathrm{e}^{\mathrm{i} \pi k/3},
\end{equation}
and, for $m \! \in \! \lbrace 0 \rbrace \cup \mathbb{N} \! =: \! \mathbb{Z}_{+}$,
\begin{align} \label{thmk6}
\mathfrak{u}_{2(m+5)}(k) \! =& \, \dfrac{1}{27} \! \left(\frac{c_{0,k}}{b} \right)^{2} \!
\left(\vphantom{M^{M^{M^{M^{M}}}}} \mathfrak{w}_{2(m+3)}(k) \! - \! 2
\mathfrak{u}_{0}(k) \mathfrak{w}_{2(m+2)}(k) \! + \! \eta_{2(m+2)}(k) \! - \!
\mathfrak{u}_{0}(k) \eta_{2(m+1)}(k) \right. \nonumber \\
+&\left. \, \sum_{p=0}^{2m} \eta_{p}(k) \mathfrak{w}_{2(m+1)-p}(k) \right) \! - \!
\dfrac{1}{3} \sum_{p=0}^{2(m+4)}(\mathfrak{u}_{p}(k) \! + \! \mathfrak{w}_{p}(k))
\mathfrak{u}_{2(m+4)-p}(k) \nonumber \\
-& \, \frac{1}{3} \! \left(\frac{c_{0,k}}{b} \right)^{2} \! \left(\frac{2m \! + \! 7}{3}
\right)^{2} \mathfrak{u}_{2(m+3)}(k),
\end{align}
\begin{equation} \label{thmk7}
\mathfrak{u}_{2(m+5)+1}(k) \! = \! 0,
\end{equation}
where
\begin{equation} \label{thmk8}
\mathfrak{w}_{0}(k) \! = \! -\mathfrak{u}_{0}(k), \quad \quad \mathfrak{w}_{1}(k) \! = \! 0,
\quad \quad \mathfrak{w}_{n+2}(k) \! = \! -\mathfrak{u}_{n+2}(k) \! - \! \sum_{p=0}^{n}
\mathfrak{w}_{p}(k) \mathfrak{u}_{n-p}(k), \quad n \! \in \! \mathbb{Z}_{+},
\end{equation}
with
\begin{gather} \label{thmk10}
\eta_{j}(k) \! := \! -2(j \! + \! 3) \mathfrak{u}_{j+2}(k) \! + \! \sum_{p=0}^{j}(p \! + \! 1)
(j \! - \! p \! + \! 1) \mathfrak{u}_{p}(k) \mathfrak{u}_{j-p}(k), \quad j \! \in \! \mathbb{Z}_{+},
\end{gather}
and
\begin{equation} \label{thmk12}
\vartheta (\tau) \! := \! \dfrac{3 \sqrt{3}}{2}(\varepsilon b)^{1/3} \tau^{2/3} \,, \qquad
\qquad \beta (\tau) \! := \! \dfrac{9}{2}(\varepsilon b)^{1/3} \tau^{2/3}.
\end{equation}
Let the auxiliary function $f_{-}(\tau)$ (corresponding to $u(\tau)$ above) defined by
Equation~\eqref{hatsoff7} solve the second-order non-linear {\rm ODE}~\eqref{thmk13},
and let the auxiliary function $f_{+}(\tau)$ (corresponding to $u(\tau)$ above) defined by
Equation~\eqref{pga3} solve the second-order non-linear {\rm ODE}~\eqref{yooplus3}.
Then, for $s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! \neq
\! \mathrm{i} \mathrm{e}^{(-1)^{1+\varepsilon_{2}} \pi a}$,
\begin{align} \label{thmk17}
2f_{-}(\tau) \underset{\tau \to +\infty \mathrm{e}^{\mathrm{i} \pi \varepsilon_{1}}}{=} f_{0,k}^{\ast}(\tau)
- &\dfrac{(-1)^{\varepsilon_{1}}k(\varepsilon b \mathrm{e}^{-\mathrm{i} \pi \varepsilon_{2}})^{1/6}
\mathrm{e}^{\mathrm{i} \pi k/4} \mathrm{e}^{\mathrm{i} \pi k/3}(s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},
m(\varepsilon_{2}) \vert \ell) \! - \! \mathrm{i} \mathrm{e}^{(-1)^{1+\varepsilon_{2}} \pi a})}{\sqrt{\pi} \,
2^{k/2}3^{1/4}(\sqrt{3} \! + \! 1)^{-k}(2 \! + \! \sqrt{3})^{\mathrm{i} k(-1)^{1+\varepsilon_{2}}a}}
\nonumber \\
\times& \, \tau^{1/3} \mathrm{e}^{-\mathrm{i} k(-1)^{\varepsilon_{2}} \vartheta (\tau)}
\mathrm{e}^{(-1)^{1+\varepsilon_{2}} \beta (\tau)} \! \left(1 \! + \! \mathcal{O}(\tau^{-1/3})
\right), \quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{equation} \label{thmk14}
f_{0,k}^{\ast}(\tau) \! := \! -\mathrm{i} \! \left((-1)^{\varepsilon_{2}}a \! - \! \mathrm{i}/2 \right)
\! + \! \dfrac{\mathrm{i} (-1)^{\varepsilon_{2}}(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{2}
\tau^{2/3} \! \left(-2 \! + \! \tau^{-2/3} \sum_{m=0}^{\infty} \dfrac{\mathfrak{r}_{m}
(k)}{((-1)^{\varepsilon_{1}} \tau^{1/3})^{m}} \right),
\end{equation}
and
\begin{align} \label{efhpls1}
\frac{(-1)^{\varepsilon_{2}} \mathrm{i} 4}{\varepsilon b}f_{+}(\tau) \underset{\tau \to
+\infty \mathrm{e}^{\mathrm{i} \pi \varepsilon_{1}}}{=} \mathfrak{f}_{0,k}^{\ast}(\tau) +
&\dfrac{(-1)^{\varepsilon_{1}}(\varepsilon b \mathrm{e}^{-\mathrm{i} \pi \varepsilon_{2}})^{1/6}
\mathrm{e}^{\mathrm{i} \pi k/4} \mathrm{e}^{\mathrm{i} \pi k/3}(2^{(k+1)/2} \! - \! k(\sqrt{3} \! + \! 1)^{k})}{
\sqrt{\pi} \, 2^{k/2}3^{1/4}(2 \! + \! \sqrt{3})^{\mathrm{i} k(-1)^{1+\varepsilon_{2}}a}}
\nonumber \\
\times& \, (s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2})
\vert \ell) \! - \! \mathrm{i} \mathrm{e}^{(-1)^{1+\varepsilon_{2}} \pi a}) \tau^{1/3}
\mathrm{e}^{-\mathrm{i} k(-1)^{\varepsilon_{2}} \vartheta (\tau)} \nonumber \\
\times& \, \mathrm{e}^{(-1)^{1+\varepsilon_{2}} \beta (\tau)} \! \left(1 \! + \!
\mathcal{O}(\tau^{-1/3}) \right), \quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{equation} \label{efhpls2}
\mathfrak{f}_{0,k}^{\ast}(\tau) \! := \! \mathrm{i} \! \left((-1)^{\varepsilon_{2}}a \!
+ \! \mathrm{i}/2 \right) \! + \! \mathrm{i} (-1)^{\varepsilon_{2}}(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} 2 \pi k/3} \tau^{2/3} \! \left(1 \! + \! \tau^{-2/3} \sum_{m=0}^{
\infty} \dfrac{(\frac{1}{2} \mathfrak{r}_{m}(k) \! + \! 2 \mathfrak{w}_{m}(k))}{
((-1)^{\varepsilon_{1}} \tau^{1/3})^{m}} \right),
\end{equation}
with
\begin{equation} \label{thmk15}
\mathfrak{r}_{0}(k) \! = \! \frac{a \! - \! \mathrm{i} (-1)^{\varepsilon_{2}}/2}{3 \alpha_{k}^{2}},
\, \quad \, \mathfrak{r}_{1}(k) \! = \! 0, \, \quad \, \mathfrak{r}_{2}(k) \! = \! \frac{\mathrm{i}
(-1)^{\varepsilon_{2}}a(1 \! + \! \mathrm{i} (-1)^{\varepsilon_{2}}a)}{18 \alpha_{k}^{4}}, \,
\quad \, \mathfrak{r}_{3}(k) \! = \! 0,
\end{equation}
\begin{align} \label{thmk16}
\mathrm{i} 2 \alpha_{k}^{2} \mathfrak{r}_{m+4}(k) \! =& \, \sum_{p=0}^{m} \left(
\mathrm{i} 4 \alpha_{k}^{2}(\mathfrak{u}_{m+2-p}(k) \! - \! \mathfrak{u}_{0}(k)
\mathfrak{u}_{m-p}(k)) \! - \! \dfrac{(-1)^{\varepsilon_{2}}}{3}(m \! - \! p \!
+ \! 2) \mathfrak{u}_{m-p}(k) \right) \! \mathfrak{w}_{p}(k) \nonumber \\
+& \, \mathrm{i} 4 \alpha_{k}^{2}(\mathfrak{u}_{m+4}(k) \! - \! \mathfrak{u}_{0}(k)
\mathfrak{u}_{m+2}(k)) \! - \! \dfrac{(-1)^{\varepsilon_{2}}}{3}(m \! + \! 4)
\mathfrak{u}_{m+2}(k), \quad m \! \in \! \mathbb{Z}_{+}.
\end{align}
Let the Hamiltonian function $\mathcal{H}(\tau)$ (corresponding to $u(\tau)$
above) be defined by Equation~\eqref{eqh1}. Then, for $s_{0}^{0}(\varepsilon_{1},
\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! \neq \! \mathrm{i} \mathrm{e}^{(-1)^{1+
\varepsilon_{2}} \pi a}$,
\begin{align} \label{thmk21}
\mathcal{H}(\tau) \underset{\tau \to +\infty \mathrm{e}^{\mathrm{i} \pi \varepsilon_{1}}}{=}
\mathcal{H}_{0,k}^{\ast}(\tau) - &\dfrac{(-1)^{\varepsilon_{1}}(\varepsilon b
\mathrm{e}^{-\mathrm{i} \pi \varepsilon_{2}})^{1/6} \mathrm{e}^{\mathrm{i} \pi k/4} \mathrm{e}^{\mathrm{i} \pi k/3}
(s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! - \! \mathrm{i}
\mathrm{e}^{(-1)^{1+\varepsilon_{2}} \pi a})}{\sqrt{\pi} \, 2^{k/2}3^{3/4}(\sqrt{3} \! +
\! 1)^{-k} (2 \! + \! \sqrt{3})^{\mathrm{i} k(-1)^{1+\varepsilon_{2}}a}} \nonumber \\
\times& \, \tau^{-2/3} \mathrm{e}^{-\mathrm{i} k(-1)^{\varepsilon_{2}} \vartheta (\tau)}
\mathrm{e}^{(-1)^{1+\varepsilon_{2}}\beta (\tau)} \! \left(1 \! + \! \mathcal{O}
(\tau^{-1/3}) \right), \quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{align} \label{thmk18}
\mathcal{H}_{0,k}^{\ast}(\tau) \! :=& \, 3(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}
\tau^{1/3} \! + \! 2(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3}(a \! - \! \mathrm{i}
(-1)^{\varepsilon_{2}}/2) \tau^{-1/3} \! + \! \dfrac{1}{6} \! \left((a \! - \! \mathrm{i}
(-1)^{\varepsilon_{2}}/2)^{2} \right. \nonumber \\
-&\left. \, 1/3 \right) \! \tau^{-1} \! + \! \alpha_{k}^{2}(\tau^{-1/3})^{5}
\sum_{m=0}^{\infty} \left(\vphantom{M^{M^{M^{M}}}} \! -4(a \! - \! \mathrm{i}
(-1)^{\varepsilon_{2}}/2) \mathfrak{u}_{m+2}(k) \! + \! \alpha_{k}^{2}
\mathfrak{d}_{m}(k) \right. \nonumber \\
+&\left. \, \sum_{p=0}^{m} \left(\tilde{\mathfrak{h}}_{p}(k) \! - \! 4(a \! - \!
\mathrm{i} (-1)^{\varepsilon_{2}}/2) \mathfrak{u}_{p}(k) \right) \! \mathfrak{w}_{m-p}
(k) \right) \! \left((-1)^{\varepsilon_{1}} \tau^{-1/3} \right)^{m},
\end{align}
with
\begin{align} \label{thmk19}
\mathfrak{d}_{m}(k) \! :=& \, \sum_{p=0}^{m+2}(8 \mathfrak{u}_{p}(k)
\mathfrak{u}_{m+2-p}(k) \! + \! (4 \mathfrak{u}_{p}(k) \! - \!
\mathfrak{r}_{p}(k)) \mathfrak{r}_{m+2-p}(k)) \nonumber \\
-& \, \sum_{p_{1}=0}^{m} \sum_{m_{1}=0}^{p_{1}} \mathfrak{r}_{m_{1}}(k)
\mathfrak{r}_{p_{1}-m_{1}}(k) \mathfrak{u}_{m-p_{1}}(k), \quad m \! \in \!
\mathbb{Z}_{+},
\end{align}
and
\begin{equation} \label{thmk20}
\tilde{\mathfrak{h}}_{0}(k) \! = \! -\dfrac{(12a^{2} \! + \! 1) \mathrm{e}^{\mathrm{i} \pi k/3}}{18
(\varepsilon b)^{1/3}}, \quad \quad \tilde{\mathfrak{h}}_{1}(k) \! = \! 0, \quad
\quad \tilde{\mathfrak{h}}_{m+2}(k) \! = \! \alpha_{k}^{2} \mathfrak{d}_{m}(k).
\end{equation}
Let the auxiliary function $\sigma (\tau)$ (corresponding to $u(\tau)$ above)
defined by Equation~\eqref{thmk23} solve the second-order non-linear
{\rm ODE}~\eqref{thmk22}. Then, for $s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},
m(\varepsilon_{2}) \vert \ell) \! \neq \! \mathrm{i} \mathrm{e}^{(-1)^{1+\varepsilon_{2}} \pi a}$,
\begin{align} \label{thmk25}
\sigma (\tau) \underset{\tau \to +\infty \mathrm{e}^{\mathrm{i} \pi \varepsilon_{1}}}{=}
\sigma_{0,k}^{\ast}(\tau) - &\dfrac{(-1)^{\varepsilon_{1}}(\varepsilon b
\mathrm{e}^{-\mathrm{i} \pi \varepsilon_{2}})^{1/6} \mathrm{e}^{\mathrm{i} \pi k/4} \mathrm{e}^{\mathrm{i} \pi k/3}
(s_{0}^{0}(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! - \!
\mathrm{i} \mathrm{e}^{(-1)^{1+\varepsilon_{2}} \pi a})}{\sqrt{\pi} \, 2^{k/2}3^{3/4}(\sqrt{3}
\! + \! 1)^{-k}(1 \! + \! k \sqrt{3})^{-1}(2 \! + \! \sqrt{3})^{\mathrm{i} k(-1)^{1+
\varepsilon_{2}}a}} \nonumber \\
\times& \, \tau^{1/3} \mathrm{e}^{-\mathrm{i} k(-1)^{\varepsilon_{2}} \vartheta (\tau)}
\mathrm{e}^{(-1)^{1+\varepsilon_{2}} \beta (\tau)} \! \left(1 \! + \! \mathcal{O}
(\tau^{-1/3}) \right), \quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{align} \label{thmk24}
\sigma_{0,k}^{\ast}(\tau) \! :=& \, 3(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3} \tau^{4/3}
\! - \! \mathrm{i} (-1)^{\varepsilon_{2}}2(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3}(1 \! +
\! \mathrm{i} (-1)^{\varepsilon_{2}}a) \tau^{2/3} \! + \! \dfrac{1}{3} \! \left((1 \! + \! \mathrm{i}
(-1)^{\varepsilon_{2}}a)^{2} \right. \nonumber \\
+&\left. 1/3 \right) \! + \! \alpha_{k}^{2} \tau^{-2/3} \sum_{m=0}^{\infty} \left(
-4(a \! - \! \mathrm{i} (-1)^{\varepsilon_{2}}/2) \mathfrak{u}_{m+2}(k) \! + \! \alpha_{k}^{2}
\mathfrak{d}_{m}(k) \! + \! \sum_{p=0}^{m}(\tilde{\mathfrak{h}}_{p}(k) \right. \nonumber \\
-&\left. 4(a \! - \! \mathrm{i} (-1)^{\varepsilon_{2}}/2) \mathfrak{u}_{p}(k)) \mathfrak{w}_{m-p}
(k) \vphantom{M^{M^{M^{M}}}} \! + \! \mathrm{i} (-1)^{\varepsilon_{2}} \mathfrak{r}_{m+2}(k)
\right) \! \left((-1)^{\varepsilon_{1}} \tau^{-1/3} \right)^{m}.
\end{align}
\end{ddddd}
\begin{eeeee} \label{remgrom}
For $\mathrm{i} a \! \in \! \mathbb{Z}$, a separate analysis based on B\"{a}cklund
transformations is required in order to generate the analogue of the sequence
of $\mathbb{C}$-valued expansion coefficients $\lbrace \mathfrak{u}_{m}(k)
\rbrace$, $m \! \in \! \lbrace 0 \rbrace \cup \mathbb{N}$, $k \! = \! \pm 1$,
and the corresponding function $u_{0,k}^{\ast}(\tau)$; this comment applies,
\emph{mutatis mutandis}, to the $\mathbb{C}$-valued expansion coefficients
$\lbrace \hat{\mathfrak{u}}_{m}(k) \rbrace$ and the corresponding function
$\hat{u}_{0,k}^{\ast}(\tau)$ given in Theorem~\ref{appen} below. In fact, as
discussed in Section~1 of \cite{a1}, for fixed values of $\mathrm{i} a \! = \! n \! \in
\! \mathbb{Z}$, $\varepsilon$, and $b$, there is only one algebraic solution
(rational function of $\tau^{1/3}$) of the DP3E~\eqref{eq1.1} which is a
multi-valued function with three branches (see, also, \cite{yomura}): this
solution can be derived via the $\lvert n \rvert$-fold iteration of the B\"{a}cklund
transformations given in Subsection~6.1 of \cite{a1} to the simplest solution of the
DP3E~\eqref{eq1.1} (for $a \! = \! 0$), namely, $u(\tau) \! = \! b^{2/3} \tau^{1/3}/2
\varepsilon$. The case $\mathrm{i} a \! \in \! \mathbb{Z}$ will be considered elsewhere.
\hfill $\blacksquare$
\end{eeeee}
\begin{ddddd} \label{appen}
Let $\hat{\varepsilon}_{1} \! \in \! \lbrace \pm 1 \rbrace$, $\hat{\varepsilon}_{2}
\! \in \! \lbrace 0,\pm 1 \rbrace$, $\hat{m}(\hat{\varepsilon}_{2}) \! = \!
\left\{
\begin{smallmatrix}
0, \, \, \hat{\varepsilon}_{2} \in \lbrace \pm 1 \rbrace, \\
\pm \hat{\varepsilon}_{1}, \, \, \hat{\varepsilon}_{2}=0,
\end{smallmatrix}
\right.$ $\hat{\ell} \! \in \! \lbrace 0,1 \rbrace$, $\varepsilon b \! = \! \vert \varepsilon
b \vert \mathrm{e}^{\mathrm{i} \pi \hat{\varepsilon}_{2}}$, and $u(\tau)$ be a solution of the
{\rm DP3E}~\eqref{eq1.1} corresponding to the monodromy data $(a,s^{0}_{0},
s^{\infty}_{0},s^{\infty}_{1},g_{11},g_{12},g_{21},g_{22})$. For $k \! = \! +1$, let
\begin{equation*}
\hat{g}_{11}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2})
\vert \hat{\ell}) \hat{g}_{12}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \hat{g}_{21}(\hat{\varepsilon}_{1},
\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! \neq \! 0 \quad
\text{and} \quad \hat{g}_{22}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! = \! 0,
\end{equation*}
and, for $k \! = \! -1$, let
\begin{equation*}
\hat{g}_{11}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2})
\vert \hat{\ell}) \! = \! 0 \quad \text{and} \quad \hat{g}_{12}(\hat{\varepsilon}_{1},
\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \hat{g}_{21}
(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert
\hat{\ell}) \hat{g}_{22}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! \neq \! 0.
\end{equation*}
Then, for $\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! \neq \! \mathrm{i} \mathrm{e}^{(-1)^{\hat{\varepsilon}_{2}}
\pi a}$,\footnote{Recall that (cf. Remark~\ref{stokmxx}) $\hat{s}_{0}^{0}
(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert
\hat{\ell}) \! = \! s_{0}^{0}$. For $\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},
\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! = \! \mathrm{i}
\mathrm{e}^{(-1)^{\hat{\varepsilon}_{2}} \pi a}$, the exponentially small correction terms
in Asymptotics~\eqref{appen9}, \eqref{appen14}, \eqref{efhpls3}, \eqref{appen18},
and~\eqref{appen20} are absent.}
\begin{align} \label{appen9}
u(\tau) \underset{\tau \to +\infty \mathrm{e}^{\mathrm{i} \pi \hat{\varepsilon}_{1}/2}}{=}
\hat{u}_{0,k}^{\ast}(\tau) - &\dfrac{\mathrm{i} \mathrm{e}^{-\mathrm{i} \pi \hat{\varepsilon}_{1}/2}
\varepsilon (\varepsilon b \mathrm{e}^{-\mathrm{i} \pi \hat{\varepsilon}_{2}})^{1/2}
\mathrm{e}^{\mathrm{i} \pi k/4}(\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! - \! \mathrm{i} \mathrm{e}^{(-1)^{\hat{\varepsilon}_{2}}
\pi a})}{\sqrt{\pi} \, 2^{3/2}3^{1/4}(2 \! + \! \sqrt{3})^{\mathrm{i} k(-1)^{\hat{\varepsilon}_{2}}a}}
\nonumber \\
\times& \, \mathrm{e}^{-\mathrm{i} k (-1)^{\hat{\varepsilon}_{2}} \vartheta (\tau_{\ast})}
\mathrm{e}^{(-1)^{1+\hat{\varepsilon}_{2}} \beta (\tau_{\ast})} \! \left(1 \! + \!
\mathcal{O}(\tau^{-1/3}) \right), \quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{equation} \label{appen1}
\hat{u}_{0,k}^{\ast}(\tau) \! := \! \mathrm{e}^{-\mathrm{i} \pi \hat{\varepsilon}_{1}/2}c_{0,k}
\tau_{\ast}^{1/3} \! \left(1 \! + \! \tau_{\ast}^{-2/3} \sum_{m=0}^{\infty}
\dfrac{\hat{\mathfrak{u}}_{m}(k)}{(\tau_{\ast}^{1/3})^{m}} \right),
\end{equation}
with $c_{0,k}$ defined by Equation~\eqref{thmk2},
\begin{gather}
\tau_{\ast} \! := \! \tau \mathrm{e}^{-\mathrm{i} \pi \hat{\varepsilon}_{1}/2}, \label{teas} \\
\hat{\mathfrak{u}}_{0}(k) \! = \! -\dfrac{a \mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{3(\varepsilon
b)^{1/3}} \! = \! -\dfrac{a}{6 \alpha_{k}^{2}}, \qquad \hat{\mathfrak{u}}_{1}
(k) \! = \! \hat{\mathfrak{u}}_{2}(k) \! = \! \hat{\mathfrak{u}}_{3}(k) \!
= \! \hat{\mathfrak{u}}_{5}(k) \! = \! \hat{\mathfrak{u}}_{7}(k) \! = \!
\hat{\mathfrak{u}}_{9}(k) \! = \! 0, \label{appen2} \\
\hat{\mathfrak{u}}_{4}(k) \! = \! \dfrac{a(a^{2} \! + \! 1)}{3^{4}(\varepsilon b)},
\qquad \hat{\mathfrak{u}}_{6}(k) \! = \! \dfrac{a^{2}(a^{2} \! + \! 1)
\mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{3^{5}(\varepsilon b)^{4/3}}, \qquad \hat{\mathfrak{u}}_{8}
(k) \! = \! -\dfrac{a(a^{2} \! + \! 1) \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{3^{5}(\varepsilon b)^{5/3}},
\label{appen3}
\end{gather}
where $\alpha_{k}$ is defined by Equation~\eqref{thmk5}, and, for $m \! \in \!
\mathbb{Z}_{+}$,
\begin{align} \label{appen4}
\hat{\mathfrak{u}}_{2(m+5)}(k) \! =& \, \frac{1}{27} \! \left(\frac{c_{0,k}}{b}
\right)^{2} \! \left(\vphantom{M^{M^{M^{M}}}} \hat{\mathfrak{w}}_{2(m+3)}(k)
\! - \! 2 \hat{\mathfrak{u}}_{0}(k) \hat{\mathfrak{w}}_{2(m+2)}(k) \! + \!
\hat{\eta}_{2(m+2)}(k) \! - \! \hat{\mathfrak{u}}_{0}(k) \hat{\eta}_{2(m+1)}(k)
\right. \nonumber\\
+&\left. \, \sum_{p=0}^{2m} \hat{\eta}_{p}(k) \hat{\mathfrak{w}}_{2(m+1)-p}(k)
\right) \! - \! \frac{1}{3} \sum_{p=0}^{2(m+4)}(\hat{\mathfrak{u}}_{p}(k) \!
+ \! \hat{\mathfrak{w}}_{p}(k)) \hat{\mathfrak{u}}_{2(m+4)-p}(k)\nonumber \\
-& \, \frac{1}{3} \! \left(\frac{c_{0,k}}{b} \right)^{2} \! \left(\dfrac{2m \! + \! 7}{3}
\right)^{2} \hat{\mathfrak{u}}_{2(m+3)}(k),
\end{align}
\begin{equation} \label{appen5}
\hat{\mathfrak{u}}_{2(m+5)+1}(k) \! = \! 0,
\end{equation}
where
\begin{equation} \label{appen6}
\hat{\mathfrak{w}}_{0}(k) \! = \! -\hat{\mathfrak{u}}_{0}(k), \quad \quad
\hat{\mathfrak{w}}_{1}(k) \! = \! 0, \quad \quad \hat{\mathfrak{w}}_{n+2}(k) \!
= \! -\hat{\mathfrak{u}}_{n+2}(k) \! - \! \sum_{p=0}^{n} \hat{\mathfrak{w}}_{p}
(k) \hat{\mathfrak{u}}_{n-p}(k), \quad n \! \in \! \mathbb{Z}_{+},
\end{equation}
with
\begin{gather}
\hat{\eta}_{j}(k) \! := \! -2(j \! + \! 3) \hat{\mathfrak{u}}_{j+2}(k) \! + \! \sum_{p=0}^{j}
(p \! + \! 1)(j \! - \! p \! + \! 1) \hat{\mathfrak{u}}_{p}(k) \hat{\mathfrak{u}}_{j-p}(k),
\quad j \! \in \! \mathbb{Z}_{+}, \label{appen8}
\end{gather}
and where $\vartheta (\tau)$ and $\beta (\tau)$ are defined in Equation~\eqref{thmk12}.
Let the auxiliary function $f_{-}(\tau)$ (corresponding to $u(\tau)$ above) defined by
Equation~\eqref{hatsoff7} solve the second-order non-linear {\rm ODE}~\eqref{thmk13},
and let the auxiliary function $f_{+}(\tau)$ (corresponding to $u(\tau)$ above) defined by
Equation~\eqref{pga3} solve the second-order non-linear {\rm ODE}~\eqref{yooplus3}. Then,
for $\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2})
\vert \hat{\ell}) \! \neq \! \mathrm{i} \mathrm{e}^{(-1)^{\hat{\varepsilon}_{2}} \pi a}$,
\begin{align} \label{appen14}
2f_{-}(\tau) \underset{\tau \to +\infty \mathrm{e}^{\mathrm{i} \pi \hat{\varepsilon}_{1}/2}}{=}
\hat{f}_{0,k}^{\ast}(\tau) - &\dfrac{k(\varepsilon b \mathrm{e}^{-\mathrm{i} \pi
\hat{\varepsilon}_{2}})^{1/6} \mathrm{e}^{\mathrm{i} \pi k/4} \mathrm{e}^{\mathrm{i} \pi k/3}
(\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! - \! \mathrm{i} \mathrm{e}^{(-1)^{
\hat{\varepsilon}_{2}} \pi a})}{\sqrt{\pi} \, 2^{k/2}3^{1/4}(\sqrt{3} \! + \! 1)^{-k}
(2 \! + \! \sqrt{3})^{\mathrm{i} k(-1)^{\hat{\varepsilon}_{2}}a}} \nonumber \\
\times& \, \tau_{\ast}^{1/3} \mathrm{e}^{-\mathrm{i} k(-1)^{\hat{\varepsilon}_{2}}
\vartheta (\tau_{\ast})} \mathrm{e}^{(-1)^{1+\hat{\varepsilon}_{2}} \beta (\tau_{\ast})} \!
\left(1 \! + \! \mathcal{O}(\tau^{-1/3}) \right), \quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{equation} \label{appen11}
\hat{f}_{0,k}^{\ast}(\tau) \! := \! -\mathrm{i} \! \left((-1)^{1+\hat{\varepsilon}_{2}}a \! - \!
\mathrm{i}/2 \right) \! + \! \dfrac{\mathrm{i} (-1)^{\hat{\varepsilon}_{2}}(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} 2 \pi k/3}}{2} \tau_{\ast}^{2/3} \! \left(-2 \! + \! \tau_{\ast}^{-2/3}
\sum_{m=0}^{\infty} \dfrac{\hat{\mathfrak{r}}_{m}(k)}{(\tau_{\ast}^{1/3})^{m}} \right),
\end{equation}
and
\begin{align} \label{efhpls3}
\frac{(-1)^{\hat{\varepsilon}_{2}} \mathrm{i} 4}{\varepsilon b}f_{+}(\tau) \underset{\tau \to
+\infty \mathrm{e}^{\mathrm{i} \pi \hat{\varepsilon}_{1}/2}}{=} \hat{\mathfrak{f}}_{0,k}^{\ast}(\tau) +
&\dfrac{(\varepsilon b \mathrm{e}^{-\mathrm{i} \pi \hat{\varepsilon}_{2}})^{1/6} \mathrm{e}^{\mathrm{i} \pi k/4}
\mathrm{e}^{\mathrm{i} \pi k/3}(2^{(k+1)/2} \! - \! k(\sqrt{3} \! + \! 1)^{k})}{\sqrt{\pi} \, 2^{k/2}
3^{1/4}(2 \! + \! \sqrt{3})^{\mathrm{i} k(-1)^{\hat{\varepsilon}_{2}}a}} \nonumber \\
\times& \, (\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! - \! \mathrm{i} \mathrm{e}^{(-1)^{\hat{\varepsilon}_{2}}
\pi a}) \tau_{\ast}^{1/3} \mathrm{e}^{-\mathrm{i} k(-1)^{\hat{\varepsilon}_{2}} \vartheta
(\tau_{\ast})} \nonumber \\
\times& \, \mathrm{e}^{(-1)^{1+\hat{\varepsilon}_{2}} \beta (\tau_{\ast})} \! \left(1 \!
+ \! \mathcal{O}(\tau^{-1/3}) \right), \quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{equation} \label{efhpls4}
\hat{\mathfrak{f}}_{0,k}^{\ast}(\tau) \! := \! \mathrm{i} \! \left((-1)^{1+\hat{\varepsilon}_{2}}
a \! + \! \mathrm{i}/2 \right) \! + \! \mathrm{i} (-1)^{\hat{\varepsilon}_{2}}(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} 2 \pi k/3} \tau_{\ast}^{2/3} \! \left(1 \! + \! \tau_{\ast}^{-2/3} \sum_{m=0}^{
\infty} \dfrac{(\frac{1}{2} \hat{\mathfrak{r}}_{m}(k) \! + \! 2 \hat{\mathfrak{w}}_{m}(k))}{
(\tau_{\ast}^{1/3})^{m}} \right),
\end{equation}
with
\begin{equation} \label{appen12}
\hat{\mathfrak{r}}_{0}(k) \! = \! -\dfrac{(a \! + \! \mathrm{i} (-1)^{\hat{\varepsilon}_{2}}/2)}{
3 \alpha_{k}^{2}}, \, \quad \, \hat{\mathfrak{r}}_{1}(k) \! = \! 0, \, \quad \,
\hat{\mathfrak{r}}_{2}(k) \! = \! \dfrac{\mathrm{i} a((-1)^{1+\hat{\varepsilon}_{2}} \!
+ \! \mathrm{i} a)}{18 \alpha_{k}^{4}}, \, \quad \, \hat{\mathfrak{r}}_{3}(k) \! = \! 0,
\end{equation}
\begin{align} \label{appen13}
\mathrm{i} 2 \alpha_{k}^{2} \hat{\mathfrak{r}}_{m+4}(k) \! =& \, \sum_{p=0}^{m} \left(
\mathrm{i} 4 \alpha_{k}^{2}(\hat{\mathfrak{u}}_{m+2-p}(k) \! - \! \hat{\mathfrak{u}}_{0}
(k) \hat{\mathfrak{u}}_{m-p}(k)) \! - \! \dfrac{(-1)^{\hat{\varepsilon}_{2}}}{3}(m \! - \!
p \! + \! 2) \hat{\mathfrak{u}}_{m-p}(k) \right) \! \hat{\mathfrak{w}}_{p}(k) \nonumber \\
+& \, \mathrm{i} 4 \alpha_{k}^{2}(\hat{\mathfrak{u}}_{m+4}(k) \! - \! \hat{\mathfrak{u}}_{0}
(k) \hat{\mathfrak{u}}_{m+2}(k)) \! - \! \dfrac{(-1)^{\hat{\varepsilon}_{2}}}{3}(m \! +
\! 4) \hat{\mathfrak{u}}_{m+2}(k), \quad m \! \in \! \mathbb{Z}_{+}.
\end{align}
Let the Hamiltonian function $\mathcal{H}(\tau)$ (corresponding to $u(\tau)$ above)
be defined by Equation~\eqref{eqh1}. Then, for $\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},
\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! \neq \! \mathrm{i}
\mathrm{e}^{(-1)^{\hat{\varepsilon}_{2}} \pi a}$,
\begin{align} \label{appen18}
\mathcal{H}(\tau) \underset{\tau \to +\infty \mathrm{e}^{\mathrm{i} \pi \hat{\varepsilon}_{1}/2}}{=}
\hat{\mathcal{H}}_{0,k}^{\ast}(\tau) - &\dfrac{\mathrm{e}^{-\mathrm{i} \pi \hat{\varepsilon}_{1}/2}
(\varepsilon b \mathrm{e}^{-\mathrm{i} \pi \hat{\varepsilon}_{2}})^{1/6} \mathrm{e}^{\mathrm{i} \pi k/4}
\mathrm{e}^{\mathrm{i} \pi k/3}(\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! - \! \mathrm{i} \mathrm{e}^{(-1)^{\hat{\varepsilon}_{2}}
\pi a})}{\sqrt{\pi} \, 2^{k/2}3^{3/4}(\sqrt{3} \! + \! 1)^{-k}(2 \! + \! \sqrt{3})^{\mathrm{i} k
(-1)^{\hat{\varepsilon}_{2}}a}} \nonumber \\
\times& \, \tau_{\ast}^{-2/3} \mathrm{e}^{-\mathrm{i} k(-1)^{\hat{\varepsilon}_{2}} \vartheta
(\tau_{\ast})} \mathrm{e}^{(-1)^{1+\hat{\varepsilon}_{2}} \beta (\tau_{\ast})} \! \left(1 \!
+ \! \mathcal{O}(\tau^{-1/3}) \right), \quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{align} \label{appen15}
\hat{\mathcal{H}}_{0,k}^{\ast}(\tau) \! :=& \, \mathrm{e}^{-\mathrm{i} \pi \hat{\varepsilon}_{1}/2}
\! \left(3(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3} \tau_{\ast}^{1/3} \! + \!
(-1)^{\hat{\varepsilon}_{2}}2(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3} \! \left(
(-1)^{1+\hat{\varepsilon}_{2}}a \! - \! \mathrm{i}/2 \right) \! \tau_{\ast}^{-1/3} \right.
\nonumber \\
+&\left. \, \dfrac{1}{6} \! \left(((-1)^{1+\hat{\varepsilon}_{2}}a \! - \! \mathrm{i}/2)^{2} \! -
\! 1/3 \right) \! \tau_{\ast}^{-1} \! + \! (-1)^{\hat{\varepsilon}_{2}} \alpha_{k}^{2}
(\tau_{\ast}^{-1/3})^{5} \sum_{m=0}^{\infty} \left(-4((-1)^{1+\hat{\varepsilon}_{2}}
a \! - \! \mathrm{i}/2) \right. \right. \nonumber \\
\times&\left. \left. \hat{\mathfrak{u}}_{m+2}(k) \! + \! (-1)^{\hat{\varepsilon}_{2}}
\alpha_{k}^{2} \hat{\mathfrak{d}}_{m}(k) \! + \! \sum_{p=0}^{m} \left(
\hat{\mathfrak{h}}_{p}^{\ast}(k) \! - \! 4((-1)^{1+\hat{\varepsilon}_{2}} a \!
- \! \mathrm{i}/2) \hat{\mathfrak{u}}_{p}(k) \right) \! \hat{\mathfrak{w}}_{m-p}(k)
\right) \right. \nonumber \\
\times&\left. \left(\tau_{\ast}^{-1/3} \right)^{m} \right),
\end{align}
with
\begin{align} \label{appen16}
\hat{\mathfrak{d}}_{m}(k) \! :=& \, \sum_{p=0}^{m+2}(8 \hat{\mathfrak{u}}_{p}
(k) \hat{\mathfrak{u}}_{m+2-p}(k) \! + \! (4 \hat{\mathfrak{u}}_{p}(k) \! - \!
\hat{\mathfrak{r}}_{p}(k)) \hat{\mathfrak{r}}_{m+2-p}(k)) \nonumber \\
-& \, \sum_{p_{1}=0}^{m} \sum_{m_{1}=0}^{p_{1}} \hat{\mathfrak{r}}_{m_{1}}(k)
\hat{\mathfrak{r}}_{p_{1}-m_{1}}(k) \hat{\mathfrak{u}}_{m-p_{1}}(k), \quad m \!
\in \! \mathbb{Z}_{+},
\end{align}
and
\begin{equation} \label{appen17}
\hat{\mathfrak{h}}_{0}^{\ast}(k) \! = \! \dfrac{(-1)^{1+\hat{\varepsilon}_{2}}
(12a^{2} \! + \! 1) \mathrm{e}^{\mathrm{i} \pi k/3}}{18(\varepsilon b)^{1/3}},
\quad \quad \hat{\mathfrak{h}}_{1}^{\ast}(k) \! = \! 0, \quad \quad
\hat{\mathfrak{h}}_{m+2}^{\ast}(k) \! = \! (-1)^{\hat{\varepsilon}_{2}}
\alpha_{k}^{2} \hat{\mathfrak{d}}_{m}(k).
\end{equation}
Let the auxiliary function $\sigma (\tau)$ (corresponding to $u(\tau)$ above) defined
by Equation~\eqref{thmk23} solve the second-order non-linear {\rm ODE}~\eqref{thmk22}.
Then, for $\hat{s}_{0}^{0}(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}
(\hat{\varepsilon}_{2}) \vert \hat{\ell}) \! \neq \! \mathrm{i}
\mathrm{e}^{(-1)^{\hat{\varepsilon}_{2}} \pi a}$,
\begin{align} \label{appen20}
\sigma (\tau) \underset{\tau \to +\infty \mathrm{e}^{\mathrm{i} \pi \hat{\varepsilon}_{1}/2}}{=}
\hat{\sigma}_{0,k}^{\ast}(\tau) - &\dfrac{(\varepsilon b \mathrm{e}^{-\mathrm{i} \pi
\hat{\varepsilon}_{2}})^{1/6} \mathrm{e}^{\mathrm{i} \pi k/4} \mathrm{e}^{\mathrm{i} \pi k/3}(\hat{s}_{0}^{0}
(\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{m}(\hat{\varepsilon}_{2}) \vert
\hat{\ell}) \! - \! \mathrm{i} \mathrm{e}^{(-1)^{\hat{\varepsilon}_{2}} \pi a})}{\sqrt{\pi} \, 2^{k/2}
3^{3/4}(\sqrt{3} \! + \! 1)^{-k}(1 \! + \! k \sqrt{3})^{-1}(2 \! + \! \sqrt{3})^{\mathrm{i}
k(-1)^{\hat{\varepsilon}_{2}}a}} \nonumber \\
\times& \, \tau_{\ast}^{1/3} \mathrm{e}^{-\mathrm{i} k(-1)^{\hat{\varepsilon}_{2}} \vartheta
(\tau_{\ast})} \mathrm{e}^{(-1)^{1+\hat{\varepsilon}_{2}} \beta (\tau_{\ast})} \! \left(1 \!
+ \! \mathcal{O}(\tau^{-1/3}) \right), \quad k \! \in \! \lbrace \pm 1 \rbrace,
\end{align}
where
\begin{align} \label{appen19}
\hat{\sigma}_{0,k}^{\ast}(\tau) \! :=& \, 3(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}
\tau_{\ast}^{4/3} \! - \! \mathrm{i} (-1)^{\hat{\varepsilon}_{2}}2(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} 2 \pi k/3}(1 \! + \! \mathrm{i} (-1)^{1+\hat{\varepsilon}_{2}}a) \tau_{\ast}^{2/3}
\nonumber \\
+& \, \dfrac{1}{3} \! \left((1 \! + \! \mathrm{i} (-1)^{1+\hat{\varepsilon}_{2}}a)^{2} \! + \!
1/3 \right) \! + \! (-1)^{\hat{\varepsilon}_{2}} \alpha_{k}^{2} \tau_{\ast}^{-2/3}
\sum_{m=0}^{\infty} \left(-4((-1)^{1+\hat{\varepsilon}_{2}}a \! - \! \mathrm{i}/2)
\hat{\mathfrak{u}}_{m+2}(k) \right. \nonumber \\
+&\left. (-1)^{\hat{\varepsilon}_{2}} \alpha_{k}^{2} \hat{\mathfrak{d}}_{m}(k)
\! + \! \sum_{p=0}^{m} \left(\hat{\mathfrak{h}}_{p}^{\ast}(k) \! - \! 4((-1)^{1
+\hat{\varepsilon}_{2}}a \! - \! \mathrm{i}/2) \hat{\mathfrak{u}}_{p}(k) \right) \!
\hat{\mathfrak{w}}_{m-p}(k) \right. \nonumber \\
+&\left. \mathrm{i} \hat{\mathfrak{r}}_{m+2}(k) \vphantom{M^{M^{M^{f}}}} \right) \!
\left(\tau_{\ast}^{-1/3} \right)^{m}.
\end{align}
\end{ddddd}
\section{Asymptotic Solution of the Direct Problem of Monodromy Theory} \label{sec3}
In this section, the monodromy data introduced in Subsection~\ref{sec1d} is
calculated as $\tau \! \to \! +\infty$ for $\varepsilon b \! > \! 0$ (corresponding
to $(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! = \! (0,0,0
\vert 0)$; cf. Section~\ref{sec2}): this constitutes the first step towards the proof
of the results stated in Theorems~\ref{theor2.1}, \ref{appen}, \ref{pfeetotsa},
and~\ref{pfeetotsb}.
The aforementioned calculation consists of three components: (i) the matrix WKB
analysis for the $\mu$-part of the System~\eqref{newlax3}, that is,
\begin{equation} \label{eq3.1}
\partial_{\mu} \Psi (\mu) \! = \! \widetilde{\mathscr{U}}(\mu,\tau) \Psi (\mu),
\end{equation}
where $\Psi (\mu) \! = \! \Psi (\mu,\tau)$ (see Subsection~\ref{subsec3.1} below);
(ii) the approximation of $\Psi (\mu)$ in the neighbourhoods of the turning points
(see Subsection~\ref{sec3.2} below); and (iii) the matching of these asymptotics
(see Subsection~\ref{sec3.3} below).
Before commencing the asymptotic analysis, the notation used throughout this work
is introduced:
\begin{enumerate}
\item[(1)] $\mathrm{I} \! = \! \diag (1,1)$ is the $2 \times 2$ identity
matrix, $\sigma_{1} \! = \!
\left(
\begin{smallmatrix}
0 & 1 \\
1 & 0
\end{smallmatrix}
\right)$, $\sigma_{2} \! = \!
\left(
\begin{smallmatrix}
0 & -\mathrm{i} \\
\mathrm{i} & 0
\end{smallmatrix}
\right)$ and $\sigma_{3} \! = \!
\left(
\begin{smallmatrix}
1 & 0 \\
0 & -1
\end{smallmatrix}
\right)$ are the Pauli matrices, $\sigma_{\pm} \! := \! \tfrac{1}{2}(\sigma_{1}
\! \pm \! \mathrm{i} \sigma_{2})$, $\mathbb{Z}_{+} \! := \! \lbrace 0 \rbrace \cup
\mathbb{N}$, $\mathbb{R}_{\pm} \! := \! \lbrace \mathstrut x \! \in \!
\mathbb{R}; \, \pm x \! > \! 0 \rbrace$, and $\mathbb{C}_{\pm} \! := \!
\lbrace \mathstrut z \! \in \! \mathbb{C}; \, \pm \Im (z) \! > \! 0 \rbrace$;
\item[(2)] for $(\varsigma_{1},\varsigma_{2}) \! \in \! \mathbb{R} \times
\mathbb{R}$, the function $(z \! - \! \varsigma_{1})^{\mathrm{i} \varsigma_{2}}
\colon \mathbb{C} \setminus (-\infty,\varsigma_{1}] \! \to \! \mathbb{C}$,
$z\! \mapsto \! \exp (\mathrm{i} \varsigma_{2} \ln (z \! - \! \varsigma_{1}))$, with
the branch cut taken along $(-\infty,\varsigma_{1}]$ and the principal branch
of the logarithm chosen (that is, $\lvert \arg (z \! - \! \varsigma_{1}) \rvert
\! < \! \pi)$;
\item[(3)] for $\omega_{o} \! \in \! \mathbb{C}$ and $\widehat{\Upsilon}
\! \in \! \mathrm{M}_{2}(\mathbb{C})$, $\omega_{o}^{\ad (\sigma_{3})}
\widehat{\Upsilon } \! := \! \omega_{o}^{\sigma_{3}} \widehat{\Upsilon}
\omega_{o}^{-\sigma_{3}}$;
\item[(4)] for $\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \mathfrak{I}(z)$,
$(\mathfrak{I}(z))_{ij}$ or $\mathfrak{I}_{ij}(z)$, $i,j \! \in \! \lbrace 1,2
\rbrace$, denotes the $(i \, j)$-element of $\mathfrak{I}(z)$;
\item[(5)] for $\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \hat{\mathfrak{Y}}(z)$,
$\hat{\mathfrak{Y}}(z) \! =_{z \to z_{0}} \! \mathcal{O}(\pmb{\ast})$ (resp.,
$o(\pmb{\ast}))$ means $\hat{\mathfrak{Y}}_{ij}(z) \! =_{z \to z_{0}} \!
\mathcal{O}(\pmb{\ast}_{ij})$ (resp., $o(\pmb{\ast}_{ij}))$, $i,j \! \in \! \lbrace
1,2 \rbrace$;
\item[(6)] for $\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \hat{\mathfrak{B}}(z)$,
$\lvert \lvert \hat{\mathfrak{B}}(\pmb{\cdot}) \rvert \rvert \! := \! (\sum_{i,j=1}^{2}
\hat{\mathfrak{B}}_{ij}(\pmb{\cdot}) \overline{\hat{\mathfrak{B}}_{ij}
(\pmb{\cdot})})^{1/2}$ denotes the Hilbert-Schmidt norm, where
$\overline{\pmb{\star}}$ denotes complex conjugation of $\pmb{\star}$;
\item[(7)] for $\delta_{\ast} \! > \! 0$, $\mathscr{O}_{\delta_{\ast}}(z_{0})$
denotes the (open) $\delta_{\ast}$-neighbourhood of the point $z_{0}$,
that is, for $z_{0} \! \in \! \mathbb{C}$, $\mathscr{O}_{\delta_{\ast}}(z_{0})
\! := \! \lbrace \mathstrut z \! \in \! \mathbb{C}; \, \lvert z \! - \! z_{0}
\rvert \! < \! \delta_{\ast} \rbrace$, and, for $z_{0}$ the point at infinity,
$\mathscr{O}_{\delta_{\ast}}(\infty) \! := \! \lbrace \mathstrut z \! \in
\! \mathbb{C}; \, \lvert z \rvert \! > \! \delta_{\ast}^{-1} \rbrace$;
\item[(8)] the `symbol(s)' (`notation(s)') $c_{1},c_{2},c_{3},\dotsc$, with or
without subscripts, superscripts, underscripts, overscripts, etc., appearing
in the various error estimates are not equal but are all $\mathcal{O}(1)$.
\end{enumerate}
\subsection{Matrix WKB Analysis} \label{subsec3.1}
This subsection is devoted to the WKB analysis of Equation~\eqref{eq3.1} as
$\tau \! \to \! +\infty$ for $\varepsilon b \! > \! 0$.
In order to transform Equation~\eqref{eq3.1} into a form amenable to WKB analysis,
one uses the result of Proposition~4.1.1 in \cite{a1} (see, also, Proposition~3.2.1 in
\cite{av2}), which is summarised here for the reader's convenience.
\begin{bbbb}[\textrm{\cite{a1,av2}}] \label{prop3.1.1}
In the System~\eqref{newlax3}, let
\begin{equation} \label{eq3.2}
\begin{gathered}
A(\tau) \! = \! a(\tau) \tau^{-2/3}, \, \quad \, B(\tau) \! = \! b(\tau) \tau^{-2/3},
\, \quad \, C(\tau) \! = \! c(\tau) \tau^{-1/3}, \, \quad \, D(\tau) \! = \! d(\tau)
\tau^{-1/3}, \\
\widetilde{\mu} \! = \! \mu \tau^{1/6}, \, \quad \, \quad \, \widetilde{\Psi}
(\widetilde{\mu}) \! := \! \tau^{-\frac{1}{12} \sigma_{3}} \Psi (\widetilde{\mu}
\tau^{-1/6}),
\end{gathered}
\end{equation}
where $\widetilde{\Psi}(\widetilde{\mu}) \! = \! \widetilde{\Psi}(\widetilde{\mu},
\tau)$. Then, the $\mu$-part of the System~\eqref{newlax3} transforms as follows:
\begin{equation} \label{eq3.3}
\partial_{\widetilde{\mu}} \widetilde{\Psi}(\widetilde{\mu}) \! = \! \tau^{2/3}
\mathcal{A}(\widetilde{\mu},\tau) \widetilde{\Psi}(\widetilde{\mu}),
\end{equation}
where
\begin{equation} \label{eq3.4}
\mathcal{A}(\widetilde{\mu},\tau) \! := \! -\mathrm{i} 2 \widetilde{\mu} \sigma_{3}
\! + \!
\begin{pmatrix}
0 & -\frac{\mathrm{i} 4 \sqrt{\smash[b]{-a(\tau)b(\tau)}}}{b(\tau)} \\
-2d(\tau) & 0
\end{pmatrix} \! - \! \dfrac{1}{\widetilde{\mu}} \dfrac{\mathrm{i} r(\tau)(\varepsilon
b)^{1/3}}{2} \sigma_{3} \! + \! \dfrac{1}{\widetilde{\mu}^{2}} \!
\begin{pmatrix}
0 & \frac{\mathrm{i} (\varepsilon b)}{b(\tau)} \\
\mathrm{i} b(\tau) & 0
\end{pmatrix},
\end{equation}
with
\begin{equation} \label{eq3.5}
\dfrac{\mathrm{i} r(\tau)(\varepsilon b)^{1/3}}{2} \! = \! (\mathrm{i} a \! + \! 1/2) \tau^{-2/3} \! + \!
\dfrac{2a(\tau)d(\tau)}{\sqrt{\smash[b]{-a(\tau)b(\tau)}}}.
\end{equation}
\end{bbbb}
As in Subsection~3.2 of \cite{av2}, define the functions $h_{0}(\tau)$, $\hat{r}_{0}(\tau)$,
and $\hat{u}_{0}(\tau)$ via the relations
\begin{gather}
\sqrt{\smash[b]{-a(\tau)b(\tau)}} + \! c(\tau)d(\tau) \! + \! \dfrac{a(\tau)d(\tau)
\tau^{-2/3}}{2 \sqrt{\smash[b]{-a(\tau)b(\tau)}}} \! - \! \dfrac{1}{4}(a \! - \! \mathrm{i}/2)^{2}
\tau^{-4/3} \! = \! \dfrac{3}{4}(\varepsilon b)^{2/3} \! - \! h_{0}(\tau) \tau^{-2/3},
\label{iden1} \\
r(\tau) \! = \! -2 \! + \! \hat{r}_{0}(\tau), \label{iden3oldr} \\
\sqrt{\smash[b]{-a(\tau)b(\tau)}} = \! \dfrac{(\varepsilon b)^{2/3}}{2}
(1 \! + \! \hat{u}_{0}(\tau)). \label{iden4oldu}
\end{gather}
As follows {}from the First Integral~\eqref{aphnovij} (cf. Remark~\ref{newlax6}), the
functions $a(\tau)$, $b(\tau)$, $c(\tau)$, and $d(\tau)$ are related via the formula
\begin{equation} \label{iden6}
a(\tau)d(\tau) \! + \! b(\tau)c(\tau) \! + \! \mathrm{i} a \sqrt{\smash[b]{-a(\tau)b(\tau)}}
\tau^{-2/3} \! = \! -\mathrm{i} \varepsilon b/2, \quad \varepsilon \! \in \! \lbrace \pm
1 \rbrace.
\end{equation}
\begin{eeee} \label{remfooy6}
\textsl{It is worth noting that Equations~\eqref{iden1}--\eqref{iden6} are self-consistent;
in fact, a calculation reveals that they are equivalent to}
\begin{align}
a(\tau)d(\tau) \! =& \, \dfrac{(\varepsilon b)^{2/3}}{2}(1 \! + \! \hat{u}_{0}(\tau)) \!
\left(-\dfrac{\mathrm{i} (\varepsilon b)^{1/3}}{2} \! + \! \dfrac{\mathrm{i} (\varepsilon b)^{1/3}
\hat{r}_{0}(\tau)}{4} \! - \! \dfrac{\mathrm{i}}{2}(a \! - \! \mathrm{i}/2) \tau^{-2/3} \right),
\label{iden7} \\
b(\tau)c(\tau) \! =& \, \dfrac{(\varepsilon b)^{2/3}}{2}(1 \! + \! \hat{u}_{0}(\tau))
\! \left(-\dfrac{\mathrm{i} (\varepsilon b)^{1/3}}{2} \! + \! \mathrm{i} (\varepsilon b)^{1/3} \!
\left(\dfrac{\hat{u}_{0}(\tau)}{1 \! + \! \hat{u}_{0}(\tau)} \! - \! \dfrac{\hat{r}_{0}
(\tau)}{4} \right) \! - \! \dfrac{\mathrm{i}}{2}(a \! + \! \mathrm{i}/2) \tau^{-2/3} \right),
\label{iden8} \\
-h_{0}(\tau) \tau^{-2/3} \! =& \, \dfrac{(\varepsilon b)^{2/3}}{2} \! \left(
\dfrac{(\hat{u}_{0}(\tau))^{2} \! + \! \frac{1}{2} \hat{u}_{0}(\tau) \hat{r}_{0}
(\tau)}{1 \! + \! \hat{u}_{0}(\tau)} \! - \! \dfrac{(\hat{r}_{0}(\tau))^{2}}{8} \right)
\! + \! \dfrac{(\varepsilon b)^{1/3}(a \! - \! \mathrm{i}/2) \tau^{-2/3}}{2(1 \! + \!
\hat{u}_{0}(\tau))}; \label{iden9}
\end{align}
\textsl{moreover, via Equations~\eqref{iden4oldu}, \eqref{iden7}, and~\eqref{iden8},
one deduces that}
\begin{align}
-c(\tau)d(\tau) =& \, \left(\dfrac{\mathrm{i} (\varepsilon b)^{1/3}}{2} \! - \! \mathrm{i}
(\varepsilon b)^{1/3} \! \left(\dfrac{\hat{u}_{0}(\tau)}{1 \! + \! \hat{u}_{0}
(\tau)} \! - \! \dfrac{\hat{r}_{0}(\tau)}{4} \right) \! + \! \dfrac{\mathrm{i}}{2}
(a \! + \! \mathrm{i}/2) \tau^{-2/3} \right) \nonumber \\
\times& \, \left(\dfrac{\mathrm{i} (\varepsilon b)^{1/3}}{2} \! - \! \dfrac{\mathrm{i}
(\varepsilon b)^{1/3} \hat{r}_{0}(\tau)}{4} \! + \! \dfrac{\mathrm{i}}{2}
(a \! - \! \mathrm{i}/2) \tau^{-2/3} \right). \label{iden10}
\end{align}
\end{eeee}
In this work, in lieu of the functions $h_{0}(\tau)$, $\hat{r}_{0}(\tau)$, and
$\hat{u}_{0}(\tau)$, it is more convenient to work with the functions
$\hat{h}_{0}(\tau)$, $\tilde{r}_{0}(\tau)$, and $v_{0}(\tau)$, respectively,
which are defined as follows: for $k \! = \! \pm 1$,
\begin{gather}
h_{0}(\tau) \! := \! \left(\dfrac{3(\varepsilon b)^{2/3}}{4} \! \left(1 \! - \!
\mathrm{e}^{-\mathrm{i} 2 \pi k/3} \right) \! + \! \hat{h}_{0}(\tau) \right) \! \tau^{2/3},
\label{iden2} \\
-2 \! + \! \hat{r}_{0}(\tau) \! := \! \mathrm{e}^{\mathrm{i} 2 \pi k/3} \! \left(-2 \! + \!
\tilde{r}_{0}(\tau) \tau^{-1/3} \right), \label{iden3} \\
1 \! + \! \hat{u}_{0}(\tau) \! := \! \mathrm{e}^{-\mathrm{i} 2 \pi k/3} \! \left(1 \! + \!
v_{0}(\tau) \tau^{-1/3} \right). \label{iden4}
\end{gather}
The WKB analysis of Equation~\eqref{eq3.3} is predicated on the assumption that
the functions $\hat{h}_{0}(\tau)$, $\tilde{r}_{0}(\tau)$, and $v_{0}(\tau)$ satisfy
the---asymptotic---conditions
\begin{equation}
\begin{gathered}
\lvert \hat{h}_{0}(\tau) \rvert \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-2/3}),
\qquad \lvert \tilde{r}_{0}(\tau) \rvert \underset{\tau \to +\infty}{=} \mathcal{O}
(\tau^{-1/3}), \qquad \lvert v_{0}(\tau) \rvert \underset{\tau \to +\infty}{=}
\mathcal{O}(\tau^{-1/3}). \label{iden5}
\end{gathered}
\end{equation}
\begin{eeee} \label{abeych}
The function $\hat{h}_{0}(\tau)$ defined by Equation~\eqref{iden2} plays a prominent
r\^{o}le in the asymptotic estimates of this work; for further reference, therefore, a
compact expression for it, which simplifies several of the ensuing estimates, is
presented here: via Equation~\eqref{iden9} and the Definition~\eqref{iden2}, one
shows that
\begin{equation} \label{expforeych}
\hat{h}_{0}(\tau) \! = \! \alpha_{k}^{2} \tau^{-2/3} \! \left(\dfrac{\varkappa_{0}^{2}
(\tau)}{4} \! - \! \dfrac{(a \! - \! \mathrm{i}/2)}{1 \! + \! v_{0}(\tau) \tau^{-1/3}} \right),
\quad k \! = \! \pm 1,
\end{equation}
where $\alpha_{k}$ is defined by Equation~\eqref{thmk5}, and the function
$\varkappa_{0}^{2}(\tau)$ has the following equivalent representations:
\begin{align} \label{expforkapp}
\left(\dfrac{\varkappa_{0}(\tau)}{\tau^{1/3}} \right)^{2} =& \, \left(2 \alpha_{k}
\! + \! \dfrac{(\varepsilon b)^{1/3}r(\tau)}{2 \alpha_{k}} \right)^{2} \! + \!
\left(\dfrac{1}{\alpha_{k}^{2}} \! + \! \dfrac{r(\tau)}{(\varepsilon b)^{1/3}
(1 \! + \! \hat{u}_{0}(\tau))} \right) \! \left(-2(\varepsilon b)^{2/3}(1 \! + \!
\hat{u}_{0}(\tau)) \! + \! \dfrac{(\varepsilon b)}{\alpha_{k}^{2}} \right) \nonumber \\
=& \, -\left(2 \alpha_{k} \! - \! \dfrac{(\varepsilon b)^{1/3}r(\tau)}{2
\alpha_{k}} \right) \! \left(2 \alpha_{k} \! + \! \dfrac{(\varepsilon b)^{1/3}
r(\tau)}{2 \alpha_{k}} \right) \nonumber \\
+& \, \dfrac{1}{\alpha_{k}^{2}} \! \left(\dfrac{2(\varepsilon b)}{\alpha_{k}^{2}}
\! + \! (\varepsilon b)^{2/3} \! \left(-2(1 \! + \! \hat{u}_{0}(\tau)) \! + \!
\dfrac{r(\tau)}{1 \! + \! \hat{u}_{0}(\tau)} \right) \right) \nonumber \\
=& \, -\dfrac{(\varepsilon b)}{8 \alpha_{k}^{4}} \! \left(\dfrac{(8v_{0}^{2}
(\tau) \! + \! 4 \tilde{r}_{0}(\tau)v_{0}(\tau) \! - \! (\tilde{r}_{0}(\tau))^{2})
\tau^{-2/3} \! - \! (\tilde{r}_{0}(\tau))^{2}v_{0}(\tau) \tau^{-1}}{1 \! + \!
v_{0}(\tau) \tau^{-1/3}} \right).
\end{align}
It follows {}from the Conditions~\eqref{iden5} that $\lvert \varkappa_{0}^{2}(\tau)
\rvert \! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{-2/3})$. \hfill $\blacksquare$
\end{eeee}
From Proposition~\ref{prop1.2}, the Definitions~\eqref{newlax2}, Equations~\eqref{eq3.2},
Equation~\eqref{iden4oldu}, and the Definition~\eqref{iden4}, one deduces that, in terms
of the function $v_{0}(\tau)$, the solution of the DP3E~\eqref{eq1.1} is given by
\begin{equation} \label{iden217}
u(\tau) \! = \! c_{0,k} \tau^{1/3}(1 \! + \! \tau^{-1/3}v_{0}(\tau)), \quad k \! = \! \pm 1,
\end{equation}
where $c_{0,k}$ is defined by Equation~\eqref{thmk2}. As per the argument at the end
of Subsection~\ref{sec1a} regarding the particular form of asymptotics for $u(\tau)$ as
$\tau \! \to \! +\infty$ with $\varepsilon b \! > \! 0$ (cf. Equation~\eqref{mainyoo}
and Remark~\ref{remlndaip34}), it follows, in conjunction with the
Representation~\eqref{iden217}, that the function $v_{0}(\tau)$ can be presented in
the following form:
\begin{equation} \label{tr1}
v_{0}(\tau) \! := \! v_{0,k}(\tau) \! \underset{\tau \to +\infty}{=}
\sum_{m=0}^{\infty} \dfrac{\mathfrak{u}_{m}(k)}{(\tau^{1/3})^{m+1}} \!
+ \! \mathrm{A}_{k} \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)} \!
\left(1 \! + \! \mathcal{O}(\tau^{-1/3}) \right), \quad k \! = \! \pm 1,
\end{equation}
where the sequence of $\mathbb{C}$-valued expansion coefficients
$\lbrace \mathfrak{u}_{m}(k) \rbrace_{m=0}^{\infty}$ are determined in
Proposition~\ref{recursys} below, $\vartheta (\tau)$ and $\beta (\tau)$ are
defined in Equations~\eqref{thmk12}, and, in the course of the ensuing
analysis, it will be established that $\mathrm{A}_{k}$ depends on the Stokes
multiplier $s_{0}^{0}$ (see Section~\ref{finalsec}, Equations~\eqref{geek109}
and~\eqref{geek111}, below).\footnote{In fact, it will be shown that, as
$\tau \! \to \! +\infty$ for $\varepsilon b \! > \! 0$, if $s_{0}^{0} \! = \!
\mathrm{i} \mathrm{e}^{-\pi a}$, then $\mathrm{A}_{k} \! = \! 0$, $k \! = \! \pm 1$.}
\begin{bbbb} \label{recursys}
Let the function $v_{0}(\tau) \! := \! v_{0,k}(\tau)$, $k \! = \! \pm 1$, have the
asymptotic expansion stated in Equation~\eqref{tr1}, and let $u(\tau)$ denote
the corresponding solution of the {\rm DP3E}~\eqref{eq1.1}. Then, the expansion
coefficients $\mathfrak{u}_{m}(k)$, $m \! \in \! \mathbb{Z}_{+}$, are determined
{}from Equations~\eqref{thmk2}--\eqref{thmk10}.\footnote{For the case $\mathrm{i}
a \! \in \! \mathbb{Z}$, see Remark~\ref{remgrom}.}
\end{bbbb}
\emph{Proof}. {}From Equation~\eqref{iden217} and the Expansion~\eqref{tr1}, it
follows that the associated solution of the DP3E~\eqref{eq1.1} has asymptotics
\begin{equation} \label{recur15}
u(\tau) \underset{\tau \to +\infty}{=} c_{0,k} \tau^{1/3} \! \left(1 \! + \!
\sum_{m=0}^{\infty} \dfrac{\mathfrak{u}_{m}(k)}{(\tau^{1/3})^{m+2}} \! + \!
\mathrm{A}_{k} \tau^{-1/3} \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)} \!
\left(1 \! + \! \mathcal{O}(\tau^{-1/3}) \right) \right), \quad k \! = \! \pm 1.
\end{equation}
As the exponentially small correction term does not contribute to the algebraic
determination of the coefficients $\mathfrak{u}_{m}(k)$, $m \! \in \! \mathbb{Z}_{+}$,
$k \! = \! \pm 1$, hereafter, only the following `truncated' (and differentiable)
asymptotics of $u(\tau)$ will be considered (with abuse of notation, also denoted
by $u(\tau))$:
\begin{equation} \label{recur16}
u(\tau) \underset{\tau \to +\infty}{=} c_{0,k} \tau^{1/3} \! \left(1 \!
+ \! \tau^{-2/3} \sum_{m=0}^{\infty} \dfrac{\mathfrak{u}_{m}(k)}{
(\tau^{1/3})^{m}} \right), \quad k \! = \! \pm 1.
\end{equation}
Via the Asymptotics~\eqref{recur16}, one shows that
\begin{equation} \label{recur17}
\dfrac{1}{u(\tau)} \underset{\tau \to +\infty}{=} \dfrac{\tau^{-1/3}}{c_{0,k}}
\! \left(1 \! + \! \tau^{-2/3} \sum_{m=0}^{\infty} \dfrac{\mathfrak{w}_{m}
(k)}{(\tau^{1/3})^{m}} \right), \quad k \! = \! \pm 1,
\end{equation}
where $\mathfrak{w}_{m}(k)$, $m \! \in \! \mathbb{Z}_{+}$, are determined
iteratively {}from Equations~\eqref{thmk8}; in particular (this will be required
for the ensuing proof), for $k \! = \! \pm 1$,
\begin{align}
\mathfrak{w}_{0}(k) \! =& -\mathfrak{u}_{0}(k), \label{recur18} \\
\mathfrak{w}_{1}(k) \! =& -\mathfrak{u}_{1}(k), \label{recur19} \\
\mathfrak{w}_{2}(k) \! =& -\mathfrak{u}_{2}(k) \! + \! \mathfrak{u}_{0}^{2}
(k), \label{recur20} \\
\mathfrak{w}_{3}(k) \! =& -\mathfrak{u}_{3}(k) \! + \! 2 \mathfrak{u}_{0}
(k) \mathfrak{u}_{1}(k), \label{recur21} \\
\mathfrak{w}_{4}(k) \! =& -\mathfrak{u}_{4}(k) \! + \! 2 \mathfrak{u}_{0}(k)
\mathfrak{u}_{2}(k) \! + \! \mathfrak{u}_{1}^{2}(k) \! - \! \mathfrak{u}_{0}^{3}
(k), \label{recur22} \\
\mathfrak{w}_{5}(k) \! =& -\mathfrak{u}_{5}(k) \! + \! 2 \mathfrak{u}_{0}
(k) \mathfrak{u}_{3}(k) \! + \! 2 \mathfrak{u}_{1}(k) \mathfrak{u}_{2}(k)
\! - \! 3 \mathfrak{u}_{0}^{2}(k) \mathfrak{u}_{1}(k), \label{recur23} \\
\mathfrak{w}_{6}(k) \! =& -\mathfrak{u}_{6}(k) \! + \! 2 \mathfrak{u}_{0}(k)
\mathfrak{u}_{4}(k) \! + \! 2 \mathfrak{u}_{1}(k) \mathfrak{u}_{3}(k) \! + \!
\mathfrak{u}_{2}^{2}(k) \! - \! 3 \mathfrak{u}_{0}^{2}(k) \mathfrak{u}_{2}(k)
\! - \! 3 \mathfrak{u}_{0}(k) \mathfrak{u}_{1}^{2}(k) \! + \! \mathfrak{u}_{0}^{4}(k),
\label{recur24} \\
\mathfrak{w}_{7}(k) \! =& -\mathfrak{u}_{7}(k) \! + \! 2 \mathfrak{u}_{0}(k)
\mathfrak{u}_{5}(k) \! + \! 2 \mathfrak{u}_{1}(k) \mathfrak{u}_{4}(k) \! + \!
2 \mathfrak{u}_{2}(k) \mathfrak{u}_{3}(k) \! - \! 3 \mathfrak{u}_{3}(k)
\mathfrak{u}_{0}^{2}(k) \! - \! 6 \mathfrak{u}_{0}(k) \mathfrak{u}_{1}(k)
\mathfrak{u}_{2}(k) \nonumber \\
+& \, 4 \mathfrak{u}_{1}(k) \mathfrak{u}_{0}^{3}(k) \! - \! \mathfrak{u}_{1}^{3}(k).
\label{recur25}
\end{align}
{}From Equations~\eqref{thmk8} and the Asymptotics~\eqref{recur16}
and~\eqref{recur17}, one shows that (cf. DP3E~\eqref{eq1.1}), for $k \! = \! \pm 1$,
\begin{equation} \label{recur26}
\dfrac{b^{2}}{u(\tau)} \underset{\tau \to +\infty}{=} \dfrac{b^{2}
\tau^{-1/3}}{c_{0,k}} \! \left(1 \! - \! \mathfrak{u}_{0}(k) \tau^{-2/3} \!
- \! \mathfrak{u}_{1}(k)(\tau^{-1/3})^{3} \! - \! (\tau^{-1/3})^{4}
\sum_{m=0}^{\infty} \lambda_{m}(k)(\tau^{-1/3})^{m} \right),
\end{equation}
where $\lambda_{j}(k) \! := \! -\mathfrak{w}_{j+2}(k)$, $j \! \in \! \mathbb{Z}_{+}$,
{\fontsize{10pt}{11pt}\selectfont
\begin{align}
\dfrac{1}{\tau} \! \left(-8 \varepsilon u^{2}(\tau) \! + \! 2ab \right)
\underset{\tau \to +\infty}{=}& \, -8 \varepsilon c_{0,k}^{2} \tau^{-1/3} \! + \!
(2ab \! - \! 16 \varepsilon c_{0,k}^{2} \mathfrak{u}_{0}(k))(\tau^{-1/3})^{3} \! -
\! 16 \varepsilon c_{0,k}^{2} \mathfrak{u}_{1}(k)(\tau^{-1/3})^{4} \nonumber \\
-& \, 8 \varepsilon c_{0,k}^{2}(\tau^{-1/3})^{5} \sum_{m=0}^{\infty} \!
\left(2 \mathfrak{u}_{m+2}(k) \! + \! \sum_{p=0}^{m} \mathfrak{u}_{p}
(k) \mathfrak{u}_{m-p}(k) \right) \! (\tau^{-1/3})^{m}, \label{recur27}
\end{align}}
\begin{equation}
\dfrac{u^{\prime}(\tau)}{\tau} \underset{\tau \to +\infty}{=}
\dfrac{1}{3}c_{0,k}(\tau^{-1/3})^{5} \! \left(1 \! - \! \tau^{-2/3}
\sum_{m=0}^{\infty}(m \! + \! 1) \mathfrak{u}_{m}(k)(\tau^{-1/3})^{m}
\right), \label{recur28}
\end{equation}
{\fontsize{10pt}{11pt}\selectfont
\begin{align}
\dfrac{(u^{\prime}(\tau))^{2}}{u(\tau)} \underset{\tau \to +\infty}{=}& \,
\dfrac{1}{9}c_{0,k}(\tau^{-1/3})^{5} \! \left(1 \! - \! 3 \mathfrak{u}_{0}
(k) \tau^{-2/3} \! - \! 5 \mathfrak{u}_{1}(k)(\tau^{-1/3})^{3} \! + \!
(2 \mathfrak{u}_{0}^{2}(k) \! - \! \lambda_{0}(k) \! + \! \eta_{0}(k))
(\tau^{-1/3})^{4} \vphantom{M^{M^{M^{M^{M^{M^{M}}}}}}} \right.
\nonumber \\
+&\left. \, (6 \mathfrak{u}_{0}(k) \mathfrak{u}_{1}(k) \! - \! \lambda_{1}
(k) \! + \! \eta_{1}(k))(\tau^{-1/3})^{5} \! + \! (4 \mathfrak{u}_{1}^{2}(k)
\! - \! \lambda_{2}(k) \! + \! 2 \mathfrak{u}_{0}(k) \lambda_{0}(k) \! + \!
\eta_{2}(k) \right. \nonumber \\
-&\left. \, \mathfrak{u}_{0}(k) \eta_{0}(k))(\tau^{-1/3})^{6} \! + \!
(-\lambda_{3}(k) \! + \! 2 \mathfrak{u}_{0}(k) \lambda_{1}(k) \! + \!
4 \mathfrak{u}_{1}(k) \lambda_{0}(k) \! + \! \eta_{3}(k) \! - \!
\mathfrak{u}_{0}(k) \eta_{1}(k) \right. \nonumber \\
-&\left. \, \mathfrak{u}_{1}(k) \eta_{0}(k))(\tau^{-1/3})^{7} \! + \!
(\tau^{-1/3})^{8} \sum_{m=0}^{\infty} \! \left(-\lambda_{m+4}(k) \! +
\! 2 \mathfrak{u}_{0}(k) \lambda_{m+2}(k) \! + \! 4 \mathfrak{u}_{1}(k)
\lambda_{m+1}(k) \vphantom{M^{M^{M^{M^{M^{M^{M}}}}}}} \right.
\right. \nonumber \\
+&\left. \left. \eta_{m+4}(k) \! - \! \mathfrak{u}_{0}(k) \eta_{m+2}(k)
\! - \! \mathfrak{u}_{1}(k) \eta_{m+1}(k) \! - \! \sum_{p=0}^{m} \eta_{p}
(k) \lambda_{m-p}(k) \right) \! (\tau^{-1/3})^{m} \right), \label{recur29}
\end{align}}
where $\eta_{m}(k)$ is defined by Equation~\eqref{thmk10}, and
\begin{equation} \label{recur30}
u^{\prime \prime}(\tau) \underset{\tau \to +\infty}{=} -\dfrac{2}{9}
c_{0,k}(\tau^{-1/3})^{5} \! \left(1 \! - \! \tau^{-2/3} \sum_{m=0}^{\infty}
\dfrac{(m \! + \! 1)(m \! + \! 4)}{2} \mathfrak{u}_{m}(k)(\tau^{-1/3})^{m}
\right).
\end{equation}
Substituting, now, the Expansions~\eqref{recur26}--\eqref{recur30} into
the DP3E~\eqref{eq1.1}, and equating coefficients of like powers of
$(\tau^{-1/3})^{m}$, $m \! \in \! \mathbb{N}$, one arrives at, for $k \!
= \! \pm 1$, the following system of non-linear recurrence relations for
the expansion coefficients $\mathfrak{u}_{m^{\prime}}(k)$, $m^{\prime}
\! \in \! \mathbb{Z}_{+}$:
\begin{align}
&\mathcal{O} \! \left(\tau^{-1/3} \right) \colon & \quad 0 \! =&
-8 \varepsilon c_{0,k}^{2} \! + \! b^{2}c_{0,k}^{-1}, \label{recur31} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{3} \right) \colon & \quad 0 \! =&
-16 \varepsilon c_{0,k}^{2} \mathfrak{u}_{0}(k) \! + \! 2ab \! - \!
b^{2}c_{0,k}^{-1} \mathfrak{u}_{0}(k), \label{recur32} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{4} \right) \colon & \quad 0 \! =&
-16 \varepsilon c_{0,k}^{2} \mathfrak{u}_{1}(k) \! - \! b^{2}c_{0,k}^{-1}
\mathfrak{u}_{1}(k), \label{recur33} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{5} \right) \colon & \quad 0 \! =&
\mathfrak{t}_{k}(2,0), \label{recur34} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{6} \right) \colon & \quad 0 \! =&
\mathfrak{t}_{k}(3,1), \label{recur35} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{7} \right) \colon & \quad
\dfrac{4}{9}c_{0,k} \mathfrak{u}_{0}(k) \! =& \mathfrak{t}_{k}
(4,2), \label{recur36} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{8} \right) \colon & \quad c_{0,k}
\mathfrak{u}_{1}(k) \! =& \mathfrak{t}_{k}(5,3), \label{recur37} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{9} \right) \colon & \quad c_{0,k}
\mathfrak{u}_{2}(k) \! =& \dfrac{1}{9}c_{0,k} \! \left(2 \mathfrak{u}_{0}^{2}
(k) \! - \! \lambda_{0}(k) \! + \! \eta_{0}(k) \right) \nonumber \\
& & +& \, \mathfrak{t}_{k}(6,4), \label{recur38} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{10} \right) \colon & \quad \left(
\dfrac{4}{3} \right)^{2}c_{0,k} \mathfrak{u}_{3}(k) \! =& \dfrac{1}{9}
c_{0,k} \! \left(6 \mathfrak{u}_{0}(k) \mathfrak{u}_{1}(k) \! - \!
\lambda_{1}(k) \! + \! \eta_{1}(k) \right) \nonumber \\
& & +& \, \mathfrak{t}_{k}(7,5), \label{recur39} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{11} \right) \colon & \quad \left(
\dfrac{5}{3} \right)^{2}c_{0,k} \mathfrak{u}_{4}(k) \! =& \dfrac{1}{9}
c_{0,k} \! \left(4 \mathfrak{u}_{1}^{2}(k) \! - \! \lambda_{2}(k) \! + \!
2 \mathfrak{u}_{0}(k) \lambda_{0}(k) \right. \nonumber \\
& & +&\left. \eta_{2}(k) \! - \! \mathfrak{u}_{0}(k) \eta_{0}(k) \right)
\! + \! \mathfrak{t}_{k}(8,6), \label{recur40} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{12} \right) \colon & \quad \left(
\dfrac{6}{3} \right)^{2}c_{0,k} \mathfrak{u}_{5}(k) \! =& \dfrac{1}{9}
c_{0,k} \! \left(-\lambda_{3}(k) \! + \! 2 \mathfrak{u}_{0}(k) \lambda_{1}
(k) \! + \! 4 \mathfrak{u}_{1}(k) \lambda_{0}(k) \right. \nonumber \\
& & +&\left. \eta_{3}(k) \! - \! \mathfrak{u}_{0}(k) \eta_{1}(k) \! - \!
\mathfrak{u}_{1}(k) \eta_{0}(k) \right) \! + \! \mathfrak{t}_{k}(9,7),
\label{recur41} \\
&\mathcal{O} \! \left((\tau^{-1/3})^{m+13} \right) \colon & \quad \left(
\dfrac{m \! + \! 7}{3} \right)^{2}c_{0,k} \mathfrak{u}_{m+6}(k) \! =&
\dfrac{1}{9}c_{0,k} \! \left(-\lambda_{m+4}(k) \! + \! 2 \mathfrak{u}_{0}
(k) \lambda_{m+2}(k) \right. \nonumber \\
& & +&\left. 4 \mathfrak{u}_{1}(k) \lambda_{m+1}(k) \! + \! \eta_{m+4}
(k) \! - \! \mathfrak{u}_{0}(k) \eta_{m+2}(k) \right. \nonumber \\
& & -&\left. \mathfrak{u}_{1}(k) \eta_{m+1}(k) \! - \! \sum_{p=0}^{m}
\eta_{p}(k) \lambda_{m-p}(k) \right) \nonumber \\
& & +& \, \mathfrak{t}_{k}(m \! + \! 10,m \! + \! 8), \quad m \! \in \!
\lbrace 0 \rbrace \cup \mathbb{N}, \label{recur42}
\end{align}
where
\begin{equation} \label{teekayjayell}
\mathfrak{t}_{k}(j,l) \! := \! -8 \varepsilon c_{0,k}^{2} \! \left(2
\mathfrak{u}_{j}(k) \! + \! \sum_{p=0}^{l} \mathfrak{u}_{p}(k)
\mathfrak{u}_{l-p}(k) \right) \! - \! b^{2}c_{0,k}^{-1} \lambda_{l}(k).
\end{equation}
Noting that (cf. Definition~\eqref{thmk2}) Equation~\eqref{recur31} is identically
true, the algorithm, hereafter, is as follows: (i) one solves Equation~\eqref{recur32}
for $\mathfrak{u}_{0}(k)$ in order to arrive at the first of Equations~\eqref{thmk3};
(ii) via the formula for $\mathfrak{u}_{0}(k)$, the definitions of $c_{0,k}$,
$\lambda_{i}(k)$, and $\eta_{m}(k)$ given heretofore, and
Equations~\eqref{recur18}--\eqref{recur25}, one solves
Equations~\eqref{recur33}--\eqref{recur41}, in the indicated order, to arrive at the
expressions for the coefficients $\mathfrak{u}_{j}(k)$, $j \! = \! 1,2,\dotsc,9$, given
in Equations~\eqref{thmk3} and~\eqref{thmk4}; and (iii) using the fact that
$\mathfrak{u}_{1}(k) \! = \! 0$ (cf. Equations~\eqref{thmk3}), and the definition of
$\lambda_{i}(k)$, one solves Equation~\eqref{recur42} for $\mathfrak{u}_{m+10}
(k)$, $m \! \in \! \mathbb{Z}_{+}$, and, after an induction argument, arrives at
Equations~\eqref{thmk6} and~\eqref{thmk7}. \hfill $\qed$
It follows {}from Equations~\eqref{hatsoff9}, \eqref{eq3.2}, \eqref{eq3.5},
and~\eqref{iden3oldr} that
\begin{equation} \label{textfeqn1}
\dfrac{u^{\prime}(\tau) \! - \! \mathrm{i} b}{u(\tau)} \! = \! \frac{2}{\tau^{1/3}} \! \left(
\frac{2a(\tau)d(\tau)}{\sqrt{\smash[b]{-a(\tau)b(\tau)}}} \! + \! \tau^{-2/3}
(\mathrm{i} a \! + \! 1/2) \right) \! = \! \mathrm{i} (\varepsilon b)^{1/3} \tau^{-1/3}r(\tau)
\! = \! \mathrm{i} (\varepsilon b)^{1/3} \tau^{-1/3}(-2 \! + \! \hat{r}_{0}(\tau));
\end{equation}
thus, via the Definition~\eqref{iden3}, it follows that
\begin{equation} \label{tr2}
\tilde{r}_{0}(\tau) \! = \! 2 \tau^{1/3} \! - \! \dfrac{\mathrm{i} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}
\tau^{2/3}}{(\varepsilon b)^{1/3}} \! \left(\dfrac{u^{\prime}(\tau) \! - \! \mathrm{i} b}{
u(\tau)} \right), \quad k \! = \! \pm 1.
\end{equation}
\begin{bbbb} \label{proprr}
Let the function $\tilde{r}_{0}(\tau)$ be given by Equation~\eqref{tr2}, and let
$u(\tau)$ denote the corresponding solution of the {\rm DP3E}~\eqref{eq1.1}
having the differentiable asymptotics~\eqref{recur15}, with $\mathfrak{u}_{m}(k)$,
$m \! \in \! \mathbb{Z}_{+}$, $k \! = \! \pm 1$, given in Proposition~\ref{recursys}.
Then, the function $\tilde{r}_{0}(\tau)$ has the following asymptotic expansion:
\begin{equation} \label{tr3}
\tilde{r}_{0}(\tau) \! := \! \tilde{r}_{0,k}(\tau) \underset{\tau \to +\infty}{=}
\sum_{m=0}^{\infty} \dfrac{\mathfrak{r}_{m}(k)}{(\tau^{1/3})^{m+1}}
\! + \! 2(1 \! + \! k\sqrt{3}) \mathrm{A}_{k} \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)}
\mathrm{e}^{-\beta (\tau)} \! \left(1 \! + \! \mathcal{O}(\tau^{-1/3}) \right), \quad
k \! = \! \pm 1,
\end{equation}
where the expansion coefficients $\mathstrut \mathfrak{r}_{m}(k)$, $m \! \in \!
\mathbb{Z}_{+}$, are given in Equations~\eqref{thmk15} and~\eqref{thmk16}.
\end{bbbb}
\emph{Proof}. Substituting the differentiable asymptotics~\eqref{recur15} for
$u(\tau)$ into Equation~\eqref{tr2} and using the expressions for the coefficients
$c_{0,k}$, $\mathfrak{u}_{m}(k)$, and $\mathfrak{w}_{m}(k)$, $k \! = \! \pm 1$,
$m \! \in \! \mathbb{Z}_{+}$, stated in the proof of Proposition~\ref{recursys},
one arrives at, after a lengthy, but otherwise straightforward, algebraic calculation,
the asymptotics for $\tilde{r}_{0}(\tau) \! := \! \tilde{r}_{0,k}(\tau)$ stated in the
proposition. \hfill $\qed$
\begin{eeee} \label{remforkay}
Hereafter, explicit $k$ dependencies for the subscripts of the functions $v_{0}
(\tau)$ and $\tilde{r}_{0}(\tau)$ (cf. Equations~\eqref{tr1} and~\eqref{tr3},
respectively) will be suppressed, except where absolutely necessary and/or
where confusion may arise. \hfill $\blacksquare$
\end{eeee}
In certain domains of the complex $\widetilde{\mu}$-plane (see the discussion
below), the leading term of asymptotics (as $\tau \! \to \! +\infty$ for
$\varepsilon b \! > \! 0)$ of a fundamental solution of Equation~\eqref{eq3.3}
is given by the following matrix WKB formula (see, for example, Chapter~5 of
\cite{F}),\footnote{Hereafter, for simplicity of notation, explicit $\tau$
dependencies will be suppressed, except where absolutely necessary.}
\begin{equation} \label{eq3.16}
T(\widetilde{\mu}) \exp \! \left(-\sigma_{3} \mathrm{i} \tau^{2/3}
\int_{}^{\widetilde{\mu}}l(\xi) \, \mathrm{d} \xi \! - \! \int_{}^{\widetilde{\mu}}
\text{diag}(T^{-1}(\xi) \partial_{\xi} T(\xi)) \, \mathrm{d} \xi \right) \! :=
\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB}}(\widetilde{\mu}),
\end{equation}
where
\begin{equation}
l(\widetilde{\mu}) \! := \! (\det (\mathcal{A}(\widetilde{\mu})))^{1/2},
\label{eq3.17}
\end{equation}
and the matrix $T(\widetilde{\mu})$, which diagonalizes $\mathcal{A}
(\widetilde{\mu})$, that is, $T^{-1}(\widetilde{\mu}) \mathcal{A}
(\widetilde{\mu})T(\widetilde{\mu}) \! = \! -\mathrm{i} l(\widetilde{\mu})
\sigma_{3}$, is given by
\begin{equation}
T(\widetilde{\mu}) \! = \! \dfrac{\mathrm{i}}{\sqrt{\smash[b]{2 \mathrm{i} l(\widetilde{\mu})
(\mathcal{A}_{11}(\widetilde{\mu}) \! - \! \mathrm{i} l(\widetilde{\mu}))}}} \left(
\mathcal{A}(\widetilde{\mu}) \! - \! \mathrm{i} l(\widetilde{\mu}) \sigma_{3} \right)
\sigma_{3}. \label{eq3.18}
\end{equation}
\begin{bbbb}[\textrm{\cite{av2}}] \label{prop3.1.2}
Let $T(\widetilde{\mu})$ be given in Equation~\eqref{eq3.18}, with
$\mathcal{A}(\widetilde{\mu})$ and $l(\widetilde{\mu})$ defined by
Equations~\eqref{eq3.4} and \eqref{eq3.17}, respectively. Then,
$\det (T(\widetilde{\mu})) \! = \! 1$, and $\tr (T^{-1}(\widetilde{\mu})
\partial_{\widetilde{\mu}}T(\widetilde{\mu})) \! = \! 0$$;$ moreover,
\begin{equation}
\diag \! \left(T^{-1}(\widetilde{\mu}) \partial_{\widetilde{\mu}}T(\widetilde{\mu})
\right) \! = \! -\dfrac{1}{2} \! \left(\dfrac{\mathcal{A}_{12}(\widetilde{\mu})
\partial_{\widetilde{\mu}} \mathcal{A}_{21}(\widetilde{\mu}) \! - \!
\mathcal{A}_{21}(\widetilde{\mu}) \partial_{\widetilde{\mu}} \mathcal{A}_{12}
(\widetilde{\mu})}{2l(\widetilde{\mu})(\mathrm{i} \mathcal{A}_{11}(\widetilde{\mu})
\! + \! l(\widetilde{\mu}))} \right) \! \sigma_{3}. \label{eq3.23}
\end{equation}
\end{bbbb}
\begin{ffff} \label{cor3.1.1}
Let $\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB}}(\widetilde{\mu})$
be defined by Equation~\eqref{eq3.16}, with $l(\widetilde{\mu})$ defined
by Equation~\eqref{eq3.17} and $T(\widetilde{\mu})$ given in
Equation~\eqref{eq3.18}. Then, $\det (\widetilde{\Psi}_{\scriptscriptstyle
\mathrm{WKB}}(\widetilde{\mu})) \! = \! 1$.
\end{ffff}
The domains in the complex $\widetilde{\mu}$-plane where Equation~\eqref{eq3.16}
gives the---leading---asymptotic approximation of solutions to Equation~\eqref{eq3.3}
are defined in terms of the \emph{Stokes graph} (see, for example, \cite{F,mlaud,W}).
The vertices of the Stokes graph are the singular points of Equation~\eqref{eq3.3},
that is, $\widetilde{\mu} \! = \! 0$ and $\widetilde{\mu} \! = \! \infty$, and the
\emph{turning points\/}, which are the roots of the equation $l^{2}(\widetilde{\mu})
\! = \! 0$. The edges of the Stokes graph are the \emph{Stokes curves\/}, defined as
$\Im (\int_{\widetilde{\mu}_{\scriptscriptstyle \mathrm{TP}}}^{\widetilde{\mu}}l(\xi)
\, \mathrm{d} \xi) \! = \! 0$, where $\widetilde{\mu}_{\scriptscriptstyle \mathrm{TP}}$
denotes a turning point. \emph{Canonical domains\/} are those domains in the
complex $\widetilde{\mu}$-plane containing one, and only one, Stokes curve and
bounded by two adjacent Stokes curves.\footnote{Note that the restriction of any
branch of $l(\widetilde{\mu})$ to a canonical domain is a single-valued function.}
In each canonical domain, for any choice of the branch of $l(\widetilde{\mu})$, there
exists a fundamental solution of Equation~\eqref{eq3.3} which has asymptotics
whose leading term is given by Equation~\eqref{eq3.16}. {}From the definition of
$l(\widetilde{\mu})$ given by Equation~\eqref{eq3.17}, one arrives at
\begin{equation} \label{eq3.19}
l^{2}(\widetilde{\mu}) \! := \! l_{k}^{2}(\widetilde{\mu}) \! = \! \frac{4}{\widetilde{\mu}^{4}}
\! \left(\left(\widetilde{\mu}^{2} \! - \! \alpha_{k}^{2} \right)^{2} \left(\widetilde{\mu}^{2}
\! + \! 2 \alpha_{k}^{2} \right) \! + \! \widetilde{\mu}^{2} \hat{h}_{0}(\tau) \! + \!
\widetilde{\mu}^{4} \! \left(a \! - \! \mathrm{i}/2 \right) \tau^{-2/3} \right), \quad
k \! = \! \pm 1,
\end{equation}
where $\alpha_{k}$ is defined by Equation~\eqref{thmk5}. It follows {}from
Equation~\eqref{eq3.19} that there are six turning points. For $k \! = \! \pm 1$, the
Conditions~\eqref{iden5} imply that one pair of turning points coalesce at $\alpha_{k}$
with asymptotics $\mathcal{O}(\tau^{-1/3})$, another pair has asymptotics $-\alpha_{k}
\! + \! \mathcal{O}(\tau^{-1/3})$, and the two remaining turning points have the asymptotic
behaviour $\pm \mathrm{i} \sqrt{2} \alpha_{k} \! + \! \mathcal{O}(\tau^{-2/3})$. For simplicity
of notation, denote by $\widetilde{\mu}_{1}(k)$ any one of the turning points coalescing
at $\alpha_{k}$, and denote by $\widetilde{\mu}_{2}(k)$ the turning point approaching
$\mathrm{i} k \sqrt{2} \alpha_{k}$. Let $\mathscr{G}_{\scriptscriptstyle \mathbb{S}}(k)$, $k \!
= \! \pm 1$, be the part of the Stokes graph that consists of the vertices $0,\infty,
\widetilde{\mu}_{1}(k)$ and $\widetilde{\mu}_{2}(k)$, and the union of
the---oriented---edges $\operatorname{arc}(\mathrm{i} k \infty,\widetilde{\mu}_{2}(k))$,
$\operatorname{arc}(\widetilde{\mu}_{2}(k),0)$ and $\operatorname{arc}
(\widetilde{\mu}_{2}(k),-\infty)$, and $\operatorname{arc}(\mathrm{i} k \infty,
\widetilde{\mu}_{1}(k))$, $\operatorname{arc}(\widetilde{\mu}_{1}(k),0)$,
$\operatorname{arc}(0,\widetilde{\mu}_{1}(k))$ and $\operatorname{arc}
(\widetilde{\mu}_{1}(k),+\infty)$; denote by
$\mathscr{G}^{\ast}_{\scriptscriptstyle \mathbb{S}}(k)$, $k \! = \! \pm 1$, the mirror
image of $\mathscr{G}_{\scriptscriptstyle \mathbb{S}}(k)$ with respect to the real and
the imaginary axes of the complex $\widetilde{\mu}$-plane: the complete Stokes
graph is given by $\mathscr{G}_{\scriptscriptstyle \mathbb{S}}(k) \cup
\mathscr{G}^{\ast}_{\scriptscriptstyle \mathbb{S}}(k)$.
\begin{bbbb} \label{prop3.1.3}
Let $l_{k}^{2}(\widetilde{\mu})$, $k \! = \! \pm 1$, be given in Equation~\eqref{eq3.19}.
Then,
\begin{equation}
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}l_{k}(\xi) \, \mathrm{d} \xi \underset{
\tau \to +\infty}{=} \varUpsilon_{k}(\widetilde{\mu}) \! - \! \varUpsilon_{k}
(\widetilde{\mu}_{0,k}) \! + \! \mathcal{O}(\mathscr{E}_{k}(\widetilde{\mu}))
\! + \! \mathcal{O}(\mathscr{E}_{k}(\widetilde{\mu}_{0,k})), \label{eq3.21}
\end{equation}
where, for $\delta \! > \! 0$, $\widetilde{\mu},\widetilde{\mu}_{0,k} \! \in \!
\mathbb{C} \setminus (\mathscr{O}_{\tau^{-1/3+ \delta}}(\pm \alpha_{k})
\cup \mathscr{O}_{\tau^{-2/3+2 \delta}}(\pm \mathrm{i} \sqrt{2} \alpha_{k}) \cup
\lbrace 0,\infty \rbrace)$ and the path of integration lies in the corresponding
canonical domain,
\begin{align}
\varUpsilon_{k}(\xi) :=& \, (\xi \! + \! 2 \alpha_{k}^{2} \xi^{-1})(\xi^{2} \! + \!
2 \alpha_{k}^{2})^{1/2} \! + \! \tau^{-2/3}(a \! - \! \mathrm{i}/2) \ln \! \left(\xi \! + \!
(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2} \right) \nonumber \\
+& \, \dfrac{\tau^{-2/3}}{2 \sqrt{3}} \! \left((a \! - \! \mathrm{i}/2) \! + \!
\frac{\tau^{2/3}}{\alpha_{k}^{2}} \hat{h}_{0}(\tau) \right) \! \ln \! \left(\left(
\dfrac{3^{1/2}(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2} \! - \! \xi \! + \! 2
\alpha_{k}}{3^{1/2}(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2} \! + \! \xi \! +
\! 2 \alpha_{k}} \right) \! \left(\dfrac{\xi \! - \! \alpha_{k}}{\xi \! + \!
\alpha_{k}} \right) \right), \label{equpsi}
\end{align}
and
\begin{equation} \label{eqeel}
\tau^{4/3} \mathscr{E}_{k}(\xi) \! := \!
\begin{cases}
{\fontsize{9pt}{11pt}\selectfont
\begin{aligned}[b]
&\frac{((a \! - \! \mathrm{i}/2) \! + \! \frac{\tau^{2/3}}{\alpha_{k}^{2}}
\hat{h}_{0}(\tau))^{2}}{192 \sqrt{3} (\xi \! \mp \! \alpha_{k})^{2}}
\! + \! \mathcal{O} \! \left(\frac{c_{1,k} \! + \! c_{2,k} \tau^{2/3}
\hat{h}_{0}(\tau) \! + \! c_{3,k}(\tau^{2/3} \hat{h}_{0}(\tau))^{2}}{
\xi \! \mp \! \alpha_{k}} \right),
\end{aligned}} &\text{$\xi \! \in \! \mathbb{U}_{k}^{1}$,} \\
{\fontsize{8pt}{11pt}\selectfont
\begin{aligned}[b]
&\frac{((a \! - \! \mathrm{i}/2) \! - \! \frac{\tau^{2/3}}{2 \alpha_{k}^{2}} \hat{h}_{0}
(\tau))^{2}}{d_{0,k}(\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{1/2}} \! + \!
\mathcal{O} \! \left((\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{1/2}(c_{4,k} \! + \!
c_{5,k} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! c_{6,k}(\tau^{2/3} \hat{h}_{0}
(\tau))^{2}) \right),
\end{aligned}} &\text{$\xi \! \in \! \mathbb{U}_{k}^{2}$,} \\
{\fontsize{10pt}{11pt}\selectfont
\begin{aligned}[b]
&\mathfrak{f}_{1,k}(\xi^{-1}) \! + \! \tau^{2/3} \hat{h}_{0}
(\tau) \mathfrak{f}_{2,k}(\xi^{-1}) \! + \! (\tau^{2/3} \hat{h}_{0}
(\tau))^{2} \mathfrak{f}_{3,k}(\xi^{-1}),
\end{aligned}} &\text{$\xi \! \to \! \infty$,} \\
{\fontsize{10pt}{11pt}\selectfont
\begin{aligned}[b]
\mathfrak{f}_{4,k}(\xi) \! + \! \tau^{2/3} \hat{h}_{0}
(\tau) \mathfrak{f}_{5,k}(\xi) \! + \! (\tau^{2/3} \hat{h}_{0}
(\tau))^{2} \mathfrak{f}_{6,k}(\xi),
\end{aligned}} &\text{$\xi \! \to \! 0$,}
\end{cases}
\end{equation}
where $\mathbb{U}_{k}^{1} \! := \! \mathscr{O}_{\tau^{-1/3 + \delta_{k}}}
(\pm \alpha_{k})$, $\mathbb{U}_{k}^{2} \! := \! \mathscr{O}_{\tau^{-2/3
+2 \delta_{k}}}(\pm \mathrm{i} \sqrt{2} \alpha_{k})$, the parameter $\delta_{k}$ satisfies
(see Corollary~\ref{cor3.1.2} below) $0 \! < \! \delta \! < \! \delta_{k} \! < \! 1/9$,
$d_{0,k}^{-1} \! := \! 2^{-1/4} \mathrm{e}^{\mp \mathrm{i} 3 \pi/4} \alpha_{k}^{-3/2}/27$,
$\mathfrak{f}_{j,k}(z)$, $j \! = \! 1,2,\dotsc,6$, are analytic functions of $z$,
with $k$-dependent coefficients, in a neighbourhood of $z \! = \! 0$ given by
Equations~\eqref{alpbet1}--\eqref{alpbet6} below, and $c_{m,k}$, $m \! = \!
1,2,\dotsc,6$, are constants.
\end{bbbb}
\emph{Proof}. Let $l_{k}^{2}(\widetilde{\mu})$, $k \! = \! \pm 1$, be given in
Equation~\eqref{eq3.19}, with $\alpha_{k}$ defined by Equation~\eqref{thmk5}.
Recalling {}from the Conditions~\eqref{iden5} that $\lvert \hat{h}_{0}(\tau) \rvert
\! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{-2/3})$, set
\begin{equation}
l_{k,\infty}^{2}(\widetilde{\mu}) \! = \! 4 \widetilde{\mu}^{-4}(\widetilde{\mu}^{2}
\! - \! \alpha_{k}^{2})^{2}(\widetilde{\mu}^{2} \! + \! 2 \alpha_{k}^{2}).
\label{eqlsquared}
\end{equation}
Define
\begin{equation}
\Delta_{k,\tau}(\widetilde{\mu}) \! := \! \dfrac{l_{k}^{2}(\widetilde{\mu})
\! - \! l_{k,\infty}^{2}(\widetilde{\mu})}{l_{k,\infty}^{2}(\widetilde{\mu})}
\! = \! \dfrac{\widetilde{\mu}^{2} \hat{h}_{0}(\tau) \! + \! \widetilde{\mu}^{4}
(a \! - \! \mathrm{i}/2) \tau^{-2/3}}{(\widetilde{\mu}^{2} \! - \! \alpha_{k}^{2})^{2}
(\widetilde{\mu}^{2} \! + \! 2 \alpha_{k}^{2})}; \label{eqdellt}
\end{equation}
hence, presenting $l_{k}(\widetilde{\mu})$ as $l_{k}(\widetilde{\mu}) \! = \!
l_{k,\infty}(\widetilde{\mu})(1 \! + \! \Delta_{k,\tau}(\widetilde{\mu}))^{1/2}$,
a straightforward calculation, via the Conditions~\eqref{iden5}, shows that,
for $k \! = \! \pm 1$,
\begin{align}
l_{k}(\widetilde{\mu}) \underset{\tau \to +\infty}{=}& \, l_{k,\infty}
(\widetilde{\mu}) \! \left(1 \! + \! \Delta_{k,\tau}(\widetilde{\mu})/2 \! + \!
\mathcal{O}(-(\Delta_{k,\tau}(\widetilde{\mu}))^{2}/8) \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, 2(1 \! - \! \alpha_{k}^{2}/\widetilde{\mu}^{2})
(\widetilde{\mu}^{2} \! + \! 2 \alpha_{k}^{2})^{1/2} \! + \! \dfrac{\hat{h}_{0}(\tau)
\! + \! \widetilde{\mu}^{2}(a \! - \! \mathrm{i}/2) \tau^{-2/3}}{(\widetilde{\mu}^{2} \!
- \! \alpha_{k}^{2})(\widetilde{\mu}^{2} \! + \! 2 \alpha_{k}^{2})^{1/2}} \! + \!
\mathcal{O} \! \left(-\dfrac{\widetilde{\mu}^{2}(\hat{h}_{0}(\tau) \! + \!
\widetilde{\mu}^{2}(a \! - \! \mathrm{i}/2) \tau^{-2/3})^{2}}{4(\widetilde{\mu}^{2} \! - \!
\alpha_{k}^{2})^{3}(\widetilde{\mu}^{2} \! + \! 2 \alpha_{k}^{2})^{3/2}} \right).
\label{eqforl}
\end{align}
Integration of the first two terms in the second line of Equation~\eqref{eqforl}
gives rise to the leading term of asymptotics in Equation~\eqref{eq3.21},
and integration of the error term in the second line of Equation~\eqref{eqforl}
leads to an explicit expression for the error function, $\mathscr{E}_{k}
(\pmb{\cdot})$, whose asymptotics at the turning and the singular points read:
(i) for $\xi \! \in \! \mathscr{O}_{\tau^{-1/3+ \delta_{k}}}(\pm \alpha_{k})$,
$0 \! < \! \delta \! < \! \delta_{k} \! < \! 1/9$,
\begin{align} \label{terrbos1}
\tau^{4/3} \mathscr{E}_{k}(\xi) \underset{\tau \to +\infty}{=}& \,
\dfrac{((a \! - \! \mathrm{i}/2) \! + \! \alpha_{k}^{-2} \tau^{2/3} \hat{h}_{0}
(\tau))^{2}}{192 \sqrt{3} (\xi \! \mp \! \alpha_{k})^{2}} \! + \!
\dfrac{\hat{d}_{-1,k}(\tau)}{\xi \! \mp \! \alpha_{k}} \! + \! \hat{d}_{0,k}
(\tau) \ln (\xi \! \mp \! \alpha_{k}) \nonumber \\
+& \, \sum_{m \in \mathbb{Z}_{+}} \hat{d}_{m+1,k}(\tau)(\xi \! \mp \!
\alpha_{k})^{m},
\end{align}
where
\begin{equation*}
\hat{d}_{m,k}(\tau) \! := \! \hat{\mathfrak{c}}^{\flat}_{m,k} \! + \!
\hat{\mathfrak{c}}^{\natural}_{m,k} \tau^{2/3} \hat{h}_{0}(\tau) \! + \!
\hat{\mathfrak{c}}^{\sharp}_{m,k}(\tau^{2/3} \hat{h}_{0}(\tau))^{2},
\quad m \! \in \! \lbrace -1 \rbrace \cup \mathbb{Z}_{+},
\end{equation*}
with $\hat{\mathfrak{c}}^{r}_{m,k}$, $r \! \in \! \lbrace \flat,\natural,\sharp
\rbrace$, constants, and thus, retaining only the first two terms of the
Expansion~\eqref{terrbos1}, one arrives at the representation for
$\mathscr{E}_{k}(\xi)$ stated in the first line of Equation~\eqref{eqeel};
(ii) for $\xi \! \in \! \mathscr{O}_{\tau^{-2/3+ 2 \delta_{k}}}(\pm \mathrm{i} \sqrt{2}
\alpha_{k})$,
\begin{equation} \label{terrbos2}
\tau^{4/3} \mathscr{E}_{k}(\xi) \underset{\tau \to +\infty}{=} \dfrac{2^{-1/4}
((a \! - \! \mathrm{i}/2) \! - \! \alpha_{k}^{-2} \tau^{2/3} \hat{h}_{0}(\tau)/2)^{2}}{27
\mathrm{e}^{\pm \mathrm{i} 3 \pi/4} \alpha_{k}^{3/2}(\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{1/2}}
\! + \! (\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{1/2} \sum_{m \in \mathbb{Z}_{+}}
\tilde{d}_{m,k}(\tau)(\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{m},
\end{equation}
where
\begin{equation*}
\tilde{d}_{m,k}(\tau) \! := \! \tilde{\mathfrak{c}}^{\flat}_{m,k} \! + \!
\tilde{\mathfrak{c}}^{\natural}_{m,k} \tau^{2/3} \hat{h}_{0}(\tau) \! + \!
\tilde{\mathfrak{c}}^{\sharp}_{m,k}(\tau^{2/3} \hat{h}_{0}(\tau))^{2},
\quad m \! \in \! \mathbb{Z}_{+},
\end{equation*}
with $\tilde{\mathfrak{c}}^{r}_{m,k}$, $r \! \in \! \lbrace \flat,\natural,
\sharp \rbrace$, constants, and thus, keeping only the first two terms
of the Expansion~\eqref{terrbos2}, one arrives at the representation for
$\mathscr{E}_{k}(\xi)$ stated in the second line of Equation~\eqref{eqeel};
(iii) as $\xi \! \to \! \infty$, one arrives at the representation for
$\mathscr{E}_{k}(\xi)$ stated in the third line of Equation~\eqref{eqeel},
where
\begin{gather}
\mathfrak{f}_{1,k}(z) \! = \! \dfrac{(a \! - \! \mathrm{i}/2)^{2}}{12}z^{3} \! + \!
(a \! - \! \mathrm{i}/2)^{2}z^{7} \sum_{m \in \mathbb{Z}_{+}}
\hat{\mathfrak{c}}^{\circ,1}_{m,k}z^{2m}, \label{alpbet1} \\
\mathfrak{f}_{2,k}(z) \! = \! \dfrac{(a \! - \! \mathrm{i}/2)}{10}z^{5} \! + \!
(a \! - \! \mathrm{i}/2)z^{9} \sum_{m \in \mathbb{Z}_{+}}
\hat{\mathfrak{c}}^{\circ,2}_{m,k}z^{2m}, \label{alpbet2} \\
\mathfrak{f}_{3,k}(z) \! = \! \dfrac{1}{28}z^{7} \! + \! z^{11} \sum_{m
\in \mathbb{Z}_{+}} \hat{\mathfrak{c}}^{\circ,3}_{m,k}z^{2m}, \label{alpbet3}
\end{gather}
with $\hat{\mathfrak{c}}^{\circ,r}_{m,k}$, $r \! = \! 1,2,3$, $m \! \in \! \mathbb{Z}_{+}$,
constants; and (iv) as $\xi \! \to \! 0$, one arrives at the representation for
$\mathscr{E}_{k}(\xi)$ stated in the fourth line of Equation~\eqref{eqeel}, where
\begin{gather}
\mathfrak{f}_{4,k}(z) \! = \! -\dfrac{(a \! - \! \mathrm{i}/2)^{2}}{14 \sqrt{2}
\alpha_{k}^{9}}z^{7} \! + \! (a \! - \! \mathrm{i}/2)^{2}z^{9} \sum_{m \in \mathbb{Z}_{+}}
\tilde{d}^{\circ,4}_{m,k}z^{2m}, \label{alpbet4} \\
\mathfrak{f}_{5,k}(z) \! = \! -\dfrac{(a \! - \! \mathrm{i}/2)}{5 \sqrt{2} \alpha_{k}^{9}}
z^{5} \! + \! (a \! - \! \mathrm{i}/2)z^{7} \sum_{m \in \mathbb{Z}_{+}}
\tilde{d}^{\circ,5}_{m,k}z^{2m}, \label{alpbet5} \\
\mathfrak{f}_{6,k}(z) \! = \! -\dfrac{1}{6 \sqrt{2} \alpha_{k}^{9}}z^{3} \! + \! z^{5}
\sum_{m \in \mathbb{Z}_{+}} \tilde{d}^{\circ,6}_{m,k}z^{2m}, \label{alpbet6}
\end{gather}
with $\tilde{d}^{\circ,r}_{m,k}$, $r \! = \! 4,5,6$, $m \! \in \! \mathbb{Z}_{+}$,
constants. \hfill $\qed$
\begin{ffff} \label{cor3.1.2}
Set $\widetilde{\mu}_{0,k} \! = \! \alpha_{k} \! + \! \tau^{-1/3} \widetilde{
\Lambda}$, $k \! = \! \pm 1$, where $\widetilde{\Lambda} \! =_{\tau \to
+\infty} \! \mathcal{O}(\tau^{\delta_{k}})$, $0 \! < \! \delta \! < \! \delta_{k}
\! < \! 1/9$. Then,
\begin{align} \label{eq3.22}
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}l_{k}(\xi) \, \mathrm{d} \xi
\underset{\tau \to +\infty}{=}& \, \varUpsilon_{k}(\widetilde{\mu}) \! + \!
\varUpsilon_{k}^{\sharp} \! + \! \mathcal{O}(\mathscr{E}_{k}(\widetilde{\mu}))
\! + \! \mathcal{O}(\tau^{-1} \widetilde{\Lambda}^{3}) \! + \! \mathcal{O}
(\tau^{-1} \widetilde{\Lambda}) \nonumber \\
+& \, \mathcal{O} \! \left(\dfrac{\tau^{-1}}{\widetilde{\Lambda}} \left(
\mathfrak{c}_{1,k} \! + \! \mathfrak{c}_{2,k} \tau^{2/3} \hat{h}_{0}(\tau)
\! + \! \mathfrak{c}_{3,k}(\tau^{2/3} \hat{h}_{0}(\tau))^{2} \right) \right),
\end{align}
where $\varUpsilon_{k}(\widetilde{\mu})$ and $\mathscr{E}_{k}(\widetilde{\mu})$
are defined by Equations~\eqref{equpsi} and~\eqref{eqeel}, respectively,
\begin{align}
\varUpsilon_{k}^{\sharp} :=& \, \mp 3 \sqrt{3} \alpha_{k}^{2} \! \mp \! 2 \sqrt{3}
\tau^{-2/3} \widetilde{\Lambda}^{2} \! - \! \tau^{-2/3}(a \! - \! \mathrm{i}/2) \ln \!
\left((\sqrt{3} \pm \! 1) \alpha_{k} \mathrm{e}^{\mathrm{i} \pi (1 \mp 1)/2} \right) \nonumber \\
\mp& \, \dfrac{\tau^{-2/3}}{2 \sqrt{3}} \! \left((a \! - \! \mathrm{i}/2) \! + \! \alpha_{k}^{-2}
\tau^{2/3} \hat{h}_{0}(\tau) \right) \! \left(\ln \widetilde{\Lambda} \! - \!
\dfrac{1}{3} \ln \tau \! - \! \ln (3 \alpha_{k}) \right), \label{equpsisharp}
\end{align}
with the upper (resp., lower) signs taken according to the branch of the square-root
function $\lim_{\xi^{2} \to +\infty}(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2} \! = \!
+\infty$ (resp., $\lim_{\xi^{2} \to +\infty}(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2}
\! = \! -\infty)$, and $\mathfrak{c}_{m,k}$, $m \! = \! 1,2,3$, are constants.
\end{ffff}
\emph{Proof}. Substituting $\widetilde{\mu}_{0,k}$, as given in the corollary,
for the argument of the functions $\varUpsilon_{k}(\xi)$ and $\mathscr{E}_{k}
(\xi)$ (cf. Equation~\eqref{equpsi} and the first line of Equation~\eqref{eqeel},
respectively) and expanding with respect to the `small parameter' $\tau^{-1/3}
\widetilde{\Lambda}$, one arrives at the following estimates:
\begin{equation} \label{terrbos3}
-\varUpsilon_{k}(\widetilde{\mu}_{0,k}) \underset{\tau \to +\infty}{=}
\varUpsilon_{k}^{\sharp} \! + \! \mathcal{O}(\tau^{-1} \widetilde{
\Lambda}^{3}) \! + \! \mathcal{O}(\tau^{-1} \widetilde{\Lambda}) \! + \!
\mathcal{O} \! \left(\tau^{-1} \widetilde{\Lambda}((a \! - \! \mathrm{i}/2) \! + \!
\alpha_{k}^{-2} \tau^{2/3} \hat{h}_{0}(\tau)) \right),
\end{equation}
where $\varUpsilon_{k}^{\sharp}$ is defined by Equation~\eqref{equpsisharp}, and
\begin{align} \label{terrbos4}
\mathcal{O}(\mathscr{E}_{k}(\widetilde{\mu}_{0,k})) \underset{\tau \to +\infty}{=}&
\, \mathcal{O} \! \left(\dfrac{\tau^{-2/3}}{\widetilde{\Lambda}^{2}}((a \! - \! \mathrm{i}/2)
\! + \! \alpha_{k}^{-2} \tau^{2/3} \hat{h}_{0}(\tau))^{2} \right) \nonumber \\
+& \, \mathcal{O} \! \left(\dfrac{\tau^{-1}}{\widetilde{\Lambda}} \left(c_{1,k} \! + \!
c_{2,k} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! c_{3,k}(\tau^{2/3} \hat{h}_{0}(\tau))^{2}
\right) \right),
\end{align}
where $c_{m,k}$, $m \! = \! 1,2,3$, are constants. {}From Equations~\eqref{iden9},
\eqref{iden2}, \eqref{iden3}, and~\eqref{iden4}, one shows that
\begin{equation} \label{terrbos5}
-\tau^{2/3} \hat{h}_{0}(\tau) \! = \! \dfrac{\alpha_{k}^{2}(a \! - \! \mathrm{i}/2)}{1 \!
+ \! v_{0}(\tau) \tau^{-1/3}} \! + \! \dfrac{\alpha_{k}^{4}(8v_{0}^{2}(\tau) \! +
\! 4 \tilde{r}_{0}(\tau)v_{0}(\tau) \! - \! (\tilde{r}_{0}(\tau))^{2} \! - \! v_{0}(\tau)
(\tilde{r}_{0}(\tau))^{2} \tau^{-1/3})}{4(1 \! + \! v_{0}(\tau) \tau^{-1/3})},
\end{equation}
whence, via the Conditions~\eqref{iden5},
\begin{align} \label{terrbos6}
(a \! - \! \mathrm{i}/2) \! + \! \frac{\tau^{2/3}}{\alpha_{k}^{2}} \hat{h}_{0}(\tau)
\underset{\tau \to +\infty}{=}& \, -\dfrac{\alpha_{k}^{2}}{4} \left(8v_{0}^{2}
(\tau) \! + \! 4v_{0}(\tau) \tilde{r}_{0}(\tau) \! - \! (\tilde{r}_{0}(\tau))^{2}
\right) \! + \! (a \! - \! \mathrm{i}/2) v_{0}(\tau) \tau^{-1/3} \nonumber \\
+& \, \mathcal{O}((2v_{0}^{2}(\tau) \! + \! v_{0}(\tau) \tilde{r}_{0}(\tau))
v_{0}(\tau) \tau^{-1/3}) \! + \! \mathcal{O}(v_{0}^{2}(\tau) \tau^{-2/3}).
\end{align}
Note {}from the Conditions~\eqref{iden5} and the Expansion~\eqref{terrbos6} that
\begin{equation*}
(a \! - \! \mathrm{i}/2) \! + \! \frac{\tau^{2/3}}{\alpha_{k}^{2}} \hat{h}_{0}(\tau)
\underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-2/3}) \! \quad \! \text{and}
\! \quad \! c_{1,k} \! + \! c_{2,k} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! c_{3,k}
(\tau^{2/3} \hat{h}_{0}(\tau))^{2} \underset{\tau \to +\infty}{=} \! \mathcal{O}(1):
\end{equation*}
{}from the Expansions~\eqref{terrbos3} and~\eqref{terrbos4} and the latter two
estimates, it follows that
\begin{gather}
-\varUpsilon_{k}(\widetilde{\mu}_{0,k}) \underset{\tau \to +\infty}{=}
\varUpsilon_{k}^{\sharp} \! + \! \mathcal{O}(\tau^{-1} \widetilde{
\Lambda}^{3}) \! + \! \mathcal{O}(\tau^{-1} \widetilde{\Lambda}) \!
+ \! \mathcal{O}(\tau^{-5/3} \widetilde{\Lambda}), \label{terrbos7} \\
\mathcal{O}(\mathscr{E}_{k}(\widetilde{\mu}_{0,k})) \underset{\tau \to +\infty}{=}
\mathcal{O}(\tau^{-1} \widetilde{\Lambda}^{-1}) \! + \! \mathcal{O}
(\tau^{-2} \widetilde{\Lambda}^{-2}), \label{terrbos8}
\end{gather}
whence, introducing the inequality $0 \! < \! \delta \! < \! \delta_{k}
\! < \! 1/9$ in order to guarantee that the error estimates in the
Expansions~\eqref{terrbos7} and~\eqref{terrbos8} are $o(1)$
after multiplication by the `large parameter' $\tau^{2/3}$ (cf.
Equation~\eqref{eq3.16}), retaining only leading-order contributions,
one arrives at
\begin{align*}
-\varUpsilon_{k}(\widetilde{\mu}_{0,k}) \! + \! \mathcal{O}(\mathscr{E}_{k}
(\widetilde{\mu}_{0,k})) \underset{\tau \to +\infty}{=}& \, \varUpsilon_{k}^{\sharp}
\! + \! \mathcal{O} \! \left(\dfrac{\tau^{-1}}{\widetilde{\Lambda}} \left(
c_{1,k} \! + \! c_{2,k} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! c_{3,k}
(\tau^{2/3} \hat{h}_{0}(\tau))^{2} \right) \right) \nonumber \\
+& \, \mathcal{O}(\tau^{-1} \widetilde{\Lambda}^{3}) \! + \! \mathcal{O}
(\tau^{-1} \widetilde{\Lambda}),
\end{align*}
which, via Equation~\eqref{eq3.21}, implies the result stated in the corollary.
\hfill $\qed$
\begin{ffff} \label{cor3.1.4}
Let the conditions stated in Corollary~{\rm \ref{cor3.1.2}} be valid. Then, for
the branch of $l_{k}(\xi)$, $k \! = \! \pm 1$, that is positive for large and small
positive $\xi$,
\begin{align} \label{eq3.37}
-\mathrm{i} \tau^{2/3} \int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}l_{k}(\xi) \, \mathrm{d}
\xi \underset{\underset{\widetilde{\mu} \to \infty}{\tau \to +\infty}}{=}& \, -\mathrm{i}
(\tau^{2/3} \widetilde{\mu}^{2} \! + \! (a \! - \! \mathrm{i}/2) \ln \widetilde{\mu}) \!
+ \! \mathrm{i} 3(\sqrt{3}- \! 1) \alpha_{k}^{2} \tau^{2/3} \! + \! \mathrm{i} 2 \sqrt{3} \,
\widetilde{\Lambda}^{2} \! + \! C^{\scriptscriptstyle \mathrm{WKB}}_{\infty,k}
\nonumber \\
-& \, \dfrac{\mathrm{i}}{2 \sqrt{3}}((a \! - \! \mathrm{i}/2) \! + \! \alpha_{k}^{-2} \tau^{2/3}
\hat{h}_{0}(\tau)) \! \left(\dfrac{1}{3} \ln \tau \! - \! \ln \widetilde{\Lambda}
\! + \! \ln \left(\dfrac{6 \alpha_{k}}{(\sqrt{3} \! + \! 1)^{2}} \right) \right)
\nonumber \\
+& \, \mathcal{O} \! \left(\dfrac{\tau^{-1/3}}{\widetilde{\Lambda}}(\mathfrak{c}_{1,k}
\! + \! \mathfrak{c}_{2,k} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! \mathfrak{c}_{3,k}
(\tau^{2/3} \hat{h}_{0}(\tau))^{2}) \right) \nonumber \\
+& \, \mathcal{O}(\tau^{-1/3} \widetilde{\Lambda}^{3}) \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}) \! + \! \mathcal{O}(\tau^{-2/3}
\widetilde{\mu}^{-3}),
\end{align}
where
\begin{equation} \label{eq3.38}
C^{\scriptscriptstyle \mathrm{WKB}}_{\infty,k} \! := \! \mathrm{i} (a \! - \! \mathrm{i}/2)
\ln (2^{-1}(\sqrt{3} \! + \! 1) \alpha_{k}),
\end{equation}
and
\begin{align} \label{eq3.39}
-\mathrm{i} \tau^{2/3} \int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}l_{k}(\xi) \,
\mathrm{d} \xi \underset{\underset{\widetilde{\mu} \to 0}{\tau \to +\infty}}{=}& \,
\dfrac{1}{\widetilde{\mu}} \mathrm{i} 2 \sqrt{2} \alpha_{k}^{3} \tau^{2/3}
\! - \! \mathrm{i} 3 \sqrt{3} \alpha_{k}^{2} \tau^{2/3} \! - \! \mathrm{i} 2 \sqrt{3} \,
\widetilde{\Lambda}^{2} \! + \! \dfrac{\mathrm{i}}{2 \sqrt{3}} \left((a \! - \! \mathrm{i}/2)
\right. \nonumber \\
+&\left. \, \alpha_{k}^{-2} \tau^{2/3} \hat{h}_{0}(\tau) \right) \! \left(\dfrac{1}{3}
\ln \tau \! - \! \ln \widetilde{\Lambda} \! + \! \ln (3 \alpha_{k} \mathrm{e}^{-\mathrm{i} \pi k})
\right) \! + \! C^{\scriptscriptstyle \mathrm{WKB}}_{0,k} \nonumber \\
+& \, \mathcal{O} \! \left(\dfrac{\tau^{-1/3}}{\widetilde{\Lambda}}
(\mathfrak{c}_{4,k} \! + \! \mathfrak{c}_{5,k} \tau^{2/3} \hat{h}_{0}(\tau)
\! + \! \mathfrak{c}_{6,k}(\tau^{2/3} \hat{h}_{0}(\tau))^{2}) \right) \nonumber \\
+& \, \mathcal{O}(\tau^{-1/3} \widetilde{\Lambda}^{3}) \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}) \! + \! \mathcal{O}(\tau^{2/3}(\hat{h}_{0}
(\tau))^{2} \widetilde{\mu}^{3}),
\end{align}
where
\begin{equation} \label{eq3.40}
C^{\scriptscriptstyle \mathrm{WKB}}_{0,k} \! := \! -\mathrm{i} (a \! - \! \mathrm{i}/2)
\ln (2^{-1/2}(\sqrt{3} \! + \! 1)),
\end{equation}
with $\mathfrak{c}_{m,k}$, $m \! = \! 1,2,\dotsc,6$, constants.
\end{ffff}
\emph{Proof}. Consequence of Corollary~\ref{cor3.1.2}, Equation~\eqref{eq3.22},
upon choosing consistently the corresponding br\-a\-n\-c\-h\-e\-s in
Equations~\eqref{equpsi} and~\eqref{equpsisharp} and taking the limits
$\widetilde{\mu} \! \to \! \infty$ and $\widetilde{\mu} \! \to \! 0$: the error
estimate $\mathcal{O}(\mathscr{E}_{k}(\xi))$ in Equation~\eqref{eq3.22} is
given in Equation~\eqref{eqeel}; in particular, {}from the last two lines of
Equation~\eqref{eqeel},
\begin{equation*}
\mathcal{O}(\tau^{2/3} \mathscr{E}_{k}(\widetilde{\mu}))
\underset{\underset{\widetilde{\mu} \to \infty}{\tau \to +\infty}}{=}
\mathcal{O}(\tau^{-2/3} \widetilde{\mu}^{-3}) \, \qquad \, \text{and}
\, \quad \, \mathcal{O}(\tau^{2/3} \mathscr{E}_{k}(\widetilde{\mu}))
\underset{\underset{\widetilde{\mu} \to 0}{\tau \to +\infty}}{=}
\mathcal{O}(\tau^{2/3}(\hat{h}_{0}(\tau))^{2} \widetilde{\mu}^{3}),
\end{equation*}
which implies the results stated in the corollary. \hfill $\qed$
\begin{bbbb} \label{prop3.1.4}
Let $T(\widetilde{\mu})$ be given in Equation~\eqref{eq3.18}, with $\mathcal{A}
(\widetilde{\mu})$ defined by Equation~\eqref{eq3.4} and $l_{k}^{2}
(\widetilde{\mu})$, $k \! = \! \pm 1$, given in Equation~\eqref{eq3.19}. Then,
\begin{equation}
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \diag (T^{-1}(\xi) \partial_{\xi}
T(\xi)) \, \mathrm{d} \xi \underset{\tau \to +\infty}{=} \left(\mathcal{I}_{\tau,k}
(\widetilde{\mu}) \! + \! \mathcal{O}(\mathcal{E}_{\scriptscriptstyle T,k}
(\widetilde{\mu})) \! + \! \mathcal{O}(\mathcal{E}_{\scriptscriptstyle T,k}
(\widetilde{\mu}_{0,k})) \right) \! \sigma_{3}, \label{eqintTminus1T}
\end{equation}
where, for $\delta \! > \! 0$, $\widetilde{\mu},\widetilde{\mu}_{0,k} \! \in \!
\mathbb{C} \setminus (\mathscr{O}_{\tau^{-1/3+ \delta}}(\pm \alpha_{k})
\cup \mathscr{O}_{\tau^{-2/3+2 \delta}}(\pm \mathrm{i} \sqrt{2} \alpha_{k}) \cup
\lbrace 0,\infty \rbrace)$ and the path of integration lies in the corresponding
canonical domain,
\begin{equation} \label{eqItee}
\mathcal{I}_{\tau,k}(\widetilde{\mu}) \! = \! \mathfrak{p}_{k}(\tau)
(\digamma_{\tau,k}(\widetilde{\mu}) \! - \! \digamma_{\tau,k}
(\widetilde{\mu}_{0,k})),
\end{equation}
with
\begin{equation}
\mathfrak{p}_{k}(\tau) \! := \! \dfrac{\alpha_{k}^{2} \left(-2 \! +
\! \tilde{r}_{0}(\tau) \tau^{-1/3} \! + \! 2(1 \! + \! v_{0}(\tau)
\tau^{-1/3})^{2} \right) \! - \! (a \! - \! \mathrm{i}/2) \tau^{-2/3}}{8
(-2 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3})(1 \! + \! v_{0}(\tau)
\tau^{-1/3})}, \label{eqpeetee}
\end{equation}
\begin{equation}
\digamma_{\tau,k}(\xi) \! := \! \dfrac{2}{\xi^{2} \! - \! \alpha_{k}^{2}}
\! + \! \dfrac{2}{3 \sqrt{3} \alpha_{k}^{2}} \ln \! \left(\left(\dfrac{3^{1/2}
(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2} \! - \! \xi \! + \! 2 \alpha_{k}}{
3^{1/2}(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2} \! + \! \xi \! + \! 2 \alpha_{k}}
\right) \! \left(\dfrac{\xi \! - \! \alpha_{k}}{\xi \! + \! \alpha_{k}} \right)
\right) \! - \! \dfrac{2}{3 \alpha_{k}^{2}} \dfrac{\xi (\xi^{2} \! + \! 2
\alpha_{k}^{2})^{1/2}}{\xi^{2} \! - \! \alpha_{k}^{2}}, \label{eqFtee}
\end{equation}
and
\begin{equation} \label{eqee2}
\mathcal{E}_{\scriptscriptstyle T,k}(\xi) \! := \!
\begin{cases}
\mathfrak{p}_{k}(\tau) \! \left(\frac{\mathfrak{c}_{1,k}^{\blacklozenge}
\tilde{r}_{0}(\tau) \tau^{-1/3} + \mathfrak{c}_{2,k}^{\blacklozenge}
\hat{\mathfrak{f}}_{1,k}(\tau)}{(\xi \mp \alpha_{k})^{2}} \! + \! \frac{
\mathfrak{c}_{3,k}^{\blacklozenge} \tilde{r}_{0}(\tau) \tau^{-1/3}}{\xi
\mp \alpha_{k}} \right), & \text{$\xi \! \in \! \mathbb{U}_{k}^{1}$,} \\
\mathfrak{p}_{k}(\tau) \hat{\mathfrak{f}}_{3,k}(\tau) \! \left(
\frac{\mathfrak{c}_{4,k}^{\blacklozenge}}{(\xi \mp \mathrm{i} \sqrt{2}
\alpha_{k})^{1/2}} \! + \! \mathfrak{c}_{5,k}^{\blacklozenge}
\ln (\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k}) \right), & \text{$\xi \!
\in \! \mathbb{U}_{k}^{2}$,} \\
\mathfrak{p}_{k}(\tau) \xi^{-4} \! \left(\mathfrak{c}_{6,k}^{\blacklozenge}
\tilde{r}_{0}(\tau) \tau^{-1/3} \! + \! \mathcal{O}(
(\mathfrak{c}_{7,k}^{\blacklozenge} \tilde{r}_{0}(\tau) \tau^{-1/3}
\! + \! \mathfrak{c}_{8,k}^{\blacklozenge} \tau^{-2/3}) \xi^{-2})
\right), & \text{$\xi \! \to \! \infty$,} \\
\mathfrak{p}_{k}(\tau) \tilde{r}_{0}(\tau) \tau^{-1/3} \xi^{2}(
\mathfrak{c}_{9,k}^{\blacklozenge} \! + \! \mathcal{O}(\xi)),
& \text{$\xi \! \to \! 0$,}
\end{cases}
\end{equation}
where $\mathbb{U}_{k}^{1} \! := \! \mathscr{O}_{\tau^{-1/3+ \delta_{k}}}
(\pm \alpha_{k})$, $\mathbb{U}_{k}^{2} \! := \! \mathscr{O}_{\tau^{-2/3+
2 \delta_{k}}}(\pm \mathrm{i} \sqrt{2} \alpha_{k})$, the parameter $\delta_{k}$ satisfies
(cf. Corollary~\ref{cor3.1.2}$)$ $0 \! < \! \delta \! < \! \delta_{k} \! < \! 1/9$,
the functions $\hat{\mathfrak{f}}_{1,k}(\tau)$ and $\hat{\mathfrak{f}}_{3,k}(\tau)$
are given in Equation~\eqref{mfkeff} below, and $\mathfrak{c}_{m,k}^{\blacklozenge}$,
$m \! = \! 1,2,\dotsc,9$, are constants.
\end{bbbb}
\emph{Proof}. {}From Equations~\eqref{eq3.4}, \eqref{iden3}, and
\eqref{eqlsquared}--\eqref{eqforl}, one shows that
\begin{align}
2l_{k}(\xi)(\mathrm{i} \mathcal{A}_{11}(\xi) \! + \! l_{k}(\xi)) \underset{\tau \to +\infty}{=}&
\, \mathcal{P}_{\infty,k}(\xi) \! + \! \mathcal{P}_{1,k}(\xi) \Delta_{k,\tau}(\xi) \! +
\! \mathcal{O} \! \left(l_{k,\infty}^{2}(\xi) \Delta^{2}_{k,\tau}(\xi) \right) \nonumber \\
+& \, \mathcal{O} \! \left(l_{k,\infty}(\xi) \Delta^{2}_{k,\tau}(\xi) \! \left(2 \xi
\! + \! \dfrac{(\varepsilon b)^{1/3}}{2 \xi}(-2 \! + \! \hat{r}_{0}(\tau)) \right)
\right), \label{eqdenomTminus1T}
\end{align}
where
\begin{align}
\mathcal{P}_{\infty,k}(\xi) :=& \, 2l_{k,\infty}^{2}(\xi) \! + \! 2l_{k,\infty}(\xi)
\! \left(2 \xi \! + \! \dfrac{(\varepsilon b)^{1/3}}{2 \xi}(-2 \! + \! \hat{r}_{0}
(\tau)) \right), \label{eq3.24} \\
\mathcal{P}_{1,k}(\xi) :=& \, 2l_{k,\infty}^{2}(\xi) \! + \! l_{k,\infty}(\xi) \!
\left(2 \xi \! + \! \dfrac{(\varepsilon b)^{1/3}}{2 \xi}(-2 \! + \! \hat{r}_{0}
(\tau)) \right), \label{eq3.25}
\end{align}
and, via Equations~\eqref{eq3.4}, \eqref{iden7}, \eqref{iden3}, and~\eqref{iden4},
\begin{equation} \label{eqnumTminus1T}
\mathcal{A}_{12}(\xi) \partial_{\xi} \mathcal{A}_{21}(\xi) \! - \!
\mathcal{A}_{21}(\xi) \partial_{\xi} \mathcal{A}_{12}(\xi) \! = \!
-\dfrac{4(\varepsilon b)^{2/3}}{\xi^{3}} \left(\dfrac{2(1 \! + \! \hat{u}_{0}
(\tau))^{2} \! + \! (-2 \! + \! \hat{r}_{0}(\tau))}{2(1 \! + \! \hat{u}_{0}(\tau))}
\right) \! + \! \dfrac{4(\varepsilon b)^{1/3}(a \! - \! \mathrm{i}/2) \tau^{-2/3}}{
\xi^{3}(1 \! + \! \hat{u}_{0}(\tau))}.
\end{equation}
Substituting Equations~\eqref{eqdenomTminus1T} and~\eqref{eqnumTminus1T}
into Equation~\eqref{eq3.23} and expanding $(2l_{k}(\xi)(\mathrm{i} \mathcal{A}_{11}
(\xi) \! + \! l_{k}(\xi)))^{-1}$ into a series of powers of $\Delta_{k,\tau}(\xi)$,
one arrives at (cf. Equation~\eqref{eq3.16})
\begin{equation} \label{iden31}
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \diag (T^{-1}(\xi) \partial_{\xi}
T(\xi)) \, \mathrm{d} \xi \underset{\tau \to +\infty}{=} \left(\varkappa_{k}(\tau)
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \dfrac{1}{\xi^{3} \mathcal{P}_{
\infty,k}(\xi)} \, \mathrm{d} \xi \! + \! \mathcal{O} \! \left(\varkappa_{k}(\tau) \int_{
\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \dfrac{\xi^{3} \mathcal{P}_{1,k}(\xi)
\Delta_{k,\tau}(\xi)}{(\xi^{3} \mathcal{P}_{\infty,k}(\xi))^{2}} \, \mathrm{d} \xi \right)
\right) \! \sigma_{3},
\end{equation}
where
\begin{equation} \label{iden30}
\varkappa_{k}(\tau) \! := \! (\varepsilon b)^{2/3} \! \left(\dfrac{2(1 \! +
\! \hat{u}_{0}(\tau))^{2} \! + \! (-2 \! + \! \hat{r}_{0}(\tau))}{1 \! + \!
\hat{u}_{0}(\tau)} \right) \! - \! \dfrac{2(\varepsilon b)^{1/3}(a \! - \!
\mathrm{i}/2) \tau^{-2/3}}{1 \! + \! \hat{u}_{0}(\tau)}.
\end{equation}
Via Equations~\eqref{eqlsquared} and~\eqref{eq3.24}, a calculation reveals
that
\begin{equation}
\dfrac{\varkappa_{k}(\tau)}{\xi^{3} \mathcal{P}_{\infty,k}(\xi)} \! = \!
\mathfrak{p}_{k}(\tau) \! \left(\dfrac{\xi \left(\xi (4 \xi^{2} \! + \!
(\varepsilon b)^{1/3}(-2 \! + \! \hat{r}_{0}(\tau))) \! - \! 4(\xi^{2} \! - \!
\alpha_{k}^{2})(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2} \right)}{(\xi^{2} \!
- \!\alpha_{k}^{2})(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2}(\xi^{2} \! + \!
\hat{\mathfrak{z}}_{k}^{+}(\tau))(\xi^{2} \! + \! \hat{\mathfrak{z}}_{k}^{-}
(\tau))} \right), \label{iden32}
\end{equation}
where $\mathfrak{p}_{k}(\tau)$ is defined by Equation~\eqref{eqpeetee}, and
\begin{equation} \label{iden33}
\hat{\mathfrak{z}}_{k}^{\pm}(\tau) \! := \! \dfrac{(\varepsilon b)^{1/3}}{4
(-2 \! + \! \hat{r}_{0}(\tau))} \! \left(\left(\dfrac{-2 \! + \! \hat{r}_{0}(\tau)}{2}
\right)^{2} \! - \! 3 \mathrm{e}^{\mathrm{i} \pi k/3} \! \mp \! \sqrt{\left(\left(\dfrac{-2 \!
+ \! \hat{r}_{0}(\tau)}{2} \right)^{2} \! - \! 3 \mathrm{e}^{\mathrm{i} \pi k/3} \right)^{2} \!
+ \! 8(-2 \! + \! \hat{r}_{0}(\tau))} \, \right).
\end{equation}
One shows {}from Equations~\eqref{iden3} and~\eqref{iden4}, the
Conditions~\eqref{iden5}, and the Definition~\eqref{iden33} that
\begin{equation} \label{iden34}
\hat{\mathfrak{z}}_{k}^{\pm}(\tau) \underset{\tau \to +\infty}{=}
\dfrac{(\varepsilon b)^{1/3} \mathrm{e}^{-\mathrm{i} \pi k/3}}{2} \left(1 \! + \! \left(
\dfrac{1 \! \pm \! \sqrt{3}}{4} \right) \tilde{r}_{0}(\tau) \tau^{-1/3} \!
+ \! \left(\dfrac{3\sqrt{3} \! \pm \! 5}{16 \sqrt{3}} \right)(\tilde{r}_{0}
(\tau) \tau^{-1/3})^{2} \! + \! \mathcal{O}((\tilde{r}_{0}(\tau) \tau^{-1/3})^{3}) \right),
\end{equation}
whence, via Equation~\eqref{iden32}, the first term on the right-hand side
of Equation~\eqref{iden31} can be presented as follows:
\begin{equation} \label{iden35}
\varkappa_{k}(\tau) \int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \dfrac{1}{
\xi^{3} \mathcal{P}_{\infty,k}(\xi)} \, \mathrm{d} \xi \underset{\tau \to +\infty}{=}
\mathcal{I}_{\tau,k}(\widetilde{\mu}) \! + \! \mathcal{I}_{{\scriptscriptstyle A},k}
(\widetilde{\mu}) \! + \! \mathcal{O}(\mathcal{I}_{{\scriptscriptstyle B},k}
(\widetilde{\mu})),
\end{equation}
where
\begin{align}
\mathcal{I}_{\tau,k}(\widetilde{\mu}) \! :=& \, \mathfrak{p}_{k}(\tau)
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \left(\dfrac{4 \xi^{2}(\xi^{2}
\! + \! 2 \alpha_{k}^{2})^{1/2}}{(\xi^{2} \! + \! 2 \alpha_{k}^{2})(\xi^{2}
\! - \! \alpha_{k}^{2})^{2}} \! - \! \dfrac{4 \xi}{(\xi^{2} \! - \!
\alpha_{k}^{2})^{2}} \right) \mathrm{d} \xi, \label{iden36} \\
\mathcal{I}_{{\scriptscriptstyle A},k}(\widetilde{\mu}) \! :=& \, \mathfrak{p}_{k}
(\tau) \tilde{r}_{0}(\tau) \tau^{-1/3} \int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}
\left(\dfrac{4 \alpha_{k}^{2} \xi^{2}(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2}}{
(\xi^{2} \! + \! 2 \alpha_{k}^{2})(\xi^{2} \! - \! \alpha_{k}^{2})^{3}} \! - \!
\dfrac{2 \alpha_{k}^{2} \xi}{(\xi^{2} \! - \! \alpha_{k}^{2})^{3}} \right) \mathrm{d} \xi,
\label{iden37} \\
\mathcal{I}_{{\scriptscriptstyle B},k}(\widetilde{\mu}) \! :=& \, \mathfrak{p}_{k}
(\tau)(\tilde{r}_{0}(\tau) \tau^{-1/3})^{2} \int_{\widetilde{\mu}_{0,k}}^{
\widetilde{\mu}} \left(\dfrac{\alpha_{k}^{4} \xi^{2}(\xi^{2} \! + \! 2
\alpha_{k}^{2})^{1/2}}{(\xi^{2} \! + \! 2 \alpha_{k}^{2})(\xi^{2} \! - \!
\alpha_{k}^{2})^{4}} \! - \! \dfrac{4 \xi^{3}}{(\xi^{2} \! - \! \alpha_{k}^{2})^{4}}
\! + \! \dfrac{4 \xi^{4}(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2}}{(\xi^{2} \!
+ \! 2 \alpha_{k}^{2})(\xi^{2} \! - \! \alpha_{k}^{2})^{4}} \right) \mathrm{d} \xi.
\label{iden38}
\end{align}
A partial fraction decomposition shows that
\begin{equation} \label{iden39}
\dfrac{\xi^{2}}{(\xi^{2} \! + \! 2 \alpha_{k}^{2})(\xi^{2} \! - \! \alpha_{k}^{2})^{2}}
\! = \! \dfrac{\alpha_{k}^{-3}}{36} \dfrac{1}{\xi \! - \! \alpha_{k}} \! + \!
\dfrac{\alpha_{k}^{-2}}{12} \dfrac{1}{(\xi \! - \! \alpha_{k})^{2}} \! - \!
\dfrac{\alpha_{k}^{-3}}{36} \dfrac{1}{\xi \! + \! \alpha_{k}} \! + \!
\dfrac{\alpha_{k}^{-2}}{12} \dfrac{1}{(\xi \! + \! \alpha_{k})^{2}} \! - \!
\dfrac{2 \alpha_{k}^{-2}}{9} \dfrac{1}{\xi^{2} \! + \! 2 \alpha_{k}^{2}};
\end{equation}
substituting Equation~\eqref{iden39} into Equation~\eqref{iden36} and
integrating, one arrives at Equations \eqref{eqItee}--\eqref{eqFtee}.
Equations~\eqref{iden37} and~\eqref{iden38} contribute to the error
function, $\mathcal{E}_{\scriptscriptstyle T,k}(\mathbf{\cdot})$, in
Equation~\eqref{eqintTminus1T}; therefore, only its asymptotics at the
turning and the singular points are requisite. Evaluating the integrals
in Equations~\eqref{iden37} and~\eqref{iden38}, one shows that
\begin{equation} \label{iden40}
\mathcal{I}_{{\scriptscriptstyle A},k}(\widetilde{\mu})
\underset{\tau \to +\infty}{=}
\begin{cases}
\mathfrak{p}_{k}(\tau) \tilde{r}_{0}(\tau) \tau^{-1/3}
(\hat{\mathfrak{h}}_{1,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{1,k}
(\widetilde{\mu}_{0,k})), & \text{$\widetilde{\mu} \! \in \!
\mathscr{O}_{\tau^{-1/3+ \delta_{k}}}(\pm \alpha_{k})$,} \\
\mathfrak{p}_{k}(\tau) \tilde{r}_{0}(\tau) \tau^{-1/3}
(\hat{\mathfrak{h}}_{2,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{2,k}
(\widetilde{\mu}_{0,k})), & \text{$\widetilde{\mu} \! \in \!
\mathscr{O}_{\tau^{-2/3+ 2 \delta_{k}}}(\pm \mathrm{i} \sqrt{2} \alpha_{k})$,} \\
\mathfrak{p}_{k}(\tau) \tilde{r}_{0}(\tau) \tau^{-1/3}
(\hat{\mathfrak{h}}_{3,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{3,k}
(\widetilde{\mu}_{0,k})), & \text{$\widetilde{\mu} \! \to \! \infty$,} \\
\mathfrak{p}_{k}(\tau) \tilde{r}_{0}(\tau) \tau^{-1/3}
(\hat{\mathfrak{h}}_{4,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{4,k}
(\widetilde{\mu}_{0,k})), & \text{$\widetilde{\mu} \! \to \! 0$,}
\end{cases}
\end{equation}
where
\begin{gather*}
\hat{\mathfrak{h}}_{1,k}(\xi) \! := \! c_{1,k}^{\flat}(\xi \! \mp \! \alpha_{k})^{-2}
\! + \! c_{2,k}^{\flat}(\xi \! \mp \! \alpha_{k})^{-1} \! + \! c_{3,k}^{\flat}
\ln (\xi \! \mp \! \alpha_{k}) \! + \! \sum_{m \in \mathbb{Z}_{+}}d_{m,k}^{\flat}
(\xi \! \mp \! \alpha_{k})^{m}, \\
\hat{\mathfrak{h}}_{2,k}(\xi) \! := \! (\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{1/2}
\sum_{m \in \mathbb{Z}_{+}}c_{m,k}^{\natural}(\xi \! \mp \! \mathrm{i} \sqrt{2}
\alpha_{k})^{m} \! + \! \sum_{m \in \mathbb{Z}_{+}}d_{m,k}^{\natural}
(\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{m}, \\
\hat{\mathfrak{h}}_{3,k}(\xi) \! := \! \xi^{-4} \sum_{m \in \mathbb{Z}_{+}}
c_{m,k}^{\sharp,\infty} \xi^{-2m}, \, \quad \, \quad \, \hat{\mathfrak{h}}_{4,k}
(\xi) \! := \! \xi^{2} \sum_{m \in \mathbb{Z}_{+}}c_{m,k}^{\sharp,0} \xi^{m},
\end{gather*}
with $c_{1,k}^{\flat}$, $c_{2,k}^{\flat}$, $c_{3,k}^{\flat}$, $d_{m,k}^{\flat}$,
$c_{m,k}^{\natural}$, $d_{m,k}^{\natural}$, $c_{m,k}^{\sharp,\infty}$, and
$c_{m,k}^{\sharp,0}$ constants, and
\begin{equation} \label{iden41}
\mathcal{I}_{{\scriptscriptstyle B},k}(\widetilde{\mu}) \underset{\tau \to
+\infty}{=}
\begin{cases}
\mathfrak{p}_{k}(\tau)(\tilde{r}_{0}(\tau) \tau^{-1/3})^{2}
(\hat{\mathfrak{h}}_{5,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{5,k}
(\widetilde{\mu}_{0,k})), & \text{$\widetilde{\mu} \! \in \!
\mathscr{O}_{\tau^{-1/3+ \delta_{k}}}(\pm \alpha_{k})$,} \\
\mathfrak{p}_{k}(\tau)(\tilde{r}_{0}(\tau) \tau^{-1/3})^{2}
(\hat{\mathfrak{h}}_{6,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{6,k}
(\widetilde{\mu}_{0,k})), & \text{$\widetilde{\mu} \! \in \!
\mathscr{O}_{\tau^{-2/3+ 2 \delta_{k}}}(\pm \mathrm{i} \sqrt{2} \alpha_{k})$,} \\
\mathfrak{p}_{k}(\tau)(\tilde{r}_{0}(\tau) \tau^{-1/3})^{2}
(\hat{\mathfrak{h}}_{7,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{7,k}
(\widetilde{\mu}_{0,k})), & \text{$\widetilde{\mu} \! \to \! \infty$,} \\
\mathfrak{p}_{k}(\tau)(\tilde{r}_{0}(\tau) \tau^{-1/3})^{2}
(\hat{\mathfrak{h}}_{8,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{8,k}
(\widetilde{\mu}_{0,k})), & \text{$\widetilde{\mu} \! \to \! 0$,}
\end{cases}
\end{equation}
where
\begin{align*}
\hat{\mathfrak{h}}_{5,k}(\xi) \! :=& \, \hat{c}_{1,k}^{\flat}(\xi \! \mp \!
\alpha_{k})^{-3} \! + \! \hat{c}_{2,k}^{\flat}(\xi \! \mp \! \alpha_{k})^{-2}
\! + \! \hat{c}_{3,k}^{\flat}(\xi \! \mp \! \alpha_{k})^{-1} \! + \!
\hat{c}_{4,k}^{\flat} \ln (\xi \! \mp \! \alpha_{k}) \! + \! \sum_{m \in
\mathbb{Z}_{+}} \hat{d}_{m,k}^{\flat}(\xi \! \mp \! \alpha_{k})^{m}, \\
\hat{\mathfrak{h}}_{6,k}(\xi) \! :=& \, (\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{1/2}
\sum_{m \in \mathbb{Z}_{+}} \hat{c}_{m,k}^{\natural}(\xi \! \mp \! \mathrm{i} \sqrt{2}
\alpha_{k})^{m} \! + \! \sum_{m \in \mathbb{Z}_{+}} \hat{d}_{m,k}^{\natural}
(\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{m}, \\
\hat{\mathfrak{h}}_{7,k}(\xi) \! :=& \, \xi^{-6} \sum_{m \in \mathbb{Z}_{+}}
\hat{c}_{m,k}^{\sharp,\infty} \xi^{-2m}, \, \quad \, \quad \, \hat{\mathfrak{h}}_{8,k}
(\xi) \! := \! \xi^{3} \sum_{m \in \mathbb{Z}_{+}} \hat{c}_{m,k}^{\sharp,0} \xi^{m},
\end{align*}
with $\hat{c}_{1,k}^{\flat}$, $\hat{c}_{2,k}^{\flat}$, $\hat{c}_{3,k}^{\flat}$,
$\hat{c}_{4,k}^{\flat}$, $d_{m,k}^{\flat}$, $\hat{c}_{m,k}^{\natural}$,
$\hat{d}_{m,k}^{\natural}$, $\hat{c}_{m,k}^{\sharp,\infty}$, and
$\hat{c}_{m,k}^{\sharp,0}$ constants.
One now estimates the second term on the right-hand side of
Equation~\eqref{iden31}. {}From Equations~\eqref{eqlsquared}---\eqref{eqforl},
it follows, after simplification, that
\begin{align} \label{iden43}
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \dfrac{\xi^{3}
\mathcal{P}_{1,k}(\xi) \Delta_{k,\tau}(\xi)}{(\xi^{3} \mathcal{P}_{\infty,k}
(\xi))^{2}} \, \mathrm{d} \xi \! = \! \int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}&
\, \dfrac{\xi \left(\xi (4 \xi^{2} \! + \! (\varepsilon b)^{1/3}(-2 \! + \!
\hat{r}_{0}(\tau))) \! + \! 8(\xi^{2} \! - \! \alpha_{k}^{2})(\xi^{2} \! + \! 2
\alpha_{k}^{2})^{1/2} \right)}{\left(\xi (4 \xi^{2} \! + \! (\varepsilon b)^{1/3}
(-2 \! + \! \hat{r}_{0}(\tau))) \! + \! 4(\xi^{2} \! - \! \alpha_{k}^{2})(\xi^{2}
\! + \! 2 \alpha_{k}^{2})^{1/2} \right)^{2}} \nonumber \\
\times& \, \dfrac{(\xi^{2} \hat{h}_{0}(\tau) \! + \! \xi^{4}(a \! - \! \mathrm{i}/2)
\tau^{-2/3})}{4(\xi^{2} \! - \! \alpha_{k}^{2})^{3}(\xi^{2} \! + \! 2
\alpha_{k}^{2})^{3/2}} \, \mathrm{d} \xi.
\end{align}
Evaluating the integral in Equation~\eqref{iden43}, a lengthy calculation shows
that its asymptotics at the turning and the singular points are given by
\begin{equation} \label{iden47}
\varkappa_{k}(\tau) \int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}
\dfrac{\xi^{3} \mathcal{P}_{1,k}(\xi) \Delta_{k,\tau}(\xi)}{(\xi^{3}
\mathcal{P}_{\infty,k}(\xi))^{2}} \, \mathrm{d} \xi \underset{\tau \to +\infty}{=}
\begin{cases}
\hat{\mathfrak{h}}_{9,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{9,k}
(\widetilde{\mu}_{0,k}), & \text{$\widetilde{\mu} \! \in \!
\mathscr{O}_{\tau^{-1/3+ \delta_{k}}}(\pm \alpha_{k})$,} \\
\hat{\mathfrak{h}}_{10,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{10,k}
(\widetilde{\mu}_{0,k}), & \text{$\widetilde{\mu} \! \in \!
\mathscr{O}_{\tau^{-2/3+ 2 \delta_{k}}}(\pm \mathrm{i} \sqrt{2} \alpha_{k})$,} \\
\hat{\mathfrak{h}}_{11,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{11,k}
(\widetilde{\mu}_{0,k}), & \text{$\widetilde{\mu} \! \to \! \infty$,} \\
\hat{\mathfrak{h}}_{12,k}(\widetilde{\mu}) \! - \! \hat{\mathfrak{h}}_{12,k}
(\widetilde{\mu}_{0,k}), & \text{$\widetilde{\mu} \! \to \! 0$,}
\end{cases}
\end{equation}
where
\begin{align*}
\hat{\mathfrak{h}}_{9,k}(\xi) :=& \, \tilde{c}_{1,k}^{\sharp} \mathfrak{p}_{k}(\tau)
\hat{\mathfrak{f}}_{1,k}(\tau)(\xi \! \mp \! \alpha_{k})^{-2} \! + \!
\mathfrak{p}_{k}(\tau)(\tilde{c}_{2,k}^{\sharp} \hat{\mathfrak{f}}_{2,k}
(\tau) \! + \! \tilde{c}_{3,k}^{\sharp} \tilde{r}_{0}(\tau) \tau^{-1/3}
\hat{\mathfrak{f}}_{1,k}(\tau))(\xi \! \mp \! \alpha_{k})^{-3}, \\
\hat{\mathfrak{h}}_{10,k}(\xi) :=& \, \tilde{c}_{4,k}^{\sharp} \mathfrak{p}_{k}(\tau)
\hat{\mathfrak{f}}_{3,k}(\tau)(\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k})^{-1/2} \!
+ \! \tilde{c}_{5,k}^{\sharp} \mathfrak{p}_{k}(\tau) \hat{\mathfrak{f}}_{3,k}
(\tau) \ln (\xi \! \mp \! \mathrm{i} \sqrt{2} \alpha_{k}), \\
\hat{\mathfrak{h}}_{11,k}(\xi) :=& \, \mathfrak{p}_{k}(\tau) \tau^{-2/3} \xi^{-6}
\! \left(\tilde{c}_{6,k}^{\sharp} \! + \! \xi^{-2}(\tilde{c}_{7,k}^{\sharp} \! + \!
\tilde{c}_{8,k}^{\sharp} \tau^{2/3} \hat{h}_{0}(\tau)) \right. \\
+&\left. \, \mathcal{O} \! \left(\tilde{r}_{0}(\tau) \tau^{-1/3}(\tilde{c}_{9,k}^{\sharp}
\! + \! \xi^{-2}(\tilde{c}_{10,k}^{\sharp} \! + \! \tilde{c}_{11,k}^{\sharp} \tau^{2/3}
\hat{h}_{0}(\tau))) \right) \right), \\
\hat{\mathfrak{h}}_{12,k}(\xi) :=& \, \mathfrak{p}_{k}(\tau) \tau^{-2/3} \xi^{4}
\! \left(\tilde{c}_{12,k}^{\sharp} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! \xi
\tilde{c}_{13,k}^{\sharp} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! \xi^{2}
(\tilde{c}_{14,k}^{\sharp} \! + \! \tilde{c}_{15,k}^{\sharp} \tau^{2/3}
\hat{h}_{0}(\tau)) \right. \\
+&\left. \, \mathcal{O} \! \left(\tilde{r}_{0}(\tau) \tau^{-1/3}
(\tilde{c}_{16,k}^{\sharp} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! \xi
\tilde{c}_{17,k}^{\sharp} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! \xi^{2}
(\tilde{c}_{18,k}^{\sharp} \! + \! \tilde{c}_{19,k}^{\sharp} \tau^{2/3}
\hat{h}_{0}(\tau))) \right) \right),
\end{align*}
with $\tilde{c}_{m,k}^{\sharp}$, $m \! = \! 1,2,\dotsc,19$, constants, and
\begin{equation} \label{mfkeff}
\hat{\mathfrak{f}}_{j,k}(\tau) \! = \! \tau^{-2/3} \! \left((a \! - \! \mathrm{i}/2) \!
+ \! \dfrac{2 \hat{s}(j) \hat{h}_{0}(\tau) \tau^{2/3}}{(3 \! + \! (-1)^{j+1})
\alpha_{k}^{2}} \right), \quad j \! = \! 1,2,3,
\end{equation}
where $\hat{s}(1) \! = \! \hat{s}(2) \! = \! +1$ and $\hat{s}(3) \! = \! -1$. Thus,
assembling the error estimates~\eqref{iden40}, \eqref{iden41}, and~\eqref{iden47},
and retaining only leading-order terms, one arrives at the error function defined
by Equation~\eqref{eqee2}. \hfill $\qed$
\begin{ffff} \label{cor3.1.3}
Set $\widetilde{\mu}_{0,k} \! = \! \alpha_{k} \! + \! \tau^{-1/3} \widetilde{
\Lambda}$, $k \! = \! \pm 1$, where $\widetilde{\Lambda} \! =_{\tau
\to +\infty} \! \mathcal{O}(\tau^{\delta_{k}})$, $0 \! < \! \delta \! < \!
\delta_{k} \! < \! 1/9$. Then,
\begin{align}
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \diag (T^{-1}(\xi) \partial_{\xi}
T(\xi)) \, \mathrm{d} \xi \underset{\tau \to +\infty}{=}& \, \left(\mathfrak{p}_{k}
(\tau)(\digamma_{\tau,k}(\widetilde{\mu}) \! + \! \digamma_{\tau,k}^{\sharp}
(\tau)) \! + \! \mathcal{O}(\mathcal{E}_{\scriptscriptstyle T,k}(\widetilde{\mu}))
\right. \nonumber \\
+&\left. \, \mathcal{O} \! \left((\mathfrak{c}_{3,k} \tau^{-1/3} \! + \!
\mathfrak{c}_{4,k}(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau))) \right. \right.
\nonumber \\
\times&\left. \left. \left(\dfrac{\mathfrak{c}_{1,k} \tau^{-1/3} \! + \!
\mathfrak{c}_{2,k} \tilde{r}_{0}(\tau)}{\widetilde{\Lambda}^{2}} \right)
\right) \right) \! \sigma_{3}, \label{eq3.36}
\end{align}
where $\mathfrak{p}_{k}(\tau)$, $\digamma_{\tau,k}(\xi)$ and
$\mathcal{E}_{\scriptscriptstyle T,k}(\xi)$ are defined by
Equations~\eqref{eqpeetee}, \eqref{eqFtee}, and~\eqref{eqee2}, respectively,
\begin{equation}
\digamma_{\tau,k}^{\sharp}(\tau) \! := \! -\dfrac{\tau^{1/3}}{\alpha_{k}
\widetilde{\Lambda}} \! \left(\dfrac{\sqrt{3} \! \mp \! 1}{\sqrt{3}} \right) \!
\mp \! \dfrac{2}{3 \sqrt{3} \alpha_{k}^{2}} \! \left(-\dfrac{1}{3} \ln \tau
\! + \! \ln \widetilde{\Lambda} \right) \! \pm \! \dfrac{(5 \! \pm \!
3 \sqrt{3})}{6 \sqrt{3} \alpha_{k}^{2}} \! \pm \! \dfrac{2}{3 \sqrt{3}
\alpha_{k}^{2}} \ln (3 \alpha_{k}), \label{eqfteesharp}
\end{equation}
with the upper (resp., lower) signs taken according to the branch of the
square-root function $\lim_{\xi^{2} \to +\infty}(\xi^{2} \! + \! 2
\alpha_{k}^{2})^{1/2} \! = \! +\infty$ (resp., $\lim_{\xi^{2} \to +\infty}
(\xi^{2} \! + \! 2 \alpha_{k}^{2})^{1/2} \! = \! -\infty)$, and $\mathfrak{c}_{m,k}$,
$m \! = \! 1,2,3,4$, are constants.
\end{ffff}
\emph{Proof}. Substituting $\widetilde{\mu}_{0,k}$, as given in the corollary,
for the argument of the functions $\digamma_{\tau,k}(\xi)$ and
$\mathcal{E}_{\scriptscriptstyle T,k}(\xi)$ (cf. Equation~\eqref{eqFtee} and the
first line of Equation~\eqref{eqee2}, respectively) and expanding with respect
to the small parameter $\tau^{-1/3} \widetilde{\Lambda}$, one arrives at the
following estimates:
\begin{equation} \label{asympforf1}
-\digamma_{\tau,k}(\widetilde{\mu}_{0,k}) \underset{\tau \to +\infty}{=}
\digamma_{\tau,k}^{\sharp}(\tau) \! + \! \mathcal{O}(\tau^{-1/3}
\widetilde{\Lambda}),
\end{equation}
where $\digamma_{\tau,k}^{\sharp}(\tau)$ is defined by
Equation~\eqref{eqfteesharp}, and
\begin{equation} \label{asympforf2}
\mathcal{O}(\mathcal{E}_{\scriptscriptstyle T,k}(\widetilde{\mu}_{0,k}))
\underset{\tau \to +\infty}{=} \mathcal{O} \! \left(\dfrac{\mathfrak{p}_{k}
(\tau) \tilde{r}_{0}(\tau)}{\tau^{-1/3} \widetilde{\Lambda}^{2}} \right) \! +
\! \mathcal{O} \! \left(\dfrac{\mathfrak{p}_{k}(\tau) \hat{\mathfrak{f}}_{1,k}
(\tau)}{\tau^{-2/3} \widetilde{\Lambda}^{2}} \right) \! + \! \mathcal{O} \!
\left(\dfrac{\mathfrak{p}_{k}(\tau) \tilde{r}_{0}(\tau)}{\widetilde{\Lambda}} \right).
\end{equation}
{}From the Conditions~\eqref{iden5} and the Definitions~\eqref{eqpeetee}
and~\eqref{mfkeff} (for $j \! = \! 1)$, one shows that
\begin{equation} \label{asympforf3}
\mathfrak{p}_{k}(\tau) \underset{\tau \to +\infty}{=} \mathfrak{p}_{k}^{\infty}
(\tau) \! + \! \mathcal{O}((\tilde{r}_{0}(\tau) \! - \! 2v_{0}(\tau)) \tau^{-1}) \! +
\! \mathcal{O}(((\tilde{r}_{0}(\tau) \! - \! 2v_{0}(\tau))(\tilde{r}_{0}(\tau) \! + \!
4v_{0}(\tau)) \! + \! 4v_{0}^{2}(\tau)) \tau^{-2/3}),
\end{equation}
where
\begin{equation} \label{peekayity}
\mathfrak{p}_{k}^{\infty}(\tau) \! := \! \frac{\tau^{-1/3}}{16} \! \left(-\alpha_{k}^{2}
(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau)) \! + \! (a \! - \! \mathrm{i}/2) \tau^{-1/3} \right),
\end{equation}
and
\begin{align} \label{asympforf4}
\hat{\mathfrak{f}}_{1,k}(\tau) \underset{\tau \to +\infty}{=}
\tau^{-2/3} \! \left(\dfrac{1}{2}(a \! - \! \mathrm{i}/2) \! + \! \mathcal{O}(v_{0}(\tau)
\tau^{-1/3}) \! + \! \mathcal{O} \! \left(8v_{0}^{2}(\tau) \! + \! 4v_{0}(\tau)
\tilde{r}_{0}(\tau) \! - \! (\tilde{r}_{0}(\tau))^{2} \right) \right);
\end{align}
thus, {}from the Conditions~\eqref{iden5} and the
Asymptotics~\eqref{asympforf2}--\eqref{asympforf4}, it follows that, for constants
$c_{m,k}$, $m \! = \! 1,2,\dotsc,6$,
\begin{align} \label{asympforf5}
\mathcal{O}(\mathcal{E}_{\scriptscriptstyle T,k}(\widetilde{\mu}_{0,k}))
\underset{\tau \to +\infty}{=}& \, \mathcal{O} \! \left(\left(\dfrac{c_{1,k}
\tau^{-1/3} \! + \! c_{2,k} \tilde{r}_{0}(\tau)}{\widetilde{\Lambda}^{2}}
\right) \! \left(c_{3,k} \tau^{-1/3} \! + \! c_{4,k}(\tilde{r}_{0}(\tau)
\! + \! 4v_{0}(\tau)) \right) \right) \nonumber \\
+& \, \mathcal{O} \! \left(\dfrac{\tau^{-1/3}}{\widetilde{\Lambda}} \!
\left(c_{5,k} \tilde{r}_{0}(\tau) \tau^{-1/3} \! + \! c_{6,k} \tilde{r}_{0}
(\tau)(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau)) \right) \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathcal{O}(\tau^{-2/3}
\widetilde{\Lambda}^{-2}) \! + \! \mathcal{O}(\tau^{-1}
\widetilde{\Lambda}^{-1}).
\end{align}
{}From the Conditions~\eqref{iden5}, Equation~\eqref{eqItee}, and the
asymptotics~\eqref{asympforf1} and~\eqref{asympforf3}, it follows that
\begin{equation} \label{asympforf6}
\mathcal{I}_{\tau,k}(\widetilde{\mu}) \underset{\tau \to +\infty}{=}
\mathfrak{p}_{k}(\tau)(\digamma_{\tau,k}(\widetilde{\mu}) \! +
\! \digamma_{\tau,k}^{\sharp}(\tau)) \! + \! \mathcal{O}((\tilde{r}_{0}(\tau)
\! + \! 4v_{0}(\tau)) \tau^{-2/3} \widetilde{\Lambda}) \! + \! \mathcal{O}
(\tau^{-1} \widetilde{\Lambda}).
\end{equation}
Therefore, via the asymptotic estimates~\eqref{asympforf5}
and~\eqref{asympforf6}, and the fact that $\widetilde{\Lambda} \!
=_{\tau \to +\infty} \! \mathcal{O}(\tau^{\delta_{k}})$, $0 \! < \! \delta
\! < \! \delta_{k} \! < \! 1/9$, the result stated in the corollary (cf.
Equation~\eqref{eq3.36}) is a consequence of Proposition~\ref{prop3.1.4}
(cf. Equation~\eqref{eqintTminus1T}), upon retaining only leading-order
contributions. \hfill $\qed$
\begin{ffff} \label{cor3.1.5}
Let the conditions stated in Corollary~{\rm \ref{cor3.1.3}} be valid. Then, for
the branch of $l_{k}(\xi)$, $k \! = \! \pm 1$, that is positive for large and
small positive $\xi$,
\begin{align} \label{eq3.41}
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \diag (T^{-1}(\xi) \partial_{\xi}
T(\xi)) \, \mathrm{d} \xi \underset{\underset{\widetilde{\mu} \to \infty}{\tau \to
+\infty}}{=}& \, \left(\mathfrak{p}_{k}(\tau) \digamma_{\tau,k}^{\sharp,\infty}
(\tau) \! + \! \mathcal{O} \! \left(\left(\dfrac{\mathfrak{c}_{1,k} \tau^{-1/3}
\! + \! \mathfrak{c}_{2,k} \tilde{r}_{0}(\tau)}{\widetilde{\Lambda}^{2}} \right)
\right. \right. \nonumber \\
\times&\left. \left. (\mathfrak{c}_{3,k} \tau^{-1/3} \! + \! \mathfrak{c}_{4,k}
(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau))) \right) \right. \nonumber \\
+&\left. \, \mathcal{O}(\widetilde{\mu}^{-2} \tau^{-1/3}(\mathfrak{c}_{5,k}
\tau^{-1/3} \! + \! \mathfrak{c}_{6,k}(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau))))
\right) \! \sigma_{3},
\end{align}
where $\mathfrak{p}_{k}(\tau)$ is defined by Equation~\eqref{eqpeetee},
\begin{equation} \label{asympforf7}
\digamma_{\tau,k}^{\sharp,\infty}(\tau) \! := \! -\dfrac{(\sqrt{3} \! - \! 1)
\tau^{1/3}}{\sqrt{3} \alpha_{k} \widetilde{\Lambda}} \! - \! \dfrac{2}{3
\sqrt{3} \alpha_{k}^{2}} \! \left(-\dfrac{1}{3} \ln \tau \! + \! \ln
\widetilde{\Lambda} \right) \! + \! \dfrac{5 \! - \! \sqrt{3}}{6 \sqrt{3}
\alpha_{k}^{2}} \! + \! \dfrac{2}{3 \sqrt{3} \alpha_{k}^{2}} \ln (3(2 \! -
\! \sqrt{3}) \alpha_{k}),
\end{equation}
and
\begin{align} \label{eq3.43}
\int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}} \diag (T^{-1}(\xi) \partial_{\xi}
T(\xi)) \, \mathrm{d} \xi \underset{\underset{\widetilde{\mu} \to 0}{\tau \to
+\infty}}{=}& \, \left(\mathfrak{p}_{k}(\tau) \digamma_{\tau,k}^{\sharp,0}
(\tau) \! + \! \mathcal{O} \! \left(\left(\dfrac{\mathfrak{c}_{7,k} \tau^{-1/3}
\! + \! \mathfrak{c}_{8,k} \tilde{r}_{0}(\tau)}{\widetilde{\Lambda}^{2}}
\right) \right. \right. \nonumber \\
\times&\left. \left. (\mathfrak{c}_{9k} \tau^{-1/3} \! + \! \mathfrak{c}_{10,k}
(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau))) \right) \right. \nonumber \\
+&\left. \, \mathcal{O}(\widetilde{\mu}^{2} \tau^{-1/3}(\mathfrak{c}_{11,k}
\tau^{-1/3} \! + \! \mathfrak{c}_{12,k}(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau))))
\right) \! \sigma_{3},
\end{align}
where
\begin{equation} \label{asympforf8}
\digamma_{\tau,k}^{\sharp,0}(\tau) \! := \! -\dfrac{(\sqrt{3} \! + \! 1)
\tau^{1/3}}{\sqrt{3} \alpha_{k} \widetilde{\Lambda}} \! + \! \dfrac{2}{3
\sqrt{3} \alpha_{k}^{2}} \! \left(-\dfrac{1}{3} \ln \tau \! + \! \ln
\widetilde{\Lambda} \right) \! - \! \dfrac{(5 \! + \! 9 \sqrt{3})}{6 \sqrt{3}
\alpha_{k}^{2}} \! + \! \dfrac{2}{3 \sqrt{3} \alpha_{k}^{2}} \ln
(\mathrm{e}^{\mathrm{i} k \pi}/3 \alpha_{k}),
\end{equation}
with constants $\mathfrak{c}_{m,k}$, $m \! = \! 1,2,\ldots,12$.
\end{ffff}
\emph{Proof}. Choosing consistently the corresponding branches in
Equations~\eqref{eqFtee} and~\eqref{eqfteesharp}, and via the third and
fourth lines of Equation~\eqref{eqee2}, respectively, one shows, via the
Conditions~\eqref{iden5} and the asymptotics~\eqref{asympforf3}, that
(cf. Equation~\eqref{eq3.36})
\begin{align}
\digamma_{\tau,k}(\widetilde{\mu}) \underset{\underset{\widetilde{\mu}
\to \infty}{\tau \to +\infty}}{=}& \, -\dfrac{2}{3 \alpha_{k}^{2}} \! + \!
\dfrac{2}{3 \sqrt{3} \alpha_{k}^{2}} \ln (2 \! - \! \sqrt{3}) \! + \!
\mathcal{O}(\widetilde{\mu}^{-2}), \label{iden48} \\
\digamma_{\tau,k}(\widetilde{\mu}) \underset{\underset{\widetilde{\mu}
\to 0}{\tau \to +\infty}}{=}& \, -\dfrac{2}{\alpha_{k}^{2}} \! + \! \dfrac{2}{3
\sqrt{3} \alpha_{k}^{2}} \ln (\mathrm{e}^{\mathrm{i} k \pi}) \! + \! \mathcal{O}
(\widetilde{\mu}^{2}), \label{iden49} \\
\mathcal{O}(\mathcal{E}_{\scriptscriptstyle T,k}(\widetilde{\mu})) \underset{
\underset{\widetilde{\mu} \to \infty}{\tau \to +\infty}}{=}& \, \mathcal{O}
(\widetilde{\mu}^{-4} \tilde{r}_{0}(\tau)(\tilde{r}_{0}(\tau) \! + \! 4v_{0}
(\tau)) \tau^{-2/3}) \! + \! \mathcal{O} \! \left(\widetilde{\mu}^{-4}
\tilde{r}_{0}(\tau) \tau^{-1} \right), \label{iden50} \\
\mathcal{O}(\mathcal{E}_{\scriptscriptstyle T,k}(\widetilde{\mu})) \underset{
\underset{\widetilde{\mu} \to 0}{\tau \to +\infty}}{=}& \, \mathcal{O}
(\widetilde{\mu}^{2} \tilde{r}_{0}(\tau)(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau))
\tau^{-2/3}) \! + \! \mathcal{O} \! \left(\widetilde{\mu}^{2} \tilde{r}_{0}
(\tau) \tau^{-1} \right). \label{iden51}
\end{align}
Via the Conditions~\eqref{iden5}, Equation \eqref{eqfteesharp}, and the
Asymptotics~\eqref{asympforf3} and~\eqref{iden48}--\eqref{iden51}, it
follows that (cf. Equation~\eqref{eq3.36})
\begin{gather}
\mathfrak{p}_{k}(\tau)(\digamma_{\tau,k}(\widetilde{\mu}) \! + \!
\digamma_{\tau,k}^{\sharp}(\tau)) \underset{\underset{\widetilde{\mu}
\to \infty}{\tau \to +\infty}}{=} \mathfrak{p}_{k}(\tau) \digamma_{
\tau,k}^{\sharp,\infty}(\tau) \! + \! \mathcal{O}(\widetilde{\mu}^{-2}
(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau)) \tau^{-1/3}) \! + \! \mathcal{O}
(\widetilde{\mu}^{-2} \tau^{-2/3}), \label{asympforf9} \\
\mathfrak{p}_{k}(\tau)(\digamma_{\tau,k}(\widetilde{\mu}) \! + \!
\digamma_{\tau,k}^{\sharp}(\tau)) \underset{\underset{\widetilde{\mu}
\to 0}{\tau \to +\infty}}{=} \mathfrak{p}_{k}(\tau) \digamma_{\tau,k}^{
\sharp,0}(\tau) \! + \! \mathcal{O}(\widetilde{\mu}^{2}(\tilde{r}_{0}(\tau)
\! + \! 4v_{0}(\tau)) \tau^{-1/3}) \! + \! \mathcal{O}(\widetilde{\mu}^{2}
\tau^{-2/3}), \label{asympforf10}
\end{gather}
where $\digamma_{\tau,k}^{\sharp,\infty}(\tau)$ and
$\digamma_{\tau,k}^{\sharp,0}(\tau)$ are defined by Equations~\eqref{asympforf7}
and~\eqref{asympforf8}, respectively. The results stated in the corollary are now a
consequence of the Conditions~\eqref{iden5}, Equation~\eqref{eq3.36}, and the
asymptotic expansions~\eqref{iden50}--\eqref{asympforf10}, upon retaining
only leading-order terms. \hfill $\qed$
\begin{bbbb} \label{prop3.1.5}
Let $T(\widetilde{\mu})$ be given in Equation~\eqref{eq3.18}, with
$\mathcal{A}(\widetilde{\mu})$ defined by Equation~\eqref{eq3.4}
and $l_{k}^{2}(\widetilde{\mu})$, $k \! = \! \pm 1$, given in
Equation~\eqref{eq3.19}, with the branches defined as in
Corollary~{\rm \ref{cor3.1.4}}. Then,
\begin{align} \label{eq3.51}
T(\widetilde{\mu}) \underset{\underset{\widetilde{\mu} \to \infty}{\tau
\to +\infty}}{=}& \, (b(\tau))^{-\frac{1}{2} \ad (\sigma_{3})} \! \left(
\mathrm{I} \! + \! \dfrac{1}{\widetilde{\mu}}
\begin{pmatrix}
0 & -\frac{(\varepsilon b)^{2/3}}{2}(1 \! + \! \hat{u}_{0}(\tau)) \\
\frac{2(a-\mathrm{i}/2) \tau^{-2/3} - (\varepsilon b)^{1/3}(-2+\hat{r}_{0}
(\tau))}{4(\varepsilon b)^{2/3}(1+ \hat{u}_{0}(\tau))} & 0
\end{pmatrix} \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left(\dfrac{1}{\widetilde{\mu}^{2}} \!
\begin{pmatrix}
\mathfrak{c}_{1}(\tau) & 0 \\
0 & \mathfrak{c}_{1}(\tau)
\end{pmatrix} \right) \right),
\end{align}
and
\begin{equation} \label{eq3.52}
T(\widetilde{\mu}) \underset{\underset{\widetilde{\mu} \to 0}{\tau \to
+\infty}}{=} \dfrac{1}{\sqrt{2}} \! \left(\dfrac{b(\tau)}{\sqrt{\smash[b]{
\varepsilon b}}} \right)^{-\frac{1}{2} \ad (\sigma_{3})} \! \left(
\begin{pmatrix}
1 & -1 \\
1 & 1
\end{pmatrix} \! + \! \widetilde{\mu} \, \dfrac{(-2 \! + \! \hat{r}_{0}
(\tau))}{4(\varepsilon b)^{1/6}}
\begin{pmatrix}
-1 & -1 \\
1 & -1
\end{pmatrix} \! + \! \mathcal{O} \! \left(\widetilde{\mu}^{2} \!
\begin{pmatrix}
\mathfrak{c}_{2}(\tau) & \mathfrak{c}_{3}(\tau) \\
\mathfrak{c}_{4}(\tau) & \mathfrak{c}_{2}(\tau)
\end{pmatrix}
\right) \right),
\end{equation}
where $\mathfrak{c}_{1}(\tau)$, $\mathfrak{c}_{2}(\tau)$, $\mathfrak{c}_{3}
(\tau)$, and $\mathfrak{c}_{4}(\tau)$, respectively, are defined by
Equations~\eqref{tempsea1}--\eqref{tempsea4} below.
\end{bbbb}
\emph{Proof}. The proof is presented for the Asymptotics~\eqref{eq3.51}.
Let the conditions stated in the proposition be valid. Then, via
Equations~\eqref{iden7}, \eqref{iden3}, and~\eqref{iden4}, and the
Conditions~\eqref{iden5}, one shows that
\begin{align}
l_{k}(\widetilde{\mu}) \underset{\underset{\widetilde{\mu} \to \infty}{\tau \to
+\infty}}{=}& \, 2 \widetilde{\mu} \! + \! \dfrac{1}{\widetilde{\mu}}(a \! - \! \mathrm{i}/2)
\tau^{-2/3} \! + \! \mathcal{O}(\widetilde{\mu}^{-3} \hat{\lambda}_{1}(\tau)),
\label{iden52} \\
\mathrm{i} (\mathcal{A}(\widetilde{\mu}) \! - \! \mathrm{i} l_{k}(\widetilde{\mu}) \sigma_{3})
\sigma_{3} \underset{\underset{\widetilde{\mu} \to \infty}{\tau \to +\infty}}{=}&
\, 4 \widetilde{\mu} \, \mathrm{I} \! + \!
\begin{pmatrix}
0 & -\frac{4 \sqrt{\smash[b]{-a(\tau)b(\tau)}}}{b(\tau)} \\
-\mathrm{i} 2d(\tau) & 0
\end{pmatrix} \! + \! \dfrac{1}{\widetilde{\mu}} \mathfrak{d}_{0,0}^{\lozenge}
(\tau) \mathrm{I} \nonumber \\
+& \, \dfrac{1}{\widetilde{\mu}^{2}} \!
\begin{pmatrix}
0 & \frac{(\varepsilon b)}{b(\tau)} \\
-b(\tau) & 0
\end{pmatrix} \! + \! \mathcal{O} \! \left(\widetilde{\mu}^{-3}
\hat{\lambda}_{1}(\tau) \!
\begin{pmatrix}
c_{1,k} & 0 \\
0 & c_{2,k}
\end{pmatrix} \right), \label{iden53} \\
\dfrac{1}{\sqrt{\smash[b]{2 \mathrm{i} l_{k}(\widetilde{\mu})(\mathcal{A}_{11}
(\widetilde{\mu}) \! - \! \mathrm{i} l_{k}(\widetilde{\mu}))}}}
\underset{\underset{\widetilde{\mu} \to \infty}{\tau \to +\infty}}{=}& \,
\dfrac{1}{4 \widetilde{\mu}} \! \left(1 \! - \! \dfrac{1}{\widetilde{\mu}^{2}}
\dfrac{\mathfrak{d}_{1,0}^{\lozenge}(\tau)}{8} \! + \! \mathcal{O}
(\widetilde{\mu}^{-4} \hat{\lambda}_{2}(\tau)) \right), \label{iden54}
\end{align}
where
\begin{gather*}
\mathfrak{d}_{m,j}^{\lozenge}(\tau) \! := \! \dfrac{(\varepsilon b)^{1/3}}{2}
(-2 \! + \! \hat{r}_{0}(\tau)) \! + \! (-1)^{j}(2m \! + \! 1)(a \! - \! \mathrm{i}/2)
\tau^{-2/3}, \quad m,j \! \in \! \lbrace 0,1 \rbrace, \\
\hat{\lambda}_{1}(\tau) \! := \! -3 \alpha_{k}^{4} \! + \! \hat{h}_{0}
(\tau) \! - \! \dfrac{1}{4} \left(a \! - \! \mathrm{i}/2 \right)^{2} \tau^{-4/3}, \\
\hat{\lambda}_{2}(\tau) \! := \! c_{3,k} \hat{\lambda}_{1}(\tau) \! + \!
c_{4,k}(\mathfrak{d}_{1,0}^{\lozenge}(\tau))^{2} \! + \! c_{5,k}
\tau^{-2/3} \mathfrak{d}_{0,0}^{\lozenge}(\tau),
\end{gather*}
and $c_{m,k}$, $m \! = \! 1,2,\dotsc,5$, are constants; thus, via the
Conditions~\eqref{iden5}, Equation~\eqref{eq3.18}, and the
Expansions~\eqref{iden52}--\eqref{iden54}, one arrives at the
Asymptotics~\eqref{eq3.51}, where
\begin{equation} \label{tempsea1}
\mathfrak{c}_{1}(\tau) \! := \! \mathfrak{d}_{0,1}^{\lozenge}(\tau)/8.
\end{equation}
Proceeding analogously, one arrives at the Asymptotics~\eqref{eq3.52}, where
\begin{gather}
\mathfrak{c}_{2}(\tau) \! := \! -\dfrac{(-2 \! + \! \hat{r}_{0}(\tau))^{2}}{32
(\varepsilon b)^{1/3}}, \label{tempsea2} \\
\mathfrak{c}_{3}(\tau) \! := \! \dfrac{-3 \alpha_{k}^{4} \! + \! \hat{h}_{0}(\tau)}{4
\alpha_{k}^{6}} \! - \! \dfrac{3(-2 \! + \! \hat{r}_{0}(\tau))^{2}}{32(\varepsilon
b)^{1/3}} \! + \! \dfrac{2(1 \! + \! \hat{u}_{0}(\tau))}{(\varepsilon b)^{1/3}},
\label{tempsea3} \\
\mathfrak{c}_{4}(\tau) \! := \! \dfrac{3 \alpha_{k}^{4} \! - \! \hat{h}_{0}(\tau)}{4
\alpha_{k}^{6}} \! + \! \dfrac{3(-2 \! + \! \hat{r}_{0}(\tau))^{2}}{32(\varepsilon
b)^{1/3}} \! + \! \dfrac{2 \mathfrak{d}_{0,1}^{\lozenge}(\tau)}{(\varepsilon
b)^{2/3}(1 \! + \! \hat{u}_{0}(\tau))}, \label{tempsea4}
\end{gather}
with $\mathfrak{d}_{0,1}^{\lozenge}(\tau)$ defined above. \hfill $\qed$
\begin{bbbb} \label{prop3.1.6}
Let $T(\widetilde{\mu})$ be given in Equation~\eqref{eq3.18}, with $\mathcal{A}
(\widetilde{\mu})$ defined by Equation~\eqref{eq3.4} and $l_{k}^{2}(\widetilde{\mu})$,
$k \! = \! \pm 1$, given in Equation~\eqref{eq3.19}. Set $\widetilde{\mu}_{0,k} \!
= \! \alpha_{k} \! + \! \tau^{-1/3} \widetilde{\Lambda}$, where $\widetilde{\Lambda}
\! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{\delta_{k}})$, $0 \! < \! \delta \! < \!
\delta_{k} \! < \! 1/9$. Then,
\begin{align} \label{iden55}
T(\widetilde{\mu}_{0,k}) \underset{\tau \to +\infty}{=}& \, \dfrac{(b(\tau))^{
-\frac{1}{2} \ad (\sigma_{3})}}{(2 \sqrt{3}(\varpi \! + \! \sqrt{3}))^{1/2}} \left(
\begin{pmatrix}
\varpi \! + \! \sqrt{3} & (2 \varepsilon b)^{1/2} \varpi \\
-\frac{\sqrt{2} \varpi}{(\varepsilon b)^{1/2}} & \varpi \! + \! \sqrt{3}
\end{pmatrix} \! + \!
\begin{pmatrix}
\frac{\varpi}{3 \alpha_{k}} & -\frac{(2 \varepsilon b)^{1/2}
(2 \varpi + \sqrt{3}) \varpi}{3(\varpi + \sqrt{3}) \alpha_{k}} \\
\frac{\sqrt{2}(2 \varpi + \sqrt{3}) \varpi}{3(\varepsilon b)^{1/2}
(\varpi + \sqrt{3}) \alpha_{k}} & \frac{\varpi}{3 \alpha_{k}}
\end{pmatrix} \! \tau^{-1/3} \widetilde{\Lambda} \right.
\nonumber \\
+&\left. \,
\begin{pmatrix}
\mathbb{T}_{11,k}(\varpi;\tau) & \mathbb{T}_{12,k}(\varpi;\tau) \\
\mathbb{T}_{21,k}(\varpi;\tau) & \mathbb{T}_{22,k}(\varpi;\tau)
\end{pmatrix} \! \dfrac{1}{\widetilde{\Lambda}} \! + \!
\mathcal{O} \! \left(
\begin{pmatrix}
\mathfrak{c}_{1,k} & \mathfrak{c}_{2,k} \\
\mathfrak{c}_{3,k} & \mathfrak{c}_{1,k}
\end{pmatrix} \! (\tau^{-1/3} \widetilde{\Lambda})^{2} \right) \right),
\end{align}
where
\begin{align}
\mathbb{T}_{11,k}(\varpi;\tau) \! =& \, \mathbb{T}_{22,k}(\varpi;\tau) \!
:= \! \dfrac{\varpi}{4} \! \left(\dfrac{\alpha_{k} \tilde{r}_{0}(\tau)}{2} \! - \!
\frac{\tau^{-1/3} \hat{\mathfrak{g}}^{\ast}_{k}(\tau)}{3 \alpha_{k}} \right),
\label{unikay} \\
\mathbb{T}_{12,k}(\varpi;\tau) \! :=& \left(\frac{\varepsilon b}{2} \right)^{1/2}
\! \left(\varpi \alpha_{k}v_{0}(\tau) \! - \! \dfrac{\alpha_{k} \tilde{r}_{0}(\tau)}{4
(\varpi \! + \! \sqrt{3})} \! - \! \dfrac{(1 \! + \! 2 \sqrt{3} \varpi) \tau^{-1/3}
\hat{\mathfrak{g}}^{\ast}_{k}(\tau)}{6(\varpi \! + \! \sqrt{3}) \alpha_{k}} \right),
\label{unikby} \\
\mathbb{T}_{21,k}(\varpi;\tau) \! :=& \, \dfrac{\varpi}{(2 \varepsilon b)^{1/2}}
\! \left(\dfrac{(\varepsilon b)^{1/3}(\tilde{r}_{0}(\tau) \! + \! 2v_{0}(\tau)) \! + \!
2(a \! - \! \mathrm{i}/2) \mathrm{e}^{\mathrm{i} \pi k/3} \tau^{-1/3}}{2^{3/2}(\varepsilon b)^{1/6}
\mathrm{e}^{-\mathrm{i} \pi k/3}(1 \! + \! v_{0}(\tau) \tau^{-1/3})} \! + \! \dfrac{\alpha_{k}
\tilde{r}_{0}(\tau) \! + \! \frac{2(1+2 \sqrt{3} \varpi) \tau^{-1/3}
\hat{\mathfrak{g}}^{\ast}_{k}(\tau)}{3 \alpha_{k}}}{4(\varpi \! + \! \sqrt{3})
\varpi} \right), \label{unikcy}
\end{align}
with $\hat{\mathfrak{g}}^{\ast}_{k}(\tau) \! := \! \tau^{2/3} \hat{\mathfrak{f}}_{1,k}
(\tau)$, where $\hat{\mathfrak{f}}_{1,k}(\tau)$ is given in Equation~\eqref{mfkeff}
(for $j \! = \! 1)$, $(\widetilde{\Lambda}^{2})^{1/2} \! := \! \varpi
\widetilde{\Lambda}$, $\varpi \! = \! \pm 1$, and $\mathfrak{c}_{m,k}$,
$m \! = \! 1,2,3$, are constants.
\end{bbbb}
\emph{Proof}. Set $T(\widetilde{\mu}) \! = \! (T(\widetilde{\mu}))_{i,j=1,2}$.
{}From the formula for $T(\widetilde{\mu})$ given in Equation~\eqref{eq3.18},
with $\mathcal{A}(\widetilde{\mu})$ defined by Equation~\eqref{eq3.4} and
$l_{k}^{2}(\widetilde{\mu})$, $k \! = \! \pm 1$, given in Equation~\eqref{eq3.19},
one shows that
\begin{equation} \label{iden56}
\begin{gathered}
T_{11}(\widetilde{\mu}) \! = \! T_{22}(\widetilde{\mu}) \! = \! \dfrac{\mathrm{i}
(\mathcal{A}_{11}(\widetilde{\mu}) \! - \! \mathrm{i} l_{k}(\widetilde{\mu}))}{\sqrt{
\smash[b]{2 \mathrm{i} l_{k}(\widetilde{\mu})(\mathcal{A}_{11}(\widetilde{\mu})
\! - \! \mathrm{i} l_{k}(\widetilde{\mu}))}}}, \, \, \quad \, \, T_{12}(\widetilde{\mu})
\! = \! -\dfrac{\mathrm{i} \mathcal{A}_{12}(\widetilde{\mu})}{\sqrt{\smash[b]{2
\mathrm{i} l_{k}(\widetilde{\mu})(\mathcal{A}_{11}(\widetilde{\mu}) \! - \! \mathrm{i}
l_{k}(\widetilde{\mu}))}}}, \\
T_{21}(\widetilde{\mu}) \! = \! \dfrac{\mathrm{i} \mathcal{A}_{21}
(\widetilde{\mu})}{\sqrt{\smash[b]{2 \mathrm{i} l_{k}(\widetilde{\mu})
(\mathcal{A}_{11}(\widetilde{\mu}) \! - \! \mathrm{i} l_{k}(\widetilde{\mu}))}}}.
\end{gathered}
\end{equation}
{}From Equations~\eqref{eq3.4}, \eqref{iden7}, \eqref{iden3}, and~\eqref{iden4},
the Conditions~\eqref{iden5}, and Equation~\eqref{mfkeff} for $\hat{\mathfrak{f}}_{1,k}
(\tau)$ (with associated asymptotics~\eqref{asympforf4}), one shows, upon taking
$\widetilde{\mu}_{0,k}$ as stated in the proposition, that
\begin{align}
\dfrac{1}{\sqrt{\smash[b]{2 \mathrm{i} l_{k}(\widetilde{\mu}_{0,k})(\mathcal{A}_{11}
(\widetilde{\mu}_{0,k}) \! - \! \mathrm{i} l_{k}(\widetilde{\mu}_{0,k}))}}}
\underset{\tau \to +\infty}{=}& \, \dfrac{(\varpi \tau^{-1/3}
\widetilde{\Lambda})^{-1}}{4(2 \sqrt{3}(\varpi \! + \! \sqrt{3}))^{1/2}}
\! \left(1 \! + \! \dfrac{(5 \varpi \! + \! 7 \sqrt{3})}{6(\varpi \! + \!
\sqrt{3}) \alpha_{k}} \tau^{-1/3} \widetilde{\Lambda} \right. \nonumber \\
-&\left. \, \left(\dfrac{\alpha_{k} \tilde{r}_{0}(\tau) \! + \! 2(1 \! + \! 2 \sqrt{3}
\varpi)(3 \alpha_{k})^{-1} \hat{\mathfrak{g}}^{\ast}_{k}(\tau) \tau^{-1/3}}{8
\varpi (\varpi \! + \! \sqrt{3})} \right) \! \dfrac{1}{\widetilde{\Lambda}} \right.
\nonumber \\
+&\left. \, \mathcal{O}((\tau^{-1/3} \widetilde{\Lambda})^{2})
\vphantom{M^{M^{M^{M^{M^{M}}}}}} \right), \label{iden59}
\end{align}
\begin{align}
\mathrm{i} \mathcal{A}_{11}(\widetilde{\mu}_{0,k}) \! + \! l_{k}(\widetilde{\mu}_{0,k})
\underset{\tau \to +\infty}{=}& \, 4 \varpi (\varpi \! + \! \sqrt{3}) \tau^{-1/3}
\widetilde{\Lambda} \! \left(1 \! - \! \dfrac{\sqrt{3}(7 \! + \! \sqrt{3} \varpi)}{
6(\varpi \! + \! \sqrt{3}) \alpha_{k}} \tau^{-1/3} \widetilde{\Lambda} \! + \!
\mathcal{O}((\tau^{-1/3} \widetilde{\Lambda})^{2}) \right. \nonumber \\
+&\left. \, \left(\dfrac{\alpha_{k} \tilde{r}_{0}(\tau) \! + \! 2 \varpi (\sqrt{3}
\alpha_{k})^{-1} \hat{\mathfrak{g}}^{\ast}_{k}(\tau) \tau^{-1/3}}{4 \varpi
(\varpi \! + \! \sqrt{3})} \right) \! \dfrac{1}{\widetilde{\Lambda}}
\vphantom{M^{M^{M^{M^{M^{M}}}}}} \right), \label{iden60} \\
-\mathrm{i} \mathcal{A}_{12}(\widetilde{\mu}_{0,k}) \underset{\tau \to +\infty}{=}&
\, (b(\tau))^{-1} \! \left(-2(\varepsilon b) \alpha_{k}^{-3} \tau^{-1/3}
\widetilde{\Lambda} \! + \! 3(\varepsilon b) \alpha_{k}^{-4}(\tau^{-1/3}
\widetilde{\Lambda})^{2} \! + \! \mathcal{O}((\tau^{-1/3} \widetilde{\Lambda})^{3})
\right. \nonumber \\
-&\left. \, 2(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}v_{0}(\tau) \tau^{-1/3}
\vphantom{M^{M^{M^{M^{M}}}}} \right), \label{iden61} \\
\mathrm{i} \mathcal{A}_{21}(\widetilde{\mu}_{0,k}) \underset{\tau \to +\infty}{=}&
\, b(\tau) \! \left(2 \alpha_{k}^{-3} \tau^{-1/3} \widetilde{\Lambda} \! - \! 3
\alpha_{k}^{-4}(\tau^{-1/3} \widetilde{\Lambda})^{2} \! + \! \mathcal{O}
((\tau^{-1/3} \widetilde{\Lambda})^{3}) \right. \nonumber \\
+&\left. \, \dfrac{\mathrm{e}^{\mathrm{i} \pi k/3} \tau^{-1/3} \left((\varepsilon b)^{1/3}
(\tilde{r}_{0}(\tau) \! + \! 2v_{0}(\tau)) \! + \! 2(a \! - \! \mathrm{i}/2) \mathrm{e}^{\mathrm{i} \pi k/3}
\tau^{-1/3} \right)}{(\varepsilon b)^{2/3}(1 \! + \! v_{0}(\tau) \tau^{-1/3})}
\right), \label{iden62}
\end{align}
where $\hat{\mathfrak{g}}^{\ast}_{k}(\tau)$ and $\varpi$ are defined in the
proposition. Substituting expansions~\eqref{iden59}---\eqref{iden62} into
Equations~\eqref{iden56} (with $\widetilde{\mu} \! = \! \widetilde{\mu}_{0,k})$,
one arrives at the asymptotics for $T(\widetilde{\mu}_{0,k})$ stated in the
proposition. \hfill $\qed$
\subsection{The Model Problem: Asymptotics Near the Turning Points} \label{sec3.2}
The matrix WKB formula (cf. Equation~\eqref{eq3.16}) doesn't provide an
approximation for solutions of Equation~\eqref{eq3.3} in shrinking (as $\tau \! \to
\! +\infty$ with $\varepsilon b \! > \! 0$) neighbourhoods of the turning points,
where a more refined approximation must be constructed. There are two simple
turning points approaching $\pm \mathrm{i} \sqrt{2} \alpha_{k}$, $k \! = \! \pm 1$: the
approximate solution of Equation~\eqref{eq3.3} in the neighbourhoods of these
turning points is expressed in terms of Airy functions. There are, additionally, two
pairs of double-turning points, one pair coalescing at $-\alpha_{k}$, and another
pair coalescing at $\alpha_{k}$: in neighbourhoods of $\pm \alpha_{k}$, the approximate
solution of Equation~\eqref{eq3.3} is expressed in terms of parabolic-cylinder functions
(see, for example, \cite{F,a5,a18,a2,W}). In order to obtain asymptotics for $u(\tau)$
and the associated, auxiliary functions $f_{\pm}(\tau)$, $\mathcal{H}(\tau)$, $\sigma
(\tau)$, and $\hat{\varphi}(\tau)$, it is sufficient to study a subset of the complete set
of the monodromy data, which can be calculated via the approximation of the general
solution of Equation~\eqref{eq3.3} in a neighbourhood of the double-turning point
$\alpha_{k}$, because the remaining monodromy data can be calculated via
Equations~\eqref{monoeqns}, which define the monodromy manifold.\footnote{More
precisely, Equations~\eqref{monok2} (resp., Equations~\eqref{monok3}) for
$k \! = \! +1$ (resp., $k \! = \! -1$).} For the asymptotic Conditions~\eqref{iden5}
on the functions $\hat{h}_{0}(\tau)$, $\tilde{r}_{0}(\tau)$, and $v_{0}(\tau)$, this
parametrix (approximation) is given in Lemma~\ref{nprcl} below.
\begin{cccc} \label{nprcl}
Set
\begin{equation} \label{prpr1}
\nu (k) \! + \! 1 \! := \! -\dfrac{p_{k}(\tau)q_{k}(\tau)}{2 \mu_{k}(\tau)}, \quad
k \! = \! \pm 1,
\end{equation}
where $\mu_{k}(\tau)$, $p_{k}(\tau)$, and $q_{k}(\tau)$ are defined
by Equations~\eqref{prcy54}, \eqref{prcy57}, and~\eqref{prcy58},
respectively,\footnote{See, also, the corresponding Definitions~\eqref{prcy10},
\eqref{prcy15}--\eqref{prcy20}, \eqref{prcy22}, \eqref{prcy33}--\eqref{prcy35},
\eqref{prcy40}, \eqref{prcy45}, \eqref{prcy46}, and~\eqref{prcy53}.} and
let $\widetilde{\mu} \! = \! \widetilde{\mu}_{0,k} \! = \! \alpha_{k} \!
+ \! \tau^{-1/3} \widetilde{\Lambda}$, where $\widetilde{\Lambda}
\! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{\delta_{k}})$,
$0 \! < \! \delta \! < \! \delta_{k} \! < \! 1/9$. Concomitant with
Equations~\eqref{iden1}--\eqref{iden6}, the Definitions~\eqref{iden2}--\eqref{iden4},
and the Conditions~\eqref{iden5}, impose the following restrictions:
\begin{equation} \label{pc4}
\begin{gathered}
0 \underset{\tau \to +\infty}{<} \Re (\nu (k) \! + \! 1) \underset{\tau \to
+\infty}{<} 1, \, \, \qquad \, \, \Im (\nu (k) \! + \! 1) \underset{\tau \to +\infty}{
\leqslant} \mathcal{O}(1), \\
0 \underset{\tau \to +\infty}{<} \delta_{k} \underset{\tau \to +\infty}{<}
\dfrac{1}{6(3 \! + \! \Re (\nu (k) \! + \! 1))}, \quad k \! = \! \pm 1.
\end{gathered}
\end{equation}
Then, there exists a fundamental solution of Equation~\eqref{eq3.3},
$\widetilde{\Psi}(\widetilde{\mu}) \! = \! \widetilde{\Psi}_{k}(\widetilde{\mu},
\tau)$, $k \! = \! \pm 1$, with asymptotics
\begin{align} \label{prpr2}
\widetilde{\Psi}_{k}(\widetilde{\mu},\tau) \underset{\tau \to +\infty}{=}& \,
(b(\tau))^{-\frac{1}{2} \sigma_{3}} \mathcal{G}_{0,k} \mathfrak{B}_{k}^{
\frac{1}{2} \sigma_{3}}
\begin{pmatrix}
1 & 0 \\
(\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)} \! - \! 1) \mathfrak{A}_{k} & 1
\end{pmatrix} \! \left(\mathrm{I} \! + \! \gimel_{{\scriptscriptstyle A,k}}(\tau)
\widetilde{\Lambda} \! + \! \gimel_{\scriptscriptstyle{B,k}}(\tau)
\widetilde{\Lambda}^{2} \right) \nonumber \\
\times& \left(\mathrm{I} \! + \! \mathcal{O} \! \left(\tilde{\mathfrak{C}}_{k}
(\tau) \lvert \nu (k) \! + \! 1 \rvert^{2} \lvert p_{k}(\tau) \rvert^{-2}
\tau^{-(\frac{1}{3}-2(3+ \Re (\nu (k)+1)) \delta_{k})} \right) \right) \!
\Phi_{M,k}(\widetilde{\Lambda}),
\end{align}
where
\begin{gather}
\gimel_{{\scriptscriptstyle A,k}}(\tau) \! := \!
\begin{pmatrix}
\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{
\chi_{k}(\tau)} & \ell_{0,k}^{+} \\
(\frac{4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)})^{2}
\ell_{0,k}^{+} \! + \! \ell_{1,k}^{+} \! + \! \ell_{2,k}^{+} & -\frac{\mathrm{i} 4
\sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)}
\end{pmatrix}, \label{prpr3} \\
\gimel_{{\scriptscriptstyle B,k}}(\tau) \! := \! \ell_{0,k}^{+}
(\ell_{1,k}^{+} \! + \! \ell_{2,k}^{+}) \!
\begin{pmatrix}
1 & 0 \\
-\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)} & 0
\end{pmatrix}, \label{prpr4}
\end{gather}
with $\mathcal{G}_{0,k}$, $\mathcal{Z}_{k}$, $\mathfrak{A}_{k}$,
$\mathfrak{B}_{k}$, $\ell_{0,k}^{+}$, $\ell_{1,k}^{+}$, $\chi_{k}(\tau)$,
and $\ell_{2,k}^{+}$ defined by Equations~\eqref{prcy9}, \eqref{prcy10},
\eqref{prcy15}, \eqref{prcy16}, \eqref{prcy40}, \eqref{prcy45}, \eqref{prcy46},
and~\eqref{prcy53}, respectively,\footnote{See, also, the corresponding
Definition~\eqref{prcy5}.} $\mathrm{M}_{2}(\mathbb{C}) \! \ni \!
\tilde{\mathfrak{C}}_{k}(\tau) \! =_{\tau \to +\infty} \! \mathcal{O}(1)$,
and $\Phi_{M,k}(\widetilde{\Lambda})$ is a fundamental solution of
\begin{equation} \label{prcy1}
\dfrac{\partial \Phi_{M,k}(\widetilde{\Lambda})}{\partial \widetilde{\Lambda}}
\! = \! \left(\mu_{k}(\tau) \widetilde{\Lambda} \sigma_{3} \! + \! p_{k}
(\tau) \sigma_{+} \! + \! q_{k}(\tau) \sigma_{-} \right) \! \Phi_{M,k}
(\widetilde{\Lambda}):
\end{equation}
$\Phi_{M,k}(\widetilde{\Lambda})$ has the explicit representation
\begin{equation} \label{prcy2}
\Phi_{M,k}(\widetilde{\Lambda}) \! = \!
\begin{pmatrix}
D_{-\nu (k)-1}(\mathrm{i} (2 \mu_{k}(\tau))^{1/2} \widetilde{\Lambda})
& D_{\nu (k)}((2 \mu_{k}(\tau))^{1/2} \widetilde{\Lambda}) \\
\mathbb{D}_{k}^{\ast}(\tau,\widetilde{\Lambda}) D_{-\nu (k)-1}
(\mathrm{i} (2 \mu_{k}(\tau))^{1/2} \widetilde{\Lambda}) &
\mathbb{D}_{k}^{\ast}(\tau,\widetilde{\Lambda})D_{\nu (k)}
((2 \mu_{k}(\tau))^{1/2} \widetilde{\Lambda})
\end{pmatrix},
\end{equation}
where $\mathbb{D}_{k}^{\ast}(\tau,\widetilde{\Lambda}) \! := \! \frac{1}{p_{k}
(\tau)} \! \left(\frac{\partial}{\partial \widetilde{\Lambda}} \! - \! \mu_{k}(\tau)
\widetilde{\Lambda} \right)$, and $D_{\pmb{\ast}}(\boldsymbol{\cdot})$ is the
parabolic-cylinder function {\rm \cite{a24}}.
\end{cccc}
\emph{Proof}. The derivation of the parametrix~\eqref{prpr2} for a
fundamental solution of Equation~\eqref{eq3.3} consists of applying
the sequence of invertible linear transformations $\mathfrak{F}_{j}$,
$j \! = \! 1,2,\dotsc,11$; for $k \! = \! \pm 1$,
\begin{align*}
\text{\pmb{(i)}} \quad \mathfrak{F}_{1} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \widetilde{\Psi}(\widetilde{\mu}) \! \mapsto \!
\widetilde{\Psi}_{k}(\widetilde{\Lambda}) \! := \! \widetilde{\Psi}
(\alpha_{k} \! + \! \tau^{-1/3} \widetilde{\Lambda}), \\
\text{\pmb{(ii)}} \quad \mathfrak{F}_{2} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \widetilde{\Psi}_{k}(\widetilde{\Lambda}) \! \mapsto
\! \tilde{\Phi}_{k}(\widetilde{\Lambda}) \! := \! (b(\tau))^{\frac{1}{2}
\sigma_{3}} \widetilde{\Psi}_{k}(\widetilde{\Lambda}), \\
\text{\pmb{(iii)}} \quad \mathfrak{F}_{3} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \tilde{\Phi}_{k}(\widetilde{\Lambda}) \! \mapsto
\! \Phi_{k}^{\sharp}(\widetilde{\Lambda}) \! := \! \mathcal{G}_{0,k}^{-1}
\tilde{\Phi}_{k}(\widetilde{\Lambda}), \\
\text{\pmb{(iv)}} \quad \mathfrak{F}_{4} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \Phi_{k}^{\sharp}(\widetilde{\Lambda}) \! \mapsto
\! \hat{\Phi}_{k}(\widetilde{\Lambda}) \! := \! \mathcal{G}_{1,k}^{-1}
\Phi_{k}^{\sharp}(\widetilde{\Lambda}), \\
\text{\pmb{(v)}} \quad \mathfrak{F}_{5} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \hat{\Phi}_{k}(\widetilde{\Lambda}) \! \mapsto \!
\hat{\Phi}_{0,k}(\widetilde{\Lambda}) \! := \! \tau^{-\frac{1}{6} \sigma_{3}}
\hat{\Phi}_{k}(\widetilde{\Lambda}), \\
\text{\pmb{(vi)}} \quad \mathfrak{F}_{6} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \hat{\Phi}_{0,k}(\widetilde{\Lambda}) \! \mapsto \!
\Phi_{0,k}(\widetilde{\Lambda}) \! := \! (\mathrm{I} \! + \! \mathrm{i} \omega_{0,k}
\sigma_{-}) \hat{\Phi}_{0,k}(\widetilde{\Lambda}), \\
\text{\pmb{(vii)}} \quad \mathfrak{F}_{7} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \Phi_{0,k}(\widetilde{\Lambda}) \! \mapsto \!
\Phi_{0,k}^{\flat}(\widetilde{\Lambda}) \! := \! (\mathrm{I} \! - \! \ell_{0,k}
\widetilde{\Lambda} \sigma_{+}) \Phi_{0,k}(\widetilde{\Lambda}), \\
\text{\pmb{(viii)}} \quad \mathfrak{F}_{8} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \Phi_{0,k}^{\flat}(\widetilde{\Lambda}) \! \mapsto \!
\Phi_{0,k}^{\sharp}(\widetilde{\Lambda}) \! := \! (\mathrm{I} \! - \! \ell_{1,k}
\widetilde{\Lambda} \sigma_{-}) \Phi_{0,k}^{\flat}(\widetilde{\Lambda}), \\
\text{\pmb{(ix)}} \quad \mathfrak{F}_{9} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \Phi_{0,k}^{\sharp}(\widetilde{\Lambda}) \! \mapsto
\! \Phi_{0,k}^{\natural}(\widetilde{\Lambda}) \! := \! \mathcal{G}_{2,k}^{-1}
\Phi_{0,k}^{\sharp}(\widetilde{\Lambda}), \\
\text{\pmb{(x)}} \quad \mathfrak{F}_{10} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \Phi_{0,k}^{\natural}(\widetilde{\Lambda}) \! \mapsto
\! \Phi_{k}^{\ast}(\widetilde{\Lambda}) \! := \! (\mathrm{I} \! - \! \ell_{2,k}
\widetilde{\Lambda} \sigma_{-}) \Phi_{0,k}^{\natural}(\widetilde{\Lambda}), \\
\text{\pmb{(xi)}} \quad \mathfrak{F}_{11} \colon& \operatorname{SL}_{2}
(\mathbb{C}) \! \ni \! \Phi_{k}^{\ast}(\widetilde{\Lambda}) \! \mapsto \!
\Phi_{M,k}(\widetilde{\Lambda}) \! := \! \hat{\chi}_{k}^{-1}(\widetilde{\Lambda})
\Phi_{k}^{\ast}(\widetilde{\Lambda}) \! \in \! \mathrm{M}_{2}(\mathbb{C}),
\end{align*}
where the $\mathrm{M}_{2}(\mathbb{C})$-valued, $\tau$-dependent
functions $\mathcal{G}_{0,k}$, $\mathcal{G}_{1,k}$, $\mathrm{I} \! + \!
\mathrm{i} \omega_{0,k} \sigma_{-}$, $\mathcal{G}_{2,k}$, and $\hat{\chi}_{k}
(\widetilde{\Lambda})$, and the $\tau$-dependent parameters
$\ell_{0,k}$, $\ell_{1,k}$, and $\ell_{2,k}$ are described in steps \pmb{(iii)},
\pmb{(iv)}, \pmb{(vi)}, \pmb{(ix)}, \pmb{(xi)}, \pmb{(vii)}, \pmb{(viii)},
and \pmb{(x)}, respectively, below, and $\mathrm{M}_{2}(\mathbb{C}) \!
\ni \! \Phi_{M,k}(\widetilde{\Lambda})$ is given in Equation~\eqref{prcy2}.
\pmb{(i)} The gist of this step is to simplify the System~\eqref{eq3.3} in a proper
neighbourhood of the (coalescing) double-turning point $\alpha_{k}$, $k \!
\in \! \lbrace \pm 1 \rbrace$. Let $\widetilde{\Psi}(\widetilde{\mu})$ solve
Equation~\eqref{eq3.3}; then, using Equations~\eqref{iden3oldr}, \eqref{iden4oldu},
\eqref{iden7}, \eqref{iden3}, and~\eqref{iden4}, the Conditions~\eqref{iden5},
and applying the transformation $\mathfrak{F}_{1}$, one shows that, for
$k \! = \! \pm 1$,
\begin{equation} \label{prcy3}
\dfrac{\partial \widetilde{\Psi}_{k}(\widetilde{\Lambda})}{\partial
\widetilde{\Lambda}} \underset{\tau \to +\infty}{=} (b(\tau))^{-\frac{1}{2}
\ad (\sigma_{3})} \! \left(\hat{\mathcal{P}}_{0,k}(\tau) \! + \!
\hat{\mathcal{P}}_{1,k}(\tau) \widetilde{\Lambda} \! + \! \hat{\mathcal{P}}_{2,k}
(\tau) \widetilde{\Lambda}^{2} \! + \! \mathcal{O}(\hat{\mathbb{E}}_{k}(\tau)
\widetilde{\Lambda}^{3}) \right) \! \widetilde{\Psi}_{k}(\widetilde{\Lambda}),
\end{equation}
where
\begin{align}
\hat{\mathcal{P}}_{0,k}(\tau) \! :=&
\begin{pmatrix}
\hat{\mathcal{A}}_{0} & \hat{\mathcal{B}}_{0} \\
\hat{\mathcal{C}}_{0} & -\hat{\mathcal{A}}_{0}
\end{pmatrix} \nonumber \\
=&
\begin{pmatrix}
-\mathrm{i} \alpha_{k} \tilde{r}_{0}(\tau) & -\mathrm{i} 2(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}
v_{0}(\tau) \\
-\frac{(\mathrm{i} (\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} \pi k/3}(\tilde{r}_{0}(\tau)+2v_{0}
(\tau))+ \mathrm{i} 2 (a - \mathrm{i}/2) \mathrm{e}^{\mathrm{i} 2 \pi k/3} \tau^{-1/3})}{(\varepsilon b)^{2/3}
(1+v_{0}(\tau) \tau^{-1/3})} & \mathrm{i} \alpha_{k} \tilde{r}_{0}(\tau)
\end{pmatrix}, \label{prcy4} \\
\hat{\mathcal{P}}_{1,k}(\tau) \! :=&
\begin{pmatrix}
\hat{\mathcal{A}}_{1} & \hat{\mathcal{B}}_{1} \\
\hat{\mathcal{C}}_{1} & -\hat{\mathcal{A}}_{1}
\end{pmatrix} \! = \!
\begin{pmatrix}
\mathrm{i} (-4 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3}) & \mathrm{i} 4 \sqrt{2}(\varepsilon b)^{1/2} \\
\mathrm{i} 4 \sqrt{2}(\varepsilon b)^{-1/2} & -\mathrm{i} (-4 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3})
\end{pmatrix}, \label{prcy5} \\
\hat{\mathcal{P}}_{2,k}(\tau) \! :=&
\begin{pmatrix}
\hat{\mathcal{A}}_{2} & \hat{\mathcal{B}}_{2} \\
\hat{\mathcal{C}}_{2} & -\hat{\mathcal{A}}_{2}
\end{pmatrix} \nonumber \\
=&
\begin{pmatrix}
\frac{\mathrm{i} \sqrt{2} \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{(\varepsilon b)^{1/6}}
(-2 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3}) \tau^{-1/3} & -\mathrm{i} 12
(\varepsilon b)^{1/3} \mathrm{e}^{-\mathrm{i} \pi k/3} \tau^{-1/3} \\
-\mathrm{i} 12 (\varepsilon b)^{-2/3} \mathrm{e}^{-\mathrm{i} \pi k/3} \tau^{-1/3} &
-\frac{\mathrm{i} \sqrt{2} \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{(\varepsilon b)^{1/6}}
(-2 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3}) \tau^{-1/3}
\end{pmatrix}, \label{prcy6}
\end{align}
and
\begin{equation} \label{prcy7}
\hat{\mathbb{E}}_{k}(\tau) \! = \!
\begin{pmatrix}
\mathrm{i} \alpha_{k}^{-2}(-2 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3}) \tau^{-2/3}
& -\mathrm{i} 32 \alpha_{k} \tau^{-2/3} \\
-\mathrm{i} 4 \alpha_{k}^{-5} \tau^{-2/3} & -\mathrm{i} \alpha_{k}^{-2}(-2 \! + \!
\tilde{r}_{0}(\tau) \tau^{-1/3}) \tau^{-2/3}
\end{pmatrix}.
\end{equation}
Observe that $\tr (\hat{\mathcal{P}}_{0,k}(\tau)) \! = \! \tr (\hat{\mathcal{P}}_{1,k}
(\tau)) \! = \! \tr (\hat{\mathcal{P}}_{2,k}(\tau)) \! = \! \tr (\hat{\mathbb{E}}_{k}
(\tau)) \! = \! 0$.
\pmb{(ii)} This intermediate step removes the scalar-valued function $b(\tau)$
{}from Equation~\eqref{prcy3}. Let $\widetilde{\Psi}_{k}(\widetilde{\Lambda})$
solve Equation~\eqref{prcy3}; then, applying the transformation
$\mathfrak{F}_{2}$, one shows that, for $k \! = \! \pm 1$,
\begin{equation} \label{prcy8}
\dfrac{\partial \tilde{\Phi}_{k}(\widetilde{\Lambda})}{\partial \widetilde{\Lambda}}
\underset{\tau \to +\infty}{=} \left(\hat{\mathcal{P}}_{0,k}(\tau) \! + \!
\hat{\mathcal{P}}_{1,k}(\tau) \widetilde{\Lambda} \! + \! \hat{\mathcal{P}}_{2,k}
(\tau) \widetilde{\Lambda}^{2} \! + \! \mathcal{O}(\hat{\mathbb{E}}_{k}(\tau)
\widetilde{\Lambda}^{3}) \right) \! \tilde{\Phi}_{k}(\widetilde{\Lambda}).
\end{equation}
\pmb{(iii)} The essence of this step is to transform the coefficient matrix
$\hat{\mathcal{P}}_{1,k}(\tau)$ (cf. Definition~\eqref{prcy5}) into diagonal
form. Let $\tilde{\Phi}_{k}(\widetilde{\Lambda})$ be a solution of
Equation~\eqref{prcy8}; then, applying the transformation $\mathfrak{F}_{3}$,
where
\begin{equation} \label{prcy9}
\mathcal{G}_{0,k} \! := \! \left(\dfrac{\hat{\mathcal{C}}_{1}}{2 \lambda^{\ast}_{1}
(k)} \right)^{1/2}
\begin{pmatrix}
\frac{\hat{\mathcal{A}}_{1}+ \lambda^{\ast}_{1}(k)}{\hat{\mathcal{C}}_{1}} &
\frac{\hat{\mathcal{A}}_{1}- \lambda^{\ast}_{1}(k)}{\hat{\mathcal{C}}_{1}} \\
1 & 1
\end{pmatrix}, \quad k \! = \! \pm 1,
\end{equation}
with $\hat{\mathcal{A}}_{1}$ and $\hat{\mathcal{C}}_{1}$ given in
Equation~\eqref{prcy5}, and
\begin{equation} \label{prcy10}
\lambda^{\ast}_{1}(k) \! := \! \mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! = \! \mathrm{i} 4 \sqrt{3}
\left(1 \! - \! \dfrac{1}{6} \tilde{r}_{0}(\tau) \tau^{-1/3} \! + \! \dfrac{1}{48}
(\tilde{r}_{0}(\tau) \tau^{-1/3})^{2} \right)^{1/2},
\end{equation}
one shows that
\begin{equation} \label{prcy11}
\dfrac{\partial \Phi_{k}^{\sharp}(\widetilde{\Lambda})}{\partial \widetilde{\Lambda}}
\underset{\tau \to +\infty}{=} \left(\mathcal{P}_{0,k}^{\vartriangle}(\tau)
\! + \! \mathcal{P}_{1,k}^{\vartriangle}(\tau) \widetilde{\Lambda} \!
+ \! \mathcal{P}_{2,k}^{\vartriangle}(\tau) \widetilde{\Lambda}^{2}
\! + \! \mathcal{O}(\mathcal{G}_{0,k}^{-1} \hat{\mathbb{E}}_{k}(\tau)
\mathcal{G}_{0,k} \widetilde{\Lambda}^{3}) \right) \! \Phi_{k}^{\sharp}
(\widetilde{\Lambda}),
\end{equation}
where
\begin{gather}
\mathcal{P}_{0,k}^{\vartriangle}(\tau) \! := \! \mathcal{G}_{0,k}^{-1}
\hat{\mathcal{P}}_{0,k}(\tau) \mathcal{G}_{0,k} \! = \!
\mathfrak{A}_{k} \sigma_{3} \! + \! \mathfrak{B}_{k} \sigma_{+} \! + \!
\mathfrak{C}_{k} \sigma_{-}, \label{prcy12} \\
\mathcal{P}_{1,k}^{\vartriangle}(\tau) \! := \! \mathcal{G}_{0,k}^{-1}
\hat{\mathcal{P}}_{1,k}(\tau) \mathcal{G}_{0,k} \! = \! \mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \sigma_{3}, \label{prcy13} \\
\mathcal{P}_{2,k}^{\vartriangle}(\tau) \! := \! \mathcal{G}_{0,k}^{-1}
\hat{\mathcal{P}}_{2,k}(\tau) \mathcal{G}_{0,k} \! = \!
\mathfrak{A}_{0,k}^{\sharp} \sigma_{3} \! + \! \mathfrak{B}_{0,k}^{\sharp}
\sigma_{+} \! + \! \mathfrak{C}_{0,k}^{\sharp} \sigma_{-}, \label{prcy14}
\end{gather}
with
\begin{align}
\mathfrak{A}_{k} \! =& \, \dfrac{1}{(6 \varepsilon b)^{1/2} \mathcal{Z}_{k}} \!
\left(-\dfrac{\mathrm{i} \alpha_{k} (\varepsilon b)^{1/2}}{2 \sqrt{2}} \tilde{r}_{0}(\tau)
(-4 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3}) \! - \! \mathrm{i} 2 (\varepsilon b)^{2/3}
\mathrm{e}^{-\mathrm{i} 2 \pi k/3}v_{0}(\tau) \right. \nonumber \\
-&\left. \mathrm{i} (\varepsilon b)^{1/3} \! \left(\dfrac{(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} \pi k/3}(\tilde{r}_{0}(\tau) \! + \! 2v_{0}(\tau)) \! + \! 2(a \! - \!
\mathrm{i}/2) \mathrm{e}^{\mathrm{i} 2 \pi k/3} \tau^{-1/3}}{1 \! + \! v_{0}(\tau) \tau^{-1/3}}
\right) \right), \label{prcy15} \\
\mathfrak{B}_{k} \! =& \, \dfrac{1}{(6 \varepsilon b)^{1/2} \mathcal{Z}_{k}} \!
\left(-\dfrac{\mathrm{i} \alpha_{k} (\varepsilon b)^{1/2}}{2 \sqrt{2}} \tilde{r}_{0}(\tau)
(-4 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3} \! - \! 4 \sqrt{3} \mathcal{Z}_{k}) \!
- \! \mathrm{i} 2 (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}v_{0}(\tau) \right.
\nonumber \\
+&\left. \mathrm{i} (\varepsilon b)^{1/3} \! \left(\dfrac{(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} \pi k/3}(\tilde{r}_{0}(\tau) \! + \! 2v_{0}(\tau)) \! + \! 2(a \! - \!
\mathrm{i}/2) \mathrm{e}^{\mathrm{i} 2 \pi k/3} \tau^{-1/3}}{1 \! + \! v_{0}(\tau) \tau^{-1/3}}
\right) \right. \nonumber \\
\times&\left. \left(1 \! + \! \frac{1}{16}(-4 \! + \! \tilde{r}_{0}(\tau)
\tau^{-1/3})(-4 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3} \! - \! 4 \sqrt{3}
\mathcal{Z}_{k}) \right) \right), \label{prcy16} \\
\mathfrak{C}_{k} \! =& \, \dfrac{1}{(6 \varepsilon b)^{1/2} \mathcal{Z}_{k}} \!
\left(\dfrac{\mathrm{i} \alpha_{k} (\varepsilon b)^{1/2}}{2 \sqrt{2}} \tilde{r}_{0}(\tau)
(-4 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3} \! + \! 4 \sqrt{3} \mathcal{Z}_{k})
\! + \! \mathrm{i} 2 (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}v_{0}(\tau) \right.
\nonumber \\
-&\left. \mathrm{i} (\varepsilon b)^{1/3} \! \left(\dfrac{(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} \pi k/3}(\tilde{r}_{0}(\tau) \! + \! 2v_{0}(\tau)) \! + \! 2(a \! - \!
\mathrm{i}/2) \mathrm{e}^{\mathrm{i} 2 \pi k/3} \tau^{-1/3}}{1 \! + \! v_{0}(\tau) \tau^{-1/3}}
\right) \right. \nonumber \\
\times&\left. \left(1 \! + \! \dfrac{1}{16}(-4 \! + \! \tilde{r}_{0}(\tau)
\tau^{-1/3})(-4 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3} \! + \! 4 \sqrt{3}
\mathcal{Z}_{k}) \right) \right), \label{prcy17} \\
\mathfrak{A}_{0,k}^{\sharp} \! =& \, -\dfrac{\mathrm{i} (\varepsilon b)^{1/3}
\mathrm{e}^{-\mathrm{i} \pi k/3} \tau^{-1/3}}{2(6 \varepsilon b)^{1/2} \mathcal{Z}_{k}}
\! \left(48 \! + \! (-2 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3})(-4 \! + \!
\tilde{r}_{0}(\tau) \tau^{-1/3}) \right), \label{prcy18} \\
\mathfrak{B}_{0,k}^{\sharp} \! =& \, \dfrac{\mathrm{i} (\varepsilon b)^{1/3}
\mathrm{e}^{-\mathrm{i} \pi k/3} \tau^{-1/3}}{2(6 \varepsilon b)^{1/2} \mathcal{Z}_{k}}
(-4 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3} \! - \! 4 \sqrt{3} \mathcal{Z}_{k})
(-4 \! + \! \tfrac{1}{2} \tilde{r}_{0}(\tau) \tau^{-1/3}), \label{prcy19} \\
\mathfrak{C}_{0,k}^{\sharp} \! =& \, -\dfrac{\mathrm{i} (\varepsilon b)^{1/3}
\mathrm{e}^{-\mathrm{i} \pi k/3} \tau^{-1/3}}{2(6 \varepsilon b)^{1/2} \mathcal{Z}_{k}}
(-4 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3} \! + \! 4 \sqrt{3} \mathcal{Z}_{k})
(-4 \! + \! \tfrac{1}{2} \tilde{r}_{0}(\tau) \tau^{-1/3}). \label{prcy20}
\end{align}
Observe that $\tr (\mathcal{P}_{0,k}^{\vartriangle}(\tau)) \! = \! \tr
(\mathcal{P}_{1,k}^{\vartriangle}(\tau)) \! = \! \tr (\mathcal{P}_{2,k}^{\vartriangle}
(\tau)) \! = \! \tr (\mathcal{G}_{0,k}^{-1} \hat{\mathbb{E}}_{k}(\tau)
\mathcal{G}_{0,k}) \! = \! 0$. For the requisite estimates in step~\pmb{(xi)} below,
the asymptotics of the functions $\mathcal{G}_{0,k}$, $\mathfrak{A}_{k}$,
$\mathfrak{B}_{k}$, $\mathfrak{C}_{k}$, $\mathfrak{A}_{0,k}^{\sharp}$,
$\mathfrak{B}_{0,k}^{\sharp}$, and $\mathfrak{C}_{0,k}^{\sharp}$ are
essential; via the Conditions~\eqref{iden5}, the Asymptotics~\eqref{tr1}
and~\eqref{tr3}, the Definitions~\eqref{prcy5}, \eqref{prcy9},
and~\eqref{prcy10}, and Equations~\eqref{prcy15}--\eqref{prcy20}, a
lengthy, but otherwise straightforward, algebraic calculation shows that
\begin{equation} \label{prcyg1}
\mathcal{G}_{0,k} \underset{\tau \to +\infty}{=} \mathcal{G}_{0,k}^{\infty}
\! + \! \Delta \mathcal{G}_{0,k}, \quad k \! = \! \pm 1,
\end{equation}
where
\begin{equation} \label{prcyg2}
(6 \varepsilon b)^{1/4} \mathcal{G}_{0,k}^{\infty} \! = \!
\begin{pmatrix}
\frac{(\varepsilon b)^{1/2}(\sqrt{3}-1)}{\sqrt{2}} &
-\frac{(\varepsilon b)^{1/2}(\sqrt{3}+1)}{\sqrt{2}} \\
1 & 1
\end{pmatrix},
\end{equation}
and
\begin{equation} \label{prcyg3}
\Delta \mathcal{G}_{0,k} \! := \! \mathcal{G}_{0,k} \! - \! \mathcal{G}_{0,k}^{\infty}
\! = \!
\begin{pmatrix}
(\Delta \mathcal{G}_{0,k})_{11} & (\Delta \mathcal{G}_{0,k})_{12} \\
(\Delta \mathcal{G}_{0,k})_{21} & (\Delta \mathcal{G}_{0,k})_{22}
\end{pmatrix},
\end{equation}
with
{\fontsize{10pt}{11pt}\selectfont
\begin{align}
(6 \varepsilon b)^{1/4}(\Delta \mathcal{G}_{0,k})_{11} \! :=& \,
\dfrac{(\varepsilon b)^{1/2}}{4 \sqrt{2}} \! \left(\dfrac{(\sqrt{3} \! -
\! 1)(2 \sqrt{3} \! + \! 1)}{6} \tilde{r}_{0}(\tau) \tau^{-1/3} \! + \!
\dfrac{1}{12 \sqrt{3}} \! \left(1 \! + \! \dfrac{(\sqrt{3} \! - \! 1)
(4 \sqrt{3} \! - \! 1)}{8 \sqrt{3}} \right) \right. \nonumber \\
\times&\left. (\tilde{r}_{0}(\tau) \tau^{-1/3})^{2} \! + \! \mathcal{O}
((\tilde{r}_{0}(\tau) \tau^{-1/3})^{3}) \right), \label{prcyg4} \\
(6 \varepsilon b)^{1/4}(\Delta \mathcal{G}_{0,k})_{12} \! :=& \,
\dfrac{(\varepsilon b)^{1/2}}{4 \sqrt{2}} \! \left(\dfrac{(\sqrt{3} \! +
\! 1)(2 \sqrt{3} \! - \! 1)}{6} \tilde{r}_{0}(\tau) \tau^{-1/3} \! + \!
\dfrac{1}{12 \sqrt{3}} \! \left(-1 \! + \! \dfrac{(\sqrt{3} \! + \! 1)
(4 \sqrt{3} \! + \! 1)}{8 \sqrt{3}} \right) \right. \nonumber \\
\times&\left. (\tilde{r}_{0}(\tau) \tau^{-1/3})^{2} \! + \! \mathcal{O}
((\tilde{r}_{0}(\tau) \tau^{-1/3})^{3}) \right), \label{prcyg5} \\
(6 \varepsilon b)^{1/4}(\Delta \mathcal{G}_{0,k})_{21} \! =& \, (6 \varepsilon
b)^{1/4}(\Delta \mathcal{G}_{0,k})_{22} \! := \! \dfrac{1}{24} \tilde{r}_{0}
(\tau) \tau^{-1/3} \! - \! \dfrac{1}{2(24)^{2}}(\tilde{r}_{0}(\tau) \tau^{-1/3})^{2}
\! + \! \mathcal{O}((\tilde{r}_{0}(\tau) \tau^{-1/3})^{3}), \label{prcyg6}
\end{align}}
and
{\fontsize{9pt}{10pt}\selectfont
\begin{align}
\mathfrak{A}_{k} \underset{\tau \to +\infty}{=}& \, \dfrac{\mathrm{i} (a \! - \!
\mathrm{i}/2) \tau^{-1/3}}{\sqrt{3} \alpha_{k}} \! + \! \dfrac{\mathrm{i} \tau^{-1/3}}{4
\sqrt{3}} \! \left(\alpha_{k}(4v_{0}(\tau)(\tilde{r}_{0}(\tau) \! + \! 2v_{0}
(\tau)) \! - \! (\tilde{r}_{0}(\tau))^{2}) \! - \! \dfrac{(a \! - \! \mathrm{i}/2)(12
v_{0}(\tau) \! - \! \tilde{r}_{0}(\tau)) \tau^{-1/3}}{3 \alpha_{k}} \right)
\nonumber \\
+& \, \mathcal{O} \! \left((6 \varepsilon b)^{-1/2} \! \left(-\mathrm{i} (\varepsilon
b)^{1/3}((\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} \pi k/3}(\tilde{r}_{0}(\tau) \! + \!
2v_{0}(\tau)) \! + \! 2(a \! - \! \mathrm{i}/2) \mathrm{e}^{\mathrm{i} 2 \pi k/3} \tau^{-1/3})
(v_{0}(\tau) \tau^{-1/3})^{2} \right. \right. \nonumber \\
+&\left. \left. \dfrac{\mathrm{i} (\varepsilon b)^{1/3}}{12} \! \left(-\dfrac{
(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} \pi k/3}}{4}(\tilde{r}_{0}(\tau))^{2}
\tau^{-1/3} \! + \! ((\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} \pi k/3}(\tilde{r}_{0}
(\tau) \! + \! 2v_{0}(\tau)) \! + \! 2(a \! - \! \mathrm{i}/2) \mathrm{e}^{\mathrm{i} 2 \pi k/3}
\tau^{-1/3}) \right. \right. \right. \nonumber \\
\times&\left. \left. \left. v_{0}(\tau) \tau^{-1/3} \right)
\tilde{r}_{0}(\tau) \tau^{-1/3} \right) \right), \label{prcyak1} \\
\mathfrak{B}_{k} \underset{\tau \to +\infty}{=}& \, \mathrm{i} (\sqrt{3} \! + \! 1) \!
\left(\dfrac{\alpha_{k}}{2}(4v_{0}(\tau) \! + \! (\sqrt{3} \! + \! 1) \tilde{r}_{0}
(\tau)) \! - \! \dfrac{(\sqrt{3} \! + \! 1)(a \! - \! \mathrm{i}/2) \tau^{-1/3}}{2
\sqrt{3} \alpha_{k}} \right) \! + \! \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1)^{2}
\tau^{-1/3}}{4 \sqrt{3}} \nonumber \\
\times& \left(-\dfrac{\alpha_{k}}{2}((\tilde{r}_{0}(\tau))^{2} \! + \!
2(\sqrt{3} \! + \! 1)v_{0}(\tau) \tilde{r}_{0}(\tau) \! + \! 8v_{0}^{2}(\tau))
\! + \! \dfrac{(a \! - \! \mathrm{i}/2)(12v_{0}(\tau) \! + \! (2 \sqrt{3} \! - \! 1)
\tilde{r}_{0}(\tau)) \tau^{-1/3}}{6 \alpha_{k}} \right) \nonumber \\
+& \, \mathcal{O} \! \left((6 \varepsilon b)^{-1/2} \! \left(-\dfrac{\mathrm{i} (\sqrt{3}
\! + \! 1)^{2}(a \! - \! \mathrm{i}/2)(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{12}
\tilde{r}_{0}(\tau)(\tau^{-1/3})^{3} \left(v_{0}(\tau) \! + \! \tilde{r}_{0}
(\tau)/2 \sqrt{3} \right) \right. \right. \nonumber \\
-&\left. \left. \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1)(\varepsilon b)^{2/3}
\mathrm{e}^{\mathrm{i} \pi k/3}}{48 \sqrt{3}} \tilde{r}_{0}(\tau)(\tau^{-1/3})^{2}
((\tilde{r}_{0}(\tau))^{2} \! + \! (\sqrt{3} \! + \! 1)(\tilde{r}_{0}(\tau) \! + \!
2 \sqrt{3} \, v_{0}(\tau))(\tilde{r}_{0}(\tau) \! + \! 2v_{0}(\tau))) \right. \right.
\nonumber \\
+&\left. \left. \dfrac{\mathrm{i} \alpha_{k}(\varepsilon b)^{1/2}}{24 \sqrt{6}}
(\tilde{r}_{0}(\tau))^{3}(\tau^{-1/3})^{2} \! + \! \left(\dfrac{\mathrm{i} (\varepsilon
b)^{1/3}(\sqrt{3} \! + \! 1)^{2}}{2}(v_{0}(\tau) \tau^{-1/3})^{2} \! + \!
\dfrac{\mathrm{i} (\varepsilon b)^{1/3}(3 \sqrt{3} \! + \! 4)}{48 \sqrt{3}}(\tilde{r}_{0}
(\tau) \tau^{-1/3})^{2} \right. \right. \right. \nonumber \\
+&\left. \left. \left. \dfrac{\mathrm{i} (\varepsilon b)^{1/3}(2 \! + \! \sqrt{3})}{2
\sqrt{3}}v_{0}(\tau) \tilde{r}_{0}(\tau)(\tau^{-1/3})^{2} \right) \! \left(
(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} \pi k/3}(\tilde{r}_{0}(\tau) \! + \! 2v_{0}(\tau))
\! + \! 2(a \! - \! \mathrm{i}/2) \mathrm{e}^{\mathrm{i} 2 \pi k/3} \tau^{-1/3} \right) \right)
\right), \label{prcybk1} \\
\mathfrak{C}_{k} \underset{\tau \to +\infty}{=}& \, -\mathrm{i} (\sqrt{3} \! - \! 1)
\! \left(\dfrac{\alpha_{k}}{2}(4v_{0}(\tau) \! - \! (\sqrt{3} \! - \! 1) \tilde{r}_{0}
(\tau)) \! - \! \dfrac{(\sqrt{3} \! - \! 1)(a \! - \! \mathrm{i}/2) \tau^{-1/3}}{2 \sqrt{3}
\alpha_{k}} \right) \! + \! \dfrac{\mathrm{i} (\sqrt{3} \! - \! 1)^{2} \tau^{-1/3}}{4
\sqrt{3}} \nonumber \\
\times& \left(\dfrac{\alpha_{k}}{2}((\tilde{r}_{0}(\tau))^{2} \! - \! 2
(\sqrt{3} \! - \! 1)v_{0}(\tau) \tilde{r}_{0}(\tau) \! + \! 8v_{0}^{2}(\tau))
\! - \! \dfrac{(a \! - \! \mathrm{i}/2)(12v_{0}(\tau) \! - \! (2 \sqrt{3} \! + \! 1)
\tilde{r}_{0}(\tau)) \tau^{-1/3}}{6 \alpha_{k}} \right) \nonumber \\
+& \, \mathcal{O} \! \left((6 \varepsilon b)^{-1/2} \! \left(\dfrac{\mathrm{i} (\sqrt{3}
\! - \! 1)^{2}(a \! - \! \mathrm{i}/2)(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{12}
\tilde{r}_{0}(\tau)(\tau^{-1/3})^{3} \left(v_{0}(\tau) \! - \! \tilde{r}_{0}
(\tau)/2 \sqrt{3} \right) \right. \right. \nonumber \\
+&\left. \left. \dfrac{\mathrm{i} (\sqrt{3} \! - \! 1)(\varepsilon b)^{2/3}
\mathrm{e}^{\mathrm{i} \pi k/3}}{48 \sqrt{3}} \tilde{r}_{0}(\tau)(\tau^{-1/3})^{2}
((\tilde{r}_{0}(\tau))^{2} \! + \! (\sqrt{3} \! - \! 1)(2 \sqrt{3} \, v_{0}(\tau) \!
- \! \tilde{r}_{0}(\tau))(\tilde{r}_{0}(\tau) \! + \! 2v_{0}(\tau))) \right. \right.
\nonumber \\
+&\left. \left. \dfrac{\mathrm{i} \alpha_{k}(\varepsilon b)^{1/2}}{24 \sqrt{6}}
(\tilde{r}_{0}(\tau))^{3}(\tau^{-1/3})^{2} \! - \! \left(\dfrac{\mathrm{i} (\varepsilon
b)^{1/3}(\sqrt{3} \! - \! 1)^{2}}{2}(v_{0}(\tau) \tau^{-1/3})^{2} \! + \!
\dfrac{\mathrm{i} (\varepsilon b)^{1/3}(3 \sqrt{3} \! - \! 4)}{48 \sqrt{3}}
(\tilde{r}_{0}(\tau) \tau^{-1/3})^{2} \right. \right. \right. \nonumber \\
-&\left. \left. \left. \dfrac{\mathrm{i} (\varepsilon b)^{1/3}(2 \! - \! \sqrt{3})}{2
\sqrt{3}}v_{0}(\tau) \tilde{r}_{0}(\tau)(\tau^{-1/3})^{2} \right) \! \left(
(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} \pi k/3}(\tilde{r}_{0}(\tau) \! + \! 2v_{0}
(\tau)) \! + \! 2(a \! - \! \mathrm{i}/2) \mathrm{e}^{\mathrm{i} 2 \pi k/3} \tau^{-1/3} \right)
\right) \right), \label{prcyck1}
\end{align}}
\begin{gather}
\mathfrak{A}_{0,k}^{\sharp} \underset{\tau \to +\infty}{=}
-\dfrac{\mathrm{i} 14 \tau^{-1/3}}{\sqrt{3} \alpha_{k}} \! - \! \dfrac{\mathrm{i}
\tilde{r}_{0}(\tau)(\tau^{-1/3})^{2}}{4 \sqrt{3} \alpha_{k}} \! \left(-
\dfrac{4}{3} \! + \! \dfrac{1}{2} \tilde{r}_{0}(\tau) \tau^{-1/3} \! + \!
\mathcal{O}((\tilde{r}_{0}(\tau) \tau^{-1/3})^{2}) \right), \label{prcya0k1} \\
\mathfrak{B}_{0,k}^{\sharp} \underset{\tau \to +\infty}{=} \dfrac{\mathrm{i}
4(\sqrt{3} \! + \! 1) \tau^{-1/3}}{\sqrt{3} \alpha_{k}} \! + \! \dfrac{\mathrm{i}
\tilde{r}_{0}(\tau)(\tau^{-1/3})^{2}}{4 \sqrt{3} \alpha_{k}} \! \left(-
\dfrac{2(3 \sqrt{3} \! + \! 7)}{3} \! + \! \mathcal{O}((\tilde{r}_{0}(\tau)
\tau^{-1/3})^{2}) \right), \label{prcyb0k1} \\
\mathfrak{C}_{0,k}^{\sharp} \underset{\tau \to +\infty}{=} \dfrac{\mathrm{i}
4(\sqrt{3} \! - \! 1) \tau^{-1/3}}{\sqrt{3} \alpha_{k}} \! + \! \dfrac{
\mathrm{i} \tilde{r}_{0}(\tau)(\tau^{-1/3})^{2}}{4 \sqrt{3} \alpha_{k}} \! \left(
-\dfrac{2(3 \sqrt{3} \! - \! 7)}{3} \! + \! \mathcal{O}((\tilde{r}_{0}(\tau)
\tau^{-1/3})^{2}) \right). \label{prcyc0k1}
\end{gather}
\pmb{(iv)} The idea behind the transformation for Equation~\eqref{prcy11}
that is subsumed in this step is to put the coefficient matrix
$\mathcal{P}_{0,k}^{\vartriangle}(\tau)$ (cf. Definition~\eqref{prcy12})
into a particular Jordan canonical form, namely, to find a unimodular,
$\tau$-dependent function $\mathcal{G}_{1,k}$ such that
\begin{equation} \label{prcy21}
\mathcal{G}_{1,k}^{-1} \mathcal{P}_{0,k}^{\vartriangle}(\tau) \mathcal{G}_{1,k}
\! = \! \mathrm{i} \omega_{0,k} \sigma_{3} \! + \! \tau^{1/3} \sigma_{+}, \quad
k \! = \! \pm 1,
\end{equation}
where (cf. Equations~\eqref{expforeych}, \eqref{expforkapp},
and~\eqref{prcy15}--\eqref{prcy17})
\begin{align} \label{prcy22}
\omega_{0,k}^{2} \! := \! \det (\mathcal{P}_{0,k}^{\vartriangle}(\tau))
\! =& \, \varkappa_{0}^{2}(\tau) \! + \! \dfrac{4(a \! - \! \mathrm{i}/2)v_{0}(\tau)
\tau^{-1/3}}{1 \! + \! v_{0}(\tau) \tau^{-1/3}} \! = \! 4 \! \left((a \! - \! \mathrm{i}/2)
\! + \! \alpha_{k}^{-2} \tau^{2/3} \hat{h}_{0}(\tau) \right) \nonumber \\
=& \, -\alpha_{k}^{2} \! \left(\dfrac{8v_{0}^{2}(\tau) \! + \! 4v_{0}(\tau)
\tilde{r}_{0}(\tau) \! - \! (\tilde{r}_{0}(\tau))^{2} \! - \! v_{0}(\tau)(\tilde{r}_{0}
(\tau))^{2} \tau^{-1/3}}{1 \! + \! v_{0}(\tau) \tau^{-1/3}} \right) \nonumber \\
+& \, \dfrac{4(a \! - \! \mathrm{i}/2)v_{0}(\tau) \tau^{-1/3}}{1 \! + \! v_{0}(\tau)
\tau^{-1/3}};
\end{align}
the following lower-triangular solution for $\mathcal{G}_{1,k}$ is chosen:
\begin{equation} \label{prcy23}
\mathcal{G}_{1,k} \! = \! \mathfrak{B}_{k}^{\frac{1}{2} \sigma_{3}}
\tau^{-\frac{1}{6} \sigma_{3}} \! \left(\mathrm{I} \! + \! (\mathrm{i} \omega_{0,k}
\! - \! \mathfrak{A}_{k}) \tau^{-1/3} \sigma_{-} \right), \quad k \! = \! \pm 1.
\end{equation}
Let $\Phi_{k}^{\sharp}(\widetilde{\Lambda})$ solve Equation~\eqref{prcy11};
then, applying the transformation $\mathfrak{F}_{4}$, one shows that
\begin{equation} \label{prcy24}
\dfrac{\partial \hat{\Phi}_{k}(\widetilde{\Lambda})}{\partial \widetilde{\Lambda}}
\underset{\tau \to +\infty}{=} \left(\mathcal{P}_{0,k}^{\triangledown}(\tau)
\! + \! \mathcal{P}_{1,k}^{\triangledown}(\tau) \widetilde{\Lambda} \! + \!
\mathcal{P}_{2,k}^{\triangledown}(\tau) \widetilde{\Lambda}^{2} \! + \!
\mathcal{O}(\mathcal{G}_{1,k}^{-1} \mathcal{G}_{0,k}^{-1} \hat{\mathbb{E}}_{k}
(\tau) \mathcal{G}_{0,k} \mathcal{G}_{1,k} \widetilde{\Lambda}^{3}) \right) \!
\hat{\Phi}_{k}(\widetilde{\Lambda}),
\end{equation}
where
\begin{align}
\mathcal{P}_{0,k}^{\triangledown}(\tau) \! :=& \, \mathcal{G}_{1,k}^{-1}
\mathcal{P}_{0,k}^{\vartriangle}(\tau) \mathcal{G}_{1,k} \! = \! \mathrm{i}
\omega_{0,k} \sigma_{3} \! + \! \tau^{1/3} \sigma_{+}, \label{prcy25} \\
\mathcal{P}_{1,k}^{\triangledown}(\tau) \! :=& \, \mathcal{G}_{1,k}^{-1}
\mathcal{P}_{1,k}^{\vartriangle}(\tau) \mathcal{G}_{1,k} \! = \! \mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \sigma_{3} \! - \! \mathrm{i} 8 \sqrt{3}(\mathrm{i} \omega_{0,k} \! - \!
\mathfrak{A}_{k}) \mathcal{Z}_{k} \tau^{-1/3} \sigma_{-}, \label{prcy26} \\
\mathcal{P}_{2,k}^{\triangledown}(\tau) \! :=& \, \mathcal{G}_{1,k}^{-1}
\mathcal{P}_{2,k}^{\vartriangle}(\tau) \mathcal{G}_{1,k} \nonumber \\
=&
\begin{pmatrix}
\mathfrak{A}_{0,k}^{\sharp} \! + \! \frac{(\mathrm{i} \omega_{0,k}-\mathfrak{A}_{k})
\mathfrak{B}_{0,k}^{\sharp}}{\mathfrak{B}_{k}} & \frac{\mathfrak{B}_{0,k}^{
\sharp} \tau^{1/3}}{\mathfrak{B}_{k}} \\
\frac{2(\mathrm{i} \omega_{0,k}-\mathfrak{A}_{k})(\mathfrak{A}_{k}
\mathfrak{B}_{0,k}^{\sharp}-\mathfrak{B}_{k} \mathfrak{A}_{0,k}^{\sharp})
+(\mathfrak{B}_{k} \mathfrak{C}_{0,k}^{\sharp}-\mathfrak{C}_{k}
\mathfrak{B}_{0,k}^{\sharp}) \mathfrak{B}_{k}}{\mathfrak{B}_{k}
\tau^{1/3}} & -(\mathfrak{A}_{0,k}^{\sharp} \! + \! \frac{(\mathrm{i} \omega_{0,k}
-\mathfrak{A}_{k}) \mathfrak{B}_{0,k}^{\sharp}}{\mathfrak{B}_{k}})
\end{pmatrix}. \label{prcy27}
\end{align}
Note that, at this stage, the matrix $\mathcal{P}_{1,k}^{\triangledown}(\tau)$
is not diagonal; rather, it now contains an additional, lower off-diagonal
contribution. For the requisite estimates in step~\pmb{(xi)} below, the
asymptotics of the function $\omega_{0,k}^{2}$ is essential; via the
Conditions~\eqref{iden5}, the Asymptotics~\eqref{tr1} and~\eqref{tr3},
and the Definition~\eqref{prcy22}, one shows that, for $k \! = \! \pm 1$,
\begin{align} \label{prcyomg1}
\omega_{0,k}^{2} \underset{\tau \to +\infty}{=}& \, -\alpha_{k}^{2}
(8v_{0}^{2}(\tau) \! + \! 4v_{0}(\tau) \tilde{r}_{0}(\tau) \! - \! (\tilde{r}_{0}
(\tau))^{2}) \! + \! 4(a \! - \! \mathrm{i}/2)v_{0}(\tau) \tau^{-1/3} \nonumber \\
+& \, (4 \alpha_{k}^{2}v_{0}(\tau)(\tilde{r}_{0}(\tau) \! + \! 2v_{0}(\tau))
\! - \! 4(a \! - \! \mathrm{i}/2)v_{0}(\tau) \tau^{-1/3})v_{0}(\tau) \tau^{-1/3}
\nonumber \\
+& \, \mathcal{O} \! \left((-4 \alpha_{k}^{2}v_{0}(\tau)(\tilde{r}_{0}(\tau) \!
+ \! 2v_{0}(\tau)) \! + \! 4(a \! - \! \mathrm{i}/2)v_{0}(\tau) \tau^{-1/3})(v_{0}
(\tau) \tau^{-1/3})^{2} \right).
\end{align}
\pmb{(v)} This step entails a straightforward $\tau$-dependent scaling.
Let $\hat{\Phi}_{k}(\widetilde{\Lambda})$ solve Equation~\eqref{prcy24};
then, applying the transformation $\mathfrak{F}_{5}$, one shows that,
for $k \! = \! \pm 1$,
\begin{align} \label{prcy28}
\dfrac{\partial \hat{\Phi}_{0,k}(\widetilde{\Lambda})}{\partial \widetilde{
\Lambda}} \underset{\tau \to +\infty}{=}& \, \left(\tilde{\mathcal{P}}_{0,k}^{
\blacktriangle}(\tau) \! + \! \tilde{\mathcal{P}}_{1,k}^{\blacktriangle}(\tau)
\widetilde{\Lambda} \! + \! \tilde{\mathcal{P}}_{2,k}^{\blacktriangle}(\tau)
\widetilde{\Lambda}^{2} \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left(\tau^{-\frac{1}{6} \sigma_{3}} \mathcal{G}_{1,
k}^{-1} \mathcal{G}_{0,k}^{-1} \hat{\mathbb{E}}_{k}(\tau) \mathcal{G}_{0,k}
\mathcal{G}_{1,k} \tau^{\frac{1}{6} \sigma_{3}} \widetilde{\Lambda}^{3}
\right) \right) \! \hat{\Phi}_{0,k}(\widetilde{\Lambda}),
\end{align}
where
\begin{align}
\tilde{\mathcal{P}}_{0,k}^{\blacktriangle}(\tau) \! :=& \, \tau^{-\frac{1}{6}
\sigma_{3}} \mathcal{P}_{0,k}^{\triangledown}(\tau) \tau^{\frac{1}{6}
\sigma_{3}} \! = \! \mathrm{i} \omega_{0,k} \sigma_{3} \! + \! \sigma_{+},
\label{prcy29} \\
\tilde{\mathcal{P}}_{1,k}^{\blacktriangle}(\tau) \! :=& \, \tau^{-\frac{1}{6}
\sigma_{3}} \mathcal{P}_{1,k}^{\triangledown}(\tau) \tau^{\frac{1}{6}
\sigma_{3}} \! = \! \mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \sigma_{3} \! - \! \mathrm{i}
8 \sqrt{3}(\mathrm{i} \omega_{0,k} \! - \! \mathfrak{A}_{k}) \mathcal{Z}_{k}
\sigma_{-}, \label{prcy30} \\
\tilde{\mathcal{P}}_{2,k}^{\blacktriangle}(\tau) \! :=& \, \tau^{-\frac{1}{6}
\sigma_{3}} \mathcal{P}_{2,k}^{\triangledown}(\tau) \tau^{\frac{1}{6}
\sigma_{3}} \nonumber \\
=&
\begin{pmatrix}
\mathfrak{A}_{0,k}^{\sharp} \! + \! \frac{(\mathrm{i} \omega_{0,k}-
\mathfrak{A}_{k}) \mathfrak{B}_{0,k}^{\sharp}}{\mathfrak{B}_{k}} &
\frac{\mathfrak{B}_{0,k}^{\sharp}}{\mathfrak{B}_{k}} \\
\frac{2(\mathrm{i} \omega_{0,k}-\mathfrak{A}_{k})(\mathfrak{A}_{k}
\mathfrak{B}_{0,k}^{\sharp}-\mathfrak{B}_{k} \mathfrak{A}_{0,k}^{\sharp})
+(\mathfrak{B}_{k} \mathfrak{C}_{0,k}^{\sharp}-\mathfrak{C}_{k}
\mathfrak{B}_{0,k}^{\sharp}) \mathfrak{B}_{k}}{\mathfrak{B}_{k}} &
-(\mathfrak{A}_{0,k}^{\sharp} \! + \! \frac{(\mathrm{i} \omega_{0,k}-
\mathfrak{A}_{k}) \mathfrak{B}_{0,k}^{\sharp}}{\mathfrak{B}_{k}})
\end{pmatrix}. \label{prcy31}
\end{align}
\pmb{(vi)} The purpose of this step is to transform the coefficient matrix
$\tilde{\mathcal{P}}_{0,k}^{\blacktriangle}(\tau)$ (cf. Equation~\eqref{prcy29})
into off-diagonal form. Let $\hat{\Phi}_{0,k}(\widetilde{\Lambda})$ solve
Equation~\eqref{prcy28}; then, applying the transformation $\mathfrak{F}_{6}$,
one shows that, for $k \! = \! \pm 1$,
\begin{equation} \label{prcy32}
\dfrac{\partial \Phi_{0,k}(\widetilde{\Lambda})}{\partial \widetilde{\Lambda}}
\underset{\tau \to +\infty}{=} \left(
\begin{pmatrix}
0 & 1 \\
-\omega_{0,k}^{2} & 0
\end{pmatrix} \! + \!
\begin{pmatrix}
\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} & 0 \\
\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} & -\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}
\end{pmatrix} \! \widetilde{\Lambda} \! + \!
\begin{pmatrix}
\mathfrak{P}_{0,k}^{\ast} & \mathfrak{Q}_{0,k}^{\ast} \\
\mathfrak{R}_{0,k}^{\ast} & -\mathfrak{P}_{0,k}^{\ast}
\end{pmatrix} \! \widetilde{\Lambda}^{2} \! + \! \mathcal{O}
(\mathbb{E}_{k}^{\ast}(\tau) \widetilde{\Lambda}^{3}) \right) \!
\Phi_{0,k}(\widetilde{\Lambda}),
\end{equation}
where
\begin{gather}
\mathfrak{P}_{0,k}^{\ast} \! := \! \mathfrak{A}_{0,k}^{\sharp} \! - \!
\mathfrak{B}_{0,k}^{\sharp} \mathfrak{A}_{k} \mathfrak{B}_{k}^{-1},
\label{prcy33} \\
\mathfrak{Q}_{0,k}^{\ast} \! := \! \mathfrak{B}_{0,k}^{\sharp}
\mathfrak{B}_{k}^{-1}, \label{prcy34} \\
\mathfrak{R}_{0,k}^{\ast} \! := \! -\mathfrak{B}_{0,k}^{\sharp}
\mathfrak{A}_{k}^{2} \mathfrak{B}_{k}^{-1} \! + \! 2 \mathfrak{A}_{k}
\mathfrak{A}_{0,k}^{\sharp} \! + \! \mathfrak{B}_{k}
\mathfrak{C}_{0,k}^{\sharp}, \label{prcy35}
\end{gather}
and
\begin{gather} \label{prcy36}
\mathbb{E}_{k}^{\ast}(\tau) \! := \! \left(\mathrm{I} \! + \! \mathrm{i} \omega_{0,k}
\sigma_{-} \right) \tau^{-\frac{1}{6} \sigma_{3}} \mathcal{G}_{1,k}^{-1}
\mathcal{G}_{0,k}^{-1} \hat{\mathbb{E}}_{k}(\tau) \mathcal{G}_{0,k}
\mathcal{G}_{1,k} \tau^{\frac{1}{6} \sigma_{3}} \left(\mathrm{I} \! - \!
\mathrm{i} \omega_{0,k} \sigma_{-} \right).
\end{gather}
\pmb{(vii)} This step, in conjunction with steps~\pmb{(viii)} and~\pmb{(x)}
below, is precipitated by the fact that, in order to derive a (canonical)
model problem solvable in terms of parabolic-cylinder functions (see
step~\pmb{(xi)} below), one must eliminate the coefficient matrix of the
$\widetilde{\Lambda}^{2}$ term {}from Equation~\eqref{prcy32}; in particular,
this step focuses on the excision of the $(1 \, 2)$-element. Let $\Phi_{0,k}
(\widetilde{\Lambda})$ solve Equation~\eqref{prcy32}; then, applying the
transformation $\mathfrak{F}_{7}$, with $\tau$-dependent parameter
$\ell_{0,k}$, one shows, via the Conditions~\eqref{iden5}, that, for
$k \! = \! \pm 1$,
\begin{align} \label{prcy37}
\dfrac{\partial \Phi_{0,k}^{\flat}(\widetilde{\Lambda})}{\partial
\widetilde{\Lambda}} \underset{\tau \to +\infty}{=}& \, \left(
\begin{pmatrix}
0 & -\ell_{0,k} \! + \! 1 \\
-\omega_{0,k}^{2} & 0
\end{pmatrix} \! + \!
\begin{pmatrix}
\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k} & 0 \\
\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} & -\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \! - \! \omega_{0,k}^{2} \ell_{0,k}
\end{pmatrix} \! \widetilde{\Lambda} \right. \nonumber \\
+&\left. \,
\begin{pmatrix}
-\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k} \! + \!
\mathfrak{P}_{0,k}^{\ast} & \omega_{0,k}^{2} \ell_{0,k}^{2} \! + \! \mathrm{i}
8 \sqrt{3} \mathcal{Z}_{k} \ell_{0,k} \! + \! \mathfrak{Q}_{0,k}^{\ast} \\
\mathfrak{R}_{0,k}^{\ast} & \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k} \! - \! \mathfrak{P}_{0,k}^{\ast}
\end{pmatrix} \! \widetilde{\Lambda}^{2} \right. \nonumber \\
+&\left. \, \mathcal{O}(\mathbb{E}^{\triangledown}_{k}(\ell_{0,k};\tau)
\widetilde{\Lambda}^{3}) \right) \! \Phi_{0,k}^{\flat}(\widetilde{\Lambda}),
\end{align}
where
\begin{equation} \label{prcy38}
\mathbb{E}^{\triangledown}_{k}(\ell_{0,k};\tau) \! := \!
\mathbb{E}^{\ast}_{k}(\tau) \! + \!
\begin{pmatrix}
-\mathfrak{R}_{0,k}^{\ast} \ell_{0,k} & -\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{2} \! + \! 2 \mathfrak{P}_{0,k}^{\ast} \ell_{0,k} \\
0 & \mathfrak{R}_{0,k}^{\ast} \ell_{0,k}
\end{pmatrix},
\end{equation}
with $\mathbb{E}_{k}^{\ast}(\tau)$ defined by Equation~\eqref{prcy36}. One now
chooses $\ell_{0,k}$ so that the $(1 \, 2)$-element of the coefficient matrix of
the $\widetilde{\Lambda}^{2}$ term in Equation~\eqref{prcy37} is zero, that is,
$\omega_{0,k}^{2} \ell_{0,k}^{2} \! + \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \ell_{0,k}
\! + \! \mathfrak{Q}_{0,k}^{\ast} \! = \! 0$; the roots are given by
\begin{equation} \label{prcy39}
\ell_{0,k}^{\pm} \! = \! \dfrac{-\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \! \pm
\! \sqrt{(\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k})^{2} \! - \! 4 \omega_{0,k}^{2}
\mathfrak{Q}_{0,k}^{\ast}}}{2 \omega_{0,k}^{2}}, \quad k \! = \! \pm 1.
\end{equation}
Noting {}from the Conditions~\eqref{iden5}, the Asymptotics~\eqref{tr1}
and~\eqref{tr3}, Equations~\eqref{prcy16} and \eqref{prcy19}, and the
Definitions~\eqref{prcy10}, \eqref{prcy22}, and~\eqref{prcy34} that
$\mathcal{Z}_{k} \! =_{\tau \to +\infty} \! 1 \! + \! \mathcal{O}
(\tau^{-2/3})$, $\omega_{0,k}^{2} \! =_{\tau \to +\infty} \! \mathcal{O}
(\tau^{-2/3})$, and $\mathfrak{Q}_{0,k}^{\ast} \! =_{\tau \to +\infty} \!
\mathcal{O}(1)$, it follows that, for the class of functions consistent with the
Conditions~\eqref{iden5}, the `$+$-root' in Equation~\eqref{prcy39} is chosen:
\begin{equation} \label{prcy40}
\ell_{0,k} \! := \! \ell_{0,k}^{+} \! = \! \dfrac{-\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\! + \! \sqrt{(\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k})^{2} \! - \! 4 \omega_{0,k}^{2}
\mathfrak{Q}_{0,k}^{\ast}}}{2 \omega_{0,k}^{2}}.
\end{equation}
Via the formula for the $\tau$-dependent parameter $\ell_{0,k} \! := \! \ell_{0,k}^{+}$
given in Equation~\eqref{prcy40}, one re-writes Equation~\eqref{prcy37} as follows:
for $k \! = \! \pm 1$,
\begin{align} \label{prcy41}
\dfrac{\partial \Phi_{0,k}^{\flat}(\widetilde{\Lambda})}{\partial
\widetilde{\Lambda}} \underset{\tau \to +\infty}{=}& \, \left(
\begin{pmatrix}
0 & -\ell_{0,k}^{+} \! + \! 1 \\
-\omega_{0,k}^{2} & 0
\end{pmatrix} \! + \!
\begin{pmatrix}
\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k}^{+} & 0 \\
\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} & -\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \! - \! \omega_{0,k}^{2} \ell_{0,k}^{+}
\end{pmatrix} \! \widetilde{\Lambda} \right. \nonumber \\
+&\left. \,
\begin{pmatrix}
-\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+} \! + \!
\mathfrak{P}_{0,k}^{\ast} & 0 \\
\mathfrak{R}_{0,k}^{\ast} & \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+} \! - \! \mathfrak{P}_{0,k}^{\ast}
\end{pmatrix} \! \widetilde{\Lambda}^{2} \! + \! \mathcal{O}
(\mathbb{E}_{k}^{\triangledown}(\ell_{0,k}^{+};\tau) \widetilde{\Lambda}^{3})
\right) \! \Phi_{0,k}^{\flat}(\widetilde{\Lambda}).
\end{align}
For the requisite estimates in step~\pmb{(xi)} below, the asymptotics of
the $\tau$-dependent parameter $\ell_{0,k}^{+}$ is essential; via the
Conditions~\eqref{iden5}, the Asymptotics~\eqref{tr1} and~\eqref{tr3},
and the Definitions~\eqref{prcy10}, \eqref{prcy22}, \eqref{prcy34},
and~\eqref{prcy40}, one shows that, for $k \! = \! \pm 1$,
\begin{align} \label{prcyellok1}
\ell_{0,k}^{+} \underset{\tau \to +\infty}{=}& \, \dfrac{\mathrm{i}}{8 \sqrt{3}} \!
\left(1 \! + \! \dfrac{\tilde{r}_{0}(\tau) \tau^{-1/3}}{12} \! + \! \mathcal{O}
((\tilde{r}_{0}(\tau) \tau^{-1/3})^{3}) \right) \! \dfrac{\mathfrak{B}_{0,k}^{
\sharp}}{\mathfrak{B}_{k}} \nonumber \\
-& \, \dfrac{\mathrm{i} \omega_{0,k}^{2}}{(8 \sqrt{3})^{3}} \! \left(1 \! + \!
\dfrac{\tilde{r}_{0}(\tau) \tau^{-1/3}}{12} \! + \! \mathcal{O}((\tilde{r}_{0}
(\tau) \tau^{-1/3})^{3}) \right)^{3} \! \left(\dfrac{\mathfrak{B}_{0,k}^{
\sharp}}{\mathfrak{B}_{k}} \right)^{2} \nonumber \\
+& \, \mathcal{O} \! \left(\omega_{0,k}^{4} \! \left(1 \! + \! \dfrac{\tilde{r}_{0}
(\tau) \tau^{-1/3}}{12} \! + \! \mathcal{O}((\tilde{r}_{0}(\tau) \tau^{-1/3})^{3})
\right)^{5} \! \left(\dfrac{\mathfrak{B}_{0,k}^{\sharp}}{\mathfrak{B}_{k}}
\right)^{3} \right),
\end{align}
where the asymptotics of the functions $\mathfrak{B}_{k}$ and
$\mathfrak{B}_{0,k}^{\sharp}$ are given by the Expansions~\eqref{prcybk1}
and~\eqref{prcyb0k1}, respectively.
\pmb{(viii)} This step focuses on the excision of the $(2 \, 1)$-element
{}from the coefficient matrix of the $\widetilde{\Lambda}^{2}$ term in
Equation~\eqref{prcy41}. Let $\Phi_{0,k}^{\flat}(\widetilde{\Lambda})$
solve Equation~\eqref{prcy41}; then, under the action of the transformation
$\mathfrak{F}_{8}$, with $\tau$-dependent parameter $\ell_{1,k}$, one
shows that, for $k \! = \! \pm 1$,
\begin{align} \label{prcy42}
\dfrac{\partial \Phi_{0,k}^{\sharp}(\widetilde{\Lambda})}{\partial
\widetilde{\Lambda}} \underset{\tau \to +\infty}{=}& \, \left(
\begin{pmatrix}
0 & -\ell_{0,k}^{+} \! + \! 1 \\
-\omega_{0,k}^{2} \! - \! \ell_{1,k} & 0
\end{pmatrix} \! + \!
\left((\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k}^{+}
\! + \! \ell_{1,k}(-\ell_{0,k}^{+} \! + \! 1)) \sigma_{3} \right. \right.
\nonumber \\
+&\left. \left. \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \sigma_{-}
\right) \! \widetilde{\Lambda} \! + \! \left((\mathfrak{R}_{0,k}^{\ast} \! - \!
2(\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k}^{+})
\ell_{1,k} \! - \! \ell_{1,k}^{2}(-\ell_{0,k}^{+} \! + \! 1)) \sigma_{-} \right.
\right. \nonumber \\
-&\left. \left. (\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}
\! - \! \mathfrak{P}_{0,k}^{\ast}) \sigma_{3} \right) \! \widetilde{\Lambda}^{2}
\! + \! \mathcal{O}
(\overset{\ast}{\mathbb{E}}_{k}^{\raise-6.75pt\hbox{$\scriptstyle \blacktriangle$}}
(\ell_{0,k}^{+},\ell_{1,k};\tau) \widetilde{\Lambda}^{3}) \right) \! \Phi_{0,k}^{\sharp}
(\widetilde{\Lambda}),
\end{align}
where
\begin{equation} \label{prcy43}
\overset{\ast}{\mathbb{E}}_{k}^{\raise-6.75pt\hbox{$\scriptstyle \blacktriangle$}}
(\ell_{0,k}^{+},\ell_{1,k};\tau) \! := \! \mathbb{E}_{k}^{\triangledown}(\ell_{0,k}^{+};
\tau) \! + \! 2 \ell_{1,k}(-\mathfrak{P}_{0,k}^{\ast} \! + \! \mathrm{i} 8 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}) \sigma_{-}.
\end{equation}
One now chooses $\ell_{1,k}$ so that the $(2 \, 1)$-element of the coefficient matrix
of the $\widetilde{\Lambda}^{2}$ term in Equation~\eqref{prcy42} vanishes, that is,
$(-\ell_{0,k}^{+} \! + \! 1) \ell_{1,k}^{2} \! + \! 2(\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! +
\! \omega_{0,k}^{2} \ell_{0,k}^{+}) \ell_{1,k} \! - \! \mathfrak{R}_{0,k}^{\ast} \! = \!
0$; the roots are given by
\begin{equation} \label{prcy44}
\ell_{1,k}^{\pm} \! = \! \dfrac{-(\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \!
\omega_{0,k}^{2} \ell_{0,k}^{+}) \! \pm \! \sqrt{(\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k}^{+})^{2} \! + \!
\mathfrak{R}_{0,k}^{\ast}(-\ell_{0,k}^{+} \! + \! 1)}}{-\ell_{0,k}^{+} \! + \! 1},
\quad k \! = \! \pm 1.
\end{equation}
Noting {}from the Conditions~\eqref{iden5}, the Asymptotics~\eqref{tr1}
and~\eqref{tr3}, Equations~\eqref{prcy15}--\eqref{prcy20}, and the
Definition~\eqref{prcy35} that $\mathfrak{R}_{0,k}^{\ast} \! =_{\tau \to +\infty}
\! \mathcal{O}(\tau^{-2/3})$, and, recalling ({}from step~\pmb{(vii)} above) the
asymptotics $\mathcal{Z}_{k} \! =_{\tau \to +\infty} \! 1 \! + \! \mathcal{O}
(\tau^{-2/3})$, $\omega_{0,k}^{2} \! =_{\tau \to +\infty} \! \mathcal{O}
(\tau^{-2/3})$, and $\mathfrak{Q}_{0,k}^{\ast} \! =_{\tau \to +\infty} \!
\mathcal{O}(1)$, it follows {}from the Definition~\eqref{prcy40} for $\ell_{0,k}^{+}$
that, for the class of functions consistent with the Conditions~\eqref{iden5},
the `$+$-root' in Equation~\eqref{prcy44} is taken:
\begin{equation} \label{prcy45}
\ell_{1,k} \! := \! \ell_{1,k}^{+} \! = \! \dfrac{-(\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}
\! + \! \omega_{0,k}^{2} \ell_{0,k}^{+}) \! + \! \chi_{k}(\tau)}{-\ell_{0,k}^{+}
\! + \! 1},
\end{equation}
where
\begin{equation} \label{prcy46}
\chi_{k}(\tau) \! := \! \left((\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2}
\ell_{0,k}^{+})^{2} \! + \! \mathfrak{R}_{0,k}^{\ast}(-\ell_{0,k}^{+} \! + \! 1)
\right)^{1/2}.
\end{equation}
Via the formula for the $\tau$-dependent parameter $\ell_{1,k} \! := \!
\ell_{1,k}^{+}$ defined by Equations~\eqref{prcy45} and \eqref{prcy46},
one re-writes Equation~\eqref{prcy42} as follows: for $k \! = \! \pm 1$,
\begin{align} \label{prcy47}
\dfrac{\partial \Phi_{0,k}^{\sharp}(\widetilde{\Lambda})}{\partial
\widetilde{\Lambda}} \underset{\tau \to +\infty}{=}& \, \left(
\begin{pmatrix}
0 & -\ell_{0,k}^{+} \! + \! 1 \\
-\omega_{0,k}^{2} \! - \! \ell_{1,k}^{+} & 0
\end{pmatrix} \! + \! \left(\chi_{k}(\tau) \sigma_{3} \! + \! \mathrm{i} 8 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \sigma_{-} \right) \! \widetilde{\Lambda}
\right. \nonumber \\
+&\left. \, \left(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{+} \right) \! \widetilde{\Lambda}^{2} \sigma_{3}
\! + \! \mathcal{O}
(\overset{\ast}{\mathbb{E}}_{k}^{\raise-6.75pt\hbox{$\scriptstyle \blacktriangle$}}
(\ell_{0,k}^{+},\ell_{1,k}^{+};\tau) \widetilde{\Lambda}^{3}) \right) \!
\Phi_{0,k}^{\sharp}(\widetilde{\Lambda}).
\end{align}
For the requisite estimates in step~\pmb{(xi)} below, the asymptotics of the
function $\chi_{k}(\tau)$ and the $\tau$-dependent parameter $\ell_{1,k}^{+}$
are essential; via the Conditions~\eqref{iden5}, the Asymptotics~\eqref{tr1}
and~\eqref{tr3}, and the Definitions~\eqref{prcy10}, \eqref{prcy22}, \eqref{prcy34},
\eqref{prcy35}, \eqref{prcy40}, \eqref{prcy45}, and~\eqref{prcy46}, one shows
that, for $k \! = \! \pm 1$,
\begin{equation} \label{prcychik1}
\chi_{k}(\tau) \underset{\tau \to +\infty}{=} \mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}
\! + \! \omega_{0,k}^{2} \ell_{0,k}^{+} \! + \! \dfrac{\mathfrak{R}_{0,k}^{\ast}
(-\ell_{0,k}^{+} \! + \! 1)}{2(\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2}
\ell_{0,k}^{+})} \! - \! \dfrac{(\mathfrak{R}_{0,k}^{\ast}(-\ell_{0,k}^{+} \! + \!
1))^{2}}{8(\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k}^{+})^{3}}
\! + \! \mathcal{O} \! \left(\dfrac{(\mathfrak{R}_{0,k}^{\ast}(-\ell_{0,k}^{+} \! + \!
1))^{3}}{(\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k}^{+})^{5}}
\right),
\end{equation}
where
\begin{equation} \label{prcyzeek1}
\mathcal{Z}_{k} \underset{\tau \to +\infty}{=} 1 \! - \! \dfrac{\tilde{r}_{0}
(\tau) \tau^{-1/3}}{12} \! + \! \left(\dfrac{\tilde{r}_{0}(\tau) \tau^{-1/3}}{12}
\right)^{2} \! + \! \mathcal{O} \! \left((\tilde{r}_{0}(\tau) \tau^{-1/3})^{3} \right),
\end{equation}
with the asymptotics for $\omega_{0,k}^{2}$ and $\ell_{0,k}^{+}$ given by
the Expansions~\eqref{prcyomg1} and~\eqref{prcyellok1}, respectively, and
\begin{equation} \label{prcyell1k1}
\ell_{1,k}^{+} \underset{\tau \to +\infty}{=} \dfrac{\mathfrak{R}_{0,k}^{\ast}}{
2(\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k}^{+})} \! -
\! \dfrac{(\mathfrak{R}_{0,k}^{\ast})^{2}(-\ell_{0,k}^{+} \! + \! 1)}{8(\mathrm{i} 4
\sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k}^{+})^{3}} \! + \!
\mathcal{O} \! \left(\dfrac{(\mathfrak{R}_{0,k}^{\ast})^{3}(-\ell_{0,k}^{+} \! +
\! 1)^{2}}{(\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \! + \! \omega_{0,k}^{2} \ell_{0,k}^{
+})^{5}} \right).
\end{equation}
\pmb{(ix)} This step is necessitated by the fact that the coefficient matrix
of the $\widetilde{\Lambda}$ term in Equation~\eqref{prcy47} remains to
be re-diagonalised. Let $\Phi_{0,k}^{\sharp}(\widetilde{\Lambda})$ solve
Equation~\eqref{prcy47}; then, under the action of the transformation
$\mathfrak{F}_{9}$, where
\begin{equation} \label{prcy48}
\mathcal{G}_{2,k} \! := \!
\begin{pmatrix}
1 & 0 \\
\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)} & 1
\end{pmatrix}, \quad k \! = \! \pm 1,
\end{equation}
with $\mathcal{Z}_{k}$, $\mathfrak{A}_{k}$, and $\chi_{k}(\tau)$ defined by
Equations~\eqref{prcy10}, \eqref{prcy15}, and~\eqref{prcy46}, respectively,
one shows that
\begin{align} \label{prcy49}
\dfrac{\partial \Phi_{0,k}^{\natural}(\widetilde{\Lambda})}{\partial
\widetilde{\Lambda}} \underset{\tau \to +\infty}{=}& \, \left(
\begin{pmatrix}
\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)}
(-\ell_{0,k}^{+} \! + \! 1) & -\ell_{0,k}^{+} \! + \! 1 \\
-(\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)})^{2}
(-\ell_{0,k}^{+} \! + \! 1) \! - \! \ell_{1,k}^{+} \! - \! \omega_{0,k}^{2} &
-\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)}
(-\ell_{0,k}^{+} \! + \! 1)
\end{pmatrix} \right. \nonumber \\
+&\left. \, \chi_{k}(\tau) \widetilde{\Lambda} \sigma_{3} \! + \!
(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{+})
\begin{pmatrix}
1 & 0 \\
-\frac{\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)} & -1
\end{pmatrix} \! \widetilde{\Lambda}^{2} \right. \nonumber \\
+&\left. \, \mathcal{O}(\mathcal{G}_{2,k}^{-1}
\overset{\ast}{\mathbb{E}}_{k}^{\raise-6.75pt\hbox{$\scriptstyle \blacktriangle$}}
(\ell_{0,k}^{+},\ell_{1,k}^{+};\tau) \mathcal{G}_{2,k} \widetilde{\Lambda}^{3})
\right) \! \Phi_{0,k}^{\natural}(\widetilde{\Lambda}).
\end{align}
\pmb{(x)} This penultimate step focuses on the annihilation of the
nilpotent coefficient sub-matrix of the $\widetilde{\Lambda}^{2}$ term in
Equation~\eqref{prcy49}. Let $\Phi_{0,k}^{\natural}(\widetilde{\Lambda})$
solve Equation~\eqref{prcy49}; then, under the action of the transformation
$\mathfrak{F}_{10}$, with $\tau$-dependent parameter $\ell_{2,k}$, one shows
that, for $k \! = \! \pm 1$,
\begin{align} \label{prcy50}
\dfrac{\partial \Phi_{k}^{\ast}(\widetilde{\Lambda})}{\partial \widetilde{\Lambda}}
\underset{\tau \to +\infty}{=}& \, \left(
\begin{pmatrix}
\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)}
(-\ell_{0,k}^{+} \! + \! 1) & -\ell_{0,k}^{+} \! + \! 1 \\
-(\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)})^{2}
(-\ell_{0,k}^{+} \! + \! 1) \! - \! \ell_{1,k}^{+} \! - \! \ell_{2,k} \! - \!
\omega_{0,k}^{2} & -\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{
\chi_{k}(\tau)}(-\ell_{0,k}^{+} \! + \! 1)
\end{pmatrix} \right. \nonumber \\
+&\left.
\begin{pmatrix}
\chi_{k}(\tau) \! + \! \ell_{2,k}(-\ell_{0,k}^{+} \! + \! 1) & 0 \\
-\frac{\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)}
\ell_{2,k}(-\ell_{0,k}^{+} \! + \! 1) & -(\chi_{k}(\tau) \! + \! \ell_{2,k}
(-\ell_{0,k}^{+} \! + \! 1))
\end{pmatrix} \! \widetilde{\Lambda} \right. \nonumber \\
+&\left. \left( \left(-\ell_{2,k}^{2}(-\ell_{0,k}^{+} \! + \! 1) \! - \! 2 \ell_{2,k}
\chi_{k}(\tau) \! - \! \frac{\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{
\chi_{k}(\tau)}(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{+}) \right) \! \sigma_{-} \right. \right. \nonumber \\
+&\left. \left. (\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{+}) \sigma_{3} \right) \! \widetilde{\Lambda}^{2}
\! + \! \mathcal{O}
(\overset{\ast}{\mathbb{E}}_{k}^{\raise-6.75pt\hbox{$\scriptstyle \ast$}}
(\ell_{0,k}^{+},\ell_{1,k}^{+},\ell_{2,k};\tau) \widetilde{\Lambda}^{3}) \right)
\! \Phi_{k}^{\ast}(\widetilde{\Lambda}),
\end{align}
where
\begin{equation} \label{prcy51}
\overset{\ast}{\mathbb{E}}_{k}^{\raise-6.75pt\hbox{$\scriptstyle \ast$}}
(\ell_{0,k}^{+},\ell_{1,k}^{+},\ell_{2,k};\tau) \! := \! \mathcal{G}_{2,k}^{-1}
\overset{\ast}{\mathbb{E}}_{k}^{\raise-6.75pt\hbox{$\scriptstyle \blacktriangle$}}
(\ell_{0,k}^{+},\ell_{1,k}^{+};\tau) \mathcal{G}_{2,k} \! - \! 2 \ell_{2,k}
(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+}) \sigma_{-}.
\end{equation}
One now chooses $\ell_{2,k}$ so that the $(2 \, 1)$-element of the nilpotent
coefficient matrix of the $\widetilde{\Lambda}^{2}$ terms in Equation~\eqref{prcy50}
is zero, that is, $(-\ell_{0,k}^{+} \! + \! 1) \ell_{2,k}^{2} \! + \! 2 \chi_{k}(\tau)
\ell_{2,k} \! + \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \chi_{k}^{-1}(\tau)
(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+}) \! = \! 0$; the roots are given by
\begin{equation} \label{prcy52}
\ell_{2,k}^{\pm} \! = \! \dfrac{-\chi_{k}(\tau) \! \pm \! \sqrt{\chi_{k}^{2}
(\tau) \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \chi_{k}^{-1}(\tau)
(-\ell_{0,k}^{+} \! + \! 1)(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+})}}{-\ell_{0,k}^{+} \! + \! 1},
\quad k \! = \! \pm 1.
\end{equation}
Arguing as in steps~\pmb{(vii)} and~\pmb{(viii)} above, for the class of functions
consistent with the Conditions~\eqref{iden5}, the `$+$-root' in Equation~\eqref{prcy52}
is taken:
\begin{equation} \label{prcy53}
\ell_{2,k} \! := \! \ell_{2,k}^{+} \! = \! \dfrac{-\chi_{k}(\tau) \! + \! \mu_{k}
(\tau)}{-\ell_{0,k}^{+} \! + \! 1},
\end{equation}
where
\begin{equation} \label{prcy54}
\mu_{k}(\tau) \! := \! \left(\chi_{k}^{2}(\tau) \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \chi_{k}^{-1}(\tau)(-\ell_{0,k}^{+} \! + \! 1)(\mathfrak{P}_{0,k}^{\ast}
\! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}) \right)^{1/2},
\end{equation}
with $\chi_{k}(\tau)$ defined by Equation~\eqref{prcy46}. Via the formula for the
$\tau$-dependent parameter $\ell_{2,k} \! := \! \ell_{2,k}^{+}$ defined by
Equations~\eqref{prcy53} and~\eqref{prcy54}, one simplifies Equation~\eqref{prcy50}
to read
\begin{equation} \label{prcy55}
\dfrac{\partial \Phi_{k}^{\ast}(\widetilde{\Lambda})}{\partial \widetilde{\Lambda}}
\underset{\tau \to +\infty}{=} \left(\daleth_{k}(\tau,\widetilde{\Lambda}) \!
+ \! \mathcal{O}(\beth_{k}(\tau,\widetilde{\Lambda})) \right) \! \Phi_{k}^{\ast}
(\widetilde{\Lambda}), \quad k \! = \! \pm 1,
\end{equation}
where
\begin{equation} \label{prcy56}
\daleth_{k}(\tau,\widetilde{\Lambda}) \! := \! \mu_{k}(\tau) \widetilde{\Lambda}
\sigma_{3} \! + \! p_{k}(\tau) \sigma_{+} \! + \! q_{k}(\tau) \sigma_{-},
\end{equation}
with
\begin{gather}
p_{k}(\tau) \! := \! -\ell_{0,k}^{+} \! + \! \hat{\mathbb{L}}_{k}(\tau) \! + \! 1,
\label{prcy57} \\
q_{k}(\tau) \! := \! (4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \chi_{k}^{-1}
(\tau))^{2}(-\ell_{0,k}^{+} \! + \! 1) \! - \! \ell_{1,k}^{+} \! - \! \ell_{2,k}^{+}
\! - \! \omega_{0,k}^{2}, \label{prcy58}
\end{gather}
and
\begin{align} \label{prcy59}
\beth_{k}(\tau,\widetilde{\Lambda}) \! :=& \,
\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)}(-\ell_{0,k}^{+}
\! + \! 1) \sigma_{3} \! - \! \hat{\mathbb{L}}_{k}(\tau) \sigma_{+} \! - \!
\dfrac{\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathcal{A}_{k}}{\chi_{k}(\tau)} \ell_{2,k}^{+}
(-\ell_{0,k}^{+} \! + \! 1) \widetilde{\Lambda} \sigma_{-} \nonumber \\
+& \, (\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{+}) \widetilde{\Lambda}^{2} \sigma_{3} \! + \!
\overset{\ast}{\mathbb{E}}_{k}^{\raise-6.75pt\hbox{$\scriptstyle \ast$}}
(\ell_{0,k}^{+},\ell_{1,k}^{+},\ell_{2,k}^{+};\tau) \widetilde{\Lambda}^{3},
\end{align}
where the yet-to-be-determined scalar function $\hat{\mathbb{L}}_{k}(\tau)$
is chosen in the proof of Lemma~\ref{ginversion} below (see, in particular,
Equations~\eqref{nueeq11}--\eqref{nueeq15}).\footnote{It will be shown that
$\hat{\mathbb{L}}_{k}(\tau) \! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{-2/3})$,
$k \! \in \! \lbrace \pm 1 \rbrace$: this fact will be used throughout the remainder
of the proof of Lemma~\ref{nprcl}.} For the requisite estimates in step~\pmb{(xi)}
below, the asymptotics of the function $\mu_{k}(\tau)$ and the $\tau$-dependent
parameter $\ell_{2,k}^{+}$ are essential; via the Conditions~\eqref{iden5}, the
Asymptotics~\eqref{tr1} and~\eqref{tr3}, and the Definitions~\eqref{prcy10},
\eqref{prcy15}, \eqref{prcy33}, \eqref{prcy40}, \eqref{prcy46}, \eqref{prcy53},
and~\eqref{prcy54}, one shows that, for $k \! = \! \pm 1$,
\begin{align} \label{prcymuk1}
\mu_{k}(\tau) \underset{\tau \to +\infty}{=}& \, \chi_{k}(\tau) \! - \! \dfrac{
\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}(-\ell_{0,k}^{+} \! + \! 1)
(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+})}{2 \chi_{k}^{2}(\tau)} \nonumber \\
-& \, \dfrac{(\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}(-\ell_{0,k}^{+}
\! + \! 1)(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{+}))^{2}}{8 \chi_{k}^{5}(\tau)} \nonumber \\
+& \, \mathcal{O} \! \left(\dfrac{(\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
(-\ell_{0,k}^{+} \! + \! 1)(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}))^{3}}{\chi_{k}^{8}(\tau)} \right),
\end{align}
where the asymptotics of $\mathfrak{A}_{k}$, $\ell_{0,k}^{+}$, $\chi_{k}
(\tau)$, and $\mathcal{Z}_{k}$ are given by the Expansions~\eqref{prcyak1},
\eqref{prcyellok1}, \eqref{prcychik1}, and~\eqref{prcyzeek1}, respectively,
and
\begin{align} \label{prcyell2k1}
\ell_{2,k}^{+} \underset{\tau \to +\infty}{=}& \, -\dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k}(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8
\sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+})}{\chi_{k}^{2}(\tau)}
\! - \! \dfrac{(-\ell_{0,k}^{+} \! + \! 1)(\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+}))^{2}}{8 \chi_{k}^{5}(\tau)} \nonumber \\
+& \, \mathcal{O} \! \left(\dfrac{(-\ell_{0,k}^{+} \! + \! 1)^{2}(\mathrm{i} 8 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k}(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}))^{3}}{\chi_{k}^{8}(\tau)} \right).
\end{align}
\pmb{(xi)} The rationale for this---final---step is to transform
Equation~\eqref{prcy55} into a `model' matrix linear ODE describing the coalescence
of turning points. Let $\Phi_{M,k}(\widetilde{\Lambda})$, $k \! = \! \pm 1$, be
a fundamental solution of Equation~\eqref{prcy1}; then, changing variables
according to $\widetilde{\Lambda} \! = \! \widetilde{\Lambda}(z) \! = \!
a_{k}^{\ast}(\tau)b^{\ast}z$, where $a_{k}^{\ast}(\tau) \! := \! (4 \sqrt{3}
\mathrm{e}^{\mathrm{i} \pi/2} \mu_{k}^{-1}(\tau))^{1/2}$ and $b^{\ast} \! := \! 2^{-3/2}
3^{-1/4} \mathrm{e}^{-\mathrm{i} \pi/4}$, and defining $\phi_{M,k}(z) \! := \! \Phi_{M,k}
(\widetilde{\Lambda}(z))$, one shows that $\phi_{M,k}(z)$ solves the canonical
matrix ODE
\begin{equation} \label{prcy60}
\partial_{z} \phi_{M,k}(z) \! = \! \left(\dfrac{z}{2} \sigma_{3} \! + \! P_{k}
(\tau) \sigma_{+} \! + \! Q_{k}(\tau) \sigma_{-} \right) \! \phi_{M,k}(z),
\quad k \! = \! \pm 1,
\end{equation}
where $P_{k}(\tau) \! := \! a_{k}^{\ast}(\tau)b^{\ast}p_{k}(\tau)$ and
$Q_{k}(\tau) \! := \! a_{k}^{\ast}(\tau)b^{\ast}q_{k}(\tau)$, with fundamental
solution expressed in terms of the parabolic-cylinder function, $D_{\star}
(\pmb{\pmb{\cdot}})$,\footnote{See, for example, \cite{a5,a18,a2}.}
\begin{equation} \label{prcy61}
\phi_{M,k}(z) \! = \!
\begin{pmatrix}
D_{-\nu (k)-1}(\mathrm{i} z) & D_{\nu (k)}(z) \\
\frac{1}{P_{k}(\tau)}(\frac{\partial}{\partial z} \! - \! \frac{z}{2})D_{-\nu (k)-1}
(\mathrm{i} z) & \frac{1}{P_{k}(\tau)}(\frac{\partial}{\partial z} \! - \! \frac{z}{2})
D_{\nu (k)}(z)
\end{pmatrix},
\end{equation}
where $-(\nu (k) \! + \! 1) \! := \! P_{k}(\tau)Q_{k}(\tau)$. Inverting the
dependent- and independent-variable linear transformations given above,
one arrives at the formula for the parameter $\nu (k) \! + \! 1$ defined
by Equation~\eqref{prpr1} and the representation for $\Phi_{M,k}
(\widetilde{\Lambda})$ given in Equation~\eqref{prcy2}.\footnote{{}From the
results subsumed in the proof of Lemma~\ref{ginversion} below, it will be
deduced \emph{a posteriori} that (cf. Definition~\eqref{prcy54}) $\mu_{k}
(\tau)$ possesses the asymptotics $\mu_{k}(\tau) \! =_{\tau \to +\infty} \!
\mathrm{i} 4 \sqrt{3} \! + \! \sum_{\scriptscriptstyle \underset{m_{1}+m_{2}+m_{3}
\geqslant 2}{m_{1},m_{2},m_{3} \in \lbrace 0 \rbrace \cup \mathbb{N}}}c_{m_{1},
m_{2},m_{3}}(k)(\tilde{r}_{0}(\tau))^{m_{1}}(v_{0}(\tau))^{m_{2}}(\tau^{-1/3})^{m_{3}}
\! + \! c_{\infty}(k) \tau^{-1/3} \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}
(1 \! + \! \mathcal{O}(\tau^{-1/3}))$, $k \! = \! \pm 1$, where $c_{m_{1},m_{2},
m_{3}}(k) \! \in \! \mathbb{C}$, and $\vartheta (\tau)$ and $\beta (\tau)$ are
defined in Equations~\eqref{thmk12}; via this fact, and the Definitions~\eqref{prpr1},
\eqref{prcy22}, \eqref{prcy57}, and~\eqref{prcy58}, a lengthy, circuitous calculation
reveals that the asymptotic expansion of $\nu (k) \! + \! 1$, $k \! = \! \pm 1$, can
be presented in the following form:
\begin{align*}
-(\nu (k) \! + \! 1) \underset{\tau \to +\infty}{=}& \, \frac{\mathrm{i}}{8 \sqrt{3}} \!
\left(\dfrac{-\alpha_{k}^{2}(8v_{0}^{2}(\tau) \! + \! 4v_{0}(\tau) \tilde{r}_{0}
(\tau) \! - \! (\tilde{r}_{0}(\tau))^{2} \! - \! v_{0}(\tau)(\tilde{r}_{0}(\tau))^{2}
\tau^{-1/3})\! + \! 4(a \! - \! \mathrm{i}/2)v_{0}(\tau) \tau^{-1/3}}{1 \! + \! v_{0}
(\tau) \tau^{-1/3}} \right) \\
+& \, \frac{2 \mathfrak{p}_{k}(\tau)}{3 \sqrt{3} \alpha_{k}^{2}} \! + \!
\sum_{m=2}^{\infty} \hat{\mu}_{m}^{\ast}(k)(\tau^{-1/3})^{m} \! + \!
\hat{c}_{\infty}(k) \tau^{-1/3} \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)}
\mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}(\tau^{-1/3})),
\end{align*}
where $\mathfrak{p}_{k}(\tau)$ is defined by Equation~\eqref{eqpeetee}. {}From
the Asymptotics~\eqref{tr1} and~\eqref{tr3}, and Propositions~\ref{recursys}
and~\ref{proprr}, in conjunction with the formulae for the
monodromy-data-dependent expansion coefficients $\mathrm{A}_{k}$, $k \! = \!
\pm 1$, derived in the proof of Lemma~\ref{ginversion} below (see, in particular,
Equations~\eqref{geek109} and~\eqref{geek111}), it will be shown that the sum of
the coefficients of each term $(\tau^{-1/3})^{j}$, $\mathbb{N} \! \ni \! j \! \geqslant
\! 2$, and $\tau^{-1/3} \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}$ on
the right-hand side of the latter asymptotic expansion for $\nu (k) \! + \! 1$
are equal to zero (e.g., $\hat{\mu}_{2}^{\ast}(k) \! = \! -\frac{\mathrm{i}}{24 \sqrt{3}
\alpha_{k}^{2}}((a \! - \! \mathrm{i}/2)^{2} \! - \! 1/6))$, resulting, finally, in the
asymptotics $\nu (k) \! + \! 1 \! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{-2/3}
\mathrm{e}^{-\beta (\tau)})$, $k \! = \! \pm 1$ (see Asymptotics~\eqref{geek13} below).}
Finally, in order to establish the asymptotic representation~\eqref{prpr2}, one
has to estimate the unimodular function $\hat{\chi}_{k}(\widetilde{\Lambda})$
defined in the transformation $\mathfrak{F}_{11}$. Under the action of the
transformation $\mathfrak{F}_{11}$, one re-writes Equation~\eqref{prcy55} as follows:
\begin{equation} \label{prcy62}
\dfrac{\partial \hat{\chi}_{k}(\widetilde{\Lambda})}{\partial \widetilde{\Lambda}}
\underset{\tau \to +\infty}{=} \beth_{k}(\tau,\widetilde{\Lambda}) \hat{\chi}_{k}
(\widetilde{\Lambda}) \! + \! \left[\daleth_{k}(\tau,\widetilde{\Lambda}),\hat{\chi}_{k}
(\widetilde{\Lambda}) \right], \quad k \! = \! \pm 1,
\end{equation}
where $\daleth_{k}(\tau,\widetilde{\Lambda})$ is defined by
Equations~\eqref{prcy56}--\eqref{prcy58}, and $\beth_{k}(\tau,\widetilde{\Lambda})$
is defined by Equation \eqref{prcy59}. The normalised solution of Equation~\eqref{prcy62},
that is, the one for which $\hat{\chi}_{k}(0) \! = \! \mathrm{I}$, is given by
\begin{equation} \label{prcy63}
\hat{\chi}_{k}(\widetilde{\Lambda}) \! = \! \mathrm{I} \! + \!
\int_{0}^{\widetilde{\Lambda}} \Phi_{M,k}(\widetilde{\Lambda}) \Phi_{M,k}^{-1}
(\xi) \beth_{k}(\tau,\xi) \hat{\chi}_{k}(\xi) \Phi_{M,k}(\xi) \Phi_{M,k}^{-1}
(\widetilde{\Lambda}) \, \mathrm{d} \xi, \quad k \! = \! \pm 1.
\end{equation}
In order to prove the required estimate for $\hat{\chi}_{k}(\widetilde{\Lambda})$,
one uses the method of successive approximations, namely,
\begin{equation*}
\hat{\chi}_{k}^{(m)}(\widetilde{\Lambda}) \! = \! \mathrm{I} \! + \! \int_{0}^{
\widetilde{\Lambda}} \Phi_{M,k}(\widetilde{\Lambda}) \Phi_{M,k}^{-1}(\xi)
\beth_{k}(\tau,\xi) \hat{\chi}_{k}^{(m-1)}(\xi) \Phi_{M,k}(\xi) \Phi_{M,k}^{-1}
(\widetilde{\Lambda}) \, \mathrm{d} \xi, \quad k \! = \! \pm 1, \quad m \! \in \!
\mathbb{N},
\end{equation*}
with $\hat{\chi}^{(0)}_{k}(\widetilde{\Lambda}) \! \equiv \! \mathrm{I}$, to
construct a Neumann series solution for $\hat{\chi}_{k}(\widetilde{\Lambda})$
($\hat{\chi}_{k}(\widetilde{\Lambda}) \! := \! \lim_{m \to \infty} \hat{\chi}^{(m)}_{k}
(\widetilde{\Lambda})$); in this instance, however, it suffices to estimate the
matrix norm of the associated resolvent kernel. Via the above iteration argument,
a calculation shows that, for $k \! = \! \pm 1$,
\begin{equation} \label{prcy64}
\lvert \lvert \hat{\chi}_{k}(\widetilde{\Lambda}) \! - \! \mathrm{I} \rvert \rvert
\underset{\tau \to +\infty}{\leqslant} \exp \! \left(\int_{0}^{\widetilde{\Lambda}}
\lvert \lvert \Phi_{M,k}(\widetilde{\Lambda}) \rvert \rvert \lvert \lvert
\Phi_{M,k}^{-1}(\xi) \rvert \rvert \lvert \lvert \beth_{k}(\tau,\xi) \rvert
\rvert \lvert \lvert \Phi_{M,k}(\xi) \rvert \rvert \lvert \lvert \Phi_{M,k}^{-1}
(\widetilde{\Lambda}) \rvert \rvert \lvert \mathrm{d} \xi \rvert \right) \! - \! 1,
\end{equation}
where $\lvert \mathrm{d} \xi \rvert$ denotes integration with respect to arc length.
Noting that (see Remark~\ref{leedasyue} below) $\det (\Phi_{M,k}(z)) \! = \!
-\mathrm{e}^{-\mathrm{i} \pi (\nu (k)+1)/2}(2 \mu_{k}(\tau))^{1/2}p^{-1}_{k}(\tau)$, it
follows {}from the Estimate~\eqref{prcy64} that, for $k \! = \! \pm 1$,
\begin{equation} \label{prcy65}
\lvert \lvert \hat{\chi}_{k}(\widetilde{\Lambda}) \! - \! \mathrm{I} \rvert
\rvert \underset{\tau \to +\infty}{\leqslant} \exp \! \left(\dfrac{\lvert p_{k}(\tau)
\rvert^{2} \lvert \lvert \Phi_{M,k}(\widetilde{\Lambda}) \rvert \rvert^{2}}{
\lvert 2 \mu_{k}(\tau) \rvert (\mathrm{e}^{\pi \Im (\nu (k)+1)/2})^{2}} \int_{0}^{
\widetilde{\Lambda}} \lvert \lvert \Phi_{M,k}(\xi) \rvert \rvert^{2} \lvert
\lvert \beth_{k}(\tau,\xi) \rvert \rvert \, \lvert \mathrm{d} \xi \rvert \right) \! - \! 1.
\end{equation}
One now proceeds to estimate the respective norms in Equation~\eqref{prcy65}.
One commences with the estimation of the norm $\lvert \lvert \beth_{k}(\tau,\xi)
\rvert \rvert$ appearing in Equation~\eqref{prcy65}. Via Equations \eqref{prcy7},
\eqref{prcy11}, \eqref{prcy24}, \eqref{prcy28}, \eqref{prcy32}, \eqref{prcy36},
\eqref{prcy37}, \eqref{prcy38}, \eqref{prcy41}, \eqref{prcy42}, \eqref{prcy43},
\eqref{prcy47}, \eqref{prcy49}, \eqref{prcy50}, \eqref{prcy51}, and~\eqref{prcy59},
one shows that, for $k \! = \! \pm 1$, in terms of the composition of the linear
transformations $\mathfrak{F}_{j}$, $j \! = \! 1,2,\dotsc,11$,
\begin{align} \label{prcy81}
\beth_{k}(\tau,\widetilde{\Lambda}) \! :=& \, \left(\mathfrak{F}_{11} \circ
\mathfrak{F}_{10} \circ \mathfrak{F}_{9} \circ \mathfrak{F}_{8} \circ
\mathfrak{F}_{7} \circ \mathfrak{F}_{6} \circ \mathfrak{F}_{5} \circ
\mathfrak{F}_{4} \circ \mathfrak{F}_{3} \circ \mathfrak{F}_{2} \circ
\mathfrak{F}_{1} \right) \! (\widetilde{\Psi}(\widetilde{\mu},\tau) \! - \!
\widetilde{\Psi}_{k}(\widetilde{\mu},\tau)) \nonumber \\
=& \, \dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)}
(-\ell_{0,k}^{+} \! + \! 1) \sigma_{3} \! - \! \hat{\mathbb{L}}_{k}(\tau) \sigma_{+}
\! - \! \dfrac{\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)}
\ell_{2,k}^{+}(-\ell_{0,k}^{+} \! + \! 1) \widetilde{\Lambda} \sigma_{-}
\nonumber \\
+& \, (\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{+}) \widetilde{\Lambda}^{2} \sigma_{3} \! +
\! \left(-2 \ell_{2,k}^{+}(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}) \sigma_{-} \right. \nonumber \\
+&\left. \mathcal{G}_{2,k}^{-1}
\left(
\begin{pmatrix}
1 & 0 \\
\mathrm{i} \omega_{0,k} & 1
\end{pmatrix} \tau^{-\frac{1}{6} \sigma_{3}} \mathcal{G}_{1,k}^{-1}
\mathcal{G}_{0,k}^{-1} \hat{\mathbb{E}}_{k}(\tau) \mathcal{G}_{0,k}
\mathcal{G}_{1,k} \tau^{\frac{1}{6} \sigma_{3}}
\begin{pmatrix}
1 & 0 \\
-\mathrm{i} \omega_{0,k} & 1
\end{pmatrix} \right. \right.
\nonumber \\
+&\left. \left.
\begin{pmatrix}
-\mathfrak{R}_{0,k}^{\ast} \ell_{0,k}^{+} & \ell_{0,k}^{+}(2 \mathfrak{P}_{0,k}^{
\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}) \\
-2 \ell_{1,k}^{+}(\mathfrak{P}_{0,k}^{\ast} \! - \! \mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{+}) & \mathfrak{R}_{0,k}^{\ast} \ell_{0,k}^{+}
\end{pmatrix} \right) \! \mathcal{G}_{2,k} \right) \! \widetilde{\Lambda}^{3},
\end{align}
whence, via the Definitions~\eqref{prcy9}, \eqref{prcy23},
\eqref{prcy33}--\eqref{prcy35}, \eqref{prcy40}, and~\eqref{prcy48}, and
a matrix-multipli\-c\-a\-t\-i\-o\-n argument, one arrives at, for $k \! = \!
\pm 1$,
\begin{align} \label{prcy82}
\beth_{k}(\tau,\widetilde{\Lambda}) \! =& \, \dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k}}{\chi_{k}(\tau)}(-\ell_{0,k}^{+} \! + \! 1) \sigma_{3} \! - \!
\hat{\mathbb{L}}_{k}(\tau) \sigma_{+} \! - \! \dfrac{\mathrm{i} 8 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k}}{\chi_{k}(\tau)} \ell_{2,k}^{+}(-\ell_{0,k}^{+} \! + \! 1)
\widetilde{\Lambda} \sigma_{-} \nonumber \\
+& \, (\mathfrak{A}_{0,k}^{\sharp} \! + \! \mathfrak{A}_{k} \omega_{0,k}^{2}
(\ell_{0,k}^{+})^{2}) \widetilde{\Lambda}^{2} \sigma_{3} \! + \!
\begin{pmatrix}
\mathcal{N}_{11}^{\ast}(\tau) \! + \! \mathcal{M}_{11}^{\ast}(\tau) &
\mathcal{N}_{12}^{\ast}(\tau) \! + \! \mathcal{M}_{12}^{\ast}(\tau) \\
\mathcal{N}_{21}^{\ast}(\tau) \! + \! \mathcal{M}_{21}^{\ast}(\tau) &
-(\mathcal{N}_{11}^{\ast}(\tau) \! + \! \mathcal{M}_{11}^{\ast}(\tau))
\end{pmatrix} \! \widetilde{\Lambda}^{3},
\end{align}
where
{\fontsize{10pt}{11pt}\selectfont
\begin{align}
\mathcal{N}_{11}^{\ast}(\tau) \! :=& \, \ell_{0,k}^{+} \mathfrak{A}_{k} \!
\left(\dfrac{\mathfrak{A}_{k} \mathfrak{B}_{0,k}^{\sharp}}{\mathfrak{B}_{k}}
\! - \! 2 \mathfrak{A}_{0,k}^{\sharp} \right) \! \left(1 \! - \! \dfrac{\mathrm{i}
4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)} \right) \! - \! \ell_{0,k}^{+} \!
\left(\mathfrak{B}_{k} \mathfrak{C}_{0,k}^{\sharp} \! - \! \dfrac{\mathrm{i}
4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)} \mathfrak{A}_{k}^{2}
\omega_{0,k}^{2}(\ell_{0,k}^{+})^{2} \right), \label{prcy83} \\
\mathcal{N}_{12}^{\ast}(\tau) \! :=& \, \ell_{0,k}^{+} \! \left(2
\mathfrak{A}_{0,k}^{\sharp} \! + \! \mathfrak{A}_{k} \omega_{0,k}^{2}
(\ell_{0,k}^{+})^{2} \! - \! \dfrac{\mathfrak{A}_{k} \mathfrak{B}_{0,k}^{
\sharp}}{\mathfrak{B}_{k}} \right), \label{prcy84} \\
\mathcal{N}_{21}^{\ast}(\tau) \! :=& \, -\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k}}{\chi_{k}(\tau)} \! \left(\ell_{0,k}^{+} \mathfrak{A}_{k} \!
\left(\dfrac{\mathfrak{A}_{k} \mathfrak{B}_{0,k}^{\sharp}}{\mathfrak{B}_{k}}
\! - \! 2 \mathfrak{A}_{0,k}^{\sharp} \right) \! \left(2 \! - \! \dfrac{\mathrm{i} 4
\sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)} \right) \! - \! \ell_{0,k}^{+} \!
\left(2 \mathfrak{B}_{k} \mathfrak{C}_{0,k}^{\sharp} \! - \! \dfrac{
\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)} \right. \right. \nonumber \\
\times&\left. \left. \, \mathfrak{A}_{k}^{2} \omega_{0,k}^{2}(\ell_{0,k}^{+})^{2}
\right) \right) \! - \! 2(\mathfrak{A}_{0,k}^{\sharp} \! + \! \mathfrak{A}_{k}
\omega_{0,k}^{2}(\ell_{0,k}^{+})^{2})(\ell_{1,k}^{+} \! + \! \ell_{2,k}^{+}),
\label{prcy85} \\
\mathcal{M}_{11}^{\ast}(\tau) \! :=& \, \dfrac{\hat{\mathcal{C}}_{1}}{2
\lambda_{1}^{\ast}(k) \mathfrak{B}_{k}} \! \left((\hat{\mathbb{E}}_{k}(\tau))_{11}
\! \left(\hat{\mathfrak{g}}_{11} \mathfrak{B}_{k} \! + \! \hat{\mathfrak{g}}_{12}
\mathfrak{A}_{k} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)}
\! - \! 1 \right) \! + \! \hat{\mathfrak{g}}_{12} \! \left(\mathfrak{B}_{k} \! + \!
\mathfrak{A}_{k} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)} \!
- \! 1 \right) \right) \right) \right. \nonumber \\
+&\left. \, (\hat{\mathbb{E}}_{k}(\tau))_{12} \! \left(\mathfrak{B}_{k} \! + \!
\mathfrak{A}_{k} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)}
\! - \! 1 \right) \right) \! - \! (\hat{\mathbb{E}}_{k}(\tau))_{21}
\hat{\mathfrak{g}}_{12} \! \left(\hat{\mathfrak{g}}_{11} \mathfrak{B}_{k} \!
+ \! \hat{\mathfrak{g}}_{12} \mathfrak{A}_{k} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k}}{\chi_{k}(\tau)} \! - \! 1 \right) \right) \right), \label{prcy86} \\
\mathcal{M}_{12}^{\ast}(\tau) \! :=& \, \dfrac{\hat{\mathcal{C}}_{1}}{2
\lambda_{1}^{\ast}(k) \mathfrak{B}_{k}} \left(2(\hat{\mathbb{E}}_{k}(\tau))_{11}
\hat{\mathfrak{g}}_{12} \! + \! (\hat{\mathbb{E}}_{k}(\tau))_{12} \! - \!
(\hat{\mathbb{E}}_{k}(\tau))_{21}(\hat{\mathfrak{g}}_{12})^{2} \right),
\label{prcy87} \\
\mathcal{M}_{21}^{\ast}(\tau) \! :=& \, \dfrac{\hat{\mathcal{C}}_{1}}{2
\lambda_{1}^{\ast}(k) \mathfrak{B}_{k}} \! \left(-2(\hat{\mathbb{E}}_{k}
(\tau))_{11} \! \left(\mathfrak{B}_{k} \! + \! \mathfrak{A}_{k} \! \left(
\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)} \! - \! 1 \right) \right) \!
\left(\hat{\mathfrak{g}}_{11} \mathfrak{B}_{k} \! + \! \hat{\mathfrak{g}}_{12}
\mathfrak{A}_{k} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)}
\! - \! 1 \right) \right) \right. \nonumber \\
-&\left. \, (\hat{\mathbb{E}}_{k}(\tau))_{12} \! \left(\mathfrak{B}_{k} \! + \!
\mathfrak{A}_{k} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)}
\! - \! 1 \right) \right)^{2} \! + \! (\hat{\mathbb{E}}_{k}(\tau))_{21} \! \left(
\hat{\mathfrak{g}}_{11} \mathfrak{B}_{k} \! + \! \hat{\mathfrak{g}}_{12}
\mathfrak{A}_{k} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)}
\! - \! 1 \right) \right)^{2} \right), \label{prcy88}
\end{align}}
with
\begin{equation} \label{prcy89}
\hat{\mathfrak{g}}_{11} \! := \! \dfrac{\hat{\mathcal{A}}_{1} \! + \!
\lambda_{1}^{\ast}(k)}{\hat{\mathcal{C}}_{1}} \, \quad \, \text{and} \,
\quad \, \hat{\mathfrak{g}}_{12} \! := \! \dfrac{\hat{\mathcal{A}}_{1}
\! - \! \lambda_{1}^{\ast}(k)}{\hat{\mathcal{C}}_{1}}.
\end{equation}
Via Equations~\eqref{prcy5}, \eqref{prcy7}, \eqref{prcy10}, \eqref{prcy15},
\eqref{prcy17}, \eqref{prcy18}--\eqref{prcy20}, \eqref{prcy22}, \eqref{prcy40},
\eqref{prcy45}, \eqref{prcy46}, \eqref{prcy53}, \eqref{prcy54},
and~\eqref{prcy83}--\eqref{prcy89}, a tedious calculation shows that
{\fontsize{10pt}{11pt}\selectfont
\begin{align}
\mathcal{N}_{11}^{\ast}(\tau) \underset{\tau \to +\infty}{=}& \, \dfrac{\mathrm{i}
4(\tau^{-1/3})^{2}}{3 \sqrt{3} \alpha_{k}^{2}} \! - \! \dfrac{\mathrm{i} \omega_{0,
k}^{2,\infty}(\tau^{-1/3})^{3}}{108 \alpha_{k}^{3} \mathfrak{B}_{k}^{\infty}}
\! \left(1 \! + \! \dfrac{7(a \! - \! \mathrm{i}/2) \tau^{-1/3}}{\sqrt{3} \alpha_{k}
\mathfrak{B}_{k}^{\infty}} \! + \! \dfrac{(a \! - \! \mathrm{i}/2)^{2}(\tau^{-1/3})^{2}}{
6 \alpha_{k}^{2}(\mathfrak{B}_{k}^{\infty})^{2}} \! + \! \mathcal{O}(\tau^{-2/3})
\right), \label{prcy90} \\
\mathcal{N}_{12}^{\ast}(\tau) \underset{\tau \to +\infty}{=}& \, \dfrac{\mathrm{i}
\tau^{-1/3}}{6 \alpha_{k} \mathfrak{B}_{k}^{\infty}} \! \left(-\dfrac{\mathrm{i} 4
\tau^{-1/3}}{\sqrt{3} \alpha_{k}} \! \left(7 \! + \! \dfrac{(a \! - \! \mathrm{i}/2)
\tau^{-1/3}}{\sqrt{3} \alpha_{k} \mathfrak{B}_{k}^{\infty}} \right) \! +
\! \dfrac{\mathrm{i} (14 \! - \! \sqrt{3}) \tilde{r}_{0}(\tau)(\tau^{-1/3})^{2}}{6
\alpha_{k}} \! - \! \dfrac{\mathrm{i} (\tau^{-1/3})^{2} \mathcal{A}_{k}^{1}}{3
\alpha_{k} \mathfrak{B}_{k}^{\infty}} \right. \nonumber \\
+&\left. \, \dfrac{\mathrm{i} 2(a \! - \! \mathrm{i}/2)(\sqrt{3} \! + \! 1)(\tau^{-1/3})^{3}
\mathfrak{B}_{k}^{1}}{3 \sqrt{3} \alpha_{k}^{2}(\mathfrak{B}_{k}^{\infty})^{2}}
\! + \! \dfrac{\mathrm{i} 2(a \! - \! \mathrm{i}/2) \tilde{r}_{0}(\tau)(\tau^{-1/3})^{3}}{3
\sqrt{3} \alpha_{k}^{2} \mathfrak{B}_{k}^{\infty}} \! + \! \dfrac{\mathrm{i} 7
\omega_{0,k}^{2,\infty}(\tau^{-1/3})^{2}}{36 \alpha_{k}^{2}
\mathfrak{B}_{k}^{\infty}} \right. \nonumber \\
+&\left. \, \dfrac{\mathrm{i} 7(\sqrt{3} \! + \! 1)(\tau^{-1/3})^{2}
\mathfrak{B}_{k}^{1}}{3 \alpha_{k} \mathfrak{B}_{k}^{\infty}} \! + \!
