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\section{Introduction\label{sect: intr}}
The Internet-of-Things (IoT), which enables the connections of massive devices through the internet, is one of the key technologies of the fifth-generation (5G) wireless communications \cite{mei2019survey, wong2017key}. With the rapid development of wireless sensor technologies, a huge number of devices are connected via the IoT. However, the power consumption of the associated wireless devices becomes a critical problem \cite{palattella2016internet, chen2020massive, zhang2020prospective}. For example, most of IoT devices are expected to be powered by batteries with limited energy storage capacity and lifetime. As a result, various advanced technologies have been proposed in the literature to tackle the issues \cite{wu2017overview}.
One of the promising solutions is ambient backscatter communication (AmBC), a low-cost and energy-efficient communication scheme which enables passive backscatter devices (e.g., tags, sensors) to transmit their information bits to readers over ambient radio-frequency (RF) signals (e.g., Wi-Fi signals, cellular base station signals, and TV tower radio signals) \cite{van2018ambient, yang2017modulation}. Since ambient RF signals always exist in modern cities, the communication between passive devices is possible without requiring additional power for generating a carrier wave \cite{xie2014managing}. In particular, an AmBC tag of an AmBC system could transmit its binary tag symbols by choosing whether to backscatter the ambient RF signals or not. Thus, the main task of an AmBC system is to perform tag signal detection, i.e., recovering the tag signal at the reader, which has attracted tremendous attention from both academia and industry \cite{van2018ambient}, respectively.
Generally, there are two main challenges for tag detection: (1) since both the direct link signal from the RF source and the backscatter link signal from the tag could be received by the reader simultaneously, the received direct link signal generally causes severe interference to the received backscatter link signal; (2) in contrast to the traditional wireless communication systems, estimating the channel state information (CSI) in AmBC systems is challenging due to the lack of pilot signals sent from the ambient RF source.
Recently, various effective algorithms and schemes have been proposed for tag signal detection in AmBC systems. For example, Liu \emph{et al.} \cite{liu2013ambient} implemented practical ambient backscatter devices and proposed an energy detector adopting a differential coding scheme to decode the tag signals, which paves the way for realizing AmBC systems. Lu \emph{et al.} \cite{lu2015signal} developed an improved energy detection method which however requires the knowledge of perfect CSI. To overcome this problem, Qian \emph{et al.} \cite{qian2017semi} designed a semi-coherent detection method requiring only a few pilots and unknown data symbols, which substantially reduces the signaling overhead and decoding computational complexity.
Furthermore, in order to eliminate the process of channel estimation, Wang \emph{et al.} \cite{wang2016ambient} adopted a differential encoding scheme on tag signals and proposed a minimum BER detector with the corresponding optimal detection threshold. On top of \cite{wang2016ambient}, Qian \emph{et al.} in \cite{qian2016noncoherent} provided fundamental studies of the BER performance for non-coherent detectors.
Nevertheless, the above methods either require channel estimation explicitly or lead to an unsatisfactory detection performance. As a remedy, machine learning (ML)-based methods have been proposed recently which aim to directly recover the tag signals with only a few (training) pilots without the need of estimating relevant channel parameters explicitly. For example, Zhang \emph{et al.} \cite{zhang2018constellation} proposed a clustering method to extract the features of constellation symbols and designed two constellation learning-based signal detection methods, which achieve satisfying BER performance. However, these proposed methods were designed for systems when the ambient RF source signal is a constellation modulated signal.
Furthermore, Hu \emph{et al.} \cite{hu2019machine} transformed the task of tag signal detection into a classification task and designed the support vector machine (SVM)-based energy detection method to improve the BER performance.
However, the proposed ML-based method requires a large number of training pilots and there exists a large gap between the proposed method and the optimal method.
In contrast, different from traditional ML methods, deep learning technology, which adopts a neural network to intelligently explore features in a data-driven manner, has been shown to be able to achieve outstanding performance in many areas, such as natural language processing \cite{young2018recent}, computer vision \cite{goodfellow2016deep}, and also wireless communications \cite{liu2020deepresidualL, liu2020deep, liu2020location, yuan2020learning, liu2020deepresidual}. Nevertheless, practical wireless channels change over time with a huge dynamic range. Besides, transmission in wireless systems can be bursty and the transmission pattern is time varying.
In this case, a well-trained deep neural network (DNN) can hardly perform well in the detection tasks under different channel coefficients.
To overcome these limitations, a more practical approach which can dynamically adjust the weights in DNNs adapting the changing of channel environment is expected.
In contrast to the traditional ML methods, deep transfer learning (DTL) is proposed to adopt a DNN to extract the time-varying features with a few online training data by transferring knowledge from a source domain to a target domain \cite{pan2009survey, weiss2016survey, tan2018survey}.
{However, there are two main challenges when applying DTL to signal detection in AmBC systems:
(1) since the received signal power from the backscatter link is much weaker than that from the direct link, it leads to a small difference in the two hypotheses (i.e., the tag chooses to backscatter the ambient RF source signals or not), which brings difficulties to DTL in completing the classification task \cite{goodfellow2016deep};
(2) different from the general classification tasks in the field of computer vision in which an error rate of around $10\%$ is acceptable \cite{goodfellow2016deep}, the classification task in AmBC systems, i.e., the tag signal detection task, has a much lower bit error rate (BER) requirement, which is generally lower than $10^{-2}$ \cite{3GPPTS2018Quality, van2018ambient}.}
Motivated by this, in this paper, we try to overcome these challenges and propose a framework based on the DTL approach for tag signal detection, which adopts a DNN to transfer the knowledge learned from one tag detection task under the offline channel coefficients to another different but related tag detection task in real-time. The proposed framework can adaptively fine-tune the detector for different channel environments to further improve the detection performance.
To our best knowledge, this is the first study applying DTL to tag signal detection for AmBC systems.
The main contributions of this paper are as follows:
\begin{enumerate}[(1)]
\item We introduce a universal DTL-based tag signal detection framework which consists of offline learning, transfer learning, and online detection. Different from conventional methods which require explicit channel estimation, our proposed scheme adopts a DNN to extract the features of channel and directly recover the tag symbols, providing a more practical approach for tag signal detection.
Specifically, according to the minimum error probability (MEP) criterion, a DTL-based likelihood ratio test (LRT) is derived for tag signal detection, which enables the design of an effective detector.
\item Under the developed framework, we creatively adopt a convolutional neural network (CNN) to explore the features of the sample covariance matrix. Inspired by the novel covariance matrix aware-CNN (CM-CNN) structure \cite{liu2019deep}, \cite{xie2019activity}, we develop a DTL-oriented covariance matrix aware neural network (CMNet) for tag signal detection and propose the related tag signal detection algorithm. Exploiting the powerful capability of CNN in exploring features of data in a matrix form, the proposed method could extract more discriminative features to further improve the BER performance.
\item Although it is intractable to directly analyze the performance of a CNN, we formulate the proposed CMNet as a non-linear function and derive an asymptotic explicit expression to characterize its properties when the number of samples is sufficiently large.
\item Extensive experiments have been conducted using both the modulated and complex Gaussian ambient sources. The results show that the proposed algorithm can achieve an outstanding BER performance which is close to that of the optimal LRT method with perfect CSI.
\end{enumerate}
The remainder of this paper is organized as follows. Section \Rmnum{2} formulates the AmBC system model. In Section \Rmnum{3}, we develop a DTL-based detection framework for the AmBC system. As a realization of the developed framework, Section \Rmnum{4} proposes a novel CMNet-based tag signal detection algorithm and provides the corresponding theoretical analysis. Extensive simulation results are presented in Section \Rmnum{5}, and Section \Rmnum{6} finally concludes the work of this paper.
The notations used in this paper are shown in the following. The superscripts $T$ and $H$ indicate the transpose and conjugate transpose, respectively. Term ${\mathcal{CN}}( \bm{\mu},\mathbf{\Sigma} )$ represents a circularly symmetric complex Gaussian (CSCG) distribution with a mean vector $\bm{\mu}$ and a covariance matrix $\mathbf{\Sigma}$. Term ${\bf{I}}_M$ is used to denote the $M$-by-$M$ identity matrix and ${\mathbf{0}}$ is used to denote the zero vector. $(\cdot)^{-1}$ indicates the matrix inverse operation. $\max(a,b)$ is the maximum value of $a$ and $b$. Subscripts $S$ and $T$ represent the source domain and the target domain, respectively. $|\cdot|$ denotes the cardinality of a set. $\mathrm{Re}(\cdot)_{i,j}$ and $\mathrm{Im}(\cdot)_{i,j}$ represent the $i$-th row and the $j$-th column elements of the real part and imaginary part, respectively. $\det(\cdot)$ is the determinant operator. $P(\cdot)$ and $E(\cdot)$ represent the probability of an event and the statistical expectation, respectively. $\|\cdot\|^2$ denotes the norm of an input vector. $\exp(\cdot)$ represents the exponential function.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{AmBC_scenario.pdf}\vspace{-0.6cm}
\caption{ An illustration of the considered ambient backscatter communication system. }\vspace{-1cm}
\end{figure}
\section{System Model}
In this paper, we consider a general AmBC system, which consists of an ambient RF source, a tag, and a reader, as depicted in Fig. 1. Both the RF source and the passive tag are equipped with a single antenna, while the reader is equipped with an $M$-element antenna array for signal detection. Due to the broadcasting nature of the RF source, the transmitted RF signal is received by both the reader and the tag simultaneously. Although the tag is a passive device, it can transmit its binary tag symbols by choosing whether to reflect the ambient RF signals to the reader. In this case, the reader can then decode the tag symbols through sensing the changes from the received signals.
The frame structure of the signal model at the reader is shown in Fig. 2, which consists of $P$ pilot symbols{{\footnotemark}}\footnotetext{{Note that the pilots are not used for acquiring the channel information explicitly, while they are exploited to build the online training dataset for the learning algorithm to capture the features of the real-time channel to facilitate the tag signal detection. By doing this, the proposed framework not only eliminates the requirement of explicit channel estimation, but also further improves the system performance.}} and $T-P$ $(T>P)$ data symbols in one frame. For the pilots, the tag symbols are known by the reader, and the remaining tag symbols are used for data transmission.
{The tag frame structure considered in this paper is designed for slow fading channel models, i.e., the channel remains unchange during each frame, which is commonly adopted in the literature for AmBC and RFID applications, e.g., \cite{van2018ambient, finkenzeller2010rfid}. For the case of fast fading channel, since the pilot-based training data and the test data come from different channel environments, the learned features by the neural network from the training data do not match with the test data and thus they cannot be used for the tag signal detection. As a result, a new tag frame structure is desired which will be left for future work.} In addition, the tag transmits its bits at a rate $N$ times lower than the sampling rate of the RF source signal. Thus, the source-to-tag ratio (STR) which is defined as the number of RF source symbols in one tag symbol period is $N$.
In other words, each tag symbol remains the same within the $N$ RF source symbol periods.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\linewidth]{AmBC_frame_structure.pdf}\vspace{-0.6cm}
\caption{ The tag frame structure of the considered AmBC system. The shaded parts are the pilot signals labelled and the remaining parts are the test data.}\vspace{-1cm}
\end{figure}
Denote by $c^{(t)}\in \mathcal{C} = \{0,1\}$ the $t$-th tag symbol with the binary on-off keying modulation, i.e., $c^{(t)} = 0$ refers to that the tag does not reflect the RF source signal; otherwise, the tag chooses to reflect the RF source signal. Correspondingly, we use $s_n^{(t)}$ to represent the $n$-th RF source signal sample within the tag symbol $c^{(t)}$.
Let $ \mathbf{x}_n^{(t)}={{[x_{n,1}^{(t)},x_{n,2}^{(t)},\cdots ,x_{n,M}^{(t)}]}^{T}}, n \in \{ 0,1,\cdots,N-1\}$, represent the $n$-th observation vector within the $t$-th, $t\in \{ 1,\cdots,T \}$, tag symbol period, where $x_{n,m}^{(t)}$, $m \in \{1,2, \cdots, M\}$, denotes the $n$-th discrete-time sample observed at the $m$-th antenna element.
Therefore, the received signal at the reader can be expressed as \cite{guo2018exploiting, yang2015multi}
\begin{equation}\label{sr}
\mathbf{x}_n^{(t)} = \mathbf{h}s_n^{(t)} + \alpha f\mathbf{g}s_n^{(t)}c^{(t)} + \mathbf{u}_n^{(t)}, \forall n,t.
\end{equation}
Here, $\mathbf{h}=[h_1, h_2, \cdots, h_M ]^T$, where $h_m \in \mathbb{C}$ is the channel coefficient from the RF source to the $m$-th antenna at the reader. Similarly, $\mathbf{g}=[g_1, g_2, \cdots, g_M]^T$, where $g_m \in \mathbb{C}$ is the channel coefficient from the tag to the $m$-th antenna at the reader. Variables $f$, $\alpha\in \mathbb{C}$ are the channel coefficient from the RF source to tag and the reflection coefficient of the tag, respectively. In addition, $\mathbf{u}_n^{(t)}\in \mathbb{C}^{M\times1}$ is assumed to be an independent and identically distribution (i.i.d.) CSCG random noise vector with $\mathbf{u}_n^{(t)}\sim \mathcal{CN}( \mathbf{0},\sigma _u^2{{\mathbf{I}}_M} )$, where $\sigma_u^2$ represents the noise variance at each antenna of the reader. Considering a tag symbol is time invariant within $N$ RF source signal sampling periods, let
\begin{equation}\label{observation matrix} \vspace{-0.05cm}
\mathbf{X}^{(t)}=[\mathbf{x}_1^{(t)},\mathbf{x}_2^{(t)},\cdots,\mathbf{x}_N^{(t)}], \forall t, \vspace{-0.05cm}
\end{equation}
denote a sampling matrix collecting the $N$ observations of the $t$-th tag symbol at the reader. The task of the tag signal detection is to recover $c^{(t)}$ based on $\mathbf{X}^{(t)}$.
Thus, the tag signal detection can be further formulated as a binary hypothesis testing problem:
\begin{equation}\label{sensing model}
\begin{split}
& {{H}_{1}}:\mathbf{x}_n^{(t)} = \mathbf{w}s_n^{(t)} + \mathbf{u}_n^{(t)}, \\
& {{H}_{0}}:\mathbf{x}_n^{(t)} = \mathbf{h}s_n^{(t)} + \mathbf{u}_n^{(t)}, \\
\end{split}
\end{equation}
where $\mathbf{w} = \mathbf{h}+\alpha f\mathbf{g}$, and $H_1$ and $H_0$ denote the hypotheses that $c^{(t)}=1$ and $c^{(t)}=0$, respectively.
Note that the ambient source signal in practice may arise from either an unknown (or indeterminate) ambient RF source or a known and discrete constellation ambient RF source \cite{qian2017semi}. Therefore, we consider two different ambient sources:
\begin{enumerate}[(a)]
\item \emph{Complex Gaussian Ambient Source}: $s_n^{(t)}$ is assumed to be a CSCG random variable{\footnotemark}\footnotetext{In the absence of any priori knowledge of $s_n^{(t)}$, a general approach is to model $s_n^{(t)}$ by a Gaussian random variable \cite{liu2019maximum, qian2017semi}.} with $s_n^{(t)}\sim \mathcal{C}\mathcal{N}(0,\sigma_s^2)$;
\item \emph{Modulated Ambient Source}: $s_n^{(t)}$ is assumed to be a Q-ary modulated symbol with power $\sigma_s^2$ drawn from a constellation set $\mathcal{S}=\{{S}_1,{S}_2, \cdots, {S}_Q\}$, with an equal probability.
\end{enumerate}
For the sake of presentation, we first define the received signal-to-noise ratio (SNR) of the direct link as
\begin{equation}\label{SNR}
\mathrm{SNR} = \frac{E(||\mathbf{h}s_n^{(t)}||^2)}{E(||\mathbf{u}_n^{(t)}||^2)}.
\end{equation}
Besides, the relative coefficient between the direct signal path and the backscattered signal path is defined as a ratio of their average channel gains which is given by
\begin{equation}\label{relative_SNR}
\zeta = \frac{E(||\alpha f \mathbf{g}||^2)}{E(||\mathbf{h}||^2)}.
\end{equation}
Then, in the following, we will introduce the optimal likelihood ratio test for above two cases and the corresponding results serve as a benchmark to the considered AmBC system.
\emph{1) \underline{Optimal LRT with Complex Gaussian Ambient Source:}}
When $s_n^{(t)}$ is a complex Gaussian signal as defined in (a), we have
\begin{equation}\label{random-x(n)}
\mathbf{x}_n^{(t)}\sim
\bigg\{ \begin{matrix}
\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{\Sigma}_1),\;{{H}_{1}} \\
\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{\Sigma}_0),\;{{H}_{0}} \\
\end{matrix},
\end{equation}
where $\mathbf{\Sigma}_1 = \sigma_s^2\mathbf{w}\mathbf{w}^{H}+\sigma _{u}^{2}{{\mathbf{I}}_{M}}$ and $\mathbf{\Sigma}_0 = \sigma_s^2\mathbf{h}\mathbf{h}^{H}+\sigma _{u}^{2}{{\mathbf{I}}_{M}}$.
According to \cite{kay1998fundamentals}, if perfect CSI, e.g., $\mathbf{w}$ and $\mathbf{h}$, are known at the reader, we can then derive the logarithmic form of likelihood ratio test (LRT) under the complex Gaussian distributed ambient source:
\begin{equation}\label{L_R}
L_{\mathrm{CG}}({\mathbf{X}}^{(t)}) = \sum\limits_{n = 0}^{N - 1} \ln \left({\frac{{p\left( {\mathbf{x}_n^{(t)}|{H_1};\mathbf{0},{\mathbf{\Sigma}_1}} \right)}}{{p\left( {\mathbf{x}_n^{(t)}|{H_0};\mathbf{0},{\mathbf{\Sigma}_0}} \right)}}}\right),
\end{equation}
where
\begin{equation}\label{}
p\left( {\mathbf{x}_n^{(t)}|{H_1};\mathbf{0},{\mathbf{\Sigma} _1}} \right) = \frac{1}{{{\pi ^M}\det ({\mathbf{\Sigma} _1})}}\exp \left( { - (\mathbf{x}_n^{(t)})^H\mathbf{\Sigma} _1^{ - 1}{{\mathbf{x}_n^{(t)}}}} \right)
\end{equation}
and
\begin{equation}\label{}
p\left( {\mathbf{x}_n^{(t)}|{H_0};\mathbf{0},{\mathbf{\Sigma} _0}} \right) = \frac{1}{{{\pi ^M}\det ({\mathbf{\Sigma} _0})}}\exp \left( { - ( \mathbf{x}_n^{(t)}) ^H\mathbf{\Sigma} _0^{ - 1}{{\mathbf{x}_n^{(t)}}}} \right).
\end{equation}
\emph{2) \underline{Optimal LRT with Modulated Ambient Source:}}
When $s_n^{(t)}$ is a discrete modulated signal as defined in (b), we have
\begin{equation}\label{modulated-x(n)}
\mathbf{x}_n^{(t)}\sim
\bigg\{ \begin{matrix}
\mathcal{C}\mathcal{N}(\mathbf{w}S_q,\sigma _{u}^{2}{\mathbf{I}}_{M}),\;{{H}_{1}} \\
\mathcal{C}\mathcal{N}(\mathbf{h}S_q,\sigma _{u}^{2}{\mathbf{I}}_{M}),\;{{H}_{0}} \\
\end{matrix}.
\end{equation}
Similarly, we can then derive the logarithmic form of LRT under modulated ambient source:
\begin{equation}\label{L_M}
L_\mathrm{M}(\mathbf{X}^{(t)}) = \sum\limits_{n = 0}^{N - 1} \ln \left({\frac{\sum\limits_{q = 1}^{Q}{p\left( {\mathbf{x}_n^{(t)}|{H_1};\mathbf{w}S_q,\sigma _{u}^{2}{\mathbf{I}}_{M}} \right)}}{\sum\limits_{q = 1}^{Q}{p\left( {\mathbf{x}_n^{(t)}|{H_0};\mathbf{h}S_q,\sigma _{u}^{2}{\mathbf{I}}_{M}} \right)}}}\right),
\end{equation}
where
\begin{equation}\label{}
p\left( {\mathbf{x}_n^{(t)}|{H_1};\mathbf{w}S_q,{\sigma _{u}^{2}{\mathbf{I}}_{M}}} \right) = \frac{1}{{({\pi}\sigma _{u}^2)^{M}}}\exp \left( { - \frac{1}{\sigma_u^2}(\mathbf{x}_n^{(t)} - \mathbf{w}S_q )^H{{(\mathbf{x}_n^{(t)} - \mathbf{w}S_q)}}} \right)
\end{equation}
and
\begin{equation}\label{}
p\left( {\mathbf{x}_n^{(t)}|{H_0};\mathbf{h}S_q,{\sigma _{u}^{2}{\mathbf{I}}_{M}}} \right) = \frac{1}{{({\pi}\sigma _{u}^2)^{M}}}\exp \left( { - \frac{1}{\sigma_u^2}(\mathbf{x}_n^{(t)} - \mathbf{h}S_q )^H{{(\mathbf{x}_n^{(t)} - \mathbf{h}S_q)}}} \right).
\end{equation}
Based on the above analysis, we can find that although the LRT can achieve the optimal detection performance, it requires the availability of perfect CSI which is not always possible in practical AmBC systems due to the non-cooperation between the legacy transceiver and the reader. On the other hand, the covariance matrix captures various distinguishable features (such as energies and eigenvalues \cite{liu2019deep}) implicitly, it has been adopted to design detectors, as shown in (\ref{L_R}) and (\ref{L_M}). Inspired by this, we propose a tag detection framework based on a DNN which intelligently explores the features of sample covariance matrix.
In the following, we will first introduce the framework and then propose a covariance matrix aware neural network as a realization of the developed framework.
\begin{figure}[t]
\centering
\includegraphics[width=0.98\linewidth]{TL_framework.pdf}\vspace{-0.6cm}
\caption{ The proposed DTL framework for tag signal detection. }\vspace{-1cm}
\end{figure}
\section{Deep Transfer Learning-based Tag Signal Detection Framework}
Note that practical channel coefficients change over time due to the time-varying nature of the environment. Based on this, we can adopt a DTL approach to transfer the knowledge obtained from one detection task via offline channel coefficients to another different but related detection task under the real-time channel coefficients. The advanced DTL approach allows the designed detector to adapt itself properly to different channel environments for improving the system performance.
As shown in Fig. 3, we propose a DTL framework, which consists of offline learning, transfer learning, and online detection. {In the proposed framework, we first establish a pre-trained DNN to extract the common features of invariant channel models through offline learning. We then freeze{\footnotemark} \footnotetext{Layer freezing means that the layer weights of a pre-trained neural network keep unchanged during training in a subsequent task, i.e., they remain frozen.} partial layers of the pre-trained DNN and only fine-tune the remaining layers to adjust the DNN to the current channel coefficients through (online) transfer learning. Finally, we can apply the well-trained network for online detection: decoding the tag signals.} {Therefore, the proposed framework does not require the channel information. Instead, it exploits the pilot-based training dataset to capture the features of the real-time channel to facilitate the tag signal detection.} In the following, we will introduce the initialization of deep transfer learning, offline learning module, transfer learning module, and online detection module, respectively.
\subsection{Initialization for Deep Transfer Learning}
The unified dataset adopted in this paper is given by
\begin{equation}\label{unified_dataset}
(Y,Z) = \{({{y}}^{(1)},{{{z}}^{(1)}}),({{y}}^{(2)},{{{z}}^{(2)}}), \cdots ,({{y}}^{(K)},{{{z}}^{(K)}})\},
\end{equation}
where $({{y}}^{(k)},{{{z}}^{(k)}})$ represents the $k$-th, $k\in\{1,2,\cdots,K\}$, example of the training set $(Y,Z)$. In particular, $y^{(k)}$ denotes the input feature which is the test statistic of raw samples and can be regarded as a function of $\mathbf{X}^{(t)}$, as defined in (\ref{observation matrix}). Term $z^{(k)}\in\{1,0\}$ is a label, where ${z}^{(k)}=1$ and ${z}^{(k)}=0$ denote the hypotheses of $H_1$ and $H_0$, respectively.
According to \cite{pan2009survey}, a domain can be defined as $\mathcal{D} = \{\mathcal{Y},P(Y)\}$, where $\mathcal{Y}$ denotes the feature space and $P(Y)$ indicates the marginal probability distribution with $Y = \{ {y}^{(1)}, {y}^{(2)}, \cdots, {y}^{(K)} \} \in \mathcal{Y}$, as defined in (\ref{unified_dataset}).
Correspondingly, the task of $\mathcal{D}$ is defined by $\mathcal{T} = \{\mathcal{Z},P(Z|Y)\}$, where $\mathcal{Z}$ denotes the label space and $P(Z|Y)$ is the posterior probability distribution for $\mathcal{D}$ with $Z = \{ {z}^{(1)}, {z}^{(2)}, \cdots, {z}^{(K)} \} \in \mathcal{Z}$, as defined in (\ref{unified_dataset}). Thus, $P(Z|Y)$ can be regarded as a prediction function $\lambda(\cdot)$ which is used to predict the labels of the inputs.
Note that in this paper, the source domain data and the target domain data arise from the training sets of offline learning and transfer learning, respectively. We can then define the source domain and the target domain as \cite{pan2009survey}
\begin{equation}\label{}
\mathcal{D}_S = \{\mathcal{Y}_S,P(Y_S)\}
\end{equation}
and
\begin{equation}\label{}
\mathcal{D}_T = \{\mathcal{Y}_T,P(Y_T)\},
\end{equation}
respectively.
Here, $\mathcal{Y}_S$ and $\mathcal{Y}_T$ are the feature spaces of the source domain and the target domain, respectively. Terms $P(Y_S)$ and $P(Y_T)$ are the marginal probability distributions of the source domain data and the target domain data, respectively, where $Y_S =\{y_S^{(1)}, y_S^{(2)}, \cdots , y_S^{(K_S)}\} \in \mathcal{Y}_S$ and $Y_T =\{y_T^{(1)}, y_T^{(2)}, \cdots , y_T^{(K_T)}\}$ $\in \mathcal{Y}_T$. Correspondingly, we denote
\begin{equation}\label{}
\mathcal{T}_S = \{\mathcal{Z}_S,P(Z_S|Y_S)\}
\end{equation}
and
\begin{equation}\label{}
\mathcal{T}_T = \{\mathcal{Z}_T,P(Z_T|Y_T)\}
\end{equation}
as the task for $\mathcal{D}_S$ and $\mathcal{D}_T$, respectively. Here, $\mathcal{Z}_S$ and $\mathcal{Z}_T$ denote the label spaces for $\mathcal{D}_S$ and $\mathcal{D}_T$, respectively. Terms $P(Z_S|Y_S)$ and $P(Z_T|Y_T)$ are the posterior probability distributions for $\mathcal{D}_S$ and $\mathcal{D}_T$, respectively, where $Z_S =\{z_S^{(1)}, z_S^{(2)}, \cdots , z_S^{(K_S)}\} \in \mathcal{Z}_S$ and $Z_T =\{z_T^{(1)}, z_T^{(2)}, \cdots , z_T^{(K_T)}\} \in \mathcal{Z}_T$.
Let $\mathbf{\Omega}_{\mathbf{h}_S}$ and $\mathbf{\Omega}_{\mathbf{w}_S}$ (or $\mathbf{\Omega}_{\mathbf{h}_T}$ and $\mathbf{\Omega}_{\mathbf{w}_T}$) denote the sets of channel coefficients $\mathbf{h}$ and $\mathbf{w}$ in the source domain (or the target domain), respectively. In this paper, we assume that the channel coefficients of $\mathcal{D}_S$ and $\mathcal{D}_T$ are i.i.d. with distinct values{\footnotemark}\footnotetext{This i.i.d. assumption is applicable to many practical channel environments. In fact, the proposed framework in this paper is also valid for the case which the channel coefficients of $\mathcal{D}_S$ and $\mathcal{D}_T$ are drawn from different distributions. In this case, it will take more time for the pre-trained network in the source domain to adjust itself to the target domain compared to that of the i.i.d. case.}.
Thus, $\forall i\in [1,|\mathbf{\Omega}_{\mathbf{h}_S}|],j\in [1,|\mathbf{\Omega}_{\mathbf{w}_S}|], p\in [1,|\mathbf{\Omega}_{\mathbf{h}_T}|], q\in [1,|\mathbf{\Omega}_{\mathbf{w}_T}|]$, where $\mathbf{\Omega}(i)$ represents the $i$-th element of the set, we have $\mathbf{\Omega}_{\mathbf{h}_S}(i) \neq \mathbf{\Omega}_{\mathbf{h}_T}(j)$, and $\mathbf{\Omega}_{\mathbf{w}_S}(p) \neq \mathbf{\Omega}_{\mathbf{w}_T}(q)$.
Based on this, it is obvious that $P(Y_S) \neq P(Y_T)$ and we have $\mathcal{D}_S \neq \mathcal{D}_T$.
Therefore, the objective of transfer learning is to improve the learning of the detection problem in $\mathcal{D}_T$ exploiting the knowledge in $\mathcal{D}_S$. To the end, according to \cite{tan2018survey}, we provide the definition of deep transfer learning as follows:
\textbf{Definition 1}. \emph{Deep Transfer Learning (DTL)}. Given a transfer learning task defined by $\langle\mathcal{D}_S, \mathcal{T}_S,$ $\mathcal{D}_T, \mathcal{T}_T, \lambda_T(\cdot)\rangle$. Deep transfer learning aims to improve the performance of predictive function $\lambda_T(\cdot)$ for learning task $\mathcal{T}_T$ by using a deep neural network to discover and transfer latent knowledge from $\mathcal{D}_S$ and $\mathcal{T}_S$, where $\mathcal{D}_S \neq \mathcal{D}_T$ or/and $\mathcal{T}_S \neq \mathcal{T}_T $. \hfill \rule{8pt}{8pt}
According to the definition of DTL and the proposed DTL framework in Fig. 3, the dataset of the offline learning is drawn from the source domain, and the datasets of the transfer learning and the online detection both come from the target domain. In the following, we will systematically introduce the proposed DTL framework based on the definition of DTL.
\subsection{Offline Learning: Pre-trained DNN}
Given the training set of offline learning:
\begin{equation}\label{training_set_expression}
D_S = (Y_S,Z_S) = \{(y_S^{(1)},{z_S^{(1)}}),(y_S^{(2)},{z_S^{(2)}}), \cdots ,(y_S^{(K_S)},{z_S^{(K_S)}})\},
\end{equation}
where $D_S$ denotes the source domain training data, ${y}_S^{(k)}\in \mathcal{D}_S$ and ${{z}_S^{(k)}}\in \mathcal{T}_S$ represent the input and label of the $k$-th, $k\in\{1,2,\cdots,K_S\}$, example, respectively.
Note that the tag detection for AmBC system is actually a binary hypothesis testing problem \cite{qian2017semi}. In this case, the training process becomes a binary classification problem and thus the label is encoded as a one-hot vector \cite{wang2017deep}
\begin{equation}\label{}
{{{\bf{z}}_S}^{(k)}} = \Bigg\{ {\begin{array}{*{20}{c}}
{{{[1,0]}^T},{\kern 1pt} {\kern 1pt} {\kern 1pt} {z_S^{(k)}=1}}\\
{{{[0,1]}^T},{\kern 1pt} {\kern 2pt} {z_S^{(k)}=0}}
\end{array}}.
\end{equation}
Correspondingly, the output of the DNN is a $2 \times 1$ class score{\footnotemark} vector \cite{goodfellow2016deep}: \footnotetext{{The class score created by a DNN represents the likelihood of each category.}}
\begin{equation}\label{feature_vector}
{f_{\theta_S}}({y}_S^{(k)}) = \left[ {\begin{array}{*{20}{c}}
{{f_{\theta_S|{H_1}}}({y}_S^{(k)})}\\
{{f_{\theta_S|{H_0}}}({y}_S^{(k)})}
\end{array}} \right],
\end{equation}
where ${f_{\theta_S|{H_1}}}({y}_S^{(k)}) + {f_{\theta_S|{H_0}}}({y}_S^{(k)}) = 1$, $0\leq f_{\theta_S}(\cdot) \leq 1$ is the expression of the DNN under the parameter of $\theta_S$, and ${f_{\theta_S|{H_i}}}({y}_S^{(k)})$ represents the $H_i$ class score of the input ${y}_S^{(k)}$ under $f_{\theta_S}(\cdot)$.
In fact, the class score can be rewritten as the following probability expressions \cite{goodfellow2016deep}:
\begin{equation}\label{conditional_pro_single_example}
\begin{array}{l}
{H_1}:{f_{\theta_S|{H_1}}}({y}_S^{(k)}) = P( z_S^{(k)} = 1|{y}_S^{(k)};\theta_S ),\\
{H_0}:{f_{\theta_S|{H_0}}}({y}_S^{(k)}) = P( z_S^{(k)} = 0|{y}_S^{(k)};\theta_S ),
\end{array}
\end{equation}
where $P( z_S^{(k)}|{y}_S^{(k)};\theta_S )$ denotes the conditional probability under $\theta_S$.
Therefore, the goal of the offline learning is to maximize the likelihood
\begin{equation}\label{}
L(\theta_S ) = P(Z_S|Y_S;\theta_S )=\prod\limits_{k = 1}^{K_S} {{{({f_{\theta_S|{H_1}} }({y_S^{(k)}}))}^{{z_S^{(k)}}}}{{( {f_{\theta_S|{H_0}} }({y_S^{(k)}}))}^{1 - {z_S^{(k)}}}}},
\end{equation}
or equivalently the log-likelihood which is given as
\begin{equation}\label{}
l(\theta_S ) = \ln L(\theta_S )= \sum\limits_{k = 1}^{K_S} {z_S^{(k)}}\ln {f_{\theta_S|{H_0}}}({y_S^{(k)}}) + (1 - {z_S^{(k)}})\ln({f_{\theta_S|{H_0}}}({y_S^{(k)}})).
\end{equation}
Therefore, the target of offline learning is to find parameter $\theta_S^*$ to maximize $P(Z_S|Y_S)$, i.e.,
\begin{equation}\label{theta_MAP}
{\theta_S ^*} = \arg {\kern 1pt} {\kern 1pt} \mathop {\max }\limits_{\theta_S} P( Z_S|Y_S;\theta_S ),
\end{equation}
which is also equivalent to minimizing the following cost function
\begin{equation}\label{cost_function}
J(\theta_S ) = - \frac{1}{K_S}l(\theta_S )= - \frac{1}{K_S} \sum\limits_{k = 1}^{K_S} {{z_S^{(k)}}\ln {f_{\theta_S|{H_1}} }({y_S^{(k)}})} + {(1 - {z_S^{(k)}})\ln({f_{\theta_S|{H_0}} }({y_S^{(k)}}))}.
\end{equation}
Based on (\ref{cost_function}), we can then obtain a well-trained DNN through the backpropagation (BP) algorithm \cite{Yi2018Unifying}. In this case, the output of the well-trained DNN can be expressed as
\begin{equation}\label{fv_function}
f_{\theta_S^*}( y ) = \left[ {\begin{array}{*{20}{c}}
{{f_{\theta_S^*|{H_1}}}( y )}\\
{{f_{\theta_S^*|{H_0}}}( y )}
\end{array}} \right],
\end{equation}
where $y$ denotes arbitrary input and $f_{\theta_S^*}( \cdot )$ denotes the well-trained DNN with the well-trained parameter $\theta_S^*$. Correspondingly, ${f_{\theta_S^*|{H_i}}}(y)$ represents the $H_i$ class score of the input $y$ under $f_{\theta_S^*}( \cdot )$.
Since the well-trained DNN is achieved in $\mathcal{D}_S$, we also call it as the pre-trained DNN.
In next subsection, we will introduce transfer learning stage based on the pre-trained DNN.
\subsection{Transfer Learning: Fine-tuning the Pre-trained DNN}
Given the pilots-based set for transfer learning:
\begin{equation}\label{training_set_expression}
D_T = (Y_T,Z_T) = \{(y_T^{(1)},z_T^{(1)}),(y_T^{(2)},z_T^{(2)}), \cdots ,(y_T^{(K_T)},z_T^{(K_T)})\},
\end{equation}
where $D_T$ denotes the target domain training data{\footnotemark\footnotetext{{Since the pilot symbols and the test data in the same tag frame experience the same channel coefficients in slow fading channels, they have an identical statistical distribution. Thus, the target-domain training data is chosen from the pilot-based data, from which the neural network can learn the features of the test data to facilitate the tag signal detection.}}}, ${y}_T^{(k)}\in \mathcal{D}_T$ and ${z}_T^{(k)}\in \mathcal{T}_T$ represent the input and the label of the $k$-th, $k\in\{1,2,\cdots,K_T\}$, example, respectively. Correspondingly, $y_T^{(k)}$ and $z_T^{(k)}\in\{1,0\}$ are the input of the network and the label, respectively.
Considering both the source task and the target task handle the same tag detection problem, we can reuse the partial feature layers of the pre-trained DNN in the source domain for performance improvement in the target domain, i.e., transferring and transforming partial layers of the pre-trained DNN to be a part of the DNN in the target domain. Based on this, we can then express the DNN in the target domain as
\begin{equation}\label{DNN_target}
{f_{{\theta _T}}}(y) = f_{{\theta _T}}^{\mathrm{FL}}(f_{\theta _S^*}^{\mathrm{TL}}(y)),
\end{equation}
where ${f_{{\theta _T}}}(\cdot)$ is the target domain's DNN expression with parameter $\theta _T$, $f_{{\theta _T}}^{\mathrm{FL}}(\cdot)$ represents the fine-tuning layers of the target domain's DNN, and $f_{\theta _S^*}^{\mathrm{TL}}(\cdot)$ indicates the transfer layers of the well-trained DNN from the source domain. To achieve (\ref{DNN_target}), we freeze the transfer layers and only update the parameters of the fine-tuning layers during the training process. Similar to (\ref{cost_function}), we define the cost function for the target domain as
\begin{equation}\label{cost_function_target}
J(\theta_T ) = - \frac{1}{K_T}l(\theta_T ) = - \frac{1}{K_T} \sum\limits_{k = 1}^{K_T} {{z_T^{(k)}}\ln {f_{\theta_T|{H_0}} }({y_T^{(k)}})} + {(1 - {z_T^{(k)}})\ln(1 - {f_{\theta_T|{H_0}}}({y_T^{(k)}}))}.
\end{equation}
By using the BP algorithm in the training process, we can obtain the well-trained parameter $\theta_T^*$ for the target domain's DNN. Note that $\theta_T^*$ contributes to $\theta_S^*$, hence, the well-trained DNN for the target domain can be expressed as
\begin{equation}\label{fv_function}
f_{\theta_T^*}( y ) = f_{{\theta _T^*}}^{\mathrm{FL}}(f_{\theta _S^*}^{\mathrm{TL}}(y)) = \left[ {\begin{array}{*{20}{c}}
{{f_{\theta_T^*|{H_1}}}( y )}\\
{{f_{\theta_T^*|{H_0}}}( y )}
\end{array}} \right],
\end{equation}
where $f_{\theta_T^*}( \cdot )$ denotes the expression of the well-trained DNN with the well-trained parameter $\theta_T^*$. Function ${f_{\theta_T^*|{H_i}}}(y)$ represents the $H_i$ class score of input $y$ under $f_{\theta_T^*}( \cdot )$.
From a probabilistic viewpoint, if $y \in \mathcal{D}_T$, we can then rewrite the outputs of the well-trained DNN as the following:
\begin{equation}\label{}
\begin{split}
{H_1}: f_{\theta_T^*|{H_1}}(y) = P({H_1}|y), \\
{H_0}: f_{\theta_T^*|{H_0}}(y) = P({H_0}|y), \\
\end{split}
\end{equation}
where $P({H_i}|y)$ denotes the posterior probability expression.
Based on Bayes' theorem, we can obtain the likelihood expressions:
\begin{equation}\label{ConPro_H1}
L({H_1}|y) = \frac{{P({H_1}|y)}\cdot P(y)}{{P({H_1})}} = \frac{{f_{\theta_T^* |{H_1}}^*(y)}\cdot P(y)}{{P({H_1})}}
\end{equation}
and
\begin{equation}\label{ConPro_H0}
L({H_0}|y) = \frac{{P({H_0}|y)}\cdot P(y)}{P({H_0})} = \frac{{f_{\theta_T^* |{H_0}}^*(y)}\cdot P(y)}{{P({H_0})}},
\end{equation}
where $P(H_i)$ is the priori probability of $H_i$, and $P(y)$ is the marginal probability. Note that we always set $P(H_1)=P(H_0)=0.5$ for binary communication systems. Therefore, according to the minimum error probability (MEP) criterion, we can then derive the DTL-based LRT (DTL-LRT):
\begin{equation}\label{T_DNN}
{L}_{\mathrm{DTL}}(y) = \frac{L({H_1}|y)}{L({H_0}|y)} = \frac{f_{\theta_T^*|{H_1}}(y)}{f_{\theta_T^*|{H_0}}(y)} \gtrless 1,
\end{equation}
where we make a decision that $H_1$ holds if ${L}_{\mathrm{DTL}}(y) > 1$, otherwise $H_0$ holds.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{TL_DTL_LRT.pdf}\vspace{-0.6cm}
\caption{ A flowchart of the derivation of the DTL-based LRT. }\vspace{-1cm}
\end{figure}
For further understanding, we summarize the derivation of DNN-LRT as a flow-chart in Fig. 4. Given the training set, we can first obtain the posterior probability expressions through DTL. Then, based on the Bayes' theorem, we can get the expressions of likelihood of two hypothesis. Finally, according to the minimum error probability (MEP) criterion, we can derive the expression of the DTL-LRT.
\subsection{Online Detection: DTL-LRT using Well-trained DNN }
Given an arbitrary test data $\tilde{y} \in \mathcal{D}_T$, we can directly send it to the well-trained DNN to operate the DTL-LRT, i.e.,
\begin{equation}\label{T_DNN}
{L}_{\mathrm{DTL}}(\tilde{y}) = \frac{f_{\theta_T^*|{H_1}}(\tilde{y})}{f_{\theta_T^*|{H_0}}(\tilde{y})} \gtrless 1,
\end{equation}
where if ${L}_{\mathrm{DTL}}(\tilde{y})>1$, $H_1$ holds, otherwise, $H_0$ holds.
\textbf{Remark 1}: Note that the proposed DTL scheme is a universal DTL workflow for tag signal detection, which is suitable for any neural network and can be realized by any kind of DNN structure, e.g., the multi-layer perceptron, recurrent neural network, and CNN \cite{lecun2015deep}. The proposed DTL scheme does not require to estimate the channel coefficients explicitly. Instead, we only need a few tag signal pilots for transfer learning, which is a practical detection approach for AmBC systems.
{In addition, the proposed method can be extended to the case of $M$-ary tag modulation \cite{van2018ambient, qian2016noncoherent} by extending the current binary classification task to an $M$-class classification task \cite{goodfellow2016deep}, which can further improve the system performance and is left for future work.}
\section{CNN-based Deep Transfer Learning for Tag Signal Detection}
In this section, we provide a realization of the proposed DTL workflow. Note that the sample covariance matrix is a versatile statistic capturing rich distinguishable features. Meanwhile, CNN has a powerful capability in extracting features of matrix-formed data. Therefore, we apply the novel CM-CNN \cite{liu2019deep}, \cite{xie2019activity} to explore the features of the covariance matrix by proposing a covariance matrix-based neural network (CMNet).
In the following, we will introduce the CMNet structure, design the CMNet-based detection algorithm{\footnotemark}{\footnotetext{{Since the backscatter link signal strength from the tag is much weaker than that of the direct link signal from the RF source, the received signals from the tag can be ignored at the legacy receiver \cite{van2018ambient, liu2013ambient, wang2016ambient}.
Thus, the proposed tag signal detection algorithm adopted at the AmBC system does not affect the performance of source signal detection in the legacy system.}}}, and provide the related theoretical analysis.
\begin{figure}[t]
\centering
\includegraphics[width=0.96\linewidth]{CMCNNv2_structure0226.jpg}\vspace{-0.6cm}
\caption{ The designed CMNet structure for tag signal detection. }\vspace{-1cm}
\end{figure}
\subsection{CMNet Structure}
Note that the training set for transfer learning is extracted from pilots with very few samples which may lead to overfitting issue of neural networks \cite{goodfellow2016deep}. To overcome this problem, we introduce two dropout layers and redesign the CM-CNN with a modified structure, as shown in Fig. 5. In particular, CMNet consists of two convolutional layers, one pooling layer, one flattening layer, two dropout layers, and two fully connected layers. The convolutional, pooling, and flattening layers are used for extracting features from input. Then, the dropout layers offer a computationally cheap manner to reduce the possibility of overfitting \cite{srivastava2014dropout}. Finally, the fully connected layers learn the non-linear combinations of these extracted features to further improve the performance of the task.
The corresponding hyperparameters are introduced in Table \Rmnum{1}, where we first set these hyperparameters based on the CM-CNN structure \cite{liu2019deep} and then fine-tune these hyperparameters to search the appropriate values for the network empirically.
The details of each layer will be introduced in the following.
a) \textbf{Input Layer} (${\emph{S}_0}$).
\begin{table}[t]
\normalsize
\caption{Hyperparameters of the proposed CMNet}
\vspace{-0.3cm}
\centering
\small
\renewcommand{\arraystretch}{0.95}
\begin{tabular}{c c}
\hline
\vspace{-0.6cm} \\
\multicolumn{2}{l}{\textbf{Input}: Sample Covariance Matrix (Dimension: $8 \times 8 \times 2$) } \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} \textbf{Layer} & \hspace{0.6cm} \textbf{Filter Size} \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} ${S_0}$ & \hspace{0.6cm} -- \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} ${C_1}$ + ReLU & \hspace{0.6cm} $ 32 \times ( 3 \times 3 \times 2 ) $ \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} ${C_2}$ + ReLU & \hspace{0.6cm} $ 32 \times ( 3 \times 3 \times 32 ) $ \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} $S_1$ (Max-Pooling) & \hspace{0.6cm} $ 2 \times 2 $ \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} $C_3$ (Flattening) & \hspace{0.6cm} $ 32 \times (4 \times 4) $ \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} $D_1$ ($\rho = 0.5$) & \hspace{0.6cm} -- \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} ${F_1}$ + ReLU & \hspace{0.6cm} $ 128 \times 512 $ \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} $D_2$ ($\rho = 0.25$) & \hspace{0.6cm} -- \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\hspace{0.6cm} ${F_2}$ + Softmax & \hspace{0.6cm} $ 2 \times 128 $ \\
\vspace{-0.6cm} \\
\hline
\vspace{-0.6cm} \\
\multicolumn{2}{l}{\textbf{Output}: Score Vector (Dimension: $2 \times 1$)} \\
\vspace{-0.6cm} \\
\hline
\end{tabular}
\vspace{-0.6cm}
\end{table}
According to (\ref{observation matrix}), there are $N$ RF source signal sampling periods within one tag symbol, it is naturally to compute the sample covariance matrix of the $t$-th tag symbol as:
\begin{equation}\label{Rx(N)}
{{\bf{R}}_{\bf{x}}^{(t)}}(N) = \frac{1}{N}{\mathbf{X}^{(t)}}{(\mathbf{X}^{(t)})}^{H}=\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{\bf{x}}_n^{(t)}{({\bf{x}}_n^{(t)})^H}}.
\end{equation}
Considering ${{\bf{R}}_{\bf{x}}^{(t)}}(N)$ is a complex-valued matrix, it is necessary to adopt two input channels to represent the real part and imaginary part, respectively. Let $S_0(i; j; \beta) \in \mathbb{R}^{M \times M}$ represent the $i$-th row and the $j$-th column element of the $\beta$-th channel in $S_0$ layer, we have
\begin{equation}\label{model_S0}
S_0(i; j; 0) = (\mathrm{Re}({{\bf{R}}_{\bf{x}}^{(t)}}(N))_{i,j}
\end{equation}
and
\begin{equation}\label{}
S_0(i; j; 1) = (\mathrm{Im}({{\bf{R}}_{\bf{x}}^{(t)}}(N))_{i,j}.
\end{equation}
b) \textbf{Convolutional Layers} (${{\emph{C}_1}}$,${{\emph{C}_2}}$, and ${{\emph{C}_3}}$).
We adopt three convolutional layers, denoted by ${{\emph{C}_1}}$,${{\emph{C}_2}}$, and ${{\emph{C}_3}}$, to further explore the features for detection. Each feature map is the convolution results of the last layer. As shown in Table \Rmnum{1}, $p \times ( q \times q )$ denotes that there are $p$ kernels with kernel size of $ q \times q $.
Take $C_1$ as an example, if we adopt a convolution size of $ L \times L $, the $i$-th row and the $j$-th column element of the $\beta$-th feature map in $C_1$ can be expressed as
{\begin{equation}\label{model_C1}
{C_1}{\rm{(}}i,j,\beta {\rm{)}} = {f_R}\big( \sum\limits_{r = 0}^{1} \sum\limits_{{i_0} = 0}^{L-1} \sum\limits_{{j_0} = 0}^{L-1} [{S_0}(i + {i_0},j + {j_0},r) \cdot {K_\beta^{C_1}}(L - {i_0},L - {j_0},1 - r)] \big), \\
\end{equation}
where $f_R(t)=\max(0,t)$ represents the ReLU function and ${K_\beta^{C_1}}(\cdot,\cdot,\cdot)$ indicates the corresponding kernel of the $\beta$-th feature map in $C_1$. In Table \Rmnum{1}, we adopt ``ReLU'' and ``Softmax'' to denote the activation functions using rectified linear unit and normalized exponential function \cite{goodfellow2016deep}, respectively. In addition, ``flattening'' means that we flatten the outputs of the convolutional layer to create a single long feature vector for the following fully connected layers.
c) \textbf{Pooling Layer} (${{\emph{S}_1}}$).
There is one pooling layer, i.e., ${{\emph{S}_1}}$, which is obtained by the maximum polling operations of the last convolutional layer. As shown in Table \Rmnum{1}, the pooling size of $S_1$ is $2 \times 2$, and ``Max-Pooling'' refers to the the maximum polling operation.
Based on above analysis, the $i$-th row and the $j$-th column element of the $\beta$-th feature map in $S_1$ can be expressed as
\begin{equation}\label{S1_pooling}
S_1(i;j;\beta) = \max(C_1(2i-1,2j-1,\beta), C_1(2i-1,2j,\beta),C_1(2i,2j-1,\beta), C_1(2i,2j,\beta)).
\end{equation}
d) \textbf{Dropout Layers} (${{\emph{D}_1}}$ and ${{\emph{D}_2}}$).
Considering that there are limited training examples for transfer learning, we introduce two dropout layers $D_1$ and $D_2$ to overcome the overfitting problem. In particular, the dropout rate, denoted by $\rho$ in Table \Rmnum{1}, is a common dropout hyperparameter, which refers to the probability of training a given node in a layer. For example, $\rho = 0$ indicates no dropout and $\rho = 1.0$ means that there are no outputs from the layer.
e) \textbf{Fully Connected Layers} (${{\emph{F}_1}}$ and ${{\emph{F}_2}}$).
To fully exploit of the feature maps, one fully connected layer $F_1$ is connected with flattening layer $C_3$, and the number of neurons in $F_1$ is set as $128$. Finally, a two-neuron fully connected layer $F_2$ with a softmax function is connected as the output of CMNet. Therefore, we can regard the network as a non-linear function and express the CMNet as
\begin{equation}\label{expression_CMNet}
{h}_{\theta_S}( {{\bf{R}}_{\bf{x}}^{(t)}}(N) ) = \left[ {\begin{array}{*{20}{c}}
{{{h}_{\theta_S |{H_1}}}( {{\bf{R}}_{\bf{x}}^{(t)}}(N) )}\\
{{{h}_{\theta_S |{H_0}}}( {{\bf{R}}_{\bf{x}}^{(t)}}(N) )}
\end{array}} \right],
\end{equation}
where ${{h}_{\theta_S}( \cdot )}$ is the expression of the CMNet and ${{h}_{\theta_S | {H_i}}( \cdot )}$ is the class score of $H_i$ by the CMNet.
\textbf{Remark 2}: Note that although the dimension of the sample covariance matrix generally changes with the number of antennas, the high scalability of the proposed CMNet structure can be easily extended to different shapes accordingly, as will be verified in the simulation section.
\begin{figure}[t]
\centering
\includegraphics[width=0.96\linewidth]{CMNet_TL.pdf}\vspace{-0.6cm}
\caption{ The proposed CMNet-based deep transfer learning scheme for tag signal detection (A realization of the developed DTL-based workflow in Fig. 3). }\vspace{-1cm}
\end{figure}
\subsection{ CMNet-based Detection Algorithm }
In this section, we adopt the designed CMNet as a core DNN of the developed framework in Fig. 3 to perform tag detection. The details are summarized in Fig. 6. To start with, we first perform offline learning.
\subsubsection{Offline Learning}
Given $K_S$ labelled tag symbols{\footnotemark}\footnotetext{{Historically, researchers
have designed many effective channel models which can well characterize the actual channels in terms of channel statistics. With these well-developed channel models, the training data of offline learning can be obtained by simulation \cite{ye2017power}.}}, we adopt $\mathbf{X}_S^{(k)}=[{\mathbf{x}^{(k)}_{S_1}},{\mathbf{x}^{(k)}_{S_2}},\cdots,{\mathbf{x}^{(k)}_{S_N}}] \in \mathbb{C}^{M\times N}$ to denote the sampling matrix of the $k$-th, $k\in\{1,2,\cdots,K_S\}$, tag symbol in the source domain. According to (\ref{Rx(N)}), the sample covariance matrix of the $k$-th tag symbol is written as
\begin{equation}\label{R_S(N)}
{{\bf{R}}_{{\bf{x}}_S}^{(t)}}(N) = \frac{1}{N}{\mathbf{X}_S^{(t)}}{(\mathbf{X}_S^{(t)})}^{H}.
\end{equation}
Based on this, we build the training set of the source domain:
\begin{equation}\label{}
D_S =(\mathbf{\Omega }_S,Z_S)=\{({\bf{R}}_{\bf{x}}^{(1)}(N),{{{z}}_S^{(1)}}),({\bf{R}}_{\bf{x}}^{(2)}(N),{{{z}}_S^{(2)}}), \cdots ,({\bf{R}}_{\bf{x}}^{(K_S)}(N),{{{z}}_S^{(K_S)}})\},
\end{equation}
where ${{{z}_S}^{(k)}} \in \{0,1\}$ is the corresponding label.
According to (\ref{cost_function}), we can then derive the cost function of CMNet as
\begin{equation}\label{CMCNNv2_cost_function_Source}
{J_{\mathrm{CMNet}}}(\theta_S ) = - \frac{1}{K_S}\sum\limits_{k = 1}^{K_S} {{z_S^{(k)}}\ln ({{h}_{\theta_S|{H_1} }}({{{\bf{R}}_{{\bf{x}}_S}^{(k)}(N)}}))} + (1 - {z_S^{(k)}})\ln({{h}_{\theta_S|{H_0}} }({{\bf{R}}_{{\bf{x}}_S}^{(k)}(N)})).
\end{equation}
Then, exploiting the BP algorithm, we can obtain the pre-trained CMNet:
\begin{equation}\label{fv_CMNet}
{h}_{\theta_S^*}( \mathbf{R} ) = \left[ {\begin{array}{*{20}{c}}
{{{h}_{\theta_S^* |{H_1}}}( \mathbf{R} )}\\
{{{h}_{\theta_S^* |{H_0}}}( \mathbf{R} )}
\end{array}} \right],
\end{equation}
where $\mathbf{R}$ is the input matrix which can be an arbitrary sample covariance matrix and ${\hat{h}_{\theta_S^*}( \cdot )}$ is the expression of the pre-trained CMNet.
\subsubsection{Transfer Learning}
According to the frame structure in Fig. 2, there are $P$ pilots for transfer learning. For any pilot, we use $\mathbf{X}_T^{(k)}=[{\mathbf{x}^{(k)}_{T_1}},{\mathbf{x}^{(k)}_{T_2}},\cdots,{\mathbf{x}^{(k)}_{T_N}}]
$ to denote the target domain's sampling matrix of the $k$-th, $k\in\{1,2,\cdots,K_T\}$ tag symbol. Similar to (\ref{Rx(N)}), we have
\begin{equation}\label{R_S(N)}
{{\bf{R}}_{{\bf{x}}_T}^{(k)}}(N) = \frac{1}{N}{\mathbf{X}_T^{(k)}}{(\mathbf{X}_T^{(k)})}^{H}.
\end{equation}
We can then build the training set of the target domain:
\begin{equation}\label{}
D_T =(\mathbf{\Omega }_T,Z_T)=\{({\bf{R}}_{{\bf{x}}_T}^{(1)}(N),{{{z}}_T^{(1)}}),({\bf{R}}_{{\bf{x}}_T}^{(2)}(N),{{{z}}_T^{(2)}}), \cdots ,({\bf{R}}_{{\bf{x}}_T}^{(K_T)}(N),{{{z}}_T^{(K_T)}})\},
\end{equation}
where ${{z}_T^{(k)}} \in \{0,1\}$ is the corresponding label.
Similar to (\ref{cost_function}), we define the CMNet's cost function for target domain as
\begin{equation}\label{CMCNNv2_cost_function_Target}
{J_{\mathrm{CMNet}}}(\theta_T ) = - \frac{1}{K_T}\sum\limits_{k = 1}^{K_T} {{z_T^{(k)}}\log ({{h}_{\theta_T|{H_1} }}({{{\bf{R}}_{{\bf{x}}_T}^{(k)}(N)}}))} + (1 - {z_T^{(k)}})\log({{h}_{\theta_T|{H_0}} }({{\bf{R}}_{{\bf{x}}_T}^{(k)}(N)})).
\end{equation}
Based on the cost function, we can then operate the training process. As analyzed in (\ref{DNN_target}), we need to freeze the convolutional layers and only update the parameters of the fully connected layers by using the backpropagation algorithm, as analyzed in Section \Rmnum{3}. Finally, we can obtain the well-trained CMNet:
\begin{equation}\label{fv_function}
h_{\theta_T^*}( \mathbf{R} ) = h_{{\theta _T^*}}^{\mathrm{FCL}}(h_{\theta _S^*}^{\mathrm{CL}}(\mathbf{R})) = \left[ {\begin{array}{*{20}{c}}
{{h_{\theta_T^*|{H_1}}}( \mathbf{R} )}\\
{{h_{\theta_T^*|{H_0}}}( \mathbf{R} )}
\end{array}} \right],
\end{equation}
where $h_{\theta_T^*}( \cdot )$ denotes the expression of the well-trained CMNet with the well-trained parameter $\theta_T^*$, $h_{{\theta _T^*}}^{\mathrm{FCL}}(\cdot)$ denotes the fully connected layers ($F_1\rightarrow F_2$), and $f_{\theta _S^*}^{\mathrm{CL}}(\cdot)$ represents the convolutional layers ($C_1\rightarrow C_3$).
\subsubsection{Online Detection}
Given the $t$-th tag symbol's sampling matrix for testing, denoted by $\mathbf{\tilde{X}}^{(t)}\in\mathbb{C}^{M \times N}$, we can obtain the corresponding sample covariance matrix $\mathbf{\tilde{R}_x}^{(t)}(N) = \frac{1}{N}\mathbf{\tilde{X}}^{(t)}(\mathbf{\tilde{X}}^{(t)})^{H}$. From the well-trained CMNet, we then derive the CMNet-based LRT as
\begin{equation}\label{L-CMNet}
{L}_{\mathrm{CMNet}}(\mathbf{\tilde{R}_x}^{(t)}(N)) = \frac{h_{\theta_T^*|{H_1}}(\mathbf{\tilde{R}_x}^{(t)}(N))}{h_{\theta_T^*|{H_0}}(\mathbf{\tilde{R}_x}^{(t)}(N))} \mathop \gtrless\limits_{{c^{(t)}=0}}^{{c^{(t)}=1}} 1,
\end{equation}
where we make a decision that $c^{(t)}=1$ if ${L}_{\mathrm{CMNet}}(\mathbf{\tilde{R}_x}^{(t)}(N))>1$, otherwise, $c^{(t)}=0$.
\subsubsection{Algorithm Steps}
Based on the analysis above, we propose a novel CMNet-based detection algorithm, which is summarized in \textbf{Algorithm 1}, where $i_S$ and $i_T$ are iteration indices, and $I_S$ and $I_T$ denote the maximum numbers of iterations of the source domain and the target domain, respectively.
\begin{table}[t]
\normalsize
\vspace{-0.3cm}
\centering
\renewcommand{\arraystretch}{1.0}
\begin{tabular}{l}
\vspace{-0.35cm}\\
\toprule[1.8pt] \vspace{-0.85cm}\\
\hspace{-0.1cm} \textbf{Algorithm 1} \hspace{0.6cm} {CMNet-based Detection Algorithm} \hspace{0.1cm} \\
\toprule[1.8pt] \vspace{-0.85cm}\\
\textbf{Initialization:} \\
\hspace{1.8cm} $i_S = 0$, $i_T = 0$, $\theta_S$ with random weights, $\theta_T$ with random weights \\
\textbf{Offline Learning:} \\
1:\hspace{1.35cm}\textbf{Input:} Training set $D_S =(\mathbf{\Omega }_S,Z_S)$\\
2:\hspace{1.7cm}\textbf{while} $i_S \leq I_S $ \textbf{do} \\
3:\hspace{2.2cm}update $\theta_S$ by BP algorithm on $J_{\mathrm{CMNet}}(\theta_S)$ in (\ref{CMCNNv2_cost_function_Source})\\
\hspace{2.4cm} $i_S = i_S + 1$ \\
4:\hspace{1.7cm}\textbf{end while} \\
5:\hspace{1.35cm}\textbf{Output}: ${h}_{\theta_S^*}( \cdot ) $\\
\textbf{Transfer Learning:} \\
6:\hspace{1.35cm}\textbf{Input:} Training set $D_T =(\mathbf{\Omega }_T,Z_T)$\\
7:\hspace{1.7cm}\textbf{while} $i_T \leq I_T $ \textbf{do} \\
8:\hspace{2.2cm}update $\theta_T$ by BP algorithm on $J_{\mathrm{CMNet}}(\theta_T)$ in (\ref{CMCNNv2_cost_function_Target})\\
\hspace{2.4cm} $i_T = i_T + 1$ \\
9:\hspace{1.7cm}\textbf{end while} \\
10:\hspace{1.2cm}\textbf{Output:} ${h}_{\theta_T^*}( \cdot ) $\\
\textbf{Online Detection:} \\
11:\hspace{1.2cm}\textbf{Input:} Test data $\mathbf{\tilde{R}}_x$ \\
12:\hspace{1.55cm}\textbf{do} CMNet-LRT by (\ref{L-CMNet}) \\
13:\hspace{1.2cm}\textbf{Output:} Decision: $c^{(t)}=1$ or $c^{(t)}=0$ \\
\bottomrule[1.8pt]
\end{tabular}\vspace{-0.6cm}
\end{table}
\subsection{Theoretical Analysis}
{In general, a practical neural network consists of a large number of non-linear units and parameters, which is intractable for analysis \cite{goodfellow2016deep, wang2017deep, mao2018deep}.
As an alternative, we adopt an indirect approach to analyze the performance of the proposed approach.
In particular, to shed light on the performance of the considered system, in this paper, we formulate the proposed CMNet as a non-linear function and analyze the output of CMNet asymptotically to characterize its properties when the number of samples is sufficiently large under a richly scattered multipath environment{\footnotemark}\footnotetext{This assumption is merely made for convenience of the following analysis and the proposed method is valid for any channel models satisfying the previously stated conditions.}.}
In this case, the received sampling vector at the multi-antenna reader can be formulated as \cite{tse2005fundamentals}
\begin{equation}\label{}
\begin{split}
& {{H}_{1}}:\mathbf{x}_n^{(t)} = \tilde{{\mathbf{s}}}_{\mathbf{w}n}^{(t)} + \mathbf{u}_n^{(t)}, \\
& {{H}_{0}}:\mathbf{x}_n^{(t)} = \tilde{{\mathbf{s}}}_{\mathbf{h}n}^{(t)} + \mathbf{u}_n^{(t)}, \\
\end{split}
\end{equation}
where $\tilde{{\mathbf{s}}}_{\mathbf{w}n}^{(t)} = \mathbf{w}s_n^{(t)} \in \mathbb{C}^{M\times1}$ (or $\tilde{{\mathbf{s}}}_{\mathbf{h}n}^{(t)} = \mathbf{h}s_n^{(t)} \in \mathbb{C}^{M\times1}$) can be modelled by i.i.d. Gaussian random vector with $\mathcal{CN} (\mathbf{0}, \sigma_{sw}^2\mathbf{I}_M)$ (or $\mathcal{CN} (\mathbf{0}, \sigma_{sh}^2\mathbf{I}_M)$). Based on this, the distribution of $\mathbf{x}_n^{(t)}$ can be expressed by
\begin{equation}\label{}
\mathbf{x}_n^{(t)}\sim
\bigg\{ \begin{matrix}
\mathcal{C}\mathcal{N}(\mathbf{0},{\sigma_1^2{{\bf{I}}_M}}),{{H}_{1}}, \\
\mathcal{C}\mathcal{N}(\mathbf{0},{\sigma_0^2{{\bf{I}}_M}}),{{H}_{0}}, \\
\end{matrix}
\end{equation}
where $\sigma_1^2 = \sigma_{sw}^2 + \sigma_u^2$ and $\sigma_0^2 = \sigma_{sh}^2 + \sigma_u^2$. In this case, if we ignore the irrelevant items in (\ref{L_R}), the optimal LRT can be expressed as
\begin{equation}\label{T-EN}
T_{\mathrm{LRT}} = \sum\limits_{n = 0}^{N - 1} {{{\left\| {{\bf{x}}(n)} \right\|}^2}}.
\end{equation}
Therefore, when considering a richly scattered environment, the optimal LRT detector behaves the same as an energy detector (ED) with perfect CSI.
In the following, we will analyze the output of the CMNet.
When the number of samples is sufficiently large, the sample covariance matrix has the following expression:
\begin{equation}\label{Sigma_IM}
{{\mathbf{R}}_{\mathbf{x}}^{(t)}}(N) \mathop \approx \limits^{N \rightarrow \infty } \bigg\{ {\begin{array}{*{20}{l}}
{\sigma_1^2{{\bf{I}}_M},\hspace{0.2cm}{H_1}}, \\
{\sigma_0^2{{\bf{I}}_M}, \hspace{0.2cm}{H_0}}.
\end{array}}
\end{equation}
Hence, the sample covariance matrix approaches a real-valued diagonal matrix, that is, the real part and imaginary part of ${{\mathbf{R}}_{\mathbf{x}}^{(t)}}(N)$ become a diagonal matrix and a zero matrix, respectively. According to (\ref{model_S0}), we can express the element of input layer $S_0$ as \vspace{-0.1cm}
\begin{equation}\label{}
{S_0}(i,j,0){\rm{ = }}\bigg\{ {\begin{array}{*{20}{l}}
{{\sigma ^2},i = j},\\
{0\;\;,i \neq j},
\end{array}}
\end{equation}
where $\sigma^2$ can be either $\sigma_1^2$ or $\sigma_0^2$. Similar to (\ref{model_C1}), if we select a kernel size of $L \times L$, the element at the $i$-th row and the $j$-th column of the $\beta$-th feature map in $C_1$ can be expressed as \vspace{-0.1cm}
\begin{equation}\label{C1_sigma}
\begin{array}{l}
{C_1}{\rm{(}}i,j,\beta {\rm{)}} = {f_R}\left( {\sum\limits_{{i_0} = 0}^{L-1} {\sum\limits_{{j_0} = 0}^{L-1} {[{S_0}(i + {i_0},j + {j_0},0) \cdot {K_\beta^{C_1} }(L - {i_0},L - {j_0},0)]} } } \right) \vspace{0.2cm}\\
{\kern 9pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ = }}{f_R}\left( {\sum\limits_{d = 1}^M {\sum\limits_{\scriptstyle i + {i_0} = j + {j_0} = d,\hfill\atop
\scriptstyle0 \le {i_0},{j_0} \le L-1\hfill} {[{S_0}(d,d,0) \cdot {K_\beta^{C_1} }(L - {i_0},L - {j_0},0)]} } } \right)\vspace{0.2cm}\\
{\kern 9pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ = }}{\sigma ^2}{f_R}\left( {\sum\limits_{d = 1}^M {\sum\limits_{\scriptstyle i + {i_0} = j + {j_0} = d,\hfill\atop
\scriptstyle0 \le {i_0},{j_0} \le L-1\hfill} {{K_\beta^{C_1} }(L - {i_0},L - {j_0},0)} } } \right).
\end{array}
\end{equation}
For a well-trained CMNet, the parameters are fixed and thus (\ref{C1_sigma}) can be rewritten as \vspace{-0.2cm}
\begin{equation}\label{}
{C_1}(i,j,\beta) = \eta \sigma ^2,
\end{equation}
where
\begin{equation}\label{}
\eta = f_R \left( {\sum\limits_{d = 1}^M {\sum\limits_{\scriptstyle i + {i_0} = j + {j_0} = d,\hfill\atop
\scriptstyle0 \le {i_0},{j_0} \le L-1\hfill} {{K_\beta^{C_1} }(L - {i_0},L - {j_0},0)} } } \right)
\end{equation}
is a constant term.
Therefore, if we use ${h}_{\theta_T^*}^{C_1}(\cdot)$ to represent the expression of $C_1$ of the well-trained CMNet, we have the homogeneity property: ${h}_{\theta_T^*}^{C_1} ({\sigma}^2\mathbf{I}_M) = {\sigma}^2 {h}_{\theta_T^*}^{C_1} (\mathbf{I}_M)$.
Based on the above discussions, if we consider $C_1$ to $F_1$ as a subnetwork ${h}_{\theta_T^*}^{{C_1}{F_1}} (\cdot)$, we have
\begin{equation}\label{}
{h}_{\theta_T^*}^{{C_1}{F_1}}({\sigma ^2}{{\mathbf{I}}_M})= {\sigma ^2}{h}_{\theta_T^*}^{{C_1}{F_1}}({{\mathbf{I}}_M}).
\end{equation}
Denote $\tilde{\theta} = \{\tilde{\theta}_1,\tilde{\theta}_2\}$ as the weights between $F_1$ and $F_2$ of the well-trained CMNet, where $\tilde{\theta}_1$ and $\tilde{\theta}_2$ indicate the weights corresponding to the the 1-th row and 2-th row elements of $F_2$, respectively. The output of the well-trained CMNet can be finally expressed as
\begin{equation}\label{h_theo}
\begin{split}
{{h}_{\theta_T^*} }({\sigma ^2}{{\mathbf{I}}_M}) & = {f_{\mathrm{softmax}}}\left( {{\sigma ^2}{{\tilde \theta }^{T}}{h}_{\theta_T^*}^ {{C_1}{F_1}}({{\mathbf{I}}_M})} \right)\\
& =
\frac{1}{ \exp({\sigma ^2} \tilde{\theta}_1^{T}{{h}_{\theta_T^*}^ {{C_1}{F_1}}({{\mathbf{I}}_M})}) + \exp( {\sigma ^2} \tilde{\theta}_2^{T} {
{h}_{\theta_T^*}^ {{C_1}{F_1}}({{\mathbf{I}}_M})} ) }
\begin{bmatrix}
\exp( {\sigma ^2} \tilde{\theta}_1^{T} {{h}_{\theta_T^*}^ {{C_1}{F_1}}({{\mathbf{I}}_M})} ) \\
\exp( {\sigma ^2} \tilde{\theta}_2^{T} {{h}_{\theta_T^*}^ {{C_1}{F_1}}({{\mathbf{I}}_M})} ) \\
\end{bmatrix},
\end{split}
\end{equation}
where
\begin{equation}\label{}
\begin{split}
f_{\mathrm{softmax}}(x) &=
\frac{1}{ \exp(\tilde{\theta}_1^{T}x) + \exp( \tilde{\theta}_2^{T} x ) }
\begin{bmatrix}
\exp( \tilde{\theta}_1^{T} x ) \\
\exp( \tilde{\theta}_2^{T} x )
\end{bmatrix}
\end{split}
\end{equation}
denotes the softmax function.
According to (\ref{L-CMNet}), the CMNet-based LRT can be expressed as
\begin{equation}\label{L-CMNet-theo}
L_{\mathrm{CMNet}} = \frac{\exp( {\sigma ^2} \tilde{\theta}_1^{T} {{h}_{\theta_T^*}^ {{C_1}{F_1}}({{\mathbf{I}}_M})} )}{\exp( {\sigma ^2} \tilde{\theta}_2^{T} {{h}_{\theta_T^*}^ {{C_1}{F_1}}({{\mathbf{I}}_M})} )}= \exp({\sigma^2(\tilde{\theta}_1^{T} - \tilde{\theta}_2^{T}) {h}_{\theta_T^*}^ {{C_1}{F_1}} (\mathbf{I}_M)
}) \gtrless 1.
\end{equation}
Note that when $N$ is large enough, $\sigma^2 \approx \sum\limits_{n = 0}^{N - 1} {{{\left\| {{\bf{x}}(n)} \right\|}^2}} / {MN}$ \cite{kay1998fundamentals}.
Discard the irrelevant constants, we can rewrite (\ref{L-CMNet-theo}) as
\begin{equation}\label{L-CMNet-EN}
\sum\limits_{n = 0}^{N - 1} {{{\left\| {{\bf{x}}(n)} \right\|}^2}} \gtrless \gamma,
\end{equation}
where $\gamma$ is the threshold. {Therefore, when $N$ is sufficiently large, the ratio of the two elements in the output vector of a well-trained CMNet, as shown in (\ref{L-CMNet-EN}), is identical to the expression of the optimal LRT exploiting the perfect CSI, as defined in (\ref{T-EN}), i.e., the proposed CMNet method is equivalent to the optimal LRT detector.}
\section{Numerical Results}
In this section, we provide extensive simulation results to evaluate the performance of the proposed algorithm. Without special notes, we consider an AmBC system which consists of a single-antenna RF source, a single-antenna tag, and a 8-element ($M = 8$) multi-antenna reader{{\footnotemark}}\footnotetext{
{Considering the requirement of the coverage area and the adopted carrier frequency of the RFID reader \cite{van2018ambient, finkenzeller2010rfid}, we assume that $M=8$ antennas are equipped at the reader.}}. According to the framework structure in Fig. 2, we set the length of the framework as $NT = 5000$, and the number of the pilots as $P = 10$, which is based on a practical protocol \cite{coleri2002channel}. To evaluate the BER performance, we compare the proposed CMNet method with the optimal LRT method \cite{kay1998fundamentals}, the classical ED method with perfect CSI\cite{qian2017semi}, and the SVM method \cite{hu2019machine}. {The hyperparameters of CMNet are given in Table \Rmnum{1}, where the optimizer used for training is Adam \cite{goodfellow2016deep}.
In addition, the numbers of examples in offline dataset and online dataset are $60,000$ and $2,000$, respectively.
The sizes of the input and the label of each example are shown in Table \Rmnum{1}.
The online dataset is generated by using the data augmentation technique \cite{shorten2019survey}, i.e., adding Gaussian noise variables with a distribution of $\mathcal{CN}(0,0.001)$ to the received $10$ pilot symbols to generate $2,000$ examples.}
{Note that although the proposed CMNet has large-size layers with large number of parameters, the online detection time can be greatly reduced through the parallelization of graphics processing unit (GPU) \cite{goodfellow2016deep}. For example, when we execute the proposed CMNet algorithm on a desktop computer with an i7-6700 3.4 GHz central processing unit (CPU) and a Nvidia GeForce GTX 1080 GPU, the online detection time of the CMNet algorithm is only 2.6 milliseconds, which is shorter than the coherence time of a general slow fading environment \cite{tse2005fundamentals} and is acceptable for the practical AmBC systems \cite{van2018ambient}.}
In addition, the SNRs of the simulation results are defined in (\ref{SNR}), and the relative coefficient is set as $\zeta = -20$ dB, as defined in (\ref{relative_SNR}).
{Each point in the simulation results is obtained by averaging over $10^6$ Monte Carlo realizations.}
{Fig. \ref{Fig_BER_SNR}(a) presents the BER curves versus SNRs for different algorithms when the ambient source transmits QPSK modulated signals. Note that the ED method presented in the simulations is not the same as the conventional ED method without CSI. Instead, it is a genie-aided ED method where the perfect CSI is available, denoted by ``ED with perfect CSI''.
In addition, the SVM method is an energy-based machine learning method, whose input is the signal energy received at each antenna of the reader.
The simulation results show that the BER performance of the SVM method is slightly worse than that of the ED method with perfect CSI.
The reason is that although the SVM method can exploit more distinguishable features to further improve the BER performance, it does not know the perfect CSI for detection and thus there is still a performance gap compared with the method of ``ED with perfect CSI''.}
In contrast to the conventional methods, the proposed CMNet method outperforms both the ED and SVM-based methods substantially, achieving almost the same BER performance as the optimal LRT method. For example, the proposed CMNet method achieves a SNR gain of 4 dB at BER$\,\approx10^{-2}$ compared with the traditional SVM-based method. The reason is that the SVM-based method only makes decisions based on the features capturing the characteristics of energy of received signals. In contrast, the proposed method makes decisions depending on the inherent features of the covariance matrix including rich distinguishable features without explicitly estimating the channels, which are acquired through a DTL approach.
\begin{figure}[t]
\centering
\renewcommand{\arraystretch}{1.0}
\setlength{\tabcolsep}{1pt}
\begin{tabular}{c c}
\includegraphics[width=8cm,height=7cm]{response_BER_SNR_QPSK.pdf}
&
\includegraphics[width=8cm,height=7cm]{response_BER_SNR_CSCG.pdf} \vspace{-0.3cm}\\
{\scriptsize{(a) QPSK ambient source. }} &
{\scriptsize{(b) Complex Gaussian ambient source.}} \vspace{-0.2cm}\\
\end{tabular}
\caption{BER versus SNR with $M=8$, $N=20$, and $\zeta=-20~\mathrm{dB}$.}\label{Fig_BER_SNR} \vspace{-0.73cm}
\end{figure}
In addition, Fig. \ref{Fig_BER_SNR}(b) shows the BER-SNR curve under the complex Gaussian ambient source. {It is shown that the BER performance of all the presented algorithms under the complex Gaussian ambient source is slightly worse than that under the QPSK ambient source, as it is more difficult to distinguish the two hypotheses in the former case.} Similar to the results in Fig. \ref{Fig_BER_SNR}(a), our proposed CMNet also presents a satisfying BER performance which is almost the same as the performance of the optimal LRT method. Thus, the proposed method could achieve almost the optimal BER performance under both the QPSK and the complex Gaussian ambient sources.
\begin{figure}[t]
\centering
\renewcommand{\arraystretch}{1.0}
\setlength{\tabcolsep}{1pt}
\begin{tabular}{c c}
\includegraphics[width=8cm,height=7cm]{response_BER_N_QPSK.pdf}
&
\includegraphics[width=8cm,height=7cm]{response_BER_M_QPSK.pdf} \vspace{-0.3cm}\\
{\scriptsize{(a) $M=8$, $\mathrm{SNR}=6~\mathrm{dB}$, and $\zeta = -20~\mathrm{dB} $. }} &
{\scriptsize{(b) $N=25$, $\mathrm{SNR}=5~\mathrm{dB}$, and $\zeta = -20~\mathrm{dB} $. }} \vspace{-0.2cm}\\
\end{tabular}
\caption{BER versus SNR under a QPSK ambient source.}\label{Fig_BER_MN}
\vspace{-0.73cm}
\end{figure}
Note that the sample covariance matrix heavily depends on the STR, $N$, and $M$. To evaluate the scalability of the proposed CMNet method, we fix the SNRs and vary the values of $N$ and $M$, presenting the curves of BER versus $N$ and BER versus $M$ in Fig. \ref{Fig_BER_MN}. From the results of Fig. \ref{Fig_BER_MN}(a), we can find that the BER of each detection algorithm decreases with an increasing $N$. Among all the detection algorithms, the BER performances of LRT and CMNet methods improve more dramatically than that of the ED and SVM-based algorithms.
Specifically, with the increasing of $N$, the BER of the proposed CMNet method scales with the same slope as the optimal LRT method, which demonstrates the scalability of the proposed method. {Similar results can also be found in Fig. \ref{Fig_BER_MN}(b), which show that the proposed method is able to achieve outstanding BER performance under different numbers of antennas. This is because the proposed method can learn more distinguishable features from a larger scale input matrix to improve the accuracy of detection. Therefore, the proposed method is scalable for different STRs and numbers of antennas.}
{In addition, Fig. \ref{Fig_BER_MN}(b) shows that all algorithms perform closely when $M=6$ and the proposed CMNet achieves a significant performance gain when $M$ is large. This is because when $M$ is small, the differences between the two hypotheses become small, which hinders the signal detection in all the algorithms. In contrast, for a reasonably large $M$, the proposed CMNet can efficiently exploit the spatial degrees of freedom, which facilitates the signal detection.}
\begin{figure}[t]
\centering
\renewcommand{\arraystretch}{1.0}
\setlength{\tabcolsep}{1pt}
\begin{tabular}{c c}
\includegraphics[width=8cm,height=7cm]{response_BER_alpha_QPSK.pdf}
&
\includegraphics[width=8cm,height=7cm]{response_BER_distance_QPSK.pdf} \vspace{-0.3cm}\\
{\scriptsize{(a) BER versus relative coefficient with $N=10$ and $\mathrm{SNR}=2~\mathrm{dB}$. }} &
{\scriptsize{(b) BER versus distance with $N=20$ and $\mathrm{SNR}_{\mathrm{tag}}=28~\mathrm{dB}$.}} \vspace{-0.2cm}\\
\end{tabular}
\caption{BER curves with different relative coefficients and tag-to-reader distances under a QPSK ambient source with $M=8$.}\label{Fig_BER_rlink}
\vspace{-0.73cm}
\end{figure}
{Finally, we turn to the study of the impact of reflection link on the system BER performance. As shown in Fig. \ref{Fig_BER_rlink}(a), a practical range of relative coefficient $\zeta \in [-20,-5]$ is investigated \cite{van2018ambient}. It is shown that the BER of each detection algorithm decreases with the increase of the relative coefficient $\zeta$, i.e., increasing the value of reflected coefficient contributes to the improvement of BER performance. This is because the improved strength of the reflected path makes the reader easier to distinguish the tag signals from the signals of the direct path.
In addition, although both the ED and SVM-based methods obtain some performance improvements by increasing the value of $\zeta$, the proposed CMNet method still outperforms both the SVM and the ED methods, achieving almost the same performance as the optimal LRT detector.}
{Note that the supported tag-to-reader distance is of great importance for practical implementation of AmBC systems. We then present the BER curves with different tag-to-reader distances in Fig. \ref{Fig_BER_rlink}(b).
In the simulations, a QPSK ambient RF source with a carrier frequency of $900~\mathrm{MHz}$ is adopted \cite{finkenzeller2010rfid} and the received SNR at the tag is set as $\mathrm{SNR}_{\mathrm{tag}} = 28~\mathrm{dB}$ \cite{van2018ambient}. In addition, a path loss model \cite{cho2010mimo} is introduced to characterize the large-scale fading of the tag-to-reader link, i.e., $\zeta = \beta(d/d_0)^{-\gamma}$, where $d$ is the tag-to-reader distance, $d_0=1~\mathrm{m}$ is the reference distance, $\gamma=2.7$ \cite{cho2010mimo} denotes the path loss exponent, and $\beta = (\lambda/(4\pi d_0))^2$ denotes the path loss of the signal with a wavelength of $\lambda$ at $d_0$.
It is shown from Fig. \ref{Fig_BER_rlink}(b) that a supported tag-to-reader distance of $2~\mathrm{m}$ is achieved by our proposed method with a BER requirement of $10^{-2}$, which is sufficient for the ambient backscatter communications in many practical scenarios \cite{van2018ambient}.
In addition, the BER performance of the proposed method approaches that of the optimal LRT method.
This is because the proposed method can adapt itself to different channel environments by the knowledge transfer from the source domain to the target domain.}
\section{Conclusions}
This paper studied the tag signal detection problem for AmBC systems adopting the DTL technology.
Firstly, we designed a universal DTL-based tag signal detection framework, which uses a DNN to implicitly extract the features of communication channels and directly recover the tag symbols.
Based on the established pre-trained DNN and a few pilots, a DTL-LRT was obtained through transfer learning, which enables the design of an effective detector. Furthermore, exploiting the advantages of the CNN's powerful capability in exploring features of data in a matrix form, we then designed a CMNet for the sample covariance matrix and proposed a CMNet-based detection algorithm. In particular, theoretical analysis of the proposed CNN-based method was provided correspondingly. Finally, simulation results showed that the proposed CMNet method can achieve a close-to-optimal performance without explicitly obtaining the CSI, despite the ambient source transmits modulated signals or complex Gaussian signals.
\bibliographystyle{ieeetr}
\setlength{\baselineskip}{10pt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,224 |
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John Lawrence - our Indian driver, w/ his car, in Kerala (198 KB)
the beach at Kovalum, Kerala (150 KB)
Gopuram at the Meenakshi shine in Madurai, Tamil Nadu (205 KB)
monkeys hanging out at Kabini Lodge, Karnataka (146 KB)
the main arch at Mysore Palace, Karnataka (157 KB)
evening sky from the Lalitha Mahal in Mysore, Karnataka (115 KB)
an irrigated rice field between Mysore and Somnathpur, Karnataka (195 KB)
a sacred cow, festively painted yellow, in Mysore, Karnataka (150 KB)
Sravanabelagola: Jain statue - large and in charge, Karnataka (96 KB)
the entry gate at Brihadishwara Temple, Thanjavur, Tamil Nadu (141 KB)
evening sky at Brihadishwara Temple, Thanjavur, Tamil Nadu (91 KB)
fresco and lingam at Brihadishwara Temple, Thanjavur, Tamil Nadu (197 KB)
:# 5
Subj: India Report: Back in the USA
Date: 97-11-29 16:19:07 EST
From: TomForest
To: SewNSusan
Dear Family and Friends,
I'm at the Internet C@fe in Honolulu, listening to Steely Dan ("Ricky Don't Lose That Number"), taking some time off from cruising around the tropics. My AOL account has been seriously spammed; any mail sent before 11/4 has been deleted already because I guess my mailbox was full. "Call me: hot hot girl" or "get rich quick" or "you too can spam people" are the favorite spam topics. Before I forget:
We're at the Ilima Hotel in Waikiki: (808) 923-1877, through at least Dec. 2. For those of you in Hawaii, please call us! Other callers are welcome, but it won't be cheap for you. I'll check email (TomForest (at) aol.com) again on Monday night
We return to Boston Thursday (scheduled) or Friday. We'll be staying with Carol's sister Barbie for a few days, collecting ourselves for the drive across the US. Barbie's address: <address deleted 3/1/98>
Tom Clarke & Alison Hodges have kindly agreed to host an open house for us a week from today from 1 PM, until ?, a week from today, i.e., December 6. They are at <address deleted 3/1/98>
India was some of what we expected, and a lot of what we didn't.
Expected:
The food was great, and we didn't get tired of it. We gained weight (that part was a surprise).
The weather was warm.
The music was great - especially the 2.5 hour concert of ragas the evening of the wedding.
The people were friendly and easy to get along with. Whether it was because people we know had horror stories or our Egypt experience was tough I don't know, but we expected to be constantly harassed by unhappy people. Maybe our being in the South, not North, of India had something to do with it, but Indians were low-key and easy to laugh.
The only illness we got was that we each got a cold. We expected to lose weight, not gain it.
We felt safe. Even at the $20/night places, we were in places with gates & guards, and we felt safe walking the streets.
Access to the National Parks was terrible - not worth the trouble. The jeep rides at Nagarahole were marginal, and the 30 minute drives at Mudumalai and speed walk (nerd walk) through Periyar was the nadir of our India trip. I wouldn't have minded the leeches if we'd actually seen some wildlife.
Though at times surrounded by bad smells, hovels, and a few beggars, I was never shocked by anything I saw.
We felt a little too safe, a little too insulated. We'd have liked more contact with Indians. But after Cairo, we retreated upmarket in India - a little too far.
The drivers are insane, like Cairo, yet kind - unlike Cairo: Indians stop for people, cows, dogs, goats, sheep, and anything else in the road). See below.
On a related note, the roads are terrible. Speeds greater than 40 mph are ill-advised, and in many places 20 mph is pushing things. Also see below.
Saris are beautiful, elegant, and everywhere. It looks like all Indian women are always in evening dresses - even when breaking rocks at the side of the road or harvesting rice.
India Highlights
Hoysala-style temple at Halebid. The profusion of detail and different figures was a gorgeously mind-expanding experience.
Worshippers at a temple in Srirangpatnam. It was striking to see so many people fervent about something so ancient (Hindu worship) about which I have so little insight, exposure, or understanding.
At Ranganthoo Bird Sanctuary near Mysore, we saw dozens of storks and ibises, several crocodiles, and hundreds of fruit bats hanging from the trees. Ranganthoo is a tiny reserve and popular with the locals. The abundance were a pleasant surprise, and our private rowboat tour got us quite close.
Our first night at Kabini River Lodge. Our game drive was cut to 10 minutes by a tropical downpour, but we didn't mind. It was the first heavy rain we'd seen in months, and was beautiful to watch. Lightning knocked out the power (a diesel-powered generator at the camp) for an hour until they got the backup generator running. So we lit candles and sat on the porch, watching the rain and seeing frogs hop onto the porch - big hand-sized ones and little thumbnail sized ones.
Borrowing a guitar at Bokkapuram. I was talking to a staffer who was playing a Pink Floyd CD. I asked for Santana (which he didn't have) and the Grateful Dead and Clapton (which he had). It was nice to hear the lyric "been all around the world" in a different light. So I asked him if he had a guitar, which he did. After tuning it & putting on some new strings, I got to play for a couple of hours. I'm out of practice, so it hurt more than usual, but it was great to run through "Imagine", "Get Together", "Good Lovin'", and especially "Ripple" from the porch looking up at the Western Ghats (which looked there rather like Crawford Notch, NH, with teak instead of pine).
The drive to & from Ooty. Ooty's at 7200', so we had a lot of climbing up switchbacks to do. The road was terrible (like a logging road paved then left for 5 years in a 120" rain/year state), the traffic insane (no curve or cliff to steep to be passed on), and the car under-powered (a 1981 Ambassador capable of no more than 40 km/hr), but it meant that we lingered on every curve & overlook. It was very easy on the eyes, and we were in no hurry.
Kathakali dance in Kochi. This was a classical costumed dance portraying a scene from the Mahabhrata. It was beautiful, and the explanation/color commentary by the MC gave me at least a glimmer of understanding of Hinduism amidst an interesting soliloquy on beauty, meaning, and faith.
Murals at Mattancherry Palace, Kochi. These 16th century wall paintings looked like extended meditations on the cover of Hendrix's "Axis: Bold as Love" - which is an anachronistic attribution, but that's the order I saw them. I also have a new appreciation of Krishna after seeing HIS appreciation for milkmaids. Having eight arms could be useful.
Dhandu's wedding in Trivandrum. It was an honor to be invited, and we felt most welcomed. Some family members provided live interpretation (color commentary). This was as exotic as we got, yet the feelings of the people involved were strong and clear to us.
Meenakshi Temple, Madurai. It's another over-the-top temple like Halebid (India does over-the-top well), but unlike Halebid wonderfully painted and a current house (temple complex) of worship. It's statuary is stylistically quite different, and has an almost Quincy Market feel to the merchandising around it (10,000 tourists a day visit, mostly Hindu Indians).
Gandhi Museum, Madurai. It's an atypically understated presentation of India's struggle for independence and Gandhi's life. The Lonely Planet guide is right on the money when they characterize it as 'oddly moving'. It reminded me most of the Kennedy Memorial Library in Boston: the hairs on the back of my neck stood up.
We're well & happy to be back in America: people speak English as a first language, we can get the breakfast we like (our first morning at Denny's was euphoric), and everything works! Thanks to Howard, Heather, Crystal, Tony, Joe, and Christiana for their messages during our trip. Until next time,
Tom (w/ Carol)
© 1995-2007 Tom Lum Forest :: Forest Grove, Oregon USA :: Home | {
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{"url":"https:\/\/docs.eyesopen.com\/toolkits\/python\/oechemtk\/OEChemClasses\/OEVectorBindings.html","text":"# OEVectorBindings\u00b6\n\nclass OEVectorBindings\n\n\nThis class represents OEVectorBindings.\n\nThe OEVectorBindings class is used to store a set of vector bindings or lexigraphic replacements that may appear in a SMARTS pattern.\n\nA vector binding is a SMARTS pattern bound to a name. For example, the name [$HALO] can be used to represent the pattern [$(F,Cl,Br,I)] in order to make a pattern more human readable. The OEVectorBindings class converts SMARTS patterns written using vector bindings (human readable form) to the corresponding machine readable form by performing a name to pattern replacement.\n\n## Constructors\u00b6\n\nOEVectorBindings()\nOEVectorBindings(const OEVectorBindings &)\n\n\nDefault and copy constructors.\n\n## operator=\u00b6\n\nOEVectorBindings &operator=(const OEVectorBindings &)\n\n\n## Add\u00b6\n\nbool Add(const char *label, const char *pattern)\n\n\nAdds one vector binding to an OEVectorBindings object. The \u2018label\u2019 given as the first method argument is the name that appears in the human readable version of a SMARTS, while the \u2018pattern\u2019 given as the second method argument is the SMARTS that should be used to replace the \u2018\u2019label\u2019\u2018. If the \u2018pattern\u2019 is parsed correctly and the association is made in the OEVectorBindings object, the method will return true. If the \u2018pattern\u2019 is invalid or the association cannot be made the method will return false. All attempts to associate a SMARTS pattern with a particular \u2018label\u2019 after the first association succeeds will result subsequent failures to create new associations.\n\n## Get\u00b6\n\nbool Get(const OEExprBase *&expr, const char *&label, const char *smarts) const\n\n\nRetrieves the \u2018label\u2019 and corresponding OEExprBase pointer reference as the first and second method arguments, respectively, given a pointer to a position within a smarts string given as the final argument. If the \u2018smarts\u2019 string position points to the beginning of a vector bound \u2018label\u2019 contained in the OEVectorBindings object then the method will return true. If the vector binding cannot be identified then the method will return false.","date":"2021-05-06 22:55:11","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\": 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\": 0.24232836067676544, \"perplexity\": 4310.445277907689}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"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-2021-21\/segments\/1620243988763.83\/warc\/CC-MAIN-20210506205251-20210506235251-00471.warc.gz\"}"} | null | null |
Xenochrophis es un género de serpientes de la familia Colubridae. Sus especies se distribuyen por Asia (sudeste de Asia Central y región indomalaya) y la Wallacea.
Especies
Se reconocen las siguientes:
Xenochrophis asperrimus (Boulenger, 1891)
Xenochrophis bellula (Stoliczka, 1871)
Xenochrophis cerasogaster (Cantor, 1839)
Xenochrophis flavipunctatus (Hallowell, 1860)
Xenochrophis maculatus (Edeling, 1864)
Xenochrophis melanzostus (Gravenhorst, 1807)
Xenochrophis piscator (Schneider, 1799)
Xenochrophis punctulatus (Günther, 1858)
Xenochrophis sanctijohannis (Boulenger, 1890)
Xenochrophis schnurrenbergeri Kramer, 1977
Xenochrophis trianguligerus (Boie, 1827)
Xenochrophis tytleri (Blyth, 1863)
Xenochrophis vittatus (Linnaeus, 1758)
Referencias
Reptiles de Asia
Reptiles de la Wallacea | {
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Q: MessageDigest in Rust I'm not really that strong in cryptography, so I'm striving to understand what this Java code exactly does:
MessageDigest md = MessageDigest.getInstance("SHA-256");
md.update("some string".getBytes("UTF-8"));
byte[] digest = md.digest();
and convert it to Rust.
So how can I do the same thing in Rust, what traits should I use? Is it http://doc.rust-lang.org/rustc/util/sha2/struct.Sha256.html or something else?
A: rustc::util::sha2::Sha256 is part of the Rust compiler and is not meant for external use. You should use the rust-crypto crate instead. Its sha2 module implements SHA-256.
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\section*{Main}
Elucidating how changes in gene frequencies within species drive ecological dynamics, and how ecological dynamics, in turn, affect gene frequencies is a central goal in ecology and evolutionary biology (\citealt{yoshida03, post09, schoener11}, Hendry \textit{in press}\nocite{hendry16}). Such eco-evolutionary feedbacks can potentially be quite complex, especially when the important ecological processes (for instance, inter- and intra-specific interactions) are underlain by quantitative traits influenced by several loci. Purely phenotypic models that do not describe the underlying genetic details (e.g., classical quantitative genetics) have greatly enhanced our understanding of evolutionary processes. Yet rapidly growing knowledge of the genetic architecture of ecologically important traits (e.g., \citealt{stapley10, hendry13}) necessitates incorporating such genetic details into our models to advance our understanding of eco-evolutionary dynamics. Such models are needed to guide developing effective responses to anthropogenic perturbations of natural communities, improve agricultural practices and manage global health risks such as emergent pathogens.
Due to their inherent complexity, models coupling genetic and ecological dynamics must be analyzed through numerical simulation, often using individual-based models (IBMs) that explicitly characterize the fates of individuals and the alleles they carry (e.g., \citealt{deangelis05} and \citealt{carvajal10}). IBMs are attractive for several reasons. First, the same processes that drive changes in allele frequencies in nature (e.g., selection, recombination and mutation acting through reproduction, survivorship, dispersal, etc.) have a one-to-one correspondence in the model (e.g., \citealt{hoban12}). Second, their inherent stochasticity allows us to rigorously conduct statistical inference to test theory with data using likelihood functions based on biological principles, rather than mathematical convenience (e.g., \citealt{hartig11}). Finally, parameterizing IBMs is often easier than parameterizing alternative population models (e.g., \citealt{pacala96}). Thus, IBMs provide both a mechanistic explanation for the observed patterns of individual phenotypic variation, as well as a framework for forecasting how selection, drift and gene flow and constraints (e.g., pleiotropy; also, \citealt{futuyma10}) can drive evolutionary change.
However, some of these advantages make using IBMs computationally challenging. Because they model realistic properties (e.g., age or size structure, species interactions) over evolutionary time scales (e.g., large number of generations), an individual run of a particular simulation can take a very long time to complete (\citealt{deangelis05}). Moreover, stochastic IBMs require several replicate simulations per parameter combination, and that several such combinations be analyzed.
These performance issues constitute non-trivial concerns for investigators who face constraints on their time and/or funding. Modern central processing units (CPUs), the most commonly used type of processor by most biologists, are unlikely to see meaningful performance improvements in the near future (\citealt{fuller11}). This means IBMs need to be simulated simultaneously on multiple computers, which often requires considerable financial investment and institutional support.
Here we introduce an open-source software framework, sPEGG (\underline{s}imulating \underline{P}henotypic \underline{E}volution on \underline{G}eneral Purpose \underline{G}raphics Processing Units (GPGPUs)). sPEGG aims to markedly accelerate the analysis of eco-evolutionary dynamics using individual-based, forward-time population genetics models levereging cost-effective GPGPUs. sPEGG allows investigators to simulate the evolution of multi-locus traits under a wide array of ecological (e.g., age or spatially structured populations, multi-species communities) and genetic (e.g., mutation, variable linkage maps) scenarios. Below, we describe the overall design and key features of sPEGG. We then illustrate its performance gains relative to equivalent CPU implementations for three case studies.
\newpage
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\linewidth] {Fig1.pdf}
\caption*{Figure 1. Calculations performed in sPEGG during each time step. Parents are identified (A), and genetic information is transmitted to offspring by subroutines implementing recombination and mutation (B). User-supplied functions map the resulting phenotypic distribution to newborns (C). A selective environment (here, substrate color in each panel) updates individual phenotypes (including mortality risk) (D) and survivors are identified (E). The relevant environmental variables variables are updated to reflect the phenotype distribution (F), thereby closing the eco-evolutionary feedback loop. Though not illustrated, gene flow resulting from migration among patches can also be modelled in sPEGG.}
\end{figure}
In sPEGG, the following calculations are performed at each step on each individual in each of the $n$ species. First, the eligible parents are identified according to rules supplied by the user, and the number of neonates are determined (Fig 1A). Second, parents are assigned to neonates, and genetic information is transmitted from the assigned parents to their offspring (Fig 1B). Third, based on the genetic information, the offspring's phenotype at birth is determined, based on a user-supplied genotype-phenotype map (Fig 1C). We highlight that sPEGG allows users to specify any genotype-phenotype map. Thus, in addition to an additive effect among loci, non-additive genetic effects such as dominance and epistasis, as well as maternal effects, environmental effects and gene $\times$ environment effects which characterize the trait of interest (\citealt{hendry13}) can be readily accommodated. Fourth, the individual's phenotypes (including their probability of survivorship) are updated according to the selective environment (which includes ecological interactions) that the user specifies (Fig 1D). Fifth, the survivors are identified and dead individuals are removed from the model (Fig 1E). A further step may involve updating the prevailing environmental conditions according to the phenotypic distribution of survivors, thereby completing the eco-evolutionary feedback loop (Fig 1F). The ability to simulate migration and gene flow in spatially-structured environments is also supported. sPEGG could be viewed as analogous to a numerical ordinary differential equation solver. As in those solvers, sPEGG still requires the user to provide some code describing the species-specific calculations, but the core computations above are carried out by the software. The online Methods describe sPEGG's design in greater detail.
sPEGG accommodates alternative mating systems (random, shared preference across individuals, or assortative; monogamy or polygynandry). The transfer of genetic information can mimic recombination and sexual reproduction in diploid species or asexual reproduction, and can involve mutation and arbitrary linkage patterns. Importantly, all of these calculations, and more, are done across individuals using parallel algorithms, as described in the online Methods.
To illustrate the versatility and performance benefits of sPEGG, we use it to build IBMs to investigate three classic problems in evolutionary ecology. We first consider a very simple model of a randomly mating, diploid species with overlapping generations, in which the per-capita density-independent birth and death rate are quantitative traits whose numerical values are determined additively by five independently segregating loci subject to mutation in their allelic values (case study 1). Here, the population simply evolves to minimize density-independent mortality and maximize the density-independent birth rate (Supplementary Figure S1). Figure 2A shows that, as more individuals are simulated the total simulation speed increases approximately 10-fold compared to an equivalent simulation running entirely on a single CPU core.
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth] {Fig2.pdf}
\caption*{Figure 2. Performance comparisons as functions of the problem size (number of patches simulated) in terms of the number of seconds it takes to execute the simulations for the CPU version of sPEGG (dotted lines, open circles) and the GPU version (dashed lines, filled circles), as well as the speed up of sPEGG relative to a serial implementation (solid lines, asterisks). Performance comparisons are based on 250 iterations of steps represented in Figure 1 in the main text for both implementations. (A) Results from case study 1, which models evolution in fecundity and density-independent survivorship. (B) Results from case study 2, which models the evolution of neonate size in a physiologically structured model for a species that experiences an ontogenetic niche shift. (C) Results from case study 3, which models coevolving consumer-resource metacommunities linked by migration. In all panels, each patch contains approximately 50,000 individuals of each species; thus, a simulation of 100 patches simulates approximately $5\times10^6$ individuals in (A), (B) and approximately $10^7$ individuals in (C).}
\end{figure}
Next, we use sPEGG to model a size- and physiologically-structured (e.g., \citealt{persson98}), sexually reproducing species in which individuals of different sizes complete for two dynamic, biological resources (case study 2). Individuals transition from utilizing one resource type to utilizing another as they increase in size. We model the evolution of offspring size at birth, a trait which we assume (i) is inversely related to clutch size, and (ii) is controlled additively by a finite number of loci. Offspring size is then modeled to evolve to maximize individual fitness, subject to the individual-level constraints of size-specific survivorship, resource-use, and fecundity (Supplementary Information S2). Figure 2B illustrates how when the evolutionary question is addressed by a complex IBM, sPEGG can accelerate the simulation on a GPGPU by a factor of over 200x.
As a final case study, we model a spatially-structured, consumer-resource system in which the resource species and the consumer species coevolve (case study 3). We also model anti-predator defense to be costly for the resource. We apply sPEGG to analyze coevolution in such a system, investigating how a spatial gradient in the cost of anti-predator defense drives coevolutionary dynamics (Supplementary Information S3). Figure 2c shows that the performance improvements provided by sPEGG are higher for case study 3 than for case study 1 (up to approximately a factor of 44x), and not as high as the performance improvements for case study 2.
The GPGPU versions of all case studies show accelerations of up to 1-2 orders of magnitude compared to the serial implementations. The differences in performance between the case studies reflect differences in their biological complexity. The fraction of the code's execution time spent performing calculations evaluating fitness, as opposed to reading and writing data from memory (albeit within the confines of GPU memory), is higher for case studies 2 and 3 than for case study 1 (where the speed of the simulation is limited instead by the speed of memory access). We note, however, that even in complicated IBMs, memory access may eventually prove increasingly taxing as more patches are simulated (Fig. 2B). Our comparative analysis of the three case studies thus identifies the biological circumstances under which sPEGG is likely to be significantly more efficient than serial implementations.
Our results show that the performance-improvements provided by sPEGG can best be exploited when calculating how ecological processes affect individual fitness, delivering performance improvements comparable to a small cluster for our most complicated case-study model. In addition to being the first general framework for constructing and accelerating evolutionary individual-based models on GPGPUs, sPEGG also represents the first resource using a generalized, forward time population genetics model to study multi-species evolution in ecological communities. Thus, sPEGG may be an especially attractive framework for using system-specific, mechanistic models to elucidate the interplay between ecological and evolutionary dynamics.
\section*{Acknowledgments}
This research was funded in part by a Chair's Fellowship from the Department of Ecology and Evolutionary Biology at the University of California, Los Angeles to K.W.O and a Complex Systems Scholar Award from the James S. McDonnell Foundation and NSF grant DEB 1457815 to P.A. We would like to thank G. Lin for discussion and assistance with some of the programming.
\bibliographystyle{cbe}
\section*{Online Methods}
\subsection*{Application overview}
The core parallelization routines in sPEGG are implemented using Thrust (\citealt{thrust}), a highly optimized parallel algorithms library loosely resembling the widely used C++ standard template library (STL) (\citealt{stl}). To exploit GPGPU hardware, the present generation of Thrust uses CUDA (\citealt{nickolls08}). Thus, the GPGPU version of sPEGG can currently only run on CUDA-enabled NVIDIA hardware, although this may change as Thrust expands to support other GPGPU programming languages (e.g., OpenCL). Thrust also allows us to use multi-core CPU processors for parallellization. Currently, most consumer desktops have only a small number (generally 2-8) of CPU cores (e.g., \citealt{hruska12}). However, if CPU core counts increase dramatically at affordable prices while GPGPU hardware improvements stagnate, sPEGG's use of the Thrust library enables it to be well-positioned to exploit such hardware developments.
In general, sPEGG relies on an object-oriented design (e.g., \citealt{booch82}, \citealt{rumbaugh90} and \citealt{martin02}). For instance, sPEGG models each species as a separate object, consisting of thrust vectors stored on GPGPU memory that describe the phenotypes and genotypes of individuals. The calculations are then performed in parallel on individual-level data stored in the species objects. Models created with sPEGG can then be written in standard C++. Supplementary Figure S4 illustrates how the species objects of sPEGG could be used in an IBM.
sPEGG is released under the GNU Public License v3 (\citealt{gpl3}). The most recent version of sPEGG, further documentation, source code for the case studies (including their parameter values and initial conditions) and a short tutorial are available at \nolinkurl{https://github.com/kewok/spegg}.
\subsection*{Approach to GPGPU parallelization}
The simulation of dynamical models in ecology and evolution is inherently serial (as the state variables are often Markovian). Thus, parallelization in sPEGG occurs within a time-step of the model across individuals. For many modeling problems, the benefits of parallelization, particularly on GPGPUs, are often greatest when the number of calculations are large (e.g., \citealt{harish07}, \citealt{nyland07}, and \citealt{owens08}). Hence, the performance gains due to sPEGG are largest when calculations within each time-step can be carried out over a large number (typically $>$ 10$^6$) of individuals. This may present a potential problem for accelerating simulations that model the eco-evolutionary dynamics of a smaller number of individuals, as can often be the case in some systems (e.g., \citealt{pelletier09}).
A single simulation run of a given model may only simulate a small number of individuals - for instance, on the order of hundreds to tens of thousands, which are well below abundances at which the benefits of parallelization on the GPGPU may be apparent. However, sPEGG's power lies in its ability to simulate a large number of individuals across several simulation runs simultaneously. Consider simulating a model of 1000 individuals across 10,000 different parameter combinations. This exercise requires performing calculations on 10$^7$ individuals, which is well within the range of the number of calculations whose performance can benefit from parallelization on a GPGPU.
To accomplish this, sPEGG organizes individuals into discrete patches. The population of individuals within a given patch is referred to as a deme. Individuals can potentially migrate between patches, although the user can also prohibit migration to and from patches. In sPEGG, patches that are not linked by migration can therefore be thought of as representing independent, replicate simulations. For instance, in the example above, sPEGG will simulate a model of 10,000 patches consisting of 1000 individuals each, where the model's dynamics within each patch are governed by a unique parameter combination. Supplementary Figure S5 illustrates an example of how sPEGG uses patches to enable parallel calculations to be performed simultaneously for all individuals. A given patch can contain multiple species, and, between migration events (if they are allowed), individuals only interact with other individuals within the same patch.
\subsection*{Customizing sPEGG}
As noted in the main text, the sPEGG code-base for simulating a species-specific model requires some level of user customization. Functions initializing the data, simulating mating and reproduction, and updating the trait values/phenotypes of individuals are completely general and easily customizable for researchers using models to address a variety of questions. sPEGG includes built-in alternative routines and classes enabling researchers to readily implement code for diverse systems, such as different mating systems and inheritance mechanisms (see the main text for details).
For situations that are harder to generalize, such as the rules describing how individual phenotypes are updated in response to heterospecific individuals during each time step, sPEGG provides methods, as well as readily usable classes, to facilitate the updating of phenotypes. The case studies illustrate the use of these classes for performing calculations between individuals of different species as well as for modeling resource dynamics and individual consumption behavior. Finally, the individual-based models used in the case studies can be explored further by merely changing the numerical values in the corresponding configuration (text) files.
\subsection*{Comparison Between Serial and Parallel Versions of the Case Studies}
To assess the performance advantages of using sPEGG in each of the case studies in the main text, each IBM was re-coded by hand using equivalent C++ classes, iterators and functions (invoking functions from the widely-used GNU scientific library - \citealt{gsl} - which relies on inherently serial algorithms) for the functions, classes, and model-specific code in sPEGG. We then applied optimization techniques, which are known to improve execution time in a serial context, to the resulting code base. To compare the serial and parallel versions of our models, we simulated our serial IBMs on a single 3.6 Ghz (Intel core i7 3820) CPU processing core and used one half of an NVIDIA GTX 690 GPU (restricting program access to 2GB of GPU RAM and 1536 CUDA cores) to assess the parallel version's performance. All optimizing transformations provided by the compilers (g++ and nvcc) were enabled.
\bibliographystyle{cbe}
\section*{Supplementary Information S2 - Model details for simulation of evolving neonate mass in the face of ontogenetic niche shifts}
In many species, the ability of individuals to exploit different resources changes as individuals grow in size or develop across life stages, a process referred to as an ontogenetic niche shift (\citealt{werner84}). When consumption by conspecifics causes resource limitation, ontogenetic niche shifts can impose distinct patterns of density-dependence during an individual's life cycle. Such density-dependence could affect the ability of newborns to survive, grow and mature, potentially affecting patterns of life history evolution (e.g., \citealt{mueller91}). For instance, evolutionary changes in neonate mass are subject to a trade-off between selection for larger clutch sizes and increased offspring survivorship (e.g. \citealt{lack66}, \citealt{parker86} and \citealt{roff01}). However, the strength of this trade-off itself can depend upon prevailing patterns of resources available for juvenile growth (necessary for escaping vulnerable size classes) and adult reproduction (which partially governs the number of offspring a parent can birth or sire). These patterns of resource availability at different life stages may, in turn, depend on the strength of an ontogenetic niche shift individuals experience.
To understand the effect of ontogenetic niche shifts on the evolution of neonate mass, we use sPEGG to model a physiologically-structured (e.g., \citealt{persson98} and \citealt{deroos01}), sexually reproducing consumer population whose individuals utilize and compete over two dynamic, biological resources. We assume maximum resource consumption rates increase allometrically (\citealt{brown04}), but that individuals transition from utilizing one resource type to utilizing another as they increase in size. We model offspring size at birth as an evolvable trait controlled additively by a finite number of loci of large effect. We assume survivorship increases monotonically with individual size (e.g., \citealt{werner84}), but that parents with larger neonates also birth fewer offspring (e.g., \citealt{parker86}).
\subsection*{Model overview}
We consider a size-structured, sexually reproducing consumer population. Our model has two key components - an ecological component and a genetic component. The ecological component describes the individual's interactions with their environment. We model how differential reproductive success and survivorship (fitness) among individuals emerge from these diverse interactions. The genetic component specifies the genetic distribution for neonate mass, and how the genetic distribution changes between generations.
The model's dynamics are iterated on a discrete time step. At each time step, the model cycles through all individuals to determine their fates. We assume that the organisms reproduce once every $T$ time-steps; thus, each time-step can be thought of, for instance, as a day for a specie's with an annual breeding season. Our model applies to organisms with overlapping generations and seasonal reproduction such as insects, marine invertebrates, and vertebrates that live in seasonal environments. We also partition individual somatic mass $W$ into reversible ($Y$) and irreversible ($X$) mass to explicitly study how energetic constraints affect individual reproductive decisions.
The model's parameters characterize the selective pressures and the genetic, energetic and ecological constraints on the population. Its ecological dynamics are driven by four key processes: resource consumption, somatic growth, mating, and mortality. Below, we describe how each process is modelled in further detail. Table S2-1 provides a summary of the model's ecological components.
\subsection*{Resource Consumption}
We consider two dynamic, biological resources which the consumers utilize and for which they compete. Resource availability can regulate individual somatic growth, and ultimately affects individual survivorship and reproduction. Thus, considering two resources allows us to model situations where newborns and large adults potentially experience different regulation regimes, and hence permit different forms of resource scarcity to operate during an individual's lifetime.
The instantaneous resource-consumption rate $E_{i,{\tau},j}$ of individual $i$ of resource $j = 1,2$ at time $\tau$ is a function of its body size (somatic mass $W_{i,\tau}$) and resource density $R_{\tau,j}$:
\begin{linenomath}
\begin{equation}
E_{i,\tau} (R_{\tau,j}, K_j, W_{i,\tau}) = h(R_{\tau,j},K_j) \pi(W_{i,\tau}, j) \alpha W_{i,\tau}^{\gamma}
\end{equation}
\end{linenomath}
where $\pi(W_{i,\tau}, j)$ describes the proportion of a individual $i$'s diet that consist of resource type $j$, and $h(R_{\tau,j}, K_j)$ describes the proportion by which an individual's resource consumption decreases when the resource density $R_{\tau,j}$ falls short of the maximum daily consumption rate (attained when the resource density is at carrying capacity $K_j$).
We assume that if the resources are at their carrying capacities, then an individual's instantaneous resource consumption rate depends allometrically on its body size, where the parameters $\gamma$ and $\alpha$ are an allometric exponent and an allometric constant scaling consumption rates, respectively (e.g., \citealt{west01}).
The size-independent function $h(R_{\tau,j},K_j)$ describes how much an individual's consumption of resource $j$ decreases as the becomes more scarce. For example, if resource $j$ has carrying capacity $K_j$ and consumer individuals display a Holling type-II functional response with attack rate $a$ and handling time $T_H$, then $h(R_{\tau,j},K_j)$ describes how resource limitation constrains consumption, i.e., $h(R_{\tau,j},K_j) =\frac{a R_{\tau,j}/K_j}{1 + a T_H R_{\tau,j}/K_j}$.
We assume that the amount of a given resource of type $j$ a consumer utilizes depends on the consumer's size. Such ontogenetic niche shifts are common in nature (e.g., \citealt{werner84}). For example, many fish are planktivorous when small, but shift to piscivory as they increase in size (e.g., \citealt{claessen02}). The function $\pi(X_{i,\tau}, j)$ describes the proportion of individual $i$'s diet that consist of resource type $j$. The function $\pi(X_{i,\tau},j)$ is modeled to depend on both anatomical structures such as gape-size and body length that contribute to irreversible mass (e.g., \citealt{werner84}). It is also determined by a constant $u$ that governs what fraction of the adult resource base very small individuals consume, and the steepness $p$ of the curve as individuals grow. Thus, $\pi(X_{i,\tau},j)$ is defined by the following logistic equations:
\begin{linenomath}
\begin{equation}
\label{eqn:pi}
\pi(W_{i, \tau}, j) = \left\{ \begin{array}{ll}
\frac{1}{1 + \exp(- p(l[W_{\tau,i}] - u l[W_{max}]))}, & \textrm{if } j = 1 \\
1 - \frac{1}{1 + \exp(- p(l[W_{\tau,i}] - u l[W_{max}]))}, & \textrm{if } j = 2
\end{array} \right.
\end{equation}
\end{linenomath}
l[] is a function mapping irreversible mass to body length(e.g., \citealt{agresti02} and \citealt{claessen02}). We highlight that when $\pi(X_{i, \tau}, j)$ is invariant with $X_{i,\tau}$ (e.g., $p$ small), this describes a situation where resource scarcity affects all individuals similarly during their life-time.