\mathcal{O}(\tau^{-5/3}) \right), \label{prcy91} \\
\mathcal{N}_{21}^{\ast}(\tau) \underset{\tau \to +\infty}{=}& \, \dfrac{2
(\tau^{-1/3})^{2}}{3 \sqrt{3} \alpha_{k}^{2}} \! \left(\dfrac{4(a \! - \! \mathrm{i}/2)
\tau^{-1/3}}{\sqrt{3} \alpha_{k}} \! - \! \dfrac{(a \! - \! \mathrm{i}/2)^{3}(\tau^{-
1/3})^{3}}{3 \sqrt{3} \alpha_{k}^{3}(\mathfrak{B}_{k}^{\infty})^{2}} \! - \!
14 \mathfrak{B}_{k}^{\infty} \right) \! + \! \dfrac{\tilde{r}_{0}(\tau)
(\tau^{-1/3})^{3}}{18 \sqrt{3} \alpha_{k}^{2}} \nonumber \\
\times& \, \left(-\sqrt{3}(14 \! + \! \sqrt{3}) \mathfrak{B}_{k}^{\infty}
\! + \! \dfrac{2(a \! - \! \mathrm{i}/2)^{3}(\tau^{-1/3})^{3}}{3 \alpha_{k}^{3}
(\mathfrak{B}_{k}^{\infty})^{2}} \right) \! + \! \dfrac{(\tau^{-1/3})^{3}
\mathcal{A}_{k}^{1}}{9 \alpha_{k}^{2}} \! \left(2 \! - \! \dfrac{(a \! - \!
\mathrm{i}/2)^{2}(\tau^{-1/3})^{2}}{2 \alpha_{k}^{2}(\mathfrak{B}_{k}^{\infty})^{2}}
\right) \nonumber \\
+& \, \dfrac{(\sqrt{3} \! + \! 1)(\tau^{-1/3})^{3}\mathfrak{B}_{k}^{1}}{9
\alpha_{k}^{2}} \! \left(-7 \! + \! \dfrac{(a \! - \! \mathrm{i}/2)^{3}(\tau^{-1/3})^{3}}{3
\sqrt{3} \alpha_{k}^{3}(\mathfrak{B}_{k}^{\infty})^{3}} \right) \! + \!
\dfrac{(\tau^{-1/3})^{3} \omega_{0,k}^{2,\infty}}{54 \alpha_{k}^{3}
\mathfrak{B}_{k}^{\infty}} \left(7 \mathfrak{B}_{k}^{\infty} \right.
\nonumber \\
+&\left. \, \dfrac{41(a \! - \! \mathrm{i}/2) \tau^{-1/3}}{2 \sqrt{3} \alpha_{k}}
\! - \! \dfrac{7(a \! - \! \mathrm{i}/2)^{2}(\tau^{-1/3})^{2}}{4 \alpha_{k}^{2}
\mathfrak{B}_{k}^{\infty}} \! + \! \dfrac{(a \! - \! \mathrm{i}/2)^{3}(\tau^{-1/3})^{3}}{3
\sqrt{3} \alpha_{k}^{3}(\mathfrak{B}_{k}^{\infty})^{2}} \right) \! + \! \mathcal{O}
(\tau^{-7/3}), \label{prcy92} \\
\mathcal{M}_{11}^{\ast}(\tau) \underset{\tau \to +\infty}{=}& \, \dfrac{\mathrm{i}
2 \sqrt{3}(\tau^{-1/3})^{2}}{\alpha_{k}^{2}} \! \left(3 \! + \! \dfrac{(a \! - \!
\mathrm{i}/2) (\tau^{-1/3})^{2} \omega_{0,k}^{2,\infty}}{72 \alpha_{k}^{2}
(\mathfrak{B}_{k}^{\infty})^{2}} \! + \! \mathcal{O}(\tau^{-4/3}) \right),
\label{prcy93} \\
\mathcal{M}_{12}^{\ast}(\tau) \underset{\tau \to +\infty}{=}& \, \dfrac{
(\tau^{-1/3})^{2}}{2 \sqrt{3} \alpha_{k}^{2} \mathfrak{B}_{k}^{\infty}} \!
\left(-12 \! + \! (\sqrt{3} \! + \! 1) \tilde{r}_{0}(\tau) \tau^{-1/3} \! +
\! \dfrac{\sqrt{3}(\sqrt{3} \! + \! 1) \tau^{-1/3} \mathfrak{B}_{k}^{1}}{
\mathfrak{B}_{k}^{\infty}} \! + \! \mathcal{O}(\tau^{-4/3}) \right),
\label{prcy94} \\
\mathcal{M}_{21}^{\ast}(\tau) \underset{\tau \to +\infty}{=}& \, \dfrac{
(\tau^{-1/3})^{2} \mathfrak{B}_{k}^{\infty}}{\sqrt{3} \alpha_{k}^{2}} \!
\left(12 \! + \! (\sqrt{3} \! - \! 1) \tilde{r}_{0}(\tau) \tau^{-1/3} \! +
\! \dfrac{\sqrt{3}(\sqrt{3} \! + \! 1) \tau^{-1/3} \mathfrak{B}_{k}^{1}}{
\mathfrak{B}_{k}^{\infty}} \right. \nonumber \\
-&\left. \, \dfrac{(a \! - \! \mathrm{i}/2) (\tau^{-1/3})^{2} \omega_{0,k}^{2,\infty}}{2
\alpha_{k}^{2}(\mathfrak{B}_{k}^{\infty})^{2}} \! + \! \mathcal{O}(\tau^{-4/3})
\right), \label{prcy95}
\end{align}}
where
\begin{align}
\omega_{0,k}^{2,\infty} \! :=& \, -\alpha_{k}^{2}(8v_{0}^{2}(\tau) \!
+ \! 4v_{0}(\tau) \tilde{r}_{0}(\tau) \! - \! (\tilde{r}_{0}(\tau))^{2}) \!
+ \! 4(a \! - \! \mathrm{i}/2)v_{0}(\tau) \tau^{-1/3}, \label{prcy96} \\
\mathfrak{B}_{k}^{\infty} \! :=& \, \dfrac{\alpha_{k}}{2}(4v_{0}(\tau) \! +
\! (\sqrt{3} \! + \! 1) \tilde{r}_{0}(\tau)) \! - \! \dfrac{(\sqrt{3} \! + \! 1)
(a \! - \! \mathrm{i}/2) \tau^{-1/3}}{2 \sqrt{3} \alpha_{k}}, \label{prcy97} \\
\mathfrak{B}_{k}^{1} \! :=& \, -\dfrac{\alpha_{k}}{2}(8v_{0}^{2}(\tau) \! +
\! 2(\sqrt{3} \! + \! 1)v_{0}(\tau) \tilde{r}_{0}(\tau) \! + \! (\tilde{r}_{0}
(\tau))^{2}) \nonumber \\
+& \, \dfrac{(a \! - \! \mathrm{i}/2)(12v_{0}(\tau) \! + \! (2 \sqrt{3} \! - \! 1)
\tilde{r}_{0}(\tau)) \tau^{-1/3}}{6 \alpha_{k}}, \label{prcy98} \\
\mathcal{A}_{k}^{1} \! :=& \, \alpha_{k}(8v_{0}^{2}(\tau) \! + \! 4v_{0}
(\tau) \tilde{r}_{0}(\tau) \! - \! (\tilde{r}_{0}(\tau))^{2}) \! - \! \dfrac{
(a \! - \! \mathrm{i}/2)(12v_{0}(\tau) \! - \! \tilde{r}_{0}(\tau)) \tau^{-1/3}}{
3 \alpha_{k}}. \label{prcy99}
\end{align}
{}From the asymptotics~\eqref{tr1}, \eqref{tr3},
\eqref{prcyak1}--\eqref{prcyc0k1}, \eqref{prcyomg1}, \eqref{prcyellok1},
\eqref{prcychik1}--\eqref{prcyell1k1}, \eqref{prcymuk1}, \eqref{prcyell2k1},
and \eqref{prcy90}--\eqref{prcy95}, the Definitions~\eqref{prcy33},
\eqref{prcy35}, \eqref{prcy57}, and~\eqref{prcy96}--\eqref{prcy99}, and
Equation~\eqref{prcy82}, one arrives at, after a laborious calculation,
\begin{equation} \label{estbet1}
\lvert \lvert \beth_{k}(\tau,\xi) \rvert \rvert \underset{\tau \to +\infty}{=}
\mathcal{O}(\tau^{-1/3} \lvert \xi \rvert^{3}), \quad k \! = \! \pm 1.
\end{equation}
In order to estimate, now, the norm of the unimodular function $\hat{\chi}_{k}
(\xi)$, one has to derive a uniform approximation for $\hat{\chi}_{k}(\xi)$ on
$\mathbb{R} \cup \mathrm{i} \mathbb{R} \! \ni \! \xi$; towards this goal, one uses the
following integral representation for the parabolic-cylinder function (see, for
example, \cite{EMOT}): for $k \! = \! \pm 1$,
\begin{equation} \label{prcy66}
D_{\nu (k)}(z) \! = \! \dfrac{2^{\nu (k)/2} \mathrm{e}^{-\frac{z^{2}}{4}}}{\Gamma
(-\nu (k)/2)} \int_{0}^{+\infty} \mathrm{e}^{-\frac{\xi z^{2}}{2}} \xi^{-\frac{\nu (k)}{2}-1}
(1 \! + \! \xi)^{\frac{\nu (k)-1}{2}} \, \mathrm{d} \xi, \quad \Re (\nu (k)) \! < \! 0, \quad
\lvert \arg (z) \rvert \! \leqslant \! \pi/4,
\end{equation}
where $\Gamma (\pmb{\cdot})$ is the (Euler) gamma function. As the integral
representation~\eqref{prcy66} will be applied simultaneously to the entries of
the $\mathrm{M}_{2}(\mathbb{C})$-valued function (cf. Equation~\eqref{prcy2})
$\Phi_{M,k}(\xi)$ in order to arrive at a uniform approximation for $\hat{\chi}_{k}
(\xi)$ on the Stokes rays $\arg (\xi) \! = \! 0,\pm \pi/2,\pm \pi,\dotsc$, $0 \!
\leqslant \! \lvert \xi \rvert \! < \! +\infty$, it implies the restrictions~\eqref{pc4}
on $\nu (k) \! + \! 1$; in fact, for the purposes of this work, it is sufficient to have
a uniform approximation for $\hat{\chi}_{k}(\xi)$ on, say, the Stokes rays $\arg
(\xi) \! \in \! \lbrace 0,-\pi/2,-\pi,-3 \pi/2 \rbrace$, $0 \! \leqslant \! \lvert \xi
\rvert \! < \! +\infty$. Towards the above-mentioned goal, using the following
functional relations and values for the (Euler) gamma function (see, for example,
\cite{a24}),
\begin{gather*}
\Gamma (z \! + \! 1) \! = \! z \Gamma (z), \, \quad \, \Gamma (z) \Gamma
(1 \! - \! z) \! = \! \dfrac{\pi}{\sin (\pi z)}, \, \quad \, \sqrt{\pi} \, \Gamma (2z)
\! = \! 2^{2z-1} \Gamma (z) \Gamma (z \! + \! 1/2), \\
\Gamma (1/2) \! = \! \sqrt{\pi}, \, \quad \, \int_{0}^{+\infty} \dfrac{t^{x-1}}{(1 \!
+ \! t)^{x+y}} \, \mathrm{d} t \! = \! \dfrac{\Gamma (x) \Gamma (y)}{\Gamma (x \! + \! y)},
\quad \Re (x),\Re (y) \! > \! 0,
\end{gather*}
the linear relations relating any three of the four parabolic-cylinder functions (cf.
Equation~\eqref{prcy61}) $D_{-\nu (k)-1}(\pm \mathrm{i} z)$ and $D_{\nu (k)}(\pm z)$,
\begin{align*}
\sqrt{2 \pi}D_{\nu (k)}(z) =& \, \Gamma (\nu (k) \! + \! 1) \! \left(\mathrm{e}^{\mathrm{i} \pi
\nu (k)/2}D_{-\nu (k)-1}(\mathrm{i} z) \! + \! \mathrm{e}^{-\mathrm{i} \pi \nu (k)/2}D_{-\nu (k)-1}
(-\mathrm{i} z) \right), \\
D_{\nu (k)}(z) =& \, \mathrm{e}^{-\mathrm{i} \pi \nu (k)}D_{\nu (k)}(-z) \! + \! \dfrac{
\sqrt{2 \pi} \mathrm{e}^{-\mathrm{i} \pi (\nu (k)+1)/2}}{\Gamma (-\nu (k))}D_{-\nu (k)-1}
(\mathrm{i} z), \\
D_{\nu (k)}(z) =& \, \mathrm{e}^{\mathrm{i} \pi \nu (k)}D_{\nu (k)}(-z) \! + \! \dfrac{\sqrt{2
\pi} \mathrm{e}^{\mathrm{i} \pi (\nu (k)+1)/2}}{\Gamma (-\nu (k))}D_{-\nu (k)-1}(-\mathrm{i} z),
\end{align*}
and the fact that (see the Asymptotics~\eqref{geek13} below) $\nu (k) \! + \! 1 \!
\to \! 0$ as $\tau \! \to \! +\infty$, one arrives at, via the restrictions~\eqref{pc4}
on $\nu (k) \! + \! 1$, Equation~\eqref{prcy2}, and the integral
representation~\eqref{prcy66}, the following uniform estimates: \pmb{(a)} for
$\arg (\xi) \! = \! 0 \! + \! \mathcal{O}(\tau^{-2/3})$,\footnote{The asymptotic
estimate $\mathcal{O}(\tau^{-2/3})$ appears on the Stokes rays because of the
factor $(2 \mu_{k}(\tau))^{1/2}$ in the arguments of the various parabolic-cylinder
functions in Equation~\eqref{prcy2} and the fact that (cf. Expansions~\eqref{prcychik1},
\eqref{prcyzeek1}, and~\eqref{prcymuk1}) $\arg (\mu_{k}(\tau)) \! =_{\tau \to +\infty}
\! \tfrac{\pi}{2}(1 \! + \! \mathcal{O}(\tau^{-2/3}))$.}
{\fontsize{10pt}{11pt}\selectfont
\begin{align*}
\lvert (\Phi_{M,k}(\xi))_{11} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\left(\dfrac{2^{3/2} \mathrm{e}^{\pi \Im (\nu (k)+1)/2}2^{\Re (\nu (k))/2} \cosh^{3}
(\frac{\pi}{2} \Im (\nu (k) \! + \! 1)) \Gamma (-\Re (\nu (k)))}{\Gamma
(\frac{1}{2} \! - \! \frac{\Re (\nu (k))}{2}) \sin (-\frac{\pi}{2} \Re (\nu (k)))} \right. \\
+&\left. \, \dfrac{\sqrt{\pi} \, \mathrm{e}^{\pi \Im (\nu (k)+1)}2^{-\Re (\nu (k)+1)/2}
\lvert \sin (\frac{\pi}{2}(\nu (k) \! + \! 1)) \rvert}{\Gamma (\frac{1}{2} \! + \!
\frac{\Re (\nu (k)+1)}{2}) \sin (\frac{\pi}{2} \Re (\nu (k) \! + \! 1))} \right) \!
\left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right), \\
\lvert (\Phi_{M,k}(\xi))_{12} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{\sqrt{\pi} \, 2^{\Re (\nu (k))/2} \cosh (\frac{\pi}{2} \Im (\nu (k) \! + \! 1))}{
\Gamma (\frac{1}{2} \! - \! \frac{\Re (\nu (k))}{2}) \sin (-\frac{\pi}{2} \Re (\nu (k)))}
\! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right), \\
\lvert (\Phi_{M,k}(\xi))_{21} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{4 \sqrt{3} \, \lvert \xi \rvert \Re (\nu (k) \! + \! 1)}{\lvert p_{k}(\tau) \rvert}
\! \left(\dfrac{\mathrm{e}^{\pi \Im (\nu (k)+1)}2^{\Re (\nu (k)+1)/2} \lvert \sin
(\frac{\pi}{2}(\nu (k) \! + \! 1)) \rvert \Gamma (\frac{\Re (\nu (k)+1)}{2})}{\sin
(\frac{\pi}{2} \Re (\nu (k) \! + \! 1)) \Gamma (\Re (\nu (k) \! + \! 1))} \right. \\
+&\left. \, \dfrac{2^{3/2} \mathrm{e}^{\pi \Im (\nu (k)+1)/2}2^{-\Re (\nu (k))/2}
\cosh^{3}(\frac{\pi}{2} \Im (\nu (k) \! + \! 1)) \Gamma (-\frac{\Re (\nu (k))}{2})}{
\sqrt{\pi} \sin (-\frac{\pi}{2} \Re (\nu (k)))} \right) \\
\times& \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right), \\
\lvert (\Phi_{M,k}(\xi))_{22} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{4 \sqrt{3} \, \lvert \xi \rvert \Re (\nu (k) \! + \! 1)2^{-\Re (\nu (k))/2}
\cosh (\frac{\pi}{2} \Im (\nu (k) \! + \! 1)) \Gamma (-\frac{\Re (\nu (k))}{2})}{
\lvert p_{k}(\tau) \rvert \sin (-\frac{\pi}{2} \Re (\nu (k))) \Gamma (-\Re (\nu (k)))} \\
\times& \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right);
\end{align*}}
\pmb{(b)} for $\arg (\xi) \! = \! -\pi/2 \! + \! \mathcal{O}(\tau^{-2/3})$,
\begin{align}
\lvert (\Phi_{M,k}(\xi))_{11} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{\sqrt{\pi} \, 2^{-\Re (\nu (k)+1)/2} \lvert \sin (\frac{\pi}{2}(\nu (k) \! + \! 1))
\rvert}{\Gamma (\frac{1}{2} \! + \! \frac{\Re (\nu (k)+1)}{2}) \sin (\frac{\pi}{2} \Re
(\nu (k) \! + \! 1))} \! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right) \nonumber \\
=:& \, \hat{\varrho}_{0}(k) \! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right),
\label{prcy67} \\
\lvert (\Phi_{M,k}(\xi))_{12} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{\sqrt{\pi} \, 2^{\Re (\nu (k))/2} \cosh (\frac{\pi}{2} \Im (\nu (k) \! + \! 1))}{
\Gamma (\frac{1}{2} \! - \! \frac{\Re (\nu (k))}{2}) \sin (-\frac{\pi}{2} \Re (\nu (k)))}
\! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right) \nonumber \\
=:& \, \hat{\varrho}_{1}(k) \! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right),
\label{prcy68} \\
\lvert (\Phi_{M,k}(\xi))_{21} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{4 \sqrt{3} \, \lvert \xi \rvert \Re (\nu (k) \! + \! 1)2^{\Re (\nu (k)+1)/2}
\Gamma (\frac{\Re (\nu (k)+1)}{2}) \lvert \sin (\frac{\pi}{2}(\nu (k) \! + \! 1))
\rvert}{\lvert p_{k}(\tau) \rvert \Gamma (\Re (\nu (k) \! + \! 1)) \sin
(\frac{\pi}{2} \Re (\nu (k) \! + \! 1))} \nonumber \\
\times& \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right) \! =: \! \hat{\varrho}_{2}
(k) \dfrac{\lvert \xi \rvert \Re (\nu (k) \! + \! 1)}{\lvert p_{k}(\tau) \rvert} \! \left(
1 \! + \! \mathcal{O}(\tau^{-2/3}) \right), \label{prcy69} \\
\lvert (\Phi_{M,k}(\xi))_{22} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{4 \sqrt{3} \, \lvert \xi \rvert \Re (\nu (k) \! + \! 1)2^{-\Re (\nu (k))/2}
\cosh (\frac{\pi}{2} \Im (\nu (k) \! + \! 1)) \Gamma (-\frac{\Re (\nu (k))}{2})}{
\lvert p_{k}(\tau) \rvert \sin (-\frac{\pi}{2} \Re (\nu (k))) \Gamma (-\Re (\nu (k)))}
\nonumber \\
\times& \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right) \! =: \! \hat{\varrho}_{3}(k)
\dfrac{\lvert \xi \rvert \Re (\nu (k) \! + \! 1)}{\lvert p_{k}(\tau) \rvert} \! \left(
1 \! + \! \mathcal{O}(\tau^{-2/3}) \right); \label{prcy70}
\end{align}
\pmb{(c)} for $\arg (\xi) \! = \! -\pi \! + \! \mathcal{O}(\tau^{-2/3})$,
{\fontsize{10pt}{11pt}\selectfont
\begin{align*}
\lvert (\Phi_{M,k}(\xi))_{11} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{\sqrt{\pi} \, 2^{-\Re (\nu (k)+1)/2} \lvert \sin (\frac{\pi}{2}(\nu (k) \! + \!
1)) \rvert}{\Gamma (\frac{1}{2} \! + \! \frac{\Re (\nu (k)+1)}{2}) \sin (\frac{\pi}{2}
\Re (\nu (k) \! + \! 1))} \! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right), \\
\lvert (\Phi_{M,k}(\xi))_{12} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\left(\dfrac{2^{3/2} \mathrm{e}^{\pi \Im (\nu (k)+1)/2} \lvert \cos (\frac{\pi}{2}
(\nu (k) \! + \! 1)) \rvert \lvert \sin (\frac{\pi}{2}(\nu (k) \! + \! 1)) \rvert^{2}
\Gamma (\Re (\nu (k) \! + \! 1))}{2^{\Re (\nu (k)+1)/2} \Gamma (\frac{1}{2} \!
+ \! \frac{\Re (\nu (k)+1)}{2}) \sin (\frac{\pi}{2} \Re (\nu (k) \! + \! 1))} \right. \\
+&\left. \, \dfrac{\sqrt{\pi} \, \mathrm{e}^{\pi \Im (\nu (k)+1)}2^{\Re (\nu (k))/2} \cosh
(\frac{\pi}{2} \Im (\nu (k)+1))}{\Gamma (\frac{1}{2} \! - \! \frac{\Re (\nu (k))}{2})
\sin (-\frac{\pi}{2} \Re (\nu (k)))} \right) \! \left(1 \! + \! \mathcal{O}
(\tau^{-2/3}) \right), \\
\lvert (\Phi_{M,k}(\xi))_{21} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{4 \sqrt{3} \, \lvert \xi \rvert \Re (\nu (k) \! + \! 1)2^{\Re (\nu (k)+1)/2}
\lvert \sin (\frac{\pi}{2}(\nu (k) \! + \! 1)) \rvert \Gamma (\frac{\Re (\nu (k)
+1)}{2})}{\lvert p_{k}(\tau) \rvert \sin (\frac{\pi}{2} \Re (\nu (k) \! + \! 1))
\Gamma (\Re (\nu (k) \! + \! 1))} \\
\times& \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right), \\
\lvert (\Phi_{M,k}(\xi))_{22} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{4 \sqrt{3} \, \lvert \xi \rvert \Re (\nu (k) \! + \! 1)}{\lvert p_{k}(\tau)
\rvert} \! \left(\dfrac{\mathrm{e}^{\pi \Im (\nu (k)+1)} \cosh (\frac{\pi}{2} \Im
(\nu (k) \! + \! 1)) \Gamma (-\frac{\Re (\nu (k))}{2})}{2^{\Re (\nu (k))/2}
\sin (-\frac{\pi}{2} \Re (\nu (k))) \Gamma (-\Re (\nu (k)))} \right. \\
+&\left. \, \dfrac{2^{3/2} \mathrm{e}^{\pi \Im (\nu (k)+1)/2} \lvert \cos
(\frac{\pi}{2}(\nu (k) \! + \! 1)) \rvert \lvert \sin (\frac{\pi}{2}(\nu (k)
\! + \! 1)) \rvert^{2} \Gamma (\frac{\Re (\nu (k) +1)}{2})}{\sqrt{\pi}
2^{-\Re (\nu (k)+1)/2} \sin (\frac{\pi}{2} \Re (\nu (k) \! + \! 1))} \right) \\
\times& \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right);
\end{align*}}
and \pmb{(d)} for $\arg (\xi) \! = \! -3 \pi/2 \! + \! \mathcal{O}(\tau^{-2/3})$,
{\fontsize{10pt}{11pt}\selectfont
\begin{align}
\lvert (\Phi_{M,k}(\xi))_{11} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\left(\dfrac{2^{3/2} \mathrm{e}^{-\pi \Im (\nu (k)+1)/2}2^{\Re (\nu (k))/2} \cosh^{3}
(\frac{\pi}{2} \Im (\nu (k) \! + \! 1)) \Gamma (-\Re (\nu (k)))}{\Gamma (\frac{1}{2}
\! - \! \frac{\Re (\nu (k))}{2}) \sin (-\frac{\pi}{2} \Re (\nu (k)))} \right. \nonumber \\
+&\left. \, \dfrac{\sqrt{\pi} \, \mathrm{e}^{-\pi \Im (\nu (k)+1)}2^{-\Re (\nu (k)+1)/2}
\lvert \sin (\frac{\pi}{2}(\nu (k) \! + \! 1)) \rvert}{\Gamma (\frac{1}{2} \! + \!
\frac{\Re (\nu (k)+1)}{2}) \sin (\frac{\pi}{2} \Re (\nu (k) \! + \! 1))} \right)
\! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right) \nonumber \\
=:& \, \tilde{\varrho}_{0}(k) \! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right),
\label{prcy71} \\
\lvert (\Phi_{M,k}(\xi))_{12} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\left(\dfrac{2^{3/2} \mathrm{e}^{-\pi \Im (\nu (k)+1)/2} \lvert \cos (\frac{\pi}{2}
(\nu (k) \! + \! 1)) \rvert \lvert \sin (\frac{\pi}{2}(\nu (k) \! + \! 1)) \rvert^{2}
\Gamma (\Re (\nu (k) \! + \! 1))}{2^{\Re (\nu (k)+1)/2} \Gamma (\frac{1}{2}
\! + \! \frac{\Re (\nu (k)+1)}{2}) \sin (\frac{\pi}{2} \Re (\nu (k) \! + \! 1))}
\right. \nonumber \\
+&\left. \, \dfrac{\sqrt{\pi} \, \mathrm{e}^{-\pi \Im (\nu (k)+1)}2^{\Re (\nu (k))/2}
\cosh (\frac{\pi}{2} \Im (\nu (k) \! + \! 1))}{\Gamma (\frac{1}{2} \! - \! \frac{
\Re (\nu (k))}{2}) \sin (-\frac{\pi}{2} \Re (\nu (k)))} \right) \! \left(
1 \! + \! \mathcal{O}(\tau^{-2/3}) \right) \nonumber \\
=:& \, \tilde{\varrho}_{1}(k) \! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right),
\label{prcy72} \\
\lvert (\Phi_{M,k}(\xi))_{21} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{4 \sqrt{3} \, \lvert \xi \rvert \Re (\nu (k) \! + \! 1)}{\lvert p_{k}(\tau)
\rvert} \! \left(\dfrac{\mathrm{e}^{-\pi \Im (\nu (k)+1)/2} \lvert \sin (\frac{\pi}{2}
(\nu (k) \! + \! 1)) \rvert \Gamma (\frac{\Re (\nu (k)+1)}{2})}{2^{-\Re (\nu
(k)+1)/2} \sin (\frac{\pi}{2} \Re (\nu (k) \! + \! 1)) \Gamma (\Re (\nu (k)
\! + \! 1))} \right. \nonumber \\
+&\left. \, \dfrac{2^{3/2} \mathrm{e}^{-\pi \Im (\nu (k)+1)/2} \cosh^{3}(\frac{\pi}{2}
\Im (\nu (k) \! + \! 1)) \Gamma (-\frac{\Re (\nu (k))}{2})}{2^{\Re (\nu (k))/2}
\sin (-\frac{\pi}{2} \Re (\nu (k)))} \right) \! \left(1 \! + \! \mathcal{O}
(\tau^{-2/3}) \right) \nonumber \\
=:& \, \tilde{\varrho}_{2}(k) \dfrac{\lvert \xi \rvert \Re (\nu (k) \! + \! 1)}{
\lvert p_{k}(\tau) \rvert} \! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right),
\label{prcy73} \\
\lvert (\Phi_{M,k}(\xi))_{22} \rvert \underset{\tau \to +\infty}{\leqslant}& \,
\dfrac{4 \sqrt{3} \, \lvert \xi \rvert \Re (\nu (k) \! + \! 1)}{\lvert p_{k}(\tau)
\rvert} \! \left(\dfrac{\mathrm{e}^{-\pi \Im (\nu (k)+1)} \cosh (\frac{\pi}{2} \Im (\nu
(k) \! + \! 1)) \Gamma (-\frac{\Re (\nu (k))}{2})}{2^{\Re (\nu (k))/2} \sin
(-\frac{\pi}{2} \Re (\nu (k))) \Gamma (-\Re (\nu (k)))} \right. \nonumber \\
+&\left. \, \dfrac{2^{3/2} \mathrm{e}^{-\pi \Im (\nu (k)+1)/2} \lvert \cos
(\frac{\pi}{2}(\nu (k) \! + \! 1)) \rvert \lvert \sin (\frac{\pi}{2}(\nu (k) \!
+ \! 1)) \rvert^{2} \Gamma (\frac{\Re (\nu (k)+1)}{2})}{\sqrt{\pi}2^{-\Re
(\nu (k)+1)/2} \sin (\frac{\pi}{2} \Re (\nu (k) \! + \! 1))} \right) \nonumber \\
\times& \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right) \! =: \!
\tilde{\varrho}_{3}(k) \dfrac{\lvert \xi \rvert \Re (\nu (k) \! + \! 1)}{\lvert p_{k}
(\tau) \rvert} \! \left(1 \! + \! \mathcal{O}(\tau^{-2/3}) \right). \label{prcy74}
\end{align}}
To eschew redundant technicalities, consider, say, the case $k \! = \! +1$,
and, without loss of generality, $\arg (\widetilde{\Lambda}) \! = \! \pm
\pi/2$:\footnote{The pair of values $\arg (\widetilde{\Lambda}) \! = \! \pm
\pi/2$ on the Stokes rays are chosen for illustrative purposes only, in order
to present the general scheme of the calculations: for any of the remaining
$\binom{4}{2} \! - \! 1 \! = \! 5$ pairs of values of $\arg (\widetilde{\Lambda})$ on
the Stokes rays, one arrives at the same estimate (see Equation~\eqref{prcy100}
below) for $\lvert \lvert \hat{\chi}_{k}(\widetilde{\Lambda}) \! - \! \mathrm{I}
\rvert \rvert$, $k \! = \! \pm 1$, but with \textbf{different} $\mathcal{O}(1)$
constants.} the case $k \! = \! -1$ is analogous. Using the asymptotic expansions
for the parabolic-cylinder functions (see Remark~\ref{leedasyue} below), one
shows that: \pmb{(a)} for $\arg (\widetilde{\Lambda}) \! = \! \pi/2 \! + \!
\mathcal{O}(\tau^{-2/3})$,
\begin{gather} \label{prcy75}
\begin{split}
\lvert (\Phi_{M,1}(\widetilde{\Lambda}))_{11} \rvert \underset{\tau \to +\infty}{=}&
\mathcal{O} \! \left(\tilde{\rho}_{0} \lvert \widetilde{\Lambda} \rvert^{-\Re
(\nu (1)+1)} \right), \quad \lvert (\Phi_{M,1}(\widetilde{\Lambda}))_{12} \rvert
\underset{\tau \to +\infty}{=} \mathcal{O} \! \left(\tilde{\rho}_{1} \lvert
\widetilde{\Lambda} \rvert^{-\Re (\nu (1)+1)} \right), \\
\lvert (\Phi_{M,1}(\widetilde{\Lambda}))_{21} \rvert \underset{\tau \to +\infty}{=}&
\mathcal{O} \! \left(\tilde{\rho}_{2} \dfrac{\lvert \widetilde{\Lambda} \rvert^{
\Re (\nu (1)+1)}}{\lvert p_{1}(\tau) \rvert} \right), \quad \lvert (\Phi_{M,1}
(\widetilde{\Lambda}))_{22} \rvert \underset{\tau \to +\infty}{=} \mathcal{O}
\! \left(\tilde{\rho}_{3} \dfrac{\lvert \widetilde{\Lambda} \rvert^{\Re (\nu (1)+
1)}}{\lvert p_{1}(\tau) \rvert} \right),
\end{split}
\end{gather}
where
\begin{gather*}
\tilde{\rho}_{0} \! := \! \eta_{+} \mathrm{e}^{-3 \pi \Im (\nu (1)+1)/2},
\qquad \tilde{\rho}_{3} \! := \! \eta_{+}^{-1}2^{3/2}3^{1/4}, \\
\tilde{\rho}_{1} \! := \! \dfrac{\eta_{+}}{\sqrt{\pi}}2^{3/2} \mathrm{e}^{-\pi \Im
(\nu (1)+1)} \lvert \cos (\tfrac{\pi}{2}(\nu (1) \! + \! 1)) \rvert \lvert \sin
(\tfrac{\pi}{2}(\nu (1) \! + \! 1)) \rvert \Gamma (\Re (\nu (1) \! + \! 1)), \\
\tilde{\rho}_{2} \! := \! \dfrac{8 \eta_{+}^{-1}}{\sqrt{\pi}}3^{1/4} \mathrm{e}^{\pi
\Im (\nu (1)+1)/2} \lvert \cos (\tfrac{\pi}{2}(\nu (1) \! + \! 1)) \rvert \lvert
\sin (\tfrac{\pi}{2}(\nu (1) \! + \! 1)) \rvert \Gamma (-\Re (\nu (1))),
\end{gather*}
with $\eta_{+} \! = \! (2^{3/2}3^{1/4})^{-\Re (\nu (1)+1)} \mathrm{e}^{3 \pi \Im
(\nu (1)+1)/4}$; and \pmb{(b)} for $\arg (\widetilde{\Lambda}) \! = \!
-\pi/2 \! + \! \mathcal{O}(\tau^{-2/3})$,
\begin{gather} \label{prcy76}
\begin{split}
\lvert (\Phi_{M,1}(\widetilde{\Lambda}))_{11} \rvert \underset{\tau \to +\infty}{=}&
\mathcal{O} \! \left(\hat{\rho}_{0} \lvert \widetilde{\Lambda} \rvert^{-\Re (\nu
(1)+1)} \right), \quad \lvert (\Phi_{M,1}(\widetilde{\Lambda}))_{12} \rvert
\underset{\tau \to +\infty}{=} \mathcal{O} \! \left(\hat{\rho}_{1} \dfrac{\lvert
\widetilde{\Lambda} \rvert^{\Re (\nu (1)+1)}}{\lvert \widetilde{\Lambda}
\rvert} \right), \\
\lvert (\Phi_{M,1}(\widetilde{\Lambda}))_{21} \rvert \underset{\tau \to +\infty}{=}&
\mathcal{O} \! \left(\hat{\rho}_{2} \dfrac{\lvert \widetilde{\Lambda} \rvert^{-\Re
(\nu (1)+1)}}{\lvert p_{1}(\tau) \rvert \lvert \widetilde{\Lambda} \rvert} \right),
\quad \lvert (\Phi_{M,1}(\widetilde{\Lambda}))_{22} \rvert \underset{\tau \to
+\infty}{=} \mathcal{O} \! \left(\hat{\rho}_{3} \dfrac{\lvert \widetilde{\Lambda}
\rvert^{\Re (\nu (1)+1)}}{\lvert p_{1}(\tau) \rvert} \right),
\end{split}
\end{gather}
where
\begin{equation*}
\hat{\rho}_{0} \! := \! \eta_{-} \mathrm{e}^{\pi \Im (\nu (1)+1)/2}, \! \! \quad
\! \! \hat{\rho}_{1} \! := \! \eta_{-}^{-1}2^{-3/2}3^{-1/4}, \! \! \quad
\! \! \hat{\rho}_{2} \! := \! \eta_{-} \mathrm{e}^{\pi \Im (\nu (1)+1)/2} \lvert
\nu (1) \! + \! 1 \rvert, \! \! \quad \! \! \hat{\rho}_{3} \! := \!
\eta_{-}^{-1}2^{3/2}3^{1/4},
\end{equation*}
with $\eta_{-} \! = \! (2^{3/2}3^{1/4})^{-\Re (\nu (1)+1)} \mathrm{e}^{-\pi \Im (\nu
(1)+1)/4}$. Hence, via the elementary inequalities $\lvert \Re (\nu (1) \! + \!
1) \rvert \! \leqslant \! \lvert \nu (1) \! + \! 1 \rvert$ and $\lvert \Im (\nu (1)
\! + \! 1) \rvert \! \leqslant \! \lvert \nu (1) \! + \! 1 \rvert$, it follows {}from
the Estimates~\eqref{prcy71}--\eqref{prcy74} and~\eqref{prcy75} that, for
$\arg (\widetilde{\Lambda}) \! = \! \pi/2 \! + \! \mathcal{O}(\tau^{-2/3})$,
\begin{gather}
\lvert \lvert \Phi_{M,1}(\xi) \rvert \rvert^{2} \underset{\tau \to +\infty}{=}
\mathcal{O}(\tilde{\mathfrak{c}}_{M}^{\sharp}) \! + \! \mathcal{O} \! \left(
\dfrac{\tilde{\mathfrak{c}}_{M}^{\sharp} \lvert \nu (1) \! + \! 1 \rvert^{2}
\lvert \xi \rvert^{2}}{\lvert p_{1}(\tau) \rvert^{2}} \right), \label{prcy77} \\
\lvert \lvert \Phi_{M,1}(\widetilde{\Lambda}) \rvert \rvert^{2} \underset{\tau
\to +\infty}{=} \mathcal{O} \! \left(\lvert \widetilde{\Lambda} \rvert^{2
\Re (\nu (1)+1)} \! \left(\dfrac{\tilde{\mathfrak{c}}_{M}}{\lvert p_{1}(\tau)
\rvert^{2}} \! + \! \mathcal{O} \! \left(\dfrac{\tilde{\mathfrak{c}}_{M}}{\lvert
\widetilde{\Lambda} \rvert^{4 \Re (\nu (1)+1)}} \right) \right) \right),
\label{prcy78}
\end{gather}
where $\tilde{\mathfrak{c}}_{M}^{\sharp} \! := \! 2 \max_{m=0,1,2,3} \lbrace
(\tilde{\varrho}_{m}(1))^{2} \rbrace$, and $\tilde{\mathfrak{c}}_{M} \! := \! 2
\max_{m=0,1,2,3} \lbrace \tilde{\rho}_{m}^{2} \rbrace$, and, {}from the
Estimates~\eqref{prcy67}--\eqref{prcy70} and~\eqref{prcy76}, it follows that,
for $\arg (\widetilde{\Lambda}) \! = \! -\pi/2 \! + \! \mathcal{O}(\tau^{-2/3})$,
\begin{gather}
\lvert \lvert \Phi_{M,1}(\xi) \rvert \rvert^{2} \underset{\tau \to +\infty}{=}
\mathcal{O}(\hat{\mathfrak{c}}_{M}^{\sharp}) \! + \! \mathcal{O} \! \left(
\dfrac{\hat{\mathfrak{c}}_{M}^{\sharp} \lvert \nu (1) \! + \! 1 \rvert^{2}
\lvert \xi \rvert^{2}}{\lvert p_{1}(\tau) \rvert^{2}} \right), \label{prcy79} \\
\lvert \lvert \Phi_{M,1}(\widetilde{\Lambda}) \rvert \rvert^{2} \underset{\tau \to
+\infty}{=} \mathcal{O} \! \left(\lvert \widetilde{\Lambda} \rvert^{2 \Re (\nu
(1)+1)} \! \left(\dfrac{\hat{\mathfrak{c}}_{M}}{\lvert p_{1}(\tau) \rvert^{2}} \! + \!
\mathcal{O} \! \left(\dfrac{\hat{\mathfrak{c}}_{M}}{\lvert \widetilde{\Lambda}
\rvert^{2 \min \lbrace 1,2 \Re (\nu (1)+1) \rbrace}} \right) \right) \right),
\label{prcy80}
\end{gather}
where $\hat{\mathfrak{c}}_{M}^{\sharp} \! := \! 2 \max_{m=0,1,2,3} \lbrace
(\hat{\varrho}_{m}(1))^{2} \rbrace$, and $\hat{\mathfrak{c}}_{M} \! := \!
\max_{m=0,1,2,3} \lbrace \hat{\rho}_{m}^{2} \rbrace$. Assembling
the Asymptotics~\eqref{prcy77}--\eqref{prcy80} and invoking the
restriction~\eqref{pc4} on $\delta_{k}$ (for $k \! = \! +1)$, one deduces
{}from asymptotics~\eqref{prcy65} and~\eqref{estbet1} that, for $\arg
(\widetilde{\Lambda}) \! = \! \pm \pi/2 \! + \! \mathcal{O}(\tau^{-2/3})$,
\begin{equation} \label{prcy100}
\lvert \lvert \hat{\chi}_{k}(\widetilde{\Lambda}) \! - \! \mathrm{I} \rvert
\rvert \underset{\tau \to +\infty}{\leqslant} \mathcal{O} \! \left(\mathfrak{c}_{
k}^{\Ydown}(\tau) \lvert \nu (k) \! + \! 1 \rvert^{2} \lvert p_{k}(\tau)
\rvert^{-2} \tau^{-(\frac{1}{3}-2(3+ \Re (\nu (k)+1)) \delta_{k})} \right),
\quad k \! = \! +1,
\end{equation}
where, for $\arg (\widetilde{\Lambda}) \! = \! \pi/2 \! + \! \mathcal{O}
(\tau^{-2/3})$, $\mathfrak{c}_{1}^{\Ydown}(\tau) \! := \! \tilde{\mathfrak{
c}}_{M}^{\sharp} \tilde{\mathfrak{c}}_{M}(2^{3/2}3^{1/4} \mathrm{e}^{\pi \Im (\nu
(1)+1)/2})^{-2} \! =_{\tau \to +\infty} \! \mathcal{O}(1)$, and, for $\arg
(\widetilde{\Lambda}) \! = \! -\pi/2 \! + \! \mathcal{O}(\tau^{-2/3})$,
$\mathfrak{c}_{1}^{\Ydown}(\tau) \! := \! \hat{\mathfrak{c}}_{M}^{\sharp}
\hat{\mathfrak{c}}_{M}(2^{3/2}3^{1/4} \mathrm{e}^{\pi \Im (\nu (1)+1)/2})^{-2} \!
=_{\tau \to +\infty} \! \mathcal{O}(1)$ (see Remark~\ref{aspvals} below).
Via an analogous series of calculations, one arrives at a similar estimate
(cf. asymptotics~\eqref{prcy100}) for the case $k \! = \! -1$.
Forming the composition of the inverses of the linear transformations
$\mathfrak{F}_{j}$, $j \! = \! 1,2,\dotsc,11$, that is,
\begin{align}
\widetilde{\Psi}_{k}(\widetilde{\mu},\tau) \! :=& \, \left(\mathfrak{F}_{1}^{-1} \circ
\mathfrak{F}_{2}^{-1} \circ \mathfrak{F}_{3}^{-1} \circ \mathfrak{F}_{4}^{-1} \circ
\mathfrak{F}_{5}^{-1} \circ \mathfrak{F}_{6}^{-1} \circ \mathfrak{F}_{7}^{-1} \circ
\mathfrak{F}_{8}^{-1} \circ \mathfrak{F}_{9}^{-1} \circ \mathfrak{F}_{10}^{-1}
\circ \mathfrak{F}_{11}^{-1} \right) \! \Phi_{M,k}(\widetilde{\Lambda}) \nonumber \\
=& \, (b(\tau))^{-\frac{1}{2} \sigma_{3}} \mathcal{G}_{0,k} \mathcal{G}_{1,k}
\tau^{\frac{1}{6} \sigma_{3}}
\begin{pmatrix}
1 & 0 \\
-\mathrm{i} \omega_{0,k} & 1
\end{pmatrix} \!
\begin{pmatrix}
1 & \ell_{0,k}^{+} \widetilde{\Lambda} \\
0 & 1
\end{pmatrix} \!
\begin{pmatrix}
1 & 0 \\
\ell_{1,k}^{+} \widetilde{\Lambda} & 1
\end{pmatrix} \! \mathcal{G}_{2,k} \!
\begin{pmatrix}
1 & 0 \\
\ell_{2,k}^{+} \widetilde{\Lambda} & 1
\end{pmatrix} \nonumber \\
\times& \, \hat{\chi}_{k}(\widetilde{\Lambda}) \Phi_{M,k}(\widetilde{\Lambda}),
\quad k \! = \! \pm 1,
\label{prcy101}
\end{align}
one arrives at the asymptotic representation for $\widetilde{\Psi}_{k}(\widetilde{\mu},
\tau)$ given in Equation~\eqref{prpr2}. \hfill $\qed$
\begin{eeee} \label{delrestsn}
Heretofore, it was assumed that (cf. Corollaries~\ref{cor3.1.2}--\ref{cor3.1.5})
$0 \! < \! \delta \! < \! \delta_{k} \! < \! 1/9$, $k \! = \! \pm 1$; however, the
set of restrictions~\eqref{pc4} implies the following, more stringent restriction
on $\delta_{k}$:\footnote{Note: $18 \! <_{\tau \to +\infty} \! 6(3 \! + \! \Re
(\nu (k) \! + \! 1)) \! <_{\tau \to +\infty} \! 24$.}
\begin{equation} \label{restr1}
0 \underset{\tau \to +\infty}{<} \delta_{k} \underset{\tau \to +\infty}{<} 1/24,
\quad k \! = \! \pm 1.
\end{equation}
Since $(0,1/24) \subset (0,1/9)$, the latter restriction~\eqref{restr1} on $\delta_{k}$
implies, and is consistent with, the earlier one; henceforth, the restriction~\eqref{restr1}
on $\delta_{k}$ will be enforced. \hfill $\blacksquare$
\end{eeee}
\begin{eeee} \label{aspvals}
Using the fact that (see the Asymptotics~\eqref{geek13} below) $\nu (k) \! +
\! 1 \! \to \! 0$ as $\tau \to +\infty$, $k \! = \! \pm 1$, one shows, via the
expansion for the (Euler) gamma function \cite{a24}
\begin{equation*}
\dfrac{1}{\Gamma (z \! + \! 1)} \! = \! \sum_{j=0}^{\infty} \mathfrak{d}_{j}^{\ast}
z^{j}, \quad \lvert z \rvert \! < \! 1,
\end{equation*}
where $\mathfrak{d}_{0}^{\ast} \! = \! 1$ and $\mathfrak{d}_{n+1}^{\ast} \! = \!
(n \! + \! 1)^{-1} \sum_{j=0}^{n}(-1)^{j}s_{j+1} \mathfrak{d}_{n-j}^{\ast}$, $n \!
\in \! \mathbb{Z}_{+}$, with $s_{1} \! = \! -\psi (1)$ $(:= \! \left. \tfrac{\mathrm{d}}{\mathrm{d} x}
\ln \Gamma (x) \right\vert_{x=1})$ Euler's constant,\footnote{$-\psi (1) \! = \!
0.577215664901532860606512 \ldots$.} and $s_{m} \! = \! \zeta (m)$,
$\mathbb{N} \! \ni \! m \! \geqslant \! 2$, where $\zeta (z)$ is the Riemann Zeta
function, and well-known inequalities for complex-valued trigonometric functions,
that the auxiliary parameters introduced in step~\pmb{(xi)} of the proof of
Lemma~\ref{nprcl} have (for the case $k \! = \! +1)$ the following asymptotics:
\pmb{(1)} for $\arg (\widetilde{\Lambda}) \! = \! \pi/2 \! + \! \mathcal{O}
(\tau^{-2/3})$,
\begin{gather*}
(\tilde{\varrho}_{0}(1))^{2} \underset{\tau \to +\infty}{=} (2 \! + \! \vert \sec
\theta \rvert)^{2}(1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)), \\
(\tilde{\varrho}_{1}(1))^{2} \underset{\tau \to +\infty}{=} \dfrac{\pi}{2}(1 \! +
\! 2 \sec^{2} \theta)^{2}(1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)), \\
(\tilde{\varrho}_{2}(1))^{2} \underset{\tau \to +\infty}{=} 192(2 \sqrt{\pi} \! + \!
\vert \sec \theta \rvert)^{2}(1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)), \\
(\tilde{\varrho}_{3}(1))^{2} \underset{\tau \to +\infty}{=} 96 \pi (1 \! + \! 2
\sec^{2} \theta)^{2}(1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)), \\
\tilde{\rho}_{0}^{2} \underset{\tau \to +\infty}{=} 1 \! + \! \mathcal{O}
(\lvert \nu (1) \! + \! 1 \rvert), \quad \quad \tilde{\rho}_{1}^{2}
\underset{\tau \to +\infty}{=} 2 \pi \sec^{2}(\theta)(1 \! + \! \mathcal{O}
(\lvert \nu (1) \! + \! 1 \rvert)), \\
\tilde{\rho}_{2}^{2} \underset{\tau \to +\infty}{=} 16 \sqrt{3} \pi \lvert \nu
(1) \! + \! 1 \rvert^{2}(1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)),
\quad \quad \tilde{\rho}_{3}^{2} \underset{\tau \to +\infty}{=} 8 \sqrt{3}
(1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)),
\end{gather*}
where $\theta \! := \! \arg (\nu (1) \! + \! 1)$, whence $\tilde{\mathfrak{c}}_{M}^{
\sharp} \! := \! 2 \max_{m=0,1,2,3} \lbrace (\tilde{\varrho}_{m}(1))^{2} \rbrace
\! =_{\tau \to +\infty} \! \mathcal{O}(1)$ and $\tilde{\mathfrak{c}}_{M} \! := \!
2 \max_{m=0,1,2,3} \lbrace \tilde{\rho}_{m}^{2} \rbrace \! =_{\tau \to +\infty}
\! \mathcal{O}(1)$ $\Rightarrow$ $\mathfrak{c}_{1}^{\Ydown}(\tau) \! := \!
\tilde{\mathfrak{c}}_{M}^{\sharp} \tilde{\mathfrak{c}}_{M}(2^{3/2}3^{1/4}
\mathrm{e}^{\pi \Im (\nu (1)+1)/2})^{-2} \! =_{\tau \to +\infty} \! \mathcal{O}(1)$
(as claimed); and \pmb{(2)} for $\arg (\widetilde{\Lambda}) \! = \! -\pi/2 \!
+ \! \mathcal{O}(\tau^{-2/3})$,
\begin{gather*}
(\hat{\varrho}_{0}(1))^{2} \underset{\tau \to +\infty}{=} \sec^{2}(\theta)
(1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)), \quad \quad
(\hat{\varrho}_{1}(1))^{2} \underset{\tau \to +\infty}{=} \dfrac{\pi}{2}
(1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)), \\
(\hat{\varrho}_{2}(1))^{2} \underset{\tau \to +\infty}{=} 192 \sec^{2}
(\theta)(1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)), \quad
\quad (\hat{\varrho}_{3}(1))^{2} \underset{\tau \to +\infty}{=} 96
\pi (1 \! + \! \mathcal{O}(\lvert \nu (1) \! + \! 1 \rvert)), \\
\hat{\rho}_{0}^{2} \underset{\tau \to +\infty}{=} 1 \! + \! \mathcal{O}
(\lvert \nu (1) \! + \! 1 \rvert), \quad \quad \hat{\rho}_{1}^{2}
\underset{\tau \to +\infty}{=} \dfrac{1}{8 \sqrt{3}}(1 \! + \! \mathcal{O}
(\lvert \nu (1) \! + \! 1 \rvert)), \\
\hat{\rho}_{2}^{2} \underset{\tau \to +\infty}{=} \mathcal{O}
(\lvert \nu (1) \! + \! 1 \rvert^{2}), \quad \quad \hat{\rho}_{3}^{2}
\underset{\tau \to +\infty}{=} 8 \sqrt{3}(1 \! + \! \mathcal{O}
(\lvert \nu (1) \! + \! 1 \rvert)),
\end{gather*}
whence $\hat{\mathfrak{c}}_{M}^{\sharp} \! := \! 2 \max_{m=0,1,2,3}
\lbrace (\hat{\varrho}_{m}(1))^{2} \rbrace \! =_{\tau \to +\infty} \!
\mathcal{O}(1)$ and $\hat{\mathfrak{c}}_{M} \! := \! \max_{m=0,1,2,3}
\lbrace \hat{\rho}_{m}^{2} \rbrace \! =_{\tau \to +\infty} \! \mathcal{O}
(1)$ $\Rightarrow$ $\mathfrak{c}_{1}^{\Ydown}(\tau) \! := \!
\hat{\mathfrak{c}}_{M}^{\sharp} \hat{\mathfrak{c}}_{M}(2^{3/2}3^{1/4}
\mathrm{e}^{\pi \Im (\nu (1)+1)/2})^{-2} \! =_{\tau \to +\infty} \! \mathcal{O}(1)$
(as claimed). The case $k \! = \! -1$ is analogous. \hfill $\blacksquare$
\end{eeee}
\begin{eeee} \label{leedasyue}
In Lemma~\ref{nprcl} and hereafter, the function $\Phi_{M,k}(\pmb{\cdot})$ plays
a crucial r\^{o}le; therefore, its asymptotics are presented here: for $m \! \in \!
\lbrace -1,0,1,2 \rbrace$ and $k \! \in \! \lbrace \pm 1 \rbrace$,
\begin{equation*}
\Phi_{M,k}(z) \underset{\underset{\arg (z)=\frac{m \pi}{2}+\frac{\pi}{4}-
\frac{1}{2} \arg (\mu_{k}(\tau))}{\mathbb{C} \ni z \to \infty}}{=} \left(
\mathrm{I} \! + \! \sum_{j=1}^{\infty} \hat{\psi}_{j,k}(\tau)z^{-j} \right)
\! \mathrm{e}^{\left(\frac{1}{2} \mu_{k}(\tau)z^{2} - (\nu (k)+1) \ln ((2 \mu_{k}
(\tau))^{1/2}z) \right) \sigma_{3}} \mathcal{R}_{m}(k),
\end{equation*}
where
\begin{gather*}
\mathcal{R}_{-1}(k) \! := \!
\begin{pmatrix}
\mathrm{e}^{-\mathrm{i} \pi (\nu (k)+1)/2} & 0 \\
0 & -\frac{(2 \mu_{k}(\tau))^{1/2}}{p_{k}(\tau)}
\end{pmatrix}, \\
\mathcal{R}_{0}(k) \! := \!
\begin{pmatrix}
\mathrm{e}^{-\mathrm{i} \pi (\nu (k)+1)/2} & 0 \\
-\frac{\mathrm{i} \sqrt{2 \pi}(2 \mu_{k}(\tau))^{1/2} \mathrm{e}^{-\mathrm{i} \pi
(\nu (k)+1)/2}}{p_{k}(\tau) \Gamma (\nu (k)+1)} & -\frac{(2 \mu_{k}
(\tau))^{1/2}}{p_{k}(\tau)}
\end{pmatrix}, \\
\mathcal{R}_{1}(k) \! := \!
\begin{pmatrix}
\mathrm{e}^{\mathrm{i} 3 \pi (\nu (k)+1)/2} & \frac{\sqrt{2 \pi} \mathrm{e}^{\mathrm{i} \pi
(\nu (k)+1)}}{\Gamma (-\nu (k))} \\
-\frac{\mathrm{i} \sqrt{2 \pi}(2 \mu_{k}(\tau))^{1/2} \mathrm{e}^{-\mathrm{i} \pi
(\nu (k)+1)/2}}{p_{k}(\tau) \Gamma (\nu (k)+1)} & -\frac{(2 \mu_{k}
(\tau))^{1/2}}{p_{k}(\tau)}
\end{pmatrix}, \\
\mathcal{R}_{2}(k) \! := \!
\begin{pmatrix}
\mathrm{e}^{\mathrm{i} 3 \pi (\nu (k)+1)/2} & \frac{\sqrt{2 \pi} \mathrm{e}^{\mathrm{i} \pi
(\nu (k)+1)}}{\Gamma (-\nu (k))} \\
0 & -\frac{(2 \mu_{k}(\tau))^{1/2} \mathrm{e}^{-2 \pi \mathrm{i} (\nu (k)+1)}}{p_{k}(\tau)}
\end{pmatrix},
\end{gather*}
and $\hat{\psi}_{j,k}(\tau)$, $j \! \in \! \mathbb{N}$, are off-diagonal (resp., diagonal)
$\mathrm{M}_{2}(\mathbb{C})$-valued functions for $j$ odd (resp., $j$ even); e.g.,
\begin{gather*}
\hat{\psi}_{1,k}(\tau) \! = \! -\dfrac{1}{2 \mu_{k}(\tau)} \!
\begin{pmatrix}
0 & p_{k}(\tau) \\
-q_{k}(\tau) & 0
\end{pmatrix}, \, \, \qquad \, \, \hat{\psi}_{2,k}(\tau) \! = \!
\dfrac{(\nu (k) \! + \! 1)}{4 \mu_{k}(\tau)} \!
\begin{pmatrix}
1 \! + \! (\nu (k) \! + \! 1) & 0 \\
0 & 1 \! - \! (\nu (k) \! + \! 1)
\end{pmatrix}, \\
\hat{\psi}_{3,k}(\tau) \! = \! \dfrac{1}{8(\mu_{k}(\tau))^{2}} \!
\begin{pmatrix}
0 & (1 \! - \! (\nu (k) \! + \! 1))(2 \! - \! (\nu (k) \! + \! 1))p_{k}(\tau) \\
(1 \! + \! (\nu (k) \! + \! 1))(2 \! + \! (\nu (k) \! + \! 1))q_{k}(\tau) & 0
\end{pmatrix}.
\end{gather*}
These asymptotics can be deduced {}from the asymptotics of the parabolic-cylinder
functions \cite{EMOT}. \hfill $\blacksquare$
\end{eeee}
\subsection{Asymptotic Matching} \label{sec3.3}
In this subsection, the connection matrix is calculated asymptotically (as $\tau \! \to
\! +\infty$ with $\varepsilon b \! > \! 0)$ in terms of the matrix elements of the
function $\mathcal{A}(\widetilde{\mu},\tau)$ (cf. Equation~\eqref{eq3.4}) that are
defined via the set of functions $\hat{h}_{0}(\tau)$, $\tilde{r}_{0}(\tau)$, $v_{0}
(\tau)$,\footnote{Equivalently, the set of functions (cf. Equations~\eqref{iden2},
\eqref{iden3}, and~\eqref{iden4}, respectively) $h_{0}(\tau)$, $\hat{r}_{0}(\tau)$, and
$\hat{u}_{0}(\tau)$.} and $b(\tau)$ concomitant with the Conditions~\eqref{iden5}. Thus,
the direct monodromy problem for Equation~\eqref{eq3.3} is solved asymptotically.
\begin{cccc} \label{linfnewlemm}
Let $\widetilde{\Psi}_{k}(\widetilde{\mu},\tau)$, $k \! = \! \pm 1$, be the
fundamental solution of Equation~\eqref{eq3.3} with asymptotics given
in Lemma~\ref{nprcl}, and let $\mathbb{Y}^{\infty}_{0}(\widetilde{\mu},
\tau)$ be the canonical solution of Equation~\eqref{eq3.1}.\footnote{See
Proposition~\ref{prop1.4}.} Define {}\footnote{Since $\tau^{-\frac{1}{12}
\sigma_{3}} \mathbb{Y}^{\infty}_{0}(\tau^{-1/6} \widetilde{\mu},\tau)$
(cf. Equations~\eqref{eq3.2}) is also a fundamental solution of
Equation~\eqref{eq3.3}, it follows, therefore, that $\mathfrak{L}^{\infty}_{k}
(\tau)$ is independent of $\widetilde{\mu}$.}
\begin{equation} \label{ellinfk1}
\mathfrak{L}^{\infty}_{k}(\tau) \! := \! (\widetilde{\Psi}_{k}(\widetilde{\mu},
\tau))^{-1} \tau^{-\frac{1}{12} \sigma_{3}} \mathbb{Y}^{\infty}_{0}
(\tau^{-1/6} \widetilde{\mu},\tau), \quad k \! = \! \pm 1.
\end{equation}
Assume that the parameters $\nu (k) \! + \! 1$ and $\delta_{k}$ satisfy the
restrictions~\eqref{pc4} and~\eqref{restr1}, and, additionally, the following
conditions are valid:\footnote{The Conditions~\eqref{iden5} and~\eqref{restr1}
are consistent with the Conditions~\eqref{ellinfk2a} and~\eqref{ellinfk2b}.}
\begin{gather}
p_{k}(\tau) \mathfrak{B}_{k} \exp \! \left(-\mathrm{i} \tau^{2/3}3 \sqrt{3}(\varepsilon
b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3} \right) \underset{\tau \to +\infty}{=} \mathcal{O}
\! \left((\nu (k) \! + \! 1)^{\frac{1-k}{2}} \right), \label{ellinfk2a} \\
b(\tau) \tau^{\mathrm{i} a/3} \exp \! \left(\mathrm{i} \tau^{2/3}3(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} 2 \pi k/3} \right) \underset{\tau \to +\infty}{=} \mathcal{O}(1),
\label{ellinfk2b}
\end{gather}
where $p_{k}(\tau)$ and $\mathfrak{B}_{k}$ are defined in Lemma~\ref{nprcl}.\footnote{The
Conditions~\eqref{ellinfk2a} and~\eqref{ellinfk2b} will be validated \emph{a posteriori};
see, in particular, the proof of Lemma~\ref{ginversion} below, where it will be shown that
(cf. Definition~\eqref{prpr1}) $\nu (k) \! + \! 1 \! =_{\tau \to +\infty} \! \mathcal{O}
(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)})$, $k \! = \! \pm 1$, with $\vartheta (\tau)$ and
$\beta (\tau)$ defined in Equations~\eqref{thmk12}. Hereafter, whilst reading the text,
the reader should be cognizant of the latter asymptotics for $\nu (k) \! + \! 1$, as all
asymptotic expansions, estimates, orderings, etc., rely on this fact.} Then,
\begin{align} \label{ellinfk3}
\mathfrak{L}^{\infty}_{k}(\tau) \underset{\tau \to +\infty}{=}& \, \mathrm{i}
(\mathcal{R}_{m_{\infty}}(k))^{-1} \mathrm{e}^{\tilde{\mathfrak{z}}_{k}^{0}(\tau)
\sigma_{3}} \! \left(\dfrac{(\varepsilon b)^{1/4}(\sqrt{3} \! + \! 1)^{1/2}}{2^{1/4}
\sqrt{\smash[b]{\mathfrak{B}_{k}}} \sqrt{\smash[b]{b(\tau)}}} \right)^{\sigma_{3}}
\sigma_{2} \mathrm{e}^{-\Delta \tilde{\mathfrak{z}}_{k}(\tau) \sigma_{3}}
\begin{pmatrix}
\hat{\mathbb{B}}_{0}^{\infty}(\tau) & 0 \\
0 & \hat{\mathbb{A}}_{0}^{\infty}(\tau)
\end{pmatrix} \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle \mathcal{N}},
k}^{\infty}(\tau) \right) \! \left(\mathrm{I} \! + \! \mathcal{O} \! \left(
\mathbb{E}^{\infty}_{k}(\tau) \right) \right),
\end{align}
where $\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \mathcal{R}_{m_{\infty}}
(k)$, $m_{\infty} \! \in \! \lbrace -1,0,1,2 \rbrace$, are defined in
Remark~\ref{leedasyue},\footnote{The precise choice for the value
of $m_{\infty}$ is given in the proof of Theorem~\ref{theor3.3.1}
below.}
\begin{align}
\tilde{\mathfrak{z}}_{k}^{0}(\tau) \! :=& \, -\dfrac{\mathrm{i} a}{6} \ln \tau \! + \! \mathrm{i}
\tau^{2/3}3(\sqrt{3} \! - \! 1) \alpha_{k}^{2} \! + \! \mathrm{i} (a \! - \! \mathrm{i}/2) \ln
((\sqrt{3} \! + \! 1) \alpha_{k}/2), \label{ellinfk4} \\
\Delta \tilde{\mathfrak{z}}_{k}(\tau) \! :=& \, -\left(\dfrac{5 \! - \! \sqrt{3}}{6
\sqrt{3} \alpha_{k}^{2}} \right) \! \mathfrak{p}_{k}(\tau) \! + \! (\nu (k) \! + \! 1)
\ln (2 \mu_{k}(\tau))^{1/2} \! + \! \dfrac{1}{3}(\nu (k) \! + \! 1) \ln \tau
\nonumber \\
+& \, (\nu (k) \! + \! 1) \ln (6(\sqrt{3} \! + \! 1)^{-2} \alpha_{k}), \label{ellinfk5}
\end{align}
with $\mathfrak{p}_{k}(\tau)$ defined by Equation~\eqref{eqpeetee}, and
$\mu_{k}(\tau)$ defined in Lemma~\ref{nprcl},
\begin{gather}
\hat{\mathbb{A}}_{0}^{\infty}(\tau) \! := \! 1 \! + \! \dfrac{2^{1/4}(\Delta
G_{k}^{\infty}(\tau))_{12}}{(\varepsilon b)^{1/4}(\sqrt{3} \! + \! 1)^{1/2}},
\label{ellinfk6} \\
\hat{\mathbb{B}}_{0}^{\infty}(\tau) \! := \! 1 \! - \! \dfrac{(\varepsilon b)^{1/4}
(\sqrt{3} \! + \! 1)^{1/2}}{2^{1/4}} \! \left((\Delta G_{k}^{\infty}(\tau))_{21} \!
- \! \dfrac{\mathfrak{A}_{k}}{\mathfrak{B}_{k}} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k}}{\chi_{k}(\tau)} \! - \! 1 \right) \! (\Delta G_{k}^{\infty}(\tau))_{11}
\right), \label{ellinfk7}
\end{gather}
with $\mathcal{Z}_{k}$, $\mathfrak{A}_{k}$, and $\chi_{k}(\tau)$ defined in
Lemma~\ref{nprcl}, and
\begin{equation} \label{ellinfk8}
\Delta G_{k}^{\infty}(\tau) \! := \! \dfrac{1}{(2 \sqrt{3}(\sqrt{3} \! + \! 1))^{1/2}}
\begin{pmatrix}
(\Delta G_{k}^{\infty}(\tau))_{11} & (\Delta G_{k}^{\infty}(\tau))_{12} \\
(\Delta G_{k}^{\infty}(\tau))_{21} & (\Delta G_{k}^{\infty}(\tau))_{22}
\end{pmatrix},
\end{equation}
with
\begin{gather*}
(\Delta G_{k}^{\infty}(\tau))_{11} \! = \! (\sqrt{3} \! + \! 1)(\Delta \mathcal{G}_{0,k})_{22}
\! + \! (2/\varepsilon b)^{1/2}(\Delta \mathcal{G}_{0,k})_{12}, \\
(\Delta G_{k}^{\infty}(\tau))_{12} \! = \! -(\sqrt{3} \! + \! 1)(\Delta \mathcal{G}_{0,k})_{12}
\! + \! (2 \varepsilon b)^{1/2}(\Delta \mathcal{G}_{0,k})_{22}, \\
(\Delta G_{k}^{\infty}(\tau))_{21} \! = \! -(\sqrt{3} \! + \! 1)(\Delta \mathcal{G}_{0,k})_{21}
\! - \! (2/\varepsilon b)^{1/2}(\Delta \mathcal{G}_{0,k})_{11}, \\
(\Delta G_{k}^{\infty}(\tau))_{22} \! = \! (\sqrt{3} \! + \! 1)(\Delta \mathcal{G}_{0,k})_{11}
\! - \! (2 \varepsilon b)^{1/2}(\Delta \mathcal{G}_{0,k})_{21},
\end{gather*}
where $(\Delta \mathcal{G}_{0,k})_{i,j=1,2}$ are defined by
Equations~\eqref{prcyg4}--\eqref{prcyg6},
\begin{align} \label{ellinfk9}
\mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{\infty}(\tau) :=& \,
\left(\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{b(\tau)}}}
\right)^{\ad (\sigma_{3})} \! \left(-\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+}}{\chi_{k}(\tau)} \! \left(\dfrac{(\varepsilon b)^{1/2}(\sqrt{3} \! + \! 1)
(\nu (k) \! + \! 1)}{\sqrt{2}p_{k}(\tau) \mathfrak{B}_{k}} \sigma_{+} \right. \right.
\nonumber \\
+&\left. \left. \dfrac{p_{k}(\tau) \mathfrak{B}_{k}}{\sqrt{2}(\varepsilon b)^{1/2}
(\sqrt{3} \! + \! 1) \mu_{k}(\tau)} \sigma_{-} \right) \! \sigma_{3} \! + \! \dfrac{1}{2
\sqrt{3}(\sqrt{3} \! + \! 1)} \right. \nonumber \\
\times&\left.
\begin{pmatrix}
\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)} &
-\frac{(\varepsilon b)^{1/2}(\sqrt{3}+1)}{\sqrt{2} \mathfrak{B}_{k}}((\frac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k}}{\chi_{k}(\tau)})^{2} \ell_{0,k}^{+} \! - \! \ell_{1,k}^{+}
\! - \! \ell_{2,k}^{+}) \\
\frac{\sqrt{2} \mathfrak{B}_{k} \ell_{0,k}^{+}}{(\varepsilon b)^{1/2}(\sqrt{3}+1)}
& -\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)}
\end{pmatrix} \right. \nonumber \\
\times&\left.
\begin{pmatrix}
\sqrt{3} \! + \! 1 & -(2 \varepsilon b)^{1/2} \\
(2/\varepsilon b)^{1/2} & \sqrt{3} \! + \! 1
\end{pmatrix} \!
\begin{pmatrix}
\mathbb{T}_{11,k}(1;\tau) & \mathbb{T}_{12,k}(1;\tau) \\
\mathbb{T}_{21,k}(1;\tau) & \mathbb{T}_{22,k}(1;\tau)
\end{pmatrix} \right),
\end{align}
with $\ell_{0,k}^{+}$, $\ell_{1,k}^{+}$, and $\ell_{2,k}^{+}$ defined in Lemma~\ref{nprcl},
$(\mathbb{T}_{ij,k}(1;\tau))_{i,j=1,2}$ defined in Proposition~\ref{prop3.1.6}, and
$\tilde{\beta}_{k}(\tau)$ defined by Equation~\eqref{ellinfk18} below, and
\begin{equation} \label{ellinfk10}
\mathcal{O}(\mathbb{E}^{\infty}_{k}(\tau)) \underset{\tau \to +\infty}{:=}
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+3\delta_{k}}) & \mathcal{O}(\tau^{-\frac{1}{3}
(\frac{1+k}{2})-\delta_{k}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}(\frac{1-k}{2})-\delta_{k}}) & \mathcal{O}
(\tau^{-\frac{1}{3}+3\delta_{k}})
\end{pmatrix}.
\end{equation}
\end{cccc}
\emph{Proof}. Denote by $\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}
(\widetilde{\mu},\tau)$, $k \! = \! \pm 1$, the solution of Equation~\eqref{eq3.3}
that has leading-order asymptotics given by Equations~\eqref{eq3.16}--\eqref{eq3.18}
in the canonical domain containing the Stokes curve approaching, for $k \! = \! +1$
(resp., $k \! = \! -1)$, the positive real $\widetilde{\mu}$-axis {}from above
(resp., below) as $\widetilde{\mu} \! \to \! +\infty$. Let $\mathfrak{L}_{k}^{\infty}
(\tau)$, $k \! = \! \pm 1$, be defined by Equation~\eqref{ellinfk1}; re-write
$\mathfrak{L}_{k}^{\infty}(\tau)$ in the following, equivalent form:
\begin{equation} \label{ellinfk11}
\mathfrak{L}_{k}^{\infty}(\tau) \! = \! \left((\widetilde{\Psi}_{k}(\widetilde{\mu},
\tau))^{-1} \widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}(\widetilde{\mu},
\tau) \right) \! \left((\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}
(\widetilde{\mu},\tau))^{-1} \tau^{-\frac{1}{12} \sigma_{3}} \mathbb{Y}^{\infty}_{0}
(\tau^{-1/6} \widetilde{\mu},\tau) \right).
\end{equation}
Taking note of the fact that $\widetilde{\Psi}_{k}(\widetilde{\mu},\tau)$,
$\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}(\widetilde{\mu},\tau)$,
and $\tau^{-\frac{1}{12} \sigma_{3}} \mathbb{Y}^{\infty}_{0}(\tau^{-1/6}
\widetilde{\mu},\tau)$ are all solutions of Equation~\eqref{eq3.3}, it follows that
they differ on the right by non-degenerate, $\widetilde{\mu}$-independent,
$\mathrm{M}_{2}(\mathbb{C})$-valued factors: via this observation, one evaluates,
asymptotically, each of the factors appearing in Equation~\eqref{ellinfk11} by
considering separate limits, namely, $\widetilde{\mu} \! \to \! \alpha_{k}$ and
$\widetilde{\mu} \! \to \! +\infty$, respectively; more specifically, for $k \! =
\! \pm 1$,
\begin{align} \label{ellinfk12}
&(\widetilde{\Psi}_{k}(\widetilde{\mu},\tau))^{-1} \widetilde{\Psi}_{\scriptscriptstyle
\mathrm{WKB},k}(\widetilde{\mu},\tau) \underset{\tau \to +\infty}{:=} \nonumber \\
&\underbrace{\left((b(\tau))^{-\frac{1}{2} \sigma_{3}} \mathcal{G}_{0,k}
\mathfrak{B}_{k}^{\frac{1}{2} \sigma_{3}} \mathbb{F}_{k}(\tau) \Xi_{k}(\tau;
\widetilde{\Lambda}) \hat{\chi}_{k}(\widetilde{\Lambda}) \Phi_{M,k}
(\widetilde{\Lambda}) \right)^{-1}T(\widetilde{\mu})
\mathrm{e}^{\scriptscriptstyle \mathrm{W}_{k}(\widetilde{\mu},\tau)}}_{\widetilde{\mu}
=\widetilde{\mu}_{0,k}, \, \, \, \, \widetilde{\Lambda} \underset{\tau \to +\infty}{=}
\mathcal{O}(\tau^{\delta_{k}}), \, \, 0< \delta < \delta_{k}<\frac{1}{24}, \, \, \, \,
\arg (\widetilde{\Lambda})=\frac{\pi m_{\infty}}{2}+\frac{\pi}{4}-\frac{1}{2} \arg
(\mu_{k}(\tau)), \, \, m_{\infty} \in \lbrace -1,0,1,2 \rbrace},
\end{align}
where (cf. Lemma~\ref{nprcl})
\begin{gather}
\mathbb{F}_{k}(\tau) \! := \!
\begin{pmatrix}
1 & 0 \\
(\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}}{\chi_{k}(\tau)} \! - \! 1) \mathfrak{A}_{k} & 1
\end{pmatrix}, \label{ellinfk13} \\
\Xi_{k}(\tau;\widetilde{\Lambda}) \! := \! \mathrm{I} \! + \!
\gimel_{\scriptscriptstyle A,k}(\tau) \widetilde{\Lambda} \! + \!
\gimel_{\scriptscriptstyle B,k}(\tau) \widetilde{\Lambda}^{2}, \label{ellinfk14}
\end{gather}
and
\begin{equation} \label{ellinfk15}
\hat{\chi}_{k}(\widetilde{\Lambda}) \underset{\tau \to +\infty}{=} \mathrm{I}
\! + \! \mathcal{O} \! \left(\tilde{\mathfrak{C}}_{k}(\tau) \lvert \nu (k) \! + \!
1 \rvert^{2} \lvert p_{k}(\tau) \rvert^{-2}
\tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)} \right),
\end{equation}
with $\nu (k) \! + \! 1$, $p_{k}(\tau)$, $\widetilde{\mu}_{0,k}$,
$\mathcal{G}_{0,k}$, $\mathfrak{A}_{k}$, $\mathfrak{B}_{k}$, $\mathcal{Z}_{k}$,
$\gimel_{\scriptscriptstyle A,k}(\tau)$, $\gimel_{\scriptscriptstyle B,k}(\tau)$,
$\mu_{k}(\tau)$, and $\chi_{k}(\tau)$ defined in Lemma~\ref{nprcl}, $\mathrm{W}_{k}
(\widetilde{\mu},\tau) \! := \! -\sigma_{3} \mathrm{i} \tau^{2/3} \int_{\widetilde{\mu}_{0,k}}^{
\widetilde{\mu}}l_{k}(\xi) \, \mathrm{d} \xi \! - \! \int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}
\diag (T^{-1}(\xi) \partial_{\xi}T(\xi)) \, \mathrm{d} \xi$, $\epsilon_{\mathrm{\scriptscriptstyle TP}}
(k) \! := \! \tfrac{1}{3} \! - \! 2(3 \! + \! \Re (\nu (k) \! + \! 1)) \delta_{k}$ $(> \! 0)$, and
$\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \tilde{\mathfrak{C}}_{k}(\tau) \! =_{\tau \to +\infty}
\! \mathcal{O}(1)$, and
\begin{equation} \label{ellinfk16}
(\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}(\widetilde{\mu},\tau))^{-1}
\tau^{-\frac{1}{12} \sigma_{3}} \mathbb{Y}^{\infty}_{0}(\tau^{-1/6} \widetilde{\mu},
\tau) \underset{\tau \to +\infty}{:=} \lim_{\underset{\arg (\widetilde{\mu})=0}{
\Omega_{0}^{\infty} \ni \widetilde{\mu} \to \infty}} \left((T(\widetilde{\mu})
\mathrm{e}^{\scriptscriptstyle \mathrm{W}_{k}(\widetilde{\mu},\tau)})^{-1} \tau^{-
\frac{1}{12} \sigma_{3}} \mathbb{Y}^{\infty}_{0}(\tau^{-1/6} \widetilde{\mu},
\tau) \right).
\end{equation}
One commences by considering the asymptotics subsumed in the
Definition~\eqref{ellinfk16}. {}From the asymptotics for $\mathbb{Y}^{\infty}_{0}
(\tau^{-1/6} \widetilde{\mu},\tau)$ stated in Proposition~\ref{prop1.4},
Equations~\eqref{iden3}, \eqref{iden4}, \eqref{expforeych}, \eqref{expforkapp},
\eqref{eq3.37}, \eqref{eq3.38}, \eqref{eqpeetee}, \eqref{eq3.41},
\eqref{asympforf7}, \eqref{eq3.51}, \eqref{prcy22}, and~\eqref{prcyomg1}, one
arrives at, via the Conditions~\eqref{iden5} and the Asymptotics~\eqref{terrbos6}
and \eqref{asympforf3},
\begin{equation} \label{ellinfk17}
\lim_{\underset{\arg (\widetilde{\mu})=0}{\Omega_{0}^{\infty} \ni \widetilde{\mu}
\to \infty}} \left((T(\widetilde{\mu}) \mathrm{e}^{\scriptscriptstyle \mathrm{W}_{k}
(\widetilde{\mu},\tau)})^{-1} \tau^{-\frac{1}{12} \sigma_{3}} \mathbb{Y}^{\infty}_{0}
(\tau^{-1/6} \widetilde{\mu},\tau) \right) \underset{\tau \to +\infty}{=}
\exp (\tilde{\beta}_{k}(\tau) \sigma_{3}), \quad k \! = \! \pm 1,
\end{equation}
where
\begin{align} \label{ellinfk18}
\tilde{\beta}_{k}(\tau) :=& \, \dfrac{\mathrm{i} a}{6} \ln \tau \! - \! \mathrm{i} \tau^{2/3}
3(\sqrt{3} \! - \! 1) \alpha_{k}^{2} \! - \! \mathrm{i} 2 \sqrt{3} \, \widetilde{\Lambda}^{2}
\! - \! \mathrm{i} (a \! - \! \mathrm{i}/2) \ln ((\sqrt{3} \! + \! 1) \alpha_{k}/2) \! + \!
\dfrac{(5 \! - \! \sqrt{3}) \mathfrak{p}_{k}(\tau)}{6 \sqrt{3} \alpha_{k}^{2}}
\nonumber \\
+& \, \left(\dfrac{\mathrm{i}}{2 \sqrt{3}} \! \left((a \! - \! \mathrm{i}/2) \! + \! \alpha_{k}^{-2}
\tau^{2/3} \hat{h}_{0}(\tau) \right) \! + \! \dfrac{2 \mathfrak{p}_{k}(\tau)}{3
\sqrt{3} \alpha_{k}^{2}} \right) \! \left(\dfrac{1}{3} \ln \tau \! - \! \ln
\widetilde{\Lambda} \! + \! \ln \left(\dfrac{6 \alpha_{k}}{(\sqrt{3} \! + \! 1)^{2}}
\right) \right) \nonumber \\
-& \, \dfrac{(\sqrt{3} \! - \! 1) \mathfrak{p}_{k}(\tau)}{\sqrt{3} \alpha_{k} \tau^{-1/3}
\widetilde{\Lambda}} \! + \! \mathcal{O} \! \left(\left(\dfrac{\mathfrak{c}_{1,k}
\tau^{-1/3} \! + \! \mathfrak{c}_{2,k} \tilde{r}_{0}(\tau)}{\widetilde{\Lambda}^{2}}
\right) \! \left(\mathfrak{c}_{3,k} \tau^{-1/3} \! + \! \mathfrak{c}_{4,k}
(\tilde{r}_{0}(\tau) \! + \! 4v_{0}(\tau)) \right) \right) \nonumber \\
+& \, \mathcal{O}(\tau^{-1/3} \widetilde{\Lambda}^{3}) \! + \!
\mathcal{O}(\tau^{-1/3} \widetilde{\Lambda}) \! + \! \mathcal{O} \! \left(
\dfrac{\tau^{-1/3}}{\widetilde{\Lambda}} \left(\mathfrak{c}_{5,k} \! + \!
\mathfrak{c}_{6,k} \tau^{2/3} \hat{h}_{0}(\tau) \! + \! \mathfrak{c}_{7,k}
(\tau^{2/3} \hat{h}_{0}(\tau))^{2} \right) \right),
\end{align}
and $\mathfrak{c}_{m,k}$, $m \! = \! 1,2,\dotsc,7$, are constants.
One now derives the asymptotics defined by Equation~\eqref{ellinfk12}.
{}From Asymptotics~\eqref{iden55} for $\varpi \! = \! +1$,
Equation~\eqref{prcy2} for $\Phi_{M,k}(\widetilde{\Lambda})$ (in
conjunction with its large-$\widetilde{\Lambda}$ asymptotics stated
in Remark~\ref{leedasyue}), the Definitions \eqref{ellinfk13}
and~\eqref{ellinfk14} (concomitant with the fact that $\det (\Xi_{k}(\tau;
\widetilde{\Lambda})) \! = \! 1)$, and the Asymptotics~\eqref{ellinfk15},
one shows, via the relation $(\mathrm{W}_{k}(\widetilde{\mu}_{0,k},
\tau))_{i,j=1,2} \! = \! 0$ and Definition~\eqref{ellinfk12}, that, for
$k \! = \! \pm 1$,
{\fontsize{10pt}{11pt}\selectfont
\begin{align} \label{ellinfk19}
(\widetilde{\Psi}_{k}(\widetilde{\mu},\tau))^{-1} \widetilde{\Psi}_{\scriptscriptstyle
\mathrm{WKB},k}(\widetilde{\mu},\tau) \underset{\tau \to +\infty}{:=}& \,
\Phi_{M,k}^{-1}(\widetilde{\Lambda}) \hat{\chi}_{k}^{-1}(\widetilde{\Lambda})
\Xi_{k}^{-1}(\tau;\widetilde{\Lambda}) \mathbb{F}_{k}^{-1}(\tau) \mathfrak{B}_{k}^{
-\frac{1}{2} \sigma_{3}} \mathcal{G}_{0,k}^{-1}(b(\tau))^{\frac{1}{2} \sigma_{3}}
T(\widetilde{\mu}_{0,k}) \nonumber \\
\underset{\tau \to +\infty}{=}& \, (\mathcal{R}_{m_{\infty}}(k))^{-1}
\mathrm{e}^{-\mathcal{P}_{0}^{\ast} \sigma_{3}} \mathfrak{Q}_{\infty,k}(\tau) \! \left(
\mathrm{I} \! + \! \dfrac{1}{\widetilde{\Lambda}} \mathfrak{Q}_{\infty,k}^{-1}
(\tau) \hat{\psi}_{1,k}^{-1}(\tau) \mathfrak{Q}_{\infty,k}(\tau) \right. \nonumber \\
+&\left. \, \dfrac{1}{\widetilde{\Lambda}^{2}} \mathfrak{Q}_{\infty,k}^{-1}
(\tau) \hat{\psi}_{2,k}^{-1}(\tau) \mathfrak{Q}_{\infty,k}(\tau) \! + \! \mathcal{O}
\! \left(\dfrac{1}{\widetilde{\Lambda}^{3}} \mathfrak{Q}_{\infty,k}^{-1}(\tau)
\hat{\psi}_{3,k}^{-1}(\tau) \mathfrak{Q}_{\infty,k}(\tau) \right) \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \mathcal{O} \! \left(\lvert \nu (k) \! + \! 1 \rvert^{2}
\lvert p_{k}(\tau) \rvert^{-2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}
\mathfrak{Q}_{\infty,k}^{-1}(\tau) \tilde{\mathfrak{C}}_{k}(\tau) \mathfrak{Q}_{\infty,k}
(\tau) \right) \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \widetilde{\Lambda} \mathfrak{Q}_{\infty,k}^{-1}
(\tau) \gimel_{\scriptscriptstyle A,k}^{-1}(\tau) \mathfrak{Q}_{\infty,k}(\tau) \! + \!
\widetilde{\Lambda}^{2} \mathfrak{Q}_{\infty,k}^{-1}(\tau)
\gimel_{\scriptscriptstyle B,k}^{-1}(\tau) \mathfrak{Q}_{\infty,k}(\tau) \right)
\nonumber \\
\times& \, \left(\mathrm{I} \! + \! \widetilde{\Lambda} \tau^{-1/3}
\mathbb{P}_{\infty,k}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}}
\widehat{\mathbb{E}}_{\infty,k}(\tau) \! + \! \mathcal{O} \! \left((\tau^{-1/3}
\widetilde{\Lambda})^{2} \widetilde{\mathbb{E}}_{\infty,k}(\tau) \right) \right),
\end{align}}
where $\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \mathcal{R}_{m_{\infty}}(k)$, $m_{\infty}
\! \in \! \lbrace -1,0,1,2 \rbrace$, are defined in Remark~\ref{leedasyue},
\begin{gather}
\mathcal{P}_{0}^{\ast} \! := \! \dfrac{1}{2} \mu_{k}(\tau) \widetilde{\Lambda}^{2}
\! - \! (\nu (k) \! + \! 1) \ln \widetilde{\Lambda} \! - \! (\nu (k) \! + \! 1)
\ln (2 \mu_{k}(\tau))^{1/2}, \label{ellinfk20} \\
\mathfrak{Q}_{\infty,k}(\tau) \! := \! \mathbb{F}_{k}^{-1}(\tau) \! \left(
\left(\dfrac{(\varepsilon b)^{1/4}(\sqrt{3} \! + \! 1)^{1/2}}{2^{1/4}
\sqrt{\smash[b]{\mathfrak{B}_{k}}} \sqrt{\smash[b]{b(\tau)}}} \right)^{\sigma_{3}}
\mathrm{i} \sigma_{2} \! + \! \mathfrak{B}_{k}^{-\frac{1}{2} \sigma_{3}} \Delta G_{k}^{\infty}
(\tau)(b(\tau))^{\frac{1}{2} \sigma_{3}} \right), \label{ellinfk21}
\end{gather}
with $\Delta G_{k}^{\infty}(\tau)$ defined by Equation~\eqref{ellinfk8},
\begin{align}
\hat{\psi}_{1,k}^{-1}(\tau) \! :=& \, \dfrac{1}{2 \mu_{k}(\tau)} \!
\begin{pmatrix}
0 & p_{k}(\tau) \\
-q_{k}(\tau) & 0
\end{pmatrix}, \label{ellinfk22} \\
\hat{\psi}_{2,k}^{-1}(\tau) \! :=& \, \dfrac{(\nu (k) \! + \! 1)}{4 \mu_{k}(\tau)} \!
\begin{pmatrix}
1 \! - \! (\nu (k) \! + \! 1) & 0 \\
0 & 1 \! + \! (\nu (k) \! + \! 1)
\end{pmatrix}, \label{ellinfk23} \\
\hat{\psi}_{3,k}^{-1}(\tau) \! :=& \, -\dfrac{1}{8(\mu_{k}(\tau))^{2}} \!
\begin{pmatrix}
0 & (1 \! - \! (\nu (k) \! + \! 1))(2 \! - \! (\nu (k) \! + \! 1))p_{k}(\tau) \\
(1 \! + \! (\nu (k) \! + \! 1))(2 \! + \! (\nu (k) \! + \! 1))q_{k}(\tau) & 0
\end{pmatrix}, \label{ellinfk24} \\
\mathbb{P}_{\infty,k}(\tau) \! :=& \, (b(\tau))^{-\frac{1}{2} \ad (\sigma_{3})} \!
\begin{pmatrix}
0 & -\frac{(\varepsilon b)^{1/2}}{3 \sqrt{2} \alpha_{k}} \\
\frac{(\varepsilon b)^{-1/2}}{3 \sqrt{2} \alpha_{k}} & 0
\end{pmatrix}, \label{ellinfk25} \\
\widehat{\mathbb{E}}_{\infty,k}(\tau) \! :=& \, \dfrac{1}{2 \sqrt{3}(\sqrt{3}
\! + \! 1)}(b(\tau))^{-\frac{1}{2} \ad (\sigma_{3})} \!
\begin{pmatrix}
\sqrt{3} \! + \! 1 & -(2 \varepsilon b)^{1/2} \\
(2/\varepsilon b)^{1/2} & \sqrt{3} \! + \! 1
\end{pmatrix} \!
\begin{pmatrix}
\mathbb{T}_{11,k}(1;\tau) & \mathbb{T}_{12,k}(1;\tau) \\
\mathbb{T}_{21,k}(1;\tau) & \mathbb{T}_{22,k}(1;\tau)
\end{pmatrix}, \label{ellinfk26} \\
\widetilde{\mathbb{E}}_{\infty,k}(\tau) \! :=& \, \dfrac{1}{2 \sqrt{3}
(\sqrt{3} \! + \! 1)}(b(\tau))^{-\frac{1}{2} \ad (\sigma_{3})} \!
\begin{pmatrix}
\sqrt{3} \! + \! 1 & -(2 \varepsilon b)^{1/2} \\
(2/\varepsilon b)^{1/2} & \sqrt{3} \! + \! 1
\end{pmatrix} \! \tilde{\mathfrak{C}}_{k}^{\lozenge}, \label{ellinfk27}
\end{align}
$\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \tilde{\mathfrak{C}}_{k}(\tau) \!
=_{\tau \to +\infty} \! \mathcal{O}(1)$, $(\mathbb{T}_{ij,k}(1;\tau))_{i,j=1,2}$
defined in Proposition~\ref{prop3.1.6}, and $\mathrm{M}_{2}(\mathbb{C})
\! \ni \! \tilde{\mathfrak{C}}_{k}^{\lozenge}$ a constant.
Recalling the Definitions~\eqref{ellinfk12} and~\eqref{ellinfk16}, and
substituting the Expansions \eqref{ellinfk17}, \eqref{ellinfk18},
and \eqref{ellinfk19} into Equation~\eqref{ellinfk11}, one shows,
via the Conditions~\eqref{iden5}, the Definition \eqref{prpr1}, the
restrictions~\eqref{pc4}, the Asymptotics~\eqref{prcychik1},
\eqref{prcyzeek1}, \eqref{prcymuk1}, and (cf. step~\pmb{(xi)} in the
proof of Lemma~\ref{nprcl}) $\arg (\mu_{k}(\tau)) \! =_{\tau \to +\infty}
\! \tfrac{\pi}{2}(1 \! + \! \mathcal{O}(\tau^{-2/3}))$, and the
restriction~\eqref{restr1}, that
\begin{align} \label{ellinfk28}
\mathfrak{L}^{\infty}_{k}(\tau) \underset{\tau \to +\infty}{=}& \, \mathrm{i}
(\mathcal{R}_{m_{\infty}}(k))^{-1} \mathrm{e}^{\tilde{\mathfrak{z}}_{k}^{0}(\tau)
\sigma_{3}} \! \left(\dfrac{(\varepsilon b)^{1/4}(\sqrt{3} \! + \! 1)^{1/2}}{
2^{1/4} \sqrt{\smash[b]{\mathfrak{B}_{k}}} \sqrt{\smash[b]{b(\tau)}}}
\right)^{\sigma_{3}} \sigma_{2} \mathrm{e}^{-\Delta \tilde{\mathfrak{z}}_{k}
(\tau) \sigma_{3}} \nonumber \\
\times& \, \operatorname{diag}(\hat{\mathbb{B}}_{0}^{\infty}(\tau),
\hat{\mathbb{A}}_{0}^{\infty}(\tau)) \overset{\Yup}{\mathbb{E}}_{\scriptscriptstyle
\mathfrak{L}^{\infty}_{k}}^{\raise-6.75pt\hbox{$\scriptstyle \leftslice$}}
(\tau), \quad k \! = \! \pm 1,
\end{align}
where $\tilde{\mathfrak{z}}_{k}^{0}(\tau)$, $\Delta \tilde{\mathfrak{z}}_{k}(\tau)$,
$\hat{\mathbb{A}}_{0}^{\infty}(\tau)$, and $\hat{\mathbb{B}}_{0}^{\infty}(\tau)$
are defined by Equations~\eqref{ellinfk4}--\eqref{ellinfk7}, respectively, and
\begin{align} \label{ellinfk29}
\overset{\Yup}{\mathbb{E}}_{\scriptscriptstyle
\mathfrak{L}^{\infty}_{k}}^{\raise-6.75pt\hbox{$\scriptstyle \leftslice$}}(\tau)
\underset{\tau \to +\infty}{:=}& \, \left(\mathrm{I} \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}^{3} \sigma_{3}) \right) \! \left(\mathrm{I}
\! + \! \left(\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{b(\tau)}}}
\right)^{\ad (\sigma_{3})} \right. \nonumber \\
\times&\left. \,
\begin{pmatrix}
0 & -\frac{(\varepsilon b)^{1/4}(\sqrt{3}+1)^{1/2} \hat{\mathbb{D}}_{0}^{\infty}
(\tau)}{2^{1/4} \hat{\mathbb{B}}_{0}^{\infty}(\tau)} \\
\frac{2^{1/4} \hat{\mathbb{C}}_{0}^{\infty}(\tau)}{(\varepsilon b)^{1/4}
(\sqrt{3}+1)^{1/2} \hat{\mathbb{A}}_{0}^{\infty}(\tau)} & 0
\end{pmatrix} \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \dfrac{1}{\widetilde{\Lambda}}
\hat{\psi}_{1,k}^{-1,\sharp}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}^{2}}
\hat{\psi}_{2,k}^{-1,\sharp}(\tau) \! + \! \mathcal{O} \! \left(\dfrac{1}{
\widetilde{\Lambda}^{3}} \hat{\psi}_{3,k}^{-1,\sharp}(\tau) \right) \right)
\nonumber \\
\times& \, \left(\mathrm{I} \! + \! \mathcal{O} \! \left(\dfrac{\lvert \nu (k) \! + \! 1
\rvert^{2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert p_{k}(\tau)
\rvert^{2}} \mathrm{e}^{-\tilde{\beta}_{k}(\tau) \ad (\sigma_{3})} \mathfrak{Q}_{\infty,k}^{-1}
(\tau) \tilde{\mathfrak{C}}_{k}(\tau) \mathfrak{Q}_{\infty,k}(\tau) \right) \right)
\nonumber \\
\times& \, \left(\mathrm{I} \! + \!
\widetilde{\Lambda} \gimel_{\scriptscriptstyle A,k}^{\sharp}(\tau) \! + \!
\widetilde{\Lambda}^{2} \gimel_{\scriptscriptstyle B,k}^{\sharp}(\tau) \right)
\! \left(\mathrm{I} \! + \! \widetilde{\Lambda} \tau^{-1/3} \mathbb{P}_{\infty,
k}^{\sharp}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}} \widehat{\mathbb{E}}_{
\infty,k}^{\sharp}(\tau) \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left((\tau^{-1/3} \widetilde{\Lambda})^{2}
\widetilde{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \right) \right),
\end{align}
where
\begin{gather}
\hat{\mathbb{C}}_{0}^{\infty}(\tau) \! := \! (\Delta G_{k}^{\infty}(\tau))_{11},
\label{ellinfk30} \\
\hat{\mathbb{D}}_{0}^{\infty}(\tau) \! := \! (\Delta G_{k}^{\infty}(\tau))_{22}
\! - \! \dfrac{\mathfrak{A}_{k}}{\mathfrak{B}_{k}} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k}}{\chi_{k}(\tau)} \! - \! 1 \right) \! \left(\dfrac{(\varepsilon b)^{1/4}
(\sqrt{3} \! + \! 1)^{1/2}}{2^{1/4}} \! + \! (\Delta G_{k}^{\infty}(\tau))_{12}
\right), \label{ellinfk31} \\
\hat{\psi}_{m,k}^{-1,\sharp}(\tau) \! := \! \mathrm{e}^{-\tilde{\beta}_{k}(\tau) \ad
(\sigma_{3})} \mathfrak{Q}_{\infty,k}^{-1}(\tau) \hat{\psi}_{m,k}^{-1}(\tau)
\mathfrak{Q}_{\infty,k}(\tau), \quad m \! = \! 1,2,3, \label{ellinfk32} \\
\gimel_{\scriptscriptstyle A,k}^{\sharp}(\tau) \! := \! \mathrm{e}^{-\tilde{\beta}_{k}
(\tau) \ad (\sigma_{3})} \mathfrak{Q}_{\infty,k}^{-1}(\tau)
\gimel_{\scriptscriptstyle A,k}^{-1}(\tau) \mathfrak{Q}_{\infty,k}(\tau),
\label{ellinfk33} \\
\gimel_{\scriptscriptstyle B,k}^{\sharp}(\tau) \! := \! \mathrm{e}^{-\tilde{\beta}_{k}
(\tau) \ad (\sigma_{3})} \mathfrak{Q}_{\infty,k}^{-1}(\tau)
\gimel_{\scriptscriptstyle B,k}^{-1}(\tau) \mathfrak{Q}_{\infty,k}(\tau),
\label{ellinfk34} \\
\mathbb{P}_{\infty,k}^{\sharp}(\tau) \! := \! \mathrm{e}^{-\tilde{\beta}_{k}(\tau)
\ad (\sigma_{3})} \mathbb{P}_{\infty,k}(\tau), \label{ellinfk35} \\
\widehat{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \! := \! \mathrm{e}^{-\tilde{\beta}_{k}
(\tau) \ad (\sigma_{3})} \widehat{\mathbb{E}}_{\infty,k}(\tau), \label{ellinfk36} \\
\widetilde{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \! := \! \mathrm{e}^{-\tilde{\beta}_{k}
(\tau) \ad (\sigma_{3})} \widetilde{\mathbb{E}}_{\infty,k}(\tau). \label{ellinfk37}
\end{gather}
Via the Conditions~\eqref{iden5}, the restrictions~\eqref{pc4} and~\eqref{restr1},
the Definitions~\eqref{eqpeetee}, \eqref{peekayity}, \eqref{prpr1}, \eqref{prpr3},
\eqref{prpr4}, \eqref{prcy57}, \eqref{prcy58}, \eqref{ellinfk6}--\eqref{ellinfk8},
\eqref{ellinfk13}, \eqref{ellinfk21}--\eqref{ellinfk27}, and
\eqref{ellinfk30}--\eqref{ellinfk37}, and the Asymptotics \eqref{tr1}, \eqref{tr3},
\eqref{asympforf3}, \eqref{prcyg4}--\eqref{prcybk1}, \eqref{prcyomg1},
\eqref{prcyellok1}, \eqref{prcychik1}--\eqref{prcyell1k1}, \eqref{prcymuk1},
\eqref{prcyell2k1}, and~\eqref{ellinfk18}, upon imposing the
Conditions~\eqref{ellinfk2a} and~\eqref{ellinfk2b}, and defining
\begin{gather*}
J_{k}^{\infty} \! := \!
\begin{pmatrix}
\sqrt{3} \! + \! 1 & -(2 \varepsilon b)^{1/2} \\
(2/\varepsilon b)^{1/2} & \sqrt{3} \! + \! 1
\end{pmatrix}, \, \, \, \qquad \, \, \, \pmb{\mathbb{T}}_{\infty,k}^{\sharp} \!
:= \! (\mathbb{T}_{ij,k}(1;\tau))_{i,j=1,2}, \\
\mathbb{D}_{\infty,k}^{\sharp} \! := \! \mathfrak{B}_{k}^{-\frac{1}{2} \sigma_{3}} \!
\begin{pmatrix}
0 & -\frac{(\varepsilon b)^{1/4}(\sqrt{3}+1)^{1/2}}{2^{1/4}} \\
\frac{2^{1/4}}{(\varepsilon b)^{1/4}(\sqrt{3}+1)^{1/2}} & 0
\end{pmatrix},
\end{gather*}
one shows that (cf. Definition~\eqref{ellinfk29}), for $k \! = \! \pm 1$,
\begin{align*}
\overset{\Yup}{\mathbb{E}}_{\scriptscriptstyle
\mathfrak{L}^{\infty}_{k}}^{\raise-6.75pt\hbox{$\scriptstyle \leftslice$}}(\tau)
\underset{\tau \to +\infty}{=}& \, \left(\mathrm{I} \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}^{3} \sigma_{3}) \right) \! \left(\mathrm{I}
\! + \! \left(\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{b(\tau)}}}
\right)^{\ad (\sigma_{3})} \right. \nonumber \\
\times&\left. \,
\begin{pmatrix}
0 & -\frac{(\varepsilon b)^{1/4}(\sqrt{3}+1)^{1/2} \hat{\mathbb{D}}_{0}^{\infty}
(\tau)}{2^{1/4} \hat{\mathbb{B}}_{0}^{\infty}(\tau)} \\
\frac{2^{1/4} \hat{\mathbb{C}}_{0}^{\infty}(\tau)}{(\varepsilon b)^{1/4}
(\sqrt{3}+1)^{1/2} \hat{\mathbb{A}}_{0}^{\infty}(\tau)} & 0
\end{pmatrix} \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \dfrac{1}{\widetilde{\Lambda}}
\hat{\psi}_{1,k}^{-1,\sharp}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}^{2}}
\hat{\psi}_{2,k}^{-1,\sharp}(\tau) \! + \! \mathcal{O} \! \left(\dfrac{1}{\widetilde{
\Lambda}^{3}} \hat{\psi}_{3,k}^{-1,\sharp}(\tau) \right) \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \mathcal{O} \! \left(\dfrac{\lvert \nu (k) \! + \!
1 \rvert^{2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert p_{k}
(\tau) \rvert^{2}} \mathrm{e}^{-\tilde{\beta}_{k}(\tau) \ad (\sigma_{3})}
\mathfrak{Q}_{\infty,k}^{-1}(\tau) \tilde{\mathfrak{C}}_{k}(\tau)
\mathfrak{Q}_{\infty,k}(\tau) \right) \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \gimel_{\scriptscriptstyle A,k}^{\sharp}(\tau)
\widehat{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}}
\widehat{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \! + \! \widetilde{\Lambda} \!
\left(\tau^{-1/3} \mathbb{P}_{\infty,k}^{\sharp}(\tau) \! + \!
\gimel_{\scriptscriptstyle A,k}^{\sharp}(\tau) \right. \right. \nonumber \\
+&\left. \left. \, \gimel_{\scriptscriptstyle B,k}^{\sharp}(\tau) \widehat{\mathbb{E}}_{
\infty,k}^{\sharp}(\tau) \right) \! + \! \widetilde{\Lambda}^{2} \! \left(\tau^{-1/3}
\gimel_{\scriptscriptstyle A,k}^{\sharp}(\tau) \mathbb{P}_{\infty,k}^{\sharp}(\tau)
\! + \! \gimel_{\scriptscriptstyle B,k}^{\sharp}(\tau) \! + \! \mathcal{O} \! \left(
\tau^{-2/3} \widetilde{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \right) \right)
\right. \nonumber \\
+&\left. \, \widetilde{\Lambda}^{3} \! \left(\tau^{-1/3}
\gimel_{\scriptscriptstyle B,k}^{\sharp}(\tau) \mathbb{P}_{\infty,k}^{\sharp}(\tau)
\! + \! \mathcal{O} \! \left(\tau^{-2/3} \gimel_{\scriptscriptstyle A,k}^{\sharp}(\tau)
\widetilde{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \right) \right) \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \left(\mathrm{I} \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}^{3} \sigma_{3}) \right) \! \left(\mathrm{I}
\! + \! \left(\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{b(\tau)}}}
\right)^{\ad (\sigma_{3})} \right. \nonumber \\
\times&\left. \,
\begin{pmatrix}
0 & -\frac{(\varepsilon b)^{1/4}(\sqrt{3}+1)^{1/2} \hat{\mathbb{D}}_{0}^{\infty}
(\tau)}{2^{1/4} \hat{\mathbb{B}}_{0}^{\infty}(\tau)} \\
\frac{2^{1/4} \hat{\mathbb{C}}_{0}^{\infty}(\tau)}{(\varepsilon b)^{1/4}
(\sqrt{3}+1)^{1/2} \hat{\mathbb{A}}_{0}^{\infty}(\tau)} & 0
\end{pmatrix} \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \dfrac{1}{\widetilde{\Lambda}}
\hat{\psi}_{1,k}^{-1,\sharp}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}^{2}}
\hat{\psi}_{2,k}^{-1,\sharp}(\tau) \! + \! \mathcal{O} \! \left(\dfrac{1}{\widetilde{
\Lambda}^{3}} \hat{\psi}_{3,k}^{-1,\sharp}(\tau) \right) \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \gimel_{\scriptscriptstyle A,k}^{\sharp}(\tau)
\widehat{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}}
\dfrac{1}{2 \sqrt{3}(\sqrt{3} \! + \! 1)} \! \left(\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}(\tau)}}{
\sqrt{\smash[b]{b(\tau)}}} \right)^{\ad (\sigma_{3})} J_{k}^{\infty}
\pmb{\mathbb{T}}_{\infty,k}^{\sharp} \right. \nonumber \\
+&\left. \, \widetilde{\Lambda} \dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+}}{\chi_{k}(\tau)} \sigma_{3} \! + \! \mathcal{O} \! \left(\dfrac{\lvert \nu
(k) \! + \! 1 \rvert^{2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert
p_{k}(\tau) \rvert^{2}} \! \left(\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{
b(\tau)}}} \right)^{\ad (\sigma_{3})} \right. \right. \nonumber \\
\times&\left. \left. \, \mathbb{D}_{\infty,k}^{\sharp} \tilde{\mathfrak{C}}_{k}
(\tau)(\mathbb{D}_{\infty,k}^{\sharp})^{-1} \right) \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \left(\mathrm{I} \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}^{3} \sigma_{3}) \right) \! \left(\mathrm{I}
\! + \! \left(\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{b(\tau)}}}
\right)^{\ad (\sigma_{3})} \right. \nonumber \\
\times&\left. \,
\begin{pmatrix}
0 & -\frac{(\varepsilon b)^{1/4}(\sqrt{3}+1)^{1/2} \hat{\mathbb{D}}_{0}^{\infty}
(\tau)}{2^{1/4} \hat{\mathbb{B}}_{0}^{\infty}(\tau)} \\
\frac{2^{1/4} \hat{\mathbb{C}}_{0}^{\infty}(\tau)}{(\varepsilon b)^{1/4}
(\sqrt{3}+1)^{1/2} \hat{\mathbb{A}}_{0}^{\infty}(\tau)} & 0
\end{pmatrix} \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \gimel_{\scriptscriptstyle A,k}^{\sharp}(\tau)
\widehat{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \! + \! \dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)} \hat{\psi}_{1,k}^{
-1,\sharp}(\tau) \sigma_{3} \! + \! \widetilde{\Lambda} \dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)} \sigma_{3}
\right. \nonumber \\
+&\left. \, \dfrac{1}{\widetilde{\Lambda}} \! \left(\hat{\psi}_{1,k}^{-1,\sharp}
(\tau) \! + \! \dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+}}{\chi_{k}(\tau)} \hat{\psi}_{2,k}^{-1,\sharp}(\tau) \sigma_{3}
\! + \! \dfrac{1}{2 \sqrt{3}(\sqrt{3} \! + \! 1)} \! \left(\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}
(\tau)}}{\sqrt{\smash[b]{b(\tau)}}} \right)^{\ad (\sigma_{3})} \right. \right.
\nonumber \\
\times&\left. \left. \, J_{k}^{\infty} \pmb{\mathbb{T}}_{\infty,k}^{\sharp}
\! + \! \hat{\psi}_{1,k}^{-1,\sharp}(\tau) \gimel_{\scriptscriptstyle A,k}^{\sharp}
(\tau) \widehat{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \right) \! + \! \dfrac{1}{
\widetilde{\Lambda}^{2}} \! \left(\hat{\psi}_{2,k}^{-1,\sharp}(\tau) \! + \!
\dfrac{1}{2 \sqrt{3}(\sqrt{3} \! + \! 1)} \right. \right. \nonumber \\
\times&\left. \left. \, \hat{\psi}_{1,k}^{-1,\sharp}(\tau) \! \left(
\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{b(\tau)}}} \right)^{\ad
(\sigma_{3})} J_{k}^{\infty} \pmb{\mathbb{T}}_{\infty,k}^{\sharp} \! + \!
\hat{\psi}_{2,k}^{-1,\sharp}(\tau) \gimel_{\scriptscriptstyle A,k}^{\sharp}
(\tau) \widehat{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \right) \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left(\dfrac{\lvert \nu (k) \! + \! 1 \rvert^{2} \tau^{-
\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert p_{k}(\tau) \rvert^{2}} \!
\left(\dfrac{\mathrm{e}^{-\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{b(\tau)}}} \right)^{
\ad (\sigma_{3})} \mathbb{D}_{\infty,k}^{\sharp} \tilde{\mathfrak{C}}_{k}(\tau)
(\mathbb{D}_{\infty,k}^{\sharp})^{-1} \right) \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left(\dfrac{1}{\widetilde{\Lambda}} \dfrac{\lvert \nu (k)
\! + \! 1 \rvert^{2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert
p_{k}(\tau) \rvert^{2}} \hat{\psi}_{1,k}^{-1,\sharp}(\tau) \! \left(\dfrac{\mathrm{e}^{-
\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{b(\tau)}}} \right)^{\ad (\sigma_{3})}
\mathbb{D}_{\infty,k}^{\sharp} \tilde{\mathfrak{C}}_{k}(\tau)(\mathbb{D}_{
\infty,k}^{\sharp})^{-1} \right) \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left(\dfrac{1}{\widetilde{\Lambda}^{2}} \dfrac{\lvert \nu
(k) \! + \! 1 \rvert^{2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert
p_{k}(\tau) \rvert^{2}} \hat{\psi}_{2,k}^{-1,\sharp}(\tau) \! \left(\dfrac{\mathrm{e}^{-
\tilde{\beta}_{k}(\tau)}}{\sqrt{\smash[b]{b(\tau)}}} \right)^{\ad (\sigma_{3})}
\mathbb{D}_{\infty,k}^{\sharp} \tilde{\mathfrak{C}}_{k}(\tau)(\mathbb{D}_{
\infty,k}^{\sharp})^{-1} \right) \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left(\dfrac{1}{\widetilde{\Lambda}^{2}}
\dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)}
\hat{\psi}_{3,k}^{-1,\sharp}(\tau) \sigma_{3} \right) \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathrm{I} \! + \! \gimel_{\scriptscriptstyle A,k}^{
\sharp}(\tau) \widehat{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \! + \! \dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)} \hat{\psi}_{1,k}^{-1,
\sharp}(\tau) \sigma_{3} \! + \! \mathcal{O}(\tau^{-\frac{1}{3}+3 \delta_{k}}
\sigma_{3}) \nonumber \\
+& \,
\begin{pmatrix}
0 & \mathcal{O}(\tau^{-2/3}) \\
\mathcal{O}(\tau^{-2/3}) & 0
\end{pmatrix} \! + \!
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+\delta_{k}}) & 0 \\
0 & \mathcal{O}(\tau^{-\frac{1}{3}+\delta_{k}})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
0 & \mathcal{O}(\tau^{-\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}}) \\
\mathcal{O}(\tau^{-\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) & 0
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}-\delta_{k}}(\nu (k) \! + \! 1)) & 0 \\
0 & \mathcal{O}(\tau^{-\frac{1}{3}-\delta_{k}}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}-\delta_{k}}) & \mathcal{O}(\tau^{-\frac{1}{3}
-\delta_{k}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}-\delta_{k}}) & \mathcal{O}(\tau^{-\frac{1}{3}
-\delta_{k}})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{4}{3}-\delta_{k}}) & \mathcal{O}(\tau^{-\frac{2}{3}
-\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}}) \\
\mathcal{O}(\tau^{-\frac{2}{3}-\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}})
& \mathcal{O}(\tau^{-\delta_{k}}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-2\delta_{k}}(\nu (k) \! + \! 1)) & 0 \\
0 & \mathcal{O}(\tau^{-2\delta_{k}}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}})
& \mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}})
& \mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{2}{3}-2\delta_{k}}(\nu (k) \! + \! 1)) & \mathcal{O}
(\tau^{-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{3+k}{2}}) \\
\mathcal{O}(\tau^{-\frac{4}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}})
& \mathcal{O}(\tau^{-\frac{2}{3}-2\delta_{k}}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-2-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}) & \mathcal{O}
(\tau^{-1-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k) \! + \! 1)^{\frac{1
+k}{2}}) \\
\mathcal{O}(\tau^{-3-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k) \! + \!
1)^{\frac{1-k}{2}}) & \mathcal{O}(\tau^{-2-\epsilon_{\mathrm{\scriptscriptstyle TP}}
(k)})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-3-\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu
(k) \! + \! 1)) & \mathcal{O}(\tau^{-2-\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}
(k)}(\nu (k) \! + \! 1)^{\frac{1+k}{2}}) \\
\mathcal{O}(\tau^{-2-\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}
(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) & \mathcal{O}(\tau^{-1-\delta_{k}-
\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-2-2\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k)
\! + \! 1)) & \mathcal{O}(\tau^{-1-2\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}
(k)}(\nu (k) \! + \! 1)^{\frac{3+k}{2}}) \\
\mathcal{O}(\tau^{-3-2\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}
(\nu (k) \! + \! 1)^{\frac{3-k}{2}}) & \mathcal{O}(\tau^{-2-2\delta_{k}-
\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
0 & \mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) & 0
\end{pmatrix} \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathrm{I} \! + \! \gimel_{\scriptscriptstyle A,k}^{
\sharp}(\tau) \widehat{\mathbb{E}}_{\infty,k}^{\sharp}(\tau) \! + \! \dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)} \hat{\psi}_{1,k}^{-1,\sharp}
(\tau) \sigma_{3} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+3 \delta_{k}}) & \mathcal{O}(\tau^{-2/3}) \\
\mathcal{O}(\tau^{-2/3}) & \mathcal{O}(\tau^{-\frac{1}{3}+3 \delta_{k}})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+\delta_{k}}) & \mathcal{O}(\tau^{-\frac{1}{3}
(\frac{1+k}{2})-\delta_{k}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}(\frac{1-k}{2})-\delta_{k}}) & \mathcal{O}
(\tau^{-\frac{1}{3}+\delta_{k}})
\end{pmatrix} \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathrm{I} \! + \!
\underbrace{\gimel_{\scriptscriptstyle A,k}^{\sharp}(\tau) \widehat{\mathbb{E}}_{
\infty,k}^{\sharp}(\tau) \! + \! \dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)} \hat{\psi}_{1,k}^{-1,\sharp}(\tau)
\sigma_{3}}_{=: \, \mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{\infty}(\tau)}
\nonumber \\
+& \, \underbrace{\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+3\delta_{k}}) & \mathcal{O}(\tau^{-\frac{1}{3}
(\frac{1+k}{2})-\delta_{k}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}(\frac{1-k}{2})-\delta_{k}}) & \mathcal{O}
(\tau^{-\frac{1}{3}+3\delta_{k}})
\end{pmatrix}}_{=: \, \mathcal{O}(\mathbb{E}^{\infty}_{k}(\tau))} \nonumber \\
\underset{\tau \to +\infty}{=}& \, (\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle
\mathcal{N}},k}^{\infty}(\tau)) \! \left(\mathrm{I} \! + \! \underbrace{(\mathrm{I}
\! + \! \mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{\infty}(\tau))^{-1}}_{= \,
\mathcal{O}(1)} \, \mathcal{O}(\mathbb{E}^{\infty}_{k}(\tau)) \right) \quad
\Rightarrow \nonumber
\end{align*}
\begin{equation} \label{ellinfk38}
\overset{\Yup}{\mathbb{E}}_{\scriptscriptstyle
\mathfrak{L}^{\infty}_{k}}^{\raise-6.75pt\hbox{$\scriptstyle \leftslice$}}(\tau)
\underset{\tau \to +\infty}{=} (\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle
\mathcal{N}},k}^{\infty}(\tau))(\mathrm{I} \! + \! \mathcal{O}(\mathbb{E}^{\infty}_{k}
(\tau))).
\end{equation}
Thus, via Asymptotics~\eqref{ellinfk28} and~\eqref{ellinfk38}, one arrives at the
result stated in the lemma. \hfill $\qed$
\begin{cccc} \label{lzernewlemm}
Let $\widetilde{\Psi}_{k}(\widetilde{\mu},\tau)$, $k \! = \! \pm 1$, be the fundamental
solution of Equation~\eqref{eq3.3} with asymptotics given in Lemma~\ref{nprcl}, and
let $\mathbb{X}^{0}_{1-k}(\widetilde{\mu},\tau)$ be the canonical solution of
Equation~\eqref{eq3.1}.\footnote{See Proposition~\ref{prop1.4}.} Define {}\footnote{Since
$\tau^{-\frac{1}{12} \sigma_{3}} \mathbb{X}^{0}_{1-k}(\tau^{-1/6} \widetilde{\mu},
\tau)$, $k \! = \! \pm 1$, (cf. Equations~\eqref{eq3.2}) is also a fundamental solution
of Equation~\eqref{eq3.3}, it follows, therefore, that $\mathfrak{L}^{0}_{k}(\tau)$ is
independent of $\widetilde{\mu}$.}
\begin{equation} \label{ellohk1}
\mathfrak{L}^{0}_{k}(\tau) \! := \! (\widetilde{\Psi}_{k}(\widetilde{\mu},\tau))^{-1}
\tau^{-\frac{1}{12} \sigma_{3}} \mathbb{X}^{0}_{1-k}(\tau^{-1/6} \widetilde{\mu},
\tau), \quad k \! = \! \pm 1.
\end{equation}
Assume that the parameters $\nu (k) \! + \! 1$ and $\delta_{k}$ satisfy the
restrictions~\eqref{pc4} and~\eqref{restr1}, and, additionally, the
Conditions~\eqref{ellinfk2a} and~\eqref{ellinfk2b} are valid. Then,
\begin{align} \label{ellohk3}
\mathfrak{L}^{0}_{k}(\tau) \underset{\tau \to +\infty}{=}& \, (\mathcal{R}_{m_{0}}
(k))^{-1} \mathrm{e}^{\hat{\mathfrak{z}}_{k}^{0}(\tau) \sigma_{3}} \! \left(\dfrac{\mathrm{i}
2^{1/4}}{(\sqrt{3} \! - \! 1)^{1/2} \sqrt{\smash[b]{\mathfrak{B}_{k}}}}
\right)^{\sigma_{3}} \mathrm{e}^{\Delta \hat{\mathfrak{z}}_{k}(\tau) \sigma_{3}}
\begin{pmatrix}
\hat{\mathbb{A}}_{0}^{0}(\tau) & 0 \\
0 & \hat{\mathbb{B}}_{0}^{0}(\tau)
\end{pmatrix} \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle \mathcal{N}},
k}^{0}(\tau) \right) \! \mathbb{S}_{k}^{\ast} \! \left(\mathrm{I} \! + \!
\mathcal{O} \left(\mathbb{E}^{0}_{k}(\tau) \right) \right),
\end{align}
where $\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \mathcal{R}_{m_{0}}(k)$,
$m_{0} \! \in \! \lbrace -1,0,1,2 \rbrace$, are defined in
Remark~\ref{leedasyue},\footnote{The precise choice for the value of
$m_{0}$ is given in the proof of Theorem~\ref{theor3.3.1} below.}
\begin{align}
\hat{\mathfrak{z}}_{k}^{0}(\tau) \! :=& \, \mathrm{i} \tau^{2/3}3 \sqrt{3} \alpha_{k}^{2}
\! + \! \mathrm{i} (a \! - \! \mathrm{i}/2) \ln (2^{-1/2}(\sqrt{3} \! + \! 1)), \label{ellohk4} \\
\Delta \hat{\mathfrak{z}}_{k}(\tau) \! :=& \, -\left(\dfrac{5 \! + \! 9 \sqrt{3}}{6
\sqrt{3} \alpha_{k}^{2}} \right) \! \mathfrak{p}_{k}(\tau) \! + \! (\nu (k) \! + \! 1)
\ln (2 \mu_{k}(\tau))^{1/2} \! + \! \dfrac{1}{3}(\nu (k) \! + \! 1) \ln \tau
\nonumber \\
-& \, (\nu (k) \! + \! 1) \ln (\mathrm{e}^{\mathrm{i} k \pi}/3 \alpha_{k}), \label{ellohk5}
\end{align}
with $\mathfrak{p}_{k}(\tau)$ defined by Equation~\eqref{eqpeetee}, and
$\mathfrak{B}_{k}$ and $\mu_{k}(\tau)$ defined in Lemma~\ref{nprcl},
\begin{gather}
\hat{\mathbb{A}}_{0}^{0}(\tau) \! := \! 1 \! + \! \dfrac{(\varepsilon b)^{1/4}
(\sqrt{3} \! - \! 1)^{1/2}(\Delta G_{k}^{0}(\tau))_{11}}{2^{1/4}}, \label{ellohk6} \\
\hat{\mathbb{B}}_{0}^{0}(\tau) \! := \! 1 \! + \! \dfrac{2^{1/4}}{(\varepsilon
b)^{1/4}(\sqrt{3} \! - \! 1)^{1/2}} \! \left((\Delta G_{k}^{0}(\tau))_{22} \! -
\! \dfrac{\mathfrak{A}_{k}}{\mathfrak{B}_{k}} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k}}{\chi_{k}(\tau)} \! - \! 1 \right) \! (\Delta G_{k}^{0}(\tau))_{12} \right),
\label{ellohk7}
\end{gather}
with $\mathcal{Z}_{k}$, $\mathfrak{A}_{k}$, and $\chi_{k}(\tau)$ defined in
Lemma~\ref{nprcl}, and
\begin{equation} \label{ellohk8}
\Delta G_{k}^{0}(\tau) \! := \! \dfrac{1}{(2 \sqrt{3}(\sqrt{3} \! - \! 1))^{1/2}}
\begin{pmatrix}
(\Delta G_{k}^{0}(\tau))_{11} & (\Delta G_{k}^{0}(\tau))_{12} \\
(\Delta G_{k}^{0}(\tau))_{21} & (\Delta G_{k}^{0}(\tau))_{22}
\end{pmatrix},
\end{equation}
with
\begin{align*}
(\Delta G_{k}^{0}(\tau))_{11} \! :=& \, (\sqrt{3} \! - \! 1)(\Delta \mathcal{G}_{0,k})_{22}
\! - \! (2/\varepsilon b)^{1/2}(\Delta \mathcal{G}_{0,k})_{12}, \\
(\Delta G_{k}^{0}(\tau))_{12} \! :=& \, -(\sqrt{3} \! - \! 1)(\Delta \mathcal{G}_{0,k})_{12}
\! - \! (2 \varepsilon b)^{1/2}(\Delta \mathcal{G}_{0,k})_{22}, \\
(\Delta G_{k}^{0}(\tau))_{21} \! :=& \, -(\sqrt{3} \! - \! 1)(\Delta \mathcal{G}_{0,k})_{21}
\! + \! (2/\varepsilon b)^{1/2}(\Delta \mathcal{G}_{0,k})_{11}, \\
(\Delta G_{k}^{0}(\tau))_{22} \! :=& \, (\sqrt{3} \! - \! 1)(\Delta \mathcal{G}_{0,k})_{11}
\! + \! (2 \varepsilon b)^{1/2}(\Delta \mathcal{G}_{0,k})_{21},
\end{align*}
where $(\Delta \mathcal{G}_{0,k})_{i,j=1,2}$ are defined by
Equations~\eqref{prcyg4}--\eqref{prcyg6},
\begin{equation} \label{ellohk9}
\mathbb{S}_{k}^{\ast} \! := \!
\begin{pmatrix}
1 & -(1 \! + \! k)s_{0}^{0}/2 \\
(1 \! - \! k)s_{0}^{0}/2 & 1
\end{pmatrix},
\end{equation}
\begin{align} \label{ellohk10}
\mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{0}(\tau) :=& \,
\mathrm{e}^{-\hat{\beta}_{k}(\tau) \ad (\sigma_{3})} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)} \! \left(
\dfrac{(\sqrt{3} \! - \! 1)p_{k}(\tau) \mathfrak{B}_{k}}{2^{3/2} \mu_{k}(\tau)}
\sigma_{+} \! + \! \dfrac{\sqrt{2}(\nu (k) \! + \! 1)}{(\sqrt{3} \! - \! 1)p_{k}
(\tau) \mathfrak{B}_{k}} \sigma_{-} \right) \! \sigma_{3} \right. \nonumber \\
+&\left. \, \dfrac{1}{2 \sqrt{3}(\sqrt{3} \! - \! 1)} \!
\begin{pmatrix}
-\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)}
& \frac{(\sqrt{3}-1) \mathfrak{B}_{k} \ell_{0,k}^{+}}{\sqrt{2}} \\
-\frac{\sqrt{2}}{(\sqrt{3}-1) \mathfrak{B}_{k}}((\frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k}
\mathfrak{A}_{k}}{\chi_{k}(\tau)})^{2} \ell_{0,k}^{+} \! - \! \ell_{1,k}^{+} \! - \!
\ell_{2,k}^{+}) & \frac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+}}{\chi_{k}(\tau)}
\end{pmatrix} \right. \nonumber \\
\times&\left. \,
\begin{pmatrix}
\sqrt{3} \! - \! 1 & (2 \varepsilon b)^{1/2} \\
-(2/\varepsilon b)^{1/2} & \sqrt{3} \! - \! 1
\end{pmatrix} \!
\begin{pmatrix}
\mathbb{T}_{11,k}(-1;\tau) & \mathbb{T}_{12,k}(-1;\tau) \\
\mathbb{T}_{21,k}(-1;\tau) & \mathbb{T}_{22,k}(-1;\tau)
\end{pmatrix} \right),
\end{align}
with $\ell_{0,k}^{+}$, $\ell_{1,k}^{+}$, and $\ell_{2,k}^{+}$ defined in Lemma~\ref{nprcl},
$(\mathbb{T}_{ij,k}(-1;\tau))_{i,j=1,2}$ defined in Proposition~\ref{prop3.1.6}, and
$\hat{\beta}_{k}(\tau)$ defined by Equation~\eqref{ellohk16} below, and
\begin{equation} \label{ellohk11}
\mathcal{O}(\mathbb{E}^{0}_{k}(\tau)) \underset{\tau \to +\infty}{:=}
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+3\delta_{k}}) & \mathcal{O}
(\tau^{-\frac{1}{3}(\frac{1-k}{2})-\delta_{k}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}(\frac{1+k}{2})-\delta_{k}}) & \mathcal{O}
(\tau^{-\frac{1}{3}+3\delta_{k}})
\end{pmatrix}.
\end{equation}
\end{cccc}
\emph{Proof}. Denote by $\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}
(\widetilde{\mu},\tau)$, $k \! = \! \pm 1$, the solution of Equation~\eqref{eq3.3}
that has leading-order asymptotics given by Equations~\eqref{eq3.16}--\eqref{eq3.18}
in the canonical domain containing the Stokes curve approaching, for $k \! = \! +1$
(resp., $k \! = \! -1)$, the real $\widetilde{\mu}$-axis {}from above (resp., below)
as $\widetilde{\mu} \! \to \! 0$. Let $\mathfrak{L}_{k}^{0}(\tau)$, $k \! = \! \pm 1$,
be defined by Equation~\eqref{ellohk1}; re-write $\mathfrak{L}_{k}^{0}(\tau)$ in
the following, equivalent form:
\begin{equation} \label{ellohk12}
\mathfrak{L}_{k}^{0}(\tau) \! = \! \left((\widetilde{\Psi}_{k}(\widetilde{\mu},
\tau))^{-1} \widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}(\widetilde{\mu},
\tau) \right) \! \left((\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}
(\widetilde{\mu},\tau))^{-1} \tau^{-\frac{1}{12} \sigma_{3}} \mathbb{X}^{0}_{1}
(\tau^{-1/6} \widetilde{\mu},\tau) \right) \! \mathbb{S}_{k}^{\ast},
\end{equation}
where $\mathbb{S}_{k}^{\ast}$ is defined by Equation~\eqref{ellohk9}.
Taking note of the fact that $\widetilde{\Psi}_{k}(\widetilde{\mu},\tau)$,
$\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}(\widetilde{\mu},\tau)$, and
$\tau^{-\frac{1}{12} \sigma_{3}} \mathbb{X}^{0}_{1}(\tau^{-1/6} \widetilde{\mu},
\tau)$ are all solutions of Equation~\eqref{eq3.3}, it follows that they differ on the right
by non-degenerate, $\widetilde{\mu}$-independent, $\mathrm{M}_{2}
(\mathbb{C})$-valued factors: via this observation, one evaluates, asymptotically,
each of the factors appearing in Equation~\eqref{ellohk12} by considering separate
limits, namely, $\widetilde{\mu} \! \to \! \alpha_{k}$ and $\widetilde{\mu} \! \to \! 0$,
respectively; more precisely, for $k \! = \! \pm 1$,
\begin{align} \label{ellohk13}
&(\widetilde{\Psi}_{k}(\widetilde{\mu},\tau))^{-1} \widetilde{\Psi}_{\scriptscriptstyle
\mathrm{WKB},k}(\widetilde{\mu},\tau) \underset{\tau \to +\infty}{:=} \nonumber \\
&\underbrace{\left((b(\tau))^{-\frac{1}{2} \sigma_{3}} \mathcal{G}_{0,k}
\mathfrak{B}_{k}^{\frac{1}{2} \sigma_{3}} \mathbb{F}_{k}(\tau) \Xi_{k}(\tau;
\widetilde{\Lambda}) \hat{\chi}_{k}(\widetilde{\Lambda}) \Phi_{M,k}(\widetilde{\Lambda})
\right)^{-1}T(\widetilde{\mu}) \mathrm{e}^{\scriptscriptstyle \mathrm{W}_{k}(\widetilde{\mu},
\tau)}}_{\widetilde{\mu}=\widetilde{\mu}_{0,k}, \, \, \, \, \widetilde{\Lambda}
\underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{\delta_{k}}), \, \, 0< \delta <
\delta_{k}<\frac{1}{24}, \, \, \, \, \arg (\widetilde{\Lambda})=\frac{\pi m_{0}}{2}+
\frac{\pi}{4}-\frac{1}{2} \arg (\mu_{k}(\tau)), \, \, m_{0} \in \lbrace -1,0,1,2 \rbrace},
\end{align}
where (cf. Lemma~\ref{linfnewlemm}) $\mathbb{F}_{k}(\tau)$ and $\Xi_{k}
(\tau;\widetilde{\Lambda})$ are defined by Equations~\eqref{ellinfk13}
and~\eqref{ellinfk14}, respectively, $\mathrm{W}_{k}(\widetilde{\mu},\tau) \! :=
\! -\sigma_{3} \mathrm{i} \tau^{2/3} \int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}
l_{k}(\xi) \, \mathrm{d} \xi \! - \! \int_{\widetilde{\mu}_{0,k}}^{\widetilde{\mu}}
\diag (T^{-1}(\xi) \partial_{\xi}T(\xi)) \, \mathrm{d} \xi$, and $\hat{\chi}_{k}
(\widetilde{\Lambda})$ has the asymptotics~\eqref{ellinfk15}, and
\begin{equation} \label{ellohk14}
(\widetilde{\Psi}_{\scriptscriptstyle \mathrm{WKB},k}(\widetilde{\mu},\tau))^{-1}
\tau^{-\frac{1}{12} \sigma_{3}} \mathbb{X}^{0}_{1}(\tau^{-1/6} \widetilde{\mu},
\tau) \underset{\tau \to +\infty}{:=} \lim_{\underset{\arg (\widetilde{\mu})=
\pi}{\Omega_{1}^{0} \ni \widetilde{\mu} \to 0}} \left((T(\widetilde{\mu})
\mathrm{e}^{\scriptscriptstyle \mathrm{W}_{k}(\widetilde{\mu},\tau)})^{-1}
\tau^{-\frac{1}{12} \sigma_{3}} \mathbb{X}^{0}_{1}(\tau^{-1/6}
\widetilde{\mu},\tau) \right).
\end{equation}
One commences by considering the asymptotics subsumed in the
Definition~\eqref{ellohk14}. {}From the asymptotics for $\mathbb{X}^{0}_{1}
(\tau^{-1/6} \widetilde{\mu},\tau)$ stated in Proposition~\ref{prop1.4},
Equations~\eqref{iden3}, \eqref{iden4}, \eqref{expforeych}, \eqref{expforkapp},
\eqref{eq3.39}, \eqref{eq3.40}, \eqref{eqpeetee}, \eqref{eq3.43}, \eqref{asympforf8},
\eqref{eq3.52}, \eqref{prcy22}, and~\eqref{prcyomg1}, one arrives at,
via the Conditions~\eqref{iden5} and the Asymptotics~\eqref{terrbos6} and
\eqref{asympforf3},
\begin{equation} \label{ellohk15}
\lim_{\underset{\arg (\widetilde{\mu})=\pi}{\Omega_{1}^{0} \ni \widetilde{\mu}
\to 0}} \left((T(\widetilde{\mu}) \mathrm{e}^{\scriptscriptstyle \mathrm{W}_{k}
(\widetilde{\mu},\tau)})^{-1} \tau^{-\frac{1}{12} \sigma_{3}} \mathbb{X}^{0}_{1}
(\tau^{-1/6} \widetilde{\mu},\tau) \right) \underset{\tau \to +\infty}{=}
\left(\dfrac{\mathrm{i} (\varepsilon b)^{1/4}}{\sqrt{\smash[b]{b(\tau)}}} \right)^{\sigma_{3}}
\exp (\hat{\beta}_{k}(\tau) \sigma_{3}), \quad k \! = \! \pm 1,
\end{equation}
where
\begin{align} \label{ellohk16}
\hat{\beta}_{k}(\tau) :=& \, \mathrm{i} \tau^{2/3}3 \sqrt{3} \alpha_{k}^{2} \! +
\! \mathrm{i} 2 \sqrt{3} \, \widetilde{\Lambda}^{2} \! + \! \mathrm{i} (a \! - \! \mathrm{i}/2)
\ln ((\sqrt{3} \! + \! 1)/\sqrt{2}) \! - \! \dfrac{(5 \! + \! 9 \sqrt{3})
\mathfrak{p}_{k}(\tau)}{6 \sqrt{3} \alpha_{k}^{2}} \nonumber \\
+& \, \left(\dfrac{\mathrm{i}}{2 \sqrt{3}} \! \left((a \! - \! \mathrm{i}/2) \! + \! \alpha_{k}^{-2}
\tau^{2/3} \hat{h}_{0}(\tau) \right) \! + \! \dfrac{2 \mathfrak{p}_{k}(\tau)}{3 \sqrt{3}
\alpha_{k}^{2}} \right) \! \left(-\dfrac{1}{3} \ln \tau \! + \! \ln \widetilde{\Lambda}
\! + \! \ln (\mathrm{e}^{\mathrm{i} k \pi}/3 \alpha_{k}) \right) \nonumber \\
-& \, \dfrac{(\sqrt{3} \! + \! 1) \mathfrak{p}_{k}(\tau)}{\sqrt{3} \alpha_{k}
\tau^{-1/3} \widetilde{\Lambda}} \! + \! \mathcal{O} \! \left(\left(
\dfrac{\tilde{\mathfrak{c}}_{1,k} \tau^{-1/3} \! + \! \tilde{\mathfrak{c}}_{2,k}
\tilde{r}_{0}(\tau)}{\widetilde{\Lambda}^{2}} \right) \! \left(\tilde{\mathfrak{c}}_{3,k}
\tau^{-1/3} \! + \! \tilde{\mathfrak{c}}_{4,k}(\tilde{r}_{0}(\tau) \! + \! 4v_{0}
(\tau)) \right) \right) \nonumber \\
+& \, \mathcal{O}(\tau^{-1/3} \widetilde{\Lambda}^{3}) \! + \!
\mathcal{O}(\tau^{-1/3} \widetilde{\Lambda}) \! + \! \mathcal{O} \! \left(
\dfrac{\tau^{-1/3}}{\widetilde{\Lambda}} \left(\tilde{\mathfrak{c}}_{5,k}
\! + \! \tilde{\mathfrak{c}}_{6,k} \tau^{2/3} \hat{h}_{0}(\tau) \! + \!
\tilde{\mathfrak{c}}_{7,k}(\tau^{2/3} \hat{h}_{0}(\tau))^{2} \right) \right),
\end{align}
and $\tilde{\mathfrak{c}}_{m,k}$, $m \! = \! 1,2,\dotsc,7$, are constants.
One now derives the asymptotics defined by Equation~\eqref{ellohk13}. {}From
Asymptotics~\eqref{iden55} for $\varpi \! = \! -1$, Equation~\eqref{prcy2} for
$\Phi_{M,k}(\widetilde{\Lambda})$ (in conjunction with its large-$\widetilde{\Lambda}$
asymptotics stated in Remark~\ref{leedasyue}), the Definitions \eqref{ellinfk13}
and~\eqref{ellinfk14} (concomitant with the fact that $\det (\Xi_{k}(\tau;
\widetilde{\Lambda})) \! = \! 1)$, and the Asymptotics~\eqref{ellinfk15}, one shows,
via the relation $(\mathrm{W}_{k}(\widetilde{\mu}_{0,k},\tau))_{i,j=1,2} \! = \! 0$
and Definition~\eqref{ellohk13}, that, for $k \! = \! \pm 1$,
{\fontsize{10pt}{11pt}\selectfont
\begin{align} \label{ellohk17}
(\widetilde{\Psi}_{k}(\widetilde{\mu},\tau))^{-1} \widetilde{\Psi}_{\scriptscriptstyle
\mathrm{WKB},k}(\widetilde{\mu},\tau) \underset{\tau \to +\infty}{:=}& \,
\Phi_{M,k}^{-1}(\widetilde{\Lambda}) \hat{\chi}_{k}^{-1}(\widetilde{\Lambda})
\Xi_{k}^{-1}(\tau;\widetilde{\Lambda}) \mathbb{F}_{k}^{-1}(\tau) \mathfrak{B}_{k}^{
-\frac{1}{2} \sigma_{3}} \mathcal{G}_{0,k}^{-1}(b(\tau))^{\frac{1}{2} \sigma_{3}}
T(\widetilde{\mu}_{0,k}) \nonumber \\
\underset{\tau \to +\infty}{=}& \, (\mathcal{R}_{m_{0}}(k))^{-1}
\mathrm{e}^{-\mathcal{P}_{0}^{\ast} \sigma_{3}} \mathfrak{Q}_{0,k}(\tau) \! \left(
\mathrm{I} \! + \! \dfrac{1}{\widetilde{\Lambda}} \mathfrak{Q}_{0,k}^{-1}
(\tau) \hat{\psi}_{1,k}^{-1}(\tau) \mathfrak{Q}_{0,k}(\tau) \right. \nonumber \\
+&\left. \, \dfrac{1}{\widetilde{\Lambda}^{2}} \mathfrak{Q}_{0,k}^{-1}(\tau)
\hat{\psi}_{2,k}^{-1}(\tau) \mathfrak{Q}_{0,k}(\tau) \! + \! \mathcal{O} \!
\left(\dfrac{1}{\widetilde{\Lambda}^{3}} \mathfrak{Q}_{0,k}^{-1}(\tau)
\hat{\psi}_{3,k}^{-1}(\tau) \mathfrak{Q}_{0,k}(\tau) \right) \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \mathcal{O} \! \left(\lvert \nu (k) \! + \! 1 \rvert^{2}
\lvert p_{k}(\tau) \rvert^{-2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}
\mathfrak{Q}_{0,k}^{-1}(\tau) \tilde{\mathfrak{C}}_{k}(\tau)
\mathfrak{Q}_{0,k}(\tau) \right) \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \widetilde{\Lambda} \mathfrak{Q}_{0,k}^{-1}
(\tau) \gimel_{\scriptscriptstyle A,k}^{-1}(\tau) \mathfrak{Q}_{0,k}
(\tau) \! + \! \widetilde{\Lambda}^{2} \mathfrak{Q}_{0,k}^{-1}(\tau)
\gimel_{\scriptscriptstyle B,k}^{-1}(\tau) \mathfrak{Q}_{0,k}(\tau) \right)
\nonumber \\
\times& \, \left(\mathrm{I} \! + \! \widetilde{\Lambda} \tau^{-1/3}
\mathbb{P}_{0,k}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}}
\widehat{\mathbb{E}}_{0,k}(\tau) \! + \! \mathcal{O} \! \left((\tau^{-1/3}
\widetilde{\Lambda})^{2} \widetilde{\mathbb{E}}_{0,k}(\tau) \right) \right),
\end{align}}
where $\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \mathcal{R}_{m_{0}}(k)$, $m_{0}
\! \in \! \lbrace -1,0,1,2 \rbrace$, are defined in Remark~\ref{leedasyue},
$\mathcal{P}_{0}^{\ast}$, $\hat{\psi}_{1,k}^{-1}(\tau)$, $\hat{\psi}_{2,k}^{-1}
(\tau)$, and $\hat{\psi}_{3,k}^{-1}(\tau)$ are defined by Equations~\eqref{ellinfk20},
\eqref{ellinfk22}, \eqref{ellinfk23}, and~\eqref{ellinfk24}, respectively,
\begin{equation} \label{ellohk18}
\mathfrak{Q}_{0,k}(\tau) \! := \! \mathbb{F}_{k}^{-1}(\tau) \! \left(\left(
\dfrac{2^{1/4} \sqrt{\smash[b]{b(\tau)}}}{(\varepsilon b)^{1/4}(\sqrt{3} \!
- \! 1)^{1/2} \sqrt{\smash[b]{\mathfrak{B}_{k}}}} \right)^{\sigma_{3}} \!
+ \! \mathfrak{B}_{k}^{-\frac{1}{2} \sigma_{3}} \Delta G_{k}^{0}(\tau)
(b(\tau))^{\frac{1}{2} \sigma_{3}} \right),
\end{equation}
with $\Delta G_{k}^{0}(\tau)$ defined by Equation~\eqref{ellohk8},
\begin{align}
\mathbb{P}_{0,k}(\tau) \! :=& \, (b(\tau))^{-\frac{1}{2} \ad (\sigma_{3})} \!
\begin{pmatrix}
0 & -\frac{(\varepsilon b)^{1/2}}{3 \sqrt{2} \alpha_{k}} \\
\frac{(\varepsilon b)^{-1/2}}{3 \sqrt{2} \alpha_{k}} & 0
\end{pmatrix}, \label{ellohk19} \\
\widehat{\mathbb{E}}_{0,k}(\tau) \! :=& \, \dfrac{1}{2 \sqrt{3}
(\sqrt{3} \! - \! 1)}(b(\tau))^{-\frac{1}{2} \ad (\sigma_{3})} \!
\begin{pmatrix}
\sqrt{3} \! - \! 1 & (2 \varepsilon b)^{1/2} \\
-(2/\varepsilon b)^{1/2} & \sqrt{3} \! - \! 1
\end{pmatrix} \!
\begin{pmatrix}
\mathbb{T}_{11,k}(-1;\tau) & \mathbb{T}_{12,k}(-1;\tau) \\
\mathbb{T}_{21,k}(-1;\tau) & \mathbb{T}_{22,k}(-1;\tau)
\end{pmatrix}, \label{ellohk20} \\
\widetilde{\mathbb{E}}_{0,k}(\tau) \! :=& \, \dfrac{1}{2 \sqrt{3}
(\sqrt{3} \! - \! 1)}(b(\tau))^{-\frac{1}{2} \ad (\sigma_{3})} \!
\begin{pmatrix}
\sqrt{3} \! - \! 1 & (2 \varepsilon b)^{1/2} \\
-(2/\varepsilon b)^{1/2} & \sqrt{3} \! - \! 1
\end{pmatrix} \tilde{\mathfrak{C}}_{k}^{\lozenge}, \label{ellohk21}
\end{align}
$\mathrm{M}_{2}(\mathbb{C}) \! \ni \! \tilde{\mathfrak{C}}_{k}(\tau) \!
=_{\tau \to +\infty} \! \mathcal{O}(1)$, $(\mathbb{T}_{ij,k}(-1;\tau))_{i,j=1,2}$
defined in Proposition~\ref{prop3.1.6}, and $\mathrm{M}_{2}(\mathbb{C}) \!
\ni \! \tilde{\mathfrak{C}}_{k}^{\lozenge}$ a constant.
Recalling the Definitions~\eqref{ellohk13} and~\eqref{ellohk14}, and substituting
the Expansions \eqref{ellohk15}, \eqref{ellohk16}, and~\eqref{ellohk17} into
Equation~\eqref{ellohk12}, one shows, via the Conditions~\eqref{iden5}, the
Definition~\eqref{prpr1}, the restrictions~\eqref{pc4}, the Asymptotics~\eqref{prcychik1},
\eqref{prcyzeek1}, \eqref{prcymuk1}, and (cf. step~\pmb{(xi)} in the proof of
Lemma~\ref{nprcl}) $\arg (\mu_{k}(\tau)) \! =_{\tau \to +\infty} \! \tfrac{\pi}{2}
(1 \! + \! \mathcal{O}(\tau^{-2/3}))$, and the restriction~\eqref{restr1}, that
\begin{equation} \label{ellohk22}
\mathfrak{L}^{0}_{k}(\tau) \underset{\tau \to +\infty}{=} (\mathcal{R}_{m_{0}}(k))^{-1}
\mathrm{e}^{\hat{\mathfrak{z}}_{k}^{0}(\tau) \sigma_{3}} \! \left(\dfrac{\mathrm{i} 2^{1/4}}{(\sqrt{3}
\! - \! 1)^{1/2} \sqrt{\smash[b]{\mathfrak{B}_{k}}}} \right)^{\sigma_{3}} \mathrm{e}^{\Delta
\hat{\mathfrak{z}}_{k}(\tau) \sigma_{3}} \operatorname{diag}(\hat{\mathbb{A}}_{0}^{0}
(\tau),\hat{\mathbb{B}}_{0}^{0}(\tau)) \overset{\Ydown}{\mathbb{E}}_{\scriptscriptstyle
\mathfrak{L}^{0}_{k}}^{\raise-6.75pt\hbox{$\scriptstyle \leftslice$}}(\tau)
\mathbb{S}_{k}^{\ast}, \quad k \! = \! \pm 1,
\end{equation}
where $\hat{\mathfrak{z}}_{k}^{0}(\tau)$, $\Delta \hat{\mathfrak{z}}_{k}(\tau)$,
$\hat{\mathbb{A}}_{0}^{0}(\tau)$, and $\hat{\mathbb{B}}_{0}^{0}(\tau)$ are
defined by Equations~\eqref{ellohk4}--\eqref{ellohk7}, respectively, and
{\fontsize{10pt}{11pt}\selectfont
\begin{align} \label{ellohk23}
\overset{\Ydown}{\mathbb{E}}_{\scriptscriptstyle
\mathfrak{L}^{0}_{k}}^{\raise-6.75pt\hbox{$\scriptstyle \leftslice$}}(\tau)
\underset{\tau \to +\infty}{:=}& \, \left(\mathrm{I} \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}^{3} \sigma_{3}) \right) \! \left(
\mathrm{I} \! + \! \mathrm{e}^{-\hat{\beta}_{k}(\tau) \ad (\sigma_{3})} \!
\begin{pmatrix}
0 & -\frac{\mathrm{i} (\sqrt{3}-1)^{1/2} \hat{\mathbb{C}}_{0}^{0}(\tau)}{2^{1/4}
\hat{\mathbb{A}}_{0}^{0}(\tau)} \\
\frac{\mathrm{i} 2^{1/4} \hat{\mathbb{D}}_{0}^{0}(\tau)}{(\sqrt{3}-1)^{1/2}
\hat{\mathbb{B}}_{0}^{0}(\tau)} & 0
\end{pmatrix} \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \dfrac{1}{\widetilde{\Lambda}}
\hat{\psi}_{1,k}^{-1,\natural}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}^{2}}
\hat{\psi}_{2,k}^{-1,\natural}(\tau) \! + \! \mathcal{O} \! \left(\dfrac{1}{\widetilde{
\Lambda}^{3}} \hat{\psi}_{3,k}^{-1,\natural}(\tau) \right) \right) \! \left(
\mathrm{I} \! + \! \mathcal{O} \! \left(\dfrac{\lvert \nu (k) \! + \! 1 \rvert^{2}
\tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert p_{k}(\tau) \rvert^{2}}
\right. \right. \nonumber \\
\times&\left. \left. \, \mathfrak{Q}_{\ast,k}^{-1}(\tau) \tilde{\mathfrak{C}}_{k}
(\tau) \mathfrak{Q}_{\ast,k}(\tau) \right) \right) \! \left(\mathrm{I} \! + \!
\widetilde{\Lambda} \gimel_{\scriptscriptstyle A,k}^{\natural}(\tau) \! + \!
\widetilde{\Lambda}^{2} \gimel_{\scriptscriptstyle B,k}^{\natural}(\tau) \right)
\! \left(\mathrm{I} \! + \! \widetilde{\Lambda} \tau^{-1/3} \mathbb{P}_{0,k}^{
\natural}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}} \widehat{\mathbb{E}}_{0,
k}^{\natural}(\tau) \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left((\tau^{-1/3} \widetilde{\Lambda})^{2}
\widetilde{\mathbb{E}}_{0,k}^{\natural}(\tau) \right) \right),
\end{align}}
where
\begin{gather}
\hat{\mathbb{C}}_{0}^{0}(\tau) \! := \! -\mathrm{i} (\varepsilon b)^{-1/4}(\Delta G_{k}^{0}
(\tau))_{12}, \label{ellohk24} \\
\hat{\mathbb{D}}_{0}^{0}(\tau) \! := \! \mathrm{i} (\varepsilon b)^{1/4}(\Delta G_{k}^{0}
(\tau))_{21} \! - \! \dfrac{\mathfrak{A}_{k}}{\mathfrak{B}_{k}} \! \left(\dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k}}{\chi_{k}(\tau)} \! - \! 1 \right) \! \left(\dfrac{\mathrm{i} 2^{1/4}}{(\sqrt{3}
\! - \! 1)^{1/2}} \! + \! \mathrm{i} (\varepsilon b)^{1/4}(\Delta G_{k}^{0}(\tau))_{11}
\right), \label{ellohk25} \\
\hat{\psi}_{m,k}^{-1,\natural}(\tau) \! := \! \mathfrak{Q}_{\ast,k}^{-1}(\tau)
\hat{\psi}_{m,k}^{-1}(\tau) \mathfrak{Q}_{\ast,k}(\tau), \quad m \! = \! 1,2,3,
\label{ellohk26} \\
\mathfrak{Q}_{\ast,k}(\tau) \! := \! \mathfrak{Q}_{0,k}(\tau)(\mathrm{i} (\varepsilon b)^{
1/4})^{\sigma_{3}}(b(\tau))^{-\frac{1}{2} \sigma_{3}} \mathrm{e}^{\hat{\beta}_{k}(\tau)
\sigma_{3}}, \label{ellohk27} \\
\gimel_{\scriptscriptstyle A,k}^{\natural}(\tau) \! := \! \mathfrak{Q}_{\ast,k}^{-1}
(\tau) \gimel_{\scriptscriptstyle A,k}^{-1}(\tau) \mathfrak{Q}_{\ast,k}(\tau),
\label{ellohk28} \\
\gimel_{\scriptscriptstyle B,k}^{\natural}(\tau) \! := \! \mathfrak{Q}_{\ast,k}^{-1}
(\tau) \gimel_{\scriptscriptstyle B,k}^{-1}(\tau) \mathfrak{Q}_{\ast,k}(\tau),
\label{ellohk29} \\
\mathbb{P}_{0,k}^{\natural}(\tau) \! := \! (\mathrm{i} (\varepsilon b)^{1/4})^{-\ad
(\sigma_{3})}(b(\tau))^{\frac{1}{2} \ad (\sigma_{3})} \mathrm{e}^{-\hat{\beta}_{k}
(\tau) \ad (\sigma_{3})} \mathbb{P}_{0,k}(\tau), \label{ellohk30} \\
\widehat{\mathbb{E}}_{0,k}^{\natural}(\tau) \! := \! (\mathrm{i} (\varepsilon b)^{1/4})^{
-\ad (\sigma_{3})}(b(\tau))^{\frac{1}{2} \ad (\sigma_{3})} \mathrm{e}^{-\hat{\beta}_{k}
(\tau) \ad (\sigma_{3})} \widehat{\mathbb{E}}_{0,k}(\tau), \label{ellohk31} \\
\widetilde{\mathbb{E}}_{0,k}^{\natural}(\tau) \! := \! (\mathrm{i} (\varepsilon b)^{1/4})^{
-\ad (\sigma_{3})}(b(\tau))^{\frac{1}{2} \ad (\sigma_{3})} \mathrm{e}^{-\hat{\beta}_{k}
(\tau) \ad (\sigma_{3})} \widetilde{\mathbb{E}}_{0,k}(\tau). \label{ellohk32}
\end{gather}
Via the Conditions \eqref{iden5}, the restrictions \eqref{pc4} and \eqref{restr1},
the Definitions \eqref{eqpeetee}, \eqref{peekayity}, \eqref{prpr1}, \eqref{prpr3},
\eqref{prpr4}, \eqref{prcy57}, \eqref{prcy58}, \eqref{ellinfk13},
\eqref{ellinfk22}--\eqref{ellinfk24}, \eqref{ellohk6}--\eqref{ellohk8},
\eqref{ellohk18}--\eqref{ellohk21}, and \eqref{ellohk24}--\eqref{ellohk32},
and the Asymptotics \eqref{tr1}, \eqref{tr3}, \eqref{asympforf3},
\eqref{prcyg4}--\eqref{prcybk1}, \eqref{prcyomg1}, \eqref{prcyellok1},
\eqref{prcychik1}--\eqref{prcyell1k1}, \eqref{prcymuk1}, \eqref{prcyell2k1},
and \eqref{ellohk16}, upon imposing the Conditions \eqref{ellinfk2a}
and \eqref{ellinfk2b}, and defining
\begin{equation*}
J_{k}^{0} \! := \!
\begin{pmatrix}
\sqrt{3} \! - \! 1 & (2 \varepsilon b)^{1/2} \\
-(2/\varepsilon b)^{1/2} & \sqrt{3} \! - \! 1
\end{pmatrix}, \quad \pmb{\mathbb{T}}_{0,k}^{\natural} \! := \!
(\mathbb{T}_{ij,k}(-1;\tau))_{i,j=1,2}, \quad \mathbb{D}_{0,k}^{\natural}
\! := \! \mathfrak{B}_{k}^{\frac{1}{2} \sigma_{3}} \left(\dfrac{\mathrm{i} 2^{1/4}}{
(\sqrt{3} \! - \! 1)^{1/2}} \right)^{-\sigma_{3}},
\end{equation*}
one shows that (cf. Definition~\eqref{ellohk23}), for $k \! = \! \pm 1$,
\begin{align*}
\overset{\Ydown}{\mathbb{E}}_{\scriptscriptstyle
\mathfrak{L}^{0}_{k}}^{\raise-6.75pt\hbox{$\scriptstyle \leftslice$}}(\tau)
\underset{\tau \to +\infty}{=}& \, \left(\mathrm{I} \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}^{3} \sigma_{3}) \right) \! \left(
\mathrm{I} \! + \! \mathrm{e}^{-\hat{\beta}_{k}(\tau) \ad (\sigma_{3})} \!
\begin{pmatrix}
0 & -\frac{\mathrm{i} (\sqrt{3}-1)^{1/2} \hat{\mathbb{C}}_{0}^{0}(\tau)}{2^{1/4}
\hat{\mathbb{A}}_{0}^{0}(\tau)} \\
\frac{\mathrm{i} 2^{1/4} \hat{\mathbb{D}}_{0}^{0}(\tau)}{(\sqrt{3}-1)^{1/2}
\hat{\mathbb{B}}_{0}^{0}(\tau)} & 0
\end{pmatrix} \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \dfrac{1}{\widetilde{\Lambda}}
\hat{\psi}_{1,k}^{-1,\natural}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}^{2}}
\hat{\psi}_{2,k}^{-1,\natural}(\tau) \! + \! \mathcal{O} \! \left(\dfrac{1}{
\widetilde{\Lambda}^{3}} \hat{\psi}_{3,k}^{-1,\natural}(\tau) \right) \right)
\nonumber \\
\times& \, \left(\mathrm{I} \! + \! \mathcal{O} \! \left(\dfrac{\lvert \nu (k) \! +
\! 1 \rvert^{2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert p_{k}
(\tau) \rvert^{2}} \mathfrak{Q}_{\ast,k}^{-1}(\tau) \tilde{\mathfrak{C}}_{k}(\tau)
\mathfrak{Q}_{\ast,k}(\tau) \right) \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \!
\gimel_{\scriptscriptstyle A,k}^{\natural}(\tau) \widehat{\mathbb{E}}_{0,k}^{
\natural}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}} \widehat{\mathbb{E}}_{0,
k}^{\natural}(\tau) \! + \! \widetilde{\Lambda} \! \left(\tau^{-1/3} \mathbb{P}_{0,
k}^{\natural}(\tau) \! + \! \gimel_{\scriptscriptstyle A,k}^{\natural}(\tau) \right.
\right. \nonumber \\
+&\left. \left. \, \gimel_{\scriptscriptstyle B,k}^{\natural}(\tau) \widehat{
\mathbb{E}}_{0,k}^{\natural}(\tau) \right) \! + \! \widetilde{\Lambda}^{2} \! \left(
\tau^{-1/3} \gimel_{\scriptscriptstyle A,k}^{\natural}(\tau) \mathbb{P}_{0,k}^{
\natural}(\tau) \! + \! \gimel_{\scriptscriptstyle B,k}^{\natural}(\tau) \! + \!
\mathcal{O} \! \left(\tau^{-2/3} \widetilde{\mathbb{E}}_{0,k}^{\natural}(\tau)
\right) \right) \right. \nonumber \\
+&\left. \, \widetilde{\Lambda}^{3} \! \left(\tau^{-1/3}
\gimel_{\scriptscriptstyle B,k}^{\natural}(\tau) \mathbb{P}_{0,k}^{\natural}
(\tau) \! + \! \mathcal{O} \! \left(\tau^{-2/3} \gimel_{\scriptscriptstyle A,k}^{
\natural}(\tau) \widetilde{\mathbb{E}}_{0,k}^{\natural}(\tau) \right) \right)
\right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \left(\mathrm{I} \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}^{3} \sigma_{3}) \right) \! \left(
\mathrm{I} \! + \! \mathrm{e}^{-\hat{\beta}_{k}(\tau) \ad (\sigma_{3})} \!
\begin{pmatrix}
0 & -\frac{\mathrm{i} (\sqrt{3}-1)^{1/2} \hat{\mathbb{C}}_{0}^{0}(\tau)}{2^{1/4}
\hat{\mathbb{A}}_{0}^{0}(\tau)} \\
\frac{\mathrm{i} 2^{1/4} \hat{\mathbb{D}}_{0}^{0}(\tau)}{(\sqrt{3}-1)^{1/2}
\hat{\mathbb{B}}_{0}^{0}(\tau)} & 0
\end{pmatrix} \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \dfrac{1}{\widetilde{\Lambda}}
\hat{\psi}_{1,k}^{-1,\natural}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}^{2}}
\hat{\psi}_{2,k}^{-1,\natural}(\tau) \! + \! \mathcal{O} \! \left(\dfrac{1}{
\widetilde{\Lambda}^{3}} \hat{\psi}_{3,k}^{-1,\natural}(\tau) \right) \right)
\nonumber \\
\times& \, \left(\mathrm{I} \! + \!
\gimel_{\scriptscriptstyle A,k}^{\natural}(\tau) \widehat{\mathbb{E}}_{0,k}^{
\natural}(\tau) \! + \! \dfrac{1}{\widetilde{\Lambda}} \dfrac{1}{2 \sqrt{3}
(\sqrt{3} \! - \! 1)}(\mathrm{i} (\varepsilon b)^{1/4} \mathrm{e}^{\hat{\beta}_{k}(\tau)})^{
-\ad (\sigma_{3})} J_{k}^{0} \pmb{\mathbb{T}}_{0,k}^{\natural} \right. \nonumber \\
-&\left. \, \widetilde{\Lambda} \dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+}}{\chi_{k}(\tau)} \sigma_{3} \! + \! \mathcal{O} \! \left(\dfrac{\lvert \nu
(k) \! + \! 1 \rvert^{2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert
p_{k}(\tau) \rvert^{2}} \mathrm{e}^{-\hat{\beta}_{k}(\tau) \ad (\sigma_{3})}
\mathbb{D}_{0,k}^{\natural} \right. \right. \nonumber \\
\times&\left. \left. \, \tilde{\mathfrak{C}}_{k}(\tau)
(\mathbb{D}_{0,k}^{\natural})^{-1} \right) \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \left(\mathrm{I} \! + \! \mathcal{O}
(\tau^{-1/3} \widetilde{\Lambda}^{3} \sigma_{3}) \right) \! \left(
\mathrm{I} \! + \! \mathrm{e}^{-\hat{\beta}_{k}(\tau) \ad (\sigma_{3})} \!
\begin{pmatrix}
0 & -\frac{\mathrm{i} (\sqrt{3}-1)^{1/2} \hat{\mathbb{C}}_{0}^{0}(\tau)}{2^{1/4}
\hat{\mathbb{A}}_{0}^{0}(\tau)} \\
\frac{\mathrm{i} 2^{1/4} \hat{\mathbb{D}}_{0}^{0}(\tau)}{(\sqrt{3}-1)^{1/2}
\hat{\mathbb{B}}_{0}^{0}(\tau)} & 0
\end{pmatrix} \right) \nonumber \\
\times& \, \left(\mathrm{I} \! + \! \gimel_{\scriptscriptstyle A,k}^{\natural}
(\tau) \widehat{\mathbb{E}}_{0,k}^{\natural}(\tau) \! - \! \dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)} \hat{\psi}_{1,k}^{
-1,\natural}(\tau) \sigma_{3} \! - \! \widetilde{\Lambda} \dfrac{\mathrm{i} 4 \sqrt{3}
\mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)} \sigma_{3}
\right. \nonumber \\
+&\left. \, \dfrac{1}{\widetilde{\Lambda}} \! \left(\hat{\psi}_{1,k}^{-1,\natural}
(\tau) \! - \! \dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k}
\ell_{0,k}^{+}}{\chi_{k}(\tau)} \hat{\psi}_{2,k}^{-1,\natural}(\tau) \sigma_{3}
\! + \! \dfrac{(\sqrt{3} \! + \! 1)}{4 \sqrt{3}}(\mathrm{i} (\varepsilon b)^{1/4}
\mathrm{e}^{\hat{\beta}_{k}(\tau)})^{-\ad (\sigma_{3})} \right. \right. \nonumber \\
\times&\left. \left. \, J_{k}^{0} \pmb{\mathbb{T}}_{0,k}^{\natural} \! + \!
\hat{\psi}_{1,k}^{-1,\natural}(\tau) \gimel_{\scriptscriptstyle A,k}^{\natural}
(\tau) \widehat{\mathbb{E}}_{0,k}^{\natural}(\tau) \right) \! + \! \dfrac{1}{
\widetilde{\Lambda}^{2}} \! \left(\hat{\psi}_{2,k}^{-1,\natural}(\tau) \! + \!
\dfrac{1}{2 \sqrt{3}(\sqrt{3} \! - \! 1)} \right. \right. \nonumber \\
\times&\left. \left. \, \hat{\psi}_{1,k}^{-1,\natural}(\tau)(\mathrm{i} (\varepsilon b)^{1/4}
\mathrm{e}^{\hat{\beta}_{k}(\tau)})^{-\ad (\sigma_{3})}J_{k}^{0} \pmb{\mathbb{T}}_{0,k}^{
\natural} \! + \! \hat{\psi}_{2,k}^{-1,\natural}(\tau) \gimel_{\scriptscriptstyle A,k}^{
\natural}(\tau) \widehat{\mathbb{E}}_{0,k}^{\natural}(\tau) \right) \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left(\dfrac{\lvert \nu (k) \! + \! 1 \rvert^{2} \tau^{-
\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert p_{k}(\tau) \rvert^{2}}
\mathrm{e}^{-\hat{\beta}_{k}(\tau) \ad (\sigma_{3})} \mathbb{D}_{0,k}^{\natural}
\tilde{\mathfrak{C}}_{k}(\tau)(\mathbb{D}_{0,k}^{\natural})^{-1} \right) \right.
\nonumber \\
+&\left. \, \mathcal{O} \! \left(\dfrac{1}{\widetilde{\Lambda}} \dfrac{\lvert \nu (k)
\! + \! 1 \rvert^{2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert
p_{k}(\tau) \rvert^{2}} \hat{\psi}_{1,k}^{-1,\natural}(\tau) \mathrm{e}^{-\hat{\beta}_{k}
(\tau) \ad (\sigma_{3})} \mathbb{D}_{0,k}^{\natural} \tilde{\mathfrak{C}}_{k}
(\tau)(\mathbb{D}_{0,k}^{\natural})^{-1} \right) \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left(\dfrac{1}{\widetilde{\Lambda}^{2}} \dfrac{\lvert \nu
(k) \! + \! 1 \rvert^{2} \tau^{-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}}{\lvert
p_{k}(\tau) \rvert^{2}} \hat{\psi}_{2,k}^{-1,\natural}(\tau) \mathrm{e}^{-\hat{\beta}_{k}
(\tau) \ad (\sigma_{3})} \mathbb{D}_{0,k}^{\natural} \tilde{\mathfrak{C}}_{k}
(\tau)(\mathbb{D}_{0,k}^{\natural})^{-1} \right) \right. \nonumber \\
+&\left. \, \mathcal{O} \! \left(\dfrac{1}{\widetilde{\Lambda}^{2}} \dfrac{\mathrm{i} 4
\sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)}
\hat{\psi}_{3,k}^{-1,\natural}(\tau) \sigma_{3} \right) \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathrm{I} \! + \! \gimel_{\scriptscriptstyle A,
k}^{\natural}(\tau) \widehat{\mathbb{E}}_{0,k}^{\natural}(\tau) \! - \! \dfrac{\mathrm{i}
4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)}
\hat{\psi}_{1,k}^{-1,\natural}(\tau) \sigma_{3} \! + \! \mathcal{O}
(\tau^{-\frac{1}{3}+3 \delta_{k}} \sigma_{3}) \nonumber \\
+& \,
\begin{pmatrix}
0 & \mathcal{O}(\tau^{-2/3}) \\
\mathcal{O}(\tau^{-2/3}) & 0
\end{pmatrix} \! + \!
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+\delta_{k}}) & 0 \\
0 & \mathcal{O}(\tau^{-\frac{1}{3}+\delta_{k}})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
0 & \mathcal{O}(\tau^{-\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) \\
\mathcal{O}(\tau^{-\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}}) & 0
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}-\delta_{k}}(\nu (k) \! + \! 1)) & 0 \\
0 & \mathcal{O}(\tau^{-\frac{1}{3}-\delta_{k}}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}-\delta_{k}}) & \mathcal{O}(\tau^{-\frac{1}{3}
-\delta_{k}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}-\delta_{k}}) & \mathcal{O}(\tau^{-\frac{1}{3}
-\delta_{k}})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\delta_{k}}(\nu (k) \! + \! 1)) & \mathcal{O}(\tau^{-
\frac{2}{3}-\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) \\
\mathcal{O}(\tau^{-\frac{2}{3}-\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}})
& \mathcal{O}(\tau^{-\frac{4}{3}-\delta_{k}})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-2\delta_{k}}(\nu (k) \! + \! 1)) & 0 \\
0 & \mathcal{O}(\tau^{-2\delta_{k}}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) &
\mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}}) &
\mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{2}{3}-2\delta_{k}}(\nu (k) \! + \! 1)) & \mathcal{O}
(\tau^{-\frac{4}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) \\
\mathcal{O}(\tau^{-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{3+k}{2}}) &
\mathcal{O}(\tau^{-\frac{2}{3}-2\delta_{k}}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-2-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}) & \mathcal{O}
(\tau^{-3-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k) \! + \! 1)^{\frac{1
-k}{2}}) \\
\mathcal{O}(\tau^{-1-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k) \! + \!
1)^{\frac{1+k}{2}}) & \mathcal{O}(\tau^{-2-\epsilon_{\mathrm{\scriptscriptstyle TP}}
(k)})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-1-\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k)
\! + \! 1)) & \mathcal{O}(\tau^{-2-\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}
(k)}(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) \\
\mathcal{O}(\tau^{-2-\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}
(\nu (k) \! + \! 1)^{\frac{1+k}{2}}) & \mathcal{O}(\tau^{-3-\delta_{k}-
\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-2-2\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k)
\! + \! 1)) & \mathcal{O}(\tau^{-3-2\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}
(k)}(\nu (k) \! + \! 1)^{\frac{3-k}{2}}) \\
\mathcal{O}(\tau^{-1-2\delta_{k}-\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}
(\nu (k) \! + \! 1)^{\frac{3+k}{2}}) & \mathcal{O}(\tau^{-2-2\delta_{k}-
\epsilon_{\mathrm{\scriptscriptstyle TP}}(k)}(\nu (k) \! + \! 1))
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
0 & \mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1-k}{2}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}-2\delta_{k}}(\nu (k) \! + \! 1)^{\frac{1+k}{2}}) & 0
\end{pmatrix} \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathrm{I} \! + \! \gimel_{\scriptscriptstyle A,
k}^{\natural}(\tau) \widehat{\mathbb{E}}_{0,k}^{\natural}(\tau) \! - \! \dfrac{\mathrm{i}
4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{\chi_{k}(\tau)}
\hat{\psi}_{1,k}^{-1,\natural}(\tau) \sigma_{3} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+3 \delta_{k}}) & \mathcal{O}(\tau^{-2/3}) \\
\mathcal{O}(\tau^{-2/3}) & \mathcal{O}(\tau^{-\frac{1}{3}+3 \delta_{k}})
\end{pmatrix} \nonumber \\
+& \,
\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+\delta_{k}}) & \mathcal{O}(\tau^{-\frac{1}{3}
(\frac{1-k}{2})-\delta_{k}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}(\frac{1+k}{2})-\delta_{k}}) & \mathcal{O}
(\tau^{-\frac{1}{3}+\delta_{k}})
\end{pmatrix} \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathrm{I} \! + \!
\underbrace{\gimel_{\scriptscriptstyle A,k}^{\natural}(\tau) \widehat{\mathbb{E}}_{0,k}^{
\natural}(\tau) \! - \! \dfrac{\mathrm{i} 4 \sqrt{3} \mathcal{Z}_{k} \mathfrak{A}_{k} \ell_{0,k}^{+}}{
\chi_{k}(\tau)} \hat{\psi}_{1,k}^{-1,\natural}(\tau) \sigma_{3}}_{=: \,
\mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{0}(\tau)} \nonumber \\
+& \, \underbrace{\begin{pmatrix}
\mathcal{O}(\tau^{-\frac{1}{3}+3\delta_{k}}) & \mathcal{O}(\tau^{-\frac{1}{3}
(\frac{1-k}{2})-\delta_{k}}) \\
\mathcal{O}(\tau^{-\frac{1}{3}(\frac{1+k}{2})-\delta_{k}}) & \mathcal{O}
(\tau^{-\frac{1}{3}+3\delta_{k}})
\end{pmatrix}}_{=: \, \mathcal{O}(\mathbb{E}^{0}_{k}(\tau))} \nonumber \\
\underset{\tau \to +\infty}{=}& \, (\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle
\mathcal{N}},k}^{0}(\tau)) \! \left(\mathrm{I} \! + \! \underbrace{(\mathrm{I} \! + \!
\mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{0}(\tau))^{-1}}_{= \, \mathcal{O}(1)}
\, \mathcal{O}(\mathbb{E}^{0}_{k}(\tau)) \right) \quad \Rightarrow \nonumber
\end{align*}
\begin{equation} \label{ellohk33}
\overset{\Ydown}{\mathbb{E}}_{\scriptscriptstyle
\mathfrak{L}^{0}_{k}}^{\raise-6.75pt\hbox{$\scriptstyle \leftslice$}}(\tau)
\underset{\tau \to +\infty}{=} (\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle
\mathcal{N}},k}^{0}(\tau))(\mathrm{I} \! + \! \mathcal{O}(\mathbb{E}^{0}_{k}
(\tau))).
\end{equation}
Thus, via Asymptotics~\eqref{ellohk22} and~\eqref{ellohk33}, one arrives at the
result stated in the lemma. \hfill $\qed$
\begin{dddd} \label{theor3.3.1}
Assume that the Conditions~\eqref{iden5}, \eqref{pc4}, \eqref{restr1},
\eqref{ellinfk2a}, and~\eqref{ellinfk2b} are valid. Then, the connection matrix has
the following asymptotics:
\begin{equation} \label{jeek1}
G_{k} \underset{\tau \to +\infty}{=} \widetilde{G}(k) \widehat{\mathscr{G}}(k)
(\mathrm{I} \! + \! \mathcal{O}(\mathbb{E}_{k}^{\scriptscriptstyle G_{k}}(\tau))),
\quad k \! = \! \pm 1,
\end{equation}
where
\begin{gather}
\widetilde{G}(k) \! := \! (\mathbb{S}_{k}^{\ast})^{-1} \mathrm{G}^{\ast}(k), \label{jeek2} \\
\widehat{\mathscr{G}}(k) \! := \! (\mathrm{G}^{\ast}(k))^{-1}(\mathrm{I} \! + \!
\mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{0}(\tau))^{-1} \mathrm{G}^{\ast}
(k)(\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{\infty}
(\tau)), \label{jeek3}
\end{gather}
with $\mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{\infty}(\tau)$,
$\mathbb{S}_{k}^{\ast}$, and $\mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{0}
(\tau)$ defined by Equations~\eqref{ellinfk9}, \eqref{ellohk9}, and~\eqref{ellohk10},
respectively, and
\begin{equation} \label{jeek4}
\mathrm{G}^{\ast}(k) \! = \!
\begin{pmatrix}
\frac{\hat{\mathbb{G}}_{11}(k) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{
0}^{0}(\tau)} \mathrm{e}^{-\Delta \tilde{\mathfrak{z}}_{k}(\tau)-\Delta \hat{\mathfrak{z}}_{k}
(\tau)} & \frac{\hat{\mathbb{G}}_{12}(k) \hat{\mathbb{A}}_{0}^{\infty}(\tau)}{\hat{
\mathbb{A}}_{0}^{0}(\tau)} \mathrm{e}^{\Delta \tilde{\mathfrak{z}}_{k}(\tau)-\Delta \hat{
\mathfrak{z}}_{k}(\tau)} \\
\frac{\hat{\mathbb{G}}_{21}(k) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{B}}_{
0}^{0}(\tau)} \mathrm{e}^{-\Delta \tilde{\mathfrak{z}}_{k}(\tau)+\Delta \hat{\mathfrak{z}}_{k}
(\tau)} & \frac{\hat{\mathbb{G}}_{22}(k) \hat{\mathbb{A}}_{0}^{\infty}(\tau)}{\hat{
\mathbb{B}}_{0}^{0}(\tau)} \mathrm{e}^{\Delta \tilde{\mathfrak{z}}_{k}(\tau)+\Delta \hat{
\mathfrak{z}}_{k}(\tau)}
\end{pmatrix},
\end{equation}
where
\begin{gather}
\hat{\mathbb{G}}_{11}(k) \! := \! -\dfrac{\mathrm{i} \sqrt{2 \pi} \, p_{k}(\tau) \mathfrak{B}_{k}
\sqrt{\smash[b]{b(\tau)}} \, \mathrm{e}^{\mathrm{i} \pi (\nu (k)+1)}}{(\varepsilon b)^{1/4}
(2 \! + \! \sqrt{3})^{1/2}(2 \mu_{k}(\tau))^{1/2} \Gamma (-\nu (k))}
\exp \! \left(-\tilde{\mathfrak{z}}_{k}^{0}(\tau) \! - \! \hat{\mathfrak{z}}_{k}^{0}
(\tau) \right), \label{jeek5} \\
\hat{\mathbb{G}}_{12}(k) \! := \! -\dfrac{\mathrm{i} (\varepsilon b)^{1/4}}{\sqrt{
\smash[b]{b(\tau)}}} \exp \! \left(\tilde{\mathfrak{z}}_{k}^{0}(\tau) \! - \!
\hat{\mathfrak{z}}_{k}^{0}(\tau) \right), \label{jeek6} \\
\hat{\mathbb{G}}_{21}(k) \! := \! -\dfrac{\mathrm{i} \sqrt{\smash[b]{b(\tau)}} \, \mathrm{e}^{-2
\pi \mathrm{i} (\nu (k)+1)}}{(\varepsilon b)^{1/4}} \exp \! \left(-\tilde{\mathfrak{z}}_{k}^{0}
(\tau) \! + \! \hat{\mathfrak{z}}_{k}^{0}(\tau) \right), \label{jeek7} \\
\hat{\mathbb{G}}_{22}(k) \! := \! -\dfrac{\sqrt{2 \pi} \, (\varepsilon b)^{1/4}(2 \!
+ \! \sqrt{3})^{1/2}(2 \mu_{k}(\tau))^{1/2} \mathrm{e}^{-2 \pi \mathrm{i} (\nu (k)+1)}}{p_{k}
(\tau) \mathfrak{B}_{k} \sqrt{\smash[b]{b(\tau)}} \, \Gamma (\nu (k) \! + \! 1)}
\exp \! \left(\tilde{\mathfrak{z}}_{k}^{0}(\tau) \! + \! \hat{\mathfrak{z}}_{k}^{0}
(\tau) \right), \label{jeek8}
\end{gather}
with $\tilde{\mathfrak{z}}_{k}^{0}(\tau)$, $\Delta \tilde{\mathfrak{z}}_{k}(\tau)$,
$\hat{\mathbb{A}}_{0}^{\infty}(\tau)$, $\hat{\mathbb{B}}_{0}^{\infty}(\tau)$,
$\hat{\mathfrak{z}}_{k}^{0}(\tau)$, $\Delta \hat{\mathfrak{z}}_{k}(\tau)$,
$\hat{\mathbb{A}}_{0}^{0}(\tau)$, and $\hat{\mathbb{B}}_{0}^{0}(\tau)$ defined
by Equations~\eqref{ellinfk4}, \eqref{ellinfk5}, \eqref{ellinfk6}, \eqref{ellinfk7},
\eqref{ellohk4}, \eqref{ellohk5}, \eqref{ellohk6}, and~\eqref{ellohk7}, respectively,
and
\begin{equation} \label{jeek9}
\mathcal{O}(\mathbb{E}_{k}^{\scriptscriptstyle G_{k}}(\tau)) \underset{\tau \to
+\infty}{:=} \mathcal{O} \! \left(\mathbb{E}_{k}^{\infty}(\tau) \right) \!
+ \! \mathcal{O} \! \left((\widetilde{G}(k) \widehat{\mathscr{G}}(k))^{-1}
\mathbb{E}_{k}^{0}(\tau) \widetilde{G}(k) \widehat{\mathscr{G}}(k) \right),
\end{equation}
with the asymptotics $\mathcal{O}(\mathbb{E}_{k}^{\infty}(\tau))$ and
$\mathcal{O}(\mathbb{E}_{k}^{0}(\tau))$ defined by Equations~\eqref{ellinfk10}
and~\eqref{ellohk11}, respectively.
\end{dddd}
\emph{Proof}. Mimicking the calculations subsumed in the proof of Theorem~3.4.1
of \cite{av2}, one shows that
\begin{equation} \label{jeek10}
G_{k} \! = \! (\mathfrak{L}_{k}^{0}(\tau))^{-1} \mathfrak{L}_{k}^{\infty}(\tau),
\quad k \! = \! \pm 1.
\end{equation}
{}From Equations~\eqref{ellinfk3}--\eqref{ellinfk10}, \eqref{ellohk3}--\eqref{ellohk11},
and \eqref{jeek10}, one arrives at
\begin{align} \label{jeek11}
G_{k} \underset{\tau \to +\infty}{=}& \, (\mathrm{I} \! + \! \mathcal{O}
(\mathbb{E}_{k}^{0}(\tau)))(\mathbb{S}_{k}^{\ast})^{-1}(\mathrm{I} \!
+ \! \mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{0}(\tau))^{-1}
\mathrm{e}^{-\Delta \hat{\mathfrak{z}}_{k}(\tau) \sigma_{3}} \operatorname{diag}
((\hat{\mathbb{A}}_{0}^{0}(\tau))^{-1},(\hat{\mathbb{B}}_{0}^{0}(\tau))^{-1})
\nonumber \\
\times& \left(\dfrac{\mathrm{i} 2^{1/4}}{(\sqrt{3} \! - \! 1)^{1/2} \sqrt{\smash[b]{
\mathfrak{B}_{k}}}} \right)^{-\sigma_{3}} \mathrm{e}^{-\hat{\mathfrak{z}}_{k}^{0}
(\tau) \sigma_{3}} \mathcal{R}_{m_{0}}(k)(\mathcal{R}_{m_{\infty}}(k))^{-1}
\mathrm{e}^{\tilde{\mathfrak{z}}_{k}^{0}(\tau) \sigma_{3}} \left(\dfrac{(\varepsilon b)^{1/4}
(\sqrt{3} \! + \! 1)^{1/2}}{2^{1/4} \sqrt{\smash[b]{\mathfrak{B}_{k}}}
\sqrt{\smash[b]{b(\tau)}}} \right)^{\sigma_{3}} \nonumber \\
\times& \, \mathrm{i} \sigma_{2} \mathrm{e}^{-\Delta \tilde{\mathfrak{z}}_{k}(\tau) \sigma_{3}}
\operatorname{diag}(\hat{\mathbb{B}}_{0}^{\infty}(\tau),\hat{\mathbb{A}}_{0}^{
\infty}(\tau))(\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle \mathcal{N}},
k}^{\infty}(\tau))(\mathrm{I} \! + \! \mathcal{O}(\mathbb{E}_{k}^{\infty}(\tau))):
\end{align}
taking $(m_{\infty},m_{0}) \! = \! (0,2)$, that is, $\Delta \arg (\widetilde{\Lambda})
\! := \! \pi (m_{0} \! - \! m_{\infty})/2 \! = \! \pi$, and using the definitions of
$\mathcal{R}_{0}(k)$ and $\mathcal{R}_{2}(k)$ given in Remark~\ref{leedasyue},
one arrives at, via Equation \eqref{jeek11} and the reflection formula $\Gamma (z)
\Gamma (1 \! - \! z) \! = \! \pi/\sin (\pi z)$, the result stated in the theorem.
\hfill $\qed$
\section{The Inverse Monodromy Problem: Asymptotic Solution} \label{finalsec}
In Subsection~\ref{sec3.3}, the corresponding connection matrices, $G_{k}$, $k \!
\in \! \lbrace \pm 1 \rbrace$, were calculated asymptotically (as $\tau \! \to \!
+\infty$ with $\varepsilon b \! > \! 0)$ under the assumption of the validity of
the Conditions~\eqref{iden5}, \eqref{pc4}, \eqref{restr1}, \eqref{ellinfk2a},
and~\eqref{ellinfk2b}. Using these conditions, one can derive the $\tau$-dependent
class(es) of functions $G_{k}$ belongs to: this, most general, approach will not be
adopted here; rather, the isomonodromy condition will be evoked on $G_{k}$, that
is, $g_{ij} \! := \! (G_{k})_{ij}$, $i,j \! \in \! \lbrace 1,2 \rbrace$, are $\mathcal{O}(1)$
constants, and then the formula for $G_{k}$ will be inverted in order to derive the
coefficient functions of Equation~\eqref{eq3.3}, after which, it will be verified that
they satisfy all of the imposed conditions for this isomonodromy case. The latter
procedure gives rise to explicit asymptotic formulae for the coefficient functions of
Equation~\eqref{eq3.3}, leading to asymptotics of the solution of the system of
isomonodromy deformations~\eqref{newlax8},\footnote{Via the
Definitions~\eqref{newlax2}, also the asymptotics of the solution of
the---original---system of isomonodromy deformations~\eqref{eq1.4}.} and, in
turn, defines asymptotics of the solution $u(\tau)$ of the DP3E~\eqref{eq1.1} and
the related, auxiliary functions $\mathcal{H}(\tau)$, $f_{\pm}(\tau)$, $\sigma
(\tau)$,\footnote{See the Definitions~\eqref{eqh1}, \eqref{hatsoff7}, \eqref{pga3},
and~\eqref{thmk23}, respectively.} and $\hat{\varphi}(\tau)$.
\begin{ccccc} \label{ginversion}
Let $g_{ij} \! := \! (G_{k})_{ij}$, $i,j \! \in \! \lbrace 1,2 \rbrace$, $k \! = \! \pm 1$,
denote the matrix elements of the corresponding connection matrices. Assume that
all of the conditions stated in Theorem~\ref{theor3.3.1} are valid. For $k \! = \! +1$,
let $g_{11}g_{12}g_{21} \! \neq \! 0$ and $g_{22} \! = \! 0$, and, for $k \! = \! -1$,
let $g_{12}g_{21}g_{22} \! \neq \! 0$ and $g_{11} \! = \! 0$. Then, for $0 \! < \! \delta
\! < \! \delta_{k} \! < \! 1/24$ and $k \! \in \! \lbrace \pm 1 \rbrace$, the functions
$v_{0}(\tau)$, $\tilde{r}_{0}(\tau)$,\footnote{See the Asymptotics~\eqref{tr1}
and~\eqref{tr3}, respectively.} and $b(\tau)$ have the following asymptotics:
\begin{align}
v_{0}(\tau) \! := \! v_{0,k}(\tau) \underset{\tau \to +\infty}{=} \sum_{m=0}^{\infty}
\dfrac{\mathfrak{u}_{m}(k)}{(\tau^{1/3})^{m+1}}
+ &\dfrac{\mathrm{i} \mathrm{e}^{\mathrm{i} \pi k/4} \mathrm{e}^{-\mathrm{i} \pi k/3}(\mathscr{P}_{a})^{k}(s_{0}^{0}
\! - \! \mathrm{i} \mathrm{e}^{-\pi a})}{\sqrt{\smash[b]{2 \pi}} \, 3^{1/4}(\varepsilon b)^{1/6}}
\mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)} \nonumber \\
\times& \left(1 \! + \! \mathcal{O}(\tau^{-1/3}) \right), \label{geek1} \\
\tilde{r}_{0}(\tau) \! := \! \tilde{r}_{0,k}(\tau) \underset{\tau \to +\infty}{=}
\sum_{m=0}^{\infty} \dfrac{\mathfrak{r}_{m}(k)}{(\tau^{1/3})^{m+1}} +
&\dfrac{\mathrm{i} k(\sqrt{3} \! + \! 1)^{k} \mathrm{e}^{\mathrm{i} \pi k/4} \mathrm{e}^{-\mathrm{i} \pi k/3}
(\mathscr{P}_{a})^{k}(s_{0}^{0} \! - \! \mathrm{i} \mathrm{e}^{-\pi a})}{\sqrt{\smash[b]{\pi}}
\, 2^{(k-2)/2}3^{1/4}(\varepsilon b)^{1/6}} \nonumber \\
\times& \, \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)} \! \left(
1 \! + \! \mathcal{O}(\tau^{-1/3}) \right), \label{geek2}
\end{align}
and
\begin{equation}
\sqrt{\smash[b]{b(\tau)}} \underset{\tau \to +\infty}{=} \mathfrak{b}(k)
(\varepsilon b)^{1/4}(2^{-1/2} \alpha_{k})^{\mathrm{i} (a-\mathrm{i}/2)} \tau^{-\mathrm{i} a/6} \exp
\! \left(\frac{3}{4}(k \sqrt{3} \! + \! \mathrm{i})(\varepsilon b)^{1/3} \tau^{2/3} \! + \!
\mathcal{O}(\tau^{-\delta_{k}}) \right), \label{geek3}
\end{equation}
where $\vartheta (\tau)$ and $\beta (\tau)$ are defined in Equations~\eqref{thmk12},
\begin{gather}
\mathscr{P}_{a} \! := \! (2 \! + \! \sqrt{3})^{\mathrm{i} a}, \label{geek4} \\
\mathfrak{b}(k) \! = \!
\begin{cases}
g_{11} \mathrm{e}^{\pi a}, &\text{$k \! = \! +1$,} \\
-g_{22}^{-1} \mathrm{e}^{-\pi a}, &\text{$k \! = \! -1$,}
\end{cases} \label{eequeb}
\end{gather}
and the expansion coefficients $\mathfrak{u}_{m}(k)$ (resp., $\mathfrak{r}_{m}(k))$,
$m \! \in \! \mathbb{Z}_{+}$, are given in Equations~\eqref{thmk2}--\eqref{thmk10}
(resp., \eqref{thmk15} and~\eqref{thmk16}$)$.\footnote{Trans-series asymptotics
(as $\tau \! \to \! +\infty$ with $\varepsilon b \! > \! 0$) for $b(\tau)$ are
given in the proof of Theorem~\ref{pfeetotsa} below; see, in particular,
Equations~\eqref{totsa8}, \eqref{totsa9}, and~\eqref{totsa25}.}
\end{ccccc}
\emph{Proof}. The scheme of the proof is, \emph{mutatis mutandis}, similar for both
cases $(k \! = \! \pm 1)$; therefore, without loss of generality, the proof for the case
$k \! = \! +1$ is presented: the case $k \! = \! -1$ is proved analogously.
It follows {}from the Asymptotics~\eqref{tr1}, \eqref{tr3}, and~\eqref{prcybk1},
the Conditions~\eqref{ellinfk2a} and \eqref{ellinfk2b}, and the
Definitions~\eqref{ellinfk4} and~\eqref{ellohk4} that, for $k \! = \! +1$, $p_{1}
(\tau) \! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{1/3} \mathrm{e}^{-\beta (\tau)})$ and
$\sqrt{\smash[b]{b(\tau)}} \! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{-\frac{\mathrm{i} a}{6}}
\mathrm{e}^{\frac{3 \sqrt{3}}{4}(\varepsilon b)^{1/3} \tau^{2/3}})$, where $\vartheta
(\tau)$ and $\beta (\tau)$ are defined in Equations~\eqref{thmk12}. {}From the
Definitions~\eqref{prpr1}, \eqref{prcy46}, \eqref{prcy54}, \eqref{prcy57}, and \eqref{prcy58},
and the Asymptotics~\eqref{tr1}, \eqref{tr3}, \eqref{prcyak1}, \eqref{prcyomg1},
\eqref{prcychik1}--\eqref{prcyell1k1}, \eqref{prcymuk1}, and \eqref{prcyell2k1}, it
follows, via a lengthy linearisation and inversion argument,\footnote{That is, retaining
only those terms that are $\mathcal{O}(\tau^{-1/3})$.} in conjunction with the latter
asymptotics for $p_{1}(\tau)$, that, for $k \! = \! +1$,
\begin{align}
\mathfrak{r}_{0}(1) \tau^{-1/3} \! + \! \mathcal{O}(\tau^{-2/3})
\underset{\tau \to +\infty}{=}& \, \dfrac{1}{2 \sqrt{3}} \! \left(\dfrac{2
(a \! - \! \mathrm{i}/2) \tau^{-1/3}}{\sqrt{3} \alpha_{1}^{2}} \! - \! \dfrac{48
\sqrt{3}(p_{1}(\tau) \! - \! 1)(\nu (1) \! + \! 1)}{p_{1}(\tau) \tau^{-1/3}} \! - \!
\dfrac{\mathrm{i} p_{1}(\tau) \tau^{-1/3}}{3 \alpha_{1}^{2}(p_{1}(\tau) \! - \! 1)} \right),
\label{geek5} \\
\mathfrak{u}_{0}(1) \tau^{-1/3} \! + \! \mathcal{O}(\tau^{-2/3})
\underset{\tau \to +\infty}{=}& \, \dfrac{1}{8 \sqrt{3}} \! \left(\dfrac{4(a \! - \!
\mathrm{i}/2) \tau^{-1/3}}{\sqrt{3} \alpha_{1}^{2}} \! + \! \dfrac{48 \sqrt{3}(\sqrt{3}
\! + \! 1)(p_{1}(\tau) \! - \! 1)(\nu (1) \! + \! 1)}{p_{1}(\tau) \tau^{-1/3}}
\right. \nonumber \\
+&\left. \, \dfrac{\mathrm{i} \tau^{-1/3}}{3 \alpha_{1}^{2}} \! \left(\sqrt{3} \! +
\! 1 \! - \! \dfrac{(\sqrt{3} \! - \! 1)}{p_{1}(\tau) \! - \! 1} \right) \right),
\label{geek6}
\end{align}
where
\begin{equation} \label{geek7}
-\dfrac{(\nu (1) \! + \! 1)}{p_{1}(\tau)} \! = \! \dfrac{q_{1}(\tau)}{2 \mu_{1}(\tau)},
\end{equation}
with
\begin{gather}
q_{1}(\tau) \underset{\tau \to +\infty}{=} c_{q}^{\ast}(1) \tau^{-2/3} \! + \!
\mathcal{O}(\tau^{-1}), \label{geek8} \\
2 \mu_{1}(\tau) \underset{\tau \to +\infty}{=} \mathrm{i} 8 \sqrt{3}(1 \! + \! \mathcal{O}
(\tau^{-2/3})), \label{geek9}
\end{gather}
where $c_{q}^{\ast}(1)$ is some to-be-determined coefficient. Recalling {}from
Propositions~\ref{recursys} and~\ref{proprr}, respectively, that $\mathfrak{u}_{0}
(1) \! = \! a/6 \alpha_{1}^{2}$ and $\mathfrak{r}_{0}(1) \! = \! (a \! - \! \mathrm{i}/2)/3
\alpha_{1}^{2}$, it follows via the asymptotic relations~\eqref{geek5}
and~\eqref{geek6}, Equation~\eqref{geek7}, the Asymptotics~\eqref{geek8}
and~\eqref{geek9}, and the asymptotics for $p_{1}(\tau)$ stated above that
\begin{gather}
\dfrac{(a \! - \! \mathrm{i}/2) \tau^{-1/3}}{3 \alpha_{1}^{2}} \! + \! \mathcal{O}
(\tau^{-2/3}) \underset{\tau \to +\infty}{=} \dfrac{\tau^{-1/3}}{2 \sqrt{3}}
\! \left(\dfrac{2(a \! - \! \mathrm{i}/2)}{\sqrt{3} \alpha_{1}^{2}} \! + \! \mathrm{i}
6c_{q}^{\ast}(1) \right) \! + \! \mathcal{O}(\tau^{-2/3}), \label{geek10} \\
\dfrac{a \tau^{-1/3}}{6 \alpha_{1}^{2}} \! + \! \mathcal{O}(\tau^{-2/3})
\underset{\tau \to +\infty}{=} \dfrac{\tau^{-1/3}}{8 \sqrt{3}} \! \left(
\dfrac{4a}{\sqrt{3} \alpha_{1}^{2}} \! - \! \mathrm{i} 6 (\sqrt{3} \! + \! 1)
c_{q}^{\ast}(1) \right) \! + \! \mathcal{O}(\tau^{-2/3}), \label{geek11}
\end{gather}
whence
\begin{equation} \label{geek12}
c_{q}^{\ast}(1) \! = \! 0.
\end{equation}
Thus, {}from Equation~\eqref{geek7}, the Asymptotics~\eqref{geek8}
and~\eqref{geek9}, the Relation~\eqref{geek12}, and the asymptotics
(see above) $p_{1}(\tau) \! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{1/3}
\mathrm{e}^{-\beta (\tau)})$, one deduces that, for $k \! = \! +1$,\footnote{Even though
this realisation is not utilised anywhere in this work, it turns out that $\nu (k) \!
+ \! 1$ has the asymptotic trans-series expansion
\begin{equation*}
\nu (k) \! + \! 1 \underset{\tau \to +\infty}{=} \sum_{j \in \mathbb{Z}_{+}} \,
\sum_{m \in \mathbb{N}} \hat{\mathfrak{s}}_{j,k}(m)(\tau^{-1/3})^{j}
(\mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)})^{m}, \quad k \! = \! \pm 1,
\end{equation*}
for certain coefficients $\hat{\mathfrak{s}}_{j,k}(m) \colon \mathbb{Z}_{+} \times
\{\pm 1\} \times \mathbb{N} \! \to \! \mathbb{C}$, where, in particular,
$\hat{\mathfrak{s}}_{0,k}(1) \! = \! \hat{\mathfrak{s}}_{1,k}(1) \! = \! 0$.}
\begin{equation} \label{geek13}
\nu (1) \! + \! 1 \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-2/3}
\mathrm{e}^{-\beta (\tau)}).
\end{equation}
{}From the corresponding $(k \! = \! +1)$ Asymptotics~\eqref{tr1}
and~\eqref{tr3}, the Definitions~\eqref{eqpeetee}, \eqref{ellinfk5},
and~\eqref{ellohk5}, the expansion $\mathrm{e}^{z} \! = \! \sum_{m=0}^{\infty}
z^{m}/m!$, and the leading-order Asymptotics~\eqref{geek9}
and~\eqref{geek13}, one shows that, for $k \! = \! +1$,
\begin{gather}
\mathrm{e}^{\pm \Delta \tilde{\mathfrak{z}}_{1}(\tau)} \underset{\tau \to +\infty}{=}
1 \! + \! \tau^{-2/3} \sum_{m=0}^{\infty} \tilde{\zeta}_{m}^{\pm}(1)(\tau^{-1/3})^{m}
\! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}), \label{geek14} \\
\mathrm{e}^{\pm \Delta \hat{\mathfrak{z}}_{1}(\tau)} \underset{\tau \to +\infty}{=}
1 \! + \! \tau^{-2/3} \sum_{m=0}^{\infty} \hat{\zeta}_{m}^{\pm}(1)(\tau^{-1/3})^{m}
\! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}), \label{geek15}
\end{gather}
for certain coefficients $\tilde{\zeta}_{m}^{\pm}(1)$ and $\hat{\zeta}_{m}^{\pm}
(1)$. {}From the corresponding $(k \! = \! +1)$ Asymptotics~\eqref{tr1}
and~\eqref{tr3}, \eqref{prcyomg1}, \eqref{prcychik1}, \eqref{prcyzeek1},
and~\eqref{prcymuk1}, the Definition~\eqref{prcy57}, and the asymptotics $p_{1}
(\tau) \! =_{\tau \to +\infty} \! \mathcal{O}(\tau^{1/3} \mathrm{e}^{-\beta (\tau)})$, it
follows that, for $k \! = \! +1$,
\begin{equation} \label{geek16}
\dfrac{1}{(2 \mu_{1}(\tau))^{1/2}} \underset{\tau \to +\infty}{=} \dfrac{
\mathrm{e}^{-\mathrm{i} \pi/4}}{2^{3/2}3^{1/4}} \! \left(1 \! + \! \tau^{-2/3} \sum_{m
=0}^{\infty} \alpha_{m}^{\sharp}(1)(\tau^{-1/3})^{m} \! + \! \mathcal{O}
(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}) \right),
\end{equation}
for certain coefficients $\alpha_{m}^{\sharp}(1)$. {}From the corresponding $(k \!
= \! +1)$ Asymptotics~\eqref{tr1}, \eqref{tr3}, \eqref{prcyg4}--\eqref{prcybk1},
\eqref{prcyomg1}, \eqref{prcychik1}, and~\eqref{prcyzeek1}, the
Definitions~\eqref{prcy57}, \eqref{ellinfk6}--\eqref{ellinfk8}, and
\eqref{ellohk6}--\eqref{ellohk8}, and the above asymptotics for $p_{1}(\tau)$,
one deduces that, for $k \! = \! +1$,\footnote{Recall that $v_{0}(\tau) \! :=
\! v_{0,1}(\tau)$ and $\tilde{r}_{0}(\tau) \! := \! \tilde{r}_{0,1}(\tau)$.}
\begin{align}
\hat{\mathbb{A}}_{0}^{\infty}(\tau) \underset{\tau \to +\infty}{=}& \, 1 \! - \!
\dfrac{\tilde{r}_{0,1}(\tau) \tau^{-1/3}}{8 \sqrt{3}} \! \left(1 \! + \! \mathcal{O}
(\tilde{r}_{0,1}(\tau) \tau^{-1/3}) \right) \underset{\tau \to +\infty}{=} 1 \! + \!
\mathcal{O}(\tau^{-2/3}), \label{geek17} \\
\hat{\mathbb{B}}_{0}^{\infty}(\tau) \underset{\tau \to +\infty}{=}& \, 1 \! + \!
\dfrac{\tilde{r}_{0,1}(\tau) \tau^{-1/3}}{8 \sqrt{3}} \! \left(1 \! + \! \mathcal{O}
(\tilde{r}_{0,1}(\tau) \tau^{-1/3}) \right) \! \left(1 \! - \! \dfrac{(a \! - \! \mathrm{i}/2)
\tau^{-2/3}}{72 \sqrt{3} \alpha_{1}^{2}} \right. \nonumber \\
\times&\left. \dfrac{(-\alpha_{1}^{2}(8(v_{0,1}(\tau))^{2} \! + \! 4v_{0,1}(\tau)
\tilde{r}_{0,1}(\tau) \! - \! (\tilde{r}_{0,1}(\tau))^{2}) \! + \! 4(a \! - \! \mathrm{i}/2)
v_{0,1}(\tau) \tau^{-1/3})}{\left(\frac{\alpha_{1}}{2}(4v_{0,1}(\tau) \! + \! (\sqrt{3}
\! + \! 1) \tilde{r}_{0,1}(\tau)) \! - \! \frac{(\sqrt{3}+1)(a-\mathrm{i}/2) \tau^{-1/3}}{2
\sqrt{3} \alpha_{1}} \right)^{2}} \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, 1 \! + \! \mathcal{O}(\tau^{-2/3}), \label{geek18} \\
\hat{\mathbb{A}}_{0}^{0}(\tau) \underset{\tau \to +\infty}{=}& \, 1 \! - \!
\dfrac{\tilde{r}_{0,1}(\tau) \tau^{-1/3}}{8 \sqrt{3}} \! \left(1 \! + \! \mathcal{O}
(\tilde{r}_{0,1}(\tau) \tau^{-1/3}) \right) \underset{\tau \to +\infty}{=} 1 \! + \!
\mathcal{O}(\tau^{-2/3}), \label{geek19} \\
\hat{\mathbb{B}}_{0}^{0}(\tau) \underset{\tau \to +\infty}{=}& \, 1 \! + \!
\dfrac{\tilde{r}_{0,1}(\tau) \tau^{-1/3}}{8 \sqrt{3}} \! \left(1 \! + \! \mathcal{O}
(\tilde{r}_{0,1}(\tau) \tau^{-1/3}) \right) \! \left(1 \! - \! \dfrac{(a \! - \! \mathrm{i}/2)
\tau^{-2/3}}{72 \sqrt{3} \alpha_{1}^{2}} \right. \nonumber \\
\times&\left. \dfrac{(-\alpha_{1}^{2}(8(v_{0,1}(\tau))^{2} \! + \! 4v_{0,1}(\tau)
\tilde{r}_{0,1}(\tau) \! - \! (\tilde{r}_{0,1}(\tau))^{2}) \! + \! 4(a \! - \! \mathrm{i}/2)
v_{0,1}(\tau) \tau^{-1/3})}{\left(\frac{\alpha_{1}}{2}(4v_{0,1}(\tau) \! + \! (\sqrt{3}
\! + \! 1) \tilde{r}_{0,1}(\tau)) \! - \! \frac{(\sqrt{3}+1)(a-\mathrm{i}/2) \tau^{-1/3}}{2
\sqrt{3} \alpha_{1}} \right)^{2}} \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, 1 \! + \! \mathcal{O}(\tau^{-2/3}). \label{geek20}
\end{align}
Via the Definitions~\eqref{ellinfk9} and~\eqref{ellohk10}, one argues as in the
proof of Lemmata~\ref{linfnewlemm} and~\ref{lzernewlemm}, respectively, to
show that, for $k \! = \! \pm 1$, to leading order,
\begin{gather}
\mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{\infty}(\tau) \underset{\tau \to
+\infty}{=}
\begin{pmatrix}
\mathcal{O}(\tau^{-2/3}) & \mathcal{O} \! \left(\tau^{-1/3}(\mathrm{e}^{-\beta
(\tau)})^{\frac{1+k}{2}} \right) \\
\mathcal{O} \! \left(\tau^{-1/3}(\mathrm{e}^{-\beta (\tau)})^{\frac{1-k}{2}} \right) &
\mathcal{O}(\tau^{-2/3})
\end{pmatrix}, \label{geek21} \\
\mathbb{E}_{{\scriptscriptstyle \mathcal{N}},k}^{0}(\tau) \underset{\tau \to
+\infty}{=}
\begin{pmatrix}
\mathcal{O}(\tau^{-2/3}) & \mathcal{O} \! \left(\tau^{-1/3}(\mathrm{e}^{-\beta
(\tau)})^{\frac{1-k}{2}} \right) \\
\mathcal{O} \! \left(\tau^{-1/3}(\mathrm{e}^{-\beta (\tau)})^{\frac{1+k}{2}} \right) &
\mathcal{O}(\tau^{-2/3})
\end{pmatrix}, \label{geek22}
\end{gather}
whence, via the Asymptotics~\eqref{geek13}, \eqref{geek21}, and~\eqref{geek22},
and the relation $\det (\mathrm{I} \! + \! \mathbb{J}) \! = \! 1 \! + \! \tr (\mathbb{J})
\! + \! \det (\mathbb{J})$, $\mathbb{J} \! \in \! \mathrm{M}_{2}(\mathbb{C})$, it
follows that, for $k \! = \! +1$, to all orders,
\begin{gather}
\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle \mathcal{N}},1}^{\infty}(\tau)
\underset{\tau \to +\infty}{=} \mathrm{I} \! + \! \sum_{m=1}^{\infty} \zeta_{m}^{
\flat}(1)(\tau^{-1/3})^{m} \! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}),
\label{geek23} \\
(\mathrm{I} \! + \! \mathbb{E}_{{\scriptscriptstyle \mathcal{N}},1}^{0}(\tau))^{-1}
\underset{\tau \to +\infty}{=} \mathrm{I} \! + \! \sum_{m=1}^{\infty} \zeta_{m}^{
\natural}(1)(\tau^{-1/3})^{m} \! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}),
\label{geek24}
\end{gather}
for certain coefficients $\zeta_{m}^{\flat}(1)$ and $\zeta_{m}^{\natural}(1)$. It now
follows {}from the corresponding $(k \! = \! +1)$ Conditions~\eqref{ellinfk2a}
and~\eqref{ellinfk2b}, that is, $p_{1}(\tau) \mathfrak{B}_{1} \! =_{\tau \to +\infty}
\! \mathcal{O}(\mathrm{e}^{2 \hat{\mathfrak{z}}_{1}^{0}(\tau)})$ and $\sqrt{\smash[b]{
b(\tau)}} \! =_{\tau \to +\infty} \! \mathcal{O}(\mathrm{e}^{\tilde{\mathfrak{z}}_{1}^{0}
(\tau)-\hat{\mathfrak{z}}_{1}^{0}(\tau)})$, respectively, where
$\tilde{\mathfrak{z}}_{1}^{0}(\tau)$ and $\hat{\mathfrak{z}}_{1}^{0}(\tau)$ are
defined by Equations~\eqref{ellinfk4} and~\eqref{ellohk4}, respectively, the expansion
$\mathrm{e}^{z} \! = \! \sum_{m=0}^{\infty}z^{m}/m!$, the reflection formula $\Gamma (z)
\Gamma (1 \! - \! z) \! = \! \pi/\sin \pi z$, the Definitions~\eqref{jeek5}--\eqref{jeek8},
and the Asymptotics~\eqref{geek13} and \eqref{geek16}, that, for $k \! = \! +1$,
\begin{equation} \label{geek25}
\hat{\mathbb{G}}(1) \! := \!
\begin{pmatrix}
\hat{\mathbb{G}}_{11}(1) & \hat{\mathbb{G}}_{12}(1) \\
\hat{\mathbb{G}}_{21}(1) & \hat{\mathbb{G}}_{22}(1)
\end{pmatrix} \underset{\tau \to +\infty}{=}
\begin{pmatrix}
\mathcal{O}(1) & \mathcal{O}(1) \\
\mathcal{O}(1) & \mathcal{O}(\nu (1) \! + \! 1)
\end{pmatrix},
\end{equation}
and, {}from Equation~\eqref{jeek4} and the Asymptotics~\eqref{geek14},
\eqref{geek15}, \eqref{geek17}--\eqref{geek20}, and~\eqref{geek25},
\begin{equation} \label{geek26}
\mathrm{G}^{\ast}(1) \underset{\tau \to +\infty}{=}
\begin{pmatrix}
\mathcal{O}(1) & \mathcal{O}(1) \\
\mathcal{O}(1) & \mathcal{O}(\nu (1) \! + \! 1)
\end{pmatrix},
\end{equation}
whence, via the Definitions~\eqref{ellohk9}, \eqref{jeek2}, and~\eqref{jeek3},
and the Asymptotics~\eqref{geek23} and~\eqref{geek24},
\begin{gather}
\widetilde{G}(1) \underset{\tau \to +\infty}{=}
\begin{pmatrix}
\mathcal{O}(1) & \mathcal{O}(1) \\
\mathcal{O}(1) & \mathcal{O}(\nu (1) \! + \! 1)
\end{pmatrix}, \label{geek27} \\
\widehat{\mathscr{G}}(1) \underset{\tau \to +\infty}{=}
\begin{pmatrix}
\mathcal{O}(1) & \mathcal{O}(1) \\
\mathcal{O}(1) & \mathcal{O}(1)
\end{pmatrix}. \label{geek28}
\end{gather}
{}From the Asymptotics~\eqref{ellinfk10} and~\eqref{ellohk11}, the
Definition~\eqref{jeek9}, the Asymptotics~\eqref{geek27} and~\eqref{geek28},
and the relations $\max \lbrace z_{1},z_{2} \rbrace \! = \! (z_{1} \! + \! z_{2}
\! + \! \lvert z_{1} \! - \! z_{2} \rvert)/2$, $\min \lbrace z_{1},z_{2} \rbrace \!
= \! (z_{1} \! + \! z_{2} \! - \! \lvert z_{1} \! - \! z_{2} \rvert)/2$, $z_{1},z_{2}
\! \in \! \mathbb{R}$, and $\max_{k= \pm 1} \lbrace 3 \delta_{k} \! - \! 1/3,
-\delta_{k} \! - \! (1 \! + \! k)/6,-\delta_{k} \! - \! (1 \! - \! k)/6 \rbrace \! =
\! -\delta_{k}$, it follows that, for $k \! = \! +1$,
\begin{equation} \label{geek29}
\mathbb{E}_{1}^{\scriptscriptstyle G_{1}}(\tau) \underset{\tau \to +\infty}{=}
\mathcal{O}(\tau^{-\delta_{1}}).
\end{equation}
Finally, {}from the Asymptotics~\eqref{jeek1} and~\eqref{geek27}--\eqref{geek29},
one arrives at $(G_{1})_{i,j=1,2} \! =_{\tau \to +\infty} \! \mathcal{O}(1)$ (for $k \!
= \! +1$), which is, in fact, the isomonodromy condition for the corresponding
connection matrix.
{}From the Definition~\eqref{ellohk9}, the Asymptotics~\eqref{jeek1}, the
Definitions~\eqref{jeek2} and~\eqref{jeek3}, Equation~\eqref{jeek4}, the
Definitions~\eqref{jeek5}--\eqref{jeek8}, the Asymptotics~\eqref{geek23},
\eqref{geek24}, and~\eqref{geek29}, and the isomonodromy condition for the
corresponding connection matrix, $G_{1}$, it follows that, for $k \! = \! +1$,
upon setting $g_{ij} \! := \! (G_{1})_{ij}$, $i,j \! \in \! \lbrace 1,2 \rbrace$,
\begin{equation} \label{geek30}
\begin{pmatrix}
g_{11} & g_{12} \\
g_{21} & g_{22}
\end{pmatrix} \underset{\tau \to +\infty}{=}
\begin{pmatrix}
1 & s_{0}^{0} \\
0 & 1
\end{pmatrix} \!
\begin{pmatrix}
\mathrm{G}^{\ast}_{11}(1) & \mathrm{G}^{\ast}_{12}(1) \\
\mathrm{G}^{\ast}_{21}(1) & \mathrm{G}^{\ast}_{22}(1)
\end{pmatrix} \!
\begin{pmatrix}
1 \! + \! \eta_{11}(\tau) & \eta_{12}(\tau) \\
\eta_{21}(\tau) & 1 \! + \! \eta_{22}(\tau)
\end{pmatrix} \! (\mathrm{I} \! + \! \mathcal{O}(\tau^{-\delta_{1}})),
\end{equation}
where
\begin{equation} \label{geek31}
\eta_{ij}(\tau) \underset{\tau \to +\infty}{:=} \sum_{m=1}^{\infty}(\mathbb{H}_{m}
(1))_{ij}(\tau^{-1/3})^{m} \! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}),
\quad i,j \! \in \! \lbrace 1,2 \rbrace,
\end{equation}
for certain coefficients $(\mathbb{H}_{m}(1))_{ij}$. It follows {}from the
Asymptotics~\eqref{geek30} that
\begin{align} \label{geek32}
g_{12}g_{21} \underset{\tau \to \infty}{=}& \, \left(\mathrm{G}_{21}^{\ast}(1)
(1 \! + \! \eta_{11}(\tau)) \! + \! \mathrm{G}_{22}^{\ast}(1) \eta_{21}(\tau)
\right) \! \left(\mathrm{G}_{12}^{\ast}(1) \! + \! s_{0}^{0} \mathrm{G}_{22}^{\ast}
(1) \right. \nonumber \\
+&\left. \, (\mathrm{G}_{12}^{\ast}(1) \! + \! s_{0}^{0} \mathrm{G}_{22}^{\ast}(1))
\eta_{22}(\tau) \! + \! (\mathrm{G}_{11}^{\ast}(1) \! + \! s_{0}^{0} \mathrm{G}_{21}^{\ast}
(1)) \eta_{12}(\tau) \right) \! (1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})).
\end{align}
{}From the corresponding $(k \! = \! +1)$ Conditions~\eqref{ellinfk2a}
and~\eqref{ellinfk2b}, that is, $p_{1}(\tau) \mathfrak{B}_{1} \! =_{\tau \to
+\infty} \! \mathcal{O}(\mathrm{e}^{2 \hat{\mathfrak{z}}_{1}^{0}(\tau)})$ and
$\sqrt{\smash[b]{b(\tau)}} \! =_{\tau \to +\infty} \! \mathcal{O}
(\mathrm{e}^{\tilde{\mathfrak{z}}_{1}^{0}(\tau)-\hat{\mathfrak{z}}_{1}^{0}(\tau)})$,
respectively, where $\tilde{\mathfrak{z}}_{1}^{0}(\tau)$ and $\hat{\mathfrak{z}}_{1}^{0}
(\tau)$ are defined by Equations~\eqref{ellinfk4} and~\eqref{ellohk4}, respectively,
Equation~\eqref{jeek4}, the Definitions~\eqref{jeek5}--\eqref{jeek8}, the
expansion $\mathrm{e}^{z} \! = \! \sum_{m=0}^{\infty}z^{m}/m!$, and the
Asymptotics~\eqref{geek13}--\eqref{geek20}, one shows that, for $k \! = \! +1$,
\begin{align}
\mathrm{G}_{21}^{\ast}(1) \eta_{11}(\tau) \! =&\, \eta_{11}(\tau)
\dfrac{\hat{\mathbb{G}}_{21}(1) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{
\hat{\mathbb{B}}_{0}^{0}(\tau)} \mathrm{e}^{-\Delta \tilde{\mathfrak{z}}_{1}
(\tau) + \Delta \hat{\mathfrak{z}}_{1}(\tau)} \underset{\tau \to +\infty}{=}
\mathcal{O}(\tau^{-1/3}), \label{geek33} \\
\mathrm{G}_{22}^{\ast}(1) \eta_{21}(\tau) \! =& \, \eta_{21}(\tau)
\dfrac{\hat{\mathbb{G}}_{22}(1) \hat{\mathbb{A}}_{0}^{\infty}(\tau)}{
\hat{\mathbb{B}}_{0}^{0}(\tau)} \mathrm{e}^{\Delta \tilde{\mathfrak{z}}_{1}
(\tau) + \Delta \hat{\mathfrak{z}}_{1}(\tau)} \underset{\tau \to +\infty}{=}
\mathcal{O}(\tau^{-1} \mathrm{e}^{-\beta (\tau)}), \label{geek34}
\end{align}
\begin{align}
(\mathrm{G}_{12}^{\ast}(1) \! + \! s_{0}^{0} \mathrm{G}_{22}^{\ast}(1)) \eta_{22}
(\tau) =& \, \eta_{22}(\tau) \! \left(\dfrac{\hat{\mathbb{G}}_{12}(1) \hat{\mathbb{A}}_{
0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau)} \mathrm{e}^{\Delta \tilde{\mathfrak{z}}_{1}
(\tau)-\Delta \hat{\mathfrak{z}}_{1}(\tau)} \! + \! s_{0}^{0} \dfrac{\hat{\mathbb{G}}_{22}
(1) \hat{\mathbb{A}}_{0}^{\infty}(\tau)}{\hat{\mathbb{B}}_{0}^{0}(\tau)} \right.
\nonumber \\
\times&\left. \mathrm{e}^{\Delta \tilde{\mathfrak{z}}_{1}(\tau)+\Delta \hat{\mathfrak{z}}_{1}
(\tau)} \right) \! \underset{\tau \to +\infty}{=} \! \mathcal{O}(\tau^{-1/3})(\mathcal{O}
(1) \! + \! \mathcal{O}(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)})) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathcal{O}(\tau^{-1/3}), \label{geek35} \\
(\mathrm{G}_{11}^{\ast}(1) \! + \! s_{0}^{0} \mathrm{G}_{21}^{\ast}(1)) \eta_{12}(\tau)
=& \, \eta_{12}(\tau) \! \left(\dfrac{\hat{\mathbb{G}}_{11}(1) \hat{\mathbb{B}}_{0}^{\infty}
(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau)} \mathrm{e}^{-\Delta \tilde{\mathfrak{z}}_{1}(\tau)-
\Delta \hat{\mathfrak{z}}_{1}(\tau)} \! + \! s_{0}^{0} \dfrac{\hat{\mathbb{G}}_{21}(1)
\hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{B}}_{0}^{0}(\tau)} \right. \nonumber \\
\times&\left. \mathrm{e}^{-\Delta \tilde{\mathfrak{z}}_{1}(\tau)+\Delta \hat{\mathfrak{z}}_{1}
(\tau)} \right) \! \underset{\tau \to +\infty}{=} \! \mathcal{O}(\tau^{-1/3})(\mathcal{O}
(1) \! + \! \mathcal{O}(1)) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathcal{O}(\tau^{-1/3}), \label{geek36}
\end{align}
whence (cf. Asymptotics~\eqref{geek32})
\begin{align}
g_{12}g_{21} \underset{\tau \to +\infty}{=}& \, \left(\mathrm{G}_{21}^{\ast}(1) \! + \!
\mathcal{O}(\tau^{-1/3}) \! + \! \mathcal{O}(\tau^{-1} \mathrm{e}^{-\beta (\tau)}) \right) \!
(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \nonumber \\
\times& \, \left(\mathrm{G}_{12}^{\ast}(1) \! + \! \mathcal{O}(\tau^{-1/3}) \! + \!
\mathcal{O}(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}) \right) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathrm{G}_{12}^{\ast}(1) \mathrm{G}_{21}^{\ast}
(1)(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \underset{\tau \to +\infty}{=}
\hat{\mathbb{G}}_{12}(1) \hat{\mathbb{G}}_{21}(1) \dfrac{\hat{\mathbb{A}}_{0}^{\infty}
(\tau) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau)
\hat{\mathbb{B}}_{0}^{0}(\tau)}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \nonumber \\
\underset{\tau \to +\infty}{=}& \, -\mathrm{e}^{-2 \pi \mathrm{i} (\nu (1)+1)}(1 \! + \! \mathcal{O}
(\tau^{-2/3}))(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \underset{\tau \to +\infty}{=}
-(1 \! + \! \mathcal{O}(\nu (1) \! + \! 1)) \nonumber \\
\times& \, (1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \underset{\tau \to +\infty}{=}
-(1 \! + \! \mathcal{O}(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}))(1 \! + \! \mathcal{O}
(\tau^{-\delta_{1}})) \quad \Rightarrow \nonumber \\
-g_{12}g_{21} \underset{\tau \to +\infty}{=}& \, 1 \! + \! \mathcal{O}(\tau^{-\delta_{1}});
\label{geek37}
\end{align}
analogously,
\begin{align}
g_{21} \underset{\tau \to +\infty}{=}& \, \left(\mathrm{G}_{21}^{\ast}(1)(1 \! +
\! \eta_{11}(\tau)) \! + \! \mathrm{G}_{22}^{\ast}(1) \eta_{21}(\tau) \right) \!
(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \left(\mathrm{G}_{21}^{\ast}(1) \! + \! \mathcal{O}
(\tau^{-1/3}) \! + \! \mathcal{O}(\tau^{-1} \mathrm{e}^{-\beta (\tau)}) \right) \!
(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \nonumber \\
\underset{\tau \to +\infty}{=}& \, \mathrm{G}_{21}^{\ast}(1)(1 \! + \! \mathcal{O}
(\tau^{-\delta_{1}})) \underset{\tau \to +\infty}{=} \hat{\mathbb{G}}_{21}(1)
\dfrac{\hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{B}}_{0}^{0}(\tau)} \mathrm{e}^{-
\Delta \tilde{\mathfrak{z}}_{1}(\tau)+\Delta \hat{\mathfrak{z}}_{1}(\tau)}(1 \! + \!
\mathcal{O}(\tau^{-\delta_{1}})) \nonumber \\
\underset{\tau \to +\infty}{=}& \, -\dfrac{\mathrm{i} \sqrt{\smash[b]{b(\tau)}}}{(\varepsilon
b)^{1/4}} \mathrm{e}^{-\tilde{\mathfrak{z}}_{1}^{0}(\tau)+\hat{\mathfrak{z}}_{1}^{0}(\tau)}
\mathrm{e}^{-2 \pi \mathrm{i} (\nu (1)+1)}(1 \! + \! \mathcal{O}(\tau^{-2/3}))(1 \! + \! \mathcal{O}
(\tau^{-2/3}))(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \nonumber \\
\underset{\tau \to +\infty}{=}& \, -\dfrac{\mathrm{i} \sqrt{\smash[b]{b(\tau)}}}{(\varepsilon
b)^{1/4}} \mathrm{e}^{-\tilde{\mathfrak{z}}_{1}^{0}(\tau)+\hat{\mathfrak{z}}_{1}^{0}(\tau)}
(1 \! + \! \mathcal{O}(\nu (1) \! + \! 1))(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}}))
\nonumber \\
\underset{\tau \to +\infty}{=}& \, -\dfrac{\mathrm{i} \sqrt{\smash[b]{b(\tau)}}}{(\varepsilon
b)^{1/4}} \mathrm{e}^{-\tilde{\mathfrak{z}}_{1}^{0}(\tau)+\hat{\mathfrak{z}}_{1}^{0}(\tau)}
(1 \! + \! \mathcal{O}(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}))(1 \! + \! \mathcal{O}
(\tau^{-\delta_{1}})) \quad \Rightarrow \nonumber \\
g_{21} \underset{\tau \to +\infty}{=}& \, -\dfrac{\mathrm{i} \sqrt{\smash[b]{b(\tau)}}}{
(\varepsilon b)^{1/4}} \mathrm{e}^{-\tilde{\mathfrak{z}}_{1}^{0}(\tau)+\hat{\mathfrak{z}}_{1}^{0}
(\tau)}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})). \label{geek38}
\end{align}
It follows, upon inversion, {}from the Asymptotics~\eqref{geek37} and~\eqref{geek38}
that, for $k \! = \! +1$,
\begin{equation} \label{geek39}
\sqrt{\smash[b]{b(\tau)}} \underset{\tau \to +\infty}{=} \mathrm{i} g_{21}(\varepsilon b)^{1/4}
\mathrm{e}^{\tilde{\mathfrak{z}}_{1}^{0}(\tau)-\hat{\mathfrak{z}}_{1}^{0}(\tau)}(1 \! + \!
\mathcal{O}(\tau^{-\delta_{1}})) \underset{\tau \to +\infty}{=} -\mathrm{i} g_{12}^{-1}
(\varepsilon b)^{1/4} \mathrm{e}^{\tilde{\mathfrak{z}}_{1}^{0}(\tau)-\hat{\mathfrak{z}}_{1}^{0}
(\tau)}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})),
\end{equation}
whence, via Equations~\eqref{monok2} and the Definitions~\eqref{ellinfk4}
and~\eqref{ellohk4}, one arrives at the corresponding $(k \! = \! +1)$ asymptotics
for $\sqrt{\smash[b]{b(\tau)}}$ stated in Equation~\eqref{geek3} of the
lemma.\footnote{Note that the Asymptotics~\eqref{geek39} is consistent with the
corresponding $(k \! = \! +1)$ Condition~\eqref{ellinfk2b}.}
Recall the following formula (cf. Equations~\eqref{monoeqns}), which is one of the
defining relations for the manifold of the monodromy data, $\mathscr{M}$:
\begin{equation} \label{geek40}
g_{21}g_{22} \! - \! g_{11}g_{12} \! + \! s_{0}^{0}g_{11}g_{22} \! = \! \mathrm{i}
\mathrm{e}^{-\pi a}.
\end{equation}
Substituting Equation~\eqref{jeek4}, the Definitions~\eqref{jeek5}--\eqref{jeek8},
and the Asymptotics~\eqref{geek30} into Equation~\eqref{geek40}, one shows
that, for $k \! = \! +1$,
\begin{align} \label{geek41}
&\left(\mathrm{G}_{21}^{\ast}(1) \mathrm{G}_{22}^{\ast}(1) \! - \!
\mathrm{G}_{11}^{\ast}(1) \mathrm{G}_{12}^{\ast}(1) \! - \! s_{0}^{0}
\mathrm{G}_{12}^{\ast}(1) \mathrm{G}_{21}^{\ast}(1) \right) \! (1 \!
+ \! \eta_{11}(\tau))(1 \! + \! \eta_{22}(\tau)) \nonumber \\
+& \left(\mathrm{G}_{21}^{\ast}(1) \mathrm{G}_{21}^{\ast}(1) \! - \!
\mathrm{G}_{11}^{\ast}(1) \mathrm{G}_{11}^{\ast}(1) \! - \! s_{0}^{0}
\mathrm{G}_{11}^{\ast}(1) \mathrm{G}_{21}^{\ast}(1) \right) \! (1 \!
+ \! \eta_{11}(\tau)) \eta_{12}(\tau) \nonumber \\
+& \left(\mathrm{G}_{22}^{\ast}(1) \mathrm{G}_{22}^{\ast}(1) \! - \!
\mathrm{G}_{12}^{\ast}(1) \mathrm{G}_{12}^{\ast}(1) \! - \! s_{0}^{0}
\mathrm{G}_{12}^{\ast}(1) \mathrm{G}_{22}^{\ast}(1) \right) \! (1 \!
+ \! \eta_{22}(\tau)) \eta_{21}(\tau) \nonumber \\
+& \left(\mathrm{G}_{22}^{\ast}(1) \mathrm{G}_{21}^{\ast}(1) \! - \!
\mathrm{G}_{12}^{\ast}(1) \mathrm{G}_{11}^{\ast}(1) \! - \! s_{0}^{0}
\mathrm{G}_{11}^{\ast}(1) \mathrm{G}_{22}^{\ast}(1) \right) \! \eta_{12}
(\tau) \eta_{21}(\tau) \! - \! \mathrm{i} \mathrm{e}^{-\pi a} \! + \! \mathcal{O}
(\tau^{-\delta_{1}}) \underset{\tau \to +\infty}{=} 0,
\end{align}
where
\begin{align}
\mathrm{G}_{21}^{\ast}(1) \mathrm{G}_{22}^{\ast}(1) \! =& \, \dfrac{\mathrm{i}
\sqrt{2 \pi}(2 \! + \! \sqrt{3})^{1/2}(2 \mu_{1}(\tau))^{1/2} \mathrm{e}^{-\mathrm{i} 4 \pi
(\nu (1)+1)}}{p_{1}(\tau) \mathfrak{B}_{1} \mathrm{e}^{-2 \hat{\mathfrak{z}}_{1}^{0}
(\tau)} \Gamma (\nu (1) \! + \! 1)} \dfrac{\hat{\mathbb{A}}_{0}^{\infty}
(\tau) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{B}}_{0}^{0}(\tau)
\hat{\mathbb{B}}_{0}^{0}(\tau)} \mathrm{e}^{2 \Delta \hat{\mathfrak{z}}_{1}(\tau)},
\label{geek42} \\
\mathrm{G}_{11}^{\ast}(1) \mathrm{G}_{12}^{\ast}(1) \! =& \, -\dfrac{\sqrt{2 \pi}
p_{1}(\tau) \mathfrak{B}_{1} \mathrm{e}^{-2 \hat{\mathfrak{z}}_{1}^{0}(\tau)} \mathrm{e}^{
\mathrm{i} \pi (\nu (1)+1)}}{(2 \! + \! \sqrt{3})^{1/2}(2 \mu_{1}(\tau))^{1/2} \Gamma
(-\nu (1))} \dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau) \hat{\mathbb{B}}_{0}^{\infty}
(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau) \hat{\mathbb{A}}_{0}^{0}(\tau)} \mathrm{e}^{-2
\Delta \hat{\mathfrak{z}}_{1}(\tau)}, \label{geek43} \\
s_{0}^{0} \mathrm{G}_{12}^{\ast}(1) \mathrm{G}_{21}^{\ast}(1) \! =& \, -s_{0}^{0}
\mathrm{e}^{-\mathrm{i} 2 \pi (\nu (1)+1)} \dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau)
\hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau)
\hat{\mathbb{B}}_{0}^{0}(\tau)}, \label{geek44} \\
\mathrm{G}_{21}^{\ast}(1) \mathrm{G}_{21}^{\ast}(1) \! =& \, g_{21}^{2} \!
\left(\dfrac{\hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{B}}_{0}^{0}(\tau)}
\right)^{2} \mathrm{e}^{-2(\Delta \tilde{\mathfrak{z}}_{1}(\tau)-\Delta
\hat{\mathfrak{z}}_{1}(\tau))}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})),
\label{geek45} \\
\mathrm{G}_{11}^{\ast}(1) \mathrm{G}_{11}^{\ast}(1) \! =& \, \left(\dfrac{
\sqrt{2 \pi}g_{21}p_{1}(\tau) \mathfrak{B}_{1} \mathrm{e}^{-2 \hat{\mathfrak{z}}_{1}^{0}
(\tau)} \mathrm{e}^{\mathrm{i} \pi (\nu (1)+1)}}{(2 \! + \! \sqrt{3})^{1/2}(2 \mu_{1}(\tau))^{1/2}
\Gamma (-\nu (1))} \dfrac{\hat{\mathbb{B}}_{0}^{\infty}(\tau)}{
\hat{\mathbb{A}}_{0}^{0}(\tau)} \right)^{2} \mathrm{e}^{-2(\Delta \tilde{\mathfrak{z}}_{1}
(\tau)+ \Delta \hat{\mathfrak{z}}_{1}(\tau))} \nonumber \\
\times& \, (1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})), \label{geek46} \\
s_{0}^{0} \mathrm{G}_{11}^{\ast}(1) \mathrm{G}_{21}^{\ast}(1) \! =& \,
\dfrac{s_{0}^{0} \sqrt{2 \pi}g_{21}^{2}p_{1}(\tau) \mathfrak{B}_{1} \mathrm{e}^{-2
\hat{\mathfrak{z}}_{1}^{0}(\tau)} \mathrm{e}^{\mathrm{i} \pi (\nu (1)+1)}}{(2 \! + \! \sqrt{3})^{1/2}
(2 \mu_{1}(\tau))^{1/2} \Gamma (-\nu (1))} \dfrac{\hat{\mathbb{B}}_{0}^{\infty}
(\tau) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau)
\hat{\mathbb{B}}_{0}^{0}(\tau)} \mathrm{e}^{-2 \Delta \tilde{\mathfrak{z}}_{1}(\tau)}
\nonumber \\
\times& \, (1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})), \label{geek47} \\
\mathrm{G}_{22}^{\ast}(1) \mathrm{G}_{22}^{\ast}(1) \! =& \, \left(\dfrac{\mathrm{i}
\sqrt{2 \pi}(2 \! + \! \sqrt{3})^{1/2}(2 \mu_{1}(\tau))^{1/2} \mathrm{e}^{-\mathrm{i} 2 \pi
(\nu (1)+1)}}{g_{21}p_{1}(\tau) \mathfrak{B}_{1} \mathrm{e}^{-2 \hat{\mathfrak{z}}_{1}^{0}
(\tau)} \Gamma (\nu (1) \! + \! 1)} \dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau)}{
\hat{\mathbb{B}}_{0}^{0}(\tau)} \right)^{2} \mathrm{e}^{2(\Delta \tilde{\mathfrak{z}}_{1}
(\tau)+ \Delta \hat{\mathfrak{z}}_{1}(\tau))} \nonumber \\
\times& \, (1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})), \label{geek48} \\
\mathrm{G}_{12}^{\ast}(1) \mathrm{G}_{12}^{\ast}(1) \! =& \, g_{21}^{-2} \!
\left(\dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{0}
(\tau)} \right)^{2} \mathrm{e}^{2(\Delta \tilde{\mathfrak{z}}_{1}(\tau)-\Delta
\hat{\mathfrak{z}}_{1}(\tau))}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})),
\label{geek49} \\
s_{0}^{0} \mathrm{G}_{12}^{\ast}(1) \mathrm{G}_{22}^{\ast}(1) \! =& \,
-\dfrac{\mathrm{i} s_{0}^{0} \sqrt{2 \pi}(2 \! + \! \sqrt{3})^{1/2}(2 \mu_{1}(\tau))^{1/2}
\mathrm{e}^{-\mathrm{i} 2 \pi (\nu (1)+1)}}{g_{21}^{2}p_{1}(\tau) \mathfrak{B}_{1} \mathrm{e}^{-
2 \hat{\mathfrak{z}}_{1}^{0}(\tau)} \Gamma (\nu (1) \! + \! 1)} \dfrac{
\hat{\mathbb{A}}_{0}^{\infty}(\tau) \hat{\mathbb{A}}_{0}^{\infty}(\tau)}{
\hat{\mathbb{A}}_{0}^{0}(\tau) \hat{\mathbb{B}}_{0}^{0}(\tau)}
\mathrm{e}^{2 \Delta \tilde{\mathfrak{z}}_{1}(\tau)} \nonumber \\
\times& \, (1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})), \label{geek50} \\
s_{0}^{0} \mathrm{G}_{11}^{\ast}(1) \mathrm{G}_{22}^{\ast}(1) \! =& \, \mathrm{i}
2s_{0}^{0} \sin (\pi (\nu (1) \! + \! 1)) \mathrm{e}^{-\mathrm{i} \pi (\nu (1)+1)} \dfrac{
\hat{\mathbb{A}}_{0}^{\infty}(\tau) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{
\hat{\mathbb{A}}_{0}^{0}(\tau) \hat{\mathbb{B}}_{0}^{0}(\tau)}. \label{geek51}
\end{align}
Let
\begin{equation} \label{geek52}
x \! := \! \dfrac{\sqrt{2 \pi} \, p_{1}(\tau) \mathfrak{B}_{1}
\mathrm{e}^{-2 \hat{\mathfrak{z}}_{1}^{0}(\tau)} \mathrm{e}^{\mathrm{i} \pi (\nu (1)+1)}}{(2 \!
+ \! \sqrt{3})^{1/2}(2 \mu_{1}(\tau))^{1/2} \Gamma (-\nu (1))} \dfrac{
\hat{\mathbb{A}}_{0}^{\infty}(\tau) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{
\hat{\mathbb{A}}_{0}^{0}(\tau) \hat{\mathbb{A}}_{0}^{0}(\tau)} \mathrm{e}^{-2
\Delta \hat{\mathfrak{z}}_{1}(\tau)}(1 \! + \! \eta_{11}(\tau))(1 \! + \!
\eta_{22}(\tau));
\end{equation}
in terms of the newly-defined variable $x$, an algebraic exercise reveals
that the Asymptotics~\eqref{geek41} can be recast in the following form:
\begin{equation} \label{geek53}
y_{1}x^{-2} \! + \! (y_{2} \! + \! y_{3} \! + \! y_{4})x^{-1} \! + \! (1 \! + \!
y_{5} \! + \! y_{6})x \! + \! y_{7}x^{2} \! + \! y_{8} \! + \! y_{9} \! + \! y_{10}
\! + \! y_{11} \! - \! \mathrm{i} \mathrm{e}^{-\pi a} \! + \! \mathcal{O}(\tau^{-\delta_{1}})
\underset{\tau \to +\infty}{=} 0,
\end{equation}
where
\begin{align}
y_{1} \! :=& \, \left(\mathrm{i} 2g_{21}^{-1} \sin (\pi (\nu (1) \! + \! 1)) \mathrm{e}^{-\mathrm{i} \pi
(\nu (1)+1)} \dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau) \hat{\mathbb{B}}_{0}^{
\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau) \hat{\mathbb{A}}_{0}^{0}(\tau)}
\right)^{2} \! \left(\dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau)}{\hat{\mathbb{B}}_{
0}^{0}(\tau)} \right)^{2} \mathrm{e}^{2(\Delta \tilde{\mathfrak{z}}_{1}(\tau)-\Delta
\hat{\mathfrak{z}}_{1}(\tau))} \nonumber \\
\times& \, (1 \! + \! \eta_{11}(\tau))^{2}(1 \! + \! \eta_{22}(\tau))^{3}
\eta_{21}(\tau)(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})), \label{geek59} \\
y_{2} \! :=& \, \mathrm{i} 2 \sin (\pi (\nu (1) \! + \! 1)) \mathrm{e}^{-\mathrm{i} 3 \pi (\nu (1)+1)} \!
\left(\dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau) \hat{\mathbb{B}}_{0}^{\infty}
(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau) \hat{\mathbb{B}}_{0}^{0}(\tau)}(1 \!
+ \! \eta_{11}(\tau))(1 \! + \! \eta_{22}(\tau)) \right)^{2}, \label{geek54} \\
y_{3} \! :=& \, \mathrm{i} 2 s_{0}^{0}g_{21}^{-2} \sin (\pi (\nu (1) \! + \! 1)) \mathrm{e}^{-\mathrm{i}
\pi (\nu (1)+1)} \! \left(\dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau)}{\hat{\mathbb{
A}}_{0}^{0}(\tau)} \right)^{3} \dfrac{\hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{
\mathbb{B}}_{0}^{0}(\tau)} \mathrm{e}^{2(\Delta \tilde{\mathfrak{z}}_{1}(\tau)-\Delta
\hat{\mathfrak{z}}_{1}(\tau))} \nonumber \\
\times& \, (1 \! + \! \eta_{11}(\tau))(1 \! + \! \eta_{22}(\tau))^{2} \eta_{21}
(\tau), \label{geek61} \\
y_{4} \! :=& \, \mathrm{i} 2 \sin (\pi (\nu (1) \! + \! 1)) \mathrm{e}^{-\mathrm{i} 3 \pi (\nu (1)+1)}
\! \left(\dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau) \hat{\mathbb{B}}_{0}^{\infty}
(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau) \hat{\mathbb{B}}_{0}^{0}(\tau)}
\right)^{2}(1 \! + \! \eta_{11}(\tau)) \nonumber \\
\times& \, (1 \! + \! \eta_{22}(\tau)) \eta_{12}(\tau) \eta_{21}(\tau), \label{geek62} \\
y_{5} \! :=& \, -s_{0}^{0}g_{21}^{2} \dfrac{\hat{\mathbb{A}}_{0}^{0}(\tau)
\hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{\infty}(\tau)
\hat{\mathbb{B}}_{0}^{0}(\tau)} \mathrm{e}^{-2(\Delta \tilde{\mathfrak{z}}_{1}(\tau)-
\Delta \hat{\mathfrak{z}}_{1}(\tau))} \dfrac{\eta_{12}(\tau)}{1 \! + \! \eta_{22}
(\tau)}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})), \label{geek58} \\
y_{6} \! :=& \, \dfrac{\eta_{12}(\tau) \eta_{21}(\tau)}{(1 \! + \! \eta_{11}
(\tau))(1 \! + \! \eta_{22}(\tau))}, \label{geek63} \\
y_{7} \! :=& \, -g_{21}^{2} \! \left(\dfrac{\hat{\mathbb{A}}_{0}^{0}(\tau)}{
\hat{\mathbb{A}}_{0}^{\infty}(\tau)} \right)^{2} \mathrm{e}^{-2(\Delta \tilde{\mathfrak{z}}_{1}
(\tau)-\Delta \hat{\mathfrak{z}}_{1}(\tau))} \dfrac{\eta_{12}(\tau)}{(1 \! + \! \eta_{11}
(\tau))(1 \! + \! \eta_{22}(\tau))^{2}}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})),
\label{geek57} \\
y_{8} \! :=& \, s_{0}^{0} \mathrm{e}^{-\mathrm{i} 2 \pi (\nu (1)+1)} \dfrac{\hat{\mathbb{
A}}_{0}^{\infty}(\tau) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{
0}^{0}(\tau) \hat{\mathbb{B}}_{0}^{0}(\tau)}(1 \! + \! \eta_{11}(\tau))(1 \! +
\! \eta_{22}(\tau)), \label{geek55} \\
y_{9} \! :=& \, g_{21}^{2} \! \left(\dfrac{\hat{\mathbb{B}}_{0}^{\infty}(\tau)}{
\hat{\mathbb{B}}_{0}^{0}(\tau)} \right)^{2} \mathrm{e}^{-2(\Delta \tilde{\mathfrak{z}}_{1}
(\tau)-\Delta \hat{\mathfrak{z}}_{1}(\tau))}(1 \! + \! \eta_{11}(\tau)) \eta_{12}
(\tau)(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})), \label{geek56} \\
y_{10} \! :=& \, -g_{21}^{-2} \! \left(\dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau)}{
\hat{\mathbb{A}}_{0}^{0}(\tau)} \right)^{2} \mathrm{e}^{2(\Delta \tilde{\mathfrak{z}}_{1}
(\tau)-\Delta \hat{\mathfrak{z}}_{1}(\tau))}(1 \! + \! \eta_{22}(\tau)) \eta_{21}
(\tau)(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})), \label{geek60} \\
y_{11} \! :=& \, -\mathrm{i} 2 s_{0}^{0} \sin (\pi (\nu (1) \! + \! 1)) \mathrm{e}^{-\mathrm{i} \pi
(\nu (1)+1)} \dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau) \hat{\mathbb{B}}_{0}^{
\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau) \hat{\mathbb{B}}_{0}^{0}(\tau)}
\eta_{12}(\tau) \eta_{21}(\tau). \label{geek64}
\end{align}
Via the Asymptotics~\eqref{geek13}--\eqref{geek20} and~\eqref{geek31},
and the expansion $\mathrm{e}^{z} \! = \! \sum_{m=0}^{\infty}z^{m}/m!$, it
follows {}from the Definitions~\eqref{geek59}--\eqref{geek64} that
\begin{gather}
y_{1} \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-5/3} \mathrm{e}^{-2 \beta
(\tau)}), \qquad \quad y_{2} \underset{\tau \to +\infty}{=} \mathcal{O}
(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}), \label{geek65} \\
y_{3} \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-1} \mathrm{e}^{-\beta (\tau)}),
\quad \quad y_{4} \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-4/3}
\mathrm{e}^{-\beta (\tau)}), \quad \quad y_{5} \underset{\tau \to +\infty}{=}
\mathcal{O}(\tau^{-1/3}), \label{geek66} \\
y_{6} \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-2/3}), \quad \quad
y_{7} \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-1/3}), \quad \quad
y_{8} \underset{\tau \to +\infty}{=} s_{0}^{0}(1 \! + \! \mathcal{O}(\tau^{-1/3})),
\label{geek67} \\
y_{9} \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-1/3}), \quad \quad
y_{10} \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-1/3}), \quad \quad
y_{11} \underset{\tau \to +\infty}{=} \mathcal{O}(\tau^{-4/3} \mathrm{e}^{-\beta (\tau)}).
\label{geek68}
\end{gather}
One notes that---the asymptotic---Equation~\eqref{geek53} is a quartic
equation for the indeterminate $x$, which can be solved explicitly: via a study
of the four solutions to the quartic equation (see, for example, \cite{antvsnon}),
in conjunction with the Asymptotics~\eqref{geek65}--\eqref{geek68} and a
method-of-successive-approximations argument, it can be shown that the
sought-after solution, that is, the one for which $x \! =_{\tau \to +\infty} \!
\mathcal{O}(1)$, can be extracted as one of the two solutions to the quadratic
equation
\begin{equation} \label{geek76}
(1 \! + \! \upsilon_{1}^{\ast})x^{2} \! + \! \left(y_{8} \! + \! \upsilon_{2}^{\ast}
\! - \! \mathrm{i} \mathrm{e}^{-\pi a} \! + \! \mathcal{O}(\tau^{-\delta_{1}}) \right) \! x \!
+ \! \upsilon_{3}^{\ast} \underset{\tau \to +\infty}{=} 0,
\end{equation}
where
\begin{gather}
\upsilon_{1}^{\ast} \! := \! y_{5} \! + \! y_{6} \underset{\tau \to +\infty}{=}
\mathcal{O}(\tau^{-1/3}), \, \, \, \quad \quad \, \, \, \upsilon_{2}^{\ast} \! :=
\! y_{9} \! + \! y_{10} \! + \! y_{11} \underset{\tau \to +\infty}{=} \mathcal{O}
(\tau^{-1/3}), \label{geek77} \\
\upsilon_{3}^{\ast} \! := \! y_{2} \! + \! y_{3} \! + \! y_{4} \underset{\tau \to
+\infty}{=} \mathcal{O}(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}). \label{geek79}
\end{gather}
The roots of the quadratic Equation~\eqref{geek76} are
\begin{equation} \label{geek80}
x \underset{\tau \to +\infty}{=} \dfrac{-(y_{8} \! + \! \upsilon_{2}^{\ast} \! -
\! \mathrm{i} \mathrm{e}^{-\pi a} \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \! \pm \! \sqrt{
(y_{8} \! + \! \upsilon_{2}^{\ast} \! - \! \mathrm{i} \mathrm{e}^{-\pi a} \! + \! \mathcal{O}
(\tau^{-\delta_{1}}))^{2} \! - \! 4(1 \! + \! \upsilon_{1}^{\ast})
\upsilon_{3}^{\ast}}}{2(1 \! + \! \upsilon_{1}^{\ast})};
\end{equation}
of the two solutions given by Equation~\eqref{geek80}, the one that is consistent
with the corresponding $(k \! = \! +1)$ Condition~\eqref{ellinfk2a} reads
\begin{equation} \label{geek81}
x \underset{\tau \to +\infty}{=} \dfrac{-(y_{8} \! + \! \upsilon_{2}^{\ast} \!
- \! \mathrm{i} \mathrm{e}^{-\pi a} \! + \! \mathcal{O}(\tau^{-\delta_{1}})) \! - \! \sqrt{
(y_{8} \! + \! \upsilon_{2}^{\ast} \! - \! \mathrm{i} \mathrm{e}^{-\pi a} \! + \! \mathcal{O}
(\tau^{-\delta_{1}}))^{2} \! - \! 4(1 \! + \! \upsilon_{1}^{\ast})
\upsilon_{3}^{\ast}}}{2(1 \! + \! \upsilon_{1}^{\ast})}:
\end{equation}
via the Definition~\eqref{geek52}, and the Asymptotics~\eqref{geek65},
\eqref{geek77}, and~\eqref{geek79}, it follows {}from Equation~\eqref{geek81}
and an application of the Binomial Theorem that, for $s_{0}^{0} \! \neq \! \mathrm{i}
\mathrm{e}^{-\pi a}$,
\begin{align}
&\dfrac{\sqrt{2 \pi} \, p_{1}(\tau) \mathfrak{B}_{1} \mathrm{e}^{-2 \hat{\mathfrak{z}}_{1}^{0}
(\tau)} \mathrm{e}^{\mathrm{i} \pi (\nu (1)+1)}}{(2 \! + \! \sqrt{3})^{1/2}(2 \mu_{1}(\tau))^{1/2}
\Gamma (-\nu (1))} \dfrac{\hat{\mathbb{A}}_{0}^{\infty}(\tau) \hat{\mathbb{B}}_{
0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{0}(\tau) \hat{\mathbb{A}}_{0}^{0}(\tau)}
\mathrm{e}^{-2 \Delta \hat{\mathfrak{z}}_{1}(\tau)}(1 \! + \! \eta_{11}(\tau))(1 \! + \!
\eta_{22}(\tau)) \nonumber \\
&\underset{\tau \to +\infty}{=} -(s_{0}^{0} \! - \! \mathrm{i} \mathrm{e}^{-\pi a}) \! + \!
\mathcal{O}(\tau^{-\delta_{1}}). \label{geek82}
\end{align}
{}From the Asymptotics~\eqref{tr1}, \eqref{tr3}, \eqref{geek13}, \eqref{geek15},
\eqref{geek17}--\eqref{geek20}, and~\eqref{geek31}, the reflection formula
$\Gamma (z) \Gamma (1 \! - \! z) \! = \! \pi/\sin \pi z$, the expansion
$\mathrm{e}^{z} \! = \! \sum_{m=0}^{\infty}z^{m}/m!$, and the Asymptotics (cf.
Remark~\ref{aspvals}) $(\Gamma (-\nu (1)))^{-1} \! =_{\tau \to +\infty} \! 1 \!
+ \! \mathcal{O}(\nu (1) \! + \! 1) \! =_{\tau \to +\infty} \! 1 \! + \! \mathcal{O}
(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)})$, one shows that, for $k \! = \! +1$,
\begin{gather}
\dfrac{\mathrm{e}^{\mathrm{i} \pi (\nu (1)+1)}}{\Gamma (-\nu (1))} \dfrac{\hat{\mathbb{A}}_{
0}^{\infty}(\tau) \hat{\mathbb{B}}_{0}^{\infty}(\tau)}{\hat{\mathbb{A}}_{0}^{0}
(\tau) \hat{\mathbb{A}}_{0}^{0}(\tau)} \underset{\tau \to +\infty}{:=} 1 \! + \!
\tau^{-2/3} \sum_{m=0}^{\infty} \alpha_{m}(1)(\tau^{-1/3})^{m} \! + \!
\mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}), \label{geek83} \\
\mathrm{e}^{-2 \Delta \hat{\mathfrak{z}}_{1}(\tau)} \underset{\tau \to +\infty}{:=} 1 \!
+ \! \tau^{-2/3} \sum_{m=0}^{\infty} \alpha_{m}^{\natural}(1)(\tau^{-1/3})^{m}
\! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}), \label{geek84} \\
(1 \! + \! \eta_{11}(\tau))(1 \! + \! \eta_{22}(\tau)) \underset{\tau \to +\infty}{:=}
1 \! + \! \tau^{-1/3} \sum_{m=0}^{\infty} \alpha_{m}^{\flat}(1)(\tau^{-1/3})^{m}
\! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}), \label{geek85}
\end{gather}
for certain coefficients $\alpha_{m}(1)$, $\alpha_{m}^{\natural}(1)$, and
$\alpha_{m}^{\flat}(1)$. Via the Asymptotics~\eqref{geek16}
and~\eqref{geek83}--\eqref{geek85}, upon defining
\begin{align}
&\left(1 \! + \! \sum_{m_{1}=0}^{\infty} \dfrac{\alpha_{m_{1}}^{\flat}(1)}{(\tau^{1/3})^{
m_{1}+1}} \! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}) \right) \! \left(1 \! +
\! \sum_{m_{2}=0}^{\infty} \dfrac{\alpha_{m_{2}}(1)}{(\tau^{1/3})^{m_{2}+2}} \! + \!
\mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}) \right) \nonumber \\
&\times \left(1 \! + \! \sum_{m_{3}=0}^{\infty} \dfrac{\alpha_{m_{3}}^{\natural}(1)}{
(\tau^{1/3})^{m_{3}+2}} \! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)})
\right) \! \left(1 \! + \! \sum_{m_{4}=0}^{\infty} \dfrac{\alpha_{m_{4}}^{\sharp}
(1)}{(\tau^{1/3})^{m_{4}+2}} \! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)})
\right) \nonumber \\
&\underset{\tau \to +\infty}{=:} 1 \! + \! \sum_{m=0}^{\infty} \dfrac{\hat{\epsilon}_{m}^{
\sharp}(1)}{(\tau^{1/3})^{m+1}} \! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}),
\label{geek86}
\end{align}
it follows {}from the corresponding $(k \! = \! +1)$ Definition~\eqref{ellohk4} and
the Asymptotics~\eqref{geek82} and~\eqref{geek86} that, for $s_{0}^{0} \! \neq
\! \mathrm{i} \mathrm{e}^{-\pi a}$,
\begin{align} \label{geek87}
p_{1}(\tau) \mathfrak{B}_{1} \! \left(1 \! + \! \sum_{m=0}^{\infty} \frac{
\hat{\epsilon}_{m}^{\sharp}(1)}{(\tau^{1/3})^{m+1}} \! + \! \mathcal{O}(\tau^{-1/3}
\mathrm{e}^{-\beta (\tau)}) \right) \underset{\tau \to +\infty}{=}& \, -\frac{2^{3/2}
3^{1/4} \mathrm{e}^{\mathrm{i} \pi/4}(2 \! + \! \sqrt{3}) \mathscr{P}_{a}(s_{0}^{0} \! - \!
\mathrm{i} \mathrm{e}^{-\pi a})}{\sqrt{2 \pi}} \nonumber \\
\times& \, \mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}
(\tau^{-\delta_{1}})),
\end{align}
where $\mathscr{P}_{a}$ is defined by Equation~\eqref{geek4}.\footnote{{}From the
leading term of asymptotics for $\mathfrak{B}_{1}$ given in Equation~\eqref{prcybk1},
that is, $\mathfrak{B}_{1} \! =_{\tau \to +\infty} \! -\tfrac{(\sqrt{3}+1) \tau^{-1/3}}{6
\alpha_{1}} \! + \! \mathcal{O}(\tau^{-1})$, and the Asymptotics~\eqref{geek87},
it follows that $p_{1}(\tau) \! =_{\tau \to +\infty} \! \mathfrak{D}_{1} \tau^{1/3}
\mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}}))$,
where $\mathfrak{D}_{1} \! := \! 6(\sqrt{3} \! + \! 1)3^{1/4} \mathrm{e}^{\mathrm{i} \pi/4} \alpha_{1}
\mathscr{P}_{a}(s_{0}^{0} \! - \! \mathrm{i} \mathrm{e}^{-\pi a})/\sqrt{\pi}$, whence $p_{1}(\tau)
\mathfrak{B}_{1} \! =_{\tau \to +\infty} \! \mathcal{O}(\mathrm{e}^{-\beta (\tau)})$, which is
consistent with the corresponding $(k \! = \! +1)$ Condition~\eqref{ellinfk2a}.} Via the
Asymptotics~\eqref{prcyellok1} and the Definition~\eqref{prcy57}, a multiplication
argument reveals that
\begin{align} \label{geek88}
p_{1}(\tau) \mathfrak{B}_{1} \underset{\tau \to +\infty}{=}& \, -\dfrac{\mathrm{i}
\mathfrak{B}_{0,1}^{\sharp}}{8 \sqrt{3}} \! + \! \mathfrak{B}_{1}(1 \! + \!
\hat{\mathbb{L}}_{1}(\tau)) \! - \! \dfrac{\mathrm{i} \tilde{r}_{0,1}(\tau) \tau^{-1/3}}{96
\sqrt{3}} \! \left(1 \! + \! \mathcal{O}((\tilde{r}_{0,1}(\tau) \tau^{-1/3})^{2}) \right)
\! \mathfrak{B}_{0,1}^{\sharp} \nonumber \\
+& \, \dfrac{\mathrm{i} \omega_{0,1}^{2}}{(8 \sqrt{3})^{3}} \! \left(1 \! + \!
\dfrac{\tilde{r}_{0,1}(\tau) \tau^{-1/3}}{12} \! + \! \mathcal{O}((\tilde{r}_{0,1}
(\tau) \tau^{-1/3})^{3}) \right)^{3} \! \left(\dfrac{\mathfrak{B}_{0,1}^{\sharp}}{
\mathfrak{B}_{1}} \right)^{2} \! \mathfrak{B}_{1} \nonumber \\
+& \, \mathcal{O} \! \left(\omega_{0,1}^{4} \! \left(1 \! + \! \dfrac{\tilde{r}_{0,1}
(\tau) \tau^{-1/3}}{12} \! + \! \mathcal{O}((\tilde{r}_{0,1}(\tau) \tau^{-1/3})^{3})
\right)^{5} \! \left(\dfrac{\mathfrak{B}_{0,1}^{\sharp}}{\mathfrak{B}_{1}} \right)^{3}
\! \mathfrak{B}_{1} \right);
\end{align}
{}from the corresponding $(k \! = \! +1)$ Asymptotics~\eqref{tr1}, \eqref{tr3},
\eqref{prcybk1}, \eqref{prcyb0k1}, and~\eqref{prcyomg1}, the various terms
appearing in the Asymptotics~\eqref{geek88} can be presented as
follows:\footnote{Note, in particular, that $\mathfrak{B}_{0,1}^{\sharp}/
\mathfrak{B}_{1} \! =_{\tau \to +\infty} \! -\mathrm{i} 8 \sqrt{3}(1 \! + \! o(1))$.}
{\fontsize{10pt}{11pt}\selectfont
\begin{gather}
-\dfrac{\mathrm{i} \mathfrak{B}_{0,1}^{\sharp}}{8 \sqrt{3}} \underset{\tau \to +\infty}{=}
\dfrac{(\sqrt{3} \! + \! 1) \tau^{-1/3}}{6 \alpha_{1}} \! + \! \sum_{m=0}^{\infty}
\dfrac{\mathfrak{b}_{m}^{\flat}(1)}{(\tau^{1/3})^{m+3}} \! + \! \mathcal{O}
(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}), \label{geek89} \\
-\dfrac{\mathrm{i} \tilde{r}_{0,1}(\tau) \tau^{-1/3}}{96 \sqrt{3}} \! \left(1 \! + \! \mathcal{O}
((\tilde{r}_{0,1}(\tau) \tau^{-1/3})^{2}) \right) \! \mathfrak{B}_{0,1}^{\sharp}
\underset{\tau \to +\infty}{=} \sum_{m=0}^{\infty} \dfrac{\mathfrak{b}_{m}^{\natural}
(1)}{(\tau^{1/3})^{m+3}} \! + \! \mathcal{O}(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}),
\label{geek90}
\end{gather}
\begin{align}
&\dfrac{\mathrm{i} \omega_{0,1}^{2}}{(8 \sqrt{3})^{3}} \! \left(1 \! + \! \dfrac{\tilde{r}_{0,1}
(\tau) \tau^{-1/3}}{12} \! + \! \mathcal{O}((\tilde{r}_{0,1}(\tau) \tau^{-1/3})^{3})
\right)^{3} \! \left(\dfrac{\mathfrak{B}_{0,1}^{\sharp}}{\mathfrak{B}_{1}} \right)^{2}
\! \mathfrak{B}_{1} \! + \! \mathcal{O} \! \left(\omega_{0,1}^{4} \! \left(1 \! + \!
\dfrac{\tilde{r}_{0,1}(\tau) \tau^{-1/3}}{12} \right. \right. \nonumber \\
+&\left. \left. \, \mathcal{O}((\tilde{r}_{0,1}(\tau) \tau^{-1/3})^{3}) \right)^{5} \!
\left(\dfrac{\mathfrak{B}_{0,1}^{\sharp}}{\mathfrak{B}_{1}} \right)^{3} \! \mathfrak{B}_{1}
\right) \underset{\tau \to +\infty}{=} \sum_{m=0}^{\infty} \dfrac{\mathfrak{b}_{m}^{
\sharp}(1)}{(\tau^{1/3})^{m+3}} \! + \! \mathcal{O}(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}),
\label{geek91}
\end{align}}
for certain coefficients $\mathfrak{b}_{m}^{\flat}(1)$, $\mathfrak{b}_{m}^{\natural}
(1)$, and $\mathfrak{b}_{m}^{\sharp}(1)$, whence (cf. Asymptotics~\eqref{geek88})
\begin{equation} \label{geek92}
p_{1}(\tau) \mathfrak{B}_{1} \underset{\tau \to +\infty}{=} \mathfrak{B}_{1}(1 \! + \!
\hat{\mathbb{L}}_{1}(\tau)) \! + \! \dfrac{(\sqrt{3} \! + \! 1) \tau^{-1/3}}{6 \alpha_{1}}
\! + \! \sum_{m=0}^{\infty} \dfrac{\mathfrak{b}_{m}^{\dagger}(1)}{(\tau^{1/3})^{m+3}}
\! + \! \mathcal{O}(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}),
\end{equation}
for certain coefficients $\mathfrak{b}_{m}^{\dagger}(1)$; for example,
\begin{equation} \label{nueeq1}
\mathfrak{b}_{0}^{\dagger}(1) \! = \! \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1)}{48 \sqrt{3} \alpha_{1}}
\! \left(\mathrm{i} 6 \mathfrak{r}_{0}(1) \! + \! 4(a \! - \! \mathrm{i}/2) \mathfrak{u}_{0}(1) \! - \!
\alpha_{1}^{2}(8 \mathfrak{u}_{0}^{2}(1) \! + \! 4 \mathfrak{u}_{0}(1) \mathfrak{r}_{0}
(1) \! - \! \mathfrak{r}_{0}^{2}(1)) \right).
\end{equation}
One shows {}from the corresponding $(k \! = \! +1)$ Asymptotics~\eqref{tr1},
\eqref{tr3}, and~\eqref{prcybk1} that
\begin{equation} \label{nueeq2}
\mathfrak{B}_{1} \underset{\tau \to +\infty}{=} \llfloor \mathfrak{B}_{1} \rrfloor \!
+ \! \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1) \alpha_{1}}{2}(4 \mathrm{A}_{1} \! + \! (\sqrt{3}
\! + \! 1) \mathrm{B}_{1}) \mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}(1 \! +
\! \mathcal{O}(\tau^{-1/3})),
\end{equation}
where
\begin{equation} \label{nueeq3}
\mathrm{B}_{1} \! := \! 2(1 \! + \! \sqrt{3}) \mathrm{A}_{1},
\end{equation}
and
\begin{equation} \label{nueeq4}
\llfloor \mathfrak{B}_{1} \rrfloor \! := \! -\dfrac{(\sqrt{3} \! + \! 1) \tau^{-1/3}}{6
\alpha_{1}} \! + \! \sum_{m=0}^{\infty} \dfrac{b_{m}(1)}{(\tau^{1/3})^{m+3}},
\end{equation}
for certain coefficients $b_{m}(1)$; for example,
\begin{align}
b_{0}(1) \! =& \, \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1)^{2}}{2} \! \left(\alpha_{1} \mathfrak{r}_{2}
(1) \! + \! \dfrac{1}{2 \sqrt{3}} \! \left(-\dfrac{\alpha_{1}}{2}(\mathfrak{r}_{0}^{2}(1)
\! + \! 2(\sqrt{3} \! + \! 1) \mathfrak{r}_{0}(1) \mathfrak{u}_{0}(1) \! + \! 8
\mathfrak{u}_{0}^{2}(1)) \right. \right. \nonumber \\
+&\left. \left. \dfrac{(a \! - \! \mathrm{i}/2)}{6 \alpha_{1}}(12 \mathfrak{u}_{0}(1) \! + \!
(2 \sqrt{3} \! - \! 1) \mathfrak{r}_{0}(1)) \right) \right), \label{nueeq5} \\
b_{1}(1) \! =& \, 0. \label{nueeq6}
\end{align}
{}From the Expansions~\eqref{geek92} and~\eqref{nueeq2}, and the
Definition~\eqref{nueeq4}, it follows that
\begin{align} \label{nueeq7}
p_{1}(\tau) \mathfrak{B}_{1} \underset{\tau \to +\infty}{=}& \, \tau^{-1}
\sum_{m=0}^{\infty} \dfrac{d_{m}^{\ast}(1)}{(\tau^{1/3})^{m}} \! + \!
\hat{\mathbb{L}}_{1}(\tau) \! \left(\llfloor \mathfrak{B}_{1} \rrfloor \!
+ \! \mathcal{O}(\mathrm{e}^{-\beta (\tau)}) \right) \nonumber \\
+& \, \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1) \alpha_{1}}{2}(4 \mathrm{A}_{1} \! + \!
(\sqrt{3} \! + \! 1) \mathrm{B}_{1}) \mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta
(\tau)}(1 \! + \! \mathcal{O}(\tau^{-1/3})),
\end{align}
for coefficients $d_{m}^{\ast}(1) \! := \! \mathfrak{b}_{m}^{\dagger}(1)
\! + \! b_{m}(1)$, $m \! \in \! \mathbb{Z}_{+}$; for example,
\begin{align} \label{nueeq8}
d_{0}^{\ast}(1) =& \, \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1)}{48 \sqrt{3} \alpha_{1}} \!
\left(\mathrm{i} 6 \mathfrak{r}_{0}(1) \! + \! 4(a \! - \! \mathrm{i}/2) \mathfrak{u}_{0}(1)
\! - \! \alpha_{1}^{2}(8 \mathfrak{u}_{0}^{2}(1) \! + \! 4 \mathfrak{u}_{0}(1)
\mathfrak{r}_{0}(1) \! - \! \mathfrak{r}_{0}^{2}(1)) \right) \nonumber \\
+& \, \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1)^{2}}{2} \! \left(\alpha_{1} \mathfrak{r}_{2}
(1) \! + \! \dfrac{1}{2 \sqrt{3}} \! \left(-\dfrac{\alpha_{1}}{2}(\mathfrak{r}_{0}^{2}
(1) \! + \! 2(\sqrt{3} \! + \! 1) \mathfrak{r}_{0}(1) \mathfrak{u}_{0}(1)
\! + \! 8 \mathfrak{u}_{0}^{2}(1)) \right. \right. \nonumber \\
+&\left. \left. \dfrac{(a \! - \! \mathrm{i}/2)}{6 \alpha_{1}}(12 \mathfrak{u}_{0}(1)
\! + \! (2 \sqrt{3} \! - \! 1) \mathfrak{r}_{0}(1)) \right) \right).
\end{align}
Thus, via the Asymptotics~\eqref{geek87} and~\eqref{nueeq7}, one arrives at
\begin{align} \label{nueeq9}
&\left(\sum_{m=0}^{\infty} \dfrac{d_{m}^{\ast}(1)}{(\tau^{1/3})^{m+3}} \! +
\! \hat{\mathbb{L}}_{1}(\tau) \! \left(\llfloor \mathfrak{B}_{1} \rrfloor \! + \!
\mathcal{O}(\mathrm{e}^{-\beta (\tau)}) \right) \! + \! \frac{\mathrm{i} (\sqrt{3} \! + \! 1)
\alpha_{1}}{2}(4 \mathrm{A}_{1} \! + \! (\sqrt{3} \! + \! 1) \mathrm{B}_{1})
\mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)} \right. \nonumber \\
&\times \left. (1 \! + \! \mathcal{O}(\tau^{-1/3})) \right) \! \left(1 \! + \!
\sum_{m=0}^{\infty} \dfrac{\hat{\epsilon}_{m}^{\sharp}(1)}{(\tau^{1/3})^{m+1}}
\! + \! \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}) \right) \underset{\tau \to
+\infty}{=} -\mathcal{Q}_{1} \mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}
(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})),
\end{align}
where
\begin{equation} \label{geek94}
\mathcal{Q}_{1} \! := \! \dfrac{2^{3/2}3^{1/4} \mathrm{e}^{\mathrm{i} \pi/4}(2 \! + \! \sqrt{3})
\mathscr{P}_{a}(s_{0}^{0} \! - \! \mathrm{i} \mathrm{e}^{-\pi a})}{\sqrt{2 \pi}}.
\end{equation}
One now chooses $\hat{\mathbb{L}}_{1}(\tau)$ so that the---divergent---power
series on the left-hand side of Equation~\eqref{nueeq9} is identically equal to zero:
\begin{equation} \label{nueeq10}
\left(\tau^{-1} \sum_{m=0}^{\infty} \dfrac{d_{m}^{\ast}(1)}{(\tau^{1/3})^{m}} \!
+ \! \hat{\mathbb{L}}_{1}(\tau) \llfloor \mathfrak{B}_{1} \rrfloor \right) \! \left(
1 \! + \! \tau^{-1/3} \sum_{m=0}^{\infty} \dfrac{\hat{\epsilon}_{m}^{\sharp}
(1)}{(\tau^{1/3})^{m}} \right) \! = \! 0;
\end{equation}
via the Definition~\eqref{nueeq4}, one solves Equation~\eqref{nueeq10} for
$\hat{\mathbb{L}}_{1}(\tau)$ to arrive at
\begin{equation} \label{nueeq11}
\hat{\mathbb{L}}_{1}(\tau) \! = \! \tau^{-2/3} \sum_{m=0}^{\infty}
\dfrac{\hat{\mathfrak{l}}_{m+2}(1)}{(\tau^{1/3})^{m}},
\end{equation}
where the coefficients $\hat{\mathfrak{l}}_{m^{\prime}}(1)$, $m^{\prime} \! \in \!
\mathbb{Z}_{+}$, are determined according to the recursive prescription
\begin{gather}
\hat{\mathfrak{l}}_{0}(1) \! = \! \hat{\mathfrak{l}}_{1}(1) \! = \! 0, \, \, \quad \quad
\, \, \hat{\mathfrak{l}}_{2}(1) \! = \! \dfrac{6 \alpha_{1}d_{0}^{\ast}(1)}{\sqrt{3} \!
+ \! 1}, \label{nueeq12} \\
\hat{\mathfrak{l}}_{m+3}(1) \! = \! \dfrac{6 \alpha_{1}}{\sqrt{3} \! + \! 1} \! \left(
d_{m+1}^{\ast}(1) \! + \! \sum_{p=0}^{m}d_{p}^{\ast}(1) \hat{\epsilon}_{m-p}^{
\sharp}(1) \! + \! \sum_{j=0}^{m+2} \hat{\mathfrak{l}}_{j}(1) \hat{d}_{m+4-j}(1)
\right), \quad m \! \in \! \mathbb{Z}_{+}, \label{nueeq13}
\end{gather}
with
\begin{gather}
\hat{d}_{0}(1) \! = \! 0, \! \quad \! \hat{d}_{1}(1) \! = \! -\dfrac{(\sqrt{3} \! + \!
1)}{6 \alpha_{1}}, \! \quad \! \hat{d}_{2}(1) \! = \! -\dfrac{(\sqrt{3} \! + \! 1)
\hat{\epsilon}_{0}^{\sharp}(1)}{6 \alpha_{1}}, \! \quad \! \hat{d}_{3}(1) \! = \! b_{0}
(1) \! - \! \dfrac{(\sqrt{3} \! + \! 1) \hat{\epsilon}_{1}^{\sharp}(1)}{6 \alpha_{1}},
\label{nueeq14} \\
\hat{d}_{m+4}(1) \! = \! b_{m+1}(1) \! - \! \dfrac{(\sqrt{3} \! + \! 1)
\hat{\epsilon}_{m+2}^{\sharp}(1)}{6 \alpha_{1}} \! + \! \sum_{p=0}^{m}b_{p}(1)
\hat{\epsilon}_{m-p}^{\sharp}(1), \quad m \! \in \! \mathbb{Z}_{+}. \label{nueeq15}
\end{gather}
{}From the Condition~\eqref{nueeq10}, Equation~\eqref{nueeq11}, and the
Asymptotics~\eqref{nueeq9}, it follows that
\begin{equation} \label{nueeq16}
\dfrac{\mathrm{i} (\sqrt{3} \! + \! 1) \alpha_{1}}{2}(4 \mathrm{A}_{1} \! + \! (\sqrt{3}
\! + \! 1) \mathrm{B}_{1}) \mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}
\underset{\tau \to +\infty}{=} -\mathcal{Q}_{1} \mathrm{e}^{-\mathrm{i} \vartheta (\tau)}
\mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})),
\end{equation}
whence, via the Definitions~\eqref{geek4}, \eqref{nueeq3}, and~\eqref{geek94},
one arrives at
\begin{equation} \label{geek109}
\mathrm{A}_{1} \! = \! \dfrac{\mathrm{i} \mathrm{e}^{\mathrm{i} \pi/4} \mathrm{e}^{-\mathrm{i} \pi/3}(2 \! + \!
\sqrt{3})^{\mathrm{i} a}(s_{0}^{0} \! - \! \mathrm{i} \mathrm{e}^{-\pi a})}{\sqrt{2 \pi} \, 3^{1/4}
(\varepsilon b)^{1/6}}.
\end{equation}
Alternatively, one may proceed as follows. Substituting the Asymptotics~\eqref{geek92}
and~\eqref{nueeq2} into Equation~\eqref{geek87}, one shows, via the
Definition~\eqref{nueeq4} and the definition $d_{m}^{\ast}(1) \! := \!
\mathfrak{b}_{m}^{\dagger}(1) \! + \! b_{m}(1)$, $m \! \in \! \mathbb{Z}_{+}$, that
\begin{align}
&\mathfrak{B}_{1} \! + \! \dfrac{(\sqrt{3} \! + \! 1) \tau^{-1/3}}{6 \alpha_{1}}
\! + \! \tau^{-1} \sum_{m=0}^{\infty} \dfrac{d_{m}(1)}{(\tau^{1/3})^{m}} \!
+ \! \hat{\mathbb{L}}_{1}(\tau) \mathfrak{B}_{1} \! \left(1 \! + \! \tau^{-1/3}
\sum_{m=0}^{\infty} \dfrac{\hat{\epsilon}_{m}^{\sharp}(1)}{(\tau^{1/3})^{m}}
\right. \nonumber \\
+&\left. \mathcal{O}(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}) \right) \! + \! \mathcal{O}
(\tau^{-1/3} \mathrm{e}^{-\beta (\tau)}) \underset{\tau \to +\infty}{=} -\mathcal{Q}_{1}
\mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}
(\tau^{-\delta_{1}})), \label{nueeq17}
\end{align}
where $\mathcal{Q}_{1}$ is defined by Equation~\eqref{geek94},
\begin{equation} \label{nueeq18}
d_{0}(1) \! = \! \mathfrak{b}_{0}^{\dagger}(1), \quad \quad d_{m+1}(1) \! =
\! \mathfrak{b}_{m+1}^{\dagger}(1) \! + \! \sum_{p=0}^{m}d_{p}^{\ast}(1)
\hat{\epsilon}_{m-p}^{\sharp}(1), \quad m \! \in \! \mathbb{Z}_{+}.
\end{equation}
{}From the Condition~\eqref{nueeq10}, Equation~\eqref{nueeq11}, the
Asymptotics~\eqref{nueeq17}, the definition $d_{m}^{\ast}(1) \! := \!
\mathfrak{b}_{m}^{\dagger}(1) \! + \! b_{m}(1)$, $m \! \in \! \mathbb{Z}_{+}$,
and Equations~\eqref{nueeq18}, it follows that
\begin{equation} \label{nueeq19}
\mathfrak{B}_{1} \underset{\tau \to +\infty}{=} -\dfrac{(\sqrt{3} \! + \! 1)
\tau^{-1/3}}{6 \alpha_{1}} \! + \! \tau^{-1} \sum_{m=0}^{\infty} \dfrac{b_{m}
(1)}{(\tau^{1/3})^{m}} \! - \! \mathcal{Q}_{1} \mathrm{e}^{-\mathrm{i} \vartheta (\tau)}
\mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})).
\end{equation}
It follows {}from the corresponding $(k \! = \! +1)$ Asymptotics~\eqref{tr1},
\eqref{tr3}, and~\eqref{prcybk1} that the function $\mathfrak{B}_{1}$ can also
be presented in the form
\begin{align} \label{nueeq20}
\mathfrak{B}_{1} \underset{\tau \to +\infty}{=}& \, \mathrm{i} (\sqrt{3} \! + \! 1) \!
\left(\dfrac{\alpha_{1}}{2}(4v_{0,1}(\tau) \! + \! (\sqrt{3} \! + \! 1) \tilde{r}_{0,1}
(\tau)) \! - \! \dfrac{(\sqrt{3} \! + \! 1)(a \! - \! \mathrm{i}/2)}{2 \sqrt{3} \alpha_{1}
\tau^{1/3}} \right) \! + \! \sum_{m=0}^{\infty} \dfrac{\hat{b}_{m}^{\ast}(1)}{
(\tau^{1/3})^{m+3}} \nonumber \\
+& \, \mathcal{O}(\tau^{-2/3} \mathrm{e}^{-\beta (\tau)}),
\end{align}
for certain coefficients $\hat{b}_{m}^{\ast}(1)$ (see, for example,
Equations~\eqref{nueeq22} and~\eqref{nueeq23} below); hence, {}from the
Asymptotics~\eqref{nueeq19} and~\eqref{nueeq20}, one deduces that
\begin{align} \label{geek97}
4v_{0,1}(\tau) \! + \! (\sqrt{3} \! + \! 1) \tilde{r}_{0,1}(\tau) \underset{\tau \to
+\infty}{=}& \, \dfrac{(\sqrt{3} \! + \! 1)(\sqrt{3}a \! - \! \mathrm{i}/2)}{3 \alpha_{1}^{2}
\tau^{1/3}} \! + \! \sum_{m=0}^{\infty} \dfrac{\iota_{m}^{\ast}(1)}{(\tau^{1/3})^{m+3}}
\nonumber \\
+& \, \dfrac{\mathrm{i} 2 \mathcal{Q}_{1} \mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta
(\tau)}}{(\sqrt{3} \! + \! 1) \alpha_{1}}(1 \! + \! \mathcal{O}(\tau^{-\delta_{1}})),
\end{align}
where
\begin{equation} \label{nueeq21}
\iota_{m}^{\ast}(1) \! := \! -\dfrac{\mathrm{i} 2(b_{m}(1) \! - \! \hat{b}_{m}^{\ast}(1))}{
(\sqrt{3} \! + \! 1) \alpha_{1}}, \quad m \! \in \! \mathbb{Z}_{+}.
\end{equation}
Combining the corresponding $(k \! = \! +1)$ Equations~\eqref{iden217}
and~\eqref{tr2}, it follows that, in terms of the corresponding $(k \! = \! +1)$ solution
of the DP3E~\eqref{eq1.1},
\begin{equation} \label{geek98}
4v_{0,1}(\tau) \! + \! (\sqrt{3} \! + \! 1) \tilde{r}_{0,1}(\tau) \! = \! \dfrac{8
\mathrm{e}^{2 \pi \mathrm{i}/3}u(\tau)}{\varepsilon (\varepsilon b)^{2/3}} \! - \! \dfrac{\mathrm{i}
(\sqrt{3} \! + \! 1) \mathrm{e}^{-\mathrm{i} 2 \pi/3} \tau^{2/3}}{(\varepsilon b)^{1/3}} \!
\left(\dfrac{u^{\prime}(\tau) \! - \! \mathrm{i} b}{u(\tau)} \right) \! + \! 2(\sqrt{3}
\! - \! 1) \tau^{1/3};
\end{equation}
finally, {}from the Asymptotics~\eqref{geek97} and Equation~\eqref{geek98}, one
arrives at the---asymptotic---Riccati differential equation
\begin{equation} \label{geek99}
u^{\prime}(\tau) \underset{\tau \to +\infty}{=} \tilde{\mathfrak{a}}(\tau) \! + \!
\tilde{\mathfrak{b}}(\tau)u(\tau) \! + \! \tilde{\mathfrak{c}}(\tau)(u(\tau))^{2},
\end{equation}
where
\begin{equation} \label{ricc1}
\begin{gathered}
\tilde{\mathfrak{a}}(\tau) \! := \! \mathrm{i} b, \qquad \qquad \tilde{\mathfrak{c}}(\tau) \!
:= \! \frac{\mathrm{i} \varepsilon 8 \sqrt{2} \alpha_{1} \tau^{-2/3}}{(\sqrt{3} \! + \! 1)
(\varepsilon b)^{1/2}}, \\
\tilde{\mathfrak{b}}(\tau) \! := \! -\frac{\mathrm{i} 8 \alpha_{1}^{2} \tau^{-1/3}}{(\sqrt{3}
\! + \! 1)^{2}} \! + \! \frac{\mathrm{i} 2(\sqrt{3}a \! - \! \mathrm{i}/2)}{3 \tau} \! + \! \frac{\mathrm{i} 2
\alpha_{1}^{2}}{(\sqrt{3} \! + \! 1)} \sum_{m=0}^{\infty} \frac{\iota_{m}^{\ast}(1)}{
(\tau^{1/3})^{m+5}} \! - \! \frac{4 \alpha_{1} \mathcal{Q}_{1} \mathrm{e}^{-\mathrm{i} \vartheta
(\tau)} \mathrm{e}^{-\beta (\tau)}}{(\sqrt{3} \! + \! 1)^{2} \tau^{2/3}}(1 \! + \! \mathcal{O}
(\tau^{-\delta_{1}})).
\end{gathered}
\end{equation}
Incidentally, changing the dependent variable according to $w(\tau)
\! = \! \tfrac{1}{2} \tilde{\mathfrak{b}}(\tau) \! + \! \tfrac{1}{2}
\tfrac{\tilde{\mathfrak{c}}^{\prime}(\tau)}{\tilde{\mathfrak{c}}(\tau)} \! + \!
\tilde{\mathfrak{c}}(\tau)u(\tau)$,\footnote{See Section~4.6 of \cite{ntsnmz};
see, also, Chapter~5 of \cite{eillh}.}
it follows that the Riccati differential Equation~\eqref{geek99} transforms into
\begin{equation} \label{ricc2}
w^{\prime}(\tau) \underset{\tau \to +\infty}{=} \Xi (\tau) \! + \! (w(\tau))^{2},
\end{equation}
where
\begin{equation} \label{ricc3}
-\Xi (\tau) \! := \! -\tilde{\mathfrak{a}}(\tau) \tilde{\mathfrak{c}}(\tau) \! + \! \frac{1}{4}
(\tilde{\mathfrak{b}}(\tau))^{2} \! - \! \frac{1}{2} \tilde{\mathfrak{b}}^{\prime}(\tau) \!
+ \! \frac{1}{2} \frac{\tilde{\mathfrak{b}}(\tau) \tilde{\mathfrak{c}}^{\prime}(\tau)}{
\tilde{\mathfrak{c}}(\tau)} \! - \! \frac{1}{2} \frac{\tilde{\mathfrak{c}}^{\prime \prime}
(\tau)}{\tilde{\mathfrak{c}}(\tau)} \! + \! \frac{3}{4} \! \left(\frac{\tilde{\mathfrak{c}}^{
\prime}(\tau)}{\tilde{\mathfrak{c}}(\tau)} \right)^{2}.
\end{equation}
Substituting the corresponding $(k \! = \! +1)$ differentiable
Asymptotics~\eqref{recur15} into either the Riccati differential Equation~\eqref{geek99}
or its dependent-variable-transformed variant~\eqref{ricc2}, and recalling that $c_{0,1}
\! = \! \tfrac{1}{2} \varepsilon (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi/3}$, one shows that
\begin{align} \label{geek100}
&\dfrac{\varepsilon 8 \mathrm{e}^{\mathrm{i} 2 \pi/3}}{(\varepsilon b)^{2/3}} \! \left(c_{0,1}^{2}
\tau^{2/3} \! + \! 2c_{0,1}^{2} \sum_{m=0}^{\infty} \dfrac{\mathfrak{u}_{m}(1)}{
(\tau^{1/3})^{m}} \! + \! c_{0,1}^{2} \tau^{-2/3} \sum_{m=0}^{\infty}
\sum_{m_{1}=0}^{m} \mathfrak{u}_{m_{1}}(1) \mathfrak{u}_{m-m_{1}}(1)
(\tau^{-1/3})^{m} \right. \nonumber \\
&+ \left. 2c_{0,1} \mathbb{P} \tau^{1/3} \mathrm{e}^{-\mathrm{i} \vartheta (\tau)}
\mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}(\tau^{-1/3})) \right) \! - \!
\dfrac{\mathrm{i} (\sqrt{3} \! + \! 1) \mathrm{e}^{-\mathrm{i} 2 \pi/3} \tau^{2/3}}{(\varepsilon b)^{1/3}}
\! \left(-\mathrm{i} b \! + \! \dfrac{c_{0,1}}{3} \tau^{-2/3} \right. \nonumber \\
&- \left. \dfrac{c_{0,1}}{3} \sum_{m=0}^{\infty} \dfrac{(m \! + \! 1)
\mathfrak{u}_{m}(1)}{(\tau^{1/3})^{m+4}} \! + \! \mathrm{i} 2 \sqrt{3}(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} 2 \pi/3} \mathbb{P} \tau^{-1/3} \mathrm{e}^{-\mathrm{i} \vartheta (\tau)}
\mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}(\tau^{-1/3})) \right) \nonumber \\
&+ 2(\sqrt{3} \! - \! 1) \tau^{1/3} \! \left(c_{0,1} \tau^{1/3} \! + \! c_{0,1}
\sum_{m=0}^{\infty} \dfrac{\mathfrak{u}_{m}(1)}{(\tau^{1/3})^{m+1}} \! + \!
\mathbb{P} \mathrm{e}^{-\mathrm{i} \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}
(\tau^{-1/3})) \right) \nonumber \\
& \underset{\tau \to +\infty}{=} \left(\dfrac{(\sqrt{3} \! + \! 1)(\sqrt{3}a \! - \!
\mathrm{i}/2)}{3 \alpha_{1}^{2} \tau^{1/3}} \! + \! \sum_{m=0}^{\infty} \frac{\iota_{m}^{\ast}
(1)}{(\tau^{1/3})^{m+3}} \! + \! \dfrac{\mathrm{i} 2 \mathcal{Q}_{1} \mathrm{e}^{-\mathrm{i} \vartheta
(\tau)} \mathrm{e}^{-\beta (\tau)}}{(\sqrt{3} \! + \! 1) \alpha_{1}}(1 \! + \! \mathcal{O}
(\tau^{-\delta_{1}})) \right) \nonumber \\
& \, \times \left(c_{0,1} \tau^{1/3} \! + \! c_{0,1} \sum_{m=0}^{\infty}
\dfrac{\mathfrak{u}_{m}(1)}{(\tau^{1/3})^{m+1}} \! + \! \mathbb{P} \mathrm{e}^{-\mathrm{i}
\vartheta (\tau)} \mathrm{e}^{-\beta (\tau)}(1 \! + \! \mathcal{O}(\tau^{-1/3})) \right),
\end{align}
where
\begin{equation} \label{geek101}
\mathbb{P} \! := \! c_{0,1} \mathrm{A}_{1}.
\end{equation}
Equating coefficients of terms that are $\mathcal{O}(\tau^{1/3} \mathrm{e}^{-\mathrm{i}
\vartheta (\tau)} \mathrm{e}^{-\beta (\tau)})$, $\mathcal{O}(\tau^{2/3})$, $\mathcal{O}(1)$,
$\mathcal{O}(\tau^{-1/3})$, $\mathcal{O}(\tau^{-2/3})$, and $\mathcal{O}
(\tau^{-1})$, respectively, in Equation~\eqref{geek100}, one arrives at, in the
indicated order:
\begin{align}
&\left(\dfrac{16 \mathrm{e}^{\mathrm{i} 2 \pi/3}c_{0,1}}{\varepsilon (\varepsilon b)^{2/3}}
\! + \! 2 \sqrt{3}(\sqrt{3} \! + \! 1) \! + \! 2(\sqrt{3} \! - \! 1) \right) \!
\mathbb{P} \! = \! \dfrac{\mathrm{i} 2 \mathcal{Q}_{1}c_{0,1}}{(\sqrt{3} \! + \! 1)
\alpha_{1}}, \label{geek102} \\
&\dfrac{8 \mathrm{e}^{\mathrm{i} 2 \pi/3}c_{0,1}^{2}}{\varepsilon (\varepsilon b)^{2/3}} \!
- \! \dfrac{(\sqrt{3} \! + \! 1) \mathrm{e}^{-\mathrm{i} 2 \pi/3}b}{(\varepsilon b)^{1/3}} \!
+ \! 2(\sqrt{3} \! - \! 1)c_{0,1} \! = \! 0, \label{geek103} \\
&\dfrac{16 \mathrm{e}^{\mathrm{i} 2 \pi/3}c_{0,1} \mathfrak{u}_{0}(1)}{\varepsilon (\varepsilon
b)^{2/3}} \! - \! \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1) \mathrm{e}^{-\mathrm{i} 2 \pi/3}}{3(\varepsilon
b)^{1/3}} \! + \! 2(\sqrt{3} \! - \! 1) \mathfrak{u}_{0}(1) \! = \! \dfrac{(\sqrt{3}
\! + \! 1)(\sqrt{3}a \! - \! \mathrm{i}/2)}{3 \alpha_{1}^{2}}, \label{geek104} \\
&\left(\dfrac{16 \mathrm{e}^{\mathrm{i} 2 \pi/3}c_{0,1}}{\varepsilon (\varepsilon b)^{2/3}} \! +
\! 2(\sqrt{3} \! - \! 1) \right) \! \mathfrak{u}_{1}(1) \! = \! 0, \label{geek105} \\
&\dfrac{8 \mathrm{e}^{\mathrm{i} 2 \pi/3}c_{0,1}}{\varepsilon (\varepsilon b)^{2/3}} \! \left(
2 \mathfrak{u}_{2}(1) \! + \! \mathfrak{u}_{0}^{2}(1) \right) \! + \! \dfrac{\mathrm{i}
(\sqrt{3} \! + \! 1) \mathrm{e}^{-\mathrm{i} 2 \pi/3} \mathfrak{u}_{0}(1)}{3(\varepsilon
b)^{1/3}} \! + \! 2(\sqrt{3} \! - \! 1) \mathfrak{u}_{2}(1) \nonumber \\
&= \! \dfrac{(\sqrt{3} \! + \! 1)(\sqrt{3}a \! - \! \mathrm{i}/2)}{3 \alpha_{1}^{2}} \!
+ \! \iota_{0}^{\ast}(1), \label{geek106} \\
&\dfrac{16 \mathrm{e}^{\mathrm{i} 2 \pi/3}c_{0,1}}{\varepsilon (\varepsilon b)^{2/3}} \! \left(
\mathfrak{u}_{3}(1) \! + \! \mathfrak{u}_{0}(1) \mathfrak{u}_{1}(1) \right) \! +
\! \dfrac{\mathrm{i} 2(\sqrt{3} \! + \! 1) \mathrm{e}^{-\mathrm{i} 2 \pi/3} \mathfrak{u}_{1}(1)}{3
(\varepsilon b)^{1/3}} \! + \! 2(\sqrt{3} \! - \! 1) \mathfrak{u}_{3}(1)
\nonumber \\
&= \! \dfrac{(\sqrt{3} \! + \! 1)(\sqrt{3}a \! - \! \mathrm{i}/2) \mathfrak{u}_{1}(1)}{3
\alpha_{1}^{2}} \! + \! \iota_{1}^{\ast}(1). \label{geek107}
\end{align}
Using the corresponding $(k \! = \! +1)$ coefficients~\eqref{thmk3}, in particular,
$\mathfrak{u}_{0}(1) \! = \! a/6 \alpha_{1}^{2}$ and $\mathfrak{u}_{1}(1) \! = \!
\mathfrak{u}_{2}(1) \! = \! \mathfrak{u}_{3}(1) \! = \! 0$, one analyses
Equations~\eqref{geek102}--\eqref{geek107}, in the indicated order, in order
to arrive at the following conclusions: (i) solving Equation~\eqref{geek102}
for $\mathbb{P}$, one deduces that
\begin{equation} \label{geek108}
\mathbb{P} \! = \! -\dfrac{\mathrm{i} \varepsilon (\varepsilon b)^{1/2} \mathrm{e}^{\mathrm{i} \pi/4}
\mathscr{P}_{a}(s_{0}^{0} \! - \! \mathrm{i} \mathrm{e}^{-\pi a})}{\sqrt{\pi} \, 2^{3/2}3^{1/4}},
\end{equation}
whence, {}from the Definition~\eqref{geek101}, one arrives at
Equation~\eqref{geek109}; (ii) Equations~\eqref{geek103}--\eqref{geek105} are
identically true; and (iii) solving Equations~\eqref{geek106} and~\eqref{geek107}
for $\iota_{0}^{\ast}(1)$ and $\iota_{1}^{\ast}(1)$, respectively, one concludes that
\begin{equation} \label{geek110}
\iota_{0}^{\ast}(1) \! = \! \dfrac{\mathrm{i} a (1 \! + \! \mathrm{i} a)(\sqrt{3} \! + \! 1)}{18
\alpha_{1}^{4}} \qquad \text{and} \qquad \iota_{1}^{\ast}(1) \! = \! 0;
\end{equation}
moreover, {}from Equations~\eqref{nueeq5} and~\eqref{nueeq6}, the
Definition~\eqref{nueeq21}, and Equations~\eqref{geek110}, it also follows that
\begin{gather}
\hat{b}_{0}^{\ast}(1) \! = \! \dfrac{\mathrm{i} (\sqrt{3} \! + \! 1)^{2}}{4 \sqrt{3}} \!
\left(-\dfrac{\alpha_{1}}{2}(\mathfrak{r}_{0}^{2}(1) \! + \! 2(\sqrt{3} \! + \! 1)
\mathfrak{r}_{0}(1) \mathfrak{u}_{0}(1) \! + \! 8 \mathfrak{u}_{0}^{2}(1)) \! + \!
\dfrac{(a \! - \! \mathrm{i}/2)}{6 \alpha_{1}}(12 \mathfrak{u}_{0}(1) \! + \! (2 \sqrt{3} \!
- \! 1) \mathfrak{r}_{0}(1)) \right), \label{nueeq22} \\
\hat{b}_{1}^{\ast}(1) \! = \! 0. \label{nueeq23}
\end{gather}
Finally, {}from the Asymptotics~\eqref{tr1} and~\eqref{tr3} (for $k \! = \! +1)$
and Equation~\eqref{geek109}, one arrives at the corresponding asymptotics
for $v_{0}(\tau) \! := \! v_{0,1}(\tau)$ and $\tilde{r}_{0}(\tau) \! := \! \tilde{r}_{
0,1}(\tau)$ stated in Equations~\eqref{geek1} and~\eqref{geek2}, respectively,
of the lemma.
Similarly, proceeding as delineated above, one deduces that, for $k \! = \! -1$,
\begin{equation} \label{geek111}
\mathrm{A}_{-1} \! = \! \dfrac{\mathrm{i} \mathrm{e}^{-\mathrm{i} \pi/4} \mathrm{e}^{\mathrm{i} \pi/3}(2 \! + \!
\sqrt{3})^{-\mathrm{i} a}(s_{0}^{0} \! - \! \mathrm{i} \mathrm{e}^{-\pi a})}{\sqrt{2 \pi} \, 3^{1/4}
(\varepsilon b)^{1/6}};
\end{equation}
thus, {}from the Asymptotics~\eqref{tr1} and~\eqref{tr3} (for $k \! = \! -1)$
and Equation~\eqref{geek111}, one arrives at the corresponding asymptotics for
$v_{0}(\tau) \! := \! v_{0,-1}(\tau)$ and $\tilde{r}_{0}(\tau) \! := \! \tilde{r}_{0,-1}
(\tau)$ stated in Equations~\eqref{geek1} and~\eqref{geek2}, respectively, of the
lemma. \hfill $\qed$
{}From Equation~\eqref{iden217}, the Asymptotics~\eqref{geek1},
Definition~\eqref{geek4}, and recalling that $c_{0,k} \! = \! \tfrac{1}{2} \varepsilon
(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}$, $k \! = \! \pm 1$, one arrives at the
corresponding $(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! =
\! (0,0,0 \vert 0)$ asymptotics (as $\tau \! \to \! +\infty$ with $\varepsilon b \!
> \! 0)$ for the solution $u(\tau)$ of the DP3E~\eqref{eq1.1} stated in
Theorem~\ref{theor2.1}.
Via the Definitions~\eqref{hatsoff7} and~\eqref{pga3} and Equations~\eqref{pga5}
and~\eqref{tr2}, one deduces that, for $k \! = \! \pm 1$,
\begin{gather}
2f_{-}(\tau) \! = \! -\mathrm{i} (a \! - \! \mathrm{i}/2) \! + \! \frac{\mathrm{i} (\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} 2 \pi k/3}}{2} \tau^{2/3} \! \left(-2 \! + \! \tilde{r}_{0}(\tau) \tau^{-1/3}
\right), \label{textfeqn2} \\
\frac{\mathrm{i} 4}{\varepsilon b}f_{+}(\tau) \! = \! \mathrm{i} (a \! + \! \mathrm{i}/2) \! + \! \frac{\mathrm{i}
(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{2} \tau^{2/3} \! \left(-2 \! + \! \tilde{r}_{0}
(\tau) \tau^{-1/3} \right) \! + \! \frac{\mathrm{i} b \tau}{u(\tau)}; \label{efhpls5}
\end{gather}
thus, {}from the Asymptotics~\eqref{geek1} and~\eqref{geek2}, the Definition~\eqref{geek4},
and Equations~\eqref{textfeqn2} and~\eqref{efhpls5}, one arrives at the corresponding
$(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! = \! (0,0,0 \vert 0)$
asymptotics (as $\tau \! \to \! +\infty$ with $\varepsilon b \! > \! 0)$ for the principal
auxiliary functions $f_{\pm}(\tau)$ (corresponding to $u(\tau)$) stated in
Theorem~\ref{theor2.1}.
It was shown in Equation~(4.25) of \cite{av2} that, in terms of the function
$h_{0}(\tau)$, the Hamiltonian function $\mathcal{H}(\tau)$ (corresponding
to $u(\tau))$ defined by Equation~\eqref{eqh1} is given by
\begin{equation} \label{eqnhtext1}
\mathcal{H}(\tau) \! = \! 3(\varepsilon b)^{2/3} \tau^{1/3} \! + \!
\dfrac{1}{2 \tau}(a \! - \! \mathrm{i}/2)^{2} \! - \! 4 \tau^{-1/3}h_{0}(\tau):
\end{equation}
via the Definition~\eqref{iden2}, and Equation~\eqref{eqnhtext1}, it follows
that, in terms of the function $\hat{h}_{0}(\tau) \! := \! \hat{h}_{0,k}(\tau)$
studied herein,
\begin{equation} \label{eqnhtext2}
\mathcal{H}(\tau) \! = \! 3(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3} \tau^{1/3}
\! + \! \dfrac{1}{2 \tau}(a \! - \! \mathrm{i}/2)^{2} \! - \! 4 \tau^{1/3} \hat{h}_{0,k}
(\tau), \quad k \! = \! \pm 1;
\end{equation}
consequently, {}from Equation~\eqref{expforeych}, the third relation of
Equations~\eqref{expforkapp}, and Equation \eqref{eqnhtext2}, upon
recalling that (cf.~Lemma~\ref{ginversion}) $v_{0}(\tau) \! := \! v_{0,k}(\tau)$
and $\tilde{r}_{0}(\tau) \! := \! \tilde{r}_{0,k}(\tau)$, one shows that the
Hamiltonian function, $\mathcal{H}(\tau)$, is given by
\begin{align} \label{eqnhtext3}
\mathcal{H}(\tau) \! =& \, 3(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3} \tau^{1/3}
\! + \! \dfrac{1}{2 \tau}(a \! - \! \mathrm{i}/2)^{2} \! + \! \frac{\alpha_{k}^{2} \tau^{-1/3}}{1
\! + \! \tau^{-1/3}v_{0,k}(\tau)} \! \left(\vphantom{M^{M^{M^{M^{M}}}}} \!
\alpha_{k}^{2} \left(8v_{0,k}^{2}(\tau) \! + \! (4v_{0,k}(\tau) \right. \right. \nonumber \\
-&\left. \left. \tilde{r}_{0,k}(\tau)) \tilde{r}_{0,k}(\tau) \! - \! \tau^{-1/3}
v_{0,k}(\tau)(\tilde{r}_{0,k}(\tau))^{2} \right) \! + \! 4(a \! - \! \mathrm{i}/2) \right),
\quad k \! = \! \pm 1.
\end{align}
Finally, {}from the Asymptotics~\eqref{geek1} and~\eqref{geek2},
Definition~\eqref{geek4}, and Equation~\eqref{eqnhtext3}, one arrives at,
after a lengthy, but otherwise straightforward, calculation, the corresponding
$(\varepsilon_{1},\varepsilon_{2},m(\varepsilon_{2}) \vert \ell) \! = \! (0,0,0
\vert 0)$ asymptotics (as $\tau \! \to \! +\infty$ with $\varepsilon b \! > \! 0)$
for the Hamiltonian function $\mathcal{H}(\tau)$ stated in Theorem~\ref{theor2.1}.
Via Definition~\eqref{thmk23} and the asymptotics (as $\tau \! \to \! +\infty$
with $\varepsilon b \! > \! 0)$ for $f_{-}(\tau)$ and $\mathcal{H}(\tau)$ stated
above, one arrives at the corresponding $(\varepsilon_{1},\varepsilon_{2},
m(\varepsilon_{2}) \vert \ell) \! = \! (0,0,0 \vert 0)$ asymptotics for the
function $\sigma (\tau)$ stated in Theorem~\ref{theor2.1}.
\begin{bbbbb} \label{isomonoabcd}
Under the conditions of Lemma~\ref{ginversion}, the functions $a(\tau)$, $b(\tau)$,
$c(\tau)$, and $d(\tau)$, defining, via Equations~\eqref{eq3.2}, the solution of the
corresponding system of isomonodromy deformations~\eqref{newlax8}, have the
following asymptotic representations: for $k \! = \! \pm 1$,
{\fontsize{10pt}{11pt}\selectfont
\begin{align}
\sqrt{\smash[b]{-a(\tau)b(\tau)}} \underset{\tau \to +\infty}{=}& \, \frac{
(\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{2} \! \left(1 \! + \! \sum_{m=
0}^{\infty} \frac{\mathfrak{u}_{m}(k)}{(\tau^{1/3})^{m+2}} \right) \! - \!
\frac{\mathrm{i} (\varepsilon b)^{1/2} \mathrm{e}^{\mathrm{i} \pi k/4}(\mathscr{P}_{a})^{k}
(s_{0}^{0} \! - \! \mathrm{i} \mathrm{e}^{-\pi a})}{\sqrt{\pi} \, 2^{3/2}3^{1/4} \tau^{1/3}}
\nonumber \\
\times& \, \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)} \! \left(1 \! + \!
\mathcal{O}(\tau^{-1/3}) \right), \label{isomk1} \\
a(\tau)d(\tau) \underset{\tau \to +\infty}{=}& \, -\frac{\mathrm{i} (\varepsilon b)}{4}
\! - \! \frac{\mathrm{i} (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{4}(a \! - \! \mathrm{i}/3)
\tau^{-2/3} \! + \! \frac{\mathrm{i} (\varepsilon b)}{8} \left(\mathfrak{r}_{1}(k)
\! - \! 2 \mathfrak{u}_{1}(k) \right) \! \tau^{-1} \nonumber \\
+& \, (\tau^{-1/3})^{4} \sum_{m=0}^{\infty} \left(\frac{\mathrm{i} (\varepsilon b)}{8}
\left(\mathfrak{r}_{m+2}(k) \! - \! 2 \mathfrak{u}_{m+2}(k) \right) \! - \!
\frac{\mathrm{i} (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{4}(a \! - \! \mathrm{i}/2)
\mathfrak{u}_{m}(k) \right. \nonumber \\
+&\left. \, \frac{\mathrm{i} (\varepsilon b)}{8} \sum_{p=0}^{m} \mathfrak{u}_{p}(k)
\mathfrak{r}_{m-p}(k) \right) \! (\tau^{-1/3})^{m} \! - \! \frac{k(\varepsilon
b)^{5/6}3^{1/4} \mathrm{e}^{\mathrm{i} \pi k/4}(\mathscr{P}_{a})^{k}(s_{0}^{0} \! - \! \mathrm{i}
\mathrm{e}^{-\pi a})}{4 \sqrt{2 \pi} \, \mathrm{e}^{\mathrm{i} \pi k/3} \tau^{1/3}} \nonumber \\
\times& \, \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)} \! \left(1 \! + \!
\mathcal{O}(\tau^{-1/3}) \right), \label{isomk2} \\
b(\tau)c(\tau) \underset{\tau \to +\infty}{=}& \, -\frac{\mathrm{i} (\varepsilon b)}{4}
\! - \! \frac{\mathrm{i} (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{4}(a \! + \! \mathrm{i}/3)
\tau^{-2/3} \! - \! \frac{\mathrm{i} (\varepsilon b)}{8} \left(\mathfrak{r}_{1}(k)
\! - \! 2 \mathfrak{u}_{1}(k) \right) \! \tau^{-1} \nonumber \\
+& \, (\tau^{-1/3})^{4} \sum_{m=0}^{\infty} \left(-\frac{\mathrm{i} (\varepsilon b)}{8}
\left(\mathfrak{r}_{m+2}(k) \! - \! 2 \mathfrak{u}_{m+2}(k) \right) \! - \!
\frac{\mathrm{i} (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{4}(a \! + \! \mathrm{i}/2)
\mathfrak{u}_{m}(k) \right. \nonumber \\
-&\left. \, \frac{\mathrm{i} (\varepsilon b)}{8} \sum_{p=0}^{m} \mathfrak{u}_{p}(k)
\mathfrak{r}_{m-p}(k) \right) \! (\tau^{-1/3})^{m} \! + \! \frac{k(\varepsilon
b)^{5/6}3^{1/4} \mathrm{e}^{\mathrm{i} \pi k/4}(\mathscr{P}_{a})^{k}(s_{0}^{0} \! - \! \mathrm{i}
\mathrm{e}^{-\pi a})}{4 \sqrt{2 \pi} \, \mathrm{e}^{\mathrm{i} \pi k/3} \tau^{1/3}} \nonumber \\
\times& \, \mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)} \! \left(1 \! + \!
\mathcal{O}(\tau^{-1/3}) \right), \label{isomk3} \\
-c(\tau)d(\tau) \underset{\tau \to +\infty}{=}& \, \frac{(\varepsilon b)^{2/3}
\mathrm{e}^{\mathrm{i} \pi k/3}}{4} \! - \! \frac{a(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{3}
\tau^{-2/3} \! - \! \frac{(\varepsilon b)^{2/3} \mathrm{e}^{\mathrm{i} \pi k/3}}{2}
\mathfrak{u}_{1}(k) \tau^{-1} \nonumber \\
-& \, \left(\frac{1}{6}(a^{2} \! + \! 1/6) \! + \! \dfrac{(\varepsilon b)^{2/3}
\mathrm{e}^{\mathrm{i} \pi k/3}}{2} \mathfrak{u}_{2}(k) \right) \! (\tau^{-1/3})^{4} \! + \!
(\tau^{-1/3})^{4} \sum_{m=1}^{\infty} \left(-\frac{(\varepsilon b)^{2/3}
\mathrm{e}^{\mathrm{i} \pi k/3}}{2} \right. \nonumber \\
\times&\left. \, \mathfrak{u}_{m+2}(k) \! + \! \frac{\mathrm{i} (\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} 2 \pi k/3}}{8} \mathfrak{r}_{m}(k) \! - \! \dfrac{(\varepsilon b)^{1/3}
\mathrm{e}^{\mathrm{i} 2 \pi k/3}}{2}(a \! - \! \mathrm{i}/2) \mathfrak{w}_{m}(k) \! - \!
\frac{(\varepsilon b)^{2/3} \mathrm{e}^{\mathrm{i} \pi k/3}}{2} \right. \nonumber \\
\times&\left. \, \sum_{p=0}^{m} \left(\left(\mathfrak{u}_{p}(k) \! + \!
\frac{1}{2} \mathfrak{r}_{p}(k) \right) \! \mathfrak{w}_{m-p}(k) \! + \!
\frac{1}{8} \mathfrak{r}_{p}(k) \mathfrak{r}_{m-p}(k) \right) \right) \!
(\tau^{-1/3})^{m} \nonumber \\
-& \, \dfrac{\mathrm{i} (\varepsilon b)^{1/2} \mathrm{e}^{\mathrm{i} \pi k/4}(\mathscr{P}_{a})^{k}
(s_{0}^{0} \! - \! \mathrm{i} \mathrm{e}^{-\pi a})}{\sqrt{\pi} \, 2^{3/2}3^{1/4} \tau^{1/3}}
\mathrm{e}^{-\mathrm{i} k \vartheta (\tau)} \mathrm{e}^{-\beta (\tau)} \! \left(1 \! + \! \mathcal{O}
(\tau^{-1/3}) \right), \label{isomk4}
\end{align}}
where the expansion coefficients $\mathfrak{u}_{m}(k)$ (resp., $\mathfrak{r}_{m}(k))$,
$m \! \in \! \mathbb{Z}_{+}$, are given in Equations~\eqref{thmk2}--\eqref{thmk10}
(resp., \eqref{thmk15} and~\eqref{thmk16}$)$.
\end{bbbbb}
\emph{Proof}. If, for $k \! = \! \pm 1$, $g_{ij}$, $i,j \! \in \! \lbrace 1,2 \rbrace$, are
$\tau$ dependent, then, functions whose asymptotics (as $\tau \! \to \! +\infty$
with $\varepsilon b \! > \! 0)$ are given by Equations~\eqref{geek1}--\eqref{geek3}
satisfy the Conditions~\eqref{iden5}, \eqref{pc4}, \eqref{restr1}, \eqref{ellinfk2a},
and~\eqref{ellinfk2b}; therefore, one can use the justification scheme suggested in
\cite{a20} (see, also, \cite{a22}). {}From Equations~\eqref{iden4oldu}, \eqref{iden7},
\eqref{iden8}, and~\eqref{iden10}, respectively, one shows, via the
Definitions~\eqref{iden3} and~\eqref{iden4}, that, for $k \! = \! \pm 1$,\footnote{Recall
that (cf. Lemma~\ref{ginversion}) $v_{0}(\tau) \! := \! v_{0,k}(\tau)$ and $\tilde{r}_{0}
(\tau) \! := \! \tilde{r}_{0,k}(\tau)$, $k \! = \! \pm 1$.}
\begin{align}
\sqrt{\smash[b]{-a(\tau)b(\tau)}} =& \, \frac{(\varepsilon b)^{2/3}
\mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{2}(1 \! + \! \tau^{-1/3}v_{0,k}(\tau)), \label{isomk5} \\
a(\tau)d(\tau) =& \, \frac{\mathrm{i} (\varepsilon b)}{8}(1 \! + \! \tau^{-1/3}v_{0,k}
(\tau))(-2 \! + \! \tau^{-1/3} \tilde{r}_{0,k}(\tau)) \nonumber \\
-& \, \frac{\mathrm{i} (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{4}(a \! - \! \mathrm{i}/2)
(1 \! + \! \tau^{-1/3}v_{0,k}(\tau)) \tau^{-2/3}, \label{isomk6} \\
b(\tau)c(\tau) =& \, -\frac{\mathrm{i} (\varepsilon b)}{2} \! - \! \frac{\mathrm{i}
(\varepsilon b)}{8}(1 \! + \! \tau^{-1/3}v_{0,k}(\tau))(-2 \! + \!
\tau^{-1/3} \tilde{r}_{0,k}(\tau)) \nonumber \\
-& \, \frac{\mathrm{i} (\varepsilon b)^{2/3} \mathrm{e}^{-\mathrm{i} 2 \pi k/3}}{4}(a \! + \! \mathrm{i}/2)
(1 \! + \! \tau^{-1/3}v_{0,k}(\tau)) \tau^{-2/3}, \label{isomk7} \\
-c(\tau)d(\tau) =& \, -\frac{(\varepsilon b)^{2/3} \mathrm{e}^{\mathrm{i} \pi k/3}}{4} \!
\left(\frac{-2 \! + \! \tau^{-1/3} \tilde{r}_{0,k}(\tau)}{1 \! + \! \tau^{-1/3}v_{0,k}
(\tau)} \right) \! - \! \frac{(\varepsilon b)^{2/3} \mathrm{e}^{\mathrm{i} \pi k/3}}{16}
(-2 \! + \! \tau^{-1/3} \tilde{r}_{0,k}(\tau))^{2} \nonumber \\
-& \, \frac{1}{4}(a \! - \! \mathrm{i}/2)(a \! + \! \mathrm{i}/2) \tau^{-4/3} \! + \!
\frac{(\varepsilon b)^{1/3} \mathrm{e}^{\mathrm{i} 2 \pi k/3}}{2} \! \left(\mathrm{i} (-2 \! + \!
\tau^{-1/3} \tilde{r}_{0,k}(\tau))/4 \right. \nonumber \\
-&\left. \, \frac{(a \! - \! \mathrm{i}/2)}{1 \! + \! \tau^{-1/3}v_{0,k}(\tau)} \right)
\! \tau^{-2/3}. \label{isomk8}
\end{align}
Via the Asymptotics~\eqref{geek1} and~\eqref{geek2}, and
Equations~\eqref{isomk5}--\eqref{isomk8}, one arrives at the asymptotics
(as $\tau \! \to \! +\infty$ with $\varepsilon b \! > \! 0)$ for the functions
$\sqrt{\smash[b]{-a(\tau)b(\tau)}}$, $a(\tau)d(\tau)$, $b(\tau)c(\tau)$, and
$-c(\tau)d(\tau)$ stated in Equations~\eqref{isomk1}--\eqref{isomk4},
respectively. \hfill $\qed$
\begin{eeeee} \label{intofmot}
It is important to note that Asymptotics~\eqref{isomk1}--\eqref{isomk4}
are consistent with Equation~\eqref{iden6}$;$ moreover, via the
Definitions~\eqref{newlax2}, Equations~\eqref{eq3.2}, and the
Asymptotics~\eqref{geek3} and~\eqref{isomk1}--\eqref{isomk4}, one arrives
at the asymptotics (as $\tau \! \to \! +\infty$ with $\varepsilon b \! > \! 0)$
for the solution of the---original---system of isomonodromy
deformations~\eqref{eq1.4}. \hfill $\blacksquare$
\end{eeeee}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,723 |
# As We Know
## Poems
### John Ashbery
###
# Contents
Publisher's Note
Litany
Sleeping in the Corners of Our Lives
Silhouette
Many Wagons Ago
As We Know
Figures in a Landscape
Statuary
Otherwise
Five Pedantic Pieces
Flowering Death
Haunted Landscape
My Erotic Double
I Might Have Seen It
The Hills and Shadows of a New Adventure
Knocking Around
Not Only / But Also
Train Rising Out of the Sea
Late Echo
And I'd Love You To Be in It
Tapestry
The Preludes
A Box and Its Contents
The Cathedral Is
I Had Thought Things Were Going Along Well
Out Over the Bay the Rattle of Firecrackers
We Were on the Terrace Drinking Gin and Tonics
Fallen Tree
The Picnic Grounds
A Sparkler
The Wine
A Love Poem
There's No Difference
Distant Relatives
Histoire Universelle
Hittite Lullaby
In a Boat
Variations on an Original Theme
Homesickness
This Configuration
Metamorphosis
Their Day
A Tone Poem
The Other Cindy
No, But I Seen One You Know You Don't Own
The Shower
Landscapeople
The Sun
The Plural of "Jack-in-the-Box"
About the Author
# Publisher's Note
Long before they were ever written down, poems were organized in lines. Since the invention of the printing press, readers have become increasingly conscious of looking at poems, rather than hearing them, but the function of the poetic line remains primarily sonic. Whether a poem is written in meter or in free verse, the lines introduce some kind of pattern into the ongoing syntax of the poem's sentences; the lines make us experience those sentences differently. Reading a prose poem, we feel the strategic absence of line.
But precisely because we've become so used to looking at poems, the function of line can be hard to describe. As James Longenbach writes in _The Art of the Poetic Line_ , "Line has no identity except in relation to other elements in the poem, especially the syntax of the poem's sentences. It is not an abstract concept, and its qualities cannot be described generally or schematically. It cannot be associated reliably with the way we speak or breathe. Nor can its function be understood merely from its visual appearance on the page." Printed books altered our relationship to poetry by allowing us to see the lines more readily. What new challenges do electronic reading devices pose?
In a printed book, the width of the page and the size of the type are fixed. Usually, because the page is wide enough and the type small enough, a line of poetry fits comfortably on the page: What you see is what you're supposed to hear as a unit of sound. Sometimes, however, a long line may exceed the width of the page; the line continues, indented just below the beginning of the line. Readers of printed books have become accustomed to this convention, even if it may on some occasions seem ambiguous—particularly when some of the lines of a poem are already indented from the left-hand margin of the page.
But unlike a printed book, which is stable, an ebook is a shape-shifter. Electronic type may be reflowed across a galaxy of applications and interfaces, across a variety of screens, from phone to tablet to computer. And because the reader of an ebook is empowered to change the size of the type, a poem's original lineation may seem to be altered in many different ways. As the size of the type increases, the likelihood of any given line running over increases.
Our typesetting standard for poetry is designed to register that when a line of poetry exceeds the width of the screen, the resulting run-over line should be indented, as it might be in a printed book. Take a look at John Ashbery's "Disclaimer" as it appears in two different type sizes.
Each of these versions of the poem has the same number of lines: the number that Ashbery intended. But if you look at the second, third, and fifth lines of the second stanza in the right-hand version of "Disclaimer," you'll see the automatic indent; in the fifth line, for instance, the word _ahead_ drops down and is indented. The automatic indent not only makes poems easier to read electronically; it also helps to retain the rhythmic shape of the line—the unit of sound—as the poet intended it. And to preserve the integrity of the line, words are never broken or hyphenated when the line must run over. Reading "Disclaimer" on the screen, you can be sure that the phrase "you pause before the little bridge, sigh, and turn ahead" is a complete line, while the phrase "you pause before the little bridge, sigh, and turn" is not.
Open Road has adopted an electronic typesetting standard for poetry that ensures the clearest possible marking of both line breaks and stanza breaks, while at the same time handling the built-in function for resizing and reflowing text that all ereading devices possess. The first step is the appropriate semantic markup of the text, in which the formal elements distinguishing a poem, including lines, stanzas, and degrees of indentation, are tagged. Next, a style sheet that reads these tags must be designed, so that the formal elements of the poems are always displayed consistently. For instance, the style sheet reads the tags marking lines that the author himself has indented; should that indented line exceed the character capacity of a screen, the run-over part of the line will be indented further, and all such runovers will look the same. This combination of appropriate coding choices and style sheets makes it easy to display poems with complex indentations, no matter if the lines are metered or free, end-stopped or enjambed.
Ultimately, there may be no way to account for every single variation in the way in which the lines of a poem are disposed visually on an electronic reading device, just as rare variations may challenge the conventions of the printed page, but with rigorous quality assessment and scrupulous proofreading, nearly every poem can be set electronically in accordance with its author's intention. And in some regards, electronic typesetting increases our capacity to transcribe a poem accurately: In a printed book, there may be no way to distinguish a stanza break from a page break, but with an ereader, one has only to resize the text in question to discover if a break at the bottom of a page is intentional or accidental.
Our goal in bringing out poetry in fully reflowable digital editions is to honor the sanctity of line and stanza as meticulously as possible—to allow readers to feel assured that the way the lines appear on the screen is an accurate embodiment of the way the author wants the lines to sound. Ever since poems began to be written down, the manner in which they ought to be written down has seemed equivocal; ambiguities have always resulted. By taking advantage of the technologies available in our time, our goal is to deliver the most satisfying reading experience possible.
# I
# LITANY
_Author's Note: "Litany" consists of two independent monologues meant to be experienced simultaneously. In traditional print format, the two monologues are presented side by side on facing pages, allowing the reader to experience their simultaneity, but this arrangement is not possible with the current generation of ebook devices. To download a PDF of "Litany" as it was originally meant to be laid out on the page, please visitwww.openroadmedia.com/john-ashbery/litany. To listen to a 1980 recording of John Ashbery and Ann Lauterbach reading the poem's two monologues simultaneously, visit the PennSound website at writing.upenn.edu/pennsound/x/Ashbery.php_.
## I
For someone like me
The simple things
Like having toast or
Going to church are
Kept in one place.
Like having wine and cheese.
The parents of the town
Pissing elegantly escape knowledge
Once and for all. The
Snapdragons consumed in a wind
Of fire and rage far over
The streets as they end.
The casual purring of a donkey
Rouses me from my accounts:
What given, what gifts. The air
Stands straight up like a tail.
He spat on the flowers.
Also for someone
Like me the time flows round again
With things I did in it.
I wish to keep my differences
And to retain my kinship
To the rest. That is why
I raise these flowers all around.
They do not stand for flowers or
Anything pretty they are
Code names for the silence.
And just as it
Always keeps getting sorted out
And there is still the same amount to do
I wish to remain happily among these islands
Of rabbit-eared leaved plants
And sand and lava rock
That is so little tedious.
My way shall run from there
And not mind the pain
Of getting there. This is an outburst.
The last rains fed
Into the newly opened canal.
The dust blows in.
The disturbance is
Nonverbal communication:
Meaningless syllables that
Have a music of their own,
The music of sex, or any
Nameless event, something
That can only be taken as
Itself. This rules ideas
Of what else may be there,
Which regroup farther on,
Standing around looking at
The hole left by the great implosion.
It is they who carry news of it
To other places. Therefore
Are they not the event itself?
Especially since it persists
In dumbness which isn't even
A negative articulation—persists
And collapses into itself.
I had greatly admired
The shirt.
He looks fairly familiar.
The pancake
Is around in idea.
Today the wisteria is in league
With the Spanish minstrels.
Who come to your house
To serenade it
All or in part.
The windows are open again
The dust blows through
A diagram of a room.
This is where it all
Had to take place,
Around a drum of living,
The motion by which a life
May be known and recognized,
A shipwreck seen from the shore,
A puzzling column of figures.
The dark shirt dragged frequently
Through the bayou.
Your luggage
Is found
Upon the plane.
If I could plan how
To remember what had indeed once
Been there
Without reference to professions,
Medical school,
Etc.,
Being there indeed once
(Everyday occurrence),
We stopped at the Pacific Airport
To hear the rush of disguises
For the elegant truth, notwithstanding
Some in underwear stood around
Puddles in the darkened
Cement and sodium lights
Beyond the earthworks
Beyond the chain-link fence
Until dawn touched with her cool
Stab of grace nobody deserved (but
It's always that way isn't it)
_Le charme du matin_
You and Sven-Bertil must
At some point have overridden
The barriers real or fancied
Blowing like bedcurtains later
In the oyster light—
Something I saw once
Reminded me of it:
That old, evil, not-so-secret
Formula
Now laundered, made to look
Transparent. Surely
There is a shoulder there,
Some high haunch half-sketched, a tremor
And intent to the folds that shower from the sky.
And must
At some earlier time
Seem the garter
The cow in the trees.
What was green before
Is homeless.
The mica on the front
Of the prefecture spells out
"Coastline"—a speedboat
Would alter even at a distance
But they shift anyway
Come round
To my idea
My hat
As it would be
If I were you
In dreams and in business
Only, in supper meetings
On the general line of progress
If I had a talking picture of you.
You are
So perversely evasive:
The ticking of a clock in the
Background could be
Only the plait.
We must learn to read
In the dark, to enjoy the long hills
Of studious celebrity.
The long Chinese shadow that
Hooks over a little
At the top
The stone that sinks
To the bottom of the aquarium:
All this mummified writing
As the dusting of new light
In the hollow collar of a hill
That never completes its curve
Or the thought of what
It was going to say: our going in.
The hedges are nice and it's too bad
That one bad axe stroke could fell
Whatever needed to advertise its
Very existence.
And then cars strut forth on the highway
Singly and in groups
Of three and four: orange,
Flamingo, blue-pencil blue,
The gray of satisfaction, the red
Of discussion, and now, moved, the sky
Calls itself up.
As leaves are seen in mirrors
In libraries
Half-noticed, the sound
Half-remembered and the
Continuing chapter half-sketched—
O were we wrong to notice
To remember so much
When so little else has survived?
All were moments big with particulars
An elaborate pastry concocted in the wings
In darkness, and each
Has vanished on the carrousel
Of rage, along the coast
Like a chameleon's hide.
The suffering, the pleasure that broke
Over it like a wave,
Are these fixed limits, off-limits
To the game as darkness confounds
The two teams, makes it one with chance?
Still, somewhere wings are
Being slowly lifted,
Over and over again.
The point must have been made.
But out of so much color
It still does come again
The colors of tiger lilies and around
And down, remembered
Now as dirty colors, the color
Of forgetting-grass, of
Old rags or sleep, buoyed
On the small zephyrs
That keep the hour and remind each boy
To turn home from school past the sheep
In the paper meadow and to wind the clock.
An old round is being passed out,
The players take their places.
How nice that in the stalls
Is still room for certain boys to stand,
The main song is successfully
Programmed and the others too in part:
Enough gets through to make the occasion
A glottal one full of success
And coated with the film of success
In which are reflected
Many a bright occasion
Lads who go out with girls
In the numb prime of springtime
For instance.
Except for that, the camera sighs,
Is no hollow behind the black backing.
That was short-lived.
A sheaf of selected odes
Bundled on the waters.
A superior time
Of blueberries and passion flowers,
Of a four-poster.
The thirties light
Has infested the blond
Hairdo from the grooves up
But we must not treasure
It less in the magnesium
Flare that is manna to all things
In the here and now. You were saying
How she is coming along, praying
For it to be better
Day by day.
And some of these days the waning
Silver lashes out
Like a trussed alligator:
Mother and the kids standing around
The bowl that is portal,
Hitching post, tufted
Mattress and field of wild
Scruffy flowers are removed
One by one as a demonstration.
See, there is only light.
Nothing to live at,
To worry.
It is the old sewer of our resources
Disguised again as a corridor.
There is some anthropology here
It seems, and then
The dust on the jamb is warning
And intrigue enough. The summer day is put by.
The bells in the shower
Are outnumbered by plain queries
Whose answer is their falling echo.
Birds in modish, corporeal
Gear take off at the
Scallops of the umbrella.
This past is sampled and is again
The right one, and in testing
For the zillionth time we are
As built into the fixed wall of water
That indicates where the present leaves off
And the past begins, whose transparencies
Admit impressions of traceries of leaves
And shallow birds among memories.
The climate seceded then,
The glad speculation about what clothes
They wore stacked like leaves,
Speckled behind the eye of what
Consumer, what listener?
And the praise is lascivious
To the onyx ear at evening
But not forwarded
Into the ring with the other shouting,
The desperate competitions willed
Until darkness, dripping toward death
By late morning.
She circles plainly away
From it in wider and wider loops,
And what have you to say? What account
To give? Of the season's vast
Storehouse of agendas, bales
Of items for discussion dwindling
Down to a last seed on the stone doorstep?
If this was the season only of death
That licorice blast would not keep only
In its retelling the unfurled
Question-mark of the shaved future but redound
To us waiting here against the spike fence
In pleasant attitudes from which the waiting
Is forgotten like thorns in the memory
Of laced paths merging on
Extinct, ultimate slopes,
But trap us in the game of two flavors
(A rising shout some distance away,
The tabac alike in resisting
_Terribilità_
Yet basing it on us, all the same
A knowledge of its measure, its
Proportion, until the end is sought
Dryly, among stringent grasses).
To have sought it any more, mining
Its anfractuosities, is to bear witness,
The living getting trampled
Underfoot always the same way
And as surely one desiccated spike of
Sea-oats rises quizzically after the
Hordes have passed over, the film
Slips over the cogs
That brought us to this unearthly spot.
So death is really an appetite for time
That can see through the haze of blue
Smoke-rings to the turquoise ceiling.
She said this once and turned away
Knowing we wanted to hear it twice,
But knowing also as we knew that speculation
Raves and raves as on a mirror
To the outlandish accompaniment of its own death
That reads as life to the toilers
And potboys who make up these blond
Coils of citizenry which are life in the abstract.
What it was like to be mouthing those
Solemn abstractions that were crimson
And solid as beefsteak. One
Shouldn't be surprised by
The smell of mignonette and the loss
As each stands still, and the softness
Of the land behind each one,
Where each one comes from.
Because it is the way of the personality of each
To blush and act confused, groping
For the wrong words so that the
_Coup de théâtre_
Will unfold all at once like shaken-out
Lightning and no one
Will have heard anything. The gray,
Fake Palladian club buildings will
Still stand the next moment, at their grim
Business: empty entablatures, _oeils-de-boeuf_ ,
Gun-metal laurels, the eye
Revolving slowly in the empty socket
That the bronze visor shades: there was
Never anything but this,
No footfalls on the mat-polished marble floor,
No bird-dropping, no fates, no sanctuary.
The sheet slowly rises to greet you.
The asters are reflected
Simultaneously in ruby drops of the wine
The morning after the great storm
That swept our sky away, leaving
A new muscle in its place: a relaxed, far-away
Tissue of scandal and dreams like noon smoke
Lingering above horizon roofs.
But what difference did any of it make
Woven on death's loom as indeed
All of it was though divided into
Chapters each with its ornamental
Capital at the beginning, and its polished
Sequel? You knew
You were coming to the end by the way the other
Would be beginning again, so that nobody
Was ever lonesome, and the story never
Came to its dramatic conclusion, but
Merely leveled out like linen close up
In the mirror. So that the roundness
Was all around to be appreciated, yet somehow flat
As well, and could never be trusted
Even though the rushes slanted all one way
In the autumn wind, and the leaves
And branches tried to slant with them
In a poem of harmonious dejection, but it was
Only picture-making. Under
The intimate light of the lantern
One really felt rather than saw
The thin, terrifying edges between things
And their terrible cold breath.
And no one longed for the great generalities
These seemed to preclude. Each thought only
Of his private silence, and hungered
For the promised moment of rest.
## II
I photographed all things,
All things as happening
As prelude, as prelude to the impatience
Of enormous summer nights opening
Out farther and farther, like the billowing
Of a parachute, with only that slit
Of starlight. The old, old
Wonderful story, and it's all right
As far as it goes, but impatience
Is the true ether that surrounds us.
Without it everything would be asphalt.
Now that the things of autumn
Have been sequestered too in their chain
The other part of the year become
Visible
And the summer night is like a goldfish bowl
With everything in full view, yet only parts
Are what is actually seen, and these supply
The rest. It's not like cheating
Since it _is_ all there, but more like
Helping the truth along a little:
The artifice lets it become itself,
Nestling in truth. These are long days
And we need all the help we can get.
We are to become ashamed only much later,
Much later on, under the long bench.
And it is not like the old days
When we used to sing off-key
For hours in the rain-drenched schoolroom
On purpose. Here, whatever is forgotten
Or stored away is imbued with vitality.
Whatever is to come is too.
How can I explain?
No matter how raffish
The new clients moving slowly along,
Taking in the sights, placing bets,
There comes a time when the moment
Is full of, knows only itself.
Like a moment when a tree
Is seen to tower above everything else,
To know itself, and to know everything else
As well, but only in terms of itself
Without knowing or having a clear concept
Of itself. This is a moment
Of fast growing, of compounding myths
As fast as they can be thrown off,
Trampled under, forgotten. The moment
Not made of itself or any other
Substance we know of, reflecting
Only itself. Then there are two moments,
How can I explain?
It was as though this thing—
More creature than person—
Lumbered at me out of the storm,
Brandishing a half-demolished beach umbrella,
So that there might be merely this thing
And me to tell about it.
It was awful. And I too have no rest
From the storm that is always something
To worry about. Really. My unworthiness
Like a loose garment or cape of some sort
Constantly sliding off the shoulders,
Around the elbows ... I cannot keep it on,
Even as I am invisible in the eye
Of the storm, we two are blind,
And blind to the inaudible repercussions,
The strange woody aftertaste.
After that the wave came
And left no mark on the shore.
The waves advanced as the tide withdrew.
There was nothing for it but to
Retreat from the edge of the earth,
In that time, that climate expecting rain,
Behind some brackish business
On the margin intuiting cataclysms of light.
All that fall I wanted to be with you,
Tried to catch up to you in the streets
Of that time. Needless to say,
Although we were together a good part of the time
I never quite made it to the thunder.
The boy who cried "wolf" used to live there.
This place of islands and slow reefs,
Like petals of mercury, that fold up
Whenever that allusion is made.
It falls off the others like
Water off piled-up stones at the base
Of a waterfall, and the petals
Curl up, injured, into themselves.
Only the frozen emphasis
On a single thing that was out of sight
When the allusion was made, remains.
We all bought tickets to the allusion
And are disappointed, of course.
But what can you do? Events have
A way of snapping off like that, like
The glassblower's striped candy canes
Of glass at a moment he knows is coming,
Is there, even. _The old_ ,
_Wonderful story_. Not yet ended.
You who approach me,
All grace and linearity,
With my new crayons I think I'll
Do a series of box-sprays—stippled
Cobalt on the gold
Of a sun-pure afternoon
In October when things change over.
There is no longer time for a line
Or rather there are no lines in the time
Of ripeness that is past,
Yet still pausing on the ridge
Stealing into permanence.
It was all French horns
And oboes and purple vetch:
That was what it was all about, but
What it came to be came later
And other—a scene, a
Simple situation, something as
Basic as two people sitting in the sun
With no thought of the morrow, or of today,
As the whispers mingled in a choir outlining them
And we took a lesson away from this,
A lesson like a piece of cloth.
It's going to be different in the future
But now the now is what matters,
Knowing itself old, and open to vengeance,
And, in short, up to nobody's expectations
For it, as dank and empty
As an old Chevy parked under the trees
Amid dead leaves and dogshit, everybody's
Idea of what was coming true for them
Which is now burning in lava-like letters
In the sky, a piece of good news
If you agree that good news is what
Is happening at this very instant.
The California sun turned its back on us
So we chose New England and the more vibrant
Violet light of tame tempests,
Dreams of sleeping watchdogs,
And the whole house was full of people
Having a good time, and though
No one offered you a drink and there were no
Clean glasses and the supper
Never appeared on the table, it was
Strangely rewarding anyway.
It gave one an idea of what they thought of one:
Even the ocean that came crashing almost
Into the back yard did not seem ill-disposed
And that was something. Presently
Out of this near-chaos an unearthly
Radiance stood like a person in the room,
The memory of the host, perhaps. And all
Fell silent, or stayed at their musings, silent
As before, and no one any longer
Offered words of advice or misgiving, but drank
The silence that had been silence before,
On this scant strip of slag,
Basking in the same light as before,
Inhabiting the same thought:
A shelf of breasts and underwear packaging
Rumored in the dark ages.
These people, you see,
Had to come to appear to thrive
And somewhat later sidestep the destiny
That pretended not to see them.
It was all necessary so that some source,
An origin of the present, might
In the scent of verbena and dreams of
Combat locked in the sky over the mid-ocean
Gradually give less and less of itself
And in so dying bequeath the manner
Of its being to the sidewalk shrubbery
And so enable it to become itself
Even though that self is only the sometimes-noticed
Backdrop for ourselves and all
We wondered whether we would become,
Pockmarked flecks of polluted matter
Infrequently visible in the hail of ventilated indifference
Or seconds of radiation, our own very special
Thing we had been trying to get our hands
On for so many years.
Honey, it's all Greek to me, I—
(And just to make sure you get
It: _the thought crossed my mind_
_That I would do well to take up my studies again_ ,
_I seemed to have become less averse to laughter_
_And less disinclined for certain small pleasures_ ,
_And I began quietly to reason with myself_
_About this matter, as I usually do about others_ ,
_So that I regretfully concluded_
_That I would soon again be the same man as before_ —)
Meaning: _the same nausea when I heard cheerful talk_ ,
_The same grief, the same deep and prolonged meditation_ ,
_And almost the same frenzy and oppression_.
Supposing that you are a wall
And can never contribute to nature anything
But the feeling of being alongside it,
A certain luxury, and now,
They come to you with the old matter
Of your solidity, that firmness,
That way you have of squaring off
The maps of distant hills, so that nature
Seems farther apart from itself because of you.
Is it this you have done?
And a certain grassy look, the color
Of old semiprecious stones, has to be
What's coming out of you, for the two of you.
And the mechanical reverie is cut up by fits
Of blaring trumpets and alarms, in the night.
Forward then into the yellow villages.
Despite the eerie setbacks
Of our subpolar ambience, _we are_
_Living, we are dwelling_ on a network
Of insane desires handled frugally.
Passport in hand, we arrive in the morning
At the station, the dumb train
Vaults you along into forests of
Broccoli, or tracts of leathery
Tundra, one eye on the digital watch.
The tonal purity grows, and dissipates,
But meanwhile the plateau remains staunch,
It's only the towers that dot it that tend
To look pierced by the sky
Or fade away absentmindedly, altogether.
The naked report arrived vividly
In the night.
_Groaning for the latter day_ brought us
To this place, a trough of silent chatter
Between two notable waves. And we must arrange
These filaments of silence as an elephant trap
Over the grid of city conversations and background doings.
The quietude
Of the future to be built, beside which
Today's valors and sighs must appear
As vanished suburbs beside some eighteenth-century
Metropolis, or stairs rolling down to a sea
Of urgent scrolls and torsades:
A Baltic commonplace riven by tremendous
Hairline fissures as deep as the heavens.
In other words, leave it alone.
That's interesting. In my diary
I have noted down all kinds of exceptional
Things to go with the rest
As one who naps beside a chasm
Swollen with the hellish sound of wind
And torrents, and never chooses
To play back the tape. Waking
Refreshed if not alert, he steps forth
Into the centuries that grew like shadows
Under tall trees while he slept;
The days rub off like scales, the years
Like burrs or briars plucked
Patiently from the sleeve, and never sees
Or hears the havoc wrought by his passing,
Abysses that open up behind
His perilous, beribboned journey, the jalopy
Disappearing deep into vales
To re-emerge suddenly on heights, through
The tunnel of a giant sequoia. And always
An old-time mannerliness and courtesy informs
The itinerary, leaving us
Without much to go on.
Once it becomes fatality,
Of course,
The journey is at an end, and it is just beginning—
Innate—
A moody performance.
The critics hated it.
Now one borrows money from his friends,
In double time, the consequences
Blur the motives. The contours of the figures
Are curved and fat. He goes out among the trees,
Sees the lights in the valley far below.
Up here the air is black, ice-cold, of a
Terrifying purity, doubled over somehow.
But your story isn't getting boring,
On the contrary, the slowing-down speeds up the
Afterthought. We are perverse spelling and punctuation.
It could not be confirmed
That the recent violent storms were a part of the pattern
Of civil calamity that had overtaken the outpost.
Perhaps they were fatal but parallel,
Wounds inflicted on a corpse, footnotes
To the desert, the explosion
That a quiet, mediocre career is. We read
Through some Haydn quartet movements last night
But this morning my hand and heart are heavy, heavy alack.
The day before yesterday it seemed to me
That my cherished sorrow was about to depart,
And yesterday morning too. And now, fatality
Has overtaken it. The end
Has been quiet, and no one has told the rabbits
And dying bees. Finally some warmth
From the death floated downstream to us,
Saving a few moments of mildness
Among the by-now unmanageably thick grease-crayon
Outline that coagulates like a ball of soot in the air
Watched by hemophiliac princes, like an orange.
And as mushrooms spring up
After great rains have purged the heavens
Of their terrible delight, so the weight of event
And counterevent conspired to shift the focus
Of the scenery away from the action:
It was always wartime Britain, or some other place
Dictated by the circumstances, never
The road leading over the hill
To yet another home. Rudeness, shabbiness—
We could have put up with more than a little
Of these in the hope of getting some bed-rest,
But a measured calm, maddening in
Its insularity, always prevailed at the window,
Priming the hour with anguish, and yet
It was never any later, there was never anything
More to do, everybody kept telling you
To relax until you were ready to scream,
And now this patient night has infused,
In whose folds only one soul is awake, in the whole wide world.
Feeling no need to look at the world through rose-colored glasses,
To get by on "cuteness,"
To create large new forms and people them with space,
You thwart any directions, right or wrong.
The _séduction de l'âme_ will not take place.
The long rains in November, November
Of long rains, silent woods,
Open like a compass to receive the anomaly,
Press it back into the damp earth,
The shadow of a whisper on someone's lips.
You can neither define
Nor erase it, and, seen by torchlight,
Being cloaked with the shrill
Savage drapery of non-being, it
Stands out in the firelight.
It is more than anything was meant to be.
Yet somehow mournful, as though
The three-dimensional effect had been achieved
At the cost of a crisp vagueness
That raised one twig slightly higher than the
Morass of leafless branches that supported it,
And now, eager, fatigued, it had sunk back
Below the generally satisfying
Contours of the rest. It had eaten
The food you gave it, and kept to itself
Mainly, in a corner of the pen.
You never spoke to it except in the kindest
Tones, and it replied sadly,
If somewhat politely, and how much, now
You wish you had kept a record of those exchanges!
One thing is sure: nothing
Can replace it; as fatally
As it was given to you, so now
It has been removed from you, for your comfort,
And nothing stands in its place.
It is not a question of emptiness, only
Of a place the others never seem to venture,
A sunken Parnassus.
There is a slight change, a chance rather
Of its coming to life at the reunion,
Amid the automatic greetings, summonses
From a brazen tongue:
"And so you thought this
Was where he brought you, the
Updated silhouette, late sunlight
Developed on the tallest slope, to the assignation
Rumored so often, to a corral
Shaped like a snowflake, and love
Blurring each of the points. Yet you
Stand fast and cannot see
Where it is leading. And the seducer remains at home."
Yet whereto, with damaged wing
Assay th'empyrean? Scalloped horizon
Of Cloud-Cuckoo-Land? O land
Of recently boiling water, witches'
Misgivings, ships
Pulling away from piers,
Already slipping deep into the norm
Of blue worsted seas? Yet that is just what I did.
There are always those who think you ought to
Turn back from dull autumn sunsets like whey in the breeze that escorts
Us up inclined planes whose appearance, dull too
At first, is experienced
As if bathed in magic, when its density,
"A flash of lightning, seen in passing and very faintly,"
Stuns the apprehending faculties
With the perfection of its desire
Like the scream of the rising moon.
It is best to abide with minstrels, then,
To play at least one game
Seriously. The old-timers will
Let you take over the old lease.
One of them will be in you.
If there were concerts on the water there
We could turn back. Tar floated upriver
In the teeth of the gulls' outlandish manifestations;
The banks pocked with flowers whose names
I used to know,
Before poetic license took over and abolished everything.
People shade their eyes and wave
From the strand: to us or someone behind us?
Just as everything seemed about to go wrong
The music began; later on, the missing
Refreshments would be found and served,
The road turn caramel just as the first stars
Were putting in a timid appearance, like snowdrops.
And somehow you found the strength
To be carried irresistibly away from all this.
But in the scrapbooks and postcard albums
Of the land, you are remembered,
Although you do not figure there,
And because a train once passed near where
You spent a night, a tall, translucent
Monument like a spike has been erected to your memory,
Only do not go there. One can live
In the land like a spy without ever
Trespassing on the mortal, forgotten frontier.
In the psalms of the invisible chorus
There is a germ of you that lives like a coal
Amid the hostile indifference of the land
That merely forgets you. Your hand
Is at the heart of its weavings and nestlings.
You are its guarantee.
At that moment, fatality
Or some woman resembling her, angel,
Goddess, whatever: "the Beautiful Lady"
Arrives to announce the Brass Age—
"You are being asked to believe
No more in the subtle possibilities of silver,
Which, like the tintinnabulation of an ethereal
Silver chime, marking an unknown hour
From a remote, dismal room, no longer
Promises harvests, only the translucent melancholy
Of the skies which follow in their wake,
Pale, greenish blue, with magnificent
Clouds like overloaded schooners, that dip
To rise again, higher, and seem
Endlessly on the move, until they round—
What? Is there some cape, some destination,
Some port of debarkation in all this?
There is only the slow but febrile motion
Of sky and cloud, a toast, a promise,
A new diary, until one gets too close
And becomes oneself part of the meaningless
Rolling and lurching, so hard to read
Or hear, and never closer
To the end or to the beginning: the mimesis
Of death, without the finality—is
There anything in this for you?
Sad, browning flowers, tokens
Of the wind's remembering you, damp, rotting
Nostalgia under a head of twigs or at the end
Of some log spangled with brand-new, ice-green lichens,
Dead pine-needles, worthy
Objects of contemplation if you wish, but there is
Less comfort but more interest in the drab
Clear moment that enshrines us
Now, in this place. No one
Could mistake this for morning, or afternoon,
Or the specious perfection of twilight, yet
It is within us, and the substance
Of your latest interventions. Therefore, begone!"
The voice
Straddled the stone canyon like vapors.
In the distance one could see oneself, drawn
On the air like one of Millet's "Gleaners," extracting
This or that from the vulgar stubble, with the roistering
Of harvesters long extinct, dead for the ear, and in the middle
Distance, one's new approximation of oneself:
A seated figure, neither imperious nor querulous,
No longer invoking the riddle of the skies, of distance,
Nor yet content with the propinquity
Of strangers and admirers, all rapt,
In attitudes of fascination at your feet, waiting
For the story to begin.
All right. Let's see—How about "The outlook wasn't brilliant
For the Mudville nine that day"? No,
That kind of stuff is too old-hat. Today
More than ever readers are looking for
Something upbeat, to sweep them off their feet.
Something candid but also sophisticated
With an unusual slant. A class act
That doesn't _look_ like a class act
Is more like ...
It goes without saying
That I enjoy
You as you are,
The pleasant taste of you.
You are with me as the seasons
Circle with us around the sun
That dates back to the seventeenth century,
We circling with them,
United with ourselves and directly linked
To them, changing as they change,
Only their changes are always the same, and we,
We are always a little different with each change.
But in the end our changes make us into something,
Bend us into some shape maybe
No one we would recognize,
And it is ours, anyway, beyond understanding
Or even beyond our perception:
We may never perceive the thing we have become.
But that's all right—we have to be it
Even as we are ourselves. Anyway,
That's the way I like you and the way
Things are going to be increasingly,
With the seasons a mirror of our indeterminate
Activities, so that they do end
In burgeoning leaves and buds and then
In bare twigs against a Pater-painted
Sky of gray, expecting snow ...
How can we know ourselves through
These excrescences of time that take
Their cues elsewhere? Whom
Should I refer you to, if I am not
To be of you? But you
Will continue in your own way, will finish
Your novel, and have a life
Full of happy, active surprises, curious
Twists and developments of character:
A charm is fixed above you
And everything you do, but you
Must never make too much of it, nor
Take it for granted, either. Anyway, as
I said, I like you this way, understood
If under-appreciated, and finally
My features come to rest, locked
In the gold-filled chain of your expressions,
The one I was always setting out to be—
Remember? And now it is so.
Yet—whether it wasn't all just a little,
Well, silly, or whether on the other hand this
Wasn't a welcome sign of something
Human at last, like a bird
After you've been sailing on and on for days:
How could we tell
The serene and majestic side of nature
From the other one, the mocking and swearing
And smoke billowing out of the ground?
Because they are so closely and explicitly
Intertwined that good
Oftentimes seems merely the necessary
Attractive side of evil, which in turn
Can be viewed as the less appealing but more
Human side of good, something at least
Which can be appreciated?
But poetry is making things in the past;
The past tense transcends and excuses these
Grimy arguments which fog over as soon as
You begin to contemplate them. Poetry
Has already happened. And the agony
Of looking steadily at something isn't
Really there at all, it's something you
Once read about; its narrative thrust
Carries it far beyond what it thought it was
All het up about; its charm, no longer
A diversionary tactic, is something like
Grace, in the long run, which is what poetry is.
Musing on these things he turned off the
Great high street which is like a too-busy
Harbor full of boats knocking against each
Other, a blatantly cacophonous if stirring
Symphony, with all its most
Staggeringly beautiful aspects jammed against
The lowest motives and inspirations that ever
Infected the human spirit, into a
Small courtyard continued by an alley as
Though a sudden hush or drop in the temperature
Suddenly fell across him, like steep
Building-shadows, and he wondered
What it had all been leading up to. Up there
Wisps of smoke raced away from grimy
Chimney pots as though pursued by demons;
Down here all was yellowing silence and
Melancholy though not without a secret
Feeling of satisfaction at having escaped
The rat race, if only for a time, to plunge
Into profitless meditations, as threadbare
As the old mohair coat he had worn from
Earliest times, and which no one
Had ever seen him doff, no matter
What the prevailing meteorological conditions were.
These were now the fabric
Of his existence, and fabric was precisely
What he felt that existence to be: something old
And useful, useful and useless at the same time.
I was waiting for a taxi.
It seemed there were fewer
Of us now, and suddenly a
Whole lot fewer. I was afraid
I might be the only one.
Then I spotted a young man
With a guitar over his leg
And next to him, a young girl
Seated on the pavement, sitting
Merely. Not even
Lost in thought she seemed, but
Accepting the waiting for it
Or whatever else might be in the channel
Of time we were being ferried across.
Her face was totally devoid of expression
Yet wore a somehow kind look, so I was glad
Of it in the deepening fever of the day.
No sign
Did she make of interest to her companion
Who ever and anon did searchingly
Regard her face, as though to ascertain
That the signs he wished to read there
Were indeed not there, that there was nothing
In her aspect to cause him to change
And from time to time
Would stare at his guitar, as though
Rapt in concentration of what it would be like
To play something on it, yet
No stealthy movement of his hand
Was e'er discerned, no fandango or urgent
Serenade compelled his trusting back
To arch in expectation of an air
Which might have refreshed us all, given
The gloom of that moment, made us think
Of past scenes of cheerfulness, and remember
That they could easily happen again, unless
The mechanism had jammed, and we
Were to be tenants forever of a time
With little to hold the interest, and no
Promise of relief in movement.
And afterwards it was as though decay
Or senility of time had set in.
The scene changed, of course, and nothing
Was, again, as once it had been.
And therefore I do not see how I
Shall ever be able to acquire again
My old love of study, for it seems to me
That even when this infirmity of time
Has passed, the knowledge
Will always remain with me that there is one
Thing more delightful than study, and that once
I experienced it. And though it was not joy
But rather something more like the concept of joy,
I was able to experience it like a fruit
One peels, then eats. It's no secret
That I have learned the things that are
Truly impossible, and left alone much
That might have been of profit, and use.
One destroys so much merely by pausing
To get one's bearings, and afterwards
The scent is lost. To use it
I must forget the clouds and turn to my book,
Whose shifting characters, like desert sand
Betray my own fatigue, and loss
Of time, that ever, with nervous, accurate fingers
Cross-hatches the shade in the corner
Of the piazza where I stand, and leave
The lighted areas scarcely perforated, almost
Pristine. Lovers in parked cars
Undulated like the sensibility that refrigerates
Me at those times: and who
Could pick up the pieces, over and over?
Yes, it was a fine gift that you sent
Me, your book, wherein I could read
The very syllables of your soul, as dark-arched
And true as any word
You ever grunted, and whose truant
Punctuation resumed again the thread
Of what is outside, outdoors, and brought
It all ingeniously around to the beginning again
As a fountain swipes and never misses
The basin's fluted edge. But how in
Heck can I get it operating again? Only
Yesterday it was in perfect working order
And now the thing has broken down again.
Autumn rains rust it. And their motion
Attacks my credulity also, and all seems lost.
Yet fences were not ever built to last:
A year or two and all is blown away
And no trace can be found.
As a last blessing
Bestow this piece of shrewd, regular knowledge
On me who hungers so much for something
To calm his appetite, not food necessarily—
The pattern behind the iris that lights up
Your almost benevolent eyelash: turn
All this anxious scrutiny into some positive
Chunk to counteract the freedom
Of too much speculation. Tell me
What is on your mind, and do not explain it away.
"The egrets are beginning their annual migration.
From the banks of the Hag River a desolate
Convoy issues, like a directional pointing hand.
There is a limit to what the wilderness
Can accomplish on its own, and meanwhile,
Back in civilization, you don't seem to be
Doing too well either: those flying
Bits of newspaper and plastic bags scarce
Bode better for him who sits and picks at
The secret, when suddenly
The meaning knocks him down, a light bulb
Appears in a balloon above his head: it had nothing
To do with what the others were thinking, what
Energies they poured into the mould of their
Collective statement. It was only
As a refugee from all this that living
Were possible if at all, but it cast no shadow,
No reflection in the mirror, and was nervous
And waifed, so strong was the shuttle
Of accurate presentiment plying directly
Between it and the discarded past. Playing
A game is the only way to see it through, and have it
Finally integral, but the matter is that
This is somewhere else: its rails
Run deep into the leafy wilderness, sink
And disappear under moss and slime
Long before the end is reached. It's a crime,
And meanwhile your velvet portrait presides,
Benevolent as Queen Anne, over the scene
Below, and at no point
Do reality and your joyous truth coincide."
So sang one who was in prison, and the erosion
Process duly left its mark
On the wall:
Only a wan, tainted shadow leaned
Down from the place where it had been.
The eroding goes on constantly in the brain
Where its music is softest, a lullaby
On the edge of a precipice where the whole movement
Of the night can be seen:
How it begins, undresses, and disappears
In hollows before the level is seen to rise.
And then we are in a full, static music,
Violent and spongy as bronze, but
There is no need, no chance to examine
The accidents of the surface that stretches away
Forever, toward the ultramarine gates
Of the horizon of this tidal basin, and beyond,
Pouring silently into the vast concern
Of heaven, in which the greatest explanation
Is but a drop in the bucket of eternity;
_Mon rêve_.
But why, in that case,
Whispered the petitioner, pushing her
Magenta lips close to the thick wire mesh
That separated them, rubbing
Her gloved hand athwart it as though
Devoured now by curiosity, can God
Let the eroding happen at all, since it is all,
As you say, horizontal, without
Beginning or end, and seamless
At the horizon where it bends
Into a past which has already begun? In
Truth, then, if we are particles of anything
They must belong to our conception
Of our destiny, and be as complete as that.
It's like we were children again: the bicycle
Sighs and the stars pecking at the sky
Are unconstrained in spite of the distance:
The blanket buries us in a joyous tumult
Of indifference when night is
Blackest
So that we grow up again as we were taught to do
Before that. With the increase of joy
The sorrow is precipitated out, and life takes on
An uncanny resemblance to the photograph of me
That everybody said was terrible, only now it is real
And cannot be photographed.
It was nice of you to love me
But I must be thinking about getting back
Over the mountain
That divides day from night:
Visions more and more restless
All now sunk in black of Egypt.
The enduring obloquy of a gaze struck
The new year, cracking it open
At the point where people and animals, each busy
With his own thoughts, wandered away
In unnamed directions. If there is a fire,
I thought, why single out the glares
Impaling those least near it
In such a way as to reflect them back
On its solid edifice? But here
In a tissue of starlight, each is alone and valid.
You can stand up to breathe
And the garment falling around you is history,
Someone's, anyway, some perfectly accessible,
Reasonable assessment of the recent past, which
With its pattern dips into the shadow of the folds
To re-emerge and be striking on the crest
Of them somewhere, and thus serves
Twice over, as plan and decoration,
A garden plunged in sun seen through a fixed lattice
Of regrets and doubts, pinned there
For a variety of good reasons, alive, stupid
As a sail stunned in a vast haze,
Perfect for you. And you rise
Imperfect and beautiful as a second, a continent
Whose near coast alone can be seen, but
Which makes up for that in the strength of the confusion
Building behind it, and is at rest.
And I'll tell you why:
The elaborate indifference of some people, of some person
Far out on the curve
Is always rescued by another person
And this will be some forgotten day three years ago
At today's prices. The tensions, overlaid,
Superimposed, produce an effect of "character"
And quizzical harmony, like the outdoors.
But on death's dark river,
On the demon's charcoal-colored heaths
Where the luscious light never falls, but fluffy
Cinders are falling everywhere, the persons
Gesture hurriedly at each other from a distance.
Surely this is no time to play dumb, or dead, but
A directive has not been issued.
At the plant they know no more about it than you do
Here, and in the dump behind
They are singing of something else, trilling surely
But no one any longer can make any sense of it.
It is as though you had paid the bills
But the sun keeps writhing: "For this
I gave apples unto the tawny couch-grass, kept ledgers
In my time, as you do in yours?
That a badger with a trumpet on a far tussock
May rake in the calls, and none of it
Ever gets distributed to the poor, which I had stipulated
As being part of the deal? And who are we poor workers?
Not much surely, but we were
Just getting over the shock of dispossession
When this happened, and now this on top of it.
Who is any the wiser? What are we to make of
What now appears to be our lot, though we did nothing
To deserve it? Our efforts were in some way
Directed at a greater good, though we never forgot
Our own interests, as long as they harmed no one.
And now we are cast out like a stone. Surely
The sun knows something I do not know
Although I am the sun."
And slowly
The results are brought in, and are found disappointing
As broken blue birds'-eggs in a nest among rushes
And we fall away like fish from the Grand Banks
Into the inky, tepid depths beyond. It is said
That this is our development, but no one believes
It is, but no one has any authority to proceed further.
And we keep chewing on darkness like a rind
For what comfort it can give in the crevices
Between us, like those between your eyes
When you speak sideways to me, and I cannot
Hear you, though farther out there are those
Who hear you and are encouraged, and their effort
Brightens on the side of the mountain.
"I haven't seen him since I've been here"—and I,
All liking and no indifference, transfixed
By the macaronic, like a florist, weary and slippy-eyed,
Athwart blooms, compose, out of what the day provides,
Mindful of teasing and subtle pressures put,
Yet careful to seize the pen first. "What
Have you been up to?" Well, this time has been very good
For my working, the work is progressing, and so
I assume it's been good for you too, whose work
Is also doubtless coming along, indeed, I know so
From the sudden aging visible in both of us, tired
And cozy around the eyes, as the work prepares to take off.
Anyway, I am the author. I want to
Talk to you for a while, teach you
About some things of mine, some things
I've put away, more still that I remember
With a tinge of sadness, even
Regret around the sunset hour, that puts these
Things away, jettisons 'em, pulls the plug
On 'em, the carpet out from under their feet:
Even such, they say, as stand in narrow lanes
Wanly soliciting passersby, but without much
Hope of interest. Nevertheless, the
Things I want to visit with you about
Are important to me. I've kept them so long!
Zephyrs are one. How
Idly they played around me, around
My wrists, even in the bygone time!
And pictures—
Pictures of capes and peninsulas
With big clouds moving down on them,
Pressing with a frightening weight—
And shipwrecks barely seen (sometimes
Not seen at all) through the snow
In the foreground, and howling, ravenous gales
In the background. Almost all landscapes
Are generous, well proportioned, hence
Welcome. We feel we have more in common with a
Landscape, however shifty and ill-conceived,
Than with a still-life: those oranges
And apples, and dishes, what have they to do
With us? Plenty, but it's a relief
To turn away from them. Portraits, on the other
Hand, are a different matter—they have no
Bearing on the human shape, their humanitarian
Concerns are foreign to us, who dream
And know not we are humane, though, as seen
By others, we are. But this is about people.
Right. That's why landscapes are more
Familiar, more what it's all about—we can see
Into them and come out on the other side. With
People we just see another boring side of ourselves,
One we may not know too well, but on the other
Hand why should we be interested in it? Better
The coffee pot and sewing basket of a still-life—
It's more human, if you want, I mean something
A human is more likely to be interested in
Than pictures of human beings, no matter how well drawn
And sympathetic-looking. However, as the author
Of this, I want to buy a certain picture,
A still-life in fact, from a man who has one
And need the permission of the man
In order to do so. Unless I can acquire it
I can never feel the point of any of this. Oh,
I can see it intellectually, all right, but to really
Feel it, experience it, I have to have the picture.
That's all. I'd hate to give it up.
To be consigned to this world
Of life, a sea-world
Which forms, shapes,
Faces probably decorate—
It is all as you had suspected
All along, my dear.
They proliferate slowly, build,
Then clog, and in weathering
Become a foundation of sorts
For what is afterwards to be erected
On this plot of unfinal ecstasies—
Benign, in sum. They don't just go away, either.
But like a hollow tower
Let in some sun, and keep the wind
Far hence; whatever can destroy
Us loses, but it's pretty hard to say
How far we have come, how much accomplished
And whether there's a lot more to be said:
But for stretches at a time of life the outlined
Masks and scabbards which are our vague
Impression of what is probably going on
All around us, keep us distracted,
From playing and working too hard.
And yet life is not really for the squeamish either.
The hyacinths are dying
At the end of a broad blue day
Whose words somehow have not touched you.
Mad to sacrifice next to them
In late life, you were "just looking"
Instead when the uneasy feeling that a jewel
Might someday be around crossed you
But I can't figure out
What ever happened. You treasured it,
I contain you, and there are a few clouds
Down near the baseboard of the room that prevent
Us from ever continuing our conversation
About the terrible lake that exists behind us.
Piss and destruction
Are the order of the day, the office blues,
The Monday morning smiling through tears
That never come.
Partly because you always expect the impossible,
But also because here, on the level of personal
Life, it becomes easier to say, nay, think
The transversals that haven't stopped
Defining our locus, have indeed only begun
To, you are invited, and cannot refuse,
To share this wall
Of painted wooden tulips, the wooden clouds
In the sky behind it, to feel the intensity
As it is there. Good news travels fast
But what about the news you forgot
To tell until now, so we can't tell
All that much about it? Well, it joins us.
The ground is soaked with tears.
The tears of centuries are being wiped away.
The tower is beaded with sweat that
Has smiled down on our effort
For so long.
The lovers saunter away.
It is a mild day in May.
With music and birdsong alway
And the hope of love in the way
The sleeve detaches itself from the body
As the two bodies do from the throng of gay
Lovers on the prowl that do move and sway
In the game of sunrise they play
For stakes no higher than the gray
Ridge of loam that protects the way
Around the graveyard that sexton worm may
Take to the mound Death likes to stay
Near so as to be able to slay
The lovers who humbly come to pray
Him to pardon them yet his stay
Of execution includes none and they lay
Hope aside and soon disappear.
Yet none is in disrepair
And soon, no longer in fear
Of the flowers their arrears
Vanish and each talks gaily of his fear
That is in the past whose ear
Has been pierced by the flowers and the air
Is now contagious to him
He walks by the sea wall
With a mate or lover and all
The waves stand on tiptoe around the ball
Of land where they all are.
Thus, by giving up much,
The lovers have lost less than
The average man.
No bird of paradise flies up
With an explosive cry at his touch,
The lover's, yet all
Are made whole in the circle that rounds
Him, filled the whole time with sweet sounds.
It is not the disrepair of these lives
Where we may find the key to all that gives
Eloquence and truth to our passing thoughts,
And shapes them as a shipwright shapes
The staves for the hull of some desolate
Ship; rather, it is in the disrepair
Of these lives that we not find despair
But all that nourishes and comforts death
In life and causes people to gather round
As when they hear a good story is being told
And makes us wish we were younger but also cherishes
Our advancing years, and to find there no fears.
The tower was more a tower inside a house.
Even its outside (tendril-clogged crannies)
Was shaded from the view of most.
It grew chaste, and slim, like a prism
In a protected, secular environment
That overlooked the torment, fogs and crevasses
Of orderly religion. That house
Grew all alone in a desolate avenue
(Avenue so shady)
That people began to forget coming to
Long before its present state
Of patched-up oblivion, and even
In those days were those who remembered back
To what seemed a state of true freedom:
Bopping down the valleys wild, beaks
Tearing the invisible ear to shreds
But was actually a rudimentary stage
Of serfdom dating from the Silver Age.
Now, however, that house was as it was
Never going to be: a modest yet firmly
Rooted pure excrescence, a spiritual
Rubber plant:
A grave no one wanted to visit
Which remained popular and holy down to the present afternoon,
Something which nobody in particular
Was interested in, yet which mattered more
To the earth's population in general
Than practically anything they could think of.
It was history just as it disappears in the
Twilight of yesterday and before it
Materializes today as everything that is
Fresh, young, and strange, and almost
Out of the house and halfway down the street—
An index, in other words, of everything
That is not going to and is going to happen
To us once we forget about its progress
And actually begin to feel better
For having done so.
It goes without saying that
To have it make sense you
Would have to belong to all who are asleep
Making no sense, and then
Flowers of the desert begin, peep by peep,
To emerge and you are saved
Without having taken a step, but I
Don't know how you're going to get
Another person to do that. It all boils down to
Nothing, one supposes. There is a central crater
Which is the word, and around it
All the things that have names, a commotion
Of thrushes pretending to have hatched
Out of the great egg that still hasn't been laid.
These one gets to know, and by then
They have formed tightly compartmented, almost feudal
Societies claiming kinship with the word:
(If on a priority basis however
It takes longer to catch them)
And their age flows out of time, is left
Like a bluish deposit on the brown ploughed fields
That surround our century: like the note of a harp.
The phosphorescent spring fails, and newer,
Numbered days come up. The wind pulls at
The leaves of the calendar, peels them off one by one
In a fitful expression of what time is like
As it goes by, that's like a look
Out of a window, and then the moment has gone away
From the window The vast quantities of scum
Did not materialize. Only the sterile minuet
Proceeds at an always altered rate
Leading to bad feelings here and there
But the main feeling is safe and out of reach.
Love is different.
It moves, or grows, at the same rate
As time does, yet within time:
The waxing is invisible, and can never be felt
Outside time, as a few things—happiness,
For instance—can. As perennial as time
Is, and as insipid to the tongue, yet it
Is built in another street; such luminescence
As it has, it takes from the idea of itself
Each of us has, and knows not, except
To recognize, and feel secure again about its growing:
I mean that it is a replica
Of itself, which is itself the replica,
Counterfeited from itself, which is something
False, yet true, like the moon, and whose
Earthly reflection is of a truly
Hair-raising solidity, like the earth
Dissolved in the sun, suffused with a kinetic
Purpose it could never have for us
Unless we dreamed it. It is, then,
Gigantic, yet life-size. And
Once it has lived, one has lived with it. The astringent,
Clear timbre is, having belonged to one,
One's own, forever, and this
Despite the green ghetto that intrudes
Its blighted charm on each of the moments
We called on love for, to lead us
To farther tables and new, surprised,
Suffocated chants just beyond the range
Of simple perception. These, brown
Motes, may unclasp themselves like
Japanese paper flowers at any moment,
Rending themselves into a final
Fixed appreciation of themselves and whatever
They were going to be confronted with
Lest the politicians despair of its ever
Becoming a diamond that gives back the night
Into its smallest box and learns to live
With itself, like a true feeling.
## III
But, what is time, anyway? Not,
Not certainly, the faces and pleasures
Encrusted in it, the "beautifully varied streets,"
The wicked taunting us to some kind of action,
Any kind, with hands partially covering
Their faces, to hide or to mock us, or both.
No, these things are part of time,
Or are rather a kind of parallel tide,
A related activity. And the markings?
Some say that the measuring of time
Is a recognition of what it is, but
I think the things that are in it
Are more like it, though not quite it.
Actually what is in it is controlled
And colored by the units of measuring it.
That summer jog you had
A long time ago
Is probably it, it fits so
Neatly over it anyway, nobody
Could ever tell the difference.
And what was said
All afternoon, long afternoons
Ago, whatever it was, and it
_Was_ something special, you know
You really can remember it.
I wanted to forget it but it was like
Not remembering it and having the whole
Force of it brought home to you, and who
Wants that? Who cares, anyway, about
What it is or what it was like?
You must be mad to care. Yes,
I am mad, I think, and I do care.
I can't help it. I am mad,
And don't care. But it will not remain
Any more outside of me for all that.
It is the marrow of my thought
That all night I stand up chewing,
Trying to remember things, mostly things
I'd forgotten, and who
Remembers these? And also
Some things I
Actually remembered, and here I am
Trying to remember them all over again, to have
Them live up to me.
And it is as it was when I was a kid:
The moment stays on, but is
Lacing up its shoelaces or engaged
In some other form of maddening and hard to
Notice activity, but it gets its work done,
And still it can stay it has stayed
Around long enough to count for that
So that it is I who have aged without
Having done anything, certainly nothing
To deserve it, like a lost cause.
I would just love to go
Would love it
And you too want to go, with me,
And there is no reason not to, nothing
Keeping us here, we
Can go out into the street
Where nobody is, no dirt
Any more, and climb to the lower edge of the sky
And wait there, and soon
Someone will come to take care of us.
All I want
Is for someone to take care of me,
I have no other thought in mind,
Have never entertained any.
When that day comes I'll go gladly
Into whatever situation or room you want me in
To take care of.
And meanwhile I'll wait, obligingly, full
Of manna and joy, for that to take place
Which it will, soon.
But why you
May ask do I want someone to take care of me
So much? This is why:
I can do it better than anyone, and have
All my life, and now I am tired
And a little bored with taking care of myself
And would like to see how somebody else might
Do it, even if that person falls on their face
In the attempt.
When leaves pass over, and then ice
And finally warm, bottled-up breezes
I'll notice how it has all seemed the same until now,
This very moment, and as a
Duck takes off into the nether blue,
Find my rationale or whatever, something
Inside these movements all around me that
Enclose me loosely like a cage with the bars
Wide enough apart to walk through
Into the open air, onto God's road, in the blond,
Shambling sunlight, and look back
After all that, thinking how fortunate
It has all been on the whole, and how, though joy
Has been lacking, and that severely on occasion,
Happiness has not. I must
Make do with happiness, and am glad
To do so, as long as everyone
Is happy and doesn't mind. The car
Drove back to get me, through miles and miles
Of mud ruts and mangrove swamps, and stopped
And I got in and it drove away
To a slightly less flat land where you
And I can build a new life together on the shore
Several inches above sea level as the blue
Whitecaps on the charging waves come foaming in.
The Americans, with a sigh, never call it
By another word than its name. O
People who loiter by the Pacific,
Whose swaggering insouciance might convince
If left to play, and who can never lie,
Not even from the truth, how is it
With you, nestling all of you on one side?
The buildup predicted by others never
Quite matriculated, and now some of you
Are in this impasse, preparing to stay, while
Others straggle here and there, finding
Food, shelter, deserts, and in the tall
Tales some kindling, an advantage, and
You never look down.
_The narrator:_
Something you would want here is the
Inexpressible, rage of form
Vs. content, to show how the latter,
The manner, vitiates the thing-in-
Itself that the poem is actually about
And which, for this reason, cannot
Be considered the subject. Living
On the tranquil slope of an inactive volcano
All these days which group themselves
Into decades, consuming
The egg puddings of each one of these days
Is like unto form as subject matter
Perceives it through the cracks in its
Makeshift cell, and knows
There is light and activity outdoors to which
It can never contribute, but of which
It must needs always be aware, and this
Oozing sore is progress, slow
And miserable at times but magnificent
In its conception, in theory, and may never
Be anything more than this, but knows
About itself. Luckily, the object
Keeps making itself known to the opinions
About form and remains strong and warm
Long after it has gone out of fashion
And so never ceases, even in its earliest
Days preceding its demise, to be a runic
Maquette of the ideal poem-construct
Even after it has finally washed its hands of all
Notion of form, pleads ignorance or conflict
Of interest, and releases Barabbas to the
Delighted distraction of the rabble whose
Destiny is always to be of two minds
About everything and will end up on your doorstep
If you don't watch out:
You private yet public excuse for a still
Active poetasting writer but whether what
Is lasting in your work will last is the
Big question: it's poetry, it's extraordinary,
It makes a great deal of sense. It starts out
With some notion and switches to both, yet
The object will be partially perceived by the forms
Around it it is responsible for.
Note that, in the liturgical sense
Of history, the way I see it, we are falling down
In our duty toward the dustman's spasms, derelict
And decrepit as regards the outside world.
Deduce a spasm? Aye, a very
Insomniac'd tear it down so as to rebuild
And resell it. Tear his tattered ensign
Down? I don't know, I thought it looked nice
Hanging overhead, though I could
Be wrong. Valentine, I need you,
The mice in the plaster disturb all my reasoning
On this vale, this slope. The outer districts
Were succinct, full of enough plans,
But on the interior was the abysm, no
Invitation available, nothing about
The plodding fever that grew him, and the worries
That came after. No clue.
In industry we are persuaded that we may in some
Connection contribute a certain stone or effort
And this lazily winds away over the hill.
Or say that between the effort and the screws
Some scorpion intruded, and to top
It off a storm interfered with the rescue efforts
Blurring them? What then? What do you make
Of the red traffic light turning green to admit
A few cars farther on in the shuffle when night
Binds the tubing with rain and you
Can see yourself only as you used to be in college?
Make you mine
Valentine
Feelin' fine too if consumed
With energy to be mad and go on
Confessing even if it means that the sought-after
Absolution be rescinded after a time and those who
Looked silently at you for a while direct
Their gaze downward to the sunlit
Tundra. And you go out to the party
As toes slip into shoes
And I am not just left on the corner
But am as the traveling salesman of a joke
With a permanent hard-on and no luck and
All these samples in this here suitcase. Wanna see 'em?
Otherwise, why, we don't know too much. Fellow was over
Here recently from the British Isles,
Wanted to see something of how the life goes
On. He never made it back. Well some of us
Enjoy that way too as though we knew
Life was a picnic or parade down under the
Hassles and disrobing, the dust,
But now well we pretend to see otherwise
Into the great blue eyes of concrete that best
Our city, in the time of industry, and so
Panic slowly in the vegetal heart of things
Until told to disconnect the operation.
No wonder so many of us
Get discouraged, know not where to turn.
The truth is that nowhere in Europe,
India or America is this a straight line
Drawn, vertically, from one point to another
So as to connect them and in so doing
Provide a lot of fun and refreshment
For the students so they may never
Feel insecure again. Such a line may exist
But it would be horizontal, like the Northwest Passage,
And not connect people up with anything else.
It's a wager, and emptiness, and though warm
And the color of baked loaves in the sun
It has no idea of nourishment or where
You should go.
Its idea is that the Latin text
Might also have existed in German or be so close
It doesn't matter any more and the cottage
Be shut up at the end of summer and be there
Come early or mid-spring, but this
Presupposes a helpless mankind pigeonholed
With a rival deity so that neither can make
The hands of the clock move and it all goes down
In darkness, with the sun. To the supreme
Moment then, but it spreads out in sullenness
Over a vast tidal plain to dissipate in what
It is not even sure is horizon, is nothing but
Images. Earthly inadequacy
Is indescribable, and heavenly satisfaction
Needs no description, but between
Them, hovering like Satan on airless
Wing, is the matter at hand:
The essence of it is that all love
Is imitative, creative, and that we can't hear it.
Oh, once
A long time ago, in towns and cities
The line was different. We lived
Indifferently then, but perhaps more accurately,
And once it was over we knew
What to do with it. We carried out
Our neighbors' lives and they had our
Instructions about where to go. We lived
Inadequately, blushing, but we knew we were
On the outside and that only one thing
Prevented us from traveling inward, and that
Thing was our knowledge of how little we imagined
Everything. As though a door
Were enough to stop the average person and he
Would just curl up on the doormat forever.
But this
Person turned out to be mass-produced. He was funny
And knew about elegance, how to dress
For an occasion, yet the error that incites us
To duplication was missing, or inexact. We have
Not spoken to him. It should be outrageous
To do so. Yet to ignore him will bring no light.
But to get it right
We might ask this once: how goes it
Down there? What objects
Have you found recently?
"There are no trade winds. The ocean too
Is someone's idea. The pleasant banter of
The elements cannot disguise this basically
Thin concept, nor remove us from
Contemplation of it, and that is the best
Answer that may precede the question. Until later
When the shooting fires light up the sides
Of the volcano and each task and catastrophe
Become clear and succinct. By that time kindness
Will have replaced effort."
Why keep on seeding the chairs
When the future is night and no one knows what
He wants? It would probably be best though
To hang on to these words if only
For the rhyme. Little enough,
But later on, at the summit, it won't
Matter so much that they fled like arrows
From the taut string of a restrained
Consciousness, only that they mattered.
For the present, our not-knowing
Delights them. Probably they won't be devoured
By the lions, like the others, but be released
After a certain time. Meanwhile, keep
Careful count of the rows of windows overlooking
The deep blue sky behind the factory: we'll need them.
## _I_
_So this must be a hole_
_Of cloud_ ,
_Mandate or trap_
_But haze that casts_
_The milk of enchantment_
_Over the whole town_ ,
_Its scenery, whatever_
_Could be happening_
_Behind tall hedges_
_Of dark, lissome knowledge_.
_The brown lines persist_
_In explicit sex_
_Matters like these_
_No one can care about_ ,
_"Noone." That is I've said it_
_Before and no one_
_Remembers except that elf_.
_Around us are signposts_
_Pointing to the past_ ,
_The old-fashioned, pointed_
_Wooden kind. And nothing directs_
_To the present that is_
_About to happen_.
_These traumas_
_That sped us on our way_
_Are to be linked with the invisible damage_
_Resulting in the future_
_From too much direction_ ,
_Too many coils_
_Of remembrance, too much arbitration_.
_And the sun shines_
_On all of it_
_Fairly and equitably_.
_It was a way of getting to see the world_
_At minimal cost and without_
_Risk_
_But it can no longer stand up to_
_That_.
_The fences are barrel staves_
_Surrounding, encroaching on_
_The pattern of the city_ ,
_The formula that once made sense to_
_A few of us until it became_
_The end_.
_The magic has left the_
_Drawings finally_.
_They blow around the rest—tumbleweed_
_In a small western ghost town_
_That sometimes hits and sometimes misses_.
_That tower of lightning high over_
_The Sahara Desert could have missed you_ ,
_An experience_
_Unlike any other, leaching_
_Back into the lore of_
_The songs and sagas_ ,
_The warp of knowledge_.
_But now it's_
_Come close_
_Strict identities form it_ ,
_Build it up like sheaves_
_Of nerves, articulate_ ,
_Defiant of itself_.
_The posse had seen them_
_Pass by like a caravan_
_In slow motion_ ,
_Elephants and wolves_
_Painted bright colors_ ,
_Hardly visible_
_Through the cistern of shade_
_Of a hand held up to the eye_.
_Now that they are gone and_
_To be dreamed of_
_A new alertness changes_
_Into the look of things_
_Placed on the railing_
_Of this terrace:_
_The beheld with all the potential_
_Of the visible, acting_
_To release itself_
_Into the known_
_Dust under_
_The sky_.
_Hands where it took place_
_Moving over the nebulous_
_Keyboard: the heft_
_Now invisible, only the fragments_
_Of the echo are left_
_Intruding into the color_ ,
_How we remember them_.
_How quickly the years pass_
_To next year's sun_
_In the mountain family_.
_All the barriers are loaded_
_With fruit and flowers_
_At the same time_.
_The leaves stumble up to_
_Intercept the light one last time_
_Outnumbering the sheaves_ ,
_Even the ants on the anthill_ ,
_Black line leading to_
_The cake of disasters_ ,
_Leading outward to encircle the profit_
_Of laughter and ending of all the tales_
_In an explosion of surprise and marbled_
_Opinions as the sun closes in_
_Building darkness_.
_In later editions you_
_Were called, casual, harsh_ ,
_Dispensing arbitrary edicts_
_Under present law_
_Timed and always sunk in the_
_Gnat-embroiled shade_.
_It was in fact a colossal_
_Desert full of valleys and_
_Melting canyons and soared_
_Under the heaving of sighs_
_Knowing it would all end_
_But never end, but exist_
_In the memory of itself turned to flesh_
_Of ice cream and sting_
_Without obliteration_.
_But as I see it you_
_Can only amble on, not free_
_Nor on a journey, appearing_
_Though at some later_
_Juncture_
_Of our tepid and insidious_
_Greeting:_
_The shock of the path_
_Worn like this_
_Never scaled_
_Beyond a certain point_
_And returning and returning_
_Like a pole pointed to the sky_.
_In some Greek_
_Coves barely under the water_
_Or barely inundated (you might say)_
_A ball was found, and stated_
_The body's predilection to it:_
_There is no more history you_
_Seem to say no more June_.
_The blue wraith that stands_
_Straight above each chimney: forget it!_
_It is almost gone_ ,
_Has almost departed_.
_Now the dry, half-seen pods_
_Are layered, and the beating_
_Of an old man in some dungeon_.
_No one sees how fast its processes_
_Whiz, until some day_
_When things are better_.
_Who can elicit these possible_ ,
_Rubbery spirals? Return of all that's new_ ,
_Antithesis chirping_
_To antithesis: let's climb_
_The roof, look out over all_
_That was so near and is:_
_Vanity of the dishpan_ ,
_The radio chortling succor to moved_
_Behemoths of sense shredding_
_Underwear and ulcers alike_
_In a past of no mean confection:_
_This wound like a small wall_
_Of ceramic intent:_
_It is meant to hound you_
_With its brothers in the afterlight_
_Of forest prisms, the brown sky sweeping_
_Unusually_
_Away. The cavern this time is big enough to fit in:_
_The broken apse_
_Wind slams through, the snail-sexton_
_With rheumy specs, dung beetle bringing up the rear:_
_Who could explain it?_
_Who could have explained it?_
_"Only pluralism ..." but we get_
_Far less for our money that way_.
_Aye, and fewer replies too_
_To sopping prayer-strips_
_Hanging like dejected plumage from that_
_Rafter over the porch swing_.
_They are anxious to be done with us_ ,
_For the interview to be over, and we_ ,
_We have just begun_.
_Yet I too_
_Was once captured this way_.
_How it became a delight_
_To think about it and when_
_Pain intervened, as usual_ ,
_The calm remained, held over_
_From the other time_
_And no broken trace was seen_.
_Now houses have been razed_
_Where once fields of vegetables_
_Stood; nothing's there_
_That cannot truly be_
_And was all along_
_Yet never was for the seeing_ ,
_The tasting that jabs back_
_Into the past as well_ ,
_For what is present savoring?_
_Mouthing of initials, of a career?_
_There is no case_
_For samurai, or witches' coattails_ ,
_But so long as the buoyant opening_
_Of a vacant career stand around healthily_
_There is no need to ascertain_
_The pink and red paper stratosphere_
_Balloons pasted a little crazily_
_Against a teetering sky_
_Where color cannot have ever been_.
_There was another photograph_
_In that album, but not so amusing_
_To remember or to describe:_
_Three dark women_
_On a swerving path that saucily_
_Pulled the rug out from under the spectator_.
_And the three expressions faded or_
_Were never there to begin with, picking_
_Up a little strength perhaps from the exhausted_
_Eye that watched them, guardedly_.
_And all it said was, we are stones_
_To be like this and never to be able_
_To reveal, being forward like this, but we can say_
_How repellent was the adumbration_
_That lodged us here, around_
_Our holes, and did not_
_Shove us away, but rather_
_As with brave looks out to sea_
_Left everything here to crumble_ ,
_Whether new and fine, or old_
_Or like us, not new nor old_
_Having no share in the time-cusp_
_That keeps you and they running here to imagined_
_Meetings as though some sense were here_
_In the fences and the privileged_
_Omissions of the frolic grass_.
_A close one_.
_I haven't seen him_
_Since I've been here_.
_Only an aftertaste of medicine_
_And subtle pressures put_
_Beyond this lattice that is_
_As narrow as the visible universe_.
_A whisper directs:_
_How many homeless_ ,
_Wandering, improvisatory_
_As new deserts move up_
_Into the constellation that was_
_Only a moment ago_.
_Straggling players reverse_
_The indications:_
_Lutes, feathers, hard_
_Leather berries fall:_
_The autumn in the spring_
_Again with July sandwiched_
_In the middle, lament_
_Of all the days from the least popular_
_To the most sought after, the play_
_Forever turning on itself:_
_Refrains, the spirit of sorrow_
_Begin it; duration_
_Only conjugates, the last happening_
_Is seen as inadequate only after the passing_
_Of much else varied stuff_
_Only in being turned inside out_
_Can it deny itself so that the meaning_
_Pierces in any given point_
_And in the texture of the sea, O_
_Sky-blue-violet raiment given_
_Not to be heeded_
_Only as an oblique arch through which sails_
_Perpendicular_
_The speeding hollow bullet of these times_
_Of mud and velvet, these_
_Choreographed intrusions_.
_Farther from far away_
_No more the colored echoes ring_
_On the afternoon groundswell already dissolved_
_In the thousands of hastening_
_Feet of birds and raindrops_
_In wasted penitence sucked back_
_Up to the crest again_
_From which the view is fine as views go_
_From low, stubby towers_
_Of which there aren't too many_
_Here_
_Like cash registers in a darkened store_
_Even as afresh dawn approaches, before_
_The winds come_.
_Further on up only birches_
_Grow and the red sweater_
_Is for you. You breathing_
_Into the angle of shadow in sunlight_
_Of the frosted kiosk that was taken_
_By men with tools and a surveying kit_.
_That was long after_
_The night out on the glacier_.
_In the morning the children and kittens ran around_.
_It wasn't necessary to remind us_
_Once we were seated at our desks in the school_
_Under the giant tree-roots sheathed_
_In moss about the quartz lightning_
_Tumbling down the bed of the stream_
_As on a stair. We were quick and ready_
_For level plant-games in the sun_
_That arrived just at noon as a horizontal line_.
_The error was in the hollowed-out, weed-choked_
_Afternoon and even it was only confession_
_Of too many strands of vagueness, neuters_
_Too independent of each other and yet_
_Abashed with the other heretics like ourselves:_
_Clusters of black inkberries sweeping the horizon_
_And we always prepared for a fight_
_Yet so innocent we have no place to go_.
_The spaces between the teeth told you_
_That the smile hung like an aria on the mind_
_And all effort came into being_
_Only to yank it away_
_Came at it_
_Hard as the lines of citrus planted_
_In firm yet wavering rows_
_All across the land to the water_.
_Bells were rung_
_For some members of the family only_ ,
_These relatives like scarlet trees who infested_
_The background but were not much more than_
_The dust as it is seen_
_In folds of the furniture_ ,
_These were the ones who were always_
_Pushing out toward the Pacific coast—what_
_A time we all had of it, but all that part_
_Is over, in a chapter_
_That somehow has passed us by. And yet, I wonder_.
_Certainly the academy has performed_
_A useful function. Where else could_
_Tiny flecks of plaster float almost_
_Forever in innocuous sundown almost_
_Fashionable as the dark probes again_.
_An open beak is shadowed against the_
_Small liturgical opera this time_.
_It is nobody's fault. And the academy_
_Has saved it all for remembering_.
_It performs another useful function:_
_Pointing out the way at the beginning_
_When everybody giggled nervously and_
_Got lost against the peach-fuzz sky_
_Where too many nice miracles were always_
_Happening and the blood-colored ground_
_Grasped them like straws, for a minute_.
_There was a smoother, less ambiguous way_
_To be determined and its banners shook like smoke_
_To become an arch of the bridge_
_And the bridge was acknowledged in good time_
_But never to this day_
_As its echo in the sky performing to meet it_
_Behind invisible cataracts and cloud catafalques_
_And yet, the carrion still_
_Steams here, the mote_
_Pursues the eye, and all is other and the same_
_Of which the rite dismantles bit by bit_
_The blind empathy_
_Of a homeland. It emerges as a firm_
_Enigma, burnished, filled in_.
_Furthermore, there was nothing like_
_Shadows of oranges_
_In the new game, nothing fanciful_
_And abstract one step away from foggy_
_Reality. The series were all sisters_
_Back in the fifties when more of this_
_Sort of thing was allowed. Two could_
_Go on at once without special permission_
_And the dreams were responsible to no base_
_Of authority but could wander on for_
_Short distances into the amazing nearness_
_That the world seemed to be. Sometimes_
_We would all sing together_
_And at night people would take leave of each other_
_And go into their houses, singing_.
_It was a time of rain and Hawaii_
_And tears big as crystals. A time_
_Of reading and listening to the wireless_.
_We never should have parted, you and me_.
## _II_
_Something I read once_
_In some poem reminded me of it:_
_The dark, wet street_
_(It gets dark at seven now)_
_Gleaming, ecstatic, with the thin spear_
_Of faerie trumpet-calls. A lullaby_
_That is an exclamation_.
_It cannot be found_
_As when the whole sky shifts and stays_
_Where it is until the next time_.
_Like a summer job in a department store_
_It stays on and on_ ,
_Breaking up the moments, hiding_
_The kissing_ ,
_Taking whatever is there away from us_.
_Its temperature is darkness_ ,
_Its taste, the silent, bitter welcome_
_On the edge of the forest_
_When you were starting to reach home_.
_Also, too much is written_
_About it, as though each time_
_Were starting from zero toward an imaginary_
_Number. No one sees it's_
_Just the evening news, mostly_ ,
_A translation into the light of day_ ,
_Or two fiddles scraping along_
_Out of kindness, you think, but_
_To whom? In short, any kind of tame_
_Manifestation against the straw_
_Of darkness and the darkening trees_
_Until the aftertaste claimed it_.
_Nothing here is like the_
_Wet, hot vigil_
_That loneliness erected:_
_There is nothing here that can be seen_
_The way that city could be seen_ ,
_Most precisely at night, perhaps_
_When thousands of tongues inspect it_
_And the outline of its state of mind_
_Tapers off hard and clear_
_Until the next time_.
_The noises in the bedroom dissolve slowly_
_And at last the thread holds_
_So that the lining adheres strictly_
_Or as a plumb line erected straight into the air_
_To stand for all vertical constructions_
_That chide and quietly amaze_
_The pale blue of the sky_.
_The shops here don't sell anything_
_One would want to buy_.
_It's even hard to tell exactly what_
_They're selling—in one, you might_
_Find a pile of ventilators next_
_To a lot of cuckoo-clock parts_ ,
_Plus used government documents and stacks_
_Of cans of brine shrimp, and an_
_Extremely elegant saleslady, in_
_Printed chiffon, seeming to be from a different_
_World entirely. But that's—_ que voulez-vous?—
_Par for the course, I guess. You_
_Pick up certain things here, where_
_You need them, and_
_Do without the others for the moment_ ,
_Essential though they may be_.
_Every collection is as notable for its gaps_
_As for what's there. The wisest among us_
_Collect gaps, knowing it's the only way_
_To realize a more complete collection_
_Than one's neighbor's. It's also cheaper_
_And easier to show off to advantage_.
_At night rain whips the collection_ ,
_The plunge, the surge of the tide_
_Drowns the memory of it. Only a dark field remains_
_But with the return of morning, the same_
_Familiar sticks and pieces poke_
_Their extremities out of the dewy mound of straw_.
_The collection, at least for some people_ ,
_Is still there. And it matters_
_To them, and to tax collectors_
_And taxation buffs, because_
_Now none of it will get lost_
_Any more than it already has. A_
_Garage can contain it_.
_All_
_Evening I have waited for your call_.
_The early period was never like this_.
_Even birds are happier than this_.
_You have_
_No right to take something out of life_
_And then put it back, knowingly, beside_
_Its double, from whom_
_The original tensions unwittingly came_.
_The collection matures_.
_Amateurs flock to it, to get a look at it_.
_And some day the idea_
_Will have been removed, extracted_ ,
_From the flurry of particulars_
_From numbered exhibits_ ,
_And the collected will have no end_.
_A few always stay behind mechanically_
_On a glimpsed piece of scaffolding_.
_There are many of us to choose from:_
_Blowhards, barnacles, old fogeys_
_Rushing up from under the earth_
_Into the sun!_
_It doesn't matter that the fruit is greenish_ ,
_Or that the ill-defined sidewalks seem to lead nowhere_
_As long as the clock is stowed in somebody's luggage_.
_The round smile of celebration_
_Is always there_ ,
_Is part of the permanent scenery_
_Of this age's accumulation_
_And seeps, or drifts, only a little_.
_My dear yesterday_ ,
_You were ugly and full of promise_
_And today the delta is forming:_
_The water, or is it sandbars, stretching away_
_Almost too far for them to mean to each other_
_What they still mean to us_.
_Another thing they can do to you_
_Is also celebration, but of another kind:_
_The dance that is a brown study_
_Under the skylight_ ,
_The music of eternal moping_
_As far as it goes, since eternity_
_Is an eye, and some things elude the eye:_
_Polite gestures, timid farewells_
_Alongside a flooded creek in April_ ,
_The false sparkle, the finish, the edge_.
_These permutate, combine_
_In a gentle ellipse of spoken vagaries_
_That pester nobody, and yet_
_How few invitations are received!_
_They say they're having trouble with the mails_
_And so many people have moved as_
_We become an increasingly mobile populace_
_In the deep shade of a quiet trailer park_
_Where nobody minds waiting_
_For one to finish examining the elaborate_
_Mechanical toys of the last century_
_Or playing warped, scratched 78 records_
_Of the great coloraturas of the past_.
_One is always free to sink into history_
_Up to the waist, and the mountains are_
_Now so breathtakingly close to the city_
_That it's like taking a vacation_
_Just to stay home and look at them_.
_That's all one can do_.
_Inhaling the while the extremely cold_
_Fresh cement smell which you must pass_
_On your way to school_.
_For all those with erysipelas_
_And the wrinkles on the forehead_
_And the cheeks that come from within, like reverse scars_
_For all those wearing old clothes_
_With the dormant look of expectation about them_
_For the women ironing_
_And who cut into lengths of white cloth_
_The glass stopper has been removed_
_We can breathe! The ocean has been pulled away_.
_I was over to the dog show the other day and_
_Noticed a nice-looking girl gazing around_
_As if puzzled. I went over to her and said:_
_"Pardon me, but can't you find the kennel_
_You wish?_
_If not, I shall be glad to assist you."_
_"Oh, thank you!" she replied. "Would you_
_Mind showing me where they are exhibiting the ocean greyhounds?"_
_I came out here originally I_
_Came to this flat place_
_On the side away from the sun_ ,
_I think my stain must be cauterized_.
_I have touched no drink_
_For an elevenmonth, yet my head_
_Seems stuck in my collar. I have_
_No friends because I move too rapidly_
_From place to place, only an assistant_.
_The time is always false dawn_
_In Indian Summer. Faded markings on_
_The floor where I walk could have_
_Been produced by me, or at best_
_Some outside agency. I have no reason_
_To rejoice in my mummy condition, yet_
_Am fairly happy from day to day_
_Like a steeple rejoicing in the sun_
_It is the last to shake hands with_.
_I wear my weather_
_With a good-natured air of secrecy_ ,
_And have no trouble finding my way home_
_Once the fun is done. I can sleep_.
_I can stand up. The buzzing in the vault_
_Of the temple disturbs me only insofar_
_As I consult my pocket watch and replace it_
_Affably in my breast-pocket. But_
_There is a time and a light_
_Which do not approach, which leave me_
_In the years_.
_Don't flog it. Remember how_
_Insane your other undertakings seemed to you_ ,
_How hopeless your desires, how tortured_
_The ambience, or riddled_
_With the stuff of hazard_.
_The orgy_
_Bubbles away, the vapors weep their burthen to the ground_.
_But in that hotel_
_The night is ongoing, the rain_
_Continues. Too much of a philosophy_
_Is about all it can stand, and we wait_
_For the men and ducks to go away, and still_
_Most everything stays with us_ ,
_Rooted in thoughtful soil_.
_The elephant's-foot umbrella stand_
_That used to be over there, why_ ,
_Somebody must have changed it, or the last_
_Catastrophe fished it up out of the depths_
_Beyond heaven, or it is here_ ,
_For us to see, yet absent for a while_.
_Or perhaps someone merely heard of it_
_Or it got written down the wrong way_
_In a page of an account book that got mailed_
_In a letter by mistake. Perhaps the dust_ ,
_That emptiness on the outside of air, ate it_.
_Or in the bin of odd-size and discontinued_
_Artifacts it holds its own while seen_
_Only partially because the surrounding_
_Knobs and hues rob it of a full presence_.
_Or a photograph was taken, after which_
_It could be destroyed, and now_
_The photograph and the negative are lost_
_Up ahead in one of the strands_
_Where one shall encounter this and all the_
_Other deviating forms of momentary life_
_In a contradiction which shall make its point_.
_I like to imagine though_
_That nothing so awkward as the stand ever_
_Existed. It must have been_
_The trunk of an old apple tree_
_And bees hollowed it out to make honey_ ,
_Itself now gone, a remnant_
_Of a memory, a gesture time made_
_To no one in particular, to itself_
_Or not even to itself, a tic_ ,
_A twinge long invisible now_
_On the low-pressure area_
_On the weather map. A tremor_
_Far removed from the individual man_
_And his daily wants, a number_
_To be looked up in a book, or the catalogue number_
_Of that book, or both_ ,
_The number in the book and the catalogue number_
_In white guano on the brilliant cranberry binding_ ,
_Concerns galore_
_Under both headings, the identical twin numbers_.
_Ours, actually, is an "age on ages telling,"_
_Once it has become finality. Afterwards_ ,
_It drifts like a stalagmite, advancing_
_Pea-brained arguments an inch forward_.
_Of course all this has to go on_
_Parallel to the hoping, so as to display_
_The ancestral linkage, and, more importantly, to drown out_
_Any rumors of competing loyalties_.
_It is merely a question of avoiding the shadow_
_And the starched patch of light_ ,
_At the same time deferring to no sun_ ,
_No shore. No half-naked limit_ ,
_And, in the orange light that the sun succeeds nevertheless_
_In shedding all over this terrestrial ball, to avert_
_One's gaze no longer and no less time than is intended_
_By the illuminating party to be your account_
_Of yourself, here on earth and for all time_.
_A grand army of fatality succeeding_
_One after the other like a phylloxera_
_Never succeeded in erasing intimate_
_Knowledge of how long that was supposed to be_
_Despite ferocious efforts from age to age the same_
_From the minds of those men in which it had been planted_
_Originally, and who continued to keep up_
_With the changing time and modes while retaining_
_With no effort at all_ ,
_As though all were elegy and toccata_
_(Which happens to be the case)_ ,
_The guidelines. Once given_
_They can be forgotten in the sad joy of life_ ,
_Reverence for which is almost incumbent_
_On each contestant, and no one, including them_ ,
_Will ever be wider for it. Yet_
_Thereby hangs a tale, of starving musicians_ ,
_Strolling players, grasshopper and the ant_
_Whose contemptible fireside contrasts so untellingly_
_With the barren outdoors. Just to play an instrument_ ,
_It seems, is to have to come round one_
_Day to the impossibility of making a living on it_ ,
_To being forced to prostitute oneself, innocently_ ,
_For the greater pleasure which is as the damage_
_Succeeding on the small first pleasure_.
_And there's no way out, unless_
_The sound of harps is sufficient distraction_
_Against the thunder of the fray "for which_
_Gog and Magog are said to be continually preparing_ ,
_Or loss of memory (which cannot, by definition_ ,
_Take place) render one oblivious to the traffic_
_And all it implies. That loss of memory_
_Which is itself a music_ ,
_A kind of music_.
_And meanwhile, growing older like leaves that lean back_
_Against the trees, is an accomplishment_
_Without comfort_.
_Back home from the beauty contest_
_And its attendant squalors, she doesn't feel_
_Like much. The world_
_Is vaguer and less pejorative, a time_
_Of stressful headache but also_
_Of architectonic inklings and inspiration:_
_Agony for a day, and then the refreshing dream_
_Bubbles up like an artesian well in all its_
_Wealth of accurately observed detail_ ,
_Its truth of being, on the surface_
_But striking long, pointed roots into the dull earth_
_Behind the mask. Yet like a pain_
_That went away, its immanence_
_Is very much an ongoing thing, its present_
_Departed in the greater interest of the whole_.
_A coronet of dark red jewels_
_Like winter berries was slowly lowered_
_Onto the snow-white curls, and the dream became_
_A person, a beautiful princess unable to stand_
_Or sit. And the older guests remembered_
_How none of it had been predicted, though the mystery word_ ,
_"Magic," had been imagined_
_Many years before. How_
_Do we live from the beginning of the tale_
_To its inevitable, momentary end, where all_
_Its pocket's treasures are summarily emptied_ ,
_On the mirroring tabletop? And wait_
_For someone to whisper the word that restores them_
_To their velvet hummock, sets all right again?_
_Only the cartoon animals know_
_How hard it was to get inside the frame, and then_
_To make a noise, or eventually to place_
_An inky paw-print on the wide, blinding white_
_Damask desert as the company was leaving_
_In twos and threes. Someone_
_Projects a shriek of recognition far up, into the civilized_
_But dim world of the farthest chandelier_.
_A commercial airliner streaked by. Once again_
_The prize will not be awarded_.
_The distant plains match up with_
_The pictures of them on these transparent walls_ ,
_And that is all. No children_
_To relieve the tensions of the adult business_ ,
_No new funny animals, only the vocal abstractions_
_Of the solemn, imaginary world of transportation_
_And commerce. No one_
_Laughs at the brilliant errors any more_.
_Yet we who came to know them_ ,
_Castaways of middle life, somehow_
_Grew aware through the layers of numbing comfort_ ,
_The eiderdown of materialism and space, how much meaning_
_Was there languishing at the roots, and how_
_To take some of it home before it melts (as all_
_Will, dreams and mica-sparkling sidewalks, clouds_
_And office buildings, the conversation_
_And the trance, until_
_A day when they can do no more, and the mass_
_Of the scenery wanders partially_
_Over the defunct terrain of broken fences_
_And windows stuffed with rags) while the ballad_
_Still rings in the seller's ear_.
_In the beginning of speech the question_
_Of frontiers is taken up again_.
_And the trees and buildings are porous_
_And the dome of heaven_.
_The talk leads nowhere but is_
_Inside its space_.
_It is contracting, it is observed ._..
_Breath we wanted, to build and lie down_
_In slumber at night, under the tattered shade_
_Of the trees, open to the rain, rustling of night_.
_And the wet, doggy smell_ ,
_The pealing of church bells interspersed with thunder_
_And lightning, the distress_
_And tiny triumphs of the field_.
_Everything is a shaft_
_Sunk far too deep into the body, opening landscapes_ ,
_New people, mingling in new conversations_ ,
_Yet distant, as the back of one's head is distant_.
_It all seems like 2½ years ago_
_To the impatient sun trapped in the attic_
_When all it wants is to be able to write about mathematics and the word_ ,
_For although a few wind-chime notes filter down_
_From heaven in the small hours, one cannot help_
_But note the frequent fanfare of hoofbeats_
_In the wet, empty street_.
_No one said it would turn out this way_
_But of course, no one knew, and now most of them_
_Are dead_.
_One, however, still looms_ ,
_Billboard-size in the picaresque_
_Night sky of eleven years ago. And whose_
_Hand is it, placed comically against your throat_ ,
_Emerging from a checkered cuff? Because a long time ago_
_You were promised safe-conduct_
_From a brief, mild agony_
_To these not-uninteresting pangs of birth_
_And so, and so, a landscape always seen through black lace_
_Became this institution_
_For you, inflected, as we shall see_ ,
_From time to time by discreet nautical allusions_
_And shreds of decor, to amount_
_To these handfuls and no other: a reminder_
_To keep it soft and straight forever as long_
_As no other pick up your ringing phone_.
_Play it on any instrument. It is in whack_
_And ready to do your bidding, though sunk_
_In the rat-infested heap of rose embers_
_Of the terminating day. A keepsake_.
_This has been a remarkable afternoon:_
_The sky turned pitch-black at some point though there_
_Was still enough light to see things by_.
_Everything looked very festive and elegant_
_Against the inky backdrop. But who cares?_
_Isn't it normal for things to happen this way_
_During the Silver Age, which ours is?_
_Motifs like the presentation of the Silver Rose_
_Abound, and no one really pays much attention_
_To anything at all. People_
_Are either too stunned or too engrossed_
_In their own petty pursuits to bother with_
_What is happening all around them, even_
_When that turns out to be extremely interesting_
_As is now so often the case_.
_You will see them buying tickets_
_To this or that opera, but how many times_
_Will they tell you whether they enjoyed it_
_Or anything? Sometimes_
_I think we are being punished for the overabundance_
_Of things to enjoy and appreciate that we have_ ,
_By being rendered less sensitive to them_.
_Just one minute of contemporary existence_
_Has so much to offer, but who_
_Can evaluate it, formulate_
_The appropriate apothegm, show us_
_In a few well-chosen words of wisdom_
_Exactly what is taking place all about us?_
_Not critics, certainly, though that is precisely_
_What they are supposed to be doing, yet how_
_Often have you read any criticism_
_Of our society and all the people and things in it_
_That really makes sense, to us as human beings?_
_I don't mean that a lot that is clever and intelligent_
_Doesn't get written, both by critics_
_And poets and men-of-letters in general_
_But exactly whom are you aware of_
_Who can describe the exact feel_
_And slant of a field in such a way as to_
_Make you wish you were in it, or better yet_
_To make you realize that you actually are in it_
_For better or for worse, with no_
_Conceivable way of getting out?_
_That is what_
_Great poets of the past have done, and a few_
_Great critics as well. But today_
_Nobody cares or stands for anything_ ,
_Not even the handful of poets one admires, though_
_You don't see them quitting the poetry business_ ,
_Far from it. It behooves_
_Our critics to make the poets more aware of_
_What they're doing, so that poets in turn_
_Can stand back from their work and be enchanted by it_
_And in this way make room for the general public_
_To crowd around and be enchanted by it too_ ,
_And then, hopefully, make some sense of their lives_ ,
_Bring order back into the disorderly house_
_Of their drab existences. If only_
_They could see a little better what was going on_
_Then this desirable effect might occur_ ,
_But today's artists and writers won't have it_ ,
_That is they don't see it that way_.
_They_ do _see a certain way, and that way_
_Is interesting to them, but_
_Doesn't help your average baker or cheerleader_
_To see precisely the same way, which_
_Is the only thing that could rescue them_
_From the desperate, tangled muddle of their_
_Frustrated, unsatisfactory living. Seeing things_
_In_ approximately _the same way as the writer or artist_
_Doesn't help either, in fact, if anything, it makes things worse_
_Because then the other person thinks he_
_Or she has found out whatever it is that makes_
_Art interesting to them, the reason_
_For those diamond tears on the scarlet_
_Velvet of the banquette at the opera_ ,
_And goes on a rampage, featuring his or her emotions_
_As the banners with a strange device of a new revolution_
_Of the senses, but it's doomed_
_To end in failure, unless that person happens to be_
_Exactly the same person as the artist who is doing_
_All this to them, which of course is impossible_ ,
_Impossible at any rate in a Silver Age_
_Wherein a multitude of glittering, interesting_
_Things and people attack one_
_Like a blizzard at every street crossing_
_Yet remain unseen, unknown and undeveloped_
_In the electrical climate of sensitivities that ask_
_Only for self-gratification_ ,
_Not for outside or part-time help_
_In assimilating and enjoying whatever it is_.
_Therefore a new school of criticism must be developed_.
_First of all, the new_
_Criticism should take into account that it is we_
_Who made it, and therefore_
_Not be too eager to criticize us: we_
_Could do that for ourselves, and have done so_.
_Nor_
_Should it take itself as a fitting subject_
_For critical analysis, since it knows_
_Itself only through us, and us_
_Only through being part of ourselves, the bark_
_Of the tree of our intellect. What then_
_Shall it criticize, in order to dispel_
_The quaint illusions that have been deluding us_ ,
_The pictures, the trouvailles, the sallies_
_Swallowed up in the howl? Whose subjects_
_Are these? Yet all_
_Is by definition subject matter for the new_
_Criticism, which is us: to inflect_
_It is to count our own ribs, as though Narcissus_
_Were born blind, and still daily_
_Haunts the mantled pool, and does not know why_.
_It's sad the way they feel about it—_
_Poetry—_
_As though it could synchronize our lives_
_With our feelings about ourselves_ ,
_And form a bridge between them and "life"_
_As we come to think about it_.
_No one has ever really done a good piece_
_On all the things a woman carries inside her pocketbook_ ,
_For instance, and there are other ways_
_Of looking out over wide things_.
_And yet the sadness is already built into_
_The description. Who can begin_
_To describe without feeling it?_
_So many points of view, so many details_
_That are probably significant. And when_
_We have finished writing our novel or_
_Critical essay, what it does say, no matter_
_How good it is, it merely mocks the idea_
_Of a whole comprised by all those now mostly invisible_
_Ideas, ghosts_
_Of things and reasons for them_ ,
_So that it takes over, seizes the glitter_
_And luminosity of what ought to have been our_
_Creative writing, even though it is dead_
_Or was never called to life, and could not be_
_Anything living, like what we managed_
_Somehow to get down on the page_.
_And the afternoon backs off_ ,
_Won't have anything to do with all of this_.
_Yet the writing that doesn't offend us_
_(Keats' "grasshopper" sonnet for example)_
_Soothes and flatters the easier, less excitable_
_Parts of our brain in such a way as to set up a_
_Living, vibrant turntable of events_ ,
_A few selected ones, that nonetheless have_
_Their own veracity and their own way of talking_
_Directly into us without any effort so_
_That we can ignore what isn't there—_
_The death patterns, swirling ideas like_
_Autumn leaves in the teeth of an insane gale_ ,
_And can end up really reminding us_
_How big and forceful some of our ideas can be—_
_Not giants or titans, but strong, firm_
_Human beings with a good sense of humor_
_And a grasp of a certain level of reality that_
_Is going to be enough—will have to be_ ,
_And so lead us gradually back to words_
_With names we had forgotten, old friends from_
_Childhood, and then everything_
_Is forgiven at last, and we_
_Can sit and talk quietly with them for hours_ ,
_Words ourselves, so that when sleep comes_
_No one is to blame, and no reproach_
_Can finally be uttered as the lamp_
_Is trimmed. The tales_
_Live now, and we live as part of them_ ,
_Caring for them and for ourselves, warm at last_.
_All life_
_Is as a tale told to one in a dream_
_In tones never totally audible_
_Or understandable, and one wakes_
_Wishing to hear more, asking_
_For more, but one wakes to death, alas_ ,
_Yet one never_
_Pays any heed to that, the tale_
_Is still so magnificent in the telling_
_That it towers far above life, like some magnificent_
_Cathedral spire, far above the life_
_Pullulating around it (what_
_Does it care for that, after all?) and not_
_Even aiming at the heavens far above it_
_Yet seemingly nearer, just because so_
_Vague and. pointless: the spire_
_Outdistances these, and the story_
_With its telling, which is like gothic_
_Architecture seen from a great distance_ ,
_Booms on in such a way_
_As to make us forget the prodigious_
_Distance of the waking from the_
_Thing that was going on, in the novel_
_We had been overhearing, all that time_.
_Not that writing can transcend life_ ,
_Any more than the act of writing can_
_Outdistance the imagination it feeds on and_
_Imitates in its ductility, its swift_
_Garrulity, jumping from line to line_ ,
_From page to page: it is both_
_Too remote and too near to transcend it_ ,
_It_ is _it, probably, and this is what_
_We have awakened only to hear: maybe just_
_A long list of complaints or someone's_
_Half-formed notions of what they thought_
_About something, too greedy_
_Even to feed on itself, and therefore_
_Lost in the muck_
_Of sleep and all that is forever outside_ ,
_Condemned to be told, and never_
_To hear of itself_.
_Sometimes a pleasant, dimpling_
_Stream will seem to flow so slowly all of a_
_Sudden that one wonders if it was this_
_Rather than the other that one was supposed to read_.
_In the charmèd air one_
_Imagines one hears waltzes, ländler, and écossaises_
_And concludes that it is literature_
_That is doing it, and that therefore_
_It must do it all the time. It works out too well_ ,
_The ending is too happy_
_For it to be life, and therefore it must_
_Be the product of some deluded poet's brain: life_
_Could never be this satisfactory, nor indulge_
_That truly human passion to be all alone_.
_And I too am concerned that it_
_Be this way for you. That you_
_Get something out of it too_.
_Otherwise the night has no end_.
_Otherwise the weeping messiah_
_Who comforts us on those nights_
_When truth has flown out the window_
_Would never place an asterisk_
_On your heart. Tour whole life_
_Would be like walking through a field_
_Of tall grasses, in time with the wind_
_As it blows. And in old age_
_There will have been no jump to the barefaced_
_Old man you then are, only a nudge_
_And promise of more suppers: some things I have to do_.
_How is it that you get from this place_
_To that one only a little distance away_
_Without anybody's seeing you do it?_
_The trip to the basement_
_Performed unseen, unknown ._..
_Uncle Fred and his cigars_
_All my old Mildred Bailey records_
_And a highly intelligent kangaroo_
_Riding with me, all of us in the back seat_
_In our old Hudson_.
_It doesn't explain much—_
_Rituals don't—_
_But as frantic as the commotion in nature_
_Now is, the grand impermanence_
_Of this storm, impatience_
_Of the calm skies to start again_ ,
_The house stays much the same_.
_One day a little bit of rust_
_At the eaves, a bit of tape removed_
_And its story will have been elsewhere_ ,
_Soon removed, like a porch, and the head_
_Must again sneeze out an idea of flowers_.
_That music, the same old one, will be born again_.
_So much for the resident way_
_Of adding up the drawbacks and the satisfactions_
_If any are to be found, and_
_I salute you so as you enjoy_
_The mellow fecund death of that past_.
_Ah'm impressed. And should we_
_Never get together, the deal stands_.
_We want it for them and we and us_
_More than ever now that it has dwindled_
_To a sticky, unsightly root. But now_
_The present has dried out in front of the fire_
_And we must resume the flight again_.
_Someone who likes you first_
_Comes along. The act is open_
_And a nation of stargazers begins_
_To unwrap the fever of forgetting, the while_
_You sidle next to each other and never_
_Afterward shall it be a question of these blooms_
_In that time, of speech heard_
_In that apartment. Nowhere that the light comes_
_Can you and he argue the subtle hegemony_
_Of guilt that loops you together_
_In the continual crisis of a rood-screen_
_Pierced here and there with old commercials_
_Shimmering and shining in the sun_.
_You are cast down into the lowest place_
_In the universe, and you both love it_.
_All this time larger and I may say graver_
_Destinies were being unfurled on the political front_
_And in the marketplace, important issues_
_That you are unable and unwilling to understand_ ,
_Though you know you ignore them at your peril_ ,
_That any schoolchild can recite them now_.
_Yet somehow it doesn't bode well that_
_In your sophistication you choose to disregard_
_What is so heavy with potential tragic consequences_
_Hanging above you like a storm cloud_
_And cannot know otherwise, even by diving_
_Into the shallow stream of your innocence_
_And wish not to hear news of_
_What brings the world together and sets fire to it_.
_It wasn't innocence even then, but a desire to_
_Keep the severe sparkle of childhood for_
_The sudden moments of maturity that come_
_Surprisingly in the night, dazzling_
_By the very singleness of their passage_
_Like white blossoming trees glimpsed_
_In the May night, before the tempests of summer_
_Put an end to all dreams of sailing and hoped-for_
_Good weather and luck, before the frosts come_
_Like magic garments. And so_
_I say unto you: beware the right margin_
_Which is unjustified; the left_
_Is justified and can take care of itself_
_But what is in between expands and flaps_
_The end sometimes past the point_
_Of conscious inquiry, noodling in the near_
_Infinite, off-limits. Therefore_
_All your story should be phrased so that_
_Tinkers and journeymen may inspect it_
_And find it all in place, and pass on_
_Or suddenly on a night of profound sleep_
_The thudding of a moth's body will awaken you_
_And drag you with it_ vers la flamme,
_Kicking and screaming. And then_
_What might have been written down is seen_
_To have been said, and heard, and silence_
_Has flowed around the place again and covered it_.
_"The morning cometh, and also the night."_
_I'll dampen you_
_As I celebrate you, but first_
_I'll turn your feet over_
_And enjoy you with this ever slenderer_
_Aspen climate, as one in the know would do_.
_I'll mouth expressions of yours_
_And replay your tricycle in the formal walks_
_And garden beds. Some very pretty views_
_Can be ascertained now. I'll not_
_Put a glove on so you may see the snake_
_With the cobalt eyes, and bring you offerings_
_Of olives, bananas, guavas, Japanese persimmons. Furthermore_ ,
_I will await you in indolence, so that_
_The view of the sea will move in slowly_
_And become the walls of this room_.
_But it was on this day that_
_I wanted to do something_ ,
_Commemorate something_ ,
_Not "never" or that day coming up_.
_So I offer you everything_
_You may ever want, not_
_Knowing how I'll pay the bills, just_
_Keeping to the memory of it like larkspur_
_Or a bird's head I once saw in a forest at dusk_.
_Lots of them are coming to prepare you_
_For this, and if I can't have you_
_I'll figure out some way out of this_
_Until the hour tolls its distinction_
_Amid great bravery and truth_
_Where men are seen running in and around from all over_
_And the rendition of great sonatas_
_May then be seen to give back some fitful_ ,
_Momentary spark of "the" truth_
_As cedars blacken against the fence and the sky_
_Just before slipping through the buttonhole of truth—_
_The commonplace, casual occurrence_.
_An honest killer would have caught you_
_And told you that way, and gone away_.
_But the basin of remorse is so vast_
_No drop ever increases it, and telling_
_Only makes it reverberate_
_Inward upon itself, toward the center that is not there_.
_And whether you search for nightingales_
_Or distress signals on the earth's clay lid, all_
_Is much the same: your face at morning_
_And your blue-plaid face at evening with no_
_Expression are nevertheless the same_
_Until the code is ventilated, and we who have_
_Come down with you, to the same root or comma_ ,
_Are new now, but with no difference_.
_He would cook up these goulashes_
_Make everything shipshape_
_And then disappear, like Hamlet, in a blizzard_
_Of speculation that comes to occupy_
_The forefront for a time, until_
_Nothing but the forefront exists, like a forehead_
_Of the times, speechless, drunk, imagined_
_In all its five shapes, and never in one state_
_Of repose, though always disclosed_
_And disclosing, keeping itself like a chance_
_In the dark, living wholly in a dream_
_Sweet reality discovers_.
_I wander through each dirty street_
_Knowing how painted rooms are bonny_ ,
_Remembering feather beds are soft, and Jack_ ,
_Eating rotten cheese. As the babble_
_Of apes in an orchard are the slogans_
_That solicit us like pennants in the sky:_
_Fools rush into my head, and so I write_.
_I'll wipe away all trivial fond records_
_From the interstices of my desirings_
_And imaginings, and find the whiteness_
_That was there. Already the colors of sleep_
_Are fading, a blankness_
_Is taking shape, and its magnificent outline_
_Washes true like the sound of a French horn_ ,
_And then somehow, sqwunched or_
_Scrunched down in the corner, in the folds_
_It collects itself, again, and all the differences_
_Are differences among rainbows, or adhesions_
_In the dance, that dissolve and strengthen_
_As it reaches its pitch. Again, ambition is seen_
_As no idle thing. Reading the papers_
_We are inflamed to emulate it, even as_
_There seems nothing wrong with it, and finally_
_Vote for it. Impetuously_
_We travel on, life seems full of promise_ ,
_And ambition is so recent as to be almost_
_Stronger than living, and makes its own_
_Definitions and pays for them. Surely_
_Life is meant to be this way, solemn_
_And joyful as an autumn wood rent by the hunters'_
_Horns and their dogs, unmixed with pleasure_ ,
_Turned inside out, violating_
_The very name of intimacy, but assured_
_Of an easy victory. Time was when it seemed_
_Too rich, too filling, but now the lean_
_Bones of the November wind are seen as dainty_ ,
_And just sufficient_ ,
_Emblems of the famished year_.
_O sun, God's creation_ ,
_Shine hot for one hour, confounding my enemies_
_Or else make them like me. I want to write_
_Poems that are as inexact as mathematics. I have been_
_Sitting making mudpies, in the sparkling sunlight_ ,
_And the difficulty of giving them away_
_Doesn't matter so long as I want you_
_To enjoy them. Enjoy these! You are busy, I know_ ,
_But could find time for this. Some day_
_People will remember them—this always happens—_
_And you'll be caught with your pants down_.
_Besides, how many streams can you rake_
_With your copper rake, without counting;_
_How much pouring fog chase away, larks_
_And ploughmen delight? In the occupied countries_
_You are raised to the statute of a god, no one_
_Questions your work, its validity, all_
_Are eager only to support it, to give of themselves_
_So as to push your crowning effort over the top:_
_Never_
_Had any such a plebiscite, but you must earn it_
_Even so, prepare, purify yourself to be worthy of it_
_Although no one will notice. Then, when you_
_Are setting, in a blaze of glory, you'll find_
_You have already written about this, about all_
_That's already happened, and everything that could be_
_In the future, and won't mind_
_About disappearing behind yon crag_
_Which already is grown silent, erect_
_With waiting, tense and eager as a bridegroom_
_For you to fall alongside its spine:_
_"The protector_
_Came from the tussock, the son rose up from the bottom."_
_I have heard that in spring the mountains change_
_And seldom pay any mind to the sun (who continues_ ,
_Nonetheless, to do good deeds, bringing_
_Cowslips and other small plants out of the mould, changing_
_The barren shale to faerie, coaxing_
_Mica glints out of the flat, unappreciative sidewalks_ ,
_Turning everything around but making it_
_Delightful), occupied as they are_
_With furthering their own desires, spreading_
_Their dominion over the flat, quiet land around them_.
_But no one is punished for inattention any more:_
_It seems, in fact, to further the enjoyable_
_Side of the world's activities. What seemed_
_Reckless, incoherent, even filthy at times_
_Is now the shortest distance; everything gets done_
_And, more important, ought to be done_
_This way, and only in this way_ ,
_For happiness to sustain, and fish to remain_
_In the depths, not elbowing the birds of the skies;_
_For it all to come right and not be noticed_
_Until just after it has slipped by, for the noble_
_And wonderful thing it is, so that the other_
_Visions may arise and occupy the same space_.
_Before long they too_
_Turn up in your mind_.
_You wonder what the original uses_
_Of famine were, after_
_We saw the film about it_.
_The brine shrimp were brought_
_And the fairy pudding placed next to them_.
_It's good though—_
_It has meat on it_.
_We fucked too long_ ,
_Though, you see_.
_Now it's too late to stay home_
_Or go anywhere except to that film_
_We've both seen a dozen or more times_.
_Of course it's good—that's why_
_We saw it so often—_
_But after a while one feels one has lived it_
_And wants to get on with other living experiences_.
_Yet we keep returning to it—_
_It is good, after all, and we know the plot_
_And the characters by now, which makes it_
_Ours in a way that living our own lives_
_Never does. We know ourselves_
_And each other only too well; on the other_
_Hand the action is always new, though plotless_ ,
_The same. Toes are again pointed_
_Down a sidewalk, spring_
_Is in the air and the word "brothel" floats_
_Like a ribbon in the sunset, upsetting_
_The teen balance that was never anything_
_But a continuing collapse, that brought_
_Music and minor pleasures, and some_
_Nourishment, but always rolled back the conditions_
_To that flat, narrow time before the beginning_ ,
_Kind of a sample of the horizon_
_Before there was any place for it_
_And now that it exists it seems_
_Almost tame, or not as ripe_
_As we always imagined it would be_.
_In the sea of the farm_
_The dream of hay whirls us toward_
_Horizons like those only_
_Imagined, with no space, no groove_
_Between the sky and the earth, metallic_ ,
_Unfleshed, as though, as children_ ,
_Each of us might say how good_
_He or she is, and afterwards it is forgotten_ ,
_The thought, the very words_.
_But there are times when darkness_
_Hides this not very real horizon, and it turns_
_Steadfast for us. Sprays_
_Of trees are imagined there, and they endure_
_For a while once daylight has come_ ,
_The stubborn, sticky mixture of daylight_.
_If all the retinues of all_
_The archdukes stretched away into a powdery_
_Infinity, and you stood_
_On the top step but one, waiting to advance_
_Your argument into the aura, and time suddenly_
_At that moment seemed to sag, and the staircase_
_Became a giant hammock littered with dead leaves_
_And ants, and the horizon of the universe_
_Raised it up into something bald and filled_
_With unexpressed and inexpressible menace_ ,
_No word of which would ever_
_Attest to the configuration of desires_
_That had gone into its construction, dark now_ ,
_Absent-minded flowers, reticent birds, and much_
_Else that is scarcely present, needing_
_No avenue, no way to be born_ ,
_Who would greet you? Which might be_
_What you want to tell me: open the door_.
_Your hopes and fears, ambitions, inspirations_
_Are a closed book to me. And your_
_Uneasy acceptance of what doesn't really matter_ ,
_Like a makeshift latrine, is, well_ ,
_Changed, back into remoteness by your verbs_
_Like winking dragonflies that officiate_
_So far down near the bottom of "caring"_
_As to seem interlopers, themselves_
_Displaced by later arrivals_
_That fell off the others, are part_
_And parcel, but that merely, of_
_The old, old wonderful story:_
_Grace and linearity_
_That take us up and bathe us, changing_
_The dirty colors of the little zephyrs_
_Into the next best thing: short gaffer_ ,
_Very short roses_.
_It goes without saying that I can't_ ,
_"For the life of me," figure out why we were both_
_Here. You are again listening to Haydn quartets_ ,
_Following them with the score. Afterwards_
_I wander all over you. Anyway that is the_
_Way I want you, the way things are_
_Going to be increasingly_.
"Now to my tragic business."
_The moon, in a coma, listens nevertheless_
_To all that is said. Any word we_
_May have ever uttered gets recorded and_
_Catalogued, and anybody can go and look them up_.
_The storms don't matter; even when the wind_
_Is about to demolish the roof, and the sea_
_Is banging on the front door, our words_ ,
_Even whispers, even unuttered thoughts are_
_Channeled into this cesspool of oral history_.
_You may be wondering about what comes next_.
_Never change when love has found its home_.
_Compliments of "a friend."_
_But not in our day. It sits_
_Open and limited like the yard_.
_Yet there are silent beginnings of beginnings_ ,
_Nothing but prayers, though it seems_
_That we can now feel with our minds_
_Which is someplace between prayers_
_And the answer to prayers_.
_In all these_
_Accessories of going down into day, though polished_
_And bristling, the telling of the way_
_Still fails to appear. Stopping everyone_
_Along the way for news of a long list_
_Of people, the field of folk_
_Is full of people in gentle raiment_
_Of the sun woven with the moon, and smiles_
_Half hilarious and half tragic, so that they_
_Seem specters of some cosmic romance_
_Beyond comedy and tragedy, and their love pours_
_Over the dikes and barriers that are no more_
_Now that the flood has occurred_
_And stopped, a broad and quiet ocean_
_Woven of the sun and wind and true_
_Kisses that are heartier than love_.
_Kind words are like apples of gold_
_And pitchers of silver_.
_I thought I thought I thought_
_In vain_
_At last I thought with my name_.
_Remember me now_
_Remember me ever_
_And think of the fun_
_We had together_.
_A friend_.
_I will tell you lovers, it is the little boy or sire_
_That has a present smell or word_
_For all their meat_.
_A little boy was running away_
_To be seen no more, who is now seen_
_As before, in the abstract and the particular_ ,
_The flesh and the appearance of flesh_ ,
_Who is not unlike the little boy_
_Of love, with his mama_
_The lady of love, who arranged all this_
_And who is good beyond the shadows_
_Of evil and corruption others throw_
_Into our corner but we are always beside them_.
_Some think him mean-tempered and gruff_
_But actually his is an occasion for all occasions_
_And one can get by calling anything love_
_As long as it 's locked up in the Finis_
_Of the end, and still come out ahead_.
_(This is probably the fourth most important kind of love_
_But as long as lovers still look at the moon_
_In June, weaving fingers under the moon_
_We cannot know what happens here_ ,
_Whether or not we should go away.)_
_But I'm against all forms of physical_
_Sexual activity—against billy goats, too_ ,
_Never could stand 'em. Which is why_
_It's difficult to get up in public and proclaim_
_About my cherished sorrow departing_ ,
_My appetite coming back, since all lovers_
_Are shadows projected from behind on the screen_
_Of my collective unconscious, eidolons_
_That won't say yes or no, but keep prodding_
_The ground for the treasure buried there_.
_One or two a year is all right_
_But more than that releases the shadow_
_Of throngs of passersby, of the correct object_
_And the precise moment in the sea-level street_.
_So later we come to abide_
_By the state as we remember it_
_And in dreams overhear it_
_And all our richness of invention_
_Is as physic to the evil of the surround_
_Which can't exist until we go after it_ ,
_Prove it by default_.
_Therefore I can't advance too much_
_Toward the packed, glittering crowd:_
_It dematerializes too soon and my oblivion_
_Is the cost of the precise definition there_
_Besides which no one would ever want to see it_
_In that much detail (warts and all)_
_Knowing he would have to come out that way_
_Himself one day, and turn his back on all_
_He had with such difficulty become_ ,
_A pejorative lover, alone and palely_
_Loitering, having forgotten what the object_
_Of his affection was, with only the Pavlovian_
_Reflex of loving left to try to remind him_
_What it was all like one day, how it could have_
_Been. And as we realize this_ , they
_Grow paler but more fixed, more sovereign_
_For this day and this hour, are what_
_Has been bearing down all along, the sleep_
_We have tried without success to ward off_
_All day, until the trap_
_Of night caves in under us and we emerge_
_Pellucid and dry-eyed as the others, beings made of_
_Love and time, who are to each other_
_What each is to himself_.
_I cry in the daytime_ ,
_And in the night season, and am not silent_.
_But what shall clean me within?_
_The way to nothing_
_Is the way to all things. The thoroughfare_
_That kept me inside_
_Is blocked with thurifers_
_That would lead to a different kind of life_.
_Yet all behaviors_
_Are equal in the eyes of a jade leaf_
_Prodded into history_
_But with a sense of itself and of society_
_Unequal to history_.
_History is a forest_
_In which a separate, positioned leaf_
_Could not occur_
_Leading to storms as multitudinous and varied_
_As the drops in a single storm_
_That flowers by the roadside_
_In winter, as white if taken this way_
_As an object which the mind can never_
_Control, leading to frosted silence_
_And cold unregard_.
_It is a landmark in a chain of landmarks_ ,
_Never to be harvested_.
_The_ atrocious _accident, as ascribed_
_In columns of print, refreshes_ ,
_And briefly, but the memory_
_Of its signification does not go away_.
_Instead of forgetting, we become nicer_.
_After which it is time to play_.
_The Yellow River (the river_ ,
_Not the novel by I. P. Daly) has suffered a_
_Decline in popularity, though it_
_Passes through one of the world's most_
_Populous regions. Think about it_.
_On the heights, jammed with pagodas_
_And temples, the light_
_Is starting to recede, the popularity_
_That no one wants. But in the flat_
_Depths of the gorges, the river is waning_
_On. Now no one comes_
_To disturb the murk, and the profoundest_
_Tributaries are silent with the smell_
_Of being alone. How it_
_Dances alone, in winter shine_
_Or autumn filth. It is become_
_Ingrown, and with this_
_Passes out of our existence, as we enter_
_A new chapter, confused and possibly excited_ ,
_Yet a new one, all the same_.
# _III_
_But I want him here_.
_Something is changed without him_ ,
_Something we will go on understanding_
_Until he returns to us_.
_The sunset is no reflection_
_Of its not knowing—even its knowing_
_Can be known but is not_
_A reflection_.
_Sometimes when we see another person_
_Walking down a street or_
_Standing to one side, we feel_
_We ought to go up and speak to that person_
_Because they expected to die_.
_But we do not, or seldom, speak to strangers_.
_It is forbidden_
_To have much to do with strangers_.
_We can lie, and get along in short periods_
_That way, we can go out of our house_
_To see what is there, but we can_
_Turn around and go back and not speak_
_To the others who were there_
_No matter who they were_.
_We could feel ashamed, on some days_
_That it was all brought before_
_And we in it_ ,
_That we have not known an edict, and that_
_Person knows it too. We are seldom_
_Invited by friends, and even less by strangers_.
_That is the problem of having too many friends:_
_We forget most of them, and just_
_When we need them most, they are gone_.
_We have no friends at any given_
_Moment, or they are gone away_.
_However, we do have friends when we need them_.
_They are almost always around, the shore_
_Has them. The lake recedes_
_Toward the close, pale horizon like a bench_.
_We were not asked any more_
_And now we feel we have given up on them_.
_They will never rely on us_
_Even if we were to go down, all the way down_ ,
_To them. They might not like us any more_.
_But the sunset sees its reflection, and_
_In the curve_
_Is cured. People, not all, come back_
_To us in pairs or threes. And so_
_Are festive, the light in the face_
_And all people shoo_
_You, they are back on the place_
_Of the temple, and nothing seems rustic any more_.
_They have their own perfume though_
_And it keeps growing through the mist_.
_The trees—excuse me—keep smiling—are grown_
_In the comprehensive materials_
_That swim alternately over and under_
_Never appreciating any more_
_Never stopping to think_
_Or ask why things are this way_
_And not the way you thought they_
_Were going to be_
_which would have been nicer_.
_The light of some forgotten hell_
_Leaves them in a new state of mind, begging_
_The question of growth_ ,
_Of additional dampers_.
_The prettiness urges_
_Far into the body, deep_
_Into the coffin of reactions, splitting light_
_Into two unequal portions. One_
_For me, the other for my things_
_Like my memories and the changes I'd_
_Want to introduce each time I'd come to a_
_Particular one but would turn over instead_ ,
_Disappointed with the other way it'd_
_Turn out shoveling no matter what_
_Into the boiler to keep that engine going_
_And it would all reduce to this or that other_
_Blackened memory, always the same, always_
_Healthy in spite of it. O who_
_Can judge their memories lest they have_
_Already been sized up by them?_
_But it is April now_ ,
_An air of commerce in things, and I should_
_Forget the past and think about_
_The flutes and premises of the future, and whether_
_A satisfactory sex life was one of the things_
_Included in the agenda or somebody forgot it_
_Again—just like them—_
_And the life of art_
_Matters a lot now too, is seen_
_To be perhaps the most important of all, slightly_
_Overtopping that other, and joy_
_Is after all predestined. Isn't it? I mean_ ,
_Otherwise, what the fuck are we doing_
_Here, worrying about it, having it all collapse_
_On our heads trying to dig our way out_
_Of this sand pit? No_ ,
_It's got to be preordained, in some way, by_
_Someone, otherwise we wouldn't like it_ ,
_Recognize it as it flies, and sit down casually_
_Again, knowing that, as the truth knows_
_A true story when it hears one, so we, wandering_
_Along the lake again shall hear blossoms_
_And imagine radiant blue flamingoes against the sacred sky_.
_As for those others, citizens_
_Of the great night, freaks, weirdos_ ,
_Commies and pimps: once it was all hers_ ,
_The Queen of Diamonds_ ,
_As they called her. Her real name_
_Was Rosine Esterhazy. That's what she thought_.
_Then the war was postponed_.
_The boyfriends flooded the fields_.
_She thought it was some protection_
_Nor was the great night considered especially_
_Dangerous_.
_The flower fields thriving_
_On craft items which can be made_
_At night. And for a few years, there is peace_.
_We can use this time for changing, shifting_
_Back to be a better way_
_Into ourselves. These years have become_
_A masquerade. Fine! We'll use that too_ ,
_Drinking toasts to perfect strangers_.
_When the winter is over, and the sodden spring_
_That goes on even longer, a pitcher of water_
_Drawn deep from the well is to be_
_The reward and the end of just about everything_ ,
_And joy invades all this. Makes it_
_Hard to write about_.
_Just a few letters lately, in fact_ ,
_Choruses of praise from outsiders, and I keep_
_Dropping my diary different places, forgetting_
_What I was talking about, letting it combine_
_With the loam and humus, and maybe a quick_
_Star-shape of a flower is produced. If not_ ,
_Each of us still has all our work to be done_
_In the joy of working so that the even greater joy_
_Of the hammock may be tasted later on, and so much_
_Of the padding may be appreciated then for what it is_ ,
_Just stuffing, of the kind that is needed_
_Everywhere, that keeps the Mozart symphonies_
_Apart and gradually leads us, each of us_ ,
_Back to the fragment of sense which is the place_
_We started out from. Isn't it strange_
_That this was home all along, and none of us_
_Knew it? What could our voyages_
_Have been like, that we forgot them so soon?_
_What galleons, what freighters were made to appear_
_And as sullenly to vanish in the thick foam bearing_
_Down from the horizon? What kind of a school_
_Is this, that they teach you these things_ ,
_And neglect whatever was important, that we were made to feel_
_Around for and so lost our names_
_And our dogs and were coming back, back_
_Into the commotion under the waterwheels_
_So that everything is spinning now, bears_
_Very little resemblance to what was supposed to be the entrance to the port, but is now_
_Whittled away to almost nothing?_
_But I wouldn't want you to think I_
_Cared for anything rather than go home_
_In the rain to the crafty islet_
_With the gasoline under the cellar_
_Roof. Yet betimes_
_In the morning stuck with the_
_Magic of turning into everything_
_Insane amid chimes he breathes and preaches_ ,
_Envy of all but himself Silent_ ,
_The parishioners file out, leaving the last man_
_On the quasi-tropical islet; he is left_
_As if alone again. No one cares_
_For its train—what greasy pebbles and rocks_
_It slithered over occupy_
_No one's attention any more and much_
_More is in store for the hyenas coupling_
_In the wallpaper and much less will have been_
_Noted down about this once he returns_ ,
_If ever. We clarify everything_ ,
_Throw it away and then the ranch comes_
_To devour our after-need, and what_
_Is left is of the kind no one uses_.
_Some certified nut_
_Will try to tell you it's poetry_ ,
_(It's extraordinary, it makes a great deal of sense)_
_But watch out or he'll start with some_
_New notion or other and switch to both_
_Leaving you wiser and not emptier though_
_Standing on the edge of a hill_.
_We have to worry_
_About systems and devices, there is no_
_Energy here no spleen either_.
_We have to take over the sewer plans—_
_Otherwise the coursing clear water, planes_
_Upon planes of it, will have its day_
_And disappear. Same goes for business:_
_Holed up in some office skyscraper it's_
_Often busy to predict the future for business plans_
_But try doing it from down_
_In the street and see how far it gets you! You_
_Really have to sequester yourself to see_
_How far you have come but I'm_
_Not going to talk about that_.
_I'm fairly well pleased_
_With the way you and I have come around the hill_
_Ignoring and then anointing its edge even if_
_We felt it keenly in the backwind_.
_You were a secretary at first until it_
_Came time to believe you and then the black man_
_Replaced your headlights with fuel_
_You seemed to grow from no place. And now_ ,
_Calmed down, like a Corinthian column_
_You grow and grow, scaling the high plinths_
_Of the sky_.
_Others, the tenor, the doctor_ ,
_Want us to walk about on it to see how we feel_
_About it before they attempt anything, yet_
_In whose house are we? Must we not sit_
_Quietly, for we would not do this at home?_
_A splattering of trumpets against the very high_
_Pockmarked wall and a forgetting of spiny_
_Palm trees and it is over for us all_ ,
_Not just us, and yet on the inside it was_
_Doomed to happen again, over and over, like a_
_Wave on a beach, that thinks it's had this_
_Tremendous idea, coming to crash on the beach_
_Like that, and it's true, it has, yet_
_Others have gone before, and still others will_
_Follow, and far from undermining the spiciness_
_Of this individual act, this knowledge plants_
_A seed of eternal endeavor for fear of_
_Happening just once, and goes on this way_ ,
_And yet the originality should not deter_
_Our vision from the drain_
_That absorbs, night and day, all our equations_ ,
_Makes us brittle, emancipated, not men in a word_.
_Dying of fright_
_In the violet night you come to understand how it_
_Looked to the ancestors and what there was about it_
_That moved them and are come no closer_
_To the divine riddle which is aging_ ,
_So beautiful in the eternal honey of the sun_
_And spurs us on to a higher pitch_
_Of elocution that the company_
_Will not buy, and so back to our grandstand_
_Seat with the feeling of having mended_
_The contrary principles with the catgut_
_Of abstract sleek ideas that come only once in_
_The night to be born and are gone forever after_
_Leaving their trace after the stitches have_
_Been removed but who is to say they are_
_Traces of what really went on and not_
_Today's palimpsest? For what_
_Is remarkable about our chronic reverie (a watch_
_That is always too slow or too fast)_
_Is the lively sense of accomplishment that haloes it_
_From afar. There is no need_
_To approach closely, it will be done from here_
_And work out better, you'll see_.
_So the giant slabs of material_
_Came to be, and precious little else, and_
_No information about them but that was all right_
_For the present century. Later on_
_We'd see how it might be in some other_
_Epoch, but for the time being it was neither_
_Your nor the population's concern, and may_
_Have glittered as it declined but for now_
_It would have to do, as any magic_
_Is the right kind at the right time_.
_There is no soothsaying_
_Yet it happens in rows, windrows_
_You call them in your far country_.
_But you are leaving:_
_Some months ago I got an offer_
_From Columbia Tape Club, Terre_
_Haute, Ind., where I could buy one_
_Tape and get another free. I accept-_
_Ed the deal, paid for one tape and_
_Chose a free one. But since I've been_
_Repeatedly billed for my free tape_.
_I've written them several times but_
_Can't straighten it out—would you_
_Try?_
# _Sleeping in the Corners of Our Lives_
So the days went by and the nickname caught on.
It became a curiosity, but it wasn't curious.
Afternoon leaves blew against the stale brick
Surface. Just an old castle. Enjoy it
While you're here. And in looking for a more convenient way
To save one's soul, one is led up to it like a season,
And in looking all around, and about, its tome
Becomes legible in the interstices. A great biography
That is also a good autobiography, at the station;
A honeycomb of pages with listings
Of the tried and true, that radiates
Out into what is there, that averages up as wind,
And settles back into a tepid, modest
Chamber with its mouse-gray furniture, its redundant pictures.
This is tall sleeping
To prepare you for the soup and the ruins
In giving the very special songs of the first meaning,
The ones incorporating the changes.
# _Silhouette_
Of how that current ran in, and turned
In the climate of the indecent moment
And became an act,
I may not tell. The road
Ran down there and was afterwards there
So that no further borrowing
Of criticism or the desire to add pleasure
Was ever seen that way again.
In the blank mouths
Of your oppressors, however, much
Was seen to provoke. And the way
Though discontinuous, and intermittent, sometimes
Not heard of for years at a time, did,
Nonetheless, move up, although, to his surprise
It was inside the house,
And always getting narrower.
There is no telling to what lengths,
What mannerisms and fictitious subterranean
Flowerings next to the cement he might have
Been driven. But it all turned out another way.
So cozy, so ornery, tempted always,
Yet not thinking in his 1964 Ford
Of the price of anything, the grapes, and her tantalizing touch
So near that the fish in the aquarium
Hung close to the glass, suspended, yet he never knew her
Except behind the curtain. The catastrophe
Buried in the stair carpet stayed there
And never corrupted anybody.
And one day he grew up, and the horizon
Stammered politely. The sky was like muslin.
And still in the old house no one ever answered the bell.
# _Many Wagons Ago_
At first it was as though you had passed,
But then no, I said, he is still here,
Forehead refreshed. A light is kindled. And
Another. But no I said
Nothing in this wide berth of lights like weeds
Stays to listen. Doubled up, fun is inside,
The lair a surface compact with the night.
It needs only one intervention,
A stitch, two, three, and then you see
How it is all false equation planted with
Enchanting blue shrubbery on each terrace
That night produces, and they are backing up.
How easily we could spell if we could follow,
Like thread looped through the eye of a needle,
The grooves of light. It resists. But we stay behind, among them,
The injured, the adored.
# _As We Know_
All that we see is penetrated by it—
The distant treetops with their steeple (so
Innocent), the stair, the windows' fixed flashing—
Pierced full of holes by the evil that is not evil,
The romance that is not mysterious, the life that is not life,
A present that is elsewhere.
And further in the small capitulations
Of the dance, you rub elbows with it,
Finger it. That day you did it
Was the day you had to stop, because the doing
Involved the whole fabric, there was no other way to appear.
You slid down on your knees
For those precious jewels of spring water
Planted on the moss, before they got soaked up
And you teetered on the edge of this
Calm street with its sidewalks, its traffic,
As though they are coming to get you.
But there was no one in the noon glare,
Only birds like secrets to find out about
And a home to get to, one of these days.
The light that was shadowed then
Was seen to be our lives,
Everything about us that love might wish to examine,
Then put away for a certain length of time, until
The whole is to be reviewed, and we turned
Toward each other, to each other.
The way we had come was all we could see
And it crept up on us, embarrassed
That there is so much to tell now, really now.
# _Figures in a Landscape_
What added note, what responsibility
Do you bring? Inserted around us like birdcalls
With an insistent fall. But the body
Builds up a resistance. The signs
Are no longer construed as they could have been.
The yellow chevron sails against the blue block
Of the sky, and is off. It turns tail and disappears.
Moving through much tepid machinery,
It makes more sense as it goes along.
Father and the others will be there
In their wooden jewelry, under the trees,
Since it makes sense not to quarrel
About the hole. You will perhaps see us dancing
Whom no one could ever figure out until you settled
At our feet like bushes and in the new glare
Several of the old features returned.
Without that we'd shoot back into the hills.
# _Statuary_
The prevailing winds lied in intent
The day she was given up.
The long cloth cawed from the cough cave:
First shallow groping outward, thirsty bites, more
Than heart can bestow.
You tell me I missed the most interesting part
But I think I found the most interesting part:
An unheralded departure by extinguished torchlight
Whose decorative patina
Is everything to the group—wind, fire, breathing, snores.
I was not there I was aware of Yogi Bear
There where I found a most interesting port
Crying wares to millennial crossings of voyagers
But this space is a checkerboard,
Whether it be land, sea or art
Trapped in the principle of the great beyond
Lacking only the expertise to
"Make a statement."
# _Otherwise_
I'm glad it didn't offend me
Not astral rain nor the unsponsored irresponsible musings
Of the soul where it exists
To be fed and fussed over
Are really what this trial is about.
It is meant to be the beginning
Yet turns into anthems and bell ropes
Swaying from landlocked clouds
Otherwise into memories.
Which can't stand still and the progress
Is permanent like the preordained bulk
Of the First National Bank
Like fish sauce, but agreeable.
# _Five Pedantic Pieces_
An idea I had and talked about
Became the things I do.
The poem of these things takes them apart,
And I tremble. Sparse winter, less vulnerable
Than deflated summer, the nests of words.
Some of the tribes believe the spirit
Is immanent in a person's nail parings.
They gather up their dead swiftly,
At sundown. And this will be
Some forgotten day three years ago:
Startling evidence of light after death.
Another person. The yellow-brick and masonry
Wall, deeper, duller all afternoon
And a voice waltzing, fabricating works
Of sentimental gadgetry—messes he'd cook up.
And the little hotel looked all right
And well lit, in the dark, on the flat
Beach behind the breakers, stiff, harmless.
And you are amazed that so much flimsy stuff
Stays erect, trapped in our mummery.
# _Flowering Death_
Ahead, starting from the far north, it wanders.
Its radish-strong gasoline fumes have probably been
Locked into your sinuses while you were away.
You will have to deliver it.
The flowers exist on the edge of breath, loose,
Having been laid there.
One gives pause to the other,
Or there will be a symmetry about their movements
Through which each is also an individual.
It is their collective blankness, however,
That betrays the notion of a thing not to be destroyed.
In this, how many facts we have fallen through
And still the old façade glimmers there,
A mirage, but permanent. We must first trick the idea
Into being, then dismantle it,
Scattering the pieces on the wind,
So that the old joy, modest as cake, as wine and friendship,
Will stay with us at the last, backed by the night
Whose ruse gave it our final meaning.
# _Haunted Landscape_
Something brought them here. It was an outcropping of peace
In the blurred afternoon slope on which so many picnickers
Had left no trace. The hikers then always passed through
And greeted you silently. And down in one corner
Where the sweet william grew and a few other cheap plants
The rhythm became strained, extenuated, as it petered out
Among pots and watering cans and a trowel. There were no
People now but everywhere signs of their recent audible passage.
She had preferred to sidle through the cane and he
To hoe the land in the hope that some day they would grow happy
Contemplating the result: so much fruitfulness. A legend.
He came now in the certainty of her braided greeting,
Sunlight and shadow, and a great sense of what had been cast off
Along the way, to arrive in this notch. Why were the insiders
Secretly amused at their putting up handbills at night?
By day hardly anyone came by and saw them.
They were thinking, too, that this was the right way to begin
A farm that would later have to be uprooted to make way
For the new plains and mountains that would follow after
To be extinguished in turn as the ocean takes over
Where the glacier leaves off and in the thundering of surf
And rock, something, some note or other, gets lost,
And we have this to look back on, not much, but a sign
Of the petty ordering of our days as it was created and led us
By the nose through itself, and now it has happened
And we have it to look at, and have to look at it
For the good it now possesses which has shrunk from the
Outline surrounding it to a little heap or handful near the center.
Others call this old age or stupidity, and we, living
In that commodity, know how only it can enchant the dear soul
Building up dreams through the night that are cast down
At the end with a graceful roar, like chimes swaying out over
The phantom village. It is our best chance of passing
Unnoticed into the dream and all that the outside said about it,
Carrying all that back to the source of so much that was precious.
At one of the later performances you asked why they called it a "miracle,"
Since nothing ever happened. That, of course, was the miracle,
But you wanted to know why so much action took on so much life
And still managed to remain itself, aloof, smiling and courteous.
Is that the way life is supposed to happen? We'll probably never know
Until its cover turns into us: the eglantine for duress
And long relativity, until it becomes a touch of red under the bridge
At fixed night, and the cries of the wind are viewed as happy, salient.
How could that picture come crashing off the wall when no one was in the room?
At least the glass isn't broken. I like the way the stars
Are painted in this one, and those which are painted out.
The door is opening. A man you have never seen enters the room.
He tells you that it is time to go, but that you may stay,
If you wish. You reply that it is one and the same to you.
It was only later, after the house had materialized elsewhere,
That you remembered you forgot to ask him what form the change would take.
But it is probably better that way. Now time and the land are identical,
Linked forever.
# _My Erotic Double_
He says he doesn't feel like working today.
It's just as well. Here in the shade
Behind the house, protected from street noises,
One can go over all kinds of old feeling,
Throw some away, keep others.
The wordplay
Between us gets very intense when there are
Fewer feelings around to confuse things.
Another go-round? No, but the last things
You always find to say are charming, and rescue me
Before the night does. We are afloat
On our dreams as on a barge made of ice,
Shot through with questions and fissures of starlight
That keep us awake, thinking about the dreams
As they are happening. Some occurrence. You said it.
I said it but I can hide it. But I choose not to.
Thank you. You are a very pleasant person.
Thank you. You are too.
# _I Might Have Seen It_
The person who makes a long-distance phone call
Is talking into the open receiver at the other end
The mysterious discourse also emerges as pointed
In his ear there are no people in the room listening
As the curtain bells out majestically in front of the starlight
To whisper the words This has already happened
And the footfalls on the stair turn out to be real
Those of your neighbor I mean the one who moved away
# _The Hills and Shadows of a New Adventure_
Even the most finicky would find
Some way to stand in the way.
He looked down at the ledge,
Grappling with more serious, better times.
A lady's leg crossed his mind.
Far out at sea the gulls shifted like weights.
This freshness was only a chore. In other words
The screen of lights is always there, calling
A name of vowels and then there is silence,
A burnt-out moon, our old Franklin
Parked in the yard
Under the final shade.
If there was a way to separate these objects
We feel, from these lived eventualities
That torment our best intentions
With a vision of a man bent over his desk,
Writing, communicating with the pad
Which becomes dream velvet the next time,
A moonlit city in which minorities
Fluctuate, drawing out the cultural medium
As fine as floating threads of cobwebs
Around the one ambiguous space:
Its own discoverer and name,
Named after itself,
Which is its name, and all these go into cities
Like ships behind a sea wall.
You cannot know them
Yet they are a part of you, the cold reason part
You do know about.
You were not present at the beginning
But this is not so difficult to figure out:
Messengers crying your name
In the streets of all the principal cities.
Morning. An old tractor.
It seems strange that there is no name for these
And that the night passages now seem so clear
Where you thought were only telephone wires
And the birds of strange rented buildings
In a place close to the north yet not north
With a strong smell of burlap,
A place to wait for, not in.
# _Knocking Around_
I really thought that drinking here would
Start a new chain, that the soft storms
Would abate, and the horror stories, the
Noises men make to frighten themselves,
Rest secure on the lip of a canyon as day
Died away, and they would still be there the next morning.
Nothing is very simple.
You must remember that certain things die out for awhile
So that they can be remembered with affection
Later on and become holy. Look at Art Deco
For instance or the "tulip mania" of Holland:
Both things we know about and recall
With a certain finesse as though they were responsible
For part of life. And we congratulate them.
Each day as the sun wends its way
Into your small living room and stays
You remember the accident of night as though it were a friend.
All that is forgotten now. There are no
Hard feelings, and it doesn't matter that it will soon
Come again. You know what I mean. We are wrapped in
What seems like a positive, conscious choice, like a bird
In air. It doesn't matter that the peonies are tipped in soot
Or that a man will come to station himself each night
Outside your house, and leave shortly before dawn,
That nobody answers when you pick up the phone.
You have all lived through lots of these things before
And know that life is like an ocean: sometimes the tide is out
And sometimes it's in, but it's always the same body of water
Even though it looks different, and
It makes the things on the shore look different.
They depend on each other like the snow and the snowplow.
It's only after realizing this for a long time
That you can make a chain of events like days
That more and more rapidly come to punch their own number
Out of the calendar, draining it. By that time
Space will be a jar with no lid, and you can live
Any way you like out on those vague terraces,
Verandas, walkways—the forms of space combined with time
We are allowed, and we live them passionately,
Fortunately, though we can never be described
And would make lousy characters in a novel.
# _Not Only / But Also_
Having transferred the one to the other
And living on the plain of insistent self-knowledge
Just outside the great city, I see many
Who come and go, and being myself involved in distant places
Ask how they adjust to
The light that rains on the traveler's back
And pushes out before him. It is always "the journey,"
And we are never sure if these are preparations
Or a welcome back to the old circle of stone posts
That was there before the first invention
And now seems a place of vines and muted shimmers
And sighing at noon
As opposed to
The terrain of stars, the robe
Of only that journey. You adjusted to all that
Over a long period of years. When we next set out
I had spent years in your company
And was now turning back, half amused, half afraid,
Having in any case left something important back home
Which I could not continue without,
An invention so simple I could never figure out
How they spent so many ages without discovering it.
I would have found it, altered it
To be my shape, probably in my own lifetime,
In a decade, in just a few years.
# _Train Rising Out of the Sea_
It is written in the Book of Usable Minutes
That all things have their center in their dying,
That each is discrete and diaphanous and
Has pointed its prow away from the sand for the next trillion years.
After that we may be friends,
Recognizing in each other the precedents that make us truly social.
Do you hear the wind? It's not dying,
It's singing, weaving a song about the president saluting the trust,
The past in each of us, until so much memory becomes an institution,
Through sheer weight, the persistence of it, no,
Not the persistence: that makes it seem a deliberate act
Of duration, much too deliberate for this ingenuous being,
Like an era that refuses to come to an end or be born again.
We need more night for the sky, more blue for the daylight
That inundates our remarks before we can make them
Taking away a little bit of us each time
To be deposited elsewhere
In the place of our involvement
With the core that brought excessive flowering this year
Of enormous sunsets and big breezes
That left you feeling too simple
Like an island just off the shore, one of many, that no one
Notices, though it has a certain function, though an abstract one
Built to prevent you from being towed to shore.
# _Late Echo_
Alone with our madness and favorite flower
We see that there really is nothing left to write about.
Or rather, it is necessary to write about the same old things
In the same way, repeating the same things over and over
For love to continue and be gradually different.
Beehives and ants have to be reexamined eternally
And the color of the day put in
Hundreds of times and varied from summer to winter
For it to get slowed down to the pace of an authentic
Saraband and huddle there, alive and resting.
Only then can the chronic inattention
Of our lives drape itself around us, conciliatory
And with one eye on those long tan plush shadows
That speak so deeply into our unprepared knowledge
Of ourselves, the talking engines of our day.
# _And I'd Love You To Be in It_
Playing alone, I found the wall.
One side was gray, the other an indelible gray.
The two sides were separated by a third,
Or spirit wall, a coarser gray. The wall
Was chipped and tarnished in places,
Polished in places.
I wanted to put it behind me
By walking beside it until it ended.
This was never done. Meanwhile
I stayed near the wall, touching the two ends.
With all of my power of living
I am forced to lie on the floor.
To have reached the cleansing end of the journey,
Appearances put off forever, in my new life
There is still no freedom, but excitement
Turns in our throats like woodsmoke.
In what skyscraper or hut
I'll finish? Today there are tendrils
Coming through the slats, and milky, yellowy grapes,
A mild game to divert the doorperson
And we are swiftly inside, the resurrection finished.
# _Tapestry_
It is difficult to separate the tapestry
From the room or loom which takes precedence over it.
For it must always be frontal and yet to one side.
It insists on this picture of "history"
In the making, because there is no way out of the punishment
It proposes: sight blinded by sunlight.
The seeing taken in with what is seen
In an explosion of sudden awareness of its formal splendor.
The eyesight, seen as inner,
Registers over the impact of itself
Receiving phenomena, and in so doing
Draws an outline, or a blueprint,
Of what was just there: dead on the line.
If it has the form of a blanket, that is because
We are eager, all the same, to be wound in it:
This must be the good of not experiencing it.
But in some other life, which the blanket depicts anyway,
The citizens hold sweet commerce with one another
And pinch the fruit unpestered, as they will,
As words go crying after themselves, leaving the dream
Upended in a puddle somewhere
As though "dead" were just another adjective.
# _The Preludes_
The difficulty with that is
I no longer have any metaphysical reasons
For doing the things I do.
Night formulates, the rest is up to the scribes and the eunuchs.
The reasons though were not all that far away,
In the ultramarine well under the horizon,
And they were—why not admit it?—real,
If not all that urgent.
And night too was real. You could step up
Into the little balloon carriage and be conducted
To the core of bland festival light.
And you mustn't forget you can sleep there.
Over near somewhere else there is the problem
Of the difficulty. They weave together like dancers
And no one knows anything about the problem any more
Only the problem, like the outline
Of a housewife closing her door in the face of a traveling salesman
Throbs on the air for some time after.
Perhaps for a long time after that.
O we are all ushered in—
Into the presence that explains.
# _A Box and Its Contents_
Even better than summer, but I no longer
Aim a poem at you, center of the forest at night,
One shoe off and one shoe on, half-nubile, old.
The excited ashes of your tale, always telling, more telling
Until the day we get it right,
A day of thoughtful joy. You said if it's all right
To do it then there will be animals sleeping under the trees anyway.
You come out of love. But are. The treasure they
Were firing at was always yours anyway, you meant
To stand for it. Now there is no way down. But we
Children of that particular time, we always get back down.
You see, only some of the others were crying
And how your broad smile paints in the wilderness
A scene of happiness, with balloons and cars.
It was always yours to dig into, and you can't, loving us.
# _The Cathedral Is_
Slated for demolition.
# _I Had Thought Things Were Going Along Well_
But I was mistaken.
# _Out Over the Bay the Rattle of Firecrackers_
And in the adjacent waters, calm.
# _We Were on the Terrace Drinking Gin and Tonics_
When the squall hit.
# _Fallen Tree_
We do not have it, and they
Who have it are plunged in confusion:
It is so easy not to have it, the gold coin, we know
The contour of having it, a pocket
Around space that is an endless library
Where each book follows in a divinely ordered procession,
Like the rays of the sun.
Yet it was the pageant that you never wanted
But which you need now to make sense of the strengthening
Of the mounting days that begin to form a vault
Above this ancient red stage.
The days proceed.
Each is good in his role,
Very clever, in fact. But it is up to you
To make sense of what each has done.
Otherwise, in the rain-washed fiasco—
Twilight? A coming triumph? Or some other
Diversion you haven't yet learned to recognize?—
We shall never recognize our true reflections,
Speaking to them as strangers, scolding,
Asking the time of day.
And the love that has happened for us
Will not know us
Unless you climb to a median kingdom
Of no climate
Where day and night exist only for themselves
And the future is our table and chairs.
# _The Picnic Grounds_
Let the music tell it:
It came here, was around for a little while,
And left, like the campers,
Leaving fire-blackened brick, wrappers of things
And especially monster mood and emptiness
Of those who were here and are gone.
A complex, but optional, experience.
Will the landscape mean anything new now?
But even if it doesn't, the charge
Is up ahead somewhere, in the near future,
Squashing even the allegory of the grass
Into the mould of its aura, a lush patina.
So we, with all our high-minded notions
Of the self and the eventually winged purpose
Of that self, are now meaning
The raw material of the days and the ways that came over.
The shadow has been indefinitely postponed.
And the shape it takes in the process
Of definition of the evolving
Delta of shapes is too far, far in the milky limpid
Future of things. Too far to care, yet
There are those who do care for that
Kind of outline, distant, yes, but warm,
Full of the traceable meaning that never
Gets adopted. Well, isn't that truth?
# _A Sparkler_
The simple things I notice:
That they were coming at us, were at us, and _were_ us
In this night like rotten mayonnaise I am afraid of
(It is helping me out) and steady boys
I want no one to latch onto
This time it has a special snap
And how it curved outward that time was more elaborate
But in the end got fuzzier
And at the same time more deducible
An illuminated word entered its crucible
But just once come back see it the way
I now see it
Sit fooling with your hair
Looking at me out of the corner of your eye
I'm so sorry
For what we haven't done in the time we've known each other.
Then it's back to school
Again yes the sales are on.
What do you need? We'll try ...
Or is it all just a symbol of bad taste,
Of a bad taste in the mouth? I tried,
Not hard but pretty regular. But the pitch was
Elsewhere, parallel. The habitués would have
Had it, entertained it anyway,
But I was in disgrace. I lived in disgrace.
I was no one on that lawn.
But, lasting by lasting,
And by no other moment, we have come down
At last to where the plumbing is.
We had hoped for a dialogue.
But they're rusty.
Then is it too late for me?
The wide angle that seeks to contain
Everything, as a sea, is an eye.
What is beheld is whatever lives,
Is wildly unappetizing and inappropriate,
And sits, and fits us.
# _The Wine_
It keeps a large supply of personal pronouns
On hand. They awaken to see
Themselves being used as it grows up,
Confused, in a rush of fluidity.
Once men came back here to rot.
Now the salt banners only interrupt the sky—
Black crystals, quartzite. The balm of not
Knowing living filters to the bottom of each eye.
The telephone was involved in it. And bored
Glances, boring questions about the hem no
One wanted to look at, or would admit having seen.
These things came after it was a place to go.
Yet nothing was its essence. The core
Remained as elusive as ever. Until the day you
Fitted the unlikely halves together, and they clicked.
So its wholeness was an order. But it had seemed not to
Be part of the original blueprint, the way
It had appeared in intermittent dreams, stretching
Over several nights, like that. But that was okay,
Providing the noise factor didn't suddenly loom
Too large, as was precisely happening just now.
Where have I seen that face before? And I see
Just what it means to itself, and how it came
Down to me. And so, in like manner, it came to be.
# _A Love Poem_
And they have to get it right. We just need
A little happiness, and when the clever things
Are taken up (O has the mouth shaped that letter?
What do we have bearing down on it?) as the last thin curve
("Positively the last," they say) before the dark:
(The sky is pure and faint, the pavement still wet) and
The dripping is in the walls, within sleep
Itself. I mean there is no escape
From me, from it. The night is itself sleep
And what goes on in it, the naming of the wind,
Our notes to each other, always repeated, always the same.
# _There's No Difference_
In pendent tomes the unalterable recipe
Is decoded. Then, a space,
And another space. I was consulting
The surface of the wand
While you in white painter's pants adored
A sunflower, hoping it would shit across the nation.
The explosion taught us to read again.
Do not remember why everything is unsavory
That in the night a pineapple came
For this poster is nominally a conjecture.
# _Distant Relatives_
Six o'clock. The fast fragrance
Is clawing past me, frantic to be let out,
Not competent to stand trial.
Like trees on a golf course
These hours propose themselves, one by one,
And each comes to terms with roundness.
The bobbed heads bob. The silence
For once is melony, sweet as the light
Off parked cars.
I don't need one of the hand-held jobs,
A heavy machine will do. And I must put across
Right now my idea of what it will do for me, before
It too founders in the tolling of leaves
(If all the tongues of all the bells
In this city fluttered silently)
As in that movie we saw where Mouloudji ...
What will he do with it?
1. I don't get it.
2. It may not be worth it.
However the distances, it so happens, come to seem
Like partitions, both near and far:
Near, starting where my shoe is, and far
Ahead in the perspective, but connected
As the hours are connected to minutes
And I still feel the absence of you
As a thing that is both negative and positive
Like the broken mould of a lost
Statue
As the din becomes an uproar.
# _Histoire Universelle_
As though founded by some weird religious sect
It is a paper disk, partially lit up from behind
With testaments to its cragginess, many of them
Illegible, covering most of its surface. In the hours
Between midnight and 4 A.M. it assumes a fitful
But calm sedentary existence, and it is then that
You may reach in and take out a name, any name,
And it will be your own, at least while
The walls of Bill's villa resonate with the intermittent,
Migraine-like drone of motorized gondolas and the distant
Murmur of cats. To be treated, at times like these,
To free speech is an aspect of the dream and of Dreamland
In general that asserts an even larger
View of the universe pinned on the midnight-blue
Backcloth of the universe that can't understand
Who all these people are, and about what
So much fuss is being made: it ignores its own entrails
And we love it even more for it until we too
Are parted like curtains across the empty stage of its memory.
The house was for living in,
So much was sure. But when the ways split
And we saw out over what was after all
Water and dawn, and prayed to the rocks
Overhead, and no answer was forthcoming,
It was then that the cosmic relaxer released us.
We were together on such a day. You, oddly
But becomingly dressed, pointed out that that
Day is today, the moral. All that.
# _Hittite Lullaby_
This time for you
The hair-blackened beans
And next semester the shouting
In carpeted corridors
More letters from the Sphinx
About what it was like
I greet you. I call to you
To release me from the contract
Morning flaps like a garment
Over a corner of the city
In mistrust with tears streaming
I can see clearly to know precisely
What is meant. My tact merely
A delaying stratagem
Is all I have. The sunlight
On your broad feet today
Withheld smiling.
Why did we board the ocean liner
Of lust signals out into the fog
Knowing there were excursions
But not this big one? My dog
Has died, I think. I come on you
All aspirations in the teeth
Of some pedantic ritual.
You take me where we were born.
# _In a Boat_
Even when confronted by the small breakwater
That juts out from the pebbled shore of truth
You arch your eyebrows toward the daytime stars
And remind me, "This is how I was. This was the last
Part of me you were to know." And I can see the lot
Ending in the wood of general indifference to hostility
That wants to know how with two such people around
So much is finishing, so much rushing through the present.
There was a tag on the little sailboat
That idled there, all its sails rolled up
As tightly as umbrellas. What difference?
The orange shine stood off, just far enough away
Not to catch the commas and puns as you spoke
This time in defense of riders of the squall,
Of open-faced daring, not just to the empty seas
But for the people swathed in oilcloth on the beach.
"It is no great matter to take this in hand, convince
The tips of the trees they were rubbing against each other
All along. Each contrives to slip into his own hall of fame
And my common touch has triumphed. The doorpost shall turn again and again."
# _Variations on an Original Theme_
Our humblest destinies amount to this:
A maze of leaves, and one who sat
Within them dreaming of plants and their syrups
(Because of the yellow rings and zigzags
Visited on the moss-grown turret walls)
And a hare running far away, in the blond night.
And to dream of having sex with my beloved
Brings the figured wall no closer:
A fleet of pleasure boats and shadow
Dipping over them, lost
To the righteous eye brooding expensively
On tomorrow's fabric, how it overflows
Where there are no kick-pleats, and thins,
And what is wasted comes back anyway.
A ride in common variety
Was all it ever got to be; there are no friends
To make it serve. Only sometimes, a promised
Stranger makes us see it in another light
As though we have been standing here always,
Lake to the right, and the house, a Manichaean
Presence between the two widely spaced trees
On the backed-up, rusted gold of the grass.
And setting out in the punt on a larger
Stream and returning just in time
For the oracle, these things had not yet
Begun to dream, and there was thus no questioning
Of them yet. What was one day to be
Removed itself as far as possible from scrutiny.
We got down to the business of preparing
For the night only to find it prepared
For us as a bride, a flag rolled in the darkness,
Now no longer comfort, a spirit only.
# _Homesickness_
The deep water in the travel poster finds me
In the change as I was about to back away
From the idea of the comedy around us—
In the chairs. And you too knew how to do the job
Just right. Trumpets in the afternoon
And you first get down to business and
The barges disappear, one by one, up the river.
One of them must be saved for a pirate. But no,
The park continues. There is no space between the leaves.
Once when there was more furniture
It seemed we moved more freely not noticing things
Or ourselves: our relationships were wholly articulate
And direct. Now the air between them has thinned
So that breathing becomes a pleasure, an unconscious act.
Then when you had finished talking about the trip
You had planned, and how many days you were to be away
I was looking into the night forests as I held
The receiver to my ear, replying correctly
As I always do, to everything, having become the sleeper in you.
It no longer mattered that I didn't want you to go away,
That I wanted you to return as quickly as possible
To my house, not yours this time, except
This house is yours when we sleep in it.
And you will be chastised and purified
Once we are both inside the world's lean-to.
Our words will rise like cigarette smoke, straight to the stars.
# _This Configuration_
This movie deals with the epidemic of the way we live now.
What an inane cardplayer. And the age may support it.
Each time the rumble of the age
Is an anthill in the distance.
As he slides the first rumpled card
Out of his dirty ruffled shirtfront the cartoon
Of the new age has begun its ascent
Around all of us like a gauze spiral staircase in which
Some stars have been imbedded.
It is the modern trumpets
Who decide the mood or tenor of this cross-section:
Of the people who get up in the morning,
Still half-asleep. That they shouldn't have fun.
But something scary will come
To get them anyway. You might as well linger
On verandas, enjoying life, knowing
The end is essentially unpredictable.
It might be soldiers
Marching all day, millions of them
Past this spot, like the lozenge pattern
Of these walls, like, finally, a kind of sleep.
Or it may be that we are ordinary people
With not unreasonable desires which we can satisfy
From time to time without causing cataclysms
That keep getting louder and more forceful instead of dying away.
Or it may be that we and the other people
Confused with us on the sidewalk have entered
A moment of seeming to be natural, expected,
And we see ourselves at the moment we see them:
Figures of an afternoon, of a century they extended.
# _Metamorphosis_
The long project, its candling arm
Come over, shrinks into still-disparate darkness,
Its pleasaunce an urn. And for what term
Should I elect you, O marauding beast of
Self-consciousness? When it is you,
Around the clock, I stand next to and consult?
You without a breather? Testimonials
To its not enduring crispness notwithstanding,
You can take that out. It needs to be shaken in the light.
To be delivered again to its shining arm—
O farewell grief and welcome joy! Gosh! So
Unexpected too, with much else. Yet stay,
Say how we are to be delivered from the fair content
If all is in accord with the morning—no prisms out of order—
And the nutty context isn't just there on a page
But rolling toward you like a pig just over
The barges and light they conflict with against
The sweep of low-lying, cattle-sheared hills,
Our plight in progress. We can't stand the crevasses
In between sections of feeling, but knowing
They come once more is a blessed decoction—
Is their recessed cry.
The penchant for growing and giving
Has left us bereft, and intrigued, for behind the screen
Of whatever vanity he chose to skate on, it was
Us and our vigilance who outlined the act for us.
We were perhaps afraid, and less purposefully benevolent
Because the chair was placed outside, the chair
No one would come to sit in, except the storm,
If it ever came. No shame, meanwhile,
To sit in the hammock, or wherever straw was
To see it and acclaim the differences as they were born.
And we were drunk as flowers
That should someday be, or could be,
We weren't keeping track, but just then
It all turned the corner into a tiny want ad:
Someone with something to sell someone
And the stitches ceased to make sense.
They climb now, gravely, with each day's decline
Farther into the unmapped sky over the sunset
And prolong it indelicately. With maps and whips
You came eagerly, we were obedient, and then, just then
The real big dark business got abated, and I
Awoke stretched out on a ladder lying on the cold ground,
Too upset and confused to imagine how you
Had built the colossal staircase in my flesh that armies
Were using now, their command a curse
As all my living swept by, the flags curved with stars.
# _Their Day_
Each act of criticism is general
But, in cutting itself off from all the others,
Explicit enough.
We know how the criticism must be done
On a specific day of the week. Too much matters
About this day. Another day, and the criticism is thrown down
Like trash into a dim, dusty courtyard.
It will be built again. That's all the point
There is to it. And it is built,
In sunlight, this time. All look up to it.
It has changed. It is different. It is still
Cut off from all the other acts of criticism.
From this it draws a tragic strength. Its greatness.
They are constructing pleasure simultaneously
In an adjacent chamber
That occupies the same cube of space as the critic's study.
For this to be pleasure, it must also be called criticism.
It is the very expensive kind
That comes sealed in a bottle. It is music of the second night
That winds up as if to say: Well, you've had it,
And in doing so, you have it.
From these boxed perimeters
We issue forth irregularly. Sometimes in fear,
But mostly with no knowledge of knowing, only a general
But selective feeling that the world had to go on being good to us.
As long as we don't know that
We can live at the square corners of the streets.
The winter does what it can for its children.
# _A Tone Poem_
It is no longer night. But there is a sameness
Of intention, all the same, in the ways
We address it, rude
Color of what an amazing world,
As it goes flat, or rubs off, and this
Is a marvel, we think, and are careful not to go past it.
But it is the same thing we are all seeing,
Our world. Go after it,
Go get it boy, says the man holding the stick.
Eat, says the hunger, and we plunge blindly in again,
Into the chamber behind the thought.
We can hear it, even think it, but can't get disentangled from our brains.
Here, I am holding the winning ticket. Over here.
But it is all the same color again, as though the climate
Dyed everything the same color. It's more practical,
Yet the landscape, those billboards, age as rapidly as before.
# _The Other Cindy_
A breeze came to the aid of that wilted day
Where we sat about fuming at projects
With the funds running out, and others
Too simple and unheard-of to create pressure that moment,
Though it was one of these, lurking in the off-guard
Secrecy of a mind like a magazine article, that kept
Proposing, slicing, disposing, a truant idea even
In that kingdom of the blind, that finally would have
Reined in the mad hunt, quietly, and kept us there,
Thinking, not especially dozing any more, until
The truth had revealed itself the way a natural-gas
Storage tank becomes very well known sometime after
Dawn has slipped in
And seems to have been visible all along
Like a canoe route across the great lake on whose shore
One is left trapped, grumbling not so much at bad luck as
Because only this one side of experience is ever revealed.
And that meant something.
Sure, there was more to it
And the haunted houses in those valleys wanted to congratulate
You on your immobility. Too often the adventurous acolyte
Drops permanently from sight in this beautiful country.
There is much to be said in favor of the danger of warding off danger
But if you ever want to return
Though it seems improbable on the face of it
You must master the huge retards and have faith in the slow
Blossoming of haystacks, stairways, walls of convolvulus,
Until the moon can do no more. Exhausted,
You get out of bed. Your project is completed
Though the experiment is a mess. Return the kit
In the smashed cardboard box to the bright, bland
Cities that gave rise to you, you know
The one with the big Woolworth's and postcard-blue sky.
The contest ends at midnight tonight
But you can submit again, and again.
# _No, But I Seen One You Know You Don't Own_
Only sometimes is the seam in the way
Of space broken and three schedules cross:
The seasonable cold raging to be pliant and tit
Of gold.
He walks backward on the conveyor belt
As the blue powder of the day is dismissed
And he might pull the switch that would release
The immobile Niagaras that hover in the background.
There is no need, finally,
To inspissate the corded torsades
Of his loon voice. The dragees arrive in fumes:
The reprisal spinning through the air
Like an incandescent boomerang
As small flowers spring up at the feet
Of the near beasts, and in the distance
The hills are shrouded like shoulders
Behind the definitive errand of this glance.
# _The Shower_
The water began to fall quite quickly
Just wanting to be friendly.
It's too macho, and the sides and the plains get worse.
What are you writing?
Thus incurring a note for the milkman
City unit buses pass through. A laborer
Dragging luggage after cashing the king and ace of
It sifted slowly along the map, trying the lips,
The defender's last trick.
Somewhere in the grotto it festered,
The summer was cast in a circle. Knots
There were to see, knot by knot
But almost as much as is your punishment again.
By ruffing the third club defender would be
Just a fat man in sunglasses
That knots caress, moving
Through shine—the uncle in the mirror—
As it is beginning again these are the proportions.
Instead the place,
Where we had been before, got tangled
Within us, forced
To break out so that no one knew
The stalks from the knot of pleasure
And it would be determined to happen again—
Said this, through rain and the shine
That comes after, so many opinions
And words later, so many dried tears
Loitering at the sun's school shade.
# _Landscapeople_
Long desired, the journey is begun. The suppliants
Climb aboard the damaged carrousel:
Some have been hacked to death, one has learned
Some new thing, and all are touched
With the same blight, like a snowfall
Of moments as they are read back to the monitor
Which only projects.
Some can decipher it,
The outline of an eddy that traced itself
Before moving on, yet its place had to be,
Such was the appetite of those times. A ring
Of places existed around the central one,
And of course these died away eventually.
Everything has turned out for the best,
The "eggs of the sun" have been returned anonymously,
And the new ways are as simple as the old ones,
Only more firmly anchored to the spectacle
Of the madness of the seasons as it unfolds
With iron-clad rigidity, filling the sky with light.
We began in an anonymous sensuality
And lived most of it out before the difference
Of time got in the way, filling up the margins of the days
With pictures of fruit, light, colors, music, and vines,
Until it ceases to be a problem.
# _The Sun_
The watermark said it was alone with us,
"To do your keeping and comparing." But there were bushes
On the horizon shaped like hearts, spades, clubs and diamonds.
They were considered
To belong to a second class, to which lower standards
Were applied, as called for in the original rule,
And these standards were now bent inward to become
The invariable law, to which exceptions
Were sometimes apposite, and they liked the new clime,
So bracing here on the indigo slopes
To which families of fathers and daughters have come
Summer after summer, decade after decade, and it never stops
Being refreshing. It is a sign of maturity,
This stationary innocence, and a proof
Of our slow, millennial growth, ring after ring
Just inside the bark. Yet we get along well without it.
Water boils more slowly, and then faster
At these altitudes, and slowness need never be something
To criticize, for it has an investment in its own weight,
Rare bird. We know we can never be anything but parallel
And proximate in our relations, but we are linked up
Anyway in the sun's equation, the house from which
It steals forth on occasion, pretending, isn't
It funny, to pass unnoticed, until the deeply shelving
Darker pastures project their own reflection
And are caught in history,
Transfixed, like caves against the sky
Or rotting spars sketched in phosphorus, for what we did.
# _The Plural of "Jack-in-the-Box"_
How quiet the diversion stands
Beside my gate, and me all eager and no grace:
Until tomorrow with sifting hands
Uncode the sea that brought me to this place,
Discover people with changing face
But the way is wide over stubble and sands,
Wider and not too wide, as a dish in space
Is excellent, conforming to demands
Not yet formulated. Let certain trends
Believe us, and that way give chase
With hounds, and with the hare erase
All knowledge of its coming here. The lands
Are fewer now under the plain blue blanket whose
Birthday keeps them outside at the end.
# About the Author
John Ashbery was born in 1927 in Rochester, New York, and grew up on a farm near Lake Ontario. He studied English at Harvard and at Columbia, and along with his friends Frank O'Hara and Kenneth Koch, he became a leading voice in what came to be called the New York School of poets. Ashbery's poetry collection _Some Trees_ was selected by W. H. Auden as the winner of the Yale Series of Younger Poets prize in 1955—the first of over twenty-five critically admired works Ashbery has published in a career spanning more than six decades. His book _Self-Portrait in a Convex Mirror_ (1975) received the Pulitzer Prize for Poetry, the National Book Critics Circle Award, and the National Book Award, and since then Ashbery has been the recipient of a MacArthur Fellowship, a National Humanities Medal, the Ruth Lilly Poetry Prize, and a Gold Medal for Poetry from the American Academy of Arts and Letters, among other honors.
For years, Ashbery taught creative writing at Brooklyn College and Bard College in New York, working with students and codirecting MFA programs while continuing to write and publish award-winning collections of poetry—all marked by his signature philosophical wit, ardent honesty, and polyphonic explorations of modern language. His most recent book of poems is _Quick Question_ , published in 2012. He lives in New York.
All rights reserved, including without limitation the right to reproduce this ebook or any portion thereof in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.
Grateful acknowlegment is made to the following publications, in which some of the poems in this book originally appeared : _American Poetry Review_ : "Litany" (Part II). _Broadway_ : "Otherwise." _Cincinnati Poetry Review:_ "And I'd Love You to Be in It." _Contemporary Poets Calendar_ (1980): "A Tone Poem." _Harvard Magazine_ : "The Picnic Grounds" and "Their Day." _New York Review of Books_ : "My Erotic Double" and " _Histoire Universelle_." _The New Yorker:_ "Haunted Landscape," "Knocking Around," and "Tapestry." _Paris Review_ : "Homesickness" and "This Configuration." _Ploughshares:_ "No, But I Seen One You Know You Don't Own," "The Shower," "Landscapeople," and "The Plural of 'Jack-in-the-Box'" (as part of the sequence " _Kannst du die alten Lieder noch Spielen?"). Poetry:_ "Many Wagons Ago," "The Sun," "Five Pedantic Pieces," "Flowering Death," "Not only /but also," "Train Rising out of the Sea," and "Late Echo." _Vogue_ : "As We Know." _The World_ : "Sleeping in the Corners of Our Lives" and "In a Boat." _Zero_ : "Variations on an Original Theme" and "The Other Cindy."
"The Preludes," "A Box and Its Contents," "A Sparkler," "The Wine," "There's No Difference," "Hittite Lullaby," "No, But I Seen One You Know You Don't Own," "The Shower," "Landscapeople," and "The Pleural of 'Jack-in-the-Box'" appear in _Solitary Travelers_ , a volume of Mellon Lectures published by Cooper Union.
Copyright © 1979 by John Ashbery
Cover design by Mimi Bark
978-1-4804-5905-2
This edition published in 2014 by Open Road Integrated Media, Inc.
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New York, NY 10014
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\section{Introduction}\label{s:intro}
Fluid flows are predominantly multiscale phenomena occurring over a wide range of length and time scales such as transition \cite{Edstrand:18parallel}, turbulence \cite{Wu:17transitional} and flow separation \cite{Deem:18Exp}. Direct numerical simulation (DNS) of such realistic high Reynolds number flows even in their canonical forms is a challenge even with current computing capacity. On the other hand, advances in experimental techniques for visualization and data acquisition have led to an abundance of fluid flow measurement data, but these measurements are often sparse and in many cases the underlying phenomenology or governing model is not known. In both these cases, there is a need for efficient data-driven models to serve the twin goals of (i) system modulation to achieve desired effects, i.e. flow control \cite{Kim:07ARev,Brunton:15CLT} or (ii) forecasting for informed decision making\cite{Cao:07Reduced,Fang:09POD,Benner:15survey} or both.
Additionally, data-driven models also allow for extraction of dynamical and physical characteristics to generate insight~\cite{Bagheri:13,Rowley:09} into the system behavior. In flow control applications linear operator based control is often preferred so that one can leverage the expertise accumulated from the past~\cite{Rowley:17ARev}. Consequently, learning a linear system model is attractive as evidenced by voluminous recent literature in this area~\cite{SchmidDMD:10,Williams:15} including that from our team \cite{Lu:18sparse}. However, such methods have their inherent limitations and perform inadequately with small amounts of data. In this paper, we explore the potential of machine learning frameworks for nonlinear function representations to extend the horizon of prediction for canonical fluid flows. Particularly for this article, we explore bluff body wake flows and buoyancy-driven mixing.
A good data-driven model should perform well in both system identification and prediction using limited amounts of data. In addition, these models need to be computationally tractable which makes dimensionality reduction essential. System identification enables learning of stability and physical characteristics such as unstable modes and coherent structures. For example, proper orthogonal decomposition (POD) \cite{Lumley:70POD} via singular value decomposition (SVD) \cite{Trefethen:97} and its close cousin, the Dynamic mode decomposition (DMD) \cite{SchmidDMD:10} are well known methods to extract such relevant spectral information. However, the capacity of DMD for long-term prediction is underwhelming~\cite{Lu:17,Lu:18sparse}. POD-based methods that use Galerkin projection onto the flow governing equations are more successful as long as the basis remains relevant to the flow evolution, but require knowledge of the system. In this study, we focus on purely data-driven scenarios without knowledge of governing equations. By long-time predictions, we imply evolving the system model over multiple characteristic time-scales beyond the training regime. The other prominent use of such models is to forecast the system evolution along different trajectories. Obviously, the precise definitions of 'long-time' prediction or forecasting is physics dependent. For example, a limit-cycle system evolving on a stable attractor will be more amenable to prediction from limited data as compared to more complex nonlinear mixing dynamics. In the case of cylinder wake flow explored in this study, forecasting represents predicting the limit cycle \cite{Berkooz:93POD,Noack:03hierarchy} dynamics using limited data in the transient unstable wake region. We explore such cases as they are sensitive to error growth and hence, used to evaluate a given model. Errors in model learning can be attributed to limited training data, measurement noise, model over fitting and insufficient validation~\cite{bishop1995neural,christopher2016pattern}. The contribution from this paper is a systematic exploration and assessment of how nonlinear regression-based data-driven models perform relative to commonly used linear regression models for dynamically evolving fluid flows.
There are two classes of approaches for modeling dynamical systems from limited data, namely Markov and non-Markov models. For a given current state $\pmb{x}_t$ and future state $\pmb{x}_{t+T}$ of a dynamical system, a Markov model \cite{Wu:17var}, under some transformation $g,\ h$, evolves the system state as $ g(\pmb{x}_{t+T}) = \mathcal{K} h(\pmb{x}_t)$. Learning such an operator $\mathcal{K}$ is the key to building such models. Markovian processes are minimally memory dependent and popular approaches for modeling such systems include dynamic mode decomposition (DMD) \cite{SchmidDMD:10,Rowley:09} and Feed forward neural networks (FFNNs). Recently, linear operator \cite{Rowley:17ARev,Mezic:05,Williams:15,Lu:18sparse} methods for modeling nonlinear dynamics have been related to the Koopman operator \cite{Koopman:31} theoretic framework.
The Koopman approximation-based methods are a special case of Markov models that employ symmetric transformations of the input and out to the same feature space (i.e. $g = h$). On the other hand, if the model incorporates copious amounts of memory of the state variables to predict a future state, then it is considered non-Markovian. Recurrent neural networks(RNN) are good examples of non-Markovian models and have been employed for learning dynamical systems both in the past~\cite{Hopfield:82PNAS,Hochreiter:97LSTM} and in recent times~\cite{Soltani:16,yu:learning}. Although these have shown success, they are very hard to build and train~\cite{Bengio:94IEEE} as compared to standard feed forward neural networks (FFNNs)~\cite{bengio2015deep}. This is because, the standard backpropagation-based algorithms can lead to exploding or vanishing gradient problems.
\par While Markov models are popular, especially the linear variants, their success is often tied to two aspects: (i) the ability of the projection or maps to the feature space \cite{Rowley:17ARev,Taira:17aiaa,Lu:18sparse} to accurately map data without loss of information while incorporating the appropriate degree of nonlinearity and (ii) their ability to capture the evolution of the dynamics in the feature space \cite{Lu:18sparse}. This renders many such learning methodologies into an exercise in identifying the optimal `magic' feature maps. A common approach to building such nonlinear map operators is to layer multiple `elementary' maps \cite{Williams:14arXiv,Williams:15,Lu:18sparse}. While DMD \cite{SchmidDMD:10,Rowley:09} employs a single-layer map operator based on singular value decomposition (SVD) of the training data, its multilayer variant EDMD \cite{Williams:15} layers a second nonlinear functional map over the SVD. This approach is effective if one knows the nature of the nonlinearity {\it a priori}, but often results in a high-dimensional feature space. The kernel variant of this method, KDMD \cite{Williams:14arXiv} helps reduce dimension, but limited by the approximation capabilities of the kernel function.
A major limitation of all such multilayer methods is related to the sequential learning of the feature maps independent of each other, i.e. learning occurs through local features as a one time-measure in a specified direction and the upstream map isnot adjusted for a downstream map. This `one-way and one-time' learning process limits the representational capacity of the model for handling nonlinear fluid flows. Deep neural networks (DNNs) have been employed to identify multilayer maps without such limitations. Particularly, such DNN-based multilayer maps have been used to embed the nonlinear dynamical system into a Koopman function space~\cite{Shiva:18AIAA,Otto:17KDN,Lusch:17KDN} governed by linear dynamics \cite{Mezic:05}. They are expected to provide improved performance due to the `two-way and iterative' process of learning the model so that the optimal nonlinear multilayer map can be discovered instead of the assumed structure. In this paper we term the former as \emph{Multilayer Sequential Maps} or (MSMs) and the latter as \emph{Multilayer End-to-end Maps} (MEMs).
In this work, we carefully and systematically asses the predictive performance of both the sequential (MSM) and end-to-end (MEM) learning of Markov models of complex nonlinear dynamics using limited data. In particular, for the sequential maps (MSMs) we restrict ourselves to the popular class of Koopman approximation methods such as Dynamic Mode Decomposition~\cite{SchmidDMD:10,Rowley:09} and its extensions~\cite{Williams:15}. For the end-to-end learning architectures (MEMs) we focus on different types of feed forward neural networks (FFNN), a robust approach for learning the embedded nonlinearity in the dynamics from data. In all the case studies considered, the maps are carefully chosen so as to minimize variability so that we can focus purely on the effect of the \emph{sequential versus end-to-end optimization} on the learning of the dynamics. Proper orthogonal decomposition (POD) is used as the first layer in all the above architectures in order to operate in a low-dimensional feature space.
The outcomes of our study indicate that MSMs in spite of including multiple layers behave more like shallow neural networks while MEMs carry the advanced function approximation capabilities of deep learning tools. It is well known that while shallow NN are known to possess universal function approximation properties \cite{Hornik:89}, it usually requires exponentially more neurons (features) for accurate prediction as compared to deeper architectures. Deep neural networks (DNNs) offer a low-dimensional (short) and layered (deep) alternative for high (almost exponential) representational capacity of complex data. This low-dimensional feature space also helps limit overfitting in a relative sense, i.e. as compared to MSMs. In particular, we observed that for a similar architecture, i.e same number of layers and feature dimension, MEMs offer robust and accurate learning performance using the same training data as compared to MSMs by leveraging an extended learning parameter space with elements estimated concurrently using nonlinear regression techniques. Similar performance from MSMs require very `tall' layers that cause overfitting. These ideas are illustrated using different flow case studies including transient dynamical evolution of a cylinder wake towards a limit-cycle attractor and a transient buoyancy-driven mixing layer. The organization of this paper is as follows.
In section \ref{s:methods} we present an overview of data-driven Markov models for transient dynamical systems and their connections to neural networks (section~\ref{ss:markovNN}) and Koopman theoretic methods (section \ref{ss:markovkoopman}). In section \ref{ss:markovLOC}, we describe multilayer sequential maps (MSM) for Markov modeling and its two variants in subsections \ref{sss:DMDLOC} and \ref{sss:EDMDLOC}. In section~\ref{ss:markovGOC} we introduce feed forward neural network based Markov representations. The numerical examples and discussion of the modeling performance is presented in section \ref{s:results} and the various outcomes are summarized with discussion in section \ref{s:conclusions}.
\section{Data-driven Markov Models for Transient Dynamical Systems} \label{s:methods}
\par Extraction of high-fidelity Markov models from snapshot (time) data of nonlinear dynamical systems is a major need in science and engineering, where measurement data can be the only available piece of information. It is advantageous to learn the model in a low-dimensional feature space to both simplify the learning process and also improve efficiency. Most Markov models are built as linear operators in the feature space to take advantage of the powerful linear systems machinery for control \cite{Kim:07ARev}, optimization and spectral analysis \cite{Rowley:09} although this is not necessary for the following formulation.
Given a discrete-time dynamical fluid system that evolves as below:
\begin{equation}
\pmb y = \pmb x_{t+T}= \pmb{F}(\pmb x_t)= \pmb{F}(\pmb x)
\label{e:nonlinear}
\end{equation}
where ${\pmb x},{\pmb y} \in \mathcal{M}$ are $N$-dimensional state vectors ($\mathbb{R}^N$), e.g., velocity components at discrete locations in a flow field at a current instant $t$, and separated by an appropriate unit of time $T$. To be explicit, ${\bm x} \triangleq {\bm x_t}$ and ${\pmb y} \triangleq {\pmb x}_{t+T}$. Operator $\pmb {F}$ evolves the dynamical system nonlinearly from $\pmb x$ to $\pmb y$, i.e. $\pmb {F}: \mathcal{M}\to \mathcal{M}$. This representation can easily be made relevant to continuous time systems as well in the limit $t\rightarrow0$. A general (linear) Markov description of such a dynamical system is given in eqn.~\eqref{e:genMarkov} :
\begin{equation}
{\bm g}(\pmb y)={\bm g}(\pmb x_{t+T})=\mathcal{D}{\bm h}(\pmb x_t)=\mathcal{D}{\bm h}(\pmb x).
\label{e:genMarkov}
\end{equation}
Here, ${\bm g}(\pmb y)$ and ${\bm h}(\pmb y)$ are vector-valued transformations (components of $\bm {g,h}$ are scalar-valued) to a feature space.
In general, $\bm {g,h} \in \mathcal{F}$ (where $\mathcal{F}$ is a function space) are infinite-dimensional, but approximated into a finite-dimensional vector in practice and $\mathcal{D}: \mathcal{F} \to \mathcal{F}$.
Without loss of generality we use the first order Markov process approximation of the dynamical system, i.e. $\bm g(\bm x_{t+T})=\mathcal{D}{\bm h}(\bm x_{t})$ in the above discussion.
That said, the algorithms presented here can easily be generalized to $n^{th}$ order processes. Mathematically, we can represent such as system as $\bm g(\bm x_{t+T})=\mathcal{D}{\bm h}({\bm x_t},{\bm x_{t-T}},{\bm x_{t-2T}},{\bm x_{t-3T}}...{\bm x_{t-(n-1)T}})$.
In subsection \ref{ss:markovNN} we explore how feedforward neural networks and the popular Koopman approximation-based methods build such Markov representations.
\subsection{ Markov Model Approximation using Feed-forward Neural Networks (FFNNs)}\label{ss:markovNN}
The key to developing a model for the Markovian dynamics is to learn the transformations $\bm g,\bm h$ and the operator $\mathcal{D}$. As mentioned earlier, each of ${\bm g,\bm h}$ and $\mathcal{D}$ can be either linear or nonlinear. FFNNs are excellent function approximators~\cite{Hornik:89} that one can use to learn these maps ${\bm g,\bm h} \textnormal{ or }\mathcal{D}$ for a given training data. A standard FFNN architecture as shown in fig.~\ref{f:6levelFFNN} involves a linear map, $\Theta_l$ applied to the features ($\bar{X}^l$) at any given layer followed by the application of a nonlinear activation function, $\mathcal{N}_l$. Usually, $l=1\dots L$, indicating a $L$-layered network governed by the recursive relationship as shown in eqn.~\eqref{eq:DNN1}. $\bar{X}^1$ and $\bar{X}^L$ represent the input ($\pmb{x}^t$) and output ($\pmb x^{t+T}$) features respectively. It is common to include a bias term inside the parentheses in eqn.~\eqref{eq:DNN1}, i.e. $\bar{X}^{l+1}=\mathcal{N}_l\left(\Theta_l \bar{X}^l + b^l \right)$ for improved approximation properties.
\begin{equation}
\bar{X}^{l+1}=\mathcal{N}_l\left(\Theta_l \bar{X}^l \right)
\label{eq:DNN1}
\end{equation}
\begin{equation}
{\pmb x}^{t+T}=\bar{X}^{L}=\mathcal{N}_{L-1}\left(\Theta_{L-1}\mathcal{N}_{L-2}\left(\Theta_{L-2}
\mathcal{N}_{L-3} \left(\cdots\left(\Theta_1 {\pmb x}^t \right)\right)\right)\right)=FFNN({\pmb x}^t)
\label{eq:DNN2}
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[width=0.81\columnwidth]{methods/6-layer-FFNN.pdf}
\caption{A six-level Feed forward Neural Network architecture for building a Markov model. \label{f:6levelFFNN}}
\end{center}
\end{figure}
Using this framework, we can develop an evolutionary model that approximates $\pmb g,\ \pmb h \textnormal{ and } \mathcal{D}$ as shown below in eqn.~\eqref{eq:DNN2}.
The above represents a nonlinear regression model of the data which can be interpreted in many ways.
A convenient interpretation adopted in this article is that $\bm{g}$ and $\mathcal{D}$ are identity maps, i.e. $\bm g(a)=\mathcal{I}a=a$ and $\mathcal{D}a=Ia=a$ for any $a\in \mathbb{R}^N$ and $\bm{h}$ is given by the `layering' of $\Theta_l,\ \mathcal{N}_l$, i.e. $\bm h(a)={\it FFNN}(a)$. For this interpretation, the feature space coincides with the input state space. Other interpretations are possible by splitting the FFNN into a combination of $\bm h$ and $\mathcal{D}$. A key takeaway here is the use of a layered map to approximate $\bm g$ or $\bm h$ or both, which is nonlinear and can be asymmetric, i.e. $\bm h \neq \bm g$. Typically, it is not possible to have $\bm g=\bm h$ using a standard FFNN, but one can use machine learning tools such as deep autoencoder networks~\cite{hinton2006reducing} to accomplish this as reported in~\cite{Shiva:18AIAA}. Irrespective of the chosen interpretation, another key takeaway is that the learning process is end-to-end, i.e., it includes estimating the entire set of $\Theta_l$ (with $\mathcal{N}_l$ specified) by inverting the nonlinear system in eqn.~\eqref{eq:DNN2} requiring significant training cost. For this reason, it is not uncommon to reduce of the dimension of input and output features using projections onto some sparse basis such as the singular vectors of the data matrix as used in this study.
\subsection{Markov Model Approximation using Koopman Framework}\label{ss:markovkoopman}
In the earlier section, we interpreted FFNNs as an asymmetric Markov model with a linear transition operator, identity map, $\pmb g$ and a nonlinear map $\pmb h$.
In this section, we explore how first order Markov models can be represented through the class of Koopman operator-theoretic frameworks\cite{Mezic:05,Koopman:31} for modeling nonlinear dynamics. A Markov process is approximated by the Koopman operator \cite{Mezic:05,Koopman:31,Rowley:09} under conditions of $\bm g=\bm h$ and $\mathcal{D}=\mathcal{K}$ with $\mathcal{K}$ being linear.
In the Koopman framework, the feature space is the space characterized by a vector of observables $\pmb g(\pmb x),\pmb h (\pmb x)$ and the feature maps, $\pmb g,\pmb h$ represent a vector of observable functions. The operator theoretic view\cite{Mezic:05,Williams:15} interprets $\mathcal{K}$ as operating on the space of functions $\mathcal{K}: \mathcal{F} \to \mathcal{F}$.
When $\pmb g$ and $\pmb h$ are identical, then the linear operator $\mathcal{K}$ evolves the Markovian dynamics in Eq. (\ref{e:genMarkov}) as a Koopman evolutionary model given by:
\begin{equation}
\mathcal{K}\bm g(\pmb x)=\bm g(\pmb y)=g({\pmb F}(\pmb x)).
\label{e:koopman1}
\end{equation}
This representation is exact when $\bm {g,h} \in \mathcal{F}$ span the infinite-dimensional function space, $\mathcal{F}$ in which the koopman operator, $\mathcal{K}$ acts. However, one often uses a finite-dimensional approximation in practice.
Since, the Koopman operator has the effect of operating on the functions of state space as shown in eqn. (\ref{e:koopman1}), it is commonly referred to as a composition operator where $\circ$ represents the composition between $\bm g$ and the exact model describing the dynamical system, $\pmb F$.
\begin{equation}
\mathcal{K}{\bm g}={\bm g}{\circ}{\pmb F}.
\label{e:koopman2}
\end{equation}
Being a linear operator, the products of Koopman spectral analysis such as the eigenfunctions ($\phi_{j}$), eigenmodes ({$v_{j}$) and eigenvalues ({$\mu_{j}$) can be leveraged to reconstruct the transformation $\bm g(\pmb x)$ as shown in eqn. \eqref{e:evolution1} provided the elements of $\pmb g$ lie in the span of $\phi$.
If this is true, then the evolutionary model can be represented as in eqn.~\eqref{e:evolution2}.
\begin{equation}
\bm g(\pmb x)=\sum_{j=1}^{\infty} \phi_{j}{\pmb v}_{j}
\label{e:evolution1}
\end{equation}
\begin{equation}
\bm g(\pmb y)=\mathcal{K}{\bm g}(\pmb x)=\sum_{j=1}^{\infty} \phi_{j}{\pmb v}_{j}\mu_{j}
\label{e:evolution2}
\end{equation}
In practice, a temporal sequence of data, $(\pmb x^{T}, \pmb x^{t+T}\dots)$ generated by a nonlinear dynamical system as in eqn.~\eqref{e:nonlinear} needs to be represented using a Koopman Markov framework as in eqn.~\eqref{e:koopman_sys} where, $\pmb g,\pmb h$ is yet to be identified (or modeled).
\begin{equation}\label{e:koopman_sys}
\bm g(\pmb{x}^{t+T})=\bm g(\pmb{F}(\pmb{x}^{t}))=\mathcal{K} \bm g(\pmb{x}^{t})
\end{equation}
Arranging the data into snapshot pairs as $X=(\pmb x^{t} \dots\ \pmb x^{t+(M-2)T}, \pmb x^{t+(M-1)T})$ and $Y=(\pmb x^{t+T}\dots\ \pmb x^{t+(M-1)T}, \pmb x^{t+MT})$ such that $(X,Y) \in \mathbb{R}^{N \times M}$, eqn.~\eqref{e:nonlinear} can be recast as $Y=\pmb{F}{X}$ with a corresponding quasi-linear form given by ${Y} \approx \pmb A(X)X$ with $N, M$ representing the dimensions of instantaneous system state and data snapshots respectively.
The observable function $\bm g$ is unknown and modeled as a finite-dimensional map $C \in \mathcal{R}^{N \times K}$ that can either be functional or data-driven. It is common to treat the map as a projection of the input state onto an appropriate basis such that the dynamics evolve in a feature space that is low-dimensional ($K\ll N$). This would require $\pmb x^t$ be spanned accurately by the basis forming the columns of $C$.
A finite-dimensional approximation of $\mathcal{K}$ (in eqn.~\eqref{e:koopman_sys}) corresponding to the choice of $C \in \mathbb{R}^{N \times K}$ can be obtained using the following method. The approximation of $\pmb g$ is given by the relationship $C \pmb g(\pmb x^t)\approx \pmb x^t$ for a single snapshot and $C \pmb g(X)=X=C\bar{X}$ for a collection of snapshots. Substituting $X = C\bar{X}=C \pmb g(X)$ and $Y = C\bar{Y}=C \pmb g(Y)$ in the linearized model for the dynamical system $Y \approx A(X)X$, we get
\begin{equation}
C^{+} \pmb A(X)C\bar{X} = \bar{Y},
\label{e:conv4}
\end{equation}
where $C^{+}$, is the left Moore-Penrose pseudo-inverse. $C^{+}$ should be computable as $C^TC$ is not likely to be rank deficient for $K\ll N$. Relating eqn.~\eqref{e:koopman1} to eqn.~\eqref{e:conv4} for $\pmb g(X)=\bar{X}$ and $\pmb g(Y)=\bar{Y}$, we get a linear finite-dimensional approximation for the Koopman operator, $\mathcal{K}$ given by $\mathcal{K}\approx {\Theta} \triangleq C^{+} \pmb A(X)C$ that governs the evolution of the dynamics in the feature space. $\bar{X} \in \mathbb{R}^{K \times M}$ and $\bar{Y} \in \mathbb{R}^{K \times M}$ are the representations of the state in the feature space.
Naturally, the fidelity of the above approximation to the dynamical system in eqn.~\eqref{e:nonlinear} depends on the choice of $C$ as an approximation to $\bm g$. Further, for ${\Theta}$ to be truly linear, it is easy to infer that $C$ will have to evolve with the state $\pmb x^t$ as $C(\pmb x^t)$. For detailed discussion on the architecture and choice of map maps we refer to Lu and Jayaraman \cite{Lu:18sparse}. For a chosen $C$ and given $X,Y$, we can learn ${\Theta}$ by minimizing the frobenius norm of $\|{{\bar Y}} - {\Theta}{\bar X}\|_{F}$ via $\Theta = \bar Y {\bar X}^+$. In principle one could minimize the $2$-norm $\|{{\bar Y}} - {\Theta}{\bar X}\|_{2}$ at the risk of added complexity, but the Frobenius norm serves an efficient alternative.
\section{Modeling the Feature Maps $g,h$}\label{ss:markovLOC}\label{s:featuremaps}
In the previous section, we interpreted FFNNs as learning one side of a feature map, i.e., $\pmb h$ in a Markov model with $\pmb g,\ \mathcal{D}$ being modeled as identity maps. Contrastingly, the Koopman theoretic methods assume a model for the feature maps, i.e. $\pmb g=\pmb h = C$ and learns the linear transition or Koopman operator $\mathcal{D}=\mathcal{K}\approx \Theta$ using data. As the Koopman framework is symmetric it allows one to learn a low-dimensional linear transition operator that can be used for spectral analysis. The FFNN's offer no such luxury \cmnt{as the data-driven learning is focused on identifying the feature map}. The key to the success of both approaches for extended predictions relies on the accuracy of the feature map approximations either using data (FFNNs) or through models (Koopman). A prominent approach to improving the fidelity in Koopman approximation methods is to sequentially layer elementary maps (both functions and basis projections) in a supervised fashion and then approximate the Koopman operator in the resulting feature space. While this layering approach is similar to FFNNs, there exist key differences, We will explore these in the following subsections.
\subsection{Multilayer Sequential Maps (MSMs) for Koopman Approximation}\label{ss:markovLOC}
For Koopman approximation methods, the basis space onto which the input state is mapped should evolve with the state itself, i.e., $C$ should be $C(X)$ so that $\Theta \approx \mathcal{K}$ can be linear. However, this often leads to a futile search for `magic' basis \cmnt{when the nature of the dynamical system is unknown}. Alternative approaches such as Extended Dynamic Mode Decomposition or EDMD~\cite{Williams:15} include approximating the functional form of the map from data using a dictionary of basis functions. However, the dependence of $C$ on the choice of functions populating the dictionary and the relative ease with which the feature dimension grows, limits these approaches. In \cite{Lu:18sparse}, Jayaraman and collaborators propose an alternate approach to building complex and efficient maps through layering of elementary operators based on the hypothesis that \emph{deeper and shorter is better than taller and shallow} operators. This approach is similar to the kernel DMD framework~\cite{Williams:14arXiv} that combined EDMD with a kernel principal component analysis (KPCA) to improve efficiency. This evolution of methods align with the recent successes in \emph{deep learning} ideas for artificial intelligence~\cite{bengio2007challenge} in spite of the need to tune many hyperparameters. It is worth noting that both strategies increase the number of model parameters to be learned, but layering offers a systematic way to build model complexity for data-driven learning as compared to shallow architectures~\cite{Hornik:89}. A generalized way of building $C$ is to layer recursively multiple mapping operators such as:
\begin{equation}
X = C_{1}C_{2}{\bar{X}^2} = {\mathcal{C}}_{ML}{\bar{X}}^2,
\label{e:mconv1}
\end{equation}
\begin{equation}
Y = C_{1}C_{2}{\bar{Y}^2} = {\mathcal{C}}_{ML}{\bar{Y}^2},
\label{e:mconv2}
\end{equation}
where ${\bar{X}}^2$ and ${\bar{Y}}^2$ represent the features at the $2^{nd}$ layer and ${\mathcal{C}}_{ML}$ represents the multilayer sequential map. A schematic of such a model is presented in fig~\ref{f:MLCF} where the state $X$ is operated by a two-layer map $C^{+}_{1}C^{+}_{2}$ to yield ${\bar{X}}^2$. Similarly, $C^{+}_{1}C^{+}_{2}$ operate on $Y$, $\bar{Y}^1$ respectively to yield ${\bar{Y}}^2$ . An approximate linear Koopman operator $\Theta$ is then learned from ${\bar{X}}^2$ and ${\bar{Y}}^2$.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth]{methods/ARSCH1.pdf}
\caption{Schematic of a six-layer representation of the multilayer sequential map (MSM) framework to approximate the Koopman operator. $X,\ Y$ represent state space matrices and $C^{+}_{i}$, $C_{i}$, represent the elemental maps and its inverse operation respectively. The arrows indicate direction of the maps, i.e., $C^{+}_{1}$ acts on $X$ to yield $\bar{X}^1$ and $C_{1}$ acts on $\bar{Y}^1$ to yield $Y$. $\Theta$ represents the approximation Koopman operator shown in Eq.(\ref{e:koopman_sys}). The size of data matrices in the high ($X$) and low dimensional($\bar{X}$ or $\bar{\bar{X}}$ ) space is also shown.}
\label{f:MLCF}
\end{center}
\end{figure}
Substituting eqns.~\eqref{e:mconv1} and \eqref{e:mconv2} into eqn.\eqref{e:conv4}, we have:
\begin{equation}
{{\mathcal{C}}_{ML}}^{+}A{{\mathcal{C}}_{ML}}{\bar {X}^2} = \Theta{\bar {X}^2}= {\bar {Y}^2},
\label{e:mconv4}
\end{equation}
with $\Theta \triangleq {{\mathcal{C}}_{ML}}^{+}A{{\mathcal{C}}_{ML}}$. Instead of the two-layer map, we can have a deep architecture with ${{\mathcal{C}}_{ML}}^{+}=C_{L}^{+}...C_{1}^{+}C_{2}^{+}C_{3}^{+}$ and ${{\mathcal{C}}_{ML}}=C_{3}C_{2}C_{1}...C_{L}$ where $2(L+1)$ represent the total number of layers in the design. The encoder map ${{\mathcal{C}}_{ML}}^{+}$ can be computed as long as the elemental maps, $C_i$, are invertible in a generalized sense. Although this MSM formulation is designed for Koopman approximations, i.e. $\bm g = \bm h = \mathcal{C}_{ML}$, one can build generalized Markov versions of this model i.e. $\bm g = \mathcal{C}_{ML1}$ \& $ \bm h = \mathcal{C}_{ML2}$. A key limitation of such MSM frameworks is that $C_i$ and consequently $\mathcal{C}_{ML}$ are usually predetermined maps (or functions) and the Koopman approximation relies only on the local features. In the following subsections \ref{sss:DMDLOC} and \ref{sss:EDMDLOC} we present some of the Koopman approximation methods in the MSM context.
\subsubsection{DMD as a Four-level Multilayer Sequential Map (MSM) based Markov Model}\label{sss:DMDLOC}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{methods/ARSCH2.pdf}
\end{center}
\caption{Dynamic Mode Decomposition (DMD): four-level MSM framework with linear maps for Koopman approximation.\label{f:MLCF_4}}
\end{figure}
There exist many methods to approximate the Koopman tuples including DMD \cite{Rowley:17ARev,SchmidDMD:10}, EDMD \cite{Williams:15} and its kernel variant, Kernel DMD~ \cite{Williams:14arXiv} and generalized Laplace analysis (GLA) \cite{Mezic:05}. DMD~\cite{SchmidDMD:10} employs observables which are linear functions of the state. The multilayer architecture for DMD is shown in fig.\ref{f:MLCF_4} as a four-layer framework containing with both forward and backward maps, i.e. $X\rightarrow \bar{X}^1\Leftrightarrow\bar{Y}^1\leftarrow Y$ consisting of POD basis (via SVD~\cite{Trefethen:97} of the training data ($X$)) projections. Given data snapshots separated in time $X,Y$ as before, we use $X=C_{POD}{\bar X}^1$ and $Y=C_{POD}{\bar Y}^1$ to generate a Koopman approximation subject to $\Theta \bar X^1 = \bar Y^1$.
The pairs ${\bar X}^1$ and ${\bar Y}^1$ have the structure shown in eqn.~\eqref{e:PODfeatures} where $a_i^j$ represent the $i^{th}$ POD coefficient of the $j^{th}$ snapshot.
\begin{equation}\label{e:PODfeatures}
{\bar X}^1 = \begin{bmatrix}
$$a_{1}^{1}$$ & $$a_{1}^{2}$$ & $. . . . .$ & $$a_{1}^{M-1}$$ & $$a_{1}^{M}$$\\
$$a_{2}^{1}$$ & $$a_{2}^{2}$$ & $. . . . .$ & $$a_{2}^{M-1}$$ & $$a_{2}^{M}$$\\
$.$&$.$&$. . . . .$&$.$&$.$\\
$.$&$.$&$. . . . .$&$.$&$.$\\
$$a_{K}^{1}$$ & $$a_{K}^{2}$$ & $. . . . .$ & $$a_{K}^{M-1}$$ & $$a_{K}^{M}$$\\
\end{bmatrix}_{(K\textrm{x}M)}
\textrm{and}\ \ \ \
{\bar Y}^1 = \begin{bmatrix}
$$a_{1}^{2}$$ & $$a_{1}^{3}$$ & $. . . . .$ & $$a_{1}^{M}$$ & $$a_{1}^{M+1}$$\\
$$a_{2}^{2}$$ & $$a_{2}^{3}$$ & $. . . . .$ & $$a_{2}^{M}$$ & $$a_{2}^{M+1}$$\\
$.$&$.$&$. . . . .$&$.$&$.$\\
$.$&$.$&$. . . . .$&$.$&$.$\\
$$a_{K}^{2}$$ & $$a_{K}^{3}$$ & $. . . . .$ & $$a_{K}^{M}$$ & $$a_{K}^{M+1}$$\\
\end{bmatrix}_{(K\textrm{x}M)}
\end{equation}
Knowing $\Theta$ allows one to model the Markovian evolution of this dynamical system in the feature space. In this MSM framework, the data-driven learning is accomplished through a local optimization as shown in eqn.~\eqref{e:LDMD}, i.e. through minimizing the mapping error between the two immediate layers constituted by pairs of features $\pmb{a}^t,\pmb{a}^{t+1}$ which are the column vectors in ${\bar X}^1,{\bar Y}^1$. Estimating $\Theta$ that minimizes the Frobenius norm $\| \bar{Y}^1 - \Theta \bar{X}^1\|_F$ requires computing a least squares solution to eqn.~\eqref{e:LDMD} as $\Theta = \bar Y \ (\bar{X}+ \lambda I)^{+} $ where $()^{+}$ denotes the generalized Moore-Penrose pseudo-inverse \cite{Trefethen:97}. $\lambda$ is a $l_2$ regularization~\cite{Scholkopf:01} parameter to generate a unique solution for this overdetermined system (with $K < M$).
\begin{equation}\label{e:LDMD}
{\bar Y}^1 =\begin{bmatrix}
$$\pmb{a}^{2}$$ & $$\pmb{a}^{3}$$ & $$\dots$$ &$$\pmb{a}^{i+1}$$ & $\dots$ & $$\pmb{a}^{M+1}$$
\end{bmatrix} = \Theta
\begin{bmatrix}
$$\pmb{a}^{1}$$ & $$\pmb{a}^{2}$$ & $$\dots$$ &$$\pmb{a}^{i}$$ & $\dots$ & $$\pmb{a}^{M}$$
\end{bmatrix} =\Theta {\bar X}^1
\end{equation}
In this $4-$layer MSM architecture, the maps between any two layers are layer-wise optimal and the sequence of application, i.e. map direction becomes critical as $\Theta$ depends on $C_{POD}$, but not vice versa.
\subsubsection{Extended DMD: A Six-level Multilayer Sequential Map (MSM) based Markov Model}\label{sss:EDMDLOC}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth]{methods/ARSCH3.pdf}
\caption{Extended Dynamic Mode Decomposition (DMD): A six-level Koopman approximation MSM framework with nonlinear maps ($\mathcal{N}, \ \mathcal{N}^{-1}$). $I$ represents the identity linear operator and $\mathcal{I}$, the identity function. }
\label{f:MLCF_6}
\end{center}
\end{figure}
In the earlier DMD multilayer framework, the elemental maps $C$ were linear functions of the state (i.e. the POD features were computed form training data and not the the instantaneous flow state) which has difficulty modeling nonlinear dynamics\cite{Rowley:17ARev,Lu:18sparse}. Extensions to DMD such as EDMD \cite{Williams:15,Rowley:17ARev} help alleviate this problem to some extent by layering nonlinear maps ($\bar{X}^2=\mathcal{N}\left( I\bar{X}^1 \right)$ and $\bar{Y}^2=I\mathcal{N}\left(\bar{Y}^1 \right)$) over linear ones ($\bar{X}^1=\mathcal{I}\left(C_{POD}^+X\right)$ and $\bar{Y}^1=C_{POD}^+\mathcal{I}\left(Y\right)$) as shown in fig.~\ref{f:MLCF_6}.
The architecture for the EDMD in fig.~\ref{f:MLCF_6} is a $6-$level framework ($4-$ layer without the POD-map for dimensionality reduction) with both forward and backward maps, i.e. $X\rightarrow \bar{X}^1\rightarrow \bar{X}^2\Leftrightarrow\bar{Y}^2\leftarrow \bar{Y}^1\leftarrow Y$ with the first and fifth layers representing a linear-map made up of POD-basis of the training data while the $2^{nd}$ and $4^{th}$ layers represent nonlinear functional maps operating on the corresponding features. In the schematic, we present a generalized representation where each map consists of linear operators, i.e. $C_{POD},I$ and functional maps, $\mathcal{N},\mathcal{I}$ with $I,\mathcal{I}$ representing the operator and functional forms of the identity map respectively. However, the architecture represented in fig.~\ref{f:MLCF_6} has a `forward' direction with the nonlinear mapping in the $4^{th}$ layer denoted by an inverse function $\mathcal{N}^{-1}$ that may not always be well behaved. In practice, the layers six to five to four flow backward, i.e. $Y\rightarrow \bar{Y}^1\rightarrow \bar{Y}^2$ ($\bar{Y^2}=I\mathcal{N}\left(\bar{Y}^1 \right)$ and $\bar{Y}^1=C_{POD}^+\mathcal{I}\left(Y\right)$) which helps bypass such issues. The approximation to the Koopman operator, $\Theta$ is estimated as the optimal linear operator that relates the features $\bar{X}^2$ and $\bar{Y}^2$ in a least squares sense (i.e. find the $\Theta$ that minimizes the Frobenius norm $\|{\bar Y}^2- \Theta {\bar X}^2\|_F$) as was shown for the DMD framework. In this study, we present two variants of this method corresponding to different choices of $\mathcal{N}$, namely, EDMD-P \cite{Williams:15} which uses polynomial functions (eqn.\eqref{e:nonlinconv1}) of the features ($\bar{X^2}=\mathcal{N}\left(I \bar{X}^1 \right)$) and EDMD-TS which uses a tan-sigmoid nonlinearity (eqn.\eqref{e:nonlinconv2}) .
\begin{eqnarray}
\label{e:nonlinconv1}
\bar{\pmb {a}}=\mathcal{N}(\pmb{a})&=&\begin{bmatrix}
\pmb{a} \\
\pmb{a}\otimes \pmb{a}
\end{bmatrix}\\
\bar{\pmb {a}}=\mathcal{N}(\pmb{a}) &=& \tanh{\left(\pmb a\right)}
\label{e:nonlinconv2}
\end{eqnarray}
Here $\bar{ \pmb{a}}$ represents the features in the $\mathcal{N}$ space, i.e. columns of $\bar{X}^1,\bar{Y}^1$. It is easily seen that EDMD-P with $2^{nd}$ order polynomials leads to a quadratic growth in the feature dimension and even worse when using higher order polynomials. On the other hand, EDMD-TS does not lead to increase in the number of features. Just as in DMD, the EDMD MSM architecture optimizes the maps only between the two immediate layers and direction of the map (sequence of application of operators) strongly influences the model, i.e. $C_{POD}$ and $\mathcal{N}$ determine $\Theta$ but not vice versa. As a consequence of this layer-wise treatment and symmetric formulation (i.e. $\pmb g = \pmb h$), the map is bi-directional which makes learning the linear Koopman operator efficient. In the following section, we will focus on end-to-end learning of the map using neural networks.
\subsection{Feed Forward Neural Networks (FFNN): Multilayer End-to-End Map (MEM) based Markov Models}\label{ss:markovGOC}
In principle, multilayer map increases the number of design variables, i.e. the choice of nonlinear functions ($\mathcal{N}$), depth ($L$) and dimension ($K,R$) of the layers in the model. For the MSM framework described in section~\ref{ss:markovLOC}, we observe that the direction of the map, choice of the elemental operators and order of layering can generate different representations. This is also true in the case of a standard feed forward neural networks (FFNNs) as depicted in fig.~\ref{f:MGCF_6}. The figure shows a six-layer FFNN architecture so as to compare against the six-layer MSM framework (EDMD) in fig.~\ref{f:MLCF_6}.
Here, each interior map between any two layers includes a linear map $\Theta_i,\ \left(i=1..5\right)$ and nonlinear transfer functions $\mathcal{N}_i,\ \left(i=1..5\right)$ with the latter predetermined for a given model.
One can mimic the EDMD exactly using the FFNN framework by setting $\Theta_1=C_{POD}$, $\Theta_5=C_{POD}^+$ and $\Theta_2=\Theta_4=I$ where $I$ is the identity tensor, $\mathcal{N}_2=\mathcal{N}$, $\mathcal{N}_4=\mathcal{N}^{-1}$, $\mathcal{N}_1=\mathcal{N}_3=\mathcal{N}_5=\mathcal{I}$ where $\mathcal{I}$ is the identity map along with $\Theta_3=\Theta$, the Koopman operator.
For this FFNN architecture that only supports forward maps, building a map with $\mathcal{N}^{-1}$ is not explored currently. This is because, for many common choices of $\mathcal{N}$, $\mathcal{N}^{-1}$ is not always bounded. It is for this reason, even in the MSM architectures, the backward operation is preferred. In this study, we use a tansigmoid function for $\mathcal{N}_{2,3,4}=\mathcal{N}$. Further, since we are dealing with high-dimensional flow datasets, we set $\mathcal{N}_{1,5}=\mathcal{I}$, $\Theta_1=C_{POD}$ and $\Theta_5=C_{POD}^+$ to reduce dimensionality of the interior layers. This leaves $\Theta_1,\Theta_2 \textrm{ and }\Theta_3$ to be determined from data. \cmnt{In addition to reducing the training cost, this also helps reduce overfitting.} In addition, we did include a bias term to facilitate better comparison with conventional MSM architectures such as DMD and EDMD.
While similar in architecture, a key difference between the EDMD/DMD (MSM) and FFNN approaches is how they leverage the extended model hyperparameter space (e.g. elements of the Koopman operator) for learning from data. In the MSM framework, the linear parts of the map are either precomputed (i.e. $C_{POD}$) or assumed (i.e. $I$) for a given model design which allows estimation of the layerwise features before solving for the unknown Koopman operator $\Theta$ (size $R\times R$) using linear regression techniques. In the FFNN framework, the linear operators $\Theta_1,\Theta_2 \textrm{ and }\Theta_3$ are all unknowns while the nonlinear activation functions $\mathcal{N}_1,\mathcal{N}_2 \textrm{ and } \mathcal{N}_3$ are specified in the design. To learn the optimal solutions for $\Theta_{i}, i=1,2,3$, one needs constrain the resulting Markov model to the training data and solve a nonlinear regression problem~\cite{bengio2015deep}. In this way, the FFNN architecture takes advantage of the extended model hyperparameter space offered by the multilayer map by learning $K\times R+R\times R+R \times K$ parameters in $\Theta_{i}, i=1,2,3$ as against just $R \times R$ parameters in $\Theta$. Such frameworks that incorporate `end-to-end' learning can be characterized as \emph{Multilayer End-to-end Map (MEM)} based Markov models. It is anticipated that MEM frameworks can offer improved representations of nonlinear dynamics as compared to MSM frameworks. It is well known that MSMs work well for predicting select dynamics but fail to model highly transient nonlinear systems. The downside of such FFNN/MEM framework includes: (i) increased training cost to estimate more unknowns than the MSM framework; (ii) propensity to generate non-unique solutions that require regularization and (iii) propensity to overfit data by learning more parameters, especially when using deeper networks which requires careful monitoring.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth]{methods/ARSCH4.pdf}
\caption{Feedforward Neural Network (FFNN) as a six-level Multilayer end-to-end map. $\Theta_l, \ \mathcal{N}_l$ with the arrow represents the application of a linear operator followed by a nonlinear function.}
\label{f:MGCF_6}
\end{center}
\end{figure}
We briefly summarize the algorithm used for training the FFNN/MEM architecture. As before, $X,Y$ are the time dependent flow snapshot pairs and ${\bar X}^1,{\bar Y}^1$ represent the snapshots of time-dependent POD features as shown in eqn.~\eqref{e:LDMD} with columns $\pmb{a}^i$ and $\pmb{a}^{i+1}$ respectively. The effective nonlinear map is trained between $\bar{Y}^1$ and $\bar{X}^1$ as shown below in eqns.~\eqref{e:FNN1}-\eqref{e:FNN3}:
\begin{equation}\label{e:FNN1}
{\bar{X}^2}=\mathcal{N}_2(\Theta_2 \bar{X}^1)
\end{equation}
\begin{equation}\label{e:FNN2}
{\bar Y}^2=\mathcal{N}_3(\Theta_3 {\bar X}^2)
\end{equation}
\begin{equation}\label{e:FNN3}
{\bar Y}_p^1=\mathcal{N}_4(\Theta_4 {\bar Y}^2)
\end{equation}
In general, a multilayer network is characterized by the recursive relationship $X_l=\mathcal{N}_l(\Theta_l X_{l-1})$ where $X_l,\Theta_l$ and $\mathcal{N}_l$ represent the mapped features, linear operator and nonlinear map relating the $l^{th}$ and $l+1^{th}$ layers. In this specific example, $\bar{Y}^1_{p}$ is the predicted features at the fifth layer to be compared with the ground truth, $\bar{Y}^1$, obtained from the training data as $\bar{Y}^1=C_{POD}Y$. The linear operator $\Theta_l,\ \textrm{with} \ l=2\dots (L-2)$, for a $L-$layer framework is estimated by minimizing the overall cost function as in eqn.~\eqref{e:regcost}:
\begin{equation}\label{e:regcost}
\begin{split}
\mathcal{J}(\Theta) = & \underbrace{\frac{1}{2M} \sum_{i = 1}^{M} \sum_{j = 1}^{K} (\bar{Y}_{p}(j,i) -
\bar{Y}(j,i) )^2}_\text{Feed forward Cost} \\
+ & \underbrace{\left(\frac{\lambda}{2M} \sum_{l =
2}^{L-2} \sum_{s = 1}^{S} \sum_{q = 1}^{Q} \left(\Theta_{l}(s,q)\right)^2
\right)}_\text{Regularization term}
\end{split}
\end{equation}
In the above, \cmnt{$\bar{Y}^1$ is the original data,} $S,Q$ represent the dimension of the features in layers $l \textrm{ and } l+1$ respectively.
The optimal solution for $\Theta_l, \ l=2 \dots (L-2)$ is obtained using backpropagation with a gradient descent framework employing a Polack-Ribiere conjugate gradient algorithm~\cite{Golub:12matrix} that employs a Wolfe-Powell stopping criteria. This nonlinear inversion to estimate the $\Theta$'s is the most important distinction between MSM and MEM methods. The gradient descent framework requires $\mathcal{N}$ to be infinitely differentiable which is not always guaranteed when choosing $\mathcal{N}_l=\mathcal{N}^{-1}$.
To minimize overfitting, we use $\mathcal{L}_2$ norm based regularization in the cost function in eqn.\eqref{e:regcost} with $\lambda$ as the tuning parameter. To characterize the dimension of intermediate layer features we use a factor $(N_f)$ that is multiplied with the input feature dimension, i.e., $S,Q=N_f\times \textrm{input feature dimension}$. For such FFNN architectures, designing a forward-backward map to learn the Koopman operator (as in MSM frameworks such as DMD \& EDMD) is hard to realize using regular backpropagation training. As shown in ~\cite{Shiva:18AIAA}, incorporating special feedback networks with some similarity to recurrent neural network offer a way forward. However, these aspects are beyond the scope of this article.
\begin{table}[]
\centering
\label{tab:compare}
\begin{tabular}{|c|c|c|}
\hline
& \textbf{\begin{tabular}[c]{@{}c@{}}Multilayer Sequential Map\\ (MSM)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Multilayer End-to-end Map\\ (MEM)\end{tabular}} \\ \hline
\textbf{Interdependence of layers} & One-way dependence & Two-way dependence \\ \hline
\textbf{Learning Paradigm} & \begin{tabular}[c]{@{}c@{}}All the linear maps except \\ one are precomputed\end{tabular} & \begin{tabular}[c]{@{}c@{}}All the linear maps across the layers \\ are computed simultaensouly\end{tabular} \\ \hline
\textbf{Real-time Learning Cost} & \begin{tabular}[c]{@{}c@{}}Very efficient to train as most \\ layers are learnt offline\end{tabular} & \begin{tabular}[c]{@{}c@{}}Requires significantly more\\ time to train\end{tabular} \\ \hline
\textbf{Map direction} & Bi-directional (symmetric) & Unidirectional (can be asymmetric) \\ \hline
\end{tabular}
\caption{Assessment of the sequential and end-to-end learning maps for generating Markov models }
\end{table}
\section{Numerical Experiments and Discussion}\label{s:results}
In this section we compare the predictive capabilities of MSM with MEM Markov models. While it is to be expected that learning an extended set of parameters by minimizing the training error cost function allows for improved predictions of time-series flow data, the dimension of this parameter set depends on nature of the model architecture. Consistent with the earlier sections, we adopt the nomenclature `L-Method-$\mathcal{N}-N_f$' to denote the different architectures and their respective parameters, where the `L' represents the total number of layers used to map from one flow state to another ($X\rightarrow Y$), $\mathcal{N}$ defines the choice of nonlinear function and $N_f$ represents the feature growth factor. For example, we can easily describe the EDMD framework in fig.~\ref{f:MLCF_6} as a 6-level multilayer-sequential map with a polynomial nonlinearity of order two as 6-MSM-P2-$N_f$ (EDMD-P2), while a 6-level EDMD with a tansigmoid nonlinearity is denoted by 6-MSM-TS-1 (EDMD-TS) where the number followed by TS represents the feature growth factor $(N_f)$ from the first layer to the next. A 4-level MSM representing the DMD architecture is denoted by 4-MSM-$\mathcal{I}$-1 (DMD), where $\mathcal{I}$ defines identity mapping and $M=1$ defines the feature growth factor.
In this study, we have used FFNN as the MEM architecture with four different designs for comparative assessment. They are 6-MEM-TS-$N_f$ with $N_f\ = \ 1,\ 3,\ 9,\ 20$. The various model possibilities are delineated in section~\ref{ss:analframework}. Section~\ref{ss:dataGen} details the generation of flow data from high fidelity computations for use in this study, namely the cylinder wake flow (sec.~\ref{sss:cylwake}) and the buoyancy-driven mixing flow (sec.~\ref{sss:buoymix}).
\subsection{Data generation}\label{ss:dataGen}
To assess the different modeling architectures and the learning algorithms, we build a database of snapshots of transient flow field data generated from high fidelity CFD simulations of a bluff body wake flow and a buoyancy-driven mixing layer. Both these flows are transient in their own way. The cylinder wake flow evolves on a stable attractor and approaches limit-cycle behavior rather quickly while the buoyancy-driven flow is a transient mixing problem with dynamics that dies out in the long-time limit. The former is an example of `data-rich' situation where the training data requirement to predict the dynamics is limited. On the other hand, the latter represents a `data-sparse' situation where any amount of training data may not be sufficient to predict future evolution. We explore the performance of MSM and MEM architectures for both these situations. In the following section, we summarize the data generation process.
\subsubsection{Transient Wake Flow of a Cylinder}\label{sss:cylwake}
Studies of cylinder wakes~\cite{roshko54,williamson89,Noack:03hierarchy,Rowley:17ARev} have attracted considerable interest from the flow system learning community for its particularly rich physics that encompass many of the complexities of nonlinear dynamical systems and yet easy to compute. For this exploration into the performance of different data-driven modeling frameworks we leverage both the unstable transient and the stable limit-cycle dynamics of two-dimensional cylinder wake flow at a Reynolds numbers of hundred, i.e. $Re=100$. To generate two-dimensional cylinder flow data, we adopt a spectral Galerkin method~\cite{Cantwell:15nektar++} to solve incompressible Naiver-Stokes equations, as shown in Eq.~\eqref{eq:cylinderflow} below:
\begin{subequations} \label{eq:cylinderflow}
\begin{eqnarray}
\frac{\partial{u}}{\partial{x}}+\frac{\partial{u}}{\partial{y}}&=&0,\\
\frac{\partial{u}}{\partial{t}}+u\frac{\partial{u}}{\partial{x}}+v\frac{\partial{u}}{\partial{y}}&=&-\frac{\partial{P}}{\partial{x}}+\nu\nabla^2{u},\\
\frac{\partial{v}}{\partial{t}}+u\frac{\partial{v}}{\partial{x}}+v\frac{\partial{v}}{\partial{y}}&=&-\frac{\partial{P}}{\partial{y}}+\nu\nabla^2{v},
\end{eqnarray}
\end{subequations}
In the above system of equations, $u$ and $v$ are horizontal and vertical velocity components. $P$ is the pressure field, and $\nu$ is the fluid viscosity. The rectangular domain used for this flow problem is $-25D<x<45D$ and $-20D<y<20D$, where $D$ is the diameter of the cylinder. For the purposes of this study data is extracted from a reduced domain, i.e., $-2D<x<10D$ and $-3D<y<3D$, where the dynamics occur. The mesh with $\approx 24,000$ points was designed to sufficiently resolve the thin shear layers near the surface of the cylinder and transit wake physics downstream.
The computational method employed fourth order spectral expansions within each element in each direction. The data snapshots were sampled at $\Delta t = 0.2$ non-dimensional time units, arranged as described in section \ref{ss:markovkoopman} and SVD of the flow state matrix was performed to obtain POD coefficients along the modes. The most dominant POD coefficients correspond to $St = 0.16 $ for $Re = 100$, from which we deduced that a single cycle corresponds to approximately $31$ data points in time.
For this study we denote normalized time in as the number cycles to specify the width of the training regime.
Although, more than $15$ POD modes are required for capturing nearly $100 \%$ of the energy at $Re=100$, the large scale coherent structures which govern the flow dynamics are adequately represented within the first $3$ modes and account for approximately $95 \%$ energy as shown in fig.\ref{f:energyRE100}(a)\cmnt{ and \ref{f:energyRE1000}(a)}. The eigenfunctions corresponding to these three modes are presented in fig.~\ref{f:energyRE100}(b)\cmnt{and \ref{f:energyRE1000}(b) respectively and show qualitatively similar flow structures for both $Re=100$ and $Re=1000$}. In fig.~\ref{f:energyRE100}(c) \cmnt{and \ref{f:energyRE1000}(c)} we show the phase portrait for the flow dynamical system, wherein the flow transitions from a steady wake through an unstable growth phase and settles into a limit cycle regime.
\begin{figure}
\begin{center}
\includegraphics[width=0.33\textwidth]{RE100/EnergyContent3_100.pdf}
\includegraphics[width=0.31\textwidth]{RE100/POD_RE100.pdf}
\includegraphics[width=0.33\textwidth]{RE100/phase3_100.pdf}
\caption{Energy content in POD features selected (a) 3 coefficients (b) eigen modes/functions corresponding to 3 POD features (c) phase portrait of $Re=100$ flow.}
\label{f:energyRE100}
\end{center}
\end{figure}
\subsubsection{2D Buoyant Boussinesq Mixing Flow}\label{sss:buoymix}
The above discussion pertains to a nonlinear wake flow dynamical system that transitions from a steady wake into a stable limit-cycle attractor. Such systems have seen success in prediction from data-driven models with the availability of limited data as demonstrated in ~\cite{Lu:18sparse} . The instability-driven Bousinesq buoyant mixing flow~\cite{weinan98,liu03} exhibits strong shear and Kelvin-Helmholtz instabilities driven by thermal gradients. The convective dynamics in such a system cannot be efficiently represented by data-driven POD modes. Further, the data-driven basis representing the low-dimensional manifold itself evolves temporally indicative of highly transient physics. Such systems are sensitive to noise in the initial state that produce very different trajectories and consequently, a very different dynamical system with its own basis space. This renders such dynamical systems hard to predict even if one were to leverage equation-driven models such as POD-Galerkin \cite{Noack:03hierarchy}. Earlier work from our team~\cite{Lu:18sparse} has shown that such problems are difficult to model accurately using MSM-based models. In this work, we compare these outcomes with those of the MEM-based Markov models.
The data is generated by modeling the dimensionless form of the two-dimensional incompressible flow transport equations\cite{liu03} augmented with buoyancy terms and thermal transport equations, as shown in Eq.~\ref{eq:bseq} on a rectangular domain that is $0<x<8$ and $0<y<1$. To achieve this,we use a $6^{th}$-order compact scheme~\cite{lele1992compact} in space and $4^{th}$-order Runge-Kutta method for the time-integration~\cite{gottlieb2001strong}.
\begin{subequations} \label{eq:bseq}
\begin{eqnarray}
\frac{\partial{u}}{\partial{x}}+\frac{\partial{u}}{\partial{y}}&=&0,\\
\frac{\partial{u}}{\partial{t}}+u\frac{\partial{u}}{\partial{x}}+v\frac{\partial{u}}{\partial{y}}&=&-\frac{\partial{P}}{\partial{x}}+\frac{1}{Re}\nabla^2{u},\\
\frac{\partial{v}}{\partial{t}}+u\frac{\partial{v}}{\partial{x}}+v\frac{\partial{v}}{\partial{y}}&=&-\frac{\partial{P}}{\partial{y}}+\frac{1}{Re}\nabla^2{v}+Ri\theta,\\
\frac{\partial{\theta}}{\partial{t}}+u\frac{\partial{\theta}}{\partial{x}}+v\frac{\partial{\theta}}{\partial{y}}&=&\frac{1}{{Re}{Pr}}\nabla^2{\theta},\\
\end{eqnarray}
\end{subequations}
In the above system, $u$, $v$, and $\theta$ represent the horizontal, vertical velocity, and temperature field, respectively. The system is characterized by the following dimensionless parameters: Reynolds number, $Re$, Richardson number $Ri$, and Prandtl number, $Pr$ with values of $1000$, $4.0$, and $1.0$ respectively. The grid resolution employed is $256 \times 33$. The initial condition for the simulation is designed by vertically segregating the fluids at two different temperatures (uniformly distributed) at the middle of the domain. All the boundaries are adiabatic and friction generating walls. The thermal field evolution over the simulation duration of 32 non-dimensional time units as shown in fig. \ref{fig:buoyantmixing_time_evolution} illustrates the highly transient dynamics. To represent the system in a low-dimensional feature space, POD modes were computed from the entire $1600$ snapshots corresponding to $64$ time units.
The reduced feature set consisting of three POD features (capturing nearly 80\% of the total energy) representing a low resolution measurement is shown in fig. \ref{fig:buoyantmixing_PODWeights} is used to train the model and predict the trajectory.
\begin{figure}
\centering
\begin{subfigure}{0.495\columnwidth}
\begin{center}
\includegraphics[width=\textwidth,clip]{Figures/BM/Temperature0.pdf}
\caption{Time $=0$}
\end{center}
\end{subfigure}
\begin{subfigure}{0.495\columnwidth}
\begin{center}
\includegraphics[width=\textwidth,clip]{Figures/BM/Temperature200.pdf}\\
\caption{Time $=4$}
\end{center}
\end{subfigure}
\begin{subfigure}{0.495\columnwidth}
\begin{center}
\includegraphics[width=\textwidth,clip]{Figures/BM/Temperature400.pdf}
\caption{Time $=8$}
\end{center}
\end{subfigure}
\begin{subfigure}{0.495\columnwidth}
\begin{center}
\includegraphics[width=\textwidth,clip]{Figures/BM/Temperature600.pdf} \\
\caption{Time $=12$}
\end{center}
\end{subfigure}
\begin{subfigure}{0.495\columnwidth}
\begin{center}
\includegraphics[width=\textwidth,clip]{Figures/BM/Temperature800.pdf}
\caption{Time $=16$}
\end{center}
\end{subfigure}
\begin{subfigure}{0.495\columnwidth}
\begin{center}
\includegraphics[width=\textwidth,clip]{Figures/BM/Temperature1000.pdf}\\
\caption{Time $=20$}
\end{center}
\end{subfigure}
\begin{subfigure}{0.495\columnwidth}
\begin{center}
\includegraphics[width=\textwidth,clip]{Figures/BM/Temperature1200.pdf}
\caption{Time $=24$}
\end{center}
\end{subfigure}
\begin{subfigure}{0.495\columnwidth}
\begin{center}
\includegraphics[width=\textwidth,clip]{Figures/BM/Temperature1400.pdf}\\
\caption{Time $=28$}
\end{center}
\end{subfigure}
\begin{subfigure}{0.495\columnwidth}
\begin{center}
\includegraphics[width=\textwidth,clip]{Figures/BM/Temperature1600.pdf}
\caption{Time $=32$}
\end{center}
\end{subfigure}
\caption{Time evolution of the isocontours of the temperature field in the 2D buoyant Boussinesq mixing layer is shown over a period of $32$ non-dimensional time units. }
\label{fig:buoyantmixing_time_evolution}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=1.0\columnwidth, scale=1.0]{lockexchange/full/FP1600.pdf}
\caption{Time evolution of the POD weight features, $a_i^t$ for the buoyant mixing flow.}
\label{fig:buoyantmixing_PODWeights}
\end{figure}
\subsection{Analysis Framework}\label{ss:analframework}
In this section we summarize the different candidate model architectures and learning algorithms. Table~\ref{t:methods}, lists the different MSM architectures and the comparable MEM architectures along with the total number of learning parameters ($\mathcal{LP}$) to be estimated. The first column under each class of sequential and end-to-end map in table~\ref{t:methods} verbalizes the multilayer structure and the second column represents the feature dimension of the different layers. For readability and conciseness, we have excluded the first and last layers corresponding to the input and output state vectors whose dimension is reduced by projecting onto a POD basis. Here we remind the reader of the nomenclature used to denote the different architectures as `L-Method-$\mathcal{N}-N_f$'.
As noted earlier, $N_f$ represents the feature dimension growth factor from the $2^{nd}$ to the $3^{rd}$ layers in the FFNN architectures. For the MSM maps, we invariably denote $N_f=1$ as the
dimension of the inner layer features are determined by the choice of nonlinear map $\mathcal{N}$.
The rightmost column represents the total number of learning parameters ($\mathcal{LP}$) to be estimated from training. For example, when using 6-MSM-P2-1 or EDMD-P2 in table~\ref{t:methods} we learn an operator ($\Theta$) with $81\ (9\textrm{x}9)$ parameters and similarly, when using 6-MEM-TS-3 we learn operators($[\Theta_1, \Theta_3, \Theta_3]$ as in section~\ref{ss:markovGOC}) totaling $135\ (27+81+27)$ parameters. The six-layer EDMD-P (i.e. 6-MSM-P2-1 with quadratic polynomial features) method generates $9$ features in the intermediate layer which is then used to learn a linear map between the $9$ features at the next intermediate layer followed by reverse map to the penultimate layer with $3$ POD features. A similar construct is designed for the FFNN (MEM) framework using 6-MEM-TS-3 (i.e. $N_f = 3$). While the EDMD-P requires learning $81$ parameters, the FFNN with 6-MEM-TS-3 requires estimating $135$ parameters. In the following analysis of the predictive performance, we find that using just $3$ POD features with a 2nd order polynomial expansion in EDMD-P does not produce accurate results. So, in addition to P2, we also explore higher order polynomial basis explore if better predictions can be realized.
\begin{table}
\begin{center}
\ra{1.6}
\begin{tabular}{@{}lcccccccccc@{}}
\toprule
\phantom{a}&\phantom{a}&\multicolumn{2}{c}{\bf \it Sequential Maps}& \phantom{a}& $\mathcal{LP}$ & \phantom{a}& \multicolumn{2}{c}{\bf \it End-to-End Maps}& \phantom{a}& $\mathcal{LP} $\\
\cmidrule{3-4} \cmidrule{6-6} \cmidrule{8-9} \cmidrule{11-11}
\addlinespace
$1$&& \multicolumn{2}{c}{\bf DMD}&& && \multicolumn{2}{c}{\bf FFNN-Linear} && \\
&& 4-MSM-$\mathcal{I}-1$ & 3-3 && 9 && 6-MEM-$\mathcal{I}-1$ & 3-3-3-3 && 27 \\
\midrule
\addlinespace
$2$&& \multicolumn{2}{c}{\bf EDMD-TS} && && \multicolumn{2}{c}{\bf FFNN ($N_f=1$)}&& \\
&& 6-MSM-TS-1 & 3-3-3-3 && 9 && 6-MEM-TS-1 & 3-3-3-3 &&27\\
\addlinespace
$3$&& \multicolumn{2}{c}{\bf EDMD-P} && &&\multicolumn{2}{c}{\bf FFNN ($N_f=3,9,20$)}&& \\
&& 6-MSM-P2-1 & 3-9-9-3 && 81 && 6-MEM-TS-3 & 3-9-9-3&& 135 \\
&& \multirow{2}{*}{6-MSM-P7-1} & \multirow{2}{*}{3-125-125-3}&& \multirow{2}{*}{15,625} && 6-MEM-TS-9 & 3-27-27-3&& 891 \\
&& && & && 6-MEM-TS-20 & 3-60-60-3&& 3960 \\
\bottomrule
\end{tabular}
\caption{Overview of the different model architectures used as part of this analysis. The dimensions of different layers correspond to that used for cylinder wake flow. }
\label{t:methods}
\end{center}
\end{table}
\begin{figure}[H]
\begin{center}
\begin{subfigure}{\columnwidth}
\begin{center}
\includegraphics[width=0.6\columnwidth]{RE100/Data_window_100.pdf}
\caption{Times series plot of the weights corresponding to the three most energetic POD modes with different training regions (a) Limit cycle ($16-20$): 124 data points, (b) transient region-I ($8-20$): 372 data points and (c) Transient region-II ($4-16$): 372 data points, where each cycle consists of $31$ data points.}
\label{f:datawindRE100}
\end{center}
\end{subfigure}\\
\caption{Schematic showing the different training regions chosen for prediction using the different models. }
\end{center}
\end{figure}
\subsection{Training, Validation and Error Quantification in \emph{Posteriori} Predictions}\label{ss:trainValidate}
A key aspect of data-driven modeling is to minimize overfitting so that realistic learning can be realized. To achieve this the data generated from computer simulations described above are separated into training and testing regimes. The training data set is used for learning the optimal $\Theta$'s using which \emph{a posteriori} predictions are computed with the earlier prediction(s) alone as the input to mimic a practical usage of the model. For this study , we assess model performance based on both qualitative representation of the dynamics and \emph{posteriori} prediction errors unlike the \emph{a priori} error estimates used in machine learning community. We quantify model errors using the $\mathcal{L}_{2}$ norm of the \emph{posteriori} prediction error from the data-driven model relative to the truth which requires accurate specification of only the initial condition $\pmb{a}_{0}$. To bypass the complexities of computing the $2-$norm, we instead compute the Forbenius norm of the error as in eqn.~\eqref{e:l2error}.
\begin{equation}
\mathcal{E}_{t,p} = \frac{1}{2M_{i}} \| \bar{Y}^1_{p} - \bar{Y}^1 \|_{2}^{2}.
\label{e:l2error}
\end{equation}
In the above equation $\bar{Y}^1_{p}$ represents the posteriori prediction of the data-driven model and $\bar{Y}^1$ the true data. We make separate quantifications of the posteriori error in the training region where the data-driven model is operating in reconstruction mode and in the testing region where the model operates in a prediction or extrapolation role. The posteriori error in the training region is denoted by $\mathcal{E}_t$ and combined error in both the training and testing regions is denoted by $\mathcal{E}_p$.
To assess and characterize the robustness of the different architectures (table \ref{t:methods}) we train the models across various data regimes corresponding to different dynamics of the flow, i.e. transient unstable wake or stable limit cycle regime with periodic vortex shedding. To this end, we identified three different training regions (see fig.~\ref{f:datawindRE100}) highlighted by windows shaded in grey. A stiff test for any data-driven model is to learn the underlying dynamics using information from the steady wake regime as shown in fig.~\ref{f:datawindRE100} and predict the growth of instability which ultimately stabilizes into limit cycle. From our experience models that use only training information from the steady wake region to predict the vortex shedding dynamics are highly unstable. Consequently, we designed two different training regions (TR I and TR II) where the flow transitions across flow regimes, but with different proportions of limit-cycle (vortex shedding) and steady wake content.
The figures in row (a) represent a training region in the limit-cycle regime and rows (b) and (c) correspond to regions in the transition part of the dynamics and denoted by region I (or TR-I) and region II (or TR-II).
In the following sections we will highlight and discuss the key results from our data-driven aposteriori predictions.
These training regions are further divided into training data ($\approx 70\%$) and validation data ($\approx 30\%$) uniformly as shown in fig.\ref{f:TrainTestdata} to assess and validate (check for overfitting) learning performance. In fig.\ref{f:Cost8_20}, we show the learning cost evolution for TR-I obtained for the FFNN with a design specified by 6-MEM-TS and $N_f = 1,3,9,20$. We see that the learning cost for training dataset and validation dataset are same, which signifies model generalization. A similar trend was observed for all the MEM models used in this study. We also note that for all these posteriori predictions, a regularization parameter in range $(1e^{-12} - 1e^{-8})$ was used. The FFNN models learned in this study show difference between \emph{a priori} and \emph{a posteriori} predictions as shown in figs.~\ref{f:predictions} and \ref{f:Timeseriespredictions}. For the \emph{a priori} predictions, we see that the dynamics are predicted accurately while for the \emph{a posteriori} case there exists deviations from the true data. It is worth reminding that while the \emph{a priori} analysis (Fig.\ref{f:apriori}) is directly correlated to the learning cost, the posteriori analysis(fig.\ref{f:apriori}) shows how the accumulated error interacts with the learned model. From the timeseries of priori and posteriori predictions in figs.\ref{f:aprioriTimeseries} and \ref{f:aposterioriTimeseries}, we see that the posteriori error growth impacts the shift POD mode the most. The shift mode~\cite{Noack:03hierarchy} represents the shift in trajectory of the system from an unstable regime to a neutrally stable regime. We have seen that including a bias term in the FFNN (6-MEM-TS-3) models decrease this prediction error (see Appendix.\ref{s:Appendix1}). In the following sections we highlight the key results from our data-driven \emph{a posteriori} predictions.
\begin{figure}[ht!]
\begin{center}
\begin{subfigure}{0.485\columnwidth}
\begin{center}
\includegraphics[width=0.9\columnwidth]{RE100/8_20/TrainTest_8_20_RE100.pdf}
\caption{TR-I region uniformly divided into training and \\ validation data. The green dots denote the validation \\ dataset and black line with dots denote training dataset.}
\label{f:TrainTestdata}
\end{center}
\end{subfigure}%
\begin{subfigure}{0.485\columnwidth}
\begin{center}
\includegraphics[width=0.9\columnwidth]{RE100/8_20/Cost_8_20_RE100.pdf}
\caption{Learning cost evolution with respect to epochs on TR-I using FFNN (6-MEM-TS-$N_f$ with $N_f = 1,3,9,20$). The symbols represent the error cost of the test dataset.}
\label{f:Cost8_20}
\end{center}
\end{subfigure}
\caption{Schematic showing training performance of FFNNs. (a) Separation of input data into training and testing sets on a phase plot; (b) Comparison of learning performance for the transient regime training region I. }
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\begin{subfigure}{0.485\columnwidth}
\begin{center}
\includegraphics[width=0.9\columnwidth]{RE100/8_20/apriori_8_20_RE100.pdf}
\caption{\emph{a priori} prediction of TR-I using 6-MEM-TS3.}
\label{f:apriori}
\end{center}
\end{subfigure}%
\begin{subfigure}{0.485\columnwidth}
\begin{center}
\includegraphics[width=0.9\columnwidth]{RE100/8_20/aposteri_8_20_RE100.pdf}
\caption{\emph{a posteriori} prediction of TR-I using 6-MEM-TS3. The black dots denote the entire set of features $[a_1,a_2,a_3]$}
\label{f:aposteriori}
\end{center}
\end{subfigure}
\caption{\emph{a Priori} vs \emph{a Posteriori} predictions using FFNNs. }
\label{f:predictions}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\begin{subfigure}{\columnwidth}
\begin{center}
\includegraphics[width=1.05\columnwidth]{RE100/8_20/FP_apriori_RE100.pdf}
\caption{\emph{a priori} prediction of TR-I using 6-MEM-TS3.}
\label{f:aprioriTimeseries}
\end{center}
\end{subfigure}\\
\begin{subfigure}{\columnwidth}
\begin{center}
\includegraphics[width=1.05\columnwidth]{RE100/8_20/FP_aposteriori_RE100.pdf}
\caption{\emph{a posteriori} prediction of TR-I using 6-MEM-TS3.}
\label{f:aposterioriTimeseries}
\end{center}
\end{subfigure}
\caption{Timeseries of features from \emph{a priori} and \emph{a posteriori} prediction compared with the true data. }
\label{f:Timeseriespredictions}
\end{center}
\end{figure}
\subsection{\emph{a Priori} Learning and \emph{a Posteriori} Prediction of Limit-cycle Cylinder Wake Dynamics}\label{ss:limitcycle}
The focus of this section is to learn from limit-cycle training data and predict the corresponding limit-cycle physics over long durations. Successful prediction of this case is considered a benchmark for data-driven models. The underlying theme in this article is to explore whether iterative end-to-end learning of the model parameters ($\mathcal{LP}$) can outperform one-time sequential learning of the model parameters for predictions of fluid flows. To verify this we compare the following four models namely: DMD (4-MSM-$\mathcal{I}$-1, FFNN-linear (6-MEM-$\mathcal{I}$-1), EDMD-TS (6-MSM-TS-1) and FFNN-TS (6-MEM-TS-1). The FFNN-linear architecture can be viewed as a multilayer neural network analogue of linear map-based methods such as DMD. Similarly, EDMD-TS can be viewed as a MSM analogue of the standard FFNN architecture. Therefore, these set of two pairs of architectures can provide useful insight into the role of learning methodology (sequential vs end-to-end) and nonlinear functions in multilayer maps. It is well known that DMD performs well in the limit cycle region as shown in \cite{Lu:18sparse,Rowley:17ARev} and under performs in the strongly nonlinear transient regimes on account of being a linear model of the state. In fig.~\ref{f:ltcyc}, the time series posteriori predictions of the first three POD features are shown with rows $1-4$ (top-to-bottom) representing outcomes from the learned parameters ($\Theta$s) obtained using the DMD, FFNN-linear (6-MEM-$\mathcal{I}$-1), EDMD-TS (6-MSM-TS-1) and FFNN (6-MEM-TS-1) architectures respectively.
Specifically, we assess the role of sequential versus end-to-end optimization of the parameters as well as the impact of nonlinear mapping on model prediction.
The first major observation is that both the sequential and end-to-end models with linear mapping predict the overall dynamics relatively accurately while the sequential model with nonlinear sigmoid mapping damps the POD features over time.
The second observation is that all the models show gradual error growth with time except the standard FFNN (6-MEM-TS-1) architecture. The plots in fig.~\ref{f:ltcyc} convey that a nonlinear mapping is not essential to capturing the limit-cycle dynamics, but if used, should be carefully designed. For example, it was shown in \cite{Lu:18sparse} that EDMD-P2 can predict such dynamics very well while the current results (fig.\ref{f:ltcyc}(c)) show that the same architecture with a tansigmoid function (EDMD-TS) produces errors. The TS function is primarily used in machine learning for classification and has a \emph{squashing} nature to it, i.e. it has the effect of compressing the features which explains its inability to predict the dynamics.
A plausible reason could be that the TS nonlinearity does not extend the space of learning parameters in contrast to polynomial basis. Nevertheless, when the TS nonlinearity (using the $\mathcal{N}$) is combined with an end-to-end framework such as the well known FFNN, the prediction drastically improves as learning the parameters in $\Theta_1,\Theta_2,\Theta_3$ simultaneously while applying the TS nonlinearity produces a compensatory and powerful outcome. Further, this FFNN model can predict over long times without growth in error as seen from the evolution of third POD feature (shift mode) in fig.\ref{f:ltcyc}(d).
\begin{figure}
\includegraphics[width=0.33\columnwidth]{RE100/16_20/A1_16_20_RE100.pdf}\hfill
\includegraphics[width=0.32\columnwidth]{RE100/16_20/A2_16_20_RE100.pdf}\hfill
\includegraphics[width=0.32\columnwidth]{RE100/16_20/A3_16_20_RE100.pdf}
\caption{Times series of \emph{posteriori} predicted POD features (\xdash[0.5em] \xdash[0.5em] \xdash[0.5em]) obtained from (a) DMD (4-MSM-$\mathcal{I}$-1), (b) FFNN-linear (6-MEM-$\mathcal{I}$-1), (c) EDMD-P (6-MSM-TS1-1) and (d) FFNN (6-MEM-TS-1) are plotted with their respective original data (\xdash[1.5em]) in the limit cycle regime.}
\label{f:ltcyc}
\end{figure}
We had mentioned earlier that the success of the FFNN/MEM frameworks possibly comes from learning an extended parameter ($\mathcal{LP}$) space, but the following discussion shows that this is true only in the presence of a nonlinear function as part of the mapping. In the DMD framework, there are $9$ learning parameters in $\Theta$ to predict the limit cycle dynamics as compared to $27$ parameters for FFNN-linear \cmnt{6-MEM-$\mathcal{I}$-1} architecture. However, in the absence of an nonlinear function in the map, the linear operator computed from the two methods turned out to be the same, i.e. the product of the different $\Theta_l\ \textrm{for } l=1-3$ from FFNN-linear is same as the $\Theta$ learned from DMD.
In fig.\ref{f:ltcyc}, we use 4-cycles of (124 points) data in the limit cycle region for training and predict upto 17 cycles (527 data points). We see that the predictions obtained using DMD and FFNN-linear \cmnt{(6-MEM-$\mathcal{I}$)} in fig.\ref{f:ltcyc}(a) and (b) are similar as the same linear transition operator is estimated. However, with limited training data, the predictions start to diverge from the truth over large times as is clearly seen from the evolution of the third POD feature, $a_3$.
While the addition of nonlinear functions in the map aids the prediction of nonlinear dynamics, employing this formulation with a local optimization of the $\mathcal{LP}$ does not always guarantee good results. We see an illustration of this in the performance of the EDMD-TS \cmnt{(6-MSM-TS-1}} architecture as seen from fig.\ref{f:ltcyc}(c), where all the three input features are incorrectly predicted in contrast to predictions by the FFNN \cmnt{(6-MEM-TS1)} in fig.\ref{f:ltcyc}(d). The \emph{a posteriori} prediction error quantifications for the limit-cycle regime in the training and testing regions are shown in the first two rows of the table \ref{t:predstatRe100}. These show that the DMD and FFNN \cmnt{(6-MEM-TS-1}} produce error magnitudes of $7.4\times 10^{-3}$ and $1.6\times 10^{-2}$ respectively outside the training region. These errors are higher than the $O(1e^{-4})$ values in the training region as one would expect. In spite of generating more prediction errors, the MEM models cap their growth which is a desirable feature. As additional benchmarks we also include testing region errors for other architectures including EDMD-P2 \cmnt{(6-MSM-P2-1)}, FFNN \cmnt{(6-MEM-TS-1)} and FFNN with $N_f=3$ \cmnt{(6-MEM-TS-3)} which generate comparable prediction accuracy with EDMD-P2 \cmnt{(6-MSM-P2-1)} being the smallest. In summary, except for the EDMD-TS \cmnt{(6-EDMD-TS-1)} all the other models display reasonable accuracy for this limit-cycle dynamics in both the training and prediction regimes. However, we observe gradual error growth \cmnt{of the third feature} in all the models except for the FFNN \cmnt{(6-MEM-TS-1)} which has implications for long-time predictions.
\cmnt{It is for this reason that we consider the MEM architectures to perform the best within this regime.}
\begin{table*}
\begin{center}
{
\ra{1.6}
\begin{tabular}{@{}lccccccccccccc@{}}
\toprule
Train & \phantom{a}& \multicolumn{1}{c}{DMD}& \phantom{a}&\multicolumn{1}{c}{EDMD-TS} & \phantom{a}& \multicolumn{2}{c}{EDMD-P} & \phantom{a} & \multicolumn{4}{c}{FFNN}\\
arch & \phantom{a}& \multicolumn{1}{c}{(4-MSM-$\mathcal{I}$-$N_f$)}& \phantom{a}&\multicolumn{1}{c}{(6-MSM-TS-$N_f$)} & \phantom{a}& \multicolumn{2}{c}{(6-MSM-Pp-$N_f$)} & \phantom{a} & \multicolumn{4}{c}{(6-MEM-TS-$N_f$)}\\
\cmidrule{3-3} \cmidrule{5-5} \cmidrule{7-8} \cmidrule{10-13}
cycles & $\mathcal{E}$ & $N_f = 1$ && $N_f = 1$ && p = $2$ & p =$7$ &&$N_f =1$ & $N_f =3$ & $N_f =9$ & $N_f =20$ \\ \midrule\\
\multirow{2}{*}{$16-20$}&$\mathcal{E}_t$ &$1.6e^{-4}$ &&$3.3e^{-2}$ &&$6.7e^{-5}$ &$--$ &&$2.7e^{-4}$ &$2.1e^{-4}$ &$--$ &$--$\\
{(LC)}&$\mathcal{E}_p$ &$7.4e^{-3}$ &&$0.269$ &&$3.5e^{-4}$ &$--$ &&$1.6e^{-2}$ &$8.1e^{-3}$ &$--$ &$--$\\
\multirow{2}{*}{$08-20$}&$\mathcal{E}_t$ &$0.417$ &&$0.467$ &&$0.475$ &$6.4e^{-6}$ &&$0.320$ &$3.7e^{-2}$ &$1.9e^{-2}$ &$2.1e^{-2}$\\
{(TR-I)}&$\mathcal{E}_p$ &$0.513$ &&$0.483$ &&$0.776$ &$3.9e^{-4}$ &&$0.686$ &$0.146$ &$0.153$ &$0.148$\\
\multirow{2}{*}{$04-16$}&$\mathcal{E}_t$ &$0.246$ &&$0.238$ &&$0.223$ &$0.191$ &&$--$ &$0.106$ &$0.182$ &$0.385$\\
{(TR-II)}&$\mathcal{E}_p$ &$0.551$ &&$0.530$ &&$0.683$ &$0.977$ &&$--$ &$0.883$ &$0.948$ &$0.720$\\
\\
\bottomrule
\end{tabular}}
\caption{\emph{a Posteriori} Prediction error estimates for the different MSM and MEM architectures for $Re = 100$ data across training regimes. \label{t:predstatRe100}}
\end{center}
\end{table*}
\subsection{\emph{a Priori} Learning and \emph{a Posteriori} Prediction of Transient Cylinder Wake Dynamics }\label{ss:transientdata}
In the earlier section, we highlighted the importance of nonlinearity in the map and its combination with a MEM framework for stable long-time predictions. In this section, we focus on learning from transient wake flow data and predict the resulting limit-cycle system. It is well known that DMD performs better on limit cycle problems and underperforms in the transient regime due to its inability to handle the enhanced nonlinear instability growth that characterizes the underlying dynamical system. In particular, if the limit-cycle dynamics represents a nonlinearity of order $k$ then the transient wake regime corresponds to a nonlinearity of order $\geq k+1$~\cite{Noack:03hierarchy}.
Consequently, models that incorporate nonlinearity in the map such as the EDMD-P with polynomial basis~\cite{Williams:15} or the corresponding kernel representation~\cite{Williams:14arXiv} perform better for such problems, but only when using significant number of input features. In this section, we show that end-to-end learning of a nonlinear multilayer map provides much better prediction capabilities from as little input data as three features which is the minimum needed to capture the wake instability behind a cylinder~\cite{Noack:03hierarchy}.
\begin{figure}
\begin{center}
\includegraphics[width=0.33\columnwidth]{RE100/8_20/F1_A1_8_20_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/8_20/F1_A2_8_20_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/8_20/F1_A3_8_20_RE100.pdf}
\caption{Times series of predicted POD features obtained from (a) DMD (4-MSM-$\mathcal{I}$-1), (b) EDMD-TS (6-MSM-TS1) and (c) FFNN (6-MEM-TS-1) for TR-I as training region.}
\label{f:MT_F1}
\end{center}
\end{figure}
\subsubsection{Choice of Training Data:} For this analysis, we used two training regions in the unstable transition regime, namely transient region-I (TR-I) and transient region-II (TR-II) as shown in figure \ref{f:datawindRE100} corresponding to $8-20$ and $4-16$ cycles respectively with both regions consisting of 372 data points. TR-I is relatively less challenging as almost all of the training data incorporates vortex shedding, but with an amplitude that is growing. In TR-II the first $30\%$ of the training data includes a stable wake with onset of instability that grows in amplitude all through the regime. This has implications for predictions using machine learning models where the training data almost always determines what kind of dynamics the model can predict. If one were to rank the level of difficulty in predicting the resulting limit-cycle dynamics from different sets of training data then the most difficult would be TR-II followed by TR-I and lastly, the limit-cycle training data\cmnt{ used in the previous section}.
\subsubsection{Posteriori Predictions with Training Region I (TR I)}
\paragraph{\underline{Posteriori Predictions with Insufficient Nonlinearity and $\mathcal{LP}$ Dimension} :} Figure~\ref{f:MT_F1} shows the predictions obtained from the different multilayer sequentially maps such as DMD and EDMD-TS and multilayer end-to-end FFNN for the TR-I training region. We see that all these methods fail to learn the nonlinear dynamics and predict the resulting limit-cycle system to varying levels of inaccuracy with MEM being the least. This can be attributed to the lack of sufficient nonlinearity in the models and insufficient learning parameters to capture the dynamics. Highly transient systems with instability do not adhere to a point spectrum and require many eigenmodes to represent the unstable growth phase of the dynamics. However, once it settles into a limit cycle, a discrete spectrum is sufficient to represent the system. This correlates with a need for nonlinearity and increase in learning parameters in the data-driven architecture for modeling such systems. It is worth pointing out that the EDMD-TS does not extend the $\mathcal{LP}$ space as against its polynomial variant, EDMD-P2. Also, the choice of P2 basis is physics-driven to account for the quadratic nonlinearity of the POD features as embedded within the Navier-Stokes equations that describe the flow. On the other hand, for the MEM architectures, a logical way to extend the $\mathcal{LP}$ space is to increase the number of features in the intermediate layers by increasing $N_f$. Consequently, we use EDMD-P2 as the baseline case and design a MEM architecture with similar sized $\mathcal{LP}$ space with feature factor, $N_f=3$. This approach of choosing $N_f$ based on the dimension of the quadratic polynomial features is a logical way to design MEM architectures as against more \emph{ad hoc} choices.
For EDMD-P2 (6-MSM-P2-1), the three input POD features are mapped onto a polynomial basis space with nine features. In the FFNN (6-MEM-TS-3), the three input features are mapped onto an unknown basis space, but guaranteed to be optimal for the chosen architecture and given training data. In this spirit of exploration, we also try a $7^{th}$-order polynomial feature map, i.e. a EDMD-P7 (6-MSM-P7-1) and corresponding MEM architectures with an increased $\mathcal{LP}$ dimension ($N_f=9$ and $20$) to assess the effect of $\mathcal{LP}$ dimensionality on the predictions. A downside to increasing the $\mathcal{LP}$ dimension is a tendency to overfit the data which we will address.
\par Figure \ref{f:MT_F2} shows the predictions from EDMD-P2\cmnt{(6-MSM-P2-1)} and FFNN ($N_f=3$)\cmnt{(6-MEM-TS-3)} using TR-I data. In spite of the embedded quadratic nonlinearity, the EDMD-P2 fails to the predict the correct limit-cycle dynamics using just three input features. On the other hand, FFNN ($N_f=3$)\cmnt{(6-MEM-TS-3)} with a similar architecture but with end-to-end learning predicts the dynamical evolution of the more accurately. These prediction error trends are quantified in table \ref{t:predstatRe100}. This is consistent with our expectation that a increasing $\mathcal{LP}$ dimension improves predictions as FFNN ($N_f=3$)\cmnt{(6-MEM-TS-3)} learns $135$ parameters compared to $27$ for the FFNN ($N_f=1$)\cmnt{(6-MEM-TS-1)} case. On the other hand, the EDMD-P2\cmnt{(6-MSM-P2-1)} case with $\mathcal{LP}=81$ fails to even predict qualitatively accurate results in spite of the added nonlinearity through the quadratic features. In a related work by Jayaraman et al.~\cite{Lu:18sparse}, we have observed that EDMD-P2 with nearly fifty input features (with $1325$ quadratic nonlinear features and $\mathcal{LP}=1.7\times10^6$) can predict this transient instability driven growth of the wake. It is also worth noting that $N_f=3$)\cmnt{(6-MEM-TS-3)} predicts the first two POD features accurately (see fig.~\ref{f:MT_F2}), but the third coefficient is biased towards a zero magnitude. We have found that this can mitigated through the use of a bias term which when incorporated into the MEM architectures corrects for this systematic deviation as discussed and shown in fig.~\ref{f:WBias_820Re100} included in Appendix \ref{s:Appendix1}.
\begin{figure}
\begin{center}
\includegraphics[width=0.33\columnwidth]{RE100/8_20/F2_A1_8_20_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/8_20/F2_A2_8_20_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/8_20/F2_A3_8_20_RE100.pdf}
\caption{Times series of predicted POD features obtained from (a) 6-EDMD-P2, (b) 6-MEM-TS3 for TR-I as training region. }
\label{f:MT_F2}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.33\columnwidth]{RE100/8_20/F3_A1_8_20_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/8_20/F3_A2_8_20_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/8_20/F3_A3_8_20_RE100.pdf}
\caption{Times series of predicted POD features obtained from (a) 6-EDMD-P7, (b) 6-MEM-TS9 and (c) 6-MEM-TS20 for TR-I as training region.}
\label{f:MT_F3}
\end{center}
\end{figure}
\paragraph{\underline{Effect of Increased $\mathcal{LP}$ Dimension on Posteriori Predictions:}}
Here, we explore the effect of expanding the $\mathcal{LP}$ dimension with just $3$ input features on the model performance. We accomplish this by increasing the order of polynomial to $7^{th}$-degree for the MSM i.e. we consider a EDMD-P7 with the architecture denoted by 6-MSM-P7-1. For this method, using just $3$ POD features results in an $\mathcal{LP}$ dimension of $15,625$- a nearly $\approx 200$ time increase as compared EDMD-P2. With just $372$ snapshots being used, learning a linear operator $\Theta$ of size $15,625\times 15,625$ leads to overfitting and is expected to generate good results. In fact, increasing the $\mathcal{LP}$ dimension by a couple of orders of magnitude produces accurate predictions of the nonlinear dynamics as shown in fig.~\ref{f:MT_F3}a. We note that choices of polynomial smaller than degree seven did not produce accurate predictions although there may exist an isolated regularized solution that is reasonably accurate.
We also explore the effect of increasing the $\mathcal{LP}$ dimension for the MEM architectures by changing $N_f$ as shown in table~\ref{t:methods}. The predictions obtained using FFNN ($N_f=9$)\cmnt{ (6-MEM-TS-9)} and FFNN ($N_f=20$)\cmnt{ (6-MEM-TS-20)} (see figs.\ref{f:MT_F3}(b) and (c)) with $\mathcal{LP}$ dimension of $891$ and $3960$ respectively (factors of $\approx 10 \ \& \ 40$) also showed improved performance and compare favorably to the outcomes from the EDMD-P7 architecture. To address concerns of overfitting associated with these large $\mathcal{LP}$ dimension, we performed validation of the learning process by splitting the data into training and testing sets as discussed in section~\ref{ss:trainValidate}. The outcomes shown in fig.~\ref{f:Cost8_20} clearly indicate that the error cost between training and testing remain consistent indicative of little overfitting for the MEM models with $N_f=1,3,9,20$. Another indication of how MEM models reduce overfitting as compared to the MSM (i.e. EDMD-P7) is how the prediction saturates as one increases the $\mathcal{LP}$ dimension.
In summary, both the sequential and end-to-end architectures work better by increasing the $\mathcal{LP}$ dimension and introducing nonlinearity. However, MEM requires relatively modest increases in $\mathcal{LP}$ dimension for substantial increases in performance. Contrastingly, the MSM frameworks require large growth in features and $\mathcal{LP}$ dimension for performance improvement and is prone to overfitting the data. In a way, this result reinforces the underlying principles behind the success of deep learning architectures~\cite{bengio2015deep}. The MSM can be viewed as a two-layer shallow learning architecture requiring larger intermediate layer dimensions while the MEM is its deep learning counterpart requiring smaller number of intermediate layer features, but across multiple layers which in turn reduces overfitting.
\subsubsection{Posteriori Predictions with Training Region II (TR II)}
We use the same modeling architecture's as before for this challenging TR-II dataset and the resulting predictions of the POD features are shown in figures \ref{f:FT_F2} and \ref{f:FT_F3}. In this case both the MSM architectures, i.e. EDMD-P2 and EDMD-P7 perform inadequately in spite of the increased $\mathcal{LP}$ dimension. On the other hand, predictions obtained using FFNN (MEM) offer better qualitative results and predict the limit cycle dynamics, but display perceptible quantitative inaccuracy without a bias term and is insensitive to extension of learning parameter space (see table~\ref{t:predstatRe100}). However, as before, we observe that this quantitative inaccuracy, especially in the third POD feature is mitigated through the inclusion of a bias term as the plots clearly show in fig.~\ref{f:WBias_416Re100} in Appendix~\ref{s:Appendix1}.
\subsubsection{Analysis of Prediction Errors}
We note that computing the error metrics using a simple $L_2$ norm does not adequately represent the qualitative nature of the predictions accurately for such repetitive limit-cycle dynamics. For example, the predictions which qualitatively mimic the dynamics, but with incorrect phase tends to show larger errors than some of the non-qualitative predictions. The other aspect worth mentioning is that learning is based on \emph{a priori} prediction cost minimzation and not the \emph{a posteriori} predictions (as shown in section~\ref{ss:trainValidate} and figs.~\ref{f:apriori} \& \ref{f:apriori}) that takes into account error propagation. We can understand this clearly by studying the compilation of the error metrics in table \ref{t:predstatRe100}. While the learning cost ($\mathcal{J}$) for the different FFNN architectures is $O(1e^{-6})$ (see fig.\ref{f:Cost8_20}), the associated posteriori prediction errors are of $O(1e^{-1})$. It is well known that classical machine learning is based on \emph{a priori} prediction cost and is not designed for time-series estimation where error propagation is significant. Recent approaches~\cite{pan2018long} propose improved regularizations that account for error growth through the use of a Jacobian of the cost function. To relate the observed deviations in the POD features to the predicted flow field of interest, we show in fig.\ref{f:Reconst} the reconstructed solution (i.e. the actual predicted state vector) for $Re=100$ obtained using the different methods considered in this paper. These plots are generated based on learning and prediction using TR-I $(cycles: \ 8-20)$ data, and shown at $\approx T=86.2$ (first column) which is the midpoint of the training region. Columns 2 and 3 in fig.\ref{f:Reconst} represent predictions at $T=124$, the last point in TR-I and $T=205$, the last point in the prediction regime. These results clearly show that the MSM frameworks with low $\mathcal{LP}$ dimension such as DMD, EDMD-TS and EDMD-P2 show delayed onset of wake instability and incorrect vortex shedding while the FFNN for all the different architectures predict the instability growth accurately.
\begin{figure}
\begin{center}
\includegraphics[width=0.33\columnwidth]{RE100/4_16/F2_A1_4_16_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/4_16/F2_A2_4_16_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/4_16/F2_A3_4_16_RE100.pdf}
\caption{Times series of predicted POD features obtained from (a) 6-EDMD-P2, (b) 6-MEM-TS3 for TR-II data. }
\label{f:FT_F2}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.33\columnwidth]{RE100/4_16/F3_A1_4_16_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/4_16/F3_A2_4_16_RE100.pdf}
\includegraphics[width=0.32\columnwidth]{RE100/4_16/F3_A3_4_16_RE100.pdf}
\caption{Times series of predicted POD features obtained from an extended $\mathcal{LP}$ space (a) 6-EDMD-P7, (b) 6-MEM-TS9 and (c) 6-MEM-TS20 for TR-II data.}
\label{f:FT_F3}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.95\columnwidth]{RE100/contf/CADT_8_20_RE100.pdf}
\includegraphics[width=0.95\columnwidth]{RE100/contf/CD312_8_20_RE100.pdf}
\includegraphics[width=0.95\columnwidth]{RE100/contf/CED312TF_8_20_RE100.pdf}
\includegraphics[width=0.95\columnwidth]{RE100/contf/CED332P_8_20_RE100.pdf}
\includegraphics[width=0.95\columnwidth]{RE100/contf/CED372P_8_20_RE100.pdf}
\includegraphics[width=0.95\columnwidth]{RE100/contf/CFF312_8_20_RE100.pdf}
\includegraphics[width=0.95\columnwidth]{RE100/contf/CFF332_8_20_RE100.pdf}
\includegraphics[width=0.95\columnwidth]{RE100/contf/CFF392_8_20_RE100.pdf}
\caption{Reconstruction of $Re100$ flow field based on predicted POD features obtained from (a) Actual data, (b) DMD (4-MSM-$\mathcal{I}$-1) (c) EDMD-TS (6-MSM-TS-1) (d) EDMD-P2 (6-MSM-P2-1) (e) EDMD-P7 (6-MSM-P7-1) (f) FFNN with $N_f=1$ (6-MEM-TS-1) (g) FFNN with $N_f=3$ (6-MEM-TS-3) (h) FFNN with $N_f=9$ (6-MEM-TS-9). The contours levels are given by fifteen equally spaces values ranging between $(-0.2645,1.2963)$ .}
\label{f:Reconst}
\end{center}
\end{figure}
\subsection{\emph{A priori} Learning and \emph{A posteriori} Prediction of a Transient 2D Buoyant Boussinesq Mixing Flow }
\label{ss:evo_transientdata}
Unlike the low-dimensional limit-cycle attractor modeled in the earlier sections, here we explore a non-stationary and higher-dimensional buoyant Boussinesq mixing flow discussed in sec.~\ref{sss:buoymix}. In fact, we observed previously that prediction of the transient instability growth and subsequent stabilization of the cylinder wake dynamics is highly sensitive to the choice of training data. In addition, learning and predictability of these dynamics are also dependent on the training data including sufficient information for accurate prediction. For this study, we chose to retain just $80\%$ of the total energy of the system captured in the CFD generated data snapshots (similar to a low resolution measurement) resulting in just $3$ POD features in the $2^{nd}$ layer of the MSM and MEM architectures. Sensitivity to these aspects is stronger when trying to predict non-stationary phenomena that may settle into an unknown attractor over long times. The training data is almost always insufficient to represent all the possible dynamics for such systems and may not overtly show any evidence of the existence of such an attractor. Such instability-driven non-stationary problems are challenging for data-driven techniques that do not leverage knowledge of the underlying governing system and employ black box machine learning. Even if one were to diversify the training data-set with multiple realizations of the system, performance improvements are not guaranteed as the underlying dynamics will depend a lot on the initial state. For this study, we choose a single realization of such a data-sparse and low-dimensional representation of a system for assessment of the different MSM and MEM architectures .
In particular, we consider DMD, EDMD-TS and EDMD-P for the MSM class of methods and contrast these with different FFNN (MEM with $N_f = 1,3,5$ ) models that incorporate a growing $\mathcal{LP}$ dimension. As a preliminary step, we use the entire available data for training and assess the reconstruction performance of these models. Figure \ref{f:F1_1600} compares the results for the DMD with the nonlinear EDMD-TS1 and FFNN ($N_f=1$) models with small number of learning parameters ($3,9$ and $27$ respectively). Contrary to findings from the earlier sections for the transient cylinder wake, all the MSM frameworks including the linear DMD and nonlinear EDMD-TS1 compare favorably to the FFNN/MEM with $N_f=1$. All three models fail to predict the dynamics of the third POD feature which represents the secondary eddies from the Kelvin-Helmholtz instability generated by the mixing layer dynamics (see bottom plot in fig.~\ref{f:lockexmodes}). The MSM models generate slightly better outcomes as compared to MEM for the first two POD features that represent transverse and vertical mixing (top two plots in fig.~\ref{f:lockexmodes}). To improve the predictions of the third POD feature, we enhance the learning parameter ($\mathcal{LP}$) dimension by employing EDMD-P3 (6-MSM-P3-1), FFNN with $N_f=3$ (6-MEM-TS-3) and FFNN with $N_f=5$ (6-MEM-TS-5) as shown in fig.~\ref{f:F3_1600}. Consistent with earlier observations, this increase in $\mathcal{LP}$ improves the prediction of the third feature for both the multilayer sequential and end-to-end learning methods with the former performing better. Similar performance was also realized with the EDMD-P2 architecture and is not reported here for brevity. This shows that for reconstructing the dynamics, MSM architectures are more accurate as compared to the MEM frameworks that leverage nonlinear regression techniques. An aspect that is relatively under-explored in the study of FFNNs is role of nonlinear mapping, $\mathcal{N}$ on learning performance. For example, this current study shows that for the MSM class of methods, EDMD-TS performs inadequately relative to the different variants of EDMD-P thus hinting that a polynomial basis being better suited to approximate this data. Give this, it is only natural to speculate whether MEM architectures would perform better with other choices of nonlinear maps although such exploration is beyond the scope of this study.
\begin{figure}
\begin{center}
\includegraphics[width=0.81\columnwidth]{Figures/BM/phi1_1.pdf}
\includegraphics[width=0.81\columnwidth]{Figures/BM/phi2_1.pdf}
\includegraphics[width=0.81\columnwidth]{Figures/BM/phi3_1.pdf}
\caption{Visualization of the first three POD basis (in decreasing order of energy content) used to model the dynamics with the data-driven models.}
\label{f:lockexmodes}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.33\columnwidth]{lockexchange/full/F1_A1_F1600.pdf}
\includegraphics[width=0.32\columnwidth]{lockexchange/full/F1_A2_F1600.pdf}
\includegraphics[width=0.32\columnwidth]{lockexchange/full/F1_A3_F1600.pdf}
\caption{Comparison of the time evolution of the posteriori prediction of the $3$ POD features generated from the different modes with the true data using all the $1600$ snapshots for training. The different plots correspond to: (a)DMD, (b) EDMD-TS1 and (c) FFNN with $N_f=1$.}
\label{f:F1_1600}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.33\columnwidth]{lockexchange/full/F3_A1_F1600.pdf}
\includegraphics[width=0.32\columnwidth]{lockexchange/full/F3_A2_F1600.pdf}
\includegraphics[width=0.32\columnwidth]{lockexchange/full/F3_A3_F1600.pdf}
\caption{Comparison of the time evolution of the posteriori prediction of the $3$ POD features generated from the different modes with the true data using all the $1600$ snapshots for training. The different plots correspond to: (a) EDMD-P3 , (b) FFNN with $N_f=3$ and (c) FFNN with $N_f=5$.}
\label{f:F3_1600}
\end{center}
\end{figure}
To assess the ability of the models to learn the underlying system dynamics, we split the dataset ($1600$ snapshots) equally into training (i.e. $800$ snapshots for training) and prediction regimes. Figure \ref{f:F1_800} compares the posteriori predictions of the three POD features for EDMD-TS1, EDMD-P2 and FFNN with $N_f=3$. For all these models, we clearly observe that reconstruction is better than true prediction performance indicating that dynamics is evolving rapidly and the data-driven models are finding it hard to forecast physics that it has not seen through training.
Therefore, focusing on the predictions, the multilayer end-to-end learning FFNN outperforms the two MSM architectures considered here in terms of stability and accuracy. Particularly, the FFNN/MEM prediction using $50\%$ data (fig.\ref{f:F1_800}c ) is highly similar to that obtained from reconstruction of the entire dataset as shown in fig.~\ref{f:F3_1600}c. This shows that these models offer robust and stable performance even with limited data. In summary, we see that MSM frameworks offer competitive reconstruction performance, but MEM learning models with different architectures (at least for the different examples considered in this study) offer stable and robust model performance for long time predictions using limited data.
\begin{figure}
\begin{center}
\includegraphics[width=0.33\columnwidth]{lockexchange/half/F1_A1_F800.pdf}
\includegraphics[width=0.32\columnwidth]{lockexchange/half/F1_A2_F800.pdf}
\includegraphics[width=0.32\columnwidth]{lockexchange/half/F1_A3_F800.pdf}
\caption{Comparison of the time evolution of the posteriori prediction of the $3$ POD features generated from the different modes with the true data using $50\%$ data i.e. $800$ snapshots for training. The different plots correspond to: (a) EDMD-TS1 , (b) EDMD-P2 and (c) FFNN with $N_f=3$.}
\label{f:F1_800}
\end{center}
\end{figure}
\section{Discussion and Summary}\label{s:conclusions}
Fluid flows represent multiscale PDE dynamical systems that often require low-dimensional data-driven representations and evolutionary models for a multitude of applications.
In this article we explore the performance of multilayer sequential maps (MSMs) versus multilayer end-to-end maps (MEMs) in Markov models for learning and long-time prediction of nonlinear fluid flows using small amounts of training data. In particular, we assess the role of learning parameter dimension and nonlinear transfer functions on the ability of the architecture to reconstruct and predict over long times without overfitting to the data. The sequential multilayer frameworks (MSMs) allow for both backward and forward mapping operations in symmetric architectures and can support the estimation of the Koopman operator for spectral analysis and linear control in addition to serving as data-driven models. On the other hand, multilayer maps that incorporate end-to-end (MEMs) learning from data can support only forward maps within the framework of an asymmetric Markov model due to the use of gradient-based optimization algorithms employed to solve the nonlinear regression problem. Consequently, architectures like FFNN cannot learn the Koopman operator in base configuration although recent advancements~\cite{Shiva:18AIAA} can help bypass this limitation.
The major outcomes of the study are as follows. The success of both the MSM and MEM architectures is tied to the choice of nonlinearity in the mapping and the dimension of the learning parameter space embedded in the design of the multilayer architecture. We observe that for prediction of limit-cycle dynamics from limit-cycle data both the MSM and MEM-based models show reasonable success although MEM models such as FFNN control the growth of long-time prediction errors better than any of the MSM model considered. Further, MEM architectures generate the most accurate predictions for a given learning parameter ($\mathcal{LP}$) budget as long as the map incorporates appropriate nonlinear functions. In the absence of nonlinear functions in the map, the $\mathcal{LP}$ dimension did not impact the predictions. Further, any choice of nonlinear function will not produce good results. We observed that tansigmoid functions operate well with MEM architectures while polynomial nonlinearity fared well with MSMs.
To assess the ability of these model architectures to generalize across diverse training data regimes, we considered two different case studies with different training regimes that differ in their proportion of limit-cycle to unstable wake growth dynamics. To mimic the availability of limited resolution data as is commonly the case, we chose to train these models using their low-dimensional representation with only three POD features. With these constraints, we observed that for comparable number of learning parameters, the FFNN (MEM) architectures outperform the corresponding MSM frameworks by a significant margin in terms of accuracy and robustness.
To illustrate this, we show that the FFNN\cmnt{( 6-MEM-TS-1)} architecture with $N_f=1$ and $9$ learning parameters produce qualitatively accurate results as against the gross inaccuracy of MSM frameworks such as DMD\cmnt{ (4-MSM-$\mathcal{I}$-1)} and EDMD \cmnt{(6-MSM-P2-1 and 6-MSM-TS-1)}. With increase in $\mathcal{LP}$ dimension, both class of methods converge to the accurate predictions although the MSM reaches their slowly and results in significant overfitting as compared to MEM architectures.
The downside of MEM-based models is the added computational cost and learning time which limits the dimension of the input feature space for practical applications. The use of iterative gradient-based search algorithms impact convergence with a tendency for being stuck in local minima. However, this is compensated by more efficient learning from data i.e. requires only a relatively modest increase in $\mathcal{LP}$ dimension for improved predictions. All these perceived advantages of MEM over MSM (and vice versa) are valid only in the limit of availability of sufficient data which injects a dose of reality regarding data-driven modeling approaches. We observed this when training a model for a different flow regime (TR-II) that contained little information about the limit-cycle dynamics where both class of methods found learning and prediction harder. Yet, the MEM architectures were able to generate qualitatively accurate predictions with as little as $135$ learning parameters whereas the equivalent MSM architectures could not generate meaningful predictions.
We also explored the performance of the various data-driven modeling approaches for an instability-driven, non-stationary, buoyant mixing flow which requires unlimited amounts of data to represent all the possible dynamics of the system. While we knew the challenge faced by data-driven methods for such problems, we analyzed how the various models fared in learning-based reconstruction and learning-based prediction of such flows. While both class of methods struggle to generate accurate predictions, the MSM-based models perform well in reconstruction whereas the MEM-based models offer better predictions and model generalization.
In summary, the strategy of extending the $\mathcal{LP}$ space, learning the model parameters concurrently using a end-to-end maps and improved regularizations can help improve learning from data for robust and accurate predictions. However, as with machine learning in general, these outcomes are strongly tied to data sufficiency and quality.
\begin{acknowledgement}
We acknowledge support from Oklahoma State University start-up grant and OSU HPCC for compute resources to generate the data used in this article. The authors thank Chen Lu, a former member of the Computational, Flow and Data science research group at OSU for providing the CFD data sets used in this article. \cmnt{BJ acknowledges discussions on data-driven modeling with Prof. Karthik Duraisamy at the University of Michigan.}
\end{acknowledgement}
\section*{Author Contributions}
BJ conceptualized the work with input from SCP. SCP developed the data-driven modeling codes used in this article with input from BJ. BJ and SCP analyzed the results. BJ developed the manuscript with contributions from SCP.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,610 |
Professor Susskind is one of the country's most experienced public and environmental dispute mediators and a leading figure in the dispute resolution field. He has mediated more than fifty complex disputes related to the siting of controversial facilities, the setting of public health and safety standards, the formulation and implementation of development plans and projects, and conflicts among racial and ethnic groups — serving on occasion as a special court-appointed master.
Professor Susskind was the founder and Senior Editor of Environmental Impact Assessment Review, a peer-reviewed journal for twenty years. He was also the publisher of Consensus, a quarterly newspaper distributed to every elected body in the United States by the Public Disputes Network. He was co-Editor of the Negotiation Series published by Sage Publications, co-Editor of the Environmental Policy and Management Series published by Island Press, as well as co-editor of the International Environmental Negotiation Annual published by the Program on Negotiation at Harvard Law School. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,190 |
\section{Introduction}
PID is one of the most popular controllers in industry \cite{AR4} because it is easily implemented and tuned and has a simple structure. Despite its popularity, it is a linear controller, which is limited due to the waterbed effect \cite{Waterbed} and bode's phase gain relation \cite{AR88}. The solution to this problem for overcoming the aforementioned limitation lies in the nonlinear control theory\cite{HIGSOVERCOME}. To this end, there have been several attempts, such as variable gain \cite{AR5}, split path nonlinear (SPAN) filters \cite{AR6}, and reset controllers \cite{AR9,Clegg,FORE1}.\\
Among those, the reset controller has attracted a lot of attention since it can be easily implemented in a PID framework. J.C. Clegg was first to introduce the reset integrator in the late 1950s \cite{Clegg}, which others named after him: the Clegg integrator. This integrator can reset its state and thus has a nonlinear behavior. The Clegg integrator has only $-38.15^\circ$ phase lag but similar gain behavior compared to a linear integrator. This property is interesting because it breaks Bode's phase-gain relation. The phase lead reduces overshoot and improves the settling time of the system. In addition to Clegg, there have been several other reset controllers such as first-order reset element (FORE) \cite{FORE1}, generalized FORE \cite{FORE2},\cite{GFrORE}, constant in the gain lead in phase (CgLp) \cite{CgLp}.
Reset systems, despite their advantages, have a severe drawback, and it is the harsh jump on its output. This discontinuity is not desired as it may cause practical issues like the saturation of actuator in the motion control systems \cite{ExtendedHIGS}.\\
Hybrid Integrator Gain Systems (HIGS) has been recently proposed to avoid the unwanted jumps in the reset control systems. HIGS is presented as a piecewise linear system with no jump on its output \cite{enh}. In \cite{HIGSTr}, a linear bandpass filter has been compared to a hybrid integrator-gain-based bandpass filter to control an active vibration isolation system. In \cite{Remedy}, it has been shown that the overshoot inherent when using any stabilizing linear time-invariant feedback controller can be eliminated with a HIGS-based control strategy. In addition, the stability of HIGS has been extensively discussed in \cite{ProjStab}, \cite{LOOPHIGS}. \\
The frequency-domain analysis is preferred in the design of linear motion controllers since it allows intuitively ascertaining closed-loop performance measures. HIGS is also analyzed in the frequency domain \cite{datadriven}. To this end, the describing function of HIGS is obtained from the Fourier expansion of output signal. According to describing function, HIGS acts as a linear low pass filter in gain but leads 52$^\circ$ in phase. Unlike generalized reset control systems \cite{Freset}, this phase lead is not controllable in HIGS. By changing the after reset value in reset control systems, the phase lead is tuned, but doing the same for HIGS will cause discontinuity. Therefore, in this paper, we fill this gap in the state of art by generalizing HIGS using fractional calculus. Fractional calculus has been used for control designs like fractional-order PID \cite{podlubny1999fractional,xue2007fractional,tavazoei2012traditional} and CRONE control \cite{oustaloup1993great}. Recently, fractional-order elements have also been used within reset elements \cite{sebastian2021augmented}, \cite{karbasizadeh2021fractional} in order to regulate the higher-order harmonics in the reset control systems.\\ The principal contribution of this paper is to design a new HIGS where the integer-order integrator is replaced by the fractional-order one. Varying the order of integrator from one to zero, the proposed fractional-order HIGS varies between linear and nonlinear behavior. The design of this filter enables the extension of the reset-based CgLp element to the HIGS-based CgLp element. In addition, describing function is obtained as another contribution to analyze the controller in the frequency domain.\\
The remainder of this paper is organized as follows.
In section II, the background information of HIGS will be given, then the describing function of HIGS will be discussed, and finally, some fundamental definitions of the fractional-order derivative will be brought. In section III, the fractional-order HIGS will be introduced and formulated in state-space. Then, the describing function corresponding to fractional-order HIGS will be calculated. Finally two different approaches to compensate the slope loss in fractional integrator will be introduced and named generalized HIGS. In section IV, an illustrative example including the implementation condition, controller design, and results will be given. Section V summarizes the main conclusions of the article.\vspace{-4mm}\\
\section{Background}
\subsection{Hybrid integrator-gain system}
The hybrid integrator-gain system (HIGS) is defined in \cite{enh} and its state-space representation is given by:
\begin{equation}
H : \begin{cases}
\label{eq.HIGSSS}
\dot{x}_h(t)=\omega_h e, & \quad \text{if } (e,\dot{e},u) \in F_1\\
x_h(t)=k_h e, & \quad \text{if } (e,\dot{e},u) \in F_2\\
u=x_h,
\end{cases}
\end{equation}
where $x_h\in\mathbb{R} $ is the state variable, $u\in\mathbb{R}$ is the control output, $e\in\mathbb{R} $ is the input, $ k_h\geq 0$ is the gain value, $ \omega_h\geq 0$ is the integral frequency. Also, $F_1$ and $F_2$ denote the regions where the integrator mode or gain mode are active, and they are defined as:
\begin{subequations}
\label{HIGS domains}
\begin{align}
F_1 &:= \left\{
(e,\dot{e},u) \in R^3\mid eu\geq\frac{1}{k_h}u^2 \wedge (e,\dot{e},u) \notin F_2 \label{eq:DomF1} \right\},\\
F_2 &:= \bigg\{
(e,\dot{e},u) \in R^3\mid u=k_h e \wedge \omega_h e^2 > k_h e\dot{e} \label{eq:DomF2} \bigg\},
\end{align}
\end{subequations}
the $F_1$ and $F_2$ regions are set for three important reasons. First of all, the output stays continuous all over the time, and the output is always bounded between the input and zero. And finally, the control output $u(t)$ of the HIGS will always be in the same direction as the input signal $e(t)$, which represented as error. This is shown in Fig. \ref{fig.HIGS}. \vspace{-4mm}\\
\subsection{Describing function of HIGS}
To analyze the system in the frequency domain, describing function analysis can be done. In \cite{datadriven} the describing function of HIGS for a sinusoidal input has been given by:
\begin{align}
\label{HDF}
D(j\omega)=\frac{\omega_h}{j\omega}(\frac{\gamma}{\pi}+j\frac{e^{-2j\gamma}-1}{2\pi}-4j\frac{e^{-j\gamma}-1}{2\pi}) \\
+k_h(\frac{\pi-\gamma}{\pi}+\frac{e^{-2j\gamma}-1}{2\pi}) \label{eq:DF HIGS},\nonumber
\end{align}
where $\gamma=2 \arctan(\frac{k_h\omega}{\omega_h})$. In Fig. \ref{fig.HIGS freq}, the Bode plot of HIGS for different values of $\omega_h$ and $k_h$ is illustrated.
\subsection{Fractional-order derivative}
In this section, we define the Liouville-Caputo fractional-order derivative as an approach that we will use for fractional-order HIGS calculations \cite{exact}. Its definition is given by:
\begin{equation}
{}^{LC}_{}D^{\alpha}_x f(x)=\frac{1}{\Gamma(1-\alpha)}\int_ {-\infty}^x dt(x-t)^{-\alpha} \frac{df(t)}{dt},
\label{eq:LC}
\end{equation}
where $0\leq \alpha \leq1$ is derivative order, $x$ is upper bound of integral and $\Gamma(.)$ is the Euler Gamma function. For a sinusoidal function $f(t)$ we have:
\begin{equation}
\label{Frsin}
{}^{LC}_{}D^{\alpha}_x[\sin(\omega t)]=\omega^\alpha \sin(\omega t + \frac{\pi \alpha}{2}).
\end{equation}
For convenience in the following, writing $LC$ in ${}^{LC}_{}D^{\alpha}_x$ will be refrained.
\begin{figure}
\centering
\includegraphics[scale=0.42,trim=4 4 4 8,clip]{Figs/multisincolor2.pdf}
\setlength{\abovecaptionskip}{-5pt}
\caption{\centering Time domain response of HIGS for multi-sine input $e(t)=\sin(\omega t)+0.7\sin(3\omega t)$. Blue color shows the $F_1$ region and green color shows the $F_2$ region.}
\label{fig.HIGS}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.5,trim=4 10 4 30,clip]{Figs/HIGSfreq.pdf}
\setlength{\belowcaptionskip}{-15pt}
\caption{\centering Result of describing function for HIGS with various amount of $\omega_h$ and $k_h$.}
\label{fig.HIGS freq}
\end{figure}
\section{Generalized HIGS}
\subsection{Fractional-order HIGS}
As shown in \eqref{eq.HIGSSS}, the hybrid system under $F_1$ condition is in its integral mode, but here we are interested in changing the state space and condition regions in a way that the hybrid system takes fractional-order integrator to integrate into $F_1$ region. Therefore, for fractional-order HIGS, the state-space representation is defined as:
\begin{equation}
\mathscr{H} : \begin{cases}
\label{eq.FrHIGSSS}
D^{\alpha}_t x_h(t)=\omega_h e, & \quad \text{if } (e,\dot{e},u) \in \mathscr{F}_1,\\
x_h(t)=k_h e, & \quad \text{if } (e,\dot{e},u) \in \mathscr{F}_2,\\
u=x_h.\\
\end{cases}
\end{equation}
Being continuous and sector boundedness are two important features in HIGS. These features are also considered in fractional-order HIGS. To this respect, the domains in \eqref{HIGS domains} should be revised as follow:\\
\begin{subequations}
\label{eq:frF1F2}
\begin{align}
\mathscr{F}_1 &:= \left\{
(e,\dot{e},u) \in R^3\mid eu\geq\frac{1}{k_h}u^2 \wedge (e,\dot{e},u) \notin \mathscr{F}_2 \label{eq:FrDomF1} \right\},\\
\mathscr{F}_2 &:= \bigg\{
(e,\dot{e},u) \in R^3\mid u=k_h e \wedge \bigg( \omega_h D_t^{1-\alpha}(e)e> k_h\dot{e}e \nonumber \\
&\vee \omega_h D_t^{1-\alpha}(e)e< 0\bigg) \label{eq:FrDomF2} \bigg\}.
\end{align}
\end{subequations}
According to the above regions, it is obvious that the output is continuous because all switch points are always on the signal $k_h e$. The sector boundedness for the output is investigated in Remark 1. \vspace{-1mm}
\begin{remark}
System \eqref{eq.FrHIGSSS} satisfying conditions in \eqref{eq:frF1F2} is sector bounded (bounded in the region $[0,k_h e]$).
\end{remark}
To this end, we set some conditions so that the integral mode starts and ends inside the region $[0,k_h e]$ and the integral signal is monotonic. Due to the continuity of the output, these conditions guarantee that the output is sector bounded. These conditions is studied mathematically as follows.\\
With the condition $eu\geq\frac{1}{k_h}u^2$ in domain $\mathscr{F}_1$, the output never exits the region $[0,k_h e]$ at the end of the integral mode
and the integral is forced to start inside the region $[0,k_h e>0]$, if:
\begin{equation}
\omega_h D_t^{-\alpha}e\leq k_h e. \nonumber
\end{equation}
By differentiating from the both sides, we have:
\begin{equation}
\omega_h D_t^{1-\alpha}(e)\leq k_h\dot{e}, \nonumber
\end{equation}
and to be true for negative error values, we have:
\begin{equation}
\omega_h D_t^{1-\alpha}(e)e\leq k_h\dot{e}e. \label{eq:sect1}
\end{equation}
Also, due to the fact that the fractional-order integrator is not always accumulative, we need \eqref{eq:sect2} to ensure that the integral signal is monotonic at the beginning of the region $\mathscr{F}_1$:
\begin{equation}
\omega_h D_t^{1-\alpha}(e)e\geq 0.
\label{eq:sect2}
\end{equation}
The conditions \eqref{eq:sect1} and \eqref{eq:sect2} are simultaneously necessary at the beginning of the integral part to guarantee that the output is sector bounded. Therefore, if one of these conditions are not satisfied, the system maintain on its gain mode. To this respect, the NOT of these conditions are added to the $\mathscr{F}_2$ domain.\\
The output of the fractional-order HIGS for a sinusoidal input $e(t)=\sin(t)$ is depicted in Fig. \ref{fig:single}. It can be seen by decreasing $\alpha$ from 1 to 0 with fixed parameters $\omega_h=1$ and $ k_h=1 $, the output gradually goes from HIGS to gain mode. Also, given the importance of being sector bounded in HIGS, Fig. \ref{fig:sectore} shows that the output of the fractional-order HIGS is bounded between $0$ and $k_h \sin(t)$ in this case.
\begin{figure}
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.44,trim=20 20 20 4,clip]{Figs/singlesin3.pdf}
\caption{}
\label{fig:single}
\end{subfigure}%
~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.47,trim=22 4 2 4,clip]{Figs/sector2.pdf}
\caption{}
\label{fig:sectore}
\end{subfigure}
\setlength{\belowcaptionskip}{-15pt}
\setlength{\abovecaptionskip}{-10pt}
\caption{\centering Time domain response of fractional-order HIGS\qquad a) Sinusoidal input response. b) $e-u$ plane. }
\label{fig:Fractional}
\end{figure}
\subsection{Describing function analysis for fractional-order HIGS}
To obtain describing function, first, the time domain output signal of the fractional-order HIGS with input $e=\hat{e}\sin(\omega t)$ is obtained as:
\begin{equation}
u(t) = \begin{cases}
\label{eq.Fr HIGS TD}
\omega_h \omega^{-\alpha} \hat{e} \big(\sin(\omega t-\frac{\pi \alpha}{2})+\sin(\frac{\pi \alpha}{2})\big) & 0\leq t<\frac{\gamma}{\omega} \\
k_h \hat{e} \sin(\omega t) & \frac{\gamma}{\omega}\leq t<\frac{\pi}{\omega} \\
\omega_h \omega^{-\alpha} \hat{e} \big(\sin(\omega t-\frac{\pi \alpha}{2})-\sin(\frac{\pi \alpha}{2})\big) & \frac{\pi}{\omega}\leq t<\frac{\gamma +\pi}{\omega} \\
k_h \hat{e} \sin(\omega t). & \frac{\gamma+\pi}{\omega}\leq t<\frac{2\pi}{\omega} \\
\end{cases}
\end{equation}
The fractional modes for time intervals of $0\leq t<\frac{\gamma}{\omega}$ and $\frac{\pi}{\omega}\leq t<\frac{\gamma +\pi}{\omega}$ are obtained from \eqref{Frsin}. Switching between two modes, i.e., fractional-order integral mode and proportional mode, happens at $t=\gamma/\omega$. Since fractional-order HIGS is continuous, thus by substituting $\omega t=\gamma$ in the first two equations of \eqref{eq.Fr HIGS TD} and equalizing them we have:
\begin{equation}
\label{eq:gamma calc}
\omega_h \omega^{-\alpha} \big(\sin(\gamma -\frac{\pi \alpha}{2})+\sin(\frac{\pi \alpha}{2})\big)=k_h \sin(\gamma).\\
\end{equation}
By solving \eqref{eq:gamma calc}, we have:
\begin{align}
\label{eq:gamma}
\gamma=\arccos(X),
\end{align}
where:
\begin{align}
X=\frac{b-a}{b+a},\quad a=(B \sqrt{1-A^2}-C)^2, \quad b=B^2 A^2, \qquad \nonumber \\
A=sin(\frac{\pi \alpha}{2}), \quad B=\omega_h \omega^{-\alpha}, \quad C=k_h. \qquad \qquad \nonumber \\
\end{align}
The Fourier expansion of $u(t)$, can be obtained as:
\begin{equation}
u(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n \cos(n\omega t)+b_n \sin(n\omega t))
\label{Fourier},
\end{equation}
where $a_n$ and $b_n$ are the Fourier coefficients and described as:
\begin{subequations}
\label{eq:anbn}
\begin{align}
a_n=\frac{1}{T}\int_{0}^{T}u(t)\cos(n\omega t) dt \\
b_n=\frac{1}{T}\int_{0}^{T}u(t)\sin(n\omega t) dt.
\end{align}
\end{subequations}
Also, $n$ is a positive integer and $T$ is the period of signal $u(t)$. By substituting \eqref{eq.Fr HIGS TD} in \eqref{eq:anbn} and set $n=1$ for $a_1$ and $b_1$ we have:
\begin{subequations}
\label{a1b1}
\begin{align}
&a_1= \frac{\omega_h}{4\pi}\omega^{-\alpha}\hat{e}\bigg[-\cos(2\gamma-\frac{\pi \alpha}{2})+\cos(\frac{\pi \alpha}{2})-2\gamma \sin(\frac{\pi \alpha}{2})+ \nonumber \\
&2\cos(\gamma-\frac{\pi \alpha}{2})-2\cos(\gamma+\frac{\pi \alpha}{2})\bigg]+\frac{k_h}{4\pi}\hat{e}[\cos(2\gamma -1)] \quad \\
&b_1=\frac{\omega_h}{4\pi}\omega^{-\alpha}\hat{e}\bigg[2\gamma \cos(\frac{\pi \alpha}{2})-\sin(2\gamma -\frac{\pi \alpha}{2})+3\sin(\frac{\pi \alpha}{2})- \nonumber \\
&2\sin(\gamma + \frac{\pi \alpha}{2})+2\sin(\gamma-\frac{\pi \alpha}{2})\bigg]
+\frac{k_h}{4\pi}\hat{e}[2\pi - 2\gamma+\sin(2\gamma)].
\end{align}
\end{subequations}
According to \cite{khalil} describing function is calculated as:
\begin{equation}
\mathscr{D}(\omega , \hat{e})=\frac{b_1 + ja_1}{\hat{e}}.
\label{Df}
\end{equation}
Fig. \ref{fig.Fr HIGS freq} shows the describing function of fractional-order HIGS obtained from \eqref{a1b1} and \eqref{Df}. By substituting $\alpha=1$ in \eqref{a1b1} the describing function is equal to that of HIGS in \eqref{HDF}. We can see that by changing parameter $\alpha$, the output phase can vary between $-38^{\circ}$ and $0^{\circ}$. It represents the behavior of the system represented in \eqref{eq.FrHIGSSS}. Also, it verifies that by decreasing $\alpha$, the system behaves as a proportional gain.
\begin{figure}
\centering
\includegraphics[scale=0.6,trim=4 4 4 10,clip]{Figs/FrHIGSfreq.pdf}
\caption{\centering Describing function for fractional-order HIGS with various values of $\alpha$.}
\label{fig.Fr HIGS freq}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[scale=0.4,trim=12 12 12 4,clip]{Figs/ArchB.pdf}
\caption{}
\label{fig:ArchA}
\end{subfigure}%
~
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[scale=0.4,trim=4 90 1 4,clip]{Figs/ArchA.pdf}
\caption{}
\label{fig:ArchB}
\end{subfigure}
\setlength{\belowcaptionskip}{-15pt}
\caption{\centering Two block diagrams for generalization of HIGS\qquad \quad \quad a) Architecture $a$ ($\mathscr{H}_f$),\quad b) Architecture $b$ \qquad \qquad \quad where $\omega_r$ is the cut-off frequency.}
\label{fig:Architecture}
\end{figure}
\subsection{Architecture design for generalization of HIGS}
In this subsection, we utilize the fractional-order HIGS to construct a generalized HIGS ($\mathscr{H}_f$) where the parameter $\alpha$ can reproduce a linear low pass filter or HIGS if $\alpha$ is chosen to be 0 or 1, respectively. The architecture of the proposed generalized HIGS is shown in Fig. \ref{fig:ArchA}. Architecture $a$ utilizes a fractional-order HIGS as described in \eqref{eq.FrHIGSSS} and a complementary linear filter to set the order of generalized HIGS to 1. Any $\alpha$ between 0 to 1 generalizes the HIGS such that the phase lag of the filter varies between $-90^\circ$ to $-38^\circ$, as shown in Fig. \ref{fig.FrHIGS Imp}. With this architecture, the gain of the describing function is unchanged with respect to the variation of $\alpha$. Similar analysis can be done with the structure $b$, as shown in Fig. \ref{fig:ArchB}. Architecture $b$ is inspired by the so-called PI+CI structure in reset control \cite{banos2012reset}. This structure consists of a linear LPF and a HIGS parallel to construct the generalized HIGS. The value of $\beta$ shows the percentage of utilization of each element. Setting this value to 0 represents a linear filter, while the value of 1 results in a HIGS. Any value between 0 and 1 adapts the phase lag of the filter from $-90^\circ$ to $-38^\circ$.
\begin{figure}
\centering
\includegraphics[scale=0.6,trim=4 4 4 4,clip]{Figs/FrHIGSImp.pdf}
\setlength{\belowcaptionskip}{-15pt}
\caption{\centering Frequency response of generalized HIGS ($\mathscr{H}_f$) with fractional-order integrator approach.}
\label{fig.FrHIGS Imp}
\end{figure}
Although both architectures in Fig. \ref{fig:Architecture} generalize HIGS. Architecture $a$ is advantageous over architecture $b$. Since it utilizes a linear low pass filter after the nonlinear element, it attenuates the higher-order harmonics by $1/n^{(1-\alpha)}$ where $n$ is the order of harmonics. However, the higher-order harmonics are reduced by a constant value of $1-\beta$ in the architecture $b$.\\ To visually illustrate this, we have compared the FFT of two architectures to achieve a generalized HIGS with a phase lag of $-57^{\circ}$. To achieve this phase lag we set $\alpha=0.68$ and $\beta=0.5$. As can be seen from Fig. \ref{fig:harmonics}, not only the higher-order harmonics of architecture $a$ are smaller than $b$, they are descending with the increase of $n$. Also, in Fig. \ref{fig:3rd}, it is shown that by changing parameter $\alpha$, the amount of the third harmonic of architecture $a$ is always less than architecture $b$. A similar trend can be expected for the higher-order harmonics.
\section{Illustrative example}
In this section, in order to illustrate the time response of generalized HIGS, we control a single mass (double integrator) system with a proportional-integral-derivative (PID) controller and replace the linear integrator with an integrator made by fractional-order generalized HIGS.
\begin{figure}
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.42,trim=20 4 1 4,clip]{Figs/harmonics4.pdf}
\caption{}
\label{fig:harmonics}
\end{subfigure}%
~
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.45,trim=20 4 2 4,clip]{Figs/3rd3.pdf}
\caption{}
\label{fig:3rd}
\end{subfigure}
\caption{\centering a) Higher-order harmonics for Architecture $a$ and $b$ with $\alpha=0.68$ and $\beta=0.5$, b) The third harmonic of two architecture obtained by varying $\alpha$ and $\beta$.}
\label{fig:harmonics2}
\end{figure}
\subsection{Controller design}
The PID controller ($C_{pid}$) consists of a lead in series connection to a PI and a gain $k_p$:
\begin{equation}
C_{pid}(s)=K_p\bigg(\frac{1+\frac{s}{\omega_d}}{1+\frac{s}{\omega_t}}\bigg)\bigg(1+\omega_i \mathcal{I} \bigg),
\label{Cpid}
\end{equation}
where $\omega_t$ and $\omega_d$ are the cut-off frequencies of the lead term and $\mathcal{I}$ is the integrator term in PI, which is given by:
\begin{equation}
\label{I}
\mathcal{I}=\mathscr{H}_f\times(1+\frac{\omega_r}{s}). \qquad \qquad \\
\end{equation}
As shown in the previous section, the term $\mathscr{H}_f$ in \eqref{I} can vary between linear and nonlinear behavior. Therefore, the nonlinearity of integrator ($\mathcal{I}$) can be controlled by parameter $\alpha$. Here in the PID structure by, setting $\alpha=0$, the $\mathcal{I}$ takes the linear integral, and with $\alpha=1$, it takes nonlinear integral, and for $\alpha$ between $1$ and $0$, the $\mathcal{I}$ handles a trade-off between these two behaviors.\\
Without loss of generality, we have chosen the crossover frequency as $\omega_c=200\pi (100Hz)$, and set parameters of the controller as $K_p=\omega_c^2/1.8$, $\omega_d=\omega_c/1.8$, $\omega_t=1.8\omega_c$ and $\omega_i=\omega_c/10$.
\subsection{Architecture design}
The sequence of elements in reset control is very important. In \cite{cheng}, it has been shown for using lead or lag element in series with the reset element, the optimal sequence is Lead-Reset-Lag. Therefore, the lead filter in \eqref{Cpid}, is divided into two elements as below:
\begin{equation}
\label{divid}
\bigg(\frac{1+\frac{s}{\omega_d}}{1+\frac{s}{\omega_t}}\bigg) \implies \underbrace{\bigg({1+\frac{s}{\omega_d}}\bigg)}_{PD} \times \underbrace{\bigg(\frac{1}{1+\frac{s}{\omega_t}}\bigg)}_{LPF}
\end{equation}
According to the above decomposition, the block diagram of the controller is depicted in Fig. \ref{fig.Arch C}. Note that in an actual system, the PD term should be implemented in a proper form. \vspace{-4mm}
\begin{figure}
\centering
\includegraphics[scale=0.4,trim=4 4 4 4,clip]{Figs/ArchC2.pdf}
\setlength{\belowcaptionskip}{-15pt}
\setlength{\abovecaptionskip}{-15pt}
\caption{\centering Simplified schematic of the PID controller with integrator $\mathcal{I}$, that can be variable between linear and non-linear integrator.}
\label{fig.Arch C}
\end{figure}
\subsection{Results}
In Fig. \ref{fig.VaryAlpha}, the step response of the closed-loop system with controller $C_{pid}$ \eqref{Cpid} is depicted.
It confirms that by changing parameter $\alpha$, the output signal can vary between the responses of linear and nonlinear control systems. By looking at the step response of the system from Fig. \ref{fig.VaryAlpha}, it is clear that by increasing the nonlinearity of the controller, overshoot and settling time are decreased. Therefore, the close loop system has better performance in transient response. Also, it can be seen from the step response by choosing the $\alpha$ between $0$ and $1$, the overshoot and settling time can be decreased as desired.\\
In this example, the step response shows the improvement in transient response, where the conventional HIGS performs the best compared to generalized HIGS. It is not the purpose of this paper to show the superiority of either of these controllers rather open a path for more flexible design in the future.\\
Fractional-order HIGS can be advantageous in improving sready state response. As can be seen from Fig. \ref{fig:3rd} the amount of HIGS's third harmonic (the green dot) is always greater than the generalized one. It can be problematic, especially for more complex systems with high-frequency modes and control systems where the precision is of concern \cite{karbasizadeh2021fractional}. The generalized HIGS compromises between improving the transient response and reduction of higher-order harmonics for higher precision and tracking of motion.
\section{Conclusions}
It has been known that HIGS can overcome fundamental limitations of linear control, which leads to better results without the harmful behavior of reset controllers. Unlike reset control systems, where the reset action can be tuned and its nonlinearity level (the phase lag of describing function) is controllable, the phase lag in HIGS cannot be tuned. In this paper, we proposed a novel fractional-order HIGS that overcomes the aforementioned limitation in HIGS and extends it for more general applications. The describing function was determined for the proposed filter. According to describing function, it has been shown that the phase lag of generalized HIGS is variable between $-38.15^{\circ}$ and $-90^{\circ}$. Hence by using this system as an integrator, the output can vary between linear and nonlinear behaviors. In the end, the proposed new filter was used in form of a PID to control a double integrator (mass) system. The results validate the generality of generalized HIGS that can be utilized in the future to construct a continuous constant gain lead phase filter.
\begin{figure}
\centering
\includegraphics[scale=0.53,trim=4 4 4 4,clip]{Figs/VaryAlpha2.pdf}
\setlength{\belowcaptionskip}{-15pt}
\caption{\centering Step response of the closed-loop system with PID controller made by generalized HIGS.}
\label{fig.VaryAlpha}
\end{figure}
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,234 |
\section{Introduction}
In the last decade, ultracold gases of dipolar particles, which include atoms with a large magnetic moment and polar molecules,
attracted a great deal of interest \cite{Baranov,Pfau,Carr,Pupillo2012}. Being electrically or magnetically polarized such particles
interact with each other via long-range anisotropic dipole-dipole forces, which drastically changes the nature of quantum degenerate
regimes. Experiments with chromium atoms (magnetic moment $6\mu_B$) \cite{Pfau}, together with theoretical studies \cite{Santos,Eberlein}, have
revealed the dependence of the shape and stability diagram of trapped dipolar Bose-Einstein condensates on the trapping geometry and interaction
strength. They initiated spinor physics with quantum dipoles \cite{Santos2,pasqu}, and now also dysprosium \cite{ming} and erbium \cite{erbium} atoms (magnetic moments $10\mu_B$ and $7\mu_B$, respectively)
have entered the game.
Recently, fascinating prospects for the observation of novel quantum phases have been opened by the creation of ultracold clouds
of polar molecules and cooling them to almost quantum degeneracy \cite{kk,Carr}. In this case, the dipole-dipole forces can
be orders of magnitude larger, and one has a possibility to manipulate the molecules making use of their rotational degrees of
freedom.
For dipolar Bose-condensed gases, one of the key issues was related to the presence of the roton-maxon character of the excitation spectrum and to the possibility of obtaining supersolid states in which the condensate wavefunction is a
superposition of a uniform background and a lattice structure. The roton-maxon structure of the spectrum has been first predicted for strongly
pancaked Bose-Einstein condensates \cite{gora}. However, the idea to obtain a supersolid state when the roton touches zero and uniform BEC
becomes unstable, did not succeed because of the collapse of the system \cite{gora2,cooper}. Since that time, several proposals have been
made for the creation of supersolid states with bosons \cite{prok}. They rely on the potential of interatomic interaction which is flat at short distances and decays at large separations, and the results then indicate the presence of dense supersolid clusters \cite{cint}.
This activity brought in analogies with liquid helium, where the
studies of the roton-maxon spectrum and the attempts to observe experimentally the supersolid state have spanned for decades \cite{bali,Noz}
after the early theoretical prediction \cite{andr}. However, the most credible claim for the observation of the supersolid \cite{kim} is now withdrawn
\cite{kim1}. It is worth mentioning that the old idea of obtaining a stable density-modulated state (supersolid) in superfluid helium moving with a supercritical velocity \cite{Pit84} is now being discussed in a general context of superfluidity in Bose gases flowing with velocities larger than the Landau critical velocity \cite{Baym}.
The roton-maxon character of the excitation spectrum was also attracting large attention by itself. In relation to
liquid helium, it has been discussed how the position of the roton minimum influences the phenomenon of superfluidity \cite{Noz}.
In the context of dipolar bosons in two dimensions, numerical calculations of zero temperature phase diagram \cite{mora,bush,Astr} found that the reduction
of the condensed fraction with an increase in the density can be attributed to the appearance of the roton minimum. Finite temperature
Monte Carlo calculations \cite{prok3} have revealed that the rotonization of the spectrum can decrease the Kosterlitz-Thouless superfluid
transition temperature. Although the calculations \cite{mora,bush,Astr,prok3} were focused on fairly high densities, they raised the question of applicability of the Bogoliubov approach for dipolar bosons \cite{Astr}. At the same time, there was
an extended activity on the static and dynamical properties of dilute trapped dipolar Bose-Einstein condensates on the basis of the Bogoliubov approach \cite{uwe,bohn1,bohn2,bohn3}.
It is therefore instructive to identify the validity criterion of the Bogoliubov approach for Bose-condensed dipolar gases with the roton-maxon excitation spectrum,
and this is the subject of the present paper. We show that at zero temperature the
density fluctuations originating from the presence of the roton minimum, lead to a significant depletion of the condensate
and modify thermodynamic quantities if the roton minimum is sufficiently close to zero.
At finite temperatures exceeding the roton energy, thermal density fluctuations may have a much stronger influence, leading to a large increase of the normal fraction and compressibility.
We consider a dilute Bose-condensed gas of dipolar bosons (tightly) confined in one direction $(z)$ to zero point oscillations and assume that in the
$x,y$ plane the translational motion is free (see Fig.1). The dipole moments are oriented perpendicularly to the $x,y$ plane, which for electric dipoles (polar molecules) can be done applying an electric field, and for magnetic atoms by using a magnetic field. In this quasi-2D geometry, at large interparticle separations $r$ the interaction potential is
\begin{equation}\label{dd}
V(r) = \frac{d^2}{r^3}=\frac{\hbar^2r_*}{mr^3},
\end{equation}
with $d$ being the dipole moment, $m$ the particle mass, and $r_*=md^2/\hbar^2$ the characteristic dipole-dipole distance. The short-range part of the potential is assumed to be such that there is a roton-maxon excitation spectrum.
\begin{figure}[htb1]
\includegraphics[scale=0.5]{ddi.eps}
\caption{Dipolar Bose-Einstein condensate tightly confined in one direction.}
\end{figure}
In the ultracold limit where the particle momenta satisfy the inequality $kr_*\ll1$, the off-shell scattering amplitude defined as
$f({\vec k},{\vec k'})=\int \exp(-i{\vec k'} {\vec r'}) V(r)\psi_{\vec k}({\vec r})d^2r$ ($\psi_{\vec k}({\vec r})$ is the wavefunction
of the relative motion with momentum $\vec{k}$), is given by (see \cite{Pikgora} and refs. therein):
\begin{equation}\label{scam1}
f({\vec k},{\vec k'})=\frac{\hbar^2}{m}\left[\frac{2\pi}{\ln(\kappa/k)+i\pi/2}-2\pi r_*\vert \vec k-\vec k'\vert\right],
\end{equation}
where we omit higher order terms in $k$.
The second term in the right hand side represents the so-called anomalous contribution coming from distances of the order of the de Broglie wavelength of particles \cite{Landlif}. It takes into account all partial waves and is obtained using a perturbative approach in $V(r)$.
The first term in the right hand side of Eq.(\ref{scam1}) describes the short-range contribution. It is obtained by putting $k'=0$
and proceeding along the lines of the 2D scattering theory \cite{Landlif}. The parameter $\kappa$ depends on the behavior of $V(r)$ at short distances.
In the quasi-2D geometry it also depends on the confinement length in the $z$-direction, $l_0=\sqrt{\hbar/m\omega_0}$,
where $\omega_0$ is the confinement frequency.
One then can express $\kappa$ through the 3D coupling constant $g_{3D}$.
If the 3D s-wave scattering length, $ a_{3D}=mg_{3D}/4\pi \hbar^2\ll l_0$, then $\kappa$ is exponentially small
\cite{ petr1,petr2} and we may omit the k-dependence under logarithm in Eq.(\ref{scam1}), as well as $i\pi/2$ in the denominator of the first term. This gives $f({\vec k},{\vec k'})=g(1-C\vert \vec k-\vec k'\vert)$, where the 2D short-range coupling constant is $g=g_{3D}/\sqrt{2}l_0$ and $C =2\pi \hbar^2r_*/mg=2\pi d^2/g$.
Employing this result in the secondly quantized Hamiltonian \cite{gora2,cooper}, we obtain
\begin{eqnarray}\label{he3}
\!\!\!\!\hat H\!\!=\!\!\sum_{\vec k}\!E_k\hat a^\dagger_{\vec k}\hat a_{\vec k}\!+\!\frac{g}{2S}\!\!\sum_{\vec k,\vec q,\vec p}\!\!
(1\!\!-\!C\vert \vec q\!-\!\vec p\vert)\hat a^\dagger_{\vec k\!+\!\vec q} \hat a^\dagger_{\vec k\!-\!\vec q}\hat a_{\vec k\!+\!\vec p}\hat a_{\vec k\!-\!\vec p}
\end{eqnarray}
where $S$ is the surface area, $E_k=\hbar^2k^2/2m$, and $\hat a_{\bf k}^\dagger$, $\hat a_{\bf k}$ are the creation and annihilation operators of particles.
At zero temperature there is a true Bose-Einstein condensate in 2D, and we may use the standard Bogoliubov approach.
Assuming the weakly interacting regime where $mg/2\pi\hbar^2\ll 1$ and $r_*\ll \xi$, with $\xi=\hbar/\sqrt{mng}$ being the healing length,
we reduce the Hamiltonian (\ref{he3}) to a bilinear form, use the Bogoliubov transformation $\hat a^\dagger_{\vec k}= u_k \hat b^\dagger_{\vec k}-v_k \hat b_{-\vec k}$,
and obtain the diagonal form $\hat H = E_0+\sum_{\vec k} \varepsilon_k\hat b^\dagger_{\vec k}\hat b_{\vec k}$ in terms of operators
$b^\dagger_{\vec k}$, $\hat b_{\vec k}$ of elementary excitations. The Bogoliubov functions $ u_k,v_k$ are expressed in a standard way:
$ u_k,v_k=(\sqrt{\varepsilon_k/E_k}\pm\sqrt{E_k/\varepsilon_k})/2$, and the Bogoluibov excitation energy is given by
$\varepsilon_k=\sqrt{E_k^{2}+2ngE_k(1-Ck)}$.
To zero order the chemical potential is $\mu=ng$.
For small momenta the excitations are sound waves, $\varepsilon_k=\sqrt{ng/m}k$. The dependence of $\varepsilon_k$
on $k$ remains monotonic with increasing $k$ if $C\leq \sqrt{8}\xi/3$ (see Fig.2 ).
For the constant $C$ in the interval
\begin{equation}\label{pump3}
\frac{\sqrt{8}}{3}\xi\leq C\leq\xi,
\end{equation}
the excitation spectrum has a roton-maxon structure.
It is then convenient to represent $\varepsilon_k$ in the form:
\begin{equation}\label{pump4}
\varepsilon_k= \frac{\hbar^2 k}{2m}\sqrt{ (k-k_r)^2 +k_{\Delta}^2},
\end{equation}
where $k_r=2C/\xi^2$ and $k_{\Delta}=\sqrt{4/\xi^2-k_r^2}$.
If the roton is close to zero, then $k_r$ is the position of the roton,
and
\begin{equation} \label{Delta}
\!\!\Delta\!=\!\hbar^2 k_rk_{\Delta}/2m\!=\!2ngC\sqrt{mng/\hbar^2-C^2(mng/\hbar^2)^2}\!\!
\end{equation}
is the height of the roton minimum (see Fig.2).
For $C=\xi$ the roton minimum touches zero, and at larger $C$ the uniform Bose condensate becomes dynamically unstable.
It should be noted that the coupling constant $g$ can be tuned by using Feshbach resonances or by modifying the frequency of the
tight confinement $\omega_0$. Therefore, although the range of $C$ given by Eq.(\ref{pump3}) is rather narrow, it can be reached without serious
difficulties. The condition $C=2\pi d^2/g=\xi$ is reduced to $(mg/2\pi\hbar^2)=a_{3D}/\sqrt{2}l_0\simeq 2\pi nr_*^2$. For dysprosium atoms we have the dipole-dipole distance $r_*\simeq 200$ \AA, and at 2D densities $\sim 10^9$ cm$^{-2}$ the roton-maxon spectrum is realized for the 3D scattering length $a_{3D}$ of several tens of angstroms at the frequency of the tight confinement of $10$ kHz leading to the confinement length $l_0$ about 1000 \AA.
\begin{figure}[htb]
\includegraphics[scale=0.7, angle=0]{spectrum.eps}
\caption{Excitation energy $\varepsilon_k$ of the quasi-2D dipolar BEC as a function of momentum $k$ for several values of $k_r$. The solid curve ($k_r\xi=1.84$) shows a monotonic dependence $\varepsilon_{k}$,
the dotted curve ($k_r\xi=1.96$) is $\varepsilon_k$ with the roton-maxon structure, and the dashed curve ($k_r\xi=2.08$) corresponds to dynamically unstable BEC.}
\end{figure}
The Bogoliubov approach assumes that the density and phase fluctuations are small. In the 2D case at $T=0$, the presence of the roton does not significantly change the phase fluctuations and they remain small. However, the situation with the density fluctuations is different. Writing the operator of the density fluctuations as (see \cite{LL9}) $\delta\hat n=\sqrt{n}\sum_{\bf k}(u_k-v_k)\exp(i{\bf kr})\hat b_{\bf k}+h.c.$,
we obtain for the density-density correlation function:
\begin{equation} \label{deltan1}
\frac{\langle \delta\hat n({\bf r})\delta\hat (0)\rangle}{n^2}=\frac{1}{n}\int\frac{d^2k}{(2\pi)^2}\frac{E_k}{\varepsilon_k}(1+2N_k)\exp(i{\bf kr}),
\end{equation}
where $N_k=[\exp(\varepsilon_k/T)-1]^{-1}$ are occupation numbers for the excitations. It is instructive to single out the roton contribution to the correlation function (\ref{deltan1}). Asuming that the roton is close to zero and the roton energy is $\Delta\ll ng$, we have the cofficient $C$ close to $\xi$, and $k_r\simeq 2/\xi$. Then, using Eqs.(\ref{pump4}) and (\ref{deltan1}), for the contribution of momenta near the roton minimum at $T=0$ we obtain:
\begin{equation} \label{deltanfin}
\!\!\frac{\langle \delta\hat n({\bf r})\delta\hat n(0)\rangle_r}{n^2}\!=\!\frac{2mg}{\pi\hbar^2}\ln\left(\frac{2ng}{\Delta}\right)J_0(2r/\xi);\,\,\Delta\ll ng,
\end{equation}
where $J_0$ is the Bessel function.