Resource variability is driven by both extrinsic and intrinsic sources of variability. Extrinsic sources of variability, such as climactic fluctuations, affect the per-capita intrinsic growth rate $r_j^{\prime}$ of resource $j$. By contrast, temporal variation in resource utilization by the consumer generates variability in resource levels that is intrinsic to the system.
We assume that the dynamics of resource $j$ are described by intrinsic, density-dependent growth and predation by the consumer (e.g., \citealt{claessen00}) and are governed by the following Beverton-Holt-like growth curve:
\begin{linenomath}
\begin{equation}
R_{\tau+1,j} = \frac{r_j^{\prime} R_{\tau,j}}{1+(R_{\tau,j})/K_j} - \sum_{i=1}^{N_{\tau}} E_{i,\tau,j} (R_{\tau,j})
\end{equation}
\end{linenomath}
where $E_{i,0,j} (R_{0,j})$ is the consumption rate of resource $j$ by the individual consumer $i$ at the start of the time step, and $N_{\tau}$ is the population size of consumers at the beginning of the time step.
We model stochastic fluctuations in the resource dynamics due to extrinsic factors, such as climactic variability, by allowing $r_j^{\prime}$ to vary from time step to time step by drawing the actual value of $r_j^{\prime}$ used in the evaluation of $R_{t,j}^{\star}$ from a normal distribution with mean $r_j$ and standard deviation $e_{r,j}$. Our model does not consider externally imposed deterministic fluctuations. Hence, temporal variability in resource abundance and the consumer's population dynamics are emergent properties of the underlying ecological processes we model, as well as the stochastic nature of the model itself.
\subsection*{Somatic Growth}
In our model, somatic growth generates size structure. We model two components of somatic growth: growth in irreversible mass $X$ and growth in reversible mass $Y$. An individual's irreversible, or structural, mass $X$ consists of compounds such as organ and skeletal tissue that cannot be starved away (e.g, \citealt{broekhuizen94} and \citealt{deroos01}). Irreversible mass can be viewed as a surrogate for body length, which does not decrease even under starvation conditions (e.g., \citealt{broekhuizen94}). By contrast, an individual's reversible mass is determined by energy reserves such as lipids and gonadal tissue in mature individuals that can be starved away. Reversible mass is partitioned into the mass of storage tissue $Y$, and, in mature individuals, the mass of gonadal tissue, $G$ (e.g., \citealt{broekhuizen94}, \citealt{persson98}, \citealt{deroos01}) (Fig. 1B). Hence,
\begin{linenomath}
\begin{equation}
W = X + Y + G.
\end{equation}
\end{linenomath}
In females, gonadal mass $G$ consists largely of reproductive tissue, while in males, gonadal mass $G$ is interpreted as the amount of reversible mass allocated to reproduction through the loss of somatic mass incurred by competing with other males for access to mature females as well as the production of reproductive tissue.
We assume that individuals are born with the maximum reversible mass to irreversible mass ratio. We define an individual's condition, $c$, as the ratio of storage tissue mass to total somatic mass, i.e., $c=\frac{Y}{W}$.
An individual consumer's size at maturity and the fraction of resources it allocates towards reproduction affects how it allocates its energetic intake between irreversible, reversible, and gonadal mass. Here, we seek to link resource consumption to an individual's physiological state mechanistically. An individual's physiological state (e.g., body size, condition, etc...) determines its performance (in particular, survivorship and reproduction). Therefore, explicating how metabolic constraints affect somatic growth is key to describing the feedback between an individual's environment and its physiological state.
A consumer with mass $W_{i,\tau}$ at time $\tau$ grows according to the following growth equation (e.g., \citealt{west01})
\begin{linenomath}
\begin{equation}
\label{eqn:growth1}
W_{i,\tau+1} = H(R_{\tau}, W_{i,\tau}) \alpha W_{i,\tau}^{\gamma} - \delta W_{i,\tau},
\end{equation}
\end{linenomath}
where $\delta$ describes the metabolic rate per unit mass of consumer tissue, $H(R_{\tau}, W_{i,\tau}) = \sum_{i=1,2} (h(R_{\tau,j}, K_j) \pi(W_{i,\tau}))$ describes how resource scarcity ($R_{\tau}$) at time $\tau$ reduces the growth rate of an individual of size $W_{i,\tau}$, and the parameters $\gamma$ and $\alpha$ are an allometric constant and an allometric exponent scaling consumption. We therefore assume that when the mass derived from consumption exceeds metabolic costs, any surplus mass is invested in somatic growth. For taxa with indeterminate growth, $W_{max}$ represents an asymptotic body size which individuals approach at a decelerating rate. For taxa with determinate growth, $W_{max}$ simlpy defines the body size at which the somatic growth rate is zero. We assume $\gamma \leq 1 $ so that there is an asymptotic maximum size $W_{max}$ beyond which matabolic costs render further somatic growth impossible (for empirical values of $\gamma$, see, e.g., \citealt{moses08}).
Consumer individuals are assumed to be born with the maximum ratio $q_J$ between reversible mass and irreversible mass. We note that at birth, an individual's reversible mass does not include gonadal mass - instead, reversible mass consists of storage tissue such as fats. Upon maturation, reversible mass includes the mass of gonadal tissue as well as storage tissue. Hence, the corresponding maximum ratio between reversible mass and irreversible mass for adults is given by $q_A > q_J$. Because reversible mass can be starved away, it represents a reserve that individuals can utilize during periods of resource limitation. Thus, when resources are not limiting, we assume consumer individuals will maintain the maximum ratio of reversible to irreversible mass. The fraction $\kappa$ of $ \Delta W_i = W_{i,\tau+1} - W_{i,\tau}$ that is allocated to irreversible mass depends on the ratio of reversible to irreversible mass; thus:
\begin{linenomath}
\begin{equation}
\label{eqn:growth1}
X_{i,\tau+1} = \kappa \frac{Y_{i,\tau}}{X_{i,\tau}} \Delta W_i + X_{i,\tau+1},
Y_{i,\tau+1} = (1-\kappa \frac{Y_{i,\tau}}{X_{i,\tau}}) \Delta W_i + Y_{i,\tau+1}.
\end{equation}
\end{linenomath}
Hence, $\kappa$ describes a constraint on how much an individual can allocate resource intake towards fending off starvation risk (by improving condition) instead of reducing other forms of density-independent mortality by increasing irreversible mass.
In mature individuals, a fixed proportion $\rho_i$ of $1-\kappa \frac{Y_{i,\tau}}{X_{i,\tau}}) \Delta W_i$ is set aside for reproduction. For example, in female animals this takes the form of allocation to reproductive tissue mass $G_i$, while in males this represents, for example, the mass loss associated with searching and securing female mates as well as the energy expended in producing gonadal tissue. For simplicity, we assume the mass of reproductive tissue in juveniles is negligible. Thus, at the end of each time step, reproductive tissue mass for individual $i$ is given by:
\begin{linenomath}
\begin{eqnarray}
\label{eqn:allocations}
G_{i,\tau+1} &=& I_f \times (Y_{i,\tau} - X_{i,\tau} q_A)
\end{eqnarray}
\end{linenomath}
where $I_f = 0$ if the individual is immature, $I_f = 1$ if the individual is mature. The growth equations above are based on the assumption that all reproductive tissue mass from the previous time step was spent on reproduction during that time step. Following reproduction, the gonadal mass $G_{i,\tau+1}$ is deducted from the individual's reversible mass $Y_{i,\tau}$.
Equations (\ref{eqn:allocations}) summarize how energetic constraints govern the feedback between an individual's environment and its physiological state. In particular, they describe how resource use affects an individual's reproductive success (which depends on gonadal mass $G$) and survivorship (which depends on irreversible mass $X$ and condition $c=Y/(X+Y+G)$).
\subsection*{Reproduction}
Individual consumers breed once every $T$ time steps, and we assume random mating between males and females, conditioned on gonadal mass. If, prior to mating, an individual consumer $i$'s irreversible mass $X_i$ is larger than its size at maturity ($\sigma_i$), then individual $i$ is considered to be mature.
The number $F$ of fertilized eggs in the population is determined by three variables: (1) the average neonate mass of offspring of female $i$, $\Omega_i + q_J \Omega_i$, where $q_J$ is the maximum ratio of reversible mass to irreversible mass, (2) the reproductive tissue mass of female $i$ $G_i$, and (3) $N_{f,t}$, the total number of mature females in time step $t$. Then,
\begin{linenomath}
\begin{equation}
\label{eqn:fecundity}
F = \sum_{i=1}^{N_{f,t}} \frac{G_i}{\Omega_i + q_J \Omega_i}
\end{equation}
\end{linenomath}
The present modeling framework applies to both viviparous and oviparous organisms. However, for simplicity, we discuss model development in terms of an oviparous organism.
For each egg, the mother and father are drawn at random from mature males and females in the population. The probability that the egg comes from female $i$ is a function of the female's relative gonadal mass $G_i$ in the population,
\begin{linenomath}
\begin{equation}
\label{eqn:Femalerepr}
\Pr(\textrm{female $i$ produces an egg} | G_i) = (\frac{\frac{G_i}{\Omega_i + q_J \Omega_i }}{F})^{v_f},
\end{equation}
\end{linenomath}
where $v_f$ describes the severity fo reproductive skew among females.
\indent Similarly, the probability that a mature male $i$ fertilizes a given egg is a function of its irreversible mass, $X_i$, relative to the mass of other mature males in the population. In particular, our model reflects the common observation that body size (often measured as length, which depends on irreversible mass) is positively correlated with reproductive success in males (e.g., \citealt{trivers72}, \citealt{blanckenhorn05}), and a male's relative structural mass is incorporated into calculating the probability of fertilization as:
\begin{linenomath}
\begin{equation}
\Pr(\textrm{male $i$ fertilizes an egg} |X_i, G_i) = \frac{X_i^{v_m}}{\sum_{j=1}^{N_{m,t}}(X_j)^{v_m}}
\end{equation}
\end{linenomath}
where $N_{m,t}$ is the total number of mature males, and $v_m$ determines how strongly a male's reproductive value increases with its relative irreversible mass. The parameter $v_m$ can describe a host of biological processes, including female preference for larger males and the superior fighting ability of larger males. High values of $v_m$ characterize populations where reproductive success is strongly correlated with male body size, while lower values of $v_m$ characterize populations where male reproductive success is similar across a range of body sizes.
\subsection*{Mortality}
\indent At any given time step, the number $\Phi$ of newborns that survive to the juvenile stage depends on the population's total egg production (e.g., \citealt{wootton98}). Several mechanisms can contribute to such density-dependent mortality at the earliest life stages (\citealt{shepherd80}). For instance, in many organisms early life stages (e.g., larvae) are particularly vulnerable to predators that can be attracted to large aggregations of eggs or hatchlings (\citealt{martin93}, \citealt{bellinato95}, and \citealt{white07}). Furthermore, when hatchlings or early larvae form dense aggregations over small spatial scales, such aggregations can lead to the rapid exhaustion of locally available resources (e.g., \citealt{leggett94}, \citealt{arino98}). Finally, in many organisms, dispersal occurs at very early stages of the life cycle. In such situations, the ability of newborns to become established and survive to maturation at suitable sites depends, in part, on the fraction of sites already occupied by breeding adults, which in turn affects the value of $F$ (eq. (\ref{eqn:fecundity}) e.g., \citealt{caley96}). For example, in several coral reef fishes, the density of adults is inversely correlated with the survivorship of juveniles, possibly because of competition faced by juveniles for suitable shelter sites that protect individuals from predation or the availability of desirable feeding sites (e.g., \citealt{sale76}, \citealt{forrester95}, \citealt{caley96}, and \citealt{wilson02}). When these mechanisms interact, as when limited resource availability stunts somatic growth and thereby keeps new borns from growing out of vulnerable size classes, the relationship between the number of surviving newborns and total fecundity is described by classical stock-recruitment curves (\citealt{shepherd80}).
Here we model the density-dependent recruitment $\Phi$ according to the function:
\begin{linenomath}
\begin{equation}
\Phi = \frac{\nu}{1+N_f},
\end{equation}
\end{linenomath}
where $\nu$ is a constant and $N_f$ is the total number of reproductive females. For viviparous organisms, $\Phi$ can be interpreted as the fraction of offspring that survive past a weaning period.
\indent In addition to density-dependent mortality of newborns, individual survivorship from one time step to the next is a function of the individual's irreversible mass. We assume that the probability that an individual survives from one time step to the next is described as
\begin{linenomath}
\begin{eqnarray}
\label{eqn:survivorship1}
\Pr(\textrm{individual $i$ survives} | X_{i,t}) = \frac{1}{1+\exp(-\beta_2 (X_{i,t} - \beta_1))}
\end{eqnarray}
\end{linenomath}
This functional form results from using standard logistic regression to model survivorship probability as a function of size, as is frequently recommended (e.g., \citealt{morris02}, \citealt{ellner06}, and \citealt{hesse08}). The parameter $\beta_1$ represents the mass (scaled by the maximum irreversible mass $W_{max}$) at which the mortality risk is equal to 1/2. The parameter $\beta_2$ characterizes the steepness of the survivorship curve around this point. This functional form is quite flexible, and depending on the values of $\beta_1$ and $\beta_2$ can describe an exponential increase in survival with increased irreversible mass, a sigmoidal increase in survival with larger irreversible mass, a relatively linear increase in mortality risk with irreversible mass, or a rate of mortality that is largely independent of body size.
We further assume that individuals also suffer an additional source of mortality due to starvation. We assume that as individual condition worsens, starvation risk increases and survival decreases at an exponential rate. The survivorship function thus depends on individual condition and somatic mass as:
\begin{linenomath}
\begin{eqnarray}
\label{eqn:survivorship}
\Pr(\textrm{individual $i$ survives at time $t$} | X_{i,t},c_{i,t}) = (1-\exp(-\beta_s Y_{i,t}/X_{i,t})) \times \frac{1}{1+\exp(-\beta_2 (X_{i,t} - \beta_1))}.
\end{eqnarray}
\end{linenomath}
The parameter $\beta_s$ characterizes how survivorship increases exponentially with improving condition. Because resource availability governs somatic growth rates and determines an individual's body size and condition, eq. (17) accounts for the potential for density-dependent processes to affect survivorship throughout an individual's life cycle.
Size-specific mortality links an individual's physiological state to its performance. Size-specific mortality can directly select for different neonate sizes, as well as generate population fluctuations and induce variability in resource availability (e.g., \citealt{stearns76}, \citealt{costantino97}, \citealt{deRoos031} and \citealt{ernande04}). In turn, these patterns of temporal variability could subsequently affect the evolution of life history syndromes (e.g., \citealt{winemiller92}).
\subsection*{Genetics}
In our model, the fitness of individuals emerges from their interactions with other individuals and their environment. The evolutionary response of the population to selection on individuals depends on the genetic distribution among individuals. Moreover, the genetic distribution, in turn, is determined by the combined effects of mutation, recombination, and selection. In individual-based models, describing the population's genetic distribution requires we specify the distribution of genotypes among individuals. Thus, specifying how multiple life history traits evolve requires explicit consideration of the genetics underlying the life history traits. Below, we describe how we model the genetics and transmission of the life history traits.
We analyze evolution in the mean irreversible mass $\Omega$ of offspring at birth (neonate mass). While survivorship often increases monotonically with individual size (e.g., \citealt{werner84}), parents that give birth to larger neonates give birth to fewer offspring (e.g., \citealt{parker86}). Thus, focusing on neonate mass allows us to study a trait that directly evolves in response to a trade-off between clutch size and offspring survivorship (e.g. \citealt{lack66} and \citealt{roff01}).
We assume that offspring size at birth $\Omega$ is a quantitative traits whose genetic values are additively determined by $N_\Omega$ trait-specific loci. The genome is diploid, individuals reproduce sexually, and there is free recombination between all loci. Pleiotropy is absent from the model, and the allelic values of the loci vary continuously over the biologically feasible range of the life history trait. To describe the evolution of this life history trait, we track the dynamics of the alleles and loci underlying the trait explicitly. Such an explicit, multi-locus framework allows us to characterize and account for changes in the genetic distribution of the trait over time. All loci were treated as autosomal and freely recombining with other loci. The allelic values of the loci could change by mutations and the effect of mutations on the allelic values are assumed to be Gaussian distributed. We note that when the number of trait-specific loci approaches infinity and individual allelic effects decline to zero, these assumptions allow our model to recover the classical infinitesimal model used in quantitative genetics (e.g., \citealt{bulmer85b}).
Prior to reproduction, each parent produces haploid gametes consisting of half the parent's alleles (e.g., \citealt{van06}). Mutations occur with a probability $\mu$ at each locus. If a mutation occurs at a locus, the new allelic value is drawn from a normal distribution with the mean at the allelic value prior to mutation and a standard deviation given by $\varpi \times$ the mean initial allelic value. The gametes from both parents fuse to produce an offspring's diploid genome. The offspring's genetic value at each locus is given by the midpoint of the parental gamete's allelic values at that locus. Thus, the offspring's genetic value for $\Omega$ is the sum of the genetic values across all $N_\Omega$ loci.
Once an individual's genotypic value is additively determined, its phenotypic value is determined first by drawing a random variable, $z_J$ from a normal distribution with the mean given by the genotypic value and a trait-specific standard deviation $\varrho_J$. In effect, determining the phenotypic value in this way is analogous to specifying the residual variance for the trait (i.e., the difference between the trait's phenotypic and additive genetic variances - e.g., \citealt{houle96}). Thus, if $A_{i,f}, A_{i,m}$ denote the allelic value at locus $i$ an individual inherited from its female and male parent, respectively, and $N(M,\varrho)$ describes a normal distribution with mean $M$ and standard deviation $\varrho$, the individual's phenotypic value $z_J$ is a random variable given as:
\begin{linenomath}
\begin{eqnarray}
\label{eqn:phenval}
z_J \sim N(\displaystyle\sum_{i=1}^{N_J} \frac{1}{2} (A_{i,f} + A_{i,m}), \varrho_J).
\end{eqnarray}
\end{linenomath}
Life history traits appear to generally have low heritabilities (e.g., \citealt{mousseau87} and \citealt{price91}). Nevertheless, there is some question about the relative importance of dominant and epistatic effects among loci for life history traits (e.g., \citealt{roff06}) or whether environmental, gene-by-environment interactions, and developmental processes account for the low heritability in life history traits (e.g, \citealt{price91}, \citealt{houle96}, \citealt{burger00}). Because $\varrho_J$ is shared across individuals, our model formulation effectively assumes most of the residual variance reflects the effects of developmental and (unspecified) environmental effects rather than nonadditive genetic components.
Finally, the values of $z_J$ were transformed using standard approaches to ensure the expressed phenotypic values were biologically realistic. In particular, we set
\begin{linenomath}
\begin{eqnarray}
\label{eqn:survivorship}
z_J^{\prime} = \exp(z_J),
\end{eqnarray}
\end{linenomath}
for neonate mass at birth. Thus, when the genetic values of these traits follow a normal distribution, the genetic component of their phenotypic values follow a lognormal distribution (e.g., \citealt{lynch97}).
\subsection*{Analysis and Results}
We vary the density-independent, per-capita recruitment rate of the resource consumed by smaller individuals to assess how shifting the relative availability of juvenile and adult resources affects the evolution of offspring size at birth. The parameter values used in the simulations of this model employed in the main text are available at \nolinkurl{https://github.com/kewok/spegg}. We find that depending on the relative availability of these resources, the consumer population can evolve towards two potential evolutionary endpoints. The first endpoint occurs where individuals give birth to a large number of small offspring. The second endpoint occurs where individuals give birth to a small number of large offspring (Supplementary Fig. S2). If the juvenile resource is relatively scarce, then this selects against parents that give birth to small offspring, who can remain in vulnerable size-categories until they acquire sufficient resources for growth. By contrast, if the juvenile resource is more abundant, this permits relatively rapid juvenile growth even by small offspring. This relaxes the offspring survivorship-clutch size trade-off, favoring the evolution of smaller body sizes at birth. The model illustrates how differences in a phenotype of the resource species (in this case, per-capita density-independent recruitment) can cascade through ontogenetic changes in the consumer to select for distinct consumer life-history strategies.
\bibliographystyle{cbe}
\section*{Supplementary Information S3 - Model details for simulating a co-evolving resource-consumer metacommunity}
We model a $P$ patch meta-community where each patch is linked by migration to its neighboring patch. Within each patch, individuals of the consumer species $C$ interact with individuals of the resource species $R$. We model coevolutionary dynamics in systems where consumers with a given phenotypic value are most effective at exploiting resources with a particular resource trait value, but may be poor at exploiting resources with other trait values (i.e., phentoypic matching - e.g., \citealt{nuismer07}). For instance, an herbivorous insect may evolve specific enzymes permitting it to exploit a novel plant resource, while the plant species may evolve alternative chemical defenses against the herbivore (\citealt{rasmann11a}). Alternatively, the ability of a resource to defend itself, and the ability of a consumer to exploit the resource, may itself be a composite trait of many distinct phenotypes (e.g., behavioral as well as morphological defenses), with consumer individuals varying in their ability to exploit different facets of resource defense (e.g., some consumers may be particularly good at identifying resource hiding sites, while others may have greater ability to puncture resource defenses such as shells).
In our model, the potential individual fecundity $w_{F,C}(t)$ of an individual consumer varies over time. An individual's breeding value determines the genetic component of this trait. $w_{F,C}(t)$ is also affected by how successfully an individual consumer is able to exploit the resource species. A consumer's ability to successfully exploit the resource, in turn, depends on two quantities: first, the individual's genetically determined exploitation ability, $\alpha$, and second, the phenotypic distribution of a resource defense trait $\delta$. This adds a further source of genetic variation in individual fecundity, as well as an environmental effect that varies according to the prevailing distribution of the resource's phenotypes. At a given time $t$, the consumer's fecundity can be given by:
\begin{align}
w_{F,C}(t) &= \xi k \int_{y \in \{\delta\}} \frac{\exp(-(\alpha-y)^2)}{1 + \exp(-(\alpha-y)^2)} dy, \label{eqn:consumer_fecundity}
\end{align}
where $\xi$ is the breeding value characterizing the consumer's baseline fecundity and $k$ scales the effect of the interaction on consumers (it could thus represent, e.g., something like a conversion efficiency). Figure S3-A illustrates the per-capita effect of encounters between a given consumer and resource individual based on their phenotypic values. We note that removing the squared expression inside the exponential would describe a consumer-resource interaction with monotonic arms-race dynamics.
The consumer's per-capita mortality rate $w_{M,C}$ is assumed to be constant during an individual consumer's's life; thus, we do not model reduction in survivorship through a failure to acquire resources (e.g., via starvation).
Encounters with consumers are assumed to increase the mortality risk of the resource; we describe changes in the resource's mortality rate $w_{M,R}$ to depend on the extent to which the resource's trait is dissimilar to the distribution of consumer exploitation traits:
\begin{align}
w_{M,R}(t) &= \gamma v \int_{x \in \{\alpha\}} \frac{\exp(-(x-\delta)^2)}{1 + \exp(-(x - \delta)^2)} dx,\label{eqn:resource_mortality}
\end{align}
where $\gamma$ is the breeding value characterizing the resource's baseline survivorship and $v$ scales the effect of the interactions on resource mortality.
We also model implicit exploitative competition among resource individuals. The outcome of such resource-resource competition determines the distribution of resource fecundities. We assume that the the resource's ability to defend itself against consumers comes at a cost to reproduction. Such a trade-off could arise, for instance, if energetic reserves need to be allocated to producing secondary compounds instead of fruit (e.g., \citealt{bazzaz87}; also \citealt{bohannan97}). Changes to resource fecundity are then modeled as:
\begin{align}
w_{F,R}(t) &= w_{F,R}(t) n_i \exp(-\rho \alpha^2),
\label{eqn:resource_fecundity}
\end{align}
where $n_i$ describes the aggregate effect of any encounters a resource individual $i$ may have with other resource individuals, and $\rho$ describes the severity of the trade-off between resource defense and resource fecundity. This entails that greater investment by the resource in any given anti-consumer strategy can entail greater fecundity costs.
Eqns. (\ref{eqn:consumer_fecundity} - \ref{eqn:resource_fecundity}) are mathematical abstractions in which a given individual of the focal species is assumed to encounter all possible individuals in the other species. In nature, however, encounters between individuals often result from Poisson processes (e.g., \citealt{hassell78}). Therefore, to capture the effects of contingent interspecific interactions on trait evolution, for a given individual of either species we instead simulate a random subset of possible encounters out of all possible species. The specific life history traits therefore change according to the cumulative interactions an individual experiences during a given time step, rather than summing over all possible interactions.
For the simulations of this model reported in the main text, we assumed that the phenotype governing consumer exploitation ability is controlled by three loci which we assume have additive effects on the consumer trait of interest. Similarly, the resource's defensive trait was assumed to be additively controlled by five loci. For both the consumer and resource species, genetic variations in the life history traits at birth were assumed to be minimal.
Figure S3-A illustrate how the strength $\rho$ of the tradeoff between resource defense and resource fecundity can govern the eco-evolutionary outcome of consumer-resource coevolution. When the tradeoff is weak, the resource can readily evolve anti-consumer defenses and drive the consumer extinct; when the tradeoff is strong, the resource can ill-afford to invest too heavily in a particular strategy and thus the manner in which it avoids predation shifts, enabling coevolutionary cycling. The consumer can only persist when the tradeoff faced by the resource is weak when there is sufficient migration from patches in which the tradeoff is strong. Figure S3-B show how the extent of migration between patches governs consumer persistence. Figure S3-B shows how when the tradeoff is weak, the consumer population struggles to persist. However, when cross patch dispersal is relatively high, the inflow of maladapted consumers prevents successful local adaptation and consumer persistence is somewhat more limited. The parameter values used in the simulations of this model employed in the main text are available at \nolinkurl{https://github.com/kewok/spegg}.
\bibliographystyle{cbe}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,990 |
A month or so ago South Calgary Dreambuilders asked their clients and followers on Facebook, Linked In and Twitter, as well as our Blog followers, to weigh in with their thoughts on the importance of Better Business Bureau accreditation for renovation companies. While some responded that it did not matter much to them, others stated that it played a role in their decision making.
Fair enough! Our clients and potential clients had spoken, and we listened. We are pleased to announce that, as of today, we have been upgraded from standard BBB members to accredited members. What this means is that the Better Business Bureau has researched our methods of doing business, our professionalism and our conduct within the industry and we have passed with flying colours!
We look forward to our relationship with the Better Business Bureau, and are pleased that our future clients will now be able to check with the BBB as a reference of South Calgary Dreambuilders.
We would like to thank our clients and followers for their feedback, as it certainly helped shed light on the importance of the BBB's role to the consumer. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,931 |
{"url":"http:\/\/www.ma.utexas.edu\/mp_arc-bin\/mpa?yn=98-132","text":"98-132 C. Remling\nBounds on embedded singular spectrum for one-dimensional Schroedinger operators (30K, LaTeX 2e) Mar 4, 98\nAbstract , Paper (src), View paper (auto. generated ps), Index of related papers\n\nAbstract. Consider the 1D Schroedinger equation -y''+Vy=Ey with potential V bounded by |V(x)| < C(1+x)^{-c}. We prove that the solutions y satisfy the WKB asymptotic formulae off a set of energies E of Hausdorff dimension \\le 2(1-c). This gives restrictions on the structure of possible embedded singular spectrum.\n\nFiles: 98-132.tex","date":"2018-07-16 01:14:22","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\": 0.9314916729927063, \"perplexity\": 9134.09693300694}, \"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-2018-30\/segments\/1531676589029.26\/warc\/CC-MAIN-20180716002413-20180716022413-00379.warc.gz\"}"} | null | null |
Abel bosum`s thoughts on the question relating to the Cree Referedum on Quebec Separation. And the wording of the referendum question that was asked to the Cree people.
3 legends about Ayashaau, Mishikaakihkiyaahkwaau and Piipiihchaau. Traditonal songs that go with the legends.
Abraham sings and discusses cree hymn which he learned from Reverend Walton many years ago.
Abraham begins to tell the story of battles that were fought among the cree and Inuk people.
Abraham sings and explains about his goose song.There is also a story that is missing its beginning. He also sings one Cree hymn.
Abraham recounts the story of Iyaashaau and his son. Iyaashaau abandoned his son on an island far from home, but with the help of the seagull and fish, Iyaashaau`s son finds his way home. When the son returns home he seeks revenge on his father and sets his mother and siblings free by turning them into birds.
Abraham shares some traditional songs about animals, specifically the goose and fox. He also mentions the introduction of the fiddle and how it was adopted.
Traditional songs about animals and hunting animals.
The first story tells about how rituals were performed to create good. It is a story about when a community nearly went into starvation and how an elder performed a shaking tent to prevent the community from this bad omen. The second story tells how an old man who was sick was brought back to health by similar rituals.
This is a story of long ago when the Crees and the Inuits were still fighting and killing each other.
Brothers playing with magic powers and how they use their skills against each other.
Stories about how to take care of the deceased.
Encounter with Spirits, the event that takes place and the teachings from it.
These stories explain the events of what happens after death.
Story about magic and the attempt to stop the person involved.
Cree stories about the boogeymen (prospectors) along the shores of James Bay. Tales on how they got along with each other and with the local guides.
Once upon a time animals were still talking. The wolverine met all kinds of other animals and talked to them as if they were his brothers and sisters.
A story about a boy who got infested with lice and was left behind.
A legend about a family whose father got kidnapped by a monster and never returned home.
Annie tells about child birth and how she was the midwife for her mother while she was in the bushes with her parents. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,072 |
<?php
/**
* This code was generated by
* \ / _ _ _| _ _
* | (_)\/(_)(_|\/| |(/_ v1.0.0
* / /
*/
namespace Twilio\Rest\Sync\V1\Service\SyncList;
use Twilio\Deserialize;
use Twilio\Exceptions\TwilioException;
use Twilio\InstanceResource;
use Twilio\Options;
use Twilio\Values;
use Twilio\Version;
/**
* PLEASE NOTE that this class contains beta products that are subject to change. Use them with caution.
*
* @property integer index
* @property string accountSid
* @property string serviceSid
* @property string listSid
* @property string url
* @property string revision
* @property array data
* @property \DateTime dateExpires
* @property \DateTime dateCreated
* @property \DateTime dateUpdated
* @property string createdBy
*/
class SyncListItemInstance extends InstanceResource {
/**
* Initialize the SyncListItemInstance
*
* @param \Twilio\Version $version Version that contains the resource
* @param mixed[] $payload The response payload
* @param string $serviceSid The unique SID identifier of the Service Instance
* that hosts this List object.
* @param string $listSid The unique 34-character SID identifier of the List
* containing this Item.
* @param integer $index The index
* @return \Twilio\Rest\Sync\V1\Service\SyncList\SyncListItemInstance
*/
public function __construct(Version $version, array $payload, $serviceSid, $listSid, $index = null) {
parent::__construct($version);
// Marshaled Properties
$this->properties = array(
'index' => Values::array_get($payload, 'index'),
'accountSid' => Values::array_get($payload, 'account_sid'),
'serviceSid' => Values::array_get($payload, 'service_sid'),
'listSid' => Values::array_get($payload, 'list_sid'),
'url' => Values::array_get($payload, 'url'),
'revision' => Values::array_get($payload, 'revision'),
'data' => Values::array_get($payload, 'data'),
'dateExpires' => Deserialize::dateTime(Values::array_get($payload, 'date_expires')),
'dateCreated' => Deserialize::dateTime(Values::array_get($payload, 'date_created')),
'dateUpdated' => Deserialize::dateTime(Values::array_get($payload, 'date_updated')),
'createdBy' => Values::array_get($payload, 'created_by'),
);
$this->solution = array(
'serviceSid' => $serviceSid,
'listSid' => $listSid,
'index' => $index ?: $this->properties['index'],
);
}
/**
* Generate an instance context for the instance, the context is capable of
* performing various actions. All instance actions are proxied to the context
*
* @return \Twilio\Rest\Sync\V1\Service\SyncList\SyncListItemContext Context
* for this
* SyncListItemInstance
*/
protected function proxy() {
if (!$this->context) {
$this->context = new SyncListItemContext(
$this->version,
$this->solution['serviceSid'],
$this->solution['listSid'],
$this->solution['index']
);
}
return $this->context;
}
/**
* Fetch a SyncListItemInstance
*
* @return SyncListItemInstance Fetched SyncListItemInstance
*/
public function fetch() {
return $this->proxy()->fetch();
}
/**
* Deletes the SyncListItemInstance
*
* @return boolean True if delete succeeds, false otherwise
*/
public function delete() {
return $this->proxy()->delete();
}
/**
* Update the SyncListItemInstance
*
* @param array|Options $options Optional Arguments
* @return SyncListItemInstance Updated SyncListItemInstance
*/
public function update($options = array()) {
return $this->proxy()->update($options);
}
/**
* Magic getter to access properties
*
* @param string $name Property to access
* @return mixed The requested property
* @throws TwilioException For unknown properties
*/
public function __get($name) {
if (array_key_exists($name, $this->properties)) {
return $this->properties[$name];
}
if (property_exists($this, '_' . $name)) {
$method = 'get' . ucfirst($name);
return $this->$method();
}
throw new TwilioException('Unknown property: ' . $name);
}
/**
* Provide a friendly representation
*
* @return string Machine friendly representation
*/
public function __toString() {
$context = array();
foreach ($this->solution as $key => $value) {
$context[] = "$key=$value";
}
return '[Twilio.Sync.V1.SyncListItemInstance ' . implode(' ', $context) . ']';
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,502 |
\section{Introduction}
Real-world datasets usually exist in the form of graph structure, such as social networks[1], citation networks[2], knowledge graphs[3], especially recommendation systems[4] where nodes and edges represent objects and relationships, respectively. Taking an example, users and items in recommendation system can be represented as nodes while relationships such as purchases and clicks can be represented as edges, so that we can turn the recommendation dataset into a graph structure. Because graph is a high-dimensional non-Euclidean structure, it is difficult to model by traditional machine learning methods. Therefore, it is helpful to represent nodes by low-dimensional dense vectors, which can be the input of other machine learning models.
There have been many graph embedding methods, which are divided into two different solutions generally. For methods based on random walk, such as Deepwalk[5], Node2vec[6] and so on, sequences generated by random walk are fed into the skip-gram model to learn node embedding. However, with the rapid development of neural networks, methods based on graph neural network(GNN) have become more widely used, which learn the node embedding using specially designed neural layers. Methods like GCN[7], GAT[8], GraphSAGE[9] and other variants, they perform convolution operations on the graph or apply attention mechanism to generate more reasonable node representations.
Although the above methods have achieved state-of-the-art results in graph embedding learning, their input is homogeneous graph, which consist one edge type and one node type. Many real-world datasets are heterogeneous graphs, which consist of various types of nodes and edges. For example, an E-commerce recommendation graph consists at least two types of nodes, namely \textit{user} and \textit{item}. At the same time, different types of nodes have different attributes. Attributes of \textit{user} node may include age, sex and address while \textit{item} node attributes may consist of price, brand, category, and so on. Due to the heterogeneity of graphs, it is a challenge for GNNs to encoder the complex information into low-dimensional vectors.
Since metapath can extract relations between different types of nodes in heterogeneous graphs, most of heterogeneous graph embedding methods are based on metapath. Metapath is an ordered sequence of node and edge types, which represents a semantic space of the graph. For example, in E-commerce recommendation graph, metapath \textit{user-item-user} means the User-based collaborative filtering, while \textit{item-user-item} means the Item-based collaborative filtering. Also, metapath can guide the way to sample the heterogeneous graph and obtain neighbors.
Currently, there are many methods using metapath to generate node representation of heterogeneous graph, but they still have some limitations. (1) Some methods do not make use of the attributes of nodes, resulting in the lack of rich information, such as Metapath2vec[10], HERec[11]. (2) Some methods do not consider local and global information of nodes, which are important to the generation of node embedding(e.g. HAN[12], GATNE[13]). (3) Although nodes in different semantics have different meanings, some methods adopt one metapath to embed the heterogeneous graph, ignoring the importance of multiple semantic spaces(e.g. Metapath2vec[10], EGES[14]).
In order to address these limitations, we propose a Metapaths-guided Neighbors-aggregated Heterogeneous Graph Neural Network(MHN) model for heterogeneous graph embedding learning. Through applying node base embedding by attributes transformation, aggregation within one metapath and aggregation among matapaths, MHN can address these limitations. Specifically, MNH first adopts type-specific linear transformations to project attributes of different types of nodes, aiming to transform them into the same latent vector space. Then, MHN applies metapath-guided neighbors aggregation for each metapath. During extracting local information from BFS neighbors and global information from DFS neighbors, MHN weighted sums them and obtains the representation of target node under the current semantic. In this way, MHN captures structural and semantical information of the heterogeneous graph. Finally, MHN conducts aggregation among metapaths using the attention mechanism, with the aim of fusing latent vectors obtained from multiple metapaths into the final node embedding. Therefore, MHN can learn the comprehensive semantics in the heterogeneous graph.
The contribution of this paper lies in three aspects:
\begin{itemize}
\item We propose an end-to-end model MHN for heterogeneous graph embedding, which is a novel metapath aggregated graph neural network.
\item MHN extracts local and global information under the guidance of a single metapath, and applies attention mechanism to fuse different semantic vectors. MHN supports both supervised and unsupervised learning.
\item We conduct extensive experiments on the DBLP dataset for node classification task, as well as on the Amazon and Alibaba datasets for link prediction task to evaluate the performance of the proposed model. Moreover, we conduct online A/B test on Alibaba mobile application. Results show that representations generated by MHN performs better than other state-of-the-art methods consistently.
\end{itemize}
The rest of this paper is organized as follows. Section 2 introduces the related work. Section 3 describes some preliminary knowledge. Then we proposed the heterogeneous graph embedding method in Section 4. Experiments and analysis are shown in Section 5. Finally, we conclude the paper in Section 6.
\section{Related Work}
In this section, we will review the related studies about graph representation learning related to the proposed model. These methods are organized into four subsections: Homogeneous Graph Embedding methods, Homogeneous GNN methods, Heterogeneous Graph Embedding methods and Heterogeneous GNN methods.
\textbf{Homogeneous Graph Embedding methods}. The goal of these methods is to learn a low-dimensional representations for each node from homogeneous graph, which can be used for many downstream task directly.
DeepWalk[5] is a model for learning latent representation, which applies random walk to obtain node sequences and feeds them into skip-gram model to generate representations.
LINE[15] learns node representations on large-scale graph, which summaries local and global information through first-order and second-order proximities.
Node2vec[6] designs a biased random walk to explore diverse neighborhoods and maximize the likelihood of preserving network neighborhoods of nodes.
SDNE[16] applies an autoencoder structure to optimize both the first-order and second-order similarities.
\textbf{Homogeneous GNN methods}. These methods are mainly built by homogeneous graph convolution and can be used for both supervised and unsupervised learning. GCN[7] generates the node embedding by graph convolution, which is performed in the graph Gourier domain.
GAT[8] introduces the attention mechanism into the graph convolution, and assigns different weights to neighboring nodes to update the node representation.
GraphSage[9] is a inductive learning method. By training the aggregation function, it can merge features of neighborhoods and generate the target node embedding.
\textbf{Heterogeneous Graph Embedding methods}. Unfortunately, most of above studies focus on the homogeneous graph and cannot be used directly in heterogeneous graph. Nowadays, there are more and more studies about heterogeneous graph embedding.
Metapath2vec[10] proposes to use metapath guided random walk to sample the heterogeneous graph and obtain several node sequences.
EGES[14] is proposed to solve above problem. Through attention mechanism, it can merge attribute information into node embedding, which have achieved good improvement in CTR prediction task.
HIN2vec[17] captures the rich semantics embedded in heterogeneous graph by predicting whether there is a metapath between nodes.
\textbf{Heterogeneous GNN methods}. Due to the complexity of heterogeneous graph, homogeneous GNN methods cannot be applied directly. There are many researches on how to introduce graph convolution into heterogeneous graph.
HAN[12] proposes a node-level attention layer to aggregate neighbors of target node features and a semantic-level attention layer to merge different semantic representations.
GATNE[13] solves the problem of embedding learning for the heterogeneous graph with attributes.
HERec[11] proposes an embedding method for heterogeneous graph and applies to the recommendation scene by matrix factorization.
However, heterogeneous graph embedding methods introduced above have the limitation of ignoring the local and global information. Although they have achieved some results in several datasets, we believe that there is still room for improvement by fully utilizing the information, which promotes us to study the optimal method of embedding for heterogeneous graph.
\section{PRELIMINARIES}
In this section, we give definitions of important terms related to heterogeneous graph and show them in Figure 1.
\textbf{Definition 3.1 Heterogeneous graph}[18]. A heterogeneous graph is denoted as a graph $G=(\mathcal{V},\epsilon)$, consisting of a node set $\mathcal{V}$ and a link set $\epsilon$, which is also associated with a node type mapping function $\varphi:\mathcal{V} \xrightarrow{} \mathcal{A}$ and a link type mapping function $\psi : \epsilon \xrightarrow{} \mathcal{R}$. $\mathcal{A}$ and $\mathcal{R}$ denote the sets of predefined node types and link types, where $|\mathcal{A}| + |\mathcal{R}| > 2$.
\textit{Example.} As shown in Figure 1(a), we construct a heterogeneous graph to model the E-commerce. It consists of two type of nodes(\textit{User}($U$) and \textit{Item}($I$)) and two relations, which are user click item relation(u-i) and the similarity of items relation(i-i).
\textbf{Definition 3.2 Metapath}[19]. A metapath $p$ is defined as a path in the form of $p=A_1\xrightarrow{R_1}A_2\xrightarrow{R_2}A_3 \cdots \xrightarrow{R_l}A_{l+1}$, which describes a composite relation $R=R_1 \circ R_2 \cdots R_l$ between objects $A_l$ and $A_{l+1}$, where $\circ$ denotes the composition operator on relations. Different metapaths represent different semantics[20].
\textbf{Definition 3.3 Metapath instance}[21]. Given a metapath $p$ of a heterogeneous graph, we can sample the graph under the guidance of $p$ and obtain several node sequences, which is defined as metapath instance.
\textit{Example.} As is shown in Figure 1(c), under the guidance of metapath $p = \{\textit{U-I-I-U}\}$, we can sample the graph and get two metapath instances $user1-item1-item2-user2$ and $user1-item3-item4-user3$.
\textbf{Definition 3.4 Metapath based BFS Neighbors}. Given a node $u$ and a metapath $p$ in a heterogeneous graph, we can sample the the metapath and obtain metapath instance for $u$, which is denoted as $P(u)$. Metapath based BFS Neighbors $N_u^p$ of node $u$ is defined as the set of nodes which connect to the node $u$ directly from $P(u)$. Note that the node's BFS neighbors does not include itself.
\textit{Example.} Taking Figure 1(d) as an example, given the metapath $p = \{\textit{U-I-I-U}\}$, the metapath based BFS neighbors of $user1$ is $\{item1,item3\}$ because we can get two metapath instances. Obviously, metapath based BFS neighbors can exploit the local information of the graph because these neighbors connect to target node directly.
\begin{comment}
\renewcommand\arraystretch{1.3}
\begin{table}[!htbp]
\centering
\caption{Notations used in paper}
\begin{tabular}{cc}
\toprule
Notation & Definition \\
\midrule
$\mathcal{V}$ & The set of nodes \\
$\epsilon$ & The set of edges \\
$G=(\mathcal{V}, \epsilon)$ & Heterogeneous Graph \\
$\mathcal{A}$ & Set of node types \\
$p$ & A metapath \\
$\mathcal{P}_A$ & Metapath set started from node type $A$ \\
$P(u)$ & Metapath instance start from node $u$ \\
$N_u^p$ & BFS neighbors of $u$ under metapath $p$ \\
$M_u^p$ & DFS neighbors of $u$ under metapath $p$ \\
$x_u$ & The feature vector of node $u$ \\
$\alpha, \beta$ & Normalized attention weight \\
$d$ & Embedding size of nodes \\
$z_u$ & Embedding of node $u$ \\
\bottomrule
\end{tabular}
\end{table}
\end{comment}
\textbf{Definition 3.5 Metapath based DFS Neighbors}. Given a node $u$ and a metapath $p$ in a heterogeneous graph, after sampling the graph and get metapth instance $P(u)$, we can randomly choose a node sequence $t \in P(u)$. Metapath based DFS Neighbors $M_u^p$ is defined as the set of nodes that appears on node sequence $t$. Note that the node's DFS neighbors dose not include the first two nodes.
\textit{Example.} Taking Figure 1(d) as an example, given a metapath $p = \{\textit{U-I-I-U}\}$, we can randomly choose a node sequence $\{user1-item1-item2-user2\}$ from metapath instances. According to the definition, we remove the first two nodes and obtain the metapath based DFS neighbors of $user1$ is $\{item2, user2\}$. Compared with BFS neighbors, DFS neighbors focus on the global information.
\textbf{Definition 3.6 Heterogeneous Graph Embedding}. Given a heterogeneous graph $G=\{ \mathcal{V}, \epsilon\}$, heterogeneous graph embedding is the task to learn the $d$-dimensional node representations $z_u \in \mathbb{R}^d, \forall u \in \mathcal{V}$, which can capture the structural and semantic information of the heterogeneous graph, where $d \ll |\mathcal{V}|$.
\begin{figure*}
\centering
\includegraphics[width=.95\linewidth]{fig1.pdf}
\caption*{ \textbf{Figure 1: Illustration of definitions. (a) A heterogeneous graph with two types of nodes(i.e., \textit{users, items}). (b) An example of metapath, User-Item-Item-User(\textit{UIIU}). (c) Metapath instances of the \textit{UIIU}. (d) The metapath based BFS and DFS neighbors of the \textit{UIIU}}}
\label{fg:state}
\end{figure*}
\section{THE PROPOSED MODEL}
In this section, we present a metapath guided Heterogeneous Graph Embedding Method, called MHN.
In order to make full use of node attributes and structure information of HIN, the proposed MHN model consists of three major components as is shown in Figure 2. Firstly, we propose a node embedding representation method combining node attributes. Then, we propose an attention based method to aggregate local and global information of HIN under a single semantic space. Finally, we propose a fusion model to merge the node embeddings under multiple semantics. We will present detailed illustration of the proposed model next.
\subsection{Node Base Embedding}
For large-scale networks, node id represents the node directly, which has a great impact on the node embedding. For example, Deepwalk[5] and Metapath2vec[10] feed node id into network to learn node embedding directly. Therefore, we apply a embedding lookup layer to get embedding from id. We have
\begin{equation}
h_u^{id} = W_e \cdot u
\end{equation}
where $W_e \in \mathbb{R}^{|\mathcal{V}| \times d}$ is the parameter matrix, $u$ is the id of node, $h_u^{id}$ is the latent vector of the node. The role of this layer is to obtain the corresponding vector according to the node id. The parameter matrix $W_e$ is updated during the training process.
In real world graph, nodes are commonly attributed. Because attributes represent the information of node, so it is important for heterogeneous graph. For example, the information of item contains characteristics such as brand, price, etc, which need to be reflected in node embedding. However, different kinds of node may have unequal feature dimension. Even for different types of nodes with the same feature dimension, their features have different meanings. So we can not simply use a matrix to transform attributes. Therefore, we need to design a method to map different types of node features into the same vector space.
In MHN, we multiple transformation matrices to map different types of node attributes into the same space. For node $u \in \mathcal{V}_A$, we have
\begin{equation}
h_u^{att} = W_A \cdot x_u
\end{equation}
where $W_A$ is the parametric weight matrix for type A's nodes, $x_u$ is the feature vector of node $u$, $h_u^{att}$ is the attribute transformed latent vector of node $u$.
Considering id and attributes, we can finally obtain the representation of the node by average these two vectors:
\begin{equation}
h_u = pooling(h_u^{id}, h_u^{att})
\end{equation}
After applying these options, we can get the node latent vector containing id and attribute information in the same dimension. Then we will explore how to aggregate under the guidance of metapath.
\subsection{Aggregation Within Metapath}
Given a single metapath $p_i \in \mathcal{P}$, the aggregation within metapath learns the local and global information through sampling the target node $u$ by BFS and DFS. Firstly, we sample the heterogeneous graph under the guidance of $p_i$ and get some paths started from $u$. Then, we let $N_u^{p_i}$ denote the BFS neighbors of $u$ under the metapath $p_i$. Through the function $f_\theta$, we can encode the neighbors and obtain the $h_{u,p_i}^{BFS}$.
\begin{equation}
h_{u,p_i}^{BFS} = f_\theta(h_v, v \in N_u^{p_i})
\end{equation}
where $f_\theta$ is the encoder function. We exam three functions:
\begin{itemize}
\item \textbf{MEAN encoder} This function takes the mean of all the neighbors, thinking that all neighbors have the same contribution.
\begin{equation}
h_{u,p_i}^{BFS} = MEAN(h_v, v \in N_u^{p_i})
\end{equation}
\item \textbf{Weighted encoder} This function assigns different weights to neighboring nodes through $\beta$.
\begin{equation}
h_{u,p_i}^{BFS} = SUM( \beta \cdot [h_v, v \in N_u^{p_i}])
\end{equation}
\item \textbf{Non-linear encoder} The above two functions focus on the linear aggregation, which has limited expressive power in modeling complex relations. So we propose a non-linear function to enhance the representation of the relations through parameter matrix $W \in \mathbb{R}^{|h^{'} \times h^{'}}$, which is updated during the process of training.
\begin{equation}
h_{u,p_i}^{BFS} = \sigma ( W \cdot [h_v, v \in N_u^{p_i}])
\end{equation}
Where $\sigma()$ is non-linear function, i.e., sigmoid or relu.
\end{itemize}
Similarity, with the DFS neighbors of $u$ under the metapath $p_i$, we can encode $v \in M_u^{p_i}$ and obtain $h_{u,p_i}^{DFS}$.
After encoding the BFS and DFS information into vector representations, we adopt a simple attention mechanism to weighted sum two vectors related to target node $u$. The key idea is that BFS neighbors and DFS neighbors have different impacts on node representation. We can model this by learning a normalized importance weight $[\alpha_{1}, \alpha_{2}]$:
\begin{equation}
\begin{split}
e_1, e_2 &= h_u^T \cdot h_{u,p_i}^{BFS}, h_u^T \cdot h_{u,p_i}^{DFS} \\
\alpha_1, \alpha_2 &= \frac{ e^{e_1}} {e^{e_1} + e^{e_2}}, \frac{e^{e_2}} {e^{e_1} + e^{e_2}} \\
h_u^{p_i} = &\alpha_1 \cdot h_{u,p_i}^{BFS} + \alpha_2 \cdot h_{u,p_i}^{DFS}
\end{split}
\end{equation}
where $h_u$ is the representation of node $u$, $\alpha_1$ and $\alpha_2$ represents the relevance of target node between BFS information and DFS information.
In general, given the heterogeneous graph $G=(\mathcal{V}, \epsilon)$, node attributes $x_u, \forall u \in V$ and a set of metapaths $\mathcal{P}=\{p_1,…,p_{|\mathcal{P}|}\}$, aggregation within metapath of MHN generates $M$ metapath guided vectors for target node $u$, denoted as \{$h_u^{p_1},…,h_u^{p_|\mathcal{P}|}$\}. Each $h_u^{p_i}$ can be interpreted as the representation of $p_i$ metapath instance of node u, which reflects the semantic information of node $u$ under the metapah $p_i$.
\begin{figure*}
\centering
\includegraphics[width=.95\linewidth]{fig2.pdf}
\caption*{ \textbf{Figure 2: The overall structure of proposed method MHN. (a) Sampling the heterogeneous graph and obtain metapath instances. (b) Encode BFS and DFS neigbors to summary local and global information, and aggregate them through weighted summation. (c) Calculate normalized attention weight for each metapath and generate embedding of node $u$. (d) Design the loss according to the downstream task and end-to-end optimize the model.}}
\label{fig:BLSTM}
\end{figure*}
\subsection{Aggregation Among Metapaths}
After aggregating DFS and BFS information to generate the final representation under a single metapath, we need to merge these different semantic information revealed by metapaths into a embedding vector. For $\forall u \in \mathcal{V}$, we have $|M|$ latent embeddings $\{h_u^{p_1},…,h_u^{p_M}\}$, where $M$ is the number of metapaths and $M=|\mathcal{P}|$. In order to obtain the final embedding, we assign different weights to different metapaths through the attention mechanism. The operation is reasonable because the optimization object may focus on different semantics.
We apply the attention mechanism to merge mebeddings of node $u$ under different semantics as follows:
\begin{equation}
\begin{split}
e_{p_i} &= q^T \cdot h_u^{p_i} \\
\beta_{p_i} &= \frac{e^{e_{p_i}}}{\sum_{p \in \mathcal{P}} e^p} \\
h_u &= \sum_{p_i \in \mathcal{P}} \beta_{p_i} \cdot h_u^{p_i}
\end{split}
\end{equation}
where $q$ is the parameterized attention vector which is updated in backpropogation, $\beta_{p_i}$ be interpreted as the importance of metapath $p_i$.
Attention mechanism can also be extended to multi-heads self attention, which helps to stabilize the learning process and reduce the high variance. We first form all the embeddings of node $u$ into matrix $H_u$ of shape $[M, d]$, where $M$ is the number of embeddings and $d$ is the embedding dimension. Then we calculate self-attention output under each head. Finally, we concatenate the output of each head as the embedding of node. Taking head 1 as an example, the calculating process is as follows:
\begin{equation}
\begin{split}
Q_1, K_1,V_1 &= H_u \cdot W^{Q_1}, H_u \cdot W^{K_1}, H_u \cdot W^{V_1}\\
h_{u,1} &= softmax(\frac{Q_1 \cdot K_1}{ \sqrt{d}}) \cdot V_1 \\
h_u &= concatenate_{k=1}^K h_{u,k}
\end{split}
\end{equation}
where $ W^{Q_0}, W^{K_0}, W^{V_0} \in \mathbb{R}^{d\times \frac{d}{K}}$, $K$ is the number of heads, $softmax = e^i / \sum_j e^j$, $d$ is the embedding size.
At last, we apply a fully connected layer to enhance the nonlinear fitting ability of the network and the output is the final embedding of node $u$:
\begin{equation}
z_u = \sigma(W \cdot h_u)
\end{equation}
\subsection{Metapaths Generation}
When using methods based on metapaths for heterogeneous graph embedding, we usually need to handcraft some metapaths which are adopted to sample the graph. However, it is not trivial for human to find useful metapaths in a complex heterogeneous graph with multiple node or edge types. Therefore, we need to design an automatic generation method of metapath that does not rely on manual intervention, which can generate the most useful metapaths and sample as many nodes as possible.
MST[25] is proposed to select metapaths by using maximal spanning tree, which is instructive but ignore the applicability of metapaths. We propose a three stage approach to generate reasonable metapaths from heterogeneous graph automatically. In the first stage, we perform random walks on heterogeneous graph under a certain length and obtain lots of metapaths instances. Secondly, through node type mapping and rule constraints, we can get hundreds of metapaths. According to unique demand, we can design scoring function and rank these metapaths to get the top K highest. Take Alibaba dataset as an example, we generate 4 million node sequences by setting sequence length to 10 and sampling 5 instances for each node. After filtering through rules, e.t. all node types must appear in the node sequence, we get almost 400 candidate metapaths. Due to the goal of training more nodes of video type, we design scoring formula in Eq. (12)
\begin{equation}
score(i) = \frac{log c(i)}{i.count(u) / i.count(v)}
\end{equation}
where $c(i)$ represents the instances sampled by the i'th metapath, $i.count(u)$ and $i.count(v)$ means the number of \textit{user} and \textit{item} in the i'th metapath.
\subsection{Training}
After finishing the above components, we generate the embedding of each node, which can be applied in different downstream tasks. According to whether there are labels of data, we mainly divide the training process into two paradigms, supervised learning and unsupervised learning.
For supervised learning with node labels, we minimize the cross entropy loss and update the network parameters through backpropagation and gradient descent. The cross entropy loss of multi-classification for supervised learning is:
\begin{equation}
L=-\frac{1} {|\mathcal{V}|} \sum_{u \in \mathcal{V}} \sum_{c=1}^C y_{u,c} \cdot log p_{u,c}
\end{equation}
where $\mathcal{V}$ is the set of training nodes, $C$ is the number of classes, $y_{u,c}$ is 1 if the label of $u$ is $c$ else 0, $p_{u,c}$ is the probability that $u^{'}$s label belongs to $c$ obtained by model.
For unsupervised learning without node labels, we can optimize the model by minimizing the following nce-loss function through negative sampling:
\begin{equation}
L=-\sum_{(u,v) \in S} log \sigma(z_u^T \cdot z_v) - \sum_{(u^{'},v^{'}) \in S^{-}}log \sigma (-z_{u^{'}}^T \cdot z_{v^{'}})
\end{equation}
where $\sigma(\cdot)$ is the sigmoid funchtion, $S$ is the positive node pairs, $S^{-}$ is the negative node pairs.
Through above model, we can not only aggregate the DFS and BFS information within a single metapath of the HIN, but also merge the different semantics represented by metapaths into the final embedding. The algorithm is shown in Algorithm1.
\begin{algorithm}[h]
\caption{MHN forward propagation}
\begin{flushleft}
\hspace*{0.0in} {\bf Input:} The heterogeneous graph $G=(\mathcal{V}, \epsilon)$ \\
\qquad \quad node features $\{x_u, \forall u \in \mathcal{V}\}$, \\
\qquad \quad node types $\mathcal{A}=(A_1,…,A_{|\mathcal{A}|})$, \\
\qquad \quad metapaths set $\mathcal{P}=\{p_1,…,p_{|\mathcal{P}|} \}$ \\
\hspace*{0.0in} {\bf Output:} The node embeddings $\{z_u, \forall u \in \mathcal{V}\}$
\end{flushleft}
\begin{algorithmic}[1]
\FOR{each node type $A \in \mathcal{A}$ }
\FOR{node $u \in \mathcal{V}_{A}$}
\STATE Get node id information $h_u^{id}$ and attributes transformation $h_u^{att}=W_A \cdot x_u$
\STATE Calculate node representation $h_u = mean(h_u^{id}+h_u^{att})$
\FOR{metapath $p \in \mathcal{P}_A$}
\STATE Aggregate nodes in $N_u^p$, $M_u^p$ to obtain vectors $h_{u,p}^{BFS}$, $h_{u,p}^{DFS}$
\STATE Calculate the weight $\alpha_1, \alpha_2$ for two vectors
\STATE Obtain $h_u^p=\alpha_1 \cdot h_{u,p}^{BFS} + \alpha_2 \cdot h_{u,p}^{DFS}$
\ENDFOR
\STATE Calculate the weight $\beta_p$ for each metapath $p \in P_A$
\STATE Merge the embeddings from all metapths:
\STATE $h_u = \sum_{p \in \mathcal{P}_A} \beta_p \cdot h_u^p$
\ENDFOR
\ENDFOR
\STATE $z_u = \sigma(W_0 \cdot h_u), \forall u \in \mathcal{V}$
\STATE return $z_u$
\end{algorithmic}
\end{algorithm}
\begin{comment}
\subsection{Model Analysis}
Here we give the analysis of the proposed model as follows:
\begin{itemize}
\item The proposed model can deal with various types of nodes and relations in a heterogeneous graph. Through aggregating neighbors and fusing different semantics, we get the final embedding of each node. Benefitted from such model, different node can be represented in a same space and we can use it in different types of downstream tasks.
\item The proposed model is highly efficient and can be easily parallelized because the computation of attention mechanism is individually. Given a node $u$, the time complexity is $O(|\mathcal{P}| \cdot 2 \cdot (m+n) \cdot d + |\mathcal{P}| \cdot d)$, where $d$ is the embedding dimension, $|\mathcal{P}|$ is the number of metapaths, $m$ and $n$ are the number of BFS and DFS neighbors. In the proposed model, $|\mathcal{P}|$ is generally small and d is at mose several hundreds, which makes the proposed model efficient in large datasets.
\item The proposed model MHN is based on attention, which is shared for the whole heterogeneous graph. It means that the number of parameters is not independent on the scale of a heterogeneous graph and can be used for inductive problems[22], which means MHN can generate embedding for previous unseen nodes.
\end{itemize}
\end{comment}
\section{experiments}
In this section, we present several experiments to demonstrate the effectiveness of the model we proposed in this paper. We verify the model on both offline and online datasets.
\subsection{Datasets}
In order to evaluate the performance of MHN as compared to state-of-the-art baselines, we adopt two widely used heterogeneous graph datasets and collect a real-world dataset from Alibaba mobile application from Android and IOS online. Specifically, the DBLP dataset is used in the experiments for node classification and visualization. Amazon and Alibaba datasets are used in the experiment for link prediction. The details of these datasets are shown in Table 1.
\begin{itemize}
\item \textbf{DBLP} is a computer science bibliography website, which we adopt a subset of DBLP extracted by [23]. The heterogeneous graph contains 4057 author nodes, 14328 paper nodes, 20 conference nodes, 19645 pa(paper to author) links and 14328 pc(paper to conference) links. These nodes are divided into four classes(Database, Data Mining, Artificial Intelligence and Information Retrieval). Paper's feature is made by its terms. Author's and publication's feature is described by a bag-of-words representation of their papers' terms. For supervised learning tasks, we divide author nodes into training, validation, test sets of 3245(80.00\%), 406(10.01\%), 406(10.01\%).
\item \textbf{Amazon} includes product metadata and links between products. We adopt a subset of Amazon extracted by [24], in which we only use the product metadata of Electronics category. We build a heterogeneous graph including the co-viewing and co-purchasing links between products, and the product attributes include the price, sales-rank, brand and category with one-hot processing. For unsupervised learning tasks, we divide the dataset into training, validation, test sets of 3475(74.46\%), 398(8.53\%), 794(17.01\%).
\item \textbf{Alibaba} consists of four types of links including user-click-item, user-click-video, similarity relation between items, parallelism relation between item and video with three node types \textit{user, item, video}, which is sampled from the log of Alibaba mobile application. We build a heterogeneous graph by sampling several active users and their behaviors with items and videos. Under the guidance of section 4.4, we generate three metapaths, who's sampled nodes can cover 96\% all nodes. For unsupervised learning tasks, we divide the dataset into training, validation, test sets of 5800(80.00\%), 725(10.00\%), 725(10.00\%).
\end{itemize}
\begin{table}[!htbp]
\centering
\caption{Datasets Statics}
\begin{tabular}{@{}cccc@{}}
\toprule
Dataset & Node & Edge & Metapath \\ \midrule
DBLP &
\begin{tabular}[c]{@{}c@{}}author(A):4057\\ paper(P):14328\\ conference(C):20\end{tabular} &
\begin{tabular}[c]{@{}c@{}}P-A:19645\\ P-C:14328\end{tabular} &
\begin{tabular}[c]{@{}c@{}}APA\\ APCPA\end{tabular} \\ \midrule
Amazon & Product(P):3475 & \begin{tabular}[c]{@{}c@{}}P$\stackrel{viewing}{\longrightarrow}$P:2683\\ P$\stackrel{purchasing}{\longrightarrow}$P:791\end{tabular} & \begin{tabular}[c]{@{}c@{}}P$\stackrel{1}{\longrightarrow}$P\\ P$\stackrel{2}{\longrightarrow}$P\end{tabular} \\ \midrule
Alibaba &
\begin{tabular}[c]{@{}c@{}}user(U):2785\\ item(I):2780\\ video(V):2716\end{tabular} &
\begin{tabular}[c]{@{}c@{}}U-I:2935\\ I-V:1380\\ U-V:2935 \\ I-I:8569 \end{tabular} &
\begin{tabular}[c]{@{}c@{}}UIU\\ UIVIU\\ UVU\\ IUI\\ IUVUI\\ IVI \\ IIUIUVUVUI \\ IVIVIVIUII \\ IUIIUVUVUI \end{tabular} \\ \bottomrule
\end{tabular}
\end{table}
\subsection{Comparing Methods}
We categorize different graph embedding methods into four groups and compare MHN against these methods. The overall embedding size is set to 200.
\textbf{Homogeneous Graph Embedding Methods.} The compared methods include Deepwalk[5], LINE[15] and node2vec[6]. As these methods can only deal with Homogeneous graph, so we ignore the heterogeneity of graph and treat datasets as homogeneous.
\begin{itemize}
\item \textbf{Deepwalk} is a approach for learning latent representations of vertices in a network, using random walks to learn embeddings by treating walks as the equivalent of sentences.
\item \textbf{LINE} optimizes a objective function that preserves both the local and global network structures using proposed edge-sampling algorithm.
\item \textbf{Node2vec} is an algorithm framework for learning continuous feature representations for nodes in homogeneous network. We unify the feature of different kinds of nodes into the same dimension.
\end{itemize}
\begin{table*}[!htbp]
\centering
\caption{Performance comparison (\%) on DBLP dataset for node classification task}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}*{Dataset} & \multirow{2}*{Metrics} & \multirow{2}*{Train(\%)} & \multicolumn{4}{|c|}{Unsupervised} & \multicolumn{4}{|c|}{Supervised}
\\
\cline{4-11}
~ & ~ & ~ & Deepwalk & Node2vec & LINE & Metapath2vec & GCN & GAT & HAN & MHN
\\
\hline
\multirow{8}{*}{DBLP} & \multirow{4}{*}{Micro-F1} & 20 & 84.35 & 89.71 & 88.61 & 89.49 & 89.76 & 90.31 & 91.23 & \textbf{92.16} \\
\cline{3-11}
~ & ~ & 40 & 86.35 & 89.85 & 89.12 & 90.31 & 90.43 & 90.91 & 91.75 & \textbf{92.48} \\
\cline{3-11}
~ & ~ & 60 & 86.49 & 90.13 & 89.68 & 90.53 & 90.82 & 91.05 & 92.01 & \textbf{92.94} \\
\cline{3-11}
~ & ~ & 80 & 86.86 & 90.88 & 89.75 & 91.01 & 90.83 & 91.18 & 92.37 & \textbf{93.29} \\
\cline{2-11}
~ & \multirow{4}{*}{Macro-F1} & 20 & 82.49 & 89.27 & 88.36 & 88.97 & 89.61 & 89.68 & 90.75 & \textbf{91.37} \\
\cline{3-11}
~ & ~ & 40 & 82.59 & 89.96 & 88.52 & 90.03 & 89.74 & 89.74 & 90.96 & \textbf{91.65} \\
\cline{3-11}
~ & ~ & 60 & 82.97 & 90.06 & 89.23 & 90.17 & 90.19 & 89.75 & 91.37 & \textbf{92.02} \\
\cline{3-11}
~ & ~ & 80 & 85.27 & 90.25 & 89.42 & 90.62 & 90.56 & 90.69 & 91.89 & \textbf{92.31} \\
\hline
\end{tabular}
\end{table*}
\textbf{Homogeneous GNNs.} These models focus on the structure of network and use graph convolution method to obtain information, including GCN[7] and GAT[8]. We test these models on metapath-based homogeneous graph and select the best result.
\begin{itemize}
\item \textbf{GCN} obtains information through convolutional operations in Fourier domain for semi-supervised learning.
\item \textbf{GAT} adopts attention mechanism to perform convolution in the homogeneous graph through masked self attention layer.
\end{itemize}
\textbf{Heterogeneous Graph Embedding Methods.} The compared methods include Metapath2vec[10]. These methods focus on the generation of node embedding in heterogeneous graph.
\begin{itemize}
\item \textbf{Metapath2vec} adopts metapath-guided random walk to generate train sentences and feed them into skip-gram model and generate node embedding. We test multiple metapaths and select the best result.
\end{itemize}
\textbf{Heterogeneous GNNs.} These methods include HAN[12] and GATNE[13]. Due to the the precise capture of heterogeneous information, they perform well in heterogeneous graph.
\begin{itemize}
\item \textbf{HAN} uses attention mechanism to combine embeddings from different metapath-guided graph into one vector, capturing information from different emtapath.
\item \textbf{GATNE} focuses on the combination of different types of edges using attention mechanism. We test several GATNE variants and choose the best model.
\end{itemize}
For skip-gram based models, like Deepwalk, LINE, Node2vec, Metapath2vec, we set the window size to 5, walk length
to 10, walks per node to 20, and number of negative samples to
5. For GNNs, including GCN, GAT, HAN, GATNE, we apply Adam optimization with learning rate set to 0.01. These models are trained for 100 epochs and early stopping is set to 5. For models with attention mechanism, we set the number of attention head to 8 and attention dimension to 100.
\subsection{Node Classification}
We conduct experiment on the DBLP dataset to compare the performance of different methods on node classification task. We send node embeddings generated by different learning models into the Logistic Regression(LR) classifier with varying training proportions. In order to ensure fairness, all the data used for comparison comes from the test set, which both supervised learning and unsupervised methods have not trained. We compare the average \textit{Macro-F1} and \textit{Micro-F1} of different methods in Table 2.
As shown in Table 2, under different training proportions of DBLP dataset, MHN can achieve the best results over other learning methods. It is worth noting that, whether it is supervised or unsupervised learning, the method based on random walk performs better than the these based on GNN. This is because the DBLP dataset pays more attention to the connections between nodes, rather than the nodes themselves. Our method not only fully considers the global information, but also premeditates the local information, which ensures that the node embedding contains rich semantic information. The performance gain obtained by MHN over the best baseline(HAN) is about 0.42\%-0.93\% absolutely.
\subsection{Link Prediction}
Link prediction task is widely used to evaluate the quality of graph embeddings in both academia and industry. We also conduct experiments on the Amazon and Alibaba datasets. We hide a set of edges as the test set and train on the remaining graph. For unsupervised learning models, we treat the connected links as positive node pairs and unconnected links as negative node pairs by minimizing the objective function described in Equation 14. Given the embedding $z_u$ and $z_v$, we calculate the probability that $u$ and $v$ are linked as following:
\begin{equation}
p_{u,v} = \sigma(z_u \cdot z_v)
\end{equation}
We use some commonly used metrics like \textit{the ROC curve}(ROC-AUC), \textit{the PR curve}(PR-AUC), \textit{the average precision}(AP) and \textit{the F1 score}.
\begin{table*}[!htbp]
\centering
\caption{Experiment results (\%) on Amazon and Alibaba datasets for link prediction task}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}*{} & \multicolumn{4}{|c|}{Amazon} & \multicolumn{4}{|c|}{Alibaba}
\\
\cline{2-9}
~ & ROC-AUC & PR-AUC & F1 & AP & ROC-AUC & PR-AUC & F1 & AP
\\
\hline
Deepwalk & 89.01 & 87.35 & 64.76 & 59.28 & 73.69 & 73.11 & 66.46 & 73.14 \\
\hline
Node2vec & 88.96 & 87.29 & 66.12 & 57.83 & 73.83 & 72.95 & 67.01 & 71.29 \\
\hline
LINE & 88.83 & 86.49 & 62.28 & 62.75 & 67.18 & 72.93 & 61.98 & 70.56 \\
\hline
Metapath2vec & 90.56 & 88.69 & 71.95 & 70.65 & 77.92 & 73.54 & 70.94 & 75.17 \\
\hline
GCN & 87.09 & 86.11 & 67.74 & 66.35 & 76.38 & 72.56 & 67.12 & 73.14 \\
\hline
GAT & 88.73 & 88.64 & 69.91 & 67.33 & 76.84 & 72.26 & 67.54 & 73.87 \\
\hline
HAN & 89.32 & 88.66 & 70.51 & 70.12 & 73.14 & 73.03 & 68.04 & 74.01 \\
\hline
GATNE & 89.27 & 88.04 & 69.79 & 70.06 & 74.11 & 72.89 & 68.23 & 73.91 \\
\hline
MNH & \textbf{92.02} & \textbf{89.81} & \textbf{73.15} & \textbf{71.46} & \textbf{79.37} & \textbf{74.92} & \textbf{71.85} & \textbf{75.38} \\
\hline
\end{tabular}
\end{table*}
From Table 3, we can see that MHN performs better than other comparison algorithms. The strongest traditional method here is Metapath2vec, which learns embedding from node sequences generated by random walk guided by one metapath. MHN achieves better scores than Metapath2vec, proving the importance of multiple semantics of heterogeneous graph. Based on the idea of multi-semantics fusion, MHN considers the influence of BFS and DSF neighbors of the target node, which helps achieve a relative improvement of around 8\% on Alibaba dataset over HAN. This result supports our claim that local and global information are critical to the node embeddings.
\subsection{Parameter Sensitivity}
We investigate the sensitivity of hyper-parameter in MHN, mainly the effect of embedding dimension. Figure 3 illustrates the performance of different methods when the embedding dimension changes. We can see that as the embedding dimension increases, the performance of models also increases, but the it drops when embedding dimension is either too small or too large. It can be conclude that the performance of MHN is relatively stable within the range of embedding dimensions. Since the heterogeneous information cannot be identified, Deepwalk and GCN performs the worst.
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{figure_embedding_dimension.png}
\caption*{ \textbf{Figure 3: The performance of different methods on Amazon dataset when changing embedding dimensions.}}
\label{fg:BLSTM}
\end{figure}
\subsection{Visualization}
In addition to quantitative analysis of node embedding, we also adopt visualization method to qualitatively assess node embedding results. We randomly select four categories from DBLP dataset with 25 items under each category, and then project the embeddings of these nodes into a 2-dimensional space using t-SNE. We illustrate the visualization results of Deepwalk, Metapath2vec, HAN and MHN in Figure 4, where red and blue points indicate different category respectively.
Through visualization, we can intuitively tell the differences among learning ability of graph embedding methods for heterougeneous graph. As a traditional homogeneous graph representation learning method, Deepwalk cannot effectively divide these nodes into four groups. On the contrary, the heterogeneous model Metapah2vec can roughly distinguish these nodes. Because HAN fuse multiple semantics into node embedding, it achieves better performance. MHN method proposed in paper obtains the best embedding results, in which only a few nodes have errors and most of them are completely separated.
\begin{figure}
\centering
\subfigure[Deepwalk]{
\centering
\label{fig:subfig:a}
\includegraphics[width=0.48\linewidth]{deepwalk.png}}
\centering
\subfigure[Metapath2vec]{
\centering
\label{fig:subfig:b}
\includegraphics[width=0.48\linewidth]{Metapath2vec.png}}
\subfigure[HAN]{
\centering
\label{fig:subfig:c}
\includegraphics[width=0.48\linewidth]{HAN.png}}
\subfigure[MHN]{
\centering
\label{fig:subfig:d}
\includegraphics[width=0.48\linewidth]{MHN.png}}
\caption*{\textbf{Figure 4: Embedding visualization of nodes in DBLP.} }
\label{fig:BLSTM}
\end{figure}
\subsection{Online A/B Test}
We deploy our inductive model MHN on Alibaba mobile application for it's recall process of recommendation system. The training dataset has about half a million users and videos, a million items, with 3 million interactions among these nodes. We adopt MHN to generate embedding of each node under several metapaths. For each item, we apply K nearest neighbor (KNN) with Euclidean distance to obtain the top-50 videos that are most similar to the current item. Taking top-50 hit-rate as a goal, we compared the original method based on item-CF, Metapath2vec and MHN. The results demonstrate that MHN improves hit-rate by \textbf{2.93\%} and \textbf{6.71\%} compared to Metapath2vec and item-CF methods, respectively.
\section{Conclusion}
In this paper, we propose a metapaths-guided neighbors-aggregated Heterogeneous Graph Neural Network(MHN) method for heterogeneous graph node embedding learning, which can address three limitations mentioned above. MHN applies node base embedding to transform node attributes and enrich node representation. In addition, aggregation within metapath can merge BFS and DFS neighbors to obtain local and global information of the target node, respectively. Finally, MHN adopts attention based algorithm in aggregation among metapaths to capture information in different semantics. Especially, we put forward several encode functions for neighbors aggregation and self-attention mechanism for vectors aggregation. In experiments, MHN achieves best results on three real-world datasets in node classification and link prediction task. Parameter sensitivity analysis illustrates the effect of embedding dimension. Visualization analysis shows the quality of node representation obtained by different methods directly. Online test in Alibaba mobile application proves the feasibility and effectiveness of MHN.
For the future work, we will consider about the dynamic heterogeneous graph node representation learning methods to adapt the changes in graph.
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"redpajama_set_name": "RedPajamaC4"
} | 2,963 |
{"url":"http:\/\/math.stackexchange.com\/questions\/139673\/inductive-proof-of-a-countable-set-cartesian-product?answertab=votes","text":"# Inductive Proof of a countable set Cartesian product [duplicate]\n\nPossible Duplicate:\nProving $\\mathbb{N}^k$ is countable\n\nI would like to prove that if S is countable then for any positive integer n the set $S^n$ (the n-fold Cartesian product of S with itself) is countable using mathematical induction.\n\nI think I should initialize it at n=0 but I don't know where to go from there.\n\nThanks so much for the help\n\n-\n\n## marked as duplicate by Asaf Karagila, Martin Sleziak, Chris Eagle, t.b., Guess who it is.Aug 18 '12 at 1:29\n\nThe result is trivial for $n=0$ and $n=1$, so your first step should be to prove it for $n=2$. Then you can use that result in your induction step to go from countability of $S^n$ to countability of $S^{n+1}$, since $S^{n+1}$ clearly admits a bijection with $S^n\\times S$, a product of two countable sets. \u2013\u00a0 Brian M. Scott May 1 '12 at 23:08\nBrian gave me some excellent advise and I found a way to do it. Showing that the cartesian $S^n \\times S$","date":"2015-05-24 06:09:06","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\": 0.8890313506126404, \"perplexity\": 269.9652771721204}, \"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\/1432207927843.59\/warc\/CC-MAIN-20150521113207-00036-ip-10-180-206-219.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
A positive integer $n$ is called a {\em congruent number} if it is the area of a right-angled triangle, all of whose
sides have rational lengths. The congruent number problem, which is the oldest unsolved major problem in number theory,
is the question of finding an algorithm for deciding in finite number of steps whether or not a given integer is a congruent number.
In this paper, based on an ideal of Tian \cite{Tian} we will establish a new {\em sufficient} condition for $n$ to be congruent
in terms of the Legendre symbols $\sfrac pq$, with $p$ and $q$ running over the prime factors of $n$.
This type of criterion was first given by Heegner \cite{Heegner} and Birch and Stephens \cite{BS} for some $n$ with a single odd prime factor,
and by Monsky \cite{Monsky} for some $n$ with two odd prime factors, and finally Tian \cite{Tian} saw how to extend it to $n$ with an
arbitrary number of prime factors.
Our criterion generalizes all of this work, and we believe that it has potential applications to the following {\em distribution conjecture of congruent numbers}:
{\em all $n\equiv 5, 6, 7\mod 8$ are congruent and all but density $0$ of $n\equiv 1,2,3$ are not congruent.}
Note that in \cite{Tu2}, Tunnell gave a {\em necessary} condition for $n$ to be
congruent in terms of numbers of solutions of some equations $n=Q(x, y, z)$ with positive definite quadratic forms $Q(x, y, z)$ over ${\mathbb {Z}}$.
Tunnell's criterion is also sufficient if the rank part of the BSD conjecture is assumed.
In the following, we would like to describe our main results. Let us first consider the elliptic curve
$$E_n: \quad ny^2=x^3-x,$$
where $n$ is assumed to be a \emph{square-free positive} integer. Then it is well known that $n$ is congruent if and only if $E_n ({\mathbb {Q}})$ has positive
rank. This is equivalent to the vanishing of $L(E_n, 1)$ under the rank part of BSD conjecture
$$
{\mathrm{rank}}\ E_n({\mathbb {Q}})={\mathrm{ord}}_{s=1}L(E_n, s).
$$
By Birch--Stephens \cite{BS}, the root number
$$
\epsilon(E_n)=
\begin{cases}
1
& \mbox{if } n\equiv 1, 2, 3\pmod 8, \\
-1
& \mbox{if } n\equiv 5, 6, 7\pmod 8.
\end{cases}
$$
It follows that ${\mathrm{ord}} _{s=1}L(E_n, s)$ is even (resp. odd) if and only if $n\equiv 1, 2, 3\mod 8$ (resp. $n\equiv 5, 6, 7\mod 8$).
The density conjecture of congruent numbers follows from rank part of the BSD and the following {\em density conjecture of $L$-functions}:
${\mathrm{ord}} _{s=1}L(E_n, s)\le 1$ for all but density $0$ of $n$'s.
By work of Coates--Wiles \cite{Coates-Wiles}, Rubin \cite{Rubin}, Gross--Zagier \cite{GZ} and Kolyvagin \cite{K1}, the rank part of the BSD conjecture holds for $E_n$
if ${\mathrm{ord}}_{s=1}L(E_n, s)\leq 1$ and the Tate--Shafarevich group $\hbox{\cyr X}(E_n)$ is finite.
Thus we define an invariant ${\mathcal {L}}(n)$ of $E_n$ as follows:
$$
{\mathcal {L}}(n):=
\begin{cases} \left[L(E_n, 1)/(2^{2k(n)-2-a(n)} \Omega _{n, \infty})\right]^{1/2}
& \mbox{if } {\mathrm{ord}} _{s=1}L(E_n, s)=0, \\
\left[L'(E_n, 1)/(2^{2k(n)-2-a(n)}\cdot \Omega _{n, \infty}R_n)\right]^{1/2}
& \mbox{if } {\mathrm{ord}} _{s=1}L(E_n, s)=1, \\
0 & \mbox{if } {\mathrm{ord}} _{s=1}L(E_n, s)>1.
\end{cases}
$$
Here
\begin{itemize}
\item $k(n)$ is the number of odd prime factors of $n$;
\item $a(n)=0$ if $n$ is even, and $1$ if $n$ is odd;
\item the real period
$$ \Omega _{n, \infty}=\frac 2{\sqrt {n}}\int_1^\infty \frac {dx}{\sqrt{x^3-x}},$$
\item $R_n$ is twice of the N\'{e}ron--Tate height of a generator of $E_n({\mathbb {Q}})/E_n({\mathbb {Q}})_{\mathrm{tor}}$ (in the case of rank one).
\end{itemize}
The definition is made so that the full BSD conjecture for $E_n$ in the case ${\mathrm{ord}} _{s=1}L(E_n, s)\leq 1$ writes as
\begin{equation}\label{bsd}
\#\hbox{\cyr X}(E_n)= {\mathcal {L}}(n)^2.
\end{equation}
The density conjecture of congruent numbers is equivalent to {\em non-vanishing ${\mathcal {L}}(n)$ for density one of $n$'s}.
The number ${\mathcal {L}} (n)$ is a priori a complex number defined up to a sign.
In this paper, we show that ${\mathcal {L}} (n)$ is an integer, and give a criterion for when it is odd in terms of the parities of the genus class numbers
$$g(d):=\#(2 {\mathrm{Cl}} ({\mathbb {Q}}(\sqrt {-d})))$$
of positive divisors $d$ of $n$.
It is clear that $g(d)$ is odd if and only if ${\mathrm{Cl}} ({\mathbb {Q}}(\sqrt {-d}))$ has no element of exact order $4$.
Thus by R\'edei \cite{Re}, the parity of $g(d)$ can be computed in terms of the R\'edei matrix of the Legendre symbols $\displaystyle\sfrac pq$ of prime factors $p, q$ of $d$.
The choice of the sign of ${\mathcal {L}} (n)$ is not an issue in this paper since we are mainly interested in its parity.
We divide our results naturally into two cases by the root number $\epsilon (E_n)$.
\begin{thm}\label{lg}
Let $n\equiv 1, 2, 3\pmod 8$ be a positive and square-free integer.
Then ${\mathcal {L}} (n)$ is an integer, and
$${\mathcal {L}} (n)\equiv \sum_{\substack {n=d_0d_1\cdots d_\ell\\
d_i\equiv 1\pmod 8,\ i>0}}
\prod_i g(d_i)
\quad\pmod 2.$$
Here all decompositions $n=d_0\cdots d_\ell$ are non-ordered with $d_i>1$ for all $i\geq 0$. The right-hand side is considered to be 1 if $n=1$.
\end{thm}
For $n\equiv 5, 6, 7$, we introduce an integer $\rho (n)\geq 0$ by
$$2^{\rho(n)} =[E_n({\mathbb {Q}}): \varphi _n (A_n({\mathbb {Q}}))+E_n[2]],$$
where $\varphi_n: A_n \to E_n$ is a 2-isogeny from
$A_n: 2nv^2=u^3+u$
to $E_n: ny^2=x^3-x$
defined by
$$\varphi _n (u, v)=\left(\frac 12 \left(u+\frac 1u\right ), \frac v{2u}\left(u-\frac 1u\right)\right).$$
\begin{thm}\label{lg'}
Let $n\equiv 5, 6, 7\pmod 8$ be a positive and square-free integer.
Then ${\mathcal {L}} (n)$ is an integer.
If $n\equiv 5, 7\pmod 8$, then $2^{-\rho(n)}{\mathcal {L}}(n)$ is even only if
$$\sum_{\substack{{n=d_0\cdots d_\ell}\\ {d_i\equiv 1\pmod 8,\ i>0}}} \prod_i
g(d_i)\quad \equiv \sum_{\substack{{n=d_0\cdots d_\ell,}\\
{d_0\equiv 5, 6, 7\pmod 8}\\ {d_1\equiv 1, 2 ,3 \pmod 8} \\ {d_i\equiv 1\pmod 8,\ i>1}}} \prod_i g(d_i) \ \equiv 0
\quad\pmod 2.$$
If $n\equiv 6\pmod 8$, then $2^{-\rho(n)}{\mathcal {L}}(n)$ is even only if
$$
\sum_{\substack{{n=d_0\cdots d_\ell,}\\
{d_0\equiv 5, 6, 7\pmod 8}\\ {d_1\equiv 1, 2 ,3 \pmod 8} \\ {d_i\equiv 1\pmod 8,\ i>1}}} \prod_i g(d_i) \ \equiv 0
\quad\pmod 2.$$
Here all decompositions $n=d_0\cdots d_\ell$ are non-ordered with all $d_i>1$.
\end{thm}
Our method of proving these theorems is to give formulae of ${\mathcal {L}}(n)$ in terms of the so-called genus periods and genus points (cf. Theorems \ref{LQ} and Theorem \ref{m3}).
These formulae are derived from the Waldspurger formula in \cite{Wa} and the generalized Gross--Zagier formula of Yuan--Zhang--Zhang \cite{YZZ} using an induction argument of Tian \cite{Tian}.
\begin{remark} For each residue class in $\{1, 2, 3, 5, 6, 7\mod 8\}$,
we believe that our formulae in Theorems \ref{lg}, \ref{lg'} give a positive density of $n$ with ${\mathcal {L}} (n)$ odd.
For $n\equiv 1, 2, 3\mod 8$, this is already implied by the BSD formula \ref{bsd} modulo $2$ and the work of Heath-Brown \cite{Heath-Brown}.
Moreover the BSD formula \ref{bsd} modulo 2 can be checked case by case. In fact, in \cite{Monsky3}, the ${\mathbb {F}}_2$-rank of
${\mathrm{Sel}} _2(E_n)/E_n ({\mathbb {Q}})[2]$ can be also calculated in terms of Legendre symbols for every $n$.
\end{remark}
In the following we want to give some some criterions of congruent and non-congruent numbers extending Tian
\cite{Tian} in terms of a single genus class number.
\begin{cor}
Let $n$ be a square-free positive integer such that ${\mathbb {Q}}(\sqrt{-n})$ has no ideal classes of exact order $4$.
For any integer $r$, let $A_r, B_r$ denote the following property of $n$:
\begin{align*}
&A_r(n): \quad \#\{p\mid n: \ p\equiv 3\pmod 4\}\le r;\\
&B_r(n):\quad \#\{p\mid n: \ p\equiv \pm 3\pmod 8\}\le r.\end{align*}
Then in the following case, $n$ is a non-congruent number:
\begin{itemize}
\item $n\equiv 1\pmod 8$ with $A_2(n)$ or $B_2(n)$,
\item $n\equiv 2\pmod{8}$ with $A_0(n)$ or $B_2(n)$,
\item $n\equiv 3\pmod 8$ with $A_1(n)$ or $B_1(n)$.
\end{itemize}
In the following case, $n$ is a congruent number:
\begin{itemize}
\item $n\equiv 5\pmod 8$ with $A_0(n)$ or $B_1(n)$,
\item $n\equiv 7\pmod 8$ with $A_1(n)$ or $B_0(n)$.
\end{itemize}
\end{cor}
\begin{proof}
By R\'edei \cite{Re}, $g(d)$ is even in any of the following cases:
\begin{itemize}
\item $d=p_1 \cdots p_k\equiv 1\pmod 8$, $p_i\equiv \pm1\pmod 8$, $k>0$;
\item $d=2p_1 \cdots p_k$, $p_i\equiv \pm1 \pmod 8 $, $k>0$;
\item $d=p_1\cdots p_k\equiv 1\pmod 8$ , $p_i\equiv 1 \pmod 4$, $k>0$.
\end{itemize}
It follows that under any of the conditions of the corollary, the following congruence holds:
$$\sum_{\substack {n=d_0d_1\cdots d_\ell\\
d_i\equiv 1 \pmod 8 ,\ i>0}}
\prod_i g(d_i)\equiv g(n)\ \pmod 2.$$
The conclusion follows from Theorem \ref{lg} and \ref{lg'}.
\end{proof}
\subsubsection*{Acknowledgements}
Ye Tian would like to acknowledge the support of the NSFC grants 11325106 and 11031004.
Xinyi Yuan would like to acknowledge the support of the National Science Foundation
under the award DMS-1330987.
Shou-Wu Zhang would like to acknowledge the support of the
National Science Foundation under the awards DMS-0970100 and DMS-1065839.
\section{Quadratic periods and genus periods} \label{section 2}
The goal of this section is to prove Theorem \ref{lg}.
Assume that $n\equiv 1,2,3 \pmod 8$ is positive and square-free throughout this section.
\subsection{Quadratic periods and genus periods} \label{section2.1}
Let $K_n={\mathbb {Q}}(\sqrt {-n})$ be the quadratic imaginary extension.
For any decomposition $n=d_1\cdot d_2$ with $d_2$
\emph{positive and odd}, we have an unramified quadratic extension
$K_n (\sqrt {d_2^*})$ of $K_n$ where $d_2^*=(-1)^{(d_2-1)/2}d_2$.
By the class field theory, the extension gives a quadratic character
$$\chi_{d_1, d_2}:{\mathrm{Cl}}_n \longrightarrow \{\pm1\}$$
on the class group ${\mathrm{Cl}}_n$ of $K_n$.
In the degenerate case $d_2=1$, we take the convention $\chi_{d_1, d_2}=1$.
Conversely, by Gauss's genus theory, any quadratic character of ${\mathrm{Cl}}_n$ comes from such a decomposition $n=d_1\cdot d_2$.
The Rankin-Selberg L-series of the elliptic curve $E:y^2=x^3-x$ twisted by $\chi_{d_1, d_2}$ is given by
$$L(E_{K_n}, \chi_{d_1, d_2}, s)=L (E_{d_1}, s)L(E_{d_2}, s).$$
In the following, we give a formula
for ${\mathcal {L}}(d_1){\mathcal {L}}(d_2)$ using the Waldspurger formula. Notice that such formulae concern the quaternion algebra determined by the local root numbers of the L-function $L(E_{K_n}, \chi_{d_1, d_2}, s)$.
Let $B$ be the quaternion algebra over ${\mathbb {Q}}$ ramified exactly at 2 and
$\infty$. In fact, $B$ is the classical
Hamiltonian quaternion (over ${\mathbb {Q}}$):
$$B={\mathbb {Q}}+{\mathbb {Q}} i+{\mathbb {Q}} j+{\mathbb {Q}} k, \quad i^2=j^2=-1,\ ij=k=-ji.$$
Let $ O_B$ be the standard maximal order of $B$:
$$ O_B:= O_B'+{\mathbb {Z}} \zeta, \qquad O_B':={\mathbb {Z}}+ {\mathbb {Z}} i+{\mathbb {Z}} j+{\mathbb {Z}} k, \qquad \zeta =(-1+i+j+k)/2.$$
Fix an embedding $\tau:K_n\hookrightarrow B$ such that the image of $O_{K_n}$ lies in
$O_B$. If $n\equiv 1\pmod8$, we further specify the embedding by
$$
\tau(\sqrt{-n})= ai+bj+ck
$$
where $n=a^2+b^2+c^2$ with $a,b,c\in {\mathbb {Z}}$ and $4|c$.
It is a classical result of Legendre that we can find integer solutions $a,b,c$ if $n$ is not of the form $4^e(8m-1)$. The more specific condition $n\equiv 1\pmod8$ implies the existence of a solution with $4|c$. See \cite[Theorem 5]{JP} for example.
By the Jacquet--Langlands correspondence, the newform $f_E\in S_2(\Gamma _0(32))$ corresponding to the elliptic curve $E: y^2=x^3-x$ defines an automorphic representation $\pi=\otimes_v \pi_v$ of $B^\times({\mathbb {A}})$.
Note that the central character of $\pi$ and
the infinite part $\pi_\infty$ are trivial, so $\pi=\otimes_v \pi_v$ is naturally
realized as a subspace of $C^\infty(B^\times\backslash
\widehat{B}^\times/\widehat{{\mathbb {Q}}}^\times)$.
Denote by $\pi_{\mathbb {Z}}$ the ${\mathbb {Z}}$-submodule of $\pi$ consisting of elements of $\pi$ which takes integral values on $B^\times\backslash\widehat{B}^\times/\widehat{{\mathbb {Q}}}^\times$.
Denote $U_n={\widehat R_n^\times\cdot K_{n, 2}^\times}$, an open subgroup of $\widehat B^\times$.
Here $R_n= O_{K_n}+4 O_B$ is an order of $B$ of conductor $32$.
Consider the $U_n$-invariant submodule $\pi^{U_n}_{\mathbb {Z}}$ of $\pi_{\mathbb {Z}}$.
We will see that $\pi^{U_n}_{\mathbb {Z}}$ is free of rank $1$ over ${\mathbb {Z}}$,
as the special case of $\chi=1$ in Theorem \ref{thm2.3}.
This is an integral example of the multiplicity one theorem of Tunnell \cite{Tu} and Saito \cite{Sa} reviewed in Theorem \ref{TS1} and Corollary \ref{TS2}.
Fix a ${\mathbb {Z}}$-generator $f_n$ of $\pi^{U_n}_{\mathbb {Z}}$, which is determined up to multiplication by $\pm1$.
Define the {\em quadratic period} $P(d_1, d_2)$ by
$$P(d_1, d_2):=\sum _{t\in {\mathrm{Cl}}_n}f_n(t)\chi_{d_1, d_2}(t).$$
\begin{thm}\label{thm2.1}
The period $P(d_1, d_2)\ne 0$ only if $d_2\equiv 1\pmod 8$.
In that case,
$$P(d_1, d_2)=\pm 2^{k-a}\cdot w_K\cdot {\mathcal {L}}(d_1){\mathcal {L}}( d_2),$$
where $2w_K$ is the number of roots of unity in $K$, $k$ is the number of odd prime factors of $n=d_1d_2$, and $a=1$ if $n$ is odd
and $a=0$ otherwise.
\end{thm}
Now we define the {\em genus period} $Q(n)$ by
$$Q(n):=\sum _{t\in 2{\mathrm{Cl}}_n}f_n(t).$$
Notice that $P(n)$ and $Q(n)$ are well-defined up to signs.
\begin{thm}\label{LQ}
The number ${\mathcal {L}} (n)$ is an integer and satisfies
$${\mathcal {L}}(n)\equiv
\sum_{\substack{n=d_0d_1\cdots d_\ell\\ d_i\equiv 1\pmod 8, \ i>0}}
\prod_i Q(d_i) \quad\pmod 2.$$
Here in the sums, all decompositions $n=d_0\cdots d_\ell$ are non-ordered with $d_i>1$ for all $i\geq 0$.
\end{thm}
Now Theorem \ref{lg} follows from Theorem \ref{LQ} and the following result.
\begin{prop}\label{odd}
One has
$$f_n\left( (\widehat{B}^{\times})^ 2\right)\subset 1+2{\mathbb {Z}}.$$
Therefore,
$$Q(n)\equiv g(n)\quad\pmod 2.$$
\end{prop}
\subsection{Primitive test vectors} \label{section2.2}
In this subsection, we give an explicit construction of the test vector $f_n$, to prepare for the proof of the result in the last subsection.
Resume the above notations related to $K_n, B$ and $\pi$.
The local components of the automorphic representation $\pi=\otimes_v \pi_v$
of $B^\times({\mathbb {A}})$ has the following properties:
\begin{itemize}
\item $\pi_\infty$ is trivial;
\item $\pi_p$ is unramified if $p\ne \infty, 2$, i.e., $\pi^{ O_{B, p}^\times}$ is one-dimensional;
\item $\pi_2$ has a conductor of exponential $4$ (cf. \cite{G2}), i.e., for a uniformizer
$\lambda$ (for example, $1+i$) of $B$ at $2$,
$$\pi_2^{1+\lambda^4 O_{B, 2}}\ne 0, \qquad \pi_2^{1+\lambda^3 O_{B, 2}}=0.$$
\end{itemize}
Let $U=\prod_p U_p$ be the open compact subgroup of
$\widehat{ O}_B^\times$ with $U_p= O_{B, p}^\times$ if $p\ne 2$, and
$$U_2={\mathbb {Z}}_2^\times(1+\lambda ^4 O_{B, 2})={\mathbb {Z}}_2^\times(1+4 O_{B, 2}).$$
Then $\pi^U\simeq\pi_2^{U_2}$ is stable under the action of $B_2^\times$, since $U_2$ is normal in $B_2^\times$. By the irreducibility of $\pi_2$, we further have
$\pi_2^{U_2}=\pi_2$.
By definition, $\pi^U$ is a subspace of $C^\infty(B^\times\backslash \widehat{B}^\times/\widehat{{\mathbb {Q}}}^\times U)$, the space of maps from (the finite set) $B^\times\backslash \widehat{B}^\times/\widehat{{\mathbb {Q}}}^\times U$ to ${\mathbb {C}}$. The following is a more detailed description.
\begin{thm} \label{thm2.4}
\begin{enumerate} [(1)]
\item The space $\pi^U$ is a $6$-dimensional irreducible
representation of $B_2^\times$, with an orthogonal basis
$$f_\delta \in C^\infty(B^\times\backslash \widehat{B}^\times/\widehat{{\mathbb {Q}}}^\times U), \qquad
\delta\in \left\{\frac {\pm i\pm j}2, \frac {\pm j \pm k}2, \frac
{\pm k\pm i}2\right\}\Big/\{\pm 1\}.$$
Here for each $\delta$, the function $f_\delta$ is determined by its restriction to $1+2 O_{B, 2}$
and
$$f_\delta(1+2x)=(-1)^{{\mathrm{Tr}}(\delta x)}, \qquad \forall x\in
O_{B,2}.$$
\item The representation $\pi^U$ of $B_2^\times$ has an integral
structure $\pi_{\mathbb {Z}}^U$ generated by
$$\qquad\qquad f_{i\pm j}:= \frac12 (f_{\frac{i+j}{2}}\pm f_{\frac{i-j}{2}}),
\quad f_{j\pm k}:=\frac12 (f_{\frac{j+k}{2}}\pm f_{\frac{j-k}{2}}),
\quad f_{k\pm i}:=\frac 12(f_{\frac{k+i}{2}}\pm
f_{\frac{k-i}{2}}).$$
Moreover, this ${\mathbb {Z}}$-basis is orthonormal with
respect to the Tamagawa measure on $B^\times\backslash B^\times({\mathbb {A}})/{\mathbb {A}}^\times$.
\item Let $\chi_0$ be the
character of $B^\times({\mathbb {A}})$ associated to the quadratic extension
${\mathbb {Q}}(i)$, i.e. the composition
$$B^\times({\mathbb {A}})\overset {\det} \longrightarrow {\mathbb {A}}^\times \simeq {\mathbb {Q}}^\times \times
(\widehat {\mathbb {Z}}^\times\times {\mathbb {R}}_+^\times) \longrightarrow \widehat {\mathbb {Z}}^\times\longrightarrow ({\mathbb {Z}}/4{\mathbb {Z}})^\times \simeq
\{\pm 1\}.$$Then $\pi\simeq \pi\otimes\chi_0$ and
$$\quad\chi _0f_{\frac{i+j}{2}}=f_{\frac{i-j}{2}}, \qquad \chi _0f_{\frac{j+k}{2}}=f_{\frac{j-k}{2}},
\qquad \chi _0f_{\frac{k+i}{2}}=f_{\frac{k-i}{2}}.$$
\end{enumerate}
\end{thm}
To deduce the theorem, we first need the following precise description of $B^\times\backslash \widehat{B}^\times/\widehat{{\mathbb {Q}}}^\times U$.
\begin{lem} The following natural maps are bijective:
$$ O_{B, 2}/({\mathbb {Z}}_2+2 O_{B, 2})\stackrel{\sim}{\longrightarrow}
(1+2 O_{B, 2})/{\mathbb {Z}}_2^\times(1+4 O_{B, 2})\stackrel{\sim}{\longrightarrow}
B^\times\backslash \widehat{B}^\times/\widehat{{\mathbb {Q}}}^\times U,$$
where the first map is
defined by $x\mapsto 1+2x$, and the second one is given by the natural inclusion
$B_2^\times \subset \widehat{B}^\times$.
Moreover, under the composition
$$
B^\times\backslash B^\times({\mathbb {A}})/\widehat{{\mathbb {Q}}}^\times
\longrightarrow
B^\times\backslash \widehat{B}^\times/\widehat{{\mathbb {Q}}}^\times U
\stackrel{\sim}{\longrightarrow}
O_{B, 2}/({\mathbb {Z}}_2+2 O_{B, 2}),$$
the Tamagawa measure on $B^\times\backslash B^\times({\mathbb {A}})/{\mathbb {A}}^\times$ transfers to the Haar measure of (the finite abelian group) $ O_{B, 2}/({\mathbb {Z}}_2+2 O_{B, 2})$ of total volume 2.
\end{lem}
\begin{proof}
We first prove the bijectivity.
The first map is clearly a group isomorphism. For the second
map, we use the class number one
property of $B$, i.e.,
$$\widehat B^\times =B^\times\cdot \widehat O_B^\times=B^\times\cdot B_2^\times \cdot U^{(2)}.$$
It follows that
$$B^\times \backslash \widehat B^\times /\widehat{\mathbb {Q}}^\times U\simeq H\backslash B_2^\times/{\mathbb {Q}}_2^\times U_2, \quad H=B^\times \cap (U^{(2)}\cdot B_2^\times).$$
It is easy to see
that $H$ is a semi-product of $\lambda^{\mathbb {Z}}$, where $\lambda\in O_B$ is an element
with reduced norm $2$, and the subgroup
$$ O_B^\times =\left\{\pm1, \quad \pm i, \quad \pm j, \quad \pm k, \quad \frac {\pm 1\pm i\pm j\pm k}2\right\}.$$
The group $ O_B^\times$ is a semi-product of $\mu_3$ generated by
$\zeta=(-1+i+j+k)/2$ and
$$( O_B' )^\times=\{\pm1, \quad \pm i, \quad \pm j, \quad \pm k\}.$$
Consider the filtration of $B_2^\times$ given by
$$B_2^\times \supset O_{B, 2}^\times \supset 1+\lambda
O_{B, 2}^\times \supset 1+2 O_{B, 2},$$
and its induced filtration
$$H\supset O_B^\times \supset ( O_B')^\times\supset \mu_2.$$
It is straight forward to check that these two exact sequences have
isomorphic sub-quotients. It follows that the map $H\to B_2^\times$
induces an exact sequence
$$1\longrightarrow \mu_2\longrightarrow H\longrightarrow B_2^\times /(1+2 O_{B, 2})\longrightarrow 1.$$
In other words, the $B_2^\times$ is generated by $H$ and the normal
subgroup $1+2 O_{B, 2}$ with intersection $H\cap (1+2 O_{B,
2})=\mu_2.$ Thus
$$H\backslash B_2^\times/{\mathbb {Q}}_2^\times U_2\stackrel{\sim}{\longleftarrow} \mu_2\backslash (1+2 O_{B, 2}/1+4 O_{B, 2}
)\stackrel{\sim}{\longleftarrow} (1+2 O_{B, 2})/{\mathbb {Z}}_2^\times (1+4 O_{B,
2}).$$ The other two relations can be verified similarly.
Now we treat the measure. Note that the Tamagawa measure gives
$B^\times\backslash B^\times({\mathbb {A}})/{\mathbb {A}}^\times$ total volume 2.
Then the induced measure on $ O_{B, 2}/({\mathbb {Z}}_2+2 O_{B, 2})$ also has total volume 2.
It suffices to check that the induced measure is uniform.
Equivalently, we need to show that
${\mathrm{vol}}(B^\times\backslash B^\times g \widehat{{\mathbb {Q}}}^\times U)$
is constant in $g\in \widehat B^\times$.
By the first part of the lemma, we can always take a representative
$g\in 1+2 O_{B, 2}$ for the double coset $B^\times g \widehat{{\mathbb {Q}}}^\times U$. The key is that $g_p=1$ for $p\neq 2$. It follows that
$$B^\times\backslash B^\times g \widehat{{\mathbb {Q}}}^\times U
=B^\times\backslash B^\times \widehat{{\mathbb {Q}}}^\times U g,$$
whose volume is independent of $g$ since the measure is invariant under the right translation.
\end{proof}
In the lemma, the right
multiplication action of $B_2^\times=H\cdot (1+2 O_{B, 2})$ on
$\widehat{B}^\times$ induces its action on $ O_{B, 2}/({\mathbb {Z}}_2+2 O_{B,
2})$ given by right conjugation of $H$ and translation of $x$, for
elements $1+2x\in 1+2 O_{B, 2}$.
Consider the space ${\mathcal {A}}_0\subseteq C^\infty(B^\times\backslash
\widehat{B}^\times/\widehat{{\mathbb {Q}}}^\times)$ of forms perpendicular to forms
$\chi\circ \det$ where $\chi$ runs over all characters of
${\mathbb {Q}}^\times\backslash \widehat{{\mathbb {Q}}}^\times$, then $\pi\subset {\mathcal {A}}_0$. Let
${\mathcal {A}}_0^U$ be the subspace of ${\mathcal {A}}_0$ of forms invariant under $U$,
then $\pi^U\subset {\mathcal {A}}_0^U$.
The restriction map
$${\mathcal {A}}_0^U\longrightarrow {\mathbb {C}}[ O_{B, 2}/({\mathbb {Z}}_2+2 O_{B, 2})], \qquad f\longmapsto (\phi_f: x\mapsto f(1+2x))$$
define an isomorphism between ${\mathcal {A}}_0^U$ and the space ${\mathcal {A}}_1$ of
functions $\phi $ on $ O_{B, 2}/({\mathbb {Z}}_2+2 O_{B, 2})$ perpendicular
to the characters $1$ and $(-1)^{\mathrm{Tr}}$ on $ O_{B, 2}/({\mathbb {Z}}_2+2 O_{B, 2})$.
Here the trace map $${\mathrm{Tr}}: O_{B,
2}/({\mathbb {Z}}_2+2 O_{B, 2})\longrightarrow {\mathbb {Z}}_2/2{\mathbb {Z}}_2$$
is induced form the reduced trace.
The vector space ${\mathcal {A}}_1$
is decomposed into the direct sum
$${\mathcal {A}}_1=\sum _{\psi\in \Psi} {\mathbb {C}} \psi,$$
where $$\Psi=\left\{\psi\in {\mathrm{Hom}} ( O_B/(2 O_B+{\mathbb {Z}}), \mu_2), \quad
\psi\ne 1, (-1)^{{\mathrm{Tr}} }\right\}$$
is a set of quadratic characters $\psi$ of
$ O_{B, 2}/({\mathbb {Z}}_2+2 O_{B, 2})$.
We have an explicit description of forms in ${\mathcal {A}}_0^U$ corresponding to $\Psi$. Let $\wp=\lambda O_{B, 2}$
be the maximal ideal of $ O_{B, 2}$. Then the trace map defines a
perfect pairing
$$ O_{B, 2}\otimes \wp^{-1}\longrightarrow {\mathbb {Z}}_p, \qquad (x, y)\longrightarrow {\mathrm{Tr}}(xy).$$
It induces a perfect pairing
$$( O_{B, 2}/2 O_{B, 2})\otimes (\wp^{-1}/2\wp^{-1})\longrightarrow \mu_2, \qquad (x, y)\mapsto (-1)^{{\mathrm{Tr}}(xy)}.$$
It is easy to see that $\Psi$ corresponds to the subset $\bar\Delta$
of elements $\bar \delta\in \wp^{-1}/2\wp^{-1}$ with the following
properties:
$${\mathrm{Tr}}(\delta)=0\pmod 2, \qquad \delta\ne 0, 1\pmod 2.$$
Note that the set
$$\Delta= \left\{\frac {\pm i\pm j}2, \frac {\pm j \pm k}2, \frac
{\pm k\pm i}2\right\}$$
in Theorem \ref{thm2.4} is contained in
$\wp^{-1}$.
Thus we can identify $\bar\Delta=\Delta/\{\pm1\}$.
For each $\delta \in \Delta/\{\pm1\}$, the corresponding form
$f_\delta$ is given by
$$f_\delta (g)=(-1)^{{\mathrm{Tr}} (\delta x)},$$for any $g=bh(1+2x)u \in \widehat B^\times$
with $b\in B^\times$, $h\in H$, $x\in O_{B, 2}$, and $u\in
\widehat{{\mathbb {Q}}}^\times U.$
Hence, the space ${\mathcal {A}}_0^U$ is $6$-dimensional with the
explicit decomposition
$${\mathcal {A}}_0^U=\sum_{\delta\in \Delta/\{\pm1\}} {\mathbb {C}} f_\delta$$
into characters of
$(1+2 O_{B,2})/(1+4 O_{B, 2})$.
\begin{proof}[Proof of Theorem \ref{thm2.4}]
For (1), it suffices to prove that ${\mathcal {A}}_0^U$ is irreducible as a representation of $G:=B_2^\times/(1+4 O_{B,
2})$. Note that $G$ contains a normal and commutative finite
subgroup $C=(1+2 O_{B, 2})/(1+4 O_{B, 2})$. Thus any invariant subspace $V$ of
${\mathcal {A}}_0^U$ is a direct sum
$$V=\oplus _{\chi \in X}V_\chi$$
over some multiset $X$ of characters of $C$.
The multiset $X$ is stable under the conjugation of
$G$. We have seen that $V_\chi$ are all one-dimensional, and $X$ is
included into $\Psi$, the set of characters induced by elements in
$\Delta$. Thus we need only prove that $G$ acts transitively on
$\Delta$ by conjugations. In fact, $\Delta$ is a principle
homogenous space of $ O_B^\times/\mu_2$ under conjugation.
Now we treat (2).
Note that for any $h\in H$, $x\in O_{B, 2}$, and $\delta\in
\Delta$, we have that $h\delta h^{-1}\in \Delta$ and
$$\pi(h(1+2x))f_\delta=\psi_\delta(x)f_{h\delta h^{-1}}
=\pm f_{h\delta h^{-1}},$$
where
$\psi_\delta\in \Psi$ denotes the character $x\mapsto (-1)^{{\mathrm{Tr}}
(\delta x)}$ on $ O_{B, 2}/({\mathbb {Z}}_2+2 O_{B, 2})$. Therefore, the action
of $B_2^\times$ on $f_{i\pm j}$ is given by
$$\pi(h)f_{i\pm j}=f_{hih^{-1}\pm hjh^{-1}},\qquad \pi(1+2x)f_{i\pm j}\in
\{\pm f_{i+j}, \pm f_{i-j}\}.$$
Similar results hold for $f_{j\pm k}$ and
$f_{k\pm i}$.
Thus $\pi_{\mathbb {Z}}^U$ is an integral structure on $\pi^U$.
The orthonormality of the basis is a simple consequence of the previous result on the measures.
For (3), it is clear that $\chi_0$ is invariant under the left action of
$B^\times \cdot H$ and the right action of $U$ and its restriction on
$1+2 O_{B, 2}$ is given by $\chi_0(1+2x)=(-1)^{{\mathrm{Tr}} x}$ for any
$x\in O_{B, 2}$. Thus for any $x\in O_{B, 2}$,
$$\chi_0f_{\frac{i+j}{2}}(1+2x)=(-1)^{{\mathrm{Tr}}(x+\frac{i+j}{2}x)}=(-1)^{{\mathrm{Tr}}(x\frac{i-j}{2})}
(-1)^{{\mathrm{Tr}}((1+j)x)}=f_{\frac{i-j}{2}}(1+2x).$$
\end{proof}
\begin{thm}\label{thm2.3}
Let $K$ be an imaginary quadratic field and $\chi$ a
quadratic character of $\widehat{K}^\times/K^\times\widehat{ O}_K^\times$
such that $L(E_{K},\chi,s)$ has root number $+1$ (so that $2$ cannot split
in $K$). Let $\varpi$ be a uniformizer of $K_2$ and $\chi_2$ the
2-component of $\chi$. Fix a ${\mathbb {Q}}$-embedding of $\tau:K\hookrightarrow B$ such
that $ O_K$ is contained in $ O_B$. Then the vector space
$$\pi^{U, \chi_2}:=\{f\in \pi^U, \ \pi(t)f=\chi_2(t)f, \ \forall t\in
K_2^\times\}$$ is one-dimensional. All the possible cases of
$(K_2, \chi_2(\varpi))$ are listed below:
$$({\mathbb {Q}}_2(\sqrt{-3}), 1), \quad ({\mathbb {Q}}_2(\sqrt{-1}), \pm 1), \quad
({\mathbb {Q}}_2(\sqrt{-2m}), (-1)^{\frac{m-1}{2}}),\ m\equiv 1, 3, 5, 7\pmod
8.$$ Let $g\in B^\times_2$ be such that $\tau_2':=g^{-1}\tau_2 g$ is
given by
$$\begin{aligned}
&(-1+\sqrt{-3})/2\longmapsto \zeta, \quad \sqrt{-1}\longmapsto
k,\qquad\qquad\
\sqrt{-2}\longmapsto i+j,\\
&\sqrt{-10}\longmapsto i-3j,\qquad \sqrt{-6}\longmapsto i+j+2k,\quad
\sqrt{-14}\longmapsto -3i+j-2k,\end{aligned}$$
respectively in the above cases. Then the vector
$f=\pi(g)f_0$ lies in $\pi^{U, \chi_2}$, where
$$f_0=\begin{cases}
f_{\frac{i-j}{2}}+f_{\frac{j-k}{2}}+f_{\frac{k-i}{2}},\quad
&\text{if $K_2={\mathbb {Q}}_2(\sqrt{-3})$},\\
f_{i\pm j}=\frac{1}{2}\left(f_{\frac{i+j}{2}}\pm
f_{\frac{i-j}{2}}\right), \quad &\text{if $(K_2,
\chi_2(\varpi))=({\mathbb {Q}}_2(\sqrt{-1}), \pm 1)$},\\
f_{\frac{i+j}{2}}, &\text{if $K_2={\mathbb {Q}}_2(\sqrt{-2m})$ with $m$ odd.}
\end{cases}$$
Moreover, $\pi_{\mathbb {Z}}^{U, \chi_2}:=\pi_{\mathbb {Z}}^U\cap \pi^{U, \chi_2}={\mathbb {Z}} f$.
\end{thm}
\begin{definition} \label{def primitive}
The automorphic forms $f$ and $-f$ described in the theorem are called \emph{primitive test vectors} for $(\pi, \chi)$.
\end{definition}
The theorem can be interpreted by the multiplicity one theorem of Tunnell \cite{Tu} and Saito \cite{Sa} reviewed in Theorem \ref{TS1} and Corollary \ref{TS2}.
In fact, the space
$$\pi^{\widehat O_B^{2,\times}, \chi_2}:=\{f\in \pi^{\widehat O_B^{2,\times}}, \ \pi(t)f=\chi_2(t)f, \ \forall t\in
K_2^\times\}$$
is at most one-dimensional by the multiplicity one theorem. The theorem confirms that it is one-dimensional and constructs an explicit generator of the integral structure.
\begin{proof}[Proof of Theorem \ref{thm2.3}]
It suffices to show that $f_0$ is $\chi_2$-invariant under the embedding
$\tau_2':K\hookrightarrow B$.
First consider the case $K_2={\mathbb {Q}}_2(\sqrt{-3})$, where
$$K_2^\times/{\mathbb {Q}}_2^\times(1+4 O_2)= O_2^\times/{\mathbb {Z}}_2^\times(1+4 O_2)$$
is cyclic of order $6$ and generated by $\zeta$ and $1+2\zeta$. Note
that $$\zeta^{-1}i\zeta=j, \quad \zeta^{-1}j \zeta=k, \quad
\zeta^{-1}k\zeta=i.$$ Thus the subspace of $\pi^U$ of forms fixed by
$\zeta\in H$ is of 2-dimensional with basis
$$f_{\frac{i-j}{2}}+f_{\frac{j-k}{2}}+f_{\frac{k-i}{2}}, \quad
f_{\frac{i+j}{2}}+f_{\frac{j+k}{2}}+f_{\frac{k+i}{2}}.$$ Moreover,
note that $\psi_\delta(\zeta)=1$ for $\delta=\frac{i-j}{2},
\frac{j-k}{2}, \frac{k-i}{2}$ and $\psi_\delta(\zeta)=-1$ otherwise.
Thus $\pi^{U, \chi_2}$ is one-dimensional with basis
$f_{\frac{i-j}{2}}+f_{\frac{j-k}{2}}+f_{\frac{k-i}{2}}$.