We thus see that the density fluctuations grow logarithmically when the roton minimum is approaching zero and they can become strong for very small $\Delta$.
In this case they lead to a significant depletion of the condensate. The non-condensed density of particles is
$n'=\int v_k^2 d^2k/(2\pi)^2$ and the integral over $dk$ is logarithmically divergent at large momenta because of the dipolar contribution to the interaction strength, $-gCk$. However, this form of the dipole-dipole contribution is valid only for $k\ll 1/r_*$. We thus may put a high momentum cut-off $1/r_*$, which leads to (see Fig.3):
\begin{equation} \label{nonBEC1}
\!\!\!\!\frac{n'}{n}\!\!=\!\!\frac{mg}{4\pi\hbar^2}\!\left[1\!\!-\!k_r\xi\!-\!\frac{3(k_r\xi)^2}{4}\!+\!\frac{(k_r\xi)^2}{2}\ln\!\!\left(\!\frac{\xi}{r_*(2\!\!-\!k_r\xi)}\!\!\right)\!\right]\!.\!\!\!\!
\end{equation}
In the absence of the dipole-dipole interaction ($r_*=0$ and $k_r=0$) we recover the usual result for the 2D BEC with short-range
interparticle repulsion, $n'=n(mg/4\pi\hbar^2)$. For $\Delta\ll ng$ we have $(2-k_r\xi)\simeq (k_{\Delta}\xi)^2/4$ and Eq.(\ref{nonBEC1})
transforms to
\begin{equation} \label{nonBECfin}
\frac{n'}{n}\simeq \frac{mg}{\pi\hbar^2}\ln\left(\frac{2ng}{\Delta}\zeta\right);\,\,\,\,\,\,\Delta\ll ng,
\end{equation}
where $\zeta=\sqrt{2\pi\hbar^2/e^2 mg}$.
As we see from Eqs.~(\ref{deltanfin}) and (\ref{nonBECfin}), for the roton minimum close to zero a small condensate depletion and small fluctuations of the density require the inequality $(mg/\pi\hbar^2)\ln(2ng/\Delta)\ll 1$. It differs only by a logarithmic factor $\ln(2ng/\Delta)$ from the small
parameter of the theory, $(mg/2\pi\hbar^2)\ll 1$, in the absence of the roton.
The same logarithmic factor appears in the fluctuation correction to the chemical potential and in the one-body density matrix $g_1(r)=\langle\hat\Psi^{\dagger}({\bf r})\hat\Psi(0)\rangle$,
where $\hat\Psi({\bf r})$ is the field operator. Assuming that the roton minimum is close to zero and taking into account only the contribution of momenta near this minimum we have for $\Delta\ll ng$:
\begin{equation} \label{g1}
g_1(r)=n_0\left[1+\frac{mg}{\pi\hbar^2}\ln\left(\frac{2ng}{\Delta}\right)J_0(2r/\xi)\right].
\end{equation}
The correction to the chemical potential due to quantum fluctuations is given by:
\begin{equation} \label{deltamu}
\frac{\delta\mu}{\mu}\simeq \frac{2mg}{\pi\hbar^2}\ln\left(\frac{2ng}{\Delta}\right);\,\,\,\,\,\,\Delta\ll ng.
\end{equation}
\begin{figure}[htb]
\centering
\includegraphics[scale=0.6, angle=0]{depletion.eps}
\caption{Non-condensed fraction as a function of $k_r\xi$ for $mg/4\pi\hbar^2=0.01$ ($\xi/r_*=100/k_r\xi$). A similar increase of the non-condensed fraction with decreasing the roton energy $\Delta$ has been found in numerical calculations of Ref. \cite{uwe}.}
\label{mcp}
\end{figure}
However, the situation changes in the calculation of the compressibility. At $T=0$ the inverse compressibility is equal to $n^2\partial\mu/\partial n$. Then, using Eqs.(\ref{deltamu}) and (\ref{Delta}), for the roton minimum close to zero we obtain at $\Delta\ll ng$:
\begin{equation} \label{comp}
\frac{\partial\mu}{\partial n}=g\left[1+\frac{2mg}{\pi\hbar^2}\ln\left(\frac{2ng}{\Delta}\right)+\frac{mg}{\pi\hbar^2}\left(\frac{2ng}{\Delta}\right)^2\right],
\end{equation}
where $g$ is the mean field contribution, and the second and third terms originate from quantum fluctuations. Small deviations of the compressibility from the mean field result require the inequality
\begin{equation} \label{Bogcrit0}
\frac{mg}{\pi\hbar^2}\left(\frac{2ng}{\Delta}\right)^2\ll 1.
\end{equation}
We thus conclude that at $T=0$ the validity of the Bogoliubov approach is guaranteed by the presence of the small parameter (\ref{Bogcrit0}). For the dysprosium example given after Eq.(\ref{Delta}) we have $ng$ about 5 nK, and the criterion (\ref{Bogcrit0}) is satisfied for the roton energy above $2$ nK.
In 2D at finite temperatures, long-wave fluctuations of the phase destroy the condensate \cite{merm,hoh,pop}. There is the so-called quasicondensate, or condensate with fluctuating phase. In this state fluctuations of the density are suppressed but the phase still fluctuates. The transition from a non-condensed state to quasiBEC is of the Kosterlitz-Thouless type and it occurs through the formation of bound vortex-antivortex pairs \cite{KT}. Somewhat below the Kosterlitz-Thouless transition temperature the vortices are no longer important, and in the weakly interacting regime that we consider the phase coherence length $l_{\phi}$ is exponentially large. Thermodynamic properties, excitations, and correlation properties on a distance scale smaller than $l_{\phi}$ are the same as in the case of a true BEC. Moreover, for realistic parameters of quantum gases, $l_{\phi}$ exceeds the size of the system \cite{GPS}, so that one can employ the ordinary BEC theory.
Irrespective of the relation between $l_{\phi}$ and the size of the system, one may act in terms of the density and phase variables (hydrodynamic approach). We now show that the rotonization of the spectrum can strongly increase thermal fluctuations of the density and destroy the Bose-condensed state even at very low $T$. Using equation (\ref{deltan1}) we calculate the density-density correlation function. Assuming that the roton energy $\Delta$ is very small (at least $\Delta\ll T$), the main contribution to the integral in Eq.(\ref{deltan1}) comes from momenta near $k_r$, and we obtain:
\begin{equation} \label{deltanT}
\frac{\langle \delta\hat n({\bf r})\delta\hat (0)\rangle}{n^2}=\frac{4mg}{\hbar^2}\frac{T}{\Delta}J_0(2r/\xi),
\end{equation}
where it is also assumed that $k_{\Delta}r\ll 1$. Comparing this result with Eq.(\ref{deltanfin}) we see that instead of the logarithmic factor we have $2\pi T/\Delta\gg 1$.
The same factor appears in the correction to the chemical potential due to thermal fluctuations:
\begin{equation} \label{deltamuT}
\frac{\delta\mu}{\mu}=\sum_{\bf k}(u_k-v_k)^2N_k\simeq\frac{2mg}{\hbar^2}\frac{T}{\Delta};\,\,\,\,\Delta\ll T.
\end{equation}
We now calculate the density of the normal component in the presence of the roton. In 2D the expression for this quantity reads (c.f. \cite{LL9}):
$$n_T=-\int\frac{\hbar^2k^2}{2m}\frac{\partial N_k}{\partial\varepsilon_k}\frac{d^2k}{(2\pi)^2}.$$
If the roton minimum is close to zero and $\Delta\ll T$, then the momenta near the roton minimum are the most important, and the integration over $dk$ yields:
\begin{equation} \label{nT}
\frac{n_T}{n}=\frac{2mg}{\hbar^2}\frac{T}{\Delta}.
\end{equation}
The employed approach requires the condition $n_T\ll n$ because we used the spectrum of excitations obtained by the Bogoliubov method. Again, at temperatures $T\gtrsim \Delta$ we should have the inequality
$(2mg/\hbar^2)T/\Delta\ll 1$.
A different small parameter appears in the calculation of the compressibility. The inverse isothermal compressibility is proportional to $(\partial P/\partial n)_T$, where the pressure is $P=-(\partial F/\partial S)_T$, with the free energy given by $F=E_0+T\sum_{\bf k}\ln[1-\exp(-\varepsilon_k/T)]$. For the roton minimum close to zero and $\{T,ng\}\gg\Delta$, we obtain:
\begin{equation} \label{comprT}
\left(\frac{\partial P}{\partial n}\right)_T=ng\left[1-\frac{mg}{\hbar^2}\left(\frac{2ng}{\Delta}\right)^2\frac{T}{\Delta}+...\right],
\end{equation}
where we omitted less important finite-temperature contributions and the zero temperature contribution proportional to the small parameter (\ref{Bogcrit0}). Eq.(\ref{comprT}) shows that at $T\gg\Delta$ the Bogoliubov approach requires the inequality
\begin{equation} \label{BogcritT}
\frac{mg}{\hbar^2}\left(\frac{2ng}{\Delta}\right)^2\frac{T}{\Delta}\ll 1,
\end{equation}
whereas for $T\lesssim \Delta$ it is sufficient to have criterion (\ref{Bogcrit0}).
For certain quantities the Bogoliubov approach may give good results at $T\gg \Delta$ if $(mg/\hbar^2)T/\Delta\ll 1$, and at $T\lesssim\Delta$ in the presence of the ordinary small parameter $mg/2\pi\hbar^2$ amplified by a logarithmic factor $\ln(2ng/\Delta)$ for $ng\gg\Delta$. However, the validity of this approach is guaranteed only if the inequalities (\ref{BogcritT}) and (\ref{Bogcrit0}) are satisfied. For $T\gg\Delta$ the compressibility following from
Eq.(\ref{comprT}) and the normal fraction given by equation (\ref{nT}) can become significant, by far exceeding similar quantities in an ordinary 2D Bose gas with short-range interactions at the same temperature, coupling constant $g$, and density.
In conclusion, we have shown that the roton-maxon structure of the excitation spectrum, which can be achieved in dipolar Bose-condensed gases in the 2D geometry, strongly enhances fluctuations of the density. We obtained the validity criterion of the Bogoliubov approach and found that at finite temperatures in the dilute regime where $nr_*^2\ll 1$, thermal fluctuations may significantly increase the compressibility and reduce the superfluid fraction even at temperatures well below the Kosterlitz-Thouless transition temperature.
We acknowledge support from CNRS, from the University of Chlef, and from the Dutch Foundation FOM. We are grateful to L.P. Pitaevskii and D.S. Petrov for stimulating discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,282 |
// merge contacts from journals
var fs = require('fs'),
sys = require('sys'),
http = require('http'),
url = require('url'),
lfs = require('../../Common/node/lfs.js'),
locker = require("../../Common/node/locker.js"),
lconfig = require("../../Common/node/lconfig.js"),
request = require("request"),
crypto = require('crypto');
var lockerInfo;
var express = require('express'),connect = require('connect');
var app = express.createServer(connect.bodyParser(), connect.cookieParser(), connect.session({secret : "locker"}));
// Process the startup JSON object
process.stdin.resume();
process.stdin.on("data", function(data) {
lockerInfo = JSON.parse(data);
if (!lockerInfo || !lockerInfo["workingDirectory"]) {
process.stderr.write("Was not passed valid startup information."+data+"\n");
process.exit(1);
}
process.chdir(lockerInfo.workingDirectory);
app.listen(lockerInfo.port, "localhost", function() {
sys.debug(data);
process.stdout.write(data);
gatherContacts();
});
});
app.set('views', __dirname);
app.get('/', function(req, res) {
res.writeHead(200, {
'Content-Type': 'text/html'
});
lfs.readObjectsFromFile("contacts.json",function(contacts){
res.write("<html><p>Found "+contacts.length+" contacts: <ul>");
for(var i in contacts) {
res.write('<li>' + (contacts[i].name? '<b>' + contacts[i].name + ': </b>' : '') +
JSON.stringify(contacts[i])+"</li>");
}
res.write("</ul></p></html>");
res.end();
});
});
app.get("/allContacts", function(req, res) {
res.writeHead(200, {
"Content-Type":"text/javascript"
});
res.write("[");
res.write(fs.readFileSync("contacts.json", "utf8"));
res.write("]");
res.end();
});
app.get("/update", function(req, res) {
gatherContacts();
res.writeHead(200);
res.end("Updating");
});
function gatherContacts(){
// This should really be timered, triggered, something else
locker.providers(["contact/facebook", "contact/twitter", "contact/google"], function(services) {
if (!services) return;
services.forEach(function(svc) {
if(svc.provides.indexOf("contact/facebook") >= 0) {
addContactsFromConn(svc.id,'/allContacts','contact/facebook');
} else if(svc.provides.indexOf("contact/twitter") >= 0) {
addContactsFromConn(svc.id,'/allContacts','contact/twitter');
} else if(svc.provides.indexOf("contact/google") >= 0) {
addContactsFromConn(svc.id, "/allContacts", "contact/google");
}
});
});
}
var contacts = {};
var debug = false;
function cadd(c, type) {
if(!c)
return;
morphContact(c, type);
var key;
if(c.name)
key= c.name.replace(/[A-Z]\./g, '').toLowerCase().replace(/\s/g, '');
else if(c.email && c.email.length > 0)
key = c.email[0].value;
else {
var m = crypto.createHash('sha1');
m.update(JSON.stringify(c));
key = m.digest('base64');
}
if (contacts[key]) {
// merge
mergeContacts(contacts[key], c);
} else {
contacts[key] = c;
}
}
function morphContact(c, type) {
if(type == 'contact/foursquare')
{
if(c.contact.email) c.email = [{'value':c.contact.email}];
if(c.contact.phone) c.phone = [{'value':c.contact.phone}];
}
}
/**
* name
* email
* phone
* address
* pic (avatar)
*/
function mergeContacts(one, two) {
mergeArrays(one,two,"_via",function(a,b){return a==b;});
mergeArrayInObjects(one, two, "email", function(obj1, obj2) {
return obj1.value.toLowerCase() == obj2.value.toLowerCase();
});
mergeArrayInObjects(one, two, "phone", function(obj1, obj2) {
return obj1.value.replace(/[^0-9]/g,'').toLowerCase() ==
obj2.value.replace(/[^0-9]/g,'').toLowerCase();
});
mergeArrayInObjects(one, two, "address", function(obj1, obj2) {
return obj1.value.replace(/[,\s!.#-()@]/g,'').toLowerCase() ==
obj2.value.replace(/[,\s!.#-()@]/g,'').toLowerCase();
});
mergeArrayInObjects(one, two, "pic", function(obj1, obj2) {return false;});
}
/**
* Merge two arrays of the name arrayName in two objects
*/
function mergeArrayInObjects(obj1, obj2, arrayName, entriesAreEqual) {
if(obj1[arrayName]) {
if(obj2[arrayName]) {
mergeArrays(obj1[arrayName], obj2[arrayName], entriesAreEqual);
}
} else if(obj2[arrayName]) {
obj1[arrayName] = obj2[arrayName];
}
}
/**
* Merge two arrays, removing duplicates that match based on equals function
*/
function mergeArrays(one, two, entriesAreEqual) {
for(var i = 0; i < two.length; i++) {
var present = false;
for(var j = 0; j < one.length; j++) {
if(entriesAreEqual(one[j], two[i]))
present = true;
}
if(!present)
one.push(two[i]);
}
}
/**
* Reads in a file (at path), splits by line, and parses each line as JSON.
* return parsed objects in an array
*/
function parseLinesOfJSON(data) {
var objects = [];
var cs = data.split("\n");
for (var i = 0; i < cs.length; i++) {
if (cs[i].substr(0, 1) != "{") continue;
if(debug) console.log(cs[i]);
objects.push(JSON.parse(cs[i]));
}
return objects;
}
function addContactsFromConn(conn, path, type) {
var puri = url.parse(lockerInfo.lockerUrl);
var httpClient = http.createClient(puri.port);
request.get({url:lconfig.lockerBase + "/Me/"+conn+path}, function(err, res, data) {
var cs = data[0] == "[" ? JSON.parse(data) : parseLinesOfJSON(data);
for (var i = 0; i < cs.length; i++) {
cs[i]["_via"] = [conn];
cadd(cs[i],type);
}
csync();
});
}
function csync()
{
var stream = fs.createWriteStream("contacts.json");
var ccount=0;
for (var c in contacts) {
stream.write(JSON.stringify(contacts[c]) + "\n");
ccount++;
}
stream.end();
console.log("saved " + ccount + " contacts");
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,516 |
\section{Introduction}
Beams of electrons carrying non--zero projection of the orbital angular momentum (OAM) upon their propagation direction currently attract much interest, both in theory and in experiment \cite{BlB07,UcT10,VeT10}. A wave--front of these electrons has a helical spatial structure which twists around the beam axis. Such \textit{twisted} beams are produced today at scanning electron microscopes with energies up to a few hundreds of keV and with the OAM projection as high as $\hbar m = 1000 \hbar$ \cite{VeT10,GrG15,MaG15,m=1000}. Owing to the non-vanishing $m$, the twisted electrons possess a magnetic moment $\mu \propto m \mu_B$ with $\mu_B$ being the Bohr magneton. This -- additional (and big when $m \gg 1$)-- magnetic moment makes the OAM beams particularly suitable for probing magnetic properties of materials at the nano-- and even atomic--scale \cite{VeT10,RuB14,ErL16} as well as for studying unusual properties of the electromagnetic radiation generated by such electrons \cite{Iv13}.
A number of material studies are planned with the twisted beams that are based on the (analysis of) electron \textit{scattering} by targets. In order to achieve high spatial resolution in these scattering experiments one would need to focus electron beams to a sub--nanometer scale \cite{ErL16}. Such tight focusing implies that the relative \textit{size} and \textit{position} of the beam and a target can be of paramount importance. Detailed theoretical analysis of the beam--size (and position) effects is demanded, therefore, for the guidance and analysis of the twisted--electron scattering experiments. First steps towards such an analysis were carried out by one of us \cite{IvS11} as well as by Van Boxem and co--workers \cite{BoP14,BoP15}. Those previous works, however, mainly dealt with the scattering amplitudes and angular distributions of the outgoing electrons. Much less attention has been paid so far to the \textit{quantitative} study of the scattering process which requires evaluation of (i) the number of events and, if possible, (ii) the cross sections. The analysis of these (quantitative) observables is a rather complicated task which requires accurate account of the beam's (and the target's) sizes, as well as of the position effects.
In this contribution we derive and study the number of events and the
generalized cross sections for potential scattering of the focused twisted electron beams.
Such beams are treated as spatially localized wave--packets that collide with targets
consisting of short--range potentials. The theoretical analysis of these collisions is
based on the generalized Born approximation which was recently elaborated by us in Ref.~\cite{KaK16} and was successfully used for describing scattering of ``the ordinary'' Gaussian packets.
In order to apply the (generalized) Born theory for twisted electrons, we first construct their wave--packets in Section~\ref{subsec:twisted_states}. The scattering amplitude for a single potential placed at a particular impact parameter is derived then in Section~\ref{subsec:amplitudes}. With the help of this amplitude and by making use of the theory from Ref.~\cite{KaK16}, we obtain the number of events and the averaged angle-differential cross sections for different ``experimental set--ups''. For example, in Section ~\ref{subsec:cross_sections_number_of_events} we consider collisions of a twisted electron
with a single scatterer as well as with the localized- (mesoscopic-) and the infinitely wide (macroscopic) targets, the latter are described as incoherent ensembles of potential centers.
The derived expressions for the number of events and the averaged cross sections can be applied for a target consisting of central short--range potentials of \textit{any} form. In Section~\ref{sec:Yukawa_and_hydrogenic_formulas}, however, we discuss the particular cases of Yukawa and hydrogenic potentials. In collision theory, these potentials are often used to approximate realistic electron--atom interactions, see for example Refs.~\cite{SaM87,Sal91}. Detailed calculations for the Yukawa-- and hydrogenic targets and for fast but still non-relativistic electrons are presented in Section~\ref{sec:results}.
Throughout these calculations, emphasis is placed, in particular, on the questions of (i) whether the number of scattering events in collision of twisted electrons is comparable to the one in the ``standard'' plane--wave case, (ii) how the averaged cross section depends on a size of the initial wave packet, and (iii) how the angular distributions of outgoing electrons depend on the impact parameters $b$ between the potential centre and the packet's axis. The summary of these results and outlook will be given in Section~\ref{sec:summary}.
Similar to the previous work~\cite{KaK16}, we use units with the Planck's constant set to unity, $\hbar = 1$.
\section{Theoretical background}
\label{sec:theory}
\subsection{Twisted electron states}
\label{subsec:twisted_states}
Before we start with the analysis of the potential scattering of twisted electrons, let us briefly remind how these electrons are described within the non--relativistic framework. First we will discuss the monochromatic Bessel solutions, and will show later how to construct the (twisted) wave--packets.
\subsubsection{Monochromatic Bessel states}
Not much has to be said about the stationary electron states with the well--defined energy $\varepsilon$, longitudinal momentum $k_z$ and projection $m$ of the orbital angular momentum onto the quantization ($z$--) axis. In the cylindrical coordinates ${\bm r} = (r_\perp, \varphi_r, z)$, these \textit{Bessel} solutions of the field--free Schr\"odinger equation are described by the wave--function \cite{JeS11,JeS11a,IvS11}:
\begin{equation}
\label{eq:wavefunction_Bessel_stationary}
\sprm{{\bm r}}{k_z \, \varkappa \, m} = {\rm e}^{-i \omega t + i k_z z} \;
\Psi_{\rm tr}^{\varkappa m}({\bm r}_\perp) \, ,
\end{equation}
with the transverse component
\begin{equation}
\label{eq:wavefunction_Bessel_stationary_transverse}
\Psi_{\rm tr}^{\varkappa m}({\bm r}_\perp) =
\frac{{\rm e}^{i m \varphi_r}}{\sqrt{2\pi}} \;
\sqrt{\varkappa}\, J_m(\varkappa\, r_\perp) \, ,
\end{equation}
where $J_m(\varkappa\, r_\perp)$ is the Bessel function and the absolute value of the transverse momentum is fixed to $|{\bm k}_\perp| = \varkappa = \sqrt{2m_e\,\varepsilon- k_z^2}$.
For practical calculations, it is often more convenient to write the function $\sprm{{\bm r}}{k_z \, \varkappa \, m}$ in the momentum representation:
\begin{equation}
\label{eq:wavefunction_Bessel_stationary_momentum}
\sprm{{\bm r}}{k_z \, \varkappa \, m} = {\rm e}^{-i \omega t + i k_z z} \, \int
\frac{{\rm d}^2{\bm k}_\perp}{(2\pi)^2} \,
\Phi_{\rm tr}^{\varkappa m}({\bm k}_\perp) \, {\rm e}^{i {\bm k}_\perp {\bm r}_\perp} \, .
\end{equation}
Here, the Fourier coefficient $\Phi_{\rm tr}^{\varkappa m}({\bm k}_\perp)$ is the transverse part of the momentum--space wave--function and is given by:
\begin{equation}
\label{eq:wavefunction_Bessel_stationary_momentum_transverse}
\Phi_{\rm tr}^{\varkappa m}({\bm k}_\perp) =
(-i)^m \, \frac{{\rm e}^{i m \varphi_k}}{\sqrt{2\pi}}
\,\frac{\delta(k_\perp - \varkappa)}{\sqrt{k_\perp}} \, .
\end{equation}
As can be seen from these two expressions, the Bessel electron state can be \textit{understood} as a coherent superposition of the plane waves, whose wave--vectors ${\bm k} = ({\bm k}_\perp, k_z)$ lay on the surface of a cone with the opening angle $\tan \theta_k = \varkappa / k_z$.
\subsubsection{Wave--packets of Bessel states}
The monochromatic Bessel state, (\ref{eq:wavefunction_Bessel_stationary})--(\ref{eq:wavefunction_Bessel_stationary_transverse}) and (\ref{eq:wavefunction_Bessel_stationary_momentum})--(\ref{eq:wavefunction_Bessel_stationary_momentum_transverse}), is a non--square--integrable solution for the Schr\"odinger equation that is spread over the \textit{entire} coordinate space, i.e., its dispersions are $\Delta z = \Delta {r_\perp} = \infty$. Such a monochromatic solution is not sufficient to describe nowadays experiments in which electron beams can be focused to a sub--nanometer scale. Therefore, we have to employ $\sprm{{\bm r}}{k_z \, \varkappa \, m}$ in order to construct a \textit{spatially--localized} wave--packet:
\begin{eqnarray}
\label{eq:wavepacket_Bessel_coordinate}
\Psi_{\varkappa_0 m p_i}({\bm r}) &=& {\rm e}^{-i \omega t} \, \Psi^{(m)}_{\rm tr}({\bm r}_\perp) \nonumber\\[0.1cm]
&\times& \int_0^\infty {\rm e}^{i k_z z} \, g_{p_i \sigma_{k_z}}(k_z) \, {\rm d}k_z \, ,
\end{eqnarray}
where the convolution of the transverse component (\ref{eq:wavefunction_Bessel_stationary_transverse}) reads as:
\begin{eqnarray}
\label{eq:transverse_component_convolution}
\Psi^{(m)}_{\rm tr}({\bm r}_\perp) &=& \int_0^\infty \Psi_{\rm tr}^{\varkappa m}({\bm r}_\perp) \, g_{\varkappa_0 \sigma_\varkappa}(\varkappa) \, {\rm d}\varkappa \nonumber \\
&=& \frac{{\rm e}^{i m \varphi_r}}{\sqrt{2\pi}} \,
R^{(m)}(r_\perp) \, ,
\end{eqnarray}
with $g_{p_i \sigma_{k_z}}(k_z)$ and $g_{\varkappa_0 \sigma_\varkappa}(\varkappa)$ being the weight functions, and
\begin{equation}
\label{eq:R_function}
R^{(m)}\left(r_\perp\right) = \int_0^\infty
\sqrt{\varkappa} \, J_m\left(\varkappa\, r_\perp \right) \, g_{\varkappa_0 \sigma_\varkappa}(\varkappa) \, {\rm d}\varkappa \, .
\end{equation}
The wave--packet $\Psi_{\varkappa_0 m p_i}({\bm r})$ is a physically--relevant solution with the definite OAM's projection $m$, with the mean values of the longitudinal momentum
$\left< k_z \right> = p_i$ and of the absolute value of the transverse one, $\left< k_\perp \right> = \varkappa_0$.
As before, we can represent the transverse part of the wave--packet (\ref{eq:transverse_component_convolution}) in terms of the plane--wave components:
\begin{eqnarray}
\label{eq:wavepacket_Bessel_momentum}
\Psi_{\varkappa_0 m p_i}({\bm r}) &=&
{\rm e}^{-i \omega t} \, \int \frac{{\rm d}^2{\bm k}_\perp}{(2\pi)^2} \,
\Phi_{\rm tr}^{(m)}({\bm k}_\perp) \, {\rm e}^{i {\bm k}_\perp {\bm r}_\perp} \nonumber\\[0.1cm]
&\times& \int_0^\infty {\rm e}^{i k_z z} \, g_{p_i \sigma_{k_z}}(k_z) \, {\rm d}k_z \, ,
\end{eqnarray}
where the convoluted Fourier coefficient is
\begin{eqnarray}
\label{eq:wave_packet_transverse}
\Phi_{\rm tr}^{(m)}({\bm k}_\perp) &=&
\int_0^\infty \Phi_{\rm tr}^{\varkappa m} ({\bm k}_\perp) g_{\varkappa_0 \sigma_\varkappa}(\varkappa) \, {\rm d}\varkappa \nonumber \\
&=& (-i)^m\,\frac{e^{im\varphi_k}}{\sqrt{2\pi k_\perp}}\, g_{\varkappa_0 \sigma_\varkappa}(k_\perp) \, .
\end{eqnarray}
The weight functions $g_{p_i \sigma_{k_z}}(k_z)$ and $g_{\varkappa_0 \sigma_\varkappa}(\varkappa)$ in Eqs.~(\ref{eq:wavepacket_Bessel_coordinate})--(\ref{eq:wave_packet_transverse}) are peaked around $p_i$ and $\varkappa_0$ and have the widths $\sigma_{k_z}$ and $\sigma_\varkappa$, respectively. For the numerical analysis below we choose these functions to be of the Gaussian form.
For example, the distribution of the transverse momentum is described by:
\begin{equation}
\label{eq:Gausian_distribution}
g_{\varkappa_0 \sigma_\varkappa}(\varkappa) =
C \, {\rm e}^{-(\varkappa - \varkappa_0)^2/(2\sigma^2_\varkappa)} \, ,
\end{equation}
where the constant $C$ is determined from the normalization condition $\int |g_{\varkappa_0 \sigma_\perp}(\varkappa)|^2 \, {\rm d}\varkappa = 1$. Eq.~(\ref{eq:Gausian_distribution}) corresponds to the dispersion $\Delta x = \Delta y \sim 1 /\sigma_{\varkappa}$ in the transverse plane. For the small values, $\sigma_\varkappa \ll 1/a$, where $a$ is the typical radius of the field action, the $\Delta x$ and $\Delta y$ become large and we call it a \textit{wide} wave--packet limit. The approximation of a wide packet will be employed below for simplifying the formulas for the scattering cross sections and the number of events.
\subsection{Scattering amplitudes}
\label{subsec:amplitudes}
Having briefly discussed construction of the Bessel electron's wave--packets, we are ready now to describe an amplitude for potential scattering. We consider the experimentally--relevant scenario in which (i) the longitudinal size of the packet $\sigma_z $ is larger than the characteristic radius of the field action, $\sigma_{z} \gg a$, and (ii) the dispersion of the packet in the transverse plane during the collision is negligible, that is, $t_{\rm dis} \sim \sigma_\perp/v_\perp \gg t_{\rm col} \sim \sigma_z/v_z$. Under these assumptions, which can be re--written as
\begin{equation}
\label{eq:assumptions_for_amplitude}
a\ll \sigma_z \ll p_i/(\varkappa_0 \sigma_\varkappa) \, ,
\end{equation}
the scattering amplitude is given by Eqs.~(29)--(30) of Ref.~\cite{KaK16}:
\begin{equation}
\label{eq:amplitude_monochromatic}
F({\bm Q}, {\bm b}) = \int f({\bm Q} - {\bm k}_\perp) \,
\Phi_{\rm tr}^{\rm (m)}({\bm k}_\perp) \, {\rm e}^{i {\bm k}_\perp {\bm b}} \, \frac{{\rm d}^2 {\bm k}_\perp}{2 \pi} \, .
\end{equation}
In this expression, the convoluted transverse component of the wave--packet is given by Eq.~(\ref{eq:wave_packet_transverse}), the momentum transfer is ${\bm Q} = {\bm p}_f - {\bm p}_i$ with ${\bm p_i} = \left< {\bm k}_i \right> = (0, 0, p_i)$ being an \textit{averaged} momentum of the incident electron, ${\bm p}_f$ the momentum of the scattered electron, and $f({\bm Q} - {\bm k}_\perp)$ is the \textit{plane--wave} scattering amplitude in the first--Born approximation:
\begin{equation}
\label{eq:amplitude_plane_wave}
f({\bm q}) = - \frac{m_e}{2 \pi} \, \int U(r) \, {\rm e}^{-i {\bm q} {\bm r}} \,
{\rm d}^3{\bm r} \, .
\end{equation}
We have assumed here that the interaction of electrons with a target is described by a central potential $U(r)$ and that the outgoing (scattered) electrons are detected as plane--waves with the wave--vector ${\bm p}_f$. In Eq.~(\ref{eq:amplitude_monochromatic}), moreover, we have introduced the exponential factor ${\rm exp}(i {\bm k}_\perp {\bm b})$ in order to specify the lateral position of the scatterer with regard to the central ($z$--) axis of the incident wave--packet. Here, ${\bm b} = (b_x, b_y, 0)$ is the impact parameter which \textit{vanishes} when the potential is placed in the center of the beam.
Along with Eq.~(\ref{eq:amplitude_monochromatic}), one can derive another representation for the scattering amplitude $F({\bm Q}, {\bm b})$ that, in some cases,
can be more convenient for theoretical analysis. By inserting the standard plane--wave amplitude (\ref{eq:amplitude_plane_wave}) into Eq.~(\ref{eq:amplitude_monochromatic}) we find:
\begin{equation}
\label{eq:amplitude_wave_packet}
F({\bm Q}, {\bm b}) = - \frac{m_e}{2 \pi} \, \int U(r) \,
\Psi_{\rm tr}^{(m)}({\bm r} + {\bm b}) \, {\rm e}^{-i {\bm Q} {\bm r}} \, {\rm d}{\bm r} \, ,
\end{equation}
where
\begin{equation}
\label{eq:wave_packet_shifted}
\Psi_{\rm tr}^{(m)}({\bm r} + {\bm b}) = \int {\rm e}^{i ({\bm r} + {\bm b}) {\bm k}_\perp} \,
\Phi_{\rm tr}^{\rm (m)}({\bm k}_\perp) \, \frac{{\rm d}^2 {\bm k}_\perp}{2 \pi} \, .
\end{equation}
This expression can help to analyze (at least qualitatively) scattering at the small impact parameters. For $b = 0$, in particular, the function $U(r) \, \Psi_{\rm tr}^{(m)}({\bm r} + {\bm b}) = U(r) \, \Psi_{\rm tr}^{(m)}({\bm r})$ under the integral in Eq.~(\ref{eq:amplitude_wave_packet}) corresponds to a state with a definite projection of the orbital angular momentum $m$. The expansion of this function into spherical harmonics $Y_{lm}(\theta_r, \varphi_r)$ may contain, therefore, \textit{only} multipoles with the OAM $l \ge |m|$. It results in a specific behaviour of the angular distributions of scattered electrons that become most pronounced for the forward direction. Indeed, by substituting the transverse component (\ref{eq:transverse_component_convolution}) of the wave--packet into the scattering amplitude (\ref{eq:amplitude_wave_packet}), we obtain:
\begin{eqnarray}
\label{eq:amplitude_wave_packet_b_0}
F({\bm Q}, {\bm b} = 0) && \nonumber \\
&& \hspace*{-2.3cm} = - \frac{m_e}{2 \pi} \int r_\perp \, {\rm d}r_\perp {\rm d}z\,
U\left(\sqrt{r_\perp^2 + z^2}\right) \, R^{(m)}\left(r_\perp \right) \,
{\rm e}^{-iQ_z z} \nonumber \\
&\times& \int_0^{2\pi} {\rm e}^{i m \varphi_r - iQ_\perp r_\perp
\cos{(\varphi_r - \varphi)}} \, \frac{{\rm d}\varphi_r}{2\pi} \, ,
\end{eqnarray}
where $Q_\perp = p_{f, \perp} = p_f (\sin\theta \cos\varphi, \sin\theta \sin\varphi, 0)$, $Q_z = p_f \cos\theta - p_i$, and the integral over $\varphi_r$ is reduced to:
\begin{eqnarray}
\int_0^{2\pi} \, {\rm e}^{i m \varphi_r - i Q_\perp r_\perp \cos(\varphi_r - \varphi)}\,
\frac{{\rm d}\varphi_r}{2 \pi} && \nonumber \\
&& \hspace*{-3cm} = {\rm e}^{i m \varphi} \, (-i)^m \, J_m(Q_\perp r_\perp) \,.
\end{eqnarray}
From these expressions and from the asymptotic behaviour of the Bessel function, $|J_m(x)| = (x/2)^{|m|}/|m|!$ for $x \to 0$, one finds that the amplitude
\begin{eqnarray}
\label{eq:amplitude_wave_packet_b_0_forward}
F({\bm Q}, {\bm b} = 0) \propto Q_{\perp}^{|m|} \propto (\sin\theta)^{|m|}
\end{eqnarray}
\textit{vanishes} for $\theta \to 0$ when the OAM projection $m \ne 0$. Since the angle--differential cross section is proportional to the square of $F({\bm Q}, {\bm b})$, the emission pattern of the scattered electrons develops a dip in the forward direction and for the central collision, $b = 0$. However behaviour of the function $U(r) \Psi_{\rm tr}^{(m)}(\bm r+\bm b)$ changes with the growth of $\bm b$, and the dip at $\theta=0$ disappears. This angular behaviour, peculiar to twisted electron beams, will be discussed in detail later.
A further analysis of $F({\bm Q}, {\bm b})$ requires knowledge of the interaction potential $U(r)$. In Section \ref{sec:Yukawa_and_hydrogenic_formulas}, for example, we will show how this scattering amplitude is calculated for the Yukawa- and hydrogenic potentials.
\subsection{Number of events and cross sections}
\label{subsec:cross_sections_number_of_events}
With the help of the amplitude $F({\bm Q}, {\bm b})$ one can calculate the number of events and the cross section for potential scattering of a twisted electron's wave--packet. The explicit form of these observables depends on a set--up of the scattering ``experiment'' and, in particular, on the composition of the target. In this section, we study scattering off (i) a single potential,
(ii) the infinitely wide (macroscopic) targets, and (iii) off the localized (mesoscopic) targets.
\subsubsection{Scattering by a single potential}
\label{subsubsect:single_potential}
We start the discussion with a \textit{single} potential $U(r)$ located at the impact parameter ${\bm b}$ from the center of the electron wave--packet. As discussed already in Ref.~\cite{KaK16}, a usual definition of the cross section is not applicable in this case. Instead, the scattering process can be described by a number of events:
\begin{equation}
\label{eq:numebr_events_single_potential_general}
\frac{{\rm d}\nu}{{\rm d}\Omega}=\frac{N_e}{\cos{\theta_k}} \,
\left| F({\bm Q}, {\bm b}) \right|^2 \, ,
\end{equation}
where $N_e$ is the number of electrons in the incident beam.
Eq.~(\ref{eq:numebr_events_single_potential_general}) can be further simplified for the so--called \textit{wide} wave--packet. Here, the distribution function $g_{\varkappa_0 \sigma_\varkappa}(k_\perp)$ from Eq.~(\ref{eq:transverse_component_convolution}) is sharply peaked at $\left< k_\perp \right> = \varkappa_0 \ne 0$, so that $1/\sigma_\varkappa \gg a$. In this case, the transverse momentum ${\bm k}_\perp = k_\perp \left(\cos{\varphi_k}, \sin{\varphi_k}, 0 \right)$ of the incident electron wave can be approximated as:
\begin{equation}
\label{eq:k_perp_0}
{\bm k}^{(0)}_\perp= \varkappa_0 \,
\left(\cos{\varphi_k}, \sin{\varphi_k}, 0 \right) \, .
\end{equation}
By substituting ${\bm k}_\perp \to {\bm k}^{(0)}_\perp$ in the plane--wave amplitude $f({\bm Q} - {\bm k}_\perp)$ and making use of Eqs.~(\ref{eq:amplitude_monochromatic}) and (\ref{eq:numebr_events_single_potential_general}), we find:
\begin{equation}
\label{eq:eq:numebr_events_single_potential_general_wide}
\frac{{\rm d}\nu}{{\rm d}\Omega} = L^{({\rm tw})} \,
\left| \int_0^{2\pi} \, f\left({\bm Q} - {\bm k}^{(0)}_\perp \right) \,
{\rm e}^{i m\varphi_k + i{\bm k}^{(0)}_\perp {\bm b}} \,
\frac{{\rm d}\varphi_k}{2\pi} \right|^2 \, ,
\end{equation}
where the quantity
\begin{equation}
\label{eq:Ltw_general}
L^{({\rm tw})} = \frac{N_e}{\cos{\theta_k}} \,
\left|\int_0^\infty g_{\varkappa_0 \sigma_\varkappa}(k_\perp) \sqrt{\frac{k_\perp}{2\pi}} \, {\rm d}k_\perp \right|^2
\,
\end{equation}
can be viewed as luminosity of the collision.
When comparing this with Eq.~(\ref{eq:transverse_component_convolution}), one sees that $L^{({\rm tw})}$ can be expressed in terms of the transverse density of the incident Bessel packet with $m = 0$ at the coordinate origin:
\begin{equation}
\label{eq:Ltw_general_PSI}
L^{({\rm tw})} = \frac{N_e}{\cos{\theta_k}} \,
\left|\Psi^{(0)}_{\rm tr}({\bm r}_\perp={\bf 0}) \right|^2\,.
\end{equation}
Together with Eq.~(\ref{eq:eq:numebr_events_single_potential_general_wide}), this expression indicates that for $m = 0$ and $\theta_k \to 0$ we obtain the number of events for the ``standard'' plane--wave case, see Eq.~(4) from Ref.~\cite{KaK16}. Therefore, the quantity
\begin{equation}
\frac{1}{L^{\rm (tw)}}\, \frac{{\rm d}\nu}{{\rm d}\Omega}\equiv \frac{d\sigma^{\rm (tw)}(\bm b)}{d\Omega}
\label{cross-single}
\end{equation}
can be considered as a cross section for a single potential, localized at the impact parame\-ter~$\bm b$. In particular, at $\varkappa_0 \ll Q$ we obtain the following simple formula:
\begin{equation}
\frac{{\rm d}\sigma^{\rm (tw)}(\bm b)}{{\rm d}\Omega}=
\frac{{\rm d}\sigma^{\rm (PW)}}{{\rm d}\Omega}\,
J_m^2(\varkappa_0 b).
\label{cross-single-up}
\end{equation}
\subsubsection{Infinitely wide (macroscopic) target}
\label{subsubsec_wide_target}
After discussion of the scattering by a single potential, let us now turn to the \textit{macroscopic} target, which consists of randomly distributed force centers and has a transverse extension ${\mathcal R} \gg 1/\sigma_{\varkappa}$. For such a target one can introduce the \textit{averaged} cross section:
\begin{eqnarray}
\label{eq:cross_section_macroscopic_target}
\frac{{\rm d}{\bar \sigma}}{\rm{d}\Omega} &=& \frac{1}{\cos\theta_k} \,
\int \left| F({\bm Q}, {\bm b}) \right|^2 \, {\rm d}^2{\bm b} \\
&& \hspace*{-1cm} = \frac{1}{\cos\theta_k} \, \int_0^{2\pi} \frac{{\rm d}\varphi_k}{2\pi} \,
\int_0^\infty {\rm d}k_\perp \left| g_{\varkappa_0 \sigma_\perp}(k_\perp)\right|^2 \,
\left|f({\bm Q} - {\bm k}_\perp)\right|^2 \, \nonumber
\end{eqnarray}
which is obtained (i) by integrating the number of events over all the impact parameters $b$, and (ii) by normalizing the result by a number of incident electrons, see Eqs. (33) and (38) in Ref.~\cite{KaK16}.
As can be seen from Eq.~({\ref{eq:cross_section_macroscopic_target}), ${\rm d}{\bar \sigma}/{\rm d}\Omega$ depends neither on the projection $m$ of the orbital angular momentum nor on the spatial structure of an incident phase front.
One can further simplify Eq.~(\ref{eq:cross_section_macroscopic_target}) in the approximation of a wide wave--packet. As mentioned already, the momentum distribution function $g_{\varkappa_0 \sigma_\varkappa}(k_\perp)$ is sharply peaked in this case at $\left< k_\perp \right> = \varkappa_0$ and the wave--packet (\ref{eq:wavepacket_Bessel_coordinate}) approaches the monochromatic limit. Similar to before, one can approximate the transverse momentum of the incident electron ${\bm k}_\perp$ by ${\bm k}_{\perp}^{(0)}$, given by Eq.~(\ref{eq:k_perp_0}). By substituting ${\bm k}_\perp = {\bm k}_{\perp}^{(0)}$ in Eq.~(\ref{eq:cross_section_macroscopic_target}) and taking the square of the (plane--wave) scattering amplitude $f({\bm Q} - {\bm k}^{(0)}_\perp)$ out of the integral over $k_\perp$, we obtain:
\begin{equation}
\label{eq:cross_section_macroscopic_target_wide_packet}
\frac{{\rm d}{\bar \sigma}}{\rm{d}\Omega} =
\frac{1}{\cos\theta_k} \, \int_0^{2\pi} \frac{{\rm d}\varphi_k}{2\pi} \,
\left|f({\bm Q} - {\bm k}^{(0)}_\perp)\right|^2 \, ,
\end{equation}
where, as usual, the momentum transfer is ${\bm Q} = {\bm p}_f - {\bm p}_i$, and $|{\bm p}_f|=\left|{\bm p}_i + {\bm k}^{(0)}_\perp\right|$.
As can be seen from Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet}), the calculation of the scattering cross section for a wide wave--packet and a macroscopic target can be traced back to the plane--wave amplitude $f({\bm Q} - {\bm k}^{(0)}_\perp)$. Therefore, based on the properties of this amplitude we can predict the main features of ${\rm d}{\bar \sigma}/\rm{d}\Omega$. It is well--known, for example, that for Yukawa and hydrogenic potentials $f({\bm q})$ depends solely on the ${\bm q}^2$, see Ref.~\cite{KaK16}. Since in Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet}) the argument of the plane--wave scattering amplitude is ${\bm Q} - {\bm k}^{(0)}_\perp$ and the square of this momentum transfer is given by:
\begin{eqnarray}
\label{eq:Q2_formula}
\left({\bm Q} - {\bm k}^{(0)}_\perp \right)^2 &&\\
&& \hspace*{-2cm} = 2p^2_f\,\left[1-\cos\theta \cos\theta_k-\sin\theta\sin\theta_k \cos(\varphi_k-\varphi)\right] \, , \nonumber
\end{eqnarray}
we can re--write Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet}) as follows:
\begin{equation}
\label{eq:cross_section_macroscopic_target_wide_packet_2}
\frac{{\rm d}{\bar \sigma}(\theta; \theta_k)}{\rm{d}\Omega} =
\frac{1}{\cos\theta_k} \, \int_0^{2\pi} \frac{{\rm d}(\varphi_k - \varphi)}{2\pi} \,
\left|f({\bm Q} - {\bm k}^{(0)}_\perp)\right|^2 \, ,
\end{equation}
where $\varphi$ is the azimuthal angle of outgoing electrons. Eqs.~(\ref{eq:Q2_formula})--(\ref{eq:cross_section_macroscopic_target_wide_packet_2}) reveal that the scattering pattern is azimuthally symmetric with respect to the beam ($z$--) axis and is peaked at $\theta = \theta_k$. The latter follows also from the fact that $f({\bm Q} - {\bm k}^{(0)}_\perp)$ is maximal at $\left({\bm Q} - {\bm k}^{(0)}_\perp\right)^2 = 0$ for the Yukawa and hydrogenic potentials, see Ref.~\cite{KaK16}.
Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet_2}) can now be used to calculate the total cross section. Namely, by integrating over the directions of outgoing electrons, we find:
\begin{eqnarray}
\label{eq:total_cross_section_wide_packet}
\bar \sigma &=& \frac{1}{\cos{\theta_k}}
\int_0^{2\pi} \frac{{\rm d}\varphi_k}{2\pi}
\int {\rm d}\Omega \left|f\left({\bm p}_f- {\bm p}_i - {\bm k}^{(0)}_\perp\right)\right|^2
\nonumber \\
&=& \frac{\sigma_{\rm pl}}{\cos{\theta_k}} \, ,
\end{eqnarray}
where $\sigma_{\rm pl}$ is the plane--wave cross section. Clearly, the cross section for the scattering of twisted electrons by a macroscopic target is generally \textit{larger} than $\sigma_{\rm pl}$.
\subsubsection{Superposition of two twisted beams}
\label{subsubsec_two_twisted_beams}
As we have seen from Eqs.~(\ref{eq:cross_section_macroscopic_target}) and (\ref{eq:cross_section_macroscopic_target_wide_packet_2}), the angle--differential cross section for the scattering by a macroscopic (infinite) target is independent of the projection of the orbital angular momentum $m$ and of the phase structure of the incident twisted beam. Later, in Section \ref{subsubsec:scattering_localized}, we will show that this OAM's-- and phase--sensitivity is restored for spatially localized scatterers and well--focused electron beams. However, even in the ``large--target---wide--packet'' regime the averaged cross section can be sensitive to the OAM if electrons are initially prepared in a coherent superposition of two states with the same kinematic parameters but different $m$:
\begin{equation}
\label{eq:superposition_two_states}
\Psi_{\varkappa_0 p_i}({\bm r}) = c_1 \Psi_{\varkappa_0 m_1 p_i}({\bm r}) + c_2 \Psi_{\varkappa_0 m_2 p_i}({\bm r})
\end{equation}
where $\Psi_{\varkappa_0 m p_i}({\bm r})$ is given by Eq.~(\ref{eq:wavepacket_Bessel_coordinate}) and the expansion coefficients are:
\begin{eqnarray}
\label{eq:c_coefficients}
c_n = \left| c_n \right| \, {\rm e}^{i \alpha_n}, \; \; \; \left|c_1\right|^2 + \left|c_2\right|^2 = 1 \, .
\end{eqnarray}
By inserting the superposition (\ref{eq:superposition_two_states}) into Eqs.~(\ref{eq:amplitude_monochromatic}) and (\ref{eq:cross_section_macroscopic_target}), and passing to the limit $\sigma_\varkappa \to 0$ we obtain:
\begin{eqnarray}
\label{eq:cross_section_macroscopic_target_wide_packet_superposition_1}
\frac{{\rm d}{\bar \sigma}^{\rm (2)}(\theta, \varphi ; \, \theta_k)}{\rm{d}\Omega} &=& \frac{1}{\cos\theta_k}
\nonumber \\
&& \hspace*{-3.0cm} \times
\int_0^{2\pi} \frac{{\rm d}\varphi_k}{2\pi} \,
\left|f({\bm Q} - {\bm k}^{(0)}_\perp)\right|^2 \, G(\varphi_k, \Delta m, \Delta \alpha) \, ,
\end{eqnarray}
with $\Delta m = m_2 - m_1$, $\Delta \alpha = \alpha_2 - \alpha_1$ and the factor:
\begin{eqnarray}
\label{eq:G_factor}
G(\varphi_k, \Delta m, \Delta \alpha) &=& 1 + 2 \left|c_1 c_2 \right| \nonumber \\
&& \hspace*{-2cm} \times \cos\Big[(m_2-m_1)(\varphi_k-\pi/2) + \alpha_2 - \alpha_1 \Big].
\end{eqnarray}
In order to further evaluate this expression, we note that the plane--wave scattering amplitude $f({\bm Q} - {\bm k}^{(0)}_\perp)$ depends on the difference between the azimuthal angles $\varphi_k - \varphi$, see Eq.~(\ref{eq:Q2_formula}). This angular dependence allows us to perform an integration over $\varphi_k$ and to find:
\begin{eqnarray}
\label{eq:cross_section_macroscopic_target_wide_packet_superposition_2}
\frac{{\rm d}{\bar \sigma}^{\rm (2)}(\theta, \varphi ; \, \theta_k)}{\rm{d}\Omega} &=&
\frac{{\rm d}{\bar \sigma}(\theta ; \, \theta_k)}{\rm{d}\Omega}
\nonumber \\[0.2cm]
&& \hspace*{-2.0cm} \times \Big[1 + A(\theta; \theta_k) \,
\cos{[\Delta m(\varphi-\pi/2)+\Delta\alpha]} \Big] \, ,
\end{eqnarray}
where ${\rm d}{\bar \sigma}/\rm{d}\Omega$ is the cross section (\ref{eq:cross_section_macroscopic_target_wide_packet_2}) for a single Bessel beam, and the azimuthal asymmetry parameter reads as follows:
\begin{eqnarray}
\label{eq:cross_section_macroscopic_target_wide_packet_superposition_interference}
A(\theta; \theta_k) &=& \left( \frac{{\rm d}{\bar \sigma}(\theta ; \, \theta_k)}{\rm{d}\Omega} \right)^{-1} \,
\frac{2 \left|c_1 c_2\right|}{\cos\theta_k} \\
&& \hspace*{-2cm} \times \int_0^{2\pi} \,
\left| f({\bm Q} - {\bm k}^{(0)}_\perp)\right |^2 \,
\cos{[\Delta m(\varphi_k-\varphi)]}\frac{{\rm d}(\varphi_k-\varphi)}{2\pi} \, . \nonumber
\end{eqnarray}
These expressions indicate that the angle--differential cross section ${\rm d}{\bar \sigma}^{\rm (2)}/\rm{d}\Omega$ exhibits an azimuthal asymmetry which depends on the difference of the OAM's projections $\Delta m = m_2 - m_1$ and of the phases $\Delta \alpha = \alpha_2 - \alpha_1$. This dependence, however, does not appear in the total cross section:
\begin{eqnarray}
\label{eq:total_cross_section_wide_packet_two beams}
{\bar \sigma}^{\rm (2)} &=& \int \frac{{\rm d}{\bar \sigma}^{\rm (2)}(\theta, \varphi ; \, \theta_k)}{\rm{d}\Omega} \, {\rm d}\Omega = \frac{\sigma^{\rm (pl)}}{\cos\theta_k} \, ,
\end{eqnarray}
since the second term in Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet_superposition_2}) vanishes identically after the integration over the scattering angles.
\subsubsection{Scattering by a localized (mesoscopic) target}
\label{subsubsec:scattering_localized}
Until now we have discussed the potential scattering for two extreme cases of either a single--potential or a macroscopic (infinitely wide) target. In a more realistic experimental scenario, a focused electron beam collides with a well--localized mesoscopic atomic target. In order to account for geometrical effects in such a scenario, we describe a target as an incoherent ensemble of potential centers. The density of the scatterers in the transverse ($xy$--) plane is characterized by a distribution function $n({\bm b})$, which is normalized as follows:
\begin{equation}
\label{eq:electron_distribution}
\int n({\bm b}) \, {\rm d}^2{\bm b} = 1 \, .
\end{equation}
With the help of this distribution, we can also obtain a form factor of the mesoscopic target:
\begin{equation}
\label{eq:form_factor_target}
{\mathcal F}({\bm k}_\perp) = \int n({\bm b}) \, {\rm e}^{-i {\bm k}_\perp {\bm b}} \,
{\rm d}^2{\bm b} \, ,
\end{equation}
which later will be employed to simplify expression for the number of scattering events.
By making use of Eqs.~(29)--(30) from Ref.~\cite{KaK16} we find the number of scattering events:
\begin{equation}
\label{eq:number_of_events_localized_target}
\frac{{\rm d} \nu}{{\rm d} \Omega}= \frac{N_e}{\cos{\theta_k}} \,
\int \left| F({\bm Q}, {\bm b}) \right|^2 \,
n({\bm b}) \, {\rm d}^2{\bm b} \, ,
\end{equation}
where $N_e$ is the total number of electrons in the incident twisted bunch.
To perform an integration over the impact parameter ${\bm b}$ we insert into this expression the amplitude $F({\bm Q}, {\bm b})$ from (\ref{eq:amplitude_monochromatic}):
\begin{eqnarray}
\label{eq:number_of_events_localized_target_2}
\frac{{\rm d} \nu}{{\rm d} \Omega} &=& \frac{N_e}{\cos{\theta_k}} \, \int {\rm d}^2{\bm b}
\, n({\bm b}) \, \Bigg[ \int \frac{{\rm d}^2{\bm k}_\perp}{2 \pi} \, f({\bm Q} - {\bm k}_\perp) \,
\Phi^{(m)}_{\rm tr}({\bm k}_\perp) \nonumber \\[0.2cm]
&\times& \int \frac{{\rm d}^2{\bm k}'_\perp}{2 \pi} \, f^*({\bm Q} - {\bm k}'_\perp) \,
\Phi^{(m) *}_{\rm tr}({\bm k}'_\perp) \, {\rm e}^{i ({\bm k}_\perp-{\bm k}'_\perp) {\bm b}}\Bigg].
\end{eqnarray}
Using the explicit form of the transverse component $\Phi^{(m)}_{\rm tr}({\bm k}_\perp)$, as given by Eq.~(\ref{eq:wave_packet_transverse}), and noting that
\begin{equation}
\label{eq:b_integral_target_density}
\int {\rm e}^{i({\bm k}_\perp - {\bm k}'_\perp) {\bm b}} \, n({\bm b}) \,{\rm d}^2{\bm b} =
{\mathcal F}({\bm k}'_\perp - {\bm k}_\perp) \, ,
\end{equation}
we arrive at the following formula:
\begin{eqnarray}
\label{eq:number_of_events_localized_target_3}
\frac{{\rm d} \nu}{{\rm d} \Omega} &=& \frac{N_e}{\cos{\theta_k}} \int
\frac{{\rm d}^2{\bm k}_\perp}{2 \pi} \, \frac{{\rm d}^2{\bm k}'_\perp}{2 \pi} \,
f({\bm Q} - {\bm k}_\perp) \, f^*({\bm Q} - {\bm k}'_\perp) \nonumber \\
&& \hspace*{-1cm} \times{\mathcal F}({\bm k}'_\perp - {\bm k}_\perp) \,
\frac{{\rm e}^{i m (\varphi_k - \varphi_{k'})}}{2\pi \sqrt{k_\perp k'_\perp}}
g_{\varkappa_0 \sigma_\varkappa}(k_\perp) g_{\varkappa_0 \sigma_\varkappa}(k'_\perp) \, .
\end{eqnarray}
Further evaluation of this number of events requires the knowledge of (i) the target density $n({\bm b})$ as well as of (ii) the plane--wave amplitude $f({\bm Q} - {\bm k}_\perp)$.
For the numerical analysis below we take $n({\bm b})$ to be a Gaussian function:
\begin{equation}
\label{eq:Gaussian_distribution_2}
n({\bm b}) = \frac{1}{2 \pi \sigma_b^2} \, {\rm e}^{-\frac{({\bm b} - {\bm b}_0)^2}{2 \sigma_b^2}}
\, .
\end{equation}
This distribution is sharply peaked at the impact parameter ${\bm b}_0$ and its form factor reads as:
\begin{equation}
\label{eq:Gaussian_form_factor_2}
{\mathcal F}({\bm k}_\perp) = {\rm e}^{-i {\bm k}_\perp {\bm b}_0 -
{\bm k}^2_\perp \sigma_b^2/2}\, .
\end{equation}
Eq.~(\ref{eq:number_of_events_localized_target_3}) can be further simplified for two limiting cases. When the target is \textit{considerably smaller} than the incident wave-packet,
\be
\sigma_b \ll 1/\sigma_\varkappa,
\ee
we get
\be
\frac{{\rm d} \nu}{{\rm d} \Omega} =
L^{\rm (tw)}\, \fr{{\rm d}\sigma^{\rm (mesos)}}{{\rm d}\Omega},
\ee
where
\bea
\fr{{\rm d}\sigma^{\rm (mesos)}}{{\rm d}\Omega}
&=& \int_0^{2\pi} \fr{d\varphi_k} {2\pi}\fr{d\varphi_{k'}}{2\pi}
f\left(\bm Q-\bm k_\perp^{(0)}\right)\,f^*\left(\bm Q-\bm k_\perp^{'(0)}\right)
\nn
\\
&&\times {\rm e}^{im(\varphi_k-\varphi_{k'})}\,
{\cal F}\left(\bm k_\perp^{'(0)}-\bm k_\perp^{(0)}\right).
\nn
\eea
If, additionally, the inequality $\varkappa_0 \ll Q$ is fulfilled, then the cross section becomes
\be
\fr{d\sigma^{\rm (mesos)}}{d\Omega}=\fr{d\sigma^{\rm (PW)}}{d\Omega}\,
\int J_m^2(\varkappa_0 b)\, n(\bm b) \,d^2 \bm b,
\ee
which results in a one-dimension integral for the Gaussian distribution~(\ref{eq:Gaussian_distribution_2}) of atoms in the target:
\bea
\label{ratioRb0}
&&\fr{d\sigma^{\rm (mesos)}(b_0)}{d\Omega}= R(b_0)\, \fr{d\sigma^{\rm (PW)}}{d\Omega}\,,
\\
&& R(b_0)=\int_0^\infty J^2_m(\varkappa_0 b) I_0(bb_0/\sigma^2_b)\,{\rm e}^{-(b^2+b_0^2)/(2\sigma^2_b)}\, \fr{b\,db}{\sigma^2_b}.
\nn
\eea
When the target is \textit{considerably larger} than the incident wave-packet,
\be
\sigma_b \gg 1/\sigma_\varkappa,
\ee
and the amplitude $f$ is a real function ($\text{Arg}\,f = 0$), we obtain
\bea
\fr{d\nu}{d\Omega}&=&\fr{N_e}{\cos\theta_k}
\int\fr{d^2 \bm k_\perp}{2\pi k_\perp}\, g^2(k_\perp)\,
\left|f\left(\bm Q-\bm k_\perp\right) \right|^2 \,
n\left(\bm b_k \right),
\nn
\\
\bm b_k&=&\fr{m}{k_\perp}\,(\sin\varphi_{k}\,, -\cos\varphi_{k}\,, 0).
\label{eq:large-beam}
\eea
The scattering amplitude $f({\bm Q} - {\bm k}_\perp)$ depends on the potential $U(r)$ and it is real in the examples of Yukawa and hydrogenic potentials given in the next Section.
\section{Scattering by Yukawa and hydrogenic potentials}
\label{sec:Yukawa_and_hydrogenic_formulas}
In Section \ref{sec:theory} we have derived the number of events and the cross sections for scattering of the Bessel electron's wave--packets by a single--potential and by the mesoscopic-- and macroscopic targets. In that analysis, however, a particular form of the central potential $U(r)$ was not specified. Below we consider two specific cases, when $U(r)$ is the Yukawa-- and hydrogenic potential,
and discuss properties of the scattered electrons.
\subsection{Yukawa potential}
\label{susec:Yukawa_potential}
The Yukawa potential
\begin{equation}
\label{eq:Yukawa_potential}
U(r) = \frac{V_0}{r} \, {\rm e}^{-\mu r}
\end{equation}
describes the field with a typical radius of action $a\sim 1/\mu$. In a theory of atomic collisions, it is used very often as an approximation to the Coulomb field of the nucleus screened by atomic electrons, see for example Ref.~\cite{Sal91,SaM87}. This potential allows one to evaluate analytically the standard plane--wave scattering amplitude within the first--Born approximation:
\begin{eqnarray}
\label{eq:amplitude_plane_wave_Yukawa}
f({\bm q}) = \frac{-2 m_e V_0}{q^2 + \mu^2} \, .
\end{eqnarray}
By inserting this expression into Eq.~(\ref{eq:amplitude_monochromatic}) and making simple algebra we derive the scattering amplitude for the twisted electron:
\begin{eqnarray}
\label{eq:amplitude_wave_packet_twisted_Yukawa}
F({\bm Q}, {\bm b}) &=& - \frac{(-i)^m \, 2 m_e \, V_0 \, {\rm e}^{i m \varphi}}{\sqrt{2\pi}} \nonumber \\[0.1cm]
&& \hspace*{-1.5cm} \times \int_0^\infty {\rm d}k_\perp \, g_{\varkappa_0 \sigma_\varkappa}(k_\perp) \, \sqrt{k_\perp} \, I_{m}(\alpha, \beta, {\bm b}) \, ,
\end{eqnarray}
where the function
\begin{equation}
\label{eq:I_function}
I_{m}(\alpha, \beta, {\bm b}) = \int_0^{2\pi}
\frac{{\rm d}\psi}{2\pi} \;
\frac{{\rm e}^{im\psi+ik_\perp b\cos(\psi + \varphi-\varphi_b)}}{\alpha-\beta\,\cos{\psi}}
\end{equation}
was introduced and studied in Ref.~\cite{SeI15}. Here both $\alpha$ and $\beta$ are positive:
\begin{eqnarray}
\label{eq:alpha_beta}
\alpha &=& Q^2 + k_\perp^2 + \mu^2 \nonumber \\
&=& p^2_f(1+\cos^2{\theta_k} - 2\cos{\theta}\cos{\theta_k})+k^2_\perp+\mu^2 \, ,
\nonumber \\[0.2cm]
\beta &=& 2 k_\perp Q_\perp =2k_\perp p_f \sin{\theta} \, ,
\end{eqnarray}
and $\alpha > \beta$. By making use of Eqs.~(\ref{eq:amplitude_wave_packet_twisted_Yukawa})--(\ref{eq:alpha_beta}) one can study the potential scattering of a twisted electron for any experimental ``scenario''. Below we treat a few such scenarios.
First, let us study collision with a single Yukawa scatterer, located at the central axis of the incident packet.
For such a \textit{central collision} the impact parameter ${\bm b} = 0$ and the function (\ref{eq:I_function}) simplifies to:
\begin{equation}
\label{eq:I_function_central_collision}
I_m(\alpha, \beta, 0) =
\left( \frac{\beta}{\alpha + \sqrt{\alpha^2-\beta^2}} \right)^{|m|}
\frac{1}{\sqrt{\alpha^2-\beta^2}} \, ,
\end{equation}
see Ref.~\cite{SeI15} for further details. With the help of this expression we can study angular distribution of the scattered electrons.
The small scattering angles are of specific interest here, for which $I_m(\alpha, \beta, 0) \propto \beta^{|m|} \propto (\sin \theta)^{|m|}$. Such a $\theta$--behaviour implies that for $m \ne 0$ the angular distribution
\begin{equation}
\label{eq:angular_distribution_single_Yukawa}
\frac{{\rm d}\nu}{{\rm d}\Omega} \propto (\sin{\theta})^{2|m|}
\; \; {\rm for} \; \; \theta \to 0 \, ,
\end{equation}
vanishes for the forward emission, $\theta \to 0$. We remind the reader that this result was predicted above on a basis of the analysis of the scattering amplitude, see Eq.~(\ref{eq:amplitude_wave_packet_b_0_forward}).
The dip in the scattering pattern at $\theta \to 0$ disappears, however, with the increase of the impact parameter $b$. Indeed, since for the forward scattering the function (\ref{eq:I_function}) reads as
\begin{equation}
\label{eq:I_function_forward_emission}
I_m(\alpha, \beta, \bm b){\Large |}_{\theta=0}= \frac{e^{-im(\varphi-
\varphi_b+\pi/2)}}{\alpha}
\;J_m(k_\perp b) \, ,
\end{equation}
and $J_m(k_\perp b) \propto b^{|m|}$ for small impact parameters, we find:
\begin{equation}
\label{eq:angular_distribution_single_Yukawa_2}
\frac{{\rm d}\nu}{{\rm d}\Omega} \propto b^{2|m|} \; \; {\rm for} \; \; b \to 0 \, .
\end{equation}
This expression predicts that the (forward) electron emission quickly increases if the scattering center is shifted from the central beam axis.
In the second scenario, we still focus on the scattering off a single potential but for a particular case of a \textit{wide} wave--packet. The number of events for this scenario can be obtained from Eq.~(\ref{eq:eq:numebr_events_single_potential_general_wide}) as follows:
\begin{equation}
\label{eq:number_of_events_Yukawa_wide_packet}
\frac{{\rm d}\nu}{{\rm d}\Omega} =
L^{({\rm tw})} \, \left| 2 m_e V_0 \, I_m(\alpha_0, \beta_0, b) \right|^2 \, ,
\end{equation}
where $\alpha_0$ and $\beta_0$ are given by Eq.~(\ref{eq:alpha_beta}) in which $k_\perp = \varkappa_0$ and $L^{({\rm tw})}$ is from Eq.~(\ref{eq:Ltw_general}).
This expression can be further simplified for the central collision, that is, for $b = 0$:
\begin{equation}
\label{eq:number_of_events_Yukawa_wide_packet_central}
\frac{{\rm d}\nu}{{\rm d} \Omega}= L^{({\rm tw})} \, \left|f_V^B \right|^2 \,,
\end{equation}
where we made use of Eq.~(\ref{eq:I_function_central_collision}) and introduced the ``amplitude'':
\begin{eqnarray}
\label{eq:fBV_amplitude}
f_V^B &=& -(-i)^m \, 2 m_e V_0 \, {\rm e}^{i m\varphi} \,
\left(\frac{v}{u+\sqrt{u^2-v^2}}\right)^{|m|} \nonumber \\
&\times& \frac{1}{\sqrt{u^2-v^2}} \, ,
\end{eqnarray}
with $u= Q^2 + \varkappa_0^2 + \mu^2$ and $v = 2 \varkappa_0 Q_\perp$. Up to unessential pre--factor this amplitude coincides with $f_V^B$ reported in Eq.~(24) of Ref.~\cite{BoP14}. In that work, however, no indication was given of how $f_V^B$ is related to the number of events or to the cross section.
Up to now we have discussed the potential scattering off a single Yukawa potential. If, in contrast, a wide wave--packet collides with an infinitely extended target consisting of randomly distributed Yukawa scatterers, one can use Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet}) to derive the cross section:
\begin{eqnarray}
\label{eq:cross_section_macroscopic_target_wide_packet_Yukawa}
\frac{{\rm d}\bar\sigma(\theta; \theta_k)}{{\rm d}\Omega}
&& \nonumber \\
&& \hspace*{-1.5cm} = \frac{(2 m_e V_0)^2}{\cos{\theta_k}}
\,\int_0^{2\pi}\frac{{\rm d}\varphi_k}{2\pi}
\frac{1}{\left[u-v\cos{(\varphi_k-\varphi)}\right]^2}
\nonumber \\[0.2cm]
&& \hspace*{-1.5cm} =
\frac{(2 m_e V_0)^2}{\cos{\theta_k}} \frac{u}{\sqrt{(u^2-v^2)^3}} \, .
\end{eqnarray}
Here, we used the same short--hand notations $u$ and $v$ as in Eq.~(\ref{eq:fBV_amplitude}). One can further modify this expression to account for the incident electron beam, prepared as a coherent superposition of two Bessel states, see Eq.~(\ref{eq:superposition_two_states}). In this case, the differential cross section ${\rm d}\bar\sigma^{(2)}(\theta, \varphi; \theta_k)/{\rm d}\Omega$ is given by the second line of Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet_Yukawa}) in which the additional factor (\ref{eq:G_factor}) should be inserted under the integral over $\varphi_k$.
\subsection{Hydrogen atom in its ground state}
\label{susec:hydrogenic_potential}
We have discussed above the scattering of twisted electrons by a Yukawa potential. A superposition of few such potentials can be employed to reproduce accurately the (static) potential of \textit{any} neutral atom \cite{Sal91,SaM87}. In this section, however, we study a particular case of atomic hydrogen in the ground $1s$ state. Its potential is represented as follows:
\begin{equation}
\label{eq:hydrogenic_potential}
U_{H}(r) = -\frac{e^2}{r} \,\left( 1+ \frac{r}{a_0} \right) \, {\rm e}^{-2r/a_0} \, ,
\end{equation}
where $a_0$ is the Bohr radius. By inserting this expression into Eqs.~(\ref{eq:amplitude_plane_wave}) and (\ref{eq:amplitude_monochromatic}), we find the scattering amplitudes for the incident plane--wave:
\begin{equation}
\label{eq:amplitude_plane_wave_hydrogenic}
f({\bm q}) = \frac{a_0}{2} \, \left(\frac{1}{1 + (q a_0 /2)^2} +
\frac{1}{\left(1 + (q a_0 /2)^2 \right)^2} \right) \, ,
\end{equation}
and for the twisted electron's wave--packet:
\begin{eqnarray}
\label{eq:amplitude_wave_packet_twisted_hydrogenic}
F({\bm Q}, {\bm b}) &=& \frac{(-i)^m a_0}{2\sqrt{2\pi}} \,
\int_0^\infty {\rm d}k_\perp \, g_{\varkappa_0 \sigma_\perp}(k_\perp) \sqrt{k_\perp} \nonumber \\
&& \hspace*{-1.5cm} \times \int_0^{2\pi} \frac{{\rm d} \varphi_k}{2\pi} \;
\left(\frac{1}{w}+\frac{1}{w^2} \right) \; {\rm e}^{i m\varphi_k + ik_\perp b \cos{(\varphi_k-\varphi_b)}} \, ,
\end{eqnarray}
where we introduced the following notations:
\begin{eqnarray}
\label{eq:w_alpha_beta_hydrogenic}
w &=& \alpha - \beta \, \cos{(\varphi_k-\varphi)} \, , \nonumber \\[0.1cm]
\alpha &=& 1 + \mbox{$\frac 14$} a_0^2 p^2_f \left[1 + \cos^2{\theta_k} - 2\cos{\theta} \cos{\theta_k}\right] + \mbox{$\frac 14$} a_0^2 k^2_\perp \, , \nonumber \\[0.1cm]
\beta&=&\mbox{$\frac 12$} a_0^2 k_\perp p_f \sin{\theta} \, .
\end{eqnarray}
In order to simplify $F({\bm Q}, {\bm b})$ further one can use the identity:
\begin{equation}
\label{eq:w_identity}
\frac{1}{w} + \frac{1}{w^2}=
\left(1 -\frac{\partial}{\partial \alpha} \right)\frac{1}{w} \, ,
\end{equation}
and then we re--write Eq.~(\ref{eq:amplitude_wave_packet_twisted_hydrogenic}) as follows:
\begin{eqnarray}
\label{eq:amplitude_wave_packet_twisted_hydrogenic_2}
F({\bm Q}, {\bm b}) &=& \frac{(-i)^m \,a_0\, {\rm e}^{im\varphi}}{2 \sqrt{2\pi}}
\nonumber \\
&& \hspace*{-2cm} \times
\int_0^\infty {\rm d}k_\perp \, g_{\varkappa_0 \sigma_\perp}(k_\perp)\sqrt{k_\perp} \,
\left(1- \frac{\partial}{\partial \alpha} \right)\,
I_m(\alpha, \beta, \bm b),
\end{eqnarray}
where the function $I_m(\alpha, \beta, \bm b)$ is given by Eq.~(\ref{eq:I_function}).
Similar to the Yukawa potential, one can employ the amplitude (\ref{eq:amplitude_wave_packet_twisted_hydrogenic_2}) to investigate the scattering of twisted electrons by various hydrogenic targets. Again, we start with a \textit{single} hydrogen atom for which the number of (scattering) events is given by:
\begin{eqnarray}
\label{eq:number_events_hydrogenic_single}
\frac{{\rm d}\nu}{{\rm d}\Omega} &=& \frac{N_e a_0^2}{8 \pi \cos\theta_k} \nonumber \\
&& \hspace{-2cm} \times
\left| \int_0^\infty {\rm d}k_\perp \, g_{\varkappa_0 \sigma_\perp}(k_\perp) \, \sqrt{k_\perp}\,
\left(1-\frac{\partial}{\partial \alpha}\right)\, I_m(\alpha, \beta,\bm b)
\right|^2 .
\end{eqnarray}
From this expression and from the properties of the function $I_m(\alpha, \beta, \bm b)$ we can again derive Eqs.~(\ref{eq:angular_distribution_single_Yukawa}) and (\ref{eq:angular_distribution_single_Yukawa_2}) which describe emission pattern of the outgoing electrons for the forward direction and the small impact parameters $b$. Moreover, in the limit of a wide wave--packet, when the momentum distribution function $g_{\varkappa_0 \sigma_\varkappa}(k_\perp)$ is sharply peaked ar $k_\perp = \varkappa_0$, we get from Eqs.~(\ref{eq:eq:numebr_events_single_potential_general_wide}) and (\ref{eq:amplitude_wave_packet_twisted_hydrogenic_2})
\begin{equation}
\label{eq:number_events_hydrogenic_single_wide}
\frac{{\rm d}\nu}{{\rm d}\Omega} = L^{({\rm tw})}
\left| \frac{a_0}{2} \, \left(1 -
\frac{\partial}{\partial \alpha_0}\right) \, I_m(\alpha_0, \beta_0, \bm b)
\right|^2 \, .
\end{equation}
Here, $\alpha_0$ and $\beta_0$ are given by Eq.~(\ref{eq:w_alpha_beta_hydrogenic}) with
$k_\perp= \varkappa_0$, and $L^{\rm tw}$ given by Eq. (\ref{eq:Ltw_general}).
With the help of the amplitude (\ref{eq:amplitude_wave_packet_twisted_hydrogenic_2}) one can also treat the scattering of the twisted wave--packet by a macroscopic hydrogenic target. For this case, the angle--differential cross section reads as follows:
\begin{eqnarray}
\label{eq:number_events_hydrogenic_single_macroscopic}
\frac{{\rm d}\bar\sigma(\theta,\theta_k,p_f)}{{\rm d}\Omega} &=&
\frac{a_0^2}{4\cos{\theta_k}}
\,\int_0^{2\pi} \frac{d\varphi_k}{2\pi} \,
\Bigg[\frac{1}{u-v\cos{(\varphi_k-\varphi)}} \nonumber \\
&+& \frac{1}{(u-v\cos{(\varphi_k-\varphi)})^2} \Bigg]^2 \nonumber \\[0.1cm]
&& \hspace*{-2cm} = \frac{a_0^2}{4\cos{\theta_k}}\left(-\frac{\partial}{\partial u}+
\frac{\partial^2}{\partial u^2}-\frac{1}{6} \frac{\partial^3}{\partial
u^3}\, \right)\frac{1}{\sqrt{u^2-v^2}}\,,
\end{eqnarray}
where $u = 1+\mbox{$\frac 12$} a_0^2 p^2_f(1 -\cos{\theta}\cos{\theta_k})$, $v = \mbox{$\frac 12$} a_0^2 p^2_f \sin{\theta}\sin{\theta_k}$, $u > v$, and the wide incident wave--packet is assumed.
\section{Results and discussions}
\label{sec:results}
In order to illustrate the theory developed in Sections~\ref{sec:theory} and \ref{sec:Yukawa_and_hydrogenic_formulas}, we intend now to present calculations for scattering of the twisted electrons by \textit{hydrogen atoms}. All the results in Figs.~1--4 are presented for the wave packets with the averaged momentum $p_i = 10/a_0$ where $a_0$ is the Bohr radius (this momentum corresponds to the kinetic energy of $1.4$ keV). Similar to before, these calculations are performed for three various scenarios in which electrons collide with a single H atom, as well as with infinitely extended (macroscopic) and localized (mesoscopic) atomic target.
\begin{figure}[t]
\hspace*{-0.2cm}
\includegraphics[width=0.99\linewidth]{Fig_1_last}
\caption{(Color online) The averaged cross section (\ref{eq:cross_section_macroscopic_target}), (\ref{eq:number_events_hydrogenic_single_macroscopic}) for the scattering of twisted electrons by a macroscopic target, consisting of hydrogen atoms in their ground state. Results are presented for the incident wave packets with the width
$\sigma_\varkappa = \varkappa_0/3$ (blue dashed line) and $\sigma_\varkappa \ll \varkappa_0$ (black solid line), and the opening angle $\theta_k =$~15~deg (upper panel) and 30~deg (lower panel). Results of calculations are compared, moreover, with the prediction obtained for the plane--wave electrons (red dotted line).
\label{Fig1}}
\end{figure}
\subsection{Scattering by a macroscopic target}
\label{subsec:results_macroscopic}
We start our calculations from the most experimentally accessible scenario of a macroscopic target. In this case, the transverse extension of an electron beam is assumed to be much smaller than the size of the target, $\mathcal{R} \gg 1/\sigma_{\varkappa}$, and the scattering process is described by the cross section (\ref{eq:cross_section_macroscopic_target}). This cross section, \textit{averaged} over all impact parameters of individual scatterers, is independent of the electron's OAM projection but is still sensitive to the kinematic parameters and to the size of the beam. In order to illustrate such a sensitivity, we display in Fig.~\ref{Fig1} the cross section ${\rm d}{\bar \sigma}/{\rm d}\Omega$ for the scattering of twisted electrons by a (macroscopic) hydrogenic target. Calculations have been performed for two values of the beam opening angle, $\theta_k = 15$~deg (upper panel) and $\theta_k = 30$~deg (lower panel). We also compare the results obtained for the incident plane--wave electrons (red dotted line) with the predictions for the twisted wave--packet (\ref{eq:wavepacket_Bessel_momentum}) with the width $\sigma_{\varkappa} = \varkappa_0/3$ (blue dashed line) and $\sigma_{\varkappa} \ll \varkappa_0$ (black solid line). The latter result corresponds to an approximation of a wide wave--packet which, in its limit, recovers the monochromatic case.
As can be seen in the Fig.~\ref{Fig1}, the angle--differential cross section is indeed very sensitive to the opening angle $\theta_k$. While the incident plane--wave electrons are scattered predominantly in the forward direction, $\theta = 0$~deg, the ${\rm d}{\bar \sigma}/{\rm d}\Omega$ for the twisted wave--packets is peaked near $\theta = \theta_k$. As discussed already in Section~\ref{subsubsec_wide_target} for a wide packet, this behaviour is expected from Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet_2}) and from the (plane--wave) amplitude $f({\bm Q} - {\bm k}_{\perp}^{(0)})$ that is maximal at $\left({\bm Q} - {\bm k}_{\perp}^{(0)}\right)^2 = 0$. With the increase of $\sigma_\varkappa$ and, hence, decrease of the transversal size of the wave packet $\Delta x = \Delta y \sim 1/\sigma_{\varkappa}$, the ``wide--packet'' approximation is not valid any more. However, even in this ``spatially--localized--packet'' case the twisted electrons are scattered most likely under the angles near $\theta = \theta_k$, although the peak in the emission pattern becomes less sharp, see blue dashed curve in Fig.~\ref{Fig1}.
Despite the different scattering patterns of the incident plane--wave-- and twisted electrons, the \textit{total} cross sections for these two cases are generally of the same order of magnitude. For instance, by integrating the ${\rm d}{\bar \sigma}/{\rm d}\Omega$ (black solid line) and ${\rm d}{\sigma}_{\rm pl}/{\rm d}\Omega$ (red dotted line) from Fig.~\ref{Fig1} over the angle $\theta$, we confirm numerically the relation (\ref{eq:total_cross_section_wide_packet}) between ${\bar \sigma}$ and $\sigma_{\rm pl}$ for the wide twisted packet and the plane wave, respectively. For a spatially localized packet, for which $\sigma_\varkappa \sim \varkappa_0$, the total cross section is about 10--30 \% smaller than the ${\bar \sigma}({\rm wide}) = \sigma_{\rm pl}/\cos\theta_k$, which, again, makes the observation of the scattering of focused twisted beams experimentally feasible.
\begin{figure}[t]
\includegraphics[width=0.95\linewidth]{Fig_2_last}
\caption{(Color online) The azimuthal assymetry parameter (\ref{eq:cross_section_macroscopic_target_wide_packet_superposition_interference}) for the scattering of the superposition of two Bessel wave--packets by a macroscopic hydrogenic target. The calculations have been performed for two incident electron beams with the width $\sigma_{\varkappa} \ll \varkappa_0$, difference of OAM projections $\Delta m =$~2, and the opening angles $\theta_k =$~10~deg (red dotted line), 20~deg (blue dashed line) and 30~deg (black solid line).
\label{Fig2}}
\end{figure}
Until now we have discussed the potential scattering of a \textit{single} twisted wave--packet. For this case, and for a macroscopic target, the averaged cross section ${\rm d}{\bar \sigma}/{\rm d}\Omega$ appears to be insensitive to the projection of the orbital angular momentum $m$. However, the OAM--sensitivity can be restored if the incident electron beam is a coherent superposition (\ref{eq:superposition_two_states}) of \textit{two} twisted states with different $m$'s. As seen from Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet_superposition_2}), the scattering pattern for such a superposition depends not only on the polar angle $\theta$ but also on the azimuthal one, $\varphi$. This $\varphi$--dependence is parametrized as ${\rm d}{\bar \sigma}^{(2)}/{\rm d}\Omega \sim 1 + A \, \sin\left( \Delta m(\varphi-\pi/2)+\Delta\alpha \right)$ and, hence, is determined by the difference of the beam phases $\Delta \alpha$ and the OAM projections $\Delta m$. The strength of the azimuthal asymmetry is defined by the parameter $A \equiv A(\theta; \theta_k)$ that is given by Eq.~(\ref{eq:cross_section_macroscopic_target_wide_packet_superposition_interference}) and it is insensitive to $\varphi$. In Fig.~\ref{Fig2} we display this parameter $A$ as a function of the polar scattering angle $\theta$ for three opening angles: $\theta_k =$~10~deg (red dotted line), 20~deg (blue dashed line) and 30~deg (black solid line). The calculations have been performed for the \textit{wide} wave packets with $c_1 = c_2 = 1/\sqrt{2}$, $\Delta m = 2$, and the width $\sigma_{\varkappa} \ll \varkappa_0$. As seen in the figure, the $A(\theta; \theta_k)$ is peaked at the angles $\theta \approx \theta_k$ and, moreover, it increases with the growth of $\theta_k$. This implies that
the $\Delta m$-- and $\Delta \alpha$--dependences can be observed most easily for the beams with large opening angles $\theta_k$ and for electron detectors placed at $\theta \approx \theta_k$.
\begin{figure}[t]
\includegraphics[width=0.95\linewidth]{Fig_3_last}
\caption{(Color online)
The (normalized) number of events ${\rm d}\nu(\theta) / {\rm d} \Omega $ with $N_e=1$ from Eq.~(\ref{eq:number_events_hydrogenic_single})
for the scattering of the Bessel wave--packet by a single hydrogen atom, placed at a distance $b = 0$ (upper panel) and $b = a_0$ (lower panel). Results are presented for the incident packets with the width $\sigma_\varkappa = \varkappa_0/5$, the opening angle $\theta_k =$~10~deg, and the OAM projection $m = 0$ (black solid line), $m = 1$ (blue dashed line), and $m = 2$ (red dotted line).
\label{Fig3}}
\end{figure}
\subsection{Scattering by a single potential}
\label{subsec:results_single}
Until now we have discussed the most realistic (experimental) scenario is which twisted electron wave--packet collides with a macroscopic atomic target. However, in order to better understand the features of the potential scattering of Bessel packets it is more convenient to consider a collision with a \textit{single} atom, located at a particular impact parameter $b$ with respect to the beam's axis. As mentioned above, the number of (scattering) events ${\rm d}\nu / {\rm d} \Omega$ can be used to characterize the process for such a ``single--atom--case''. In Fig.~\ref{Fig3} we display, for example, the (normalized) number of events ${\rm d}\nu(\theta) / {\rm d} \Omega $ with $N_e=1$ from Eq.~(\ref{eq:number_events_hydrogenic_single}), as a function of the polar angle $\theta$ of outgoing electrons, defined with respect to the beam ($z$--) axis. Here we assume $\varphi = 0$, that is, scattered electrons are ``detected'' within the plane of the target atom. Calculations were performed for a hydrogen in the ground $1s$ state and for the incident wave packet with the width $\sigma_\varkappa = \varkappa_0/5$. We consider, moreover, two impact parameters, $b = 0$ (upper panel) and $b = a_0$ (lower panel) and three projections of the orbital angular momentum, $m = 0$ (black solid line), $m = 1$ (blue dashed line), and $m = 2$ (red dotted line).
As seen from the Fig.~\ref{Fig3}, there is a strong dependence of the electron scattering pattern on the OAM projection. This $m$--dependence is most pronounced for the central collision, $b = 0$, and small angles, $\theta \to 0$. In this case, the (forward) electron scattering is allowed only for $m = 0$, while ${\rm d}\nu (\theta=0)/{\rm d} \Omega$ vanishes identically for $m = 1$ and $m = 2$. Such a behaviour is expected from the analysis of the transverse component of the electron wave--function and of the scattering amplitude, see Eq.~(\ref{eq:amplitude_wave_packet_b_0_forward}). With the increase of the impact parameter $b$ the dip in the electron angular distribution for $\theta = 0$ (and $m \ne 0$) disappears. Indeed, for $b = a_0$ (lower panel of Fig.~\ref{Fig3}) the forward scattering is allowed and even becomes dominant as the target atom is further shifted from the center of the incident packet.
In contrast to the case of the macroscopic target, the collision of a wave--packet having one definite value of $m$ with a single well--localized atom can also result in the \textit{azimuthal asymmetry} of the electron scattering pattern.
\begin{figure}[t]
\includegraphics[width=0.9\linewidth]{Fig_4_last}
\caption{(Color online)
The azimuthal angle dependence of the normalized number of events ${\rm d}\nu (\theta, \varphi)/ {\rm d}\nu (\theta, \varphi = 0)$ for scattering of the Bessel wave--packet by a single hydrogen atom, placed at the distance $b = 2a_0$. The results are presented for the incident packets with the width $\sigma_\varkappa = \varkappa_0/5$, the opening angle $\theta_k =10$~deg, and the OAM projection $m = -2$ (red dotted line), $m = -1$ (blue dashed line), $m = 0$ (black solid line), $m = 1$ (blue solid line), and $m = 2$ (red solid line). We assumed, moreover, that outgoing photons are detected at the polar angles $\theta = 1$~deg (upper panel) and 20~deg (lower panel).
\label{Fig4}}
\end{figure}
One can expect this because the system of the incident packet \textit{plus} target atom at $b \ne 0$ does not possess the azimuthal symmetry, that is ``recovered'' only after integration over $b$. The resulting $\varphi$--distribution of outgoing electrons
(see Eq.~(\ref{eq:number_events_hydrogenic_single})) is very sensitive to the kinematic parameters and the OAM projection of Bessel electrons. In Fig.~\ref{Fig4}, for example, we display the normalized number of scattered events, ${\rm d}\nu (\theta, \varphi)/ {\rm d}\nu (\theta, \varphi=0)$, for the incident wave--packet whose parameters are $\theta_k = 10$~deg, and $\sigma_\varkappa = \varkappa_0/5$. Here, we performed calculations for the impact parameter $b = 2 a_0$, two polar scattering angles, $\theta = 1$~deg (upper panel) and $\theta = 20$~deg (lower panel), and five OAM projections:
$m = -2$ (red dotted line), $m = -1$ (blue dashed line), $m = 0$ (black solid line), $m = 1$ (blue solid line) and $m = 2$ (red solid line).
As seen from the figure, the variation of $m$ may lead to the \textit{qualitative} changes in the azimuthal angular distribution, thus suggesting that the scattering by well--localized targets can be used for diagnostics of the twisted beams.
\subsection{Scattering by a mesoscopic target}
\label{subsec:results_mesoscopic}
In Sections~\ref{subsec:results_macroscopic} and \ref{subsec:results_single} above we treated two limiting cases of scattering either by an infinitely large target or by a single atom. These calculations have clearly indicated that the size of a target strongly influences the OAM--sensitivity of the electron scattering pattern. Indeed, while the averaged cross section (\ref{eq:cross_section_macroscopic_target}) is independent of $m$ unless the superposition of two packets is considered, the number of events (\ref{eq:numebr_events_single_potential_general}) for a single potential varies significantly with the change of the OAM. In order to illustrate better the ``size--effect'' in scattering of the twisted wave--packets, here we apply Eqs.~(\ref{eq:number_of_events_localized_target}) and (\ref{eq:Gaussian_distribution_2}) which describe collisions with a target of finite sizes. By making use of these expressions, we have evaluated the number of scattering events for
two limiting cases given in Sect. II C 4.
\begin{figure}[t]
\includegraphics[width=0.9\linewidth]{Fig_5_last}
\caption{(Color online)
The relative differential cross section
$R(b_0)=d\sigma^{\rm (mesos)}(b_0)/d\sigma^{\rm (PW)}$ from Eq.~(\ref{ratioRb0}) as a function of the impact parameter $b_0$ of the target center (upper panel) for $\sigma_b=1/\varkappa_0=10$ nm, $1/\sigma_\varkappa = 50$ nm, $\theta_k\ll 1$ and for the following projections of the orbital angular momentum: $m=0$ (black solid line), $m=1$ (blue dashed line), $m=3$ (red dot-dashed line), $m=5$ (black dotted line). On the lower panel the normalized density of the incident twisted beam $\rho^{(m)}(r_\perp)/\rho^{(0)}(0)$ is shown for the same parameters.
\label{Fig5}}
\end{figure}
When the size of the target $\sigma_b$ is smaller than the transverse spread of the wave--packet $\Delta x = \Delta y \sim 1/\sigma_{\varkappa}$, we use Eq.~(\ref{ratioRb0}) and present in Fig.~\ref{Fig5} (upper panel) the relative differential cross section
\be
R(b_0)=\fr{{d\sigma^{\rm (mesos)}(b_0)}/{d\Omega}}
{{d\sigma^{\rm (PW)}}/{d\Omega}}
\ee
as a function of the impact parameter $b_0$ of the target
center. Calculations have been performed for the following parameters: $\sigma_b=1/\varkappa_0=10$ nm, $1/\sigma_\varkappa = 50$ nm, $\theta_k\ll 1$ and for four projections of the orbital angular momentum: $m=0$ (black solid line), $m=1$ (blue dashed line), $m=3$ (red dot-dashed line), $m=5$ (black dotted line). On the lower panel of Fig~\ref{Fig5} we present the normalized density of the incident twisted beam $\rho^{(m)}(r_\perp)/\rho^{(0)}(0)$ for the same parameters. Here the density itself reads as follows (see Eq.~(\ref{eq:transverse_component_convolution})):
\be
\rho^{(m)}(r_\perp)= \left|\int_0^\infty \sqrt{\varkappa/(2\pi)}\,
J_m (\varkappa r_\perp) g_{\varkappa_0 \sigma_\varkappa}\varkappa\,
{\rm d} \varkappa \right|^2.
\ee
It is clearly seen from comparison of these panels that the ``twisted'' cross section is very sensitive to variation of the incident beam's density.
When the size of the target $\sigma_b$ is larger than the transverse spread of the wave--packet $1/\sigma_{\varkappa}$, we use Eq.~(\ref{eq:large-beam}) and
present in Fig.~ 6 the number of scattering events as a function of the impact parameter for three different values of the OAM: $m=0$ (black solid line), $m=50$ (blue dashed line), $m=100$ (red dotted line). Clearly, scattering off the large target is sensitive to the spatial density of the incident wave front only for the big values of the OAM when $\sigma_b \sim m/\varkappa_0,\, m \gg 1$.
Note that electrons with the OAM quanta as high as $m=200$ have already been generated \cite{GrG15}.
\begin{figure}[t]
\includegraphics[width=0.99\linewidth]{Fig_6_last}
\caption{(Color online)
The number of events (\ref{eq:large-beam}) for a wide target as a function of the impact parameter $b_0$ for $\sigma_b = 10$ nm, $1/\sigma_\varkappa = 2$ nm, $\theta_k = \theta = 1\, \text{deg},\, \varphi = \varphi_b = 0$ and for the following projections of the OAM: $m=0$ (black solid line), $m=50$ (blue dashed line), $m=100$ (red dotted line).
\label{Fig6}}
\end{figure}
\section{Summary}
\label{sec:summary}
In summary, we have applied the generalized Born approximation, developed by us in Ref.~\cite{KaK16}, in order to investigate scattering of the vortex electrons by atomic targets. In our study, we focused especially on derivation of the physically meaningful expressions for the number of (scattering) events and for the differential cross sections. These physical observables have been obtained for different ``experimental'' scenarios in which incident wave--packet collides with (i) a single potential as well as with (ii) localized (mesoscopic) target and with (iii) an infinitely wide (macroscopic) one. Even though the developed theory can be employed for \textit{any} type of the electron--atom interaction, in the present study we described the scattering off the Yukawa- and hydrogenic potentials. For these potentials, simple expressions for the cross sections and for the numbers of scattering events are presented which can be used for analysis and guidance of the future scattering experiments.
On the basis of the developed theory we showed that the number of scattering events in collisions involving twisted electrons is comparable to that in the standard plane--wave regime. This implies that experiments with the focused twisted electrons are feasible with the present-day detectors. The outcome of these experiments will depend, however, on the relative size of an incident beam and an atomic target. For example, for targets whose width is narrower than the transverse size of the beam, the angular distribution of the scattered electron can be very sensitive not only to the opening angle, but also to the OAM projection itself. The sensitivity to the orbital momentum $m$ vanishes, however, with the increase of the target's size.
On the other hand, even in this macroscopic case the OAM--sensitivity can be recovered if one \textit{prepares} an incident beam as a coherent superposition of two twisted packets with two different $m$'s. We conclude, therefore, that the potential scattering can provide a wide range of opportunities for diagnostics of the twisted electron beams, and the corresponding experiments can be carried out at the present--day facilities.
\
D.K. wishes to thank C.\,H.\,Keitel, A.\,Di Piazza and S.\,Babacan for their hospitality
during his stay at the Max-Planck-Institute for Nuclear Physics in Heidelberg.
D.K. is supported by the Alexander von Humboldt Foundation (Germany)
and by the Competitiveness Improvement Program of the Tomsk State University. G.L.K and V.G.S. are supported by the Russian Foundation for Basic Research via grant 15-02-05868.
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{"url":"https:\/\/www.hackmath.net\/en\/math-problem\/31201","text":"# Find the 15\n\nFind the tangent line of the ellipse 9 x2 + 16 y2 = 144 that has the slope k = -1\n\nResult\n\nt1 = (Correct answer is: x + sqrt(19))\nt2 = (Correct answer is: x-sqrt(19))\n\n### Step-by-step explanation:\n\n${t}_{2}=x-\\sqrt{19}$\n\nDid you find an error or inaccuracy? Feel free to write us. Thank you!\n\nTips to related online calculators\nLine slope calculator is helpful for basic calculations in analytic geometry. The coordinates of two points in the plane calculate slope, normal and parametric line equation(s), slope, directional angle, direction vector, the length of the segment, intersections of the coordinate axes, etc.\nLooking for help with calculating roots of a quadratic equation?\nDo you have a linear equation or system of equations and looking for its solution? Or do you have a quadratic equation?\n\n#### You need to know the following knowledge to solve this word math problem:\n\nWe encourage you to watch this tutorial video on this math problem:\n\n## Related math problems and questions:\n\n\u2022 Ellipse\nEllipse is expressed by equation 9x2 + 25y2 - 54x - 100y - 44 = 0. Find the length of primary and secondary axes, eccentricity, and coordinates of the center of the ellipse.\n\u2022 Tangents to ellipse\nFind the magnitude of the angle at which the ellipse x2 + 5 y2 = 5 is visible from the point P[5, 1].\n\u2022 Speed of Slovakian trains\nRudolf decided to take the train from the station 'Ostratice' to 'Horn\u00e9 Ozorovce'. In the train timetables found train Os 5409 : km 0 Chynorany 15:17 5 Ostratice 15:23 15:23 8 Rybany 15:27 15:27 10 Doln\u00e9 Na\u0161tice 15:31 15:31 14 B\u00e1novce nad Bebravou 15:35 1\n\u2022 Prove\nProve that k1 and k2 are the equations of two circles. Find the equation of the line that passes through the centers of these circles. k1: x2+y2+2x+4y+1=0 k2: x2+y2-8x+6y+9=0\n\u2022 Curve and line\nThe equation of a curve C is y=2x\u00b2 -8x+9, and the equation of a line L is x+ y=3 (1) Find the x coordinates of the points of intersection of L and C. 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A) the line is given parametrically: x = - 4 + 2p, y = 2 - 3p B) the line is given by the slope form: y = 3x - 1 C) the line is given by two points: A [3; -3], B [-5; 2] D) t\n\u2022 Trapezoid 15\nArea of trapezoid is 266. What value is x if bases b1 is 2x-3, b2 is 2x+1 and height h is x+4\n\u2022 Find the\nFind the image A\u00b4 of point A [1,2] in axial symmetry with the axis p: x = -1 + 3t, y = -2 + t (t = are real number)\n\u2022 Parametric form\nCalculate the distance of point A [2,1] from the line p: X = -1 + 3 t Y = 5-4 t Line p has a parametric form of the line equation. ..\n\u2022 Equation\nEequation f(x) = 0 has roots x1 = 64, x2 = 100, x3 = 25, x4 = 49. How many roots have equation f(x2) = 0 ?\n\u2022 Consider 2\nConsider the following formula: y = 3 ( x + 5 ) ( x - 2 ) Which of the following formulas is equivalent to this one? A. Y=3x2+9x-30 B. Y=x2+3x-10 C. Y=3x2+3x-10 D. 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Home › Letitia Elizabeth Landon - The Poetry Of (Audiobook)
Letitia Elizabeth Landon - The Poetry Of (Audiobook)
Read by Ghizela Rowe (Unabridged: 54mins)
Letitia Elizabeth Landon was born on 14 August 1802 in Chelsea, London. A precocious child she had her first poem published is 1820 using the single 'L' as her marker.
The following year her first volume appeared and sold well. She published a further two poems that same year with just the initials 'L.E.L." It provided the basis for much intrigue. She became the chief reviewer of the Gazette and published her second collection, The Improvisatrice, in 1824. By 1826, rumours began to circulate that she had had affairs.
For several years they continued to circulate until she broke off an engagement when her betrothed, upon further investigation, found them to be unfounded. Her words reflect the lack of trust she felt "The mere suspicion is dreadful as death".
On June 7th 1838 she married George Maclean, initially in secret, and a month later they sailed to Cape Coast. However the marriage proved to be short lived as on October 15th Letitia was found dead, a bottle of prussic acid in her hand. Her reputation as a poet diminished until fairly recently; her work felt to be simplistic and too simply constructed. However when put into context it is more rightly seen as working on many levels and meanings as was needed for those more moral times.
Letitia Elizabeth Landon Biography | {
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Саккады (от французского saccade; «рывок», «толчок») — быстрые, строго согласованные движения глаз, происходящие одновременно и в одном направлении. На электроокулограмме имеют вид вертикальных прямых тонких линий.
Специалисты нередко применяют термин «микросаккады» к быстрым движениям глаз, угловая амплитуда которых не превышает 1°.
А быстрые движения глаз амплитудой более 1° называет «макросаккадами». С точки зрения В. А. Филина это деление чисто условное, так как предполагает, что эти два вида быстрых движений глаз имеют разный механизм происхождения.
В настоящее время предполагается, что любые быстрые движения глаз имеют одну природу возникновения и поэтому их целесообразно называть одним словом — «саккада».
Автоматия саккад
Автоматия саккад — это свойство глазодвигательного аппарата совершать быстрые движения глаз непроизвольно в определённом ритме. Саккады могут совершаться в бодрствующем состоянии при наличии зрительных объектов (в этом случае с помощью саккад происходит изменение точки фиксации взора, благодаря чему осуществляется рассматривание зрительного объекта), при отсутствии зрительных объектов, а также во время парадоксальной стадии сна (Филин В. А., 1987). Характер следования саккад обусловлен деятельностью центральной нервной системы, соответствующие структуры которой способны генерировать сигнал по типу автоматии, то есть способны к ритмогенезу. Каждому человеку присущ собственный паттерн следования саккад, который определяется тремя параметрами: интервалом между саккадами, их амплитудой и ориентацией. Наибольшее число саккад следует через 0,2—0,6 секунд, амплитуда саккад изменяется в большом угловом диапазоне от 2′ до 15°, ориентированы саккады практически во всех направлениях (вправо, влево, вверх, вниз), но обычно их число больше в горизонтальной плоскости.
Произвольные саккады
Саккады могут осуществляться и произвольно. Одной из наиболее распространенных методик исследования саккадических движений глаз является «антисаккадическая задача». В условиях этой задачи от испытуемого требуется подавить рефлекторную саккаду в сторону предъявляемого зрительного стимула и совершить саккаду в противоположную сторону.
Программирование саккад
Саккадические движения являются баллистическими — начавшись, саккада будет закончена независимо от того, изменила ли своё положение точка фиксации за время, прошедшее после начала саккады. В связи с этим саккады программируются заранее. Система, принимающая участие в программировании саккад, иерархически организована и включает в себя четыре уровня.
Первый уровень саккадной системы обеспечивает непосредственное выполнение саккад и включает наружные мышцы глаза и ядра III, IV и VI пар черепных нервов (Подвигин и др., 1986).
Второй уровень саккадной системы объединяет стволовые структуры надъядерного контроля движений глаз. К ним относят ядра ретикулярной формации ствола, структуры моста и некоторые ядра покрышки среднего мозга (Подвигин и др., 1986; Шульговский, 1993). Структуры второго уровня управляют целостными координированными движениями обоих глаз.
Третий уровень глазодвигательной системы представлен структурами, контролирующими работу стволового генератора саккад. К этому уровню относят верхнее двухолмие (ВД), базальные ганглии, мозжечок, мозолистое тело, латеральное коленчатое тело, область внутренней капсулы, комплекс подушки и ряд других ядер таламуса (Подвигин и др., 1986).
К четвёртому уровню глазодвигательной системы относят различные зоны коры больших полушарий, среди которых важнейшее место занимают фронтальное глазодвигательное поле и заднетеменные поля (5, 7 по Бродману). Кроме этого, в подготовке саккадических движений глаз принимают участие дополнительное глазодвигательное поле, дорсолатеральная префронтальная кора (поле 46) и др. Данный уровень необходим для осуществления произвольных саккад(Шульговский, 2004).
Саккады в психофизиологических и клинических исследованиях
Саккады играют существенную роль в целенаправленном поведении, зрительном восприятии, исследовании окружающего мира и в полной мере развиты только у приматов (включая человека) (Шульговский, 1993). С ними связано явление саккадического подавления, когда субъект не воспринимает зрительную информацию во время осуществления саккад. Кроме того, нарушения саккадических движений глаз объективно отражают нейродегенеративные процессы при физиологическом старении, психических и двигательных расстройствах. В последнем случае саккады могут опережать проявление других двигательных симптомов и служить одним из специфических маркеров заболевания.
См. также
Нистагм
Литература
Гиппенрейтер Ю. Б. Движения человеческого глаза / Ю. Б. Гиппенрейтер. — М.: Изд-во Моск. ун-та, 1978. — 256 с.
Марр Д. Зрение. Информационный подход к изучению представления и обработки зрительных образов / Д. Марр; Пер. с англ. Н. Г. Гуревич. — М.: «Радио и связь», 1987.- 400 с.
Подвигин Н. Ф., Макаров Ф. Н., Шелепин Ю. Е. Элементы структурно-функциональной организации зрительно-глазодвигательной системы./ Л.: Наука, 1986. — 252 с.
Филин В. А. Закономерности саккадической деятельности глазодвигательного аппарата // Автореф. дис. д-ра биол. наук, М.: 1987 б. 44 с.
Филин В. А. Автоматия саккад. М.: Изд-во МГУ. 2002. 240 с 113 илл.
Филин В. А., Филина Т. Ф. Автоматия саккад у младенцев в быстром сне // Журнал высшей нервной деятельности. М.: Наука, 1989. Т.39. Вып. 4. С.603-608
Шульговский В. В. Физиология целенаправленного поведения млекопитающих. — М.:Изд-во МГУ, 1993. — 224 с.
Шульговский В. В. Психофизиология пространственного зрительного внимания у человека. // Соросовский образовательный журнал. — 2004. — Т. 8. — № 1. — С. 17-23.
Примечания
Глаз
Движения глаз | {
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{"url":"http:\/\/mathhelpforum.com\/algebra\/119008-exponential-equations-questions-print.html","text":"# Exponential equations questions\n\n\u2022 Dec 6th 2009, 07:43 PM\nPaymemoney\nExponential equations questions\nHi\ni have tried to solve these equations but still cannot get the correct answer:\nThe following is what i have done:\n$2^{2x}-20(2^x)=-64$\nmake $y=2^x$\n$2y^2-20y+64=0$\n$2(y^2-10y+32)$\n$2(y^2-10y-(\\frac{10}{2})^2+32-25)$\n$2(y-5)^2+7$\n$2(y-5+\\sqrt{7})(y-5-\\sqrt{7})$\nthen:\n$2(y-5+\\sqrt{7})$\n$2y-10+2\\sqrt{7}=0$\n$2y+2\\sqrt{7}=10$\n$2y+\\sqrt{7}=5$\n$2y=25-7$\n$y=\\frac{18}{2}$\n$2^x=9$\nhas no solution but answer says it is does.\n$2(y-5-\\sqrt{7})$\n$2y-10-2\\sqrt{7}=0$\n$2y-2\\sqrt{7}=10$\n$2y=25+7$\n$y=\\frac{32}{2}$\n$2^x=16$\n$2^x=16$\n$x=4$\n\nP.S\n\u2022 Dec 6th 2009, 07:49 PM\npickslides\nQuote:\n\nOriginally Posted by Paymemoney\n$2^x=9$\nhas no solution but answer says it is does.\n\nGiven the arithmetic is correct\n\n$\\Rightarrow x = \\log_2(9)$\n\u2022 Dec 6th 2009, 07:50 PM\nVonNemo19\nQuote:\n\nOriginally Posted by Paymemoney\nHi\ni have tried to solve these equations but still cannot get the correct answer:\nThe following is what i have done:\n$2^{2x}-20(2^x)=-64$\nmake $y=2^x$\n$2y^2-20y+64=0$\n$2(y^2-10y+32)$\n$2(y^2-10y-(\\frac{10}{2})^2+32-25)$\n$2(y-5)^2+7$\n$2(y-5+\\sqrt{7})(y-5-\\sqrt{7})$\nthen:\n$2(y-5+\\sqrt{7})$\n$2y-10+2\\sqrt{7}=0$\n$2y+2\\sqrt{7}=10$\n$2y+\\sqrt{7}=5$\n$2y=25-7$\n$y=\\frac{18}{2}$\n$2^x=9$\nhas no solution but answer says it is does.\n$2(y-5-\\sqrt{7})$\n$2y-10-2\\sqrt{7}=0$\n$2y-2\\sqrt{7}=10$\n$2y=25+7$\n$y=\\frac{32}{2}$\n$2^x=16$\n$2^x=16$\n$x=4$\n\nP.S\n\n$y^2-20y+64=0$\n\nNot\n\n$2y^2...$\n\u2022 Dec 6th 2009, 07:56 PM\nwerepurple\n\" 2y^2-20y+64=0 \" is incorrect.\n\n2^(2x)-20(2^x)=-64\n\" Let 2^x be y \" is the right direction, but-\n\n2^(2x) = (2^x)(2^x) = 2^(x+x) = 2^(2x) = (y)(y) = y^2 ----( not 2y^2 )\n\nAs for the rest of the steps, I believe you know them already :D\n\u2022 Dec 6th 2009, 07:56 PM\nWilmer\nQuote:\n\nOriginally Posted by Paymemoney\n$2^{2x}-20(2^x)=-64$\n\nYikes! What are you doing? Keep it simple:\n\n2^2(2^x) - 20(2^x) = -64\n4(2^x) - 20(2^x) = -64\n-16(2^x) = -64\n2^x = -64 \/ -16\n2^x = 4\nx = 2\n\u2022 Dec 6th 2009, 08:00 PM\nwerepurple\nlols, Wilmer.\n\n2^2(2^x) = 2^(x+2) is not the same as 2^(2x) = (2^x)(2^x) = 2^(x+x) = 2^(2x)\n\u2022 Dec 6th 2009, 08:01 PM\nPaymemoney\nQuote:\n\nOriginally Posted by werepurple\n\" 2y^2-20y+64=0 \" is incorrect.\n\n2^(2x)-20(2^x)=-64\n\" Let 2^x be y \" is the right direction, but-\n\n2^(2x) = (2^x)(2^x) = 2^(x+x) = 2^(2x) = (y)(y) = y^2 ----( not 2y^2 )\n\nAs for the rest of the steps, I believe you know them already :D\n\noh oops, i think i'll tired it again and i should be able to get it.\n\u2022 Dec 6th 2009, 08:06 PM\nPaymemoney\nlol, i didn't ever need to complete the square and i answer my own question.\n$(y-16)(y-4)$\nwhich will therefore get me x=4 and x=2.\n\noh well, thanks for spotting the mistake\n\u2022 Dec 6th 2009, 08:08 PM\nWilmer\nQuote:\n\nOriginally Posted by werepurple\nlols, Wilmer.\n2^2(2^x) = 2^(x+2) is not the same as 2^(2x) = (2^x)(2^x) = 2^(x+x) = 2^(2x)\n\nright...BUT tell me why I got the correct answer (Smirk)\n\u2022 Dec 6th 2009, 08:10 PM\nwerepurple\nSheer coincidence, Wilmer. Haha, and that is why there are intersections of lines, curves, and etc; values coincide. ^^","date":"2016-10-22 07:58:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 50, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7735211253166199, \"perplexity\": 6109.825227570831}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988718840.18\/warc\/CC-MAIN-20161020183838-00039-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
est une espèce de poissons de la famille des .
Description
Ce poisson mesure jusqu'à de longueur.
Répartition
Cette espèce vit en Amérique du Sud, dans la rivière Río Paraguay et ses affluents.
Annexes
Références taxinomiques
Liens externes
Notes et références
Loricariidae
Faune endémique d'Amérique du Sud | {
"redpajama_set_name": "RedPajamaWikipedia"
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The Museum of Science and Industry (MSI) is a science museum located in Chicago, Illinois, in Jackson Park, in the Hyde Park neighborhood between Lake Michigan and The University of Chicago. It is housed in the former Palace of Fine Arts from the 1893 World's Columbian Exposition. Initially endowed by Julius Rosenwald, the Sears, Roebuck and Company president and philanthropist, it was supported by the Commercial Club of Chicago and opened in 1933 during the Century of Progress Exposition.
Among the museum's exhibits are a full-size replica coal mine, captured during World War II, a model railroad, the command module of Apollo 8, and the first diesel-powered streamlined stainless-steel passenger train (Pioneer Zephyr).
History
The Palace of Fine Arts (also known as the Fine Arts Building) at the 1893 World's Columbian Exposition was designed by Charles B. Atwood for D. H. Burnham & Co. During the fair, the palace displayed paintings, prints, drawing, sculpture, and metal work from around the world.
Unlike the other "White City" buildings, it was constructed with a brick substructure under its plaster facade.
After the World's Fair, the palace initially housed the Columbian Museum, largely displaying collections left from the fair, which evolved into the Field Museum of Natural History. When the Field Museum moved to a new building five miles north in the Near South Side in 1920, the palace was left vacant.
School of the Art Institute of Chicago professor Lorado Taft led a public campaign to restore the building and turn it into another art museum, one devoted to sculpture. The South Park Commissioners (now part of the Chicago Park District) won approval in a referendum to sell $5 million in bonds to pay for restoration costs, hoping to turn the building into a sculpture museum, a technical trade school, and other things. However, after a few years, the building was selected as the site for a new science museum.
At this time, the Commercial Club of Chicago was interested in establishing a science museum in Chicago. Julius Rosenwald, the Sears, Roebuck and Company president and philanthropist, energized his fellow club members by pledging to pay $3 million towards the cost of converting the Palace of Fine Arts (Rosenwald eventually contributed more than $5 million to the project). During its conversion into the MSI, the building's exterior was re-cast in limestone to retain its 1893 Beaux Arts look. The interior was replaced with a new one in Art Moderne style designed by Alfred P. Shaw.
Rosenwald established the museum organization in 1926 but declined to have his name on the building. For the first few years, the museum was often called the Rosenwald Industrial Museum. In 1928, the name of the museum was officially changed to the Museum of Science and Industry. Rosenwald's vision was to create a museum in the style of the Deutsches Museum in Munich, which he had visited in 1911 while in Germany with his family.
Sewell Avery, another businessman, had supported the museum within the Commercial Club and was selected as its first president of the board of directors. The museum conducted a nationwide search for the first director. MSI's Board of Directors selected Waldemar Kaempffert, then the science editor of The New York Times, because he shared Rosenwald's vision.
He assembled the museum's curatorial staff and directed the organization and construction of the exhibits. In order to prepare the museum, Kaempffert and his staff visited the Deutsches Museum in Munich, the Science Museum in Kensington, and the Technical Museum in Vienna, all of which served as models. Kaempffert was instrumental in developing close ties with the science departments of the University of Chicago, which supplied much of the scholarship for the exhibits. Kaempffert resigned in early 1931 amid growing disputes with the second president of the board of directors; they disagreed over the objectivity and neutrality of the exhibits and Kaempffert's management of the staff.
The new Museum of Science and Industry opened to the public in three stages between 1933 and 1940. The first opening ceremony took place during the Century of Progress Exposition. Two of the museum's presidents, a number of curators and other staff members, and exhibits came to MSI from the Century of Progress event.
For years, visitors entered the museum through its original main entrance, but that entrance became no longer large enough to handle an increasing volume of visitors. The newer main entrance is a structure detached from the main museum building, through which visitors descend into an underground area and re-ascend into the main building, similar to the Louvre Pyramid.
In 1992, due to increased attendance, the museum started planning its underground parking lot, located in three underground levels below the front lawn. Construction of the underground parking lot was finished in July 1998.
For over 55 years, admission to the MSI was free, although some exhibits such as the Coal Mine and U-505 required small fees. General entrance fees were first charged in the early 1990s, with general admission rates increasing from $13 in 2008 to $18 in 2015. Many "free days"—for Illinois residents only—are offered throughout the year.
On October 3, 2019, the museum announced that it intends to change its name to the Kenneth C. Griffin Museum of Science and Industry after a donation of $125 million from the now former Chicago billionaire Kenneth C. Griffin. It is the largest single gift in the museum's history, effectively doubling its endowment. However, president and chief executive officer David Mosena said the formal name change could take some time, due to the complexity of the process. He also said part of the gift will go into funding "a state-of-the-art digital gallery and performance space that will be the only experience of its kind in North America." Chevy Humphrey became president and CEO of the private, non-profit museum in January 2021.
Exhibits
The museum has over 2,000 exhibits, displayed in 75 major halls. The museum has several major permanent exhibits. Access to several of the exhibits (including the Coal Mine and U-505) requires the payment of an additional fee.
Entry Hall
The first diesel-powered, streamlined stainless-steel train, the Pioneer Zephyr, is on permanent display in the Great Hall, renamed the Entry Hall in 2008. The train was once displayed outdoors, but it was restored and placed in the former Great Hall during the construction of the museum's underground parking lot.
Lower Level
U-505
is one of just six German submarines captured by the Allies during World War II, and, since its arrival in 1954, the only one on display in the Western Hemisphere, as well as the only one in the United States. The U-boat was newly restored beginning in 2004 after 50 years of being displayed outdoors, and was then moved indoors as "The New U-505 Experience" on June 5, 2005. Displayed in an underground shed, it remains as a popular exhibit for visitors, as well as a memorial to all the casualties of the Battle of the Atlantic during World War II. Guided tours of the submarine are offered for an additional fee. Near the U-505 there is both a Mold-A-Rama machine and a penny flattening device. Both have U-505 designs.
Henry Crown Space Center
MSI's Henry Crown Space Center includes the Apollo 8 spacecraft, which flew the first mission beyond low earth orbit to the Moon, enabling its crew, Frank Borman, James Lovell and William Anders, to become the first human beings to see the Earth as a whole, as well as becoming the first to view the Moon up close (as well as the first to view its far side). Other exhibits include Scott Carpenter's Mercury-Atlas 7 spacecraft and a lunar module trainer.
Located in the Henry Crown Space Center is a domed theater, considered to be the only domed theater in Chicago. The screen of the theater is made of perforated aluminum, allowing the speakers mounted behind the screen to be heard throughout the theater.
FarmTech
The "FarmTech" exhibit showcases modern agricultural techniques and how farmers use modern technology like GPS systems to improve work on the farm, and includes a tractor and a combine harvester from John Deere. The exhibit also showcases a greenhouse, a mock up of a kitchen showcasing how much of the food we eat comes from soybeans, and how we use cows, from energy to what we drink.
Other
A transportation gallery is located on the museum's west wing, containing models of "Ships Through the Ages" and several historic racing cars.
"Future Energy Chicago" shows alternative resources, housing developments, and the future of Chicago. The exhibit requires an additional fee.
Some areas in the museum aim for younger children, including the "Swiss Jollyball", the world's largest pinball machine built by a British man from Switzerland using nothing but salvaged junk; the "Idea Factory", a toddler water table play area; and the "Circus", featuring animated dioramas of a miniature circus as well as containing a shadow garden and several funhouse mirrors. The Circus exhibit was closed in September 2022.
Silent-film star and stock-market investor Colleen Moore's Fairy Castle "doll's house" is on display.
First Level
Numbers in Nature: A Mirror Maze
Numbers in Nature: A Mirror Maze contains an interactive theater and stations to learn about patterns in nature, including the Golden Ratio, spirals, fractal branching, and Voronoi patterns.
It also contains a mirror maze to help emphasize the geometric patterns that can be utilized.
Transportation Gallery
The Transportation Zone contains several permanent exhibits.
The Great Train Story is a HO-scale model railroad and recounts the story of transportation from Chicago to Seattle.
The museum includes a replica of Stephenson's Rocket, which was the first steam locomotive to exceed 25 miles per hour.
The 999 Empire State Express steam locomotive was alleged to be the first vehicle to exceed in 1893, although no reliable measurement ever took place (and such a speed was likely impossible). Designed to win the battle of express trains to the World's Columbian Exhibition, it was donated to the museum by the New York Central in 1962. The locomotive was located outside the museum until 1993, when extensive restoration took place and it was moved indoors as an exhibit in the Transportation Zone.
A replica of the Wright Brothers first airplane, the Wright Flyer, is on display.
Two World War II warplanes are also exhibited. Both were donated by the British government: a German Ju 87 R-2/Trop. Stuka divebomber—one of only two intact Stukas left in the world—and a British Supermarine Spitfire. Also on display is the museum's Travel Air Type R Mystery Ship, nicknamed "Texaco 13", which set many world records in flying.
"Take Flight" features the first Boeing 727 jet plane in commercial service, donated by United Airlines, with one wing removed and holes cut on the fuselage to facilitate visitor access.
Science Storms
In March 2010, the museum opened "Science Storms" in the Allstate Court, as a permanent exhibit. This multilevel exhibit features a water vapor tornado, tsunami tank, Tesla coil, heliostat system, and a Wimshurst machine built by James Wimshurst in the late 19th century. Also housed are Newton's Cradle, the color spectrum, and Foucault pendulum. All artifacts allow guests to explore the physics and chemistry of the natural world.
Genetics: Decoding Life
In keeping with Rosenwald's vision, many of the exhibits are interactive. "Genetics: Decoding Life," looks at how genetics affect human and animal development as well as containing a chick hatchery composed of an incubator where baby chickens hatch from their eggs and a chick pen for those that have already hatched, as well as housing genetically modified frogs, mice, and drought resistant plants.
The chick hatchery has been part of the museum since 1956. About 20 chicks are hatched a day, around 140 hatch in a week, and up to 8000 hatch in a year. A week after emerging from their shells, the chicks are sent to the Lincoln Park Zoo to be fed to various animals, including lions, crocodiles, snakes, vultures, owls and tigers. This partnership between the museum and the zoo has been operating for decades, with about 7000 chicks being sent to the zoo each year. Some of the chicks hatched are of the Java species of chicken, and these chicks are sent to a farm in La Fox, Illinois that works to preserve the rare breed. There have been numerous efforts to shut down the exhibit, as early as 1998 and as recent as 2017.
ToyMaker 3000
"ToyMaker 3000", is a working assembly line which lets visitors order a toy top and watch as it is made. The interactive "Fab Lab MSI" is intended as an interactive lab where members can "build anything".
Coal Mine
The "Coal Mine" re-creates a working deep-shaft, bituminous coal mine inside the museum's Central Pavilion, using original equipment from Old Ben #17, circa 1933. It is one of the oldest exhibits at the museum. In this unique exhibit, visitors go underground and ride a mine train to different parts of the mine and learn the basics of its operation. The experience takes around 30 minutes and requires an additional fee.
Yesterday's Main Street
"Yesterday's Main Street" is a mock-up of a Chicago street from the early 20th century, complete with a cobblestone roadway, old-fashioned light fixtures, fire hydrants, and several shops, including the precursors to several Chicago-based businesses. Included are:
Unlike the other shops, Finnigan's Ice Cream Parlor and The Nickelodeon Cinema can be entered and are functional. Finnigan's serves an assortment of ice cream and The Cinema plays short silent films throughout the day.
Other
In spring 2013, the "Art of the Bicycle" exhibit opened, showcasing the history of the bicycle and how modern bikes continue to evolve.
"Reusable City" focuses on recycling and other methods that could cut down harmful pollution and especially climate change and the Regenstein Hall of Science, containing a giant periodic table of the elements. Other main level exhibits include: "Fast Forward", which features some aspects of how technology will change in the future; "Earth Revealed", featuring a "Science on a Sphere" holographic globe; and a "Whispering Gallery".
Second Level
YOU! The Experience
The museum is also known for unique and quirky permanent exhibits, such as a walk-through model of the human heart, which was removed in 2009 for the construction of "YOU! the Experience", which replaced it with a , interactive, 3D heart. Also well known are the "Body Slices" (two cadavers exhibited in slices) in the exhibit.
Other
Several US Navy warship models are also on display in the museum, and flight simulators including of the new F-35 Lightning II are featured.
Former exhibits
An F-104 Starfighter on loan to MSI from the US Air Force since 1978 was sent to the Mid-America Air Museum in Liberal, Kansas, in 1993.
In March 1995, Santa Fe Steam Locomotive 2903 was moved from outside the museum to the Illinois Railway Museum.
"Telefun Town", a hall dedicated to the wonders of telephone communication, sponsored by the company then known as "The Bell Telephone Company", no longer exists.
Exhibitions
In addition to its three floors of standing exhibits, the museum hosts temporary and traveling exhibitions. Exhibitions last for five months or less and usually require a separate paid admission fee. Exhibitions at MSI have included Titanic: The Exhibition, which was the largest display of relics from the wreck of RMS Titanic; Gunther von Hagens' Body Worlds, a view into the human body through use of plastinated human specimens; Game On, which featured the history and culture of video games; Leonardo da Vinci: Man, Inventor, Genius; CSI: The Experience; Robots Like Us; City of the Future; Star Wars: Where Science Meets Imagination; The Glass Experience; Harry Potter: The Exhibition; Robot Revolution, which was sponsored by Google and featured numerous hands-on demonstrations and advice from experts for prospective future robot scientists and engineers; and four installments of Smart Home: Green + Wired, featuring the work of green architect Michelle Kaufmann. The Science Behind Pixar exhibit opened May 24, 2018. The Wired to Wear exhibit opened on March 21, 2019. The touring exhibit Marvel: Universe of Super Heroes officially opened to the public on March 7, 2021.
Yearly, from late November to early January, the museum hosts its Christmas Around the World and Holidays of Light exhibit, featuring Christmas trees from different cultures from around the world. Started in 1942 with just one tree to honor soldiers fighting in World War Two, the tradition spawned into more than 50 trees.
See also
Architecture of Chicago
List of museums and cultural institutions in Chicago
References
Explanatory notes
Citations
Further reading
Kogan, Herman. A Continuing Marvel: The Story of the Museum of Science and Industry. 1st ed. Garden City, N.Y., Doubleday, 1973.
Pridmore, Jay. Inventive Genius: The History of the Museum of Science and Industry, Chicago. Chicago, Museum of Science and Industry, 1996.
Pridmore, Jay. Museum of Science and Industry, Chicago. New York, Harry N. Abrams, 1997.
External links
Museum website
Commercials and news clips at The Museum of Classic Chicago Television
High-resolution 360° Panoramas and Images at Columbia University
Museums in Chicago
Chicago Landmarks
Cultural infrastructure completed in 1893
Hyde Park, Chicago
Industry museums in Illinois
Technology museums in Illinois
World's Columbian Exposition
World's fair architecture in Chicago
Museums established in 1933
1933 establishments in Illinois
Association of Science-Technology Centers member institutions
Science museums in Illinois | {
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The following information is provided as background on how to request public records from the City of San Rafael. The California State Legislature adopted the Public Records Act in 1968. It is designed to give the public access to information in the possession of public agencies. The Act also provides that public records shall be open for inspection during regular office hours of the agency. The public can inspect or receive a copy of any record unless the record is exempted from disclosure under the act. Normally, a public agency has 10 days to respond to a public records request, but if the records are readily accessible they may be available sooner for inspection during regular office hours.
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Intersectionality 101
The aim of this primer is to provide a clear-language guide to intersectionality; exploring its key elements and characteristics, how it is distinct from other approaches to equity, and how it can be applied in research, policy, practice and teaching. | {
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Q: xfs directory quota doesn't work I have two tasks and I want to run them in different directory and limit the directory's size. I use xfs's project quota to achieve this.
I follow the steps from here:https://solidlinux.wordpress.com/2012/12/09/xfs-quota-managament/
here is detail of my two config files
/etc/projects:
11:/home/xiameng.xm/xfs_dir/task1
12:/home/xiameng.xm/xfs_dir/task2
/etc/projid:
task1:11
task2:12
and here is my step:
touch test_xfs
dd if=/dev/zero of=test_xfs bs=100M count=1
mkfs.xfs test_xfs
mkdir xfs_dir
mount test_xfs ./xfs_dir -o pquota,loop
xfs_quota -xc 'project –s task1' /home/xiameng.xm/xfs_dir
xfs_quota -xc 'project –s task2' /home/xiameng.xm/xfs_dir
xfs_quota -x -c 'limit -p bhard=2m task1' /home/xiameng.xm/xfs_dir
xfs_quota -x -c 'limit -p bhard=10m task2' /home/xiameng.xm/xfs_dir
here is the output of xfs_quota -x -c 'report /home/xiameng.xm/xfs_dir'
Project quota on /home/xiameng.xm/xfs_dir (/dev/loop1)
Blocks
Project ID Used Soft Hard Warn/Grace
---------- --------------------------------------------------
task1 0 0 2048 00 [--------]
task2 0 0 10240 00 [--------]
the question is no mater how large I create a file in ./xfs_dir/task1 or ./xfs_dir/task2, it succeeds!
The quota limit doesn't work!
A: We ran into the identical issue. The problem was that we failed to initialize the mount point with the project using:
xfs_quota -x -c 'project -s yourProjectName' yourMountPoint
As soon as we did this, the xfs_quota report successfully reported the Used space under the project path.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,696 |
Aaron Jameer Dobson (born July 23, 1991) is a former American football wide receiver. He was drafted by the New England Patriots in the second round of the 2013 NFL Draft and played college football at Marshall.
Early years
Dobson was born in Dunbar, West Virginia. He attended South Charleston High School in South Charleston, West Virginia, and played high school football and high school basketball for the South Charleston Black Eagles. He recorded 45 receptions for 1,298 yards and 17 touchdowns as a senior and added seven interceptions on defense, and finished his high school career with 108 receptions, 2,365 yards and 32 touchdowns. He had 10 interceptions, two of which he returned for touchdowns, and was a member of the 2008 MSAC Championship team and 2008 West Virginia AAA State Championship, which finished with a 14-0 record.
In 2009, he played for USA Football's U.S. Under-19 National Team that won the 2009 IFAF Under-19 World Championship in Canton, Ohio, where he was teammates with future New York Giants running back, David Wilson.
College career
Dobson attended Marshall University, where he played for the Marshall Thundering Herd football team from 2009 to 2012. During his college career, Dobson had 165 receptions for 2,398 yards and 24 touchdowns. As a junior in 2011, he was the MVP of the 2011 Beef 'O' Brady's Bowl. He ended his Marshall senior season being named 2nd team All-Conference USA and being invited to play in the Senior Bowl.
Dobson gained recognition in 2011 during a game against East Carolina, when he had a one-handed backhand catch for a touchdown in the second quarter. The play went viral and was ranked second on ESPN's Top 10 Plays of the Year.
Professional career
New England Patriots
The New England Patriots selected Dobson in the second round, with the 59th overall pick, of the 2013 NFL Draft. He signed a four-year, $3.4 million contract. His first career catch was for a touchdown against the New York Jets in Week 2. In a Week 9 win against the Pittsburgh Steelers, he had the first 100-yard game of his career. He caught five passes for 130 yards and two touchdowns. He suffered a foot injury in week 12 against the Broncos and missed weeks 13-15. He appeared in 12 games (nine starts) with 519 receiving yards and four touchdowns during his rookie campaign in 2013.
Dobson was inactive for eight of the first twelve weeks of the 2014 season before injuring his hamstring against the Green Bay Packers in Week 13. On December 4, 2014, he was placed on injured reserve. With Dobson on IR, the Patriots won Super Bowl XLIX after they defeated the defending champion Seattle Seahawks, 28-24.
Dobson was active for Week 1 against the Steelers on September 10, 2015. He played sparingly recording one reception for nine yards. In the second week, against the Buffalo Bills, he tied a career-high with seven catches, for 87 yards. He recorded a 17-yard pass from quarterback Tom Brady in a 20-13 win over the Bills in Week 11; on the play, he injured his ankle and had to leave the game. He was diagnosed with a high ankle sprain, and on November 26, 2015, the Patriots placed him on injured reserve, ending his season.
On September 3, 2016, Dobson was released by the Patriots as part of final roster cuts.
Detroit Lions
On September 21, 2016, Dobson was signed by the Detroit Lions. He was released on September 24, 2016. On September 27, 2016, he was re-signed by the Lions. He was released again on October 8, 2016.
Arizona Cardinals
On January 5, 2017, Dobson signed a reserve/future contract with the Arizona Cardinals. He was placed on injured reserve on September 2, 2017. He was released on September 6, 2017.
Career statistics
Regular season
Postseason
References
External links
Marshall Thundering Herd bio
New England Patriots bio
1991 births
Living people
American football wide receivers
Arizona Cardinals players
Detroit Lions players
Marshall Thundering Herd football players
New England Patriots players
People from Dunbar, West Virginia
Players of American football from West Virginia
South Charleston High School alumni
People from South Charleston, West Virginia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,532 |
package org.apache.sysds.runtime.matrix.operators;
import org.apache.sysds.runtime.DMLRuntimeException;
import org.apache.sysds.runtime.functionobjects.Builtin;
import org.apache.sysds.runtime.functionobjects.ValueFunction;
public class UnaryOperator extends Operator
{
private static final long serialVersionUID = 2441990876648978637L;
public final ValueFunction fn;
private final int k; //num threads
private final boolean inplace;
public UnaryOperator(ValueFunction p) {
this(p, 1, false); //default single-threaded
}
public UnaryOperator(ValueFunction p, int numThreads, boolean inPlace) {
super(p instanceof Builtin &&
(((Builtin)p).bFunc==Builtin.BuiltinCode.SIN || ((Builtin)p).bFunc==Builtin.BuiltinCode.TAN
// sinh and tanh are zero only at zero, else they are nnz
|| ((Builtin)p).bFunc==Builtin.BuiltinCode.SINH || ((Builtin)p).bFunc==Builtin.BuiltinCode.TANH
|| ((Builtin)p).bFunc==Builtin.BuiltinCode.ROUND || ((Builtin)p).bFunc==Builtin.BuiltinCode.ABS
|| ((Builtin)p).bFunc==Builtin.BuiltinCode.SQRT || ((Builtin)p).bFunc==Builtin.BuiltinCode.SPROP
|| ((Builtin)p).bFunc==Builtin.BuiltinCode.LOG_NZ || ((Builtin)p).bFunc==Builtin.BuiltinCode.SIGN) );
fn = p;
k = numThreads;
inplace = inPlace;
}
public int getNumThreads() {
return k;
}
public boolean isInplace() {
return inplace;
}
public double getPattern() {
switch( ((Builtin)fn).bFunc ) {
case ISNAN:
case ISNA: return Double.NaN;
case ISINF: return Double.POSITIVE_INFINITY;
default:
throw new DMLRuntimeException(
"No pattern existing for "+((Builtin)fn).bFunc.name());
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 96 |
{"url":"http:\/\/mattytraxx.com\/wiki\/index.php\/Hashing","text":"# Hashing\n\nGiven a key k, a good hash function h(k) should map to a unique index in table T. The hash function should exhibit uniform distribution regardless of the domain of k, therefore, the determination of where the next value will go in T should be as random as possible. If h(k) does not posess this property, the likelihood that two unique keys k1 and k2 will hash to the same index is severely increased and a collision might occur. For example, if we try to add k2 (with the same hash as k1) to T but k1 has already been added, a decision needs to be made as to where the offending key should be placed. The typical scheme to is to probe around in a methodical way to locate a vacancy in T. This probing must be methodical because it needs to be repeatable for the hashing function to re-locate this item. For this reason, once an element has been added to a hash table, it cannot be deleted unless the entire table is rehashed.\n\n## Double hashing\n\nThe double hashing function given by Cormen et al produces a somewhat pseudo-random search for vacant spots in T should a collision occur:\n\n$h(k,i)=\\left(h_1(k)+ih_2(k)\\right)\\text{mod}\\;m$\n\nwhere\n\n\\begin{align} h_1(k)&=k\\;\\text{mod}\\;m \\\\ h_2(k)&=1 + \\left(k\\;\\text{mod}\\;m^\\prime \\right) \\end{align}\n\nand ${m}\\,\\!$ is prime.\n\nThe value ${m^\\prime}\\,\\!$ should be set to ${m-1}\\,\\!$, which will produce ${m}\\left(m-1\\right)\\,\\!$ unique probe sequences. This means that any given pair $\\left(h_1(k), h_2(k)\\right)$ will only repeat on the order of $\\Theta(m^2)\\,\\!$. Additonally, each probe sequence is a unique permutation of $\\{0, 1, 2, ..., m-1\\}\\,\\!$. So while this guarantees that every possible vacancy can be located, it also minimizes primary clustering typical of linear probing methods. Because double hashing uses two hash functions, it also does not suffer from secondary clustering to the extent that quadratic probing does.\n\nOn the initial hash of k, the value h1(k) is computed. If there is a collision, then the next probe location is calculated by adding h2(k) to the previous position, and taking the (modulo m) of this sum. The process of calculating (previous_pos + h2(k))(mod m) is repeated until a vacant spot is located. If the calculation is performed m-1 times, then the probe sequence given by this function guarantees that the table is full.\n\nThe load factor is given by $\\alpha=\\frac{n}m\\,\\!$, where n is the number of elements currently in T. If this is kept under 0.5, then the average maximum number of probes required will be 2, assuming the hash function is uniform. In this case, the running time for typical hash operations is still O(n).\n\n## References\n\n\u2022 T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein. Introduction to Algorithms, 2nd ed. Cambridge, Mass: MIT Press, 2001. Print.\n\u2022 M.A. Weiss. Data Structures and Algorithm Analysis in Java, 2nd ed. Boston: Peason Addison-Wesley, 2007. Print.","date":"2017-12-13 22:43:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 10, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6938527226448059, \"perplexity\": 944.6174090050188}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948531226.26\/warc\/CC-MAIN-20171213221219-20171214001219-00431.warc.gz\"}"} | null | null |
Pet Safe Classic are for the original Pet Safe pet door series.
The Pet Safe Pet Door Classic Replacement Flap is a tinted vinyl single flap with a secure magnetic closure. The top of the flap is mounted with vinyl studs. The single replacement flap is compatible with the Classic Pet Door, Deluxe Patio Panel Door, and the Wall Entry Aluminum Pet Door. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,738 |
Q: Master-Detail Relationship - Process Builder I have a Custom Object = A, another Custom Object = B in a master detail relationship and the Account Object.
I want to be able to make checkbox = true on the Account, depending on some criteria on Custom Object = B. To mention: the checkbox is unchecked by default and based on Edit - Save, it gets automatically checked.
*
*If custom field 1 = "Inbound" and custom field 2 = not blank, to make the checkbox = true on Account
***custom field 1** = text formula looking back at picklist field on Custom Object = A
***custom field 2** = text field
***there is a lookup field from custom object B to the Account that gets populated through a before trigger.**
What I have tried to do initially:
*
*Create a Process on Custom Object B - to start only when a record is created.
*check if custom field 2 = Global Constant = .Null AND custom field = formula "inbound", but it didn't work.
So I have also tried:
*
*Create a Process on Custom Object A - to start only when a record is created.
*Check custom field 3 = picklist = inbound
*I have created an invocable process on Custom Object B to call as an action in the process on Custom Object A, but the field reference is not helpful - created by, last modified by, owner id, etc.
Any help will be much appreciated. Thank you.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,422 |
\section{Introduction}
The investigation on jets in the relativistic heavy ion collisions (RHIC) provides us insights to understand the properties of quark-gluon plasma
(QGP)
\cite{STAR,PHENIX,BRAHMS,PHOBOS}. One of the important RHIC
discoveries is the jet quenching phenomenon coming from the partonic
energy loss in QGP. In the conventional approach to the jet
quenching the perturbative (pQCD) type of energy loss
is taken into account by the channels, elastic and radiative, of one-gluon exchange between
the jet and the massless gluons and quarks (see recent discussion in \cite{radiative}). However, the large quark-gluon rapidity
density $dN_{qg}/dy\approx 2000$ which is needed to describe the RHIC jet quenching
data within this approach, seems to be in contradiction with the
restriction $dN_{qg}/dy\leq 1/4dS/dy\approx 1300$ coming from the
measured final entropy density $dS/dy\approx 5000$ \cite{muller}.
Furthermore, the lattice calculations show that even at very
high temperature
gluons and quarks still interact strongly in QGP
\cite{lattice1}.
Recently, it was suggested that the glueballs, the bound states of
gluons, can exist above deconfinement temperature and may play
an important role in the dynamic of strongly interacting QGP
\cite{vento,kochmin}. In particular, in \cite{kochmin}
it is suggested that a very light pseudoscalar glueball can
exist in QGP and might be responsible for the residual strong interaction
between gluons. The lattice results showing a change of sign of the gluon
condensate \cite{lattice1} and a small value of the topological
susceptibility \cite{lattice2} above $T_c$ can be explained in
the glueball picture as well. Furthermore, one expects that the
suppression of the mixing between glueballs and quarkonium states in the QGP leads
to a smaller width for former as compared to the vacuum \cite{vento}. This property opens the
possibility for a clear
separation of the glueball and the quark states in heavy ion collisions. Such separation
is rather difficult in other hadron reactions due to existence of strong
glueball-quarkonium mixing in the vacuum.
In this communication we report our study on the contribution of glueballs to the
energy loss by high energy partons propagated in QGP. We
will argue that significant partonic energy loss can result
from strong glueball-gluon coupling.
Our starting point is the effective pseudoscalar glueball-gluon
vertex employed in Refs.~\cite{kochmin, soni}
\begin{eqnarray}
{\cal L}_{Ggg}=\frac{1}{f_S}(\alpha_sG^a_{\mu\nu}{G}^a_{\mu\nu}S+
\xi\alpha_sG^a_{\mu\nu}\widetilde{G}^a_{\mu\nu}P), \label{lag}
\end{eqnarray}
where $G^a_{\mu\nu}$ is gluon field strength,
$\widetilde{G}^a_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}{G}^a_{\alpha\beta}/2$,
$ S $ and $P$ are scalar and pseudoscalar glueball fields,
respectively, $\xi\approx 1$, and
$f_S\approx 0.35$ GeV which is fixed by low energy theorem
\cite{kochmin}.
In Fig.1 the diagrams contributing to the energy loss in
high energy limit $s>> (-t,M^2_{S,P})$ are illustrated.
The elastic energy loss is given by Bjorken's formula
\cite{bjorken}
\begin{equation}
\frac{dE}{dx}(T)=\int d^3kn(k,T)[Flux factor]\int
dt\frac{d\sigma}{dt}\nu, \label{bj}
\end{equation}
where $\nu=E^\prime-E$ energy difference between fast incoming
and outcoming partons, $ [Flux factor]=(1-cos\theta)$, $\theta $
is the laboratory angle between the incident partons,
$d\sigma/dt$ is partonic cross section and $n(k,T)$ is density of
target parton in QGP at the temperature $T$.
\begin{figure}[h]
\centerline{\epsfig{file=qgp_neu.eps,width=6cm,height=10cm,
angle=90}}\ \caption{ The diagrams contributed to a),b),d) gluon and
c),d) quark energy losses. The g (G) denotes gluon (glueball) and
q the quark.}
\end{figure}
\begin{figure}
\begin{minipage}[c]{8cm}
\centerline{\epsfig{file=lossgluonNFLc.eps,width=8cm, angle=0}}
\caption{
The temperature dependence of gluon energy loss. The solid
(dashed) line is glueball (pQCD) contribution.}
\end{minipage}
\hspace*{0.5cm}
\begin{minipage}[c]{8cm}
\centerline{\epsfig{file=lossquarkNFc.eps,width=8cm, angle=0}}\
\vspace*{-0.5cm} \caption{ The temperature dependence of quark
energy loss. The notations are same as in Fig.2}
\end{minipage}
\end{figure}
The result of the
calculation of diagrams in Fig.1 is
\begin{eqnarray}
\frac{d\sigma_a}{dt}\approx\frac{15\alpha_s^3}{f_S^2|t|}F(t), \ \
\frac{d\sigma_b}{dt}\approx\frac{15\alpha_s^3}{4f_S^2|t|}F(t),\nonumber\\
\frac{d\sigma_c}{dt}\approx\frac{16\alpha_s^3}{3f_S^2|t|}F(t), \ \
\frac{d\sigma_d}{dt}\approx\frac{\alpha_s^3}{3f_S^2|t|}F(t),
\label{cross}
\end{eqnarray}
where $|t|=2k\nu(1-cos\theta)$, and the form factor in gluon-glueball
vertex reads \cite{kochmin}
\begin{equation}
F(t)=e^{-\Lambda^2|t|}, \label{cut}
\end{equation}
with $\Lambda\approx 0.6$ GeV$^{-1}$.
In the high energy limit we will neglect the small
effect coming from finite masses of the produced glueballs
but we will take into account the finite value of
their masses in
the densities, Eq.\ref{dens}. Furthermore,
our consideration here is restricted
by calculation to the leading order in $\alpha_s$. So the possible thermal gluon
mass effects, $m_g\propto \alpha_sT$, are not considered. Therefore,
in the case of energetic parton it is enough to keep only
leading energy independent terms of the partonic cross sections shown in
Eq.\ref{cross}.
The final result for energy loss for gluon and quark jets reads
\begin{eqnarray}
\frac{dE_{g}}{dx}(T)&=& \frac{15\alpha_s^3}{2f_S^2\Lambda^2}\int
\frac{d^3k}{k}[n_S(k,T)+n_P(k,T)+\frac{n_g(k,T)}{4}+\frac{n_q(k,T)}{45}]\nonumber\\
\frac{dE_{q}}{dx}(T)&=&\frac{8\alpha_s^3}{3f_S^2\Lambda^2}\int
\frac{d^3k}{k}[n_S(k,T)+n_P(k,T)+\frac{n_g(k,T)}{8}],
\label{loss2}
\end{eqnarray}
For estimation we will assume in QGP gluons, quarks and glueballs are
in thermodynamical equilibrium and will use the gas
approximation for gluon and glueball densities
\begin{equation}
n_{i}(k,T)=\frac{N_i}{(2\pi)^3(exp{(\sqrt{k^2+M^2_i}/T)}\pm
1)},
\label{dens}
\end{equation}
where the plus (minus) sign is for fermions (bosons) and numbers of
degrees of freedom are $N_S=1$ for scalar and $N_P=1$ for pseudoscalar glueballs,
respectively,
$N_g=16$ for gluons and $N_q=12 $ for
number of light quark flavors $N_F=2$.
In our previous paper \cite{kochmin} it was argued that in the model with Lagrangian Eq.~\ref{lag}
the behaviour of the masses of pseudoscalar and scalar glueballs above $T_c$ is very different.
Indeed, it was shown that the scalar glueball remain to be massive, $M_S\approx 1.5$ GeV,
but pseudoscalar
glueball is very light, $M_P\approx 0$, above deconfinement temperature.
Within such approximation we obtain
\begin{equation}
\frac{dE_{g,q}}{dx}(T)\approx
C^G_{g,q}\frac{\alpha_s^3T^2}{f_S^2\Lambda^2},
\label{loss3}
\end{equation}
where $C^G_g=79/24 $ is the coefficient for the glueball contribution to gluon energy loss and
$C^G_q=2/3$ is the correspondent coefficient for quark energy loss. \footnote{ We neglect the
small contribution arising from the interaction of energetic
parton with massive scalar glueball in
QGP due to small value of its density $n_S<<n_P$.}
The numerical result presented in Figs.(2,3) of the temperature dependence of elastic
energy loss due to interaction with glueballs is to be compared with the recent re-analysis of perturbative QCD
elastic contribution in the range of temperatures $T_c<T<2T_c$, which is
accessible at RHIC experiments.
The pQCD elastic contribution is as following \cite{pQCD}
\begin{equation}
\frac{dE^{pQCD}_{g,q}}{dx}(T)=
C^{p}_{g,q}\frac{8\pi^2\alpha_sT^2}{b_0}(1+\frac{N_F}{6}),
\label{lossp}
\end{equation}
where $C^p_g=3/2, C^p_q=2/3$ are coefficients for the perturbative gluon and quark energy loss,
respectively,
and $b_0=11/3N_c-2/3N_F$. We take
$T_c=170$ MeV for
$N_F=2$ \cite{lattice} and $\alpha_s\approx 0.6$ at
$T_c<T<2T_c$ \cite{latticealpha,alpha2} for the estimation of
energy loss in gluon-glueball plasma. It follows from Figs. 2,3
that glueball-induced energy loss is large for both gluon and
quark jets. In particular, for the gluon jet such contribution is about of few GeV/fm and
approximately twice larger than the perturbative elastic loss \cite{pQCD}.
In spite of the fact that
for the quark jet the glueball contribution is smaller than perturbative elastic loss, it can
not be neglected in comparison with latter one. It is
evident that the origin of such large contribution is in
strong glueball-gluon coupling in Eq.\ref{lag}.
We should point
out that more than one half of contribution to the gluon energy
loss comes from interaction of gluon with the light
pseudoscalar glueball in QGP. Therefore, existence of such light
bound state of gluons above $T_c$ is crucial for the
understanding of the large observed partonic energy loss in QGP.
In summary, we made the estimation of the energy loss induced by
interaction of an energetic parton, which was produced in the hard scattering of two heavy ion's
partons, with glueballs in hot quark-gluon
plasma. It is shown that such contribution leads to a
significant energy loss. We conclude that not only pQCD type of
energy loss but also glueball-induced loss, arising from existence
of scalar and pseudoscalar glueballs in QGP, are important for the
understanding of the RHIC results such as the jet quenching.
We should emphasize
that the main goal of our paper is to show the significance of the
glueball-induced energy loss and we left the detailed comparison with experiment,
which should
include also the consideration of the effects of both elastic and radiative pQCD losses,
for a forthcoming
publication.\\
{\bf Acknowledgments}\\
We would like to thank V.Vento for useful discussions.
This work was supported by Brain Pool program of Korea Research
Foundation through KOFST, grant 042T-1-1. NK is very grateful to
the School of Physics and Astronomy of Seoul National University
for their warm hospitality during this work.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,145 |
{"url":"https:\/\/www.physicsforums.com\/threads\/sagnac-effect-interferometer.192258\/","text":"Sagnac Effect\/Interferometer\n\n1. Oct 18, 2007\n\nn0_3sc\n\nI'm having trouble understanding the Sagnac Effect.\nSo far all I know is that a ring cavity is formed where 2 beams travel in opposite directions. My question is:\nWhy do the 2 beams form an interference pattern if the entire cavity is rotating at some angular velocity?\n\n2. Oct 19, 2007\n\nGokul43201\n\nStaff Emeritus\nThis is easier to see when you have an external light source, in which case, even thinking non-relativistically, you can see that the beams traveling in opposite directions travel through different path lengths before meeting at the detector. (This is similar to what Michelson and Morley were hoping to see).\n\nFor the case where the source is also part of the rotating setup, this requires a more careful explanation. It might be better to just refer you to a good source. I can't think of one at the moment, but I will, in a bit.\n\n3. Oct 20, 2007\n\nn0_3sc\n\nif you had an external light source, is this rotating as well? if not wouldn't that mean that all the mirrors in the cavity would have to be beam splitters?\nThanks for the help.","date":"2016-10-23 22:07:48","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8156405091285706, \"perplexity\": 422.7512331904742}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988719437.30\/warc\/CC-MAIN-20161020183839-00323-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
import sys
import os
import mmap
import ctypes
# Alias long to int on Python 3
if sys.version_info[0] >= 3:
long = int
class MMIOError(IOError):
"""Base class for MMIO errors."""
pass
class MMIO(object):
def __init__(self, physaddr, size):
"""Instantiate an MMIO object and map the region of physical memory
specified by the address base `physaddr` and size `size` in bytes.
Args:
physaddr (int, long): base physical address of memory region.
size (int, long): size of memory region.
Returns:
MMIO: MMIO object.
Raises:
MMIOError: if an I/O or OS error occurs.
TypeError: if `physaddr` or `size` types are invalid.
"""
self.mapping = None
self._open(physaddr, size)
def __del__(self):
self.close()
def __enter__(self):
pass
def __exit__(self, t, value, traceback):
self.close()
def _open(self, physaddr, size):
if not isinstance(physaddr, int) and not isinstance(physaddr, long):
raise TypeError("Invalid physaddr type, should be integer.")
if not isinstance(size, int) and not isinstance(size, long):
raise TypeError("Invalid size type, should be integer.")
pagesize = os.sysconf(os.sysconf_names['SC_PAGESIZE'])
self._physaddr = physaddr
self._size = size
self._aligned_physaddr = physaddr - (physaddr % pagesize)
self._aligned_size = size + (physaddr - self._aligned_physaddr)
try:
fd = os.open("/dev/mem", os.O_RDWR | os.O_SYNC)
except OSError as e:
raise MMIOError(e.errno, "Opening /dev/mem: " + e.strerror)
try:
self.mapping = mmap.mmap(fd, self._aligned_size, flags=mmap.MAP_SHARED, prot=(mmap.PROT_READ | mmap.PROT_WRITE), offset=self._aligned_physaddr)
except OSError as e:
raise MMIOError(e.errno, "Mapping /dev/mem: " + e.strerror)
try:
os.close(fd)
except OSError as e:
raise MMIOError(e.errno, "Closing /dev/mem: " + e.strerror)
# Methods
def _adjust_offset(self, offset):
return offset + (self._physaddr - self._aligned_physaddr)
def _validate_offset(self, offset, length):
if (offset+length) > self._aligned_size:
raise ValueError("Offset out of bounds.")
def read32(self, offset):
"""Read 32-bits from mapped physical memory, starting at the specified
`offset`, in bytes, relative to the base physical address the MMIO
object was opened with.
Args:
offset (int, long): offset from base physical address, in bytes.
Returns:
int: 32-bit value read.
Raises:
TypeError: if `offset` type is invalid.
ValueError: if `offset` is out of bounds.
"""
if not isinstance(offset, int) and not isinstance(offset, long):
raise TypeError("Invalid offset type, should be integer.")
offset = self._adjust_offset(offset)
self._validate_offset(offset, 4)
return ctypes.c_uint32.from_buffer(self.mapping, offset).value
def read16(self, offset):
"""Read 16-bits from mapped physical memory, starting at the specified
`offset`, in bytes, relative to the base physical address the MMIO
object was opened with.
Args:
offset (int, long): offset from base physical address, in bytes.
Returns:
int: 16-bit value read.
Raises:
TypeError: if `offset` type is invalid.
ValueError: if `offset` is out of bounds.
"""
if not isinstance(offset, int) and not isinstance(offset, long):
raise TypeError("Invalid offset type, should be integer.")
offset = self._adjust_offset(offset)
self._validate_offset(offset, 2)
return ctypes.c_uint16.from_buffer(self.mapping, offset).value
def read8(self, offset):
"""Read 8-bits from mapped physical memory, starting at the specified
`offset`, in bytes, relative to the base physical address the MMIO
object was opened with.
Args:
offset (int, long): offset from base physical address, in bytes.
Returns:
int: 8-bit value read.
Raises:
TypeError: if `offset` type is invalid.
ValueError: if `offset` is out of bounds.
"""
if not isinstance(offset, int) and not isinstance(offset, long):
raise TypeError("Invalid offset type, should be integer.")
offset = self._adjust_offset(offset)
self._validate_offset(offset, 1)
return ctypes.c_uint8.from_buffer(self.mapping, offset).value
def read(self, offset, length):
"""Read an array of bytes from mapped physical memory, starting at the
specified `offset` in bytes, relative to the base physical address the
MMIO object was opened with.
Args:
offset (int, long): offset from base physical address, in bytes.
length (int): number of bytes to read
Returns:
bytes: bytes read.
Raises:
TypeError: if `offset` type is invalid.
ValueError: if `offset` is out of bounds.
"""
if not isinstance(offset, int) and not isinstance(offset, long):
raise TypeError("Invalid offset type, should be integer.")
offset = self._adjust_offset(offset)
self._validate_offset(offset, length)
c_byte_array = (ctypes.c_uint8 * length).from_buffer(self.mapping, offset)
return bytes(bytearray(c_byte_array))
def write32(self, offset, value):
"""Write 32-bits to mapped physical memory, starting at the specified
`offset`, in bytes, relative to the base physical address the MMIO
object was opened with.
Args:
offset (int, long): offset from base physical address, in bytes.
value (int, long): 32-bit value to write
Raises:
TypeError: if `offset` or `value` type are invalid.
ValueError: if `offset` or `value` are out of bounds.
"""
if not isinstance(offset, int) and not isinstance(offset, long):
raise TypeError("Invalid offset type, should be integer.")
if not isinstance(value, int) and not isinstance(value, long):
raise TypeError("Invalid value type, should be integer.")
if value < 0 or value > 0xffffffff:
raise ValueError("Value out of bounds.")
offset = self._adjust_offset(offset)
self._validate_offset(offset, 4)
ctypes.c_uint32.from_buffer(self.mapping, offset).value = value
def write16(self, offset, value):
"""Write 16-bits to mapped physical memory, starting at the specified
`offset`, in bytes, relative to the base physical address the MMIO
object was opened with.
Args:
offset (int, long): offset from base physical address, in bytes.
value (int, long): 16-bit value to write
Raises:
TypeError: if `offset` or `value` type are invalid.
ValueError: if `offset` or `value` are out of bounds.
"""
if not isinstance(offset, int) and not isinstance(offset, long):
raise TypeError("Invalid offset type, should be integer.")
if not isinstance(value, int) and not isinstance(value, long):
raise TypeError("Invalid value type, should be integer.")
if value < 0 or value > 0xffff:
raise ValueError("Value out of bounds.")
offset = self._adjust_offset(offset)
self._validate_offset(offset, 2)
ctypes.c_uint16.from_buffer(self.mapping, offset).value = value
def write8(self, offset, value):
"""Write 8-bits to mapped physical memory, starting at the specified
`offset`, in bytes, relative to the base physical address the MMIO
object was opened with.
Args:
offset (int, long): offset from base physical address, in bytes.
value (int, long): 8-bit value to write
Raises:
TypeError: if `offset` or `value` type are invalid.
ValueError: if `offset` or `value` are out of bounds.
"""
if not isinstance(offset, int) and not isinstance(offset, long):
raise TypeError("Invalid offset type, should be integer.")
if not isinstance(value, int) and not isinstance(value, long):
raise TypeError("Invalid value type, should be integer.")
if value < 0 or value > 0xff:
raise ValueError("Value out of bounds.")
offset = self._adjust_offset(offset)
self._validate_offset(offset, 1)
ctypes.c_uint8.from_buffer(self.mapping, offset).value = value
def write(self, offset, data):
"""Write an array of bytes to mapped physical memory, starting at the
specified `offset`, in bytes, relative to the base physical address the
MMIO object was opened with.
Args:
offset (int, long): offset from base physical address, in bytes.
data (bytes, bytearray, list): a byte array or list of 8-bit integers to write
Raises:
TypeError: if `offset` or `data` type are invalid.
ValueError: if `offset` is out of bounds, or if data is not valid bytes.
"""
if not isinstance(offset, int) and not isinstance(offset, long):
raise TypeError("Invalid offset type, should be integer.")
if not isinstance(data, bytes) and not isinstance(data, bytearray) and not isinstance(data, list):
raise TypeError("Invalid data type, expected bytes, bytearray, or list.")
offset = self._adjust_offset(offset)
self._validate_offset(offset, len(data))
data = bytearray(data)
c_byte_array = (ctypes.c_uint8 * len(data)).from_buffer(self.mapping, offset)
for i in range(len(data)):
c_byte_array[i] = data[i]
def close(self):
"""Unmap the MMIO object's mapped physical memory."""
if self.mapping is None:
return
self.mapping.close()
self.mapping = None
self._fd = None
# Immutable properties
@property
def base(self):
"""Get the base physical address the MMIO object was opened with.
:type: int
"""
return self._physaddr
@property
def size(self):
"""Get the mapping size the MMIO object was opened with.
:type: int
"""
return self._size
@property
def pointer(self):
"""Get a ctypes void pointer to the memory mapped region.
:type: ctypes.c_void_p
"""
return ctypes.cast(ctypes.pointer(ctypes.c_uint8.from_buffer(self.mapping, 0)), ctypes.c_void_p)
# String representation
def __str__(self):
return "MMIO 0x%08x (size=%d)" % (self.base, self.size)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,906 |
Q: Graphing Fourier triangle wave So far, no success trying to recreate this in Latex:
The code I'm using is this:
\documentclass[14pt]{extarticle}
\usepackage{pgfplots}
\usepackage{tikz}
\usepackage{ifthen}
\begin{document}
\pgfplotsset{compat=1.14}
\begin{center}
\begin{tikzpicture}%[scale=1.5]
\begin{axis}[
width=\textwidth,
xmin = -1, xmax = 1,
ymin = 0, ymax = 3,
domain = -1: 1,
xlabel = $x$,
ylabel = $y$,
axis x line = center,
every axis x label/.append style = {below},
every axis y label/.append style = {left},
samples = 100,
xtick = {-1, 0, 1},
xticklabels = {$-1$, $0$, $1$},
declare function = {
s(\x) = ifthenelse(\x < pi, 1, 0);
s0(\x) = 0.5 + (2 / (pi * pi)) * (1-cos(pi)) * cos(deg(\x * pi));
s1(\x) = s0(\x) + (2 / (pi * pi * 9)) * (1-cos((\x * pi * 3)) * cos(deg(\x * pi * 3));
s2(\x) = s1(\x) + (2 / (pi * pi * 25)) * (1-cos((\x * pi * 5)) * cos(deg(\x * pi * 5));
s3(\x) = s1(\x) + (2 / (pi * pi * 49)) * (1-cos((\x * pi * 7)) * cos(deg(\x * pi * 7));
}, ]
\addplot[ultra thick, black] {s(x)};
\addplot[thick, blue] {s0(x)};
\addplot[thick, red] {s1(x)};
\addplot[thick, orange] {s2(x)};
\addplot[thick, cyan] {s3(x)};
\legend{signal, $s_0$, $s_1$, $s_2$, $s_3$};
% labels
\draw[gray, dashed] (-1, 0) -- (-1, 2);
\draw[gray, dashed] (1, 0) -- (1, 2);
\end{axis}
\end{tikzpicture}
\end{center}
\end{document}
And the current output is:
Probably my error is more of mathematical nature than Latex related, but if someone could help me I would highly appreciate it.
A: There are two main problems here, one related to pgfplots, one related to the mathematics. As I mentioned in my comment, you are a bit inconsistent when it comes to the arguments to the cosine function. pgf assumes that the input comes in degrees, so when you input radians you should convert it to degrees. You do this in some cases, but not all.
You can also tell pgfplots to assume radians for trig functions, by adding trig format plots=rad to the axis options. (trig format plots is better than trig formats which I mentioned in my comment, as the latter will also influence rotation of things like the ylabel.)
The mathematical problem is that in s1, s2 and s3 you have (1-cos((\x * pi * m)), which should be (1-cos(pi * m)), i.e. no \x* and the parenthesis are wrong as well.
In the code below there are a few other changes as well, to make it a bit closer to your sketch. I also defined a function
sN(\N, \x) = (2 / (pi * pi * \N * \N)) * (1-cos(pi * \N)) * cos(\x * pi * \N);
to reduce code duplication and make the other function definitions easier.
\documentclass[14pt]{extarticle}
\usepackage{pgfplots}
\pgfplotsset{compat=1.14}
\begin{document}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
width=\textwidth,height=0.5\textwidth,
xmin = -1, xmax = 1,
ymin = 0, ymax = 1.2, % changed from 3
domain = -1: 1,
xlabel = $x$,
ylabel = $y$,
hide y axis, % from your sketch you don't seem to want any y-axis
axis x line = bottom, % changed from center
samples = 101,
xtick = {-1, 0, 1},
trig format plots=rad, % added this, so pgfplots assumes radians for trig functions
clip=false, % to avoid clipping half of the dashed lines
declare function = {
% signal function
s = ifthenelse(\x<0, \x+1, -\x+1);
% constant term
s0 = 1/2;
% generic function for term number N in the series
sN(\N, \x) = (2 / (pi * pi * \N * \N)) * (1-cos(pi * \N)) * cos(\x * pi * \N);
s1(\x) = s0 + sN(1,\x);
s2(\x) = s1(\x) + sN(3,\x);
s3(\x) = s2(\x) + sN(5,\x);
s4(\x) = s3(\x) + sN(7,\x);
}
]
\addplot[ultra thick, black, samples at={-1,0,1}] {s};
\addplot[thick, black] {s0};
\addplot[thick, blue] {s1(x)};
\addplot[thick, red] {s2(x)};
\addplot[thick, orange] {s3(x)};
\addplot[thick, cyan] {s4(x)};
\legend{signal, $s_0$, $s_1$, $s_2$, $s_3$, $s_4$};
% just a different method for making the vertical dashed lines
% yours works fine as well
\addplot [forget plot, ycomb, dashed, gray] coordinates {(-1,1.2)(1,1.2)};
\end{axis}
\end{tikzpicture}
\end{center}
\end{document}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,914 |
using Xunit;
using AuthenticodeLint.Core.PE;
using System.Threading.Tasks;
using System.IO;
namespace AuthenticodeLint.Core.Tests
{
public class PortableExecutableTests
{
[Fact]
public async Task ShouldReadSimpleAttributesOfPE()
{
using (var pe = new PortableExecutable(PathHelper.CombineWithProjectPath("files/authlint.exe")))
{
var header = await pe.GetDosHeaderAsync();
Assert.NotEqual(0, header.e_lfanew);
var peHeader = await pe.GetPeHeaderAsync(header);
Assert.Equal(MachineArchitecture.x86, peHeader.Architecture);
Assert.Equal(16, peHeader.DataDirectories.Count);
var securityHeader = peHeader.DataDirectories[ImageDataDirectoryEntry.IMAGE_DIRECTORY_ENTRY_SECURITY];
Assert.NotEqual(0, securityHeader.Size);
Assert.NotEqual(0, securityHeader.VirtualAddress);
}
}
[Fact]
public async Task ShouldReadSecuritySection()
{
using (var pe = new PortableExecutable(PathHelper.CombineWithProjectPath("files/authlint.exe")))
{
var header = await pe.GetDosHeaderAsync();
var peHeader = await pe.GetPeHeaderAsync(header);
var securityHeader = peHeader.DataDirectories[ImageDataDirectoryEntry.IMAGE_DIRECTORY_ENTRY_SECURITY];
using (var file = new FileStream(PathHelper.CombineWithProjectPath("files/authlint.exe"), FileMode.Open, FileAccess.Read, FileShare.Read))
{
SecuritySection section;
var result = SecuritySection.ReadSection(file, securityHeader, out section);
Assert.True(result);
Assert.Equal(0x30, section.Data[0]);
}
}
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,111 |
Click here to watch livestream of Work Sessions. Agendas and background information for Work Sessions are posted on BoardDocs as they are available. Agendas are subject to change without notice and the public is encouraged to check the website in advance to confirm details or changes.
Visit the School Board Meetings web page for a listing of all upcoming meeting dates.
For additional information, contact the School Board Office at 703-228-6015 or school.board@apsva.us. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,003 |
Q: How to add a line break before and after a regex in a text file? This is an excerpt from the file I want to edit:
>chr1|-|9|S|somatic ACCACAGCCCTGTTTTACGTTGCGTCATCGCCCCGGGTGCCTGGTGACGTCACCAGCCCGCTCG >chr1|+|9|Y|somatic ACCACAGCCCTGTTTTACGTTGCGTCATCGCCCCGGGTGCCTGGTGACGTCACCAGCCCGCTCG
I would a new text file in which I add a line break before ">" and after "somatic" or after "germline", how can I do in R or Unix?
Expected output:
>chr1|-|9|S|somatic
ACCACAGCCCTGTTTTACGTTGCGTCATCGCCCCGGGTGCCTGGTGACGTCACCAGCCCGCTCG
>chr1|+|9|Y|somatic
ACCACAGCCCTGTTTTACGTTGCGTCATCGCCCCGGGTGCCTGGTGACGTCACCAGCCCGCTCG
A: (\bsomatic\b|\bgermline\b)|(?=>)
Try this.See demo.Replace by $1\n
http://regex101.com/r/tF5fT5/53
If there's no support for lookahead then try
(\bsomatic\b|\bgermline\b)
Try this.Replace by $1\n.See demo.
http://regex101.com/r/tF5fT5/50
and
(>)
Replace by \n$1.See demo.
http://regex101.com/r/tF5fT5/51
A: By the looks of your input, you could simply replace spaces with newlines:
tr -s ' ' '\n' <infile >outfile
(Some tr dialects don't like \n. Try '\012' or a literal newline: opening quote, newline, closing quote.)
If that won't work, you can easily do this in sed. If somatic is static, just hard-code it:
sed -e 's/somatic */&\n/g' -e 's/ >/\n>/g' file >newfile
The usual caveats about different sed dialects apply. Some versions don't like \n for newline, some want a newline or a semicolon instead of multiple -e arguments.
On Linux, you can modify the file in-place:
sed -i 's/somatic */&\
/g
s/ >/\
/g' file
(For variation, I'm showing how to do this if your sed doesn't recognize \n but allows literal newlines, and how to put the script in a single multi-line string.)
On *BSD (including MacOS) you need to add an argument to -i always; sed -i '' ...
If somatic is variable, but you always want to replace the first space after a wedge, try something like
sed 's/\(>[^ ]*\) /\1\n/g'
>[^ ] matches a wedge followed by zero or more non-space characters. The parentheses capture the matched string into \1. Again, some sed variants don't want backslashes in front of the parentheses, or are otherwise just ... different.
If you have very long lines, you might bump into a sed which has problems with that. Maybe try Perl instead. (Luckily, no dialects to worry about!)
perl -i -pe 's/(>[^ ]*) /$1\n/g;s/ >/\n>/g' file
(Skip the -i option if you don't want to modify the input file. Then output will be to standard output.)
A: Thank you everyone!
I used:
tr -s ' ' '\n' <infile >outfile
as suggested by tripleee and it worked perfectly!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 76 |
\section{Introduction}
The origin of four dimensional gauge symmetries is one of the deepest mysteries of physics.
The idea of Theodor Kaluza, improved by Oskar Klein (cf. for example, \cite{Chodos} and references therein)
that higher dimensional
spacetime symmetries imply low energy gauge symmetries in four dimensions provided the extra
dimensions are curled up in an appropiate way has proved quite fruitful and worth pursuing.
\par
In the simplest setting, the Einstein-Hilbert gravitational
action in a five-dimensional manifold
which is a product of four-dimensional Minkowski space-time with a one-dimensional circle
of radius $R$, looks at energies
\begin{equation}
E<< M\equiv \frac{1}{R}
\end{equation}
like four-dimensional Einstein-Hilbert coupled to an abelian Maxwell field.
\par
In order to be more precise, if we believe that extra dimensions are {\em real}, we got to
renormalize the theory. Even if we do not embed the extra-dimensional theory in some suposedly
consistent framework, such as supestrings, (which would provide a cutoff of sorts), at one loop order, the fact
that the higher dimensional (sometimes called the mother) theory is not renormalizable
is not directly relevant, in
the sense
that we still can study and classify all divergences. For example, the six-dimensional
electric charge is
dimensionful, which allows for an unbounded number
of candidate counterterms. However, to any given order in perturbation theory
this number is finite, and the theory can in principle be renormalized, although it is
still true that always appear new operators in the counterterms which were not
present in the original lagrangian. This is then essentially a low energy
approximation, because we can only expect it to be good (in the example of $QED_6$, in which
we are going to concentrate upon) when the dimensionless
quantity $\alpha_{d=6}E^2 << 1$, where $\alpha_{d=6}$ is the six-dimensional fine structure
constant. Given the fact that the six and four-dimensional coupling constants are related by
$\alpha_{d=6}M^2\equiv\alpha_{d=4}\equiv\frac{1}{137}$, in terms of the usual four-dimensional fine-structure constant, this means
$E<<\frac{M}{\sqrt{\alpha}}\sim 10 M$. It follows that one can compute reliably for energies
$E\sim M$, but not much bigger. Our viewpoint will thus be that the
theory is {\em defined} in higher dimensions by means of the necessary counterterms, in a sense
that we shall try to make more precise in what follows.
\par
At any rate, and in order to dissipate any doubts, we shall repeat in due course the same analysis on $QED_4$ on a two-torus.
In this case the extra dimensional theory is well defined (forgetting the Landau pole), and our results are
essentially the same.
\par
Besides the six-dimensional viewpoint we are going to favor, there is always the possibility of
expanding all fields in harmonics and perform the integrals over the extra dimensional compact manifold.
In that way we find a four dimensional theory, but with an infinite number of fields. It seems quite intuitive that
provided we keep track of the infinite set of modes, this four dimensional theory should be equivalent to
the full extra-dimensional one; their respective divergences, in particular, should match.
The main purpose of this paper is to check this intuition with some explicit computations.
Although we are not going to work it out in any detail, it should be possible to express
our results in
the language of effective low energy field theories. Some steps in this direction have
been already given in \cite{Oliver} \cite{Ghilencea}.
\par
There are then two complementary
viewpoints, the higher dimensional one, and the four dimensional with the Kaluza-Klein
tower, and if we want to make explicit statements on exactly when the tower begins to be
relevant, we have to relate not only the classical parts, but also the quantum contributions
on both sides.
\par
Curiously enough, in the case the fields only interact through the universal coupling to an external gravitational field,
the two viewpoints are {\em exactly} equivalent (with some qualifications).This was proved by Duff and Toms \cite{Duff}, and provided
a strong motivation for our research.
\par
We shall work to one loop order only. To this order, the effective action is
given in terms of a functional determinant. We shall regularize it through the heat kernel
approach, which respects all gauge invariances, including the geometrical ones. Let us quickly
review our notation and remark on some potential ambiguities.
\par
The geometric setting is given by a riemannian n-dimensional manifold, with a metric
$g_{MN}$.
This manifold will usually be of a factorized form: $\mathbb{R}^4\times K$
where $K$ is a compact $(n-4)$-dimensional manifold , and $\mathbb{R}^4$
represents the euclidean version of Minkowski space.
More generally, like in the models recently popularized by
Randall and Sundrum \cite{Randall}, this structure is present only locally,
i.e., we have a fiber bundle
(warped space) based on
Minkowski space.
\par
All our operators enjoy the form
\begin{equation}\label{ope}
\Delta\equiv -D_{M}D^{M} + Y
\end{equation}
with
\begin{equation}
D_M\equiv \partial_M+X_M
\end{equation}
and the operator defining the heat kernel is formally given by:
\begin{equation}
K(\tau)\equiv e^{-\tau \Delta}
\end{equation}
acting on a convenient functional space in such a way that
\begin{equation}
(K f)(x)\equiv \int \sqrt{|g|} d^n y K(x,y;\tau) f(y)
\end{equation}
The short time off-diagonal \cite{de Witt} expansion is
defined (for manifolds without
boundary) by:
\begin{equation}
K(x,y;\tau)=K_0(x,y;\tau)\sum_{p=0}b_{2p}(x,y) \tau^p
\end{equation}
where the flat space solution is given by:
\begin{equation}
K_0(x,y;\tau)=\frac{1}{(4\pi\tau)^{n/2}}e^{-\frac{\sigma^2}{4\tau}}
\end{equation}
and $\sigma$ is the geodesic distance
between the two points, given in flat space by:
\begin{equation}
\sigma^2=(x-y)^2
\end{equation}
and for consistency
\begin{equation}
b_0(x,x)=1.
\end{equation}
When boundaries are present, odd powers of $\tau^{1/2}$ do appear, which can
formally be incorporated in the
former expansion by allowing non vanishing odd coefficients, $b_{2p+1}\tau^{p+ 1/2}\neq0$.
\par
It is sometimes useful to consider the integrated quantity:
\begin{equation}
Y(\tau,f)\equiv tr\,(f e^{-\tau \Delta})=\sum_{k=0}\tau^{\frac{k-n}{2}}a_k(f)
\end{equation}
where the trace involves whatever finite rank indices the operator might posses, and
\begin{equation}
a_k(f)=\frac{1}{(4\pi)^{n/2}}\int \sqrt{|g|}d^n x\,tr\,b_k(x,x)f(x)
\end{equation}
The mass dimension of $a_k$ is $k-n$, whereas the one of $b_k$ is simply $k$.
It follows that
\begin{equation}
a_0=\frac{tr\,{\mathbb{I}}}{(4\pi)^{n/2}}V\equiv\frac{1}{(4\pi)^{n/2}}\int \sqrt{|g|}d^n x\,tr\mathbb{I}
\end{equation}
As usual, we shall denote
\begin{equation}
a_k\equiv a_k(f=1)
\end{equation}
Note in particular that
\begin{equation}
Y(\tau)\equiv Y(\tau,f=1)=tr\, e^{-\tau \Delta}=\sum_{k=0}\tau^{\frac{k-n}{2}}a_k
\end{equation}
After all these prolegomena,
the determinant is defined as:
\begin{equation}
\log\,\det\,\Delta=-\int_0^\infty\frac{d\tau}{\tau^{1+ n/2}}\sum_{p=0}a_{p}\tau^{p/2}
\end{equation}
There are several possible viewpoints on this integral. One of them is to analytically continue
on the dimension $n$.
The integral over the proper time $\tau$, cut off in the infrared by $\tau_{max}=\mu^{-2}$
produces poles in
the complex variable $n$, given by:
\begin{equation}
\log\,\det\,\Delta=- \sum_{p=0}a_{p}\frac{2 \mu^{n-p}}{p-n}+finite\,part.
\end{equation}
which when $n$ approaches the physical dimension, say, $d$,
\begin{equation}
n=d+\epsilon
\end{equation}
yields the divergent piece of the determinant (a dimensionless quantity):
\begin{equation}
\log\,\det\,\Delta |_{div}=\frac{2 \mu^\epsilon}{ \epsilon}\,a_d(\Delta).
\end{equation}
in the even dimensional case. This prescription yields
a finite answer for odd dimensions in the absence of a boundary, and is the one usually favored when
working with effective lagrangians (cf. for example \cite{Donoghue}).
\par
A different, and in some sense more physical possibility is to
introduce a cutoff in the lower end of the proper time integral,
$\Lambda/\mu\rightarrow\infty$.
In that way we get, for example \footnote{Although we shall try our best
to avoid cluttering the notation unnecessarily,
we are forced to distinguish between quantities bearing identical names,
but coming from different dimensions.} in six dimensions:
\begin{equation}\label{6d}
\log\,\det\,\Delta |_{div}=\frac{1}{3} a_0\Lambda_{(d=6)}^6+ \frac{1}{2}
a_2 \Lambda_{(d=6)}^4+ a_4\Lambda_{(d=6)}^2+ a_6 \log{\frac{\Lambda^2_{(d=6)}}{\mu^2_{(d=6)}}}
\end{equation}
Where the heat kernel coefficients are obviously in six dimensions. In four dimensions instead:
\begin{equation}\label{4d}
\log\,\det\,\Delta |_{div}=\frac{1}{2} a_0\Lambda_{(d=4)}^4+ a_2 \Lambda_{(d=4)}^2+
a_4 \log{\frac{\Lambda^2_{(d=4)}}{\mu^2_{(d=4)}}}
\end{equation}
Where now the coefficients are the corresponding ones in four dimensions. The dominant divergence (sixth power and fourth power of the cutoff) is universal and
independent of the particular operator under consideration. We shall not study it
further here.
\par
In spite of the fact that it is often pointed out that there is no way of
imposing a cutoff in a gauge invariant way, we would like to stress that, at least to the
one loop order, this procedure respects all gauge invariances, abelian and non
abelian, as well as general covariance in its case. This is obvious, because we are {\em not} cutting off the momentum,
but rather the {\em proper time}, a covariant as well as gauge invariant concept. If we remember that the proper time in the sense we are employing it,
has mass dimension $-2$, we are neglecting in the evaluation of the one loop determinants proper times smaller than $\Lambda^{-1}$. This fact, which was probably first pointed out by Schwinger
\cite{Schwinger} in 1951, has been exploited by Bryce deWitt \cite{de Witt} to get covariant expansions in quantum gravity; and also by
Fujikawa \cite{Fujikawa} to get the covariant anomaly.
\par
We shall denote these two procedures {\em dimensional regularization} and {\em cutoff},
respectively. Both respect all gauge invariances of the theory but only the cutoff theory
yields information on the divergences in the odd dimensional case.
\section{Six dimensional quantum electrodynamics compactified on a torus.}
Let us now consider an example not altogether trivial, namely quantum electrodynamics (QED) on
a six-dimensional
manifold which is
topologically four-dimensional
Minkowski space times a two-torus, that is, $\mathbb{R}^4\times S^1\times S^1$.
This example avoids the complications of interacting gravitational sectors, but in some
sense is not representative of
the whole Kaluza-Klein philosophy, because we are introducing gauge fields already in the extra dimensions. We are using it as a toy model.
\par
The metric for the
time being is assumed to be
\begin{equation}
ds^2=\delta_{\mu\nu}dx^{\mu}dx^\nu+R_5^2 d\theta_5^2 + R_6^2 d\theta_6^2
\end{equation}
that is, $y_5=R_5 \theta_5$ and $y_6=R_6 \theta_6$. We shall follow consistently the
convention that capital indices, like $M,N,\ldots$ run over the full dimensions, in our case
from $1$ to $6$; greek indices, $\mu,\nu,\ldots$ run over the ordinary Minkowski coordinates, from $1$ to $4$; and small roman letters,
$a,b,\ldots$, over the extra dimensions, that is, from $5$ to $6$.
\par
The (euclidean version of the) action then reads
\begin{equation}
S=\int d^6 x\left(\frac{1}{4}F_{MN}^2+\bar{\psi}(D\fsl+m)\psi\right)
\end{equation}
where the abelian covariant derivative is simply:
\begin{equation}
D_M\psi\equiv\left(\partial_M-eA_M\right)\psi
\end{equation}
Let us recall here that, for vanishing curvature, the general formulas \cite{Gilkey}
for the first few
coefficients of an operator of the form (\ref{ope}) are:
\begin{equation}
a_2=-\int\frac{d^n x}{(4\pi)^{\frac{n}{2}}}\,tr\, Y
\end{equation}
\begin{equation}
a_4=\int\frac{d^n x}{(4\pi)^{\frac{n}{2}}}\,tr\left(\frac{1}{12}X_{MN}^2+\frac{1}{2}Y^2-
\frac{1}{6}Y_{;MM}\right)
\end{equation}
\begin{eqnarray}\label{eq:a6}
\lefteqn{a_6=\frac{1}{360}\int\frac{d^n x}{(4\pi)^{\frac{n}{2}}}
\,tr\Big( 8X_{MN;R}^2+2X_{MN;N}^2+{} } \nonumber\\& & {}+12X_{MN;RR}X^{MN}-12X_{MN}X^{NR}
X_R^{\hspace{1ex}M}-6Y_{;MMNN}+{} \nonumber\\& & {}+60YY_{;MM}+30Y_{;M}^2-
60Y^3-30YX_{MN}^2\Big)
\end{eqnarray}
Where $;$ denotes covariant derivative, and
\begin{equation}
X_{MN}=\partial_M X_N-\partial_N X_M +\left[X_M,X_N\right]
\end{equation}
In order to perform the explicit computation, it is exceedingly useful to combine the fermionic
and bosonic sectors in a full supermatrix. Please read the appendix for a brief review of
the technique and notation.
\par
Computing the coefficients is then straightforward albeit somewhat laborious.
In terms of the background fields $\bar{A}_M,\eta,\bar{\eta}$
\begin{equation}
a_2=\int\frac{d^6 x}{(4\pi)^3}8m^2
\end{equation}
as well as
\begin{equation}
a_4=\int\frac{d^6 x}{(4\pi)^3}\left(\frac{4}{3}e^2\bar{F}_{MN}^2+4 e^2 \bar{\eta}\bar{D\fsl}\eta
+12me^2\bar{\eta}\eta\right)
\end{equation}
Finally we get (using the background equations of motion):
\begin{eqnarray}\label{sdcounter}
\lefteqn{ a_6=\int\frac{d^6 x}{(4\pi)^3}\left(-\frac{1}{12}e^4\bar{\eta}\Sigma_{MNL}\eta\bar{\eta}\Sigma^{MNL}\eta+\frac{19}{15}e^2m\bar{\eta}\bar{D}_M\bar{D}^M\eta+\frac{2}{15}e^3\bar{\eta}\gamma_N\bar{D}_M\eta\bar{F}^{MN}-\right.}\nonumber\\
& &{}-e^3m\bar{\eta}\gamma_M\gamma_N\eta\bar{F}_{MN}-2e^2m^2\bar{\eta}
\gamma^M\bar{D}_M\eta-6e^2m^3\bar{\eta}\eta-\frac{11}{45}e^2\bar{D}_R\bar{F}_{MN}\bar{D}^R\bar{F}^{MN}+{}\nonumber\\
&&\left.{}+\frac{23}{9}e^2\bar{D}_M\bar{F}^{MN}\bar{D}^R\bar{F}_{RN}-\frac{4}{3}
e^2m^2\bar{F}_{MN}\bar{F}^{MN}\right)
\end{eqnarray}
Where $\Sigma_{MNL}$ is the totally antisymmmetric product of three gammas. Remember that in dimensional regularization
\begin{equation}
\Delta S=\frac{1}{\epsilon}a_6
\end{equation}
plus a possible finite part. With a cutoff, these are the logarithmic divergences, and
we have in addition both quadratic and quartic divergences, on which more to follow.
\par
The first conclusion we can draw from this analysis is that
quantum effects, besides renormalizing the six-dimensional couplings, induce a set of
non-minimal interactions which are generated with arbitrary coefficients.
\par
Actually, due to the fact that the mass dimension of the coupling constant is $-1$, there is
no finite closed set of operators of counterterms. Let us be more specific.
\par
First of all, there is a dimension five operator, which becomes a potential
counterterm in the massive case:
\begin{equation}
{\cal O}_{(5)}=\left(\bar{\psi}\psi\right)
\end{equation}
The set of gauge-invariant dimension six operators is given by:
\begin{equation}
{\cal O}^i_{(6)}=\left(\bar{\psi} D\fsl\psi,\,F_{MN}^2\right)
\end{equation}
To the next order, that is, dimension seven, the list reads:
\begin{equation}
{\cal O}^i_{(7)}=\left(\bar{\psi} D\fsl\Dslash\psi\right)
\end{equation}
The dimension eight operators are:
\begin{equation}
{\cal O}^i_{(8)}=\left(\bar{\psi} D\fsl\DslashD\fsl\psi,\,
\bar{\psi}\sigma_{MN}\psi F^{MN},\,D^M F_{MN}D_RF^{RN},\,F_{NL}D^2F^{NL}\right)
\end{equation}
And finally, to dimension nine we have to consider:
\begin{equation}
{\cal O}^i_{(9)}=\left(\bar{\psi} \gamma_M D_N\psi F^{MN},\,\bar{\psi}D_A D_B D_C D_D\psi t^{ABCD}
\right)
\end{equation}
In the massive case the dimension of this operators can be increased by introducing powers of $m$. Amongst the operators that actually appear as counterterms only the ${\cal O}^2_{(8)}$ is absent. At any rate it should be plain
that we can claim results only to first nontrivial order in the six-dimensional fine structure constant, and
that we have really no right to keep the $e^3$ and $e^4$ terms in the counterterm.
\par
The non renormalizabity of the theory manifests itself in the fact that if we were to include
all those dimension seven and dimension eight operators, they would generate more and more
higher dimension operators as counterterms. There is no closed set, unless we assume, as is
natural to the order we are working, that the effect of all those couplings is of higher
order in the six-dimensional fine structure constant.\footnote{
Keeping in mind that we are not performing a fully consistent computation,
if we define the renormalization constants as is usually done:
\begin{eqnarray}
&&A_0=Z_3^{1/2}A\nonumber\\
&&\psi_0=Z_2^{1/2}\psi\nonumber\\
&&e_0=Z_1 Z_2^{-1} Z_3^{-1/2} e\nonumber\\
&&m_0=Z_m m
\end{eqnarray}
we easily get $Z_1=Z_2$ which conveys the fact that the theory is gauge invariant, and
\begin{eqnarray}
&&Z_2=1-\frac{ e^2 m^2}{32\pi^3\epsilon}\nonumber\\
&&Z_3=1-\frac{ e^2 m^2}{12\pi^3 \epsilon}\nonumber\\
&&Z_m=1-\frac{ e^2 m^2}{16\pi^3\epsilon}
\end{eqnarray}
A simple calculation then leads to the renormalization group functions:
\begin{eqnarray}
&&\beta_e\equiv\frac{\partial e}{\partial\log{\mu}}=-\frac{1}{24 \pi^3} e^3 m^2\nonumber\\
&&\beta_m\equiv\frac{\partial m}{\partial \log{\mu}}=\frac{1}{16\pi^3} e^2 m^3
\end{eqnarray}
The renormalization of the fermion mass is entangled with the charge renormalization.
The behavior of the coupling constants reads:
\begin{eqnarray}
&&e=e_0-\frac{1}{24\pi^3} m_0^2 e_0^3 \log{\mu/\mu_0}\nonumber\\
&&m=m_0\left(1-\frac{1}{24 \pi^3} m_0^2 e_0^2 \log{\mu/\mu_0}\right)^{- 3/2}
\end{eqnarray}
The dimensionful charge vanishes when
\begin{equation}
\mu=\mu_0 e^{\frac{24\pi^3}{m_0^2 e_0^2}}
\end{equation}
If we define the dimensionless couplings
\begin{eqnarray}
&&\hat{e}\equiv e\mu\nonumber\\
&&\hat{m}\equiv\frac{m}{\mu}
\end{eqnarray}
then the renormalization group equations read
\begin{eqnarray}
&&\beta_{\hat{e}}=\hat{e}-\frac{1}{24\pi^3}\hat{m}^2 \hat{e}^3\nonumber\\
&&\beta_{\hat{m}}=-\hat{m}+\frac{1}{16\pi^3}\hat{e}^2\hat{m}^3
\end{eqnarray}
}
\section{The four-dimensional viewpoint}
Let's consider the point of view of the reduced theory. We can now expand all fields in
Fourier series:
\begin{equation}
\phi (x,y)=\frac{1}{2\pi\sqrt{R_5 R_6}}\sum_n \phi_n(x) e^{i \frac{n}{R}.y}
\end{equation}
where $n\equiv (n_5,n_6)$, and we have included a convenient factor in front to
take care of the diference of canonical dimensions of the fields in six and four dimensions.
Real fields (such as the photon) obey
\begin{equation}
\phi^*_n(x)=\phi_{-n}(x)
\end{equation}
The six-dimensional gamma matrices can be chosen such as:
\begin{eqnarray}
&&\gamma_\mu^{(6)}=\sigma_3\otimes\gamma_\mu^{(4)}\nonumber\\
&&\gamma_5^{(6)}=\sigma_1\otimes 1\nonumber\\
&&\gamma_6^{(6)}=\sigma_2\otimes1
\end{eqnarray}
In that way, six-dimensional spinors split in two four-dimensional ones:
\begin{equation}
\psi=\left(\begin{array}{c}
\psi_1\\
\psi_2
\end{array}\right)
\end{equation}
It is a simple matter to perform the integrals over the angular variables and obtain
the gauge fixed action (still exact) in the four dimensional form:
\begin{eqnarray}\label{tower}
\lefteqn{S=\int d^4 x\sum_{n_5,n_6}\left(\bar{\psi}^1_{n}\partial\fsl \psi^1_n+\bar{\psi}^2_{n}
\partial\fsl \psi^2_n+\bar{\psi}^1_{n}(i\frac{n_5}{R_5}+\frac{n_6}{R_6})\psi^2_n-
\bar{\psi}^2_{n}(i\frac{n_5}{R_5}-\frac{n_6}{R_6})\psi^1_n+\right.{}}\nonumber\\
&&{}+m\left(\bar{\psi}^1_{n}\psi^1_n-\bar{\psi}^2_{n}\psi^2_n\right)
-\frac{1}{2}(A_\mu^n)^{*}\left(\Box-\frac{n_5^2}{R_5^2}-\frac{n_6^2}{R_6^2}\right) A^\mu_n-
\frac{1}{2}(A_5^n)^{*}\left(\Box-\frac{n_5^2}{R_5^2}-\frac{n_6^2}{R_6^2}\right)A_5^n-
\nonumber\\
&&\left.-\frac{1}{2}(A_6^n)^{*}\left(\Box-\frac{n_5^2}{R_5^2}-
\frac{n_6^2}{R_6^2}\right)A_6^n-e\sum_m\left(\bar{\psi}^1_mA\gsl _{m-n}\psi^1_n+
\bar{\psi}^2_mA\gsl _{m-n}\psi^2_n
+\right.\right.{}\nonumber\\&&{}\left.+\bar{\psi}^1_mA_5^{m-n}\psi^2_n-
\bar{\psi}^2_mA_5^{m-n}\psi^1_n-i\bar{\psi}^1_mA_6^{m-n}\psi^2_n-
i\bar{\psi}^2_mA_6^{m-n}\psi^1_n\right)\Bigg)
\end{eqnarray}
and the four-dimensional coupling constant is
\begin{equation}
e\equiv \frac{e^{(6)}}{2\pi\sqrt{R_5 R_6}}\equiv e^{(6)}M
\end{equation}
Here we see clearly a generic feature of interacting theories, namely that there is no
consistent truncation in the sense that all massive fields interact among themselves
and with the massless fields.
\subsection{Gauge symmetries of the four-dimensional action.}
Six-dimensional QED has an obvious $U(1)$ symmetry. It is interesting to see how this invariance is traduced in the lower dimensional theory. Before gauge fixing, the four-dimensional action enjoys the infinite set of symmetries:
\begin{eqnarray}
&&\delta A_\mu^n=i\partial_\mu \Lambda_n\nonumber\\
&& \delta A_5^n=-\frac{n_5}{R_5} \Lambda_n\nonumber\\
&& \delta A_6^n =-\frac{n_6}{R_6} \Lambda_n
\end{eqnarray}
Where $\Lambda_n$ are the modes of the expansion of the abelian transformation parameter. All those gauge symmetries $\Lambda_{n_5,n_6}$ are spontaneously broken, except for the
zero mode, corresponding to $\Lambda_{0,0}$. The $A_\mu^n$ are the massive vector bosons, and
the $A_5^n$ and $A_6^n$ the scalar higgses.
\par
There is a curious fact, however, and this is the appearance of two singlets in four dimensions,
namely $A_5^0$ and $A_6^0$. Those singlets are massless at tree level, but no symmetry protects
them from getting massive through quantum corrections.
\par
The same fields are protected from getting masses in six dimensions, through gauge invariance
and six dimensional Lorentz covariance. The point is that the breaking
\begin{equation}
O(1,5)\rightarrow O(1,3)\times O(2)\times O(2)
\end{equation}
of the symmetry group of the vacuum is an instance of spontaneous compactification; i.e., the
equations of motion enjoy the full $O(1,5)$ symmetry, and only the solution breaks it.
\subsection{The massless action}
The zero mode of the above action is
\begin{eqnarray}
\lefteqn{S_{zm}=\int d^4 x\left(\bar{\psi}^1\partial\fsl \psi^1+\bar{\psi}^2
\partial\fsl \psi^2+m\left(\bar{\psi}^1\psi^1-\bar{\psi}^2\psi^2\right)
-\frac{1}{2}A_\mu\Box A^\mu-\right.{}}\nonumber\\
&&\left.-\frac{1}{2}\phi^*\Box\phi-e\left(\bar{\psi}^1A\gsl \psi^1+\bar{\psi}^2A\gsl \psi^2
+\bar{\psi}^1\phi\psi^2-\bar{\psi}^2\phi^*\psi^1\right)\right)
\end{eqnarray}
where we have represented the zero modes of all fields by the same letter without any subindex:
\begin{equation}
A_5^0-iA_6^0\equiv\phi^0\equiv\phi
\end{equation}
It must be stressed that this is {\em not} a consistent truncation,(in the sense of the word usually employed in supergravity
and superstrings)
owing to the fact that both $A^0_\mu$ and $\phi$ couple diagonally to the whole fermionic tower; it is expected, however, to be a
physically sensible one at energies $E<<M$.
\par
Denoting $\bar{\phi}$ the background for $\phi$ the cuadratic part of the action is
\begin{eqnarray}
\lefteqn{S_{zm}=\int d^4x\left(\bar{\psi}^1\partial\fsl\psi^1+
\bar{\psi}^2\partial\fsl\psi^2+m\left(\bar{\psi}^1\psi^1-
\bar{\psi}^2\psi^2\right)-\frac{1}{2}\phi_\mu\Box\phi^\mu\right.-}{}\nonumber\\
& &{}\left.-\frac{1}{2}\phi^*\Box\phi-e\left(\bar{\psi}^1\bar{A\gsl }\psi^1+
\bar{\psi}^2\bar{A\gsl }\psi^2+\bar{\eta}^1\gamma^\mu\phi_\mu\psi^1+
\bar{\eta}^2\gamma^\mu\phi_\mu\psi^2+\bar{\psi}^1\gamma^\mu\phi_\mu\eta^1+
\right.\right.{}\nonumber\\
& &{}\left.\left.+\bar{\psi}^2\gamma^\mu\phi_\mu\eta^2+\bar{\psi}^1\bar{\phi}\psi^2-
\bar{\psi}^2\bar{\phi}^*\psi^1+\bar{\eta}^1\phi\psi^2-\bar{\eta}^2\phi^*\psi^1+
\bar{\psi}^1\phi\eta^2-\bar{\psi}^2\phi^*\eta^1\right)\right)
\end{eqnarray}
Where $e$ is now the four dimensional coupling. The first two coefficients
in the heat kernel expansion read:
\begin{equation}
a_2^{(zm)}=\int\frac{d^4 x}{(4\pi)^2}8\left(m^2 - e^2 |\bar{\phi}|^2\right)
\end{equation}
and
\begin{eqnarray}\label{fdcounter}
\lefteqn{a_4^{(zm)}=\int\frac{d^4 x}{(4\pi)^2}\left(\frac{4}{3}e^2\bar{F}_{\mu\nu}^2-
4e^2\bar{\phi}^*\Box\bar{\phi}+8e^2m^2|\bar{\phi}|^2-
4e^4|\bar{\phi}|^4+\right.{}}\nonumber\\
&&+4e^2\left(\bar{\eta}^1\bar{D\fsl}\eta^1+
\bar{\eta}^2\bar{D\fsl}\eta^2\right)+12me^2\left(\bar{\eta}^1\eta^1-\bar{\eta}^2\eta^2\right)+8e^3
\bar{\eta}^2\bar{\phi}^*\eta^1-8e^3\bar{\eta}^1\bar{\phi}\eta^2\Bigg)
\end{eqnarray}
This is the logarithmically divergent counterterm that arises when renormalizing the zero mode of the four dimensional
action.
\par
It should be remarked that the resulting four dimensional model is superficially
very similar to the
Coleman-Weinberg setup, in which
radiative spontaneous symmetry breaking was first discovered. There is a crucial difference
though, and this is that the scalar field is not charged, in spite of being complex.
The reason is that it remembers its gauge origin, and six-dimensional gauge invariance
manifests here as a Kac-Moody transformation acting on the full tower of
massive states. In addition to that, the quartic coupling is here a
quantum effect, because it was not present in the bare four-dimensional lagrangian. Also
the scalar field gets massive, with a mass proportional to the fermion mass (times the four-
dimensional fine structure constant)\footnote{At any rate, this yields (twice) the usual beta function for the four dimensional fine structure
constant:
\begin{equation}
\beta_e=\frac{1}{6\pi^2} e^3
\end{equation}
The behavior of the charge is:
\begin{equation}
e^2=\frac{e_0^2}{1-\frac{e_0^2}{3\pi^2}\log{\mu/\mu_0}}
\end{equation}
which blows up at a Landau pole located at
\begin{equation}
\Lambda\equiv \mu_0 e^{3\pi^2 /e_0^2}
\end{equation}
}
.
\section{A comparison of the massless sector of the full six-dimensional divergences
with the divergences of the massless sector of the four-dimensional theory.}
After all this work, we are finally in a position to study our main concern, namely, how
the divergent part of the six-dimensional effective action is related
to the corresponding four-dimensional quantity.
\subsection{The cutoff theory}
Let us first analyze the problem from the viewpoint of the cutoff theory. As we have seen in six dimensions the divergent part of the effective action is
given through the equation (\ref{6d}); while from the four-dimensional viewpoint the corresponding formula stems from (\ref{4d}).
\par
When we are interested in the zero mode, i.e., the piece in six dimensions where
all fields are independent of the extra dimensions, the measure clearly factorizes:
\begin{equation}
d^6 x\rightarrow \frac{1}{M^2}d^4 x
\end{equation}
It is plain that the divergences never coincide exactly.
The {\em only} way to make the divergences related to the fourth heat-kernel coefficient identical
in six and in four dimensions is choose different proper time cutoffs in both dimensions
in such a way that:
\begin{equation}
\frac{\Lambda_{(d=6)}^2}{M^2}\equiv\log{\frac{\Lambda^2_{(d=4)}}{\mu^2_{(d=4)}}}
\end{equation}
We choose that because those coefficients are {\em almost} identical, so that the logarithmic divergences
are as similar as possible.
This identification leads to the reinterpretation of the six-dimensional quartic divergences as
$\log^2$:
\begin{equation}
\Lambda_{(d=6)}^4\rightarrow M^4 \left(\log{\frac{\Lambda^2_{(d=4)}}{\mu^2_{(d=4)}}}\right)^2
\end{equation}
and finally, the six-dimensional logarithmic divergences appear in the guise of log\,log.
\begin{equation}
\log{\frac{\Lambda^2_{(d=6)}}{\mu^2_{(d=6)}}}\rightarrow
\log\left(\frac{M^2}{\mu^2_{(d=6)}}\log{\frac{\Lambda^2_{(d=4)}}{\mu^2_{(d=4)}}}\right)
\end{equation}
This reinterpretation gives rise to a few more four-dimensional nonstandard counterterms,
which we will comment upon in a moment.
\par
Let us stress, for the time being, that the logarithmic divergence, when renormalizing (correctly) from six dimensions
is not identical to the one (\ref{fdcounter}), but rather
\begin{eqnarray}
\lefteqn{\Delta S_{\log}\equiv\int\frac{d^4 x}{(4\pi)^3}e^2
\Bigg(4\left(\bar{\eta}^1\partial\fsl \eta^1+\bar{\eta}^2
\partial\fsl \eta^2 \right)+{}}\nonumber\\
&&+12m\left(\bar{\eta}^1\eta^1-\bar{\eta}^2\eta^2\right)
+\frac{4}{3}\left(\bar{F}^{\mu\nu}\bar{F}_{\mu\nu}-\right.\nonumber\\
&&\left.-2\bar{A}_5 \Box \bar{A}_5- 2\bar{A}_6\Box\bar{A}_6\right)-
\nonumber\\
&&-4e\left(\bar{\eta}^1\bar{A\gsl }\eta^1+
\bar{\eta}^2\bar{A\gsl }\eta^2+\bar{\eta}^1\bar{\phi}\eta^2-
\bar{\eta}^2\bar{\phi}^*\eta^1\right)\Bigg)\log{\frac{\Lambda_{d=4}^2}{\mu_{d=4}^2}}
\end{eqnarray}
The scalars $A_5$ and $A_6$ are now protected by the six dimensional symmetries, as they should be.
\subsection{Dimensional regularization}
Were we to stick to dimensional regularization, we would have to
compare the four dimensional counterterm with the massless sector of the
six-dimensional one, which was previously determined in equation (\ref{sdcounter}).
There are then
two types of terms.
\par
First of all, those terms which have negative dimension constants in front, which
are precisely the ones not present in the original six-dimensional lagrangian, yield in four
dimensions counterterms with dimension six operators, suppressed by two powers
of the Kaluza-Klein scale:
\begin{eqnarray}
\lefteqn{\Delta S_{(1)}=\frac{e^2}{64\pi^3 M^2 \epsilon}\int d^4 x\left(-\frac{1}{12}e^2\left(\bar{\eta}\Sigma_{\mu\nu\rho}\eta\right)^2+\frac{19}{15}m\bar{\eta}\bar{D}_\mu\bar{D}^\mu\eta+\right.{}}\nonumber\\
&&\left.+\frac{2}{15}e\bar{\eta}\gamma_\nu\bar{D}_\mu\eta\bar{F}^{\mu\nu}-em\bar{\eta}\gamma^\mu\gamma^\nu\eta\bar{F}_{\mu\nu}-
\frac{11}{45}\left(\bar{D}_\lambda\bar{F}_{\mu\nu}\right)^2+
\frac{23}{9}(\bar{D}_\mu\bar{F}_{\mu\nu})^2+\right.\nonumber\\
&&\left.+\ldots\right)
\end{eqnarray}
Where the dots stand for terms with contractions of index in the extra dimensions and $e$ is the four-dimensional coupling. Then, there are the usual four-dimensional counterterms in the guise
\begin{equation}
\Delta S_{(2)}=-\frac{2 e^2 m^2}{64\pi^3 M^2\epsilon}\int d^4 x\left( \bar{\eta}
\bar{D\fsl}\eta+ 3 m\bar{\eta}\eta+\frac{2}{3}
\bar{F}_{\mu\nu}^2\right)
\end{equation}
The six-dimensional mass $ m^2$ can clearly be tuned so as to survive in the
limit in
which the Kaluza-Klein scale is pushed to infinity. We simply have to tune the dimensionless
quantity
\begin{equation}
\frac{e^2 m^2}{64\pi^3 M^2 \epsilon}
\end{equation}
towards the true four-dimensional $\frac{e^2}{16\pi^2 \epsilon}$, while keeping the six-dimensional
mass $m$
in their four-dimensional value. In such a way we recover {\em almost}
all four dimensional counterterms, albeit with a different sign, which could be accounted for
by changing the direction of the analytical continuation: $\epsilon_{d=6}=-\epsilon_{d=4}$.
\par
We say almost, because it can easily be seen from these results that there is no room
for the $|\phi|^2$ and $|\phi|^4$
counterterms, which appear when working upwards from four-dimensions, but do not appear
in the zero mode of the six-dimensional counterterm.
\par
The only (dim) hope is that these
four-dimensional counterterms
are actually cancelled when the full tower of Kaluza-Klein states is considered. The next
subsection is devoted to
disipate this lingering doubt.
\par
It seems indeed strange that no quartic interaction is generated when coming
from six dimensions.
No definite conclusions can be draw, however, because those effects are of order $O(\lambda^2)$,
where $\lambda$ is que quartic coupling constant, which means order $O(e^8)$ in our case. We have
no right to keep those terms.
\par
There is a very simple mapping from six-dimensional operators to four-dimensional ones, namely
\begin{equation}
{\cal O}_{(n)}\rightarrow {\cal O}_{(n-N)}
\end{equation}
where $N$ is the number of fields involved in the operator.
\par
In that way it is seen that the reduction works at follows:
\begin{eqnarray}
&&{\cal O}_{(5)}\rightarrow {\cal O}_{(3)}\nonumber\\
&&{\cal O}_{(6)}\rightarrow {\cal O}_{(4)}\nonumber\\
&&{\cal O}_{(7)}\rightarrow {\cal O}_{(5)}\nonumber\\
&&{\cal O}_{(8)}\rightarrow {\cal O}_{(6)}
\end{eqnarray}
except for
\begin{equation}
{\cal O}^2_{(8)}\rightarrow {\cal O}^2_{(5)}
\end{equation}
In four dimensions, all operators with dimension higher than four appear necessarily
with coefficients which get inverse powers of the compactification scale, $M$. We should be
then pretty confident of all results gotten in the limit in which this scale goes
to infinity.
\par
Another question is what happens in the chiral limit. If the mass of the fermion vanishes,
then the six-dimensional counterterms do {\em not} include the four-dimensional ones. If we
think about it, the conclusion is almost unavoidable, because there is no parameter
in the lagrangian with the dimension of mass. The inverse coupling constant does not
qualify for this, because it is never going to appear in a perturbative computation.
\section{The full tower of four-dimensional divergences}
Let us consider now the problem of the divergences of the four-dimensional
theory with the whole Kaluza-Klein tower. We intend to compute the counterterm
asociated with the full four-dimensional Lagrangian (\ref{tower}).
We let the index $n=(n_5,n_6)$ run over the whole tower of each field.
Notice that the bosonic fields are now complex (except the one corresponding to $n=0$).
$N$ is the complex mass number $N=\frac{n_6}{R_6}+i\frac{n_5}{R_5}$,
and also $L=\frac{l_6}{R_6}+i\frac{l_5}{R_5}$.
We have also defined $\bar{\phi}_n\equiv\bar{A}^n_5-i\bar{A}_6^n$ and $\bar{\phi}^*_n\equiv
\bar{A}^n_5+i\bar{A}_6^n\ne(\bar{\phi}_n)^*=\bar{A}^{-n}_5+i\bar{A}_6^{-n}$.
\par
As we have said the massive ($n\ne0$) modes are complex. In order to use the algorithm explained in the appendix we have to double this modes into real and imaginary parts. However it is also possible to do the calculatios with the complex fields and introduce at the end some extra factors in the adecuate terms. After squaring the matrices and performing the supertraces we get the following
counterterms in four dimensions with some labor
\begin{equation}
a_2=\int\frac{d^4 x}{(4\pi)^2}\sum_l\left(8m^2-4|L|^2-8e^2\sum_n\bar{\phi}_n^*\bar{\phi}_{-n}+8e\left(L^*\bar{\phi}_0-L\bar{\phi}^*_0\right)\right)
\end{equation}
The mode sum can be regularized and performed with the help of a zeta function. We shall do it
in the next section, when working out the reduction of $QED_4$ on a two-torus.
The fourth heat-kernel coefficient is \footnote{Here is the explicit expression
\begin{eqnarray}
\lefteqn{a_4=\int\frac{d^4 x}{(4\pi)^2}\sum_l\left(\left(-4m^4+2|L|^4-8m^2|L|^2\right)
+\frac{4}{3}e^2\sum_n\bar{F}_{\mu\nu}^n\bar{F}^{\mu\nu}_{-n}-4e^2\sum_n\bar{\phi}^*_n
\Box\bar{\phi}_{-n}-\right.{}}\nonumber\\
&& \left.-8e\left(m^2+|L|^2\right)\left(L\bar{\phi}_0^*-
L^*\bar{\phi}_0\right)+8e^2\sum_n\left(m^2+|L+N|^2+|N|^2\right)\bar{\phi}^*_n\bar{\phi}_{-n}-\right.\nonumber\\
&& \left.-
4e^2\sum_n\left(N^*+L^*\right)L^*\bar{\phi}_n\bar{\phi}_{-n}-4e^2\sum_n\left(N+L\right)L\bar{\phi}_n^*\bar{\phi}_{-n}^*+\right.\nonumber\\
&& +\left.
8e^3\sum_{m,n}\bar{\phi}^*_{m-l}\bar{\phi}_{l-n}\left( M\bar{\phi}^*_{n-m}-
N^*\bar{\phi}_{n-m}\right)+4e^2\sum_{m,n,s}\bar{\phi}_{m-l}^*\bar{\phi}_{l-s}\bar{\phi}^*_{s-n}\bar{\phi}_{n-m}-\right.\nonumber\\
&& \left.-4e^2\sum_nN^*\partial_\mu\bar{\phi}_n\bar{A}_{-n}^\mu+4e^2\sum_nN\partial_\mu
\bar{\phi}_n^*\bar{A}^\mu_{-n}+4e^2\sum_n|N|^2\bar{A}_\mu^n\bar{A}^\mu_{-n}+\right.\nonumber\\
&& \left.+8e^2\sum_{n\ne0}\left(\bar{\eta}^1_{l-n}
\partial\fsl\eta^1_{l-n}+\bar{\eta}^2_{l-n}\partial\fsl\eta^2_{l-n}\right)-
8e^3\sum_{m\ne0,n}\left(\bar{\eta}^1_{l-m}\bar{A\gsl }_{l-n}\eta^1_{n-m}+\bar{\eta}^2_{l-m}\bar{A\gsl }_{l-n}\eta^2_{n-m}\right)+\right.\nonumber\\
&& \left.+
24me^2\sum_{n\ne0}\left(\bar{\eta}^1_{l-n}\eta^1_{l-n}-\bar{\eta}^2_{l-n}\eta^2_{l-n}\right)+
16e^3\sum_{m\ne0,n}\bar{\eta}^2_{l-m}\bar{\phi}^*_{l-n}\eta^1_{n-m}-16e^3\sum_{m\ne0,n}\bar{\eta}^1_{l-m}
\bar{\phi}_{l-n}\eta^2_{n-m}+\right.\nonumber\\&&\left.+16e^2\sum_{n\ne0}L^*\bar{\eta}^2_{l-n}\eta^1_{l-n}+16e^2\sum_{n\ne0}
L\bar{\eta}^1_{l-n}\eta^2_{l-n}+4e^2\left(\bar{\eta}^1_l
\partial\fsl\eta^1_l+\bar{\eta}^2_l\partial\fsl\eta^2_l\right)-\right.\nonumber\\
&& \left.-4e^3\sum_n\left(\bar{\eta}^1_n\bar{A\gsl }_{n-l}\eta^1_l+\bar{\eta}^2_n\bar{A\gsl }_{n-l}\eta^2_l\right)+12me^2\left(\bar{\eta}^1_l\eta^1_l-\bar{\eta}^2_l\eta^2_l\right)+
8e^3\sum_{n}\bar{\eta}^2_n\bar{\phi}^*_{n-l}\eta^1_l-\right.\nonumber\\&&\left.-8e^3\sum_{n}\bar{\eta}^1_n
\bar{\phi}_{n-l}\eta^2_l+8e^2L^*\bar{\eta}^2_l\eta^1_l+8e^2L\bar{\eta}^1_l\eta^2_l
\right)
\end{eqnarray}
}
quite messy indeed. At least, one thing is clear: there is no way to perform
a clever resummation (like the one Duff and Toms did in the free case) in order to
cancel the four dimensional counterterms for both $|\phi|^2$ and $|\phi|^4$, for the simple
reason that there is no contribution of the massive fields to them. This fact was not
obvious {\em a priori} and the doubt about it was the main reason why
this computation was performed.
\section{The true four-dimensional renormalization}
From our point of view, in which the full theory is defined in six dimensions, the true
renormalization is the one that is obtained via an harmonic expansion of the
six-dimensional counterterm(s).
\subsection{The cutoff theory}
With the interpretation of the six-dimensional cutoff we have advocated, the four-dimensional
logarithmic divergences read
\begin{eqnarray}
\lefteqn{\Delta S_{\log}\equiv\int\frac{d^4 x}{(4\pi)^3}e^2
\sum_{n}\Bigg(4\left(\bar{\eta}^1_{n}\partial\fsl \eta^1_n+\bar{\eta}^2_{n}
\partial\fsl \eta^2_n+N\bar{\eta}^1_{n}\eta^2_n+N^*
\bar{\eta}^2_{n}\eta^1_n\right)+{}}\nonumber\\
&&+12m\left(\bar{\eta}^1_{n}\eta^1_n-\bar{\eta}^2_{n}\eta^2_n\right)
+\frac{4}{3}\left(\bar{F}^{\mu\nu}_{-n}\bar{F}_{\mu\nu}^n+2|N|^2\bar{A}_{-n}^\mu\bar{A}_\mu^n-4i\partial_\mu\bar{A}^\mu_{-n}\left(\frac{n_5}{R_5}\bar{A}_5^n+\frac{n_6}{R_6}\bar{A}_6^n\right)+\right.\nonumber\\&&\left.
+2\bar{A}_5^{-n}\left(-\Box+\frac{n_6^2}{R_6^2}\right)\bar{A}_5^n+2\bar{A}_6^{-n}\left(-\Box+\frac{n_5^2}{R_5^2}\right)\bar{A}_6^n-4\frac{n_5n_6}{R_5R_6}\bar{A}_5^{-n}\bar{A}_6^n\right)-
\nonumber\\
&&-4e\sum_m\left(\bar{\eta}^1_m\bar{A\gsl }_{m-n}\eta^1_n+
\bar{\eta}^2_m\bar{A\gsl }_{m-n}\eta^2_n+\bar{\eta}^1_m\bar{\phi}_{m-n}\eta^2_n-
\bar{\eta}^2_m\bar{\phi}^*_{m-n}\eta^1_n\right)\Bigg)\log{\frac{\Lambda_{d=4}^2}{\mu_{d=4}^2}}
\end{eqnarray}
In addition to that, there are the $\log^2$ divergences, coming from the quartic
divergences in six dimensions. Those are trivial in our case, because they do not
depend on the background fields.
\par
Finally, there are the $\log\,\log$ divergences, stemming from the logarithmic divergence
in six dimensions. This divergence is suppressed by the scale of compactification.
The result of a somewhat heavy computation, keeping terms up to
cubic order in the four-dimensional electric charge, is:
\begin{eqnarray}
\lefteqn{\Delta S_{\log\log}\equiv \int\frac{d^4 x}{(4\pi)^3}
\frac{e^2}{M^2}\sum_n\left(-m\bar{\eta}^1_n\left(\frac{19}{15}\left(-\Box+|N|^2\right)+
2m\partial\fsl+6m^2\right)\eta^1_n+\right.{}}\nonumber\\
& &\left.+m\bar{\eta}^2_n\left(\frac{19}{15}\left(-\Box+|N|^2\right)-2m\partial\fsl+6m^2\right)\eta^2_n-2m^2\left(N\bar{\eta}^1_n\eta^2_n+N^*\bar{\eta}^2_n\eta^1_n
\right)+\right.\nonumber\\
&& \left.+\frac{23}{9}\partial_\mu\bar{F}^{\mu\nu}_{-n}\left(\partial^\rho\bar{F}_{\rho\nu}^n-2|N|^2\bar{A}_\nu^n\right)-
\bar{F}_{\mu\nu}^{-n}\left(\frac{11}{45}\left(-\Box+|N|^2\right)+\frac{4}{3}m^2\right)\bar{F}^{\mu\nu}_n+\right.\nonumber\\
&&\left.+|N|^2\bar{A}^\mu_{-n}\left(\frac{31}{15}\left(-\Box+
|N|^2\right)-\frac{8}{3}m^2\right)\bar{A}_\mu^n-\right.\nonumber\\
&&\left.-i\partial_\mu\bar{A}^\mu_{-n}\left(\frac{62}{15}\left(
-\Box+|N|^2\right)-\frac{16}{3}m^2\right)\left(\frac{n_5}{R_5}\bar{A}_5^n+\frac{n_6}{R_6}\bar{A}_6^n\right)
+\right.\nonumber\\
&&\left.+\bar{A}_5^{-n}\left(\frac{31}{15}\left(-\Box+|N|^2\right)-\frac{8}{3}m^2\right)\left(-\Box+\frac{n_6^2}{R_6^2}
\right)\bar{A}_5^n+\right.\nonumber\\
&&\left.+\bar{A}_6^{-n}\left(\frac{31}{15}\left(-\Box+|N|^2\right)-\frac{8}{3}m^2\right)\left(-\Box+\frac{n_5^2}{R_5^2}
\right)\bar{A}_6^n-\right.\nonumber\\
&&\left.-\frac{n_5n_6}{R_5R_6}\bar{A}_5^{-n}\left(\frac{62}{15}\left(-\Box+|N|^2\right)-\frac{16}{3}m^2\right)\bar{A}_6^n
\right)\log\left(\frac{M^2}{\mu^2_{(d=6)}}\log{\frac{\Lambda^2_{(d=4)}}{\mu^2_{(d=4)}}}\right)+\nonumber\\
&&+O(e^3)
\end{eqnarray}
\subsection{Dimensional regularization}
In that case the true divergences only come from the sixth coefficient,
which yields the $\log\,\log$ divergences we just wrote down.
\par
This means that in addition to the already mentioned counterterms to the zero modes there
are a full tower of counterterms
involving six-dimensional operators.
\par
It is of interest to specialize to the massless case ($m=0$), in which, as we have
already noticed, no ordinary dimension four operator is recovered:
\begin{eqnarray}
a_6&=&\int\frac{d^4 x}{(4\pi)^3}
\frac{e^2}{M^2}\sum_n\left(\frac{23}{9}\partial_\mu\bar{F}^{\mu\nu}_{-n}\left(\partial^\rho\bar{F}_{\rho\nu}^n-2|N|^2\bar{A}_\nu^n\right)-\frac{11}{45}\bar{F}_{\mu\nu}^{-n}\left(-\Box+|N|^2\right)\bar{F}^{\mu\nu}_n+\right.\nonumber\\
&&\left.+\frac{31}{15}|N|^2\bar{A}^\mu_{-n}\left(-\Box+
|N|^2\right)\bar{A}_\mu^n-i\frac{62}{15}\partial_\mu\bar{A}^\mu_{-n}\left(
-\Box+|N|^2\right)\left(\frac{n_5}{R_5}\bar{A}_5^n+\frac{n_6}{R_6}\bar{A}_6^n\right)
+\right.\nonumber\\
&&\left.+\frac{31}{15}\bar{A}_5^{-n}\left(-\Box+|N|^2\right)\left(-\Box+\frac{n_6^2}{R_6^2}
\right)\bar{A}_5^n+\frac{31}{15}\bar{A}_6^{-n}\left(-\Box+|N|^2\right)\left(-\Box+\frac{n_5^2}{R_5^2}
\right)\bar{A}_6^n-\right.\nonumber\\
&&\left.-\frac{62}{15}\frac{n_5n_6}{R_5R_6}\bar{A}_5^{-n}\left(-\Box+|N|^2\right)\bar{A}_6^n
\right)+O(e^3)
\end{eqnarray}
That is, in the chiral case there is no renormalization of the fermionic tower (at this order) whatsoever,
which is {\em not} what happens from the four-dimensional point of view
of the previous paragraph.
\section{Another example: $QED_4$ on a two-torus}
Let us now repeat this exercise in a situation that, although
probably much less interesting from the physical point of view, is much better defined as
a quantum theory, namely $QED_4$ on a two-torus. The reduced theory is a two-dimensional one,
where all divergences are more or less trivial (essentially normal ordering).
It is nevertheless possible to analyze it with the very same general techniques.
\subsection{The four-dimensional viewpoint}
Let us then consider $QED_4$ on a manifold $R^2\times S^1\times S^1$. The action is
\begin{equation}
S=\int d^4 x\left(\frac{1}{4}F_{\mu\nu}^2+\bar{\psi}(D\fsl+m)\psi\right)
\end{equation}
where the abelian covariant derivative is simply:
\begin{equation}
D_\mu\psi\equiv\left(\partial_\mu-eA_\mu\right)\psi
\end{equation}
The theory is renormalizable. In dimensional renormalization the counterterm is the fourth
coefficient in the small-time heat kernel expansion:
\begin{eqnarray}\label{a4}
a_4&=&\int \frac{d^4x}{(4\pi)^{2}}\left(\frac{2}{3}e^2\bar{F}_{\mu\nu}^2+
2e^2\bar{\eta}\gamma^\mu\bar{D}_\mu\eta+8e^2 m\bar{\eta}\eta\right)\end{eqnarray}
In the cutoff theory, this is precisely the coefficient of the logaritrhmic divergence, but
there is a quadratic divergence as well:
\begin{equation}
\Delta S=\int d^4 x \left(b_2 \Lambda_{d=4}^2+b_4
\log\frac{\Lambda_{d=4}^2}{\mu_{d=4}^2}\right)
\end{equation}
where
\begin{equation}
a_2=\int \frac{d^4x}{(4\pi)^{2}}4m^2
\end{equation}
\subsection{The two-dimensional viewpoint}
In order to dimensionaly reduce the theory we consider the matrices $(a=1,2)$
\begin{eqnarray}
&&\gamma_a^{(4)}=\sigma_3\otimes\sigma_a\nonumber\\
&&\gamma_3^{(4)}=\sigma_1\otimes 1\nonumber\\
&&\gamma_4^{(4)}=\sigma_2\otimes1
\end{eqnarray}
In that way, four-dimensional spinors split in two two-dimensional ones:
\begin{equation}
\psi=\left(\begin{array}{c}
\psi_1\\
\psi_2
\end{array}\right)
\end{equation}
It is a simple matter to perform the integrals over the angular variables and obtain
the gauge fixed action (still exact) in the two-dimensional form:
\begin{eqnarray}\label{tower}
\lefteqn{S=\int d^2 x\sum_{n_3,n_4}\left(\bar{\psi}^1_{n}\partial\fsl \psi^1_n+\bar{\psi}^2_{n}
\partial\fsl \psi^2_n+\bar{\psi}^1_{n}(i\frac{n_3}{R_3}+\frac{n_4}{R_4})\psi^2_n-
\bar{\psi}^2_{n}(i\frac{n_3}{R_3}-\frac{n_4}{R_4})\psi^1_n+\right.{}}\nonumber\\
&&{}+m\left(\bar{\psi}^1_{n}\psi^1_n-\bar{\psi}^2_{n}\psi^2_n\right)
-\frac{1}{2}(A_a^n)^{*}\left(\Box-\frac{n_3^2}{R_3^2}-\frac{n_4^2}{R_4^2}\right) A^a_n-
\frac{1}{2}(A_3^n)^{*}\left(\Box-\frac{n_3^2}{R_3^2}-\frac{n_4^2}{R_4^2}\right)A_3^n-
\nonumber\\
&&\left.-\frac{1}{2}(A_4^n)^{*}\left(\Box-\frac{n_3^2}{R_3^2}-
\frac{n_4^2}{R_4^2}\right)A_4^n-e\sum_m\left(\bar{\psi}^1_mA\gsl _{m-n}\psi^1_n+
\bar{\psi}^2_mA\gsl _{m-n}\psi^2_n
+\right.\right.{}\nonumber\\&&{}\left.+\bar{\psi}^1_mA_3^{m-n}\psi^2_n-
\bar{\psi}^2_mA_3^{m-n}\psi^1_n-i\bar{\psi}^1_mA_4^{m-n}\psi^2_n-
i\bar{\psi}^2_mA_4^{m-n}\psi^1_n\right)\Bigg)
\end{eqnarray}
The two-dimensional coupling constant is
\begin{equation}
e\equiv \frac{e^{(4)}}{2\pi\sqrt{R_3 R_4}}\equiv e^{(4)} M
\end{equation}
In two dimensions, gauge fields are dimensionless and so are scalar fields. Fermionic fields
enjoy mass dimension $1/2$. We hope that there would arise no confusion
for the use of the same symbol $e$ for both
coupling constants.
The zero mode of this action is
\begin{eqnarray}
\lefteqn{S=\int d^2 x\left(\bar{\psi}^1\partial\fsl \psi^1+\bar{\psi}^2
\partial\fsl \psi^2+m\left(\bar{\psi}^1\psi^1-\bar{\psi}^2\psi^2\right)
-\frac{1}{2}A_a\Box A^a-\right.{}}\nonumber\\
&&\left.-\frac{1}{2}\phi^*\Box\phi-e\left(\bar{\psi}^1A\gsl \psi^1+\bar{\psi}^2A\gsl \psi^2
+\bar{\psi}^1\phi\psi^2-\bar{\psi}^2\phi^*\psi^1\right)\right)
\end{eqnarray}
where we have represented the zero modes of all fields by the same letter without any subindex:
\begin{equation}
A_3^0-iA_4^0\equiv\phi^0\equiv\phi
\end{equation}
If we define the theory by dimensional renormalization, the
counterterm associated to the above action is
\begin{equation}
\Delta S_{zero\,mode}=\frac{1}{\epsilon}a_2^{(0)}=\frac{1}{\epsilon}\int\frac{d^2 x}{4\pi}4
\left(m^2 - e^2 |\phi|^2\right)
\end{equation}
If instead we consider the whole tower the corresponding counterterm is given in terms
of the complex mass parameter:
\begin{equation}
L\equiv \frac{l_4}{R_4}+i\frac{l_3}{R_3}
\end{equation}
\begin{equation}
\Delta S_{tower}=\frac{1}{\epsilon}a_2=\frac{1}{\epsilon}\int\frac{d^2 x}{4\pi}
\sum_l4\left(m^2-|L|^2-e^2\sum_n\bar{\phi}_n^*\bar{\phi}_{-n}+e\left(L^*\bar{\phi}_0-
L\bar{\phi}^*_0\right)\right)\end{equation}
Here we have a sum of contributions from all higher modes. This is a divergent sum which needs
regularization. In the expression for the tadpole, for example, we are forced to compute
the sum
\begin{equation}
T(R)\equiv\sum_{n\in\mathbb{Z}}\frac{n}{R}\equiv \frac{1}{R}\sum_{n\in\mathbb{Z}}n
\end{equation}
This can be regularized, for example, (\cite{Polyakov}) by imposing a cutoff
\begin{eqnarray}
\sum_{n=1}n&\equiv& \lim_{\epsilon\rightarrow 0}\sum_{n=1}ne^{-\epsilon n}=\lim_{\epsilon\rightarrow 0}\sum_{n=1}
-\frac{\partial}{\partial \epsilon}e^{-\epsilon n}=-\lim_{\epsilon\rightarrow 0}\frac{\partial}{\partial \epsilon}\sum_{n=1}e^{-\epsilon n}
=\nonumber\\
&=&-\lim_{\epsilon\rightarrow 0}\frac{\partial}{\partial \epsilon}\frac{1}{e^{\epsilon}-1}=
\lim_{\epsilon\rightarrow 0}\frac{e^{\epsilon}}{(e^\epsilon-1)^2}=\lim_{\epsilon\rightarrow 0}\left(\frac{1}{\epsilon^2}-\frac{1}{12}\right)
\end{eqnarray}
This clearly shows the divergence of the sum. When adopting a finite prescription, it is
important to keep this in mind.
One such finite prescription, quite natural, stems from a consideration of the laplacian
operator
on the extra dimensions, $\Delta_y$, whose eigenvalues are precisely
\begin{equation}
\lambda_l\equiv |L|^2
\end{equation}
and the corresponding $\zeta$ function is
\begin{equation}
\zeta(s)\equiv \sum_{l\neq 0} \left(|L|^2\right)^{-s}
\end{equation}
which happens to be a particular instance of Epstein's zeta function.
This would lead to definite values for
\begin{equation}
\sum_l 1\equiv \zeta(s=0)+1=0
\end{equation}
and
\begin{equation}
\sum_l |L|^2\equiv \zeta(-1)=0
\end{equation}
In order to evaluate the coefficient of the tadpole, it is not possible to use this same
$\zeta$ function. One possibility is to use Riemann's $\zeta$ function
\begin{equation}
\zeta_R(s)\equiv\sum_{n=1} n^{-s}
\end{equation}
so that, for example,
\begin{equation}
T(R)=\frac{1}{R}(\zeta_R(-1)-\zeta_R(-1))=0
\end{equation}
Actually this is a unavoidable consequence of any definition in which the first of Hardy's
properties of the sum of a divergent series is satisfied, namely, if
$\sum a_n=S $ then $\sum \lambda a_n=\lambda S$ (cf.\cite{Hardy}, and the discussion in
\cite{Meessen})
It has to be acknowledged that the need to use two different zeta functions greatly diminishes
the attractiveness of this whole procedure of resummation.
\par
Ay any rate, in order to eliminate the tadpole, one would have in its case to shift the field:
\begin{equation}
\bar{\phi}_0\rightarrow\bar{\phi}_0-\frac{T}{2 e}
\end{equation}
This shift would in turn affect the fermionic masses through the Yukawa couplings and
convey
another contribution to the fermion mass renormalization.
\par
When either theory is defined through a proper time cutoff, the counterterm is given precisely
by
\begin{equation}
\Delta S=a_2\log\frac{\Lambda_{d=2}^2}{\mu_{d=2}^2}
\end{equation}
\subsection{The limitations of the two-dimensional approach.}
Let us first concentrate upon dimensional renormalization. The mode expansion of the
four-dimensional counterterm (\ref{a4}) is:
\begin{eqnarray}
\lefteqn{a_4=\int\frac{d^2 x}{(4\pi)^2}\frac{e^2}{M^2}
\sum_{n}\Bigg(2\left(\bar{\eta}^1_{n}\partial\fsl \eta^1_n+\bar{\eta}^2_{n}
\partial\fsl \eta^2_n+N\bar{\eta}^1_{n}\eta^2_n+N^*
\bar{\eta}^2_{n}\eta^1_n\right)+{}}\nonumber\\
&&+8m\left(\bar{\eta}^1_{n}\eta^1_n-\bar{\eta}^2_{n}\eta^2_n\right)
+\frac{2}{3}\left(\bar{F}^{ab}_{-n}\bar{F}_{ab}^n+2|N|^2\bar{A}_{-n}^a\bar{A}_a^n-
4i\partial_a\bar{A}^a_{-n}\left(\frac{n_3}{R_3}\bar{A}_3^n+\frac{n_4}{R_4}\bar{A}_4^n\right)
+\right.\nonumber\\&&\left.
+2\bar{A}_3^{-n}\left(-\Box+\frac{n_4^2}{R_4^2}\right)\bar{A}_3^n+2\bar{A}_4^{-n}\left(-\Box+
\frac{n_3^2}{R_3^2}\right)\bar{A}_4^n-4\frac{n_3n_4}{R_3R_4}\bar{A}_3^{-n}\bar{A}_4^n\right)-
\nonumber\\
&&-2e\sum_m\left(\bar{\eta}^1_m\bar{A\gsl }_{m-n}\eta^1_n+
\bar{\eta}^2_m\bar{A\gsl }_{m-n}\eta^2_n+\bar{\eta}^1_m\bar{\phi}_{m-n}\eta^2_n-
\bar{\eta}^2_m\bar{\phi}^*_{m-n}\eta^1_n\right)\Bigg)
\end{eqnarray}
Which has a zero mode
\begin{eqnarray}
a_4^{(0)}&=&\int\frac{d^2 x}{(4\pi)^2}\frac{e^2}{M^2}\Bigg(2\left(\bar{\eta}^1\partial\fsl
\eta^1+\bar{\eta}^2
\partial\fsl \eta^2\right)+8m\left(\bar{\eta}^1\eta^1-\bar{\eta}^2\eta^2\right)+
\frac{2}{3}\bar{F}^{ab}\bar{F}_{ab}-\frac{4}{3}\bar{\phi}^*\Box\bar{\phi}-\nonumber\\
&&-2e\left(\bar{\eta}^1\bar{A\gsl }\eta^1+\bar{\eta}^2\bar{A\gsl }\eta^2
+\bar{\eta}^1\bar{\phi}\eta^2-\bar{\eta}^2\bar{\phi}^*\eta^1\right)\Bigg)
\end{eqnarray}
In that case, it is plain that there are many differences between the detailed forms of
the mode expansion
of the renormalized four dimensional theory and the renormalization of the two-dimensional
mode expansion of the bare four-dimensional theory.
In the cutoff theory we could be tempted to identify
\begin{equation}
\frac{\Lambda^2_{d=4}}{M^2}\equiv\log\frac{\Lambda^2_{d=2}}{\mu_{d=2}^2}
\end{equation}
If one is willing to do this, there are two things that happen. First of all, one never
recovers the two dimensional correction to the mass of the scalar field,
\begin{equation}
e^2|\phi|^2
\end{equation}
The reason is exactly the same as it was when reducing from six to four dimensions in our
previous paper, namely, the
spontaneously nature of the breaking of Lorentz symmetry of the mother theory:
\begin{equation}
O(1,3)\rightarrow O(1,1)\times O(2)\times O(2)
\end{equation}
It is true that this correction vanishes when one considers the full tower and one is willing
to regularize the sum using the zeta funcion approach. As we have pointed out, there is
an implicit renormalization of the scalar mass involved in this regularization. It is
nevertheless true that one can regularize the sum in such a way as to get essentially the same
result for the dominant (logarithmic) divergence in both the mother and the daughter theories..
\par
\par
The second thing that happens, and this seems unavoidable, is that there are
$\log\,\log\,\Lambda^2$ divergences coming from
the $a_4$ four-dimensional counterterm, suppressed by appropiate powers of the Kaluza-Klein
scale.
\par
To conclude, even in this example,
the two-dimensional theory never forgets its mother. This exercise fully supports the
general conclusions of our previous reduction.
\section{Conclusions}
Two radically different ways to define $QED_6$ at a one-loop level have been explored.
The lessons of this exercise seem
to be as follows.
\par
When the fundamental theory is defined
in dimension higher than four using dimensional regularization,
the divergences of the four dimensional theory do not match
the ones of the extra-dimensional (mother) one. This is true even in the zero volume limit,
when the volume of the extra dimensions is shrunk to zero, and the Kaluza-Klein scale
correspondingly goes to infinity, and this is also true even when the full Kaluza-Klein
tower is taken into account, as we have shown in detail in an explicit six-dimensional
example.
\par
In other words, the theory never forgets its higher dimensional origin. This is most clearly
seen in the chiral limit, but appears also in the massive case, with the need of taking into
account counterterms involving higher dimensional operators, whose coefficients can be computed
in an unambiguous and straightforward way. We understand that a need for those counterterms
has been hinted at in \cite{Oliver} and \cite{Ghilencea}.
\par
The full set of four-dimensional counterterms can be easily recovered from the six-dimensional
one by performing an harmonic expansion. This yields what is, in our opinion, {\em the} correct
way of renormalizing Kaluza-Klein theories.
\par
\par
In the massless case (as well as when coming from an odd number
of spacetime dimensions) the four-dimensional counterterms are simply not contained in the
higher dimensional ones. The appropiate procedure in those cases would be, from our point of
view, to compute in the mother theory (in which finite results are obtained through
the use of dimensional regularization), and then perform the mode expansion.
\par
Alternatively, when the quantum theory is defined through a proper time cutoff, we recover the four dimensional
logarithmic divergences via a tuning of the six-dimensional cutoff. There are then
calculable $\log\,\log\,\Lambda^2$ divergences coming from the six-dimensional
logarithmic divergences
as a reminder of the sicknesses of the mother theory. Those divergences are, however,
suppressed by appropiate powers of the compactification scale, which means that they are multiplied by a small coefficient
at energies at which six-dimensional perturbation theory is reliable (essentially $E/M<<\alpha_{d=4}^{-1}$).
\par
In neither case do we find from six dimensions corrections to the potential energy
of the four-dimensional singlet scalars associated to the zero modes of the
extra-dimensional legs of the gauge field. This being true for the zero mode, is clearly a low
energy effect, well within
the range of validity of the one-loop six-dimensional calculation .
Those corrections are found in four dimensions because there is no gauge symmetry
to prevent that to happen.
\par
We have repeated the analysis for $QED_4$ on a two-torus, getting
similar results. This is very important, because there is now no ambiguity as to how to define the extra-dimensional theory.
This shows, in our opinion, that our main results do not stem from the ambiguities inherent in any
practical approach to a non-renormalizable theory.
\par
\par
There are no special difficulties with either odd-dimensional spaces (cf. for example
\cite{Fradkin}) or massless fermions
from the viewpoint of the cutoff theory. Let us finally stress that the strictest equivalence
{\em does work} for free theories coupled to the gravitational field, so that all the
effects we have studied here are due
to the interaction.
\par
Our results have obvious applications to the study of the range of validity of
the low energy effective four-dimensional models when studying Kaluza-Klein theories (cf.
for example
\cite{Hatanaka})
because our framework is consistent by construction (that is, to the extent that the six-dimensional model is consistent).
\par
Although a very simple abelian model has been studied in this paper
as an example, we do not expect our main
results to change in more complicated (non abelian) situations.
Besides obvious extensions, like supersymmetry and chiral fermions, it would be interesting
to study the effects of a nontrivial gravitational field, as well as
the physics of codimension one terms in the action (like the presence of branes in it).
\par
A most interesting related issue is how the full mother theory compares with the
ultraviolet completion as implied by deconstruction (cf. for example \cite{Pokorski}).
\par
Work
is in progress in several of these matters.
\par
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,310 |
\section*{Introduction}
\addcontentsline{toc}{section}{Introduction}
\medskip
Pour chaque nombre premier $\ell$, il est avancé dans \cite{J63} la conjecture générale suivante:
\begin{CoCy}
Soient $\ell$ un nombre premier arbitraire; $K$ un corps de nombres; $K_{\si{\infty}}=\bigcup_{n\in\NN}K_n$ sa $\Zl$-extension cyclotomique; $\Lambda$ l'algèbre d'Iwasawa de $\Gamma=\Gal(K_{\si{\infty}}/K)=\gamma^{\Zl}$; et $Pl^\ell_{\si{K}}=S_{\si{K}} \sqcup T_{\si{K}}$ une partition de l'ensemble $Pl^\ell_{\si{K}}$ des places de $K$ au-dessus de $\ell$.\par
Alors le polynôme caractéristique du $\Lambda$-module $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})=\Gal(H^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})/K_{\si{\infty}})$ attaché à la pro-$\ell$-extension abélienne maximale $H^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})$ de $K_{\si{\infty}}$ qui est $S_{\si{K}}$-décomposée et $T_{\si{K}}$-ramifiée n'est pas divisible par $\omega=\gamma-1$.
Autrement dit, son sous-module des points fixes $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^\Gamma$ est fini:\smallskip
\centerline{$\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^\Gamma \sim 1$.}
\end{CoCy}
Cette conjecture cyclotomique contient celles de Leopoldt et de Gross-Kuz'min (cf. e.g. \cite{J55}), qui correspondent respectivement aux cas $(S_{\si{K}},T_{\si{K}})=(\emptyset,Pl_{\si{K}}^\ell)$ et $(S_{\si{K}},T_{\si{K}})=(Pl_{\si{K}}^\ell,\emptyset)$ et réciproquement: si $K$ est totalement réel, ou encore si $K$ est un corps à conjugaison complexe extension quadratique totalement imaginaire d'un sous-corps totalement réel, la conjecture cyclotomique ci-dessus est vraie dès lors que les ensembles $S_{\si{K}}$ et $T_{\si{K}}$ sont stables par conjugaison complexe et que $K$ vérifie à la fois la conjecture de Leopoldt et celle de Gross-Kuz'min (\cite{J63}, Th. 5). En résumé, elle est alors équivalente à la conjonction des conjectures de Leopoldt et de Gross-Kuz'min.
Pour prendre en compte cette restriction sur la stabilité par conjugaison des ensembles $S_{\si{K}}$ et $T_{\si{K}}$, restée ambiguë dans \cite{J55}, introduisons la forme affaiblie ci-après de la conjecture:
\begin{Def}[\bf Conjecture cyclotomique faible]
Nous disons qu'un corps de nombres $K$ à conjugaison complexe (i.e. extension quadratique totalement imaginaire d'un sous-corps totalement réel $K^+$) satisfait la conjecture cyclotomique faible pour un nombre premier $\ell$, lorsqu'il vérifie la conjecture cyclotomique pour toute partition $Pl^\ell_{\si{K}}=S_{\si{K}} \sqcup T_{\si{K}}$ stable par la conjugaison complexe.
\end{Def}
Avec ces conventions le Théorème 5 de \cite{J55} affirme alors qu'un tel $K$ vérifie la Conjecture cyclotomique {\em faible} pour $\ell$ si et seulement s'il vérifie simultanément la conjecture de Leopoldt et celle de Gross-Kuz'min pour ce même $\ell$; ce qui est toujours le cas pour $K$ abélien.
\medskip
Le but de la présente note est de déterminer le $\Zl$-rang du module des points fixes $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_\%)^\Gamma$,
sous l'une ou l'autre des conjectures cyclotomiques ci-dessus, pour tout couple $(S_{\si{K}},T_{\si{K}})$ d'ensembles finis disjoints de places de $K$, lors même que la condition $Pl_{\si{K}}^\ell\subset S_{\si{K}}\cup T_{\si{K}}$ n'est pas satisfaite, et d'en tirer quelques conséquences sur les modules d'Iwasawa $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_\%)$ pour $K$ à conjugaison complexe en liaison avec les résultats de \cite{J43,JMa,JMP}. Nous procédons pour cela en trois temps:\smallskip
-- Dans le cadre de la conjecture faible d'abord, qui permet de traiter le cas où $S_{\si{K}}$ et $T_{\si{K}}$ sont stables par la conjugaison complexe.
La formule de rang que nous obtenons (Th. \ref{TP1}), qui est donc vérifiée dans le cas abélien, peut être regardée comme une généralisation de la classique conjecture de Coates et Lichtenbaum démontrée par Greenberg dans ce même contexte (cf. \cite{CL,Grb1}).\smallskip
-- Dans le cadre de la conjecture forte ensuite qui ouvre sur le cas plus général où n'est pas fait d'hypothèse de stabilité. Pour cela, nous commençons par tirer quelques conséquences algébriques de la Conjecture cyclotomique généralisant le résultat de semi-simplicité de Greenberg (cf. \cite{Grb1}) qui prouve que, pour un corps abélien $K$, le polynôme minimal du $\Lambda$-module $\Gal(H_{Pl_{\si{K}}^\ell}(K_\%)/K_{\si{\infty}})$ de la plus grande pro-$\ell$-extension abélienne non ramifiée et $\ell$-décomposée $H_{Pl_{\si{K}}^\ell}(K_\%)$ de $K_{\si{\infty}}$, n'est pas divisible par $(\gamma-1)$. Il en résulte une formule de rang qui étend la précédente (Th. \ref{TP2}). \smallskip
-- Cela fait, nous abordons la comparaison des formes faible et forte de la conjecture et montrons qu'elles sont en fait équivalentes (Th. \ref{TP3}).
L'idée centrale est la suivante: on s'intéresse à un $\Zl$-module qui n'est pas {\em a priori} stable par conjugaison complexe, mais qui est canoniquement l'image d'un autre module stable par conjugaison, dont on peut donc définir les composantes réelle et imaginaire. Sous la conjecture de Leopoldt, il se trouve que sa composante réelle est pseudo-nulle. Quotientant donc le module de départ par l'image de celle-ci, on obtient un module pseudo-isomorphe au module de départ lequel est, lui, purement imaginaire.\smallskip
Enfin ans la dernière partie de cette note nous abordons directement le calcul du $\Zl$-rang à l'aide de la théorie $\ell$-adique du corps de classes. La preuve alternative particulièrement concise que nous en donnons (Cor. \ref{CF}) redonne immédiatement l'équivalence étable plus haut.
\smallskip
Signalons pour finir qu'au cours de l'élaboration de ce résultat nous avons eu connaissance d'un travail indépendant de Lee et Yu \cite{LY} complétant l'étude antérieure de Lee et Seo \cite{LS} sur une forme équivalente de la conjecture cyclotomique. Avec celles données dans la présente note, on dispose de ce fait de trois façons différentes de déterminer le $\Zl$-rang du module $\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^\Gamma$, en présence d'une conjugaison complexe sous les conjectures équivalentes ci-dessus.\medskip
\Remarque Comme expliqué dans \cite{J63}, Scolie 8, les places étrangères à $\ell$ étant sans incidence sur le $\Zl$-rang de $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_\%)^\Gamma$, il est toujours possible de supposer $S_{\si{K}} \cup T_{\si{K}} \subset Pl_{\si{K}}^\ell$ dans les démonstrations, sans restreindre aucunement la généralité.
\newpage
\section{Théorème Principal sous la conjecture faible}
Supposons maintenant que $K$ soit une extension quadratique totalement imaginaire d'un sous-corps totalement réel $K^+$. Notons $\tau$ la conjugaison complexe et $\Delta=\Gal(K/K^+)=\{1,\tau\}$.\smallskip
Rappelons que, pour $\ell$ impair, chaque $\Zl[\Delta]$-module $M$ s'écrit comme somme directe de ses composantes réelle $M^+=M^{e_{\si{+}}}$ et imaginaire $M^-=M^{e_{\si{-}}}$ via les idempotents $e_\pm= \frac{1}{2}(1 \pm \tau)$. Et, si $\ell$ vaut $2$ et si $M$ est $\Z2$-noethérien, confondre le noyau de $(1\pm\tau)$ avec l'image de $(1\mp\tau)$ donne lieu à une erreur finie.
On peut donc dans tous les cas définir composantes réelle et imaginaire de $M$ comme image et noyau respectifs de $(1+\tau)$ et écrire à un fini près: $M \sim M^+\oplus M^-$.
Pour chaque ensemble de places $S$ de $K^+$, notons de même $S^-$ le sous-ensemble de celles qui sont décomposées par la conjugaison complexe et $S^+$ son complémentaire.\smallskip
Avec ces conventions, sous la conjecture faible le {\em Théorème principal} s'énonce comme suit:
\begin{Th}\label{TP1}
Soient $\ell$ un nombre premier; $K$ une extension quadratique totalement imaginaire d'un sous-corps $K^+$ totalement réel; $S_{\si{K^+}}$ et $T_{\si{K^+}}$ deux ensembles finis disjoints de places de $K^+$; $R_{\si{K^+}}=Pl_{\si{K^+}}^\ell\setminus (S_{\si{K^+}}\cup T_{\si{K^+}})$ l'ensemble des places au-dessus de $\ell$ qui ne sont ni dans $S_{\si{K^+}}$ ni dans $T_{\si{K^+}}$.\par
Soient enfin $\,\C^T_S(K_{\si{\infty}})=\Gal(H^T_S(K_{\si{\infty}})/K_{\si{\infty}})$ le groupe de Galois attaché à la pro-$\ell$-extension abélienne maximale $H^T_S(K_{\si{\infty}})$ de $K_{\si{\infty}}$ qui est $S$-décomposée et $T$-ramifiée; et $\,\C^T_S(K_{\si{\infty}})^\Gamma$ le sous-module de $\,\C^T_S(K_{\si{\infty}})$ fixé par $\Gamma=\Gal(K_\%/K)$.
Sous la conjecture cyclotomique, il vient alors:\smallskip
\begin{itemize}
\item[(i)] La composante réelle du groupe $\,\C^T_S(K_{\si{\infty}})^\Gamma$ est finie: $\big(\C^T_S(K_{\si{\infty}})^\Gamma\big)^+\sim\, \C^T_S(K^+_{\si{\infty}})^\Gamma \sim 1$.
\item[(ii)] Sa composante imaginaire est un $\Zl$-module de rang: $\rg_{\Zl}\big(\C^T_S(K_{\si{\infty}})^\Gamma\big)^-=|R^-|$,
où $|R^-|$ désigne le nombre de places de $R$ qui sont décomposées par la conjugaison complexe.
\end{itemize}
\end{Th}
\Preuve Comme rappelé plus haut, nous pouvons supposer, sans perte de généralité, que $R \sqcup S \sqcup T$ et une partition de $Pl^\ell$. De plus, côté réel, nous avons immédiatement: $\big(\C^T_S(K_{\si{\infty}})^\Gamma\big)^+\sim\, \C^T_S(K^+_{\si{\infty}})^\Gamma$, puisque l'opérateur $1+\tau$ correspond à la norme $N_{\si{K/K^+}}$.\smallskip
Considérons alors le quotient des genres $^\Gamma\C^T_S(K^+_{\si{\infty}})=\Gal(H^T_S(K^+_\%/K^+)/K_\%^+)$, où $H^T_S(K^+_\%/K^+)$ désigne la plus grande sous-extension de $H^T_S(K^+_\%)$ qui est abélienne sur $K^+$. Par construction, $H^T_S(K^+_\%/K^+)$ est $\ell$-ramifiée sur $K_\%^+$, donc sur $K^+$ et, par conséquent, de degré fini sur $K_\%^+$ sous la conjecture de Leopoldt (qui est vérifiée ici, puisque la conjecture cyclotomique est supposée l'être).
En résumé, il vient: $^\Gamma\C^T_S(K^+_{\si{\infty}}) \sim 1$; donc, a fortiori: $\,\C^T_S(K^+_{\si{\infty}})^\Gamma \sim 1$.
\smallskip
Regardons maintenant la composante imaginaire. D'après la Proposition 2 de \cite{J63} appliquée aux étages finis $K_n/K$ de la $\Zl$-extension cyclotomique de $K$, l'isomorphisme de modules galoisiens\smallskip
\centerline{$\Cl^T_S(K_n)^\Gamma/cl^T_S\big(\D^T_S(K_n)^\Gamma\big) \simeq \big(\E^T_S(K)\cap N_{\si{K_n/K}}(\R(K_n))\big)/N_{\si{K_n/K}}(\E_S^T(K_n))$}\smallskip
\noindent identifie le quotient du pro-$\ell$-groupe des $S$-classes $T$-infinitésimales ambiges de $K_n$ par le sous-groupe des classes des $S$-diviseurs $T$-infinitésimaux ambiges à un certain quotient du groupe des $S$-unités $T$-infinitésimales; ce qui donne, par passage à la limite projective pour la norme:\smallskip
\centerline{$\C^T_S(K_\%)^\Gamma/\varprojlim cl^T_S\big(\D^T_S(K_n)^\Gamma\big) \simeq \big(\E^T_S(K)\cap \wE(K)\big)/\big(\bigcap_{n\in\NN}N_{\si{K_n/K}}(\E_S^T(K_n)\big)$.}\smallskip
Le point essentiel ici est que, sous la conjecture de Gross-Kuz'min (donc ici encore sous la conjecture cyclotomique), la composante imaginaire du groupe $\,\wE(K)=\bigcap_{n\in\NN} N_{\si{K_n/K}}(\R(K_n))$ des unités logarithmiques se réduit au $\ell$-sous-groupe des racines de l'unité (cf. e.g. \cite{J28}). Il suit:\smallskip
\centerline{$\big(\C^T_S(K_\%)^\Gamma\big)^-\!\sim \varprojlim\, cl^T_S \big(\D^T_S(K_n)^\Gamma\big)^-$.}\smallskip
\noindent Or, le pro-$\ell$-groupe $\big(\D^T_S(K_n)^\Gamma\big)^-$ des $S$-diviseurs étrangers à $T$, qui sont ambiges et imaginaires, est engendré par le sous-groupe $\D^T_S(K)^-$ étendu de $K$ (donc sans incidence sur la limite projective) et le sous-groupe $\D^T_S(K_n)^{[R]-}$ construit sur les produits $\a_n(\p)=\prod_{\p_n\mid \p}\p_n$, pour $\p\in R^-$, lesquels satisfont les identités normiques $N_{\si{K_m/K_n}} (\a_m(\p))=\a_n(\p)$, pour $m\ge n\gg 0$.
Son sous-groupe principal $\P^T_S(K_n)^{[R]-}$ étant ultimement constant, puisque les $S$-unités imaginaires $T$-infinitésimales des $K_n$ proviennent d'un $K_{n_{\si{0}}}$ aux racines de l'unité près, il vient bien finalement:\smallskip
\centerline{$\big(\C^T_S(K_\%)^\Gamma\big)^-\!\sim \varprojlim\, cl^T_S(\D^T_S(K_n)^{[R]-}) \sim \varprojlim\, \D^T_S(K_n)^{[R]-} \simeq \Zl^{|R^-|}$.}
\newpage
\section{Semi-simplicité des modules d'Iwasawa $\,\C^T_S(K_\infty)$}
Avant d'énoncer le Théorème Principal dans le cadre plus général de la conjecture forte, commençons par préciser quelques conséquences algébriques de celle-ci.\smallskip
Notons $\gamma$ un générateur topologique du groupe de galois $\Gamma=\Gal(K_\%/K)$ et $\Lambda=\Zl[[\gamma-1]]$ l'algèbre d'Iwasawa attachée à $\Gamma$. Le point essentiel est que les facteurs cyclotomiques du polynôme minimal $\Pi^{T_{\si{K}}}_{S_{\si{K}}}(\gamma-1)$ du sous-module de $\Lambda$-torsion $\,\T^{T_{\si{K}}}_{S_{\si{K}}}(K_\%)$ de $\C^{T_{\si{K}}}_{S_{\si{K}}}(K_\%)$ sont de multiplicité 1:
\begin{Prop}\label{Prop}
Etant donné un corps de nombres $K$ qui satisfait la conjecture cyclotomique forte pour $\ell$, soient $K_\%=\bigcup K_n$ sa $\Zl$-extension cyclotomique; $S_{\si{K}}$ et $T_{\si{K}}$ deux ensembles finis disjoints quelconques de places finies de $K$; puis $R_{\si{K}}= Pl_{\si{K}}^\ell\setminus(S_{\si{K}} \cup T_{\si{K}})$ le sous-ensemble des places de $K$ au-dessus de $\ell$ qui ne sont ni dans $S_{\si{K}}$ ni dans $T_{\si{K}}$.
Soient enfin $\,\C^{T_{\si{K}}}_{S_{\si{K}}}=\C^{T_{\si{K}}}_{S_{\si{K}}}(K_\%)$ le groupe de Galois $\Gal(H^{T_{\si{K}}}_{S_{\si{K}}}(K_\%)/K_\%)$ de la pro-$\ell$-extension abélienne $S_{\si{K}}$-décomposée $T_{\si{K}}$-ramifiée maximale de $K_\%$ et $\,\C^{T_{\si{K}}}_{S_{\si{K}}}[R_{\si{K}}]$ engendré par les sous-groupes de décomposition des places de $R_{\si{K}}$.
\smallskip
\begin{itemize}
\item[(i)] Le sous-module de $\,\C^{T_{\si{K}}}_{S_{\si{K}}}$ fixé par $\Gamma=\Gal(K_\%/K)$ contient $\,\C^{T_{\si{K}}}_{S_{\si{K}}}[R_{\si{K}}]^\Gamma$ avec un indice fini:\smallskip
\centerline{$\C^{T_{\si{K}}}_{S_{\si{K}}}[R_{\si{K}}]^\Gamma \sim \;(\C^{T_{\si{K}}}_{S_{\si{K}}})^\Gamma$.}\smallskip
\item[(ii)] Et celui du quotient $\,\C^{T_{\si{K}}}_{R_{\si{K}}S_{\si{K}}}=\C^{T_{\si{K}}}_{S_{\si{K}}}/\C^{T_{\si{K}}}_{S_{\si{K}}}[R_{\si{K}}]$ est fini: $(\C^{T_{\si{K}}}_{R_{\si{K}}S_{\si{K}}})^\Gamma\sim 1$.
\end{itemize}\smallskip
\noindent En particulier, le polynôme minimal $\Pi^{T_{\si{K}}}_{S_{\si{K}}}(\gamma-1)$ du sous-module de $\Lambda$-torsion $\,\T^{T_{\si{K}}}_{S_{\si{K}}}$ de $\,\C^{T_{\si{K}}}_{S_{\si{K}}}$ n'est pas divisible par $\Phi_1(\gamma)^2=(\gamma-1)^2$. Plus généralement, sous la conjecture cyclotomique dans $K_\%$, le polynôme $\Pi^{T_{\si{K}}}_{S_{\si{K}}}(\gamma-1)$ n'est divisible par aucun carré de la forme $\Phi_{\ell^n}(\gamma)^2=((\gamma^{\ell^n}-1)/(\gamma^{\ell^{n-\si{1}}}-1))^2$.
\end{Prop}
\Preuve Prenant les points fixes par $\Gamma$ dans la suite exacte courte qui définit $\,\C^{T_{\si{K}}}_{S_{\si{K}}}[R_{\si{K}}]$, \smallskip
\centerline{$1 \to \C^{T_{\si{K}}}_{S_{\si{K}}}[R_{\si{K}}] \to \C^{T_{\si{K}}}_{S_{\si{K}}} \to \C^{T_{\si{K}}}_{R_{\si{K}}S_{\si{K}}} \to 1$,}
\noindent nous obtenons la suite:
\centerline{$1 \to( \C^{T_{\si{K}}}_{S_{\si{K}}}[R_{\si{K}}])^\Gamma \to (\C^{T_{\si{K}}}_{S_{\si{K}}})^\Gamma \to (\C^{T_{\si{K}}}_{R_{\si{K}}S_{\si{K}}})^\Gamma$;}\smallskip
\noindent et la conjecture cyclotomique appliquée avec $R_{\si{K}}\cup S_{\si{K}}$ et $T_{\si{K}}$ nous donne: $(\C^{T_{\si{K}}}_{R_{\si{K}}S_{\si{K}}})^\Gamma \sim 1$.
Les deux assertions $(i)$ et $(ii)$ en résultent immédiatement.\smallskip
Il suit de là que $(\gamma-1)$ et $(\gamma-1)^2$ ont même noyau dans $\,\T^{T_{\si{K}}}_{S_{\si{K}}}$, i.e. que $\Phi_1(\gamma)=\gamma-1$ apparait dans $\Pi^{T_{\si{K}}}_{S_{\si{K}}}(\gamma-1)$ avec une multiplicité au plus 1; et, sous la conjecture cyclotomique dans $K_n$, qu'il en va de même des $\Phi_{\ell^m}(\gamma)$ pour $m\le n$.
\begin{Cor}
Sous la conjecture forte le sous-module des points fixes $(\C^{T_{\si{K}}}_{S_{\si{K}}})^\Gamma=(\T^{T_{\si{K}}}_{S_{\si{K}}})^\Gamma$ de $\,\C^{T_{\si{K}}}_{S_{\si{K}}}$ est un pseudo-facteur direct du sous-module de $\Lambda$-torsion $\,\T^{T_{\si{K}}}_{S_{\si{K}}}$ de $\,\C^{T_{\si{K}}}_{S_{\si{K}}}$.
\end{Cor}
\Preuve Le quotient $\bar\Pi^{T_{\si{K}}}_{S_{\si{K}}}(\gamma-1)=\Pi^{T_{\si{K}}}_{S_{\si{K}}}(\gamma-1)/(\gamma-1)$ étant étranger à $\gamma-1$ dans l'anneau $\Ql[\gamma-1]$, le produit de leurs noyaux respectifs dans $\,\C^{T_{\si{K}}}_{S_{\si{K}}}$ est d'indice fini dans $\,\T^{T_{\si{K}}}_{S_{\si{K}}}$ et pseudo-direct (en ce sens que leur intersection $\Ker \bar\Pi^T_S(\gamma-1) \cap\Ker (\gamma-1)$ est finie): $\Ker \bar\Pi^{T_{\si{K}}}_{S_{\si{K}}}(\gamma-1) \times \Ker (\gamma-1) \sim \,\T^T_S$.
\begin{Cor}\label{C}
Soient $S'_{\si{K}}\subset S_{\si{K}}$ et $T'_{\si{K}}\supset T_{\si{K}}$ deux autres ensembles finis disjoints de places de $K$.
Sous la conjecture cyclotomique forte, si $\,\C^{T_{\si{K}}}_{S_{\si{K}}}$ et $\,\C^{T'_{\si{K}}}_{S'_{\si{K}}}$ ont même $\Lambda$-rang, la surjection canonique $f$ de $\,\C^{T'_{\si{K}}}_{S'_{\si{K}}}$ sur $\,\C^{T_{\si{K}}}_{S_{\si{K}}}$ induit un pseudo-épimorphisme de $(\C_{S'_{\si{K}}}^{T'_{\si{K}}})^\Gamma $vers $(\C^{T_{\si{K}}}_{S_{\si{K}}})^\Gamma$.
\end{Cor}
\Preuve Notons $\,\T^T_S$ le sous-module de $\Lambda$-torsion de $\,\C^T_S$ et $\,\T^{T'}_{S'}$ celui de $\,\C^{T'}_{S'}$. La surjection canonique $f$ envoie $(\C^{T'}_{S'})^\Gamma=(\T^{T'}_{S'})^\Gamma$ vers $(\C^{T}_{S})^\Gamma=(\T^{T}_{S})^\Gamma$.
Et l'identité des $\Lambda$-rangs $\rg_\Lambda\C^{T'}_{S'}=\rg_\Lambda\C^{T}_{S}$ nous assure que $f$ envoie $\,\T^{T'}_{S'}$ {\em sur} $\,\T^{T}_{S}$.
En particulier, le polynôme minimal $\Pi(\gamma-1)$ de $\,\T^{T}_{S}$ divise donc le polynôme minimal $\Pi'(\gamma-1)$ de $\,\T^{T'}_{S'}$. Cela étant:\smallskip
\begin{itemize}
\item Si $\gamma-1$ ne divise pas $\Pi'(\gamma-1)$, le sous-module $(\C^{T}_{S})^\Gamma$ est fini; et il n'y a rien à démontrer.\smallskip
\item Sinon, soit $\bar\Pi'(\gamma-1)=\Pi'(\gamma-1)/(\gamma-1)$, avec $\bar\Pi'(\gamma-1)$ et $\gamma-1$ copremiers. Il vient:\smallskip
\centerline{$(\C^{T}_{S})^\Gamma=(\T^{T}_{S})^\Gamma \sim (\T^{T}_{S})^{\bar\Pi'(\gamma-1)}=f((\T^{T'}_{S'})^{\bar\Pi'(\gamma}) \sim f((\T^{T'}_{S'})^\Gamma) = f((\C^{T'}_{S'})^\Gamma)$.}
\end{itemize}
\newpage
\section{Théorème principal sous la conjecture forte}
Dans le Théorème \ref{TP1}, les ensembles respectifs de places de $K$ au-dessus de $S$ et $T$ sont, du fait même de leur construction, stables par la conjugaison complexe $\tau$. Mais il est facile de s'affranchir de cette restriction, ce qui donne le résultat de Lee et Yu énoncé dans \cite{LY}:
\begin{Th}\label{TP2}
Soient $\ell$ un nombre premier et $K$ une extension quadratique totalement imaginaire d'un sous-corps $K^+$ totalement réel, supposée satisfaire la Conjecture cyclotomique forte pour $\ell$. Étant donnés deux ensembles finis disjoints $S_{\si{K}}$ et $T_{\si{K}}$ de places de $K$, notons $\hat S_{\si{K}}=S_{\si{K}} \cup S_{\si{K}}^\tau$ et $\check T_{\si{K}}=T_{\si{K}} \cap\, T^\tau_{\si{K}}$ leurs saturés respectivement supérieur et inférieur pour la conjugaison complexe $\tau$; désignons par $\hat S_{\si{K^+}}$ et $\check T_{\si{K^+}}$ les ensembles de places de $K^+$ au-dessous de $\hat S_{\si{K}}$ et $\check T_{\si{K}}$; et notons enfin $\bar R_{\si{K^+}}=Pl_{\si{K^+}}^\ell\setminus (\hat S_{\si{K^+}}\cup \check T_{\si{K^+}})$ l'ensemble des places de $K^+$ au-dessus de $\ell$ qui ne sont ni dans $\hat S_{\si{K^+}}$ ni dans $\check T_{\si{K^+}}$.
Alors le $\Zl$-rang du sous-module ambige $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^\Gamma$ du groupe de Galois $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})$ attaché à la pro-$\ell$-extension abélienne $H^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})$ $S_{\si{K}}$-décomposée $T_{\si{K}}$-ramifiée maximale sur $K_{\si{\infty}}$ est:\smallskip
\centerline{$\rg_{\Zl}\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^\Gamma =
\rg_{\Zl}\,\C^{\check T_{\si{K}}}_{\hat S_{\si{K}}}(K_{\si{\infty}})^\Gamma=|\bar R_{\si{K^+}}^-|$,}\smallskip
\noindent où $|\bar R_{\si{K^+}}^-|$ désigne le nombre de places de $\bar R_{\si{K^+}}$ qui sont décomposées par la conjugaison complexe.
\end{Th}
Pour établir ce dernier résultat, nous allons nous appuyer sur l'égalité des rangs:
\begin{Lem}\label{L}
Les $\Lambda$-modules $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})$ et $\,\C^{\check T_{\si{K}}}_{\hat S_{\si{K}}}(K_{\si{\infty}})$ ont inconditionnellement même $\Lambda$-rang:\smallskip
\centerline{$\rho^{\si{T}}_{\si{S}} = \rg_\Lambda\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}}) = \rg_\Lambda\,\C^{\check T_{\si{K}}}_{\hat S_{\si{K}}}(K_{\si{\infty}}) = \deg_\ell \check T_{\si{K}}$,}\smallskip
\noindent où $\deg_\ell \check T = \sum_{\l\in(\Pl^\ell \cap \check T)} [K_\l^+:\Ql]$ désigne le degré en $\ell$ de l'ensemble de places $\check T$.
\end{Lem}
\Preuve Si $K$ contient une racine $\ell$-ième primitive de l'unité $\zeta_\ell$, c'est le Théorème 9 de \cite{JMa}. Sinon, écrivant $K=K^+[\sqrt\delta]$ avec $\delta$ totalement négatif, on peut remplacer $K^+$ par $K^+[\zeta_\ell+\bar\zeta_\ell, \sqrt\delta(\zeta_\ell-\bar\zeta_\ell)]$ et $K$ par $K'=K[\zeta_\ell]$; appliquer le Théorème 2.7 de \cite{J43} à $K'$; et redescendre le résultat dans $K$.
\medskip
\PreuveTh Comme précédemment, nous pouvons supposer sans perte de généralité $S_{\si{K}}$ et $T_{\si{K}}$ contenus dans l'ensemble $Pl^\ell_{\si{K}}$ des places de $K$ au-dessus de $\ell$. Cela étant, les sous-ensembles\smallskip
\centerline{$S_{\si{K}}=\check S_{\si{K}} \sqcup (T_{\si{K}}\cap S_{\si{K}}^\tau) \sqcup S^\circ_{\si{K}};
\quad T_{\si{K}}=\check T_{\si{K}} \sqcup (S_{\si{K}}\cap T_{\si{K}}^\tau) \sqcup T^\circ_{\si{K}};
\quad R_{\si{K}}=\check R_{\si{K}} \sqcup S_{\si{K}}^{\circ\,\tau} \sqcup T_{\si{K}}^{\circ\,\tau}$.}\smallskip
\noindent forment alors une partition de $Pl_{\si{K}}^\ell$. Il vient: $\bar R_{\si{K}}= \check R_{\si{K}} \sqcup (T^\circ_{\si{K}} \sqcup T^{\circ\,\tau}_{\si{K}})$ puis $\bar R_{\si{K}}^-= \check R_{\si{K}}^- \sqcup (T^\circ_{\si{K}} \sqcup T^{\circ\,\tau}_{\si{K}})$.\smallskip
Procédons par minoration et majoration.\smallskip
$(i)$ Le Corollaire \ref{C}, nous donne directement la minoration: $\rg_\Zl\,\C_{S_{\si{K}}}^{T_{\si{K}}}(K_\%)^\Gamma \ge \rg_\Zl\,\C_{\hat S_{\si{K}}}^{\check T_{\si{K}}}(K_\%)^\Gamma$. Et ce dernier est donné par le Théorème \ref{TP1}: $\rg_\Zl\,\C_{\hat S_{\si{K}}}^{\check T_{\si{K}}}(K_\%)^\Gamma = |(\bar R_{\si{K^+}}^-|$.\smallskip
$(ii)$ D'autre part, par la Proposition \ref{Prop}, les classes invariantes de $\,\C_{S_{\si{K}}}^{T_{\si{K}}}(K_\%)^\Gamma$ proviennent des familles projectives $\a_n(\p)$ pour $\p\in R_{\si{K}}=\check R_{\si{K}} \sqcup S_{\si{K}}^{\circ\,\tau} \sqcup T_{\si{K}}^{\circ\,\tau}$. Or, par le Théorème \ref{TP1}:
\begin{itemize}
\item pour $\p \in \check R_{\si{K}}$, on a: $\a(\p)\a(\p^\tau)=\a(\p\p^\tau)\sim 1$ dans $\,\C^{\hat T_{\si{K}}}_\ph(K_\%)$ donc, a fortiori, dans $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_\%)$;
\item pour $\p \in S_{\si{K}}^{\circ\,\tau}$, on a de même: $\a(\p)\a(\p^\tau)=\a(\p\p^\tau)\sim 1$ dans $\,\C^{\hat T_{\si{K}}}_\ph(K_\%)$ donc dans $\,\C^{T_{\si{K}}}_\ph (K_\%)$; et finalement: $\a(\p)\sim\a(\p^{-\tau})\sim 1$ dans $\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_\%)$.
\end{itemize}\smallskip
Il vient donc: $\rg_\Zl\,\C_{S_{\si{K}}}^{T_{\si{K}}}(K_\%)^\Gamma\le \frac{1}{2}|\check R_{\si{K}}^-|+|T_{\si{K}}^{\circ\,\tau}|= |\bar R_{\si{K^+}}^-|$.\smallskip
En fin de compte, les deux inégalités réunies nous donnent l'égalité attendue.
\begin{Cor}
Sous les hypothèses du Théorème, le $\Zl$-rang du quotient des genres
est donné par:\smallskip
\centerline{$\rg_\Zl{}^\Gamma \C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}}) = \deg_\ell \check T{\si{K}} + |\bar R_{\si{K^+}}^-|$.}
\end{Cor}
\Preuve Écrivant la pseudo-décomposition $\C^S_T(K_\%) \sim \Lambda^{\rho^{\si{S}}_{\si{T}}}\oplus \,\T^S_T(K_\%) $, nous avons immédiatement:\smallskip
\centerline{$^\Gamma \C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})
\sim \Zl^{\rho_{\si{S}}^{\si{T}}}\oplus \,^\Gamma \T^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})
\sim \Zl^{\rho_{\si{S}}^{\si{T}}}\oplus \, \T^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^\Gamma
= \Zl^{\rho_{\si{S}}^{\si{T}}}\oplus \, \C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^\Gamma$,}\smallskip
\noindent avec $\rho^{\si{S}}_{\si{T}}=\deg_\ell \check T_{\si{K}}$ et $\rg_\Zl\,\C_{\hat S_{\si{K}}}^{\check T_{\si{K}}}(K_\%)^\Gamma = |\bar R_{\si{K^+}}^-|$.
\newpage
\section{Équivalence des conjectures faible et forte}
Supposons encore $K$ extension quadratique totalement imaginaire d'un sous-corps $K^+$ totalement réel; mais partons cette fois d'une partition arbitraire $S_{\si{K}} \sqcup T_{\si{K}}$ de $Pl_{\si{K}}^\ell$. Notons toujours $\tau$ la conjugaison complexe et posons: $\hat S_{\si{K}}=S_{\si{K}} \cup S_{\si{K}}^\tau$ et $\check S_{\si{K}}=S_{\si{K}} \cap S_{\si{K}}^\tau$ , comme dans le Théorème \ref{TP2}; et écrivons de même: $\hat T_{\si{K}}=T_{\si{K}} \cup T_{\si{K}}^\tau$ et $\check T_{\si{K}}=T_{\si{K}} \cap T_{\si{K}}^\tau$ en omettant l'indice $K$ dans ce qui suit.\par
Observons que $\hat S_{\si{K}} \sqcup \check T_{\si{K}}$ et $\check S_{\si{K}} \sqcup \hat T_{\si{K}}$ forment des partitions de $P^\ell_{\si{K}}$ stables par la conjugaison $\tau$.\smallskip
Considérons alors la surjection canonique $f^{\si{S}}_{\si{T}}:\;\C^{\check S}_{\hat T}(K_\%)\to\,\C^{S}_{T}(K_\%)$. Son noyau $\Ker f^{\si{S}}_{\si{T}}$ est engendré conjointement par les sous-groupes de décomposition $\D_\l$ des places $\l$ de $S\setminus\check S$ et par les sous-groupes d'inertie $\I_\l$ des places $\l$ de $\hat T\setminus T$ dans $\,\C^{\check S}_{\hat T}(K_\%)$.
Regardons l'image par $f^{\si{S}}_{\si{T}}$ du sous-module réel $\,\C^{\check S}_{\hat T}(K_\%)^+=\,\C^{\check S}_{\hat T}(K_\%)^{1+\tau}\sim \,\C^{\check S}_{\hat T}(K_\%^+)$. Nous avons:\smallskip
\centerline{$\C^{S}_{T}(K_\%)/f^{\si{S}}_{\si{T}}(\C^{\check S}_{\hat T}(K_\%)^+)
\simeq\,\C^{\check S}_{\hat T}(K_\%)/\big(\Ker f^{\si{S}}_{\si{T}}\;\,\C^{\check S}_{\hat T}(K_\%)^+ \big)$.}\smallskip
Or, par construction, le quotient à droite est annulé par $1+\tau$. Ainsi, puisque les $\D_\l$ (pour $\l\in S\setminus\check S$) et les $\I_\l$ (pour $\l\in \hat T\setminus\ T$) y ont une image triviale, il en est de même de leurs conjugués respectifs $\D^\tau_\l=\D^\ph_{\l^\tau}$ et $\I^\tau_\l=\I^\ph_{\l^\tau}$. Et il vient donc:\smallskip
\centerline{$\C^{\check S}_{\hat T}(K_\%)/\big(\Ker f^{\si{S}}_{T}\,.\,\C^{\check S}_{\hat T}(K_\%)^+ \big)
\simeq \,\C^{\hat S}_{\check T}(K_\%)^-$,}\smallskip
\noindent où $\,\C^{\hat S}_{\check T}(K_\%)^-$ désigne le plus grand quotient de $\,\C^{\hat S}_{\check T}(K_\%)$ annulé par $1+\tau$.\smallskip
En résumé, nous avons la suite exacte courte:\smallskip
\centerline{$1 \longrightarrow \,\F^{\check S}_{\hat T}(K_\%)^+ \longrightarrow \,\C^{S}_{T}(K_\%) \longrightarrow \,\C^{\hat S}_{\check T}(K_\%)^- \longrightarrow 1$,}\smallskip
\noindent avec $ \,\F^{\check S}_{\hat T}(K_\%)^+= \,\C^{\check S}_{\hat T}(K_\%)^+/\big( \,\C^{\check S}_{\hat T}(K_\%)^+\cap \Ker f^{S}_{T}\big)$; puis, en prenant les points fixes par $\Gamma$:\smallskip
\centerline{$1 \longrightarrow \,\F^{\check S}_{\hat T}(K_\%)^{\Gamma\,+} \longrightarrow \,\C^{S}_{T}(K_\%)^\Gamma \longrightarrow \,\C^{\hat S}_{\check T}(K_\%)^{\Gamma\,-}$.}\smallskip
Supposons maintenant que $K$ satisfasse la conjecture cyclotomique {\em faible}, c'est à dire, comme établi dans \cite{J55}, à la fois la conjecture de Leopoldt et celle de Gross-Kuz'min pour $\ell$.
\begin{itemize}
\item À gauche, la conjecture de Leopoldt assure la finitude de $\,\C^{\check S}_{\hat T}(K_\%)^+$, donc de $\,\F^{\check S}_{\hat T}(K_\%)^{+}$; et finalement celle du sous-module des points fixes $\,\F^{\check S}_{\hat T}(K_\%)^{\Gamma\,+}$.
\item À droite, la conjecture cyclotomique {\em faible} (en fait la conjecture de Gross-Kuz'min) assure celle de $\,\C^{\hat S}_{\check T}(K_\%)^{\Gamma\,-}$.
\end{itemize}
\noindent Il suit de là que le groupe médian $\,\C^{S}_{T}(K_\%)^\Gamma$ est lui-même fini; autrement dit que $K$ vérifie la conjecture cyclotomique {\em forte} pour $\ell$. Ainsi:
\begin{Th}[\bf Équivalence des conjectures]\label{TP3}
Pour tout corps de nombres $K$ à conjugaison complexe et tout nombre premier $\ell$ fixé, les trois assertions suivantes sont équivalentes:
\begin{itemize}
\item $K$ vérifie les conjectures de Leopoldt et de Gross-Kuz'min.
\item $K$ vérifie la conjecture cyclotomique faible.
\item $K$ vérifie la conjecture cyclotomique forte.
\end{itemize}
\end{Th}
Comme vu dans les sections précédentes, la conjecture faible entraîne la validité de la formule des rangs pour les partitions $R \sqcup S \sqcup T$ de $Pl^\ell$ stables par conjugaison; et la conjecture forte l'entraîne indépendamment de cette restriction. Inversement la formule écrite pour les seules partitions vérifiant $R=\emptyset$ exprime précisément ces mêmes conjectures.\smallskip
Avec les notations de Théorèmes \ref{TP1} et \ref{TP2}, il vient ainsi:
\begin{Sco}
Sont encore équivalentes aux précédentes chacune des deux assertions suivantes:
\begin{itemize}
\item Pour toute partition $P_{\si{K}}^\ell=R_{\si{K}} \sqcup S_{\si{K}} \sqcup T_{\si{K}}$ de l'ensemble $Pl^\ell_{\si{K}}$ stable par conjugaison complexe, on a les identités de rang:\smallskip
\centerline{$\rg_{\Zl}\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K^+_{\si{\infty}})^\Gamma =0$ \qquad \& \qquad $\rg_{\Zl}\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^\Gamma =|R_{\si{K^+}}^-|$.}\smallskip
\item Pour toute partition $Pl_{\si{K}}^\ell=R_{\si{K}} \sqcup S_{\si{K}} \sqcup T_{\si{K}}$ de l'ensemble des places au-dessus de $\ell$, on a les identités de rang:
\centerline{$\rg_{\Zl}\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K^+_{\si{\infty}})^\Gamma =0$ \qquad \& \qquad $\rg_{\Zl}\,\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^\Gamma =|\bar R_{\si{K^+}}^-|$.}
\end{itemize}
\end{Sco}
\newpage
\section{Interprétation par le corps de classes $\ell$-adique}
Regardons maintenant le groupe $^\Gamma\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})$ par la théorie $\ell$-adique du corps de classes, telle qu'exposée dans \cite{J31}: le nombre premier $\ell$ et le corps $K$ étant supposés fixés, pour chaque place non complexe $\p$ de $K$ notons
$\,\U^{\phantom{*}}_\p$ et $\,\wU^{\phantom{*}}_\p$ les sous-groupes unités, respectivement au sens habituel et logarithmique, du $\ell$-adifié $\R_\p=\varprojlim \,K_\p^{\si{\times}}/K_\p^{\si{\times\ell^n}}$ du groupe multiplicatif du complété $K_\p$.
Écrivons de même $\J=\prod_\p^{\si{\rm res}}\R_\p$ le $\ell$-adifé du groupe des idèles; $\,\U=\prod_\p\U_\p$ et $\,\wU=\prod_\p\wU_\p$ ses sous-groupes unités (au sens habituel et logarithmique); et $\R=\Zl\otimes_\ZZ K^{\si{\times}}$ le sous-groupe principal de $\J$.\smallskip
Pour tout $P\subset Pl_K$, notons $\R_P=\prod_{\p\in P}\R_\p$; puis $\,\U_P=\prod_{\p\in P}\U_\p$ et $\,\U^P=\prod_{\p\notin P}\U_\p$. Écrivons de même $\,\wU_P=\prod_{\p\in P}\wU_\p$ et $\,\wU^P=\prod_{\p\notin P}\wU_\p$. Soient enfin $\D_P=\R_P/\U_P$ et $\wD_P=\R_P/\wU_P$ les groupes de diviseurs (respectivement aux sens habituel et logarithmique). Cela étant, il vient:
\begin{Th}\label{TP4}
Soient $K$ un corps de nombres et $R \sqcup S \sqcup T$ une partition de l'ensemble $L$ des places de $K$ au-dessus de $\ell$. Avec les notations précédentes le groupe de Galois $\G^T_S=\Gal(H^T_S(K_\%/K)/K)$ de la pro-$\ell$-extension $S$-décomposée $T$-ramifiée maximale de $K_\%$ qui est abélienne sur $K$ est donné à un fini près par le pseudo-isomorphisme:\smallskip
\centerline{$\G_S^T \sim \wD_R\wD_S\,\U_T/(\wt\nu_{\si{RS}}\times p_{\si{T}})(\E_S)$,}\smallskip
\noindent où $\wt\nu_{\si{RS}}=(\wi\nu_\l)_{\l\in R\cup S}$ est la famille des valuations logarithmiques aux places de $R \sqcup S$ et $p_{\si{T}}$ est le morphisme de semi-localisation aux places de $T$.
\end{Th}
\Preuve
Rappelons que $\,\wU_\p$ est le groupe de normes attaché à la $\Zl$-extension cyclotomique de $K_\p$ et qu'on a $\,\U_\p=\,\wU_\p$, pour $\p\nmid \ell$. Posons $\,\bar\U_\p=\,\U_\p \cap\,\wU_\p$. Comme $\R_S\,\U^S\R=\R_S\,\U_R\,\U_T\,\U^L\R$ est d'indice fini dans $\J$, puisque $\J/\R_S\,\U^S\R$ s'identifie au $\ell$-groupe des $S$-classes d'idéaux, il vient:\smallskip
\centerline{$\G_S^T = \J / \,\wU_S\,\bar\U^T\R \sim \R_S\,\U_R\,\U_{\,T}\,\U^L\R /\,\wU_S\,\bar\U_R\,\U^L\R
\simeq \R_S\,\U_R\,\U_{\,T} /\,\wU_S\,\bar\U_R(\R_S\,\U_R\,\U_{\,T}\cap \,\U^L\R)$;}\smallskip
\noindent ce qui donne la formule annoncée, puisque les idèles principaux qui interviennent au dénominateur sont les $S$-unités, et que l'on a, par ailleurs: $\R_S/\,\wU_S\simeq\wD_S$ et $\,\U_R/\,\bar\U_R \sim \R_R/\,\wU_R \simeq \wD_R$
\begin{Cor}\label{CF}
Lorsque $K$ est extension quadratique totalement imaginaire d'un sous-corps $K^+$ totalement réel, sous les conjectures de Leopoldt et de Gross-Kuz'min, le pseudo-isomorphisme\smallskip
\centerline{$^\Gamma\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}}) \sim \,\G^{\check T_{\si{K}}\,-}_{\hat S_{\si{K}}}
\sim \,\wD_{\bar R_{\si{K}}}^- \times \,\U_{\,\check T_{\si{K}}}^-$ \quad
donne directement: \quad $\rg_\Zl{}^\Gamma \C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}}) = \deg_\ell \check T_{\si{K^+}} + |\bar R_{\si{K^+}}^-|$,}\smallskip
\noindent avec $\bar R_{\si{K}}= L_{\si{K}}\setminus(\check T_{\si{K}} \sqcup \hat S_{\si{K}})$ pour toute partition $R_{\si{K}} \sqcup S_{\si{K}} \sqcup T_{\si{K}}$ de $L_{\si{K}}=Pl_{\si{K}}^\ell$.
\end{Cor}
\Preuve
Reprenons le schéma de démonstration du Théorème \ref{TP3} en partant cette fois de la surjection canonique $g^{T_{\si{K}}}_{S_{\si{K}}}: \,\G^{\check T_{\si{K}}}_{\hat S_{\si{K}}} \to \,\G^{T_{\si{K}}}_{S_{\si{K}}}$. Nous obtenons alors la suite pseudo-exacte courte:\smallskip
\centerline{$1 \longrightarrow \big(\G_{\check S_{\si{K}}}^{\hat T_{\si{K}}}\big)^+/ \big(\G_{\check S_{\si{K}}}^{\hat T_{\si{K}}}\cap \Ker g_{S_{\si{K}}}^{T_{\si{K}}}\big)^+ \longrightarrow \,\G_{S_{\si{K}}}^{T_{\si{K}}} \longrightarrow \big(\G_{\hat S_{\si{K}}}^{\check T_{\si{K}}}\big)^- \longrightarrow 1$,}\smallskip
\noindent où apparaissent les composantes réelles ou imaginaires des modules de droite et de gauche, ce qui permet de définir partie réelle et partie imaginaire du groupe $\,\G_{S_{\si{K}}}^{T_{\si{K}}}$ alors même qu'il n'est pas {\em a priori} stable par conjugaison.\smallskip
\begin{itemize}
\item Côté réel, la conjecture de Leopoldt nous donne immédiatement: $\big(\G_{\check S_{\si{K}}}^{\hat T_{\si{K}}}\big)^+\sim \Gamma \simeq \Zl$; i.e. $^\Gamma\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}})^+\sim 1$; et finalement: $^\Gamma\C^{T_{\si{K}}}_{S_{\si{K}}}(K_{\si{\infty}} ) \sim \big(\G_{\hat S_{\si{K}}}^{\check T_{\si{K}}}\big)^-$.\smallskip
\item Côté imaginaire, il vient, d'après le Théorème \ref{TP4}: $\G^{\check T_{\si{K}}}_{\hat S_{\si{K}}}{}^- \!\sim \wD_{\bar R_{\si{K}}}^-\wD_{\hat S_{\si{K}}}^-\,\U_{\,\check T_{\si{K}}}^-/(\wt\nu_{\si{\bar R\hat S}}\times p_{\si{\check T}})(\E_{\hat S_{\si{K}}}^-)$. Or, sous la conjecture de Gross-Kuz'min, le $\Zl$-module des $\hat S_{\si{K}}$-unités imaginaires s'envoie pseudo-injectivement dans $\wD_{\hat S_{\si{K}}}^-$, puisque les unités logarithmiques sont réelles. Il suit donc: $\,\E_{\hat S_{\si{K}}}^-\sim\wD_{\hat S_{\si{K}}}^-$; puis $\,\G^{\check T_{\si{K}}}_{\hat S_{\si{K}}}{}^- \!\sim \wD_{\bar R_{\si{K}}}^-\,\U_{\,\check T_{\si{K}}}^-$, comme annoncé.
\end{itemize}\smallskip
On retrouve ainsi très simplement l'expression du $\Zl$-rang donnée par le Théorème \ref{TP2}, ce qui fournit une démonstration alternative du Thérème d'équivalence \ref{TP3}.
\newpage
\noindent {\em Commentaires bibliographiques}\smallskip
Il est formulé dans \cite{J10} une conjecture générale sur l'indépendance $\ell$-adique de nombres algébriques, vérifiée dans les corps abéliens, qui implique en particulier les conjectures de Leopoldt et de Gross-Kuz'min et en factorise les preuves transcendantes (partielles) classiques.
Les $\ell$-groupes de $S$-classes $T$-infinitésimales sont étudiés dans \cite{J18} (Ch. II, \S 2). On y trouve notamment la suite exacte des classes ambiges évoquée dans la section 1.\par
Les $\ell$-groupes de classes logarithmiques ont été introduits dans \cite{J28}. Leur calcul est maintenant implanté dans {\sc pari} (cf. \cite{BJ}). L'interprétation logarithmique de la conjecture de Gross-Kuz'min est donnée dans \cite{J28} sous l'appellation initiale de conjecture de Gross généralisée.\par
Une formulation équivalente de la conjecture cyclotomique en termes de points fixes de $(S,T)$-modules d'Iwasawa a été avancée dans \cite{LS} par Lee et Seo. Tout récemment, Lee et Yu en ont tiré dans \cite{LY} un calcul du rang analogue à celui donné ici. Leur démonstration indépendante est plus laborieuse en l'absence des simplifications apportées par l'introduction des unités logarithmiques.\par
Les principaux résultats de la Théorie $\ell$-adique du corps de classes introduite dans \cite{J18} sont présentés dans \cite{J31}. On peut aussi se reporter au livre de Gras \cite{Gra2}.\par
Enfin, le calcul des invariants d'Iwasawa attachés aux $\ell$-groupes de $S$-classes $T$-infinitésimales est développé dans \cite{J43,JMa,JMP} en liaison avec les identités de dualité de Gras. L'article \cite{JMa} contient une erreur, reproduite dans \cite{J43} mais corrigée dans \cite{JMP}, qui ne concerne heureusement que l'invariant $\lambda^S_T$. Elle est sans incidence sur les résultats présentés ici.
\bigskip
\def\refname{\normalsize{\sc Références}}
\addcontentsline{toc}{section}{Bibliographie}
{\footnotesize
| {
"redpajama_set_name": "RedPajamaArXiv"
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Рубі́жне (Снігаровка) — село Добропільської міської громади Покровського району Донецької області, Україна.
Історія.
Заснування.
Територія села була в стародавні часи частиною Залозного шляху, по якому в часи Київської Русі переправляли сіль.
Козацький період.
У часи Запорізької Січі тут був козацький волок з річки Бик в річку Грузька. Під час будівництва другої черги каналу «Дніпро - Донбас» в районі села Золотий Колодязюь знайшли залишки «чайок» гетьмана Петра Сагайдачного. Долину і балки системи річки Казений Торець запорожці освоювали в XVII -XVIII сторічч. Село Рубіжне виникло на місці сторожової застави запорізьких козаків кінця XVII - початку XVIII сторіччя.
Російська імперія.
Село засноване приблизно в 1860 році. На території землі, оточуючої нинішнє село Рубіжне була розташована економія пана Снігірьова. Він мав маєток, який був розташований на північ літнього табору бригади №03, де ще й зараз зустрічаються залишки споруди. Села тоді не було.
Пан Грузчан, який жив на території нинішнього села Грузьке, віддав свою доньку заміж за сина пана Снігірьова і в придане доньці дав селян, яких пан Снігірьов поселив на березі річки Бик. Для садиб селянам він віділив по 4 десятини землі, але то були найгірші, самі незручні ділянки землі. Спочатку в селі налічувалося 37 дворів з 260 жителями. Село входило до складу Казенно-Торсько-Олексіївської волості, Бахмутського повіту, Катеринославської губернії.
Південніше, в Харцизькій балці в XIX сторіччі виник хутор Пастуха.
Селяни після скасування кріпацтва працювали на пана 3 дні, а 3 дні на своїх ділянках землі. Через нестачу землі й неможливість її викупити у пана селяни їхали до Сибіру. Так з села виїхали сім'ї Стонгушенко Андрія і Компанійця Михайла. Які вернулися в село після Лютневої революції, їм виділили як і всім селянам по дві десятини відібраних земель у пана.
Станом на 1908 рік, в селі Рубіжне було 42 двори, де проживало 318 осіб. Селяни користувалися 240 десятинами зручної, 11 десятинами незручної та 260 десятинами придбаної землі (дані 1909 і 1915 років).
У 1912 році Слов'янською міською управою було розроблено проєкт шлюзування річки Сіверський Донець і судноплавного каналу з річки Дон до річки Дніпро, - із використанням колишнього волоку в районі Рубіжного.
УРСР.
Після революції в селі вже було 69 дворів з 490 жителями. Ці 69 сімей користувались 240 десятинами землі . У 30 роках ХХ сторіччя в Рубіжному відкрилися дитсадок і початкова школа (зруйнована в роки другої світової війни, відновлена в 1958 році, ліквідована в 70 роках ХХ сторіччя). З 50 років ХХ сторіччя в Рубіжному функціонував сільський клуб. Також в селі є братська могила радянським воїнам.
Жертви сталінських репресій.
Дерев'янко Микола Матвійович, 1905 року народження, село Рубіжне Добропільського району Донецької області, українець, освіта 2 класи, безпартійний. Проживав в селі Рубіжне Добропільського району Сталінської (Донецької) області. Староста сільської общини. Заарештований 08 березня 1943 року. Військовим трибуналом військ НКВС по Сталінській (Донецькій) області засуджений на 10 років ВТТ з позбавленням прав на 5 років та конфіскацією майна. Реабілітований у 1968 році.
Денисенко Олексій Леонтійович, 1919 року народження, село Рубіжне Добропільського району Донецької області, українець, освіта 7 класів, безпартійний. Проживав в селі Рубіжне Добропільського району Сталінської (Донецької) області. Колгоспник колгоспу "Новий побут". Заарештований 25 грудня 1943 року. Військовим трибуналом 3 гвардійської армії засуджений на 10 років ВТТ з позбавлення прав на 3 роки. Реабілітований у 1956 році.
Парпара Герасим Дмитрович, 1887 року народження, село Велика Новосілка Великоновосілківського району Донецької області, грек, освіта початкова, безпартійний. Проживав в селі Рубіжне Добропільського району Донецької області. Бджоляр радгоспу. Заарештований 31 грудня 1937 року. Засуджений Особливою нарадою при НКВС СРСР до розстрілу. Даних про виконання вироку немає. Реабілітований у 1989 році.
Джерела.
http://pabel2007.narod.ru/kdd.htm
Подолян В.В. Слово про Добропілля: роки, події, люди. Донецьк: Престиж-party, 2009. с. 36
Добропільська районна централізована бібліотечна система Центральна бібліотека. "Береже пам'ять село" Матеріали історіко-краєзнавчих конференцій (травень, вересень 2015 року). Випуск 6 Збірник у двох частинах . Частина 2 . Добропілля 2016.
Примітки.
Села Донецької області | {
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Q: Jquery / Javascript plugin for in-browser data analysis and charting? Can anyone suggest an existing Javascript / JQuery based Analytics plugin? I spent some time searching but cannot find a quick one.
It may works like getting feed with data from external and providing basic charts to web containing element, like distribution, average, correlation of 2 data arrays, etc. Thx.
A: If you're just looking for something lightweight, you might want to check out Chartist JS. It doesn't have all of the bells and whistles of some other plugins, but it's quick to get up and running. There are even wrappers for integration with libraries/frameworks like Angular & WordPress.
For something more robust, check out Highcharts or D3, or something built on top of D3 like nvd3.
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\section{Introduction}
There has been a growing interest in studying rough volatility models \cite{gatheral2018volrough,eleuch2019char,guennoun2018asym}. Rough volatility models are stochastic volatility models whose trajectories are rougher than the paths of a standard Brownian motion in terms of the H\"older regularity. Specifically, when the H\"older regularity is less than 1/2, the stochastic path is regarded as rough. The roughness is closely related to the Hurst parameter $H$. This paper focuses on the Volterra Heston model, whose probabilistic characterization does not involve the rough paths theory \cite{eleuch2019char}.
Rough volatility models are attractive because they capture the dynamics of historical and implied volatilities remarkably well with only a few additional parameters. Investigations of the time series of the realized volatility\footnote{See, for example, Oxford-Man Institute's realized library at \url{https://realized.oxford-man.ox.ac.uk/data}} from high frequency data estimate the Hurst parameter $H$ to be near $0.1$, which is much smaller than the $0.5$ for the standard Brownian motion. The Hurst parameter is used to reflect the memoryness of a time series and is associated with the roughness of the fractional Brownian motion (fBM). The smaller the $H$, the rougher the time series model. Therefore, the empirical finding suggests a rougher realized path of volatility than the standard Brownian motion. Although previous studies have found a long memory property within realized volatility series, it is shown in \cite{gatheral2018volrough} that rough volatility models can generate the illusion of a long memory. However, the simulated paths with a small Hurst parameter resemble the realized ones.
Rough volatility models also better capture the term structure of an implied volatility surface, especially for the explosion of at-the-money (ATM) skew when maturity goes to zero. More precisely, let $\sigma_{BS}(k, \tau)$ be the implied volatility of an option where $k$ is the log-moneyness and $\tau$ is the time to expiration. The ATM skew at maturity $\tau$ is defined by
\begin{equation}
\phi(\tau) \triangleq \Big| \frac{\partial \sigma_{BS}(k, \tau)}{\partial k} \Big|_{k=0}.
\end{equation}
Empirical evidence shows that the ATM skew explodes when $\tau \downarrow 0$. However, conventional volatility models such as the Heston model \cite{heston1993closed} generate a constant ATM skew for a small $\tau$. If the volatility is modeled by a fractional Brownian motion, then the ATM skew has an asymptotic property \cite{fukasawa2011asymptotic},
\begin{equation}
\phi(\tau) \approx \tau^{H - 1/2}, \text{ when } \tau \downarrow 0,
\end{equation}
where $H$ is the Hurst parameter. Rough volatility models can fit the explosion remarkably well by simply adjusting the $H$.
Recent advances offer elegant theoretical foundations for rough volatility models. We note the martingale expansion formula for implied volatility \cite{fukasawa2011asymptotic}, asymptotic analysis of fBM \cite[Section 3.3]{fukasawa2011asymptotic}, the microstructural foundation of rough Heston models by scaling the limit of proper Hawkes processes \cite{eleuch2018micro}, the closed-form characteristic function of rough Heston models up to the solution of a fractional Riccati equation \cite{eleuch2019char}, and the hedging strategy for options under rough Heston models \cite{eleuch2018perfect}. In this paper, we are particularly interested in the affine Volterra processes \cite{abi2017affine} because these models embrace rough Heston model \cite{eleuch2019char} as a special case. The characteristic function in \cite{eleuch2019char} is extended to the exponential-affine transform formula in terms of Riccati-Volterra equations \cite{abi2017affine}. Affine Volterra processes are applied to finance problems in \cite{keller2018affine}. In addition, an alternative rough version of the Heston model is introduced in \cite{guennoun2018asym}, where some asymptotic results are derived.
While the rough volatility literature focuses on option pricing, only a few works contribute to portfolio optimization such as \cite{fouque2018fast,fouque2018fractional,bauerle2018protfolio}. All of them consider utility maximization. To the best of our knowledge, this is the first paper to consider the mean-variance (MV) portfolio selection under a rough stochastic environment. The MV criterion in portfolio selection pioneered by Markowitz's seminal work is the cornerstone of the modern portfolio theory. We cannot give a full list of research outputs related to this Nobel Prize winning work, but mention contributions in continuous-time settings \cite{zhou2000lq,lim2002complete,lim2004incomplete,cerny2008mean,jeanblanc2012mean,shen2015mean} as important references.
\subsection{Major contributions}
We formulate the MV portfolio selection under the Volterra Heston models in a reasonably rigorous manner. As pointed out by \cite{abi2017affine,keller2018affine}, the Volterra Heston model (\ref{vol})-(\ref{stock}) has a unique in law weak solution, but its pathwise uniqueness is still an open question in general. This enforces us to consider the MV problem under a general filtration $\mathbb{F}$ that satisfies the usual conditions but may not be the augmented filtration generated by the Brownian motion. A similar general setting also appears in \cite{jeanblanc2012mean}. We emphasize that the probability basis and Brownian motions are always fixed for the problem in Section \ref{Sec:MV}. Therefore, our formulation is still considered to be a {\it strong formulation}, because the filtered probability space and Brownian motions are not parts of the control.
Under such a problem formulation, we construct in Section \ref{Sec:Sol} an auxiliary stochastic process $M_t$ to solve the MV portfolio selection by completion of squares. Several properties of $M_t$ are derived in Theorem \ref{Thm:M}, which is a main result of this paper. Like \cite{eleuch2019char,eleuch2018perfect,abi2017affine}, we encounter difficulties due to the non-Markovian and non-semimartingale structure of the Volterra Heston model (\ref{vol})-(\ref{stock}). Inspired by the exponential-affine formulas in \cite{abi2017affine,eleuch2019char}, the process $M_t$ is constructed upon the forward variance under a proper alternative measure. The explicit solution for the optimal investment strategy is obtained in Theorem \ref{Thm:Sol}.
Under the rough Heston model, we investigate the impact of roughness on the optimal investment strategy $u^*$. Recently, a trading strategy has been proposed to leverage the information of roughness \cite{glasserman2019buy}. The strategy longs the roughest stocks and shorts the smoothest stocks. Excess returns from this strategy are not fully explained by standard factor models like the CAPM model and Fama-French model. We examine this trading signal under the MV setting. Our theory predicts that the effect of roughness on investment strategy is opposite under different volatility of volatility (vol-of-vol). We also discuss the roughness effect on the efficient frontier.
The rest of the paper is organized as follows. Section \ref{Sec:Model} presents the Volterra Heston model and some useful properties. We discuss a related Riccati-Volterra equation. We then formulate the MV portfolio selection problem in Section \ref{Sec:MV} and solve it explicitly in Section \ref{Sec:Sol}. Numerical illustrations are given in Section \ref{Sec:Numerical}. Section \ref{Sec:Conclusion} concludes the paper. The existence and uniqueness of the solution to Riccati-Volterra equations are summarized in Appendix \ref{Appendix}. An auxiliary result used in Theorem \ref{Thm:M} is proved in Appendix \ref{App:Pos}.
\section{The Volterra Heston model}\label{Sec:Model}
Our problem is defined under a given complete probability space $(\Omega, {\mathcal F}, \mathbb{P})$, with a filtration $\mathbb{F} = \{ {\mathcal F}_t \}_{ 0 \leq t \leq T}$ satisfying the usual conditions, supporting a two-dimensional Brownian motion $W = (W_1, W_2)$. The filtration $\mathbb{F}$ is not necessarily the augmented filtration generated by $W$; thus, it can be a strictly larger filtration. This consideration is different from some previous studies like \cite{lim2002complete,lim2004incomplete,shen2015mean} but is consistent with \cite{jeanblanc2012mean} for the MV hedging problem under a general filtration. This consideration is important because the stochastic Volterra equation (\ref{vol})-(\ref{stock}) only has a unique in law weak solution but its strong uniqueness is still an open question in general. Recall that for stochastic differential equations, $X$ is referred to as a strong solution if it is adapted to the augmented filtration generated by $W$, and a weak solution otherwise. For a weak solution, the driving Brownian motion $W$ is also a part of the solution \cite[Chapter IX]{ry1999book}. Therefore, $\mathbb{F}$ cannot be simply chosen as the augmented filtration generated by $W$, as extra information may be needed to construct a solution to (\ref{vol})-(\ref{stock}).
To proceed, we introduce a kernel $K(\cdot) \in L^2_{loc} (\mathbb{R}_+, \mathbb{R})$, where $\mathbb{R}_+ = \{ x \in \mathbb{R} | x \geq 0\}$, and make the following standing assumption throughout the paper, in line with \cite{abi2017affine,keller2018affine}. A function $f$ is called completely monotone on $(0, \infty)$, if it is infinitely differentiable on $(0, \infty)$ and $(-1)^k f^{(k)}(t) \geq 0$ for all $ t > 0 $, and $k = 0, 1, ...$.
\begin{assumption}\label{Assum:K}
$K$ is strictly positive and completely monotone on $(0, \infty)$. There is $\gamma \in(0,2]$, such that $\int_{0}^{h} K(t)^{2} d t=O\left(h^{\gamma}\right)$ and $\int_{0}^{T}(K(t+h)-K(t))^{2} d t=O\left(h^{\gamma}\right)$ for every $T<\infty$.
\end{assumption}
The convolutions $K*L$ and $L*K$ for a measurable kernel $K$ on $\mathbb{R}_+$ and a measure $L$ on $\mathbb{R}_+$ of locally bounded variation are defined by
\begin{equation}
(K*L)(t) = \int_{[0,t]} K(t-s)L(ds) \quad \text{and} \quad (L*K)(t) = \int_{[0,t]} L(ds)K(t-s)
\end{equation}
for $t>0$ under proper conditions. The integral is extended to $t=0$ by right-continuity if possible. If $F$ is a function on $\mathbb{R}_+$, let
\begin{equation}
(K*F)(t) = \int_0^t K(t-s) F(s) ds.
\end{equation}
Let $W$ be a $1$-dimensional continuous local martingale. The convolution between $K$ and $W$ is defined as
\begin{equation}
(K*dW)_t = \int_0^t K(t-s)dW_s.
\end{equation}
A measure $L$ on $\mathbb{R}_+$ is called {\em resolvent of the first kind} to $K$, if
\begin{equation}
K*L = L*K \equiv {\rm id}.
\end{equation}
The existence of a resolvent of the first kind is shown in \cite[Theorem 5.5.4]{gripenberg1990volterra} under the complete monotonicity assumption, imposed in Assumption \ref{Assum:K}. Alternative conditions for the existence are given in \cite[Theorem 5.5.5]{gripenberg1990volterra}.
A kernel $R$ is called the {\em resolvent} or {\em resolvent of the second kind} to $K$ if
\begin{equation}
K*R = R*K = K - R.
\end{equation}
The resolvent always exists and is unique by \cite[Theorem 2.3.1]{gripenberg1990volterra}.
Further properties of these definitions can be found in \cite{gripenberg1990volterra,abi2017affine}. Although the same notion can be defined for higher dimensions and in matrix form, it suffices for us to consider the scalar case. Commonly used kernels \cite{abi2017affine} summarized in Table \ref{Tab:Kernel} satisfy Assumption \ref{Assum:K} once $c>0$, $\alpha \in (1/2, 1]$, and $\beta \geq 0$.
\begin{table}[h!]
\centering
\begin{tabular}{c c c c }
\hline
& $K(t)$ & $R(t)$ & $L(dt)$ \\
\hline \\[0.5ex]
Constant & $c$ & $ce^{-ct}$ & $c^{-1} \delta_0(dt)$\\ \\
Fractional (Power-law) & $c\,\frac{t^{\alpha-1}}{\Gamma(\alpha)}$ & $ct^{\alpha-1} E_{\alpha, \alpha} (-ct^{\alpha})$ & $c^{-1}\,\frac{t^{-\alpha}}{\Gamma(1-\alpha)}dt$\\ \\
Exponential & $ce^{-\beta t}$ & $ce^{-\beta t}e^{-ct}$ & $c^{-1}(\delta_0(dt) + \beta\,dt)$ \\\\
\hline
\end{tabular}
\caption{Examples of kernels $K$ and their resolvents $R$ and $L$ of the second and first kind. $E_{\alpha,\beta}(z)=\sum_{n=0}^\infty \frac{z^n}{\Gamma(\alpha n+\beta)}$ is the Mittag--Leffler function. See \cite[Appendix A1]{eleuch2019char} for its properties. The constant $c \neq 0$.}
\label{Tab:Kernel}
\end{table}
The variance process within the Volterra Heston model is defined as
\begin{equation}\label{vol}
V_{t}=V_{0}+ \kappa \int_{0}^{t} K(t-s)\left(\phi -V_{s}\right) d s + \int_{0}^{t} K(t-s) \sigma \sqrt{V_{s}} d B_{s},
\end{equation}
where $ dB_s = \rho dW_{1s} + \sqrt{1 - \rho^2} dW_{2s} $ and $V_0, \kappa, \phi$, and $\sigma$ are positive constants. The correlation $\rho$ between stock price and variance is also constant. As documented in \cite{gatheral2018volrough}, the general overall shape of the implied volatility surface does not change significantly, indicating that it is still acceptable to consider a variance process whose parameters are independent of stock price and time. The rough Heston model in \cite{eleuch2019char,eleuch2018perfect} becomes a special case of (\ref{vol}) once $K(t) = \frac{t^{\alpha-1}}{\Gamma(\alpha)}$. Another rough version of the Heston model studied in \cite{guennoun2018asym} is adopted to investigate the power utility maximization \cite{bauerle2018protfolio}.
Following \cite{abi2017affine} and \cite{kraft2005optimal,cerny2008mean,zeng2013portfolio,shen2015square}, the risky asset (stock) price $S_t$ is assumed to follow
\begin{equation}\label{stock}
dS_t = S_t (r_t + \theta V_t) dt + S_t \sqrt{V_t} dW_{1t}, \quad S_0 > 0,
\end{equation}
with a deterministic bounded risk-free rate $r_t>0$ and constant $\theta \neq 0$. The market price of risk, or risk premium, is then given by $\theta \sqrt{V_t}$. The risk-free rate $r_t >0$ is the rate of return of a risk-free asset available in the market.
We take the existence and uniqueness result from \cite[Theorem 7.1]{abi2017affine} and restate it as follows.
\begin{theorem}\label{Thm:SVSol}
(\cite[Theorem 7.1]{abi2017affine}) Under Assumption \ref{Assum:K}, the stochastic Volterra equation (\ref{vol})-(\ref{stock}) has a unique in law $\mathbb{R}_+ \times \mathbb{R}_+$-valued continuous weak solution for any initial condition $(S_0, V_0) \in \mathbb{R}_+ \times \mathbb{R}_+$.
\end{theorem}
\begin{remark}
Our model (\ref{vol})-(\ref{stock}) is defined under the physical measure, whereas the option pricing model of \cite[Equations (7.1)-(7.2)]{abi2017affine} is under a risk-neutral measure with a zero risk-free rate. However, the proofs are almost identical because the affine structure is maintained and $S$ is determined by $V$.
\end{remark}
\begin{remark}
For strong uniqueness, we mention \cite[Proposition B.3]{abi2019multifactor} as a related result with kernel $K \in C^1([0, T], \mathbb{R})$ and \cite[Proposition 8.1]{mytnik2015uniqueness} for certain Volterra integral equations with smooth kernels. However, the strong uniqueness of (\ref{vol})-(\ref{stock}) is left open for singular kernels. For weak solutions, it is free to construct the Brownian motion as needed. However, the MV objective only depends on the mathematical expectation for the distribution of the processes. In the sequel, we will only work with a version of the solution to (\ref{vol})-(\ref{stock}) and fix the solution $(S, V, W_1, W_2)$, as other solutions have the same law.
\end{remark}
The following condition enables us to verify the admissibility of the optimal strategy. To be more precise about the constant $a$, \eqref{Eq:Const_a} gives an explicit sufficient large value needed.
\begin{assumption}\label{Assum:V}
$\mathbb{E} \Big[ \exp \big( a \int^T_0 V_s ds \big) \Big] < \infty$ for a large enough constant $a > 0$.
\end{assumption}
To verify that Assumption \ref{Assum:V} holds under reasonable conditions, we consider the Riccati-Volterra equation (\ref{Eq:g}) for $g(a, t)$ as follows:
\begin{equation}\label{Eq:g}
g(a, t) = \int^t_0 K(t-s) \big[ a - \kappa g(a,s) + \frac{\sigma^2}{2} g^2(a, s) \big] ds.
\end{equation}
The existence and uniqueness of the solution to (\ref{Eq:g}) are given in Lemmas \ref{Lem:g} and \ref{Lem:gfractional}.
\begin{theorem}\label{Thm:ExpV}
Suppose Assumption \ref{Assum:K} holds and the Riccati-Volterra equation (\ref{Eq:g}) has a unique continuous solution on $[0, T]$, then
\begin{equation}
\mathbb{E} \Big[ \exp \big( a \int^T_0 V_s ds \big) \Big] = \exp\Big[ \kappa \phi \int^T_0 g(a, s) ds + V_0 \int^T_0 \big[ a - \kappa g(a, s) + \frac{\sigma^2}{2} g^2(a, s) \big] ds \Big] < \infty.
\end{equation}
Moreover, denote $L$ as the resolvent of the first kind to $K$, then
\begin{equation}
\mathbb{E} \Big[ \exp \big( a \int^T_0 V_s ds \big) \Big] = \exp \Big[ \kappa \phi \int^T_0 g(a, s) ds + V_0 \int^T_0 g(a, T-s) L(ds) \Big].
\end{equation}
\end{theorem}
\begin{proof}
Note $g(a, t)$ in \eqref{Eq:g} corresponds to \cite[Equation (4.3)]{abi2017affine} with $u = 0$ and $f = a$. \cite[Theorem 4.3]{abi2017affine} shows the equivalence between \cite[Equation (4.4)]{abi2017affine} and \cite[Equation (4.6)]{abi2017affine}. For $t = T$, the expressions in \cite[Equation (4.4)-(4.6)]{abi2017affine} indicate that
\begin{equation}
a \int^T_0 V_s ds = Y_0 - \frac{\sigma^2}{2} \int^T_0 g^2(a, T-s) V_s ds + \sigma \int^T_0 g(a, T-s) \sqrt{V_s} dB_s,
\end{equation}
with
\begin{equation}
Y_0 = \kappa \phi \int^T_0 g(a, s) ds + V_0 \int^T_0 \big[ a - \kappa g(a, s) + \frac{\sigma^2}{2} g^2(a, s) \big] ds.
\end{equation}
As $g(a, \cdot)$ is continuous on $[0, T]$ and therefore bounded, $\exp\big( - \frac{\sigma^2}{2} \int^t_0 g^2(a, T-s) V_s ds + \sigma \int^t_0 g(a, T-s) \sqrt{V_s} dB_s\big)$ is a martingale by \cite[Lemma 7.3]{abi2017affine}. Therefore,
\begin{equation}
\mathbb{E} \Big[ \exp \big( a \int^T_0 V_s ds \big) \Big] = \exp(Y_0) = \exp\Big[ \kappa \phi \int^T_0 g(a, s) ds + V_0 \int^T_0 \big[ a - \kappa g(a, s) + \frac{\sigma^2}{2} g^2(a, s) \big] ds \Big].
\end{equation}
Note that $K * L = {\rm id} $ implies
\begin{equation}
\int^T_0 \big[ a - \kappa g(a, s) + \frac{\sigma^2}{2} g^2(a, s) \big] ds = \int^T_0 g(a, T-s) L(ds).
\end{equation}
The result follows.
\end{proof}
Theorem \ref{Thm:ExpV} recovers the same expression for $\mathbb{E} \Big[ \exp \big( a \int^T_0 V_s ds \big) \Big]$ in \cite[Theorem 3.2]{eleuch2018perfect}. We stress that the proof circumvents the use of the Hawkes processes. In addition, we mention \cite{gerhold2018moment}, which examines the moment explosions in the rough Heston model, as a related reference.
\section{Mean-variance portfolio selection}\label{Sec:MV}
Let $u_t \triangleq \sqrt{V_t} \pi_t$ be the investment strategy, where $\pi_t$ is the amount of wealth invested in the stock. Then wealth process $X_t$ satisfies
\begin{equation}\label{Eq:wealth}
d X_t = \big(r_t X_t + \theta \sqrt{V_t} u_t \big) dt + u_t dW_{1t}, \quad X_0 = x_0 > 0.
\end{equation}
\begin{definition}
An investment strategy $u(\cdot)$ is said to be admissible if
\begin{enumerate}[label={(\arabic*).}]
\item $u(\cdot)$ is $\mathbb{F}$-adapted;
\item $\mathbb{E}\Big[ \Big(\int^T_0 |\sqrt{V_t} u_t |dt \Big)^2 \Big] < \infty$ and $\mathbb{E}\Big[ \int^T_0 |u_t|^2 dt \Big] < \infty$; and
\item the wealth process (\ref{Eq:wealth}) has a unique solution in the sense of \cite[Chapter 1, Definition 6.15]{yong1999book}, with $\mathbb{P}$-$\mbox{{\rm a.s.}}$ continuous paths.
\end{enumerate}
The set of all of the admissible investment strategies is denoted as ${\mathcal U}$.
\end{definition}
\begin{remark}
In Condition (1), $\mathbb{F}$ is possibly strictly larger than the Brownian filtration of $W = (W_1, W_2)$, which means that extra information in addition to $W$ can be used to construct an admissible strategy. In general, $u$ can rely on a local $\mathbb{P}$-martingale that is strongly $\mathbb{P}$-orthogonal to $W$. See hedging strategy (3.6) in \cite[Theorem 3.1]{jeanblanc2012mean} for such examples. However, our optimal strategy $u^*$ turns out to only depend on the variance $V$ and Brownian motion $W$, as shown in Theorem \ref{Thm:Sol}.
\end{remark}
\begin{remark} We emphasize once again that the underlying probability space and Brownian motions are not parts of our control. Therefore, our formulation should still be referred to as a strong formulation. Readers may refer to \cite[Chapter 2, Section 4]{yong1999book} for discussions of the difference between strong and weak formulations of stochastic control problems.
\end{remark}
The MV portfolio selection in continuous-time is the following problem\footnote{There are several equivalent formulations.}.
\begin{equation}\label{Eq:obj}
\left\{\begin{array}{l}{
\min _{ u(\cdot) \in {\mathcal U}} J \left(x_{0} ; u(\cdot)\right) = \mathbb{E} \left[ (X_T - c)^2 \right]}, \\
\text{ subject to } \mathbb{E}[X_T] = c, \\
(X(\cdot), u(\cdot)) \text { satisfy (\ref{Eq:wealth})}.
\end{array}\right.
\end{equation}
The constant $c$ is the target wealth level at the terminal time $T$. We assume $c\geq x_0 e^{\int^T_0 r_s ds}$ following \cite{lim2002complete,lim2004incomplete,shen2015mean}. Otherwise, a trivial strategy that puts all of the wealth into the risk-free asset can dominate any other admissible strategy. The MV problem is said to be feasible for $c \geq x_0 e^{\int^T_0 r_s ds}$ if there exists a $u(\cdot) \in {\mathcal U}$ that satisfies $\mathbb{E}[X_T] = c$. Note that $r_t > 0$ is deterministic and $\mathbb{E}[\int^T_0 V_t dt] > 0$. It is then clear that the feasibility of our problem is guaranteed for any $c\geq x_0 e^{\int^T_0 r_s ds}$ by a slight modification to the proof in \cite[Propsition 6.1]{lim2004incomplete}.
As Problem (\ref{Eq:obj}) has a constraint, it is equivalent to the following max-min problem \cite{luenberger1968opt}.
\begin{equation}\label{Eq:maxminobj}
\left\{\begin{array}{l}{
\max _{\eta \in \mathbb{R} } \min _{ u(\cdot) \in {\mathcal U}} J\left(x_{0} ; u(\cdot)\right) = \mathbb{E} \left[(X_T - (c-\eta))^{2}\right] - \eta^{2}}, \\
(X(\cdot), u(\cdot)) \text { satisfy (\ref{Eq:wealth})}.
\end{array}\right.
\end{equation}
Let $\zeta = c - \eta$ and consider the inner Problem (\ref{Eq:innerobj}) of (\ref{Eq:maxminobj}) first.
\begin{equation}\label{Eq:innerobj}
\left\{\begin{array}{l}{
\min _{ u(\cdot) \in {\mathcal U}} J\left(x_{0} ; u(\cdot)\right) = \mathbb{E} \left[(X_T - \zeta )^{2}\right] - \eta^{2} }, \\
(X(\cdot), u(\cdot)) \text { satisfy (\ref{Eq:wealth})}.
\end{array}\right.
\end{equation}
\section{Optimal investment strategy}\label{Sec:Sol}
To solve Problem (\ref{Eq:innerobj}), we introduce a new probability measure $ \tilde \mathbb{P}$ by
\begin{equation}\label{Eq:tildeP}
\left. \frac{d \tilde \mathbb{P}}{d\mathbb{P}} \right|_{{\mathcal F}_t} = \exp\Big( - 2 \theta^2 \int^t_0 V_s ds - 2 \theta \int^t_0 \sqrt{V_s} dW_{1s} \Big),
\end{equation}
where the stochastic exponential is a true martingale \cite[Lemma 7.3]{abi2017affine}. Then $\tilde W_{1t} \triangleq W_{1t} + 2 \theta \int^t_0 \sqrt{V_s} ds$ is a new Brownian motion under $\tilde \mathbb{P}$. Hence,
\begin{equation}
V_{t}=V_{0}+ \int_{0}^{t} K(t-s)\left( \kappa \phi - \lambda V_{s}\right) d s + \int_{0}^{t} K(t-s) \sigma \sqrt{V_{s}} d \tilde B_{s},
\end{equation}
where $\lambda = \kappa + 2\theta \rho \sigma$ and $d \tilde B_s = \rho d\tilde W_{1s} + \sqrt{1- \rho^2} d W_{2s}$.
Denote $\tilde \mathbb{E}[\cdot]$ and $\tilde \mathbb{E}[ \cdot |{\mathcal F}_t]$ as the $\tilde \mathbb{P}$-expectation and conditional $\tilde \mathbb{P}$-expectation, respectively. The forward variance under $\tilde \mathbb{P}$ is the conditional $\tilde \mathbb{P}$-expected variance: $ \mathbb{\tilde E} \left[V_{s} | \mathcal{F}_{t}\right] \triangleq \xi_{t}(s)$. The following identity is proven in \cite[Propsition 3.2]{keller2018affine} by an application of \cite[Lemma 4.2]{abi2017affine}.
\begin{equation}\label{Eq:xi}
\xi_{t}(s) = \mathbb{\tilde E} \left[V_{s} | \mathcal{F}_{t}\right]=\xi_{0}(s)+\int_0^t \frac{1}{\lambda} R_{\lambda}(s-u) \sigma \sqrt{V_{u}} d \tilde B_{u},
\end{equation}
where
\begin{equation}
\xi_{0}(s) = \left(1-\int_{0}^{s} R_{\lambda}(u) d u\right) V_{0} + \frac{\kappa \phi}{\lambda} \int_{0}^{s} R_{\lambda}(u) du,
\end{equation}
and $R_\lambda$ is the resolvent of $\lambda K$ such that
\begin{equation}\label{Eq:R_lambda}
\lambda K * R_\lambda = R_\lambda * ( \lambda K) = \lambda K - R_\lambda.
\end{equation}
If $\lambda = 0$, interpret $R_\lambda/\lambda = K$ and $R_\lambda = 0$.
Consider the stochastic process,
\begin{equation}\label{Eq:M}
M_t = 2 \exp \Big[ \int^T_t \big(2 r_s - \theta^2 \xi_t(s) + \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-s) \xi_t(s) \big) ds \Big],
\end{equation}
where
\begin{equation}\label{Eq:psi}
\psi(t) = \int^t_0 K(t - s) \big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(s) - \lambda \psi(s) - \theta^2 \big] ds.
\end{equation}
The existence and uniqueness of the solution to (\ref{Eq:psi}) are established in Lemma \ref{Lem:psi}.
The process $M$ is the key to applying the completion of squares technique in Theorem \ref{Thm:Sol}, inspired by \cite{lim2002complete,lim2004incomplete,shen2015mean}. Heuristically speaking, the non-Markovian and non-semimartingale characteristics of the Volterra Heston model are overcome by considering $M$. The construction of $M$ is based on the following observations. To make a completion of squares, we need an auxiliary process $M$ as an additional stochastic factor in a place consistent with previous studies of MV portfolios under semimartingales. The completion of squares procedure for proving Theorem \ref{Thm:Sol} indicates that $M$ should satisfy \eqref{Eq:dM}. We then link $M$ with the conditional expectation in \eqref{Eq:TransM} via a proper transformation. The exponential-affine transform formula in \cite[Equation (4.7)]{abi2017affine} is applied to obtain \eqref{Eq:M}.
\begin{theorem}\label{Thm:M}
Assume Assumption \ref{Assum:K} holds and (\ref{Eq:psi}) has a unique continuous solution on $[0, T]$, then $M$ satisfies the following properties.
\begin{enumerate}[label={(\arabic*).}]
\item $M_t$ is essentially bounded and $0< M_t < 2e^{2\int^T_t r_s ds}$, $\mathbb{P}$-$\mbox{{\rm a.s.}}$, $\forall \; t \in [0, T)$. $M_T = 2$.
\item Apply It\^o's lemma to $M$ on $t$, then
\begin{equation}\label{Eq:dM}
dM_t = \big[ -2 r_t + \theta^2 V_t \big] M_t dt + \big[ 2 \theta \sqrt{V_t} U_{1t} + \frac{U^2_{1t}}{M_t} \big] dt + U_{1t} dW_{1t} + U_{2t} dW_{2t},
\end{equation}
where
\begin{align}
U_{1t} &= \rho \sigma M_t \sqrt{V_t} \psi(T-t), \label{Eq:U1simp}\\
U_{2t} &= \sqrt{1-\rho^2} \sigma M_t \sqrt{V_t} \psi(T-t). \label{Eq:U2simp}
\end{align}
\item
\begin{equation}
M_0 = 2 \exp \Big[ \int^T_0 2 r_s ds + \kappa \phi \int^T_0 \psi(s) ds + V_0 \int^T_0 \big[ \frac{(1-2\rho^2)\sigma^2}{2} \psi^2(s) - \lambda \psi(s) - \theta^2 \big] ds \Big].
\end{equation}
Furthermore, for fractional kernel $K(t) = \frac{t^{\alpha-1}}{\Gamma(\alpha)}$, denote the fractional integral as $I^\alpha \psi(t) = K * \psi(t)$. Then
\begin{equation}\label{Eq:fracM0}
M_0 = 2 \exp \Big[ \int^T_0 2 r_s ds + \kappa \phi I^1 \psi(T) + V_0 I^{1-\alpha}\psi (T) \Big].
\end{equation}
\item $\mathbb{E}\Big[ \big(\int^T_0 U^2_{it} dt \big)^{p/2} \Big] < \infty$ for $p \geq 1$ , $ i = 1, 2$.
\end{enumerate}
\end{theorem}
\begin{proof}
{\bf Property (1)}.
It is straightforward to see that $M_t > 0$ in (\ref{Eq:M}). As for the upper bound, if $1-2\rho^2 = 0$, note $\int^T_t \xi_t(s) ds > 0$, $\mathbb{P}$-$\mbox{{\rm a.s.}}$ by Lemma \ref{Lem:Positive}, then $M_t < 2e^{2\int^T_t r_s ds}$, $\mathbb{P}$-$\mbox{{\rm a.s.}}$. If $1-2\rho^2 \neq 0$, we claim
\begin{equation}\label{Eq:TransM}
M^{1-2\rho^2}_t = 2^{1-2\rho^2} \exp\big[2 (1-2\rho^2) \int^T_t r_s ds\big] \mathbb{\tilde E} \Big[ \exp \big(- \theta^2(1-2\rho^2) \int^T_t V_s ds\big) \Big| {\mathcal F}_t \Big].
\end{equation}
It is equivalent to show that
\begin{align}\label{Eq:Mtrans}
& \mathbb{\tilde E} \Big[ \exp \big(- \theta^2(1-2\rho^2) \int^T_t V_s ds\big) \Big| {\mathcal F}_t \Big] \\
& = \exp \Big[ \int^T_t \big( - (1-2\rho^2) \theta^2 \xi_t(s) + \frac{(1-2\rho^2)^2 \sigma^2}{2} \psi^2(T-s) \xi_t(s) \big) ds \Big]. \nonumber
\end{align}
Denote $ \tilde \psi = (1 - 2 \rho^2) \psi$. Then $\tilde \psi$ satisfies
\begin{equation}
\tilde \psi = K * \big( \frac{\sigma^2}{2} \tilde \psi^2 - \lambda \tilde \psi - (1 - 2\rho^2) \theta^2 \big).
\end{equation}
Therefore, (\ref{Eq:Mtrans}) holds for all $t \in [0, T]$ by \cite[Theorem 4.3]{abi2017affine} applying to $\tilde \psi$. The martingale assumption in \cite[Theorem 4.3]{abi2017affine} is verified by \cite[Lemma 7.3]{abi2017affine}.
If $ 1 - 2 \rho^2 > 0$, then $\mathbb{\tilde E} \Big[ \exp \big(- \theta^2(1-2\rho^2) \int^T_t V_s ds\big) \Big| {\mathcal F}_t \Big] < 1$, $\mathbb{P}$-$\mbox{{\rm a.s.}}$, which implies $M_t < 2e^{2\int^T_t r_s ds}$, $\mathbb{P}$-$\mbox{{\rm a.s.}}$. $ 1 - 2 \rho^2 < 0$ can be discussed similarly. Property (1) is proved.
\noindent {\bf Property (2).}
Denote $ M_t = 2 e^{Z_t} $ in (\ref{Eq:M}) with proper $Z_t$. We first derive the equation for $dZ_t$. From (\ref{Eq:xi}), apply It\^o's lemma to $\xi_t(s)$ on time $t$ and get
\begin{equation}
d \xi_t(s) = \frac{1}{\lambda} R_{\lambda}(s-t) \sigma \sqrt{V_t} d \tilde B_t.
\end{equation}
Then
\begin{align*}
d Z_t = &\big[ - 2 r_t + \theta^2 V_t - \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-t) V_t \big] dt \\
& - \theta^2 \int^T_t \frac{1}{\lambda} R_{\lambda}(s-t) \sigma \sqrt{V_t} d \tilde B_t ds + \frac{(1-2\rho^2) \sigma^2}{2} \int^T_t \psi^2(T-s) \frac{1}{\lambda} R_{\lambda}(s-t) \sigma \sqrt{V_t} d \tilde B_t ds \\
= &\big[ - 2 r_t + \theta^2 V_t - \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-t) V_t \big] dt \\
& - \theta^2 \int^T_t \sigma \frac{1}{\lambda} R_{\lambda}(s-t) ds \sqrt{V_t} d \tilde B_t + \frac{(1-2\rho^2) \sigma^2}{2} \int^T_t \sigma \psi^2(T-s) \frac{1}{\lambda} R_{\lambda}(s-t) ds \sqrt{V_t} d \tilde B_t \\
= &\big[ - 2 r_t + \theta^2 V_t - \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-t) V_t \big] dt \\
& + d \tilde B_t \cdot \sigma \sqrt{V_t} \int^T_t \Big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-s) - \theta^2 \Big] \frac{1}{\lambda} R_{\lambda}(s-t) ds .
\end{align*}
The second equality is guaranteed by the stochastic Fubini theorem \cite{veraar2012fubini}.
We claim the following representation for (\ref{Eq:U1simp})-(\ref{Eq:U2simp}).
\begin{align}
U_{1t} & = \sigma \rho M_t \sqrt{V_t} \int^T_t \Big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-s) - \theta^2 \Big] \frac{1}{\lambda} R_{\lambda}(s-t) ds, \label{Eq:U1}\\
U_{2t} & = \sigma \sqrt{1 - \rho^2} M_t \sqrt{V_t} \int^T_t \Big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-s) - \theta^2 \Big] \frac{1}{\lambda} R_{\lambda}(s-t) ds. \label{Eq:U2}
\end{align}
Indeed, we only have to show
\begin{equation}\label{Eq:Uequivalent}
\int^T_t \Big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-s) - \theta^2 \Big] \frac{1}{\lambda} R_{\lambda}(s-t) ds = \psi (T-t).
\end{equation}
Although one can verify \eqref{Eq:Uequivalent} in the same fashion as \cite[Lemma 4.4]{abi2017affine}, we still detail the derivation here for a self-contained paper. As
\begin{align*}
& \int^T_t \Big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-s) - \theta^2 \Big] \frac{1}{\lambda} R_{\lambda}(s-t) ds \\
= & \int^{T - t}_0 \Big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T- t - s) - \theta^2 \Big] \frac{1}{\lambda} R_{\lambda}(s) ds \\
= & \big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2 - \theta^2 \big] * \frac{1}{\lambda} R_{\lambda}(T - t),
\end{align*}
we have
\begin{align*}
& \int^T_t \Big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-s) - \theta^2 \Big] \frac{1}{\lambda} R_{\lambda}(s-t) ds - \psi (T-t)\\
= & \big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2 - \theta^2 \big] * \frac{1}{\lambda} R_{\lambda}(T - t) - K* \big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2 - \lambda \psi - \theta^2 \big](T-t) \\
= & \big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2 - \theta^2 \big] * \big[ \frac{1}{\lambda} R_{\lambda} - K \big](T - t) + \lambda K*\psi(T-t)\\
= & - R_\lambda*K*\big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2 - \theta^2 \big](T-t) + \lambda K*\psi(T-t) .
\end{align*}
The application of (\ref{Eq:psi}) leads to
\begin{equation}
R_\lambda * \psi = R_\lambda*K*\big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2 - \lambda \psi - \theta^2 \big].
\end{equation}
Consequently,
\begin{align*}
& - R_\lambda*K*\big[ \frac{(1-2\rho^2) \sigma^2}{2} \psi^2 - \theta^2 \big](T-t) + \lambda K*\psi(T-t) \\
& = \big[ \lambda K - R_\lambda - \lambda K* R_\lambda \big]* \psi(T-t) = 0.
\end{align*}
This shows that
\begin{align}
dZ_t = &\big[ - 2 r_t + \theta^2 V_t - \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-t) V_t \big] dt + \frac{U_{1t}}{M_t} d\tilde W_{1t} + \frac{U_{2t}}{M_t} d W_{2t}.
\end{align}
Applying It\^o's lemma to $M_t= 2e^{Z_t}$ with function $f(z) = 2 e^z$ yields
\begin{align*}
d M_t = & M_t d Z_t + \frac{1}{2} M_t dZ_t dZ_t \\
= & M_t \big[ - 2 r_t + \theta^2 V_t - \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-t) V_t \big] dt + \frac{U^2_{1t} + U^2_{2t}}{2 M_t} dt \\
& + U_{1t} d\tilde W_{1t} + U_{2t} d W_{2t} \\
= & \big[ -2 r_t + \theta^2 V_t \big] M_t dt + \big[ 2 \theta \sqrt{V_t} U_{1t} + \frac{U^2_{1t}}{M_t} \big] dt + U_{1t} dW_{1t} + U_{2t} dW_{2t}.
\end{align*}
\noindent {\bf Property (3)}.
The proof for the property of $Y_t$ in \cite[Theorem 4.3]{abi2017affine} indicates
\begin{align*}
& \int^T_0 \big[ - \theta^2 \xi_0(s) + \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-s) \xi_0(s) \big] ds \\
& = \int^T_0 \big[ - \theta^2 V_0 + (\kappa \phi - \lambda V_0) \psi(s) + \frac{(1-2\rho^2)\sigma^2}{2} \psi^2(s) V_0 \big] ds.
\end{align*}
Under the fractional kernel, we show by integration by parts that
\begin{equation}
\int^T_0 \big[ - \theta^2 - \lambda \psi(s) + \frac{(1-2\rho^2)\sigma^2}{2} \psi^2(s) \big] ds = I^{1 - \alpha} \psi(T).
\end{equation}
This gives the desired result.
\noindent {\bf Property (4)}.
It is sufficient to consider the case with $p > 2$. As $\psi(t)$ is continuous on $[0, T]$ and $M_t$ is essentially bounded,
\begin{align*}
\mathbb{E}\Big[ \big(\int^T_0 U^2_{it} dt \big)^{p/2} \Big] \leq C \mathbb{E}\Big[ \big(\int^T_0 V_t dt \big)^{p/2} \Big] \leq C \int^T_0 \mathbb{E} \big[ V^{p/2}_t \big] dt \leq C\sup_{t \in [0, T]} \mathbb{E}\big[ V^{p/2}_t \big] < \infty.
\end{align*}
The last term is finite by \cite[Lemma 3.1]{abi2017affine}.
\end{proof}
We first propose a candidate optimal control $u^*$. In the following theorem, we prove the admissibility of $u^*$ and the integrability of the corresponding $X^*$. Theorem \ref{Thm:X^*u^*} is in the spirit of \cite{lim2002complete,lim2004incomplete,shen2015mean}. Finally, we prove the optimality of $u^*$ in \eqref{Eq:u*} by Theorem \ref{Thm:Sol}.
\begin{theorem}\label{Thm:X^*u^*}
Assume Assumption \ref{Assum:K} holds and (\ref{Eq:psi}) has a unique continuous solution on $[0, T]$. Denote $A_t \triangleq \theta + \rho \sigma \psi(T-t)$. Suppose Assumption \ref{Assum:V} holds with constant $a$ given the following:
\begin{equation}\label{Eq:Const_a}
a = \max \Big\{ 2 p |\theta| \sup_{t \in [0, T]} | A_t |, (8p^2 - 2p) \sup_{t \in [0, T]} A^2_t \Big\}, \quad \text{for certain } p > 2.
\end{equation}
Consider
\begin{equation}\label{Eq:u*}
u^*(t) = (\theta + \rho \sigma \psi (T-t)) \sqrt{V_t} (\zeta^* e^{-\int^T_t r_s ds} - X^*_t),
\end{equation}
where $X^*_t$ is the wealth process under $u^*$ and $\zeta^* = c - \eta^*$ with
\begin{equation}\label{Eq:eta*}
\eta^* = \frac{e^{-\int^T_0 r_s ds}M_0 x_0 - e^{-\int^T_0 2 r_s ds} M_0 c}{2 - e^{-\int^T_0 2 r_s ds} M_0}.
\end{equation}
$u^*(\cdot)$ in (\ref{Eq:u*}) is admissible and $X^*$ under $u^*(\cdot)$ satisfies
\begin{equation}\label{Eq:X*Integral}
\mathbb{E} \Big[ \sup_{ t \in [0, T]} | X^*_t |^p \Big] < \infty,
\end{equation}
for $p \geq 1$. Moreover,
\begin{equation}\label{Eq:X*bound}
\zeta^* e^{-\int^T_t r_s ds} - X^*_t \geq 0, \quad \text{$\mathbb{P}$-$\mbox{{\rm a.s.}}$}, \forall \; t \in [0, T].
\end{equation}
\end{theorem}
\begin{proof}
The wealth process under $u^*$ is given by
\begin{equation}
\left\{
\begin{array}{rcl}
dX^*_t &=& \big[ r_t X^*_t + \theta A_t V_t (\zeta^* e^{-\int^T_t r_s ds} - X^*_t) \big] dt + A_t \sqrt{V_t} (\zeta^* e^{-\int^T_t r_s ds} - X^*_t) dW_{1t}, \\
X^*_0 &=& x_0.
\end{array}\right.
\end{equation}
To find a solution to $X^*$, define $Y_t$ satisfying
\begin{equation}
\left\{
\begin{array}{rcl}
dY_t &=& - r_t Y_t dt - \theta \sqrt{V_t} Y_t dW_{1t} + Y_t \sqrt{1 - \rho^2} \sigma \psi(T-t) \sqrt{V_t} dW_{2t}, \\
Y_0 &=& M_0 (\zeta^* e^{-\int^T_0 r_s ds} -x_0 ).
\end{array}\right.
\end{equation}
The unique solution of $Y_t$ is given by
\begin{align*}
Y_t =& Y_0 \exp \Big[ - \frac{1}{2} \int^t_0 \big( 2r_s + \theta^2 V_s + (1 - \rho^2) \sigma^2 \psi^2(T-s) V_s \big) ds - \int^t_0 \theta \sqrt{V_s} dW_{1s} \\
& \qquad \quad + \int^t_0 \sqrt{1 - \rho^2} \sigma \psi(T-s) \sqrt{V_s} dW_{2s} \Big].
\end{align*}
It\^o's lemma yields
\begin{equation}\label{Eq:X*2Y}
X^*_t = \zeta^* e^{-\int^T_t r_s ds} - \frac{Y_t}{M_t}
\end{equation}
as the unique solution of the wealth process. Indeed,
\begin{equation}
d \frac{Y_t}{M_t} = \Big[ r_t \frac{Y_t}{M_t} - \theta A_t V_t \frac{Y_t}{M_t} \Big] dt - A_t \sqrt{V_t} \frac{Y_t}{M_t} dW_{1t}.
\end{equation}
The existence of $u^*$ is also guaranteed by the existence of the solution $X^*$. Furthermore, $\frac{Y_t}{M_t} = \frac{Y_0}{M_0} \Phi(t)$, where
\begin{align*}
\Phi(t) \triangleq & \exp \Big[ \int^t_0 \big[ r_s - \big( \theta A_s + \frac{A^2_s}{2} \big) V_s \big] ds - \int^t_0 A_s \sqrt{V_s} dW_{1s} \Big].
\end{align*}
As $Y_t/M_t \geq 0$, (\ref{Eq:X*bound}) follows from (\ref{Eq:X*2Y}).
For (\ref{Eq:X*Integral}), note that by Doob's maximal inequality and \cite[Lemma 7.3]{abi2017affine},
\begin{align*}
& \mathbb{E} \Big[ \sup_{ t \in [0, T]} | \Phi(t) |^p \Big] \\
& \leq C \mathbb{E} \Big[ \sup_{ t \in [0, T]} \Big| e^{- \int^t_0 \theta A_s V_s ds} \Big|^{2p} \Big] + C \mathbb{E} \Big[ \sup_{ t \in [0, T]} \Big| \exp \Big(- \int^t_0 \frac{A^2_s}{2} V_s ds - \int^t_0 A_s \sqrt{V_s} dW_{1s} \Big) \Big|^{2p}\Big] \\
& \leq C \mathbb{E} \Big[ e^{2p \int^T_0 |\theta A_s| V_s ds} \Big] + C \mathbb{E} \Big[ \exp \Big(- \int^T_0 p A^2_s V_s ds - \int^T_0 2p A_s \sqrt{V_s} dW_{1s} \Big) \Big].
\end{align*}
The first term is finite by Assumption \ref{Assum:V} with constant $a = 2 p |\theta| \sup_{t \in [0, T]} | A_t |$. The second term is also finite. In fact, by H\"older's inequality and Assumption \ref{Assum:V} with a constant $a = (8p^2 - 2p) \sup_{t \in [0, T]} A^2_t$,
\begin{align*}
& \mathbb{E} \Big[ \exp \Big(- \int^T_0 p A^2_s V_s ds - \int^T_0 2p A_s \sqrt{V_s} dW_{1s} \Big) \Big] \\
& \leq \Big\{ \mathbb{E} \Big[ e^{(8p^2 - 2p) \int^T_0 A^2_s V_s ds} \Big] \Big\}^{1/2} \Big\{ \mathbb{E} \Big[ \exp \Big(- 8 p^2 \int^T_0 A^2_s V_s ds - 4 p \int^T_0 A_s \sqrt{V_s} dW_{1s} \Big) \Big] \Big\}^{1/2} \\
& < \infty.
\end{align*}
$\mathbb{E} \Big[ \sup_{ t \in [0, T]} | X^*_t |^p \Big] < \infty$ is proved. As for admissibility of $u^*$, $u^*$ is $\mathbb{F}$-adapted at first. For integrability, let $1/\hat{p} + 1/\hat{q} = 1$, $\hat{p},\, \hat{q} >1$, we have
\begin{align*}
& \mathbb{E}\Big[ \Big(\int^T_0 |\sqrt{V_t} u^*_t |dt \Big)^2 \Big] \leq C \mathbb{E}\Big[ \Big(\int^T_0 |A_t V_t \Phi(t) |dt \Big)^2 \Big] \\
& \leq C \mathbb{E}\Big[ \sup_{ t \in [0, T]} \Phi^2(t) \Big(\int^T_0 V_t dt \Big)^2 \Big] \leq C \Big\{\mathbb{E}\Big[ \sup_{ t \in [0, T]} \Phi^{2\hat{p}} (t) \Big] \Big\}^{1/\hat{p}} \Big\{\mathbb{E}\Big[ \Big(\int^T_0 V_t dt \Big)^{2 \hat{q}} \Big] \Big\}^{1/\hat{q}} \\
& \leq C \Big\{ \mathbb{E}\Big[ \sup_{ t \in [0, T]} \Phi^{2\hat{p}} (t) \Big] \Big\}^{1/\hat{p}} \Big( \sup_{t \in [0, T]} \mathbb{E}\big[ \ V_t^{2 \hat{q}} \big] \Big)^{1/\hat{q}} < \infty
\end{align*}
and
\begin{align*}
& \mathbb{E}\Big[ \int^T_0 |u^*_t|^2 dt \Big] \leq C \mathbb{E}\Big[ \int^T_0 A^2_t V_t \Phi^2(t) dt \Big] \\
& \leq C \mathbb{E}\Big[ \sup_{ t \in [0, T]} \Phi^2(t) \int^T_0 V_t dt \Big] \leq C \Big\{ \mathbb{E}\Big[ \sup_{ t \in [0, T]} \Phi^{2\hat{p}} (t) \Big] \Big\}^{1/\hat{p}} \Big\{\mathbb{E}\Big[ \Big(\int^T_0 V_t dt \Big)^{\hat{q}} \Big] \Big\}^{1/\hat{q}} \\
& \leq C \Big\{ \mathbb{E}\Big[ \sup_{ t \in [0, T]} \Phi^{2\hat{p}} (t) \Big] \Big\}^{1/\hat{p}} \Big( \sup_{t \in [0, T]} \mathbb{E}\big[ \ V_t^{\hat{q}} \big] \Big)^{1/\hat{q}} < \infty.
\end{align*}
The last terms in the two inequalities above are finite by \cite[Lemma 3.1]{abi2017affine} and take $p = 2 \hat{p}$.
\end{proof}
We are now ready to prove $u^*$ in \eqref{Eq:u*} is optimal and to derive the efficient frontier.
\begin{theorem}\label{Thm:Sol}
Suppose the assumptions in Theorem \ref{Thm:X^*u^*} hold, then the optimal investment strategy for Problem (\ref{Eq:obj}) is given by \eqref{Eq:u*}. Moreover, \eqref{Eq:u*} is unique under a given solution $(S, V, W_1, W_2)$ to (\ref{vol})-(\ref{stock}). The variance of $X^*_T$ is
\begin{equation}\label{Eq:VarX*}
\text{Var} [X^*_T] = \frac{M_0}{ 2 - e^{-\int^T_0 2 r_s ds} M_0} \big( c e^{-\int^T_0 r_s ds} - x_0 \big)^2.
\end{equation}
\end{theorem}
\begin{proof}
First, we consider the inner Problem (\ref{Eq:innerobj}) with an arbitrary $\zeta \in \mathbb{R}$. Denote $h_t = \zeta e^{-\int^T_t r_s ds}$. By It\^o's lemma with the property of $M$ and completing the square, for any admissible strategy $u$,
\begin{align*}
& d \frac{1}{2} M_t (X_t - h_t)^2 \\
& = \frac{1}{2} \big[ (X_t - h_t)^2 M_t \theta^2 V_t + 2 (X_t - h_t)^2 \theta \sqrt{V_t} U_{1t} + (X_t - h_t)^2 \frac{U^2_{1t}}{M_t} + 2 M_t (X_t - h_t) \theta \sqrt{V_t} u_t \\
& \qquad + 2 ( X_t - h_t) u_t U_{1t} + M_t u^2_t \big] dt \\
& \quad + \frac{1}{2} \big[ (X_t - h_t)^2 U_{1t} + 2 M_t (X_t - h_t) u_t \big] dW_{1t} + \frac{1}{2} (X_t - h_t)^2 U_{2t} dW_{2t} \\
& = \frac{1}{2} M_t \Big[ u_t + \big(\theta \sqrt{V_t} + \frac{U_{1t}}{M_t}\big)(X_t - h_t) \Big]^2 dt \\
& \quad + \frac{1}{2} \big[ (X_t - h_t)^2 U_{1t} + 2 M_t (X_t - h_t) u_t \big] dW_{1t} + \frac{1}{2} (X_t - h_t)^2 U_{2t} dW_{2t}.
\end{align*}
As $M_t$ and $h_t$ are bounded, $\mathbb{E}\Big[ \int^T_0 U^2_{it} dt \Big] < \infty$ for $i =1, 2$, $u_t$ is admissible, and $X_t$ has $\mathbb{P}$-$\mbox{{\rm a.s.}}$ continuous paths, then stochastic integrals
\begin{equation*}
\int^t_0 \big[ (X_s - h_s)^2 U_{1s} + 2 M_s (X_s - h_s) u_s \big] dW_{1s} \quad \text{and} \quad \int^t_0 (X_s - h_s)^2 U_{2s} dW_{2s}
\end{equation*}
are $(\mathbb{F}, \mathbb{P})$-local martingales. There is an increasing localizing sequence of stopping times $\{ \tau_k \}_{k = 1, 2, ...}$ such that $\tau_k \uparrow T$ when $ k \rightarrow \infty$. The local martingales stopped by $\{ \tau_k \}_{k = 1, 2, ...}$ are true martingales. Consequently,
\begin{equation}
\frac{1}{2} \mathbb{E}[M_{\tau_k} (X_{\tau_k} - h_{\tau_k})^2 ] = \frac{1}{2} M_0 (x_0 - h_0)^2 + \frac{1}{2} \mathbb{E} \Big[ \int^{\tau_k}_0 M_t \Big( u_t + \big(\theta \sqrt{V_t} + \frac{U_{1t}}{M_t}\big)(X_t - h_t) \Big)^2 dt \Big].
\end{equation}
From (\ref{Eq:wealth}), by Doob's maximal inequality and the admissibility of $u(\cdot)$,
\begin{equation}
\mathbb{E}[X^2_{\tau_k}] \leq C \Big[ x^2_0 + \mathbb{E} \Big[ \big( \int^T_0 | u_t \sqrt{V_t}| dt \big)^2 \Big] + \mathbb{E} \Big[ \int^T_0 u^2_t dt \Big] \Big] < \infty.
\end{equation}
Then $M_{\tau_k} (X_{\tau_k} - h_{\tau_k})^2$ is dominated by a non-negative integrable random variable for all $k$. Sending $k$ to infinity, by the dominated convergence theorem and the monotone convergence theorem, we derive
\begin{equation}\label{Eq:Square}
\mathbb{E}[ (X_T - \zeta )^2 ] = \frac{1}{2} M_0 (x_0 - h_0)^2 + \frac{1}{2} \mathbb{E} \Big[ \int^T_0 M_t \Big( u_t + \big(\theta \sqrt{V_t} + \frac{U_{1t}}{M_t}\big)(X_t - h_t) \Big)^2 dt \Big].
\end{equation}
Therefore, the cost functional $\mathbb{E}[ (X_T - \zeta )^2 ]$ is minimized when
\begin{equation}
u_t = - \big(\theta \sqrt{V_t} + \frac{U_{1t}}{M_t}\big)(X_t - h_t).
\end{equation}
Then $ \mathbb{E}[ (X_T - \zeta )^2 ] = \frac{1}{2} M_0 (x_0 - h_0)^2$. The uniqueness of $u^*$ follows directly from \eqref{Eq:Square} and $M_t > 0$, $\mathbb{P}$-$\mbox{{\rm a.s.}}$, $\forall \; t \in [0, T]$. To solve the outer maximization problem in (\ref{Eq:maxminobj}), consider
\begin{equation}
J(x_0; u(\cdot)) = \frac{1}{2} M_0 \big[x_0 - (c - \eta) e^{- \int^T_0 r_s ds} \big]^2 - \eta^2.
\end{equation}
The first and second order derivatives are
\begin{align*}
\frac{\partial J}{\partial \eta} &= M_0 \big[x_0 - (c - \eta) e^{- \int^T_0 r_s ds} \big] e^{- \int^T_0 r_s ds} - 2 \eta, \\
\frac{\partial^2 J}{\partial \eta^2} &= M_0 e^{- 2 \int^T_0 r_s ds} - 2 < 0,
\end{align*}
where we have used the strict inequality $M_0 < 2 e^{\int^T_0 2 r_s ds}$, by Theorem \ref{Thm:M}.
Then the optimal value for $\eta$ is given by (\ref{Eq:eta*}), solved from $\frac{\partial J}{\partial \eta} = 0$. $\text{Var}[X^*_T]$ is obtained by direct simplification of $J(x_0; u(\cdot))$ with $\eta^*$.
\end{proof}
Although the Volterra Heston model is non-Markovian and non-semimartingale in nature, the optimal control $u^*$ in \eqref{Eq:u*} does not rely on the whole volatility path. Moreover, the optimal amount of wealth in the stock, $\pi^*_t$, does not depend on the volatility value directly, but rather on the roughness and dynamics of volatility through parameters and the Riccati-Volterra equation \eqref{Eq:psi}. If we let kernel $K = {\rm id}$, it is then clear that the Volterra Heston model (\ref{vol}) reduces to the classic Heston model \cite{heston1993closed}. Our results in Theorem \ref{Thm:X^*u^*} and Theorem \ref{Thm:Sol} indicate that the $u^*$ in (\ref{Eq:u*}) is optimal even under a general filtration $\mathbb{F}$. It extends the corresponding result in \cite{cerny2008mean,shen2015square} where the filtration is chosen as the Brownian filtration. As a sanity check, the following corollary verifies that our solution reduces to the one under the Heston model.
\begin{corollary}
Consider the Heston model, that is, the kernel $K = {\rm id}$. Suppose other assumptions in Theorem \ref{Thm:X^*u^*} hold, then the optimal strategy \eqref{Eq:u*} is the same as the one in \cite{cerny2008mean}.
\end{corollary}
\begin{proof}
Without loss of generality, suppose $r_t = 0$, as in \cite{cerny2008mean}. We first match $M_t/2$ in (\ref{Eq:M}) with opportunity process $L_t$ in \cite[Equation (3.2)]{cerny2008mean}.
Note the resolvent in (\ref{Eq:R_lambda}) reduces to $R_\lambda (t) = \lambda e^{-\lambda t}$ and the forward variance in (\ref{Eq:xi}) is
\begin{equation}
\xi_t(s) = e^{-\lambda (s - t)} V_t + \frac{\kappa \phi}{\lambda} \left( 1 - e^{-\lambda(s - t)} \right).
\end{equation}
Therefore,
\begin{equation*}
\int^T_t \xi_t(s) ds = \frac{1 - e^{-\lambda(T-t)}}{\lambda} V_t + \frac{\kappa \phi}{\lambda} \left( T - t - \frac{1 - e^{-\lambda(T-t)}}{\lambda} \right)
\end{equation*}
and
\begin{equation*}
\int^T_t \psi^2(T-s) \xi_t(s) ds = V_t \int^T_t \psi^2(T - s) e^{-\lambda(s - t)} ds + \frac{\kappa \phi}{\lambda} \int^T_t \big[ 1 - e^{-\lambda (s - t)} \big] \psi^2(T - s) ds.
\end{equation*}
Then
\begin{equation}
\int^T_t \big[ - \theta^2 \xi_t(s) + \frac{(1-2\rho^2) \sigma^2}{2} \psi^2(T-s) \xi_t(s) \big] ds = w(T-t) V_t + y(T-t),
\end{equation}
where
\begin{align*}
w(T-t) &\triangleq \frac{(1 - 2\rho^2)\sigma^2}{2} \int^T_t \psi^2(T - s) e^{-\lambda(s - t)} ds - \theta^2 \frac{1 - e^{-\lambda(T-t)}}{\lambda}, \\
y(T-t) &\triangleq \frac{(1 - 2\rho^2)\sigma^2}{2} \frac{\kappa \phi}{\lambda} \int^T_t \big[ 1 - e^{-\lambda (s - t)} \big] \psi^2(T - s) ds - \theta^2 \frac{\kappa \phi}{\lambda} \left(T - t - \frac{1 - e^{-\lambda(T-t)}}{\lambda} \right).
\end{align*}
Replacing $t$ with $T - t$ and taking derivative on $t$ give
\begin{align*}
\dot{w}(t) &= \frac{(1 - 2\rho^2)\sigma^2}{2} \psi^2(t) - \lambda \frac{(1 - 2\rho^2)\sigma^2}{2} \int^T_{T-t} \psi^2(T - s) e^{-\lambda(s - T + t)} ds - \theta^2 e^{-\lambda t} \\
& = \frac{(1 - 2\rho^2)\sigma^2}{2} \psi^2(t) - \lambda w(t) - \theta^2.
\end{align*}
Comparing with (\ref{Eq:psi}), we find $w(t) = \psi(t)$. Moreover,
\begin{align*}
\dot{y}(t) & = \frac{(1 - 2\rho^2)\sigma^2}{2} \frac{\kappa \phi}{\lambda} \int^T_{T-t} \lambda e^{-\lambda (s - T + t)} \psi^2(T - s) ds - \theta^2 \frac{\kappa \phi}{\lambda} \left( 1 - e^{-\lambda t} \right) \\
& = \kappa \phi w(t).
\end{align*}
$y(t)$ and $w(t)$ satisfy the same ODEs as in \cite[Equations (A.1)-(A.4)]{cerny2008mean}, with our notations. Therefore, $M_t/2$ in (\ref{Eq:M}) reduces to $L_t$ in \cite[Equation (3.2)]{cerny2008mean}.
Consider the inner Problem (\ref{Eq:innerobj}). With a constant $H = \zeta$, terms in the optimal hedge $\varphi(x, H)$ \cite[p.476]{cerny2008mean} are reduced to
\begin{align}
\xi = 0, \quad a= (\theta + \psi(T-t) \rho \sigma)/ S_t, \quad V = \zeta, \quad \text{and} \quad x + \varphi(x, H) \cdot S = X^*_t.
\end{align}
Then it is clear that the optimal strategies are the same.
\end{proof}
\section{Numerical studies}\label{Sec:Numerical}
In this section, we restrict ourself to the case with $K(t) = \frac{t^{\alpha-1}}{\Gamma(\alpha)}$, $\alpha \in (1/2, 1)$, for the rough Heston model in \cite{eleuch2019char}. $\alpha = 1$ recovers the classic Heston model. We examine the effect of $\alpha$ on the optimal investment strategy and efficient frontier.
The first step is to solve the Riccati-Volterra equation (\ref{Eq:psi}) numerically. Following \cite{eleuch2019char}, we use the fractional Adams method in \cite{diethelm2002predictor,diethelm2004error}. The convergence of this numerical method is given in \cite{li2009adams}. Readers may refer to \cite[Section 5.1]{eleuch2019char} for more details about the procedure.
In Figure (\ref{Fig:psi}), $\psi$ decreases when $\alpha$ becomes smaller under certain specific parameters, close to the calibration result in \cite{eleuch2019char} with one extra risk premium parameter $\theta$. However, one cannot expect $\psi$ to be monotone in $\alpha$ in general (see Figure (\ref{Fig:psi_new})). Figures (\ref{Fig:psi})-(\ref{Fig:psi_new}) also confirm the claim that $\psi \leq 0$ when $ 1 - 2\rho^2 > 0$.
\begin{figure}[!h]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{psi}
\subcaption{$\psi$ under parameters in \cite{eleuch2019char}}\label{Fig:psi}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{psi_new}
\subcaption{$\psi$ under another setting}\label{Fig:psi_new}
\end{minipage}
\caption{Plot of $\psi$ under different $\alpha$. Other parameters are as follows. In Figure (\ref{Fig:psi}), vol-of-vol $\sigma = 0.03$, mean-reversion speed $\kappa = 0.1$, risk premium parameter $\theta = 5$, correlation $\rho = -0.7$, and time horizon $T = 1$. In Figure (\ref{Fig:psi_new}), $\sigma = 0.04$, $\kappa = 2.25$, $\theta = 0.15$, $\rho = -0.56$, and $T=1.35$.}
\end{figure}
The relationship between $u^*$ and $\alpha$ is not straightforward and may change with different combinations of parameters. We emphasize that the following analysis is based on the parameter setting detailed in the descriptions of the figures. Consider the setting in Figure (\ref{Fig:psi}) first. Interestingly, the effect of $\alpha$ on $u^*$ is significantly influenced by $\sigma$. This can be explained using (\ref{Eq:u*}). If the correlation $\rho$ between stock and volatility is negative due to the leverage effect in the equity market, $\theta + \rho \sigma \psi (T-t)$ will increase as $\alpha$ decreases, as shown in Figure (\ref{Fig:psi}). In contrast, $\zeta^* e^{-\int^T_t r_s ds} - X^*_t \geq 0$ by Theorem \ref{Thm:X^*u^*}. Note
\begin{equation}
\zeta^* = c - \eta^* = \frac{2c - e^{-\int^T_0 r_s ds}M_0 x_0}{2 - e^{-\int^T_0 2 r_s ds} M_0}.
\end{equation}
The $M_0$ in (\ref{Eq:fracM0}) is an increasing function on $\alpha$ because $\psi$ is negative. Then $\zeta^*$ will be smaller if $\alpha$ is smaller, under certain parameters. Therefore, $\zeta^* e^{-\int^T_t r_s ds} - X^*_t$ and $\theta + \rho \sigma \psi (T-t)$ move in different directions when $\alpha$ is decreasing. If $\sigma$ is small, $\zeta^* e^{-\int^T_t r_s ds} - X^*_t$ will dominate $\theta + \rho \sigma \psi (T-t)$. Then $u^*$ will decrease as $\alpha$ becomes smaller. If $\sigma$ is relatively large, $\theta + \rho \sigma \psi (T-t)$ will dominate $\zeta^* e^{-\int^T_t r_s ds} - X^*_t$. Then $u^*$ increases when $\alpha$ becomes smaller. The above effect of vol-of-vol $\sigma$ also appears under the parameters setting in Figure (\ref{Fig:psi_new}), where $\psi$ is not monotone in $\alpha$. Figures (\ref{Fig:u4smallsig})-(\ref{Fig:u4bigsig}) display the optimal investment strategy $u^*$. We make use of the open-source Python package {\bf differint}\footnote{Available at \url{https://github.com/differint/differint} } to calculate the fractional integrals $I^{1-\alpha}$ and $I^1$ in (\ref{Eq:fracM0}). Assumption \ref{Assum:V} is validated under the setting in Figures (\ref{Fig:u4smallsig})-(\ref{Fig:u4bigsig}).
\begin{figure}[!h]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{u4smallsig}
\subcaption{$u^*$ under $\sigma=0.04$}\label{Fig:u4smallsig}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{u4bigsig}
\subcaption{$u^*$ under $\sigma = 3$}\label{Fig:u4bigsig}
\end{minipage}
\caption{Optimal strategy $u^*$ with $\alpha=0.6, 0.7, 0.8, 0.9$, and $1.0$. In both subplots, we set initial wealth $x_0 = 1$, risk-free rate $r= 0.01$, initial variance $V_0 = 0.5$, long-term mean level $\phi = 0.04$, and expected terminal wealth $c = x_0 e^{(r+0.1)T}$. For simplicity, we set $V_t = 0.5$ and $X^*_t = 1$ for all time $t \in [0, T]$. The other parameters are the same as in Figure (\ref{Fig:psi_new}), namely, $\kappa = 2.25$, $\theta = 0.15$, $\rho = -0.56$, and $T=1.35$. Figures (\ref{Fig:u4smallsig})-(\ref{Fig:u4bigsig}) only differ in the vol-of-vol $\sigma$. }
\end{figure}
Figures (\ref{Fig:u4smallsig})-(\ref{Fig:u4bigsig}) are a sensitivity analysis as we keep most of the parameters unchanged, and vary a few of them. Specifically, the use of constant $V_t$ and $X^*_t$ in Figures (\ref{Fig:u4smallsig})-(\ref{Fig:u4bigsig}) has the following interpretation. We are interested in the sensitivity of the optimal control on the Hurst parameter through $\alpha$. As the other parameters being fixed, if we observe $V_t = 0.5$ and $X^*_t = 1$ at $t \in [0, T]$, Figures (\ref{Fig:u4smallsig})-(\ref{Fig:u4bigsig}) illustrate the marginal effect of the Hurst parameter on the investment strategy. The constant values of $V_t$ and $X^*_t$ are not from a realized path.
Figures (\ref{Fig:u4smallsig})-(\ref{Fig:u4bigsig}) only provide a marginal effect of $\alpha$; thus, we conduct a further numerical analysis under the settings in \cite{abi2019lifting}. Consider a realistic situation in which the investor calibrates two sets of parameters for the Heston model and rough Heston model for a given implied volatility surface. We contrast the two strategies induced from the calibrated parameters. Figure (\ref{Fig:pi_sim}) exhibits the optimal amount of wealth $\pi^*$ with one simulation path of $V_t$ in Figure (\ref{Fig:roughv}) by the lifted Heston approach \cite{abi2019lifting}. Assumption \ref{Assum:V} holds true under the setting in Figure 3. Figure (\ref{Fig:At}) plots the $A_t = \theta + \rho \sigma \psi(T-t)$. Furthermore, $\zeta^* = 30.7458$ for the rough Heston model and $\zeta^* = 21.6351$ for the classic Heston model. The optimal strategy under the rough Heston model consistently suggests holding more in the stock. We stress that this is a persistent phenomenon for all of the simulation runs and is not limited to the particular one in Figure (\ref{Fig:pi_sim}). Indeed, Figures (\ref{Fig:uRough})-(\ref{Fig:uHn}) show the mean and confidence intervals of the strategies. The rough Heston strategy has larger values during the whole investment horizon. It can be explained with Figure (\ref{Fig:At}) and $\zeta^*$ reported. A rough Heston investor has a larger $A_t \zeta^*$ but a smaller $A_t$. Moreover, Figure (\ref{Fig:wealth}) illustrates that the rough Heston strategy has an average terminal wealth closer to the target $c = 1.1163$. Finally, we emphasize that Figure (\ref{Fig:pi_sim}) and Figures (\ref{Fig:uRough})-(\ref{Fig:uHn}) do not conflict with Figures (\ref{Fig:u4smallsig})-(\ref{Fig:u4bigsig}) because the mean-reversion rate and the vol-of-vol are different for the two strategies in Figure (\ref{Fig:pi_sim}). See \cite[Table 6]{abi2019lifting} and \cite[Table 4]{abi2019lifting} for more details.
\begin{figure}[!h]
\centering
\begin{minipage}{0.33\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{At}
\subcaption{$A_t$}\label{Fig:At}
\end{minipage}%
\begin{minipage}{0.33\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{RoughV}
\subcaption{volatility}\label{Fig:roughv}
\end{minipage}%
\begin{minipage}{0.33\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{pi}
\subcaption{$\pi^*$}\label{Fig:pi_sim}
\end{minipage}
\caption{Investment strategies under the Heston and rough Heston models. The variance process is simulated with the lifted Heston model in \cite{abi2019lifting}. The parameters for simulation are specified in \cite[Equations (23) and (26)]{abi2019lifting} with $\alpha=0.6$. The path is rougher than that of the classic Heston model. Moreover, we implement the Euler scheme for the stock process. The simulation is run with $250$ time steps for one year, corresponding to the $250$ trading days in a year. The investor under the Heston model uses the calibrated parameters in \cite[Table 6]{abi2019lifting} to implement the optimal strategy with $\alpha = 0.59973346$ being the calibrated value. The investor under rough Heston model uses \cite[Table 4]{abi2019lifting} instead. We set $x_0 = 1$, $r = 0.01$, $\theta = 0.4$, $T = 1$, and $c = x_0 e^{(r+0.1)T} = 1.1163$. }
\end{figure}
\begin{figure}[!h]
\centering
\begin{minipage}{0.33\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{uRough.png}
\subcaption{$u^*$ under the rough Heston model}\label{Fig:uRough}
\end{minipage}%
\begin{minipage}{0.33\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{uHn.png}
\subcaption{$u^*$ under the Heston model}\label{Fig:uHn}
\end{minipage}%
\begin{minipage}{0.33\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{wealth.png}
\subcaption{Wealth}\label{Fig:wealth}
\end{minipage}
\caption{Statistics for strategies and wealth. Based on 3000 simulated paths, the solid line plots the mean and the shadow area is the $95\%$ confidence interval estimated by bootstrapping. The rough Heston model suggests investing more and the terminal wealth is closer to the expected value $c=1.1163$. The parameters are the same as in Figure 3.}
\end{figure}
Recently, a trading strategy has been proposed to buy the roughest stocks and sell the smoothest stocks \cite{glasserman2019buy}. This model-free strategy aims at investments in multiple assets. Although we consider a single risky asset with a specific model, it is still interesting to compare that strategy with ours. Note that a stock is rougher for a smaller $\alpha$. Figures (\ref{Fig:u4smallsig})-(\ref{Fig:u4bigsig}) indicate that $\alpha$ is not the only factor determining the investment in a stock. The trading idea in \cite{glasserman2019buy} agrees with Figure (\ref{Fig:u4bigsig}), because the optimal investment position $u^*$ is larger for a smaller $\alpha$. However, an inconsistency occurs in Figure (\ref{Fig:u4smallsig}). Indeed, if we use the VVIX index as a proxy for the vol-of-vol, then the vol-of-vol seems larger in 2007, 2008, 2010, and 2015. The buy-rough-sell-smooth strategy \cite{glasserman2019buy} performs better in 2005, 2007, 2008, 2010, and 2014 than in other years, as shown in \cite[Figure 3]{glasserman2019buy}. This consistency suggests that vol-of-vol may also be important when roughness is considered. It would be interesting to test the performance of strategies based on roughness and vol-of-vol in a future study.
\begin{figure}[!h]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{effront}
\subcaption{Efficient frontier}\label{Fig:effront}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{var}
\subcaption{Var$[X^*_T]$ under different expected value $c$}\label{Fig:Varc}
\end{minipage}
\caption{Plots of the efficient frontier and variance. Roughness parameter $\alpha \in [0.5, 1]$. We set $r=0.03$, $V_0 =0.04$, $x_0 = 1$, $\phi = 0.3$, $\sigma = 0.03$, $\kappa = 0.1$, $\theta = 0.6$, $\rho = -0.7$, $T = 1$, and $c \in [x_0 e^{(r+0.01)T}, \,x_0 e^{(r+0.5)T}]$.}
\end{figure}
In Figures (\ref{Fig:effront})-(\ref{Fig:Varc}), the efficient frontier is shown for different values of $\alpha$ and expected wealth level $c$. Their relationship is clear, and the variance of the optimal wealth is reduced if $\alpha$ decreases, as $M_0$ decreases when $\alpha$ decreases and Var$[X^*_T]$ in (\ref{Eq:VarX*}) is an increasing function on $M_0$. We have also verified Assumption \ref{Assum:V} under the setting in Figures (\ref{Fig:effront})-(\ref{Fig:Varc}).
\section{Conclusion}\label{Sec:Conclusion}
To the best of our knowledge, this is the first study of the continuous-time Markowitz's mean-variance portfolio selection problem under a rough stochastic environment. We specifically focus on the Volterra Heston model. By deriving the optimal strategy and efficient frontier, we obtain further insights into the effect of roughness on them.
There are many possible future research directions. Natural considerations are the utility maximization and time-inconsistency of the MV criterion. In addition, we have already included model ambiguity with rough volatility in our research agenda.
\section*{Acknowledgements}
The authors would like to thank two anonymous referees and the Editor for their careful reading and valuable comments, which have greatly improved the manuscript.
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The WNED Public Radio Internship is open to students who are interested in gaining hands-on experience in radio operations and production and developing their writing and broadcasting skills. Interns will work on projects that may include: news-gathering for newscasts on WNED-AM, production of radio shows, and production of promotional spots and underwriting credits. You must develop your own semi-monthly work schedule and provide regular transportation for yourself throughout the internship. You must also demonstrate: an interest in public radio, a willingness to learn, the ability to work independently and work collaboratively, the ability to write cogently, and the ability to use basic office software to be eligible for this internship. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,698 |
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use crate::core::GitReference;
use crate::util::errors::CargoResult;
use crate::util::{network, Config, IntoUrl, MetricsCounter, Progress};
use anyhow::{anyhow, Context as _};
use cargo_util::{paths, ProcessBuilder};
use curl::easy::List;
use git2::{self, ErrorClass, ObjectType};
use log::{debug, info};
use serde::ser;
use serde::Serialize;
use std::env;
use std::fmt;
use std::path::{Path, PathBuf};
use std::process::Command;
use std::time::{Duration, Instant};
use url::Url;
fn serialize_str<T, S>(t: &T, s: S) -> Result<S::Ok, S::Error>
where
T: fmt::Display,
S: ser::Serializer,
{
s.collect_str(t)
}
pub struct GitShortID(git2::Buf);
impl GitShortID {
pub fn as_str(&self) -> &str {
self.0.as_str().unwrap()
}
}
/// `GitRemote` represents a remote repository. It gets cloned into a local
/// `GitDatabase`.
#[derive(PartialEq, Clone, Debug, Serialize)]
pub struct GitRemote {
#[serde(serialize_with = "serialize_str")]
url: Url,
}
/// `GitDatabase` is a local clone of a remote repository's database. Multiple
/// `GitCheckouts` can be cloned from this `GitDatabase`.
#[derive(Serialize)]
pub struct GitDatabase {
remote: GitRemote,
path: PathBuf,
#[serde(skip_serializing)]
repo: git2::Repository,
}
/// `GitCheckout` is a local checkout of a particular revision. Calling
/// `clone_into` with a reference will resolve the reference into a revision,
/// and return an `anyhow::Error` if no revision for that reference was found.
#[derive(Serialize)]
pub struct GitCheckout<'a> {
database: &'a GitDatabase,
location: PathBuf,
#[serde(serialize_with = "serialize_str")]
revision: git2::Oid,
#[serde(skip_serializing)]
repo: git2::Repository,
}
// Implementations
impl GitRemote {
pub fn new(url: &Url) -> GitRemote {
GitRemote { url: url.clone() }
}
pub fn url(&self) -> &Url {
&self.url
}
pub fn rev_for(&self, path: &Path, reference: &GitReference) -> CargoResult<git2::Oid> {
reference.resolve(&self.db_at(path)?.repo)
}
pub fn checkout(
&self,
into: &Path,
db: Option<GitDatabase>,
reference: &GitReference,
locked_rev: Option<git2::Oid>,
cargo_config: &Config,
) -> CargoResult<(GitDatabase, git2::Oid)> {
// If we have a previous instance of `GitDatabase` then fetch into that
// if we can. If that can successfully load our revision then we've
// populated the database with the latest version of `reference`, so
// return that database and the rev we resolve to.
if let Some(mut db) = db {
fetch(&mut db.repo, self.url.as_str(), reference, cargo_config)
.context(format!("failed to fetch into: {}", into.display()))?;
match locked_rev {
Some(rev) => {
if db.contains(rev) {
return Ok((db, rev));
}
}
None => {
if let Ok(rev) = reference.resolve(&db.repo) {
return Ok((db, rev));
}
}
}
}
// Otherwise start from scratch to handle corrupt git repositories.
// After our fetch (which is interpreted as a clone now) we do the same
// resolution to figure out what we cloned.
if into.exists() {
paths::remove_dir_all(into)?;
}
paths::create_dir_all(into)?;
let mut repo = init(into, true)?;
fetch(&mut repo, self.url.as_str(), reference, cargo_config)
.context(format!("failed to clone into: {}", into.display()))?;
let rev = match locked_rev {
Some(rev) => rev,
None => reference.resolve(&repo)?,
};
Ok((
GitDatabase {
remote: self.clone(),
path: into.to_path_buf(),
repo,
},
rev,
))
}
pub fn db_at(&self, db_path: &Path) -> CargoResult<GitDatabase> {
let repo = git2::Repository::open(db_path)?;
Ok(GitDatabase {
remote: self.clone(),
path: db_path.to_path_buf(),
repo,
})
}
}
impl GitDatabase {
pub fn copy_to(
&self,
rev: git2::Oid,
dest: &Path,
cargo_config: &Config,
) -> CargoResult<GitCheckout<'_>> {
// If the existing checkout exists, and it is fresh, use it.
// A non-fresh checkout can happen if the checkout operation was
// interrupted. In that case, the checkout gets deleted and a new
// clone is created.
let checkout = match git2::Repository::open(dest)
.ok()
.map(|repo| GitCheckout::new(dest, self, rev, repo))
.filter(|co| co.is_fresh())
{
Some(co) => co,
None => GitCheckout::clone_into(dest, self, rev, cargo_config)?,
};
checkout.update_submodules(cargo_config)?;
Ok(checkout)
}
pub fn to_short_id(&self, revision: git2::Oid) -> CargoResult<GitShortID> {
let obj = self.repo.find_object(revision, None)?;
Ok(GitShortID(obj.short_id()?))
}
pub fn contains(&self, oid: git2::Oid) -> bool {
self.repo.revparse_single(&oid.to_string()).is_ok()
}
pub fn resolve(&self, r: &GitReference) -> CargoResult<git2::Oid> {
r.resolve(&self.repo)
}
}
impl GitReference {
pub fn resolve(&self, repo: &git2::Repository) -> CargoResult<git2::Oid> {
let id = match self {
// Note that we resolve the named tag here in sync with where it's
// fetched into via `fetch` below.
GitReference::Tag(s) => (|| -> CargoResult<git2::Oid> {
let refname = format!("refs/remotes/origin/tags/{}", s);
let id = repo.refname_to_id(&refname)?;
let obj = repo.find_object(id, None)?;
let obj = obj.peel(ObjectType::Commit)?;
Ok(obj.id())
})()
.with_context(|| format!("failed to find tag `{}`", s))?,
// Resolve the remote name since that's all we're configuring in
// `fetch` below.
GitReference::Branch(s) => {
let name = format!("origin/{}", s);
let b = repo
.find_branch(&name, git2::BranchType::Remote)
.with_context(|| format!("failed to find branch `{}`", s))?;
b.get()
.target()
.ok_or_else(|| anyhow::format_err!("branch `{}` did not have a target", s))?
}
// We'll be using the HEAD commit
GitReference::DefaultBranch => {
let head_id = repo.refname_to_id("refs/remotes/origin/HEAD")?;
let head = repo.find_object(head_id, None)?;
head.peel(ObjectType::Commit)?.id()
}
GitReference::Rev(s) => {
let obj = repo.revparse_single(s)?;
match obj.as_tag() {
Some(tag) => tag.target_id(),
None => obj.id(),
}
}
};
Ok(id)
}
}
impl<'a> GitCheckout<'a> {
fn new(
path: &Path,
database: &'a GitDatabase,
revision: git2::Oid,
repo: git2::Repository,
) -> GitCheckout<'a> {
GitCheckout {
location: path.to_path_buf(),
database,
revision,
repo,
}
}
fn clone_into(
into: &Path,
database: &'a GitDatabase,
revision: git2::Oid,
config: &Config,
) -> CargoResult<GitCheckout<'a>> {
let dirname = into.parent().unwrap();
paths::create_dir_all(&dirname)?;
if into.exists() {
paths::remove_dir_all(into)?;
}
// we're doing a local filesystem-to-filesystem clone so there should
// be no need to respect global configuration options, so pass in
// an empty instance of `git2::Config` below.
let git_config = git2::Config::new()?;
// Clone the repository, but make sure we use the "local" option in
// libgit2 which will attempt to use hardlinks to set up the database.
// This should speed up the clone operation quite a bit if it works.
//
// Note that we still use the same fetch options because while we don't
// need authentication information we may want progress bars and such.
let url = database.path.into_url()?;
let mut repo = None;
with_fetch_options(&git_config, url.as_str(), config, &mut |fopts| {
let mut checkout = git2::build::CheckoutBuilder::new();
checkout.dry_run(); // we'll do this below during a `reset`
let r = git2::build::RepoBuilder::new()
// use hard links and/or copy the database, we're doing a
// filesystem clone so this'll speed things up quite a bit.
.clone_local(git2::build::CloneLocal::Local)
.with_checkout(checkout)
.fetch_options(fopts)
.clone(url.as_str(), into)?;
repo = Some(r);
Ok(())
})?;
let repo = repo.unwrap();
let checkout = GitCheckout::new(into, database, revision, repo);
checkout.reset(config)?;
Ok(checkout)
}
fn is_fresh(&self) -> bool {
match self.repo.revparse_single("HEAD") {
Ok(ref head) if head.id() == self.revision => {
// See comments in reset() for why we check this
self.location.join(".cargo-ok").exists()
}
_ => false,
}
}
fn reset(&self, config: &Config) -> CargoResult<()> {
// If we're interrupted while performing this reset (e.g., we die because
// of a signal) Cargo needs to be sure to try to check out this repo
// again on the next go-round.
//
// To enable this we have a dummy file in our checkout, .cargo-ok, which
// if present means that the repo has been successfully reset and is
// ready to go. Hence if we start to do a reset, we make sure this file
// *doesn't* exist, and then once we're done we create the file.
let ok_file = self.location.join(".cargo-ok");
let _ = paths::remove_file(&ok_file);
info!("reset {} to {}", self.repo.path().display(), self.revision);
// Ensure libgit2 won't mess with newlines when we vendor.
if let Ok(mut git_config) = self.repo.config() {
git_config.set_bool("core.autocrlf", false)?;
}
let object = self.repo.find_object(self.revision, None)?;
reset(&self.repo, &object, config)?;
paths::create(ok_file)?;
Ok(())
}
fn update_submodules(&self, cargo_config: &Config) -> CargoResult<()> {
return update_submodules(&self.repo, cargo_config);
fn update_submodules(repo: &git2::Repository, cargo_config: &Config) -> CargoResult<()> {
debug!("update submodules for: {:?}", repo.workdir().unwrap());
for mut child in repo.submodules()? {
update_submodule(repo, &mut child, cargo_config).with_context(|| {
format!(
"failed to update submodule `{}`",
child.name().unwrap_or("")
)
})?;
}
Ok(())
}
fn update_submodule(
parent: &git2::Repository,
child: &mut git2::Submodule<'_>,
cargo_config: &Config,
) -> CargoResult<()> {
child.init(false)?;
let url = child.url().ok_or_else(|| {
anyhow::format_err!("non-utf8 url for submodule {:?}?", child.path())
})?;
// Skip the submodule if the config says not to update it.
if child.update_strategy() == git2::SubmoduleUpdate::None {
cargo_config.shell().status(
"Skipping",
format!(
"git submodule `{}` due to update strategy in .gitmodules",
url
),
)?;
return Ok(());
}
// A submodule which is listed in .gitmodules but not actually
// checked out will not have a head id, so we should ignore it.
let head = match child.head_id() {
Some(head) => head,
None => return Ok(()),
};
// If the submodule hasn't been checked out yet, we need to
// clone it. If it has been checked out and the head is the same
// as the submodule's head, then we can skip an update and keep
// recursing.
let head_and_repo = child.open().and_then(|repo| {
let target = repo.head()?.target();
Ok((target, repo))
});
let mut repo = match head_and_repo {
Ok((head, repo)) => {
if child.head_id() == head {
return update_submodules(&repo, cargo_config);
}
repo
}
Err(..) => {
let path = parent.workdir().unwrap().join(child.path());
let _ = paths::remove_dir_all(&path);
init(&path, false)?
}
};
// Fetch data from origin and reset to the head commit
let reference = GitReference::Rev(head.to_string());
cargo_config
.shell()
.status("Updating", format!("git submodule `{}`", url))?;
fetch(&mut repo, url, &reference, cargo_config).with_context(|| {
format!(
"failed to fetch submodule `{}` from {}",
child.name().unwrap_or(""),
url
)
})?;
let obj = repo.find_object(head, None)?;
reset(&repo, &obj, cargo_config)?;
update_submodules(&repo, cargo_config)
}
}
}
/// Prepare the authentication callbacks for cloning a git repository.
///
/// The main purpose of this function is to construct the "authentication
/// callback" which is used to clone a repository. This callback will attempt to
/// find the right authentication on the system (without user input) and will
/// guide libgit2 in doing so.
///
/// The callback is provided `allowed` types of credentials, and we try to do as
/// much as possible based on that:
///
/// * Prioritize SSH keys from the local ssh agent as they're likely the most
/// reliable. The username here is prioritized from the credential
/// callback, then from whatever is configured in git itself, and finally
/// we fall back to the generic user of `git`.
///
/// * If a username/password is allowed, then we fallback to git2-rs's
/// implementation of the credential helper. This is what is configured
/// with `credential.helper` in git, and is the interface for the macOS
/// keychain, for example.
///
/// * After the above two have failed, we just kinda grapple attempting to
/// return *something*.
///
/// If any form of authentication fails, libgit2 will repeatedly ask us for
/// credentials until we give it a reason to not do so. To ensure we don't
/// just sit here looping forever we keep track of authentications we've
/// attempted and we don't try the same ones again.
fn with_authentication<T, F>(url: &str, cfg: &git2::Config, mut f: F) -> CargoResult<T>
where
F: FnMut(&mut git2::Credentials<'_>) -> CargoResult<T>,
{
let mut cred_helper = git2::CredentialHelper::new(url);
cred_helper.config(cfg);
let mut ssh_username_requested = false;
let mut cred_helper_bad = None;
let mut ssh_agent_attempts = Vec::new();
let mut any_attempts = false;
let mut tried_sshkey = false;
let mut url_attempt = None;
let orig_url = url;
let mut res = f(&mut |url, username, allowed| {
any_attempts = true;
if url != orig_url {
url_attempt = Some(url.to_string());
}
// libgit2's "USERNAME" authentication actually means that it's just
// asking us for a username to keep going. This is currently only really
// used for SSH authentication and isn't really an authentication type.
// The logic currently looks like:
//
// let user = ...;
// if (user.is_null())
// user = callback(USERNAME, null, ...);
//
// callback(SSH_KEY, user, ...)
//
// So if we're being called here then we know that (a) we're using ssh
// authentication and (b) no username was specified in the URL that
// we're trying to clone. We need to guess an appropriate username here,
// but that may involve a few attempts. Unfortunately we can't switch
// usernames during one authentication session with libgit2, so to
// handle this we bail out of this authentication session after setting
// the flag `ssh_username_requested`, and then we handle this below.
if allowed.contains(git2::CredentialType::USERNAME) {
debug_assert!(username.is_none());
ssh_username_requested = true;
return Err(git2::Error::from_str("gonna try usernames later"));
}
// An "SSH_KEY" authentication indicates that we need some sort of SSH
// authentication. This can currently either come from the ssh-agent
// process or from a raw in-memory SSH key. Cargo only supports using
// ssh-agent currently.
//
// If we get called with this then the only way that should be possible
// is if a username is specified in the URL itself (e.g., `username` is
// Some), hence the unwrap() here. We try custom usernames down below.
if allowed.contains(git2::CredentialType::SSH_KEY) && !tried_sshkey {
// If ssh-agent authentication fails, libgit2 will keep
// calling this callback asking for other authentication
// methods to try. Make sure we only try ssh-agent once,
// to avoid looping forever.
tried_sshkey = true;
let username = username.unwrap();
debug_assert!(!ssh_username_requested);
ssh_agent_attempts.push(username.to_string());
return git2::Cred::ssh_key_from_agent(username);
}
// Sometimes libgit2 will ask for a username/password in plaintext. This
// is where Cargo would have an interactive prompt if we supported it,
// but we currently don't! Right now the only way we support fetching a
// plaintext password is through the `credential.helper` support, so
// fetch that here.
//
// If ssh-agent authentication fails, libgit2 will keep calling this
// callback asking for other authentication methods to try. Check
// cred_helper_bad to make sure we only try the git credential helper
// once, to avoid looping forever.
if allowed.contains(git2::CredentialType::USER_PASS_PLAINTEXT) && cred_helper_bad.is_none()
{
let r = git2::Cred::credential_helper(cfg, url, username);
cred_helper_bad = Some(r.is_err());
return r;
}
// I'm... not sure what the DEFAULT kind of authentication is, but seems
// easy to support?
if allowed.contains(git2::CredentialType::DEFAULT) {
return git2::Cred::default();
}
// Whelp, we tried our best
Err(git2::Error::from_str("no authentication available"))
});
// Ok, so if it looks like we're going to be doing ssh authentication, we
// want to try a few different usernames as one wasn't specified in the URL
// for us to use. In order, we'll try:
//
// * A credential helper's username for this URL, if available.
// * This account's username.
// * "git"
//
// We have to restart the authentication session each time (due to
// constraints in libssh2 I guess? maybe this is inherent to ssh?), so we
// call our callback, `f`, in a loop here.
if ssh_username_requested {
debug_assert!(res.is_err());
let mut attempts = vec![String::from("git")];
if let Ok(s) = env::var("USER").or_else(|_| env::var("USERNAME")) {
attempts.push(s);
}
if let Some(ref s) = cred_helper.username {
attempts.push(s.clone());
}
while let Some(s) = attempts.pop() {
// We should get `USERNAME` first, where we just return our attempt,
// and then after that we should get `SSH_KEY`. If the first attempt
// fails we'll get called again, but we don't have another option so
// we bail out.
let mut attempts = 0;
res = f(&mut |_url, username, allowed| {
if allowed.contains(git2::CredentialType::USERNAME) {
return git2::Cred::username(&s);
}
if allowed.contains(git2::CredentialType::SSH_KEY) {
debug_assert_eq!(Some(&s[..]), username);
attempts += 1;
if attempts == 1 {
ssh_agent_attempts.push(s.to_string());
return git2::Cred::ssh_key_from_agent(&s);
}
}
Err(git2::Error::from_str("no authentication available"))
});
// If we made two attempts then that means:
//
// 1. A username was requested, we returned `s`.
// 2. An ssh key was requested, we returned to look up `s` in the
// ssh agent.
// 3. For whatever reason that lookup failed, so we were asked again
// for another mode of authentication.
//
// Essentially, if `attempts == 2` then in theory the only error was
// that this username failed to authenticate (e.g., no other network
// errors happened). Otherwise something else is funny so we bail
// out.
if attempts != 2 {
break;
}
}
}
let mut err = match res {
Ok(e) => return Ok(e),
Err(e) => e,
};
// In the case of an authentication failure (where we tried something) then
// we try to give a more helpful error message about precisely what we
// tried.
if any_attempts {
let mut msg = "failed to authenticate when downloading \
repository"
.to_string();
if let Some(attempt) = &url_attempt {
if url != attempt {
msg.push_str(": ");
msg.push_str(attempt);
}
}
msg.push('\n');
if !ssh_agent_attempts.is_empty() {
let names = ssh_agent_attempts
.iter()
.map(|s| format!("`{}`", s))
.collect::<Vec<_>>()
.join(", ");
msg.push_str(&format!(
"\n* attempted ssh-agent authentication, but \
no usernames succeeded: {}",
names
));
}
if let Some(failed_cred_helper) = cred_helper_bad {
if failed_cred_helper {
msg.push_str(
"\n* attempted to find username/password via \
git's `credential.helper` support, but failed",
);
} else {
msg.push_str(
"\n* attempted to find username/password via \
`credential.helper`, but maybe the found \
credentials were incorrect",
);
}
}
msg.push_str("\n\n");
msg.push_str("if the git CLI succeeds then `net.git-fetch-with-cli` may help here\n");
msg.push_str("https://doc.rust-lang.org/cargo/reference/config.html#netgit-fetch-with-cli");
err = err.context(msg);
// Otherwise if we didn't even get to the authentication phase them we may
// have failed to set up a connection, in these cases hint on the
// `net.git-fetch-with-cli` configuration option.
} else if let Some(e) = err.downcast_ref::<git2::Error>() {
match e.class() {
ErrorClass::Net
| ErrorClass::Ssl
| ErrorClass::Submodule
| ErrorClass::FetchHead
| ErrorClass::Ssh
| ErrorClass::Callback
| ErrorClass::Http => {
let mut msg = "network failure seems to have happened\n".to_string();
msg.push_str(
"if a proxy or similar is necessary `net.git-fetch-with-cli` may help here\n",
);
msg.push_str(
"https://doc.rust-lang.org/cargo/reference/config.html#netgit-fetch-with-cli",
);
err = err.context(msg);
}
_ => {}
}
}
Err(err)
}
fn reset(repo: &git2::Repository, obj: &git2::Object<'_>, config: &Config) -> CargoResult<()> {
let mut pb = Progress::new("Checkout", config);
let mut opts = git2::build::CheckoutBuilder::new();
opts.progress(|_, cur, max| {
drop(pb.tick(cur, max, ""));
});
debug!("doing reset");
repo.reset(obj, git2::ResetType::Hard, Some(&mut opts))?;
debug!("reset done");
Ok(())
}
pub fn with_fetch_options(
git_config: &git2::Config,
url: &str,
config: &Config,
cb: &mut dyn FnMut(git2::FetchOptions<'_>) -> CargoResult<()>,
) -> CargoResult<()> {
let mut progress = Progress::new("Fetch", config);
network::with_retry(config, || {
with_authentication(url, git_config, |f| {
let mut last_update = Instant::now();
let mut rcb = git2::RemoteCallbacks::new();
// We choose `N=10` here to make a `300ms * 10slots ~= 3000ms`
// sliding window for tracking the data transfer rate (in bytes/s).
let mut counter = MetricsCounter::<10>::new(0, last_update);
rcb.credentials(f);
rcb.transfer_progress(|stats| {
let indexed_deltas = stats.indexed_deltas();
let msg = if indexed_deltas > 0 {
// Resolving deltas.
format!(
", ({}/{}) resolving deltas",
indexed_deltas,
stats.total_deltas()
)
} else {
// Receiving objects.
//
// # Caveat
//
// Progress bar relies on git2 calling `transfer_progress`
// to update its transfer rate, but we cannot guarantee a
// periodic call of that callback. Thus if we don't receive
// any data for, say, 10 seconds, the rate will get stuck
// and never go down to 0B/s.
// In the future, we need to find away to update the rate
// even when the callback is not called.
let now = Instant::now();
// Scrape a `received_bytes` to the counter every 300ms.
if now - last_update > Duration::from_millis(300) {
counter.add(stats.received_bytes(), now);
last_update = now;
}
fn format_bytes(bytes: f32) -> (&'static str, f32) {
static UNITS: [&str; 5] = ["", "Ki", "Mi", "Gi", "Ti"];
let i = (bytes.log2() / 10.0).min(4.0) as usize;
(UNITS[i], bytes / 1024_f32.powi(i as i32))
}
let (unit, rate) = format_bytes(counter.rate());
format!(", {:.2}{}B/s", rate, unit)
};
progress
.tick(stats.indexed_objects(), stats.total_objects(), &msg)
.is_ok()
});
// Create a local anonymous remote in the repository to fetch the
// url
let mut opts = git2::FetchOptions::new();
opts.remote_callbacks(rcb);
cb(opts)
})?;
Ok(())
})
}
pub fn fetch(
repo: &mut git2::Repository,
url: &str,
reference: &GitReference,
config: &Config,
) -> CargoResult<()> {
if config.frozen() {
anyhow::bail!(
"attempting to update a git repository, but --frozen \
was specified"
)
}
if !config.network_allowed() {
anyhow::bail!("can't update a git repository in the offline mode")
}
// If we're fetching from GitHub, attempt GitHub's special fast path for
// testing if we've already got an up-to-date copy of the repository
match github_up_to_date(repo, url, reference, config) {
Ok(true) => return Ok(()),
Ok(false) => {}
Err(e) => debug!("failed to check github {:?}", e),
}
// We reuse repositories quite a lot, so before we go through and update the
// repo check to see if it's a little too old and could benefit from a gc.
// In theory this shouldn't be too too expensive compared to the network
// request we're about to issue.
maybe_gc_repo(repo)?;
// Translate the reference desired here into an actual list of refspecs
// which need to get fetched. Additionally record if we're fetching tags.
let mut refspecs = Vec::new();
let mut tags = false;
// The `+` symbol on the refspec means to allow a forced (fast-forward)
// update which is needed if there is ever a force push that requires a
// fast-forward.
match reference {
// For branches and tags we can fetch simply one reference and copy it
// locally, no need to fetch other branches/tags.
GitReference::Branch(b) => {
refspecs.push(format!("+refs/heads/{0}:refs/remotes/origin/{0}", b));
}
GitReference::Tag(t) => {
refspecs.push(format!("+refs/tags/{0}:refs/remotes/origin/tags/{0}", t));
}
GitReference::DefaultBranch => {
refspecs.push(String::from("+HEAD:refs/remotes/origin/HEAD"));
}
GitReference::Rev(rev) => {
let is_github = || Url::parse(url).map_or(false, |url| is_github(&url));
if rev.starts_with("refs/") {
refspecs.push(format!("+{0}:{0}", rev));
} else if is_github() && is_long_hash(rev) {
refspecs.push(format!("+{0}:refs/commit/{0}", rev));
} else {
// We don't know what the rev will point to. To handle this
// situation we fetch all branches and tags, and then we pray
// it's somewhere in there.
refspecs.push(String::from("+refs/heads/*:refs/remotes/origin/*"));
refspecs.push(String::from("+HEAD:refs/remotes/origin/HEAD"));
tags = true;
}
}
}
// Unfortunately `libgit2` is notably lacking in the realm of authentication
// when compared to the `git` command line. As a result, allow an escape
// hatch for users that would prefer to use `git`-the-CLI for fetching
// repositories instead of `libgit2`-the-library. This should make more
// flavors of authentication possible while also still giving us all the
// speed and portability of using `libgit2`.
if let Some(true) = config.net_config()?.git_fetch_with_cli {
return fetch_with_cli(repo, url, &refspecs, tags, config);
}
debug!("doing a fetch for {}", url);
let git_config = git2::Config::open_default()?;
with_fetch_options(&git_config, url, config, &mut |mut opts| {
if tags {
opts.download_tags(git2::AutotagOption::All);
}
// The `fetch` operation here may fail spuriously due to a corrupt
// repository. It could also fail, however, for a whole slew of other
// reasons (aka network related reasons). We want Cargo to automatically
// recover from corrupt repositories, but we don't want Cargo to stomp
// over other legitimate errors.
//
// Consequently we save off the error of the `fetch` operation and if it
// looks like a "corrupt repo" error then we blow away the repo and try
// again. If it looks like any other kind of error, or if we've already
// blown away the repository, then we want to return the error as-is.
let mut repo_reinitialized = false;
loop {
debug!("initiating fetch of {:?} from {}", refspecs, url);
let res = repo
.remote_anonymous(url)?
.fetch(&refspecs, Some(&mut opts), None);
let err = match res {
Ok(()) => break,
Err(e) => e,
};
debug!("fetch failed: {}", err);
if !repo_reinitialized && matches!(err.class(), ErrorClass::Reference | ErrorClass::Odb)
{
repo_reinitialized = true;
debug!(
"looks like this is a corrupt repository, reinitializing \
and trying again"
);
if reinitialize(repo).is_ok() {
continue;
}
}
return Err(err.into());
}
Ok(())
})
}
fn fetch_with_cli(
repo: &mut git2::Repository,
url: &str,
refspecs: &[String],
tags: bool,
config: &Config,
) -> CargoResult<()> {
let mut cmd = ProcessBuilder::new("git");
cmd.arg("fetch");
if tags {
cmd.arg("--tags");
}
cmd.arg("--force") // handle force pushes
.arg("--update-head-ok") // see discussion in #2078
.arg(url)
.args(refspecs)
// If cargo is run by git (for example, the `exec` command in `git
// rebase`), the GIT_DIR is set by git and will point to the wrong
// location (this takes precedence over the cwd). Make sure this is
// unset so git will look at cwd for the repo.
.env_remove("GIT_DIR")
// The reset of these may not be necessary, but I'm including them
// just to be extra paranoid and avoid any issues.
.env_remove("GIT_WORK_TREE")
.env_remove("GIT_INDEX_FILE")
.env_remove("GIT_OBJECT_DIRECTORY")
.env_remove("GIT_ALTERNATE_OBJECT_DIRECTORIES")
.cwd(repo.path());
config
.shell()
.verbose(|s| s.status("Running", &cmd.to_string()))?;
cmd.exec_with_output()?;
Ok(())
}
/// Cargo has a bunch of long-lived git repositories in its global cache and
/// some, like the index, are updated very frequently. Right now each update
/// creates a new "pack file" inside the git database, and over time this can
/// cause bad performance and bad current behavior in libgit2.
///
/// One pathological use case today is where libgit2 opens hundreds of file
/// descriptors, getting us dangerously close to blowing out the OS limits of
/// how many fds we can have open. This is detailed in #4403.
///
/// To try to combat this problem we attempt a `git gc` here. Note, though, that
/// we may not even have `git` installed on the system! As a result we
/// opportunistically try a `git gc` when the pack directory looks too big, and
/// failing that we just blow away the repository and start over.
fn maybe_gc_repo(repo: &mut git2::Repository) -> CargoResult<()> {
// Here we arbitrarily declare that if you have more than 100 files in your
// `pack` folder that we need to do a gc.
let entries = match repo.path().join("objects/pack").read_dir() {
Ok(e) => e.count(),
Err(_) => {
debug!("skipping gc as pack dir appears gone");
return Ok(());
}
};
let max = env::var("__CARGO_PACKFILE_LIMIT")
.ok()
.and_then(|s| s.parse::<usize>().ok())
.unwrap_or(100);
if entries < max {
debug!("skipping gc as there's only {} pack files", entries);
return Ok(());
}
// First up, try a literal `git gc` by shelling out to git. This is pretty
// likely to fail though as we may not have `git` installed. Note that
// libgit2 doesn't currently implement the gc operation, so there's no
// equivalent there.
match Command::new("git")
.arg("gc")
.current_dir(repo.path())
.output()
{
Ok(out) => {
debug!(
"git-gc status: {}\n\nstdout ---\n{}\nstderr ---\n{}",
out.status,
String::from_utf8_lossy(&out.stdout),
String::from_utf8_lossy(&out.stderr)
);
if out.status.success() {
let new = git2::Repository::open(repo.path())?;
*repo = new;
return Ok(());
}
}
Err(e) => debug!("git-gc failed to spawn: {}", e),
}
// Alright all else failed, let's start over.
reinitialize(repo)
}
fn reinitialize(repo: &mut git2::Repository) -> CargoResult<()> {
// Here we want to drop the current repository object pointed to by `repo`,
// so we initialize temporary repository in a sub-folder, blow away the
// existing git folder, and then recreate the git repo. Finally we blow away
// the `tmp` folder we allocated.
let path = repo.path().to_path_buf();
debug!("reinitializing git repo at {:?}", path);
let tmp = path.join("tmp");
let bare = !repo.path().ends_with(".git");
*repo = init(&tmp, false)?;
for entry in path.read_dir()? {
let entry = entry?;
if entry.file_name().to_str() == Some("tmp") {
continue;
}
let path = entry.path();
drop(paths::remove_file(&path).or_else(|_| paths::remove_dir_all(&path)));
}
*repo = init(&path, bare)?;
paths::remove_dir_all(&tmp)?;
Ok(())
}
fn init(path: &Path, bare: bool) -> CargoResult<git2::Repository> {
let mut opts = git2::RepositoryInitOptions::new();
// Skip anything related to templates, they just call all sorts of issues as
// we really don't want to use them yet they insist on being used. See #6240
// for an example issue that comes up.
opts.external_template(false);
opts.bare(bare);
Ok(git2::Repository::init_opts(&path, &opts)?)
}
/// Updating the index is done pretty regularly so we want it to be as fast as
/// possible. For registries hosted on GitHub (like the crates.io index) there's
/// a fast path available to use [1] to tell us that there's no updates to be
/// made.
///
/// This function will attempt to hit that fast path and verify that the `oid`
/// is actually the current branch of the repository. If `true` is returned then
/// no update needs to be performed, but if `false` is returned then the
/// standard update logic still needs to happen.
///
/// [1]: https://developer.github.com/v3/repos/commits/#get-the-sha-1-of-a-commit-reference
///
/// Note that this function should never cause an actual failure because it's
/// just a fast path. As a result all errors are ignored in this function and we
/// just return a `bool`. Any real errors will be reported through the normal
/// update path above.
fn github_up_to_date(
repo: &mut git2::Repository,
url: &str,
reference: &GitReference,
config: &Config,
) -> CargoResult<bool> {
let url = Url::parse(url)?;
if !is_github(&url) {
return Ok(false);
}
let github_branch_name = match reference {
GitReference::Branch(branch) => branch,
GitReference::Tag(tag) => tag,
GitReference::DefaultBranch => "HEAD",
GitReference::Rev(rev) => {
if rev.starts_with("refs/") {
rev
} else if is_long_hash(rev) {
return Ok(reference.resolve(repo).is_ok());
} else {
debug!("can't use github fast path with `rev = \"{}\"`", rev);
return Ok(false);
}
}
};
// This expects GitHub urls in the form `github.com/user/repo` and nothing
// else
let mut pieces = url
.path_segments()
.ok_or_else(|| anyhow!("no path segments on url"))?;
let username = pieces
.next()
.ok_or_else(|| anyhow!("couldn't find username"))?;
let repository = pieces
.next()
.ok_or_else(|| anyhow!("couldn't find repository name"))?;
if pieces.next().is_some() {
anyhow::bail!("too many segments on URL");
}
// Trim off the `.git` from the repository, if present, since that's
// optional for GitHub and won't work when we try to use the API as well.
let repository = repository.strip_suffix(".git").unwrap_or(repository);
let url = format!(
"https://api.github.com/repos/{}/{}/commits/{}",
username, repository, github_branch_name,
);
let mut handle = config.http()?.borrow_mut();
debug!("attempting GitHub fast path for {}", url);
handle.get(true)?;
handle.url(&url)?;
handle.useragent("cargo")?;
let mut headers = List::new();
headers.append("Accept: application/vnd.github.3.sha")?;
headers.append(&format!("If-None-Match: \"{}\"", reference.resolve(repo)?))?;
handle.http_headers(headers)?;
handle.perform()?;
Ok(handle.response_code()? == 304)
}
fn is_github(url: &Url) -> bool {
url.host_str() == Some("github.com")
}
fn is_long_hash(rev: &str) -> bool {
rev.len() == 40 && rev.chars().all(|ch| ch.is_ascii_hexdigit())
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,423 |
Q: The Force Dark side posses you? When you tun to the dark side is like you have been possesed by a demon? you lost your sanity? That bring me to the mind seeing how ...
Anakin kills all the young padawans
...something that nobody (not insane) would do for most angry you are.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,562 |
Bernard Malamud: A Writer's Life
Philip Davis (Author)
FORMAT <div class="flex flex-wrap space-x-1"><span>Paperback</span><b>£20.11</b><span>(English)</span></div> <div class="flex flex-wrap space-x-1"><span>Hardback</span><span class='line-through ml-2 text-primary'>£18.99</span><b>£18.04</b><span>(English)</span></div>
Out of Stock. Usually despatches within 2 weeks.
Philip Davis tells the story of Bernard Malamud (1914-1986), the self-made son of poor Jewish immigrants who went on to become one of the foremost novelists and short-story writers of the post-war period. The time is ripe for a revival of interest in a man who at the peak of his success stood alongside Saul Bellow and Philip Roth in the ranks of Jewish American writers.
Nothing came easily to Malamud: his family was poor, his mother probably committed suicide when Malamud was 14, and his younger brother inherited her schizophrenia. Malamud did everything the second time round - re-using his life in his writing, even as he revised draft after draft. Davis's meticulous biography shows all that it meant for this man to be a writer in terms of both the uses of and the costs to his own life. It also restores Bernard Malamud's literary reputation as one of the great original voices of his generation, a writer of superb subtlety and clarity.
Bernard Malamud: A Writer's Life benefits from Philip Davis's exclusive interviews with family, friends, and colleagues, unfettered access to private journals and letters, and detailed analysis of Malamud's working methods through the examination of hitherto unresearched manuscripts. It is very much a writer's life. It is also the story of a struggling emotional man, using an extraordinary but long-worked-for gift, in order to give meaning to ordinary human life.
Biography: literary
Fred's Bookshop staff picks: Steve
by Fred's Ambleside Bookshop View all (45) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,429 |
by Amalie Ise, Junior, iPreparatory Academy
Florida is not only the "Sunshine State," but also we're the "Education State!"
According to Governor Ron DeSantis and Education Commissioner Richard Corcoran, after a US News and World Report ranking of some Florida Universities, declared in a press release on December 23, 2020, that Florida is considered the "Education State." After research from The online periodical the Florida Phoenix, a non-profit that "cover[s] state government and politics,"
The research explains how Florida was ranked third among the 50 US states. It was determined through the tuition, debt upon graduating, and the percent of students that graduated with a bachelor's in six years. It did not include factors like SAT or ACT scores or GPAs. Florida takes pride in its low tuition rates and leaving students in little to no debt. When all these factors are considered Florida ranks well compared to other states. With over twenty-one million people living in Florida, filled with diversity, there are many options for state university study: Florida State University, the University of Central Florida, and the University of Florida.
University of Florida- ranked 5th:
The University of Florida, located in Gainesville, is commonly referred to as UF. The in-state tuition and other fees such as books, room and board, and expenses come to $21,430 and out-of-state $43,708 per year. The student to faculty ratio is 17:1 similar to that of the ratio of iPrep. UF also accepts Bright Futures and Florida Prepaid. The acceptance rate is 36.6% and 53,000 students attend the school.
According to the 2022 edition of Best Colleges and National Universities, the University of Florida is ranked 5th in the country! It is the highest-ranked University in our state!
UF campus
Florida State University- ranked 55:
FSU Campus
Florida State University also known as FSU is located in Tallahassee and is composed of 16 colleges. The tuition differs from in-state to out-of-state. The in-state tuition and other fees such as books, room and board, and expenses come to $23, 486 and out-of-state $37,732 per year. According to Johnny Darrisaw, a recruitment coordinator from FAMU-FSU College of Engineering, explains that most students from Florida rely on the bright futures scholarship as well as the Florida Prepaid Plan to lower the costs. The student faculty ratio is 21:1. But lab classes can range anywhere from 10:1 to 40:1 depending on the class. The acceptance rate at FSU is 36% and the university consists of 42,000 students!
According to the 2022 edition of Best Colleges and National Universities, FSU ranks 55th!
University of South Florida- ranked 103:
University of South Florida also known as USF is located in Tampa Bay. The in-state tuition and other fees such as books, room and board, and expenses come to is $23,446 and the out-of-state is $34,360. Like the other universities, these students rely on Bright Scholars and Florida Prepaid to cover most of the costs. The student-to-faculty ratio is 23:1 and 50,000 students attend USF! The acceptance rate is 48%, nearly accepting half the applicants.
USF Campus
University of Central Florida- ranked 148:
The University of Central Florida commonly referred to as UCF is located in Orlando. Its cost of tuition also differs between in-state and out-of-state. The in-state tuition and other fees such as books, room and board, and expenses come to $22,548K and out-of-state $37,880 per year. Like FSU, UCF also accepts Florida Prepaid and Bright Futures scholarships. The students to faculty ratio is 31:1, so it's the best fit for those who enjoy and thrive in bigger class sizes.
The acceptance rate is 44.4% and nearly 60,000 students attend UCF, which puts it in the top five biggest schools in the nation! According to the 2022 edition of Best Colleges and National Universities, UCF is ranked 148th in the nation.
All of these schools are located throughout Florida and are great for someone who thrives in bigger schools. All three schools are great schools where anyone can further one's education.
UCF Campus | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,647 |
Q: Laravel 9 migration fails after switching to mysql I have a locally developed laravel project with sqlite. For deployment reasons I want to switch to mysql. Unfortunately my relation migrations do not work anymore and produce the following error (I have made sure the order in which they run is correct, all other required tables are generated first and look correct)
Can't create table `laraveltest`.`test1s_test2s` (errno: 150 "Foreign key constraint is incorrectly formed")
(SQL: alter table `test1s_test2s` add constraint `test1s_test2s_test1_id_foreign` foreign key (`suacap_id`)
references `test1s` (`id`) on delete cascade)
The migration looks like this:
test1
public function up()
{
Schema::create('test1s', function (Blueprint $table) {
$table->id();
...
test2
public function up()
{
Schema::create('test2s', function (Blueprint $table) {
$table->id();
...
relation table test1s_test2s
public function up()
{
Schema::create('test1s_test2s', function (Blueprint $table) {
$table->primary(['test1_id', 'test2_id']);
$table->string('test1_id');
$table->foreign('test1_id')
->references('id')
->on('test1s')->onDelete('cascade');
$table->string('test2_id');
$table->foreign('test2_id')
->references('id')
->on('test2s')->onDelete('cascade');
});
}
I'm guessing this is related to the primary keys not being unsigned while the bigInt id's of the other tables are? I tried modifying
$table->primary(['test1_id', 'test2_id'])->unsigned();
but that does not work.
Can someone point me in the right direction? Thanks
A: Think whenever u make something Foreign keys they should be UNSIGNED BIGINT not string , in Laravel 9 - $table->foreignId('user_id');
also read the official document Laravel Official Doc
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,101 |
Q: Python code checker for comparing a function as an attribute I occasionally spend a considerable amount of time tracking down brainfarts in my code... while I normally run pylint against it, there are some things that slip past pylint. The easiest problem for me to overlook is this...
# normally, variable is populated from parsed text, so it's not predictable
variable = 'fOoBaR'
if variable.lower == 'foobar':
# ^^^^^<------------------ should be .lower()
do_something()
Neither pylint nor Python bark about this... is there a python code-checking tool that can flag this particular issue?
A: @Mike Pennington I just want to first say that I also run into this a lot -.-
@eyquem 'lower()' is a function. 'lower' is a function pointer (if I'm not mistaken). Python will let you attempt to run this code, but it will not invoke the function.
I think the reason this is hard to catch is that you don't always know the type of the variable which you're calling methods on. For example, say I have 2 classes.
class Foo()
def func(self):
#do stuff
pass
class Bar()
self.func = "stuff"
If your code has a function in it that takes an argument 'baz' like so:
def myfunction(baz):
print baz.func
def myfunction(baz):
baz.func()
Either one of these could be valid depending on baz's type.
There is literally no way of knowing if baz is of type 'Foo' or 'Bar', though.
EDIT: I meant with static analysis...
A: How do you propose a code-checker validate this? It's perfectly legitimate syntax. Rather than checking for this kind of mistake, it would be better to get into the habit of using better patterns.
Instead of:
variable = 'fOoBaR'
if variable.lower == 'foobar':
# ^^^^^<------------------ should be .lower()
do_something()
Do this:
variable = 'fOoBaR'
sane_variable = variable.lower()
if sane_variable == 'foobar':
do_something()
This way you're always explicitly calling .lower() on the value you're comparing against, instead of relying on an in-place method-call and comparison, which leads to the very pitfall you're experiencing.
A: This is pylint ticket #65910
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,350 |
Q: How can I get the value of an attribute for a method that has duplicates? I have some code that generates a report based off the attributes of my CodedUI test project. I want to be able to add the TestCategoryAttribute to this report but I don't know how to adapt my code to allow for duplicate attributes like below:
[TestMethod]
[TestCategory("Smoke")]
[TestCategory("Feature1")]
public void CodedUITest()
{
}
The code below works when I only have one TestCategory but will not work with multiple test categories as above:
//Other code above to find all CodedUI classes and all public, nonstatic methods with the TestMethod attribute
//find method with testcategory attribute
if (attrs.Any(x => x is TestCategoryAttribute))
{
var testcategoryAttr = (TestCategoryAttribute)attrs.SingleOrDefault(x => x is TestCategoryAttribute);
string testCategories = string.Join(", ", testcategoryAttr.TestCategories.Select(v => v.ToString()));
}
A: Replace the SingleOrDefault with the Where:
var testcategoryAttrs = attrs.Where(x => x is TestCategoryAttribute)
.Select(x => ((TestCategoryAttribute)x).TestCategory);
string testCategories = string.Join(", ", testcategoryAttrs.ToArray());
I don't know the property name in the TestCategoryAttribute, so I used the TestCategory in this sample.
A: SingleOrDefault throws an exception if there is more than one item that mathes with your condition.In this case you have two attributes and that's why you are getting the exception.
If you want to get only one item, then use FirstOrDefault. It returns the first item that mathces with the condition otherwise it returns null so you should be careful when casting returning result of FirstOrDefault, you may want to add a null-check before the cast,since you are using Any method and make sure there is at least one TestCategoryAttribute exists, null-check is not necessary in this case.
var testcategoryAttr = (TestCategoryAttribute)attrs
.FirstOrDefault(x => x is TestCategoryAttribute);
A: Here is the solution that ultimately worked for me. I asked an actual developer this question instead of trying to figure it out myself (I'm QA) :) I had to add some special logic to format the string properly because the attr.TestCategories object is a List.
//find method with testcategory attribute
if (attrs.Any(x => x is TestCategoryAttribute))
{
var testCategoryAttrs = attrs.Where(x => x is TestCategoryAttribute);
if (testCategoryAttrs.Any())
{
foreach (var testCategoryAttr in testCategoryAttrs)
{
TestCategoryAttribute attr = (TestCategoryAttribute)testCategoryAttr;
testCategories += string.IsNullOrEmpty(testCategories)
? string.Join(", ", attr.TestCategories)
: string.Format(", {0}", string.Join(", ", attr.TestCategories));
}
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 742 |
{"url":"http:\/\/www.docstoc.com\/docs\/98611331\/IRAC-Instrument-Handbook","text":"# IRAC Instrument Handbook by fdh56iuoui\n\nVIEWS: 101 PAGES: 206\n\n\u2022 pg 1\n\t\t\t\t\t\t\t\t\tIRAC Instrument Handbook\n\nSpitzer Heritage Archive Documentation\n\nIRAC Instrument and Instrument Support Teams\n\nVersion 2.0.1, June 2011\nIRAC Instrument Handbook\n\n1 Introduction ..............................................................................................................................1\n1.1 DOCUMENT PURPOSE AND SCOPE..............................................................................................................................1\n1.2 BASIC DEFINITIONS ......................................................................................................................................................1\n1.3 IRA C ESSENT IALS........................................................................................................................................................2\n1.4 ST ANDARD A CKNOWLEDGMENT S FOR IRAC PUBLICAT IONS................................................................................ 2\n1.5 HOW TO CONT ACT US..................................................................................................................................................3\n2 Instrument Description..............................................................................................................4\n2.1 OVERVIEW .....................................................................................................................................................................4\n2.2 DESCRIPT ION OF OPTICS..............................................................................................................................................5\n2.2.1 Field of View (FOV)................................................................................................................................................5\n2.2.2 IRAC Image Quality................................................................................................................................................6\n2.2.3 Spectral Response....................................................................................................................................................8\n2.2.4 Distortion ................................................................................................................................................................10\n2.3 DETECT ORS..................................................................................................................................................................11\n2.3.1 Design......................................................................................................................................................................11\n2.3.2 Performance ...........................................................................................................................................................11\n2.4 ELECT RONICS ..............................................................................................................................................................13\n2.4.1 Hardware ................................................................................................................................................................13\n2.4.2 Fowler Sampling....................................................................................................................................................13\n2.4.3 Exposure Times and Frame Time .......................................................................................................................14\n2.4.4 Subarray Mode.......................................................................................................................................................15\n2.4.5 Calibration Lamps.................................................................................................................................................15\n2.4.6 Firmware.................................................................................................................................................................15\n2.5 SENSITIVIT Y AND SAT URATION................................................................................................................................16\n2.5.1 Sensitivity................................................................................................................................................................16\n2.5.2 Saturation................................................................................................................................................................24\n\n3 Operating Modes .....................................................................................................................26\n3.1 READOUT M ODES AND FRAME TIMES DURING CRYOGENIC OPERATIONS........................................................ 26\n3.2 M AP GRID DEFINITION...............................................................................................................................................27\n3.3 DITHERING PATTERNS................................................................................................................................................27\n\n4 Calibration ..............................................................................................................................31\n4.1 DARKS ..........................................................................................................................................................................31\n4.2 FLAT FIELDS................................................................................................................................................................32\n4.3 PHOT OMET RIC CALIBRATION....................................................................................................................................34\n4.4 COLOR CORRECTION..................................................................................................................................................37\n4.5 A RRAY LOCAT ION-DEPENDENT PHOTOMET RIC CORRECT IONS FOR COMPACT SOURCES WITH STELLAR\nSPECT RAL SLOPES.....................................................................................................................................................................42\n4.6 PIXEL PHASE-DEPENDENT PHOT OMETRIC CORRECTION FOR POINT SOURCES................................................. 45\n4.7 IRA C POINT SPREAD AND POINT RESPONSE FUNCTIONS .................................................................................... 46\n4.7.1 Core PRFs ..............................................................................................................................................................48\n4.7.2 Extended PRFs.......................................................................................................................................................49\n4.7.3 Point Source Fitting Photometry ........................................................................................................................50\n\nii\nIRAC Instrument Handbook\n\n4.8 CALCULATION OF IRAC ZMAGS ..............................................................................................................................50\n4.9 A ST ROMET RY AND PIXEL SCALES............................................................................................................................52\n4.9.1 Optical Distortion..................................................................................................................................................52\n4.9.2 Pixel Solid Angles..................................................................................................................................................52\n4.10 POINT SOURCE PHOTOMET RY...................................................................................................................................53\n4.11 EXT ENDED SOURCE PHOT OMETRY ..........................................................................................................................55\n4.11.1 Best Practices for Extended Sources............................................................................................................. 56\n4.11.2 Extended Source Aperture Correction.......................................................................................................... 57\n4.11.3 Low Surface Brightness Measurements and the Maximum Scaling Factors ......................................... 59\n4.11.4 Caveats & Cautionary Notes..........................................................................................................................60\n4.11.5 Faint Surface Brightness Behavior ............................................................................................................... 60\n4.11.5.1 Bi nning............................................................................................................................................................ 60\n4.11.5.2 Small Scales..................................................................................................................................................... 62\n4.11.5.3 Medium Scales................................................................................................................................................ 62\n4.11.5.4 La rge Scales..................................................................................................................................................... 62\n4.11.5.5 Increasing exposure ti me................................................................................................................................ 62\n4.12 POINTING PERFORMANCE..........................................................................................................................................64\n4.12.1 Pointing Accuracy............................................................................................................................................65\n4.12.2 Jitter and Drift ..................................................................................................................................................66\n\n5 Pipeline Processing ..................................................................................................................68\n5.1 LEVEL 1 (BCD) PIPELINE ..........................................................................................................................................68\n5.1.1 SANITY DATATYPE (parameter checking) ...................................................................................................... 68\n5.1.2 SANITY C HECK (image contents checking) .................................................................................................... 68\n5.1.4 INSBPOSDOM (InSb array sign flipping) ........................................................................................................ 71\n5.1.5 CVTI2R4 (byte type changing) ............................................................................................................................72\n5.1.6 Wraparound Correction: IRAC WRAPDET AND IRAC WRAPCORR.......................................................... 73\n5.1.7 IRACNORM (Fowler sampling renormalization)............................................................................................ 74\n5.1.8 SNESTIMATOR (initial estimate of uncertainty) ............................................................................................. 75\n5.1.9 IRACEBWC (limited cable bandwidth correction).......................................................................................... 76\n5.1.10 Dark Subtraction I: FFCORR (first frame effect correction) or LABDARKSUB (lab dark\nsubtraction) ...........................................................................................................................................................................77\n5.1.11 MUXBLEEDC ORR (electronic ghosting correction) ............................................................................... 80\n5.1.12 DARKDRIFT (readout channel\u2019\u2019 bias offset correction) ....................................................................... 81\n5.1.13 FOWLINEARIZE (detector linearization) ................................................................................................... 82\n5.1.14 BGMODEL (zodiacal background estimation) ........................................................................................... 83\n5.1.15 Dark Subtraction II: SKYDARKSUB (sky \u201cdelta-dark\u201d subtraction)..................................................... 83\n5.1.16 FLATAP (flatfielding) ......................................................................................................................................84\n5.1.17 IMFLIPROT ......................................................................................................................................................84\n5.1.18 DETEC T-RADHIT (cosmic ray detection)................................................................................................... 85\n5.1.19 DNTOFLUX (flux calibration).......................................................................................................................85\n5.1.20 Pointing Transfer (calculation of pointing information) ........................................................................... 86\n5.1.21 PREDICTSAT (HDR saturation processing)............................................................................................... 88\n5.1.22 LATIMFLAG (residual image flagging)....................................................................................................... 89\n5.2 THE A RTIFACT -CORRECTED BCD PIPELINE........................................................................................................... 89\n5.2.1 Stray Light ..............................................................................................................................................................89\n5.2.2 Saturation................................................................................................................................................................90\n5.2.3 Sky Background Estimation .................................................................................................................................91\n\niii\nIRAC Instrument Handbook\n\n5.2.4 Column Pulldown ..................................................................................................................................................91\n5.2.5 Banding Correction (Channels 3 and 4) ........................................................................................................... 91\n5.2.6 Muxstripe Correction (Channels 1 and 2) ........................................................................................................ 92\n5.3 LEVEL 2 (POST -BCD) PIPELINE................................................................................................................................94\n5.3.1 Pointing Refinement ..............................................................................................................................................95\n5.3.2 Superboresight Pointing Refinement.................................................................................................................. 95\n6 Data Products ..........................................................................................................................97\n6.1 FILE-NAMING CONVENTIONS ...................................................................................................................................97\n6.2 IRA C SPECIFIC HEADER KEYWORDS...................................................................................................................... 99\n\n7 Data Features and Artifacts................................................................................................... 103\n7.1 DARKS, FLAT S AND BAD PIXELS............................................................................................................................103\n7.1.2 Flatfield .................................................................................................................................................................106\n7.2 ELECT RONIC A RT IFACT S .........................................................................................................................................108\n7.2.1 Saturation and Nonlinearity ..............................................................................................................................108\n7.2.2 Muxbleed (InSb)...................................................................................................................................................110\n7.2.3 Bandwidth Effect (Si:As) ....................................................................................................................................112\n7.2.4 Column Pull-Down\/Pull-Up .............................................................................................................................113\n7.2.5 Row Pull-Up .........................................................................................................................................................114\n7.2.6 Full-Array Pull-Up..............................................................................................................................................114\n7.2.7 Inter-Channel Crosstalk .....................................................................................................................................115\n7.2.8 Persistent Images.................................................................................................................................................116\n7.2.8.1 Cryogeni c Mission Persistent Images ........................................................................................................... 116\n7.2.8.2 Wa rm Mission Persistent Images ................................................................................................................. 118\n7.3 OPT ICAL A RTIFACT S ................................................................................................................................................120\n7.3.1 Stray Light from Array Covers..........................................................................................................................120\n7.3.2 Optical Banding and Internal Scattering ........................................................................................................125\n7.3.3 Optical Ghosts......................................................................................................................................................126\n7.3.4 Large Stray Light Ring and Splotches .............................................................................................................129\n7.4 COSMIC RAYS AND SOLAR PROT ONS.....................................................................................................................130\n\n8 Introduction to Data Analysis................................................................................................ 134\n8.1 POST -BCD DAT A PROCESSING...............................................................................................................................134\n8.1.1 Pointing Refinement ............................................................................................................................................134\n8.1.2 Overlap Correction .............................................................................................................................................134\n8.1.3 Mosaicking of IRAC Data ..................................................................................................................................135\n8.1.3.1 Crea ting a Common Fiducial Frame.............................................................................................................. 135\n8.1.3.2 Outlier Rejection ........................................................................................................................................... 135\n8.1.3.3 Mosai cker Output Files................................................................................................................................. 136\n8.1.3.4 To Dri zzle or Not to Drizzle? ......................................................................................................................... 136\n8.1.3.5 Mosai cking Moving Ta rgets.......................................................................................................................... 136\n8.1.4 Source Extraction ................................................................................................................................................137\n8.1.4.1 Noise Es tima tion ........................................................................................................................................... 137\n8.1.4.2 PRF Es tima tion .............................................................................................................................................. 137\n8.1.4.3 Ba ckground Es ti mation................................................................................................................................. 137\n8.1.4.4 Source Extra ction.......................................................................................................................................... 137\n\niv\nIRAC Instrument Handbook\n\n8.1.4.5 Outlier Rejection ........................................................................................................................................... 137\n\nAppendix A. Pipeline History Log ........................................................................................... 138\nAppendix B. Performing Photometry on IRAC Images ........................................................... 149\nAppendix C. Point Source Fitting IRAC Images with a PRF ................................................. 153\nC.3.1 Test on Calibration Stars ...................................................................................................................................155\nC.3.2 Subpixel Response in Channels 1 and 2 ..........................................................................................................156\nC.3.3 The Serpens Test Field .......................................................................................................................................159\n\nAppendix D. IRAC BCD File Header ...................................................................................... 167\nAppendix E. Acronyms ............................................................................................................ 173\nAppendix F. Acknowledgments ............................................................................................... 177\nAppendix G. List of Figures ..................................................................................................... 189\nAppendix H. List of Tables ...................................................................................................... 195\nAppendix I. Version Log......................................................................................................... 196\nBibliography ................................................................................................................................. 197\nIndex............................................................................................................................................. 199\n\nv\nIRAC Instrument Handbook\n\n1 Introduction\n\n1.1 Document Purpose and Scope\n\nThe IRAC Instrument Handbook is one in a series of documents that explain the operations of the Spitzer\nSpace Telescope and its three instruments, the data received from the instruments and the processing\ncarried out on the data. Spitzer Space Telescope Handbook gives an overview of the entire Spitzer\nmission and it explains the operations of the observatory, while the other three handbooks document the\noperation of, and the data produced by the individual instruments (IRAC, IRS and MIPS). The IRAC\nInstrument Handbook is intended to provide all the information necessary to understand the IRAC\nstandard data products, as processed by Version S18.18 of the online pipeline system, and which are\nretrievable from the Spitzer Heritage Archive (SHA). Besides the detailed pipeline processing steps and\ndata product details, background information is provided about the IRAC instrument itself, its\nobservational modes and all aspects of IRAC data calibration. It should be stressed that this Handbook is\nnot intended to support interactive data analysis. For data analysis advice and suggested data processing\nprocedures, please refer to the separate documentation available at the documentation website, including\nthe Spitzer Data Analysis Cookbook. This Handbook serves as the reference for both the processing as\nwell as the correct interpretation of IRAC data as available from the Spitzer Heritage Archive.\n\nIn this document we present information on:\n\u2022 the IRAC instrument and its observing modes,\n\u2022 the processing steps carried out on the Level 0 (raw) data,\n\u2022 the calibration of the instrument,\n\u2022 the artifacts, features and uncertainties in the data,\n\u2022 and the final IRAC archival data products.\nAn overview of the IRAC instrument is given in Chapter 2. In Chapter 3 the modes in which IRAC could\ntake data are presented. The calibration is described in Chapter 4. Online pipeline processing is described\nin Chapter 5. The data products themselves are described in Chapter 6. Data features are presented in\nChapter 7. A brief introduction into IRAC data analysis is given in Chapter 8. Several appendices are\nattached to give more detailed information on individual subjects.\n\n1.2 Basic Definitions\n\nThis section contains a description of the most commonly used terms in this Handbook. A complete list of\nacronyms can be found in Appendix E.\n\nAn Astronomical Observing Template (AOT) is the list of parameters for a distinct Spitzer observing\nmode. There is one possible IRAC AOT for the cryogenic mission, and one for the warm mission. The\nobserving parameters and mode are described in Chapter 3. A fundamental unit of Spitzer observing is the\n\nIntroduction 1 Document Purpose and Scope\nIRAC Instrument Handbook\n\nAstronomical Observation Request (AOR), also referred to sometimes as \u201cobservation.\u201d It is an AOT\nwith all of the relevant parameters fully specified. Each AOR is identified in the Spitzer Heritage Archive\nby a unique observation identification number known as AORKEY. An AOR consists of several Data\nCollection Events (DCEs), which can be thought of as single frame exposures. The data products consist\nof Level 0 products (\u201craw data\u201d) and Level 1 data products that are also called Basic Calibrated Data\n(BCD) and which are derived from the DCEs after pipeline processing. See Chapter 5 for more\ninformation about pipeline processing. BCDs (or in the case of IRAC, corrected BCDs or CBCDs) are\ndesigned to be the most reliable data product achievable by automated pipeline processing, and should be\nthe starting point for further data processing. The pipeline also produces Level 2 data products or Post-\nBCD products, which are derived from data from the whole AOR (i.e., combination of several CBCDs).\n\n1.3 IRAC Essentials\n\nThe most relevant software for IRAC data reduction is MOPEX (mosaicking and point source extraction).\nDocumentation for it can be found in the data analysis section of the documentation website. See Chapter\n8 for a brief introduction into IRAC data analysis. The separate Data Analysis section of the\ndocumentation website provides access to tools, user\u2019s guides and data analysis recipes.\n\nBefore you start using IRAC data, we recommend that you familiarize yourself very carefully with this\ndocument, and specifically Chapter 7, which discusses the various artifacts in IRAC data. Several of these\nartifacts are at least partially corrected in the pipeline, but you should still be aware of them. The CBCD\nframes contain the artifact-corrected observations, and should usually be a starting point for your further\ndata reduction and analysis. However, you always have the option of going back to the BCD frames if\nyou are not happy with how artifacts were corrected in the CBCDs, and perform your own corrections.\n\nthe photometry. These are discussed in Chapter 4. You can achieve down to about a few percent flux\naccuracy if you carefully perform all the corrections to the data. Specifically, we recommend that you\nperform point source photometry using aperture photometry, unless your targets lie in an area of sky that\nhas an extremely high surface density and\/or strongly variable background. The extended emission fluxes\nthem in 4 and think carefully before you publish any results about extended emission fluxes and\nbrightnesses.\n\n1.4 Standard Acknowledgments for IRAC Publications\n\nAny paper published based on Spitzer data should contain the following text: \u201cThis work is based [in\npart] on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion\nLaboratory, California Institute of Technology under a contract with NASA.\u201d If you received NASA data\nanalysis funding for your research, you should use one of the templates listed under\n\nIntroduction 2 IRAC Essentials\nIRAC Instrument Handbook\n\nhttp:\/\/irsa.ipac.caltech.edu\/data\/SPITZER\/docs\/spitzermission\/publications\/ackn\/. We also ask that you\ncite at least the seminal IRAC paper (Fazio, G. G., et al. 2004, ApJS, 154, 10 [9]) in your research paper,\nand other IRAC-related papers, as appropriate.\n\nA broad collection of information about IRAC and IRAC Data Analysis is available on the Spitzer\nDocumentation website, accessible via http:\/\/irsa.ipac.caltech.edu\/data\/SPITZER\/docs\/irac. In addition\n\nIRAC Instrument Handbook\n\n2 Instrument Description\n\n2.1 Overview\n\nThe InfraRed Array Camera (IRAC) was built by the NASA Goddard Space Flight Center (GSFC) with\nmanagement and scientific leadership by the Smithsonian Astrophysical Observatory (SAO) under\nprincipal investigator Giovanni Fazio. The information in this Handbook is based on the design\nrequirements and on the characterization of the flight instrument in pre-flight ground tests and on in-flight\nperformance, including the In-Orbit Checkout (IOC)\/Science Verification (SV) period in\nAugust\u2212November 2003.\n\nFigure 2.1. IRAC Cryogenic Assembly model, wi th the top cover removed to show the inner components.\nA brief, high-level summary of IRAC for astronomers appeared in the ApJS Spitzer Special Issue,\nspecifically the paper by Fazio et al. (2004, ApJS, 154, 10, [9]) entitled \u201cThe Infrared Array Camera\n(IRAC) for the Spitzer Space Telescope\u201d and in the paper by Hora et al. (2004, SPIE, 5487, 244, [15])\nentitled \u201cIn-flight performance and calibration of the Infrared Array Camera (IRAC) for the Spitzer\nSpace Telescope.\u201d Copies of these papers are available on the Spitzer documentation website.\n\nIRAC is a four-channel camera that provides simultaneous 5.2\u2019 \u00d7 5.2\u2019 images at 3.6, 4.5, 5.8, and 8 \u00b5m.\nTwo adjacent fields of view are imaged in pairs (3.6 and 5.8 \u00b5m; 4.5 and 8.0 \u00b5m) using dichroic\nbeamsplitters. All four detector arrays in the camera are 256 \u00d7 256 pixels in size, with a pixel size of\n\nInstrument Description 4 Overview\nIRAC Instrument Handbook\n\n~1.2\u201d \u00d7 1.2\u201d. The two short wavelength channels use InSb detector arrays and the two longer wavelength\nchannels use Si:As detectors. The IRAC instrument was designed to address the four major scientific\nobjectives defining the Spitzer mission. These are (1) to study the early universe, (2) to search for and\nstudy brown dwarfs and superplanets, (3) to study ultraluminous galaxies and active galactic nuclei, and\n(4) to discover and study protoplanetary and planetary debris disks. The utility of IRAC is in no way\nlimited to these objectives, which we only mention to explain the scientific drivers for the instrument\ndesign. IRAC is a powerful survey instrument because of its high sensitivity, large field of view, mapping\ncapabilities, and simultaneous four-color imaging.\n\nIRAC consists of the Cryogenic Assembly (CA) installed in the Multiple Instrument Chamber (MIC) in\nthe CTA, and the Warm Electronics Assembly (WEA) mounted in the spacecraft. Harnesses connect the\ndetectors and calibration subsystem in the CA to the WEA. The WEA communicates with the spacecraft\nover three RS-422 serial lines that allow receiving commands from, and sending acknowledgments and\nimage data to, the spacecraft Command & Data Handling (C&DH) computer.\n\nThe IRAC Cryogenic Assembly, depicted in Figure 2.1, consists of the following major subassemblies:\nthe Pickoff Mirrors; the Shutter; the Optics Housings, which hold the doublet lenses, beamsplitters,\nfilters, and cold stops; the Focal Plane Assemblies (FPAs) that include the detector arrays and associated\ncomponents; the Transmission Calibrator with its Source and Integrating Spheres; and the Housing\nStructure, consisting of the Main Housing Assembly and the wedge-shaped MIC Adapter Plate.\n\n2.2 Description of Optics\n\n2.2.1 Field of View (FOV)\n\nThe IRAC optical layout is shown in Figure 2.2 and Figure 2.3. Light from the telescope is reflected into\nthe IRAC structure by the pickoff mirrors for the two fields of view (FOVs). Each pair of channels has a\ndoublet lens which re-images the Spitzer focal plane onto the detectors. A beamsplitter reflects the short\nwavelength light to the InSb detectors (Channels 1 and 2) and transmits the longer wavelength light to the\nSi:As detectors (Channels 3 and 4). Channels 1 and 3 view the same telescope field (within a few pixels),\nand Channels 2 and 4 view a different field simultaneously. The edges of the two IRAC fields of view are\nseparated by approximately 1.52\u2019, with no overlap on the sky. The IRAC pixel scale is nearly the same in\nall channels (~1.2\u201d per pixel), providing a 5.2\u2019\u00d75.2\u2019 FOV.\n\nInstrument Description 5 Description of Optics\nIRAC Instrument Handbook\n\nInSb Detector\n\nDoublet\nLens Lyot Stop\n\nGe Filters\nPickoff Mirror\nSi:As\nDetector\n\nLyot Stop\nGe Beamsplitter\n\n35.00 MM\n\nFigure 2.2. IRAC optical layout, top view. The layout is similar for both pairs of channels; the light enters the\ndoublet and the l ong wavelength passes through the beams plitter to the Si:As detector (Channels 3 and 4) and\nthe short wavelength light is reflected to the InSb detector (Channels 1 and 2).\n\nTelescope Beam\n\nChannels 2 and 4\n\nFiducial\n\nPickoff Mirror\n\n35.00 MM\nChannels 1 and 3\n\nFigure 2.3. IRAC optics, side view. The Si:As detectors are shown at the far right of the figure, the InSb\narrays are behind the beams plitters.\n\n2.2.2 IRAC Image Quality\n\nThe IRAC optics specifications limit the wavefront errors to < \u03bb\/20 in each channel. IRAC provides\ndiffraction-limited imaging internally, and image quality is limited primarily by the Spitzer telescope. The\nmajority of the IRAC wavefront error is a lateral chromatic aberration that is most severe at the corners of\nthe IRAC field. The aberration is due to the difficulty of producing an achromatic design with a doublet\nlens over the large bandpasses being used. The effect is small, with the total lateral chromatic dispersion\nless than a pixel in the worst case. The sky coordinates of each pixel have been accurately measured in\nflight using an astrometric solution from the ultra-deep GOODS Legacy data, resulting in distortion\n\nInstrument Description 6 Description of Optics\nIRAC Instrument Handbook\n\ncoefficients that are in the world coordinate system of each image. The main effect is that the PSF and\ndistortion may be slightly color-dependent, which may be detectable for sources with extreme color\nvariations across the IRAC bands.\n\nA much larger variation in the flux of sources measured in different parts of the array is due to the tilt of\nthe filters, which leads to a different spectral response in different parts of the field of view. The flat field\ncalibration is done with the zodiacal light, which is relatively red; blue sources have a flux variation of up\nto 10% from one side of an array to the other (see Section 4.5 in this Handbook for more details).\n\nTable 2.1: IRAC i mage quality properties.\n\nChannel Noise FWHM FWHM of Central Pixel M aximum\npixels (mean;\u201d) centered pixel flux size distortion (pixels\n(mean) PRF (\u201c) (peak; %) (\u201c) relative to square\ngrid)\n\n1 7.0 1.66 1.44 42 1.221 1.3\n\n2 7.2 1.72 1.43 43 1.213 1.6\n\n3 10.8 1.88 1.49 29 1.222 1.4\n\n4 13.4 1.98 1.71 22 1.220 2.2\n\nTable 2.1 shows some properties relating to the IRAC image quality. These numbers were derived from\nin-flight measurements of bright stars. PRF is the \u201cPoint Response Function\u201d, further discussed in Section\n4.7.\n\nThe noise pixels column in Table 2.1 gives the equivalent number of pixels whose noise contributes to a\nlinear least-squares extraction of the flux of a point source from a 13\u00d713 pixel portion of an unconfused\nimage and assuming the PRF is perfectly known. In more detail, the quantity is derived as follows.\n\nLet the PRF in pixel i be P i and the intensity of an image in pixel i be Ii . If a point source with flux F is\npresent in the image, then Ii = FP i . If we do a least-squares fit to determine F, then we minimize\n\nI i \u2212 FPi 2\n\u03c72 = \u03a3\n\u03c3 i2\n\nwhere \u03c3i is the measurement uncertainty in pixel i. We will assume here that \u03c3i is independent of pixel\nand set \u03c3i = \u03c3. Now we take the derivative of \u03c7 2 with respect to the source flux and set it to zero to find\nthe optimum value. We find\n\n0 = \u03a3(I i \u2212 FPi )Pi\n\nInstrument Description 7 Description of Optics\nIRAC Instrument Handbook\n\nsolving for F, we find\n\n\u03a3I i Pi\nF=\n\u03a3Pi 2\n\nNow we derive the uncertainty in the flux. Using the well-known theorem for propagation of errors\n\n2\n\uf8eb dF \uf8f6\n\u03c3 = \u03a3\uf8ec \uf8f7 ,\n2\nF \uf8ec dI \uf8f7\n\uf8ed i\uf8f8\n\nand applying it to the result above, we find that\n\n\uf8eb Pi \uf8f6 2 2 \u03a3Pi2\u03c3 2 \u03c32\n\u03c3 = \u03a3\uf8ec 2 \uf8f7 \u03c3 =\n2\n= 2,\n\uf8ed \u03a3Pi \uf8f8 (\u03a3Pi2 ) \u03a3Pi\nF 2\n\n1\nor, equivalently, \u03c3F = \u03c3 N where N = , which is the definition of noise pixels.\n\u03a3Pi 2\nThere are two columns for the full width at half-maximum (FWHM) of the PRF in Table 2.1. The mean\nFWHM is from observations of a star at 25 different locations on the array. The FWHM for \u201ccentered\nPRF\u201d is for cases where the star was most closely centered in a pixel. The fifth column in Table 2.1 is the\nfraction of the flux in the central pixel for a source that is well centered in a pixel. It was determined from\nthe images of the focus star (after the telescope was focused) that were the most symmetric and\nconcentrated. These values for the flux in the central pixel can be used in the saturation predictions (see\nSection 2.4 below). The flux in the central pixel for a random observation is lower, because the Spitzer\nPRF is rather undersampled at the IRAC pixel scale.\n\n2.2.3 Spectral Response\n\nThe IRAC system throughput and optical performance is governed by a combination of the system\ncomponents, including the lenses, beamsplitters, filters, mirrors, and detectors. The system response is\nbased on measurements of the final in-flight system, including the beamsplitter, filter, ZnS & ZnSe\ncoating transmissions, mirror reflectance, BaF2 and MgF2 coating transmissions, and detector quantum\nefficiency.\n\nAt each wavelength, the spectral response curve gives the number of electrons produced in the detector\nper incoming photon. While the curves provided are best estimates of the actual spectral response, it is\nrecommended that the curves are used in a relative sense for color corrections and the supplied\nphotometric scaling (implicit in Level 1 products [\u201cBCDs\u201d] and described in Reach et al. 2005, PASP,\n117, 978, [22]) is used for absolute photometric calibration. Tests during IOC\/SV showed that the out-of-\n\nInstrument Description 8 Description of Optics\nIRAC Instrument Handbook\n\nband leaks are less than the astronomical background at all locations for sources of any temperature\ndetectable in the IRAC bands.\n\nThe spectral response curves presented below reflect our best knowledge of the telescope throughput and\ndetector quantum efficiency. The response curves use measurements of filter and beamsplitter\ntransmissions over the range of angles of incidence corresponding to distribution of incident angles across\nthe fields of view of the IRAC detectors (Quijada et al. 2004, Proc. SPIE, 5487, 244, [21]).\n\nWe provide three sets of curves for each IRAC channel: an average response curve for the entire array, an\naverage curve for the subarray field of view and a data cube of the response curves on a per pixel basis.\nThe average curves are useful for making color corrections to photometry of well-dithered (four or more)\nobservations. The response cubes can be used for more rigorous color corrections on per instance basis.\nFor most purposes, the average curves are sufficient. A more detailed discussion of the spectral response\ncurves is given by Hora et al. (2008, [14]). The derived IRAC spectral response curves are shown in\nFigure 2.4. The IRAC web pages contain links to the tabulated spectral response curves.\n\nFigure 2.4. S pectral res ponse curves for all four IRAC channels. The full array average curve is displ ayed in\nbl ack. The subarray average curve is in green. The extrema of the full array per-pixel transmission curves\nare also shown for reference.\n\nInstrument Description 9 Description of Optics\nIRAC Instrument Handbook\n\nFigure 2.5. Optical i mage distorti on i n IRAC channels. The panels show the i mage distorti ons as calculated\nfrom a quadratic pol ynomial model that has been fit to in-flight data. The magnitude of the distortion and\nthe directi on to which objects have moved from their i deal tangential pl ane projected positions is shown wi th\narrows. The length of the arrows has been increased by a factor of ten for cl arity. The maxi mum positional\ndeviations across the arrays for this quadratic distortion model are less than 1.3, 1.6, 1.4 and 2.2 pi xels for\nchannels 1\u22124, respecti vel y. The deri vation of the pi xel scales that are listed in Table 2.1 fully accounted for\nthe quadratic distortion effects shown here.\n\n2.2.4 Distortion\n\nDue to the off-axis placement of IRAC in the Spitzer focal plane, there is a small amount of distortion\nover the IRAC FOV. The maximum distortion in each IRAC band is < 2.2 pixels (compared to a perfectly\nregular grid) over the full FOV. Figure 2.5 shows the distortion across all four IRAC channels, as\ndetermined from data taken during IOC\/SV.\n\nInstrument Description 10 Description of Optics\nIRAC Instrument Handbook\n\n2.3 Detectors\n\n2.3.1 Design\n\nThe IRAC detector arrays were developed by the Raytheon\/Santa Barbara Research Center (SBRC) in\nGoleta, CA, under contract to SAO (Hoffman et al. 1998, [13]; Estrada et al. 1998, [8]). Channels 1 and\n2 use InSb arrays operating at ~15 K, and channels 3 and 4 use Si:As detectors operating at ~ 6 K. Both\narray types use the CRC744 CMOS readout circuit, and the same physical pixel size of 30 \u00b5m. The arrays\nare anti-reflection coated with SiO (Channels 1, 2, and 3) and ZnS (Channel 4). The power dissipation for\neach array is <1 mW. Table 2.2 gives some of the detector properties for IRAC channels 1\u20134. The\noperability is the percentage of the pixels in an array that are within usable specifications.\n\nTable 2.2: IRAC detector characteristics.\n\nFPA Read noise for frame with specified Quantum Well Depth Operability\nEfficiency (e-) (%)\ndesignation frame time (electrons) (%)\n\n0.1s* 2s 12s 30s 100s \u2022 \u2022 \u2022\n\n1 48534\/34\n16.9 11.8 9.4 7.8 8.4 87 145,000 99.97\n(UR)\n\n2 48975\/66\n16.8 12.1 9.4 7.5 7.9 86 140,000 99.9\n(GSFC)\n\n3 30052\/41\n9.0 9.1 8.8 10.7 13.1 45 170,000 99.99\n(ARC)\n\n4 30219\/64\n8.4 7.1 6.7 6.9 6.8** 70 200,000 99.75\n(ARC)\n\n* Per single subframe (1 of 64 planes in the BCD cube).\n**Per 50 s frame.\n\n2.3.2 Performance\n\nBoth types of detectors have measurable nonlinearity. The InSb arrays are nearly linear until they reach\nsaturation. The Si:As detectors are somewhat nonlinear over most of their operating range, and above\nhalf-well capacity this will contribute noticeably to the total error budget. However, all of the arrays are\nlinearized to better than 1% up to approximately 90% of their full-well capacity (defined in electrons in\nTable 2.2, with the gain listed in Table 2.4, corresponding typically to 45,000-60,000 DN). The detector\nlinearity has been measured during ground testing and in flight. The laboratory linearity measurements,\nwith the flight instrument, are shown in Figure 2.6. The arrays were illuminated with a constant flux,\nand successively longer exposures were taken. For a perfectly linear system, the flux would be directly\n\nInstrument Description 11 Detectors\nIRAC Instrument Handbook\n\nproportional to the exposure time, and the graph would show a straight line. In fact, the arrays were\ndriven past their saturation levels, and the shape of the curve up to 90% of the saturation level was fitted\nwith a polynomial for the linearization module in the pipeline.\n\nTable 2.3: IRAC Channel characteristics.\n\nChan Effective Bandwidth Average Minimum Peak\n\u03bb (\u00b5m) transmission in-band trans-\n(\u03b7I) trans- mission\n(\u00b5m) mission\n\n1 3.550 0.750 (21%) 0.676 0.563 0.748\n\n2 4.493 1.015 (23%) 0.731 0.540 0.859\n\n3 5.731 1.425 (25%) 0.589 0.522 0.653\n\n4 7.872 2.905 (36%) 0.556 0.450 0.637\n\nInstrument Description 12 Detectors\nIRAC Instrument Handbook\n\nFigure 2.6 : Non-linearity curves for the IRAC detectors. The detector responses are fairly linear\nuntil saturation, where there is a steep drop-off in responsivity.\n\n2.4 Electronics\n\n2.4.1 Hardware\n\nIRAC has no moving parts (other than the shutter, which was not operated in flight). The instrument takes\ndata by staring at the sky and sampling the arrays between resets. IRAC was capable of operating each of\nits four arrays independently and\/or simultaneously. All four arrays were used during normal, full-array\noperation.\n\n2.4.2 Fowler Sampling\n\nMultiple (Fowler) sampling is used to reduce the effective read noise. This mode of sampling consists of\ntaking N non-destructive reads immediately after the reset, and another N non-destructive reads near the\nend of the integration. Differencing is performed in the IRAC electronics to generate one integer value\nper pixel per exposure to store on the spacecraft and transmit to the ground. The Fowler N used for an\n\nInstrument Description 13 Electronics\nIRAC Instrument Handbook\n\nobservation depends on integration time and was selected to maximize the S\/N, based on in-flight\nperformance tests.\n\n2.4.3 Exposure Times and Frame Time\n\nThe relationship between the exposure time (Tex) and frame time is shown in Figure 2.7. The exposure\ntime is defined as the time elapsed between the first pedestal sample and the first signal sample. The\nFowler samples are taken consecutively at 0.2-second intervals in each group (pedestal and signal\nsamples). The frame time (Tf - Ti ) is the total time elapsed between resets which could include multiple\nreads and dead time before and after Fowler sampling. The frames are commanded by specifying the\nnumber of Fowler samples for the pedestal (NF) and the number of wait ticks in between the pedestal\nand signal frames (NW ); then the frame time is TF = (2NF + NW)\u03c4, where \u03c4 is the readout time (0.2 sec for\nfull array, 0.01 sec for sub-array mode.\n\nS4\nS3\nTex S2\nS1\n\nP4\nP3\nP2\nVoltage P1\n\nReset\nTime\n\nTi\nFrame Time Tf\n\nFigure 2.7: Fowler sampling times for one pixel (Fowler N=4). The Pn (n=1,2,3,4) show the\n\u201cPedestal\u201d readouts, and the Sn show the \u201cSignal\u201d readouts. Tex is the effective exposure time, and\nTf \u2013 Ti is the \u201cframe time,\u201d or total time to obtain one IRAC image. The reset part of the sketch is\nnot at the same time and voltage scale as the rest of the figure.\n\nInstrument Description 14 Electronics\nIRAC Instrument Handbook\n\n2.4.4 Subarray Mode\n\nIn subarray mode, only one corner, 32\u00d732 pixels offset by 8 pixels from the edges, is read out from one\narray. Pixels (9:40, 9:40) of the array are read out. The subarray pixel size is the same as the full array\npixel size (~1.2\u201d). Fowler sampling is performed as in full array mode, but a set of 64 subarray images are\ngenerated and tiled into a single 256\u00d7256 image before data are sent from IRAC. In subarray mode,\nFowler sampling is performed at 0.01 sec intervals. Subarray mode is useful for observing very bright\nsources and for obtaining high temporal resolution.\n\n2.4.5 Calibration Lamps\n\nIRAC contained two types of internal calibration lamps. The transmission calibrator lamps were designed\nto illuminate all four arrays and provide an internal responsivity measurement. There were two\ntransmission calibrator spheres, each of which contains two lamp elements. To illuminate the arrays, the\nshutter is closed, a transmission lamp is turned on, and the light from that lamp bounces off a mirror on\nthe back of the shutter. The flood calibrators individually illuminate each detector. The flood calibrators\ncould be controlled individually, and they could be used whether the shutter is open or closed. The flood\ncalibrators were operated at the end of each IRAC campaign and used as a consistency check. Calibration\nof IRAC observations is described in Chapter 4.\n\n2.4.6 Firmware\n\nThe IRAC firmware controls the focal plane assemblies, calibration electronics, and warm electronics\nboards. Apart from autonomous fault protection, the IRAC firmware responds only to commands sent by\nthe Spitzer Command & Data Handling (C&DH) computer. The C&DH sends setup commands to\nconfigure the electronics, requests for each telemetry packet, and integration commands to generate\nimages. IRAC responds to each command with an acknowledgment. In the case of a command that\nrequests telemetry, the acknowledgment consists of the telemetry packet, which is sent on the low-speed\nconnection between IRAC and the C&DH. There are two types of engineering data: special engineering\ndata, which are collected every 4 sec, and housekeeping data, which were collected every 30 seconds.\nSpecial engineering data are used for onboard communication between IRAC and the C&DH, while\nhousekeeping data are used on the ground to monitor instrument performance. A command that generates\nimages from the arrays is acknowledged on the low-speed line, and when the frame is complete, the data-\nready signal is sent on the high-speed line. The frames (together with their ancillary data) were then\ntransferred one at a time to the C&DH. The rate of transfer is 2 seconds per frame, which limited the data\ncollection rate to a maximum of four frames every 8 seconds for IRAC observations with all four arrays.\n\nAutonomous fault protection ensures that none of the monitored voltages or currents entered into a red\nlimit. Fault protection is performed by a watchdog demon that is always running when the instrument is\non. The red limits are stored in IRAC memory. If a voltage in the focal plane assembly goes into a red\nlimit, IRAC will turn off the affected focal plane array. (The individual pixel values are not monitored, so\na bright astronomical source does not trigger a red limit.) The command sequences would continue to\nexecute; therefore, it was possible for normal completion of an IRAC observing campaign to occur with\nonly three of the four arrays returning data. If a second focal plane assembly had a telemetry datum go\n\nInstrument Description 15 Electronics\nIRAC Instrument Handbook\n\ninto a red limit, then IRAC would send the C&DH (via the special engineering data on the low-speed\nline) a request to be turned off. If a telemetry point other than one affecting a single focal plane goes into\na red limit, IRAC would also send the C&DH a request to be turned off.\n\n2.5 Sensitivity and Saturation\n\n2.5.1 Sensitivity\n\nTo estimate the sensitivity of IRAC in flight, where possible we use the measured properties of Spitzer\nand IRAC from IOC\/SV; otherwise we use the required performance based on the design specifications.\nThe sensitivity to point sources (in flux density units) is based on the following formula:\n\nN pix\n\u03c3 = BTex + ( BTex f F ) 2 + R 2 + DTex (2.1)\nSTex f p\n\nwhere the scale factor is\n\nQ\u03b7T \u03b7 I A\u2206\u03bb\nS= (2.2)\nh\u03bb\n\nthe background current is\n\nB = SI bg f S \u2126 pix f ex (2.3)\n\nand the effective exposure time is\n\nTex = TF \u2212 0.2 N F (2.4)\n\nInstrument Description 16 Sensitivity and Saturation\nIRAC Instrument Handbook\n\nIn these equations, the spectral resolving power \u03bb \/ \u2206\u03bb is from Table 2.3 ; the detector quantum\nefficiency Q (electrons per photon) is from Table 2.2; the instrumental throughput \u03b7I is from Table 2.3;\nthe telescope throughput \u03b7T =[0.889, 0.902, 0.908, 0.914] for channels 1 to 4, respectively (with Be\nprimary, Al-coated secondary, and 50 nm ice contamination); the telescope area (including obstruction)\nA=4636 cm2 ; the equivalent number of noise pixels Npix is from Table 2.1 (and defined in Section 2.2.2);\nh is the Planck constant; Ibg is the background surface brightness in MJy\/sr; f S =1.2 is the stray light\ncontribution to the background; the dark current D is <0.1, 0.28, 1, and 3.8 e-\/s for channels 1, 2, 3, and 4,\nrespectively; the read noise R is from Table 2.2; \u2126 pix is the pixel solid angle (see Table 2.1); f p is the in-\nflight estimated throughput correction for point sources (Table 2.4); f ex is the in-flight estimated\nthroughput correction for the background (Table 2.4).\n\nThe \u201cthroughput corrections\u201d f p and f ex were determined by comparing the observed to expected\nbrightness of stars and zodiacal light. Stars were measured in a 10-pixel radius aperture, and the zodiacal\nlight was measured in channels 3 and 4 for comparison to the COBE\/DIRBE zodiacal light model. (This\nmeasurement was not possible in channels 1 and 2 because we could not use the shutter for absolute\nreference.)\n\nEarly in the mission, we found that the throughput in channels 3 and 4 was lower than expected, both for\nextended emission and point sources (but more so for point sources). Measurements of diffuse Galactic\nemission confirmed the deficit, and measurements of the PRF using bright stars showed that a\nconsiderable amount of stellar flux was being spread all across the arrays. The deficit in throughput from\ndiffuse emission was due to the fact that the QE of the arrays had been overestimated. No reliable\nmeasurements of QE existed for channels 3 and 4, so the QE was based on a theoretical model that\nincorrectly assumed that all the flux was reflected at the front of the detector diode chip (the detector\narrays are backlit). Some of the light that passes through the detector is scattered widely across the array.\nMeasurements on a sister array confirmed this internal scattering, and showed that it is strongly\nwavelength dependent. There is considerable evidence that the \u201closs\u201d of QE and the scattered light are not\ndue to contamination, or damaged optical coatings, etc. Table 2.4 lists some useful combinations of IRAC\ninstrument parameters.\n\nTable 2.5 gives the background brightnesses, in useful units, for three nominal observing directions. The\nlow-background model applies near the ecliptic pole; the high-background case is in the ecliptic plane;\nand the medium-background case is intermediate. The background model includes contributions from\nemission and scattering from zodiacal dust and emission from Galactic dust. The near-infrared cosmic\ninfrared background radiation is not included because it was partially resolved by Spitzer.\n\nTable 2.4: Useful quanti ties for IRAC sensitivity calculati ons.\n\nWavelength 3.6 \u00b5m 4.5 \u00b5m 5.8 \u00b5m 8 \u00b5m\nConversion factor (electrons\/sec)\/(MJy\/sr) 25 29 14 29\nS (electrons\/sec)\/(\u00b5Jy) 0.77 0.89 0.42 0.91\nGain (electrons\/DN) 3.3 3.7 3.8 3.8\nf p (throughput correction for point sources) 1.06 0.84 0.45 0.61\n\nInstrument Description 17 Sensitivity and Saturation\nIRAC Instrument Handbook\n\nf ex (throughput correction for background) 1 1 0.72 0.88\n\nTable 2.5 : B ackground brightness in IRAC wavebands.\n\n3.6 \u00b5m 4.5 \u00b5m 5.8 \u00b5m 8 \u00b5m\n\u201clow\u201d background model\nI\u03bd f S (MJy\/sr) 0.093 0.32 1.7 6.6\nBG\nF\u03bd (\u00b5Jy) 3.2 11 57 220\nB (elec\/sec) 2.5 9.9 18 184\n\u201cmedium\u201d background model\nI\u03bd f S (MJy\/sr) 0.15 0.44 2.3 9.3\nBG\nF\u03bd (\u00b5Jy) 5.1 15 79 320\nB (elec\/sec) 4.1 14 25 260\n\u201chigh\u201d background model\nI\u03bd f S (MJy\/sr) 0.52 1.0 5.6 22\nBG\nF\u03bd (\u00b5Jy) 18 35 190 750\nB (elec\/sec) 14 32 60 620\n\nThe quantity f F is the flat field pixel-to-pixel variance, which depends on the observing strategy. In what\nfollows, we will set fF =0, which would apply strictly in the case of stable detectors with perfect flat field\nmeasurements, and should apply practically for highly-dithered observations. An observation with no\ndithering will be limited by the correlated noise. The accuracy of a flat field derived from a single\nobserving campaign was measured to be 2.4, 1.2, 1.0, and 0.3% in channels 1, 2, 3, and 4, respectively, by\ncomparing flats in several campaigns. Using combined flats (\u201csuper sky flat\u201d) from the first two years, the\nestimated f F is 0.14%, 0.09%, 0.07%, and 0.01% in channels 1\u20134, respectively. Using these values for f F\nin equation 6.1, single frames are dominated by background and read noise. When combining multiple\nframes to generate a mosaic, the background and read noises will average down (as square root of the\nnumber of frames), while the flat-field noise will only average down for dithered observations. For N\nundithered observations on the \u201cmedium\u201d background, flat-field noise dominates when the total exposure\ntime, N \u00d7 Tex , exceeds approximately 420 sec (using individual campaign flats) or 2.5 hrs (using the\nsuper sky flat). For dithered observations, the flat-field noise will also average down, and will only be\nimportant for the very deep observations of high background fields.\n\nFor the frame times used in IRAC operations in flight, Table 2.6 gives the readout mode and Fowler\nnumber. For full array readout mode, only the 2, 12, 30, and 100 sec frame times can be chosen in the\nIRAC AOT; the 0.6 and 1.2 sec frame times come as part of the \u201chigh dynamic range\u201d (HDR) sequences.\nThe 0.4 sec full frame time is only available for channels 1 and 2 in Stellar Mode. The frame sets that are\ntaken for each pointing in HDR mode are shown in Table 2.7. Long frame times at 8 \u00b5m are background-\nlimited. Therefore there is a maximum frame time of 50 sec at 8 \u00b5m, and the 100\/200 sec frames were\n\nInstrument Description 18 Sensitivity and Saturation\nIRAC Instrument Handbook\n\nautomatically converted into two\/four repeats of 50 sec frames. The last column, Th , gives the extra time\nspent taking the HDR frames (used in the observing time estimate equation below).\n\nTable 2.6: Fowler numbers for IRAC frames\n\nFrame Time (sec) Readout Mode Fowler Number Wait Ticks\n200 Full 32 936\n100 Full 16 468\n50 Full (8 \u00b5m) 16 218\n30 Full 16 118\n12 Full 8 44\n2 Full 4 2\n1.2 HDR 1 4\n0.6 HDR 1 1\n0.4 Stellar 1 0\n0.4 Subarray 8 24\n0.1 Subarray 2 6\n0.02 Subarray 1 0\n\nTable 2.7: IRAC High-Dynamic-Range (HDR) framesets\n\nLong Frame Time List of frames taken Th (sec)\n200 0.6, 12, 200 15\n100 0.6, 12, 100 15\n30 1.2, 30 3\n12 0.6, 12 2\n\nTable 2.8, Table 2.9, Table 2.10 and Figure 2.8, Figure 2.9 and Figure 2.10 show the sensitivities for the\nfour IRAC channels for each of the three background models. The sensitivities in the tables are for point\nsources extracted from single images (but perfectly flat-fielded). In the figures, the sensitivities are for\npoint sources extracted from coadded images (perfectly registered). We do not include \u201cconfusion noise\u201d\n(due to overlapping images of distant galaxies or other sources of background structure) in the sensitivity\nestimates. The detectors are assumed to perform according to the IRAC detector measurements of read\nnoise, dark current, and quantum efficiency. The first 7 rows in each table show the sensitivity for full-\narray readouts, and the last three rows show the sensitivity for subarray readouts.\n\nTable 2.8: IRAC point-source sensiti vity, low background (1 \u03c3 , \u00b5Jy).\n\nFrame Time (sec) 3.6 \u00b5m 4.5 \u00b5m 5.8 \u00b5m 8 \u00b5m\n200 0.40 0.84 5.5 6.9\n100 0.60 1.2 8.0 9.8\n30 1.4 2.4 16 18\n12 3.3 4.8 27 29\n\nInstrument Description 19 Sensitivity and Saturation\nIRAC Instrument Handbook\n\n2 32 38 150 92\n0.6a 180 210 630 250\n0.4b 360 430 1260 450\n0.4c 81 89 609 225\n0.1c 485 550 2010 690\n0.02c 7300 8600 25000 8100\n\nTable 2.9: IRAC point-source sensiti vity, medi um background (1 \u03c3 , \u00b5Jy).\n\nFrame Time (sec) 3.6 \u00b5m 4.5 \u00b5m 5.8 \u00b5m 8 \u00b5m\n200 0.49 0.97 6.4 8.2\n100 0.73 1.4 9.3 12\n30 1.6 2.8 18 21\n12 3.6 5.3 31 34\n2 32 38 150 110\n0.6a 180 210 640 260\nb\n0.4 360 430 1260 460\nc\n0.4 82 89 610 250\nc\n0.1 490 550 2020 720\nc\n0.02 7300 8600 25000 8100\n\nTable 2.10: IRAC poi nt-source sensitivity, high background (1 \u03c3 , \u00b5Jy).\n\nFrame Time (sec) 3.6 \u00b5m 4.5 \u00b5m 5.8 \u00b5m 8 \u00b5m\n200 0.89 1.5 9.8 12\n100 1.3 2.1 14 18\n30 2.5 4.1 27 32\n12 4.8 7.1 44 52\n2 34 41 180 156\n0.6a 180 220 660 330\nb\n0.4 360 430 1280 540\nc\n0.4 84 93 650 340\nc\n0.1 490 560 2100 860\nc\n0.02 7300 8600 25000 8200\na\navailable only in high-dynamic-range mode.\nb\navailable only in stellar photometry mode.\nc\nsubarray mode (set of 64 32\u00d732 images). Sensitivity is per frame, not the sensitivity of a 64-frame\n\nInstrument Description 20 Sensitivity and Saturation\nIRAC Instrument Handbook\n\nFigure 2.8: IRAC point source sensiti vity as a function of frame ti me, for low background. To convert to\nMJ y\/sr, see equation 2.8.\n\nFigure 2.8, Figure 2.9 and Figure 2.10 show the point source sensitivity as a function of integration time\nfor each background model. The \u201ctime\u201d axes in the plots represent the frame time for the images, which\ndoes not include time for moving the telescope. The IRAC full-array frame times are 0.6, 2, 12, 30, and\n100 seconds (200 seconds was also available in the early mission). Other times plotted below are assumed\nto use multiple exposures of those fixed times.\n\nFor bright sources, shot noise due to counting statistics in electrons from the source itself becomes the\ndominant source of noise. We can estimate the total noise by adding the shot noise in quadrature, so that\n\n\u03c3 tot = \u03c3 (1 + ( F \/ Fb )) (2.5)\n\nInstrument Description 21 Sensitivity and Saturation\nIRAC Instrument Handbook\n\nFigure 2.9: IRAC point source sensiti vity as a function of frame ti me, for medium background. To convert to\nMJ y\/sr, see equation 2.8.\n\nwhere \u03c3 is the noise from equation 6.1 and\n\nFb = STex\u03c3 2 (2.6)\n\nIn the bright source limit, F >> Fb , then the signal-to-noise ratio becomes\n\nS \/ N = STex F (2.7)\n\nIf the exposure time is in seconds and the source flux is in \u00b5Jy, then for IRAC channels 1, 2, 3, and 4,\nrespectively, S\/N is 0.88, 0.95, 0.65, and 0.95 times Tex F .\n\nInstrument Description 22 Sensitivity and Saturation\nIRAC Instrument Handbook\n\nFigure 2.10: IRAC point source sensitivity as a functi on of frame ti me, for high background. To convert to\nMJ y\/sr, see equation 2.8.\nIn the sensitivity figures, the dashed line at 0.6 \u00b5Jy is the confusion limit predicted by Franceschini et al.\n(1991, [10]). This does not represent a hard sensitivity limit, but rather indicates where source confusion\naffects reliability of source extractions for low background regions. Data from IOC\/SV show noise\ndecreasing as N to 0.25 \u00b5Jy (channels 1 and 2) or 0.6 \u00b5Jy (channels 3 and 4). Moderately deep source\ncounts indicate that a source density equivalent to 36 beams\/source is reached at 20.5 mag, or 1.8 and 1.1\n\u00b5Jy at 3.6 and 4.5 \u00b5m, respectively (Fazio et al. 2004, [9]). The confusion estimates by Franceschini et\nal. and Fazio et al. are for low background, extragalactic observations only. For observations of higher\nbackground or more \u201ccluttered\u201d regions (such as the Galactic Plane) the confusion noise will be much\nmore significant.\n\nFor diffuse emission, the surface brightness sensitivity per pixel (in MJy\/sr) is\n\n0.03 f p\n\u00d7 the point source sensitivity [in \u00b5Jy]. (2.8)\nf ex N pix\n\nInstrument Description 23 Sensitivity and Saturation\nIRAC Instrument Handbook\n\nThe noise pixels, Npix , are defined in Table 2.1.\n\n2.5.2 Saturation\n\nThe saturation limit for IRAC is calculated as follows. Using the same notation as earlier in this section,\n\nWf w \u2212 BTF f ex\nFsat = (2.9)\nSTF f cen f p\n\nwhere W is the well depth (Table 2.2), fW =0.9 is the fraction of the well depth to which we can linearize\nthe intensities, and fcen is the fraction of the source flux falling onto the central pixel (Table 2.1). Table\n2.11 shows the point source saturation limits of IRAC at each frame time. In an extremely bright area of\nsky, such as an H II region, the saturation limit is lower. Note that the saturation value is conservatively\ncomputed from the worst-case in which the PSF is directly centered on a pixel. To apply Table 2.11 for\nextended sources,\n\nIsat = 28.6 \u00d7 f cen Fsat for compact (diameter < 30\u2033) sources (2.10)\n\nf cen f p\nIsat = 28.6 \u00d7 Fsat for more extended sources, (2.11)\nf ex\n\nwhere Isat is the total surface brightness (in MJy\/sr) at which a pixel saturates; \u2126pix is the solid angle, in\next\n\nsr, subtended by the pixel; f p , f cen , and f ex are as defined above, and Fsat is the saturating point source flux\ndensity (in mJy) from Table 2.11, appropriate for the channel and integration time. For 8.0 \u00b5m\nobservations at low ecliptic latitude, an estimate of zodiacal light should be included in the surface\nbrightness.\n\nTable 2.11: Maxi mum unsaturated point source (in mJ y), as a function of IRAC frame ti me .\n\nFrame Time (sec) 3.6 um 4.5 \u00b5m 5.8 \u00b5m 8.0 \u00b5m\n200 1.9 1.9 14 28\n100 3.8 3.9 27 28\n30 13 13 92 48\n12 32 33 230 120\n\nInstrument Description 24 Sensitivity and Saturation\nIRAC Instrument Handbook\n\n2 190 200 1400 740\n0.6 630 650 4600 2500\n0.4* 950 980 6950 3700\n0.4** 1000 820 3100 2300\n0.1** 4000 3300 13000 9000\n0.02** 20000 17000 63000 45000\n*stellar mode; **subarray mode\n\nThe zodiacal background only makes a difference for long frames in channel 4 when observing near the\necliptic plane. If the bright extended source extends well beyond the 5.2\u00b4\u00d75.2\u00b4 FOV, then the saturation\nbrightness is lower by the factor f s .\n\nInstrument Description 25 Sensitivity and Saturation\nIRAC Instrument Handbook\n\n3 Operating Modes\n\nThe IRAC Astronomical Observation Template (AOT) consists of an (optional) dither pattern superposed\non an (optional) rectangular-grid raster.\n\n3.1 Readout Modes and Frame Times During Cryogenic Operations\n\nIn full-array readout mode, there were four selectable frame times: 2, 12, 30, and 100 sec (and a fifth, 200\nsec, during the early mission). To allow sensitive observations without losing dynamic range, there was a\nhigh dynamic range (HDR) option. When this option was selected, the IRAC AOT took extra frames,\nwith frame times shorter than the selected frame time.The HDR frame times are given in Table 2.6. No\nspacecraft repositioning was done between frames, and the frames always were taken from shortest to\nlongest. If dithers were selected, then the entire frame set was repeated at each dither position.\n\nStellar photometry mode was available for observations of objects much brighter in channels 1 and 2 than\nin 3 and 4 (typically stars). This mode took short exposures in channels 1 and 2, and long exposures in\nchannels 3 and 4. Originally developed as engineering observations for taking calibration stars, this mode\nwas available for all observers. Three framesets were available. The shortest set took a single 0.4 sec\nframe in channels 1 and 2, and a 2 sec frame in channels 3 and 4. The next set took two undithered 2 sec\nframes in channels 1 and 2, and a 12 sec frame in channels 3 and 4. The longest frame time combination\ntook two undithered 12 sec frames in channels 1 and 2, and a 30 sec frame in channels 3 and 4. The\nsensitivities of each frame are identical to those in full array mode. Dithering and mapping were also\navailable in this mode.\n\nFor very bright sources, a subarray mode was available. In this mode, only a small 32\u00d732 pixel portion of\nthe array was read out, so the field of view was only 38\u201d\u00d738\u201d. Mapping was not allowed in subarray\nmode. However, small maps could be made using a cluster target. In subarray readout mode, there were\nthree selectable frame times: 0.02, 0.1, and 0.4 sec. For one commanded image in subarray mode, a set of\n64 Fowler-sampled frames were taken in succession, so that each time an image was commanded in\nsubarray mode, a cube of 64\u00d732\u00d732 pixel images was generated. This means that the durations of a\nsingle repeat at each of the three subarray frame times were 1.28, 6.4, or 25.6 sec, respectively. The IRAC\nAOT moved the telescope to point to the subarray region of each requested channel at the target in turn.\nFor the 0.02 sec frame time, data rate limitations allowed only data in the channel actually pointing at the\ntarget to be taken. For the 0.1 sec and 0.4 sec frame times, data were taken in all four channels at each\npointing position, although only one channel at a time pointed at the target.\n\nOperating Modes 26 Readout Modes and Frame\nTimes During Cryogenic\nOperations\nIRAC Instrument Handbook\n\n3.2 Map Grid Definition\n\nIf \u201cNo mapping\u201d was selected in the AOT, the map grid consisted of a single position at the coordinates\nspecified in the Target section of the AOT. With \u201cNo mapping\u201d selected, and selecting both fields of\nview, first the 4.8\/8.0 \u00b5m field of view was pointed at the target, then the telescope repositioned so that\nthe 3.6\/5.8 \u00b5m field of view pointed at the target. In both cases, data from all 4 arrays were collected,\nwhether they were pointed at the target or not.\n\nIf the mapping mode was used, a rectangular map grid needed to be specified in either array or celestial\ncoordinates. In array coordinates, the map grid is aligned with the edges of the array, such that the map\nrows and columns correspond to rows and columns of the array. Specifically, a column is along a line of\nconstant solar elongation, and a row is along an ecliptic parallel (line of constant ecliptic latitude). It is\nworth noting that the two IRAC fields of view are at approximately constant solar elongation, so that a\nmap with 1 column and several rows made a strip along the direction of the separation between the two\nfields of view and yielded 4-array coverage along part of the strip (if it was long enough). In celestial\ncoordinates the rows and columns correspond to J2000 right ascension and declination. A position angle,\ndegrees E of N, could be specified to orient the raster in equatorial coordinates. Specifically, if the\nposition angle is zero, a column is along a line of constant right ascension, and a row is along a parallel\n(line of constant declination). The map can be offset from the specified coordinates by giving a map\ncenter offset.\n\n3.3 Dithering Patterns\n\nFor the full-array mode there were two types of dither patterns available. Five such patterns were fixed\npatterns, which were performed identically at each mapping position. The cycling pattern is a set of dither\npositions (also referred to as points), a different subset of which was performed at each map grid position.\n\nDifferent patterns were available in subarray mode, as the angular scales covered by the arrays were quite\ndifferent. Two fixed patterns were available for this mode.\n\nThe characteristics of the available dither patterns are given in Table 3.1. The Reuleaux Triangle patterns\nwere designed with the idea of optimizing the Figure of Merit of Arendt, Fixsen, & Moseley (2000, [3]).\nThey thus sample a wide range of spatial frequencies in a fairly uniform manner, and were well suited to\nthe Fixsen least-squares flat fielding technique. The 9-point and 16-point patterns were designed to be the\noptimum size for 1\/3 and \u00bc subpixel dithering, respectively. The random 9 pattern is based on a uniform\nrandom distribution. The spiral 16 pattern was designed by R. Arendt to provide a pattern which is both\ncompact and has a good figure of merit for self-calibration. The cycling patterns were designed for\nobservations (\u201cAORs\u201d) having many mapping\/dithering observations. The large and medium patterns\nwere Gaussian distributions (with dithers >128 pixels removed). The small pattern was specifically\ndesigned for mapping, where only a few dithers were taken at each map position. It was also based on a\nGaussian distribution, but the center was downweighted to decrease the fraction of small dithers in the\npattern, and it was truncated at a maximum dither of 11 pixels to ensure that maps with up to 280\u201d\n\nOperating Modes 27 Map Grid Definition\nIRAC Instrument Handbook\n\nspacing have no holes, even if there is only one dither per map point. All the patterns were constrained to\nhave no pair of dithers closer than three pixels in any run of four consecutive points. The cycling dither\ntable wraps around once the final (311th) element was reached. This pattern had a \u00bd sub-pixel sampling\npattern superposed on it, starting with point 1 and repeating continuously every four points (at point 311,\nthe final cycle was simply truncated early, thus patterns which wrap around the table missed a sub-pixel\ndither point). The five-point Gaussian pattern was a general use pattern suitable for shallow observations\nwhere the exact sub-pixel sampling is unimportant. It had a \u00bd subpixel pattern, with the 5th point at sub-\npixel (\u00bc,\u00bc). Figure 3.1 shows the dither patterns at the default (large) scale. Figure 3.2 shows the cycling\ndither patterns and the distribution of both the dithers and of the separation between dithers for each scale.\n\nTable 3.1: Characteristics of the di ther patterns.\n\nDither Pattern Scale Max dither Median dither Sub-pixel\n(pixels from (0,0)) separation (pixels) dither pattern\nCycling Small 11 10.5 \u00bd pixel\nMedium 119 53 \u00bd pixel\nLarge 161 97 \u00bd pixel\n5-point random Small 26 23 \u00bd pixel\nMedium 52 46 \u00bd pixel\nLarge 105 92 \u00bd pixel\n9-point random Small 16 14 1\/3 pixel\nMedium 34 28 1\/3 pixel\nLarge 69 59 1\/3 pixel\n12-point Reuleaux Small 13 15 \u00bd pixel\nMedium 27 30 \u00bd pixel\nLarge 55 59 \u00bd pixel\n16-point spiral Small 16 12 \u00bc pixel\nMedium 32 23 \u00bc pixel\nLarge 64 45 \u00bc pixel\n36-point Reuleaux Small 17 19 \u00bc pixel\nMedium 34 39 \u00bc pixel\nLarge 67 78 \u00bc pixel\n\nEach of the IRAC dither patterns was available in three sizes, large (default), medium, and small. For\nmost of the patterns, the scaling of the large, medium, and small patterns is approximately in the ratio\n4:2:1. Exceptions are the small cycling pattern, which is about 1\/5 of the size of the large cycling pattern\n\nOperating Modes 28 Dithering Patterns\nIRAC Instrument Handbook\n\nand has a lower-weighted inner region to reduce the numbers of small separation dithers, and the 4-point\nsubarray pattern where the scaling is 4:3:1.5. For all the patterns, the sub-pixel dithering is maintained,\nindependent of scale.\n\nSub-pixel dithering, combined with the drizzle technique (Fruchter & Hook 2002, [11]) to reconstruct the\nimages, can improve the sampling of the mosaics that are obtained from IRAC (or any other)\nobservations. Such strategies have been used for the WFPC2 and NICMOS instruments on the HST for\nsome time (for details see the HST Drizzle Handbook). Dithering is also needed to calibrate intra-pixel\nsensitivity variations, and needed for programs requiring accurate photometry and astrometry (Anderson\n& King 2000, [2]). To be effective, however, accurate pointing and low image distortion are required.\nThe offsetting accuracy of Spitzer is in the range 0.1\u201d\u22120.4\u201d. This, combined with the image distortion in\nthe IRAC arrays, places a limit of about \u00bc pixel on the sub-sampling that is likely to prove useful in\npractice. For example, the distortion of the IRAC camera is < 1% (see Figure 2.5). Thus for the largest\ndither patterns, which typically offset up to \u00b164 pixels from the starting point, the offsets will be up to\n\u00b10.6 pixels from the nominal values. Thus only in the small scale patterns, where the offsets are less than\n\u00b116 pixels, will the sub-pixel sampling work well, though even on the larger scales some improvement of\nthe images will probably be noticeable.\n\nOperating Modes 29 Dithering Patterns\nIRAC Instrument Handbook\n\nFigure 3.1 : IRAC di ther patterns for the \u201clarge\u201d scale factor.\n\nFigure 3.2: Characteristics of the cycling dither pattern, in pi xels.\n\nOperating Modes 30 Dithering Patterns\nIRAC Instrument Handbook\n\n4 Calibration\n\nThe Spitzer Science Center (SSC) performed routine calibrations of IRAC using observations of standard\nstars and other astronomical objects. The data obtained in these observations were used to construct the\nnecessary calibration inputs to the pipeline for the IRAC data processing of science observations. The\ncalibration data files are available to the general user in the Spitzer Heritage Archive maintained by\nIRSA.\n\n4.1 Darks\n\nDark current and bias offsets were calibrated via the standard ground-based technique of dark subtraction.\nAs part of routine operations, the SSC observed a dark region of the sky (skydark) near the north ecliptic\npole at least twice per campaign (at the beginning and end). These data were reduced and combined in\nsuch a way as to reject stars and other astronomical objects with size-scales smaller than the IRAC array.\nThe resulting image (Figure 4.1) of the minimal uniform sky background contains both the bias and dark\ncurrent. When subtracted from the routine science data, this eliminates both of these instrumental\nsignatures. Naturally, this also subtracts a component of the true celestial background. The SSC included\na COBE-based model estimate of the true celestial background. Note that the lack of an isolated\nmeasurement of the dark current and bias offset during shutterless operations limits the ability of IRAC to\nmeasure the true celestial background.\n\nCalibration 31 Darks\nIRAC Instrument Handbook\n\nFigure 4.1: IRAC instrument dark current i mages. These measurements were made during a normal\ncampaign producing a skydark wi th an exposure ti me of 100 seconds.\n\n4.2 Flat Fields\n\nPixel-to-pixel relative gain variations are commonly know as the \"flatfield\". To get the most accurate\nmeasurement of the flatfield, including the effects of the telescope and the IRAC pickoff mirrors, one\nmust use observations of the sky. Because the IRAC detectors are relatively large, there are few discrete\nastronomical objects large enough and bright enough to fill the detector field of view.\n\nCalibration 32 Flat Fields\nIRAC Instrument Handbook\n\nThe flatfield was derived from many dithered observations of a network of 22 high zodiacal background\nregions of the sky in the ecliptic plane, which ensured a relatively uniform illumination with sufficient\nflux on all pixels such that the observations were relatively quick to perform. One such region was\nobserved in every instrument campaign.\n\nThe data were combined with object identification and outlier rejection, creating an object-free image of\nthe uniform celestial background, further smoothed by the dither pattern. An identical observation made\nat the north ecliptic pole (the \"skydark\") was subtracted, and the result normalized to create the flatfield.\nThe resulting flatfield was divided into the science data. Pixel-to-pixel accuracy of the flat-fielding\nderived from a single observing campaign was typically 2.4%, 1.2%, 1.0%, and 0.3%, 1\u03c3, for channels 1\nthrough 4, respectively.\n\nAnalysis of the flatfield response on a campaign-wise basis showed that there were no changes\nthroughout the cryogenic mission. Based on this, all of the flatfield data were combined into a \"super\nskyflat\". The 1-sigma pixel-to-pixel accuracy of this flat is 0.14%, 0.09%, 0.07%, and 0.01% in channels\n1\u22124, respectively. This is the flatfield used for all pipeline processed data. The super skyflats are available\nfrom the \u201cIRAC calibration and analysis files\u201d section of the archival documentation website.\n\nUsers should note that the flatfield data were generated from the zodiacal background, and are appropriate\nfor objects with that color. There is a significant color term, of order 5%\u221210%, for objects with a\nRayleigh-Jeans spectrum in the mid-infrared (such as stars); see Section 4.5 for more information. Note\nthat for deep survey observations and other data sets with a large number of frames and a good dithering\nstrategy, the system gain can be determined by the actual survey frames themselves, rather than using the\nstandard set of dedicated observations of some other part of the sky.\n\nDuring warm operations the flatfield was remeasured. While it is similar in overall appearance, details are\nsufficiently different that the warm and cold flats cannot be interchanged.\n\nCalibration 33 Flat Fields\nIRAC Instrument Handbook\n\nFigure 4.2: IRAC instrument super skyflats showi ng the fl atfiel d response as measured onboard, for channels\n1\u20134.\n.\n\n4.3 Photometric Calibration\n\nA number of astronomical standard stars were observed in each instrument campaign to obtain a valid\nabsolute flux calibration. Stars with a range of fluxes were observed at a number of positions across the\n\nCalibration 34 Photometric Calibration\nIRAC Instrument Handbook\n\narray and many times throughout the mission to monitor any changes that may have occurred. Calibration\nstars with measured spectral types and accurate absolutely calibrated fluxes in the IRAC bands have been\ndetermined. These absolute calibration stars were in the continuous viewing zone (CVZ) so that they\ncould be observed at any time necessary and could be monitored throughout the mission.\n\nFour stars were observed in the CVZ at the beginning and end of each instrument campaign. These\nstandards remained the same throughout the mission, and provide the absolute flux reference for IRAC.\nAdditionally, a calibrator near the ecliptic plane (which was different for each campaign) was observed\nevery twelve hours. Its placement in the ecliptic plane minimized telescope slews. This calibrator was\nused to monitor any short-term variation in the photometric stability.\n\nAnalysis of the flux calibrator data indicates that absolute flux calibration is accurate to 3% (reflecting\nmostly the uncertainty in the models). Repeatability of measurements of individual stars is better than\n1.5% (dispersion), and can be as good as 0.01% with very careful observation design (e.g., Charbonneau\net al. 2005, [6]). The absolute calibration is derived taking several systematic effects into account. The\nsteps are described in detail by Reach et al. (2005, [22]). If this methodology is not applied, then point\nsource photometry from the Level 1 products (BCDs) can be in error by up to 10%.\n\nIRAC is calibrated using both so-called primary and secondary calibrator stars. The primary stars are used\nto monitor long-term variations in the absolute calibration. They number 11 stars, are located in the\ncontinuous viewing zone (CVZ), and were thus observable year-round. They were observed once at the\nbeginning, and once at the end of each campaign, i.e., about every 10 days whenever the instrument was\nswitched on. The primary calibrators (in decreasing brightness) are (J2000; with flux densities in mJy in\nchannels 1\u22124, respectively):\n\nNPM1+67.0536 = SAO 17718 = 2MASS J17585466+6747368 (K2III, Ks=6.4); 843.6, 482.3, 320.0,\n185.3\nHD 165459 = 2MASS J18023073+5837381 (A1V, Ks=6.6); 647.7, 421.3, 268.6, 148.1\nNPM1+68.0422 = BD+68 1022 = 2MASS J18471980+6816448 (K2III, Ks=6.8); 580.4, 335.5, 223.2,\n128.9\nKF09T1 = GSC 04212-01074 = 2MASS J17592304+6602561 (K0III, Ks=8.1); 169.9, 104.7, 67.03,\n38.75\nNPM1+66.0578 = GSC 04229-01455 = 2MASS J19253220+6647381 (K1III, Ks=8.3); 140.9, 82.37,\n54.54, 29.72\nNPM1+64.0581 = HD 180609 = 2MASS J19124720+6410373 (A0V, Ks=9.1); 63.00, 41.02, 26.18,\n14.40\nNPM1+60.0581 = BD+60 1753 = 2MASS J17245227+6025508 (A1V, Ks=9.6); 38.21, 24.74, 15.74,\n8.699\nKF06T1 = 2MASS J17575849+6652293 (K1.5III, Ks=11.0); 13.92, 7.801, 5.339, 3.089\nKF08T3 = 2MASS J17551622+6610116 (K0.5III, Ks=11.1); 11.77, 7.247, 4.642, 2.691\nKF06T2 = 2MASS J17583798+6646522 (K1.5III, Ks=11.3); 10.53, 5.989, 4.050, 2.339\n2MASS J18120956+6329423 (A3V, Ks=11.6) ; 8.681, 5.662, 3.620, 2.000\n\nCalibration 35 Photometric Calibration\nIRAC Instrument Handbook\n\nAll of the calibration data taken with these stars are public and are available in the Spitzer Heritage\nArchive. The secondary calibrator stars were used to monitor short-term variations in the absolute\nlocated near the ecliptic plane, in a tightly constrained window of about 20 degrees. Because of the\nmotion of the Earth about the Sun this window constantly moved and so any one secondary calibrator was\nvisible for only a campaign or two per year. In practice, the calibration values for IRAC appear to be quite\ntemporally stable.\n\nThe data are calibrated by means of aperture photometry, using a 10 native pixel radius (12 arcseconds)\naperture. The background was measured using a robust average in a 12\u221220 pixel annulus around the\ncentroid of the star. Unfortunately, ground-based infrared calibrators were too bright to use as calibrators\nfor IRAC. Therefore, one must use models to predict the actual flux for each channel as a function of\nspectral type (Cohen et al. 2003, [7]). Table 4.1 lists the calibration factors that are used in the final\nprocessing of all IRAC data. The absolute calibration is described in detail in Reach et al. (2005, [22]),\nwith further refinements at the 1%\u20133% level, based on better models for the calibration stars and a better\nestimate of the corrections to photometry (pixel phase, array-location dependent photometric correction,\netc.).\n\nTable 4.1: The photometric calibrati on and zero magnitude flux for IRAC.\n\n\u03bb (\u03bcm) FLUXCONV (M Jy\/sr)\/(DN\/sec) F\u03bd0 (Jy)\n\n3.6 0.1088 280.9\u00b14.1\n\n4.5 0.1388 179.7\u00b12.6\n\n5.8 0.5952 115.0\u00b11.7\n\n8.0 0.2021 64.9\u00b10.9\n\nThe absolute gain calibration is accurate to better than 3%. The stellar photometry is repeatable at the <\n1% level. The absolute fluxes of the calibration stars are known to 2% \u2013 3% (Cohen et al. 2003, [7]). To\nobtain photometry at this accuracy, photometric corrections for the location of the source within its peak\npixel, and the location of the source within the array, must be made.\n\nNote that IRAC is not an absolute background photometer, so the total brightness in IRAC images should\nbe used with great caution. There was a cold shutter in the calibration assembly, but it was not operated in\nflight, in order to minimize mission risk. Therefore, the offset level in IRAC images is referred to\nlaboratory measurements before launch, where the offset level was observed to change very significantly\nfrom one laboratory experiment to another.\n\nIn laboratory tests, the absolute offset of IRAC images was found to vary at levels that are comparable to\nthe minimum celestial background in channels 1 and 2. Furthermore, the offset level changes depending\non whether the detector was recently annealed. Thus, for diffuse surface brightness measurements, we\n\nCalibration 36 Photometric Calibration\nIRAC Instrument Handbook\n\nrecommend making differential measurements among at least two sky positions, preferably from the same\ncampaign.\n\n4.4 Color Correction\n\nIRAC is a broad-band photometer. We describe here the method used for calibrating our data in specific\nsurface brightness (MJy\/sr) or flux density (Jy) units, and we provide the prescription for how to interpret\nthe data for sources with spectral shapes other than the nominal one assumed in the calibration process.\nThe conventions used by IRAC are the same as those used by IRAS (Beichman et al. 1988, IRAS\nExplanatory Supplement, \u00a7VI.C.3, [4]), COBE\/DIRBE (Hauser et al. 1998, [12]), and ISO (Blommaert et\nal. 2003, [5]). The basic idea is to quote a flux density F\u03bdquot at a nominal wavelength \u03bb0 = c \/ \u03bd 0 that\n0\n\nwould be accurate for a source with a nominal spectrum, \u03bdF\u03bdnom =constant. Using this F\u03bdnom is merely a\nmatter of convention; in fact a wide range of spectra are expected, both redder (e.g., interstellar medium,\nasteroids) and bluer (e.g., stars) than nominal. The color correction tables given below allow observers to\nconvert the nominally-calibrated data, F\u03bdquot into more accurate estimates of the flux density at the\n0\n\nnominal wavelength.\n\nThe number of electrons collected from the nominal source in a straight integration of duration t using a\ntelescope with area A is\n\nF\u03bdnom\nN enom = tA\u222b Rd\u03bd ; (4.1)\nh\u03bd\n\nwhere R is the system spectral response. The convention for R is that it is proportional to the number of\nelectrons produced by a single photon with energy h\u03bd. If we define for convenience\n\n\u2206\u03bd = \u222b (\u03bd \/ \u03bd 0 ) \u22122 Rd\u03bd (4.2)\n\nthen the number of electrons collected from a source with the nominal spectrum is\n\nF\u03bdnom\nN nom\n= tA 0\n\u2206\u03bd (4.3)\nh\u03bd 0\ne\n\nCalibration 37 Color Correction\nIRAC Instrument Handbook\n\nThe calibration factor, by which the number of electrons, Ne from an arbitrary source must be divided in\norder to give the quoted flux density at the nominal wavelength, is\n\nN enom At\u2206\u03bd\nC = nom = (4.4)\nF\u03bd 0 h\u03bd 0\n\nThe calibration factor is measured using observations of a celestial calibrator source of known spectrum,\nF\u03bd* . The number of electrons collected from the star is\n\nF\u03bd*\nN e* = tA\u222b Rd\u03bd (4.5)\nh\u03bd\n\nCombining with equation 4.4, we can express the calibration factor in terms of the observed number of\nelectrons from the calibration source:\n\nN e* \u2206\u03bd\nC=\nF* (4.6)\nh\u03bd 0 \u222b \u03bd Rd\u03bd\nh\u03bd\n\nWe can now cast this in convenient terms as follows:\n\nN e*\nC= * * (4.7)\nF\u03bd 0 K\n\nwhere F\u03bd*0 is the flux density of the calibration source at the nominal wavelength, and K * is the color\ncorrection factor for the calibration source spectrum.\n\nThe color correction factor for a source with spectrum F\u03bd is defined as:\n\n\u222b ( F\u03bd \/ F\u03bd )(\u03bd \/\u03bd ) Rd\u03bd\n\u22121\n0\nK= (4.8)\n0\n\n\u222b (\u03bd \/\u03bd ) Rd\u03bd\n\u22122\n0\n\nCalibration 38 Color Correction\nIRAC Instrument Handbook\n\nIn this convention, the overall normalization of R is unimportant. Observers can correct the photometry to\nthe spectrum of their source by either performing the integral in this equation or looking up the color\ncorrections for sources with similar spectra. Note that our definition of the color correction looks slightly\ndifferent from that in the IRAS Explanatory Supplement [4], because we used the system spectral\nresponse R in electrons\/photon, instead of ergs\/photon.\n\nWe selected nominal wavelengths that minimize the need for color corrections, such that the quoted flux\ndensities in IRAC data products are minimally sensitive to the true shape of the source spectrum. (This\nparagraph can be skipped by most readers; the table is given below.) First, let us expand the source\nspectrum in a Taylor series about the nominal wavelength:\n\n\uf8ee \uf8eb \u03bb \u2212 \u03bb0 \uf8f6 \uf8f9\nF\u03bd = F\u03bd 0 \uf8ef1 + \u03b2 \uf8ec\n\uf8ec \u03bb \uf8f7 \uf8f7 + ...\uf8fa (4.9)\n\uf8f0 \uf8ed 0 \uf8f8 \uf8fb\n\nUsing equation 4.8, the color correction for a source with spectrum F\u03bd is\n\n1 \uf8ee \uf8eb \u03bb \u2212 \u03bb0 \uf8f6\uf8f9\nK=\n\u2206\u03bd \u222b \uf8ef1 + \u03b2 \uf8ec \u03bb0\n\uf8ef\n\uf8ec\n\uf8ed\n\uf8f7\uf8fa (\u03bd \/ \u03bd 0 ) \u22121 Rd\u03bd\n\uf8f7\n\uf8f8\uf8fa\n(4.10)\n\uf8f0 \uf8fb\n\nThe choice of \u03bb 0 that makes K minimally sensitive to \u03b2 is the one for which\n\ndK\n= 0.\nd\u03b2\n\nSolving for \u03bb 0 we get\n\n\u222b \u03bb (\u03bd \/ \u03bd ) Rd\u03bd \u222b\u03bd Rd\u03bd\n\u22121 \u22122\n0\n\u03bb0 =< \u03bb >= =C (4.11)\n\u222b (\u03bd \/ \u03bd ) Rd\u03bd \u222b\u03bd Rd\u03bd\n\u22121 \u22121\n0\n\nSo the optimum choice of \u03bb 0 for insensitivity to spectral slope is the weighted average wavelength.\nUsing the nominal wavelengths from Table 4.2, the color corrections for a wide range of spectral shapes\nare less than 3%. Thus, when comparing IRAC fluxes to a theoretical model, placing the data points on\n\nCalibration 39 Color Correction\nIRAC Instrument Handbook\n\nthe wavelength axis at \u03bb 0 takes care of most of the potential color-dependence. To place the data points\nmore accurately on the flux density axis, take the quoted flux densities derived from the images, and\ndivide by the appropriate color correction factor in the tables below:\n\nF\u03bdquot\nF\u03bd 0 = 0\n(4.12)\nK\n\nOr, calculate the color correction using equation 4.8 together with the spectral response tables, which are\navailable in the IRAC section of the documentation website.\n\nTable 4.2: IRAC nominal wavelengths and bandwi dths.\n\nChannel \u03bb0 (\u00b5m) Rmax Eff. Width \u2206\u03bd (Eq. Width \u2206\u03bd \/Rmax \u00bd-power wavelengths (\u00b5m)\n4.2; 1012 Hz) (1012 Hz) blue red\n\n1 3.550 0.651 7.57 16.23 3.18 3.92\n\n2 4.493 0.736 6.93 12.95 4.00 5.02\n\n3 5.731 0.285 1.93 11.70 5.02 6.43\n\n4 7.872 0.428 3.94 12.23 6.45 9.33\n\nTable 4.3 shows the color corrections for sources with power-law spectra, F\u03bd \u221d \u03bd \u03b1 , and Table 4.4 shows\nthe color corrections for blackbody spectra with a range of temperatures. The nominal spectrum has\n\u03b1 = \u22121 , so the color corrections are unity by definition in that column. These calculations are accurate to\n~ 1%. Note that the color corrections for a \u03bd-1 and a \u03bd0 spectrum are always unity. This is in fact a\ntheorem that is easily proven using equations 4.8 and 4.11.\n\nTable 4.3: Color corrections for power-law s pectra, F\u03bd \u221d \u03bd \u03b1\n\nColor correction for \u03b1 =\nBand -2 -1 0 1 2\n1 1.0037 1 1 1.0037 1.0111\n2 1.0040 1 1 1.0040 1.0121\n3 1.0052 1 1 1.0052 1.0155\n4 1.0111 1 1 1.0113 1.0337\n\nTable 4.4: Color corrections for bl ackbody s pectra.\n\nCalibration 40 Color Correction\nIRAC Instrument Handbook\n\nTemperature (K)\nChannel 5000 2000 1500 1000 800 600 400 200\n1 1.0063 0.9990 0.9959 0.9933 0.9953 1.0068 1.0614 1.5138\n2 1.0080 1.0015 0.9983 0.9938 0.9927 0.9961 1.0240 1.2929\n3 1.0114 1.0048 1.0012 0.9952 0.9921 0.9907 1.0042 1.1717\n4 1.0269 1.0163 1.0112 1.0001 0.9928 0.9839 0.9818 1.1215\n\nTable 4.5 gives the color corrections for the spectrum of the zodiacal light, which is the dominant diffuse\nbackground in the IRAC wavelength range. The first model is the COBE\/DIRBE zodiacal light model as\nimplemented in Spot. The zodiacal light is mostly due to thermal emission from grains at ~ 260 K over\nthe IRAC wavelength range, except in channel 1 where scattering contributes ~ 50% of the brightness.\nThe second zodiacal light spectrum in Table 4.5 is the ISOCAM CVF spectrum (5.6\u221215.9 \u00b5m; Reach et\nal. 2003, [22]) spliced with the COBE\/DIRBE model at shorter wavelengths.\n\nTable 4.5: Color corrections for zodi acal light s pectrum.\n\nCOBE\/DIRBE model ISOCAM+COBE\/DIRBE\nI \u03bd 0 ( MJy \/ sr ) I \u03bd 0 ( MJy \/ sr )\nBand K I \u03bdquot\n0\nK I \u03bdquot\n0\n\n1 0.067 1.0355 0.069 0.40 1.0355 0.42\n2 0.24 1.0835 0.26 1.44 1.0835 1.56\n3 1.11 1.0518 1.16 6.64 1.0588 7.00\n4 5.05 1.0135 5.12 25.9 1.0931 28.4\n\nFor sources with complicated spectral shape the color corrections can be significantly different from\nunity. The corrections are infinite in the case of a spectrum dominated by narrow lines, because there may\nbe no flux precisely at the nominal wavelength, which only demonstrates that such sources should be\ntreated differently from continuum-dominated sources. We calculated one illustrative example which may\nprove useful. The ISO SWS spectrum of NGC 7023 is dominated by PAH emission bands and a faint\ncontinuum over the IRAC wavelength range. Table 4.6 shows the color corrections using equation 4.8 and\nthe ISO spectrum. The large value in channel 1 is due to the presence of the 3.28 \u00b5m PAH band, which\ndominates the in-band flux relative to the weak continuum at the nominal wavelength of 3.550 \u00b5m.\nChannel 2 is mostly continuum. Then channel 3 is dominated by a PAH band at 6.2 \u00b5m. Channel 4 has\nsignificant PAH band emission throughout, with prominent peaks at 7.7 and 8.6 \u00b5m. The values in this\ntable can be used for comparison to IRAC colors of other sources by anti-color-correction, which gives\nthe predicted colors for NGC 7023 in the same units as the SSC calibrated data: F\u03bdquot = F\u03bd 0 \u00d7 K , which\n0\n\nis shown in the last column of Table 4.6. Thus, PAH-dominated sources are expected to have\n\nF\u03bdquot (8\u00b5m) \/ F\u03bdquot (5.8\u00b5m) = 599 \/ 237 = 2.5\n0 0\n(4.13)\n\n.\n\nCalibration 41 Color Correction\nIRAC Instrument Handbook\n\nTable 4.6: Color corrections for NGC 7023 (PAH-dominated) s pectrum.\n\nBand F\u03bd 0 K F\u03bdquot\n0\n\n1 17.3 2.21 38.3\n2 30.3 1.21 36.6\n3 169 1.40 237\n4 1021 0.59 599\n\nFor observations of sources dominated by spectral lines, the quoted flux densities should be converted\ninto fluxes using\n\nF\u03bdquot \u2206\u03bd\u03bb0\nF= 0\n(4.14)\nRl \u03bb\n\nwhere \u03bb is the wavelength of the spectral feature and Rl is the spectral response at that wavelength.\nBoth \u03bb0 and effective width \u2206\u03bd are in Table 4.2. The formalism used for continuum sources is\ninappropriate for spectral-line sources because it is likely that F\u03bd 0 and K = \u221e . It is important that the\nnormalization of R used to determine \u2206\u03bd and Rl is the same. In Table 4.2, the column \u2206\u03bd (effective\nwidth) was calculated with the same normalization of response function as on the documentation website\nso it is the appropriate one to use. The maximum response, Rmax is also given in that table, so the fluxes\nof lines in heart of the waveband can be estimated by simply multiplying the quoted flux densities by\n\u2206\u03bd \/ Rmax , which is listed in the table in the column \u201cWidth.\u201d\n\n4.5 Array Location-Dependent Photometric Corrections for Compact Sources\nwith Stellar Spectral Slopes\n\nPoint source photometry requires an additional correction that arises from the way in which the data are\nflat fielded. Flat-fielding is a way of removing pixel-to-pixel gain variations. The IRAC flatfield is\nderived by imaging the high surface brightness zodiacal background. The way the IRAC flatfield is\nderived has a few consequences on making photometrical measurements using IRAC data.\n\nFirst, the zodiacal background is extended and essentially uniform over the 5.2\u2019x5.2\u2019 IRAC field of view.\nThe vast majority of objects seen by IRAC are not like this. Many are compact, being either stars\nor background galaxies. IRAC has significant scattering as well as distortion. As a result, the extended\n\nCalibration 42 Array Location-Dependent\nPhotometric Corrections for\nCompact Sources with Stellar\nSpectral Slopes\nIRAC Instrument Handbook\n\nsource effective gain is slightly different from the point source effective gain. IRAC point source\nphotometry then requires a correction for the effective gain change between extended and point sources.\n\nSecond, the spectrum of the zodiacal background peaks redward of the IRAC filters. The vast majority of\nobjects seen by IRAC are not like this. Many have spectral energy distributions in the IRAC filters more\nclosely resembling stars. Stars (and many galaxies) have color temperatures that are fairly high, and peak\nblueward of the IRAC filters. Generally speaking, for these objects the IRAC filters are well on the\nRayleigh-Jeans side of the blackbody spectrum. IRAC photometry of warmer sources then requires a\ncorrection for the spectral slope change between the zodiacal light and Rayleigh-Jeans spectra.\n\nLastly, there is a variation in the effective filter bandpass of IRAC as a function of angle of incidence,\nwhich in turn depends on the exact position of an object on the array (Quijada et al. 2004, [21]). As a\nresult of this, all objects in the IRAC field of view need to be corrected based on their location on the\narray.\n\nAll three of these effects can be directly measured and a correction derived. Stars (Rayleigh-Jeans, point\nsources) were sampled at many different locations on the array, and their flux was measured from the\n(C)BCD images (see Chapter 6 for the definition of the various types of data, including BCD and CBCD).\nThe systematic variations in their measured fluxes were used to derive the corrections. The amplitude of\nthis effect is sizeable. It may reach 10% peak-to-peak, depending on the detector array. This is larger than\nany other source of uncertainty in IRAC calibration. For well-dithered data, experiments have shown that\nthis effect tends to average out so that the amplitude of the effect is very small (less than 1%). However,\ndepending on the exact details of mapping and dithering, it is not uncommon to have small areas of data\nwhere the mean correction approaches the full 10%.\n\nCorrection images may be downloaded from IRAC instrument web pages. Users should note the\nfollowing:\n\u2022 The correction images are oriented so that they apply multiplicatively to the (C)BCD images.\nAmong other things, the channel 1 and 2 arrays are flipped around their vertical axis during the\nreduction by the BCD pipeline, hence these images cannot be directly applied to the raw data.\n\u2022 The correction images are for compact, or point-like sources.\n\u2022 The correction images are for a Rayleigh-Jeans (stellar, Vega-like) spectrum. Spectral indices\ndiffering from this will have different corrections. Generally, most IRAC objects have spectral\nslopes that are bracketed by the two extremes of the red zodiacal spectrum and the blue stellar\nspectrum, so the corrections will lie between zero and that in the correction image.\n\u2022 Note that the existing flatfield flattens the zodiacal background. After correction, although the\npoint sources may be correctly measured, the background will no longer be flat.\n\nTo apply the correction from these images to photometry on a single (C)BCD image, a) perform\nphotometry on your (C)BCD image, b) measure the value from the correction images at the central pixel\nof your target for which you are performing photometry, c) multiply your photometrical flux\nmeasurement by the measured correction value for the central pixel of your target to obtain a corrected\n\nCalibration 43 Array Location-Dependent\nPhotometric Corrections for\nCompact Sources with Stellar\nSpectral Slopes\nIRAC Instrument Handbook\n\nflux density value. To apply the correction from these images to photometrical measurements made on a\nmosaic image, you will need to first mosaic the correction images in the same way as the science images.\nMaking the correction mosaic is now possible using the MOPEX tool. You can also use this IDL script.\n\nFigure 4.3. Array location-dependent photometric correction i mages. Ch 1 is in the upper left, ch 2 in the\nupper right, ch 3 in the lower left and channel 4 in the lower right.\n\nA note on correction image filenames: The filenames are in the pattern ch[1-4]_photcorr_rj.fits where\n\"rj\" means \"Rayleigh-Jeans.\u201d\n\nCalibration 44 Array Location-Dependent\nPhotometric Corrections for\nCompact Sources with Stellar\nSpectral Slopes\nIRAC Instrument Handbook\n\n.\n\n4.6 Pixel Phase-Dependent Photometric Correction for Point Sources\n\nThe flux density of a point source measured from an IRAC image depends on the exact location where the\npeak of the Point Spread Function (PSF) falls on a pixel. This effect is due to the variations in the\nquantum efficiency across a pixel, combined with the undersampling of the PSF. It is most severe in\nchannel 1, partly due to the smallest (among IRAC channels) PSF angular size. The correction for this\neffect can be as much as 4% peak to peak. The effect is graphically shown in Figure 4.4 where the\nnormalized measured flux density (y-axis) is plotted against the distance of the source centroid from the\ncenter of a pixel. The correction for channel 1 can be calculated from\n\uf8ee 1 \uf8f9\nCorrection = 1+ 0.0535 \u00d7 \uf8ef \u2212 p\uf8fa (4.15)\n\uf8f0 2\u03c0 \uf8fb\n\nwhere p is the pixel phase ( p = ( x \u2212 x 0 ) 2 + ( y \u2212 y 0 ) 2 ) , (x,y) is the centroid of the point source and x0\nand y0 are the integer pixel numbers containing the source centroid. The correction was derived from\nphotometry of a sample of stars, each star observed at many positions on the array. The \u201cratio\" on the\nvertical axis in Figure 4.4 is the ratio of the measured flux density to the mean value for the star. To\ncorrect the flux of a point source, calculate the correction from equation 4.14 and divide the source flux\nby that correction. Thus, the flux of sources well-centered in a pixel will be reduced by 2.1%.\n\nCalibration 45 Pixel Phase-Dependent\nPhotometric Correction for Point\nSources\nIRAC Instrument Handbook\n\nFigure 4.4: Dependence of point source photometry on the distance of the centroi d of a point source from the\nnearest pi xel center i n channel 1. The ratio on the vertical axis is the measured fl ux density to the mean value\nfor the star, and the quantity on the horizontal axis is the fractional distance of the centroi d from the nearest\npi xel center.\n\n4.7 IRAC Point Spread and Point Response Functions\n\nFigure 4.5 shows the IRAC point response functions (PRF) reconstructed from images of a bright star\nobtained during IOC\/SV. (Here we use the language common in the optics field; the point spread function\n[PSF] is before sampling by the detector array, and the point response function [PRF] is after sampling by\nthe detector array. The PRFs, which are undersampled at the IRAC pixel scale, were generated by\ncombining 108 individual IRAC images in each band. By offsetting each image by a fraction of a pixel\nwidth, fully sampled PRFs can be extracted from the data. The resulting PRFs are the optical point spread\nfunction projected onto the focal plane by the IRAC and telescope optics, convolved with the response\nfunction of a single detector pixel. The images were combined using a drizzle algorithm (Fruchter &\nHook 2002, [11]) to minimize smoothing of the PRF during the reconstruction process. The resulting\npixel scale was \u00bc the width of an IRAC pixel. Images of a bright star at the native IRAC pixel scale are\nalso displayed for comparison.\n\nFITS images of the IRAC PRFs are available in the IRAC section of the SSC website. The\nappropriateness of a given PRF is dependent on the observation sampling and the photometric reduction\npackage used.\n\nCalibration 46 IRAC Point Spread and Point\nResponse Functions\nIRAC Instrument Handbook\n\nFigure 4.5. The in-flight IRAC point response functi ons (PRFs) at 3.6, 4.5, 5.8 and 8 microns. The PRFs were\nreconstructed onto a gri d of 0.3\u201d pixels, \u00bc the size of the IRAC pixel, using the drizzle algorithm. We dis play\nthe PRF with both a s quare root and log arithmic scaling, to emphasize the structure in the core and wings of\nthe PRF, respecti vely. We also show the PRF as it appears at the IRAC pi xel scale of 1.2\u201d. The reconstructe d\nimages clearly show the first and second Airy rings, with the first Airy ring blending with the core in the 3.6\nand 4.5 \u00b5 m data.\n\nPoint source fitting to IRAC data has proven problematic as the PSF is undersampled, and, in channels 1\nand 2, there is a significant variation in sensitivity within pixels (Section 4.6). To deal with these\nproblems, we have developed Point Response Functions (PRFs) for IRAC. A PRF is a table (not an\nimage, though for convenience it is stored as a 2D FITS image file) which combines the information on\nthe PSF, the detector sampling and the intrapixel sensitivity variation. By sampling this table at regular\nintervals corresponding to single detector pixel increments, an estimate of the detector point source\nresponse can be obtained for a source centered at any given subpixel positition.\n\nCalibration 47 IRAC Point Spread and Point\nResponse Functions\nIRAC Instrument Handbook\n\n4.7.1 Core PRFs\n\nThe FITS files of the core PRFs are linked off the IRAC web pages. These core PRFs can be used for\nPRF-fitting photometry and source extraction in (C)BCDs for all but the brightest sources. We still\nrecommend performing aperture photometry instead of PRF fitting in all instances except in crowded\nfields and regions with a strongly varying background, because aperture photometry is much simpler,\nstraightforward and faster to do. In addition, especially in channels 1 and 2, the PSF is undersampled by\nthe native IRAC pixel size, causing futher uncertainty to PRF fitting. PRF fitting does not work in image\nmosaics where the information from the PSFs has been scrambled\u2019\u2019 together. Aperture photometry is\nthe correct way to perform point source flux density measurements in image mosaics.\n\nThe PRFs are provided in two different samplings, 1\/5th and 1\/100th native pixels. The 1\/100th native\npixel sampling PRFs have been created by interpolating the 1\/5th sampled PRFs onto a finer grid. These\nPRFs are designed to work with the photometry extraction software APEX. The 1\/5th pixel sampling\nversions are the originally derived versions and are appropriate for use with custom PRF-fitting software,\nbut not APEX. For both versions of sampling, the PRFs are provided for 25 positions in a 5x5 grid upon\nthe array for each channel. The PRFs are normalized such that the flux is unity within 12 arcsecond (10\npixel) radius around each point source with the zero pixel phase instance (centered on a pixel).\n\nCalibration 48 IRAC Point Spread and Point\nResponse Functions\nIRAC Instrument Handbook\n\nFigure 4.6. The IRAC poi nt res ponse functions (PRFs) at 3.6, 4.5, 5.8 and 8.0 microns. The PRFs were\ngenerated from models refined with in-flight cali bration test data invol vi ng a bright cali brati on star observed\nat several epochs. Central PRFs for each channel are shown above wi th a logarithmic scaling to hel p dis play\nthe entire dynamic range. The PRFs are shown as they appear with 1\/5th the nati ve IRAC pixel sampling of\n1.2 arcseconds to highlight the core structure.\n\n4.7.2 Extended PRFs\n\nThe FITS files of the extended PRFs can be obtained using the links in the IRAC web pages. In order to\ngain high signal-to-noise out to the edge of the arrays, PRFs were generated from a combination of on-\nboard calibration and science observations of stars with different brightness, joined together to produce\nextended high dynamic range (HDR) observational PRFs. These PRFs have two main components: a core\nHDR PRF created by the observations of a reference star, and the extended region from observations of a\nset of bright stars that saturated the IRAC array. They can be used to perform source extraction and PRF-\nfitting photometry of bright, highly saturated stars with extended wings. The core of the extended PRF\nwas generated using the prf_estimate module of MOPEX which has been shown to be inadequate for\nmaking high-quality PRFs for IRAC. As a result, the extended PRF should not be used for PRF-fitting\nphotometry and source extraction of non-saturated point sources. Instead, the core PRF in Section 4.7.1 is\nmore appropriate for PRF-fitting photometry. Also, note that the detailed structure of the center of\nsaturated sources fitted using the extended PRF will not be correct in detail.\n\nThese extended HDR PRFs have a pixel size of 0.2 IRAC pixels, or ~ 0.24 arcsec. The size of each PRF\nimage is 1281x1281 pixels, covering an area of ~ 5.1 arcmin x 5.1 arcmin. The PRFs are centered within\neach image. The PRFs are calibrated in MJy\/sr. The PRFs represent an unsaturated, very high S\/N image\nof Vega, and the flux density contained within a 10 native IRAC pixel aperture radius (50 HDR PRF\npixels), with the sky level estimated in a radial annulus from 12 to 20 native IRAC pixels, is equal to the\nflux density of Vega. The pedestal level of each image is set to zero in the corners of each PRF.\n\nTo produce the core portion of the HDR PRF, 300 HDR observations of a calibration star were obtained\nduring three separate epochs, each observation consisting of short exposures (0.6 sec\/1.2 sec) and long\nexposures (12 sec\/30 sec). The HDR PRFs were generated by first combining short-exposure frames and\nlong-exposure frames separately. The short frames enabled the cores to be constructed without a\nsaturation problem, while the long exposures allowed the construction of a higher signal-to-noise PRF in\nthe wings out to 15 arcseconds. The assembly required the replacement of any saturated areas in the long-\nexposure frames with unsaturated data from the same pixel area of the short-exposure frames. It also\nrequired the replacement of a few pixels in the long-exposure frames by the corresponding pixels in the\nshort-exposure frames to mitigate the non-linear bandwidth effect in channels 3 and 4. The \"stitching\" of\nthe two components of the HDR PRF was completed using a 1\/r masking algorithm requiring a\npercentage of each frame to be added together over a small annulus two IRAC pixels in width just outside\nthe saturated area. Each epoch was treated separately and then all three epochs were aligned and a median\nwas taken to remove background stars.\n\nCalibration 49 IRAC Point Spread and Point\nResponse Functions\nIRAC Instrument Handbook\n\nObservations of the stars Vega, Epsilon Eridani, Fomalhaut, Epsilon Indi and Sirius were used in the\nconstruction of the extended portion of the PRF. Each star was observed with a sequence of 12 sec IRAC\nfull frames, using a 12-point Reuleaux dither pattern with repeats to obtain the required total integration\ntime (the stars were typically observed for 20 \u2212 60 minutes during each epoch). The images were aligned,\nrescaled to the observation of Vega, and then averaged together with a sigma-clipping algorithm to reject\nbackground stars.\n\nThe core HDR PRFs were aligned and rescaled to the extended portion of the PRF by matching their\noverlapping areas. The alignment was done at best to an accuracy of ~ 0.1 arcsec. The rescaling was made\nby forcing the cores to have the same flux density, that of Vega, within a 10 native IRAC pixel radius\naperture. The stitching was made using a mask with a smooth 1\/r transition zone, 2.4 arcseconds wide,\nbetween the core (contributing where the extended portion PRF data were missing due to saturation\ncutoff), and the extended portion of the PRF. The merged PRFs were then cropped to a final 5.1 arcmin x\n5.1 arcmin size, and a pedestal level was removed in order to have a surface brightness as close as\npossible to zero in the corners of the images.\n\n4.7.3 Point Source Fitting Photometry\n\nThe PRF is not an oversampled representation of a point source. Rather it is a map of the appearance of a\npoint source imaged by the detector array at a sampling of pixel phases (positions of the source centroid\nrelative to the pixel center). For that reason, performing aperture photometry directly on the PRF is not\nstrictly correct.\n\nIRAC provides diffraction-limited imaging internally. The image quality is limited primarily by the\nSpitzer telescope. The core PRFs are provided for 25 positions in a 5x5 grid on the array for each channel.\nInterpolating to the nearest position is needed. The extended PRFs have been created at the center of the\narray. Therefore these PRFs degrade as a function of distance from the center. The PRFs will vary with\nposition on the array, including, but not limited to, the relative position of the optical ghosts in channels 1\nand 2, and the diffraction spikes in all channels.\n\nA step-by-step description of IRAC PRF-fitting photometry is given in Appendix C.\n\n4.8 Calculation of IRAC Zmags\n\nSome software packages, such as IRAF's \"phot\" task, require specifying \"zmag\". For IRAC data, you\nneed to know the pixel size of the IRAC image being analyzed in order to convert surface brightness to\nflux density. The zmag can be evaluated from 2.5xlog(F0 \/C), where F0 is the zero magnitude flux density\nin Jy for the relevant channel, tabulated in Table 4.1, and C is the conversion factor from MJy\/sr to\n\u2033\n\u00b5Jy\/pixel, e.g., 8.461595 for 0.6 x 0.6\u2033 pixels (the value of C will be different depending on the pixel\nsize).\n\nCalibration 50 Calculation of IRAC Zmags\nIRAC Instrument Handbook\n\nTo understand where the IRAC zmag comes from, you can start with the fundamental equation between\nmagnitudes and flux densities. In one incarnation, it becomes\n\nm - M0 = -2.5xlog(F\/F0 ) (4.16)\n\nHere m is the magnitude of the source you want to measure, M0 is the zero magnitude (= 0), F is the flux\ndensity in Jy of the source you want to measure and F0 is the flux density of a zero magnitude source. For\nIRAC channel 1 in cryogenic mission, F0 = 280.9 Jy. Expanded out, this becomes thus\n\nm = -2.5xlog(F) + 2.5xlog(F0 ) (4.17)\n\nHere 2.5*log(F0 ) is the same as zmag. Now, since the IRAC images are in units of MJy\/sr, we have to do\nsome manipulation to get the equation to this form. Specifically, the measurable F that we have in IRAC\nimages is the surface brightness, not the flux density. So therefore the equation becomes\n\nm = -2.5xlog(SB*C) + 2.5xlog(F0 ) (4.18)\n\nwhere SB is the measured surface brightness in the image in MJy\/sr and C is a conversion factor from\nMJy\/sr to Jy\/pixel. For IRAC channel 1 mosaics with 0.6 arcsec x 0.6 arcsec pixels it equals C =\n8.461595 x 10-6 Jy\/pixel\/(MJy\/sr). Therefore the equation becomes\n\nm = -2.5xlog(SB) + 2.5xlog(F0 \/C) (4.19)\n\nwhere zmag now corresponds to the latter term, +2.5xlog(F0 \/C). Inserting the values of F0 and C\nmentioned above, we get zmag = 2.5xlog(280.9\/8.461595E-06) = 18.80 mag in channel 1.\n\nPlease remember that this is true only for the 0.6 arcsec x 0.6 arcsec pixel scale mosaics. For other pixel\nscales you will get a different value. Also, please remember the required corrections (e.g., aperture\ncorrection) that are needed for high accuracy photometry.\n\nCalibration 51 Calculation of IRAC Zmags\nIRAC Instrument Handbook\n\n4.9 Astrometry and Pixel Scales\n\n4.9.1 Optical Distortion\n\nOptical distortion is a significant (measurable) effect in IRAC data. The ~ 1% distortion in all channels is\ndue principally to being offset from the optical axis of Spitzer, with additional components from the\ntelescope and camera optics. In addition to varying the effective pixel size, there are also higher-order\nterms such as skew (the two axes are not exactly perpendicular) and a difference in the pixel scales\nbetween the two axes. Failure to account for the distortion will lead not only to errors in photometry\n(described below), but also shifts in astrometric position approaching 1\" near the array corners.\n\nOptical distortion in each of the IRAC FOVs is described in the headers using a standard method\ndescribed by Shupe et al. (2005, [24]). This method places the center of the distortion at the center of\neach detector array, in particular at CRPIX1 and CRPIX2. The linear terms and any skew are represented\nin the CD matrix header keywords (CD1_1, CD1_2, CD2_1, and CD2_2), while the distortion keywords\nprovide the second and higher order terms. Importantly, these distortion corrections apply to the array\ncoordinates, prior to the transformation to sky coordinates. This means that all IRAC data for a given\ndetector share the same distortion keywords. In addition we also provide a separate set of keywords\nrepresenting the \u201creverse\u201d transformation from sky to array coordinates.\n\nThe form of the optical distortion that is encoded in the (C)BCDs is read properly by several \u201cstandard\u201d\ntools available to the general astronomical community: (1) the Spitzer mosaicker (MOPEX), (2)\nWCSTOOLS by Doug Mink (SAO) and (3) DS9 (except for grid overlays).\n\nThe optical distortion is fit independently for each IRAC detector. Originally a second-order fit was used,\nbut an improved fit to third order was derived from the GOODS data by S. Casertano. The (C)BCD\ncoefficients remove the distortion to 0.1\u2033 accuracy.\n\n4.9.2 Pixel Solid Angles\n\nAs a result of the optical distortion described above, the detector pixels do not all subtend the same\nprojected solid angle on the sky. The variation in projected pixel solid angle is roughly 1.5%.\n\nThis size variation is accounted for in the flat-fielding process because the flats are derived from actual\nsky measurements. As a result, after flat-fielding, the (C)BCD images are calibrated in units of true\nsurface brightness (MJy\/sr). This poses a difficulty because virtually all software assumes that the pixels\nare in units of flux per pixel, and simply sum the pixel values. In order to properly measure fluxes from an\nimage in surface brightness units, one must multiply the pixel value by the pixel size. Failure to do so\ncould induce photometric errors at the 1% level, depending on location on the array. Unfortunately, only\nthe newest photometry software can read the new FITS-standard WCS distortion keywords written in the\n(C)BCD headers and properly account for the sizes of the pixels.\n\nCalibration 52 Astrometry and Pixel Scales\nIRAC Instrument Handbook\n\nThe simplest solution to this problem is to reproject the images onto an equal area (or nearly so)\nprojection system (such as TAN-TAN) using suitable software that can understand the distortion\nkeywords in the WCS header (e.g., MOPEX). MOPEX also has the significant advantage that it\nunderstands how to properly handle surface brightness images during coaddition. After processing, the\npixels will all subtend the same solid angle, and hence any standard photometry software can produce the\ncorrect result.\n\nHowever, some observers may prefer alternative approaches, in particular if they wish to measure\nphotometry directly from the (C)BCD images. Therefore, we supply maps of the pixel size in the \u201cIRAC\ncalibration and analysis files\u201d section of the IRAC documentation website that can be used to correct the\npixel solid angles in BCD images if any measurements are being directly made on them. Note that this\ncorrection is built into the \"location-dependent photometric correction\" image, also available on the\nwebsite, so multiplying by this correction map (intended to provide correct photometry for point sources\nwith stellar-like SEDs) will also produce the correct result.\n\n4.10 Point Source Photometry\n\nPlease refer to Appendix B for a detailed description of how to achieve the highest possible accuracy\nwhen performing point source photometry. Appendix C summarizes the proper use of PRF fitting to\nobtain high accuracy point source photometry in a crowded field or in a field with highly varying\nbackground.\n\nPhotometry using IRAC data is no different from that with any other high-quality astronomical data. Both\naperture photometry and PRF-fitting work successfully. Aperture photometry is most commonly used, so\nwe will discuss it briefly. The radius of the on-source aperture should be chosen in such a way that it\nincludes as much of the flux from the star (thus, greater than 2 arcseconds) as possible, but it should be\nsmall enough that a nearby background annulus can be used to accurately subtract unrelated diffuse\nemission, and that other point sources are not contributing to the aperture. For calibration stars, an\nannulus of 12 arcseconds is used; such a wide aperture will often not be possible for crowded fields. The\ndominant background in regions of low interstellar medium (ISSA 100 \u00b5m brightness less than 10\nMJy\/sr) is zodiacal light, which is very smooth. In regions of significant interstellar emission, it is\nimportant to use a small aperture, especially in IRAC channels 3 and 4, where the interstellar PAH bands\nhave highly-structured emission. For example, an aperture in a star-forming region might have a radius of\n3 native pixels with a background annulus from 3 to 7 native pixels. The flux of a source can then be\ncalculated in the standard way, taking the average over the background annulus, subtracting from the\npixels in the on-source region, and then summing over the on-source region.\n\nIt is important to apply an aperture correction to flux densities measured through aperture photometry or\nPRF fitting, unless the exact same aperture and background radii and annuli were used as for the\ncalibration stars. The IRAC data are calibrated using aperture photometry on a set of flux calibration stars.\nThe calibration aperture has a 10 native pixel radius (12 arcsec) in all 4 channels. For flux density\nmeasurements in crowded fields, a much smaller on-source aperture should be used (or use PRF-fitting\n\nCalibration 53 Point Source Photometry\nIRAC Instrument Handbook\n\nphotometry). And in the presence of extended emission, a small off-source annulus is normally used. The\ncalibration aperture does not capture all of the light from the calibration sources, so the extended emission\nappears too bright in the data products we delivered. See the more detailed discussion under 4.11.\nSimilarly, observers will often use smaller apertures and will want to correct their photometry to match\nthe absolute calibration.\n\nUsers should note that the spatial extent of the PSF in channels 3 and 4 is much larger than the subarray\narea. In other words, a large amount of the total power in the PSF is scattered onto arcminute size scales.\nAs a result, special care needs to be taken when measuring fluxes in these channels, since accurate\nmeasurement of the \u201cbackground\" is difficult. Proper application of aperture corrections is very\nimportant.\n\nFor photometry using different aperture sizes, the aperture correction can be estimated with Table 4.7. All\ndistances in this table are in native pixels (~ 1.2\u201d). Note that the post-BCD mosaics currently available\nfrom the Spitzer data archives use pixels that correspond to exactly 0.6\u201d x 0.6\u201d. The aperture corrections\nas written will INCREASE the flux measured by the listed method, i.e., your measured brightness should\nbe MULTIPLIED by the aperture corrections in the table. The third decimal place in these numbers is\nincluded only to illustrate the trends; the accuracy of these corrections is presently ~ 1% \u2013 2%. The\naperture corrections in Table 4.7 are averages of the values derived from PSFs measured using stars at 23\ndifferent positions on the array. Standard deviations (including measurement errors and true variations\nacross the array) are less than 0.5% for all entries except the smallest aperture, in which they are still less\nthan 1%. The extended source (infinite) corrections in Table 4.7 come from Reach et al. (2005, [23]). The\nmeasured flux densities can then be converted to magnitudes, if desired, using the zero-points in Table\n4.1.\n\nTable 4.7: IRAC aperture corrections.\n\nRadius on source Background annulus Aperture correction\n(native ~1.2\u201d pixels) (native ~1.2\u201d pixels)\n3.6 \u00b5m 4.5 \u00b5m 5.8 \u00b5m 8.0 \u00b5m\n\ninfinite N\/A 0.994 0.937 0.772 0.737\n\n10 12-20 1.000 1.000 1.000 1.000\n\n5 12-20 1.049 1.050 1.058 1.068\n\n5 5-10 1.061 1.064 1.067 1.089\n\n3 12-20 1.112 1.113 1.125 1.218\n\n3 3-7 1.124 1.127 1.143 1.234\n\n2 12-20 1.205 1.221 1.363 1.571\n\n2 2-6 1.213 1.234 1.379 1.584\n\n.\n\nCalibration 54 Point Source Photometry\nIRAC Instrument Handbook\n\n4.11 Extended Source Photometry\n\nThe photometric calibration of IRAC is tied to point sources (calibration stars) measured within a\nstandard aperture with a radius of 12 arcseconds. This point-source calibration is applied to all IRAC data\nproducts during pipeline processing to put them into units of MJy\/sr (1 MJy\/sr = 1017 erg s-1 cm-2 Hz-1 sr-\n1\n). This method results in a highly accurate calibration for point sources. However, transferring this\ncalibration to extended sources requires extra thought. The discrepancy between the (standard) point\nsource calibration and the extended source calibration arises from the complex scattering of incident light\nin the array focal planes. Our best understanding is that there is a truly diffuse scattering that distributes a\nportion of the incident flux on a pixel throughout the entire array.\n\nThe surface brightness of extended emission in IRAC images will tend to appear BRIGHTER than it\nactually is. The reason for this is two-fold. First, photons that would normally scatter out of the PSF\naperture used to measure a point source are instead captured by an extended source. The scattering\ndepends on the convolution between the IRAC PSF and how the light is distributed across the focal plane,\nwhich is usually quite complex for extended sources (galaxies, ISM and nebulae). Second, photons are\nscattered into the aperture from the emission regions outside the aperture. As a thought experiment, one\ncan imagine a single point source inside an aperture, which is easy to measure. But if four point sources\nare placed around it just outside the measurement aperture, each of them scatters light into the aperture,\nwhich leads to an overestimate of the real flux. For the extended source case, we can imagine the same\nexperiment taken to the limit where all the regions have emitters in them.\n\nFor photometry of extended sources, the calculated flux inside an aperture must be scaled by the ratio of\nthe extended and point source throughputs. The scaling factors (fp \/fex) to be used are given in Table 4.7\n(the infinite aperture case). Note that these are not really \u201cthroughputs,\" in the sense that they have\nanything to do with the number of photons reaching the detector. It is more accurate to think of them as a\nspecial type of an aperture correction. The values in Table 4.7 are for a very extended, red source like the\nZodiacal light.\n\nThe most challenging case of extended source photometry is of objects with sizes on arcminute scales,\nwithin apertures of similar size or smaller. Examples might be typical observations of nearby galaxies. In\nthis case the aperture correction is related both to the aperture size and the underlying surface brightness\ndistribution of the target. To derive a set of aperture corrections more appropriate to this case, a detailed\nanalysis of early-type spheroidal galaxies (due to the relative ease of modeling the light profiles of these\nstellar-dominated sources), ranging in size from 20 arcseconds to several arcminutes, was carried out. A\nsummary of the results is given below, including aperture correction curves that may be applied to\nphotometry of all types of well-resolved galaxies. These extended source aperture corrections are\nsomewhat larger than the infinite aperture corrections given in Table 4.7.\n\nA commonly encountered problem is that of measuring the total flux of extended objects that are still\nsmaller than the standard aperture size used for the photometric calibration. For example, the background\ngalaxies seen in all IRAC images are often slightly extended on size-scales of a few arcseconds. PRF-\nfitting photometry of such objects will obviously underestimate their fluxes. One methodology for\nhandling such sources was developed by the SWIRE project; readers are referred to the data release\n\nCalibration 55 Extended Source Photometry\nIRAC Instrument Handbook\n\ndocument for SWIRE, available from the Spitzer documentation website under Legacy projects. Detailed\nanalysis by SWIRE has indicated that Kron fluxes, with no aperture corrections applied, provide\nmeasurements of small extended sources that agree closely with hand-measured fluxes. Kron fluxes are\nprovided as one of several flux measures in the popular \u201cSExtractor\" software. Note that it is important to\ndetermine that an object actually is extended before using the Kron flux, as it is ill-defined otherwise.\nThis may be determined by using the stellarity and isophotal area as defined by the SExtractor software.\nSelecting limits on these parameters based on their breakdown as a function of signal-to-noise ratio\ngenerally will mimic SExtractor's own \u201cauto\" function.\n\nTo measure absolute flux on large scales (sizes of order the field of view), consider all the sources of flux\nthat go into each pixel. The IRAC images are in surface-brightness units. The flux of an extended object\nis the integral of the surface brightness over the solid angle of the object. The value of a pixel in an IRAC\nBCD is the real sky value plus a contribution from the zodiacal light minus the dark current value at that\npixel. The dark current value is made from observations of a low background region at the north ecliptic\npole and so it contains some small amount of flux of astrophysical origin. The darks have also had an\nestimate of zodiacal light subtracted from them before use. The (theoretically) estimated zodiacal light\nbrightness during an observation is in the BCD header keyword ZODY_EST, and that for the sky dark\nobservation is listed as SKYDRKZB. While it is possible using the above keywords to recover something\nsimilar to the absolute sky surface brightness, this brightness estimate is still limited by the accuracy of\nthe underlying model of the zodiacal emission.\n\nIn practice, most extended source photometry will usually be performed with respect to a background\nregion within the image (for example, large aperture photometry of galaxies, nebulae, etc.) and one does\nnot attempt to measure the absolute sky brightness on large scales (like the zodiacal cloud). The median\nvalue of the pixels located in user-selected background regions is generally a reasonable estimator of the\nbackground.\n\n4.11.1 Best Practices for Extended Sources\n\nResolved galaxies with apertures centered on the nucleus:\nFor sources < 8\u20139 arcsec in size, treat as point source (small aperture photometry, with local annular\nbackground subtraction)\nFor sources > 8\u20139 arcsec in size, apply extended source aperture corrections (see below).\nEmission knots, embedded resolved sources\nIf the source is small (compact), treat as point source (small aperture photometry, with local annular\nbackground subtraction)\nIf the source is large and fuzzy, use the extended source aperture corrections (see below). Beware that\nbackground structure will introduce large uncertainties (~10%)\nSurface Brightness (pixel-to-pixel measurements)\n\nCalibration 56 Extended Source Photometry\nIRAC Instrument Handbook\n\nFor very extended sources (> 300 arcseconds) or flat, low surface brightness sources (e.g., Magellanic-\ntype galaxies), use the maximum scaling factors given below.\n\nCross-comparing IRAC images (e.g., channel 1 versus channel 4), we recommend that you first cross-\nconvolve the images. For the example above, convolve the channel 4 image with the channel 1 PSF, and\nconvolve the channel 1 image with the channel 4 PSF. This operation will reduce the deleterious effects\nof the light scattering, but will not completely eliminate them. Be very conservative in interpreting colors\nas surface brightness measurements can be off by 5%\u201310% in the short-wavelength channels and 30% in\nthe long-wavelength channels.\n\n4.11.2 Extended Source Aperture Correction\n\nThe following aperture corrections are intended to correct the photometry of extended sources (e.g.,\ngalaxies) whose absolute calibration is tied to point sources. These corrections not only account for the\n\"extended\" emission from the IRAC PSF itself, but also from the diffuse scattering of the emission across\nthe IRAC focal plane. The curves were derived from detailed analysis of elliptical galaxies (see related\nnotes in Section 4.11.4). The curves may be applied to all types of galaxies, but beware that significant\ndepartures can be expected for sources that are morphologically different from elliptical galaxies (e.g.,\nlate-type LSB galaxies; see surface brightness recommendations above).\n\nCalibration 57 Extended Source Photometry\nIRAC Instrument Handbook\n\nFigure 4.7. Extended source flux correction factors; solid lines represent exponenti al functi on fits to the data.\nAlso indicated are correction factors deri ved from zodi acal light tests, and Galactic HII region tests (e.g.\nMartin Cohen's GLIMPS E vs. MSX, pri vate communication).\n\nFigure 4.8. Extended source flux correction factors for galaxies (solid lines) versus the PS F aperture\ncorrection factors (dotted lines). The main difference between the two is the truly diffuse scattering internal\nto the array.\nAperture photometry should also include background subtraction; we recommend that you use an annulus\nthat is located just outside the boundary of your galaxy. Circular or elliptical apertures may be used.\n\nThe procedure for correcting extended source photometry is to apply the correction factor to the\nintegrated flux measured from the IRAC image (subject to the standard or point source calibration). The\ncorrection factor is a function of the circular aperture radius or the effective circular aperture radius (if\nusing ellipses). These corrections should be good to 10%. For convenience, we have converted the\nempirical curves into a functional form:\n\nCalibration 58 Extended Source Photometry\nIRAC Instrument Handbook\n\nB (4.21)\ncorrection_factor (radius) = true_flux \/ flux = [A x exp (-radius )] + C\n\nwhere radius is in arcsec, and A, B and C are the best fit coefficients tabulated below:\n\nTable 4.8: IRAC extended source photometrical correction coefficients.\n\nIRAC A B C\n\n3.5 \u00b5m 0.82 0.370 0.910\n\n4.5 \u00b5m 1.16 0.433 0.94\n\n5.8 \u00b5m 1.49 0.207 0.66\n\n8.0 \u00b5m 1.37 0.330 0.740\n\nThe coefficient \"C\" represents the infinite, asymptotic value.\n\n4.11.3 Low Surface Brightness Measurements and the Maximum Scaling Factors\n\nPhotometry of diffuse emission or low surface brightness objects is also subject to a large calibration\ncorrection in the IRAC 5.8 and 8.0 \u00b5m channels. The way to think about \u201cflat\u201d extended objects is that\nany aperture you use to measure the integrated flux (or surface brightness) is equivalent to an infinitely\nlarge aperture applied to a point source (or galaxy). Hence, the appropriate aperture correction (or\nequivalently, surface brightness factor) is the large radius case of the above aperture corrections:\n\nTable 4.9: IRAC surface brightness correction factors.\n\nIRAC Surface\nBrightness\nCorrection\nFactor\n\n3.5 \u00b5m 0.91\n\n4.5 \u00b5m 0.94\n\nCalibration 59 Extended Source Photometry\nIRAC Instrument Handbook\n\n5.8 \u00b5m 0.66\u22120.73\n\n8.0 \u00b5m 0.74\n\nSurface Brightness = measured surface brightness x correction_factor, where the correction factors\nrepresent the infinite aperture value. Note that for IRAC channel 3 the recommended correction is\nsomewhere between 0.66 and 0.73, depending on the downward curvature of the aperture corrections\n(which is highly uncertain). These aperture corrections should be good to 10%.\n\nExamples of LSB objects: large, late-type galaxies (e.g., NGC 300); Magellanic-type galaxies (e.g., NGC\n6822); diffuse dwarf galaxies (e.g., M81 DwA); HII regions that are larger than ~100 arcseconds and not\nvery centrally condensed.\n\n4.11.4 Caveats & Cautionary Notes\n\nAt small radii, r < 7\u20138\", the extended source aperture corrections should not be used. Instead, we\nrecommend using the point source aperture corrections for small radii.\n\nIt remains uncertain how much the spectral shape of the extended object determines the flux corrections;\nthe aperture corrections presented here were derived using relatively \"old\" spheroidal galaxies. To first\norder, the extended source aperture corrections apply to most types of galaxies.\n\nLikewise with the spectral color caveat, it remains uncertain how much the spatial distribution of the light\ndetermines the flux corrections; these corrections were derived using relatively high surface brightness\nspheroidal galaxies; it is unknown whether these corrections apply to lower surface brightness galaxies\n(e.g., late-type spirals; irregulars; Magellanic-types).\n\n4.11.5 Faint Surface Brightness Behavior\n\nNote that the discussion in this section applies only to warm IRAC data. For more detailed information,\nplease see Krick et al. (2011, [16]).\n\n4.11.5.1 Binning\n\nBinning data by essentially making larger \u201cpixels\u201d should reduce the noise in the image linearly with\nbinning length. Figure 4.9 and Figure 4.10 show a plot of noise versus binning length for a set of deep\nmapping data in the Virgo cluster (PID 60173). These data have been carefully corrected for the first\nframe effect using the data themselves. The measured noise does not achieve the expected linear relation\nwith binning length.\n\nCalibration 60 Extended Source Photometry\nIRAC Instrument Handbook\n\nFigure 4.9. Noise versus binning length in channel 1. To make this plot the surface brightness was measured\nin nine regions across an object-masked mosaic. These regions are not near the bright g alaxies, stars, or\ndi ffuse plumes. The noise is defined as the standard devi ation of those nine regions. The box size is\nincrementall y increased until the box length is many hundreds of pi xels. For reference the soli d line shows the\nexpected linear relati on.\n\nFigure 4.10. Noise versus binning length in channel 2. To make this plot the surface brightness was measured\nin six regions across an object-masked mosaic. These regions are not near the bright g alaxies, stars, or di ffuse\npl umes. The noise is defined as the standard deviation of those six regions. The box size is incrementally\n\nCalibration 61 Extended Source Photometry\nIRAC Instrument Handbook\n\nincreased until the box length is many hundreds of pi xels. For reference the soli d line shows the expected\nlinear relati on.\n\n4.11.5.2 Small Scales\n\nThere is a discrepancy between the expected linear behavior and the data at short binning length scales of\njust a few pixels (mosaics only). This discrepancy occurs because we have correlated noise on a\nmosaicked image on small pixel scales (a few pixels), so the noise does not bin down appropriately.\n\n4.11.5.3 Medium Scales\n\nOn 5\u201d\u221230\u201d scales much of the extra noise is due to sources in the image. The first level of masking used\nthe SExtractor segmentation map as a mask. The resulting noise properties are shown with asterisks.\nIncreasing the size of the masks to 1.5 (2.0) times the SExtractor-determined object radii produced noise\nproperties shown with a square (triangle) symbol. Further increases in mask size are inconsequential. The\ndiscrepancy between the observed and expected behaviors in this binning length regime is dominated by\nnoise from the wings of galaxies that are improperly masked. Even after increasing the mask sizes, extra\nsources of noise remain which prevent detection of ultra-low surface brightness. There appears to be a\nfloor to the noise at roughly 0.0005 MJy\/sr at 3.6 \u03bcm and 0.0008 MJy\/sr at 4.5 \u03bcm.\n\n4.11.5.4 Large Scales\n\nSome of the large-scale noise is caused by the mapping pattern used for the observations. On scales of\nroughly half a field of view there are differences in the total exposure time and hence the total number of\nelectrons detected (not a dominant source of noise). There are remaining sources of noise on the large\nscales, both instrumental and astronomical, which are very hard to disentangle. Uncertainties in the flat-\nfielding and removal of the first frame effect are two instrumental effects that are contributing to the noise\non large scales. The first frame effect has a column-wise dependence that requires special calibration data\nto measure. Astrophysically, there is real structure in the zodiacal light and Galactic cirrus. There is also\ndocumented diffuse intracluster light in the Virgo cluster itself, and a small signal from the extragalactic\nbackground light that are both adding to the noise at low levels. There is potentially also noise due to the\nblue infrared color of intracluster light, while the zodiacal light from which the flats are made is red in\nnear-IR (see Section 4.2). Differentiating between all of these sources of noise is difficult.\n\n4.11.5.5 Increasing exposure time\n\nThe IRAC dark field was used to study whether the noise decreases with the square root of exposure time,\nas expected. The dark field has extremely low zodiacal light and low Galactic diffuse emission. Using all\nthe warm mission dark calibration data through 2010, a mosaic was made from 300 dark frames (each\nwith 100 second frame time). Object masking was made with the SExtractor segmentation image. The\n\nCalibration 62 Extended Source Photometry\nIRAC Instrument Handbook\n\nnoise on the distribution of pixel values is the standard deviation of the Gaussian fit to that distribution.\nEach distribution has > 750 pixels in it. For comparison the same analysis was performed on the dark\nfield mosaics from the first year of the cryogenic mission. The results from both are in Figure 4.11 and\nFigure 4.12.\n\nFigure 4.11. Noise as a functi on of exposure ti me (number of frames) i n channel 1. The results from the warm\nmission data are shown with x\u2019s and the expected behavi or with the soli d line. The results from the cryogenic\nmission are shown wi th open s quares and the expected behavi or wi th the dashed line.\n\nCalibration 63 Extended Source Photometry\nIRAC Instrument Handbook\n\nFigure 4.12. Noise as a functi on of exposure ti me (number of frames) i n channel 2. The results from the warm\nmission data are shown with x\u2019s and the expected behavi or with the soli d line. The results from the cryogenic\nmission are shown wi th open s quares and the expected be havi or wi th the dashed line.\n\nThese plots show that background noise in IRAC channels 1 and 2 does decrease roughly as expected\nwith exposure time. The slight deviation at larger exposure times is likely caused by the first frame effect\nand by residual source wings.\n\n4.12 Pointing Performance\n\nPointing is controlled by Spitzer's Pointing Control System (PCS). This uses a combination of a star\ntracker and gyros to locate and control the attitude of the spacecraft. Absolute pointing is controlled by\nthe star tracker, through a filter (known as the \"observer\") which smooths the raw star tracker output.\nSlews under control of the observer take ~ 10 seconds to settle, so only the initial slew and cluster slews\nin celestial coordinates are carried out using the observer. Once the observatory has taken the initial frame\nat the starting position, attitude control is handed over to the gyros. Mapping and dithering slews are\nmade under gyro control with a shorter (~ 5 sec) settle time. The price to be paid for the shorter settle time\nis that the spacecraft attitude will slowly drift with respect to the observer attitude, at a rate ~ 1 mas\/sec.\n\nCalibration 64 Pointing Performance\nIRAC Instrument Handbook\n\nFor long integrations (100 sec frame time), attitude control is returned to the observer after 80 seconds to\nhalt the drift.\n\nIn addition, attitude resets were performed regularly (about every 30 minutes) to return the spacecraft\nattitude to the observer attitude. The system is designed to ensure that any motion to return the spacecraft\nattitude to that of the observer does not take place during an IRAC integration, to avoid smearing the\nimages. Throughout the first 18 months of the mission the PCS system and the corresponding parts of the\nIRAC AOT were being adjusted for optimal performance. Below is a guide to the astrometric accuracy\nand image quality that can be expected from a typical observation.\n\n4.12.1 Pointing Accuracy\n\nSlews under observer control settle to the accuracy to which the star tracker to IRAC pointing offset is\nknown, about 0.5\u201d. Offsets between dither\/mapping moves are accurate to 0.1\u201d relative to the commanded\nmove for small moves (~ 10\u201d), and for large moves (~ 0.5 deg) the accuracy is ~ 0.5\u201d (though this was\nimproved as of Spring 2005, and should be only ~ 0.2\u201d thereafter). An additional pointing error comes\nfrom the gyro drift which can accumulate over the 30 minute period between attitude resets. This error is\ntypically ~ 2\u201d for a \u201cworst case\" frame just before a reset.\n\nThe pointing of each frame as reported in the header keywords CRVAL1 and CRVAL2 is an average of\nthe observer attitude during the frame, and is typically accurate to ~ 0.5\u201d (though it may be slightly worse\nfor short frames where the observer has not fully settled). Other header keywords related to pointing\ninclude RA_RQST and DEC_RQST, the requested R.A. and Dec. of the frame, and PTGDIFF, the\ndifference between the requested and actual pointing. USEDBPHF should be T for all frames, if not, then\npointing transfer has failed for the frame.\n\nThe Level 2 (Post-BCD) pointing refinement module is run by default in the post-BCD pipeline to refine\nthe pointing to 2MASS accuracy (~ 0.15\u201d), and will be successful if there is a sufficient number of\n2MASS stars in the data. The module operates by matching common stars between frames (\u201crelative\nrefinement\") and a fiducial set of stars from 2MASS (\u201cabsolute refinement\"). The (R.A., Dec.) position\nand twist of each frame is then adjusted until a global minimum in the residuals is found. Application of\nthis to the Extragalactic IRAC First Look Survey (FLS) data results in a mean position error for high\nsignal-to-noise stars with respect to 2MASS positions of 0.25\u201d.\n\nThe pointing refinement module writes several new keywords to the header. RFNDFLAG is true if\npointing refinement was run and produced a refined solution. The refined position is given by keywords\nRARFND and DECRFND, and rotation by CT2RFND. A new version of the CD matrix, given by\nkeywords CD11RFND, etc., is also written to reflect the new rotation angle (note that the pixel scale and\ndistortion are not changed by pointing refinement). If pointing refinement fails, then the header keyword\nRFNDFLAG will be false and RARFND, DECRFND and CT2RFND will be set to CRVAL1, CRVAL2\nand CROTA2, respectively. Note that the refined solution may be poor if the number of astrometry stars\nin the frame, NASTROM, is low (i.e., 0, or only a few stars). The refined pointing keywords are used by\nthe post-BCD software if USE_REFINED_POINTING = 1 in the namelists. To use the refined pointing\n\nCalibration 65 Pointing Performance\nIRAC Instrument Handbook\n\nwith other software, copy the non-standard keywords to their FITS standard equivalents, e.g., RARFND\nto CRVAL1, CD11RFND to CD1_1 etc. Pointing refinement works well on most channel 1 and 2 data,\nthough short frames in fields near the Galactic poles in channels 3 and 4 will frequently have too few stars\nfor a good solution.\n\nAll data have a \"superboresight\" correction applied. Users wishing to make use of the superboresight\nsolutions need to set USE_REFINED_POINTING = 0 in MOPEX, as the superboresight pointing is\ncontained in the standard CRVAL1, CRVAL2 and CD matrix keywords (this is the recommended\npointing to be used when making mosaics etc.). Data which have had this correction applied will also\nhave the ORIG_RA and ORIG_DEC keywords present which contain the initial (uncorrected) pointing\nestimate.\n\n4.12.2 Jitter and Drift\n\nJitter is typically 0.1\u201d. It has been measured on timescales ~ 0.04 seconds to 5 minutes. In addition to high\nfrequency jitter, there are modulations ~ 0.1\u201d on timescales of 200\u2013400 seconds. These are not expected\nto noticeably affect the IRAC PSF. Gyro drift occurs for the first 80 seconds of IRAC integrations, but\nagain this should result in only ~ 0.1\u201d of motion. Some amount (< 0.4\u201d) of image smearing is expected in\nshort frames due to settling motions. Other instances of pointing glitches occur when one of the four\nreaction wheels goes through zero velocity. To reduce stiction when the speed actually hits zero, the\nwheels are given a small \u201cbump\" in torque at this point, which has been seen to result in a small (~ 0.05\u201d),\n\nshort duration (~ 10 seconds) pointing glitch. On average, only about one crossing per hour occurs, and\nthey are thought to mostly happen during slews, so they are not expected to affect many IRAC images.\nOne manifestation of settling, jitter and drift during integrations is that the pointing of HDR short,\nmedium (for 100 sec HDR) and long frames are slightly different (the same is also true of repeats taken in\nthe same position). These differences are usually ~ 0.1 arcseconds, so they should not be a problem for\nmost observers, but they are large enough to show up as residuals in difference images.\n\nCalibration 66 Pointing Performance\nIRAC Instrument Handbook\n\nFigure 4.13: Position of a star in the x (left) and y (right) axes of IRAC during a long (8 hr) observation. The ~\n3000 sec oscillation is superposed on a slow drift of the Star Tracker to telescope alignment.\n\nA slowly varying pointing oscillation is seen in long staring observations. This oscillation has an\namplitude of 0.1\" and a period ~ 3000 sec (Figure 4.13). It is believed to be related to battery heating and\ncooling cycles influencing the mechanical link between the Star Tracker and the telescope. There is also a\nsteady drift of the pointing, ~ 0.01\u201d\/hr due to other changes in the Star Tracker to telescope alignment.\nThe accumulated drift was removed using regular Star Tracker-to-telescope boresight calibrations every ~\n12 hours.\n\nCalibration 67 Pointing Performance\nIRAC Instrument Handbook\n\n5 Pipeline Processing\n\n5.1 Level 1 (BCD) Pipeline\n\nThe IRAC Level 1 (BCD; Basic Calibrated Data) pipeline is designed to take a single Level 0 (\u201craw\u201d)\nimage from a single IRAC detector and produce a flux-calibrated image which has had all well-\nunderstood instrumental signatures removed. The following describes the data reduction pipeline for\nscience data. Similar pipelines are used for reducing calibration data.\n\nThe IRAC pipeline consists of two principal parts: the data reduction software modules and the\ncalibration server. The individual modules each correct a single instrumental signature. They are written\nas standalone code executable from the UNIX command line. Each uses FITS files and text configuration\nfiles as input and produces one or more FITS files and log files as output. These modules are strung\ntogether with a single PERL script. The actual calibration data needed to reduce a given DCE is produced\nvia \u201ccalibration pipelines.\u201d A raw IRAC DCE is thus \"passed\" between successive modules, and at each\nstep becomes closer and closer to a finished, fully reduced image. The following sections describe the\nreduction steps used to produce the BCD data.\n\n5.1.1 SANITY DATATYPE (parameter checking)\n\nBefore data proceeds through the pipeline, it is checked to ensure that it is of the type of data expected. In\nparticular, ancillary keywords are checked against their expected values to ensure that they are in range\nand of the expected logical state. These include the shutter state (open\/closed), transmission and flood\ncalibrator lamp status (on\/off), and read mode (full\/subarray).\n\n5.1.2 SANITY CHECK (image contents checking)\n\nBefore pipeline processing continues, the actual image contents are checked to ensure that they contain\nvalues expected for actual image data. These tests include checking to insure that the image is not all\nzeros, that the pixels are not all identical, or that areas of the image do not have an abnormal data range.\n\nThe FITS headers delivered by JPL\/FOS are translated into a more readable format. For example,\n\nA0612D00= 14478455 \/ AINTBEG\nA0612E00= 1.4478455E5 \/ [Sec]\nA0614D00= 8 \/ AFOWLNUM\nA0614E00= \/ [NONE]\nA0615D00= 44 \/ AWAITPER\n\nPipeline Processing 68 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nA0615E00= \/ [NONE]\nA0657D00= 14479495 \/ ATIMEEND\nA0657E00= 1.4479495E5 \/ [Sec]\n\nis translated to:\n\nAINTBEG = 144784.55 \/ [Secs since IRAC turn-on] Time of integ. start\nATIMEEND= 144794.95 \/ [Secs since IRAC turn-on] Time of integ. end\nAFOWLNUM= 8 \/ Fowler number\nAWAITPER= 44 \/ [0.2 sec] Wait period\n\nPipeline Processing 69 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nFigure 5.1: Data flow for processing a raw IRAC science DCE i nto a B CD that is described i n this Chapter.\n\nPipeline Processing 70 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nIt is also at this stage that \u201cderived\" parameters, most notably the integration time, are added to the\nheaders. The integration time is related to the Fowler number (AFOWLNUM) and the number of wait\nperiods (AWAITPER) via\n\nEXPTIME = mode \u2217 (AWAITPER + AFOWLNUM) (5.1)\n\nThe integration time is stored in the header in the keyword EXPTIME. Another timescale of importance\nis the frame time. This is the actual length of time that the observation was integrating on the sky, and is\nequal to\n\nFRAMTIME = mode \u2217 (AWAITPER + 2 \u2217 AFOWLNUM) (5.2)\n\nThe factor \u201cmode\u201d is equal to 0.2 seconds for full-array mode, and 0.01 seconds for subarray mode. The\nread-mode is determined by the least significant bit of the ancillary keyword AREADMOD. If\nAREADMOD is 0 (or even) then the mode is full-array. If it is 1 (or odd) then the image is sub-array.\n\nNote that because of TRANHEAD processing, the headers of the raw data and the final BCD data\nproducts are not identical. In general, users should only need to read the BCD headers. However, if it\nbecomes somehow necessary to examine any of the camera telemetry (voltages, currents, etc.), then they\n\n5.1.4 INSBPOSDOM (InSb array sign flipping)\n\nThe IRAC InSb arrays (channels 1 and 2) were operated in such a way that flux appears \"negative\" in the\nraw data (Figure 5.2). That is, data numbers start at 65,535 (16-bit max) for zero light levels and become\nincreasingly close to 0 as light levels increase. The INSBPOSDOM module rectifies this so that\nincreasing DN yields increasing flux (0 to 65,535), as is more common. This is done by\n\nAout = (65,535 \u2212 Ain) (5.3)\n\nwhere A is the pixel intensity in DN for the two InSb arrays (ACHANID = 1 or 2). ACHANID is turned\ninto CHNLNUM in the BCD header by the last step in the pipeline.\n\nPipeline Processing 71 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nFigure 5.2: INSBPOSDOM works only on the two InS b arrays (Channels 1 & 2) and reverses the sense of\nintensities.\n\n5.1.5 CVTI2R4 (byte type changing)\n\nData are converted from the native unsigned 16-bit format used by IRAC to the 32-bit floating point\nformat used in astronomical calculations. At this point, the following DN is added to all pixels in order to\naccount for the bias introduced by the spacecraft on-board bit truncation.\n\n\u22120.5 x (1 \u2212 2\u2212ABARREL ) for channels 1 and 2\n+0.5 x (1 \u2212 2\u2212ABARREL ) for channels 3 and 4\n\nHere, \u201cABARREL\" is the barrel-shift number keyword where the bit truncation occurs (see Section\n5.1.7). Also, if the header indicates that any rows or columns are blank (usually due to data loss during\ntransmission from Spitzer to the ground), then those pixels are set equal to NaN\u2019s.\n\nPipeline Processing 72 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nFigure 5.3: Di agram of the wrapping of negati ve val ues due to truncation of the sign bi t.\n\n5.1.6 Wraparound Correction: IRACWRAPDET AND IRACWRAPCORR\n\nIRAC suffers from two kinds of \u201cwraparound\" errors, wherein DN values are actually multi-valued. That\nis, a given DN actually corresponds to more than one possible flux level.\n\nIRACWRAPDET (sign truncation wraparound)\nAs a means of data compression, IRAC discards the sign bit of its data before transmission to the ground.\nThis creates an ambiguity in that negative numbers appear in the raw data as very large positive numbers.\nHowever, by design the detector reaches physical saturation before \u201celectronic\u201d (A\/D) saturation (Figure\n5.3). That is, the maximum physical values the detectors ever have are around 45,000 DN for the InSb\narrays and 60,000 DN for the Si:As arrays, which are less than the maximum 16-bit value of 65,535.\nIRAC uses the 2s-complement storage system for negative numbers. In this system negative numbers are\ndenoted by setting the sign bit and then complementing (i.e., flipping) all the remaining bits. For example,\nin 2s-complement storage, \u22121 is represented by 65534 in unsigned integer form. Therefore, values higher\nthan the maximum saturation levels must be \u201cwrapped\" negative numbers. For each array a set of\nmaximum values has been chosen such that no pixel will be erroneously identified as wrapped. The\nmodule then flags any pixels lying in the \u201cwraparound\" DN region. Observers are strongly cautioned to\ncheck for possible saturation problems by examining the structure in their data. If a user finds that any\n\nPipeline Processing 73 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\npart of their image is near the saturation value (typically either 45,000 DN in channels 1 and 2 or 60,000\nDN in channels 3 and 4) then they should suspect surrounding pixels of being near saturation.\n\nIRACWRAPCORR (wraparound correction)\nThis module uses the flag bits set by the previous module and attempts to correct the wraparound problem\n(Figure 5.4). Note that both the sign truncation and the non-linearity wraparound, i.e., doughnuts, are\ncorrected in the pipeline. The sign truncation correction is made by\n\nAcorrected = Auncorrec ted \u2212 65535. (5.4)\n\n5.1.7 IRACNORM (Fowler sampling renormalization)\n\nIRAC data are taken with Fowler (multi) sampling in order to reduce read noise. This is done by non-\ndestructively reading the array multiple times (set by the Fowler number), and accumulating the sum into\nan internal register. Since these reads are summed, the result must be divided by the number of reads in\norder to get the actual number of DN. Additionally, when data are transmitted to the ground, a variable\nnumber of least significant bits are discarded as a means of data compression (Figure 5.5). In order to\ncorrect for the effects of bit-truncation and Fowler sampling the data are transformed by\n\nAin \u00d7 2 ABARREL\nAout = (5.5)\nAFOWLNUM\n\nwhere ABARREL is the barrel-shift keyword and AFOWLNUM is the Fowler number keyword. Note\nthat in normal usage the Fowler number and barrel shift actually used and commanded by the science\ncenter are such that they cancel, i.e.,\n\n2 ABARREL\n=1 (5.6)\nAFOWLNUM\n\nand hence observers should not be surprised if this module normally appears to do nothing.\n\nPipeline Processing 74 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nFigure 5.4: Application of IRACWRAPCORR to Channel 1 data. The many apparently \u201chot\u201d\npixels are actually wrapped negative values, which are detected on the basis of their vastly\nexceeding the physical saturation value for the detectors, and corrected by subtracting the\nappropriate value. Real hot pixels do not exceed the physical saturation value, and hence are not\nchanged.\n\nFigure 5.5: Illustration of bit truncation used by IRAC for ground transmission, necessitating\nIRACNORM. The internally stored 24-bit word in truncated to 16 bits, with a sliding window set\nby the barrel shift value. Illustrated is the case for ABARREL=4.\n\n5.1.8 SNESTIMATOR (initial estimate of uncertainty)\n\nThe module SNESTIMATOR calculates the uncertainty of each pixel based on the input image (here, the\ninput image is the output of IRACNORM). The uncertainty for each pixel is estimated as the Poisson\nnoise in electrons and the readout noise added in quadrature. The formula for the calculation is as follows:\n\n\u03c3 2 = \u03c3 readnoise + \u03c3 poisson\n2 2\n(5.7)\n\nPipeline Processing 75 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nTo obtain an expression in DN, \u03c3 is divided by Gain. This uncertainty image will be carried through the\nappropriate.\n\n5.1.9 IRACEBWC (limited cable bandwidth correction)\n\nThe cables that connect the IRAC Cold Assembly (the detectors) to the Warm Electronics Assembly (the\nreadout electronics) have a characteristic time constant similar to the rate at which individual pixels are\nread. As a result, all pixels have an \u201cecho\" or ghost in the following readout pixel (Figure 5.6). Since the\npixels are contained in four readout channels, the \u201cnext\" pixel is actually four pixels to the right. The first\npixel read out in an IRAC image is the first data byte in the image, and is situated in the lower left corner\nin most astronomical display software. This effect is corrected for by using the known readout order of\nthe pixels. Starting at the first pixel, we correct the following pixel, and so on. An additional wrinkle is\nthat the time required to go from the end of one row to the beginning of the next is slightly longer (by\n75%) than the time to go from one column to the next in the same row. As a result, a slightly different\ncoefficient must be applied. The task is simplified by two things. First, the effect is so small that it is only\nnecessary to correct the following pixel, as the next echo is below 1e-5 th of the original in intensity.\nSecond, the time of the effect is much faster than the decay time. Thus, the problem need only be solved\nin one direction. The current bandwidth coefficients are given in Table 5.1. They are applied using\n\nAn + 4 = An + 4 \u2212 \u03baAn (5.8)\n\nwhere A is the pixel intensity in DN and \u03ba is the correction coefficient for a given readout channel (of 4).\nA different value of \u03ba is used for correcting the first 4 pixels in a row, based on the pixel values of the last\nfour pixels of the previous row.\n\nPipeline Processing 76 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nTable 5.1: B andwi dth correcti on coefficients.\n\nChannel 1\n\n\u03ba 1.58e-3 2.17e-3 2.33e-3 1.64e-3\n\n\u03ba(end of row) 1.2e-5 2.1e-5 2.4e-5 1.3e-5\n\nChannel 2\n\n\u03ba 1.71e-3 3.63e-3 1.06e-3 1.11e-3\n\n\u03ba (end of row) 1.4e-5 5.2e-5 6.0e-5 6.6e-5\n\nChannel 3\n\n\u03ba 3.19e-3 1.04e-2 3.3e-3 2.09e-3\n\n\u03ba (end of row) 4.2e-5 3.3e-4 4.4e-5 2.0e-5\n\nChannel 4\n\n\u03ba 3.74e-3 3.74e-3 3.74e-3 3.74e-3\n\n\u03ba (end of row) 5.5e-5 5.5e-5 5.5e-5 5.5e-5\n\n5.1.10 Dark Subtraction I: FFCORR (first frame effect correction) or LABDARKSUB (lab dark\nsubtraction)\n\nThe true dark current in the IRAC detectors is actually very low \u2212 the most notable dark current features\nare the electronic glows seen in the Si:As arrays (channels 3 & 4). However, the IRAC arrays experience\nconsiderable pedestal offsets which are commonly of the order of tens of DN. These offsets are dependent\non the Fowler sampling, exposure time, and operation history of the arrays, and are believed to be due to\nvery small thermal changes in the internal IRAC cold electronics. The most significant of these offsets is\nthe \u201cfirst-frame\" effect: the laboratory measurements show that the dark patterns and DC levels change as\na function of the time elapsed between the end of the previous frame and the start of the current frame\n(called \u201cdelay time\"). The first frame of a series of exposures is most affected, and therefore this effect is\n\nPipeline Processing 77 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\ncalled \u201cthe first-frame effect\". Figure 5.7 shows how the DC levels of darks change as a function of\ndelay-time.\n\nFigure 5.6: Correction of cable-induced bandwidth error by IRACEBWC. The illustrated data\nshow a cosmic ray hit.\n\nDue to the decision not to use the photon-shutter on the IRAC for dark and flat measurements, we have a\nsomewhat sophisticated dark subtraction procedure. There will be two steps for the dark subtraction, one\nusing a dark from the ground-based laboratory measurements (called, \u201clab darks\"), and another using a\ndelta dark which is the difference between the lab dark and the sky dark measured at the low zodiacal\nlight region.\n\nIn the first step of dark subtraction, we subtract a calibrated lab dark from the data at this point in the\nprocessing. This lab dark subtraction occurs before the linearization of the array, so that we can linearize\nthe data as well as possible. The labdark subtraction will be handled by a combination of modules\nincluding LABDARKSUB and FFCORR depending on which kind of labdark data is needed. In some\nobserving modes (subarray mode, shortest frames within the HDR mode and the first frame of an\nobservation or AOR), not enough data are available to construct delay-time dependent darks. In such\ncases, a single mean dark has been computed using 30 sec as a delay time, and it is used as a labdark. The\nLABDARKSUB module subtracts this mean labdark. The correction of the first-frame effect for all other\nframes is handled by the FFCORR module, which interpolates the library of labdarks taken at different\nexposure times with different delay times, and creates a labdark corresponding to the particular delay time\nof the frame being calibrated. These delay-time dependent darks are then subtracted from the\nIRACEBWC-processed frame. Therefore, FFCORR requires a number of different labdarks taken with\ndifferent delay times to calibrate properly. These were taken pre-launch and have been loaded into the\ncalibration database. The IRAC pipeline determines the delay time (header keyword INTRFRDLY), and\nthe lab dark file (header keyword LBDRKFLE) that was subtracted is placed within the header keywords\nof the BCD.\n\nPipeline Processing 78 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nFigure 5.7: First-frame effect. Dark counts as a function of interval between frames. This figure is\nfor a 30 second exposure frame.\n\nThe second step of the dark subtraction uses a delta-dark found in the SKYDARKSUB module described\nbelow. This \u201cskydark\u201d, described in Section 4.1, is subtracted from the IRAC image after the linearization\n\nPipeline Processing 79 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nand should take away any additional dark features which are not present in the labdarks, but exist in the\nflight data. Note that the delta-dark includes the sky background around the low-zody region.\n\n5.1.11 MUXBLEEDCORR (electronic ghosting correction)\n\nThe InSb arrays suffer from an effect known as \u201cmuxbleed\". This is believed to be a result of operating\nthe arrays at unusually cold temperatures. When a bright source is read out, the cold electronics do not\nreturn to their quiescent state for a considerable length of time. The result is a ghosting along the pixel\nreadout channels, sometimes referred to as \u201cant trails\" (Figure 5.8). The effect is easily noticeable against\na low background (such as a dark current measurement), and can extend the full length of the array. The\nmuxbleed flux is not real \u2212 it is not \u201cborrowed\" from the actual source and as such needs to be accounted\nfor, or removed, unlike CTE smearing in CCDs.\n\nFigure 5.8: Correction of pseudo-muxbleed for channel 1. Shown is a bright source within a\ncalibration AOR and a background of sources under the muxbleed limit.\n\nThis effect is complicated. It appears that a pixel bleeds only as a result of the light falling onto it, and not\nas the sum of the value of the pixel plus the bleeding from previous pixels. Since we know the readout\norder of the pixels, we can start by correcting all pixels downstream from the first pixel, and then move\non to the next pixel. The exact shape of the mubleed pattern, obtained after examining hundreds of\nmuxbleed incidences is summarized as a modified polynomial:\n\nLog(MuxbleedIntensity) = 3.1880 \u2013 2.4973x +1.2010x2 \u2212 0.2444x 3 (5.9)\n\nPipeline Processing 80 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nwhere\n\nx = log10(pixel number +1) (5.10)\n\nPixel numbers are counted along the detector readout direction, starting from the muxbleed causing pixel\n(pixel number zero).The muxbleed pattern appears to be independent of readout channel, Fowler\nsampling, etc. Furthermore, the pattern seems to be applicable to both the channel 1 and channel 2\nmuxbleed, with a slightly different scaling factor.The severity of muxbleed depends on the brightness of\nthe bleeding pixel. Muxbleed scaling laws as a function of the bleeding pixel were obtained for channels 1\nand 2. They are\n\n\uf8eb 1 \uf8eb x \u2212 B \uf8f62 \uf8f6\nScalingFactor = A \u22c5 exp\uf8ec \u2212 \uf8ec \uf8f7\n\uf8ec 2\uf8ed C \uf8f7 \uf8f7\n(5.11)\n\uf8ed \uf8f8 \uf8f8\n\nwhere\n\nx = log10(bleeding pixel intensity in DN). (5.12)\n\nFor channel 1, A = 0.6342, B = 5.1440 and C = 0.5164. For channel 2, A = 0.3070, B = 4.9320 and C =\n0.4621. Both the scale factors and the muxbleed pattern are fixed for all pixels in a given array. Muxbleed\nfrom triggering pixels with brightnesses below 10,000 DN is not corrected, because the corrections in\nthese cases would be just a few times the read noise. Muxbleed is also not corrected in the subarray\nobservations.\n\nObservers should note that calibration darks are not muxbleed corrected. Muxbleed occurs in these\nimages due to the presence of hot pixels. However, this occurs equally both in the darks and in the science\nframes and has been found to subtract noiselessly from the science data. Thus, any dark frame muxbleed\nis simply considered a feature of the darks.\n\nNote that the muxbleed correction decribed here does not correct 100% of the muxbleed effect.\n\n5.1.12 DARKDRIFT (readout channel\u2019\u2019 bias offset correction)\n\nEach IRAC array is read out through four separate channels. The pixels read out by these channels are\narranged vertically, and repeat every four columns. Small drifts in the bias levels of these readouts,\nparticularly relative to the calibration dark data, can produce a vertical striping called the \u201cjailbar\" effect.\n\nPipeline Processing 81 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nThis is mostly noticeable in very low background conditions. This is corrected by adding to the individual\nreadout channels a common mean offset. For any image, the flux in a pixel is assumed to be\n\nAi , j = S i , j + B + DCi , j + DOi , j (5.13)\n\nwhere A is the detected intensity in DN, S is the incident \u201cscience flux\" (celestial background + objects),\nB is a constant offset in the frame, DC is the standard calibration dark, and DO is the dark offset. The first\ndark varies on a pixel by pixel basis, whereas the offsets are assumed to vary on a readout channel basis.\nIt can be assumed that the mean Si,j is the same for all readout channels i, and therefore there is a mean\nestimator function M for each readout channel\n\nMi = MeanEstimator(Si,0\u2026Si,n) (5.14)\n\nThe corrected image (post dark-subtraction) is then\n\n1 4\nAi', j = Ai , j \u2212 ( M i \u2212 \u2211Mi)\n4 i =1\n(5.15)\n\n5.1.13 FOWLINEARIZE (detector linearization)\n\nLike most detectors, the IRAC arrays are non-linear near full-well capacity. The number of read-out DN\nis not proportional to the total number of incident photons, rather it becomes increasingly small as the\nnumber of photons increases. In IRAC, if fluxes are at levels above half full-well (typically 20,000\u2013\n30,000 DN in the raw data), they can be non-linear by several percent. During processing the raw data are\nlinearized on a pixel-by-pixel basis using a model derived from ground-based test data and re-verified in\nflight. The software module that does this is called FOWLINEARIZE. FOWLINEARIZE works by\napplying a correction to each pixel based on the number of DN, the frame time, and the linearity solution.\nFor channels 1, 2 and 4, we use a quadratic solution, i.e., we model the detector response as\n\nDNobs = kmt \u2013 Ak2t2 (5.16)\n\nPipeline Processing 82 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nThe linearization solution to the above quadratic model is:\n\n\u2212 1 + 1 \u2212 4 L\u03b1DN obs\nDN = (5.17)\n2 L\u03b1\n\nwhere\n\n\u03b1 \uf8ee\uf8eb 2 n + w 2 n 2 \uf8f6 t \uf8f9\n2 \uf8ef \u2211\nL= \uf8ec i \u2212 \u2211 i \uf8f7 \u2212 2(1 \u2212 d )n(n + w)\uf8fa , (5.18)\nn( w + n) \uf8f0\uf8ed n + w+1 1 \uf8f8 tc \uf8fb\n\nA\nand \u03b1 = , n is the Fowler number, and w is the wait period. The above expression for L is the\nm2\ncorrection required to account for multi-sampling. This is required because the multi-sampling results in\nthe apparent time spent integrating not actually being equal to the real time spent collecting photons. Note\nthat td is the time between the reset and the 1st readout of the pixel. For channel 3, we use a cubic\nlinearization model:\n\nDNobs = Ckt3 + Akt2 + kmt (5.19)\n\nFor the cubic model, the solution is derived via a numerical inversion.\n\n5.1.14 BGMODEL (zodiacal background estimation)\n\nFor this module, a spacecraft-centric model of the celestial background was developed. For each image,\nthe zodiacal background will be estimated (a constant for the entire frame) based on the pointing and time\nthat the data were taken. This value is written to the header keyword ZODY_EST in units of MJy\/sr. The\nzodiacal background is also estimated for the subtracted skydark (see next module) and placed in the\n\n5.1.15 Dark Subtraction II: SKYDARKSUB (sky \u201cdelta-dark\u201d subtraction)\n\nThis module, the second part of dark subtraction, strongly resembles traditional ground-based data\nreduction techniques for infrared data. Since IRAC did not use the photon-shutter for its dark\nmeasurement, a pre-selected region of low zodiacal background in the north ecliptic cap is observed in\norder to create a \u201cskydark\". At least twice during each campaign a library of skydarks of all Fowler\nnumbers and frame times were observed, reduced, and created by the calibration pipeline. The skydarks\nhave had the appropriate labdark subtracted in their DARKCAL pipeline and are therefore a \u201cdelta-dark.\u201d\n\nPipeline Processing 83 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nThese skydarks are then subtracted from the data in the pipeline within this module, based on the\nexposure time, channel and the time of the observation.\n\n5.1.16 FLATAP (flatfielding)\n\nLike all imaging detectors, each of the IRAC pixels has an individual response function (i.e., DN\/incident\nphoton conversion). To account for this pixel-to-pixel responsivity variation, each IRAC image is divided\nby a map of these variations, called a \u201cflatfield.\u201d\n\nObservations are taken of pre-selected regions of high-zodiacal background with relatively low stellar\ncontent located in the ecliptic plane. They are dithered frames of 100 seconds in each channel. These\nobservations are processed in the same manner as science data and then averaged with outlier rejection.\nThis outlier rejection includes a sophisticated spatial filtering stage to reject the ever-present stars and\ngalaxies that fill all IRAC frames of this depth. The result is a smoothed image of the already very\nuniform zodiacal background. This \u201cskyflat\" is similar to flatfields taken during ground-based\nobservations. The flatfields are then normalized to one. The flatfield calibration pipeline produces a\nlibrary of the flat fields throughout each campaign since a flatfield is taken at the beginning and end of an\nobserving campaign. We have have found that there is no difference in flatfields from campaign to\ncampaign, so a \u201csuper skyflat,\u201d composed of five full years worth of data and therefore of very high S\/N,\nis used for processing science data in the BCD pipeline. It should be noted that this flatfield is generated\nfrom a very red target, i.e., the zodiacal background. There is considerable evidence for a spatially-\ndependent color term in the IRAC calibration (which is roughly a quadratic polynomial function across\nthe array). Objects that have color temperatures radically different from the zodiacal background require\nan additional multiplicative correction of order 5%\u201310%. This is not treated by the flat-fielding stage.\n\nSoftware module FLATAP applies the flats generated by the calibration server. This operation is\nequivalent to division by the flat.\n\n5.1.17 IMFLIPROT\n\nIRAC utilizes beamsplitters to redirect the incoming light for a given FOV through each of the two filters.\nAs a result, although for a given FOV two filters such as for channel 1 and 3 view the same piece of sky,\nthe detectors see (and hence read out) mirror images of each other. An image transposition is applied to\nensure that the two filters for each FOV are in the same orientation (Figure 5.9). Note that the images are\nnot de-rotated, that is, each is now correct relative to the other filter for a given FOV, but all of the\nimages still have the effects of spacecraft rotation.\n\nAxflipped = Ax , 255\u2212 y\n,y\noriginal\n(5.20)\n\nPipeline Processing 84 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nFigure 5.9: Transposition of an IRAC channel 1 dark image by the IMFLIPROT module.\n\nCurrently, channels 1 & 2 are flipped about their vertical axis, which is illustrated by the equation above.\nThe image flip of these two channels provides an array orientation with the E axis located to the left of N\nfor all channels. Since this image transposition is applied after skydark subtraction and flatfielding, those\ncalibration files do not have such an orientation.\n\nWithin this module, individual frames are analyzed for probable radiation hits (cosmic rays), and the\nresults appear as a flag in the imask file. This is computed by a median filtering technique. Input images\nare read in, and a median filter is applied. The difference between the input image and the median-filtered\nimage is then computed. Pixels above a specified threshold (i.e., are \u201cpointier\" than is possible for a true\npoint source) are then flagged in a mask image (bit 9 of imask is set when a pixel is suspected to be hit by\na cosmic ray).\n\n5.1.19 DNTOFLUX (flux calibration)\n\nIRAC flux calibration is tied to a system of celestial standards measured at regular intervals during each\ncampaign. The IRAC IST provides the calibration server with calibration files based upon these\nmeasurements. Because the flux calibration is determined from stellar point sources, the calibration for\nextended sources is somewhat different. For details of the photometric calibration and correction factors,\nsee Chapter 4. The IRAC data are calibrated in units of MJy\/sr in this module. This is accomplished by\nmultiplying the data image by a conversion factor provided by the calibration server. This conversion\nfactor is written to the data header as:\n\n\/ PHOTOMETRY\n\nPipeline Processing 85 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nCOMMENT 1 blank line\nBUNIT = 'MJy\/sr ' \/ Units of image data\nFLUXCONV= 0.2008 \/ Flux Conv. factor (MJy\/sr per DN\/sec)\nGAIN = 3.8 \/ e\/DN conversion.\n\n5.1.20 Pointing Transfer (calculation of pointing information)\n\nThe Pointing Transfer pipeline is a separate script from the actual data reduction pipeline script, designed\nto insert raw-pointing and distortion information into the FITS headers of BCDs. Like the reduction\npipeline, it executes on a per-BCD basis. Pointing data was acquired by the spacecraft star-tracker at a\nrate of 2 Hz, transferred to the boresight onboard, and down-linked every 12 hours as a Boresight\nPointing History File (BPHF). The BPHF is received via a separate telemetry stream from the science\ndata. The first step in pointing transfer is to aquire the portion of the BPHF which spans the integration\ntime of the BCD (getPH module). The 2 Hz sampled data are then transferred to the specific channel-\ndependent science FOV using a set of Euler transformations handled by the \"BORESIGHTTRAN\nmodule\". The Euler angles relating the boresight and FOV positions have been determined in-flight and\nare stored in a configuration file.\n\nThe pointing samples are then averaged and combined by the \"ANGLEAVG\" module to compute the\nraw-pointing for the BCD: CRVAL1 (RA), CRVAL2 (Dec) and PA (position angle). These, along with\nuncertainties and reference pixel coordinates (CRPIX1, CRPIX2), are inserted as keywords into the FITS\nheader of the BCD. The module also computes a CD matrix and transfers distortion coefficients\n(represented in the pixel coordinate frame) from a calibration file to the FITS header. The default\nprojection type for the celestial reference system (CTYPE keyword) is \"TAN-SIP\". This is a tangent\n(TAN) projection modified to make use of the Spitzer Imaging (distortion) Polynomials (SIP) in\ncoordinate mappings.\n\nThe Final Product Generator (FPG) is executed at the end. This reformats the FITS header and adds\nadditional keywords, which are most useful to the user, from the database. An example of a BCD header\ncontaining pointing and distortion information is given below.\n\nSIMPLE = T \/ Fits standard\nBITPIX = -32 \/ FOUR-BYTE SINGLE PRECISION FLOATING POINT\nNAXIS = 2 \/ STANDARD FITS FORMAT\nNAXIS1 = 256 \/ STANDARD FITS FORMAT\nNAXIS2 = 256 \/ STANDARD FITS FORMAT\nORIGIN = 'Spitzer Science Center' \/ Organization generating this FITS file\nCREATOR = 'S18.7.0 ' \/ SW version used to create this FITS file\nTELESCOP= 'Spitzer ' \/ SPITZER Space Telescope\nINSTRUME= 'IRAC ' \/ SPITZER Space Telescope instrument ID\n\n\/ TARGET AND POINTING INFORMATION\n\nOBJECT = 'NGC7479 ' \/ Target Name\nOBJTYPE = 'TargetFixedSingle' \/ Object Type\nCRPIX1 = 128. \/ Reference pixel along axis 1\n\nPipeline Processing 86 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nCRPIX2 = 128. \/ Reference pixel along axis 2\nCRVAL1 = 346.229408109806 \/ [deg] RA at CRPIX1,CRPIX2 (using Pointing Recon\nCRVAL2 = 12.3406747724052 \/ [deg] DEC at CRPIX1,CRPIX2 (using Pointing Reco\nCRDER1 = 3.47278123674643E-05 \/ [deg] Uncertainty in CRVAL1\nCRDER2 = 3.44327670377712E-05 \/ [deg] Uncertainty in CRVAL2\nRA_HMS = '23h04m55.1s' \/ [hh:mm:ss.s] CRVAL1 as sexagesimal\nDEC_DMS = '+12d20m26s' \/ [dd:mm:ss] CRVAL2 as sexagesimal\nRADESYS = 'ICRS ' \/ International Celestial Reference System\nEQUINOX = 2000. \/ Equinox for ICRS celestial coord. system\nCD1_1 = 0.000165673336023108 \/ Corrected CD matrix element with Pointing Recon\nCD1_2 = -0.000296839887227466 \/ Corrected CD matrix element with Pointing Reco\nCD2_1 = -0.000296695378233646 \/ Corrected CD matrix element with Pointing Reco\nCD2_2 = -0.00016538074830257 \/ Corrected CD matrix element with Pointing Recon\nCTYPE1 = 'RA---TAN-SIP' \/ RA---TAN with distortion in pixel space\nCTYPE2 = 'DEC--TAN-SIP' \/ DEC--TAN with distortion in pixel space\nPXSCAL1 = -1.22334117768332 \/ [arcsec\/pix] Scale for axis 1 at CRPIX1,CRPIX2\nPXSCAL2 = 1.22328355209902 \/ [arcsec\/pix] Scale for axis 2 at CRPIX1,CRPIX2\nPA = -119.12383984174 \/ [deg] Position angle of axis 2 (E of N) (was OR\nUNCRTPA = 0.000467418894131902 \/ [deg] Uncertainty in position angle\nCSDRADEC= 1.31286126610331E-06 \/ [deg] Costandard deviation in RA and Dec\nSIGRA = 0.0965180263226379 \/ [arcsec] RMS dispersion of RA over DCE\nSIGDEC = 0.0477081433171542 \/ [arcsec] RMS dispersion of DEC over DCE\nSIGPA = 0.627707783301654 \/ [arcsec] RMS dispersion of PA over DCE\nPA = -119.12383984174 \/ [deg] Position angle of axis 2 (E of N) (was OR\nRA_RQST = 346.229439557555 \/ [deg] Requested RA at CRPIX1, CRPIX2\nDEC_RQST= 12.3408384725542 \/ [deg] Requested Dec at CRPIX1, CRPIX2\nPM_RA = 0. \/ [arcsec\/yr] Proper Motion in RA (J2000)\nPM_DEC = 0. \/ [arcsec\/yr] Proper Motion in Dec (J200)\nRMS_JIT = 0.00561943353954093 \/ [arcsec] RMS jitter during DCE\nRMS_JITY= 0.00415225189801845 \/ [arcsec] RMS jitter during DCE along Y\nRMS_JITZ= 0.00378640165338011 \/ [arcsec] RMS jitter during DCE along Z\nSIG_JTYZ= -0.000574938554882557 \/ [arcsec] Costadard deviation of jitter in YZ\nPTGDIFF = 0.599601238096299 \/ [arcsec] Offset btwn actual and rqsted pntng\nPTGDIFFX= 0.460985501941048 \/ [pixels] rqsted - actual pntng along axis 1\nPTGDIFFY= -0.383877068036649 \/ [pixels] rqsted - actual pntng along axis 2\nRA_REF = 346.235833333333 \/ [deg] Commanded RA (J2000) of ref. position\nDEC_REF = 12.3227777777778 \/ [deg] Commanded Dec (J2000) of ref. position\nUSEDBPHF= T \/ T if Boresight Pointing History File was used\nBPHFNAME= 'SBPHF.0773452800.031.pntg' \/ Boresight Pointing History Filename\nFOVVERSN= 'BodyFrames_FTU_14a.xls' \/ FOV\/BodyFrames file version used\nRECONFOV= 'IRAC_Center_of_3.6umArray' \/ Reconstructed Field of View\nORIG_RA = 346.229614257812 \/ [deg] Original RA from raw BPHF (without pointi\nORIG_DEC= 12.3407106399536 \/ [deg] Original Dec from raw BPHF (without point\nORIGCD11= 0.0001656730165 \/ [deg\/pix] Original CD1_1 element (without point\nORIGCD12= -0.0002968400659 \/ [deg\/pix] Original CD1_2 element (without point\nORIGCD21= -0.0002966955653 \/ [deg\/pix] Original CD2_1 element (without point\nORIGCD22= -0.0001653804356 \/ [deg\/pix] Original CD2_2 element (without point\n\n\/ DISTORTION KEYWORDS\n\nA_ORDER = 3 \/ polynomial order, axis 1, detector to sky\nA_0_2 = 2.9656E-06 \/ distortion coefficient\nA_0_3 = 3.7746E-09 \/ distortion coefficient\n\nPipeline Processing 87 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nA_1_1 = 2.1886E-05 \/ distortion coefficient\nA_1_2 = -1.6847E-07 \/ distortion coefficient\nA_2_0 = -2.3863E-05 \/ distortion coefficient\nA_2_1 = -8.561E-09 \/ distortion coefficient\nA_3_0 = -1.4172E-07 \/ distortion coefficient\nA_DMAX = 1.394 \/ [pixel] maximum correction\nB_ORDER = 3 \/ polynomial order, axis 2, detector to sky\nB_0_2 = 2.31E-05 \/ distortion coefficient\nB_0_3 = -1.6168E-07 \/ distortion coefficient\nB_1_1 = -2.4386E-05 \/ distortion coefficient\nB_1_2 = -5.7813E-09 \/ distortion coefficient\nB_2_0 = 2.1197E-06 \/ distortion coefficient\nB_2_1 = -1.6583E-07 \/ distortion coefficient\nB_3_0 = -2.0249E-08 \/ distortion coefficient\nB_DMAX = 1.501 \/ [pixel] maximum correction\nAP_ORDER= 3 \/ polynomial order, axis 1, sky to detector\nAP_0_1 = -6.4275E-07 \/ distortion coefficient\nAP_0_2 = -2.9425E-06 \/ distortion coefficient\nAP_0_3 = -3.582E-09 \/ distortion coefficient\nAP_1_0 = -1.4897E-05 \/ distortion coefficient\nAP_1_1 = -2.225E-05 \/ distortion coefficient\nAP_1_2 = 1.7195E-07 \/ distortion coefficient\nAP_2_0 = 2.4146E-05 \/ distortion coefficient\nAP_2_1 = 6.709E-09 \/ distortion coefficient\nAP_3_0 = 1.4492E-07 \/ distortion coefficient\nBP_ORDER= 3 \/ polynomial order, axis 2, sky to detector\nBP_0_1 = -1.6588E-05 \/ distortion coefficient\nBP_0_2 = -2.3424E-05 \/ distortion coefficient\nBP_0_3 = 1.651E-07 \/ distortion coefficient\nBP_1_0 = -2.6783E-06 \/ distortion coefficient\nBP_1_1 = 2.4753E-05 \/ distortion coefficient\nBP_1_2 = 3.8917E-09 \/ distortion coefficient\nBP_2_0 = -2.151E-06 \/ distortion coefficient\nBP_2_1 = 1.7E-07 \/ distortion coefficient\nBP_3_0 = 2.0482E-08 \/ distortion coefficient\n\n5.1.21 PREDICTSAT (HDR saturation processing)\n\nPREDICTSAT is used to process data taken in the High Dynamic Range (HDR) mode by identifying\nsaturated pixels using the information obtained from the shorter exposure time frames. Specifically, if the\nshorter frame is frame 1 and the longer frame is frame 2, and they have Fowler numbers F, wait periods\nW, pixel values DN, and saturation values of DNsat , then if\n\nDN 1 (2 F2 + W2 )\n> DN sat (5.21)\nF1 + W1\n\nPipeline Processing 88 Level 1 (BCD) Pipeline\nIRAC Instrument Handbook\n\nfor any pixel, then that pixel is masked as saturated in the longer frame 2. Optionally, surrounding pixels\nmay also be masked. This information is used in the post-BCD pipeline by the mosaicker when coadding\nframes to a priori reject saturated pixels before applying any other outlier rejection. High dynamic range\ndata are received as separate DCEs by IRAC. No co-addition is done of these frames at the BCD level,\nbut they are received by the user as separate BCDs.\n\n5.1.22 LATIMFLAG (residual image flagging)\n\nThe IRAC pipeline detects and flags residual images left from imaging bright objects. A model of the\ncharge decay is used to build a time history of the DCEs and sets a mask bit to indicate that a given pixel\nin the DCE is contaminated by a residual. Currently, the algorithm works the following way: Starting\nwith the first image in each observation or AOR, LATIMFLAG computes what is called a \u201clatent-trap\"\nimage at the end of its exposure. This specifies the amount of trapped charge in every pixel. The charge-\ntrap decays and appears as a residual in subsequent images. The latent-trap image is effectively an image\nof the number of filled traps, which we label NF(t)i . The subscript i refers to the \u201ctrap-species\" or type of\nlatent trap distinguished by a characteristic decay-time and trap filling efficiency. The latent-trap image is\npropagated forward in time, and updated with each consecutive image in the sequence. The images with\npixels sustaining and exceeding a threshold above the background noise from image to image within the\ndecay time are flagged in the imask (bit 5).\n\n5.2 The Artifact-Corrected BCD Pipeline\n\nThere are several artifacts commonly seen in IRAC images. For a complete description, see Chapter 7 in\nthis Instrument Handbook. To mitigate the commonly found artifacts of stray light, saturation, muxstripe,\ncolumn pulldown, and banding, an artifact correction pipeline was created. It performs the artifact\ncorrection on the BCD files. The pipeline then creates a product called a Corrected BCD, or CBCD. The\nCBCDs are used to create the pipeline mosaic. The user receives the BCD and CBCD files in case the\nartifact correction was not completely successful or it needs to be run again more conservatively.\n\nAt each step, an attempt is made to identify the artifacts in the BCDs, adjust the imask pixel values\naccording to the identified artifact, and correct the artifact, with CBCD files being the corrected files. The\nhistory of these artifact changes is recorded within the imask file headers. The user can find all of the\nartifact correction modules on the contributed software section of the Spitzer website, and replicate or\nimprove the corrections using the BCD and imask files as input.\n\n5.2.1 Stray Light\n\nThe IRAC stray light masker was written by Mark Lacy of the IRAC Instrument Support Team with help\nfrom Rick Arendt of the IRAC Instrument Team and adapted for the artifact mitigation pipeline. The\n\nPipeline Processing 89 The Artifact-Corrected BCD\nPipeline\nIRAC Instrument Handbook\n\nprogram is designed to mask out stray or scattered light from stars outside the array location as well as\nfilter ghosts from bright stars. The module will alter each imask, corresponding to each BCD file, with\npixels likely to be affected by stray light, by turning bit 3 on for those pixels that are likely to be affected\nby scattered light. The program turns imask bit 2 on for those pixels likely to be affected by filter ghosts.\n\nThe module first queries the 2MASS database, producing a table of sources that are likely to produce\nscattered light within the field of view. Using the BCDs and the 2MASS table, including the flux of the\nbright sources, possible stray light affected areas are calculated. These pixel positions are then turned on\nwithin the imask. When the CBCDs are combined to a mosaic, the corresponding pixels in the CBCDs\nwill not be used to produce the mosaic, thereby \u201cmasking out\u201d the input pixels possibly affected by\nscattered light.\n\nIRAC data users are reminded that observations that were not adequately dithered (such as the ones made\nwith the small-scale dither patterns) will have gaps if the stray light mask is used. In these cases, the stray\nlight masking program can be downloaded from the Spitzer website and run on the BCDs in an\nunaggressive mode by setting a keyword. This disables the production of the larger masks for very bright\nstars (which produce diffuse scattered light over a large fraction of the array), avoiding gaps in mosaics.\n\n5.2.2 Saturation\n\nMany of the following artifact corrections need knowledge of the offending source\u2019s flux to work\ncorrectly. For observations of very bright sources, the signal (and even pedestal) reads can be saturated.\nTherefore, the next step in the artifact mitigation process is the saturation correction.\n\nFor a bright, strongly saturated point source, the DN will increase from some low number away from the\nsource to some maximum value between 35,000 and 47,000 DN, and then decrease to a small, usually\nnegative number, at the center. The image looks like a bright doughnut with a dark center. This inverted\n\u201ccrater\u201d peak profile indicates that a significant fraction of the light from the bright star may have been\nlost due to saturation. Recovery, or flux rectification, is possible if the point-spread-function, or PSF, of\nthe star is known. The PSF can then be scaled in flux until the the non-saturated pixels in \u201cwings\u201d of the\nstellar profile can be fit correctly.\n\nThere are several steps to rectify the inner region of the saturated star. First, the exact position of the\nsaturated star is identified using the \u201ccraters\u201d. The program then creates a sub-image around the saturated\nstar, and that is resampled on a finer grid to match the 0.24 arcsecond resampled PSF. Remaining\nartifacts, such as banding and muxbleed, are masked out. The PSF is then matched pixel-by-pixel, the\nPSF flux wings are scaled to the target wings, mean flux ratios are computed, and the best fit outside the\ninner saturated region is determined. The \u201clost flux\u201d is then calculated and the star is rectified by\nreplacing the inner, saturated pixels with flux determined from the PSF profile.\n\nThe IRAC PRF in channel 1 was found to be too narrow for stars, and so a \u201cpuffed up\u201d version was\nempirically derived and found statistically to be more accurate by testing it on stars with known flux.\n\nPipeline Processing 90 The Artifact-Corrected BCD\nPipeline\nIRAC Instrument Handbook\n\nAlso, the program will fail in fitting a PSF to saturated stars that are closer than 20 pixels to the edge of\nthe image. Such saturated stars are not corrected.\n\nA star that is saturated, or predicted to be saturated, has bit 13 flipped on in the corresponding imask.\nAfter this module has corrected the saturation, it will turned off bit 13 and turn on bit 4, meaning that the\n7.2.1.\n\n5.2.3 Sky Background Estimation\n\nThe remaining artifacts are caused by the flux in a pixel reaching a set threshold (documented in Chapter\n7). Once the saturated stars have been corrected, all of the point sources that have fluxes over the\nrespective artifact flux thresholds are detected and flagged. Each pixel that is potentially affected by the\nartifacts triggered by these high fluxes is flagged in the corresponding imask file (column pulldown has\nbit 7 set and banding has bit 6 set).\n\nAn estimated truth image of the sky background is created. To replace the masked pixels, a 5x5 pixel box\naround the corrupted pixel is used to create a weighted average for the pixel value to be replaced using a\nGaussian-weighted interpolation from the pixels within the box. If there are not enough pixels in the 5x5\npixel area around the affected pixel due masking, then an 11x11 pixel area is used in the calculation.\n\n5.2.4 Column Pulldown\n\nIn all four arrays, a bright pixel will trigger a bias shift within its respective column, creating a lower\nbackground value throughout the entire column than in the surrounding columns. The imask will have bit\n7 set, denoting the column pulldown artifact, as mentioned above.\n\nIn this module, the \u201ctruth image\u201d of the sky background is used, and for each column, a robust weighted\nDC offset is determined. This is a simple offset between the affected column and the estimated\nbackground value. This offset is determined separately above and below the triggering source. The offset\nis then applied to the affected column and saved into the CBCD image, thereby removing the bias shift\nfrom all the pixels in the column. More information about column pulldown can be found in Section\n7.2.4.\n\n5.2.5 Banding Correction (Channels 3 and 4)\n\nThe banding effect manifests itself as the rows and columns that contain a bright source having an\nenhanced level of flux. This happens only in the Si:As arrays (channels 3 and 4) and has been shown to be\ndue to internal optical scattering (inside the array). Both bright stellar sources and bright extended sources\ncause banding. It is clearly different from the optical diffraction patterns and the column pulldown effect.\n\nPipeline Processing 91 The Artifact-Corrected BCD\nPipeline\nIRAC Instrument Handbook\n\nAgain, the \u201ctruth image\u201d of the background is used to compute a robust weighted DC offset. The banding\nartifact is extra flux above the background and it will be subtracted out and saved into the CBCD image.\nTheIRAC pipeline does not model the flaring of banding towards the edges of the array. Therefore, the\n\n5.2.6 Muxstripe Correction (Channels 1 and 2)\n\nFor a very bright source, muxbleed is accompanied by a pinstripe pattern (\u201cmuxstripe\u201d; every 4th column\nfrom the bright source is affected) that may extend over part of the image preceding or following the\nbright pixel (for example, see Figure 7.2 and Figure 7.3). Stars, hot pixels, and particle hits can generate\nmuxbleed and muxstripe. In the artifact correction pipeline, a procedure was developed to mitigate the\nmuxstripe in the image.\n\nFigure 5.10. An image showing all four readout channel images side by side. These have been\nobtained by rearranging the columns in the original image. Muxbleed is apparent in the bottom\nright of the 4 th readout channel image.\n\nThe algorithm involves converting the BCD image into 256X64 pixel arrays (each of the four readout\nchannels into a separate image; every fourth column is read out by the same channel; see Figure 5.10).\n\nPipeline Processing 92 The Artifact-Corrected BCD\nPipeline\nIRAC Instrument Handbook\n\nThe muxstripe for one source contaminates only one readout channel, and therefore only one of these\nseparate arrays. The median of the four arrays is created and subtracted from each array, which allows\ndeviation from the normal background to stand out.\n\nFigure 5.11. Subtraction of the median background from the readout channel images. This makes\nthe muxstripe much more apparent in the 4th readout channel image (on the right).\n\nPipeline Processing 93 The Artifact-Corrected BCD\nPipeline\nIRAC Instrument Handbook\n\nFigure 5.12. Profiles showing the column median versus row values for identifying muxstripe. The\nmuxstripe is now identifiable between rows 125 and 200 (significantly lower values than the median\nbackground).\n\nFrom each of these median-subtracted arrays, a one-dimensional array of the values along each row is\nthen created, simply by combining the pixel values along the X-dimension. A profile that represents the\nmedian versus row is created. For each profile, statistics arre calculated to identify any muxstripe. It will\nbe identified as a deviation larger than 3% of the median value for several rows. This will miss the\nweakest muxstripe or a case where all readout channels have muxstripe in the same position (cluster of\nvery bright stars), but this will be rare. The subset of pixels that are affected by the muxstripe is identified\nand this column is corrected using the difference of the median of the \u2018clean\u2019 pixels and the median of the\naffected pixels. The readout channel arrays are then read back out to recreate the original image, and the\ncorrected image is written to the CBCD file. More information about muxstriping can be found in Section\n7.2.2.\n\n5.3 Level 2 (Post-BCD) Pipeline\n\nPipeline processing of IRAC data also includes more advanced processing of many individual IRAC\nframes together to form more \u201creduced\u201d data products. Known by the generic title of \u201cpost-BCD\u201d\nprocessing, this extended pipeline refines the telescope pointing, attempts to correct for residual bias\nvariations and produces mosaicked images. We do not attempt to improve (relative to the BCD) the point\nsource or extended emission flux calibration by automatically comparing to a reference source catalog.\nThe mosaic only includes data from a single observation or AOR.\n\nAll IRAC BCD images contain a pointing estimate based on the output of the Spitzer pointing control\nsystem (star tracker and gyros), i.e., the boresight pointing history file. This initial pointing estimate is\naccurate to about 0.5\u201d. The post-BCD pipeline performs additional pointing refinement for all IRAC\nframes. This is achieved by running the SSC point source detector on the channel 1 and 2 frames and\ncomparing the resultant list of point sources to the 2MASS catalog. The results are then averaged, and the\nknown focal plane offsets between all four channels are applied to produce a \u201csuperboresight\u201d pointing\nhistory file, which is then applied to the data during end-of-campaign reprocessing. This improves the\npointing accuracy of the frame to better than about 0.3\u201d. This refined RA, Dec appears in the header as\nthe CRVAL1, CRVAL2 keyword values.\n\nThe pipeline SSC mosaicker produces a single image (one per band) from many input images. First, the\nBCDs are corrected for overlap consistency. The parts of the images that overlap are forced to have the\nsame background value via addition of an offset. Then a \u201cfiducial frame\u201d is derived. This is the definition\nfor the output frame in terms of its physical size, projection, and orientation. Because IRAC has such a\nlarge field of view, projection effects are non-negligible, and the mosaicking and coadding process must\nreproject the data. The fiducial frame finder seeks to minimize the amount of \u201cblank\u201d area in the output\nmosaic by rotating the output projection such that it is aligned with the map axes. This is useful for long\n\nPipeline Processing 94 Level 2 (Post-BCD) Pipeline\nIRAC Instrument Handbook\n\nthin maps, where potentially the output mosaic could be very large, but with a great deal of empty space.\nThe mosaicker then reprojects all of the input data onto the output projection. It reads the SSC WCS,\nwhich contains the field pointing center, rotation, scale, and instrument distortion, and reprojects this onto\na standard TAN FITS projection. In the process, the data are undistorted. The reprojected images are\ninterpolated onto the fiducial image frame with outlier rejection, rejecting radiation hits that happen in\noverlapping observations. The outlier rejection scheme is specifically designed to work well in the case of\nintermediate coverage and may not be adequate for all observations and science programs. In addition to a\nsky map (in units of surface brightness), a noise image and coverage map are also produced.\n\nThe post-BCD pipeline modules have been made available for general public use as part of the MOPEX\ntool. They consist of a number of C-modules connected via PERL wrapper scripts. Namelists are used for\ninput. In most cases their operation simply consists of supplying the software with a list of input image\n\n5.3.1 Pointing Refinement\n\nTo improve the ~ 0.5\u201d blind pointing, a pointing refinement is run in which the point sources are\nidentified in the IRAC frames and astrometrically correlated with stars near the source position in the\n2MASS catalog. The pointing refinement typically improves the positional error to < 0.3\u201d and removes\nany systematic offsets.\n\nFirst, point sources are extracted from the pipeline-processed mosaics and transformed to RA and Dec\nusing the transformations derived from the current pointing. If there are less than five sources in an\nimage, then there will be no refinement for that BCD.\n\nA comparison is made of the position and flux of each point-source match found in the 2MASS point-\nsource catalog. The new translational and rotational reference frame can be computed from the\ndifferences and uncertainties, and a refinement is made of the celestial pointings and angles of each BCD\nin the observation or AOR used for the mosaic. These refined values are written to the end of the FITS\nheaders as RA_RFND, DEC_RFND, including many others with RFND as an indicator of \u201crefined\"\npointing.\n\n5.3.2 Superboresight Pointing Refinement\n\nPointing refinement operates on each IRAC channel independently. This often results in poor pointing\nsolutions for channels 3 and 4, in which stellar fluxes are lower and the background higher than in\nchannels 1 and 2. We have therefore developed a technique which combines the results of pointing\nrefinement in channels 1 and 2 and applies it to all four channels using the known offsets between the\nIRAC fields of view. This improved pointing solution is derived during campaign reprocessing. The\nresults of the pointing refinement from the first run of the post-BCD pipelines are averaged, and the\ncorrection derived from this is applied to the boresight pointing history file (which contains the pointing\n\nPipeline Processing 95 Level 2 (Post-BCD) Pipeline\nIRAC Instrument Handbook\n\nestimate derived from the spacecraft telemetry and which provides our initial pointing estimate). This\ncorrected pointing history file (the \u201csuperboresight\" file) is then applied to the BCD at the pointing\ntransfer stage of the BCD pipeline. The superboresight RA and Dec estimates are recorded in the\nCRVAL1 and CRVAL2 FITS keywords, and the position angle estimate is recorded in the CD matrix\nkeywords. The uncorrected RA and DEC are retained, but called ORIG_RA, ORIG_DEC, as is the\npointing refinement solution for each frame (as RARFND and DECRFND). Note that to use the\nsuperboresight solution, USE REFINED POINTING = 0 should be set in the MOPEX namelists.\n\nSuperboresight was implemented as a patch to the S13 software build, thus most (but not all) data\nprocessed with S13 or subsequent pipeline versions will have it. Users should check for the presence of\nthe ORIG_RA, ORIG_DEC keywords to see if it has been applied to their data. From S14 onwards, the\nHDR data have the long frame RA, Dec solution copied to the short frames, as the short frame pointing\nsolutions are less accurate. Neither superboresight nor pointing refinement are run on the subarray data.\n\n.\n\nUsing the refined coordinates, individual IRAC BCDs from a given observation (AOR) are reconstructed\nonto a larger field (mosaicking), and overlapping frames are averaged together to achieve a higher S\/N.\nOutlier rejection is performed on sets of overlapping pixels. Because Spitzer observations cover such a\nlarge area, individual BCDs are remapped onto a common grid with a technique similar to \u201cdrizzle\u201d\n(Fruchter & Hook 2002, [11]). The pixel size in the mosaics produced by the pipeline is exactly 0.6\narcseconds x 0.6 arcseconds in the final data processing. The masks are used in the coaddition in such a\nway that the pixels previously flagged as bad (for example, hot or dead pixels) are rejected before the\naveraging process. Cosmic rays are rejected at this point via the outlier rejection algorithm. Users will\nreceive a single coadded image per channel, and per observation (AOR). It will be accompanied by a\ncoverage map and an uncertainty file per channel per exposure time.\n\nPipeline Processing 96 Level 2 (Post-BCD) Pipeline\nIRAC Instrument Handbook\n\n6 Data Products\n\nThis section describes the basic data products the observer will receive from the Spitzer Heritage Archive.\nThe available data products consist of Level 0 (raw) data, Level 1 (BCD) data, calibration files, log files,\nand Level 2 (post-BCD) data. IRAC data are supplied as standard FITS files.\n\nEach file consists of a single data collection event (i.e., a single exposure), and contains one image\ncorresponding to one of the four IRAC arrays (the exception being post-BCD products, described below).\nThe FITS headers are populated with keywords including (but not limited to) physical sky coordinates\nand dimensions, a photometric solution, details of the instrument and spacecraft including telemetry when\nthe data were taken, and the steps taken during pipeline processing.\n\n6.1 File-Naming Conventions\n\nTable 6.1 lists the IRAC data files produced by the IRAC data reduction pipelines, together with brief\ndescriptions of these files. The Basic Calibrated Data (BCD) are the calibrated, individual images. These\nare in array orientation and have a size of 256 x 256 pixels for the full array images, and 64 planes times\n32 x 32 pixels for the subarray images. These data are fully calibrated and have detailed file headers. The\nPost-BCD pipeline combines the BCD images into mosaics (per wavelength and per frame time).\nCalibration observations designated as darks or flats go through a similar but separate pipeline that\ngenerates the products listed in the last section of Table 6.1.\n\nNote that because of the \u201cfirst frame effect,\u201d the first frame of every Astronomical Observation Request\n(AOR) has a different delay time and it cannot be calibrated correctly. Therefore, the first frame of every\nAOR with a frame time greater than 2 seconds is taken in HDR mode which causes the first frame to be\n0.6 seconds or 1.2 seconds in duration instead of the full frame time. This first frame usually has a name\nsuch as SPITZER_I1_11111111_0000_0000_2_bcd.fits and has no associated cosmic ray mask (brmsk\nfile). The observer is encouraged not to use this first short frame. The pipeline mosaicker does not use it\neither when building the mosaic.\n\nThe BCD uncertainty files (listed below) are rough uncertainty estimations and do not include all of the\nsystematic effects associated with IRAC detectors, nor do they include the absolute flux uncertainty.\nThese uncertainty images are generated as follows. They begin as an estimate of the read noise (one\nnumber in electrons for the whole image) and the shot-noise due to the sky (proportional to the square-\nroot of the number of electrons in the image). Then each module propagates the uncertainty image\nforward, including the uncertainties in dark and flat calibration files. The pipeline modules use the\nuncertainty image as a way to quantitatively estimate the quality of the sky estimate given by the value of\na pixel. In the end, the uncertainty images overestimate the formal uncertainty of the image, because the\nnet propagated uncertainty is much higher than the observed pixel-to-pixel fluctuations in the images. We\ntherefore recommend that the uncertainty images only be used for relative weights between pixels, for\nexample when performing outlier rejection or making a weighted mosaic that combines multiple input\nframes that view the same sky mosaic pixel.\n\nData Products 97 File-Naming Conventions\nIRAC Instrument Handbook\n\nPlease note that all of the calibration products (specifically, skydarks, skyflats, and linearity curves) are in\nthe raw reference frame. Hence, the subarrays are located at pixel coordinates 9:40, 9:40. In the BCDs,\nthe subarrays are located in pixel coordinates 9:40, 9:40 in channels 3 and 4, and in pixel coordinates\n9:40, 217:248 in channels 1 and 2. Also note that those coordinates are for the case in which the first pixel\nis indexed as 1,1 (i.e., IRAF convention). E.g. IDL pixel indices start from 0,0.\n\nTable 6.1 Sample IRAC file names.\n\nBrief Description\nBasic Calibrated Data (BCD)\nSPITZER_I1_0008845056_0031_0000_01 dce.fits Raw data (in units of DN)\nSPITZER_I1_8845056_0031_0000_1_bcd.fits BCD data (in units of MJy\/sr)\nSPITZER_I1_8845056_0031_0000_1_bcd.log BCD pipeline log\nSPITZER_I1_8845056_0031_0000_1_bunc.fits BCD uncertainty file\nSPITZER_I1_8845056_0031_0000_1_ptn.log BCD Pointing log\nSPITZER_I1_8845056_0031_0000_1_sub2d.fits 2D BCD image (subarray only)\nSPITZER_I1_8845056_0031_0000_1_unc2d.fits 2D BCD uncertainty file (subarray only)\nSPITZER_I1_8845056_0031_0000_1_msk2d.fits 2D BCD Imask file (subarray only)\nSPITZER_I1_8845056_0031_0000_1_cov2d.fits 2D BCD coverage file (subarray only)\nArtifact-Corrected BCD Processing (CBCD)\nSPITZER_I1_8845056_0031_0000_1_cbcd.fits Artifact-corrected BCD data\nSPITZER_I1_8845056_0031_0000_1_cbunc.fits Artifact-corrected BCD uncertainty file\nPost-BCD Processing\nSPITZER_I1_8845056_0000_1_E123458_maic.fits Mosaic\nSPITZER_I1_8845056_0000_1_A2987651_munc.fits Mosaic uncertainty file\nSPITZER_I1_8845056_0000_1_A2987653_mcov.fits Mosaic coverage file\nSPITZER_I1_8845056_0000_1_A2987654_maicm.fits HDR intermediate frame time mosaic\nSPITZER_I1_8845056_0000_1_A2987655_muncm.fits HDR intermediate frame time mosaic\nuncertainty file\nSPITZER_I1_8845056_0000_1_A2987656_mcovm.fits HDR intermediate frame time mosaic\ncoverage file\nSPITZER_I1_8845056_0000_1_A2987657_mmskm.fits HDR intermediate frame time mosaic mask\nfile\nSPITZER_I1_8845056_0000_1_A2987658_maics.fits HDR short frame time mosaic\nSPITZER_I1_8845056_0000_1_A2987659_muncs.fits HDR short frame time mosaic uncertainty file\nSPITZER_I1_8845056_0000_1_A2987660_mcovs.fits HDR short frame time mosaic coverage file\nSPITZER_I1_8845056_0000_1_A2987653_mmsks.fits HDR short frame time mosaic mask file\nSPITZER_I1_8845056_0000_1_A26871875_irsa.tbl List of 2MASS stars for pointing refinement\nSPITZER_I1_8845056_0000_1_A26871873_refptg.tbl Table of pointing refinement information\nCalibration pipeline data files\nSPITZER I1_13450853_0000_1_C92523_ sdark.fits Skydark\nSPITZER I1_13450853_0000_1_A210654_ scmsk.fits Skydark mask file\nFUL_2s_2sf4d1r1_ch[1-4]_v1.2.0_dark.fits IRAC labdark image (full array)\n\nData Products 98 File-Naming Conventions\nIRAC Instrument Handbook\n\nFUL_2s_2sf4d1r1_ch[1-4]_v1.2.0_dark_noise.fits IRAC labdark noise image\nHDR_30s_1.2sf1d1r_ ch[1-4]_v1.2.0_dark.fits IRAC labdark image (HDR short frame)\nHDR_30s_30sf16d1r_ ch[1-4]_v1.2.0_dark.fits IRAC labdark image (HDR long frame)\nSUB_0.1s_0.1sf2d1r1_ch[1-4]_v1.2.0_dark.fits IRAC labdark image (subarray)\nirac_b[1-4]_[fa\/sa]_superskyflat_finalcryo_091004.fits IRAC superskyflat image\nirac_b[1-4]_[fa\/sa]_20020921_lincal.fits linearization calibration image\nirac_b[1-4]_[fa\/sa]_cdelt12_distort.tbl Array distortion table\nirac_b[1-4]_[fa\/sa]_16_118_muxbleed_coeff_112003Muxbleed correction coefficients\nirac_b[1-4]_[fa\/sa]_16_118_muxbleed_lut_100102 Muxbleed correction look-up table\nFlipped pmask (nearest in time; not in use)\nirac_b[1-4]_fa_slmodel_v1.0.1.fits Subtracted scattered light model\nirac_b[1-4_ fluxconv_10112010.tbl Flux conversion file used\ninstrument_FOV.tbl IRAC array locations in Spitzer FOV\nirac_b[1 \u22124]_mosaicPRF.fits PRF file used for pointing refinement\nirac_b[1 \u22124]_PRF.tbl PRF used for the four quadrants of the image\nSPITZER_I1_21752576_0000_5_A41882936_avg.fits Average image of all BCDs in an AOR\nSPITZER_I1_21752576_0000_5_A41882938_avmed.fits Average image of all HDR intermediate\nframes\nSPITZER_I1_21752576_0000_5_A41882937_ashrt.fits Average image of all HDR short frames\nSPITZER_I1_21752576_0000_5_C8232029_mdn.fits Median image of all BCDs in an AOR\nSPITZER_I1_21752576_0000_5_A41882940_mdmed.fits Median image of all HDR intermediate\nframes\nSPITZER_I1_21752576_0000_5_A41882939_mshrt.fits Median image of all HDR short frames\n\nHere we describe some of the important header keywords. A complete IRAC image header description is\nincluded in Appendix D.\n\nAORLABEL is the name of the AOR as it was defined by the user in Spitzer Observation Planning Tool\nSpot when the observations were requested. The P.I. of a program under which the data were taken will\nbe listed as the OBSRVR of each project. AORKEY is a unique identification number or \u201cdigit sequence\u201d\nfor each observation; it is also part of the filename for each BCD. EXPID is an exposure counter\nincremented within a given AOR for each data-taking command. Most data-taking commands generate\nmultiple files: one per array in full array mode. The DCENUM is a counter of individual frames (per\nwavelength) from an individual command; it can be used to separate frames generated with internal\nrepeats. The only observations with non-zero DCENUM are channel 4 BCDs for 100\/200 second frame\ntime (taken as two\/four 50 second frames). In high dynamic range mode, the long and short exposures are\ngenerated with independent commands and have different EXPIDs. Thus, for example, data from 12-\nsecond high dynamic range observations can be separated into long and short frames using the odd' or\n\nData Products 99 IRAC Specific Header Keywords\nIRAC Instrument Handbook\n\neven' EXPIDs. DATE_OBS is the time at the start of the AOR. Other times in the header include the\ntime since IRAC was turned on (both for the beginning and end of the frame). FRAMTIME is the\nduration of the frame including Fowler sampling, and EXPTIME is the effective integration time.