In the case $K_2={\mathbb {Q}}(\sqrt{-1})$, let $\varpi=k-1$, we have that
$$K_2^\times/{\mathbb {Q}}_2^\times(1+4 O_2)=\varpi^{{\mathbb {Z}}/4{\mathbb {Z}}}\times \langle
1+\varpi, 1+2\varpi\rangle.$$ Note that $1+2\varpi=2k-1$, and
$\psi_\delta(k)=1$ if $\delta=\frac{i\pm j}{2}$ and
$\psi_\delta(k)=-1$ otherwise. Thus the subspace of $\pi^U$ of forms
fixed by $1+2\varpi$ is of 2-dimensional with basis
$$f_{i+j}, \quad f_{i-j},$$where $1+\varpi=k$ acts trivially
since $k^{-1}ik=-i, k^{-1}jk=-j$. Finally, since $\varpi=k-1\in H$
and
$$\varpi^{-1} i\varpi=j, \quad \varpi^{-1}j \varpi=-i, \varpi^{-1} k
\varpi=k,$$ we have that $\pi^{U, \chi_2}$ is of one-dimensional
with base $f_{i\pm j}$ for $\chi_2(\varpi)=\pm 1$.
For the case $K_2={\mathbb {Q}}_2(\sqrt{-2n})$ with $n=1, 3, 5, 7$, let
$\varpi=\sqrt{-2n}$, we have that
$K_2^\times/{\mathbb {Q}}_2^\times(1+4 O_2)$ is generated by the order $2$
element $\varpi$ and order 4 element $1+\varpi$. The embedding
$\tau_2'$ maps $1+\varpi$ to $k(1+2(\zeta +i))\mod
{\mathbb {Z}}_2^\times(1+4 O_{B, 2})$. Note that $kik^{-1}=-i, kjk^{-1}=-j$
and
$$\psi_\delta(\zeta+i)=\begin{cases}1, \quad &\text{if $\delta=\frac{i+j}{2},
\frac{i+k}{2}, \frac{j-k}{2}$},\\
-1, &\text{otherwise}.\end{cases}$$ It follows that the subspace of
$\pi^U$ fixed by $\tau_2'(1+\varpi)$ is of one-dimensional with
basis $f_{\frac{i+j}{2}}$. We have the following decompositions of
$\tau_2'(\varpi)\in B_2^\times=H\cdot (1+2 O_{B, 2}) \mod
{\mathbb {Z}}_2^\times(1+4 O_{B, 2})$:
$$\begin{aligned}
&i+j\in H, \ && i-3j\equiv (i+j)(1+2k), \\
&i+j+2k\equiv (j-i)(1+2(\zeta-k)), \ &&
-3i+j-2k=(i-j)(1+2(\zeta-k)).\end{aligned}$$ Note that
$\psi_{\frac{i+j}{2}}(k)=1$ and $\psi_{\frac{i+j}{2}}(\zeta)=-1$, we
know that $f_{\frac{i+j}{2}}$ is $\chi_2$-invariant.
\end{proof}
\subsection{Proofs of Theorems \ref{thm2.1}, \ref{LQ}
and Proposition \ref{odd} }
Resume the notations in \S \ref{section2.1}.
Especially, $f_n$ is a basis of $\pi_{\mathbb {Z}}^{U_n}$ with
$$U_n={\widehat R_n^\times\cdot K_{n, 2}^\times}, \quad R_n= O_{K_n}+4 O_B.$$
We first connect it to the primitive test vectors in \S \ref{section2.2}.
Recall that in \S \ref{section2.2}, we have introduced
$$U= \widehat O_{B}^{2,\times}\cdot U_2, \quad U_2={\mathbb {Z}}_2^\times(1+4 O_{B, 2}).$$
In Theorem \ref{thm2.3} and Definition \ref{def primitive}, we have introduced the primitive test vectors for $(\pi,\chi)$.
For the connection, it is easy to verify $U_n=U\cdot K_{n, 2}^\times$.
Hence, $f_n$ is a primitive test vector for $(\pi,\chi)$ if and only if $\chi_2=1$.
\subsubsection*{Proof of Theorem \ref{thm2.1}}
Write $K=K_n$ for simplicity. The goal is to treat
$$P(d_1, d_2)=\sum_{t\in {\mathrm{Cl}}_n} f_n(t) \chi_{d_1, d_2}(t).$$
The tool is the Waldspurger formula.
By ${\mathrm{Cl}}_n=K^\times\backslash \widehat K^\times/\widehat O_K^\times$, the summation is essentially an integration on $K^\times\backslash \widehat K^\times$.
Since $f_n$ is invariant under the action of $K_2^\times$, the integration is nonzero only if
$\chi_{d_1, d_2}$ is trivial on $K_{2}^\times$.
In other words, $K_2(\sqrt{d_2^*})$ splits into two copies of $K_2={\mathbb {Q}}_2(\sqrt{-n})$.
This is equivalent to $d_2^*\equiv 1\pmod 8$. Then $d_2\equiv \pm 1\pmod 8$.
We will exclude the case $d_2\equiv -1\pmod 8$ later.
Assume $d_2^*\equiv 1\pmod 8$.
Then $\chi_{d_1, d_2}$ is trivial on $K_{2}^\times$, and $f_n$ is a primitive test vector for
$(\pi,\chi_{d_1,d_2})$ as described in Theorem \ref{thm2.3}.
In particular,
$$(f_n, f_n)_{\mathrm{Pet}}=\begin{cases}
6, \ &\text{if $K_2={\mathbb {Q}}_2(\sqrt{-3})$},\\
1, &\text{if $K_2={\mathbb {Q}}_2(\sqrt{-1})$},\\
2, &\text{if $K_2={\mathbb {Q}}_2(\sqrt{-2m})$ with $m=1, 3, 5,
7$}.\end{cases}$$
Apply the explicit Waldspurger formula in Theorem \ref{thm4.2}.
We have
$$|P(d_1, d_2)|^2=\frac{w_K^2}{2^5 3^b \pi^3}\cdot \frac{(f_n, f_n)_{\mathrm{Pet}}}{(f', f')_{\mathrm{Pet}}}\cdot L(E_{d_1},1)L(E_{d_2},1),$$
where $b=1$ if $2$ is inert in $K$ and $b=0$ otherwise, and $f'$ is the normalized new form in the automorphic representation of ${\mathrm{GL}}_2({\mathbb {A}})$ associated to $E$.
We claim that
$$(f' f')_{\mathrm{Pet}}=\Omega_{d_1, \infty}\Omega_{d_2, \infty}\cdot \frac{|D|^{1/2}}{2^8 \pi^3 e},$$
where $D$ is the discriminant of $K$.
Let $\phi=\sum_{n=1}^\infty a_n q^n$ be the corresponding newform of weight $2$. Note that
$$(\phi, \phi)_{\Gamma_0(32)}=\iint_{\Gamma_0(32)\backslash {\mathcal {H}}} |\phi(z)|^2 dxdy$$
and
$$(f', f')_{\mathrm{Pet}}=\int_{{\mathrm{GL}}_2({\mathbb {Q}})\backslash {\mathrm{GL}}_2({\mathbb {A}})/{\mathbb {Q}}^\times } |f'(g)|^2 dg
$$
are related by
$$\frac{(\phi, \phi)_{\Gamma_0(32)}}{{\mathrm{vol}}(X_0(32))}=\frac{(f', f')_{\mathrm{Pet}}}{2}, \qquad \text{where ${\mathrm{vol}}(X_0(32))=16\pi$}.$$
Let $\varphi: X_0(32)\rightarrow E$ be a modular parametrization of degree $2$, and $\omega$ the N\'{e}ron differential on $E$, and $\Omega=\int_{E({\mathbb {R}})} \omega$.
Note that
$$\varphi^*\omega=4\pi i \phi(z) dz, \quad
2^{-1}\Omega^2=\iint_{E({\mathbb {C}})} |\omega \wedge \overline{\omega}|,$$
and thus
$$\Omega^2=32\pi^2 (\phi, \phi)_{\Gamma_0(32)}.$$
By definition,
$$\Omega_{d_1, \infty}\Omega_{d_2, \infty}=\Omega^2/\sqrt{d_1d_2}=2^{e-1}\Omega^2/\sqrt{|D|},$$
where $e=1$ if $2\nmid D$ and $e=2$ otherwise. Put all these together, we have the formula for $(f' f')_{\mathrm{Pet}}$.
Hence, we have
$$|P(d_1, d_2)|^2=2^{4+c} w_K^2 \frac{L(E_{d_1},1)}{\Omega_{d_1, \infty}}\cdot \frac{L(E_{d_2},1)}{\Omega_{d_2, \infty}},$$
where $c=0$ if $8\nmid D$ and $c=1$ otherwise.
It gives the formula of the theorem.
It remains to prove that $d_2\equiv -1\pmod 8$ implies $P(d_1, d_2)=0$.
This is a direct consequence from the formula we just proved, since $L(E_{d_1},1)=0$ by considering the root number in this case.
\subsubsection*{Proof of Theorem \ref{LQ}}
Let $h_2(n)=\dim _{{\mathbb {F}}_2}{\mathrm{Cl}}_n/2{\mathrm{Cl}}_n$.
By Gauss's genus theory, $h_2(n)+1$ is exactly equal to the number of prime factors of the discriminant of $K_n$, and any character of ${\mathrm{Cl}}_n/2{\mathrm{Cl}}_n$ is of the form $\chi_{d_1,d_2}$ for some decomposition $d=d_1d_2$ with $d_2$ positive and odd.
Moreover, a repetition $\chi_{d_1',d_2'}=\chi_{d_1,d_2}$ occurs only if
$(d_1',d_2')=(d_2,d_1)$ and $n\equiv3\pmod8$.
Hence, we have the character formula
$${\sum_{n=d_1d_2}} '\chi_{d_1, d_2}(t)=2^{h_2(n)}\delta _{2{\mathrm{Cl}} (K_n)}(t), \qquad t\in {\mathrm{Cl}}_n,$$
where the sum is over ordered (resp. non-ordered) decompositions $d=d_1d_2$ if $n\equiv 1\pmod 8$ (resp. $n\equiv 3\pmod 8$), and requires $d_2$ to be odd if $n\equiv 2\pmod 8$.
As a result, we have
\begin{equation} \label{PQ}
{\sum_{n=d_1d_2}} ' P(d_1, d_2)=2^{h_2(n)} Q(n).
\end{equation}
The summation follows the same rule as above.
The following lemma shows the symmetry in the case $n\equiv 1\pmod 8$.
\begin{lem}\label{lem2.6}
Assume $n\equiv 1\pmod 8$. Then
for any decomposition $d=d_1d_2$ with $d_1,d_2>0$,
$$P(d_1, d_2)= P(d_2, d_1).$$
\end{lem}
\begin{proof} Let $\chi_0$ be the character on $\widehat{B}^\times$
corresponding to the extension ${\mathbb {Q}}(i)$ over ${\mathbb {Q}}$, defined in
Theorem \ref{thm2.4}. The two quadratic characters are related by
$$\chi_{d_1, d_2}=\chi_{d_2, d_1} \cdot \chi_0.$$
In fact, for any $t\in \widehat{K}^\times$, we have
$$\chi_{d_1, d_2}(t)
\chi_{d_2, d_1}(t)=\frac{\sigma_t(\sqrt{d_1})}{\sqrt{d_1}}\frac{\sigma_t(\sqrt{d_2})}
{\sqrt{d_2}}=\frac{\sigma_t(i)}{i}=\chi_0(t).$$
In the notation of Theorem \ref{thm2.3}, the primitive test vector is given by
$f_n=\pi(g)f_0$ (up to $\{\pm1\}$) with $g\in B_2^\times$ and
$$f_0=\frac 12 (f_{\frac {i+j}2}+f_{\frac {i-j}2})=f_{\frac {i+j}2}\cdot (\frac {1+\chi_0}2).$$
We claim that $\chi_0(g)=1$ by our special choice of $\tau:K_n\hookrightarrow B$ at the beginning.
Assuming $\chi_0(g)=1$, then
$$\begin{aligned}
P(d_2, d_1)&=\sum_t f_{\frac{i+j}{2}}(tg)\frac {1+\chi_0(t)}2\chi_{d_2, d_1}(t)\\
&=\sum_t f_{\frac{i+j}{2}}(tg)\frac {1+\chi_0(t)}2 \chi_0(t)\chi_{d_1, d_2}(t)\\
&= \sum_t f_{\frac{i+j}{2}}(tg)\frac {1+\chi_0(t)}2\chi_{d_1, d_2}(t) \\
&=P(d_1, d_2).
\end{aligned}$$
It remains to check $\chi_0(g)=1$.
Recall that $g\in B^\times_2$ is an element such that $\tau_2'=g^{-1}\tau_2 g: K_{n,2} \hookrightarrow B_2$ gives $\tau_2'(\sqrt{-1})=k$.
Recall that the embedding $\tau:K_n\hookrightarrow B$ is defined by
$$
\tau(\sqrt{-n})= ai+bj+ck
$$
where $n=a^2+b^2+c^2$ with $a,b,c\in {\mathbb {Z}}$ and $4|c$.
Thus the equation for $g$ is just
$$g^{-1}\cdot \frac{1}{\sqrt n}(ai+bj+ck)\cdot g=k.$$
Here $\sqrt n$ denotes a square root of $n$ in $K_2$.
Explicit computation gives a solution
$$
g_0= ai+bj+(c+\sqrt n)k.
$$
For this solution, we have
$$
\det(g_0)=a^2+b^2+(c+\sqrt n)^2=2(n+c\sqrt n).
$$
Note that $n+c\sqrt n\equiv 1 \pmod 4$ by the condition $4|c$, and thus $\chi_0(g_0)=1$.
It is easy to see that any other solution is of the form $g=g_0 (u+v k)$ for $u,v\in {\mathbb {Q}}_2$. Then we have $\chi_0(u+v k)=1$ and thus $\chi_0(g)=1$.
\end{proof}
\begin{lem}
One has
$$\sum _{n=d_1d_2}\epsilon (d_1, d_2){\mathcal {L}} (d_1){\mathcal {L}}(d_2)=Q(n),$$
where $\epsilon (d_1, d_2)=\pm 1$, and the sum is over non-ordered decompositions $n=d_1d_2$ such that $d_1,d_2>0$ and $d_2\equiv 1\pmod 8$.
\end{lem}
\begin{proof}
Writing equation (\ref{PQ}) in terms of non-ordered decompositions, we have
\begin{equation*}
\sum_{n=d_1d_2} P(d_1, d_2)=2^{h_2(n)-\delta} Q(n),
\end{equation*}
where $\delta =1$ if $n\equiv 1\pmod 8$ and $\delta=0$ otherwise.
Here in the case $n\equiv 1\pmod 8$, we have used the symmetry
$P(d_1, d_2)=P(d_2, d_1)$.
Apply Theorem \ref{thm2.1}.
\end{proof}
Finally, we are ready to derive Theorem \ref{LQ}.
\begin{proof}[Proof of Theorem \ref{LQ}]
Since ${\mathcal {L}} (1)=1$, the above lemma gives a recursive formula
$$\pm {\mathcal {L}} (n)=Q(n)-
\sum_{\substack{n=d_1d_2 \\ d_2\equiv 1\pmod 8, \ d_2>1}}
\epsilon (d_1, d_2){\mathcal {L}} (d_1){\mathcal {L}}(d_2).$$
Here the sum is over non-ordered decompositions.
This formula determines ${\mathcal {L}}(n)$ uniquely.
In particular, ${\mathcal {L}}(n)$ is an integer.
Now we prove the congruence formula
$${\mathcal {L}}(n)\equiv
\sum_{\substack{n=d_0d_1\cdots d_\ell\\ d_i\equiv 1\pmod 8, \ i>0\\
d_i>1,\ i\geq0}}
\prod_i Q(d_i) \quad\pmod 2.$$
It suffices to prove that the congruence formula (applied to every $P(d_1)$ and $P(d_2)$ below) satisfies the recursive formula
$$Q(n) \equiv
\sum_{\substack{n=d_1d_2\\d_2\equiv 1\pmod 8, \ d_2>0}}
{\mathcal {L}} (d_1){\mathcal {L}}(d_2)
\quad\pmod 2.$$
Namely, we need to check that
\begin{multline*}
Q(n)\equiv
\sum_{\substack{n=d_1d_2\\d_2\equiv 1\pmod 8, \ d_2>0}}
\left(\sum_{\substack {d_1=d_0'd_1'\cdots d_{\ell'}'\\
d_j'\equiv 1\pmod 8, \ j>0\\
d_j'>1,\ j\geq0}}
\prod_{j\geq 0} g(d_j') \right)
\left(\sum_{\substack{d_2=d_0''d_1''\cdots d_{\ell''}''\\
d_k''\equiv 1 \pmod 8, \ k>0\\
d_k''>1,\ k\geq0}}
\prod_{k\geq 0} g(d_k'')\right)
\quad\pmod 2.
\end{multline*}
The right-hand side is a ${\mathbb {Z}}$-linear combination of
$$
\prod_{j=0}^{\ell'} g(d_j')
\prod_{k=0}^{\ell''} g(d_k'').
$$
We consider the multiplicity of this term in the sum.
Each appearance of such a term gives a partition
$$
\{d_1',\cdots, d_{\ell'}', d_0'',\cdots, d_{\ell''}''\}
=\{d_1',\cdots, d_{\ell'}'\}\coprod \{d_0'',\cdots, d_{\ell''}''\}.
$$
If the set on the left-hand side is non-empty, the number of such partitions is even. Then the contribution of this set in the sum is zero modulo 2.
Thus, we are only left with the empty set, which corresponds to the unique term $g(n)$ on the right. This proves the formula.
\end{proof}
\subsubsection*{Proof of Proposition \ref{odd}}
The proof easily follows from the explicit result in Theorem \ref{thm2.3}.
In fact, take $K=K_n$ and $\chi=1$ in the theorem. We see that the primitive test vector
$f_n=\pi(g)f_0$ for some $g\in B_2^\times$, where
$$f_0=\begin{cases}
f_{\frac{i-j}{2}}+f_{\frac{j-k}{2}}+f_{\frac{k-i}{2}},\quad
&\text{if $K_2={\mathbb {Q}}_2(\sqrt{-3})$},\\
f_{i+ j}=\frac{1}{2}\left(f_{\frac{i+j}{2}}+
f_{\frac{i-j}{2}}\right), \quad &\text{if $K_2={\mathbb {Q}}_2(\sqrt{-1})$},\\
f_{\frac{i+j}{2}}, &\text{if $K_2={\mathbb {Q}}_2(\sqrt{-2m})$ with $m$ odd.}
\end{cases}$$
Note that the case $(K_2, \chi_2(\varpi))=({\mathbb {Q}}_2(\sqrt{-1}), -1)$ does not occur here.
It is immediate that $f_0$ and $f$ take odd values everywhere in the first and the third cases.
Assume that we are in the case $K_2={\mathbb {Q}}_2(\sqrt{-1})$.
By Theorem \ref{thm2.4}, $\displaystyle\chi _0f_{\frac{i+j}{2}}=f_{\frac{i-j}{2}}$.
For any $h\in \widehat{B}^{\times}$, we have
$$
f_n(h^2)=f_0(h^2g)
=\frac{1}{2} f_{\frac{i+j}{2}} (h^2g)(1+\chi_0(h^2g))
=f_{\frac{i+j}{2}} (h^2g)
=\pm 1,
$$
which is odd. Here we have used the fact $\chi_0(g)=1$, which has been treated in the proof of Lemma \ref{lem2.6}.
\section{Quadratic points and genus points}
This section treats ${\mathcal {L}}(n)$ for $n\equiv 5,6,7 \pmod 8$.
The goal is to prove Theorem \ref{lg'}.
We assume $n\equiv 5,6,7 \pmod 8$ throughout this section.
The method is to construct rational points using the tower $X=\lim _UX_U$
of modular curves $X_U$.
\subsection{Quadratic points and genus points} \label{section3.1}
In the following, we will mainly work on the elliptic curve $A: 2y^2=x^3+x$ (instead of $E:y^2=x^3-x$), which is isomorphic to $(X_0(32), \infty)$.
Fix an identification $i_0:(X_0(32), \infty)\to A$.
We will introduce a morphism $f_n:X_V\to A$ from certain modular curve $X_V$ to $A$, and use this morphism to produce Heegner points on $A$.
\subsubsection*{Test vector}
Recall that the open compact subgroup $U_0(32)$ of ${\mathrm{GL}}_2(\widehat{\mathbb {Q}})$ is given by
$$
U_0(32)= \left\{ \matrixx{a}{b}{c}{d}\in {\mathrm{GL}}_2(\widehat{\mathbb {Z}}): 32|c \right\}.
$$
Define another open compact subgroup
$$
U= \left\{ \matrixx{a}{b}{c}{d}\in U_0(32): 4|(a-d) \right\}.
$$
Then $U$ is a normal subgroup of $U_0(32)$ of index two.
Denote by $f_0: X_U\to A$ the natural projection map $X_U\to X_0(32)$.
It is finite and \'etale of degree 2.
Note that the geometrically connected components of $X_U$ are parametrized by ${\mathrm{Spec}}\, {\mathbb {Q}}(i)$. Then it is easy to figure out that $X_U\cong {X_0(32)}_{{\mathbb {Q}}(i)}$ over ${\mathbb {Q}}$, and under this identification $f_0$ is the natural map by the base change. Then
$${\mathrm{Aut}}_{\mathbb {Q}}(X_U)={\mathrm{Aut}}_{{\mathbb {Q}}(i)}({X_0(32)}_{{\mathbb {Q}}(i)})\rtimes \{1, \epsilon\},$$
where $\epsilon$ is the Hecke operator given by $\matrixx{1}{}{}{-1}$, which is also the automorphism coming from the non-trivial automorphism of ${\mathrm{Spec}}\, {\mathbb {Q}}(i)$.
For $n\equiv 7\pmod 8$, let $f_n:X_0(32)\to A$ be the identity map $i_0:X_0(32)\to A$.
For $n\equiv 5,6\pmod 8$, define $f_n: X_U\rightarrow A$ by
$$f_n:=\begin{cases} f_0-f_0\circ [i], \qquad &\text{if $n\equiv 5\pmod 8$},\\
f_0\circ [i], &\text{if $n\equiv 6\pmod 8$}.
\end{cases}$$
Denote $K_n={\mathbb {Q}}(\sqrt {-n})$ as before.
Embed $K_n$ into $M_2({\mathbb {Q}})$ by
$$\sqrt{-n} \longmapsto \matrixx{-1}{1/4}{-4(n+1)}{1}, \qquad \matrixx{}{1/4}{-4n}{}, \qquad \matrixx{\delta}{2}{-(n+\delta^2)/2}{-\delta},$$
according to $n\equiv 5, 6, 7\pmod 8$ respectively. Here $\delta$ is an integer such that $\delta^2\equiv -n\pmod {128}$ in the case $n\equiv 7\pmod 8$.
The embeddings look arbitrary, but they are chosen on purpose. For $n\equiv 5, 6\pmod 8$, the embeddings make $K_{n,2}^\times$ normalize $U_2$ in ${\mathrm{GL}}_2({\mathbb {Q}}_2)$ at the place $2$, which is the basis of our treatment. For $n\equiv 7\pmod 8$, the embedding gives $\widehat{O}_K^\times\subset U_0(32)$, which makes the easiest calculation.
Similarly, the choices of $f_n$ seem artificial and technical here. However, they are obtained by some prescribed representation-theoretical properties below.
Following \cite[\S 1.2]{YZZ}, consider the representation
$$\pi={\mathrm{Hom}} ^0_\infty(X, A)=\varinjlim_V {\mathrm{Hom}}^0_\infty (X_V, A)$$
of ${\mathrm{GL}}_2(\widehat {\mathbb {Q}})$.
Here for any open compact subgroup $V$ of ${\mathrm{GL}}_2(\widehat {\mathbb {Q}})$,
$$ {\mathrm{Hom}}^0_\infty (X_V, A)={\mathrm{Hom}}_\infty (X_V, A)\otimes_{\mathbb {Z}}{\mathbb {Q}},$$
where
$${\mathrm{Hom}}_\infty(X_V, A)=\{f\in {\mathrm{Hom}} (X_V, A): \ f(\infty)\in A(\overline {\mathbb {Q}})_{{\mathrm{tor}}}\}.$$
Here $\infty$ denotes the cusp of $X_V$.
\begin{prop} \label{choice}
\begin{enumerate} [(1)]
\item
If $n\equiv 5,6\pmod 8$, the space $\pi^{{\mathrm{GL}}_2(\widehat{\mathbb {Z}}^{(2)})\cdot K_{n,2}^\times}$ is one-dimensional and contains $f_n$.
\item
If $n\equiv 7\pmod 8$, the space
$\pi^{U_0(32)}$ is one-dimensional and contains $f_n$.
\end{enumerate}
\end{prop}
The proposition explains that $f_n$ is an explicit vector in a one-dimensional space in the framework of the multiplicity one theorem of Tunnell \cite{Tu} and Saito \cite{Sa}. See Theorem \ref{TS1} and Corollary \ref{TS2}.
One can also define an integral structure $\pi_{\mathbb {Z}}$ of $\pi$ as the subgroup of elements of $\pi$ coming from ${\mathrm{Hom}}_\infty (X_V, A)$ for some $V$.
Then one can consider the primitivity of $f_n$ under this integral structure as in \S \ref{section 2}. However, this is too involved in the current setting, so we will only consider the behavior of $f_n$ in the rational structure $\pi$.
\subsubsection*{CM point2}
Note that we have chosen an explicit embedding of $K_n$ in $M_2({\mathbb {Q}})$, which induces an action of $K_n^\times$ on the upper half plane ${\mathcal {H}}$.
Let
$$P_n=[h, 1]\in X_U({\mathbb {C}})$$
be the CM point, where $h\in {\mathcal {H}}^{K_n^\times}$ is the unique fixed point of $K_n^\times$ in
${\mathcal {H}}$.
Let
$$z_n=f_n(P_n) \in A(K_n^{\mathrm{ab}}).$$
Note that $z_n$ is not necessarily defined over the Hilbert class field $H_n$ of $K_n$.
Denote by $H_n'=H_n(z_n)$ the extension of $H_n$ generated by the residue field of $z_n$.
The following result is a precise description of the field of definition of $z_n$. In the following, denote by
$$
\sigma: K_n^\times \backslash \widehat K_n^\times \longrightarrow {\mathrm{Gal}}(K_n^{\mathrm{ab}}/K_n)
$$
the geometric Artin map, normalized by sending the uniformizers to the geometric Frobenii.
So it is the reciprocal of the usual Artin map.
\begin{prop} \label{CM point1}
\begin{enumerate}[(1)]
\item Assume that $n\equiv 5\pmod 8$. Then ${\mathrm{Gal}}(H_n'/H_n)\simeq {\mathbb {Z}}/2{\mathbb {Z}}$ is generated by $\sigma_{\varpi}^2$. Here $\varpi=(\sqrt{-n}-1)_2\in K_{n,2}^\times$. The field $H_n'(\sqrt{2})$ is the ring class field of conductor $4$ over $K_n$.
The Galois group ${\mathrm{Gal}}(H_n'(\sqrt{2})/H_n)\simeq ({\mathbb {Z}}/2{\mathbb {Z}})^2$ is generated by $\sigma_{1+2\varpi}$ and $\sigma_{\varpi}^2$, and $H'_n$ is the subfield of $H_n'(\sqrt{2})$ fixed by $\sigma_{1+2\varpi}$.
\item Assume that $n\equiv 6\pmod 8$. Then ${\mathrm{Gal}}(H_n'/H_n)\simeq {\mathbb {Z}}/4{\mathbb {Z}}$ is generated by $\sigma_{1+\varpi}$. Here $\varpi=(\sqrt{-n})_2\in K_{n,2}^\times$. The subfield of $H_n'$ fixed by $\sigma_{1+\varpi}^2$ is $H_n(i)$. The field $H_n'$ is exactly the ring class field of conductor $4$ over $K_n$.
\item Assume that $n\equiv 7\pmod 8$. Then $H_n'=H_n$.
\item For any $n\equiv 5,6,7\pmod 8$, $2z_n$ is defined over $H_n$.
\end{enumerate}
\end{prop}
Denote $K_n'=K_n, K_n(i), K_n$ according to $n\equiv 5, 6, 7\pmod 8$ respectively. Set
${\mathrm{Cl}}_n={\mathrm{Gal}}(H_n/K_n)$ and ${\mathrm{Cl}}'_n={\mathrm{Gal}}(H_n'/K_n')$.
Let $\sigma$ be the unique order-two element of ${\mathrm{Gal}}(H_n'/H_n)$ in the case $n\equiv 5,6\pmod 8$, and set $\sigma=1$ in the case $n\equiv 7\pmod 8$.
Then the natural map ${\mathrm{Cl}}_n'\to {\mathrm{Cl}}_n$ induces two isomorphisms
$$
{\mathrm{Cl}}'_n/\langle\sigma\rangle\cong{\mathrm{Cl}}_n, \quad
(2{\mathrm{Cl}}'_n)/\langle\sigma\rangle\cong2{\mathrm{Cl}}_n.
$$
The least obvious case is the second isomorphism for $n\equiv 6\pmod 8$. For that, it suffices to check that $\sigma=\sigma_{1+\varpi}^2$ lies in $2{\mathrm{Cl}}'_n$.
Note that $\sigma_{1+\varpi}\notin {\mathrm{Cl}}'_n$, but we use the relations
$\sigma=(\sigma_{1+\varpi}\sigma_{\varpi})^2$ and $\sigma_{1+\varpi}\sigma_{\varpi}\in {\mathrm{Cl}}'_n$ instead. In fact, an easy calculation shows $\sigma_{\varpi}^2=1$ (on $H_n'$) and $\sigma_{\varpi}(i)=-i$, which give the new relations.
\subsubsection*{Quadratic point}
Fix a set $\Phi\subset {\mathrm{Cl}}'_n$ of representatives of ${\mathrm{Cl}}'_n/\langle\sigma\rangle\cong {\mathrm{Cl}}_n$. Let $\chi:{\mathrm{Cl}}_n\to \{\pm1\}$ be a character.
Define the quadratic point $P_\chi$ associated to $\chi$ by
$$P_\chi:=\sum_{t\in \Phi} f_n(P_n)^{t} \chi(t)\in A(H_n').$$
Here $\chi$ is also viewed as a function on $\Phi$ via the bijection $\Phi\to {\mathrm{Cl}}_n$.
To give a formula for $P_\chi$, we need to describe another algebraic point on the elliptic curve.
Recall that ${\mathcal {L}}(n)$ and $\rho(n)$ are defined in the introduction of this paper.
We will see that ${\mathcal {L}}(n)$ is a rational number.
Define
$${\mathcal {P}}(n):=2^{-1-\rho(n)}{\mathcal {L}}(n) \alpha_n \in A(K_n)^-\otimes_{\mathbb {Z}}{\mathbb {Q}},$$
where
$$A(K_n)^-:=\{\alpha\in A(K_n): \bar\alpha=-\alpha\}\ \subset\ A(K_n),$$
and
$\alpha_n\in A(K_n)^-$ is any point which generates the free part $A(K_n)^-/A(K_n)^-_{\mathrm{tor}}$ if ${\mathcal {L}}(n)\neq 0$. Note that ${\mathcal {P}}(n)=0$ if ${\mathcal {L}}(n)=0$.
\begin{thm}[Gross-Zagier formula]\label{GZ}
Let $\chi:{\mathrm{Cl}}_n\to \{\pm1\}$ be a character.
The point $P_\chi$ is non-torsion only if $\chi$ is of the form
$$\chi_{d_0, d_1},\quad n=d_0d_1,\ 0<d_0\equiv 5, 6, 7\pmod 8,\ 0<d_1\equiv 1, 2, 3\pmod 8, $$ where $\chi_{d_0, d_1}$ is the unique Hecke character over $K_n$ associated to the extension $K_n(\sqrt{d_1})$ for $n\equiv 5,6\pmod 8$ or $K_n(\sqrt{d_1^*})$ for $n\equiv 7\pmod 8$.
Here $d_1^*=(-1)^{(d_1-1)/2}d_1$ as before.
In that case, in the vector space
$A(H_n'(i))\otimes_{{\mathbb {Z}}} {\mathbb {Q}}=A(H_n'(i))\otimes_{{\mathbb {Z}}[i]} {\mathbb {Q}}[i]$,
$$P_{\chi}=\epsilon (d_0, d_1) 2^{h_2(n)}{\mathcal {L}}(d_1) {\mathcal {P}}(d_0),$$
where $\epsilon(d_0, d_1)=\pm i$ if $(d_0, d_1)\equiv (5, 3)\pmod 8$ and $\epsilon(d_0, d_1)=\pm 1$ otherwise.
\end{thm}
\subsubsection*{Genus point}
Set $\Phi_0=\Phi\cap (2{\mathrm{Cl}}_n')$ as a subset of ${\mathrm{Cl}}_n'$.
Then $\Phi_0\subset 2{\mathrm{Cl}}_n'$ is a set of representatives of $(2{\mathrm{Cl}}'_n)/\langle\sigma\rangle\cong 2{\mathrm{Cl}}_n$ in $2{\mathrm{Cl}}_n'$. Define
$$Z(n):=\sum_{t\in \Phi_0} f_n(P_n)^{t}\in A(H_n').$$
To compare $Z(d_0)$ for different divisors $d_0$ of $n$, we introduce the composite field
$${\mathbb {H}}_n':=L_n(i)\cdot\prod_{\substack{d_0|n, \ d_0>0 \\ d_0\equiv 5, 6\pmod 8}} H_{d_0}'\subset \overline{\mathbb {Q}}.$$
Here $L_n(i)={\mathbb {Q}}(i, \sqrt{d}:d|n)$. The field seems to be very large, but we will see that
$A({\mathbb {H}}_n')_{\mathrm{tor}}\subset A[4]$ in Lemma \ref{torsion2}, which is a key property in our treatment.
Note that $A({\mathbb {H}}_n')$ is a ${\mathbb {Z}}[i]$-module.
Define $P(n)\in A({\mathbb {H}}_n')$ inductively by
$$P(n):=Z(n)-\sum_{\substack{n=d_0d_1\\ d_0\equiv 5, 6, 7\pmod 8\\
d_1\equiv 1,2 ,3 \pmod 8,\ d_1>1}} \epsilon(d_0, d_1){\mathcal {L}}(d_1)P(d_0),$$
where $\epsilon(d_0, d_1)\in \mu_4$ is as in Theorem \ref{GZ}.
Note that ${\mathcal {L}}(d_1)\in {\mathbb {Z}}$ by Theorem \ref{lg}.
By definition, it is easy to verify the following congruence formula.
\begin{prop} \label{congruence}
In $A({\mathbb {H}}_n')$,
\begin{multline*}
P(n) \equiv
\sum_{\substack{n=d_0d_1\cdots d_\ell\\
d_0\equiv 5,6, 7\pmod 8\\
d_1\equiv 1, 2, 3\pmod 8\\
d_i\equiv 1 \pmod 8, \ i>1}}\epsilon(d_0, d_1) \left(\prod_{i\geq 1} g(d_i)\right) Z(d_0)
\\
+i\sum_{\substack{n=d_0d_1\cdots d_\ell\\
(d_0,d_1,d_2)\equiv (5,3,2)\pmod 8\\
d_i\equiv 1 \pmod 8, \ i>2}} \left(\prod_{i\geq 1} g(d_i)\right) Z(d_0)
\qquad\mod\ 2A({\mathbb {H}}_n').
\end{multline*}
\end{prop}
The main result of this section is as follows, which is an enhanced version of Theorem \ref{lg'}.
\begin{thm}\label{m3}
The vector ${\mathcal {P}}(n)\in A(K_n)^-\otimes_{\mathbb {Z}}{\mathbb {Q}}$ is represented by the point $P(n)\in A({\mathbb {H}}_n')$ in the sense that they are equal in $A({\mathbb {H}}_n')\otimes_{\mathbb {Z}}{\mathbb {Q}}$.
Moreover,
\begin{enumerate}[(1)]
\item The image of $2P(n)$ under any $2$-isogeny from $A$ to $E$ belongs to $E(K_n)^-$, i.e. ${\mathcal {L}}(n)$ is integral.
\item Assume that $P(n)\in A(K_n)^-+A[4]$, i.e. $2^{-\rho(n)}{\mathcal {L}}(n)$ is even.
If $n\equiv 5, 7\pmod 8$, then
$$\sum_{\substack{{n=d_0\cdots d_\ell}\\ {d_i\equiv 1\pmod 8,\ i>0}}} \prod_i
g(d_i)\equiv \sum_{\substack{{n=d_0\cdots d_\ell,}\\
{d_0\equiv 5, 6, 7\pmod 8}\\ {d_1\equiv 1,2 ,3 \pmod 8} \\ {d_i\equiv 1\pmod 8,\ i>1}}} \prod_i g(d_i) \equiv 0 \quad
\pmod 2.$$
If $n\equiv 6\pmod 8$, then
$$ \sum_{\substack{{n=d_0\cdots d_\ell,}\\
{d_0\equiv 5, 6, 7\pmod 8}\\ {d_1\equiv 1,2 ,3 \pmod 8} \\ {d_i\equiv 1\pmod 8,\ i>1}}} \prod_i g(d_i) \equiv 0\quad
\pmod 2.$$
\end{enumerate}
\end{thm}
\subsection{Test vectors}
Recall that in Proposition \ref{CM point1} we have described the field $H_n'=H_n(z_n)$.
The major goal of this section is to prove some results about Galois actions on $z_n$. We will also prove Proposition \ref{choice} and Proposition \ref{CM point1}.
To describe the results about Galois actions on $z_n$, we recall some basic facts about $X_0(32)$ and $A$, which are basic facts or results proved in \cite{Tian}.
\begin{enumerate}[(1)]
\item
There is an analytic isomorphism
$$\tau: {\mathbb {C}}/(1+i){\mathbb {Z}}[i]\longrightarrow A({\mathbb {C}}).$$
The map $\tau$ is unique up to multiplication by $\mu_4=\{\pm1, \pm i\}$.
We can adjust $\tau$ such that ${\mathbb {R}}/2{\mathbb {Z}}$ maps onto $A({\mathbb {R}})$ and
$$A[2^\infty]\ =\ {\mathbb {Q}}_2(i)/(1+i){\mathbb {Z}}_2[i]\ \subset\ {\mathbb {Q}}(i)/(1+i){\mathbb {Z}}[i] \ =\ A({\mathbb {C}})_{{\mathrm{tor}}}. $$
\item
Under the uniformization $\tau$, the Galois group $G_{\mathbb {Q}}=G_{{\mathbb {Q}}(i)}\rtimes \{1, c\}$ acts on $A[2^\infty]={\mathbb {Q}}_2(i)/(1+i){\mathbb {Z}}_2[i]$ as follows.
The induced action of $c$ on ${\mathbb {Q}}_2(i)/(1+i){\mathbb {Z}}[i]$ is still given by the conjugation $i\mapsto -i$, and the induced action of $G_{{\mathbb {Q}}(i)}$ on ${\mathbb {Q}}_2(i)/(1+i){\mathbb {Z}}[i]$ is given by multiplying by the composition
$$G_{{\mathbb {Q}}(i)}\rightarrow {\mathrm{Gal}}({\mathbb {Q}}(i)^{{\mathrm{ab}}}/{\mathbb {Q}}(i))\stackrel{\sigma_{{\mathbb {Q}}(i)}^{-1}}{\longrightarrow} {\mathbb {Q}}(i)^\times \backslash \widehat{{\mathbb {Q}}(i)}^\times \cong (1+(1+i)^3\widehat{{\mathbb {Z}}[i]})^\times\rightarrow (1+(1+i)^3{\mathbb {Z}}_2[i])^\times,$$
where $\sigma_{{\mathbb {Q}}(i)}$ denotes the Artin map and the last map is the natural projection.
\item
The identification $i_0:X_0(32)\rightarrow A$ (mapping $\infty$ to $0$) identifies the set
${\mathcal {S}}=\Gamma_0(32)\backslash {\mathbb {P}}^1({\mathbb {Q}})$ of cusps with $A[(1+i)^3]=A({\mathbb {Q}}(i))$.
Replacing $\tau$ by $-\tau$ if necessary, we can (and we will) assume that the induced bijection
$$\tau: \frac12 {\mathbb {Z}}[i]/(1+i){\mathbb {Z}}[i] \longrightarrow A[(1+i)^3]= \Gamma_0(32)\backslash {\mathbb {P}}^1({\mathbb {Q}})$$
gives
$$\tau(0)=[\infty], \ \quad \tau(1/2)=[0], \ \quad \tau(-1/2)=[1/2],$$
$$\tau(1)=[1/16], \ \quad \tau(\pm i/2)=[\pm 1/4], \ \quad \tau((1\pm i)/2)=[\pm 1/8].$$
\end{enumerate}
Now we are ready to state the main result of this subsection.
\begin{thm}\label{CM point2}
Resume the notations in Proposition \ref{CM point1}. The following are true:
\begin{enumerate}[(1)]
\item Assume that $n\equiv 5\pmod 8$. Then
$$ z_n^{\sigma_\varpi}=z_n+\tau(\frac{1+i}2), \qquad
z_n^{\sigma_{1+2\varpi}}=z_n, \qquad \bar z_n=-z_n+\tau(1).$$
Thus $z_n^{\sigma_{\varpi^2/2}}=z_n^{\sigma_{\varpi^2}}=z_n+\tau(1)$.
\item Assume that $n\equiv 6\pmod 8$. Then
$$z_n^{\sigma_{\varpi}}=z_n+\tau(-i/2),\qquad z_n^{\sigma_{1+\varpi}}=z_n+\tau(\frac{1-i}2), \qquad \bar z_n=-z_n.$$
Thus $z_n^{\sigma_{1+\varpi}^2}=z_n+\tau(1)$.
\item Assume that $n\equiv 7\pmod 8$.
Let $v_2$ and $v_2'$ be the two places of $K_n$ above $2$ such that $v_2(\sqrt{-n}-\delta)\geq 6$. Let $\varpi\in K_{n,2}$ be an element with $v_2(\varpi)=1$ and $v_2'(\varpi)=0$.
Then
$$\bar z_n+z_n^{\sigma_{\varpi^5}}=\tau(1/2).$$
\end{enumerate}
\end{thm}
Here $\bar z_n$ denotes the complex conjugate of $z_n$.
The results will be treated case by case in the following.
For simplicity, we write $K$ for $K_n$ (so that $K_2$ means the local field $K_{n,2}$ of $K_n$ at $2$).
\
\subsection*{Case $n\equiv 5\pmod 8$}
In this case, $f_n:X_U\to A$ is given by $f_n=f_0-f_0\circ [i]$, and the embedding of $K$ into $M_2({\mathbb {Q}})$ is given by
$$\sqrt{-n} \longmapsto \matrixx{-1}{1/4}{-4(n+1)}{1}.$$
The embedding gives
$(\widehat{{\mathbb {Z}}}+4\widehat{O}_{K})^\times \subset U$.
\begin{lem} \label{coset5}
Assume $n\equiv 5\pmod 8$.
\begin{enumerate}[(1)]
\item The quotient
$K_{2}^\times/{\mathbb {Q}}_2^\times (1+4 O_{K,2})$
is isomorphic to ${\mathbb {Z}}/4{\mathbb {Z}}\times {\mathbb {Z}}/2{\mathbb {Z}}$, and generated by the order-four element
$\varpi=(\sqrt{-n}-1)_2$ and the order-two element $1+2\varpi$.
\item
The multiplicative group $K_2^\times$ normalizes $U_2$.
\end{enumerate}
\end{lem}
\begin{proof}
We first check (1). Note that $K_2$ is ramified over ${\mathbb {Q}}_2$.
Then ${\mathbb {Q}}_2^\times O_{K,2}^\times$ has index two in $K_{2}^\times$.
Then $K_{2}^\times/{\mathbb {Q}}_2^\times (1+4 O_{K,2})$ has an index-two subgroup
$${\mathbb {Q}}_2^\times O_{K,2}^\times /{\mathbb {Q}}_2^\times (1+4 O_{K,2})
=O_{K,2}^\times /{\mathbb {Z}}_2^\times (1+4 O_{K,2})
=(O_{K,2}/4 O_{K,2})^\times /\{\pm 1\}
\simeq {\mathbb {Z}}/2{\mathbb {Z}}\times {\mathbb {Z}}/2{\mathbb {Z}}.
$$
It follows that $K_{2}^\times/{\mathbb {Q}}_2^\times (1+4 O_{K,2})$ is isomorphic to ${\mathbb {Z}}/4{\mathbb {Z}}\times {\mathbb {Z}}/2{\mathbb {Z}}$.
Now it is easy to check that $\varpi$ and $1+2\varpi$ generate the group.
For (2), since $1+4 O_{K,2} \subset U$, we see that ${\mathbb {Q}}_2^\times (1+4 O_{K,2})$ normalizes $U_2$. By (1), it suffices to check that $\varpi$ and $1+2\varpi$ normalize $U_2$, which can be done by explicit calculations.
\end{proof}
By the lemma, $K_2^\times$ normalizes $U_2$, and thus it acts on $X_U$ by the right multiplication. The subgroup ${\mathbb {Q}}_2^\times(1+4 O_{K_n,2})$ acts trivially and induces a homomorphism
$$K_2^\times/{\mathbb {Q}}_2^\times
(1+4 O_{K_n,2})\longrightarrow {\mathrm{Aut}}_{\mathbb {Q}}(X_U).$$
We will describe this homomorphism explicitly.
The following result contains a lot of identities in
$${\mathrm{Aut}} _{\mathbb {Q}}(X_U)={\mathrm{Aut}}_{{\mathbb {Q}}(i)}(X_0(32)_{{\mathbb {Q}}(i)})\rtimes \{1,
\epsilon\}\simeq \left( A({\mathbb {Q}}(i))\rtimes \mu _4\right) \rtimes \{1,
\epsilon\}.$$
Here $\epsilon=\matrixx{1}{}{}{-1}\in {\mathrm{GL}}_2({\mathbb {Q}})$ also normalizes $U$, and its Hecke action on $X_U$ gives the non-trivial element of ${\mathrm{Gal}}(X_U/X_0(32))$. In particular, $\epsilon$ acts on ${\mathrm{Aut}}_{{\mathbb {Q}}(i)}(X_0(32)_{{\mathbb {Q}}(i)})$ by sending $i$ to $-i$.
Recall that we have also identified
$$
A({\mathbb {Q}}(i))= \frac12 {\mathbb {Z}}[i]/(1+i){\mathbb {Z}}[i]
$$
with the set ${\mathcal {S}}$ of cusps of $X_0(32)$.
\begin{prop}\label{heckeaction5}
Assume $n\equiv 5\pmod 8$.
\begin{enumerate}[(1)]
\item For any
$Q\in X_U({\mathbb {C}})$,
$$Q^{\varpi}
=[i] Q^\epsilon+\tau(\frac12), \quad\
Q^{1+2\varpi}=Q+\tau(1).$$
\item The order-two element $j=\matrixx{1}{0}{8}{-1} \in
{\mathrm{GL}}_2({\mathbb {Q}})$ normalizes $K^\times$ such that $jxj=\overline{x}$ for all
$x\in K^\times$ and normalizes $U$ with the induced action on
$X_U$ given by
$$Q^{j}=[-i] Q^\epsilon+\tau(\frac {1+i}2), \quad \forall\ Q\in X_U({\mathbb {C}}).$$
\end{enumerate}
\end{prop}
\begin{proof}
The right translation by an element $g\in {\mathrm{GL}}_2({\mathbb {A}}_f)$ switches the two geometric
components of $X_U$ if and only if the image of $g$ under the
composition
$${\mathrm{GL}}_2(\widehat{{\mathbb {Q}}})\stackrel{\det}{\longrightarrow}
\widehat{{\mathbb {Q}}}^\times={\mathbb {Q}}^\times_+\cdot \widehat{{\mathbb {Z}}}^\times \longrightarrow
({\mathbb {Z}}_2/4{\mathbb {Z}}_2)^\times \cong \{\pm 1\}$$ is $-1$. For example, all
$\epsilon=\matrixx{1}{0}{0}{-1},\ j=\matrixx{1}{0}{8}{-1}$,\ and
$\varpi=\matrixx{-2}{1/4}{-4(n+1)}{0}$ are such elements, but
$1+2\varpi$ is not.
Hence, the actions of $\varpi \epsilon$,
$1+2\varpi$ and $j\epsilon$ take the form
$$Q^{\varpi \epsilon}=\alpha Q+R, \quad Q^{1+2\varpi}=\beta Q +S,
\quad Q^{j\epsilon}=\gamma Q+T$$ where $\alpha, \beta, \gamma\in
\mu_4$ and $R, S, T\in{\mathcal {S}}$ are cusps of $X_0(32)$. Here the right sides belong to
$${\mathrm{Aut}}_{{\mathbb {Q}}(i)}(X_0(32)_{{\mathbb {Q}}(i)})={\mathrm{Aut}}(X_U/{\mathbb {Q}}(i))\subset
{\mathrm{Aut}}_{\mathbb {Q}}(X_U).$$
To compute $R, S, T$, we take $Q=[\infty]$. In terms of the complex uniformization
$$X_0(32)({\mathbb {C}})={\mathrm{GL}}_2({\mathbb {Q}})_+ \backslash ({\mathcal {H}}\cup {\mathbb {P}}^1({\mathbb {Q}})) \times
{\mathrm{GL}}_2(\widehat{{\mathbb {Q}}})/U_0(32),$$
we have
$$
R=[\infty, \varpi \epsilon], \quad
S=[\infty, 1+2\varpi], \quad
T=[\infty, j \epsilon].
$$
We need to convert them to expressions of the form $[\theta]=[\theta,1]$ with $\theta\in {\mathbb {P}}^1({\mathbb {Q}})$.
By the complex uniformization,
$$\begin{aligned}
{\mathcal {S}}&={\mathrm{GL}}_2({\mathbb {Q}})_+\backslash {\mathbb {P}}^1({\mathbb {Q}})\times
{\mathrm{GL}}_2(\widehat{{\mathbb {Q}}})/U_0(32)=P({\mathbb {Q}})_+\backslash {\mathrm{GL}}_2(\widehat{{\mathbb {Q}}})/U_0(32)\\
&=P({\mathbb {Q}})_+\backslash P(\widehat{{\mathbb {Q}}})\cdot {\mathrm{GL}}_2(\widehat{{\mathbb {Z}}})/U_0(32)=N({\mathbb {Z}}_2)\backslash
{\mathrm{GL}}_2({\mathbb {Z}}_2)/U_0(32)_2. \end{aligned}$$
For the truth of the last identity, we refer to \cite[Lemma 4.12(2)]{YZZ}.
We have decompositions in
${\mathrm{GL}}_2({\mathbb {Q}}_2)$ as follows:
$$\begin{aligned}
\varpi\epsilon&=\matrixx{1/4}{0}{0}{8}\matrixx{-8}{-1}{1}{0}\matrixx
{-(n+1)/2}{0}{4(n+3)}{1},\\
1+2\varpi&=\matrixx{1}{1/2}{0}{1}\matrixx{1}{0}{-16}{1}\matrixx{4n+1}{0}{8(7n+1)}{1}, \\
j\epsilon&=\matrixx{1}{0}{8}{1}.
\end{aligned}$$
It follows that
$$
R=\left[\infty, \matrixx{-8}{-1}{1}{0}_2\right]=\left[\matrixx{-8}{-1}{1}{0}^{-1}\infty, 1\right]=[0]=\tau(1/2).
$$
Similarly, $S=[1/16]=\tau(1)$ and $T=[-1/8]=\tau((1-i)/2)$.
To find $\alpha, \beta, \gamma$, we only need
check the action on the cusp $[0]=\left[\infty,\matrixx{0}{1}{-1}{0}\right]$. We have decompositions:
$$\begin{aligned}
\matrixx{0}{1}{-1}{0}\varpi\epsilon
&=\matrixx{8}{0}{0}{1/4}\matrixx{1}{0}{8}{1}\matrixx{(n+1)/2}{0}{-4(n+3)}{-1},\\
\matrixx{0}{1}{-1}{0}(1+2\varpi)&=\matrixx{-2}{+1}{0}{-1/2}\matrixx{7}{-3}{-2}{1}
\matrixx{4n-17}{3}{8(n-5)}{7},\\
\matrixx{0}{1}{-1}{0}j\epsilon&=\matrixx{8}{1}{-1}{0}.
\end{aligned}$$
It follows that
$$[0]^{\varpi\epsilon}=[-1/8]=\tau((1-i)/2), \quad
[0]^{1+2\varpi}=[1/2]=\tau(-1/2), \quad
[0]^{j\epsilon}=[0]=\tau(1/2).$$
We then have the equations
$$\tau((1-i)/2)=\alpha\tau(1/2)+\tau(1/2), \quad \tau(-1/2)=\beta
\tau(1/2)+\tau(1), \quad \tau(1/2)=\gamma\tau(1/2)+\tau((1-i)/2),$$
which give $\alpha=-i$, $\beta=1$ and $\gamma=i$.
Therefore, we have
$$Q^{\varpi\epsilon}=[-i]Q+\tau(1/2), \quad\
Q^{1+2\varpi}=Q+\tau(1), \quad
Q^{j\epsilon}=[i]Q+\tau(\frac {1-i}2).$$
For the first and the the third equations, we take a further $\epsilon$-action on both sides. Then
$$
Q^{\varpi}=([-i]Q+\tau(1/2))^\epsilon
=[i](Q^\epsilon)+\tau(1/2)
$$
and
$$
Q^{j}=([i]Q+\tau(\frac {1-i}2))^\epsilon=[-i](Q^\epsilon)+\tau(\frac {1+i}2).$$
It finishes the proof.
\end{proof}
The map $K_2^\times \to {\mathrm{Aut}}(X_U)$ induces an action of $K_2^\times$ on ${\mathrm{Hom}}(X_U,A)$.
Still use $\pi$ denote this action. Now it is easy to have the action on $f_n=f_0\circ [1-i].$
\begin{cor} \label{integral5}
In ${\mathrm{Hom}}(X_U,A)$,
$$\pi(\varpi)f_n=f_n+\tau(\frac{1+i}2), \quad\ \pi(1+2\varpi)f_n=f_n.$$
\end{cor}
\begin{proof}
For any $Q\in X_U({\mathbb {C}})$,
$$
(\pi(\varpi)f_n)(Q)= f_0([1-i] Q^\varpi).
$$
By the proposition,
$$
[1-i] Q^\varpi=[1-i]([i]Q^\epsilon+\tau(1/2)) =[1+i]Q^\epsilon+\tau(\frac{1-i}2)
=( [1-i]Q+\tau(\frac{1+i}2) )^\epsilon.$$
Note that $f_0$ is invariant under $\epsilon.$
Thus
$$
(\pi(\varpi)f_n)(Q)= f_0([1-i]Q+\tau(\frac{1+i}2))
= f_0([1-i]Q)+\tau(\frac{1+i}2)
= f_n(Q)+\tau(\frac{1+i}2).
$$
The second equality is proved similarly.
\end{proof}
The corollary is an integral version of Lemma \ref{choice} for $n\equiv5\pmod8$.
Now $f_n$ lies in the space $\pi^{{\mathrm{GL}}_2(\widehat{\mathbb {Z}}^{(2)})\cdot K_{2}^\times}\simeq \pi_2^{K_{2}^\times}$, which is one-dimensional by Theorem \ref{TS1} and Theorem \ref{TS2}.
Now we are ready to prove Theorem \ref{CM point2} for $n\equiv5\pmod8$, i.e.,
$$ z_n^{\sigma_\varpi}=z_n+\tau(\frac{1+i}2), \qquad
z_n^{\sigma_{1+2\varpi}}=z_n, \qquad \bar z_n=-z_n+\tau(1).$$
For the first two equalities, the key is that the Galois action of $K_2^\times$ on $P$ (via the Artin map $\sigma$) is the same as the Hecke action of $K_2^\times$, by the special form of
$P_n=[h,1]$.
Then by Corollary \ref{integral5},
$$
z_n^{\sigma_\varpi}=f_n(P_n^{\sigma_\varpi})
=f_n(P_n^{\varpi})
=(\pi(\varpi)f_n)(P_n)
=f_n(P_n)+\tau(\frac{1+i}2)
=z_n+\tau(\frac{1+i}2).
$$
The second equality is similarly obtained.
For the third equality, the Hecke action of the element $j$ in Proposition \ref{heckeaction5} (2) gives the complex conjugation of $P_n=[h,1]$ by the condition $jxj^{-1}=\overline{x}$ for all
$x\in K^\times$.
In fact,
$$
\bar P_n=[\bar h,1], \quad P_n^j=[h,j]=[j(h),1].
$$
It suffices to check $\bar h=j(h)$. Note that $\{h, \bar h\}$ is the set of fixed points of $K^\times$ in ${\mathcal {H}}^\pm$. By $jK^\times j=K^\times$, we see that $\{h, \bar h\}=\{j(h), j(\bar h)\}$ as sets. Since $\det(j)=-1<0$, we have $j(h)\in {\mathcal {H}}^-$ and thus $j(h)=\bar h$.
Hence,
$$
\bar z_n=f_n(\bar P_n)
=f_n(P_n^{j})
=f_n([-i] P_n^\epsilon+\tau(\frac {1+i}2))
=f_0([-1-i] P_n^\epsilon+\tau(1)).
$$
By $[-1-i] P_n^\epsilon+\tau(1)=([-1+i] P_n^\epsilon+\tau(1))^\epsilon$, we have
$$
\bar z_n
=f_0([-1+i] P_n+\tau(1))
=-f_0(P_n)+\tau(1)
=-z_n+\tau(1).
$$
This proves the theorem in the current case.
Finally, we prove Proposition \ref{CM point1} for $n\equiv 5\pmod 8$.
By the reciprocity law, the point $P_n$ is defined over the abelian extension of $K$ with Galois group $\widehat K^\times/ K^\times (\widehat K^\times\cap U)$.
It is easy to see $(\widehat{{\mathbb {Z}}}+4\widehat{O}_{K})^\times \subset U$.
Then $P_n$ is defined over the ring class field $H_{n,4}$ of $K$
with Galois group $\widehat K^\times/ K^\times (\widehat{{\mathbb {Z}}}+4\widehat{O}_{K})^\times$.
We have
\begin{multline*}
{\mathrm{Gal}}(H_{n,4}/H_n)
\cong K^\times \widehat{O}_{K}^\times/
K^\times (\widehat{{\mathbb {Z}}}+4\widehat{O}_{K})^\times
= \widehat{O}_{K}^\times/ (\widehat{{\mathbb {Z}}}+4\widehat{O}_{K})^\times \\
= O_{K,2}^\times/ ({\mathbb {Z}}_2+4 O_{K,2})^\times
= O_{K,2}^\times/ {\mathbb {Z}}_2^\times(1+4 O_{K,2})
= (O_{K,2}/4 O_{K,2})^\times/ \{\pm1\}.
\end{multline*}
As in the proof of Lemma \ref{coset5}, the right-hand side is isomorphic to ${\mathbb {Z}}/2{\mathbb {Z}}\times {\mathbb {Z}}/2{\mathbb {Z}}$, and generated by $\frac12\varpi^2$ and $1+2\varpi$.
Consider $z_n=f_n(P_n)$ and $H_n'=H_n(z_n) \subset H_{n,4}$.
By Theorem \ref{CM point2},
$$z_n^{\sigma_{\varpi^2/2}}=z_n+\tau(1), \qquad
z_n^{\sigma_{1+2\varpi}}=z_n.$$
It follows that $H_n'$ is the index-two subfield of $H_{n,4}$ fixed by $\sigma_{1+2\varpi}$.
Note that $\sqrt2\in H_{n,4}$ but $\sqrt2\notin H_{n}'$ by
$\sigma_{1+2\varpi}(\sqrt2)=-\sqrt2$.
The equations also indicate that $2z_n$ is invariant under both $\sigma_{\varpi^2/2}$ and $\sigma_{1+2\varpi}$, and thus it is defined over $H_n$.
The proposition is proved in this case.
\
\subsection*{Case $n\equiv 6\pmod 8$}
Now we consider the case $n\equiv 6\pmod 8$.
The exposition is very similar to the previous case $n\equiv 5\pmod 8$, and the calculations are slightly simpler.
We still follow the process of the previous case, but only sketch some of the proofs.
In this case, $f_n:X_U\to A$ is given by $f_n=f_0\circ [i]$, and the embedding of $K$ into $M_2({\mathbb {Q}})$ is given by
$$\sqrt{-n}\longmapsto {\matrixx{}{1/4}{-4n}{}}.$$
The embedding still gives
$(\widehat{{\mathbb {Z}}}+4\widehat{O}_{K})^\times \subset U$.
\begin{lem} \label{coset6}
Assume $n\equiv 6\pmod 8$.
\begin{enumerate}[(1)]
\item The quotient
$K_{2}^\times/{\mathbb {Q}}_2^\times (1+4 O_{K,2})$
is isomorphic to ${\mathbb {Z}}/4{\mathbb {Z}}\times {\mathbb {Z}}/2{\mathbb {Z}}$, and generated by the order-two element
$\varpi=(\sqrt{-n})_2$ and the order-four element $1+\varpi$.
\item
The multiplicative group $K_2^\times$ normalizes $U_2$.
\end{enumerate}
\end{lem}
\begin{proof}
The proof is similar to that of Lemma \ref{coset5}.
\end{proof}
Now we describe the homomorphism
$$K_2^\times/{\mathbb {Q}}_2^\times
(1+4 O_{K_n,2})\longrightarrow {\mathrm{Aut}}_{\mathbb {Q}}(X_U).$$
\begin{prop}\label{heckeaction6}
Assume $n\equiv 6\pmod 8$.
\begin{enumerate}[(1)]
\item For any
$Q\in X_U({\mathbb {C}})$,
$$Q^{\varpi}=-Q^\epsilon+\tau(\frac12), \quad Q^{1+\varpi}=-Q^\epsilon+\tau(\frac{1-i}2).$$
\item The order-two element $\epsilon=\matrixx{1}{}{}{-1}\in
{\mathrm{GL}}_2({\mathbb {Q}})$ normalizes $K^\times$ such that $\epsilon x \epsilon=\overline{x}$ for all
$x\in K^\times$.
\end{enumerate}
\end{prop}
\begin{proof}
Follow the strategy of Proposition \ref{heckeaction5}.
The Hecke operators
$\varpi\epsilon $ and $(1+\varpi)\epsilon$ do not switch the two geometric
components of $X_U$.
We have the decompositions
$$\begin{aligned}
\varpi\epsilon&=\matrixx{1/4}{0}{0}{8}\matrixx{0}{1}{-1}{0}\matrixx{n/2}{0}{0}{-1},\\
(1+\varpi)\epsilon&=\matrixx{1}{1/4}{0}{1}\matrixx{-1}{0}{8}{-1}\matrixx{-(1+n)}
{0}{-8(1+n/2)}{1}.\end{aligned}$$
It follows that $\varpi\epsilon $ and $(1+\varpi)\epsilon$ maps $[\infty]$ to $[0]$ and
$[1/8]$, respectively. Thus they acts on $X_U$ take form
$$Q^{\varpi\epsilon}=\alpha Q+\tau(1/2), \quad Q^{(1+\varpi)\epsilon}=\beta Q
+\tau((1+i)/2)$$for some $\alpha, \beta\in \mu_4$.
Use the decompositions
$$\begin{aligned}
\matrixx{0}{1}{-1}{0}\varpi\epsilon &
=\matrixx{-8}{0}{0}{-1/4}\matrixx{n/2}{0}{0}{-1},\\
\matrixx{0}{1}{-1}{0}(1+\varpi)\epsilon&
=\matrixx{4}{-1}{0}{1/4}\matrixx{1}{0}{-4}{1}\matrixx{-(1+n)}{0}{-8(1+n/2)}{1}.
\end{aligned}$$
Setting $Q=[0]=\tau(1/2)$, we have the equations
$$\tau(0)=\alpha\tau(1/2)+\tau(1/2), \quad
\tau(i/2)=\beta\tau(1/2)+\tau((1+i)/2).$$ It follows that $\alpha=-1$ and
$\beta=-1$. Hence, we have
$$Q^{\varpi\epsilon}=-Q+\tau(\frac12), \quad Q^{(1+\varpi)\epsilon}=-Q+\tau(\frac{1+i}2).$$
Further actions by $\epsilon$ gives the results.
\end{proof}
We have the following integral version of Lemma \ref{choice} for $n\equiv5\pmod8$.
\begin{cor} \label{integral6}
Assume $n\equiv 6\pmod 8$.
In ${\mathrm{Hom}}(X_U,A)$,
$$\pi(\varpi)f_n=f_n+\tau(-\frac{i}2), \quad\ \pi(1+\varpi)f_n=f_n+\tau (\frac {1-i}2).$$
\end{cor}
\begin{proof}
The proof is similar to that of Corollary \ref{integral5}.
\end{proof}
Now we can prove Theorem \ref{CM point2} and Proposition \ref{CM point1} for $n\equiv6\pmod8$ similarly.
For example, the proof of $\bar z_n=-z_n$ is given by:
$$
\bar z_n=f_n(\bar P_n)
=f_n(P_n^{\epsilon})
=f_0([i] P_n^\epsilon)
=f_0(([-i] P_n)^\epsilon)
=f_0([-i] P_n)
=-z_n.
$$
\
\subsection*{Case $n\equiv 7\pmod 8$}
Now we consider the case $n\equiv 7\pmod 8$. Then $2$ is split over $K$. It is the simplest case since $f_n:X_0(32)\to A$ is just the identity map $i_0:X_0(32)\to A$. The theory does not involve the more complicated curve $X_U$. For example, Proposition \ref{choice} is true in this case since $\dim \pi^{U_0(32)}=1$ by the newform theory.
The embedding of $K$ into $M_2({\mathbb {Q}})$ is given by
$$\sqrt{-n}\longmapsto
{\matrixx{\delta}{2}{-(n+\delta^2)/2}{-\delta}},$$
where $\delta\in {\mathbb {Z}}$ satisfies $\delta^2\equiv -n\pmod {128}$.
It is easy to check that the embedding gives $\widehat{O}_K^\times\subset U_0(32)$.
Then Proposition \ref{CM point1} is automatic in this case.
The following is devoted to prove Theorem \ref{CM point2} in this case.
Recall from the theorem that $v_2$ and $v_2'$ are the two places of $K_n$ above $2$ such that $v_2(\sqrt{-n}-\delta)\geq 6$, and that $\varpi\in K_{n,2}$ is an element with $v_2(\varpi)=1$ and $v_2'(\varpi)=0$.
\begin{prop}\label{prop3.9}
Assume $n\equiv 7\pmod 8$.
Let
$$W=\matrixx{0}{1}{-32}{0}, \qquad
j=\matrixx{1}{}{-\delta}{-1}$$
be elements of ${\mathrm{GL}}_2({\mathbb {Q}})$.
Then
\begin{enumerate}[(1)]
\item
The element $W$ normalizes $U_0(32)$, and
$$Q^W=-Q+\tau (1/2), \quad\ \forall\ Q\in X_0(32)({\mathbb {C}}).$$
\item
One has $j^2=1$, $jxj=\bar x$ for any $x\in K$, and $j\varpi^5\in W\cdot U_0(32)_2$.
Therefore,
$$Q^{j\varpi^5}=-Q+\tau (1/2), \quad\ \forall\ Q\in X_0(32)({\mathbb {C}}).$$
\end{enumerate}
\end{prop}
\begin{proof}
For (1), consider the Atkin--Lehner operator $\pi(W)$.
Note that $\pi(W)f_n=-f_n$ in the ${\mathbb {Q}}$-space $\pi$ since $A$ has root number 1.
It follows that, in the ${\mathbb {Z}}$-module ${\mathrm{Hom}}(X_0(32),A)$, the sum
$\pi(W)f_n+f_n$ is a torsion point of $A$.
To figure out the torsion point, evaluate at $[\infty]$. We have
$$(\pi(W)f_n+f_n)([\infty])=f_n([\infty]^W)+f_n([\infty])=\tau (1/2).$$
This proves (1).
For (2), consider the ${\mathbb {Q}}_2$-algebra $K_2\cong K_{v_0}\times K_{v_0'}={\mathbb {Q}}_2\times {\mathbb {Q}}_2$.
Let $\alpha\in {\mathbb {Z}}_2^\times$ be such that
$$(\alpha, -\alpha)=\sqrt{-n}\longmapsto
\matrixx{\delta}{2}{-(n+\delta^2)/2}{-\delta}.$$ Then
$64|(\alpha-\delta)$ and $2\| (\alpha+\delta)$. We may take
$\varpi=(2, 1)$. Then
$$\varpi^5
=(32, 1)= \frac{31}{2\alpha} (\alpha,-\alpha)+ \frac{33}{2} (1,1)
$$
corresponds to the matrix
$$
\frac{31}{2\alpha} \matrixx{\delta}{2}{-(n+\delta^2)/2}{-\delta}
+\frac{33}{2} \matrixx{1}{}{}{1}
=\frac{1}{2\alpha} \matrixx{31\delta+33\alpha}{62}{-31(n+\delta^2)/2}{-31\delta+33\alpha}.
$$
It is now straight forward to check
$W^{-1} j\varpi^5\in U_0(32)$.
\end{proof}
Now it is easy to obtain Theorem \ref{CM point2} for $n\equiv 7\pmod 8$ which asserts
$$\bar z_n+z_n^{\sigma_{\varpi^5}}=\tau(1/2).$$
In fact, the proposition gives
$$P_n^{j\varpi^5}=-P_n+\tau (1/2).$$
As before, $j$ computes the complex conjugate of $P_n$. Then the above becomes
$$(\bar P_n)^{\varpi^5}=-P_n+\tau (1/2).$$
The Hecke action is defined over ${\mathbb {Q}}$ and thus commutes with the complex conjugation.
This finishes the proof.
\subsection{Proofs of Theorems \ref{GZ}, Proposition \ref{congruence} and Theorem \ref{m3}}
In this section, we prove our main theorems in the case $n\equiv 5, 6, 7\pmod 8$.
\subsubsection*{Proof of Theorem \ref{GZ}}
We first prove the following result, which gives the first statement of the theorem.
\begin{lem}\label{char'}
Let $\chi:{\mathrm{Cl}}_n\to \{\pm1\}$ be a character satisfying the following conditions:
\begin{enumerate} [(1)]
\item The root number of $L(A_{K_n},\chi,s)$ is $-1$;
\item If $2$ is not split in $K_n$, then the $2$-component $\chi_2:K_{n,2}^\times \to \{\pm1\}$ of $\chi$ is trivial.
\end{enumerate}
Then $\chi$ is exactly of the form
$$\chi_{d_0, d_1},\qquad n=d_0d_1,\ 0<d_0\equiv 5, 6, 7\pmod 8,\ 0<d_1\equiv 1, 2, 3\pmod 8, $$ where $\chi_{d_0, d_1}$ is the unique Hecke character over $K_n$ associated to the extension $K_n(\sqrt{d_1})$ for $n\equiv 5,6\pmod 8$ and $K_n(\sqrt{d_1^*})$ for $n\equiv 7\pmod 8$.
\end{lem}
\begin{proof}
The character $\chi$ corresponds to an extension over $K_n$ of degree dividing $2$ and inside the genus field $L_n$ of $K_n$, which must be of form $K_n(\sqrt{d})=K_n(\sqrt{-n/d})$ for some integer $d|n$ with $\sqrt{d}\in L_n$.
First, the L-function $L(A_{K_n},\chi,s)=L(A_d,s)L(A_{n/d},s)$ has root number $-1$ if and only if exactly one element of $\{|d|, \ n/|d|\}$ is congruent to $1, 2, 3$ modulo $8$ and the other one is congruent to $5, 6, 7$ modulo $8$. Thus we may assume that the extension corresponding to $\chi$ is of the form $K_n(\sqrt{\pm d_1})\subset L_n$, $0<d_1\equiv 1, 2, 3\pmod 8$, such that $d_0:=n/d_1\equiv 5, 6, 7\pmod 8$.
If $n\equiv 7\pmod 8$, then $2$ is split in $K$ and the second condition is empty. Note that $\sqrt{d_1^*}\in L_n$ but $\sqrt{-d_1^*}\notin L_n$. Thus $\chi$ is exactly of desired form.
If $n\equiv 5\pmod 8$, then $2$ is ramified in $K_n$. Both $\sqrt{d_1}$ and
$\sqrt{-d_1}$ are in $L_n$. We have $(d_0,d_1)\equiv (5,1), (7,3) \pmod8$.
The restriction that $\chi_2$ is trivial implies that the extension corresponding to $\chi$ is $K_n(\sqrt{d_1})$ in the first case or $K_n(\sqrt{-d_0})$ in the second case. Then $K_n(\sqrt{d_1})$ is a uniform way to write down the field.
If $n\equiv 6\pmod 8$, then $2$ is ramified in $K_n$.
We have $(d_0,d_1)\equiv (6,1), (7,2) \pmod8$.
Then the extension corresponding to $\chi$ is $K_n(\sqrt{d_1})$ in the first case or $K_n(\sqrt{-d_0})$ in the second case, and $K_n(\sqrt{d_1})$ is still a uniform way.
\end{proof}
Now we prove Theorem \ref{GZ}. It is an example of Theorem \ref{EGZ},
an explicit version of the Gross--Zagier formula proved by Yuan--Zhang--Zhang \cite{YZZ}. Recall that
$$
P_\chi= \sum_{t\in \Phi} f_n(P_n)^t \chi(t).
$$
The summation on $\Phi$ is not canonical, so the expression is not the exact case to apply the formula.
However, by Proposition \ref{CM point1}, $2z_n=2f_n(P_n)$ is defined over $H_n$ and thus
$$
2P_\chi= \sum_{t\in \Phi} (2f_n(P_n))^t \chi(t)
= \sum_{t\in {\mathrm{Cl}}_n} (2f_n(P_n))^t \chi(t).
$$
This is the situation to apply the Gross--Zagier formula (to the test vector $2f_n$).
First, we see that the point $P_\chi$ is non-torsion only if $\chi$ satisfies the two conditions of Lemma \ref{char'}. The first condition holds by considering the Tunnell--Saito theorem (cf. Theorem \ref{TS1}). See the remarks after \cite[Theorem 1.2]{YZZ} for example.
For the second condition, assume that $2$ is not split in $K=K_n$. By ${\mathrm{Cl}}_n=K^\times\backslash \widehat K^\times/\widehat O_K^\times$, the summation for $2P_\chi$ is essentially an integration on $K^\times\backslash \widehat K^\times$.
By Proposition \ref{choice}, $f_n$ is invariant under the action of $K_2^\times$ up to torsions, so the integration is non-torsion only if
$\chi$ is trivial on $K_{2}^\times$.
Hence, Lemma \ref{char'} implies the first statement of the theorem.
Next, assume $\chi=\chi_{d_0, d_1}$ as in the theorem.
Denote $P(d_0,d_1)=P_{\chi_{d_0,d_1}}$.
We first have the following basic result.
\begin{lem} \label{multiples1}
If $(d_0, d_1)\equiv (5, 3)\pmod 8$, then $4P(d_0, d_1)\in A({\mathbb {Q}}(\sqrt{d_0}))^-=[i] A({\mathbb {Q}}(\sqrt{-d_0}))^-$.
Otherwise, $4P(d_0, d_1)\in A({\mathbb {Q}}(\sqrt{-d_0}))^-.$
\end{lem}
\begin{proof}
Recall that
$$
2P(d_0, d_1)= \sum_{t\in {\mathrm{Cl}}_n} (2z_n)^t \chi_{d_0, d_1}(t).
$$
Then $2P(d_0, d_1)$ is invariant under the action of
$\ker(\chi_{d_0,d_1})={\mathrm{Gal}}(H_n/K_{d_0,d_1})$.
Then $2P(d_0, d_1)$ is defined over $K_{d_0,d_1}$.
Here $K_{d_0,d_1}=K_n(\sqrt{-d_1})$ if $(d_0, d_1)\equiv (5, 3)\pmod 8$, and $K_{d_0,d_1}=K_n(\sqrt{d_1})$ otherwise.
First, assume that $d_1\neq 1$, so that $[K_{d_0,d_1}:{\mathbb {Q}}]=4$.
Consider the action of ${\mathrm{Gal}}(K_{d_0,d_1}/{\mathbb {Q}})$ on $2P(d_0, d_1)$.
The group ${\mathrm{Gal}}(K_{d_0,d_1}/{\mathbb {Q}})$ has two explicit elements: the complex conjugation $c$ and the unique nontrivial element
$\tau$ of ${\mathrm{Gal}}(K_{d_0,d_1}/K_n)$.
By definition, $\chi$ takes $-1$ on any lifting of $\tau$ in ${\mathrm{Cl}}_n$. It follows that
$$
(2P(d_0, d_1))^\tau= -2P(d_0, d_1).
$$
On the other hand, the complex conjugate
$$
(2P(d_0, d_1))^c= \sum_{t\in {\mathrm{Cl}}_n} (2\bar z_n)^{1/t}
= \sum_{t\in {\mathrm{Cl}}_n} (2\bar z_n)^{t} .
$$
By Theorem \ref{CM point2}, if $n\equiv5, 6\pmod 8$, then $2\bar z_n=-2z_n$. It follows that $(2P(d_0, d_1))^c=-2P(d_0, d_1)$.
If $n\equiv7\pmod 8$, we only have
$2\bar z_n=-2z_n^{\sigma_{\varpi^5}}+\tau(1)$, which gives
$$(2P(d_0, d_1))^c=-2\chi_{d_0, d_1}(\sigma_\varpi) P(d_0, d_1)+ |{\mathrm{Cl}}_n|\tau(1).$$
Here
$$\chi_{d_0, d_1}(\sigma_\varpi)=-1
\ \Longleftrightarrow \
\sigma_\varpi(\sqrt{d_1^*})=-\sqrt{d_1^*}
\ \Longleftrightarrow \
(d_0, d_1)\equiv (5, 3)\pmod 8.$$
In summary, if $(d_0, d_1)\not\equiv (5, 3)\pmod 8$ (and $d_1\neq1$), then
$$
(4P(d_0, d_1))^\tau= -4P(d_0, d_1), \quad
(4P(d_0, d_1))^c= -4P(d_0, d_1).
$$
It follows that $4P(d_0, d_1)$ is invariant under $c\tau$, and thus defined over
$$
K_{d_0,d_1}^{c\tau}={\mathbb {Q}}(\sqrt{-d_0}, \sqrt{d_1})^{c\tau}={\mathbb {Q}}(\sqrt{-d_0}).
$$
The action of $\tau$ further gives $4P(d_0, d_1)\in A({\mathbb {Q}}(\sqrt{-d_0}))^-$.
If $(d_0, d_1)\equiv (5, 3)\pmod 8$, then
$$
(4P(d_0, d_1))^\tau= -4P(d_0, d_1), \quad
(4P(d_0, d_1))^c= 4P(d_0, d_1).
$$
It follows that $4P(d_0, d_1)$ is invariant under $c$, and thus defined over
$$
K_{d_0,d_1}^{c}={\mathbb {Q}}(\sqrt{d_0}, \sqrt{-d_1})^{c}={\mathbb {Q}}(\sqrt{d_0}).
$$
The action of $\tau$ further gives $4P(d_0, d_1)\in A({\mathbb {Q}}(\sqrt{d_0}))^-$.
In the last case $d_1=1$, we have $K_{d_0,d_1}=K_n$. Then $(4P(d_0, d_1))^c= 4P(d_0, d_1)$ implies $4P(d_0, d_1)\in A({\mathbb {Q}}(\sqrt{-d_0}))^-$.
\end{proof}
Go back to the proof of the theorem. Now we are ready to prove the formula
$$P_{\chi}=\epsilon (d_0, d_1) 2^{h_2(n)}{\mathcal {L}}(d_1) {\mathcal {P}}(d_0) \ \in\ A(H_n'(i))\otimes_{\mathbb {Z}}{\mathbb {Q}}.$$
The formula is equivalent to
$$
2 \bar\epsilon (d_0, d_1) P_{\chi}= 2^{h_2(n)+1}{\mathcal {L}}(d_1) {\mathcal {P}}(d_0).
$$
By Lemma \ref{multiples1}, this is an identity in $A(K_{d_0})^-\otimes_{\mathbb {Z}}{\mathbb {Q}}$.
We first claim that the equality is true up to a multiple in ${\mathbb {Q}}^\times$.
In fact, if $L'(A_{K_n},\chi,1)=L'(A_{d_0},1)L(A_{d_1},1)$ is zero, then the right-hand side is zero by definition, and $P_\chi$ is zero since the canonical height $\widehat h(P_\chi)=0$ by the Gross--Zagier formula (in either \cite[Theorem 1.2]{YZZ} or the explicit version Theorem \ref{EGZ}).
If $L'(A_{K_n},\chi,1)\neq 0$, then by the theorems of Gross--Zagier and Kolyvagin, $E(K_{d_0})^-\otimes_{\mathbb {Z}}{\mathbb {Q}}$ is one-dimensional, and the thus two sides of the equality are proportional.
To finish the proof, it suffices to check that the two sides of the equality have the same canonical heights.
One can do the whole computation on $A$, but we will carry it out on $E$ to be compatible with our original framework.
Let $\varphi: A\rightarrow E$ be the isogeny of degree $2$.
The desired formula becomes
$$ R_\chi= \epsilon(d_0, d_1) 2^{h_2(n)} {\mathcal {L}}(d_1)\CR(d_0) \ \in\ E(H_n'(i))\otimes_{\mathbb {Z}}{\mathbb {Q}}.$$
Here $R_\chi=\varphi(P_\chi)$ and $\CR(d_0)=\varphi({\mathcal {P}}(d_0))$.
The vector $\CR(d_0)\in E(K_{d_0})^-\otimes_{\mathbb {Z}}{\mathbb {Q}}$ has an independent description. If ${\mathcal {L}}(d_0)=0$, then $\CR(d_0)=0$. If
${\mathcal {L}}(d_0)\neq 0$, then the theorems of Gross--Zagier and Kolyvagin imply that $E(K_{d_0})^-$ is of rank one.
In this case, $\CR(d_0)=2^{-1}{\mathcal {L}}(d_0)\beta_{d_0}\in E(K_{d_0})^-_{\mathbb {Q}}$,
where $\beta_{d_0}\in E(K_{d_0})^-$ is any ${\mathbb {Z}}$-basis of the free part of $E(K_{d_0})^-$.
The height identity we need to check is
$$\widehat{h}(R_\chi)=4^{h_2(n)-1} {\mathcal {L}}(d_1)^2 {\mathcal {L}}(d_0)^2 \widehat{h}(\beta_{d_0}).$$
Assuming $L'(E_{K_n},\chi,1)\neq 0$. By the definitions of ${\mathcal {L}}(d_1)$ and ${\mathcal {L}}(d_0)$ in the introduction, the identity becomes
$$\widehat{h}(R_\chi)=
L'(E_{K_n},\chi,1)/(2^{2k(n)-2h_2(n)-2-a(n)} \Omega _{d_0, \infty} \Omega _{d_1, \infty}).
$$
Apply Theorem \ref{EGZ}, the explicit Gross-Zagier formula in the appendix, for $(E_{K_n},\chi_{d_0, d_1})$ and the morphism $\varphi \circ f_n$.
The proof is finished by computations similar to that in the proof of Theorem \ref{thm2.1}.
In the proof, we also see that ${\mathcal {L}}(n)\in {\mathbb {Q}}$. For example, the height formula
$$\widehat{h}(R_\chi)=4^{h_2(n)-1} {\mathcal {L}}(d_1)^2 {\mathcal {L}}(d_0)^2 \widehat{h}(\beta_{d_0})$$
actually implies that ${\mathcal {L}}(d_1) {\mathcal {L}}(d_0)\in {\mathbb {Q}}$. Setting $d_1=1$, we see that ${\mathcal {L}}(n)\in {\mathbb {Q}}$.
\subsubsection*{Proof of Proposition \ref{congruence}}
Here we prove Proposition \ref{congruence} which asserts that
\begin{multline*}
P(n) \equiv
\sum_{\substack{n=d_0d_1\cdots d_\ell\\
d_0\equiv 5,6, 7\pmod 8\\
d_1\equiv 1, 2, 3\pmod 8\\
d_i\equiv 1 \pmod 8, \ i>1}}\epsilon(d_0, d_1) \left(\prod_{i\geq 1} g(d_i)\right) Z(d_0)
\\
+i\sum_{\substack{n=d_0d_1\cdots d_\ell\\
(d_0,d_1,d_2)\equiv (5,3,2)\pmod 8\\
d_i\equiv 1 \pmod 8, \ i>2}} \left(\prod_{i\geq 1} g(d_i)\right) Z(d_0)
\qquad\mod\ 2A({\mathbb {H}}_n').
\end{multline*}
It suffices to prove that the above formula
(applied to every $P(d_0)$ below) and the formula in Theorem \ref{lg} (applied to every ${\mathcal {L}}(d_1)$ below) satisfies
$$Z(n)\equiv \sum_{\substack{n=d_0d_1\\ d_0\equiv 5, 6, 7\pmod 8\\
d_1\equiv 1,2 ,3 \pmod 8}} \epsilon(d_0, d_1){\mathcal {L}}(d_1)P(d_0)\qquad
\mod\ 2A({\mathbb {H}}_n').$$
We first treat the case $n\equiv 5,7\pmod8$. Then the formula simplifies as
\begin{multline*}
P(n) \equiv
\sum_{\substack{n=d_0d_1\cdots d_\ell\\
d_0\equiv 5, 7\pmod 8\\
d_1\equiv 1, 3\pmod 8\\
d_i\equiv 1 \pmod 8, \ i>1}}\epsilon(d_0, d_1) \left(\prod_{i\geq 1} g(d_i)\right) Z(d_0)
\qquad\mod\ 2A({\mathbb {H}}_n').
\end{multline*}
We need to check that
\begin{multline*}
Z(n)\equiv \sum_{\substack{n=d_0d_1\\ d_0\equiv 5, 7\pmod 8\\
d_1\equiv 1 ,3 \pmod 8}} \epsilon(d_0, d_1)
\left(\sum_{\substack {d_1=d_0'd_1'\cdots d_{\ell'}'\\
d_j'\equiv 1\pmod 8, \ j>0}}
\prod_{j\geq 0} g(d_j') \right)
\\
\left(\sum_{\substack{d_0=d_0''d_1''\cdots d_{\ell''}''\\
d_0''\equiv 5,7\pmod 8\\
d_1''\equiv 1, 3\pmod 8\\
d_k''\equiv 1 \pmod 8, \ k>1}}\epsilon(d_0'', d_1'')
\prod_{k\geq 1} g(d_k'')Z(d_0'')\right)
\qquad
\mod\ 2A({\mathbb {H}}_n').
\end{multline*}
The right-hand side is a ${\mathbb {Z}}$-linear combination of
$$
\epsilon(d_0, d_1) \epsilon(d_0'', d_1'')
\prod_{j=0}^{\ell'} g(d_j')
\prod_{k=1}^{\ell''} g(d_k'') Z(d_0'').
$$
Consider the multiplicity of this term in the sum.
Each appearance of such a terms gives a partition
$$
\{d_1',\cdots, d_{\ell'}', d_2'',\cdots, d_{\ell''}''\}
=\{d_1',\cdots, d_{\ell'}'\}\cup \{d_2'',\cdots, d_{\ell''}''\}.
$$
If this set is non-empty, the number of such partitions is even, and thus the contribution is zero in the congruence equation. Moreover, if $d_0'\equiv 1\pmod 8$ or $d_1''\equiv 1\pmod 8$, then we can put also put it into the partition deduce that the contribution of such terms is still zero.
Note that the contribution by $d_0=d_0''=n, d_1=1$ is the single term $Z(n)$.
Therefore, it is reduced to check
$$0\equiv \sum_{\substack{n=d_0d_1\\ d_0\equiv 5, 7\pmod 8\\
d_1\equiv 3 \pmod 8}} \epsilon(d_0, d_1) g(d_1)
\sum_{\substack{d_0=d_0''d_1''\\
d_0''\equiv 5, 7\pmod 8\\
d_1''\equiv 3\pmod 8}}\epsilon(d_0'', d_1'') g(d_1'') Z(d_0'')
\qquad
\mod\ 2A({\mathbb {H}}_n').$$
Rewrite it as
$$0\equiv {\sum_{n=d_0''d_1''d_1}}'
\epsilon(d_0''d_1'', d_1)
\epsilon(d_0'', d_1'') g(d_1) g(d_1'') Z(d_0'')
\qquad
\mod\ 2A({\mathbb {H}}_n').$$
Here the sum is over ordered decompositions $n=d_0''d_1''d_1$ which satisfy the original congruence conditions (with $d_0=d_0''d_1''$).
The ordered decomposition
$n=d_0''d_1''d_1$ corresponds to the ordered decomposition $n=d_0''d_1d_1''$ uniquely.
One checks in this case
$$
\epsilon(d_0''d_1'', d_1) \epsilon(d_0'', d_1'')
=\pm \epsilon(d_0''d_1, d_1'') \epsilon(d_0'', d_1).
$$
Then the sum is divisible by $2$.
Now we treat the case $n\equiv 6\pmod8$.
We need to check
\begin{multline*}
Z(n)\equiv \sum_{\substack{n=d_0d_1\\ d_0\equiv 6, 7\pmod 8\\
d_1\equiv 1, 2 \pmod 8}}
\left(\sum_{\substack {d_1=d_0'd_1'\cdots d_{\ell'}'\\
d_j'\equiv 1\pmod 8, \ j>0}}
\prod_{j\geq 0} g(d_j') \right)\cdot
\\
\left(\sum_{\substack{d_0=d_0''d_1''\cdots d_{\ell''}''\\
d_0''\equiv 5,6,7\pmod 8\\
d_1''\equiv 1, 2,3\pmod 8\\
d_k''\equiv 1 \pmod 8, \ k>1}}
\epsilon(d_0'', d_1'')
\prod_{k\geq 1} g(d_k'')Z(d_0'')
\,+\,
i\sum_{\substack{d_0=d_0''d_1''\cdots d_{\ell''}''\\
(d_0'',d_1'',d_2'')\equiv (5,3,2)\pmod 8\\
d_k''\equiv 1 \pmod 8, \ k>2}}
\prod_{k\geq 1} g(d_k'')Z(d_0'')
\right) \\
\qquad
\mod\ 2A({\mathbb {H}}_n').
\end{multline*}
Split the outer sum $d=d_0d_1$ into the case $(d_0,d_1)\equiv (6,1) \pmod8$
and the case $(d_0,d_1)\equiv (7,2) \pmod8$. We obtain three triple sums, since the conditions $(d_0,d_1)\equiv (7,2) \pmod8$ and $(d_0'',d_1'',d_2'')\equiv (5,3,2)\pmod 8$ do not hold simultaneously.
Similar to the case $n\equiv 5,7\pmod8$, the contribution of the terms with some $d_j'\equiv 1\pmod 8$ or some $d_k''\equiv 1\pmod 8$ is divisible by
$2$. In particular, for the case $d_1\equiv 1\pmod 8$, we are only left with $d_1=1$.
Then it is reduced to check
\begin{multline*}
0\equiv \sum_{\substack{n=d_0d_1\\ d_0\equiv 7\pmod 8\\
d_1\equiv 2 \pmod 8}}
\left( g(d_1) Z(d_0)
+\sum_{\substack{d_0=d_0''d_1''\\
d_0''\equiv 5\pmod 8\\
d_1''\equiv 3\pmod 8}}i\, g(d_1) g(d_1'') Z(d_0'')
\right)\\
+
\sum_{\substack{n=d_0''d_1''\\
d_0''\equiv 7\pmod 8\\
d_1''\equiv 2\pmod 8}} g(d_1'') Z(d_0'')
\, +\,
i \sum_{\substack{n=d_0''d_1''d_2''\\
(d_0'',d_1'',d_2'')\equiv (5,3,2)\pmod 8}}
g(d_1'')g(d_2'') Z(d_0'')
\qquad
\mod\ 2A({\mathbb {H}}_n').
\end{multline*}
This is true by obvious cancellations, which finishes the proof of the proposition.
\subsubsection*{Torsion points}
To prepare the proof of Theorem \ref{m3}, we present some results on torsion points of $A$. They will be the key to lower multiples of algebraic points.
Denote $F={\mathbb {Q}}(i)$.
Recall that we have fixed an identification
$A({\mathbb {C}})\cong{\mathbb {C}}/(1+i)O_F$,
which gives $A({\mathbb {C}})_{\mathrm{tor}}=A(F^{\mathrm{ab}})_{\mathrm{tor}}\cong F/(1+i)O_F$.
Under the identification, the complex conjugation on $A(F^{\mathrm{ab}})_{\mathrm{tor}}$ is given by the conjugation $i\mapsto -i$ on $F$,
The induced action of the Galois group ${\mathrm{Gal}}(F^{\mathrm{ab}}/F)$ on
$F/(1+i)O_F$ is given by multiplying by the composition
$${\mathrm{Gal}}(F^{\mathrm{ab}}/F)\stackrel{\sigma_{F}^{-1}}{\longrightarrow}
F^\times \backslash \widehat{F}^\times \cong (1+(1+i)^3\widehat O_F)^\times.$$
\begin{lem} \label{torsion1}
Over $F$, the elliptic curve $A_F$ is isomorphic to $E_F$. Moreover,
$${\mathbb {Q}}(A[4])={\mathbb {Q}}(\sqrt 2, i),\quad A({\mathbb {Q}}(i))=A[(1+i)^3].$$
\end{lem}
\begin{proof}
The results can be checked by explicit computations, but we include a theoretical proof.
For the first statement, consider the two 2-isogenies
$$
\varphi_F: A_F\longrightarrow E_F, \quad [1+i]: A_F\longrightarrow A_F.
$$
One checks that these two morphisms have the same kernel $\{0,\tau(1)\}$. It
follows that there is an isomorphism $A_F\to E_F$ carrying $[1+i]$ to
$\psi_F$.
Now we treat ${\mathbb {Q}}(A[4])$. It is easy to have $F={\mathbb {Q}}(A[2])\subset {\mathbb {Q}}(A[4])$, and thus ${\mathbb {Q}}(A[4])=F(A[4])$.
The Galois action of ${\mathrm{Gal}} (F^{\mathrm{ab}}/F)$ on $A[4]$ is given by
$$s_4:(1+(1+i)^3\widehat O_F)^\times
\longrightarrow (1+(1+i)^3O_{F_2})/(1+4O_{F_2}).$$
The field $F(A[4])$ is given by the subfield of $F^{\mathrm{ab}}$ fixed by
$\ker(s_4)=(1+4\widehat O_F)^\times$, which is the ring class field of $F$ of conductor $4$. The norm map
$$(1+(1+i)^3O_{F_2})/(1+4O_{F_2}) \simeq (1+4{\mathbb {Z}}_2)/(1+8{\mathbb {Z}}_2)$$
implies that $F(A[4])$ is equal to the ring class field ${\mathbb {Q}}(\zeta_8)$ of ${\mathbb {Q}}$.
For $A({\mathbb {Q}}(i))$, we first see that it is torsion since $A_1({\mathbb {Q}})\simeq A_{-1}({\mathbb {Q}})$ are torsion. We also have $A({\mathbb {Q}}(i))[p]=0$ for any odd prime $p$.
In fact, we can show that any non-trivial element of $A[p]$ has a residue field ramified above $p$ and cannot be defined over ${\mathbb {Q}}(i)$.
This argument will be used in Lemma \ref{torsion2} in a more complicated situation, so we omit it here.
Finally, we show that $A({\mathbb {Q}}(i))[2^\infty]=A[(1+i)^3]$.
Note that the stabilizer of any element $x_4$ of $A[4]\setminus A[(1+i)^3]$ is still $\ker(s_4)$. Then the residue field $F(x_4)$ is still ${\mathbb {Q}}(\zeta_8)$, and thus $x_4\notin A({\mathbb {Q}}(i))$. It follows that $A[4]({\mathbb {Q}}(i))=A[(1+i)^3]$.
\end{proof}
\begin{lem} \label{torsion3}
Let $\kappa\in {\mathrm{Gal}}({\mathbb {Q}}(\zeta_8)/{\mathbb {Q}})$ be the element sending $\zeta _8$ to $\zeta _8^5$. Then
$$(\kappa+1)A[4]=A[4][\kappa+1]=A[2], \qquad (\kappa+1)E[4]=E[4][\kappa+1]=E[2].$$
Here $(\kappa+1)A[4]$ and $A[4][\kappa+1]$ are respectively the image and the kernel of the map
$$\kappa+1: A[4]\longrightarrow A[4], \quad x\longmapsto x^\kappa+x.$$
\end{lem}
\begin{proof}
The results for $A$ and $E$ are equivalent since they are isomorphic over $F={\mathbb {Q}}(i)$.
Note that $\kappa$ acts on ${\mathbb {Q}}(\zeta_8)\subset F^{\mathrm{ab}}$ as $\sigma_{F,2}(\pm 1\pm 2i)=\sigma_{F,2}(\pm 2\pm i)$.
In terms of the CM theory, $\kappa$ acts on $A[4]\cong O_F/4O_F$ by multiplication by $-1\pm 2i\in 1+(1+i)^3O_{F_2}$. Then $\kappa+1$ acts by multiplication by $\pm 2i$.
The results are true.
\end{proof}
\begin{lem} \label{torsion2}
The torsion subgroup $A({\mathbb {H}}_n')_{\mathrm{tor}}=A[(1+i)^3]$ if $n$ is odd, and $A({\mathbb {H}}_n')_{\mathrm{tor}}=A[4]$ if $n$ is even.
\end{lem}
\begin{proof}
We prove the results by three steps.
\medskip
\noindent \emph{Step 1}. The group $A({\mathbb {H}}_n')[p]=0$ for any odd prime $p$.
Let $\wp$ be a prime ideal of $F$ above $p$.
The action of the Galois group on $A[\wp]$ gives a homomorphism
$$
{\mathrm{Gal}}(F^{\mathrm{ab}}/F)\longrightarrow {\mathrm{Aut}}_{O_F}(A[\wp])=(O_F/\wp)^\times.
$$
This map is surjective since it is given by
$$s_\wp:(1+(1+i)^3\widehat O_F)^\times
\longrightarrow O_{F_\wp}^\times \longrightarrow (O_F/\wp)^\times.$$
As a consequence, we have the following two properties:
\begin{enumerate}[(1)]
\item For any nonzero $x\in A[\wp]$, the residue field $F(x)=F(A[\wp])$ has degree $N(\wp)-1\geq 4$ over $F$.
\item The prime $\wp$ is totally ramified in $F(x)$.
\end{enumerate}
On the other hand, we claim that the ramification index of $\wp$ in ${\mathbb {H}}_n'$ is at most $2$.
In fact, ${\mathbb {H}}_n'$ is the composite of $L_n(i)$ and $H_{d_0}'$ for different $d_0$, where the extensions $L_n(i)/K_n$ and $H_{d_0}'/K_{d_0}$ do not involve ramification above
$p$. If follows that we only need to consider the ramification index of $p$ in
the composite of $K_n$ and $K_{d_0}$ for different $d_0$, which is at most $2$.
Combining the claim and the properties (1) and (2),
we see that $F(x)$ cannot be contained in ${\mathbb {H}}_n'$. In other words, $A({\mathbb {H}}_n')[\wp]=0$. Then $A({\mathbb {H}}_n')[p]=0$.
Hence, $A({\mathbb {H}}_n')_{\mathrm{tor}}=A({\mathbb {H}}_n')[2^\infty]$.
\medskip
\noindent \emph{Step 2}.
For any $n\equiv 5,6,7\pmod8$, $A({\mathbb {H}}_n')[2^\infty]\subset A[4]$.
Note that $A({\mathbb {H}}_n')[2^\infty]$ is a finite $O_F$-module, so it must be of the form $A[(1+i)^e]$ for some positive integer $e$. Thus it suffices to prove $|A({\mathbb {H}}_n')[2^\infty]|\leq 16$.
The idea is to use the reduction map to obtain the bound.
Take a prime number $p\nmid (2n)$, and let $v$ be a place of ${\mathbb {H}}_n'$ above $p$. Denote by $k(v)$ the residue field of $v$. The reduction map gives an injection
$$
A({\mathbb {H}}_n')[2^\infty] \longrightarrow A(k(v))[2^\infty].
$$
We will choose $p$ carefully to get an easy bound on the right-hand side.
In fact, we choose $p$ satisfying the following properties:
\begin{enumerate}[(1)]
\item $p\equiv 3\pmod8$.
\item $p$ is inert in $K_{d_0}$ for any positive factor $d_0$ of $n$ with $d_0\equiv 5,6\pmod8$.
\end{enumerate}
Assuming the existence of such $p$, we first see how it implies the desired bound. The proof consists of two steps.
The first step is to show that $k(v)={\mathbb {F}}_{p^2}$.
Denote by
$w$ the restriction of $v$ to $L_n(i)$, and
$v_{d_0}$ the restriction of $v$ to $H_{d_0}'$. It is easy to see that the residue field $k(w)={\mathbb {F}}_{p^2}$. To prove $k(v)={\mathbb {F}}_{p^2}$, it suffices prove that $k(v_{d_0})\subset {\mathbb {F}}_{p^2}$ for any $d_0\equiv 5,6\pmod8$.
Note that $p$ is inert in $K_{d_0}$. Then it suffices to check that $pO_{K_{d_0}}$ is totally split in $H_{d_0}'$.
By Lemma \ref{CM point1}, $H_{d_0}'$ is contained in the ring class field $H_{d_0,4}$ of conductor $4$.
We claim that $pO_{K_{d_0}}$ is totally split in $H_{d_0,4}$.
In fact, by the class field theory, it is equivalent to the easy fact that the image of $p$ under the composition
$$
K_{d_0, p}^\times \longrightarrow \widehat K_{d_0}^\times \longrightarrow
K_{d_0}^\times\backslash \widehat K_{d_0}^\times/(\widehat {\mathbb {Z}}+4 \widehat O_{K_{d_0}})^\times={\mathrm{Gal}}(H_{d_0,4}/K_{d_0})
$$
is trivial.
The second step is to show that $|A({\mathbb {F}}_{p^2})[2^\infty]|\leq 16$.
This is done by explicit computation. In fact, by the choice $p\equiv 3\pmod8$, we see that $A$ has supersingular reduction at $p$.
Then the eigenvalues of the absolute Frobenius $\varphi_p$ on the Tate modules of $A$ are $\pm\sqrt{-p}$, so the eigenvalues of $\varphi_p^2$
are $-p, -p$. It follows that
$$
|A({\mathbb {F}}_{p^2})|=p^2+1-(-p-p)=(p+1)^2.
$$
By the choice $p\equiv 3\pmod8$, we have $|A({\mathbb {F}}_{p^2})[2^\infty]|= 16$.
This finishes the second step.
Finally, we check the existence of the prime $p$ satisfying the two conditions.
The second condition is equivalent to $(-d_0/p)=-1$, which becomes
$(d_0/p)=1$ by the first condition. Then we choose $p$ satisfying:
\begin{enumerate}[(a)]
\item $p\equiv 3\pmod8$.
\item $(\ell/p)=1$ for any prime factor $\ell$ of $n$ with $\ell\equiv 1\pmod4$.
\item $(\ell/p)=-1$ for any prime factor $\ell$ of $n$ with $\ell\equiv -1\pmod4$.
\end{enumerate}
It is easy to check that it gives $(d_0/p)=1$ for any $d_0\equiv 5,6\pmod8$.
Now the existence of $p$ satisfying (a), (b) and (c) is just a combination of the quadratic reciprocity law, the Chinese remainder theorem, and Dirichlet's density theorem.
\medskip
\noindent \emph{Step 3}.
If $n$ is odd, then $A({\mathbb {H}}_n')[2^\infty]= A[(1+i)^3]$.
We will prove $\sqrt2\notin {\mathbb {H}}_n'$, which implies
$A({\mathbb {H}}_n')_{\mathrm{tor}}=A[(1+i)^3]$ by Lemma \ref{torsion1}.
To prove $\sqrt2\notin {\mathbb {H}}_n'$, note that ${\mathbb {H}}_n'$ is the composite of $L_n(i)$ and $H_{d_0}'$ for some $d_0\equiv 5\pmod8$. Let $v$ be a place of ${\mathbb {H}}_n'$ above 2, and $v_{d_0}$ the restriction to $H_{d_0}'$. It suffices to show $\sqrt2\notin ({\mathbb {H}}_n')_v$.
Consider the ramification of $v$ above $2$.
Note that $({\mathbb {H}}_n')_v$ is the composite of ${\mathbb {Q}}_2(i)$ and $(H_{d_0}')_{v_{d_0}}$ for all related $d_0$. By Proposition \ref{CM point1},
$(H_{d_0}')_{v_{d_0}}$ is unramified over
$N_{d_0,4}=(M_{d_0,4})^{\sigma_{1+2\varpi_{d_0}}}$, where $\varpi_{d_0}=(\sqrt{-d_0}-1)_2$ and $M_{d_0,4}$ is the ring class field of ${\mathbb {Q}}_2(\sqrt{-d_0})$ of conductor $4$.
We claim that $N_{d_0,4}$ is independent of
$d_0$.
In fact, fix an isomorphism ${\mathbb {Q}}_2(\sqrt{-d_0})\cong{\mathbb {Q}}_2(\sqrt{-5})$, which induces an isomorphism $M_{d_0,4}\cong M_{5,4}$. Note that $1+2\varpi_{d_0}$ and $1+2\varpi_{5}$ have the same image in $K_{5,2}^\times/({\mathbb {Z}}_2+4O_{K_{5,2}})^\times$, so their actions on $M_{5,4}$ are the same. It follows that $N_{d_0,4}=N_{5,4}$.
Note that $i\in N_{5,4}$ and $\sqrt2\notin N_{5,4}$.
Therefore, $({\mathbb {H}}_n')_v$ is unramified over $N_{5,4}$. To prove
$\sqrt2\notin ({\mathbb {H}}_n')_v$, it suffices to prove that $N_{5,4}(\sqrt2)=M_{5,4}$ is ramified over $N_{5,4}$. This is clear since ${\mathrm{Gal}}(M_{5,4}/N_{5,4})$ is generated by $\sigma_{1+2\varpi_{5}}$ with $1+2\varpi_{5}\in O_{K_{5,2}}^\times$.
\end{proof}
\subsubsection*{Proof of Theorem \ref{m3}: representative}
By definition, $\Phi_0\subset \Phi$. Recall that
$$P_\chi=\sum_{t\in \Phi} f_n(P_n)^{t} \chi(t).$$
Summing over all characters $\chi: {\mathrm{Cl}}_n\cong {\mathrm{Cl}}_n'/\langle\sigma\rangle\rightarrow \{\pm 1\}$.
We have
$$\sum_{\chi: {\mathrm{Cl}}_n\rightarrow \{\pm 1\}} P_\chi
=\sum_{t\in \Phi} f_n(P_n)^{t} \sum_{\chi: {\mathrm{Cl}}_n\rightarrow \{\pm 1\}} \chi(t).$$
As in the case $n\equiv 1,2,3\pmod8$, apply the character formula
$$ \sum_{\chi: {\mathrm{Cl}}_n\rightarrow \{\pm 1\}} \chi(t)
=2^{h_2(n)}\delta _{2{\mathrm{Cl}}_n}(t), \qquad t\in {\mathrm{Cl}}_n.$$
Here $h_2(n)=\dim _{{\mathbb {F}}_2}{\mathrm{Cl}}_n/2{\mathrm{Cl}}_n$.
Then we obtain
$$\sum_{\chi: {\mathrm{Cl}}_n\rightarrow \{\pm 1\}} P_\chi
=2^{h_2(n)} \sum_{t\in \Phi_0} f_n(P_n)^{t}
=2^{h_2(n)} Z(n).$$
This is an equality in $A(H_n')$.
By Theorem \ref{GZ}, the equality gives
$$\sum_{\substack{n=d_0d_1\\ d_0\equiv 5, 6, 7\pmod 8\\
d_1\equiv 1,2 ,3 \pmod 8,\ d_1>0}}
\epsilon (d_0, d_1) 2^{h_2(n)}{\mathcal {L}}(d_1) {\mathcal {P}}(d_0)
=2^{h_2(n)} Z(n)\
\in\ A(H_n'(i))\otimes_{{\mathbb {Z}}} {\mathbb {Q}}.$$
We end up with
$$\sum_{\substack{n=d_0d_1\\ d_0\equiv 5, 6, 7\pmod 8\\
d_1\equiv 1,2 ,3 \pmod 8,\ d_1>0}}
\epsilon (d_0, d_1) {\mathcal {L}}(d_1) {\mathcal {P}}(d_0)
= Z(n)\ \in\ A(H_n'(i))\otimes_{{\mathbb {Z}}} {\mathbb {Q}}.$$
Then we have
$${\mathcal {P}}(n)=Z(n)-\sum_{\substack{n=d_0d_1\\ d_0\equiv 5, 6, 7\pmod 8\\
d_1\equiv 1,2 ,3 \pmod 8,\ d_1>1}} \epsilon(d_0, d_1){\mathcal {L}}(d_1){\mathcal {P}}(d_0).$$
It follows that ${\mathcal {P}}(n)$ and $P(n)$ satisfy the same iteration formula (in different groups).
Therefore, $P(n)$ represents ${\mathcal {P}}(n)$.
This proves the first statement of the theorem.
\subsubsection*{Proof of Theorem \ref{m3}: part (1)}
Here we prove part (1) of the theorem.
Let $R(n)$
(resp. $R(d_0, d_1)$)
be the image of $P(n)$
(resp. $P(d_0, d_1)=P_{\chi_{d_0, d_1}}$)
under the $2$-isogeny from $A$ to $E$. Then $R(n)\in E({\mathbb {H}}_n')$ and
$R(d_0, d_1)\in E(H_n')$.
We need to prove that $2R(n)\in E(K_n)^-$.
Note that in Lemma \ref{multiples1} we have already checked $4P(n,1)\in A(K_n)^-$ and thus $4R(n,1)\in E(K_n)^-.$ To relate to $2R(n)$, we have the following simple connection.
\begin{lem} \label{multiples2}
$$4 P(n,1)=\pm 2^{2+h_2(n)} P(n), \qquad
4 R(n,1)=\pm 2^{2+h_2(n)} R(n).$$
\end{lem}
\begin{proof}
By Theorem \ref{GZ},
$$P(n,1)=\pm 2^{h_2(n)} P(n)\
\in\ A({\mathbb {H}}_n')\otimes_{{\mathbb {Z}}} {\mathbb {Q}}.$$
Then
$$P(n,1)\mp 2^{h_2(n)} P(n)\
\in\ A({\mathbb {H}}_n')_{\mathrm{tor}}= A({\mathbb {H}}_n')[4].$$
Here the last identity follows from Lemma \ref{torsion2}.
\end{proof}
Before proving part (1) of the theorem,
we introduce some notations on fields.
Recall that $H_n$ is the Hilbert class field of $K_n={\mathbb {Q}}(\sqrt{-n})$ and $H_n'=H_n(f_n(P_n))$. Recall that $K_n'=K_n, K_n(i), K_n$ for $n\equiv 5, 6,7\pmod 8$ respectively.
Let $L_n\subset H_n$ be the genus field of $K_n$; that is, $L_n$ is subfield of $H_n$ fixed by the subgroup $2{\mathrm{Cl}}_n$ of ${\mathrm{Cl}}_n={\mathrm{Gal}}(H_n/K_n)$. Define
$L_n'=L_n, L_n(i), L_n$ for $n\equiv 5, 6,7\pmod 8$ respectively.
Then $L_n'=L_nK_n'$.
Set $K_n''=K_n(E[4](L_n'))$, i.e. $K_n''=K_n(i), K_n(\sqrt{2}, i), K_n$ for $n\equiv 5, 6,7\pmod 8$ respectively.
First, we prove $2R(n)\in E(K_n'')$.
Consider the image of $4R(n, 1)=\pm2^{h_2(n)+2}R(n)$ under the (injective) Kummer map $$\delta: E(K_n'')/2^{h_2(n)+2}E(K_n'') \longrightarrow H^1(K_n'', E[2^{h_2(n)+2}]),$$ and the inflation-restriction exact sequence
$$1\longrightarrow {\mathrm{Hom}} ({\mathrm{Gal}}(L_n'/K_n''), E[4](K_n''))\longrightarrow H^1(K_n'', E[2^{h_2(n)+2}])
\longrightarrow H^1(L_n', E[2^{h_2(n)+2}]).$$
(Note that $E[2^\infty](L_n')=E[4](K_n'')$.)
The image of $\delta(2^{h_2(n)+2}R(n))$ in $H^1(L_n', E[2^{h_2(n)+2}])$ is 0, since it is 0 in $E(L_n')/2^{h_2(n)+2}E(L_n')$.
Then $\delta(2^{h_2(n)+2}R(n))$ lies in ${\mathrm{Hom}}({\mathrm{Gal}}(L_n'/K_n''), E[4](K_n''))$, which has exponent 2 since ${\mathrm{Gal}}(L_n'/K_n'')$ has exponent 2.
It follows that $\delta(2^{h_2(n)+3}R(n))=0$. Thus
$$2^{h_2(n)+3}R(n)\in 2^{h_2(n)+2}E(K_n''), \qquad
2R(n)\in E(K_n'')+E[2^\infty](L_n')=E(K_n'').$$
Second, we prove $2R(n)\in E(K_n)^-$ for $n\equiv 7\pmod 8$.
This is the simplest case, but it illustrates the key idea.
In this case, we already have $2R(n)\in E(K_n)$, and we need to prove
$2\overline{R(n)}=-2R(n)$.
By Lemma \ref{multiples1} and Lemma \ref{multiples2},
$$2^{h_2(n)+2}(R(n)+\overline{R(n)})=\pm (4R(n, 1)+4\overline{R(n, 1)})=0.$$
Then $R(n)+\overline{R(n)}\in E(K_n)[2^\infty]=E[2]$ is killed by 2.
The result follows.
Third, we prove $2R(n)\in E(K_n)^-$ for $n\equiv 5\pmod 8$.
It suffices to prove $2R(n)\in E(K_n)$, since the process from $E(K_n)$ to $E(K_n)^-$ is the same as the case $n\equiv 7\pmod 8$.
We already know $2R(n)\in E(K_n(i))$.
Denote by $\xi\in {\mathrm{Gal}}(K_n(i)/K_n)$ the unique non-trivial element, and take a lifting of $\xi$ to ${\mathrm{Gal}}({\mathbb {H}}_n'/K_n)$, which we still denote by $\xi$.
By Lemma \ref{multiples1} and Lemma \ref{multiples2},
$$2^{h_2(n)+2}(P(n)^\xi-P(n))=\pm (4P(n, 1)^\xi-4 P(n, 1))=0.$$
Then $P(n)^\xi-P(n)\in A(K_n(i))[2^\infty]=A[(1+i)^3]$. Note that $A[(1+i)^3]$ is
exactly killed by $2\varphi:A\to E$. We have $2R(n)^\xi-2R(n)=0$, and thus $2R(n)\in E(K_n)$.
For the case $n\equiv 6\pmod 8$, we need the following simple result.
\begin{lem}
For any $n\equiv 5,6,7\pmod 8$, $R(n)\in E(L_n(i))$.
\end{lem}
\begin{proof}
By the recursion formula, it suffices to prove $\varphi(Z(n))\in E(L_n')$ for any $n\equiv 5,6,7\pmod 8$.
By Theorem \ref{CM point2}, $\varphi(z_n)$ is invariant under the action of
$\sigma$. Here $\sigma$ is described right after Proposition \ref{CM point1}.
If $n\equiv 5,7\pmod 8$, then $\varphi(z_n)$ is defined over $H_n$, and thus $\varphi(Z(n))$ is defined over $L_n$.
If $n\equiv 6\pmod 8$, then $\varphi(z_n)$ is defined over $H_n(i)$, and thus $\varphi(Z(n))$ is defined over $L_n(i)$.
\end{proof}
Finally, we prove $2R(n)\in E(K_n)^-$ for $n\equiv 6\pmod 8$.
We already know $2R(n)\in E(K_n(\sqrt{2}, i))$.
It suffices to prove $2R(n)\in E(K_n(i))$, since the process from $E(K_n(i))$ to $E(K_n)^-$ is the same as that for the case $n\equiv 5\pmod 8$.
Let $\kappa \in {\mathrm{Gal}}(K_n(\sqrt{2}, i)/K_n(i))$ be the unique non-trivial element, and take any lifting of $\kappa$ in ${\mathrm{Gal}}(L_n(i)/K_n(i))$, still denoted by $\kappa$.
We need to show that $(2R(n))^{\kappa}=2R(n)$.
Note that $\kappa^2=1$ since ${\mathrm{Gal}}(L_n(i)/K_n(i))$ has exponent 2.
By Lemma \ref{multiples1} and Lemma \ref{multiples2},
$$2^{h_2(n)+2}(R(n)^\kappa-R(n))=\pm (4R(n, 1)^\kappa-4R(n, 1))=0,$$
so $R(n)^\kappa-R(n)$ lies in
$E[4][\kappa+1]=\{x\in E[4]:x^\kappa+x=0 \}$.
By Lemma \ref{torsion3},
$E[4][\kappa+1]=E[2]$. It follows that $2(R(n)^\kappa-R(n))=0$.
The proof of part (1) is complete.
\subsubsection*{Proof of Theorem \ref{m3}: part (2)}
We start with some Galois-theoretic preparation.
Denote by
$$r: {\mathbb {Q}}^\times \backslash {\mathbb {A}}^\times \longrightarrow {\mathrm{Gal}} ({\mathbb {Q}}^{\mathrm{ab}}/{\mathbb {Q}})$$
the Artin map over ${\mathbb {Q}}$.
Then $c=r_\infty (-1)$ is the complex conjugation.
Define $\beta_1,\beta_2\in {\mathrm{Gal}}({\mathbb {Q}}^{\mathrm{ab}}/{\mathbb {Q}})$ by
$$\beta_1=r _\infty (-1)r _2(-2), \qquad
\beta_2=r _\infty (-1)r _2 (6).
$$
Let $\beta_1', \beta_2'\in {\mathrm{Gal}}(\overline{\mathbb {Q}}/{\mathbb {Q}})$ be any liftings of $\beta_1,\beta_2$.