\nUTCS_OBS is the start of IRAC data taking sequence. Modified Julian date is in keyword MJD_OBS and\nthe corresponding heliocentric modified Julian date in HMJD_OBS. There is also a Solar System\nbarycentric modified Julian date in BMJD_OBS. See the IRAC FITS header in Appendix D for more\ntiming related keywords. ATIMEEND is the correct time of an integration end. HDRMODE tells you if\nthe frame was taken in the high dynamic range mode.\n\nBUNIT gives the units (MJy\/sr) of the images. For reference, 1 MJy\/sr = 10\u201317 erg s\u20131 cm\u20132 Hz\u20131 sr\u20131 .\nFLUXCONV is the calibration factor derived from standard star observations; its units are\n(MJy\/sr)\/(DN\/s). The raw files are in \u201cdata numbers\u201d (DN). To convert from MJy\/sr back to DN, divide\nby FLUXCONV and multiply by EXPTIME. To convert DN to electrons, multiply by GAIN.\n\nThe predicted background (using the same model as what was implemented in Spot, evaluated for the\nwavelength, date, and coordinates of observation) is contained in three keywords: ZODY_EST,\nISM_EST, CIB_EST. These are not based on the actual data from Spitzer. SKYDRKZB is the zodiacal\nbackground prediction for the skydark that was subtracted from the science image in the reduction\npipeline. Thus the predicted background in the BCD data is ZODY_EST \u2013 SKYDRKZB. DS_IDENT is a\njournal identification number for the Astrophysics Data System (ADS) to keep track of papers published\nfrom these data.\n\nAbsolute pointing information is contained in the following keywords. ORIG_RA and ORIG_DEC give\nthe coordinates of the image center constructed from the telemetry using the Boresight Pointing History\nFile, as indicated by the Boolean keyword USEDBPHF, and the file is listed in BPHFNAME. When\npointing telemetry is not available, due to a telemetry outage, the commanded positions are inserted\ninstead, USEDBPHF is false, and the coordinates will be less certain. RARFND and DECRFND are the\nrefined positions derived by matching the brightest sources in the image with the 2MASS catalog.\nPA_RFND is the refined position angle of the +y axis of the image, measured east from north (CROTA2\nis the same position angle but measured west from north). CRVAL1 and CRVAL2 give the coordinates of\nthe image center, derived from the refined positions in all channels, and are usually the most accurate\ncoordinates available.\n\nSometimes a bad pixel value (zero) was inserted in the data field. These pixels are detected and shown in\nraw frame header where ABADDATA assumes the value of 1. In the BCD FITS header you will then\nfind header keyword BADTRIG set to \u201cT\u201d (true) and the number of zero pixels in the frame listed in\nheader keyword ZEROPIX. If there is only one bad pixel, the pipeline fixes the problem and gives the bad\npixel position in header keyword ZPIXPOS.\n\nThe BCD +x-axis (bottom, or horizontal axis) is in the direction of the telescope +Y-axis, and the BCD -\ny-axis (left side or vertical axis) is in the direction of the telescope +Z-axis.\n\nNext we give an example of how an AOR file translates into final data products. A Spitzer observation is\nspecified by a small list of parameters that are listed in the \u201c.aor\" file. This file was generated when the\n\nData Products 100 IRAC Specific Header Keywords\nIRAC Instrument Handbook\n\nobservation was designed (using Spot). The \u201c.aor\" file for a planned or performed observation can be\nretrieved using the \u201cview program\" feature of Spot. (You will be prompted for the program name or ID,\nwhich can be obtained from the image header keywords PROGTITLE and PROGID). Here is an example\nAOR file:\n\n# Please edit this file with care to maintain the\n# correct format so that SPOT can still read it.\n# Generated by SPOT on: 5\/9\/2003 12:10:9\n\nAOT_TYPE: IRAC Mapping\nAOR_LABEL: IRAC-FLS-CVZ-a\nAOR_STATUS: new\nMOVING_TARGET: NO\nTARGET_TYPE: FIXED SINGLE\nTARGET_NAME: FLS-CVZ\nCOORD_SYSTEM: Equatorial J2000\nPOSITION: RA_LON=17h13m05.00s, DEC_LAT=+59d10m52.0s\nOBJECT_AVOIDANCE: EARTH = YES, OTHERS = YES\nARRAY: 3.6_5.8u=YES, 4.5_8.0u=YES\nHI_DYNAMIC: NO\nFRAME_TIME: 12.0\nDITHER_PATTERN: TYPE=Cycling, N_POSITION=5, START_POINT=1\nDITHER_SCALE: small\nN_FRAMES_PER_POINTING: 1\nMAP: TYPE=RECTANGULAR, ROWS=7, COLS=6, ROW_STEP=277.0, COL_STEP=280.0,\nORIENT=ARRAY, ROW_OFFSET=0.0,COL_OFFSET=440.0,N_CYCLE=1\nSPECIAL: IMPACT = none, LATE_EPHEMERIS = NO,SECOND_LOOK = NO\nRESOURCE_EST: TOTAL_DURATION=5848.4, SLEW_TIME=1089.0, SETTLE_TIME=1045.0,\nINTEGRATION_TIME: IRAC_3_6=60.0,IRAC_4_5=60.0,IRAC_5_8=60.0,IRAC_8_0=60.0\n\nFor this AOR, there are 210 files (6 columns x 7 rows x 5 dither positions) of each type for each channel.\nThe final data products from this AOR in channel 2, provided it got assigned the AORKEY 6213376, are\nas follows:\n\nSPITZER_I2_0006213376_0000_0000_01_dce.fits\nSPITZER_I2_6213376_0000_0000_1_bcd.fits\nSPITZER_I2_6213376_0000_0000_1_cbcd.fits\nSPITZER_I2_6213376_0000_0000_1_bcd.log\nSPITZER_I2_6213376_0000_0000_1_bunc.fits\nSPITZER_I2_6213376_0000_0000_1_cbunc.fits\nSPITZER_I2_6213376_0000_0000_1_bdmsk.fits\nSPITZER_I2_6213376_0000_0000_1_bimsk.fits\nSPITZER_I2_6213376_0000_0000_1_brmsk.fits\nSPITZER_I2_6213376_0000_0000_1_ptn.log\n....\nSPITZER_I2_0006213376_0209_0000_01_dce.fits\nSPITZER_I2_6213376_0209_0000_1_bcd.fits\n\nData Products 101 IRAC Specific Header Keywords\nIRAC Instrument Handbook\n\nSPITZER_I2_6213376_0209_0000_1_cbcd.fits\nSPITZER_I2_6213376_0209_0000_1_bcd.log\nSPITZER_I2_6213376_0209_0000_1_bunc.fits\nSPITZER_I2_6213376_0209_0000_1_cbunc.fits\nSPITZER_I2_6213376_0209_0000_1_bdmsk.fits\nSPITZER_I2_6213376_0209_0000_1_bimsk.fits\nSPITZER_I2_6213376_0209_0000_1_brmsk.fits\nSPITZER_I2_6213376_0209_0000_1_ptn.log\n\nAfter the name of the telescope, the first partition gives the instrument (\u201cI\" = IRAC), and the number after\nthe \u201cI\" gives the channel (in this case, 2). The next part gives the AORKEY, then we have the EXPID,\nDCENUM, and the version number (how many times these data have been processed through the\npipeline). One should generally use only the data from the highest version number, in case multiple\nversions have been downloaded from the archive. To verify that the data are from the latest pipeline\nversion, check the CREATOR keyword in the header (S18.18 for the final cryogenic IRAC data\nprocessing). Finally, there is a group of letters that specify what kind of data are in the file (see Table 6.1\nabove), and the file type (usually \u201cfits\" or \u201clog\"). The post-BCD file names include telescope name\n(SPITZER), \u201cI\" (for \u201cIRAC\"), the channel number, the productid (not the same as the AORKEY), the\nDCENUM, the (pipeline) version, \u201censemble product id,\u201d the type of the data and the suffix. In the case\nof an ensemble product, \u201cDCENUM\" in the filename refers to the first DCE that was used in the\nensemble creation, and \u201cversion\u201d refers to the version of that first DCE. The letter \u201cC\u201d stands for\n\u201ccalibration\" product: in the case of a calibration product, \u201cDCENUM\" refers to the first DCE that was\nused in the calibration creation, (pipeline) \u201cversion\u201d refers to the version of that first DCE, and number\nafter the \u201cC\u201d letter is the \u201ccalibration number\". Note that for a given AORKEY of science data being\nretrieved, the AORKEY for the associated calibration products is different.\n\nA list of 2MASS sources for the field of the IRAC observation is included in the data delivery as\n*irsa.tbl. Note that the 2MASS magnitudes given in the *irsa.tbl file are not meant for scientific use. For\nscientific use of the 2MASS data, query the 2MASS catalog directly from IRSA, and take into account\nthe flux quality flags.\n\nData Products 102 IRAC Specific Header Keywords\nIRAC Instrument Handbook\n\n7 Data Features and Artifacts\n\nThe common artifacts in IRAC data are discussed in this chapter. Most of these have been mitigated by\nthe pipeline processing, which produces artifact-corrected images (\u201cCBCDs\u201d). Further mitigation is often\npossible by a judicious quality inspection of the data, and\/or further processing of the BCDs. Note that\nmany of these artifacts are quite commonly seen in IRAC images.\n\nThe most common artifacts are as follows. Stray light from point sources should be masked by hand.\nPersistent images usually come from a bright source observed as part of the observation. In some cases,\nhowever, persistent images from a preceding observation may be found. One way to check this is by\ninspecting a median of all the images in an observation (AOR). Another possible flaw in the observations\nwould be an exceptionally high radiation dosage. The nominal rate is 1.5 hits per array per second, and\nthe radiation hits range from single pixels to connected streams (and occasionally small clouds of\nsecondaries). High particle hit rates occurred following one solar flare during the In-Orbit Checkout, and\none in Nominal Operations. In the latter event, several hours of science data were rendered useless\nbecause of the large number of hits in the images. Objects that are bright enough leave muxbleed trails\nand can generate pinstripe patterns over large parts of the image, and offsets along the columns and rows\ncontaining the bright source. Ghosts from internal reflections within the filters can be seen in almost\nevery channel 1 or 2 BCD, and more ghosts in all channels are noticeable from bright objects.\n\nWe begin with a discussion of the basic characteristics of the dark frames and flatfields that affect every\nimage. We follow with a discussion of electronic artifacts. These effects arise from the inherent\nnonlinearity of the detector diodes and saturation of either the detector well, transistors in the mux, or the\nanalog-to-digital converter (ADC) in the warm electronics; crosstalk within the mux or warm electronics;\nor from inductive coupling to currents in spacecraft cables. Most electronic effects have a short\npersistence, but image persistence, which is also nonlinear in photon fluence, can last seconds, minutes,\nhours, or even weeks. Next we have a section on optical artifacts, which include stray light or ghosts from\nsources within or outside the FOV. Finally, we discuss the effects of cosmic rays and solar protons on\nIRAC observations. Please note that asteroids may be \u201ccontaminants\u201d in the data as well, especially when\nthe target is close to the ecliptic plane. Asteroids can most effectively be rejected from datasets that have\nbeen taken at least several hours apart, so that the asteroids have moved in the data and can be masked out\nby temporal outlier rejection routines.\n\n7.1 Darks, Flats and Bad Pixels\n\nThe true median dark currents, due to nonzero leakage resistance or recombination in reverse-biased\ndetector diodes, are very small compared to the current from the background at the darkest part of the\ncelestial sphere. Labdarks, which were measured with the cold IRAC shutter closed, with zero photon\nflux, are not zero, and have significant pixel-dependent offsets, usually positive, that depend on the frame\ntime and the Fowler number, as well as the history of readouts and array idling over the previous several\nhours. Channel 3 is by far the most extreme case, in which, for example, a 100 second (Fowler-16) frame\ncan be offset as much as 370 DN (median), or the equivalent of 1400 electrons at the integrating node,\n\nData Features and Artifacts 103 Darks, Flats and Bad Pixels\nIRAC Instrument Handbook\n\nwith no light incident on the array. The signal from the darkest background in a 100 second frame in\nchannel 3 is only about 1000 electrons. Channel 1 has much smaller offsets, but the sky is so dark that the\noffsets are often larger than the background signal. Channel 2 has very small offsets, which are less than\nthe background signal except in short frames or certain frames immediately following a change in\nintegration time. The background in channel 4 is so large that the offsets are almost negligible except in\nvery short integrations. There is no measureable excess noise from the offset itself: the noise is not the\nsquare root of the equivalent number of charge quanta on the integrating node. This is because the offset\narises from the redistribution of charge within the mux in which the associated currents and capacitance\nare much greater than in the detector diodes. However, imperfect correction of this \u201cfirst-frame\u201d effect\ndoes increase the uncertainties in BCD frames. The uncertainty scales with the size of the offset and its\nsmall-scale spatial nonuniformity. Only in channel 3 does it significantly increase the total pixel noise.\n\nWe can break down the offset into contributions beginning with the largest spatial scale down to the\nsmallest. In this view, the largest part of the offset is uniform over the array, followed by the contribution\nof a few spatial gradients, and some pinstriping that repeats every four columns (due to the four array\noutputs), with a few columns with odd offsets (due to hot pixels or parts of the mux), and weakest of all,\npixel-to-pixel dependent offsets.\n\nThere are some very obvious features imposed on the true offset, due to a relatively small number of hot\npixels, and mux glow. Hot pixels usually appear bright, and in such cases one can see a trail of muxbleed\n(in raw images) or a pinstripe pattern in InSb arrays, or the bandwidth effect (in Si:As arrays) following\nthe hot pixel. These pixels have high dark currents and are usually isolated, but sometimes in a clump.\n\u201cDead\u201d pixels are really just very hot pixels, so hot that they saturate before the first pedestal sample. In a\nBCD image, hot pixels do not appear bright because they have been canceled by the labdark or skydark\nsubtraction. Most hot pixels appeared after launch and are the result of hits by energetic nuclei. By\nannealing the arrays, we restored most pixels that got activated. Some of them cannot be restored, and\nthus they became \u201cpermanent\u201d hot pixels. Some pixels jumped randomly from normal to high dark\ncurrent and back, dwelling in one state for anywhere from a few minutes to weeks, so they may not be\ncanceled by a skydark subtraction. These are IRAC's \u201crogue pixels.\u201d The IRAC \u201cstatic\u201d bad pixel masks\nwere updated when significant changes in the permanent bad and\/or hot pixels occurred.\n\nAreas of mux glow are visible in the labdark and images. Electrons and holes recombine in diodes in the\nmux, allowing current to flow. Photons emitted in the recombination are detected in the InSb or Si:As\ndetector above or near the source of the glow. Most prominent is the glow from the four output FETs\nvisible only in Channels 3 and 4 (the Si:As arrays). These are semicircular areas about 17 pixels in radius\nlocated near column 256, row 30 at the right edge of the images. The glow is most obvious in long\nframes. Another glow region is visible along the last few rows in all 4 channels; it comes from the unit\ncell FETs. Currents flow through all the unit cell FETs in the last row which is left selected during the\nintegration, so the glow is particularly bright in the last row itself. The 3rd and faintest glow region is\nalong the left edge (column 1) of channel 3. Detected glows have shot noise, which can exceed the\nbackground noise along the last row and in the brightest parts of the semicircular areas. Pixels are masked\nin these areas where the noise significantly degrades sensitivity in 100-second frames.\n\nData Features and Artifacts 104 Darks, Flats and Bad Pixels\nIRAC Instrument Handbook\n\nTable 7.1: Defi nition of bi ts in the \u201c pmask\u201d.\n\nBit Condition\n0 Not set\n1 Not set\n2 Not set\n3 Not set\n4 Not set\n5 Not set\n6 Not set\n7 Dark current highly variable\n8 Response to light highly variable\n9 Pixel response to light is too high (unacceptably fast saturation)\n10 Pixel dark current is too excessive\n11 Not set\n12 Not set\n13 Not set\n14 Pixel response to light is too low (pixel is dead)\n15 [reserved: sign bit]\n\nor semi-permanent bad pixels and regions, and which is the same for all BCDs in a given AOR and\nchannel, the \"imask\" which contains bad pixels specific to any one BCD, and the \u201crmask\u201d which contains\noutliers masked by the post-BCD pipeline. All of the bits set in the imask indicate pixels that have been\ncompromised in some fashion. Not all of the imask bits are set by the BCD pipeline, but some bits are\nplaceholders for post-BCD processing of data artifacts. The higher the order of bit set in the imask, the\nmore severe the effect on data quality. Mask values are set as powers of two, and summed together for\neach pixel. Any pixel with a bit set in the pmask is suspect.\n\nSeveral sets of pmasks have been produced. At the start of the mission, sets were produced at 3\u22126 month\nintervals. As the bad pixel behavior has been shown to vary little with time, these intervals were extended\nto 12\u221218 months. The masks are made from calibration data spanning three campaigns, allowing some\nshort-term bad pixels to anneal out, while retaining the ones persistent on timescales of weeks or more.\nPixels consistently noisy in the darks and\/or flats in these three campaign sets are flagged. The regions of\namplifier glow are also flagged (with bit 10). Combined masks are also available, with bit 0 set to indicate\n\nData Features and Artifacts 105 Darks, Flats and Bad Pixels\nIRAC Instrument Handbook\n\na suspect pixel. The \u201dOR\" masks contain all pixels which have been flagged in any pmask set during the\nmission, and the \u201cAND\" masks contain only those pixels set in every pmask set. Table 7.1 and Table 7.2\nshow what each bit value corresponds to. The DCE Status Mask Fatal Bit Pattern = 32520 (bits 3, 8-14; to\nbe used with the MOPEX software).\n\nTable 7.2: Defi nition of bi ts in the \u201cimask\u201d.\n\nBit Condition\n0 reserved for boolean mask (or if best practice bits set, data quality)\n\n1 reserved for future use\n2 optical ghost flag (set by post-BCD tool)\n3 stray light flag (set by post-BCD tool)\n4 saturation corrected in pipeline\n5 muxbleed flag in ch 1,2; bandwidth effect in ch 3,4 (set by post-BCD tool)\n6 banding flag (set by post-BCD tool)\n7 column pulldown flag in ch 1,2; vertical banding flag in ch 3,4 (set by post-BCD tool)\n8 crosstalk flag\n10 latent flag\n11 not flat-field corrected\n12 data not very linear\n13 saturated (not corrected in pipeline), or predicted to be saturated in long HDR frames\n15 [reserved: sign bit]\n\n7.1.2 Flatfield\n\nIndividual pixel-to-pixel gain variations are corrected by means of a pixel-to-pixel gain map commonly\nknown as a \u201cflatfield.\" IRAC flats are derived by making highly dithered observations of one of\napproximately 20 fixed locations in the ecliptic plane, specifically chosen to be as free of stars and\nextended cirrus emission as possible, and in which the zodiacal light provides a uniform illumination. The\ndata are processed much like science data and then averaged with outlier rejection. Additionally, since\nstars, asteroids and galaxies are a significant contaminant in the data, an object detector is used to find and\nthen explicitly reject them during the averaging. The flats are normalized to a median of one. New flat\nfield measurements are made every time the instrument is turned on.\n\nData Features and Artifacts 106 Darks, Flats and Bad Pixels\nIRAC Instrument Handbook\n\nAnalysis of data from the first two years of operations has shown that the flatfield response of IRAC is\nunchanging at the limit of our ability to measure. As a result, so-called \u201csuper skyflats\" were generated\nfrom the first two years of data. The super skyflats are shown in Figure 7.1.\n\nThese flats are extremely low-noise, with stochastic pixel-to-pixel uncertainties of 0.14%, 0.09%, 0.07%,\nand 0.01% in channels 1 through 4, respectively. This is smaller in amplitude than the intrinsic pixel-to-\npixel scatter in the gain. Furthermore, because the super skyflats are derived from data over many parts of\nthe sky, with many dithers and rotations of the telescope, they are substantially free of errors arising from\ngradients in the zodiacal background, or from residual contamination by stars and galaxies. Currently all\nIRAC data are reduced with the same set of super skyflats.\n\nLarge-scale gradients corrected by the flats are on the order of 10%\u221215%. Systematic errors in the flats\nare due to the gradient in the zodiacal background and straylight removal errors. The former is expected\nto be very small based on results from other missions (Abraham et al. 1997 [1], ISOPHOT 25 \u00b5m).\nDiffuse stray light is a significant contaminant in the raw images at the ~ 5%\u221210% level. This diffuse\nlight looks like a \u201cbutterfly\u201d across the top of the InSb detectors in channels 1 and 2, or a \u201ctic-tac-toe\u201d\npattern in channels 3 and 4. It is always present, resulting from scattering of the zodiacal background onto\nthe detectors. In both the skyflats and the science data, a model of the straylight has been subtracted, but\nthis leaves a residual pattern on the order of 1% which contaminates the flats. These errors are\nsubstantially ameliorated by dithering (errors will decrease as N , where N is the number of dithers, and\nwill quickly become very small relative to other uncertainties).\n\nData Features and Artifacts 107 Darks, Flats and Bad Pixels\nIRAC Instrument Handbook\n\nFigure 7.1: Super skyflats for IRAC. These were made by combi ning the fl at fiel ds from the first fi ve years of\noperations. The dark s pot in channel 4, near the left side and about hal f way up, and the dark spot i n about\nthe same pl ace in channel 2, are due to the same speck of contami nation on the channel 2\/4 pickoff mirror.\nThe darkest pi xels in the s pot are 20% below the surroundi ng area in channel 2, and 32% in channel 4. Fl at-\nfielding in the pi peline fully corrects for these dark spots in the data.\nThe left edges of channels 1 and 3 are vignetted due to misalignment of IRAC optics with the telescope.\nThe darkest pixels have 50% of the mean throughput in channel 1, and 70% in channel 3. The vignetting\nonly extends for 10-15 pixels. The vignetting is compensated for by the flat-fielding, and results primarily\nin an increase by at most 2 of the noise in the affected pixels.\n\nFinally, one should note that the flat fields are generated from a diffuse, extremely red emission source.\nWhile the resulting flats perfectly flatten the zodiacal background, they are not accurate for compact\nobjects with different spectral slopes, the most obvious examples being stars. Please see the section on\narray location-dependent corrections (Section 4.5).\n\n7.2 Electronic Artifacts\n\n7.2.1 Saturation and Nonlinearity\n\nThe IRAC detector pixels are limited in the number of photons (actually, electrons) they can accurately\naccumulate and detect. Once this maximum number is reached, the detector pixel is \"saturated\" and\n\nData Features and Artifacts 108 Electronic Artifacts\nIRAC Instrument Handbook\n\nadditional photons will not result in an increase in read-out data numbers. Prior to this, the detector\nbecomes effectively less sensitive as more photons are received, an effect referred to as \u201cnon-linearity.\u201d\n\nThe saturation value varies slightly pixel-to-pixel, and substantially from detector to detector. The IRAC\nInSb (3.6 and 4.5 microns) detectors typically have saturation values of approximately 44,000 DN in the\nraw data. The Si:As detectors (5.8 and 8 microns) have saturation values closer to 52,000 DN. The IRAC\npipeline automatically detects pixels that exceed a pre-defined threshold and marks them in the data mask.\nUnfortunately, IRAC uses a Fowler-sampling scheme where the returned DN are the difference between a\nset of readouts at the end of the integration (signal reads) and a set at the beginning (pedestal reads). Thus,\nonce a pixel has saturated the signal reads, the DN for that saturated pixel will actually start to decrease,\nand as a result of this double-valued nature the DN value alone is not a reliable saturation indicator.\nExamining the images containing very bright sources is necessary in order to evaluate saturation based on\nthe observed spatial structure of the source. Very bright sources, for example, will appear to plateau or\neven develop a dark hole in the center. For point sources, a rough estimate of the flux in the saturated\npixels can be made by fitting the wings of the PSF to the linearized pixels in the BCD image. If the data\nwere taken in the high dynamic range mode, the IRAC pipeline will automatically identify pixels in the\nlong frametimes that are saturated based on the observed flux in the short frame times. The short frame\ntime data can then be used to recover saturation in the long frame time data (this is not done\nautomatically). This replacement is accurate to about 10% at the peak of bright sources as the ~ 0.1\narcsecond jitter of the telescope coupled with pixel phasing in channels 1 and 2 and charge diffusion in all\nchannels will cause the measured flux densities between short and long frames to vary.\n\nThe IRAC arrays are slightly nonlinear at all signal levels. At levels above 30,000 DN (in the Level 0 raw\ndata) the response is low by several percent. As part of pipeline processing, the data are linearized based\non ground calibrations (which have been verified in flight) of this effect. The BCD data are linear to\nbetter than 1% up to about 90% of full well, which is defined to be the level where we no longer can fully\nlinearize the data, and at which saturation, by definition, begins. Below 20% of full well the nonlinearity\nin the raw data is negligible.\n\nIn detail, there are four places in the electronics where a pixel may saturate: the detector diode, the unit\ncell source-follower in the Read-Out Integrated Circuit (ROIC), the output source-follower in the ROIC,\nand the analog-to-digital converter (ADC) in the warm electronics. In most cases, it is the ADC that\nsaturates first, at 0 or 65,535 units. ADC saturation produces a discontinuity in the second derivative of\nthe measured Fowler DN versus the flux. The other saturations are smooth, with no discontinuity. In the\nother cases, depending on the channel, the detector diode may saturate before or after the source-\nfollowers.\n\nIn principle, for any source for which we already know the spatial variation of its intrinsic surface\nbrightness, we can determine whether the pixel is above or below saturation, and therefore, its flux. In\npractice, we do not know the gains of the source-followers very well near saturation, nor do we know\nenough about the detector diode saturation, to make a good estimate of the flux. Therefore, we flag pixels\nwhich are above the range of our linearization correction.\n\nData Features and Artifacts 109 Electronic Artifacts\nIRAC Instrument Handbook\n\n7.2.2 Muxbleed (InSb)\n\nMultiplexer bleed, or \u201cmuxbleed,\" appears in IRAC channels 1 and 2 (3.6 and 4.5 \u00b5m). It looks like a\ndecaying trail of pixels, repeating every 4th column, with enhanced output level trailing a bright spot on\nthe same array row. The effect can wrap around to subsequent rows, but it does not wrap from the last\nrow to the first. Since columns are read simultaneously in groups of four, one for each mux output, the\nnext pixel read out on any single output is four pixels to the right, in array coordinates. As the BCDs for\nchannels 1 and 2 are flipped in the y-direction when compared to the raw images, the read direction is top\nto bottom for BCDs and muxbleed-triggering pixels will affect rows beneath the source. Muxbleed is\nusually accompanied by a pinstripe pattern (every 4th column) that may extend over part of the image\npreceding or following the pixel. It is caused by a slow relaxation of the mux following the momentary\ndisequilibrium induced when a bright pixel's voltage is placed on an output FET during pedestal and\nsignal reads. Although the pixel rise and fall times are fast (2.6 and 1.0 \u00b5sec, respectively) compared to\nthe 10 \u00b5sec time to clock the next pixel onto an output, longer relaxation times are involved for an output\nFET to fully recover after the voltage from a bright pixel is briefly impressed on its gate. The decaying\ntrail has a time constant of tens of \u00b5sec, and the pinstripe, tens of seconds. In BCDs produced by pipeline\nversions prior to S13, the pinstripe pattern from muxbleed was complicated by a de-striping step in the\npipeline in the darkdrift module. This often caused pinstriping to appear over an entire image. Beginning\nwith pipeline version S13, we turned off the de-pinstriping in channels 1, 2, and 4, but left it on for\nchannel 3.\n\nStars, hot pixels, and particle hits can generate muxbleed, and the characteristics of the pinstripe depend\non frame time and Fowler number. Hot pixels may show muxbleed in a raw image, but in the BCD the\nmuxbleed induced by hot pixels may not be present because it was canceled in either the labdark\nsubtraction or in the skydark subtraction. The pinstripe pattern is nearly constant in areas of a single\nimage that do not contain a saturating star, particle hit, or hot pixel. The characteristics of muxbleed from\nparticle hits depend on when the hit occurs within the frame.\n\nMuxbleed was characterized long before the launch of Spitzer, but it is reasonably well understood and it\nis fully corrected in the final IRAC pipeline. The pinstripe is strongest in channel 2, particularly in 12\nsecond frames. In channel 2 mosaics, even with overlap correction, there may appear to be bright and\ndark patches everywhere, about the size of one frame or part of a frame. Upon close inspection, though,\nindividual patches are revealed as areas of nearly constant pinstripe pattern that runs between the edges of\nthe array, bright stars, hot pixels, and particle hits. A systematic and automated pinstripe correction\nscheme has been implemented in the pipeline.\n\nData Features and Artifacts 110 Electronic Artifacts\nIRAC Instrument Handbook\n\nFigure 7.2: Images showing the muxbleed effect (the horizontal line on both si des of a bright stellar image).\nThe pixels on the left side of the bright source are pi xels on rows following the row in which the bright source\nwas located (and have wrapped around in the readout order of the array). The vertical (white) lines are due\nto the so-called \u201ccolumn pull-down\" effect. These are 12-second B CD frames in IRAC channel 1, taken from\nIRAC program pi d = 618, AORKEY = 6880000.\n\nThe amplitude of the effect decays as one moves away from the bright spot, and this decrease can be\nnicely described by a simple function. In general, the muxbleed decays rapidly within 5\u221210 reads and\nplateaus at a roughly constant value. The functional form of the muxbleed is frame time independent.\nHowever, the amplitude does not scale linearly with the flux at the brightest pixel or the integrated flux of\nthe triggering source, and this often leaves over\/undercorrection of muxbleed in BCD frames. For this\nreason, an additional muxbleed correction by fitting the functional form of the muxbleed pattern to the\nactual muxbleed incidence is performed after the BCD frame creation (i.e., CBCD frames) and this will\ncorrect muxbleed below the rms noise level of the image.\n\nFigure 7.3: Demonstration of the S18 pi peline muxbleed removal. The i mage on the left is before and the one\non the right is after the correction. These are First Look Survey channel 1 data, taken from AORKEY =\n\nData Features and Artifacts 111 Electronic Artifacts\nIRAC Instrument Handbook\n\n4958976. Note that the brightest star in the upper-left corner is heavily saturated and the current muxbleed\nscheme can correct muxbleed from a saturated source also.\n\nFigure 7.4: A typical bandwi dth effect trail in channel 4, in a 30 second frame. These data were taken from\nprogram pi d=1154, AORKEY = 13078016.\n\nAn example of the current (S18) muxbleed correction is shown in Figure 7.3. It can be seen that at least\ncosmetically the effect can be greatly reduced without introducing new artifacts. With an additional\ncorrection to residual muxbleed during the CBCD pipeline, resultant images should be nearly muxbleed\nfree.\n\n7.2.3 Bandwidth Effect (Si:As)\n\nThe bandwidth effect appears in IRAC channels 3 and 4 (5.8 and 8.0 \u00b5m). It looks like a decaying trail of\npixels 4, 8, and 12 columns to the right of a bright or saturated spot. Only in the most highly saturated\ncases is the effect visible 12 columns to the right. A typical case for a star is shown in Figure 7.4. The\neffect is due to the fact that inside the ROIC the maximum voltage slew rate is limited, so charge on the\noutput bus can not be drained fast enough for the output to settle to the value for a dark pixel that follows\na bright pixel, or vice versa, in the 10 \u00b5sec or 20 \u00b5sec at which times the next two pixels (4 and 8\ncolumns to the right) are read out. A smaller, additional delay comes from charging or discharging the\ncables from the array to the warm electronics. The effect is nonlinear except in the weakest cases. The\noutput FETs in the Si:As arrays do not have the long recovery time that causes the long muxbleed trails\nand pinstriping in the InSb arrays, in part because the voltage swings have the opposite sign. The\nbandwidth effect presumably affects the first two or three pixels read out after a bright pixel in the InSb\narrays as well, but for InSb, we have included the bandwidth effect as part of the overall \u201cmuxbleed\"\neffect. It is much better behaved in InSb because the voltage swings are smaller and the slew rates are\nfaster. A rare case which gives rise to a bizarre image is shown in Figure 7.5. Here, an extremely\nsaturated star saturates an area in the last 4 columns. The bandwidth effect appears in the first 12 columns,\nmaking it appear as if the right edge of the image was cut and pasted onto the left side of the image.\n\nData Features and Artifacts 112 Electronic Artifacts\nIRAC Instrument Handbook\n\nBecause of the details of the array clocking, part of the unsettled signal appears in both the same row and\nthe next row.\n\nFigure 7.5: The bandwi dth effect when a bright object is in the last 4 columns. IRC+10216, strongly\nsaturated, is just off the right si de of the channel 3 array. Even the filter ghost is saturated. The bandwi dth\neffect appears on the left si de of the array. These data were taken from program pi d = 124, AORKEY =\n5033216.\n\n7.2.4 Column Pull-Down\/Pull-Up\n\nWhen a bright star or cosmic ray on the array reaches a level of approximately 35,000 DN, there is a\nchange in the intensity of the column in which the signal is found. In channels 1 and 2, the intensity is\nreduced throughout the column (thus the term \u201ccolumn pull-down\"); see Figure 7.6. When the effect\noccurs, it shifts the intensities of the pixels above and below the position of the \u201cguilty\" source, within the\nsame column. This effect is limited to the brightest sources. The amplitude of the column pull-down does\nnot scale linearly with the flux of the source or the brightest pixel. The effect appears to be constant on\neither side of the source and algorithms which fit separate DC offsets above and below the source should\nbe effective. Cosmetic corrections are partially successful. One, provided by the GOODS Legacy team,\ntakes the median of each column, identifies columns that deviate from the local average by more than\nsome threshold, and then adds back in a constant to the apparently affected columns. The code does not\ncurrently work in fields with extended emission. A more general algorithm which estimates the \u201ctrue\" sky\nvalue for affected pixels and fits DC offsets is also available for observations of more structured emission.\nThis algorithm is implemented in the BCD pipeline.\n\nData Features and Artifacts 113 Electronic Artifacts\nIRAC Instrument Handbook\n\nFigure 7.6: IRAC channel 1 (left) and channel 2 (right) observati ons of a crowded fiel d wi th column pull-\ndown apparent from the brightest sources. Note that the brighter sources affect a larger number of columns.\nThese data were taken from program pi d = 613, AORKEY = 6801408.\n\n7.2.5 Row Pull-Up\n\nIn addition to muxbleed in channels 1 and 2, there may be electronic banding, which is manifest as a\npositive offset for rows that contain bright pixels. This effect is at least an order of magnitude smaller\nthan muxbleed. Electronic banding is more significant in channels 3 and 4 but it is not as significant as\nthe optical banding in those channels (see Section 7.3.2). The BCD pipeline mitigates against these\neffects. The algorithm finds instances of pull-up and banding and fits the DC offsets on either side of the\ntriggering source to them.\n\n7.2.6 Full-Array Pull-Up\n\nIn all four arrays, there is also an effect where an entire image is uniformly offset by some amount of\nDN\u2019s that is approximately proportional to the total flux or fluence integrated over the array. It is easily\nnoticed in a mosaic when overlap correction is turned off, and when the mosaic contains areas with and\nwithout strongly saturated stars. We call this effect \"full-array pull-up,\u201d but it is also known as \"droop\" to\nthe community of users of doped silicon IBC arrays. The effect can go unnoticed when overlap correction\nis done in the mosaic. It has no significant effect on aperture photometry of point sources or extended\nsources when a good background mean can be obtained within the same 5 arcmin x 5 arcmin image as the\nsource. The effect is largest in channels 3 and 4, and if uncorrected, can lead to significant errors in the\nderived flux of extended objects, and especially in the brightness of the background itself. It is hard to\n\nData Features and Artifacts 114 Electronic Artifacts\nIRAC Instrument Handbook\n\ndistinguish the effect from the internal scattering in channels 3 and 4. The IRAC pipeline does not correct\nthis effect.\n\n7.2.7 Inter-Channel Crosstalk\n\nWe have detected electronic crosstalk between channels only in the brightest sources that have been\nobserved. All four channels are read out simultaneously, except for the 100\/200-second frames in channel\n4, for which two\/four 50-second frames are taken instead of the long integrations in the other channels,\nbecause IRAC is background-limited in channel 4. When a source falls on a pixel of one array, crosstalk\nmay occur in the same pixel location in the other arrays, or in the next pixel read out. The crosstalk\nappears as a combination of either a positive or a negative offset in the same pixel and the derivative of\nthe signal in the same or previous pixel. As the source is dithered, the crosstalk follows it, and therefore\ncrosstalk appears in the mosaics. It is so weak that we have detected it so far only in channel 3, when the\nsource is in channels 2 and 4, and in channel 4, when the source is in channels 1 and 3. Figure 7.7 shows\nparts of the mosaics from the off-beams in a dithered observation of a very bright star. The star was\nobserved in channels 1 and 3 FOV first, so there are residual images in channels 1 and 3 from the bright\nstar. The residual images appear as a diffuse glow near the center. This glow is a combination of the\nresidual images of a very strongly saturated star observed with a Reuleaux dither pattern, thus effectively\nsmoothed by outlier rejection. The crosstalk appears in channels 3 and 4 as a partial dark ring with a\nbright core.\n\nFigure 7.7: Channels 1 and 2 (top) and 3 and 4 (bottom) showi ng inter-channel crosstalk (dark s pots near the\ncenter of the l ower panels ).\n\nData Features and Artifacts 115 Electronic Artifacts\nIRAC Instrument Handbook\n\n7.2.8 Persistent Images\n\nThe terms \u201cpersistent image\", \u201cresidual image\", and \u201dlatent image\" are used interchangeably to describe\nthe contamination of an IRAC image by a bright source from a previous exposure. When a pixel is\nilluminated, a small fraction of the photoelectrons become trapped. The traps have characteristic decay\nrates, and can release a hole or electron that accumulates on the integrating node long after the\nillumination has ceased. The warm mission short-term residual images are different in character than the\ncryogenic residuals, as the behavior of the trap populations is a function of the impurity type and array\ntemperature. During the cryogenic mission, in all arrays, the longest e-folding decay time is about 1000\nsec. For the warm mission, residuals are <0.01% of the fluence of the illuminating source after 60\nseconds.\n\nFor extremely bright sources, residuals are produced even when the source is not imaged on the array.\nResiduals at 3.6 and 4.5 microns can be produced during slews from one science target to another and\nfrom one dither position to the next. These slew residuals appear as linear features streaking across IRAC\nimages. Note that the pipeline cannot flag slew residuals, as there is no reasonable way of tracking the\nappearance of bright sources relative to the moving telescope pointing.\n\nObservations contaminated by residual images can often be corrected with the data themselves. If the\nobservations were well dithered, it is likely that the persistent image artifacts will be rejected as outliers\nwhen building the mosaic. Examining the median stack images that can be downloaded from the Spitzer\nHeritage Archive together with the data is can often be used to identify pixels that are affected by residual\nimages. Residual images can often be at least partially mitigated by subtracting the normalized median\nstack image (made with object and outlier rejection).\n\n7.2.8.1 Cryogenic Mission Persistent Images\n\nTests performed during the In-Orbit Checkout (IOC) revealed that there are both short-term persistent\nimages, with time scales of order minutes and which are present in all four arrays, and longer-term\npersistent images in channels 1 and 4. The short-term persistent images were known before the launch,\nand extensive calibrations and data analysis were performed to characterize them. The pipeline produces a\nmask (bit 10 of the imask) for each image that indicates whether a bright source seen by a previous\nexposure would have left a persistent image above three times the predicted noise in the present frame. To\nidentify persistent images in your own data, we recommend doing a visual search on a median combined\nstack.\n\nThe longer-term persistent images were discovered in flight. In channel 1, the persistent images are\ngenerated by stars as faint as K = 13 (in very long stares). They can be generated by any long dwell time\nwith a bright star on the array, whether or not the array is being read out. They were first noticed during a\nhigh-gain antenna downlink, when IRAC was left at a fixed position viewing the Galactic plane (by\nchance) for 45 minutes. The persistent images do not have the same size as a direct point source; they are\nsignificantly more diffuse (looking more like the logarithm of the point-spread function). The channel 1\nlong-term persistent images have time scales of order 6 hours, and they decay gradually. The cause of\n\nData Features and Artifacts 116 Electronic Artifacts\nIRAC Instrument Handbook\n\nthese persistent images has been identified as a known feature of the flight array (broken clamp) that\ncannot be fixed. The longer-term persistent images in channel 4 are induced by bright mid-infrared\nsources or bright stars. The channel 4 persistent images have very unusual properties: they have lasted for\nas long as two weeks, they can survive instrument power cycles, they do not decay gradually, and they\ncan switch sign, as they decay, from positive to negative. The amplitude and decay rate of long-term\npersistent images is variable and no secure model exists to remove these artifacts from the data.\n\nFigure 7.8: Medi an of channel 1 i mages from a calibrati on observati on performed after observing Pol aris.\nThe 5 bright s pots are persistent images from staring at the star while observi ng, while the set of criss-\ncrossing lines were generated by slews between the vari ous pointi ngs. These observations were taken from\nAORKEY=3835904, in program pi d=19.\n\nWe instituted a proactive and highly successful method of eliminating persistent images. Channels 1 and\n4 were temporarily heated, or \u201cannealed,\" briefly, with a small current running through the detector. The\narrays were annealed after every telemetry downlink, which erased any persistent images built up during\nthe downlink or during the previous 12-hour period of autonomous operations (PAO). This strategy,\ncombined with scheduling known bright object observations immediately before downlinks, greatly\ndecreased the possibility that preceding observations produced persistent images.\n\nWe have found that stars brighter than about magnitude \u22121 at 3.6 microns, when observed for more than\nabout 6 seconds, leave a residual image that persists through an anneal, and even through multiple\n\nData Features and Artifacts 117 Electronic Artifacts\nIRAC Instrument Handbook\n\nanneals. These latents from extremely bright objects are seldom visible in a mosaic of a science\nobservation, but they appear in skydarks and other median-filtered stacked images of longer science\nobservations. In channels 2, 3, and 4, all residual images are completely removed by a single anneal, and\nsince January 2006 we annealed all 4 arrays every 12 to 24 hours.\n\nFinally, we show an example of persistent images. Note that not all cases will be this obvious. In Figure\n7.8 we see not only residual images of the star Polaris, but also residual streaks left by Polaris as the\ntelescope moved between dither positions.\n\n7.2.8.2 Warm Mission Persistent Images\n\nChannel 1 and channel 2 have different persistent image responses in the warm mission data. There are no\nlong-term residual images that last for weeks, such as those seen in channel 4 data during the cryogenic\nmission. Channel 1 residual images last for minutes to hours, depending on the brightness of the original\nsource and the background levels in the subsequent images. Figure 7.9 shows this persistent image\nbehavior for a first magnitude star (data taken from PID 1318). The residual image decay is exponential\nin character, as expected for trapped electron decay rates. The decay rate is constant for all sources, so\nthat while residual images from brighter sources take longer to decay below the background level, all the\npersistent images decay at the same rate. These rates have been implemented for residual image flagging\nin the warm mission IRAC pipeline.\n\nA consequence of the intermediate-term (hours) residual images is that it is possible for observations from\na previous AOR to produce residual images. The residual image flagging module correspondingly tracks\nresiduals from one AOR to the next. Given the original brightness of the saturation-corrected source, and\nthe decay time calculated with the exponential decay rate, the pipeline flags all residual images until their\naperture fluxes are less than three times the background noise in each image. Each image in each AOR\nobserved is checked for residual images from all previous observations within the observing campaign.\n\nChannel 2 residual images decay much faster than those in channel 1, which last only a few minutes\n(<10) for even the brightest stars. Therefore, the pipeline flagging for channel 2 does not cross AORs.\nChannel 2 residuals start out as positive, but then become negative. The timing of the switch from\npositive to negative depends on the exposure time and brightness of the source.\n\nTable 7.3: Warm mission residual i mage durations.\n\nStar magnitude Channel 1 residual duration (hours) Channel 2 residual duration (hours)\n\n1 10 0.1\n2 7 < 0.1\n3 3.5 < 0.1\n4 1.5 < 0.1\n\nData Features and Artifacts 118 Electronic Artifacts\nIRAC Instrument Handbook\n\nThe arrays are not annealed during the warm mission as there is no evidence that annealing removes\nresidual images (the arrays currently operate at nearly the old annealing temperature), and all residual\nimages decay in a reasonably short time scale compared to those mitigated by annealing in the cryogenic\nmission.\n\nTable 7.3 gives a rough idea of warm mission latent durations. Durations should not be taken as exact\nbecause they also depend on the background levels in the images that will change from one AOR to the\nnext. This example comes from bright star observations in PID 1318 and starts with 12s observations of\nthe bright stars.\n\nFigure 7.9: Residual image brightness decay as a functi on of ti me i nterval since exposure to a first magnitude\nsource at 3.6 \u03bcm. The residual is compared to three times the noise in the sky background as measured in an\nequi valent aperture. The fitted exponential decay function is plotted as the dot-dashed line. These curves have\nbeen smoothed to mitigate flux jumps due to sources at the position of the original source in subsequent\nimages.\n\nData Features and Artifacts 119 Electronic Artifacts\nIRAC Instrument Handbook\n\n7.3 Optical Artifacts\n\n7.3.1 Stray Light from Array Covers\n\nStray or scattered light on the arrays can be produced by illuminating regions off the edges of the arrays.\nStray light from outside the IRAC fields of view is scattered into the active region of the IRAC detectors\nin all four channels. The problem is significantly worse in channels 1 and 2 than in channels 3 and 4.\nStray light has two implications for observers. First, patches of stray light can show up as spurious\nsources in the images. Second, background light, when scattered into the arrays, is manifest as additions\nto the flatfields when they are derived from observations of the sky. The scattered light is an additive, not\na multiplicative term, so this will result in incorrect photometry when the flatfield is divided into the data\nunless the scattered light is removed from the flat. Stars which fall into those regions which scatter light\ninto the detectors produce distinctive patterns of scattered light on the array. We have identified scattered\nlight avoidance zones in each channel where observers should avoid placing bright stars if their\nobservations are sensitive to scattered light.\n\nFigure 7.10: An image of the M51 system, showing an overlay of the IRAC fields of view, wi th the scattered\nlight origin zones for channels 1 and 2 overlai d.\n\nData Features and Artifacts 120 Optical Artifacts\nIRAC Instrument Handbook\n\nFigure 7.10 shows the zones for channels 1 and 2 with Spot overlays. Zones 1A, 1B, 2A and 2B (which\nproduce the strongest scattered light) typically scatter about 2% of the light from a star into a scattered\nlight \u201csplatter pattern\" which has a peak value of about 0.2% of the peak value of the star. Figure 7.11 to\nFigure 7.14 show examples of stray light in channels 1\u22124. Both point sources and the diffuse background\ngenerate stray or scattered light. Stray light due to the diffuse background is removed in the pipeline by\nassuming the source of illumination is uniform and has a brightness equal to the COBE\/DIRBE zodiacal\nlight model. This assumption is not true at low Galactic latitudes or through interstellar clouds, but in the\n3.6\u22128 \u00b5m wavelength range it is nearly correct. A scaled stray light template is subtracted from each\nimage, in both the science and calibration pipelines. Before this correction was implemented, diffuse stray\nlight from scattered zodiacal background contaminated the flats, which are derived from observations of\nhigh zodiacal background fields, and led to false photometric variations of 5%\u221210% in the portions of the\narray affected by stray light; this photometric error is now estimated to be less than 2%.\n\nFigure 7.11: Channel 1 i mage showing scattered light on both si des of a bright star. The scattered light\npatches are marked wi th whi te \u201cS\" letters. The images were taken from program PID 30 data.\n\nData Features and Artifacts 121 Optical Artifacts\nIRAC Instrument Handbook\n\nExample images of scattered light are shown here to alert you in case you see something similar in your\nIRAC images. The scattered light pattern from point sources is difficult to predict, and very difficult to\nmodel for removal. To the first order, you should not use data in which scattered light from point sources\nis expected to cover or appears to cover your scientific target. Stray light masking is done in the pipeline.\nThis procedure incorporates our best understanding of the stray light producing regions. The procedure\nupdates the corresponding imask for a BCD by determining whether a sufficiently bright star is in a stray\nlight-producing region. The 2MASS point source list is used to determine the bright star positions.\n\nFigure 7.12: Channel 2 i mage showing scattered light on one si de of a bright star. The scattered light patches\nare marked with white \u201cS\" letters. The i mages were taken from program PID 30 data.\n\nFigure 7.11 to Figure 7.14 are 201 pixels (4.1\u2019) square, and have been extracted from larger mosaics\nproduced from the IRAC GTO shallow survey (from program ID 30). This survey covers 9 square\ndegrees with three 30-second images at each position. Because the mosaics cover large areas, the star\ncausing the scattered light appears in many of the images. All of the sample images have the same array\norientation as the BCD images. The sample images are mosaics of a BCD that contains the stray light and\n\nData Features and Artifacts 122 Optical Artifacts\nIRAC Instrument Handbook\n\nthe BCD that contains the star that produces the stray light. Figure 7.11 and Figure 7.12 show scattered\nlight in the two short-wavelength channels, from zones 1A, 1B, 2A, and 2B. Figure 7.13 and Figure 7.14\nshow examples of scattered light in channels 3 and 4.\n\nBecause stars are much fainter in these channels, and the scattering geometry is much less favorable,\nthese scattered light spots are much less obvious than in the short-wavelength channels. Dithering by\nmore than a few pixels will take the bright star off the channel 3 and 4 \u201cscattering strip,\" so the scattered\nlight spots should be removed from mosaics made with adequately dithered data.\n\nFigure 7.13: Channel 3 i mage showing scattered light from a scattering strip around the edge of the array\nwhere a bright star is located. The scattered light patches are marked with white \u201cS\" letters. The i mages were\ntaken from program PID 30 data.\n\nData Features and Artifacts 123 Optical Artifacts\nIRAC Instrument Handbook\n\nPlease note that Figure 7.11 to Figure 7.14 were made with no outlier rejection. A dithered observation,\ncombined with outlier rejection, will have much reduced stray light. Further, a diligent data analyst who\nrecognizes and masks stray light in the individual BCDs will be able to eliminate stray light from well-\nplanned mosaics. Observations made with little or no redundancy, or with dithers on scales smaller than\nthe size of the stray light patches, will contain stray light and should be used with caution.\n\nFigure 7.14: Channel 4 i mages showing scattered light from a scattering stri p around the edge of the array\nwhere a bright star is located. The scattered light patches are poi nted to by black arrows. The i mages were\ntaken from program PID 30 data.\n\nData Features and Artifacts 124 Optical Artifacts\nIRAC Instrument Handbook\n\n7.3.2 Optical Banding and Internal Scattering\n\nThe banding effect manifests itself as the rows and columns that contain a bright source having an\nenhanced level of brightness. This happens only in the Si:As arrays and has been shown to be due to\ninternal optical scattering (inside the array). Both bright stellar sources and bright extended sources cause\nbanding. It is clearly different from the optical diffraction patterns and the column pull-down effect. The\nSSC pipeline corrects for banding, but it does not model the flaring of banding towards the edges of the\narray. Therefore, the pipeline correction is not always perfect.\n\nBanding only appears in IRAC channels 3 and 4 (5.8 and 8 micron bands), and it is stronger in channel 3.\nBanding probably occurs at all intensity levels, but only appears obvious around bright sources that are at\nor near saturation levels. Banding is seen both in row and column directions, though their relative\nintensities are somewhat different. In addition, there is an electronic effect. Channel 4 has a strong row\npull-up, and channel 3 has a weak column pull-up. The column pull-up is uniform across the row where\nthe source is bright. The optical banding intensity falls off with distance from the bright spot. Cosmic ray\nhits cause electronic banding, but not optical banding.\n\nFigure 7.15: Typical image sections showi ng the banding effect. These are channel 3 (left) and channel 4\n(right) i mages of the same object (S140), adopted from a report by R. Gutermuth. These data were taken\nfrom program pi d 1046, AORKEY 6624768.\n\nThe optical banding is only an enhancement of the optical scattering in channels 3 and 4 near the row and\ncolumn where the source is. Approximately 25% of the light incident from a point source is scattered\nthroughout the channel 3 array. The detected scattered light falls with distance from the source. Channel 4\nhas the same problem to a smaller degree. Laboratory tests have confirmed the large amount of optical\nscattering within the Si:As arrays. At wavelengths shorter than about 10 microns, the Si:As in the channel\n3 and 4 arrays is not opaque, and most of the incident photons, especially in channel 3, reach the front\nsurface of the detector chip, where they are diffracted by the rectilinear grid of conductive pads. Many are\ndiffracted into high angles and are multiply-reflected within the detector chip, and some can travel fully\nacross the array before being absorbed (and detected). Other photons can pass through the detector chip\nand be scattered back into the detector chip where they are detected. The interference pattern tends to\n\nData Features and Artifacts 125 Optical Artifacts\nIRAC Instrument Handbook\n\nconcentrate the scattered light along the rows and columns, causing the optical banding. The pattern is\ndue to interference that depends on wavelength and the spatial extent of the source at each wavelength.\nThe banding\/scattering pattern does not vary much for point sources with a continuous spectrum, but a\nnarrow-band source has a complex banding\/scattering pattern.\n\nUsers should be aware of the uncertainties resulting from banding, specifically when attempting\nmeasurements of faint sources near the affected rows or columns. For bright sources with significant\nbanding, aperture photometry may not be successful, and it would be better to measure these sources\nusing frames of shorter exposure times. Users are encouraged to experiment with image restoration\ntechniques of their choice. Algorithms similar to the pull-down corrector may have some effectiveness in\nmitigating banding.\n\n7.3.3 Optical Ghosts\n\nThere are three types of known or potential optical ghosts visible in IRAC images. The brightest and most\ncommon ghosts are produced by internal reflections within the filters. The first-order filter ghosts (one\npair of internal reflections) in channels 1 and 2 are triangular, and in the BCD images they appear above\nand\/or to the left of the star in channel 1, and above and\/or to the right of the star in channel 2. The\nchannel 1 first order filter ghost contains ~ 0.5% of the flux of the main PSF in channel 1, and the channel\n2 ghost ~ 0.8% of the flux of the channel 2 PSF. Because of the increase in the optical path length, ghost\nimages are not in focus. The separation between the main image and its ghost is roughly proportional to\nthe distance of the main image from the Spitzer optical axis in both Y and Z directions, i.e., (DeltaY ,\nDeltaZ) = (Ay y+ By , Azz+ Bz) where (y,z) are normalized coordinates in which the FPAs span the range\n[0,1] with the axes increasing away from the Spitzer optical axis, and the coefficients are as listed in\nTable 7.4 below. The +Y direction is in the IRAC (C)BCD +x direction and the +Z direction is in the\nIRAC (C)BCD \u2013y direction. The peak intensity of the ghost is roughly 0.05% of the (unsaturated) peak\nintensity of the star. The second-order filter ghosts (two pairs of internal reflections) are much fainter (~\n25% of the flux and ~ 6% of the surface brightness of the first order ghosts), rounder, larger, and about\ntwice as far away from the star. The separation between the star and its ghosts increases with distance\nfrom the optical axis of the telescope. The channel 3 and 4 filter ghosts appear as small crosses at a larger\ndistance, mostly to the left or right of the star, respectively. They are offset from the primary image by\napproximately (+36 pix, +2 pix) and (-36 pix,+2 pix) in the Spitzer (Y,Z) directions for channels 3 and 4\nrespectively. The Z-offset varies slightly with position on the array. The channel 3 and 4 filter ghosts\ncontain < 0.2% of the flux of the main PSF in these channels. The separation and orientation are different\nfrom channels 1 and 2 because of the different orientations of the filters. Examples of filter ghosts are\nshown in Figure 7.16.\n\nTable 7.4. Coefficients for channel 1 & 2 ghost l ocations.\n\nChannel Ay By Az Bz\n1 0.04351 0.00288 0.04761 0.00211\n2 0.04956 0.00105 0.04964 0.00387\n\nData Features and Artifacts 126 Optical Artifacts\nIRAC Instrument Handbook\n\nFigure 7.16: Filter and beams plitter g hosts.\n\nSimilar ghosts are created by internal reflections within the beamsplitters. These only affect channels 3\n\n\u0394y ~ -36 pixels relative to a bright star (Figure 7.16), but are often obscured by brighter \"banding\"\nand 4 which are transmitted through the beamsplitters. They appear as a very faint, short, horizontal bar at\n\nartifacts. They are slightly fainter than the filter ghosts.\n\nData Features and Artifacts 127 Optical Artifacts\nIRAC Instrument Handbook\n\nFigure 7.17: Pupil ghost in channel 2 from V416 Lac.\n\nThe faintest identified ghosts appear as images of the Spitzer entrance pupil, i.e., the primary mirror\nshadowed by the secondary and supports. These pupil ghosts are only found in channels 2 and 4, and\nrequire an extremely bright source (e.g., a first magnitude star in a 12-second frame) to be seen due to\ntheir low surface brightnesses. The pupil image is at a fixed location on the array (but in different\nlocations in different channels). However, the pupil image is only partially illuminated by a single source,\nand the portion of the image that is illuminated depends on the source position on the array. An example\nof a pupil ghost is shown in Figure 7.17. This figure also shows some fringes in between the pupil ghost\nand the star. It is not clear if the fringes are directly related to the ghost. The total flux in these ghosts is ~\n0.05%\u22120.5% of the total flux in the PSF.\n\nCurrently we have no model for the exact shape and brightnesses of the ghosts, but we expect to develop\nmodels in the future. However, because the relative locations of the ghosts do vary with position on the\narray, sufficiently large dithering can help reduce or eliminate their effects. The stray light masking\nsoftware also will flag the filter ghosts. The PRFs that we provide on our web pages include all the\nghosts, and the apertures used in calibrating channels 1 and 2 include the filter ghosts. In performing\nphotometry for channels 1 and 2, the filter ghosts should be included.\n\n.\n\nData Features and Artifacts 128 Optical Artifacts\nIRAC Instrument Handbook\n\n7.3.4 Large Stray Light Ring and Splotches\n\nThere is a faint ring of scattered light with a mean radius of about 23.7 arcminutes in channel 1 that is\nvisible around bright objects. There is also a slightly larger, fainter ring in channel 2. They were first\nnoticed in mosaics of a SWIRE field that was adjacent to the bright star Mira in PID 181 (see Figure\n7.18). The ring was visible in channel 1 and 2 mosaics. We verified that the ring was an artifact by\nobserving the field near Beta Gru where we observed pieces of the ring in the same places relative to the\nstar. Starlight that is specularly reflected off the telescope mirrors cannot enter the MIC directly when the\nstar is more than 16 arcminutes off the telescope boresight. IRAC's pupil stop is a little oversized, so the\nring is probably light that is once or twice diffusely scattered at areas outside the secondary and\/or near\nthe top of the primary conical baffle. The mean surface brightness of the ring in channel 1 is 4.5 x 10-10\n(\u00b130%) times the mean surface brightness of the center pixel of a pixel-centered point source. Presumably\nthere are stray light rings in channels 3 and 4, but they are too faint to see.\n\nFigure 7.18: Part of the channel 1 mosaic (from observations in PID 181; AORKEYs 5838336, 5838592,\n5839872 and 5840128) of the SWIRE fiel d near Mira showi ng the 24 arcminute radi us ring of stray light from\nthe telescope.\n\nData Features and Artifacts 129 Optical Artifacts\nIRAC Instrument Handbook\n\nThe \"splotches\" (see Figure 7.19) are areas of more concentrated stray light that appear when a bright\nsource is about 20\u221232 arcminutes off the center of the FOV, to the left or right in array coordinates. The\nsplotches were seen in the SWIRE field near Mira and in the Beta Gru tests in channels 1, 2 and 4. The\npresence or absence of a splotch is very sensitive to the position relative to the bright object \u2212 a bright\nsplotch can be present in one image and absent in an image with the telescope pointed a few pixels away.\nEven fainter splotches appeared in channel 2 about 1 degree away from Beta Gru, along the same\ndirections. We thank R. Arendt for providing us with most of the information that was presented in this\nsection.\n\nFigure 7.19: Channel 2 i mages from the SWIRE map showing stray light spl otches from Mira, which was\nabout 30 arcminutes away. Successive pairs of i mages were slightly dithered. The last pair is about 5\narcminutes from the first pair, but has a similar spl otch. Note the absence of any stray light in the second\nimage, though it was centered only a few pi xels away from the first image. The images are from PID 181,\nAORKEY 5838336; EXPID 187-192, 199, and 200.\n\n7.4 Cosmic Rays and Solar Protons\n\nThe SSC mosaicker, MOPEX, identifies energetic particle hits as follows. All pixels in BCDs that\ncontribute to a given pixel in the final mosaic are identified, and significant outliers (a user-specified\nnumber of sigmas above or below the filtered mean of all overlapping pixels of overlapping BCDs) are\nrejected. This method is very similar to the outlier rejection performed by shifting and adding ground\nbased images. The rejected pixels can be inspected in the \u201cRmask\" output files (one per input image).\nOutlier rejection in MOPEX can be adjusted. The parameters used in the online pipeline-generated\n\nData Features and Artifacts 130 Cosmic Rays and Solar Protons\nIRAC Instrument Handbook\n\nmosaics rely on multiple (\u22653) sightings of each sky pixel. In general, a coverage of at least five is\nnecessary to produce optimal results with the multi-frame (standard) outlier rejection.\n\nA special outlier rejection scheme can be used for sparse (2\u22124x) coverage; this \u201cdual outlier\" mode can be\nturned on using the namelist parameter file. Dual outlier rejection identifies pixels greater than a specified\nthreshold above the background, groups these pixels and adjacent pixels above a threshold into objects,\nand compares the object to objects in overlapping frames. If the object overlaps with objects in other\nframes (in celestial coordinates), then it is not a cosmic ray. If the object is not detected in a user-specified\nfraction of overlapping images, it is flagged as a cosmic ray. This information is written into the Rmask\nfiles used for mosaicking and source extraction. The dual outlier method should also be used in\nconjunction with the multi-frame outlier rejection method. Multi-frame rejection may throw out data\naround bright sources depending on the thresholding, due to pixel phase effects between BCDs. Using the\ndual outlier rejection and the REFINE_OUTLIER=1 option in MOPEX will prevent this.\n\nAdditionally, a single-frame radiation hit detector is run and produces bit 9 in the imask, but this bit is not\nused by the SSC post-BCD software and is not recommended because radiation hits cannot be uniquely\nseparated from real sources in single images.\n\nData Features and Artifacts 131 Cosmic Rays and Solar Protons\nIRAC Instrument Handbook\n\nFigure 7.20: The central 128x128 pixels of IRAC 12-second images taken on January 20, 2005 during a major\nsolar proton event. Channels 1 and 2 are top left and top right; channels 3 and 4 are bottom left and bottom\nright. Except for the bright star i n channels 1 and 3, al most every other source in these images is a cosmic\nray. These data are from observati ons in pi d 3126.\n\nCosmic rays for channels 3 and 4 are larger and affect more pixels than the channel 1 and 2 cosmic rays\ndue to the larger width of the active layer of the Si:As detectors. Some tuning of cosmic ray detection\nparameters may be necessary when working with deep integrations, especially for channels 3 and 4.\n\nEach IRAC array receives approximately 1.5 cosmic ray hits per second, with ~ 2 pixels per hit affected\nin channels 1 and 2, and ~ 6 pixels per hit affected in channels 3 and 4. The cosmic ray flux varies\nrandomly by up to a factor of a few over time scales of minutes but does not undergo increases larger than\n\nData Features and Artifacts 132 Cosmic Rays and Solar Protons\nIRAC Instrument Handbook\n\nthat. Also, the cosmic ray flux is normally about a factor of two higher on average around solar minimum\ncompared with solar maximum. Radiation hits do increase suddenly and dramatically during some major\nsolar proton events. Historically, several such events have occurred over the course of the active part of a\nSolar cycle.\n\nTwo major solar proton events occurred during IOC, so we have experience in identifying them and their\neffects. Because of shielding around the instruments, only extremely energetic protons (> 100 MeV) of\nany origin appear as cosmic ray hits in the data. Thus, many solar weather phenomena (\u201cstorms,\" etc.)\nwhich do occasionally affect other spacecraft, or ground systems, are not of concern to Spitzer.\n\nRadiation has very little effect on the IRAC arrays beyond elevating the counts in a given pixel. Some\nhigh energy cosmic rays cause persistent images, column pull-down, and muxbleed effects.\n\nData Features and Artifacts 133 Cosmic Rays and Solar Protons\nIRAC Instrument Handbook\n\n8 Introduction to Data Analysis\n\n8.1 Post-BCD Data Processing\n\nAll processing of IRAC data beyond the individual, calibrated, 256 x 256 images produced by the BCD\nscience pipeline is called \u201cpost-BCD.\" This includes combining all images from an observation (AOR)\ninto a mosaic, detecting sources, and any cosmetic corrections, e.g., cosmic ray hits, to the images\n(individual or mosaic) that are not based on understood instrumental artifacts or detector physics. Two\nimportant post-BCD processes are performed routinely by the pipeline and generate results that are placed\nin the science archive. These include pointing refinement, wherein a set of point sources are identified in\nthe images and are astrometrically matched to the 2MASS catalog, and mosaicking, wherein the\nindividual images in an AOR are robustly combined into celestial coordinate mosaics for each IRAC\nwaveband. The post-BCD processing can (and should) be performed in different ways for different\nobserving strategies and scientific goals. The post-BCD pipeline processing was performed with a\nspecific, conservative, set of parameters. Observers and archival researchers will very likely need to do\npost-BCD processing on their own. Most common will be generating mosaics from data in multiple\nAORs. Here we discuss some IRAC-specific issues. The post-BCD software consists of a series of\nmodules linked by Perl wrapper scripts and controlled by namelists. Namelists need to be placed in a\nsubdirectory called cdf and have filenames ending .nl. A (MOPEX) GUI is available as well. The\nnamelist controls which modules are called, contain the names of the input file lists and output directories,\nand detailed parameter sets for each module. Input file lists should not have any blank lines, otherwise the\nprograms will look for non-existent files. The following subsections deal with each part of the post-BCD\npipeline in turn, starting with pointing refinement, then overlap correction, mosaicking and finally point\nsource extraction.\n\n8.1.1 Pointing Refinement\n\nPointing refinement corrects the pointing of each frame to the 2MASS sky. In the pipeline, the pointing\nrefinement solutions for channels 1 and 2 are combined and applied to all four channels to produce the\ndefault pointing via the \u201csuperboresight.\" However, if they wish to try to improve on the supplied\npointing, users may rerun the pointing refinement themselves using scripts that come with the MOPEX\nsoftware package. Pointing refinement may not always be successful in channels 3 and 4, in which there\nare few 2MASS stars per image. Note that each run of pointing refinement overwrites any previous\nsolutions (in the header keywords RARFND, DECRFND etc), so users should make copies of the BCDs\nbefore rerunning the pointing refinement if they wish to retain the old corrections.\n\n8.1.2 Overlap Correction\n\nThe post-BCD software contains an overlap correction module which matches the background levels of\noverlapping frames in a mosaic. Generating new mosaics by running MOPEX with overlap correction\n\nIntroduction to Data Analysis 134 Post-BCD Data Processing\nIRAC Instrument Handbook\n\nturned on can remove the \u201cpatchiness\" often seen in mosaics due to bias fluctuations in the array (due\nboth to the first-frame effect and bright source effects).\n\n8.1.3 Mosaicking of IRAC Data\n\n8.1.3.1 Creating a Common Fiducial Frame\n\nNote that in the following we name the settings needed for the command line version of MOPEX.\nCorresponding values need to be set if using the MOPEX GUI.\n\nAs a first step in creating a mosaic, run the mosaicker with all the modules turned off except for the\nfiducial frame module (i.e., run fiducial image frame = 1 in the namelist), and include all the files you\nwish to mosaic from all four channels in the input list. This will generate the boundaries of the mosaic and\nwill allow all channels to be mosaicked onto the exact same grid. Rename this file to, e.g., FIF_all.tbl to\nprevent it being accidently overwritten. Then, when running the mosaicker, set the fiducial frame to the\nfile created by the fiducial frame module in this initial run using FIF_FILE_NAME = (path)\/FIF_all.tbl,\nand turn off the fiducial frame module (i.e., set run_fiducial_image_frame = 0 in the namelist). The use of\nthe common fiducial frame will ensure that the mosaics from all four channels will be accurately co-\naligned. The pixel scale is controlled by the MOSAIC_PIXEL_RATIO X\/Y parameters. Set Edge\nPadding = 100 to get a good border around the image. You can also specify CROTA2 for the output\nmosaic if you wish, or set CROTA2 = 'A' to get the smallest possible mosaic. The pixel size in the mosaic\nproduced by the final pipeline is exactly 0.6 arcsec x 0.6 arcsec (CDELT1, CDELT2 =\n\u00b10.000166666667).\n\n8.1.3.2 Outlier Rejection\n\nThe mosaicker has four outlier rejection strategies: single-frame outlier rejection, dual-outlier rejection,\nmulti-frame outlier rejection and box outlier rejection. For IRAC, the most useful are the dual outlier and\nmulti-frame rejections. Be sure to set THRESH_OPTION = 1 in the namelist in the multi-frame\n&MOSAICOUTLIERIN section. Setting the thresholds too low in the outlier modules can result in\nunwanted rejection of pixels in the cores of real objects. Users of these modules should carefully check\nthe coverage maps produced by the mosaicker to ensure that the centers of real objects are not being\nmasked out. The outlier rejection modules set bits in the rmasks. Bit 0 is set by the single-frame outlier\nrejection, bit 1 by the temporal (multi-frame) rejection, bit 2 by the dual outlier rejection and bit 3 by the\nbox outlier detection. Which rmask bits are used by the mosaicker is controlled by the\nUSE_BOX_OUTLIER_FOR_RMASK control which of the outlier detection modules are used. An\nback onto the input images to determine which pixels will be masked in the final mosaic. An rmask pixel\nis divided amongst the overlapping pixels in the input image. Input image pixels with projected values of\nthe rmask mosaic above RM_THRESH have the multi-frame outlier bit (1) set in their rmask. For\nchannels 1 and 2 a fairly high value (e.g., 0.5) can be used. The more diffuse radhits in channels 3 and 4\n\nIntroduction to Data Analysis 135 Post-BCD Data Processing\nIRAC Instrument Handbook\n\ncan be more effectively rejected by setting RM_THRESH to a lower value, e.g., 0.05, which has the\neffect of growing the rmask and thus rejects the more diffuse edges of the radhit.\n\nAll the outlier rejection modules require uncertainty images. The BCD uncertainty images are adequate\nfor this purpose, and the list of them should be specified with the SIGMALIST_FILE_NAME keyword in\nthe namelist. To use them set have_uncertainties = 1 and compute uncertainties internally= 0. To compute\nyour own uncertainty images, set compute_uncertainties_internally = 1, have_uncertainties = 0, and set\nthe appropriate values in the &SNESTIMATORIN section of the namelist (see the mosaicker\ndocumentation for more details). Outlier rejection creates another set of masks, the rmasks. These indicate\nthe pixels flagged by outlier rejection, and which are used by the mosaicker. A mosaic of the rmasks can\nbe provided by setting run_mosaic_rmask = 1. As a check on the outlier rejection, it is often helpful to\nexamine the coverage maps output by the mosaicker. If the outlier rejection has been over-zealous there\nwill be reductions in coverage at the positions of real sources in the mosaic. Blinking the mosaics and\ncoverage maps in, e.g., DS9 can thus be very helpful for determining whether the outlier rejection is set\nup correctly to reject only genuine outliers.\n\n8.1.3.3 Mosaicker Output Files\n\nThe output directory structure after running the mosaicker looks like: BoxOutlier, Coadd, DualOutlier,\nInterp, ReInterp, Combine, Medfilter, Rmask, Detect, Sigma, Dmask, Outlier with \u201c-mosaic\u201d appended to\nthese names, and the files in the output directory are FIF.tbl, header_list.tbl, and a namelist file with a\ndate stamp. The directory \u201cCombine\" contains the mosaic, mosaic.fits, a coverage map, mosaic_cov.fits\nand an uncertainty map mosaic_unc.fits.\n\n8.1.3.4 To Drizzle or Not to Drizzle?\n\nThe mosaicker has three interpolation options, set by the INTERP_METHOD keyword. The default is a\nlinear interpolation. Drizzling is available as an option, as is a grid interpolation (useful for creating quick\nmosaics if the PSF quality is not important). Our experience with the drizzle option suggests that it is\neffective when used on datasets with many dithers per sky position, and it can reduce the point-response\nfunction (PRF) width by 10% \u2013 20%, though at the expense of an unevenly-weighted image. The\ncoverage map produced by the mosaicker can be used to investigate the pixel-to-pixel variation in the\ncoverage of the drizzled image.\n\n8.1.3.5 Mosaicking Moving Targets\n\nAlthough Spitzer does track moving targets to a sub-pixel accuracy, the BCD pipeline only produces\nmosaics of IRAC data in fixed celestial coordinates. The user may opt to generate his or her own mosaic\nin a moving coordinate reference frame by setting the appropriate flags in MOPEX. The individual BCDs\nor CBCDs should be overlap-corrected first and then the mosaicker should be run with the flag\nMOVING_OBJECT_MOSAIC=1 set, using outlier rejection. Stars in the frames may be removed by\noutlier rejection, and the resultant composite of a moving target will be produced.\n\nIntroduction to Data Analysis 136 Post-BCD Data Processing\nIRAC Instrument Handbook\n\n8.1.4 Source Extraction\n\nThe Spitzer source extractor, APEX, may be used to fit point sources in IRAC data. This software can be\nrun in two modes, in a single frame mode (apex_1frame) which can be run on an individual BCD, CBCD\nor a mosaic, and in a multi-frame mode which uses the mosaic to detect sources, but the individual BCDs\nor CBCDs to measure their fluxes.\n\n8.1.4.1 Noise Estimation\n\nThe accuracy of the fluxes from APEX is very sensitive to the noise estimates, as these affect the fitted\nbackground value. For crowded fields, it is essential to include an estimate of the confusion noise\n(currently not included in the BCD uncertainty image). This can be estimated by measuring the difference\nbetween the actual image RMS and the estimated RMS in the uncertainty image, and then either adding it\nto the uncertainty image, or using it as the confusion noise value when generating uncertainty images with\nthe post-BCD software (see above).\n\n8.1.4.2 PRF Estimation\n\nThe PRFs released with MOPEX should be fairly good matches to the data and a significant improvement\non the previous versions. We do not recommend using the prf estimate tool to generate PRFs from the\nmosaics.\n\n8.1.4.3 Background Estimation\n\nThe namelist parameter Background_Fit controls the type of background used for PRF fitting. If you give\nBackground_Fit = 0, a median background is computed for the whole frame. A more accurate background\nestimate for PRF fitting, local to the source, can be generated by setting Background_Fit = 1. Note that\nthe aperture fluxes reported by APEX are always made using the median background, and hence may be\ninaccurate for faint sources.\n\n8.1.4.4 Source Extraction\n\nSource detection and extraction are controlled by the parameters Detection max\/min area and detection\nthreshold. APEX will frequently try to split bright sources into several components. This tendency can be\ncontrolled by setting the Max_Number_PS parameter in &SOURCESTIMATE to a low number (2\u20133).\nTwo files are output, extract raw.tbl contains all detections, and extract.tbl, which is a subset of extract\nraw.tbl containing the objects and fields which are selected by select conditions and select columns.\nSource extraction from the BCDs or CBCDs (multiframe mode) is recommended for IRAC data.\n\n8.1.4.5 Outlier Rejection\n\nBy default, APEX will not perform outlier rejection. This can be gotten around by running the mosaicker\nwith outlier rejection turned on and keeping the intermediate products (delete_intermediate files = 0).\n\nIntroduction to Data Analysis 137 Post-BCD Data Processing\nIRAC Instrument Handbook\n\nAppendix A. Pipeline History Log\n\nS18.18\n\n1. Saturation Correction in Pipeline\nSaturation is now corrected before artifacts. Artifact correction for saturated sources is now possible. The\ncriteria for selecting sources for correction were changed. The source selection is now frame time\ndependent. Bit 13 in imask is changed to bit 4 after correction has been performed.\nvarious bits that may have been set.\nFrames with ABADDATA set in raw file headers are now corrected. This problem is due to shifting the\ndata by one pixel after an extra word was read into the data. If only one such instance occurs in a frame it\nis now corrected. Frames with multiple instances are not corrected. The BCDs will include a header\npixels. If ZEROPIX = 1, then header keyword ZPIXPOS gives the position of the pixel that was fixed and\nheader keyword BADFILL gives the value (in DN) of the fixed pixel.\nBarycentric Julian Date calculated with SCLK precision is now included in header keyword BMJD_OBS.\nAlso, a new header keyword AORHDR has been added. This keyword is true if the entire AOR in which\nthe frame was taken (not just the frame itself, such as the first frame in every AOR) was taken in the high-\ndynamic-range (HDR) mode.\n\nS18.14\n\n1. Saturation Correction Update\nOnly sources in the 2MASS Point Source Catalog are now corrected (extended sources, such as the nuclei\nof bright galaxies are not corrected). Bit 13 in imasks is now flipped to zero after the saturation has been\ncorrected.\nMosaics of all the imasks for a given frame time in a given AOR are now produced by the pipeline and\nplaced in the PBCD directory (mmsk files).\nmore robust, include flagging of various artifacts that are not present in dmasks and make full use of the\nsaturation correction made by the pipeline.\n4. Higher Accuracy Pointing Refinement\n\nPipeline History Log 138\nIRAC Instrument Handbook\n\nPointing refinement is now done with the help of the 100x sampled PRFs, leading to a more accurate\npointing solution.\n\nS18.5\n\n1. A New Saturated Source Fitter\nThe pipeline now attempts to systematically find all saturated point sources in the images and fit them\nusing an appropriate PSF that is matched to the unsaturated wings of the source. The new module replaces\nthe saturated point source with an unsaturated point source that has the correct flux density of the point\nsource. Please note that the new module does not successfully fit super-saturated point sources or point\nsources that are too close to the edge of the field of view for proper fitting. Only the CBCD files are\nsaturation corrected (not the BCD files). The saturated pixels that have been replaced are identified within\nthe bimsk (imask) files (bit 13).\n2. A New Muxstripe Remover\nMuxstriping (due to very bright objects in channel 1 or 2 fields of view, usually exhibiting itself as a\ndepressed bias level in every fourth column in a section of an IRAC image) is now fit and corrected in a\nnew pipeline module. The noise of the affected pixel area is compared to the unaffected pixels in the\nframe and a deviation is removed, without changing the background level or the flux in the pixels.\nOccasionally this correction fails and the muxstriping is unchanged. Only the CBCD files include this\ncorrection, not the BCD files.\nSee the imask bit definition in Section 7.1 of the IRAC Instrument Handbook for the definition of the\nvarious bits.\n4. Mosaic Mask Files Now Available\nThe PBCD mosaics created in the pipeline now have associated mask files. The imasks associated with\nthe CBCDs that were used for the creation of the mosaic have been combined to create the mask file, so\nthe bit values can be deduced from the imask bit definition table.\n5. Improved First-Frame Correction\nMore appropriate skydarks are now used in the pipeline, producing an improved first-frame effect\ncorrection.\n\nS18.0\n\n1. Muxbleed Correction Update\nThe correction of the muxbleed effect in the BCD frames was updated. After extensive testing a new\nfunctional form and scaling law was developed for muxbleed correction in channels 1 and 2. The new\nfunctional form and scaling law correct muxbleed better than before.\n2. MOPEX Now Using the bimsk.fits Files\n\nPipeline History Log 139\nIRAC Instrument Handbook\n\nIn creating the final mosaic image, *maic.fits, the MOPEX pipeline now uses the *bimsk.fits files as\ninput, instead of the *dmsk.fits files. The *bimsk.fits files have more relevant information useful for\nflagging when building mosaics and analyzing the images.\nWCS CD matrix keywords were added to the Post-BCD file headers. The CDELT1, CDELT2, and\nCROTA2 keywords have been preserved but were placed in comments to avoid any confusion when\nhandled by astronomical software. The following keywords were added to the BCD, CBCD, and Post-\nBCD image files: PXSCAL1, and PXSCAL2, and PA. The keywords are the pixel scale along axis 1 and\naxis 2 in arcsec\/pixel and the position angle of axis 2 (East of North) in degrees. Keywords containing\nadditional information on the AOR mapping parameters have been added in a separate section of the\n\nS17.0\n1. Artifact Mitigation Within the Pipeline\nArtifact-mitigated images from the BCD pipeline and their associated uncertainty images (*cbcd.fits and\n*cbunc.fits) are now available in the archive. These images include corrections for column pulldown and\nbanding, induced by bright sources in the images. The corrections are empirical fits to the BCDs and may\nnot always improve the data quality. The standard BCD files (*bcd.fits) remain available in the archive.\nThe mosaics (post-BCD products) are now created from the *cbcd.fits images.\n2. Muxbleed Correction Updated Again\nThe muxbleed correction has been revised to include a better empirical fit.\n3. Two-Dimensional Subarray Images\nA two-dimensional image is now generated for each subarray BCD cube. Each pixel in the 2D image\n(*sub2d.fits) is a robust (outlier-rejected) mean of the 64 samples of the *bcd.fits cube. Two-dimensional\nmasks, uncertainty images, and coverage maps are now also provided.\n4. Artifacts Now Flagged For Subarray Images\nThe subarray imasks (*_bimsk.fits) now include masking for muxbleed, column pulldown and banding,\ninduced by bright sources. The updated masks can be used to mitigate bright source artifacts the same\nway as with the full array data.\n5. Darkdrift Values Written Out in the Subarray Header\nThe pipeline darkdrift module reduces a \"jailbar\" bias effect in the IRAC images. The values used for the\nreduction within the pipeline are included in the header of the full array data, and now in the header of the\nsubarray data for all the planes. This allows the user to remove the correction, if desired.\nInstrument Handbook, Section 6.\n\nS16.0\n1. Labdark Change for 100 Second HDR Data\n\nPipeline History Log 140\nIRAC Instrument Handbook\n\nWithin 100s AORs, the channel 4 observations are split into two 50s frames. The second 50s frame\nreceived an incorrect non-HDR 50s labdark instead of an HDR labdark. This was a minor problem, and\nhas now been corrected.\n2. Muxbleed Correction Updated\nThe module that detects and corrects the muxbleed caused by bright sources has been updated. It now\nperforms a more consistent and better correction than previously.\n3. Artifacts Flagged Within the Pipeline\nThe imasks (*_bimsk.fits) now include masking for muxbleed, column pulldown and banding induced by\nbright sources on images. The updated masks can be used with existing contributed software to mitigate\nbright source artifacts, and will be used in future versions of the IRAC pipeline after mitigation\nalgorithms have been implemented. In general, observers should not flag artifacts in mosaicking unless\nthey have observations at various roll angles.\n4. Pixel Linearization\nThe handling of bad and saturated pixels has been changed - they are in most cases now left with their\noriginal values, as opposed to being set equal to NaN. The method of flagging saturation in BCD mask\nfiles was changed, and now more accurately reflects the presence of saturation.\n\nS15.0\n1. Ghost Images And Scattered Light Flagged Within Pipeline\nPipeline versions of the ghost image and scattered light detection algorithms have been integrated into the\nIRAC pipeline. The modules use the location of bright sources upon the array (ghost image) or just\noutside the array, as found in 2MASS catalogs (scattered light) to predict possible optical ghosts and\nscattered light locations, and flag these pixels within the imask. The imask is an ancillary data product\nmaking mosaics etc.\n2. Incorrect Group Ids in Header to Be Fixed\nA bug that caused a small percentage of BCDs (< 0.1%) to have an unreadable header and which were\ntherefore not pipeline-processed, has been fixed. This should significantly decrease the number of missed\nBCDs in large mapping programs.\n\nS14.0\n1. Darkdrift Module Changes\nAs mentioned below, in S13 the darkdrift module was applied only to channel 3 data. This module is used\nto adjust the bias level in the four readouts in an array, thereby removing vertical striping in the data, the\nso-called \"jailbar effect.\" After S13 reprocessing of IRAC data it was found that the jailbar effect can be\ntriggered in channels 1, 2 and 4 as well. Therefore, the darkdrift module will again be applied to all four\nchannels, and all the IRAC data will be reprocessed with pipeline version S14.\n\nWe have released a \"jailbar corrector,\u201d which may be used to correct for the jailbar effect. It produces\nsimilar results to the darkdrift corrector module in the IRAC pipeline.\n\nPipeline History Log 141\nIRAC Instrument Handbook\n\nS13.0\n1. \u201cSuper-Boresight\u201d Pointing Refinement (S13.2 And Thereafter)\nPrevious versions of the pipeline performed pointing refinement on each IRAC channel separately. The\nrefinement was performed by matching detected point sources to 2MASS stars and registering the\nastrometry to minimize the positional offset between matches. In most cases, the refinement of channels 3\nand 4 is less accurate as the number of stars detected in an individual frame is less than in channels 1 and\n2. \u201cSuper-boresight\u201d refinement corrects the astrometry for all four channels by simultaneously using\nappropriately weighted matches from all four channels and the known orientations of the FPAs. This\nmethod can dramatically improve the pointing accuracy for channels 3+4, and it removes any positional\noffsets between channels. The superboresight pointing is inserted into the CRVAL1 and CRVAL2\nkeywords in the header, while the basic (less accurate) pointing refinement remains in RARFND and\nDECRFND header keywords, and the original boresight pointing solution is placed in new header\nkeywords ORIG_RA and ORIG_DEC.\n2. First-Frame Effect\nThe interval between frames (INTRFDLY) is now maintained in a database, instead of the pipeline\nreading the previous image in an AOR to process the current image. This streamlines operations and\nhandling of missing images. It is also placed in the header as a keyword.\n3. Linearity Correction\nNew linearity corrections have been calculated from on-orbit tests and small changes will be made to\nchannel 3 full array and all channel subarray data. The effect is roughly 2% at half-well, and 8% at 90%\nfull-well in channel 3. The other channels are within specifications and the linearity corrections will not\nbe changed for them.\n4. Darkdrift Module Changes\nSmall drifts in the bias level of each of the four readouts in each array, particularly relative to the\ncalibration labdarks, can produce a vertical striping called the \"jailbar\" effect. This is corrected in the\npipeline software by applying a constant offset per readout channel (arranged in columns), derived from\nthe median of those columns such that their arithmetic mean is zero. In other words, all readout channels\nare adjusted to a common additive bias level. In in-orbit tests, the mean offset and correction was found to\nbe negligible, except in channel 3 data. Therefore, in S13 reprocessing, the darkdrift correction was only\napplied to channel 3 data. The derived correction values for each channel are located in the header in the\nkeywords DRICORR1, DRICORR2, DRICORR3, and DRICORR4. The overall background term\ndetermined is DRIBKGND.\n5. Distortion Files\nThe subarray distortion files were found to be derived from the incorrect place on the full array and have\nnow been updated with correct ones. This should only make a small, but noticeable difference in pixel\nsizes when measuring relative separations in the subarray.\n6. Super-Skyflat\nA new \"super-skyflat\" has been derived from the first two years of flatfield data on IRAC and will be\nused as the flatfield for all reprocessing and further campaigns. Uncertainties in the pixel-to-pixel\nresponsivity calibration are only 0.5%, 0.2%, 0.2%, and 0.05% for channels 1\u22124, respectively.\n\nPipeline History Log 142\nIRAC Instrument Handbook\n\n7. Flux Conversion\nThe flux conversion has been updated to reflect the derivation described in the IRAC calibration paper.\nThe currently used numbers were from a nearly complete phase of this derivation, but different by 3% in\nch 4.\n1. Median brightness of Calibration Skydark (SKYDKMED)\n2. More Precise Start time of observation (SCLK_OBS).\n\nS12.0\nSince S11.0 there have been no significant changes to the IRAC pipeline affecting calibration. The\n1. New observing mode: \u201cStellar Mode\u201d has multiple full-array short time exposures within channel\n1 and 2 and at the same time has a longer integration in channels 3 and 4. This allows for brighter\nobjects to be observed in the longer wavelength channels to higher signal-to-noise without\nsaturating in the shorter wavelength observations. Available frame times are 0.4\/2 sec, 2x2\/12 sec\nand 2x12 sec\/30 sec. The first number(s) refer to channels 1 and 2, the last number to channels 3\nand 4.\n2. The median value of the frames used to create the skydark subtracted from the data will be placed\nin the header of the BCD: keyword SKYDKMED.\n3. The name of the labdark subtracted from the data will be placed in the header: keyword\nLBDRKFLE.\n4. The time of the observation (SCLK_OBS) will be computed using telemetry only to allow for a\nmore exact timing. This keyword will be placed in the database and header. Further S13 changes\nwill include calculating the first frame correction from this more exact timing.\n5. Keywords PTGDIFFX, PTGDIFFY were inserted to refer to the pointing differences in actual\npixels along the X & Y axis.\n\nS11.0\n\n1. The EQUINOX header keyword for BCDs has been fixed.\n2. Other changes to the header include the addition of the First Frame Delay and Immediate Delay\n(FFDLAY & IMMDLAY) times, calculated from the first frame correction.\n3. Previously, a DCE with a non-zero CHECKSUM from MIPL was not allowed to process through\nthe pipeline. In S11, the CHECKSUM will now be reported within the header and the DCE\nprocessed.\n4. The first frame correction has been fixed for the high-dynamic-range observations. The only\nremaining bug is for the intermediate frame times (12 sec) when used as part of an HDR frameset.\nThis effect will not be noticeable except as a slight background DC-level offset from frame to\nframe in the 12 sec data as part of 100s or 200s HDR framesets.\n5. After study of last year's worth of flat-fields and finding no noticeable change from campaign to\ncampaign, a super skyflat has been composed of last year's worth of observations. A sub-array flat\nhas been composed of this super skyflat, and both have been loaded into the pipeline.\n\nPipeline History Log 143\nIRAC Instrument Handbook\n\n6. Overlap correction is now applied in the post-BCD pipeline.\n7. The mosaic image headers have been populated with more keywords.\n\nS10.5\n1. Updated the ffcorr module to use the correct delay time between frames for full array non-HDR\nframes. The HDR frames will be fixed in S11.\n\nS10.0\n\n1. New linearity model in channel 4 (full and sub). Change from quadratic to cubic (actually updated\nin S9.5.2).\n2. Module ffcorr set to output only one plane for interpolated correction image rather than all planes.\n3. FITS Keyword: Create and populate new FITS header keyword (DS_IDENT).\n4. Update to readnoise in initial noise image.\n5. If BCD pixel = NaN, uncertainty pixel = 0.\n6. Keyword from dark ensemble placed in BCD header (SKYDRKZB; skydark zodiacal background\n\nS9.5\n\n1. Addition of two fields, hdrmode and numrepeats, to caldata tables. Requires a backfill script to\ntransform and migrate current fallbacks and metadata to new tables. The HDRMODE field is in\ncurrent use. The NUMREPEATS field is to facilitate use of the external repeat number in future\ncalibration activities.\n2. In S9.5 the flux conversion will be delivered in an IPAC table, for example:\n\\char Comment Calibration data file for dntoflux module.\n\\char INSTRUME = 'IRAC'\n\\int CHNLNUM = 4\n\\char fluxconv = 'Conversion factor in MJy\/sr per DN\/s'\n\\char fluxconvunc = 'Uncertainty in fluxconv'\n|fluxconv |fluxconvunc |\n|float |float |\n0.195 0.020\n3. HDR skydarks are now delineated from non-HDR skydarks. Skydarks are now aware of channel\n4 repeats and pipelines fetch skydarks for the correct repeat. This is possible due to new fields in\nthe caldata tables.\n4. Scattered light removal module (\u201cslremove\u201d) added to science pipeline and calibration\npreprocessing.\n5. Calibration ensemble pipelines now use \u201cfpgen\u201d to clean up the product header.\n6. New pipeline to create subarray flats from full array flats.\n\nPipeline History Log 144\nIRAC Instrument Handbook\n\n7. Latent ensemble creates new request median and request average images.\n8. New keywords to be added to the mosaic header: AOT_TYPE, AORLABEL, FOVID,\nFOVNAME, PRIMEARR, OBJECT, PAONUM, CAMPAIGN.\n\nS9.1\n\nLess than < 0.1% of the DCEs may not have pointing reconstruction applied to the data. BCDs with\nUSEDBPHF=F indicate that the Boresight Pointing History File was not used, and the RA and DEC\nin the headers for these cases are based on pre-observation predictions which can be off by 5\u201d\u221250\".\nDo not use such data if pointing is important.\n\nS9.0\n\n1. Some AORs have been affected by long-term residual images from previous observations. For the\nmost part, observers have sufficiently dithered their data, so that the impact is minimal, on\n2. Note that the noise in the images and the sensitivity to point sources are not equal to our pre-\nlaunch predictions (e.g., as available from our website until December 19, 2003, or in the\nObserver's Manual versions before 4.0), although they are close. New sensitivity numbers are\navailable in the revised Observer's Manual (version 4.0), which was available at our website\nstarting ~December 19, 2003. For reference, the ratio of the new point source detection threshold\nto the pre-launch advertised value, for low background observations in 30 sec frames, is 0.69,\n0.75, 1.60, and 1.31 in channels 1, 2, 3, and 4, respectively.\n3. Persistent images in channel 1. When a bright source (K=13 mag or brighter) is stared at for a\nlong time, for example, during a downlink, it will leave a persistent image in channel 1 that\ndecays very slowly (persists for several hours or more). A persistent image mitigation strategy\ninvolving annealing the array after downlinks has been put in place for nominal operations. These\nanneals will erase the persistent images from the array, but do not protect against persistent\nimages from bright object observations that can accumulate on the array before the next\ndownlink. Science impact: left unmitigated, you will have extra, spurious sources in your image.\nThese sources have a PSF that is wider than the actual true source PSF. Dithering helps to get rid\nof these spurious sources.\n4. Persistent images in channel 4. These are different in nature from the channel 1 persistent images.\nA bright source leaves a persistent image that can last for more than a week and even through\nIRAC power cycles. These images keep building up on the array. However, the amplitude of the\npersistent images is rather low. Annealing has been found to erase also the channel 4 persistent\nimages. Therefore, we will anneal both channels 1 and 4 simultaneously, every 12 hours (after\neach downlink), to erase persistent images. Again, dithering helps to get rid of these spurious\nimages.\n5. Diffuse stray light: All IRAC images contain a stray light pattern, resembling a \"butterfly\" in\nchannels 1 and 2, and a \"tic-tac-toe\" board in channels 3 and 4. These artifacts are due to zodiacal\nlight scattered onto the arrays, possibly reflected from a hole in the FPA covers above the channel\n\nPipeline History Log 145\nIRAC Instrument Handbook\n\n1 and 2 arrays, and from reflective surfaces outside the edges of channel 3 and 4 arrays. The stray\nlight scales with zodiacal light, which is the light source for our flatfields, so the stray pattern\ncontaminates the flats. As a result, the flatfields will aesthetically remove the stray light rather\nwell from images but will induce systematic errors of approximately 5% in flux calibration for\npoint sources that fall in the peak stray light location. Dithering will mitigate this effect, because\nit is unlikely that a dithered observation will keep a source within the stray light lobes. Diffuse\nstray light will be removed from both the flatfields and the science frames in a future version of\nthe pipeline.\n6. Stray light from point sources. Spot allows you to overlay stray light boxes on any image; if a\nbright star is placed in those boxes during an observation, a scattered light patch will appear on\nthe array. We have found three more such boxes during testing, in channels 1 and 2. The new\nstray light boxes are included in Spot now and are also shown in the new Observer's Manual.\nChannels 3 and 4 have less stray light, and the stray light inducing regions are not the same as the\nones we guessed (by analogy to channels 1 and 2) from the lab tests, so the channel 3 and 4 boxes\nwere removed from Spot. In channels 3 and 4 the stray light arises when a star lands on a thin\nregion just outside the array (the same region that causes the \"tic-tac-toe\" pattern from diffuse\nstray light in flat fields). A redundant observing strategy will help eliminate stray light problems.\nObservers covering fields with bright sources should inspect the individual images; this is\nrequired if the depth of coverage is less than 3, to identify spurious spots and rays that could be\nmistaken for real astronomical objects.\n7. Dark spots on pick-up mirror. There is contamination on the mirror, which causes a dark spot\nabout 10 pixels wide in channels 2 and 4. This is a 15% effect. Flatfields completely correct for\nthis feature in the data.\n8. Muxbleed. We have a correction algorithm, but the coefficients need fine-tuning. Furthermore,\nfor bright sources, muxbleed does not scale linearly with source brightness, so even a\nsophisticated algorithm cannot accurately remove it. Some experiments at fitting the muxbleed\nfor bright sources indicate that the decay pattern is always the same, and only the amplitude\nappears to be variable.\n9. Banding and column pulldown. A bright source on the array will cause its column to be pulled\ndown by a small amount. An algorithm to cosmetically correct the images for column pulldown\nhas been developed and is being tested. This appears to be an additive effect. An analogous effect\nfor an extremely bright source is that the entire image appears to have a different DC level from\nthe preceding and following images.\n\nS8.9\n\n1. Some AORs have been affected by long term residual images from previous observations.\nFor the most part, observers have sufficiently dithered so that the impact is minimal, on\n2. Note that the noise in the images and the sensitivity to point sources are not equal to our\npre-launch predictions (e.g., as available from our website until December 19, or in the\nObserver's Manual versions before 4.0), although they are close. New sensitivity numbers\nare available in the revised Observer's Manual (version 4.0), which was available at our\n\nPipeline History Log 146\nIRAC Instrument Handbook\n\nwebsite starting ~December 19, 2003. For reference, the ratio of the new point source\ndetection threshold to the pre-launch advertised value, for low background observations in\n30 sec frames, is 0.69, 0.75, 1.60, and 1.31 in channels 1, 2, 3, and 4, respectively. The\napparent modest decrease in sensitivity in channels 3 and 4 is under investigation.\n3. Persistent images in channel 1. When a bright source (K=13 mag or brighter) is stared at\nfor a long time, for example, during a downlink, it will leave a persistent image in channel\n1 that decays very slowly (persists for several hours or more). A persistent image\nmitigation strategy involving annealing the array after downlinks has been put in place for\nnominal operations. These anneals will erase the persistent images from the array, but do\nnot protect against persistent images from bright object observations that can accumulate\non the array before the next downlink. Science impact: left unmitigated, you will have\nextra, spurious sources in your image. These sources have a PSF that is wider than the\nactual true source PSF. Dithering helps to get rid of these spurious sources.\n4. Persistent images in channel 4. These are different in nature from the channel 1 persistent\nimages. A bright source leaves a persistent image that can last for more than a week and\neven through IRAC power cycles. These images keep building up on the array. However,\nthe amplitude of the persistent images is rather low. Annealing has been found to erase also\nthe channel 4 persistent images. Therefore, we will anneal both channels 1 and 4\nsimultaneously, every 12 hours (after each downlink), to erase persistent images. Again,\ndithering helps to get rid of these spurious images.\n5. Diffuse stray light: All IRAC images contain a stray light pattern, resembling a \"butterfly\"\nin channels 1 and 2, and a \"tic-tac-toe\" board in channels 3 and 4. These artifacts are due to\nzodiacal light scattered onto the arrays, possibly reflected from a hole in the FPA covers\nabove the channel 1 and 2 arrays, and from reflective surfaces outside the edges of channel\n3 and 4 arrays. The stray light scales with zodiacal light, which is the light source for our\nflatfields, so the stray pattern contaminates the flats. As a result, the flatfields will\naesthetically remove the stray light rather well from images but will induce systematic\nerrors of approximately 5% in flux calibration for point sources that fall in the peak stray\nlight location. Dithering will mitigate this effect, because it is unlikely that a dithered\nobservation will keep a source within the stray light lobes. Diffuse stray light will be\nremoved from both the flatfields and the science frames in a future version of the pipeline.\n6. Stray light from point sources. Spot allows you to overlay stray light boxes on any image;\nif a bright star is placed in those boxes during an observation, a scattered light patch will\nappear on the array. We have found three more such boxes during testing, in channels 1\nand 2. The new stray light boxes are included in Spot now and are also shown in the new\nObserver's Manual. Channels 3 and 4 have less stray light, and the stray light inducing\nregions are not the same as the ones we guessed (by analogy to channels 1 and 2) from the\nlab tests, so the channel 3 and 4 boxes were removed from Spot. In channels 3 and 4 the\nstray light arises when a star lands on a thin region just outside the array (the same region\nthat causes the \"tic-tac-toe\" pattern from diffuse stray light in flat fields). A redundant\nobserving strategy will help eliminate stray light problems. Observers covering fields with\nbright sources should inspect the individual images; this is required if the depth of coverage\nis less than 3, to identify spurious spots and rays that could be mistaken for real\nastronomical objects.\n\nPipeline History Log 147\nIRAC Instrument Handbook\n\n7. Dark spots on pick-up mirror. There is contamination on the mirror, which causes a dark\nspot about 10 pixels wide in channels 2 and 4. This is a 15% effect. Flatfields completely\ncorrect for this feature in the data.\n8. Muxbleed. We have a correction algorithm, but the coefficients need fine-tuning.\nFurthermore, for bright sources, muxbleed does not scale linearly with source brightness,\nso even a sophisticated algorithm cannot accurately remove it. Some experiments at fitting\nthe muxbleed for bright sources indicate that the decay pattern is always the same, and only\nthe amplitude appears to be variable.\n9. Banding and column pulldown. A bright source on the array will cause its column to be\npulled down by a small amount. An algorithm to cosmetically correct the images for\ncolumn pulldown has been developed and is being tested. This appears to be an additive\neffect. An analogous effect for an extremely bright source is that the entire image appears\nto have a different DC level from the preceding and following images. The physical origin\nof these effects and the probably related (and already known) banding effect is not yet\nunderstood. This work is in progress.\n10. Mosaics produced by the online pipeline for HDR mode data incorrectly weight the short\nand long frame times. For long exposures (> 12s), data are effectively taken in HDR mode,\nand hence the pipeline produced mosaics will not be very useful.\n11. Cosmic ray rejection is not functioning well.\n\nPipeline History Log 148\nIRAC Instrument Handbook\n\nAppendix B. Performing Photometry on IRAC Images\nThis is a quick guide for performing point source photometry on IRAC images.\n\nA. Point Source Photometry on a Mosaic\nIf you are only interested in photometry down to about 10% accuracy and have bright point sources, you\ncan usually perform photometry on the pipeline mosaic. Set the aperture size to 10 pixels and the sky\nannulus to between 12 and 20 pixels. The IRAC calibration is based on an aperture of this size, so for this\naperture no aperture correction is necessary. For fainter stars, it is better to use a smaller aperture and then\napply an aperture correction. Remember that the units of the images are in MJy\/sr, so you need to convert\nyour measured values into flux density units in micro-Jy, by accounting for the pixel size in steradians.\nConversion into magnitudes is magnitudes = \u22122.5*log10(f\/f(0)), where f is your measured flux density\nand f(0) is the zero magnitude flux density. If using software such as \"phot\" or \"qphot\" in\nIRAF\/DAOPHOT which requires a magnitude zeropoint, the \"zmag\" keyword in photpars should be set\nto 18.80 (ch1), 18.32 (ch2), 17.83 (ch3) and 17.20 (ch4) if using a mosaic pixel scale of 0.6 arcsec\/pixel.\nOther zmag values will be needed for other pixel sizes. Note that if you require photometry to a higher\naccuracy than 10% \u2013 20%, you should follow the steps listed below.\nExamine your data (CBCDs) and identify artifacts that could affect your photometry and that need to be\ncorrected.\nFirst perform artifact mitigation on the pipeline-produced CBCDs. While the pipeline-reduced CBCD\nfiles are mostly artifact-free, some residual artifacts remain.For example, the pipeline and contributed\nsoftware have difficulty recognizing very saturated pixels that produce artifacts. As a result, they will not\nusually correct artifacts from very saturated point sources or from extended saturated regions. Data at 5.8\nand 8.0 microns exhibiting the bandwidth effect should be masked before performing photometry.\nMake a mosaic of artifact-corrected images, for example with the MOPEX package. When creating the\nmosaic, the overlap correction option should be used in MOPEX, most importantly in channels 3 and 4, to\nmatch the backgrounds. Inspect the mosaic to confirm that outlier rejection is acceptable. If not, then\nremosaic with more appropriate MOPEX parameters. Comparing mosaics of adjacent channels on a per-\npixel basis will readily identify if outliers remain in a mosaic. The mosaic coverage maps should be\ninspected to verify that the outlier rejection has not preferentially removed data from actual sources. If the\ncoverage map systematically shows lower weights on actual sources, then the rejection is too aggressive\nand should be redone.\nIf you are interested in blue point sources (sources with spectral energy distributions, SEDs, that decline\ntoward the longer wavelength IRAC passbands) you should create an array location-dependent\nphotometric correction image mosaic. If you are interested in only red sources (with SEDs that rise\ntoward the longer wavelength IRAC passbands), you do not need to apply the photometric correction\nimages and make a mosaic out of them. We recommend making a correction mosaic, instead of\nmultiplying the correction images with the CBCDs and then mosaicking these CBCDs together, since you\nmay need to iterate this a few times and\/or you may have both red and blue sources in the field, and thus\nthe correction only applies to a subset of sources. This location-dependent effect is as large as 10%. It is\nthe dominant source of uncertainty in the photometry of IRAC images. For observations that well sample\nthe array for each sky position the effect will average out. MOPEX software now is capable of creating\nthese correction mosaics for you. If you want to make the BCD-matched photometric correction images\n\nPerforming Photometry on IRAC 149\nImages\nIRAC Instrument Handbook\n\nyourself, first copy the FITS header keywords CTYPE1, CTYPE2, CRPIX1, CRPIX2, CRVAL1,\nCRVAL2, CD1_1, CD1_2, CD2_1, CD2_2 from the headers of the BCDs to the headers of the\nphotometric correction images in each channel using your favorite FITS manipulation software. Thus,\nyou make the same number of photometric correction images (otherwise identical except for the keyword\ninformation) as there are CBCDs in each channel. The correction images must be divided by the pixel\nsolid angle correction images before mosaicking them together, because the pixel solid angle effect is\nessentially corrected for already in the photometric correction images and thus needs to be \"canceled out\"\nbefore running the images through MOPEX (which corrects for this effect). Then, copy the namelist you\nused to make the CBCD mosaic images into some other name, and edit the namelist to disable all the\noutlier rejection modules. Do not run the fiducial image frame module but instead point MOPEX to the\nexisting \"FIF.tbl\" file used for generating the corresponding CBCD mosaic. Next, specify the\nRMASK_LIST file (generate a file listing the rmasks and their path, as created by the mosaicker run for\nthe corresponding CBCDs). Finally, make the correction image mosaic with MOPEX.\nPerform photometry with your favorite software. Aperture photometry is preferred over PRF-fitting\nphotometry due to the undersampled nature of the data. To properly estimate the uncertainties in your\nphotometry, the uncertainty images provided with the CBCDs can be used and mosaicked into an\nuncertainty mosaic. The CBCD uncertainties are slightly conservative as they take into account the\nuncertainties in each pipeline calibration step. For packages that estimate noise directly from the data\nassuming Poisson noise, you can convert the mosaic into electron units, so as to calculate the uncertainty\ndue to source shot noise and background correctly. The conversion from MJy\/sr is \u2217GAIN \u2217 EXPTIME \/\nFLUXCONV where GAIN, EXPTIME and FLUXCONV are the keywords from the CBCD header. In\ndetermining the noise, the coverage of the observation at the position of your target should also be taken\ninto account (e.g., by entering the correct number of frames in DAOPHOT or by dividing the noise by the\nsquare root of coverage, from the coverage mosaic at the position of each target). Your aperture\nphotometry software should of course subtract the appropriate background (usually in an annulus around\nthe source).\nApply aperture correction, found in Chapter 4 of this handbook, if you perform aperture photometry in an\naperture different from the 10 pixel radius aperture used for IRAC calibration or determine the\nbackground by other means than an annulus. Observers can determine their own aperture corrections by\nphotometry to that published in the IRAC Calibration Paper (Reach et al. 2005, [23]).\nObservers should apply the array location-dependent photometric correction for blue sources and the\nappropriate color correction for all sources (based on the spectral energy distribution of the source).\nDetermine the array location-dependent photometric correction (for blue compact sources) from the\ncorrection mosaic, constructed in step 5 above, by looking at the values of the pixels at the positions of\nthe peaks of your point sources. Apply a color correction from Chapter 4 of this handbook using the\ntabulated values, if appropriate, or calculate the color correction for a source spectral energy distribution\nas done in that chapter. To calculate a color correction, you will need the IRAC spectral response curves,\navailable in the IRAC web pages. Color corrections are typically a few percent for stellar and blackbody\nsources, but can be more significant for sources with ISM-like source functions (50% \u2013 250% depending\non spectrum and passband). Measured flux density is the flux density at the effective wavelength of the\narray: 3.550, 4.493, 5.731 and 7.872 microns, for channels 1\u20134, respectively.\nA pixel phase correction to the measured channel 1 flux densities should then be considered. More\ninformation on the pixel phase correction can be found in Chapter 4 of this Handbook. This effect is as\n\nPerforming Photometry on IRAC 150\nImages\nIRAC Instrument Handbook\n\nlarge as 4% peak-to-peak at 3.6 microns and < 1% at 4.5 microns. To apply a correction for mosaicked\ndata is difficult as the pixel phase correction depends on the placement of the source centroid on each\nCBCD. For well-sampled data the pixel phase should average out for the mosaic. For precise photometry\nin low coverage data, the source centroids on the CBCDs should be measured and the phase corrections\naveraged together and applied to the final source photometry.\n\nB. Point Source Photometry on Individual BCDs\n\nAlthough most of the time it is a good idea to use the mosaic for performing photometry, performing\nphotometry on the (C)BCD stack is important for variability studies and can be useful for faint sources as\none can measure N out of M statistics (how many times you found the source). When performing source\nprofile fitting, performing photometry on the the (C)BCD stack is better as the phase information of the\nPRF is preserved.\n\nExamine your data (CBCDs) and identify artifacts that could affect your photometry and that need to be\ncorrected.\nFirst perform artifact mitigation on the pipeline-produced CBCDs. While the pipeline-reduced CBCD\nfiles are mostly artifact-free, some residual artifacts remain. The pipeline and contributed software have\ndifficultly recognizing very saturated pixels that produce artifacts. As a result they will not usually correct\nartifacts from very saturated point sources and extended saturated regions. Data at 5.8 and 8.0 microns\nexhibiting the bandwidth effect should be masked. If performing aperture photometry on the CBCDs, a\nparticular CBCD should not be used for a source when there are masked (bad) data in the source aperture.\nMake a mosaic of artifact-corrected images, for example with the MOPEX package. This needs to be\ndone to create the proper rmask files to be applied to the CBCDs when performing the photometry on\nthem, and also to get a nice comparison of CBCD-revealed and mosaic-revealed image features. When\ncreating the mosaic, the overlap correction option should be used in MOPEX, most importantly in\nchannels 3 and 4, to match the backgrounds. Inspect the mosaic to confirm that outlier rejection is\nacceptable, if not, then remosaic with more appropriate parameters. Comparing mosaics of adjacent\nchannels on a per-pixel basis will readily identify if outliers remain in a mosaic. The mosaic coverage\nmaps should be inspected to verify that the outlier rejection has not preferentially removed data from\nactual sources. If the coverage map systematically shows lower weights on actual sources, then the\nrejection is too aggressive and should be redone. One result of making the mosaic is the production of\nrmask files which identify bad pixels in the CBCDs. One should apply the rmasks when performing the\nphotometry in the next step so that bad pixels are not included within the apertures.\nPerform photometry with your favorite software. The PRFs supplied can be used with APEX in\nmultiframe mode for point source fitting. A \"How To\" guide and details of the validation are presented in\nAppendix C. The CBCD uncertainties are slightly conservative as they take into account the uncertainties\nin each pipeline calibration step. For packages that estimate noise directly from the data assuming Poisson\nnoise, you can convert the CBCDs into electron units, so as to calculate the uncertainty due to source shot\nnoise and background correctly. The conversion from MJy\/sr is \u2217GAIN \u2217 EXPTIME \/ FLUXCONV\nwhere GAIN, EXPTIME and FLUXCONV are the keywords from the CBCD header. For accurate\nphotometry, a good background estimate is required. When performing point source fitting with APEX,\nthe parameters of the medfilter module should be tuned to ensure good background subtraction. For\n\nPerforming Photometry on IRAC 151\nImages\nIRAC Instrument Handbook\n\naperture photometry, the background estimate can be obtained from an annulus around the source (but\nnote that the radii of the background annulus will affect the aperture correction).\nApply aperture correction, found in Chapter 4 of this Handbook, if you perform aperture photometry in an\naperture different from the 10 pixel radius aperture used for IRAC calibration. Observers can determine\nHeritage Archive and comparing the photometry to that published in the IRAC Calibration Paper.\nAperture corrections for fitted fluxes are given in Appendix C.\nObservers should apply the array location-dependent photometric correction for blue sources and the\nappropriate color correction for all sources (based on the spectral energy distribution of the source). The\nphotometric array location-dependent correction images are linked from the IRAC web pages. Apply a\ncolor correction from Chapter 4 of this Handbook, using the tabulated values, if appropriate, or calculate\nthe color correction for a source spectral energy distribution as done in that chapter. To calculate a color\ncorrection, you will need the IRAC spectral response curves, which are also available on the IRAC web\npages. Color corrections are typically a few percent for stellar and blackbody sources, but can be more\nsignificant for sources with ISM-like source functions (50%-250% depending on spectrum and passband).\nThe measured flux density is the flux density at the effective wavelength of the array: 3.550, 4.493, 5.731\nand 7.872 microns, for channels 1-4, respectively.\nPixel phase corrections need to be applied in channels 1 and 2. The PRFs include the pixel phase effect,\nso the single mean correction given in Appendix C is adequate. In the case of aperture fluxes, all the\nfluxes need correction. More information on the pixel phase correction can be found in Chapter 4 of this\nHandbook. This effect is as large as 4% peak-to-peak at 3.6 microns and < 1% at 4.5 microns.\nCombine photometry from CBCDs, taking into account uncertainties, to generate a robust, weighted mean\nvalue. Verify that the dispersion in these measurements is comparable to the uncertainty of the individual\nmeasurements (if not, use the dispersion until you track down the source of extra error, e.g., bad\npixels\/cosmic rays in source).\n\nPerforming Photometry on IRAC 152\nImages\nIRAC Instrument Handbook\n\nAppendix C. Point Source Fitting IRAC Images with a\nPRF\n\nThis Appendix discusses the use of point source response functions (PRFs) for fitting sources in IRAC\ndata. For true point sources, it is possible to obtain agreement between PRF-fitted and aperture flux\nmeasurements at better than the 1% level. In this Appendix, we describe validation tests on point sources\nin IRAC data using the PRFs in combination with the MOPEX\/APEX software. The procedure for using\nthe PRFs in conjunction with MOPEX\/APEX is given in the form of a \u201cHow To'' description, and the\nnecessary corrections to the resulting flux densities are detailed.\n\nPoint source fitting is a valuable tool for characterizing images. If the image consists of true point\nsources, PRF fitting can make optimal use of the information in the image, thus improving astrometric\nand photometric results beyond what is achievable using other techniques. PRF fitting also allows point\nsources to be subtracted from an image (for example, using the apex_qa task in MOPEX\/APEX),\nenabling any diffuse background emission to be more easily characterized. Point source fitting is less\nuseful in fields containing large numbers of partially-resolved objects (as typically seen in IRAC\nextragalactic survey fields), and aperture photometry is recommended in such fields. (In principle, model\nfitting could be used for extended sources by convolving a source model with the appropriate point source\nrealizations, but such techniques lie outside the scope of this Appendix.) For isolated point sources on\nfeatureless backgrounds aperture photometry and point source fitting should give almost identical results.\nPoint source fitting to IRAC data has proven problematic as the PSF is undersampled, and, in channels 1\nand 2, there is a significant variation in sensitivity within pixels. Techniques for dealing with these\nproblems were developed for the WFPC2 and NICMOS instruments on HST (Lauer 1999 [18]; Anderson\n& King 2000, [2], see also Mighell 2005, [19]). These techniques involve building a ''point response\nfunction'' (PRF; Anderson & King use the alternative terminology ''effective PSF''), and users interested in\nthe detailed theory of the PRF should refer to these papers. In summary, the PRF is a table (not an image,\nthough for convenience it is stored as a 2D FITS image file) which combines the information on the PSF,\nthe detector sampling and the intrapixel sensitivity variation. By sampling this table at regular intervals\ncorresponding to single detector pixel increments, an estimate of the detector point source response can be\nobtained for a source at any given pixel phase.\n\nPRFs for IRAC have been created by William Hoffmann of the University of Arizona, a member of the\nIRAC instrument team. The starting point for these PRFs was the Code V optical models for\nSpitzer\/IRAC, made at the Goddard Space Flight Center. These were constructed on a 5x5 grid covering\neach of the IRAC arrays. Observations of a calibration star made during the in-orbit checkout at each of\nthese 25 positions per array were then deconvolved by their respective optical models. The results were\naveraged into a single convolution kernel per array which represents additional PRF scatter from\nunmodeled optical effects and spacecraft jitter. A paper on \u201csimfit\u201d that gives more details is included in\nthe IRAC section of the documentation website. The intrapixel sensitivity function was estimated using a\npolynomial fit as a function of pixel phase. The PRFs were then transposed, and flipped in x and y to align\nthem with the BCD coordinate system.\n\nPoint Source Fitting IRAC Images 153\nwith a PRF\nIRAC Instrument Handbook\n\nC.1 Use of the Five Times Oversampled PRFs Outside of APEX\nAs supplied in the documentation website, the PRFs are oversampled by a factor of five in Delta_x and\nDelta_y. This allows for 5x5=25 independent realizations of a point source, corresponding to 25 different\npixel phase combinations (five each in x and y). To obtain any given point source realization (PSR), the\nPRF needs to be sampled every fifth pixel in x and y at the appropriate phase, i.e.,\n\nPSR1(i,j) = PRF(5i-4,5j-4)\nPSR2(i,j) = PRF(5i-4,5j-3)\nPSR3(i,j) = PRF(5i-4,5j-2)\nPSR4(i,j) = PRF(5i-4,5j-1)\nPSR5(i,j) = PRF(5i,5j)\nPSR6(i,j) = PRF(5i-3,5j-4)\n\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\nPSR13(i,j) = PRF(5i-2,5j-2)\nPSR25(i,j) = PRF(5i,5j)\n\nwhere i,j are integers running from 1 to n in the case of a PRF table which is 5n x 5n in size. In this case,\nPSR13 corresponds to the source landing in the center of a pixel. Note that the PRF should not be block\naveraged, as this will result in the loss of the pixel phase information.\n\nThese PRFs may be implemented directly by those willing to write their own code. In IDL, for example, a\npoint source realization may be generated using the \/SAMPLE switch in REBIN, e.g,\n\npsr = rebin(phasedPRF,n,n,\/SAMPLE)\n\nwhere phasedPRF is the 5n x 5n PRF shifted to the appropriate pixel phase in both dimensions. In IRAF,\nuse:\n\nimcopy PRF.fits[1:5n-4:5,1:5n-4:5] PSR1.fits\n\nNote that the PSRs are normalized to unity at infinity, not to the IRAC 10 pixel calibration aperture.\nFluxes obtained with these thus need to be multiplied by the appropriate infinite aperture correction.\nThese have been determined to be 0.943 in channel 1 and 0.929 in channel 2, based on measurements of\nthe PRF, and can be compared to \"direct\" measurements of 0.944 and 0.937. Estimates from the PRF are\nunavailable in channels 3 and 4, but the corrections given in this Handbook are 0.772 and 0.737,\nrespectively (Table 4.7).\n\nC.2 Modifications to the IRAC PRFs for Use with APEX\nThe IRAC PRFs are centered relative to the optical axis, so they are slightly off center in array\ncoordinates due to array distortion. APEX assumes that the PRF is centered on its array, so to use the\nPRFs with APEX requires them to be re-centered. APEX also requires odd-valued axes.\n\nPoint Source Fitting IRAC Images 154\nwith a PRF\nIRAC Instrument Handbook\n\nAPEX performs PRF fitting by varying the position and flux of a source using a modified simplex\ntechnique (see the APEX manual). However, for IRAC data, particularly in channels 1 and 2, where the\nPRF is undersampled, the default 5x sampling of the PRF is insufficient to obtain a sufficiently accurate\nposition for fitting.\n\nTherefore the following transformations were applied to the PRFs:\n\ni) The PRFs were magnified (using linear interpolation) by a factor of 20 (so the resultant PRF\nsampling is x100).\nii) The last row and column were removed to give odd-valued axes.\niii) The PRF was recentered on a first-moment centroid measured using the array values within a\n250 (resampled) pixel border.\niv) The PRF was zeroed out in a 50 (resampled) pixel border (to avoid wrapping problems).\nv) Information describing the PRFs and their modifications was added to the headers.\n\nC.3 Results of Tests with PRF fitting\n\nC.3.1 Test on Calibration Stars\n\nOne sample observation (AOR) was selected for each of the nine brightest IRAC calibration stars (Reach\net al. 2005, [23]). The selected AORs were from 2005 June 05 to 2006 September. Photometry was\nperformed on the five BCDs in each AOR and the results averaged. (C)BCD uncertainties and imasks\nwere used. The pipeline versions were S14.0\u2212S14.4. The central PRF, modified for APEX use as\ndescribed above, was used as the stars were close to the center of the array in each of the images.\n\nAPEX_1frame was used with current default parameters in the namelists provided in the cdf\/ sub-\ndirectory of the MOPEX distribution, e.g., apex_1frame_I1.nl etc, with one change. A Normalization\nRadius for the PRF is needed to correspond to the IRAC calibration radius of 10 pixels. This was placed\nin the parameter block for sourcestimate: Normalization_Radius = 1000 (since it is in units of PRF pixels,\nand the sampling is 100x).\n\nWe performed aperture photometry using a 10 pixel (calibration) radius for IRAC channels 1 and 2, and a\n3 pixel radius for IRAC channels 3 and 4, and a 12\u221220 pixel background annulus for all. Aperture\ncorrections from this Handbook were applied to IRAC channels 3 and 4. The use of smaller apertures at\nlonger wavelengths is not critical but reduces the effect of background noise. No aperture corrections\nwere needed for IRAC channels 1 and 2 for this aperture\/annulus combination as it is used to define the\nflux calibration. The IRAC channel 1 aperture photometry was divided by the empirical pixel-phase flux\ncorrection from Chapter 4 in this Handbook:\n\nPoint Source Fitting IRAC Images 155\nwith a PRF\nIRAC Instrument Handbook\n\n(C.1)\n\nwhere p is the radial pixel phase, defined as the distance of the centroid of the stellar image from the\n1\ncenter of its peak pixel. This corrects to an average pixel phase of p = \u2248 0.4 pix.\n2\u03c0\nThe average PRF-fitted fluxes compared to aperture photometry are shown in Figure C.1. The weighted\naverage differences between PRF fluxes and (corrected) aperture fluxes are shown as long blue dashes.\nThere are offsets in all four channels between the aperture and fitted fluxes. In IRAC channels 3 and 4,\nthe offset is due to the fact that in these channels, the PSFs are wide and there is significant flux in the\nin its PRF normalization, so the PRF fluxes are too high. We examined the \"core\" PRFs and estimated\nthis factor. The estimated effect of the annulus on the PRF fluxes is shown in Fig. C.1 as black, short\ndashes. These are within 1% of the IRAC channel 3 and 4 estimates from the calibration stars. For IRAC\nchannels 1 and 2, these annulus terms appear to be small, so we assume zero correction for the present\ntime. The annulus correction factors (divide PRF fluxes by these) are 1.022 for IRAC channel 3, and\n1.014 for IRAC channel 4 (Table C.1).\n\nC.3.2 Subpixel Response in Channels 1 and 2\n\nThe offset for IRAC channel 1 in Figure C.1 is due to a completely different effect, namely the pixel\nphase effect described above. Aperture sums on the channel 1 IRAC PRFs match reasonably well the\npixel phase relation in Eqn. C.1 if we sum a 10 pixel radius aperture.\n\nAPEX performs normalization on the ''center-of-pixel'' (pixel phase [0,0]) PRF, and applies this\nnormalization factor to all sub-pixel positions. This results in an offset of the photometry relative to the\n1\nmean pixel phase of p = . We need to ''back out'' APEX's center normalization. Setting p=0 in Eqn.\n2\u03c0\nC.1 gives us the required factor: divide the PRF fluxes by 1.021. Similarly, using the pixel phase slope of\n0.0301 in IRAC channel 2 leads to a correction factor of 1.012.\n\nWith these corrections, the PRF fitting on single CBCDs matches aperture results with any systematics\nless than a percent in all IRAC channels (Fig. C.2). The remaining scatter is most likely due to residual\npixel phase effect not removed by the one-dimensional correction applied to the aperture photometry. The\ntrue pixel phase effect has two dimensional structure which is included in the PRF (see also Mighell et al.\n2008, [20]).\n\nPoint Source Fitting IRAC Images 156\nwith a PRF\nIRAC Instrument Handbook\n\nTable C.1. Correction factors for PRF flux densities\n\nBand PRF aperture corrections Correction to mean Total\n\nFrom Core PRFs From Cal Stars Adopted pixel phase correction\n\nIRAC1 1.004 1.000 1.021 1.021\n\nIRAC2 1.004 1.000 1.012 1.012\n\nIRAC3 1.021 1.023\u00b10.002 1.022 1.000 1.022\n\nIRAC4 1.014 1.014\u00b10.002 1.014 1.000 1.014\n\nDivide PRF fluxes by the last column.\n\nPoint Source Fitting IRAC Images 157\nwith a PRF\nIRAC Instrument Handbook\n\nFigure C.1: PRF fits vs. aperture photometry for selected IRAC calibration star CBCDs. The\nvertical axis is the fractional difference between the PRF fit and corrected aperture photometry.\nThe aperture photometry for IRAC channels 3 and 4 is in a 3 pixel radius with a 12\u201320 pixel\nbackground annulus and an aperture correction factor from this Handbook. For IRAC channels 1\nand 2, it is in a 10 pixel radius with the same annulus. Short black dashed lines are the expected\nannulus correction needed. Long blue dashed line is the offset estimated from a weighted average of\nthe data. Note this is essentially the expected value for IRAC channels 3 and 4. But IRAC channel 1\n(and IRAC channel 2 to a lesser extent) requires a pixel-phase correction (see text).\n\nPoint Source Fitting IRAC Images 158\nwith a PRF\nIRAC Instrument Handbook\n\nFigure C.2: Data from Fig. C.1, with IRAC channels 3 and 4 corrected for the annulus\ncontribution, and IRAC channels 1 and 2 corrected for the pixel-phase effect.\n\nC.3.3 The Serpens Test Field\n\nData for this test is a ''C2D'' off-cloud field (OC3) near Serpens, AORKEY 5714944 (S14.0). The\nobservation is HDR mode data (0.6 and 12 sec) from all four IRAC channels. The observation used two\nrepeats of two dithers, so the typical coverage is 4. The observation consisted of a 3x4 map. The field was\nchosen to be a crowded, predominantly stellar, field. The BCD data were run through artifact mitigation\nto correct muxbleed, column pulldown\/pullup, electronic banding and the first frame effect. No pixel\nreplacement was done. Long and short HDR data were handled separately. The tests here are with the\nlong frames.\n\nPoint Source Fitting IRAC Images 159\nwith a PRF\nIRAC Instrument Handbook\n\nAPEX multiframe was used with the Hoffmann PRFs, using a complete set of 25 array-location-\ndependent PRFs. Note that APEX does aperture photometry on the mosaic, but PRF fits on the stack\n(individual images). Final extracted sources shown are those with SNR > ~ 8.\n\nFigure C.3 shows the comparison of PRF-fitted fluxes to aperture-corrected aperture photometry in a 3\npixel radius aperture. For IRAC channels 1 and 2, this is without pixel-phase corrections; for IRAC\nchannels 3 and 4 it is with correction for the PRF aperture (Table C.1), but without correction for mosaic\nsmear. Mosaicking involves an interpolation process which smears out point sources. Aperture\ncorrections for aperture photometry off the mosaics need therefore to be made either based on point\nsources in the mosaic itself, or using values for CBCDs with a correction for mosaic smear. The amount\nof smearing depends on the pixel sampling in the final mosaic.\n\nPoint Source Fitting IRAC Images 160\nwith a PRF\nIRAC Instrument Handbook\n\nFigure C.3: APEX PRF-fitted photometry in the Serpens test field, with array-location-dependent\nPRFs vs. aperture photometry. The aperture has a 3 pixel radius, the background annulus is 12\u201320\npixels. The aperture fluxes have been corrected using the aperture corrections in this Handbook.\nThe IRAC channel 3 and 4 PRF fluxes have been corrected for annulus contribution.\n\nFigure C.4 shows the data with the remaining corrections discussed above applied. PRF fluxes for IRAC\nchannels 1 and 2 were corrected for the pixel phase effect (Table C.1). Mosaic smear corrections for the\naperture fluxes were determined empirically by comparing BCD and mosaic aperture fluxes. In IRAC\nchannels 1 and 2 they were negligible, but IRAC channel 3 and 4 fluxes were corrected by 2.8% and\n1.5%, respectively.\n\nThe results (Fig. C.4) show generally good agreement with aperture photometry, with any systematic\noffset < 1%.\n\nPoint Source Fitting IRAC Images 161\nwith a PRF\nIRAC Instrument Handbook\n\nFigure C.4: APEX PRF-fitted photometry with a PRF Map vs. aperture photometry in the Serpens\ntest field. PRF and aperture fluxes have been corrected as described in the text.\n\nC.3.4 The GLIMPSE Test Field\nWe also analyzed the GLIMPSE AORKEY 9225728 in a similar manner. This produced similarly good\nagreement between the aperture and fitted fluxes. In addition, we stacked the residuals of the brighter\nsources in an attempt to determine the size of any systematics, and plotted out the ratio of the residuals to\nthe uncertainties for the inner four pixels closest to the source position. No significant residual could be\nfound in a stack of 111 sources with channel 1 fluxes between 50 and 100 mJy, corresponding to a limit\nof ~0.1% on the size of any systematic residual. Similarly, no significant difference could be found for the\ndistribution of the ratio of residual to uncertainty between the pixels near to the peak star position and\npixels in the remainder of the image.\n\nC.3.5 Photometry of Moderately-Resolved Sources\nPoint source fitting is most appropriate for true point sources. The flux of astronomical objects that are\nextended will be underestimated by such a procedure. Nearly all fields observed by IRAC have a\nsubstantial population of faint (10s of micro-Jy) background sources, which are in fact galaxies, and in a\ntypical 100-second exposure these can approach 100 galaxies per IRAC frame at 3.6 microns. Although a\ncasual visual inspection of the IRAC data would seem to indicate that the majority of these sources are\ncompact and point-like, in fact treating them as such will lead to substantial errors in photometry, as these\nobjects are typically resolved on a scale of ~1 arcsecond (e.g., Lacy et al. 2005, [17]).\n\nThis issue has been studied in substantial detail in the IRAC Dark Field, which is the dark current\ncalibration field for IRAC. This is an extremely deep IRAC pointing of approximately 200 square\narcminutes near the north ecliptic pole, and which reaches the confusion limit in all IRAC bands. More\nimportantly, there is also deep high spatial resolution HST optical imaging over the same field, which can\nprovide prior information on true source sizes and shapes.\n\nPoint source fitting was used to extract photometry for the IRAC Dark Field. An examination of the\npoint-source subtracted residual images shows clearly that the residuals mimic the HST source\nmorphology, conclusively demonstrating that IRAC does in fact resolve the majority of the faint galaxies.\nThis result is strongest at the shorter IRAC wavelengths, where the spatial resolution is higher and the\ngalaxies may be slightly more extended. This result was hardly unexpected - calculations of expected\ngalaxy angular sizes assuming a modern cosmology indicated that most galaxies would be marginally\nresolved by IRAC almost regardless of distance, modulo changes in galaxy morphology with redshift and\nthe ability to detect faint extended emission.\n\nCurves of growth were generated for the galaxies, and when used in conjunction with the optical priors,\nthe amount of error associated with point source fitting was quantified. Sources below a few micro-Jy\nstart to be affected by confusion issues, so we describe here results for galaxies brighter that this. At 3.6\n\nPoint Source Fitting IRAC Images 162\nwith a PRF\nIRAC Instrument Handbook\n\nmicrons, roughly 50% of all galaxies are demonstrably resolved by IRAC. In 20% of the objects, the use\nof point source fitting will underestimate the true flux by a factor of two or more.\n\nA much more effective solution is to use aperture photometry for such sources. The SWIRE survey\nperformed detailed analyses to determine an \"ideal\" extraction aperture such that it minimized noise. This\naperture was 1.9 arcseconds in radius, or roughly twice the FWHM. Most other survey groups have found\nsimilar results, and this mirrors well-known ideas about aperture photometry of small sources. When such\nan aperture is used, even though some objects may be larger than this the number where the flux differs\nby a factor of 2 falls to only 3%. This improvement over the PSF-fitting reflects the fact that the\nsummation over an aperture larger than the PSF FWHM will always capture a better representation of the\ntrue flux of an extended object, even if that is more extended than the aperture itself. A more ideal\nsolution is to use Kron-like apertures (which are dynamically sized based on moments derived from the\nimage) which are either derived from the data themselves or from image priors in some other band.\n\nWe may thus conclude that for the extragalactic background, which is present in nearly all IRAC data, at\nleast half the objects are resolved by IRAC in a meaningful fashion. Ideally, measurements should\ndynamically use shape information determined from the data themselves, or from priors derived from\nother, higher resolution datasets. Barring the use of shape parameters, use of aperture photometry in\ncircular apertures somewhat larger than the PSF provides a more accurate result than PRF fitting.\n\nC.3.6 Positional Accuracy\nTests were performed on GLIMPSE AORKEY 9225728, which contained approximately 10000 point\nsources in channels 1 and 2. Comparisons were made with respect to SExtractor Gaussian-windowed\ncentroids XWIN_WORLD, YWIN_WORLD using both the pipeline mosaics, and mosaics made with the\noriginal pointing. Using the 100x oversampled PRFs recentered as previously described we found that the\nsource positions agreed with SExtractor to within ~0.1\". Systematic shifts with respect to 2MASS are\n~0.2\" in the pipeline (superboresight) pointing, and ~0.4\" in the original pointing. Recentering the PRF\nhas no effect on photometry. The shifted and unshifted PRFs gave nearly identical photometric results in\nchannels 1\u20134.\nC.3.7 A How-To-Guide for IRAC Point Source Photometry with APEX\nIt is recommended that APEX in point source fitting mode should be used only directly on the BCD data\nusing the Hoffmann PRFs modified for use with APEX as described above. Trying to fit point sources on\nthe mosaic is not recommended as the mosaicking process both blurs the undersampled point sources, and\nloses the pixel phase information. We also do not recommend using the prf_estimate tool to derive a PRF\nfrom IRAC data, as it does not deal correctly with the undersampling of the PRF.\nWe list below the steps towards producing a point source list using APEX in multiframe mode (i.e., on\nthe stack of individual [C]BCDs).\n\nSource fitting versus aperture fluxes: ask yourself if point source photometry is appropriate for your\nsources of interest. If in doubt after reading about photometry of moderately resolved sources above, use\naperture photometry with APEX or Sextractor.\n\nPoint Source Fitting IRAC Images 163\nwith a PRF\nIRAC Instrument Handbook\n\nArtifact correction: use CBCDs or preprocess your IRAC BCD data to remove or mask artifacts as\nnecessary.\nRmasks: assuming the data were taken with overlapping (C)BCDs, make a mosaic with MOPEX, doing\nAPEX.\nPRF: put the center Hoffmann 100x PRF (the one with ...col129_row129...) in your MOPEX cal\/\nsubdirectory for command-line (this will be PRF_FILE_NAME in the namelist file), or type it into the\nGUI. Although you can run APEX with just the center PRF, we recommend using the whole PRF Map\nset, as it noticeably improves the quality of the fits for sources outside of the central region of the arrays.\nTo do this, create a table like the one linked from the PSF\/PRF section of the IRAC web pages\n(substituting appropriate filenames and paths). PRF position refers to the bottom-left corner of the region\nof size NAXIS1, NAXIS2 over which the PRF is valid (in native pixels). This will be\nPRFMAP_FILE_NAME in a namelist, or you can type it into the GUI. Figure C.5 shows how the PRFs\nare distributed over the arrays.\nNormalization Radius: the Hoffmann PRFs require a normalization that matches the IRAC calibration\nradius. In the Sourcestimate block, set Normalization_Radius = 1000 (since it is in PRF pixels and the\nsampling is 100x).\nRun APEX. If doing command-line for IRAC1, edit the default namelist for your data and run: apex.pl -n\napex_I1_yourdata.nl\nPRF Flux: The PRF flux column is called ''flux'' in the extract.tbl output file, and the units are micro-Jy.\nThese need to be divided by the appropriate photometric correction factors from Table C.1: 1.021 (IRAC\n1), 1.012 (IRAC2), 1.022 (IRAC3) and 1.014 (IRAC4).\nPRF Flux Uncertainty: The column labelled ''delta_flux'' is the formal uncertainty from the least-squares\nfit. It will in general underestimate the flux uncertainty. Do not use the column labelled \u201cSNR\u201d' for IRAC,\nas it only takes into account the background noise, and ignores the Poisson (shot) noise term which\ntypically dominates the error. The best estimate is the aperture uncertainty (calculated from the data\nuncertainties) in a 3 pixel radius. This covers the majority of the PSF without going too far out. (For the\ndefault namelist, the relevant uncertainty is in column ''ap_unc2'' [microJy].)\nArray Location-Dependent Photometric Corrections: Multiply the (C)BCDs by the correction image\n(\"...photcorr...\") and run APEX on the resulting images. The fluxes will be correct for \"blue\" sources\n(where blue means having the colors of an early-type stellar photosphere). For \"red\" sources (objects with\ncolors close to that of the zodiacal light) use fluxes derived from running APEX on unmodified CBCDs.\nColor Correction: This is the correction needed to get the right monochromatic flux if your source\nspectrum is different from the reference spectrum used to calibrate the IRAC filters (\u03bdF\u03bd = constant).\nThere is a good discussion of this in Chapter 4 of this Handbook.\nIf all these steps are followed, then the systematic error in the flux measurement for bright, isolated point\nsources should be ~1%. A comparable systematic error exists in the flux density scale. Background\nestimation errors will contribute significantly to the error budget for fainter sources and in confused\nfields.\n\nPoint Source Fitting IRAC Images 164\nwith a PRF\nIRAC Instrument Handbook\n\nFigure C.5: The 25 PRF positions on an IRAC BCD.\nC.3.8 Pixel Phase\nWe define pixel phase as the offset between the centroid of a stellar image and the center of the pixel in\nwhich that centroid lies. For example, an object whose centroid has pixel coordinates 128.23, 127.85 has\na pixel phase of Dx,Dy=(-0.27, 0.35). The pixel phase effect in aperture photometry in Chapter 4 is\ncharacterized in terms of the radial pixel phase p = D x2 + D y ). To shift a PRF to a given pixel phase\n2\n\nwe have adopted the following technique:\n\ni) Magnify the PRFs by a large factor, e.g. 20, using linear interpolation (so the resultant PRF sampling is\nx100).\nii) Re-center the PRF by shifting it to its centroid. (Note that the estimate of the centroid of a source is\nitself a function of the method used to determine the centroid, so ideally you would use equivalent\nalgorithms to find centroids in the (C)BCDs as you do to centroid the PRF. Note also that some of the\nIDL centroiding functions perform poorly with the very undersampled IRAC data at 3.6 and 4.5 microns).\n\nPoint Source Fitting IRAC Images 165\nwith a PRF\nIRAC Instrument Handbook\n\nii) Shift the PRF array (in the example above a shift by 27, -35 resampled pixels would move the desired\nPRF to the center of the array).\niii) Extract the point source realization by sampling the PRF at intervals corresponding to native pixel\nincrements (every 100 oversampled pixels in this example), making sure to pick up the center of the\ncentral pixel.\n\nIn IDL the commands would be:\ni) Use rebin on the 1282 , 5x oversampled PRF to produce the magnified PRF:\nmagPRF = rebin(PRF,2560,2560).\nii) Re-center the PRF. We find that calculating the first moments is usually a robust way to find the\ncentroids. Set xmin, xmax and ymin and ymax to approximately the same values in native pixels as you\nuse to estimate the centroids in your data:\nxx = float(lindgen(2560,2560) mod 2560)\nyy = float(lindgen(2560,2560)\/2560)\nxcen =\ntotal(xx[xmin:xmax,ymin:ymax]*magPRF[xmin:xmax,ymin:ymax])\/total(magPRF[xmin:xmax,ymin:ym\nax])\nycen =\ntotal(yy[xmin:xmax,ymin:ymax]*magPRF[xmin:xmax,ymin:ymax])\/total(magPRF[xmin:xmax,ymin:ym\nax])\nxcensh = nx\/2 - round(xcen)\nycensh = ny\/2 - round(ycen)\ncenPRF = shift(magPRF,xcensh,ycensh)\nii) Shift the re-centered PRF to the center of the central PRF pixel and trim to an integer multiple of the\n100x oversampling factor such that the central pixel (1280,1280) is moved to (1200,1200), the center of\nthe trimmed array:\nshiftedPRF = shift(cenPRF,27,-35)\nphasedPRF = fltarr(2500,2500)\ntrimmedPRF = shiftedPRF[80:2559,80:2559]\nphasedPRF[0:2479,0:2479] = trimmedPRF\niii) Sample the trimmed PRF to produce the point source realization:\nPSR = rebin(trimmedPRF,25,25,\/SAMPLE)\nThe center of the zero phase PSR in this example should be IDL pixel (12,12).\n\nPoint Source Fitting IRAC Images 166\nwith a PRF\nIRAC Instrument Handbook\n\nAppendix D. IRAC BCD File Header\n\nSIMPLE = T \/ Fits standard\nBITPIX = -32 \/ FOUR-BYTE SINGLE PRECISION FLOATING POINT\nNAXIS = 2 \/ STANDARD FITS FORMAT\nNAXIS1 = 256 \/ STANDARD FITS FORMAT\nNAXIS2 = 256 \/ STANDARD FITS FORMAT\nORIGIN = 'Spitzer Science Center' \/ Organization generating this FITS file\nCREATOR = 'S18.18.0' \/ SW version used to create this FITS file\nTELESCOP= 'Spitzer ' \/ SPITZER Space Telescope\nINSTRUME= 'IRAC ' \/ SPITZER Space Telescope instrument ID\nCHNLNUM = 3 \/ 1 digit instrument channel number\nEXPTYPE = 'sci ' \/ Exposure Type\nREQTYPE = 'AOR ' \/ Request type (AOR, IER, or SER)\nAOT_TYPE= 'IracMap ' \/ Observation template type\nAORLABEL= 'IRAC_calstar_NPM1p67.0536_spt4l2 - copy' \/ AOR Label\nFOVID = 67 \/ Field of View ID\nFOVNAME = 'IRAC_Center_of_3.6&5.8umArray' \/ Field of View Name\n\n\/ PROPOSAL INFORMATION\n\nOBSRVR = 'William Reach' \/ Observer Name (Last, First)\nOBSRVRID= 125 \/ Observer ID of Principal Investigator\nPROCYCL = 3 \/ Proposal Cycle\nPROGID = 1181 \/ Program ID\nPROTITLE= 'SIRTF IRAC Calibration Program' \/ Program Title\nPROGCAT = 32 \/ Program Category\n\n\/ TIME AND EXPOSURE INFORMATION\n\nDATE_OBS= '2009-03-23T00:39:17.567' \/ Date & time (UTC) at DCE start\nUTCS_OBS= 291040757.567 \/ [sec] DCE start time from noon, Jan 1, 2000 UTC\nMJD_OBS = 54913.0272867 \/ [days] MJD in UTC at DCE start (,JD-2400000.5)\nHMJD_OBS= 54913.027367 \/ [days] Corresponding Helioc. Mod. Julian Date\nBMJD_OBS= 54913.0273674 \/ [days] Solar System Barycenter Mod. Julian Date\nET_OBS = 291040823.752 \/ [sec] DCE start time (TDB seconds past J2000)\nSCLK_OBS= 922236194.809 \/ [sec] SCLK time (since 1\/1\/1980) at DCE start\nSPTZR_X = -116152405.204261 \/ [km] Heliocentric J2000 x position\nSPTZR_Y = 87280111.04679 \/ [km] Heliocentric J2000 y position\nSPTZR_Z = 37591123.947116 \/ [km] Heliocentric J2000 z position\nSPTZR_VX= -18.879473 \/ [km\/s] Heliocentric J2000 x velocity\nSPTZR_VY= -21.032571 \/ [km\/s] Heliocentric J2000 y velocity\nSPTZR_VZ= -9.762563 \/ [km\/s] Heliocentric J2000 z velocity\nSPTZR_LT= 500.593938 \/ [sec] One-way light time to Sun's center\nAORTIME = 2. \/ [sec] Frameset selected in IRAC AOT\nSAMPTIME= 0.2 \/ [sec] Sample integration time\nFRAMTIME= 2. \/ [sec] Time spent integrating (whole array)\nCOMMENT Photons in Well = Flux[photons\/sec\/pixel] * FRAMTIME\nEXPTIME = 1.2 \/ [sec] Effective integration time per pixel\nCOMMENT DN per pixel = Flux[photons\/sec\/pixel] \/ GAIN * EXPTIME\nFRAMEDLY= 18. \/ [sec] Frame Delay Time\nFRDLYDET= 'T ' \/ Frame Delay Time Determinable (T or F)\n\nIRAC Instrument Handbook\n\nINTRFDLY= 18. \/ [sec] Inter Frame Delay Time\nIMDLYDET= 'T ' \/ Immediate Delay Time Determinable (T or F)\nAINTBEG = 1114427.07 \/ [Secs since IRAC turn-on] Time of integ. start\nATIMEEND= 1114429.03 \/ [Secs since IRAC turn-on] Time of integ. end\nAFOWLNUM= 4 \/ Fowler number\nAWAITPER= 2 \/ [0.2 sec] Wait period\nANUMREPS= 1 \/ Number of repeat integrations\nAREADMOD= 0 \/ Full (0) or subarray (1)\nAORHDR = F \/ Requested AOT is HDR mode\nHDRMODE = F \/ DCE taken in High Dynamic Range mode\nABARREL = 2 \/ Barrel shift\nAPEDSIG = 0 \/ 0=Normal, 1=Pedestal, 2=Signal\n\n\/ TARGET AND POINTING INFORMATION\n\nOBJECT = 'NPM1p67.0536' \/ Target Name\nOBJTYPE = 'TargetFixedCluster' \/ Object Type\nCRPIX1 = 128. \/ Reference pixel along axis 1\nCRPIX2 = 128. \/ Reference pixel along axis 2\nCRVAL1 = 269.646261212312 \/ [deg] RA at CRPIX1,CRPIX2 (using Pointing Recon\nCRVAL2 = 67.789551215035 \/ [deg] DEC at CRPIX1,CRPIX2 (using Pointing Reco\nCRDER1 = 6.19688778419269E-05 \/ [deg] Uncertainty in CRVAL1\nCRDER2 = 6.4255687707609E-05 \/ [deg] Uncertainty in CRVAL2\nRA_HMS = '17h58m35.1s' \/ [hh:mm:ss.s] CRVAL1 as sexagesimal\nDEC_DMS = '+67d47m22s' \/ [dd:mm:ss] CRVAL2 as sexagesimal\nRADESYS = 'ICRS ' \/ International Celestial Reference System\nEQUINOX = 2000. \/ Equinox for ICRS celestial coord. system\nCD1_1 = -0.000214831469601829 \/ Corrected CD matrix element with Pointing Reco\nCD1_2 = -0.000262955698252767 \/ Corrected CD matrix element with Pointing Reco\nCD2_1 = -0.000264515231791188 \/ Corrected CD matrix element with Pointing Reco\nCD2_2 = 0.000215085880792259 \/ Corrected CD matrix element with Pointing Recon\nCTYPE1 = 'RA---TAN-SIP' \/ RA---TAN with distortion in pixel space\nCTYPE2 = 'DEC--TAN-SIP' \/ DEC--TAN with distortion in pixel space\nPXSCAL1 = -1.22673962032422 \/ [arcsec\/pix] Scale for axis 1 at CRPIX1,CRPIX2\nPXSCAL2 = 1.22298117494211 \/ [arcsec\/pix] Scale for axis 2 at CRPIX1,CRPIX2\nPA = -50.7183862658392 \/ [deg] Position angle of axis 2 (E of N) (was OR\nUNCRTPA = 0.000575824424074831 \/ [deg] Uncertainty in position angle\nCSDRADEC= 2.27476132413713E-06 \/ [deg] Costandard deviation in RA and Dec\nSIGRA = 0.033918650300932 \/ [arcsec] RMS dispersion of RA over DCE\nSIGDEC = 0.0233826859041868 \/ [arcsec] RMS dispersion of DEC over DCE\nSIGPA = 0.0180000000057134 \/ [arcsec] RMS dispersion of PA over DCE\nPA = -50.7183862658392 \/ [deg] Position angle of axis 2 (E of N) (was OR\nRA_RQST = 269.646397245074 \/ [deg] Requested RA at CRPIX1, CRPIX2\nDEC_RQST= 67.7896310093685 \/ [deg] Requested Dec at CRPIX1, CRPIX2\nPM_RA = 0. \/ [arcsec\/yr] Proper Motion in RA (J2000)\nPM_DEC = 0. \/ [arcsec\/yr] Proper Motion in Dec (J200)\nRMS_JIT = 0.00414784151698205 \/ [arcsec] RMS jitter during DCE\nRMS_JITY= 0.00293095487000397 \/ [arcsec] RMS jitter during DCE along Y\nRMS_JITZ= 0.00293497747861887 \/ [arcsec] RMS jitter during DCE along Z\nSIG_JTYZ= -0.00102057552390795 \/ [arcsec] Costadard deviation of jitter in YZ\nPTGDIFF = 0.341735121617664 \/ [arcsec] Offset btwn actual and rqsted pntng\nPTGDIFFX= 0.339553396668735 \/ [pixels] rqsted - actual pntng along axis 1\nPTGDIFFY= 0.0374044542123624 \/ [pixels] rqsted - actual pntng along axis 2\nRA_REF = 269.727166666669 \/ [deg] Commanded RA (J2000) of ref. position\nDEC_REF = 67.7936944444455 \/ [deg] Commanded Dec (J2000) of ref. position\n\nIRAC Instrument Handbook\n\nUSEDBPHF= T \/ T if Boresight Pointing History File was used\nBPHFNAME= 'SBPHF.0922233600.041.pntg' \/ Boresight Pointing History Filename\nFOVVERSN= 'BodyFrames_FTU_14a.xls' \/ FOV\/BodyFrames file version used\nRECONFOV= 'IRAC_Center_of_5.8umArray' \/ Reconstructed Field of View\nORIG_RA = 269.646270751953 \/ [deg] Original RA from raw BPHF (without pointi\nORIG_DEC= 67.7895202636719 \/ [deg] Original Dec from raw BPHF (without point\nORIGCD11= -0.0002148321219 \/ [deg\/pix] Original CD1_1 element (without point\nORIGCD12= -0.0002629551745 \/ [deg\/pix] Original CD1_2 element (without point\nORIGCD21= -0.0002645147033 \/ [deg\/pix] Original CD2_1 element (without point\nORIGCD22= 0.0002150865184 \/ [deg\/pix] Original CD2_2 element (without point\n\n\/ DISTORTION KEYWORDS\n\nA_ORDER = 3 \/ polynomial order, axis 1, detector to sky\nA_0_2 = -4.3447E-06 \/ distortion coefficient\nA_0_3 = -1.016E-09 \/ distortion coefficient\nA_1_1 = 3.5897E-05 \/ distortion coefficient\nA_1_2 = -1.5883E-07 \/ distortion coefficient\nA_2_0 = -1.6032E-05 \/ distortion coefficient\nA_2_1 = -1.0378E-09 \/ distortion coefficient\nA_3_0 = -1.5738E-07 \/ distortion coefficient\nA_DMAX = 1.641 \/ [pixel] maximum correction\nB_ORDER = 3 \/ polynomial order, axis 2, detector to sky\nB_0_2 = 2.5424E-05 \/ distortion coefficient\nB_0_3 = -1.6169E-07 \/ distortion coefficient\nB_1_1 = -9.977E-06 \/ distortion coefficient\nB_1_2 = 7.6924E-09 \/ distortion coefficient\nB_2_0 = -7.8167E-06 \/ distortion coefficient\nB_2_1 = -1.6873E-07 \/ distortion coefficient\nB_3_0 = -1.1593E-08 \/ distortion coefficient\nB_DMAX = 1.184 \/ [pixel] maximum correction\nAP_ORDER= 3 \/ polynomial order, axis 1, sky to detector\nAP_0_1 = -2.3883E-07 \/ distortion coefficient\nAP_0_2 = 4.406E-06 \/ distortion coefficient\nAP_0_3 = 6.4348E-10 \/ distortion coefficient\nAP_1_0 = -1.5761E-05 \/ distortion coefficient\nAP_1_1 = -3.6428E-05 \/ distortion coefficient\nAP_1_2 = 1.64E-07 \/ distortion coefficient\nAP_2_0 = 1.6243E-05 \/ distortion coefficient\nAP_2_1 = -9.3393E-10 \/ distortion coefficient\nAP_3_0 = 1.5989E-07 \/ distortion coefficient\nBP_ORDER= 3 \/ polynomial order, axis 2, sky to detector\nBP_0_1 = -1.6807E-05 \/ distortion coefficient\nBP_0_2 = -2.5772E-05 \/ distortion coefficient\nBP_0_3 = 1.6546E-07 \/ distortion coefficient\nBP_1_0 = -8.8532E-07 \/ distortion coefficient\nBP_1_1 = 1.0173E-05 \/ distortion coefficient\nBP_1_2 = -8.7895E-09 \/ distortion coefficient\nBP_2_0 = 7.8383E-06 \/ distortion coefficient\nBP_2_1 = 1.7089E-07 \/ distortion coefficient\nBP_3_0 = 1.2114E-08 \/ distortion coefficient\n\n\/ PHOTOMETRY\n\nIRAC Instrument Handbook\n\nBUNIT = 'MJy\/sr ' \/ Units of image data\nFLUXCONV= 0.6074 \/ Flux Conv. factor (MJy\/sr per DN\/sec)\nGAIN = 3.8 \/ e\/DN conversion\nRONOISE = 9.1 \/ [Electrons] Readout Noise from Array\nZODY_EST= 1.977768 \/ [MJy\/sr] Zodiacal Background Estimate\nISM_EST = 0.1727364 \/ [MJy\/sr] Interstellar Medium Estimate\nCIB_EST = 0. \/ [MJy\/sr] Cosmic Infrared Background Estimate\nSKYDRKZB= 1.934454 \/ [MJy\/sr] Zodiacal Background Est of Subtracted\nSKYDKMED= -1.472912 \/ [MJy\/sr] Median of Subtracted Skydark\nSKDKRKEY= 29840640 \/ Skydark AOR Reqkey\nSKDKTIME= 2. \/ [sec] Skydark AOR Duration Time\nSKDKFDLY= 11.97 \/ [sec] Average Frame Delay Time of Skydark\nSKDKIDLY= 11.97 \/ [sec] Average Immediate Delay Time of Skydark\n\n\/ IRAC MAPPING KEYWORDS\n\n\/ INSTRUMENT TELEMETRY DATA\n\nASHTCON = 2 \/ Shutter condition (1:closed, 2: open)\nAWEASIDE= 0 \/ WEA side in use (0:B, 1:A)\nACTXSTAT= 0 \/ Cmded transcal status\nATXSTAT = 0 \/ transcal status\nACFLSTAT= 0 \/ Cmded floodcal status\nAFLSTAT = 0 \/ floodcal status\nAVRSTUCC= -3.4 \/ [Volts] Cmded VRSTUC Bias\nAVRSTBEG= -3.3878126 \/ [Volts] VRSTUC Bias at start integration\nAVDETC = -4. \/ [Volts] Cmded VDET Bias\nAVDETBEG= -3.9877046 \/ [Volts] VDET Bias at start of integration\nAVGG1C = -3.45 \/ [Volts] Cmded VGG1 Bias\nAVGG1BEG= -3.4382206 \/ [Volts] VGG1 Bias at start of integration\nAVDDUCC = -3. \/ [Volts] Cmded VDDUC Bias\nAVDDUBEG= -2.9828275 \/ [Volts] VDDUC Bias at start integration\nAVGGCLC = 0. \/ [Volts] Cmnded VGGCL clock rail voltage\nAVGGCBEG= 0. \/ [Volts] VGGCL clock rail voltage\nAHTRIBEG= 150.05788 \/ [uAmps] Heater current at start of integ\nAHTRVBEG= 1.9357675 \/ [Volts] Heater Voltage at start integ.\nAFPAT2B = 6.0626999 \/ [Deg_K] FPA Temp sensor #2 at start integ.\nAFPAT2BT= 1114416.9 \/ [Sec] FPA Temp sensor #2 time tag\nAFPAT2E = 6.0626999 \/ [Deg_K] FPA temp sensor #2, end integ.\nAFPAT2ET= 1114416.9 \/ [Sec] FPA temp sensor #2 time tag\nACTENDT = 20.925428 \/ [Deg_C] C&T board thermistor\nAFPECTE = 18.04231 \/ [Deg_C] FPE control board thermistor\nAFPEATE = 21.025127 \/ [Deg_C] FPE analog board thermistor\nASHTEMPE= 22.11418 \/ [Deg_C] Shutter board thermistor\nATCTEMPE= 23.272451 \/ [Deg_C] Temp. controller board thermistor\nACETEMPE= 20.864466 \/ [Deg_C] Calib. electronics board thermistor\nAPDTEMPE= 20.98639 \/ [Deg_C] PDU board thermistor\nACATMP1E= 1.302006 \/ [Deg_K] CA Temp, Foot 2\nACATMP2E= 1.2846904 \/ [Deg_K] CA Temp, Foot 1\nACATMP3E= 1.3177339 \/ [Deg_K] CA Temp, Shutter\nACATMP4E= 1.3142446 \/ [Deg_K] CA Temp, Top OMH\nACATMP5E= 1.3122628 \/ [Deg_K] CA Temp, Bottom OMH\n\nIRAC Instrument Handbook\n\nACATMP6E= 1.3106288 \/ [Deg_K] CA Temp, Top TxCal Sphere\nACATMP7E= 1.3093235 \/ [Deg_K] CA Temp, Bottom TxCal Sphere\nACATMP8E= 1.3022636 \/ [Deg_K] CA Temp, Foot 3\n\n\/ DATA FLOW KEYWORDS\n\nORIGIN0 = 'JPL_FOS ' \/ Site where RAW FITS file was written\nCREATOR0= 'J5.3 ' \/ SW system that created RAW FITS\nDATE = '2010-08-17T21:59:39' \/ [YYYY-MM-DDThh:mm:ss UTC] file creation date\nAORKEY = 29850368 \/ AOR or EIR key. Astrnmy Obs Req\/Instr Eng Req\nEXPID = 9 \/ Exposure ID (0-9999)\nDCENUM = 0 \/ DCE number (0-9999)\nTLMGRPS = 1 \/ expected number of groups\nFILE_VER= 1 \/ Version of the raw file made by SIS\nRAWFILE = 'IRAC.3.0029850368.0009.0000.01.mipl.fits' \/ Raw data file name\nCPT_VER = '3.1.11 ' \/ Channel Param Table FOS versioN\nCTD_VER = '3.0.94S ' \/ Cmded telemetry data version\nEXPDFLAG= T \/ (T\/F) expedited DCE\nMISS_LCT= 0 \/ Total Missed Line Cnt in this FITS\nMANCPKT = F \/ T if this FITS is Missing Ancillary Data\nMISSDATA= F \/ T if this FITS is Missing Image Data\nBADTRIG = F \/ Bad data (zero pixel) was located in raw frame\nCHECKSUM= 0 \/ MIPL computed checksum\nPAONUM = 3383 \/ PAO Number\nCAMPAIGN= 'IRAC013100' \/ Campaign\nDCEID = 117913164 \/ Data-Collection-Event ID\nDCEINSID= 24298735 \/ DCE Instance ID\nDPID = 318707046 \/ Data Product Instance ID\nPIPENUM = 107 \/ Pipeline Script Number\nSOS_VER = 2 \/ Data-Product Version\nPLVID = 6 \/ Pipeline Version ID\nCALID = 8 \/ CalTrans Version ID\nSDRKEPID= 6906663 \/ Sky Dark ensemble product ID\nPMSKFBID= 1878 \/ Pixel mask ID\nLDRKFBID= 852 \/ Fall-back lab dark ID\nLINCFBID= 1021 \/ Fall-back Linearity correction ID\nFLATFBID= 1161 \/ Fall-back flat ID\nFLXCFBID= 1801 \/ Flux conversion ID\nLBDRKFLE= 'FUL_2s_2sf4d1r1_ch3_v1.2.0_dark.txt' \/ Labdark File Used\nLBDRKTD = 'T ' \/ Labdark Time Dependent (T or F)\n\n\/ PROCESSING HISTORY\n\nHISTORY job.c ver: 1.50\nHISTORY TRANHEAD v. 13.1, ran Tue Aug 17 14:58:33 2010\nHISTORY CALTRANS v. 4.0, ran Tue Aug 17 14:58:44 2010\nHISTORY cvti2r4 v. 1.31 A61025, generated 8\/17\/10 at 14:58:47\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:58:52\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:58:55\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:58:59\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:02\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:05\n\nIRAC Instrument Handbook\n\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:08\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:11\nHISTORY FFC v. 1.0, ran Tue Aug 17 14:59:13 2010\nHISTORY FOWLINEARIZE v. 4.900000, ran Tue Aug 17 14:59:13 2010\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:16\nHISTORY BGMODEL v. 1.0, ran Tue Aug 17 14:59:16 2010\nHISTORY SLREMOVE v. 1.0, ran Tue Aug 17 14:59:17 2010\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:19\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:22\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:24\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:27\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:30\nHISTORY hdrupd8 v. 1.6 A70821, updated 8\/17\/10 at 14:59:33\nHISTORY DARKSUBNG v. 1.000, ran Tue Aug 17 14:59:34 2010\nHISTORY DARKDRIFT v. 4.1, ran Tue Aug 17 14:59:35 2010\nHISTORY FLATAP v. 1.500 Tue Aug 17 14:59:35 2010\nHISTORY DNTOFLUX v. 4.2, ran Tue Aug 17 14:59:39 2010\nHISTORY CALTRANS v. 4.0, ran Thu Aug 19 04:55:27 2010\nHISTORY PTNTRAN v. 1.4, ran Thu Aug 19 04:55:34 2010\nHISTORY FPGen v. 1.26, ran Thu Aug 19 04:55:40 2010\nHISTORY PTGADJUST v. 1.0, ran Thu Aug 19 05:01:12 2010\nHISTORY SATCORR Module version 1.7 image created Thu Aug 19 19:58:27 2010\nSATSRCFX= T \/ Saturated sources corrected with estimate\nSATSRCS = 0 \/ number of saturated sources fixed\nCLPLDNFX= T \/ Column Pulldown artifact corrected\nHISTORY pulldown_correction v. 1.2 Thu Aug 19 19:59:16 2010\nBANDNGFX= T \/ Banding artifact corrected\nHISTORY bandingcorr v. 1.1 Thu Aug 19 19:59:38 2010\nEND\n\nIRAC Instrument Handbook\n\nAppendix E. Acronyms\n\nAnalog-to-Digital Converter\n\nAOR\n\nAstronomical Observation Request - an individual observation.\n\nAORKEY\n\nAstronomical Observation Request Key - a unique numerical identification of an observation.\n\nAOT\n\nAstronomical Observation Template - the IRAC observing mode.\n\nBCD\n\nBasic Calibrated Data, the Level 1 data product from each DCE that has been pipeline-processed and\nis fully calibrated.\n\nCBCD\n\nCorrected Basic Calibrated Data. This should be the starting point for further scientific analysis in\nmost cases.\n\nCampaign\n\nUnbroken time period when an instrument is powered on. Most instrument IRAC campaigns are\nexpected to be on the order of one week in length.\n\nCTE\n\nCharge Transfer Efficiency.\n\nDCE\n\nData Collection Event, for IRAC a single 256x256 pixel image from a single detector.\n\nAcronyms 173\nIRAC Instrument Handbook\n\nDN\n\nData Number.\n\nFET\n\nField Effect Transistor.\n\nFOS\n\nFlight Operations System.\n\nFPA\n\nFocal Plane Assembly, housing one IRAC detector.\n\nInSb\n\nIndium Antimonide, the detector material used in the short wavelength channels of IRAC (1 and\n2).\n\nIOC\n\nIn-Orbit Checkout, the two-month long testing period of the telescope following its launch.\n\nISSA\n\nIRAS Sky Survey Atlas.\n\nJPL\n\nJet Propulsion Laboratory\n\nMIC\n\nMultiple Instrument Chamber.\n\nPAO\n\nAcronyms 174\nIRAC Instrument Handbook\n\nPeriod of Autonomous Operation, the interval between ground contacts for uplinking commands\nand downlinking data, normally 12\u221224 hours.\n\nPCS\n\nPointing Control System.\n\nPRF\n\nPoint Response Function. The PRF is essentially the convolution of a box the size of the image\npixel with the PSF.\n\nROIC\n\nRead-Out Integrated Circuit. It is the chip that contains the multiplexer (column and row scanners\nand the buses), the unit cell amplifiers, and the output amplifiers. The chip containing the detector\ndiodes is bonded to it. It is often called simply the \u201cmux.\"\n\nSAO\n\nSmithsonian Astrophysical Observatory, Cambridge, MA.\n\nSSC\n\nSpitzer Science Center, Caltech, Pasadena, CA.\n\nSi:As\n\nArsenic-doped Silicon, the detector material in the IRAC long-wavelength channels (3 & 4).\n\nSV\n\nScience Verification, a 35-day period after the IOC during which the science instruments and their\nobservational modes were commissioned.\n\nSWIRE\n\nSpitzer Wide Area Infrared Survey.\n\nAcronyms 175\nIRAC Instrument Handbook\n\nWCS\n\nWorld Coordinate System.\n\nAcronyms 176\nIRAC Instrument Handbook\n\nAppendix F. Acknowledgments\n\nIRAC would not have been the successful instrument it was without the enthusiastic and capable\ncontribution of many colleagues (see the lists of collaborators and laboratories below). Support for the\nIRAC instrument was provided by NASA through contract 960541 issued by JPL.\n\nPrincipal Investigator\nDr. Giovanni Fazio (SAO, Harvard)\nDr. Gary J. Melnick, Deputy Principal Investigator (SAO, Harvard)\nDr. Joseph L. Hora, Project Scientist (SAO, Harvard)\nRichard S. Taylor, Project Manager (SAO, Harvard)\n\nCo-Investigators\nThe co-investigators played a central role in defining the technical characteristics of IRAC in order to\ncarry out the science programs agreed upon by all of them. Lynne Deutsch, a co-investigator of IRAC,\ndied on April 2, 2004, after a long illness. Lynne was a dear friend and a close colleague. The IRAC team\ndeeply misses her presence.\nDr. William F. Hoffmann (University of Arizona)\nDr. Craig R. McCreight ( Ames Research Center)\nDr. S. Harvey Moseley (Goddard Space Flight Center)\nDr. Judith L. Pipher (University of Rochester)\nDr. Lori E. Allen (SAO, Harvard)\nDr. Matthew L. N. Ashby (SAO, Harvard)\nDr. Pauline Barmby (SAO, Harvard)\nDr. Lynne K. Deutsch (SAO, Harvard)\nDr. Peter Eisenhardt (JPL, Caltech)\nDr. Jiasheng Huang (SAO, Harvard)\nDr. David I. Koch (SAO, Harvard)\nDr. Massimo Marengo (SAO, Harvard)\nDr. S. Thomas Megeath (SAO, Harvard; University of Toledo)\nDr. Michael Pahre (SAO, Harvard)\nDr. Brian Patten (SAO, Harvard)\nDr. Howard Smith (SAO, Harvard)\nDr. John R. Stauffer (SAO, Harvard; SSC, Caltech)\nDr. Eric V. Tollestrup (SAO, Harvard)\nDr. Zhong Wang (SAO, Harvard)\nDr. Steven P. Willner (SAO, Harvard)\nDr. Edward L. Wright (UCLA)\nDr. William F. Hoffmann (U. Arizona)\nDr. William J. Forrest (University of Rochester)\nDr. Daniel Gezari (GSFC)\n\nAcknowledgments 177\nIRAC Instrument Handbook\n\nCollaborators\n\nConstruction and ground calibration phase\n\nPeople listed here include those that participated in the mechanical, optical and cryo-mechanical studies\nthat led to the definition of IRAC, as well as those who designed and built the on-board electronic\nsubsystems, developed the on-board software and worked on the Ground Support Equipment. Others\nparticipated in the development of the extensive software systems and procedures later used for the in-\nflight calibration, system tests and uplink subsystems or in the Off-Line pipeline.\n\nDr. Jon Chappell, Data Systems Analyst (SAO, Harvard)\nDr. Martin Cohen, Calibration Scientist (SAO, Harvard)\nDr. Steven Kleiner, IT Specialist (SAO, Harvard)\nDr. John Spitzak, Data Systems Analyst (SAO, Harvard)\nSAO IRAC PROJECT OFFICE\nJo-Ann Campbell-Cameron, Group Secretary\nRalph Paganetti, Management Support\nSAO CENTRAL ENGINEERING\nJohn P. Polizotti, IRAC Systems Engineer\nVaman S. Bawdekar, Quality Assurance Engineer\nDavid A. Boyd, Thermal Engineer\nJohn Boczenowski, Quality Assurance Manager\nKathy Daigle, Documentation Control Engineer\nLeslie Frazier, Quality Assurance Engineer\nThomas Gauron, Mechanical Engineer\nJoaquim J. Gomes, Electrical Engineer\nEverett Johnston, Electrical Engineer\nMaggie Kanouse, Documentation Specialist\nWarren Martell, Quality Assurance Engineer\nPaul Okun, Electrical Engineer\nJoel Rosenberg, Electrical Engineer\n\nAMES RESEARCH CENTER\nDr. Craig R. McCreight, Lead Si:As Scientist\nRoy R. Johnson, Detector Test Engineer\nRoderick N. McHugh, Electronic Technician\nMark E. McKelvey, Detector Test Lead\nRobert E. McMurray, Jr., Detector Scientist\nNicolas N. Moss, Programmer\n\nAcknowledgments 178\nIRAC Instrument Handbook\n\nWilliam I. Ogilvie, Programmer\nNicholas N. Scott, Mechanical and Electrical Tech\nSteven I. Zins, Programmer\n\nUNIVERSITY OF ARIZONA\nDr. William F. Hoffmann\nThomas J. Tysenn, Research Specialist\nPatrick M. Woida, Staff Technician, Sr.\n\nUNIVERSITY OF ROCHESTER\nDr. Judith L. Pipher, Lead InSb Scientist\nDr. William J. Forrest\n\nInSb ARRAY DEVELOPMENT\nHao Chen, Senior Engineer\nDr. James D. Garnett, Research Associate\nDr. William J. Glaccum, Research Associate\nDr. Zoran Ninkov, Research Associate\nJian Wu, Senior Engineer\n\nInSb ARRAY TESTING\nNathaniel Cowen, Programmer\/Analyst\nD. Michael Myers, Programmer\/Analyst\nRyan Overbeck, Programmer\/Analyst\nRichard Sarkis, Programmer\/Analyst\nJustin Schoenwald, Programmer\/Analyst\nBrendan White, Programmer\/Analyst\n\nOBSERVATIONAL PLANNING AND DATA ANALYSIS\n\nAcknowledgments 179\nIRAC Instrument Handbook\n\nRAYTHEON VISION SYSTEMS\nSANTA BARBARA RESEARCH CENTER\nInSb AND Si:As DETECTOR DEVELOPMENT, FABRICATION AND TEST\n\nDr. Alan Hoffman, Project Manager\nDr. George Domingo, Si:As Development Lead, Project Manager\nConrad Anderson, IBC Detector & Test\nVirginia Bowman, Si:As Processing\nGeorge Chapman, Detector Test\nBruce Fletcher, Hybridization Engineer\nPeter Love, System Engineer\nDr. Nancy Lum, Multiplexer Designer\nSusan Morales, Production Control\nOlivia Moreno, Quality Assurance\nJoseph Rosbeck, InSb Detector Engineer\nKiomi Schartz, IBC Processing\nMichael S. Smith, Detector Test Engineer\nSteve Solomon, Array Test Engineer\nKevin Sparkman, Test Engineer\nAndrew S. Toth, IBC Detector Engineer\nPeter S. Villa, Hybridization Engineer\nSharon E. Woolaway, Hybridization Engineer\n\nGODDARD SPACE FLIGHT CENTER\n\nINSTRUMENT MANAGEMENT\nLois Workman, Instrument Manager\nFelicia Jones-Selden, Instrument Engineer\/Manager\nJuan Rivera, Instrument Engineer\/Manager\n\nAcknowledgments 180\nIRAC Instrument Handbook\n\nRich Barney, Instrument Manager, Branch Head\n\nSCIENCE TEAM\nDr. S. Harvey Moseley, Instrument Scientist\nDr. Richard Arendt, Science Support\nDr. Sean Casey, Science Support\nDr. Dale Fixsen, Science Support\nDr. Daniel Gezari, Instrument Scientist\nDr. Alexander Kutyrev, Science Support\nTim Powers, Electronics Technician\n\nREVIEW TEAM\nWilliam T. Tallant, Review Team Chairman\nSteve Bartel, Review Team\nJames Caldwell, Review Team\nMichael Dipirro, Review Team\nPam Davila, Review Team,\nGene Gochar, Review Team\nFrank Kirchman, Review Team\nRobert Martineau, Review Team\nIan Mclean, Review Team\nVern Weyers, Review Team\n\nINSTRUMENT SYSTEM ENGINEERS\nGabe Karpati, Instrument System Engineer\nNeil Martin, Instrument System Engineer\nRobert Maichle, Instrument System Engineer\nKevin Brenneman, System Engineer\nRobert Kichak, Chief Engineer, Electrical\n\nMECHANICAL SYSTEMS\nWillie Barber, Mechanical Technician\nCarlos Bernabe, Mechanical Engineer\nKen Blumenstock, Mechanical Engineer\nGary Brown, Mechanical Engineer\nDr. Philip Chen, Contamination Engineer\nRainer Fettig, Mechanical Engineer\nBryan Grammer, Designer\nPaul Haney, Mechanical Technician\nTom Hanyok, Designer\n\nAcknowledgments 181\nIRAC Instrument Handbook\n\nDarron Harris, Mechanical Technician\nMike Hersh, Mechanical Engineer\nSid Johnson, Mechanical Technician\nBen Lewit, Mechanisms Engineer\nCarlos Lugo, Mechanical Engineer\nDave Pfenning, Electro\/Mechanical Tech\nGeorge Reinhardt, Mechanical Engineer\nScott Schwinger, Mechanical Engineer\nRyan Simmons, Systems Analyst\nDr. Michael G. Ryschkewitsch, Designer, 1988-1989\nCharles Tomasevich, Mechanical Engineer, 1997-2000\nGeorge Voellmer, Mechanical Engineer, 1995-1996\nSteve Wood, Mechanical Technician, 1997-2000\n\nELECTRICAL SYSTEMS\nVicky Brocius, Parts Procurement\nRobert Clark, Parts Procurement\nTracy Clay, WEA Enclosure Supervisor\nJim Cook, PWA Assembly\nGlenn Davis, Polymerics\nMitch Davis, ESE Engineer\nBob Demme, PWA Assembly Manager\nMelissa Eberhardinger, Parts Procurement\nMajed El Alami, Parts Procurement\nPatricia Gilbertson, Parts Procurement\nSteve Graham, Electrical Engineer\nDavid Hessler, Electrical Engineer\nGina Kanares, Parts Stock\nRichard Katz, Electrical Engineer\nIgor Kleiner, BTE S\/W\nTracie Lampke, PWA Assembly\nDavid Liu, Electrical Engineer\nJim Lohr, Parts Engineer\nBill Long, WEA Enclosure Supervisor\nJack Lorenz, WEA Enclosure Designer\nJohn McCloskey, Electrical Engineer\nCharlie McClunin, WEA Enclosure Designer\nMargaret McVicker, PWA Assembly\nTim Miralles, WEA Test Engineer\nKim Moats, PWA Assembly\nTrang Nguyen, Electrical Engineer\n\nAcknowledgments 182\nIRAC Instrument Handbook\n\nJ. R. Norris, Litton Task Manager\nAllen Rucker, WEA Test Engineer\nNarenda Shukla, DC-DC Converter\nKevin Smith, WEA PWA Designer\nSteve Smith, WEA Test Engineer\nJohn Stewart, Electronics Technician\nVictor Torres, Electrical Engineer\nYen Tran, WEA PWA Designer\nSteven Van Nostrand, WEA PWA Designer\nSherry Wagner, PWA Assembly\nBanks Walker, WEA PWA Designer\nMark Walter, DC-DC Converter\nRichard Williams, Parts Engineer\n\nSOFTWARE SYSTEMS\nRaymond Whitley, Software Manager\nLouise Bashar, Software Engineer\nCraig Bearer, Ground Software Engineer\nGlenn Cammarata, Software Engineer\nJenny Geiger, Software Engineer\nBob Koehler, Software Engineer\nSteve Mann, Software Engineer\nDave McComas, Software Engineer\nJanet McDonnell, Software Engineer\nKen Rehm, Software Engineer\nJann Smith, Software Engineer\nCarlos Trujillo, Software Engineer\nDavid Vavra, Software Engineer\n\nCRYOGENICS\nDan McHugh, Cryogenics Technician,\nJohn Bichell, Cryogenics Technician\nRob Boyle, Cryogenics Engineer\nSusan Breon, Cryogenics Engineer\nMichael Dipirro, Cryogenics Engineer\nDarrell Gretz, Cryogenics Technician\nEd Quinn, Cryogenics Technician\nPeter Shirron, Cryogenics Engineer\n\nFLIGHT ASSURANCE\nTed Ackerson, Systems Assurance Manager\n\nAcknowledgments 183\nIRAC Instrument Handbook\n\nDick Bolt, Flight Assurance\nJerry Bushman, Flight Assurance\nJack Galleher, System Reliability\nSteve Hull, Parts Assurance\nRon Kolecki, Flight Assurance Manager\nNorman Lee, Flight Assurance\nShirley Paul, Flight Assurance\n\nOPTICAL SYSTEMS\nCatherine Marx, Optical Designer, Optics Lead\nPat Losch, Lead Optics, Optics I&T\nBill Eichorn, Lead Optics, Optical I&T\nJulie Crooke, Optical Engineer\nAndy Dantzler, Optical Designer\nDr. Bruce Dean, Optical Analyst\nDennis Evans, Optical Design\nThomas French, Optical Technician\nDr. David Glenar, Calibration Optics\nDr. Qian Gong, Optical Engineer\nPaul Hannan, Optical Designer\nDr. Donald Jennings, Optics Engineer\nJay Jett, Optical Technician\nLinette Kolos, Plating Technician\nDr. Ritva Keski-Kuha, Optics Engineer\nJames Lyons, Optical Technician\nEric Mentzell, Optical Analyst\nJoseph McMann, Optics Engineer\nDr. Ray Ohl, Optical Engineer\nDean Osgood, Optical Technician\nGrant Piegari, Optical Technician\nSteve Rice, Thin Films Technician\nKevin Redman, Optical Technician\nVicki Roberts, Optical Technician\nDr. Frederick Robinson, Optics Engineer\nDr. Kenneth Stewart, Optics Engineer\nCarl Strojny, Optical Technician\nFelix Threat, Thin Films Technician\nLarry White, Mechanical Technician\nMark Wilson, Optical Designer\nLou Worrel, Optics Technician\n\nTHERMAL SYSTEMS\n\nAcknowledgments 184\nIRAC Instrument Handbook\n\nMike Choi, Thermal Engineer\nRaymond Trunzo, Thermal Engineer\n\nFOCAL PLANE ASSEMBLIES\nDr. Murzy D. Jhabvala, FPA Lead\nChristine Allen, FPA Engineer\nSachi Babu, Detector Assembly Technician\nMark Cushman, Detector Assembly Tech\nJohn Godfrey\nSteve Graham, Electronics Engineer\nAnh La, Test Engineer\nGerald Lamb, FPA Engineer\nKim Moats, Electronics Technician\nTrang Nguyen, Electronics Engineer\nFrank Peters, Detector Technician\nDavid Rapchun, Detector Technician\nPeter Shu, Science Team\nRobert Stanley, Electronics Technician\nJeff Travis, Electronics Engineer\n\nINTEGRATION AND TEST\nRay Jungo, Integration and Test Manager\nMichael Alexander, Test Conductor\nMaureen Armbruster, Test Conductor\nCraig Bearer, S\/C Simulator Programmer\nMarty Brown, Test Conductor\nJamie Britt, Environmental Test Engineer\nFrank Carroll, Test Conductor\nJames E. Golden, Programmer\nPeter Gorog, Programmer\nShirley M. Jones, Harness Technician\nDon Kirkpatrick, Harness Technician\nJuli Lander, Test Conductor\nMatthew E. (Ed) Lander, Test Conductor\nJim MacLeod, Test Conductor\nRudy Manriquez, Harness Technician\nAyman Mekhail, Test Conductor\nJames Mills, Harness Fabrication Supervisor\nBrian Ottens, Environmental Test Engineer\nRamjit Ramrattan, Harness Technician\nMarco Rosales, Harness Technician\n\nAcknowledgments 185\nIRAC Instrument Handbook\n\nCharles Stone, Harness Technician\n\nINSTRUMENT TEAM SUPPORT\nJohn Anders, Configuration Manager\nKim Brecker, Web Page Manager\nWalt Carel, Transportation\nRon Colvin, Web Server\nJohn Davis, Scheduler\nRobert Dipalo, Transportation\nCristina Doria-Warner, Resource Analyst\nSteve Ford, Program Analyst\nToni Hegarty, Configuration Manager\nKen Lathan, Web Page Support\nLois Pettit, Configuration Manager\nChris Romano, Program Analyst\nSharmaine Stewart, Resource Analyst\nLynette Sullivan, Configuration Manager\nCatherine Traffanstedt, Transportation\nDebra A. Yoder, Configuration Manager\n\nOpe rations phase\n\nThe following are the people who participated in the operational phase of IRAC. The list includes those\nactively involved with the day-to-day operations within the IRAC Instrument Support Team (IST) at SSC,\nand those in the \u201chome team\u201d within the IRAC Instrument Team (IT) at SAO, Harvard, and anyone that\ncontributed to a better understanding of IRAC by producing software code used in the online and offline\npipelines.\n\nIRAC INSTRUMENT SUPPORT TEAM (IST) AT SPITZER SCIENCE CENTER, CALTECH\n\nDr. Bidushi Bhattacharya\nHeidi Brandenburg\nDr. David Cole\nDr. William Glaccum\nDr. Myungshin Im\nDr. James Ingalls\nDr. Thomas Jarrett\nIffat Khan\nDr. Jessica Krick\nDr. Mark Lacy\n\nAcknowledgments 186\nIRAC Instrument Handbook\n\nDr. Seppo Laine\nWen-Piao Lee\nDr. Patrick Lowrance\nDr. Brant Nelson\nDr. JoAnn O\u2019Linger-Luscusk\nDr. Inseok Song\nDr. Gillian Wilson\n\nIRAC INSTRUMENT TEAM AT SAO\/HARVARD\nDr. Giovanni Fazio, IRAC Principal Investigator\nDr. Joseph L. Hora, Instrument Scientist\nDr. Lori E. Allen\nDr. Matthew L. N. Ashby\nDr. Pauline Barmby\nDr. Jiasheng Huang\nDr. Massimo Marengo\nDr. S. Thomas Megeath\nDr. Michael Pahre\nDr. Brian Patten\nDr. Howard Smith\nDr. Zhong Wang\nDr. Steven P. Willner\n\nSPITZER SCIENCE CENTER OBSERVER SUPPORT\/SCIENCE USER SUPPORT\nDr. Seppo Laine\nDr. Solange Ramirez\n\nIRAC IOC\/SV\nDr. Peter Eisenhardt (JPL)\nDr. Daniel Stern (JPL)\n\nIRAC CALIBRATION\/SOFTWARE CONTRIBUTORS\nDr. Stefano Casertano (STScI)\nDr. Mark Dickinson (STScI; NOAO)\nDr. David Elliott (JPL)\nDr. Robert Gehrz (U. Minnesota)\nDr. William Hoffmann (U. Arizona)\nDr. Leonidas Moustakas (JPL)\nEdward Romana (JPL)\n\nPost-Operations phase\nThe Post Operations phase is expected to last 6 months.\n\nList of Laboratories\n\nAcknowledgments 187\nIRAC Instrument Handbook\n\nCaltech (California Institute of Technology, Pasadena, CA)\nSAO (Smithsonian Asttrophysical Observatory, Harvard, MA)\nSBRC (RaytheonVision Systems\/Santa Barbara Research Center, Santa Barbara, CA)\nGSFC (Goddard Space Flight Center, Greenbelt, MD)\nJPL (Jet Propulsion Center, Pasadena, CA)\nSteward Observatory, University of Arizona,Tucson, AZ\nNASA Ames Research Center, Moffett Field, CA\nUniversity of Rochester, Rochester, NY\n\nAcknowledgments 188\nIRAC Instrument Handbook\n\nAppendix G. List of Figures\nFIGURE 2.1. IRAC CRYOGENIC ASSEMBLY MODEL, WITH THE TOP COVER REMOVED TO\nSHOW THE INNER COMPONENTS..........................................................................................4\nFIGURE 2.2. IRAC OPTICAL LAYOUT, TOP VIEW. THE LAYOUT IS SIMILAR FOR BOTH\nPAIRS OF CHANNELS; THE LIGHT ENTERS THE DOUBLET AND THE LONG\nWAVELENGTH PASSES THROUGH THE BEAMSPLITTER TO THE SI:AS DETECTOR\n(CHANNELS 3 AND 4) AND THE SHORT WAVELENGTH LIGHT IS REFLECTED TO THE\nINSB DETECTOR (CHANNELS 1 AND 2).................................................................................6\nFIGURE 2.3. IRAC OPTICS, SIDE VIEW. THE SI:AS DETECTORS ARE SHOWN AT THE FAR\nRIGHT OF THE FIGURE, THE INSB ARRAYS ARE BEHIND THE BEAMSPLITTERS. ...........6\nFIGURE 2.4. SPECTRAL RESPONSE CURVES FOR ALL FOUR IRAC CHANNELS. THE FULL\nARRAY AVERAGE CURVE IS DISPLAYED IN BLACK. THE SUBARRAY AVERAGE\nCURVE IS IN GREEN. THE EXTREMA OF THE FULL ARRAY PER-PIXEL TRANSMISSION\nCURVES ARE ALSO SHOWN FOR REFERENCE.....................................................................9\nFIGURE 2.5. OPTICAL IMAGE DISTORTION IN IRAC CHANNELS. THE PANELS SHOW THE\nIMAGE DISTORTIONS AS CALCULATED FROM A QUADRATIC POLYNOMIAL MODEL\nTHAT HAS BEEN FIT TO IN-FLIGHT DATA. THE MAGNITUDE OF THE DISTORTION\nAND THE DIRECTION TO WHICH OBJECTS HAVE MOVED FROM THEIR IDEAL\nTANGENTIAL PLANE PROJECTED POSITIONS IS SHOWN WITH ARROWS. THE LENGTH\nOF THE ARROWS HAS BEEN INCREASED BY A FACTOR OF TEN FOR CLARITY. THE\nMAXIMUM POSITIONAL DEVIATIONS ACROSS THE ARRAYS FOR THIS QUADRATIC\nDISTORTION MODEL ARE LESS THAN 1.3, 1.6, 1.4 AND 2.2 PIXELS FOR CHANNELS 1\u22124,\nRESPECTIVELY. THE DERIVATION OF THE PIXEL SCALES THAT ARE LISTED IN\nTABLE 2.1 FULLY ACCOUNTED FOR THE QUADRATIC DISTORTION EFFECTS SHOWN\nHERE.......................................................................................................................................10\nFIGURE 2.6 : NON-LINEARITY CURVES FOR THE IRAC DETECTORS. THE DETECTOR\nRESPONSES ARE FAIRLY LINEAR UNTIL SATURATION, WHERE THERE IS A STEEP\nDROP-OFF IN RESPONSIVITY...............................................................................................13\nFIGURE 2.7: FOWLER SAMPLING TIMES FOR ONE PIXEL (FOWLER N=4). THE PN (N=1,2,3,4)\nTEX IS THE EFFECTIVE EXPOSURE TIME, AND TF \u2013 TI IS THE \u201cFRAME TIME,\u201d OR TOTAL\nTIME TO OBTAIN ONE IRAC IMAGE. THE RESET PART OF THE SKETCH IS NOT AT THE\nSAME TIME AND VOLTAGE SCALE AS THE REST OF THE FIGURE. ................................14\nFIGURE 2.8: IRAC POINT SOURCE SENSITIVITY AS A FUNCTION OF FRAME TIME, FOR LOW\nBACKGROUND. TO CONVERT TO MJY\/SR, SEE EQUATION 2.8. .......................................21\nFIGURE 2.9: IRAC POINT SOURCE SENSITIVITY AS A FUNCTION OF FRAME TIME, FOR\nMEDIUM BACKGROUND. TO CONVERT TO MJY\/SR, SEE EQUATION 2.8. ......................22\nFIGURE 2.10: IRAC POINT SOURCE SENSITIVITY AS A FUNCTION OF FRAME TIME, FOR\nHIGH BACKGROUND. TO CONVERT TO MJY\/SR, SEE EQUATION 2.8. ............................23\nFIGURE 3.1 : IRAC DITHER PATTERNS FOR THE \u201cLARGE\u201d SCALE FACTOR. ..........................30\nFIGURE 3.2: CHARACTERISTICS OF THE CYCLING DITHER PATTERN, IN PIXELS................30\nFIGURE 4.1: IRAC INSTRUMENT DARK CURRENT IMAGES. THESE MEASUREMENTS WERE\nMADE DURING A NORMAL CAMPAIGN PRODUCING A SKYDARK WITH AN EXPOSURE\nTIME OF 100 SECONDS..........................................................................................................32\n\nList of Figures 189\nIRAC Instrument Handbook\n\nFIGURE 4.2: IRAC INSTRUMENT SUPER SKYFLATS SHOWING THE FLATFIELD RESPONSE\nAS MEASURED ONBOARD, FOR CHANNELS 1\u20134................................................................34\nFIGURE 4.3. ARRAY LOCATION-DEPENDENT PHOTOMETRIC CORRECTION IMAGES. CH 1\nIS IN THE UPPER LEFT, CH 2 IN THE UPPER RIGHT, CH 3 IN THE LOWER LEFT AND\nCHANNEL 4 IN THE LOWER RIGHT. ....................................................................................44\nFIGURE 4.4: DEPENDENCE OF POINT SOURCE PHOTOMETRY ON THE DISTANCE OF THE\nCENTROID OF A POINT SOURCE FROM THE NEAREST PIXEL CENTER IN CHANNEL 1.\nTHE RATIO ON THE VERTICAL AXIS IS THE MEASURED FLUX DENSITY TO THE MEAN\nVALUE FOR THE STAR, AND THE QUANTITY ON THE HORIZONTAL AXIS IS THE\nFRACTIONAL DISTANCE OF THE CENTROID FROM THE NEAREST PIXEL CENTER......46\nFIGURE 4.5. THE IN-FLIGHT IRAC POINT RESPONSE FUNCTIONS (PRFS) AT 3.6, 4.5, 5.8 AND\n8 MICRONS. THE PRFS WERE RECONSTRUCTED ONTO A GRID OF 0.3\u201d PIXELS, \u00bc THE\nSIZE OF THE IRAC PIXEL, USING THE DRIZZLE ALGORITHM. WE DISPLAY THE PRF\nWITH BOTH A SQUARE ROOT AND LOGARITHMIC SCALING, TO EMPHASIZE THE\nSTRUCTURE IN THE CORE AND WINGS OF THE PRF, RESPECTIVELY. WE ALSO SHOW\nTHE PRF AS IT APPEARS AT THE IRAC PIXEL SCALE OF 1.2\u201d. THE RECONSTRUCTED\nIMAGES CLEARLY SHOW THE FIRST AND SECOND AIRY RINGS, WITH THE FIRST\nAIRY RING BLENDING WITH THE CORE IN THE 3.6 AND 4.5 \u00b5M DATA...........................47\nFIGURE 4.6. THE IRAC POINT RESPONSE FUNCTIONS (PRFS) AT 3.6, 4.5, 5.8 AND 8.0\nMICRONS. THE PRFS WERE GENERATED FROM MODELS REFINED WITH IN-FLIGHT\nCALIBRATION TEST DATA INVOLVING A BRIGHT CALIBRATION STAR OBSERVED AT\nSEVERAL EPOCHS. CENTRAL PRFS FOR EACH CHANNEL ARE SHOWN ABOVE WITH A\nLOGARITHMIC SCALING TO HELP DISPLAY THE ENTIRE DYNAMIC RANGE. THE PRFS\nARE SHOWN AS THEY APPEAR WITH 1\/5TH THE NATIVE IRAC PIXEL SAMPLING OF 1.2\nARCSECONDS TO HIGHLIGHT THE CORE STRUCTURE. ...................................................49\nFIGURE 4.7. EXTENDED SOURCE FLUX CORRECTION FACTORS; SOLID LINES REPRESENT\nEXPONENTIAL FUNCTION FITS TO THE DATA. ALSO INDICATED ARE CORRECTION\nFACTORS DERIVED FROM ZODIACAL LIGHT TESTS, AND GALACTIC HII REGION\nTESTS (E.G. MARTIN COHEN'S GLIMPSE VS. MSX, PRIVATE COMMUNICATION)..........58\nFIGURE 4.8. EXTENDED SOURCE FLUX CORRECTION FACTORS FOR GALAXIES (SOLID\nLINES) VERSUS THE PSF APERTURE CORRECTION FACTORS (DOTTED LINES). THE\nMAIN DIFFERENCE BETWEEN THE TWO IS THE TRULY DIFFUSE SCATTERING\nINTERNAL TO THE ARRAY...................................................................................................58\nFIGURE 4.9. NOISE VERSUS BINNING LENGTH IN CHANNEL 1. TO MAKE THIS PLOT THE\nSURFACE BRIGHTNESS WAS MEASURED IN NINE REGIONS ACROSS AN OBJECT-\nMASKED MOSAIC. THESE REGIONS ARE NOT NEAR THE BRIGHT GALAXIES, STARS,\nOR DIFFUSE PLUMES. THE NOISE IS DEFINED AS THE STANDARD DEVIATION OF\nTHOSE NINE REGIONS. THE BOX SIZE IS INCREMENTALLY INCREASED UNTIL THE\nBOX LENGTH IS MANY HUNDREDS OF PIXELS. FOR REFERENCE THE SOLID LINE\nSHOWS THE EXPECTED LINEAR RELATION. .....................................................................61\nFIGURE 4.10. NOISE VERSUS BINNING LENGTH IN CHANNEL 2. TO MAKE THIS PLOT THE\nSURFACE BRIGHTNESS WAS MEASURED IN SIX REGIONS ACROSS AN OBJECT-\nMASKED MOSAIC. THESE REGIONS ARE NOT NEAR THE BRIGHT GALAXIES, STARS,\nOR DIFFUSE PLUMES. THE NOISE IS DEFINED AS THE STANDARD DEVIATION OF\n\nList of Figures 190\nIRAC Instrument Handbook\n\nTHOSE SIX REGIONS. THE BOX SIZE IS INCREMENTALLY INCREASED UNTIL THE BOX\nLENGTH IS MANY HUNDREDS OF PIXELS. FOR REFERENCE THE SOLID LINE SHOWS\nTHE EXPECTED LINEAR RELATION. ...................................................................................61\nFIGURE 4.11. NOISE AS A FUNCTION OF EXPOSURE TIME (NUMBER OF FRAMES) IN\nCHANNEL 1. THE RESULTS FROM THE WARM MISSION DATA ARE SHOWN WITH X\u2019S\nAND THE EXPECTED BEHAVIOR WITH THE SOLID LINE. THE RESULTS FROM THE\nCRYOGENIC MISSION ARE SHOWN WITH OPEN SQUARES AND THE EXPECTED\nBEHAVIOR WITH THE DASHED LINE..................................................................................63\nFIGURE 4.12. NOISE AS A FUNCTION OF EXPOSURE TIME (NUMBER OF FRAMES) IN\nCHANNEL 2. THE RESULTS FROM THE WARM MISSION DATA ARE SHOWN WITH X\u2019S\nAND THE EXPECTED BEHAVIOR WITH THE SOLID LINE. THE RESULTS FROM THE\nCRYOGENIC MISSION ARE SHOWN WITH OPEN SQUARES AND THE EXPECTED\nBEHAVIOR WITH THE DASHED LINE..................................................................................64\nFIGURE 4.13: POSITION OF A STAR IN THE X (LEFT) AND Y (RIGHT) AXES OF IRAC DURING\nA LONG (8 HR) OBSERVATION. THE ~ 3000 SEC OSCILLATION IS SUPERPOSED ON A\nSLOW DRIFT OF THE STAR TRACKER TO TELESCOPE ALIGNMENT...............................67\nFIGURE 5.1: DATA FLOW FOR PROCESSING A RAW IRAC SCIENCE DCE INTO A BCD THAT\nIS DESCRIBED IN THIS CHAPTER. .......................................................................................70\nFIGURE 5.2: INSBPOSDOM WORKS ONLY ON THE TWO INSB ARRAYS (CHANNELS 1 & 2)\nAND REVERSES THE SENSE OF INTENSITIES.....................................................................72\nFIGURE 5.3: DIAGRAM OF THE WRAPPING OF NEGATIVE VALUES DUE TO TRUNCATION\nOF THE SIGN BIT. ..................................................................................................................73\nFIGURE 5.4: APPLICATION OF IRACWRAPCORR TO CHANNEL 1 DATA. THE MANY\nAPPARENTLY \u201cHOT\u201d PIXELS ARE ACTUALLY WRAPPED NEGATIVE VALUES, WHICH\nARE DETECTED ON THE BASIS OF THEIR VASTLY EXCEEDING THE PHYSICAL\nSATURATION VALUE FOR THE DETECTORS, AND CORRECTED BY SUBTRACTING THE\nAPPROPRIATE VALUE. REAL HOT PIXELS DO NOT EXCEED THE PHYSICAL\nSATURATION VALUE, AND HENCE ARE NOT CHANGED. ................................................75\nFIGURE 5.5: ILLUSTRATION OF BIT TRUNCATION USED BY IRAC FOR GROUND\nTRANSMISSION, NECESSITATING IRACNORM. THE INTERNALLY STORED 24-BIT\nWORD IN TRUNCATED TO 16 BITS, WITH A SLIDING WINDOW SET BY THE BARREL\nSHIFT VALUE. ILLUSTRATED IS THE CASE FOR ABARREL=4..........................................75\nFIGURE 5.6: CORRECTION OF CABLE-INDUCED BANDWIDTH ERROR BY IRACEBWC. THE\nILLUSTRATED DATA SHOW A COSMIC RAY HIT...............................................................78\nFIGURE 5.7: FIRST-FRAME EFFECT. DARK COUNTS AS A FUNCTION OF INTERVAL\nBETWEEN FRAMES. THIS FIGURE IS FOR A 30 SECOND EXPOSURE FRAME..................79\nFIGURE 5.8: CORRECTION OF PSEUDO-MUXBLEED FOR CHANNEL 1. SHOWN IS A BRIGHT\nSOURCE WITHIN A CALIBRATION AOR AND A BACKGROUND OF SOURCES UNDER\nTHE MUXBLEED LIMIT.........................................................................................................80\nFIGURE 5.9: TRANSPOSITION OF AN IRAC CHANNEL 1 DARK IMAGE BY THE IMFLIPROT\nMODULE.................................................................................................................................85\nFIGURE 5.10. AN IMAGE SHOWING ALL FOUR READOUT CHANNEL IMAGES SIDE BY SIDE.\nTHESE HAVE BEEN OBTAINED BY REARRANGING THE COLUMNS IN THE ORIGINAL\n\nList of Figures 191\nIRAC Instrument Handbook\n\nIMAGE. MUXBLEED IS APPARENT IN THE BOTTOM RIGHT OF THE 4TH READOUT\nCHANNEL IMAGE. .................................................................................................................92\nFIGURE 5.11. SUBTRACTION OF THE MEDIAN BACKGROUND FROM THE READOUT\nCHANNEL IMAGES. THIS MAKES THE MUXSTRIPE MUCH MORE APPARENT IN THE 4TH\nREADOUT CHANNEL IMAGE (ON THE RIGHT)...................................................................93\nFIGURE 5.12. PROFILES SHOWING THE COLUMN MEDIAN VERSUS ROW VALUES FOR\nIDENTIFYING MUXSTRIPE. THE MUXSTRIPE IS NOW IDENTIFIABLE BETWEEN ROWS\n125 AND 200 (SIGNIFICANTLY LOWER VALUES THAN THE MEDIAN BACKGROUND)..94\nFIGURE 7.1: SUPER SKYFLATS FOR IRAC. THESE WERE MADE BY COMBINING THE FLAT\nFIELDS FROM THE FIRST FIVE YEARS OF OPERATIONS. THE DARK SPOT IN CHANNEL\n4, NEAR THE LEFT SIDE AND ABOUT HALF WAY UP, AND THE DARK SPOT IN ABOUT\nTHE SAME PLACE IN CHANNEL 2, ARE DUE TO THE SAME SPECK OF\nCONTAMINATION ON THE CHANNEL 2\/4 PICKOFF MIRROR. THE DARKEST PIXELS IN\nTHE SPOT ARE 20% BELOW THE SURROUNDING AREA IN CHANNEL 2, AND 32% IN\nCHANNEL 4. FLAT-FIELDING IN THE PIPELINE FULLY CORRECTS FOR THESE DARK\nSPOTS IN THE DATA............................................................................................................ 108\nFIGURE 7.2: IMAGES SHOWING THE MUXBLEED EFFECT (THE HORIZONTAL LINE ON\nBOTH SIDES OF A BRIGHT STELLAR IMAGE). THE PIXELS ON THE LEFT SIDE OF THE\nBRIGHT SOURCE ARE PIXELS ON ROWS FOLLOWING THE ROW IN WHICH THE\nBRIGHT SOURCE WAS LOCATED (AND HAVE WRAPPED AROUND IN THE READOUT\nORDER OF THE ARRAY). THE VERTICAL (WHITE) LINES ARE DUE TO THE SO-CALLED\n\u201cCOLUMN PULL-DOWN\" EFFECT. THESE ARE 12-SECOND BCD FRAMES IN IRAC\nCHANNEL 1, TAKEN FROM IRAC PROGRAM PID = 618, AORKEY = 6880000. ................. 111\nFIGURE 7.3: DEMONSTRATION OF THE S18 PIPELINE MUXBLEED REMOVAL. THE IMAGE\nON THE LEFT IS BEFORE AND THE ONE ON THE RIGHT IS AFTER THE CORRECTION.\nTHESE ARE FIRST LOOK SURVEY CHANNEL 1 DATA, TAKEN FROM AORKEY =\n4958976. NOTE THAT THE BRIGHTEST STAR IN THE UPPER-LEFT CORNER IS HEAVILY\nSATURATED AND THE CURRENT MUXBLEED SCHEME CAN CORRECT MUXBLEED\nFROM A SATURATED SOURCE ALSO................................................................................ 111\nFIGURE 7.4: A TYPICAL BANDWIDTH EFFECT TRAIL IN CHANNEL 4, IN A 30 SECOND\nFRAME. THESE DATA WERE TAKEN FROM PROGRAM PID=1154, AORKEY = 13078016.\n.............................................................................................................................................. 112\nFIGURE 7.5: THE BANDWIDTH EFFECT WHEN A BRIGHT OBJECT IS IN THE LAST 4\nCOLUMNS. IRC+10216, STRONGLY SATURATED, IS JUST OFF THE RIGHT SIDE OF THE\nCHANNEL 3 ARRAY. EVEN THE FILTER GHOST IS SATURATED. THE BANDWIDTH\nEFFECT APPEARS ON THE LEFT SIDE OF THE ARRAY. THESE DATA WERE TAKEN\nFROM PROGRAM PID = 124, AORKEY = 5033216. .............................................................. 113\nFIGURE 7.6: IRAC CHANNEL 1 (LEFT) AND CHANNEL 2 (RIGHT) OBSERVATIONS OF A\nCROWDED FIELD WITH COLUMN PULL-DOWN APPARENT FROM THE BRIGHTEST\nSOURCES. NOTE THAT THE BRIGHTER SOURCES AFFECT A LARGER NUMBER OF\nCOLUMNS. THESE DATA WERE TAKEN FROM PROGRAM PID = 613, AORKEY = 6801408.\n.............................................................................................................................................. 114\nFIGURE 7.7: CHANNELS 1 AND 2 (TOP) AND 3 AND 4 (BOTTOM) SHOWING INTER-CHANNEL\nCROSSTALK (DARK SPOTS NEAR THE CENTER OF THE LOWER PANELS)................... 115\n\nList of Figures 192\nIRAC Instrument Handbook\n\nFIGURE 7.8: MEDIAN OF CHANNEL 1 IMAGES FROM A CALIBRATION OBSERVATION\nPERFORMED AFTER OBSERVING POLARIS. THE 5 BRIGHT SPOTS ARE PERSISTENT\nIMAGES FROM STARING AT THE STAR WHILE OBSERVING, WHILE THE SET OF CRISS-\nCROSSING LINES WERE GENERATED BY SLEWS BETWEEN THE VARIOUS POINTINGS.\nTHESE OBSERVATIONS WERE TAKEN FROM AORKEY=3835904, IN PROGRAM PID=19.\n.............................................................................................................................................. 117\nFIGURE 7.9: RESIDUAL IMAGE BRIGHTNESS DECAY AS A FUNCTION OF TIME INTERVAL\nSINCE EXPOSURE TO A FIRST MAGNITUDE SOURCE AT 3.6 \u039cM. THE RESIDUAL IS\nCOMPARED TO THREE TIMES THE NOISE IN THE SKY BACKGROUND AS MEASURED\nIN AN EQUIVALENT APERTURE. THE FITTED EXPONENTIAL DECAY FUNCTION IS\nPLOTTED AS THE DOT-DASHED LINE. THESE CURVES HAVE BEEN SMOOTHED TO\nMITIGATE FLUX JUMPS DUE TO SOURCES AT THE POSITION OF THE ORIGINAL\nSOURCE IN SUBSEQUENT IMAGES. .................................................................................. 119\nFIGURE 7.10: AN IMAGE OF THE M51 SYSTEM, SHOWING AN OVERLAY OF THE IRAC\nFIELDS OF VIEW, WITH THE SCATTERED LIGHT ORIGIN ZONES FOR CHANNELS 1 AND\n2 OVERLAID......................................................................................................................... 120\nFIGURE 7.11: CHANNEL 1 IMAGE SHOWING SCATTERED LIGHT ON BOTH SIDES OF A\nBRIGHT STAR. THE SCATTERED LIGHT PATCHES ARE MARKED WITH WHITE \u201cS\"\nLETTERS. THE IMAGES WERE TAKEN FROM PROGRAM PID 30 DATA. ........................ 121\nFIGURE 7.12: CHANNEL 2 IMAGE SHOWING SCATTERED LIGHT ON ONE SIDE OF A BRIGHT\nSTAR. THE SCATTERED LIGHT PATCHES ARE MARKED WITH WHITE \u201cS\" LETTERS.\nTHE IMAGES WERE TAKEN FROM PROGRAM PID 30 DATA........................................... 122\nFIGURE 7.13: CHANNEL 3 IMAGE SHOWING SCATTERED LIGHT FROM A SCATTERING\nSTRIP AROUND THE EDGE OF THE ARRAY WHERE A BRIGHT STAR IS LOCATED. THE\nSCATTERED LIGHT PATCHES ARE MARKED WITH WHITE \u201cS\" LETTERS. THE IMAGES\nWERE TAKEN FROM PROGRAM PID 30 DATA.................................................................. 123\nFIGURE 7.14: CHANNEL 4 IMAGES SHOWING SCATTERED LIGHT FROM A SCATTERING\nSTRIP AROUND THE EDGE OF THE ARRAY WHERE A BRIGHT STAR IS LOCATED. THE\nSCATTERED LIGHT PATCHES ARE POINTED TO BY BLACK ARROWS. THE IMAGES\nWERE TAKEN FROM PROGRAM PID 30 DATA.................................................................. 124\nFIGURE 7.15: TYPICAL IMAGE SECTIONS SHOWING THE BANDING EFFECT. THESE ARE\nCHANNEL 3 (LEFT) AND CHANNEL 4 (RIGHT) IMAGES OF THE SAME OBJECT (S140),\nADOPTED FROM A REPORT BY R. GUTERMUTH. THESE DATA WERE TAKEN FROM\nPROGRAM PID 1046, AORKEY 6624768. ............................................................................. 125\nFIGURE 7.16: FILTER AND BEAMSPLITTER GHOSTS. ............................................................. 127\nFIGURE 7.17: PUPIL GHOST IN CHANNEL 2 FROM V416 LAC. ................................................ 128\nFIGURE 7.18: PART OF THE CHANNEL 1 MOSAIC (FROM OBSERVATIONS IN PID 181;\nAORKEYS 5838336, 5838592, 5839872 AND 5840128) OF THE SWIRE FIELD NEAR MIRA\nSHOWING THE 24 ARCMINUTE RADIUS RING OF STRAY LIGHT FROM THE\nTELESCOPE. ......................................................................................................................... 129\nFIGURE 7.19: CHANNEL 2 IMAGES FROM THE SWIRE MAP SHOWING STRAY LIGHT\nSPLOTCHES FROM MIRA, WHICH WAS ABOUT 30 ARCMINUTES AWAY. SUCCESSIVE\nPAIRS OF IMAGES WERE SLIGHTLY DITHERED. THE LAST PAIR IS ABOUT 5\nARCMINUTES FROM THE FIRST PAIR, BUT HAS A SIMILAR SPLOTCH. NOTE THE\n\nList of Figures 193\nIRAC Instrument Handbook\n\nABSENCE OF ANY STRAY LIGHT IN THE SECOND IMAGE, THOUGH IT WAS CENTERED\nONLY A FEW PIXELS AWAY FROM THE FIRST IMAGE. THE IMAGES ARE FROM PID\n181, AORKEY 5838336; EXPID 187-192, 199, AND 200......................................................... 130\nFIGURE 7.20: THE CENTRAL 128X128 PIXELS OF IRAC 12-SECOND IMAGES TAKEN ON\nJANUARY 20, 2005 DURING A MAJOR SOLAR PROTON EVENT. CHANNELS 1 AND 2 ARE\nTOP LEFT AND TOP RIGHT; CHANNELS 3 AND 4 ARE BOTTOM LEFT AND BOTTOM\nRIGHT. EXCEPT FOR THE BRIGHT STAR IN CHANNELS 1 AND 3, ALMOST EVERY\nOTHER SOURCE IN THESE IMAGES IS A COSMIC RAY. THESE DATA ARE FROM\nOBSERVATIONS IN PID 3126............................................................................................... 132\n\nList of Figures 194\nIRAC Instrument Handbook\n\nAppendix H. List of Tables\nTABLE 2.1: IRAC IMAGE QUALITY PROPERTIES. ........................................................................7\nTABLE 2.2: IRAC DETECTOR CHARACTERISTICS. ....................................................................11\nTABLE 2.3: IRAC CHANNEL CHARACTERISTICS.......................................................................12\nTABLE 2.4: USEFUL QUANTITIES FOR IRAC SENSITIVITY CALCULATIONS..........................17\nTABLE 2.5: BACKGROUND BRIGHTNESS IN IRAC WAVEBANDS. ...........................................18\nTABLE 2.6: FOWLER NUMBERS FOR IRAC FRAMES .................................................................19\nTABLE 2.7: IRAC HIGH-DYNAMIC-RANGE (HDR) FRAMESETS................................................19\nTABLE 2.8: IRAC POINT-SOURCE SENSITIVITY, LOW BACKGROUND (1\u03c3 , \u00b5JY)....................19\nTABLE 2.9: IRAC POINT-SOURCE SENSITIVITY, MEDIUM BACKGROUND (1 \u03c3 , \u00b5JY). ............20\nTABLE 2.10: IRAC POINT-SOURCE SENSITIVITY, HIGH BACKGROUND (1 \u03c3 , \u00b5JY). ................20\nTABLE 2.11: MAXIMUM UNSATURATED POINT SOURCE (IN MJY), AS A FUNCTION OF IRAC\nFRAME TIME..........................................................................................................................24\nTABLE 3.1: CHARACTERISTICS OF THE DITHER PATTERNS....................................................28\nTABLE 4.1: THE PHOTOMETRIC CALIBRATION AND ZERO MAGNITUDE FLUX FOR IRAC..36\nTABLE 4.2: IRAC NOMINAL WAVELENGTHS AND BANDWIDTHS. .........................................40\nTABLE 4.3: COLOR CORRECTIONS FOR POWER-LAW SPECTRA, F\u03bd \u221d \u03bd \u03b1 .............................40\nTABLE 4.4: COLOR CORRECTIONS FOR BLACKBODY SPECTRA.............................................40\nTABLE 4.5: COLOR CORRECTIONS FOR ZODIACAL LIGHT SPECTRUM..................................41\nTABLE 4.6: COLOR CORRECTIONS FOR NGC 7023 (PAH-DOMINATED) SPECTRUM. .............42\nTABLE 4.7: IRAC APERTURE CORRECTIONS. ............................................................................54\nTABLE 4.8: IRAC EXTENDED SOURCE PHOTOMETRICAL CORRECTION COEFFICIENTS. ....59\nTABLE 4.9: IRAC SURFACE BRIGHTNESS CORRECTION FACTORS.........................................59\nTABLE 5.1: BANDWIDTH CORRECTION COEFFICIENTS. ..........................................................77\nTABLE 6.1 SAMPLE IRAC FILE NAMES.......................................................................................98\nTABLE 7.1: DEFINITION OF BITS IN THE \u201cPMASK\u201d.................................................................. 105\nTABLE 7.2: DEFINITION OF BITS IN THE \u201cIMASK\u201d. ................................................................. 106\nTABLE 7.3: WARM MISSION RESIDUAL IMAGE DURATIONS................................................. 118\nTABLE 7.4. COEFFICIENTS FOR CHANNEL 1 & 2 GHOST LOCATIONS. ................................. 126\n\nList of Tables 195\nIRAC Instrument Handbook\n\nAppendix I. Version Log\n\nVersion 2.0.1, June 2011:\n\nReference to Krick et al. added to Section 4.11.5\n\nVersion 2.0, April 2011:\n\nA new processing version S18.18 BCD FITS header replaced the old header file in Appendix D.\n\nDiscussion of a few new header keywords, including timing keywords and bad data value keywords,\n\nThe filenames in Table 6.1 were updated.\n\nAdded information about warm mission persistent images and edited the information on cryogenic\nmission persistent images in Section 7.2.8.\n\nAdded information about the effect of very deep surface brightness level observations in Section\n4.11.\n\nAdded an Index at the end of the document.\n\nVersion 1.0, February 2010:\n\nThe first version of the IRAC Instrument Handbook, which includes information from the old IRAC\nData Handbook, the old IRAC Pipeline Description Document, the cryogenic Spitzer Observer\u2019s\nManual (SOM), and Spitzer Science Center web pages.\n\nVersion Log 196\nIRAC Instrument Handbook\n\nBibliography\n\n[1] Abraham, P., Leinert, C., & Lemke, D. 1997, Search for Brightness Fluctuations in the Zodiacal Light\nat 25 \u00b5m with ISO, A&A, 328, 702\n[2] Anderson, J. & King, I. R. 2000, Toward High-Precision Astrometry with WFPC2. I. Deriving an\nAccurate Point-Spread Function, PASP, 112, 1360\n[3] Arendt, R. G., Fixsen, D. J., & Moseley, S. H. 2000, Dithering Strategies for Efficient Self-Calibration\nof Imaging Arrays, Ap.J., 536, 500\n\nAstronomical Satellite (IRAS) Catalogs and Atlases. Volume I: Explanatory Supplement, \u00a7VI.C.3\n[4] Beichman, C. A., Neugebauer, G., Habing, H. J., Clegg, P. E., Chester, T. J. 1988, Infrared\n\n[5] Blommaert et al. 2003, CAM \u2013 The ISO Camera ISO Handbook, Vol. 2, Ver 2.0 (Noordwijk: ESA),\n(http:\/\/iso.esac.esa.int\/manuals\/HANDBOOK\/cam_hb\/)\n[6] Charbonneau, D., et al. 2005, Detection of Thermal Emission from an Extrasolar Planet, ApJ, 626,\n523\n[7] Cohen, M., Megeath, S. T., Hammersley, P. L., Mart\u00edn-Luis, F., Stauffer, J. 2003, Spectral Irradiance\nCalibration in the Infrared. XIII. \u201cSupertemplates\u2019\u2019 and On-Orbit Calibrators for the SIRTF Infrared\nArray Camera, AJ, 125, 2645\n[8] Estrada, A. D., et al. 1998, Si:As IBC IR Focal Plane Arrays for Ground-Based and Space-Based\nAstronomy, Proc. SPIE, 3354, 99\n[9] Fazio, G. G., et al. 2004, The Infrared Array Camera (IRAC) for the Spitzer Telescope, ApJS, 154, 10\n[10] Franceschini, A., Toffolatti, L., Mazzei, P., Danese, L., & de Zotti, G., 1991, Galaxy Counts And\nContributions to the Background Radiation from 1 micron to 1000 Microns, A&AS, 89, 285\n[11] Fruchter, A. S., & Hook, R. N. 2002, Drizzle: A Method for the Linear Reconstruction of\nUndersampled Images, PASP, 114, 144\n[12] Hauser, M. G., et al. 1998, COBE Diffuse Infrared Background Experiment (DIRBE) Explanatory\nSupplement, (Washington: GSFC) (http:\/\/lambda.gsfc.nasa.gov\/product\/cobe\/dirbe_exsup.cfm)\n[13] Hoffmann, W. F., Hora, J. L., Fazio, G. G., Deutsch, L. K., & Dayal, A. 1998, MIRAC2: a Mid-\nInfrared Array Camera for Astronomy, Proc. SPIE, 3354, 24\n[14] Hora, J. L, et al. 2008, Photometry Using the Infrared Array Camera on the Spitzer Space Telescope,\nPASP, 120, 1233\n[15] Hora, J. L. et al. 2004, In-Flight Performance And Calibration of the Infrared Array Camera (IRAC)\nfor the Spitzer Space Telescope, SPIE, 5487, 244\n[16] Krick, J., et al. 2011, Spitzer IRAC Low Surface Brightness Observations of the Virgo Cluster, ApJ,\n735, 76\n[17] Lacy, M., et al. 2005, The Infrared Array Camera Component of the Spitzer Space Telescope First\nLook Survey, ApJS, 161, 41\n[18] Lauer, T. R. 1999, The Photometry of Undersampled Point-Spread Functions, PASP, 111, 1434\n[19] Mighell, K. J. 2005, Stellar Photometry And Astrometry with Discrete Point Spread Functions,\nMNRAS, 361, 861\n[20] Mighell, K. J., Glaccum, W., Hoffmann, W. 2008, Improving the Photometric Precision of IRAC\nChannel 1, Proc. SPIE, Vol 7010, p. 70102W\n\nBibliography 197\nIRAC Instrument Handbook\n\n[21] Quijada, M. A., Marx, C. T., Arendt, R. G., & Moseley, S. H. 2004, Angle-of-Incidence Effects in\nthe Spectral Performance of the Infrared Array Camera of the Spitzer Space Telescope, Proc. SPIE, 5487,\n244\n[22] Reach W. T., Morris, P., Boulanger, F., & Okumura, K. 2003, The Mid-infrared Spectrum of the\nZodiacal and Exozodiacal Light, Icarus, 164, 384\n[23] Reach, W. T., et al. 2005, Absolute Calibration of the Infrared Array Camera on the Spitzer Space\nTelescope, PASP, 117, 978\n[24] Shupe, D., Moshir, M., Li, J., Makovoz, D., Narron, R., Hook, R. N. 2005, The SIP Convention for\nRepresenting Distortion in FITS Image Headers, ASP Conf. Series, Vol 347, San Francisco: PASP, p.\n\nBibliography 198\nIRAC Instrument Handbook\n\nIndex\n\n2MASS, 35, 65, 90, 94, 95, 98, 100, 102, 122, file name, 98\n134, 138, 141, 142, 163 orientation, 85, 97\nanneal, 105, 117, 145, 147\nAOR, 2, 78, 80, 89, 94, 95, 96, 97, 99, 100, 101, calibration, 1, 2, 4, 5, 7, 8, 15, 26, 27, 31, 33, 34,\n103, 105, 118, 119, 134, 138, 140, 142, 155, 35, 36, 37, 38, 43, 49, 53, 55, 57, 58, 59, 62,\n167, 170, 171, 173, 191 68, 78, 80, 81, 82, 83, 84, 85, 86, 94, 97, 98,\nAORKEY, 2, 99, 101, 102, 111, 112, 113, 114, 99, 100, 102, 105, 117, 121, 142, 143, 144,\n117, 125, 130, 159, 162, 163, 171, 173, 191, 146, 147, 149, 150, 151, 152, 153, 154, 155,\n192, 193 156, 158, 162, 164, 178, 190, 191, 192, 194\naperture, 17\naperture correction, 51, 53, 54, 55, 58, 59, 149, zero magnitude flux density, 36\n150, 152, 154, 158, 190\ncold assembly, 76\naperture photometry, 2, 36, 48, 50, 53, 56, 114,\ncolor correction, 37, 38, 39, 40, 150, 152\n126, 150, 151, 152, 153, 155, 156, 158, 160,\ncolumn pull-down, 111, 113, 114, 125, 133, 191,\n161, 162, 163, 165\n192\nextended source, 54, 56, 114 column pull-up, 125\nextended sources, 55 columns, 8, 27, 72, 81, 91, 92, 101, 103, 104,\npoint source, 43, 53, 114, 149, 151 110, 112, 113, 114, 125, 126, 137, 142, 191,\n192\nAPEX, 48, 137, 151, 153, 154, 155, 156, 160, confusion, 19, 23\n161, 162, 163, 164 confusion noise, 23, 137\nAstronomical Observation Template (AOT), 18 convolution, 55, 153, 175\nbackground, 1, 2, 9, 16, 17, 18, 19, 20, 21, 22, cosmic rays, 85, 103, 132, 133, 152\n23, 25, 31, 33, 36, 41, 42, 43, 48, 49, 50, 53, crosstalk, 103, 106, 115, 192\n54, 55, 56, 58, 62, 64, 80, 82, 83, 84, 89, 91, dark, 17, 19\n93, 94, 95, 100, 103, 104, 107, 108, 114, 115, dark current, 31, 32, 56, 77, 80, 104, 105, 162,\n118, 119, 120, 121, 131, 134, 137, 139, 142, 189\n143, 144, 145, 147, 150, 151, 153, 155, 156, data\n158, 161, 162, 163, 164, 189, 191, 192, 194\nbad pixel, 100, 104, 105 DN, 11, 17, 36, 71, 72, 73, 74, 76, 77, 81, 82,\nbanding, 89, 90, 91, 106, 114, 125, 126, 127, 84, 86, 88, 90, 98, 100, 103, 109, 113, 114,\n140, 141, 148, 159, 192 138, 144, 167, 169, 174\nbandwidth effect, 49, 104, 106, 112, 113, 149, units, 16, 17, 37, 41, 51, 52, 55, 56, 83, 85,\n151, 191, 192 95, 98, 100, 109, 149, 150, 151, 155, 164\nBCD, 2, 11, 43, 52, 53, 54, 56, 65, 68, 70, 71,\nDCE, 68, 70, 87, 89, 102, 106, 143, 167, 168,\n78, 84, 86, 88, 89, 92, 94, 95, 97, 98, 99, 100,\n171, 173, 190\n102, 103, 104, 105, 106, 109, 110, 111, 113,\nDCENUM, 99, 102, 171\n114, 122, 126, 131, 134, 136, 137, 138, 139,\ndetector\n140, 141, 143, 144, 149, 151, 153, 155, 159,\n161, 163, 164, 165, 167, 173, 190, 191, 195\n\nIndex 199\nIRAC Instrument Handbook\n\nchannel, 4, 6, 9, 24, 25, 26, 36, 41, 43, 44, 45, BUNIT, 86, 100, 169\n46, 48, 49, 50, 51, 57, 60, 61, 63, 64, 66, CD matrix, 52, 65, 66, 86, 87, 95, 140, 168\n76, 80, 81, 82, 83, 84, 85, 86, 90, 92, 93, CRPIX, 52, 86, 150\n94, 95, 96, 99, 101, 102, 103, 104, 105, CRVAL, 65, 150\n108, 109, 110, 111, 112, 113, 114, 115, DECRFND, 65\n116, 117, 118, 120, 123, 125, 126, 128, EXPID, 99, 102, 130, 171, 193\n129, 130, 132, 139, 141, 142, 143, 144, EXPTIME, 71, 100, 150, 151, 167\n145, 146, 147, 150, 154, 155, 156, 158, FLUXCONV, 36, 86, 100, 150, 151, 169\n161, 162, 167, 189, 190, 191, 192, 194 FRAMTIME, 71, 99, 167\nFowler, 13, 14, 15, 19, 26, 69, 71, 74, 77, 81, HDRMODE, 100, 144, 168\n83, 88, 100, 103, 109, 110, 168, 189, 194 ORIG_DEC, 66, 87, 96, 100, 142, 169\nreadout, 11, 14, 18, 26, 75, 76, 80, 81, 82, 83, ORIG_RA, 66, 87, 96, 100, 142, 169\n92, 93, 94, 111, 142, 191 RARFND, 65, 66, 96, 100, 134, 142\nUSEDBPHF, 65, 87, 100, 145, 168\ndistortion, 6, 7, 10, 29, 42, 52, 53, 65, 86, 87, 88,\n94, 99, 142, 154, 168, 169, 189 High Dynamic Range (HDR), 18\ndither, 18 IDL, 44, 98, 154, 165, 166\ndithering, 26, 27, 28, 29, 30, 33, 50, 65, 90, 101, image\n115, 116, 118, 189, 194\ndrizzle, 29, 46, 47, 96, 136, 190 artifacts, 2, 89, 90, 91, 92, 103, 129, 149, 151,\nextended source, 17, 23 159, 172, 195\nfiducial frame, 94, 135 elecronic glow, 77\nfield of view, 5, 7, 9, 26, 27, 32, 42, 43, 56, 62,\n90, 94, 139 InSb, 5, 6, 11, 71, 72, 73, 80, 104, 107, 109,\nfirst-frame effect, 77, 78, 104, 135, 139 110, 112, 174, 179, 180, 189, 191\nflat, 18, 19 IOC, 4, 8, 10, 16, 23, 46, 116, 133, 174, 175,\nflatfield, 106 187\nflux calibration, 53, 85, 155 IRAF, 50, 98, 149, 154\nFowler sampling, 18, 19 jitter, 66\nghost linearization, 12, 78, 79, 82, 83, 99, 109\nmapping, 5, 26, 27, 43, 60, 62, 65, 140, 141\nbeamsplitter, 127 masks, 62, 90, 96, 104, 105, 124, 136, 140, 141\nfilter, 127\npupil, 128 imask, 85, 89, 90, 91, 105, 106, 116, 122,\n131, 138, 139, 141, 194\nghosts, 50, 89, 103, 126, 127, 128, 141, 192 pmask, 99, 105, 106, 194\ngyro drift, 65 rmask, 105, 135, 136, 151\nHDR, 18, 19, 26, 49, 50, 66, 78, 88, 96, 97, 98,\n99, 106, 138, 140, 141, 143, 144, 148, 159, mosaicking, 2, 94, 96, 131, 134, 141, 149, 163\n168, 194\ncoverage map, 95, 96, 136, 149, 151\nheader, 52, 53, 56, 65, 68, 71, 72, 78, 83, 85, 86,\n94, 99, 100, 101, 102, 134, 136, 138, 140, mosaics, 29, 48, 51, 54, 63, 66, 90, 95, 96, 97,\n141, 142, 143, 144, 145, 150, 151, 195 110, 115, 122, 123, 124, 129, 131, 134, 135,\nkeyword, 65, 78, 83, 100, 138\n\nIndex 200\nIRAC Instrument Handbook\n\n136, 137, 139, 140, 141, 148, 149, 151, 160, history file, 94, 95\n163 refinement, 65, 94, 95, 96, 98, 134, 142\nnamelist, 131, 134, 135, 136, 137, 150, 164\nnoise pixels, 17 post-BCD, 65, 94, 97, 105, 134, 137, 140\nobserving mode, 1, 143, 173 Rayleigh-Jeans, 33, 43, 44\nresidual image, 89, 116, 117, 118, 194\nHDR, 18, 148 residual images\nrepeats, 99\nslew residual, 116\noutlier detection, 135\noutlier rejection, 33, 84, 88, 95, 96, 97, 103, 106, rows, 19, 27, 91, 93, 94, 101, 103, 104, 110,\n115, 116, 124, 130, 131, 135, 136, 137, 149, 111, 114, 125, 126, 191\n150, 151, 164 saturation, 8, 11, 13, 24, 25, 49, 50, 73, 75, 88,\nOverlap correction, 144 89, 90, 91, 103, 105, 106, 109, 118, 125, 138,\nPAO, 117 139, 141, 189, 191\npipeline, 1, 2, 12, 31, 33, 43, 55, 65, 68, 71, 74, scattered light, 17, 89, 90, 99, 120, 121, 122,\n76, 78, 83, 84, 86, 88, 89, 92, 94, 95, 96, 97, 123, 124, 125, 129, 141, 146, 147, 192\n98, 100, 102, 103, 105, 106, 108, 109, 110, sensitivity, 16, 19, 21\n111, 112, 113, 114, 115, 116, 118, 121, 122, Si:As, 73, 91, 104, 109, 112, 125, 132\n125, 130, 134, 135, 136, 138, 139, 140, 141, spectral lines, 42\n142, 143, 144, 146, 147, 148, 149, 150, 151, spectral response, 7, 8, 9, 37, 39, 40, 42, 150,\n155, 163, 173, 178, 191 152\nstray light, 17\nBCD, 105, 134 subarray, 9, 15, 19, 20, 25, 26, 27, 29, 54, 68,\ncalibration, 68 71, 78, 81, 96, 97, 98, 99, 140, 142, 144, 168,\npost-BCD, 65, 94, 97, 105, 134 189\nversion, 102, 110 superboresight, 66, 94, 95, 96, 134, 142, 163\nthroughputs, 8, 9, 17, 18, 108\npoint response function (PRF), 17 uncertainty image, 76, 97, 137\npoint source, 19 vignetting, 108\npoint spread function, 46 warm electronics, 15, 103, 109, 112\npointing, 18, 64, 65, 66, 86, 87, 95, 98, 100, 134, zodiacal background, 7, 17, 24, 25, 33, 41, 42,\n138, 139, 142, 145, 168, 175 43, 53, 56, 58, 62, 78, 83, 84, 100, 106, 107,\n108, 121, 144, 145, 146, 147, 164, 190, 194\naccuracy, 94, 142\n\nIndex 201\n\n\nTo top","date":"2015-06-03 20:21:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5092017650604248, \"perplexity\": 6297.960135437057}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-22\/segments\/1433195036920.5\/warc\/CC-MAIN-20150601214356-00020-ip-10-180-206-219.ec2.internal.warc.gz\"}"} | null | null |
[uigesturerecognizer](../../index.md) / [it.sephiroth.android.library.uigestures](../index.md) / [UIScreenEdgePanGestureRecognizer](index.md) / [handleMessage](./handle-message.md)
# handleMessage
`protected open fun handleMessage(msg: `[`Message`](https://developer.android.com/reference/android/os/Message.html)`): `[`Unit`](https://kotlinlang.org/api/latest/jvm/stdlib/kotlin/-unit/index.html)
Overrides [UIGestureRecognizer.handleMessage](../-u-i-gesture-recognizer/handle-message.md)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,373 |
Interview with Brummie Artist Sarah Silverwood
By Sunny S, April 19, 2015 at 5:12 pm
Image Credit: Sarah Taylor Silverwood, Battles of Mathematics #1, 2014
Continuing my trend of Brummie activity in Chicago, I interview Sarah Silverwood who spent time in the Windy City.
She visited Chicago to make new connections and carry out research for new work on public housing and even left drawings behind to be found in and around the city!
Sarah joins me for a QnA as she tells me more about her visit and what inspired her to become an artist.
Hi Sarah, please introduce yourself?
I am a visual artist based in Birmingham UK. I have a drawing based practice, often bringing together historical material with ideas from pop culture, and playing with narrative forms, like a tabloid newspaper, a comic book or a fabric pattern.
Like myself, you have an interest in Chicago. What attracted you to the city and how did you put that into your drawings?
Ideas around place, cities and architecture have always influenced my work, in particular my comic CITY and the body of work LIVING CITY.
These projects had centred around constructions of identity within urban culture. I had been researching the differing layers of history within cities, in particular researching the idea of a city as a place for living. Chicago has an incredible history relating to its public housing in particular, which has undergone major upheaval. Although the buildings may have gone, layers of stories and identities remain in a landscape. The sheer size and scale of Chicago, and its fast paced regeneration, was an incredible starting point for a new project.
In October, you visited Chicago as an artist in residence for Chicago Sister Cities International. What is an artist in residence, and how did that come about?
An artist residency is a broad term that covers programs that organisations offer to artists and creatives, which gives time and space outside of their usual working life to produce new work, provide a time of reflection and/or undertake research.
Chicago and Birmingham are sister cities, and Chicago Sister Cities International (CSCI) supports cultural exchange between the two countries. I was invited by CSCI and the British Consulate in Chicago to be artist in residence in Chicago to expand my networks, initiate projects and make new work. They helped instigate meetings and suggest projects that might be useful to visit in relation to my work.
While there, you teamed up with web developer Appoet, who launched an app called Infused.
How does Infused work, and what pictures of yours can those who download the app expect to see?
My work has an ethos behind it inspired by an open source culture, often resulting in mass produced publications or prints as counterparts to a physical work, or free digital downloads of works. I think it is important to consider how the format of a conventional gallery hang can be expanded to reach more people and give an alternative mode for the distribution of art.
With this in mind I used Infused as a platform for exploring the idea that people could experience art in the place that the artist intends. Infused is a hyperlocal broadcasting and publishing app, allowing you to leave art in physical locations. As users pass these locations in real life, they will receive a push notification. I have left drawings to be found in Chicago, so that you can follow my walking route and download free digital artworks as you go. This allow you to experience the work in a public setting, where I intended you to see it. Also, the majority of the artworks will be only available to see in this way, so it will be a unique experience.
Describe the creative process behind your art in Chicago?
I spent a lot of time researching material for new work on architecture and housing – taking photographs, meeting people, looking at archives. I spent some time behind the scenes at the Art Institute of Chicago looking at works on paper. This research will develop into a new body of work soon. While I was there I also made a lot of drawings of buildings in the city, which fed into my work with the mobile app Infused. These are available to buy as a limited edition hand coloured print series, called Battles of Mathematics, available to buy from my website.
Both cities share a rich industrial past and both are considered the second city in their respective countries.
From an artists' perspective, what similarities and differences do you feel Birmingham and Chicago have?
The most immediate thing I noticed was the water: before coming to Chicago I hadn't realised the impact that Lake Michigan and the river have on the landscape. It feels so unusual to sail through the skyscrapers on a river taxi, and it's also one of the best ways to get around and to see the skyline. In Birmingham, the canals are so important to its heritage, in terms of trade, transport and its landscape. I often walk and cycle up along the canals and it was interesting to see another city where water is so important to its infrastructure.
Like Chicago, Birmingham is undergoing fast regeneration. There are numerous projects in Birmingham that are trying to preserve the heritage of communities through oral history projects, digital archives etc. When in Chicago I visited places like the National Public Housing Museum to discuss how they are doing similar things for their communities. For me this reinforced the importance of these projects – and also how similar themes arise in both cities: immigration, political influence, poverty, gentrification of areas and changing property values.
Did you find many opportunities to explore Chicago outside of art contemporaries?
My favourite place in the evening was Andy's Jazz Club on East Hubbard Street. I loved the Original Pancake House for a proper American breakfast. Millennium Park was brilliant place to discover an open air concert, follow a sculpture trail, or just sit and have your lunch with free wifi.
It was also great to visit students and staff at the School of the Art Institute of Chicago, as I have worked with various Universities in the UK and it was good to see how things differ in the US, and the variety of public programs that they offer.
Being in the presence of other artists there and visiting art galleries, were you inspired by any ideas from your time there which you have brought back with you to Birmingham?
I was really inspired by Mana Contemporary, an arts center in Pilsen. It houses studios, exhibition spaces, classrooms, a cafe and library. I saw the center mid-development, and learned how they could afford to provide affordable studio spaces and high quality facilities. Emerging artists work side by side with established artists, and the whole organisation had a collaborative and adaptive atmosphere. This was interesting model to see, especially as Birmingham has a studio shortage and new projects like Birmingham Production Space are trying to get off the ground.
Are there current projects that you are working on?
I am currently working on a new book called WEST POINT, charting the 55 year history of an award winning housing estate in Allesley, UK. This follows a short residency at the estate, where I am currently meeting residents and researching the architectural significance of the estate. This will be out on 12th June 2015.
This project will eventually tie in to the research I did on housing in Chicago, in particular looking at the layers of history that a piece of land can hold.
I have also just finished a new large scale commission for the New Art Gallery Walsall, which will be part of their permanent collection.
Thank you for your time Sarah and I look forward to see you continue to push out awesome artwork!
Check out Sarah's website by clicking here. You can also follow her on Twitter and like her page on Facebook to stay updated with her latest art work.
For more information on Chicago's Sister City relationship program with Birmingham and its other cities, click here.
Tags: birmingham, Sarah Silverwood, Sister Cities | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,741 |
Q: C++ Compile Error I can't figure out why I am getting these errors when I try to compile. I've never encountered the 'expected _ before _ token' error, but I believe they're common (if not feel free to enlighten me).
pe4.cpp: In function 'int main()':
pe4.cpp:18: error: expected ')' before ';' token
pe4.cpp:18: error: expected ';' before ')' token
pe4.cpp:45: error: a function-definition is not allowed here before '{' token
pe4.cpp:51: error: a function-definition is not allowed here before '{' token
pe4.cpp:57: error: a function-definition is not allowed here before '{' token
#include <iostream>
using namespace std;
void printStar(int);
void printSpace(int);
void printNewLine();
int main()
{
int side, i, j;
if (i=0; i < 2; i++)
{
cout << "Enter side: " << endl;
cin << side;
if (side < 3 || side > 20)
{
cout << "Out of Bounds!!!"
return 0;
}
printStar(side);
printNewLine();
{
printStar(1);
printSpace(side-2);
printStar(1);
printNewLine();
}
printStar(side);
printNewLine();
}
void printStar(int a)
{
for (int j = 0; j < a; j++)
cout << "*";
}
void printSpace(int a)
{
for (int j = 0; j < a; j++)
cout << " ";
}
void printNewLine()
{
cout << endl;
}
}
A: You have no ; at the end of the cout << "Out of Bounds!!!" line.
You have if (i=0; i < 2; i++); that should be for (i=0;....
You have cin << side; that should be cin >> side.
You have defined your function bodies inside main(); they should live outside.
A: You are defining your functions printStar(), etc, inside your main() definition. Move those functions outside of main()'s closing bracket.
A: The closing } of the int main() method needs to go before void printStart(int a).
Also, you need a ; at the end of cout << "Out of Bound!!!"
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,087 |
#include BOSS_WEBRTC_U_modules__desktop_capture__blank_detector_desktop_capturer_wrapper_h //original-code:"modules/desktop_capture/blank_detector_desktop_capturer_wrapper.h"
#include <algorithm>
#include <utility>
#include BOSS_WEBRTC_U_modules__desktop_capture__desktop_geometry_h //original-code:"modules/desktop_capture/desktop_geometry.h"
#include BOSS_WEBRTC_U_rtc_base__checks_h //original-code:"rtc_base/checks.h"
#include BOSS_WEBRTC_U_system_wrappers__include__metrics_h //original-code:"system_wrappers/include/metrics.h"
namespace webrtc {
BlankDetectorDesktopCapturerWrapper::BlankDetectorDesktopCapturerWrapper(
std::unique_ptr<DesktopCapturer> capturer,
RgbaColor blank_pixel)
: capturer_(std::move(capturer)), blank_pixel_(blank_pixel) {
RTC_DCHECK(capturer_);
}
BlankDetectorDesktopCapturerWrapper::~BlankDetectorDesktopCapturerWrapper() =
default;
void BlankDetectorDesktopCapturerWrapper::Start(
DesktopCapturer::Callback* callback) {
capturer_->Start(this);
callback_ = callback;
}
void BlankDetectorDesktopCapturerWrapper::SetSharedMemoryFactory(
std::unique_ptr<SharedMemoryFactory> shared_memory_factory) {
capturer_->SetSharedMemoryFactory(std::move(shared_memory_factory));
}
void BlankDetectorDesktopCapturerWrapper::CaptureFrame() {
RTC_DCHECK(callback_);
capturer_->CaptureFrame();
}
void BlankDetectorDesktopCapturerWrapper::SetExcludedWindow(WindowId window) {
capturer_->SetExcludedWindow(window);
}
bool BlankDetectorDesktopCapturerWrapper::GetSourceList(SourceList* sources) {
return capturer_->GetSourceList(sources);
}
bool BlankDetectorDesktopCapturerWrapper::SelectSource(SourceId id) {
return capturer_->SelectSource(id);
}
bool BlankDetectorDesktopCapturerWrapper::FocusOnSelectedSource() {
return capturer_->FocusOnSelectedSource();
}
void BlankDetectorDesktopCapturerWrapper::OnCaptureResult(
Result result,
std::unique_ptr<DesktopFrame> frame) {
RTC_DCHECK(callback_);
if (result != Result::SUCCESS || non_blank_frame_received_) {
callback_->OnCaptureResult(result, std::move(frame));
return;
}
RTC_DCHECK(frame);
// If nothing has been changed in current frame, we do not need to check it
// again.
if (!frame->updated_region().is_empty() || is_first_frame_) {
last_frame_is_blank_ = IsBlankFrame(*frame);
is_first_frame_ = false;
}
RTC_HISTOGRAM_BOOLEAN("WebRTC.DesktopCapture.BlankFrameDetected",
last_frame_is_blank_);
if (!last_frame_is_blank_) {
non_blank_frame_received_ = true;
callback_->OnCaptureResult(Result::SUCCESS, std::move(frame));
return;
}
callback_->OnCaptureResult(Result::ERROR_TEMPORARY,
std::unique_ptr<DesktopFrame>());
}
bool BlankDetectorDesktopCapturerWrapper::IsBlankFrame(
const DesktopFrame& frame) const {
// We will check 7489 pixels for a frame with 1024 x 768 resolution.
for (int i = 0; i < frame.size().width() * frame.size().height(); i += 105) {
const int x = i % frame.size().width();
const int y = i / frame.size().width();
if (!IsBlankPixel(frame, x, y)) {
return false;
}
}
// We are verifying the pixel in the center as well.
return IsBlankPixel(frame, frame.size().width() / 2,
frame.size().height() / 2);
}
bool BlankDetectorDesktopCapturerWrapper::IsBlankPixel(
const DesktopFrame& frame,
int x,
int y) const {
uint8_t* pixel_data = frame.GetFrameDataAtPos(DesktopVector(x, y));
return RgbaColor(pixel_data) == blank_pixel_;
}
} // namespace webrtc
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,087 |
// Generated by the protocol buffer compiler. DO NOT EDIT!
// source: google/cloud/pubsublite/v1/subscriber.proto
package com.google.cloud.pubsublite.proto;
public interface PartitionAssignmentOrBuilder
extends
// @@protoc_insertion_point(interface_extends:google.cloud.pubsublite.v1.PartitionAssignment)
com.google.protobuf.MessageOrBuilder {
/**
*
*
* <pre>
* The list of partition numbers this subscriber is assigned to.
* </pre>
*
* <code>repeated int64 partitions = 1;</code>
*
* @return A list containing the partitions.
*/
java.util.List<java.lang.Long> getPartitionsList();
/**
*
*
* <pre>
* The list of partition numbers this subscriber is assigned to.
* </pre>
*
* <code>repeated int64 partitions = 1;</code>
*
* @return The count of partitions.
*/
int getPartitionsCount();
/**
*
*
* <pre>
* The list of partition numbers this subscriber is assigned to.
* </pre>
*
* <code>repeated int64 partitions = 1;</code>
*
* @param index The index of the element to return.
* @return The partitions at the given index.
*/
long getPartitions(int index);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,966 |
{"url":"https:\/\/deepai.org\/publication\/outage-analysis-of-ambient-backscatter-communication-systems","text":"# Outage Analysis of Ambient Backscatter Communication Systems\n\nThis paper addresses the problem of outage characterization of an ambient backscatter communication system with a pair of passive tag and reader. In particular, an exact expression for the effective channel distribution is derived. Then, the outage probability at the reader is analyzed rigorously. Since the expression contains an infinite sum, a tight truncation error bound has been derived to facilitate precise numerical evaluations. Furthermore, an asymptotic expression is provided for high signal-to-noise ratio (SNR) regime.\n\n## Authors\n\n\u2022 3 publications\n\u2022 8 publications\n\u2022 10 publications\n\u2022 7 publications\n\u2022 ### A Simple Evaluation for the Secrecy Outage Probability Over Generalized-K Fading Channels\n\nA simple approximation for the secrecy outage probability (SOP) over gen...\n07\/03\/2019 \u2219 by Hui Zhao, et al. \u2219 0\n\n\u2022 ### Outage Analysis of 2\u00d72 MIMO-MRC in Correlated Rician Fading\n\nThis paper addresses one of the classical problems in random matrix theo...\n11\/16\/2018 \u2219 by Prathapasinghe Dharmawansa, et al. \u2219 0\n\n\u2022 ### Unified Composite Distribution and Its Applications to Double Shadowed \u03b1-\u03ba-\u03bc Fading Channels\n\nIn this paper, we propose a mixture Gamma shadowed (MGS) distribution as...\n11\/16\/2020 \u2219 by Hussien Al-Hmood, et al. \u2219 0\n\n\u2022 ### Reliability Analysis of Large Intelligent Surfaces (LISs): Rate Distribution and Outage Probability\n\nLarge intelligent surfaces (LISs) have been recently proposed as an effe...\n03\/27\/2019 \u2219 by Minchae Jung, et al. \u2219 0\n\n\u2022 ### Distribution of the Sum of Fisher-Snedecor F Random Variables and Its Applications\n\nThe statistical characterization of the sum of random variables (RVs) ar...\n10\/23\/2019 \u2219 by Hongyang Du, et al. \u2219 0\n\n\u2022 ### Performance Analysis of Distributed Intelligent Reflective Surfaces for Wireless Communications\n\nIn this paper, a comprehensive performance analysis of a distributed int...\n10\/23\/2020 \u2219 by Diluka Loku Galappaththige, et al. \u2219 0\n\n\u2022 ### Deep Transfer Learning for Signal Detection in Ambient Backscatter Communications\n\nTag signal detection is one of the key tasks in ambient backscatter comm...\n09\/11\/2020 \u2219 by Chang Liu, et al. \u2219 0\n\n##### This week in AI\n\nGet the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.\n\n## I Introduction\n\nBackscatter communication systems, such as Radio Frequency Identification (RFID) system, enable connecting massive small computing devices specially for applications in Internet of Things (IoT) [1]. The traditional RFID system typically consists of a tag and a reader. The reader first generates and transmits an electromagnetic wave signal to the tag, and then the tag receives and backscatters the signal to the reader.\n\nOne disadvantage for the RFID system is that the reader needs an oscillator to transmit a carrier signal, for which dedicated encoding\/decoding circuitry and power supply are required [2]. While these are essential components for a successful communication, such in-built technology may no longer be promising for small-scale devices. To overcome such overheads, ambient backscatter prototypes are proposed in [3, 4].\n\nThe ambient backscatter technology utilizes environmental wireless signals (e.g., digital TV broadcasting, cellular or Wi-Fi) for both energy harvesting and information transmission, which avoids battery as well as manual maintenance. Specifically, the tag indicates bit 1 or bit 0 through reflecting or non-reflecting state, and the reader decodes the received backscattered signal accordingly [5]. Ambient backscatter may be widely used for future applications (e.g., many applications in IoT with sensors located in dangerous spots filled with poisonous gases\/liquids, or inside building walls) that are inconvenient and unsafe for wired communications [6].\n\nThe performance analysis of the ambient backscatter communication is considered over real Gaussian channels in [7], and complex Gaussian channels in [8, 9]. The bit error rate (BER) is derived and the BER-based outage probability is obtained in [7] for ambient bakcscatter communication systems with energy detector. In addition, the outage capacity optimization problem is investigated in [8] when successive interference cancellation (SIC) method is applied. Besides, the BER-based outage probability of a semi-coherent detection scheme is calculated in [9] in the case of perfect and imperfect channel state information (CSI), respectively.\n\nTo our best knowledge, effective channel distribution for ambient backscatter communication systems has not be addressed and the outage performance based on signal-to-noise ratio (SNR) remains an open problem, which is the focus of this paper.\n\nIn this paper, we consider an ambient backscatter communication system over real Gaussian channels. We derive an exact expression for the effective channel distribution in this system. Particularly, we evaluate the outage performance and analyse its asymptotic outage performance at high transmit SNR. Moreover, since the derived outage probability is the summation of infinite items, the corresponding truncation error bound is calculated.\n\n## Ii System Model\n\nWe consider an ambient backscatter communication system comprised of an ambient RF source () and a pair of passive tag () and reader () (Fig.\u00a01). While the RF source communicates with its legacy users (e.g., smartphones, laptops, etc.), both tag and reader may also receive that source signal. The tag first harvests energy from the source signal, and then communicates with the reader via ambient backscatter. Particularly, the tag can backscatter or consume the energy of the received signal to represent two states \u201c1\u201d or \u201c0\u201d for the reader, respectively [5].\n\nThe fading channels of , , and links are denoted as , and\n\n, respectively, which are real Gaussian random variables (RVs) distributed as\n\n, and , where , and\n\nare channel variances. Further, the corresponding distances are\n\n, and , respectively. Without loss of generality, we consider time instance . The signal received at the tag can be given as\n\n yt(n)=~hst\u221ad\u03b1sts(n), (1)\n\nwhere is the source signal with the average power , and is the path-loss exponent. The signal backscattered by the tag can be written as\n\n x(n)=\u03b7B(n)yt(n), (2)\n\nwhere is a binary symbol and is the attenuation factor. Then, the received signal at the reader can be given as [5]\n\n yr(n)=~hsr\u221ad\u03b1srs(n)+~htr\u221ad\u03b1trx(n)+w(n)=hs(n)+w(n), (3)\n\nwhere is the additive white Gaussian noise (AWGN) at the reader with zero mean and variance, and is the effective channel gain which can be given for two states as\n\n h ={hsr,B(n)=0,hsr+\u03b7hsthtr,B(n)=1, (4)\n\nwhere , , and with , and .\n\n## Iii Performance Analysis\n\n### Iii-a Effective Channel and SNR Distributions\n\nWe first derive the probability density function (PDF) of the effective channel,\n\n, which can be given as\n\n (5)\n\nwhere and are the Whittaker function [10, eq.\u00a0(9.223)] and the Gamma function [10, eq.\u00a0(8.310.1)], respectively. The proof is in Appendix\u00a0-A.\n\nThe receive SNR at the reader is where may be the average transmit SNR. With a variable transformation for (5), we can derive the PDF of , , as\n\n f\u03c1(x)=[fh(\u221ax\u00af\u03c1)+fh(\u2212\u221ax\u00af\u03c1)]2\u221a\u00af\u03c1x=fh(\u221ax\u00af\u03c1)\u221a\u00af\u03c1x, (6)\n\nwhere the second equality follows as the PDF is an even function, i.e., .\n\n### Iii-B Outage Probability\n\nThe outage probability is the probability that the SNR at the reader falls below a certain predetermined threshold . Thus, it can be derived as\n\n Po =Pr[\u03c1\u2264\u03c1t]=\u222b\u03c1t0f\u03c1(x)dx =\u23a7\u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa\u23a8\u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa\u23a9erf(\u221a\u03c1t2\u00af\u03c1\u03c32sr),B(n)=0,\u221e\u2211k=022k\u03932(k+12)\u221a\u03c03(2k)!W\u2212k,0(\u03c32sr2\u03b72\u03c32st\u03c32tr)\u00d7e\u03c32sr4\u03b72\u03c32st\u03c32tr\u03b3(k+12,\u03c1t2\u00af\u03c1\u03c32sr),B(n)=1, (7)\n\nwhere and are the error function [10, eq.\u00a0(8.250.1)] and the incomplete gamma function [10, eq.\u00a0(8.350.1)], respectively. The equation (7) can be obtained by following from [10, eq.\u00a0(3.321.2)], [10, eq.\u00a0(8.250.1)] and [10, eq.\u00a0(3.381.1)].\n\n### Iii-C Truncation Error Bound\n\nSince the outage probability expression for case in (7) is with an infinite sum, it is a challenge for numerical calculation. We therefore truncate it into a finite number of terms in order to ensure a given numerical accuracy requirement. Then, we bound the truncation error as\n\n |\u03f5T|\u2264\u03a8(12,0,\u03bd)\u221a\u03c0\u03bdT![\u221a2\u03c1t\u03c32sr\u00af\u03c1\u03b3(T+1,\u03c1t2\u03c32sr\u00af\u03c1)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u22122\u03b3(T+32,\u03c1t2\u03c32sr\u00af\u03c1)], (8)\n\nwhere , and is the confluent hypergeometric function [10, eq.\u00a0(9.211.4)]. The proof is in Appendix\u00a0-B.\n\n### Iii-D Asymptotic Analysis for High SNR\n\nTo further investigate the ambient backscatter system, we approximate outage of the reader for large SNR as\n\n Po\u2248\u23a7\u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa\u23a8\u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa \u23aa\u23a9\u221a2\u03c1t\u03c0\u03c32sr1\u221a\u00af\u03c1,B(n)=0,\u221a2\u03c1te\u03c32sr4\u03b72\u03c32st\u03c32trW0,0(\u03c32sr2\u03b72\u03c32st\u03c32tr)\u221a\u03c0\u03c32sr1\u221a\u00af\u03c1,B(n)=1. (9)\n\nThe proof is in Appendix\u00a0-C.\n\nInterestingly, when transmit SNR tends to infinity, we get that the diversity gain is for both cases and . Accordingly, when power is big enough, the outage probability of the reader is inversely proportional to square root of the power.\n\n## Iv Numerical and Simulation Results\n\nThis section provides simulation results based on the system model in Section II and numerical results based on the analytical results in Section III. Here, the attenuation factor is set as 0.7. Besides, we assume and , unless otherwise specified.\n\nFor the state , we derive a truncation error bound (8). Fig.\u00a02 shows the relative error versus the number of terms when dB for dB and dB. We calculate the relative error with truncation as ; the relative error with bound as ; the exact value with numerical integration; and the truncated value for different by using (7). The relative error with truncation is less than when and for dB and dB, respectively. The relative error with bound is less than when and for dB and dB, respectively. This shows the tightness of the bound. Moreover, by observation, we may say that a very accurate value can be calculated using small .\n\nFig.\u00a03 illustrates the outage probability versus the average transmit SNR when dB and dB. The outage probability decreases with the increasing average transmit SNR. The asymptotic expressions (9) also approach the exact values asymptotically at high SNR. Since the diversity gain is 1\/2, the slope of the asymptotic outage curves is 1\/2.\n\nFig.\u00a04 depicts the outage probabilities versus the distance between the tag and the reader. We consider both cases of SNR dB and dB, respectively. We set dB, , meters and meters. Besides, the channel variances , and are set as 1, 1 and 3, respectively. The outage probability is a constant in the case of due to no transmission between the tag and the reader. However, when enlarging the distance between the tag and the reader, the outage probability witnesses a upward trend in the case of . For example, if we expect , the distance between the tag and the reader should not exceed 3 meters when dB and 7 meters when dB.\n\n## V Conclusion\n\nAmbient backscatter, a new form of wireless communication, has potential commercial value as well as a series of open problems. In this paper, we first derived the effective channel distribution for the ambient backscatter communication system. We next analyzed the outage probabilities, its truncation error bound as well as the asymptotic outage probabilities at high SNR. It was found that the asymptotic outage probabilities could well approach the exact values, and our truncation error bound could provide a reasonable estimation of the truncation terms.\n\n### -a Proof of (5)\n\nThe distribution of any real Gaussian channel is\n\n fhab(x)=1\u221a2\u03c0\u03c32abe\u2212x22\u03c32ab, (10)\n\nwhere .\n\nIn the case of , we have . The distribution of can be shown as [11]\n\n f\u03be(x)=1\u03c0\u03b7\u221a\u03c32st\u03c32trK0\u239b\u239c \u239c\u239d|x|\u03b7\u221a\u03c32st\u03c32tr\u239e\u239f \u239f\u23a0, (11)\n\nwhere is the modified Bessel function of the second kind [10].\n\nSince the two random variables and are independent, the distribution is the convolution of and . Therefore, we can obtain\n\n fh(x) =\u222b\u221e\u2212\u221efhsr(x\u2212z)f\u03be(z)dz (a)=e\u2212x22\u03c32sr\u03b4\u03c6\u222b\u221e0K0(z\u03c6)e\u2212z22\u03c32sr(exz\u03c32sr+e\u2212xz\u03c32sr)dz (b)=2e\u2212x22\u03c32sr\u03b4\u03c6\u222b\u221e0K0(z\u03c6)e\u2212z22\u03c32srcosh(xz\u03c32sr)dz (c)=2e\u2212x22\u03c32sr\u03b4\u03c6\u221e\u2211k=0x2k\u03c34ksr(2k)!\u222b\u221e0K0(z\u03c6)e\u2212z22\u03c32srz2kdz (d)=e\u03bd2\u03b4\u221e\u2211k=02k\u03932(k+12)W\u2212k,0(\u03bd)(2k)!\u03c32ksre\u2212x22\u03c32srx2k, (12)\n\nwhere\n\n \u03b4=\u221a2\u03c03\u03c32sr,\u03c6=\u03b7\u221a\u03c32st\u03c32tr,\u03bd=\u03c32sr2\u03b72\u03c32st\u03c32tr, (13)\n\n(a) follows by two integrals having identical bounds, (b) follows by using the definition of hyperbolic cosine , (c) follows by replacing with its series expression , and (d) follows from [10, eq.\u00a0(6.631.3)].\n\n### -B Proof of (8)\n\nOn the basis of (7), the truncation error with the number of terms can be bounded as\n\n \u03f5(T) = e\u03bd2\u221a\u03c0\u03c0\u221e\u2211k=T+122k\u03932(k+12)W\u2212k,0(\u03bd)(2k)!\u03b3(k+12,\u03c1t2\u03c32sr\u00af\u03c1) (a)= \u221a\u03bd\u03c0\u221e\u2211k=T+11k!\u222b\u221e0e\u2212\u03bdxxk+12(1+x)k+12dx\u222b\u03c1t2\u03c32sr\u00af\u03c10e\u2212yyk\u221212dy (b)= \u221a\u03bd\u03c0(T+1)!\u222b\u221e0\u222b\u03c1t2\u03c32sr\u00af\u03c10e\u2212\u03bdxxT+32(1+x)T+32e\u2212yyT+12 \u00d71F1(1;T+2;xyx+1)dydx (c)= \u221a\u03bd\u03c0T!\u222b\u221e0\u222b\u03c1t2\u03c32sr\u00af\u03c10e\u2212\u03bdxx12(1+x)12e\u2212yy\u221212 \u00d7exyx+1\u03b3(T+1,xyx+1)dydx\n (d)< \u221a\u03bd\u03c0T!\u222b\u221e0e\u2212\u03bdxx12(1+x)12dx\u222b\u03c1t2\u03c32sr\u00af\u03c10\u03b3(T+1,y)\u221aydy = \u03a8(12,0,\u03bd)\u221a\u03c0\u03bdT![\u221a2\u03c1t\u03c32sr\u00af\u03c1\u03b3(T+1,\u03c1t2\u03c32sr\u00af\u03c1) \u22122\u03b3(T+32,\u03c1t2\u03c32sr\u00af\u03c1)],\n\nwhere is defined in (13), (a) is obtained from [10, eq.\u00a0(9.222.1)] and [10, eq.\u00a0(8.350.1)], (b) follows by setting and leveraging the integral representation of Hypergeometric function [10, eq.\u00a0(9.211.4)], (c) utilizes the following equation\n\n 1F1(1;T+2;xyx+1)=(T+1)(xyx+1)\u2212(T+1)exyx+1 \u00d7\u03b3(T+1,xyx+1),\n\n(d) is based on for , and is the confluent hypergeometric function [10, eq.\u00a0(9.211.4)].\n\n### -C Proof of (9)\n\nIn the case of and using series representation of the error function [10, eq.\u00a0(8.253.1)], we can expand (7) as\n\n erf\u239b\u239d\u221a\u03c1t2\u00af\u03c1\u03c32sr\u239e\u23a0=2\u221a\u03c0\u221e\u2211k=0(\u22121)k\u03c1tk+1\/2k!(2k+1)(2\u00af\u03c1\u03c32sr)k+1\/2.\n\nFor , we consider the lowest exponent for , i.e., the index . Similarly, in the case of , we can expand in (7) as\n\n \u03b3(a,x)=\u221e\u2211j=0(\u22121)jxa+jj!(a+j).\n\nWe then consider the lowest exponent for , i.e., the indies and . Thereby, when average transmit SNR tends to infinity, namely, , the asymptotic outage probability can be simplified as (9).\n\n## References\n\n\u2022 [1] A.\u00a0Al-Fuqaha, M.\u00a0Guizani, M.\u00a0Mohammadi, M.\u00a0Aledhari, and M.\u00a0Ayyash, \u201cInternet of things: A survey on enabling technologies, protocols, and applications,\u201d IEEE Commun. Surveys Tuts., vol.\u00a017, pp.\u00a02347\u20132376, Jun. 2015.\n\u2022 [2] T.\u00a0H. Lee, The design of CMOS Radio-Frequency integrated circuits. Cambridge Univ. Press, 1998.\n\u2022 [3] V.\u00a0Liu, A.\u00a0Parks, V.\u00a0Talla, S.\u00a0Gollakota, D.\u00a0Wetherall, and J.\u00a0R. Smith, \u201cAmbient backscatter: wireless communication out of thin air,\u201d in Proc. ACM SIGCOMM, Aug. 2013, pp.\u00a039\u201350.\n\u2022 [4] A.\u00a0N. Parks, A.\u00a0Liu, S.\u00a0Gollakota, and J.\u00a0R. Smith, \u201cTurbocharging ambient backscatter communication,\u201d in Proc. ACM SIGCOMM, Aug. 2014, pp.\u00a0619\u2013630.\n\u2022 [5] G.\u00a0Wang, F.\u00a0Gao, R.\u00a0Fan, and C.\u00a0Tellambura, \u201cAmbient backscatter communication systems: Detection and performance analysis,\u201d IEEE Trans. Commun., vol.\u00a064, pp.\u00a04836\u20134846, Nov. 2016.\n\u2022 [6] D.\u00a0Kuester and Z.\u00a0Popovic, \u201cHow good is your tag?: RFID backscatter metrics and measurements,\u201d IEEE Microwave Mag., vol.\u00a014, pp.\u00a047\u201355, Jul. 2013.\n\u2022 [7] Y.\u00a0Zhang, J.\u00a0Qian, F.\u00a0Gao, and G.\u00a0Wang, \u201cAmbient backscatter system with real source,\u201d in IEEE Workshop on Signal Processing Advances in Wireless Commun., 2017, to be published.\n\u2022 [8] S.\u00a0Xing, R.\u00a0Long, Y.\u00a0Liang, M.\u00a0Ding, and Z.\u00a0Lin, \u201cOutage capacity analysis for ambient backscatter communication systems,\u201d in IEEE Wireless Commun. and Networking Conf. (WCNC), 2017, to be published.\n\u2022 [9] J.\u00a0Qian, F.\u00a0Gao, G.\u00a0Wang, S.\u00a0Jin, and H.\u00a0Zhu, \u201cSemi-coherent detection and performance analysis for ambient backscatter system,\u201d IEEE Trans. Commun., 2017, to be published.\n\u2022 [10] I.\u00a0S. Gradshteyn and I.\u00a0M. Ryzhik, Table of Integrals, Series and Products. Academic Press Inc, 7th revised edition\u00a0ed., 2007.\n\u2022 [11] M.\u00a0K. Simon, Probability distributions involving Gaussian Random Variables: a handbook for engineers and scientists. Springer, 2006.","date":"2021-04-14 07:24:21","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8112616539001465, \"perplexity\": 1571.2203976637275}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038077336.28\/warc\/CC-MAIN-20210414064832-20210414094832-00249.warc.gz\"}"} | null | null |
{"url":"https:\/\/gmatclub.com\/forum\/water-is-leaking-out-from-a-cylinder-container-at-the-rate-80226.html","text":"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 20 Jan 2019, 07:30\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n## Events & Promotions\n\n###### Events & Promotions in January\nPrevNext\nSuMoTuWeThFrSa\n303112345\n6789101112\n13141516171819\n20212223242526\n272829303112\nOpen Detailed Calendar\n\u2022 ### FREE Quant Workshop by e-GMAT!\n\nJanuary 20, 2019\n\nJanuary 20, 2019\n\n07:00 AM PST\n\n07:00 AM PST\n\nGet personalized insights on how to achieve your Target Quant Score.\n\u2022 ### GMAT Club Tests are Free & Open for Martin Luther King Jr.'s Birthday!\n\nJanuary 21, 2019\n\nJanuary 21, 2019\n\n10:00 PM PST\n\n11:00 PM PST\n\nMark your calendars - All GMAT Club Tests are free and open January 21st for celebrate Martin Luther King Jr.'s Birthday.\n\n# Water is leaking out from a cylinder container at the rate\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nManager\nJoined: 04 Sep 2006\nPosts: 112\nWater is leaking out from a cylinder container at the rate\u00a0 [#permalink]\n\n### Show Tags\n\n27 Jun 2009, 21:29\n8\n00:00\n\nDifficulty:\n\n35% (medium)\n\nQuestion Stats:\n\n73% (02:18) correct 27% (02:19) wrong based on 305 sessions\n\n### HideShow timer Statistics\n\nWater is leaking out from a cylinder container at the rate of 0.31 m^3 per minute. 10 minutes later, the water level decreases 0.25 meter, what is value of the radius?\n\n(A) 0.5\n(B) 1.0\n(C) 1.5\n(D) 2.0\n(E) 2.5\nManager\nJoined: 27 Jun 2008\nPosts: 135\n\n### Show Tags\n\n22 Jul 2009, 00:17\nTotal water loss = 3.1 mtr in 10 min @.31 meter per min\n3.14XR^2X.25 = 3.1\nr^2 = 4\nr=2\n\n(D)\nIntern\nJoined: 17 Nov 2009\nPosts: 33\nSchools: University of Toronto, Mcgill, Queens\n\n### Show Tags\n\n20 Feb 2010, 11:08\nirajeevsingh wrote:\nTotal water loss = 3.1 mtr in 10 min @.31 meter per min\n3.14XR^2X.25 = 3.1\nr^2 = 4\nr=2\n\n(D)\n\nCan you please elaborate that how you did\n\n3.1 * r^2 * .25 = 3.1\n_________________\n\n--Action is the foundational key to all success.\n\nMath Expert\nJoined: 02 Sep 2009\nPosts: 52296\n\n### Show Tags\n\n20 Feb 2010, 12:00\n3\n1\nBullet wrote:\nirajeevsingh wrote:\nTotal water loss = 3.1 mtr in 10 min @.31 meter per min\n3.14XR^2X.25 = 3.1\nr^2 = 4\nr=2\n\n(D)\n\nCan you please elaborate that how you did\n\n3.1 * r^2 * .25 = 3.1\n\nIn 10 min the volume of the water which leaked out the cylinder would be $$10*0.31=3.1$$ m^3.\n\nVolume = $$\\pi*r^2*height=3.1$$ --> $$\\pi*r^2*0.25=3.1$$ --> $$\\pi=3.14$$ --> $$3.14*r^2*0.25=3.1$$ --> $$r=2$$ (approximately).\n\nHope it's clear.\n_________________\nIntern\nJoined: 17 Nov 2009\nPosts: 33\nSchools: University of Toronto, Mcgill, Queens\n\n### Show Tags\n\n20 Feb 2010, 22:39\nBunuel wrote:\nBullet wrote:\nirajeevsingh wrote:\nTotal water loss = 3.1 mtr in 10 min @.31 meter per min\n3.14XR^2X.25 = 3.1\nr^2 = 4\nr=2\n\n(D)\n\nCan you please elaborate that how you did\n\n3.1 * r^2 * .25 = 3.1\n\nIn 10 min the volume of the water which leaked out the cylinder would be $$10*0.31=3.1$$ m^3.\n\nVolume = $$\\pi*r^2*height=3.1$$ --> $$\\pi*r^2*0.25=3.1$$ --> $$\\pi=3.14$$ --> $$3.14*r^2*0.25=3.1$$ --> $$r=2$$ (approximately).\n\nHope it's clear.\n\nThanks Bunuel, I was confusing 3.14 with the volume where is was pi.\n\nCheers!\n_________________\n\n--Action is the foundational key to all success.\n\nManager\nJoined: 27 Aug 2014\nPosts: 138\nConcentration: Finance, Strategy\nGPA: 3.9\nWE: Analyst (Energy and Utilities)\nRe: Water is leaking out from a cylinder container at the rate\u00a0 [#permalink]\n\n### Show Tags\n\n13 Mar 2015, 06:48\nBunuel wrote:\nBullet wrote:\nirajeevsingh wrote:\nTotal water loss = 3.1 mtr in 10 min @.31 meter per min\n3.14XR^2X.25 = 3.1\nr^2 = 4\nr=2\n\n(D)\n\nCan you please elaborate that how you did\n\n3.1 * r^2 * .25 = 3.1\n\nIn 10 min the volume of the water which leaked out the cylinder would be $$10*0.31=3.1$$ m^3.\n\nVolume = $$\\pi*r^2*height=3.1$$ --> $$\\pi*r^2*0.25=3.1$$ --> $$\\pi=3.14$$ --> $$3.14*r^2*0.25=3.1$$ --> $$r=2$$ (approximately).\n\nHope it's clear.\n\nI have a question, that here it is assumed that the cylinder is standing on the radius and the water is being decreased in the length of the cylinder. Is this a good assumption?\nI think this should be stated in the question, as I was trying the problem with the cylinder lying on the length.\nMath Expert\nJoined: 02 Aug 2009\nPosts: 7209\nRe: Water is leaking out from a cylinder container at the rate\u00a0 [#permalink]\n\n### Show Tags\n\n13 Mar 2015, 08:04\nsantorasantu wrote:\n\nI have a question, that here it is assumed that the cylinder is standing on the radius and the water is being decreased in the length of the cylinder. Is this a good assumption?\nI think this should be stated in the question, as I was trying the problem with the cylinder lying on the length.\n\nhi,\nsince the question says the water decreased by .25m , it would mean that the height of it .. since if it is lying as you have taken, this .25m does not have same cross section at any place...\nAlso you have to take a cylinder as on base because that is what is the original shape..\nyou can take the way you have taken only when it is specified that way..\n_________________\n\n1) Absolute modulus : http:\/\/gmatclub.com\/forum\/absolute-modulus-a-better-understanding-210849.html#p1622372\n2)Combination of similar and dissimilar things : http:\/\/gmatclub.com\/forum\/topic215915.html\n3) effects of arithmetic operations : https:\/\/gmatclub.com\/forum\/effects-of-arithmetic-operations-on-fractions-269413.html\n\nGMAT online Tutor\n\nIntern\nJoined: 27 Nov 2014\nPosts: 45\nRe: Water is leaking out from a cylinder container at the rate\u00a0 [#permalink]\n\n### Show Tags\n\n14 Mar 2015, 00:59\nchetan2u\n\nhi,\nsince the question says the water decreased by .25m , it would mean that the height of it .. since if it is lying as you have taken, this .25m does not have same cross section at any place...\nAlso you have to take a cylinder as on base because that is what is the original shape..\nyou can take the way you have taken only when it is specified that way..\n_____________________________________________________________________________________________________\nChetan i am not convinced from the ans i.e 3.14*$$r^2$$*h\nwhy we took height = .25m as it is not the actual total height ??\nand total volume =3.1 as it leaked for 10 min only given.\n\nRegards\nSG\nMath Expert\nJoined: 02 Aug 2009\nPosts: 7209\nRe: Water is leaking out from a cylinder container at the rate\u00a0 [#permalink]\n\n### Show Tags\n\n14 Mar 2015, 04:00\n1\nsmartyguy wrote:\n.\n_____________________________________________________________________________________________________\nChetan i am not convinced from the ans i.e 3.14*$$r^2$$*h\nwhy we took height = .25m as it is not the actual total height ??\nand total volume =3.1 as it leaked for 10 min only given.\n\nRegards\nSG\n\nhi SG,\n\nWater is leaking out from a cylinder container at the rate of 0.31 m^3 per minute. 10 minutes later, the water level decreases 0.25 meter, what is value of the radius?\n\n(A) 0.5\n(B) 1.0\n(C) 1.5\n(D) 2.0\n(E) 2.5\n\nin one minute, the volume of water lost is .31$$m^3$$\nso in 10 minutes the volume of water lost is 3.1$$m^3$$..(1)\n\nnow, we are also said that the height lost is .25m, we are not concerned with the entire height...\nwhat is the volume of water in this .25m...\nsince it is a cylinder the volume of .25 m= pi*$$r^2$$*h=3.14*$$r^2$$*.25...\n3.14*$$r^2$$*.25=3.1...\n$$r^2$$=1\/.25=4.. so r=2 approx\n_________________\n\n1) Absolute modulus : http:\/\/gmatclub.com\/forum\/absolute-modulus-a-better-understanding-210849.html#p1622372\n2)Combination of similar and dissimilar things : http:\/\/gmatclub.com\/forum\/topic215915.html\n3) effects of arithmetic operations : https:\/\/gmatclub.com\/forum\/effects-of-arithmetic-operations-on-fractions-269413.html\n\nGMAT online Tutor\n\nIntern\nJoined: 27 Nov 2014\nPosts: 45\nRe: Water is leaking out from a cylinder container at the rate\u00a0 [#permalink]\n\n### Show Tags\n\n14 Mar 2015, 06:27\nchetan2u\n\nHmm it doesn't matter what is the exact total height and volume .\n\nThanks again chetan\n\nRegards\nSG\nIntern\nJoined: 27 Nov 2014\nPosts: 45\nRe: Water is leaking out from a cylinder container at the rate\u00a0 [#permalink]\n\n### Show Tags\n\n14 Mar 2015, 06:28\nchetan2u\n\nHmm it doesn't matter what is the exact total height and volume .\n\nThanks again chetan\n\nRegards\nSG\nTarget Test Prep Representative\nStatus: Founder & CEO\nAffiliations: Target Test Prep\nJoined: 14 Oct 2015\nPosts: 4551\nLocation: United States (CA)\nRe: Water is leaking out from a cylinder container at the rate\u00a0 [#permalink]\n\n### Show Tags\n\n18 Jan 2018, 07:51\nvcbabu wrote:\nWater is leaking out from a cylinder container at the rate of 0.31 m^3 per minute. 10 minutes later, the water level decreases 0.25 meter, what is value of the radius?\n\n(A) 0.5\n(B) 1.0\n(C) 1.5\n(D) 2.0\n(E) 2.5\n\nSince the rate of leaking is 0.31 m^3 per minute, in 10 minutes, 10 x 0.31 = 3.1 m^3 of water has been leaked. Let\u2019s let r = radius of the cylinder. Recall that the formula for the volume of a right circular cylinder is V = \u03c0r^2h. Using the volume of water leaked and the height associated with that volume, we can create the following equation:\n\n\u03c0r^2 x 0.25 = 3.1\n\n\u03c0r^2 \/ 4 = 3.1\n\nr^2 = 3.1 x 4 \/ \u03c0\n\nNotice that \u03c0 \u2248 3.1, thus we have:\n\nr^2 = 4\n\nr = 2\n\n_________________\n\nScott Woodbury-Stewart\nFounder and CEO\n\nGMAT Quant Self-Study Course\n500+ lessons 3000+ practice problems 800+ HD solutions\n\nRe: Water is leaking out from a cylinder container at the rate &nbs [#permalink] 18 Jan 2018, 07:51\nDisplay posts from previous: Sort by","date":"2019-01-20 15:30:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6812123656272888, \"perplexity\": 3882.939176839188}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-04\/segments\/1547583722261.60\/warc\/CC-MAIN-20190120143527-20190120165527-00341.warc.gz\"}"} | null | null |
{"url":"https:\/\/mathoverflow.net\/questions\/227310\/what-is-the-relationship-between-turing-machines-and-g%C3%B6dels-incompleteness-theo","text":"# What is the relationship between Turing Machines and G\u00f6del's Incompleteness Theorem?\n\nIn this article, Scott Aaronson talks about using Turing Machines for proving the Rosser Theorem.\n\nWhat is the relationship between the numbering that G\u00f6del used in his proof of incompleteness and Turing Machines?\n\n\u2022 You might want to read this MO post. \u2013\u00a0Burak Dec 30 '15 at 13:39\n\u2022 Turing machines can be numbered according to their representation for a Universal Turing Machine. \u2013\u00a0Thorbj\u00f8rn Ravn Andersen Dec 30 '15 at 21:33\n\n## 1 Answer\n\nIt's simple. If the halting problem is undecidable, then PA is not complete, since otherwise, you could solve the halting problem by searching for proofs in PA. And the same argument works for any sound computably axiomatizable theory $T$ able to express arithmetic. Given a Turing machine $M$ on input $i$, you formulate the assertion $\\sigma$ asserting that $M$ halts on $i$, and then search for a proof in $T$ of $\\sigma$ or a proof of $\\neg\\sigma$. If your theory were complete, then you'll find one or the other, and this would solve the halting problem. Since the halting problem is not solvable, there must be sentences of this form that are not settled by the theory.\n\nThis argument proves the first incompleteness theorem as an elementary consequence of the halting problem. The second incompleteness theorem takes a bit more work.\n\nThere is more discussion on Are the two meanings of undecidability related? and How undecidable is the spectral gap problem?\n\n\u2022 Goes also the other way. If halting problem would be decidable you could make PA complete. \u2013\u00a0Lucas K. Dec 30 '15 at 19:22\n\u2022 I agree with that statement; but perhaps it would paint a somewhat fuller picture to say that the completions of PA are branches through a certain computable tree. It follows that there are completions that are low, and these have strictly lower complexity in the Turing degrees than the halting problem. Meanwhile, other completions are far more complicated than the halting problem. \u2013\u00a0Joel David Hamkins Dec 30 '15 at 19:32\n\u2022 For example, true arithmetic is a completion of PA with Turing complexity $0^{(\\omega)}$. \u2013\u00a0Joel David Hamkins Dec 30 '15 at 20:49\n\u2022 @JoelDavidHamkins Another point, you say \"the same argument works for any sound computably axiomatizable theory $T$\". I'm probably mistaken, but doesn't the incompleteness theorem only assume consistency (and not soundness)? If so, is it viable to prove undecidability of the halting problem implies incompleteness of T, assuming that $T$ is only consistent? I suppose the linked blog post does this when it is proving Rosser's strengthened version of the incompleteness theorem? \u2013\u00a0gowrath Jun 17 '17 at 3:54\n\u2022 @gowrath Thinking about the converse is an interesting idea. I don't know any direct argument for proving that the incompleteness of PA implies the halting problem is undecidable. And given that Turing's proof of the undecidability of the halting problem is so quick, it would be hard to improve upon it. About soundness, the proof of incompletenes via the halting problem seems to use it, since one needs the veracity of the proof. But I suppose that you can eliminate this by re-proving the undecidability of the halting problem in T, thereby effectively making T sound enough for the purpose. \u2013\u00a0Joel David Hamkins Jun 17 '17 at 11:25","date":"2019-10-14 19:21:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8459585905075073, \"perplexity\": 316.6672744606935}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986654086.1\/warc\/CC-MAIN-20191014173924-20191014201424-00185.warc.gz\"}"} | null | null |
Q: Convert ArrayList to ArrayList I am fairly new to ArrayList and I'm just trying to use different ArrayList. The ArrayList<xyz> is of type float. I use this to store values in a database in Sqlite. Later, I try to move the stored values to an Arraylist<float>. I know both are of different types but is there way to do conversion. Any kind of help is appreciated.
The following is the code:
public class DatabaseHandler extends SQLiteOpenHelper {
private static final int DATABASE_VERSION = 1;
private static final String DATABASE_NAME = "TemperatureManager";
private static final String TABLE_TEMPERATURE_READING = "Temperature";
private static final String TEMPERATURE_READING = "name";
public DatabaseHandler(Context context) {
super(context, DATABASE_NAME, null, DATABASE_VERSION);
}
@Override
public void onCreate(SQLiteDatabase db) {
String CREATE_CONTACTS_TABLE = "create table " + TABLE_TEMPERATURE_READING + "("
+ TEMPERATURE_READING + " FLOAT"+")";
db.execSQL(CREATE_CONTACTS_TABLE);
}
// Upgrading database
@Override
public void onUpgrade(SQLiteDatabase db, int oldVersion, int newVersion) {
// Drop older table if existed
db.execSQL("DROP TABLE IF EXISTS " + TABLE_TEMPERATURE_READING);
// Create tables again
onCreate(db);
}
/**
* All CRUD(Create, Read, Update, Delete) Operations
*/
// Adding new contact
void addContact(Contact contact) {
SQLiteDatabase db = this.getWritableDatabase();
ContentValues values = new ContentValues();
values.put(TEMPERATURE_READING, contact.getName());
// Inserting Row
db.insert(TABLE_TEMPERATURE_READING, null, values);
db.close(); // Closing database connection
}
// Getting single contact
Contact getContact(int id) {
SQLiteDatabase db = this.getReadableDatabase();
Cursor cursor = db.query(TABLE_TEMPERATURE_READING, new String[] {
TEMPERATURE_READING},null, null, null, null, null);
if (cursor != null)
cursor.moveToFirst();
Contact contact = new Contact(cursor.getFloat(0));
// return contact
return contact;
}
// Getting All Contactsc
public ArrayList<Contact> getAllContacts() {
ArrayList<Contact> contactList = new ArrayList<Contact>();
// Select All Query
String selectQuery = "SELECT * FROM " + TABLE_TEMPERATURE_READING;
SQLiteDatabase db = this.getWritableDatabase();
Cursor cursor = db.rawQuery(selectQuery, null);
// looping through all rows and adding to list
if (cursor.moveToFirst()) {
do {
Contact contact = new Contact();
//contact.setID(Integer.parseInt(cursor.getString(0)));
contact.setName(cursor.getFloat(1));
//contact.setPhoneNumber(cursor.getString(2));
// Adding contact to list
contactList.add(contact);
} while (cursor.moveToNext());
}
// return contact list
return contactList;
}
// Getting contacts Count
public int getContactsCount() {
String countQuery = "SELECT * FROM " + TABLE_TEMPERATURE_READING;
SQLiteDatabase db = this.getReadableDatabase();
Cursor cursor = db.rawQuery(countQuery, null);
//cursor.close();
// return count
return cursor.getCount();
}
}
A: Add the following function to your class xyz.
public Float floatValue()
{
return this.f;
}
Assuming that f is the float variable of class xyz.
Now you can populate the arraylist as:Assuming list_of_objects is the arraylist of objects of class xyz.
Arraylist<float> al=new Arraylist<float>();
for(xyz o: list_of_objects)
al.add(o.floatValue());
A: You must iterate over your ArrayList and, for each value, make the transformation and store it in your ArrayList. Try the next code:
ArrayList<xyz> a1 = new ArrayList<xyz>();
//set the data in a1....
ArrayList<Float> a2 = new ArrayList<Float>();
int n = a1.size();
for(int i=0;i<n;i++)
a2.add(a1.get(i).methodXYZtoFloat());
In the method methodXYZtoFloat (it will return a Float) you must define the conversion of XYZ to Float.
Hope it helps!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 88 |
using sync_datatype_helper::test;
namespace extensions_helper {
// Returns a unique extension name based in the integer |index|.
std::string CreateFakeExtensionName(int index) {
return SyncExtensionHelper::GetInstance()->CreateFakeExtensionName(index);
}
bool HasSameExtensions(int index1, int index2) {
return SyncExtensionHelper::GetInstance()->ExtensionStatesMatch(
test()->GetProfile(index1), test()->GetProfile(index2));
}
bool HasSameExtensionsAsVerifier(int index) {
return SyncExtensionHelper::GetInstance()->ExtensionStatesMatch(
test()->GetProfile(index), test()->verifier());
}
bool AllProfilesHaveSameExtensionsAsVerifier() {
for (int i = 0; i < test()->num_clients(); ++i) {
if (!HasSameExtensionsAsVerifier(i)) {
LOG(ERROR) << "Profile " << i
<< " doesn't have the same extensions as"
" the verifier profile.";
return false;
}
}
return true;
}
bool AllProfilesHaveSameExtensions() {
for (int i = 1; i < test()->num_clients(); ++i) {
if (!SyncExtensionHelper::GetInstance()->ExtensionStatesMatch(
test()->GetProfile(0), test()->GetProfile(i))) {
LOG(ERROR) << "Profile " << i
<< " doesnt have the same extensions as"
" profile 0.";
return false;
}
}
return true;
}
std::string InstallExtension(Profile* profile, int index) {
return SyncExtensionHelper::GetInstance()->InstallExtension(
profile, CreateFakeExtensionName(index),
extensions::Manifest::TYPE_EXTENSION);
}
std::string InstallExtensionForAllProfiles(int index) {
std::string extension_id;
for (Profile* profile : test()->GetAllProfiles()) {
extension_id = InstallExtension(profile, index);
}
return extension_id;
}
void UninstallExtension(Profile* profile, int index) {
return SyncExtensionHelper::GetInstance()->UninstallExtension(
profile, CreateFakeExtensionName(index));
}
std::vector<int> GetInstalledExtensions(Profile* profile) {
std::vector<int> indices;
std::vector<std::string> names =
SyncExtensionHelper::GetInstance()->GetInstalledExtensionNames(profile);
for (std::vector<std::string>::const_iterator it = names.begin();
it != names.end(); ++it) {
int index;
if (SyncExtensionHelper::GetInstance()->ExtensionNameToIndex(*it, &index)) {
indices.push_back(index);
}
}
return indices;
}
void EnableExtension(Profile* profile, int index) {
return SyncExtensionHelper::GetInstance()->EnableExtension(
profile, CreateFakeExtensionName(index));
}
void DisableExtension(Profile* profile, int index) {
return SyncExtensionHelper::GetInstance()->DisableExtension(
profile, CreateFakeExtensionName(index));
}
bool IsExtensionEnabled(Profile* profile, int index) {
return SyncExtensionHelper::GetInstance()->IsExtensionEnabled(
profile, CreateFakeExtensionName(index));
}
void IncognitoEnableExtension(Profile* profile, int index) {
return SyncExtensionHelper::GetInstance()->IncognitoEnableExtension(
profile, CreateFakeExtensionName(index));
}
void IncognitoDisableExtension(Profile* profile, int index) {
return SyncExtensionHelper::GetInstance()->IncognitoDisableExtension(
profile, CreateFakeExtensionName(index));
}
bool IsIncognitoEnabled(Profile* profile, int index) {
return SyncExtensionHelper::GetInstance()->IsIncognitoEnabled(
profile, CreateFakeExtensionName(index));
}
void InstallExtensionsPendingForSync(Profile* profile) {
SyncExtensionHelper::GetInstance()->InstallExtensionsPendingForSync(profile);
}
} // namespace extensions_helper
ExtensionsMatchChecker::ExtensionsMatchChecker()
: profiles_(test()->GetAllProfiles()) {
DCHECK_GE(profiles_.size(), 2U);
for (Profile* profile : profiles_) {
SyncExtensionHelper::GetInstance()->InstallExtensionsPendingForSync(
profile);
CHECK(extensions::ExtensionSystem::Get(profile)
->extension_service()
->updater());
extensions::ExtensionSystem::Get(profile)
->extension_service()
->updater()
->SetUpdatingStartedCallbackForTesting(base::BindLambdaForTesting(
[self = weak_ptr_factory_.GetWeakPtr(), profile]() {
base::ThreadTaskRunnerHandle::Get()->PostTask(
FROM_HERE,
base::BindOnce(
&ExtensionsMatchChecker::OnExtensionUpdatingStarted, self,
base::Unretained(profile)));
}));
extensions::ExtensionRegistry* registry =
extensions::ExtensionRegistry::Get(profile);
registry->AddObserver(this);
}
}
ExtensionsMatchChecker::~ExtensionsMatchChecker() {
for (Profile* profile : profiles_) {
extensions::ExtensionRegistry* registry =
extensions::ExtensionRegistry::Get(profile);
registry->RemoveObserver(this);
}
}
bool ExtensionsMatchChecker::IsExitConditionSatisfied(std::ostream* os) {
*os << "Waiting for extensions to match";
auto it = profiles_.begin();
Profile* profile0 = *it;
++it;
for (; it != profiles_.end(); ++it) {
if (!SyncExtensionHelper::GetInstance()->ExtensionStatesMatch(profile0,
*it)) {
return false;
}
}
return true;
}
void ExtensionsMatchChecker::OnExtensionLoaded(
content::BrowserContext* context,
const extensions::Extension* extension) {
CheckExitCondition();
}
void ExtensionsMatchChecker::OnExtensionUnloaded(
content::BrowserContext* context,
const extensions::Extension* extension,
extensions::UnloadedExtensionReason reason) {
CheckExitCondition();
}
void ExtensionsMatchChecker::OnExtensionInstalled(
content::BrowserContext* browser_context,
const extensions::Extension* extension,
bool is_update) {
CheckExitCondition();
}
void ExtensionsMatchChecker::OnExtensionUninstalled(
content::BrowserContext* browser_context,
const extensions::Extension* extension,
extensions::UninstallReason reason) {
CheckExitCondition();
}
void ExtensionsMatchChecker::OnExtensionUpdatingStarted(Profile* profile) {
// The extension system is trying to check for updates. In the real world,
// this would be where synced extensions are asynchronously downloaded from
// the web store and installed. In this test framework, we use this event as
// a signal that it's time to asynchronously fake the installation of these
// extensions.
SyncExtensionHelper::GetInstance()->InstallExtensionsPendingForSync(profile);
CheckExitCondition();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 621 |
Visa requirements do not put a stop from visiting any country. The constraints just make the procedure tiring, which turns up upsetting the vast majority from applying.
As a traveler, we all know that the most crucial part of all our travel formalities is the visa processing. A traveler will always be in a dilemma once he has submitted visa application until he find out the status of the application from the Embassy/Consulate. This circumstance is precisely similar to the way one feels while giving an exam. Why exam? Because you don't know for sure what will happen as embassy staff are the one who is going to decide whether to grant a visa after reviewing the application. And there might be a chance your visa doesn't get approval due to something unsatisfactory as regards of that application. Thus, one can take the help of visa consultants as they have the right expertise and can identify problems, this assures us that the process is going to be smooth and hassle-free.
Here we will help you to describe the role a visa consultant plays in your visa processing. How can visa consultant be remarkably helpful to you further, what are the limitations to a visa consultant/agency?
The main role of a visa consultant is to help a client to travel from one country to another country with advice on legal and documentation to complete the process smoothly and also to increase the chances of visa approval whether travel, study, work or business purpose with professional advice.
The most important role that a consultant plays in our visa procedure is that they deal with all of the tedious manual work which generally an applicant has to do himself/herself. Right from filling the application forms, preparing the letters, booking appointment dates wherever necessary, paying the fees in the bank / getting the DD done, presenting an application, collection passports, etc. We just need to gather our important documents as per list provided by them, sign some documents and we are good to go.
We can't deny that a visa expert without a doubt has more data on the visa procedure than a voyager. Before starting the visa process one needs to take a proper information of what will be the process and what documentation is required. We put in a lot of efforts and money while planning our itinerary and all of that is at stake once we apply for the visa. Here it is very important to have an advice from an expert, who is well ahead of time if any increments or cancellations are required in the application or not. Likewise, they remain up to date with any changes to legislation. So we spare all the time that we generally would need to spend on the interconnection investigating.
Time in our most valuable asset and yet so many people are ready to misuse it. Days can be spent properly researching about visa procedure and documentation. With the amount of differing data available online, it's easy to quickly get puzzled and a single mistake can cost your visa denial. There is no doubt you will save a lot of time by hiring visa consultant.
"Yes, you read it right." This is one of the most important roles the visa consultants play for us. Be it a first-time traveler or not, we as a whole need that comfort from somebody on whom we can depend and complete the procedure. The expertise they have can give us the certainty we need and all the essential points are taken care by them. They not only just process our visas but also consult as on how the application must be showcased which will improve the chances for us to get the visas approved. If any last-minute changes come up, guidance in critical situations is something that we look for while hiring the visa consultant as we might have to provide extra documents, go for an interview anytime.
The important point we travelers need to get right. Our visa consultant is only an intermediator between us and the competent visa issuing authority. The expert can set up our application, counsel us for the procedure and documentation, yet the visa choice, all things considered, will be of the Embassy/Consulate. As a visa specialist one can't influence the visa choice, neither decidedly nor contrarily. They can assure you the visa but cannot guarantee.
They have a dependency throughout the process. Dependency on the courier services, the dependency of the submission centres, dependency of public holidays, visiting on natural calamities! Yes, they do have dependencies on natural calamities as well, the documents are shipped to the Metro cities like Mumbai, Delhi, Chennai, etc. where the embassy/consulate is located which is via air. So, any natural disturbances will affect the shipping.
By shrushti kadaganchi| 27 Feb, 2017. Posted In visa.
Thanks! we will get back to you soon. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,453 |
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Cult expert Steven Hassan was on Tuesday brought on CNN to advise on how to deprogram members of what he described in his 2019 book as the "Cult of Trump." Perhaps with an eye toward boosting his deprogramming business, however, he suggested the entire country was in need of his …
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If everyone is permitted to speak freely, their reasoning goes, people's lives will be put at risk. Those whose opinions diverge from the mainstream should not be permitted to voice those opinions, lest their words hurt people – not just people's feelings. Yet at the same time as this self-styled …
"We did what we came here to do — and so much more," Trump said in the speech on Tuesday afternoon. He argued that his administration restored American strength at home and leadership abroad, "built the greatest economy in the history of the world," and achieved a "series of historic …
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High on the endorphins of inauguration week, Siskind took to Twitter on Tuesday to emphatically exclaim: "Can't be said enough: we toppled a dictator." While it might sound like a parody, a cursory glance at the feminist's feed reveals she posted the message entirely in earnest. She even doubled down …
Page 3 of 8,359«12345 » 102030...Last » | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,070 |
\section{Introduction}
In the seminal papers of Blecher et al. \cite{blecherRead2,blecherRead,blecherRuan}, research on operators $X\in\mathcal{B}(\mathcal{H})$ with positive Hermitian parts
\begin{equation*}
\Re X=\frac{1}{2}(X+X^*)\geq 0
\end{equation*}
on a Hilbert space $\mathcal{H}$, which are called \emph{real positive operators}, has been initiated to study general operator algebras.
Such operators are also called \emph{accretive} and play an essential role in strongly continuous operator semigroups, see for instance in \cite{yosida}. Among others, further studies related to real positivity can be found in \cite{bearden,blecher,drury}. In particular, motivated in part by the paper of Kubo-Ando \cite{kubo} characterizing two-variable operator means of positive bounded linear operators on a Hilbert space, Blecher and Wang in \cite{blecher} studied root functions and the extension of the Pusz-Woronowitz geometric mean \cite{pusz}, which is given by the formula
\begin{equation*}
A\#B=\max
\left\{ X\geq 0 ~:~ \begin{bmatrix}
A & X \\
X & B
\end{bmatrix}\right\},
\end{equation*}
to the real positive setting. It is fundamental that the mean $\#$ preserves the positive definite order induced by the positive cone of bounded linear operators, and it also satisfies the arithmetic-geometric-harmonic mean inequalities \cite{bhatia,bhatiaholbrook}. It has been pointed out in \cite{blecher} that even though the usual formula
\begin{equation*}
A\#B=A^{1/2}\left(A^{-1/2}BA^{-1/2}\right)^{1/2}A^{1/2}
\end{equation*}
makes sense for two real positive operators $A,B$, it does not preserve the real positive definite order and the corresponding arithmetic-geometric-harmonic mean inequalities also fail badly.
Since, it has been somewhat surprising that no nontrivial nice real positive order preserving functions were known, even though the purely self-adjoint counterpart, the theory of (free) operator monotone functions with respect to the positive definite order is well understood in the single variable case by the classical theory of Loewner \cite{lowner,hansen3}, and now also in the non-commutative multivariable case \cite{agler,palfia2,pascoe,pascoe2} that exists in the realm of free function theory \cite{verbovetskyi}. The theory of operator means has also been extended to cover a large class of functions of probability measures on positive operators endowed with the stochastic order \cite{palfia1}.
It is obvious that the affine (arithmetic mean like) function
\begin{equation*}
F(X_1,X_2)=cI+aX_1+bX_2
\end{equation*}
for scalars $a,b\geq 0$ and $c\in\mathbb{C}$ preserves the real positive order, and apparently so do its multivariate analogues.
We prove that essentially no other locally bounded similarity invariant free function preserves the real positive order on open domains. As a precursor to this, we show that even if we consider possibly non-continuous free functions $F(X_1,\ldots,X_k)$ that are invariant just under unitary conjugations, then $F$ is real operator monotone if and only if the real part $\Re F(X_1,\ldots,X_k)$ of such functions is an operator monotone function of the real part $(\Re X_1,\ldots,\Re X_k)$ of the variables $(X_1,\ldots,X_k)$ such that $\Re F$ is independent of the skew-Hermitian part (imaginary part) of $(X_1,\ldots,X_k)$. These results show that real operator monotonicity is a rather strong property, especially rigid in the class of holomorphic functions. We further demonstrate, how our analysis generalizes to the more general case of free functions with a domain that is a free open subset of $\mathcal{B}(\mathcal{H})\otimes\mathcal{Z}$ for an arbitrary operator space $\mathcal{Z}$, not just $\mathcal{Z}=\mathbb{C}^k$ which corresponds to the set of $k$-tuples of operators above.
The paper is organized as follows.
In the second section we briefly review some necessary background material on free sets and free function theory, and also on real positivity along with basic characterizations of real monotonicity.
Then we establish intimate connections between real monotonicity and concavity with respect to the real positive definite order in Sections 3-4.
Finally, Section 5 deals with the complete characterization of real operator monotone free functions.
\section{Real monotonicity}
A bounded linear operator $X\in\mathcal{B}(E)$ is \emph{real positive} (denoted by $X\geq_{\mathrm{Re}} 0$) whenever the real part of $X$ is positive semi-definite, that is
\begin{equation*}
\Re(X):=\frac{X+X^*}{2}\geq 0
\end{equation*}
on the Hilbert space $E$. The symbol $\mathbb{P}_{\mathrm{Re}}(E)$ stands for the cone of real positive definite operators over the Hilbert space $E$ such that their real parts are invertible.
For $A,B\in\mathcal{B}(E)$ we write $A\leq_{\mathrm{Re}} B$ if and only if $A-B\geq_{\mathrm{Re}} 0$. The \emph{real positive order} for $k$-tuples $A,B\in\mathcal{B}(E)^k$ is defined similarly as $A\leq_{\mathrm{Re}} B$ exactly when $A_i-B_i\geq_{\mathrm{Re}} 0$ for all $i\in \mathbb{N}_k$. Note however that $\leq_{\mathrm{Re}}$ is just a preorder, it is not a partial order because the conditions $A\leq_{\mathrm{Re}}B$ and $B\leq_{\mathrm{Re}} A$ do not imply that $A=B$. They just imply $\Re A=\Re B$.
\begin{definition}[Free set and matrix convex set]\label{def:matrix_covexity}
A collection $(D(E))$ of sets of operators $D(E)\subseteq\mathcal{B}(E)^k$ for each Hilbert space $E$ is a called a \emph{free set} whenever for all Hilbert space $E,K$ we have the following:
\begin{itemize}
\item[1)] $U^*D(E)U\subseteq D(K)$ for all unitary $U:E\mapsto K$.
\item[2)] $D(E)\oplus D(K)\subseteq D(E\oplus K)$
\end{itemize}
where $U^*XU:=(U^{*}X_1U,\ldots,U^{*}X_kU)$ for $X\in\mathcal{B}(E)^k$.
If additionally (2) holds for any linear isometry $U:K\mapsto E$, then $(D(E))$ is a \emph{matrix convex set}.
\end{definition}
We remark that if a given free set $(D(E))$ is matrix convex, then according to \cite{helton4} each $D(E)$ is convex in the usual sense.
\begin{definition}[Free function]\label{D:freeFunction}
Let $\mathcal{L}$ be a fixed Hilbert space. A multivariate function $F:D(E)\mapsto \mathcal{B}(\mathcal{L}\otimes E)$ for a domain $D(E)\subseteq\mathcal{B}(E)^k$ defined for all Hilbert spaces $E,K$ is called a \textit{free function} whenever for all $A \in \mathcal{B}(E)^k$ and $B\in \mathcal{B}(K)^k$ in the domain of $F$, we have
\begin{itemize}
\item[1)] unitary invariance, that is
$$F(U^{*}A_1U,\ldots,U^{*}A_kU)=(I_\mathcal{L}\otimes U^{*})F(A_1,\ldots,A_k)(I_\mathcal{L}\otimes U)$$
holds for all unitaries $U\in \mathcal{B}(E)$;
\item[2)] direct sum invariance, that is
$$F\left(A_1 \oplus B_1,\ldots,A_k \oplus B_k\right)=F(A_1,\ldots,A_k) \oplus F(B_1,\ldots,B_k).$$
\end{itemize}
\end{definition}
Notice that the above extends naturally the notion of a free function given as a graded map between self-adjoint sets \cite{palfia2,pascoe,pascoe2}.
\begin{definition}[Real monotonicity and concavity]\label{D:realmon}
\end{definition}
\begin{itemize}
\item[1)] Given a free set $(B(E))$ where $B(E)\subseteq \mathcal{B}(E)^k$, a free function $F:D(E)\mapsto \mathcal{B}(\mathcal{L}\otimes E)$ is said to be \emph{real operator monotone} if we have $F(A)\leq_{\mathrm{Re}} F(B)$, whenever $A\leq_{\mathrm{Re}} B$ for $A,B\in D(E)$.
\item[2)] If each $D(E)$ is convex, then the free function $F:D(E)\mapsto \mathcal{B}(\mathcal{L}\otimes E)$ is said to be \emph{real operator concave} if for all $A,B\in D(E)$ and $\lambda\in[0,1]$, we have
\begin{equation*}
(1-\lambda)F(A)+\lambda F(B)\leq_{\mathrm{Re}} F((1-\lambda)A+\lambda B).
\end{equation*}
\item[3)] If one of the above two properties is satisfied only for finite dimensional $E$, then we say that the free function $F:D(\mathbb{C}^n)^k\mapsto \mathcal{B}(\mathcal{L}\otimes \mathbb{C}^n)$ is \emph{real $n$-monotone} or \emph{real $n$-concave}, accordingly.
\end{itemize}
\medskip
Let $\mathbb{S}(E):=\{X\in\mathcal{B}(E):X^*=X\}$ denote the set of self-adjoint bounded linear operators acting on a Hilbert space $E$.
\begin{remark}
If a free domain $(D(E))$ consists of only self-adjoint operators, then a real operator monotone free function $F:D(E)\mapsto \mathbb{S}(\mathcal{L}\otimes E)$ is operator monotone in the usual sense, that is, it preserves the positive definite order. For such functions a powerful structure theory is already available for matrix convex $(D(E))$ with nonempy interior in \cite{agler,palfia2,pascoe2}. They are essentially analytic functions of its entries such that they analytically continue to upper half-spaces, that is, operator entries with strictly positive imaginary parts. In \cite{palfia2} a widely applicable formula, based on the Schur complement, is also available through which the analytic extension can be obtained.
\end{remark}
\begin{proposition}\label{P:FrechetRealPos}
Let $D(E)$ be a matrix convex set with $D(E)\subseteq\mathcal{B}(E)^k$ and let $F:D(E)\mapsto \mathcal{B}(\mathcal{L}\otimes E)$ be a free function such that the Frech\'et-derivative $DF(X)[H]$ exists for any $H\in\mathcal{B}(E)^k$ and $X\in D(E)$. Then $F$ is real operator monotone if and only if $DF(X)[\cdot]$ is a real completely positive linear map, that is, we have
\begin{equation}\label{eq:P:FrechetRealPos}
DF(X)[H]\geq_{\mathrm{Re}}0
\end{equation}
for $H\geq_{\mathrm{Re}}0$.
\end{proposition}
\begin{proof}
$"\Rightarrow":$ Let $X\in D(E)$ and $0\leq_{\mathrm{Re}}H\in\mathcal{B}(E)^k$. Then $X+tH\geq_{\mathrm{Re}} X$ for any $t>0$, hence $F(X+tH)\geq_{\mathrm{Re}}F(X)$. This implies that
\begin{equation*}
DF(X)[H]=\lim_{t\to 0+}\frac{F(X+tH)-F(X)}{t}\geq_{\mathrm{Re}} 0.
\end{equation*}
Further $DF(X)[H]$ is also a free function of its variables $(X,H)$. Thus, the linear map $H\mapsto DF(\cdot)[H]$ satisfies the amplification formula
of completely bounded linear maps (for a proof, see Proposition 2.10. \cite{pascoe}), that is
\begin{equation*}
DF(X\otimes I)[H\otimes V]=DF(X)(H)\otimes V
\end{equation*}
for any $V\in\mathbb{S}(K)$, thus $DF(X)[\cdot]$ is also completely positive.
$"\Leftarrow":$ Let $A\leq_{\mathrm{Re}}B\in D(E)$ and $A(t):=(1-t)A+tB$ for $t\in[0,1]$. Then $A'(t)=B-A\geq_{\mathrm{Re}} 0$ and it follows that $DF(A(t))[A'(t)]\geq_{\mathrm{Re}} 0$ by the assumption. Since
\begin{equation*}
\int_{0}^{1}DF(A(t))[A'(t)]dt=F(B)-F(A),
\end{equation*}
we get that $F(B)\geq_{\mathrm{Re}} F(A)$.
\end{proof}
\begin{remark}\label{R:Steinspring}
It is known that all real completely positive linear maps satisfy the same Stinespring representation formula
as completely positive linear maps do \cite[Theorem 2.4.]{bearden}
\end{remark}
\section{Characterizations of real operator monotone functions on $\mathbb{P}_{\mathrm{Re}}$}
In this section we turn to the investigation of general properties of real operator monotone and concave functions.
We shall need the following technical lemma, which is a slight modification of \cite[Lemma 3.5.5.]{niculescu}
\begin{lemma}\label{L:concave-bounded}
Let $F$ be a concave function into $\mathbb{S}(E)$ on an open convex set $U$ in a normed linear space. If $F$ is bounded from below in a neighborhood of one point of $U$, then $F$ is locally bounded on $U$.
\end{lemma}
\begin{proof}
Suppose that $F$ is bounded from below by $MI$ for some $M\in\mathbb{R}$ on an open ball $B(a,r)$ with radius $r$ around $a$. Let $x\in U$ and choose $\rho>1$ such that $z:=a+\rho(x-a)\in U$. If $\lambda=1/\rho$, then $$V=\{v:v=(1-\lambda)y+\lambda z, y\in B(a,r)\}$$
is a neighborhood of $x=(1-\lambda)a+\lambda z$, with radius $(1-\lambda)r$. Moreover, for $v\in V$ we have
\begin{equation*}
F(v)\geq (1-\lambda)F(y)+\lambda F(z)\geq (1-\lambda)MI+\lambda F(z)\geq KI
\end{equation*}
for some $K\in\mathbb{R}$. To show that $F$ is bounded above in the same neighborhood, choose arbitrarily $v\in V$ and notice that $2x-v\in V$. By the concavity of $F$, one finds that $$F(x)\geq \frac{F(v)+F(2x-v)}{2}$$ which easily yields
\begin{equation*}
F(v)\leq 2F(x)-F(2x-v)\leq 2F(x)-KI.
\end{equation*}
\end{proof}
\begin{proposition}[see also Proposition 3.5.4 in \cite{niculescu}]\label{P:concave-cont}
A function $F:\mathbb{P}_{\mathrm{Re}}(E)^k\mapsto \mathcal{B}(E)$ with concave real part that is locally bounded from below has a continuous real part $\Re(F):\mathbb{P}_{\mathrm{Re}}(E)^k\mapsto \mathbb{S}(E)$ in the norm topology.
\end{proposition}
\begin{proof}
Let $U\subseteq\mathbb{P}_{\mathrm{Re}}(E)^k$ be an open norm bounded neighborhood with respect to the operator norm $\|\cdot\|$. Let $A\in U$ and $r>0$ such that the open ball $$B(A,2r)=\{X\in U:\|X-A\|<2r\}\subseteq U.$$
Let $X,Y\in B(A,r)$ and $X\neq Y$ such that $\alpha:=\|Y-X\|<r$. Define
\begin{equation}\label{eq:P:concave-cont-1}
Z:=Y+\frac{r}{\alpha}(Y-X).
\end{equation}
Then
\begin{equation*}
\|Z-A\|\leq\|Y-A\|+\frac{r}{\alpha}\|Y-X\|<2r,
\end{equation*}
that is, $Z\in B(A,2r)$. By \eqref{eq:P:concave-cont-1} we have
\begin{equation*}
Y=\frac{r}{r+\alpha}X+\frac{\alpha}{r+\alpha}Z,
\end{equation*}
so by the real operator concavity of $F$ we get
\begin{equation*}
\Re(F)(Y)\geq\frac{r}{r+\alpha}\Re(F)(X)+\frac{\alpha}{r+\alpha}\Re(F)(Z),
\end{equation*}
which after rearranging yields
\begin{equation*}
\begin{split}
\Re(F)(X)-\Re(F)(Y)&\leq\frac{\alpha}{r+\alpha}(\Re(F)(X)-\Re(F)(Z))\\
&\leq\frac{\alpha}{r+\alpha}2MI\leq\frac{\alpha}{r}2MI,
\end{split}
\end{equation*}
where the real number $M>0$ provides a local bound for $\Re(F)$ on $U$ in the form of
$$-2MI\leq \Re(F)(X)-\Re(F)(Z)\leq 2MI$$
in view of Lemma~\ref{L:concave-bounded}. Now exchange the role of $X$ and $Y$ in the above to obtain the reverse inequality
\begin{equation*}
\Re(F)(Y)-\Re(F)(X)\leq\frac{\alpha}{r}2MI.
\end{equation*}
From the above pair of inequalities we get
\begin{equation*}
\|\Re(F)(Y)-\Re(F)(X)\|\leq2\frac{M}{r}\|Y-X\|
\end{equation*}
proving the continuity.
\end{proof}
A net of operators $\{A_i\}_{i\in \mathcal{I}}$ is called increasing if $A_i\geq A_j$ for $i\geq j$ and $i,j\in\mathcal{I}$. Also $\{A_i\}_{i\in \mathcal{I}}$ is bounded from above if there exists some real constant $K>0$ such that $A_i\leq KI$ for all $i\in\mathcal{I}$. It is well known that any bounded from above increasing net of operators $\{A_i\}_{i\in \mathcal{I}}$ has a least upper bound $\sup_{i\in\mathcal{I}}A_i$ such that $B_j:=A_j-\sup_{i\in\mathcal{I}}A_i$ converges to $0$ in the strong operator topology. Similarly if we have a decreasing net of bounded operators that is bounded from below, then the net converges to its greatest lower bound.
The next characterization result is an extension of Theorem 2.1 in \cite{hansen3} to several variables and to the case of the real positive order. The proof is analogous to that of Theorem 2.1. We consider the finite dimensional situation, however, the proof is presented in such a way that it works also in the infinite dimensional setting as well.
\begin{proposition}\label{P:contmonotone}
Let $F:\mathbb{P}_{\mathrm{Re}}(\mathbb{C}^{2n})^k\mapsto \mathcal{B}(\mathbb{C}^{2n})$ be a real $2n$-monotone function. Then its restriction $F:\mathbb{P}_{\mathrm{Re}}(\mathbb{C}^{n})^k\mapsto \mathcal{B}(\mathbb{C}^{n})$ is real $n$-concave. Moreover, the real part $\Re(F):\mathbb{P}_{\mathrm{Re}}(\mathbb{C}^{n})^k\mapsto \mathbb{S}(\mathbb{C}^{n})$ is norm-continuous.
\end{proposition}
\begin{proof}
Let $A,B\in\mathbb{P}_{\mathrm{Re}}(\mathbb{C}^{n})^k$ and let $\lambda\in[0,1]$. Then the $2n$-by-$2n$ block matrix
\begin{equation*}
V:=\left[ \begin{array}{cc}
\lambda^{1/2}I_n & -(1-\lambda)^{1/2}I_n \\
(1-\lambda)^{1/2}I_n & \lambda^{1/2}I_n \end{array} \right]
\end{equation*}
is unitary.
Elementary calculation reveals that
\begin{equation*}
V^*\left[ \begin{array}{cc}
A & 0 \\
0 & B \end{array} \right]V=
\left[ \begin{array}{cc}
\lambda A+(1-\lambda)B & \lambda^{1/2}(1-\lambda)^{1/2}(B-A) \\
\lambda^{1/2}(1-\lambda)^{1/2}(B-A) & (1-\lambda)A+\lambda B \end{array} \right].
\end{equation*}
Set $D:=-\lambda^{1/2}(1-\lambda)^{1/2}(\Re(B)-\Re(A))$ and notice that for any given $\epsilon>0$
\begin{equation*}
\left[ \begin{array}{cc}
\lambda A+(1-\lambda)B+\epsilon I & 0 \\
0 & 2Z \end{array} \right]-V^*\left[ \begin{array}{cc}
A & 0 \\
0 & B \end{array} \right]V\geq_{\mathrm{Re}}
\left[ \begin{array}{cc}
\epsilon I & D \\
D & Z \end{array} \right]
\end{equation*}
if $Z \geq (1-\lambda)\Re(A)+\lambda \Re(B)$. The last $k$-tuple of block matrices is positive semi-definite if $Z \geq D^2/\epsilon$. So for sufficiently large positive definite $Z$ we have
\begin{equation*}
V^*\left[ \begin{array}{cc}
A & 0 \\
0 & B \end{array} \right]V\leq_{\mathrm{Re}} \left[ \begin{array}{cc}
\lambda A+(1-\lambda)B+\epsilon I & 0 \\
0 & 2Z \end{array} \right].
\end{equation*}
For such $Z>0$, by the $2n$-monotonicity of $F$ we get
\begin{equation*}
F\left(V^*\left[ \begin{array}{cc}
A & 0 \\
0 & B \end{array} \right]V\right)\leq_{\mathrm{Re}} \left[ \begin{array}{cc}
F(\lambda A+(1-\lambda)B+\epsilon I) & 0 \\
0 & F(2Z) \end{array} \right].
\end{equation*}
We also have that
\begin{equation*}
\begin{split}
&F\left(V^*\left[ \begin{array}{cc}
{A} & 0 \\
0 & {B} \end{array} \right]V\right)=
V^*\left[ \begin{array}{cc}
F({A}) & 0 \\
0 & F({B}) \end{array} \right]V\\
&=\left[ \begin{array}{cc}
\lambda F({A})+(1-\lambda)F({B}) & \lambda^{1/2}(1-\lambda)^{1/2}(F({B})-F({A})) \\
\lambda^{1/2}(1-\lambda)^{1/2}(F({B})-F({A})) & (1-\lambda)F({A})+\lambda F({B}) \end{array} \right],
\end{split}
\end{equation*}
hence we obtain that
\begin{equation}\label{eq:P:concave}
\lambda F({A})+(1-\lambda)F({B})\leq_{\mathrm{Re}} F(\lambda{A}+(1-\lambda){B}+\epsilon I).
\end{equation}
Now since $F$ is real $2n$-monotone, $\Re(F)(X+\epsilon{I})$ for $\epsilon>0$ forms a decreasing net of operators bounded from below by $\Re(F)(X)$, thus the right strong limit
$$\Re(F^{+})({X}):=\inf_{\epsilon>0}\Re(F)(X+\epsilon{I})=\lim_{\epsilon\to 0+}\Re(F)(X+\epsilon{I})$$
exists for all ${X}\in\mathbb{P}_{\mathrm{Re}}(\mathbb{C}^{2n})^k$ defining the real part of $F^{+}$. The imaginary part is defined as $\Im(F^{+})(X):=\Im(F)(X)$. Hence for any $\epsilon>0$, using \eqref{eq:P:concave}, we obtain
\[
\begin{gathered}
\lambda F^{+}({A})+(1-\lambda)F^{+}({B})\leq_{\mathrm{Re}} \lambda F({A}+\epsilon{I})+(1-\lambda)F({B}+\epsilon{I}) \\
\leq_{\mathrm{Re}} F(\lambda{A}+(1-\lambda){B}+2\epsilon{I}).
\end{gathered}
\]
Taking the limit $\epsilon\to 0+$ in the strong operator topology we conclude that
\begin{equation*}
\lambda F^{+}({A})+(1-\lambda)F^{+}({B})\leq_{\mathrm{Re}} F^{+}(\lambda{A}+(1-\lambda){B})
\end{equation*}
meaning that the free function $F^{+}$ is real $n$-concave. Also
\begin{equation*}
F({X})\leq_{\mathrm{Re}} F^{+}({X})\leq_{\mathrm{Re}} F({X}+\epsilon{I})
\end{equation*}
for all $\epsilon>0$, since $\Re(F)$ is monotone increasing. Thus, $\Re(F^{+})$ is bounded from below on order bounded sets, whence by Proposition~\ref{P:concave-cont} $\Re(F^{+})$ is norm continuous on order bounded sets because every point $A\in\mathbb{S}$ has a basis of neighborhoods in the norm topology that are order bounded sets.
As the last step, again by the real monotonicity of $F$ we have
\begin{equation*}
F^{+}({X}-\epsilon{I})\leq_{\mathrm{Re}} F({X})\leq_{\mathrm{Re}} F^{+}({X}),
\end{equation*}
and since $\Re(F^{+})$ is norm-continuous we get $F=F^{+}$ by taking the norm-limit $\epsilon\to 0+$. Hence we can also take the norm-limit $\epsilon\to 0+$ in \eqref{eq:P:concave} and conclude that $F$ is real $n$-concave and $\Re(F)$ is continuous in the norm topology.
\end{proof}
\begin{corollary}\label{P:concavemonotone}
A real operator monotone function $F:\mathbb{P}_{\mathrm{Re}}(E)^k\mapsto \mathcal{B}(E)$ is real operator concave, and it has a norm-continuous real part $\Re(F)$.
\end{corollary}
\begin{proof}
The proof goes along the lines of the previous Proposition~\ref{P:contmonotone}, where the role of $\mathbb{C}^{n}$ is taken by $E$ and using the fact that when $\dim(E)=+\infty$ we have that $E\oplus E\simeq E$.
\end{proof}
The reverse implication is also true if $F$ is bounded from below, its proof goes along the lines of Theorem 2.3 in \cite{hansen3}.
So it is worth to isolate the following result.
\begin{theorem}\label{T:concavemonotone}
Let $F:\mathbb{P}_{\mathrm{Re}}(E)^k\mapsto \mathbb{P}_{\mathrm{Re}}(E)$ be a real operator concave ($n$-concave) function. Then $F$ is real operator monotone ($n$-monotone).
\end{theorem}
\section{Hypographs and convexity}
In this section we will use the theory of matrix convex sets introduced first by Wittstock. For more on free convexity and matrix convex sets the reader is referred to \cite{effros,helton,helton2,helton3,helton4}.
Let $\Lat(E)$ denote the \emph{lattice of subspaces} of $E$. The notation $K\leq E$ means that $K$ is a closed subspace of $E$, hence a Hilbert space itself.
\begin{definition}
A graded collection $C=(C(K))$, where each $C(K)\subseteq \mathcal{B}(K)^k$, is \emph{closed with respect to reducing subspaces} if for any tuple of operators $(X_1,\ldots,X_k)\in C(K)$ and any corresponding mutually invariant subspace $N\subseteq K$, we have that $(\hat{X}_1,\ldots,\hat{X}_k)\in C(N)$, where all the $\hat{X}_i$'s are the restrictions of $X_i$ to the invariant subspace $N$ for $i\in \mathbb{N}_k$.
\end{definition}
\begin{lemma}[Lemma 2.3 in \cite{helton4}, \S 2 in \cite{helton2}]\label{lem:matrix_convexity_equivalent}
Suppose that the graded collection $C=(C(K))$, where each $C(K)\subseteq \mathcal{B}(K)^k$ respects direct sums in the sense of 1) in Definition~\ref{def:matrix_covexity} and it respects unitary conjugation in the sense of 2) in Definition~\ref{def:matrix_covexity} with $N=K$.
\begin{itemize}
\item[1)] If $C$ is closed with respect to reducing subspaces, then $C$ is matrix convex if and only if each $C(K)$ is convex in the classical sense of taking scalar convex combinations.
\item[2)] If $C$ is nonempty and matrix convex, then $0=(0,\ldots,0)\in C(1)$ if and only if $C$ is closed with respect to simultaneous conjugation by contractions.
\end{itemize}
\end{lemma}
Given a set $A\subseteq\mathcal{B}(E)$ we define its \emph{saturation} as $$\sat(A):=\{X\in\mathcal{B}(E) ~:~ \exists Y\in A, Y\geq_{\mathrm{Re}} X\}.$$
Similarly, for a graded collection $C=(C(K))$, where each $C(K)\subseteq\mathcal{B}(K)$, its \emph{saturation} $\sat(C)$ is the disjoint union of $\sat(C(K))$ for each Hilbert space $K$.
\begin{definition}[Hypographs]
Let $F:D(E)\mapsto \mathcal{B}(E)$ be a free function where $(D(E))$ is a free set. Then we define its \emph{real hypograph} $\hypo_{\mathrm{Re}}(F)$ as the graded collection of the saturation of its image, that is
$$\hypo_{\mathrm{Re}}(F)=(\hypo_{\mathrm{Re}}(F)(K)):=(\{(Y,X)\in\mathcal{B}(K)\times D(K):Y\leq_{\mathrm{Re}} F(X)\}).$$
\end{definition}
\begin{theorem}\label{T:convex_hypographs}
Let $F:D(E)\mapsto \mathcal{B}(E)$ be a free function, where $(D(E))$ is a matrix convex set which is closed with respect to reducing subspaces. Then its real hypograph $\hypo_{\mathrm{Re}}(F)$ is a matrix convex set if and only if $F$ is real operator concave.
\end{theorem}
\begin{proof}
Suppose first that $F$ is real operator concave. We will prove the matrix convexity of $\hypo_{\mathrm{Re}}(F)$ by establishing the properties in (1) of Lemma~\ref{lem:matrix_convexity_equivalent}. By the definition of real concavity, and the convexity of $\mathbb{P}_{\mathrm{Re}}$ and the real order intervals, it follows easily that for each Hilbert space $K$, $\hypo_{\mathrm{Re}}(F)(K)$ is convex in the usual sense of taking scalar convex combinations. To see that $\hypo_{\mathrm{Re}}(F)$ is closed with respect to reducing subspaces, assume that $(Y,X)\in\hypo_{\mathrm{Re}}(F)(L)$ such that $(Y,X)=(\hat{Y},\hat{X})\oplus(\overline{Y},\overline{X})$ and $$(\hat{Y},\hat{X})\in\mathcal{B}(K)\times D(K), \quad (\overline{Y},\overline{X})\in\mathcal{B}(N)\times D(N)$$
for Hilbert spaces $K\oplus N=L$. Then since $F$ is a free function, it respects direct sums, hence
$$Y\leq_{\mathrm{Re}} F(X)=F(\hat{X})\oplus F(\overline{X}).$$ Again by the definition of free functions, we have $F(\hat{X})\in\mathcal{B}(K)$ and $F(\overline{X})\in\mathcal{B}(N)$. Since $Y=\hat{Y}\oplus\overline{Y}$, it follows that $\hat{Y}\leq_{\mathrm{Re}} F(\hat{X})$ and $\overline{Y}\leq_{\mathrm{Re}} F(\overline{X})$, or in another words $(\hat{Y},\hat{X})\in\hypo_{\mathrm{Re}}(F)(K)$ and $(\overline{Y},\overline{X})\in\hypo_{\mathrm{Re}}(F)(N)$.
As for the converse, suppose that $\hypo_{\mathrm{Re}}(F)$ is a matrix convex set. First notice that $\hypo_{\mathrm{Re}}(F)$ is closed with respect to reducing subspaces. Indeed, similarly to the above assume that $(Y,X)\in\hypo_{\mathrm{Re}}(F)(L)$ with $(Y,X)=(\hat{Y},\hat{X})\oplus(\overline{Y},\overline{X})$ and $$(\hat{Y},\hat{X})\in\mathcal{B}(K)\times D(K), \quad (\overline{Y},\overline{X})\in\mathcal{B}(N)\times D(N)$$ for Hilbert spaces $K\oplus N=L$. Then since $F$ is a free function, it respects direct sums, hence $$Y\leq_{\mathrm{Re}} F(X)=F(\hat{X})\oplus F(\overline{X}).$$ Again by the definition of free functions, we have $F(\hat{X})\in\mathcal{B}(K)$ and $F(\overline{X})\in\mathcal{B}(N)$. Since $Y=\hat{Y}\oplus\overline{Y}$, it follows that $\hat{Y}\leq_{\mathrm{Re}} F(\hat{X})$ and $\overline{Y}\leq_{\mathrm{Re}} F(\overline{X})$, that is $(\hat{Y},\hat{X})\in\hypo_{\mathrm{Re}}(F)(K)$ and $(\overline{Y},\overline{X})\in\hypo_{\mathrm{Re}}(F)(N)$. So again by part 1) of Lemma~\ref{lem:matrix_convexity_equivalent} it follows that for each Hilbert space $L$, $\hypo_{\mathrm{Re}}(F)(L)$ is convex in the usual sense. It means that for all $t\in[0,1]$ and $A,B\in\mathbb{P}_{\mathrm{Re}}(L)^k$ we have that the tuple $$(Y,X):=(1-t)(F(A),A)+t(F(B),B)$$ lies in $\hypo_{\mathrm{Re}}(F)(L)$, that is $$(1-t)F(A)+tF(B)\leq_{\mathrm{Re}} F(X)=F((1-t)A+tB)$$ meaning that $F$ is real operator concave.
\end{proof}
The above Theorem~\ref{T:convex_hypographs} combined with Theorem~\ref{T:concavemonotone} leads to the following.
\begin{corollary}\label{C:convex_hypographs_monotone}
Let $F:\mathbb{P}_{\mathrm{Re}}(E)^k\mapsto \mathbb{P}_{\mathrm{Re}}(E)$ be a free function. Then its hypograph $\hypo_{\mathrm{Re}}(F)$ is a matrix convex set if and only if $F$ is real operator monotone.