In the following, we take the convention that $(\gamma+1)R$ means $\gamma(R)+1$ for any $\gamma\in {\mathrm{Gal}}(\overline{\mathbb {Q}}/{\mathbb {Q}})$.
The key of the proof is the following lemma.
\begin{lem} \label{betaaction}
\begin{enumerate}[(1)]
\item For any $n\equiv 5\pmod 8$,
$$ Z(n)^{\beta_1'+1}=Z(n)^{\beta_2'+1} \ \in \ g(n)\ \tau(\frac{1-i}2)+{\mathbb {Z}}\, \tau(1).$$
\item For any $n\equiv -2\pmod{16}$,
$$
Z(n)^{\beta_1'+1} \ \in \ g(n)\ \tau(\frac{i}2)+{\mathbb {Z}}\, \tau(1).$$
\item For any $n\equiv 6\pmod{16}$,
$$
Z(n)^{\beta_2'+1} \ \in \ g(n)\ \tau(\frac{i}2)+{\mathbb {Z}}\, \tau(1).$$
\item For any $n\equiv 7\pmod 8$,
$$Z(n)^{\beta_1'+1}=Z(n)^{\beta_2'+1}= g(n)\ \tau(\frac12).$$
\end{enumerate}
\end{lem}
\begin{proof}
Recall that after Proposition \ref{CM point1} we have introduced $\sigma\in 2{\mathrm{Cl}}_n'$ which gives
$$
{\mathrm{Cl}}'_n/\langle\sigma\rangle\cong{\mathrm{Cl}}_n, \quad
(2{\mathrm{Cl}}'_n)/\langle\sigma\rangle\cong2{\mathrm{Cl}}_n.
$$
Note that the genus field $L_n$ is the subfield of $H_n$ fixed by $2{\mathrm{Cl}}_n$.
It follows that the subfield of $H_n'$ fixed by $2{\mathrm{Cl}}_n'$ is $L_n, L_n(i), L_n$ according to $n\equiv 5,6,7\pmod 8$.
The field $L_n(i)={\mathbb {Q}}(i, \sqrt{d}:d|n)$ is a subfield of ${\mathbb {Q}}^{\mathrm{ab}}$. It is easy to check that the action of the involved $\beta_j'$ on $L_n(i)$ is the same as that of
$\sigma_{\varpi}\circ c$ in all the four cases of the lemma.
For example, if $n\equiv 5\pmod 8$, then
$\sigma_{\varpi}$ acts on $L_{n}(i)$ as
$r_2(N_{K_{n}/{\mathbb {Q}}}(\varpi))=r_2(n+1)=r_2(-2)$.
As a consequence, we claim that
$$
Z(n)^{\beta_j'}-Z(n)^{\sigma_{\varpi}\circ c} \in {\mathbb {Z}}\, \tau(1)
$$
in all four cases.
In fact, denote
$\alpha=\sigma_{\varpi}\circ c \circ \beta_j'^{-1}$, viewed as an element of
${\mathrm{Cl}}_n'={\mathrm{Gal}}(H_n'/K_n')$.
It suffices to show
$$Z(n)^{\alpha}-Z(n) \in {\mathbb {Z}}\, \tau(1).$$
Since $\alpha$ acts trivially on $L_n(i)$, we see that $\alpha\in 2{\mathrm{Cl}}_n'$.
Recall the definition
$$Z(n)=\sum_{t\in \Phi_0} z_n^{t}, \qquad
Z(n)^\alpha=\sum_{t\in \alpha\Phi_0} z_n^{t}.$$
Here $\Phi_0$ is a set of representatives of $2{\mathrm{Cl}}_n=(2{\mathrm{Cl}}'_n)/\langle\sigma\rangle$ in $2{\mathrm{Cl}}_n'$. Then $\alpha\Phi_0$ is also a set of representatives of $2{\mathrm{Cl}}_n$ in $2{\mathrm{Cl}}_n'$.
Write $\Phi_0=\{t_i:i=1,\cdots, g(n)\}$.
Then $\alpha\Phi_0=\{\sigma_i t_i: i=1,\cdots, g(n)\}$,
where each $\sigma_i\in \langle\sigma\rangle$.
By Theorem \ref{CM point2}, we see that $z_n^\sigma=z_n$ or $z_n^\sigma=z_n+\tau(1)$. It follows that
$$
Z(n)^\alpha-Z(n)=\sum_{t_i\in \Phi_0} (z_n^{\sigma_i}-z_n)^{{t_i}}
\in {\mathbb {Z}}\, \tau(1).$$
Therefore, the result for $Z(n)^{\beta_j'+1}$ becomes that for
$Z(n)^{\sigma_{\varpi}\circ c+1}$, which can be checked easily by Theorem \ref{CM point2} for $n\equiv 5,6\pmod 8$.
In the case $n\equiv 7\pmod 8$, $H_n'=H_n$ and thus $Z(n)$ is already defined over $L_n$. Then
$$
Z(n)^{\beta_j'}=Z(n)^{\sigma_{\varpi}\circ c}
=Z(n)^{\sigma_{\varpi^5}\circ c}.
$$
Here the last identity holds since the Galois group ${\mathrm{Gal}}(L_n(i)/{\mathbb {Q}})$ has exponent 2. Then the result for $Z(n)^{\beta_j'+1}$ still follows from Theorem \ref{CM point2}.
\end{proof}
Now we prove part (2) of Theorem \ref{m3}.
Assume that $P(n)=P+t$ for some $P\in A(K_n)^-$ and $t\in A[4]$.
Define $\beta\in {\mathrm{Gal}}({\mathbb {Q}}^{\mathrm{ab}}/{\mathbb {Q}})$ by
$$\beta=\begin{cases}
r _\infty (-1)r _2(-2)&\text{if $n\equiv 5,7\pmod 8$ or $n\equiv -2 \pmod{16}$},
\\
r _\infty (-1)r _2 (6)&\text{if $n\equiv 6\pmod{16}$}.
\end{cases}
$$
Let $\beta'\in {\mathrm{Gal}}(\overline{\mathbb {Q}}/{\mathbb {Q}})$ be any liftings of $\beta$.
Explicit calculation shows that $\beta$ acts on $K_n$ by $\sqrt{-n}\mapsto-\sqrt{-n}$. It follows that $P(n)^{\beta}+P(n)=t^\beta+t$.
We first treat the case $n\equiv 5,7\pmod 8$.
Then $t\in A({\mathbb {H}}_n')[4]=A({\mathbb {Q}}(i))$ by Lemma \ref{torsion2}. Note that $\beta$ acts on ${\mathbb {Q}}(i)$ trivially. Then
$P(n)^\beta+P(n)=2t \in {\mathbb {Z}}\tau(1).$
Apply $\beta'+1$ to both sides of Proposition \ref{congruence}.
By Lemma \ref{betaaction},
we have
$$\left(i^{\frac{n-1}{2}}
\sum_{\substack{n=d_0d_1\cdots d_\ell\\
d_0\equiv 5\pmod 8\\
d_1\equiv 1, 3\pmod 8\\
d_i\equiv 1 \pmod 8,\ i>1}} \prod_i g(d_i) \right)\tau(\frac{1-i}{2}) +\left(\sum_{\substack{n=d_0d_1\cdots d_\ell\\
d_0\equiv 7\pmod 8\\
d_1\equiv 1, 3\pmod 8\\
d_i\equiv 1 \pmod 8,\ i>1}}\prod_i g(d_i) \right) \tau(\frac12)\in 2A({\mathbb {H}}_n')^{\beta'+1}+{\mathbb {Z}} \tau(1).$$
It follows that the contribution from $2A({\mathbb {H}}_n')^{\beta'+1}$ is torsion, which is contained in
$$2A({\mathbb {H}}_n')_{{\mathrm{tor}}}= 2 A({\mathbb {Q}}(i)) ={\mathbb {Z}}\, \tau(1).$$
Then the left-hand side lies in ${\mathbb {Z}}\, \tau(1)$.
Thus the coefficients in both of the brackets must be even.
\
Now we treat the case $n\equiv 6\pmod 8$.
In this case we can only have the weaker result
$$P(n)^\beta+P(n)\in (\beta+1)A[4]=A[2]$$
by Lemma \ref{torsion3}.
Apply $\beta'+1$ to Proposition \ref{congruence} again.
We get
\begin{multline*}
\left(\sum_{\substack{n=d_0d_1\cdots d_\ell\\
d_0\equiv 6\pmod 8\\
d_i\equiv 1 \pmod 8,\ i>0}} \prod_i g(d_i)\right)\tau(\frac i2)+ \left(\sum_{\substack{n=d_0d_1\cdots d_\ell\\
d_0\equiv 7\pmod 8\\
d_1\equiv 2\pmod 8\\
d_i\equiv 1 \pmod 8,\ i>1}} \prod_i g(d_i)\right) \tau(\frac12)
\\
+ \left(i\sum_{\substack{n=d_0d_1\cdots d_\ell\\
(d_0,d_1,d_2)\equiv (5,3,2)\pmod 8\\
d_i\equiv 1 \pmod 8, \ i>2}}\prod_{i\geq 1} g(d_i)\right)
\tau(\frac{1-i}{2})
\ \in\ 2A({\mathbb {H}}_n')^{\beta'+1}+A[2].
\end{multline*}
The contribution of $2A({\mathbb {H}}_n')^{\beta'+1}$ is a torsion point, and thus lies in $2A[4]=A[2]$.
Then the left-hand side lies in $A[2]$.
It follows that the first two coefficients have the same parity, which is the same as the assertion of the theorem in this case.
This finishes the proof of Theorem \ref{m3}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,828 |
{"url":"http:\/\/icieve.conference.upi.edu\/pages\/abstracts1.php","text":"ICIEVE 2015 International Conference on Innovation in Engineering and Vocational Education\n 1 \u201cPengaruh Persepsi Tentang Karir Pekerjaan dan Motivasi Berprestasi Terhadap Pelaksanaan Praktik Industri Mahasiswa Program Studi Pendidikan Teknologi Agroindustri UPI\u201dSyahriandi Akbari Siregar \u201cPengaruh Persepsi Tentang Karir Pekerjaan dan Motivasi Berprestasi Terhadap Pelaksanaan Praktik Industri Mahasiswa Program Studi Pendidikan Teknologi Agroindustri UPI\u201d Oleh : Syahriandi Akbari Siregar 1302879 ABSTRAK Sebagai lembaga pendidikan kejuruan Prodi Pendidikan Teknologi Agroindustri menyusun suatu program praktik industri pada kurikulumnya. Dalam pelaksanaan praktik industri persoalan yang dihadapi oleh Prodi Pendidikan Teknologi Agroindustri terdapat beberapa kendala dalam pelaksanaan praktik industri termasuk pada suatu persepsi tentang karir pekerjaan dan motivasi berprestasi mahasiswa. Tujuan penelitian ini (1) mengetahui pengaruh persepsi tentang karir pekerjaan mahasiswa terhadap keberhasilan pelaksanaan praktik industri pada Program Studi Pendidikan Teknologi Agroindustri, (2 mengetahui pengaruh motivasi berprestasi mahasiswa terhadap keberhasilan pelaksanaan praktik industri, (3), mengetahui apakah persepsi tentang karir pekerjaan mahasiswa serta motivasi berprestasi mahasiswa berpengaruh secara bersama-sama terhadap pelaksanaan praktik industri dan (4) mengetahui hubungan persepsi tentang karir pekerjaan mahasiswa dan motivasi berprestasi mahasiswa. Desain penelitian yang digunakan adalah penelitian deskriptif dengan menggunakan teknik penelitian regresi dan korelasional dengan penekanan pengaruh dan hubungan antara dua atau lebih variabel. Sampel penelitian adalah mahasiswa Prodi Pendidikan Teknologi Agroindustri UPI angkatan 2011 yaitu sebanyak 50 mahasiswa. Instrumen yang digunakan untuk pengumpulan data menggunakan kuesioner (angket). Dari hasil pengolahan data penelitian didapatkan hasil analisis linieritas regresi dan korelasi variabel persepsi tentang karir pekerjaan ( X_1) dan pelaksanaan praktek industri (Y) berpengaruh signifikan dengan persamaan regresi (Y ) \u0302= 28,51 + 0.43 dan tingkat hubungan sebesar 0,43, analisis linieritas regresi dan korelasi variabel motivasi berprestasi( X_2) dan pelaksanaan praktek industri (Y) berpengaruh siginifikan dengan persamaan linieritas (Y ) \u0302= 24,42 + 0.51 dan tingkat hubungannya sebesar 0,52, analisis regresi ganda dan korelasi ganda variabel persepsi tentang karir pekerjaan ( X_1) motivasi berprestasi (X_2) pelaksanaan praktek industri (Y) berpengaruh siginifikan dengan persamaan regresi (Y ) \u0302= 0,35 + 0,44X_1 + 0,56 X_2 dan tingkat hubungannya sebesar 0,59. analisis korelasi variabel persepsi tentang karir pekerjaan( X_1) dan motivasi berprestasi (X_2) pengaruh siginifikan dan tingkat hubungannya sebesar 0,31 Kesimpulan penelitian ini adalah persepsi tentang karir pekerjaan mahasiswa berpengaruh secara signifikan terhadap praktik industri mahasiswa Prodi Pendidikan Teknologi Agroindustri dengan kategori hubungan cukup kuat , motivasi berprestasi mahasiswa berpengaruh secara signifikan terhadap pelaksanaan praktik industri mahasiswa Prodi Pendidikan Teknologi Agroindustri dengan kategori hubungan cukup kuat, persepsi tentang karir pekerjaan dan motivasi berprestasi berpengaruh secara bersama-sama terhadap pelaksanaan praktik industri mahasiswa Prodi Pendidikan Teknologi Agroindustri kategori hubungan cukup kuat dan persepsi tentang karir pekerjaan berhubungan dengan motivasi berprestasi mahasiswa Prodi Pendidikan Teknologi Agroindustri kategori hubungan rendah. Kata Kunci : Persepsi Tentang Karir Pekerjaan, Motivasi Berprestasi dan Pelaksanaan Praktik Industri. Topic: Engineering Education 2 A Comparative of Business Process Modelling TechniquesIrene R.H.T. Tangkawarow, Jimmy Waworuntu In this era, there is a lot of business process modeling techniques. This article is the research about differences of business process modeling techniques. It will cover 4 different techniques known as: Data Flow Diagrams (DFD), Business Process Modeling Notation (BPMN), Activity Diagrams, and Integration Definition for Function Modeling (IDEF0). For each technique will explain about the definition and the structure. To support, the research will provide also for each technique an example related to the case in Somerleyton Animal Park. Each technique will end with the advantages and disadvantages. The final conclusion will give which business process modeling technique is recommended. Topic: Engineering and Technology Innovation 3 A Model of Trait and Factor Career Counseling to Enhance Students Career Maturity State Vocational Schools in BandungDr. Sudjani, M.Pd. ABSTRAK Model Konseling Karir Trait and Factor untuk Meningkatkan Kematangan Karir Siswa SMKN di Kota Bandung Dr. Sudjani, M.Pd. Masalah penelitian ini dilatarbelakangi oleh kenyataan bahwa sebagian besar siswa SMKN di Kota Bandung mengalami kesulitan dalam memilih dan menentukan pilihan karir. Kegiatan bimbingan dan konseling karir belum membantu meningkatkan kematangan karir, sehingga siswa tidak memiliki kesiapan untuk melaksanakan tugas-tugas perkembangan karirnya. Tujuan penelitian ini menghasilkan model konseling karir trait and factor untuk meningkatkan kematangan karir siswa SMK. Metode penelitian ini menggunakan gabungan kualitatif dan kuantitatif (mixed methods design) termasuk tipe exploratory mixed methode. Penelitian ini dilakukan dengan menggunakan pendekatan penelitian dan pengembangan. Tahapan penelitian dan pengembangan meliputi: (1) studi pendahuluan, (2) pengembangan model, (3) validasi rasional model, dan (4) validasi empirik model. Disain penelitian menggunakan disain kuasi eksperimental dengan jenis rancangan Pretest-Posttest Nonequivalent Group Design tanpa acak. Teknik sampling menggunakan teknik two stage cluster sampling yang populasinya seluruh siswa SMKN Program Keahlian Teknik Bangunan di Bandung. Hasil penelitian menunjukkan sebagian besar siswa berada pada kategori belum matang. Faktor-faktor seperti lingkungan keluarga, lingkungan masyarakat, wawasan dunia kerja, lingkungan sekolah, dukungan infrastruktur, dan sikap terhadap konsepsi pekerjaan\/jabatan berpengaruh rendah terhadap kematangan karir siswa. Kegiatan diskusi dengan orang tua\/wali merupakan kegiatan yang sering dilakukan siswa dalam upaya meningkatkan kematangan karirnya. Melalui hasil uji t diperoleh p value kurang dari 0,05 berarti model konseling karir trait and factor teruji efektif untuk meningkatkan kematangan karir siswa. Berdasarkan hasil penelitian di atas direkomendasikan kepada guru bimbingan dan konseling\/konselor agar model konseling karir trait and factor ini menjadi alternatif pilihan yang dapat diimplementasikan di sekolah. Pihak sekolah mengadakan pelatihan bagi konselor untuk meningkatkan kompetensi terkait dengan model konseling karir ini. Peneliti selanjutnya dapat melakukan penelitian berkaitan dengan tingkat kesiapan siswa dan konselor untuk melaksanakan konseling karir ini. Kata kunci: model konseling karir trait and factor, kematangan karir \u2003 ABSTRACT A Model of Trait and Factor Career Counseling to Enhance Students Career Maturity State Vocational Schools in Bandung Dr. Sudjani, M.Pd. The research is motivated by the fact that most of the state vocational high school students in the city of Bandung have difficulties in choosing and determining career choices. Guidance and career counseling has not helped improve career maturity, so that students do not have the readiness to implement career development tasks. The purpose of this study is to produce a trait and factor career counseling model to increase students career maturity. This research method uses a combination of qualitative and quantitative (mixed methods design) include the type of exploratory mixed methods. The model of this research is a research and development using quasi experimental design with non randomly Pretest-Posttest Nonequivalent Group Design. The stages of the research and development include: (1) preliminary study, (2) model development, (3) rational model validation, and (4) empirical model validation. Sampling technique utilizes two stage cluster sampling technique with every students of the building techniques study program of SMKN in Bandung as its population. The result of this study shows that majority of students are in the career immature category. Factors such as family environment, community, workplace worldview, the school environment, infrastructure support, and attitudes toward conception jobs \/ positions provide low impact on students career maturity. Discussion with a parent\/guardian is an activity oftenly done by the students in their effort to increase career maturity. By conducting t-test, a p-value of lower than 0,05 was obtained, meaning the trait and factor career counseling model is proven to be effective to increase students career maturity. Based on the results above, it is recommended for guidance and counseling teachers (counselors) that career counseling model of trait and factor this into alternative options that can be implemented in schools. The school held a training for counselors to improve the competencies related to career counseling model. Researchers can then conduct research related to the level of preparedness of students and counselors to implement career counseling. . Topic: Innovation in Teaching and Learning 4 A New Tool to Fasilitate Learning Reading for Early ChildhoodC Puspitasari(*), Subiyanto This paper present a new android application creativity for early childhood learning reading. The description includes a design, development, and an evaluation experiment of an educational game for learning reading on android. The result of development game is a 2D educational game that applying Belajar Membaca Tanpa Mengeja (BMTM) method, and consist the elements of game. Twenty-six students in kindergarten have participated in this evaluation experiment. Students were divided into two groups, i.e. an experimental group that learning with the developed android educational game and a control group that learning with a reading book. The evaluation experiment results that the Gain score of the experimental group is 0.39 that in medium category, this score is higher than gain score of control group. Finally, the android educational game can increase student\u2019s reading ability, which in the medium category. So, the developed android educational game is a better alternative tool that can be used by early childhood for learning reading. Topic: Innovation in Teaching and Learning 5 A new variant of Android Educational Game as the facility introduction number for early childhoodD A Nawangnugraeni(*), Subiyanto This paper presents a new research, development and experiment evaluation of an android application game. The game is proposed as for the educational facility to introduce number for early child. This educational game teaches a childhood to learn while playing that is learning numbers 0-9 uses the media game on mobile smart phone with an operating system Android. The result of development was developed a game that consists of tutorial and game are implementing elements of the game. To evaluate the influence of educational game for learning outcomes carried on pre-kindergarten as many as 20 children and devided to the experimental group and the control group. The result of Gain score the experimental group is 0.77 (highest category) and the control group is 0.48 (medium category). Finally, the android educational game can use to be an educational facility that can improve learning outcomes introduction number of early childhood Topic: Innovation in Teaching and Learning 6 A Preliminary Study on Augmented Reality for Learning Local Wisdom on Indonesian BatikIsma Widiaty1,*, Lala Septem Riza2, Ade Gafar Abdullah3, Ana1 On October 2009, UNESCO entitled Indonesian batik as a masterpiece of oral and intangible heritage of humanity. There are over 100 variants of batik that are designed by considering on drawing techniques, regions, patterns, colors, etc. Moreover, they commonly represent culture, philosophy, and local wisdom of a place and society where batik was designed. This paper presents the preliminary study on an implementation of Augmented Reality (AR) for explaining and studying culture and local wisdom contained on batik. One of the benefits offered by this research is that it is used for learning media in vocational schools (SMK), which raises the value of local knowledge batik in an active, interactive, and fun way. The method used in designing learning media is participatory design, including interviews, paper prototyping, and usability tests. Steps being taken in the design of AR in SMK is a needs analysis device type, marker, and content used on AR Desktop and AR Mobile. Results from this study are used to improve the media and to suggest reviews directions for future works in this study. Topic: Innovation in Teaching and Learning 7 Activated Carbon-Based Working Pairs for Adsorptive Solar Cooling System: A Comparative Study with SimulationNyoman Sugiartha, I Made Sugina and Ida Ayu Anom Arsani Solar energy is considered as an environmentally benign thermal source for the operation of adsorptive cooling system with intermittent cycle. Such a system needs suitable adsorbent-refrigerant working pairs to assure good reversible desorption-adsorption process for producing cooling capacity. This paper presents a detailed comparison of the combined activated carbon adsorbents (AC, ACF and Maxsorb III) with refrigerants, i.e. ethanol, methanol, R-134a and ammonia based on adsorption isotherms data available from the published literatures. The study employs a theoretical thermodynamic approach and adsorption equilibrium model to simulate performance indicators such as adsorption capacity, coefficient of performance (COPs) and specific cooling effect (SCE) at various operating conditions. The results show that Maxsorb III-methanol delivers the highest COP and SCE for desorption temperature below 95 oC and suitable for air conditioning and food or medicine storage applications. Topic: Engineering and Technology Innovation 8 AN ANALYSIS TOWARD LIFE SKILLS OF VOCATIONAL SCHOOL STUDENTS(A study conducted toward 12th graders of Computer and Networking Engineering Major in North Jakarta)Nur Rahma Yenita This research conducted since not all of the graduated students from vocational school are able to work in business and industrial world. Many problems related to students graduated from vocational school arose since the number of offered job vacancies is not sufficient with the number of students graduated from vocational school and life skills mastered by the students are varied as a result of teaching and learning processes. Therefore, this research is aimed to analyze vocational school students\u2019 life skills formed after a process of teaching and learning. The general purpose of this research is to know the value of students\u2019 life skills. Meanwhile, the specific purpose of this research is to know the value of personal skills, social skills, academic skills, and vocational skills of vocational school students. Descriptive method along with quantitative approach is used in this research. Sampling technique used in this research is purposive sample where the number of sample is 102 students of 12th graders of Computer and Networking Engineering Major consisted of 31 students from SMK N 4 Jakarta, 34 students from SMK N 36 Jakarta and 37 students from SMK Perguruan Cikini Jakarta. Questionaire is used as the research instrument in order to collect the data. According to the result of this research, all research questions are answered. Personal skills and social skills of 12th grade of vocational school students majoring Computer and Networking Engineering in North Jakarta is categorized as good. Meanwhile, the academic skills and vocational skills of 12th grade of vocational school students majoring Computer and Networking Engineering in North Jakarta is categorized as good Topic: Vocational Education and Training 9 An Evaluation of Authentication Methods for Smartphone Based on Users\u2019 PreferencesPuspita Kencana Sari (*), Gati Sabrina Ratnasari, Adhi Prasetio This study discusses about smartphone screen lock preferences using some types of authentication method. The purpose is to determine the user behaviour based on perceived of security and perceived of convenience, as well as preferences for different types of authentication method. Variables used are consideration for locking the screen and the types of authentication methods. The population consists of smartphone users with total sample of 400 respondents with nonprobability sampling method. Data analysis method using descriptive analysis. The results showed the convenience factor is still a major consideration for locking the smartphone screen. The majority of users choose the unlock pattern as the most convenient method to use. Meanwhile, fingerprint unlock become the most secure method in users perception and method to be used in the future as well. Topic: Engineering and Technology Innovation 10 ANALYSIS OF BRAND VALUE OF HIGHER EDUCATION INSTITUTION IN WEST JAVAPuspo Dewi Dirgantari, Agus Rahayu, Disman, Ratih Hurriyati Every year universities in Indonesia continues to increase in number. many universities in Indonesia are located in the region of West Java and Banten provinces, but the number of students in West Java decreased. Moreover, at a time when competition among universities higher, the ranking universities in West Java decreased in the eyes of the world, including in Asia. Some evidence suggests that the brand value of colleges in West Java is still not optimal. The assessment process itself enrich the organization with a much more comprehensive understanding through and understand their customers, markets and channels they operate, the competitive environment, and operational capabilities. This study uses science approach to referral marketing management theory. The method used in this research is descriptive survey and explanatory survey. The data used are primary data and secondary data that were collected through questionnaires and documentation of media. Meanwhile, to measure the influence of brand element and marketing mix of educational services to brand value of higher education in West Java used Structural Equation Modelling. Results of the study revealed that elements of the brand and marketing mix of education services gave positive effect on the brand value of higher education in West Java. Topic: Innovation in Teaching and Learning 11 Analysis of Corrosion Properties of AISI 316 Based on Variation of Electrolyte Solution Type and ConcentrationIda Hamidah, Agus Solehudin, Agus Setiawan, Asep Bayu Dani Nandiyanto, Budi Mulyanti, Ade Gafar Abdullah AISI 316 has been widely used in a variety of technologies because of its corrosion resistance properties. The corrosion resistance properties are very dependent on the environment that surrounds the AISI 316. In this study, the corrosion rate of AISI 316 in various environments electrolyte solution had been analyzed. The electrolyte solution is KOH and NaOH which have alkaline properties as well as NaCl which has acidic properties. Testing is done by immersing the AISI 316 in an electrolyte solution in a glass autoclave. The corrosion test was monitored on temperature of 25oC and 336 hours intervals for a period of 1008 hours. Through the calculation using the mass loss method, it was found that the AISI 316 have the best corrosion resistance in an alkaline solution with a small solution concentration Topic: Engineering and Technology Innovation 12 ANALYSIS PROVIDING DIESEL-WIND HIBRID ELECTRICAL ENERGY SYSTEM IN TIMOR ISLAND INDONESIAGunadi Tjahjono; Tetty Setiawaty The research\u2019s purpose are: (1) analyzing the capacity needs of the wind energy system based on the parameters of energy requirements, the ability of the inverter, generator capability and the ability of the local wind; (2) generating a simulation of diesel-wind hybrid power plants for district and city in Timor island; (3) generating an optimization study of diesel-wind hybrid power plants for district and city in Timor island; and (4) generating a scheme of wind-diesel hybrid power plants for district and city in Timor island. This research applies quantitative method with the approach of simulation, optimization and sensitivity analysis using HOMER application program for diesel-wind hybrid power plant system. The results of this research are: (1) Simulation in districts and Kupang city is wind turbines 20 KW with 71.7 KW generator capacity . Results of optimization of wind-energy contribution only 7% and production costs analysis of wind turbines at US $0.47 to produce wind turbines of 20.000 kwh\/yr. (2) Simulation in South Central Timor districts is wind turbines 3 KW with installed capacity of 11.4 KW. Results of optimization of energy-wind contribution only 9% and production costs analysis of wind turbines at US$ 0.525 to produce wind turbines of 1.165 kwh\/yr. (3) Simulation in North Central Timor districts is wind turbines 1 KW with installed capacity of 7.62 KW. Results of optimization of energy-wind contribution only 4% and production costs analysis of wind turbines at US $0.53 to produce wind turbines of 1.150 kwh\/yr. (4) Simulation in Belu districts is wind turbines 10 KW with installed capacity of 12.1 KW. Results of optimization of energy-wind contribution only 22% and production costs analysis of wind turbines at US$ 0.48 to produce wind turbines of 1.210 kwh\/yr. Topic: Engineering and Technology Innovation 13 ANALYZE THE IMPACT OF CLIMATE FACTORS ON THE TOTAL OF ENERGY EXPENDITURE DURING STATIC BICYCLE EXERCISEsugiono How to manage energy expenditure for cyclist is very crucial part to achieve a good performance. As the tropical situation, the differences of temperature level might be contributed in energy expenditure and durability. The purpose of the paper is to estimate and to analysis the configuration of energy expenditure for static cycling activity based on heart rate value in room with (AC)\/no AC treatment. The research is started with study literatures of climate factors, temperature impact on human body, and definition of energy expenditure. The next step is design the experiment for 5 participants in 2 difference models for 26.80C \u2013 74 % relative humidity (room no AC) and 23,80C \u2013 54.8 % relative humidity (room with AC). The participants\u2019 heart rate and blood pressure are measured in rest condition and in cycling condition to know the impact of difference temperature in energy expenditure profile. According to the experiment results, the reducing of the temperature has significantly impact on the decreasing of energy expenditure at average 0.3 Kcal\/minute for all 5 performers. Finally, the research shows that climate condition (temperature and relative humidity) are very important factors to manage and to reach a higher performance of cycling sport. Topic: Engineering and Technology Innovation 14 Analyzing Traffic Source Impact on Returning Visitors Ratio in Information Provider WebsiteAdhi Prasetio, Puspita Kencana Sari, Osa Omar Sharif, Endang Sofyan Web site performance, especially returning visitor is an important metric for an information provider web site. Since high returning visitor is a good indication of a web site\u2019s visitor loyalty, it is important to find a way to improve this metric. This research investigated if there is any difference on returning visitor metric among three web traffic sources namely direct, referral and search. Monthly returning visitor and total visitor from each source is retrieved from Google Analytics tools and then calculated to measure returning visitor ratio. The period of data observation is from July 2012 to June 2015 resulting in a total of 108 samples. These data then analysed using One-Way Analysis of Variance (ANOVA) to address our research question. The results showed that different traffic source has significantly different returning visitor ratio especially between referral traffic source and the other two traffic sources. On the other hand, this research did not find any significant difference between returning visitor ratio from direct and search traffic sources. Topic: Engineering and Technology Innovation 15 ANIMATION IN THE PROBLEM SOLVING ON LEARNING OF FIELD OF SHEAR OF ATOMS THAT DETERMINANT MECHANICAL PROPERTIES OF MATERIALM Komaro, A Djohar, A Setiawan, B Hasan, HP Kusuma. Difficulty understanding abstract concepts, complexity and dynamics are obstacles faced by students in the learning materials engineering, especially in the materials slide sector. Data from study material engineering indicates 43% of students could solve problems related to the slide sector causes of metallic properties. Considering this, research was conducted on the use of multimedia animation on the material plane shear. This study aims to illustrate the problem solving ability of students, and document student response after the process of learning by using multimedia animation on the material plane shear engineering course materials. The research method was quasi-experimental methods ( quasi experimental research). Data collection was conducted using post - test and student questionnaire responses. The results after comparing the pre and post test showed that students using MMA in the learning material shear field in Materials Engineering courses did MMA better than the class that used pictures and handouts. Application of MMA helped students more easily understand material content and facilitate their learning process. The respones from the questionaire showed student reacted positive to the use of MMA, and that they reguard it as a high quality learning tool. Topic: Innovation in Teaching and Learning 16 Applying Cognitive Load Theory in Example-Problem-Based Learning: Effects on Performance, Effort and Efficiency of Vocational StudentsNoor Hisham Jalani, Lai Chee Sern. Affero Ismail Cognitive Load Theory (CLT) suggests that learning takes place best in a situation that equivalent to individual cognitive design. Thus, this article proposes a learning model called Example-Problem-Based Learning (EPBL) which is a combination of two learning strategies: worked-examples and problem-solving. This teaching method guides students to go through several cognitive developments. At the early stages of knowledge acquisition, novice students benefit more from worked-examples, which is a model of problem-solving. After they have gained sufficient knowledge, worked-examples may no longer be appropriate because the positive effects of worked-examples will be lost. Therefore, learning through problem-solving should be applied since students have already been equipped with profound domain knowledge. A quasi-experimental study with pre-test post-test design was conducted to investigate the effect of EPBL in Electrical Circuit Theory teaching and learning on Malaysian first year vocational diploma-level students\u2019 performance and mental effort as well as the efficiency. The experiment was carried out for eight weeks with 38 students. The EPBL was used in the experimental group, while the existing teaching method based on Teacher-Centered Learning (TCL) is maintained for the control group. The students have completed the Electrical Circuit Theory Performance Test and the 9-Point Mental Effort Rating Scale. Established in a conducted experiment, the EPBL teaching method enhances vocational students\u2019 performance and mental effort during learning as well as increasing the learning efficiency. Topic: Vocational Education and Training 17 APPLYING PARTICLE SWARM OPTIMIZATION IN INCREASING PERFORMANCE OF PID CONTROLLER ON AVR SYSTEMMUHAMMAD FALAH ARANZA, JAJA KUSTIKA, BAMBANG TRISNO, DADANG LUKMAN HAKIM In this paper explains applying Particle Swarm Optimization algorithm and analyzing performance of PID controller on AVR (Automatic Voltage Regulator) system. Performance of PID controller is depend on Kp (Proportional Gain), Ki (Integral Gain) and Kd (Derivative Gain). These gains can be got by using method Ziegler-Nichols (ZN), Coheen COON, gain-phase margin, Root Locus, Minimum Variance dan Gain Scheduling, however these methods are not optimal to control systems that nonlinear and have high-orde, in addition, in calculating of these methods relative difficult. To solve those obstacles, particle swarm optimization (PSO) algorithm is proposed to get Kp, Ki and Kd which optimal. Choosing PSO is caused PSO has result that convergence and not require many iterations, so that in calculating relative sufficient fast. Based on result of analyzing transient, stability Root Locus and frequency response, show that using PSO algorithm has performance of PID controller better than Ziegler-Nichols and system without PID controller. Topic: Engineering and Technology Innovation 18 Augmented Reality for Introduction of Unit System in Personal ComputerV R Palilingan; Gladly C Rorimpandey; Irsan I Kapoh There are lots of technologies that used to develop a learning media. Creative and attractive are some of reasons the teachers want to develop a learning media. Technology Augmented Reality is the one of user interface technology. It is not just attractive and creative, but it is also can reduce the cost for purchasing a practical tool. Introduction of Unit System in Personal Computer is the part of Computer Assembly subject. So, aim of this research is to develop the learning media with augmented reality for Introduction of Unit System in Personal Computer. This research is used Multimedia Development Life Cycle (MDLC). An analysis is done to determine the user and system requirements so that a list of requirements for the new system may derive. The learning media has main feature that will show the object visualization in Augmented Reality. Other feature is visualization object in 2 Dimension. The learning media is then tested to ensure that all functionalities provided are running well. Finally a conclusion is made that the learning media has accomplished the project purposes and fulfilled the requirements. Topic: Innovation in Teaching and Learning 19 Bandung Land-use Management toward a Compact CityBeta Paramita The concept of a compact city has been introduced since 1973. It is an utopian vision largely driven by a desire to see more efficient uses of resources. In 1980s, the reconfiguration of the physical urban form of metropolitan areas was increasingly debated by both theorists and practitioners. Recently, the concept of a compact city is more focused on developed countries in which the population tends to decrease. However, in Asia, except Japan which contains many dense cities, it has become a concept which promotes relatively high residential density with mixed land uses, though rather only in population and density. This paper has two aims: first, to collect data on densities and other correlated data of cities similar to Bandung in developing countries. Secondly, to assess the land-use of each Bandung\u2019s district. Finally, this paper explores the implications of land use management and describes challenges faced and possible approaches especially in land-use management strategies to be implemented in Bandung. Topic: Engineering and Technology Innovation 20 BATIK LESSON PLAN DESIGN BASED ON LOCAL WISDOM VALUE IN VOCATIONAL HIGH SCHOOLAdhani Nurul Hasanah, Tati, Ana, Isma Widiaty The implementation of the 2013 curriculum reflects the scientific approach with the learner as the learning center. Batik lesson plan scenarios that are based on the value of local wisdom require validation results from expertise. This study aims to develop a batik lesson plan design based on local wisdom value in Vocational High School. The development of the lesson plan is using scientific learning. The method used in this study is the Research and Development (R & D) method, adopting the R & D stages developed by Plomp. The R & D stages conducted in this study are the investigation stage, design stage, realization stage, and product evaluation stage. The data was collected using observation, interview, documentation study and expert judgement. The expert judgement made by seven expert validators, including four teaching and learning experts, and three batik prominent figures in West Java. This study resulted in a batik lesson plan based on local wisdom value, and received a very good score from the expert validators in the field of teaching and learning, and the expert in the field of learning materials based on local wisdom values. Topic: Vocational Education and Training 21 Building Education Facilities, an Assessment of FPBS - UPI on reducing overheat temperatureElkandi, Iqbal, Beta Paramita UPI as the largest land area among universities in Bandung, lies in geographical location within tropical climate zone. FPBS is one of faculty in UPI contain of three building groups. The measurement which conducted during the hottest day in October has shown the phenomena between the high outdoor temperature and low indoor temperature. The highest mean radiant temperature in the plaza shows 43C meanwhile the average of indoor air temperature shows 24-25C. Giving the respect of this phenomenon, this paper describes the climatic aspect of FPBSs building on reducing overheat temperature. Topic: Engineering and Technology Innovation 22 Challenges in Developing Engineering Class Design at Middle School Classroom to Improve Science, Technology, Engineering, And Mathematics (STEM) EducationIda Kaniawati1, 1, Irma Rahma Suwarma1, 2, Lilik Hasanah1, 3, Nuryani Y Rustaman1, 4, Elah Nurlaelah2,.5 Abstract. This paper discuss challenges in designing engineering class that conducted in a middle school to improve STEM education in Indonesia. The class emerged engineering process design with technology to improve creativity skill, problem solving skill, and science concept mastery. Engineering class design was implemented in several schools in US to embed STEM education into school curriculum that emphasized on engineering practice. However, the class design is newly introduced in Indonesia. In fact, the education system differences become a crucial challenge. The class design was created through STEM professional development program. Seven teachers from Muhammadiyah 8 Secondary School in Bandung followed this program. Designing method was initiated by giving STEM education knowledge review to improve teacher\u2019s knowledge and perception. Furthermore, Teachers were divided into two groups to analyse the contents in different level. They analysed content curriculum in science, mathematics, and computer technology subjects, then created big themes that cover several concepts in the contents. Finally, they plan students\u2019 STEM based project in a worksheet format that adapted engineering process design. Students in this class will be divided into six groups that have two different tasks; two group acts as a supplier, and the rest acts as engineer. The result implied that teachers created six STEM based projects plan that emphasizes engineering (E) practice design and integrates science (S), technology (T) and mathematics (M) to trigger students\u2019 creativity, problem-solving skills, and improve students\u2019 concept mastery. However, they meet challenges in using STEM context that reflect on time consuming, number of participant, activity location, products, occurred problems, and attained agreement among participant. Topic: Innovation in Teaching and Learning 23 Characterisctic of Fly Ash and Coconut Fibre Ash as Cement Replacement Materials on Cement Paste StrengthRidho Bayuaji, Riky Wahyu Kurniawan, Abdul Karim Yasin, Herta Ahsani Takwim Fatoni, Fa\u2019izah Maulidya Afifah Lutfi The wealth of local materials Indonesia have not been fully utilized in concrete technology. Concrete is the backbone material in the construction field. The main concept of the concrete material is composed of a binder and filler. Cement, concrete main binder highlighted by environmentalists as one of the industry are not environmentally friendly because of the burning of cement raw materials in the kiln requires energy up to a temperature of 1450o C and the output air waste CO2. On the other hand, the compound content of cement that can be utilized in innovation is Calcium Hydroxide (CaOH), this compound will react with pozzolan material and produces additional strength and durability of concrete, Calcium Silicate Hydrates (CSH). Pozzolan materials used in this study were coconut fibers ash and fly ash. This material was used as cement replacement materials on cement paste. SNI-03-1974-1990 is standard used to clarify the compressive strength of cement paste at the age of 3 and 7 days. To sum up of this study that the optimum composition of coconut fiber ash and fly ash to substitute 30% of cement with 25% and 5% for coconut fibers ash and fly ash. Topic: Engineering and Technology Innovation 24 Classroom Blog Use in Teaching Mathematics Engneering Related to Students Comprehenssion AbilityDedi Rohendi The use of blog in any other activities is increasing but there is limited research assessing whether blogs use in teaching learning process. Moreover, the contribution of student comprehenssion ability to the relationship between learning and blogs use has not been thoroughly investigated. To bridge this gap, students in mechanical engineering department class were investigated of their use of blog in the classroom. This study aims to 1) investigated the used of blog to increase the students comprehension abilities in mathematicaal engineerring.2) know the students compreenssion abilities by using classroom blogging. 3) know the students responses of using classroom blogging. The method used is a quasi-experiments method. Samples taken from the students of mechanical engineerring departement. The instrument used in this study are test kits (pretest and posttest) and questionnaires. Tests on classroom blogging media obtained results that the media deemed feasible for use, while based on the results of testing hypothesis that there is an increase in students comprehension concept skill of mathematics engineerring for the students using classroom blogging media in teaching learning process. It is shown from the normalized gain index was 0.69 in the classroom of students who use classroom blogging media on a middle criteria. While in questionnaries study give the result that students responds in average is agree with using classroom blogging. Topic: Innovation in Teaching and Learning 25 Comparison of Performance of Partial Prestressed Beam-Column Subassemblages Made of Reactive Powder Concrete and Normal Concrete Materials Using Finite Element ModelsS A Nurjannah , B Budiono , I Imran2, S Sugiri2 Research on concrete material continues in several countries and had produced a concrete type of Ultra High Performance Concrete (UHPC) which has a better compressive strength, tensile strength, flexural strength, modulus of elasticity, and durability than normal concrete (NC) namely Reactive Powder Concrete (RPC). Researches on structures using RPC material showed that the RPC structures had a better performance than the NC structures in resisting gravity and lateral cyclic loads. In this study, an experiment was conducted to apply combination of constant axial and lateral cyclic loads to a prototype of RPC interior partial prestressed beam-column subassemblage (prototype of BCS-RPC) with a value of Partial Prestressed Ratio (PPR) of 31.72% on the beam. The test results were compared with finite element model of beam-column subassemblage made of RPC by PPR of 31.72% (BCS-RPC-31.72). Furthermore, there was BCS-RPC modeling with PPR of 21.39% (BCS-RPC-21.39) and beam-column subassemblages made of NC materials modeling with a value of PPR at 21.09% (BCS-NC-21.09) and 32.02% (BCS-NC-32.02). The purpose of this study was to determine the performance of the BCS-RPC models compared to the performance of the BCS-NC models with PPR values below and above 25%, which is the maximum limit of permitted PPR. The results showed that all models of BCS-RPC had a better performance than all models of BCS-NC and the BCS-RPC model with PPR above 25% still behaved ductile and was able to dissipate energy well. Topic: Engineering and Technology Innovation 26 Comparison of the Calculation QRS Angle for Bundle Branch Block DetectionLeonard Goeirmanto QRS angle represent condition of blood circulation in the heart. Normally QRS angle is between -30 until 90 degree. Left Axis Defiation (LAD) and Right Axis Defiation (RAD) are abnormality conditions that lead to Bundle Branch Block. QRS angle is calculated using common method from physicians and compared to mathematical method using difference amplitudos and difference areas. We analyzed the standard 12 lead electrocardiogram data from MITBIH physiobank database. All methods using lead I and lead avF produce similar QRS angle and right QRS axis quadrant. QRS angle from mathematical method using difference areas is close to common method from physician. Mathematical method using difference areas can be used as a trigger for detecting heart condition. Topic: Engineering and Technology Innovation 27 Compliance of Indonesian Qualification Framework (IQF) towards ARQF: Challenges and Opportunities of the Referencing to Regional Qualification FrameworkAgus Setiawan Recently, TVET (Technical and Vocational Education and Training) become attract more attention because of its important role for the world economics development. In order to face global market and ASEAN Economics Community, standardization and harmonization in TVET sector in the region are crucial issues to overcome. Some efforts have been taken by government level and regional TVET organization to standardize and harmonize TVET sector. This paper address IQF in conjunction with ASEAN Reference Qualification Framework (ARQF) in the TVET sector. Since 2012, Indonesia has the IQF with a legal endorsement in the form of Presidential Decree no. 8\/2012. The stages of the IQF\u2019s implementation are currently being designed and partly applied. The development of IQF was stimulated by the fact that at present, education and training provision in Indonesia is fragmented and often poor in quality. The IQF consists of 9 levels characterized by both learning outcomes and job-specific competencies. The IQF is potentially useful mechanism for guiding curriculum reform, increasing the relevance of education and training, engaging with employers and articulating the knowledge and abilities expected at each level and developing opportunities for recognition of prior learning. The AQRF, a common reference framework, will function as a translation device to enable comparisons of qualifications across participating ASEAN countries.The AQRF purpose to enable comparison of qualifications across countries that will (a) Support recognition of qualifications, (b) Facilitate lifelong learning, (c) Promote and encourage credit transfer and learner mobility, (d) Promote worker mobility, (e) Lead to better understood and higher quality qualifications systems. The AQRF will support and enhance each country\u2019s national qualifications framework or qualifications system while providing a mechanism to facilitate comparison and transparency. The ARQF will link the participating ASEAN NQFs or qualification systems and become the ASEAN\u2019s mechanism for recognition of its qualifications against other regional and international qualifications systems. Comparison between IQF and ARQF, we find following points: (a) In term of descriptor, ARQF accommodates informal and prior learning while IQF tends to focus on formal education only, (b) ARQF derives qualification into 8 level of complexity of learning outcome while IQF defines 9 level, (c) Both ARQF and IQF refer learning outcome as a key feature of qualifications, (d) The level descriptors of ARQF include three domains: knowledge and skills, application and responsibility and accountability, while the level of descriptor of IQF include general descriptors which cover personality, working attitude and ethics and the specific descriptors which describe the knowledge and skills. In order to facilitate linking NQF levels against the levels in the AQRF, NQF or qualifications systems should have qualifications \u2018demonstrably based on learning outcomes\u2019. For national qualifications frameworks that are not based on learning outcomes, the referencing process and report should demonstrate progress towards a learning outcomes based approach. As a part of ASEAN, Indonesia has to refer AQRF. Therefore, Indonesia has to actively involve in development and referencing process of ARQF to avoid rigidity and lack of flexibility. In addition, the uniqueness of educational system of Indonesia and multicultural characteristics of Indonesia also has to be considered in the development and referencing process and strategy of implementation of ARQF. Topic: Vocational Education and Training 28 Concept Development House Living Traditional Mandailaing in North SumatraPutri Lynna A. Luthan Mandailing is one of the Batak ethnic in North Sumatra were classified as holding strong indigenous culture moress Mandailings contained in parts of the traditional houses. However, with the rapid development of modern houses will be possible extinction of the traditional residences containing cultural values. This study aimed to obtain the architecture of traditional houses Mandailing towards the development of the concept of living house modern while still containing cultural values. Mandailings containing values: 1) religion or beliefs, 2) kinship, 3) philosophy of life, 4) leadership and social contained the structure of traditional houses. The development concept is based on the concept of a traditional house Mandailaing Godang and Mandailing Julu contained in Mandailing region. The method used is a qualitative research method. The results were: 1) for the needs of people whose activities is high, then use the characteristics of the house that extends laterally, 2) for the needs of people whose activities are low, then use the characteristics of the house that extends backward. The second characteristic of the house still contains the values Mandailing culture. Topic: Engineering and Technology Innovation 29 Conceptual Design of Passive Safety System For Lead-Bismuth Cooled Fast ReactorA.G. Abdullah, A.B.D. Nandiyanto This paper presents the results of the conceptual design of passive safety systems for reactor power 225 MWth using Pb-Bi coolant. Main purpose of this research is to design of heat removal system from the reactor wall. The heat from the reactor wall is removed by RVACS system using the natural circulation from the atmosphere around the reactor at steady state. The calculation is performed numerically using Newton-Raphson method. The analysis involves the heat transfer systems in a radiation, conduction and natural convection. Heat transfer calculations is performed on the elements of the reactor vessel, outer wall of guard vessel and the separator plate. The simulation results conclude that the conceptual design is able to remove heat 1.33% to 4.67% from the thermal reactor power. It can be hypothesized if the reactor had an accident, the system can still overcome the heat due to decay. Topic: Engineering and Technology Innovation 30 CONCEPTUAL MODEL A TOOL TO ANALYSIS OF RELEVANCE CURRICULUM : Fulfillment Index ApproachIwa Kuntadi, Isma Widiaty, Lilis Widaningsih, Ana This research is motivated by the need for a software to analyze the relevance of curriculum of vocational schools (SMK) in accordance with the needs of the creative industries batik. Analysis of the relevance of the curriculum used is the approach fullfilment index which is a mathematical model of the analysis of the demand side of the creative industries batik. The method used in this design through the analysis stage of the existing demand side, the design of the conceptual model, and validation. Research results illustrate the conceptual model of fullfilment index consists of four dimensions of harmonization that aspects of quantity, quality, location, and time. Topic: Engineering and Technology Innovation 31 Conseptual Design Pedagogy Model of Hybrid Project Based Learning-PEPP&ER in Computer Networking LearningRudi Haryadi, Ade Gafar Abdullah Learning process in vocational school should be able to prepare students with professional skills fit the demands of industrial standard. Professional activities in the workplace should be integrated in the learning process in the classroom. To realize this, we need a design model of pedagogy that can facilitate learning process in order to produce students with ability to work in accordance with the demands of the industry. Development of model design has done with collaboration of Project-based learning models and PEPPER model through mix method. This research resulted in the syntax of pedagogical hybrid models and learning tools that have characteristics of industrial standard which quantitatively tested into Learning of Computer Networking , so that the application of this model can improve the assessment of vocational education \/ vocational quality through quality graduates. Topic: Engineering Education 32 Contacless Inductive Chargers Design for Hand Phone Re-charging SystemPola Risma (a)*, Yurni Oktarina (a), M. Taufik Roseno (a) The current trends are the development of contact less recharging method called inductive charging. Inductive charging is the hand phone recharging method using magnetic field instead of connecting the battery to the electric source. Components used for this system is a pad integrated to the hand phone that is functioning as transceiver of electric source to receiver through the air Topic: Engineering and Technology Innovation 33 Continuing Professional Development of vocational teacher-Based Local CultureYadi Mulyadi Continuous professional development of vocational teachers in the context of lifelong education is a series of the comprehensive system continuously performed, either formally or informally. Life experience is the structure of teacher built the perfect combination of culture process verbal skills, mastery of teaching materials and teacher professional mastery learning methods. The process of continuous professional development is not a natural process that is routine, traditional and runs by itself. However, have a process that requires strategic planning, clear and actual in a system structured in order to achieve better education. In fact, occur dichotomous debate the pros and cons, on the one hand, the continuous professional development to be one savior god improvement in the quality of education but on the other hand, criticism is also emerging. In general, the process of continuous professional development is an activity that is fragmented, coherent meeting that decontextualized and isolated from the real classroom situation. Approach to teacher professional development program continuous can be done with a variety of patterns and models. In this, paper will be developed five variables that affect the continuing professional development of teachers are competence mastery of a subject matter of knowledge, teaching and learning, personality, ethnopedagogy and self-efficacy. Topic: Vocational Education and Training 34 Contrast Enhancement For Satellite Image Segmentation With Fuzzy Cluster Means Based Using Morphological Filtering TechniqueHarsiti, Tb Ai Munandar, Akip Suhendar, Ade Gafar Abdullah, Dedi Rohendi Image segmentation divides an image into some homogeneous regions based on the value of a certain similarity between the gray level of a pixel by pixel gray neighbors. The quality of image segmentation is generally influenced by the characteristics and handling of images to be processed. This research generally aims to perform segmentation of satellite images of an area using fuzzy cluster means (FCM). Improving the quality of segmentation results of this research was done by using morphological filtering (top-hat and boots-hat filtering) to increase the contrast of the image before segmentation. Four scenarios research was conducted to compare the results of image segmentation of a region. Namely, (1) pure image segmentation with FCM; (2) image segmentation with FCM after the top-hat techniques applied; (3) image segmentation with FCM after bot-hat techniques applied; and (4) image segmentation with FCM after being applied a combination of a top-hat and bot-hat filtering. Satellite imagery analysis carried out on 5 districts namely Pandeglang, Pagelaran, Mandalawangi, Menes and Saketi. The results of four scenarios show that, the use of pure FCM able to show the segmentation of objects analyzed, but the details of the edges of objects, either from the surface or side of the object is not visible. While the use of a top-hat fileting able to increase the contrast of the image before segmentation of the top surface of the image. On the other hand, the use of bot-hat filtering was able to detect the edges of objects by increasing the contrast of the background object, but was unable to reveal details of the surface of the object. While the use of morphological filtering can improve the contrast of the image before segmentation level, but the results of segmentation is difficult to identify the actual object to be analyzed. Topic: Engineering and Technology Innovation 35 Coordination Hydrothermal Interconnection Java-Bali Using Simulated AnnealingBagus Wicaksono, Ade Gafar A, Wasimudin Surya S Coordination of hyrdrothermal aim to minimize the total cost of the operating system that is represented by fuel costs and constraints for optimization. For further optimization, there are several methods that can be used. Simulated Annealing (SA) is one of the methods that can be used to solve the optimization problem. This method is inspired of annealing or cooling process in the manufacture of the materials composed of the crystals. On hydrothermal coordination has basic principle that the hydro generating as cantilever base load while the thermal generating as a cantilever load of the rest. In this paper, use 2 units hydro generating and 6 units thermal generating, 25 bus, with calculate transmission loss and load balance from each unit generating with software MATLAB to solve the calculating. Result total cost of hydrothermal coordination using simulated annealing for 24 hours amounted to $13288508.01. Topic: Engineering and Technology Innovation 36 Coordination Hydrothermal Power System Study in Java-Bali Network System Using Minimax Optimization AlgorithmAris Primadi Alparisi, Ade Gafar Abdullah, Wasimudin Surya Saputra In power system operation, coordination between power plants is an important things. The objective of hydrotermal power system coordination is to reduce the operational fuel cost. Theree are many optimization method had been depeloved,one of them is Minimax Optimization method. This method works by analyzing every possible combination formed of the hydrothermal combinatorial problem. Therefore, the algorithm will remove a combination of power plants that are not economically optimal. The optimization process involves six thermal power plants and two hydro plants with transmition loss considering. Optimization results using mnimax optimization sho, the fuel cost generation$ 18,401,423.23 is more economical 209,491.20 then PLN calculation. Topic: Engineering and Technology Innovation 37 Coordination System of Hydro-Thermal Generation with Calculate Transmission Loss based Genetic AlgorithmNoor Achmad Albar*, Ade Gaffar Abdullah Hydro-thermal coordination system is an important procedure in the operation of electric power systems. This study proposes to implement the Genetic algorithm in solving the unit commitment problem in the Java-Bali interconnection system. Simulations performed on two hydro plants and six thermal plants. The results indicated that the hydro-thermal coordination with Genetic algorithm provides results that are economical compared with the real data system. Topic: Engineering and Technology Innovation 38 CURRICULUM DEVELOPMENT AGROINDUSTRY ENGINEERING EDUCATION STUDY PROGRAM BASED ON NEED STAKEHOLDERSElih Mulyana, Ati Sugiaarti, Siti Mujdalipah Agro-Industry Engineering Education Studi Program ( PTAG ), Indonesian education university ( UPI ) is a new study program and the only formal education institutions, higher education vocational agriculture in Indonesia As a new study program, the development and adjustment of the curriculum that had been developed at this time needs to be adjusted to the demand and need of stakeholders in order to achieve the competencies of graduates who are ready to work The design used in this research is descriptive qualitative comparative approach to content analysis and data analysis technique use approach a percentage . Suistability curriculum Agroindustrial Engineering Education Studi Program with Curriculum SMKN AHP ( Agribusiness Agricultural Products ) are 85.42 % . And Suistability curriculum Agroindustrial Engineering Education Studi Program with Curriculum SMKN TPHP ( Agricultural Products Processing Engineering ) contained 62.5 %. Topic: Vocational Education and Training 39 DESIGN AND PRODUCTION UNIT MODEL WITH THEAPPROACH TO KNOWLEDGE BASED INDUSTRY IN SMK COMPETENCE EXPERTISE CLOTHING DESAINMally Maeliah, Yoyoh Jubaedah, Neni Rohaeni Abstract This study was based on the fact that it is important to synchronize the competences of vocational school students or graduates and workplace requirements. The students should actually be equipped with abundant learning experiences tailored with the industrial requirements. The aim of this study was to produce a model production unit with the approach of Knowledge Based Industry Vocational Skills Competency in dressmaking. Specific targets to be achieved from this research is to produce: (1) The findings concerning the characteristics of the business management field of fashion in the opinion of the expert and practitioner, (2) Findings of the routine activities of the production units of clothing in Vocational Skills Competencing dressmaking, (3) Design models Unit Production Approach Knowledge with Based Industry Vocational Skills Competency in dressmaking, (4) Strengthening training programs in vocational competency fashion expertise. This research uses a descriptive method with the approach of Research and Development. This development studies are carried out in three stages, comprising: a preliminary study phase, the model development and validation of test models. The subjects consist of the students and master teachers on entrepreneurship subjects and productive skills in the vocational field of dressmaking. The data were collected through interviews, observation and documentation study. The results showed that, the design model of a production unit with the approach of Knowledge Based Industry in Vocational Skills Competency is designed to combine dressmaking boutiques and mass produk system adapted to the adjustment in production processes adapted to the demands of working in the fashion industry and consumers needs. To implement the model developed should be supported by adequate competence, in order to obtain the findings and strengthen the need for training of fashion competency skills for learners in business practices at the production unit that pioneered SMK. The outcomes of this study are: (1) Design Approach models Production Unit with Knowledge Based Industry Vocational Skills Competency in dressmaking, (2) training program strengthening competence in vocational fashion expertise, (3) the results of the research article is to be published in international journals. Topic: Vocational Education and Training 40 Design of SCADA System Simulation in Suralaya Steam Power PlantIndra Karim Nugraha, Ade Gafar Abdullah, Dadang Lukman Hakim Simulation SCADA system (Supervisory Control and Data Acquisition) is a software-based HMI (Human Machine Interface) that is able to visualize the process plant. This paper contains simulations scada system used for Suralaya Steam Power Plant. This generation system consists of three main processes, the processes of fuel such as coal, process of water as boiler filler (Steam source) and water as a coolant. This water is taken from seawater. This simulation uses technical data from Suralaya Steam Power Plant and software developed with Wonderware Intouch 10. This software comes with the component images, animation, display control, alarm system, historical trand, historical real-time and security system. This simulation illustrates the flow of energy and energy conversion in the plant. This simulation can be used as operator training before operating the real plant. Topic: Engineering and Technology Innovation 41 Design of Virtual SCADA Simulation System for Pressurized Water ReactorUmar Wijaksono Ade Gafar Abdullah Dadang Lukman Hakim The Virtual SCADA system is a software-based Human Machine Interface that is able to visualize the process of a plant. This paper described the results of the virtual SCADA system design that aims to recognize the principle of the Nuclear Power Plant type Pressurized Water Reactor. This simulation uses technical data of the Nuclear Power Plant Unit Olkiluoto 3 in Finland. This device was developed using Wonderware Intouch which is equipped with manual book for each component, animation links, alarm systems, real time and historical trending, and security system. The results showed that in general this device can demonstrate clearly the principles of energy flow and energy conversion processes in PWR reactors. This virtual SCADA simulation system can used as instructional media to recognize the principle of PWR reactor. Topic: Engineering and Technology Innovation 42 Design Optimization of Hyperboloid Geometry in ArchitectureRendy Perdana Khidmat In order to assist architect in choosing an alternative of hyperboloid geometry in the early stage process of designing, this paper tries to demonstrate the use of parametric system with the genetic algorithm rules to perform multi objective optimization process. The experiments demonstrate the ability to utilize multi-objective optimization to map out the potential of each alternative toward the performance targeted. It is also to find the relation between the geometry and volume of materials used. Form Finding and Performance comparison is simulated in single system of parametric and optimization process. The result of simulation shows the variety of the designs individual that spread on the population field as a map of optimization process. Topic: Engineering and Technology Innovation 43 Design Simulator of SCADA System Geothermal Power Plant KamojangHafizh Tri Januar*, Ade Gaffar Abdullah, Dadang Lukman Hakim SCADA has an important in a power plant. SCADA is a modern control system based on a computer network used for control a process. Geothermal power plants one of the industries that use SCADA. Kamojang Geothermal Power Plants used as research objects and its already use a SCADA system. But the human machine interface (HMI) of the SCADA system used analog form and not all parts of the plant system can be controlled via the control room. Therefore needed a modern SCADA system so that the working efficiency of geothermal power generation systems may increase. This paper explain the design SCADA simulator Kamojang Geothermal Power Plant. The method used for design of this simulator is the experimental method. The design of this simulator used Wonderware Intouch 10.1 applications. This SCADA design can demonstrate flow diagram Kamojang Geothermal Power Plants with an attractive appearance and easy to understand. Historical trends, real-time trend and alarm systems are some parts of this SCADA design the display. Topic: Engineering and Technology Innovation 44 Designing A SCADA System Simulator For Nuclear Power Plant type Fast Breeder ReactorEka Nugraha, Ade Gafar Abdullah, Dadang Lukman Hakim SCADA (Supervisory Control and Data Acquisition) system simulator is a Human Machine Interface-based software that is able to visualize the process of a plant. This study describes the results of the designing a SCADA system simulator that aims to facilitate the operator in monitoring, controlling, handling the alarm, accessing historical data and historical trends in Nuclear Power Plant (NPP) type Fast Breeder Reactor (FBR). This simulation uses technical data from NPP Kalpakkam, India. This simulator was developed using Wonderware Intouch software 10 and is equipped with main menu, plant overview, area graphics, control displays, setpoint display, alarm system, real-time trending, historical trending and security system. This simulator can simulate both the principle of energy flow and energy conversion process on NPP type FBR. This SCADA system simulator can be used as training media for plant operators. Topic: Engineering and Technology Innovation 45 Designing Media Flip Chart-Based Local WisdomDetria Tisna Putri, Tati, Isma Widiaty This research is motivated by the lack of instructional media batik based on local wisdom in Vocational High School (SMK). The research objective is to design learning media on the subjects of batik based on local wisdom at SMK Negeri 14 Bandung in media forms flip chart. The method used is the Research and Development (R & D) model ADDIE (Analysis, Design, Development, Implementation, and Evaluation), which has been adapted to the needs of the research includes the analysis, design and development or production. Samples were obtained from three experts learning batik, two instructional media experts, and two experts in West Java batik. The results obtained in the form of a product of learning media in the form of media flip chart for subjects based batik local wisdom. Media sheets behind on the subjects of batik based on local wisdom that has been designed showing the results of feasibility aspects as a learning medium is at a very decent criteria whereas the feasibility aspects of local knowledge content are in decent criteria. Recommendations can be delivered to the subject teachers are expected to use the media batik batik learning based on local wisdom that has researchers created. For further research to be followed up in digging the content of West Java batik from other regions or to develop other types of learning media. Topic: Vocational Education and Training 46 DEVELOPING PROJECT-BASED LEARNING ON ENTREPRENEURSHIP SUBJECT MATTER AT FASHION DESIGN STUDY PROGRAMSicilia Sawitri1, Ade Novi Nurul Ichsani2, Siti Nurrohamah3, The research aims to: (1) develop Project-Based Learning on Entrepreneur material at Home Economics Department, (2) produce kits of the Entrepreneurship Project-based Learning Model. Research and Development method was use in the research. The development stage consisted of 3 phases, i. e, (1) Need analyzed, (2) design the model, (3) validation. The instrument were used in the research Data analyzed were used percentage descriptive. The mean validity of the kits models from 3 expert were: 3.5, it was in good validity. The model kits can be used in the instructional activity in Entrepreneur Subject Matter. The result was, (1) the developed Project-based Learning entrepreneur model, (2) The validity of the model by expert, (3) the developed model satisfies validity criterion to be used on Entrepreneurship Subject Matter at Home Economics Departent. Topic: Innovation in Teaching and Learning 47 DEVELOPING BASIC TEACHING MATERIALS AND ELECTRICAL MEASUREMENT FOR VOCATIONAL HIGH SCHOOLALIANGGA KUSUMA, MUKHIDIN, BACHTIAR HASAN This research is aimed to develop teaching materials of basic principle and electrical measurement, and to figure out the feasibility of teaching materials which have been developed in SMK Negeri 1 Koba Kab. Bangka Tengah Bangka Belitung. This research method using the research & development (R & D) that begins with a preliminary study followed the development phase and product assessment. Research subjects are determined by using purposive sampling technique that consists of three expert judgment and 28 students electricity utilization installation techniques. Collecting data with tecnic documentation study, interview and questionnaire in the form of a questionnaire. The procedure of this research is the process of preparing the basic teaching materials and electrical measurements through the stages as follows: (a) A preliminary study, by conducting interviews with the head of the program and teachers. Studies syllabus analysis and analyzing core competencies in materials development. (b) Collect a source of learning and literature as well as the main points of the material to be prepared. (c) Preparation of draft basic teaching materials and electrical measurements. (d) The trial is limited products. (e) More extensive product testing. (f) Data processing and evaluation. The conclusion of this study is the development of teaching materials and the basic subjects, namely electrical measurements of curriculum materials developed in 2013 and syllabus basic subjects and electrical measurements, so that the teaching materials developed starting from the material of electric current and electron current, electrical materials, passive elements, active elements, resistive circuits direct current, two poles theorem, power and effort, switching circuits, system of units of measurement, measuring instruments and electrical measurements , and the type of measuring device. The validation of the feasibility of teaching materials and assessed by expert judgment, whereas for readability and use basic teaching materials and electrical measurements on the test in a number of students with results in both categories. Topic: Vocational Education and Training 48 DEVELOPING PERFORMANCE ASSESSMENT INSTRUMENT TO EVALUATE THE COMPETENCE ACHIEVEMENT IN PATISSERIE LEARNINGMauren Gita, Ana, Sudjani The lack of standardized assessment instruments which can assess student achievement competence in learning patisserie made the researchers motivated to create it. The purpose of this study is to create the design of assessment instrument which can be applied in teaching patisserie. This study used Developmental Research method. The development of performance assessment instrument consists of assessment indicators and assessment rubric. The development of assessment indicators must refer to the patisserie learning curriculum, which is the standard competence in \u201cMaking Bread Products\u201d and the basis of competence in \u201cMaking Hard Roll and Soft Roll Bread\u201d. The samples of this study were the eleventh graders and the subject teachers of SMK Negeri 9 Bandung, East Java. This study was conducted into two steps: (1) the preliminary stage, which consists of the identification problem, and (2) the formative evaluation stage, which consists of assessment theory, constructing the initial design model of instrument and rubrics, validation and revision, limited testing, and making report. The content and evaluation experts had validated well the assessment indicators to be used in field test. The results showed that the performance assessment instrument was suitable for assessing the student achievement competence and assisting teachers in assessing student achievement. Topic: Innovation in Teaching and Learning 49 DEVELOPING THE EMERGING SKILLS FOR STUDENTS THROUGH WORK BASED LEARNING AT HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATIONOanh D.T.K In Vietnam, the lack of emerging skills of young workers is becoming a huge barrier to workers to meet the requirements of employers. To remedy this situation, higher education institutions in Vietnam is being encouraged to apply learner-centered methods in teaching and learning. Work Based Learning is one of many active teaching methods are widely applied to implement the training program built CDIO approach at Ho Chi Minh City University of Technology and Education (HCMUTE). Practical products concerning System Thinking subject were created by the sophomore students at HCMUTE through Work Based Learning showing that students connect theoretical knowledge with specific work contexts and develop their emerging skills. Topic: Innovation in Teaching and Learning 50 DEVELOPMENT OF TEACHING MATERIALS FOR ELECTROMECHANICAL BASIC WORK SUBJECT FOR VOCATIONAL HIGH SCHOOLHILDA KHUSNUL KHARIMAH, MUKHIDIN, BACHTIAR HASAN This research aims to develop teaching materials for basic electromechanical work subject and determine the feasibility of teaching materials that have been developed by teachers and students. The method used in this research is quantitative research methods using research design and development. Teaching materials developed in this study is the book for Basic Electromechanical Works subject. This text book is for class X SMK Power Engineering Program with a thick 221 pages, which consists of five chapters. The materials contained in the book are based on core competencies and competency as well as the syllabus of basic electromechanical work subject in accordance with the curriculum of 2013. Data processing results obtained from the testing of teachers which are seen from the aspect of feasibility assessment of textbook content, language feasibility and feasibility of graphic aspects of limited trials, are included in good category, while the broader test resulted in very good category. Student eligibility test results seen from the results of the questionnaire limited trial resulted in good categories and the results of more extensive trials are included in good categories. Topic: Vocational Education and Training 51 Development of teaching materials to interactive media using Adobe Flash to enhance the comprehension of Algorithms and Programming course on students of Telkom UniversityH Vidyaningtyas1 , L V Yovita1 , R Mayasari1 , B Hudiono2 and Y Jamiah2 The goal of this study is to see the effect of the use of interactive media on Algorithms and Programming course. Based on preliminary studies, the causes of sub-optimal student learning outcomes are the inadequate both of teaching material and media that are used in the learning. Related to these findings, it is necessary to do a development of teaching materials with interactive media on Algorithms and Programming to support the learning in order to optimize both conceptual understanding and ability of programming language implementation. The population in this study is a first year student of School of Electrical Engineering Telkom University, with a total sample of 30 people. Testing is done by giving the pretest and posttest to the entire sample, which is the pretest is given to students by using teaching materials in the form of powerpoint while the post-test was given to students after trying teaching materials coupled with interactive media. The obtained average value of posttest for each chapter is 94,17% for the Sorting Chapter and 96,67% for the Searching Chapter. Topic: Innovation in Teaching and Learning 52 Development Technology And Engineering Literacy Through STEM-Based Science EducationHarry Firman, Nuryani Rustaman, Irma R Suwarma This article discusses the development of Technology and Engineering literacy (TEL) that needed in facing 21st Century demands. It describes the sample of STEM-based activities to improve TEL that conducted in other country and Indonesia. It promotes further implementation of STEM-based activities in order to develop TEL in Indonesia. Topic: Engineering Education 53 Disaster-friendly Sundanese Traditional Building ConstructionJohar Maknun, Tjahyani Busono, Nuryanto Indonesia is a disaster-prone country. Local communities in particular areas have local knowledge for facing disasters. Such knowledge is commonly kept by members of the communities and applied to their environments, including houses. This research aims to describe disaster-friendly Sundanese traditional building construction. It employs the evaluation method, by comparing Sundanese traditional construction to the standards of disaster-friendly construction. The results indicate that the Sundanese traditional building constructions have been qualified as disaster-friendly buildings. This is supported by the evidence that in the event of an earthquake in 2009, no Sundanese traditional buildings collapsed. Topic: Engineering and Technology Innovation 54 e-agriculture based on wireless sensor network with xbee connection for monitoring soilahmad sumarudin (a*) Adnan Hasyim (a) Agung Efendi (a) we propose e-agriculture system for monitoring soil. this system can monitoring soil status. monitoring system based on wireless sensor mote that sensing soil status. this sensor for monitoring utilize soil moisture, humidity and temperature. system monitoring design with mote based on microcontroler and xbee connection. data sensing send to gateway with star topology with 1 gateway. gateway utilize with mini personal computer and connect to xbee cordinator mode. on gateway, gateway include apache server for store data based on My-SQL. system web base with YII framework. system done implementation and can show soil status real time. Topic: Engineering and Technology Innovation 55 Economic Scheduling for Thermal Power Plants Considering Transmission Losses using Pattern SearchEki Nandang Supriatna, Yadi Mulyadi, Ade Gafar Abdullah In a thermal power plant, fuel is the heaviest item of operating cost. Therefore, it is need scheduling and dispatching the committed power generating outputs so as to meet the load demand at minimum operating cost, while satisfying all unit and system equality and inequality constraints. This paper presents a solution of the economic load dispatch problem including transmission losses using a pattern search algorithm. Transmission losses are presented explicitly in the equation so that the total energy consumption, consisting of both the generator\u2019s operation costs and also the transmission losses, can be well considered. The B-coefficients are used to evaluate transmission loss in the system. The effectiveness of the proposed algorithm has been tested on a thermal power plants system of 500 kV Java-Bali. A result of proposed method is compare with real condition of system. Based on the optimization results, it was found that the pattern search method is more economical than the real condition of system. Topic: Engineering and Technology Innovation 56 EDUCATIONAL LIFE SKILL MODEL WITH INTEGRATED LEARNING APPROACH IN ENHANCING LEARNERS\u2019 INTERESTNeni Rohaeni, Supandi, and Mirna Purnama Ningsih This research is based on the lack of attention of life skill education in formal, non formal and informal education. The purpose of this study is to develop a model education life skill with integrated learning approach to improve the school tuition. The model was developed using the research and development approach, through : 1) the preliminary study, 2) the development of model, and 3) validation of the model. The preliminary study was conducted: a) study literature of the research\u2019s problem, b) identification of the characteristics of life skill education which is integrated in the interest of program based on opinion experts and practitioners, c) the implementation of interest\u2019s program learning, and d) design a model education life skill with integrated learning approach in improving school tuition. The stage of the development of model was conducted: a) development of learning materials of life skill education; b) trials of the design model. The validation was tested to produced a final model through empirical, evaluation and improvements study. Finnaly, the result of this study can provide both the model of life skill education with integrated learning approach in improving learners\u2019 interest, and the teaching materials of life skill education. Topic: Innovation in Teaching and Learning 57 Effect of Added Yam (Dioscorea esculenta) Flour Modified on Some Physico-chemical and Sensory Properties of Synbiotic YoghurtMustika Nuramalia Handayani, Sri Handayani, Dewi Cakrawati This research aims to knowing characteristics of yam (Dioscorea esculenta) flour modified; knowing effect of added yam flour modified on some physico-chemical and sensory properties of synbiotic yoghurt and determining the concentration level of yam flour modified to produce synbiotic yoghurt with the most preferred characteristics of panelists. The method used was a completely randomized design of experimental design which is the treatment factor is the concentration level of addition yam flour modified to produce synbiotic yoghurt with three concentration levels namely 2%, 4%, 6%. Observed characteristics of synbiotic yoghurt include some physico-chemical characteristics (pH, lactic acid levels) and sensory properties (color, flavour, aroma, texture) of synbiotic yoghurt. The results showed that yam (Dioscorea esculenta) flour modified characteristics are follows 7.72% moisture content, ash content of 1.42%, total dietary fiber content of 10.16%, 7.49% inulin content, total starch content of 71.78%, solubility is 77.63%, water absorption 136.65%, 618.86% swelling power, and yield of 10.54%. The addition of yam flour modified in produce synbiotic yoghurt provide a significantly different effect on hedonic quality test (texture), but does not give a significantly different effect on the color, flavour and aroma. Based on the analysis of hedonic quality, synbiotic yoghurt with the addition of yam flour modified 2% is the most preferred characteristics of panelists from the aspect of color, aroma, flavor, texture. It has a pH value of 4.48, lactic acid levels is 1.7%. Topic: Engineering and Technology Innovation 58 EFFECT OF LEARNING ACHIEVEMENT AND FAMILY BACKGROUND OF MOTIVATION ENTREPRENEURSHIP AT UPI DPTA BANDUNGRISKHA MARDIANA ; M. SYAOM BARLIANA ; BACHTIAR HASAN This study aims to determine the influence of learning achievement and family background to the motivation of entrepreneurship in students of Architecture and Architectural Education at UPI DPTA. The study took place in Indonesian Education University (UPI) Study program Architecture and Architectural Education. The population in this study amounted to 66 students and the sample taken is student class of 2011, amounting to 61 students. This study uses a quantitative approach with descriptive methods correlation. Testing requirements analysis including normality test and simple regression. Menungkapkan research results that: (1) learning achievement against the background of the family gave a low influence on the study program Architectural Engineering, while the department of architectural engineering education learning achievement against the background of the family gives the same effect is low. (2) Achievement of learning on entrepreneurship motivation is very low influence on architectural engineering study program, nor is the case with the education department of architectural engineering is very low among members influence learning achievement towards entrepreneurship motivation. (3) In order to influence the architectural engineering department of backgrounds with low entrepreneurial motivation, while the education department of architectural engineering background on entrepreneurship motivation influence is very low. (4) The learning achievement and family background has a low influence on the department of architectural engineering, architectural engineering education for the department of learning achievement and family background and motivation for entrepreneurship members of different influences which is very low. Topic: Vocational Education and Training 59 Effect of Student Participation in Production Unit to the Student Work Readiness in the Field of Agro-IndustryPuji Rahmawati Nurcahyani, Fika Awalia Rizki, Yatti Sugiarti Vocational high school students (SMK) is prepared to enter the workforce and become profesional employee. But not all vocational students are ready to enter the working world. Therefore, the school seeks to develop students skills in the activities of the unit produksi. But not all vocational students are ready to enter the working world. Therefore, the school seeks to develop students skills in the activities of production units. The purpose of this study was to determine: 1) the influence of students participation in the production unit to the job readiness of students of agricultural processing (the field of Agro-Industry); 2) the influence of student participation in the production unit to the job readiness of students of agricultural processing (in the field of Agro-Industry). The research method used is descriptive regression analysis with quantitative approach. The study population was students of agricultural processing SMK 1 Kuningan. The sampling technique using probability sampling. The results showed: 1) the participation of students in the production unit and job readiness of students in the category is quite high; 2) The influence of positive and significant correlation between student participation on the readiness of the students work in the field of Agro-industry; 3) The regression coefficient generated is 741 (high or strong); 4) Contributions participation of students in the production unit to the job readiness of students is 54,9% (high enough). Results of the study are that the higher participation of students in the production unit support to the higher job readiness of students. Topic: Vocational Education and Training 60 Effects of computer simulation-based learning on students\u2019 academic achievement and skill performance in woodwork technology: A study of Federal University of Technology, Minna, NigeriaRobert Ogbanje okwori phD This study was carried out to determine the effects of computer simulation- based learning on students\u2019 academic achievement in woodwork technology. The study was conducted at Federal University of Technology, Minna, Nigeria and it used all the students in 300 level where there was high number of students in woodwork technology option. The study adopted quasi-experimental research design which involved groups of students in their intact class assigned to treatment groups. The entire population of 300 level made up of 25 students specializing in woodwork technology was used for the study of which 13 students were assigned to Woodwork Technology Computer simulation Package (WWTCSP) and other 12 students assigned to conventional method. These individual groups were examined after using these methods to learn for two weeks. Two research questions and two null hypotheses were tested at 0.05 level of significance. The instruments used for data collection were Woodwork Technology Achievement Test (WWTAT) and Woodwork Technology Computer Simulation Practical Test (WWTCSPT). The instruments were subjected to validation by three experts in woodwork technology section of the above university. The instrument was also was subjected to pilot study and a test- retest method was employed. Pearson Product Moment correction was used to determine the reliability coefficient of the instrument and it was found to be 0.86. Mean was used to answer the research questions while t-test statistics and ANCOVA were employed to test the hypotheses. The study found out that computer simulation- based learning is more effective in improving students\u2019 academic achievement and skill performance than conventional method of learning. The study found out that there is no significant interaction effect between learning methods and gender of students in the test on woodwork technology achievement. It is therefore, recommended among others that woodwork technology lectures should emphasize computer simulation- based learning towards effective understanding of the topic taught in woodwork technology and the university should provide adequate computer for effective participation of students in computer simulation-based learning. Topic: Vocational Education and Training 61 Effects of MgO buffer annealing on optical and electrical quality of P-MBE grown ZnO filmsAgus Setiawan Zinc oxide (ZnO) has been attracting much attention because of its potential applications in photonic and optoelectronic devices. The notable properties of ZnO include direct band energy gap (Eg=3.37 eV at RT), large exciton binding energy of 60 meV, and strong cohesive energy of 1.89 eV. In this present study, we investigated the effect of MgO buffer annealing on the optical and electrical quality of P-MBE grown ZnO films on c-sapphire with MgO buffer layer. The optical quality of the ZnO samples were observed by low-temperature PL (photoluminescence) measurement in the near band edge emission region measured at 10K and at 77K. The emission line located at 3.368eV dominates the spectrum in both samples at 10K. This emission can be divided into two peaks, 3.367eV and 3.363eV and assigned as I2 (ionized donor bound excitons emission) and I4 (Hydrogen donor related emission), respectively. These lines are clearly resolved into two lines at 77K in the sample with annealing. The relative intensity of these donor bound exactions to free exaction emission of the sample without MgO buffer annealing is greater than that of the sample with MgO buffer annealing. At the lower energy region, a sharp emission located at 3.335eV and longitudinal optical (LO) phonon assisted bound exaction emission were observed in both samples. For the temperature dependence of PL spectra, the intensity of the emission line at 3.335eV reduces like donor bound exactions with increasing temperature. Accordingly, this line can be attributed to exaction nature. At 77K, free exciton-A emission located at 3.374eV becomes dominant instead of donor bound excitons. The free exciton-B emission appears at 3.380eV with 6 meV energy spacing to the free exciton-A emission. LO-phonon replica of the free exciton emission clearly appeared at 3.312eV, 3.239eV, and 3.166eV. As each spectrum of excitonic emission lines can be resolved, the emission lines from the sample with MgO buffer annealing have narrower spectral width compared to those of ZnO without MgO buffer annealing. This indicates better crystal quality of ZnO film with MgO-buffer annealing. From the room temperature PL spectra measurement, we found that the free exciton emission at 3.31eV and a broad deep-level emission at around 1.75eV were observed in both the samples. Comparison of the PL spectra of ZnO with and without annealing revealed that the intensity of free exciton emission from the sample with MgO buffer annealing is twice of that from the sample without annealing. We also found that the intensity of deep-level broad emission is reduced by about 1\/3 by MgO-buffer annealing. Hence, the decrease of deep level emission intensity and the increase of free exciton emission intensity by annealing of MgO buffer corresponds to the reduction of defects of the ZnO film. The PL properties suggest that there are fewer nonradiative recombination centers in ZnO layers with MgO buffer annealing than those in ZnO layers grown without MgO buffer annealing. The electrical quality was measured by room temperature Hall measurements. We found that the samples have a background n-type carrier concentration. The ZnO samples with MgO buffer annealing has carrier concentration of 1.17x1017 cm-3 and Hall mobility of 120 cm2\/V.s, while the ZnO sample without MgO buffer annealing has carrier concentration of 2.63 x 1016 cm-3 and Hall mobility of 105 cm2\/V.s. The improvement of electron mobility of ZnO films by MgO buffer annealing is due to a decrease in dislocation density. We conclude that annealing of MgO buffer layer increases the optical and electrical quality of the ZnO films. These results agree with the structural quality as observed by HRXRD. Topic: Engineering and Technology Innovation 62 Effects of Using Passive Filter for Reduce Electrical Load HarmonicsTasma Sucita Due to the use of electrical current load that uses a lot of electronic components (passive non-linear electrical loads), so the impact will cause harmonics in the electrical network system. These harmonics can unwittingly cause a relatively large loss in electrical energy consumption and can lower the power factor of an electrical installation. Limits how much the harmonic distortion that is installed on the load adjusted to the IEEE 519-1992 standard. The study was conducted by taking data on a network of electrical installation of a building using measuring devices Fluke 43B Power Quality Analyzer. The data is then processed and consulted with the standard IEEE 519-1992. Once the data has a discrepancy with the standard, further made the filter design using linear passive components. The design is then installed on the network installation by means of simulated order harmonic losses can be overcome so that the circuit meets the IEEE standard installation by changing the parameters of the linear load L and C. The results of this study indicate that THDi value decreased after the installation of filters for phase R fell by 9.39%, the S phase decreased by 7.54% and for the T phase decreased by 16.88%. So that meets the IEEE standard by 15%. Topic: Engineering and Technology Innovation 63 Electric Differential at HYVO15 Car using Hub Motor Front Drive for Energy Efficient when Turn Left and RightI Wayan Adiyasa (a*), Aan Yudianto (b) Front-wheel drive electric car using a hub motor has a weakness when the car turn right or left is very difficult to control and needs a lot of energy is wasted due to tire slippage. The study used cars Oddyse Hybrid Vehicle 15 with the microcontroller as a test of the number of revolutions per corner and the current time of loading. The method used is research and development. When turning left and right wheel speed is set with an ATmega328 microcontroller based on input from steering wheel angle and the right and left motor current. Input the angle of the steering wheel will adjust the speed rotary wheel right and left. Input motor current is used to determine the amount of current that must be distributed to adjust the motor speed when turning. The results showed savings of electric current on the vehicle so that the more efficient use of current. HYVO control when turning easier while using Electric Differential. A reduction in front tire skid when turning to the right and to the left. Topic: Engineering and Technology Innovation 64 Embodied Energy Analysis of Wall Construction BuildingSri Novianthi Pratiwi, Trias Megayanti Assessment of environmentally friendly building materials can be analyzed from the life cycle of building materials, ranging from how the building materials were made to how the materials are can be decomposed in environment. The lower harmful emissions to environment, thus the building materials could be considered as an environmentally friendly material. In fact, most of the main resources of the energy that use in the cycle of building materials are from fossil energy. Fossil energy source known for its CO2 emission that is harmful to environment. This aim of this study is to analyze the embodied energy in the materials of wall construction, such as concrete block, cement, sand, and paint. In addition it is expected that the architectural design substance, particularly on type of construction wall could also control the value of embodied energy. Topic: Engineering and Technology Innovation 65 Enhancement of Kumar\u2019s Sentiment Analysis Algorithm with Additional of Target VariableArry Akhmad Arman (a), Adi Bhasyudewo Kawi (b), Ratih Hurriyati (c) There is a lot of research about sentiment out there that the source may come from Twitter or product review on the internet. It is very unfortunate that Indonesia, one of the most active country on the internet does not have many research in their language. This research addressed to develop sentiment classification technique in Indonesian language. Algorithm that we use inspired by research from Kumar and Sebastian [1] with different approach. The technique that we use in this research is considering sentiment\u2019s target that deliberately ignored in classifying sentiment. While the addition of the parameter obtained no significant increase in accuracy, algorithm development shown promising result in providing additional benefit in sentiment classification and get the sentiment\u2019s target. Topic: Engineering and Technology Innovation 66 Enhanching student achievement in MRSM felda (trolak) Malaysia with cooperative learning method type numbered head together (NHT) using smart card and multimedia based learning of endangered ecosystemHeru Setiawan, and Wiwi Isnaeni The aim of this study is determine the effectivity of cooperative learning methods Numbered Head Together (NHT) using Smart Card and multimedia based learning on student achievement in the subject matter Endangered ecosystem in Form 4 MRSM Felda (Trolak) Malaysia in academic year 2014\/2015. The method used was experimental method. Data collection techniques of cognitive learning achievement using tests while affective achievement of students using a questionnaire. Data analysis techniques for hypothesis testing using t- test. From the results of this study concluded that cooperative learning methods Numbered Head Together (NHT) using Smart Card and multimedia media effective to enhanching student achievement of Endangered ecosystem in Form 4 MRSM Felda (Trolak) Malaysia. This is evident from the increase of cognitive average of learning achievement in experimental class is higher than the increase in the cognitive aspects of learning achievement and the control class average value of affective for experimental class is higher than the average value of affective control class. It can be seen from the results of the t test for cognitive and affective learning achievement that greater than t table. Topic: Innovation in Teaching and Learning 67 Enhancing Human Capital High Quality through Technology Revolution and Transformation of Technical and Vocational Education And Training (TVET) in MalaysiaMohamad Amin Hamat & Mahadzir Ahmad Technical and Vocational Education and Training (TVET) is concerned with the acquisition of knowledge and skills for the world of work. This paper discusses the transformation in the field of TVET in workforce in most countries especially Malaysia. Additionally, this paper reviews the evidence, impact and discusses the importance of the TVET transformation for human capital development in Malaysia influence by the technology. In addition to discussing the importance of technology to the TVET actors, especially in the field of TVET this paper also highlight the development of human capital high quality. Topic: Vocational Education and Training 68 Entrepreneurial Model Based Technology Creative Industries Sector Software through the Use of Free Open Source Software (FOSS) for Students UPIBachtiar Hasan, Hasbullah, Wawan Purnama, Agus Heri S Creative industry development areas of software (software) by using free open source software (FOSS) is expected to be one of the solutions to foster new entrepreneurs from the students who can open up job opportunities for them and contribute to economic development in Indonesia. This study aims to create entrepreneurial coaching model based on the creative industries by utilizing FOSS software field as well as provide understanding and fostering entrepreneurial creative industries based field software for students UPI. This activity phase begins with identifying entrepreneurs or business software technology that will be developed, training and mentoring, apprenticeship process at industrial partners, creation of business plans and monitoring and evaluation. This activity involves 30 students UPI which has the motivation to self-employment and have competence in the field of information technology. The results and outcomes expected from these activities is the birth of a number of new entrepreneurs from the students engaged in software and software in the world of commerce (e-commerce) or education or learning (e-learning \/ LMS). Topic: Engineering and Technology Innovation 69 ENTREPRENEURSHIP EDUCATION LEARNING MODEL IN VOCATIONAL SECONDARY SCHOOLSarwa and M. Syaom Barliana The research objective was to develop a learning model of Entrepreneurship Education (EE) in vocational secondary school (SMK). Development of a learning model was designed to ELT (Experiential Learning Theory), which are aligned with best practices in SMK. The research methode was used by R & D model in three stages : preliminary study, model development, and classroom experiment. The results of a preliminary study of 6 samples SMK in Medan, Binjai and Deli Serdang obtained: learning materials on theoritical entrepreneurship, learning methods with lectures and discussions; learning evaluation was conducted using a written test. The design of teaching materials \"Craft and Entrepreneurship\" incompatible with competency of vocational students. Teachers difficult to apply the teaching materials and implementing the scientific method in learning EE. EE learning model that is suitable for vocational students are practical entrepreneurship of technopreneur, Experiential Learning methode, portfolio and observation base evaluation Topic: Innovation in Teaching and Learning 70 ENTREPRENEURSHIP EDUCATION LEARNING MODEL IN VOCATIONAL SECONDARY SCHOOLSarwa, M.Syaom Barliana, Asari Djohar, and Suryana The research objective was to develop a learning model of Entrepreneurship Education (EE) in vocational secondary school (SMK). Development of a learning model was designed to Experiential Learning Theory (ELT), which are aligned with best practices in SMK. The research method was used by Research and Development (R & D) model in three stages : preliminary study, model development, and classroom experiment. The results of a preliminary study of 6 samples SMK in Medan, Binjai and Deli Serdang obtained: learning materials on theoretical entrepreneurship, learning methods with lectures and discussions; learning evaluation was conducted using a written test. The design of teaching materials \"Craft and Entrepreneurship\" incompatible with competency of vocational students. EE learning model that is suitable for vocational students are practical entrepreneurship of technopreneur, Experiential Learning Method, portfolio and observation base evaluation. Topic: Vocational Education and Training 71 Evaluation of Expert System Application Based On Usability AspectsCindy P. C. Munaiseche, Olivia E. S. Liando Artificial Intelligence (AI) is the part of computer science concerned with designing intelligent computer systems, that is, systems that exhibit the characteristics we associate with intelligence in human behavior. AI programs that achieve expert-level competence in solving problems in task areas by bringing to bear a body of knowledge about specific tasks are called knowledge-based or expert systems. Today, expert systems are widely applied in medicine for prognosis, diagnosis, checkup, and treatment of various kinds of diseases. This paper presents an expert system that diagnosing skin diseases at early stage can enable to overcome and treat them appropriately. Usability testing conducted to determine whether or not this expert system satisfies the acceptance criteria and to enable the user, customers or other authorized entity to determine whether or not to accept the system. This research used quastionnaire as the primary method of usability evaluation with affect learnability, efficiency, memorability, errors, and satisfaction as evaluation criteria. The software evaluation result of this application indicates that the value of usability acceptance by the user is over the number of 3 (over the median) in the scale of 5 or has an average value of 4. In general, the web application software has a good value of usability, i.e. learnability, efficiency, memorability, errors and satisfaction. Topic: Engineering and Technology Innovation 72 EXPLORATION OF MAKING DATE SEEDS\u2019S FLOUR AND ITS NUTRITIONAL CONTENS ANALYSISMeda Wahini Date palm (Phoenix dactylifera L) is a palm plant that is identical with peoples lives the Middle East and the Arabic kingdom. Date palm is known and consumed by most of the people in the form of pulp, while the seeds are discarded, whereas palm seeds are rich in nutrition. Therefore, it need to be explored the potential of date seeds through innovation production of foodstuffs with a high nutritional value. The aims of this study were to 1) trial how making flour from the date seeds, and 2) determine the nutrient content of the date seeds flour. This study was experiment in the making of the date seeds\u2019s flour treated with different drying process on seeds. It conducted in July, 2015 at the laboratory of food technology, Family Welfare Education department, The State University of Surabaya. The results showed that 1) the processing of date seeds into flour is washing, soaking in sodium bisulfite for 24 hours, washing over, boiling at 100 \u00b0 C temperature for 60 minutes, draining, drying in the oven at 100 \u00b0 C temperature for 6 hours and drying under the sun for 2 days, grinding through the destruction with a hammer and milling machines, sieving, and flour ready for use; 2) date seeds Flour with sun dried contains 33.84% of carbohydrates; 26.52% of Protein; 31.54% of Fat; Minerals (102.42 mg \/100gr of Ca, 81.05 mg\/100gr of P, 16.54 mg\/110gr of Fe), also vitamins ( 3.08 mg\/100gr of B, 23.55 mg\/100gr of C); whereas the nutritional value of date seeds flour through a drying oven was 5.03% of protein; 12.37% of fat; Minerals (64 mg\/100gr of CA, 62 mg\/100gr of P, 16.54 mg\/100g of Fe), and vitamins (5% of A, no C, 2% K). This study explain that date seeds which is considered of waste, but in fact it has high-value, that can be alternative ingredient which subtitute wheat flour Topic: Engineering and Technology Innovation 73 Extract Transformation Loading From OLTP Data to OLAP Data using Pentaho Data IntegrationReynaldo J. Salaki, Jimmy Waworuntu, Irene R.H.T. Tangkawarow Support data warehouse expected to solve the evaluation problems of teaching and learning outcomes as well as the information relevance received as a support in decision-making by leadership (executive) level. The data warehouse is important to designed utilize the existing information resources. Data warehouse for student' scores can help the process of evaluation, decision making even further planning of PTIK study program. Diversity of data source in PTIK study program made decision-making and evaluation not easy. Pentaho Data Integration used for integrate data in PTIK easily. This data warehouse design with multidimensional database modeling approach by using dimension table and fact table. Topic: Engineering and Technology Innovation 74 FACTORS AFFECTING THE COMPETENCE OF WORKING STUDENTS SMK STATE TOURISM IN HOTEL AND RESTAURANT BANDUNGNIA LESTARI; DANNY MEIRAWAN; ANA This study aims to determine \"the factors suspected to affect the competence of the authors work to limit to two variables: motivation and work attitude. The study took place in SMK Negeri 9 Bandung. Population in this research is the students who have done the work practices of the industry (prakerin), the number of members of the study sample totaled 52 people, sample testing of instruments made to students who have done prakerin outside the sample. This study using inferential method with quantitative approach. Data were collected through questionnaires, documentation, and testing. The raw data obtained later in the reduction, are summarized and grouped by category of existing and tested the validity, reliability, and hypothesis. Results of the study revealed that: (1) There is influence that significance between work motivation to work competence, among others include the interest in the work can be categorized as very good and perseverance in carrying out work practices, tenacity in working and sharpness of attention at work, a desire to excel in work practices, and independence in the following aspects of employment practices that are in the category of good. (2) There is no significant effect, work attitude toward work competence. Working attitude towards work competence include: be able to keep objects \/ equipment entrusted, aspects of the discipline in the form of attending regularly and on time, and obey the rules \/ regulations, liable to the task given, accept the risk of the action taken and maintaining and storing equipment \/ goods, cooperation be able to work together in groups, maintaining effective working relationships, and provide assistance and support to others, initiatives such as looking for new challenges, develop themselves and the opportunity to learn, and have an idea \/ action and innovative solutions. (3) work motivation and work attitude does not affect the competence of work. Topic: Vocational Education and Training 75 Factors that affect of Leadership Effectiveness in Vocational High SchoolNathanael Sitanggang The objective of the research were to investigate the effects of interpersonal skills, task structure and directive leader behaviour on leadership effectiveness. This research was conducted at Vocational Schools (SMK) in Medan, using the survey method with 132 principals as population and the sample of 60 principals as respondents who were selected by applying proportional random sampling. The hypotheses were tested by path analysis. This research findings were as follows (1) There was a significantly direct positive effect of interpersonal skills on directive leader behaviour; (2) There was a significantly direct positive effect of task structure on directive leader behaviour; (3) There was a significantly direct positive effect of interpersonal skills leadership effectiveness; (4) There was a significantly direct positive effect of directive leader behaviour on leadership effectiveness. Topic: Vocational Education and Training 76 First Principles Molecular Dynamics Simulation of Graphene Growth on Nickel (111) SurfaceRizal Arifin(a,c*), Yasushi Shibuta(b), Kohei Shimamura(a), Fuyuki Shimojo(a) The mechanism of the graphene growth on the nickel (111) surface is studied using first principles molecular dynamics (MD) simulation. The effects of growth temperature and concentration of carbon atoms on the nickel (111) surface are investigated. It is found from the MD simulation that the optimum temperature for the graphene growth is approximately around 1000 K, although the graphene can also be grown at lower temperatures (~800 K) in a high concentration of carbon atoms which is in good agreement with the experiment. Topic: Engineering and Technology Innovation 77 FPMIPA Building form and layout, their role on campus microclimateMurry, Shinta, Beta FPMIPA is one of the greatest architecture design in Indonesia University of Education. Design and built under the provision of JICA, Japan has made this building as one of the good sample in building education facility. The role of building form and layout of FPMIPA then going to describe their impact toward campus microclimate. The measurement during the hottest day in October describes the implication of physical aspect such as building height, distance between them and green open space toward the microclimate condition. Topic: Engineering and Technology Innovation 78 Going back to the past : The Design of Historical Museum of Indonesia\u2019s People StruggleR. Arry Swaradhigraha, R. Irawan Surasetja, Trias Megayanti History is a legacy that must understood for the young generations, especially for history of people struggle. Citizen of Indonesia have to regrow the values of struggle that has been by our heroes to building this people and country. The values will cast apathetic attitudes that cause people\u2019s problems at this moment . The values is important in developing the nation\u2019s character. Although the country has stood for almost 70 years, it is still struggling to be a developed country due to apathetic. Normally, history only packed in books and movies. Through architecture could changes be a space form, hence the citizen could understanding with feels conditions as the time it happened. Through the space, values, and essence of pass time could delivered to citizens. The concept of museum is aim visitors for understanding the history with flow, hence visitors could understand the whole history. The visitors will flowed of the first time is National Awakening until the time of Revolution, hence the visitors could Interpret the heroes\u2019 struggle in built this country.Through such historical structure, it is expected that the lessons in the history, which are usually delivered in books or papers, could be conveyed through real-life experience in a simpler manner. Finally, the visitors\u2019 mindset could change because they\u2019ve interpreted heroes\u2019 struggle. Hence, the structure could contribute more on building the nation\u2019s character. Topic: Engineering and Technology Innovation 79 Green and Sustainable Development in Asia Pacific CountriesProfessor Dr. Ramlee B. Mustapha, Ph.D Green paradigm is emerging in Asia. In order to achieve sustainability, green development is critical. The growing significance of sustainability is having a major impact on business, industry, and society as a whole. Hence, preparing the future workforce for the coming green economy is a challenging task for many Asian countries especially in TVET sector in the post-2015 agenda. As a ground work, transforming TVET in Asia to meet the challenges of the green economy for the purpose of sustainability should begin now. The aim of this paper is to map the sustainable development in terms of green mindset, creativity, lifestyle, economy, education, training, employability and sustainability in selected Asian countries. A country\u2019s quantum leap or leap-frogging in sustainable economy is dependent of its transformation of human resources especially in TVET sector. Thus, TVET sector should be transformed to fit the requirements of the sustainable green paradigm. The results posit the country\u2019s policies, best practices, and challenges toward green economy in order to achieve sustainable development. Finally, the implication of green paradigm on TVET system in selected countries in the Asia Pacific will be discussed. Topic: Vocational Education and Training 80 Growth Mechanism of GaAs1-xSbx Ternary Alloy Thin Film on MOCVD Reactor using Metal Organic Sources TMGa, TDMAAs and TDMASbA. Suhandi1), Y. R. Tayubi1), P. Arifin2) Metal Organic Chemical Vapor Deposition (MOCVD) is a method for growing a solid material (in the form of thin films, especially for semiconductor materials) using vapor phase metal organic sources. Studies on the growth mechanism of GaAs1-xSbx ternary alloy thin solid film in the range of miscibility-gap using metal organic sources trimethylgallium (TMGa), trisdimethylaminoarsenic (TDMAAs), and trisdimethylaminoantimony (TDMASb) on MOCVD reactor has been done to understand the physical and chemical processes involved. Knowledge of the processes that occur during alloy formation is very important to determine the couple of growth condition and growth parameters are appropriate for yield high quality GaAs1-xSbx alloy. The mechanisms has been studied include decomposition of metal organic sources and chemical reactions that may occur, the incorporation of the alloy elements forming and the contaminants element that are formed in the gown thin film. This paper describes the results of studies on the visibility and growth mechanism of GaAs1-xSbx alloy on MOCVD reactor by using a regular-solution model. As evidence of the potency of the MOCVD technique in growing GaAs1-xSbx ternary alloy in the range of miscibility gap, this exposure is also supported by the data resulted from the experimental of growing GaAs1-xSbx thin film using MOCVD technique. Tipe of MOCVD Reactor used in this study is a vertical type contained in Laboratory for Electronic Material Physics at Departement of Physics ITB Topic: Engineering and Technology Innovation 81 Growth Mechanism of Co:TiO2 Thin Film Deposited by Metal Organic Chemical Vapor Deposition TechniqueAip Saripudin and Pepen Arifin Cobalt-doped TiO2 thin films were grown on n-type silicon substrate. The films were grown by metal organic chemical vapor deposition method. The growth temperature was varied of 325oC \u2013 450oC. The films were characterized by SEM. In this research, we focus on the growth mechanism of film. Using Arhenius equation, it is known that the activation energy value of film growth is positive at the range of temperature of 325oC \u2013 400oC and negative at the range of temperature of 400oC \u2013 450oC. These results show that the decomposition rate at the range of temperature of 325oC \u2013 400oC is due to diffusion phase of precursor gaseous. On the other hand, the decomposition rate decreased because the precursor gaseous decreased and the surface chemical reaction was high. Topic: Engineering and Technology Innovation 82 HALAL EDUCATION IN TVET: ROLES OF MALAYSIAN POLYTECHNICS IN CREATING HALAL COMPETENT WORKFORCEAHMAD SAHIR JAIS, ZULIANA ALIMAN, KAMAL ALI This paper will discuss on the position and function played by Malaysian Polytechnics in halal education, in the context of Technical, Vocational Education and Training (TVET). This paper tries objectively to examine the position of halal in TVET framework, discuss and examine the role of polytechnics in producing halal competent graduate and finally to seek the significance of dietary halal in TVET framework holistically.Dietary halal in Malaysia has gained prominence exposure in recent years, due to the heighten awareness among Muslim consumers. Therefore, this has contributed to a surge in demand for halal food. The growth in halal sub-sectors have a consequent and significant effect on the demand for halal competent workforce, resulting in inadequate supplies of workforce towards the industries that cannot be matched by the educational institution.A critical review of previous literature as well as documents analysis of the curriculum structure, highlighted several themes concerning dietary halal in the Malaysia\u2019s educational system, as well as the depth of halal education ingrained in the Malaysian polytechnics education system.The results indicate that halal education was not given prominent spots and due recognition in the overall TVET framework in Malaysia. Halal education was sparsely highlighted in the education system. In tertiary level, Malaysian Polytechnics has taken up the role of producing halal competent workforce at a low and supervisory level by introducing halal related and focused programs and courses, fostering collaborative effort with the industry and developing halal curriculum that match industry requirements. Halal market are expected to grow exponentially in later years, which seems highly significant to place and introduce halal as part of TVET educational content. The role of polytechnics in producing halal competent workforce cannot be denied. It filled a gap of producing halal competent workforce at the middle and supervisory level, whereby the demand for this position is huge as compared to a managerial position. In a larger TVET Framework, halal should be included or at least embedded as part of the TVET structure since the demands for halal competent workforce are in demand. Topic: Vocational Education and Training 83 Impact of Changing Influence Range Fuzzy Subtractive Clustering for Anomalous Load Forecasting AccuracyFirna Anindyaputri Respati, Ade Gafar Abdullah, Yadi Mulyadi Short term load forecasting (STLF) has important role for reliability and economic operation of electrical power system. In this paper , fuzzy subtractive clustering (FSC) method is used in STLF of electrical power system for spesial days in anomalous load conditions. These anomalous loads occuring during national holidays. This method is applied on dataset of Region 2 Java-Bali to forecast the load demand on half-hour in national holidays (anomalous load). The proposed methodology has been to decrease the forecasted error value. Finally, the result shows that FSC implementation for STLF of regional load have more acuraccy and better outcomes. Topic: Engineering and Technology Innovation 84 Impact of Changing Influence Range Fuzzy Subtractive Clustering for Anomalous Load Forecasting AccuracyFirna Anindyaputri Respati, Ade Gafar Abdullah, Yadi Mulyadi Short term load forecasting (STLF) has important role for reliability and economic operation of electrical power system. In this paper , fuzzy subtractive clustering (FSC) method is used in STLF of electrical power system for spesial days in anomalous load conditions. These anomalous loads occuring during national holidays. This method is applied on dataset of Region 2 Java-Bali to forecast the load demand on half-hour in national holidays (anomalous load). The proposed methodology has been to decrease the forecasted error value. Finally, the result shows that FSC implementation for STLF of regional load have more acuraccy and better outcomes. Topic: Engineering and Technology Innovation 85 Implementation of Cloud Computing in Higher EducationAsniar (a*), Reza Budiawan (b) Cloud computing research is a new trend in distributed computing, where people have developed service and SOA (Service Oriented Architecture) based application. This technology is very useful to be implemented, especially for higher education. This research is studied the need and feasibility for the suitability of cloud computing in higher education in Indonesia. Besides, we propose a model to determine the cloud model for cloud computing that can be implemented in order to support academic activities. Topic: Engineering and Technology Innovation 86 Implementation of Life Skills Oriented Education Through Learning Reorientation and School Reform on SMKTasma Sucita, Sumarto, Janulis Purba, Bachtiar Hasan This study aimed to achieving the implementation pattern of implementation of life skills education through a reorientation of learning and practice reforms implemented in vocational education. Research conducted on the CMS are in the region of West Java province with four samples SMK SMK and one private. The method used in this research is quantitative method with a problem-solving approach descriptive, associative causal and associative reciprocal. Research conducted on vocational leaders, teachers productive subjects, and students in grade 3 (XII) which has competence Power Installation Engineering expertise with random sampling techniques. Findings from this study indicate that the implementation study conducted at SMK in West Java Provincial Education Department is in conformity with the goal of life skills-oriented education. This can be evidenced by the achievement of an average implementation reorientation of learning undertaken by prolific teacher showed significant support is quite good, and the reform of educational practices contribute to the achievement of the competence of graduates achieve a significance level sufficient. The achievement of competencies possessed by students in grade 3 (XII) before graduating based on analysis of data obtained from five vocational samples showed average figures either. Competence in both categories that have been owned by the vocational graduates is the initial capital that can be used as a provision of life skills to enter the workforce. With the results of this study may be able to contribute to education, especially regarding the implementation of system-oriented education in the field of vocational life skills. Topic: Vocational Education and Training 87 IMPLEMENTATION OF OCCUPATIONAL HEALTH AND SAFETY TOWARDS LEARNING THE PRACTICE OF PRODUCTIVE IN WORKSHOP SMKAGUS HARIS ABADI, BACHTIAR HASAN, WOWO SUNARYO KUSWANA, This study aims to explore, describe and analyze the K3 planning, implementation K3, K3 barriers, students cognitive abilities, the ability of the student affective, psychomotor abilities and influence the implementation of K3 students to study of productive practices in vocational workshop. The study took place in SMK Ciwaringin. The main data of this study were obtained from students and head of the study program as a key informant, while supporting data were collected from various documents at the workshop practices and student of SMK PGRI Plumbon Cirebon as sample testing instruments. This study uses a exploration and quasi experimentation with times series design. Data were collected through questionnaires, interviews, observation, documentation, and testing. The raw data obtained later in the reduction, are summarized and grouped by category of existing and tested the validity, reliability, level of difficulty, and hypothesis. Results of the study revealed that: (1) the exploration stage; a) the safety and health plan is generally categorized enough; b) the implementation of K3 in the learning practices in the category of less productive; c) K3 barriers to learning such productive practice practitioner awareness of the K3 and limited cost-tools to buy new equipment and personal protective equipment. (2) the experiment stage; a) the students cognitive abilities, including the medium category with an increase in the average value gain of 55% after the treatment; b) affective abilities of students, including the medium category with an increase in the average value gain of 45%; c) the ability of psychomotor skills in students include assemble and disassemble a component in accordance K3 in the category with an average gain of 65%. Topic: Vocational Education and Training 88 Improvement of Competence Students Through the Implementation of Problem Based Instruction on Laboratory of Building Materials\u2019 CourseNurmi Frida Dorintan B.P. This research is a classroom action research to students of Building Technique Education Unesa. Subjects were 38 students who programmed the course of laboratory building materials. The research objective is to improve the competence of students, including cognitive, psychomotor and affective through the implementation of teaching based problems. Teaching is planned so that students are actively engaged and active to accomplish something on their own, or guided, and conducted in three cycles with the subject matter: 1) testing of bricks; 2) testing of tile materials, and 3) testing of ceramics. Indicators of success in cognitive and psychomotor aspects of competence can be achieved when 75% of all students obtained a score of at least 75. And the affective aspect is achieved when 75% of all students were on a scale of 4 and 5. The results showed that: 1) cognitive competence of students in cycle 1 was obtained by 39.47%, amounting to 57.89% cycle 2, and cycle 3 amounted to 76.32%. It means in the third cycle of the expected cognitive competency has been achieved, that is more than 75% of the students have achieved a minimum score of 75; 2) psychomotor competence in the form of performance tests show that, in cycle 1 was obtained 55.26%, 68.42% for cycle 2 and cycle 3 amounted to 78.95%. That is, in the third cycle of learning has reached a psychomotor competencies expected because more than 75% of the students have achieved a minimum score of 75; 3) affective competencies derived from the Likert scale observation shows that in cycle 1 was obtained 60.53%, 71.05% for cycle 2 and cycle 3 amounted to 81.58%. That is, in the third cycle of learning subjects building materials laboratory has achieved the expected results because more than 75% the number of students has reached a scale at least 4. The conclusion of this study are: problem-based instruction includes practice solving problems and providing measures for solving problems can increase student competency either cognitive, psychomotor, and affective on course materials laboratory building with the subject material testing bricks, tiles and ceramics. Topic: Engineering Education 89 IMPROVING EMPLOYABILITY SKILLS OF TOURISM VOCATIONAL HIGH SCHOOL STUDENTS THROUGH COLLABORATIVE INQUIRY LEARNING MODELa1Krisna Primanti, a2Ana The research aims to know the improvement of Employability Skills of Tourism Vocational High School students through collaborative inquiry learning model application. The employable skills of vocational high school students are observed from problem solving skill aspect, collaborative skill and communicative skill that must be mastered by vocational high school students to be skilled workers. Syntax in collaborative inquiry learning model consists of Framing the Problems, Collecting the Evidence, Analyzing Evidence, and Documenting, Sharing and Celebrating. The research used quasi experimental design \"Single \u2013 Group Interrupted Time \u2013 Series Design\". The technic uses : 1) Objective Test to measure students problem solving skill; 2) Observation sheet to observe the syntax implementation of collaborative inquiry learning model and to measure students collaborative and communicative skills; 3) Questionnaire to know students perceptions of collaborative inquiry learning model implementation towards student\u2019s employability skills improvement. The result shows that through collaborative inquiry learning model, students employability skills such as problem solving skill, collaborative skill, and communicative skills are improved in every meetings, students perception of collaborative inquiry learning model gets positive responses. Topic: Innovation in Teaching and Learning 90 In-Memory Business Intelligent: Concepts and Performance ResultsDr. James. J. Sumayku, M.Eng (a), Vivi.P.Rantung, ST, MISD (*b) Memory as temporary storage has functions to supplying data and saving data helps the work of CPU to be faster. Growth in technology innovation influences the memory related with data management. Business Intelligent as part of data management area takes opportunity and adopts the concepts of In-Memory Database into In-Memory BI. Implement of In-Memory BI takes to solve problem that company faces to improve decisions of enterprise. This paper reviews the benefits of using in-memory BI and gives performance results of in-memory BI application. Topic: Engineering and Technology Innovation 91 INFLUENCE OF DEMOGRAPHIC VARIABLES ON ENVIRONMENTAL CONSCIOUSNESS OF THE LOW INCOME COMMUNITIESM. Syaom Barliana, Tetsu Kubota, Usep Surahman The global society is currently faced to problems an constraints in energy saving measures, one of them is a lack of environmental awareness. Therefore, prior it enters the technical problems of energy saving as part of sustainable architecture, the required first of all the banging and raise public awareness to be more sensitive to the environment. After awareness was raised, then they will be able to apply the cultural attitudes and behaviors that support sustainable architecture. Hence, as a first step, this study wanted to study how far the awareness level of urban communities in Indonesia, and what the external variables (demographic) which influence dominantly. These demographic variables include: gender, education, income, marriage, employment, health, religiosity, leisure, competence, culture\/ethnic, and social life. This pilot research is using a combination of quantitative and qualitative methods. Quantitative methods are used as the primary method of research, while only qualitative methods to support. The research will conduct at lower class residential area in Bandung. The results of the pilot survey showed that education, religious, social meeting, and public spaces variables highly positive correlated, and significant to the environmental consciousness. However, further research could explore the external side ofeducation or religious aspect, ie for example regarding the influence of training in non formal education, community empowerment, religious leaders, community leaders, and other communication media, which can provide environmental knowledge and environmental concern. Topic: Engineering Education 92 INNOVATION IN TECHNICAL AND VOCATIONAL TRAINING IN VIETNAM FOR INTEGRATION WITH REGIONAL AND INTERNATIONALTRUONG MINH TRI INNOVATION IN TECHNICAL AND VOCATIONAL TRAINING IN VIETNAM FOR INTEGRATION WITH REGIONAL AND INTERNATIONAL TRUONG MINH TRI (HCMC University of Technology and Education Vietnam) Abstract In the context of the world have changed, the process of internationalization of production and application of science and technology and division of labor occurs deeper, the quality of human resources is considered to be determinants victory competition in economic development - economic development of each country. Vietnam country is entering a period of integration and strong growth across all sectors of economy, culture, society. The industrialization and modernization of the country is posing an urgent demand for human resources, especially technical personnel of high quality. Resolution of the 8th, the Central Executive Board XI (Resolution No. 29 - NQ \/ TW) about: Innovation basic, comprehensive education and training to meet the requirements of industrialization and modernization in conditions of market economy socialist orientation and international integration \"; including vocational training and this is the opportunity for vocational training for young people. Economic development strategy - social phase 2011 - 2020 to request economic restructuring and growth model innovation towards improving quality, set the task and also provide an opportunity for innovation and promote the development of vocational training, improving the quality of education, especially in training high quality manpower in the context of international integration. Keywords: vocational training innovation, vietnam, the context of integration. 1. The concept of technical and vocational training in Vietnam in the context of integration 1.1 Technical - Vocational Vocational training is a form of learning in which knowledge, skills, and habits of a group of people that are passed from generation to generation through teaching, training, or research. Vocational Training usually takes place under the guidance of others, but also through self-study. Anyone experiencing any significant impact on the way people think, feel, or act can be seen as vocational training. Engineering is the creative application of scientific principles to design or develop structures, machines, tools, or manufacturing processes, or the use of their works individually or together; or in the construction or operation of the above mentioned subjects with the full consciousness of their design; or to forecast their performance under certain operating conditions; all the things just mentioned with attention to the functional characteristics of the operating economics, or the safety of lives and property. Technical vocational training and acquire knowledge about the tools, machinery, engineering, career skills, systems, and methods of organization, to solve a problem, improve a solution already exists, reaching a goal, or perform a specific function. Technical Vocational significant impact on the ability to control and adapt the human to the natural environment in order to develop economic structures, national technical Vocational training has a very important position in human resources development strategy of the country. These years, vocational training has made great progress in terms of both quantity and quality and achieve a stable result: - The system was enacted law guarantees legal framework for stable operation of vocational training and development. - The network of vocational colleges and vocational secondary schools were held in all provinces and cities; network of vocational training centers was down to the district; network of vocational courses spread down to the commune, to the villages, and in the business ... has created conditions and opportunities for employees wishing to gain access to vocational training services more favorable job. As of May 6-2011, the country had 125 vocational colleges, 310 vocational schools and 864 vocational training centers, 5 universities, colleges and technical teachers and 20 vocational pedagogy, gradually meet Teacher training needs. - Scale enrollment rise year over year and an average increase of 17% \/ year, in 2010 reached 1.707 million vocational school enrollment people. Quality training is gradually improving as requested by the labor market; ratio of apprentices employed after graduation at 70% - 80%; quality vocational training to be confirmed by the merits of the candidates selected group participation skills exams ASEAN and World Skills Competition. - Socialization is promoting vocational training. We have over 30% of private vocational training institutions ensure 1\/3 of total annual enrollment size. - The organization and management of vocational training from the central to the ministries, localities and economic groups, corporations and vocational training institutions have been established and operate relatively uniform, effective Results. At present there are 53 Department of Labour - Invalids and Social Affairs with vocational training. Department of Labor - Invalids and Social Affairs districts with specialized staff training management; Commune with specialized staff training - employment. - Vocational training scheme for rural workers until 2020 and the national target program on new rural construction phase from 2010 to 2020 conducted training for farmers effectively. [1] 2.2 The context of international integration International integration is an essential process of development, due to the social nature of labor and relations between people. The birth and development of a market economy as well as the leading impetus to accelerate the process of integration. Integration occurs in many forms, on many levels and different areas, according to progress from low to high. Integration has become a major trend of the modern world, a strong impact on international relations and national life. Today, international integration is the policy choices of most developing countries. From theory and practice mentioned above, we need to define a consistent approach to the concept of \"international integration\" to build strategic platform of international integration of Vietnam in the new period. We believe that the most appropriate approach is to consider integration as a social process with comprehensive internal and often movement toward a certain goal. Accordingly, international integration is understood as the process of conducting water activities to strengthen their engagement with each other based on shared interests, goals, values, resources, power (authority authority to determine policy) and comply with the general rule in the framework of the institutions or international organizations. Thus, the other with international cooperation (acts of international actors meet the interests and aspirations of the other, not against each other), international integration beyond the usual international cooperation: it requires asked the sharing and discipline of the actors. [9] 2. The targets research on vocational training innovation in Vietnam and World 2.1 Three policy breakthrough quality vocational training in Vietnam Quality problems do not now have careers Education Law (EL) we do, that this is the core issue of vocational training has a long history of development, associated with the emergence, survival in the wet rice civilization, of the traditional villages and the process of industrialization and modernization of the country to be deployed for years. But if talking about changing the nudge they must speak to the quality of the three events have a decisive impact on the breakthrough quality vocational education. First: in 2011 by Prime Minister: \"Human Resource Development Strategy for Vietnam period 2011-2020\". Then in 2012 the Prime Minister approved the \"Strategy for development of vocational training in the 2011 - 2020\", has set a target of specific quality. [1] Secondly: in 2014 the Prime Minister issued Decision 761 \/ QD-TTg approving: \"Project development of high quality vocational schools by 2020\"; with the goal: Strive to 2020, approximately 40 high quality vocational schools capable of training some occupations are the advanced countries in the ASEAN region or internationally recognized, contributing to a comprehensive innovation training basics profession in Vietnam and meet the requirements of high-quality human resources for socio-economic development of the country. In particular targets in 2014 - 2016: Step by step 34 pilot vocational training under the vocational training program is transferred from abroad under the scheme had been approved by the Prime Minister with a minimum size of 25 students , vocational students each year. Priority investment support focused, synchronized to the selected vocational schools have good training capacity, close to the criteria of high quality vocational schools. Period 2017 - 2020: Gradually expanding vocational training pilot training students, graduates are assessed, recognized diploma or certificate by the training institutions prestige advanced countries in the ASEAN region or internationally. Strive to 2018, about 15 cases were tested, evaluated, recognized as meeting the criteria of high quality vocational schools; 2019 has added about 15 schools and 2020 with about 40 high-quality schools. [10] Third: Decision 1982 \/ QD - TTg dated 10\/31\/2014 of the Prime Minister approved the \"Scheme information technology applications in management and vocational training by 2020\". With the goal of information technology applications (IT) to manage training activities and vocational training in the most advanced software in the world. Modernization of information infrastructure towards digitization, simulation machinery of the vocational training, to change fundamentally, comprehensive management and vocational training towards modernization, creating a breakthrough the quality of training, contributing to increasing national competitiveness and regional integration and international. With these fundamental changes, so comprehensive, the next year we will have a workforce trained highly skilled, accredited in ASEAN. Contributing to improve national competitiveness in the context of integration. [11] 2.2 Development Trainers team Directive 40-CT \/ TW dated 15-6-2004 of the Party Central Committees Secretariat (IX) has stated: \"The goal is to build a contingent of teachers and education managers are standardized, ensuring quality, in sufficient quantity, synchronous structure, ... expeditiously training, supplementing and raising the level of teachers, trainers, managers of education in vocational schools... Expanding international cooperation to enhance the quality of training and retraining of teachers ...\" Plenum Resolution 6 (6th) Party stressed:\" Focus on training and upgrading High-quality vocational teachers ... consolidate and expand teacher training schools and vocational training as areas on a national scale. \"In 2011, the Congress of the Eleventh National Party questioned \"Innovation fundamental, comprehensive education,\" according to the spirit that there should be synchronization solution with an overall vision. Contributing to implement the policy of \"Innovation fundamental, comprehensive education in Vietnam Male \", article please mention the development team vocational teachers (VT), problems have long been interested but still shortcomings and challenges in the new context. [8] In recent years, with the development of vocational training, team (VT) increased in number 33,000 (VT) 2010, an increase of almost 4 times compared to 2001), the quality gradually improved on our qualification standards creation, skills and pedagogical competence. Basically, teachers in vocational training institutions have standardized level of training; 85% of teachers teaching vocational college level, 75% of teachers in vocational secondary level and 49% of teachers in vocational centers reached pedagogical qualification. There are around 46.3% of the teachers integrate both theory and practice. Business training (VT) has many innovations and expanding the training facility (VT). Until now, 4 university technical teachers and some technical pedagogy of the university, was established nearly 30 vocational pedagogy in some prestigious professional colleges for professional training vocational pedagogy; improve skills for (VT). Policies towards (VT) gradually attention. Currently, (VT) enjoy the general policy towards teachers in the national education system. In addition, there are several modes, particularly for (VT) policy such as working mode, modes of use, fostering professional; policy on allowances for teachers in teaching practice of heavy, toxic, hazardous and special allowances for disabled (VT) handicapped. Besides the results and progress achieved, work development and innovation policies and mechanisms for team (VT) inadequate stretching, very slowly overcome. Although the number (VT) recent years but increased significantly compared with the requirements of renewal and development of vocational training, the number (VT) still severely lacking. Currently, the percentage of students in teacher exchange reached 26 students, student \/ teacher, meanwhile, the target set was 20 students, student \/ teacher in 2010. With this goal , number 2015 (VT) need about 51,000 peoples and 77,000 peoples in 2020. Industry structure is irrational (VT) training, some vocational teachers do not have the basic training, vocational skills are limited, the rate integrate teachers are low compared to the requirements of the training program. Foreign language and informatics of (VT) weak, limiting the ability to update new technology, (IT) applications and the modern pedagogy. Ability to develop programs, compiling textbooks and materials of (VT) vocational limited. The university technical teachers only pedagogical and technical training, vocational skills training for about 30 out of more than 400 occupations, accounting for 7.5% of the total portfolio vocational training, creating the balance (VT) source redundancy for this occupation while other jobs are very large deficits. Policies towards (VT) still many shortcomings, not encourage, attract capable people to do (VT), not create loyalty, dedication to the profession. [6] Regarding salary regime, the operation of (VT) -specific, on the one hand, they have to be an educator, on the other hand is a \"technician\" wage policy has not reflected the peculiar preferences dirty. No separate quota (VT) salary, but receive wages according to class of secondary school teachers (under Decree No. 204\/2004 \/ ND-CP, dated 14-12-2004). Teachers teach college level vocational (CLV) are not entitled to the salary as teachers of other colleges. Its one of the reasons leading to the failure to attract qualified people, skilled and experienced in manufacturing moved to do (VT). Conversely, many (VT) skilled qualified to want to move into production in the business to have higher incomes. Also, no policies to encourage teachers to strive to improve themselves level; No mechanism and policies to enterprises and vocational training institutions facilitate practical (VT) gone manufacturing facilities, sales, annual service. However, in developing (VT), we are also faced with challenges. International integration depth, width create favorable conditions for vocational training access to new knowledge, new technology, modern vocational training model, extend the exchange of experience, have the opportunity to reach and attract external resources for the development of vocational training, but also requires (VT) must adapt quickly. The trend to diversify the type and mode of education - training, development of distance learning, online; shift functions and models of vocational training institutions as well as challenges for (VT) in new contexts. 2.3 Some solutions development team Trainers (VT) have a decisive role in the quality of vocational training, but with the fact that policies to attract, as the current treatment is hard to get a team (VT) sufficient quantity, quality assurance to avail the opportunities and overcome the challenges in the new context. To develop team (VT) ensure both quantity and quality, to make good some of the core solutions in multiple solutions. First: The issue today is the most inadequate of (VT) income. While working stress, strenuous (both a classroom teacher and technician) but income from wages and allowances under salaries are very low, not make for themselves and their families a decent standard of living, \u00e1 so hard to require (VT) wholehearted, full consultation with the profession. This fact is a fundamental cause leading to difficulty retaining qualified (VT) stay working in vocational training institutions. Perniciously, it also causes difficulty attracting skilled people, skilled people do (VT) and attract outstanding students training schools to become (VT). Therefore, first need to change treatment policy to teacher salaries and living by professional allowances. Develop a policy framework and mechanisms to encourage and honor motivate social status of teachers, for teachers of honors. Use the gray matter of effective teachers, and determine the requirements of their responsibility. Construction and improvement of professional standards, the labor norms of (VT). [2] Second: At present, the training institutions, fostering (VT) not reflected the professionalism of a vocational school. Condition inadequacies in the training, capacity development for student teachers practice is limited by the quantity and quality of teaching, while facing many difficulties in the organization is, practice. Besides, the actual ability of (VT) scientific studies are currently not focused properly, leading to scientific research activities of the vocational training institutions in general and the ability to engage in scientific research of (VT) in particular is very limited. To address these shortcomings needed to rearrange, reorganize the training facilities, vocational teacher training; diversification of forms of training and retraining (VT); innovation activities of technical teacher training colleges; the vocational pedagogy for training and retraining of pedagogical training and improvement of professional skills for (VT); encourages scientific research institutions involved in the training and retraining of pedagogic profession for GVDN thereby develop scientific research of the vocational training institutions as well as capacity building for scientific research of (VT). Third: The State plays a key role in ensuring that development resources (VT) team for the whole system, mobilizing contributions from learners as stipulated by law, to mobilize the resources of socialization, early private organizations and individuals, domestic and foreign enterprises and other legal sources. Strengthening international cooperation in developing (VT) team. (VT) decisive role in ensuring the quality of vocational training, motivation, is an important factor in ensuring competitive capacity of our human. Investment development (VT) investment can be considered the \"source\" for the development of human resources. Therefore, in the process of implementing the policy of \"Innovation fundamental, comprehensive education in Vietnam\", have placed the renewal of mechanisms and policies to attract and treatment (VT) a focus in qualitative breakthrough vocational training. [3] 2.4 Experience vocational training development in countries around the world Based on survey experience labor market development of some countries in Asia, we can draw some valuable lessons for reference in Vietnam. Specifically may generalize into the following areas: - Firstly, the need for a breakthrough in thinking and planning mechanisms and policies accordingly. - Secondly, the need to promote democracy in the rural areas in the development process. - Third, the development of mutual chain link urban - rural industry - agriculture. - Fourth, capacity building and technical education, productive capacity in rural areas. - Fifth, maintain economic sustainability - social and ecological environment in rural areas. [4] Labour together with labor materials and labor object is three important physical elements of any production process yet. Three elements that in any era, in any other country also indispensable to conduct the production process. In this era, when resources become scarce, labor is considered the most important element of the production process. The role of labor in general and the technical workforce in rural areas in particular are very important in the economic development process of the country, especially for the developing countries have a key economic sector is agriculture as Vietnam. [5] [12] In the process of urbanization, implementing industrialization and modernization, is going strong in many countries around the world, development of rural human resources Governments considered a range of issues important priority. Every country has its own policy on training and development of human resources for rural engineering to meet the requirements of the process of industrialization and modernization. However, due to be governed by the laws of motion so countries generally have the following common characteristics: Urbanization, industrialization and modernization characterized by rapidly shifting economic structure and labor structure, so the shortage of trained manpower, particularly skilled manpower and lack of management is common (may include countries such as Japan, Taiwan, Thailand,...). Meanwhile, in rural areas, the population and workers with low education level, proportion of untrained workers high. [1] The strong economic growth of the United States, Japan, the West in the mid-twentieth century, the phenomenon of the \"tiger\", \"dragon\" in East Asia largely thanks to resources qualified. Therefore, Vietnam is implementing growth paradigm shift from extensive to intensive necessarily learn in training the human resources of these countries. 2.4.1 The experience of the US America has defined the motto \"Human capital is the center of all development\". To keep superpower status economically and science, technology, human resource strategy focusing on training human resources and attract talent. In the training of human resources, the US has built an educational system characterized both public and liberal, the US education system attaches special importance to higher education in the US has 4,200 universities colleges, to ensure that all people in need are able to participate in training programs and colleges and universities. In America, the system of colleges, community colleges thrive ensure mass in higher education, these schools focus on job skills training for employees, 78% in the US currently have population about graduating from college, colleges and secondary vocational schools. In the US, developed both the community college and university studies. Proportion of research institutions and community colleges are 1\/30, means that 1 research university, there are 30 community colleges. In higher education in the US, competition among schools is fierce. The university asserted themselves with the quality of teaching and build your own brand. If students get into good universities, famous and good learning, employment opportunities will increase a lot. Along with heavy investment from the State budget for personnel training, the US also mobilize more resources from society to work in human resource training. The American company is also focused on developing human resources and labor training. In 1992, the cost of training employees in the companys 210 billion; 1995 costs amounted to $600 billion, in 2000 was over$ 800 billion and to date nearly \\$ 1,000 billion. [7] 2.4.2 The experience of Germany The development of education and training in Germany was led by a cross-cutting perspective is \"Only those who are educated and trained new lead Germany into the leading position in the global race, and copper time to enroll yourself in the best way in this development, education and training Giap is the key to the future of the difficult economic development of the country and society.\" Germany attaches great importance to the training of human resources, budget, education accounts for 6% of GDP from 2010. In Germany, the ramification early school students is done right from the junior high school level, lower secondary level (secondary) is designed to equip students with qualifications to meet the requirements of the school Common lead to a professional qualification. Students graduating from secondary school types are further education according to the priority stream high school, vocational high school (secondary education combined with vocational education) and vocational education mainly. The link between human resource training for economic development, the needs of the labor market in Germany is also very tight. Labor needs of the company are met appropriately through contract training, vocational training for students and workers. [7] The German government also mobilized the active participation and effectiveness of social forces in human resource training. In Germany the factory, now voluntarily participate in vocational training in parallel systems. Private enterprises as well as agencies and organizations involved in training outside the enterprise is widely practiced NGE training but must comply with the provisions of State law inscribed in vocational training. 2.4.3 The experience of Japan Postsecondary system include general universities, colleges and colleges of technology and specialized training institutions directly responsible for the training of human resources for the country. Colleges of technology and specialized training takes the graduating junior high. The institution of postsecondary education must adhere to the regulations of the Ministry of Education on standards established colleges, colleges of technology and universities. To mobilize social resources for the training of manpower, the government encourages and creates favorable conditions for the formation of the educational system of vocational training in companies and enterprises. [7] Not only is a country with a developed education system that Japan attaches great importance to acquire the experience, achievements in human resource training and development of other countries, to send people to study abroad State-focused, encouraging, with many different funding sources and use of funds of the State, who go to school, of all employers, other foreign partners. 2.1.4 The experience of South Korea As a resource-rich countries do not, Korea would soon determine the development of human resources is the key determinant of economic growth of the country. In fact, education has moved Korea into a country with abundant human resources, well-educated, highly disciplined and skilled skills and causes make up the magic of the Korean economy, accumulation knowledge through education and training to contribute 73% to the economic growth of Korea. [12] The South Korean government regarded manpower training are priority tasks in education to ensure that human resources meet the requirements of industrialization. Education is performed in parallel with the process of industrialization. In the early stages of industrialization, in the 60s to the 70s of XX century, Korea focused on developing light industry and electronics, Korea has focused on completion of primary education, development of basic secondary education, encourage vocational and technical schools, limited education indicators. Vocational Training Act 1967 was introduced, to encourage organizations and businesses participated in training to produce the workforce have the skills businesses need. Schools, vocational training centers developed fast and ever-expanding scale. Into the 80s, while the technology shift from medium to high technology, Korea focused on expanding the size of school education, promote vocational training and wider university quota towards development vocational colleges and technical. The degree to secondary vocational colleges, universities, graduate frequent adjustments in scale and quality to suit the requirements of the process of human resource industrialization. Currently, in order to meet the requirements of the knowledge economy, South Korea has the proportion of people graduated from high school compared to other countries in the Organization for Economic Cooperation and Development else, 2000, the ratio college South Koreas population is 78%. However, Korea still attention consolidating general education foundation for manpower training. Reforming education is considered routine tasks, continuity, now Korea is still ongoing education reform 6th. Korea conception of education and training of human resources to be in tune with the requirements of economic development. Education vocational education and technical attention right from high school. Even in the general education program, practical prevail over academic requirements threading is done urgently. After graduating from junior high school, students were streaming in secondary schools and vocational schools (including the schools of the industrial goods), 2005, 70% in high school and 30 % in vocational schools. With the introduction of the Act to promote industrial education, vocational schools, industrial training programs and training in plant growing very strong in Korea. [7] 2.1.5 The experience of Singapore Singapores leaders in the race to win conception of education will prevail in the race for economic development. Therefore, the government has spent a significant investment to develop education, from 3% to 5% of GDP in the first decade of the eleventh century, the current investment in education and training accounts for 10% of GDP Singapore. Singapore to achieve the diversification of students early, 6 years primary education, four years beginning with the general curriculum and oriented 2-year period. Seniors take the exam finished elementary school, high school placement consists of special base, Express, Normal (culture), Normal (Technical). After 4 years of junior high school students qualified class N ordinary, special and intensive student proficiency level O. The N if they wish and ability to learn 1 years to take O levels . The students have certificate level training N in skills and techniques in engineering academy where vocational training for students who have completed lower secondary school, with O level can Study technical colleges or prep for college. The classification of a junior high school programs in which the normal program aims to prepare students with knowledge before attending the vocational school or technical college after high school graduation basis. Technical education and vocational training play an important role in reforming education constantly Singapore. Engineering and technology has always been a top priority in training, English, mathematics and science subjects are compulsory subjects occupy 1\/3 of the duration and the state program of construction investment and technical institutes vocational training. Singapore also encourage companies to participate in training human resources for the country. State to apply multiple policies to encourage companies themselves organize training courses or job training for staff and workers in the work process. Technical Educational Institute in conjunction with the company performing parallel apprenticeship model, the students will participate and paid internships at the company, while the lectures will take place at the school Vocational training institutes. Singapore state investment only in very few public schools to have exemplary quality. For non-public sector, the government creates conditions for development, encourage inter-association with foreign universities invite reputable international affiliates set ... to train human resources quality for the country. [12] Singapores education strategy on the one hand has to meet the changes of global economic conditions, and the tool to build and maintain the cultural identity of national, bilingual program that is applied globally. The school offers training in English language and one of three to represent the three major ethnic groups are Chinese, Malay and Tamil. The introduction of English in the curriculum required for connecting Singapore to the world but to teach the mother tongue to preserve national identity. National education subjects are to be taught at universities. In essence, this course aims to equip young people with the basic behaviors, values and orientations make Singapore a true citizens. 2.5 Renewal of vocational training, improving the competitiveness of technical manpower in Vietnam 2.5.1 Vocational education law effective from 01.07.2015 law To address these challenges in training and development of human resources, on 11\/27\/2014 at the 8th session of the National Assembly adopted the Law XIII Vocational education, with effect from the day 01\/7 \/ 2015. Law passed with many innovative content to meet the requirements for training, human resource development in the period of industrialization - Modernization and ASEAN integration, as agreed three levels: primary, Range and colleges; organization and management innovation, training towards training yearly, accumulated modules, accumulating credits; renewal of time training to suit the needs and abilities of learners, including Vocational training time for those graduating from secondary school was 1 to 2 years to focus on knowledge, professional skills; autonomy for educational institutions were built and career development programs; for graduates college degrees will be awarded diplomas and accompanying title of engineer or bachelors practice, depending on the business practices training; policy renewal with professional educational institutions, without distinction of public and private, are participating in the bidding put cave training, preferential loans from the programs and projects at home and abroad, participation Teacher training management personnel at home and abroad with funding from the State budget; strengthening incentives to encourage and attract learners; clearly defined positions for teachers in all vocational education establishments, honoring policy, prolonging working with qualified teachers, distance learning degrees, skilled and allowances Teacher incentives for integrated teaching both theory and practice; defining roles, position, duties, and especially the rights of enterprises joining vocational education activities ... These new provisions of the Education Law profession, opportunities for vocational education institutions autonomous, dynamic, flexible training organization based businesses linked to the training of human resources at various levels according to the method of flexibility to meet the needs of businesses and markets labor, especially the training needs of high quality human resources for the industrial parks, export processing zones, the key sectors and dynamic economic region and ASEAN regional integration and international. [1] Student HCMC University of Technology and Education practice workshops According to the Law on vocational education has been approved by Parliament, college graduates will be awarded a college degree and is recognized as practicing engineers or Bachelors practice. This is good news for students attending college degree. On 11\/27\/2014 8th Session XIII National Assembly adopted the Law on professional education. This can be seen as a turning point, ushering in a new roasting, new way, modern and integration activities and vocational training. Change the whole structure of education systems and vocational training: From 07\/01\/2015 vocational education law began to take effect. Law launched vocational education has restructured the national education system of our country towards integration with regional and international. Under this law, our countrys education system is divided into two branches: heavy academic training branch of the theory by the higher education system implementation, professional training arm weighs about practice and skill vocational education system will be performing career. The design of the training system in two branches as are appropriate to the structure of training of most countries, creating conditions for education and training of our country has many advantages in exchange and cooperate with education Training of the countries in the world. An education and training system has similarities, clear and easy to understand with regional and international will facilitate mutual recognition between the education, this is especially important in the context of international integration deeper economic. [1] 2.5.2 Renewal of vocational and technical human resources in Vietnam As part of the national education system, vocational training tasked direct human resources in production, sales and service. In recent years, vocational training has grown in both size and quality, and better meet the manpower needs of the labor market, the rapid change of technology and diverse needs workers vocational training, career. So far, the system of vocational training network formed diversified across the country with 2,052 vocational training establishments (including 55 vocational colleges, 242 vocational schools, 632 vocational training centers and 1,123 educational institutions, vocational training courses in enterprises in the villages ... have functions and tasks of vocational training). The network of vocational training institutions with training scale around 1,700,000 people (scale enrollment in 2007 was 1.436 million people). Business is promoting socialization, vocational training institutions accounted for 38% of non-public and non-public student ratio up 35.6%, resources mobilized from the State budget accounted for approximately 37%. Many models of vocational training varied and creative as vocational training in enterprises, vocational training for industrial areas, rural workers, demobilized soldiers ... have contributed to poverty alleviation, job creation for workers action. Vocational training quality has gradually meet the requirements of the labor market due to the quality assurance conditions significantly improved (the percentage of students graduating from vocational training to find jobs for 60% -70%; this rate in some occupations and in some cases of over 90% now). Education Law, Vocational Training Law and the bylaws were enacted creating important legal basis established vocational training system with the primary level vocational and vocational secondary schools and vocational training colleges and create a legal framework vocational training for sustainable development, put into place; The law also important policies on investment, in terms of corporate responsibility with vocational training, to socialize, to support the development of vocational training in areas of socio-economic conditions of society particularly hard towels, vocational training support for the poor, ethnic minorities, women, disabled, handicapped and other policy objects; regulations on quality assurance and accreditation ... Although vocational training has a new development and achieved many important results but still some exist, failing to meet the needs of technical personnel directly in economic activity in both quantity and quality, qualification structure, industry structure, regional structure of the labor market at home and abroad in the process of international economic integration. [1] To develop vocational training, first of all need innovative thinking about vocational training. Trained labor accounts for about 65-70% of its workforce training so that trainees direction mainly to workers, youth vocational training and career conditions to continue learning throughout life their labor life. In current conditions, vocational training is a strategic task to develop the labor market, the breakthrough solution to achieve the objective of achieving a 50% acceleration of trained labor by 2010. Vocational longer being social benefits and aims to implement social justice for the majority present apprentices are poor, they have fewer opportunities than in access to services and vocational training. Investment in training is the development investment, so the State must prioritize investment and focus on building infrastructure, land and credit for students and vocational training institutions; Investment for teachers and managers, investment development of vocational training in areas of socio-economic conditions of society particularly difficult. Both society and the community must take care to vocational training, it is the responsibility of the older generation worry about the next generation; vocational socialization based on pre-prepared workforce, land, credit and tax policies. Vocational must perform mount now, mobilize and encourage all employers to participate in vocational training activities to meet the needs of the enterprise itself. Along with expanding the scale necessary to ensure the quality concentrate adapt vocational training to the requirements of the labor market and abroad through the development system and accreditation standards, which are measures protect the legitimate rights for apprentices and employers. Vocational training system is facing challenges in shaping the paradigm shift in the future, namely: - Delivery system towards providing vocational training system to meet the requirements of the direction of the labor market and society. - Go to the vocational training system focusing on the formal sector, public vocational training system to develop both formal vocational training and regular training. - Go to the vocational training system is managed centrally, investing primarily from the State budget to the vocational training system is decentralized management, decentralization to the base; mobilize all social resources for investment and development of vocational training. - Switch the system less flexible vocational training and hard mold in schools to flexible training system with multiple entrances, exits create favorable opportunities for learners. - Switch the system of vocational training diplomas assessed after examination and not recognize the results of the previous study to evaluate vocational training system based on the carrying capacity and recognition of learning outcomes in any Where, in any way. - Switch the system of vocational training that vocational training institutions are led and supported from the top to the vocational training system that the vocational training institutions should be responsible, in conjunction with the steering support from superiors. - Switch the system of vocational training programs with emphasis on theoretical continuity between training levels to vocational training system development program towards practical skills, continuity between training levels. To innovate and develop the vocational, Resolution Tenth Party Congress has set the development direction of vocational training in the future is: \"Rapidly increasing scale vocational training colleges and vocational secondary schools for the industrial parks and economic zones for export dynamics and labor. Expanding the network of vocational training institutions, development of vocational training centers of districts. To create a basic vocational training quality reach the advanced level of the region and the world. Promote socialization and encourage the development of diverse forms of flexible vocational training: non-public vocational training, in business, in the village .., creating conditions conducive to worker job training, career. Organize vocational training and technology transfer, production technology suitable for farmers, ethnic minorities \". To accomplish the above directions, the task of vocational training in the coming period is very heavy, namely: - For scale enrollment: Increase enrollment scale (formal training and regular training) was about 6% \/ year to 2010 the proportion of trained workers reached 26% of total employment 40% trained. Among workers with vocational training, vocational college level reached 7.5%, the level at 22.5% vocational secondary, vocational elementary level reached 70%. Thus, by 2010 the system of vocational training will receive about 2.5 million people, including vocational training colleges was about 130,000 people, approximately 420,000 vocational secondary and elementary occupations at about 1,900,000 people. [1] - In terms of network development: According to Decision No. 07\/2006 \/ QD-MOLISA dated 02\/6\/2006 on \"Network Planning vocational colleges and vocational secondary schools, vocational training centers by 2010 and oriented to 2020 \", then to 2010, 90 professional colleges (including 40 high-quality schools, three schools have access to the advanced level in the region); 270 vocational schools and 750 vocational training centers. Each province (city) has at least one vocational secondary schools or vocational colleges; each county, district, town has at least one vocational training center or cluster vocational schools districts to create favorable conditions for workers, especially in vocational areas, remote areas, islands and people minorities and rural areas. To innovation and development of vocational training should focus on the following three basic contents: One is: Scaling, create favorable opportunities for workers and young people access to vocational services. - Raising the awareness of society and workers in order to create a consensus in society about the role and position of vocational training in human resource development, contributing to increase the competitiveness of the economy and social development Assembly. - To increase investment in developing a network of vocational training institutions, strengthening of the vocational training institutions for facilities, equipment and training facilities, priorities and public land funds for training institutions crafts; - Promote vocational training in enterprises and vocational training development of private sector; Also there is now a mechanism to teach practical support, building programs, scholarships, ... - Develop strong regular vocational training system, create favorable opportunities for the youth and apprentice workers, career. -Make And streamlined policy deployment and succession in the vocational training system and linked with other training qualifications in the national education system. - Development of the system audits and grant diplomas and certificates of national vocational skills enabling workers vocational training in various career paths and different methods. Second: Improving the quality of vocational training to meet the growing demands of the labor market and international economic integration. - Renewal of the structure of vocational training qualifications and structure regions adapt to the rapidly changing needs of engineering and technology; meet the requirements of each locality, each economic sector, the domestic labor market and labor export. - Develop standard system (Standard school, teacher professional standards, standardized infrastructure) as a basis to ensure the quality and implementation of accreditation. - Development of teaching staff and training managers to adapt to the requirements of vocational training innovation. - Develop quality assessment system and accreditation policy in vocational training to contribute to improving the competitiveness of the workforce on the labor market in the country and internationally. - Develop key school system as the core of high quality for the quality of the vocational training system and gradually reach the advanced level of training in the region and the world. Thirdly, management capacity building of the vocational training system - Orientation training innovation oriented labor market, labor market and international economic integration. - Establish a system of labor market information, policy development and coordination of user training; policies and encourage cooperation with business training. - Concentrate on building management staff training at all levels, improve organization apparatus, vocational functions and responsibilities from central to local (Organizational structure, functions and duties of the units under General Department of Vocational Training, establishment of vocational training room at the ministries, corporations and at all Department of Labour, Invalids and Social Affairs). - Decentralization of vocational training by increasing the autonomy and self-responsibility of vocational training institutions; capacity management and planning of the management staff of vocational training at all levels. - Institutionalize the functional role of these organizations including enterprises joining vocational socialization. Build collaborative relationships and accountability with vocational associations and professional associations to participate in the construction program, vocational standards, vocational skills training for students; participating organizations assessment and certification of national vocational skills. - Expand international relations multilateral and bilateral, connected with some regional countries to share experiences and cooperate in the field of vocational training; recognition of diplomas and certificates equivalent. - Develop policies and tax policies to attract organizations and individual domestic and foreign investment in the field of vocational training; policies and mechanisms for transfer of vocational training institutions under the administrative and operational mechanisms to provide services. [11] 2.6 Inspection training program for human resources of regional integration and international 2.6.1 Testing Center of education quality accreditation Vietnam 2.6.1.1 Quality Inspection Center Education - National University of Hanoi) (VNU-CEA) On 05.09.2013, the Minister of Education and Training issued Decision No. 3568 \/ QD-BGD\u0110T established Center for Quality Education - National University of Hanoi. English trade name: CEA Center for Education Accreditation VNU- (VNU-CEA). [1] 2.6.1.2 Quality Inspection Center Education - National University of Ho Chi Minh City (VNU-HCM) Decision No. 5570 \/ QD-BGD\u0110T, dated 11.22.2013 established the Center for Educational Quality inspection - National University of Ho Chi Minh City. English trade name: VNU-HCM Center for Education Accreditation (VNU-HCM CEA). [1] 2.6.1.3 Center for Educational Quality - UD (CEA-UD) Decision No. 1100 \/ QD-BGD\u0110T, dated 04.06.2015 established the Center for Educational Quality inspection - UD. English trade name: VNU-DN Center for Education Accreditation The University of Da Nang (CEA-UD). [1] Education quality accreditation under the criteria of the standard testing program by the Center for Education Quality Accreditation Vietnam undertake. The recognition we achieved national accreditation standards of training and qualifications of educational institutions in the country worth. The organization of non-public inspection may be established in Vietnam after 2015 quality assurance system of higher education in Vietnam has created trust in institutions of higher education. This confidence is supported institutions of higher education are working together in the credit transfer, student exchange and mutual recognition of qualifications. [6] 2.6.2 Education Quality Accreditation standards AUN Network of Southeast Asian universities AUN (ASEAN University Network) was established in November 1995, with the aim of promoting the development of human resources through education in the ASEAN region. As of May 7\/2013, there were 31 universities from 11 ASEAN countries to become members of the AUN. Assessing the quality of higher education in accordance with the quality standards to ensure common (AUN) is to promote quality assurance within the schools, and enhancing mutual trust on training quality standard schools in the area as well as with partner universities in the world, gradually contribute to the recognition of academic achievement and development of cooperation between the universities in the region. To improve the quality of higher education, the organization evaluate the quality of training programs by the standard unit accreditation of international organizations be regarded as a recognized quality standards. AUN is standardized accreditation system for universities ASEAN has been implemented since 1995 with many activities and achievements in the evaluation and accreditation of higher education programs. Quality inspection AUN standards aimed at improving the quality of training of regional and national level. The school inspection achieve accreditation standards of the Network of Southeast Asian universities, accreditation and qualifications valid in countries in Southeast Asia. Thus, students who graduate from the university standards will ease the integration in the ASEAN region. [2] 2.6.3 Education Quality Accreditation ABET accreditation standards ABET (Accreditation Board on Engineering and Technology - Control Council for Engineering and Technology); was established in 1932, is a non-profit organization, non-governmental accreditation programs and university undergraduate disciplines in the fields of Engineering (Engineering), Technology (Technology), Computing (Computing) and Applied Science (Applied Science). ABET accreditation is widely recognized in the United States. In many cases, universities have engineering programs accredited by ABET is eligible to operate. Besides, many professional bodies require engineers to graduate from accredited programs can ABET engineering practice. ABET accreditation more than 3,100 programs at 670 colleges and universities in 23 countries. ABET provide specialized, quality inspection and assessment program a personal program of study, rather than assess an organization as a whole. ABET accreditation, which is voluntary and achieved through a process of evaluation, provide assurance that a college program or university meets the quality standards established by the profession in which the chapters preparing for his students. Since 2007, this organization official accreditation certificate for the training programs of universities outside the United States. The university tested quality education ABET accreditation standards, accreditation and qualifications have international value. The graduates from the University of ABET accreditation standards would ease international integration. [8] 3. Conclusion Vocational training has an important role in the strategic development of human resources of the country. These years, the countrys vocational training has made great progress in terms of both quantity and quality and achieve a stable result. Innovation and development of vocational training have access to innovative trends in the market economy of the country, international economic integration, and must be based on long-term stability, inheritance, promotion the results and experience gained in recent years. That is an important prerequisite for innovation in technical and vocational training in Vietnam to sustainable development to integration with the region and the world. References [1]. Discussing the economic development strategy of socio - Vietnam during the next, 2007, the Ministry of Planning and Investment. [2]. Rural Development Strategy An Giang 2020, 2004, An Giang province. [3]. Area, population and population density in 2011 by province, 2012, The General Statistics Office of Vietnam. [4]. Development of agriculture, farmers and rural areas in Vietnam, 2008, the Vietnam Economic Times. [5]. Institute of Economic Management Central - UNDP, 2004, The establishment of economic development: experience and lessons of China, volume 2, Publisher Transport Hanoi. [6]. Nguyen Thi Hang, 2013, The Innovation and Development of Vocational Training to improve the Quality of Human Resources, http:\/\/www.molisa.gov.vn. [7]. Trinh Xuan Thang, 2014, Personnel training experience of some countries in the world and lessons refer to Vietnam, http \/\/ www.tuyengiao.vn. [8]. Hong Hanh, 2014, seven solutions to enhance the quality of higher education, Reuters. [9]. Mai Trong Nhuan, 2005, Report on results of Missions in Singapore, Vietnam National University, Hanoi. [10]. Pham Quoc Tru, 2011, International integration, a number of theoretical issues and practical, the Diplomatic Academy. [11]. Nguyen Minh Tho MA, 2014, Solutions enhance training quality human resources area tourism Mekong Delta, College Tho tourism. [12]. Truong Minh Tri, 2013, The training and development of high quality human resources in Vietnam during the integration period, Proceedings of the Institute of Educational Sciences of Vietnam. Topic: Vocational Education and Training 93 INSTRUCTIONAL DESIGN BASED LEARNING CULINARY CULTUREPRENEURSHIP EDUCATED_LOCAL WISDOM IN VOCATIONAL HIGH SCHOOLIsma Widiaty1,* Ana1 Unemployed graduates of vocational schools (SMK) is still ranked highest compared to other educational levels. This condition is very alarming, given the vocational goal is to produce graduates who are ready to work. Intense competition from vocational school graduates as well as the number of jobs that are not proportional to the number of vocational graduates constitutes the main factor. One effort that can be done is to develop entrepreneurship education is expected to prepare students vocational graduates who are able to create jobs creatively. The purpose of this research is to develop instructional design culturepreneurship culinary wisdom educated_local based on production units in SMK Expertise Catering. Learning model developed not only teaches entrepreneurial competence but also includes the educational value of local wisdom local cuisine (traditional food), which is currently a trend. Elements of education based on local wisdom into a learning model on the basis of the development of the production unit at SMK Catering. The research approach used to develop a model of learning is research and development through four main stages, namely stages of analysis, design, developtment, and evaluation. Results culturepreneurship culinary instructional design development is done through - step identification of objectives, analysis of learner and the environment, formulating objectives, developing materials, strategies, and assessment. Topic: Vocational Education and Training 94 Interactive Media in Learning Japanese Language Vocabulary for Vocational SchoolNuria Haristiani, Rizka S. Rahmawati, Desri Sofiani, Asep Bayu Dani Nandiyanto This research aimed to develop a model for an interactive media on Japanese learning and to examine its impact on student ability for remembering Japanese language vocabulary. Different from other methods, this media was user friendly for students who want to learn independently. Further, this interactive media was technology-based and equipped by some features. To confirm the performance of this media, the study was completed with pre-experimental method, one group pretest-posttest design, and questionnaires. The research subjects were tens senior high school students (class X and XI) of national vocational school in Indonesia (i.e. SMK Negeri 1 Cimahi) who take Japanese language as an elective subject. The results of pretest and posttest showed that the use of this interactive media improved student ability on understanding and using Japanese language vocabulary. The questionnaires showed positive responses from students. The media was attractive, easy to be used, and useful for remembering Japanese language vocabulary. Topic: Vocational Education and Training 95 Invertigation Size Nugget Spot Welding Using Non Destructive Test (NDT) Ultrasonic Testing MethodeTito Endramawan and Agus Sifa Spot welding including electrical resistance welding. Current concentration is determined by the contact area between the electrode and the workpiece, and it is clear that the size of the weld or nugget of melting metal is associated with this contact area. Shear of strength nugget generally should ensure that when a connection is broken then the rated voltage up to occur nugget that surrounds the sheet. Material testing results of casting, forging, extrusion and welding can be checked visually, but to find more optimal results and accurate test can be done with the Non Destructive Testing (NDT) NDT is one method Ultrasonic Testing. Testing was done using 96% Cu electrode, and the worksheet Fe 98%, with a diameter of 6 mm electrode surface, and a thickness of 0.8 mm worksheet, obtained diameter weld on the worksheet with the testing Non Destructive Test (NDT) using Ultrasonic probe (UT) with the results of the average size nugget 2.9 mm. Topic: Engineering and Technology Innovation 96 KONSEP BUSANA ETNIK SUNDA SEBAGAI CULTURAL HARITAGE DALAM UPAYA PENANAMAN NILAI ARTEFAK BUDAYA BANGSAMila Karmila (Prodi Pendidikan Tata Busana Departemen PKK FPTK UPI Jl Dr. Setiabudhi No 227 Bandung) Ruang lingkup penelitian ini berkaitan dengan dokumentasi berbagai busana etnik Sunda yang ada di Jawa Barat, dengan tujuan untuk mendokumentasikan dan mengembangkan busana etnik Sunda dalam upaya pengembangan artefak budaya Sunda di Jawa Barat. Dokumentasi meliputi aspek visual(jenis dan karakteristik busana etnik Sunda) dan nilai filosofis busana etnik Sunda. Pada pembahasan busana etnik Sunda digunakan metode etnografi (visual) dan metode estetik. Metode visual digunakan untuk mengupas nilai kesundaan dalam berbagai dimensi di dalam visualisasi busana etnik Sunda: desain, model, penamaan, dan warna yang dapat mewakilinya, kemudian diarahkan kepada refleksi konsep estetik kesundaan pada masyarakat Priangan. Metode estetik digunakan untuk mengkaji sampai sejauh mana nilai-nilai estetik memiliki peran dalam upaya pemberdayaan budaya visual pada suatu masyarakat sebagai pemahaman konsep estetik kesundaan. Hasil penelitian menunjukkan bahwa tata cara berpakaian pada masyarakat Sunda erat kaitannya dengan falsafah hidup, konsep nilai, makna simbolik dan nilai-nilai religiusitas masyarakat Sunda. Sejak abad ke-19 busana etnik Sunda dibatasi pada Sunda Priangan, dan busana yang terdapat dan dipergunakan dalam kehidupan masyarakat Priangan sesuai dengan strata sosial di masyarakat Priangan yang terbagi menjadi : a. Menak Gede (menak pangluhurna : bupati), memakai gelar raden, b. Menak Leutik\/Santana ( asisten wedana), gelarnya : asep, mas, agus, ujang, (Nyimas untuk wanita), c. Somah\/Cacah : masyarakat pada umumnya\/kelas bawah. Busana etnik Sunda merupakan salah satu artefak budaya Sunda, yang harus senantiasa dijaga dan dilestarikan keberadaannya. Karena melalui artefak tersebut dapat ditelusuri nilai-nilai budaya masa lalu, dimana nilai-nilai positif yang ada pada saat itu dapat diwariskan pada generasi sekarang dan mendatang. Topic: Vocational Education and Training 97 LEARNING FROM CIREUNDEU: Prototype Rural Vocational Food Security Based Local WisdomYani Achdiani (a), Isma Widiaty (b), Elis Endang Nikmawati (c), Fitri Rahmafitria (d) The purpose of this study was to explore the potential of local wisdom as a prototype traditional village Cireundeu vocational village food security based on local wisdom. Cireundeu traditional village can be used as a learning resource associated with the processing of \"constellation\" as a staple food that is unique in the village. \"Rasi became the main source of raw material for the process of diversification variety of food products in the region. The approach used in this study is qualitative. The research method is survey data collection techniques such as literature study, deepth interviews, and observation. Results showed that Indigenous Village Cireundeu entered as vocational village category of food security based on local wisdom that is unique and inspiring. Cireundeu traditional village is also the prototype of vocational village that develops creative industries culinary field. The potential of the creative economy in Cireundeu entered in the category of 14 types of creative industries based local cultural wealth Topic: Vocational Education and Training 98 Learning model of PBL integrated toward the enhancement soft skills of high-order thinking for vocational student in patisserie vocationS Hamidah(a*), Wijiningsih(a), Yuriani(a), and S Palupi(a) There are many educational institutions, including Yogyakarta State University, realize that soft skills should be integrated into learning as an integral part with proficiency of hard skills. Soft skills needed graduates for success in the workplace. Soft skills are 1) the essential tools that contribute to grow self potential, 2) the power to change, 3) driving all effort, 4) difference among graduates. Soft skills learning in the Program of Food Education, especially patisserie deemed has strategic value to create graduates who are able to develop self potential, and able to compete in workplace. Therefore, it is necessary to analyze soft skills demand that can provide reinforcement of hard skills for graduates in patisserie. Topic: Vocational Education and Training 99 Life cycle assessment of energy and CO2 emissions for residential buildings in Jakarta, Indonesia.U Surahman1 *, T Kubota2 and A Wijaya3 In order to develop low energy and low carbon residential buildings, it is important to understand their detailed energy profiles. This study provides the results of life cycle assessment of energy and CO2 emissions for residential buildings in Jakarta, Indonesia. A survey was conducted in the city in 2012 to obtain both material inventory and household energy consumption data within the selected residential buildings (n=300), which are classified into three categories, namely simple, medium and luxurious houses. The results showed that the average embodied energy of simple, medium and luxurious houses was 58.5, 201.0, and 559.5 GJ, respectively. It was found that total embodied energy of each house can be explained by its total floor area alone with high accuracy in respective house categories. Meanwhile, it was seen that operational energy usage patterns varied largely among house categories as well as households especially in the simple and medium houses. The energy consumption for cooling was found to be the most significant factor of the increase in operational energy from simple to luxurious houses. Further, in the life cycle energy, the operational energy accounted for much larger proportions of about 86-92% than embodied energy regardless of the house categories. The life cycle CO2 emissions for medium and luxurious houses were larger than that of simple houses by 2 and 6 times on average. In the simple houses, cooking was the largest contributor to the CO2 emissions (25%), while the emissions caused by cooling increased largely with the house category and became the largest contributors in the medium (26%) and luxurious houses (41%). Topic: Engineering and Technology Innovation 100 Lightning Strike Risk Analysis On Base Transceiver Station (BTS) Using Lightning Strike Risk Factor CalculatorArjuni Budi Pantjawati1, Teguh Budiarto1, Syarif Hidayat2 This research describes the risk assessment of lightning strikes in the Base Transceiver Stations (BTS) which have different tower heights and geographical conditions. The analysis was performed by a software calculator based on IEC standard 62305-2 on the general principle of a lightning protection system. The parameter measured in this research is the value of the risk of lightning strikes in the BTS. Further, the calculator result compared to analytic calculation to measure the calculator accuracy. The comparison showed that both methods of calculation gives the same result. The result showed that the amount of the risk of lightning strikes on BTS with a height of 72 m higher than BTS with a height of 42 m, while the amount of the risk of lightning strikes on BTS in rural geographical conditions is higher than the other three geographical conditions. From this research it can be confirmed that height and geographical conditions of the BTS affect the amount of the risk of lightning strikes. It can also be concluded that this calculator can be used to help improving the BTS security system against the risk of lightning strikes. Topic: Engineering and Technology Innovation 101 LOCAL WISDOM OF TRADITIONAL HOUSE IN EARTHQUAKE RISK MITIGATION (Comparison of Traditional House in Kampung Naga,West Java and Minka Gassho-Zukuri Architecture in Shirakawa Village, Gifu Perfecture, Japan)Ahmad Yani; Lilis Widaningsih; Rosita Long before information and technology are developed, Japanese people have been aware that they live in an earthquake area. They have local wisdom how to build houses in the attempt of mitigating earthquake risk. Japanese traditional house such as Minka style in Shirakawa Village, Gifu Prefecture is an example. Indonesia also has local wisdom in disaster mitigation. Kampung Naga in West Java Province is the example. Based on the facts above, the researchers want to learn more the architecture of Minka traditional house in Shirakawa Village, Gifu Prefecture, Japan and Kampung Naga traditional house in West Java, Indonesia. The study focuses on the best of traditional technology to adapt to earthquake. The research aims to find local wisdom values as learning resource for the next generation of each culture. In detail, the study intends to (1) describe the architecture of Minka traditional house in Shirakawa Village, Japan and Kampung Naga traditional house in West Java, Indonesia; (2) depict the traditional norms of building Minka traditional house in Shirakawa Village, Japan and Kampung Naga traditional house in West Java, Indonesia; and (3) describe local wisdom values which can be found in traditional houses as learning resource in schools in West Java, Indonesia. The study employs descriptive method and intends to describe the values of local wisdoms in traditional houses in adapting to earthquake. Location of the research is in Kampung Naga West Java Indonesia, Shirakawa Village, and at some primary schools in West Java, Indonesia and around Tokyo. Methods of data collection are interviews and field observations. The results are research article published in the journal, textbooks used as a learning resource in elementary and junior high school, and reports to funders are Sumitomo foundations. Topic: Innovation in Teaching and Learning 102 MENGEMBANGKAN KOMPETENSI CALON GURU VOKASIONAL MELALUI IMPLEMENTASI MODEL PROGRAM PENDIDIKAN GURU TRANS-NASIONALDr. Amay Suherman Dr. Sudjani Dr. Dadang Hidayat M. Perkembangan Sekolah Menengah Kejuruan (SMK) di Indonesia yang pesat berdampak pada peran dan kesiapan Lembaga Pendidikan Tenaga Kependidikan (LPTK), khususnya Pendidikan Teknologi dan Kejuruan (PTK), sebagai penghasil calon guru profesional SMK. LPTK-PTK perlu mengembangkan Model Program Pendidikan Guru yang memenuhi tuntutan peraturan terkait dengan program pendidikan guru vokasional. Salah satu model program pendidikan guru vokasional yang dapat memenuhi harapan adalah melalui Program Pendidikan Guru Trans-Nasional (PPGTN). Tujuan jangka panjang dari penelitian ini adalah mengembangkan kompetensi calon guru vokasional melalui implementasi model program pendidikan guru trans-nasional. Target yang ingin dicapai pada penelitian tahun pertama menekankan pada pengembangan model implementasi PPGTN. Pendekatan penelitian adalah research and development, ditempuh dalam dua tahapan. Hasil pada tahap pertama berupa Model PPGTN meliputi: Sistem rekruitmen mahasiswa; Pola perkuliahan; Pelaksanaan perkuliahan. Sistem rekruitmen mahasiswa menerapkan skema kerja sama, agar dapat mengakomodir keperluan lapangan. Pola perkuliahan menerapkan \u201csandwich system\u201d antara di kampus dan di daerah asal. Perkuliahan di kampus tatap muka,tugas terstruktur dan belajar mandiri dijadualkan di kelas; Perkuliahan di daerah asal dengan memanfaatkan ICT. Pelaksanaan perkuliahan menerapkan sistem termin, dengan penerapan sistem SKS secara \u201cfull days\u201d. Model PPGTN tersebut sudah melalui tahan evaluasi dan pengembangan melalui Focus Group Discusion, dengan melibatkan ahli pendidikan dan praktisi pendidikan. Topic: Vocational Education and Training 103 Microscopic Virtual Media (MVM) in Physics Learning to Reduce Misconceptions: A Case Study on Students Understanding of the ElectricityFirmanul Catur Wibowo1,a), Andi Suhandi2,b), Dadi Rusdiana3,c) Dina Rahmi Darman4,d) A domain of research in physics teaching is focused on the study of the effects of different types of learning interventions aimed to help students build a scientific conception. Microscopic Virtual Media (MVM) is the application of a particular interest in physics learning because they can support microscopic powerful modeling involves physics concepts and processes. In this study, one group (experimental) of 19-20 year old students were studied to determine the MVM role in the development of a functional understanding of the concept of electricity. The experimental group used MVM using DLSM (Dual Situated Learning Model). The results presented here indicate that students work with virtual media exhibited significantly higher scores in research tasks. Our findings strongly support that MVM can be used as an alternative instructional tool, in order to help students confront reduce their scientific misconceptions and develop an understanding of physics concepts Topic: Innovation in Teaching and Learning 104 MODEL GUIDANCE STUDENT ENTREPRENEURSHIP THROUGH TECHNOLOGY-BASED BUSINESS INCUBATOR WEB 2.0 (Technological Incubator Based Learning \/ TIBL)Ratih Hurriyati Lisnawati Sulastri Indonesian youth unemployment rates are among the highest when compared with other countries. Unemployed youth in Indonesia reached 25.1 percent of the total workforce. Therefore, there must be a serious effort to reduce the unemployment rate. The general objective of this study is to provide solutions to the problems of educated unemployment realists in Indonesia, which reached 41.8 percent of total national unemployment, of which 12.78 percent came from college, one of the graduates of the University of Education Indonesia. While the specific goal is to follow up the results of learning innovation in 2013 where, during the two years of the entrepreneurial development program that has been run by the Business Management Education Program by applying TIBL (technology incubator based learning) through the subjects of small and medium-scale business. The methodology used is done through a coaching approach andragogy, which prioritizes the participation of attendees through the Business Incubator based on web 2.0 technologies are developed using the prototype through four phases, namely Phase Understanding Entreupreneurial Process, Stages being The Entreupreuneur, Stages be the entrepreneur, and Stages Scale Up. The fourth stage is carried out through 16 times personally consisting of Creativity Program (creativity program), Program Preparation Basic (foundation program), Entrepreneurship Program (entrepreneurship establishing the program), Hatching Program (hatchery program). Coaching participants are students UPI, which has a high commitment to entrepreneurship is taken from the selection process through the awarding Edupreneur activities that have been undertaken by the Business Management Education Program in the Year 2013 and 2014. The result of this activity is the availability of web 2.0 with the name http:\/\/business-incubator.pkm.upi.edu UPI place for students to carry out entrepreneurial activities. Topic: Innovation in Teaching and Learning 105 Modification of Cilembu Sweet Potato Starch With Ethanoic AcidAi Mahmudatussa\u2019adah1*, Yulia Rahmawati1, Sudewi 1 Cilembu sweet potato harvest was abundant, its use was still limited. Starch was required by various industries. Starch is generally beige, and requires a long time for the drying process. The purpose of this research was to produce a modified starch with ethanoic acid which has a white color. The method used in this study was the experimental method. The results showed acid modified starch yield was 18%, with the color characteristics of L *: 96.38 \u00b1 0.82; a *: -0.86 \u00b1 0:06; b *: 2.93 \u00b1 0:04. Native starch yield was 16%, with the color characteristics of L *: 93.55 \u00b1 0.91; a *: -0.70 \u00b1 0:02; b *: 2.70 \u00b1 0:03. The conclusion of this study was modified starch of Cilembu sweet potato using ethanoic acid have higher yield and more white bright than native starch. Topic: Engineering and Technology Innovation 106 Naturally surveilled space: The design of a male drug rehabilitation centerAkbar Raditya Permana, Tutin Aryanti, Fauzi Rahmanullah The increase of drug addicts in Indonesia has not been supported by adequate facilities, both quantitatively and qualitatively. Despite being treated in a rehabilitation center, drug addicts may still use drugs surreptitiously and put themselves in danger. Architectural design may contribute to this either positively or negatively. This article elaborates a therapeutic design of a male rehabilitation center in the borderland of Bandung city, Indonesia. Employing the notion of natural surveillance, the rehabilitation center is designed to allow continual control over attendees without them feeling suppressed. The center design uses a behavioral approach to consider both attendees\u2019 physical and psychological comforts, as well as their security. Building masses are designed in a way that forms an inward orientation and are laid out circularly according to the therapy processes that attendees must undertake. Moreover, rooms are planned differently in response to attendees\u2019 unique conditions and restrictive physical requirements, such as their restriction on lighting and requirement of water for treatment. The landscape uses shady trees and vegetations as natural borders to demarcate the private zone, where attendees live, from the public area, where visitors may enter. The design is intended to provide a model for a responsive drug rehabilitation center that facilitates drug addicts\u2019 recovery. Topic: Engineering and Technology Innovation 107 Novel Binary PSO Algorithm Based Optimization of Transmission Expansion Planning Considering Power LossesAstuty and T Haryono Abstract. Transmission expansion planning (TEP) is one of the issue that have to be faced caused by addition of large scale power generation into the existing power system. Optimization need to be conducted to get optimal solution technically and economically. Several mathematic methods have been applied to provide optimal allocation of new transmission line such us genetic algorithm, particle swarm optimization and tabu search. This paper proposed novel binary particle swarm optimization (NBPSO) to determine which line should be added to the existing power system. There are two scenerio in simulation. First, considering transmission power losses and the second is regardless transmission power losses. Compare to the first scenario, the number of new line in second scenario is less but produces high power losses that cause the cost becoming extremely expensive Topic: Engineering and Technology Innovation 108 Numerical Simulation of Marine Currents in the Bunaken Strait, North Sulawesi, IndonesiaParabelem Tinno Dolf Rompas (a*), Jenly Dyliep Isria Manongko (a) This study is intended for the generation of hydroelectric power at suitable area of the strait in order to provide electric current to a close environment. The project uses a three-dimensional model of taking flow into account the variation of hydrostatic pressure in the liquid vertical layers. We are thus brought back to a two-dimensional calculation using the shallow water equations. The objective of the study is the simultaneous obtaining of the current velocities and the power availabilities of the tides per unit of the strait horizontal area. The Bunaken strait is 5280 m width for an average depth of 130 m. Numerical calculation is simulated using horizontal meshes of 60 side meters. The numerical solutions were obtained by using a time step of one second. It was found that there was no great difference between 2D and 3D numerical simulations because the effect of flow velocity in the vertical direction is very small. The kinetic energy ranged from 0.01 to 1.00 kW\/m2 when the tide and low tide in the Bunaken strait area at discharge of 1 Sv, whereas at discharge 2 Sv, 0.41-17.40 kW\/m2 (low tide currents) and 0.11-2.77 kW\/m2 (high tide currents). These results can be used in the design of turbines for power generation ocean currents in the Bunaken strait at depths below 60 meters. Topic: Engineering and Technology Innovation 109 Optimization Production Flakes Sweet Potato (Ipomoea batatas L) as Diversification of Sweet Potato ProcessingAi Mahmudatussaadah, Yulia Rahmawati, Sudewi Yogha Sweet potatoes are rich in carbohydrates, vitamins, minerals, fiber and anthocyanins. Sweet potato processing still limited into flour, keremes, dodol, cheese stick, or chips. Flakes as pre-cooked meals made through the stages of manufacture of pasta and drying. The purpose of this study was to optimize the production of sweet potato flakes at the stage of making pasta and drying. Making the pasta is done through techniques steamed or roasted. Pasta drying using a drum dryer or dryer cabinet. This research was done by experimental method. The indicator of optimization is the number of monomeric anthocyanins, \u03b2-carotene and the color resulting from flakes. The results showed that the number of monomeric anthocyanin flakes with steamed techniques, drum dryer (0:03 \u00b1 3.83 mg CYE \/ g bk), flakes with steamed technique, dryer cabinet (3:03 \u00b1 0:02 CYE mg \/ g bk), flakes with roast technique, drum dryer (2:49 \u00b1 0:05 CYE mg \/ g bk), flakes with roast technique, dryer cabinet (0:03 \u00b11.98 mg CYE \/ g bk). The Color of purple sweet potato flakes produced through techniques steamed was bright purple, while the color purple sweet potato flakes produced through techniques roast give a brownish purple color. The amount of \u03b2-carotene Flakes yellow sweet potato steamed cooking stage, drum dryer (152 mg \/ Kg bk), grilled drum dryer (136 mg \/ kg bk), yellow sweet potato flakes roasted stages cabinet dryer (140 mg \/ Kg bk), and grilled stage dryer cabinet (122 mg \/ kg bk). In conclusion sweet potato flakes production techniques through the stages of process steam, and the dryer drum has a number of anthocyanin, or \u03b2-carotene bigger and brighter colors than the toasted flakes techniques and dryer cabinet. Topic: Engineering and Technology Innovation 110 PEMETAAN KUE JAWA BARAT BERBASIS JENIS KUE, BAHAN DASAR DAN TEKNIK PENGOLAHANAtat Siti Nurani, Sri Subekti, Ana Penelitian ini dilatarbelakangi oleh kurangnya literature tentang kue nusantara khususnya jawa Barat. Penelitian ini bertujuan untuk menggali identifikasi jenis keragaman kue ,teknik pengolahan, bahan dasar yang digunakan dengan menyusun bahan ajar kue nusantara khususnya Jawa Barat. Metodologi yang digunakan observasi deskriptif, dengan metode wawancara , sampel ditentukan dengan acak dari setiap wilayah bagian di Jawa Barat. Temuan Hasil yang diperoleh, mengidentifikasi kue tradisional khas Jawa Barat meliputi 1. jenis kue ; terdapat jenis kue yang sama disetiap wilayah penelitian yaitu opak, rangginang, nagasari, aliagrem, cuhcur, keripik, semprong, wajit, dodol, kecimpring, combro, tape ketan, serabi. Jenis kue yang khas yaitu burayot (garut), simping kaum (Purwakarta), surabi hejo (Karawang), papais cisaat (Subang), Papais moyong, opak bakar (Kuningan), opak oded, ranggesing (Sumedang), gapit, tapel (Cirebon), gulampo, kue aci (Tasikmalaya), Wajit Cililin, gurilem ( Bandung Barat), borondong (Kabupaten Bandung), 2. teknik pengolahan yang beragram yaitu teknik mengukus, merebus, menggoreng, membesta, memanggang, membakar, menyangan, menggulai. 3. Bahan yang digunakan sangat beragam yaitu beras biasa, beras ketan local, tepung beras, tepung beras ketan, tepung kanji, tepung terigu, tepung hunkue, singkong, ubi jalar, pisang, kacang tanah, jagung. 4. Terdapat pengelompokan kue di jawa barat yaitu (1) kue tradisional, (2). kue hasil mencipta, (3). kue hasil modifikasi, (4). kue pengaruh dari luar. Topic: Vocational Education and Training 111 Penerapan Model Pembelajaran Project Base Learning (PjBL) pada Pelajaran Praktik Pemesinan untuk Meningkatkan Penguasaan Prosedur Kerja Siswa SMKhendy kustendy, prof. DR. H Ash Ari Jhohar, MPd. Mastery of the working procedure of practice learning has now is low so that for learning output. Education has now become a serious issue which will increase the quality of human resources is a prerequisite for achieving development goals, one way to increase the human resource is the quality education and master competencies. This study describes the application of learning models peningakatan project based learning in peningakatan student mastery of the working procedure students of class XI TPM I Machene Technical Skills Competency SMK Merdeka Soreang by the number of students by 30 people. The data were taken that relate to work procedures taken using a written test and performance tests conducted three sessions in accordance with the implementation of experimental learning. During the learning process the experimental class each test was collected using LJK and observation assessment form. The findings showed that (1) there is a significant difference in the achievement of mastery of work procedures on the treatment class, (2) there is a significant increase control of working procedures each treatment session on the class, (3) there is a significant difference in the achievement of mastery of working procedures in class treatment and control class after the application of the model learning Project Based learning, it can be concluded that the Project Based learning model can improve the working procedures of machining lathe mastery of grade XI TPM 1 machining engineering competence SMK Merdeka Soreang. Topic: Innovation in Teaching and Learning 112 Perhitungan Harga FSR (Free Spectral Range) pada Microring Resonator (MRR)Tunggal dengan Sepasang Pandu Gelombang Silikon di atas Insulator dengan Metode Semi-numerik1Budi Mulyanti, 2Lilik Hasanah, 1Tommi Hariyadi, 1Arjuni B. Pantjawati, 1Nurhidayatulloh, 3Heru Yuwono Karakteristik suatu microring resonator (MRR) baik dalam konfigurasi tunggal maupun jamak sangat ditentukan oleh faktor FSR (free spectral range) dan Q-factor. FSR adalah jarak dua puncak intensitas yang berdekatan. Untuk keperluan aplikasi sensor berbasis MRR, harga FSR sangat mempengaruhi performansi\/unjuk kerja divais tersebut. Dalam penelitian ini dilakukan perhitungan harga FSR menggunakan 2 (dua) metode yaitu metode analitik dan semi-numerik berbasis metode transfer matriks. Adapun parameter-parameter yang digunakan dalam perhitungan adalah panjang gelombang datang, radius cincin pandu gelombang, lebar celah antar pandu-gelombang, dimensi pandu-gelombang, serta indeks bias silikon, SiO2 dan udara. Hasil kedua perhitungan tersebut kemudian dibandingkan dengan hasil simulasi menggunakan simulator elektromagnetik 3 dimensi berbasis FIT (finite integration technique). Dapat disimpulkan adanya perbedaan (mismatch) dari ketiga hasil tersebut dan hasil perhitungan semi numerik lebih mendekati simulasi dengan rata-rata perbedaan sebesar 11, 13%. Kata kunci: microring resonator, free spectral range, silikon, SiO2, finite integration technique Topic: Engineering and Technology Innovation 113 Ply thickness FiberGlass on Windmill Drive Salt Water PumpAgus Sifa,Badruzzaman, Dedi Suwandi Factors management of salt-making process needs to be considered selection of the location and the season is very important to support the efforts of marinating. One factor is the salt water pumping windmill is used, the windmill is helpful to remove the water and pumping the water [1]. At this time showed strong fiber composite material either continuous or noncontinuous surrounded by the matrix material is weak. The matrix serves to distribute the fiber and also transmit the load to the fiber. [2], composite or fiberglass are used for windmills mixture of E-glass and Epoxy. The mechanical characteristics of the power of his blade one of determining the materials used and the thickness of the blade, which needed a strong and lightweight. The calculation result thick fiberglass with a composition of 60% fiber and 40% epoxy, at a wind speed of area salt fields 9 m \/ s, the drag force that occurs at 11,56 kg, then the calculation result by 0,19 mm thick with a layer of 10, the total thickness of 1,9 mm, with a density of 1760 kg \/ m3, mechanical character of elongated elastic modulus of 46200 MPa, modulus of transverse elasticity of 10309,6 MPa, shear modulus of 3719 MPa and Poisson ratio of 0,31, then the calculation using the finite element ABAQUS obtained critical point at the confluence of the blade to the value of Von Mises tension was happening 1,158e9 MPa maximum and minimum 2,123e5 MPa, for a maximum value of displacement occurred condition at the tip of the blade. The performance test results windmills at a wind speed of 5,5 m \/ s wind power shows that occur 402,42 watts and power turbines produced 44,21 watt, and TSR 0,095 and the value Cp of 0,1. Topic: Engineering and Technology Innovation 114 Popular Government Website Hacking: Cause, Chronologies, Investigation, Ethics and Law Review, and Preventions Case Study: Hacking on KPU Website and President WebsiteYana R. Sopian, Joko Widiarto, Prima Febriansyah, Hermawan Prasetyo, Farniwati Fattah Development of Information Technology (IT) leads the negative effects, which include the misuse of IT against ethic and law as known as cybercrime. One kind of cybercrime is website hacking, which can aim at destruction, theft, or other criminal activities on the website. This paper explains the popular government website attacks on its motivation, scenario, and investigation, as well as the review based on Indonesian regulation, with the case studies on Komisi Pemilihan Umum (KPU), i.e. an Indonesian National Election Commission and President Susilo Bambang Yudhoyono (SBY) websites. This paper also elaborates the prevention of a system from such attack. Topic: Engineering and Technology Innovation 115 Preliminary Study: Kinetics of Oil Extraction from Sandalwood by Microwave-assisted HydrodistillationHeri Septya Kusuma, Mahfud Mahfud Sandalwood and its oil, is one of the oldest known perfume materials and has a long history (more than 4000 years) of use as mentioned in Sanskrit manuscripts. Sandalwood oil plays an important role as an export commodity in many countries and its widely used in the food, perfumery and pharmaceuticals industries. The aim of this study is to know and verify the kinetics and mechanism of microwave-assisted hydrodistillation of sandalwood based on a second-order model. In this study, microwave-assisted hydrodistillation is used to extract essential oils from sandalwood. The extraction was carried out in ten extraction cycles of 15 min to 2.5 hours. The initial extraction rate, the extraction capacity and the second-order extraction rate constant were calculated using the model. Kinetics of oil extraction from sandalwood by microwave-assisted hydrodistillation proved that the extraction process was based on the second-order extraction model as the experimentally done in three different steps. The initial extraction rate, h, was 0.0232 g L-1 min-1, the extraction capacity, CS, was 0.6015 g L-1, the second-order extraction rate constant, k, was 0.0642 L g-1 min-1 and coefficient of determination, R2, was 0.9597. Topic: Engineering and Technology Innovation 116 PREPARING TVET TEACHER COMPETENCE THROUGH INDUSTRY PRACTICE PROCESSI WAYAN RATNATA Abstract Entering the year 2015 all ASEAN countries have to be ready to face the MEA era, namely era for competition for various sectors, certainly through healthy competition, and fair. Indonesia constitutes one of ASEAN countries and with the number of 250 millions populations should be ready to deal with various competitions. Currently, trending issues developing in Indonesia relating to still low the skills of workforce and worrying they will be difficult to compete with other workforce around ASEAN countries. Relating to increase the skills for human resources, vocational teachers\u2019 role constitutes crucial aspects that should be paid attention by government. Through this research, attempted to provide solutions to enhance competence the vocational teachers\u2019 candidates through appropriate process during they are still at university. The research done in Department of Electrical Engineering Education, Faculty of vocational and technology education, Indonesia university of education; and in some industries in Bandung Indonesia, where students doing industrial practice. This research using qualitative research method, and the aim of the research is to produce vocational teachers having good knowledge and skills according to their study field. Associated to the industry practice process to enhance TVET teacher students, university (the world of education) should build good cooperation to the world of industry through three stipulations that should be met, namely: memorandum of understanding (MoU) between the world of education and the world of industry; university possesses study program related to the world of industry; and industry possesses enough facilities and equipment for conducting industrial practice. Topic: Vocational Education and Training 117 Principal Leadership to Alleviate Poverty at Lagging and Social Crisis Area in District of South-west Sumba East Nusa TenggaraTetty Setiawaty & Gunadi Tjahjono This research\u2019s purpose are: (1) produce guidelines models of principal leadership in alleviate of poverty at lagging and social crisis area in district of South-west Sumba East Nusa Tenggara; and (2) produce indicator models of principal leadership in people empower through education. Research method applies Research and Development (R&D), data collecting consisted of two stages, i.e. (1) pre-development stage, comprising: preliminary study, design and validation phase; and (2) development stage, comprising: limited test, expended test, evaluation and revision phase. The subjects of this research are principals, vice principals and teachers of senior high school (SMA) and vocational high school (SMK) in South-west Sumba. The results of this research are: (1) principal apply the model of visionary leadership to alleviate poverty through education; (2) empowering people to improve learning quality; (3) involving community leaders, religious leaders and churches to increase student motivation. Topic: Innovation in Teaching and Learning 118 Principal Leadership to Alleviate Poverty at Lagging and Social Crisis Area in District of South-west Sumba East Nusa TenggaraTetty Setiawaty & Gunadi Tjahjono This research\u2019s purpose are: (1) produce guidelines models of principal leadership in alleviate of poverty at lagging and social crisis area in district of South-west Sumba East Nusa Tenggara; and (2) produce indicator models of principal leadership in people empower through education. Research method applies Research and Development (R&D), data collecting consisted of two stages, i.e. (1) pre-development stage, comprising: preliminary study, design and validation phase; and (2) development stage, comprising: limited test, expended test, evaluation and revision phase. The subjects of this research are principals, vice principals and teachers of senior high school (SMA) and vocational high school (SMK) in South-west Sumba. The results of this research are: (1) principal apply the model of visionary leadership to alleviate poverty through education; (2) empowering people to improve learning quality; (3) involving community leaders, religious leaders and churches to increase student motivation; (4) Involve churches and religious leaders in motivating student learning; and (5) Involving community leaders in keeping schools and school facilities Topic: Innovation in Teaching and Learning 119 Problem based Learning to Promote Vocational Students Questioning and Problem Solving SkillsLeo Muhammad Taufik (a*), Saefudin (b) This study was conducted to analyze the increase of questioning and problem solving of environmental polution through problem-based learning (PBL). The method is weak experiment by using \u201cthe one group pretest-post-test design\u201d involved 31 vocational students majoring automotive engineering. The result showed that percentage cognitive level of questions that asked by students after PBL has decreased on C1 level (17.7%) and C2 level (0.8%) but the cognitive level of questions after has increased on C3, C4 ,and C5 level respectively 1.6%; 11.3%; 5.7%. Student\u2019s problem solving skills has increased with medium category (N-gain = 0.5). Based on questionnaire, PBL can raise student\u2019s interest and motivation of learning the material, but the teacher has problem to managed the class and time in application of PBL. Topic: Innovation in Teaching and Learning 120 Profession Allowance, Work Motivation and Teaching Performance of Vocational TeachersD.L.Hakim (a*) ; U. Syaefuddin Sa\u2019ud (b): A.A. Wahab(c): A. Komariah(d) The purpose of this research is to reveal and analize the teaching performance of vocational teachers which is affected by the profession allowance and work motivation. Profession allowance and work motivation are two important variable in developing the teaching performance along with teacher\u2019s competence. Profession allowance is becoming a great motivator for a teacher in developing his teaching performance. Nevertheless, empirical studies shown that there are still teachers who work with low motivation and those who attend the teacher\u2019s certification process in order to gain the profession allowance only thus their performance is far from optimal. The method of this research was using a survey research with correlational approach. The population were taken from all certified teachers in public vocational schools in West Java. The result shown the teachers motivation and profession allowance had positive impact and significant with the teachers teaching performance. The corelation value of teachers motivation to their performance was at the average position, whilst the corelation value of profession allowance to teachers performance was at the low position. Based on the result shown, there are several suggestions proposed by the researchers to the related institutions. Topic: Innovation in Teaching and Learning 121 PROFESSIONALISM, LEADERSHIP, ORGANIZATIONAL CLIMATE TO COMPETITIVENESS PERFORMANCE IN HIGHER EDUCATIONBambang Trisno bangino2012@gmail.com The study aimed to describe the impact of professional lecturer , leadership and organizational climate to get performance competitiveness and also getting the recognition from national and international accreditation at higher education in Bandung west java. In this paper put forward some of the main variables that play an important role in building competitiveness performance on the program of study in higher education. In focus discussion studying the contributions of professional lecturer, organizational climate and from Leadership Chairman of the department from three (3) public universities in Bandung ; with the objective of identifying patterns of development in building on performance competitiveness of the higher education institutions by describing the characteristics and typical behaviors of each performance study program in the three universities within the scope of domestic and international. The research process explored through a qualitative approach with the support of various documents, references relating to the performance pattern of higher education institutions in the program of study in the field of social and technical (engineering). From the results of this study showed that a representative picture of the behavior of the typical performance of each program of study in the building of the institutions of higher education that are competitive, especially on the level of study programs in the higher education environment. Topic: Engineering Education 122 Profile of Clean Healthy Lifestyle (Perilaku Hidup Bersih Sehat-PHBS) Elementary School Students in BandungR. Patriasih Instill healthy behavior habits are fundamentally foundation that must be held at an early age. Students as one of the targets in order Clean Healthy Lifestyle (Perilaku Hidup Bersih Sehat-PHBS) educational institutions apparently still need to get the attention given to school-age children are also a vulnerable period prone to various diseases and the emergence of various diseases. The purpose of research is conducted to know the profile of PHBS of elementary school students from the knowledge, attitude and behavior aspect. The research was conducted by the cross-sectional study with a sample of primary school students with category UKS and non UKS school, consisting of 150 students from six elementary schools in Bandung. Results of the study revealed that students knowledge in general are in good criteria with an average score of 84 but still needs to be improved, especially on how to wash hands properly. Students PHBS attitudes are at moderate criteria scores by an average of 71. Students PHBS behavior are in the moderate criteria with an average score of 63. However, the aspects relating to the use of clean toilets, measure a body height and weight every 6 months, regular exercise, maintain themselves and the environment of mosquito larvae, done very well because it deals with school programs conducted regularly and none of the students who smoke at school. While the aspects of the eating snacks habit, how to wash their hands and behavior of littering are still lacking. The correlation between knowledge and the attitude of PHBS (Spearmans rho) showed the significant positive correlation (r=0389, p <0.01). Similarly, between knowledge and behavior of PHBS showed significant positive correlation (r=421, p<0.01). Topic: Innovation in Teaching and Learning 123 Rancang Bangun Multimedia Animasi untuk Meningkatkan Kualitas Pembelajaran Penguatan Logam Guna Meningkatkan Sifat Mekanik MaterialAriyano, MT.; Dr. Amay Suherman Kesulitan mahasiswa Pendidikan Teknik Mesin pada mata kuliah Material Teknik diantaranya terkait dengan materi yang berhubungan dengan struktur kristal atom yang abstrak. Struktur kristal atom yang abstrak tersebut justru menentukan sifat-sifat material, secara khusus sifat mekanik atau kekuatan material. Pembelajaran teoritis berupa simbol-simbol verbal yang ada tidak cukup representatif menjelaskan konsep sistem yang diperlukan, sehingga tidak terjangkau (inaccessible) oleh peserta didik. Kondisi tersebut mengakibatkan efek pengalaman belajar yang tidak optimal. Hal ini berimplikasi terhadap kesalahan penguasaan konsep sistem material teknik yang utuh baik sistem struktur mikro atom maupun sistem perubahan sifat mekanik atau kekuatan material sesuai aplikasi realnya. Penelitian ini berkaitan erat dengan usaha meningkatkan kualitas pembelajaran Material Teknik, khususnya penguatan logam. Target khusus penelitian ini adalah menghasilkan multi media animasi untuk media pembelajaran penguatan logam dan sifat mekanik material. Pendekatan penelitian adalah research and development, ditempuh dalam dua tahapan . Hasil pada tahap pertama ini berupa Multi Media Animasi (MMA) tentang materi penguatan logam serta pengaruhnya terhadap sifat mekanik. MMA tersebut sudah melalui tahap evaluasi dan pengembangan melalui Focus Group Discusion (FGD) dengan melibatkan ahli media dan praktisi pendidikan. Topic: Engineering Education 124 SCADA System Design of Boiler Operation in Cirebon Steam Power PlantRillo Indra Prakoso*, Ade Gafar Abdullah, and Dadang Lukman Hakim SCADA is a system that works using the HMI (Human Machine Interface) as the medium of interaction between man and machine operator or an interaction between the equipment with other equipment. This paper aims to provide an understanding and recognition of how the working principle of a boiler steam power plant using SCADA systems. This design data is taken from the steam power plant in Cirebon. To assist this SCADA system design, it\u2019s used Wonderware Intouch software version 2007. This software using animation script to move an object and rely on logic-based programming languages such as IF, THEN, ELSE, ELSE IF, AND, OR, NOT, and ENDIF to give the command to an object actuated. So that the software can display and running processes on the boiler steam rate. Topic: Engineering and Technology Innovation 125 Self Organizing Map Algorithm Application for Anomalous Load ForecastingRetno Wibowo*, Ade Gafar Abdullah, Yadi Mulyadi Short-term load forecasting has an important role in the operation and planning of electric power systems. The last few years, this method is more often used for forecasting electricity load. Artificial neural network algorithm Kohonen maps is one form of Unsupervised Artificial Neural Network topology, where the training process does not require supervision\/target output. Load forecasting have different characteristics, such as characteristic on weekdays, weekends and public holidays are also a national holiday. Characteristics on weekdays and holidays have a pattern of repetitive loads(linear), while the characteristics of the national holidays that tend to have a different load patterns(non-linear), and therefore a national holiday known as the anomalous days. Topic: Engineering and Technology Innovation 126 Self-Designed Project Based Learning- in the Lathe Machining FieldA. Hamdani1), A Djohar1), Wahyudin2), D. Hidayat1) The problem in this research is the lack of direct experience of the atmosphere experienced by students in the industrial atmosphere that can enhance the students competence in the field of machining.The purpose of this research is to find a model learning design which can provide industrial atmosphere in the school and can improve work skills in the field of machining, and find the model implementation.The research methodology used was a research and development (R & D), which include 1) development of materials and instructional design that includes schools, industrial and Professional Certification Agency, 2) performed limited testing and extensive testing, 3) validation test of the products produced by conducting experiments in learning.The results showed that 1) the self designed learning model this project is an alternative learning which provide direct experience with industrial atmosphere in the school and is able to improve the skills demanded by industry. 2) This alternative learning more insightful works that include product planning, manufacturing work steps, cost planning and quality control of products and employment.Implications of this study are 1) Through this learning model, students can portray himself as a worker \/ operator and gain direct experience working atmosphere at school. And able to make grow the entrepreneurship spirit.2) implement this model is a challenge and to improve the professionalism of teachers, because in this model the teacher acting as supervisor as well as assessors. 3) implementation of this model can help develop learning and utilize the facilities, teacher resources and industrial relations in creating students who have industry competence. Topic: Innovation in Teaching and Learning 127 Separation of Glycerolysis Product Using HexaneS Mujdalipah1, A H Sasmita2, I K Amalia3, and A Suryani3 Oil or fat can be soluble in the polar and non-polar organic solvent. Hexane is non-polar solvent and it can dissolve triglyceride and free fatty acid. The aim of this research is to get the hexane ratio to the glycerolysis product that can produce product that contains rich of monoglyceride and diglyceride that has the best characteristics. The study was conducted in five stages. In the first stage, the physicochemical properties of palm oil were analyzed. In the second stage, palm oil glycerolysis was conducted. In the third stage, glycerolysis product was extracted by using hexane in various concentrations. In the following stage, physicochemical properties of extracted products were analyzed. In the last stage, data were subjected to an analysis of variance. Results showed that ratio between glycerolysis product to hexane did not significantly (\u03b1=0.05) affect the yield, the free glycerol content, and the ability of extracted products to reduce the surface tension of water. It was also found that the best treatment had yield of 28.5%. It was able to lower water surface tension from 72.00 to 31.80 dynes\/cm and had free glycerol content of 2.40%. Topic: Engineering and Technology Innovation 128 Short Term Load Forecasting of Anomalous Load using Artificial Neural NetworkRohmah, K. A., Abdullah, A. G., Mulyadi, Y. Short Term Load Forecasting (STLF) has always been an important instrument in the power system operation. The various of operation decision is determined by the load forecasting, such as scheduling settings generator capacity, analytical testing, and maintenance planning generator. This study aims to optimize the methods STLF using feed forward back propagation methods which is one of the algorithms of Artificial Neural Network (ANN). Studies conducted on the data usage electrical load by consumers on holidays, so-called anomalous load, because it has a different load patterns with normal days. The result of optimal forecasting tested by changing the value of learning rate, number of hidden layers and the number of input learning. Performance of this method was tested by comparing simulation results with real data load from Indonesia Power Company, West Java region. There is a significant difference in performance accuracy due to changes in learning rate and the number of input learning. While the impact of increasing the number of hidden layer only have an impact on the computing speed. Topic: Engineering and Technology Innovation 129 Short Term Load Forecasting of Anomalous Load using Hybrid Soft Computing MethodsSopian Al Rasyid*, Ade Gafar Abdullah, Yadi Mulyadi Load forecast accuracy will have an impact on the generation cost is more economical.The use of electrical energy by consumers on holiday, show the tendency of the load patterns are not identical, it is different from the pattern of the load on a normal day. It is then defined as a anomalous load. In this paper, the method of hybrid ANN-Particle Swarm proposed to improve the accuracy of anomalous load forecasting that often occur on holidays. The proposed methodology has been used to forecast the half-hourly electricity demand for power systems in the Indonesia National Electricity Market in West Java region. Experiments were conducted by testing various of learning rate and learning data input. Performance of this methodology will be validated with real data from the national of electricity company. The result of observations show that the proposed formula is very effective to short-term load forecasting in the case of anomalous load. Hybrid ANN-Swarm Particle relatively simple and easy as a analysis tool by engineers. Topic: Engineering and Technology Innovation 130 Simple Science Media to Improve Understanding Concept of Student in Secondary School at Central SulawesiMuhammad Ali, Supriyatman and Sahrul Saehana It has been developed simple science media from second goods for student of junior high school in donggala. The simple science media is used to explain expansion concept, hot transfer and pressure of liquid. This research used research and development methods. The student of class VIII SMP 1 Sigi and SMP 2 Dongala were a subject of this research. Development phase consist of phase of investigation of early, design phase, realization phase, implementation phase. The indicator of this research include: validity, practicality and effectiveness. The improvement of study results of student become key indicator in this research. The early investigation shows study result and enthusiasm of student were low. After the simple science media was implementated, study result and enthusiasm of students were higher than before. The mean score at post test and pre test were 10.00 and 14.00, respectively. According to the students that the applying of the media is very good (90%), the design of simple science media is also very well of (80%). Topic: Innovation in Teaching and Learning 131 Skills Training Model Installation and Maintenance of Electrical Installations House For Youth in Village Sukajaya Kabupaten BandungJaja Kustija, Hasbullah, Elih Mulyana, Bambang Trisno The aim of this study was to develop a model training on installation and maintenance of electrical installations of residential houses for the youth in the village Sukajaya Kabupaten Bandung. The research activities carried out by using participatory approaches Demand Responsive Approach. Training model developed is based on the needs and potential of the community and local village in the context of development and sustainability, so expect the results of this training can provide job opportunities for youth in an effort to improve peoples lives. Skills training model developed include: (1) the planning stage (need asesment), (2) the implementation phase training, (3) mentoring phase and (4) monitoring and evaluation. Results of the training model of the installation of electrical developers are opening business opportunities for the youth in the field of electrical installation so as to improve the standard of living and well-being as well as the establishment of a model village built supported by the village government officials with PKM team-based partnership UPI. Topic: Vocational Education and Training 132 Social Skills and Social Values in Malaysian Dual Training System ApprenticeshipNorhayati Yahaya (a*), Mohamad Sattar Rasul (b), Ruhizan Mohamad Yasin (c) The mismatch between the skills required by employers and the skills acquired by graduates is one of the major causes of unemployment among the graduates. The efforts to reinforce advanced technology skills training must be synchronized with the instilling of social skills and social values among the Malaysian Dual Training System apprentices. The purpose of the study was to identify the social skills and social values\u2019 problems faced by the Malaysian Dual Training System apprentices based on the perspective of the employers. The study is based on the elements of social skills and social values outlined in the Handbook on Social Skills and Social Values in Technical Education and Vocational Training by the Ministry of Human Resources. The qualitative approach applied in this study was focused group discussion (FGD). The respondents were employers from various fields of the manufacturing sector and altogether, five groups of FGD were carried out. The data were analyzed with the Atlas.ti software using the thematic analysis strategy, and a few themes were identified from the analysis. The findings illustrate that a majority of the employers provided positive feedbacks towards the technical skills of the apprentices. On the other hand, a few concerns were raised by the employers regarding the social skills and social values of the Malaysian Dual Training System apprentices. Topic: Vocational Education and Training 133 Socio Cultural Phenomenon And Citizenship Rights Of Indigenous Communities In Remote Island BuruFatimah Sialana This is the result of research on society tribal or indigenous community remote Buru island of Maluku Province. Many of the problems encountered related to social life as well as the rights of citizenship are not being met. Some indigenous peoples rights which have not been fulfilled is the right to obtain a proper education, livelihood rights such as appropriate housing, health facilities, sanitation, electric lighting, roads and transportation facilities. Construction of socio-cultural aspects of culture puts women in a weak position, and the male in a superior position. Mating foster culture and polygamy still exists today despite the smaller presentation, cultural understanding narrow strips communities of mindset is not easy to accept the changes that come from the outside and from the government itself. The efforts made by the government to improve the livelihood and quality of life people often encounter obstacles in the form of refusal to the argument is not allowed by the local culture. Ironically the parties that have been commissioned by the government often does not perform well as a variety of facilities and infrastructure constraints are insufficient to support its performance. The informants are traditional leaders, community leaders, local governments Buru and civil society groups that have concern for the remote indigenous communities of the island of Buru. The sampling technique is purposive sampling and research methods are observation, interview and documentation. Topic: Engineering Education 134 SOFT SKILL DEVELOPMENT OF STUDENTS SMK ENGINEERING SKILLS REFRIGERATION AND AIR CONDITIONING COMPETENCE THE IMPLEMENTATION OF TF-6M MODELSugeng Rifqi Mubaroq, Amay Suherman, Dadang Hidayat M. This study aims to find the implementation TF-6M model in changing the school environment into industrial climate in vocational competencies Technical Expertise Refrigeration and air conditioning, and also to develop the soft skills of vocational students Competency Technical Expertise Refrigeration and air conditioning. This research was conducted by using Quasi Experiment with Time Series Design. The instrument used in this study a questionnaire and observation sheet. The findings obtained in this study is the changing weather patterns learned in school from becoming a regular pattern of using industrial climate. Patterns and findings observed were a) visit the industry small, medium and large, b) differences in assessment in the industry are Go-No Go and schools are Range, c) differences in discipline that forms in the industry is stronger than in school, d) Equation climate industry practices in the industry and schools in terms of equipment, materials and workmanship. In this study, the students are also trained to use industrial climate in terms of communication, analysis and processing orders, as well as the implementation of the social agreement in the industrial climate of the school. Topic: Vocational Education and Training 135 Spatial Layout and Womens Participation in the MosqueTutin Aryanti; Sri Handayani Women\u2019s frequent attendance in mosque\u2019s social and religious activities does not always relate to their intense participation in the mosque\u2019s management. Neither does it benefit women to have a broad, if not unlimited, access to the mosque\u2019s resources, such as information and spaces. The low number of female representatives in the mosque management committee often results in their silence, and thus do not represent women as the mosque\u2019s users. This article investigates the way in which mosque\u2019s spatial arrangement contributes to women\u2019s participation in the mosque\u2019s activities and management. The qualitative research was conducted in Masjid Daarut Tauhid, a well-known community mosque in Bandung (Indonesia) where a number of routine Islamic learning activities are held and attended by local residents and visitors. Data was collected using participant observation, questionnairs distributed to female congregations, and interview with select participants, and analyzed using the psychology of space theory. It shows that the mosque\u2019s spatial arrangement, which locates women in a separate room, contributes to women\u2019s low participation in a mixed-sex Islamic teachings and mosque management. The results suggest architect\u2019s gender awareness in designing space and mosque committee\u2019s attention to involve more women in the management. Topic: Engineering and Technology Innovation 136 Standard Implementation of Learning Teaching Factory 6 Step Model (TF-6M)Dadang Hidayat Martawijaya, Amay Suherman, dan Sudjani. Learning Teaching Factory 6 Steps Model (Model TF-6M) is a learning model that creates a climate in the school industry, can improve their competence and at the same time form the soul Entrepreneur vocational students. Model TF-6M as a concept has been refined through the research that is strengthening the concept, so that the resulting formulation of the operational implementation of the model. Formulation is that in implementing the Model TF-6M at the Skill Competency particular integrating one or several subjects Productive with subjects Entrepreneurship, held in blocks of time so as well as Industry Work Practices (Prakerin) followed by a competency test and ends with the development of the craft Entrepreneur based Web rehearsal or rehearsal map. Through the implementation, the definition is implemented on the competence and expertise pastry and refrigeration and air conditioning for class XI with each of the two study groups. Pastry competency skills in SMK 3 Garut while competency of Refrigeration and Air Conditioning expertise in SMK TI Cimahi. Of the implementation of the Model TF-6M in both competency skills, the indicators of competence soft skills and hard skills the students can be achieved perfectly, including the competence entrepreneur, once generated Implementation Standards of Model TF-6M which is a common reference in implementing the Model TF-6M them. Implementations Standards of Model TF-6M consists of: 1. Preparation of Implementation; include: - change management \/ school climate into industrial climate; - Practice communicate; and - analyzing the exercise order. 2. Implementation; include: the preliminary stage; - The core stage; and the concluding phase \/ evaluation. Standard implementation of the Model TF-6M is obtained strategies, approaches and learning methods: problem solving; student learning is active (CBSA); scientific approach; inquiry discovery; question and answer method; discussions; play a role; Contextual Teaching Learning (CTL); social agreement; reel job; reel teaching; authentic learning and authentic evaluation, problem base learning; production based learning. Implementation Model TF-6M requires seriousness because, for the early stages should be able to overcome obstacles and challenges: 1. The ability to convince the principal for the implementation of the Model TF-6M policy requires principals; 2. Facing resistance from teachers because - do not like \/ feel disturbed by the change; - Over \/ under estimate of Model TF-6M; - Shows unprofessional attitude. Teachers who enjoyed and needed to run this model is a prolific teacher who has her love vocational competencies and the teaching profession. Topic: Vocational Education and Training 137 STUDENTS\u2019 MISCONCEPTION AND MENTAL MODEL ON HEAT CONCEPT: IMPACT OF PHYSICS CURRICULUM IN VOCATIONAL SCHOOLSIka Mustika Sari, Duden Saepuzzaman, Melisa Cahyadi Physics is one of basic sciences that must be mastered by vocational-students. The aims of this study were analyze the physics curriculum in vocational high school with the implication to students\u2019 misconception and mental model in heat and heat transfer concept. This study is a case study in one of vocational high school in Bandung City. The research method is descriptive. The data of students\u2019 misconception were taken by test using three-tier instruments. Mental model was probed by semi-structured interview. The result shows that Physics only have two hours per week to be taught. On the other hand, only the students in class X and class XI who have been taught physics. However, in physics syllabus analysis there are 18 basic competencies for students class X, four of basic competencies are about Heat and Thermodynamic. Whereas, Heat and Thermodynamic concept that should be mastered by vocational students. Thus, the conditions bring implication to students\u2019 misconception in heat and heat transfer concept. The biggest misconception was found in Black principle. Moreover, Student\u2019 mental model in heat convection was found as na\u00efve mental model or it could be said as mental model that not scientifically accepted. The selected student was the student who got the good achievement. He could not explain why the hot water rise and the cold water sink. He also could not explain the current that flow in water heating processes. Topic: Vocational Education and Training 138 Sustainable Simulation Beta wave Learning in Letter Literacy To The Achievement of Pupils With Learning Disabilities In Level One Primary School1Yeap. T. W , 2Salleh Abd Rashid, 3Mohd Fareq Abd Malek , 4Mohd Hafizi Omar and 5Nazeri Mohammad This study aims to measure the effects of Simulation Beta wave in letter literacy to the achievement of pupils with learning disabilities in level one of Malaysian primary school. This research designed using Quasi-experimental method. There were 34 respondents selected as the study sample in Sekolah jenis Kebangsaan Cina (SJKC) in Perlis area North Malaysia. This intervention with 34 respondents would be charged with intervention of Simulation Beta Wave Green Technology in six weeks period and without any controlled sample treatment group . A cognitive test was used to collect data after the study intervention was done. Amidst that, the pre and post test achievement tests carried out after intervention of Multimedia Simulation Beta wave Green Technology. The findings of this study will be analyzed using SPSS19 with descriptive in mean, frequency and %. The finding of this study show that Multimedia Simulation Beta Green Technology acts as a stimulant cure and gives positive impact on learning achievement in alphabet literacy of Year-One pupils. Topic: Innovation in Teaching and Learning 139 Sweet potatoes are rich in carbohydrates, vitamins, minerals, fiber and anthocyanins. Sweet potato processing still limited into flour, keremes, dodol, cheese stick, or chips. Flakes as pre-cooked meals made through the stages of manufacture of pasta and drying. The purpose of this study was to optimize the production of sweet potato flakes at the stage of making pasta and drying. Making the pasta is done through techniques steamed or roasted. Pasta drying using a drum dryer or dryer cabinet. This research was done by experimental method. The indicator of optimization is the number of monomeric anthocyanins, \u03b2-carotene and the color resulting from flakes. The results showed that the number of monomeric anthocyanin flakes with steamed techniques, drum dryer (0:03 \u00b1 3.83 mg CYE \/ g bk), flakes with steamed technique, dryer cabinet (3:03 \u00b1 0:02 CYE mg \/ g bk), flakes with roast technique, drum dryer (2:49 \u00b1 0:05 CYE mg \/ g bk), flakes with roast technique, dryer cabinet (0:03 \u00b11.98 mg CYE \/ g bk). The Color of purple sweet potato flakes produced through techniques steamed was bright purple, while the color purple sweet potato flakes produced through techniques roast give a brownish purple color. The amount of \u03b2-carotene Flakes yellow sweet potato steamed cooking stage, drum dryer (152 mg \/ Kg bk), grilled drum dryer (136 mg \/ kg bk), yellow sweet potato flakes roasted stages cabinet dryer (140 mg \/ Kg bk), and grilled stage dryer cabinet (122 mg \/ kg bk). In conclusion sweet potato flakes production techniques through the stages of process steam, and the dryer drum has a number of anthocyanin, or \u03b2-carotene bigger and brighter colors than the toasted flakes techniques and dryer cabinet.Ai Mahmudatussa\u2019adah, Yulia Rahmawati, Sudewi Yogha Sweet potatoes are rich in carbohydrates, vitamins, minerals, fiber and anthocyanins. Sweet potato processing still limited into flour, keremes, dodol, cheese stick, or chips. Flakes as pre-cooked meals made through the stages of manufacture of pasta and drying. The purpose of this study was to optimize the production of sweet potato flakes at the stage of making pasta and drying. Making the pasta is done through techniques steamed or roasted. Pasta drying using a drum dryer or dryer cabinet. This research was done by experimental method. The indicator of optimization is the number of monomeric anthocyanins, \u03b2-carotene and the color resulting from flakes. The results showed that the number of monomeric anthocyanin flakes with steamed techniques, drum dryer (0:03 \u00b1 3.83 mg CYE \/ g bk), flakes with steamed technique, dryer cabinet (3:03 \u00b1 0:02 CYE mg \/ g bk), flakes with roast technique, drum dryer (2:49 \u00b1 0:05 CYE mg \/ g bk), flakes with roast technique, dryer cabinet (0:03 \u00b11.98 mg CYE \/ g bk). The Color of purple sweet potato flakes produced through techniques steamed was bright purple, while the color purple sweet potato flakes produced through techniques roast give a brownish purple color. The amount of \u03b2-carotene Flakes yellow sweet potato steamed cooking stage, drum dryer (152 mg \/ Kg bk), grilled drum dryer (136 mg \/ kg bk), yellow sweet potato flakes roasted stages cabinet dryer (140 mg \/ Kg bk), and grilled stage dryer cabinet (122 mg \/ kg bk). In conclusion sweet potato flakes production techniques through the stages of process steam, and the dryer drum has a number of anthocyanin, or \u03b2-carotene bigger and brighter colors than the toasted flakes techniques and dryer cabinet. Topic: Engineering and Technology Innovation 140 Synthesis of silica particles from rice straw waste using a simple extraction methodAsep Bayu Dani Nandiyanto1,*, Taufik Rahman1, Muhammad Abqori Fadhlulloh1, Ade Gafar Abdullah2, Ida Hamidah3, and Budi Mulyanti2 The purpose of this study was to synthesize silica particles from rice straw waste using a simple extraction method. The experiment was conducted by heating and extracting rice straw waste into basic solution. To get silica particles, the extracted solution was then put into acid solution and heated to remove the remained solvent. The experimental results showed that the aggregated silica particles with sizes of about 200 nm were successfully produced. The XRD and FTIR analysis showed that the final product was silica and free of graphite. However, we found that some KCl component in the final product in which this was possibly from the use of KOH as the extraction agent. Therefore, further studies are still required to synthesize high purity silica particles from rice straw waste. Topic: Engineering and Technology Innovation 141 SYSTEM MONITORING OF LIQUID VOLUME INSIDE TANKER BASED ON ULTRASONIC SENSOR AND ARDUINO MICROCONTROLLERMuchammad Husni; Daniel O. Siahaan; Henning Titi Ciptaningtyas; Hudan Studiawan; Yoga Pratama Aliarham Incident of oil leakage and theft in oil tank often happens. To prevent it, the liquid volume insides the tank needs to be monitored continuously. This research use some ultrasonic sensors to monitor the fluid height, Bluetooth modules to sent data from the sensors to the Arduino microcontroller, Arduino microcontroller to calculate the liquid volume, and also GPS \/ GPRS \/ GSM Shield module to get location of vehicle and sent it to the Server. The experimental results show that the accuracy rate of monitoring liquid volume inside tanker while the vehicle is in the flat road is 99.33% and the one while the vehicle is in the road with elevation angle is 84%. This system can monitor the position and the liquid volume via web application to prevent illegal theft. Topic: Engineering and Technology Innovation 142 THE ANALYSIS OF CHARACTER EDUCATION IN TEACHING PHYSICAL EDUCATIONAdang Suherman The phenomenon of the low character of the students demand the teacher to be able to integrate character education in every subject at school. The purpose of this study is to find out some factors supporting the integration of character education in physical education. This study was conducted through a survey technique to 40 elementary schools and 69 elementary school physical education teachers in Bandung, west Java. Instruments used are Value Orientation Inventory (VOI), questionnaire and observation. Through statistical and qualitative analysis showed that the teachers\u2019 value orientation, open space and sport equipments, students physical and motor fitness, and the teaching structure provide the teachers opportunities to integrate character education in teaching physical education. Results of this study recommend further study of the importance of making a hypothetical model of the integrated character education in teaching physical education Topic: Innovation in Teaching and Learning 143 The development of higher order thinking lab laboratory to improve transferable skills of studentsAdam Malik and Agus Setiawan Results of a preliminary study conducted at the Physical Education Program course on Advanced School of Physics Laboratory showed most students when practical implementation is still not able to develop higher order thinking skills. Its main activity is still limited low-level thinking skills. In an effort to increase the transferable skills of students then applied the higher order thinking laboratory practicum in this study.The purpose of this research is to develop practical models which can increase the transferable skills of students. Methods used in the form of qualitative research. Research conducted at the Physical Education program. Its population is physical education student. Data on practical implementation that have implemented student sheet obtained through observation and interviews to lecturers and students taking courses School of Advanced Physics Laboratory. Observation results show the practical implementation models used in the form verification lab to melatihkan basic level thinking skills and interview stating thinking skills are trained during the practicum is still a basic level. Thus the need to develop practical models which can increase the transferable skills of students so that they can be applied in everyday life according to the demands of the National Qualifications Framework Indonesia and the world of work. Topic: Innovation in Teaching and Learning 144 The development of instructional video to improve the competence standards of teacher candidatesMeini Sondang Sumbawati and Munoto The State University of Surabaya yield the teacher candidates who are having four standards of competencies namely pedagogy, personality, social, and professional competence. Teacher competencies are taught through courses to complete teaching plan and be able to teach with innovative learning model. The aims of this study were to (1) develop learning video about how teachers teach with innovative learning model, (2) determine how the students\u2019 responses after watching the video, and (3) determine achievement of standards of professional and pedagogy students competence. The sample of this study were 30 students. The data is collected by using video validation sheets, observation sheets of teacher competence in the field of pedagogy and professional, and student responses questionnaire to the video; and it were analyzed descriptively. The result indicated that (1) there were three instructional videos, which are direct learning model (DLM), cooperative learning model (CLM), and problem based learning model (PBLM). These videos have been validated by the two experts, and it feasible in learning; (2) the DLM, CLM, and PBLM are good; (3) about 53% students\u2019 achievement are very good. Topic: Vocational Education and Training 145 THE DEVELOPMENT OF LEARNING MANAGEMENT SYSTEM MODEL TO IMPROVE STUDENTS\u2019 LEARNING OUTCOMESHasbullah This study aimed to develop and to implement Learning Management System (LMS) model in improving learning outcomes in Mastering Electronic Components Measurement (MPKE) subjects for students of vocational high school (SMK) in Bandung. LMS is an online learning system or e-learning, which is a web-based learning management application that can facilitate students\/learners to learn in a virtual classroom. This research employed research and development (R & D) method which led to a cycle based on studies and research findings were then developed to produce a model of the product. To find an increase in students\u2019 learning outcomes in the cognitive aspects, the quantitative approach was done with a technique that used a quasi-experimental pretest-posttest group design. The experimental group was given a learning treatment with LMS implementation, while in the control group was treated with conventional learning models approach in the classroom. The results of this study indicated that vocational students\u2019 achievement in MPKE subject by using LMS applications model were higher than the students\u2019 learning outcomes by using conventional learning models. Based on this analysis, it can be concluded that Learning Management System model can improve students\u2019 learning outcomes in MPKE subject at SMK in Bandung. Topic: Innovation in Teaching and Learning 146 The Development of Vocational Curriculum Implementation Evaluation ModelDr. Dedy Suryadi ABSTRACT This study aims to produce a model for the evaluation of the implementation of the curriculum of vocational Skills Competency SMK Stone Concrete Construction Engineering. The method used consists of literature study, survey, documentation studies and Delphi technique. The results of the study, completion of a vocational curriculum model of comprehensive evaluation on the dimensions of the planning and execution of the implementation curriculum. Vocational Curriculum Implementation Evaluation Model (EIKK) developed from the discrepancy evaluation model that aims to provide information to make the assessment and improvement of curriculum implementation considerations. Application by comparing the performance of the implementation of the standards in order to obtain the information gap on those aspects that are evaluated. In practice, the model EIKK using Excel as a software application programs that provide convenience in carrying out the evaluation. EIKK model has advantages views of practicality, flexibility, and accuracy of the evaluation findings. The EIKK model has limitations, so remembering designed to evaluate the components of the curriculum and for use on a single competence in vocational skills. Recommended, EIKK model with its software can be used and perfected the use of the evaluation of the implementation of the curriculum other competency skills. EIKK model can serve as a model of self-evaluation, given the internal evaluators conducted an evaluation of vocational Skills Competency Stone Concrete Construction Engineering. Topic: Vocational Education and Training 147 The Effect of Flood Caused By Climate Change to Porous Asphalt PavementFirdaus Chairuddin Abstract : The test Indirect Tensile Strength for asphalt quality 3%, 4%, 5% are 0.0673, 0.325, 0.2370 subsequently. Cantabro test, loss weight for asphalt quality 3%, 4%, 5% are 77.10, 14.56, 9.70 subsequently. Coefficient vertical test permeability 0.1795 for asphalt 3%, 0.2029 for asphalt 4%, and 0.1596 for asphalt 5%, Unconfined Compressive Strength, Modulus elasticity 146.543 and ratio poisson 0.095831 for asphalt 3%, Modulus elasticity 91.450 and Ratio poisson 0.206009 for asphalt 4%, and, Modulus elasticity 32.119 and radio poisson 0.778059 for asphalt 5%, Scanning Electron Microscope (SEM) Oxygen 58.46%, Silicon 2.60%, Aluminium 2.91%, Sodium 5.75%, Calcium 19.11%, Sulfur 7.83%, Magnesium 3.52%, SiO2 6.73%, Al2O3 6.40%, Na2O 7.45%, CaO 46.25%, SO3 27.05%, MgO 6.12%, The results of EverStressFE analysis for multilayer soil-rigid are vertical deflection 0.5 mm, vertical microstrain (\u03b5z) + 0 s\/d 200 on deepness 150 mm, and for multilayer soil-rigid-asphalt results vertical deflection (Uz) + 0.64 mm on the surface and + 0.4 mm on the deepness of 50 mm, and vertical microstrain vertical (\u03b5z) + -6400 s\/d -7200 on the surface, + -4800 s\/d -5600 on the deepness of 150 mm. As the result of laboratory test soil-rigid are vertical deflection each point 1.535 mm, 1.535 mm, 4.505 mm, 2.45 mm, 4.19 mm, dan 3.61 mm, and microstrain C1 to C4 0.36, -37.68, 44.44, 43.48, and the results of test Multilayer soil-rigid-asphalt are vertical deflection each point 1.576 mm, 0.075 mm, 3.7 mm, 1.985 mm, 2.48 mm, 0.986 mm, and the value of asphalt course microstrain is 655. Topic: Engineering and Technology Innovation 148 THE EFFECT OF PROJECT BASED LEARNING AND INTELLIGENCE QUOTIENT ON THE IMPROVEMENT OF LEARNING MOTIVATION STUDENT AT VOCATIONAL HIGH SCHOOL (SMK)Ahmad Yani (a*), Adang Suherman(b) The purpose of this study was to knows the effect of learning model and Intellegence Quotient (IQ) on the improvement of learning motivation in physical education at vocational High School students. The method used is Quasy experiment with Non equivalent Control Group Design and samples in this study were 40 students of class XI SMK 2 Padjadjaran Bandung.The processing data used a statistical approach by analysis of Covariance (ANCOVA). According to data analysis result, The effect of Project Based Learning and Direct Instruction model of physical education has a significance value of 0.004 and for the results of significance test the effect of intellegence Qoutient on learning motivation in physical education has 0.012 sig Value. From these results it can be concluded that there are significant effect of learning model on learning motivation in physical education at vocational high school. and also intellegence Quotient on improving learning motivation in physical education at vocational high schoo student. While significant results learning model and Intelectual Quotient together on improving learning motivation in the physical education of vocational students have a significance value of 0.003. So as we can conclude that there are significant effect of learning model and intellegence Quotient together on improving learning motivation in physical education at vocational high school students .According to the result\u2019s analysis, the authors suggest that teachers in the learning process, especially in teaching physical education at Vocational High School. Researchers recommend using a project-based learning model learning proven effect on student motivation, and also pay attention to other factors such as Intelligence Quotient that can also effect on learning motivation in physical education. Topic: Innovation in Teaching and Learning 149 THE EFFECT OF TEACHERS\u2019 COMMITMENT TO THE IMPLEMENTATION OF QUALITY MANAGEMENT SYSTEM ISO 9001:2008 AT SMK TEKNOLOGI INDUSTRI PEMBANGUNAN CIMAHIIMAN SETIAWAN The study was motivated by the regulation number 20 about national education system in which the quality of education should be improved. sekkeluhan dari industri. From the interview, questionnaire, and observation of the implementation of ISO at SMK Teknologi Industri Pembangunan Cimahi, it was found that there were still many obstacles that were encountered. The study was conducted to investigate the percentage of the effect of teachers commitment to the implementation of quality management system ISO 9001:2008. The research method that was used was analytical descriptive using path analysis. The result of the study showed that the effect of teachers commitment to the implementation of quality management system ISO 9001:2008 was significant and reached 0,237 or 23,7%. Topic: Vocational Education and Training 150 THE EFFECT OF TEACHING MODELS AND TEACHING MATERIALS TOWARDS SITUATIONAL INTERESTHilda Ilmawati, M.Pd (a). Prof. Dr. Adang Suherman,MA. (b) The purpose of this research is to find out the effect of teaching models (inquiry and direct instruction) and teaching materials towards situational interest. The sample of the research was Class X students at SMAN 1 Batujaya Karawang which was taken as random cluster sampling from 4 classes. The study uses a 2 x 2 factorial experimental design conducted over 16 meetings. Student\u2019s Situational Interest measured by using Situational Interest scale from Chen et al (2001) which had validity and reliability coefficients 0,718. Data was analysed using ANOVA. The findings showed that (1). Teaching models were not significantly giving the positive effect on situational interest, the significance values is 0,24. (2). Teaching materials were significantly giving positive effect on student\u2019s situational interest, the significance value is 0,002. (3). There was no interaction between teaching models and teaching materials on student\u2019s situational interest, the significance value is 0,410. Based on the result, it can be concluded that teaching materials were more able to arise student\u2019s interest rather than teaching models, meanwhile game material which had been taught by using inquiry model was teaching physical education which students more like. It needed to be more research on mixture models between teaching materials and teaching models. Topic: Innovation in Teaching and Learning 151 The Effect of Various Types of Milks on Soyghurt\u2019s CharacteristicsMustika Nuramalia Handayani, Putri Wulandari Soyghurt is a kind of yogurt as functional food product obtained from the fermentation of lactic acid bacteria in soy milk. The fermentation process of soy milk is different from cows milk fermentation, because of different carbohydrate content. Carbohydrates in soy milk is composed of oligosaccharide species that can not function optimally as a substrate of lactic acid bacteria. Therefore, fermented soy milk requires other carbohydrates, including sucrose and lactose of different types of milk. The aim of this study was to determine the effect of various types of milk in the manufacture soyghurt and determine the sensory and physicochemical characteristics of soyghurt. The method used in this study was a completely randomized design of experimental design which is the treatment factor is various types of milk used in the manufacture of soyghurt with three types of milk are skim milk, full cream milk, sweetened condensed milk. Research procedure consists of the manufacture of soy milk; manufacture soyghurt with the addition of various types of milk; soyghurt sensory and physicochemical analysis. Soyghurt incubation in the fermentation process carried out at a temperature of 45 C for 5 hours. The results showed that there were significant differences in sensory characteristics of soyghurt with the addition of various types of milk. Soyghurt with the addition of sweetened condensed milk, is the most preferred panelists. It has viscosity 0.30 dPass, pH 3.97, protein content 7.44%, and fat content 7.07%. Topic: Engineering and Technology Innovation 152 The Implementation of Inquiry Based Self Control Learning Model to Improve Self Control of Cullinary Vocational Students through Sanitation Hygiene Course.R. Patriasih, A Djohar, M Sumantri, T Ruhimat The application of hygiene for a food handler is an important thing to do because many risks that can occur when these principles are ignored. Results of studies on vocational Cullinary students shows that the hygiene self-control of food handlers on the moderate criteria but ideally they should have good self-control. It relates to realize its position as an aspiring professional food handlers to maintain the quality of food in order to stay healthy and safe for consumption. For that we need an effort to improve they self-control in the hygiene food handlers. One was conducted by applying the learning model of Inquiry-Based Self Control (IBSC) on Sanitation Hygiene course. The research aims to determine the implemetation of IBSC learning models to increase self-control on vocational students through subjects Sanitation Hygiene. Method quasi experimental study design with non-equivalent control group pretest posttes used in this study. Subjects were students of class 10th as much as two group. The research results showed that self-control the students in the experimental class with indicators: control of behavior, stimulus control, in anticipation of the event, the interpretation of events and decision making has overall increased at 23.34%. Results of t-test analysis revealed a significant difference between the experimental and control postest classes, with t-count (6.547) > t-table (2032). N-Gain analysis results reveal the effectiveness of IBSC learning model to self-control of students to be at moderate criterion (0.58) compared with the control class with low effectiveness (0.29). Based on the research results, the researcher recommends that teachers of subjects especially Sanitation Hygiene for applying the learning model IBSC. Topic: Innovation in Teaching and Learning 153 The Implementation of Integrated Course Ware on Pneumatics Valve in Vocational EducationPurnawan, Sumarto, Wahyudin, Wahid M. The aims of this research was to gain empirical experience of Integrated Course Ware (ICW) on Pneumatic Valve product implementation in vocational high school. This research use quasi-experimental methode with non-equivalent control group design. This research involved six vocational schools and the sample consisted of 330 students. The implementation of ICW Pneumatic Valve as an empirical learning aid was assumed effective to improve students learning outcome. In vocational school, this product was best used as presentation media, followed by hand book and interactive media. Therefore, as presentation media, this product was best applied consecutively for mechanical control program, mechatronics, aircraft technology and machining techniques. Topic: Innovation in Teaching and Learning 154 THE IMPLEMENTATION OF LEARNING STRATEGIES \u201cLISTEN-SEE-DO\u201d BASED ON MINI LABORATORY FOR STUDENT\u2019S SCIENCE PROCESS SKILL (KPS) IN SENIOR HIGH SCHOOlsohibun Has conducted research that aims to obtain a condition of the use of learning strategies listen-see-do based on mini labraboratory for student\u2019s science process skills (KPS) on the material quantities and units. The method used is a quasi-experimental design with a \"one shot case study\" conducted in class X MIA 1 Senior high school in Rokan Hulu district of Riau in the academic year 2014\/2015. Data collected by using science process skills test. Based on the analysis of data obtained by the KPS indicator: observing 89.23%, 59.49% communicating, predicting 10.26%, 33.33% interpret observations and apply the concept of 85.32 %. It was concluded from the results that learning with learning strategies hear-see-do based on mini laboratory, impact both on students science process skills (KPS). Topic: Innovation in Teaching and Learning 155 THE IMPLEMENTATION OF NUTRITION EDUCATION \u201cATHENA\u201d FOR IMPROVING NUTRITION KNOWLEDGE OF COLLEGE ATHLETE STUDENT FROM INDONESIA UNIVERSITY OF EDUCATIONCica Yulia 1, Ira Purnamasari 2, Ali Khomsan 3, Dadang Sukandar 3 Athlete\u2019s Nutritional status was one contributing factor for athlete endurance. Nutritional status influenced by many factor such as nutritional knowledge. Based on preliminary research, more than a half respondent ( 69%) of college athlete student from Indonesia University of Education has a poor nutrition knowledge. The aimed of this study was to implemented nutrition education \u201cATHENA\u201d for improving nutrition knowledge of college athlete student. The design of this study used quasi experiment with pretest-posttest control group. Sampling technique was done by purposive, there were 46 college athlete student divided in two group, treatment group and control group. Nutrition education program conducted in six weeks. The program includes nutrition for athletes, Eating disorder in Athlete, effective exercise, handling defresive during exercise, drugs use in sports. The program delivered after orduring the athlete exercise. the result showed that \u201cATHENA \u201c program was succesfull improved nutrition knowledge of treatment group. Topic: Innovation in Teaching and Learning 156 The Influence of Multiple Representation in Physics Learning to Students Ability in Understanding of Physics Material and Scientific ConsistencySidik Nulhaq (a*), Agus Setiawan (b) This research was to obtain an overview the ratio of ability increase in understanding and scientific consistency between students who get Physics learning using multiple representation and students who get Physics learning without using multiple representation. Physics learning using multiple representation requires students to interpret a concept in some representations so that can improve the ability of students in understanding of Physics material and scientific consistency. This research is using a quasi-experimental method with nonequivalent pre-test and post-test control-group design. The test instruments based on aspects of understanding ability according to Anderson and scientific consistency tests were adapted from the form R-FCI. Subjects of this research were students of senior high school grade X in senior high school consists of two classes with total of 82 students. The research results showed Physics learning using multiple can more improve ability in understanding with mean score of by 0.79 (high criteria) than Physics learning without using multiple representation with mean score of by 0.60 (medium criteria). Moreover, scientific consistency and representation showed students who get Physics learning using multiple representation increased in consistency quantity by 74.8 % for scientific consistency and 72.3 % for representational consistency higher than students who get Physics learning without using multiple representation with consistency quantity 55.3 % for scientific consistency and 53.92 % for representational consistency. It can be concluded that Physics learning using multiple representation can more improve the ability of students in understanding of Physics material and scientific consistency significantly than Physics learning without using multiple representation. Topic: Innovation in Teaching and Learning 157 THE INFLUENCE OF READINESS TEACHER IN IMPLEMENTATION CURRICULUM 2013 TO ACHIEVEMENT LEARNING RESULTAlan Agus Mulya This study was conducted at Vocational High School of Pandeglang Banten and especially for automotive program, which implemented of the Curriculum 2013. This study is conducted to determine the influence readiness of teacher in aspect perception, skill and knowledge to achievement of learning result implementation the Curriculum 2013, Use of descriptive method and correlational quantitative design study. The result of data analysis statistic showed that, (1) there is influence readiness of teacher for aspect perception (X1) is good, (2) influence readiness of teacher for aspect skill (X2) is good, (3) influence readiness of teacher for aspect knowledge (X3) is very good, (4) the achievement of learning result in implementation the Curriculum 2013 (Y) is good, (5) There isn\u2019t positive influence readiness of teacher for aspect perception to achievement of learning result, (6) There is positive influence readiness of teacher for aspect skill to achievement of learning result, (7) There is positive influence readiness of teacher for aspect knowledge to achievement of learning result, (8) There are positive influence readiness of teacher for aspect perception (X1), skill (X2), and knowledge (X3) to achievement of learning result (Y). Summary: influence readiness of teacher in implementation the Curriculum 2013 to achievement of learning result is good. Topic: Innovation in Teaching and Learning 158 The influence of supporting learning towards learnings Motivation in increasing learning achievement in program studi Teknik Ketenagalistrikan SMKN 2 Garut (Research on the lesson Dasar dan Pengukuran Listrik)ella rahmi fatah This research has a goal to know how big the influence of supporting learning towards learning motivation of the students in the lesson Dasar dan Pengukuran Listrik. Population in this research is102 students of first grade program study Teknik Ketenagalistrikan paket keahlian Teknik Instalasi Tenaga Listrik of SMKN 2 Garut, 2014\/ 2015, with 81 students sample. The methods which used in this research is quantitative methods with descriptive\u2019s correlation method. Analysis requirement\u2019s testing consist of normality, linierity, multicolinierty, heteroskedastisity and auto correlation\u2019s testing. Data analysis technique which was used in this research is question\u2019s form (angket) to explore supporting learning and student\u2019s motivation, whereas the increasing of learning\u2019s achievement of the lesson Dasar dan Pengukuran Listrik was achieved through first semester score. The result of the processing data shows: supporting learning has strong influence towards learning\u2019s motivation, whereas the influence of learning motivation towards the increasing of learning\u2019s achievement in the lesson Dasar Pengukuran is in average catagory. The Conclussion of this reseach is there is a positive influence and significant supporting learning towards student\u2019s learning motivation. The increasing of supporting learning will increase student\u2019s learning motivation and student\u2019s learning achievement of program study Teknik Ketenagalstrikan SMK N 2 Garut. Topic: Innovation in Teaching and Learning 159 THE INFLUENCE OF THE EMPOWERMENT OF PROVINCE SERTIFIED COACHES TOWARDS THE ACHIEVEMENT IN SPORTS COMPETITIONheri yusuf muslihin Abstract This study investigates the influence of the empowerment of coach that has been certified by the province towards the achievement of the sport competition. This study is aimed at 1) finding and analyzing the influence of information management toward the sport achievement, 2) finding and analyzing the influence of decision making toward the sport achievement, 3) finding and analyzing the skill of project planning, organizing, and integrating sport achievement system, 4) finding and analyzing the skill of internal controlling and evaluating system towards sport achievement, 5) finding and analyzing the skill of leadership , motivating and honor system towards sport achievement, and 6) finding and analyzing the skill of selecting, placing and developing of human resource toward sport achievement. This study employed exploratory survey toward 50 referee in West Java by using purposive sampling. Besides that, 50 coaches that has participate actively in developing of athlete. the result of the study shows that 1) the information management has influence toward the sport achievement, 2) the decision making has influenced toward the sport achievement, 3) the skill of project planning has also influence toward achievement, 4) the skill of evaluation and internal control was also influenced toward the sport achievement, 5) the skill of leadership and honor system has also influence toward the sport achievement, and the last (6 the skill to select and to place and HRD has influenced towards the sport achievement. Topic: Vocational Education and Training 160 The Integration of Remote Sensing and Geographical Information Systems in Monitoring of Coastal AreaNandi A coastal lagoon area is a landform that is influenced by natural processes and human activities. All human activities at the upstream, particularly agriculture and cultivation bring soil, waste, and other materials to the downstream area through the river drains into the sea. The coastal lagoon area will be the depository place for sedimentation from the upland area. The lagoon, which is located in Indonesia, is an example of a coastal tropical area, which has a unique biophysical characteristic. The region has a great natural ability to ensure the sustainability of the interrelationships between terrestrial, estuarine and marine ecosystems in harmony and balance as a habitat for flora and fauna. The region is an area of migration of various types of protected animals and it is a place of breeding for diverse species of the shrimp and fish, which have a highly economical value. The lagoon, currently experiencing acceleration narrowed on its area due to a very intensive sedimentation from the mainland. The research aims to answer the question of how the integration of remote sensing technology with geographical information system in monitoring of the coastal environmental dynamics. Achieving the objectives of this study required examining the morphology and land use changes with multitemporal remote sensing approaches. After assessment of remote sensing data, the results show that there were serious morphological, coastline changes, and land use changes in the lagoons over the period of the years of observations. The using of multitemporal remote sensing Landsat images is possible to analyze the morphological and land use changes with different time and sensors. Topic: Engineering and Technology Innovation 161 The Material Performance of HSS Chisels and Its Relation with Chemical Composition and Carbide DistributionBambang Darmawan, Maman Kusman, R. Aam Hamdani Materials used for cutters must have the following characteristics: (1) have good hardness, (2) have tenacity that can withstand shock loads, (3) have endurance toward thermal shock loads, (4) have low adhesive nature, (5) have low solubility of elements\/components of chisel materials. One material that meets such characteristics and is widely used for making chisels is High Speed Steel (HSS). Two types of HSS are available in the market: HSS from Germany (Bohler) and HSS from China. This research employed the pure experimental design. It consists of two stages. Stage 1 aims to test\/operate lathe machines to determine the lifetime and performance of chisels based on specified wear criteria. In stage 2, characterization of microstructure using SEM-EDS was conducted. Firstly, grinding of chisels was done so that the chisels could be used for cutting metal in the turning process. Grinding processes of the two types of chisels were done at the same geometry, that is side rake angle (12\u00b0-18\u00b0), angle of keenness (60\u00b0-68\u00b0), and side relief angle (10\u00b0-12\u00b0). Likewise, machining parameters were set in the same machining conditions. Based on the results of the tests, it is found that to reach 0.2 mm wear point, chisels made of HSS from Germany needed 24 minutes, while chisels made of HSS from China needed 8 minutes. Next, microstructure tests using SEM\/EDS were done. The results of the SEM tests indicate that the carbide particles of HSS from Germany were more evenly distributed than the carbide particles of HSS from China. Carbide compounds identified in HSS from China were Cr23C6 and Fe4Mo2C. Oxide impurity of Al2O3 was also found in the material. On the other hand, in HSS from Germany, no impurity and other carbide compounds were identified, except Cr23C6 and Fe4Mo2C, also Fe4W2C, and VC or V4C3. Topic: Engineering and Technology Innovation 162 The Model of Private Sectors\u2019 Contribution for Improving the Existence of Technical Vocational Education and TrainingMoh. Khairudin This study was aimed at identifying private sectors\u2019 contributions to improve the existence of technical vocational education and training. The study explored a co-op program that is conducted between vocational high school and industries. The study was description qualitative involving the vocational high schools and related industries. The subjects were 32 final year students of vocational high schools that were sampled using a proportional purposive technique. Data were collected using questionnaires and analysed qualitatively using descriptive statistict in the form of percentages. Findings of the study showed that (1) mostly students with a number of 30 students said that work experience of industrial Practice in very good category with a percentage of 93.75 %. (2) Students\u2019 competence based on the assesment of industry instructor can be expalined that some students with number of 58 % have a competence to make a coorporation with others, ready for under pressure work and have a good soft skill. Topic: Vocational Education and Training 163 The Need of Transversal Skills in Indonesia Tourism EducationHeri Puspito Diyah Setiyorini, Elly Malihah, Rini Andari The challenge of globalization and the rapid growth of information highway should be responded by enhancing the creativity for the students as future generation to make use the information and global relations wisely for better life. Hence, the young generation, especially at the higher education, should also be introduced by the transversal skill. It is the skill that consists of the capability in the field of entrepreneurship, digital skills, and multilingualism. Those skills have already existed in the tourism education curriculum at the higher education level. Tourism education has been influenced by the demand of the industry to cater the need of global context and international standard perspective. Thus the ability of digital skills, multilingualism, as well as, entrepreneurship are embedded in the tourism education curriculum to cater those demand. Hence, this research has been directed to discover the industrial perspective on the need of transversal skills in Indonesia Tourism Education. The method of this research was qualitative research by using the in-depth interview to some tourism industry professionals in Bandung City, Indonesia, such as hotel, restaurant, and tour operators. The study has found that it was important to enhance the transversal skills to develop the industry to face the tight competition and global era. Thus, the implication for the tourism education was for developing the transversal skills as one of the learning outcome, especially in tourism higher education level. Topic: Vocational Education and Training 164 The physical characteristics of semi refined carrageenan (SRC) edible filmDesi Setyorini, Puji Rahmawati Nurcahyani Currently the seaweed is processed flour and Semi Refined Carraagenan (SRC). However, total production is small, but both of these products have a high value and are used in a wide variety of products such as cosmetics, processed foods, medicines, and edible film. The purpose of this study were (1) Knowing the physical characteristics of the SRC edible film, (2) Determine the best treatment of SRC edible film. SRC flour divided into three concentrations of 1.5%; 3%; and 4.5% with the addition of glycerol 3% and 0.6% arabic gum. The physical properties of the film using a universal testing machine Orientec Co. Ltd., while the water vapor transmission rate testing using the gravimetric method Dessicant modified. The experimental design used was completely randomized design with a further test of Duncan. The calculations show the best edible film is edible film with the addition of SRC 4.5%. Edible film has an average thickness of 0.162 mm, the average tensile strength of 7.635 MPa, an average of 13.252 mm elongation, modulus of elasticity of 162.854 MPa, and the water vapor transmission rate of 0.975 g \/ m2.8 hours. Topic: Engineering and Technology Innovation 165 The Potential of Thermoelectric Generator for Engine Exhaust Heat Recovery ApplicationsNyoman Sugiartha, I Putu Sastra Negara and Sudirman Thermoelectric generator (TEG) module composes of integrated p-n semiconductors as hot and cold side junctions and uses Seebeck effect between them to directly convert heat into electrical power. Exhaust heat from engines as otherwise wasted to the atmosphere is one of the heat sources freely available to drive the TEG. This paper evaluates the potential of energy recovery of such kind of waste heat by the TEG modules. An experimental apparatus has been setup to simulate real conditions of automobile engine exhaust piping system. It includes a square section aluminium ducting, an aluminium fin heat sink and two Peltier thermoelectric TEC1 12706 modules. A heater and a cooling fan are employed to simulate hot exhaust gas and ambient air flows, respectively. Electrical loading is controlled by resistors. Dependent variables measured during the test are cold and hot side temperatures, open and loaded circuit output voltages and electrical current. The tests are designed under various loads and temperature difference operation in purpose of characterising the electrical power produced by the TEG. The results are used to identify the potential application of the TEG as an alternative electricity source in automotive sector. Topic: Engineering and Technology Innovation 166 The Profile and Implementation of Accounting Sciences Affective Aspects on Lecturer of LPTK1) Meta Arief, 2) Umar Faruk, 3) Leni Yulianti Comes from the sense of concern over the financial misapplication cases in Indonesian society. There are some factors that lead to financial misapplication, such as the nescience or inability to manage financial sector, and limited set of rules and procedures that lead to financial fraud. The discipline that closely relates to finance is Accounting, a knowledge which has been taught since Senior High School, particularly in Business and Financial Major on Vocational School. Accounting is a discipline that teaches the process of making financial information system. The learning process requires discipline, orderliness, integrity, and thoroughness, a positive trait that would be able to prevent someone from making mistakes, including errors in financial terms. These characters that come from the values of Accounting need to be implanted and nurtured from the beginning to Senior High School\u2019s students in order to reduce financial misapplication cases in Indonesia. Based on those ideas, the position of LPTK lecturers in Accounting Education major has a strategic role in implementing value philosophy of accounting to their student\u2019s daily life through learning process in university. They also act as the role model for prospective Senior High School accounting teacher in designing affective learning process. Topic: Innovation in Teaching and Learning 167 The Prototype of an Affordable Control Process Laboratory Kits for Use in the ClassroomA.G. Abdullah, D.L. Hakim, T. Gunawan, M.A. Auliya, M.A. Fahrurizal This paper describes the results of the design, development and implementation of a low cost laboratory kits that can be used outside the laboratory. This device consists of a programmable logic controller (PLC) integrated SCADA system with the plant in case of a water tank filling system, aims to provide an overview of fundamentals to advance on the practical concept of discrete control systems. This device has a simple i\/o plug-in board, controller module, plant module and power supply module. For measuring its performance, the equipment is trialed in the automation industry learning with problem based learning approach. The implementation results provide evidence, this device capable of providing real examples the process control in industry, and can improve students ability in designing of the process control systems. Problem-based learning provides learning experiences for students to work in teams, respect the opinion of other students, sharpening the ability to argue and present the results of the project. This device very suitable for use in lectures with problem-based learning. Features of this device can be implemented as instructional media for undergraduate students from Electrical Engineering Faculties. Topic: Engineering Education 168 The study of building form and layout on Education Facilities, an assessment of FPTK - UPI toward urban microclimateChitra, Beta Paramita The role of building form and layout on education facilities give the significant impact to the campus microclimate. UPI is the largest with land area among universities in Bandung that occupied around 61ha, meanwhile other universities\u2019 area are around 25ha or less. FPTK is one of the faculty in UPI contain with 3 building groups facing west north and south. The aspect of building form and layout, such as building coverage, green area ratio, building height and distance between them will be connected to the outdoor temperature. The measurement of outdoor temperature in October during the hottest day gives the description about building form and layout toward campus microclimate. Topic: Engineering and Technology Innovation 169 The Travel Industry Competency Development in IndonesiaHeri Puspito Diyah Setiyorini, Any Noor, Ina Veronica Travel industry has faced new challenges due to the changing of tourist\u2019s lifestyles in choosing destinations, their travel motivation, and the way of travelling. Other challenges has also existed in the rapid change of information, technology, and communication in tourism industry and the tight competition within the industry that works virtually, as well as, conventionally. Hence, the industry faces the challenge of market turbulence, ICT advancement for travel management and tight competition. In response to this situation, the higher education, at the vocational as well as academic degree has developed its curriculum so that they could cater the need of the industry. This research aimed to discover the travel industry needs on the skills produced by the higher education institution of tourism. The first step of this research was comparing curriculum at some major tourism education institutions that was related to the need of skills for the travel industry. Then, the next step was to gather information from the industry regarding to the curriculum developed by the tourism education institution. The implication of the research was intended to develop the better curriculum that could be complied with the travel industry needs. Topic: Vocational Education and Training 170 The use of hot gas by-pass refrigerant in domestic refrigeratorE T Berman, S Hasan and Mutaufiq In this paper will be presented performance data of domestic refrigerator which use hot gas bypass refrigerant. Studied performance data consisting of refrigeration effect (RE), the work of compression (Wk) and COP. Test data measured at cooling load of 3 C to -3 C and the working fluid is used as the primary refrigerant is R-134a. The results showed that the installation of hot gas by-pass refrigerant can improved the RE and COP and lower the discharge temperature of the compressor so that it will lighten the work of the compressor and saves power consumption required refrigerator. Topic: Engineering and Technology Innovation 171 Traits-Personality as Predictors of Authentic LeadershipAan Komariah The purpose of this study was to investigate the predictability of traits personality towards authentic leadership madrasah principal and also to examine the relation of five factors of personality with dimensions of their authentic leadership. Participants for this study consisted of madrasah principal of madrasah aliyah at Tasikmalaya distric. This study found that traits-personality made significant contributions to authentic leadership. More over, it was determined that two of five factors of personality, conscientiousness and openness to experience were predictors for authentic leadership dimensions of madrasah principal. Topic: Vocational Education and Training 172 TRANSFORMATION OF ARCHITECTURAL VALUES OF TRADITIONAL VILLAGE IN SITE MANAGEMENT AND BUILDING TECHNIQUE IN THE MODERN SUSTAINABLE ARCHITECTURE DESIGNLilis Widaningsih, Diah Cahyani P ABSTRACT This study is aim to explore the local wisdom of traditional villages as well as how these values can be an architecture precedent in the modern society in the sustainable architecture discourse. Therefore, the expected result is a description and application of the community education values of architecture. The method used is Research and Development with Participatory Action Research (PAR) model. The main data collection techniques are observation, focuss group discusson (FGD), and face-to-face questionnaire survey to households. Aspects of implementation of architectural value are including building site management and environmental management, as well as building technique (building material usage and construction systems). Environmental education can be developed related to site and environmental management is to strengthen the habit of local people who develop the natural potential for everyday life. Traditional village precedent can be applied extent of the pattern of building technique, togetherness, as well as residential land use patterns. Meanwhile, technological transformation in the application building technique is how the design concept, the building functional needs and artisan work patterns can be communicated through the empowerment approach. The building technique aspects, in general, people prefer to build buildings that are technically easier and low cost, no longer consider aspects traditionality custom inherited. The use of natural materials with traditional building construction in addition considered hard to find and expensive, the selection of manufacturing materials with a permanent building is considered more indicative of socioeconomic status. Topic: Engineering Education 173 Typology of Building Shading Elements On Jalan Sudirman Corridor in PekanbaruGun Faisal (a), Pedia Aldy (b*) In 2013, temperature in Pekanbaru was between 22.60\u2103 and 34.6\u2103with humidity 79.14 percent. This condition has increase the concern of energy utilization to building comfort. Buildings have the biggest energy consuming due to the use of air conditioner in Pekanbaru. One effort to reduced energy is shading devices application. Application of air conditioner need huge energy, replaced natural circulation with architecture elements to reduced building thermal. This research study about system and building shading devices types that influence building thermal in Pekanbaru so that knowing characteristics and elements form. This study aims to determine and identify of systems and building elements types in Pekanbaru, which the element forms to conquer in climate condition. Qualitative method with rationalistic-paradigm has used to identify typology of building shading devices on JalanSudirman corridor. The research orientation on typology theory, thermal theory, and building shading device to identification of building shading device types on Jalan Sudirman corridor. Based on the survey result, there are 2 type of building shading devices on Jalan Sudirman Pekanbaru which is based on forms and quantity of shading. The types are building shading devices based on shading quantity and building shading devices based on shading forms. Topic: Engineering and Technology Innovation 174 Understanding Residential Preferences for More Sustainable Residential Development in Riparian Musi, PalembangMaya Fitri and Sugeng Triyadi Riparian is wetland on riverbank that has an important ecological functions as the transition area between land and river. It is maintains the balance between it. Supposedly, this riparian area is not built area. But, there are some city that founded in riparian area, so its development must pay attention to the sustainability of riparian ecology. However, the sustainable development principes are not easy to be accepted by the residents in these settlements. The paper study the preferences of a sample of households along the Musi riparian settlements in Palembang. The paper addresses on the question of the residents acceptability of principle of sustainable development to minimize built area in the riparian. It does so by examining the acceptability of compactness housing mass and greater green space. The findings suggest that most households prefer a row house rather with a medium-size riverbanks open space than an apartment with large open space. Topic: Engineering and Technology Innovation 175 USC-Basic Education Department\u2019s Collaborative SHS ProgramDr. Felino Borgueta Javines Jr. USC-Basic Education Department\u2019s Collaborative SHS Program (A Hybrid Grades 11 and 12 Model) Felino B. Javines, Jr.,SVD, DM Vice President for Academic Affairs-Basic Education Department University of San Carlos, Cebu City, Philippines jun javines Abstract When President Benigno Aquino III signed into law Republic Act 10533 or the Enhanced Basic Education Act of 2013, this signals an insistent educational reform that will make academic stakeholders conscious of the quality of graduates they produce. This goes to show that the government is assertive in improving the quality of our educational system to be at par with international standards. Thus, Education was significantly prioritized and given importance. One of the most challenging features of K to 12 is the establishment of the Senior High School program. It is then the purpose of this paper is to present the Grade 11 and 12 model of the University of San Carlos Basic Education Department under the Technical Vocational (Tech-Voc) Track. The project is collaboration between two institutions with distinct orientation. Tech-Voc Track is one of the four identified tracks of the K to 12 Program under Senior High School with Academic Track, Sports Track and Arts and Design completing the list. This collaborative technical vocational track is also rooted in the country\u2019s education strategy which is anchored on the National Education for All (EFA) 2015 Plan and attainment of the Millennium Development Goals (MDG\u2019s) which aim to provide an overarching policy framework for basic education with a vision that all Filipinos will acquire basic competencies. Significantly it is also pegged in the objectives of the K to 12 Program specifically the items that state \u2013 \u201c be adequately prepared for the world of work or entrepreneurship or higher education and \u201cbe legally employable with potential for better earnings. Topic: Vocational Education and Training 176 Use of The Malcolm Baldrige Method to Formulate Strategic Planning in Technological and Vocational EducationSuharno (a*) This article describes the results of a research study evaluating the performance of Technological and Vocational Education (TVE) by using the Malcolm Baldrige method. Data of the performance were used to know its excellence and weakness. Based on the excellence and weakness in performance, a competitive strategy could be formulated to improve TVE quality. First, performance measurements using the Malcolm Baldrige criteria were done on seven study programs from different universities. Second, results of the performance measurements were analyzed and described. With the data resulting from the performance measurements as basis, the excellence and weakness in TVE performance could be known. Third, a strategy was developed. Based on the performance excellence and weakness, a performance improvement strategy could be formulated to raise the quality level in education at TVE. The research results indicate that for level of performance the seven universities measured achieve scores ranging from 526 to 711 points. It shows that the performance of study programs in TVE in Indonesia puts them in the categories of education leader and emerging education leader. On the basis of those categories, each study program could formulate its own competitive strategy to improve TVE performance so that the education conducted also rises in level of quality. Topic: Vocational Education and Training 177 Utilization of Industrial Dairy Waste for Microalgae Cultivation Medium : A Potential Study for Sustainable Energy ResourcesSari Nurmayani, Rosdiana Heryanto Putra, Yatti Sugiarti Biodiesel is an alternative fuel made from vegetable oils and animal fats. One of potential feedstock of biodiesel is microalgae. High oil content in microalgae is one of consideration for the development of biodiesel or third generation biofuels from microalgae in Europe. Generally, production process of biodiesel from microalgae have three stages, preparation, harvesting and extraction. In this paper we explain the potential use of dairy waste from industry as a cultivation medium of microalgae for biodiesel production. Dairy waste from dairy industry contains 34.98% protein, 4.42% lactose, 9.77% fiber, 11.04% fat, 2.33% calcium, 1.05% phosfor, and 0.4 % magnesium, meaning that the dairy waste from dairy industry has a relatively high nutrient content and complete from a source of carbon, nitrogen and phosphorus as macro nutrients. Then supported with minerals and vitamins as micro nutrients, so it can be used as nutrients for plants and microorganisms decomposers. Based on the fact, the dairy waste from dairy industry has potency to be used as cultivation medium of Botryococcus braunii in the production of biodiesel, replacing the conventional cultivation medium. Topic: Engineering and Technology Innovation 178 Utilization of Some Sweet Potato Baking Puree and Vegetables on Patisserie ProductAna, Sri Subekti, Sudewi, Elda Novita Perdani, Farida Hanum, Tititri Suciani, Vivi Tania The research is an experimental study in Green Skill Patisserie Course by using Project Based Learning model. It aims to complete the project development of pie named Guramnis Rainbow Pie. Several experiments were carried out to produce a pie dough crust mixed with puree sweet potato baking and added with vegetable extract as food coloring in order to make a better appearence or an attractive shape and more nutritious. In addition, the pie is filled with a mixture of sweet and sour Gurame as Indonesian food. By applying organoleptic test to 10 respondents, the result shows the ratio between flour substituted by puree sweet potato baking is 2:1 and adding extract vegetables (carrots, beets and celery) as a color additive for crust pie, produce color pie more interesting (90%) and the texture of the pie with a good level of crispness (60%). In summary, pie taste is good (70%) and its flavour (70%). Nutritional analysis results show that for each piece Guramnis Rainbow Pie contains energy as much as 81.72 calories, carbohydrates 12.5 grams, fat 2.32 grams and 2.77 grams of protein. We recommend a further research to be done in order to make pie crust sweet potato baking substitution and vegetable extracts that have optimal level of crispness. Topic: Engineering and Technology Innovation 179 VISUAL EFFECTS KEBAYA DRESS OF FIRST LADY AGAINST IMAGE OF APPEARANCES INDONESIAN WOMANSuciati1, Agus Sachari2, Kahfiati Kahdar3, Ahmad Syarief2 Indonesian women image of international level is influenced in part by the appearance of self-Lady. The role and position of First Lady is the representation of women in Indonesia, because basically the First Lady as: wife accompanied President (head of state), has a background grip strong culture, high intellectual and personality both in lifestyle in dialy included in a dress, as well as ambassador of culture and design. Lady fashion style has always been a community of praise and criticism. The purpose of this study reveal visualization effects kebaya dress style Lady Indonesia in various state occasions on the image of a woman's appearance Indonesia. This study is a qualitative study of visual data that emphasizes discussion in the study semiological Clothing Kebaya (meaning the connotation and denotation of meaning) that led to the imaging. The results showed Clothing style kebaya lady in every period of the presidency as her husband has a good hallmark of the style of clothing or hairstyle, which indicates self-image. The conclusions of this study indicate that the Clothing Kebaya (National costume) Lady interpreted has an implied message because clothing can be observed visually. The implications made on the construction of the learning patterns of clothing fashion design national and ethnic clothing design archipelago. Topic: Innovation in Teaching and Learning 180 VOCATIONAL LEARNING DESIGN FOR WOMEN IN RURAL AREAS IN INDONESIAK Sumardi and A Djohar This research aims to generate a vocational learning design for women in rural areas. The vocational learning in the design includes knowledge, attitudes, and skills which are needed by learners to be applied in their lives. Three methods were used in the design of the vocational learning, namely: regenerated frerian literacy through empowering community techniques (REFLECT), language experience approach (LEA) and participatory rural appraisal (PRA). The research used quasi-experiment with the research and development approach. Four research instruments were employed in the study: direct observations, interviews, documentations and tests. Thirty women from rural areas were taken as participants of the research. The findings of the research indicate that vocational education has encouraged the participants to have motivations and awareness of the importance of learning. Thus, they become more active in classrooms, understand learning materials better, and have lifeskills. Such learning is useful for their lives. Based on the results, it can be argued that vocational learning is effective for the learning of women in rural areas. It inspires them to be better individuals. The vocational learning meets the learning need of women in rural areas by giving them lifeskils. Topic: Vocational Education and Training 181 VOCATIONAL TRAINING OF BATIK BASED ON LOCAL WISDOMGheafani Fikria Zaki, Tati, Isma Widiaty This study is based on the problem of how to develop a program training of batik based on local wisdom in batik industry. The purpose of this study was to design batik training development program based on local wisdom that is suitable for teenagers in the batik industry. Data collection techniques used were interviews to the manager of the batik industry in Cimahi, observation of training activities batik, documentary studies and expert judgment to validate the design of the programs created by the researcher. Implementation of expert judgment made to the five expert validator, which consists of three managers of the batik industry Cimahi and two experts in the field of batik. Results the training program design components of identity, purpose, materials, methodologies, media and scenario training activities which are highly recommended to apply. Components of the training fee are on the of worth to be implemented. Recommended in this study results to industry managers batik to be able to use batik training programs based on local wisdom that has been created as a product of this study with the aim that teens can understand the values of local wisdom contained in batik Cimahi. Topic: Vocational Education and Training 182 Voltage profile improvement using Static Var Compensator (SVC) in transmission systemGia M R (a*), Yadi Mulyadi (b), Hasbullah (c) In transmission system named \u2018Subsistem Bandung Selatan dan New Ujungberung\u2019 there are the voltage drop which relatively high and the voltage profile at the receiving ends below 0.95 p.u. Therefore, this research proposed a method to improve the voltage profile in the transmission system using one of Flexible Alternating Current Transmission System (FACTS) technology which is Static Var Compensator (SVC) and \u2018Subsistem Bandung Selatan and New Ujungberung\u2019 as the object. The main purpose of this research is to get the voltage profile in \u2018Subsistem Bandung Selatan dan New Ujungberung\u2019 before and after connected to SVC and the other purpose is to set optimal location and rating of SVC to maintain the voltage profile at the system that has desire range (0.95 p.u \u2013 1.05 p.u). To get the result in accordance with these objects, Newton \u2013Raphson power flow solution is applied to the system. The result of Newton-Raphson power flow solution of the system shows the voltage profile before connecting to SVC are averagely 140.95 kV or 0.94 p.u while after connecting to SVC are 145.28 kV or 0.97 p.u. The SVC installation is connected to \u2018Bandung Utara I\u2019 as the weakest bus, and the SVC rating is -250 Mvar to 300 Mvar. Topic: Engineering and Technology Innovation 183 WEB-BASED ENTREPRENEUR DEVELOPMENT CRAFT SMK COMPETENCE FOR STUDENTS THROUGH THE IMPLEMENTATION OF MODEL TEACHING FACTORY SIX STEPSPupun Suwangsih, Sudjani, Dadang Hidayat M. Learning Teaching Model Factory 6M an integrated learning model. Soul 'entrepreneur' is also indispensable for vocational students, because it is through this spirit, educators will have a more efficient work orientation, creative, innovative, productive and independent. In line with that then it is proper entrepreneurial learning can also be done with a web-based learning supported by the Digital Simulation. Starting from this understanding, the web-based learning craft entrepreneurs for vocational students would be able to support the achievement of competence especially competence patiseri which includes cognitive, affective, and psychomotor and is also supported by the application of learning models Teaching Factory Six zangkah. This thesis is the result of research on Effect of Entrepreneur Development craft Web-Based Competency for Students Smk patiseri Through Implementation of Teaching Model Factory. The object of research is the students of class XI Productive Catering SMKN 9 Bandung, Bandung BPP SMK and SMK 3 Garut. The method used in this research is the method Quasi Experiment. Topic: Vocational Education and Training 184 Welding Technology and metallurgy of Superalloy MaterialSuharno (a*), Yuyun Estriyanto (a), Budi Harjanto (a) Has done research on aircraft turbine blades made of Inconel (nickel-based). This study aims to describe the characteristics of crack on the failure of turbine blade made from Inconel and efforts to rectify the failure of the material. Inconels are nickel-base superalloy material that has high strength and creep resistance at temperatures near Reviews their melting point. This material is commonly used in aircraft gas turbine (jet) engines, where parts or components are subjected to high temperature and high stress. Research methods include testing the chemical composition, hardness with HVN, microstructure, and SEM. The results show that the turbine blades are attached to the APU classified material nickel-based super alloy Inconel 792. This kind indicated by the content of 63% Ni and 18% Cr. This material includes a very hard material that is characterized by its hardness is 412 HVN numbers so it is susceptible to cracking. Based on SEM test show that structure formed at the grain boundary carbides and Gamma-prime precipitates are also available within the gamma matrix. Topic: Engineering and Technology Innovation 185 WORK MOTIVATION VOCATIONAL HIGH SCHOOL GRADUATES ( VHS )Elih Mulyana VHS aims to prepare graduates to enter the world of work and have the competence to work in a particular field ( Permen.No 23, 2006 ) . Vocational graduates as generally work in Electrical Contractors . They work for their needs . The requirement is an encouragement or motivation possessed by graduates . Motivation is an internal process that activates , guides and maintains behavior over time ( Robert Slavin , 2009) .Worker motivation stability over time can be investigated by the behavior of their work translated into life and social norms inherent in their work ( Coetsier and Claes . R 1990). Motivation is the process of encouragement , direction and persistence of behavior , meaning that motivated behavior is behavior that is full of energy. ( Santrock 2007) . Motivation consists of intrinsic motivation , extrinsic motivation and achievement motivation ( Winkel 2009) . Motivation is a service work performed by employees at work in the company . Employment services by employees could include work motivation ( Parasuraman , A. , Zeithaml , V. & Berry , L 2005) . Motivation of employees will provide carrying capacity on the companys work , high motivation will speed up the work in the company , so the company will have a greater advantage . The purpose of this study was to determine employee motivation assessed by the company Electrical Contractors . The study was conducted on electrical contractor in West Java with a sample of 200 contractor . Instruments made as many as 15 items that refer to indicators of work motivation : A desire to live ; The desire to occupy a position; The desire for power; The desire for recognition ( Patterson , PG & Johnson , LW , 1993) . Based on data analysis , the motivation vocational graduates who were working in Electrical Contractors have different characteristics. 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{"url":"http:\/\/exxamm.com\/QuestionSolution6\/Aptitude\/+38+of+first+number+is+52+of+the+second+number+What+is+the+respective+ratio+of+the+first+number+to+the+second+nu\/2222356231","text":"38% of first number is 52% of the second number. What is the respective ratio of the first number to the second nu\n\n### Question Asked by a Student from EXXAMM.com Team\n\nQ 2222356231. \u00a0 \u00a0 38% of first number is 52% of the second number. What is\nthe respective ratio of the first number to the second\nnumber?\nIBPS-CLERK 2017 Mock Prelims\nA\n\n5:4'\n\nB\n\n16:9\n\nC\n\n26: 19\n\nD\n\nCannot be determined\n\nE\n\nNone of these\n\n#### HINT\n\n(Provided By a Student and Checked\/Corrected by EXXAMM.com Team)\n\n#### Access free resources including\n\n\u2022 100% free video lectures with detailed notes and examples\n\u2022 Previous Year Papers\n\u2022 Mock Tests\n\u2022 Practices question categorized in topics and 4 levels with detailed solutions\n\u2022 Syllabus & Pattern Analysis","date":"2018-11-15 04:02:11","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\": 0.17115136981010437, \"perplexity\": 5485.426579490289}, \"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-2018-47\/segments\/1542039742483.3\/warc\/CC-MAIN-20181115033911-20181115055911-00501.warc.gz\"}"} | null | null |
Welcome to Digital Duval. Our mission is to eliminated the "digital divide" for individuals without access to computer technology or individuals that do not know how to capitalize on their computer access.