\end{corollary}
\section{Representation and rigidity of real operator monotone functions}
In this section we establish some further characterizations of real operator monotone free functions in terms of operator monotone free functions. This will imply by \cite{palfia2,pascoe2} that the real parts of such functions must be analytic with respect to the real parts of their variables. Further rigidity is derived if we assume free holomorphicity for the function $F$, which according to \cite{verbovetskyi} is equivalent to a mild local boundedness condition on $F$ along with that in Definition~\ref{D:freeFunction} property 1) is strengthened to cover invariance by similarities, that is
\begin{itemize}
\item[1')] $F(S^{-1}A_1S,\ldots,S^{-1}A_kS)=(I_\mathcal{L}\otimes S^{-1})F(A_1,\ldots,A_k)(I_\mathcal{L}\otimes S)$ for every invertible $S:E\mapsto K$.
\end{itemize}
\begin{theorem}\label{T:DependsOnFirstVar}
Let $(D(E))$ be a free domain where $D(E)\subseteq\mathcal{B}(E)^k$ is defined for all Hilbert spaces $E$. Let $F:D(E)\mapsto\mathcal{B}(E)$ be a free function. Define the free function $\Re F:\Re D(E)\times \Im D(E)\mapsto\mathbb{S}(E)$ by the decomposition $F(X)=\Re F(\Re X,\Im X)+i\Im F(\Re X,\Im X)$. Then $F$ is real operator monotone if and only if $\Re F$ is independent of its second variable $\Im X$ and it is operator monotone in its first variable $\Re X$.
\end{theorem}
\begin{proof}
Assume first that $F$ is real operator monotone. Let $X\in D(E)$ and let $W\in\Im D(E)\subseteq\mathbb{S}(E)$ be arbitrary. Then $\Re X+iW\leq_{\mathrm{Re}}\Re X+i\Im X\leq_{\mathrm{Re}}\Re X+iW$, so by the real monotonicity
\begin{equation*}
F(\Re X,W)\leq_{\mathrm{Re}}F(\Re X,\Im X)\leq_{\mathrm{Re}}F(\Re X,W)
\end{equation*}
where $W$ is arbitrary for any $X$. Hence we conclude that $\Re F:\Re D(E)\times \Im D(E)\mapsto\mathbb{S}(E)$ is independent of its second variable. By the real operator monotonicity of $F$, it then follows that its real part $\Re F$ is operator monotone in its first variable as a map of self-adjoint operators into self-adjoint operators, it is also not difficult to check that it respects direct sums and simultaneous unitary conjugations, whence a free function itself.
For the converse assume that $\Re F(\Re X,\Im X)=G(\Re X)$ where $G:\Re D(E)\mapsto\mathbb{S}(E)$ is a free operator monotone function. Then clearly $F:D(E)\mapsto\mathcal{B}(E)$ is real operator monotone.
\end{proof}
An immediate consequence is the following representation.
\begin{corollary}\label{C:DependsOnFirstVar}
Let $(D(E))$ be a free domain where $D(E)\subseteq\mathcal{B}(E)^k$ is defined for all Hilbert spaces $E$. Let $F:D(E)\mapsto\mathcal{B}(E)$ be a free function. Then $F$ is real operator monotone if and only if
\begin{equation}\label{eq:C:1}
F(\Re X,\Im X)=G(\Re X)+\mathrm{i}H(\Re X,\Im X)
\end{equation}
where $H:\Re D(E)\times \Im D(E)\mapsto\mathbb{S}(E)$ is a free function and $G:\Re D(E)\mapsto\mathbb{S}(E)$ is an operator monotone free function.
\end{corollary}
\begin{remark}
It is clear that for a free function $F$, its imaginary part $\Im F$ does not have any influence on the real operator monotonicity of $F$.
It can be arbitrary and thus there are real operator monotone functions which are not free holomorphic or analytic. However their real part is always analytic or even holomorphic as a free function of self-adjoint operators, see characterizations of free operator monotonicity in \cite{palfia1}.
\end{remark}
From this point on, we shall assume that $\dim(E)<\infty$ in all statements, in order to avoid delving deeply into topological subtleties. Given a free function $F:D(E) \mapsto \mathcal{B}(E)$ on a free set $(D(E))$ we say that it is also \emph{free holomorphic} if it satisfies (1') and for each norm continuous linear functional $h:\mathcal{B}(E)\to \mathbb{C}$ the multivariable complex valued function $h(F(X))$ is holomorphic, or equivalently G\^{a}teaux-differentiable, see \cite{verbovetskyi}. Notice that (1') forces the free domain $(D(E))$ to be closed under simultaneous similarity transformations as well, not just simultaneous unitary conjugations. Also we note again that according to the main results in \cite{verbovetskyi}, for a free function $F$, (1') and a mild local boundedness condition on $F$ implies that $F$ is free holomorphic.
\begin{theorem} \label{T:main}
Given a free set $({\mathcal X}(E))$, let $F:{\mathcal X}(E) \to \mathcal{B}(E)$ be a free holomorphic function where each ${\mathcal X}(E)\subseteq \mathcal{B}(E)^k$ is open. Then $F$ is real operator monotone if and only if it admits an expression
\[
F(X)=a_0\otimes I+ \sum_{j=1}^k a_j\otimes X_j
\]
where $a_j\in \mathbb{C}$, with $a_j \geq 0$ for $j\in \mathbb{N}_k$.
\end{theorem}
We emphasize in advance that the result is still new in the single variable case as well.
Its proof rests heavily on an auxiliary lemma concerning multivariate complex functions.
\begin{definition}[Pluriharmonic function]
Let $\Omega \subseteq \mathbb{C}^m$ be a complex domain for some $m\in \mathbb{N}$.
A function $u:\Omega \mapsto \mathbb{C}$ is called \emph{pluriharmonic} whenever for any complex line
\[
L_{a,b}:=\{ a+bz : z\in \mathbb{C} \}
\]
formed by every couple of complex tuples $a,b\in \mathbb{C}$ the function $z\mapsto f(a+bz)$ is harmonic on the segment $\Omega \cap L_{a,b}$.
\end{definition}
Introducing the Wirtinger derivatives
\[
\frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-\mathrm{i}\frac{\partial}{\partial y}\right), \quad
\frac{\partial}{\partial \overline{z}}=\frac{1}{2}\left(\frac{\partial}{\partial x}+\mathrm{i}\frac{\partial}{\partial y}\right)
\]
and their corresponding multivariate counterparts
\[
\partial=
\begin{pmatrix}
\frac{\partial}{\partial z_1}\\
\vdots\\
\frac{\partial}{\partial z_m}\\
\end{pmatrix} ,\quad
\overline{\partial}=
\begin{pmatrix}
\frac{\partial}{\partial \overline{z}_1}\\
\vdots\\
\frac{\partial}{\partial \overline{z}_m}\\
\end{pmatrix}
\]
the pluriharmonic functions can be characterized by the following system of partial differential equations.
\begin{equation} \label{pluriharmonic}
\partial \overline\partial u =0 \quad \mbox{ throughout } \Omega.
\end{equation}
\begin{definition}[Levi form]
The \emph{Levi form} associated to a $\mathcal{C}^2(\Omega)$ function at the footpoint $z\in \mathbb{C}$ is the Hermitian form
\[
\mathcal{L}(z;c,d)=\sum_{j=1}^m\sum_{k=1}^m \frac{\partial^2u(z)}{\partial z_{j}\partial \overline{z}_k}c_j\overline{d}_k
\]
for any $c,d\in \mathbb{C}^m$.
\end{definition}
\begin{lemma} \label{L:pluriharmonic}
Assume that the holomorphic function $f: \Omega \mapsto \mathbb{C}^k$ on a complex domain $\Omega\subseteq\mathbb{C}^m$ admits the form $f(z)=u(\Re z)+\mathrm{i}v(\Re z,\Im z)$ and satisfies $f(0)=0$. Then $f$ is linear.
\end{lemma}
\begin{proof}
Assume, as we may, that $k=1$.
The function $u$, being a real part of a holomorphic function, is pluriharmonic (see, for instance \cite{gunning}, page 102.)
Since the function $u$ depends only on its real part, at every $z\in \Omega$ the quadratic form $\mathcal{L}(z;c,c)$ for $c\in \mathbb{R}^m$ reduces to the (real) Hessian of $u$. In virtue of \eqref{pluriharmonic} the Hessian of $u$ vanishes on the whole $\Omega$. This means that $u$ is both convex and concave. As $u(0)=0$, we conclude that the function $u$ is linear.
It follows directly from the Cauchy-Riemann equations that the function $v$ is also linear as well.
\end{proof}
The forthcoming lemma describes the structure of linear free functions.
\begin{lemma} \label{L:linearmaps}
Let $F:D(E)\subseteq\mathcal{B}(E)^k \mapsto \mathcal{B}(E)$ be a linear free function where $(D(E))$ is a free set such that each $D(E)$ contains an open neighborhood of $0$ for each Hilbert space $E$. Then there exist $a_j \in \mathbb{C}$ for $j\in \mathbb{N}_k$ such that
\[
F(X)=\sum_{j=1}^k a_j\otimes X_j.
\]
\end{lemma}
\begin{proof}
Let $H\in D(E)$ so that $H=\Re H+i\Im H$, where $\Re H=(H+H^*)/2$ and $\Im H=(H-H^*)/(2i)$. Since $D(E)$ contains an open neighborhood of $0$ there exists an $r>0$ such that the open ball $B(0,r)\subseteq D(E)$. Thus, there exists an $\epsilon>0$ such that $\epsilon H\in B(0,r)$ and also $\epsilon\Re H, \epsilon\Im H\in B(0,r)$. By linearity of $F$ we conclude that
\[
\epsilon F(H)=F(\epsilon H)=F(\epsilon\Re H+i\epsilon\Im H)=F(\epsilon\Re H)+iF(\epsilon\Im H),
\]
so it is sufficient to determine $F(H)$ for all self-adjoint $H\in B(0,r)$. To this end, consider a self-adjoint operator $H=(H_1,\ldots, H_k)\in B(0,r)$ and observe that $e_j\otimes H_j\in B(0,r)$.
As each the $H_j$'s are unitary similar to some diagonal matrices, there exist unitaries $U_j\in \mathcal{B}(E)$ such that $H_j=U_j(\oplus_{m=1}^n d_{jm})U_j^*$ with some real numbers $d_{jm}$ for $m\in \mathbb{N}_n$ and $j\in \mathbb{N}_k$.
Denote $a_j:=F(e_j\otimes 1)$.
By linearity of $F$ and elementary properties of free functions, we deduce
\[
\begin{gathered}
F(H)=\sum_{j=1}^{k} U_j F(e_j \otimes (\oplus_{m=1}^n d_{jm})) U_j^* = \\
\sum_{j=1}^{k} U_j \left(\oplus_{m=1}^n F( e_j \otimes d_{jm}) \right) U_j^*=
\sum_{j=1}^{k} U_j \left(\oplus_{m=1}^n d_{jm}F( e_j \otimes 1) \right) U_j^* \\
=\sum_{j=1}^{k}F( e_j \otimes 1) U_j \left(\oplus_{m=1}^n d_{jm}\right) U_j^*=
\sum_{j=1}^{k}a_j\otimes H_j
\end{gathered}
\]
which completes the proof of the lemma.
\end{proof}
After all these preparations, we are in a position to prove the main result of the section.
\begin{proof}[Proof of Theorem~\ref{T:main}]
The 'if' part is apparent, so we are concerned with verifying the exciting 'only if' part.
Since $F$ is holomorphic, according to \cite{verbovetskyi} $F$ commutes with similarities. This yields that $F(0)$ is associated to the center of $\mathcal{B}(E)$. Therefore, there exists some $a_0\in \mathbb{C}$ such that $F(0)=a_0\otimes I$. Moreover, an application of Lemma~\ref{L:pluriharmonic} furnishes that the free function
\[
X\mapsto F(X)-a_0\otimes I
\]
is linear, whence the result follows directly from Lemma~\ref{L:linearmaps} and the fact that the above linear map is positive, for the latter see Lemma 2.3. in \cite{bearden}.
\end{proof}
\begin{remark}
So far we have studied functions in Definition~\ref{D:freeFunction} with $\mathcal{L}=\mathbb{C}$. Our results generalize to the setting when $\mathcal{L}$ is an arbitrary Hilbert space, since given a free function $F:D(E)\mapsto \mathcal{B}(\mathcal{L}\otimes E)$ for a domain $D(E)\subseteq\mathcal{B}(E)^k$, we can reduce to the case when $\mathcal{L}=\mathbb{C}$ by looking at the free function $F_h:D(E)\mapsto \mathcal{B}(E)$ defined as $F_h(X)=(h\otimes I)(F(X))$ where $h$ is a state on $\mathcal{B}(\mathcal{L})$. In this way the constants $a_j$ in Lemma~\ref{L:linearmaps} will become bounded linear operators in $\mathcal{B}(\mathcal{L})$ and furthermore $a_j\geq 0$ for $j\in \mathbb{N}_k$ accordingly in Theorem~\ref{T:main}.
\end{remark}
\begin{remark}
Another natural generalization is to consider more general domains $D(E)\subseteq\mathcal{B}(E)\otimes\mathcal{Z}$ for an operator space $\mathcal{Z}$ as in \cite{verbovetskyi}. Then for a given $X\in D(E)$, simultaneous unitary conjugation with $U\in\mathcal{B}(E)$ is to be understood as
\begin{equation*}
U^*XU:=(U^*\otimes I_\mathcal{Z})X(U\otimes I_\mathcal{Z})
\end{equation*}
and we get back to Definition~\ref{D:freeFunction} by choosing $\mathcal{Z}=\mathbb{C}^k$. Then Theorem~\ref{T:DependsOnFirstVar} and Corollary~\ref{C:DependsOnFirstVar} are still true and we can use Lemma~\ref{L:pluriharmonic} as well, since we may restrict to finite dimensional subspaces of the domain due to general properties of free holomorphic functions considered in \cite{verbovetskyi}. Then the statement of Theorem~\ref{T:main} reads that $F$ is real operator monotone if and only if it is affine linear with its nonconstant part being a (real) completely positive linear map. Here complete positivity is derived essentially from Proposition~\ref{P:FrechetRealPos}, since the Fr\'echet-derivative $DF(X)(Y)$ of an affine linear free function $F$ is a linear map of $Y$ that is independent of $X$. Thus the expression in Theorem~\ref{T:main} becomes
\begin{equation*}
F(X)=C\otimes I+\phi(X)
\end{equation*}
where $C\in\mathcal{B}(\mathcal{L})$ and $\phi:\mathcal{Z}\mapsto\mathcal{B}(\mathcal{L})$ is a completely positive linear map. This provides an alternative proof of the expression in Theorem~\ref{T:main} as well when $\mathcal{Z}=\mathbb{C}^k$ due to the structure theory of completely positive linear maps \cite{paulsen}.
\end{remark}
\subsection*{Acknowledgments}
The current research was partially supported by the National Research, Development and Innovation Office -- NKFIH Reg. No.'s K-115383 and K-128972, and by the Ministry of Human Capacities, Hungary through grant 20391-3/2018/FEKUSTRAT.
The work of Ga\'al was supported by the DAAD-Tempus PPP Grant 57448965.
The work of P\'alfia was supported by the National Research Foundation of Korea (NRF) grants founded by the Korea government (MEST) No.2015R1A3A2031159, No.2016R1C1B1011972 and No.2019R1C1C1006405.
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package ecs
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//
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//distributed under the License is distributed on an "AS IS" BASIS,
//WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
//See the License for the specific language governing permissions and
//limitations under the License.
//
// Code generated by Alibaba Cloud SDK Code Generator.
// Changes may cause incorrect behavior and will be lost if the code is regenerated.
import (
"github.com/aliyun/alibaba-cloud-sdk-go/sdk/requests"
"github.com/aliyun/alibaba-cloud-sdk-go/sdk/responses"
)
// CreateAutoProvisioningGroup invokes the ecs.CreateAutoProvisioningGroup API synchronously
func (client *Client) CreateAutoProvisioningGroup(request *CreateAutoProvisioningGroupRequest) (response *CreateAutoProvisioningGroupResponse, err error) {
response = CreateCreateAutoProvisioningGroupResponse()
err = client.DoAction(request, response)
return
}
// CreateAutoProvisioningGroupWithChan invokes the ecs.CreateAutoProvisioningGroup API asynchronously
func (client *Client) CreateAutoProvisioningGroupWithChan(request *CreateAutoProvisioningGroupRequest) (<-chan *CreateAutoProvisioningGroupResponse, <-chan error) {
responseChan := make(chan *CreateAutoProvisioningGroupResponse, 1)
errChan := make(chan error, 1)
err := client.AddAsyncTask(func() {
defer close(responseChan)
defer close(errChan)
response, err := client.CreateAutoProvisioningGroup(request)
if err != nil {
errChan <- err
} else {
responseChan <- response
}
})
if err != nil {
errChan <- err
close(responseChan)
close(errChan)
}
return responseChan, errChan
}
// CreateAutoProvisioningGroupWithCallback invokes the ecs.CreateAutoProvisioningGroup API asynchronously
func (client *Client) CreateAutoProvisioningGroupWithCallback(request *CreateAutoProvisioningGroupRequest, callback func(response *CreateAutoProvisioningGroupResponse, err error)) <-chan int {
result := make(chan int, 1)
err := client.AddAsyncTask(func() {
var response *CreateAutoProvisioningGroupResponse
var err error
defer close(result)
response, err = client.CreateAutoProvisioningGroup(request)
callback(response, err)
result <- 1
})
if err != nil {
defer close(result)
callback(nil, err)
result <- 0
}
return result
}
// CreateAutoProvisioningGroupRequest is the request struct for api CreateAutoProvisioningGroup
type CreateAutoProvisioningGroupRequest struct {
*requests.RpcRequest
LaunchConfigurationDataDisk *[]CreateAutoProvisioningGroupLaunchConfigurationDataDisk `position:"Query" name:"LaunchConfiguration.DataDisk" type:"Repeated"`
ResourceOwnerId requests.Integer `position:"Query" name:"ResourceOwnerId"`
LaunchConfigurationSystemDiskCategory string `position:"Query" name:"LaunchConfiguration.SystemDiskCategory"`
AutoProvisioningGroupType string `position:"Query" name:"AutoProvisioningGroupType"`
LaunchConfigurationSystemDiskPerformanceLevel string `position:"Query" name:"LaunchConfiguration.SystemDiskPerformanceLevel"`
LaunchConfigurationHostNames *[]string `position:"Query" name:"LaunchConfiguration.HostNames" type:"Repeated"`
LaunchConfigurationSecurityGroupIds *[]string `position:"Query" name:"LaunchConfiguration.SecurityGroupIds" type:"Repeated"`
ResourceGroupId string `position:"Query" name:"ResourceGroupId"`
LaunchConfigurationImageId string `position:"Query" name:"LaunchConfiguration.ImageId"`
LaunchConfigurationResourceGroupId string `position:"Query" name:"LaunchConfiguration.ResourceGroupId"`
LaunchConfigurationPassword string `position:"Query" name:"LaunchConfiguration.Password"`
PayAsYouGoAllocationStrategy string `position:"Query" name:"PayAsYouGoAllocationStrategy"`
DefaultTargetCapacityType string `position:"Query" name:"DefaultTargetCapacityType"`
LaunchConfigurationKeyPairName string `position:"Query" name:"LaunchConfiguration.KeyPairName"`
SystemDiskConfig *[]CreateAutoProvisioningGroupSystemDiskConfig `position:"Query" name:"SystemDiskConfig" type:"Repeated"`
DataDiskConfig *[]CreateAutoProvisioningGroupDataDiskConfig `position:"Query" name:"DataDiskConfig" type:"Repeated"`
ValidUntil string `position:"Query" name:"ValidUntil"`
LaunchTemplateId string `position:"Query" name:"LaunchTemplateId"`
OwnerId requests.Integer `position:"Query" name:"OwnerId"`
LaunchConfigurationSystemDiskSize requests.Integer `position:"Query" name:"LaunchConfiguration.SystemDiskSize"`
LaunchConfigurationInternetMaxBandwidthOut requests.Integer `position:"Query" name:"LaunchConfiguration.InternetMaxBandwidthOut"`
LaunchConfigurationHostName string `position:"Query" name:"LaunchConfiguration.HostName"`
MinTargetCapacity string `position:"Query" name:"MinTargetCapacity"`
MaxSpotPrice requests.Float `position:"Query" name:"MaxSpotPrice"`
LaunchConfigurationArn *[]CreateAutoProvisioningGroupLaunchConfigurationArn `position:"Query" name:"LaunchConfiguration.Arn" type:"Repeated"`
LaunchConfigurationPasswordInherit requests.Boolean `position:"Query" name:"LaunchConfiguration.PasswordInherit"`
ClientToken string `position:"Query" name:"ClientToken"`
LaunchConfigurationSecurityGroupId string `position:"Query" name:"LaunchConfiguration.SecurityGroupId"`
Description string `position:"Query" name:"Description"`
TerminateInstancesWithExpiration requests.Boolean `position:"Query" name:"TerminateInstancesWithExpiration"`
LaunchConfigurationUserData string `position:"Query" name:"LaunchConfiguration.UserData"`
LaunchConfigurationCreditSpecification string `position:"Query" name:"LaunchConfiguration.CreditSpecification"`
LaunchConfigurationSystemDisk CreateAutoProvisioningGroupLaunchConfigurationSystemDisk `position:"Query" name:"LaunchConfiguration.SystemDisk" type:"Struct"`
LaunchConfigurationInstanceName string `position:"Query" name:"LaunchConfiguration.InstanceName"`
LaunchConfigurationInstanceDescription string `position:"Query" name:"LaunchConfiguration.InstanceDescription"`
SpotAllocationStrategy string `position:"Query" name:"SpotAllocationStrategy"`
TerminateInstances requests.Boolean `position:"Query" name:"TerminateInstances"`
LaunchConfigurationSystemDiskName string `position:"Query" name:"LaunchConfiguration.SystemDiskName"`
LaunchConfigurationSystemDiskDescription string `position:"Query" name:"LaunchConfiguration.SystemDiskDescription"`
ExcessCapacityTerminationPolicy string `position:"Query" name:"ExcessCapacityTerminationPolicy"`
LaunchTemplateConfig *[]CreateAutoProvisioningGroupLaunchTemplateConfig `position:"Query" name:"LaunchTemplateConfig" type:"Repeated"`
LaunchConfigurationRamRoleName string `position:"Query" name:"LaunchConfiguration.RamRoleName"`
LaunchConfigurationInternetMaxBandwidthIn requests.Integer `position:"Query" name:"LaunchConfiguration.InternetMaxBandwidthIn"`
SpotInstanceInterruptionBehavior string `position:"Query" name:"SpotInstanceInterruptionBehavior"`
LaunchConfigurationSecurityEnhancementStrategy string `position:"Query" name:"LaunchConfiguration.SecurityEnhancementStrategy"`
LaunchConfigurationTag *[]CreateAutoProvisioningGroupLaunchConfigurationTag `position:"Query" name:"LaunchConfiguration.Tag" type:"Repeated"`
LaunchConfigurationDeploymentSetId string `position:"Query" name:"LaunchConfiguration.DeploymentSetId"`
ResourceOwnerAccount string `position:"Query" name:"ResourceOwnerAccount"`
OwnerAccount string `position:"Query" name:"OwnerAccount"`
SpotInstancePoolsToUseCount requests.Integer `position:"Query" name:"SpotInstancePoolsToUseCount"`
LaunchConfigurationInternetChargeType string `position:"Query" name:"LaunchConfiguration.InternetChargeType"`
LaunchTemplateVersion string `position:"Query" name:"LaunchTemplateVersion"`
LaunchConfigurationIoOptimized string `position:"Query" name:"LaunchConfiguration.IoOptimized"`
PayAsYouGoTargetCapacity string `position:"Query" name:"PayAsYouGoTargetCapacity"`
TotalTargetCapacity string `position:"Query" name:"TotalTargetCapacity"`
SpotTargetCapacity string `position:"Query" name:"SpotTargetCapacity"`
ValidFrom string `position:"Query" name:"ValidFrom"`
AutoProvisioningGroupName string `position:"Query" name:"AutoProvisioningGroupName"`
}
// CreateAutoProvisioningGroupLaunchConfigurationDataDisk is a repeated param struct in CreateAutoProvisioningGroupRequest
type CreateAutoProvisioningGroupLaunchConfigurationDataDisk struct {
PerformanceLevel string `name:"PerformanceLevel"`
KmsKeyId string `name:"KmsKeyId"`
Description string `name:"Description"`
SnapshotId string `name:"SnapshotId"`
Size string `name:"Size"`
Device string `name:"Device"`
DiskName string `name:"DiskName"`
Category string `name:"Category"`
DeleteWithInstance string `name:"DeleteWithInstance"`
Encrypted string `name:"Encrypted"`
}
// CreateAutoProvisioningGroupSystemDiskConfig is a repeated param struct in CreateAutoProvisioningGroupRequest
type CreateAutoProvisioningGroupSystemDiskConfig struct {
DiskCategory string `name:"DiskCategory"`
}
// CreateAutoProvisioningGroupDataDiskConfig is a repeated param struct in CreateAutoProvisioningGroupRequest
type CreateAutoProvisioningGroupDataDiskConfig struct {
DiskCategory string `name:"DiskCategory"`
}
// CreateAutoProvisioningGroupLaunchConfigurationArn is a repeated param struct in CreateAutoProvisioningGroupRequest
type CreateAutoProvisioningGroupLaunchConfigurationArn struct {
Rolearn string `name:"Rolearn"`
RoleType string `name:"RoleType"`
AssumeRoleFor string `name:"AssumeRoleFor"`
}
// CreateAutoProvisioningGroupLaunchConfigurationSystemDisk is a repeated param struct in CreateAutoProvisioningGroupRequest
type CreateAutoProvisioningGroupLaunchConfigurationSystemDisk struct {
Encrypted string `name:"Encrypted"`
KMSKeyId string `name:"KMSKeyId"`
EncryptAlgorithm string `name:"EncryptAlgorithm"`
}
// CreateAutoProvisioningGroupLaunchTemplateConfig is a repeated param struct in CreateAutoProvisioningGroupRequest
type CreateAutoProvisioningGroupLaunchTemplateConfig struct {
VSwitchId string `name:"VSwitchId"`
MaxPrice string `name:"MaxPrice"`
Priority string `name:"Priority"`
InstanceType string `name:"InstanceType"`
WeightedCapacity string `name:"WeightedCapacity"`
}
// CreateAutoProvisioningGroupLaunchConfigurationTag is a repeated param struct in CreateAutoProvisioningGroupRequest
type CreateAutoProvisioningGroupLaunchConfigurationTag struct {
Key string `name:"Key"`
Value string `name:"Value"`
}
// CreateAutoProvisioningGroupResponse is the response struct for api CreateAutoProvisioningGroup
type CreateAutoProvisioningGroupResponse struct {
*responses.BaseResponse
AutoProvisioningGroupId string `json:"AutoProvisioningGroupId" xml:"AutoProvisioningGroupId"`
RequestId string `json:"RequestId" xml:"RequestId"`
LaunchResults LaunchResults `json:"LaunchResults" xml:"LaunchResults"`
}
// CreateCreateAutoProvisioningGroupRequest creates a request to invoke CreateAutoProvisioningGroup API
func CreateCreateAutoProvisioningGroupRequest() (request *CreateAutoProvisioningGroupRequest) {
request = &CreateAutoProvisioningGroupRequest{
RpcRequest: &requests.RpcRequest{},
}
request.InitWithApiInfo("Ecs", "2014-05-26", "CreateAutoProvisioningGroup", "ecs", "openAPI")
request.Method = requests.POST
return
}
// CreateCreateAutoProvisioningGroupResponse creates a response to parse from CreateAutoProvisioningGroup response
func CreateCreateAutoProvisioningGroupResponse() (response *CreateAutoProvisioningGroupResponse) {
response = &CreateAutoProvisioningGroupResponse{
BaseResponse: &responses.BaseResponse{},
}
return
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,021 |
Q: Can't identify this data structure / command doesn't work on structure I'm trying to split my data:
np.split(array, np.where(np.diff(array[:,1]))[0]+1)
My original dataset looks like this (it comes from an attribute table that was converted to a NumPy array by the TableToNumPyArray command):
[(1.0, 3.0, 1, 427338.4297000002, 4848489.4332)
(1.0, 3.0, 2, 427344.7937000003, 4848482.0692)
(1.0, 3.0, 3, 427346.4297000002, 4848472.7469)
(1.0, 1.0, 7084, 427345.2709999997, 4848796.592)
(1.0, 1.0, 7085, 427352.9277999997, 4848790.9351)
(1.0, 1.0, 7086, 427359.16060000006, 4848787.4332)]
However, when I run the command: this error comes up:
Runtime error
Traceback (most recent call last):
File "<string>", line 1, in <module>
IndexError: too many indices for array
However, when I run the command on this dataset (same numbers, different format), it works:
[[ 1.00000000e+00 3.00000000e+00 1.00000000e+00 4.27338430e+05
4.84848943e+06]
[ 1.00000000e+00 3.00000000e+00 2.00000000e+00 4.27344794e+05
4.84848207e+06]
[ 1.00000000e+00 3.00000000e+00 3.00000000e+00 4.27346430e+05
4.84847275e+06]
[ 1.00000000e+00 1.00000000e+00 7.08400000e+03 4.27345271e+05
4.84879659e+06]
[ 1.00000000e+00 1.00000000e+00 7.08500000e+03 4.27352928e+05
4.84879094e+06]
[ 1.00000000e+00 1.00000000e+00 7.08600000e+03 4.27359161e+05
4.84878743e+06]]
I don't know what the first data structure is, so I'm having trouble with conversion into a structure that works. Can anyone help me either convert the data or run the command on my original dataset? Thanks!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,189 |
Kazuhiko Shingyoji (født 5. februar 1986) er en tidligere japansk fodboldspiller.
Han har spillet for flere forskellige klubber i sin karriere, herunder Mito HollyHock og Blaublitz Akita.
Eksterne henvisninger
Fodboldspillere fra Japan | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,832 |
\section{Limitations and Future Work}
Users were sometimes confusing the visualization of sounds and words and not all dialects could be distinguished. This may improve with users learning the specifics of the method but could also be further supported by making the method more intuitive and provide higher information content with more powerful face generators.
In the future, we plan to integrate realistic lip motion for spoken sounds that are captured in the lip motion while maintaining the proposed general encoding for those that are not for environment sounds.
Environment sounds lose higher frequencies since trained on speech.
\comment{
* no semantics.
* no temporal smoothness since trained on images not videos.
* not validated for deaf people.
Not just totally deaf persons, also many dyslexic deficits can be associated to missing subtleties in hearing. Perhaps, visual representations could help. Further life science studies are needed to explore these possible directions enabled by our AudioViewer{} translation system.
Future work ideas:
• Language specific models or general ones? (different for mandarin and English?)
* real-time?
* supervised with phenom labels
* higher capacity / modern video generators
* stereo video output / 3D output
* stereo audio input
* low latency generation
* different temporal granularity
* using more sophisticated perceptual metrics
• Medical / psychological / linguistic studies
• How can models be updated after deployment. A user learns the meaning of images,
replacement of the image synthesis model would require learning the correspondence anew
with each update.
* disentanglement of style and content
* joint latent space with SWD of different batch size
* visualize as hand written use letters/Phonetic Alphabet
- I feel like figures are easier to
* Additional users studies can feature live demos, and having participants trying to decode sequences of video via pronouciation feedback
* perhaps adding some constraints /loss on the visuals might be good ie. that different sounds should look different
}
\section{Conclusion}
We presented AudioViewer{}, a new approach for visualizing audio via generative models coupled with cycle consistency and smoothness constraints. We have explored different generative models and target domains and found that while visualizations based on CelebA faces have richer features and a higher SNR, the visualizations based on the MNIST digits translated with VAEs give the most consistent mapping that is best recognized by human participants as indicated per our user study. Because our method is self-supervised, personalized models for any languages or sound domain can be learned. We hope that our approach will catalyze the development of future tools for assisting with handicaps linked to perceiving subtle audio differences.
\comment{
\section*{Broader Impact}
When used as an assistive tool, AudioViewer{} has the potential to have a positive impact on society by assisting and better connecting people with hearing impairment. However, such technology should be treated with care. It will require careful clinical trials to rule out negative effects; as with any tool or mechanism used for medical training, particularly when working with children. We train and test our model on spoken English and general sounds, but biases in the data distribution of such datasets can carry over to our trained model. For example, the former dataset only represents American voices, where male speakers are over-represented. One has to be careful to ensure that such biases are mitigated before such a tool can be reliably used by the public.
However, since our method is self-supervised, only speech samples are required as input and can be re-trained for languages for which no annotated training sets exist.
\begin{ack}
We thank all participants of the user study for their time and contribution and Farnoosh Javadi for improving this manuscript.
\end{ack}
}
\section{Experiments}
In the following, we show qualitatively and quantitatively that AudioViewer{} conveys important audio features via visualizations of portraits or numbers and that it applies to speech and environment sounds.
We provide additional qualitative results in the supplemental videos and document.
\parag{Quantitative metrics.} We compare latent embeddings by the Euclidean distance, mel spectrogram with the signal to noise ratio (SNR),
Smoothness is measured as the change in latent space position.
\parag{Perceptual study.} We perform a user study to analyze the human capability to perceive the translated audio content. Some results may be subjective and we therefore report statistics over 29 questions answered by 19 users. The study details are given in the supplemental document.
\parag{Baselines.} We compare the proposed coupled audio-visual model and ablate using individual VAEs for audio and speech generation as well as when disabling the style-content disentangling. Furthermore, we ablate the latent space priors and training strategies introduced in section~\ref{sec:priors} and \ref{sec:cycle}.
As a further baseline, we experiment with principal component analysis (PCA) for building a common latent space. PCA yields a projection matrix $\mathbf{W}$ that rotates the training samples $\mathbf{u}_i$ by $\mathbf{z}_i = \mathbf{W} \mathbf{u}_i$ to have diagonal covariance matrix $\mathbf{\Sigma}$ and maximal variance.
We can use PCA for sound and images, with $\mathbf{u}$ being either the flattened mel-segment $\VEC(M_i)$ or equivalently the flattened video frame $\VEC(I_j)$. We use the reduced PCA version, where $\mathbf{W}$ maps to a $Z_D$ dimensional space. %
We further scale rows of $\mathbf{W}$ such that $\mathbf{\Sigma}$ is the identity matrix $I$. By this construction, the latent space has the same prior covariance as the VAE spaces. This encoding can be decoded with the pseudo-inverse $\mathbf{W}^\dagger$.
\parag{Audio Datasets.} We use the TIMIT dataset~\cite{garofolo1993timit} for learning speech embeddings. It contains 5.4 hours of audio recordings (16 bit, 16 kHz) as well as time-aligned orthographic, phonetic and word transcriptions for
630 speakers of eight major dialects of American English, each reading ten phonetically rich sentences. We use the training split (462/50/24 non-overlapping speakers) of the KALDI toolkit~\cite{povey2011kaldi}. The phonetic annotation is only used at training time and as ground truth for the user study. In addition we report how well a model trained on speech generalizes to other sound on the ESC-50 dataset of environmental sounds \cite{piczak2015esc} in the supplemental videos.
\parag{Image Datasets.} To test what kind of visualization is best for humans to perceive the translated audio, we train and test our model on two image datasets: The face attributes dataset CelebA~\cite{liu2015faceattributes} (162770/19962 images for train/val respectively), the MNIST~\cite{lecun1998gradient} datasets (60000/10000 images for train/val respectively).
\parag{Training and runtime.} Speech models are trained for 300 epochs with batch size 128 (65 pairs), CelebA models for 38 epochs and MNIST models for 24 epochs with batch size 144, and joint models are fine-tuned jointly on audio and image examples for 10 audio epochs. AudioViewer{} is real-time capable and we showcase a demonstration in the supplemental video, the amortized inference time for a frame is 5 ms on an i7-9700KF CPU @ 3.60GHz with a single NVIDIA GeForce RTX 2080 Ti.
\subsection{Visual Quality and Phonetics}
\input{tex/fig_words}
\input{tex/fig_phonemes.tex}
A meaningful audio translation should encode similar sounds with similar visuals. Figure~\ref{fig:phonemes content} depicts such similarity and dissimilarity on the first 200 ms of a spoken word since these can be visualized with a single video frame. This figure compares the content encoding to MNIST and faces, both are well suited to distinguish sounds.
The same word (column) has high visual similarity across different speakers (F\#: female, M\#: male; \# the speaker ID) and dialects (D\#) while words starting with different sounds are visually different. Multiple frames of entire words are shown in Fig.~\ref{fig:words}.
Our user study and supplemental document analyzes their differences in more detail and on entire sentences.
Although trained entirely on speech, our low-level formulation at the level of sounds enables AudioViewer to also visualize environment sounds. Figure~\ref{fig:environment} gives two examples.
\subsection{Information Throughput}
\input{tex/fig_throughput}
It is difficult to quantify the information throughput from audio to video as no ground truth is available and user studies are expensive. However, we can use the learned encoder and decoder to map from audio to video and back as a lower bound. This cyclic pattern lets us quantify the loss of information as the distance of the starting point to the reconstructed audio. Figure \ref{fig:throughput} gives an example and Table~\ref{tab:audio_and_video} summarizes relations quantitatively. The difference can be quantified and analyzed qualitatively by listening and comparing the original and reconstructed audio samples.
PCA attains the highest reconstruction accuracy but has poor smoothness properties and does not strike along the other dimensions.
Interpretability in the disentangled space (lower half of Table~\ref{tab:audio_and_video}) has a relatively low toll on the SNR, a reduction from $4.43$ to $4.16$. MNIST proved less stable to train and does not fair well with the cycle loss, perhaps due to having a lower dimensionality compared to the audio encoding.
We further analyze the effect of the additional constraints on the audio autoencoder by measuring its throughput individually.
Out of the models with smoothness, $\mathcal{L}_{p, \log MSE}$ with $\lambda_p=10^3$ attains the highest reconstruction quality (SNR) and lowest latent space velocity calculated with finite differences. The disentanglement reduces audio reconstruction significantly. However, this has limited effect on the overall throughput, as analyzed before, likely due to bottlenecks in the visual model dominating the overall accuracy.
\input{tex/tab_audio.tex}
\subsection{Temporal Smoothness Effectiveness}
\input{tex/fig_smoothness}
Mapping from audio to video with VAEs without constraints and PCA leads to choppy results, with mean latent space velocities above 300 s$^{-1}$. We visualize the gain in smoothness in Figure \ref{fig:embedding_curves} by plotting the high-dimensional latent trajectory embedded into three dimensions using
multi-dimensional scaling (MDS) \cite{cox2008multidimensional}. The gain is similar for all of the proposed smoothness constraints. The supplemental videos show how the smoothness eases information perception.
\subsection{Visual Domain Influence}
Earlier work~\cite{chernoff1973use,jacob1976face} suggests that faces are suitable for representing high-dimensional data. Our experiments support this finding in that the measurable information content is larger. The SNR of the jointly trained CelebA models is larger than the MNIST models. Also the Figure \ref{fig:phonemes content} indicates that facial models are richer. The distinctiveness is further analyzed in the user study evaluation.
\subsection{User study}
We analyze the gain from a disentangled latent space vs.~a combined one and the difference of using faces or digits for visualization with a user study among 19 participants answering 29 questions, with the same question repeated for the three visualization techniques mentioned above and with questions randomized.
\input{tex/tab_user_study_main}
\paragraph{Identifying and distinguishing sounds.}
Our user study analyzes the ability to distinguish between visualizations of different sounds, similar to the ones show in Figure~\ref{fig:phonemes content} and the results are show in Table~\ref{tab:user_study}. Overall, the users were able to correctly recognize visualizations for the same sound with an accuracy of 85.4\% for the CelebA content model and 79.7\% for the MNIST content model. Broken down by tasks, users for the CelebA model achieve the better accuracy of 84.6\% in correctly matching one of two visualizations against a reference (random guessing yields 50\%), while for the task of grouping four visualizations into pairs, users achieved an accuracy of 85.8 \% (random guessing yields 33.3\%).
\parag{Latent space disentanglement.}
Visualizing the style and content part separately with our full model significantly improves recognition scores. The disentangled face model increases the accuracy for distinguishing between speakers of different sex from $41.5 \pm 9.0\%$ to $ 77.1 \pm 10.7\%$ and speakers of different dialects from $15.4 \pm 16.3\%$ to $57.1 + 22.8\%$. An AudioViewer prototype should therefore use the full model and show separate visual decodings side by side or have the option to switch between content and style visualization.
\comment{
\parag{Structural similarities.}
We analyzed whether structural similarities in audio is carried over to the visualizations. We found that when comparing visualizations of words that share more phones against words which share fewer (or no) phones, users always (100\%) found shared phones to be more similar for the CelebA visualizations while they were more similar 95\% of the time for the MNIST visualizations. In addition, when trying to assess similarity between phone-pairs that do or do not share a common phone, users reported phone-pairs that do share a phone as more similar 73.3\% and 69.4\% of the time for the MNIST and CelebA visualizations respectively.
}
The user study is reported in full in the supplemental document.
\section{Introduction}
Humans perceive their environment through diverse channels, including vision, hearing, touch, taste, and smell. Impairment in any of these modals of perception can lead to drastic consequences, such as challenges in learning and communication. Various approaches have been purposed to substitute lost senses, going all the way to recently popularized attempts to directly interface with neurons in the brain (e.g., NeuraLink \cite{musk2019integrated}) .
One of the least intrusive approaches is to substitute audio and video, which is, however, challenging due to their high throughput and different modality.
In this paper, make a first step to visualize audio with natural images in real time, forming a live video that
characterizes the entire audio content. It could be seen as an automated form of sign language translation, with its own throughput, abstraction, automation, and readability trade-offs.
Figure~\ref{fig:teaser} shows an example.
Such a tool could help to perceive sound and also be used to train deaf speech, as immediate and high-throughput feedback on pronunciations is difficult to learn from traditional visual tools for the hearing-impaired speech learning that rely on spectrogram representations \cite{elssmann1987speech,yang2010speech,xu2008speech,kroger2010audiovisual}.
Our long-term goal is to show that hearing-impaired
persons can learn to perceive sound through the visual system. Similar to learning to understand sign language and likewise an unimpaired child learns to associate meaning to seen objects and perceived sounds and words, the user will have to learn a mapping from our sound visualization to its semantic. Still, articulation
could be practiced by exploring the visual feedback space and speaking learned by articulating
sounds that reproduce the same visual feedback as produced by the teacher or parent.
Translating an audio signal to video is difficult as there is no canonical one-to-one mapping across these domains with vastly different dimensions (one-dimensional audio to high-dimensional video) and characteristics (time-frequency representation to spatial structure). It is an open question of which visual abstraction level is best.
Methods for digital dubbing and lip-syncing facial expressions from spoken audio \cite{duarte2019wav2pix,sadoughi2019speech,wiles2018x2face} would be natural, allowing deaf people to lip read. However, natural lip motion is rather a result of speaking and only contains a fraction of the audio information and does not apply to environment sounds. Another direction could be to translate speech to words with a recognition system \cite{noda2015audio}, for example the spoken 'dog' would be translated to the text 'dog'. A conceivable extension is to further translate the word into an image using conditional generative adversarial networks (GANs) \cite{goodfellow2014generative}. A dog image could be shown when the word 'dog' is recognized in the audio. This is intuitive, however, such a translation is still indirect and does not contain any vocal feedback or style differences between male and female speakers~\cite{stewart1976a,zaccagnini1993effects,oster1995teaching}.
Moreover, events and ambient sounds that can not be indicated by a single word are ill-represented, such as the echo of a dropped object or the repeating beep of an alarm clock.
To overcome these limitations, we seek for an intuitive and immediate visual representation of the high-dimensional audio data and are inspired by early techniques from the 1970s that visualize general high-dimensional data with facial line drawings with varying length of the nose and curvature of the mouth~\cite{chernoff1973use,jacob1976face}.
\comment{
Our solution is a direct, low-level translation from audio to video
via a linked latent space, learned and structured with variational auto encoders (VAEs)~\cite{kingma2013auto} that capture natural statistics and which we equip with additional priors. %
For instance, when mapping from audio to portrait visuals, \TODO{we noticed that speakers of the same sex result in uniform changes in background and hair style across words.} This directness comes at the expense of loosing high-level semantics, such as lip motion, and requires the human user to do a significant amount of learning. A trade-off that we deem worth exploring, with many directions of future work.
}
In this paper, we design an immediate audio to video mapping leveraging ideas from computer vision and audio processing for unpaired domain translation.
The technical difficulty lies in finding a mapping that is expressive yet easy to perceive.
We base AudioViewer{} on four principles.
First, humans are good at recognizing patterns and objects that they see in their environment, particularly human faces \cite{aguirre1998area,kanwisher1999fusiform,tarr2000ffa,chernoff1973use,jacob1976face}. We therefore map to videos that have the image statistics of natural images and faces.
Second, humans are able to perceive complex spatial structure but quick and non-natural (e.g., flickering) temporal changes lead to disruption and tiredness \cite{sperling1960information}. Hence, we enforce smoothness constraints on the learned mapping.
Third, frequent audio features should be mapped to frequent video features. We therefore exploit cycle consistency to learn a joint structure between audio and video modalities. Fourth, we disentangle factors of variations, to separate style, such as gender and dialect, from content, individual phones and sounds.
Our core contributions are a new approach for enabling people to see what they can not hear; a method for mapping from audio to video via linked latent spaces; a quantitative analysis on the type of target videos, different latent space models, and the effect of smoothness constraints; and a first perceptual user study on the recognition capabilities.
We demonstrate the feasibility of this AudioViewer{} approach with a working prototype and analyze the success by quantifying the loss of information content as well as showing that words and pronunciation can be distinguished from the generated video features in a user study.
\comment{
The task at hand is a domain translation problem, such as translating speech to text. One could
attempt to translate speech to pictures semantically. For instance, the spoken word dog could be
mapped to a picture of a dog. However, this is difficult to attain for thousands of words and not
practical for precise vocal feedback that captures how a word is said. Furthermore, feedback at a word
level gives no feedback on intonation and introduces a delay for parsing the entire word. By contrast,
our goal is to map instantaneous sounds sequentially into a sequence of images. The representation
can capture spoken language, but also other sounds such as music, wind, and collision sounds.
We will use variational autoencoders and GANs to capture the natural distributions of sounds and
images, respectively, and will enforce temporal continuity and proximity through additional
constraints. The different intonation and tonal patterns of languages may require language-specific
models by training the audio model with speech recordings of a single language — for instance,
separate ones for English and Mandarin to model their very different characteristics. For the visual
model, we will experiment with unstructured and structured datasets, such as ImageNet. It is an open
question of how to connect the audio and visual domain. We will first train individual models that map
examples of both domains to the same space (same dimension and multivariate Gaussian distribution
over training examples). At inference time, the projection of a sound to the learned representation
using the audio model and its reconstruction by the visual generator yields a simple yet efficient
translation model. Figure 2 shows this process. Later, stronger connections can be enforced by
introducing constraints across domains, such as with cycle GAN or semantic correspondence.}
\section{Method}
Our goal is to translate a one-dimensional audio signal $\mathbf{A} = (\mathbf{a}_0, \cdots, \mathbf{a}_{T_A})$ into a video visualization $\mathbf{V} = (\mathbf{I}_0, \cdots, \mathbf{I}_{T_V})$, where $\mathbf{a}_i \in \mathbb{R}$ are sound wave samples recorded over $T_A$ frames and $\mathbf{I}_i$ are images representing the same content over $T_V$ frames. Supervised training is not possible in the absence of paired labels.
Instead, we utilize individual images and audio snippets without correspondence and learn a latent space $\mathcal{Z}$ that captures the importance of image and audio features and translates between them with minimal information loss.
Figure \ref{fig:overview} shows the individual mapping steps. The audio encoder $E_A$ yields latent code $\mathbf{z}_i \in \mathcal{Z}$ and the visual decoder $D_V(\mathbf{z}_i)$ outputs a corresponding image $\mathbf{I}_i$. This produces a video representation of the audio when applied sequentially.
We start by learning individual audio and video models that are subsequently linked with a translation layer.
\parag{Matching sound and video representation.} A first technical problem lies in the higher audio sampling frequency (16 kHz), that prevents a one-to-one mapping to 25 Hz video. We follow common practice and represent the one-dimensional sound wave with a perceptually motivated mel-scaled spectrogram, $\mathbf{M} = (\mathbf{m}_0, \dots, \mathbf{m}_{T_M})$, $\mathbf{m}_i \in \mathbb{R}^F$, where $F=80$ is the number of filter banks. It is computed via the short-time Fourier transform with a 25ms Hanning window with 10 ms shifts. The resulting coefficients are converted to decibel units flooring at -80dB. Because the typical mel spectrogram still has a higher sampling rate (100 Hz) than the video, we chunk it into overlapping segments of length $T_M=20$ covering $200$ ms. In the following, we explain how to map from mel segment to video frame.
\input{tex/fig_overview}%
\subsection{Audio Encoding}
\label{sec:latent models}
Given unlabelled audio and video sequences, we start by learning independent encoder-decoder pairs $(E_A,D_A)$ for sound and $(E_V,D_V)$ for video. We use probabilistic VAEs since these do not only learn a compact representation of the latent structure, but also allow us to control the shape of the latent distribution to be a standard normal distribution.
Let $\mathbf{x}$ be a sample from the unlabelled audio set. We optimize over all samples the VAE objective~\cite{kingma2013auto}:
\begin{align}
{\mathcal {L}}(\mathbf{x} )= -D_{\mathrm {KL} }(q_\phi(\mathbf {\mathbf{z}} |\mathbf{x} )\Vert p_{\theta }(\mathbf{z} ))+\mathbb {E} _{q_{\phi }(\mathbf {\mathbf{z}} |\mathbf{x} )}{\big (}\log p_{\theta }(\mathbf{x} |\mathbf{z} ){\big )},
\end{align}
with $D_{\mathrm {KL} }$, the Kullback-Leibler divergence, and $q_{\phi }(\mathbf{z} |\mathbf{x} )$ and $p_{\theta }(\mathbf{x} |\mathbf{z} )$, the latent code and output domain posterior respectively. These have a parametric form, with
\begin{align}
q_{\phi }(\mathbf{z} |\mathbf{x} ) &={\mathcal {N}}({\boldsymbol {\rho }}(\mathbf{x} ),{\boldsymbol {\omega }}^{2}(\mathbf{x} )\mathbf {I} )\text{ and }\\
p_{\theta }(\mathbf{x} |\mathbf{z} ) &={\mathcal {N}}({\boldsymbol {\mu }}(\mathbf{z} ),{\boldsymbol {\sigma }}^{2}(\mathbf{z} )\mathbf {I} ),
\end{align}
where ${\boldsymbol {\rho }}$ and $\boldsymbol {\omega }$ the output of the encoder and $\boldsymbol {\mu }$ and $\boldsymbol {\sigma }$ the output of the decoder.
\parag{Audio network architecture.}
\label{sec:sound_VAE}
We use the SpeechVAE model from Hsu et al.~\cite{hsu2017learning} that is widely used for generative sound models. Fig.~\ref{fig:overview}, left, sketches how the mel spectrogram is encoded with an encoder $E_A$ of three convolutional layers followed by a fully connected layer which flattens the spatial dimensions. The mel spectrogram $\mathbf{M}$ is a two-dimensional array, spanning the time and frequency, respectively. To account for the structural differences of the two, convolutions are split into separable $1\times F$ and $3\times 1$ filters, where F is the number of frequencies captured by $\mathbf{M}$ and striding is applied only on the temporal axis. The decoder is symmetric to the encoder. We use ReLU activation and batch normalization layers in the encoder and decoder.
\subsection{Structuring the Audio Encoding}
\label{sec:priors}
We desire our latent space to change smoothly in time and disentangle the style, such as gender and dialect, from the content conveyed in phones.
\paragraph{Disentangling content and style.}
We construct a SpeechVAE that disentangles the style (speaker identity) content (phonemes) in the latent encodings, i.e., the latent encoding $\mathbf{z} = [z_1,\cdots,z_d]^T \in \mathcal{R}^d$ can be separated as a style part $\mathbf{z}_s = [z_1,\cdots,z_{m}]^T$ and a content part $\mathbf{z}_c = [z_{m+1},\cdots,z_{d}]^T$, where $d$ is the whole audio latent space dimension and $m$ in the audio style latent space dimension.
\input{tex/fig_disentanglement}
To encourage such disentanglement we use an audio dataset with phone and speaker ID annotation. As it shows in Figure~\ref{fig:disentanglement}, at training time, we feed triplets of mel spectogram segments
$\{\mathbf{M}_{a,i},\mathbf{M}_{b,i},\mathbf{M}_{a,j}\}$, where $\mathbf{M}_{a,i}$ and $\mathbf{M}_{b,i}$ are the same phoneme sequence $p_i$ spoken by different speakers $s_a$ and $s_b$ respectively, and $\mathbf{M}_{a,j}$ shares the speaker $s_a$ with the first segment but a different phoneme sequence. Each element of the input triplet is encoded individually by $E_A$, forming latent triplet $\{\mathbf{z}_{a,i},\mathbf{z}_{b,i},\mathbf{z}_{a,j}\} = \{[\mathbf{z}_{s_a},\mathbf{z}_{c_i}]^T, [\mathbf{z}_{s_b},\mathbf{z}_{c_i}']^T,[\mathbf{z}_{s_a}',\mathbf{z}_{c_j}]^T\}$, instead of reconstructing the inputs from the corresponding latent encodings in an autoencoder, we reconstructed the first sample $ \mathbf{M}_{a,i}$ from a recombined latent encoding of the other two, $\mathbf{z}_{a,i}'=[\mathbf{z}_{s_a}',\mathbf{z}_{c_i}']^T $. Formally, we replaced the reconstruction loss term in the VAE objective by a recombined reconstruction loss term,
\begin{align}
{\mathcal{L}_{rr}(\mathbf{T}_{a,b,i,j}) = \mathbb {E} _{q_{\phi }(\mathbf {\mathbf{z}_{a,i}'} |\mathbf{M}_{b,i},\mathbf{M}_{a,j} )}{\big (}\log p_{\theta }(\mathbf{M}_{a,i} |\mathbf{z}_{a,i}' ){\big )}}.
\end{align}
This setup forces the model to learn separate encodings for the style and phoneme information while not requiring additional loss terms.
Note that we could alternatively enforce $\mathbf{z}_{a,i}$ to be close to $\mathbf{z}_{a,i}'$ without decoding (the unused $\mathbf{z}_{a,i}$ in Figure~\ref{fig:disentanglement}). However, an additional L2 loss on the latent space led to a bias towards zero and lower reconstruction scores than the proposed mixing strategy that works with the original VAE objective.
\parag{Temporal smoothness.} The audio encoder has a small temporal receptive field, encoding time segments of 200 ms. This lets encodings of subsequent sounds be encoded to distant latent codes leading to quick visual changes in the decoding.
To counter act, we add an additional smoothness prior. We experiment with the following negative log likelihoods.
For training the audio encoder, we sample a pair of input mel spectrogram segments $\{\mathbf{M}_i,\mathbf{M}_j\}$ at random time steps $\{t_i ,t_j\}$, spaced at most 800ms apart, from the same utterance. We test the three different pair loss functions to enforce temporal smoothness in the embedded content vectors. First, by making changes in latent space proportional to changes in time,
\begin{equation}
\mathcal{L}_{p, MSE} = \textstyle
\frac{1}{N}\sum_{i}^{N}\left(\hat{\Delta t_i}- \|t_{i,1}-t_{i,2}\|\right)^2,
\end{equation}
where the latent space dimension scale is learned by scalar parameter $s$, with $ \hat{\Delta t_i} = s\cdot\|E_A(\mathbf{M}_{i})-E_A(\mathbf{M}_{j})\| = s\cdot\|\mathbf{z}_{i}-\mathbf{z}_{j}\|$ for models without style disentanglement, and $\hat{\Delta t_i} = s\cdot\|\mathbf{z}_{c,i}-\mathbf{z}_{c,j}\|$ for the disentangled ones, where the predicted time difference only calculated with the content part of the latent encodings. Second, we try whether enforcing the ratio of velocities to be constant is better,
\begin{equation}
\mathcal{L}_{p, Q} = \textstyle \frac{1}{N}\sum_{i}^{N}\left(\frac{\hat{\Delta t_i}}{\|t_{i,1}-t_{i,2}\|}-1\right)^2,
\end{equation}
with learned scale as before. The same quotient constraint can also be measured in logarithmic scale,
\begin{equation}
\mathcal{L}_{p, \log MSE} =\textstyle \frac{1}{N}\sum_{i}^{N}\left(\log{\hat{\Delta t_i}}- \log{\|t_{i,1}-t_{i,2}\|}\right)^2.
\end{equation}
Our final model uses $\mathcal{L}_{p, \log MSE}$, which behaves better than quotients in our experiments. All objective terms are learned jointly, with weights $\lambda_\text{cycle} =10$ and $\lambda_p=10^3$ balancing their influence.
\subsection{Image Encoding for Video Generation}
\label{sec:video_VAE}
Audio encoders and video decoders can be made compatible across modalities by using two VAEs with matching latent dimension and prior distribution $p(\mathbf{z})$ over $\mathbf{z} \in \mathcal{Z}$.
We apply the per-frame image DFC-VAE model from Hou et al.~\cite{Hou2017}. The input image is passed through four 4x4 convolutional layers followed by a fully-connected layer to arrive at the latent code. The decoder uses four 3x3 convolutions. Batch norm and LeakyReLU layers are applied between convolutions. Besides the per-pixel loss of the classical VAE, the DFC-VAE uses a perceptual loss in the feature space of a trained VGG-19 network for high-level feature consistency.
This model is trained on individual videos but is applied sequentially to create the output video frames.
\subsection{Linking Audio and Visual Spaces}
\label{sec:cycle}
By learning individual audio and video VAEs with the same latent space prior, we can concatenate the audio encoder with the image decoder for our desired AudioViewer{} translation. However, the learned encoders are only approximations to the true distribution and not all points in $\mathcal{Z}$ are modelled equally well, leading to information loss. To bridge different latent space structures, we introduce an additional cycle constraint that works without image correspondence and ensures that samples from the audio latent space posterior are reconstructed well, similar in spirit to \cite{jha2018disentangling,yook2020many}. Figure~\ref{fig:cycle} shows the cyclic chaining from audio latent code to video and back. It is implemented on the content space as
\begin{equation}
\mathcal{L}_{cycle} = \left| E_V(D_V(\mathbf{z}_c)))-\mathbf{z}_c \right|,
\end{equation}
where $\mathbf{z}_c$ is drawn from the posterior of the content part $E_A(\mathbf{M})$ of the audio encoder and the video modules operate on mean values minimizing point-wise differences in position. Similar to the smoothness loss, the cycle loss can be applied to the disentangled models and visualize only the style part or content part as long as the latent dimension matches, so that the visualization can be content-agnostic or speaker-agnostic.
\input{tex/fig_cycle}
\section{Related Work}
In the following section we first review the literature on audio and video generation, with a particular focus on cross-modal models. We then put our approach in context with existing assistive systems.
\parag{Classical audio to video mappings.} Audio to video translation methods have mostly been designed for digital dubbing or lip-syncing facial expressions to spoken audio \cite{duarte2019wav2pix,sadoughi2019speech,wiles2018x2face}. Other approaches reconstruct facial attributes, such as gender and ethnicity, from audio and generating matching facial images \cite{wen2019reconstructing} and map
music to facial or body animation \cite{taylor2017deep, karras2017audio, suwajanakorn2017synthesizing, shlizerman2018audio}.
By contrast to our setting, all of these tasks can be supervised by learning from videos with audio lines, e.g., a talking person where the correspondence of lip motion, expressions and facial appearance to the spoken language is used to train the relation between sound and mouth opening. We go beyond spoken sounds and map to non-facial features such as background and brightness with an unsupervised translation mechanism.
\parag{Audio and video generation models.}
Image generation models predominantly rely on GAN \cite{goodfellow2014generative} and VAE \cite{kingma2013auto,Hou2017} formulations. The highest image fidelity is attained with hierarchical models that inject noise and latent codes at various network stages by changing their feature statistics \cite{huang2017arbitrary,karras2019style}. For audio, only few methods operate on the raw waveform~\cite{kamper2019truly}. It is more common use spectrograms and to apply convolutional models inspired by the ones used for image generation \cite{hsu2017learning,dong2018musegan,briot2017deep}. We use cross-modal VAE models as a basis to learn the important audio and image features.
\parag{Cross-modal latent variable models.}
CycleGAN and its variations~\cite{zhu2017unpaired, taigman2016unsupervised} have seen considerable success in performing cross-modal unsupervised domain transfer, for medical imaging~\cite{hiasa2018cross,tmenova2019cyclegan} and audio to visual translation~\cite{hao2018cmcgan}, but often encode information as high frequency signal that is invisible to the human eye and susceptible to adversarial attacks \cite{chu2017cyclegan}. An alternative approach involves training a VAE, subject to a cycle-consistency condition \cite{jha2018disentangling,yook2020many}, but these works were restricted to domain transfers within a single modality. Most similar is the joint audio and video model proposed by Tian et al.~\cite{tian2019latent}, which uses a VAE to map between two incompatible latent spaces using supervised alignment of attributes, however, it operates on a word not phoneme level, and has no mechanism to ensure temporal smoothness nor information throughput. Our contributions address these shortcomings. Relatedly, encoder-decoder and GAN models have been applied to generating video reconstructions of lip movements based on audio data \cite{chung2017you, chen2018lip, vougioukas2018end, zhou2019talking}, however due to mapping ambiguities between phonemes and visemes, lip movements are not a reliable source of feedback for learning sound production \cite{lidestam2006visual, newman2010limitations, cappelletta2012phoneme, fernandez2017optimizing}.
\parag{Deaf speech support tools.}
Improvements in speech production for the hearing impaired have been achieved through non-auditory aids and these improvements persist beyond learning sessions and extend to words not encountered during the sessions \cite{stewart1976a,zaccagnini1993effects,oster1995teaching}. While electrophysiological \cite{hardcastle1991visual, katz2015visual} and haptic learning aids \cite{eberhardt1993omar} have demonstrated efficacy for improving speech production, such techniques can be more invasive, especially for young children, as compared to visual aids.
Elssmann et al.~\cite{elssmann1987speech} demonstrate visual feedback from the Speech Spectrographic Display (SSD)~\cite{stewart1976a} is equally effective at improving speech production as compared with feedback from a speech-language pathologist. Alternative graphical plots generated from transformed spectral data have been explored by~\cite{yang2010speech, xu2008speech, kroger2010audiovisual}, which aim at improving upon spectrograms by creating plots which are more distinguishable with respect to speech parameters. Other methods aim at providing feedback by explicitly estimating vocal tract shapes~\cite{park1994integrated}. In addition, Levis et al.~\cite{levis2004teaching} demonstrate that distinguishing between discourse-level intonation (intonation in conversation) and sentence-level intonation (intonation of sentences spoken in isolation) is possible through speech visualization and argues that deaf speech learning could be further improved by incorporating the former. Commercially, products as such IBM's Speech Viewer \cite{ibm2004speech} are available to the public.
Our image generation approach extends these spectrogram visualization techniques by leveraging the generative ability of VAEs in creating a mapping to a more intuitive and memorable video representation. It is our hope that improved visual aids will lead to more effective learning in the future.
\parag{Sensory substitution and audio visualization.}
Related to our work is the field of sensory substitution, whereby information from one modality is provided to an individual through a different modality. While many sensory substitution methods focus on substituting visual information into other senses like tactile or auditory stimulation to help visual rehabilitation \cite{maidenbaum2014sensory, Hu_2019_CVPR, goldish1974optacon}, few methods target substituting auditory modal with visualization.
Audio visualization is another field related to our work. Music visualization works generate visualizations of songs, so that user can browsing songs more efficiently without listen to them~\cite{yoshii2008music,takahashi2018instrudive}. On the learning side, \cite{yuan2019speechlens} visualize the intonation and volume of each word in speech by the font size, so that user can learn narration strategies.
Different from the works mentioned above, our model tries to visualize speech in phoneme level with deep learning models instead of hand-crafted features.
\subsection{Style}
Papers to be submitted to NeurIPS 2020 must be prepared according to the
instructions presented here. Papers may only be up to eight pages long,
including figures. Additional pages \emph{containing only a section on the broader impact, acknowledgments and/or cited references} are allowed. Papers that exceed eight pages of content will not be reviewed, or in any other way considered for
presentation at the conference.
\paragraph{Preprint option}
If you wish to post a preprint of your work online, e.g., on arXiv, using the
NeurIPS style, please use the \verb+preprint+ option. This will create a
nonanonymized version of your work with the text ``Preprint. Work in progress.''
in the footer. This version may be distributed as you see fit. Please \textbf{do
not} use the \verb+final+ option, which should \textbf{only} be used for
papers accepted to NeurIPS.
At submission time, please omit the \verb+final+ and \verb+preprint+
options. This will anonymize your submission and add line numbers to aid
review. Please do \emph{not} refer to these line numbers in your paper as they
will be removed during generation of camera-ready copies.
}
\bibliographystyle{splncs04}
\section{Results on Environment Sounds}
With the style disentangling training of our AudioViewer{} method, the audio model is more specific to human speech than general sounds. In the following, we test the generalization capability of our method (trained on the speach TIMIT dataset \cite{garofolo1993timit}) on the ESC-50 environment sound dataset \cite{piczak2015esc}.
\input{tex/tab_ESC-50_recon_eval}
Table~\ref{tab:cycle_esc} shows the reconstruction accuracy when going via the audio and video VAEs (see Information Throughput section in the main document). The SNR for reconstructed Mel spectrum is generally lower than the speech dataset, which is expected since it was trained on the latter. The analysis of the reconstruction ability of the audio VAE in isolation (without going through the video VAE) reported in Table~\ref{tab:audio_only_esc} shows that a large fraction of this loss of accuracy stems from the learning of speech specific features of the audio VAE.
Moreover, with a recombined reconstruction loss term on the human speech dataset, the model was fitted to speech features and tended to loss high pitch information.
Still, according to the face visualization of the content encoding as we showed in the HTML file, our AudioViewer{} can generate consistent visualization to given environment sounds.
\section{Additional ablation study on $\lambda_c$ and $\lambda_p$}
\input{tex/tab_TIMIT_recon_eval}
Due to the low SNR of the disentangled model, here we report the ablation experiments on $\lambda_c$ and $\lambda_p$ without the disentangled training to better illustrate the performance of the two loss terms.
Table \ref{tab:cycle} shows the reconstruction error on TIMIT, when reconstructing the Mel spectrum via the audio and video VAEs (see Information Throughput section in the main document). We compare versions where the visual VAEs are trained independently or refined with the audio ones using different weights $\lambda_c$ for the cycle loss. Irrespective of the visual domain (CelebA or MNIST), $\lambda_c=10$ works best together with the both temporal losses and is selected for the disentangled version. Moreover, as we have expected, the higher cycled reconstruction SNR indicates a higher capacity of information of face visualization than the figure ones.
Table \ref{tab:audio_only} shows the reconstruction SNR (encoding followed by decoding with the audio VAE) on the test set with models trained with different loss functions, as in the main document but with a larger number of tested weights $\lambda_p$.
The SpeechVAE w/ $\mathcal{L}_{p, log MSE}$ with $\lambda_p=10^3$ is the best performing model that was selected for disentangled model.
\section{User Study}
Our user study required participants to complete one of 4 versions of a questionnaire: disentangled content model with CelebA visualizations (CelebA-content), disentangled style model with CelebA visualizations (CelebA-style), disentangled content model with MNIST visualizations (MNIST-content) and a combined model with CelebA visualizations (CelebA-combined). Each version of the questionnaire asked the same set of 29 questions with randomized ordering of answers within each question. The study was conducted with \NEW{14, 15}, 12, and 14 participants for the CelebA-content, CelebA-style, Celeba-combined and MNIST-content questionnaires, respectively. Amongst all the questionnaires, there were \NEW{27} unique participants. It took participants between 10-15 minutes to complete the each questionnaire. In addition, the questionnaire did not indicate the purpose of the underlying research and only asked participant to perform two possible tasks: matching and grouping visualizations. The format of the questionnaire is outlined in Table~\ref{tab:user_study_format}. The questions tested for two factors: sound content, sounds that share the same phoneme sequences, and sound style, sounds produced by speakers of the same sex or speaker dialect. This purely visual comparison allows us analyze different aspects individually.
\input{tex/fig_match_example}
\input{tex/fig_group_example}
\input{tex/tab_user_study_format}
\paragraph{Matching questions} Matching questions asked the participants to choose which of two possible visualizations which is most visually similar to a given reference visual. Figure~\ref{fig:match_example} shows examples of matching questions. Matching questions were used to assess the viability for users to distinguish between the same sounds produced by speakers possessing different speaker traits as well as determining whether structural similarities in the underlying audio translated into similarities in the visualization. In particular, the questionnaire contained 6 questions for evaluating the ability to distinguish between sound content, which compared visualizations of sounds of different phoneme sequences (3 for phoneme-pairs and 3 for words). Phone-pairs are short in length and therefore the corresponding visualisation was a single frame image, whereas visualisations of words were videos. In order to evaluate the ability to distinguish between sound style, 6 questions compared visualizations of the same phoneme sequence between male and female speakers and 4 questions for distinguishing between speakers of different dialects. In total there were 16 matching questions. Since each question has two options, the expected mean accuracy for random guessing is 50\%.
\paragraph{Grouping questions} Grouping questions asked the participants to group 4 visualizations into two pairs of similar visualizations. Figure~\ref{fig:group_example} shows examples of grouping questions. Grouping questions were used to assess the degree to which visualizations of different words are distinguishable and visualizations of the same word are similar. In particular, the study required users to group visualizations of two pairs of sounds, whereby different pairs are sound clips with shared factors of the same sound content or same sound style. In total, the user study consisted of 4 grouping questions based on phone-pairs and 9 grouping questions based on words. Since there are three possible options, the expected mean accuracy for random guessing is 33.3\%.
\subsection{Results}
\input{tex/tab_user_study_results}
\input{tex/fig_phoneme_comparison_1}
\input{tex/fig_style_comparison}
For each of the three models, we tested for sound content: phoneme sequences, and sound style: speaker dialect and speaker sound. We generated the mean accuracy and standard deviation for each tested factor and each question sub type as shown in Table~\ref{tab:user_study_results}. The results of the disentangled models with CelebA visualizations (CelebA-disentangled) is aggregated by taking the results of CelebA-content on the questions which tested for sound content and the results of CelebA-style on questions which tested for sound style. Since we only evaluated the disentangled content model for the MNIST visualizations, it is only compared for the content questions. Significance comparing model means were calculated using a two-sample two-tailed t-test with unequal variance and without any outlier rejection.
Figure~\ref{fig:phoneme_comparison} shows users achieve the highest overall accuracy on the CelebA-disentangled model with $\NEW{85.0} \pm 6.8$\% (significant with $p < 0.05$) for distinguishing between visualizations of different content. The MNIST-content model has the highest accuracy for distinguishing between different phone pairs with $91.8 \pm 9.2$\%, \NEW{although not significantly higher than the CelebA-disentangled one with ($p>0.05$)}, but has a much lower accuracy for distinguishing between different words, suggesting that the MNIST visualizations may be better suited for representing shorter sounds. The CelebA-disentangled model outperforms the CelebA-combined model for distinguishing between speakers of different sex with $\NEW{78.0 \pm 10.8}$\% (significant with $\NEW{p < 0.001}$) and between speakers of different dialects with $\NEW{56.7 \pm 22.1}$\% (marginally significant with $0.05<p<0.10$) as shown in Figure~\ref{fig:style_comparison}.
The task of distinguishing between different speakers of different dialects is much more difficult than distinguishing between phoneme sequences since there are 8 categories of dialects in the dataset and differences in dialects are much more subtle and can contain often contain overlaps. This translates to lower accuracy in this question category.
\comment{
\subsection{Style}
Papers to be submitted to NeurIPS 2020 must be prepared according to the
instructions presented here. Papers may only be up to eight pages long,
including figures. Additional pages \emph{containing only a section on the broader impact, acknowledgments and/or cited references} are allowed. Papers that exceed eight pages of content will not be reviewed, or in any other way considered for
presentation at the conference.
\paragraph{Preprint option}
If you wish to post a preprint of your work online, e.g., on arXiv, using the
NeurIPS style, please use the \verb+preprint+ option. This will create a
nonanonymized version of your work with the text ``Preprint. Work in progress.''
in the footer. This version may be distributed as you see fit. Please \textbf{do
not} use the \verb+final+ option, which should \textbf{only} be used for
papers accepted to NeurIPS.
At submission time, please omit the \verb+final+ and \verb+preprint+
options. This will anonymize your submission and add line numbers to aid
review. Please do \emph{not} refer to these line numbers in your paper as they
will be removed during generation of camera-ready copies.
}
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 439 |
#if !defined( __IPPCH_H__ ) || defined( _OWN_BLDPCS )
#define __IPPCH_H__
#ifndef __IPPDEFS_H__
#include "ippdefs.h"
#endif
#ifdef __cplusplus
extern "C" {
#endif
#if !defined( _OWN_BLDPCS )
#if defined (_WIN32_WCE) && defined (_M_IX86) && defined (__stdcall)
#define _IPP_STDCALL_CDECL
#undef __stdcall
#endif
typedef struct {
void *pFind;
int lenFind;
} IppRegExpFind;
struct RegExpState;
typedef struct RegExpState IppRegExpState;
typedef enum {
ippFmtASCII = 0,
ippFmtUTF8
} IppRegExpFormat;
#endif /* _OWN_BLDPCS */
/* /////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// Functions declarations
////////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////// */
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippchGetLibVersion
// Purpose: getting of the library version
// Returns: the structure of information about version
// of ippCH library
// Parameters:
//
// Notes: not necessary to release the returned structure
*/
IPPAPI( const IppLibraryVersion*, ippchGetLibVersion, (void) )
/* /////////////////////////////////////////////////////////////////////////////
// String Functions
///////////////////////////////////////////////////////////////////////////// */
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsFind_8u ippsFind_16u
// ippsFindC_8u ippsFindC_16u
// ippsFindRev_8u ippsFindRev_16u
// ippsFindRevC_8u ippsFindRevC_16u
//
// Purpose: Finds the match for string of elements or single element
// within source string in direct or reverse direction
//
// Arguments:
// pSrc - pointer to the source string
// len - source string length
// pFind - pointer to the searching string
// lenFind - searching string length
// valFind - searching element
// pIndex - pointer to the result index:
// *pIndex = index of first occurrence ;
// *pIndex = -1 if no match;
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc, pFind or pIndex are NULL
// ippStsLengthErr len or lenFind are negative
*/
IPPAPI (IppStatus, ippsFind_8u, (const Ipp8u* pSrc, int len,
const Ipp8u* pFind, int lenFind,
int *pIndex))
IPPAPI (IppStatus, ippsFind_16u, (const Ipp16u* pSrc, int len,
const Ipp16u* pFind, int lenFind,
int *pIndex))
IPPAPI (IppStatus, ippsFindC_8u, (const Ipp8u* pSrc, int len,
Ipp8u valFind, int *pIndex))
IPPAPI (IppStatus, ippsFindC_16u, (const Ipp16u* pSrc, int len,
Ipp16u valFind, int *pIndex))
IPPAPI (IppStatus, ippsFindRev_8u, (const Ipp8u* pSrc, int len,
const Ipp8u* pFind, int lenFind,
int *pIndex))
IPPAPI (IppStatus, ippsFindRev_16u, (const Ipp16u* pSrc, int len,
const Ipp16u* pFind, int lenFind,
int *pIndex))
IPPAPI (IppStatus, ippsFindRevC_8u, (const Ipp8u* pSrc, int len,
Ipp8u valFind, int *pIndex))
IPPAPI (IppStatus, ippsFindRevC_16u, (const Ipp16u* pSrc, int len,
Ipp16u valFind, int *pIndex))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsFind_Z_8u ippsFind_Z_16u
// ippsFindC_Z_8u ippsFindC_Z_16u
//
// Purpose: Finds the match for zero-ended string of elements or single element
// within source zero-ended string in direct or reverse direction
//
// Arguments:
// pSrcZ - pointer to the source zero-ended string
// pFindZ - pointer to the searching zero-ended string
// valFind - searching element
// pIndex - pointer to the result index:
// *pIndex = index of first occurrence;
// *pIndex = -1 if no match;
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrcZ, pFindZ or pIndex are NULL
*/
IPPAPI (IppStatus, ippsFind_Z_8u, (const Ipp8u* pSrcZ,
const Ipp8u* pFindZ, int *pIndex))
IPPAPI (IppStatus, ippsFind_Z_16u, (const Ipp16u* pSrcZ,
const Ipp16u* pFindZ, int *pIndex))
IPPAPI (IppStatus, ippsFindC_Z_8u, (const Ipp8u* pSrcZ,
Ipp8u valFind, int *pIndex))
IPPAPI (IppStatus, ippsFindC_Z_16u, (const Ipp16u* pSrcZ,
Ipp16u valFind, int *pIndex))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsCompare_8u ippsCompare_16u
//
// Purpose: Compares two strings element-by-element
//
// Arguments:
// pSrc1 - pointer to the first string
// pSrc2 - pointer to the second string
// len - string length to compare
// pResult - pointer to the result:
// *pResult = 0 if src1 == src2;
// *pResult = >0 if src1 > src2;
// *pResult = <0 if src1 < src2;
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc1, pSrc2 or pResult are NULL
// ippStsLengthErr len is negative
*/
IPPAPI (IppStatus, ippsCompare_8u, (const Ipp8u* pSrc1, const Ipp8u* pSrc2,
int len, int *pResult))
IPPAPI (IppStatus, ippsCompare_16u, (const Ipp16u* pSrc1, const Ipp16u* pSrc2,
int len, int *pResult))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsEqual_8u ippsEqual_16u
//
// Purpose: Compares two strings element-by-element
//
// Arguments:
// pSrc1 - pointer to the first string
// pSrc2 - pointer to the second string
// len - string length to compare
// pResult - pointer to the result:
// *pResult = 1 if src1 == src2;
// *pResult = 0 if src1 != src2;
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc1, pSrc2 or pResult are NULL
// ippStsLengthErr len is negative
*/
IPPAPI (IppStatus, ippsEqual_8u, (const Ipp8u* pSrc1, const Ipp8u* pSrc2,
int len, int *pResult))
IPPAPI (IppStatus, ippsEqual_16u, (const Ipp16u* pSrc1, const Ipp16u* pSrc2,
int len, int *pResult))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsTrimC_8u_I ippsTrimC_16u_I
//
// Purpose: Deletes an odd symbol at the end and the beginning of a string
// in-place
//
// Arguments:
// pSrcDst - pointer to the string
// pLen - pointer to the string length:
// *pLen = source length on input;
// *pLen = destination length on output;
// odd - odd symbol
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrcDst or pLen are NULL
// ippStsLengthErr *pLen is negative
*/
IPPAPI (IppStatus, ippsTrimC_8u_I, (Ipp8u* pSrcDst, int* pLen, Ipp8u odd ))
IPPAPI (IppStatus, ippsTrimC_16u_I, (Ipp16u* pSrcDst, int* pLen, Ipp16u odd ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsTrimC_8u ippsTrimC_16u
//
// Purpose: Deletes an odd symbol at the end and the beginning of a string
//
// Arguments:
// pSrc - pointer to the source string
// srcLen - source string length
// odd - odd symbol
// pDst - pointer to the destination string
// pDstLen - pointer to the destination string length:
// *pDstLen doesn't use as input value;
// *pDstLen = destination length on output;
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrcDst, pDst or pDstLen are NULL
// ippStsLengthErr srcLen is negative
*/
IPPAPI (IppStatus, ippsTrimC_8u, (const Ipp8u* pSrc, int srcLen, Ipp8u odd,
Ipp8u* pDst, int* pDstLen ))
IPPAPI (IppStatus, ippsTrimC_16u, (const Ipp16u* pSrc, int srcLen, Ipp16u odd,
Ipp16u* pDst, int* pDstLen ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsUppercase_16u_I
// ippsLowercase_16u_I
// ippsUppercase_16u
// ippsLowercase_16u
//
// Purpose: Forms an uppercase or lowercase version of the Unicode string
//
// Arguments:
// pSrc - pointer to the source string
// pDst - pointer to the destination string
// pSrcDst - pointer to the string for in-place operation
// len - string length
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc, pDst or pSrcDst are NULL;
// ippStsLengthErr len is negative;
*/
IPPAPI (IppStatus, ippsUppercase_16u_I,( Ipp16u* pSrcDst, int len ))
IPPAPI (IppStatus, ippsLowercase_16u_I,( Ipp16u* pSrcDst, int len ))
IPPAPI (IppStatus, ippsUppercase_16u, (const Ipp16u* pSrc, Ipp16u* pDst, int len))
IPPAPI (IppStatus, ippsLowercase_16u, (const Ipp16u* pSrc, Ipp16u* pDst, int len))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsUppercaseLatin_8u_I ippsUppercaseLatin_16u_I
// ippsLowercaseLatin_8u_I ippsLowercaseLatin_16u_I
// ippsLowercaseLatin_8u ippsUppercaseLatin_16u
// ippsUppercaseLatin_8u ippsLowercaseLatin_16u
//
// Purpose: Forms an uppercase or lowercase version of the ASCII string
//
// Arguments:
// pSrc - pointer to the source string
// pDst - pointer to the destination string
// pSrcDst - pointer to the string for in-place operation
// len - string length
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc, pDst or pSrcDst are NULL;
// ippStsLengthErr len is negative;
*/
IPPAPI (IppStatus, ippsUppercaseLatin_8u_I, ( Ipp8u* pSrcDst, int len ))
IPPAPI (IppStatus, ippsLowercaseLatin_8u_I, ( Ipp8u* pSrcDst, int len ))
IPPAPI (IppStatus, ippsUppercaseLatin_16u_I,( Ipp16u* pSrcDst, int len ))
IPPAPI (IppStatus, ippsLowercaseLatin_16u_I,( Ipp16u* pSrcDst, int len ))
IPPAPI (IppStatus, ippsLowercaseLatin_8u, (const Ipp8u* pSrc, Ipp8u* pDst, int len))
IPPAPI (IppStatus, ippsUppercaseLatin_8u, (const Ipp8u* pSrc, Ipp8u* pDst, int len))
IPPAPI (IppStatus, ippsUppercaseLatin_16u, (const Ipp16u* pSrc, Ipp16u* pDst, int len))
IPPAPI (IppStatus, ippsLowercaseLatin_16u, (const Ipp16u* pSrc, Ipp16u* pDst, int len))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsHash_8u32u ippsHash_16u32u
//
// Purpose: Calculates hashed value so that different strings yield different
// values:
// for (i=0; i<len; i++) hash = (hash << 1) ^ src[i];
//
// Arguments:
// pSrc - pointer to the source string
// len - source string length
// pHashVal - pointer to the result value
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc or pHashVal are NULL
// ippStsLengthErr len is negative
*/
IPPAPI (IppStatus, ippsHash_8u32u, (const Ipp8u* pSrc, int len, Ipp32u* pHashVal ))
IPPAPI (IppStatus, ippsHash_16u32u, (const Ipp16u* pSrc, int len, Ipp32u* pHashVal ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsHashSJ2_8u32u ippsHashSJ2_16u32u
//
// Purpose: Calculates hashed value so that different strings yield different
// values:
// for (i=0; i<len; i++) hash =
//
// Arguments:
// pSrc - pointer to the source string
// len - source string length
// pHashVal - pointer to the result value
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc or pHashVal are NULL
// ippStsLengthErr len is negative
*/
IPPAPI (IppStatus,ippsHashSJ2_8u32u, (const Ipp8u* pSrc, int len, Ipp32u* pHashVal))
IPPAPI (IppStatus,ippsHashSJ2_16u32u, (const Ipp16u* pSrc, int len, Ipp32u* pHashVal))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsHashMSCS_8u32u ippsHashMSCS_16u32u
//
// Purpose: Calculates hashed value so that different strings yield different
// values:
// for (i=0; i<len; i++) hash =
//
// Arguments:
// pSrc - pointer to the source string
// len - source string length
// pHashVal - pointer to the result value
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc or pHashVal are NULL
// ippStsLengthErr len is negative
*/
IPPAPI (IppStatus,ippsHashMSCS_8u32u, (const Ipp8u* pSrc, int len, Ipp32u* pHashVal))
IPPAPI (IppStatus,ippsHashMSCS_16u32u, (const Ipp16u* pSrc, int len, Ipp32u* pHashVal))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsConcat_8u ippsConcat_16u
//
// Purpose: Concatenates two strings together
//
// Arguments:
// pSrc1 - pointer to the first source string
// len1 - first source string length
// pSrc2 - pointer to the second source string
// len2 - second source string length
// pDst - pointer to the destination string
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc1, pSrc2 or pDst are NULL
// ippStsLengthErr len1 or len2 are negative
*/
IPPAPI (IppStatus, ippsConcat_8u, (const Ipp8u* pSrc1, int len1,
const Ipp8u* pSrc2, int len2,
Ipp8u* pDst))
IPPAPI (IppStatus, ippsConcat_16u, (const Ipp16u* pSrc1, int len1,
const Ipp16u* pSrc2, int len2,
Ipp16u* pDst))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsConcat_8u_D2L ippsConcat_16u_D2L
//
// Purpose: Concatenates several strings together
//
// Arguments:
// pSrc - pointer to the array of source strings
// srcLen - pointer to the array of source strings' lengths
// numSrc - number of source strings
// pDst - pointer to the destination string
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc, srcLen or pDst are NULL;
// pSrc[i] is NULL for i < numSrc
// ippStsLengthErr srcLen[i] is negative for i < numSrc
// ippStsSizeErr numSrc is not positive
*/
IPPAPI (IppStatus, ippsConcat_8u_D2L, (const Ipp8u* const pSrc[], const int srcLen[], int numSrc,
Ipp8u* pDst ))
IPPAPI (IppStatus, ippsConcat_16u_D2L, (const Ipp16u* const pSrc[], const int srcLen[], int numSrc,
Ipp16u* pDst ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsConcatC_8u_D2L ippsConcatC_16u_D2L
//
// Purpose: Concatenates several strings together and separates them
// by the symbol delimiter
//
// Arguments:
// pSrc - pointer to the array of source strings
// srcLen - pointer to the array of source strings' lengths
// numSrc - number of source strings
// delim - delimiter
// pDst - pointer to the destination string
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc, srcLen or pDst are NULL;
// pSrc[i] is NULL for i < numSrc
// ippStsLengthErr srcLen[i] is negative for i < numSrc
// ippStsSizeErr numSrc is not positive
*/
IPPAPI (IppStatus, ippsConcatC_8u_D2L, (const Ipp8u* const pSrc[], const int srcLen[], int numSrc,
Ipp8u delim, Ipp8u* pDst ))
IPPAPI (IppStatus, ippsConcatC_16u_D2L, (const Ipp16u* const pSrc[], const int srcLen[], int numSrc,
Ipp16u delim, Ipp16u* pDst ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsSplitC_8u_D2L ippsSplitC_16u_D2L
//
// Purpose: Splits source string to several destination strings
// using the symbol delimiter; all delimiters are significant,
// in the case of double delimiter empty string is inserted.
//
// Arguments:
// pSrc - pointer to the source string
// srcLen - source string length
// delim - delimiter
// pDst - pointer to the array of destination strings
// dstLen - pointer to the array of destination strings' lengths
// pNumDst - pointer to the number of destination strings:
// *pNumDst = initial number of destination strings on input;
// *pNumDst = number of splitted strings on output;
//
// Return:
// ippStsNoErr Ok
// ERRORS:
// ippStsNullPtrErr pSrc, pDst, dstLen or pNumDst are NULL;
// pDst[i] is NULL for i < number of splitted strings
// ippStsLengthErr srcLen is negative;
// dstLen[i] is negative for i < number of splitted strings
// ippStsSizeErr *pNumDst is not positive
// WARNINGS:
// ippStsOvermuchStrings the initial number of destination strings is less
// than the number of splitted strings;
// number of destination strings is truncated to
// initial number in this case
// ippStsOverlongString the length of one of destination strings is less than
// length of corresponding splitted string;
// splitted string is truncated to destination length
// in this case
*/
IPPAPI (IppStatus, ippsSplitC_8u_D2L, (const Ipp8u* pSrc, int srcLen, Ipp8u delim,
Ipp8u* pDst[], int dstLen[], int* pNumDst))
IPPAPI (IppStatus, ippsSplitC_16u_D2L, (const Ipp16u* pSrc, int srcLen, Ipp16u delim,
Ipp16u* pDst[], int dstLen[], int* pNumDst))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsFindCAny_8u
// ippsFindCAny_16u
// ippsFindRevCAny_8u
// ippsFindRevCAny_16u
//
// Purpose: Reports the index of the first/last occurrence in
// the vector of any value in a specified array.
//
// Arguments:
// pSrc - The pointer of vector to find.
// len - The length of the vector.
// pAnyOf - A pointer of array containing one or more values to seek.
// lenFind - The length of array.
// pIndex - The positive integer index of the first occurrence in
// the vector where any value in pAnyOf was found;
// otherwise, -1 if no value in pAnyOf was found.
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr Any of pointers is NULL.
// ippStsLengthErr len or lenAnyOf are negative.
*/
IPPAPI (IppStatus, ippsFindCAny_8u, ( const Ipp8u* pSrc, int len,
const Ipp8u* pAnyOf, int lenAnyOf, int* pIndex ))
IPPAPI (IppStatus, ippsFindCAny_16u, ( const Ipp16u* pSrc, int len,
const Ipp16u* pAnyOf, int lenAnyOf, int* pIndex ))
IPPAPI (IppStatus, ippsFindRevCAny_8u, ( const Ipp8u* pSrc, int len,
const Ipp8u* pAnyOf, int lenAnyOf, int* pIndex ))
IPPAPI (IppStatus, ippsFindRevCAny_16u, ( const Ipp16u* pSrc, int len,
const Ipp16u* pAnyOf, int lenAnyOf, int* pIndex ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsReplaceC_8u
// ippsReplaceC_16u
//
// Purpose: Replaces all occurrences of a specified value in
// the vector with another specified value.
//
// Arguments:
// pSrc - The pointer of vector to replace.
// pDst - The ponter of replaced vector.
// len - The length of the vector.
// oldVal - A value to be replaced.
// newVal - A value to replace all occurrences of oldVal.
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr Any of pointers is NULL.
// ippStsLengthErr len is negative.
*/
IPPAPI (IppStatus, ippsReplaceC_8u, ( const Ipp8u* pSrc, Ipp8u* pDst, int len,
Ipp8u oldVal, Ipp8u newVal ))
IPPAPI (IppStatus, ippsReplaceC_16u, ( const Ipp16u* pSrc, Ipp16u* pDst, int len,
Ipp16u oldVal, Ipp16u newVal ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsTrimCAny_8u
// ippsTrimCAny_16u
// ippsTrimEndCAny_8u
// ippsTrimEndCAny_16u
// ippsTrimStartCAny_8u
// ippsTrimStartCAny_16u
//
// Purpose: Removes all occurrences of a set of specified values
// from:
// TrimCAny - the beginning and end of the vector.
// TrimEndCAny - the end of the vector.
// TrimStartCAny - the beginning of the vector.
//
// Arguments:
// pSrc - The pointer of src vector to remove.
// srcLen - The length of the src vector.
// pTrim - An array of values to be removed.
// trimLen - The length of the array values.
// pDst - The pointer of dst vector to result save.
// pDstLen - The result length of the dst vector.
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr Any of pointers is NULL.
// ippStsLengthErr srcLen or trimLen are negative.
//
// Note:
// The length of the pDst should be sufficient;
// if values not found, *pDstLen = srcLen.
*/
IPPAPI (IppStatus, ippsTrimCAny_8u, ( const Ipp8u* pSrc, int srcLen,
const Ipp8u* pTrim, int trimLen, Ipp8u* pDst, int* pDstLen ))
IPPAPI (IppStatus, ippsTrimCAny_16u, ( const Ipp16u* pSrc, int srcLen,
const Ipp16u* pTrim, int trimLen, Ipp16u* pDst, int* pDstLen ))
IPPAPI (IppStatus, ippsTrimEndCAny_8u, ( const Ipp8u* pSrc, int srcLen,
const Ipp8u* pTrim, int trimLen, Ipp8u* pDst, int* pDstLen ))
IPPAPI (IppStatus, ippsTrimEndCAny_16u, ( const Ipp16u* pSrc, int srcLen,
const Ipp16u* pTrim, int trimLen, Ipp16u* pDst, int* pDstLen ))
IPPAPI (IppStatus, ippsTrimStartCAny_8u, ( const Ipp8u* pSrc, int srcLen,
const Ipp8u* pTrim, int trimLen, Ipp8u* pDst, int* pDstLen ))
IPPAPI (IppStatus, ippsTrimStartCAny_16u, ( const Ipp16u* pSrc, int srcLen,
const Ipp16u* pTrim, int trimLen, Ipp16u* pDst, int* pDstLen ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsCompareIgnoreCase_16u
//
// Purpose: Compares two Unicode strings element-by-element
//
// Arguments:
// pSrc1 - pointer to the first string
// pSrc2 - pointer to the second string
// len - string length to compare
// pResult - pointer to the result:
// *pResult = 0 if src1 == src2;
// *pResult > 0 if src1 > src2;
// *pResult < 0 if src1 < src2;
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc1, pSrc2 or pResult is NULL
// ippStsLengthErr len is negative
*/
IPPAPI (IppStatus, ippsCompareIgnoreCase_16u, (const Ipp16u* pSrc1, const Ipp16u* pSrc2, int len,
int *pResult))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsCompareIgnoreCaseLatin_8u
// ippsCompareIgnoreCaseLatin_16u
//
// Purpose: Compares two ASCII strings element-by-element
//
// Arguments:
// pSrc1 - pointer to the first string
// pSrc2 - pointer to the second string
// len - string length to compare
// pResult - pointer to the result:
// *pResult = 0 if src1 == src2;
// *pResult > 0 if src1 > src2;
// *pResult < 0 if src1 < src2;
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc1, pSrc2 or pResult is NULL
// ippStsLengthErr len is negative
*/
IPPAPI (IppStatus, ippsCompareIgnoreCaseLatin_8u, (const Ipp8u* pSrc1, const Ipp8u* pSrc2, int len,
int *pResult))
IPPAPI (IppStatus, ippsCompareIgnoreCaseLatin_16u, (const Ipp16u* pSrc1, const Ipp16u* pSrc2,
int len, int *pResult))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsInsert_8u_I ippsInsert_16u_I
// ippsInsert_8u ippsInsert_16u
//
// Purpose: Inserts one string at a specified index position in other string
//
// Arguments:
// pSrc - pointer to the source string
// srcLen - source string length
// pInsert - pointer to the string to be inserted
// insertLen - length of the string to be inserted
// pDst - pointer to the destination string
// pSrcDst - pointer to the string for in-place operation
// pSrcDstLen - pointer to the string length:
// *pSrcDstLen = source length on input;
// *pSrcDstLen = destination length on output;
// startIndex - index of start position
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc, pInsert, pDst, pSrcDst or pSrcDstLen is NULL
// ippStsLengthErr srcLen, insertLen, *pSrcDstLen or startIndex is negative Or
// startIndex is greater than srcLen or *pSrcDstLen
*/
IPPAPI (IppStatus, ippsInsert_8u_I, (const Ipp8u* pInsert, int insertLen, Ipp8u* pSrcDst,
int* pSrcDstLen, int startIndex))
IPPAPI (IppStatus, ippsInsert_16u_I, (const Ipp16u* pInsert, int insertLen, Ipp16u* pSrcDst,
int* pSrcDstLen, int startIndex))
IPPAPI (IppStatus, ippsInsert_8u, (const Ipp8u* pSrc, int srcLen, const Ipp8u* pInsert,
int insertLen, Ipp8u* pDst, int startIndex))
IPPAPI (IppStatus, ippsInsert_16u, (const Ipp16u* pSrc, int srcLen, const Ipp16u* pInsert,
int insertLen, Ipp16u* pDst, int startIndex))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRemove_8u_I ippsRemove_16u_I
// ippsRemove_8u ippsRemove_16u
//
// Purpose: Deletes a specified number of characters from the string
// beginning at a specified position.
//
// Arguments:
// pSrc - pointer to the source string
// srcLen - source string length
// pDst - pointer to the destination string
// pSrcDst - pointer to the string for in-place operation
// pSrcDstLen - pointer to the string length:
// *pSrcDstLen = source length on input;
// *pSrcDstLen = destination length on output;
// startIndex - index of start position
// len - number of characters to be deleted
//
// Return:
// ippStsNoErr Ok
// ippStsNullPtrErr pSrc, pDst, pSrcDst or pSrcDstLen are NULL
// ippStsLengthErr srcLen, *pSrcDstLen, len or startIndex is negative Or
// (startIndex + len) is greater than srcLen or *pSrcDstLen
*/
IPPAPI (IppStatus, ippsRemove_8u_I, (Ipp8u* pSrcDst, int* pSrcDstLen, int startIndex, int len))
IPPAPI (IppStatus, ippsRemove_16u_I, (Ipp16u* pSrcDst, int* pSrcDstLen, int startIndex, int len))
IPPAPI (IppStatus, ippsRemove_8u, (const Ipp8u* pSrc, int srcLen, Ipp8u* pDst, int startIndex,
int len))
IPPAPI (IppStatus, ippsRemove_16u, (const Ipp16u* pSrc, int srcLen, Ipp16u* pDst, int startIndex,
int len))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpInitAlloc
// Purpose: Allocates necessary memory, compiles a pattern into the
// internal form consideration corresponding options and
// writes it into the pRegExpState
//
// Parameters:
// pPattern Pointer to the pattern of regular expression
// pOptions Pointer to options for compiling and executing
// regular expression (possible values 'i','s','m','x','g')
// It should be NULL if no options are required.
// pRegExpState Pointer to the structure containing internal form of
// a regular expression.
// pErrOffset Pointer to offset into the pattern if compiling is break
//
// Return:
// ippStsNoErr No errors
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsMemAllocErr Can't allocate memory for pRegExpState
// ippStsRegExpOptionsErr Options are incorrect
// ippStsRegExpQuantifierErr Error caused by using wrong quantifier
// ippStsRegExpGroupingErr Error caused by using wrong grouping
// ippStsRegExpBackRefErr Error caused by using wrong back reference
// ippStsRegExpChClassErr Error caused by using wrong character class
// ippStsRegExpMetaChErr Error caused by using wrong metacharacter
//
*/
IPPAPI(IppStatus, ippsRegExpInitAlloc, ( const char* pPattern, const char* pOptions,
IppRegExpState** ppRegExpState, int* pErrOffset ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpGetSize
// Purpose: Computes the size of necessary memory (in bytes) for
// structure containing internal form of regular expression
//
// Parameters:
// pPattern Pointer to the pattern of regular expression
// pRegExpStateSize Pointer to the computed size of structure containing
// internal form of regular expression
//
// Return:
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsNoErr No errors
//
*/
IPPAPI(IppStatus, ippsRegExpGetSize, ( const char* pPattern, int* pRegExpStateSize ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpInit
// Purpose: Compiles a pattern into the internal form consideration
// corresponding options and writes it into the pRegExpState
//
// Parameters:
// pPattern Pointer to the pattern of regular expression
// pOptions Pointer to options for compiling and executing
// regular expression (possible values 'i','s','m','x','g')
// It should be NULL if no options are required.
// pRegExpState Pointer to the structure containing internal form of
// a regular expression.
// pErrOffset Pointer to offset into the pattern if compiling is break
//
// Return:
// ippStsNoErr No errors
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsRegExpOptionsErr Options are incorrect
// ippStsRegExpQuantifierErr Error caused by using wrong quantifier
// ippStsRegExpGroupingErr Error caused by using wrong grouping
// ippStsRegExpBackRefErr Error caused by using wrong back reference
// ippStsRegExpChClassErr Error caused by using wrong character class
// ippStsRegExpMetaChErr Error caused by using wrong metacharacter
//
*/
IPPAPI(IppStatus, ippsRegExpInit, ( const char* pPattern, const char* pOptions,
IppRegExpState* pRegExpState, int* pErrOffset ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpSetMatchLimit
// Purpose: Changes initial value of the matches kept in stack
//
// Parameters:
// matchLimit New value of the matches kept in stack
// pRegExpState Pointer to the structure containing internal form of
// a regular expression
//
// Return:
// ippStsNullPtrErr Pointer is NULL
// ippStsNoErr No errors
//
*/
IPPAPI(IppStatus, ippsRegExpSetMatchLimit, ( int matchLimit, IppRegExpState* pRegExpState ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpSetFormat
// Purpose: Changes initial value of the matches kept in stack
//
// Parameters:
// fmt New source encoding mode
// pRegExpState Pointer to the structure containing internal form of
// a regular expression
//
// Return:
// ippStsNullPtrErr Pointer is NULL.
// ippStsRangeErr When
// ippStsNoErr No errors
//
*/
IPPAPI(IppStatus, ippsRegExpSetFormat, ( IppRegExpFormat fmt, IppRegExpState* pRegExpState ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpFree
// Purpose: Frees allocated memory for the structure containing
// internal form of regular expression
//
// Parameters:
// pRegExpState Pointer to the structure containing internal form of
// a regular expression.
//
*/
IPPAPI(void, ippsRegExpFree, ( IppRegExpState* pRegExpState ))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpFind_8u
// Purpose: Looks for the occurrences of the substrings matching
// the specified regular expression.
//
// Parameters:
// pSrc Pointer to the source string
// srcLen Number of elements in the source string.
// pRegExpState Pointer to the structure containing internal form of
// a regular expression
// pFind Array of pointers to the matching substrings
// pNumFind Size of the array pFind on input,
// number of matching substrings on output.
//
// Return:
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsSizeErr Length of the source vector is less zero or or
// pNumFind is less than or equal to 0
// ippStsRegExpErr The structure pRegExpState contains wrong data
// ippStsRegExpMatchLimitErr The match limit has been exhausted
// ippStsNoErr No errors
//
*/
IPPAPI(IppStatus, ippsRegExpFind_8u, ( const Ipp8u* pSrc, int srcLen, IppRegExpState* pRegExpState,
IppRegExpFind* pFind, int* pNumFind ))
#if !defined( _OWN_BLDPCS )
struct RegExpMultiState;
typedef struct RegExpMultiState IppRegExpMultiState;
typedef struct {
Ipp32u regexpDoneFlag;
Ipp32u regexpID;
Ipp32s numMultiFind;
IppStatus status;
IppRegExpFind* pFind;
} IppRegExpMultiFind;
#endif /* _OWN_BLDPCS */
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpMultiGetSize
// Purpose: Computes the size of necessary memory (in bytes) for
// structure containing internal form of multi patterns search engine.
//
// Parameters:
// maxPatterns Maximum number of pattern.
// pSize Pointer to the computed size of structure containing
// internal form of search engine
//
// Return:
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsSizeErr When maxPatterns is less or equal 0.
// ippStsNoErr No errors
//
*/
IPPAPI(IppStatus, ippsRegExpMultiGetSize, ( Ipp32u maxPatterns, int *pSize))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpMultiInit
// Purpose: Initialize internal form of multi patterns search engine.
//
// Parameters:
// maxPatterns Maximum number of pattern.
// pState Pointer to the structure containing internal form of
// multi patterns search engine.
//
// Return:
// ippStsNoErr No errors
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsSizeErr When maxPatterns is less or equal 0.
//
*/
IPPAPI(IppStatus, ippsRegExpMultiInit, ( IppRegExpMultiState* pState, Ipp32u maxPatterns))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpMultiInitAlloc
// Purpose: Allocates necessary memory, initialize internal form of multi patterns search engine
//
// Parameters:
// maxPatterns Maximum number of pattern.
// pState Double pointer to the structure containing internal form of
// multi patterns search engine.
//
// Return:
// ippStsNoErr No errors
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsSizeErr When maxPatterns is less or equal 0.
*/
IPPAPI(IppStatus, ippsRegExpMultiInitAlloc, ( IppRegExpMultiState** ppState, Ipp32u maxPatterns))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpMultiFree
// Purpose: Frees allocated memory for the structure containing
// internal form of multi patterns search engine
//
// Parameters:
// pState Pointer to the structure containing internal form of
// multi patterns search engine.
//
*/
IPPAPI(void, ippsRegExpMultiFree, (IppRegExpMultiState* pState))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpMulti*
// Purpose: Controls multi patterns database. Add, remove or modify patterns.
//
// Parameters:
// pRegExpState Pointer to the structure containing internal form of a
// compiled regular expression.
// regexpID Pattern ID. 0 is invalid ID.
// pState Pointer to the structure containing internal form of
// multi patterns search engine.
//
// Return:
// ippStsNoErr No errors
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsSizeErr When ID is equal 0.
// ippStsMemAllocErr When number of patterns exceeded its maximum value.
//
*/
IPPAPI(IppStatus, ippsRegExpMultiAdd, ( const IppRegExpState* pRegExpState, Ipp32u regexpID, IppRegExpMultiState* pState))
IPPAPI(IppStatus, ippsRegExpMultiModify, ( const IppRegExpState* pRegExpState, Ipp32u regexpID, IppRegExpMultiState* pState))
IPPAPI(IppStatus, ippsRegExpMultiDelete, (Ipp32u regexpID, IppRegExpMultiState* pState))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpMultiFind_8u
// Purpose: Looks for the occurrences of the substrings matching
// the specified patterns.
//
// Parameters:
// pSrc Pointer to the source string
// srcLen Number of elements in the source string.
// pState Pointer to the structure containing internal form of
// multi patterns search engine
// pDstMultiFind Array of pointers to the matching patterns
//
// Return:
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsSizeErr Length of the source vector is less or equal zero.
// ippStsNoErr No errors
//
*/
IPPAPI(IppStatus, ippsRegExpMultiFind_8u, ( const Ipp8u* pSrc, int srcLen, IppRegExpMultiFind *pDstMultiFind, const IppRegExpMultiState* pState))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsConvertUTF
// Purpose: Convert UTF16BE or UTF16LE format to UTF8 and vice versa.
//
// Parameters:
// pSrc Pointer to the input vector.
// pSrcLen Length of the pSrc vector on input and its used length on output.
// BEFlag Flag to indicate UTF16BE format. 0 means UTF16LE format.
// pDst Pointer to the output vector.
// pDstLen Length of the pDst vector on input and its used length on output.
//
// Return:
// ippStsNullPtrErr One or several pointer(s) is NULL
// ippStsNoErr No errors
*/
IPPAPI(IppStatus, ippsConvertUTF_8u16u,( const Ipp8u* pSrc, Ipp32u *pSrcLen, Ipp16u* pDst, Ipp32u *pDstLen, int BEFlag))
IPPAPI(IppStatus, ippsConvertUTF_16u8u,( const Ipp16u* pSrc, Ipp32u *pSrcLen, Ipp8u* pDst, Ipp32u *pDstLen, int BEFlag))
#if !defined( _OWN_BLDPCS )
struct RegExpMultiState;
typedef struct RegExpReplaceState IppRegExpReplaceState;
#endif /* _OWN_BLDPCS */
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpReplaceGetSize
// Purpose: Computes the size the find and replace engine memory.
//
// Parameters:
// pSrcReplacement Pointer to the input null-terminated replace pattern.
// pSize Pointer to the computed size of the replace engine memory.
//
// Return:
// ippStsNullPtrErr pSize pointer is NULL
// ippStsNoErr No errors
//
*/
IPPAPI(IppStatus, ippsRegExpReplaceGetSize, ( const Ipp8u* pSrcReplacement, Ipp32u *pSize))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpReplaceInit
// Purpose: Initialize internal form of find and replace engine.
//
// Parameters:
// pSrcReplacement Pointer to the input null-terminated replace pattern.
// pReplaceState Pointer to the memory allocated for the find and replace
// engine structure of size after ippsRegExpReplaceGetSize function.
//
// Return:
// ippStsNoErr No errors
// ippStsNullPtrErr pReplaceState pointer is NULL
//
*/
IPPAPI(IppStatus, ippsRegExpReplaceInit, ( const Ipp8u* pSrcReplacement, IppRegExpReplaceState *pReplaceState))
/* /////////////////////////////////////////////////////////////////////////////
// Name: ippsRegExpReplace_8u
// Purpose: Performs find and replace.
//
// Parameters:
// pSrc Pointer to the source string.
// pSrcLenOffset Length of the pSrc vector on input and its used length on output.
// pRegExpState Pointer to the compiled pattern structure.
// pReplaceState Pointer to the memory allocated for the find and replace engine structure.
// pDst Pointer to the destination string.
// pDstLen Length of the pDst vector on input and its used length on output.
// pFind Array of pointers to the matching substrings.
// pNumFind Size of the array pFind on input, number of matching substrings on output.
//
// Return:
// ippStsNoErr Indicates no error.
// ippStsNullPtrErr One or several pointer(s) is NULL.
// ippStsSizeErr Indicates an error when value in pSrcLen or pDstLen is less or equal to zero.
//
*/
IPPAPI(IppStatus, ippsRegExpReplace_8u, ( const Ipp8u* pSrc, int *pSrcLenOffset, Ipp8u *pDst, int *pDstLen, IppRegExpFind* pFind, int* pNumFind, IppRegExpState* pRegExpState,
IppRegExpReplaceState *pReplaceState ))
#if defined (_IPP_STDCALL_CDECL)
#undef _IPP_STDCALL_CDECL
#define __stdcall __cdecl
#endif
#ifdef __cplusplus
}
#endif
#endif /* __IPPCH_H__ */
/* ////////////////////////////// End of file /////////////////////////////// */
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,012 |
{"url":"https:\/\/pypi.org\/project\/formulaparser\/","text":"Parse an arythmic formula from a string\n\n## Project description\n\nA formula parser for python.\n\nCopyright (C) 2017 Lars van de Kerkhof\n\nThis program is free software: you can redistribute it and\/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.\n\nThis program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.\n\nYou should have received a copy of the GNU General Public License along with this program. If not, see <http:\/\/www.gnu.org\/licenses\/>.\n\n## what it does\n\nParses simple arythmetic formula\u2019s containing variables. Can then be used to compute values resolving the variables from a context object.\n\n>>> from argparse import Namespace as KeyedObject\n>>> from formulaparser import Formula\n>>>\n>>> context = KeyedObject()\n>>> context.four = 4\n>>> context.three = 3\n>>> deepercontext = KeyedObject()\n>>> deepercontext.ten = 10\n>>> deepercontext.twelve = 12\n>>> context.nextlevel = deepercontext\n>>>\n>>> Formula(\"((1 + 2 + 3) + 4 + (3 + 7)) + 5\").calculate_value()\n25\n>>> Formula(\"4!\").calculate_value()\n24\n>>> Formula(\"3.287 \/ 6\").calculate_value()\n0.5478333333333333\n>>> Formula(\"2 ^ 8\").calculate_value()\n256\n>>> Formula(\"4 - (-4)\").calculate_value()\n8\n>>> Formula(\"(4 * 6) - 8 + 7 - 4 + 3\").calculate_value()\n22\n>>> Formula(\"((1 + four + 3) + nextlevel.ten + (3 + 7)) + 5\").calculate_value(context)\n33\n>>> Formula(\"((four!) - 6) \/ nextlevel.twelve\").calculate_value(context)\n1.5\n>>> Formula(\"2 ^ three\").calculate_value(context)\n8\n>>> Formula(\"nextlevel.twelve - (-four)\").calculate_value(context)\n16\n>>> Formula(\"(four * 6) - nextlevel.ten + 7 - 4 + 3\").calculate_value(context)\n20\n\n## Project details\n\nUploaded source","date":"2022-11-28 09:50:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4206550717353821, \"perplexity\": 3456.210288601884}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710488.2\/warc\/CC-MAIN-20221128070816-20221128100816-00839.warc.gz\"}"} | null | null |
Q: managing xcode 5 config files in git (and github)
Likely Dupe: "How to use a "private" .xcconfig in a "Shared" (git-managed) Xcode project?"
I'm running an ios project with several remote developers. Our code is checked in to a repository on github - including our xcode 5 configuration file directory and its contents ("myproj.xcodeproj") Everyone has their own version of local xcode configuration settings. The project settings are fairly complicated so I would like to keep a default version of settings that a developer can pull down when they first join the project but then I would like git to ignore the changes they make to configure for their local environment
Right now, the config files are just checked in and tracked on github. Everyone also keeps a local backup of their personal config files outside of the repository. When they pull, they can overwrite the config files that come down from github with their personal versions if necessary. Hopefully people remember not to check in their xcode project settings, but I need to deal with people who just use commit -A
I imagine this is a fairly common issue. What are the best practices for getting this done?
A: If you want overrides of a set of defaults and to have everything in git, I would recommend using .xcconfig files for your settings. You can make it possible to select the current settings through a per-developer configuration to further separate environments.
In addition to build settings, this allows you to have separate app configurations (like server urls or api tokens) for each developer, test, release, etc.
See https://github.com/distiller/DIConfiguration for an implementation and further explanation.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,866 |
The amount Does It Cost to Name a Star?
January 3, 2022 by Lewis Carroll
What amount does it cost to purchase a star? Likely significantly short of what you think. Stars may be one of the most significant gifts you can give somebody, yet they're in no way, shape or form the most costly.
On the off chance that you're looking for an extraordinary, essential gift, a star is awesome. Assuming you're searching for an extraordinary, essential gift that won't leave you crying over your next financial record, it's significantly more great.
A star may cause somebody to feel 1,000,000 bucks, however luckily, they don't cost anyplace close so a lot.
Putting a precise figure on an inquiry like 'how much does it cost to purchase a star' is difficult. A fast google search of 'the amount to purchase a star?' will turn up numerous outcomes and various responses. You may see a few stars on offer for $30; you may see others promoted for $100.
At last, there is certainly not a conclusive response to questions like how much is a star. A few venders offer a sensible help at a sensible cost, others charge the earth for something significantly more dull. A few dealers incorporate extra highlights (enlistment declarations, applications, and so on), others offer a significantly more fundamental bundle.
It may appear to be that a basic inquiry like 'how much does it cost to purchase a star' ought to have a basic response. However, with such countless factors included, it gets much more convoluted.
It settles the score more convoluted when you consider that naming a star includes more than basically settling on a name and sending your installment. There are bundles to pick between, brilliance levels to settle on, heavenly bodies to consider.
To work out a conclusive solution to the topic of how much does it cost to purchase a star, you'll initially have to settle on…
What Kind of Star You Want
All stars are something similar, correct? Wrong. Stars come in different kinds, all with their own attractions, their own advantages, and their own sticker prices.
There are standard stars, there are parallel stars, and there are Zodiac stars. We'll address the contrasts between the kinds in more detail instantly. Until further notice, it merits remembering that a few stars will cost an extensive sum more than others.
Who You're Going to Buy From
Nothing an affects the topic of how much does it cost to buy a star your decision of merchant. No two stars are by and large something very similar, nor are any two merchants.
A few merchants charge a colossal cost for a not exactly extraordinary help. Others figure out how to consolidate an exceptional cost with a remarkable item. In any case, costs can change significantly.
What Package Do You Want
Not many merchants offer only one sort of bundle. Most market something like a few, with each bundle offering a marginally unique minor departure from the subject.
Similarly as with your decision of dealer, bundle decision can hugely affect deciding a solution to how much is it to purchase a star. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 765 |
Dunnes Stores shop stewards from 100 stores across the Republic of Ireland met yesterday (Sunday, 26th April) to decide the next actions in the Decency for Dunnes Workers campaign. The shop stewards voted unanimously for a national protest march to Dunnes Stores Head Office in Dublin which will take place on Saturday, June 6th.
The protest will be preceded by a series of local protests and actions, further details of which will be announced shortly.
The meeting also endorsed a claim for a 3 percent pay increase lodged by their union Mandate.
The shop stewards at the meeting made it clear they are reserving their right to take industrial action in the future. Furthermore, they are committed to potentially escalating the industrial action beyond a one day strike.
"Once again our members in Dunnes are acting in a very responsible manner. They are calling for this day of action because they, unlike their employer, are concerned about potential long term irreparable damage to the business. The support from the public has been tremendous and this march will give them the opportunity to show Dunnes workers that they are not on their own in this struggle.
Mr Light added: "We know that the Dunnes family have very deep pockets, with the Sunday Times listing them yesterday as having more than €1.7bn in combined wealth. That enormous wealth has only been derived due to the hard work of their loyal members of staff. It's about time the company acknowledged that hard work by meaningfully engaging with their workers through their trade union.
"It is apparent that instead of fairly distributing some of this massive accumulated wealth with their workers the Dunnes owners are intent in using it in a battle of financial means with our members, many of whom are only guaranteed 15 hours work per week," he said.
Further details on the national day of protest will be announced shortly. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,248 |
\section{Introduction}
A common statistical method in analysing high frequency financial data is to take a time series view and consider the data as a sequence of ordered vectors. However, from this perspective one may lose some of the important structure which characterises useful features within the data; in particular, we may overlook the \textit{order} of events in different coordinates and their consequent effects on one another. Moreover, time series analysis is typically highly dependent on the underlying model chosen.
By taking an alternative philosophical view of the discrete points being part of a continuous \textit{path} in a multidimensional space (say some Euclidean space $\mathbb{R}^d$), one may lift the path to its signature and apply the powerful theory of rough paths in order to analyse the original signal. The idea that the signature can be thought of as an informative \textit{transform} of a multidimensional time series was first introduced by B. Hambly and T. Lyons \cite{hambly2005uniqueness}. We use the term \textit{transform} in the sense that no information is lost (modulo the notion of \textit{tree-like} equivalence) by taking the signature of a bounded variation path.
With this view in mind, D. Levin et al in \cite{learning} first introduced the possibility of using the signature of a time series to understand financial data. In this work the authors exploit the fact that any polynomial of iterated integrals can be represented by a linear combination of iterated integrals \cite[Theorem 2.15]{lyons2004differential}. This makes the signature a natural candidate for linear regression analysis, which is precisely what the authors do with numerical examples using classical time series models. Their transform of the time series to a signature of a bounded variation path closely resembles our definition of a \textit{Hoff} process below.
The recent paper \cite{gergely2013extracting} also takes the viewpoint of defining a process (which is precisely our definition of the Hoff process) out of the discrete data points and applies standard regression analysis to extract features out of WTI crude oil futures market data among other asset classes.
In this paper we assume that the underlying signal is a continuous semimartingale $X: [0,1] \to \mathbb{R}^d$ with canonical decomposition $X=M+V$, ($M$ being a local martingale and $V$ being a bounded variation path). Since the signature is invariant under translation we make the simplifying assumption that $X_0=0$. Let us establish some notation first:
\begin{Definition}[\textbf{Partition}]
A \textit{partition} is a collection of distinct points in $[0,1]$ always containing the endpoints $\seq{0,1}$. Let $\mathcal{D}_{[0,1]}$ denote the set of all partitions of $[0,1]$ and we set $\abs{D}:=\max_{t_i\in D} \abs{t_{i+1}-t_i}$ to be the \textit{mesh} of $D$.
\end{Definition}
As our original time series set up, suppose we are given the values of $X$ at times given by a partition $D=(t_i)^n_{i=0} \in \mathcal{D}_{[0,1]}$; that is we have a sequence of ordered pairs $\seq{\bra{t_i,X_{t_i}}}_{t_i\in D}$.
At this point we follow the cited papers \cite{gergely2013extracting,learning} and depart from traditional time series analysis by constructing a $(2d)$-dimensional process $X^D : [0,1]\to\mathbb{R}^{2d}$ comprised of the lead and lag components of the data. We call this the \textit{Hoff process}. Using the notation $t_i^* := 1/2(t_i + t_{i+1})$, we give the following precise definition:
\begin{Definition}[\textbf{Hoff process}]
Given a time series $\bra{X_{t_i}}_{t_i\in D}$ we construct the continuous \textit{axis} process $X^D : [0,1]\to\mathbb{R}^{2d}$,
\[
X^D = \bra{X^{D,\theta;i}}_{\substack{i \in \seq{1,\ldots, d}}, \\ \theta \in \seq{b,f}} = \bra{X^{D,b;1}, \ldots, X^{D,b,d}, X^{D,f,1}, \ldots, X^{D,f,d}},
\]
which we define as follows: for $D=(t_i)_{i=0}^n \in \mathcal{D}_{[0,1]}$;
\[
X_{u}^{D}=\begin{cases}
\left(X_{t_{i-1}}+\frac{u-t_{i}}{t_{i}^{*}-t_{i}}\left(X_{t_{i}}-X_{t_{i-1}}\right);X_{t_{i+1}}\right) & \text{\,\,\,\, if }u\in[t_{i},t_{i}^{*});\text{\,\,\,\,\,\, for }i=1,\ldots,n-1,\\
\left(X_{t_{i}};X_{t_{i+1}}+\frac{u-t_{i}^{*}}{t_{i+1}-t_{i}^{*}}\left(X_{t_{i+2}}-X_{t_{i+1}}\right)\right) & \text{\,\,\,\, if }u\in[t_{i}^{*},t_{i+1});\text{\,\, for }i=0,1,\ldots,n-2,
\end{cases}
\]
and set the first and last intervals of $X^D$ to be:
\[
X_{u}^{D}=\begin{cases}
\left(0;\frac{u}{t_{0}^{*}}X_{t_{1}}\right) & \text{\,\,\,\,}\text{if }u\in[0,t_{0}^{*});\\
\left(X_{t_{n-1}}+\frac{u-t_{n-1}^{*}}{t_{n}-t_{n-1}^{*}}\left(X_{t_{n}}-X_{t_{n-1}}\right);X_{t_{n}}\right) & \text{\,\,\,\,}\text{if }u\in[t_{n-1}^{*},t_{n}].
\end{cases}
\]
We call the process $X^D$ the \textbf{Hoff process} associated with the time stamped series $(X_{t_i})_{t_i\in D}$.
We denote the L\'{e}vy area process associated with $X^D$ by
\begin{align}\label{e-stokes-area}
A^{D, \theta, \lambda ; i , j}_{s,t} :&= \frac{1}{2} \int^t_s \bra{ X^{D, \theta; i}_{s,r}\, dX^{D,\lambda ; j}_r - X^{D, \lambda; j}_{s,r}\, dX^{D,\theta ; i}_r}
= \int^t_s X^{D, \theta; i}_{s,r}\, dX^{D,\lambda ; j}_r - \frac{1}{2} X^{D,\theta ; i}_{s,t} X^{D,\lambda ; j}_{s,t}.
\end{align}
\end{Definition}
Since $X^D$ is piecewise linear we are able to switch between the It\^{o} and Stratonovich integrals in the Stokes' area formula (\ref{e-stokes-area}) without incurring an additional drift term.
\begin{Remark}
To our knowledge, B. Hoff was the first to consider the area between rough paths and their delay within the context of moving Brownian frames in his Ph.D thesis \cite{hoff2005brownian}. Thus for this reason we use the nomenclature of a \textit{Hoff} process for $X^D$.
\end{Remark}
We call $X^{D,b}$ and $X^{D,f}$ the \textit{lag} and \textit{lead} components of the Hoff process respectively. Since \textit{lag} and \textit{lead} both share a common first letter we use \textit{b} and \textit{f} instead, which can be thought of as \textit{backward} and \textit{forward} respectively.
At this point we find it helpful in building intuition about $X^D$ to include a diagram of a possible trajectory (see Figure \ref{f-hoff-greg}).
\begin{figure}\label{f-hoff-greg}
\centering
\includegraphics[width=16cm]{plot_hoff_02.pdf}
\caption{An example of a Hoff process trajectory}
\end{figure}
The first result of this paper concerns the convergence of the rough path lift $\rp{X}^D$ of the Hoff process $X^D$ as our partition mesh $\abs{D} \to 0$. The second result is a corollary of the first; we prove that certain natural random ODEs driven by $X^D$ converge to the corresponding It\^{o} integral limit and not the usual Stratonovich stochastic integral as predicted by classical Wong-Zakai theorem \cite{eugene1965relation}. The theory of rough paths allows us to prove otherwise and refer to \cite[Theorem 17.20]{FV} and \cite{friz2009rough} for similar results concerning lead-lag driven random ODE convergence.
Interest in recovering the It\^{o} integral and the usual It\^{o} formula in a rough path context is an active area of research. From a much more theoretical perspective, D. Yang and T. Lyons in their recent paper \cite{lyons2013differential} proved that the It\^{o} integral can be recovered in a \textit{pathwise} sense. Their proof introduces the novel idea of concatenating a mean of Stratonovich solutions to add as a ``polluting'' noise to the driving martingale signal.
The perturbed signal allows the precise recovery of the It\^{o} integral, which is the main idea behind the proof of the recovery theorem in the present paper.
Throughout we denote the class of $L^p(\mathbb{P})$-bounded martingales by $\mathcal{M}^p$; that is
\[
\norm{M}_p := \abs{ \sup_{t\in [0,1]} \abs{M_t}}_{L^p(\mathbb{P})} = \EE{ \sup_{t\in [0,1]} \abs{M_t}^p}^{1/p} < \infty.
\]
We have attempted to make the paper as self-contained as possible.
Before presenting the main results of the paper in Sections \ref{s-main} and \ref{s-ito-recover}, we establish the necessary rough path theory and notation in the next section. Particular emphasis is placed on rough differential equations.
The remainder of the paper deals with establishing auxiliary regularity results required for the proof of the main convergence theorem; in particular, the pointwise convergence of the rough Hoff process lift and establishing corresponding maximal $p$-variation bounds. We also prove a tightness property on a given collection of rough Hoff processes.
\begin{Remark}
Throughout the paper, $C_1,C_2,\ldots$ denote various deterministic constants (which may vary from line to line). Their dependence on $\norm{M}_p$ and other similar quantities will be omitted for brevity within proofs but made explicit within theorem statements.
\end{Remark}
\section{Rough path concepts and notation}
In this section we provide a tailored overview of relevant rough path theory and take the opportunity to establish the notation we will need. For a more detailed overview of the theory we direct the reader to
\cite{friz2013short,FV,lejay2003introduction,lyons2002system,lyons1994differential,lyons2004differential} among many others.
\textit{Rough path analysis} introduced by T. Lyons in the seminal article \cite{lyons1994differential} provides a method of constructing solutions to differential equations driven by paths that are not of bounded variation but have controlled \textit{roughness}. The measure of this roughness is given by the $p$-variation of the path (see (\ref{e-p-variation-def}) below).
\subsection{Rough path overview}
Since this paper deals with continuous $\mathbb{R}^d$-valued (and $\mathbb{R}^{2d}$) paths on $[0,1]$ we restrict this brief overview to the finite dimensional case (mainly adopting the notation found in \cite{FV}). For the extension to rough paths over infinite dimensional Banach spaces we refer to \cite{lyons2002system}. We denote the space of such functions by $C\bra{[0,1],\mathbb{R}^d}$. We write $x_{s,t}=x_t-x_s$ as a shorthand for the increments of the path when $x\in C\bra{[0,1],\mathbb{R}^d}$. For $p\geq 1$ we define the following metrics:
\[
\abs{x}_\infty := \sup_{t\in [0,1]} \abs{x_t}, \; \; \;\gap \abs{x}_{p\textup{-var};[0,1]} := \bra{\sup_{D=(t_j)\in\mathcal{D}_{[0,1]}} \sum_{t_j\in D} \abs{x_{t_j, t_{j+1}}}^p}^{1/p},
\]
which we call the uniform norm and $p$-variation semi-norm respectively. We denote by $C^{p\textup{-var}}\bra{[0,1],\mathbb{R}^d}$ the linear subspace of $C\bra{[0,1],\mathbb{R}^d}$ consisting of paths of finite $p$-variation.
In the case of $x\in C^{p\textup{-var}}\bra{[0,1],\mathbb{R}^d}$ where $p\in [1,2)$, the \textit{iterated integrals} of $x$ are canonically defined via Young integration \cite{young1936inequality}. The collection of all these iterated integrals as an object in itself is called the \textit{signature} of the path, given by
\[
S(x)_{s,t} := 1 + \sum^\infty_{k=1} \int_{s<t_1<t_2<\ldots<t_k<t} dx_{t_1} \otimes dx_{t_2} \otimes \ldots \otimes dx_{t_k} \in \oplus^\infty_{k=0} (\mathbb{R}^d)^{\otimes k},
\]
for all $(s,t) \in \Delta_{[0,1]}:= \seq{(s,t) : 0 \leq s \leq t \leq 1}$.
With the convention that $(\mathbb{R}^d)^{\otimes 0} := \mathbb{R}$, we define the tensor algebras:
\[
T^\infty(\mathbb{R}^d):= \bigoplus^\infty_{k=0} (\mathbb{R}^d)^{\otimes k}, \; \; \;\gap\; \; \; T^N(\mathbb{R}^d):= \bigoplus^N_{k=0} (\mathbb{R}^d)^{\otimes k}.
\]
Thus the signature takes values in $T^\infty(\mathbb{R}^d)$. Defining the canonical projection mappings $\pi_N : T^\infty(\mathbb{R}^d) \to T^N(\mathbb{R}^d)$, we can also consider the \textit{truncated signature}:
\begin{align*}
S_N(x)_{s,t} :&= \pi_N\bra{S(x)_{s,t}}
= 1+ \sum^N_{k=1} \int_{s<t_1<t_2<\ldots<t_k<t} dx_{t_1} \otimes dx_{t_2} \otimes \ldots \otimes dx_{t_k} \in T^N(\mathbb{R}^d),
\end{align*}
and thus view $S_N$ as a continuous mapping from $\Delta_{[0,1]}$ into $T^N(\mathbb{R}^d)$.
Throughout this paper we will also reserve the notation $\pi_N$ for the canonical projection of $T^M(\mathbb{R}^d)$ to $T^N(\mathbb{R}^d)$ when $M > N$.
It is a well-known fact that the path $S_N(x)$ takes values in the step-$N$ free nilpotent Lie group with $d$ generators, which we denote by $G^N(\mathbb{R}^d)$. Indeed, defining the free nilpotent step-$N$ Lie algebra $g^N(\mathbb{R}^d)$ by
\[
g^N(\mathbb{R}^d) := \underbrace{\left[ \mathbb{R}^d, \left[ \ldots, \left[\mathbb{R}^d, \mathbb{R}^d \right] \right] \right]}_{(N-1) \textup{ brackets}},
\]
and the natural exponential map $\exp_N : T^N(\mathbb{R}^d) \to T^N(\mathbb{R}^d)$ by
\[
\exp_N(a) = 1 + \sum^N_{k=1} \frac{a^{\otimes k}}{k!},
\]
we define $G^N(\mathbb{R}^d) := \exp_N\bra{g^N(\mathbb{R}^d)}$.
The following characterization establishes the well-known fact (a proof can be found in \cite[Theorem 7.30]{FV}).
\begin{Theorem}[\textbf{Chow's Theorem}]
We have
\[
G^N(\mathbb{R}^d) = \seq{ S_N(x)_{0,1} : x\in C^{1\textup{-var}}\bra{[0,1],\mathbb{R}^d}}.
\]
\end{Theorem}
More generally, if $p\geq 1$ we can consider the set of such group-valued paths
\[
\RP{x}_t = \bra{1,\RP{x}^1_t,\ldots,\RP{x}^{\floor{p}}_t} \in G^{\floor{p}}(\mathbb{R}^d).
\]
Importantly, the group structure provides a natural notion of increment for the signature, namely that $\RP{x}_{s,t} := \RP{x}^{-1}_{s} \otimes \RP{x}_{t}$.
\begin{Example}
Take a path $x\in C^{1\textup{-var}}\bra{[0,1],\mathbb{R}^d}$ and set $\RP{x}:=S_2(x)\in C^{[0,1],G^2(\mathbb{R}^d)}$ for its level-$2$ signature, which we call the (\textbf{level-}$2$) \textbf{rough path lift} of $x$.
An exercise in algebra and calculus confirms that
\[
\RP{x}_{s,t}= 1+ x_{s,t} + \frac{1}{2}x_{s,t}\otimes x_{s,t} + A_{s,t} =\exp_2\bra{x_{s,t}+A_{s,t}},
\]
where $A:\Delta_{0,1}\to\left[\mathbb{R}^d,\mathbb{R}^d\right]$ is the L\'{e}vy area process of $x$. Note that we have used the truncated exponential notation defined above.
\end{Example}
We can describe the set of \textit{norms} on $G^{\floor{p}}(\mathbb{R}^d)$ which are homogeneous with respect to the natural dilation operation on the tensor algebra (see \cite{FV} for more definitions and details).
The subset of these so-called homogeneous norms which are symmetric and sub-additive gives rise to genuine metrics on $G^{\floor{p}}(\mathbb{R}^d)$. We work with the Carnot-Caratheodory norm on $G^N(\mathbb{R}^d)$ given by
\[
\norm{g} := \inf\seq{ \int^1_0 \abs{d\gamma} : \gamma \in C^{1\textup{-var}}\bra{[0,1],\mathbb{R}^d} \text{ and } S_N(\gamma)_{0,1}=g},
\]
which is well-defined by Chow's theorem.
This in turn gives rise to the notion of a homogeneous metric on $G^N(\mathbb{R}^d)$:
\[
d\bra{g,h} = \norm{g^{-1}\otimes h}.
\]
Moreover, this metric gives rise to the following $p$-variation and $\alpha$-H\"{o}lder metrics on the set of $G^{\floor{p}}(\mathbb{R}^d)$-valued paths:
\begin{align}
\dvar{p}\bra{\RP{x},\RP{y}} :&= \bra{ \sup_{D=(t_j)\in \mathcal{D}_{[0,1]}} \sum_{t_j \in D} d\bra{\RP{x}_{t_j,t_{j+1}}, \RP{y}_{t_j,t_{j+1}}}^p}^{1/p}\label{e-p-variation-def}\\
\dhol{\alpha}\bra{\RP{X},\RP{y}} :&= \sup_{0\leq s < t \leq 1} \frac{d\bra{\RP{x}_{s,t}, \RP{y}_{s,t}}}{\abs{t-s}^{1/p}}\notag.
\end{align}
We also define the metrics
\[
d_{0\textup{;}[0,1]}(\RP{x},\RP{y}) := \sup_{0 \leq s < t \leq 1} d\bra{\RP{x}_{s,t}, \RP{y}_{s,t}}, \; \; \;\gap\; \; \; d_{\infty\textup{;}[0,1]}(\RP{x},\RP{y}) := \sup_{t\in [0,1]} d\bra{\RP{x}_{t},\RP{y}_t}.
\]
If no confusion may arise we will often drop the $[0,1]$ appearance in these metrics.
If (\ref{e-p-variation-def}) is finite, then $\omega(s,t):=\norm{\RP{x}}^p_{p\textup{-var;}[s,t]}$ is a control in the following sense:
\begin{Definition}
A mapping $\omega: \Delta_{[0,1]} \to [0,\infty)$ is a \textbf{control} if it is continuous, bounded, vanishes on the diagonal and is super-additive; that is, for all $s<t<u$ in $[0,1]$:
\[
\omega(s,t) + \omega(t,u) \leq \omega(s,u).
\]
\end{Definition}
The space of \textit{weakly geometric} $p$-rough paths, denoted by $WG\Omega_p(\mathbb{R}^d)$, is the set of paths with values in $G^{\floor{p}}(\mathbb{R}^d)$ such that (\ref{e-p-variation-def}) is finite. A refinement of this notion is the space of \textit{geometric} $p$-rough paths, denoted $G\Omega_p(\mathbb{R}^d)$, which is the closure of
\[
\seq{S_{\floor{p}}(x) : x\in C^{1\textup{-var}}\bra{[0,1],\mathbb{R}^d}},
\]
with respect to the topology induced by the rough path metric $\dvar{p}$. Certainly we have the inclusion $G\Omega_p(\mathbb{R}^d) \subset WG\Omega_p(\mathbb{R}^d)$ and it turns out that this inclusion is strict (see \cite[\S3.2.2]{lyons2004differential}).
This paper is concerned with semimartingales (which almost surely have finite $p$-variation for all $p\in (2,3)$ \cite[Theorem 14.9]{FV}). Thus $\floor{p}=2$ and so we are dealing with $p$-rough paths in the step-$2$ group $G^2(\mathbb{R}^d)$. Given a stochastic process $X$ we denote its corresponding rough path lift as $\rp{X}$ (as opposed to the rough path lift of $x$ denoted by $\RP{x}$).
\subsection{Rough differential equations}
In this subsection we introduce rough differential and integral equations. For the more technical topic of rough differential equations \textit{with drift} we refer to the exhaustive \cite[Chapter 12]{FV}. The theory quoted here can be found in greater detail in \cite[Chapter 10]{FV}.
For now let $x\in C^{1\textup{-var}}\bra{[0,1],\mathbb{R}^d}$.
We denote the solution $y$ of the (controlled) ordinary differential equation (ODE)
\begin{equation}\label{e-ode}
dy=V(y)\, dx := \sum^d_{i=1} V_i(y)\, dx^i, \; \; \;\gap y_0 \in \mathbb{R}^e,
\end{equation}
by $\pi_{(V)}\bra{0;y_0,x}$. The notation $\pi_{(V)}\bra{s,y_s;x}$ stands for solutions of (\ref{e-ode}) started at time $s$ from a point $y_s\in\mathbb{R}^e$.
\begin{Definition}[\textbf{RDE}]
Let $\RP{x}\in WG\Omega_p(\mathbb{R}^d)$ for some $p\geq 1$. We say that $y\in C\bra{[0,1],\mathbb{R}^e}$ is a solution to the \textbf{rough differential equation} (shorthand: \textbf{RDE solution}) driven by $\RP{x}$ along the collection of $\mathbb{R}^e$-vector fields $V=\bra{V_i}_{i=1,\ldots,d}$ and started at $y_0$ if there exists a sequence $(x_n)_n$ in $C^{1\textup{-var}}\bra{[0,1],\mathbb{R}^d}$ such that:
\begin{enumerate}[label=\textit{(\roman{enumi})}, ref=(\roman{enumi})]\addtolength{\itemsep}{0.4\baselineskip}
\item $\lim_{n\to\infty} d_{0;[0,1]}\bra{S_{\floor{p}}(x_n),\RP{x}}=0$;
\item $\sup_n \pnorm{S_{\floor{p}}\bra{x_n}}{p}<\infty$;
\end{enumerate}
and ordinary differential equations (ODE) solutions $y_n=\pi_{(V)}\bra{0,y_0;x_n}$ such that
\[
y_n \to y \text{ uniformly on } [0,1] \text{ as } n\to\infty.
\]
We denote this situation with the (formal) equation:
\[
dy=V(y)\, d\RP{x},\; \; \;\gap y_0\in \mathbb{R}^e,
\]
which we refer to as a rough differential equation.
\end{Definition}
\begin{Remark}
The RDE solution map
\[
\RP{x} \in C^{p\textup{-var}}\bra{[0,1],G^{\floor{p}}(\mathbb{R}^d)} \mapsto \pi_{(V)}\bra{0,y_0;\RP{x}} \in C\bra{[0,1],G^{\floor{p}}(\mathbb{R}^e)}
\]
is known in the rough path literature as the \textbf{It\^{o}-Lyons map}.
\end{Remark}
Armed with the definition of a rough differential equation, it is natural to ask what a rough \textit{integration} definition would be. To this end we follow Lyons' original approach in \cite{lyons1994differential}.
To be precise, we wish to make sense of the (formal) integral equation
\begin{equation}\label{e-integral-e}
\int^\cdot_0 \varphi(z)\, d\RP{z} \text{ with } z=\pi_1\bra{\RP{z}},
\end{equation}
where $z\in WG\Omega_p(\mathbb{R}^d)$.
Certainly in the classical case of $\RP{z}=S_{\floor{p}}(z)$ for some $z\in C^{1\textup{-var}}\bra{[0,1],\mathbb{R}^d}$, we want (\ref{e-integral-e}) to coincide with $S_{\floor{p}}\bra{\xi}$, where $\xi$ is the classical Riemann-Stieltjes integral $\int^\cdot_0 \varphi(z)\, dz$. We take the following definition straight from \cite[Definition 10.44]{FV}:
\begin{Definition}[\textbf{Rough integration}]
Let $\RP{x}\in WG\Omega_p(\mathbb{R}^d)$ and $\varphi=(\varphi_i)_{i=1}^d$ be a collection of maps from $\mathbb{R}^d$ to $\mathbb{R}^e$.
We say that $\RP{y}\in C\bra{[0,1],G^{\floor{p}}(\mathbb{R}^e)}$ is a \textbf{rough path integral} of $\varphi$ along $\RP{x}$ if there exists a sequence $(x^n)\subset C^{1\textup{-var}}\bra{[0,1],\mathbb{R}^d}$ such that:
\begin{enumerate}[label=\textit{(\roman{enumi})}, ref=(\roman{enumi})]\addtolength{\itemsep}{0.4\baselineskip}
\item $x^n_0=\pi_1(\RP{x}_0)$ for all $n$;
\item\label{i-two} $\lim_{n\to\infty} d_{0;[0,1]}\bra{S_{\floor{p}}\bra{x^n},\RP{x}}=0$;
\item\label{i-three} $\sup_n \pnorm{S_{\floor{p}}\bra{x^n}}{p}<\infty$;
\item $\lim_{n\to\infty} d_\infty\bra{S_{\floor{p}}\bra{\int^\cdot_{0} \varphi\bra{x^n_u}\, dx^n_u}, \RP{y} } = 0$.
\end{enumerate}
We write $\int \varphi(x)\, d\RP{x}$ for the set of rough path integrals of $\varphi$ along $\RP{x}$ (without additional regularity assumptions there could be more than one suitable candidate).
\end{Definition}
From RDE solutions we can construct \textit{full RDE solutions}; in particular, we construct a ``full'' solution as a weak geometric $p$-rough path in its own right. This in turn will allows us to use a solution to a first RDE as the driving signal for a second RDE. Correspondingly, RDE solutions can then be used as ``integrators'' in rough integrals. This has more practical implications than mere aesthetics of the theory. In Lyons' original work \cite{lyons1994differential} the existence and uniqueness of RDEs was established via Picard iteration which critically relied on the notion of ``full'' RDE solutions.
\begin{Definition}[\textbf{Full RDEs}]
As usual, let $\RP{x}\in WG\Omega_p(\mathbb{R}^d)$ for some $p\geq 1$.
We define $\RP{y}\in C\bra{[0,1],G^{\floor{p}}(\mathbb{R}^e)}$ to be the solution of the \textbf{full RDE} driven by $\RP{x}$ along the vector fields $V=(V_i)^d_{i=1}$ and started at $\RP{y}_0\in G^{\floor{p}}(\mathbb{R}^e)$ if there exists a sequence $(x^n)_n \subset C^{1\textup{-var}}\bra{[0,1],\mathbb{R}^d}$ such that \ref{i-two} and \ref{i-three} hold, and ODE solutions $y_n \in \pi_{(V)}\bra{0,\pi_1\bra{\RP{y}_0};x^n}$ such that
\[
\RP{y}_0 \otimes S_{\floor{p}}\bra{y^n} \text{ converges uniformly to } \RP{y} \text{ as } n\to\infty.
\]
We write the corresponding (formal) equation as $d\RP{y} = V(y)\, d\RP{x}$.
\end{Definition}
\begin{Remark}
A note on regularity: consider RDEs driven by a path $\RP{x}$ in $WG\Omega_p(\mathbb{R}^d)$ along a collection of vector fields $V=(V^1,\ldots,V^d)$ on $\mathbb{R}^e$. Considerations of existence and uniqueness of such RDEs show that the appropriate way to measure the regularity of the collection $V$ turns out to be notion of $\gamma$-Lipschitzness (denoted $\textup{Lip}^\gamma$) in the sense of Stein. See \cite{FV,lyons2004differential} to note the contrast with the classical notion of Lipschitzness.
\end{Remark}
\section{Rough Hoff process convergence}\label{s-main}
The next theorem presents the first main result of the paper, namely the convergence of the rough path lift $\rp{X}^D$ of the Hoff process $X^D$ under the $p$-variation topology. Define the process $\bar{X}_t = (X_t,X_t)$ and use the standard notation of $\qv{M} := \bra{ \qv{M^i,M^j} : i,j = 1,\ldots,d}$ for the quadratic variation of $M$ as a $\mathbb{R}^{d\times d}$-valued process.
\begin{Theorem}[\textbf{Rough Hoff process convergence}]\label{t-hoff-process-convergence}
Let $X=M+V:[0,1]\to\mathbb{R}^d$ be a continuous semimartingale with associated Hoff process $X^D$ for a given partition $D\in \mathcal{D}_{[0,1]}$. Then for all $p>2$ we have
\[
\dvar{p}\bra{\rp{X}^{D}, \rp{X}^\infty} \to 0 \text{ in probability as } \abs{D}\to 0,
\]
where $\rp{X}^\infty \in C\bra{[0,1],G^2(\mathbb{R}^{2d})}$ is defined by
\[
\rp{X}^\infty_{s,t} = \exp_2\bra{\bar{X}_{s,t} +
\left[\begin{array}{cc}
A^X_{s,t} & A^X_{s,t}-1/2\qv{X}_{s,t}\\
A^X_{s,t}+1/2\qv{X}_{s,t} & A^X_{s,t}
\end{array}\right]
}.
\]
If in addition the finite variation process $V$ is bounded in $L^q(\mathbb{P})$ ($q\geq 1$) and $M\in\mathcal{M}^p$, then the convergence also holds in $L^{p\wedge q}(\mathbb{P})$.
\end{Theorem}
\begin{proof}
First we prove the theorem for $V=0$. Assuming $M\in\mathcal{M}^p$ it follows from Proposition \ref{p-maximal-p-variation-bound} that the rough path collection $\seq{\rp{M}^D: D\in \mathcal{D}_{[0,1]}}$ satisfies uniform $p$-variation bounds in $L^p(\mathbb{P})$.
Since $\abs{D}\to 0$, the collection also satisfies the tightness property of Corollary \ref{c-tightness}. Thus the second convergence statement follows from an application of \cite[Corollary 50]{friz2007differential} . In the case that $M\notin \mathcal{M}^p$ we obtain convergence in probability by a simple localization argument (as done in \cite[Theorem 14.16]{FV}).
We now turn to the general case where $V\neq 0$ and define $\bar{V} :[0,1]\to\mathbb{R}^{2d}$ by $\bar{V}_t=(V_t,V_t)$. In contrast to the martingale rough path convergence, the corresponding convergence proof for the finite variation process is straightforward:
\begin{Lemma}\label{l-fv-convergence}
For any $\delta>0$:
\[
\dvar{(1+\delta)}\bra{V^{D},\bar{V}} \to 0 \text{ in probability as } \abs{D}\to 0.
\]
If in addition $V$ is bounded in $L^q(\mathbb{P})$ for some $q\geq 1$ then the convergence also holds in $L^q(\mathbb{P})$.
\end{Lemma}
\begin{proof}
The first convergence follows readily from \cite[Proposition 1.28]{FV} and interpolation (see \cite[\S14.1]{FV} for details). The stronger convergence in $L^q(\mathbb{P})$ is a consequence of the uniform $L^q(\mathbb{P})$-bound assumption \cite[Theorem 4.14]{chung2001course}.
\end{proof}
Returning to the proof of Theorem \ref{t-hoff-process-convergence}, it is a straight-forward exercise in algebra to confirm that $\rp{X}^\infty_{s,t} = T_{\bar{V}}\bra{\rp{X}^\infty_{s,t}}$ using the translation operator defined in \cite[\S9.4.6]{FV}. The theorem then follows immediately from the continuity of the translation operator \cite[Corollary 9.35]{FV}.
\end{proof}
\begin{Remark}
The quadratic variation of $M$ naturally appears as we consider the limiting area of the rough path lift $\rp{X}^D$. This non-linear phenomena was first observed by Hoff in \cite{hoff2005brownian}.
\end{Remark}
\begin{Remark}
A natural question to ask is whether the convergence result of Theorem \ref{t-hoff-process-convergence} will hold using the stronger $\alpha$-H\"{o}lder variation metric ($\alpha <1/2$), rather than the $p$-variation metric.
The answer is no in general; a consequence of the behaviour of the Hoff process $X^D$ over consecutive intervals. In particular, over a given interval $[t_i,t_{i+1}] \subset D \in \mathcal{D}_{[0,1]}$, $X^D$ is defined by the values of $X$ at the previous, present and future times: $\seq{X_{t_{i-1}},X_{t_i}, X_{t_{i+1}},X_{t_{i+2}}}$. Thus for particular sequences of partitions $(D_n)_{n\in\mathbb{N}}$ with $\abs{D_n} \to 0$ but
\begin{equation}\label{e-david-kelly}
\sup_{n\in\mathbb{N}} n \abs{D_n} = \infty,
\end{equation}
(where $D_n$ has $n$ non-zero points), the resulting Hoff process sequence may not be uniformly bounded in H\"{o}lder norm (under expectation in $L^p(\mathbb{P})$). Thus we can certainly not expect its rough path lift $\rp{X}^D$ to converge to $\rp{X}^\infty$ in $\alpha$-H\"{o}lder norm ($\alpha < 1/2$), especially since the latter process can be shown to satisfy $\EE{\normhol{\rp{X}^\infty}{\alpha}^p} < \infty$ for all $p\geq 1$.
It remains to give an example where (\ref{e-david-kelly}) holds and the H\"{o}lder-norm of $X^{D_n}$ explodes as $n\to\infty$. To this end suppose that $X$ is just a Brownian motion.
Denoting the $n$th level dyadic partition by $\mathcal{D}_n = \bra{t^n_{k} = k/2^n}_{k=0}^{2^n}$, we set
\[
D_n := \bra{ [0,1/2] \cap \mathcal{D}^{2^n}} \cup \bra{[1/2,1] \cap \mathcal{D}_n}.
\]
We find that for $\alpha < 1/2$:
\[
\EE{\normhol{\rp{X}^{D_n}}{\alpha}^p} \geq \frac{ \EE{\abs{X_{1/2+2^{-n}} - X_{1/2}}^p}}{2^{-p2^n}} \to \infty,
\]
as $n\to\infty$.
It is well known in rough path circles that in order to use the H\"{o}lder metric on discrete rough path sequences, one must insist that the quantity (\ref{e-david-kelly}) is finite (see the recent paper \cite{kelly2014rough}).
\end{Remark}
\section{Recovery of the It\^{o} integral}\label{s-ito-recover}
We now consider the second main result of the paper:
\begin{Theorem}[\textbf{Recovery of the It\^{o} integral}]\label{t-ito-sde}
Let $X:[0,1]\to\mathbb{R}^d$ be a continuous semimartingale with canonical decomposition $X=M+V$.
Let $f=(f_i)_{i=1}^d$ be a collection of $\textup{Lip}^\gamma(\mathbb{R}^d,\mathbb{R}^e)$ mappings where $\gamma > 2$.
For each $n$ let $Y^n$ be the solution to the random ODE:
\begin{equation}\label{e-delay}
dY^n_t = f\bra{X_t^{n,b}}\, dX^{n,f}_t = \sum^d_{i=1} f_i\bra{X^{n,b}_t}\, dX^{n,f;i}_t, \; \; \;\gap Y^n_0 = y_0 \in \mathbb{R}^e,
\end{equation}
and let $Y$ be the standard It\^{o} integral
\[
dY_t = f\bra{X_t}\, dX_t = \sum^d_{i=1} f_i\bra{X_t}\, dX^i_t, \; \; \;\gap Y_0 = y_0 \in \mathbb{R}^e.
\]
Then for all $p\in (2,\gamma)$,
\[
\dvar{p}\bra{Y^n,Y}\to 0 \text{ in probability as } n\to\infty.
\]
If in addition the finite variation process $V$ of $X$ is bounded in $L^q(\mathbb{P})$ ($q\geq 1$) and $M\in\mathcal{M}^p$, then the convergence also holds in $L^{p\wedge q}(\mathbb{P})$.
\end{Theorem}
Before presenting the proof of Theorem \ref{t-ito-sde}, we set up some additional notation. Denote the rough path lift of $\bar{X_t}=(X_t,X_t)$ by $\rp{\bar{X}}$. Thus $\rp{\bar{X}}: \Delta_{[0,1]} \to G^2(\mathbb{R}^{2d})$ with
\begin{align*}
\rp{X}_{s,t} :&= 1 +
\left[\begin{array}{c}
X_{s,t} \\ X_{s,t}
\end{array}\right]
+ \frac{1}{2} \left[\begin{array}{c}
X_{s,t} \\ X_{s,t}
\end{array}\right] \otimes \left[\begin{array}{c}
X_{s,t} \\ X_{s,t}
\end{array}\right]
+
\left[\begin{array}{cc}
A_{s,t} & A_{s,t}\\
A_{s,t} & A_{s,t}
\end{array}\right],
\end{align*}
where $A : \Delta_{[0,1]} \to \left[\mathbb{R}^d,\mathbb{R}^d\right]$ is the L\'{e}vy area process of $X$.
Importantly we have $\rp{X}^\infty = \rp{\bar{X}}^{\psi}$; that is $\rp{X}^\infty$ is the corollary from perturbing $\rp{\bar{X}}$ by the map $\psi:\Delta_{[0,1]} \to [\mathbb{R}^{2d}, \mathbb{R}^{2d}]$:
\begin{align*}
\rp{X}^\infty = \rp{\bar{X}}_{s,t} + \psi_{s,t}
&= \exp_2\bra{\bar{X}_{s,t} +
\left[\begin{array}{cc}
A_{s,t} & A_{s,t}\\
A_{s,t} & A_{s,t}
\end{array}\right]
} + \psi_{s,t}
= \exp_2\bra{\bar{X}_{s,t} +
\left[\begin{array}{cc}
A_{s,t} & A_{s,t}\\
A_{s,t} & A_{s,t}
\end{array}\right]
+ \psi_{s,t} },
\end{align*}
where
\[
\psi_{s,t} := \left[\begin{array}{cc}
0 & -\frac{1}{2}\qv{X}_{s,t}\\
\frac{1}{2}\qv{X}_{s,t} & 0
\end{array}\right] \in \left[ \mathbb{R}^{2d}, \mathbb{R}^{2d} \right].
\]
The appearance of the covariation term $\psi$ in the limit $\rp{X}^\infty$ of Theorem \ref{t-hoff-process-convergence} allows us to recover the It\^{o} integral from the Stratonovich integral limit; the drift terms cancel out.
We use the notation $\rho : \mathbb{R}^d \oplus \mathbb{R}^e \to \mathbb{R}^e$ for the canonical projection.
\begin{proof}[Proof of Theorem \ref{t-ito-sde}]
We consider a more complicated SDE on the larger space $\mathbb{R}^d \oplus \mathbb{R}^e$ and denote the standard bases of $\mathbb{R}^d$ and $\mathbb{R}^e$ by $(\hat{e}_i)_{i=1}^d$ and $(\bar{e}_j)_{j=1}^e$ respectively. In particular, set $z=(\hat{z}, \bar{z}) \in \mathbb{R}^d \oplus \mathbb{R}^e$ and define the vector fields on $\mathbb{R}^d \oplus \mathbb{R}^e$ by
\[
Q_i(z)=(\hat{e}_i,0), \; \; \;\gap W_k(z)= \sum^e_{j=1} (0, \bar{e}_j)f^j_k(\hat{z}),
\]
for $i,k=1,\ldots,d$ (we have used the standard notation convention: $f_i(x)=(f_i^1(x), \ldots, f_i^e(x)) \in \mathbb{R}^e$ for $x\in \mathbb{R}^d$).
Let $z^n=(\hat{z}^{n,1},\ldots, \hat{z}^{n,d}, \bar{z}^{n,1}, \ldots, \bar{z}^{n,e})$ be the solution to the SDE
\[
dz^n_t= \sum^d_{i=1}\bra{ Q_i(z^n_t)\, dX^{n,b;i}_t + W_i(z^n_t)\, dX^{n,f;i}_t}, \; \; \;\gap z^n_0 = (0,y_0) \in \mathbb{R}^d \oplus \mathbb{R}^e.
\]
Since the axis path $X^n$ is piecewise linear, we could equivalently formulate the above SDE into its Stratonovich form without incurring a corrective drift term.
It follows that the projection of $z^n$ onto $\mathbb{R}^e$, $\bar{z}^n := \rho(z^n)$, satisfies the SDE:
\[
d\bar{z}^{n,j}_t = \sum^d_{i=1} W_i^j\bra{z^n_t}\, dX^{n,f;i}_t = \sum^d_{i=1} f^j_i\bra{\hat{z}^n_t}\, dX^{n,f;i}_t.
\]
Moreover, \cite[Theorem 17.3]{FV} tells us that $z^n=\pi_{(Q,W)}\bra{0, (0,y_0); \rp{X}^n}$ almost surely.
A simple computation confirms that the only non-zero Lie brackets between the vector fields $Q_{i_1}, Q_{i_2}, W_{k_1}$ and $W_{k_2}$ are of the form
\[
\left[Q_i, W_k\right](\hat{z}) = Q_iW_k(\hat{z}) = \sum^e_{j=1} (0,\bar{e}_j) \partial_i f^j_k(\hat{z}),
\]
for all $i,k=1,\ldots, d$. We denote this collection of vector fields by
\[
L:=\bra{Q_iW_k : i, k \in \seq{1,\ldots, d}}.
\]
Consider the SDE:
\begin{align*}
dz_t = (Q+W)(z_t)\, dX_t &= (Q+W)(z_t) \circ dX_t - \frac{1}{2}\sum^d_{i,k=1} {Q_iW_k(z_t)\, d\ip{X^i}{X^k}_t}\\
&= (Q+W)(z_t)\circ dX_t - \frac{1}{2}\sum^d_{i,k=1} \left[ Q_i, W_k\right](z_t)\, d\ip{X^i}{X^k}_t,
\end{align*}
with $z_0 = (0, y_0) \in \mathbb{R}^d \oplus \mathbb{R}^e$.
It follows from \cite[Theorem 17.3]{FV} that almost surely:
\[
z=\pi_{(Q,W;L)}\bra{0,(0,y_0); (\rp{X}, \psi)}.
\]
As noted above, $\bar{\rp{X}}^\psi=\rp{X}^\infty$ and so the perturbation theorem of \cite[Theorem 12.14]{FV} gives the equality:
\begin{align*}
z&=\pi_{(Q,W;L)}\bra{0, (0,y_0); (\rp{X}, \psi)}
= \pi_{(Q,W)}\bra{0, (0,y_0); \bar{\rp{X}}^\psi}
= \pi_{(Q,W)}\bra{0, (0,y_0); \rp{X}^\infty}.
\end{align*}
In light of the convergence result of Theorem \ref{t-hoff-process-convergence}, we conclude that
\[
\forall p> 2,\,\,\, \dvar{p}\bra{z^n,z} \to 0 \text{ in probability (or in } L^{p\wedge q}(\mathbb{P}) \text{) as } n\to\infty.
\]
Indeed, as noted in the above remark, almost sure equality and convergence in probability and $L^p(\mathbb{P})$ are all preserved under continuous mappings. Thus the It\^{o}-Lyons map is continuous with respect to the driving signal under the $p$-variation topology.
The proof is then finished by noting that $Y^n=\rho(z^n)$ and $Y=\rho(z)$.
\end{proof}
\begin{Remark}
Such random ODEs have a natural interpretation in the context of financial mathematics and similar lines of research have been considered in the recent papers \cite{friz_examples,gergely2013extracting,learning}.
Indeed, we could consider $X$ to be a $d$-dimensional stream of continuous financial data (which we assume is in the form of a semimartingale indexed over the unit interval $[0,1]$).
As noted by Friz in \cite{friz_examples}, the ODE (\ref{e-delay}) could be specifically interpreted as the investment strategy of an agent with \textit{delayed} access to a discrete sampling of the market data.
An interesting question to ask then is what the limiting behaviour of this SDE is when our agent learns more and more information on smaller timescales? This reduced latency means the agent would have access to increasingly more data points, thus reducing the delay between the backward and forward processes of $X^D$.
Theorem \ref{t-ito-sde} tells us that the limiting solution is the It\^{o} integral and not the Stratonovich formulation suggested by the classical Wong-Zakai theorem \cite{eugene1965relation}.
\end{Remark}
The remainder of the paper is devoted to establishing the technical results referred to in the proof of Theorem \ref{t-hoff-process-convergence}.
\section{Pointwise convergence of $\rp{M}^D$ in $L^p(\mathbb{P})$}
In this section we prove the pointwise convergence of $\rp{M}^D$ to $\rp{M}^\infty$ in $L^p(\mathbb{P})$ for $p>2$ as $\abs{D}\to 0$. We also provide the rate of this convergence.
We first prove the pointwise convergence in $L^p(\mathbb{P})$ of the increment process $M^{D}$ to $\bar{M}$ as $\abs{D}\to 0$, where $\bar{M}_t=(M_t,M_t)\in\mathbb{R}^{2d}$. But before this we give a small technical lemma which we will apply throughout this section.
\begin{Lemma}\label{l-holder-play}
Let $X\in \mathcal{M}^c_{0,loc}\bra{[0,1],\mathbb{R}}$ and $D=(t_i)\in\mathcal{D}_{[0,1]}$. For all $p>2$
\begin{equation}\label{e-the-bound}
\EE{ \sum_{t_i\in D} \qv{X}^{p/2}_{t_i,t_{i+1}}}
\leq
\abs{\qv{X}_{0,1}}_{L^{p/2}(\mathbb{P})} \EE{\sup_{t_i\in D}\qv{X}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{p}}
\end{equation}
\end{Lemma}
\begin{proof}
Applying the H\"{o}lder inequality with $q=p/2$ and $r=p/(p-2)$ (so that $1/q+1/r=1$) gives:
\begin{align*}
\EE{ \sum_{t_i\in D} \qv{X}^{p/2}_{t_i,t_{i+1}}}
\leq \EE{ \qv{X}_{0,1} \sup_{t_i\in D} \qv{X}_{t_i,t_{i+1}}^{p/2-1}}
&\leq \EE{\qv{X}_{0,1}^{p/2}}^{2/p} \EE{\sup_{t_i\in D}\qv{X}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{p}}\\
&= \abs{\qv{X}_{0,1}}_{L^{p/2}(\mathbb{P})} \EE{\sup_{t_i\in D}\qv{X}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{p}}.
\end{align*}
Recalling that $\EE{\qv{X}_{0,1}^{p/2}}^{1/p} = \norm{M}_p$, the proof is complete.
\end{proof}
\begin{Remark}
If $p=2$ in (\ref{e-the-bound}) then the lemma reduces to a trivial equality. For our purposes of proving pointwise convergence we need the right-hand side of (\ref{e-the-bound}) to converge to $0$ as $\abs{D}\to 0$ and so it would appear that we really do need $M\in\mathcal{M}^p\bra{[0,1],\mathbb{R}^d}$.
\end{Remark}
\begin{Lemma}[\textbf{Pointwise convergence of } $M^D$]\label{l-increment-convergence}
For all $(s,t)\in\Delta_{[0,1]}$, $M^D_{s,t} \to \bar{M}_{s,t}$ in $L^p(\mathbb{P})$ as $\abs{D}\to 0$. Moreover, there exists a constant $C=C(p,\norm{M}_p)$ such that the rate of convergence is given by
\[
\abs{ M^D_{s,t}-\bar{M}_{s,t}}_{L^p(\mathbb{P})} \leq C \EE{ \sup_{t_i\in D} \qv{M}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{p^2}}.
\]
\end{Lemma}
\begin{proof}
The proof is quick: suppose $D=(t_i)$ and select a coordinate $j\in\seq{1,\ldots d}$. For $t\in [t_i,t_{i+1})$ we find that Lemma \ref{l-holder-play} gives
\begin{align*}
\EE{\abs{M_t^{D,b;j}-M_t^j}^p}
&\leq 2^{p-1}\EE{ \sup_{r\in [t_i,t_{i+1}]} \abs{M^j_r-M^j_{t_i}}^p + \sup_{s\in [t_{i-1},t_i]} \abs{M^j_s-M^j_{t_i}}^p }\\
&\leq C_1 \sum_{t_i \in D} \EE{ \sup_{s\in [t_i,t_{i+1}]} \abs{M^j_s-M^j_{t_i}}^p}
= C_1 \EE{\sum_{t_i \in D} \qv{M^j}^{p/2}_{t_i,t_{i+1}}}\\ &\leq C_2 \EE{ \sup_{t_i\in D} \qv{M^j}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{p}}
\end{align*}
The identical bound holds for the forward process.
The lemma follows.
\end{proof}
The next proposition establishes the pointwise convergence of the area process of $\rp{M}^D$.
\begin{Proposition}[\textbf{Pointwise convergence of }$A^D$]\label{p-area-pointwise-convergence}
Fix $\theta,\lambda \in \seq{b,f}$ and $i,j\in \seq{1,\ldots,d}$. For all $(s,t)\in\Delta_{[0,1]}$ we have the following convergence in $L^{p/2}(\mathbb{P})$ as $\abs{D}\to0$:
\[
A_{s,t}^{D,\theta,\lambda;i,j}\to \hat{A}^{\theta,\lambda;i,j}_{s,t} := \begin{cases}
A_{s,t}^{M;i,j} & \;\;\;\;\;\;\;\text{ if } \theta = \lambda;\\
A_{s,t}^{M;i,j}-\frac{1}{2}\ip{M^{i}}{M^{j}}_{s,t} & \;\;\;\;\;\;\;\text{ if } \theta = b, \; \lambda=f;\\
A_{s,t}^{M;i,j} + \frac{1}{2}\ip{M^{i}}{M^{j}}_{s,t} & \;\;\;\;\;\;\;\text{ if } \theta = f, \; \lambda=b.
\end{cases}
\]
Moreover, there exists a constant $C=C(p,\norm{M}_p)$ such the rate of convergence is given by
\begin{align*}
\abs{\abs{A_{s,t}^{D} - \hat{A}_{s,t}}^{1/2}}_{L^{p}(\mathbb{P})} &\leq
C \EE{ \sup_{t_i\in D} \qv{M}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{2p^2}}.
\end{align*}
\end{Proposition}
The main result of this section is the following:
\begin{Corollary}[\textbf{Pointwise convergence in $L^p(\mathbb{P})$ of $\rp{M}^{D}$}]\label{c-pointwise-convergence}
For all $(s,t)\in\Delta_{[0,1]}$ we have
\[
d\bra{{\rp{M}^{D}_{s,t}}, \rp{M}^\infty_{s,t}} \to 0 \text{ in } L^p(\mathbb{P}) \text{ as } \abs{D} \to 0.
\]
Moreover, there exists a constant $C=C(p,\norm{M}_p)$ such that
\begin{align*}
\abs{d\bra{\rp{M}^{D}_{s,t}, \rp{M}^\infty_{s,t}} }_{L^p(\mathbb{P})}
&\leq
C \bra{ \EE{ \sup_{t_i\in D} \qv{M}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{2p^2}} \vee \EE{ \sup_{t_i\in D} \qv{M}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{p^2}}}.
\end{align*}
\end{Corollary}
\begin{proof}
The claim follows immediately from combining Lemma \ref{l-increment-convergence} and Proposition \ref{p-area-pointwise-convergence}. Indeed,
\begin{align*}
d\bra{\rp{M}^{D}_{s,t}, \rp{M}^\infty_{s,t}} &\simeq \abs{\pi_1\bra{\rp{M}^{D}_{s,t} - \rp{M}^\infty_{s,t}}} \vee \abs{\pi_2\bra{\rp{M}^{D}_{s,t},\rp{M}^\infty_{s,t}}}^{1/2}
\simeq \abs{M^{D}_{s,t} - \bar{M}_{s,t}} \vee \abs{A^{D}_{s,t}-\hat{A}_{s,t}}^{1/2}.
\end{align*}
The proof follows.
\end{proof}
It remains to prove Proposition \ref{p-area-pointwise-convergence} and we devote the remainder of this section to its proof. To this end we now consider two small technical lemmas.
\begin{Lemma}\label{l-technical-lemma}
Let $X,Y \in \mathcal{M}^c_{0,loc}\bra{[0,1],\mathbb{R}}$ and fix a partition $D=(t_i)_{i=0}^n\in\mathcal{D}_{[0,1]}$. Then there exists a constant $C=C(p, \norm{X}_p, \norm{Y}_p))$ for all $p>2$,
\begin{align*}
&\EE{ \abs{ \sum^{n-2}_{i=0} (\X{i+1}-\X{i})(\Y{i+2}-\Y{i+1})}^{p/2}}
\leq C \EE{ \sup_{t_i \in D} \qv{X}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{2p}}.
\end{align*}
\end{Lemma}
\begin{proof}
The claimed inequality is a consequence of the Burkholder-Gundy-Davis and Cauchy-Schwarz inequalities.
Define the previsible process
\[
Z_r=\sum^{n-2}_{i=0} (\X{i+1}-\X{i})\textbf{\textup{1}}\seq{r\in (t_{i+1},t_{i+2}]}.
\]
We have
\[
\sum^{n-2}_{i=0} (\X{i+1}-\X{i})(\Y{i+2}-\Y{i+1}) = \int^{1}_{0} Z_r\, dY_r,
\]
which is a well defined local martingale since $Z$ is previsible. It follows from the aforementioned inequalities that
\begin{align*}
\EE{ \abs{ \int^{1}_{0} Z_r\, dY_r }^{p/2}}
\leq C_1 \EE{ \abs{ \int^{1}_{0} \abs{Z_r}^2\, d\qv{Y}_r}^{p/4}}
&\leq C_1 \EE{ \qv{Y}_{0,1}^{p/4} \sup_{r\in [0,1]} \abs{Z_r}^{p/2}}\\
&\leq C_1 \EE{\qv{Y}_{0,1}^{p/2}}^{1/2}\EE{\sup_{r\in [0,1]}\abs{Z_r}^p}^{1/2}.
\end{align*}
Since $\sup_{r\in [0,1} \abs{Z_r}^p \leq \sum^{n-1}_{i=0} \abs{\X{i+1} - \X{i}}^p$, applying Lemma \ref{l-holder-play} gives
\begin{align*}
\EE{\sup_{r\in [0,1]}\abs{Z_r}^p}
&\leq \EE{ \sum^{n-1}_{i=0} \abs{\X{i+1}-\X{i}}^p}
\leq C_2 \EE{ \sum^{n-1}_{i=0} \qv{X}_{t_i,t_{i+1}}^{p/2}}\\
&\leq C_2 \abs{\qv{X}_{0,1}}_{L^{p/2}(\mathbb{P})} \EE{\sup_{t_i\in D}\qv{X}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{p}}.
\end{align*}
The result follows at once.
\end{proof}
The next lemma uses the same techniques to derive the same rate of convergence in $L^{p/2}(\mathbb{P})$ for discrete approximations to the It\^{o} integral.
\begin{Lemma}\label{l-new-technical-lemma}
Suppose $X,Y\in\mathcal{M}^c_{0,loc}\bra{[0,1],\mathbb{R}}$. There exists a constant $C=C(p,\norm{X}_p,\norm{Y}_p)$ such that for any given partition $D=(t_i) \in \mathcal{D}_{[0,1]}$, we have
\begin{align*}
&\EE{ \abs{ \sum_{t_i \in D} \X{i}(\Y{i+1}-\Y{i}) - \int^1_0 X_r \, dY_r}^{p/2}} \leq C \EE{ \sup_{t_i\in D} \qv{X}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{2p}}.
\end{align*}
\end{Lemma}
\begin{proof}
Defining $Z_r := \sum_{t_i \in D} \X{i}\textbf{\textup{1}}\seq{r\in (t_i,t_{i+1}]}$, the standard argument gives
\begin{align*}
&\EE{ \abs{ \sum_{t_i \in D} \X{i}(\Y{i+1}-\Y{i}) - \int^1_0 X_r \, dY_r}^{p/2}}
=\EE{ \abs{ \int^1_0 (Z_r - X_r)\, dY_r}^{p/2}}\\
&\; \; \;\leq C_1 \EE{ \abs{\int^1_0 \abs{Z_r-X_r}^2\, d\qv{Y}_r}^{p/4}}
\leq C_1 \EE{ \sup_{r\in [0,1]} \abs{Z_r-X_r}^{p/2} \qv{Y}_{0,1}^{p/4}}\\
&\; \; \;\leq C_1 \EE{ \qv{Y}_{0,1}^{p/2}}^{1/2} \EE{ \sup_{r\in [0,1]} \abs{Z_r-X_r}^p}^{1/2}\\
&\; \; \;= C_2 \EE{ \sup_{r\in [0,1]} \sum_{t_i\in D} \abs{\X{i}-X_r}^p\textbf{\textup{1}}\seq{r\in (t_i, t_{i+1}]}}^{1/2}.
\end{align*}
Noting the inequality
\[
\sup_{r\in [0,1]} \sum_{t_i\in D} \abs{\X{i}-X_r}^p\textbf{\textup{1}}\seq{r\in (t_i, t_{i+1}]} \leq \sum_{t_i\in D} \sup_{r\in [t_i,t_{i+1}]} \abs{\X{i}-X_r}^p,
\]
we have
\begin{align*}
\EE{ \sup_{r\in [0,1]} \sum_{t_i\in D} \abs{\X{i}-X_r}^p\textbf{\textup{1}}\seq{r\in (t_i, t_{i+1}]}}
&\leq \sum_{t_i\in D} \EE{ \sup_{r\in [t_i,t_{i+1}]} \abs{\X{i}-X_r}^p}
\leq C_3 \sum_{t_i\in D} \EE{ \qv{X}^{p/2}_{t_i,t_{i+1}}}
\end{align*}
The proof then follows from Lemma \ref{l-holder-play}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{p-area-pointwise-convergence}]
Recall the standard area decomposition formula; for $s<t<u$ we have
\[
A_{s,u} = A_{s,t} + A_{t,u} + \frac{ \left[ M_{s,t} \wedge M_{u,t}\right]}{2},
\]
where for vectors $a,b\in\mathbb{R}^d$, $a\wedge b$ denotes the $d\times d$ antisymmetric matrix with entries $(a\wedge b)_{i,j}=a_ib_j - a_jb_i$.
From this formula and the fact that the increment process $M^D$ converges pointwise in $L^p(\mathbb{P})$ to $\bar{M}$ (Lemma \ref{l-increment-convergence}), it suffices to only consider the convergence of $A^D_{0,t_k}$ for $t_k\in D$, as $\abs{D}\to 0$.
For ease of notation we select a pair of coordinates $i,j\in\seq{1,\ldots,d}$ and set $X=M^i$ and $Y=M^j$ (this also frees up $i,j$ for indices notation).
We first consider the situation when $\theta=\lambda=b$. For $D = (t_i) \in \mathcal{D}_{[0,1]}$ we have
\[
A^{D,b,b;i,j}_{0,t_k} = \frac{1}{2}\sum^{k-1}_{i=1} \bra{ \X{i-1}(\Y{i}-\Y{i-1}) - \Y{i-1}(\X{i}-\X{i-1})},
\]
and thus Lemma \ref{l-new-technical-lemma} gives
\[
A^{D,b,b;i,j}_{0,t_k} \to \frac{1}{2}\int^t_0 \bra{X_{r}\, dY_r - Y_{r}\, dX_r} = A^{M;i,j}_{0,t} \text{ in } L^{p/2}(\mathbb{P}) \text{ as } \abs{D} \to 0.
\]
The case of $\theta=\lambda=f$ is almost identical and thus omitted for brevity.
Next we consider the case of $\theta=b\neq f = \lambda$. Since $A^{D,f,b;i,j}=-A^{D,b,f;j,i}$ we need only prove the convergence for $A^{D,b,f;i,j}$ to establish corresponding results for the case of $\theta=f\neq b=\lambda$.
To this end, using the previous partition $D=(t_i)$ we find
\[
A^{D,b,f;i,j}_{0,t_k} = \sum^{k-1}_{i=0} \X{i}(\Y{i+2}-\Y{i+1}) - \frac{1}{2}X^{D, b;i}_{0,t_k} Y^{D,f;j}_{0,t_k}.
\]
We have the decomposition:
\begin{align*}
\sum^{k-1}_{i=0} \X{i}(\Y{i+2}-\Y{i+1})
&= \sum^{k-1}_{i=0} \X{i+1}(\Y{i+2}-\Y{i+1})
- \sum^{k-1}_{i=0} (\X{i+1}-\X{i})(\Y{i+2}-\Y{i+1}).
\end{align*}
To deal with the second sum, we refer to the inequality of Lemma \ref{l-technical-lemma} which gives
\begin{equation}\label{e-convergence-2}
\EE{ \abs{ \sum^{k-1}_{i=0} (\X{i+1}-\X{i})(\Y{i+2}-\Y{i+1})}^{p/2}}
\leq
C_1 \EE{ \sup_{i=j,\ldots,k-1} \qv{X}^{p/2}_{t_i,t_{i+1}}}^{\frac{p-2}{2p}}.
\end{equation}
From our regularity assumptions on the martingale $M$ we have $\abs{\qv{M}_{0,1}}_{L^{p/2}(\mathbb{P})}<\infty$, and thus dominated convergence combined with uniform continuity ensure that
\[
\EE{\sup_{t_i\in D} \qv{X}^{p/2}_{t_i,t_{i+1}}} \to 0 \text{ as } \abs{D}\to 0.
\]
Therefore, in light of Lemma \ref{l-new-technical-lemma}, we deduce that
\[
A^{D,b,f;i,j}_{0,t} \to \int^t_0 X_r\, dY_r - \frac{1}{2}X_t Y_t \text{ in } L^{p/2}(\mathbb{P}) \text{ as } \abs{D}\to 0.
\]
Rearranging this last term using the integration by parts formula\footnote{For continuous semimartingales $X,Y$: $d(XY) = XdY + YdX + d\ip{X}{Y}$ \cite[Proposition IV.3.1]{FV}.} we arrive at the identity
\begin{align*}
\int^t_0 X_r\, dY_r - \frac{1}{2}X_t Y_t &= \int^t_0 X_r\, dY_r - \frac{1}{2}\bra{ \int^t_0 X_r\, dY_r + \int^t_0 Y_r\, dX_r + \ip{X}{Y}_t}\\
&= \frac{1}{2} \int^t_0 \bra{X_r\, dY_r - Y_r\, dX_r} - \frac{1}{2}\ip{X}{Y}_t
= A^{M;i,j}_{0,t} - \frac{1}{2}\ip{X}{Y}_t.
\end{align*}
For the rate of convergence claim we simply combine (\ref{e-convergence-2}) with Lemma \ref{l-new-technical-lemma} concerning the convergence of the discrete It\^{o} approximation sums.
The proof is complete.
\end{proof}
\section{Maximal $p$-variation norm for $\seq{\rp{M}^D : D\in\mathcal{D}_{[0,1]}}$}
The objective of this section is to prove the following maximal $p$-variation bound:
\begin{Proposition}[\textbf{Maximal $p$-variation bounds}]\label{p-maximal-p-variation-bound}
For all $p>2$, there exists a constant $C=C(p,\norm{M}_p)$ such that
\[
\sup_{D\in\mathcal{D}_{[0,1]}} \EE{ \pnorm{ \rp{M}^{D}}{p}^p} = C < \infty.
\]
\end{Proposition}
We split the proof into two propositions concerning $M^{D,\theta,\lambda}$; the first dealing with $\theta=\lambda$ and the second $\theta\neq \lambda$, where $\theta,\lambda\in\seq{b,f}$.
\begin{Proposition}\label{p-first-prop}
For all $p>2$ and $\theta\in\seq{b,f}$, there exists a finite constant $C=C(p,\norm{M}_p)$ such that
\[
\sup_{D\in\mathcal{D}_{[0,1]}} \EE{ \pnorm{S_2\bra{M^{D,\theta,\theta}}}{p}^p} \leq C
\]
\end{Proposition}
\begin{proof}
For all $\theta\in \seq{b,f}$, $M^{D,\theta}$ is a reparameterization of the standard piecewise linear approximation $\tilde{M}^D$ of $M$ based on the partition $D$. Since $p$-variation is invariant under reparameterization, it follows from the Burkholder-Gundy-Davis rough path result of \cite[Theorem 14.15]{FV} (using the moderate function $F(x)=x^p$) that
\[
\EE{ \pnorm{S_2\bra{M^{D,\theta}}}{p}^p} = \EE{ \pnorm{S_2\bra{\tilde{M}^D}}{p}^p} \leq C \EE{ \qv{M}^{p/2}_{0,1}}.
\]
The proof is complete.
\end{proof}
\begin{Remark}
The use of \cite[Theorem 14.15]{FV} in the proof of Proposition \ref{p-first-prop} critically requires that $p>2$.
\end{Remark}
We now consider the case of $\theta\neq \lambda$. As in the proof of Proposition \ref{p-area-pointwise-convergence}, by symmetry it suffices to only consider the case of $\theta=b\neq \lambda=f$.
\begin{Proposition}
For all $p>2$, there exists a constant $C=C(p,\norm{M}_p)$ such that
\[
\sup_{D\in\mathcal{D}_{[0,1]}} \EE{ \pnorm{S_2\bra{M^{D,b,f}}}{p}^p} \leq C.
\]
\end{Proposition}
Unlike the case of $\theta=\lambda$, the proof is slightly more involved and requires a technical result for its proof. This is presented in the next lemma which is purely deterministic (that is it involves no probability).
Suppose we are given the points $(x_i,y_i)^n_{i=0} \subset \mathbb{R}^2$ and for simplicity we assume that $(x_0,y_0)=(0,0)$. From these points we construct the piecewise-linear, axis-directed path $z:[0,2n]\to \mathbb{R}^2$ such that $z_{2k}=(x_k,y_k)$ and $z_{2k+1}=(x_{k+1},y_k)$ with linear interpolation between these times. Moreover, we also construct the path $w:[0,2n]\to\mathbb{R}^2$ as the standard piecewise linear interpolation between these points; that is, $w_{2k}=(x_k,y_k)$ and $w_{2k+2}=(x_{k+1},y_{k+1})$ with lines between these points.
We include an example of $z$ and $w$ in Figure \ref{f-path-diagram}.
The level-$2$ rough path lifts of these paths are given by
\[
\RP{z}:= S_2\bra{z}, \; \; \; \RP{w}:=S_2\bra{w} \in C\bra{[0,2n],G^2(\mathbb{R}^2)}.
\]
\begin{figure}\label{f-path-diagram}
\begin{large}
\begin{center}
\begin{tikzpicture}
\node (A) {$\bra{x_0,y_0}$};
\node (B) [node distance=4cm, right of=A] {$\bra{x_1,y_0}$};
\node (C) [node distance=2.5cm, below of=B] {$\bra{x_1,y_1}$};
\node (D) [node distance=3.5cm, right of=C] {$\bra{x_2,y_1}$};
\node (E) [node distance=4.5cm, above of=D] {$\bra{x_2,y_2}$};
\node (F) [node distance=2.5cm,left of=E] {$\bra{x_3,y_2}$};
\node (G) [node distance=6cm,below of=F] {$\bra{x_3,y_3}$};
\node (FAKE) [node distance=1cm,below of=G] {};
\node (X) [node distance=1.41cm,left of=FAKE] {};
\node (H) [node distance=2cm,left of=G] {};
\draw[->] (A) to node {} (B);
\draw[->] (B) to node {} (C);
\draw[->] (C) to node {} (D);
\draw[->] (D) to node {} (E);
\draw[->] (E) to node {} (F);
\draw[->] (F) to node {} (G);
\draw[->] (G) to node {} (H);
\draw[->, dashed] (A) to node {} (C);
\draw[->, dashed] (C) to node {} (E);
\draw[->, dashed] (E) to node {} (G);
\draw[->, dashed] (G) to node {} (X);
\end{tikzpicture}
\end{center}
\end{large}
\caption{An example of the paths $z$ and $w$ (dashed)}
\end{figure}
\begin{Lemma}\label{l-tricky-technical}
For all $p>2$ there exists a constance $C=C(p)$ such that
\begin{align*}
&\pnormcust{\RP{z}}{p}{2n}^p
\leq \pnormcust{\RP{w}}{p}{2n}^p + \bra{\sum^{n-1}_{i=0} \seq{\bra{x_{i+1}-x_i}^2 + \bra{y_{i+1}-y_i}^2}}^{p/2}.
\end{align*}
\end{Lemma}
\begin{proof}
Fix a partition $R = (s_k)\in \mathcal{D}_{[0,2n]}$ and set
\[
s_{k+1}^{*}:=\begin{cases}
2m & \text{ if there exists an integer }m\text{ such that }s_{k}\leq2m\leq s_{k+1};\\
s_{k+1} & \text{ otherwise;}
\end{cases}.
\]
Certainly $s_k \leq s^*_{k+1} \leq s_{k+1}$ for all $k$.
By the subadditivity of the Carnot-Caratheodory norm,
\begin{align}
&2^{1-p} \sum_{s_k\in R} \norm{\RP{z}_{s_k,s_{k+1}}}^p
\leq \sum_{s_k\in R} \bra{ \norm{\RP{w}_{s_k,s_{k+1}}}^p + \norm{\RP{w}^{-1}_{s_k,s_{k+1}} \otimes \RP{z}_{s_k,s_{k+1}}}^p}\\
&\leq 2^{p-1}\sum_{s_k\in R} \bra{
\norm{\RP{w}_{s_k,s_{k+1}}}^p + \norm{\RP{w}_{s^*_{k+1},s_{k+1}}}^p + \norm{\RP{z}_{s^*_{k+1},s_{k+1}}}^p + \norm{\RP{w}^{-1}_{s_k,s^*_{k+1}} \otimes \RP{z}_{s_k,s^*_{k+1}}}^p}.\label{e-end}
\end{align}
Certainly we have
\[
\sum_{s_k\in R} \norm{\RP{w}_{s_k,s_{k+1}}}^p, \sum_{s_k \in R} \norm{\RP{w}_{s^*_{k+1},s_{k+1}}}^p \leq \pnormcust{\RP{w}}{p}{2n}^p,
\]
so we focus on the two remaining terms of (\ref{e-end}).
To this end, denoting the L\'{e}vy area process of $z$ by $A^z$, it follows immediately that
\begin{align*}
&\sum_{s_k\in R} \norm{\RP{z}_{s^*_{k+1},s_{k+1}}}^p\\
&\simeq \sum_{s_k\in R} \bra{ \abs{z_{s^*_{k+1},s_{k+1}}} \vee \abs{A^z_{s^*_{k+1},s_{k+1}}}^{1/2}}^p
\leq \sum_{s_k\in R} \bra{ \abs{z_{s^*_{k+1},s_{k+1}}}^p + \abs{A^z_{s^*_{k+1},s_{k+1}}}^{p/2}}\\
&\leq \sum_{s_k\in R} \abs{z_{s^*_{k+1},s_{k+1}}}^p + \bra{\sum_{s_k\in R} \abs{A^z_{s^*_{k+1},s_{k+1}}}}^{p/2}
\leq \pabscust{z}{p}{2n}^p + \bra{\sum_{s_k\in R} \abs{A^z_{s^*_{k+1},s_{k+1}}}}^{p/2}.
\end{align*}
Each coordinate $z^i$ of $z$ is simply a reparameterization of $w^i$ and thus the invariance of $p$-variation under reparameterization implies that
\[
\pabscust{z}{p}{2n}=\pabscust{w}{p}{2n} \leq \pnormcust{\RP{w}}{p}{2n}^p.
\]
By definition $\left[s^*_{k+1},s_{k+1}\right] \subset [2m,2m+2]$ for some integer $m\in \seq{0,\ldots,n-1}$. Thus the path $z$ restricted to $[s^*_{k+1},s_{k+1}]$ is contained within an open right-angled triangle. Each corner-point of $z$ can be contained in at most one of these intervals; therefore,
\begin{align*}
\sum_{s_k\in R} \abs{A^z_{s^*_{k+1},s_{k+1}}} &\leq \frac{1}{2}\sum_{i=0}^{n-1} \abs{x_{i+1}-x_i}\abs{y_{i+1}-y_i}
\leq \frac{1}{4}\sum^{n-1}_{i=0}\bra{\abs{x_{i+1}-x_i}^2 + \abs{y_{i+1}-y_i}^2},
\end{align*}
using the inequality $ab\leq 1/2(a^2+b^2)$ for $a,b\geq 0$.
It remains to bound the third term of (\ref{e-end}). We have:
\begin{align*}
\sum_{s_k\in R} \norm{\RP{w}_{s_k,s^*_{k+1}}^{-1} \otimes \RP{z}_{s_k,s^*_{k+1}}}^p
&\leq \sum_{s_k\in R} \seq{ \abs{z_{s_k,s^*_{k+1}} - w_{s_k,s^*_{k+1}}}^p + \abs{A^\phi_{s_k,s^*_{k+1}}}^{p/2} },
\end{align*}
where $A^\phi_{s_k,s^*_{k+1}}$ is the absolute area enclosed by the path $\phi$, where $\phi$ is the path given by the concatenation of $(z_t)_{t\in [s_k,s^*_{k+1}]}$ and the path $(w_{s^*_{k+1}-t})_{t\in [0,s^*_{k+1}-s_k]}$.
Evidently,
\[
\sum_{s_k\in R} \abs{z_{s_k,s^*_{k+1}} - w_{s_k,s^*_{k+1}}}^p \leq 2^{p-1} \bra{\pabscust{z}{p}{2n}^p + \pabscust{w}{p}{2n}^p}.
\]
The area $A^\phi_{s_k,s^*_{k+1}}$ is precisely the absolute sum of the triangles enclosed by the path $z$ along its corner points plus a partial triangle area. Excluding these partial triangles, every triangle is enclosed precisely once by $\phi$. Thus it follows that
\begin{align*}
\sum_{s_k\in R} \abs{A^\phi_{s_{k},s^*_{k+1}}}^{p/2}
\leq \bra{\sum_{s_k\in R} \abs{A^\phi_{s_k,s^*_{k+1}}}}^{p/2}
&\leq \bra{\frac{1}{2}\sum_{i=0}^{n-1} \abs{x_{i+1}-x_i}\abs{y_{i+1}-y_i} }^{p/2}\\
&\leq C \bra{\sum^{n-1}_{i=0} \bra{\abs{x_{i+1}-x_i}^2 + \abs{y_{i+1}-y_i}^2}}^{p/2},
\end{align*}
again using the inequality $ab\leq 1/2(a^2+b^2)$.
Putting the above inequalities together and noting that the resultant bound is independent of the chosen partition $R\in \mathcal{D}_{[0,2n]}$ completes the proof.
\end{proof}
We now apply the previous lemma to our process $M^D$:
\begin{Proposition}
For all $p>2$, there exists a constant $C=C(p,\norm{M}_p)$ such that
\[
\sup_{D\in\mathcal{D}_{[0,1]}} \EE{ \pnormcust{S_2\bra{M^{D,b,f}}}{p}{2n}^p} \leq C.
\]
\end{Proposition}
\begin{proof}
It suffices to prove the claim for $S_2\bra{M^{D,b,f;i,j}}$, where $i,j\in\seq{1,\ldots,d}$.
First fix a partition $D = (t_k)^n_{k=0}\in \mathcal{D}_{[0,1]}$.
We set the values $x_0=0$, $x_{n}=M^i_{t_n}$,
\[
x_k=M^i_{t_{k-1}} \text{ for } k=1,\ldots,n-1,
\]
and $y_{0}=0$, $y_n=M^j_{t_n}$ with
\[
y_k=M^j_{t_{k+1}} \text{ for } k=0,\ldots, n-1.
\]
Based on this discrete data $(x_k,y_k)_{k=0}^n$ we define $z$ and $w$ as above and note that $z$ is precisely a reparameterization of the process $M^{D,b,f;i,j}$ but instead defined over the interval $[0,2n]$. Similarly, $w$ is precisely a reparameterization of the standard piecewise linear approximation of $(M^i,M^j)$ with respect to the partition $D$, which is itself a reparameterization of $M^{D,\theta,\theta;i,j}$ for any $\theta\in\seq{b,f}$.
Thus Lemma \ref{p-first-prop} informs us that
\[
\EE{\pnormcust{\RP{w}}{p}{2n}^p} = \EE{\pnorm{S_2\bra{M^{D,b,b;i,j}}}{p}^p} \leq C_1.
\]
It follows from Lemma \ref{l-tricky-technical} that
\begin{align*}
&\EE{\pnorm{S_2\bra{M^{D,b,f;i,j}}}{p}^p}
= \EE{ \pnormcust{\RP{z}}{p}{2n}^p}\\
&\leq C_1 \bra{ \EE{\pnormcust{\RP{w}}{p}{2n}^p} + \EE{\bra{\sum^{n-1}_{k=0} \seq{\bra{M^i_{t_{k+1}}-M^i_{t_k}}^2+\bra{M^j_{t_{k+1}}-M^j_{t_k}}^2}}^{p/2}}}\\
&\leq C_2 + C_3 \EE{\bra{\sum^{n-1}_{k=0} \bra{M^i_{t_{k+1}}-M^i_{t_k}}^2}^{p/2} + \bra{\sum^{n-1}_{k=0} \bra{M^i_{t_{k+1}}-M^i_{t_k}}^2}^{p/2}}.
\end{align*}
From the discrete Burkholder-Davis-Gundy (BDG) inequality (c.f. \cite[Chapter 14]{FV}) followed by both sides of the continuous BDG inequality, we have
\begin{align*}
\EE{\bra{\sum^{n-1}_{k=0} \bra{M^i_{t_{k+1}}-M^i_{t_k}}^2}^{p/2}}
&\leq C_4 \EE{ \sup_{t_k\in D} \abs{M^i_{t_{k}}-M^i_{t_0}}^p }
\leq C_5 \EE{ \qv{M^i}^{p/2}_{0,1}}
\leq C_5\norm{M}^p_p.
\end{align*}
Thus there exists some constant $C=C\bra{p,\norm{M}_p}$ such that
\[
\EE{ \pnormcust{S_2\bra{M^{D,b,f}}}{p}{2n}^p} \leq C.
\]
As this bound is independent of the original partition choice of $D$, the proof is complete.
\end{proof}
\begin{Remark}
Initially it was thought that the \textit{dyadic trick} of bounding the $p$-variation of a level-$2$ rough path by a weighted sum of its moments and areas over every dyadic interval in $[0,1]$ (see \cite{ledoux2002levy}) could be employed.
To the present author it appears that this technique enforces restrictions on the rate of convergence of the partition mesh $\abs{D}$ and demands that the partitions be nested. Our algebraic proof above is much longer and involved but places no restrictions on the sequence of parititons. Indeed, the maximal bound holds for \textit{any} partition $D\in\mathcal{D}_{[0,1]}$.
\end{Remark}
\section{A tightness property}
We arrive at a tightness result on a countable rough path collection $\seq{\rp{M}^{D_n}}_{n=1}^\infty$ satisfying $\abs{D_n}\to 0$ as $n\to\infty$. But first a simple lemma.
\begin{Lemma}\label{l-last-l}
Fix $D\in\mathcal{D}_{[0,1]}$. If $\delta\geq\abs{D}$ then
\[
\sup_{\abs{t-s}\leq\delta} \abs{M^D_{s,t}} \leq 3\sup_{\abs{t-s}\leq 2\delta} \abs{M_{s,t}}.
\]
\end{Lemma}
\begin{proof}
Denoting the standard piecewise-linear approximation of $M$ based on the partition $D$ by $\tilde{M}^D$, we certainly have
\[
\sup_{\abs{t-s}\leq\delta} \abs{\tilde{M}^D_{s,t}} \leq \sup_{\abs{t-s}\leq\delta} \abs{M_{s,t}}.
\]
Suppose $D=(t_i)$.
The process $M^D$ is a reparameterization of $\tilde{M}^D$. Indeed, $M^D$ runs at twice the speed of $\tilde{M}^D$ over $[t_i,t^*_i]$ then stops over $[t^*_i,t_{i+1}]$. Considering the endpoints it follow that
\[
\sup_{\abs{t-s}\leq\delta} \abs{M^D_{s,t}} \leq 2\sup_{\abs{t-s}\leq\abs{D}} \abs{\tilde{M}^D_{s,t}} + \sup_{\abs{t-s}\leq 2 \delta} \abs{\tilde{M}^D_{s,t}}.
\]
The proof is complete.
\end{proof}
\begin{Corollary}[\textbf{Tightness property}]\label{c-tightness}
Suppose $M\in\mathcal{M}^p\bra{[0,1],\mathbb{R}^d}$ for some $p>2$ and let $(D_n)_{n=1}^\infty$ be a sequence of partitions with $\abs{D_n}\to 0$ as $n\to\infty$. Then
\begin{equation}\label{e-tightness-limit}
\lim_{\delta \downarrow 0} \sup_{n} \EE{ \sup_{\abs{t-s}\leq \delta} \norm{\rp{M}^{D_n}_{s,t}}^p} = 0.
\end{equation}
\end{Corollary}
\begin{proof}
%
%
%
%
%
Fix $\theta,\lambda\in\seq{b,f}$, $i,j\in\seq{1,\ldots,d}$ and consider the two-dimensional path $t\in [0,1] \mapsto \bra{M^{D,\theta;i}_t,M^{D,\lambda;j}}$ with L\'{e}vy area $A^{D,\theta,\lambda;i,j} : \Delta_{[0,1]} \to \left[\mathbb{R}^2,\mathbb{R}^2\right]$. Since this path is piecewise linear and axis-directed, basic isoperimetry theory tells us that for $\delta\leq\abs{D}$:
\[
\sup_{\abs{t-s}\leq \delta} \abs{A^{D,\theta,\lambda;i,j}_{s,t}} \leq \frac{1}{2} \bra{ \sup_{\abs{t-s}\leq \delta} \abs{M^{D,\theta;i}_{s,t}} + \sup_{\abs{v-u}\leq \delta} \abs{M^{D,\lambda;j}_{u,v}} }^2.
\]
That is, the area must be bounded above by the area of the right-angled triangle with sidelengths given by sum of the supremum of each increment processes.
Using the two classical inequalities: $(a+b)^{q} \leq 2^{q-1}(a^{q}+ b^q)$, $q\geq 1$ and $ab\leq 1/2(a^2+b^2)$ for $a,b\geq 0$, a quick computation confirms that for the case of $\delta\leq\abs{D}$:
\[
\EE{\sup_{\abs{t-s}\leq\delta} \abs{A^{D,\theta,\lambda;i,j}_{s,t}}^{p/2}} \leq C \bra{\EE{\sup_{\abs{t-s}\leq\delta} \abs{M^{D,\theta;i}_{s,t}}^p} + \EE{\sup_{\abs{v-u}\leq\delta} \abs{M^{D,\lambda;j}_{u,v}}^p}}
\]
for some universal constant $C$ depending only on $p$.
In the case of $\delta\geq\abs{D}$, the proof of Lemma \ref{l-tricky-technical} gives
\[
\sup_{\abs{t-s}\leq \delta} \abs{A^{D,\theta,\lambda;i,j}_{s,t}}^{p/2} \leq { \sum_{t_k \in D} \bra{ \abs{M^{D,\theta;i}_{t_k,t_{k+1}}}^2 + \abs{M^{D,\lambda;j}_{t_k,t_{k+1}}}^2 }^{p/2} }.
\]
Appealing to Lemma \ref{l-holder-play}, we have:
\begin{align*}
\EE{\sup_{\abs{t-s}\leq \delta} \abs{A^{D,\theta,\lambda;i,j}_{s,t}}^{p/2}}
&\leq C_1 \EE{\sum_{t_k\in D} \bra{ \abs{M^{D,\theta;i}_{t_k,t_{k+1}}}^p + \abs{M^{D,\lambda;j}_{t_k,t_{k+1}}}^{p}}}\\
&= C_1 \EE{\sum_{t_k\in D} \bra{ \abs{M^i_{t_k,t_{k+1}}}^p + \abs{M^j_{t_k,t_{k+1}}}^{p}}}\\
&\leq C_1 \sup_{l=1,\ldots,d} \EE{\sup_{t_k \in D} \qv{X^l}_{t_k,t_{k+1}}^{p/2}}^{\frac{p-2}{p}}
\leq C_1 \sup_{l=1,\ldots,d} \EE{\sup_{\abs{t-s}\leq \delta} \qv{X^l}_{s,t}^{p/2}}^{\frac{p-2}{p}},
\end{align*}
which certainly converges to zero as $\delta\to 0$.
Thus it suffices to prove the limit (\ref{e-tightness-limit}) only for the increment process $M^D$ and not the corresponding rough path lift $\rp{M}^D$.
For the purpose of creating a contradiction, suppose that the conclusion of the corollary is false; that is, there exist some constant $\epsilon>0$ and an increasing sequence $(k_n)_{n=1}^\infty \subset \mathbb{Z}_+$ such that for all $n$:
\begin{equation}\label{e-cuba-1}
\EE{\sup_{\abs{t-s}\leq \abs{D_n}} \abs{M^{D_{k_n}}_{s,t}}^p} > \epsilon.
\end{equation}
It follows that there exists an infinite subsequence $(k_{n_j})_{j=1}^\infty$ of $(k_n)_{n=1}^\infty$ such that $n_j \leq k_{n_j}$.
Indeed, if it were otherwise we could find an infinite subsequence $(k_{m_j})_{j=1}^\infty$ of $(k_n)_{n=1}^\infty$ such that $m_j>k_{m_j}$ with
\[
\EE{ \sup_{\abs{t-s}\leq \abs{D_{m_j}}} \abs{M^{D_{k_{m_j}}}_{s,t}}^p} > \epsilon.
\]
Since $m_j>k_{m_j}$, $\abs{D_{m_j}}\leq \abs{D_{k_{m_j}}}$ and so by Lemma \ref{l-last-l}:
\begin{align*}
\epsilon < \EE{ \sup_{\abs{t-s}\leq \abs{D_{m_j}}} \abs{M^{D_{k_{m_j}}}_{s,t}}^p}
&\leq \EE{ \sup_{\abs{t-s}\leq \abs{D_{k_{m_j}}}} \abs{M_{s,t}^{D_{k_{m_j}}}}^p}
\leq 3^p \EE{\sup_{\abs{t-s} \leq 2 \abs{D_{k_{m_j}}}} \abs{M_{s,t}}^p}
\end{align*}
but this last quantity converges to zero as $j\to\infty$ by the dominated convergence theorem. Indeed, since we have $M\in\mathcal{M}^p$ for some $p>2$, the classical BDG inequality gives us the necessary dominatation in order to apply the dominated convergence theorem..
Thus there exists such a subsequence $(k_{n_j})_{j=1}^\infty$ of $(k_n)_{n=1}^\infty$ such that $n_j\leq k_{n_j}$ for each $j$.
This means $\abs{D_{n_j}}\geq \abs{D_{k_{n_j}}}$ and so appealing to Lemma \ref{l-last-l} again, it follows that
\[
\sup_{\abs{t-s}\leq \abs{D_{n_j}}} \abs{M^{D_{k_{n_j}}}_{s,t}}^p
\leq 3^p \sup_{\abs{t-s}\leq 2\abs{D_{n_j}}} \abs{M_{s,t}}^p.
\]
Combined with (\ref{e-cuba-1}), we conclude:
\[
\epsilon < \EE{ \sup_{\abs{t-s}\leq \abs{D_{n_{j}}}} \abs{M^{D_{k_{n_j}}}_{s,t}}^p } \leq 3^p \EE{ \sup_{\abs{t-s}\leq 2\abs{D_{n_j}}} \abs{M_{s,t}}^p} \to 0 \text{ as } j\to\infty,
\]
and so the desired contradiction is established.
\end{proof}
\begin{Remark}
Corollary \ref{c-tightness} will not hold for an arbitrary sequence of partitions $(D_n)_{n=1}^\infty \subset \mathcal{D}_{[0,1]}$. The condition $\abs{D_n} \to 0$ as $n\to\infty$ is necessary.
To show this we provide a simple counterexample:
suppose we have a one dimensional Brownian motion $B$ on $[0,1]$ and set the partition sequence $D_n=\seq{0,1/n,1}$. Then $B^{D_n}_{1/n}= (0,X_1) \; \; \; \text{ for all } n$,
and thus for all $k$:
\[
\sup_n \EE{\sup_{\abs{t-s}\leq \delta} \abs{B^{D_n}_{s,t}}^p} \geq 1.
\]
\end{Remark}
\section*{Acknowledgements}
The first author would like to thank Dr. Weijun Xu for his helpful comments and Drs. Zhongmin Qian and Danyu Yang for their careful reading of earlier drafts.
The research of all authors is supported by the European Research Council under the European Union's
Seventh Framework Programme (FP7-IDEAS-ERC, ERC grant agreement nr. 291244). The research
of Terry Lyons is supported by EPSRC grant EP/H000100/1. The authors are grateful for the support
of the Oxford-Man Institute.
\bibliographystyle{plain}
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{"url":"https:\/\/forum.suricata.io\/t\/suricata-exe-entry-point-not-found\/3271","text":"Hi, i have installed on win10 getting this error (The procedure entry point pcap_dump_hopen not be located inthe dynamic link libarary. Please adivse.\n\nHi,\n\nWhich MSI instal is that , which suricata version ?\n\nSuricata version 6.0.9-1 (Windows 10)\n\nDo you have npcap installed https:\/\/npcap.com\/ on the machine ?\n\nHi,\nYes, i installed npcap,\n\nnow I am getting this error\n\nTry to supply the sniffing interface in Windows like so:\n-i 10.2.0.20 instead of -i eth0 (which is Linux style)\nwhere this is actually the sniffing interface IP\n\nI am a newbie, and I appreciate your support.\n\nok, in that case you should use -i 192.168.0.106\n\nNow getting this.\n\nTo start it and confirm functionality you would only need:\n\nsuricata -c suricata.yaml -i 192.168.0.104\n\n\nI pressed the ctrl C to break\nI want to generate the log file, regarding the error I can send you my Suricata.yaml so can check what else I need to do in it?\n\nThe log location lacks write permission so you can either add -l \\path\\to\\writable\\log-directory or update the Suricata configuration file \u2013 change the value for default-log-dir to a path to a writable directory.","date":"2023-03-30 05:22:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3369373679161072, \"perplexity\": 11142.733598215475}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949097.61\/warc\/CC-MAIN-20230330035241-20230330065241-00421.warc.gz\"}"} | null | null |
Q: Find unknown coefficient of a matrix with known rank. Let there be a matrix A = $\begin{bmatrix} 8 & 7 & 5 \\ 4 & 5 & 6 \\ 7 & 8 & λ \end{bmatrix}$.
Find the λ value, for which matrix A has a rank(A) = 2.
First I need to do the reduced row echelon form (I guess). Should I start by dividing the first row by 8 or by substracting the third row from the first? Will the result be the same? How should I proceed with the information rank(A) = 2 given?
A: Hint: You need row $3$ to be a linear combination of rows $1$ and $2$. Write $a(8,7,5)+b(4,5,6)=(7,8,\lambda) $.
Solve the system $\begin{cases}{8a+4b=7 \\7a+5b=8}\end{cases}$ for $a$ and $b$.
Then $\lambda =5a+6b$.
If we let $M=\begin{pmatrix}8&4\\7&5\end{pmatrix}$, then $M^{-1}=\frac1{12}\begin{pmatrix}5&-4\\-7&8\end{pmatrix}$.
Thus
$\begin{pmatrix} a\\b\end{pmatrix}=M^{-1}\begin{pmatrix}7\\8\end{pmatrix}=\begin{pmatrix}\frac14\\\frac54\end{pmatrix}\,\therefore \lambda =\frac{35}4$
A: You want to process by Gauss-Jordan elimination.
So you want to find the set of elementary operations to find the rank.
I recommend starting with the following operations:
*
*Line 2 becomes line 2 - half of line 1
*Line 3 becomes line 3 - (7/8) of line 1
See more here: https://en.wikipedia.org/wiki/Gaussian_elimination
A: The two first rows are linearly independent, therefore the rank is 2 or 3. To have the rank 2, one can solve $\det A = 0.$ It is a linear equation in $\lambda$ with unique solution.
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\section{Introduction}
\label{sec:intro}
In the recent years, distributed denial of service (DDoS) attacks has been growing and always seen the upward trend \cite{RealSecurity2019EvolutionSecurity}. Work from home and increased use of cloud technologies owing to the Covid pandemic in the first quarter of 2020 has increased the volume and intensity of DDoS attacks in 2020. For example, launching various amplification and UDP-based attacks to flood target networks increased 570 percent for the second quarter of 2020 in comparison with the previous year for the same time period \cite{Nexusguard2020DDoSQ2}; the traditional threshold-based mitigation methods are insufficient to detect these attacks and the machine learning models are able to accurately detect as long as the attack pattern follows the trained data model and if any new attack pattern can easily evade these models \cite{Nexusguard2020DDoSQ2}. Although the DDoS attack vectors existed for years and many solutions proposed for handling the attacks, it is still an important problem to be addressed as the new technologies increases the attack surface and exploitable vulnerabilities.
As the number of devices connected to the internet increases and new network protocol vulnerabilities are uncovered, e.g., the UDP Memcached vulnerability \cite{Newman2018AWIRED}, DDoS attack rates have increased exponentially over the last decade, as shown in Figure \ref{Attackgrowth1}. A nominal enterprise organization may not be able to effectively handle or mitigate the current terabit rate sized attacks, and it's already late to bring up the network Operators and internet service providers to react and mitigate DDoS attacks when attackers target these enterprises. However, as mentioned in Table \ref{AttacksHistory}, we can see that the cloud service providing organizations like Amazon Web Services (AWS) and Google Cloud Platform (GCP) were handled approximately more than 2 Tbps attack rate at the edge level and served the public cloud application customers with no performance or service impact in the last two years. In 2016, the IOT devices such as routers and cameras connected to the internet were compromised, and attack code deployed to launch mirai bot reflection attacks to generate attack traffic rates in excess of 1 Tbps targeting DYN (a dynamic DNS service provider), OVH (cloud service provider), and security blogger Brian Krebs's website \cite{BrianKrebs2016KrebsOnSecuritySecurity} \cite{Kennedy2016OVHWebcams}\cite{Vaughan-Nichols2016TheZDNet}.
The emerging technologies such as cloud Computing, Internet of Things (IoT), Software Defined Networking (SDN) change the internet network architecture and offers new opportunities for the attackers finding the loopholes and perform Denial of service attacks. The challenge of large-scale DDoS attacks is to mitigate them within a short span of time and avoid the loss of business and reputation for the enterprise organizations involved in the attack. Therefore, a rapid coordination and response required between the stakeholders like network operators, edge protection providers, Internet service providers, impacted organizations, third party DDoS mitigation services etc. Authenticating and establishing trust among the parties involved is essential to execute the legitimate actions for stopping the attacks.
A blockchain is a distributed ledger that can record the transactions in an efficient and permanent way. It is managed by peer-to-peer (P2P) network nodes with standard protocols designed for internode communication to approve the transaction records and validate the blocks. Owing to the inherent security by design and unalterable transaction records in the chain of blocks, a blockchain can be used for many applications including finance, healthcare, supply chain, cryptocurrency, cybersecurity, smart contacts in particular validating the identity, providing the user anonymity \cite{Zheng2019}\cite{BlockchainWikipedia}. The blockchain utility for cybersecurity application has been growing with demand to build secured systems and applications. The decentralized consortium blockchain implementation for industrial IoT \cite{Li2018} \cite{Wan2019}, credit based consensus mechanism for approving the transactions in industrial IoT \cite{Huang2019} and implementing blockchain based data storage and protection mechanism for defending the security attacks in IoT systems \cite{Liang2019} \cite{Li2019} are some of the applications of the blockchain in IoT. Additionally, blockchain is leveraged for security in other areas like secured storage of the data in mobile ad hoc networks \cite{Yazdinejad2020}, decentralized DNS database for DNS attacks mitigation such as cache poisoning attacks \cite{Li2021}, secured data storage in cloud and defend against the keyword guessing attacks \cite{Zhang2020}. Furthermore, based on the blockchain exhibiting security properties, we could see that the potential to utilize the blockchain for security threat information sharing among the key stakeholders.
\begin{figure}[!h]
\centering
\includegraphics[width=8 cm, height=6cm]{image/LargeDDoSAttack.jpg}
\caption{DDoS attack rate growth trend in the last decade \cite{Menscher2020IdentifyingBlog}.\label{Attackgrowth1}}
\end{figure}
Recently, a few researchers proposed blockchain based solutions for threat information sharing like malicious IP address for blocklist, identifying the IOT bots in the network at the network gateway level, enabling content distribution network (CDN) nodes near the victim using private blockchain when denial of service is identified, security operating center threat sharing to users accessed in private blockchain is investigated in several recent works \cite{Rodrigues2017EnablingBloSS} \cite{Kim2018DDoSBlockchain} \cite{Badruddoja2020IntegratingSensors} \cite{Yeh2020SOChain:Blockchain} \cite{Tariq2019}. But there is a knowledge gap between network security experts, who aim to mitigate DDoS attacks in real time and blockchain experts, who develop decentralized applications but may not be experts in network attacks. Our prior art research shows that there is no significant work on investigating blockchain's role to mitigate the DDoS attacks. Therefore, we believe that there is a need for a systematic thorough review of the blockchain technology to handle the denial of service attacks. In addition, the blockchain based solutions are categorized based on the DDoS mitigation deployment location in internet. To the end,
the main contributions of this paper are as follows:
\begin{itemize}
\item We performed systematic review and classification of the role of blockchain technology in DDoS attack detection and blockchain based DDoS mitigation solutions.
\item We discussed the open challenges and future directions to implement and propose new solutions for handling DDoS attacks using blockchain.
\item We categorized and described the existing blockchain related DDoS solutions based on the solution deployment location in the internet architecture.
\item Our findings show that secured collaboration among the stakeholders to share the DDoS threat indicators with blockchain is achievable while addressing the limitations.
\end{itemize}
The abbreviations used in the paper are given in Table \ref{Abbreviations}. The remainder of this paper is organized as follows: Section \ref{sec:background} discusses the key concepts such as DDoS attacks, Blockchains and Emerging technology network architecture paradigms and related work in association with our topic in the paper. Section \ref{sec:ddosusingbc} presents the Blockchain based solutions to mitigate the DDoS attacks. Section \ref{sec:openchallenges} presents the current open challenges to utilize the blockchain in the context of DDoS attacks. Section \ref{sec:futuredirection} depicts the future directions in accordance with advancement with Blockchain technology. Section \ref{sec:conclusion} concludes the paper.
\begin{table}[!h]
\caption{List of Abbreviations used in the paper.\label{Abbreviations}}
\begin{tabular}{|c|l|}
\hline
ACK & TCP Acknowledgement Flag \\ \hline
AMQP & Advanced Message Queuing Protocol \\ \hline
AMP & Asynchronous Messaging Protocol \\ \hline
API & Application Programming interface \\ \hline
AWS & Amazon Web Services \\ \hline
AS & Autonomous System \\ \hline
BFT & Byzantine Fault-Tolerant \\ \hline
BGP & Border Gateway Protocol \\ \hline
CDN & content distribution network \\ \hline
CoAP & Constrained Application Protocol \\ \hline
CIDS & Collaborative Intrusion Detection System \\ \hline
CLDAP & Connection-less Lightweight Directory Access \\ \hline
CPU & Central processing unit \\ \hline
DDoS & Distributed Denial of Service \\ \hline
DNS & Domain Name System \\ \hline
DOTS & DDoS Open Threat Signaling \\ \hline
DoS & Denial of Service \\ \hline
DOS & Decentralized Oracle Service \\ \hline
DPOS & Delegated Proof of Stake \\ \hline
EVM & Ethereum Virtual Machine \\ \hline
GRE & Generic Routing Encapsulation \\ \hline
GCP & Google Cloud Services \\ \hline
HTTPS & Hypertext Transfer Protocol Secure \\ \hline
HTTP & Hypertext Transfer Protocol \\ \hline
ICMP & Internet Control Message Protocol \\ \hline
IoT & Internet of Things \\ \hline
IPFS & InterPlanetary File System \\ \hline
IP & Internet Protocol \\ \hline
ISP & Internet Service provider \\ \hline
KNN & k-nearest neighbor \\ \hline
LSTM & Long short-term memory \\ \hline
MLP & Multi-Layer Perceptron \\ \hline
ML & Machine learning \\ \hline
MQTT & Message Queuing Telemetry Transport \\ \hline
NDP & Neighbor Discovery Protocol \\ \hline
NTP & Network Time Protocol \\ \hline
OF & Open Flow \\ \hline
PBFT & Practical Byzantine fault tolerance \\ \hline
PCA & Principal component analysis \\ \hline
PoS & Proof of Stake \\ \hline
PoW & Proof of Work \\ \hline
PSH & TCP Push flag \\ \hline
P2P & Peer to Peer \\ \hline
P4 & Programming protocol-independent packet processor \\ \hline
RAM & Random-access memory \\ \hline
SDN & Software Defined Network \\ \hline
RNN & Recurrent neural network \\ \hline
SMTP & Simple Mail Transfer Protocol \\ \hline
SNMP & Simple Network Management Protocol \\ \hline
SOC & Security Operating Center \\ \hline
SYN & TCP Synchronization Flag \\ \hline
TCP & Transmission Control Protocol \\ \hline
SVM & Support Vector Machine \\ \hline
TX & Transaction \\ \hline
UDP & User Datagram Protocol \\ \hline
UTX & Unspent Transaction Unit \\ \hline
XMPP & Extensible Messaging and Presence Protocol \\
\hline
\end{tabular}
\end{table}
\section{Key Concepts and Related Work}
\label{sec:background}
In this section, we review DDoS attack types and the solutions proposed to mitigate them, describe the main fundamental and terminology of blockchain technology, and describe the emerging technologies such as internet of things and software defined networking paradigm. These are essential and play a significant role in the understanding of recent DDoS attack variants and their mitigation solutions using blockchain.
\subsection{DDoS Attack Types and Known Solutions}
Distributed Denial of Service (DDoS) Attack is a well-known and major concern in cybersecurity area violating the security principle "Availability" of services. DDoS attack vectors exploit various features of the internet protocols, most of which were designed decades ago when security was not a concern. The relationship between an attacker exploiting the protocol features such as TCP connection setup using 3-way handshake and its victim is asymmetric in nature. DDoS attacks are mainly classified into two categories: bandwidth depletion and resources depletion attacks \cite{Douligeris2003DDoSClassification}. In the former attack, high volumes of traffic that looks legitimate but not intended for communication is directed to a victim. In the latter attack, the victim is inundated with bogus service requests that deplete its resources and prevent it from serving legitimate requests.
Multiple bots (network nodes compromised and controlled by an attacker) are often used to launch DDoS attacks. Direct attacks on a victim typically use flooding in which many packets are sent from multiple bots to the victim; examples include TCP SYN floods, UDP floods, ICMP floods, and HTTP floods \cite{Swami2018SoftwareMechanisms}.
Another tactic used in DDoS attacks is amplification: the attacker sends requests to network service providers such as Domain Name System (DNS) servers or network time providers (NTP) spoofing victim's IP address as the source IP address so that the responses, which are typically several times larger than the queries/requests, are sent to the victim and overwhelm the victim's network and resources. Examples of amplification attacks include Smurf, Fraggle, SNMP, NTP, DNS amplification \cite{Swami2018SoftwareMechanisms}.
In addition, protocol exploitation attacks like TCP SYN flooding can be performed on the victim infrastructure by taking advantage of TCP connection establishment mechanism and sending the flood of TCP SYN packets with no ACK responses to consume the victim machine resources \cite{Srivastava20AMechanisms}. The adversary may also use automated scripts to send TCP flags ACK, PUSH, RST, FIN packet floods to saturate the communication channel along the victim infrastructure. Another category of DDoS attack are ping of death and land attack. Ping of death attack focused on sending Ping command with packet size greater than maximum packet size 65536 bytes to crash the victim the system. In land attack, An attacker may send forged packets with same sender and destination IP address to target the victim to send the packet to itself forming an infinite loop and crashing the victim machine \cite{Srivastava20AMechanisms}. A zero-day can vulnerability also be leveraged to compromise the legit machines and successfully lunch the denial of service attack \cite{CatalinCimpanu2020DDoSZDNet}.
Significant research work is done on the detection and mitigation of DDoS attacks for the last two decades. The proposed mitigation solutions differ in the location and timing of deployment \cite{Zargar2013AAttacks}. The deployment location-based solutions are categorized into four types
\begin{itemize}
\item Source-based defense implemented in the attack source edge routers or source Autonomous Systems.
\item Destination-based implemented at the victim edge routers or victim AS level.
\item Network-based defense implemented by the ISP and core networks and usually required to respond the attacks at the intermediate network level and \item Hybrid defense : the combination of the source, destination and network based mechanisms.
\end{itemize}
Although the source-based defenses aim to detect and mitigate the attacks in early stages of the attack, it is very difficult to distinguish the legitimate and malicious DDoS traffic at the source level owing to the use of bots to distribute the attack traffic generation.
\begin{table*}[!h]
\captionsetup{justification=centering}
\caption{Major DDoS attacks in the history.\label{AttacksHistory}}
\begin{tabular}{|p{2.3cm}|p{0.5cm}|p{2.6cm}|p{1.7cm}|p{1cm}|p{1.4cm}|p{2.5cm}|p{2cm}|} \hline
\textbf{DDoS Attack} & \textbf{Year} & \textbf{Attack Type} & \textbf{Attack Rate} & \textbf{Duration} & \textbf{Amp Ratio} & \textbf{Protocols Involved} & \textbf{Impact} \\ \hline
AWS Attack \cite{Amazon2020AWS2020}[18] & 2020 & Reflection Attack & 2.3 Tbps & 3 days & 56 - 70 & UDP, CLDAP & No \\ \hline
Google Attack \cite{Menscher2020IdentifyingBlog} & 2017 & Reflection & 2.5 Tbps & 6 months & 6-70 & CLDAP, DNS, SMTP & No \\ \hline
Mirai Krebs \cite{BrianKrebs2016KrebsOnSecuritySecurity} & 2016 & Mirai, TCPSYN, ACK, ACK+PSH & Krebs \- 620Gbps & 2-7 days & - & TCP, GRE, HTTP & Krebs Offline \\ \hline
OVH \cite{Kennedy2016OVHWebcams} & 2016 & Mirai, TCPSYN, ACK, ACK+PSH & OVH \- 1.1 Tbps & 2-7 days & - & TCP, GRE, HTTP & OVH minimal \\ \hline
Mirai Dyn \cite{Vaughan-Nichols2016TheZDNet} & 2016 & Mirai, Reflection & 1.5Tbps & 1 day & Up to 100 & DNS & Internet Outage \\ \hline
GitHub Attack \cite{Newman2018AWIRED} & 2018 & Memcached Reflection & 1.35Tbps & 20 min & ~51000 & UDP & Service Outage \\ \hline
Six Banks \cite{Constantin2012DDoSCIO} & 2012 & Brobot & 60 Gbps & \~ 2 days & - & HTTP, HTTPS, DNS, TCP & Web Service Outage \\ \hline
Hongkong Central \cite{RUSSELL2014HongAttack} & 2014 & Brobot,TCP SYN, HTTPS Flood & 500Gbps & - & - & TCP,HTTPS & Minimal \\ \hline
Spamhaus \cite{JaikumarVijayan2013Update:Computerworld} & 2013 & Reflection Attack & 300 Gbps & - & Up to 100 & DNS,TCP & Offline \\ \hline
Cloudflare \cite{Thompson2014Record-breakingNetwork} & 2014 & Reflection Attack & 400 Gbps & - & Up to 206 & NTP & No \\
\hline
\end{tabular}
\end{table*}
The destination-based defense mechanisms are easier and cheaper to implement since the attack traffic will be concentrated closer to the victim. However, before they are detected; the attack traffic consumes the resources on the paths leading to the victim. The network-based defense solutions detects and mitigate the DDoS attacks at the Autonomous System (AS) or Internet Service Provider (ISP) levels, which are closer to the attack sources. But they incur storage and processing overhead at the network infrastructure level, for example, by the edge or ISP routers, or might need additional DDoS protection devices like middle boxes to process the traffic. Also, the attack detection will be difficult owing to lack of aggregation of traffic destined to the victim. However, attack mitigation in the internet core has the advantage of not passing the traffic till the victim network and preventing congestion of communication channel with attack network traffic as well as saving the victim's computing and network resources. The hybrid defense approach promises to be more robust since it allows to use the combination of defensive mechanism to defend against DDoS attacks. Furthermore, detection and mitigation can be implemented more efficiently. For instance, the detection can occur at the destination or network level and the mitigation technique can be applied near the source to effectively handle the DDoS attacks. However, its implementation is more challenging because it requires collaboration and cooperation between different entities to exchange attack information without receiving sufficient incentives for some of the participants like service providers \cite{Zargar2013AAttacks} and there needs to be trust between the stakeholders, given the fact that the service providers are diverse and not easy to trust the entities.
For descriptions of various DDoS mitigation techniques such as anomaly or signature-based detection, machine learning algorithms to attack detection, scrubbing, rerouting, and filtering/blocking techniques, see Zargar et al. \cite{Zargar2013AAttacks}.
\subsection{Blockchain Technology and Their Types}
A blockchain is a digital, public ledger that records list of transactions and maintains the integrity of the transactions by encrypting, validating and permanently recording transactions \cite{BlockchainBankrate.com}. Blockchain technology has emerged as a potential digital technology disrupting many areas including financial sector, security, data storage, internet of things and more. One of the best known uses of blockchains is the design of cryptocurrencies such as Bitcoin \cite{Nakamoto2009Bitcoin:System,Nakamoto2009Bitcoin:System,2021CryptocurrencyCoinMarketCap}.
A blockchain is typically managed by a peer-to-peer network and uses peer-to-peer protocol such as the Distributed Hash Table (DHT) for internode communication as well as validating new transactions. Figure ~\ref{BCcomponents} illustrates the typical structure of a block: a linked list of blocks with a header block. Each block comprises a set of transactions, a count of the transactions in the block, and a header. The block header includes block version, which tells the current version of block structure, a merkle tree root hash to incorporate the uniqueness of the transaction set in the block by determining the final hash value achieved from all the transactions in the block as well as maintain the integrity between the transactions in the block. Therefore, the transactions secured in a blockchain and cannot be tampered. The block header also contains Timestamp, i.e. the time at which the block is created and it plays an important role in extending a blockchain to record new transactions. There is a special data structure that points to the most recent block in a chain. Using the back pointers other blocks in the chain can be accessed.
\begin{figure}[!h]
\centering
\includegraphics[width=8cm, height = 8cm]{image/F2BCcomponents.png}
\centering
\caption{Blockchain Internal Components \label{BCcomponents}}
\end{figure}
Blockchain exhibits properties like decentralization, persistency, anonymity, and auditability. The essential property of anonymity is achieved using asymmetric cryptography like RSA algorithm and digital signature \cite{Albertorio2018PublicMedium}. Each user has a private and public key pair for applying an asymmetric cryptography algorithm. The hash values obtained from the existing transactions will be utilized to get the digital signature and validate the user's authenticity. The user validation is a two-step process: signing and verification. Figure ~\ref{BCcrypto} shows the asymmetric cryptography and digital signature calculation steps during the validation process \cite{Golosova2018TheTechnology}. The peer-to-peer blockchain system has no centralized node and uses consensus algorithms, which typically require participating entities to win a computing challenge, to authorize an entity to create the next block of verified transactions and append to the exiting blockchain.
\begin{figure}[!h]
\centering
\captionsetup{justification=centering}
\includegraphics[width=8.4cm, height = 6.7cm]{image/F3BCcrypto.png}
\centering
\caption{Basic cryptographic operations in blockchain \cite{Golosova2018TheTechnology}. \label{BCcrypto}}
\end{figure}
A consensus algorithm, as indicated above, is used to select nodes in peer-to-peer blockchains to add a block of new transactions to the existing blockchain. Some of the widely used algorithms are proof of work (POW), proof of stake (POS), practical Byzantine fault tolerance (PBFT), ripple consensus algorithm and delegated proof of stake (DPOS) \cite{Zheng2017AnTrends}. In POW, used by Bitcoin, every node computes the hash value of the block header and the computed value should be less than the specific value, according to the algorithm. The successfully computed node will be verified by the other nodes and selected as an authorized node to add the transaction in the block; the update is propagated to all other nodes of the blockchain. Computation of the hash value within the constraints requires requires extensive computing, which is called mining. In POS, the users that have more currency can get an authority to add the transactions in the blockchain. So, richer entities will become richer, and, potentially, a few participants dominate the blockchain management and extension; on the other hand, this method does not require extensive computing power, and is likely to more efficient. The consensus algorithm based on PBFT requires that a significant majority of the nodes participating in the blockchain should approve the transaction to be appended in the network and can tolerate 1/3rd of the node failures. The consensus process starts by choosing a primary node to process all the transactions in a block. It is a three-step process i.e. pre-prepare, prepare and commit; If 2/3rds of the nodes accept the request, then the transaction is appended to the block. Hyperledger's fabric is an example of using PBFT as a consensus mechanism to complete the transactions in the network. In Delegated Proof of Stake(DPOS), the delegated maximum currency stakeholder is chosen for adding the transactions. Some platforms like Tendermint operates on the combination of the algorithms (DPoS+PBFT) \cite{Zheng2017AnTrends}.
With decentralized consensus methods such as POW, branching, in which competing entities may propose different sets of transactions to create a new block and extend a current blockchain, can occur due to the decentralized nature for mining to approve the transaction as well as having a delay to validate the 51\% of the blockchain nodes or participants prior to adding the transaction to blockchain; nBits, which signifies the difficulty level that is being used for miner computations to add the transactions to the block; nonce, which represents a random number created by the creator of the block and can be used only once; parent block hash, which is a cryptographic hash value of the parent block to maintain the integrity between the two consecutive blocks and maintain the non-tampered chain of blocks \cite{Zheng2017AnTrends} \cite{Zheng2019}.
In general, blockchain platforms are typically classified into three types. Public blockchain, in which the existing transactions can be read by anyone in public and open to join for public. But the transactions cannot be tampered and provide high level security, even though its computation delay is high. Bitcoin is a classic example of public blockchain. Anyone can read the user account balance and the transactions that the user account involved, given the fact that the user bitcoin wallet address is known. In consortium Blockchain, only selected nodes are participated in transactional operations and a good example multiple organization in a particular sector want to use the blockchain for business applications. Each node represents a member from the organization. The consensus process is fast, and only privileged users can read the information from the blockchain. Private Blockchain requires permission to join the network and usually maintained within the organization. The nodes can be the participants from the same organization to share the data within the organization or storing the data records securely and more. The private blockchain usually becomes centralized in nature and the transaction can be tampered if untrustworthy nodes participate in the mining process. The detailed comparison of the blockchain types is described in Table \ref{blockchaintypes}.
\begin{table*}[!h]
\caption{Types of Blockchain and their Properties \cite{Zheng2017AnTrends} \label{blockchaintypes}}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
\textbf{Property} & \textbf{Public} & \textbf{Consortium} & \textbf{Private}\\ \hline
Consensus participants & All mining nodes & Selected nodes & Nodes within the organization \\ \hline
Efficiency & Low & High & High \\ \hline
Readability & Anyone & Anyone or restricted members & Members within the organization \\ \hline
Decentralized & Yes & Partial & No \\ \hline
Consensus authorization & Permissionless & Permissioned & Permissioned \\ \hline
Example & Bitcoin & R3 & Hyperledger \\ \hline
Application & Bitcoin currency, voting & Banking, payments & Supply chain, health care, retail \\ \hline
Immutability & Nearly impossible to tamper & Possibly tampered & Possibly tampered \\ \hline
\end{tabular}
\end{table*}
Since the existence of the Bitcoin, there are number of coins developed by the blockchain community focusing on specific industry application. Some of the major notable coins are Ethereum, Litecoin and Ripple \cite{REIFF2020TheBitcoin}. The second popular and largest market capitalization cryptocurrency is Ethereum, which works on smart contract functionality. Ethereum has been proposed to address some limitations in Bitcoin scripting language. Ethereum supports the turing complete programming language meaning that we can perform all computations including the loops. This is achieved by smart contracts functionality, which runs cryptographic rules when certain conditions are met. The smart contracts in the nodes are translated into EVM code and then the nodes execute the code to complete the transaction (can be creating a user account, the result of code execution).
There has been a lot of attention on Hyperledger recently owing to the applicability of enterprise standard version blockchain deployment capabilities and known to be rigorously used in academic research community for research activities. Hyperledger is an open source community contributed suite, which comprises tools, frameworks, and libraries for enterprise blockchain application deployments. One of the notable tool is the Hyperledger fabric \cite{Hyperledger/fabric:Privacy.}, a distributed ledger user for developing blockchain applications and can have private blockchain for serving the applications to specific services. The fabric consists of model file, script file, access file and query file and all zipped together to form business network archive. Fabric has a concept called "Chaincode", which is similar to Ethereum smart contract for performing secured blockchain transactions. We can also include the distributed file storage i.e. Interplanetary File System (IPFS), which store the data and the data can be shared across the nodes in the blockchain. For example, A decentralized web application can be hosted with content stored in IPFS for serving web content to users. Overall, Hyperledger is very useful platform for blockchain technology and have been widely using for developing the applications including DDoS mitigation.
\subsection{Emerging Technology Network Architectures}
Some of the notable recent technologies such as IoT, SDN and cloud computing essentially changed network paradigm. It is important to review these advanced network architectures to study the advanced DDoS attacks exploiting the architecture limitations and propose the new solutions to mitigate these attacks using blockchain technology.
\subsubsection{IOT Architecture}
IoT is a system of computing devices including the physical objects with network connectivity to connect to internet and transfer the data over the network with or without requiring the human interaction. The tremendous progress towards smart homes, smart cities, smart transportation, and smart grid applications in recent years shows that rapid advancements in Internet of Things (IOT) technology. Gartner predicted that there will be 65 billion IOT devices connected to the internet by 2025 and the current statistics show that around 31 billion IOT devices deployed and connected to internet \cite{Girad2020TheToday}. Figure ~\ref{IoTArch} depicts a typical IoT architecture with main components. The IoT devices can be sensors, actuators or other appliance installed in home, industry, person body, vehicle, farming platform to monitor or sense the current state or activity and pass the information to the nearest IoT gateway through wireless communication like Bluetooth, Wi-Fi, NFC and ZigBee. The IoT gateways connected to the public internet for sending the information to IoT service provider for data analytics, tracking the status, display in user console etc. Using IoT network protocols such as MQTT, AMP, HTTP and CoAP but not limited. Owing to the limited CPU, memory, and power capabilities of IoT devices and the existence of the multivendor IoT platforms, conventional security solutions are not compatible in IoT environment and securing IoT devices is challenging.
\begin{figure}[!h]
\centering
\captionsetup{justification=centering}
\includegraphics[width=8.5cm, height =7cm]{image/F4IoTArch.png}
\centering
\caption{A typical IoT Architecture. \label{IoTArch}}
\end{figure}
\subsubsection{SDN Architecture}
Recent advances in wide area networks (WAN) and data center networks are the culmination of the SDN paradigm. SDN enable logically the centralized management of network layer 2 and layer 3 devices such as Switches and Routers, including the management of wide area networks of the organizations where the network devices located from multiple sites are monitored/controlled using an SDN controller \cite{Govindarajan2014ASolutions}. As depicted in Figure \ref{SDNArch}, the central controller monitors manage all the network device in data plane layer and communicated through southbound API like Openflow standard. A network administrator can develop the applications on top of the control layers to perform network management operations. SDN technology can be used at the autonomous system level, internet service provider level or data center level for network monitoring and management. Although SDN provides lot of advantages including programmability, centralized control, and security, it also inherits security vulnerabilities due to the new architecture paradigm. For instance, an adversary may target the controller with TCP SYN flooding attack and other protocol exploitation techniques to saturate the controller and shutdown the whole network \cite{Boppana2020AnalyzingNetworks}. Leveraging the blockchain technology open up new research possibilities to secure the Software defined network itself from malicious denial of service attempts \cite{Huo2020ANetworking} as well as mitigation of the denial of service attacks in conventional networks.
\begin{figure}[!h]
\centering
\captionsetup{justification=centering}
\includegraphics[width=8 cm, height=8cm]{image/F5SDNArch.png}
\centering
\caption{A typical SDN Architecture} \label{SDNArch}
\end{figure}
\subsection{Related Work}
Technologies such as machine learning (ML), blockchain, IoT, and SDN are well suited to improve the security in digital world but also exhibit new security concerns and issues \cite{Lin2017AChallenges} \cite{Ahmad2019AThings} \cite{Muthanna2019} \cite{Wang2019TheSurvey}\cite{Gao2018SecurityNetworks}\cite{Boppana2020AnalyzingNetworks}\cite{Qu2018}\cite{DantasSilva2020}\cite{Dwivedi2019}\cite{Rathore2019}. Some researchers also used combinations of these technologies to address security challenges ranging from malware analysis, DNS Security, to network security as well as privacy issues \cite{Hussain2020MachineChallenges} \cite{daCosta2019InternetApproaches} \cite{Dong2019AEnvironments}\cite{Liu2020ALearning} \cite{Elagin2020}. Our focus in this paper is specific to DDoS-attack detection and mitigation techniques in conventional networks, software defined networks, cloud environments and internet of things \cite{Srivastava20AMechanisms} \cite{Dong2019AEnvironments} \cite{BeslinPajila2020DetectionSurvey} \cite{Sonar2014AThings} \cite{Park2017}. The currently known techniques include machine learning or deep learning (ML/DL) algorithms to classify the attacks, anomaly-based detection, and signature-based detection.
A recent advancement in peer to peer networks with blockchain technology enabled utilization of decentralized network concepts for multiple application areas like finance, healthcare, real estate, supply chain management, security \cite{BlockchainInsider}. Although blockchain mainly provides the anonymity, privacy and secured data storage in security applications, researchers also explored the applicability of blockchain technology in DDoS attack information sharing, threat intelligence information sharing to quickly respond to the DDoS attacks. Singh et al. \cite{Singh2020UtilizationAttacks} present a survey of DDoS mitigation techniques using blockchain technology. The authors considered four known blockchain based DDoS mitigation approaches for comparison; highlighted the operation of these mitigation mechanisms and assessed the practical applicability of these implementations \cite{Rodrigues2017AContracts} \cite{BurgerZurich2017CollaborativeBlockchains} \cite{Javaid2018MitigatingBlockchain}\cite{Kataoka2018TrustSDN}. Wani et al. \cite{Wani2021} discussed the prior art distributed denial of service attack mitigation using blockchain by describing the methodology on how the related papers are collected and proposing the taxonomy based on the technologies like artificial intelligence, information sharing capability and blockchain types. However, a comprehensive and systematic review of the state-of-the-art work with classification based on the solution implementation location by leveraging the blockchain technology to detect and mitigate the DDoS attacks in digital world and also detail description of DDoS attacks targeting Blockchain platforms to protect decentralized networks is not covered in the prior art. Our motivation for this work is to bridge the knowledge gap between network security researchers and the blockchain developing community, and enable the researchers to access this article as a reference point to continue the research of using blockchain technology in network security.
\section{RELATED WORK}
\label{sec:relatedwork}
Technologies such as machine learning (ML), blockchain, internet of things (IoT), and software defined networking (SDN) are well suited to improve the security in digital world but also exhibit new security concerns and issues \cite{Lin2017AChallenges} \cite{Ahmad2019AThings} \cite{Muthanna2019} \cite{Wang2019TheSurvey}\cite{Gao2018SecurityNetworks}\cite{Boppana2020AnalyzingNetworks}\cite{Qu2018}\cite{DantasSilva2020}\cite{Dwivedi2019}\cite{Rathore2019}. Some researchers also used combinations of these technologies to address security challenges ranging from malware analysis, Domain Name System (DNS) Security, to network security as well as privacy issues \cite{Hussain2020MachineChallenges} \cite{daCosta2019InternetApproaches} \cite{Dong2019AEnvironments}\cite{Liu2020ALearning} \cite{Elagin2020}. Our focus in this paper is specific to DDoS-attack detection and mitigation techniques in conventional networks, software defined networks, cloud environments and internet of things \cite{Srivastava20AMechanisms} \cite{Dong2019AEnvironments} \cite{BeslinPajila2020DetectionSurvey} \cite{Sonar2014AThings} \cite{Park2017}. The currently known techniques include machine learning or deep learning (ML/DL) algorithms to classify the attacks, anomaly-based detection, and signature-based detection.
A recent advancement in peer to peer networks with blockchain technology enabled utilization of decentralized network concepts for multiple application areas like finance, healthcare, real estate, supply chain management, security \cite{BlockchainInsider}. Although blockchain mainly provides the anonymity, privacy and secured data storage in security applications, researchers also explored the applicability of blockchain technology in DDoS attack information sharing, threat intelligence information sharing to quickly respond to the DDoS attacks. Singh et al. \cite{Singh2020UtilizationAttacks} present a survey of DDoS mitigation techniques using blockchain technology. The authors considered four known blockchain based DDoS mitigation approaches for comparison; highlighted the operation of these mitigation mechanisms and assessed the practical applicability of these implementations \cite{Rodrigues2017AContracts} \cite{BurgerZurich2017CollaborativeBlockchains} \cite{Javaid2018MitigatingBlockchain}\cite{Kataoka2018TrustSDN}. Wani et al. \cite{Wani2021} discussed the prior art distributed denial of service attack mitigation using blockchain by describing the methodology on how the related papers are collected and proposing the taxonomy based on the technologies like artificial intelligence, information sharing capability and blockchain types. However, a comprehensive and systematic review of the state-of-the-art work with classification based on the solution implementation location by leveraging the blockchain technology to detect and mitigate the DDoS attacks in digital world and also detail description of DDoS attacks targeting Blockchain platforms to protect decentralized networks is not covered in the prior art. Our motivation for this work is to bridge the knowledge gap between network security researchers and the blockchain developing community and enable the researchers to access this article as a reference point to continue the research of using blockchain technology in network security.
\section{DDoS Attacks Mitigation using Blockchain}
\label{sec:ddosusingbc}
In this section, the existing research works on solving the DDoS attack detection and mitigation problem using blockchain technology is presented and discussed. In addition to blockchain, the role of technologies such as SDN, IoT and ML/DL in addressing DDoS attacks near the attacker domain location, the internet core, or near the victim network domain are reviewed.
We discuss the existing DDoS mitigation blockchain solutions based on the location of solution deployment in internet architecture.
\subsection{Network level mitigation}
The network level mitigation DDoS mitigation schemes using blockchain technology is deployed at the Internet service provider (ISP) level on the internet, which may be far from attacker or victim location. The Table \ref{DDoSusingBC} illustrates the blockchain key concepts used, technologies involved in the research works proposed for DDoS mitigation using blockchain. We can clearly see that smart contract based Ethereum network is used for implementing the DDoS mitigation solutions for most of the previous contributions, as shown in the Table \ref{DDoSnearNW}. The blockchain access level policy is controlled by the owners to make the transactions accessible for public or private. \\
Tayyab et al. \cite{Tayyab2020ICMPv6} take the approach that each IDS in the network acts as a blockchain node and collaborate with other blockchain IDS nodes to share the attack information like correlated alarms. This decentralized correlated information sharing is used for the detection of ICMP6 based DDoS attacks. Although IDS collaboration improves DDoS attack detection capability, the practical implementation of collaboration can may have difficulties. For example, the IDS vendor interoperability to support the blockchain technology is needed in enterprise environment. Denial of service attacks detection at the IDS level is too late and might already congest the edge network communication channels or the content delivery network communications. \\
The following papers \cite{Yeh2020SOChain:Blockchain} \cite{Pavlidis2020OrchestratingCollaborations} \cite{Abou2019Co-IoT:SDN}\cite{Yeh2019ATechnology} \cite{Essaid2019ARNN-LSTM} \cite{Yang2019AServices} \cite{Abou2019Co-IoT:SDN} \cite{Kataoka2018TrustSDN} \cite{Rodrigues2017Multi-domainBlockchains} \cite{BurgerZurich2017CollaborativeBlockchains}
\cite{Rodrigues2017AContracts}
\cite{Rodrigues2017EnablingBloSS} focused on utilizing the SDN and blockchain technologies in the autonomous system (AS) level to detect the denial of service attempts and activating the DDoS mitigation mechanisms at the network level. The authors considered the autonomous system consists of SDN architecture, controlled by SDN controller. The core concept in these papers include leveraging the centralized controller application of the SDN to manage how the network devices in the autonomous system should handle the traffic (whitelist/blocklist) originating from malicious IP addresses, which are used to launch the DDoS attacks on the autonomous system. The SDN controller node also acts as a blockchain node running decentralized application like Ethereum to store or validate the attack IP address list, and their blocklist/whitelist status as a transaction in the blockchain, and distribute the added transactions to all the nodes (SDN controller in other autonomous systems) in the blockchain. Ethereum smart contracts were used to store the IP addresses with malicious flag status as a transaction. The DDoS detection/mitigation mechanism was tested in Ethereum testing platform Rapsten testing network and also used Ganache for testing in local blockchain network \cite{Paul2018DeployMedium}. \\
\begin{table*}[!h]
\captionsetup{justification=centering}
\caption{DDoS mitigation near network using Blockchain\label{DDoSnearNW}}
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Title} & \textbf{Blockchain} & \textbf{Type} & \textbf{Consensus} & \textbf{Technologies} \\
\hline
Yeh et al. \cite{Yeh2020SOChain:Blockchain} & Ethereum & Consortium & Proof of Work & Smart contracts, Swarm, DOS, Bloom filter \\ \hline
Yang et al. \cite{Yang2019AServices} & Ethereum & Permission & Proof of work & Smart Contract \\ \hline
Yeh et al. \cite{Yeh2019ATechnology} & Ethereum & Consortium & Proof of work & Smart contract, Swarm, Oracle \\ \hline
Rodrigues et al. \cite{Rodrigues2017Multi-domainBlockchains} & Ethereum & Public & Proof of Work & Smart Contract, SDN and VNF. \\ \hline
Burger et al. \cite{BurgerZurich2017CollaborativeBlockchains} & Ethereum & Public & Proof of Work & Smart Contract, Bloom filter \\ \hline
Rodrigues et al. \cite{Rodrigues2017AContracts} & Ethereum & Public & Proof of Work & SmartContract, SDN \\ \hline
Rodrigues et al. \cite{Rodrigues2017EnablingBloSS} & Ethereum & Consortium & Proof of Work & Smart Contract, IPFS, SDN \\ \hline
Hajizadeh et al. \cite{Hajizadeh2020} & Hyperledger Fabric & Private & Kafka & Chain code, SDN,
Threat Platform \\ \hline
Essaid et al. \cite{Essaid2019ARNN-LSTM} & Ethereum & Public & Proof of work & Smart Contract, Deep learning(LSTM), SDN\\ \hline
Aujla et al. \cite{Aujla2020BlockSDN:Applications} & Generic & Private & - & SDN \\ \hline
Shafi et al. \cite{Shafi2019DDoSThings} & Hyperledger & - & Kafka & SDN, IoT \\ \hline
Pavlidis et al. \cite{Pavlidis2020OrchestratingCollaborations} & Ethereum & Public, Private & Proof-of-Authority & Smart Contract \\ \hline
Abou et al. \cite{Abou2019Co-IoT:SDN} & Ethereum & Public & Proof of work & Smart Contract, Software Defined Networking \\ \hline
\end{tabular}
\end{table*}
Yeh et al. \cite{Yeh2020SOChain:Blockchain}, Yeh et al. \cite{Yeh2019ATechnology}, Shafi et al. \cite{Shafi2019DDoSThings} and Hajizadeh et al. \cite{Hajizadeh2020} discussed the threat information sharing including DDoS threat data among the collaborators for secure data sharing using blockchain based smart contracts technology and decentralized data storage. The Security operation centers can be upload the threat data and ISP act as verifier to confirm the illegitimacy of the threat data prior to adding to the blockchain transaction in \cite{Yeh2020SOChain:Blockchain}, \cite{Yeh2019ATechnology}. The Ethereum based smart contract implementation for DDoS data sharing is performed for evaluation. But, in \cite{Hajizadeh2020} and \cite{Shafi2019DDoSThings}, the Hyperledger caliper is used to implement the threat information sharing among the organizations. Each organization may have the SDN controller to run the blockchain application and act as a blockchain node for updating the threat information in other nodes. \\
Rodrigues et al. \cite{Rodrigues2017Multi-domainBlockchains} \cite{Rodrigues2017AContracts} \cite{Rodrigues2017EnablingBloSS} proposed the Ethereum based architecture for DDoS mitigation and their hardware implementation to allow or block the malicious IP addresses in the ISP level. Each transaction may include the IP address and their status to detect the malicious IP address performing the denial of service attacks. The main limitation of the IP address data storage in the transactions may have limitations. But, Burger et al. \cite{BurgerZurich2017CollaborativeBlockchains} discussed that Ethereum is not an ideal technology for DDoS attack IP based signaling using blockchain due to the scalability issue. The authors also mention that Ethereum smart contracts can be applicable for small number of IP addresses space related applications. They recommend that storing the list of IP address in a file storage like IPFS, and the URL of the storage location can be pointed to the blockchain transactions, and the location integrity is verified using hash value. \\
Pavlidis et al. \cite{Pavlidis2020OrchestratingCollaborations} proposed a blockchain based network provider collaboration for DDoS mitigation. The AS's are selected based on the reputation scores to participate in the DDoS mitigation plan. The programmable data planes are used to implement the mitigation mechanism for DDoS attacks, which is in contrast to most of the works using SDN Openflow protocol.
\begin{table*}[!h]
\captionsetup{justification=centering}
\caption{Advantages and limitations of near network based Blockchain solutions} \label{DDoSusingBC}
\begin{tabular}{|l|p{4.5cm}|p{4.5cm}|p{3.4cm}|l|}
\hline
\textbf{Title} & \textbf{Objective} & \textbf{Advantage} & \textbf{Limitations}\\
\hline
Yeh et al. \cite{Yeh2020SOChain:Blockchain} & Decentralized DDoS info sharing
& SOC may use DDoS data among peers & Selecting the data certifier is challenging \\ \hline
Yang et al. \cite{Yang2019AServices} & Blockchain based DDoS mitigation services & Client validation and provider authentication & Spoofed IP's are ignored \\ \hline
Yeh et al. \cite{Yeh2019ATechnology} & Collaborative DDoS info sharing & SOC info share platform & Spoofed IP's are ignored \\ \hline
Rodrigues et al. \cite{Rodrigues2017Multi-domainBlockchains} & Blockchain based DDoS mitigation architecture & First architecture for DDoS and Blockchain & Spoofed IP's are ignored \\ \hline
Burger et al. \cite{BurgerZurich2017CollaborativeBlockchains} & Scalable Ethereum based DDoS detection & Practical implementation & Questions on Ethereum usage \\ \hline
Rodrigues et al. \cite{Rodrigues2017AContracts} & Blockchain architecture and design for DDoS & Detection and mitigation also included & not for spoofed IP \\ \hline
Rodrigues et al. \cite{Rodrigues2017EnablingBloSS} & Ethereum testbed for DDoS mitigation & Tested on hardware & Scalability \\ \hline
Hajizadeh et al. \cite{Hajizadeh2020} & Blockchain based threat intelligent platform & Important security application & Fault tolerance \\ \hline
Shafi et al. \cite{Shafi2019DDoSThings} & Mitigate the IoT based DDoS attempts in SDN & - & Not support for non-SDN \\ \hline
Essaid et al. \cite{Essaid2019ARNN-LSTM} & DL and smart contract DDoS detection & DL based & Standard dataset \\ \hline
Pavlidis et al. \cite{Pavlidis2020OrchestratingCollaborations} & collaborative DDoS mitigation at the AS level & Network level DDoS mitigation & Difficult to identify slow DDoS attacks \\ \hline
Abou et al. \cite{AbouElHouda2019Cochain-SC:Contract} & Intra-domain and inter-domain DDoS mitigation & Effective DDoS mitigation & Spoofed IP's are ignored \\ \hline
\end{tabular}
\end{table*}
In the papers \cite{Manikumar2020BlockchainTechniques} \cite{Essaid2019ARNN-LSTM}, the machine learning algorithms such as K-nearest neighbors (KNN), decision tree and random forest as well as deep learning technique long short-term memory (LSTM) are applied to the network traffic to determine the DDoS attack and considered blockchain technology to whitelist/blocklist the IP addresses at the autonomous system level of the network. But, the machine learning application on the network traffic requires infrastructure and computation capabilities, and ownership responsibility to allocate the resources need to be addressed. Any specific entity like ISP, security service providers will not be interested to perform data analytics unless they have any monetary benefits or business advantages. \\
Overall, we can clearly see that the combination of SDN in AS level and Ethereum smart contract can be implemented to track the IP addresses status and update all the nodes across the internet to mitigate the DDoS attacks. However, there are some limitations like blockchain integration with legacy networks, handling spoofed IP addresses need to be solved for adopting the blockchain based DDoS mitigation in the network level.\\
\subsection{Near attack domain location}
The DDoS attacks mitigation at the attacker network is an effective way to handle DDoS attacks, as the attack traffic will not be propagated to the internet network. Most of the latest DDoS botnets are formed by compromising the legitimate IoT devices located all over the internet and target the victims to send malicious network traffic. So, detection and mitigation of IoT botnets at the source network in essential. Chen et al. \cite{Chen2020ADevices} focused on detecting and mitigating IoT based DDoS attacks or botnets in IoT environment using blockchain. The edge devices or IoT gateways acts as a blockchain node to perform transactions when a network anomaly or attack detected in the IoT environment. The techniques used for network traffic analysis in the paper include statistical analysis, conventional bot detection techniques like community detection. The smart contracts are used to write attack alerts data in transactions and Ethereum network distribute the data across the IoT nodes. But, the IoT gateway nodes are not usually customer-centric and deploying the blockchain client application in the gateway is challenging for real-time production environment. \\
\begin{table*}[!h]
\captionsetup{justification=centering}
\caption{DDoS mitigation near attack location using Blockchain.\label{DDoSNearattack1}}
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|p{3.2cm}|l|l|l|l|}
\hline
\textbf{Title} & \textbf{Blockchain} & \textbf{Type} & \textbf{Consensus} & \textbf{Technologies}\\
\hline
Chen et al. \cite{Chen2020ADevices} & Ethereum & Public
& Proof of work & Smart contract, IOT \\ \hline
Javaud et al. \cite{Javaid2018MitigatingBlockchain} & Ethereum & Public & Proof of work & Smart Contract, IoT \\ \hline
Sagirlar et al. \cite{Sagirlar2018AutoBotCatcher:Things} & Hyperledger
(Future work) & permission & BFT & IoT, Chaincode \\ \hline
Spathoulas et al. \cite{Spathoulas2019CollaborativeBotnets}
& Ethereum (Future work) & Public & Proof of work & IoT, Smart Contract \\ \hline
Abou et al. \cite{Abou2019Co-IoT:SDN} & Ethereum & Permission & Proof of work & SDN, IOT \\ \hline
Kataoka et al. \cite{Kataoka2018TrustSDN} & Ethereum & Public, Private & Proof of work & Smart Contract, SDN, IoT \\ \hline
\end{tabular}%
}
\end{table*}
Javaid et al. \cite{Javaid2018MitigatingBlockchain} discussed the blockchain based DDoS attack detection on the servers connected to the IoT devices. The IoT devices sending data to the server is approved by the Ethereum network with an expense of gas cost. When a rogue IoT device trying to send the malicious network traffic, the IoT device is penalized with high gas cost and only trusted devices are approved for connecting to the network. The integration of the IoT with Ethereum enables the denial of service mitigation on the IoT device connected servers. Sagirlar et al. \cite{Sagirlar2018AutoBotCatcher:Things} proposed a blockchain solution for detecting the IoT related peer to peer botnets. The assumption is that botnets frequently communicate to each other to perform malicious activity. The authors mentioned that the network traffic between the botnet nodes are considered as blockchain transactions in permissioned Byzantine Fault Tolerant (BFT) and use these transactions to identify the botnet IoT devices. The proposal method may not be a viable solution, as the network traffic flows are enormous and blockchain may not accommodate the transaction capacity needed for storing in blockchain nodes. \\
Spathoulas et al. \cite{Spathoulas2019CollaborativeBotnets} presented an outbound network traffic sharing among the blockchain enabled IoT gateways to detect the IoT botnet. The authors performed simulations on the proposed solution and showed the promising results using detection efficiency parameter. But, the solution is not tested in the real blockchain nodes installed in the gateway and mentioned that Ethereum smart implementation is one of their future work. But, in general, the IoT gateways are multivendor devices and interoperability among the devices is an issue.\\
Abou et al. \cite{Abou2019Co-IoT:SDN} discussed collaboration among the autonomous systems to detect the DDoS attacks. Each AS contain SDN controller, in which blockchain application like Ethereum client is installed to distribute the malicious IP addresses among other AS's. Whenever a malicious IP address is identified in the AS, the SDN controller updates to the Ethereum client and then Ethereum clients update to all the SDN controller in the AS's for DDoS detection and mitigation. To implement this solution, the AS's should support the same SDN controller and agree to collaboratively work for DDoS mitigation. Kataoka et al. \cite{Kataoka2018TrustSDN} presented a similar \cite{Abou2019Co-IoT:SDN} blockchain and SDN based architecture for whitelisting the IoT devices in the network. The trusted profile consist of IoT devices will be stored in smart contract based blockchain transaction and the SDN controller will update all the switches and routers in the SDN network. This implementation enable the malicious or IoT botnets will be blocked in the attack network itself and protect the networks. Considering there is a huge number of IoT devices connected to internet approximately 31 billion devices as of 2020, the implementation of the blockchain for each gateway in IoT environment is challenging and practically impossible. In addition, the IoT gateway vendors interoperability and supporting the blockchain nodes just for the sake of DDoS detection and mitigation may not seem to be reasonable with the current state-of-the-art technology. \\
\begin{table*}[!h]
\captionsetup{justification=centering}
\caption{Advantages and limitations of near attack location based blockchain solutions} \label{DDoSNearattack}
\begin{tabular}{|p{2.4cm}|p{5cm}|p{4.5cm}|p{4cm}|}
\hline
\textbf{Title} & \textbf{Objective} & \textbf{Advantage} & \textbf{Limitations}\\ \hline
Chen et al. \cite{Chen2020ADevices} & IoT based DDoS detection using blockchain & The Attacks can be stopped at the source network & Practically may not be viable \\
\hline
Javaid et al. \cite{Javaid2018MitigatingBlockchain} & Ethereum and IoT integration for DDoS & Automated control of the server IoT inbound traffic & Only applicable to server DDoS \\ \hline
Sagirlar et al. \cite{Sagirlar2018AutoBotCatcher:Things} & IoT botnet detection using BFT. & First blockchain-based IoT botnet detection & May not be scalable \\ \hline
Spathoulas et al. \cite{Spathoulas2019CollaborativeBotnets} & IoT botnets detection using blockchain & Outbound traffic exchange using IOT gateway & Not practically implemented \\ \hline
Abou et al. \cite{Abou2019Co-IoT:SDN} & AS level SDN and blockchain solution & Network level DDoS detection & AS legacy networks issue \\ \hline
Kataoka et al. \cite{Kataoka2018TrustSDN} & IoT botnets detection using SDN and blockchain & Attacker location based detection & Not applicable to non SDN based IoT \\ \hline
\end{tabular}
\end{table*}
\subsection{Near Victim Location}
Yang et al. \cite{Yang2019AServices} proposed a real-time DDoS mitigation service leveraging a consortium based or permissioned blockchain. Each DDoS service provider has an account in the permission blockchain to provide DDoS mitigation service. The victim looks for the attacker IP-AS mapping in the blockchain, and the trusted service provider IP tagged with AS is authorized to provide the DDoS mitigation service. The authors also proposed the reputation or credibility validation mechanism of the service providers. However, if the attack IP is spoofed, the author's proposed blockchain based DDoS mitigation service is not applicable. Kyoungmin Kim et al. \cite{Kim2018DDoSBlockchain} proposed a decentralized CDN service to mitigate the DDoS attacks with the help of private blockchain and particularly used by government and military agencies to protect their service. The victims usually the service providers hosting the web content servers. They can protect the servers using the decentralized the CDN services.
The context of the attacker and victim location may be changed based on the attack type and how the attack is conducted. For example, an attacker may use their infrastructure to send the malicious traffic. In this case, the blockchain based solutions proposed in the attacker domain can be considered as near attacker based solutions. Additionally, the attacker compromise the legitimate IoT devices and use them as a botnet to attack another victim. Here, the solutions deployed in the IoT device locations also comes under near attacker based solutions. The solutions solely implemented in the main victim (not the legitimate IoT bot owner victim) are considered under the Near victim location based solutions. We can say that near the victim based solution research articles are far too less than the network based and near attacker based solutions. It is too late to mitigate the DDoS attacks near the victim. So, the existing solutions mainly focused on the network level or near attacker.
\subsection{Hybrid solutions}
The hybrid DDoS detection and mitigation solution can be the combination of the network based, near attacker location and the near victim location based solution. For effective mitigation of the DDoS attacks, the multi level mitigation solutions are needed. But, the implementation of these solutions require the collaboration among stakeholders. Abou et al. \cite{AbouElHouda2019Cochain-SC:Contract} proposed intra domain and inter domain DDoS detection and mitigation solution using blockchain. The intra-domain detection include near the victim based solution and inter domain detection meaning that network based solution. The Ethereum smart contract is deployed in each AS to distribute the DDoS threat information and the SDN controller is used to update the AS network traffic filtering rule to block the malicious traffic for inter domain DDoS mitigation. On the other hand, the traffic from switches and routers in the same domains are monitored using SDN controller applications and apply the flow control rules in switches/routers using open flow switch protocol. This mechanism mitigate the internal attacks originating from the same domain. Based on our research, there is limited work done on proposing solutions in multi levels of internet architecture and scope for new research contributions in this area.
\section{Future Directions}
\label{sec:futuredirection}
In this section, the future directions of dealing with DDoS attacks using blockchain technology is explored. We have presented the research directions in terms of the advancements in blockchain and how these advancements can be used to address the DDoS attacks.
\subsection{Internet of Blockchain}
The current blockchain technologies like Bitcoin or Ethereum smart contracts transaction process is sequential and hence, it is very slow to add the transactions in the blockchain. To solve the scalability and interoperability issue between blockchain nodes, internet connected blockchain has been proposed and can concurrently process the transactions from different blockchains. Paralism \cite{2019ParalismBlog} built the blockchain infrastructure with unlimited scalability and digital economy platform supported by parallel blockchain. Customized script and chain virtualization make paralism support any amount of sub-chains and independently operated chain-based applications and also become the backbone of the internet in decentralized world. This technology is in the early stages of the development and lot of scope to work on utilizing parallel blockchain to share the threat data across the blockchain applications and protect denial of service attacks. We also think that the parallel blockchain surfaces new security issues including leaking the information between the blockchain applications and will be the topic to focus for researchers while building the blockchain internet backbone. Another notable advancement in the blockchain is Xrouter, which acts as blockchain router to communicate one blockchain like bitcoin to smart contracts, supporting interchain and multichain services \cite{2019IntroducingRouter}.
\subsection{Programmable data planes (P4) for Blockchain based DDoS Solutions}
The network paradigms keep changing as the new technology trends emerged in the enterprises. The Internet of Things supports IP protocol and IoT application protocols MQTT, XMPP, AMQP etc. The denial of service attacks can be carried by leveraging the weaknesses in the protocol and flooding the traffic on the victim machine. The combination of Programmable data planes at the gateway level and the blockchain technology for sharing the attack data is effective for mitigation of the attacks. The P4 device in the switch level that can parse any type of network protocol and makes easy for applying the blockchain technology. We envision that the future work would be proposing new architecture with P4 for mitigation of attacks, developing smart contracts for the gateway level device to monitor and mitigate the attacks using Programmable data planes.
\subsection{Threat Information Sharing using Blockchain}
Consortium or private based blockchains are most compatible for sharing the threat information among the Blockchain participants. Numerous Ethereum based techniques has applied to share the information with integrity and anonymity. Leveraging the decentralized file storage such as swarm, IPFS enables to store the information rather than keeping the data in transactions and causing time delay to process the sequential transactions. We believe that the information sharing field using blockchain requires improvement and architecture changes to implement secured information sharing network.
\subsection{Ethereum 2.0 Network for DDoS mitigation}
DDoS solutions implemented using Ethereum network \cite{Yeh2019ATechnology} \cite{Abou2019Co-IoT:SDN}faces scalability, speed challenges, in particular transactions refer to allow or block attack IP addresses. Ethereum 2.0 has been proposed and implemented for the last few years \cite{ReneMillman2020WhatDecrypt}. From August 2020, the upgradation to Ethereum 2.0 is initiated with three phases to complete the process. ETH 2.0 works-based proof of stake (POS) rather than POW, which is a major change and the upgradation supports the drastic increase in network bandwidth, Lower Gas Costs and benefit for scalability of the network. We envision implementing the DDoS mitigation scheme in Ethereum 2.0 in the near future.
\section{Open Challenges}
\label{sec:openchallenges}
In this section, we discuss the research challenges to leverage the blockchain technology for DDoS attack detection and mitigation solutions. The detail description of the decentralized technologies adoption in conventional network issues are presented to handle the DDoS attacks.
\subsection{Integration with Legacy Network}
Distributed denial of service attacks mitigation involves the network operators, internet service providers and edge network service providers to respond and block the malicious actor traffic. These stakeholders run the network services in legacy platforms and has been providing services for decades and adapting to the decentralized blockchain technology is a major concern. The reasons could be the lack of memory and computation requirements for blockchain in legacy networks \cite{Hajizadeh2020}, trust on the technology, unavailability of blockchain professional workforce, fear of failure to protect customers while using blockchain. In addition, a collaboration between the ISP's is required to share the malicious data indicators among the ISP's and all the stakeholder's may not be comfortable, as there is no monetization aspect for the internet service providers and usually only benefited by the attack victims. So, a responsible organization or service provider should be stepped up to coordinate among the stakeholders and make sure the involved stakeholders get benefited.
\subsection{Bitcoin/Ethereum P2P Network Zero-Day Vulnerabilities}
The Blockchain transactions process include the network traffic passing through the internet from one node and other nodes in the network; the cryptocurrency exchanges can also act as a blockchain node on behalf of the client and perform the transactions in the exchange conventional network. The attack vector for the blockchain is quite broader and the cost of a single vulnerability in the applications is in millions of dollars. For instance, a parity check vulnerability in Ethereum causes lost \$300 million dollars \cite{2017HowNoon} and a small bug found in cryptocurrencies has a huge impact on the decentralized network. It is also important to note that the cryptocurrency exchanges having conventional network will have a major consequence to impact the P2P applications. We envision that there is a scope to progress for developing the flawless applications and monitoring the traffic for illegitimate activity detection.
\subsection{Lack of Blockchain P2P Network Datasets}
Monitoring the anomalous behavior of the blockchain network traffic and transactions dataset using machine learning and deep learning techniques is one of the solutions for detecting the DDoS attacks proposed in the prior art \cite{Tayyab2020ICMPv6} \cite{Rathore2019}. But there are very few datasets available in public for continuing research and improving the detection metrics. Mt.Gox exchange trading activity data from 2011 to 2013 is available for public to use for research purpose \cite{VasekMarie2014}. The quality of the data and how older the data is questionable for testing and detecting the real time attacks. We believe that having standard datasets and application of big data analytics in the future is a must requirement for research progress in DDoS detection in cryptocurrency networks.
\subsection{Spoofed IP DDoS Attacks Detection}
The proposed solutions for DDoS attacks detection mainly identifies the source IP address and use blockchain technology to store the transactions and share the IP address among the stakeholders to block/whitelist the IP address with trust and validation at the network level \cite{Abou2019Co-IoT:SDN}\cite{Yeh2019ATechnology} \cite{Essaid2019ARNN-LSTM} \cite{Yang2019AServices} \cite{Abou2019Co-IoT:SDN} \cite{Kataoka2018TrustSDN} \cite{Rodrigues2017Multi-domainBlockchains} \cite{BurgerZurich2017CollaborativeBlockchains}. These solutions assume that the originating malicious IP addresses are not spoofed, and this condition is not always true. In most of the scenarios, as seen in Table \ref{AttacksHistory}, the attacker performs a reflection attack, in which the spoofed traffic is being sent to the victim to consume the communication capacity or saturating the CPU or memory resources for successful DDoS attack. The researchers also not addressed the IPv6 traffic and can be critical storing the IP version 6 data in blockchain in terms of memory consumption.
\subsection{IOT and SDN Vendor Interoperability}
The existing state-of-art essentially utilized the software defined networks and internet of things technology to address the denial of service attacks either at the victim level or network level. Even though those solutions prove that the attacks can be mitigated, there is a real challenge when trying to adopt the techniques in industry. The IoT device or gateway vendors are quite diversified and there are multitude of SDN supporting network device providers for enterprise solution. We tend to see incompatibility issue and also supporting blockchain node issues in these network paradigms and deploying a decentralized application across their stakeholder network is impractical. It is desirable to depend on the Blockchain based DDoS mitigation as a service solution like Gladius \cite{2017GladiusMedium}.
\section{Conclusion}
\label{sec:conclusion}
Blockchain is emerged as a disruptive technology in recent times and the blockchain application capabilities are promising to use in the field of cybersecurity. DDoS attacks are well known and still considered as a major threat to disrupt the businesses. We have performed a detailed review of the blockchain based solutions for DDoS attacks detection and mitigation including the consideration of the different network environments such as SDN, IoT, cloud or conventional network. The solutions are categorized based on the solution deployment location such as network based, near attack location, near victim location and hybrid solutions. We determined that most of the existing solutions focused on storing the malicious IP addresses in blockchain transactions implemented using smart contract and distribute the IP addresses across the AS's in the network level. However, limited research is performed to propose near victim location and hybrid solutions. Finally, we described the open challenges based on the existing research contributions and the future directions based on the advancements in blockchain technologies like parallel blockchain, Xroute, Ethereum 2.0 to effectively handle the DDoS attacks.
We believe that our review will be a great reference resource for readers and the future researchers interested to pursue the research in the combination of Blockchain and DDoS attacks domain.
\section{Proof}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\end{document}
| {
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} | 9,452 |
Dear Science: Why can't we just get rid of all the mosquitoes?
(Rachel Orr/Washington Post illustration; iStock)
By Rachel Feltman and Sarah Kaplan
August 1, 2016 at 11:00 AM EDT
Dear Science: I seem to be a total mosquito magnet, and nothing I try keeps the biters at bay. With serious illnesses like Zika to worry about, that has me wondering why we don't just get rid of the pests once and for all. What good do mosquitoes do?
Here's what science has to say:
Mosquitoes are our most deadly adversaries in the animal kingdom. While the "factoid" that floats around about malaria causing "half of all human deaths" throughout history is pretty silly, it's true that we have the insects to thank for some devastating diseases. Over 400,000 people were killed by malaria alone in 2015 – a death toll that until recently was nearly doubled. Warming climates and increased global travel have helped previously obscure viruses like Zika and chikungunya gain footholds across the globe. And even if you never face a life-threatening disease because of a mosquito bite, swelling and itching is hardly a pleasant experience.
So why don't we just zap them? Unfortunately, you shouldn't expect to read about the total eradication of mosquitoes anytime soon.
Dear Science: How many germs are actually on a toilet seat — and should I care?
Getting rid of all mosquitoes would be pretty nonsensical: You might only care to notice the species that bite you, but that's just 200 or so out of 3,000 varieties of mosquito, which inhabit every continent except for Antarctica. Even those biting species are largely harmless, and only certain species are even capable of carrying deadly illnesses. Aedes aegypti and Aedes albopictus alone are responsible for the transmission of Zika, dengue and chikungunya, while a few members of the genus Anopheles carry malaria.
Only female mosquitoes bite, even among these potentially deadly species, and only for a very brief portion of their life cycle – when they need nutrients to create eggs.
"They're trying to do the right thing," Ole Vielemeyer, infectious disease expert at Weill Cornell Medicine and NewYork-Presbyterian, told The Post. "They're just trying to have enough food for their offspring."
But during that stage of life, they might bite multiple victims (including those of different species). If a mosquito bites someone infected with an illness that can replicate inside the insect's gut instead of simply being digested, that virus or parasite can then sneak into a new host when the bug's saliva slips into a bite. It's not the infected blood itself being passed along, which is why you can't get HIV from a bug bite. Still, plenty of pathogens have evolved to thrive in mosquito saliva.
"Most of mosquitoes are harmless, but when they transmit diseases, horrible things can happen," Vielemeyer added. "The ones that carry disease have caused human suffering for millennia. So it's kind of a mixed bag."
Dear Science answers your questions about evolution
Many mosquitoes – harmless and otherwise – serve an important biological purpose. They can help pollinate plants as they feed on nectar (their usual food source, outside of that crucial blood meal period) and provide a vital source of food for larger animals. They're vital components of a complex ecosystem, just like every other living thing. Researchers in the Arctic worried that climate change would lead more animals to feast upon local mosquitoes, throwing the food web out of whack and leaving plants unpollinated. As it turns out, warmer water has actually produced an Arctic mosquito boom – but that's not great news for the baby caribou they feed upon.
The point is that just because humans hate mosquitoes doesn't mean that they can be wiped out without consequences.
"We see the mosquito as sort of this evil thing," Vielemeyer said. "But there are lots of positive things that they do in the ecosystem."
That doesn't mean that scientists have a live-and-let-live philosophy about these insects: Researchers are trying to eradicate mosquitoes. But they're focusing on the ones that cause the most harm. The aegypti mosquitoes that cause Zika and the like – mosquitoes that have evolved to live in urban areas – are a particularly attractive target.
"There is no visible end to this except a war against aegypti," Jo Lines of the London School of Hygiene and Tropical Medicine recently told the Guardian. "Otherwise this is going to go on for a thousand years."
Dear Science: When you lose weight, where does it actually go?
Many scientists are working on creating genetically modified males mosquitoes that can only father sterile offspring. If released into the population, these males could trick deadly mosquitoes into wasting their blood meals on useless eggs. Over the course of a few generations, the targeted species would dwindle. Trials in the Cayman Islands where several million of these modified mosquitoes were released resulted in a 96 percent population reduction.
Some have suggested simultaneously releasing bacteria that target males of the species (yes, that's a thing that exists), which would help cut down on the number of fertile mates available. Other studies are working on using bacteria to help mosquitoes digest these deadly viruses and parasites instead of giving them a place to grow.
Some argue that there's more work to be done before genetically modified mosquitoes are ready to save the day. But some more "natural" methods show promise: One recent study found that a species of jumping spider in East Africa likes to go after mosquitoes that have just taken a blood meal. Dumping hordes of arachnids into your home isn't going to save you from all mosquito-borne illness, but allowing natural mosquito predators to thrive certainly doesn't hurt.
Dear Science: Why does the hair on my head grow longer than the hair on my body?
All of this is little consolation if you're prone to probing proboscises. Lots of factors – from blood type to clothing color to body size – can make you a particularly appealing blood meal. If you're in an area where mosquitoes carry disease, you should always protect yourself with a repellant that contains one of the active ingredients recommended by the Centers for Disease Control and Prevention. Cover up as much as your skin as possible, and never sleep in a room with open doors or windows if you don't have access to screens or a mosquito net.
"It's always prudent to avoid getting bitten," Vielemeyer said, even if your region isn't known to host dangerous bugs. There's no knowing when a species might expand its historic stomping grounds, or when a disease might adapt to living in a new host. "Prevention is always better than dealing with the after effects," he said.
Have a question for Dear Science? Ask it here. | {
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} | 9,745 |
Zonsmalbi (Lasioglossum zonulum) är en biart som först beskrevs av Smith 1848. Zonsmalbi ingår i släktet smalbin, och familjen vägbin. Inga underarter finns listade.
Beskrivning
Ett slankt, svart bi med clypeus (munsköld) och panna upphöjda, mest tydligt hos hanen. Denne har dessutom en blekgul spets på munskölden, och en mörkbrun överläpp. Antennerna är mörka, även om de kan ha en svagt gulaktig undersida hos hanen. Båda könen har gula vingbaser. Honan har rödbrun behåring på mellankroppen. Hon har dessutom breda, vita hårband på tergit 2 till 4. Hanen har också hårband, men de är mindre, och finns bara på sidorna av tergit 2 och 3. Hanen har även lång, grå behåring på buken. Honans kroppslängd är 9 till 10 mm, hanens 7 till 10 mm. Fibblesmalbi är en förväxlingsart, men honorna kan skilljas åt på mellankroppens behåring: Hos fibblesmalbiet är den inte rödbrun, utan gråaktig. Hanarna är svårare att skilja åt, men det första bakfotssegmentet är mörkt hos zonsmalbiet, blekgult hos fibblesmalbiet.
Ekologi
Zonsmalbiet förekommer i habitat som skogsbryn, skogsstigar, ängar, trädgårdar och även kustområden. Vad gäller näringsväxter är arten polylektisk; den flyger till blommande växter från många familjer, som korgblommiga växter, korsblommiga växter, klockväxter, solvändeväxter, liljeväxter, väddväxter, johannesörtsväxter, vallmoväxter, rosväxter, grobladsväxter och ranunkelväxter. Flygtiden i Palearktis varar från tidigt i april till oktober för honor, från juni till september för hanar;
Fortplantning
Arten är en solitär, icke-samhällsbildande art; honan gräver sina larvbon i solexponerad, glesbevuxen mark. Boet utgörs av en mer eller mindre vertikal tunnel ner till ungefär 20 cm djup, som därefter vidgas till en blindgång. Från tunneln utgår förseglade larvceller som var och en innehåller ett ägg och näring i form av pollen. Efter parningen övervintrar den unga honan i det bo där hon föddes. hon kan leva upp till två år, och lägga ägg även det andra året. Det förekommer att boet angrips av blodbina ängsblodbi, eventuellt även Sphecodes scabricollis, vilkas larver lever på den lagrade näringen efter det att värdägget eller -larven dödats.
Utbredning
I Palearktis finns arten från Spanien i söder till mellersta Finland i norr, och från södra England och Wales i väster över Grekland och Turkiet till Azerbajdzjan, Iran, Sibirien och Kina i öster. I Nordamerika förekommer den i de nordöstra delarna, från Nova Scotia i Kanada till Minnesota, New England, New York, Michigan och Wisconsin i USA.
I Sverige är arten inte särskilt vanlig och ganska fragmenterad. Den förekommer främst på Öland, Gotland och på västkusten i Halland, men finns även i Svealand och norra Götaland.
I Finland finns arten främst på Åland och i södra delen av landet. Enstaka fynd har gjorts längre norrut, ett fynd 2012 i Norra Österbotten är det nordligaste.
Kommentarer
Källor
Externa länkar
Smalbin
Insekter i palearktiska regionen
Insekter i nearktiska regionen | {
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Warth is a town in the district of Neunkirchen in the Austrian state of Lower Austria.
It consists of seven localities (in brackets population as of 1 January 2016):
Haßbach (227)
Kirchau (294)
Kulm (70)
Petersbaumgarten (266)
Steyersberg (75)
Thann (42)
Warth (547)
Population
Notable people
The philosopher Ludwig Wittgenstein briefly taught at a secondary school in Haßbach in the autumn of 1922.
References
Cities and towns in Neunkirchen District, Austria
Bucklige Welt | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,973 |
Today's writing prompt is to write about a time I benefited from the kindness of a stranger or a time when I was the one extending a helping hand as a good Samaritan.
Have I ever done something kind for a stranger? I'm sure I have, but not in a big way—just little things that every human being should do without keeping score.
The good Samaritan is unique in that he helped a stranger. I'm sure all of us try to do kind things for those whom we know and love, but reaching out to help a stranger requires another level of selflessness.
Acts of kindness by a good Samaritan don't always need to be big acts.I believe that one can be a good Samaritan by simply giving the gift of a kind word. In some of my most overwhelmed moments as a mother with postpartum depression, strangers have said some of the kindest things to me and it encouraged me, probably more than they know.
I have a grandbaby and I understand.
Why don't you check out before me, I know standing in line with a baby is hard.
I remember specific moments when women have stopped to say something kind to me and I can't even remember their words—but I remember how they made me feel. Some of these women offered kind words and a smile when I was almost ready to break down and cry and their kindness strengthened me. I am grateful for those moments.
I don't know that the opportunity to help a stranger arises all that often—but we know that it does. Sometimes I have seen a stranger and felt compassion and have wanted to help or offer a kind word, but I didn't because I have questioned whether it's the right thing to do or not, because I didn't know how the recipient would react, and sometimes just because I was too scared to do so. If it is safe and reasonable, I want to be more willing to extend a helping hand or offer a kind word without fear—and I think we know when these times arise.
The important thing about being a good Samaritan is that we do so without expecting a reward or any recognition for it. Is there anything more tacky than the person who does something kind for a stranger and then posts about it on Facebook? In my opinion, a good Samaritan does good things simply because it's the right thing to do and he/she is moved by their conscience to do so—not because he/she is crafting the perfect social media post in their head while doing the kind act.
Be kind. You never know who you might strengthen.
I guess my biggest opportunity is to let someone go in line in front of me when they have only a few items. I don't work outside the home, so the grocery store is mostly when I interact with the public. But I know it always means a lot to people who are holding two or three items to be told to go in front of me - the lady with a buggy load to feed 6 people. Lol I know that I love when others let me in front of them, too, in the same situation. | {
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\section{Introduction}
All groups in this paper are finite. In this note our main result is the following.
\begin{thm}\label{t: psl2}
Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle isomorphic to a linear group $\SLq$, or to a Suzuki group $\suzuki$, or to a Ree group $\ree$, or to a unitary group $\PSU_3(q)$. Then, either $G$ is not binary, or $G=\Sym(\Omega)\cong \Sym(5)\cong \PGammaL_2(4)\cong \PGL_2(5)$, or $G=\Sym(\Omega)\cong \Sym(6)\cong \mathrm{P}\Sigma\mathrm{L}_2(9)$.
\end{thm}
Theorem~\ref{t: psl2} is a contribution towards a proof of a conjecture of Cherlin \cite{cherlin1}. This conjecture asserts that a primitive binary permutation group lies on a short explicit list of known actions.
The precise definition of ``binary'' and ``binary action'' is given in Section~\ref{s: bin back} below. An equivalent definition, couched in terms of ``relational structures'', can be found in \cite{cherlin2}; the connection between this conjecture and Lachlan's theory of sporadic structures can be found in \cite{cherlin1}. It is this connection that really enlivens the study of binary permutation groups, and provides motivation to work towards a proof of Cherlin's conjecture.
Let us briefly describe the status of this conjecture. By work of Cherlin \cite{cherlin2} and Wiscons \cite{wiscons}, this very general conjecture has been reduced to the following statement concerning almost simple groups.
\begin{conj}\label{conj: cherlin}
If $G$ is a binary almost simple primitive permutation group on the set $\Omega$, then $G=\symme(\Omega)$.
\end{conj}
One sees immediately that Theorem~\ref{t: psl2} settles Conjecture~\ref{conj: cherlin} for almost simple primitive permutation groups with socle isomorphic to $\SLq$, or $\suzuki$, or $\ree$, or $\PSU_3(q)$, that is, for each Lie type group of twisted Lie rank $1$. Theorem~\ref{t: psl2} is the third recent result of this type; in recent work, the first and third authors settled Conjecture~\ref{conj: cherlin} for groups with alternating socle, and for the $\mathcal{C}_1$ primitive actions of groups with classical socle \cite{gs_binary}.
A brief word about our methods: the aforementioned work on groups with alternating or classical socle was based on the study of so-called ``beautiful subsets''. These objects are defined below, and their usefulness is explained by Lemma~\ref{l: forbidden} and Example~\ref{ex: snba1} below, which together imply that whenever an action admits a beautiful subset the action is not binary.
In the current note our approach is different for the reason that the family of actions under consideration -- the primitive actions of almost simple groups with socle a Lie group of Lie rank $1$-- very often do not have beautiful subsets.
To deal with this situation we need to develop a more general theory: Suppose that we have a group $G$ acting on a set $\Omega$, and we want to show that this action is non-binary. The key property of beautiful subsets that makes them useful is that they allow us to argue ``inductively'', in the sense that if we can find a subset $\Lambda$ of $\Omega$ that is ``beautiful'', then the full action of $G$ on $\Omega$ is non-binary. In order to deal with the absence of beautiful subsets, we have studied this inductive property more formally via the notion of a ``strongly non-binary subset''. The theory of such subsets is developed in \S\ref{s: bin back} and allows us to apply an inductive argument in a more general setting.
The advantages of Theorem~\ref{t: psl2} and of this theory are several: firstly, Theorem~\ref{t: psl2} is a material advance towards a proof of Conjecture~\ref{conj: cherlin}; secondly, it demonstrates the possibility of obtaining results in situations where one cannot use the notion of a beautiful subset, as in~\cite{gs_binary}; thirdly, it turns out that the rank $1$ groups tend to be a sticking point when making general arguments concerning binary groups. We hope, therefore, that by disposing of this case here, we will be able to deal more easily with the remaining cases required for a proof of Cherlin's conjecture. Investigation in this direction is in progress, see~\cite{gls_binary}
\subsection{Structure of the paper}
The proof of Theorem~\ref{t: psl2} is split into several parts. First, in \S\ref{s: bin back}, after giving a number of definitions, we prove some general results about binary actions; in particular Lemma~\ref{l: forbidden} is vital.
In \S\ref{s: structure} we give some basic information concerning groups with socle isomorphic to $\PSL_2(q)$; then in \S\ref{s: fp} we calculate the size of the fixed set for various elements of $\PGammaL_2(q)$ in various primitive actions; these results are then used to prove Lemmas~\ref{l: handy q odd} and \ref{l: handy q even}; it is worth remarking that these fixed point calculations yield the required conclusions almost immediately for the groups $\SLq$ and $\PGL_2(q)$, however a finer analysis is required to deal with those almost simple groups that contain field automorphisms.
The three lemmas just mentioned -- Lemmas~\ref{l: forbidden}, \ref{l: handy q odd} and \ref{l: handy q even} -- directly imply Theorem~\ref{t: psl2} for $\PSL_2(q)$ when $q\geq 9$. The remaining small cases, when $q\in\{4,5,7,8\}$, can be verified directly using GAP \cite{GAP} or by referencing the calculations of Wiscons \cite{wiscons2}.
In \S\ref{s: suzuki}, \S\ref{s: ree} and \S\ref{s: psu}, we give a proof of Theorem~\ref{t: psl2} for groups with socle $\suzuki$, $\ree$ and $\PSU_3(q)$, respectively. In the first two cases the theorems are easy consequences of propositions asserting that the primitive actions in question admit strongly non-binary subsets (see \S\ref{s: bin back} for the definition of a strongly non-binary subset). The final case -- socle $\PSU_3(q)$ -- is dealt with somewhat differently.
\section{Binary actions and strongly non-binary actions}\label{s: bin back}
Throughout this section $G$ is a finite group acting (not necessarily faithfully) on a set $\Omega$ of cardinality $t$. Here, our job is to give a definition of ``binary action'', and of ``strongly non-binary action'', and to connect these definitions to earlier work on ``beautiful sets''. Given a subset $\Lambda$ of $\Omega$, we write $G_\Lambda:=\{g\in G\mid \lambda^g\in\Lambda,\forall \lambda\in \Lambda\}$ for the set-wise stabilizer of $\Lambda$, $G_{(\Lambda)}:=\{g\in G\mid \lambda^g=\lambda, \forall\lambda\in \Lambda\}$ for the point-wise stabilizer of $\Lambda$, and $G^\Lambda$ for the permutation group induced on $\Lambda$ by the action of $G_\Lambda$. In particular, $G^\Lambda\cong G_\Lambda/G_{(\Lambda)}$.
Given a positive integer $r$, the group $G$ is called \textit{$r$-subtuple complete} with respect to the
pair of $n$-tuples $I, J \in \Omega^n$, if it contains elements that
map every subtuple of size $r$ in $I$ to the corresponding subtuple in
$J$ i.e. $$\textrm{for every } k_1, k_2, \dots, k_r\in\{ 1,
\ldots, n\}, \textrm{ there exists } h \in G \textrm{ with }I_{k_i}^h=J_{k_i}, \textrm{ for every }i \in\{
1, \ldots, r\}.$$ Here $I_k$ denotes the $k^{\text{th}}$ element of tuple
$I$ and $I^g$ denotes the image of $I$ under the action of $g$.
Note that $n$-subtuple completeness simply requires the existence of
an element of $G$ mapping $I$ to $J$.
The group $G$ is said to be of {\it arity $r$} if, for all
$n\in\mathbb{N}$ with $n\geq r$ and for all $n$-tuples $I, J \in \Omega^n$, $r$-subtuple
completeness (with respect to $I$ and $J$) implies $n$-subtuple completeness (with respect to $I$ and $J$). When $G$ has arity 2, we say that $G$ is {\it binary}. A pair $(I,J)$ of $n$-tuples of $\Omega$ is called a {\it non-binary witness for the action of $G$ on $\Omega$}, if $G$ is $2$-subtuple complete with respect to $I$ and $J$, but not $n$-subtuple complete, that is, $I$ and $J$ are not $G$-conjugate.
To show that the action of $G$ on $\Omega$ is non-binary it is sufficient to find a non-binary witness $(I,J)$.
We say that the action of $G$ on $\Omega$ is \emph{strongly non-binary} if there exists a non-binary witness $(I,J)$ such that
\begin{itemize}
\item $I$ and $J$ are $t$-tuples where $|\Omega|=t$;
\item the entries of $I$ (resp. $J$) are distinct entries of $\Omega$.
\end{itemize}
\begin{example}\label{ex: snba1}{\rm
If $G$ acts $2$-transitively on $\Omega$ with kernel $K$ and $G/K\cong G^\Omega\not\cong\Sym(\Omega)$, then $G$ is strongly non-binary.
Indeed, by $2$-transitivity, any pair $(I,J)$ of $t$-tuples of distinct elements from $\Omega$ is $2$-subtuple complete. Since $G/K\cong G^\Omega\not\cong\Sym(\Omega)$, we can choose $I$ and $J$ in distinct $G$-orbits. Thus $(I,J)$ is a non-binary witness.}
\end{example}
\begin{example}\label{ex: snba2}{\rm
Let $G$ be a subgroup of $\Sym(\Omega)$, let $g_1, g_2,\ldots,g_r$ be elements of $G$, and let $\tau,\eta_1,\ldots,\eta_r$ be elements of $\Sym(\Omega)$ with
\[
g_1=\tau\eta_1,\,\,g_2=\tau\eta_2,\,\,\ldots,\,\,g_r=\tau\eta_r.
\]
Suppose that, for every $i\in \{1,\ldots,r\}$, the support of $\tau$ is disjoint from the support of $\eta_i$; moreover, suppose that, for each $\omega\in\Omega$, there exists $i\in\{1,\ldots,r\}$ (which may depend upon $\omega$) with $\omega^{\eta_i}=\omega$. Suppose, in addition, $\tau\notin G$.
Now, writing $\Omega=\{\omega_1,\dots, \omega_t\}$, observe that
\[
((\omega_1,\omega_2,\dots, \omega_t), (\omega_1^{\tau},\omega_2^{\tau}, \ldots,\omega_t^{\tau}))
\]
is a non-binary witness. Thus the action of $G$ on $\Omega$ is strongly non-binary.}
\end{example}
The notion of a strongly non-binary action allows us to ``argue inductively'' using suitably chosen set-stabilizers. The following lemma (which was first stated in~\cite{gs_binary} and which, in any case, is virtually self-evident) clarifies what we mean by this.
\begin{lem}\label{l: forbidden}
Suppose that there exists a subset $\Lambda \subseteq \Omega$ such that $G^\Lambda$ is strongly non-binary. Then $G$ is not binary.
\end{lem}
In what follows a \emph{strongly non-binary subset} is a subset $\Lambda$ of $\Omega$ such that $G^\Lambda$ is strongly non-binary.
We are ready for the third main concept of this section, that of a ``beautiful subset''; this is closely related to the example of a strongly non-binary action given in Example~\ref{ex: snba1}. Specifically, we say that a subset $\Lambda\subseteq \Omega$ is a \emph{$G$-beautiful subset} if $G^\Lambda$ is a $2$-transitive subgroup of $\symme(\Lambda)$ which is neither $\alter(\Lambda)$ nor $\symme(\Lambda)$. Note that we will tend to drop the ``$G$'' in $G$-beautiful, so long as the context is clear (for instance, when $G$ is a permutation group on $\Omega$, that is, $G\le\Sym(\Omega)$).
In the light of Example~\ref{ex: snba1}, the curious reader may be wondering why the definition of a beautiful subset excludes also the possibility that $G^\Lambda=\alter(\Lambda)$. This exclusion is explained by the following lemma, which is~\cite[Corollary 2.3]{gs_binary}.
\begin{lem}\label{l: beautiful}
Suppose that $G$ is almost simple with socle $S$. If $\Omega$ contains an $S$-beautiful subset, then $G$ is not binary.
\end{lem}
In what follows we will see a number of examples of strongly non-binary actions of the types given in Examples~\ref{ex: snba1} and \ref{ex: snba2}, as well as examples of beautiful subsets. To study these examples we will make use of the fact that the finite faithful 2-transitive actions are all known thanks to the Classification of Finite Simple Groups.
One naturally wonders whether other examples of strongly non-binary witnesses exist. This is indeed the case and the existence of a strongly non-binary witness is related to the classic concept of $2$-closure introduced by Wielandt~\cite{Wielandt}. Given a permutation group $G$ on $\Omega$, the \emph{$2$-closure of $G$} is the set $$G^{(2)}:=\{\sigma\in \Sym(\Omega)\mid \forall (\omega_1,\omega_2)\in \Omega\times \Omega, \textrm{there exists }g_{\omega_1\omega_2}\in G \textrm{ with }\omega_1^\sigma=\omega_1^{g_{\omega_1\omega_2}}, \omega_2^\sigma=\omega_2^{g_{\omega_1\omega_2}}\},$$
that is, $G^{(2)}$ is the largest subgroup of $\Sym(\Omega)$ having the same orbitals as $G$. The group $G$ is said to be $2$-closed if and only if $G=G^{(2)}$. We claim that $G$ is not $2$-closed if and only if $G$ has a strongly non-binary witness. Write $\Omega:=\{\omega_1,\ldots,\omega_t\}$. If $G$ is not $2$-closed, then there exists $\sigma\in G^{(2)}\setminus G$. Now, it is easy to verify that $I:=(\omega_1,\ldots,\omega_t)$ and $J:=I^\sigma=(\omega_1^\sigma,\ldots,\omega_t^\sigma)$ are $2$-subtuple complete (because $\sigma\in G^{(2)}$) and are not $G$-conjugate (because $g\notin G$). Thus $(I,J)$ is a strongly non-binary witness. The converse is similar.
\section{Groups with socle isomorphic to \texorpdfstring{$\PSL_2(q)$}{PSL2(q)}}\label{s: structure}
In this section we start by studying some of the basic properties of involutions and Klein $4$-subgroups of the almost simple groups $G$ with socle $\PSL_2(q)$. (In particular, $\PSL_2(q)\le G\le \PGammaL_2(q)$.) All of these properties are well-known and/or easy to verify by direct calculation. We also set up some basic notation for what follows.
For a group $J$, write $m_2(J)$ for the \emph{$2$-rank} of $J$, i.e.\ the maximum rank of an abelian $2$-subgroup of $J$. If $q$ is odd and $J$ is a section of $G$ (i.e.\ a quotient of a subgroup of $G$), then $m_2(J)\leq 3$. What is more, $m_2(J)\leq 2$ unless $q$ is a square and $G$ contains a field automorphism of order $2$.
\begin{lem}\label{l: quotients split}
Let $L$ be a subgroup of $\PGL_2(q)$ with $q$ odd, and let $K$ be a subgroup of $\nor {\PGL_2(q)} L $ with $K$ isomorphic to a Klein $4$-group and with $K\cap L=1$.
Then $|L|$ is odd.
\end{lem}
\begin{proof}
Let $P$ be a Sylow $2$-subgroup of $\langle K,L\rangle=K\ltimes L$ containing $K$. Then $P=K\ltimes Q$, for some Sylow $2$-subgroup $Q$ of $L$. If $Q\ne 1$, then $K$ centralises a non-identity element of $Q$ and hence $m_2(\PGL_2(q))\ge m_2(P)=m_2(K\ltimes Q)\ge m_2(K)+m_2(\cent Q K)\ge 2+1=3$, a contradiction.
\end{proof}
Suppose that $q$ is odd. There is exactly one $\PGL_2(q)$-conjugacy class of Klein $4$-subgroups of $\PSL_2(q)$, and one can check directly that $\cent{\PGL_2(q)} K =K$ for each Klein $4$-subgroup of $\PSL_2(q)$. When $q\equiv \pm 3\pmod 8$, a Sylow $2$-subgroup of $\PSL_2(q)$ is a Klein $4$-subgroup and, by Sylow's theorems, there is exactly one $\PSL_2(q)$-conjugacy class of Klein $4$-subgroups of $\PSL_2(q)$; in this case $\nor {\PSL_2(q)} K \cong \Alt(4)$. When $q\equiv \pm 1\pmod 8$, there are two $\PSL_2(q)$-conjugacy classes of Klein $4$-subgroups of $\PSL_2(q)$ and these are fused in $\PGL_2(q)$; in this case $\nor {\PSL_2(q)} K \cong\Sym(4)$.
We need information concerning involutions in $\PGammaL_2(q)\setminus\PGL_2(q)$ -- such involutions must be field automorphisms, as defined in~\cite{gls3}. The following result is a special case of \cite[Prop. 4.9.1]{gls3}.
\begin{lem}\label{l: fields}
Let $f_1,f_2\in\PGammaL_2(q)\setminus\PGL_2(q)$ be of order $t$ for some prime $t$, and suppose that $f_1\PGL_2(q)=f_2\PGL_2(q)$. Then $f_1$ and $f_2$ are $\PGL_2(q)$-conjugate.
\end{lem}
\subsection{Fixed point calculations}\label{s: fp}
We let $G$ be a group having socle $S$ with $S\cong \PSL_2(q)$. Using the classification of the maximal subgroups of $G$ (see for example~\cite{bhr}), it is important to observe that, for every maximal subgroup $M$ of $G$ there exists a maximal subgroup $H$ of $S$ with $M=\nor G H$; in particular, this allows us to identify (up to permutation isomorphism) each primitive $G$-set $\Omega$ with the set of $G$-conjugates of some maximal subgroup $H$ of $S$.
Therefore, we let $H$ be a maximal subgroup of $S$ with $\nor G H $ maximal in $G$, and set $\Omega$ to be $H^G:=\{H^g\mid g\in G\}$, the set of all conjugates of $H$ in $G$. All possibilities for $H$ and $|\Omega|$ are given in the first and in the third column of Tables~\ref{t: inv q odd} and \ref{t: inv q even}, where in Table~\ref{t: inv q odd} the symbol $\zeta$ is defined by
\begin{equation}\label{e: zeta}
\zeta:=\begin{cases}
2 & \textrm{if }G\not\le \mathrm{P}\Sigma L_2(q) \textrm{ and }q \textrm{ is odd}, \textrm{ or }q\textrm{ is even},\\
1 & \textrm{if }G\le \mathrm{P}\Sigma L_2(q) \textrm{ and }q \textrm{ is odd}.
\end{cases}
\end{equation}
(See \cite{bhr} to verify this. The conditions that are listed in Table~\ref{t: inv q odd} are necessary for the action of $G$ on $\Omega$ to be primitive, but they are not necessarily sufficient.) Finally, we write $\mathcal{P}(H)$ for the power set of $H$.
In what follows, we calculate the number of fixed points of an involution $g\in S$, and (when $q$ is odd) of a Klein $4$-subgroup $K\leq S$, for the action of $G$ on $\Omega$. (Given a subset $Y$ of a permutation group $X$ on $\Omega$, we write $\Fix_\Omega(Y):=\{\omega\in \Omega\mid \omega^y=\omega,\forall y\in Y\}$ and simply $\Fix_\Omega(y)$ when the set $Y$ consists of the single element $y$.)
To calculate the number of fixed points of $g$ and of $K$, we make use of the well-known formulas (see for instance~\cite[Lemma~$2.5$]{LiebeckSaxl})
\begin{equation}\label{e: fora}
|\Fix_\Omega(g)| = \frac{|\Omega|\cdot |H\cap g^G|}{|g^G|},\qquad
|\Fix_\Omega(K)| = \frac{|\Omega|\cdot |\mathcal{P}(H)\cap K^G|}{|K^G|}.
\end{equation}
Given an involution $g\in S$, from~\cite{gls3} we obtain
\[
|g^G|=\begin{cases}
\frac12q(q-1), & \textrm{if } q\equiv 3\pmod 4,\\
\frac12q(q+1), & \textrm{if }q\equiv 1\pmod 4, \\
q^2-1, &\textrm{if } q \textrm{ even}.
\end{cases}
\]
Using this information, Eq.~\eqref{e: fora} and the fact that $\PSL_2(q)$ has a unique conjugacy class of involutions, it is a straightforward computation to verify the fourth and fifth column in Table~\ref{t: inv q odd} and the third and fourth column in Table~\ref{t: inv q even}.
\begin{table}
\begin{adjustbox}{angle=90}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$H$ & Conditions & $|\Omega|$ & $|H\cap g^G|$ & $|\Fix_\Omega(g)|$&$|\mathcal{P}(H)\cap K^G|$&$|\Fix_\Omega(K)|$\\
\hline
$[q]:(\frac{q-1}{2})$ & None & $q+1$ & $\begin{cases}
0, & q\equiv 3(4) \\
q, & q\equiv 1(4)
\end{cases}$ & $\begin{cases}
0, & q\equiv 3(4) \\
2, & q\equiv 1(4)
\end{cases}$ &$0$&$0$\\
$D_{q-1}$ & None & $\frac{q(q+1)}{2}$ & $\begin{cases}
\frac{q-1}{2}, & q\equiv 3(4) \\
\frac{q+1}{2}, & q\equiv 1(4)
\end{cases}$ & $\frac{q+1}{2}$&$\begin{cases}0,&q\equiv 3(8)\\ \frac{q-1}{4},&q\equiv 5(8)\\\frac{\zeta(q+1)}{8},&q\equiv 1(8)\\0,&q\equiv 7(8)\end{cases}$&$\begin{cases}0,&q\equiv 3(8)\\ 3,&q\equiv 5(8)\\3,&q\equiv 1(8)\\0,&q\equiv 7(8)\end{cases}$\\
$D_{q+1}$ & None & $\frac{q(q-1)}{2}$ & $\begin{cases}
\frac{q+3}{2}, & q\equiv 3(4) \\
\frac{q+1}{2}, & q\equiv 1(4)
\end{cases}$ & $\begin{cases}
\frac{q+3}{2}, & q\equiv 3(4) \\
\frac{q-1}{2}, & q\equiv 1(4)
\end{cases}$ &$\begin{cases}\frac{q+1}{4},&q\equiv 3(8)\\0,&q\equiv 5(8)\\0,&q\equiv 1(8)\\\frac{\zeta(q+1)}{8},&q\equiv 7(8)\end{cases}$&$\begin{cases}3,&q\equiv 3(8)\\ 0,&q\equiv 5(8)\\0,&q\equiv 1(8)\\3,&q\equiv 7(8)\end{cases}$\\
$\PSL_2(q_0)$ & $q=q_0^a$, $a$\textrm{ odd } & $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ & $\begin{cases}
\frac{q_0(q_0-1)}{2}, & q_0\equiv 3(4)\\
\frac{q_0(q_0+1)}{2}, & q_0\equiv 1(4) \\
\end{cases}$ & $\begin{cases}
\frac{q+1}{q_0+1}, & q_0\equiv 3(4)\\
\frac{q-1}{q_0-1}, & q_0\equiv 1(4) \\
\end{cases}$ &$\begin{cases}\frac{q_0(q_0^2-1)}{24},&q\equiv \pm 3(8)\\\frac{\zeta q_0(q_0^2-1)}{48},&q\equiv \pm 1(8)\end{cases}$&$1$\\
$\PGL_2(q_0)$ & $q=q_0^2$, $\zeta=1$ & $\frac{\sqrt{q}(q+1)}{2}$ & $q$ & $\sqrt{q}$ &$\frac{q_0(q_0^2-1)}{24}$ or $\frac{q_0(q_0^2-1)}{8}$&$1$ or $3$\\
$\Alt(4)$ & $q=p\equiv \pm 3(8)$ & $\frac{q(q^2-1)}{24}$ & $3$ & $\begin{cases}
\frac{q+1}{4}, & q\equiv 3(8) \\
\frac{q-1}{4}, & q\equiv 5(8)
\end{cases}$ &$1$&$1$\\
$\Sym(4)$ & $q=p\equiv \pm 1(8)$, $\zeta=1$ & $\frac{q(q^2-1)}{48}$ & $9$ & $\begin{cases}
\frac{3(q+1)}{8}, & q\equiv 7(8) \\
\frac{3(q-1)}{8}, & q\equiv 1(8)
\end{cases}$ &$1$ or $3$&$1$ or $3$\\
$\Alt(5)$ & $\begin{array}{l}q=p, q\equiv \pm 1 (10), \textrm{ or}\\ q=p^2, p\equiv \pm 3(10) \end{array}$ & $\frac{\zeta q(q^2-1)}{120}$ & $15$ & $\begin{cases}
\frac{\zeta(q+1)}{4}, & q\equiv 3(4) \\
\frac{\zeta(q-1)}{4}, & q\equiv 1(4)
\end{cases}$&$5$&$\begin{cases}\zeta,&q\equiv \pm 3(8)\\2,&q\equiv \pm 1(8)\end{cases}$ \\
\hline
\end{tabular}
\end{adjustbox}
\caption{Fixed points of involutions in $S$ and of a Klein $4$-subgroup of $S$, for $q$ odd. The symbol $\zeta$ is defined in~\eqref{e: zeta}.}\label{t: inv q odd}
\end{table}
\begin{table}
\begin{tabular}{|c|c|c|c|}
\hline
$H$ & $|\Omega|$ & $|H\cap g^G|$ & $|\Fix_\Omega(g)|$\\
\hline
$[q]:(q-1)$ & $q+1$ & $q-1$ & $1$ \\
$D_{2(q-1)}$ & $\frac12q(q+1)$ & $q-1$ & $\frac12q$ \\
$D_{2(q+1)}$ & $\frac12q(q-1)$ & $q+1$ & $\frac12q$ \\
$\SL_2(q_0)$ & $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ & $q_0^2-1$ & $\frac{q}{q_0}$ \\
\hline
\end{tabular}
\caption{Fixed points of involutions in $S$ for $q$ even.}\label{t: inv q even}
\end{table}
Suppose that $q\equiv \pm 3\pmod 8$ and let $K$ be a Klein $4$-subgroup of $S$. As we mentioned above, $K$ is a Sylow $2$-subgroup of $S$, all Klein $4$-subgroups of $S$ are conjugate and $\nor {\PSL_2(q)}K\cong \Alt(4)$. Therefore $|K^G|=\frac{1}{24}q(q^2-1)$. Using this and Eq.~\eqref{e: fora}, it is easy to confirm (when $q\equiv \pm 3\pmod 8$) the veracity of the sixth and seventh column in Table~\ref{t: inv q odd}. (Note that the $\PGL_2(q_0)$ and $\Sym(4)$ rows do not apply when $q\equiv \pm3\pmod 8$.)
Suppose now that $q\equiv \pm 1\pmod 8$ and let $K$ be a Klein $4$-subgroup of $S$. In this case, there are two $S$-conjugacy classes of Klein $4$-subgroups and, regardless of the $S$-conjugacy class on which $K$ lies, we have $\nor G K\cong \Sym(4)$. In particular,
\[
|K^G|=\frac{1}{48}\zeta q(q^2-1),
\]
where $\zeta$ is the parameter that was defined in \eqref{e: zeta}. As above, using this and Eq.~\eqref{e: fora}, it is easy to confirm (when $q\equiv \pm 1\pmod 8$) the veracity of the sixth and seventh column in Table~\ref{t: inv q odd}. (Note that the $\Alt(4)$ row does not apply when $q\equiv \pm1\pmod 8$.)
For the proof of Theorem~\ref{t: psl2}, we also need to compute the number of fixed points of field involutions of $G$ only for certain primitive actions when $q$ is odd: this information is tabulated in Table~\ref{t: f field}. Of course, here we assume that $q$ is a square and that $G$ does contain a field automorphism of order $2$. Now observe that
\[
|f^G|=
\begin{cases}\frac{\zeta}{2}\sqrt{q}(q+1),&\textrm{if }q \textrm{ is odd},\\
\sqrt{q}(q+1),&\textrm{if }q \textrm{ is even}.
\end{cases}
\]
From this and~\eqref{e: fora}, the veracity of Table~\ref{t: f field} follows from easy calculations (which we omit).
Note that Lemma~\ref{l: fields} means that it is convenient to assume that $G\geq \PGL_2(q)$ where this makes no difference; however for the final action in Table~\ref{t: f field}, we must assume that $G$ does {\bf not} contain $\PGL_2(q)$ since otherwise the action is not primitive. To make this clear we state the assumed value of $\zeta$ in the ``Conditions'' column in each case.
\begin{table}
\begin{tabular}{|c|c|c|c|c|}
\hline
$H$ & Conditions & $|\Omega|$ & $|\nor GH\cap f^G|$ & $|\Fix_\Omega(f)|$\\
\hline
$D_{q-1}$ & $\zeta=2$ & $\frac12q(q+1)$ & $2\sqrt{q}$ & $q$ \\
$D_{q+1}$ & $\zeta=2$ & $\frac12q(q-1)$ & 0 & 0 \\
$\PSL_2(q_0)$ & $q=q_0^a$, $a$\textrm{ odd }, $\zeta=2$, & $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ & $\sqrt{q_0}(q_0+1)$ & $\frac{\sqrt{q}(q-1)}{\sqrt{q_0}(q_0-1)}$ \\
$\Alt(5)$ & $q=p^2\equiv \pm1\pmod{10}$, $\zeta=1$ & $\frac1{120}q(q^2-1)$ &
$10$ & $\frac{1}{6} \sqrt{q}(q-1)$ \\
\hline
\end{tabular}
\caption{Fixed points of field automorphisms of order $2$ for selected primitive actions of $G$ with $q$ odd.}\label{t: f field}
\end{table}
We are now ready to prove the two lemmas that together yield Theorem~\ref{t: psl2} for groups with socle $\PSL_2(q)$.
\begin{lem}\label{l: handy q odd}
Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle isomorphic to $\SLq$ with $q$ odd. If $q>9$, then $\Omega$ contains a strongly non-binary subset.
\end{lem}
\begin{proof}
Our notation here is consistent with that established above. For instance, we identify $\Omega$ with the set of $G$-conjugates of $H$. We must consider the actions corresponding to the first column of Table~\ref{t: inv q odd}.
\noindent\textsc{Line 1: $H$ is a Borel subgroup of $S$}. In this case $G$ acts $2$-transitively on $\Omega$, but $ \alter(\Omega)\nleq G$. Thus $\Omega$ itself is a beautiful subset and hence strongly non-binary (of the type given in Example~\ref{ex: snba1}).
\noindent\textsc{Line 5: $H\cong \PGL_2(q_0)$ where $q=q_0^2$}. We regard $S$ as the projective image of those elements of $\GL_2(q)$ that have square determinant, and we may assume that $H$ consists of the projective image of those elements in $\GL_2(q_0)$ whose entries are all in $\mathbb{F}_{q_0}$. Let $T$ be the set of diagonal elements in $S$; let $T_0:=H\cap T$, a maximal split torus in $H$; let $\alpha$ be an element of $\Fq$ that does not lie in any proper subfield of $\Fq$; and define
\[
N_0:=\left\{\begin{pmatrix}
1 & \alpha b \\ 0 & 1
\end{pmatrix} \mid b\in\mathbb{F}_{q_0}\right\}.
\]
Clearly, $N_0$ is a subgroup of $G$, $T_0$ normalizes $N_0$ and $T_0\cap N_0=\{1\}$. Thus we can form the semidirect product $X:=N_0\rtimes T_0$ and we observe that $X\cap H=T_0$. Let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $q_0$ on which $X$ acts $2$-transitively. If $q_0>5$, then $G$ does not contain a section isomorphic to $\alter(q_0)$ and we conclude that $\Lambda$ is a beautiful subset for the action of $G$ on $\Omega$. If $q_0=5$, then $q=5^2$ and we can check the result directly using \texttt{magma}~\cite{magma}.
\noindent\textsc{Line 4: $H\cong \PSL_2(q_0)$ where $q=q_0^a$ for some odd prime $a$}. We consider first the special situation where $q$ is a square. We consider $S$ as before, with $H$ the projective image of those elements in $\GL_2(q_0)$ whose entries are all in $\mathbb{F}_{q_0}$ and which have square determinant in $\mathbb{F}_{q_0}$; finally, we know that $H$ has a subgroup $H_1$ isomorphic to $\PGL_2(\sqrt{q_0})$ (since $q_0$ is a square by assumption). We take $H_1$ to be the projective image of those elements in $\GL_2(\sqrt{q_0})$ whose entries are all in $\mathbb{F}_{\sqrt{q_0}}$. Let $T$ be the set of diagonal elements in $S$; let $T_0:=H_1\cap T$, a maximal split torus in $H_1$; let $\alpha$ be an element of $\Fq$ that does not lie in any proper subfield of $\Fq$; and define
\[
N_0:=\left\{\begin{pmatrix}
1 & \alpha b \\ 0 & 1
\end{pmatrix} \mid b\in\mathbb{F}_{\sqrt{q_0}}\right\}.
\]
As above, $N_0$ is a subgroup of $G$, $T_0$ normalizes $N_0$ and $T_0\cap N_0=\{1\}$. Thus we can form the semidirect product $X:=N_0\rtimes T_0$ and we observe that $X\cap H=T_0$. Let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $\sqrt{q_0}$ on which $X$ acts $2$-transitively. If $\sqrt{q_0}>5$, then $G$ does not contain a section isomorphic to $\alter(q_0)$ and we conclude that $\Lambda$ is a beautiful subset for the action of $G$ on $\Omega$. The outstanding cases (that is, $\sqrt{q_0}\le 5$ or $q$ is not a square) will be dealt with below.
\noindent\textsc{Lines 2,3,4,6,7,8}. Here we will show that in every case we can find a strongly non-binary subset $\Lambda$ for which $G^\Lambda$ is as in Example~\ref{ex: snba2}. We let $g$ be an involution in $S$ and $h\in g^G$ with $K:=\langle g,h\rangle$ a Klein $4$-subgroup of $S$ and we let
\[
\Lambda=\Fix(g)\cup\Fix(h)\cup\Fix(gh).
\]
Observe that $\Lambda$, $\Fix(g)$, $\Fix(h)$ and $\Fix(gh)$ are $g$-invariant and $h$-invariant. Write $\tau_1$ for the permutation induced by $g$ on $\Fix(gh)$ and $\tau_2$ for the permutation induced by $g$ on $\Fix(h)$, and observe that the supports of $\tau_1$ and $\tau_2$ are disjoint, and that $g$ induces the permutation $\tau_1\tau_2$ on $\Lambda$. Observe, furthermore, that $h$ induces the permutation $\tau_1$ on $\Fix(gh)$; now write $\tau_3$ for the involution induced by $h$ on $\Fix(g)$, and observe that the supports of $\tau_1$ and $\tau_3$ are disjoint, and that $h$ induces the permutation $\tau_1\tau_3$ on $\Lambda$.
Observe, finally, that the supports of $\tau_2$ and $\tau_3$ are disjoint and that, since $g,h$ and $gh$ are conjugate, the permutations $\tau_1,\tau_2$ and $\tau_3$ all have support of equal size.
Comparing the entries in the fifth and seventh column of Table~\ref{t: inv q odd}, we see that $|\Fix(g)|\geq |\Fix(K)|+2$. (Here we are using our assumption that $q>9$.) This implies, in particular, that $\tau_1,\tau_2$ and $\tau_3$ are non-trivial permutations of order $2$. Observe that either there exists $f\in G_\Lambda$ inducing the permutation $\tau_1$ on $\Lambda$ or else $\Lambda$ is a strongly non-binary subset of $\Omega$ (it corresponds to Example~\ref{ex: snba2}).
Suppose that $G$ does not contain a field automorphism of order $2$, and suppose that $f\in G_\Lambda$ induces the permutation $\tau_1$ on $\Lambda$. This would imply that $G_{\Lambda}$ contained an elementary-abelian subgroup of order $8$. But, as we observed earlier, $m_2(Q)\leq 2$ for any section $Q$ in $G$, which is a contradiction. We conclude that $\Lambda$ is a strongly non-binary subset of $\Omega$.
Note that this argument disposes of Lines 6 and 7 of Table~\ref{t: inv q odd}. It also deals with one of the outstanding cases for Line 4, namely the situation where $q$ is not a square.