We will achieve our mission by using technology to educate Seniors, children and business owners on the advantages available technology. This site is filled with information to coach and tutor our targeted stakeholders.
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"redpajama_set_name": "RedPajamaC4"
} | 7,443 |
Full Moon Reviews - Horror, Sci-Fi, Action, B-Movies: Midnight Confessions Ep. 102: "I Was A Teenage Podcast"
Midnight Confessions Ep. 102: "I Was A Teenage Podcast"
October has arrived and we're getting into the Halloween spirit with two b&w horror classics: I WAS A TEENAGE WEREWOLF (1957) and I WAS A TEENAGE FRANKENSTEIN (1957). Plus we discuss the clown epidemic, Rob Zombie's 31, PHANTASM and Herschell Gordon Lewis. Music this week by: The Keytones, Lord Luther, Murderock, The Cramps, Alice Cooper and Big Bee Kornegay. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,418 |
Q: How to determine currently sorted header column in data grid? I want to know which header the user clicked on to give the currently sorted view.
Is there an API in flex framework that I can use to achieve this? Hopefully I can get back a column index so I know how it is currently sorted.
Thanks,
Mike
A: The mx.controls.DataGrid has a property named columns. Each column in this collection is an object of type mx.controls.dataGridClasses.DataGridColumn with a boolean property named sortDescending.
Otherwise, you can receive and handle the DataGrid event headerRelease. This event is transmitted when the user releases the mouse button on a column header, causing the column to become sorted.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,533 |
A656 är en motorväg i Baden-Württemberg, Tyskland. Den är 11,5 kilometer lång och går mellan Mannheim och Heidelberg. Den utgör förbindelseväg mellan motorvägarna A6 och A5.
Motorvägar i Tyskland
Vägar i Baden-Württemberg | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,377 |
\section{Introduction}
Suppose a differential equation describes the evolution of some feature of a system (e.g., heat conduction), depending on its initial value (at time $t = 0$). We observe the feature at time $T > 0$, in the presence of noise or measurement errors,
and the aim is to recover the initial condition.
Inverse problems of this type are often ill-posed in the sense that the solution operator of the differential equation, which maps the function describing the initial state to the function that describes the state at the later time $T > 0$ at which we observe the system, does typically not have a well-behaved, continuous inverse.
This means that in many cases some form of regularization is necessary to solve the inverse problem and to deal with the noise.
In this paper we study a Bayesian approach to this problem for the particular example of recovering the initial condition
for the heat equation. Specifically, we assume we have noisy observations of the solution $u$ to the
Dirichlet problem for the heat equation
\begin{equation}\label{Heat}
\frac{\partial}{\partial t}u(x,t) = \frac{\partial^2}{\partial x^2} u(x,t), \quad u(x,0) = \mu(x), \quad u(0,t)=u(1,t) = 0,
\end{equation}
where $u$ is defined on $[0,1]\times[0,T]$ and the function $\mu \in L^2[0,1]$ satisfies $\mu(0)=\mu(1)=0$. The solution to (\ref{Heat})
is given by
\[
u(x,t) = \sqrt{2}\sum_{i=1}^\infty \mu_i e^{-i^2\pi^2t}\sin(i\pi x),
\]
where $(\mu_i)$ are the coordinates of $\mu$ in the basis $e_i = \sqrt{2}\sin(i\pi x)$, for $i \geq 1$. In other words,
it holds that $u(\cdot, T) = K\mu$, for $K$ the linear operator on $L^2[0,1]$ that is
diagonalized by the basis $(e_i)$ and that has corresponding eigenvalues
$\kappa_i = \exp({-i^2\pi^2 T})$, for $i \geq 1$.
We assume we observe the solution $K\mu$ in white noise of intensity $1/n$. By expanding in the basis $(e_i)$ this is equivalent to
observing the sequence of noisy, transformed Fourier coefficients
$Y = (Y_1, Y_2, \ldots)$ satisfying
\begin{equation}\label{Model}
Y_i = \kappa_i \mu_{i} + \frac{1}{\sqrt{n}}Z_i, \qquad i = 1, 2, \ldots,
\end{equation}
for $(\mu_i)$ and $(\kappa_i)$ as above, and $Z_1, Z_2, \ldots$ independent, standard normal random variables.
The aim is to recover the coefficients $\mu_i$, or equivalently, the initial condition $\mu = \sum_{i=1}^\infty \mu_ie_i$, under the assumption that the signal-to-noise ratio tends to infinity (so $n \to \infty$).
This heat conduction inverse problem has been studied in frequentist literature \citep[see, e.g.,][]{Bissantz, Cavalier, Cavalier2, Golubev, Mair2, Mair} and has also been addressed in Bayesian framework (with additional assumptions on the noise), cf.\ \cite{Stuart}. For more background on how this backward heat conduction problem arises in practical problems, see for instance \cite{Beck} or \cite{Engl}, and the references therein. Since the $\kappa_i$ decay in a sub-Gaussian manner, the estimation of $\mu$ is very hard in general. It is well known for instance that
the minimax rate of estimation for $\mu$ in a Sobolev ball of regularity $\beta$ (see Sec. 1.1) relative to the $\ell^2$-loss is only $(\log n)^{-\beta/2}$. This rate is attained by various methods, including generalized \emph{Tikhonov regularization} and spectral cut-off \citep{Bissantz, Mair2, Mair, Golubev}.
Convergence rates for Bayesian methods for problems like (\ref{Model})
have only been studied for the case that $\kappa_i$ decays like a power of $i$, see \cite{KVZ}.
In this paper, like in \cite{KVZ}, we put product priors of the form
\begin{equation}\label{Prior}
\Pi = \bigotimes_{i=1}^\infty N(0, \lambda_i)
\end{equation}
on the sequence $(\mu_i)$ and study the corresponding sequence of posterior distributions.
The results we obtain are different from the ones in \cite{KVZ} in a number of ways however.
First of all, it is in this case not true that to obtain optimal contraction rates for the posterior, we need
to match the regularities of the true sequence $\mu_0$ and the prior exactly. Any degree of oversmoothing
will do as well. Moreover, if the prior variances $\lambda_i$ are
chosen sub-Gaussian, then we obtain the optimal rate $(\log n)^{-\beta/2}$
for {\em any} $\beta$-regular $\mu_0$, i.e.,\ we obtain a rate-adaptive procedure.
Unfortunately however, these very smooth priors behave badly from another point of view. We show that asymptotically, the frequentist coverage
of credible sets based on these priors is $0$ for a very large class of true $\mu_0$'s. As in \cite{KVZ}
we see that asymptotic coverage $1$ is obtained when the prior is less regular than the truth.
The radius of a credible set is in that case however of a strictly larger order than the radius of the corresponding frequentist
credible set, which is another difference with the findings in \cite{KVZ} for polynomial $\kappa_i$.
These statements are made precise and are refined to include the possibility of rescaling the priors in Sec. 2.
On a qualitative level, the conclusion of the results must be that in the severely ill-posed case that we study in this paper
it is advisable to use a prior that is slightly less regular than the truth, just as in the mildly ill-posed case of \cite{KVZ}.
Unfortunately, the
corresponding Bayesian credible sets can be very large in the present setting and hence of limited use.
The results in Sec.~2 all deal with the recovery of the full parameter $\mu$.
In Sec.~3 we derive the analogous results for the problem of estimating
linear functionals of $\mu$. The results are numerically illustrated in Sec.~4. Sec.~5 contains proofs of the results presented in Secs.~2 and 3. Auxiliary lemmas are presented in Sec.~6.
\subsection{Notation}
\noindent For $\beta > 0$, the Sobolev norm $\|\mu\|_\beta$ and the $\ell^2$-norm $\|\mu\|$ of
an element $\mu \in \ell^2$ are defined by
\[
\|\mu\|_\beta^2 = \sum_{i=1}^\infty \mu_i^2i^{2\beta}, \qquad \|\mu\|^2 = \sum_{i=1}^\infty \mu_i^2,
\]
and the corresponding Sobolev space by $S^\beta = \{\mu \in \ell^2: \|\mu\|_\beta < \infty\}$.
For two sequences $(a_n)$ and $(b_n)$ of numbers, $a_n \asymp b_n$ means that $|a_n/b_n|$ is bounded away from zero and infinity as $n \to \infty$, $a_n \lesssim b_n$ means that $a_n/b_n$ is bounded, $a_n \sim b_n$ means that $a_n/b_n \to 1$ as $n \to \infty$, and $a_n \ll b_n$ means that $a_n / b_n \to 0$ as $n \to \infty$. For two real numbers $a$ and $b$, we denote by $a \vee b$ their maximum, and by $a \wedge b$ their minimum.
\section{Recovering the full parameter}
\noindent Under the model \eqref{Model} and the prior \eqref{Prior} the coordinates $(\mu_{0,i}, Y_i)$ of the vector $(\mu_0, Y)$ are independent, and hence the conditional distribution of $\mu_0$ given $Y$ factorizes over the coordinates as well. Thus the computation of the posterior distribution reduces to countably many posterior computations in conjugate normal models. It is straightforward to verify that the posterior distribution $\Pi_n(\,\cdot \mid Y)$ is given by
\begin{equation}\label{PosteriorFull}
\Pi_n(\,\cdot \mid Y) = \bigotimes_{i=1}^\infty N\biggl(\frac{n\lambda_i\kappa_i}{1+n\lambda_i\kappa_i^2}Y_i,\frac{\lambda_i}{1+n\lambda_i\kappa_i^2}\biggr).
\end{equation}
Our first theorem shows that the posterior contracts as $n \to \infty$ to the true parameter at a rate $\eps_n$
and quantifies how this rate depends on the behavior of the sequence $(\lambda_i)$ of prior variances and the
regularity $\beta$ of the true parameter $\mu_0$. We say the posterior {\em contracts around $\mu_0$ at the rate
$\eps_n$} if
\[
\E_{\mu_0}\Pi_n(\mu:\|\mu-\mu_0\|\geq M_n\eps_n\mid Y)\to 0
\]
for every $M_n \to \infty$, where the expectation is under the true model governed by the parameter $\mu_0$.
\begin{theorem}\label{RateFullS}
Suppose the true parameter $\mu_0$ belongs to $S^\beta$ for $\beta > 0$.
If $\lambda_i = \tau_n^2i^{-1-2\alpha}$ for some $\alpha > 0$ and $\tau_n >0$ such that $n\tau_n^2\to \infty$, then
the posterior contracts around $\mu_0$ at the rate
\begin{equation}\label{RateF1}
\eps_n = (\log n\tau_n^2)^{-\beta/2} + \tau_n(\log n\tau_n^2)^{-\alpha/2}.
\end{equation}
The rate is uniform over $\mu_0$ in balls in $S^\beta$. In particular:
\begin{itemize}
\item[(i)] If $\tau_n \equiv 1$, then $\eps_n = (\log n)^{-(\beta\wedge \alpha)/2}$.
\item[(ii)] If $n^{-1/2+\delta} \lesssim \tau_n \lesssim (\log n)^{(\alpha-\beta)/2}$, for some $\delta > 0$, then $\eps_n = (\log n)^{-\beta/2}$.
\end{itemize}
If $\lambda_i = e^{-\alpha i^2}$ for some $\alpha > 0$ then
the posterior contracts around $\mu_0$ at the rate
\begin{equation}\label{RateF2}
\eps_n = (\log n)^{-\beta/2}.
\end{equation}
The rate is uniform over $\mu_0$ in balls in $S^\beta$.
\end{theorem}
We think of the parameters $\beta$ and $\alpha$ as the regularity of the true parameter $\mu_0$ and the prior, respectively. The first is validated by the fact that in the heat equation case $(e_i)$ is the (sine) Fourier basis of $L^2[0,1]$. Therefore $\beta$
quantifies the smoothness of $\mu_0$ in Sobolev sense. In case of the polynomial decay of the variances of the prior (later referred to as the polynomial prior), the parameter $\alpha$ is also closely related to Sobolev regularity.
The minimax rate of convergence over a Sobolev ball $S^\beta$ is of the order $(\log n)^{-\beta/2}$.
Now consider the case $\lambda_i = \tau_n^2i^{-1-2\alpha}$. By statement (i) of the theorem the posterior
contracts at the optimal minimax rate if the regularity of the prior is at least the regularity of the truth ($\alpha \geq \beta$) and the scale $\tau_n$ is fixed. Alternatively, the optimal rate is also attained by appropriately scaling a prior of any regularity. Note that if $\alpha \geq \beta$ scaling is redundant.
The theorem shows that `correct' specification of the prior regularity gives the optimal rate. In contrast to \cite{KVZ}
however, the regularity of the prior does not have to match exactly the regularity of the truth. Moreover, even though rough priors still need to be scaled to give the optimal rate, there is no restriction on the `roughness'.
The second assertion of the theorem shows that for very smooth priors (where we take $\lambda_i = e^{-\alpha i^2}$) the contraction rate is always optimal. Since the prior does not depend on the unknown regularity $\beta$,
the procedure is \emph{rate-adaptive} in this case.
Both choices of priors lead to the conclusion that oversmoothing yields the optimal rate, and this has been noted also in the frequentist literature \citep[see][]{Mair2}. A fully adaptive frequentist method is presented in \cite{Bissantz}, and in both situations
the optimal performance is caused by the dominating bias. However, in Bayesian inference one often takes the spread in the posterior distribution as a quantification of uncertainty. If $\lambda_i = e^{-\alpha i^2}$ this spread is much smaller than the minimax rate.
To understand the implications, we next consider the frequentist coverage of credible sets. As the posterior is Gaussian, it is natural to center a credible region at the posterior mean. Different shapes of such a set could be considered, but the natural counterpart of the preceding theorem is to consider balls. The study of linear functionals in the next section makes it possible to consider pointwise credible bands as well.
A credible ball centered at the posterior mean $\hat\mu$, where $\hat \mu_i = n\lambda_i\kappa_i(1+n\lambda_i\kappa_i^2)^{-1}Y_i$, takes the form
\begin{equation}\label{CredibleBall}
\hat\mu + B(r_{n,\gamma}):= \bigl\{\mu \in \ell^2 : \|\mu-\hat\mu\| < r_{n,\gamma}\bigr\},
\end{equation}
where $B(r)$ denotes an $\ell^2$-ball of radius $r$ around $0$
and the radius $r_{n,\gamma}$ is determined such that
\begin{equation}\label{Credibility}
\Pi_n\bigl(\hat\mu + B(r_{n,\gamma})\mid Y\bigr) = 1-\gamma.
\end{equation}
Because the spread of the posterior is not dependent on the data, neither is the radius $r_{n,\gamma}$. The frequentist \emph{coverage} or confidence of the set \eqref{CredibleBall} is, by definition,
\begin{equation}\label{AsymptoticCov}
\p_{\mu_0}\bigl(\mu_0\in \hat\mu + B(r_{n,\gamma})\bigr),
\end{equation}
where under the probability measure $\p_{\mu_0}$ the variable $Y$ follows \eqref{Model} with $\mu = \mu_0$. We shall consider the coverage as $n \to \infty$ for fixed $\mu_0$, uniformly in Sobolev balls, and also along sequences $\mu_0^n$ that change with $n$.
The following theorem shows that the relation of the coverage to the credibility level $1-\gamma$ is mediated by
the regularity of the true $\mu_0$ and the two parameters controlling the regularity of the prior---$\alpha$ and the scaling $\tau_n$---for both types of priors. For further insight, the credible region is also compared to the `correct' frequentist confidence ball $\hat\mu +B(\tilde r_{n,\gamma})$ chosen so that the probability in~\eqref{AsymptoticCov} is exactly equal to $1-\gamma$.
\begin{theorem}\label{CredibilityFull}
Suppose the true parameter $\mu_0$ belongs to $S^\beta$ for $\beta > 0$.
If $\lambda_i = \tau_n^2i^{-1-2\alpha}$ for some $\alpha > 0$ and $\tau_n >0$ such that $n\tau_n^2\to \infty$, then asymptotic coverage of the credible region~\eqref{CredibleBall} is
\begin{itemize}
\item[(i)] 1, uniformly in $\mu_0$ with $\|\mu_0\|_\beta \leq 1$, if $\tau_n \gg (\log n)^{(\alpha - \beta)/2}$; in this case $r_{n,\gamma} /\tilde r_{n,\gamma} \to \infty$.
\item[(ii)] 1, uniformly in $\mu_0$ with $\|\mu_0\|_\beta \leq r$ for $r$ small enough, if $\tau_n \asymp (\log n)^{(\alpha - \beta)/2}$;\\
1, for every fixed $\mu_0 \in S^\beta$, if $\tau_n \asymp (\log n)^{(\alpha - \beta)/2}$.
\item [(iii)] 0, along some $\mu_0^n$ with $\sup_n \bigl\|\mu_0^n\bigr\|_\beta <\infty$, if $\tau_n \lesssim (\log n)^{(\alpha - \beta)/2}$.
\end{itemize}
If $\lambda_i = e^{-\alpha i^2}$ for some $\alpha > 0$, then the asymptotic coverage of the credible region~\eqref{CredibleBall} is
\begin{itemize}
\item[(iv)] 0, for every $\mu_0$ such that $|\mu_{0,i}| \gtrsim e^{-ci^2/2}$ for some $c < \alpha$.
\end{itemize}
If $\tau_n \equiv 1$, then the cases (i), (ii), and (iii) arise if $\alpha < \beta$, $\alpha = \beta$ and $\alpha \geq \beta$, respectively. If $\alpha > \beta$ in case $(iii)$ the sequence $\mu_0^n$ can then be chosen fixed.
\end{theorem}
The easiest interpretation of the theorem is in the situation without scaling $(\tau_n\equiv 1)$. Then oversmoothing the prior (case (iii): polynomial prior with $\alpha > \beta$, and case (iv): exponential prior) has disastrous consequences for the coverage of the credible sets, whereas undersmoothing (case (i): polynomial prior with $\alpha < \beta$) leads to (very) conservative sets. Choosing a prior of correct regularity (case (ii) and (iii): polynomial prior with $\alpha = \beta$) gives mixed results, depending on the norm of the true $\mu_0$. These conclusions are analogous to the ones that can be drawn from Theorem 4.2 in \cite{KVZ} for the mildly ill-posed case.
There is one crucial difference, namely the radius of the conservative sets in case (i) are not of the correct order of magnitude. It means that the radius $\tilde r_{n,\gamma}$ of the `correct' frequentist confidence ball is of strictly smaller order than the radius of the Bayesian credible ball.
By Theorem~\ref{RateFullS} the optimal contraction rate is obtained by smooth priors. Combining the two theorems leads to the conclusion that polynomial priors that slightly undersmooth the truth might be preferable. They attain a nearly optimal rate of contraction and the spread of their posterior gives a reasonable sense of uncertainty. Slightly undersmoothing is only possible however if an assumption about the regularity of the unknown true function is made. It is an important open problem to devise methods that achieve this automatically, without knowledge about the true regularity. Exponential priors, although adaptive and rate-optimal,
often lead to very bad pointwise credible bands.
\section{Recovering linear functionals of the parameter}
\noindent In this section we consider the posterior distribution of a linear functional $L\mu$ of the parameter. In the Bayesian setting we consider \emph{measurable linear functionals relative to the prior}, covering the class of continuous functionals, but also certain discontinuous functionals (for instance point evaluation), following the definition of \cite{Skorohod}. Let $(l_i) \in \R^\infty$ satisfy $\sum_{i=1}^\infty l_i^2\lambda_i < \infty$. Then it can be shown that $L\mu := \lim_{n\to \infty} \sum_{i=1}^n l_i\mu_i$ exists for all $\mu = (\mu_i)$ in a (measurable) subspace of $\ell^2$ with $\bigotimes_{i=1}^\infty N(0, \lambda_i)$-probability one. We define $L\mu = 0$ if the limit does not exist.
The posterior of the linear functional $L\mu$ can be obtained from \eqref{PosteriorFull} and the definition given above \citep[see also][]{KVZ}
\begin{equation}\label{PosteriorMargin}
\Pi_n(\mu: L\mu \in \cdot\mid Y) = N\biggl(\sum_{i=1}^\infty \frac{nl_i\lambda_i\kappa_i}{1+n\lambda_i\kappa_i^2}Y_i,
\sum_{i=1}^\infty \frac{l_i^2\lambda_i}{1+n\lambda_i\kappa_i^2}\biggr).
\end{equation}
We measure the smoothness of the functional $L$ by the size of the coefficients $l_i$, as $i \to \infty$. It is natural to assume that the sequence $(l_i)$ is in the Sobolev space $S^q$ for some $q$, but also more controlled behavior will be assumed in following theorems.
We say that the {\em marginal posterior of $L\mu$ contracts around $L\mu_0$ at the rate $\eps_n$} if
\[
\E_{\mu_0}\Pi_n(\mu:|L\mu-L\mu_0|\geq M_n\eps_n\mid Y)\to 0
\]
as $n \to \infty$, for every sequence $M_n \to \infty$.
\begin{theorem}\label{RateFunctS}
Suppose the true parameter $\mu_0$ belongs to $S^\beta$ for $\beta > 0$.
If $\lambda_i = \tau_n^2 i^{-1-2\alpha}$ for some $\alpha > 0$ and $\tau_n > 0$ such that $n\tau_n^2\to \infty$, and the representer $(l_i)$ of the linear functional $L$ is contained in $S^q$, or $|l_i| \lesssim i^{-q-1/2}$ for some $q \geq -\beta$,
then the marginal posterior of $L\mu$ contracts around $L\mu_0$ at the rate
\begin{equation}\label{RateL1}
\eps_n = (\log n\tau_n^2)^{-(\beta+q)/2} + \tau_n(\log n\tau_n^2)^{-(1/2+\alpha+q)/2}.
\end{equation}
The rate is uniform over $\mu_0$ in balls in $S^\beta$. In particular:
\begin{itemize}
\item[(i)] If $\tau_n \equiv 1$, then $\eps_n = (\log n)^{-(\beta\wedge (1/2+\alpha)+q)/2}$.
\item[(ii)] If $n^{-1/2+\delta} \lesssim \tau_n \lesssim (\log n)^{(1/2+\alpha-\beta)/2}$, for some $\delta > 0$, then $\eps_n = (\log n)^{-(\beta+q)/2}$.
\end{itemize}
If $\lambda_i= e^{-\alpha i^2}$ for some $\alpha > 0$ then the marginal posterior of $L\mu$ contracts around $L\mu_0$ at the rate
\begin{equation}\label{RateL2}
\eps_n = (\log n)^{-(\beta+q)/2}.
\end{equation}
The rate is uniform over $\mu_0$ in balls in $S^\beta$.
\end{theorem}
The minimax rate over a ball in the Sobolev space $S^\beta$ is known to be bounded above by $(\log n)^{-(\beta+q)/2}$ (for the case of $q = -1/2$ see \citealp{GoldDeconv}, and for general $q$ in a closely related model see \citealp{Butucea}). In view of Theorem~\ref{RateFullS}, it is not surprising that exponential priors yield this optimal rate. In case of polynomial prior this rate is attained without scaling if and only if the prior smoothness $\alpha$ is greater than or equal to $\beta$ minus 1/2. Here we observe a similar phenomenon as in \cite{KVZ}, where the `loss' in smoothness by $1/2$ is discussed. The regularity of the parameter in the Sobolev scale is not the appropriate type of regularity to consider for estimating a linear functional $L\mu$.
If the polynomial prior is too rough, then the minimax rate may still be attained by scaling the prior. The upper bound on the scaling is the same as in the global case (see Theorem~\ref{RateFullS}.(ii)) \emph{after} decreasing $\beta$ by 1/2. So the `loss in regularity' persists in the scaling.
Because the posterior distribution for the linear functional $L\mu$ is the one-dimensional normal distribution $N(\widehat{L\mu}, s_n^2)$,
where $s_n^2$ is the posterior variance in (\ref{PosteriorMargin}),
the natural \emph{credible interval} for $L\mu$ has endpoints $\widehat{L\mu} \pm z_{\gamma/2}s_n$, for $z_\gamma$ the (lower) standard normal $\gamma$-quantile. The \emph{coverage} of this interval is
\[
\p_{\mu_0}\bigl(\widehat{L\mu} + z_{\gamma/2}s_n \leq L\mu_0 \leq \widehat{L\mu} - z_{\gamma/2}s_n\bigr),
\]
where $Y$ follows \eqref{Model} with $\mu = \mu_0$. In the following theorem we restrict $(l_i)$ to sequences that behave polynomially.
\begin{theorem}\label{CredibilityLin}
Suppose the true parameter $\mu_0$ belongs to $S^\beta$ for $\beta > 0$. Let $\tilde\tau_n = (\log n)^{(1/2+\alpha-\beta)/2}$.
If $\lambda_i = \tau_n^2i^{-1-2\alpha}$ for some $\alpha > 0$ and $\tau_n > 0$ such that $n\tau_n^2 \to \infty$, and
$|l_i| \asymp i^{-q-1/2}$, then the asymptotic coverage of the interval $\widehat{L\mu} \pm z_{\gamma/2}s_n$ is:
\begin{itemize}
\item[(i)] 1, uniformly in $\mu_0$ such that $\|\mu_0\|_\beta\leq 1$ if $\tau_n \gg \tilde\tau_n$,
\item[(ii)] 1, uniformly in $\mu_0$ with $\|\mu_0\|_\beta \leq r$ for $r$ small enough, if $\tau_n \asymp \tilde\tau_n$;\\
1, for every fixed $\mu_0 \in S^\beta$, if $\tau_n \asymp \tilde\tau_n$,
\item[(iii)] 0, along some $\mu_0^n$ with $\sup_n \bigl\|\mu_0^n\bigr\|_\beta < \infty$, if $\tau_n \lesssim \tilde\tau_n$.
\end{itemize}
If $\lambda_i = e^{-\alpha i^2}$ for some $\alpha > 0$, then the asymptotic coverage of the interval $\widehat{L\mu} \pm z_{\gamma/2}s_n$ is:
\begin{itemize}
\item[(iv)] 0, for every $\mu_0$ such that $\mu_{0,i}l_i\gtrsim e^{-ci^2/2}i^{-q-1/2}$ for some $c < \alpha$.
\end{itemize}
In case (iii) the sequence $\mu_0^n$ can be taken a fixed element $\mu_0$ in $S^\beta$ if $\tau_n \leq \tilde\tau_n (\log n)^{-\delta}$ for some $\delta > 0$. Furthermore, if $\tau_n\equiv 1$, then the cases (i), (ii) and (iii) arise if $\alpha < \beta-1/2$, $\alpha = \beta-1/2$ and $\alpha \geq \beta-1/2$, respectively. If $\alpha > \beta-1/2$ in case (iii) the sequence $\mu_0^n$ can then be chosen fixed.
\end{theorem}
Similarly as in the problem of full recovery of the parameter $\mu$ oversmoothing leads to coverage 0, while undersmoothing gives (extremely) conservative intervals. In the case of a polynomial prior without scaling the cut-off for under- or oversmoothing is at $\alpha = \beta-1/2$, while the cut-off for scaling is at the optimal rate $\tilde\tau_n$. Exponential priors are bad even for very smooth $\mu_0$, and the asymptotic coverage in this case is always 0. It should be noted that too much undersmoothing is also undesirable, as it leads to very wide credible intervals, and may cause that $\sum_{i=1}^\infty l_i^2\lambda_i$ is no longer finite.
In contrast with the analogous theorem in \cite{KVZ}, the conservativeness in case of undersmoothing is extreme, as the coverage is 1. Since it holds for every linear functional that can be considered in this setting, we do not have a Bernstein--von Mises theorem. The linear functionals considered in this section are not smooth enough to cancel the ill-posedness of the problem \citep[cf. discussion after Theorem 5.4 in][]{KVZ}.
\section{Simulation example}
\noindent To illustrate our results with simulated data we fix a time $T = 0.1$ and a true function $\mu_0$, which we expand as $\mu_0 = \sum_{i=1}^\infty \mu_{0,i}e_i$ in the basis $(e_i)$. The simulated data are the noisy and transformed coefficients
\[
Y_i = \kappa_i\mu_{0,i}+\frac{1}{\sqrt{n}}Z_i.
\]
The (marginal) posterior distribution for the function $\mu$ at a point $x$ is obtained by expanding $\mu(x) = \sum_{i=1}^\infty \mu_i e_i(x)$, and applying the framework of linear functionals $L\mu = \sum_{i=1}^\infty l_i\mu_i$ with $l_i = e_i(x)$ (so $l_i \lesssim 1$ and $q = -1/2$). Recall
\[
\mu(x)\mid Y \sim N\biggl(\sum_{i=1}^\infty \frac{n\lambda_i\kappa_ie_i(x)}{1+n\lambda_i\kappa_i^2}Y_i, \sum_{i=1}^\infty \frac{e_i(x)^2\lambda_i}{1+n\lambda_i\kappa_i^2}\biggr).
\]
We obtained (marginal) posterior pointwise credible bands by computing for every $x$ a central 95\% interval for the normal distribution on the right side of the above display. We considered both types of priors.
Figure~1 illustrates these bands for $n=10^4$ and the polynomial prior. In every of 10 panels in the figure the black curve represents the function $\mu_0$, defined by
\begin{equation}
\label{True}
\mu_0(x) = 4x(x-1)(8x-5), \qquad \mu_{0,i} =\frac{8\sqrt{2}(13+11(-1)^i)}{\pi^3i^3},
\end{equation}
where $\mu_{0,i}$ are the coefficients relative to $e_i$, thus $\mu_0 \in S^\beta$ for every $\beta < 2.5$. The 10 panels represent 10 independent realizations of the data, yielding 10 different realizations of the posterior mean (the red curves) and the posterior pointwise credible bands (the green curves). In the left five panels the prior is given by $\lambda_i = i^{-1-2\alpha}$ with $\alpha = 1$, whereas in the right panels the prior corresponds to $\alpha = 3$. Each of the 10 panels also shows 20 realizations from the posterior distribution. This is also valid for Figure~2, with the exponential prior, so $\lambda_i = e^{-\alpha i^2}$. In the left panels $\alpha = 1$, and in the right panels $\alpha = 5$.
\begin{figure}
\centerline{\includegraphics[width=9cm]{10_4,a_1,a_3.eps}}
\caption{Polynomial prior. Realizations of the posterior mean (red) and (marginal)
posterior credible bands (green), and 20 draws from
the posterior (dashed curves). In all ten panels $n=10^4$.
Left 5 panels: $\alpha=1$; right 5 panels: $\alpha=3$. True curve
(black) given by \eqref{True}.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=9cm]{e10_4,a_1,a_5.eps}}
\caption{Exponential prior. Realizations of the posterior mean (red) and (marginal)
posterior credible bands (green), and 20 draws from
the posterior (dashed curves). In all ten panels $n=10^4$.
Left 5 panels: $\alpha=1$; right 5 panels: $\alpha=5$. True curve
(black) given by \eqref{True}.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=9cm]{10_4,10_8,a_,5-10.eps}}
\caption{Polynomial prior. Realizations of the posterior mean (red) and (marginal)
posterior credible bands (green), and 20 draws from
the posterior (dashed curves).
Left 5 panels: $n=10^4$ and $\alpha=0.5,1,2,5,10$ (top to bottom);
right 5 panels: $n=10^8$ and $\alpha=0.5,1,2,5,10$ (top to bottom). True curve
(black) given by \eqref{True}.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=9cm]{e10_4,10_8,a_,5-10.eps}}
\caption{Exponential prior. Realizations of the posterior mean (red) and (marginal)
posterior credible bands (green), and 20 draws from
the posterior (dashed curves).
Left 5 panels: $n=10^4$ and $\alpha=0.5,1,2,5,10$ (top to bottom);
right 5 panels: $n=10^8$ and $\alpha=0.5,1,2,5,10$ (top to bottom). True curve
(black) given by \eqref{True}.}
\end{figure}
A comparison of the left and right panels in Figure~1 shows that the rough polynomial prior ($\alpha = 1$) is aware of the difficulty of inverse problem: it produces wide pointwise credible bands that in (almost) all cases contain nearly the whole true curve. Figure~1 together with Figure~2 show that smooth priors (polynomial with $\alpha = 3$ and both exponential priors) are overconfident: the spread of the posterior distribution poorly reflects the imprecision of estimation. Our theoretical results show that the inaccurate quantification of the estimation error (by the posterior spread) remains even as $n \to \infty$.
The reconstruction, by the posterior mean or any other posterior quantiles, will eventually converge to the true curve.
The specification of the prior influences the speed of this convergence. This is illustrated in Figures~3~and~4. Every of 10 panels in each of the figures is similarly constructed as before, but now with $n=10^4$ and $n=10^8$ for the five panels on the left and right side, respectively, and with $\alpha = 1/2, 1, 2, 5, 10$ for the five panels from top to bottom ($\lambda_i = i^{-1-2\alpha}$ in Figure~3, and $\lambda_i = e^{-\alpha i^2}$ in Figure~4). As discussed above, all exponential priors give the optimal rate, but lead to bad pointwise credible bands. Also smooth polynomial priors give the optimal rate. This can be seen in Figure~3 for $n=10^8$ and $\alpha = 2$ or $5$, where pointwise credible bands are very close to the true curve. However, for $\alpha = 5$ it should be noted that the true curve is mostly outside the pointwise credible band.
\section{Proofs}\label{Proofs}
\subsection{Proof of Theorem~\ref{RateFullS}}
\noindent Let $s_{i,n}$ and $t_{i,n}$ be such that the posterior distribution in \eqref{PosteriorFull} can be denoted by $\bigotimes_{i=1}^\infty N\bigl(\sqrt{nt_{i,n}} Y_i,s_{i,n}\bigr)$. Because the posterior is Gaussian, it follows that
\begin{equation}\label{NormSquared}
\int\|\mu - \mu_0\|^2\, d\Pi_n(\mu \mid Y) = \|\hat\mu-\mu_0\|^2+\sum_{i=1}^\infty s_{i,n},
\end{equation}
where $Y$ follows \eqref{Model} with $\mu = \mu_0$, and
\[
\hat\mu = \biggl(\frac{n\lambda_i\kappa_i}{1+n\lambda_i\kappa_i^2}Y_i\biggr)_i = \biggl(\frac{n\lambda_i\kappa_i^2\mu_{0,i}}{1+n\lambda_i\kappa_i^2} + \frac{\sqrt{n}\lambda_i\kappa_iZ_i}{1+n\lambda_i\kappa_i^2}\biggr)_i =: \E_{\mu_0}\hat\mu + \bigl(\sqrt{t_{i,n}}Z_i\bigr)_i.
\]
By Markov's inequality the left side of~\eqref{NormSquared} is an upper bound to $M_n^2\eps_n^2\Pi_n\bigl(\mu:\|\mu -\mu_0\|\geq M_n\eps_n\mid Y)$. Therefore, it suffices to show that the expectation under $\mu_0$ of the right side of the display is bounded by a multiple of $\eps_n^2$. The expectation of the first term is the mean square error of the posterior mean $\hat\mu$, and can be written as the sum $\|\E_{\mu_0}\hat\mu-\mu_0\|^2 + \sum_{i=1}^\infty t_{i,n}$ of its square bias and `variance'. The second term $\sum_{i=1}^\infty s_{i,n}$ is deterministic. If $\lambda_i = \tau_n^2i^{-1-2\alpha}$ the three quantities are given by:
\begin{align}
\label{SqBias} \|\E_{\mu_0}\hat\mu-\mu_0\|^2 &= \sum_{i=1}^\infty \frac{\mu_{0,i}^2}{(1+n\lambda_i\kappa_i^2)^2} = \sum_{i=1}^\infty \frac{\mu_{0,i}^2}{(1+n\tau_n^2i^{-1-2\alpha}e^{-2\pi^2 T i^2})^2}\\
\label{Var} \sum_{i=1}^\infty t_{i,n} &= \sum_{i=1}^\infty \frac{n\lambda_i^2\kappa_i^2}{(1+n\lambda_i\kappa_i^2)^2} = \sum_{i=1}^\infty \frac{n\tau_n^4i^{-2-4\alpha}e^{-2\pi^2 T i^2}}{(1+n\tau_n^2i^{-1-2\alpha}e^{-2\pi^2 T i^2})^2}\\
\label{PostSpr} \sum_{i=1}^\infty s_{i,n}&= \sum_{i=1}^\infty \frac{\lambda_i}{1+n\lambda_i\kappa_i^2} = \sum_{i=1}^\infty \frac{\tau_n^2i^{-1-2\alpha}}{1+n\tau_n^2i^{-1-2\alpha}e^{-2\pi^2 T i^2}}.
\end{align}
By Lemma~\ref{Norm} (applied with $q=\beta$, $t=0$, $r=0$, $u = 1+2\alpha$, $p = 2\pi^2T$, $v =2$, and $N = n\tau_n^2$) the first term can be bounded by $\log(n\tau_n^2)^{-\beta}$, which accounts for the first term in the definition of $\eps_n$ in \eqref{RateF1}. By Lemma~\ref{Series} (applied with $t=2+4\alpha$, $r= 2\pi^2T$, $u = 1+2\alpha$, $p=2\pi^2T$, $v=2$, and $N=n\tau_n^2$) the second expression is of the order $\tau_n^2(\log n\tau_n^2)^{-1/2-\alpha}$. The third expression is of the order the square of the second term in the definition of $\eps_n$ in \eqref{RateF1}, by Lemma~\ref{Series} (applied with $t=1+2\alpha$, $r= 0$, $u = 1+2\alpha$, $p=2\pi^2T$, $v=1$, and $N=n\tau_n^2$).
The consequences (i)--(ii) follow by verification after substitution of $\tau_n$ as given.
In case of $\lambda_i =e^{-\alpha i^2}$, we replace $i^{-1-2\alpha}$ by $e^{-\alpha i^2}$ and set $\tau_n\equiv 1$ in \eqref{SqBias}--\eqref{PostSpr}. We then apply Lemma~\ref{Norm} (with $q=\beta$, $t=0$, $r=0$, $u = 0$, $p = 2\pi^2T+\alpha$, $v =2$, and $N = n$) and see that the first term can be bounded by $(\log n)^{-\beta}$, which accounts for the first term in the definition of $\eps_n$ in \eqref{RateF2}. By Lemma~\ref{Series} (applied with $t=0$, $r= 2\alpha+2\pi^2T$, $u = 0$, $p=2\pi^2\alpha$, $v=2$, and $N=n$), and again Lemma~\ref{Series} (applied with $t=0$, $r= \alpha$, $u = 0$, $p=\alpha+2\pi^2T$, $v=1$, and $N=n$) the latter two are of the order $n^{-\alpha/(\alpha+2\pi^2T)}$.
\subsection{Proof of Theorem~\ref{CredibilityFull}}
\noindent Because the posterior distribution is $\bigotimes_{i=1}^\infty N(\sqrt{n t_{i,n}}Y_i,s_{i,n})$, by~\eqref{PosteriorFull}, the radius $r_{n,\gamma}$ in \eqref{Credibility} satisfies $\p(U_n < r_{n,\gamma}^2) = 1-\gamma$, for $U_n$ a random variable distributed as the square norm of an $\bigotimes_{i=1}^\infty N(0, s_{i,n})$-variable. Under \eqref{Model} the variable $\hat\mu$ is $\bigotimes_{i=1}^\infty N\bigl((\E_{\mu_0}\hat\mu)_i, t_{i,n}\bigr)$-distributed, and thus the coverage \eqref{AsymptoticCov} can be written as
\begin{equation}\label{eq: cov}
\p\bigl(\|W_n+\E_{\mu_0}\hat\mu -\mu_0\|\leq r_{n,\gamma}\bigr),
\end{equation}
for $W_n$ possessing a $\bigotimes_{i=1}^\infty N(0, t_{i,n})$-distribution. For ease of notation let $V_n = \|W_n\|^2$.
The variables $U_n$ and $V_n$ can be represented as $U_n = \sum_{i=1}^\infty s_{i,n}Z_i^2$ and $V_n = \sum_{i=1}^\infty t_{i,n}Z_i^2$, for $Z_1, Z_2, \ldots$ independent standard normal variables, and $s_{i,n}$ and $t_{i,n}$ are as in the proof of Theorem~\ref{RateFullS}. By Lemma~\ref{Series} (cf. previous subsection)
\begin{align*}
\E U_n &= \sum_{i=1}^\infty s_{i,n} \asymp \tau_n^2(\log n\tau_n^2)^{-\alpha} &\quad&
\sd U_n = \sqrt{2\sum_{i=1}^\infty s_{i,n}^2} \asymp \tau_n^2(\log n\tau_n^2 )^{-1/4-\alpha}\\
\E V_n &= \sum_{i=1}^\infty t_{i,n} \asymp \tau_n^2(\log n\tau_n^2)^{-1/2-\alpha} &\quad&
\sd V_n =\sqrt{2\sum_{i=1}^\infty t_{i,n}^2} \asymp \tau_n^2(\log n\tau_n^2)^{-1/2-\alpha}.
\end{align*}
It follows that
\[
r_{n,\gamma}^2 \asymp \tau_n^2(\log n\tau_n^2)^{-\alpha} \asymp \E U_n \gg \E V_n \asymp \sd V_n,
\]
and therefore
\begin{equation}\label{Partial}
\p\bigl(V_n \leq \delta r_{n,\gamma}^2\bigr)= \p\biggl(\frac{V_n-\E V_n}{\sd V_n}\leq \frac{\delta r_{n,\gamma}^2-\E V_n}{\sd V_n}\biggr) \to 1,
\end{equation}
for every $\delta > 0$.
The square norm of the bias $\E_{\mu_0}\hat\mu - \mu_0$ is given in~\eqref{SqBias}, where it was noted that
\[
B_n := \sup_{\|\mu_0\|_\beta \lesssim 1} \|\E_{\mu_0}\hat\mu - \mu_0\| \asymp (\log n\tau_n^2)^{-\beta/2}.
\]
The bias $B_n$ is decreasing in $\tau_n$, whereas $\E U_n$ is increasing. The scaling rate $\tilde\tau_n \asymp (\log n)^{(\alpha - \beta)/2}$ balances the square bias $B_n^2$ with the posterior spread $\E U_n$, and hence with $r_{n,\gamma}^2$.
Case (i). In this case $B_n \ll r_{n,\gamma}$. Hence $\p\bigl(\|W_n+\E_{\mu_0}\hat\mu -\mu_0\|\leq r_{n,\gamma}\bigr) \geq \p\bigl(\|W_n\|\leq r_{n,\gamma}-B_n\bigr) = \p\bigl(V_n\leq r_{n,\gamma}^2(1+o(1))\bigr) \to 1$, uniformly in the set of $\mu_0$ in the supremum defining $B_n$. Note that $\tilde r_{n,\gamma}$ is such that the coverage in \eqref{eq: cov} is exactly $1-\gamma$. Since $\|W_n\|^2 = V_n$, we have that $\tilde r_{n,\gamma}^2$ is of the order $B_n^2 + \tau_n^2(\log n\tau_n^2)^{-1/2-\alpha}$, so of strictly smaller order than $r_{n,\gamma}^2$, and therefore $r_{n,\gamma}/\tilde r_{n,\gamma} \to \infty$.
Case (ii). In this case $B_n \asymp r_{n,\gamma}$. By the second assertion of Lemma~\ref{Series} the bias $\|\E_{\mu_0}\hat\mu - \mu_0\|$ at a fixed $\mu_0$ is of strictly smaller order than the supremum $B_n$. The argument of (i) shows that the asymptotic coverage then tends to 1. The maximal bias $B_n(r)$ over $\|\mu_0\|_\beta \leq r$ is of the order $r_{n,\gamma}$ and proportional to the radius $r$. Thus for small enough $r$ we have that $r_{n,\gamma}-B_n(r) \gtrsim r_{n,\gamma} \to \infty$. Then $\p\bigl(\|W_n +\E_{\mu_0}\hat\mu-\mu_0\| \leq r_{n,\gamma}\bigr)\geq \p\bigl(\|W_n\|\leq r_{n,\gamma}-B_n(r)\bigl) \geq \p\bigl(V_n \lesssim r_{n,\gamma}^2\bigr)\to 1$.
Case (iii). In this case $B_n \gtrsim r_{n,\gamma}$. Hence any sequence $\mu_0^n$ that (nearly) attains the maximal bias over a sufficiently large ball $\|\mu_0\|_\beta\leq r$ such that $B_n(r)-r_{n,\gamma}\gtrsim r_{n,\gamma}$ satisfies $\p\bigl(\|W_n+\E_{\mu_0}\hat\mu -\mu_0\|\leq r_{n,\gamma}\bigr) \leq \p\bigl(\|W_n\|\geq B_n(r) - r_{n,\gamma}\bigr) \leq \p\bigl(V_n\gtrsim r_{n,\gamma}^2 \bigr)\to 0$.
If $\tau_n \equiv 1$, then $B_n$ and $r_{n,\gamma}$ are both powers of $1/\log n$ and hence $B_n \gg r_{n,\gamma}$ implies that $B_n \gtrsim r_{n,\gamma}(\log n)^\delta$, for some $\delta > 0$. The preceding argument then applies for a fixed $\mu_0$ of the form $\mu_{0,i} \asymp i^{-1/2-\beta-\eps}$, for small $\eps>0$, that gives a bias that is much closer than $(\log n)^\delta$ to $B_n$.
Case (iv). In the proof of Theorem~\ref{RateFullS}, we obtained $\E U_n \asymp \E V_n \asymp n^{-\alpha/(\alpha+2\pi^2T)}$. It can be shown that $\sd U_n \asymp n^{-\alpha/(\alpha+2\pi^2T)}$, so also $r_{n,\gamma}^2 \asymp n^{-\alpha/(\alpha+2\pi^2T)}$. If $|\mu_{0,i}| \gtrsim e^{-ci^2/2}$ for some $c < \alpha$, we have
\[
\|\E_{\mu_0}\hat\mu-\mu_0\|^2 = \sum_{i=1}^\infty \frac{\mu_{0,i}^2}{(1+n\lambda_i\kappa_i^2)^2} \gtrsim \sum_{i=1}^\infty \frac{e^{-ci^2}}{(1+ne^{-(\alpha+2\pi^2 T) i^2})^2} \asymp n^{-c/(\alpha+2\pi^2T)} \gg n^{-\alpha/(\alpha+2\pi^2T)},
\]
by Lemma~\ref{Series} (applied with $t=0$, $r = c$, $u=0$, $p=\alpha+2\pi^2 T$, $v=2$, and $N=n$). Hence $\p\bigl(\|W_n + \E_{\mu_0}\hat\mu - \mu_0\| \leq r_{n,\gamma}\bigr) \leq \p\bigl(V_n \geq \|\E_{\mu_0}\hat\mu-\mu_0\|^2 - r_{n,\gamma}^2\bigr) \to 0$.
\subsection{Proof of Theorem~\ref{RateFunctS}}
\noindent By \eqref{PosteriorMargin} the posterior distribution is $N(\widehat{L\mu}, s_n^2)$, and hence similarly as in the proof of Theorem~\ref{RateFullS} it suffices to show that
\[
\E_{\mu_0} |\widehat{L\mu} - L\mu_0|^2 + s_n^2 = |\E_{\mu_0}\widehat{L\mu} -L\mu_0|^2 + \sum_{i=1}^\infty \frac{l_i^2n\lambda_i^2\kappa_i^2}{(1+n\lambda_i\kappa_i^2)^2} + s_n^2
\]
is bounded above by a multiple of $\eps_n^2$. If $\lambda_i = \tau_n^2i^{-1-2\alpha}$ the three quantities are given by
\begin{align}
\label{LinBias} |\E_{\mu_0}\widehat{L\mu}-L\mu_0| = \biggl|\sum_{i=1}^\infty \frac{l_i\mu_{0,i}}{1+n\lambda_i\kappa_i^2}\biggr| &\leq \sum_{i=1}^\infty \frac{|l_i\mu_{0,i}|}{1+n\tau_n^2i^{-1-2\alpha}e^{-2\pi^2 T i^2}}\\
\label{LinVar} t_n^2:= \sum_{i=1}^\infty \frac{l_i^2n\lambda_i^2\kappa_i^2}{(1+n\lambda_i\kappa_i^2)^2} &= n\tau_n^4\sum_{i=1}^\infty \frac{l_i^2i^{-2-4\alpha}e^{-2\pi^2 T i^2}}{(1+n\tau_n^2i^{-1-2\alpha}e^{-2\pi^2 T i^2})^2}\\
\label{LinPostSpr} s_n^2 = \sum_{i=1}^\infty \frac{l_i^2\lambda_i}{1+n\lambda_i\kappa_i^2} &= \tau_n^2\sum_{i=1}^\infty \frac{l_i^2i^{-1-2\alpha}}{1+n\tau_n^2i^{-1-2\alpha}e^{-2\pi^2 T i^2}}.
\end{align}
By the Cauchy--Schwarz inequality the square of the bias~\eqref{LinBias} satisfies
\begin{equation}\label{LinSqBias}
|\E_{\mu_0}\widehat{L\mu}-L\mu_0|^2 \leq \|\mu_0\|^2_\beta \sum_{i=1}^\infty \frac{l_i^2i^{-2\beta}}{(1+n\tau_n^2i^{-1-2\alpha}e^{-2\pi^2 T i^2})^2}.
\end{equation}
Consider $(l_i) \in S^q$. By Lemma~\ref{Norm} (applied with $q=q$, $t=2\beta$, $r=0$, $u=1+2\alpha$, $p = 2\pi^2T$, $v=2$, and $N = n\tau_n^2$) the right side of this display can be further bounded by $\|\mu_0\|^2_\beta\|l\|_q^2$ times the square of the first term in the sum of two terms that defines $\eps_n$. By Lemma~\ref{Norm} (applied with $q=q$, $t=2+4\alpha$, $r=2\pi^2T$, $u=1+2\alpha$, $p=2\pi^2T$, $v=2$, and $N = n\tau_n^2$), and again by Lemma~\ref{Norm} (applied with $q = q$, $t = 1+2\alpha$, $r= 0$, $u=1+2\alpha$, $p = 2\pi^2T$, $v=1$, and $N=n\tau_n^2$) the right sides of \eqref{LinVar} and \eqref{LinPostSpr} are bounded above by $\|l\|_q^2$ times the square of the second term in the definition of $\eps_n$.
Consider $l_i \lesssim i^{-q-1/2}$. This follows the same lines as in the case of $(l_i) \in S^q$, except that we use Lemma~\ref{Series} instead of Lemma~\ref{Norm}. In this case the upper bound for the standard deviation of the posterior mean $t_n$ is of the order $\tau_n(\log n\tau_n^2 )^{-(1+\alpha+q)/2}$.
Consequences (i)--(ii) follow by substitution.
If $\lambda_i = e^{-\alpha i^2}$, then in case $(l_i)\in S^q$ we use Lemma~\ref{Series} (with $q=q$, $t=2\beta$, $r=0$, $u=0$, $p=\alpha + 2\pi^2 T$, $v=2$, and $N = n$), and Lemma~\ref{Series} (with $q=q$, $t=0$, $r=2\alpha+2\pi^2T$, $u=0$, $p=\alpha + 2\pi^2 T$, $v=2$, and $N = n$), and again Lemma~\ref{Series} (with $q=q$, $t=0$, $r=\alpha$, $u=0$, $p=\alpha + 2\pi^2 T$, $v=2$, and $N = n$) to bound \eqref{LinSqBias} by a multiple of $(\log n)^{-(\beta+q)}$, and \eqref{LinVar}--\eqref{LinPostSpr} by a multiple of $n^{-\alpha/(\alpha+2\pi^2 T)}(\log n)^{-q}$.
If $l_i \lesssim i^{-q-1/2}$, we use Lemma~\ref{Norm} (with $t=1+2q+2\beta$, $r = 0$, $u=0$, $p=\alpha+2\pi^2 T$, $v=2$, and $N = n$), and Lemma~\ref{Norm} (with $t=1+2q$, $r = 2\alpha+2\pi^2T$, $u=0$, $p=\alpha+2\pi^2 T$, $v=2$, and $N = n$), and again Lemma~\ref{Norm} (with $t=1+2q$, $r = \alpha$, $u=0$, $p=\alpha+2\pi^2 T$, $v=1$, and $N = n$) to bound \eqref{LinSqBias} by a multiple of $(\log n)^{-(\beta+q)}$, and \eqref{LinVar}--\eqref{LinPostSpr} by a multiple of $n^{-\alpha/(\alpha+2\pi^2 T)}(\log n)^{-1/2-q}$.
\subsection{Proof of Theorem~\ref{CredibilityLin}}
\noindent Under \eqref{Model} the variable $\widehat{L\mu}$ is $N(\E_{\mu_0}\widehat{L\mu}, t_n^2)$-distributed, for $t_n^2$ given in~\eqref{LinVar}. It follows that the coverage can be written, with $W$ a standard normal variable,
\begin{equation}\label{CovSplit}
\p\bigl(|Wt_n+\E_{\mu_0}\widehat{L\mu}-L\mu_0|\leq -s_nz_{\gamma/2}\bigr).
\end{equation}
The bias $|\E_{\mu_0}\widehat{L\mu} - L\mu_0|$ and posterior spread $s_n^2$ are expressed as series in \eqref{LinBias} and \eqref{LinPostSpr}.
Because $W$ is centered, the coverage \eqref{CovSplit} is largest if the bias $\E_{\mu_0}\widehat{L\mu}-L\mu_0$ is zero. It is then at least $1-\gamma$, because $t_n \leq s_n$, and tends to exactly $1$, because $t_n \ll s_n$.
The supremum of the bias satisfies
\begin{equation}\label{SupBias}
B_n := \sup_{\|\mu_0\|_\beta\lesssim 1} |\E_{\mu_0}\widehat{L\mu}-L\mu_0| \asymp (\log n\tau_n^2)^{-(\beta+q)/2}.
\end{equation}
The maximal bias $B_n$ is a decreasing function of the scaling parameter $\tau_n$, while the root spread $s_n$ increases with $\tau_n$. The scaling rate $\tilde\tau_n = (\log n)^{(1/2+\alpha-\beta)/2}$ in the statement of the theorem balances $B_n$ with $s_n$.
Case (i). If $\tau_n \gg \tilde\tau_n$, then $B_n \ll s_n$. Hence the bias $|\E_{\mu_0}\widehat{L\mu}-L\mu_0|$ in \eqref{CovSplit} is negligible relative to $s_n$, uniformly in $\|\mu_0\|_\beta\lesssim 1$, and $\p\bigl(|Wt_n+\E_{\mu_0}\widehat{L\mu}-L\mu_0|\leq -s_nz_{\gamma/2}\bigr)\geq \p\bigl(|Wt_n| \leq -s_nz_{\gamma/2} - |\E_{\mu_0}\widehat{L\mu}-L\mu_0|\bigr)\to 1$.
Case (ii). If $\tau_n \asymp \tilde \tau_n$, then $B_n \asymp s_n$. If $b_n = |\E_{\mu_0^n}\widehat{L\mu} - L\mu_0^n|$ is the bias at a sequence $\mu_0^n$ that nearly assumes the supremum in the definition of $B_n$, we have that $\p\bigl(|Wt_n + db_n|\leq -s_n z_{\gamma/2}\bigr) \geq \p\bigl(|Wt_n|\leq s_n |z_{\gamma/2}|-db_n\bigr) \to 1$ if $d$ is chosen sufficiently small. This is the coverage at the sequence $d\mu_0^n$, which is bounded in $S^\beta$. On the other hand, using Lemma~\ref{CSbound} it can be seen that the bias at a fixed $\mu_0 \in S^\beta$ is of strictly smaller order than the supremum $B_n$, and hence the coverage at a fixed $\mu_0$ is as in case (i).
Case (iii). If $\tau_n \lesssim \tilde\tau_n$, then $B_n \gtrsim s_n$. If $b_n = |\E_{\mu_0^n}\widehat{L\mu} - L\mu_0^n|$ is again the bias at a sequence $\mu_0^n$ that (nearly) attains the supremum in the definition of $B_n$, we we have that $\p\bigl(|Wt_n + db_n|\leq -s_n z_{\gamma/2}\bigr) \leq \p\bigl(|Wt_n|\geq db_n-s_n|z_{\gamma/2}|\bigr) \to 0$ if $d$ is chosen sufficiently large. This is the coverage at the sequence $d\mu_0^n$, which is bounded in $S^\beta$. By the same argument the coverage also tends to zero for a fixed $\mu_0$ in $S^\beta$ with bias $b_n = |\E_{\mu_0}\widehat{L\mu}-L\mu_0| \gg s_n\gg t_n$. For this we choose $\mu_{0,i}=i^{-\beta-1/2-\delta'}$ for some $\delta' > 0$. By another application of Lemma~\ref{Series}, the bias at $\mu_0$ is of the order
\[
\sum_{i=1}^\infty \frac{l_i\mu_{0,i}}{1+n\tau_n^2i^{-1-2\alpha}e^{-2\pi^2 T i^2}} \asymp \sum_{i=1}^\infty \frac{i^{-\beta-q-\delta'-1}}{1+n\tau_n^2i^{-1-2\alpha}e^{-2\pi^2 T i^2}} \asymp (\log n\tau_n^2)^{-(\beta+q+\delta')/2}.
\]
Therefore if $\tau_n \leq \tilde\tau_n (\log n)^{-\delta}$ for some $\delta>0$, then $B_n \gtrsim s_n (\log n\tau_n^2)^{\delta''}$ for some $\delta'' > 0$, and hence taking $\delta' = \delta''$ we have $b_n \asymp B_n (\log (n\tau_n^2))^{-\delta''/2} \gg s_n \gg t_n$.
Case (iv). In the proof of Theorem~\ref{RateFunctS}, we obtained $s_n \asymp t_n \asymp n^{-\alpha/(\alpha+2\pi^2 T)}(\log n)^{-q}$. If $\mu_{0,i}l_i \gtrsim e^{-ci^2/2}i^{-q-1/2}$ for some $c < \alpha$, we have
\begin{align*}
|\E_{\mu_0}\widehat{L\mu}-L\mu_0| &= \biggl|\sum_{i=1}^\infty \frac{l_i\mu_{0,i}}{1+n\lambda_i\kappa_i^2}\biggr| \gtrsim \sum_{i=1}^\infty \frac{e^{-ci^2}i^{-2q-1}}{(1+ne^{-(\alpha+2\pi^2 T) i^2})^2}\\
&\asymp n^{-c/(\alpha+2\pi^2T)}(\log n)^{-1/2-q} \gg n^{-\alpha/(\alpha+2\pi^2T)}(\log n)^{-1/2-q},
\end{align*}
by Lemma~\ref{Series} (applied with $t=1+2q$, $r = c$, $u=0$, $p=\alpha+2\pi^2 T$, $v=2$, and $N=n$). Hence $\p\bigl(|Wt_n+\E_{\mu_0}\widehat{L\mu}-L\mu_0|\leq -s_nz_{\gamma/2}\bigr)\leq \p\bigl(|Wt_n| \geq |\E_{\mu_0}\widehat{L\mu}-L\mu_0| -s_nz_{\gamma/2} \bigr)\to 0$.
If the scaling rate is fixed to $\tau_n \equiv 1$, then it can be checked from \eqref{SupBias} and the proof of Theorem~\ref{RateFunctS} that $B_n \ll s_n, B_n \asymp s_n$ and $B_n \gg s_n$ in the three cases $\alpha < \beta-1/2$, $\alpha = \beta-1/2$ and $\alpha \geq \beta-1/2$, respectively. In the first and third cases the maximal bias and the root spread differ by more than a logarithmic term $(\log n)^\delta$. It follows that the preceding analysis (i), (ii), (iii) extends to this situation.
\section{Appendix}\label{Appendix}
\begin{lemma}\label{Norm}
For any $q \in \mathbb{R}$, $u, v \geq 0$, $t\geq -2q$, $p > 0$, and $0 \leq r < vp$, as $N\to \infty$,
\[
\sup_{\|\xi\|_q\leq 1}\sum_{i=1}^\infty \frac{\xi_i^2i^{-t}e^{-ri^2}}{(1+Ni^{-u}e^{-pi^2})^v} \asymp N^{-r/p}(\log N)^{-t/2-q+ur/(2p)}.
\]
Moreover, for every fixed $\xi \in S^q$, as $N \to \infty$,
\[
N^{r/p}(\log N)^{t/2+q-ur/(2p)}\sum_{i=1}^\infty \frac{\xi_i^2i^{-t}e^{-ri^2}}{(1+Ni^{-u}e^{-pi^2})^v} \to 0.
\]
\end{lemma}
\begin{proof}
Let $I_N$ be the solution to $Ni^{-u}e^{-pi^2} = 1$. In the range $i \leq I_N$ we have $Ni^{-u}e^{-pi^2}\leq 1+ Ni^{-u}e^{-pi^2} \leq 2 Ni^{-u}e^{-pi^2}$, while $1\leq 1+ Ni^{-u}e^{-pi^2} \leq 2$ in the range $i \geq I_N$. Thus
\[
\sum_{i \leq I_N} \frac{\xi_i^2i^{-t}e^{-ri^2}}{(1+Ni^{-u}e^{-pi^2})^v} \asymp \sum_{i \leq I_N} \xi_i^2i^{2q}\frac{i^{uv-t-2q}e^{(vp-r)i^2}}{N^v} \leq \|\xi\|_q^2N^{-r/p}I_N^{-t-2q+ur/p},
\]
since for $N$ large enough all terms $i^{uv-t-2q}e^{(vp-r)i^2}$ in this range will be dominated by $I_N^{uv-t-2q}e^{(vp-r)I_N^2}$ and $I_N$ solves the equation $Ni^{-u}e^{-pi^2} = 1$. Similarly for the second range, we have
\[
\sum_{i \geq I_N} \frac{\xi_i^2i^{-t}e^{-ri^2}}{(1+Ni^{-u}e^{-pi^2})^v} \asymp \sum_{i \geq I_N} \xi_i^2i^{2q}i^{-t-2q}e^{-ri^2} \leq N^{-r/p}I_N^{-t-2q+ur/p} \sum_{i \geq I_N} \xi_i^2i^{2q}.
\]
Lemma~\ref{Split} yields the upper bound for the supremum.
The lower bound follows by considering the sequence $(\xi_i)$ given by $\xi_i = i^{-q}$ for $i \sim I_N$ and $\xi_i = 0$ otherwise, showing that the supremum is bigger than $N^{-r/p}(\log N)^{-t/2-q+ur/(2p)}$.
The preceding display shows that the sum over the terms $i \geq I_N$ is $o\bigl(N^{-r/p}(\log N)^{-t/2-q+ur/(2p)}\bigr)$. Furthermore
\[
N^{r/p}(\log N)^{t/2+q-ur/(2p)}\sum_{i\leq I_N} \frac{\xi_i^2i^{-t}e^{-ri^2}}{(1+Ni^{-u}e^{-pi^2})^v} \asymp \sum_{i \leq I_N} \xi_i^2i^{2q} \frac{i^{uv-t-2q}e^{(vp-r)i^2}}{N^vI_N^{-t-2q+ur/p}e^{-rI_N^2}},
\]
and this tends to zero by dominated convergence. Indeed, as noted before, for $N$ large enough all terms $i^{uv-t-2q}e^{(vp-r)i^2}$ in the range $i \leq I_N$ are upper bounded by $I_N^{uv-t-2q}e^{(vp-r)I_N^2} = N^{v-r/p}I_N^{-t-2q+ur/p}$, and by Lemma~\ref{Split} $N^{v-r/p}I_N^{-t-2q+ur/p} \asymp N^{v-r/p}(\log N)^{-t/2-q+ur/(2p)}\to \infty$, since $v-r/p > 0$.
\end{proof}
\begin{lemma}\label{Series}
For any $t, u, v\geq 0$, $p > 0$, and $0 \leq r < vp$, as $N \to \infty$,
\[
\sum_{i=1}^\infty \frac{i^{-t}e^{-ri^2}}{(1+Ni^{-u}e^{-pi^2})^v} \asymp \begin{cases} N^{-r/p}(\log N)^{-t/2+ur/(2p)} & \text{if } r\neq 0,\\
(\log N)^{-(t+1)/2} & \text{if } r = 0.
\end{cases}
\]
\end{lemma}
\begin{proof}
As in the preceding proof we split the infinite series in the sum over the terms $i \leq I_N$ and $i \geq I_N$. For the first part of the sum we get
\[
\sum_{i \leq I_N} \frac{i^{-t}e^{-ri^2}}{(1+Ni^{-u}e^{-pi^2})^v} \asymp \sum_{i \leq I_N} \frac{i^{uv-t}e^{(vp-r)i^2}}{N^v}.
\]
Most certainly $N^v\cdot I_N^{-t}e^{-rI_N^2} = {I_N}^{uv-t}e^{(vp-r){I_N}^2} \leq \sum_{i \leq I_N} i^{uv-t}e^{(vp-r)i^2}$. If $i^{uv-t}e^{(vp-r)i^2}$ as a function of $i$ is strictly increasing, then the sum is upper bounded by the integral in the same range, and the value at the right end-point. Otherwise $i^{uv-t}e^{(vp-r)i^2}$ first decreases, and then increases, and therefore the sum is upper bounded by the integral, and values at both endpoints:
\begin{align*}
\sum_{i \leq I_N} i^{uv-t}e^{(vp-r)i^2} &\leq \int_{1}^{I_N} x^{uv-t}e^{(vp-r)x^2}\, dx + e^{vp-r}+ {I_N}^{uv-t}e^{(vp-r){I_N}^2} \\
&= \frac{1}{2(vp-r)}{I_N}^{uv-t-1}e^{(vp-r){I_N}^2}\bigl(1+o(1)\bigr) + e^{vp-r} + {I_N}^{uv-t}e^{(vp-r){I_N}^2}\\
&\asymp {I_N}^{uv-t}e^{(vp-r){I_N}^2}\bigl(1+o(1)\bigr),
\end{align*}
by Lemma~\ref{Integrals}. Therefore by Lemma~\ref{Split}
\[
\sum_{i \leq I_N} \frac{i^{uv-t}e^{(vp-r)i^2}}{N^v} \asymp I_N^{-t}e^{-rI_N^2} = N^{-r/p}I_N^{-t+ur/p} \asymp N^{-r/p}(\log N)^{-t/2+ur/(2p)}.
\]
The other part of the sum satisfies
\[
\sum_{i \geq I_N} \frac{i^{-t}e^{-ri^2}}{(1+Ni^{-u}e^{-pi^2})^v} \asymp \sum_{i \geq I_N} i^{-t}e^{-ri^2}.
\]
Suppose $r > 0$. Again, the latter sum is lower bounded by $I_N^{-t}e^{-rI_N^2} \asymp N^{-r/p}(\log N)^{-t/2+ur/(2p)}$. Since $i^{-t}e^{-ri^2}$ is decreasing, we get the following upper bound
\begin{align*}
\sum_{i \geq I_N} i^{-t}e^{-ri^2} &\leq I_N^{-t}e^{-rI_N^2} + \int_{I_N}^\infty x^{-t}e^{-rx^2}\, dx \leq I_N^{-t}e^{-rI_N^2} + \frac{1}{2r}I_N^{-t-1}e^{-rI_N^2}\\
&\asymp I_N^{-t}e^{-rI_N^2}\bigl(1+o(1)\bigr) \asymp N^{-r/p}\bigl(\log N\bigr)^{-t/2+ur/(2p)},
\end{align*}
where the upper bound for the integral follows from Lemma~\ref{Integrals}.
In case $r = 0$, we get $\sum_{i>I_N} i^{-t} \asymp I_N^{-t+1} \asymp (\log N)^{-(t+1)/2}$ \citep[see Lemma 8.2 in][]{KVZ}.
\end{proof}
\begin{lemma}\label{CSbound}
For any $t \geq 0$, $u, p > 0$, $\mu \in S^{t/2}$, and $q > -t/2$, as $N\to \infty$
\[
\sum_{i=1}^\infty \frac{\bigl|\mu_ii^{-q-1/2}\bigr|}{1+Ni^{-u}e^{-pi^2}}\ll (\log N)^{-t/2-q}.
\]
\end{lemma}
\begin{proof}
We split the series in two parts, and bound the denominator $1+Ni^{-u}e^{-pi^2}$ by $Ni^{-u}e^{-pi^2}$ or $1$. By the Cauchy--Schwarz inequality, for any $r > 0$,
\begin{align*}
\biggl|\sum_{i\leq I_N} \frac{\bigl|\mu_ii^{-q-1/2}\bigr|}{Ni^{-u}e^{-pi^2}}\biggr|^2 &\leq \frac{1}{N^2}\sum_{i\leq I_N} \frac{i^r}{i}\sum_{i\leq I_N}\mu_i^2i^{2u-2q-r}e^{2pi^2}\\
&\leq \frac{1}{N^2}I_N^r \sum_{i\leq I_N}\mu_i^2i^t\frac{i^{2u-2q-r-t}e^{2pi^2}}{I_N^{2u-2q-r-t}e^{2pI_N^2}}I_N^{2u-2q-r-t}e^{2pI_N^2}\\
&=I_N^{-t-2q} \sum_{i\leq I_N}\mu_i^2i^t\frac{i^{2u-2q-r-t}e^{2pi^2}}{I_N^{2u-2q-r-t}e^{2pI_N^2}}.
\end{align*}
The terms in the remaining series in the right side are bounded by a constant times $\mu_i^2i^t$ for large enough $N$ and all $i$ bigger than a fixed number, and tend to zero pointwise as $N\to \infty$, and the sum tends to zero by the dominated convergence theorem. Therefore the first part of the sum in the assertion is $o(I_N^{-2q-t})$. As for the other part we have
\[
\biggl|\sum_{i > I_N} |\mu_ii^{-q-1/2}|\biggr|^2 \leq \sum_{i >I_N} i^{-2q-1} \sum_{i>I_N}\mu_i^2 \leq I_N^{-t-2q}\sum_{i>I_N}\mu_i^2i^{t},
\]
which completes the proof as $\mu \in S^{t/2}$, and $I_N^{-t-2q}\asymp (\log N)^{-t/2-q}$ by Lemma~\ref{Split}.
\end{proof}
\begin{lemma}\label{Split}
Let $I_N$ be the solution for $1=Ni^{-u}e^{-pi^2}$, for $u \geq 0$ and $p > 0$. Then
\[
I_N \sim \sqrt{\frac{1}{p}\log N}.
\]
\end{lemma}
\begin{proof}
If $u = 0$ the assertion is obvious. Consider $u > 0$. The Lambert function $W$ satisfies the following identity $z = W(z)\exp W(z)$. The equation $1=Ni^{-u}e^{-pi^2}$ can be rewritten as
\[
\frac{2p}{u}N^{2/u} = \exp\Bigl(\frac{2p}{u}i^2\Bigr) \frac{2p}{u}i^2
\]
and therefore by definition of $W(z)$
\[
I_N = \sqrt{\frac{u}{2p} W\Bigl(N^{2/u}\frac{2p}{u}\Bigr)}.
\]
By \cite{Lambert} $W(x) \sim \log(x)$, which completes the proof.
\end{proof}
\begin{lemma}\label{Integrals}
\begin{itemize}
\item[1.] For $\gamma \in \R$, $\zeta > 0$ we have, as $K \to \infty$,
\[
\int_{1}^Ke^{\zeta x^2}x^\gamma\, dx \sim \frac{1}{2\zeta}e^{\zeta K^2}K^{\gamma-1}.
\]
\item[2.] For $K>0$, $\gamma > 0$, $\zeta > 0$ we have
\[
\int_K^\infty e^{-\zeta x^2}x^{-\gamma}\, dx \leq \frac{1}{2\zeta}e^{-\zeta K^2}K^{-\gamma-1}.
\]
\end{itemize}
\end{lemma}
\begin{proof}
First integrating by substitution $y = x^2$ and then by parts proves the lemma, with the help of the dominated convergence theorem in case 1.
\end{proof}
\def\cprime{$'$}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,197 |
This NRA Recruitment Video Is So Divisive, Even Gun Owners Are Angry
"They use their schools to teach children that their president is another Hitler."
06/29/2017 11:13am EDT | Updated July 5, 2017
A new National Rifle Association recruitment ad appears to have outraged both gun control advocates and gun owners alike, even leading critics to launch a petition urging Facebook to delete the "inflammatory" message for "inciting violence."
The minute-long clip, posted on the NRA's FB page earlier this month, features Dana Loesch of TheBlaze who begins: "They use their media to assassinate real news. They use their schools to teach children that their president is another Hitler. They use their movie stars and singers and comedy shows and award shows to repeat their narrative over and over again."
The ad continues with Loesch declaring that "their" former president advocated resistance, leading to protests that "bully and terrorize the law-abiding."
"The only way we stop this, the only way we save our country, and our freedom, is to fight this violence of lies with a clenched fist of truth," the spokeswoman concludes. "I'm the National Rifle Association of America and I'm freedom's safest place."
Many commenters, including some who say they are gun owners, blasted the ad for being "incendiary" and "divisive" while "encouraging violence."
A 50-year-old former Republican from the Midwest replied to the video on Facebook, calling it "Orwellian nonsense designed to make you cheer and fist pump for your 'freedom' like dogs drooling when the bell gets rung."
The commenter, who asked not to be named because he feared for his safety, said he owns firearms for his own protection and for occasional target practice. He told HuffPost that despite the negative reaction to the video in some corners, he suspects it won't damage the NRA.
"I don't think much of anything can actually backfire on them, to be honest," he said. "Much of their core membership seems impervious to logic and reason, sadly."
The NRA did not immediately return a request for comment on the criticism.
U.S. NewsArts and EntertainmentGun Controlnational rifle associationNational Rifle Association of America | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,732 |
using BulletML.Enums;
namespace BulletML.Nodes
{
public class ChangeDirectionNode : BulletMLNode
{
public ChangeDirectionNode() : base(NodeName.changeDirection)
{
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 965 |
Fox Island Lighthouse Association
Legalese Stuff
The Fox Island Lighthouse Association (FILA) is a non-profit organization (501(c)(3)) founded in 2004 by Sandy Bradshaw, John McKinney and H. Joerg Rothenberger. Their goal was to rekindle the efforts of former groups for rescuing the South Fox Island light station, which had been abandoned in 1959 and has been threatened by gradual decay ever since.
The FILA board
John McKinney
Traverse City, MI.
Former Seagrant Agent with the Michigan State University, co-founder of the Grand Traverse Bay Watershed Initiative (now Watershed Center Grand Traverse Bay) and the Maritime Heritage Alliance; expert in just about anything concerning Great Lakes ecology.
FILA co-founder and President.
Catherine F. Allchin
Suttons Bay, Leelanau County, MI
Active with South Fox Light Station since 2002. Yacht broker, member of Maritime Heritage Alliance, Traverse Area Community Sailing (board member for 6 yrs), Recipient of Robert Allor award 2002, Working Mother of the Year 2000. UWGB BA in marketing and graphic design.
FILA Vice President.
Phil von Voigtlander
Northport, Leelanau County, MI.
PhD, retired pharmaceutical research director, avid sailor of the Great Lakes. Former chair, now vice chair of the Watershed Center Grand Traverse Bay. Designer and skipper of the Baykeeper tug boat.
Captain of the Islander. FILA Island Project Manager.
Sandy Bradshaw
Suttons Bay, Leelanau County, MI.
Writer and feature journalist, specializing in the history of the Great Lakes, environmental and maritime matters. Experienced public relations manager. Active in attempts to preserve the South Fox Island Lighthouse since 1991. FILA co-founder, Past Vice President, Webmaster's Assistant.
John Nelson
US Naval Academy graduate, Master of Science in Education (U of M), chairman of the Northern Michigan Environmental Action Council and the Coalition for Sensible Growth, Baykeeper for the Watershed Center Grand Traverse Bay (retired 2016).
Pamela Nickerson
Grawn, Grand Traverse County, MI
Computer software technical analyst/developer. Family owned and logged S. Fox Island in the 1950's and 1960's. FILA Secretary.
It is our sad duty to inform you of Pamela's death on February 16, 2018. Our hearts go out to her family and friends.
Hans Joerg Rothenberger
Suttons Bay, Leelanau County, MI & Tarasp, Switzerland.
Oral surgeon (retired), physicist, electronics and computer buff. FILA Website putter-togetherer. Seasoned skipper, involved in maritime nature conservation and lighthouse preservation since 1988.
Captain of the Lightkeeper. FILA co-founder.
Needless to say that, with the object of our efforts being situated 24 nautical miles off the nearest mainland harbors (Charlevoix and Northport), a seaworthy boat is of utmost importance. The Lightkeeper is an old gal but well maintained, albeit a bit too thirsty.
Occasionally we also use boats provided by volunteers. Transportation offers by skilled skippers of well-equipped boats are always welcome.
Last but not least, the National Park Service usually does a work trip with really heavy materials every summer. Thank you very much folks!
The Lightkeeper is a 29' Cruisers Villa Vee. With her ample loading capacity and twin inboard engines, she is a workhorse but nice to ride. She is owned by two FILA board members. Her home port is Omena Harbor.
The Fox Island Lighthouse Association, Inc. is organized exclusively for scientific, educational and charitable purposes.
The purposes of the Fox Island Lighthouse Association, Inc. include preserving and protecting the historic character and landscape of the South Fox Island Light Station [...]; protecting the natural environment of the light station property; the future use of the light station property for education programs of island and maritime history and culture, arts and the study of the ecology of the island of the Great Lakes for the benefit of its membership and the general public; promoting historic preservation, land protection and stewardship on the South Fox Island.
The Fox Island Lighthouse Association, Inc.'s purpose also includes broader activities related to historic preservation, cultural promotion and preservation, environmental education, natural resources conservation, scholarship, environmental policy and scientific research in and around the Great Lakes.
The Fox Island Lighthouse Association, Inc. seeks active participation in all of these efforts from people of all ages, backgrounds and lifestyles.
Our Mission is ....
".... to preserve and protect the historic South Fox Island Light Station and its natural environment by encouraging community involvement:"
The Inevitable Legalese Fine Print
The Fox Island Lighthouse Association (FILA) aims at including only correct and current information. Although we make every effort to keep this information accurate, we cannot guarantee it. We disclaim any liability for the use of this site or any site it links to.
Neither FILA nor any other party involved in producing or publishing this Web site or any Web site it links to, shall be made accountable in any way whatsoever for any direct, circumstantial, consequential, indirect or punitive damage resulting from your access, use or inability to use this Web site or any Web site it links to.
The FILA Web pages may contain forward-looking information, such as oncoming events. Such information may be subject to various uncertainties. Therefore actual results may differ from the presentation on this site.
We have placed all links to external Web sites in good faith. Although we check most of them periodically, links can become dead when sites are removed or when servers are temporarily down. We try to provide links to sites with useful content, however, we cannot be held responsible for any dissatisfaction on the part of the user, let alone changes made to those Web sites that are not controlled by us.
Unless stated otherwise, the copyright of the texts, photos, maps, logo designs etc. on this site is owned by FILA or the FILA members who contributed them. The FILA Website and the information contained are for informational purposes only. Requests for permission to reproduce any information contained on this Web site have to be sent to .
Please help maintain the quality of this site by . Your comments will always be welcome.
Latest update March 22, 2019 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,857 |
Cities on the Move - Contemporary Asian Art
26. Nov 1997 18. Jan 1998
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Wiener Secession
plan route show map
secession.at +43-1-587 53 07 office@secession.at
artists & participants
Arahmaiani, Nobuyoshi Araki, Duangrit Bunnag, Edge-Michael Chan, Gary Chang, Huang Chin-Ho, Choi Jeong Hwa, Charles Correa, Heri Dono, David D´ Heilly, Simryn Gill, Dominique Gonzalez-Foerster, Cai Guo Qiang, Seung H-Sang, Hanayo, Itsuko Hasegawa, Herzog & de Meuron, Tao Ho, Richard Ho, Oscar Ho, Takashi Homma, Wong Hoy-Cheong, Arata Isozaki, Toyo Ito, Koo Jeong-A, Geng Jianyi, Liang Ju-Hui, Sumet Jumsai, Wong Kar-Wai, Chitti Kasemkitvatana, Tay Kheng Soon, Kiyonori Kikutake, Jinai Kim, Yun-Tae Kim, Kimsooja, Kisho Kurokawa, Takeshi Kitano, Karl-Heinz Klopf, Aglaia Konrad, Rem Koolhaas, Liew Kung Yu, Surasi Kusolwong, Lee Bul, William Lim Associates, Ken Lum, Greg Lynn, Fumihiko Maki, Andar Manik, Fiona Meadows, Sohn-Joo Minn, Rudi Molacek, Mariko Mori, Takashi Murakami, Frederic Nantois, Matthew Ngui, Tsuyoshi Ozawa, Ellen Pau, Harvard Project, Navin Rawanchaikul, Kazuo Sejima, Chen Shaoxiong, Shen Yuan, Shi Yong, Judy Freya Sibayan, Marintan Sirait, Ho Siu Kee, Yutaka Sone, Sarah Sze, Fiona Tan, Aaron Tan, Takahiro Tanaka, Chandraguptha Tenuwara, Liu Thai Ker, Chi Ti-Nan, Rirkrit Tiravanija, Tsang Tsou-Choi, Jun-Jieh Wang, Wang Du, Wang Jianwei, Wong & Ouyang Associates, Xu Tan, Riken Yamamoto, Miwa Yanagi, Ken Yeang, Lin Yi Lin, Yin Xiuzhen, Huang Yong Ping, Yung Ho Chang, Zhan Wang, Zhang Peili, Chen Zhen, Zheng Guogu, Zhou Tiehai, Zhu Jia
"An increasing number of cities are on the move - everything is in a state of perpetual change. Economic, social, political and cultural life develops at breakneck speed. This kind of progress has produced new hybrid forms of modernity. Urban diffusion and density, improvised cities, the mobile city, post-urban city, Glux City, Sim City, Fragmented City and threatening "social decadence" that Itsuko Hasegwa describes critically in the wake of Cardboard Constructions, that pile up in the shade of skyscrapers. Rem Koolhaas states that cities are "non-stable configurations". Post-urban cities are something hybrid and do not concern themselves too much about questions regarding their own identity. This gives rise to "a new aesthetic of the casual contrast of units that have nothing in common apart from their own co-existence". Koolhaas in S, M, L, XL expresses his conviction that if the centre no longer exists, then the suburb does not exist either, and consequently everything becomes city and belongs to the city. He mentions a new pervasiveness that includes landscape, park, industry, rust belt, parking lot, housing tract, single family house, desert, airport, beach, river, sky, slope, even downtown ...
This topic constitutes the theme of the exhibition CITIES ON THE MOVE which Hou Hanru and Hans-Ulrich Obrist have conceived for the Vienna Secession (November 1997), and whose key cities are: Bangkok, Guangzhou, Hanoi, Hong Kong, Jakarta, Kuala Lumpur, Manila, Osaka, Beijing, Seoul, Shanghai, Shen Zhen, Singapore, Tokyo, ..." (Hans-Ulrich Obrist)
The urban explosion in Asia is generating a great number of new Global Cities. These new global cities represent the erection of new economic, cultural and even political powers which are bringing about a new world order and new visions of our planet in the coming century. Apart from classical characteristics of global cities, such as being active elements of the world market and communication, various and multicultural urban culture,"internationalized" modes of life, inter-connectivity, etc. these new, Non-Western global cities also have their own specific characteristics: their own cultural traditions, historical backgrounds, which are mostly connected with the Colonial past and neo-colonial present, and hence new claims for developments. But, the most important is that, with their specific legacies, they become a new and original spaces in which new visions and understandings of Modernity, and new possibilities of "Utopian/dystopian" imaginations, can be elaborated and invented. It is certainly one of the most decisive factors of the global mutation that we are experiencing at the turn of the millenium. Several generations of artists, architects, urban planners, film makers and intellectuals from Asia have been contributing inventively to the formation of such new urban visions. They represent a raising force in the restructuring of our global urban/cultural order. An exhibition which presents such a new force in a Western context today, is not only necessary but also essential since the East and West are approaching each other unprecedentedly in the process of Globalisation. It is also particularly significant to celebrate the Centenary of the Vienna Secession with such an event before touring to several international institutions of contemporary art and architecture. (Hou Hanru)
{postcode} {city}
find exhibitions in | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,428 |
{"url":"https:\/\/www.gamekiller.net\/threads\/vip-keystroke-bot.3286326\/","text":"1. ### NavehVeteran Hacker\n\nPost Count:\n139\n2,597\nStats\nThis is an improved version of: that I am selling.\n\nInfo\nMade a new version to the bot, v2.0. Better UI, UI works on all sizes, Bug fixes, Auto saves events now, auto save script, better performance!\n\nInfo\n25$- Gives you access to the program to 30 days, to 3 different devices. Accepting Paypal, you can purchase the program from my website . Info 100$ - Gives you access to the program to 180 days, to 5 different devices.\n(Added this as requested by active users who want a long term plan.)\n\nWarning\nRun the program as Administrator in order to work!!\n\nInfo\nFor contact if needed, my discord is: Naveh#3118\n\nMapleStory VIP Keystroke Bot is a Windows key sender, auto clicker\nand game data reader that automates actions in any MapleStory version or server.\nIncluding SEA.\n\nAnything you do with a keystroke or mouse click,\nVIP Keystroke Bot can automatically do for you!\n\nMAPLESTORY DATA\n\nWhat so special about this program compare to other macros? Other macros you set your keystrokes and it loops them again and again. The special addition to this program, is our picture detecting system, the program can detect from the game screen your player position, your hp, your mp, and if there other players in the map.\n\nCONTROLLING THE GAME\n\nWith the data you can get you can CONTROL your player. The program as something called events, you can create how many events you want, there 5 types of events (HP, MP, Player Position, Other Players In The Map, Timer). Every event you make, you are telling the program, \"if what im setting here happens, do those keystrokes\". For example: If the player gets to a specific position, go right. Or if the player's hp is below 40%, raise my hp. If my player is below a certain height and his width bigger from X, go up and teleport. This give us endless possibilities of options to set our character movement in the map.\n\nSETTING THE BOT\n\nYou got two ways to set your bot. First, creating your own script, VIP keystroke bot offers the developers an option to script their bot (Basic C#) which can be easier to them to apply their own logic to the bot and their player (API can be seen in the thread). Second, an option to set your bot without the need to know how to code, with the events method.\n\nSAFE\n\nVIP Keystroke Bot is a key sender and auto clicker. It is NOT a bot, even if it is written in the name, it just acts like one. It works independently from the games it is configured to be used with. It does not alter game files, memory or CPU processes. It simply \u201cpresses\u201d keys and mouse buttons... Just like you! Plus, the program can be used on a PC that is remote from the game. The program is guaranteed to be free of third-party addons and applications, viruses and malware, and does not record or log key actions or inject anything to the game.\n\nQuestions people ask me a lot.\n\nQuestion: What is the difference between the free version and VIP version?\nAnswer: First read the thread, it will explain everything. Anyway basic idea, the free version only clicks keystrokes in an order you define. The VIP gets data from the game itself by screenshots of the mini map and hp\\mp bars. With this data, you got the ability to tell the bot WHEN and WHERE you want things to happened. Plus, the VIP has the script option also.\n\nQuestion: I didn't get the Screenshot position setting idea?\nAnswer: You define in the program, where on your screen it can find the mini-map, hp bar and mp bar. So it can read data from it while botting. Which means, if you change your maplestory client location a bit, you will need to re-set the positions of the minimap\\hp\\mp. Usually I just set the position of the pictures on the default place where your client loads\\default place when you make your client not on all screen (Those are the same place)\n\nQuestion: Can I open the program on 3 different computers, on the same time?\n\nQuestion: Should I use this to try to jump on ropes or use teleport\\flash jump only?\nAnswer: Do NOT use this program to try to go on ropes and stuff, its hella annoying to set it and will fail to go on the rope a lot of times! I always bot on places I can teleport\\flash jump from platform to platform, and to go down I go from the sides or use DOWN + ALT. A lot easier. As you can see in the videos.\n\nQuestion: Can I share with someone the bots I've made?\n\nQuestion: I just used my friend Script, to him it works good and to me it doesn't, why?\nAnswer: Its possible to make it work, but its because you both have different player positions on the map. What cause it? Because the position is calculated from the mini-map screenshot, to each computer the mini-map size can be bigger\\smaller\\different resolution. So if your friend has different screensize\\different resolution of screen\\different resolution of game\\maybe he just set the mini-map a bit to the side from you, so it can effect. Or find the correct way to get the same size as your friend, or just edit the positions in the script by yourself (Not a big deal).\n\nQuestion: Is there auto rune?\nAnswer: No, if the game updates and they change something in the rune system and you bot afterwards, you can get banned. And I don't want to take that responsibility. (Plus I didn't manage to make one anyway).\n\nQuestion: How the Right\\Left\\Up\\Down works? I don't see spam time?\nAnswer: When you set it to go to a direction, for exmaple left, it won't stop walking left, until you change its direction to a different one, OR until you call STOP MOVING special key.\n\nQuestion: There refunds?\nAnswer: No sorry, if you have a technical problem in the program, talk to me.\n\nQuestion: Can you purchase for example for 6 months at once? Not only 1 month?\nAnswer: I don't want to create those options. Try for a month, if you would like to buy again, and for more than 1 month, talk to me we will think something out.\n\nVIP Keystroke Bot Pages\n\nSaved Bots (Main Page)\nHere you can find all the bots you saved.\n\nRegular Bot (Non-Script Method)\n- Client Details Tab: Writing the bot data to detect it and to save it.\n\n- Picture Detect Settings: You set where on your screen are your Mini-Map, HP bar and MP bar. So the program will know where it is to get data while botting.\n\n- Keys Tab: Where you can set keys, that will work all the time without an event running. (I prefer doing all my keys in events, depends what I'm trying to do).\n\n- Events Tab: Where all the events you create are saved and edited.\n\n- Run Bot Tab: Where you run\/stop your bot and you can see all the data while running it.\n\n- \"Live Data\" Button:\nUsually while creating events, you want to see the player position all the time live, without the character actually botting. For this you have the live data button which will create another window that shows data from the picture detect settings calculations.\n\nScript Bot Page\n- Client Details Tab: Writing the bot data to detect it and to save it.\n\n- Picture Detect Settings: You set where on your screen are your Mini-Map, HP bar and MP bar. So the program will know where it is to get data while botting.\n\n- Script Tab: Here you write your code, the Main() methods already loops everything you put insides, so your code will act as a bot.\n\nYou can also use it on all screen, the UI works on all sizes.\n\n- \"Script API\" Button: Here you can find all of the methods and vars you can use on the bot.\n\n- Timer Tab: You can add timer events, each timer events get called every X time you set. Which is good for skills execute for exmaple.\n\n- Run Bot Tab: Where you run\/stop your bot and you can see all the data while running it.\n\n- \"Live Data\" Button:\nUsually while creating events, you want to see the player position all the time live, without the character actually botting. For this you have the live data button which will create another window that shows data from the picture detect settings calculations.\n\nVIP Keystroke Bot Example (Script Method)\n\nAs you can see in the video, I've created my own script to bot in the map, the script language is C#. Knowing basic C# is enough in my opinion, mostly using ifs and stuff.\n\nHere is the script I did in the video:\n\nCode:\npublic void Main()\n{\nPotionsManage();\nFirstPlat();\nSecondPlat();\nThirdPlat();\n}\n\n\/\/Dealing with HP and MP (I only needed 1 pot).\nprivate void PotionsManage() {\nif (MyHp < 50)\ncm.PressKey(\"DELETE\", 0);\nif (MyMp < 70)\ncm.PressKey(\"END\" , 0);\n}\n\n\/\/Just a shortcut to do attack and teleport in one call\nprivate void AttackTeleCombo()\n{\ncm.PressKey(\"CONTROL\", 0);\ncm.PressKey(\"KEY_X\", 0);\ncm.Delay(0.2);\n}\n\n\/\/What players do in first platform\nprivate void FirstPlat() {\nif (MyPosition.Y < 30) {\nif (MyPosition.X < 100) {\ncm.GoRight();\nAttackTeleCombo();\n}\nelse if (MyPosition.X >= 100 && MyPosition.X < 120) {\ncm.GoUp();\ncm.PressKey(\"KEY_X\", 0);\n}\nelse {\ncm.GoLeft();\nAttackTeleCombo();\n}\n}\n}\n\n\/\/What player do in second plat\nprivate void SecondPlat() {\nif (MyPosition.Y > 30 && MyPosition.Y < 50) {\nif (MyPosition.X > 90) {\ncm.GoLeft();\nAttackTeleCombo();\n}\nelse if (MyPosition.X >= 73 && MyPosition.X < 90) {\ncm.GoUp();\ncm.PressKey(\"KEY_X\", 0);\n}\nelse {\ncm.GoRight();\nAttackTeleCombo();\n}\n}\n}\n\n\/\/What player do in third platform\nprivate void ThirdPlat() {\nif (MyPosition.Y > 50) {\ncm.GoRight();\nAttackTeleCombo();\n}\n}\n\n\nVIP Keystroke Bot Example Explained and Guidance(Non-Script Method)\n\n(Suggesting to read first the SET BOT OPTION AND MORE IMPORTANT DATA spoiler first).\n\nIn the end, this program is all basic logic and a little math. In this example I'm going to show you the right way of setting the map I want and my way of thinking.\n\nFirst thing, let me show you what we are trying to achieve here, we are a mage, cleric type in Ghost Ship 6, we want to use our heal, teleport to attack monsters and we want to go up and down the platforms without stopping. As can be seen in this video:\n\nLets get started!!\n\nName: Lets name our bot something we can recognize later, like \"Ghost Ship 6 Mage\".\n\nName of Client: How can we find our client's name? When your game is minimized, you can see the name of the program on top, that is the name of your client.\n\nMaple Version: The current version you are playing at. (Important so it will recognize the right UI of the version as it changes around v92 and again in v170 I think(?), something around).\n\nSave: Save! We want to save our program so we will be able to enter events. Now, when saving without keys it adds a Delay 0 key, which basically when you bot your character does nothing.\n\nPicture detect settings: Set your settings of your map only for now. (Explained in the spoiler I mentioned at the start). I'm repeating again what I wrote there, those settings save the location of your screen!! Not the game!! If you move your client you will need to set those settings again! My suggestion is to always set them on the default place your client is minimized after full screen. So you won't need to set it again and again all the time.\n\nEvents: Now we got into the important part of this guide. Events are made so the program will do what you want. We are going to focus on Player Position events, which are the most important event. You're basically telling the program, I want you to do Z when the player is between this and this location without stopping. We are going to create 5 events, I think you can understand what each one do by its name atm. In the end, we want the program to know what to do in every spot in the map, so the player won't get stuck, it will do what we want and it will make the character loop its action in the map.\n\nPosition Event Types: Lets explain some basic stuff first before we start creating events.\n\nYou can create event based only on width, event based only on height, and events based on both. We are going to create events on both platforms, so we need both width and height to check each event so there will not be places mixed with more than 1 event, so the program won't get confused.\n\nBefore creating the events, we need to think where are the places we want to create each event and based on that, what Position points we need in order to create those events?\n\nHere are the places I think we want to create our position events. I will explain my thinking for each place:\n\n1 \u2013 In this place, we want to go left, teleport and use heal as attack.\n\n2 \u2013 We need to get somewhere to tell the program to get down, in this spot we will do movement down, and click alt so the player will go down.\n\n3 \u2013 In this place, we want to go right, teleport and use heal as attack.\n\n4 \u2013 We can use the stairs to go up and use teleport as a mage. Here we will do movement up and use teleport until we are up.\n\n5 \u2013 But Naveh, what happens if the player passes the stairs? Will the bot get stuck? NO, because we can define all the time where we want the player to go in a specific unset spot. If the player passes the stairs, we will just make the player go left, so it will return to the stairs event.\n\nNow that we understand what places we want to set our position events, we need to get the positions to set those events. Here are the positions we need, and I'll explain why (Each red dot we need to get width and height):\n\n1 \u2013 We need the most right top point to know where the map ends.\n\n2 \u2013 We need the point where the player can go down from here to the bottom platform.\n\n3 \u2013 We need the most left top point to know where the map ends.\n\n4 \u2013 We need the most left bottom point to know where the map ends.\n\n5 \u2013 We need to know where we want to stop attacking and teleporting, to go up the stairs with teleport. Plus, this location is where the player CAN teleport on the stairs.\n\n6 \u2013 We need to know where the player cannot teleport anymore in the stairs, because he passed them. So, if he pass this point we want to go back, so the player will go left.\n\n7 \u2013 We need the most right bottom point to know where the map ends.\n\nNow that we got everything we need and know what we want to do, lets create the position events, lets start:\n\nGo left (TOP):\n\nThis event is like number 1 we explained. Always name the event with a name you will understand exactly what it does. We want all our events to be activated based on width and height. We want to set the width between point 1 to 2. Now when the player stand on the top platform, his height is 21! So we want to activate this event only when his height is bigger from 20.\n\nKey left goes left (goes left without stopping until we change direction or do stop moving option), key delete does heal, key_x does teleport, and most importantly, we have a delay of 0.4. Why delay is so important? The program clicks keys REALLY FAST, we want to let the program chill between keys sometimes or before some keys, for example, if we click 3 skills fast will all of them register?\n\nNo. We need some time before or between each skill so it will register. Here the delay is 0.4 second, which lets the program wait 0.4 seconds before doing teleport and heal again.\n\nGo down (Left Side):\n\nThis event is like number 2 we explained. We want to get the player down here. Lets set the width between point 2 and 3, and height bigger from 20. (If you notice another option, good job! You can also do if player width is smaller from 20 (which is point 2 width), go down!\n\nKey down makes the player go down, key LMENU is the left alt (jump), which will cause it to jump down.\n\nGo right (BOTTOM):\n\nWe want to go right, attack, teleport until we reach the stairs. Between points 4 and 5. So the width we want is between 8 and 89 and height smaller from 21 as it is the lower platform.\n\nKeys are like the left side, just that he goes right this time.\n\nGo up (Right side, stairs):\n\nSo, we want our player to go up the stairs using teleport. Which is between points 5 and 6, in this point our player can teleport up the stairs. Our width is between 90 to 98 and we want the player to keep trying to go up as long he is not standing on the top platform, so if his height is smaller from 21(The position where the player is standing on the top platform) it will keep trying to go up.\n\nKeys are up, and teleport key.\n\nBehind stairs (Stuck):\n\nWhat happens if the player passes the stairs? Which are points 6 to 7, is our bot doomed? No, we just tell the player to go left so it will return to the stairs and try to go up again. So, I did if he passed point 6, which is width 98, if the player width is bigger from 98 it will trigger. And of course, if his width is smaller from 21.\n\nKeys are left only.\n\nAnd we are done! We have a working bot as we wanted! In this explanation I showed you how the bot works and the logic behind it. Now you're ready to make your own different maps and use your own imagination.\n\n(Written on the older UI v1 bot, but still the points are imporant)\nThis bot send keystrokes, like you're clicking yours. It gets information like HP\\MP\\Player Location\\Other players are in the map, from taking screenshots of your game.\n\nYou can do whatever you want ingame with your logic and imagination.\n\nThis bot works on all versions(Maple Global and private servers), its sends keystrokes to your focused MapleStory Client. (Yes client has to be focused, only like that the character will be able to move).\n\nThe main botting used on the Keys Order table which contains the keystrokes you set, which loops the list again and again until the botting is stopped. If you entered events, if the event occurs, it will stop the key order and do the event's keys, after it finished it will continue the main keys.\n\nThere 5 types of events you can create as you can see in the picture.\n\nHp: You can do for exmaple when you have 30% HP, it will use keystroke DELETE where your HP is.\n\nMP: You can do for exmaple when you have 40% MP, it will use 2 keystrokes INSERT keys where your blue potions are.\n\nPlayer Position:\nThis one I create a lot of those, with this you can pretty much go wild with your logic and imagination and make the player move where ever you want.\n\nPlayer In Map: This event happens when there other detected players in the map. I for exmaple did when someone in my map, it will log off the account and stop botting. Do what ever you want or what you can, change channel? speak something to the player? Only stop botting? go to a specific place in the map?\n\nTimer: This event, unlike the others doesn't request data from your gameplay screenshot, with this you can do actions every X time. For exmaple, do skills every X time, do HS auto every X time ;), use pet food, go wild.\n\nIn this page you can view all of the bots you saved, you can save different bots of different client, different versions, everything is saved!\n\nThere some extra functions added to keys that are not keys.\n\n- STOP BOTTING: Its like you clicked stop botting, it stops all botting and events, stops everything! I use it for example when there is someone else in my map.\n- DELAY: delay is made like it sounds, to do delay, to make your character click nothing for X time (Except walking), you did an action and you want your character to wait 3 seconds before his next action, you add DELAY for 3 seconds. I use it for exmaple on timer event to do my skills, so the player will buff all skills before clicking a different key.\n- STOP MOVING: When you set your character to move right\\left\\up\\down. IT WONT STOP UNTIL STOP MOVING is called, or you told the character to move to a different location.\n\n- Your program has to be FOCUSED so the character will be able to move and that the program will get data from your game screen. Means you can use only one client.\n- The SPAM time, spams the keystrokes like clicking again and again fast. There is no option for long click. So if you have a skill that require LONG CLICK, you won't be able to use it right (For exmaple phantom's main skill). Don't know if your skill will work good? Try it first on the free version linked in the start of the thread before deciding to buy the VIP version!\n- The HP\\MP is not 100% precise! it based on pictures of your screen, it will show your HP percent (1-100%). But even it not precise, it does the work needed.\n- If you use events, you should do it on not full screen.\n-Changing the position of your client on screen will result in you need to reSET your event picture settings as it saved based on location. My suggestion to use set the pictures when the client on the default mini client position (On the side)\n\nIn order to get data of HP and such, you need to set specific factors POSITION in your screen.\n\nhow to do that? With the Picutre Detect Settings\n\nAfter you choose to set HP\\MP\\MAP, you get to this screen\n\nHere you define the position of each required data, for exmaple if its a MAP, you click Choose Image, after that your screen will become light yellow color, you drag click the position of your game minimap!, and it pick the dragged image. (Works like Windows Snipping Tool!).\nExample of the needed image of MAP:\n\nBe the most precise you can! based on the yellow character dot it gets your character position!\nAfter you drag click, this will pop up with the picture:\n\nYou MAP is set!\nNow Important:\n- IF YOU MOVE YOUR CLIENT POISITION, IT WILL RUIN THE SETTING YOU DID BECAUSE IT WILL NOT MANAGE TO DETECT THE MAP AS IT MOVED. You will need to reSET the position of the mini map. What I do? I always set them on the default position of the mini client, so I won't need to set it all the time.\n- If you can't see all of the map in the minimap, the position of your character won't work well.\n\nHP EXMAPLE: Do like the map, but you need to set the image location of the HP like this:\n\nso it will recognize the % of your HP.\n\nMP Exmaple:\n\nAfter every type you set, it will notify you there is a saved location to the type of event.\n\nWarning\n\n- Your program has to be FOCUSED so the character will be able to move and that the program will get data from your game screen. Means you can use only one client.\n- The SPAM time, spams the keystrokes like clicking again and again fast. There is no option for long click. So if you have a skill that require LONG CLICK, you won't be able to use it right (For exmaple phantom's main skill). Don't know if your skill will work good? Try it first on the free version linked in the start of the thread before deciding to buy the VIP version!\n- The HP\\MP is not 100% precise! it based on pictures of your screen, it will show your HP percent (1-100%). But even it not precise, it does the work needed.\n- If you use events, you should do it on not full screen.\n-Changing the position of your client on screen will result in you need to reSET your event picture settings as it saved based on location. My suggestion to use set the pictures when the client on the default mini client position (On the side)\n\nNew Scripting feature video exmaples:\n\nv214:\n\nv83:\n\nv178 set bot exmaple:\n\nv83 set bot exmaple:\n\nLast edited: Oct 6, 2020\n\nPost Count:\n1\n0\nStats\n\n3. ### NavehVeteran Hacker\n\nPost Count:\n139\n2,597\nStats\n\nMade an example of the program.\n\n4. ### EssencesThe New Guy\n\nPost Count:\n13\n1\nStats\nThis won\u2019t bypass the lie detector in KastiaMS though right?\n\n5. ### NavehVeteran Hacker\n\nPost Count:\n139\n2,597\nStats\nNope. Just stop the bot and do the lie detector. \u00af\\_(\u30c4)_\/\u00af\n\nPost Count:\n139\n2,597\nStats\n\n7. ### Lmaokiller11Lurker\n\nPost Count:\n1\n1\nStats\n+rep naveh the bot ran really smooth and works really well in v83. Also very helpful he answer all my question and make sure the bot was was running smooth on my end.\n\nNaveh likes this.\n8. ### JumbleMumbleThe New Guy\n\nPost Count:\n10\n2\nStats\n+1 Very helpful and patient especially with newbies. All functions work as specified\n\nNaveh likes this.\n\nPost Count:\n1\n0\nStats\n\nPost Count:\n139\n2,597\nStats\n\nPost Count:\n8\n0\nStats\n+1\n\n12. ### NavehVeteran Hacker\n\nPost Count:\n139\n2,597\nStats\nWhat you mean? This is a keystroke bot you define yourself, and keys that happened on specific events on the same time.\n\n13. ### wkwkwkwkLurker\n\nPost Count:\n1\n1\nStats\nBot works great for me, flawless and Naveh was really patient in helping me fix bugs and setting the bot up. 100% legit\n\nNaveh likes this.\n\nPost Count:\n1\n1\nStats\nKeystroke Bot smooth, take me quite some time to understand how to navigate around the Bot. Naveh was patient and helping while attending to my issues.\n\nNaveh likes this.\n15. ### monobitoLurker\n\nPost Count:\n1\n0\nStats\nIs this a one-time purchase or subscription?\n\n16. ### NavehVeteran Hacker\n\nPost Count:\n139\n2,597\nStats\nSubscription as written in the thread.\n\nPost Count:\n8\n1\nStats\n\n18. ### GrassballThe New Guy\n\nPost Count:\n35\n0\nStats\ncan this ban you? Thinking about using this to farm one my reboot character.\n\nPost Count:\n230\n90\nStats\nIf you're unlucky and get caught by a GM, you might get banned due to repetitive movements and if you don't respond to whispers or anything like that.\n\nPost Count:\n139","date":"2020-10-23 23:47:13","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\": 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\": 0.17996661365032196, \"perplexity\": 2110.972253057823}, \"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-2020-45\/segments\/1603107881551.11\/warc\/CC-MAIN-20201023234043-20201024024043-00148.warc.gz\"}"} | null | null |
{"url":"https:\/\/doc.cgal.org\/5.4-beta1\/Surface_mesher\/classImplicitSurfaceTraits__3.html","text":"CGAL 5.4 - 3D Surface Mesh Generation\nImplicitSurfaceTraits_3 Concept Reference\n\n## Definition\n\nThe concept ImplicitSurfaceTraits_3 describes the requirements of the traits class to be plugged as Traits in CGAL::Implicit_surface_3<Traits, Function>.\n\nWhen make_surface_mesh is called with a surface of type CGAL::Implicit_surface_3<Traits,Function>, the surface mesher traits generator generates automatically a traits class that is a model of SurfaceMeshTraits_3. Actually, the concept ImplicitSurfaceTraits_3 provides the types, predicates and constructors that are passed to the generated model of SurfaceMeshTraits_3.\n\nHas Models:\nAny CGAL Kernel.\nCGAL::Implicit_surface_3<Traits, Function>\nCGAL::make_surface_mesh()\n\n## Types\n\ntypedef unspecified_type\u00a0FT\nThe numerical type. More...\n\ntypedef unspecified_type\u00a0Point_3\nThe point type. More...\n\ntypedef unspecified_type\u00a0Line_3\nThe line type.\n\ntypedef unspecified_type\u00a0Ray_3\nThe ray type.\n\ntypedef unspecified_type\u00a0Segment_3\nThe segment type.\n\ntypedef unspecified_type\u00a0Vector_3\nThe vector type.\n\ntypedef unspecified_type\u00a0Sphere_3\nThe sphere type.\n\ntypedef unspecified_type\u00a0Compute_scalar_product_3\nA function object providing the operator. More...\n\ntypedef unspecified_type\u00a0Compute_squared_distance_3\nA function object providing the operator. More...\n\nA function object providing the operator. More...\n\ntypedef unspecified_type\u00a0Construct_center_3\nA function object providing the operator. More...\n\ntypedef unspecified_type\u00a0Construct_midpoint_3\nA function object providing the operator. More...\n\ntypedef unspecified_type\u00a0Construct_point_on_3\nA function object providing the following operators: More...\n\ntypedef unspecified_type\u00a0Construct_segment_3\nA function object providing the operator. More...\n\ntypedef unspecified_type\u00a0Construct_scaled_vector_3\nA function object providing the operator. More...\n\ntypedef unspecified_type\u00a0Construct_translated_point_3\nA function object providing the operator. More...\n\ntypedef unspecified_type\u00a0Construct_vector_3\nA function object providing the operator. More...\n\ntypedef unspecified_type\u00a0Has_on_bounded_side_3\nA function object providing the operator. More...\n\n## Operations\n\nCompute_scalar_product_3\u00a0compute_scalar_product_3_object ()\n\nCompute_squared_distance_3\u00a0compute_squared_distance_3_object ()\n\nConstruct_center_3\u00a0construct_center_3_object ()\n\nConstruct_midpoint_3\u00a0construct_midpoint_3_object ()\n\nConstruct_point_on_3\u00a0construct_point_on_3_object ()\n\nConstruct_scaled_vector_3\u00a0construct_scaled_vector_3_object ()\n\nConstruct_segment_3\u00a0construct_segment_3_object ()\n\nConstruct_translated_point_3\u00a0construct_translated_point_3_object ()\n\nConstruct_vector_3\u00a0construct_vector_3_object ()\n\nHas_on_bounded_side_3\u00a0has_on_bounded_side_3_object ()\n\n## \u25c6\u00a0Compute_scalar_product_3\n\nA function object providing the operator.\n\nFT operator()(Vector_3 v, Vector_3 w) which returns the scalar (inner) product of the two vectors v and w.\n\n## \u25c6\u00a0Compute_squared_distance_3\n\nA function object providing the operator.\n\nFT operator()(Point_3, Point_3) which returns the squared distance between two points.\n\nA function object providing the operator.\n\nFT operator()(const Sphere_3& s) which returns the squared radius of s.\n\n## \u25c6\u00a0Construct_center_3\n\nA function object providing the operator.\n\nPoint_3 operator()(const Sphere_3& s) which computes the center of the sphere s.\n\n## \u25c6\u00a0Construct_midpoint_3\n\nA function object providing the operator.\n\nPoint_3 operator()(const Point_3& p, const Point_3& q) which computes the midpoint of the segment pq.\n\n## \u25c6\u00a0Construct_point_on_3\n\nA function object providing the following operators:\n\nPoint_3 operator()(const Line_3& l,int i) which returns an arbitrary point on l. It holds point(i) == point(j), iff i==j. Furthermore, is directed from point(i) to point(j), for all i $$<$$ j.\n\nPoint_3 operator()(const Ray_3& r,int i) which returns a point on r. point(0) is the source, point(i), with $$i>0$$, is different from the source.\n\nPrecondition\n$$i \\geq0$$.\n\nPoint_3 operator()(const Segment_3& s,int i) which returns source or target of s: point(0) returns the source of s, point(1) returns the target of s. The parameter i is taken modulo 2, which gives easy access to the other end point.\n\n## \u25c6\u00a0Construct_scaled_vector_3\n\nA function object providing the operator.\n\nVector_3 operator()(const Vector_3 &v, const FT& scale) which returns the vector v scaled by a factor scale.\n\n## \u25c6\u00a0Construct_segment_3\n\nA function object providing the operator.\n\nSegment_3 operator()(const Point_3 &p, const Point_3 &q) which returns a segment with source p and target q. It is directed from the source towards the target.\n\n## \u25c6\u00a0Construct_translated_point_3\n\nA function object providing the operator.\n\nPoint_3 operator()(const Point_3& p, const Vector_3& v) which returns the point obtained by translating p by the vector v.\n\n## \u25c6\u00a0Construct_vector_3\n\nA function object providing the operator.\n\nVector_3 operator()(const Point_3 &a, const Point_3 &b) which returns the vector b-a.\n\n## \u25c6\u00a0FT\n\nThe numerical type.\n\nIt must be model of FieldWithSqrt and constructible from a double.\n\n## \u25c6\u00a0Has_on_bounded_side_3\n\nA function object providing the operator.\n\nbool operator()(const Sphere_3&s, const Point_3&p) which returns true iff p lies on the bounded side of s.\n\n## \u25c6\u00a0Point_3\n\nThe point type.\n\nThis point type must have a constructor Point_3(FT, FT, FT).","date":"2023-01-29 16:40:42","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\": 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\": 0.193635493516922, \"perplexity\": 12241.4320000388}, \"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-2023-06\/segments\/1674764499744.74\/warc\/CC-MAIN-20230129144110-20230129174110-00858.warc.gz\"}"} | null | null |
La Cruz is a village in the Florida Department of Uruguay.
Geography
It is located in the centre of Florida Department, on the east side of Ruta 5, northwest of Florida, in an area known as Santa Teresa.
History
On 23 November 1929, the status of the group of houses here was elevated to "Pueblo" (village) by the Act of Ley Nº 8.497.
Population
In 2011 La Cruz had a population of 747.
Source: Instituto Nacional de Estadística de Uruguay
Places of worship
Exaltation of the Cross Chapel (Roman Catholic)
References
External links
INE map of La Cruz
Populated places in the Florida Department | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,332 |
{"url":"https:\/\/tex.stackexchange.com\/questions\/578252\/how-to-draw-grid-lines-with-contour-gnuplot","text":"# How to draw grid lines with contour gnuplot?\n\nThis is a follow up question of Tick labels not showing when using contour gnuplot and axis line = middle. The difference is that I added the grid lines in the code. However it is not drawing it.\n\n\\documentclass[tikz,border=3.14mm]{standalone}\n\\usepackage{pgfplots}\n\\pgfplotsset{width=7cm,compat=1.16}\n\\begin{document}\n\\begin{tikzpicture} \\begin{axis}[\naxis lines = middle,\ntitle={$x^2-x\\,y$},\nenlarge x limits,\nview={0}{90},\nxlabel=$x$, ylabel=$y$,\nsmall,\n% grid\ngrid = both,\ngrid style = {line width = .1pt, draw = gray!10},\nmajor grid style = {line width = .2pt, draw = gray!50},\nticks = both,\nminor tick num = 4,\n]\ndomain y=-3:3,\ncontour gnuplot={levels={-1,1},labels=false},\nthick,samples=50,samples y=50,\n] {x^2-x*y};\n\\end{axis}\n\\end{tikzpicture}\n\\end{document}\n\n\nI tried to move the order of the lines, but it doesn't solve it. What can I do to draw the grid lines?\n\nHow about a plain Asymptote solution? PS: I am sure there is a pgfplots way if you look at its 571-page documentation carefully enough.\n\n\\documentclass[border=5mm]{standalone}\n\\usepackage{asymptote}\n\\begin{document}\n\\begin{asy}\nimport math; \/\/ for grids\nimport contour;\nunitsize(1cm);\n\/\/ grid and subgrid\n\n\/\/ axes, dashed line and labels\ndraw(Label(\"$x$\",EndPoint,align=SW,Fill(white)),(-3,0)--(3,0));\ndraw(Label(\"$y$\",EndPoint,align=SE,Fill(white)),(0,-3)--(0,3));\ndraw((1,0)--(1,2)--(0,2)^^(-1,0)--(-1,-2)--(0,-2),dashed);\nlabel(\"$1$\",(1,0),SE); label(\"$-1$\",(-1,0),NW);\nlabel(\"$2$\",(0,2),W); label(\"$-2$\",(0,-2),NW);\n\n\/\/ plotting graph of implicit function\nreal f(real x, real y){return x^2-x*y;}\npair A=(-3,-3), B=(3,3);\nreal[] c1={1}, c2={-1}, c={0};\ndraw(contour(f,A,B,c,300),purple); \/\/ 2 asymptote straight lines\ndraw(contour(f,A,B,c1),blue);\ndraw(contour(f,A,B,c2),red);\nlabel(\"The graph of $x^2-xy=C$\",truepoint(S)+(0,-.5));\n\n\\end{asy}\n\\end{document}\n\n\nUpdate For \"auto ticks\", I add the grid and subgrid using Step=1, step=.2 in LeftTicks, RightTicks of the xaxis and yaxis command. The module graph must be loaded. Compiling time seems a bit slower.\n\n\/\/ http:\/\/asymptote.ualberta.ca\/\nunitsize(1cm);\nimport graph;\nimport contour;\n\nreal f(real x, real y){return x^2-x*y;}\npair A=(-3,-3), B=(3,3);\nreal[] c1={1}, c2={-1};\ndraw(A--B^^(0,A.y)--(0,B.y),purple); \/\/ 2 asymptote straight lines\ndraw(contour(f,A,B,c1),cyan+linewidth(1pt));\ndraw(contour(f,A,B,c2),magenta+linewidth(1pt));\n\npen thin=gray+linewidth(.2pt);\npen verythin=lightgray+linewidth(.2pt);\nxaxis(\"$x$\",BottomTop,LeftTicks(begin=false,end=false,Step=1,step=.2,extend=true, ptick=verythin,pTick=thin));\nyaxis(\"$y$\",LeftRight,RightTicks(begin=false,end=false,Step=1,step=.2,extend=true,ptick=verythin,pTick=thin));\n\nlabel(\"The graph of $x^2-xy=C$\",truepoint(N)+(0,.5));\n\n\u2022 do you need to make the tick labels manually? Jan 10 at 19:50\n\u2022 @FacebFaceb I like to make tick manually, to get exactly what I like to tick and label. Of course, auto ticks and labels can be made using import graph; see here asymptote.sourceforge.io\/FAQ\/section6.html if you are interested. (I though you are still waiting a pgfplots solution) Jan 10 at 19:58\n\u2022 yes I am waiting for a solution with contour gnuplot and pgfplots, but I was just curious about this package Jan 10 at 20:21\n\u2022 @FacebFaceb this may be helpful for your pgfplots figure (grids) latexdraw.com\/linear-regression-in-latex-using-tikz Jan 20 at 15:36","date":"2021-10-17 08:51:58","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\": 1, \"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\": 0.5629366040229797, \"perplexity\": 7995.477879450014}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"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-2021-43\/segments\/1634323585171.16\/warc\/CC-MAIN-20211017082600-20211017112600-00186.warc.gz\"}"} | null | null |
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THE ART OF
VINYASA
_Awakening Body and Mind through the Practice of Ashtanga Yoga_
RICHARD FREEMAN & MARY TAYLOR
SHAMBHALA
BOULDER
2016
Shambhala Publications, Inc.
4720 Walnut Street
Boulder, Colorado 80301
www.shambhala.com
Cover photos by Robert Muratore
Cover design by Jim Zaccaria
© 2016 by Richard Freeman and Mary Taylor
All rights reserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.
LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
Names: Freeman, Richard, 1950– author. | Taylor, Mary (Yoga teacher)
Title: The art of vinyasa: awakening body and mind through the practice of ashtanga yoga / Richard Freeman and Mary Taylor.
Description: Boulder: Shambhala, 2016. | Includes index.
Identifiers: LCCN 2016002192 | eISBN 9780834840409 | ISBN 9781611802795 (paperback)
Subjects: LSCH: Aṣṭāṅga yoga. | BISAC: HEALTH & FITNESS / Yoga. | PHILOSOPHY / Hindu.
Classification: LCC RA781.68 .F74 2016 | DDC 613.7/046—dc23
LC record available at <https://lccn.loc.gov/2016002192>
_To Sri K. Pattabhi Jois and his wife, Ammaji._
_And with deep gratitude to Sarasvathi, Manju, and Sharath,_
_who continue to inspire us all in this practice._
This book is designed to shed light on establishing an internally rooted yoga practice that can last a lifetime. It also looks deeply at āsana practice as an external expression of the internal forms that give a profound and direct experience to the awakening of body and mind through the practice.
CONTENTS
Publisher's Note
Introduction
PART ONE
_Foundation: The Roots and Depth of Yoga_
1. Natural Alignment: The Internal Forms of the Practice
2. Aligning Intention and Action: Where the Rubber Meets the Road
3. Fluid Movement: Alignment, Form, and Imagination
4. Mechanics: Essential Anatomical Perspectives
PART TWO
_Āsana: Movements and Poses Strung Together Like Jewels on the Thread of the Breath_
5. Building Sūrya Namaskāra
6. Standing Poses
7. Forward Bends
8. Backbends
9. Twists
10. Balancing Poses
11. Finishing Poses
Acknowledgments
Appendix 1. Ancient Wisdom, Contemporary Circumstances
Appendix 2. Invocation
Appendix 3. Sequencing
Appendix 4. Illustrations of Mūlabandha, Kidney Wings, and Cobra Hood
Appendix 5. Sūrya Namaskāra A and B
Index
About the Authors
E-mail Sign-Up
PUBLISHER'S NOTE
This book contains diacritics and special characters. If you encounter difficulty displaying these characters, please set your e-reader device to publisher defaults (if available) or to an alternate font.
INTRODUCTION
YOGA IS A LIVING ART. IT IS A MEANS OF MOVING, breathing, thinking, expanding and contracting, evolving and interacting within the complex, ever-changing landscape of the world within and around us. As with any art form, yoga nurtures seeds of aesthetic satisfaction that stimulate flashes of understanding and compassion. For many practitioners, a keen truth and meaning spontaneously arise as insight into the vast, interconnected nature of all things.
When embodied, these aesthetic sparks and seeds of insight are experienced as feelings of resonating with our surroundings. They occur in yoga when we're not looking for them—just as they may when we're standing in front of a great work of art or enjoying the perfect sunset. Somehow (possibly by chance) our perception of self is released just long enough for us to feel intimately connected to everything and everyone else, and the underlying field of kind, openheartedness that is our true nature naturally arises. Clarity or conscious awareness is the fallout—the residue—from practicing yoga in this way, as an art rather than as a means of attaining this thing or that.
This approach to practice requires a willingness to invite and _be with_ not knowing. It encourages us to show up ready and eager to meet whatever arises. Perhaps most important, it demands the mental and emotional agility to be comfortable with the paradox of simultaneously holding two or more points of view with equal attentiveness.
We learn to be focused and disciplined while letting go, surrendering the ego while steadying the mind, and all the while remaining tuned in to the complexity of whatever is arising. By cultivating open-minded states of inquisitiveness and acceptance in the controlled structure of a practice, encountering paradoxes and the unknown begins to feel safe, interesting, and exciting rather than tedious or frightening. Gradually habitual patterns of behavior and thinking dissipate as preconceptions dissolve and freedom unfolds.
This isn't what most of us sign up for when we walk into our first yoga class, but at some point in our practice, it happens: that seed within is awakened. As if by accident, the art in the form is revealed, along with a glimpse of what it is like to be satisfied and fully awake. Of course this arising of free or conscious mind, like so many mysterious and important things in life, is illusive. The moment we recognize how wonderful it is, how we feel liberated and free as if swimming effortlessly in an ocean of equanimity, the mind grabs hold of the idea of feeling good and tries to package it, make a formula that will ensure we can hold on to it, or concoct a plan to duplicate (and possibly profit from) the state. And already it has vanished.
With time—possibly many years—we may realize that the awakening of this seed of reality is not something we can create but something we invite; we can simply do the work required to arrange things so that perhaps it arises again. The "work" we must do is to practice with dedication, consistency, and an open mind. We practice as a means of waking up to and seeing clearly the process of whatever is presented. At the same time we constantly remind ourselves to let go of our expectations about what we may attain or acquire through our efforts. The work is never done. It is as if, again and again, we prepare a meal and set the table for honored guests who may or may not arrive. Whether they do or not, the next day we set the table again with unyielding enthusiasm. When yoga is approached in this way, as an art form and an offering containing an endless mixture of complementary, interwoven oppositions, then the practice itself is fully satisfying. Any residual benefits we may feel or insights we may happen upon as a result of the practice are icing on the cake.
Yoga practice can take the form of _āsanas_ (poses), _prāṇāyāma_ (breathing), meditation, chanting, or philosophical inquiry. Each of these methods grows from an understanding and a trust in relationship. Everything that comes up in practice is a reflection of the interaction of oppositions, interactions, and contexts both within and without. To adapt, respond, and find balance in this web of relationship is the key to practicing yoga in a contemplative manner. The practice of _vinyāsa_ reframes our perception of any particular thing by equalizing the relational background in which it exists.
The Sanskrit word _vinyāsa_ can be broken down into its two components. _Nyāsa,_ meaning to sanctify and draw one's full attention into a particular meditative focus and then release the content of the focus. _Vi_ means to arrange or sanctify in a specific way in response to context or a lack of context. This implies a sequence of steps and countersteps. Vinyāsa, then, means the focused, intentional sequence of form, thought movement, and breath that frees the mind by recontextualizing the body, sensations, form, and all objects of attention. It can be a specific form of yoga practice, but in a broader sense, vinyāsa is the mindful process that naturally occurs when we arrange any circumstances correctly.
Most of the time, in our attempts to focus the mind (vinyāsa on its way to nyāsa), the chosen pattern is incomplete. Normally a large part of the intelligence in the rest of the body rises up in the background as a distraction. For example, if you are meditating with attention on feeling the relaxed upper portions of the sinuses, it can be a very exciting and luminous experience. But soon, some parts of your body are likely to become tense, and your thoughts will scatter; these changes in composition seep in quietly and unseen, gradually coming into full bloom. The vinyāsa process is to allow the arising of oppositional forces, contexts, and perspectives, and at just the right moment—before a story line or movement pattern is allowed to fully manifest and wander off—to consciously introduce the balancing counterstep to harmonize the field. In this way, the current focus and action merge with and digest the residue of the previous step.
Perhaps the most obvious example of vinyāsa is the constant movement of our breathing: inhaling and exhaling. This pattern is one that has been with us since we drew that initial breath during the first moments after birth and will remain as a consistent, background sensation until our final exhalation at the time of death. We all share the oppositional patterns associated with the breath—the constant rising and falling, ebbing and flowing, spreading and contracting, stimulating and relaxing union of opposites within breath, body, and mind. The fluctuation of breath is a natural manifestation of a complete and pristine intelligence that is always there, if we choose to pay attention. When we carefully observe and feel our breath, eventually both patterns—the inhalation and the exhalation—remain simultaneously awakened in our nervous system and our awareness, the embodiment of paradox. With practice, we experience a singular focus within the field of awareness, and our thoughts settle effortlessly as the mind becomes calm and steady—the result of paying attention to composite, opposing, and paradoxical patterns as they arise. This is the essence of vinyāsa.
In this book, we will be exploring the idea of yoga as an art form and as an interlacing of opposites—a vinyāsa—within the context of what is known as the Aṣṭāṅga Vinyāsa system of yoga. Aṣṭāṅga Vinyāsa yoga is a form of āsana codified by Sri T. Krishnamacharya and his student, Sri K. Pattabhi Jois, in the early twentieth century. In this form, specific series of poses are practiced in a dynamic, flowing sequence that coordinates movement with the gaze and the breath. The Aṣṭāṅga Vinyāsa form also includes breathing and meditation practices and the union and synthesis of perspectives within each of the eight limbs of yoga described by Patañjali in the Yoga Sūtra. In Aṣṭāṅga Vinyāsa yoga, four underlying threads are always at play, harmonizing relationships through vinyāsa to bring balance, depth, and integrity. These threads, or "internal forms," are breath, _bandha_ (bonding), _mudrā_ (sealing), and _dṛṣṭi_ (gazing) (discussed in Chapter 1). They are woven together in a seamless stream of form, movement, and awareness that automatically awakens the intelligence.
Here we focus on vinyāsa as it relates both to the external forms of the practice, such as correct alignment within poses, and to the sequencing of subtle movements within the postures that truly ignites and liberates the mind. We explore the idea of vinyāsa and yoga as relationship: sometimes joining, sometimes separating; interpenetrating, communicating, or merging together. Through practice, this sense of interconnectedness gradually becomes familiar within every field of experience—inhalation with exhalation, focus with horizon, text with context, foreground with background, inner world with outer world, given or physical world with creative or imaginary world. On the deepest level then, we can see that yoga is relationships with other beings, with friends, family, coworkers, pets, bugs, the world at large, our community, and the environment. Though the struggle to balance rotations in our hip joints so we may sit effortlessly in Lotus Pose may seem important at times, it is clear that most important is relationship with others because that's where the emotions—our ability to use and see through self-image and those things we value and that give life meaning—come into play. Half the world is given to us, but the other half is created by how we frame it.
PART ONE
FOUNDATION
_The Roots and Depth of Yoga_
AS MODERN YOGA PRACTITIONERS WE FIND OURSELVES in an exciting though sometimes confusing time, as long-standing yoga lineages and traditions are coming face to face with modern interpretations and spin-offs inspired by individuals and cultural waves of style. It's easy to treat your own school of yoga as the final word and then ignore the wonder and variety of other traditions and perspectives. Juxtaposing different metaphors and images is a great way to let go of our rigidity and to slide into a more open-minded experience of yoga in our search for truth.
In the next three chapters we explore subtle internal techniques and forms traditionally used in classical systems of yoga to transmute emotions and to wake up creativity and our imagination into the joy of relationship. We explore different types of images that blend together the concentration of attention with movements of Praṇa and sensation into an unfolding of embodied form. Visualization and imagery help us to harmonize our internal experiences with other beings and with the entire world around us. Finally, in Chapter 4 we take an introductory look at a more classical anatomical perspective that also underscores the necessity of good form, alignment, and imagination within a healthy and balanced yoga practice.
_1_
Natural Alignment
_The Internal Forms of the Practice_
IMAGINE THAT YOU GET A JOB AS A MODEL FOR AN artist who's going to carve a statue of Avalokiteśvara, the buddha of infinite compassion. Avalokiteśvara is to be seated holding the wish-fulfilling gem in front of the lotus flower ( _padma_ ) of the heart, and your alignment must be perfect! All you have to do is sit in that pose and not move.
It takes extraordinary focus to picture what Avalokiteśvara looks like, bringing your attention again and again to rest along the plumb line of your body. Releasing the palate in silent contact with a softening tongue and feeling a smooth, steady breath unfold, you begin to experience all the physical patterns associated with inhaling. You then drop even more deeply in, observing as the breath effortlessly turns around; the exhalation dissolves all those endless forms back to their roots, like petals falling from a flower. The centers of your ears are directly over the centers of the shoulder joints, so they're aligned exactly on the coronal plane of the body; your hip joints are centered in that same precise line. The back of the diaphragm spreads, and you notice that right around the twelfth thoracic vertebra, a radiant point of awareness is forming a warm, vibrant circle. You envision yourself having four arms, but you know not to pinch any of the shoulder blades together or the artist will kick you out and hire someone else as the model. So you drop back into the breath and feel more arms growing—just a few at first, but then an infinite number sprout and reach up out of that warm, vibrant area in the middle of your lower back. The center of each palm tingles, and you realize you can actually _see_ through the palms as you reach out to all other sentient beings, but you're not distracted by this visual stimulation. It's hard work and you start to sweat, but if you release the palate and the muscles in the back of the tongue, your mind clears; you feel an extension along the spine, out through sides of your body, and then up through the crown of the head as if you are growing bigger and taller. The pose feels easy, steady, and buoyant. You cultivate a vivid sense of concentration and form and, at the same time, the ability to dissolve and let go.
This is how alignment was taught in ancient times before the study of anatomy and theories of biomechanics and postural alignment became the norm. In those days, alignment was embodied through visualizing deity forms, which brought the finer qualities of the emotions, sensations, and thought patterns into the breath and body. Artists trained for generations in a highly disciplined manner to reproduce in their sculptures and drawings exactly what sages had discovered to be, through lifetimes of practice and visualizations, optimal forms of alignment. Forms that would facilitate a physiologically awake and open, integrated, and finely tuned state of being that is perfectly suited for contemplative practice. Symbolic representation of this kind of esoteric knowledge followed prescribed patterns and proportions that were described in minute detail so that one could meditate on a deity form and _feel_ correct alignment. In those days, teachers didn't bother describing the alignment of joints or any of that dry, boring anatomical stuff. Instead, they went right for the source—the deity form—and breathed right into it.
Visualizations would often start with simple geometric forms like squares, circles, and triangles—something easy to imagine. Using the sequences of a vinyāsa, practitioners gradually would learn to interface simple forms with sensation patterns in their own bodies; this trained the imagination-body to link sensation points throughout the physical body with patterns of breath and movement. This method of learning alignment is still extraordinarily viable and valid today when studying yoga and yoga anatomy. Of course, visualization and the study of classical anatomy are by no means mutually exclusive; in fact, they inform and complement each other well. Studying anatomy gives a tangible context for the intuitive, internal feelings and sensations that fine artistic representations of breath and movement can stimulate within. With practice, you can visualize the deities in endless detail and begin to experience the inner sensations of waking up and harmonizing your own internal awareness as a means of balancing the nervous system and the subtle body. Then when you check an anatomy text, you learn why certain movements feel better than others. Visualization helps you to organize sensations and perceptions so you can release habitual, self-centered perspectives on these sensations and relate to the world as a composition of interconnected parts. Diety visualization (or any symbolic deity construction, such as a maṇḍala) follows the step-by-step process of vinyāsa. At a certain point of balance, contextualization, and harmony the visualization or construction is placed down as an offering to the whole world as pure consciousness. We then let it go. In placing the content of the mind down, or nyāsa, insight arises into the impermanant nature of all gross and subtle structures.
Deity visualization can, of course, be taken too literally by the ego. A student may become infatuated with a deity as a true independent entity—thinking, for example, that the forms and mythology of Gaṇeśa or Garuda actually exist as absolute and separate. At this point, the student's own ego absorption has gotten in the way of truly deepening insight through study. When these kinds of challenges arise, they are a sign to smile softly and lighten up so as not to lose balance. The key to visualizing a deity is to foster the trust that at the right moment you will look at the deity, embody it, and just "get it." This takes patience and an open, inquisitive mind. Deity visualization is akin to abstract thought, such as exploration of the idea of infinity, but visualizations can be embodied to give direct, visceral experience of what otherwise might be a complex construct.
Yoga alignment can be and usually is approached from the point of view of classical anatomy, physical form, and biomechanics. And this is good. We look at similarities and differences in the structure of bones, muscles, and interconnected patterns of breath and movement. Visualization bridges a gap in understanding movement between what happens in the mind and what happens in the body. As we breathe and move in and out of postures, we simply allow a visualized form to rest in the background as a subconscious context from which to experience whatever actually arises as feeling, sensation, or thought. The visualization provides a reference point for our experience so we can be there with full attention.
Between the externally oriented perspective of studying form and movement from a classical anatomical perspective and the more abstract perspective of visualization lies another method for insight into alignment and form. That is an understanding of the internal forms of the practice. Each of these perspectives is important and may be an effective means of establishing a context for understanding our practice. Merging visualization, abstract thought, and classical anatomy through an embodiment of the internal forms gives a full understanding of form and alignment. This really is what the Aṣṭāṅga Vinyāsa system of yoga is all about: taking a multidimensional view of what happens when we practice yoga. Cultivating the simplest of circumstances in a context of open-minded awareness and a full range of movement, we invite the yoga practice to unfold like a flower in bloom.
This blossoming naturally occurs through the consistent practices of meditation, prāṇāyāma, and āsana, each of which has specific external forms that serve as gateways to understanding. We gradually learn to focus the mind by sitting straight and cultivating a stable base in the body while practicing meditation and prāṇāyāma; we move with attention to the actions and counteractions within the muscular and skeletal systems while practicing āsana. As the practice deepens, more subtle levels of movement and awareness (what we call the internal forms) emerge, providing depth and insight. These forms are integral to practicing yoga as a meditative art rather than as a gymnastic exercise or an uninspired, autopilot-directed ritual.
We define the internal forms of the practice as breathing, dṛṣṭi, bandha, and mudrā. Although they are all intimate parts of our physical experience, unlike the more external forms of practice, the internal forms are initially less obvious. It takes time and patience to tune in to their subtle physical cues, but when we do, they provide an opportunity for insight into the threshold between the outer world of experience—our physical body—and the internal, intuitive, sometimes even mystical experience that arises as we practice yoga. When we open our imagination enough to consider what internal forms are, we may—on rare occasion—spontaneously feel the alignment of a deity as a representation of the interpenetrating nature of all things.
Cultivating an understanding of and connection to the internal forms is a paradoxical practice. It is at once elusive and simple, abstract and extraordinarily clear, impossible to "do" yet simple to experience. Due to their subtle nature, we begin to explore the internal forms from the most familiar—the breath—and work our way to the subtlest form—the mudrā—so we can experience their interrelationship as well as their profound impact.
The internal forms can be revealed through visualization, but it is also important to study them by establishing an embodied context for them. We call this context internal or "subtle body" anatomy, and the most direct route to understanding it is through an interfacing of the breath ( _Prāṇa_ ) and imagination ( _citta_ ). Just as we might imagine ourselves to be Avalokiteśvara and then suddenly find that we are sitting in perfect physical alignment, so too we may start wherever we can with internal alignment, imagining—and therefore _feeling_ —something that is already in the nervous system as a pattern sensation. Through mindful attention, the pattern unfolds in the imagination and the understanding. We start with whatever is present and build from there.
We construct and embody the internal forms by imagining that we have a central channel, the _suṣumṇā nāḍī._ It runs from the area between the pituitary and pineal glands down through the "heart center" and the core of the body to open at a point about the width of two fingers above the middle of the pelvic floor. This channel of awareness can also be imagined as extending in both directions (even beyond the top and lowest boundaries of the body). The pelvic floor, also called the pelvic diaphragm, is the fanlike muscular structure that lies along the bottom of the pelvis and immediately above the urogenital triangle. Its flat, toned surface supports, coordinates, and completes the larger muscular movement patterns of the abdominal wall, hip joints, and urogenital triangle in combination with the complementary breathing patterns of inhaling and exhaling. When feeling and visualizing the pelvic floor, an area of particular interest and importance is immediately in front of the anus, behind the genitals, and above the perineal body. This area is called the _mūla_ (root). It is the center of the sacred maṇḍala or circle of the pelvic diaphragm.
All of the internal forms of yoga practice are generated through the intricate, internal core structures embedded in the mūla. Over time and with a great deal of practice, we become familiar with the core root pattern that allows the internal forms to awaken and expand outward endlessly. We can then experience our practice as vast and unbounded without becoming ungrounded or disembodied.
BREATH
Yoga practice, like life itself, begins with the breath. Breath, or Prāṇa, provides an endless, all-pervading background, a continuous ebb and flow of sound and perception that unifies, sustains, and informs us on the physical, mental, and emotional levels. Prāṇa (with a capital P) refers to the internal breath as a whole. We experience it as a vibratory quality of pure sensation and perception within every nook and cranny of the body. It is often referred to as "life breath," or the immediately distinguished characteristic of sensations, in particular the awareness of touch—what you feel within your body as the tissues expand and contract. Prāṇa is the perception of temperature or texture through the skin or the feeling of a flow from one group of sensations to another. It is associated with the visual sense of light and darkness that we comprehend as part of our visual perception, just as it is associated with the immediate quality of sound and the sensations of smell and taste. Prāṇa is perceived in all fields of perception and, like intelligence, reveals and creates context for patterns that arise. A basic axiom of yoga is that Prāṇa and citta (the mind) move together like two fish swimming in tandem. Move one, and the other automatically follows.
The broad category of Prāṇa, or breath, has many subcategories but five major subdivisions. The first is _prāṇa_ (with a lowercase p), which refers to the breath as it rises and spreads and is based in the heart. _Apāna_ —based in the pelvic floor—goes down, contracts, and squeezes things out of the body. _Samāna_ is based in the navel; it spreads out evenly and is related to the digestive process and assimilation of everything—absorbing food as well as subtleties of awareness on all levels of perception. When awakened, the seed-point of samāna is said to shine like the sun at the root of the navel. _Udāna_ is based in the throat and rises up the middle of the head and out through the top. The all-pervasive form of breath known as _vyāna_ is felt throughout the body, most notably in the skin.
The most immediate and workable patterns of breath are apāna, which controls the exhalation pattern, and prāṇa, which controls the inhalation pattern. Awareness of these complementary patterns of breath opens a door to a direct experience of the sacred nature of all perceptions and mental patterns. When we inhale, it is easy to feel the characteristic expanding, blossoming, and rising sensations that spread throughout the body. The ribs naturally expand as we draw in the breath; the heart floats; and there is an ascendant, alert feeling. At the peak of the inhalation—in the gap before the exhalation begins—there is a sense of limitlessness, interconnectedness, and perhaps even joy at the unfolding of endless forms. When we exhale, it is easy to feel the grounding, squeezing-out, and dissolving pattern of the apāna as the edges of the diaphragm automatically drop and the ribs contract to accommodate the reduction in the size of the lungs. At the end of an exhale, the pelvic floor naturally tones. When the pelvic floor tones the mind easily turns inward as we experience the visceral sensations associated with the dissolution of forms—which for some may cause feelings of fear or panic to arise.
Most people tend to prefer one of these phases of the breath; there are those who love inhaling and those who favor exhaling. Some of us are too _prāṇic,_ avoiding the sensations associated with exhaling and getting lost in imagination, floating off the face of the earth in our minds. Others favor the exhalation and become too _apānic,_ with a rigid, boring, or depressed point of view. After some practice, we are able to enjoy the entire breath by inviting the opposite physical patterns of inhaling and exhaling to support one another within the body and mind; in this way, we can experience their interdependence.
In the Aṣṭāṅga Vinyāsa system of yoga, breath is the foundation for the internal forms of the practice. We begin with _ujjāyī_ breathing, which serves as the basis for practice whether you're a beginner or an advanced practitioner. In fact, Aṣṭāṅga Vinyāsa yoga _is_ just this: simple ujjāyī breathing with a little movement tossed in. More advanced practitioners can also learn Ujjāyī Prāṇāyāma, which is a very concentrated form of ujjāyī breathing and in which there is breath retention and complete internal focus.
To learn what ujjāyī breathing is, sit really straight so the belly is happy and not compressed. Bring your awareness internally. Imagine the central line of the body as vividly as possible like an imaginary plumb line running from the crown of the head down through the middle of the chest and abdomen, through the center of the pelvic floor, all the way down to the earth's core. The plumb line serves as a reference point for balance and stability within the body and may also be imagined as any stabilizing configuration within the body's core, say as breath, a shaft of light, or a pattern of energy. To establish the ujjāyī breath, imagine that the heart floats on this central axis like a lotus flower floats in a pool of still water. The base of the plumb line is stabilized as the sitting bones, coccyx, and pubic bone drop, causing the pelvic floor muscles to tone. If you are grounded through the root of the body this way and the heart area feels free and open, then breathing is easy. If you are disconnected at any point along that central line—if the heart is constricted, tense, or closed, or if the pelvic floor is asleep—then ujjāyī breathing is not happening.
Keeping the lips lightly closed, simply begin to breathe in and out. By closing your lips, the breath moves through the nose, and the mind can focus more clearly. At this point, the eyes become steady in dṛṣṭi (gazing). This automatically releases the palate and softens the tongue, so it is easy to focus on sensations in the mouth and the stream of breath going in and out through the nostrils. The waves of the breath start the process of alignment, which is merely the intelligence waking up in the center of the body. Just keep listening, allowing the breath to unfold.
Ujjāyī breath is characterized by a sound that results from closing the vocal cords a tiny bit while continuing to keep the tongue quiet and the lips softly closed. The breath makes a smooth, aspirate sound both as you inhale _and_ as you exhale. It sounds almost as if you were whispering the word _ah_ with your lips closed. As we know, whispering can be intimate. When you're close to someone, you don't shout, and the ujjāyī breath has that same intimate quality, as if you were whispering to your beloved. Listen for that sound and strengthen the breath—directing it with intention to be smooth, easy, and even. It is not simply a matter of letting the breath come and go in whatever pattern it happens to take; correct ujjāyī breathing depends on a nonforced effort that, like a metronome, keeps the pace and tone of the breath consistent. This resonance of the breath is the mantra of ujjāyī breathing and Ujjāyī Prāṇāyāma, and 99 percent of this breathing technique is opening the ears and listening to that sound. It sounds very much like gas leaking through a valve—which technically it is. You could imagine the sound to be water running through pipes, wind in the trees, or the distant breaking of ocean waves on the shore.
_Ujjāyī_ means an upwardly triumphant or victorious breath. Esoterically, ujjāyī is when the internal Prāṇa wakes up— _ut_ —and victoriously shoots up the central channel of the body to stand on the crown of your head as in Samasthitiḥ (Equal Standing). That's true Ujjāyī Prāṇāyāma, which may come to you one day. In the meantime, just make some noise with your breath.
Every time we inhale, we concentrate on the residue of the exhalation, and every time we exhale, we focus on the essence of the inhalation. The heart is the radiant point of the inhaling breath, and the overall physical pattern of the inhalation is an upward, spreading, and floating pattern. The exhaling pattern is perceived in the body as a downward, contracting, grounding feeling—like water running to earth—with its seed-point at the center of the pelvic floor. When we inhale, we pull the attention of the mind like a thread up through the seed of the exhalation at the middle of the pelvic floor. When we exhale, we release the upper back of the palate to keep the heart open. We concentrate on the smooth, whispering sound and the physical feelings and sensations associated with each phase of this pattern of breath, in addition to the moments of transition—the gaps—as we move from in-breath to out-breath and back again. Feeling the pouring of the apāna into the prāṇa and the prāṇa into the apāna is contemplative, subtle, and wonderful. This is prāṇāyāma practice; the whole thing is right there.
During our āsana practice, having established the form, flow, and sound of the ujjāyī breath, we learn to move in conjunction with the breath. While transitioning in and out of āsanas, we consciously practice the inhalation during expansive movements like lifting the arms overhead or moving up and out of a pose. Conversely, we practice more rooted and contracting movements like folding forward, curling the spine, or twisting in conjunction with the exhalation. After getting into a pose, we typically hold it for five full rounds of breath (though for finishing poses and under certain circumstances, we may hold for longer) before again moving on the wave of the breath to initiate a smooth shift as we inhale into the next movement within the vinyāsa of the pose and sequence. After practicing āsanas for some time in this manner, the patterns of breath and the physical movements associated with both inhaling and exhaling deep within the body become intuitively felt, and these flowing patterns reflexively manifest in our external movements. The effect is astonishing. As we move in and out of poses in union with the breath, we may experience a sense of seamlessly joining our inner world of experience with the external world of perception and our interactions with others. As we do, the practice takes on a deeply meditative quality.
SUPERIMPOSING MANTRA ON BREATH
To intensify the mind's ability to focus on the breath, we may superimpose a mantra on the aspirate sound. Using a mantra gives the mind, which sometimes needs to be employed, a task that reminds it of the bigger job at hand—to concentrate on listening—so that its natural tendency to wander is quelled. After some practice, the mantra can then fall away, and the underlying mantra—the actual sound of the breath itself—can capture and eventually liberate the mind.
Probably the most famous mantra associated with the breath within the yoga traditions is the SĀ-HAṀ mantra. You may imagine the sound _sā,_ a seed sound for the apāna, as you inhale—as if whispering inside your head so only you can hear. There is a natural pause at the top of the inhalation; at that moment you can imagine or silently whisper the seed sound for the prāṇa, _haṁ_ , as you exhale. In the gap at the end of the out-breath, the mind can again return to thinking the sound _sā,_ and so on. This mantra is sometimes referred to as the swan mantra because _haṁsa_ means "swan" in Sanskrit, and if you try to superimpose SĀ-HAṀ on the breath, you'll notice that after a few rounds it starts to sound like HAṀ-SA.
The mantra can also be considered divine sound associated with the _paramātman_ (pure consciousness beyond any individual or particular being) and it may be whispered internally as either SĀ-HAṀ (feminine "I am she") or -SO'HAṀ (masculine "I am he"). She or he is the paramātman, which is experienced as the openness of the central channel. So through this mantra we remind ourselves that "I am she/he," or "I am the paramātman, the goddess or the god who is pure consciousness." The idea that you are the beloved other, the paramātman, and are located in your own central channel and heart is usually enough of a conundrum (a self-reference paradox) to stun an active mind and make it willing to just listen to the background of the sound of the breath.
Whatever meaning you choose to associate with the mantra, placing it on the breath is pretty simple—make the sound _sā_ as you breath in and _haṁ_ as you breath out. Now you may wonder, "Who is saying SĀ-HAṀ? Is it me, or is it the breath?" Or it may cross your mind to ask, "If I am she/he, does that mean I am divine? After all, what do I mean by the word _I_?" These are the types of things people who think a lot can't help fretting over. It's natural. Don't worry about it. It becomes clear after a few breaths.
With practice focusing on the breath, a smooth tone and an even extension of the breath unfold effortlessly. This causes all of the little pockets in the sense fields to become filled with Prāṇa so the mind can rest, and the body and mind can become integrated rather than splintered. In this manner, we can perceive feelings, thoughts, and sensations clearly and respond to them with kindness and equanimity. Bringing awareness to and cultivating the sound of ujjāyī breathing is a vital, underlying thread of the internal practice of Aṣṭāṅga Vinyāsa yoga. It trains the mind to listen and become absorbed by sound so that movement is transformed from a self-directed, formulaic effort into a moving meditation.
DṚṢṬI
Another internal form that facilitates a deepening of the practice is the dṛṣṭi, or gaze. The term _dṛṣṭi_ refers both to a particular place on which the eyes rest during practice and to a quality or feeling-sense associated with the gaze. The eyes are fully alert while resting on a specific point. At the same time, unlike many other situations in life when we look attentively at a particular thing, the quality of dṛṣṭi is steady, calm, and spacious—there is no physical or mental tension; no sense of drawing a conclusion with the mind; no grasping, avoiding, or naming the object upon which we gaze. No strain is created within the sensory or mental fields in proper dṛṣṭi, in spite of the fact that we look with a clear gaze and not a sleepy or dreamy gaze.
Of course, this is much easier said than done. Releasing the palate, softening the tongue, and listening to the sound of the ujjāyī breath can facilitate proper dṛṣṭi. Good dṛṣṭi is awake, innocent, and attentive. Think of how a young infant looks intensely at something: he does not identify himself as being separate from the rest of the world, and he has few if any "names" for objects, so he is just looking. Dṛṣṭi may feel as though we are gazing from a place of perception that lies behind the eyeball, somewhere along the line of the temples within the skull, and there is a felt sense of "nobody looking at nothing."
You may notice that the quality of your own gaze, even in everyday situations, can affect your state of mind. When your eyes scan or dart around the room, attention tends to be on alert, and the mind may be either highly attentive and agitated or unfocused and distracted. If the gaze is too soft or the eyes feel drowsy, attention is usually foggy. The context and content of our thoughts form patterns of tension and movement in and around the eyes. Throughout the day, the mind is naturally dragged along behind or interfaced with the quality of our gaze and the movement of our eyes. It is quite natural when we subconsciously want to avoid something (thinking too deeply about it or moving to the edge of sensation) for the eyes to flicker, causing our concentration to shift. This lets off just enough of the mental or emotional pressure so we don't actually have to examine whatever is about to arise.
Within the āsana practice, we cultivate dṛṣṭi, training the eyes to stay steady, clear, and focused while we move in conjunction with the breath, no matter what is arising. Each movement during transitions between poses is associated with a particular breath and a specific dṛṣṭi, and each pose has a specific gazing point. This helps to focus the mind. For example, while practicing Sūrya Namaskāra (Sun Salutation), during Ekam (the first movement of Sūrya Namaskāra), we lift the arms overhead as we inhale and remember something more important we're "supposed" to be doing. Rather than dashing out of the room to take care of it, we open the ears, gaze up at our thumbs, and ride the breath like a wave of sensation and thought as the exhalation draws our awareness, gaze, and sensations back down to earth, and we fold into Dve (the second movement of Sūrya Namaskāra). Moving in conjunction with the dṛṣṭi and the breath provides a powerful environment for the mind to release its presuppositions, conclusions, and opinions, allowing us to concentrate on whatever arises moment to moment so that opposite patterns of movement, thought, feeling, and sensation can be woven together intelligently.
There are eight traditional dṛṣṭi or gazing points used during an Aṣṭānġa Vinyāsa practice—or nine if you include the internal one called _antara_ dṛṣṭi. The eight are _añguṣṭha_ (the middle of the thumb); _bhrūmadhya_ (between the eyebrows); _nāsāgra_ (the end of the nose); _hastāgra_ (the tip of the hand); _pārśva_ (the side—right or left); _ūrdhvā_ (upward); _nābhi cakra_ (the navel); and _pādayoragra_ (the tip of the foot). Each pose has a prescribed dṛṣṭi that is sustained throughout the five counts of breath during which the pose is held. The practitioner or teacher may modify the dṛṣṭi for particular circumstances or desired effects in a pose. By holding steady to a particular gazing point and reducing physical and mental tension, we create an underlying, unifying field of experience within which mind, breath, and body can naturally adjust to accommodate the specific mental, emotional, and physical circumstances that arise.
_Dṛṣṭi_ also means "view," as in a philosophical view that extends even to our moment-by-moment propositions about ourselves, others, and the world. The type of gazing, or dṛṣṭi, that we practice during āsanas will eventually give you a correct view—one that is kind and compassionate. Since you are just gazing, there's no strain, no aggression, no need to formulate ideas or pull out from the background of perception as separate any one particular object in your visual field. Your view (both actual and theoretical) is crystal clear, and the mind is suspended so it is only gazing. When you're not practicing dṛṣṭi, the mind will establish a gazer (you) and then make up something (an individual field or object) to identify as separate and the object of the gaze. Once this separation starts, the process of wandering mind is triggered.
Physiologically, we feel dṛṣṭi as a softening tension in and behind the eyeballs while we release the palate. Releasing the palate invites the dṛṣṭi to become integrated into the entire structure of the body, because the root of the palate is the kingpin of all the Prāṇa movements—that is, all the sensations throughout the body. Releasing the palate is a true art in that it corresponds to the letting go of technique. To begin to release, you can subtly smile, listen closely as if to a distant sound, or imagine that you feel the palate and nasal septum as pleasantly luminous. Releasing the palate is also simply suspending the language-making function and the unencumbered flooding of the sense fields with Prāṇa. During breathing practices or āsana practices, while practicing dṛṣṭi, or in everyday life situations, if you get confused, feel tension or anger, become lost in thought, or doubt yourself, simply release your palate. See what happens.
Symbolically, releasing the palate can be thought of as the act of allowing the nectar of compassion to penetrate every cell of your body. From a yogic perspective, the root of the palate, just behind the pituitary gland, is called the _candra_ (moon). This moon is visualized as lying at the base of the thousand-petaled lotus flower, the _sahasrāra,_ that rests in the crown of the head. Like a mirror, the moon has a pure, reflective openness with no form of its own—a selfless, discriminating awareness. It serves as a reservoir for the nectar generated by the mind when intelligence is finally applied. The nectar is called _amṛta_ (literally, "no death"), truly the nectar of immortality. Its main ingredient is _karuṇā_ (compassion). We each have a limitless stash of this nectar at the root of our own palate, if only we knew how to access it!
BANDHA
Bandhas are sometimes referred to as points of contraction, binding, or bonding within the body; however, thinking of them as merely some muscle squeezings can stimulate subtle levels of tension. Instead, it is better to consider bandhas as areas within the body where complementary patterns join or bond. Bandhas are specific areas where we focus on the concentrated organization of opposing patterns that flow throughout the whole body. In a manner of speaking, bandhas serve as internal gazing points; they are seed-points of clear attention from which integrated movement, thought, emotion, and Prāṇa unfold all around.
As with correct ujjāyī breathing and proper dṛṣṭi, the cultivation of bandhas is a powerful way of setting the flow of Prāṇa into patterns that pull the mind toward meditation. Bandhas involve distinct muscular patterns and equally distinct patterns of sensation that are organized as a background to the area of focus. When the bandha is perceived as the linking together of complementary patterns, then the foreground and background come into full relationship. The sense of effort and tension initially involved in the production of the bandha melts away, and the bandha transforms into a whole-body pattern and movement.
Three interrelated bandhas are most frequently associated with Aṣṭāṅga Vinyāsa yoga: _Jālandhara_ , _Uḍḍīyāna,_ and _Mūlabandhas_. When learning about and beginning practices that allow you to cultivate the bandhas, it is important to be attentive, calm, and patient and—particularly when first starting—to visualize muscular patterns that you associate with the specific bandha. Closely tuning in to the general area of the body that is related to the bandha is the first step, and imagining that you are doing the bandha correctly (even if you're not sure) is an excellent place to start. Over time, your familiarity with the individual areas of the body associated with each of the bandhas and your ability to tune in to and control muscular patterns in these areas will improve. Keeping the palate empty and released, the tongue quiet, the breath smooth with a clear gentle sound, and the dṛṣṭi steady helps to create an internal environment in which the bandha may manifest. In addition, it is vital to weave the appropriate phase of the breath into the bandha practice—paying attention to the physical and mental impact of both inhaling and exhaling flow patterns, as well as noticing shifts in thought and sensation that arise during any gap at the transition when one breath turns into the next.
_Jālandhara Bandha_
In some ways, Jālandhara Bandha, which is usually practiced in a seated position, is the easiest of the bandhas to approximate and therefore is a good starting place for bandha practice. It is the only one of the three classic bandhas that has a blatantly obvious external form.
1. To begin experimenting with Jālandhara Bandha, sit straight in a comfortable position either on the floor or in a chair. Bring awareness to the central axis of the body and the sensation of being grounded by focusing on the parts of the body that touch the floor or the chair.
2. The spine should be elongated, with no strain in the muscles that support it. Notice how easily the heart area floats upward on an inhalation. In the gap at the top of the inhalation, lower the head forward, bowing so that the chin nestles down into the sternum. If dropping the chin to the sternum is not possible, a small scarf or washcloth may be rolled up and tucked under the chin to allow for support.
3. There is a feeling of spaciousness and softness in front of the throat—almost as if the chin were being lifted gently over and around a small, soft ball placed in front of the Adam's apple. Do not draw the chin back toward the cervical spine or press it down to make it touch the sternum. The buoyancy of the heart is so bright and distinct that the chin comes adoringly down as if to crown the heart.
4. The middle line of the palate feels bright and high within the head. Although the lower portion of the neck is flexed, the upper cervical area near the head is not. Sitting this way is called "Jālandhara Bandha position," but it is not yet the actual bandha just as ujjāyī breathing is not quite full Ujjāyī Prāṇāyāma. Both of these distinctions can cause some confusion for beginning students. The position serves as the physical scaffolding for Jālandhara Bandha itself, and the position may be practiced while doing smooth ujjāyī breathing in Padmāsana (Lotus Pose), Daṇḍāsana (Staff Pose), and other seated poses.
5. Jālandhara Bandha holds all of the physical and breathing patterns associated with the Jālandhara position, but it has the additional element of retaining the breath at the crest of the inhalation and, occasionally (during the full Uḍḍīyāna Bandha), at the end of the exhalation. When you are first learning to practice Jālandhara Bandha, you should hold the retention for just a few seconds. After much practice, you may carefully extend the length of the retention.
6. In correct Jālandhara Bandha, the muscles in front of the neck are toned, the palate is released, and there is absolutely no strain in the tongue or jaw. The gaze is downcast along the nose, either internally or with the eyes half-open. The ears are open, and the body is quite still. The key to this bandha is in the throat, where the exaggerated, full, and expansive prāṇa pattern is allowed to meet the full forward-curling pattern of the apāna. This makes the shape of the upper body, when practicing Jālandhara Bandha, resemble that of a swan sleeping with its head resting peacefully on its large chest.
_Mūlabandha_
On the other end of the spine, the pelvic floor (which by no coincidence, is related reflexively to the palate) is the territory of Mūlabandha. Although Mūlabandha is talked about frequently, it is possibly the most misunderstood and difficult of the three bandhas to practice. For this reason, Chapter 3 explores it in detail. Simply stated, Mūlabandha is a subtle toning and lifting at the center of the pelvic floor. It is not something that can be grabbed and held by contracting the pelvic floor muscles, although you can start there to bring awareness to the correct general area of the body. Instead, Mūlabandha is invited to appear. It is the contracting pattern of the apāna that pulls into a seed-point at the center of the pelvic floor, ignited like a flame, and lifted by the process of Uḍḍīyāna Bandha. The muscular contraction pattern of the pelvic floor becomes a meditation on sensation flow patterns as the pelvic floor muscles fluctuate from strong to subtle and the bandha is cultivated.
Working to perfect Mūlabandha is key to enhancing an āsana practice (and to beginning and deepening your prāṇāyāma and meditation practices), because all poses ground through the seed-point of Mūlabandha and, on a more external level, the pelvic floor. Mūlabandha is the core of the core of all integrated movement. It is the process of communication across the pelvic floor from side to side and from front to back, and it becomes the platform for an engaged, meditative flow of poses. For most of us, Mūlabandha is initially understood through the practice and development of Uḍḍīyāna Bandha.
_Uḍḍīyāna Bandha_
Uḍḍīyāna Bandha, which literally means the "flying up bandha," comes in two forms. The full form is used on retention of the exhalation and involves passively drawing the entire abdomen, above and below the navel, back and up by contracting the auxiliary muscles used for inhaling.
1. Stand with the feet about hip-width apart. Place the hands on the upper thighs and exhale as you bend the knees slightly to fold forward.
2. As the groins deepen, push with the hands and elongate the spine slightly, but do not give in to the temptation to overly arch or overly straighten the spine. The arms should be fairly straight as you push into the femurs, with moderate pressure.
3. Placing the hands farther down near the knees tends to favor the prāṇa pattern, making the lumbar spine extend more. Placing the hands near the tops of the thighs is more effective, allowing the addition of Mūlabandha and other patterns in the abdomen.
4. At this point, after completing the exhalation, the auxiliary muscles that normally fire when inhaling— _but not the respiratory_ _diaphragm_ —are engaged. No air is drawn into the lungs. The result is that the muscular pattern associated with inhaling manifests: the ribcage expands and the abdominal cavity is automatically sucked up and back without effort. Hold this for just a few seconds. (The time of the retention, _kumbhaka,_ can be extended very gradually over many months of practice.)
5. Maintain this position and the retention of the exhalation as you release the muscles associated with inhaling. This will allow the belly to drop. Give a slight puff to punctuate the end of the exhalation. Slowly begin to inhale smoothly as you return to standing.
This same action can also be practiced while lying on your back and eventually while sitting. In full Uḍḍīyāna Bandha, we can imagine that we're scooping out the entire abdominal cavity with the spoon of the mind, as if we are cleaning the front surface of the iliopsoas muscles and the pelvic floor itself, as well as hollowing out the area around the diaphragm. This is both stimulating and balancing to these myofascial structures.
_Mini Uḍḍīyāna Bandha_
Once you have developed the full Uḍḍīyāna, a more subtle and profound form eventually manifests. It is like a miniature version of this full form. The "mini" Uḍḍīyāna Bandha takes time to cultivate and is therefore considered more advanced than the full form already described. Mini Uḍḍīyāna Bandha is practiced during prāṇāyāma and while working in āsana by distinctly toning the myofascial structures that lie deep within the body beneath the pot of the belly. The pattern of Uḍḍīyāna Bandha comes up no higher than two inches below the navel and is practiced during the course of inhaling and the retention of the breath at the top of the inhalation. It creates a space above the pelvic floor in the cave of the sacrum and allows you to feel that the center point of the perineum is being drawn up and back. This muscular pattern stimulates the appearance of Mūlabandha. Mini Uḍḍīyāna Bandha is accomplished by keeping the lowest horizontal band of the transversus abdominis muscle toned and, at the same time, keeping the psoas and quadratus lumborum (QL) muscles relaxed throughout the inhaling movement. The action is part of a scooping pattern in the front of the body that causes a feeling of a cobra hood pattern on the back of the body. It induces a distinct drawing together and ascension at the mūla, or the center of the perineum.
This description of the mini Uḍḍīyāna Bandha is not used by most surviving lineages of hatha yoga; its action is taught as part of Mūlabandha alone. Because mini Uḍḍīyāna and Mūlabandha are central to the teaching of Aṣṭāṅga Vinyāsa yoga, this distinction has been a source of confusion for many practitioners who have tried to learn from books rather than personal instruction from a teacher—which is the preferred method for learning the bandhas.
Bandha practice stimulates an internal focus and is a good place to start as you develop the skill of contemplating the central channel of the body. Within this contemplation and when all three bandhas are practiced in the correct manner, the prāṇa and apāna fully unite. When this happens, you may perceive the whole body and the entire world as balanced, empty, or open, like a flower in full bloom.
NAULI
From the practice of full Uḍḍīyāna Bandha, there gradually evolves an ability to add some distinctly opposite and complementary movements that balance the patterns of the pelvic floor, the respiratory diaphragm, and the abdominal wall. These abdominal movements are called _nauli,_ a rolling wave in the abdomen:
1. While standing, lean forward after a full exhalation, placing your hands on the upper thighs. Suck the entire abdomen back toward the spine. (See the earlier instructions for Uḍḍīyāna Bandha.)
2. Holding the well-developed and clean Uḍḍīyāna Bandha, press the hands down on the thighs to contract only the rectus abdominis muscle in the abdominal wall. This paired muscle will stand out as two parallel columns. After some weeks of practice, when the form is distinct, move on to the next step.
3. Start releasing the pressure from the hands, and then press harder with one hand than with the other hand. This will cause the muscles on that side only to stand out, and the oblique muscles on the same side will start to tone. Practice this until you can contract both sides independently.
4. Next, practice rolling the contraction from side to side by alternating the pressure of the hands on the thighs. One direction is usually easier than the other, but practice rolling in both directions diligently.
5. Further practice allows you to isolate the oblique muscles. At that point, more precise patterns of rolling can evolve.
6. Whenever you release Uḍḍīyāna Bandha and/or nauli, first relax the auxiliary inhaling muscles that created the full Uḍḍīyāna Bandha. This allows the belly to fall (like jelly) to its soft, normal, ball-like form. Then exhale a little puff of air to reset the perineum to its end-of-exhalation seed of apāna tone. Inhale as smoothly and evenly as possible, using the underbelly scooping of Mūlabandha.
7. Fill up to the brim with breath, release the palate, and enjoy keeping the center of the heart open throughout the next exhalation.
MUDRĀ
The most subtle internal form of the practice is mudrā. _Mudrā_ can have many meanings, including a seal, a mark, a ritualized gesture of joining together, or specific hand or finger positions used to focus the mind. In the context of this yoga practice, mudrā is the internal pattern that occurs within the body when a bandha is practiced flawlessly, and like bandhas, mudrās must be carefully, patiently attended to and invited to appear.
Two classic mudrās associated with an āsana practice are _Khecarī Mudrā_ and _Yoni Mudrā,_ both of which are points of union at opposite ends along the central channel of the body—the tongue and the pelvic floor, respectively. As we work toward these two mudrās in our yoga practice, we initially begin by exploring bandhas. As the bandhas become more established, the mudrās appear. But at first we're just practicing and trying to get some stream of sensation that grabs our attention in the general physical areas associated with the mudrās. After years—if not lifetimes—of practice, the bandhas may transmute into the deeper level of experience we call mudrā.
_Khecarī Mudrā_
At the top of the central channel where we find Khecarī Mudrā, we allow the energy of the tongue—the tip of which is placed near the root of the palate—to move into open space or emptiness. In this mudrā, all the fluctuations of the tongue and Prāṇa cease, and Prāṇa is then established in the central channel. Of course, true Khecarī Mudrā position—placing the tip of the tongue way back and up behind the soft palate right under the sphenoid sinus—is nearly impossible. Some extreme practitioners work at it by stretching the tongue muscle and slowly, over time, severing the frenulum (the tendon beneath the tongue) to achieve this position. Others consider those measures to be extreme, because you can attain the benefits of Khecarī Mudrā without actually cutting the tendon and placing the tip of the tongue against the sphenoid sinus behind the palate. Such zealous efforts are likely to carry negative consequences that make "achieving" the mudrā ineffective. For most people, it is better to find Khecarī Mudrā in a more subtle way.
One method is by practicing what is called _Jihvā Bandha._ Open your mouth wide, while firmly pressing the tip of your tongue into the hard palate. This action stretches the frenulum, and if you can make scary noises or demonic faces while you do this bandha, then you're doing it right. With practice, Jihvā Bandha can stimulate the same internal response as Khecarī Mudrā: complete concentration and absorption of mind, emotion, and thought, as well as a feeling of clarity into the top part of the central channel. Another approach to Khecarī Mudrā is to focus on the feeling of compassion or the experience of aesthetic satisfaction and simply feel how the tongue is already in natural contact with the palate as if there were a slight electrical current or magnetic attraction at the area of contact. Combined with soft dṛṣṭi, a release at the root of the palate occurs, and the Khecarī Mudrā practice is well under way.
_Yoni Mudrā_
pādamūlena sampīḍya gudāmārgaṁ suyantritam
balādapānamākṛṣya kramādūrdhvaṁ sucārayet
kalpito 'yaṁ mūlabandho jarāmaraṇa nāśanaḥ
apānaprāṇayoraikyaṁ prakarotya vikampitam
bandhenānena sutarāṁ yonimudrā prasiddhyati
siddhāyāṁ yonimudrāyām kiṁ na siddhyati bhūtale
Completely press the anus with the heel. Slowly and strongly pull the apāna up in steps. This Mūlabandha destroys the decay of old age and death and insures a firm union of apāna and prāṇa. By means of this bandha the perfection of Yoni Mudrā comes without effort. What in this world cannot be achieved with the accomplishment of Yoni Mudrā.
_—Śiva Saṁhitā,_ ch. 4, v. 64–66
At the other end of the central channel, we have Yoni Mudrā, or the perfection of Mūlabandha and the bonding together of prāṇa and apāna in the epicenter of the pelvic floor—the mūla. These two complementary patterns, which are two ends of the same stick, can be represented as male and female; the prāṇa governing inhalation creates certain expansive body patterns that we associate with the feminine, and the apāna governing exhalation creates flexion in the spine that we associate with the masculine. Once you begin to fully understand this union of opposites on all different levels, you may begin to feel it in every pore of the skin, and that's when Yoni Mudrā starts to work. You feel a sort of humming in the pelvic floor that is almost a short-circuiting of logical thought, a paradoxical feeling of two complementary opposites arising simultaneously and conjoining across the pelvic floor.
When you practice (and practice and practice) Mūlabandha and Yoni Mudrā, and you start to feel them in relation to the structure of the whole body—even through the head, shoulders, hands, feet, front and back, inner body and outer body—you begin to experience how all of those patterns flow into and back out of the pelvic floor. This experience is illusive, especially if you think about it too much, and that is why many practitioners find visualization practices powerful; these practices are not directed by the mind, but arise spontaneously. All of the patterns of Prāṇa in the body bounce off of or root into the pelvic floor and come back out, just like a tree roots into the earth in order to grow. This happens moment by moment within the field of Prāṇa. When you start to feel that union of rooting and expansiveness and add to it the quality of compassion that comes from Khecarī Mudrā, _then_ Mūlabandha can be called Yoni Mudrā.
_Tāḍāgī Mudrā_
Tāḍāgī Mudrā, the pond mudrā, is one of a number of little-known almost-secret gems. It is like a very pleasant stiffened Corpse Pose and is used within a number of Vinyāsa sequences.
1. Lie down flat on your back with the legs together. Lightly press the sides of the big toes together as if holding a coin between them. Turn the palms down so the entire thumb and index finger of each hand is on the floor. This broadens the back to enhance the apāna pattern.
2. Tilt the chin down slightly, as if the skin on the back of the head were being spread and pulled up like a cobra's hood. In this form, smooth, high-quality ujjāyī breathing is initially used to produce full breath waves with natural gaps at the ends of the breaths.
3. Keep the dṛṣṭi steady, soft, and downcast to release the palate, allowing the proper rhythms and proportions throughout the body.
4. At the top of each inhalation, notice that the prāṇa pattern lightly pushes the head back into the floor. If you choose to add full Uḍḍīyāna Bandha at the end of an exhalation, notice the same slight pushing back of the head during the retention of the breath. Remember to relax the abdomen and exhale a tiny puff more before inhaling smoothly.
These classic mudrās demonstrate why _mudrā_ means "seal," or the perfection of bandha, and cannot be achieved through force, but require strong, focused, and consistent practice. Bandha (and eventually mudrā) practice _must_ be carried out for the sake of the practice itself, with no sense of striving, if it is to evolve into full form—there must be no attachment to the fruit of the practice. Otherwise the mind gets involved, makes overly simple reductionist formulas, and then tries too hard. Ego slips in and co-opts the situation. Bandha and mudrā practice is what a vinyāsa practice really is: the sequential joining together and separating of complementary opposites as a means of staying present, mindful, and alert to the feelings, thoughts, sensations, and insights that may—or may not—arise.
This joining together might seem an abstract and confusing task, yet it is actually something we do all the time, whether or not we're even aware of it. For instance, you may think, "I want to be wide awake and eager, and I want to be relaxed too." For most of us, these mind states are total opposites. So throughout the week you drink coffee, and Friday night you drink beer! Through yoga, we learn to manifest both mind states simultaneously.
INTERNAL CHANNELS: THE NĀḌĪS
By bringing awareness to the nuances of alignment that are revealed through the internal forms of practice, we discover a gateway to understanding the _nāḍī_ system, part of the innermost structure and scaffolding of the practice. _Nāḍī_ means "channel" or "little river" in Sanskrit, and from a yogic perspective, the nāḍīs are an intricate system of rivulets of Prāṇa and energy that flow through and penetrate every area of the body. From a Western perspective, the nāḍī system could be considered somewhat parallel to the combination of the nervous and circulatory systems. The nāḍīs bring a vibratory quality of breath and awareness to every point of sensation within the body.
As mentioned earlier, there is one central nāḍī, the suṣumṇā nāḍī, from which all other smaller nāḍīs (or rivers of Prāṇa) flow. Again and again in our yoga practice, we bring awareness into this central channel of the body so there is gradually a feeling of the practice spontaneously arising as an internalized perspective from this area of the subtle body. When we begin practicing yoga, the imagination wakes up and we may envision actually _having_ a central channel. Whatever image sparks a sense of something vibrant in the core of the body will do—a hollow reed, like you might see growing up out of a glassy pond; a beam of light; or even an amorphous yet somehow contained feeling of spaciousness in the core of the body up through the heart area. Anything that helps you cultivate inner awareness along what you perceive as a "central channel" is what you're after.
In more esoteric forms of practice, practitioners also imagine that Prāṇa is behaving as the serpent ( _Kuṇḍalinī_ ) that coils, sleeping on the pelvic floor at the base of the suṣumṇā nāḍī. It is said that until the serpent is awakened and begins to move in the central channel, the entire system of nāḍīs is unbalanced, with some nāḍīs being overstimulated and others blocked. But as beginners, we are content to imagine a simple central channel into which we may occasionally find the capacity to tap.
In addition to the suṣumṇā nāḍī, there are two other large, accessible nāḍīs. Both are said to connect to and open into their respective side of the pelvic floor alongside the suṣumṇā nāḍī. They ascend through the head and cross behind the eyes (not unlike the optic nerves) to join the sides of the _ājñā_ (command/understand) cakra, behind the middle of the eyebrows. They correspond directly to the flow of the breath in the nostrils, and they wax and wane in relation to each other as different moods and thoughts come and go. Some systems of imagery cross them or channel them into the sides of whatever cakra is to be contemplated, while other systems keep them parallel to each other on opposite sides of the suṣumṇā. Paying attention to the streams of sensation associated with breath flow in these channels is initially an excellent way to calm and focus the mind for meditation.
The right channel, or sun channel, is called _piñgālā,_ which means "bright, hot, warm, or sun." Piñgālā has a fixed, confident quality associated with the solar attitude of "Yes, I know the way. I know what to do!" You need to have this quality to function throughout the day and take practical actions. But this piñgālā quality can create anxiety in situations where you cannot know what something ultimately is, like the beauty of a butterfly or the intentions of another. An overly dominant piñgālā can make it difficult to rest in a state of not knowing. Of course in terms of questions like "Should I eat now or later?" you want to be able to make a decision and so piñgālā energy serves an important, perfectly divine function. In a way, waking up the energy of this channel is somewhat like pouring courage into your right nostril. But we all know that the downside of being too confident is that you may start to skip over the plurality, the multiplicity, the depth of what is going on; you cease to appreciate all the different viewpoints and different levels and gradations of beings that are participating in this entire process of life.
The moon channel on the left is called the _idā._ It cools you off and leaves you stunned at the beauty of multiplicity. The two channels, the piñgālā and the idā, are just like day and night. In the day, one star called the sun takes over, a single story line, a dominant point of view. Ah, but then night comes, other stories, other viewpoints and contexts gradually appear. Soon a million stars fill the sky and _you_ disappear in the starry night. You start to contemplate enormous distances and spans of time, intuitively understanding the relativity of everything. You feel tiny, and you realize that even our solar system is in the middle of nowhere; you disappear into the vastness of that plurality, that multiplicity. It's beautiful, but of course it can also be dysfunctional at the wrong time of day.
Within the nāḍī system, there is a clear oscillation between opposites, two things that are interdependent: the singularity of structure and the plurality that reveals the openness of that structure. In the center, when the channels are balanced and the invitation for awareness (or Kuṇḍalinī) to move in the central channel is received, there is brilliance as well as balance.
CAKRAS
Another brilliant aspect of traditional imagery that invites an internalized, contemplative mind, is the cakra system; through this, we meditate on various stations along the central channel that correspond to distinct sensation patterns and perceptual modes. Cakras (wheels) are usually represented and felt as lotus flowers or _padmas._ They are strung together like a garland along the suṣumṇā nāḍī. They are imagined to be sacred spaces ranging in detail from simple geometrical yantras to elaborate maṇḍalas, temples, islands, and whole worlds populated with gods, goddesses, and (potentially) all beings. Cakras or padmas function to capture and absorb our attention fully and then to balance and deepen our insight into the actual nature of what we are experiencing. Each petal or segment of every cakra needs to be interlinked with its complementary opposites and then with its deeper background.
Smooth ujjāyī breathing introduces the natural vinyāsa of the attention to balance and illuminate the cakras. Evenly illuminated, brought to life and vibrancy, they open into the middle path of the suṣumṇā nāḍī at the center of each padma where the nectar from the root of the palate can be felt. In normal distracted breathing, it is likely to feel as though half the petals are wilted while others are overinflated. But with mindful ujjāyī breathing practice, there is a sense of calm alertness within the body and mind, and the garland along the central channel feels alive, awakened, and evenly innervated.
TRUSTING THE PROCESS
We may harbor a lot of resistance toward dropping in and practicing from an internal perspective. The fear of feeling—deep inside the body—certain things like infinity, impermanence, emptiness, or the fact that there is no ultimate frame of reference, can be terrifying. Yet it is for this very reason that the practices emphasizing the internal forms are so important and why we must take them slowly and work at them with great patience and kindness toward ourselves. This is why a teacher encourages beginning students simply to have a direct experience within their own bodies of what it actually _feels_ like to take a full, deep inhale and a smooth, long exhale, and why approaching these internal forms directly through āsana practice is vital.
As an eager student, you may start with Mūlabandha practice and then sometime later realize there is more subtlety to the practice than just squeezing the anal sphincter muscle. Over the course of years and years, you finally realize that it's really just a matter of paying attention to a process that is already there. With even more time, you find yourself back at the beginning, and you see that prāṇa and apāna (or Śiva and Śakti) are already united at the center of the pelvic floor. Then Mūlabandha—experienced as a flame of pure _cit,_ pure attention, or pure intelligence—spontaneously arises. It's not a contraction or the release of a contraction. It's not a noncontraction, it's not both, and it's not neither. It's just pure intelligence, and it refuses to reduce the goddess to a theory of the goddess. The internal forms, particularly Mūlabandha, support the practice inside and out and are awakening and transformative.
There are basically two approaches in hatha yoga. One is the approach that works on Mūlabandha and tries to collect and control everything. This approach is called _bindu dhāraṇa,_ or a single-pointed focus on a specific seed or droplet (bindu) of awareness. In this practice, you hold the bindu still. Many schools of hatha yoga are based on this approach, and in fact, Aṣṭāṅga Vinyāsa yoga is initially presented this way. Fortunately a little of the opposite approach to hatha yoga can be mixed into a seasoned Aṣṭāṅga Vinyāsa practice to offer even deeper insight. This second approach is called the _amṛta plavana,_ or the flooding of the whole system with nectar, and this is what eventually occurs when we practice bandha and mudrā. All of the minor canals, nāḍīs, and rivers of Prāṇa throughout the body are awakened and balanced. It's a flood of compassion. Actually you can't do a complete yoga practice without embracing both bindu dhāraṇa and amṛta plavana. You have to work with form, discipline, and the tight interweaving of reductionist theories of technique. And you must then also be able to let go, loosening so that a whole living reality can flow between them.
_2_
Aligning Intention and Action
_Where the Rubber Meets the Road_
MANY OF US FIND OUR WAY TO YOGA IN A SEARCH for reality—looking for answers to questions such as "Who am I?" and "What is the meaning of life?" As we tune in again and again to the ends of the breath or the vastness of the gaze, a flash of insight may occasionally arise and reveal the interpenetrating nature of all things: _everything_ relates to everything else and consequently is in constant flux. That is the nature of reality, of who we are and why we're here. Understanding this, we see that many important questions are ultimately unanswerable. That doesn't mean they are not worth asking; in fact, quite the opposite. A steady and stable practice shows us how vital it is to maintain a sense of inquisitiveness so that we remain inspired by posing and re-posing questions to which we think we already know the answers. As we patiently stay the course and practice for the sake of practice, the boundless joy that resides deep within each of us spontaneously arises. This is what it means to practice yoga and to experience the cohesive nature of reality.
The view of relationship as the essence of life and reality is something we have all intimately experienced firsthand. We enter this world tied to our mother through the umbilical cord. Shortly after birth, that deepest of all bonds—the primordial physical relationship of literally being tied through a lifeline to another human being—is severed. So in some ways, from that moment forth, we might say that as we navigate our way through life, there is an instinctual interest in regaining our natural homeostasis of being intimately in relationship. We are programmed to connect and reconnect, to form bonds with other beings on a core level. Breathing—as if from the root of our navel—becomes a means of unscrambling the puzzle of how all the working parts of this world come together to make sense.
UMBILICAL BREATHING
Being connected to life and the universe through the navel is an image that figures prominently in Indian mythology and iconography. It is said that Brahmā was born from the navel of Viṣṇu, and Viṣṇu is often pictured reclining on Ādiśeṣa (the primodial residue, the serpent of infinity) with a lotus flower—symbolizing unbounded creation and clear mind—growing out of his navel, the _nābhi cakra_ which is also called the maṇipūra cakra. A visualization practice that can awaken this primal sense of connection is to imagine yourself, in exquisite detail, as Viṣṇu; relaxed, content, and happy, you recline on the comfortable sofa of support offered by your loyal attendant and vehicle, Ādiśeṣa. Deep in the core of the body, between your belly button and your spine, there is a warm, solidly rooted feeling that pulsates smoothly up and down the central channel and also radiates out. As you breathe, this feeling gently penetrates your skin, and delicate extensions of creative energy, like the petals of a lotus flower, bloom and automatically connect, as if through a strong magnetic attraction, to the world's vast and intricate web of existence. The feeling is organic and deeply emotional. It could be called "umbilical breathing." Resting with such an image for a moment, you might experience a feeling of connection unfolding, dissolving, re-forming, and disappearing, an ever-changing sense of relating to others from deep within your own core. Of course, like other visualization practices, this is one to visit and release, since you know full well that if you did find yourself with a lotus flower growing out of your navel (or worse yet, believed and proclaimed such a thing in its absence), you'd be rushed to an emergency room for immediate attention!
You can also feel this depth of connection by simply _visualizing_ that the yoga practices actually work, that you have insight into relationship as the core of existence, and that you celebrate the opportunity to ask the important questions again and again. Through this practice, we find that even though we repeatedly ask the difficult questions with our intellect and emotions, we're also inquiring on a deeper, embodied level. This means we keep the experience of the "other" in the foreground of our awareness, and although we draw conclusions, we continuously let go of even those theories and techniques that seem useful and contextually correct, as if the root of the navel is the unfolding ground for the intense radiance of true relationship and the actual nature of being.
In a yoga practice inspired from this depth within our core, we can step down off the cloud of illusion that pictures self as separate from other and walk around _seeing_ everything and having insight into the true nature of things. We are able to trust in not knowing exactly what we are encountering, comfortable in upgrading our methodology and understanding. Without a deep, visceral rooting (as if from our own navel), yoga and relationships gradually break off from reality and float away as disembodied, abstract stories and dreams.
Through practice, we find that focal points consciously selected for meditation pass through the window of awareness and are met with an inquiring mind. In this way, each time they arise, they are brand-new, yet there is always something familiar about them that we know to be true, and it becomes clear that everything is built on this open awareness.
THE YAMAS AND NIYAMAS
We practice in an intelligent and disciplined manner in the service of truth. This ensures that the practices remain in the practical realm of relating to the world and other sentient beings in a joyful and unselfish way. In texts such as the Yoga Sūtra, the Upaniṣads, and the _Haṭha Yoga Pradīpikā,_ ethical underpinnings or guidelines for behavior on which other practices rest are called the _yamas_ and the _niyamas._ The precise list and number of yamas and niyamas vary from text to text, with some identifying fifteen yamas, and others fewer. In the Yoga Sūtra, the yamas and niyamas are identified as the first two limbs of the eight-limbed path of Aṣṭāṅga yoga, indicating their importance and their underlying purpose of providing a grounded and non-self-centered context based on relationship with others, from which our practice may grow.
_The Yamas_
Having a taste of the nature of reality and our intrinsic intimacy with all other beings, certain imperatives, or yamas, present themselves. From a truly yogic perspective, each of the thousand and one choices we must make every day has to be based on the necessity of love and relationship and a clear perception of reality. Many of the major choices we face must be calculated around our self-interest or the interests of our family, friends, or community. Consequently, conflicts and ethical dilemmas are inevitable, and poor choices at these times can have deleterious effects on many fronts.
The yamas address the issues that arise in difficult situations that cause suffering and confusion. They provide guidelines for decision making and actions that not only allow us to consider our self-interest but also set a context for understanding the impact that our choices and behaviors may have on others. When, in our everyday perspective, we embrace the notion that everything we do is in relationship and therefore has some impact on others, then all our choices—especially those that have ethical nuances—hit us deeply on the visceral, subtle levels of the body that are designed to inform us when we are acting in alignment with what we know to be true (our "gut" feelings). The yamas offer a full, wholesome context within which we learn to act ethically from a place of truth, kindness, and compassion. Equally, we see that if the yamas are taken as steadfast rules and regulations, if they are applied blindly or with our own self-interest in the forefront, then they may cause harm.
_Ahiṁsā,_ the first of the yamas, means to not kill or harm. On a simpler level, ahiṁsā just means to be nice. This refers to being nice to ourselves as well as to other beings. Because of the nature of biological life, we are largely unaware of the effect of our actions on many species, so we do the best we can, seeing ourselves in all beings and acting accordingly.
Like everything else, the yamas are embedded in an understanding of relationship and, in fact, have a special link with one another—as each is built on and stems from this first yama and the notion of nonharming. So even if you are a dedicated and sincere yoga practitioner, if you apply ahiṁsā (or any of the other yamas) without considering the bigger picture within which the specific situation has arisen and how your actions may affect yourself and others, then you run the risk of acting unskillfully or harmfully. Many ethical dilemmas arise from strict rules around nonviolence. Imagine the difficult choices made when having to protect one being from another, from having to protect innocent children from aggressors like germs, psychopaths, or even terrorists. You might have to strike or even kill the aggressor (or pay someone to do it for you). A strict nonviolence rule will have to be violated. We must choose the least harmful way and forgive all involved. Often there is little time to make calculations before having to act in situations where inaction will bring huge suffering. Similar ethical dilemmas can arise in the field of medicine in emergency rooms and in situations where life and death lie in the balance. Dilemmas can even arise when we are dining with others—faced with dietary choices different from our own.
Just as we constantly refine yoga āsanas (hopefully in a way that is nonaggressive and nonharmful to our own body and noncompetitive with others), we have endlessly subtle aspects to practice in all the yamas. Ahiṁsā, when approached with a clear and kind mind, must be practiced while carefully weighing all aspects of any situation. The point is that you see the context of the situation, and considering all the available information, you make the best, kindest, least harmful choice of what action to take. When practiced in this way, ahiṁsā serves as the foundational yama from which all others unfold. Skillfully practicing ahiṁsā allows us to apply it and all of the yamas deftly rather than robotically.
The next yama, _satyam,_ means honesty, though it is often translated as "truthfulness." Satyam asks us to face things as they are and to honestly assess the nature of how we _know_ things to be "as they are." It also requires a willingness to make adjustments when we see our assessment is incorrect. When practicing satyam, we do not pretend to know what we don't know, taking action on the pretense of knowledge. By the same token, we do not plead ignorance and step aside when, in fact, we have enough information to allow us to act skillfully.
Honesty may be differentiated from truthfulness by an insight into how our actions may impact others and, given this insight, whether or not we should take action; this is how satyam builds on ahiṁsā. Practicing satyam is mostly a matter of common sense, though sometimes the appropriate choice can be unclear. Obviously, if a friend with an unfortunately large nose asks how he looks, we don't practice "truthfulness" by telling him his nose is huge. Instead, we practice honesty in the face of kindness and nonharming. When we practice satyam, we can face our own and others' suffering. It is particularly important in terms of satyam to keep questioning ourselves and our motives.
_Asteya,_ the next yama, means to not steal. This, of course, means not forcibly taking from others that which they believe to be their own. But there are different degrees of stealing, and sometimes the mind is all too eager to rationalize why this or that isn't really "stealing." Taking something that is not rightfully our own can take many forms and may also cause varying degrees of damage. Asteya may be blatant, like stealing a car, or it can be subtle, like allowing your insecurity and greed to drive your actions within a friendship. It can also be more theoretical, like stealing ideas—plagiarism—or taking credit for thoughts that are not really your own.
The next yama, _Brahmacarya,_ literally means "to act within Brahman," or to treat all beings as sacred or Brahman. By practicing Brahmacarya, we cultivate respect for others and maintain the perspective of not knowing, not assuming, and not introducing ego (even in the form of preconceptions) into conclusions, behaviors, and all forms of relationships. Traditionally Brahmacarya means to approach life as a student or a monk, someone who follows a disciplined and austere routine. Being a monk, of course, includes celibacy, and that is another sense of the word _Brahmacarya._ Esoterically, Brahmacarya means to move in the _Brahmā nāḍī,_ which is the subtle channel inside the suṣumṇā nāḍī. Moving Prāṇa in the Brahmā nāḍī precludes the object-making reductionism created by an inflated ego and cancels any need or desire to seek satisfaction in exploitive relationships.
In a practical sense, Brahmacarya means that we do not separate ourselves from the rest of the world and the universe; rather, we experience ourselves as living, breathing organisms that are part of the bigger whole. We practice Brahmacarya as a demonstration of our desire to live in harmonious relationship with others. In terms of sexual activity, it is practiced within the context of nonharming, truthfulness, and not taking what is not ours. Brahmacarya implies that we never treat the sexual act or our partner from an egocentric perspective as an object for our personal gratification.
Brahmacarya is a particularly important yama to be addressed and carefully considered by any good yoga teacher. Sexual intimacy is an incredibly powerful act of sensation and emotion (not to mention relationship). Under these circumstances, the ego-driven mind has a powerful attraction to the pleasure involved in the act as well as the ego boost that results from being found attractive. So the ego is prone to overlook, warp, or rationalize a person's own belief system, or ethical standards, to experience the gratification of the sexual act. It is imperative that we be aware of the complexity of this specific yama, because those of us who are yoga teachers are in a position of power and trust in relation to our students. A teacher may claim that the sexual act with a consenting student is not outside the bounds of ethical behavior. However, just like with a priest, gang leader, or politician—anyone in a position of power—consent from someone who looks to you as a leader is not coming from a free, autonomous, integrated being. That form of consent is from an actor in the circle of the teacher's narcissism.
_Aparigraha,_ the last of the yamas put forth by Patañjali, means not grasping, whether inside, outside on the gross level, or on the subtle levels. This is the constant discipline of cutting through desire and misappropriation in everything we think and all that we do. It is the ability to let go of our comparisons between self and others that can result in jealousy and instead to remain deeply rooted in the internal experience of being part of the vibrant, interpenetrating matrix of relationship called life. Contrary to what the ego-based mind thinks, our ability to let go of everything within the fabric of our experience by seeing it as empty of separate self provides true satisfaction.
_The Niyamas_
The niyamas are disciplined ways of taking action that facilitate our interaction within the world. They set a tone for understanding our own experiences and providing insight into relationship.
The first niyama, _śauca,_ means purity or cleanliness. Not only does this refer to the obviously important act of cleaning our living space, but it also refers to cleaning our own bodies, both in terms of washing and caring for the body itself and eating a clean diet and maintaining good mental habits.
Purity or cleanliness on all levels is really a matter of taking care of loose ends. We resolve our misperceptions and unskillful actions so we can perceive everything in the world around us or inside us as part of life's background rather than pulling things out and giving them an emotional charge that perpetuates patterns of attachment and avoidance. In the language of yoga, this is the arising of _sattva,_ or the harmonized, luminous state of intelligence that allows us to see things as they actually are. "Cleaning up" when taken literally, as in mopping the floor, or figuratively, as in improving our attitude, will often resolve ambiguous feelings of confusion or anxiety.
_Saṁtoṣa,_ or contentment, is a pure and excellent form of happiness that spontaneously arises when we free ourselves from the mind's constant nagging about unfulfilled desires. This niyama allows us to appreciate things as they are rather than being disappointed, frustrated, or angry about our current circumstances. This is really the secret to moving on with our lives rather than being stuck in and trapped by a specific situation. Saṁtoṣa arises when the mind lets go of its iron grip of a situation long enough to let us simply observe with great interest but without drawing conclusions or making judgments and assumptions. Letting go, we automatically tap into an endless reservoir of kindness and compassion that lies within.
_Tapas,_ which is usually translated as "austerity," really means to burn or to shine. It is the ability to reflect on and contemplate what is actually arising in our experience and to stay with the immediate experience, repeatedly returning to it with a state of open-mindedness. This role of a deeply engaged observer prevents us from slipping into the mental habit of projecting our fears, unconscious doubts, and shadows out onto the world and onto others. Negative thoughts and emotions as well as the positive ones are experienced without going along with their story lines. It is like sealing a container or pot when you're cooking. Cultivating the role of fully engaged observer is the basic technique used in any contemplative practice, including that of tapas.
As a result of tapas, the deeper and brighter intelligence is awakened. This creates the perfect environment to nourish the next niyama, _svādhyāya,_ or self-inquiry. Svādhyāya is the willingness to make an open-minded exploration and assessment of our own desires, mental habits, and ultimate nature. Of course, to actually practice svādhyāya rather than the often much more comfortable acts of rationalization, denial, and self-delusion, we must always practice satyam—we need to remain unflinchingly honest and truthful with ourselves.
One good beginning step when cultivating the practice of svādhyāya is to start to notice without laying blame or trying to change, when we justify our actions or twist our thoughts to make a situation or interaction fit more comfortably with our self-image or preconceptions. Svādhyāya can seem brutal—having to admit (to ourselves, no less) our shortcomings and mistakes and to stop covering up our imperfections. However, once we watch closely, looking past our presumptions, judgments, actions, and feelings, it is not harsh at all. A deeply satisfying feeling of living in alignment with ourselves, as if we've come home, emerges. It is said that svādhyāya culminates in a yoga, or union, with our _iṣṭa devatā,_ or beloved deity—or the truth that lies in the core of each of our hearts and connects us to the seed of truth in others.
The final niyama is the surrender to Īśvara, or God, and is called _Īśvara-praṇidhāna._ Īśvara must be understood not in the normal theistic sense as a God separate from or above all else, but as that which is the true being and the true nature of all beings. So in relation to the yamas, surrender can constitute an active giving or rendering of service to Īśvara or the interpenetrating pattern of the world by perceiving the true nature of all beings (ourselves included) and then by contributing in whatever way possible that we and others may awaken.
The yamas and niyamas afford us a context in which to practice yoga both on and off the mat. They provide an underlying framework that the wandering mind can use as a reference point for clear thinking when doubts, questions, and complicated situations arise. Working with them over time, we move from their literal application to their more subtle aspects.
THE KLEŚAS
Although you may not always be able to avoid difficult situations, you can modify the extent to which you can suffer by how you choose to respond to the situation.
—Dalai Lama XIV, _The Art of Happiness_
Why is it that so much of this world seems to be about suffering? Of course we see the obvious roots of suffering: injustice, hatred, killing, and domination of one person or group by another (which from a geopolitical perspective seem endless). But equally, for many of us, every day seems to present infinite opportunities for suffering: not getting what we "need" or want; being in the company of others who contribute to our unhappiness; having our bodies forsake us with the aging process or illness; or, the worst insult of all, doing everything "right" and dying anyway.
The First Noble Truth of the Buddha, _sarvam duḥkham,_ or "all is suffering," reflects this and, on first pass, sounds dismal at best. In fact, for many beginning students of yoga and Buddhism, this tenet causes alarm, if not the perfect excuse to abandon their studies and head straight for the door! But a second look reveals the teaching (both in Buddhism and yoga) that there is a cause for suffering, there can be an end to suffering, and there is a path out of suffering.
In the Yoga Sūtra, the _kleśas_ are identified as the causes of suffering, all of them arising from the first, which is _avidyā,_ or ignorance. This ignorance is our confusion when faced with the paradox of existence—being a distinct, individual human being with emotions, thoughts, and sensations that seem to be encased within our "own" sack of skin that separates us from everything else but at the same time, understanding intuitively that we are somehow nonseparate from everything else. For beginners, learning to feel both separate and nonseparate at the same time may feel rather like wrapping the mind in sandpaper.
Yet as we have seen, the ability to be comfortable in the face of paradox is foundational to a liberating yoga practice. In yoga, we learn to experience the seemingly distinct patterns of inhaling and exhaling by returning to the breath when the mind wanders. Through this practice, we gradually absorb and assimilate the nature of paradox into the subtle layers of body and mind. According to the Yoga Sūtra, losing sight of this paradoxical perspective is the first cause of suffering. Over and over we have a flash of insight into the true nature of things, and perhaps just as quickly the story of "we" returns, encapsulated by our own separate little universe, and we lose sight of the union of opposites.
The second kleśa, _asmitā,_ or "I am-ness," stems directly from avidyā. In a modern or psychological sense, asmitā would be the formation of ego or the concept of separate self in which image is confused with what it represents. To complicate matters, cultivating a healthy ego is an essential skill if we are to navigate through the world and through our yoga practice. The ego function keeps us safe; noticing the boundaries between ourselves and a mountain lion is essential for obvious reasons. Ego helps us to function in society—to work, play, love, and grieve in the context of others. But when it is not softened by intelligence and compassion, an unhealthy ego derails our understanding and happiness. So again, a paradoxical comprehension is imperative: we must see the ego's construction of our self-image as an amazing organization of the mind so it can make sense of things and work efficiently, but we must constantly be alert, eager, and ready to dissolve and update the images and stories the ego spins out.
When the ego function kicks in without an ability to merge into the unknown, it leads us to believe that we are indeed separate, special, and independent from others. This quickly sets up an opportunity to feel either better or worse ourselves—smarter, richer, more timid, or more miserable. The mind leaps at these kinds of extreme comparisons with others and conclusions about self based on the misbelief that we (as an ego) are the center of the universe. This is a natural deduction, since everything we know and all of our perceptual fields are uniquely processed through our individual body and ego structure. The mind must be gently and repeatedly reminded to let go of this conclusion, which is exactly what we do when we practice yoga.
Once we've fallen prey to the ego and the belief that we are separate, then the next two kleśas quickly arise. We find things we like, "need," or want and immediately have a great attraction to them; this is called _rāga._ Or _dveṣa,_ an aversion to things we hold to be unpleasant or disdainful, surfaces. The natural response is to grasp at the things we find pleasant and push away those we find unpleasant; both of these mind states cause suffering. We suffer because it seems we can never get exactly what we want, or things we don't want are forever bombarding us. Our dreams and expectations don't live up to what's happening—the perfect recipe for suffering.
The last kleśa is called _abhiniveśa,_ which is usually translated as "fear of death." In the Yoga Sūtra, Patañjali says that abhiniveśa is something that every living being experiences—from a tiny bug being swept away by a flood to the most practiced yogis and learned sages facing death. As a living organism, there is an instinctual fear of death, so in what we perceive to be threatening situations, we cling to life and also resist change.
Another way to look at abhiniveśa is as the fear of the ego dissolving—which is somewhat the same thing and possibly as scary as death; in both cases, we disappear. Or we might even say that abhiniveśa is a fear of yoga itself, because a natural by-product of a consistent practice is that longer and longer periods of dissolution—letting go of who we believe ourselves to be and annihilating the ego—naturally occur.
THE FOUR BOUNDLESS ABODES
Keeping the yamas, niyamas, and kleśas in the wings of our awareness, kindness and compassion begin to manifest. At some point, the question naturally arises, "What good is it for me to have these insights and become filled with joy, if others around me still suffer?" Compassion leads us to see clearly that since we are not separate from the fabric of the world, we are not truly liberated and happy until all beings are free. This is the bodhisattva vow: to forego our own liberation and keep helping others until all sentient beings have become enlightened (and if you've looked around recently, you know there are a _lot_ of sentient beings).
The Yoga Sūtra and other yoga texts offer practices that help us to fulfill this desire to help others skillfully and perhaps endlessly. These texts teach us how to stay in healthy relationship with others while we follow the path on which we ourselves may awaken from this dream state we call life. Many traditions of Buddhism also offer the same basic practices, which are called the four boundless abodes. They are a means of reining in our emotions so we may see clearly and act intelligently and kindly within the construct of our own emotional and mind states.