Suppose from here on that $G$ contains a field automorphism of order $2$. In particular $q$ is a square and $q\equiv 1\pmod 8$. Now, the previous argument implies that $G^\Lambda$ is strongly non-binary unless $G_\Lambda$ contains a field automorphism that induces the element $\tau_1$, so assume that this is the case. There are two possibilities:
\begin{enumerate}
\item[(a)] there is a field automorphism $f$ of order $2$ that induces the element $\tau_1$ on $\Lambda$;
\item[(b)] there is a field automorphism $f$ of order $2$ that fixes $\Lambda$ point-wise (and some element of $G_\Lambda\cap (G\setminus \PGL_2(q))$ of order divisible by $4$ induces the element $\tau_1$ on $\Lambda$).
\end{enumerate}
Note first that Line 3 of Table~\ref{t: inv q odd} is immediately excluded since field automorphisms of order $2$ have no fixed points in this action (see Table~\ref{t: f field}). We are left only with Lines 2 and 8, as well as Line 4 with $q_0\in\{9,25\}$.
Assume that Case~(a) holds. Observe that, the action on $\Lambda$ gives a natural homomorphism $\langle S_{(\Lambda)},f,g,h\rangle \to \Sym(\Lambda)$ whose image is elementary abelian of order $8$, and whose kernel is $S_{(\Lambda)}$. What is more, by Lemma~\ref{l: quotients split}, $S_{(\Lambda)}$ has odd order, and we conclude that $\langle f,g,h\rangle$ is elementary abelian of order $8$.
Since $f$ centralizes $\langle g,h\rangle$ we may consider the action of $\langle g,h\rangle$ on $\Fix(f)$. Observe that if $\gamma\not\in\Fix(g)\cup \Fix(h)\cup\Fix(gh)$, then $\gamma^g\neq \gamma^h$, and so $\langle g, h\rangle$ acts semi-regularly on $\Fix(f)\setminus(\Fix(f)\cap\Lambda)$ and so
\[
|\Fix(f)\setminus(\Fix(f)\cap\Lambda)|\equiv 0 \pmod 4.
\]
Now in this case $\Fix(f)\cap \Lambda = \Fix(g)\cup\Fix(h)$ and we conclude that
\begin{equation}\label{eq: f}
|\Fix(f)|-2|\Fix(g)|+|\Fix(K)|\equiv 0\pmod 4.
\end{equation}
Let us consider the remaining actions, one by one.
\noindent\textsc{Line 2: $H\cong D_{q-1}$}. In this case \eqref{eq: f} implies that
\[
|\Fix(f)|-2|\Fix(g)|+|\Fix(K)|=q-(q+1)+3\equiv 0\pmod 4
\]
which is a contradiction.
\noindent\textsc{Line 4: $H\cong \PSL_2(q_0)$ with $q=q_0^a$ and $a$ an odd prime}. Note first that we may assume that $q_0\in\{9,25\}$, with $p=\sqrt{q_0}$. Choose $g\in S$ to be an element of order $p$; an easy calculation using \eqref{e: fora} confirms that $g$ fixes $\frac{q}{q_0}$ points of $\Omega$. Now choose $h\in S$ to be an element of order $p$ (hence also fixing the same number of points of $\Omega$) such that $\langle g,h\rangle$ is an elementary-abelian group of order $q_0$. We require, moreover, that $\langle g,h\rangle$ fixes no points of $\Omega$: for this we just make sure that $\langle g,h\rangle$ is not conjugate to a Sylow $p$-subgroup of $H$. As usual we set $\Lambda=\Fix(g)\cup\Fix(h)\cup \Fix(gh)$. We define $\tau_1, \tau_2, \tau_3$ exactly as in the argument for \textsc{Lines 2,3,4,6,7,8}.
Now if $f$ is an element inducing the permutation $\tau_1$, then $f$ has order divisible by $p$, and $f$ fixes at least $\frac{2q}{q_0}$ elements of $\Omega$. This implies immediately that $f\not\in S$, and we conclude that $a=p$. Now, referring to Lemma~\ref{l: fields}, we see that $f$ must be a field automorphism of degree $a=p$ and an easy calculation with \eqref{e: fora} implies that such an element fixes $\frac12p(p^2+1)$ points of $\Omega$ and so cannot induce the permutation $\tau_1$. Now, referring to Example~\ref{ex: snba2}, we conclude that $\Lambda$ is a strongly non-binary subset.
\noindent\textsc{Line 8: $H\cong A_5$.} In this case we assume that $G\leq {\rm P\Sigma L}_2(q)$, otherwise the action on $\Omega$ is not primitive. In particular $\zeta=1$ and \eqref{eq: f} implies that
\[
|\Fix(f)|-2|\Fix(g)|+|\Fix(K)|=\frac{1}{6}\sqrt{q}(q-1)-\frac{1}{2}(q-1)+2\equiv 0\pmod 4.
\]
which is a contradiction.
\smallskip
We are left with Case~(b). Note in this case that $q=p^a$ where $a$ is divisible by $4$. This immediately excludes Line 8 of the table (since $q=p^2$ here) as well as the remaining cases for Line $4$ (since here $q=9^a$ or $25^a$ where $a$ is an odd prime). Thus the only line left to consider is Line 2. But note that, for Case (b) to hold, $\Fix(f)$ must contain $\Lambda$ and so
\[
|\Fix(f)|\geq 3|\Fix(g)|-2|\Fix(K)|.
\]
But Tables~\ref{t: inv q odd} and \ref{t: f field} then give that
\[
q\geq \frac{3}{2}(q+1)-6.
\]
This is a contradiction for $q>9$ and we are done.
\end{proof}
\begin{lem}\label{l: handy q even}
Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle isomorphic to $\SLq$ with $q=2^a$. If $a>3$, then $\Omega$ contains a strongly non-binary subset.
\end{lem}
\begin{proof}
Our notation here is consistent with that established above. We must consider the actions corresponding to the first column of Table~\ref{t: inv q even}.
\noindent\textsc{Line 1: $H$ is a Borel subgroup of $S$}. In this case $G$ acts 2-transitively on $\Omega$, but $G\not\cong \alter(\Omega)$ or $\symme(\Omega)$. Thus $\Omega$ itself is a beautiful subset (and hence strongly non-binary).
\noindent\textsc{Line 2: $H\cong D_{2(q-1)}$}. We may assume that $H$ contains $T$, the set of diagonal elements in $S$. We define
\[
N:=\left\{\begin{pmatrix}
1 & \alpha \\ 0 & 1
\end{pmatrix} \mid \alpha\in\mathbb{F}_{q}\right\}.
\]
Now it is clear that $T$ normalizes $N$ and that $T\cap N=\{1\}$. Thus we can form the semidirect product $X=N\rtimes T_0$ and we observe that $X\cap H=T$.
Using the identification of $\Omega$ with the set of $G$-conjugates of $H$, we let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $q\geq 8$ on which $X$ acts 2-transitively. Since $G$ does not contain a section isomorphic to $\alter(q)$ we conclude that $\Lambda$ is a beautiful subset for the action of $G$ on $\Omega$ and we are done.
\noindent\textsc{Line 3: $H\cong D_{2(q+1)}$}. We proceed similarly to the case where $q$ is odd in Lemma~\ref{l: handy q odd}: let $g$ be an involution in $S$ and $h\in g^G$ with $K:=\langle g,h\rangle$ a Klein $4$-subgroup of $S$ and we let
\[
\Lambda=\Fix(g)\cup\Fix(h)\cup\Fix(gh).
\]
Observe that $\Lambda$, $\Fix(g)$, $\Fix(h)$ and $\Fix(gh)$ are $g$-invariant and $h$-invariant. Observe, furthermore, that $\Fix(g)$, $\Fix(h)$ and $\Fix(gh)$ are all disjoint and, by Table~\ref{t: inv q even}, are of size $\frac12 q$. Write $\tau_1$ for the permutation induced by $g$ on $\Fix(gh)$, $\tau_2$ for the permutation induced by $g$ on $\Fix(h)$, and $\tau_3$ for the permutation induced by $g$ on $\Fix(gh)$.
Then $g$ induces the permutation $\tau_1\tau_2$ on $\Lambda$, while $h$ induces the permutation $\tau_1\tau_3$ on $\Lambda$. Then $\Lambda$ is a strongly non-binary subset provided there is no element $f\in G_\Lambda$ that induces the permutation $\tau_1$. Such an element must have even order and must fix at least $q$ elements of $\Omega$. Now Table~\ref{t: inv q even} implies that $f\not\in S$. On the other hand, if $f$ is a field-automorphism of order $2^c$, then it does not fix any elements of $\Omega$. We conclude that $\Lambda$ is a strongly non-binary subset and we are done.
\noindent\textsc{Line 4: $H\cong \SL_2(q_0)$ where $q=q_0^b$ for some prime $b$.} Note that, using \cite{bhr}, we can exclude the possibility that $q_0=2$. Suppose first that $q_0>4$, and take $\beta \in \mathbb{F}_q\setminus \mathbb{F}_{q_0}$.
We may assume that $H$ consists of those elements in $S=\SL_2(q)$ whose entries are all in $\mathbb{F}_{q_0}$. Let $T$ be the set of diagonal elements in $S$; let $T_0=S\cap T$, a maximal split torus in $S$; and define
\[
N_0:=\left\{\begin{pmatrix}
1 & \beta \alpha \\ 0 & 1
\end{pmatrix} \mid \alpha\in\mathbb{F}_{q_0}\right\}.
\]
Now it is clear that $T_0$ normalizes $N_0$ and that $T_0\cap N_0=\{1\}$. Thus we can form the semidirect product $X=N_0\rtimes T_0$ and we observe that $X\cap H=T_0$.
Using the identification of $\Omega$ with the set of $G$-conjugates of $H$, we let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $q_0\geq 8$ on which $X$ acts 2-transitively. Since $G$ does not contain a section isomorphic to $\alter(q_0)$ we conclude that $\Lambda$ is a beautiful subset for the action of $G$ on $\Omega$ and we are done.
The only remaining case is when $q_0=4$. As $q=2^a$, we have $a=2b$. In this case we make use of the fact that the number of $S$-conjugacy classes of subgroups of $S$ isomorphic to a Klein 4-subgroup is $(q+2)/6$.
Since $H$ contains a unique conjugacy class of Klein 4-subgroups, there exists a Klein $4$-subgroup $K:=\langle g,h\rangle$ of $S$ with $K\nleq H^g$, for every $g\in S$, that is, $\Fix(K)=\emptyset$ for the action on cosets of $H$.
Observe that $q$ is a square. We choose $K$ so that, not only does it not lie in a conjugate of $H$, it also doesn't lie in a conjugate of $\SL_2(\sqrt{q})=\SL_2(2^b)$, the centralizer of a field automorphism of order $2$. Define
\[
\Lambda=\Fix(g)\cup\Fix(h)\cup\Fix(gh).
\]
Observe that $g$ acts on $\Lambda$, and on $\Fix(h)$, and on $\Fix(gh)$. Write $\tau_1$ for the involution induced by $g$ on $\Fix(gh)$ and $\tau_2$ for the permutation induced by $g$ on $\Fix(h)$, and observe that the supports of $\tau_1$ and $\tau_2$ are disjoint, and that $g$ induces the permutation $\tau_1\tau_2$ on $\Lambda$.
Exactly as in the case when $H=D_{2(q-1)}$, $\Lambda$ is either strongly non-binary (and we are done), or else there exists $f\in G^\Lambda$ such that $f$ induces the permutation $\tau_1$ on $\Lambda$. Suppose that this latter possibility occurs, and observe that $\Fix(f)$ contains $\Fix(g)\cup\Fix(h)$ and so $|\Fix(f)|\geq \frac{q}{2}$. If $f\in S$, then $f$ is conjugate to $g$ and $|\Fix(g)|=\frac{q}{4}$, so we have a contradiction.
Suppose that $f\not\in S$. The subgroup structure of $\SL_2(q)$ implies that if a subgroup $X$ is normalized by a Klein $4$-group, then $X$ is elementary abelian of even order. Thus $S_{(\Lambda)}$ is elementary abelian of even order. But if $S_{(\Lambda)}$ is non-trivial, then an involution fixes at least $\frac{3q}{4}$ points which is a contradiction. Thus $S_{(\Lambda)}$ is trivial.
Now, note that since $q=4^b$, where $b$ is prime, either $q=16$, or else we may assume that $f$ is a field automorphism of order $2$. Thus $\langle K, f\rangle$ is elementary-abelian. But, since $K$ does not lie in a conjugate of $\SL_2(\sqrt{q})$ we have a contradiction here. In the case $q=16$, a moment's thought shows that either $f$ is a field automorphism of order $2$, or else there is a field automorphism of order $2$ that fixes $\Lambda$ point-wise. Either way one concludes that there is a field automorphism $f_1$ such that $\langle K, f_1\rangle$ is elementary-abelian and, again, we have a contradiction. Thus in all cases we have a strongly non-binary subset of the type given in Example~\ref{ex: snba2} and we are done.
\end{proof}
We remark again that Theorem~\ref{t: psl2} for groups with socle $\PSL_2(q)$ is an immediate consequence of Lemmas~\ref{l: forbidden}, \ref{l: handy q odd} and \ref{l: handy q even}.
\section{Groups with socle isomorphic to \texorpdfstring{$^{2}B_2(q)$}{2B2(q)}}\label{s: suzuki}
In this section we prove Theorem~\ref{t: psl2} for groups with socle $\suzuki$. This theorem follows, {\it \`a la} the other main results, from Lemma~\ref{l: suzuki} combined with Lemma~\ref{l: forbidden}. In what follows $G$ is an almost simple group with socle $S\cong\suzuki$, where $q=2^a$ and $a$ is an odd integer with $a\geq 3$. We write $r:=2^{\frac{a+1}{2}}$ and define $\theta$ to be the following field automorphism of $\Fq$:
\[
\theta: \Fq\to \Fq, \,\,\, x \mapsto x^r.
\]
We need some basic facts, all of which can be found in \cite{suzuki}. First, ${\rm Out}(S)$ is a cyclic group of odd order $a$. Second, $G$ contains a single conjugacy class of involutions; writing $g$ for one of these involutions we note that
\[
|g^G|=(q^2+1)(q-1).
\]
Third, the maximal subgroups of $S$ fall into three families: Borel subgroups, normalizers of maximal tori, and subfield subgroups. For the second of these families, we need some fixed point calculations, and these are given in Table~\ref{t: suz} (making use of \eqref{e: fora}). Each line of this table corresponds to a conjugacy class of maximal tori in $S$; we write $H$ for a maximal subgroup of $S$ and $\Omega$ for the set of right cosets of $H$ in $S$; in the final column we write $K$ for a Klein $4$-subgroup of $S$.
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$H$ & $|\Omega|$ & $|H\cap g^G|$ & $|\Fix_\Omega(g)|$ & $|\mathcal{P}(H)\cap K^S|$&$|\Fix_\Omega(K)|$\\
\hline
$D_{2(q-1)}$ & $\frac12q^2(q^2+1)$ & $q-1$ & $\frac12q^2$&$0$ & $0$\\
$(q+r+1)\rtimes 4$ & $\frac14q^2(q-1)(q-r+1)$ & $q+r+1$ & $\frac14q^2$ &$0$ & $0$\\
$(q-r+1)\rtimes 4$ & $\frac14q^2(q-1)(q+r+1)$ & $q-r+1$ & $\frac14q^2$ &$0$ &$0$\\
\hline
\end{tabular}
\caption{Fixed points of involutions and Klein $4$-subgroups for selected primitive actions of almost simple Suzuki groups.}\label{t: suz}
\end{table}
\begin{lem}\label{l: suzuki}
Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle $S\cong\suzuki$. Then $\Omega$ contains a strongly non-binary subset.
\end{lem}
\begin{proof}
Note that $|S|$ is not divisible by $3$ and hence $G$ does not contain a section isomorphic to an alternating group $\Alt(n)$ with $n\geq 3$.
Referring to \cite{suzuki}, we see that a maximal subgroup of $G$ is necessarily the normalizer in $G$ of a maximal subgroup $H$ of $S$. Thus we can identify $\Omega$ with the set of right cosets of $H$ in $S$. We split into three families, as per the discussion above.
First, if $H$ is a Borel subgroup, then the action of $G$ on $\Omega$ is $2$-transitive and, since $G$ contains no alternating sections, we obtain immediately that $\Omega$ itself is a beautiful subset.
Second, if $H$ is the normalizer in $S$ of a maximal torus, then we set $K$ to be a Klein 4-subgroup of $S$, and we let $g,h$ be distinct involutions in $K$. Referring to Table~\ref{t: suz}, we see that $g$ and $h$ fix at least $16$ points of $\Omega$, while $K$ fixes none. We set $\lambda_3$ to be one of the fixed points of $g$ and write $\lambda_4$ for the point $\lambda_3^h$. Similarly $\lambda_5\in\Fix(h)$ and $\lambda_6=\lambda_5^g$. Finally pick $\lambda_1\in\Fix(gh)$ and let $\lambda_2=\lambda_1^g$; observe that $\lambda_2=\lambda_1^h$. Now let $\Lambda=\{\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6\}$ and observe that $K$ acts on this set with the element $g$ inducing the permutation $(\lambda_1,\lambda_2)(\lambda_5,\lambda_6)$ while the element $h$ induces the permutation $(\lambda_1,\lambda_2)(\lambda_3,\lambda_4)$. Suppose that $f\in G_\Lambda$ induces the permutation $(\lambda_1,\lambda_2)$ on $\Lambda$. This would imply that $f$ and $g$ fix the point $\lambda_3$ and so the stabilizer of $\lambda_3$ must contain a section isomorphic to a Klein 4-subgroup. This is impossible: the Sylow $2$-subgroups of $H$ are cyclic of order $2$ or $4$ and, since $|{\rm Out}(S)|$ is odd, this is true of the stabilizer in $G$ of $\lambda_3$. Therefore $\Lambda$ is a strongly non-binary subset of $\Omega$: it corresponds to Example~\ref{ex: snba2}.
Third, suppose that $H$ is a subfield subgroup of $S$. It is convenient to take $S$ to be the set of $4\times 4$ matrices over $\Fq$ described on \cite[p.133]{suzuki}; then we take $H$ to be the subgroup of $S$ consisting of matrices with entries over $\mathbb{F}_{q_0}$ with $q=q_0^b$ for some prime $b$, and $q_0>2$. The following set forms a Sylow $2$-subgroup of $S$:
\[
P_2(q):= \left\{\begin{pmatrix}
1 & 0 & 0 & 0 \\ \alpha & 1 & 0 & 0 \\ \alpha^{1+\theta}+\beta & \alpha^\theta & 1 & 0 \\ \alpha^{2+\theta}+\alpha\beta+\beta^\theta & \beta&\alpha & 1
\end{pmatrix} \mid
\alpha, \beta \in \Fq \right\}.
\]
The subgroup $P_2(q)$ is normalized by the following subgroup of $S$,
\[
K(q):= \left\{\begin{pmatrix}
\zeta_1 & 0 & 0 & 0 \\ 0 & \zeta_2 & 0 & 0 \\ 0 & 0 & \zeta_3 & 0 \\ 0 & 0 & 0 & \zeta_4
\end{pmatrix} \mid
\exists\kappa \in \Fq\setminus\{0\}, \zeta_1^\theta = \kappa^{1+\theta}, \zeta_2^\theta=\kappa, \zeta_3=\zeta_2^{-1}, \zeta_4=\zeta_1^{-1} \right\}.
\]
The group $P_2(q)\rtimes K(q)$ is a maximal Borel subgroup of $S$, while $P_2(q_0)\rtimes K(q_0)$ is a maximal Borel subgroup of $H$. Observe that the center $\Zent {P_2(q)}$ of $P_2(q)$ consists of those matrices for which $\alpha=0$.
Let $\zeta\in\Fq\setminus\mathbb{F}_{q_0}$ and consider the group
\[
ZP_2(\zeta,q_0):= \left\{\begin{pmatrix}
1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \zeta\beta & 0 & 1 & 0 \\ (\zeta\beta)^\theta & \zeta\beta & 0 & 1
\end{pmatrix} \mid
\beta \in \mathbb{F}_{q_0} \right\}.
\]
Observe that $K(q_0)$ normalizes $ZP_2(\zeta,q_0)$, that $K(q_0)<H$, that $ZP_2(\zeta,q_0)\cap H=\{1\}$ and that $K(q_0)$ acts fixed-point-freely on $ZP_2(\zeta,q_0)$. Let $X:=ZP_2(\zeta,q_0)\rtimes K(q_0)$; identifying $\Omega$ with the set of $G$-conjugates of $H$, we let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $q_0\geq 8$ on which $X$ acts 2-transitively.
The absence of alternating sections implies that $\Lambda$ is a beautiful subset.
\end{proof}
\section{Groups with socle isomorphic to \texorpdfstring{$^{2}G_2(q)$}{2G2(q)}}\label{s: ree}
In this section we prove Theorem~\ref{t: psl2} for groups with socle $\ree$. This theorem follows, {\it \`a la} the other main results, from Lemma~\ref{l: ree} combined with Lemma~\ref{l: forbidden}. In what follows $G$ is an almost simple group with socle $S\cong\ree$, where $q=3^a$ and $a$ is an odd integer with $a\geq 3$. We write $r:=3^{\frac{a+1}{2}}$ and define $\theta$ to be the following field automorphism of $\Fq$:
\[
\theta: \Fq\to \Fq, \,\,\, x \mapsto x^r.
\]
We need some basic facts, all of which can be found in \cite{kleidman}. First, ${\rm Out}(S)$ is a cyclic group of odd order $a$. Second, $G$ contains a single conjugacy class of involutions; writing $g$ for one of these involutions we note that
\[
|g^G|=q^2(q^2-q+1).
\]
Third, the order of $S$ is not divisible by $5$, and so $G$ does not contain a section isomorphic to $\alter(n)$ with $n\geq 5$.
Fourth, the maximal subgroups of $G$ fall into four families: Borel subgroups, normalizers of maximal tori, involution centralizers, and subfield subgroups. For all but the first of these families, we need some fixed point calculations, and these are given in Table~\ref{t: ree} (making use of \eqref{e: fora}). In each line of this table we write $H$ for a maximal subgroup of $S$ and $\Omega$ for the set of right cosets of $H$ in $S$; in the final column we write $K$ for a Klein $4$-subgroup of $S$.
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$H$ & $|\Omega|$ & $|H\cap g^G|$ & $|\Fix_\Omega(g)|$ & $|K^G\cap \mathcal{P}(H)|$ & $|\Fix_\Omega(K)|$\\
\hline
$(2^2\times D_{\frac{q+1}{2}})\rtimes 3$ & $\frac{q^3(q^2-q+1)(q-1)}{6}$ & $q+4$ & $\frac{q(q-1)(q+4)}{6}$ & $\frac{3q+5}{2}$ & $\frac{3q+5}{2}$\\
$(q+r+1)\rtimes 6$ & $\frac{q^3(q^2-1)(q-r+1)}{6}$ & $q+r+1$ & $\frac{q(q^2-1)}{6}$ & $0$ & $0$ \\
$(q-r+1)\rtimes 6$ & $\frac{q^3(q^2-1)(q+r+1)}{6}$ & $q-r+1$ & $\frac{q(q^2-1)}{6}$ & $0$ & $0$ \\
$2\times \PSL_2(q)$ & $q^2(q^2-q+1)$ & $q^2-q+1$ & $q^2-q+1$ & $\frac{(q+4)q(q-1)}{6}$ & $q+4$\\
${^2G_2}(q_0)$ & $\frac{q^3(q^3+1)(q-1)}{q_0^3(q_0^3+1)(q_0-1)}$ & $q_0^2(q_0^2-q_0+1)$ & $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ & $\frac16q_0^3(q_0^2-q_0+1)(q_0-1)$ & $\frac{q+1}{q_0+1}$ \\
\hline
\end{tabular}
\caption{Fixed points of involutions and Klein $4$-subgroups for selected primitive actions of almost simple Ree groups.}\label{t: ree}
\end{table}
The calculations given in Table~\ref{t: ree} make use of the fact there is a unique class of involutions and a unique class of Klein $4$-subgroups in $S$; their normalizers are maximal subgroups. In particular the normalizer of a Klein $4$-subgroup in $S$ is the group $H$ in the first line of Table~\ref{t: ree}; combined with the fact that a Sylow $2$-subgroup of $S$ is elementary abelian of order $8$, we are able to complete the final entry in that first row. The other entries in the table follow from easy calculations.
\begin{lem}\label{l: ree}
Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle $S\cong\ree$. Then $\Omega$ contains a strongly non-binary subset.
\end{lem}
\begin{proof}
Referring to \cite{kleidman}, we see that a maximal subgroup of $G$ is necessarily the normalizer in $G$ of a maximal subgroup $H$ of $S$. Thus we can identify $\Omega$ with the set of right cosets of $H$ in $S$. We split into two cases.
First, if $H$ is a Borel subgroup, then the action of $G$ on $\Omega$ is $2$-transitive and, since $G$ contains no sections isomorphic to $\alter(n)$ with $n\geq 5$, we obtain immediately that $\Omega$ itself is a beautiful subset.
Second, if $H$ is not a Borel subgroup, then we set $K$ to be a Klein 4-subgroup of $S$, we let $g,h$ be distinct involutions in $K$, and we let
\[
\Lambda=\Fix(g)\cup\Fix(h)\cup\Fix(gh).
\]
Observe that $\Lambda$, $\Fix(g)$, $\Fix(h)$ and $\Fix(gh)$ are $g$-invariant and $h$-invariant. Write $\tau_1$ for the involution induced by $g$ on $\Fix(gh)$ and $\tau_2$ for the permutation induced by $g$ on $\Fix(h)$, and observe that the supports of $\tau_1$ and $\tau_2$ are disjoint, and that $g$ induces the permutation $\tau_1\tau_2$ on $\Lambda$. Observe, furthermore, that $h$ induces the permutation $\tau_1$ on $\Fix(gh)$; now write $\tau_3$ for the involution induced by $h$ on $\Fix(g)$, and observe that the supports of $\tau_1$ and $\tau_3$ are disjoint, and that $h$ induces the permutation $\tau_1\tau_3$ on $\Lambda$.
Observe, finally, that the supports of $\tau_2$ and $\tau_3$ are disjoint. Now, suppose that $f\in G_\Lambda$ induces the permutation $\tau_1$ on $\Lambda$. This would imply that $f$ fixes more points than $g$. Since $f$ has even order and all involutions in $G$ are conjugate, some odd power of $f$ is a conjugate of $g$, which is a contradiction. Thus $\Lambda$ is a strongly non-binary subset of $\Omega$ (it corresponds to Example~\ref{ex: snba2}).
\end{proof}
\section{Groups with socle isomorphic to \texorpdfstring{$\PSU_3(q)$}{PSU(3,q)}}\label{s: psu}
In this section we prove Theorem~\ref{t: psl2} for groups with socle $\PSU_3(q)$. Strictly speaking, this theorem does not follow {\it \`a la} the other main results. Firstly, we do not prove the existence of beautiful subsets or of strongly non-binary subsets: we simply prove that the primitive groups under consideration are not binary. Second, for some primitive actions we make use of computer aided computations. The basic ideas for these computations are inspired from a deeper analysis in~\cite{DV_NG_PS}, where Conjecture~\ref{conj: cherlin} is proved for most almost simple groups with socle a sporadic simple group.
The following lemmas are taken from~\cite{DV_NG_PS} and are stated in a form tailored to our needs in this paper.
\begin{lem}\label{l: again0}Let $G$ be a transitive group on a set $\Omega$, let $\alpha$ be a point of $\Omega$ and let $\Lambda\subseteq \Omega$ be a $G_\alpha$-orbit. If $G$ is binary, then $G_\alpha^\Lambda$ is binary.
\end{lem}
\begin{proof}Assume that $G$ is binary. Let $\ell\in\mathbb{N}$ and let $I:=(\lambda_1,\lambda_2,\ldots,\lambda_\ell)$ and $J:=(\lambda_1',\lambda_2',\ldots,\lambda_\ell')$ be two tuples in $\Lambda^\ell$ which are $2$-subtuple complete for the action of $G_\alpha$ on $\Lambda$. Clearly, $I_0:=(\alpha,\lambda_1,\lambda_2,\ldots,\lambda_\ell)$ and $J_0:=(\alpha,\lambda_1',\lambda_2',\ldots,\lambda'_\ell)$ are $2$-subtuple complete for the action of $G$ on $\Omega$; as $G$ is binary, $I_0$ and $J_0$ are in the same $G$-orbit; hence $I$ and $J$ are in the same $G_\alpha$-orbit. From this we deduce that $G_\alpha^\Lambda$ is binary.
\end{proof}
We caution the reader that in the next lemma when we write $\Lambda$ we \emph{are not} referring to a subset of $\Omega$ -- here the set $\Lambda$ is allowed to be any set whatsoever that satisfies the listed suppositions.
\begin{lem}\label{l: again}
Let $G$ be a primitive group on a set $\Omega$, let $\alpha$ be a point of $\Omega$, let $M$ be the stabilizer of $\alpha$ in $G$ and let $d$ be an integer. Suppose $M\ne 1$ and, for each transitive action of $M$ on a set $\Lambda$ satisfying:
\begin{enumerate}
\item $|\Lambda|>1$, and
\item every composition factor of $M$ is isomorphic to some section of $M^\Lambda$, and
\item $M$ is binary in its action on $\Lambda$,
\end{enumerate}
we have that $d$ divides $|\Lambda|$. Then either $d$ divides $|\Omega|-1$ or $G$ is not binary.
\end{lem}
\begin{proof}
Suppose that $G$ is binary. Since $\{\beta\in\Omega\mid \beta^m=\beta,\forall m\in M\}$ is a block of imprimitivity for $G$ and since $G$ is primitive, we obtain that either $M$ fixes each point of $\Omega$ or $\alpha$ is the only point fixed by $M$. The former possibility is excluded because $M\neq 1$ by hypothesis. Therefore $\alpha$ is the only point fixed by $M$. Let $\Lambda\subseteq\Omega\setminus\{\alpha\}$ be an $M$-orbit. Thus $|\Lambda|>1$ and (1) holds. Since $G$ is a primitive group on $\Omega$, from~\cite[Theorem~3.2C]{dixon_mortimer}, we obtain that every composition factor of $M$ is isomorphic to some section of $M^\Lambda$ and hence (2) holds. From Lemma~\ref{l: again0}, the action of $M$ on $\Lambda$ is binary and hence (3) also holds. Therefore, $d$ divides $|\Lambda|$ and hence each orbit of $M$ on $\Omega\setminus\{\alpha\}$ has cardinality divisible by $d$. Thus $|\Omega|-1$ is divisible by $d$.
\end{proof}
\begin{proof}[Proof of Theorem~$\ref{t: psl2}$ for almost simple groups with socle $\PSU_3(q)$.]
Let $G$ be an almost simple primitive group on the set $\Omega$ with socle $S$ isomorphic to $\PSU_3(q)$. Observe that $q\ge 3$ because $\PSU_3(2)$ is soluble. When $q\le 9$, we can check directly with \texttt{magma} the veracity of our statement by constructing all the primitive actions under consideration and checking one-by-one that none is binary (in each case we are able to exhibit a non-binary witness). For the rest of the proof we assume that $q>9$, that is, $q\ge 11$: among other things, this will allow us to exclude some ``novelties'' in dealing with the maximal subgroups of $G$. Moreover, we let $V:=\mathbb{F}_{q^2}^3$ be the natural $3$-dimensional Hermitian space over the field $\mathbb{F}_{q^2}$ of cardinality $q^2$ for the appropriate covering group of $G$.
Let $\alpha\in \Omega$ and let $M:=G_\alpha$ be the stabilizer in $G$ of the point $\alpha$. We subdivide the proof according to the structure of $M$ as described in~\cite[Section~8, Tables~8.5,~8.6]{bhr}. In this proof we use~\cite{bhr} as a crib.
\smallskip
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_1$. }This case is completely settled in~\cite{gs_binary}, where the authors have proved Cherlin's conjecture for almost simple classical groups acting on the cosets of a maximal subgroup in the Aschbacher class $\mathcal{C}_1$.
\smallskip
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_2$.} From~\cite{bhr}, we get that the action of $G$ on $\Omega$ is permutation equivalent to the natural action of $G$ on
\[
\{\{V_1,V_2,V_3\}\mid \dim_{\mathbb{F}_{q^2}}(V_1)=
\dim_{\mathbb{F}_{q^2}}(V_2)=\dim_{\mathbb{F}_{q^2}}(V_3)=1,
V=V_1\perp V_2\perp V_3, V_1,V_2,V_3 \textrm{ non isotropic}\}.
\]
Therefore we identify $\Omega$ with the latter set. Let $e_1,e_2,e_3$ be the canonical basis of $V$ and, replacing $G$ by a suitable conjugate, we may assume that the matrix associated to the Hermitian form on $V$ with respect to the basis $e_1,e_2,e_3$ is the identity matrix. Thus $\omega_0:=\{\langle e_1\rangle,\langle e_2\rangle,\langle e_3\rangle\}\in\Omega$. Consider $\Omega_0:=\{\{V_1,V_2,V_3\}\in \Omega\mid V_1=\langle e_1\rangle\}$. Clearly, $G_{\Omega_0}=G_{\langle e_1\rangle}$, $G_{\langle e_1\rangle}$ is a classical group, $G_{\Omega_0}/\Zent {G_{\Omega_0}}$ is almost simple with socle isomorphic to $\PSL_2(q)$ (here we are using $q>3$), and the action of $G_{\Omega_0}$ on $\Omega_0$ is permutation equivalent to the action of $G_{\langle e_1\rangle}$ on $\Omega_0':=\{\{W_1,W_2\}\mid \dim(W_1)=\dim(W_2), \langle e_1\rangle^\perp=W_1\perp W_2, W_1,W_2 \textrm{ non degenerate}\}$. Therefore $G^{\Omega_0}$ is an almost simple primitive group with socle isomorphic to $\PSL_2(q)$ and having degree $|\Omega_0|=q(q-1)/2$. Applying Theorem~\ref{t: psl2} to $G^{\Omega_0}$, we obtain that $G^{\Omega_0}$ is not binary and hence there exist two $\ell$-tuples $(\{W_{1,1},W_{1,2}\},\ldots,\{W_{\ell,1},W_{\ell,2}\})$ and $(\{W'_{1,1},W'_{1,2}\},\ldots,\{W'_{\ell,1},W'_{\ell,2}\})$ in $\Omega_0^\ell$ which are $2$-subtuple complete for the action of $G_{\Omega_0}$ but not in the same $G_{\Omega_0}$-orbit. By construction the two $\ell$-tuples
\begin{align*}
I& :=(\{\langle e_1\rangle,W_{1,1},W_{1,2}\},\{\langle e_1\rangle,W_{2,1},W_{2,2}\},\ldots,\{\langle e_1\rangle,W_{\ell,1},W_{\ell,2}\}), \\
J&:=(\{\langle e_1\rangle,W'_{1,1},W'_{1,2}\},\{\langle e_1\rangle,W'_{2,1},W'_{2,2}\},\ldots,\{\langle e_1\rangle,W'_{\ell,1},W'_{\ell,2}\})
\end{align*}
are in $\Omega^\ell$ and are $2$-subtuple complete. Moreover, a moment's thought yields that $I$ and $J$ are not in the same $G$-orbit. Thus $G$ is not binary.
\smallskip
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_3$. }Here $M$ is the normalizer in $G$ of a maximal non-split torus $T$ of $S$ of order $(q^2-q+1)/\gcd(q+1,3)$. From~\cite{bhr}, we infer that $\nor S T$ is a split extension of $T$ by a cyclic group $\langle x\rangle$ of order $3$ (arising from an element of order $3$ in the Weyl group of $S$), thus $M=C\rtimes K$, with $C=\langle c\rangle$ cyclic such that $C\cap S=T$ and with $K$ abelian. (The group $K$ is the direct product of a cyclic group of order $3$ and a cyclic group of order $|G:G\cap\PGU_3(q)|$.) An inspection of the maximal subgroups of $\PSU_3(q)$ reveals that there exists $g\in \nor S K\setminus M$. Set $\beta:=\alpha^g$. Since $g\notin M$, we get $\beta\neq \alpha$ and, since $g\in \nor S K$, we get $G_\alpha\cap G_\beta=M\cap M^g\ge K$. Therefore $G_\alpha\cap G_\beta=C'\rtimes K$, for some cyclic subgroup $C'$ of $C$.
Set $\Lambda:=\beta^M$. Now, the action induced by $M$ on the $M$-orbit $\Lambda$ is permutation isomorphic to the action of $M=C\rtimes K$ on the right cosets of $M\cap M^g=C'\rtimes K$. We use the ``bar'' notation and denote by $\bar{M}$ the group $M^\Lambda$. Thus $\bar{M}=\langle\bar{c}\rangle\rtimes \bar{K}$ and the action of $\bar{M}$ on $\Lambda$ is permutation isomorphic to the natural action of $\langle\bar{c}\rangle\rtimes\bar{K}$ on $\langle\bar{c}\rangle$: with $\langle\bar{c}\rangle$ acting on $\langle\bar{c}\rangle$ via its regular representation and with $\bar{K}\cong K$ acting on $\langle\bar{c}\rangle$ via conjugation. Now, $\bar{c}^{\bar{x}}=\bar{c}^\kappa$, for some $\kappa\in\mathbb{Z}$ with $\kappa^3\equiv 1\pmod {|\bar{c}|}$ and $\kappa\not\equiv 1\pmod {|\bar{c}|}$. Consider the two triples $I:=(1,\bar{c},\bar{c}^{1+\kappa^2})$ and $J:=(1,\bar{c},\bar{c}^{1+\kappa})$. Now $(1,\bar{c})^{id_{\bar{M}}}=(1,\bar{c})$, $(1,\bar{c}^{1+\kappa^2})^{\bar{x}}=(1,\bar{c}^{\kappa+\kappa^3})=(1,\bar{c}^{\kappa+1})$ and $$(\bar{c},\bar{c}^{1+\kappa^2})^{\bar{c}^{-1}\bar{x}^2\bar{c}}=(\bar{c}^{\bar{c}^{-1}\bar{x}^2\bar{c}},(\bar{c}^{1+\kappa^2})^{\bar{c}^{-1}\bar{x}^2\bar{c}})=(\bar{c},\bar{c}^{\kappa^4+1})=(\bar{c},\bar{c}^{\kappa+1}).$$
Thus $I$ and $J$ are $2$-subtuple complete for the action of $\bar{M}$ on $\langle\bar{c}\rangle$. Observe that $I$ and $J$ are not in the same $\bar{M}$-orbit because the only element of $\bar{M}$ fixing $1$ and the generator $\bar{c}$ of $\langle\bar{c}\rangle$ is the identity, but $\bar{c}^{1+\kappa^2}\ne \bar{c}^{1+\kappa}$ because $\kappa\not\equiv 1\pmod {|\bar{c}|}$. Therefore $\bar{M}$ is not binary. From Lemma~\ref{l: again0}, we deduce that $G$ is not binary.
\smallskip
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_5$.} Let $H$ be the stabilizer in $M$ of a non-isotropic $1$-dimensional subspace $\langle v\rangle$ of $V$ and let $K$ be the stabilizer of $\langle v\rangle$ in $G$. Thus $K$ is a maximal subgroup of $G$ in the Aschbacher class $\mathcal{C}_1$; moreover, using~\cite{bhr} and $q>8$, we see that there exists $g\in \Zent K\setminus M$. Set $\beta:=\alpha^g$. Since $g\notin M$, we get $\beta\neq \alpha$ and, since $g\in \Zent K$, we get $G_\alpha\cap G_\beta=M\cap M^g\ge M\cap K=H$. Since $H$ is maximal in $M$, we obtain $G_\alpha\cap G_\beta=H$ and hence the action induced by $G_\alpha=M$ on the $G_\alpha$-orbit $\beta^{G_\alpha}$ is permutation isomorphic to the action of $M$ on the right cosets of $H$. By construction, this latter action is the natural action of the classical group $M$ on the cosets of a maximal subgroup in its $\mathcal{C}_1$-Aschbacher class. From~\cite[Theorem~B]{gs_binary}, this action is not binary. Therefore, $G$ is not binary by Lemma~\ref{l: again0}.
\smallskip
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_6$ or in the Aschbacher class $\mathcal{S}$.} Now the isomorphism class of $M$ is explicitly given in~\cite{bhr}. Since $|M|$ is very small (actually $|M|\le 720$), with the invaluable help of the computer algebra system \texttt{magma}, we compute all the transitive $M$-sets and we select the $M$-sets $\Lambda$ with $|\Lambda|>1$, with every composition factor of $M$ isomorphic to some section of $M^\Lambda$ and with $M^\Lambda$ binary. In all cases, we see that $|\Lambda|$ is even. Therefore, applying Lemma~\ref{l: again}, we obtain that either $G$ is not binary or $|\Omega|-1$ is even. In the
latter case, $|\Omega|$ is odd and hence $M$ contains a Sylow $2$-subgroup of $G$. From~\cite{bhr}, we see that a Sylow $2$-subgroup of $M\cap S$ has size $8$, but this is a contradiction because $|\PSU_3(q)|$ is always divisible by $16$.
\end{proof}
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Virginia: A Photographic Journey
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Virginia's best scenery is on display! Explore Old Virginia in Farcountry Press' newest release, Virginia: A Photographic Journey. Featuring the striking photography of Richard Nowitz and Virginia natives Pat and Chuck Blackley, the book celebrates a true passion for all things Virginia.
"We wanted to make a book dedicated to all the people of Virginia, who make living here such a blessing." Says photographers and Virginia natives Pat and Chuck Blackley.
Virginia's varied landscapes rise from the tidewater area on the Atlantic Ocean through the rolling piedmont to the Blue Ridge Mountains, the Valley of Virginia, and up into the Appalachian Mountains. Jam-packed with 113 full color photographs and informative text, this tour showcases what makes Old Dominion such a beloved and exceptional place.
Including elegant shots of Richmond and Norfolk to the colonial architecture of Williamsburg, University of Virginia, and Mount Vernon, Virginia: A Photographic Journey celebrates the rich, full-color landscapes from Old Dominion making it an excellent souvenir or remembrance.
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Pat and Chuck Blackley are a photographic and writing team born and raised in Virginia. Although they work throughout North America, their concentration is on the eastern United States. With a love of history, they find a wealth of subjects throughout the mid-Atlantic, particularly in Virginia where they have enjoyed traveling for over thirty years.
Their work appears in numerous books and magazines addressing regional and national audiences as well as in many commercial projects. Their one-photographer books include Shenandoah National Park Impressions, Blue Ridge Parkway Impressions, Shenandoah Valley Impressions, Outer Banks Impressions, Backroads from the Beltway, Our Virginia, Blue Ridge Parkway Simply Beautiful, and Virginia's Historic Homes and Gardens.
Richard Nowitz began freelance photography professionally in 1975 in Israel working for American and European publications. He returned to Washington, D.C. in 1989 and began his association with National Geographic in 1991. His career as a travel photographer has taken him to numerous countries around the world, and has produced over forty-five large-format gift books and travel guides for Insight Guides and National Geographic, and also five titles for Farcountry Press. Richard lives in Maryland with his wife.
See Inside Virginia: Photographic Journey—
Watching spectacular sunsets from Shenandoah National Park's lofty overlooks is a favorite visitor experience. We enjoyed a particularly lovely one at The Point overlook, as the setting sun lit up the clouds and soft light bathed the hillsides. – Pat and Chuck Blackley
Above: ThePioneer Farmstead, adjacent to the Humpback Rocks Visitor Center on the Blue Ridge Parkway, is an outdoor nineteenth-century farm museum housing a log cabin and other period outbuildings that were collected in the area and assembled here. Along with costumed interpreters who provide demonstrations, including weaving and basket making, these buildings help to provide a glimpse of early life in the Blue Ridge Mountains. – Pat and Chuck Blackley Left: Mountain music l ls the air at a concert at the Blue Ridge Parkway's Humpback Rocks Visitor Center near Afton. – Pat and Chuck Blackley Facing page: During the Civil War, the Shenandoah Valley was called "the Breadbasket of the Confederacy," since it served as a valuable source of foodstuffs for the Southern armies. The valley is still home to four of the top five agricultural counties in Virginia. At the top of the list is Rockingham County, where picturesque and productive farms like this one near Bridgewater are abundant. – Pat and Chuck Blackley
Right: James Monroe studied law in Williamsburg under then-governor Thomas Jefferson, and afterward the two became lifelong friends. In 1793, Monroe, the fth U.S. president, purchased Highland Plantation, which was adjacent to Jefferson's Monticello. His family lived at Highland for twenty-four years, from 1799 until 1823. Interestingly, Monroe and Jefferson, as well as John Adams, three of the rst ve U.S. presidents, died on July 4, Independence Day. Jefferson and Adams died on the same day in 1826, and Monroe died five years later, in 1831. – Pat and Chuck Blackley Facing page: Designated as a World Heritage Site, Monticello, Italian for "Little Mountain," was the plantation home of the nation's third president and author of the Declaration of Independence, Thomas Jefferson. Standing on a mountaintop outside Charlottesville, Monticello re ects the pure genius of the Renaissance man who designed it. It remained a work in progress, as Jefferson redesigned and rebuilt the home over a forty-year period. – Richard Nowitz Below: On April 5, 1856, Booker T. Washington was born a slave on the 207-acre farm of James Burroughs. After the Civil War, Washington became the first principal of Tuskegee Normal and Industrial School. Later as an adviser, author, and orator, his past would influence his philosophies as the most influential African American of his era. His birthplace is now a national monument -Pat and Chuck Blackley
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\section{\label{sec:level1}Introduction}
Monolayer graphene (MLG) is a two-dimensional carbon layer that has been of increasing interest for the scientific community since its first experimental realization in 2004 \cite{Novoselov04,Berger04,Hashimoto04}. Indeed, its linear dispersion at low energies is responsible for its fascinating properties \cite{DasSarma11} such as Klein tunneling \cite{Katsnelson2006}, quantum Hall effect \cite{Novoselov2005} and their potential applications in electronic devices, graphene-based nanocomposites or chemical sensors \cite{Schedin2007,Wu2008,Rafiee2009,Stankovich2006,Huang20,chen19}. However, these applications are severely limited due to the gap zero value.
Hence, the band-gap opening and the control of graphene bilayer become essential for the applications in various electronic devices.
One way to create a gap in graphene is selective functionalization, which has been used, for example, with hydrogen adsorption on a moir\'e of Graphene-Ir(111) \cite{Jorgensen16}.
Since graphene is a zero gap material with a bipartite lattice, such a functionalization
creates states so-called midgap states at Dirac energy $E_D$.
Bernal bilayer
graphene (BLG) is a system formed by two layers of MLG.
One of its advantage is the control of its gap by applying an external gate voltage \cite{Castro07,McCann06,McCann13}, which opens the way to multiple applications for nanodevices \cite{Zhang09,Overweg18,Kurzmann19}.
On the other hand, the operational of BLG devices can be based on changes in their electrical conductivity, which can be performed with using the influence of substrate \cite{Zhou07}, vacancies, ad-atoms or ad-molecules adsorbed on the surface of BLG \cite{Leenaerts09,Mapasha12,VanTuan16,Missaoui17,Missaoui18,Katoch18,Leconte11b,VanTuan16,Yang18,PINTO20}.
The unit cell of Bernal BLG contains four Carbon atoms, A$_1$, B$_1$ in layer 1 and A$_2$, B$_2$ in layer 2 (figure \ref{Fig_bilayer}).
Atoms A have three B first neighbors in the same layer and one A neighbor in the other layer, while atoms B have only three A first neighbors in the same layer.
Thus the local environment of A and B atoms are different, and
then the probability that an atom or molecule will stick to an atom A or an atom B should be different,
and it is reasonable to think that functionalization of B atoms is favored.
Such an asymmetric adsorption properties between sublattice A and sublattice B has been recently suggested by experimental measurements \cite{Katoch18}, where the distribution of hydrogen adsorbates on the sublattices is well controlled.
Overall, BLG lattice is a bipartite lattice of the two sublattices $\alpha$ \{A$_1$,B$_2$\} and $\beta$ \{A$_2$,B$_1$\}, which allows to expect very specific electronic properties produced by selective functionalization.
Since BLG is metallic, an isolated functionalization creates an isolated state that is a kind of ``mid-band'' states, so-called midgap states by analogy with MLG.
In a previous paper \cite{Missaoui18}, we have consider the limiting cases where adsorbates randomly distributed only on A sublattice or B sublattice of layers 1 while the layer 2 remains pristine.
On one hand, such a selective functionalization leads to the creation of a gap when sublattice B$_1$ is functionalized.
This gap is a
fraction of one eV and of at least 0.5 eV for a concentration $c$ of adsorbates larger than 1\% of the total number of atoms.
On the other hand, functionalization of sublattice A$_1$ decreases the effective coupling between layers, and thus the conductivity increases when $c$ increases, since the pristine layer is less perturbed by the disordered layer when $c$ increases.
These two types of selective functionalization exhibit very different and unusual behaviors.
This opens the way to the control of electronic properties through selective functionalization, which is experimentally feasible \cite{Katoch18}.
However, these extreme cases (A$_1$ or B$_1$ functionalization only)
seem too simple to correspond to the experimental sample.
Indeed the complexity of the bipartite BLG lattice requires further theoretical studies of other selective adsorbate distributions.
This is why it is necessary to study a combined functionalization of several sublattices.
In particular, we have to consider cases where midgap states are coupled each other and thus form a midgap band,
leading to new diffusivity properties that are not a simple combination of the extreme situations studied in Ref. \cite{Missaoui18}, where midgap states are not coupled together.
\begin{figure}
\includegraphics[width=8cm]{Fig_biAB_v9}
\caption{ \label{Fig_bilayer} Sketch of the crystal structure of AB stacked (Bernal) bilayer graphene (BLG). Atoms A$_1$ and B$_1$ are on the lower layer; A$_2$ , B$_2$ on the upper layer.
}
\end{figure}
In this paper, we present a detailed study of the electronic structure and quantum transport in BLG with adsorbates located on two different sublattices among the four sublattices A$_1$, A$_2$, B$_1$ and B$_2$.
We analyze how the symmetry is broken between sublattices under this selectivity, which may causes either a gap or an abnormal behavior of the conductivity.
We will pay particular attention to cases where B atoms are preferentially functionalized, as these cases should be energetically favorable.
For instance, under some specific conditions (adsorbates on B$_1$ and B$_2$ sublattices), a spectacular increase of diffusivity of charge carrier of midgap states band edge is obtained when the concentration $c$ of adsorbates increases.
The study of conductivity -taking into account all the effects of quantum interference- requires to distinguish several cases, depending on the value of the inelastic mean-free path $L_i$, mainly due to temperature.
At high temperature (typically room temperature), we calculate the microscopic conductivity $\sigma_M$; then we will analyze the quantum corrections at low temperature when the mean inelastic free path is very large, i.e. at the localization regime.
In the latter regime, we also study how localized states due to defects (midgap states), are at the origin of a particular quantum conductivity that cannot be explained by the Boltzmann's transport theory, and which is similar to the one found in quasicrystals \cite{Trambly06,Trambly17}, twisted bilayer graphene \cite{Trambly16} and recently graphene with defects inducing flat bands \cite{Bouzerar20,Bouzerar21}.
The remainder of this paper is organized as follows.
Section \ref{sec:level2} introduces the model and the formalism to compute density of states and the conductivity.
Section \ref{sec:level3} and
\ref{sec:level4} focus on selective distributions of vacancies distributed in layer 1 only, and the two layers, respectively.
Localization effects on conductivity are discussed in Section \ref{sec:level5}.
Finally, Section \ref{Sec_Conclusion} provides a summary and conclusions.
\section{\label{sec:level2}Electronic structure and numerical methods}
\subsection{TB Hamiltonian}
The tight binding (TB) Hamiltonian model for BLG with the $p_z$ orbitals only is given by:
\begin{equation}
H = \sum_{(i,j)}t_{ij}|i\rangle\langle j| ,
\label{eq_H}
\end{equation}
where $i$ is the index of $p_z$ orbitals,
the sum runs over neighbor sites $i$, $j$
and $t_{ij}$ is the hopping element matrix between site $i$ and site $j$.
In this paper, we consider only the coupling between first neighbors orbitals.
There are thus two types of coupling
(Fig. \ref{Fig_bilayer}):
an intralayer coupling term between first neighboring orbitals A$_i$ and B$_i$, $i=1,2$, equals to $\gamma_0=2.7$\,eV, and an interlayer coupling term between first neighboring orbitals A$_1$ and A$_2$ equals to $\gamma_1=0.34$\,eV \cite{Trambly12,Missaoui18}.
For that kind of calculations
a more realistic TB model with coupling terms above first neighbors leads qualitatively to similar results \cite{Trambly14,Missaoui17,Missaoui18}.
We have also checked that such a TB model leads to the results presented here are similar, but the first neighbors TB model allows to better analyze and discuss the physical mechanisms involved as it preserves the electron-hole symmetry.
In the Hamiltonian (equation (\ref{eq_H})), the on-site energies are taken equal to zero so that the Dirac energy $E_D$ is therefore equal to zero.
\subsection{Adsorbate simulation}
We consider that resonant adsorbates are a simple atoms or molecules --such as
H, OH, CH$_3$-- that create a covalent bond with carbon atom of the BLG. To simulate this covalent bond,
we assume that the $p_z$ orbital of the carbon, just below the adsorbate, is removed \cite{Pereira08a,Robinson08,Wehling10}.
In our calculations the vacancies are randomly distributed in two of the four sublattices A$_1$, A$_2$, B$_1$, and B$_2$, with finite concentration $c$ with respect to the total number of atoms. Here we study all possible cases of the double type of vacancies:
\begin{itemize}
\item A$_1$B$_1$-Va: Vacancies randomly distributed on sublattices A$_1$ and $B_1$.
An asymmetric distribution, A$^x_1$B$^{1-x}_1$-Va, where $x$
is the proportion of vacancies in the sublattice A$_1$, is also considered.
\item A$_1$A$_2$-Va: Vacancies randomly distributed on sublattices A$_1$ and $A_2$.
\item A$_1$B$_2$-Va: Vacancies randomly distributed on sublattices A$_1$ and $B_2$.
\item B$_1$B$_2$-Va: Vacancies randomly distributed on sublattices B$_1$ and $B_2$.
\end{itemize}
In the following, we call $X$-midgap states the states create by a random distribution of vacant atoms on the $X$-sublattice, with $X = $\,A$_1$, A$_2$, B$_1$, B$_2$, A$_1$B$_1$, A$_1$A$_2$, A$_1$B$_2$, or B$_1$B$_2$.
\subsection{Quantum transport calculation}
\label{Sec_QT}
We used the
Real Space Kubo-Greenwood (RSKG) method \cite{Mayou88,Mayou95,Roche97,Roche99,Triozon02} that has already been used to study quantum transport in disordered graphene, chemically doped graphene and bilayer (see for instance \cite{Lherbier12,Roche12,Roche13,Trambly11,
Trambly13,Trambly14,Missaoui17,Missaoui18,Omid20}),
functionalized carbon nanotubes
\cite{Latil2004,IshiiInelastic2010,Jemai19},
and many other systems \cite{Fan20} such as
quasicrystals \cite{Triozon02,Trambly17}, 2D molecular semiconductors \cite{Fratini17,Missaoui19}, and Perovskites \cite{Lacroix20}.
This numerical method connects the dc-conductivity $\sigma$,
$\sigma = e^2 n D$,
with the density of states $n$ and the diffusion coefficient,
\begin{equation}
D(E,t)=\frac{\Delta X^2(E,t)}{t},
\end{equation}
where the average square spreading $\Delta X^2$
is calculated at every energy $E$ and time $t$ by using the polynomial expansion method \cite{Mayou88,Mayou95,Roche97,Roche99,Triozon02},
\begin{equation}
\Delta X^2(E,t) = \frac{{\rm Tr} \left( [X,U(t)]^{\dag} \delta(E-H) [X,U(t)] \right)}{{\rm Tr} \,\delta(E-H)},
\label{eq_DX}
\end{equation}
where $U(t)$ is the evolution operator at time $t$,
$\delta$ is the Dirac function and
${\rm Tr}$ is the trace.
This numerical approach has the advantage of using efficiently the method in real space which allows to take into account all quantum effects and a realistic distribution of static scatters in a very large super-cell containing more than $10^7$ orbitals.
Considering such a huge cell, it is possible to evaluated the traces, ${\rm Tr} A$, in equation (\ref{eq_DX}) by the average $<$$A$$>$ on a random phase state \cite{Triozon02}.
Such a calculation may be done by the recursion method (Lanczos algorithm) where the Hamiltonian is written as a tridiagonal matrix in real-space \cite{Pettifor} of dimension $N_r$.
Here we use $N_r = 1500$ and we checked that presented results do not change significantly when $N_r$ increases. Lanczos method leads to a convolution results by a Lorentzian function which a small width $\epsilon$ (that is a kind of energy resolution of the calculation) that has been used in our previous paper on transport in BLG \cite{Missaoui17,Missaoui18,Omid20}.
But for systems with gap, to avoid the tail expansion of the Lorentzian function in the gap, it is more efficient to diagonalize the tridiagonal Hamiltonian of dimension $N_r$ and compute the DOS by Gaussian broadening of the spectrum \cite{Lacroix20}.
In the present work a Gaussian broadening is used with the Gaussian standard deviation of 5 meV. Note that for energies that are not close to the gap the two methods give almost the same results.
The Hamiltonian $H$ (equation (\ref{eq_H})), written in a supercell, takes into account the effects of elastic collisions (static defects), whereas inelastic collisions caused by temperature such as electron-phonon interactions are implanted by using the approximation of Relaxation Time Approximation (RTA). Thus the conductivity in the $x$-direction is given by \cite{Trambly13}:
\begin{eqnarray}
\sigma(E_{F},\tau_{i}) &=& e^{2}n(E_{F})D(E_{F},\tau_{i}) , \\
D(E_{F},\tau_{i}) &=& \frac{L_{i}^{2}(E_{F},\tau_{i})}{2\tau_{i}} ,\label{Eq_D_tau}\\
L_{i}^{2}(E_{F},\tau_{i}) &=& \frac{1}{\tau_{i}}\int_{0}^{\infty}\Delta X^{2}(E_F,t)e^{{-t}/{\tau_{i}}}dt,
\end{eqnarray}
where $E_F$ is the Fermi energy, $n(E_{F}) = {\rm Tr}\, \delta(E_F-H)$ is the total density of states (total DOS) and $L_{i}$ is the inelastic mean-free path conductivity along the $x$-axis. $L_{i}(E_F,\tau_{i})$ is the typical distance of propagation during the time interval $\tau_{i}$ for electrons at energy $E$.
At each energy, the microscopic diffusivity $D_M$ (microscopic conductivity $\sigma_M$) is defined as the maximum value of $D(\tau_i)$ ($\sigma(\tau_i)$).
We compute also the elastic mean-free path $L_{e}$ along the $x$-axis, from the relation \cite{Trambly13},
\begin{equation}
\label{le}
L_e(E) = \frac{1}{V_{0}(E)}\, {\rm Max}_{\tau_i} \left\{ \frac{L_i^{2}(E,\tau_i)}{\tau_i} \right\} = \frac{2 D_M(E)}{V_{0}(E)}.
\end{equation}
Roughly speaking $L_e$ is the average distance between two elastic scattering events.
Microscopic diffusivity and conductivity correspond to the situation where $L_e \simeq L_i$ with is a large (or room) temperature limit.
But at low temperature, $L_i \gg L_e$, which corresponds to
localization regime.
\begin{figure*}
\includegraphics[width=7cm]{DOS_A1B1_config_3pc_0_50} ~~~~
\includegraphics[width=7cm]{DOS_A1B1_config_3pc_50_100}
\vspace{.2cm}
\includegraphics[width=7cm]{sigma_conf_A1B1_0_50}~~~~
\includegraphics[width=7cm]{sigma_conf_A1B1_50_100}
\caption{\label{Fig_A1B1_conf}
BLG with A$^x_1$B$^{1-x}_1$-Va for different distributions $x$ of vacancies between A$_1$ and B$_1$ sites:
(a-b) $x \in [0;0.5]$ (mainly B$_1$-Va)
and (c-d) $x \in [0.5;1]$ (mainly A$_1$-Va).
(a-c) Density of states $n(E)$, the integrated density of states is represented on the left insert while the density of states around the Dirac energy E$_D$ on the right insert.
(b-d) The microscopic conductivity $\sigma(E)$ for the same disorder configurations.
The total concentration of vacancies is 3$\%$.
$G_0 = 2e^2/h$.
}
\end{figure*}
\section{\label{sec:level3} Vacancies in one layer only}
In this section, we are focusing on the impact of the vacancies distributed on one layer (layer 1) of the BLG.
It should simulate adsorbates or defects that come from the preparation process \cite{Yang18} or induced by the substrate \cite{Ordered}.
For example, in epitaxial graphene on Pt(111) \cite{Ordered}, the authors have shown the appearance of covalent bonds between the carbon atoms of graphene and the atoms of Pt.
Since the B$_1$ atoms of layer 1 do not have a first neighbor in layer 2, it is likely that their functionalization is favored, although the experimental results \cite{Katoch18} do not show functionalization only on B atoms.
It is thus important to study an asymmetric functionalization of B$_1$- or A$_1$-sublattice. We first consider a majority functionalization of the B$_1$ atoms (A$_1$ atoms), and we analyze the effect of defect concentrations for a symmetric distribution of vacancies.
\subsection{A$_1$B$_1$-Va asymmetrically distributed}
\label{Sec_A1B1_Assym}
We consider an asymmetric distribution of vacancies:
A$^x_1$B$^{1-x}_1$-Va, where $x$ ($1-x$) is the proportion of vacancies on sublattice A$_1$ (B$_1$).
Considering the cases with a total number of vacancies corresponding to a concentration $c = 3\%$ with respect to the total number of atoms,
the density of states $n(E)$ and the microscopic conductivity $\sigma_M(E)$ are shown in Fig. \ref{Fig_A1B1_conf} for different $x$ values.
As presented Fig. \ref{Fig_A1B1_conf_05pc} in the Supplemental Material \cite{SupMat}, results for $c = 0.5 \%$ show very similar behaviors.
The different disorder distributions, i.e. value of $x$ between $x=0$ (B$_1$ vacancies only) and $x=1$ (A$_1$ vacancies only) affect strongly the regime around the Dirac energy $E_D$.
Midgap states at $E_D$ always appeared in both layers.
Indeed, each A$_1$ missing orbital of layer 1 produces a A$_1$-midgap state at Dirac energy E$_D$ that spreads on B$_1$ sublattice (layer 1) only, and B$_1$ missing orbital produces a B$_2$-midgap states that spreads on A$_1$ (layer 1) and B$_2$ (layer 2) sublattices \cite{Missaoui18}. A$_1$-midgap states and B$_1$-midgap states are coupled by the Hamiltonian and form a band of midgap states with specific transport properties.
In the extreme cases of vacancies distributed over a single sublattice B$_1$ ($x=0$),
we have shown \cite{Missaoui18} that a gap around the Dirac energy E$_D$ is created. This gap is a consequence of the reduction of the average number of neighboring atoms of the atoms of a sublattice in a bipartite lattice.
For intermediate $x$ values, the gap disappears under the effect of the interactions between midgap states.
Depending on $x$ values, two scenarios emerge:
(i) For $x \in [0;0.3]$ and $ x \in [0.7;1]$,
the number of A$_1$-midgap states and $B_1$-midgap states are rather different, and many of those states are not coupled to each other and remain isolated with energy $E_D$. The small number of mixed midgap states leads to a small DOS at intermediates energies
(Fig. \ref{Fig_A1B1_conf} (a)).
Concerning the conductivity, two different behaviors are obtained according to the dominant concentration of B$_1$ vacancies $(x \in [0;0.3])$ or A$_1$ vacancies $(x \in [0.7;1])$.
The behavior of $\sigma_{m}(E)$ around Dirac energy for $x \in [0;0.3]$ is determined mainly by the effects of the B$_1$ vacancies. For energies $E$ in the intermediate regime with $E \leq \gamma_1=0.34$ eV, $\sigma_{m}$ increases when the coupling between midgap states increases i.e. when A$_1$ and B$_1$ vacancy concentrations are close each other.
For $x \in [0.7;1]$, configuration is very sensitive to the concentration of A$_1$ vacancies in the structure of BLG. $\sigma_M$ increases when $x$ increases. This effect of A$_1$ vacancies affects the microscopic conductivity on a range of energy that not exceeds $1$\,eV as it is shown in Fig. \ref{Fig_A1B1_conf}(b).
In the extreme case $x=1$ a gap is appeared in the average DOS for the layer with defects (layer 1) \cite{Missaoui18}. It is proportional to the concentration $c$ of vacancies
and the layer 2 behaves more and more like a pristine MLG which gives the ballistic behavior. When $x \lesssim 1$, the gap in layer 1 disappeared which explains the progressive decrease of the microscopic conductivity when $x$ is decreasing from the value $1$.
(ii) The interactions between midgap states is important for $x \in [0.4;0.6]$, and it is maximum for $x=0.5$. Therefore $n(E)$ is larger for energy $E \ne E_D$ (right insert of Fig. \ref{Fig_A1B1_conf}(a)).
The conductivity behavior is similar to that for $x=0.5$ studied in the following section.
\subsection{A$_1$B$_1$-Va symmetrically distributed}
\label{Sec_A1B1_sym}
\begin{figure}
\includegraphics[width=7cm]{DOS_A1B1}
\vskip .2cm
\includegraphics[width=7cm]{LDOS_A1B1}~~
\includegraphics[width=7cm]{sigma_A1B1}
\caption{ \label{Fig_A1B1}
Electronic properties in BLG with A$_1$B$_1$ vacant atoms (A$_1$B$_1$-Va), with equal distribution of vacancies between A$_1$ and B$_1$ sublattices:
(a) total DOS (dashed line is the total DOS without vacancies),
(b) average local DOS on A$_1$, B$_1$, A$_2$, B$_2$ atoms for $c = 0.25$\%
(dashed line and dot line are LDOS on A and B atom without vacancies),
(c) microscopic conductivity $\sigma_M(E)$.
$c$ is the concentration of vacancies with respect to the total number of atom in BLG.
$G_0 = 2e^2/h$.
}
\end{figure}
We now study a random distribution of defects equally distributed in sublattice A$_1$ and B$_1$, labeled A$_1$B$_1$-Va. Total DOS $n(E)$, LDOS and microscopic conductivity $\sigma_M(E)$ are shown in Fig. \ref{Fig_A1B1} for several values of vacancies concentration $c$ with respect to the total number of atoms.
Since the electron transport through the BLG is mainly determined by the electrons which have an energy close to the Dirac point, the conductivity is displayed within a small energy region around the charge neutrality energy $E_D = 0$.
By inspecting Figs \ref{Fig_A1B1}(a-b-c), one can identify several important features.
(i) For all concentrations $c$ and energy around $E_D$, $0.02\,{\rm eV} < |E-E_D|<0.1$\,eV, $\sigma_M$ presents a minimum plateau at conductivity $\sigma_M \simeq 1.2$\,G$_0$, with $G_0 = 2e^2/h$.
Thus $\sigma_M \simeq 2 \sigma_M^{mono}$, where
$\sigma_M^{mono} \simeq 0.6$\,G$_0$ is the monolayer graphene (MLG) microscopic conductivity \cite{Yuan10,Gonzalez10,Trambly13,Missaoui17}.
This shows that the defects affect both planes similarly, although one of the two planes is defect-free.
Moreover, the presence of a plateau almost independent on the concentration, shows that the microscopic quantities in the BLG are not affected directly by interlayer coupling terms, which gives them a behavior similar to MLG.
That behavior is understandable since the elastic mean-free path $L_e$ (see Supplemental Material \cite{SupMat} Fig. \ref{Fig_A1B1_Le}) is smaller than the traveling distance $l_1$ in a layer between two interlayer hoppings,
$l_1 \simeq 1-2$\,nm \cite{Missaoui17}.
(ii) For energies far from $E_D$, $|E-E_D| > 0.1$\,eV, two behaviors of the conductivity is observed: for $c \leq 2\%$, $\sigma_{M}\simeq \sigma_{B}$, where $\sigma_{B}$ is calculated with the Bloch-Boltzmann approach \cite{Castro09_RevModPhys,McCann13}, and then conductivity is proportional to $1/c$.
While for $c \geq 2\%$, $\sigma_{M}$ seems to depends less on $c$, and even slightly increases when $c$ increases like for
A$_{1}$ vacancies alone or B$_{1}$ vacancies alone \cite{Missaoui18}.
\section{\label{sec:level4} Vacancies in both layers}
In this section we study the combined effect of vacancies distributed in two sublattices that do not belong to the same layer.
The case B$_1$B$_2$-Va, that seems to be the most favored cases for functionalization, is consider first. These midgap states are coupled each-other and form a midgap band characterized by a very unusual quantum diffusion of charge carriers.
After, we study the cases of
A$_1$A$_2$-Va and A$_1$B$_2$-Va, that both produce orthogonal midgap states at energy $E=E_D= 0$.
\subsection{B$_1$B$_2$-Va cases}
\begin{figure}
\includegraphics[width=7cm]{DOS_B1B2}
\includegraphics[width=7cm]{LDOS_B1B2}
\includegraphics[width=7cm]{sigma_B1B2}
\caption{ \label{Fig_B1B2}
(color online)
Electronic properties in BLG with B$_1$B$_2$ vacant atoms:
(a) total DOS (dashed line is the total DOS without vacancies),
(b) average local DOS on A$_1$, B$_1$, A$_2$, B$_2$ atoms for $c = 0.25$\%
(dashed line and dot line are LDOS on A and B atom without vacancies). The average local DOS on A$_2$, B$_2$ atoms is obtained by a symmetry with relative atoms A$_1$, B$_1$ respectively.
(c) microscopic conductivity $\sigma_M(E)$.
$c$ is the concentration of vacancies with respect to the total number of atom in BLG.
$G_0 = 2e^2/h$. }
\end{figure}
B$_1$- and B$_2$-midgap states are distributed over all the structure with different weights on the atoms A$_1$, A$_2$, B$_1$, and B$_2$ (Fig. \ref{Fig_B1B2}(b)).
They form a band since B$_1$-Va midgap states and B$_2$-Va midgap states are coupled by the Hamiltonian.
Their electronic properties are thus very different than those
of B$_1$ vacancies in BLG for which a gap proportional to $c$ is formed around $E_D$ \cite{Missaoui18}; while with B$_1$B$_2$-Va, the B$_1$- and B$_2$-midgap states are coupled, and thus the gap is filled or partially filled by a midgap states band.
Several regimes are present depending on both energy $E$ and vacancy concentration $c$.
For small $c$ concentration, typically $c \le 1\%$,
there is no gap in the DOS (Fig. \ref{Fig_B1B2}(a)) and states around $E_D$ form a narrow midgap states band.
The corresponding microscopic conductivity $\sigma_M$ presents a plateau (see the insert Fig. \ref{Fig_B1B2}(c)) at a value independent on $c$,
$\sigma_{m}\simeq 2\sigma_M^{mono}$.
For high concentration $c$, the density of states (Fig. \ref{Fig_B1B2}(a)) around $E_D$ increases significantly, and as a direct consequence, the plateau of conductivity increases $\sigma_{m}> 2\sigma_M^{mono}$.
As explain above (Sec. \ref{Sec_A1A2_A1B2}), in each layer the gap due to B-Va increases when $c$ increases, therefore
for large $c$ the midgap states band width becomes smaller than the gap, and the midgap states band becomes isolated from other states by small gaps at $|E| \gtrsim \gamma_1 $ (Fig. \ref{Fig_B1B2}(a)).
The width of this isolated bands is $\Delta w \simeq 2\gamma_1$, i.e. $E \in [-\gamma_1,\gamma_1]$.
For large $c$ concentration, the edge states ($E\simeq\pm \gamma_1$) have a very exotic conductivity $\sigma_M$ which strongly increases when $c$ increases, whereas DOS does not change too much.
Roughly speaking this spectacular behavior can be explained by considering the coupling between the B$_1$-Va monolayer midgap states and the B$_2$-Va monolayer midgap states.
In monolayer, a B-Va midgap states are located on A orbital sublattice around the B vacancy.
B-Va midgap states of each layer are orthogonal to each other.
But, since each A orbital are coupled with an A orbital of the other layer, a B$_1$-Va midgap state is coupled with a B$_2$-Va midgap state, with a hopping term $\gamma_{B_1-B_2}$.
$\gamma_{B_1-B_2} \simeq \gamma_1$, for the smallest $d_{B_1-B_2}$ distance between the B$_1$-Va and the B$_2$-Va (typically first neighbor $B_1$-$B_2$), and $\gamma_{B_1-B_2}$ decreases when $d_{B_1-B_2}$ increases.
When $c$ increases, the average $d_{B_1-B_2}$
distance decreases and thus the average $\gamma_{B_1-B_2}$ value increases. As a result, by a kind of percolation between monolayer B-midgap states of the two layers, the conductivity through the BLG increases strongly when $c$ increases.
Finally, the presence of the conductivity plateau for all concentrations (insert Fig. \ref{Fig_B1B2}(c)) can
be understood considering the
elastic mean free path $L_e$ shown
in Supplemental Material \cite{SupMat} (Fig. \ref{Fig_A1B1_Le}).
Around $E_D$ energy ($E \in [-0.2;0.2]$\,eV), $L_e<l_1$,
where $l_1 \simeq 1-2$\,nm is the traveling distance between two interlayer hopping events \cite{Missaoui17}.
Thus, the diffusion of the charge carriers is not affected by the interlayer coupling. The diffusive regime is reached in each layer independently,
and it takes the MLG minimum value in each layer.
Note that like for other type of vacancies, for energy away from Dirac energy, $|E-E_D| \gg \gamma_1$, Boltzmann behavior is always found.
\subsection{A$_1$A$_2$-Va and A$_1$B$_2$-Va cases}
\label{Sec_A1A2_A1B2}
\begin{figure}
\includegraphics[width=7cm]{DOS_mGauss_A1A2}
\includegraphics[width=7cm]{LDOS_mGauss_A1A2}
\includegraphics[width=7cm]{sigma_mGauss_A1A2}
\caption{ \label{Fig_A1A2}
(color online)
Electronic properties in BLG with A$_1$A$_2$ vacant atoms:
(a) total DOS (dashed line is the total DOS without vacancy),
(b) average local DOS on A$_1$, B$_1$ atoms for $c = 0.25$\%
(dashed line and dot line are LDOS on A and B atom without vacancy). The average local DOS on A$_2$, B$_2$ atoms is obtained by a symmetry with relative atoms A$_1$, B$_1$ respectively.
(c) microscopic conductivity $\sigma_M(E)$.
$c$ is the concentration of vacancies with respect to the total number of atoms in BLG.
For clarity the midgap states at $E_D=0$ are not shown (see text).
$G_0 = 2e^2/h$. }
\end{figure}
\begin{figure}
\includegraphics[width=7cm]{DOS_mGauss_A1B2}
\includegraphics[width=7cm]{LDOS_mGauss_A1B2}
\includegraphics[width=7cm]{sigma_mGauss_A1B2}
\caption{ \label{Fig_A1B2}
(color online)
Electronic properties in BLG with A$_1$B$_2$ vacant atoms:
(a) total DOS (dashed line is the total DOS without vacancy),
(b) average local DOS on A$_1$, B$_1$, A$_2$, B$_2$ atoms for $c = 0.25$\%
(dashed line and dot line are LDOS on A and B atom without vacancy),
(c) microscopic conductivity $\sigma_M(E)$.
$c$ is the concentration of vacancies with respect to the total number of atom in BLG.
For clarity the midgap states at $E_D=0$ are not shown (see text).
$G_0 = 2e^2/h$.}
\end{figure}
The double-type vacancies: A$_1$A$_2$-Va (vacancies randomly distributed on A$_1$ and A$_2$ sublattices) and A$_1$B$_2$-Va (vacancies randomly distributed on A$_1$ and B$_2$ sublattices) are characterized by the absence of coupling between midgap states
and thus all midgap states remain at energy $E_D=0$.
Indeed, in the case of A$_1$A$_2$-Va, $N$ vacancies on atoms A$_1$ (A$_2$) sublattice produce a set of $N$ orthogonal midgap states at Dirac energy $E_D=0$ that are located on the orbitals B$_1$ (B$_2$) of the same layer \cite{Missaoui18}. As B$_1$ orbitals and B$_2$ orbitals are not directly coupled by the Hamiltonian, midgap states located on B$_1$ and B$_2$ sublattices are not coupled together.
In the case A$_1$B$_2$-Va, vacancies are vacant atoms of the same sublattice $\alpha$ of the BLG lattice. Corresponding midgap states are thus orthogonal states at $E_D$, located on the $\beta$ sublattice with a greater weight on the B$_1$ atoms.
For clarity those isolated states at $E_{D}=0$ are not shown in the DOSs drawn Figs. \ref{Fig_A1A2} and \ref{Fig_A1B2} (see Supplemental Material \cite{SupMat} Sec. \ref{Sec_SupMat_gaussian}).
\begin{figure*}
\includegraphics[width=7cm]{sig_li_A1B1_ale}~~~~ \includegraphics[width=7cm]{sig_li_A1A1}
\includegraphics[width=7cm]{B1B2_loc} ~~~~~\includegraphics[width=7cm]{sig_li_A1B2}
\caption{ \label{locA1B1} Conductivity $\sigma$ as a function of inelastic scattering length $L_i$ for 1\%. (a) Vacancies randomly distributed on the atoms A$_{1}$ and B$_{1}$, (b) Vacancies randomly distributed on the atoms B$_{1}$ and B$_{2}$,(c) Vacancies randomly distributed on the atoms A$_{1}$ and A$_{2}$. (d) Vacancies randomly distributed on the atoms A$_{1}$ and B$_{2}$.
$G_0 = 2e^2/h$.}
\end{figure*}
\begin{figure}
\includegraphics[width=7cm]{sig_A1A2_e0}
\includegraphics[width=7cm]{sig_A1B2_e0}
\caption{ \label{loc_midgap} Conductivity $\sigma(E=E_D=0)$ as a function of inelastic scattering time $\tau_i$. (a) Vacancies randomly distributed on the atoms A$_{1}$ and A$_{2}$, (b) Vacancies randomly distributed on the atoms 1$_{1}$ and B$_{2}$. In both cases midgap states are orthogonal states at $E_D=0$ isolated by gaps.
$G_0 = 2e^2/h$.}
\end{figure}
In A$_1$A$_2$-Va case, A$_1$ vacancies and A$_2$ vacancies act on both layers symmetrically and independently because midgap states of a layer is not coupled with midgap states of the other layer.
Thus the results is simply the sum of two independent MLG.
In MLG, vacancies in sublattice A (resp. B) produce midgap states at $E_D$ that are located in sublattice B (resp. A).
As shown in our previous paper \cite{Missaoui18} by a an analysis of the spectrum of bipartite Hamiltonian,
when the concentration $c$ of vacancies increases, a gap increases occurs around the Dirac energy.
This gap is a consequence of the reduction of the average number of neighboring atoms of the atoms of the sublattice which do not contain vacancies.
Thus, A$_1$ vacancies (A$_2$ vacancies) create a gap in layer 1 (layer 2) as
it is clearly shown on the local DOS of atoms A$_1$ and B$_1$ (Fig. \ref{Fig_A1A2}(b)).
The total DOS has a gap proportional to the concentration of vacancies $c$ around the dirac energy $E_D$ (Fig. \ref{Fig_A1A2}(a)).
A$_1$B$_2$-Va create also a gap because they are distributed randomly on the same sublattice
$\alpha$ $\{\rm A_1B_2\}$ of BLG. Total and local DOSs (Fig. \ref{Fig_A1B2}(a-b)) confirm the presence of a gap around Dirac energy E$_D$.
The microscopic conductivity $\sigma_M(E)$ for both types of vacancies A$_1$A$_2$-Va and A$_1$B$_2$-Va are shown in the Figs. \ref{Fig_A1A2}(c) and \ref{Fig_A1B2}(c), respectively.
The midgap states at energies $E=E_D$ do not contribute to the conductivity $\sigma_M$ since they are isolated localized states around each vacancies.
Beyond the gap, $\sigma_M$ decreases when $c$ increases, following a typical Boltzmann behavior \cite{Castro09_RevModPhys}.
\section{\label{sec:level5} Quantum correction of conductivity: Anderson localization}
In the framework of Relaxation Time Approximation (RTA) it is possible to compute the inelastic mean-free path $L_i(E,\tau_i)$ at every energies $E$ and inelastic scattering times $\tau_i$ (Sec. \ref{Sec_QT}).
Figure \ref{locA1B1} shows the conductivity $\sigma$ drawn versus $L_i$ for different types of vacancies and different energies close to $E_D$.
The microscopic conductivity $\sigma_M(E)$ is the maximum value of the curves $\sigma(\tau_i)$ at the corresponding energy $E$.
Each curve $\sigma(L_i)$ has three parts.
For small $L_i$, typically $L_i \ll L_e$, diffusivity is ballistic, the defects have not a direct effect and $\sigma \propto L_i$.
Diffusive regime, $\sigma(L_i) \simeq \sigma_M$, is reached for $L_i \simeq L_e$.
For a small defect concentrations $c$, that regime can be found for a wide range of $L_i$ values.
For large $L_i$ values, $L_e \ll L_i$,
localization regime is reached and $\sigma(L_i)$ decreases when $L_i$ increases.
Inelastic scattering collisions can be caused by temperature through electron-phonon interactions.
At room temperature, $L_i$ is expected to be close to $L_e$, $\sigma(L_i) \simeq \sigma_M$, and the quantum correction is negligible. $L_i$ increases when temperature decreases, and at low temperatures, i.e. when $L_i \gg L_e$, quantum interferences dominate the transport properties.
At low values of temperature ($L_i\gg L_e$), in 2D materials, the Anderson localization due to quantum interferences leads to a conductivity varying linearly with $\ln L_i$, \cite{Lee85} and can be written,
\cite{Trambly11, Trambly13, Trambly14},
\begin{equation}
\label{localisation}
\sigma(E,L_i) = \sigma_0(E) - \alpha G_0 \ln \left(\frac{L_i}{L_e(E)} \right),
\end{equation}
where $G_0 = {2 e^2}/{h}$,
and $\sigma_0$ values are on the range of $\sigma_M$ values.
The second term of the right side of equation (\ref{localisation}) is the
quantum correction of the conductivity.
The linear behavior of $\sigma(L_i)$ is clearly seen for cases A$_1$B$_1$-Va and B$_1$B$_2$-Va (Fig. \ref{locA1B1}(a-b)).
The extrapolation of $\sigma(L_i)$ curves allows to obtain the $\alpha$ value, $\alpha \simeq 0.34$. These values are close to the result found in MLG \cite{Trambly13}, BLG with random vacancies \cite{Missaoui17},
twisted bilayer graphene \cite{Omid20}, and close to the prediction of the perturbation theory of 2D Anderson localization \cite{Lee85} for which $\alpha=1/\pi$.
The localization length $\xi$ can be extracted from the expression (\ref{localisation}) by extrapolation
of $\sigma(L_i)$ curves (Fig. \ref{locA1B1}(a-b)) when $\sigma(L_i=\xi)=0$, giving at the end the following expression,
\begin{equation}
\xi(E) = L_e(E) \exp \left( \frac{\sigma_0(E)}{\alpha G_0} \right).
\end{equation}
Since $\alpha$ is a constant this leads to a simple relationship between $\xi$ and $L_e$, $\xi \simeq 50 L_e$,
which is between monolayer graphene values with
random vacancy distributions ($13 L_e$) \cite{Trambly13} and that of BLG
in the same case ($13^2 L_e$) \cite{Missaoui17}.
For A$_1$A$_2$-Va and A$_1$B$_2$-Va cases,
at energies around the edge of the gap
(Figs. \ref{locA1B1}(c-d)), the decrease of $\sigma(L_i)$ does not follow the equation (\ref{localisation}).
This behavior is more similar to what is generally expected for the conduction by midgap states of graphene \cite{Trambly13}, which are very localized states with abnormal diffusion behavior.
It is also interesting to focus on the conduction by flat-band midgap state themselves i.e., here, midgap states at energy $E_D=0$ that are not coupled each-other by the Hamiltonian (cases A$_1$A$_2$-Va and A$_1$B$_2$-Va cases). In these midgap states the average velocity is zero but the conduction is possible due to the inherent quantum fluctuations of the position. Indeed in the presence of inelatic scattering these fluctuations are modified \cite{Bouzerar21} and do not cancel completely at large times which allows the electronic diffusion. It results a non-Boltzmann conductivity, similar to that found in quasicrystals \cite{Trambly06,Trambly17}, twisted bilayer graphene at the magic angle \cite{Trambly16}, and graphene with particular defects inducing flat bands \cite{Bouzerar20,Bouzerar21}.
In cases A$_1$A$_2$-Va and A$_1$B$_2$-Va, the
microscopic conductivity, i.e. small inelastic mean-free time $\tau_i$, at midgap-states energy is negligible. But at large $\tau_i$, i.e. large $L_i$, the Kubo-Greenwood conductivity of midgap states is, \cite{Bouzerar21}
\begin{equation}
\sigma(E,\tau_i) = e^2 n_i(E,\tau_i)D(E,\tau_i),
\end{equation}
where $n_i$ and $D$ are the DOS and the diffusivity (Eq. (\ref{Eq_D_tau})) in presence of inelastic scattering.
Since midgap states are non-dispersive states at $E=0$, isolated by gaps (cases A$_1$A$_2$-Va and A$_1$B$_2$-Va), $n_i$ is the broadening of the Delta function, $c \delta(E)$, by a Lorentzian with a width at half maximum $\eta$, $\eta = \hbar/\tau_i$. Thus at Dirac energy $E_D=0$,
\begin{equation}
\sigma(E=0,\tau_i) = \frac{16}{S} G_0 c \tau_i D(E=0,\tau_i),
\label{Eq_sigma_FTaui_E0}
\end{equation}
where $S$ is the surface of the unit cell.
As shown Fig. \ref{loc_midgap}, for large $\tau_i$, $\sigma(E=0,\tau_i)$ reaches a constant universal value, independent on the defect concentration $c$, which is twice that of graphene \cite{Bouzerar21}: $\sigma(E=0) \simeq 1.3 G_0$.
\section{Conclusion}
\label{Sec_Conclusion}
We have studied the effects on the electronic properties of vacant atoms (vacancies) selectively distributed over two sublattices of the Bernal bilayer graphene (BLG).
Vacancies sketch the effects of adsorbates such as atoms or molecules that are covalently bound to a C atom of the BLG.
A wide variety of behaviors has been found and classified according to the functionalized sublattices, the adsorbates concentration $c$, and the energy.
Thus we prove theoretically that controlled functionalization can be an excellent way to tune BLG conductivity, in agreement with recent experimental results \cite{Katoch18}
showing that it is possible to control the functionalization with an adsorbate rate of the order of 1\% of the total number of atoms.
It is thus possible to predict the conditions for opening a mobility gap of several $100$\,meV at the energy of charge neutrality.
Moreover,
in the specific case of B$_1$B$_2$ vacancies with $c>1\%$, an isolated midgap states band is predicted.
Spectacularly, its edge states have a high electrical conductivity due to large diffusivity of charge carriers that increases when the number of defects ($c$ value) increases.
The latter case shows a very particular and original effect of the coupling between midgap states in a bipartite lattice.
Since the covalent functionalization of B atoms seems to be favorable energetically, its experimental realization should be possible in the context of a sensor device for example.
\section*{Acknowledgments}
The authors wish to thank G.\ Bouzerar, L.\ Magaud, P.\ Mallet, G.\ Jema\"{i}, and J.-Y.\ Veuillen for fruitful discussions. Calculations
have been performed at the Centre de Calculs (CDC), CY Cergy Paris
Universit\'e.
We thank Y.\ Costes and B.\ Mary, CDC, for computing assistance. This work
was supported by the ANR project J2D (ANR-15-CE24-0017) and the Paris//Seine excellence initiative (grant 2019-055-C01-A0).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,187 |
Q: Disable recording IP address in google app engine logs Is it possible to make configurations in Google app engine for not recording IP addresses from clients to the Java application in stackdriver logs?
A: No, you cannot configure or opt-out from request logs, where the client IP addresses are recorded. From Using Stackdriver Logging in App Engine apps:
The App Engine standard environment produces the following logs:
*
*Request logs, appengine.googleapis.com/request_log, called request_log in the Logs Viewer. This log records requests sent to
all App Engine apps. The request log is provided by default and you
cannot opt out of receiving it. For more details, see the RequestLog
type.
...
The App Engine flexible environment produces the following logs:
*
*Request logs record requests sent to all App Engine apps. The request log is provided by default and you cannot opt out of receiving
it.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,771 |
El formigueret fumat (Myrmotherula schisticolor) és una espècie d'ocell de la família dels tamnofílids (Thamnophilidae).
Hàbitat i distribució
Habita el sotabosc de la selva humida i vegetació secundària als turons i muntanyes, i localment a les terres baixes, des de Mèxic, al nord de Chiapas, i Guatemala cap al sud fins l'oest de Panamà. Des de Colòmbia i oest i nord de Veneçuela, cap al sud, per l'oest dels Andes, fins l'oest d'Equador i, per l'est dels Andes, a través de l'est d'Equador fins al nord i est de Perú.
Referències
fumat | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,237 |
Q: D3 sorting with an ordinal scale and selecting top 5 I am trying to sort then select the top 5 in a D3 chart.
I have two series: "OL" and "NOL". The first thing I want to be able to do is to sort and have the labels match on the y axis. The second thing I want to do is to only show the top 5 with the two series still together udner one name in the y axis.
jsfiddle
Data is like so:
var values = feature.properties;
var data = [
{key: "N11", name:"N11OL",value:values["N11OL"]},
{key: "N11", name:"N11NOL",value:values["N11NOL"]},
{key: "N21", name:"N21OL",value:values["N21OL"]},
{key: "N21", name:"N21NOL",value:values["N21NOL"]},
{key: "N22", name:"N22OL",value:values["N22OL"]},
{key: "N22", name:"N22NOL",value:values["N22NOL"]},
{key: "N23", name:"N23OL",value:values["N23OL"]},
{key: "N23", name:"N23NOL",value:values["N23NOL"]},
{key: "N31-33", name:"N31_33OL",value:values["N31_33OL"]},
{key: "N31-33", name:"N31_33NOL",value:values["N31_33NOL"]},
{key: "N41", name:"N41OL",value:values["N41OL"]},
{key: "N41", name:"N41NOL",value:values["N41NOL"]}
];
Right now, the chart works before I sort the bars. When I sort the data with the code below, the ordinal domain doesn't match the bars anymore.
data.sort(function (a, b) {
return d3.ascending(a.value, b.value);
});
I could use
domain(data.map(function(d) { return d.name; }))`
});
to get the names from the data to show on the axis but I want "OL" and "NOL" series to be under one name: N11, etc. not show with all bars. That is why I have set a domain with :
.domain(["NAICS11", "NAICS21", "NAICS22", "NAICS23", "NAICS31-33", "NAICS41"])
That is half of the problem. Once I have sorted and the names are matching with the axis, I would like to have only the top 5 for both series.
I would be very grateful for any suggestions on how to do this.
EDIT I added a key to the data for the pairs and modified the D3 code. I feel this will make it easier going forward to get this chart where I want it.
A: How about:
// create temporary array with data grouped by
// the first part of our name (cut off the OL/NOL)
var tmp = d3.nest().key(function(d){
return d.name.substring(0, 3);
}).entries(data);
// sort this array descending with the max of our
// now "paired" entries
tmp.sort(function(a,b){
return (d3.max([a.values[0].value,
a.values[1].value]) >
d3.max([b.values[0].value,
b.values[1].value])) ? -1 : 1;
});
// loop our tmp array and
// flatten the data back out
// also use our substring names to for the domain
data = [];
domain = [];
tmp.forEach(function(d){
domain.push(d.key);
data.push(d.values[0]);
data.push(d.values[1]);
});
Updated fiddle.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,227 |
Q: Convert a jQuery statement to simple javascript I do not wish to use the jQuery plugin or the . I wish to make all inputs' autocomplete off.
I have this code-
$("input, select, textarea").attr("autocomplete", "off");
Can this be put in normal javascript, if possible?
A: Use querySelectorAll() with forEach()
*
*Get dom elements using querySelectorAll()
*Convert it to array using Array.from()
*Iterate over elements using forEach() iterator
*Set the attribute with help of setAttribute()
Array.from(document.querySelectorAll("input, select,textarea")).forEach(function(ele) {
ele.setAttribute("autocomplete", "off");
});
<textarea></textarea>
<input />
<select></select>
For older browser check polyfill options of forEch and Array.from methods.
Even you can simplify the code using call() with forEach()
[].forEach.call(document.querySelectorAll("input, select,textarea"), function(ele) {
ele.setAttribute("autocomplete", "off");
});
<textarea></textarea>
<input />
<select></select>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,020 |
Q: Need of Virtual memory? I was recently asked a question that in a computer system, if the primary memory(RAM) is comparable to the secondary memory (HDD) then is there a need for virtual memory to be implemented in such a computer system ?
Since paging and segmentation require context switching, which is purely processing overhead, would the benefits of virtual memory overshoot the processing overhead it requires ?
Can someone help me with this question ?
Thanku
A: It is true that with virtual memory, you are able to have your programs commit (i.e. allocate) a total of more memory that physically available. However, this is only one of many benefits if having virtual memory and it's not even the most important one. Personally, when I use a PC, I periodically check task manager to see how close I come to using my actual RAM. If I constantly go over, I go and I buy more RAM.
The key attribute of all OSes that use virtual memory is that every process has its own isolated address space. That means you can have a machine with 1GB of RAM and have 50 processes running but each one will still have 4GB of addressable memory space (32-bit OS assumed). Why is it important? It's not that you can "fake things out" and use RAM that isn't there. As soon as you go over and swapping starts, your virtual memory manager will begin thrashing and performance will come a halt. A much more important implication of this is that if each program has it's own address space, there's no way it can write to any random memory location and affect another program.
That's the main advantage: stability/reliability. In Windows 95, you could write an application that would crash entire operating system. In W2K+, it is simply impossible to write a program that paves all over its own address space and crashes anything other than self.
There are few other advantages as well. When executables and DLLs are loaded into RAM, virtual memory manager can detect when the same binary is loaded more than once and it will make multiple processes share the same physical RAM. At virtual memory level, it appears as if each process has its own copy, but at a lower level, it all gets mapped to one spot. This speeds up program startup and also optimizes memory usage since each DLL is only loaded once.
Virtual memory managers also allow you to perform file I/O by simply mapping files to pages in the virtual address space. In addition to introducing interesting alternative to working with files, this also allows for implementations of shared memory segments which is when physical RAM with read/write pages is intentionally shared between processes for extremely efficient inter-process communications (IPC).
With all these benefits, if we consider that most of the time you still want to shoot for having more physical RAM than total commit size and consider that modern CPUs have support for virtual address mapping built directly into the hardware, the overhead of having virtual memory manager is actually very minimal. On the other hand, in environments where many applications from many different vendors run concurrently, process address space is priceless.
A: Virtual memory working
It may not ans your whole question. But it seems the ans to me
A: I'm going to dump my understanding of this matter, with absolutely no background credentials to back it up. Gonna get downvoted? :)
First up, by saying primary memory is comparable to secondary memory, I assume you mean in terms of space. (Afterall, accessing RAM is faster than accessing storage).
Now, as I understand it,
Random Access Memory is limited by Address Space, which is the addresses which the operating system can store stuff in. A 32bit operating system is limited to roughly 4gb of RAM, while 64bit operating systems are (theoretically) limited to 2.3EXABYTES of RAM, although Windows 7 limits it to 200gb for Ultimate edition, and 2tb for Server 2008.
Of course, there are still multiple factors, such as
*
*cost to manufacture RAM. (8gb on a single ram thingie(?) still in the hundreds)
*dimm slots on motherboards (I've seen boards with 4 slots)
But for the purpose of this discussion let us ignore these limitations, and talk just about space.
Let us talk about how applications nowadays deal with memory. Applications do not know how much memory exists - for the most part, it simply requisitions it from the operating system. The operating system is the one responsible for managing which address spaces have been allocated to each application that is running. If it does not have enough, well, bad things happen.
But, surely with theoretical 2EXABYTES of RAM, you'd never run out?
Well, a famous person long ago once said we'd never need more than 64kBs of RAM.
Because most Applications nowadays are greedy (they take as much as the operating system is willing to give), if you ran enough applications, on a powerful enough computer, you could theoretically exceed the storage limits of the physical memory. In that case, Virtual Memory would be required to make up the extra required memory.
So to answer your question: (in my humble opinion formed from limited knowledge on the matter,) yes you'd still need to implement virtual memory.
Obviously take all this and do your own research. I'm turning this into a community wiki so others can edit it or just delete it if it is plain wrong :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,751 |
export { default as ElementCard } from './card'
export { default as ElementCardTitle } from './card--title'
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,341 |
{"url":"https:\/\/en.wikipedia.org\/wiki\/Beam_emittance","text":"# Beam emittance\n\nSamples of a bivariate normal distribution, representing particles in phase space, with position horizontal and momentum vertical.\n\nEmittance is a property of a charged particle beam in a particle accelerator. It is a measure for the average spread of particle coordinates in position-and-momentum phase space and has the dimension of length (e.g., meters) or length times angle (meters times radians). As a particle beam propagates along magnets and other beam-manipulating components of an accelerator, the position spread may change, but in a way that does not change the emittance. If the distribution over phase space is represented as a cloud in a plot (see figure), emittance is the area of the cloud. A more exact definition handles the fuzzy borders of the cloud and the case of a cloud that does not have an elliptical shape.\n\nA low-emittance particle beam is a beam where the particles are confined to a small distance and have nearly the same momentum. A beam transport system will only allow particles that are close to its design momentum, and of course they have to fit through the beam pipe and magnets that make up the system. In a colliding beam accelerator, keeping the emittance small means that the likelihood of particle interactions will be greater resulting in higher luminosity. In a synchrotron light source, low emittance means that the resulting x-ray beam will be small, and result in higher brightness.\n\n## Definition\n\nEmittance has units of length, but is usually referred to as \"length \u00d7 angle\", for example, \"millimeter \u00d7 milli-radians\". It can be measured in all three spatial dimensions. The dimension parallel to the motion of the particle is called the longitudinal emittance and the other two dimensions are referred to as the transverse emittances.\n\n### Geometric Emittance\n\nThe arithmetic definition of a transverse emittance (${\\displaystyle \\varepsilon }$) is:\n\n${\\displaystyle \\varepsilon ={\\frac {6\\pi \\left({\\text{width}}^{2}-D^{2}\\left({\\frac {\\mathrm {d} p}{p}}\\right)^{2}\\right)}{B}}}$\n\nWhere:\n\n\u2022 width is the width of the particle beam\n\u2022 dp\/p is the momentum spread of the particle beam\n\u2022 D is the value of the dispersion function at the measurement point in the particle accelerator\n\u2022 B is the value of the beta function at the measurement point in the particle accelerator\n\nSince it is difficult to measure the full width of the beam, either the RMS width of the beam or the value of the width that encompasses a specific percentage of the beam (for example 95%) is measured. The emittance from these width measurements is then referred to as the \"RMS emittance\" or the \"95% emittance\", respectively.\n\nOne should distinguish the emittance of a single particle from that of the whole beam. The emittance of a single particle is the value of the invariant quantity\n\n${\\displaystyle \\epsilon =\\gamma x^{2}+2\\alpha xx'+\\beta x'^{2}}$\n\nwhere x and x are the position and angle of the particle respectively and ${\\displaystyle \\beta ,\\alpha ,\\gamma }$ are the Twiss parameters. (In the context of Hamiltonian dynamics, one should be more careful to formulate in terms of a transverse momentum instead of x.) This is the single particle emittance.\n\n### RMS Emittance\n\nIn some particle accelerators, Twiss parameters are not commonly used and the emittance is defined by the beam's second order phase space statistics instead. Here, the RMS emittance (${\\displaystyle \\varepsilon _{\\text{RMS}}}$) is defined to be,[1]\n\n${\\displaystyle \\varepsilon _{\\text{RMS}}={\\sqrt {\\langle x^{2}\\rangle \\langle x^{\\prime 2}\\rangle -\\langle x\\cdot x^{\\prime }\\rangle ^{2}}}}$\n\nwhere ${\\displaystyle \\langle x^{2}\\rangle }$ is the variance of the particle's position, ${\\displaystyle \\langle x^{\\prime 2}\\rangle }$ is the variance of the angle a particle makes with the direction of travel in the accelerator (${\\displaystyle x^{\\prime }={\\frac {\\mathrm {d} x}{\\mathrm {d} z}}}$ with ${\\displaystyle z}$ along the direction of travel), and ${\\displaystyle \\langle x\\cdot x^{\\prime }\\rangle }$ represents an angle-position correlation of particles in the beam. This definition reverts to the prior listed definition of geometric emittance in the case of a periodic accelerator lattice where the Twiss parameters can be defined.\n\nThe emittance may also be expressed as the determinant of the variance-covariance matrix of the beam's phase space coordinates where it becomes clear that quantity describes an effective area occupied by the beam in terms of its second order statistics.\n\n${\\displaystyle \\varepsilon _{\\text{RMS}}={\\sqrt {\\begin{vmatrix}\\langle x\\cdot x\\rangle &\\langle x\\cdot x^{\\prime }\\rangle \\\\\\langle x\\cdot x^{\\prime }\\rangle &\\langle x^{\\prime }\\cdot x^{\\prime }\\rangle \\end{vmatrix}}}}$\n\nDepending on context, some may also add a scaling factor in front of the equation for RMS emittance so that it will correspond to the area of uniformly filled ellipse shaped distribution in phase space.\n\n#### RMS Emittance in Higher Dimensions\n\nIt is sometimes useful to talk about phase space area for either four dimensional transverse phase space (IE ${\\displaystyle x}$, ${\\displaystyle x^{\\prime }}$, ${\\displaystyle y}$, ${\\displaystyle y^{\\prime }}$) or the full six dimensional phase space of particles (IE ${\\displaystyle x}$, ${\\displaystyle x^{\\prime }}$, ${\\displaystyle y}$, ${\\displaystyle y^{\\prime }}$, ${\\displaystyle \\Delta z}$, ${\\displaystyle \\Delta z^{\\prime }}$). It is now clear from the matrix definition of RMS emittance how the definition may generalize into higher dimensions.\n\n${\\displaystyle \\varepsilon _{{\\text{RMS}},6D}={\\sqrt {\\begin{vmatrix}\\langle x\\cdot x\\rangle &\\langle x\\cdot x^{\\prime }\\rangle &\\langle x\\cdot y\\rangle &\\langle x\\cdot y^{\\prime }\\rangle &\\langle x\\cdot z\\rangle &\\langle x\\cdot z^{\\prime }\\rangle \\\\\\langle x^{\\prime }\\cdot x\\rangle &\\langle x^{\\prime }\\cdot x^{\\prime }\\rangle &\\langle x^{\\prime }\\cdot y\\rangle &\\langle x^{\\prime }\\cdot y^{\\prime }\\rangle &\\langle x^{\\prime }\\cdot z\\rangle &\\langle x^{\\prime }\\cdot z^{\\prime }\\rangle \\\\\\langle y\\cdot x\\rangle &\\langle y\\cdot x^{\\prime }\\rangle &\\langle y\\cdot y\\rangle &\\langle y\\cdot y^{\\prime }\\rangle &\\langle y\\cdot z\\rangle &\\langle y\\cdot z^{\\prime }\\rangle \\\\\\langle y^{\\prime }\\cdot x\\rangle &\\langle y^{\\prime }\\cdot x^{\\prime }\\rangle &\\langle y^{\\prime }\\cdot y\\rangle &\\langle y^{\\prime }\\cdot y^{\\prime }\\rangle &\\langle y^{\\prime }\\cdot z\\rangle &\\langle y^{\\prime }\\cdot z^{\\prime }\\rangle \\\\\\langle z\\cdot x\\rangle &\\langle z\\cdot x^{\\prime }\\rangle &\\langle z\\cdot y\\rangle &\\langle z\\cdot y^{\\prime }\\rangle &\\langle z\\cdot z\\rangle &\\langle z\\cdot z^{\\prime }\\rangle \\\\\\langle z^{\\prime }\\cdot x\\rangle &\\langle z^{\\prime }\\cdot x^{\\prime }\\rangle &\\langle z^{\\prime }\\cdot y\\rangle &\\langle z^{\\prime }\\cdot y^{\\prime }\\rangle &\\langle z^{\\prime }\\cdot z\\rangle &\\langle z^{\\prime }\\cdot z^{\\prime }\\rangle \\\\\\end{vmatrix}}}}$\n\nIn the absences of correlations between different axes in the particle accelerator, most of these matrix elements become zero and we are left with a product of the emittance along each axis.\n\n${\\displaystyle \\varepsilon _{{\\text{RMS}},6D}={\\sqrt {\\begin{vmatrix}\\langle x\\cdot x\\rangle &\\langle x\\cdot x^{\\prime }\\rangle &0&0&0&0\\\\\\langle x^{\\prime }\\cdot x\\rangle &\\langle x^{\\prime }\\cdot x^{\\prime }\\rangle &0&0&0&0\\\\0&0&\\langle y\\cdot y\\rangle &\\langle y\\cdot y^{\\prime }\\rangle &0&0\\\\0&0&\\langle y^{\\prime }\\cdot y\\rangle &\\langle y^{\\prime }\\cdot y^{\\prime }\\rangle &0&0\\\\0&0&0&0&\\langle z\\cdot z\\rangle &\\langle z\\cdot z^{\\prime }\\rangle \\\\0&0&0&0&\\langle z^{\\prime }\\cdot z\\rangle &\\langle z^{\\prime }\\cdot z^{\\prime }\\rangle \\\\\\end{vmatrix}}}={\\sqrt {\\begin{vmatrix}\\langle x\\cdot x\\rangle &\\langle x\\cdot x^{\\prime }\\rangle \\\\\\langle x^{\\prime }\\cdot x\\rangle &\\langle x^{\\prime }\\cdot x^{\\prime }\\rangle \\\\\\end{vmatrix}}}{\\sqrt {\\begin{vmatrix}\\langle y\\cdot y\\rangle &\\langle y\\cdot y^{\\prime }\\rangle \\\\\\langle y^{\\prime }\\cdot y\\rangle &\\langle y^{\\prime }\\cdot y^{\\prime }\\rangle \\\\\\end{vmatrix}}}{\\sqrt {\\begin{vmatrix}\\langle z\\cdot z\\rangle &\\langle z\\cdot z^{\\prime }\\rangle \\\\\\langle z^{\\prime }\\cdot z\\rangle &\\langle z^{\\prime }\\cdot z^{\\prime }\\rangle \\\\\\end{vmatrix}}}=\\varepsilon _{x}\\varepsilon _{y}\\varepsilon _{z}}$\n\n### Normalized Emittance\n\nAlthough the previous definitions of emittance remain constant for linear beam transport, they do change when the particles undergo acceleration (an effect called adiabatic damping). In some applications, such as for linear accelerators, photoinjectors, and the accelerating sections of larger systems, it becomes important to compare beam quality across different energies. For this purpose we define normalized emittance which is invariant under acceleration.\n\n${\\displaystyle \\varepsilon _{n}={\\sqrt {\\langle x^{2}\\rangle \\langle p_{x}^{2}\\rangle -\\langle x\\cdot p_{x}\\rangle ^{2}}}={\\sqrt {\\begin{vmatrix}\\langle x\\cdot x\\rangle &\\langle x\\cdot p_{x}\\rangle \\\\\\langle x\\cdot p_{x}\\rangle &\\langle p_{x}\\cdot p_{x}\\rangle \\end{vmatrix}}}}$\n\nwhere the angle ${\\displaystyle x^{\\prime }={\\frac {\\mathrm {d} x}{\\mathrm {d} z}}}$ has been replaced with a transverse momentum ${\\displaystyle p_{x}}$which does not depend on longitudinal momentum.\n\nNormalized emittance is related to the previous definitions of emittance through the Lorentz factor (${\\displaystyle \\gamma }$) and relativistic velocity in direction of the beam's travel (${\\displaystyle \\beta _{z}}$).[2]\n\n${\\displaystyle \\epsilon _{n}=\\beta _{z}\\gamma \\epsilon }$\n\nThe normalized emittance does not change as a function of energy and so can track beam degradation if the particles are accelerated. If \u03b2 is close to one then the emittance is approximately inversely proportional to the energy and so the physical width of the beam will vary inversely to the square root of the energy.\n\nHigher dimensional versions of the normalized emittance can be defined in analogy to the RMS version by replacing all angles with their corresponding momenta.\n\n## Emittance of electrons versus heavy particles\n\nTo understand why the RMS emittance takes on a particular value in a storage ring, one needs to distinguish between electron storage rings and storage rings with heavier particles (such as protons). In an electron storage ring, radiation is an important effect, whereas when other particles are stored, it is typically a small effect. When radiation is important, the particles undergo radiation damping (which slowly decreases emittance turn after turn) and quantum excitation causing diffusion which leads to an equilibrium emittance.[3] When no radiation is present, the emittances remain constant (apart from impedance effects and intrabeam scattering). In this case, the emittance is determined by the initial particle distribution. In particular if one injects a \"small\" emittance, it remains small, whereas if one injects a \"large\" emittance, it remains large.\n\n## Acceptance\n\nThe acceptance, also called admittance,[4] is the maximum emittance that a beam transport system or analyzing system is able to transmit. This is the size of the chamber transformed into phase space and does not suffer from the ambiguities of the definition of beam emittance.\n\n## Conservation of emittance\n\nLenses can focus a beam, reducing its size in one transverse dimension while increasing its angular spread, but cannot change the total emittance. This is a result of Liouville's theorem. Ways of reducing the beam emittance include radiation damping, stochastic cooling, and electron cooling.\n\n## Emittance and brightness\n\nEmittance is also related to the brightness of the beam. In microscopy brightness is very often used, because it includes the current in the beam and most systems are circularly symmetric.[clarification needed]\n\n${\\displaystyle B={\\frac {{\\eta }I}{{\\epsilon _{x}}{\\epsilon _{y}}}}}$\n\nwith ${\\displaystyle \\eta ={\\frac {1}{8\\pi ^{2}}}}$","date":"2021-05-18 06:16:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 32, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.844282865524292, \"perplexity\": 636.8862558843557}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989820.78\/warc\/CC-MAIN-20210518033148-20210518063148-00520.warc.gz\"}"} | null | null |
Art Of Dying Essay
Jay Butler
HIS 2800
Dr. Phipps
Precis: the Art of Dying
In "The Civil War Soldier and the Art of Dying," Drew Gilpin Faust examines how the impact and meaning of the war's death toll went beyond the numbers of Americans who died. Faust asserts that death's significance for the Civil War generation changed dramatically from its previous prevailing assumptions about life's proper end—about who should die, when and where, and under what circumstances (Faust, 4). Although mid-19th-century Americans endured a high rate of infant mortality, life expectancy ofmost individuals who ...view middle of the document...
Laura2013-11-23T22:52:00Wordy and slightly confusing, consider rewording
The perception of how life should end says a great deal about how an individual values life. Faust uses this perspective as an analytical tool in her discussion of the changing preconceptions of the 'Good Death,' a notion of concern across religious and secular milieus. Laura2013-11-23T22:54:00Great sentence
Mid-19th-century America was overwhelmingly Protestant, and death was understood within the context of Christian faith in salvation and immortality (Faust, 8). Death acted as an equalizer among religions and spawned ecumenical relationships between Protestants, Catholics and Jews. A 'Good Death,' which ultimately defined the life that had preceded it and forecast the life to come, occurred amidst one's family and required a readiness to die and to embrace salvation. Soldiers' distance from home and kin and the circumstances of war made such deaths nearly impossible. Nonetheless, men struggled to create conditions in hospitals and camps, or with comrades on the field, that affirmed these fundamental principles of how to die, even as the realities of wartime challenged the very foundations of their beliefs.
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Advocacy Role Paper | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 450 |
Q: How to get the value of this.id in string? I am trying to add a character d with the value of this.id given by below code as like : dclicki25 . How I can get this?
$('.divclasss').click(function(){
console.log(this.id); //clicki25 , clicki26 based on the clicking element
});
A: just Concatenatewith d like this
$('.divclasss').click(function(){
console.log('d'+ this.id); //clicki25 , clicki26 based on the clicking element
});
A: You can concatenate this.id with the string 'd' very easily with the below small change:
$('.divclasss').click(function(){
console.log('d'+ this.id); //clicki25 , clicki26 based on the clicking element
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,309 |
Trofeo Mirabilandia Youth Festival
14º Trofeo Mirabilandia Youth Festival
Sunday 23 April 2023 - Tuesday 25 April 2023
www.mirabilandiayouthfestival.it
Welcome to Cesenatico!
The tournament, now in its 14th edition, is organized in collaboration with Mirabilandia where it will be possible to live a few days in complete joy. Spectacular, as always, will be the inauguration inside the Park with the parade of all the participating teams and a fantastic exclusive show! The tournament is approved by F.I.G.C. All matches will take place at the beautiful sports center of ASD Virtus Cesena with 5 new generation fields and at other excellent neighboring sports centers. Stay in a 3 * hotel of excellent quality located in the Cesenatico area!
OFFER: FREE ENTRANCE TICKET TO THE MIRABILANDIA PARK FOR TWO DAYS.
An unforgettable weekend !!! COME WITH US TO LIVE THE FUN OF MIRABILANDIA !!
Our figures for Trofeo Mirabilandia Youth Festival the past season
9º Umbria Cup
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,264 |
{"url":"https:\/\/distro.tube\/man-org\/man1\/gpg.1.html","text":"# Man1 - gpg.1\n\n## NAME\n\ngpg - OpenPGP encryption and signing tool\n\n## SYNOPSIS\n\ngpg [*\u2013homedir* dir\/] [*\u2013options* \/file\/] [\/options\/] \/command [\/args\/]\n\n## DESCRIPTION\n\ngpg is the OpenPGP part of the GNU Privacy Guard (GnuPG). It is a tool to provide digital encryption and signing services using the OpenPGP standard. gpg features complete key management and all the bells and whistles you would expect from a full OpenPGP implementation.\n\nThere are two main versions of GnuPG: GnuPG 1.x and GnuPG 2.x. GnuPG 2.x supports modern encryption algorithms and thus should be preferred over GnuPG 1.x. You only need to use GnuPG 1.x if your platform doesn\u2019t support GnuPG 2.x, or you need support for some features that GnuPG 2.x has deprecated, e.g., decrypting data created with PGP-2 keys.\n\nIf you are looking for version 1 of GnuPG, you may find that version installed under the name gpg1.\n\n## RETURN VALUE\n\nThe program returns 0 if there are no severe errors, 1 if at least a signature was bad, and other error codes for fatal errors.\n\nNote that signature verification requires exact knowledge of what has been signed and by whom it has beensigned. Using only the return code is thus not an appropriate way to verify a signature by a script. Either make proper use or the status codes or use the gpgv tool which has been designed to make signature verification easy for scripts.\n\n## WARNINGS\n\nUse a good password for your user account and make sure that all security issues are always fixed on your machine. Also employ diligent physical protection to your machine. Consider to use a good passphrase as a last resort protection to your secret key in the case your machine gets stolen. It is important that your secret key is never leaked. Using an easy to carry around token or smartcard with the secret key is often a advisable.\n\nIf you are going to verify detached signatures, make sure that the program knows about it; either give both filenames on the command line or use \u2018-\u2019 to specify STDIN.\n\nFor scripted or other unattended use of gpg make sure to use the machine-parseable interface and not the default interface which is intended for direct use by humans. The machine-parseable interface provides a stable and well documented API independent of the locale or future changes of gpg. To enable this interface use the options \u2013with-colons and \u2013status-fd. For certain operations the option \u2013command-fd may come handy too. See this man page and the file \u2018\/DETAILS\/\u2019 for the specification of the interface. Note that the GnuPG info\u2019\u2019 pages as well as the PDF version of the GnuPG manual features a chapter on unattended use of GnuPG. As an alternative the library GPGME can be used as a high-level abstraction on top of that interface.\n\n## INTEROPERABILITY\n\nGnuPG tries to be a very flexible implementation of the OpenPGP standard. In particular, GnuPG implements many of the optional parts of the standard, such as the SHA-512 hash, and the ZLIB and BZIP2 compression algorithms. It is important to be aware that not all OpenPGP programs implement these optional algorithms and that by forcing their use via the \u2013cipher-algo, \u2013digest-algo, \u2013cert-digest-algo, or \u2013compress-algo options in GnuPG, it is possible to create a perfectly valid OpenPGP message, but one that cannot be read by the intended recipient.\n\nThere are dozens of variations of OpenPGP programs available, and each supports a slightly different subset of these optional algorithms. For example, until recently, no (unhacked) version of PGP supported the BLOWFISH cipher algorithm. A message using BLOWFISH simply could not be read by a PGP user. By default, GnuPG uses the standard OpenPGP preferences system that will always do the right thing and create messages that are usable by all recipients, regardless of which OpenPGP program they use. Only override this safe default if you really know what you are doing.\n\nIf you absolutely must override the safe default, or if the preferences on a given key are invalid for some reason, you are far better off using the \u2013pgp6, \u2013pgp7, or \u2013pgp8 options. These options are safe as they do not force any particular algorithms in violation of OpenPGP, but rather reduce the available algorithms to a \u201cPGP-safe\u201d list.\n\n## COMMANDS\n\nCommands are not distinguished from options except for the fact that only one command is allowed. Generally speaking, irrelevant options are silently ignored, and may not be checked for correctness.\n\ngpg may be run with no commands. In this case it will print a warning perform a reasonable action depending on the type of file it is given as input (an encrypted message is decrypted, a signature is verified, a file containing keys is listed, etc.).\n\nIf you run into any problems, please add the option \u2013verbose to the invocation to see more diagnostics.\n\n### Commands not specific to the function\n\n\u2013version\nPrint the program version and licensing information. Note that you cannot abbreviate this command.\n(no term)\n\u2013help\n-h :: Print a usage message summarizing the most useful command-line options. Note that you cannot arbitrarily abbreviate this command (though you can use its short form -h).\n\u2013warranty\nPrint warranty information.\n\u2013dump-options\nPrint a list of all available options and commands. Note that you cannot abbreviate this command.\n\n### Commands to select the type of operation\n\n\u2022 \u2013sign\n-s :: Sign a message. This command may be combined with \u2013encrypt (to sign and encrypt a message), \u2013symmetric (to sign and symmetrically encrypt a message), or both \u2013encrypt and \u2013symmetric (to sign and encrypt a message that can be decrypted using a secret key or a passphrase). The signing key is chosen by default or can be set explicitly using the \u2013local-user and \u2013default-key options.\n\u2022 \u2013clear-sign\n\u2013clearsign :: Make a cleartext signature. The content in a cleartext signature is readable without any special software. OpenPGP software is only needed to verify the signature. cleartext signatures may modify end-of-line whitespace for platform independence and are not intended to be reversible. The signing key is chosen by default or can be set explicitly using the \u2013local-user and \u2013default-key options.\n\u2022 \u2013detach-sign\n-b :: Make a detached signature.\n\u2022 \u2013encrypt\n-e :: Encrypt data to one or more public keys. This command may be combined with \u2013sign (to sign and encrypt a message), \u2013symmetric (to encrypt a message that can be decrypted using a secret key or a passphrase), or \u2013sign and \u2013symmetric together (for a signed message that can be decrypted using a secret key or a passphrase). \u2013recipient and related options specify which public keys to use for encryption.\n\u2022 \u2013symmetric\n-c :: Encrypt with a symmetric cipher using a passphrase. The default symmetric cipher used is AES-128, but may be chosen with the \u2013cipher-algo option. This command may be combined with \u2013sign (for a signed and symmetrically encrypted message), \u2013encrypt (for a message that may be decrypted via a secret key or a passphrase), or \u2013sign and \u2013encrypt together (for a signed message that may be decrypted via a secret key or a passphrase). gpg caches the passphrase used for symmetric encryption so that a decrypt operation may not require that the user needs to enter the passphrase. The option \u2013no-symkey-cache can be used to disable this feature.\n\u2022 Store only (make a simple literal data packet).\n\u2022 \u2013decrypt\n-d :: Decrypt the file given on the command line (or STDIN if no file is specified) and write it to STDOUT (or the file specified with \u2013output). If the decrypted file is signed, the signature is also verified. This command differs from the default operation, as it never writes to the filename which is included in the file and it rejects files that don\u2019t begin with an encrypted message.\n\u2022 Assume that the first argument is a signed file and verify it without generating any output. With no arguments, the signature packet is read from STDIN. If only one argument is given, the specified file is expected to include a complete signature.\n\nWith more than one argument, the first argument should specify a file with a detached signature and the remaining files should contain the signed data. To read the signed data from STDIN, use \u2018-\u2019 as the second filename. For security reasons, a detached signature will not read the signed material from STDIN if not explicitly specified.\n\nNote: If the option \u2013batch is not used, gpg may assume that a single argument is a file with a detached signature, and it will try to find a matching data file by stripping certain suffixes. Using this historical feature to verify a detached signature is strongly discouraged; you should always specify the data file explicitly.\n\nNote: When verifying a cleartext signature, gpg verifies only what makes up the cleartext signed data and not any extra data outside of the cleartext signature or the header lines directly following the dash marker line. The option \u2013output may be used to write out the actual signed data, but there are other pitfalls with this format as well. It is suggested to avoid cleartext signatures in favor of detached signatures.\n\nNote: Sometimes the use of the gpgv tool is easier than using the full-fledged gpg with this option. gpgv is designed to compare signed data against a list of trusted keys and returns with success only for a good signature. It has its own manual page.\n\n\u2013multifile\nThis modifies certain other commands to accept multiple files for processing on the command line or read from STDIN with each filename on a separate line. This allows for many files to be processed at once. \u2013multifile may currently be used along with \u2013verify, \u2013encrypt, and \u2013decrypt. Note that \u2013multifile \u2013verify may not be used with detached signatures.\n\u2013verify-files\nIdentical to \u2013multifile \u2013verify.\n\u2013encrypt-files\nIdentical to \u2013multifile \u2013encrypt.\n\u2013decrypt-files\nIdentical to \u2013multifile \u2013decrypt.\n(no term)\n\u2013list-keys\n-k\n\u2013list-public-keys :: List the specified keys. If no keys are specified, then all keys from the configured public keyrings are listed.\n\nNever use the output of this command in scripts or other programs. The output is intended only for humans and its format is likely to change. The \u2013with-colons option emits the output in a stable, machine-parseable format, which is intended for use by scripts and other programs.\n\n\u2022 \u2013list-secret-keys\n-K :: List the specified secret keys. If no keys are specified, then all known secret keys are listed. A # after the initial tags sec or ssb means that the secret key or subkey is currently not usable. We also say that this key has been taken offline (for example, a primary key can be taken offline by exporting the key using the command \u2013export-secret-subkeys). A > after these tags indicate that the key is stored on a smartcard. See also \u2013list-keys.\n\u2022 \u2013check-signatures\n\u2013check-sigs :: Same as \u2013list-keys, but the key signatures are verified and listed too. Note that for performance reasons the revocation status of a signing key is not shown. This command has the same effect as using \u2013list-keys with \u2013with-sig-check.\n\nThe status of the verification is indicated by a flag directly following the \u201csig\u201d tag (and thus before the flags described below. A \u201c!\u201d indicates that the signature has been successfully verified, a \u201c-\u201d denotes a bad signature and a \u201c%\u201d is used if an error occurred while checking the signature (e.g. a non supported algorithm). Signatures where the public key is not available are not listed; to see their keyids the command \u2013list-sigs can be used.\n\nFor each signature listed, there are several flags in between the signature status flag and keyid. These flags give additional information about each key signature. From left to right, they are the numbers 1-3 for certificate check level (see \u2013ask-cert-level), \u201cL\u201d for a local or non-exportable signature (see \u2013lsign-key), \u201cR\u201d for a nonRevocable signature (see the \u2013edit-key command \u201cnrsign\u201d), \u201cP\u201d for a signature that contains a policy URL (see \u2013cert-policy-url), \u201cN\u201d for a signature that contains a notation (see \u2013cert-notation), \u201cX\u201d for an eXpired signature (see \u2013ask-cert-expire), and the numbers 1-9 or \u201cT\u201d for 10 and above to indicate trust signature levels (see the \u2013edit-key command \u201ctsign\u201d).\n\n\u2022 \u2013locate-keys\n\u2013locate-external-keys :: Locate the keys given as arguments. This command basically uses the same algorithm as used when locating keys for encryption and may thus be used to see what keys gpg might use. In particular external methods as defined by \u2013auto-key-locate are used to locate a key if the arguments comain valid mail addresses. Only public keys are listed.\n\nThe variant \u2013locate-external-keys does not consider a locally existing key and can thus be used to force the refresh of a key via the defined external methods. If a fingerprint is given and and the methods defined by \u2013auto-key-locate define LDAP servers, the key is fetched from these resources; defined non-LDAP keyservers are skipped.\n\n\u2013show-keys\nThis commands takes OpenPGP keys as input and prints information about them in the same way the command \u2013list-keys does for locally stored key. In addition the list options show-unusable-uids, show-unusable-subkeys, show-notations and show-policy-urls are also enabled. As usual for automated processing, this command should be combined with the option \u2013with-colons.\n\u2013fingerprint\nList all keys (or the specified ones) along with their fingerprints. This is the same output as \u2013list-keys but with the additional output of a line with the fingerprint. May also be combined with \u2013check-signatures. If this command is given twice, the fingerprints of all secondary keys are listed too. This command also forces pretty printing of fingerprints if the keyid format has been set to \u201cnone\u201d.\n\u2013list-packets\nList only the sequence of packets. This command is only useful for debugging. When used with option \u2013verbose the actual MPI values are dumped and not only their lengths. Note that the output of this command may change with new releases.\n(no term)\n\u2013edit-card\n\u2013card-edit :: Present a menu to work with a smartcard. The subcommand \u201chelp\u201d provides an overview on available commands. For a detailed description, please see the Card HOWTO at https:\/\/gnupg.org\/documentation\/howtos.html#GnuPG-cardHOWTO .\n\u2013card-status\nShow the content of the smart card.\n\u2013change-pin\nPresent a menu to allow changing the PIN of a smartcard. This functionality is also available as the subcommand \u201cpasswd\u201d with the \u2013edit-card command.\n\u2013delete-keys name\nRemove key from the public keyring. In batch mode either \u2013yes is required or the key must be specified by fingerprint. This is a safeguard against accidental deletion of multiple keys. If the exclamation mark syntax is used with the fingerprint of a subkey only that subkey is deleted; if the exclamation mark is used with the fingerprint of the primary key the entire public key is deleted.\n\u2013delete-secret-keys name\nRemove key from the secret keyring. In batch mode the key must be specified by fingerprint. The option \u2013yes can be used to advise gpg-agent not to request a confirmation. This extra pre-caution is done because gpg can\u2019t be sure that the secret key (as controlled by gpg-agent) is only used for the given OpenPGP public key. If the exclamation mark syntax is used with the fingerprint of a subkey only the secret part of that subkey is deleted; if the exclamation mark is used with the fingerprint of the primary key only the secret part of the primary key is deleted.\n\u2013delete-secret-and-public-key name\nSame as \u2013delete-key, but if a secret key exists, it will be removed first. In batch mode the key must be specified by fingerprint. The option \u2013yes can be used to advise gpg-agent not to request a confirmation.\n\u2013export\nEither export all keys from all keyrings (default keyring and those registered via option \u2013keyring), or if at least one name is given, those of the given name. The exported keys are written to STDOUT or to the file given with option \u2013output. Use together with \u2013armor to mail those keys.\n\u2013send-keys keyIDs\nSimilar to \u2013export but sends the keys to a keyserver. Fingerprints may be used instead of key IDs. Don\u2019t send your complete keyring to a keyserver \u2014 select only those keys which are new or changed by you. If no keyIDs are given, gpg does nothing.\n\nTake care: Keyservers are by design write only systems and thus it is not possible to ever delete keys once they have been send to a keyserver.\n\n\u2022 \u2013export-secret-keys\n\u2013export-secret-subkeys :: Same as \u2013export, but exports the secret keys instead. The exported keys are written to STDOUT or to the file given with option \u2013output. This command is often used along with the option \u2013armor to allow for easy printing of the key for paper backup; however the external tool paperkey does a better job of creating backups on paper. Note that exporting a secret key can be a security risk if the exported keys are sent over an insecure channel.\n\nThe second form of the command has the special property to render the secret part of the primary key useless; this is a GNU extension to OpenPGP and other implementations can not be expected to successfully import such a key. Its intended use is in generating a full key with an additional signing subkey on a dedicated machine. This command then exports the key without the primary key to the main machine.\n\nGnuPG may ask you to enter the passphrase for the key. This is required, because the internal protection method of the secret key is different from the one specified by the OpenPGP protocol.\n\n\u2013export-ssh-key\nThis command is used to export a key in the OpenSSH public key format. It requires the specification of one key by the usual means and exports the latest valid subkey which has an authentication capability to STDOUT or to the file given with option \u2013output. That output can directly be added to ssh\u2019s \u2018\/authorized_key\/\u2019 file.\n\nBy specifying the key to export using a key ID or a fingerprint suffixed with an exclamation mark (!), a specific subkey or the primary key can be exported. This does not even require that the key has the authentication capability flag set.\n\n\u2022 \u2013import\n\u2013fast-import :: Import\/merge keys. This adds the given keys to the keyring. The fast version is currently just a synonym.\n\nThere are a few other options which control how this command works. Most notable here is the \u2013import-options merge-only option which does not insert new keys but does only the merging of new signatures, user-IDs and subkeys.\n\n\u2013recv-keys keyIDs :: Import the keys with the given keyIDs from a keyserver.\n\u2022 Request updates from a keyserver for keys that already exist on the local keyring. This is useful for updating a key with the latest signatures, user IDs, etc. Calling this with no arguments will refresh the entire keyring.\n\u2022 Search the keyserver for the given names. Multiple names given here will be joined together to create the search string for the keyserver. Note that keyservers search for names in a different and simpler way than gpg does. The best choice is to use a mail address. Due to data privacy reasons keyservers may even not even allow searching by user id or mail address and thus may only return results when being used with the \u2013recv-key command to search by key fingerprint or keyid.\n\u2022 Retrieve keys located at the specified URIs. Note that different installations of GnuPG may support different protocols (HTTP, FTP, LDAP, etc.). When using HTTPS the system provided root certificates are used by this command.\n\u2022 Do trust database maintenance. This command iterates over all keys and builds the Web of Trust. This is an interactive command because it may have to ask for the \u201cownertrust\u201d values for keys. The user has to give an estimation of how far she trusts the owner of the displayed key to correctly certify (sign) other keys. GnuPG only asks for the ownertrust value if it has not yet been assigned to a key. Using the \u2013edit-key menu, the assigned value can be changed at any time.\n\u2022 Do trust database maintenance without user interaction. From time to time the trust database must be updated so that expired keys or signatures and the resulting changes in the Web of Trust can be tracked. Normally, GnuPG will calculate when this is required and do it automatically unless \u2013no-auto-check-trustdb is set. This command can be used to force a trust database check at any time. The processing is identical to that of \u2013update-trustdb but it skips keys with a not yet defined \u201cownertrust\u201d.\n\nFor use with cron jobs, this command can be used together with \u2013batch in which case the trust database check is done only if a check is needed. To force a run even in batch mode add the option \u2013yes.\n\n\u2013export-ownertrust\n\nSend the ownertrust values to STDOUT. This is useful for backup purposes as these values are the only ones which can\u2019t be re-created from a corrupted trustdb. Example:\n\n gpg --export-ownertrust > otrust.txt\n\n\u2013import-ownertrust\n\nUpdate the trustdb with the ownertrust values stored in files (or STDIN if not given); existing values will be overwritten. In case of a severely damaged trustdb and if you have a recent backup of the ownertrust values (e.g. in the file \u2018\/otrust.txt\/\u2019), you may re-create the trustdb using these commands:\n\n cd ~\/.gnupg\nrm trustdb.gpg\ngpg --import-ownertrust < otrust.txt\n\n\u2013rebuild-keydb-caches\nWhen updating from version 1.0.6 to 1.0.7 this command should be used to create signature caches in the keyring. It might be handy in other situations too.\n(no term)\n\u2013print-md algo\n\u2013print-mds :: Print message digest of algorithm algo for all given files or STDIN. With the second form (or a deprecated \u201c*\u201d for algo) digests for all available algorithms are printed.\n\u2013gen-random 0|1|2 count\nEmit count random bytes of the given quality level 0, 1 or 2. If count is not given or zero, an endless sequence of random bytes will be emitted. If used with \u2013armor the output will be base64 encoded. PLEASE, don\u2019t use this command unless you know what you are doing; it may remove precious entropy from the system!\n\u2013gen-prime mode bits\nUse the source, Luke :-). The output format is subject to change with ant release.\n(no term)\n\u2013enarmor\n\u2013dearmor :: Pack or unpack an arbitrary input into\/from an OpenPGP ASCII armor. This is a GnuPG extension to OpenPGP and in general not very useful.\nSet the TOFU policy for all the bindings associated with the specified keys. For more information about the meaning of the policies, see: [trust-model-tofu]. The keys may be specified either by their fingerprint (preferred) or their keyid.\n\n### How to manage your keys\n\nThis section explains the main commands for key management.\n\n\u2022 \u2013quick-generate-key user-id [\/algo\/ [\/usage\/ [\/expire\/]]]\n\u2013quick-gen-key :: This is a simple command to generate a standard key with one user id. In contrast to \u2013generate-key the key is generated directly without the need to answer a bunch of prompts. Unless the option \u2013yes is given, the key creation will be canceled if the given user id already exists in the keyring.\n\nIf invoked directly on the console without any special options an answer to a Continue?\u2019\u2019 style confirmation prompt is required. In case the user id already exists in the keyring a second prompt to force the creation of the key will show up.\n\nIf algo or usage are given, only the primary key is created and no prompts are shown. To specify an expiration date but still create a primary and subkey use default\u2019\u2019 or future-default\u2019\u2019 for algo and default\u2019\u2019 for usage. For a description of these optional arguments see the command \u2013quick-add-key. The usage accepts also the value cert\u2019\u2019 which can be used to create a certification only primary key; the default is to a create certification and signing key.\n\nThe expire argument can be used to specify an expiration date for the key. Several formats are supported; commonly the ISO formats YYYY-MM-DD\u2019\u2019 or YYYYMMDDThhmmss\u2019\u2019 are used. To make the key expire in N seconds, N days, N weeks, N months, or N years use seconds=N\u2019\u2019, Nd\u2019\u2019, Nw\u2019\u2019, Nm\u2019\u2019, or Ny\u2019\u2019 respectively. Not specifying a value, or using -\u2019\u2019 results in a key expiring in a reasonable default interval. The values never\u2019\u2019, none\u2019\u2019 can be used for no expiration date.\n\nIf this command is used with \u2013batch, \u2013pinentry-mode has been set to loopback, and one of the passphrase options (\u2013passphrase, \u2013passphrase-fd, or \u2013passphrase-file) is used, the supplied passphrase is used for the new key and the agent does not ask for it. To create a key without any protection \u2013passphrase \u2019\u2019 may be used.\n\nTo create an OpenPGP key from the keys available on the currently inserted smartcard, the special string card\u2019\u2019 can be used for algo. If the card features an encryption and a signing key, gpg will figure them out and creates an OpenPGP key consisting of the usual primary key and one subkey. This works only with certain smartcards. Note that the interactive \u2013full-gen-key command allows to do the same but with greater flexibility in the selection of the smartcard keys.\n\nNote that it is possible to create a primary key and a subkey using non-default algorithms by using default\u2019\u2019 and changing the default parameters using the option \u2013default-new-key-algo.\n\n\u2013quick-set-expire fpr expire [*|\/subfprs\/]\nWith two arguments given, directly set the expiration time of the primary key identified by fpr to expire. To remove the expiration time 0 can be used. With three arguments and the third given as an asterisk, the expiration time of all non-revoked and not yet expired subkeys are set to expire. With more than two arguments and a list of fingerprints given for subfprs, all non-revoked subkeys matching these fingerprints are set to expire.\nDirectly add a subkey to the key identified by the fingerprint fpr. Without the optional arguments an encryption subkey is added. If any of the arguments are given a more specific subkey is added.\n\nalgo may be any of the supported algorithms or curve names given in the format as used by key listings. To use the default algorithm the string default\u2019\u2019 or -\u2019\u2019 can be used. Supported algorithms are rsa\u2019\u2019, dsa\u2019\u2019, elg\u2019\u2019, ed25519\u2019\u2019, cv25519\u2019\u2019, and other ECC curves. For example the string rsa\u2019\u2019 adds an RSA key with the default key length; a string rsa4096\u2019\u2019 requests that the key length is 4096 bits. The string future-default\u2019\u2019 is an alias for the algorithm which will likely be used as default algorithm in future versions of gpg. To list the supported ECC curves the command gpg \u2013with-colons \u2013list-config curve can be used.\n\nDepending on the given algo the subkey may either be an encryption subkey or a signing subkey. If an algorithm is capable of signing and encryption and such a subkey is desired, a usage string must be given. This string is either default\u2019\u2019 or -\u2019\u2019 to keep the default or a comma delimited list (or space delimited list) of keywords: sign\u2019\u2019 for a signing subkey, auth\u2019\u2019 for an authentication subkey, and encr\u2019\u2019 for an encryption subkey (encrypt\u2019\u2019 can be used as alias for encr\u2019\u2019). The valid combinations depend on the algorithm.\n\nThe expire argument can be used to specify an expiration date for the key. Several formats are supported; commonly the ISO formats YYYY-MM-DD\u2019\u2019 or YYYYMMDDThhmmss\u2019\u2019 are used. To make the key expire in N seconds, N days, N weeks, N months, or N years use seconds=N\u2019\u2019, Nd\u2019\u2019, Nw\u2019\u2019, Nm\u2019\u2019, or Ny\u2019\u2019 respectively. Not specifying a value, or using -\u2019\u2019 results in a key expiring in a reasonable default interval. The values never\u2019\u2019, none\u2019\u2019 can be used for no expiration date.\n\n\u2022 \u2013generate-key\n\u2013gen-key :: Generate a new key pair using the current default parameters. This is the standard command to create a new key. In addition to the key a revocation certificate is created and stored in the \u2018\/openpgp-revocs.d\/\u2019 directory below the GnuPG home directory.\n\u2022 \u2013full-generate-key\n\u2013full-gen-key :: Generate a new key pair with dialogs for all options. This is an extended version of \u2013generate-key.\n\nThere is also a feature which allows you to create keys in batch mode. See the manual section Unattended key generation\u2019\u2019 on how to use this.\n\n\u2022 \u2013generate-revocation name\n\u2013gen-revoke name :: Generate a revocation certificate for the complete key. To only revoke a subkey or a key signature, use the \u2013edit command.\n\nThis command merely creates the revocation certificate so that it can be used to revoke the key if that is ever needed. To actually revoke a key the created revocation certificate needs to be merged with the key to revoke. This is done by importing the revocation certificate using the \u2013import command. Then the revoked key needs to be published, which is best done by sending the key to a keyserver (command \u2013send-key) and by exporting (\u2013export) it to a file which is then send to frequent communication partners.\n\n\u2022 \u2013generate-designated-revocation name\n\u2013desig-revoke name :: Generate a designated revocation certificate for a key. This allows a user (with the permission of the keyholder) to revoke someone else\u2019s key.\n\u2022 Present a menu which enables you to do most of the key management related tasks. It expects the specification of a key on the command line.\nuid n\nToggle selection of user ID or photographic user ID with index n. Use * to select all and 0 to deselect all.\nkey n\nToggle selection of subkey with index n or key ID n. Use * to select all and 0 to deselect all.\nsign\nMake a signature on key of user name. If the key is not yet signed by the default user (or the users given with -u), the program displays the information of the key again, together with its fingerprint and asks whether it should be signed. This question is repeated for all users specified with -u.\nlsign\nSame as \u201csign\u201d but the signature is marked as non-exportable and will therefore never be used by others. This may be used to make keys valid only in the local environment.\nnrsign\nSame as \u201csign\u201d but the signature is marked as non-revocable and can therefore never be revoked.\ntsign\nMake a trust signature. This is a signature that combines the notions of certification (like a regular signature), and trust (like the \u201ctrust\u201d command). It is generally only useful in distinct communities or groups. For more information please read the sections Trust Signature\u2019\u2019 and Regular Expression\u2019\u2019 in RFC-4880.\n\nNote that \u201cl\u201d (for local \/ non-exportable), \u201cnr\u201d (for non-revocable, and \u201ct\u201d (for trust) may be freely mixed and prefixed to \u201csign\u201d to create a signature of any type desired.\n\nIf the option \u2013only-sign-text-ids is specified, then any non-text based user ids (e.g., photo IDs) will not be selected for signing.\n\ndelsig\nDelete a signature. Note that it is not possible to retract a signature, once it has been send to the public (i.e. to a keyserver). In that case you better use revsig.\nrevsig\nRevoke a signature. For every signature which has been generated by one of the secret keys, GnuPG asks whether a revocation certificate should be generated.\ncheck\nCheck the signatures on all selected user IDs. With the extra option selfsig only self-signatures are shown.\nCreate a photographic user ID. This will prompt for a JPEG file that will be embedded into the user ID. Note that a very large JPEG will make for a very large key. Also note that some programs will display your JPEG unchanged (GnuPG), and some programs will scale it to fit in a dialog box (PGP).\nshowphoto\nDisplay the selected photographic user ID.\ndeluid\nDelete a user ID or photographic user ID. Note that it is not possible to retract a user id, once it has been send to the public (i.e. to a keyserver). In that case you better use revuid.\nrevuid\nRevoke a user ID or photographic user ID.\nprimary\nFlag the current user id as the primary one, removes the primary user id flag from all other user ids and sets the timestamp of all affected self-signatures one second ahead. Note that setting a photo user ID as primary makes it primary over other photo user IDs, and setting a regular user ID as primary makes it primary over other regular user IDs.\nkeyserver\nSet a preferred keyserver for the specified user ID(s). This allows other users to know where you prefer they get your key from. See \u2013keyserver-options honor-keyserver-url for more on how this works. Setting a value of \u201cnone\u201d removes an existing preferred keyserver.\nnotation\nSet a name=value notation for the specified user ID(s). See \u2013cert-notation for more on how this works. Setting a value of \u201cnone\u201d removes all notations, setting a notation prefixed with a minus sign (-) removes that notation, and setting a notation name (without the =value) prefixed with a minus sign removes all notations with that name.\npref\nList preferences from the selected user ID. This shows the actual preferences, without including any implied preferences.\nshowpref\nMore verbose preferences listing for the selected user ID. This shows the preferences in effect by including the implied preferences of 3DES (cipher), SHA-1 (digest), and Uncompressed (compression) if they are not already included in the preference list. In addition, the preferred keyserver and signature notations (if any) are shown.\nsetpref string\nSet the list of user ID preferences to string for all (or just the selected) user IDs. Calling setpref with no arguments sets the preference list to the default (either built-in or set via \u2013default-preference-list), and calling setpref with \u201cnone\u201d as the argument sets an empty preference list. Use gpg \u2013version to get a list of available algorithms. Note that while you can change the preferences on an attribute user ID (aka \u201cphoto ID\u201d), GnuPG does not select keys via attribute user IDs so these preferences will not be used by GnuPG.\n\nWhen setting preferences, you should list the algorithms in the order which you\u2019d like to see them used by someone else when encrypting a message to your key. If you don\u2019t include 3DES, it will be automatically added at the end. Note that there are many factors that go into choosing an algorithm (for example, your key may not be the only recipient), and so the remote OpenPGP application being used to send to you may or may not follow your exact chosen order for a given message. It will, however, only choose an algorithm that is present on the preference list of every recipient key. See also the INTEROPERABILITY WITH OTHER OPENPGP PROGRAMS section below.\n\nAdd a subkey to this key.\nGenerate a subkey on a card and add it to this key.\nkeytocard\nTransfer the selected secret subkey (or the primary key if no subkey has been selected) to a smartcard. The secret key in the keyring will be replaced by a stub if the key could be stored successfully on the card and you use the save command later. Only certain key types may be transferred to the card. A sub menu allows you to select on what card to store the key. Note that it is not possible to get that key back from the card - if the card gets broken your secret key will be lost unless you have a backup somewhere.\nbkuptocard file\nRestore the given file to a card. This command may be used to restore a backup key (as generated during card initialization) to a new card. In almost all cases this will be the encryption key. You should use this command only with the corresponding public key and make sure that the file given as argument is indeed the backup to restore. You should then select 2 to restore as encryption key. You will first be asked to enter the passphrase of the backup key and then for the Admin PIN of the card.\ndelkey\nRemove a subkey (secondary key). Note that it is not possible to retract a subkey, once it has been send to the public (i.e. to a keyserver). In that case you better use revkey. Also note that this only deletes the public part of a key.\nrevkey\nRevoke a subkey.\nexpire\nChange the key or subkey expiration time. If a subkey is selected, the expiration time of this subkey will be changed. With no selection, the key expiration of the primary key is changed.\ntrust\nChange the owner trust value for the key. This updates the trust-db immediately and no save is required.\n(no term)\ndisable\nenable :: Disable or enable an entire key. A disabled key can not normally be used for encryption.\nAdd a designated revoker to the key. This takes one optional argument: \u201csensitive\u201d. If a designated revoker is marked as sensitive, it will not be exported by default (see export-options).\npasswd\nChange the passphrase of the secret key.\ntoggle\nThis is dummy command which exists only for backward compatibility.\nclean\nCompact (by removing all signatures except the selfsig) any user ID that is no longer usable (e.g. revoked, or expired). Then, remove any signatures that are not usable by the trust calculations. Specifically, this removes any signature that does not validate, any signature that is superseded by a later signature, revoked signatures, and signatures issued by keys that are not present on the keyring.\nminimize\nMake the key as small as possible. This removes all signatures from each user ID except for the most recent self-signature.\nchange-usage\nChange the usage flags (capabilities) of the primary key or of subkeys. These usage flags (e.g. Certify, Sign, Authenticate, Encrypt) are set during key creation. Sometimes it is useful to have the opportunity to change them (for example to add Authenticate) after they have been created. Please take care when doing this; the allowed usage flags depend on the key algorithm.\ncross-certify\nAdd cross-certification signatures to signing subkeys that may not currently have them. Cross-certification signatures protect against a subtle attack against signing subkeys. See \u2013require-cross-certification. All new keys generated have this signature by default, so this command is only useful to bring older keys up to date.\nsave\nSave all changes to the keyring and quit.\nquit\nQuit the program without updating the keyring.\n\nThe listing shows you the key with its secondary keys and all user IDs. The primary user ID is indicated by a dot, and selected keys or user IDs are indicated by an asterisk. The trust value is displayed with the primary key: \u201ctrust\u201d is the assigned owner trust and \u201cvalidity\u201d is the calculated validity of the key. Validity values are also displayed for all user IDs. For possible values of trust, see: [trust-values].\n\n\u2013sign-key name\nSigns a public key with your secret key. This is a shortcut version of the subcommand \u201csign\u201d from \u2013edit.\n\u2013lsign-key name\nSigns a public key with your secret key but marks it as non-exportable. This is a shortcut version of the subcommand \u201clsign\u201d from \u2013edit-key.\n(no term)\n\u2013quick-sign-key fpr [\/names\/]\n\u2013quick-lsign-key fpr [\/names\/] :: Directly sign a key from the passphrase without any further user interaction. The fpr must be the verified primary fingerprint of a key in the local keyring. If no names are given, all useful user ids are signed; with given [\/names\/] only useful user ids matching one of theses names are signed. By default, or if a name is prefixed with a \u2019*\u2019, a case insensitive substring match is used. If a name is prefixed with a \u2019=\u2019 a case sensitive exact match is done.\n\nThe command \u2013quick-lsign-key marks the signatures as non-exportable. If such a non-exportable signature already exists the \u2013quick-sign-key turns it into a exportable signature. If you need to update an existing signature, for example to add or change notation data, you need to use the option \u2013force-sign-key.\n\nThis command uses reasonable defaults and thus does not provide the full flexibility of the \u201csign\u201d subcommand from \u2013edit-key. Its intended use is to help unattended key signing by utilizing a list of verified fingerprints.\n\nThis command adds a new user id to an existing key. In contrast to the interactive sub-command adduid of \u2013edit-key the new-user-id is added verbatim with only leading and trailing white space removed, it is expected to be UTF-8 encoded, and no checks on its form are applied.\n\u2013quick-revoke-uid user-id user-id-to-revoke\nThis command revokes a user ID on an existing key. It cannot be used to revoke the last user ID on key (some non-revoked user ID must remain), with revocation reason User ID is no longer valid\u2019\u2019. If you want to specify a different revocation reason, or to supply supplementary revocation text, you should use the interactive sub-command revuid of \u2013edit-key.\n\u2013quick-revoke-sig fpr signing-fpr [\/names\/]\nThis command revokes the key signatures made by signing-fpr from the key specified by the fingerprint fpr. With names given only the signatures on user ids of the key matching any of the given names are affected (see \u2013quick-sign-key). If a revocation already exists a notice is printed instead of creating a new revocation; no error is returned in this case. Note that key signature revocations may be superseded by a newer key signature and in turn again revoked.\n\u2013quick-set-primary-uid user-id primary-user-id\nThis command sets or updates the primary user ID flag on an existing key. user-id specifies the key and primary-user-id the user ID which shall be flagged as the primary user ID. The primary user ID flag is removed from all other user ids and the timestamp of all affected self-signatures is set one second ahead.\n(no term)\n\u2013change-passphrase user-id\n\u2013passwd user-id :: Change the passphrase of the secret key belonging to the certificate specified as user-id. This is a shortcut for the sub-command passwd of the edit key menu. When using together with the option \u2013dry-run this will not actually change the passphrase but check that the current passphrase is correct.\n\n## OPTIONS\n\ngpg features a bunch of options to control the exact behaviour and to change the default configuration.\n\nLong options can be put in an options file (default \u201c~\/.gnupg\/gpg.conf\u201d). Short option names will not work - for example, \u201carmor\u201d is a valid option for the options file, while \u201ca\u201d is not. Do not write the 2 dashes, but simply the name of the option and any required arguments. Lines with a hash (\u2019#\u2019) as the first non-white-space character are ignored. Commands may be put in this file too, but that is not generally useful as the command will execute automatically with every execution of gpg.\n\nPlease remember that option parsing stops as soon as a non-option is encountered, you can explicitly stop parsing by using the special option --.\n\n### How to change the configuration\n\nThese options are used to change the configuration and most of them are usually found in the option file.\n\n\u2013default-key name\nUse name as the default key to sign with. If this option is not used, the default key is the first key found in the secret keyring. Note that -u or \u2013local-user overrides this option. This option may be given multiple times. In this case, the last key for which a secret key is available is used. If there is no secret key available for any of the specified values, GnuPG will not emit an error message but continue as if this option wasn\u2019t given.\n\u2013default-recipient name\nUse name as default recipient if option \u2013recipient is not used and don\u2019t ask if this is a valid one. name must be non-empty.\n\u2013default-recipient-self\nUse the default key as default recipient if option \u2013recipient is not used and don\u2019t ask if this is a valid one. The default key is the first one from the secret keyring or the one set with \u2013default-key.\n\u2013no-default-recipient\nReset \u2013default-recipient and \u2013default-recipient-self. Should not be used in an option file.\n-v, \u2013verbose\nGive more information during processing. If used twice, the input data is listed in detail.\n\u2013no-verbose\nReset verbose level to 0. Should not be used in an option file.\n-q, \u2013quiet\nTry to be as quiet as possible. Should not be used in an option file.\n(no term)\n\u2013batch\n\u2013no-batch :: Use batch mode. Never ask, do not allow interactive commands. \u2013no-batch disables this option. Note that even with a filename given on the command line, gpg might still need to read from STDIN (in particular if gpg figures that the input is a detached signature and no data file has been specified). Thus if you do not want to feed data via STDIN, you should connect STDIN to \u2018\/\/dev\/null\/\u2019.\n\nIt is highly recommended to use this option along with the options \u2013status-fd and \u2013with-colons for any unattended use of gpg. Should not be used in an option file.\n\n\u2013no-tty\nMake sure that the TTY (terminal) is never used for any output. This option is needed in some cases because GnuPG sometimes prints warnings to the TTY even if \u2013batch is used.\n\u2013yes\nAssume \u201cyes\u201d on most questions. Should not be used in an option file.\n\u2013no\nAssume \u201cno\u201d on most questions. Should not be used in an option file.\n\u2013list-options parameters\nThis is a space or comma delimited string that gives options used when listing keys and signatures (that is, \u2013list-keys, \u2013check-signatures, \u2013list-public-keys, \u2013list-secret-keys, and the \u2013edit-key functions). Options can be prepended with a no- (after the two dashes) to give the opposite meaning. The options are:\nshow-photos\nCauses \u2013list-keys, \u2013check-signatures, \u2013list-public-keys, and \u2013list-secret-keys to display any photo IDs attached to the key. Defaults to no. See also \u2013photo-viewer. Does not work with \u2013with-colons: see \u2013attribute-fd for the appropriate way to get photo data for scripts and other frontends.\nshow-usage\nShow usage information for keys and subkeys in the standard key listing. This is a list of letters indicating the allowed usage for a key (*E*=encryption, *S*=signing, *C*=certification, *A*=authentication). Defaults to yes.\nshow-policy-urls\nShow policy URLs in the \u2013check-signatures listings. Defaults to no.\n(no term)\nshow-notations\nshow-std-notations\nshow-user-notations :: Show all, IETF standard, or user-defined signature notations in the \u2013check-signatures listings. Defaults to no.\nshow-keyserver-urls\nShow any preferred keyserver URL in the \u2013check-signatures listings. Defaults to no.\nshow-uid-validity\nDisplay the calculated validity of user IDs during key listings. Defaults to yes.\nshow-unusable-uids\nShow revoked and expired user IDs in key listings. Defaults to no.\nshow-unusable-subkeys\nShow revoked and expired subkeys in key listings. Defaults to no.\nshow-keyring\nDisplay the keyring name at the head of key listings to show which keyring a given key resides on. Defaults to no.\nshow-sig-expire\nShow signature expiration dates (if any) during \u2013check-signatures listings. Defaults to no.\nshow-sig-subpackets\nInclude signature subpackets in the key listing. This option can take an optional argument list of the subpackets to list. If no argument is passed, list all subpackets. Defaults to no. This option is only meaningful when using \u2013with-colons along with \u2013check-signatures.\nshow-only-fpr-mbox\nFor each user-id which has a valid mail address print only the fingerprint followed by the mail address.\n\u2013verify-options parameters\nThis is a space or comma delimited string that gives options used when verifying signatures. Options can be prepended with a no-\u2019 to give the opposite meaning. The options are:\nshow-photos\nDisplay any photo IDs present on the key that issued the signature. Defaults to no. See also \u2013photo-viewer.\nshow-policy-urls\nShow policy URLs in the signature being verified. Defaults to yes.\n(no term)\nshow-notations\nshow-std-notations\nshow-user-notations :: Show all, IETF standard, or user-defined signature notations in the signature being verified. Defaults to IETF standard.\nshow-keyserver-urls\nShow any preferred keyserver URL in the signature being verified. Defaults to yes.\nshow-uid-validity\nDisplay the calculated validity of the user IDs on the key that issued the signature. Defaults to yes.\nshow-unusable-uids\nShow revoked and expired user IDs during signature verification. Defaults to no.\nshow-primary-uid-only\nShow only the primary user ID during signature verification. That is all the AKA lines as well as photo Ids are not shown with the signature verification status.\npka-lookups\nEnable PKA lookups to verify sender addresses. Note that PKA is based on DNS, and so enabling this option may disclose information on when and what signatures are verified or to whom data is encrypted. This is similar to the \u201cweb bug\u201d described for the \u2013auto-key-retrieve option.\npka-trust-increase\nRaise the trust in a signature to full if the signature passes PKA validation. This option is only meaningful if pka-lookups is set.\n(no term)\n\u2013enable-large-rsa\n\u2013disable-large-rsa :: With \u2013generate-key and \u2013batch, enable the creation of RSA secret keys as large as 8192 bit. Note: 8192 bit is more than is generally recommended. These large keys don\u2019t significantly improve security, but they are more expensive to use, and their signatures and certifications are larger. This option is only available if the binary was build with large-secmem support.\n(no term)\n\u2013enable-dsa2\n\u2013disable-dsa2 :: Enable hash truncation for all DSA keys even for old DSA Keys up to 1024 bit. This is also the default with \u2013openpgp. Note that older versions of GnuPG also required this flag to allow the generation of DSA larger than 1024 bit.\n\u2013photo-viewer string\nThis is the command line that should be run to view a photo ID. \u201c%i\u201d will be expanded to a filename containing the photo. \u201c%I\u201d does the same, except the file will not be deleted once the viewer exits. Other flags are \u201c%k\u201d for the key ID, \u201c%K\u201d for the long key ID, \u201c%f\u201d for the key fingerprint, \u201c%t\u201d for the extension of the image type (e.g. \u201cjpg\u201d), \u201c%T\u201d for the MIME type of the image (e.g. \u201cimage\/jpeg\u201d), \u201c%v\u201d for the single-character calculated validity of the image being viewed (e.g. \u201cf\u201d), \u201c%V\u201d for the calculated validity as a string (e.g. \u201cfull\u201d), \u201c%U\u201d for a base32 encoded hash of the user ID, and \u201c%%\u201d for an actual percent sign. If neither %i or %I are present, then the photo will be supplied to the viewer on standard input.\n\nOn Unix the default viewer is xloadimage -fork -quiet -title \u2019KeyID 0x%k\u2019 STDIN with a fallback to display -title \u2019KeyID 0x%k\u2019 %i and finally to xdg-open %i. On Windows !ShellExecute 400 %i is used; here the command is a meta command to use that API call followed by a wait time in milliseconds which is used to give the viewer time to read the temporary image file before gpg deletes it again. Note that if your image viewer program is not secure, then executing it from gpg does not make it secure.\n\n\u2013exec-path string\nSets a list of directories to search for photo viewers If not provided photo viewers use the PATH environment variable.\n\u2013keyring file\nAdd file to the current list of keyrings. If file begins with a tilde and a slash, these are replaced by the $HOME directory. If the filename does not contain a slash, it is assumed to be in the GnuPG home directory (\u201c~\/.gnupg\u201d unless \u2013homedir or$GNUPGHOME is used).\n\nNote that this adds a keyring to the current list. If the intent is to use the specified keyring alone, use \u2013keyring along with \u2013no-default-keyring.\n\nIf the option \u2013no-keyring has been used no keyrings will be used at all.\n\n\u2013primary-keyring file\nThis is a varian of \u2013keyring and designates file as the primary public keyring. This means that newly imported keys (via \u2013import or keyserver \u2013recv-from) will go to this keyring.\n\u2013secret-keyring file\nThis is an obsolete option and ignored. All secret keys are stored in the \u2018\/private-keys-v1.d\/\u2019 directory below the GnuPG home directory.\n\u2013trustdb-name file\nUse file instead of the default trustdb. If file begins with a tilde and a slash, these are replaced by the $HOME directory. If the filename does not contain a slash, it is assumed to be in the GnuPG home directory (\u2018\/~\/.gnupg\/\u2019 if \u2013homedir or$GNUPGHOME is not used).\n\u2013homedir dir\nSet the name of the home directory to dir. If this option is not used, the home directory defaults to \u2018\/~\/.gnupg\/\u2019. It is only recognized when given on the command line. It also overrides any home directory stated through the environment variable \u2018\/GNUPGHOME\/\u2019 or (on Windows systems) by means of the Registry entry HKCU\\Software\\GNU\\GnuPG:HomeDir.\n\nOn Windows systems it is possible to install GnuPG as a portable application. In this case only this command line option is considered, all other ways to set a home directory are ignored.\n\nTo install GnuPG as a portable application under Windows, create an empty file named \u2018\/gpgconf.ctl\/\u2019 in the same directory as the tool \u2018\/gpgconf.exe\/\u2019. The root of the installation is then that directory; or, if \u2018\/gpgconf.exe\/\u2019 has been installed directly below a directory named \u2018\/bin\/\u2019, its parent directory. You also need to make sure that the following directories exist and are writable: \u2018\/ROOT\/home\/\u2019 for the GnuPG home and \u2018\/ROOT\/usr\/var\/cache\/gnupg\/\u2019 for internal cache files.\n\n\u2013display-charset name\nSet the name of the native character set. This is used to convert some informational strings like user IDs to the proper UTF-8 encoding. Note that this has nothing to do with the character set of data to be encrypted or signed; GnuPG does not recode user-supplied data. If this option is not used, the default character set is determined from the current locale. A verbosity level of 3 shows the chosen set. This option should not be used on Windows. Valid values for name are:\niso-8859-1\nThis is the Latin 1 set.\niso-8859-2\nThe Latin 2 set.\niso-8859-15\nThis is currently an alias for the Latin 1 set.\nkoi8-r\nThe usual Russian set (RFC-1489).\nutf-8\nBypass all translations and assume that the OS uses native UTF-8 encoding.\n(no term)\n\u2013utf8-strings\n\u2013no-utf8-strings :: Assume that command line arguments are given as UTF-8 strings. The default (\u2013no-utf8-strings) is to assume that arguments are encoded in the character set as specified by \u2013display-charset. These options affect all following arguments. Both options may be used multiple times. This option should not be used in an option file.\n\nThis option has no effect on Windows. There the internal used UTF-8 encoding is translated for console input and output. The command line arguments are expected as Unicode and translated to UTF-8. Thus when calling this program from another, make sure to use the Unicode version of CreateProcess.\n\n\u2013options file\nRead options from file and do not try to read them from the default options file in the homedir (see \u2013homedir). This option is ignored if used in an options file.\n\u2013no-options\nShortcut for \u2013options \/dev\/null. This option is detected before an attempt to open an option file. Using this option will also prevent the creation of a \u2018\/~\/.gnupg\/\u2019 homedir.\n(no term)\n-z n\n\u2013compress-level n\n\u2013bzip2-compress-level n :: Set compression level to n for the ZIP and ZLIB compression algorithms. The default is to use the default compression level of zlib (normally 6). \u2013bzip2-compress-level sets the compression level for the BZIP2 compression algorithm (defaulting to 6 as well). This is a different option from \u2013compress-level since BZIP2 uses a significant amount of memory for each additional compression level. -z sets both. A value of 0 for n disables compression.\n\u2013bzip2-decompress-lowmem\nUse a different decompression method for BZIP2 compressed files. This alternate method uses a bit more than half the memory, but also runs at half the speed. This is useful under extreme low memory circumstances when the file was originally compressed at a high \u2013bzip2-compress-level.\n(no term)\n\u2013mangle-dos-filenames\n\u2013no-mangle-dos-filenames :: Older version of Windows cannot handle filenames with more than one dot. \u2013mangle-dos-filenames causes GnuPG to replace (rather than add to) the extension of an output filename to avoid this problem. This option is off by default and has no effect on non-Windows platforms.\n(no term)\n\u2013no-ask-cert-level :: When making a key signature, prompt for a certification level. If this option is not specified, the certification level used is set via \u2013default-cert-level. See \u2013default-cert-level for information on the specific levels and how they are used. \u2013no-ask-cert-level disables this option. This option defaults to no.\n\u2013default-cert-level n\nThe default to use for the check level when signing a key.\n\n0 means you make no particular claim as to how carefully you verified the key.\n\n1 means you believe the key is owned by the person who claims to own it but you could not, or did not verify the key at all. This is useful for a \u201cpersona\u201d verification, where you sign the key of a pseudonymous user.\n\n2 means you did casual verification of the key. For example, this could mean that you verified the key fingerprint and checked the user ID on the key against a photo ID.\n\n3 means you did extensive verification of the key. For example, this could mean that you verified the key fingerprint with the owner of the key in person, and that you checked, by means of a hard to forge document with a photo ID (such as a passport) that the name of the key owner matches the name in the user ID on the key, and finally that you verified (by exchange of email) that the email address on the key belongs to the key owner.\n\nNote that the examples given above for levels 2 and 3 are just that: examples. In the end, it is up to you to decide just what \u201ccasual\u201d and \u201cextensive\u201d mean to you.\n\nThis option defaults to 0 (no particular claim).\n\n\u2013min-cert-level\nWhen building the trust database, treat any signatures with a certification level below this as invalid. Defaults to 2, which disregards level 1 signatures. Note that level 0 \u201cno particular claim\u201d signatures are always accepted.\n\u2013trusted-key long key ID or fingerprint\nAssume that the specified key (which should be given as fingerprint) is as trustworthy as one of your own secret keys. This option is useful if you don\u2019t want to keep your secret keys (or one of them) online but still want to be able to check the validity of a given recipient\u2019s or signator\u2019s key. If the given key is not locally available but an LDAP keyserver is configured the missing key is imported from that server.\n\u2013trust-model {pgp|classic|tofu|tofu+pgp|direct|always|auto}\n\nSet what trust model GnuPG should follow. The models are:\n\npgp\nThis is the Web of Trust combined with trust signatures as used in PGP 5.x and later. This is the default trust model when creating a new trust database.\nclassic\nThis is the standard Web of Trust as introduced by PGP 2.\ntofu\nTOFU stands for Trust On First Use. In this trust model, the first time a key is seen, it is memorized. If later another key with a user id with the same email address is seen, both keys are marked as suspect. In that case, the next time either is used, a warning is displayed describing the conflict, why it might have occurred (either the user generated a new key and failed to cross sign the old and new keys, the key is forgery, or a man-in-the-middle attack is being attempted), and the user is prompted to manually confirm the validity of the key in question.\n\nBecause a potential attacker is able to control the email address and thereby circumvent the conflict detection algorithm by using an email address that is similar in appearance to a trusted email address, whenever a message is verified, statistics about the number of messages signed with the key are shown. In this way, a user can easily identify attacks using fake keys for regular correspondents.\n\nWhen compared with the Web of Trust, TOFU offers significantly weaker security guarantees. In particular, TOFU only helps ensure consistency (that is, that the binding between a key and email address doesn\u2019t change). A major advantage of TOFU is that it requires little maintenance to use correctly. To use the web of trust properly, you need to actively sign keys and mark users as trusted introducers. This is a time-consuming process and anecdotal evidence suggests that even security-conscious users rarely take the time to do this thoroughly and instead rely on an ad-hoc TOFU process.\n\nIn the TOFU model, policies are associated with bindings between keys and email addresses (which are extracted from user ids and normalized). There are five policies, which can be set manually using the \u2013tofu-policy option. The default policy can be set using the \u2013tofu-default-policy option.\n\nThe TOFU policies are: auto, good, unknown, bad and ask. The auto policy is used by default (unless overridden by \u2013tofu-default-policy) and marks a binding as marginally trusted. The good, unknown and bad policies mark a binding as fully trusted, as having unknown trust or as having trust never, respectively. The unknown policy is useful for just using TOFU to detect conflicts, but to never assign positive trust to a binding. The final policy, ask prompts the user to indicate the binding\u2019s trust. If batch mode is enabled (or input is inappropriate in the context), then the user is not prompted and the undefined trust level is returned.\n\ntofu+pgp\nThis trust model combines TOFU with the Web of Trust. This is done by computing the trust level for each model and then taking the maximum trust level where the trust levels are ordered as follows: unknown < undefined < marginal < fully < ultimate < expired < never.\n\nBy setting \u2013tofu-default-policy=unknown, this model can be used to implement the web of trust with TOFU\u2019s conflict detection algorithm, but without its assignment of positive trust values, which some security-conscious users don\u2019t like.\n\ndirect\nKey validity is set directly by the user and not calculated via the Web of Trust. This model is solely based on the key and does not distinguish user IDs. Note that when changing to another trust model the trust values assigned to a key are transformed into ownertrust values, which also indicate how you trust the owner of the key to sign other keys.\nalways\nSkip key validation and assume that used keys are always fully valid. You generally won\u2019t use this unless you are using some external validation scheme. This option also suppresses the \u201c[uncertain]\u201d tag printed with signature checks when there is no evidence that the user ID is bound to the key. Note that this trust model still does not allow the use of expired, revoked, or disabled keys.\nauto\nSelect the trust model depending on whatever the internal trust database says. This is the default model if such a database already exists. Note that a tofu trust model is not considered here and must be enabled explicitly.\n(no term)\n\u2013auto-key-locate mechanisms\n\u2013no-auto-key-locate :: GnuPG can automatically locate and retrieve keys as needed using this option. This happens when encrypting to an email address (in the \u201cuser@example.com\u201d form), and there are no \u201cuser@example.com\u201d keys on the local keyring. This option takes any number of the mechanisms listed below, in the order they are to be tried. Instead of listing the mechanisms as comma delimited arguments, the option may also be given several times to add more mechanism. The option \u2013no-auto-key-locate or the mechanism \u201cclear\u201d resets the list. The default is \u201clocal,wkd\u201d.\ncert\nLocate a key using DNS CERT, as specified in RFC-4398.\npka\nLocate a key using DNS PKA.\ndane\nLocate a key using DANE, as specified in draft-ietf-dane-openpgpkey-05.txt.\nwkd\nLocate a key using the Web Key Directory protocol.\nldap\nUsing DNS Service Discovery, check the domain in question for any LDAP keyservers to use. If this fails, attempt to locate the key using the PGP Universal method of checking \u2018ldap:\/\/keys.(thedomain)\u2019.\nntds\nLocate the key using the Active Directory (Windows only). This method also allows to search by fingerprint using the command \u2013locate-external-key.\nkeyserver\nLocate a key using a keyserver. This method also allows to search by fingerprint using the command \u2013locate-external-key if any of the configured keyservers is an LDAP server.\nkeyserver-URL\nIn addition, a keyserver URL as used in the dirmngr configuration may be used here to query that particular keyserver. This method also allows to search by fingerprint using the command \u2013locate-external-key if the URL specifies an LDAP server.\nlocal\nLocate the key using the local keyrings. This mechanism allows the user to select the order a local key lookup is done. Thus using \u2018\u2013auto-key-locate local\u2019 is identical to \u2013no-auto-key-locate.\nnodefault\nThis flag disables the standard local key lookup, done before any of the mechanisms defined by the \u2013auto-key-locate are tried. The position of this mechanism in the list does not matter. It is not required if local is also used.\nclear\nClear all defined mechanisms. This is useful to override mechanisms given in a config file. Note that a nodefault in mechanisms will also be cleared unless it is given after the clear.\n(no term)\n\u2013auto-key-import\n\u2013no-auto-key-import :: This is an offline mechanism to get a missing key for signature verification and for later encryption to this key. If this option is enabled and a signature includes an embedded key, that key is used to verify the signature and on verification success that key is imported. The default is \u2013no-auto-key-import.\n\nOn the sender (signing) site the option \u2013include-key-block needs to be used to put the public part of the signing key as \u201cKey Block subpacket\u201d into the signature.\n\n\u2022 \u2013auto-key-retrieve\n\u2013no-auto-key-retrieve :: These options enable or disable the automatic retrieving of keys from a keyserver when verifying signatures made by keys that are not on the local keyring. The default is \u2013no-auto-key-retrieve.\n\nThe order of methods tried to lookup the key is:\n\n1. If the option \u2013auto-key-import is set and the signatures includes\n\nan embedded key, that key is used to verify the signature and on verification success that key is imported.\n\n1. If a preferred keyserver is specified in the signature and the option\n\nhonor-keyserver-url is active (which is not the default), that keyserver is tried. Note that the creator of the signature uses the option \u2013sig-keyserver-url to specify the preferred keyserver for data signatures.\n\n1. If the signature has the Signer\u2019s UID set (e.g. using \u2013sender\n\nwhile creating the signature) a Web Key Directory (WKD) lookup is done. This is the default configuration but can be disabled by removing WKD from the auto-key-locate list or by using the option \u2013disable-signer-uid.\n\n1. If the option honor-pka-record is active, the legacy PKA method is\n\nused.\n\n1. If any keyserver is configured and the Issuer Fingerprint is part of\n\nthe signature (since GnuPG 2.1.16), the configured keyservers are tried.\n\nNote that this option makes a \u201cweb bug\u201d like behavior possible. Keyserver or Web Key Directory operators can see which keys you request, so by sending you a message signed by a brand new key (which you naturally will not have on your local keyring), the operator can tell both your IP address and the time when you verified the signature.\n\n\u2013keyid-format {none|short|0xshort|long|0xlong}\nSelect how to display key IDs. \u201cnone\u201d does not show the key ID at all but shows the fingerprint in a separate line. \u201cshort\u201d is the traditional 8-character key ID. \u201clong\u201d is the more accurate (but less convenient) 16-character key ID. Add an \u201c0x\u201d to either to include an \u201c0x\u201d at the beginning of the key ID, as in 0x99242560. Note that this option is ignored if the option \u2013with-colons is used.\n\u2013keyserver name\nThis option is deprecated - please use the \u2013keyserver in \u2018\/dirmngr.conf\/\u2019 instead.\n\nUse name as your keyserver. This is the server that \u2013receive-keys, \u2013send-keys, and \u2013search-keys will communicate with to receive keys from, send keys to, and search for keys on. The format of the name is a URI: scheme:[\/\/]keyservername[:port]\u2019 The scheme is the type of keyserver: \u201chkp\u201d\u201chkps\u201d for the HTTP (or compatible) keyservers or \u201cldap\u201d\u201cldaps\u201d for the LDAP keyservers. Note that your particular installation of GnuPG may have other keyserver types available as well. Keyserver schemes are case-insensitive.\n\nMost keyservers synchronize with each other, so there is generally no need to send keys to more than one server. The keyserver hkp:\/\/keys.gnupg.net uses round robin DNS to give a different keyserver each time you use it.\n\n\u2013keyserver-options {name\/=\/value}\nThis is a space or comma delimited string that gives options for the keyserver. Options can be prefixed with a no-\u2019 to give the opposite meaning. Valid import-options or export-options may be used here as well to apply to importing (\u2013recv-key) or exporting (\u2013send-key) a key from a keyserver. While not all options are available for all keyserver types, some common options are:\ninclude-revoked\nWhen searching for a key with \u2013search-keys, include keys that are marked on the keyserver as revoked. Note that not all keyservers differentiate between revoked and unrevoked keys, and for such keyservers this option is meaningless. Note also that most keyservers do not have cryptographic verification of key revocations, and so turning this option off may result in skipping keys that are incorrectly marked as revoked.\ninclude-disabled\nWhen searching for a key with \u2013search-keys, include keys that are marked on the keyserver as disabled. Note that this option is not used with HKP keyservers.\nauto-key-retrieve\nThis is an obsolete alias for the option auto-key-retrieve. Please do not use it; it will be removed in future versions..\nhonor-keyserver-url\nWhen using \u2013refresh-keys, if the key in question has a preferred keyserver URL, then use that preferred keyserver to refresh the key from. In addition, if auto-key-retrieve is set, and the signature being verified has a preferred keyserver URL, then use that preferred keyserver to fetch the key from. Note that this option introduces a \u201cweb bug\u201d: The creator of the key can see when the keys is refreshed. Thus this option is not enabled by default.\nhonor-pka-record\nIf \u2013auto-key-retrieve is used, and the signature being verified has a PKA record, then use the PKA information to fetch the key. Defaults to \u201cyes\u201d.\ninclude-subkeys\nWhen receiving a key, include subkeys as potential targets. Note that this option is not used with HKP keyservers, as they do not support retrieving keys by subkey id.\n(no term)\ntimeout\nhttp-proxy=\/value\/\nverbose\ndebug\ncheck-cert ::\nca-cert-file\nThese options have no more function since GnuPG 2.1. Use the dirmngr configuration options instead.\n\nThe default list of options is: \u201cself-sigs-only, repair-keys, repair-pks-subkey-bug, export-attributes, honor-pka-record\u201d. However, if the actual used source is an LDAP server \u201cno-self-sigs-only\u201d is assumed unless \u201cself-sigs-only\u201d has been explictly configured.\n\n\u2013completes-needed n\nNumber of completely trusted users to introduce a new key signer (defaults to 1).\n\u2013marginals-needed n\nNumber of marginally trusted users to introduce a new key signer (defaults to 3)\nThe default TOFU policy (defaults to auto). For more information about the meaning of this option, see: [trust-model-tofu].\n\u2013max-cert-depth n\nMaximum depth of a certification chain (default is 5).\n\u2013no-sig-cache\nDo not cache the verification status of key signatures. Caching gives a much better performance in key listings. However, if you suspect that your public keyring is not safe against write modifications, you can use this option to disable the caching. It probably does not make sense to disable it because all kind of damage can be done if someone else has write access to your public keyring.\n(no term)\n\u2013auto-check-trustdb\n\u2013no-auto-check-trustdb :: If GnuPG feels that its information about the Web of Trust has to be updated, it automatically runs the \u2013check-trustdb command internally. This may be a time consuming process. \u2013no-auto-check-trustdb disables this option.\n(no term)\n\u2013use-agent\n\u2013no-use-agent :: This is dummy option. gpg always requires the agent.\n\u2013gpg-agent-info\nThis is dummy option. It has no effect when used with gpg.\n\u2013agent-program file\nSpecify an agent program to be used for secret key operations. The default value is determined by running gpgconf with the option \u2013list-dirs. Note that the pipe symbol (|) is used for a regression test suite hack and may thus not be used in the file name.\n\u2013dirmngr-program file\nSpecify a dirmngr program to be used for keyserver access. The default value is \u2018\/\/usr\/bin\/dirmngr\/\u2019.\n\u2013disable-dirmngr\nEntirely disable the use of the Dirmngr.\n\u2013no-autostart\nDo not start the gpg-agent or the dirmngr if it has not yet been started and its service is required. This option is mostly useful on machines where the connection to gpg-agent has been redirected to another machines. If dirmngr is required on the remote machine, it may be started manually using gpgconf \u2013launch dirmngr.\n\u2013lock-once\nLock the databases the first time a lock is requested and do not release the lock until the process terminates.\n\u2013lock-multiple\nRelease the locks every time a lock is no longer needed. Use this to override a previous \u2013lock-once from a config file.\n\u2013lock-never\nDisable locking entirely. This option should be used only in very special environments, where it can be assured that only one process is accessing those files. A bootable floppy with a stand-alone encryption system will probably use this. Improper usage of this option may lead to data and key corruption.\n\u2013exit-on-status-write-error\nThis option will cause write errors on the status FD to immediately terminate the process. That should in fact be the default but it never worked this way and thus we need an option to enable this, so that the change won\u2019t break applications which close their end of a status fd connected pipe too early. Using this option along with \u2013enable-progress-filter may be used to cleanly cancel long running gpg operations.\n\u2013limit-card-insert-tries n\nWith n greater than 0 the number of prompts asking to insert a smartcard gets limited to N-1. Thus with a value of 1 gpg won\u2019t at all ask to insert a card if none has been inserted at startup. This option is useful in the configuration file in case an application does not know about the smartcard support and waits ad infinitum for an inserted card.\n\u2013no-random-seed-file\nGnuPG uses a file to store its internal random pool over invocations. This makes random generation faster; however sometimes write operations are not desired. This option can be used to achieve that with the cost of slower random generation.\n\u2013no-greeting\n\u2013no-secmem-warning\nSuppress the warning about \u201cusing insecure memory\u201d.\n\u2013no-permission-warning\nSuppress the warning about unsafe file and home directory (\u2013homedir) permissions. Note that the permission checks that GnuPG performs are not intended to be authoritative, but rather they simply warn about certain common permission problems. Do not assume that the lack of a warning means that your system is secure.\n\nNote that the warning for unsafe \u2013homedir permissions cannot be suppressed in the gpg.conf file, as this would allow an attacker to place an unsafe gpg.conf file in place, and use this file to suppress warnings about itself. The \u2013homedir permissions warning may only be suppressed on the command line.\n\n\u2022 \u2013require-secmem\n\u2013no-require-secmem :: Refuse to run if GnuPG cannot get secure memory. Defaults to no (i.e. run, but give a warning).\n\u2022 \u2013require-cross-certification\n\u2013no-require-cross-certification :: When verifying a signature made from a subkey, ensure that the cross certification \u201cback signature\u201d on the subkey is present and valid. This protects against a subtle attack against subkeys that can sign. Defaults to \u2013require-cross-certification for gpg.\n\u2022 \u2013expert\n\u2013no-expert :: Allow the user to do certain nonsensical or \u201csilly\u201d things like signing an expired or revoked key, or certain potentially incompatible things like generating unusual key types. This also disables certain warning messages about potentially incompatible actions. As the name implies, this option is for experts only. If you don\u2019t fully understand the implications of what it allows you to do, leave this off. \u2013no-expert disables this option.\n\n### Key related options\n\n\u2022 \u2013recipient name\n-r :: Encrypt for user id name. If this option or \u2013hidden-recipient is not specified, GnuPG asks for the user-id unless \u2013default-recipient is given.\n\u2022 \u2013hidden-recipient name\n-R :: Encrypt for user ID name, but hide the key ID of this user\u2019s key. This option helps to hide the receiver of the message and is a limited countermeasure against traffic analysis. If this option or \u2013recipient is not specified, GnuPG asks for the user ID unless \u2013default-recipient is given.\n\u2022 \u2013recipient-file file\n-f :: This option is similar to \u2013recipient except that it encrypts to a key stored in the given file. file must be the name of a file containing exactly one key. gpg assumes that the key in this file is fully valid.\n\u2022 \u2013hidden-recipient-file file\n-F :: This option is similar to \u2013hidden-recipient except that it encrypts to a key stored in the given file. file must be the name of a file containing exactly one key. gpg assumes that the key in this file is fully valid.\n\u2022 Same as \u2013recipient but this one is intended for use in the options file and may be used with your own user-id as an \u201cencrypt-to-self\u201d. These keys are only used when there are other recipients given either by use of \u2013recipient or by the asked user id. No trust checking is performed for these user ids and even disabled keys can be used.\n\u2022 Same as \u2013hidden-recipient but this one is intended for use in the options file and may be used with your own user-id as a hidden \u201cencrypt-to-self\u201d. These keys are only used when there are other recipients given either by use of \u2013recipient or by the asked user id. No trust checking is performed for these user ids and even disabled keys can be used.\n\u2022 Disable the use of all \u2013encrypt-to and \u2013hidden-encrypt-to keys.\n\u2022 Sets up a named group, which is similar to aliases in email programs. Any time the group name is a recipient (-r or \u2013recipient), it will be expanded to the values specified. Multiple groups with the same name are automatically merged into a single group.\n\nThe values are key IDs or fingerprints, but any key description is accepted. Note that a value with spaces in it will be treated as two different values. Note also there is only one level of expansion \u2014 you cannot make an group that points to another group. When used from the command line, it may be necessary to quote the argument to this option to prevent the shell from treating it as multiple arguments.\n\n\u2013ungroup name\nRemove a given entry from the \u2013group list.\n\u2013no-groups\nRemove all entries from the \u2013group list.\n(no term)\n\u2013local-user name\n-u :: Use name as the key to sign with. Note that this option overrides \u2013default-key.\n\u2013sender mbox\nThis option has two purposes. mbox must either be a complete user id with a proper mail address or just a mail address. When creating a signature this option tells gpg the user id of a key used to make a signature if the key was not directly specified by a user id. When verifying a signature the mbox is used to restrict the information printed by the TOFU code to matching user ids.\n\u2013try-secret-key name\nFor hidden recipients GPG needs to know the keys to use for trial decryption. The key set with \u2013default-key is always tried first, but this is often not sufficient. This option allows setting more keys to be used for trial decryption. Although any valid user-id specification may be used for name it makes sense to use at least the long keyid to avoid ambiguities. Note that gpg-agent might pop up a pinentry for a lot keys to do the trial decryption. If you want to stop all further trial decryption you may use close-window button instead of the cancel button.\n\u2013try-all-secrets\nDon\u2019t look at the key ID as stored in the message but try all secret keys in turn to find the right decryption key. This option forces the behaviour as used by anonymous recipients (created by using \u2013throw-keyids or \u2013hidden-recipient) and might come handy in case where an encrypted message contains a bogus key ID.\n(no term)\n\u2013skip-hidden-recipients\n\u2013no-skip-hidden-recipients :: During decryption skip all anonymous recipients. This option helps in the case that people use the hidden recipients feature to hide their own encrypt-to key from others. If one has many secret keys this may lead to a major annoyance because all keys are tried in turn to decrypt something which was not really intended for it. The drawback of this option is that it is currently not possible to decrypt a message which includes real anonymous recipients.\n\n### Input and Output\n\n\u2022 \u2013armor\n-a :: Create ASCII armored output. The default is to create the binary OpenPGP format.\n\u2022 Assume the input data is not in ASCII armored format.\n\u2022 \u2013output file\n-o file :: Write output to file. To write to stdout use - as the filename.\n\u2022 This option sets a limit on the number of bytes that will be generated when processing a file. Since OpenPGP supports various levels of compression, it is possible that the plaintext of a given message may be significantly larger than the original OpenPGP message. While GnuPG works properly with such messages, there is often a desire to set a maximum file size that will be generated before processing is forced to stop by the OS limits. Defaults to 0, which means \u201cno limit\u201d.\n\u2022 This option can be used to tell GPG the size of the input data in bytes. n must be a positive base-10 number. This option is only useful if the input is not taken from a file. GPG may use this hint to optimize its buffer allocation strategy. It is also used by the \u2013status-fd line PROGRESS\u2019\u2019 to provide a value for total\u2019\u2019 if that is not available by other means.\n\u2022 gpg can track the origin of a key. Certain origins are implicitly known (e.g. keyserver, web key directory) and set. For a standard import the origin of the keys imported can be set with this option. To list the possible values use \u201chelp\u201d for string. Some origins can store an optional url argument. That URL can appended to string after a comma.\n\u2022 This is a space or comma delimited string that gives options for importing keys. Options can be prepended with a no-\u2019 to give the opposite meaning. The options are:\nimport-local-sigs\nAllow importing key signatures marked as \u201clocal\u201d. This is not generally useful unless a shared keyring scheme is being used. Defaults to no.\nkeep-ownertrust\nNormally possible still existing ownertrust values of a key are cleared if a key is imported. This is in general desirable so that a formerly deleted key does not automatically gain an ownertrust values merely due to import. On the other hand it is sometimes necessary to re-import a trusted set of keys again but keeping already assigned ownertrust values. This can be achieved by using this option.\nrepair-pks-subkey-bug\nDuring import, attempt to repair the damage caused by the PKS keyserver bug (pre version 0.9.6) that mangles keys with multiple subkeys. Note that this cannot completely repair the damaged key as some crucial data is removed by the keyserver, but it does at least give you back one subkey. Defaults to no for regular \u2013import and to yes for keyserver \u2013receive-keys.\n(no term)\nimport-show\nshow-only :: Show a listing of the key as imported right before it is stored. This can be combined with the option \u2013dry-run to only look at keys; the option show-only is a shortcut for this combination. The command \u2013show-keys is another shortcut for this. Note that suffixes like \u2019#\u2019 for \u201csec\u201d and \u201csbb\u201d lines may or may not be printed.\nimport-export\nRun the entire import code but instead of storing the key to the local keyring write it to the output. The export options export-pka and export-dane affect the output. This option can be used to remove all invalid parts from a key without the need to store it.\nmerge-only\nDuring import, allow key updates to existing keys, but do not allow any new keys to be imported. Defaults to no.\nimport-clean\nAfter import, compact (remove all signatures except the self-signature) any user IDs from the new key that are not usable. Then, remove any signatures from the new key that are not usable. This includes signatures that were issued by keys that are not present on the keyring. This option is the same as running the \u2013edit-key command \u201cclean\u201d after import. Defaults to no.\nself-sigs-only\nAccept only self-signatures while importing a key. All other key signatures are skipped at an early import stage. This option can be used with keyserver-options to mitigate attempts to flood a key with bogus signatures from a keyserver. The drawback is that all other valid key signatures, as required by the Web of Trust are also not imported. Note that when using this option along with import-clean it suppresses the final clean step after merging the imported key into the existing key.\nrepair-keys\nAfter import, fix various problems with the keys. For example, this reorders signatures, and strips duplicate signatures. Defaults to yes.\nimport-minimal\nImport the smallest key possible. This removes all signatures except the most recent self-signature on each user ID. This option is the same as running the \u2013edit-key command \u201cminimize\u201d after import. Defaults to no.\n(no term)\nrestore\nimport-restore :: Import in key restore mode. This imports all data which is usually skipped during import; including all GnuPG specific data. All other contradicting options are overridden.\n\u2022 \u2013import-filter {name\/=\/expr}\n\u2013export-filter {name\/=\/expr} :: These options define an import\/export filter which are applied to the imported\/exported keyblock right before it will be stored\/written. name defines the type of filter to use, expr the expression to evaluate. The option can be used several times which then appends more expression to the same name.\n\nThe available filter types are:\n\nkeep-uid\nThis filter will keep a user id packet and its dependent packets in the keyblock if the expression evaluates to true.\ndrop-subkey\nThis filter drops the selected subkeys. Currently only implemented for \u2013export-filter.\ndrop-sig\nThis filter drops the selected key signatures on user ids. Self-signatures are not considered. Currently only implemented for \u2013import-filter.\n\nFor the syntax of the expression see the chapter \u201cFILTER EXPRESSIONS\u201d. The property names for the expressions depend on the actual filter type and are indicated in the following table.\n\nThe available properties are:\n\nuid\nA string with the user id. (keep-uid)\nmbox\nThe addr-spec part of a user id with mailbox or the empty string. (keep-uid)\nkey_algo\nA number with the public key algorithm of a key or subkey packet. (drop-subkey)\n(no term)\nkey_created\nkey_created_d :: The first is the timestamp a public key or subkey packet was created. The second is the same but given as an ISO string, e.g. \u201c2016-08-17\u201d. (drop-subkey)\nfpr\nThe hexified fingerprint of the current subkey or primary key. (drop-subkey)\nprimary\nBoolean indicating whether the user id is the primary one. (keep-uid)\nexpired\nBoolean indicating whether a user id (keep-uid), a key (drop-subkey), or a signature (drop-sig) expired.\nrevoked\nBoolean indicating whether a user id (keep-uid) or a key (drop-subkey) has been revoked.\ndisabled\nBoolean indicating whether a primary key is disabled. (not used)\nsecret\nBoolean indicating whether a key or subkey is a secret one. (drop-subkey)\nusage\nA string indicating the usage flags for the subkey, from the sequence ecsa?\u2019\u2019. For example, a subkey capable of just signing and authentication would be an exact match for sa\u2019\u2019. (drop-subkey)\n(no term)\nsig_created\nsig_created_d :: The first is the timestamp a signature packet was created. The second is the same but given as an ISO date string, e.g. \u201c2016-08-17\u201d. (drop-sig)\nsig_algo\nA number with the public key algorithm of a signature packet. (drop-sig)\nsig_digest_algo\nA number with the digest algorithm of a signature packet. (drop-sig)\n\u2013export-options parameters\nThis is a space or comma delimited string that gives options for exporting keys. Options can be prepended with a no-\u2019 to give the opposite meaning. The options are:\nexport-local-sigs\nAllow exporting key signatures marked as \u201clocal\u201d. This is not generally useful unless a shared keyring scheme is being used. Defaults to no.\nexport-attributes\nInclude attribute user IDs (photo IDs) while exporting. Not including attribute user IDs is useful to export keys that are going to be used by an OpenPGP program that does not accept attribute user IDs. Defaults to yes.\nexport-sensitive-revkeys\nInclude designated revoker information that was marked as \u201csensitive\u201d. Defaults to no.\n(no term)\nbackup\nexport-backup :: Export for use as a backup. The exported data includes all data which is needed to restore the key or keys later with GnuPG. The format is basically the OpenPGP format but enhanced with GnuPG specific data. All other contradicting options are overridden.\nexport-clean\nCompact (remove all signatures from) user IDs on the key being exported if the user IDs are not usable. Also, do not export any signatures that are not usable. This includes signatures that were issued by keys that are not present on the keyring. This option is the same as running the \u2013edit-key command \u201cclean\u201d before export except that the local copy of the key is not modified. Defaults to no.\nexport-minimal\nExport the smallest key possible. This removes all signatures except the most recent self-signature on each user ID. This option is the same as running the \u2013edit-key command \u201cminimize\u201d before export except that the local copy of the key is not modified. Defaults to no.\nexport-pka\nInstead of outputting the key material output PKA records suitable to put into DNS zone files. An ORIGIN line is printed before each record to allow diverting the records to the corresponding zone file.\nexport-dane\nInstead of outputting the key material output OpenPGP DANE records suitable to put into DNS zone files. An ORIGIN line is printed before each record to allow diverting the records to the corresponding zone file.\n\u2013with-colons\nPrint key listings delimited by colons. Note that the output will be encoded in UTF-8 regardless of any \u2013display-charset setting. This format is useful when GnuPG is called from scripts and other programs as it is easily machine parsed. The details of this format are documented in the file \u2018\/doc\/DETAILS\/\u2019, which is included in the GnuPG source distribution.\n\u2013fixed-list-mode\nDo not merge primary user ID and primary key in \u2013with-colon listing mode and print all timestamps as seconds since 1970-01-01. Since GnuPG 2.0.10, this mode is always used and thus this option is obsolete; it does not harm to use it though.\n\u2013legacy-list-mode\nRevert to the pre-2.1 public key list mode. This only affects the human readable output and not the machine interface (i.e. \u2013with-colons). Note that the legacy format does not convey suitable information for elliptic curves.\n\u2013with-fingerprint\nSame as the command \u2013fingerprint but changes only the format of the output and may be used together with another command.\n\u2013with-subkey-fingerprint\nIf a fingerprint is printed for the primary key, this option forces printing of the fingerprint for all subkeys. This could also be achieved by using the \u2013with-fingerprint twice but by using this option along with keyid-format \u201cnone\u201d a compact fingerprint is printed.\n\u2013with-icao-spelling\nPrint the ICAO spelling of the fingerprint in addition to the hex digits.\n\u2013with-keygrip\nInclude the keygrip in the key listings. In \u2013with-colons mode this is implicitly enable for secret keys.\n\u2013with-key-origin\nInclude the locally held information on the origin and last update of a key in a key listing. In \u2013with-colons mode this is always printed. This data is currently experimental and shall not be considered part of the stable API.\n\u2013with-wkd-hash\nPrint a Web Key Directory identifier along with each user ID in key listings. This is an experimental feature and semantics may change.\n\u2013with-secret\nInclude info about the presence of a secret key in public key listings done with \u2013with-colons.\n\n### OpenPGP protocol specific options\n\n\u2022 -t, \u2013textmode\n\u2013no-textmode :: Treat input files as text and store them in the OpenPGP canonical text form with standard \u201cCRLF\u201d line endings. This also sets the necessary flags to inform the recipient that the encrypted or signed data is text and may need its line endings converted back to whatever the local system uses. This option is useful when communicating between two platforms that have different line ending conventions (UNIX-like to Mac, Mac to Windows, etc). \u2013no-textmode disables this option, and is the default.\n\u2022 \u2013force-v3-sigs\n\u2013no-force-v3-sigs ::\n\u2022 \u2013force-v4-certs\n\u2013no-force-v4-certs :: These options are obsolete and have no effect since GnuPG 2.1.\n\u2022 \u2013force-mdc\n\u2013disable-mdc :: These options are obsolete and have no effect since GnuPG 2.2.8. The MDC is always used. But note: If the creation of a legacy non-MDC message is exceptionally required, the option \u2013rfc2440 allows for this.\n\u2022 By default the user ID of the signing key is embedded in the data signature. As of now this is only done if the signing key has been specified with local-user using a mail address, or with sender. This information can be helpful for verifier to locate the key; see option \u2013auto-key-retrieve.\n\u2022 This option is used to embed the actual signing key into a data signature. The embedded key is stripped down to a single user id and includes only the signing subkey used to create the signature as well as as valid encryption subkeys. All other info is removed from the key to keep it and thus the signature small. This option is the OpenPGP counterpart to the gpgsm option \u2013include-certs.\n\u2022 Set the list of personal cipher preferences to string. Use gpg \u2013version to get a list of available algorithms, and use none to set no preference at all. This allows the user to safely override the algorithm chosen by the recipient key preferences, as GPG will only select an algorithm that is usable by all recipients. The most highly ranked cipher in this list is also used for the \u2013symmetric encryption command.\n\u2022 Set the list of personal digest preferences to string. Use gpg \u2013version to get a list of available algorithms, and use none to set no preference at all. This allows the user to safely override the algorithm chosen by the recipient key preferences, as GPG will only select an algorithm that is usable by all recipients. The most highly ranked digest algorithm in this list is also used when signing without encryption (e.g. \u2013clear-sign or \u2013sign).\n\u2022 Set the list of personal compression preferences to string. Use gpg \u2013version to get a list of available algorithms, and use none to set no preference at all. This allows the user to safely override the algorithm chosen by the recipient key preferences, as GPG will only select an algorithm that is usable by all recipients. The most highly ranked compression algorithm in this list is also used when there are no recipient keys to consider (e.g. \u2013symmetric).\n\u2022 Use name as the cipher algorithm for symmetric encryption with a passphrase if \u2013personal-cipher-preferences and \u2013cipher-algo are not given. The default is AES-128.\n\u2022 Use name as the digest algorithm used to mangle the passphrases for symmetric encryption. The default is SHA-1.\n\u2022 Selects how passphrases for symmetric encryption are mangled. If n is 0 a plain passphrase (which is in general not recommended) will be used, a 1 adds a salt (which should not be used) to the passphrase and a 3 (the default) iterates the whole process a number of times (see \u2013s2k-count).\n\u2022 Specify how many times the passphrases mangling for symmetric encryption is repeated. This value may range between 1024 and 65011712 inclusive. The default is inquired from gpg-agent. Note that not all values in the 1024-65011712 range are legal and if an illegal value is selected, GnuPG will round up to the nearest legal value. This option is only meaningful if \u2013s2k-mode is set to the default of 3.\n\n### Compliance options\n\nThese options control what GnuPG is compliant to. Only one of these options may be active at a time. Note that the default setting of this is nearly always the correct one. See the INTEROPERABILITY WITH OTHER OPENPGP PROGRAMS section below before using one of these options.\n\n\u2013gnupg\nUse standard GnuPG behavior. This is essentially OpenPGP behavior (see \u2013openpgp), but with some additional workarounds for common compatibility problems in different versions of PGP. This is the default option, so it is not generally needed, but it may be useful to override a different compliance option in the gpg.conf file.\n\u2013openpgp\nReset all packet, cipher and digest options to strict OpenPGP behavior. Use this option to reset all previous options like \u2013s2k-*, \u2013cipher-algo, \u2013digest-algo and \u2013compress-algo to OpenPGP compliant values. All PGP workarounds are disabled.\n\u2013rfc4880\nReset all packet, cipher and digest options to strict RFC-4880 behavior. Note that this is currently the same thing as \u2013openpgp.\n\u2013rfc4880bis\nEnable experimental features from proposed updates to RFC-4880. This option can be used in addition to the other compliance options. Warning: The behavior may change with any GnuPG release and created keys or data may not be usable with future GnuPG versions.\n\u2013rfc2440\nReset all packet, cipher and digest options to strict RFC-2440 behavior. Note that by using this option encryption packets are created in a legacy mode without MDC protection. This is dangerous and should thus only be used for experiments. See also option \u2013ignore-mdc-error.\n\u2013pgp6\nSet up all options to be as PGP 6 compliant as possible. This restricts you to the ciphers IDEA (if the IDEA plugin is installed), 3DES, and CAST5, the hashes MD5, SHA1 and RIPEMD160, and the compression algorithms none and ZIP. This also disables \u2013throw-keyids, and making signatures with signing subkeys as PGP 6 does not understand signatures made by signing subkeys.\n\nThis option implies \u2013escape-from-lines.\n\n\u2013pgp7\nSet up all options to be as PGP 7 compliant as possible. This is identical to \u2013pgp6 except that MDCs are not disabled, and the list of allowable ciphers is expanded to add AES128, AES192, AES256, and TWOFISH.\n\u2013pgp8\nSet up all options to be as PGP 8 compliant as possible. PGP 8 is a lot closer to the OpenPGP standard than previous versions of PGP, so all this does is disable \u2013throw-keyids and set \u2013escape-from-lines. All algorithms are allowed except for the SHA224, SHA384, and SHA512 digests.\n\u2013compliance string\nThis option can be used instead of one of the options above. Valid values for string are the above option names (without the double dash) and possibly others as shown when using \u201chelp\u201d for value.\n\n### Doing things one usually doesn\u2019t want to do\n\n\u2022 -n\n\u2013dry-run :: Don\u2019t make any changes (this is not completely implemented).\n\u2022 Changes the behaviour of some commands. This is like \u2013dry-run but different in some cases. The semantic of this option may be extended in the future. Currently it only skips the actual decryption pass and therefore enables a fast listing of the encryption keys.\n\u2022 -i\n\u2013interactive :: Prompt before overwriting any files.\n\u2022 Select the debug level for investigating problems. level may be a numeric value or by a keyword:\nnone\nNo debugging at all. A value of less than 1 may be used instead of the keyword.\nbasic\nSome basic debug messages. A value between 1 and 2 may be used instead of the keyword.\nMore verbose debug messages. A value between 3 and 5 may be used instead of the keyword.\nexpert\nEven more detailed messages. A value between 6 and 8 may be used instead of the keyword.\nguru\nAll of the debug messages you can get. A value greater than 8 may be used instead of the keyword. The creation of hash tracing files is only enabled if the keyword is used.\n\nHow these messages are mapped to the actual debugging flags is not specified and may change with newer releases of this program. They are however carefully selected to best aid in debugging.\n\n\u2013debug flags\nSet debugging flags. All flags are or-ed and flags may be given in C syntax (e.g. 0x0042) or as a comma separated list of flag names. To get a list of all supported flags the single word \u201chelp\u201d can be used.\n\u2013debug-all\nSet all useful debugging flags.\n\u2013debug-iolbf\nSet stdout into line buffered mode. This option is only honored when given on the command line.\n\u2013faked-system-time epoch\nThis option is only useful for testing; it sets the system time back or forth to epoch which is the number of seconds elapsed since the year 1970. Alternatively epoch may be given as a full ISO time string (e.g. \u201c20070924T154812\u201d).\n\nIf you suffix epoch with an exclamation mark (!), the system time will appear to be frozen at the specified time.\n\n\u2013enable-progress-filter\nEnable certain PROGRESS status outputs. This option allows frontends to display a progress indicator while gpg is processing larger files. There is a slight performance overhead using it.\n\u2013status-fd n\nWrite special status strings to the file descriptor n. See the file DETAILS in the documentation for a listing of them.\n\u2013status-file file\nSame as \u2013status-fd, except the status data is written to file file.\n\u2013logger-fd n\nWrite log output to file descriptor n and not to STDERR.\n(no term)\n\u2013log-file file\n\u2013logger-file file :: Same as \u2013logger-fd, except the logger data is written to file file. Use \u2018\/socket:\/\/\/\u2019 to log to a socket. Note that in this version of gpg the option has only an effect if \u2013batch is also used.\n\u2013attribute-fd n\nWrite attribute subpackets to the file descriptor n. This is most useful for use with \u2013status-fd, since the status messages are needed to separate out the various subpackets from the stream delivered to the file descriptor.\n\u2013attribute-file file\nSame as \u2013attribute-fd, except the attribute data is written to file file.\n(no term)\n\u2013comment string\n\u2013no-comments :: Use string as a comment string in cleartext signatures and ASCII armored messages or keys (see \u2013armor). The default behavior is not to use a comment string. \u2013comment may be repeated multiple times to get multiple comment strings. \u2013no-comments removes all comments. It is a good idea to keep the length of a single comment below 60 characters to avoid problems with mail programs wrapping such lines. Note that comment lines, like all other header lines, are not protected by the signature.\n(no term)\n\u2013emit-version\n\u2013no-emit-version :: Force inclusion of the version string in ASCII armored output. If given once only the name of the program and the major number is emitted, given twice the minor is also emitted, given thrice the micro is added, and given four times an operating system identification is also emitted. \u2013no-emit-version (default) disables the version line.\n(no term)\n\u2013sig-notation {name\/=\/value}\n\u2013cert-notation {name\/=\/value}\n-N, \u2013set-notation {name\/=\/value} :: Put the name value pair into the signature as notation data. name must consist only of printable characters or spaces, and must contain a \u2019@\u2019 character in the form keyname@domain.example.com (substituting the appropriate keyname and domain name, of course). This is to help prevent pollution of the IETF reserved notation namespace. The \u2013expert flag overrides the \u2019@\u2019 check. value may be any printable string; it will be encoded in UTF-8, so you should check that your \u2013display-charset is set correctly. If you prefix name with an exclamation mark (!), the notation data will be flagged as critical (rfc4880:5.2.3.16). \u2013sig-notation sets a notation for data signatures. \u2013cert-notation sets a notation for key signatures (certifications). \u2013set-notation sets both.\n\nThere are special codes that may be used in notation names. \u201c%k\u201d will be expanded into the key ID of the key being signed, \u201c%K\u201d into the long key ID of the key being signed, \u201c%f\u201d into the fingerprint of the key being signed, \u201c%s\u201d into the key ID of the key making the signature, \u201c%S\u201d into the long key ID of the key making the signature, \u201c%g\u201d into the fingerprint of the key making the signature (which might be a subkey), \u201c%p\u201d into the fingerprint of the primary key of the key making the signature, \u201c%c\u201d into the signature count from the OpenPGP smartcard, and \u201c%%\u201d results in a single \u201c%\u201d. %k, %K, and %f are only meaningful when making a key signature (certification), and %c is only meaningful when using the OpenPGP smartcard.\n\n\u2013known-notation name\nAdds name to a list of known critical signature notations. The effect of this is that gpg will not mark a signature with a critical signature notation of that name as bad. Note that gpg already knows by default about a few critical signatures notation names.\n(no term)\n\u2013sig-policy-url string\n\u2013cert-policy-url string\n\u2013set-policy-url string :: Use string as a Policy URL for signatures (rfc4880:5.2.3.20). If you prefix it with an exclamation mark (!), the policy URL packet will be flagged as critical. \u2013sig-policy-url sets a policy url for data signatures. \u2013cert-policy-url sets a policy url for key signatures (certifications). \u2013set-policy-url sets both.\n\nThe same %-expandos used for notation data are available here as well.\n\n\u2013sig-keyserver-url string\nUse string as a preferred keyserver URL for data signatures. If you prefix it with an exclamation mark (!), the keyserver URL packet will be flagged as critical.\n\nThe same %-expandos used for notation data are available here as well.\n\n\u2013set-filename string\nUse string as the filename which is stored inside messages. This overrides the default, which is to use the actual filename of the file being encrypted. Using the empty string for string effectively removes the filename from the output.\n(no term)\n\u2013for-your-eyes-only\n\u2013no-for-your-eyes-only :: Set the for your eyes only\u2019 flag in the message. This causes GnuPG to refuse to save the file unless the \u2013output option is given, and PGP to use a \u201csecure viewer\u201d with a claimed Tempest-resistant font to display the message. This option overrides \u2013set-filename. \u2013no-for-your-eyes-only disables this option.\n(no term)\n\u2013use-embedded-filename\n\u2013no-use-embedded-filename :: Try to create a file with a name as embedded in the data. This can be a dangerous option as it enables overwriting files. Defaults to no. Note that the option \u2013output overrides this option.\n\u2013cipher-algo name\nUse name as cipher algorithm. Running the program with the command \u2013version yields a list of supported algorithms. If this is not used the cipher algorithm is selected from the preferences stored with the key. In general, you do not want to use this option as it allows you to violate the OpenPGP standard. \u2013personal-cipher-preferences is the safe way to accomplish the same thing.\n\u2013digest-algo name\nUse name as the message digest algorithm. Running the program with the command \u2013version yields a list of supported algorithms. In general, you do not want to use this option as it allows you to violate the OpenPGP standard. \u2013personal-digest-preferences is the safe way to accomplish the same thing.\n\u2013compress-algo name\nUse compression algorithm name. \u201czlib\u201d is RFC-1950 ZLIB compression. \u201czip\u201d is RFC-1951 ZIP compression which is used by PGP. \u201cbzip2\u201d is a more modern compression scheme that can compress some things better than zip or zlib, but at the cost of more memory used during compression and decompression. \u201cuncompressed\u201d or \u201cnone\u201d disables compression. If this option is not used, the default behavior is to examine the recipient key preferences to see which algorithms the recipient supports. If all else fails, ZIP is used for maximum compatibility.\n\nZLIB may give better compression results than ZIP, as the compression window size is not limited to 8k. BZIP2 may give even better compression results than that, but will use a significantly larger amount of memory while compressing and decompressing. This may be significant in low memory situations. Note, however, that PGP (all versions) only supports ZIP compression. Using any algorithm other than ZIP or \u201cnone\u201d will make the message unreadable with PGP. In general, you do not want to use this option as it allows you to violate the OpenPGP standard. \u2013personal-compress-preferences is the safe way to accomplish the same thing.\n\n\u2013cert-digest-algo name\nUse name as the message digest algorithm used when signing a key. Running the program with the command \u2013version yields a list of supported algorithms. Be aware that if you choose an algorithm that GnuPG supports but other OpenPGP implementations do not, then some users will not be able to use the key signatures you make, or quite possibly your entire key.\n\u2013disable-cipher-algo name\nNever allow the use of name as cipher algorithm. The given name will not be checked so that a later loaded algorithm will still get disabled.\n\u2013disable-pubkey-algo name\nNever allow the use of name as public key algorithm. The given name will not be checked so that a later loaded algorithm will still get disabled.\n(no term)\n\u2013throw-keyids\n\u2013no-throw-keyids :: Do not put the recipient key IDs into encrypted messages. This helps to hide the receivers of the message and is a limited countermeasure against traffic analysis. ([Using a little social engineering anyone who is able to decrypt the message can check whether one of the other recipients is the one he suspects.]) On the receiving side, it may slow down the decryption process because all available secret keys must be tried. \u2013no-throw-keyids disables this option. This option is essentially the same as using \u2013hidden-recipient for all recipients.\n\u2013not-dash-escaped\nThis option changes the behavior of cleartext signatures so that they can be used for patch files. You should not send such an armored file via email because all spaces and line endings are hashed too. You can not use this option for data which has 5 dashes at the beginning of a line, patch files don\u2019t have this. A special armor header line tells GnuPG about this cleartext signature option.\n(no term)\n\u2013escape-from-lines\n\u2013no-escape-from-lines :: Because some mailers change lines starting with \u201cFrom \u201d to \u201c>From \u201d it is good to handle such lines in a special way when creating cleartext signatures to prevent the mail system from breaking the signature. Note that all other PGP versions do it this way too. Enabled by default. \u2013no-escape-from-lines disables this option.\n\u2013passphrase-repeat n\nSpecify how many times gpg will request a new passphrase be repeated. This is useful for helping memorize a passphrase. Defaults to 1 repetition; can be set to 0 to disable any passphrase repetition. Note that a n greater than 1 will pop up the pinentry window \/n\/+1 times even if a modern pinentry with two entry fields is used.\n\u2013passphrase-fd n\nRead the passphrase from file descriptor n. Only the first line will be read from file descriptor n. If you use 0 for n, the passphrase will be read from STDIN. This can only be used if only one passphrase is supplied.\n\nNote that since Version 2.0 this passphrase is only used if the option \u2013batch has also been given. Since Version 2.1 the \u2013pinentry-mode also needs to be set to loopback.\n\n\u2013passphrase-file file\nRead the passphrase from file file. Only the first line will be read from file file. This can only be used if only one passphrase is supplied. Obviously, a passphrase stored in a file is of questionable security if other users can read this file. Don\u2019t use this option if you can avoid it.\n\nNote that since Version 2.0 this passphrase is only used if the option \u2013batch has also been given. Since Version 2.1 the \u2013pinentry-mode also needs to be set to loopback.\n\n\u2013passphrase string\nUse string as the passphrase. This can only be used if only one passphrase is supplied. Obviously, this is of very questionable security on a multi-user system. Don\u2019t use this option if you can avoid it.\n\nNote that since Version 2.0 this passphrase is only used if the option \u2013batch has also been given. Since Version 2.1 the \u2013pinentry-mode also needs to be set to loopback.\n\n\u2013pinentry-mode mode\nSet the pinentry mode to mode. Allowed values for mode are:\ndefault\nUse the default of the agent, which is ask.\nForce the use of the Pinentry.\ncancel\nEmulate use of Pinentry\u2019s cancel button.\nerror\nReturn a Pinentry error (No Pinentry\u2019\u2019).\nloopback\nRedirect Pinentry queries to the caller. Note that in contrast to Pinentry the user is not prompted again if he enters a bad password.\n\u2013no-symkey-cache\nDisable the passphrase cache used for symmetrical en- and decryption. This cache is based on the message specific salt value (cf. \u2013s2k-mode).\n\u2013request-origin origin\nTell gpg to assume that the operation ultimately originated at origin. Depending on the origin certain restrictions are applied and the Pinentry may include an extra note on the origin. Supported values for origin are: local which is the default, remote to indicate a remote origin or browser for an operation requested by a web browser.\n\u2013command-fd n\nThis is a replacement for the deprecated shared-memory IPC mode. If this option is enabled, user input on questions is not expected from the TTY but from the given file descriptor. It should be used together with \u2013status-fd. See the file doc\/DETAILS in the source distribution for details on how to use it.\n\u2013command-file file\nSame as \u2013command-fd, except the commands are read out of file file\n(no term)\n\u2013allow-non-selfsigned-uid\n\u2013no-allow-non-selfsigned-uid :: Allow the import and use of keys with user IDs which are not self-signed. This is not recommended, as a non self-signed user ID is trivial to forge. \u2013no-allow-non-selfsigned-uid disables.\n\u2013allow-freeform-uid\nDisable all checks on the form of the user ID while generating a new one. This option should only be used in very special environments as it does not ensure the de-facto standard format of user IDs.\n\u2013ignore-time-conflict\nGnuPG normally checks that the timestamps associated with keys and signatures have plausible values. However, sometimes a signature seems to be older than the key due to clock problems. This option makes these checks just a warning. See also \u2013ignore-valid-from for timestamp issues on subkeys.\n\u2013ignore-valid-from\nGnuPG normally does not select and use subkeys created in the future. This option allows the use of such keys and thus exhibits the pre-1.0.7 behaviour. You should not use this option unless there is some clock problem. See also \u2013ignore-time-conflict for timestamp issues with signatures.\n\u2013ignore-crc-error\nThe ASCII armor used by OpenPGP is protected by a CRC checksum against transmission errors. Occasionally the CRC gets mangled somewhere on the transmission channel but the actual content (which is protected by the OpenPGP protocol anyway) is still okay. This option allows GnuPG to ignore CRC errors.\n\u2013ignore-mdc-error\nThis option changes a MDC integrity protection failure into a warning. It is required to decrypt old messages which did not use an MDC. It may also be useful if a message is partially garbled, but it is necessary to get as much data as possible out of that garbled message. Be aware that a missing or failed MDC can be an indication of an attack. Use with great caution; see also option \u2013rfc2440.\n\u2013allow-weak-digest-algos\nSignatures made with known-weak digest algorithms are normally rejected with an invalid digest algorithm\u2019\u2019 message. This option allows the verification of signatures made with such weak algorithms. MD5 is the only digest algorithm considered weak by default. See also \u2013weak-digest to reject other digest algorithms.\n\u2013weak-digest name\nTreat the specified digest algorithm as weak. Signatures made over weak digests algorithms are normally rejected. This option can be supplied multiple times if multiple algorithms should be considered weak. See also \u2013allow-weak-digest-algos to disable rejection of weak digests. MD5 is always considered weak, and does not need to be listed explicitly.\n\u2013allow-weak-key-signatures\nTo avoid a minor risk of collision attacks on third-party key signatures made using SHA-1, those key signatures are considered invalid. This options allows to override this restriction.\n\u2013no-default-keyring\nDo not add the default keyring to the list of keyrings. Note that GnuPG needs for almost all operations a keyring. Thus if you use this option and do not provide alternate keyrings via \u2013keyring, then GnuPG will still use the default keyring.\n\u2013no-keyring\nDo not use any keyring at all. This overrides the default and all options which specify keyrings.\n\u2013skip-verify\nSkip the signature verification step. This may be used to make the decryption faster if the signature verification is not needed.\n\u2013with-key-data\nPrint key listings delimited by colons (like \u2013with-colons) and print the public key data.\n(no term)\n\n\u2013list-signatures\n\u2013list-sigs :: Same as \u2013list-keys, but the signatures are listed too. This command has the same effect as using \u2013list-keys with \u2013with-sig-list. Note that in contrast to \u2013check-signatures the key signatures are not verified. This command can be used to create a list of signing keys missing in the local keyring; for example:\n\n gpg --list-sigs --with-colons USERID | \\\nawk -F: '$1==\"sig\" &&$2==\"?\" {if($13){print$13}else{print $5}}' \u2013fast-list-mode Changes the output of the list commands to work faster; this is achieved by leaving some parts empty. Some applications don\u2019t need the user ID and the trust information given in the listings. By using this options they can get a faster listing. The exact behaviour of this option may change in future versions. If you are missing some information, don\u2019t use this option. \u2013no-literal This is not for normal use. Use the source to see for what it might be useful. \u2013set-filesize This is not for normal use. Use the source to see for what it might be useful. \u2013show-session-key Display the session key used for one message. See \u2013override-session-key for the counterpart of this option. We think that Key Escrow is a Bad Thing; however the user should have the freedom to decide whether to go to prison or to reveal the content of one specific message without compromising all messages ever encrypted for one secret key. You can also use this option if you receive an encrypted message which is abusive or offensive, to prove to the administrators of the messaging system that the ciphertext transmitted corresponds to an inappropriate plaintext so they can take action against the offending user. \u2022 \u2013override-session-key string \u2013override-session-key-fd fd :: Don\u2019t use the public key but the session key string respective the session key taken from the first line read from file descriptor fd. The format of this string is the same as the one printed by \u2013show-session-key. This option is normally not used but comes handy in case someone forces you to reveal the content of an encrypted message; using this option you can do this without handing out the secret key. Note that using \u2013override-session-key may reveal the session key to all local users via the global process table. Often it is useful to combine this option with \u2013no-keyring. \u2022 \u2013ask-sig-expire \u2013no-ask-sig-expire :: When making a data signature, prompt for an expiration time. If this option is not specified, the expiration time set via \u2013default-sig-expire is used. \u2013no-ask-sig-expire disables this option. \u2022 The default expiration time to use for signature expiration. Valid values are \u201c0\u201d for no expiration, a number followed by the letter d (for days), w (for weeks), m (for months), or y (for years) (for example \u201c2m\u201d for two months, or \u201c5y\u201d for five years), or an absolute date in the form YYYY-MM-DD. Defaults to \u201c0\u201d. \u2022 \u2013ask-cert-expire \u2013no-ask-cert-expire :: When making a key signature, prompt for an expiration time. If this option is not specified, the expiration time set via \u2013default-cert-expire is used. \u2013no-ask-cert-expire disables this option. \u2022 The default expiration time to use for key signature expiration. Valid values are \u201c0\u201d for no expiration, a number followed by the letter d (for days), w (for weeks), m (for months), or y (for years) (for example \u201c2m\u201d for two months, or \u201c5y\u201d for five years), or an absolute date in the form YYYY-MM-DD. Defaults to \u201c0\u201d. \u2022 This option can be used to change the default algorithms for key generation. The string is similar to the arguments required for the command \u2013quick-add-key but slightly different. For example the current default of \u201crsa2048\/cert,sign+rsa2048\/encr\u201d (or \u201crsa3072\u201d) can be changed to the value of what we currently call future default, which is \u201ced25519\/cert,sign+cv25519\/encr\u201d. You need to consult the source code to learn the details. Note that the advanced key generation commands can always be used to specify a key algorithm directly. \u2022 This option modifies the behaviour of the commands \u2013quick-sign-key, \u2013quick-lsign-key, and the \u201csign\u201d sub-commands of \u2013edit-key by forcing the creation of a key signature, even if one already exists. \u2022 This is an obsolete option and is not used anywhere. \u2022 Allow processing of multiple OpenPGP messages contained in a single file or stream. Some programs that call GPG are not prepared to deal with multiple messages being processed together, so this option defaults to no. Note that versions of GPG prior to 1.4.7 always allowed multiple messages. Future versions of GnUPG will remove this option. Warning: Do not use this option unless you need it as a temporary workaround! \u2013enable-special-filenames This option enables a mode in which filenames of the form \u2018\/-&n\/\u2019, where n is a non-negative decimal number, refer to the file descriptor n and not to a file with that name. \u2013no-expensive-trust-checks Experimental use only. \u2013preserve-permissions Don\u2019t change the permissions of a secret keyring back to user read\/write only. Use this option only if you really know what you are doing. \u2013default-preference-list string Set the list of default preferences to string. This preference list is used for new keys and becomes the default for \u201csetpref\u201d in the edit menu. \u2013default-keyserver-url name Set the default keyserver URL to name. This keyserver will be used as the keyserver URL when writing a new self-signature on a key, which includes key generation and changing preferences. \u2013list-config Display various internal configuration parameters of GnuPG. This option is intended for external programs that call GnuPG to perform tasks, and is thus not generally useful. See the file \u2018\/doc\/DETAILS\/\u2019 in the source distribution for the details of which configuration items may be listed. \u2013list-config is only usable with \u2013with-colons set. \u2013list-gcrypt-config Display various internal configuration parameters of Libgcrypt. \u2013gpgconf-list This command is similar to \u2013list-config but in general only internally used by the gpgconf tool. \u2013gpgconf-test This is more or less dummy action. However it parses the configuration file and returns with failure if the configuration file would prevent gpg from startup. Thus it may be used to run a syntax check on the configuration file. ### Deprecated options \u2022 \u2013show-photos \u2013no-show-photos :: Causes \u2013list-keys, \u2013list-signatures, \u2013list-public-keys, \u2013list-secret-keys, and verifying a signature to also display the photo ID attached to the key, if any. See also \u2013photo-viewer. These options are deprecated. Use \u2013list-options [no-]show-photos and\/or \u2013verify-options [no-]show-photos instead. \u2022 Display the keyring name at the head of key listings to show which keyring a given key resides on. This option is deprecated: use \u2013list-options [no-]show-keyring instead. \u2022 Identical to \u2013trust-model always. This option is deprecated. \u2022 \u2013show-notation \u2013no-show-notation :: Show signature notations in the \u2013list-signatures or \u2013check-signatures listings as well as when verifying a signature with a notation in it. These options are deprecated. Use \u2013list-options [no-]show-notation and\/or \u2013verify-options [no-]show-notation instead. \u2022 \u2013show-policy-url \u2013no-show-policy-url :: Show policy URLs in the \u2013list-signatures or \u2013check-signatures listings as well as when verifying a signature with a policy URL in it. These options are deprecated. Use \u2013list-options [no-]show-policy-url and\/or \u2013verify-options [no-]show-policy-url instead. ## EXAMPLES gpg -se -r *Bob file* sign and encrypt for user Bob gpg \u2013clear-sign file make a cleartext signature gpg -sb file make a detached signature gpg -u 0x12345678 -sb file make a detached signature with the key 0x12345678 gpg \u2013list-keys user_ID show keys gpg \u2013fingerprint user_ID show fingerprint (no term) gpg \u2013verify pgpfile gpg \u2013verify *sigfile [*datafile*]* :: Verify the signature of the file but do not output the data unless requested. The second form is used for detached signatures, where sigfile is the detached signature (either ASCII armored or binary) and datafile are the signed data; if this is not given, the name of the file holding the signed data is constructed by cutting off the extension (\u201c.asc\u201d or \u201c.sig\u201d) of sigfile or by asking the user for the filename. If the option \u2013output is also used the signed data is written to the file specified by that option; use - to write the signed data to stdout. ## HOW TO SPECIFY A USER ID There are different ways to specify a user ID to GnuPG. Some of them are only valid for gpg others are only good for gpgsm. Here is the entire list of ways to specify a key: By key Id. This format is deduced from the length of the string and its content or 0x prefix. The key Id of an X.509 certificate are the low 64 bits of its SHA-1 fingerprint. The use of key Ids is just a shortcut, for all automated processing the fingerprint should be used. When using gpg an exclamation mark (!) may be appended to force using the specified primary or secondary key and not to try and calculate which primary or secondary key to use. The last four lines of the example give the key ID in their long form as internally used by the OpenPGP protocol. You can see the long key ID using the option \u2013with-colons. 234567C4 0F34E556E 01347A56A 0xAB123456 234AABBCC34567C4 0F323456784E56EAB 01AB3FED1347A5612 0x234AABBCC34567C4 By fingerprint. This format is deduced from the length of the string and its content or the 0x prefix. Note, that only the 20 byte version fingerprint is available with gpgsm (i.e. the SHA-1 hash of the certificate). When using gpg an exclamation mark (!) may be appended to force using the specified primary or secondary key and not to try and calculate which primary or secondary key to use. The best way to specify a key Id is by using the fingerprint. This avoids any ambiguities in case that there are duplicated key IDs. 1234343434343434C434343434343434 123434343434343C3434343434343734349A3434 0E12343434343434343434EAB3484343434343434 0xE12343434343434343434EAB3484343434343434 gpgsm also accepts colons between each pair of hexadecimal digits because this is the de-facto standard on how to present X.509 fingerprints. gpg also allows the use of the space separated SHA-1 fingerprint as printed by the key listing commands. By exact match on OpenPGP user ID. This is denoted by a leading equal sign. It does not make sense for X.509 certificates. =Heinrich Heine <heinrichh@uni-duesseldorf.de> By exact match on an email address. This is indicated by enclosing the email address in the usual way with left and right angles. <heinrichh@uni-duesseldorf.de> By partial match on an email address. This is indicated by prefixing the search string with an @. This uses a substring search but considers only the mail address (i.e. inside the angle brackets). @heinrichh By exact match on the subject\u2019s DN. This is indicated by a leading slash, directly followed by the RFC-2253 encoded DN of the subject. Note that you can\u2019t use the string printed by gpgsm \u2013list-keys because that one has been reordered and modified for better readability; use \u2013with-colons to print the raw (but standard escaped) RFC-2253 string. \/CN=Heinrich Heine,O=Poets,L=Paris,C=FR By exact match on the issuer\u2019s DN. This is indicated by a leading hash mark, directly followed by a slash and then directly followed by the RFC-2253 encoded DN of the issuer. This should return the Root cert of the issuer. See note above. #\/CN=Root Cert,O=Poets,L=Paris,C=FR By exact match on serial number and issuer\u2019s DN. This is indicated by a hash mark, followed by the hexadecimal representation of the serial number, then followed by a slash and the RFC-2253 encoded DN of the issuer. See note above. #4F03\/CN=Root Cert,O=Poets,L=Paris,C=FR By keygrip. This is indicated by an ampersand followed by the 40 hex digits of a keygrip. gpgsm prints the keygrip when using the command \u2013dump-cert. &D75F22C3F86E355877348498CDC92BD21010A480 By substring match. This is the default mode but applications may want to explicitly indicate this by putting the asterisk in front. Match is not case sensitive. Heine *Heine . and + prefixes These prefixes are reserved for looking up mails anchored at the end and for a word search mode. They are not yet implemented and using them is undefined. Please note that we have reused the hash mark identifier which was used in old GnuPG versions to indicate the so called local-id. It is not anymore used and there should be no conflict when used with X.509 stuff. Using the RFC-2253 format of DNs has the drawback that it is not possible to map them back to the original encoding, however we don\u2019t have to do this because our key database stores this encoding as meta data. ## FILTER EXPRESSIONS The options \u2013import-filter and \u2013export-filter use expressions with this syntax (square brackets indicate an optional part and curly braces a repetition, white space between the elements are allowed): #+begin_quote [lc] {[{flag}] PROPNAME op VALUE [lc]} #+end_quote The name of a property (PROPNAME) may only consist of letters, digits and underscores. The description for the filter type describes which properties are defined. If an undefined property is used it evaluates to the empty string. Unless otherwise noted, the VALUE must always be given and may not be the empty string. No quoting is defined for the value, thus the value may not contain the strings && or ||, which are used as logical connection operators. The flag -- can be used to remove this restriction. Numerical values are computed as long int; standard C notation applies. lc is the logical connection operator; either && for a conjunction or || for a disjunction. A conjunction is assumed at the begin of an expression. Conjunctions have higher precedence than disjunctions. If VALUE starts with one of the characters used in any op a space after the op is required. The supported operators (op) are: =~ Substring must match. !~ Substring must not match. = The full string must match. <> The full string must not match. == The numerical value must match. != The numerical value must not match. <= The numerical value of the field must be LE than the value. < The numerical value of the field must be LT than the value. > The numerical value of the field must be GT than the value. >= The numerical value of the field must be GE than the value. -le The string value of the field must be less or equal than the value. -lt The string value of the field must be less than the value. -gt The string value of the field must be greater than the value. -ge The string value of the field must be greater or equal than the value. -n True if value is not empty (no value allowed). -z True if value is empty (no value allowed). -t Alias for \u201cPROPNAME != 0\u201d (no value allowed). -f Alias for \u201cPROPNAME == 0\u201d (no value allowed). Values for flag must be space separated. The supported flags are: -- VALUE spans to the end of the expression. -c The string match in this part is done case-sensitive. -t Leading and trailing spaces are not removed from VALUE. The optional single space after op is here required. The filter options concatenate several specifications for a filter of the same type. For example the four options in this example: #+begin_quote --import-filter keep-uid=\"uid =~ Alfa\" --import-filter keep-uid=\"&& uid !~ Test\" --import-filter keep-uid=\"|| uid =~ Alpha\" --import-filter keep-uid=\"uid !~ Test\" #+end_quote which is equivalent to #+begin_quote --import-filter \\ keep-uid=\"uid =~ Alfa\" && uid !~ Test\" || uid =~ Alpha\" && \"uid !~ Test\" #+end_quote imports only the user ids of a key containing the strings \u201cAlfa\u201d or \u201cAlpha\u201d but not the string \u201ctest\u201d. ## TRUST VALUES Trust values are used to indicate ownertrust and validity of keys and user IDs. They are displayed with letters or strings: \u2022 - unknown :: No ownertrust assigned \/ not yet calculated. \u2022 e expired :: Trust calculation has failed; probably due to an expired key. \u2022 q undefined, undef :: Not enough information for calculation. \u2022 n never :: Never trust this key. \u2022 m marginal :: Marginally trusted. \u2022 f full :: Fully trusted. \u2022 u ultimate :: Ultimately trusted. \u2022 r revoked :: For validity only: the key or the user ID has been revoked. \u2022 ? err :: The program encountered an unknown trust value. ## FILES There are a few configuration files to control certain aspects of gpg\u2019s operation. Unless noted, they are expected in the current home directory (see: [option \u2013homedir]). gpg.conf This is the standard configuration file read by gpg on startup. It may contain any valid long option; the leading two dashes may not be entered and the option may not be abbreviated. This default name may be changed on the command line (see: [gpg-option \u2013options]). You should backup this file. Note that on larger installations, it is useful to put predefined files into the directory \u2018\/\/etc\/skel\/.gnupg\/\u2019 so that newly created users start up with a working configuration. For existing users a small helper script is provided to create these files (see: [addgnupghome]). For internal purposes gpg creates and maintains a few other files; They all live in the current home directory (see: [option \u2013homedir]). Only the gpg program may modify these files. ~\/.gnupg This is the default home directory which is used if neither the environment variable GNUPGHOME nor the option \u2013homedir is given. ~\/.gnupg\/pubring.gpg The public keyring using a legacy format. You should backup this file. If this file is not available, gpg defaults to the new keybox format and creates a file \u2018\/pubring.kbx\/\u2019 unless that file already exists in which case that file will also be used for OpenPGP keys. Note that in the case that both files, \u2018\/pubring.gpg\/\u2019 and \u2018\/pubring.kbx\/\u2019 exists but the latter has no OpenPGP keys, the legacy file \u2018\/pubring.gpg\/\u2019 will be used. Take care: GnuPG versions before 2.1 will always use the file \u2018\/pubring.gpg\/\u2019 because they do not know about the new keybox format. In the case that you have to use GnuPG 1.4 to decrypt archived data you should keep this file. ~\/.gnupg\/pubring.gpg.lock The lock file for the public keyring. ~\/.gnupg\/pubring.kbx The public keyring using the new keybox format. This file is shared with gpgsm. You should backup this file. See above for the relation between this file and it predecessor. To convert an existing \u2018\/pubring.gpg\/\u2019 file to the keybox format, you first backup the ownertrust values, then rename \u2018\/pubring.gpg\/\u2019 to \u2018\/publickeys.backup\/\u2019, so it won\u2019t be recognized by any GnuPG version, run import, and finally restore the ownertrust values: $ cd ~\/.gnupg\n$gpg --export-ownertrust >otrust.lst$ mv pubring.gpg publickeys.backup\n$gpg --import-options restore --import publickeys.backups$ gpg --import-ownertrust otrust.lst\n\n~\/.gnupg\/pubring.kbx.lock\nThe lock file for \u2018\/pubring.kbx\/\u2019.\n~\/.gnupg\/secring.gpg\nThe legacy secret keyring as used by GnuPG versions before 2.1. It is not used by GnuPG 2.1 and later. You may want to keep it in case you have to use GnuPG 1.4 to decrypt archived data.\n~\/.gnupg\/secring.gpg.lock\nThe lock file for the legacy secret keyring.\n~\/.gnupg\/.gpg-v21-migrated\nFile indicating that a migration to GnuPG 2.1 has been done.\n~\/.gnupg\/trustdb.gpg\nThe trust database. There is no need to backup this file; it is better to backup the ownertrust values (see: [option \u2013export-ownertrust]).\n~\/.gnupg\/trustdb.gpg.lock\nThe lock file for the trust database.\n~\/.gnupg\/random_seed\nA file used to preserve the state of the internal random pool.\n~\/.gnupg\/openpgp-revocs.d\/\nThis is the directory where gpg stores pre-generated revocation certificates. The file name corresponds to the OpenPGP fingerprint of the respective key. It is suggested to backup those certificates and if the primary private key is not stored on the disk to move them to an external storage device. Anyone who can access theses files is able to revoke the corresponding key. You may want to print them out. You should backup all files in this directory and take care to keep this backup closed away.\n\nOperation is further controlled by a few environment variables:\n\nHOME\nUsed to locate the default home directory.\nGNUPGHOME\nIf set directory used instead of \u201c~\/.gnupg\u201d.\nGPG_AGENT_INFO\nThis variable is obsolete; it was used by GnuPG versions before 2.1.\nPINENTRY_USER_DATA\nThis value is passed via gpg-agent to pinentry. It is useful to convey extra information to a custom pinentry.\n(no term)\nCOLUMNS\nLINES :: Used to size some displays to the full size of the screen.\nLANGUAGE\nApart from its use by GNU, it is used in the W32 version to override the language selection done through the Registry. If used and set to a valid and available language name (langid), the file with the translation is loaded from gpgdir\/\/gnupg.nls\/\/langid.mo. Here gpgdir is the directory out of which the gpg binary has been loaded. If it can\u2019t be loaded the Registry is tried and as last resort the native Windows locale system is used.\n\nWhen calling the gpg-agent component gpg sends a set of environment variables to gpg-agent. The names of these variables can be listed using the command:\n\n gpg-connect-agent 'getinfo std_env_names' \/bye | awk '$1==\"D\" {print$2}'\n\n\n## BUGS\n\nOn older systems this program should be installed as setuid(root). This is necessary to lock memory pages. Locking memory pages prevents the operating system from writing memory pages (which may contain passphrases or other sensitive material) to disk. If you get no warning message about insecure memory your operating system supports locking without being root. The program drops root privileges as soon as locked memory is allocated.\n\nNote also that some systems (especially laptops) have the ability to suspend to disk\u2019\u2019 (also known as safe sleep\u2019\u2019 or hibernate\u2019\u2019). 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La saison 2018-2019 du Bayern Munich est la saison sans interruption en Bundesliga, avec 28 titres de champion d'Allemagne et 18 Coupes d'Allemagne c'est le club le plus titré en Allemagne, ce qui lui vaut le surnom de Rekordmeister. Le Bayern Munich démarre cette saison avec Niko Kovač comme nouvel entraîneur.
Préparation d'avant-saison et matchs amicaux
Transferts
Transferts estivaux
Transfert hivernaux
Compétitions
Supercoupe d'Allemagne
Sur le terrain de l'Eintracht Francfort, le Bayern Munich a pu compter sur un grand Robert Lewandowski pour décrocher dimanche la septième Supercoupe d'Allemagne de son histoire (5-0) en écrasant l'Eintracht.
Championnat
Journées 1 à 5
Le Bayern Munich ouvre officiellement la Bundesliga le à l'Allianz Arena face au TSG Hoffenheim. Il s'impose sur le score de 3-1 mais perd Kingsley Coman après une déchirure ligamentaire. Le septembre, le Bayern affronte le VfB Stuttgart, la recrue Leon Goretzka est titulaire et inscrit son premier but, le Bayern s'impose 3-0 et prend la tête du championnat. Le match suivant, le Bayern reçoit le Bayer Leverkusen, une nouvelle fois, le Bayern met trois buts à son adversaire comme c'est le cas depuis le début de la saison en Bundesliga, le Bayern s'impose 3-1 mais perd deux joueurs importants, Corentin Tolisso qui subit une rupture des ligaments croisés et une déchirure du ménisque et sera indisponible pour 6 mois minimum ainsi que Rafinha moins gravement touché. Quatrième match de Bundesliga et le Bayern affronte son dauphin de la saison passée qui vit une période difficile, le Bayern s'impose sur le score de 2-0 et enfonce le club de Gelsenkirchen. Le cinquième match ouvre une période particulièrement importante pour le Bayern puisqu'il s'agit de la fête de la bière période durant laquelle le Bayern ne perd plus depuis quelques années mais laisse échapper les premiers points de la saison à cause d'un but de Felix Götze pour Augsburg, les deux équipes se quittent sur le score de 1-1.
Journées 6 à 10
Lors de la sixième journée, le Bayern affronte le Hertha Berlin dans un Olympiastadion presque complet et s'incline pour la première fois de la saison . La septième journée arrive, le Bayern doit réagir compte tenu de sa défaite précédente, mais il n'en est rien, le Bayern s'incline de nouveau, sur sa pelouse, face à Mönchengladbach . Lors de la huitième journée, le Bayern se relance grâce à une victoire encourageante face aux loups de Wolfsbourg, et ce malgré un carton rouge d'Arjen Robben. Sorti de la crise, le Bayern est allé s'imposer sur la pelouse de Mayence lors de la neuvième journée, s'installant provisoirement à la place de Bundesliga. Lors de la dixième journée, le Bayern cale à domicile et laisse filer le Borussia Dortmund. Surpris par un but à deux minutes de la fin du temps réglementaire, le Bayern a concédé un décevant nul à domicile face à Fribourg. À une semaine du choc face au Borussia Dortmund, il y avait assurément meilleur moyen de se rassurer pour le Bayern de Munich que de partager les points avec Fribourg.
Journées 11 à 15
Klassiker oblige, le Bayern doit s'imposer, ces derniers ont eu deux fois l'avantage à la marque, mais ils sortent finalement bredouille de cette confrontation. Après avoir passé la trêve internationale à ruminer sa défaite contre le Borussia Dortmund, le Bayern recevait le Fortuna Düsseldorf avec comme seul mot d'ordre : gagner, mais, le Bayern a encore fauté ce samedi en partageant les points avec Düsseldorf à domicile. Lors de la treizième, Serge Gnabry, double buteur face à son ancienne équipe du Werder, a permis au Bayern de reprendre du poil de la bête en Bundesliga. Quatrième de Bundesliga avec 24 points au compteur, le Bayern Munich recevait Nuremberg à l'Allianz Arena. Un adversaire largement à sa portée car au classement. De ce fait, les Bavarois se devaient de l'emporter, avec la manière si possible, pour confirmer leur redressement après deux succès de rang contre Benfica en Ligue des champions, puis à Brême en Bundesliga. Le Bayern Munich n'avait pas le droit à l'erreur du côté d'Hanovre, cette pression a porté les hommes de Niko Kovac puisqu'il n'y a tout simplement pas eu de débats ce samedi après-midi, avec des réalisations d'Alaba, Kimmich ou encore Lewandowski, les Bavarois reviennent provisoirement à six points du leader, le Borussia Dortmund.
Journées 16 et 17
Lors de la seizième journée, le Bayern s'impose 1-0 face à Leipzig au terme d'un match compliqué et très disputé, grâce à une réalisation de Franck Ribéry. Les Bavarois rejoignent Mönchengladbach à la deuxième place, et à six points de Dortmund. Grâce à un Franck Ribéry en forme, le Bayern accroche un cinquième succès consécutif en championnat sur le terrain de l'Eintracht Francfort (0-3). Les Bavarois reprennent la deuxième place à Mönchengladbach et restent à 6 points de Dortmund. Lors de la journée, le Bayern Munich met un terme à sa première partie de saison en Bundesliga.
Journées 18 à 22
Le Bayern Munich débute sa deuxième partie de saison lors de la journée de Bundesliga. Le contrat a bien été rempli puisque le champion en titre s'est imposé chez le TSG avec notamment un doublé de Leon Goretzka. Souvent discret depuis son arrivée en Bavière, l'international allemand a lancé les siens sur le chemin de la victoire. Le Bayern Munich a ensuite enchaîné un septième succès de rang en championnat en dominant largement Stuttgart à l'Allianz Arena pour le compte de la journée. Il reste à 6 points du leader. Après sept victoires d'affilée, les Munichois sont retombés dans leur travers et ont chuté sur le terrain de Leverkusen. Le Bayern de Munich avait l'obligation de se racheter lors de la réception de Schalke 04. Les champions d'Allemagne n'ont pas manqué à leur tâche. Face à la formation de Gelsenkirchen, ils ont retrouvé le jeu qui fait leur force et le succès récolté était amplement mérité. Un succès qui leur permet de revenir à cinq points du leader. Le Borussia Dortmund a beau lui laisser quelques opportunités, le Bayern Munich apparaît toujours fragile dans la course au titre en Bundesliga. Sur la pelouse d'Augsbourg, les hommes de Niko Kovac ont livré une prestation collective très inégale, passant la majeure partie de la soirée à courir après le score pour ne pas laisser échapper un précieux succès.
Journées 23 à 27
Pour au moins 24 heures, le Bayern de Munich se retrouve enfin à la hauteur du Borussia Dortmund au classement de la Bundesliga. Les Bavarois ont réduit l'écart qu'ils avaient sur la formation de la Ruhr en prenant la mesure du Hertha Berlin. La victoire face à l'équipe de la capitale a été cependant poussive, et il y avait encore moins de raisons de s'extasier à la suite de la sortie sur blessure de Kingsley Coman. Le Bayern a frappé très fort lors d'un déplacement à Monchengladbach en symposant largement sur le score de 1-5 ce qui leur permet de revenir à auteur de Dortmund. Après 5 mois d'attente, le Bayern prend la tête de la Bundesliga, face à la meilleure équipe à l'extérieur, les Bavarois ont réussi une démonstration de force face à Wolfsburg, de plus la différence de but est en faveur des bavarois, un très bon match pour la préparation du match contre Liverpool en Ligue des champions. Contre Mayence, le bayern récidive et atomise les joueurs de Sandro Schwarz, et garde sa place de leader devant Dormund grâce à la différence de but. Hélas lors de la vingt-septième journée le Bayern pert deux point important et sa place de leader au passage à cause d'un match nul face au SC Fribourg. Si le bayern veut récupéré sa place de leader cela se fera lors du Klassiker.
Journées 28 à 32
Le classique du championnat allemand a tourné à une véritable correction. Le Bayern n'a fait qu'une bouchée de son grand rival de Dortmund. Avec cette victoire, le Bayern repasse devant son adversaire du jour au classement. À six journées de l'épilogue de la campagne, l'équipe de Niko Kovac semble avoir pris les rênes au meilleur moment. Mais le FC Hollywood se fait vite rattrapé puisqu'une bagarre entre Robert Lewandowski et Kingsley Coman intervient à l'entrainement le jeudi suivant la victoire contre Dortmund. Avec un nouveau carton et un nouveau retour sur le trône, le Bayern Munich a vécu un après-midi quasi parfait face à Düsseldorf, tout juste ternie par la sortie sur blessure de son gardien Manuel Neuer. Il y a des succès qui, même obtenus au forceps, sont plus précieux que d'autres, et celui récolté lors de la trentième journée par le Bayern aux dépens du Werder, dans un classique du championnat allemand, en fait assurément partie. Sans inspiration, les Bavarois ont été tenus en échec sur la pelouse de Nuremberg. Avec seulement deux points d'avance sur Dortmund au coup d'envoi, le Bayern se devait de l'emporter. C'est chose faite grâce à une victoire acquise à l'Allianz Arena lors de la journée de Bundesliga, face à Hanovre, lanterne rouge au classement.
Journées 33 et 34
Le Bayern va devoir attendre avant de fêter un éventuel nouveau titre. Les Bavarois ont été tenus en échec par Leipzig. Une rencontre durant laquelle les joueurs de Niko Kovač ont joué sur un courant alternatif face à un adversaire qui n'a pas démérité. Le jour temps décisif est arrivé, pour le derniers matchs de Robben, Ribéry et Rafinha, le bayern remporte le titre en s'imposant 5 but à 1 face à Francfort, de quoi finir en beauté. Lors de la journée, le Bayern Munich met un terme à sa saison en Bundesliga.
Classement
Évolution du classement et des résultats
Ligue des champions
La Ligue des champions 2018-2019 est la de la Ligue des champions, la plus prestigieuse des compétitions européennes inter-clubs. Elle est divisée en deux phases, une phase de groupes, qui consiste en huit mini-championnats de quatre équipes par groupe, les deux premiers poursuivant la compétition et le troisième étant repêché en seizièmes de finale de la Ligue Europa.
Parcours en Ligue des champions
Solide et redoutable en attaque, le Bayern s'impose très logiquement à Lisbonne, avec un Renato Sanches à son meilleur niveau pour son retour au Estadio da Luz. À la peine collectivement et défensivement, le Bayern Munich, malgré le but rapide d'Hummels, s'est fait très peur face à une séduisante équipe de l'Ajax. Tenu en échec à domicile par l'Ajax lors de la précédente journée, le Bayern connaît une saison qui est loin de ressembler au long fleuve tranquille de performances probantes auxquelles le club munichois nous avait habitués. Co-leaders du groupe E avec l'Ajax Amsterdam, les pensionnaires de l'Allianz Arena voulaient faire le plein et éviter le match piège. Kovac avait donc aligné un onze dans un système en 4-3-3 avec un trio offensif Robben, Gnabry et Lewandowski. Touché aux vertèbres, Franck Ribéry était forfait. En face, Marinos Ouzounidis faisait évoluer les Grecs en 4-2-3-1 avec Ponce en pointe. Dans une chaude ambiance, les Allemands prenaient d'entrée les commandes de la rencontre, n'hésitant pas à mettre le pied sur le ballon.
Phase de groupes
Classement et résultats du groupe E de la Ligue des champions 2018-2019
Phase finale
Coefficient UEFA
Coupe d'Allemagne
Le Bayern Munich s'est qualifié pour le deuxième tour de la DFB Pokal après sa victoire 1 à 0 face au SV Drochtersen/Assel, ce qui a presque fini par tourné à la peau de banane pour les champions de Bundesliga. Bien qu'ils aient obtenu la majorité du ballon comme prévu, leur qualité dans le dernier tiers était catastrophique compte tenu de l'équipe qu'ils ont formée. Le Bayern Munich a atteint les huitièmes de finale de la DFB-Pokal sans aucune splendeur. Le finaliste de l'année dernière a gagné à Osnabrück contre un club de ligue régionale, SV Rödinghausen, 2-1. Le recordman a gagné au deuxième tour de la DFB-Pokal au Hertha Berlin malgré une erreur défensive après et s'est qualifié pour la douzième fois de suite dans en huitièmes de finale. Le rêve d'un nouveau doublé national reste de mise pour le Bayern de Munich. Toutefois, il ne s'en est pas fallu de beaucoup ce mercredi pour que l'équipe bavaroise passe à la trappe en DFB-Pokal. Et c'eut été une grosse humiliation puisque les hommes de Niko Kovac se produisaient contre le de la Bundesliga 2. Le Bayern Munich se qualifie pour la finale après sa victoire sur le Werder Brême. Les Bavarois ont contrôlé le match pendant plus d'une heure avant cinq minutes de folie qui ont vu le Werder égaliser avant que Coman n'obtienne un penalty décisif. En finale, ils affronteront Leipzig. Après un début de saison mitigé qui augurait d'une saison compliquée pour les Bavarois, ces derniers se sont adjugés de la Coupe d'Allemagne de haute lutte en finale face au RB Leipzig, au grand bonheur de Frank Ribéry qui va donc s'offrir une dernière saison en beauté en Bavière.
Matchs officiels de la saison
Joueurs et encadrement technique
Encadrement technique
Effectif professionnel
Le tableau suivant liste l'effectif professionnel du Bayern Munich pour la saison 2018-2019.|-
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En grisé, les sélections de joueurs internationaux chez les jeunes mais n'ayant jamais été appelés aux échelons supérieur une fois l'âge limite dépassé.
Joueurs prêtés
Statistiques
Statistiques collectives
Statistiques individuelles
(Mis à jour le )
Onze de départ (toutes compétitions)
Récompenses et distinctions
Affluence
Affluence du Bayern Munich à domicile
Équipe réserve et Centre de formation
Équipe Réserve
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U19
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U17
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Notes et références
Saison du Bayern Munich
Munich | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,954 |
\section{Introduction}
\label{intro}
\begin{figure}[htbp!]
\includegraphics[width=0.5\textwidth, trim = 5mm 0mm 10mm 10mm, clip]{Log_constraints_axions_s4.pdf}
\caption{\textbf{Projected CMB-S4 sensitivity to the axion energy density as a function of axion mass, compared with Fisher-matrix {\sl Planck} sensitivity}: Vertical bars show $1\sigma$ errors at fixed neutrino mass $\Sigma m_\nu = 0.06~\mathrm{eV}$ while the shaded bars show the errors marginalizing over $\Sigma m_{\nu}$. We classify axions as DE-like if $m_a < 10^{-29}\,\mathrm{eV}$, `DM-like' if $m_a > 10^{-25} \mathrm{eV}$ and `fuzzy' DM for masses in between. In the `fuzzy' DM region, CMB-S4 will allow for percent-level sensitivity to the axion mass fraction, improving significantly on current constraints. For {\sl Planck} data alone, neutrino degeneracies significantly degrade sensitivity to axions, even at the 1$\sigma$ level. In contrast, CMB-S4 constraints remain robust to varying neutrino mass in the `fuzzy' region. The solid and dashed lines show the $2\sigma$ and $1\sigma$ exclusion limits, i.e. the lowest axion fraction that could be excluded at those masses. \label{fig:axions}}
\end{figure}
Identifying dark matter (DM) remains one of the outstanding cosmological challenges of the current age.
While searches for direct or indirect evidence of a dark matter candidate continue \cite{Buckley:2013bha,Cushman:2013zza}, the effect of dark matter on cosmological observables provides a complementary approach to constraining the dark sector.
In the face of increasingly strict experimental limits to Weakly Interacting Massive Particle (WIMP) DM, axions are re-emerging as a popular alternative (see Ref.~\cite{Marsh:2015xka} for an extensive review of axions). Cosmological axion production can proceed through decays of exotic particles (e.g. moduli) or topological defects, thermal production from the standard-model plasma, or coherent oscillation around a misaligned (from the vacuum state) initial value, known as vacuum realignment. If axions are also produced because of non-vanishing matter couplings, a relativistic population can be produced, contributing to the relativistic energy density in the early universe (parameterized by a generic parameter $N_\mathrm{eff}$, describing the number of \textit{relativistic degrees of freedom}). Constraints on these axion models were presented in Refs. \cite{Acharya:2010zx,Weinberg:2013kea,Iliesiu:2013rqa,Baumann:2016wac}.
Vacuum realignment is the only axion production mechanism that occurs independent of assumptions about axion couplings or inflationary physics, and produces an extremely cold population of axions, in contrast with other mechanisms. Here, we consider only axions produced by vacuum realignment.\footnote{We do vary $N_\mathrm{eff}-3.046$, but without bias as to its physical nature.} Ultralight axions (ULAs) produced via vacuum realignment with masses in the range $10^{-33}~\mathrm{eV}\leq m_{a}\leq 10^{-20}~\mathrm{eV}$ are well motivated by string theory, and can contribute to either the dark matter or dark energy components of the Universe, depending on their masses \cite{Marsh:2015xka}.
They are distinguishable from standard dark energy (DE) and cold dark matter (CDM) using cosmological observables such as the cosmic microwave background (CMB) temperature and polarization power spectra, the matter power spectrum (as probed using the correlations of galaxy positions and shapes) and the weak gravitational lensing of the CMB. Constraints on the allowed contribution of ULAs to the total DM component using these observables provide a test of the CDM scenario.
A key goal of future cosmological experiments is to measure the sum of the neutrino masses, $\Sigma m_\nu$ (see Ref.~\cite{Lesgourgues:2012uu} for a review of neutrino cosmology). The current bound on $\Sigma m_\nu$ from ground-based oscillation experiments is $\Sigma m_\nu \gtrsim 0.06~\mathrm{eV}$ \cite{Otten:2008zz}. Current cosmological neutrino bounds indicate that $\Sigma m_\nu < 0.23~\mathrm{eV}$ at 95\% confidence, using data from Planck \cite{Ade:2015xua} and measurements of Baryon Acoustic Oscillations (BAO) from the Baryon Oscillation Spectroscopic Survey \cite[BOSS,][]{Beutler:2014yhv}.
Forecasted constraints for neutrino masses are that $\sigma(\Sigma m_\nu)= 15~\mathrm{meV}$ for a fiducial model with $\Sigma m_\nu = 60~\mathrm{meV}$, for a CMB-S4-like experiment and BAO measurements from a `DESI-like' survey \cite{Abazajian:2013oma}, promising a 4$\sigma$ detection of neutrino mass \cite{Allison:2015qca}. Much of this improvement is driven by weak gravitational lensing of the CMB, in particular at high multipoles $\ell\gtrsim 1000,$ although the change in the lensing convergence power is of order 25\% even at low multipoles. The lensing deflection power-spectrum is determined from $4$-pt functions of CMB maps, extracting a factor of $\sim\sqrt{3}$ as much information from CMB experiments~\cite{Scott:2016fad}.
The promise of CMB experiments in probing neutrino masses motivates us to wonder: will future CMB experiments offer dramatic improvements in sensitivity to axion parameters? Given the known similarity of ULA and massive neutrino imprints \cite{Marsh:2011bf} on cosmological observables at low mass ($m_{a}\mathrel{\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}} 10^{-29}~{\rm eV}$), how significant are ULA-neutrino degeneracies at CMB-S4 sensitivity levels and will they degrade our ability to do fundamental physics with the CMB? To answer these questions, we conduct a Fisher-matrix analysis to explore the sensitivity of future CMB experiments to ULA masses, densities, and $\Sigma m_{\nu}$. We find that CMB-S4 will allow a $2-5\sigma$ detection of axion mass fractions that agree with pure {\sl Planck} limits, covering an axion mass range of $10^{-32}~{\rm eV}\mathrel{\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}} m_{a}\lsim10^{-24}~{\rm eV}$.
Near the top of this range, CMB-S4 will break the degeneracy of axions and CDM. Sensitivity persists (but tapers off) towards higher axion masses of $m_{a}\sim 10^{-23}~{\rm eV}$. CMB-S4 will push CMB tests of the ULA hypothesis towards the mass range probed by subtle observables, like the size of DM-halo cores and the number of missing Milky-Way satellites. In the ``dark-energy-like" (``DE-like" ULAs henceforth) ULA regime ($m_{a}\mathrel{\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}} 10^{-29}~{\rm eV}$) we find that the the ULA mass fraction is degraded by degeneracies with the sum of the neutrino masses, but that this degeneracy disappears at higher masses. We find also that future measurements of the Hubble constant could break this degeneracy. We denote ULAs in the mass range $10^{-29}~{\rm eV}\mathrel{\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}} m_{a}\mathrel{\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}} 10^{-25}~{\rm eV}$ as ``fuzzy DM", and those with $m_{a}\mathrel{\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em \raise.4ex \hbox{$>$}}} 10^{-25}~{\rm eV}$ as ``dark-matter-like" (or DM-like).
We find that measurements of the lensing-convergence power spectrum $C_{\ell}^{\kappa \kappa}$ drive much of the improvement in sensitivity; if lensing is omitted, the fractional error bar on the axion mass fraction degrades by a factor of $\sim 3-5$ in the `fuzzy' regime. Finally, we explore the dependence of our results on CMB-S4's experimental design parameters.
We begin this paper by summarizing the physics and cosmology of ULAs and neutrinos in Section~\ref{sec:axion_cosmo}. In Section~\ref{observables}, we discuss the effects of ULAs and neutrinos on cosmological observables (e.g., the CMB's primary anisotropies and its lensing-deflection power spectrum), as well as the degeneracies between axions and cosmic neutrinos. Our assumptions about future data, forecasting techniques, and key science results are presented in Section \ref{sec:results}. We conclude in Section \ref{conclusion}.
All power spectra presented here were computed using the \textsc{AxionCAMB} code, a modification to the CMB anisotropy code \textsc{CAMB} \cite{cambnotes}, which is described in Appendix \ref{code_appendix}, is publicly available, and was used to obtain the ULA constraints of Ref.~\cite{Hlozek:2014lca}.\footnote{The code may be downloaded from \url{http://github.com/dgrin1/axionCAMB}.} In Appendix \ref{nonlinear_appendix}, we discuss the computation of the nonlinear matter power-spectrum (relevant for understanding the effect on weak lensing on the CMB).
\section{Review of Axion and Neutrino Cosmology \label{sec:axion_cosmo}}
This section provides a brief introduction to axion physics, as well as the cosmology of axions and neutrinos (reviewed in greater depth by Refs.~\cite{Marsh:2015xka,Hlozek:2014lca} and \cite{Lesgourgues:2012uu}, respectively).
In this work we model the axion as a scalar field $\phi$. The dynamics of the scalar field are set by its potential, which we assume for simplicity to be a $V(\phi) \simeq \frac{1}{2}m^2\phi^2$ potential. Hence the equation of motion for the homogeneous ULA is:
\begin{align}
\ddot{\phi}_0+2\mathcal{H}\dot{\phi}_0+m_a^2 a^2 \phi_0&=0,\label{homo_eom}
\end{align}
where the conformal Hubble parameter is $\mathcal{H}=\dot{a}/a=aH$, and dots denote derivatives with respect to conformal time.
At early times the axion is slowly rolling and has an equation-of-state of $w_{a}\equiv P_{a}/\rho_{a}\simeq -1$. It therefore behaves like a cosmological constant, with roughly constant proper energy density as a function of time. $H$ decreases with the expansion of the universe and at a time $a_{\rm osc}$ such that $m_a\approx 3H(a_{\rm osc})$ the axion field begins to coherently oscillate about the potential minimum.
The relic-density parameter $\Omega_{\rm a}$ is given by
\begin{equation}
\Omega_{\rm a}=\left[\frac{a^{-2}}{2}\dot{\phi}_0^2 + \frac{m_a^2}{2}\phi_0^2 \right]_{m_a=3H}a_{\rm osc}^{3}/\rho_{\rm crit},\label{homorelic}
\end{equation}
where $\rho_{\rm crit}$ is the cosmological critical density today. This production mode is known as the vacuum realignment, or misalignment, mechanism.
In the early universe, neutrinos, like other weakly interacting particles, are coupled to the cosmological fluid until the weak interaction rate falls below the temperature of the universe, which is decreasing due to its expansion. This occurs at around $T\approx 1$ MeV. At this time, the neutrinos then decouple from the plasma. Massive neutrinos contribute to the energy density of the Universe as
\begin{equation}
\Omega_\nu h^2 = \frac{\Sigma m_\nu }{93.14\mathrm{eV}}. \label{eqn:mass_neu}
\end{equation}
Massive neutrinos behave as radiation at early times (energy density scaling as $a^{-4}$). When the temperature drops below the neutrinos mass, they behave like matter (energy density scaling as $a^{-3}$). Thus, depending on the mass, massive neutrinos can change the time of matter-radiation equality, and alter the matter density at late times. Upper bounds on the mass of standard model neutrinos imply that they have a cosmologically non-negligible free-streaming length caused by their relativistic motion at early times. For wavenumbers $k>k_{\rm fs}$, neutrino clustering is suppressed relative to that of ordinary matter, leading to decreased structure formation for larger $\sum m_\nu$ (given a fixed late-time DM content).
ULAs also suppress structure formation at large wavenumbers, $k\gtrsim k_{\rm m}$, through their scale-dependent sound speed \cite{Marsh:2010wq,Hlozek:2014lca}:
\begin{eqnarray}
c_a^{2}=\left\{\begin{array}{ll}
\frac{k^2}{4m_a^2a^2}&\mbox{if $k\ll k_{m}\equiv 2m_{a}a$},\\
1&\mbox{if $k\gg k_{m}$}.\end{array}\right.\label{heuristic_cs}\end{eqnarray}
The wave number $k_{\rm m}$ is mass dependent, moving to large length scales as the axion mass decreases. It is important to note that the axion suppression of structure and the suppression from neutrinos have very different physical origins: ULAs suppress structure growth below the Jeans length due to their wave-like nature, while neutrinos do so because of their large thermal velocities.
In addition to the contribution of a massive neutrino species, we will investigate the degeneracies between vacuum-alignment ULAs and additional massless neutrinos and other ``dark radiation'' through the \textit{relativistic degrees of freedom} ($N_\mathrm{eff}$), parameterized relative to the photon energy density, $\rho_\gamma$, as:
\begin{equation}
\rho = N_\mathrm{eff}\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\rho_\gamma.
\end{equation}
For useful descriptions of the physics of $N_{\rm eff}$ on the CMB, see Refs.~\cite{2013PhRvD..87h3008H,2015PhRvL.115i1301F}.
As noted above, ULAs produced by vacuum realignment do \emph{not} contribute to $N_\mathrm{eff}$. Axions produced by other mechanisms, however (such as thermal freeze-out or heavy particle-decay) constitute a separate population of relativistic axions, and \emph{do} contribute to $N_\mathrm{eff}$ \cite{Acharya:2010zx,Weinberg:2013kea,Iliesiu:2013rqa,Baumann:2016wac}. It is important to note that $N_\mathrm{eff}$ does not distinguish between fermions and bosons (although other cosmological observables could. See, for example Ref. \cite{Hannestad:2005bt}), nor on the production mechanism of the additional radiation. Thus additional relativistic neutrinos and axions are completely degenerate in cosmological terms: because of this we consider varying $N_\mathrm{eff}$ completely generically.
The lightest vacuum-realignment ULAs ($m_a < 10^{-30}~\mathrm{eV}$) are degenerate with a DE-like component in the universe, and generate a late-time integrated Sachs-Wolfe (ISW) \cite{Sachs:1967er} effect in the CMB \cite{Crittenden:1995ak,Coble:1996te,Hlozek:2014lca,Boughn:2004zm}. They also change the background expansion rate of the universe, altering the angular diameter distance to the last-scattering surface. This affects the position of the peak in a similar manner to how $N_\mathrm{eff}$ alters the position of the peak. Hence we expect a partial degeneracy between ULAs and $N_\mathrm{eff}$ for the lightest ULAs.
\section{CMB observables \label{observables}}
The main effects of ULAs in the temperature power in the multipole range relevant to {\sl Planck}, and in the linear galaxy power spectrum, were discussed in detail in Ref.~\cite{Hlozek:2014lca}. Primary CMB power spectra, matter power-spectra, and lensing convergence power-spectra for ULAs are all computed using the \textsc{AxionCAMB} code, which was used to obtain the results of Ref.~\cite{Hlozek:2014lca} and is described in the Appendix~\ref{code_appendix} of this paper.
\subsection{The CMB-damping tail, distance measures, and neutrino degeneracies}
In order to interpret forecasts on the allowed values of the energy density in ULAs and the degeneracies with neutrinos, we highlight the similarities and differences between the two components at the level of effects on the cosmological observables. This comparison was made for galaxy surveys in Ref.~\cite{Marsh:2011bf}, and was also discussed in Ref.~\cite{Amendola:2005ad}.
\begin{figure*}[htbp!]
\begin{center}
$\begin{array}{ll}
\includegraphics[width=\columnwidth,trim = 0mm 0mm 0mm 60mm, clip]{diff_test.pdf}&
\includegraphics[width=\columnwidth,trim = 0mm 0mm 0mm 60mm, clip]{neff_de_diffs.pdf}
\end{array}$
\caption{\textbf{Relative differences between axion effects and other cosmological parameters:} The error bars shown are for a `CMB-S4-like' survey as described in Table~\ref{table:net}. \textit{Left:} Comparing DM-like ULAs to massive neutrinos, holding the total matter density and sound horizon fixed. For each neutrino model, there is a corresponding ULA model that produces almost degenerate effects in all observables. Both massive neutrinos and ULAs produce the largest effects in the lensing convergence power, where effects of order $\Delta C^{\kappa\kappa}_\ell/C^{\kappa\kappa}_\ell \simeq 25\%$ occur even at low multipoles. \textit{Right:} Comparing DE-like ULAs (with $\log_{10}(m_a/\mathrm{[eV]}) < -30$) to additional massless neutrinos, holding only the sound horizon fixed. The ULA energy density is $\Omega_a h^2=0.002$. These types of models do not display any significant degeneracies. Note that the $\ell$-axis of the right panel is shown in log scale, while the left panel is linear. \label{fig:observables_massive}}
\end{center}
\end{figure*}
ULAs and neutrinos affect the expansion rate, changing the angular size of the sound horizon, $\theta_s$, at fixed Hubble constant, $h$. Consider the case of one additional massive neutrino eigenstate with $\Sigma m_\nu=0.06~\mathrm{eV}$ and $N_\mathrm{massive}= 1$, $N_\mathrm{massless} = 2.046$. This neutrino is relativistic throughout the radiation era, but behaves like matter at late times. The main effect of this on the high-$\ell$ acoustic peaks is to increase the angular size of the sound horizon. This can be compensated by \textit{reducing} the Hubble constant from $h=0.6715$ in to $h=0.6685$, in order to hold $\theta_s$ fixed (relative to a $\Sigma m_\nu=0$ model). ULAs also change the expansion rate relative to pure CDM due to the early $w_a=-1$ behaviour: holding $\theta_s$ fixed requires a reduction in $h$ just as for neutrinos~\cite{Marsh:2011bf,Hlozek:2014lca}.
In Figure~\ref{fig:observables_massive} we show the relative difference in CMB auto power spectra for temperature, T, E-mode polarization, and lensing convergence, $\kappa$, for ULA and neutrino models compared to a reference $\Lambda$CDM model:
\begin{equation}
\frac{\Delta C_\ell}{C_\ell} = \frac{(C_{\ell}^{\rm model}-C_\ell^{\rm ref.})}{C_\ell^{\rm ref.}} \, .
\end{equation}
The reference model contains $N_{\rm eff}=3.046$ massless neutrinos, and no ULAs. Massive neutrinos are introduced as a single massive eigenstate, i.e. $N_{\rm massive}=1$, $N_{\rm massless}=2.046$, with the energy density today fixed by the mass in as in Eq.~(\ref{eqn:mass_neu}). ULAs are introduced with a free mass and energy density, and are chosen to mimic as closely as possible the neutrino models in the observables.
The ULA and neutrino models are chosen to keep the total matter density, $\Omega_m h^2=\Omega_c h^2+\Omega_b h^2+\Omega_a h^2+\Omega_\nu h^2$, and sound horizon, $\theta_s$, fixed. Under these conditions, the effects of ULAs and massive neutrinos on the CMB observables are remarkably similar, and it is clear that there are parts of parameter space where significant degeneracies exist. Were one also to vary the number of massive neutrinos, $N_{\rm massive}$, even more degeneracies would open up~\cite{Marsh:2011bf}.
For example, we observe that a ULA model with $m_a=10^{-30}\text{ eV}$ and $\Omega_a h^2=0.0005$ is degenerate with the standard fiducial neutrino model with $m_\nu=0.06\rm{ eV}$. This ULA energy density occurs naturally (i.e. the axion misalignment angle $\theta \ll 1$) for $f_a\approx 3\times 10^{-2}M_{pl}\approx 7\times 10^{16}\text{ GeV}$: GUT-scale ULAs can be constrained by the CMB, but also have significant degeneracies with other cosmological components.
In the most massive neutrino model shown in Figure~\ref{fig:observables_massive}, $\Sigma m_\nu=0.7\text{ eV}$, holding the sound horizon fixed requires decreasing the Hubble constant to $h=0.6415$, while the corresponding axion model only requires $h=0.6635$. For the other reference models with lighter neutrinos, the change in $h$ required for ULAs and neutrinos is the same. Thus, in the case of relatively heavy ULAs and neutrinos, a local measure of $H_0$ can help break degeneracies.
ULAs and massive neutrinos can produce $\mathcal{O}(10\%)$ effects in the temperature power at $\ell\gtrsim 3000$. This comes from the lensing-induced temperature power, which at high $\ell$ is approximately~\cite{Lewis:2006fu}:
\begin{equation}
C_{\ell}^{\rm TT}\approx \ell^2 C_\ell^{\phi\phi}\int \frac{d\ell'}{\ell'}\frac{\ell'^4}{4\pi}\tilde{C}_{\ell'}^{\rm TT}\, ,
\end{equation}
where $\tilde{C}_\ell$ is the unlensed power, and $C_{\ell}^{\phi\phi}$ is the power spectrum of the lensing potential.
The lensed temperature power in ULA and massive neutrino cosmologies is reduced compared to pure CDM by the suppression of clustering (free streaming for neutrinos, the Jeans instability for ULAs) and consequent reduction of the lensing contribution to $C_\ell^{\rm TT}$. This effect is likely of little importance observationally, as temperature power at such high multipoles becomes dominated by other secondaries, such as galactic foregrounds, and the Sunyaev-Zel'dovich effect, making the direct lensing contribution hard to measure. A similar effect is also seen in the E-mode polarization, which suffers less from foregrounds at high multipoles. The effects of massive neutrinos and ULAs on the lensed E-mode power at high-$\ell$ are relatively small, however, compared to the forecasted CMB-S4 error bars.
Both massive neutrinos and ULAs produce the largest effects at relatively low multipoles in the lensing convergence power, and this offers a very powerful observable to constrain the properties of DM beyond CDM. The lensing convergence power spectrum, $C_\ell^{\kappa\kappa}$, is a direct measurement of the DM distribution, and its scale dependence at high-$\ell$ measures the clustering properties of sub-dominant components of the DM. In Ref.~\cite{Allison:2015qca}, it was shown that the lensing convergence power drives the ability of future CMB experiments to measure the sum of neutrino masses. Figure~\ref{fig:observables_massive} shows that the lensing convergence power also provides a powerful method to constrain other departures from CDM, and measures the composition and clustering properties of DM over a wide range of scales. We will quantify this in detail in Section~\ref{constraints}, showing the gains in sensitivity given by CMB-S4 over {\sl Planck}, and how much of this gain is driven by lensing.
Now consider the effect of additional massless neutrinos, parameterized by $\Delta N_{\rm eff}$, and DE-like ULAs (i.e. those for which $w_a=-1$ for some period during the matter dominated era). The effects of these models on CMB observables are also shown in Figure~\ref{fig:observables_massive}. We notice the well-known effect that $\Delta N_{\rm eff}\neq 0$ increases the amount of damping in the CMB at high-$\ell$. Since we include radiation in the closure budget, there is also reduced overall matter power, and consequently reduced lensing power. DE-like ULAs affect the lensing largely through the expansion rate and scale-dependence of the growth at low-$z$. This has a knock-on effect of slightly reducing TT and EE power at large $\ell\gtrsim 1000$ from reduced lensing, and in some cases creates a partial degeneracy with $N_{\rm eff}$ on these scales.
There are $\mathcal{O}(1\%)$ effects in the ${\rm EE}$ power for $N_{\rm eff}$ and DE-like ULAs at $\ell\approx 10$, the ``reionization bump'', caused by the different expansion histories and matter budgets in these models. The low-$\ell$ effects of $\Delta N_{\rm eff}$ and DE-like ULAs in TT and EE are opposite in sense, which predicts the degeneracy direction if such multipoles are included - here combining temperature and polarization data helps break the degeneracy. We also notice $\mathcal{O}(1\%)$ effects of $N_{\rm eff}$ at $\ell\approx 100$ in $EE$ at the ``recombination bump'', similarly caused by effects on the expansion rate. DE-like ULAs do not affect recombination relative to $\Lambda$CDM, since they behave entirely like the cosmological constant $\Lambda$ at this epoch by definition.
For $\Delta N_{\rm eff}\neq 0$ and DE-like ULAs, we have adjusted $H_0$ to hold the sound-horizon fixed. This serves to further physically distinguish the models. Massless neutrinos decrease $\theta_s$ and require an increase in $H_0$ to hold it fixed: hence a preference for $\Delta N_{\rm eff}\neq 0$ is sometimes found to reconcile CMB (lower) and other (higher) measures of $H_0$ \citep[e.g.][]{MacCrann:2014wfa}. On the other hand, we introduce DE-like ULAs with constant $\Omega_c h^2$, and as such they come out of the DE budget. As described in detail in Ref.~\cite{Hlozek:2014lca}, they require reduced $H_0$ to hold $\theta_s$ fixed, and lead to a non-$\Lambda$ effect on the late-time ISW effect at low $\ell$. In the most extreme cases shown, $\Delta N_{\rm eff}=0.1$, $m_a=2\times 10^{-32}\text{ eV}$ the change in $h=\pm 0.1$ respectively. Accurate local measures of $H_0$ can improve constraints on DE-like ULAs substantially \citep[e.g.][]{Freedman:2010xv, Riess:2016jrr}, but high-$\ell$ CMB experiments such as CMB-S4 will add little to constraints on them compared to {\sl Planck}. We discuss quantitatively the inclusion of a prior on $H_0$, in addition to CMB-S4, in Section~\ref{constraints}.
In conclusion on this topic, we do not expect significant degeneracies between additional massless neutrinos and DE-like ULAs, while we expect significant degeneracies between heavier ULAs and massive neutrinos. Via lensing, CMB-S4 should allow detection of neutrino mass, and greatly improve constraints on intermediate mass ULAs. CMB-S4 should also substantially improve constraints on $\Delta N_{\rm eff}$ by more precise measures of the damping tails. Including $H_{0}$ measurements should improve limits on DE-like ULAs, and break remaining degeneracies.
\subsection{Lensing deflection power and non-linear clustering }
\label{sec:nonlinear}
The largest deviation from standard $\Lambda$CDM caused by ULAs in the lensing deflection power occurs on small scales. Here one must take some care as both non-Gaussian noise in the experimental setup, and the theoretical modeling of nonlinear lensing add a systematic error to any inferred constraints on DM properties. This problem is particularly acute for more massive ULAs ($m_a\gtrsim 10^{-25}\text{ eV}$), which undergo non-linear clustering on observationally relevant scales or redshifts and can contribute a large fraction to the total DM abundance.
The lensing deflection power, $C_\ell^{\kappa\kappa}$, depends on the integral along the line-of-sight of the Newtonian potential power spectrum, $\mathcal{P}_{\Psi}(k,z)$~\cite{Lewis:2006fu}. These non-linear clustering contributions such that non-linear effects before important on larger angular scales in $C_\ell^{\kappa \kappa}$ than they do for $C_{\ell}^{\rm TT}$.
The lensing power on all multipoles is dominated by effects at $z\lesssim 10$. For multipoles $\ell\approx 1000$ the integral kernel peaks at $z\approx 2$. In terms of wavenumber, $k$, multipoles $\ell\gtrsim 1000$ are dominated by contributions from, $k\gtrsim 0.1\text{ Mpc}^{-1}$, where density perturbations are becoming non-linear. On these sub-horizon scales, the power spectrum of the Newtonian potential is determined from the matter power spectrum via Poisson's equation. Non-linearities in the matter clustering in this range of redshifts and wavenumber lead to $\mathcal{O}(10\%)$ effects in the lensing power for $\ell\gtrsim 1000$.
The non-linear gravitational potential power spectrum (needed to compute $C_{\ell}^{\kappa\kappa}$ including nonlinear effects) is computed in \textsc{camb} using the expression (see Ref. \cite{2010GReGr..42.2197H} and references therein):
\begin{align}
\mathcal{P}_{\Psi,{\rm non-lin}}(k,z) &= \frac{P_{m,{\rm non-lin}}(k,z)}{P_{m,{\rm lin}(k,z)}}\mathcal{P}_{\Psi,{\rm lin}}(k,z) \\
&\equiv \mathcal{R}_{\rm nl}(k,z)\mathcal{P}_{\Psi,{\rm lin}}(k,z)\, ,\label{eqn:psi_non-lin}
\end{align}
where $P_m(k,z)$ is the matter power spectrum, and non-linearities are computed using \textsc{halofit}~\cite{Smith:2002dz}, a code based on a fitting function, which is calibrated to N-body simulations of CDM (with Ref. \cite{Bird:2011rb} including massive neutrinos). One must therefore take extra care when exploring constraints on non-standard models from high-multipole lensing.\footnote{This does not just apply to non-standard DM models, such as ULAs. \textsc{halofit} is calibrated using power law initial conditions, and so care must also be taken for models with features in the primordial power spectrum at high-$k$.} We discuss the non-linear modeling of the power spectrum further in Appendix~\ref{nonlinear_appendix}.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=1.05\columnwidth]{diff_KK.pdf}
\caption{\textbf{Comparison of ULAs to CDM in lensing deflection power for different models of non-linearities, where ULAs with $\mathbf{m_a=10^{-23}}\text{ eV}$ constitute all the DM.} The unphysical power increase in the \textsc{halofit} power for ULAs, seen in Figure~\ref{fig:tk_halofit_halomodel}, causes a similar unphysical increase in lensing power compared to the halo model. On the other hand, linear theory captures the sign and approximate magnitude of the effect seen in the halo model. Thus when forecasting constraints at high ULA mass we choose to use linear theory lensing as a reasonable approximation for the Fisher matrix derivative. \label{fig:lensing_halofit_halomodel}}
\end{center}
\end{figure}
We now assess how the non-linear modeling affects the lensing deflection power of ULAs. Figure~\ref{fig:lensing_halofit_halomodel} shows the lensing power ratio $(\Delta C_\ell/C_\ell)^{\kappa\kappa}$ for $m_a=10^{-23}\text{ eV}$ assuming that either ULAs or CDM (but not both) constitute all of the DM. We compare linear theory, \textsc{halofit}, and the halo model for ULAs of Ref.~\cite{Marsh:2016vgj}. For illustration, we consider the lensing deflection power from the halo model under the Limber approximation (which is accurate for high-$\ell$ where non-linearities become important) \cite{2010GReGr..42.2197H}:
\begin{equation}
C_{\ell}^{\phi\phi} = \frac{8\pi^2}{\ell^3}\int_0^{z_{\rm rec}}dz \mathcal{P}_{\Psi}(\ell/x,z) x \frac{dx}{dz}\left(\frac{x_{\rm rec}-x}{x_{\rm rec}x} \right)^2 \,.
\end{equation}
where $x=x(z)$ is the comoving distance to redshift $z$.\footnote{We emphasize that the halo model for ULAs is \emph{not} yet incorporated into \textsc{axionCAMB}. The halo model for CDM has only recently been incorporated into \textsc{camb}. Our comparisons here attempt to use the halo model to motivate approximations appropriate to forecasting. Proper inclusion of non-linearities in real data analysis of CMB-S4 will be crucial to avoid bias caused by incorrect modeling.}
Figure~\ref{fig:tk_halofit_halomodel} shows the overdensity ratio of ULAs to CDM, $\sqrt{P_{\rm ULA}(k,z)/P_{\rm CDM}(k,z)},$ over a range of scales and redshifts for a pure ULA DM model with $m_a=10^{-23}\text{ eV}$. In this model, perturbations in the axion energy density go non-linear for $z<3$, where non-linear collapse reduces the power suppression relative to CDM for $k\gtrsim 1\, h\text{ Mpc}^{-1}$.
\begin{figure*}[htbp!]
\includegraphics[width=1.8\columnwidth]{halo_transfer.pdf}
\caption{\textbf{Comparison of power spectrum ratios for \textsc{halofit} and the halo model of Ref.~\cite{Marsh:2016vgj}\footnote{Available online at: https://github.com/DoddyPhysics/HMcode.} for $m_a=10^{-23}\text{ eV}$ and ULAs as all the DM.} The halo model is cut to set the power to linear if $\sigma^2<1$ to make a fair comparison. Non-liner clustering begins at $z=2$. \textsc{halofit} applied to non-CDM models gives an unphysical boost in power at the onset of non-linearities, which is passed on to the lensing power, Figure~\ref{fig:lensing_halofit_halomodel}. Differences between the halo model and \textsc{halofit} at high $z$ are due to the quantitiative differences between the \textsc{axionCAMB} transfer function and the combination of Refs.~\cite{Eisenstein:1997ik,Hu:2000ke} analytic fits used in the halo model. \label{fig:tk_halofit_halomodel}}
\end{figure*}
We notice that \textsc{halofit} introduces a large feature, increasing the power at the non-linear scale, $(k_{\rm nl},z_{\rm nl})$. Such a feature is not seen in the halo model, and is thus suspected to be an unphysical artifact introduced purely by the fitting functions of \textsc{halofit} -- calibrated to CDM and not a good description of ULAs at this scale. This unphysical boost in the matter power caused by \textsc{halofit} leads to a similarly unphysical increase in the lensing deflection power in Figure~\ref{fig:lensing_halofit_halomodel}. The effect seen in the (presumably more correct) halo model is that ULAs always decrease the lensing deflection power relative to CDM. Furthermore, perhaps surprisingly, the sign and approximate magnitude of the \emph{relative} effect of ULAs compared to CDM on the lensing deflection power in the full halo model is well captured by linear theory.
The above observation - that linear theory captures the relative effects of high mass ULAs on weak lensing better than \textsc{halofit} - determines how we decide to treat non-linear modeling in our forecasts (see also Appendix~\ref{nonlinear_appendix}). We choose by default to \emph{perform all forecasts with non-linear lensing turned off}. This choice is expected to give the right sign and approximate magnitude for Fisher-matrix derivatives for high mass ULAs, while non-linear modeling is not expected to be important at low mass, where ULAs do not non-linearly cluster on the relevant redshift range.
\section{Results}\label{sec:results}
This section contains our assumptions, methodology, and key science results. In Sec.~\ref{data}, we lay out the assumptions made about CMB-S4 and Fisher-matrix techniques used to obtain our results. In Sec.~\ref{constraints}, we present our conclusions about the sensitivity of CMB-S4 to ULAs, the improvement over {\sl Planck}, the role of CMB weak lensing in driving sensitivity improvements, and explore degeneracies with neutrinos. Finally in Sec.~\ref{survey_optimize}, we explore how varying potential CMB-S4 survey parameters (sky coverage, noise level, and beam width) affects the conclusions of Sec.~\ref{constraints}.
\subsection{Data and Surveys\label{data}}
The current best constraints on the axion fraction comes from a combination of the primary CMB ({\sl Planck}, SPT, and ACT TT power spectra, as well as low-$\ell$ {\sl WMAP} polarization data) with the WiggleZ galaxy redshift survey \cite{Hlozek:2014lca}.
We consider future constraints from a `CMB-S4-like' survey as discussed in the recent Snowmass proposal \cite{Abazajian:2013oma}, with observational parameters specified in Table~\ref{table:net}. The exact specification of a CMB-S4
experiment is still under development. Provided it covers a significant fraction of the sky with reasonable noise levels, CMB-S4 promises to be an incredible instrument with which to test the dark sector.
In Section~\ref{survey_optimize} we test for the dependence of the constraints on the survey parameters.
We forecast assuming a fiducial set of cosmological parameters:
\begin{equation}
\mathbf{\Xi} = \left\{\Omega_bh^2, \Omega_dh^2, H_0, A_s, n_s, \tau, m_a \Omega_a/\Omega_d \right\},
\end{equation}
where $\Omega_bh^2$ parameterizes the physical baryon density of the universe, $\Omega_dh^2$ is the energy density of the dark sector including axions, $H_0$ is the Hubble parameters in units of ${\rm km\, s}^{-1}{\rm Mpc}^{-1}$, $A_s, n_s$ are the amplitude and spectral index of the scalar density fluctuations and $\tau$ is the optical depth to decoupling. As described above, the fraction of the dark sector made of axions (at a specified fixed axion mass $m_a$ in units of ${\rm eV}$) is given by $\Omega_a/\Omega_d$. The fiducial values and step sizes used for this model are shown in Table~\ref{tab:fiducial}.
\begin{table}[t!]
\begin{center}
\begin{tabular}{ccc}
\hline
\hline
Parameter&Fiducial value& Step size\\
\hline
$\Omega_bh^2$ & 0.02222 & 0.0001\\
$\Omega_dh^2$ &0.1197 & 0.001 \\
$H_0\,[{\rm km\, s}^{-1}{\rm Mpc}^{-1}]$ & 69.0& 0.1\\
$A_s $ &$ 2.1955\times 10^{-9}$& $2.0\times 10^{-11}$\\
$n_s$ & 0.9655& 0.005 \\
$\tau$ & 0.06 & 0.01 \\
$m_a\,[\mathrm{eV}]$ & $ 10^{-32} < m_a< 10^{-22}$ & [fixed per run]\\
$\Omega_a/\Omega_d $ & 0.02 & 0.005 \\
\hline
\end{tabular}
\caption{\textbf{Fiducial model and Fisher Matrix step sizes:} The base model considered and the step sizes used to compute the Fisher derivatives. The above model was also supplemented in parts by including the additional extensions of the parameters $\Sigma m_\nu=60\,\mathrm{meV}$ and $N_\mathrm{eff}=3.046$ which were varied with step sizes of $20\,\mathrm{ meV}$ and $0.05$ respectively. \label{tab:fiducial}}
\end{center}
\end{table}
Where necessary we include $\Sigma m_\nu\,\mathrm{[eV]}$ and $N_\mathrm{eff}$ as additional parameters in the model space.
We use Fisher-matrix techniques to forecast constraints on the parameters of interest \cite{Eisenstein:1998hr,Taylor:1997ag,Matsubara:2004fr,Bassett:2009uv}. The Fisher matrix translates uncertainties on observed quantities such as the lensing deflection or the CMB power spectrum into constraints on parameters of interest in the underlying model.
The Fisher matrix is the expectation value of the second derivatives of the logarithm of the data likelihood with respect to the parameters $\mathbf{\Xi}:$
\begin{equation}
\mathscr{F}_{ij} = -\left\langle \frac{\partial^2 \ln P(\mathbf{D}|\mathbf{\Xi})}{\partial\Xi_i\partial\Xi_j}\right\rangle,
\end{equation}
where $\mathbf{D}$ is the data vector of either CMB measurements or lensing deflection, for example.
For independent experiments (or if one has prior knowledge of the uncertainties on a parameter from a separate experiment) one can add individual Fisher matrices together to get a final Fisher matrix. In order to obtain 1- or 2-dimensional constraints on parameters (i.e. 1-D likelihoods or 2-D error ellipses), one marginalizes over the other nuisance parameters in the larger parameter space under consideration.
The Fisher matrix code ({\sc OxFish}) used to forecast the full set of observables including the lensing deflection is described in Ref.~\cite{Allison:2015qca}, modified to include the axion parameters, as described in Ref.~\cite{Hlozek:2014lca}.
We compared a five-point numerical derivative, \begin{eqnarray}
f'(x) &=& \left[8f(x+h)-8f(x-h) \right. \nonumber \\
&-& \left. f(x+2h) + f(x-2h)\right]/12h,
\end{eqnarray} to the standard two-sided finite-difference derivative method and checked that the resulting parameter uncertainties were stable to the choice of derivative method. In addition, we demanded that the derivatives of the axion fraction converged to 0.1\% precision to set the step size used for finite-difference calculations.
We forecast the combination of our `CMB-S4-like' survey with{\sl Planck} temperature and polarization spectra that match the current sensitivities between the multipoles of $30< \ell < 2500$. This also allows us to assess the gains possible when moving from {\sl Planck} to {\sl Planck}+S4: Fisher-matrix forecasts are often somewhat more optimistic than sensitivities obtained in real experiments, and so we use Fisher forecasts for both {\sl Planck} and {\sl Planck}+S4 in order to conduct a fair comparison. For CMB-S4 we assume measurements of the TT,EE,TE primordial CMB spectra with an $\ell_\mathrm{min}=30$ and an $\ell_\mathrm{max}=4000$ for the EE, TE spectra and $\ell_\mathrm{max}=3000$ for the TT spectra. We include the lensing deflection power spectrum from both surveys between $30 < \ell < 3000$.
For the low-$\ell$ data we use {\sl Planck} HFI `lowP' specifications, with slightly modified noise levels to ensure a prior on the optical depth of $\tau=0.01$.
We assume that the noise has a white power-spectrum, using the standard treatment \cite{Knox:1995dq}:
\begin{equation}
N^{\alpha\alpha}_\ell = (\Delta \alpha)^2\exp{\left(\frac{\ell(\ell+1)\theta_\mathrm{FWHM}^2}{8\ln 2}\right)},
\end{equation} where $\alpha = {\rm T}$ or ${\rm E}$, labels the field of interest. $\theta_\mathrm{FWHM}$ is the beam full width half maximum, and the lensing deflection noise is estimated assuming a minimum-variance quadratic estimate of the lensing field as described in Ref.~\cite{Allison:2015qca}. We assume that relevant foregrounds have been removed on all scales up to $\ell = \ell_\mathrm{max}$. We don't include information from the BB lensing power-spectrum, as the assumption of nearly Gaussian fields (required for the validity of the Fisher-matrix formalism) breaks down for B-modes from lensing, which are produced by a scalar modulation of primordial E-modes, and is thus a higher order (and non-Gaussian) effect.
\begin{table}[t!]
\begin{center}
\begin{tabular}{cccc}
\hline
\hline
$f_\mathrm{sky}$& Beam size& $\Delta T$ & $\Delta E,B$ \\
& (arcmin)& ($\mu$K-arcmin)&($\mu$K-arcmin) \\
\hline
0.4& 1& 1&1.4\\
\hline
\end{tabular}
\caption{\textbf{Survey parameters considered for axion forecasts:} Survey sensitivity, assumed beam size and sky fraction for a possible `CMB-S4-like' survey. We test the dependence of the axion constraints on these parameters in Section~\ref{survey_optimize}.\label{table:net}}
\end{center}
\end{table}
\subsection{Forecasted sensitivity to dark-sector densities and particle masses}
\label{constraints}
We show the forecasted constraints on the axion energy density from CMB-S4 including lensing in Figure~\ref{fig:axions}. We compare $1\sigma$ errors for {\sl Planck} and {\sl Planck}+S4 (where {\sl Planck} is used on a reduced part of the sky as described in Section~\ref{data}) around a fiducial axion fraction $\Omega_a/\Omega_d=2\times 10^{-2}$, and demonstrate the effect of fixing or marginalizing over neutrino mass. We also show forecasted 1 and 2$\sigma$ exclusion lines on Figure~\ref{fig:axions}. In all other error ellipse plots we show $2\sigma$ contours, unless otherwise specified.
Figure~\ref{fig:axion_cdm_degen} shows the power of a `CMB-S4-like' survey to distinguish ULAs from CDM, by comparing constraints for {\sl Planck}+S4 (solid lines) to constraints assuming only {\sl Planck} specifications (dashed lines). CMB-S4 will not only tighten the constraints on the total DM content, but closes in on the axion parameter space as well. In particular for some masses (most notably $m_a = 10^{-25}\, \mathrm{eV}$), CMB-S4 breaks the degeneracy between ULAs and CDM even at very low axion fraction. CMB-S4 will allow for a multi-$\sigma$ detection of percent level departures from CDM for all masses in the range $10^{-30}\,\mathrm{eV}<m_a<10^{-24}\,\mathrm{eV}$. Thus CMB-S4 presents an ability to test the composition of DM, and thus the CDM paradigm, at the percent level.
For these most DM-like ULAs $(m_a\geq 10^{-25}~\mathrm{eV})$ the current data (i.e. {\sl Planck}, see Ref.~\cite{Hlozek:2014lca}) do not bound the axion fraction at the percent level. As shown in Figure~\ref{fig:axion_cdm_degen}, {\sl Planck} has essentially no constraining power for $m_a=10^{-24}\,\mathrm{eV}$, when $\Omega_a$ and $\Omega_c$ are totally degenerate. As the axion mass changes, the degeneracy goes from complete (horizontal in this representation), with the error on the total dark content unchanged irrespective of the axion fraction, to one where the axion fraction is tightly constrained (e.g. $m_a=10^{-29}\,\mathrm{eV}$). The degeneracy direction continues to change for the lighter axions as they become more DE-like.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=0.5\textwidth]{s4_frac_omdah2.pdf}
\caption{\textbf{The degeneracy between ULAs and CDM for fixed ULA masses:} The fiducial value of the axion fraction, $\Omega_a/\Omega_d=0.02$, is chosen to be consistent with current upper bounds from {\sl Planck}. The dashed lines show forecast constraints based on the {\sl Planck} `Blue Book' \cite{BlueBook2005} sensitivities, and reproduce the constraints using the actual data (see Ref.~\cite{Hlozek:2014lca} for details). The solid lines show constraints for a `CMB-S4-like' survey. At the highest masses considered, $m_a \geq 10^{-24}~\mathrm{eV}$ the axion is completely degenerate with the CDM density: the total dark matter density is well constrained, but the error on the axion fraction becomes larger. The degeneracy direction between axions and CDM rotates as the axion mass changes, with CMB-S4 breaking some strong degeneracies present in {\sl Planck}. In all cases $M_\nu$ has been fixed at its fiducial value, although the constraints in Fig.~\ref{fig:axions} shows that the error on the axion fraction is only degraded for the most degenerate masses in the `fuzzy DM' regime. CMB-S4 would detect a fraction of $\Omega_a/\Omega_d=0.02$ at $>2\sigma$ in the mass range $10^{-30}\,\mathrm{eV} < m_a < 10^{-24.5}\,\mathrm{eV}$. \label{fig:axion_cdm_degen}}
\end{center}
\end{figure}
\begin{figure*}[htbp!]
\begin{center}
$\begin{array}{ll}
\includegraphics[width=1\columnwidth]{errors_frac_mass_planck.pdf} &
\includegraphics[width=1\columnwidth]{errors_frac_mass_s4.pdf}
\end{array}$
\caption{\textbf{Forecast detection significance of `dark-matter-like' ULAs:} The \textit{left panel} shows the current constraints from a forecast {\sl Planck} survey (which is consistent with the results from Ref.~\cite{Hlozek:2014lca}). The right panel shows the forecast constraints from a `CMB-S4-like' survey over the same fixed masses ranging from $\log_{10}(m_a) = -26~\mathrm{eV}$ to $\log_{10}(m_a) = -22~\mathrm{eV}$. (For ease of viewing a random scatter has been placed in the $x$-direction for each mass, the dashed line gives the central mass value.) The $y-$axis shows the assumed fiducial axion fractions of $0.05, 0.1, 0.2, 0.5$ and $0.9$, with the forecast error on the fraction. The size of the marker is proportional to the significance with which we would detect such a fiducial axion fraction (the size is fixed for all detections $>5\sigma$). For the Planck survey, the constraints are eroded for masses heavier than around $\log_{10}(m_a) = -26~\mathrm{eV}$. CMB-S4 will push this boundary of ignorance by two orders of magnitude. A `CMB-S4-like' survey will allow a detection of an axion fraction as low as $5\%$ at $> 5\sigma$ for $\log_{10}(m_a) = -25~\mathrm{eV}$, and a fraction of $20\%$ at $> 3\sigma$ for $\log_{10}(m_a) = -24~\mathrm{eV}$. \label{fig:frac}}
\end{center}
\end{figure*}
At the largest axion masses, the near-perfect degeneracy between axions and CDM leaves us without a meaningful upper limit to saturate when choosing fiducial values for $\Omega_{a}/\Omega_{d}$. To test how a `CMB-S4-like' survey might place a tighter upper limit on the fraction of DM made up of ULAs, we instead forecast the significance of a CMB detection of ULAs while varying the fiducial fraction, and consider the detection significance. The results are shown in Figure~\ref{fig:frac}.
We fix the total DM energy density to the fiducial value of $\Omega_dh^2 = 0.1197$ (marginalizing over this and all other parameters) and vary the axion fraction as parameter of interest. We consider a range of fixed axion masses logarithmically spaced between $m_a=10^{-26}\,\mathrm{eV}$ and $m_a=10^{-22}\,\mathrm{eV}$. At each mass we use a range of fiducial fractions $(\Omega_a/\Omega_d = 0.05, 0.1, 0.2, 0.5, 0.9)$ and show the marginalized error on the fraction centred at the fiducial value. In Figure~\ref{fig:frac}, the size of the detection significance (in units of $\sigma$) is illustrated by the size of the marker, and we compare {\sl Planck} to {\sl Planck}+CMB-S4.
For axion masses of $\log_{10}(m_a)=-24~\mathrm{eV}$ using CMB-S4 an axion fraction as low as $20\%$ could be detected at $>3\sigma$, a vast improvement over {\sl Planck}, which has essentially no constraining power at this mass. We see that {\sl Planck} alone places only $\sim 1\sigma$ limits at high fraction for $m_a=10^{-25}\text{ eV}$ (consistent with the analysis of real data in Ref.~\cite{Hlozek:2014lca}), while this `wall of ignorance' is moved to $m_a=10^{-23}\text{ eV}$ with {\sl Planck}+CMB-S4.
The solid and dashed lines in Figure~\ref{fig:axions} show a different approach to the same issue of setting upper bounds. They show the fiducial models one could \textit{rule out} with $1\sigma$ (dashed) or $2\sigma$ (solid) significance. While the highest mass ULAs, $m\geq 10^{-22}\text{ eV}$, remain completely degenerate with CDM, one could rule out a fraction of $>15\%$ at $2\sigma$ confidence at $m_a=10^{-24}\,\mathrm{eV}$ and one could rule out an axion fraction of $>64\%$ at $1\sigma$ confidence at $m_a=10^{-23}\,\mathrm{eV}$. Figs~\ref{fig:axions} and~\ref{fig:frac} show how CMB-S4 could improve the lower limit on DM particle mass from the CMB alone by approximately 2 orders of magnitude compared with {\sl Planck}.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=0.5\textwidth, height=0.47\textwidth]{s4_frac_omnuh2.pdf}
\caption{\textbf{The degeneracy of ULAs with massive neutrinos:} The lighter ULAs show a significant degeneracy with neutrino mass for a `CMB-S4-like' survey, as summarized in Table~\ref{table:net}. The error bars increasing towards lighter mass - as these DE-like ULAs are less constrained with future data. Adding a prior on the expansion rate will reduce the errors on these parameters, as shown in Figure~\ref{fig:axions_prior} \label{fig:axions_mnu}}.
\end{center}
\end{figure}
The degeneracies of the ULAs with other cosmological parameters, such as $N_\mathrm{eff}$ or $\Sigma m_\nu$, also varies depending on the axion mass (see Figs.~\ref{fig:axions_mnu} and~\ref{fig:axions_prior}). As described already, DE-like ULAs with masses around $10^{-33}~\mathrm{eV}$ change the late-time expansion rate and therefore the sound horizon, changing the location of the acoustic peaks. This has degeneracies with the matter and curvature content. Heavier ULAs ($m_a \gtrsim 10^{-26}~\mathrm{eV}$) affect the expansion rate in the radiation era and reduce the angular scale of the diffusion distance, leading to a boost in the higher acoustic peaks, which has a degeneracy with $N_\mathrm{eff}$.
Consider the degeneracy between $\Sigma m_\nu$ and axion fraction, varying the axion mass (Fig.~\ref{fig:axions_mnu}). Certain axion masses are more degenerate with the fiducial neutrino model than others, making for example, a $m_a = 10^{-29}\, \mathrm{eV}$ axion more prone to masquerading as a massive neutrino than an axion of mass $m_a = 10^{-25}\, \mathrm{eV}$ (for a $m_a=10^{-29}\,\mathrm{eV}$ axion, the error on $\Sigma m_\nu$ is halved relative to the $m_a = 10^{-25}\, \mathrm{eV}$ case). The degeneracy is not total, however, and we will still be able to make a significant detection of a small axion fractions, using CMB-S4. Additionally, this degeneracy can be broken by local measurements of $H_{0}$.
As a test of how $H_0$ measurements can change constraints on the lightest ULAs, we added a prior of $1\%$ on $H_0$ to our forecasts. Current local measurements provide a $2-3\%$ constraint \cite{Riess:2016jrr}, while future efforts like DESI \cite{Font-Ribera:2013rwa} will provide roughly percent-level measurements from BAO. The addition of this prior changes the error on the axion fraction for an axion of mass $m_a = 10^{-32}\, \mathrm{eV}$ (assuming a fiducial fraction of 0.02) from $0.03$ to $0.005$ - allowing a $>4\sigma$ detection of the axion fraction even at the lowest masses. Local measurements of $H_0$ constrain the effects that these ULAs have on the low-$z$ expansion rate.
Figure~\ref{fig:axions_prior} shows how adding a $H_0$ prior to the precise measurement of the temperature and polarization power with CMB-S4 leads to an improvement in the error on $\Omega_a/\Omega_d$ at low ULA mass ($m_a\leq 10^{-30}\text{ eV}$). We show how the $H_0$ prior affects ULA degeneracy with $\Sigma m_\nu$ (left panel) and $N_\mathrm{eff}$ (right panel). In both cases the inclusion of a $H_0$ prior does not have a large effect on the error in the neutrino parameters ($\Sigma m_\nu$ or $N_\mathrm{eff}$), but it greatly reduces the degeneracy between light ULAs and neutrinos. The $H_0$ prior reduces the uncertrainty on $\Omega_a/\Omega_d$ by a factor of $\approx 3$ where both $\Sigma m_\nu$ and the axion fraction are varied, and a factor of $\approx 5$ when $N_\mathrm{eff}$ is varied with the axion fraction.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=0.45\textwidth]{s4_frac_omnuh2_prior.pdf} \\
\includegraphics[width=0.45\textwidth]{s4_frac_neff_prior.pdf}
\caption{\textbf{Priors on the expansion rate improve constraints on the lightest ULAs:} The degeneracies between the ULAs with mass $m_a < 10^{-30}~\mathrm{eV}$ and massive neutrinos (\textit{top panel}) and massless species (\textit{bottom panel}) are shown for a `CMB-S4-like' experiment (as specified in Table~\ref{table:net}), with the solid lines showing the constraints without any additional prior on the Hubble constant (there is some repetition with the left panel here and in Figure~\ref{fig:axions_mnu}). The dashed lines show the improvement when adding a prior of 1\% on $H_0$ from a `DESI-like' experiment \cite{Font-Ribera:2013rwa}. \label{fig:axions_prior}}
\end{center}
\end{figure}
The power of CMB-S4 lensing to break the degeneracy between ULAs and CDM is shown in Figure~\ref{fig:lensing_error}, which compares the error bar with and without adding in the lensing deflection measurements (solid to dashed line comparison) for different fiducial models. The largest reduction in the error including lensing deflection measurements comes in the mass range $10^{-29}\,\mathrm{eV} <m_a < 10^{-24}\,\mathrm{eV}$.
For CMB-S4 and an axion mass of $m_a = 10^{−26}\,\mathrm{eV}$, the percent-level measurement of the lensing power at multipoles $\ell > 1000$ leads to an improvement in the uncertainty on the axion energy density of a factor of eight relative to case where lensing information is excluded. Lensing also plays a key role in the ability of CMB-S4 to improve constraints on ULAs in the range $10^{-24}\,\mathrm{eV} <m_a < 10^{-22}\,\mathrm{eV}$.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=0.5\textwidth]{errorbar_compare_lensing_S4_v2.pdf}\\ [0.0cm]
\caption{\textbf{Constraints on the axion fraction with and without lensing:} For a `CMB-S4-like' survey, the $1\sigma$ marginalized error bar on the axion fraction, $\Omega_a/\Omega_d$, for the ranges of masses considered: $10^{-32}<m_a< 10^{-22}\text{ eV}$. For masses $\log(m_a/\mathrm{eV}) > -28$, lensing more than halves the error bar for the same survey parameters where the lensing deflection is not included. The improvement is also sensitive to the fiducial model of ULAs assumed. This is particularly relevant given that for the heaviest masses the ULAs are currently indistinguishable from a standard DM component. \label{fig:lensing_error}}
\end{center}
\end{figure}
\subsection{Survey optimization}\label{survey_optimize}
The specifications of a `CMB-S4-like' survey are shown in Table~\ref{table:net}. One might ask what survey parameters might be most suitable to maximise constraints on the axion parameter space.
We show the results of some choices for the beam size and noise sensitivity in Figure~\ref{fig:s4optimise}. In each case we either vary the beam and sensitivity separately (solid and dashed lines), or we change the sky area at fixed 1 arcminute beam resolution, while adjusting the sensitivity assuming fixed total number of detectors and observing time. In the case where we \emph{reduce} the amount of sky observed by S4, we adjust the correponding area used from the {\sl Planck} satellite to include the fraction \emph{not} observed by S4. This is shown in the Figure with a dot-dashed line.
As discussed in Section~\ref{observables}, ULAs affect largely the high-$\ell$ damping tail of the CMB lensing deflection power, and so improvements in the noise properties at small angular scales tightens constraints on ULAs.
Moving to small, deep patches of the sky does not reduce the error: to constrain ULAs we need larger sky area given a total noise budget.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=0.5\textwidth]{ALP_density_mass_m28_frac_sensitivity_resolution.pdf}
\caption{\textbf{Constraints on the axion fraction as a function of survey parameters:} We vary the resolution and sensitivity for a range of `CMB-S4' survey parameters, around the baseline parameters of $1~\mu$K-arcmin, a resolution of 1 arcmin and a baseline sky fraction for CMB-S4 of $f_\mathrm{sky}=0.4,$ which is supplemented with a correspondingly reduced area of the {\sl Planck} sky. The error degrades slowly with worse resolution (solid line) and sensitivity (dashed line). The dot-dashed line shows the constraints for fixed observing time, changing the fraction of sky and accordingly modifying the sensitivity of the `CMB-S4-like' survey (and the amount of sky covered in corresponding {\sl Planck} maps). Since the ULAs affect the small-scale damping tail and the lensing deflection most strongly, moving to small, sensitive patches of the sky increases the error on the axion density (as opposed to having a fixed value of $f_\mathrm{sky}$ but pushing for lower instrumental noise levels). Conversely, tripling the beam size does not have a strong effect on the error on the axion fraction. \label{fig:s4optimise}}
\end{center}
\end{figure}
\section{Conclusions \label{conclusion}}
We live in the age of precision cosmology. Future experiments like the proposed CMB-S4 will significantly improve constraints on the composition of the dark sector. We have shown in detail how this is achieved in the case of ultra-light axions, including degeneracies with dark radiation and massive neutrinos. CMB-S4 will move the wall of ignorance for the heaviest axion candidates from $m_a=10^{-26}~\mathrm{eV}$ to $m_a=10^{-24}~\mathrm{eV}$ (detection with an axion fraction of 20\% at $>3\sigma$).
The lower limit on the dominant DM particle mass will be increased from $m_a=10^{-25}~\mathrm{eV}$ to $m_a=10^{-23}~\mathrm{eV}$ ($1\sigma$ constraints rule out large fractions). This begins to make contact with the much more systematic-laden upper bounds on the axion mass and fraction from high-$z$ galaxies and reionization: $\Omega_a/\Omega_d<0.5$ for $m_a=10^{-23}\text{ eV}$ and $m_a\gtrsim 10^{-22}\text{ eV}$ for the dominant component~\cite{Bozek:2014uqa,Schive:2015kza,Sarkar:2015dib}. This value approaches the mass range needed to explain dwarf galaxy cores and missing Milky Way satellites~(e.g. Refs.~\cite{Hu:2000ke,Marsh:2013ywa,Schive:2014dra,Marsh:2015wka}).
Perhaps more impressively, the constraints on the axion energy density at intermediate mass could improve by an order of magnitude. CMB-S4 could detect an axion fraction as low as $0.02$ at $>13\sigma$ for an axion mass of $10^{-27}\, \mathrm{eV}$.
Given the power of these future efforts, it will be possible to probe the degeneracies between ULAs and other potential DM components, such as massive neutrinos, and light species such as massless sterile neutrinos.
Improved independent constraints on measurements of the expansion rate (through measurements of the Hubble constant, for example) will improve sensitivity to the lightest, DE-like axions, and reduce the degeneracy between these species and both $\Sigma m_\nu$ and $N_\mathrm{eff}$. Even when marginalizing over the neutrino mass, the error on the axion fraction for a mass of $m_a = 10^{-32}\, \mathrm{eV}$ improves by a factor of three with a prior on the expansion rate.
As $\Omega_a\propto f_a^2$ the improved sensitivity to the axion energy density improve the axion decay constant which could be detected from $10^{17}\mathrm{ GeV}$ with {\sl Planck} to $10^{16}\mathrm{ GeV}$ with CMB-S4 (over the relevant range of ULA masses). The improved sensitivity to $f_a$ will begin to test the predictions of the string axiverse scenario~\cite{Arvanitaki:2009fg}.
Axions are a well motivated dark matter candidate, and future CMB experiments suggest an exciting opportunity to explore the rich complexity of their parameter space, moving towards sub-percent level sensitivity to the axion energy density or a $10\sigma$ detection if current limits to $\Omega_{a}$ are saturated by the true axion density, all over for a wide range of masses. As a spectator field during the inflationary era, axions would also carry isocurvature perburbations (see Ref. \cite{Hlozek:2014lca} and references therein), leading to distinct imprints on CMB observables and providing a unique new lever arm on the inflationary energy scale, which is otherwise only accessible through measurements of primordial CMB B-mode polarization \cite{Marsh:2014qoa}. In future work, we will extend {\sl Planck} constraints and CMB-S4 forecasts to include the impact of isocurvature.
Unraveling the mystery of dark matter is an important goal for cosmology in the coming decades. The axion represents the lowest mass DM-candidate, and a `CMB-S4-like' survey will help identify (or rule out) these models of DM. Constraints on the light, DE-like axions are improved by independent measurements of the expansion rate of the Universe, thereby probing our knowledge of the cosmological constant, quintessence, and cosmic acceleration in general.
In this work, we have illustrated that future CMB experiments will shed new light on the nature or existence of the axion and usher axiverse cosmology into a new era.
\acknowledgements{We thank Pedro Ferreira, Wayne Hu, Alexander Mead and Simeon Bird for useful discussions. RH acknowledges funding from the Dunlap Institute. DJEM acknowledges funding from the Royal Astronomical Society. DG is funded at the University of Chicago by a National Science Foundation Astronomy and Astrophysics Postdoctoral Fellowship under Award AST-1302856. This work was supported in part by the Kavli Institute for Cosmological Physics at the University of Chicago through grant NSF PHY-1125897 and
an endowment from the Kavli Foundation and its founder Fred Kavli. RA and JD are supported by ERC grant 259505, and EC is funded by the STFC Ernest Rutherford Fellowship.}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,288 |
Marc-Antoine de La Rovère, ou Antonio della Rovere, mort à Turin en 1538, est un prélat italo-français du . Il est un neveu de son prédécesseur Léonard de La Rovère, par sa mère, et le frère de l'archevêque de Turin, Giovanni Francesco della Rovere.
Biographie
Il est nommé au siège d'Agen sur la résignation de son oncle, le . Son oncle, dans son acte de résignation de l'évêché, s'était réservé l'administration de celui-ci, et par précaution avait obtenu une bulle du pape qui le nommait à cet évêché si son résignataire venait à mourir avant lui.
Marc-Antoine de la Rovère avait été auparavant prévôt de la cathédrale de Turin.
Aussitôt nommé, le nouvel évêque signa une procuration, le , nommant deux procureurs chargés de prendre possession de l'évêché en son nom. L'un d'eux, Jean Valeri s'est alors rendu à Villeneuve-sur-Lot où le sénéchal d'Agenais et le chapitre s'étaient réfugiés pour se protéger de la peste qui sévissait à Agen. Le chapitre fit savoir que les documents présentés n'étaient pas habituels en France et que leur réception serait examinée à la prochaine réunion capitualire.
Après la mort de son oncle, Marc-Antoine a envoyé de Rome une nouvelle procuration datée du à Jean Valeri. Il a alors pu entrer en fonction le : Moi Jean Valeri, clerc et notaire du diocèse d'Yvrée, Piémontais de nation ... j'ai commencé à administrer les affaires de l'évêché, le mercredi vingt-unième jour du mois de novembre de la dite année mil cinq cent vingt.
Le nouvel évêque a fait son entrée solennelle à Agen le . Il avait alors 25 ans.
Il n'a pas laissé de marques particulières. Joseph Scaliger a noté dans la vie de son père Jules César Scaliger arrivé à Agen en 1525, qu'il était très bon et très pieux.
Il est finalement resté peu de temps présent à Agen. Pour Argenton, il ne serait resté que 6 ans, il était présent au concile de Bordeaux du . L'abbé Barrère a constaté sa présence à Hautefage en 1530.
Il est mort à Turin en 1538.
Sources
Le Clergé de France, Tome II
A. Durengues, Antoine de la Rovère, évêque d'Agen (1519-1538), , Revue de l'Agenais, 1929, tome 56 ( lire en ligne )
Personnalité italienne du XVIe siècle
Évêque d'Agen
Décès en 1538
M | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,140 |
The Truth (1920), een Amerikaanse stomme film uit 1920
The Truth (2012) (ook bekend als The Dark Truth), een Amerikaanse thriller uit 2012
The Truth (2019) (ook bekend als La Vérité), een Franse dramafilm uit 2019
The Truth (Prince), een album van Prince uit 1998
The Truth (Beanie Sigel), een album van Beanie Sigel uit 1999
De Waarheid (Pratchett) (Engels The Truth), het 25e boek uit de schijfwereld-serie (Engels: Discworld), geschreven door Terry Pratchett, uit 2000 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,149 |
Is going to court the panacea some think it is?
It is absolutely natural that when you find yourself being denied contact with your grandchildren, that you consider every option to regain contact.
As we all know the obvious one that others continually tell us is, "Well just talk to them," how I wish that was obtainable. If we were all able to sit down and communicate , to discuss it to put things right, of course we all would.
So as a last resort some grandparents choose to go to court to obtain a contact order. And it really is the last resort.
I think you know my views on that particular route, but it is my personal view, I have had several grandparents recently who have been talking about their experiences.
Lawyers often make very misleading comments in articles for newspapers and magazines, giving grandparents false optimism on the outcomes of going to court.
It may be that as stated recently in a report that 7 grandparents a day are taking those steps, what is not clear is, out of those 7 grandparents a day, who have to firstly apply for permission to then apply for a contact order, actually get permission and secondly how many are successful in obtaining a contact order and most importantly how many have a contact order breached?
From the experience of others, here are some of the facts, not the fiction.
One grandfather told me that the toll it had taken on him and his wife was enormous. They had never sought legal advice in their lives and walking into their lawyers office was very difficult. The whole process has left them broken physically and financially.
Don't believe any lawyer who tells you that it is an inexpensive process, it isn't. This particular grandparent paid out thousands of pounds, it was suggested that they set up a standing order to help!
The whole process took a couple of years.
Finally they were overjoyed when they were given a contact order, it would allow then to see their grandchild for an hour once a month. It all seemed as though it had been worthwhile.
You can imagine how the excitement built as the day approached, they arrived leaden with belated presents of birthdays and Christmases past, as the time ticked away they began to realise all was not well.
As they returned home with heavy hearts a message had been left to say their grandchild was ill so couldn't see them. That was just the first time that happened it continued that way, always a reason why it couldn't take place.
They returned for more legal advice and it became clear that they were back to square one and back to court.
As they had spent all their life savings there was nothing left and so they had to walk away.
This particular case is not a one off, it is very common.
Going to court is not the fairytale ending that some believe it is.
Think about it very carefully, are you strong enough to go through the process? Go into it with both eyes open.
Expect nothing and then anything is a bonus. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,038 |
In an unprecedented move, Randwick City Council is looking to put laws in place that would see domestic cats living indoors permanently.
The council's new Labor mayor Kathy Neilson has proposed a motion to stop cats going outside and defecating anywhere but in a litter tray.
Cr Neilson wants fines for cats that "run free or defecate in public places of neighbouring properties".
She also wants to raise registration fees for cat owners.
"It's very unfair to be targeting the owners of cats," he said.
"It just outrageous. It will not be able to be police these archaic and laughable laws and it will be a huge waste of time.
"There are a lot of issues impacting on our native plants and animals," she said, according to the Southern Courier.
"Roaming domestic cats kill about 60 million birds a year," she told the Weekend Australian in June. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,484 |
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