The boundless abodes may be practiced whenever we observe certain states of mind and of being arising in ourselves or others. The first is that when we meet someone who is _sukha,_ or happy, we practice _maitrī,_ which is friendliness or kindness—essentially, we demonstrate love. Unless you're in a really bad mood, this one is easy. If you're open to the presence of someone who is happy, it's difficult not to feel the same. Think of a baby, so full of enthusiasm and busting with excitement as she wiggles and kicks and coos while being cradled in her mother's arms. Or imagine a kitten or puppy or baby chimp and notice how hard it is not to, at least momentarily, feel an inward smile!
When we practice maitrī, there is a noticeable impact on our own nervous system that allows the mind and emotions to clear (even if ever so slightly) and our innate sense of happiness to radiate out to others. In a situation where maitrī or love is arising, it is good to notice the tendency for the mind and ego to jump in and immediately become attached to the circumstances or feelings, or to become fearful or unhappy when the situation changes. So just as it is important to practice maitrī, it is equally important to release attachment to the residue so the mind and emotions remain clear and open.
On the other hand, when we encounter a situation where someone is _duḥkha_ (suffering), we do not reflexively run away from the situation, nor do we dovetail onto the suffering, becoming somber and depressed ourselves. Instead, we practice karuṇā (compassion), the second of the boundless abodes. Compassion is a complex state of being that arises naturally when, in the face of suffering, we tap into our own nature as loving beings and tune so deeply in to another's suffering that our own ego function begins to dissolve. In this state, we can _feel_ the suffering of the other without confusing the boundaries that separate us from the other. By dropping into our own physical experience, checking in on deep visceral levels within ourselves as to who we are and what we perceive the other to be, we are able to offer the necessary level of clarity of intention to help without our ego function—and our own needs—interfering. If we detach ourselves from the situation, we cannot connect to our intelligence and see what action needs to be taken. Compassion arises naturally in the face of suffering when we are in a state of complete presence and open-mindedness, skills we hone in our yoga practice.
The third boundless abode is called _muditā,_ which may be called sympathetic joy. We practice this when we are in the presence of someone who is _puṇya._ Puṇya is often translated as "pious" or "holy," but it is better to consider it as virtuous, clear, and truthful. Someone who is truly puṇya radiates a sense of wholeness, clarity, and great presence. This stimulates a similar feeling within others. It is said that when we meet someone who is puṇya, we respond with sympathetic joy—delight in their experience, even if we are not quite able to feel the extent of it fully. Experiencing sympathetic joy is partially dependent on your own attitude and state of mind. If you feel that things are limited or limiting, that there is never enough to go around, or that others are out to best you, then fear and aversion get in the way of connecting fully to the joy that resides within each of us as the basis of life, that which is felt in the presence of someone who is puṇya.
The fourth boundless abode is practiced in the presence of someone who or a situation that is _apuṇya,_ or in a traditional translation is "non-holy." Someone who is apuṇya is not virtuous, clear, or truthful and may even be toxic. To varying degrees, we run into people and situations that are apuṇya on a regular basis. A tyrant would be considered apuṇya, and we encounter less vile versions of tyrants in those who manipulate or have a total disregard for others. The yogic advice for dealing with apuṇya is to act with what is called _upekṣā,_ which can be translated as "indifference," "nonattachment," or "equanimity." A vivid metaphor for such nonattachment (pure _vairāgyam_ ) is in the _Aparokṣānubhuti_ :
brahmādisthāvaranteṣu
vairāgyaṁ viṣayeṣvanu
yathaiva kāka viṣṭhāyāṁ
vairāgyaṁ tad dhi nirmalam
The indifference with which one treats the excreta of a crow—such an indifference to all objects of enjoyment from the realm of Brahmā to this world (in view of their perishable nature) is verily called pure vairāgya.
_—Aparokṣānubhuti,_ v. 4
Crows are notorious for eating the worst of the worst—decaying flesh, garbage, sewage-laden scraps, and so on. So the excrement of a crow is considered to be particularly disgusting and foul. Nonetheless, from the yogic perspective, _everything_ that manifests, including crow excrement, is regarded as Brahman or God. Instead of rejecting the excrement, we see it in a level-headed, neutral way, with equanimity. This graphic description powerfully demonstrates the vision of interconnectedness with godliness at the center of every manifestation. It points to the necessity, when searching for truth, not only of seeing God in all beings, situations, and manifestations, but of seeing all beings in God. The example demands we see through concepts of mind, not making one thing more "holy" than another (our concept of God, for example) or making something of lesser value (the excrement). This is the same as discriminating awareness or enlightenment, and it is absolutely necessary for true relationship and happiness to occur.
VIVEKA KHYĀTIḤ
_Viveka khyātiḥ_ is a term used to describe the skill of discriminating awareness, or the ability to see the truth clearly. It is the capacity to cut through the illusions of mind. Viveka khyātiḥ is the fruit of the practice of meditation, when the mind stays with its content long enough to see the subject matter in context and as part of its background. The content of mind is then observed in its open, sacred, irreducible form, which gives us the insight that everything exists within relationship. Viveka khyātiḥ allows us to use the ability of the mind to make symbols, categories for perceptions and thoughts and games out of concepts, conclusions, and goals without mistaking the symbol for the thing symbolized or the map for the territory. With this clear vision, we begin to see through the ego function and can experience, in an embodied way, the emptiness and "nonabsoluteness" of our beliefs—the importance and incompleteness of theory in finding reality or true relationship.
Viveka khyātiḥ is an essential skill for any serious yoga student to cultivate and maintain. Due to the abusive ways in which some people in this world discriminate against one another, a beginning student may misunderstand the term "discriminating awareness" and think of it as a bad thing. Discriminating without a sense of truth and compassion, without taking full account of the context of a situation or the relationship of another, or discriminating through a lens of prejudice or preconception is when the process of discernment goes wrong. But discriminating awareness is the ability to remain fully aware as we determine the relative truth of a given situation. It is the capacity to see through preconceptions, illusions, and prejudices. Viveka khyātiḥ is a direct path to seeing the essence of situations—even those that are difficult, complex, or distasteful—and experiencing compassion.
EMBODIMENT, SAṀSKĀRA, AND AWARENESS OF PRĀṆA
Prāṇa links what you're thinking and perceiving into its background. As embodied beings, all that we experience is processed through Prāṇa (breath) and citta (mind). These two distinctly different layers of our experience are inseparable, though we are prone to separate them in our mind's eye. Sometimes they are joined in our awareness into one powerful experience. Most of the time, however, the citta wanders around, and the physiological background (Prāṇa) is less apparent in our awareness. In a contemplative practice, we take the attention (citta) and turn it straight to Prāṇa. It is said that Prāṇa and citta move together like two fish swimming in tandem; where Prāṇa goes, citta follows, and where citta goes, so too flows Prāṇa. When we are aware that they move together like this, we experience our whole body (from subtle to gross) as unified, alert, and harmonious, giving us a tangible, embodied understanding of healthy relationship. When our experience of breath and mind is fragmented, so too is our embodied state and our visceral understanding of healthy relationship.
Tapping into the internal forms of the practice stimulates the subtle layers of body and mind in a profound trust of this process of vinyāsa, or the joining together of Prāṇa and citta. It is impossible to separate mind from Prāṇa on the subtle level, and the inner forms of practice gradually awaken an awareness of this unity. This deep level of experience is rooted in what is traditionally called the subtle body. Mind and Prāṇa serve inseparable functions in the process of perception, but the immediate perception is embedded in Prāṇa (sensation). Due to the overlaying of Prāṇa and mind, the subtle body is not a matter of pure abstraction and data processing but a virtual storehouse of deeply rooted clumps and knots of unconscious emotions, internal sensations, concepts, memories, tendencies, and stories. To affect and transform the subtle body as we practice, we use special forms, images, and ideas to take us immediately back to a pure perception of whole, balanced patterns of Prāṇa. These internal forms of embodied perception are delicate and can easily become destabilized and distorted by conclusions of the mind, even simple conclusions such as the identification of the subtle states themselves.
At the same time, without mental process, there is no understanding. The mind works by creating contingent dualisms in which symbols for things (including those we experience through our sense fields) are usually understood to be separate from the things they symbolize. Though we experience mind all the time, there is nothing in the content of our experience that we can point to as being the mind or the self, although we inevitably try. To feel or even imagine this deep experience in which our attention (citta) stays and merges with the immediate sensations (Prāṇa) is nearly impossible, yet it is so precious and important. In fact, the initial definition of yoga in Patañjali's Yoga Sūtra reflects this:
Yogaḥ citta vṛtti nirodhaḥ
Yoga is the suspension (nirodha) of the modifications (vṛtti) of the citta.
_—_ Yoga Sūtra, Samādhi Pāda, v. 2
In yoga, when there is a presentation, a _vṛtti,_ of some content to the conscious awareness, the habitual, unconscious response is suspended. The vṛtti is not accepted, rejected, or further wrapped in concept or category, but is perceived in and of itself. This has a radical and wonderful effect on both our understanding and our deep unconscious store of memory and conditioning.
SAṀSKĀRAS
As yoga practitioners, we meditate to understand and come into Prāṇa as it is. Surrendering to this process of merging an unbiased awareness of body and mind marks the point at which the subtle body starts to be purified by dismantling what are called _saṁskāras._ Saṁskāras are old overlays of memories, emotions, and conditioning onto the sensations within the subtle body that are experienced through the senses, or Prāṇa. In other words, saṁskāras are what we would call unconscious habits, memories, and conditioning. When we unravel saṁskāras, direct perception is no longer automatically reduced to a memory, an idea, or a theory that elicits an automatic, unconscious response. Breaking the habitual loop of "knowing" what a perception is allows a fuller grasp of the content and context of the perception and possibly a glimpse of infinity. The nature of reality comes into the foreground, and we see the interdependence and relationship of all things rather than having a perception stimulate an involuntary journey into the self, memory, and imagination. These kinds of involuntary journeys are our habitual way of being in the world—the hub of the wheel of saṁsāra.
Most saṁskāras are neutral. They fuel the ways we perceive and interact day to day. Some saṁskāras are bad; they perpetuate ignorance and cause suffering. Some are initially good and later get in the way, like a good scientific hypothesis that oversimplifies. Others are profound and sometimes religious saṁskāras founded in a one-time encounter of a deep, possibly mystical experience. In this situation, it is not uncommon to subconsciously attempt to make even these deep experiences into something familiar by reducing them to a theory about the occurrence so we can hold on to and possibly reproduce it. The reductionist mind mistakes the joyous experience itself for the particular content of the mind at that moment.
Perhaps nowhere in Western literature has this experience been more beautifully and famously captured than in Marcel Proust's _Remembrance of Things Past,_ in which he described his experience of a cup of tea and a petite madeleine served to him by his aunt:
No sooner had the warm liquid mixed with the crumbs touched my palate than a shudder ran through me and I stopped, intent upon the extraordinary thing that was happening to me. An exquisite pleasure had invaded my senses, something isolated, detached, with no suggestion of its origin. And at once the vicissitudes of life had become indifferent to me, its disasters innocuous, its brevity illusory—this new sensation having had on me the effect which love has of filling me with a precious essence; or rather this essence was not in me, it was me.
Since the body is a sensing organism and the mind is the interpreter, what most frequently happens when we have a brief encounter with the subtle body realm (like Proust's) is that this dichotomy of understanding immediately kicks in, and the mind "understands." It labels the experience as good, bad, radical, mundane, and so on. Unless we consciously intervene, our memories and associations with the perception live encased in our story and our myofascial webbing within the subtle body. When we have an experience, like that of eating a madeleine, overlapping yet unrelated patterns of sensation and thought (saṁskāras) are established. We may have a moment of insight, but then the mind leaps back into "understanding" (its job), the ego function perks up to identify with the insight (its function), and we latch on to the story line; this process reinforces the saṁskāra. When we happen into the deep levels of experience within the subtle body, the feeling is so powerful and enticing that we may try to re-create the same situation in order to experience that unspeakable feeling again. But because the mind has reduced, identified, categorized, and codified the inexpressible, the experience is never the same. As Proust, after describing attempts to re-create his experience, put it, "The truth I am seeking lies not in my cup, but in myself."
We learn exactly this through a yoga practice. Although it is perfectly natural to look outside ourselves for the source of profound insights, the truth lies not externally but within. Once we experience and evaluate each moment fresh, there is a sense of vastness and harmony merging patterns of citta into Prāṇa.
LOOKING CLOSELY
When we perceive an external or internal object either correctly or incorrectly, our subtle body produces a vṛtti, an image or idea of that object comprising a variety of sensual and abstract categories combined with our memories and associated body patterns. This patterning of perception then resonates through the nāḍīs. In a contemplative practice, the whole phenomenon is observed through to the end and clearly exposes the phenomenon itself.
Every perception, or vṛtti, is a combination of an "external" form overlaid by internal categorization that determines our actions and the structure and form of our subtle body. If the external object is "brought closer" and held more delicately, as in meditation, we are at the borderline of internal/external name and form. The ability to be "awake" enough at this instant to consciously participate in the process of perception as it occurs and to pause—even for just a single breath—results in a moment of insight. This, of course, is an advanced skill. It is easier to become aware of the circuitry of the flow of Prāṇa as it occurs in the body and then to balance that circuitry through a natural vinyāsa to keep the perception open and hold the object with clear mind. That's a fine place to start!
Using the brilliant sword of discriminating awareness to cut through and eliminate overlays of mind and to understand what is right in front of us allows insight and compassion to spontaneously arise. The middle path allows the mind, body, and environment to come together and work in harmonious relationship. They then do their processes in a way that creates brilliant, self-manifesting truth and also reveals self-dissolving, nondual patterns.
_3_
Fluid Movement
_Alignment, Form, and Imagination_
THE BODY, BOTH SUBTLE AND GROSS, IS A STOREHOUSE of unconscious and semiconscious tensions, retractions, expansions, and discrete movements of Prāṇa. All these movements are based on past and current thinking—a form of misunderstood imagination that picks out objects, constructs self-images, makes goals, and charts out theories soaked in the emotions of attachment and repulsion. Through the integration of Prāṇa and citta, intelligent movement effortlessly manifests within this maze of conditions that arise, and we are slowly freed from preconditioned patterns that keep us both mentally and physically entangled.
For yoga āsanas to be a truly useful aspect of this unraveling, they must be grounded and properly aligned. Then the imagination is used intelligently so poses have an open, pleasant quality that allows meditative states of mind to arise easily and so that strong, balanced, delightful currents of Prāṇa flow effortlessly. A healthy āsana practice is liberating. It ultimately supports insight that frees us from a misunderstood imagination and its habitual patterns and suffering. However, it is essential to use the imagination with boundless creativity to understand and free ourselves from it!
In the yoga traditions, imagination has been used extensively to embody idealized forms and functions among gods, goddesses, and heroes as a way of breaking our habitual patterns of perceiving our own body and feeling required movements, attitudes, and characteristics within yoga poses. Embodiment practices have contributed to the evolution of yoga, uniting subtle and esoteric teachings within the everyday physical realities of having a body. Profound and deep insights we might access through our imagination can be gateways for understanding how to live on the practical plane—a merging of practice, tradition, lineage, and mythology into "real" life.
The viability of yoga as a living art rests in the fact that even in the imagination, it never becomes stagnant. Although we use preconceptions and formulaic thinking in vinyāsa practice, the practice is not based on them; rather, it balances, exposes, and contextualizes preconceptions. Herein lies the value of visualizing whole-body patterns that can prove more insightful than attempting to understand movement from a dry, analytical approach. Whole-body patterns of alignment are associated with deep feelings, emotions, and sensations that we perceive directly rather than patterns fragmented by conceptual ideas. Whole-body patterns unite subtle-body patterns to support and inform movements. Of course, they typically do not describe a local anatomical structure in detail, so it is important to embrace scientific forms of study along with visualization; both approaches need each other to evolve.
Mindful use of the imagination in yoga reveals that impressions and stories that come into our awareness depend on context and are therefore not absolute and exclusive of other impressions and stories. This means there are potentially unlimited brilliant metaphors and images that can be used to describe or imply a yogic state in the body. On the other hand, it does _not_ mean that we should fall into a relativistic mind-set in which any metaphor or myth is an adequate (or even decent) description of alignment or yogic states. Limitless stories, forms, and metaphors correspond to deluded states of mind and can induce those unhappy states. Good metaphors are rare, like brilliant art and insight. Metaphors must be precise, expandable, visceral, and clear to really do the trick. All of them are only metaphors and must eventually be released and allowed to dissolve so they can do their job. The importance of lineage in image making is tantamount. All effective imagery shares a similar flavor, though the forms it takes may be radically diverse. Tradition is like a flame of intelligence passed on and shared by countless practitioners over centuries, using many versions of metaphor, philosophy, and technique.
Visualizations can be helpful with alignment in specific poses too, such as imagining the heart "floating," which can stimulate a full feeling of expansiveness in a pose. But where subtle anatomy is most useful is in shedding light on levels of alignment and form that govern obscure aspects of the practice, such as Mūlabandha, and whole-body patterns that connect us from top to bottom. By practicing āsana with some of these patterns in the nervous system, the poses are enhanced, and perhaps more important, the affected parts of the nervous system are primed for meditation.
OVERVIEW OF SUBTLE ANATOMY
During the Italian Renaissance, an understanding of human form came to life as great artists of the time became anatomists, peeling back the skin of dead bodies and dissecting corpses to study the intricacies of form and structure in fine detail. Some, such as Leonardo da Vinci and Michelangelo, were inspired to explore and broaden their understanding of the body in action, and their art became infused with a level of realism never before imagined. Their direct study of anatomy, merged with their innovative imaginations and artistic skill, transformed the face of art forever. As a yoga practitioner, fusing together a knowledge of anatomy with the art form of your imagination—deeply feeling movement and sensation, along with patterns of connection and breath—may not make you the Michelangelo of yoga, but it will add a uniquely clear means of inhabiting your own skin.
To begin experiencing our own subtle anatomy, it is helpful to have studied classical anatomy and artistic renderings of the body so we have a general idea of the landscape of human form. Imagining our own structure as an overlay to clear images of anatomy, while focusing on feelings and sensations as they arise, provides an embodied, broad-spectrum experience of the subtle body. In this context, it is helpful to establish a vocabulary specific to these elusive layers of understanding. Doing so is referred to as _sādhanā bhāṣā,_ or practice language. Every group or school (and ultimately every practitioner) has a unique and often abstruse sādhanā bhāṣā; words, images, and myths that give us markers of understanding and serve as memory cues so we may easily return to and build on the insight that inspired the vocabulary.
As an example, to assimilate an understanding of what it feels like to sit with balanced, well-aligned posture that connects you from the crown of your head to your pelvic floor, you might visualize a detailed image of this form and then assign some words to the experience to serve as anchors within your nervous system so your body can remember the experience and reproduce it more easily. Imagine, for instance, that you can _feel_ sensations (from subtle to gross) within your body of what it would be like if you were royalty—sitting up straight on a throne, wearing a pair of bright gold earrings and a jewel-studded crown with a golden feather rising out of the top. The more details you can invoke, the better; imagine what you must do to remain feeling regal. What minor adjustments do you need to make in your feet or sitting bones to establish a stable and balanced base from which your spine can feel strong, elongated, and supported so that your heart has an open, expansive feeling? How does this buoyant feeling in your upper body provide a complement to the weight of the crown and the tug of the earrings on your earlobes? Imagine how you actually _feel_ open and light thanks to the sense of a feather shooting up and out from the top of the crown and your head. You notice that a steady gaze helps your tongue to quiet and soften, which automatically releases your palate and allows your mind to settle; with time, you feel fully aligned and tapped in to the central channel of your body. Even though you haven't instructed yourself to expand your rib cage, spread the skin on your lower back, or release tension in your feet, you notice that these things are indeed occurring. The natural intelligence of your body, triggered by the image, has taken a front seat to preconceived concepts about what you should be doing to achieve proper alignment. After practicing with the image for a while, you find that if you simply think, "bright gold earrings" (or another sādhanā bhāṣā), it is enough to activate the entire response in your nervous system and body.
This is how subtle anatomy works. It taps into the body beneath the level of conceptual mind and makes full use of the mind's ability and agility to connect and assimilate all echelons of information. Subtle anatomy is a way of getting our sense fields involved in the process of understanding and creating stunning moments of direct perception of what is really happening in and around us.
ASPECTS OF SUBTLE FORM
This section includes thirteen important visceral visualizations that describe internal forms of the practice. They are used as sādhanā bhāṣā instructions for various āsanas so that complex movements may be described in fewer words. They can be contemplated separately when not "officially" practicing āsanas and are also essential to certain integrating yet complex movements within poses. All of these imagination-based expressions of internal forms are related to actual anatomical structures, but they have a generous overlay of imagery to produce an experience of alignment. With practice, they will eventually produce distinct patterns within body, mind, and movement that somehow seem familiar.
These imaginary forms require attention and repetition over a long period of time, because each brings together at least two different and complementary movements as well as the feelings, thoughts, and emotions that come with them. Although they are rich with imagery, releasing all grasping and neediness triggered by them makes them more effective because they then resonate within us. At the same time, we return to the specifics of the forms repeatedly as a means of rekindling the connection of imagination and structure so we may fully embody a pose or movement.
_Mūlabandha_
yanmūlaṁ sarvabhutānāṁ
yanmūlaṁ citta bandhanam
mūlabandhaḥ sadā sevyo
yogyo 'sau rājayoginām
That which is the root of all beings and the complete binding (cessation) of thought is Mūlabandha. It should always be served and attended to since it is fit for Rāja yogins.
_—Aparokṣānubhuti,_ v. 114
To some Aṣṭāṅga Vinyāsa practitioners, trying to master Mūlabandha is like attempting to grab the brass ring on a carousel ride. You "win" when you get it, but trying for it can become an obsession that ruins the ride. Grabbing the ring takes opportunity and skill, and it is a treasure worth striving for if you like merry-go-rounds because the reward is usually another ride on the carousel. On a quest for Mūlabandha, however, striving too hard can be counterproductive. Under the correct circumstances, Mūlabandha will manifest; when it does, the most glorious reward—like on a carousel—is the opportunity to try it again in another round of practice. The most appreciated gift for any yoga practitioner _is_ , after all, more yoga!
In search of Mūlabandha, we may begin with some aspects of a form that are not particularly subtle yet are still observed with an open, focused mind so that deeper patterns connect through the body. It's likely you can contract your anal sphincter muscles; many organisms do this on a daily basis, and the feeling is probably familiar. You could stop right here and say, "Ah! I know Mūlabandha." As encouragement to beginners, a teacher would say, "That's good, but keep practicing." By observing those sensations and the attitudes and thoughts associated with them, you are able to isolate the supporting muscles, then the rest of the body starts responding, creating a three-dimensional map within your awareness around the focus of attention on the area of the mūla, or pelvic floor. Eventually other images emerge through which you can chart the experience. The attention becomes pure and undistracted, and the mind stops feeling compelled to draw conclusions. Like magic, all levels of the body and mind fall into line, and we then say that Mūlabandha is coming along. (See an illustration of Mūlabandha in Appendix 4.)
If you treat Mūlabandha like a trophy to be attained, it will disappear immediately. Instead, treat it with great devotion, as a deity invited to appear and rise up like a flame at the center of the pelvic floor. Take all the necessary steps and make all the proper preparations to greet an honored guest and then wait. Perhaps one day she will emerge, or perhaps not. More important than "mastering" Mūlabandha is the joyful, unending process of keeping the heart and mind connected and open for the moment compassion emerges in the form of Mūlabandha. This kind of practice is very advanced and profound, and it erases the subtext of ego striving.
Mūlabandha is a highly intimate and individualized part of any āsana practice, yet it is key to integrating the body, mind, and emotions. You must be patient. Finding images that work for you are paramount, as are constantly rethinking, reexamining, releasing, and redrawing your imagery and preconceptions about what exactly Mūlabandha is and what you are experiencing. You feel Mūlabandha in your mouth and then up and down the central line of your body, even down into its base in the pelvic floor, where it is actually occurring. In a broad sense, Mūlabandha is the basic pattern of conjoining opposites like prāṇa and apāna, expansion and contraction, or focus and horizon. These unions can take place at any station (any cakra) along the central axis of the body, facilitating meditation on that specific spot and eventually pulling in patterns of unification from all the other stations.
One method for bringing focus to Mūlabandha is first simply to _imagine_ you can feel your pelvic floor. You might envision the bony structure of your pelvis and the muscles that connect from side to side and front to back to form the "floor" of the bowl. Onto this you could overlay an image that makes the territory come to life. For example, you could imagine the four corners of the pelvic floor as four distinct flower petals that make up a type of dais in the base of the body. You might imagine yourself as a deity seated right in the middle, and as you settle into your pelvic floor, you experience the perfection of integration. In this case, the deity is the Mūlabandha, and it is naturally effortless in a fully awakened state that bypasses preconceptions of Mūlabandha in which "you" have to "do" something to "make it happen."
Another of many indirect ways to feel Mūlabandha is to meditate on the seed mantra ṬHAṀ with your mind focusing on your palate. As you say the mantra repeatedly, it resonates through your nasal septum, releases the upper back of the palate, and moves down into the heart. At the same time, you can visualize the iṣṭa devatā (beloved deity) sitting in your heart and expand the image to include a nanoform of the deity in the shape of a subtle flame or mark within the various cakras. After a time, the relaxation associated with the imagery clarifies and descends through the body and the image returns through the pelvic floor in the form of Mūlabandha.
Kinesthetically we can also tap into Mūlabandha by imagining that the breath moves by being pulled up like a thread out of the fabric of the center of the pelvic diaphragm. Just as in needlepoint where the cloth must be smooth with an even tone and secured to a ring so the movement of the thread is precise, the toning of the "cloth" of the pelvic floor contributes to a detailed establishment of Mūlabandha. Then the thread of the breath can be drawn up slightly toward the back of the body as if feeding it into a straw—the central channel. As the image of the breath is pulled up and back, the center of the pelvic floor cloth lifts delicately and evenly to let the thread through. This causes a stimulation of the pelvic floor muscles in a way that isolates them from the other muscles around the hip joint area so there is an unusual sense of symmetry front to back, left to right, and up to down. The drawing up of the center of the pelvic floor corresponds to the even release of the sitting bones down toward the earth or the release of the coccyx and pubic bone down (or all four at once). Keeping this pattern and residual tone while canceling the peripheral asymmetries in the surrounding muscle area is the basis of the Mūlabandha action.
_Releasing the Palate_
ata ūrdhvaṁ tālumūle
sahasrāraṁ saroruham
asti yatra suṣumṇāyā
mūlaṁ savivaraṁ sthitam
Above the ājñā lotus at the root of the palate is the thousand-petaled lotus in which is the root opening of the suṣumṇā.
_—Śiva Saṁhitā,_ ch. 5, v. 156
At the opposite end of the central channel from Mūlabandha is its complementary subtle-body form—a release of the palate. When we release the palate, the _tālu,_ we feel a pattern along the entire core of the body, and that pattern penetrates out and beyond.
From an ordinary perspective, the palate, situated at the roof of the mouth, is associated with taste. Gastronomes and artists speak of having a good palate or good taste, and indeed, from a yogic point of view, the palate is associated with _rasa,_ which translates as "flavor" or "taste." Rasa also means juice, and the palate deals with the juices associated with flavor as well as other fluids from deep within the body, such as tears, saliva, and mucus. Rasa is not restricted to taste sensations from food; it is also associated with refined emotional flavor.
From a more esoteric perspective, it is said that the basic emotions we all experience can be transmuted through yoga when we meditate on them as rasa, a metaphorical flavor in the palate. Once transformed through observation, the "juice" of an emotion is transformed along the central axis so the emotion is not projected out into the world. With the release of the palate, emotion can become a seed for insight and skillful action rather than a reactionary state of being. For example, anger in its raw state may feel hot, fiery, and intense. If we identify with the sensations and our associations with the feelings that arise—thereby activating the anger—we are likely to be swept away by a constellation of feeling, thought, and sensation into what is usually less than skillful action. But if we focus the mind to release the palate, observing condensed droplets of the sensations of anger dripping toward the back of the palate, we can sense the essential rasa of that emotion. It is then automatically transmuted into a powerful quality of awakened, clear mind. This process is the same for any emotion: we observe it and release the palate, and its rasa transforms into a form that is beneficial rather than a root of suffering. Awareness in the palate, therefore, is associated with the sattva, or the harmonizing mode of energy that spontaneously arises when we have the pleasure of knowing, understanding, or even allowing something to transform.
Physically the palate is like an exchange center between the inner world of the organism of the body and the outer environment. It is also a storehouse for our past and present feelings, thoughts, and sensations—an epicenter connecting the mind and physical makeup from top to bottom. As a sounding board for language expression, the palate connects our abstract and subjective thoughts into the world. It is where the input we take in from all around is quickly sorted. With a released palate, this data can be channeled into a pattern or form that frees the mind and body for meditation.
If you look at the anatomy of the palate in an anatomy book—or even better, and perhaps scarier, open your mouth and look in a mirror—it's remarkable to see that you carry such a complex combination of strange and urchin-like structures around with you all the time. With your mouth wide open, you probably first see the shape-shifting muscle of the tongue resting beneath the bony, dome-shaped hard palate. The delicate area in the back of the mouth forms the soft palate and includes the uvula. The hard palate extends up into the skull to form the nasal septum, which is not visible when you open your mouth, but you can feel the intersection of the hard palate and nasal septum because its juncture vibrates when you talk, sing, or hum. Seeing, feeling, or just imagining the anatomical relationship of these structures allows you to tune in to the sensations within them and gives you a base for subtle-body visualizations involving the relationship of the tongue to the different sections of the palate. Awareness of the palate helps you understand the sinus passages and allows you to feel how their extensions can be experienced as reaching on and on out in patterns of sensation throughout the whole body.
To release the palate, begin by relaxing the tension in your jaw and mouth as if you were ceasing all desire and effort to talk in order to fully listen. Allow the gaze to be soft and clear along the line of the nose. Relaxation of the palatal muscles creates a distinct sensation pattern that is open and full and can be felt in the back of the mouth and head, up inside the skull, and down through the tongue and throat. Eventually there is a feeling of nectar, an elixir made of kindness and compassion, dripping down to fill the nāḍīs from what can be visualized as a moon-like cup at the base of the thousand-petaled lotus flower that spreads at the crown of the head.
The simplest way to start feeling this release of the palate is to meditate on the sensations associated with the contact of your tongue with the upper part of your mouth and, at the same time, bring awareness to the sensations of the breath flow in your nostrils. This technique is similar to just listening closely to sound; gazing with gentle, empty eyes; or smiling softly while focusing on any associated feelings that arise in the head or face. Perhaps you can induce the same feeling by imagining you are looking closely at the _Mona Lisa_ and tuning in to the feelings so vividly captured in her timeless smile. Contemplating the face of someone who is extremely kind or compassionate can have a similar effect. There are many ways to find compassion and to tune in to the sensation of the palate releasing.
Releasing the palate is also a reflexive physical action associated with being moved deeply by an aesthetic experience, having a great insight, being in love, or simply getting a joke. In all of these situations, our philosophical inquiry into beauty, deep thought and emotion, or an understanding of root paradoxes (as in a good joke) is the mind attempting to understand the whole of reality. Yet in instinctively "getting" paradox or beauty, we know that we will fully understand only through a moment of not knowing. In this instant of insight, the mind is stunned, and the Prāṇa stops moving asymmetrically in all the various paths to which it is accustomed. When Prāṇa stops moving, the mind stops too, and there is a spontaneous, deeply satisfying sense of release at the root of the palate that spreads throughout the body.
The objective in all of these meditations and visualizations is to create an optimal environment in your mind and body for you to be able to establish feelings associated with deep understanding and compassion. With no conscious technique of physical manipulation, when you feel kind or compassionate—or when you have an "aha" moment because you understand the interdependence of things—the root of your palate automatically begins to soften and release. While cultivating a release of the palate, or any of these subtle-body forms of alignment, we must define, redefine, and then redefine again that specific form so the ego doesn't think that it has mastered or achieved the action. Then we may go deep and disappear into the sensation patterns as they arise. The actions themselves involve dissolving the sense of ego function, and this is why they are illusive.
_Psoas Line_
The psoas muscles flow like ribbons on each side of the body, just to the side of the pubic bone through the groin; they are attached to the uppermost inner back edges of the femurs, or thighbone. The ribbons move out through the pelvis and up the sides of the lumbar spine, where they attach to the upper lumbar vertebrae all the way to the twelfth thoracic vertebra (T12) at the base of the rib cage (see illustration on page 92). This is where the diaphragm attaches as well. As muscles go, the psoas are among the most elegant due to their length and smooth, even fibers, as well as the fact that they are unifying muscles, connecting the top half of the body to the bottom half. Possibly because they are structurally designed to act as go-betweens, the psoas muscles seem to have an emotional quality to them. Like the "mother" muscles of the body, they often facilitate movements and make things work in a unified manner from top to bottom, even in some situations when it is structurally impossible for the muscles themselves to physically connect and work in this way.
With just a little imagination, we can extend the image of these unifying muscles and feel continuous lines that stretch on each side from the fingertips through the arms, the sides of the torso, the pelvis, the legs, and the feet. We call these lines of awareness the "psoas lines," and they become a remarkable tool for stimulating the experience of interconnectedness as we dissolve into subtle-body feelings and sensations that unify breath, body, and mind.
If we release and stretch the psoas muscle on one side while inhaling fully, it can feel as if there is an empty tube in the approximate area of the released muscle. This sense of a cylindrical shape deep in the abdomen can be imagined to extend up and down the side like a "pelvic nostril," and creates a pathway for Prāṇa to flow connecting the top half of the body to the bottom half. Maintaining a feeling of openness and release along the line, we can then stretch the psoas muscle itself, which creates a subtle-body pattern that, when properly organized, wakes up the pelvic floor. The muscles that oppose the psoas—such as the hamstrings—are also awakened by this awareness to create a sort of void or spacious feeling behind and to the sides of the belly and back in the pelvic basin. This feeling of spaciousness is a primary component used in meditation on the idā nāḍī, the piñgālā nāḍī, and the pelvic floor.
To trace the entire inhalation in this way, while keeping associated psoas patterns awake, is challenging because the tendency of both the inhaling and exhaling patterns of breath is to break away from each other, particularly when they reach peak expression. For example, when we inhale, the expansive feeling in the core of the heart naturally occurs, yet at the top of the inhalation, it is difficult to maintain the sense of spreading and expansion of the ribs in the lower back—a primary pattern associated with the exhalation. When the physical patterns represented by the ends of the breath break apart—one becoming dominant and overshadowing the other—then the smooth flow of Prāṇa is interrupted. Disruption in the Prāṇa automatically manifests as asymmetrical tension in the psoas muscles themselves, as well as tightness in the other hip flexors and possibly an uprooting of one of the sitting bones when seated. Therefore, learning to release the psoas muscles and follow the psoas lines as they extend top to bottom in the body is extremely important, though extraordinarily subtle and illusive.
Unlike some muscles, like the biceps, that thrive on being prodded and pumped to become strong and effective, if we strive too hard to release or stretch the psoas, then misaligned muscular tension or imbalances in the pelvic floor can occur, resulting in an uneven flow of Prāṇa. An uneven flow can also occur due to a conceptual misunderstanding of or poor instruction about deep internal breathing patterns. Remember, good form and technique feel really good. Poor alignment results in a sense of holding, making it almost impossible to trace the simple psoas lines smoothly with the stream of the breath.
If our movements become confused, and the psoas muscles become habituated to trying to help us release elsewhere when they can't actually _do_ anything, then the psoas muscles themselves can become chronically engaged. Though it is challenging, we should keep practicing—adjusting tiny details of uniting prāṇa and apāna at the ends of the breath, while intelligently stretching and imagining a release in the psoas muscles. Once we understand these muscles and let go of our desire for them to release, if we just breathe smoothly, they _will_ let go and establish the pattern deep within that reveals the Mūlabandha.
Releasing the psoas muscles also stabilizes T12, so as we stretch from top to bottom, the vertebra will stay positioned back, allowing full, deep flow and movement of the breath. When this subtle pattern of expansion in the back of the body on the inhalation is achieved, the psoas' companions, the quadratus lumborum (QL) muscles, will also stay released. This pattern is key in protracting the shoulder blades, which facilitates the action of reaching through the entire arm and in turn feeds back into the stretch of the psoas line.
Within the practice, it is paramount to experience the fully expressive sense of reaching "as if to infinity" from foot to fingertips along the psoas line. As a pattern of movement, this psoas line stretch is key to getting the juice out of many movements and poses; it automatically begins to facilitate a release of the palate, so there is no straining or grasping at images and technique within our movements. Of course, one more element is essential in this integrating movement, and that is the dṛṣṭi. Without proper gazing, which stops the tongue and also helps to release the palate, these whole-body patterns are unavailable. If we grasp too tightly with the literal mind at any technique, then we miss the unifying quality of the whole-body pattern within a pose.
**P SOAS STRETCH**
1. Lie on your back in _Tāḍāgī Mudrā_ (Pond Mudrā) (see page 28) and hold for five breaths.
2. On an inhalation, draw the left knee up and catch the top of the shin with your left hand. Let the leg fall out to the side at about 30 degrees from the midline.
3. Reach over your head along the floor with the right arm and gently pull the left knee toward the outer left shoulder. Already the psoas muscle on your right side is lengthening and stretching.
4. Next, firm up the right leg as you reach farther and farther with the right arm. Turn the right palm in toward the midline to protract your right shoulder blade. Be sure to stay with the breath so that sensations around the edges of the diaphragm, the pelvic floor, and the pelvic basin are included in the pattern.
5. During the inhalation, continue to reach through the right hand and arm. The right collarbone should feel as if it is rotating slightly and falling back; the front edge of the right armpit, the pectoralis major muscle, will spread and flow back toward the floor as it continues to spread away from the midline. The outwardly spiraling line of the right shoulder blade and arm links to the inward rotation of the right leg as both stretch with no perceived limit.
6. Begin to point your right foot slightly and push the heel firmly into the floor so the quadriceps and hamstrings of the right leg engage. Keep reaching in opposite directions throughout the entire inhalation. When you exhale, release the sense of reaching and allow the awareness to dissolve into the midline of the body. Then again inhale and stretch.
7. Each time you inhale, turn on the reaching pattern through the right leg and right arm. As you do this, imagine that the twelfth rib on the right side is falling a little closer to the floor and that your whole back area is spreading as it moves up your body. At the same time as you draw the left knee toward the outer left shoulder, the pelvis automatically does a posterior tilt (rolls back). Imagining the connection of your body from fingertips to pelvic floor and foot facilitates the whole-body pattern.
8. After stretching and releasing the whole-body pattern three to five times, on the final exhalation gradually release the stretch and let the brilliance of the psoas line pattern soften and dissolve into the subtle background layers of the breath, body, and mind. Bring the right arm down to your side, and allow the left leg to return to the floor next to the right.
9. Repeat the same stretch at least three times on the other side.
The psoas stretch allows you to embody a feeling of infinite expansiveness, reaching as if forever. An image that can stimulate this feeling is that of Viṣṇu in the _Trivikrama_ ("Three Steps") myth: he reclaims the whole cosmos from the egotistical demon Bali by stretching the length of his legs infinitely in both directions to step across the entire expanse of the universe. His third step places his lotus foot on the crown of Bali's head. When you tap into the psoas line, it feels as if you might be able to do such a thing. Not only does this afford an integrating whole-body pattern to emerge top to bottom, it also lengthens the actual psoas muscle on the side being stretched while giving awareness of the participation of the pelvic floor in organizing the movement. We find that as we deepen our āsana practice, attention to the psoas line in many poses and phases of movement can be exciting and liberating because it allows the Prāṇa to flow fully and without interruption.
_Cave of the Sacrum_
Awareness of the _cave of the sacrum_ is another way of beginning to feel the pelvic floor, Mūlabandha, and the lower underbelly version of Uḍḍīyāna Bandha. The sacrum, to which the coccyx is attached, is set in the sacroiliac (SI) joints in the back of the pelvic basin. Together, the sacrum and coccyx form a contour that resembles a deep cave—almost a separate chamber—below the overall abdominal cavity. The bladder, rectum, and uterus or prostate are housed in this area of the pelvis.
To feel and articulate the pelvic floor, we need to develop a sense of emptiness or spaciousness, a suction sensation in this cave, as if we had spooned its contents back and up toward the lower lumbar vertebrae. This feeling depends on having some feeling of tone in the pelvic floor muscles attached to the coccyx—as if we were holding the coccyx in place to provide stability so we could scoop the spoon of the mind back and up along the front surface of the coccyx and sacrum to "clean out the cave." Cultivating this cavelike feeling under the belly helps to fully integrate the internal form of nearly all poses and the movements between them.
Energetically the cave of the sacrum is the origin, the womb, and meditating there allows you to relax into the great irreducible mystery beyond thought. Be aware that clean and healthy bowels, as well as some of the less popular and strange _kriya_ (practices) in hatha yoga, facilitate this ability to sense the cave of the sacrum. Most of these esoteric practices, like the ability to suck water up the anus, are actually rooted in training the same muscles of the pelvic floor that establish a sense of the cave of the sacrum. Overzealous practitioners are sometimes tempted to take the kriyas—like anything extreme or strange—too far, practicing the exercises to excess or believing they are the answer to everything when in fact they are simply another type of perspective or tool among many for connecting to the subtle layers of awareness in the body.
_Kidney Wings_
The _kidney wings_ are initially defined by the position and movement of the twelfth ribs, the small ribs on the back of the body that are referred to as "floating" because, unlike other ribs, they do not attach to the sternum. The twelfth ribs lie directly behind the center of the kidney on each side of the back. Because they attach the back edges of the diaphragm to the QL muscles, these ribs are intimately involved in respiratory patterns.
When inhaling, you can feel not only the lifting and spreading of the twelfth ribs themselves but also a sense of expansion that originates at the level of the kidneys as this part of body opens back and up. By overlaying an image of wings on this kinesthetic pattern, when you fully inhale and reach up while engaging and spinning the arms, it's easy to imagine that this is what a bird must feel when beginning to spread its back to open its wings and fly. There is an integrating pattern all the way down the sides and back of the body to the kidney area, and the stretch extends on into the psoas muscles to create a feeling of suction in the cave of the sacrum. With the arms, hands, and back of the body awake like this, you may get the feeling of a smaller repeating pattern of this same kidney wing form occurring in the outer half of each hand (from the middle finger out through the little finger), and this image can serve to integrate the pattern throughout the entire body. Imagining the kidney wings while inhaling can encourage this particular illusive and unifying trail of sensation and movement. Normally the pattern is difficult to access and is seldom used in the nonintegrated body, but with imagery, the pattern automatically presents itself. (See an illustration of kidney wings in Appendix 4.) Instead of visualizing wings, you might be able to access the feeling by imagining that you have multiple arms rooted in the kidney area of your back, like a deity form. The arms are reaching up and out—as if reaching with infinite kindness in all directions so that the back of the body and diaphragm are luminous, expansive, and spread. This sense of reaching is a reminder of your supportive relationship to the world, to others, to someone outside yourself.
The kidney wing pattern is difficult to find and maintain because there is a tendency, when inhaling, to unconsciously contract the hip flexors, which reflexively closes the back in the area of the kidney wings. This contraction is a natural part of the inhaling (prāṇa) pattern, when the mind is distracted and wandering in discursive thought. When this happens, the inhaling (prāṇa) pattern can break away from the complementary exhaling apāna pattern, thereby tightening and collapsing the ribs in the lower back and interrupting the kidney wing pattern. It is common, when our minds wander, for prāṇa to separate from apāna this way, allowing the sensation patterns and our thoughts to drift away from the present moment. As we inhale, if we deliberately maintain an awareness of the seed of the complementary apāna pattern—which stabilizes the coccyx, expands the kidney area, and requires that the psoas and QL muscles refrain from contracting—then the prāṇa pattern opens symmetrically from the central axis without separating and wandering off. Preserving the smooth, rising, and spreading kidney wing pattern creates this union; fully spreads and lifts the back of the diaphragm; inflates the lower lobes of the lungs; and stimulates an expansive feeling of being awake, grounded, and symmetrical.
_Psoas Buttons_
_Psoas buttons_ are two of a number of points on the abdomen from which we can practice retracting or pulling in as a way of encouraging an awareness of the mechanics of inflating the kidney wings. Attention to the psoas buttons can also increase our awareness of the muscles in the pelvic floor and aid in the cultivation of Mūlabandha. We can imagine psoas buttons as small, circular areas on either side of the lower belly that root into the body and connect to the actual psoas muscles, which lie deep beneath the points. The "buttons" are located immediately in front of the medial edge of each of the psoas muscles, about four or five finger-widths below the navel, and about three or four finger-widths out to the sides. They lie a couple of finger-widths above the outer edges of the pubic bone and just in front of the medial border of the psoas muscles. The fibers of connective tissue from the lowest segments of the transversus abdominis muscles cross from one side of the lower belly to the other to form a band across the top edge of the pubic bone just at the point where we feel the psoas buttons.
Like buttons you might find on a wall when calling an elevator, you can imagine the psoas buttons as substantial and secure yet easy to push. Touch them lightly with the tip of a finger or fingernail in a way that causes them to pull back a bit. It should feel as if the tip of whatever touches the buttons is actually an irritant, causing the retraction, rather than that you must press hard on the buttons. By stimulating the psoas button points, the fibers of this lower section of the transversus abdominis are given a signal to tone in such a way that causes the abdomen to assume a kind of pot shape that floats up and out of the pelvic basin. We can use the image of touching the psoas buttons to train the muscles for an effective inhaling pattern.
1. Sit straight and bring the awareness to the breath. Place the fingertips of each hand on the low belly just over the line of the psoas muscles. The pressure on the belly when pushing the psoas button does not need to be forceful.
2. Instead of your fingertips, you could also place the tips of small dowels or chopsticks on the points to focus your attention.
3. During the course of a smooth, deep inhalation, keep touching the psoas button points as a reminder to keep this lowest part of the abdominal wall slightly retracted. This will tone the pelvic floor while keeping the apāna pattern stimulated at the same time. When you have found and activated the psoas buttons, it will feel as if the sitting bones are moving down as the floating ribs behind the kidneys spread.
4. As you exhale, lift the fingertips off the low belly, keeping the mind focused on the sensations that arise during the smooth and even breathing pattern. At the very end of the exhalation, bring particular attention to any sense of movement, toning, or activation in the pelvic floor muscles. This is like depositing the seed of the entire apāna pattern at the center point of the pelvic floor.
5. Gently place the fingertips back on the psoas buttons to stimulate an even retraction of the lower band of muscles that cross the abdomen as you inhale again.
6. Repeat this breathing pattern, using the fingers on the buttons as a reminder to contract without gripping on the inhalation, lifting the fingers on the exhalation, and cultivating the sense of the pot of the belly floating during the entire wave pattern of breath.
The image of a psoas button implies that lightly touching the button can begin this deeper and more complex process, which is actually the cultivation of Uḍḍīyāna Bandha and Mūlabandha. The line of tone in the transversus abdominis is distinct and does not go any higher than two finger-widths below the navel. If you overcontract the abdomen so that toning travels too far up the abdominal wall muscles, Mūlabandha will not manifest distinctly, and extraneous tension in the upper part of the abdomen will prevent a sense of being fully grounded. These two aspects of movement that are activated through the psoas button practice are precursors for a deep transformation of emotions that can occur as the inhaling and exhaling patterns of breath unite and spread throughout the body from the lower stations of the central axis.
For the psoas button areas to actually retract on the inhalation, the pubococcygeus (PC) muscle needs to tone distinctly as you initiate the action. To help in finding this action, you might imagine that you were shrinking the tissues of the pelvic floor, as if pulling a zipper closed from the front edge of the anus back and up the front surface of the coccyx until it reaches the middle of the sacrum. The entire area of the pelvic floor from front to back and up the inside of the cave of the sacrum will feel tidy and awake. You can augment this sensation by imagining that the action of inhaling is being initiated from the area immediately above the center of the pelvic floor.
With practice, even without actually touching the psoas buttons, you can maintain a sense of responsive toning in the pelvic floor and low belly throughout all of the phases of the inhalation. This should be true even at the crest of the inhalation when the scalene muscles of the neck tone so that the upper lobes of the lungs fill with breath. At this point, the mind tends to flicker and lose touch with the bodily sensations associated with the apāna at the root of the exhaling pattern in the central channel and pelvic floor. This unlinking of prāṇa and apāna happens all the time, so we study it from slightly different angles again and again. Using the psoas buttons is an effective method of pointing out this subtle and illusive internal form.
_Gaṇeśa Belly_
Imagining what we call a _Gaṇeśa belly_ is an inclusive whole-body image. Gaṇeśa, the elephant-headed deity, is considered to be a loving, jolly, relaxed, and extremely intelligent holder of the mysteries of the central channel. He represents the idea of paradoxical thinking within the esoteric yogas. One of his notable features is that his abdomen is expansive, side to side and front to back, in an even and toned way at the plane of his navel. In the most beautiful and finer artistic representations of Gaṇeśa, his lower abdomen, right at the top of the pubic bone, is hollowed out slightly. Often he is depicted with a cloth wrapped around his hips, draping around his body in such a way as to suggest a convergence of energies under the floating "pot" of his belly. Often this cloth is tied in a knot in what would on our bodies be a point around four or five finger-widths below the navel. Sometimes a fine jewel is installed there.
Gaṇeśa is definitely cued into his kidney wings, the cave of his sacrum, and his psoas buttons! He _is_ the Mūlabandha. The sense of a floating, happy potbelly, which instantly stimulates an integrating pattern that loops through the entire body, is represented in immaculate detail in Gaṇeśa's form: his large elephant ears, which are tuned to the infinity of divine sound, or _nāda;_ his deep, bright eyes that reflect his intelligent, humorous, and compassionate view of all circumstance and beings; and his long nose representing the complete cleansing of the nāḍīs. Of course, Gaṇeśa also has immaculate shoulder alignment, often manifesting with at least four arms, so he is clearly tuned in to the finer details that occur throughout the breath and body when one unites complementary opposites with a trusting, intelligent, open mind.
It is said that with much practice the breath can be united by drawing the apāna (the downward contracting pattern) up the middle line of the body into the roots of the navel, while at the same time allowing the prāṇa (which by nature is an expansive, spreading pattern) to be pulled down to the plane of the navel. This union comes naturally to Gaṇeśa. In our yoga practice when the two patterns of breath unite at the nābhi cakra (the root of the navel), you may experience the distinct contracting, shrinking pull of the apāna united with the expansive flower-like spreading of the prāṇa. This awakens the _samāna vāyu,_ the equalizing form of Prāṇa that is in the navel. Samāna vāyu is related to digestive fire and functions to synthesize opposite patterns of sensation, which then helps in synthesizing conceptual opposites and complementary movement patterns throughout the body. Allowing yourself to imagine Gaṇeśa's embodied form of this relaxed yet intentional and unstrained union of opposites at the nābhi cakra can automatically facilitate a distinct feeling of the samāna vāyu and its penetrating form. This may take a number of lifetimes to occur.
apane ūrdhvage jāte prayāte vahnimaṇḍalam
tadā'nalaśikhā dīrghā jāyate vāyunā 'hatā (v. 65)
tato yāto vahnyapānau prāṇamuṣṇasvarūpakam
tenātyantapradīptastu jvalano dehajastathā (v. 66)
tena kuṇḍalinī suptā saṁtaptā saṁprabudhyate
daṇḍāhatā bhujaṅgīva niḥśvasya ṛjutāṁ vrajet (v. 67)
bilaṁ praviṣṭeva tato brahmanāḍyantaraṁ vrajet
tasmānnityaṁ mūlabandhaḥ kartavyo yogibhiḥ sadā (v. 68)
When the apāna, tending downward, is turned upward and reaches the circle of fire (at the navel); when the apāna touches the fire, the flames lengthens. When the apāna and the fire reach the prāṇa, which is hot by nature, the heat in the body is intensified. The Kuṇḍalinī, who has been sleeping being heated completely, wakes up like a snake that has been hit by a stick and straightens herself. Then, like a serpent she enters her hole, the Brahmā Nāḍī. Therefore the yogi should always practice Mūlabandha.
_—Haṭha Yoga Pradīpikā,_ ch. 3, v. 65–68
_Cobra Hood_
_Cobra hoods_ are extremely visceral and useful images for developing essential yet illusive internal movements and conjunctions that facilitate whole-body patterns of movement. If you look at a cobra from behind when it is alert or charmed, it is easy to see the importance of being fully grounded in order to have a stable, expansive, and articulated upper body. As the snake sits on the ground, lifting up and out of the nest of its coil, its upper back and neck (referred to as its "hood") spread and broaden, connecting a line of energy from the sides of its body up and out to form an acoustic band-shell shape just behind the snake's head. This image is useful to our yoga practice as it intuitively suggests the sense of joining together through vinyāsa—rooting down on the exhaling pattern in order to expand up and out through the inhaling pattern. Cobra hoods even describe the broad feeling in the back as well as the back portion of the shoulder joint when the arms are overhead in āsanas. You can feel this in many poses, from the Downward Dog Pose to Ūrdhvā Dhanurāsana. (See an illustration of cobra hoods in Appendix 4.)
If you meditate deeply on embodying this form, imagining that you are grounded, awake, and alert while listening carefully to the acoustic space at the edges of the spreading hood, you may feel a sense of broadening and lifting throughout the entire back of your body—as if you were a cobra waking up. The sense of spreading like a snake will start in the area of your kidney wings, which will stimulate the dropping apāna pattern that coils as a tail in the pelvic floor and rises to expand the back. The cobra hood pattern continues to extend evenly over the crown of the head and can come to life if you relax extraneous tension in the mouth, release the palate, and soften the eyes as if smiling. Rather than losing focus in a dream state, it is possible to stay alert—as would any snake worth its salt—by again listening deeply. The natural counteraction, the vinyāsa, is that the middle of the heart opens to become the focus of the cobra hood pattern. The cobra hood shelters and adores the precious one at the center of the heart.
Cobra hoods can be visualized in many ways, from the simple sense of something expansive that arches over the crown of the head to precise and detailed complex images. You might imagine, for example, that from the root of a single coiled tail on which you sit, there are thousands of cobra heads extending up and out across your back, curving gently over your head to form a sort of canopy. Perhaps each head of this infinitely expansive upper part of the snake is wearing a crown of effulgent jewels, and each head is singing scriptures or hymns in an endless variety of languages from infinite points of view. Meanwhile, your upper body remains stable, strong, and expansive, yet it is also fluid and firmly rooted down through the base of the snake. Your ears and the hearing function stay with the pure cacophony of sound (without the mind picking out objects) that opens, unfolds, and soothes your nervous system. With your awareness returning to focus on the geometric wholeness stimulated by the form of serpent heads arching through and above you, the integrity settles into your entire body as it softly dissolves back down to the center of the heart and all through the central axis of the body. From the core of the heart up through the crown, it feels as if the heads are sheltering and adoring this sacred space, and from the navel down, there is a feeling of rooting deeply into the earth.
maṇi bhrātphaṇā sahasraravighṛtaviśvam
bharāmaṇḍalāyānantāya nāgarājāya namaḥ
salutations to the king of the _nagas,_
to the infinite, to the bearer of the maṇḍala,
who spreads out the universe with thousands
of hooded heads, set with blazing, effulgent jewels.
_—Kūrma Purāṇa,_ ch. 9, v. 5
Embodying this, like other images, is important to our āsana practice because it helps elicit feelings that slow us down and suspend interruptions of mind. Rather than jumping to habitual story lines and conclusions about our circumstances, our alignment and form, or our yoga practice, there is a sense of both mental and physical spaciousness that allows us to feel engulfed by a trust in not knowing. We can stay open to the arising of fresh viewpoints and a feeling of deep respect for the process of transformation within the mind, Prāṇa, other beings, and the universe at large. Feelings of safety within a vastness are enabled through the visualization of being crowned by a cobra hood as we gently hold what is most precious in the core of the heart. If you hold the center of the heart too tightly—with a motive to get or achieve something from the precious content of that space within the heart—then you just have a cobra on the loose! To really embody and enjoy the sense of a cobra hood, there is an element of embodied nonattachment and sincerity.
_Skin Flow_
The skin, which develops in utero from the same layer of embryonic tissue as the brain (ectoderm), is well innervated. When we pay attention to specific areas of the skin, we switch on its sensitivity, and this can allow us to feel structural alignment as well as existing tension in the myofascial layers that lie beneath the skin. One of the hallmarks of good alignment is that the skin seems more awake and vibrant, as if it's glowing. When we talk about _skin flow,_ we're really talking about a particular pattern of release or stretch in the myofascial sheathes under the skin. Real skin does not flow very far; if it did, a person with tattoos would find them moving all around the body.
To feel the sense of skin moving, you must focus on the sensation of air touching the outer layer of skin and at least an imaginary sense of the skin's ability to move and slide around on the layer of adipose tissue (fat) just beneath it. For example, when you sit or stand with good shoulder alignment, with close attention you can feel the shoulder blades and the muscles and skin that cover them spreading and descending down the back in just the right way. At the same time, you can feel the skin on the chest spreading and flowing up and out over the collarbones, as the skin on the front edges of the armpits flows up, back, and out. This requires a subtle level of awareness or toning that, when complemented by attentiveness and a relaxed feeling on the skin's surface, results in a gentle stretching of the pectoralis major muscles. The whole-body feeling of this movement is one of the skin being guided or smeared up and out across the fronts of your shoulder joints. Simultaneously the backs of the armpits feel as though they are flowing down and forward in space as they spread out. This allows the skin behind the floating ribs to expand and releases the extraneous tension in the rhomboid and upper trapezius muscles. If you focus on these sensations, you may feel as though the skin, starting from the back of the skull, softens and flows down the sides of the spine.
In many poses, when good alignment of the head, neck, and shoulders finally manifests, extraneous tension releases in all these areas, and there is a spontaneous feeling in the front of the throat that the skin is soft and luminous. This is a great opportunity to breathe into the sensations and imagine the skin flowing upward toward the back of the throat, which automatically softens tension in the jaw and releases the root of the palate. Simply focusing on the movement of skin in this way can stimulate an integrating whole-body pattern of awareness.
The manifestation of Prāṇa, or intelligence, in the skin is one of the functions of the _vyāna vāyu._ It has no center or home of its own, but it does integrate the vast peripheral structures of the body with their environment and then back into the central core of the body, serving an all-pervading function to connect all systems. Prāṇa and the intelligence—which have no self-form—love to link systems, to support, to hold, and to communicate. Skin flow is a secret ingredient in this alignment and form.
_Holding the Tail of a Serpent_
An image that facilitates integrated whole-body patterns of extension via a sense of grounding through the edges of the pelvic floor is that of _holding the tail of a serpent._ In this image, holding our "tailbone" steady is the precondition to initiating the inhaling pattern of the body. When we talk about the tailbone we are referring to the coccyx, the small triangular bone that attaches to the base of the sacrum and that, in most people, naturally curls down and in slightly toward the front of the pelvic basin. We sometimes speak of the coccyx dropping or moving, and there _can_ be a tiny bit of movement associated with the bone as the muscles of the pelvic floor tone. But because the coccyx is fused to the sacrum, that movement is actually very slight. So the use of visualization—imagining that instead of a little stump of a tail, we actually have a long, powerful tail like some other members of the animal kingdom—can help to bring movements associated with the coccyx to life and may be enough to initiate the patterns of Prāṇic flow we want to stimulate.
Imagine what it would feel like to have the dynamic tail of a dragon, a snake, or a monkey. Take that feeling into the patterns of movement related to the coccyx in the pelvic floor so those movements become distinct in your nervous system. Of course, for most of us, just _finding_ the coccyx and muscles of the pelvic floor—the pubococcygeus, the iliococcygeus, and the coccygeus—can be difficult. You can begin to locate the general area by repeatedly squeezing and releasing the anal sphincter muscles or by doing what are referred to as Kegel exercises. With practice, you will be able to connect to the pelvic floor on a more refined and intelligent level by paying close attention to what happens in that region when you complete an exhalation—especially if at the end of an exhale you give a short puff, almost like a punctuation mark. A connection to the pelvic floor makes it easier to use the imagination to refine movements and to map out that area of the body. Imagining that you have a tail can help; it can trigger awareness of the coccyx and its subtle ability to move in conjunction with the pelvic floor muscles, and this awareness eventually begins to wake up Mūlabandha.
Holding the tail of a serpent is a two-way movement, as if you are pulling a curved sword from its sheath; the movement of two intimately related structures flowing in opposite curves. The two-way action is a function of being able to separate an awareness of the sacrum from an awareness of the coccyx. The sacrum is like the "mother bone" of the coccyx; it is always right there above the coccyx, ready to help. In an extreme, purely apānic movement, the sacrum tends to follow the coccyx's direction of movement, so that as you tuck your tail, you may find that you also contract the abdomen and buttocks muscles and flex the spine. This causes the entire pelvis to tilt or tuck and potentially shuts down the complementary prāṇa pattern throughout the body. In sitting up straight, the sacrum tends to move up and in while the coccyx moves down and in. Separating the movement of the coccyx from that of the sacrum allows the cultivation of Mūlabandha and the refinement of an awareness of the pelvic floor.
The movement of the tail inevitably involves communication with the pubic bone. If we are able to use the tail without too much involvement from the sacrum, we also want to maintain the presence of the prāṇa pattern in the right proportion throughout the body. The pubic bone belongs to the prāṇa family and is reflected as an open heart. When "holding a pose by the tail," we often aim to maintain the complementary openhearted quality of the spine rather than the tuck and flexion associated with the extreme apāna pattern. In these cases, even though we are cultivating a sense of the coccyx dropping down and slightly forward, the sacrum feels as if it is floating up and into the body (moving into the lower abdomen), while the movement of the pubic bone is down and back toward the coccyx. These are all very subtle movements that create a balance of tone in the pelvic floor from front to back and side to side. When done correctly, they can wake up our psoas muscles, alleviate stress in the SI joints, set a base for Mūlabandha to appear, and encourage integrated and full-body movements in the spine and out through the limbs and the entire body.
_Feet Reflecting the Pelvic Floor_
When working standing poses or even when sitting and articulating our feet and toes into combinations of positions and movements, the pelvic floor comes online to help coordinate the effort. Since it is difficult, especially for beginners, to connect to the pelvic floor, focusing on the different combinations of complementary and opposing muscular and structural tones and forces in the legs and feet—which are far more accessible—can stimulate a spontaneous awakening of the pelvic floor. This is what we refer to as the _feet reflecting the pelvic floor._ It is as if the feet are the embassy of the pelvic floor—an accessible structure through which we can get a message to the homeland of the pelvic floor, which then informs subtle movement and alignment throughout the body.
The various families of muscles around the hip joints flex, extend, abduct, adduct, and rotate inwardly and outwardly through an amazing range and variety of movements. The pelvic floor coordinates all of these movements in a fine balance with each other and with the movements of the spine to produce competent, continuous, and sometimes microscopic adjustments and adaptations. With endless practice, it gradually becomes clear that the feet and pelvic floor are intimately and directly related, so one method for cultivating an ability to articulate the pelvic floor and facilitate integrated movement in all yoga poses is to "wake up" the feet.
In standing poses during which the movement in and out of the poses is almost as important as the "final" form, how the feet respond to the structural needs of the body is essential. Many yoga teachers can diagnose, prevent, or help to heal many structural problems by simply observing how students use their feet. Students sometimes consider standing poses to be basically static warm-up exercises, so they often wander away from fine-tuning the structural lines needed to go in and out of the pose with integrity. Lazy, amorphous, nonradiant feet and legs can cause the entire pose (not to mention the practitioner) to suffer. One way of waking up the feet and legs is to engage the quadriceps without switching off the hamstrings. This can be done by bringing focused attention and a sense of grounding into the sole of the foot; by rooting down through the big toes without abandoning the sense of dropping through the heels. This will bring intelligence into the pelvic floor, and with close observation, you may notice slightly altered patterns of tone in the different quadrants of the pelvic floor.
Each foot has three arches: a powerful medial arch that runs from the first joint of the big toe (near the mound) to the front edge of the heel on the inside of the foot; a smaller but important lateral arch that runs from the first joint of the little toe to the outer edge of the heel; and a transverse arch that runs from side to side across the middle of the foot. These arches give the foot a springlike, bouncing potential. With the feet together in Samasthitiḥ and attention on the feeling of the arches lifting (even if you have flat feet), it is possible to imagine a relationship between the arches of the feet and an awakened pelvic floor. Subjectively, in the practice of Mūlabandha, you can imagine the middle point of the pelvic floor being drawn up in a similar fashion to the arches of the feet, giving it a domelike feeling from side to side as well as from front to back. Of course, distinguishing the pelvic floor from all the nearby muscles is easier said than done, but most of us first get a sense of it by articulating and using the feet intelligently and enthusiastically.
You can use your imagination to tap into this feeling of waking up the feet by imagining in the standing poses that you are putting on a pair of long, high socks in which you have to push each foot and leg out away from the body as you pull the sheath of the sock up toward the hip joint. Imagining the sheath to be your skin as you activate the foot to push out and down can create a radiant, awakened state in the leg and an attentive tone in both the feet and the pelvic floor.
In āsanas in which the legs are up in the air—such as Headstand, Shoulderstand, and arm balances—or even when lying on the back with feet up the wall, the tone and structure of the feet and legs are key to bringing the pose into a full expression. For example, when practicing Piñcha Mayūrāsana (Peacock Feather Pose), it is possible to balance precariously upside down with inactive feet and legs and, almost as if by chance, not fall over. But for a fully inspired pose, one in which the spine feels as if it is being extended up toward the ceiling and the base of the pose feels stable and steady, the feet must be fully articulated with the legs active and awake.
Like so many aspects of the practice, you must work slowly and patiently to feel the immediate and more generalized benefits of connecting the feet to the pelvic floor. Gradually subtle patterns of Prāṇa and connections between muscles and fascia lines will automatically come to life. Outside of your own thought process or your attempts to apply technique to poses, movements will become inspired and integrated on their own.
_Palate-Perineum Reflex_
During a vinyāsa practice in which the movement in and out of poses is just as important as the poses themselves, we can easily find a simple yet profound relationship between the palate (roof of the mouth) and the perineum (pelvic floor). This _palate-perineum reflex_ is an essential element that integrates wavelike patterns of movement between the outer and inner body. In any pose, adjusting even a small corner of the pose—like the shape of the hand, the tone in the tongue, or the movement of the collarbones—can have a ripple effect throughout the entire body. One of the most fundamental of these interpenetrating patterns is along the spine from top to bottom and is associated with the prāṇic inhaling pattern of rising, extending, and spreading and its complementary opposite, the apānic exhaling pattern of descending, flexing, and contracting. In prāṇāyāma practice, keeping the essential pattern of the prāṇa when the apāna is active, and vice versa, is the key to calming and clarifying the mind. So too in āsana, we discover there is a reflexive relationship between these two patterns that inspires a more integrated movement within the practice.
For example, while we are bending forward into Dve position (which is an apānic movement) in Sūrya Namaskāra, we keep the spine relatively straight and the area of the heart open. This is easily accomplished by releasing the upper back of the palate, as if silently saying _ah_. During that phase of the pose, there is a sense of riding the breath as if it were a wave. The palate becomes like a surfboard on a wave, and our attentiveness to the details of movement makes us surfers. When we pay attention to every small, informative shift that arises, it makes the ride smooth and even. As the exhalation comes to an end, the wavelike pattern shifts, and an apānic flexion and slight coiling pattern appears in the body as we detect a toning of the pelvic floor. This happens because the condensed seed of the apānic pattern is in the center of the pelvic floor and can be felt at the end of a full prāṇāyāma-like exhalation. As the breath shifts and the inhalation begins, we refine the emphasis structurally to stay in touch with the seed-point of the exhalation; we do so by cultivating the feeling of "scooping up" in front of the sacrum during the inhalation. At the very top of the in-breath, as the wave of the breath crests, the palate is released and the pelvic floor resets. Then the wave pattern begins again as breathing continues.
This palate-perineum relationship is a whole-body pattern. Focusing the attention on the perineum and palate is never an exclusive focus, but it serves as a gateway within the field of awareness into the entire body. It is extremely pleasant and often relieves strain in a pose when we become too dogmatic and attempt to force the body into some contrived ideal form. Practicing this way—riding the wave of the breath and connecting palate to perineum—leaves a pleasant meditative residue within any form of yoga practice.
_Plumb Line_
After we get warmed up in our āsana, prāṇāyāma, or meditation practice, it is easy to experience a central axis, a central channel, or a plumb line running top to bottom within the body. The plumb line provides a reference point within the subtle body for feelings, thoughts, sensations, and the movement of Prāṇa to be organized and understood as a reflection of the body's natural intelligence. The plumb, which is easy to sense when the mind is still, gives our structure a clear axis of orientation so that through smooth and even breathing, movements fall spontaneously into place.
A good time to begin feeling and understanding the plumb line is while practicing āsana, particularly while moving in and out of poses. When we twist from side to side or do forward bends followed by backbends, for example, we gradually come to a point where opposite yet interdependent physical patterns become balanced and integrated. As these patterns sync with one another, we no longer fall into habitual patterns that may favor one direction of movement over another. We feel balanced left to right and front to back, with both patterns fully awake so that Prāṇa (and therefore the mind) comes to a natural pause. In that pause, the physiology of the yogic state of nirodha, or suspension of the process of mind, which is the state of pure, unbiased attention, arises. Within this state, it is easy to feel the central line of the body as being the channel for the flow of our attention. When standing or sitting straight, this central channel corresponds to the plumb line; when lying down, we refer to it merely as the middle path or central axis.
Āsana practice can reveal the plumb line, and you can augment an awareness of it through visualization. Imagine, for instance, that you were attempting to balance a pole on your nose like a circus performer might. You would have to make many compensatory movements to the front, to the back, to the left, to the right, and all around. With much practice, you would occasionally get to a point where the pole is balanced; all the minor adjustments would be temporarily suspended, and you would be captured by the process of balancing, finding yourself in a highly attentive, peaceful space. The technique involved is that of appropriate compensation blended with an equalization of present-moment attention and nonattachment to outcome. Finding the plumb line is actually an embodiment practice of the process of effort and letting go, as described in the Yoga Sūtra:
abhyāsa-vairāgyābhyāṁ tan nirodhaḥ
Through practice and releasing, the fluctuations of mind are suspended.
—Yoga Sūtra, Samādhi Pāda, v. 12
_Abhyāsa_ can be translated as "practice" or "effort" and is essential whether you are balancing a pole on your nose, moving into a difficult āsana, or simply dealing with the complexities of life. At the same time, _vairāgyam,_ letting go of techniques, theories, and expectations, while staying in the present moment is essential. It simply doesn't work if you put too aggressive an effort into the process, nor does it work if you completely let go. In so many situations in life, and certainly in yoga practice, you must have a full, devoted, and directed effort but, at the same time, an absolute sense of release without a hint of laziness or the abandonment of ideals or form. You must have it all and both ways too.
The same is true when we polish the subtleties of form while standing in Samasthitiḥ. We balance the prāṇa and the apāna, sometimes squeezing them together, sometimes relaxing and letting the alignment of the pose be. The external pose is a conversation between heels and toes, inward and outward rotations, top and bottom, palate and perineum, pubic bone and coccyx, and on and on into infinitely subtle levels of perception, form, and alignment, all of which make us able to access the sense of a plumb line. Wherever the attention falls along that plumb line, a feeling floods the body and mind of an equal opening in all directions arising in conjunction with a joining of prāṇa and apāna. Much like a flower would open to the sun, this spreading of Prāṇa and the intricacies of awareness feed back into our ability to maintain a connection to the central channel through the pelvic floor and the root of the palate.
To tap into this feeling on a more visceral level, the plumb line can be visualized as a straw that extends down into the earth from the middle of the body and between the legs. The straw rises up through the center of the pelvic floor along the midline and continues on and out through the crown of the head. As this image is clarified in the nervous system, we feel that the line corresponds to a conduit through which attention can flow easily. The straw might be perceived as a generalized sensation; a movement of energy, light, or liquid; Prāṇa flowing through a tube; or what we might eventually call the suṣumṇā nāḍī.
LIBERATING AND ILLUMINATING
Visualization practices can be entire paths in and of themselves, enriching the practice as different images and meditations intersect. Different viewpoints reveal a multiplicity of self-reference paradoxes that, when looked at through an open mind, help to refine the intelligence. Watching changes in form, thought, and sensation underscores the liberating insight that, although in a very real sense we are embodied beings, we are not _this_ body. Or rather that this body is not a solid and permanent phenomenon. We are reminded again and again to keep the senses fresh and awake—opening the ears, releasing preconceptions along with the palate, tuning in to the breath, and always delving deeper into the experience at hand.
That's why you have to be extremely careful if you try to practice yoga when you're angry, unsettled, or distracted. When the mind is dominated by extreme mental or emotional imbalance, or if there are subtle levels of physical tension and resistance within the body, it is virtually impossible to truly surrender to the entirety of what is occurring and to fully examine (and possibly embrace) an opposing perspective. Misperception, misinterpretation, injury, and unskillful actions are most likely to govern your thoughts and actions when you cling to a single point of view where tension and imbalances dominate. That is not to say you cannot practice when unsettled—in fact, sometimes practice is just the antidote needed to bring balance. But especially when you are off-kilter, you must practice carefully and with a sense of surrender, of not knowing, and of starting over again and again. When approached in this way, the practice will often smooth out imbalances, especially if you practice "all day every day."
And that is the key: we must carefully continue to practice through the ups and downs of life and the practice itself. Visualizations and whole-body patterns of alignment facilitate the visceral experience of connecting, integrating, and unifying opposing patterns. There are thousands of visualizations and many whole-body patterns of sensation through which this integrating experience of letting go of the ego can be found. Any one of them can facilitate a release of your preconceptions so that "you" disappear and a significant and paradoxical understanding is revealed. Once you are fully in your body, it disappears; it becomes open and empty, and the discursive mind, cloaked in all its natural processes of avoidance and grasping, automatically falls back into meditation. There is no one path to this form of insight, you have to get into it from wherever you _really_ are—inside, outside, still mind, agitated mind—so that you find a suspension of thought processes and of habitual body patterns.
Just as in meditation, when working with these subtle levels of body, almost anyone who practices long enough may go into what feels like an altered state or a trance. That can be fascinating yet dangerous, because without proper grounding, altered mind states can take us off on tangents. We may not see that a "trance" does not mean we're special. In fact, if we are practicing honestly and openly, everything that arises makes us feel more ordinary, more normal, more like ourselves. The most important insight is not a vision of a deity shining in the center of your heart, but an understanding that you are actually not all that special in terms of standing out from others. Rather you are special because you are inspired and delighted by being nothing special at all.
_4_
Mechanics
_Essential Anatomical Perspectives_
Classical anatomy complements and greatly informs visualization and the idea of vinyāsa in yoga practice. This chapter includes a limited selection of anatomy topics that are of particular relevance to yoga poses and internal forms. Hopefully this basic information will serve as a springboard to deepen your ongoing study of anatomy.
SYSTEMS
With regard to yoga, a good starting point for studying anatomy is seeing how movement patterns within the vinyāsa of a pose relate to the skeletal, respiratory, and muscular systems.
The _skeletal system,_ made up of bones and joints, allows movement and provides both a scaffold of form and a protective framework for the soft tissues of the body. The specific shape and size of the bones and their connections at the joints through ligaments (bone-to-bone connections) and tendons (muscle-to-bone connections) contribute to range of motion and affect proper alignment for maximum support within poses. One way of practicing yoga is to visualize a pose from the perspective of either the _axial_ (vertebral column, rib cage, and skull) or the _appendicular_ (shoulder and pelvic girdles, arms, and legs) skeleton. Focusing on the axial skeleton, we can eventually _feel_ the plumb line by imagining the bones of our spine stacked on top of each other in their natural S-shaped curve, providing strength, stability, and mobility to our extended or upright form. Feeling movement in the rib cage, which encases the lungs and is attached to the diaphragm, as we breathe is highly informative for prāṇāyāma and movement. Envisioning our body from the perspective of the appendicular skeleton reveals a solid and intelligent pelvic basin and pelvic floor with connections from the core of the body that both go out into space and are grounded into the earth.
The _muscular system_ is perhaps the most tangible system within the body. We can actually see muscles as they move, as when we engage the biceps. Focusing awareness on the character, function, and form of particular muscles and how they relate to bones and the breath can immediately "wake up" a pose. Muscle tissue is composed of elongated cells that vary in size and texture (striated or smooth) according to function. There is a continuous layer of connective tissue called _fascia_ that surrounds muscle bundles, individual muscles, and groups of muscles and connects us from head to toe. As we move, envisioning muscles contracting or releasing in line with their fibers, in conjunction with the breath, and in relationship to bones—all facilitated through the fascia—is fascinating.
Visualizing and sensing the _respiratory system_ from the gross-body perspective of the organs and structures involved in breathing—the lungs, ribs, diaphragm, and so on—and from a more subtle-body level of nāḍīs and Prāṇa flow is invaluable to our āsana and prāṇāyāma practices. Visualizing the flow of breath within our movements and in harmony with the other systems of the body is central to practicing yoga as a meditative, unifying form.
BASIC TERMS
_Active_ and _passive range of motion:_ The range of motion in a joint is active when local muscle groups that impact the joint directly contract to move it to a certain degree of rotation, flexion, or extension. Passive range of motion is when the forces that move the bones at a joint do not come from the muscle groups immediately around the joint. Instead, those "local muscles" remain soft so they do not interfere with or restrict the movement. In a vinyāsa practice, it is common for active and passive movements to complement one another in support of the desired outcome.
_Agonist_ or _antagonist:_ A muscle can be described as an agonist—one that is a "prime mover," or causes movement—or an _antagonist_ —one that inhibits or counters the movement. The erector spinae and abdominal muscles, for instance, are all involved in extension or flexion of the spine. Working together, they switch roles as agonist and antagonist during a graceful movement of the spine.
_Anatomical position:_ A spatial context for structures is called their anatomical position, described when the body is facing front, arms by the sides, palms forward. Structures closer to the head are _superior,_ those closer to the feet are _inferior. Anterior_ structures are positioned more toward the front plane of the body, and _posterior_ structures are those toward the back. _Medial_ refers to structures closer to the midline of the body, and _lateral_ refers to structures farther from the midline. The terms _proximal_ and _distal_ describe structures that are either closer or more distant, respectively, from another reference point—usually the core of the body. _Superficial_ and _deep_ refer to how far a structure is from the surface of the body, with superficial being closer.
_Co-contraction:_ This is the term for the simultaneous contraction of opposing sets of muscles. In co-contraction the relative forces of the muscles involved are continuously adjusted to achieve the necessary results of movement and stability. For example, co-contraction of the hamstrings and quadriceps is imperative when balancing on one foot, allowing stability and competent reversibility of movements, a basic principle in vinyāsa practice.
_Eccentric contraction:_ When a muscle is contracting and lengthening at the same time, in cases when it is causing resistance to a chosen direction of movement, the action is called eccentric contraction. For example, in Ūrdhvā Dhanurāsana (Upward Bow Pose) one should slightly contract the rectus abdominis muscle and yet allow it to lengthen (eccentric contraction). This keeps the opposing muscles, the psoas and the erector spinae of the back, from contracting, thereby protecting the lower back from compression.
_Origin_ and _insertion:_ The origin of a muscle is a point that is more proximal (closer to the center of the body) and is therefore typically more stable. The insertion point is onto a bone that is more mobile—the one that is drawn toward the point of origin when the muscle in question contracts. In complex actions, a muscle may not follow this rule of movement. For example, when you stand up out of Dve during Sūrya Namaskāra, the hamstring muscles pull and make the sitting bones move down toward the floor. This makes the sitting bones the insertion points in this case. Whereas when swinging the leg back, as in walking, the _ischial tuberocity_ (sitting bone) attachment is considered the origin because it does not move relative to the leg that is being pulled back.
_Reciprocal inhibition:_ When two muscles (or sets of muscles) create opposite actions, they work with reciprocal inhibition. This happens automatically in the nervous system. For instance, when the quadriceps fire while closing or opening the knee joint, the opposing hamstrings automatically relax. Reciprocal inhibition can be consciously interrupted through co-contraction.
STRUCTURES
_Spine_
Sensing and understanding movements and form within the practice is greatly informed by a simple overview of the _spinal column._ Composed of thirty-three bones (vertebrae) stacked in a natural S shape running from the coccyx up to the cranium (head), the spine is the main structure that allows us to stand straight, bend, and twist. Sections of the spine are differentiated according to their shape and function as cervical (neck), thoracic (upper back), and lumbar (lower back). Shorthand for various spots on the back, such as L4 (lumbar vertebra 4), is helpful in understanding muscular and skeletal relationships when describing our poses. Between each set of vertebrae is a disk that cushions and contributes to smooth movement within the spine. Due to the natural aging process, accident, or misuse, disks can become damaged or "herniated," oozing out from between the vertebrae. In a yoga practice, this type of injury is one that should be addressed with particular care and caution by both practitioner and teacher.
_Palate and Tongue_
The _palate_ refers to the entire roof of the mouth. It includes the _hard palate,_ the bony structure that lies directly behind the front teeth, and the _soft palate_ , the soft, movable structure at the top of the back of the mouth, composed of five muscles sheathed in mucous membrane. These muscles are essential to breathing, swallowing, and shutting off or allowing airflow from the mouth to the nasal passages. From an internal forms perspective, the uvula—the teardrop-shaped projection of tissue in the back of the throat—is included as part of the soft palate because it is intimately informed by movements of those five muscles. When we speak of the root of the palate, we are actually moving our awareness back and up behind the soft palate into the hidden yet highly sensitive area near the pituitary gland.
The palate is tied intrinsically to breathing patterns and is also connected to the toning of the pelvic floor and Mūlabandha. Though the tongue is not actually part of the palate, it is an interesting and important muscle to contemplate. It represents the language function, toning even when we think, and activity in the tongue directly affects tone or tension in the palate that is then felt throughout the body.
_Pelvic Floor_
In an Aṣṭāṅga Vinyāsa practice, the pelvic floor is a place of pilgrimage. It is the seat of Mūlabandha and the trampoline from which movements of Prāṇa rebound to help relate any movement to the rest of the body.
_From each side of the coccyx the PC muscle connects across to the pubic bone. The two sides of this muscle along with other closely related muscles affect the functioning and the perception of the anus and the urogenital triangle. Also attached to the coccyx are the illiococcygeus and the coccygeus muscles, which fan out to the sides to help in creating integrated movement through the legs, hip joints, spine, abdominal wall, and the entire body. Balancing the pelvic floor, front to back and side to side, is one of the basic practices of yoga._
The pelvic floor, or pelvic diaphragm, is composed of the muscles directly attached to the coccyx. The _coccygeus_ attaches to the outermost edges of the coccyx and spreads out to attach to the lowest portion of the sacrum. The _iliococcygeus_ attaches just in front of the coccygeus and fans out to the left and right to attach to the inner rim of the pelvis. The _pubococcygeaus_ (PC) muscle attaches to the sides of the coccyx and travels across the pelvic floor, attaching to the back inner rim of the pubic bone. All of these muscles are found on both the left and the right side, which is relevant to asymmetrical movements and the position of the legs and spine. Any asymmetries in the abdominal wall, spine, pelvis, or hip joints are intimately connected to movements in the pelvic floor. Learning to feel and imagine the pelvic floor in relationship to its peripheral structures is challenging but incredibly helpful to our yoga practice.
_SI Joint_
Most people are unaware of their _sacroiliac (SI) joints_ until the lower back begins to hurt or they feel referred pain in the buttocks or along the outer leg. The SI joint is the juncture of the concave, triangular _sacrum_ and the _ilium_ (the largest of the three pelvic bones)—hence the name SI joint. The two bones are tightly bound by ligaments that allow the sacrum to tilt slightly in the joint. This tilting is called _nutation_ (forward tilt) and _counternutation_ (backward tilt). When the two sitting bones are drawn closer together, as when the thighbones are externally rotated, the top of the pelvis broadens and the sacrum counternutates. When the sitting bones are spread apart (as with an internal rotation of the legs), the opposite occurs and the sacrum nutates. In asymmetrical movements, like walking or twisting, there is a small rotation of the two halves of the pelvis—and sacrum—in opposite, complementary directions.
The SI joint, designed for stability rather than mobility, is healthiest when nutations are small. These movements influence and are influenced by pelvic floor muscles and relate directly to the full, rhythmical movement of the spine and completed movements through the legs. Proper activation of the muscles attached to the coccyx—cultivating a sense of the bone dropping down and forward to help the pelvic floor tone—is essential to keeping the SI joint happy.
Isolating movements of the coccyx is difficult. It is important to remember that, although the sacrum and coccyx are actually fused together, on a subtle level we can _imagine_ that they move independently, with the coccyx dropping down and forward toward the pubic bone and the sacrum lifting up and into the body. This flow of sacral movement is subtle and illusive, but simply imagining it can stabilize the SI joint.
_Abdominal Muscles_
The abdominal wall defines the front of the torso and the boundary of the abdominal cavity, containing the abdominal organs. Abdominal muscles integrate patterns of movement from head to toe and impact prāṇāyāma practice profoundly.
Literally front and center is the _rectus abdominis,_ a set of paired muscles joined together along the midline by a strong band of connective tissue called the _linea alba._ The fibers of the rectus abdominis follow the vertical line of the muscle along the front of the torso, attached to the _pubic symphysis_ and _pubic crest_ and running all the way up to the _xiphoid process_ (lower end of the sternum) and _costal cartilages_ of the fifth through seventh ribs. Three bands of connective tissue traverse the rectus abdominis from side to side, at approximately the levels of the xiphoid process, the navel, and about halfway between the two. This connective tissue reinforces and strengthens the length of the muscle so it can be engaged more efficiently. When highly developed, the rectus abdominis is visible from the outside and may appear to be separate, short muscles running along the front of the body, sometimes referred to as "six- (or eight-) pack abs."
The rectus abdominis tones and stabilizes the trunk and core of the body. When engaged, it contributes to flexion of the lumbar spine if the pelvis is fixed, and it helps in drawing the pelvis toward the upper body when the rib cage is stable. In most movements, the right and left sides of the rectus abdominis are contracted simultaneously; however, in yoga they are sometimes articulated separately, as when practicing nauli (see pages 23–25).
The _internal_ and _external oblique_ muscles work in conjunction with the rectus abdominis, toning in an oblique crisscross pattern across the abdomen in certain twisting movements. The fibers of the external obliques line up precisely with the fibers of the _external intercostal_ muscles on their respective side and with the fibers of the internal obliques on the opposite side to facilitate these movements.
In general the _transversus abdominis_ is the deepest muscle of the abdominal wall. From the sides of the rectus abdominis it wraps around the sides of the waist to attach to the lumbar spine behind the _quadratus lumborum_ (QL) muscles. Like a girdle, it can give various shapes to the ball-like mass of the abdominal viscera. In the lower belly in front, at about two inches below the navel, is a horizontal seam called the _accuate line_. Here an amazing reversal takes place. The rectus abdominis goes behind the tendon and connective tissue of the other abdominal wall muscles to connect to the back edge of the pubic bone, just above tendons of the pelvic floor's PC muscle. The lower horizontal bands of the transversus then appear as the most superficial. These small strips of transversus act like an elastic belt along the top edge of the pubic bone and allow the shaping of the abdomen into the "floating pot" or the "Gaṇeśa belly" forms.
_Diaphragm_
The most important muscle impacting our breathing is the _respiratory diaphragm._ It is a unique and elegant muscle that, like other skeletal muscles, can be controlled voluntarily, expanding and contracting along the lines of its fibers from the edges of the diaphragm toward the center as we breathe in and out. It is one large, thin, dome-shaped structure of muscle and fibrous connective tissue that is attached peripherally to the abdomen and ribs. The attachments converge into a central tendon located at the crest of the dome. The diaphragm curves upward like an umbrella when we exhale or are at rest and its central area drops down when we inhale.
_The respiratory diaphragm attaches all around the lower edge of the rib cage and then down onto the fronts of the lumbar vertebra. Its movement profoundly affects the shape of and the tensions in and around the rib cage, which in turn affect the entire spine, pelvic floor, throat, and head._
At the end of an exhalation, it is possible to feel a toning of the external intercostal and abdominal muscles as well as the muscles of the pelvic floor. When inhaling, we may feel the diaphragm dropping and spreading as the ribs broaden, due to activation of the interior intercostal muscles. During a deep inhalation, the scalene muscles also tone. All of this happens unconsciously to allow room for the lungs to inflate and deflate properly. We can uncover, reveal, and refine unconscious breathing patterns as we cultivate movements that support a more full yogic breath. For example, when inhaling with the mind in its normal, distracted state, the erector spinae muscles that run along the sides of the spine tend to contract to hold the spine straight. However, their involvement can be excessive, activating the hip flexors, overstimulating and stretching the front of the diaphragm, and collapsing the lower back. When exhaling with a distracted mind, the opposite pattern can occur, abdominal muscles contracting to excess and erector spinae muscles releasing, causing the chest to collapse. In yoga, we work to minimize either extreme from manifesting by keeping the opposite form awake while the dominant pattern is at work.
_Scalene Muscles_
The _scalenes_ are a set of three pairs of muscles that run along the sides of the neck (anterior, middle, and posterior scalenes). They originate along the cervical spine (from C4 to C6) and attach to the first, second, and occasionally third ribs. The scalenes are engaged when tilting the head, but, along with the _sternocleidomastoid_ muscles on the front of the neck that look like straps, the scalenes are also the primary muscles involved at the crest of an inhalation, lifting the first two ribs and the collarbones so the top lobes of the lungs can fully inflate.
In an integrated practice, scalenes release and lengthen while remaining toned to facilitate full extension of the lower cervical spine, an action integral to every Upward Facing Dog Pose, among many other poses. As they are the primary actors in maximizing the crest of the inhalation, sensing the toning of the scalenes is central to finding the plumb line, lifting the heart, and connecting to the kidney wings when we reach the arms overhead.
_Legs and Arms_
The arms and legs have interesting similarities in terms of our yoga practice. Each is attached to the trunk of the body through a single large bone (the humerus of the arm and the femur of the leg) at a ball-and-socket joint that allows for a wide range of motion. The knee and elbow—both hinge joints—connect to the lower portions of the leg and arm to two smaller bones (the tibia and fibula in the leg and the radius and ulna in the arm). The wrist and hand also have obvious similarities as do the ankle and foot. It is through the legs, feet, arms, and hands that we express the feeling of luminous extension, groundedness, and "reaching to infinity."
Articulation of the hands and feet has a direct influence on the healthy functioning of the knees, elbows, shoulders, and hips, and ultimately on the entire spine and every pose. Using the hands and feet correctly is often challenging due to semiconscious habits and how we interact with the world. For example, the inner edge of the hand, from the heel of the thumb out through the tips of the index and middle fingers, is the area that should bear weight because the radius of the forearm is thicker than the ulna (on the outside of the forearm) and strong as it comes into the inner wrist. Yet often when we're doing arm balances or even Downward Facing Dog the index finger uproots, and our weight shifts to the outer edge of the hand unconsciously. The feet are structured so that they too should ground evenly so the arches come to life and the legs engage correctly. In both hands and feet, if the weight is distributed poorly, it will eventually cause injury somewhere along the line—whether it's the wrist, elbow, shoulder, ankle, knee, or hip. These are basically rotational problems that proper alignment can resolve.
There are muscular similarities between arms and legs too. The _biceps_ and _quadriceps_ work in similar ways to wake up the legs and arms, respectively, and the _hamstrings_ and _triceps_ working to stabilize and strengthen. Of course, there are also numerous differences between the arms and legs. For instance, the _greater trochanters_ of the femur, a vital point of postural assessment and adjustment for teachers in the legs and pelvis, are not paralleled in the arm. Nonetheless, considering similarities in an overview of anatomy provides insight into many movements and poses.
_Hips_
The pelvis consists of two curved flat bones, together called the _ilium_ , that join in the front of the body at the pubic symphysis and are fused to form the pubic bone. In the back of the pelvis the sacrum connects to the ilium to form the SI joint. The coccyx, which is fused to the sacrum, curls down and forward like a tiny tail. Two protrusions at the bottom of the pelvic bowl, one on each half of the ilium, are called the ischial tuberosities, or sitting bones.
The hip joint, or _acetabulofemoral joint_ , where the femur and pelvis meet is a ball-and-socket joint. The femur is secured by a thick ligament (the _foveal ligament_ ) that extends out from the center of the top of the head of the femur, attaching to either side of the _acetabular notch_ , a rough-edged depression on the sides of the socket. This ligament and the shape of the socket make the hip stable.
Four groups of muscles facilitate hip movement, flexibility, strength, and stability. Those primarily responsible for flexing the hip and drawing the top of the femur toward the torso are the _iliopsoas, rectus femoris,_ and _sartorius_ muscles, commonly referred to as the _hip flexors._ Those primarily responsible for extending the thigh from the hip are the _gluteus maximus_ and the hamstrings. The muscles responsible for drawing the leg toward the midline of the body, adducting the leg, are located along the inner thigh. When these muscles are weak, the hamstrings may become compromised, particularly in forward bends. The abductor muscles (all of the gluteus muscles and the tensor fasciae latae muscle) are located on the outer thigh and hip and draw the leg away from the body's midline. Another important group of muscles associated with the hips are the "deep six" (the pieriformis, gemellus superior, obturator internus, gemellus inferior, quadratus femoris, and obturator externus). These muscles are responsible for helping to rotate the leg out; they stabilize and protect the hip joint and have an intimate relationship to the pelvic floor muscles.
_Psoas_
The psoas ( _psoas major_ ) is a long, smooth-fiber muscle that connects the torso and the legs, functioning to link, move, and stabilize the body. On a subtle-body level, the psoas helps to link earth, which is our embodied situation, with our emotions and our concepts about our predicament. So similarly, it links and stabilizes our mental and emotional states to our actual embodied circumstances. The psoas attaches along the lateral border of the spine from T12 down to L2, where it joins the _iliacus_ muscle (which lines the inner surface of the upper bowl of the pelvis) to form the _iliopsoas_ muscle. After passing through the pelvic bowl, the iliopsoas attaches to the posterior surface of the femur at the _lesser trochanter._ Some people also have a psoas minor muscle that lies anterior to and follows the line of the psoas major, usually attaching to the spine at T12 at one end and below the ilium at the other.
_Imbalance and unconscious tensions in the psoas muscles affect the pelvic floor and the respiratory diaphragm. Learning to release them and then to balance them side to side allows a deeper meditative awareness of the body._
The primary function of the psoas (major and minor) is as a hip flexor—responsible for the first 30 percent of lift in the leg. When we are walking, it is mostly the psoas that swings the leg forward to take a step. When contracted unilaterally, the psoas also contributes to lateral flexion of the spine. When we are lying down and engage the psoas laterally with the pelvis stable, the psoas functions to help raise the torso off the floor.
_Hamstrings_
Many yoga practitioners learn to love and respect their hamstrings, usually after a few years of fearing and cursing them for being so tight that forward bends are next to impossible. The three hamstring muscles—the _semitendinosus_ , the _semimembranosus,_ and the _biceps femoris_ —lie along the back of the femur. The biceps femoris has both a long and a short head: the short head attaches to a small ridge of bone just below the posterior side of the sitting bone, and the long head and other hamstrings attach to the pelvis just beneath the gluteus maximus at the ischial tuberosity (sitting bones). All of the hamstrings cross the knee joint and attach to the lower leg: the semitendinosus attaches to the medial surface of the tibia, the semimembranosus attaches to the medial tibial condyle, and the biceps femoris attaches to the lateral side of the head of the fibula.
Together the hamstrings function in knee flexion, hip joint extension, and stabilization of the knee joint. They are stubborn muscles; the more you tug and pull on them to demand length so that poses are accessible, the more resistant, stiff, and short they become. Pushing to overcome hamstring limitation usually results in injury. In fact, one of the most common yoga injuries is strain, or tearing, of the hamstring attachment at the sitting bones. Less common is strain at the lower attachment (insertion) points. Working with hamstring injuries takes patience and should be addressed with intelligence and the guidance of a good teacher.
_Knees_
The knee is one of the strongest and most complex joints in the body. Its function is intimately related to the hip and ankle joints and articulation of the feet. The knee connects and facilitates movement between the lower and upper leg, and it supports and distributes body weight. It is a synovial hinge joint whose primary function is to bend the leg into flexion. When the joint is closed, it also allows for a small amount of lateral "swing" in the lower leg, a movement that is key as we work safely toward poses such as Vīrāsana (Hero Pose) and Padmāsana (Lotus Pose).
The knee joint connects the tibia, _patella_ (kneecap), and femur through a complicated design of muscles, tendons, ligaments, and bones that fit together for maximum stability and ease of movement. Two concave processes, or _condyles,_ on the distal end of the femur fit exactly into corresponding concave condyles at the proximal end of the tibia. They are separated by a shock-absorbing, figure-eight-shaped piece of cartilage (the _meniscus_ ) that lies between them. The meniscus can become compromised or injured through misuse of the knee joint or accidents in life that torque the knee. Folding the leg into a strained Padmāsana without allowing for proper rotation of the upper and lower legs is one common way yoga practitioners injure their knees.
The patella lies on the front surface of the knee joint and attaches to the femur and tibia via the _patellar ligament,_ the distal portion of the quadriceps femoris. To "wake up the legs" in standing poses, we are instructed to lift the kneecap, which is partially a function of engaging the quadriceps.
Four primary ligaments serve to reinforce the knee joint. The _medial collateral ligament_ (MCL) connects to the medial side of the femur, then crosses the knee joint and attaches to the medial side of the tibia. The MCL stabilizes the joint when force is applied to the lateral side of the knee. The meniscus is attached directly to the MCL, so it is not uncommon for the meniscus to be torn when the MCL is injured. The _lateral collateral ligament_ (LCL) runs laterally along the outer seam of the leg from femur to fibula. It inhibits forces applied to the medial side of the knee when the knee moves laterally. Two cruciform ligaments cross within the joint to stabilize it front to back. The _anterior cruciate ligament_ (ACL) extends obliquely from the inner surface of the lateral condyle of the femur to the upper medial front of the tibia. The ACL is helpful in maintaining stability and preventing hyperextension of the joint. Immediately behind the ACL is the _posterior cruciate ligament_ (PCL), which extends obliquely from the inner medial condyle of the femur to the posterior intercondylar space of the tibia.
_Shoulder_
In one sense, the shoulders are extremely familiar territory—we shrug them when we're uncertain, slide them in and out of clothing without a thought, and bear the weight of the world on them day in and day out. The shoulder area of the body is complicated, so understanding and controlling shoulder movement and mechanics can be confounding, but a basic understanding of shoulder anatomy can demystify the actions we attempt in yoga.
In common usage, the term _shoulder_ usually refers to the _shoulder girdle_ —the entire part of the upper body that connects the arms to the torso. It includes three bones: the _clavicle_ (collarbone), _scapula_ (shoulder blade), and _humerus_ (upper arm bone), which are all supported and stabilized by a number of associated muscles, tendons, and ligaments. The shoulder comprises two joints—the _glenohumeral_ and the _acromioclavicular_ (AC)—but strictly speaking, when we talk about the shoulder joint, we're referring to the former. It is a ball-and-socket joint formed where the _glenoid fossa_ (a concave hollow on the lateral end of the scapula) interfaces with the convex head of the humerus. This joint has more mobility than any other joint in the human body and is also more vulnerable to injury than most due to the shallowness of the fossa and the available range of motion. The AC joint is a synovial joint at the juncture of the uppermost part of the scapula (the _acromion_ ) and the clavicle. This is the only place the scapula actually attaches directly to another bone. The _sternoclavicular_ joint, also considered part of the shoulder, is at the center of the chest where the clavicle interfaces with the upper part of the sternum to provide stability.
The shoulder joint is also stabilized and protected by the _labrum,_ a strong, cup-shaped cuff of cartilage within the joint that cups the head of the humerus. There is also a small sac of synovial fluid, the bursa of the shoulder, that cushions the tendons of the rotator cuff and protects the joint.
Four of the seven muscles that stabilize the connection of the upper arm to the torso are grouped and called the _rotator cuff_ because their tendons form a "cuff" over the end of the humerus at the joint. They are the _supraspinatus,_ the _infraspinatus,_ the _teres minor,_ and the _subscapularis._ The supraspinatus helps to abduct the arm at the shoulder. It lies in the fossa (trough) of the top section of the scapula, between the spine of the scapula and its uppermost edge, and the muscle attaches laterally to the greater tubercle of the humerus. The infraspinatus is a relatively thick, triangular muscle that lies on the back portion of the scapula, attaching the upper arm to the shoulder blade. The teres minor connects the humerus directly to the scapula with fibers running obliquely upward and laterally to a point on the upper arm. The subscapularis lines the inner surface of the scapula and shares the same tendon as the serratus anterior. It inserts on the humerus and the front of the glenohumeral joint capsule. The subscapularis works in conjunction with the serratus anterior to stabilize the shoulder and is key to many yoga poses that involve the arms, as it rotates the humerus inward and cues the inner edge of the hand to bear weight, even when the shoulder blade is protracted.
_Serratus Anterior_
Within an āsana practice, the _serratus anterior_ muscle is critical to many patterns of movement in poses from backbends and arm balances to twists and headstands. It attaches to the ribs (from the first through the ninth) and along the anterior medial border of the scapula; it holds the scapula flat on the rib cage, which stabilizes the shoulder girdle. It also helps to protract the shoulder blade and is the primary path of structural strength between the back of the diaphragm and the arm. It opposes the _rhomboid muscle,_ which also attaches along the medial border of the scapula and along the sides of the spine. The rhomboids retract and lift the shoulder blades, while the serratus anteriors protract to stabilize—they work together in well-aligned, full-shoulder action.
_The serratus anterior protracts (spreads and outwardly rotates) the shoulder blades, fixing them firmly onto the ribs. With such a solid connection, force can be transferred through the arms without strain in the neck or danger to the shoulder joints._
PART TWO
ĀSANA
_Movements and Poses Strung Together Like Jewels on the Thread of the Breath_
FOR AṢṬĀṄGA VINYĀSA PRACTITIONERS, THE SIMPLEST and possibly the "only" way to consider the poses is from the perspective of the order in which they appear in the traditional sequences: Sūrya Namaskāras, the standing poses, the specific series, and the finishing sequence. This is good. But we can also gain understanding by examining the postures in terms of some of their common features, regarding them as twists or backbends or balancing poses. Looking at postures in terms of the overall category into which they fall is informative in terms of alignment for any practitioner, and for Aṣṭāṅga practitioners, pulling poses out of the usual sequence can be a good practice for shifting perspective.
This book presents the poses first in detail within "families" to show the underlying patterns of breath and movement that tie them together. It also gives the traditional sequences as a quick reference. All poses found in the Primary and Intermediate Series and a small number of other poses that demonstrate a particular form or idea are included. Some can belong to more than one family. Yoga is a fluid, flowing, evolving system of breath, movement, and form, forever resisting our need to categorize it yet benefiting from our attempts to do so.
_5_
Building Sūrya Namaskāra
SŪRYA NAMASKĀRA (SUN SALUTATION) SERVES as the foundation of an Aṣṭāṅga Vinyāsa practice. It contains all of the subtle alignment as well as the cues for and process of joining together complementary opposites from which the internal forms of the practice unfold. When Sūrya Namaskāra is practiced carefully, with vitality and enthusiasm, it sets a tone for our entire practice and informs the movements in every other posture. Many of the traditions of yoga believe that the sun is not only far away in space, but also simultaneously in the center of our hearts. Practice Sūrya Namaskāra as if you were the radiant sun to fully feel and express internal forms and alignment.
There are two forms of Sūrya Namaskāra: form A and form B. Both begin and end standing in Samasthitiḥ. There are nine poses, or subforms, within form A and seventeen within form B. The quality of the movement in and out of each pose is a key to the effectiveness of both sequences. It is important to work slowly and patiently when first learning Sūrya Namaskāra so you can refine the movements, cultivating a strong, integrated, and meditative flow.
In this section, Sūrya Namaskāra is broken down into constituent parts. Working on them individually can help encode the coordination of breath and gaze within each form into the body, making Sūrya Namaskāra—and the entire practice—fluid and meditative. Many of these foundational movements are also beneficial for therapeutic purposes. If Sūrya Namaskāra is not appropriate for someone, the foundational movements with modifications, such as the Puppy Pose, can be explored in a way that might still be beneficial.
Once the transitions within Sūrya Namaskāra are smooth and integrated, you can use the traditional method of "counting out" the forms, using Sanskrit numbers assigned to each position as a kind of shorthand to flow continuously on the thread of the breath. Moving in a smooth and fluid manner helps to give the entire practice a truly meditative quality.
During a traditional Aṣṭāṅga Vinyāsa practice, Sūrya Namaskāra A and B are each practiced a minimum of five times before moving into the standing sequence and then on to the body of the specific series or sequence to be practiced.
SETTING UP FOR SŪRYA NAMASKĀRA
**S AMASTHITIḤ**
_Equal Standing_
1. Stand straight and tall with the feet together, sides of the big toes touching, and the arms down at your sides. Linger for a moment in Samasthitiḥ. At this point, the eyes become steady in dṛṣṭi, which releases the palate and allows you to begin ujjāyī breathing. The simple waves of the breath initiate the process of natural alignment. This is merely the intelligence waking up in the central channel.
2. It is easy to feel a sense of floating at the center of the upper chest on the inhalation. Staying with the sensations as they arise, the middle of the pelvic floor will begin to feel like it is becoming centered on the plumb line. The back of the diaphragm—along the kidney wings—lifts and floats, as does the heart. The sitting bones, pubic bone, and coccyx all drop down, while the pelvic floor tones. As you soften into the breath you may experience a constant sense of tuning and retuning—even microtuning—on the current of the breath. Everything is there if you remember to continue to inhale and exhale into the pose. This is similar to the process that manifests in sitting meditation, except here you're standing.
3. The radiant seed-point of the inhaling pattern is in the heart. Its extended pattern all through the body is experienced as a feeling of upward expanding and floating. The seed-point of the exhaling pattern is the center of the pelvic floor, and that pattern gives the feeling of downward contracting and grounding throughout the body. When you inhale, you pull your attention like a thread up through the seed of the exhalation at the middle of the pelvic floor. When you exhale, you release the upper back of the palate to keep the heart open. Every time you inhale, you concentrate on the residue of the exhalation and with each exhalation your mind rests in the feelings and sensations of the inhalation.
Hold this position for five to ten rounds of the breath, with the hands folded in _Añjali Mudrā_ (prayer position) in front of the heart. At this point, you may chant invocations to center the mind and begin the practice. (See Appendix 2.)
4. Standing like this in Samasthitiḥ is prāṇāyāma practice, and essentially that's what you do in all of the poses. Yoga āsana, when practiced in a contemplative manner with the poses strung together like jewels on the thread of the breath, is nothing less than prāṇāyāma in motion. The technique of relishing the essence of the inhalation as you exhale and delighting in the essence of the exhalation as you draw breath in is the root of all your practices.
**E KAM**
Ekam, or "one," is the first position of Sūrya Namaskāra, and it begins the vinyāsa process of moving in sync with the breath and the gaze.
1. Standing in Samasthitiḥ, inhale as you lift the arms up and out slightly in front of the body. Spin the arms by reaching up toward the ceiling and placing the palms together.
2. As you reach up through the arms, they should feel as though they are extensions of a spreading in the back of the diaphragm, the kidney wings, and by keeping awareness in the coccyx and pelvic floor you can sense into a full stretch of the psoas line. Firm up the legs—quads and hamstrings—so that as the inhalation fills the body, the edges of the diaphragm open like an umbrella.
3. The shoulder blades spread wide (protract) as the palms turn toward each other, so you can really lift the arms as you reach the peak of the inhalation.
4. Finally, tilt the head back, keeping it behind the arms and gazing to the thumbs, along the line of the nose; there should be no strain in the neck. This stretches the throat, and you get a broad, comfortable backbend in your lower neck without your head falling back on the atlas (the first vertebra in the neck). The entire head, including the chin (even if you have a beard!) should be _completely_ behind the arms.
5. In the gap at the top of the inhalation, lift the arms even more, as if you are trying to reach the ceiling. Really lifting in this way protracts the shoulder blades and stimulates the edges of the diaphragm so those edges start to sparkle.
6. The residue from those diaphragmatic sensations percolates as you exhale and allows the arms to float back down to the sides. Keep your arms toned as they descend, and work the legs to stand taller than ever right up through the crown of the head. This helps create a sense of stability and allows the unfolding of proper geometry within the body.
7. Remember, all the movements are based in the sound of the ujjāyī breath. If you get the sound, everything is easy. As you move, keep the legs engaged and reach up on the inhalation, like you're a swimmer. Then exhale with the soft, aspirate ujjāyī sound, and push on up through the crown of the head to glide back down into Samasthitiḥ.
8. Repeat these movements from Samasthitiḥ to Ekam and back to Samasthitiḥ with feeling at least three more times.
**E KAM, DVE, TRĪṆI**
_Numbers One, Two, and Three_
Next we add the second and third movements of Sūrya Namaskāra. This begins to work the whole body rhythmically in time with the gaze and the breath.
1. As you inhale, tone the quads and pull down on the sitting bones with the hamstrings. Open the wings of the kidneys extra wide as you lift and spin the arms to reach up into Ekam.
2. Keep the heads of the femurs back slightly in the hip sockets, so the groins are hollow. This keeps the center of the pelvic floor on the plumb line, which makes the center point float.
3. The shoulder blades wrap out wide and lift while the pot of the belly is lifted up and out of the pelvic basin. This will subtly awaken Mūlabandha, like a little flame in the middle of the pelvic floor, which will maintain reversibility and core awareness through the coming movements.
4. Gaze at the thumbs, using the image of cobra hoods rising and spreading in the back of the head. This keeps your head behind the arms and the lower neck gently curved into a backbend.
5. Exhale and "swan-dive" up and over. As you begin to fold forward, keep the spine straight, the arms full of breath, and the legs toned. From the sun in your heart extend through the crown of the head. This will keep the pelvic floor online and ready for counteractions.
6. Keep the chin slightly out for the first 90 percent of the fold, then just as you run out of air, look toward your nose so the spine begins to flex. At the very end of the exhalation, contract the abdomen and the PC muscle, creating a coiled feeling at the very end of the breath to complete Dve position.
7. If you are unable to touch the floor beside your feet with your hands, bend the knees as you fold. Even with the knees bent, keep the legs toned by grounding down evenly through the mound of each big toe and the other corners of the feet. Lift the kneecaps. Keeping your legs engaged, work the edge of sensation in your forward bend, taking care not to overengage or overextend and not losing integrity in the muscular patterns awakened by these movements.
8. Inhale as you lift your head and elongate the spine through the crown, moving into Trīṇi. To keep your PC muscle toned during this inhalation, spoon up under the belly and create a vacuum under the belly right above the pelvic floor, as if drawing breath in through a straw that comes up the midline of your body between your legs. Again, you'll use the end of the exhalation to set this particular intensity of the apāna, then inhale and pull the seed-point of the exhalation up inside.
9. On the next exhale, lengthen your spine and return rhythmically to full Dve position.
10. On the next inhale, feel the form of the legs, feet, and coccyx as you reach back up with vibrant arms into Ekam.
11. As you exhale, your arms float back down to your sides and the heart floats up as you return to Samasthitiḥ. Keep the gaze steady along the line of your nose to a point out in front of you.
12. Repeat this sequence of movements—Ekam, Dve, Trīṇi—at least three more times.
**E KAM, DVE, TRĪṆI VARIATION**
If you're stiff, that's good. Who wants to hit their face on their shins anyway? You can still look toward your navel at the end of the exhalation to get that apānic coil after you fold forward. Rather than being concerned about how flexible you are, you can base the action in the pelvic floor and be happy.
1. With the feet hip-width apart, inhale and open the arms as if they were extensions of the kidney wings. Grip the legs with your awareness and keep them parallel as you inhale and reach up to a modified Ekam.
2. Swan-dive up and over, looking up and out as you fold forward. Gradually bend the knees and place your elbows on your thighs just above the knees.
3. Having folded forward as far as is appropriate for you, at the very end of the exhalation tone the abdominal muscles and drop the head. This is a modified Dve.
4. As you inhale, find the line of the PC muscle and abdomen, and lift the head to Trīṇi . Extend through the crown so the spine straightens, and imagine that the pubic bone and coccyx are moving back and then together. Relax your palate. Have a good time.
5. Exhale and fold forward again to Dve, keeping the legs toned and the spine straightening, as before.
6. Lift your head and, inhaling, reach back up to the ceiling as you straighten your legs. As you exhale, allow the arms to float back down to your sides as you return to Samasthitiḥ.
ESTABLISHING A BASE FOR SŪRYA NAMASKĀRA A
For the first Sun Salutation of the day, while learning Sūrya Namaskāra, or if you have injuries that prevent you from doing the full form, it is helpful to break the sequence down into component parts. (See the full sequence of Sūrya Namaskāra A in Appendix 5.) But remember, even when separating the movements, you must still move in sync with the breath and the gaze. When you slow down movements and concentrate on particular subtleties of form or precise points of alignment, the mind may take it as an invitation to wander or to co-opt the practice and turn it into a theoretical mind game rather than an internal physical experience. So stick with the breath, the gaze, and the internal forms, flowing mindfully with whatever full body patterns you have.
By exploring modified forms, some of the restrictive or distracting external factors—such as tight hamstrings, theories of alignment, or SI joint pain—can be removed from the equation so the internal forms are easier to observe. Slowing down the movements within Sūrya Namaskāra provides an opportunity to "relearn" how to move; this can help you to recover from an injury and to avoid creating unbalanced habitual patterns of movement that may be limiting or eventually _cause_ injury.
**S AMASTHITIḤ**
1. Establishing the plumb line, steadying the gaze, softening the tongue, and releasing the palate you can enter Samasthitiḥ. Bring your awareness to the sound of the ujjāyī breath, and notice the ends of the breath as they join together deep in the belly behind the navel.
**E KAM, DVE, AND TRĪṆI**
1. As you inhale, create a vacuum under the belly right above the pelvic floor, as if drawing breath in through a straw that comes up the midline of the body between the legs. Fill the arms with breath, as you lift and spin them reaching to the sky and looking at the thumbs in Ekam.
2. On the exhalation, release the palate up and back to do a swan dive—moving up, over, and around to fold forward into Dve. Keep the toes and fingers open for guidance.
3. At the end of the exhalation, curl the spine, contract the abdominal muscles, tone your PC muscle, and look toward the nose.
4. Lift the head as you inhale and extend the spine long, out through the crown of the head, placing your hands next to the feet for Trīṇi.
**C ATVĀRI AND VARIATIONS**
_Number Four_
1. Exhale, step back, and lie down on your belly. This is an introductory form of Catvāri, the fourth movement within Sūrya Namaskāra. You will expand on this initial form as your practice evolves. The full form of Catvāri is like a well-aligned push-up position, but in the beginning as you build strength, or for educational purposes anytime, lying on the belly can be very effective.
2. Alternatively, when learning Catvāri, you may step back into a high plank position. In this form, the arms are vertical with the elbows slightly bent to about five degrees. Push the shoulders down, away from your ears, and ground your thumbs. Keep the abdomen toned and the legs fully extended and engaged so that your body is in a straight line. Direct the gaze out in front along the line of the nose.
3. For the final form of Catvāri, come into a low plank position with the sternum about four inches from the floor. Align the top edge of the shoulders just over the tips of the fingers. Position the feet about hip-width apart, strengthen the abdominal muscles, and pull back the groins. Keep the shoulders squared and elbows fairly close to your sides. Gaze out in front of you on the floor along the line of the nose.
**P AÑCA**
In Sūrya Namaskāra, the fifth full form is Ūrdhvā Mukha Śvānāsana (Upward Facing Dog Pose).
1. From the end of the exhalation in Catvāri, pull your pubic bone about one inch farther back from the floor. As you begin to inhale, allow the coccyx to dive straight down into the pelvic floor. Pulling the spine forward, roll onto the tops of the feet as you unroll the spine into a backbend. Press through the hands to straighten the arms without locking your elbows.
2. The heart area floats up high in front of the arms. The pelvic floor will feel as if it's being pulled way up into the body, as you maintain a slight inward rotation so the "eyes" of the elbows do not look straight ahead. This keeps the weight properly distributed on each hand between the heel of the thumb and the index and middle fingers.
3. When you unroll your spine, it's as if the spine were uncoiling from bottom to top, the head being the very last component to roll back as you gaze down the nose. This is Pañca, the fifth position in the Sun Salutation.
**P AÑCA VARIATION**
_Sphinx Pose_
Two important poses that can help us train for Upward Facing Dog Pose are Sphinx Pose and Bhujaṅgāsana.
1. Lie on your belly and crawl up onto the elbows, placing them approximately under your shoulders. Position the forearms and lower arms parallel to each other, with palms facing down.
2. As you inhale, pull isometrically through the forearms back along the sides of your body. Draw the sternum forward and up to gently encourage the spine to elongate into a subtle backbend. It will feel as if the collarbones are spinning backward and being drawn up and back over the shoulders. Keep the pubic bone on the mat.
3. Meanwhile, imagine the coccyx is heavy, like gold, so that it drops easily down. Keep the eyes steady, gazing at a point out in front of you on the floor along the line of the nose. Roll the tops of the arms up, back, and down to cue the shoulder blades to slide down the back, the collarbones to spread, and the heart to float. Keep the head steady, kidney wings wide, palate released, and the crown of the head light.
4. When you inhale, you should feel the weight of the coccyx as if it were a slightly heavy tail flowing into the PC muscle. As you exhale, release the pull through your forearms and soften into the pose. On the next inhalation, again pull the spine forward and feel the breath as it spreads the kidney wings. Drop the coccyx but keep the buttocks soft.
5. If you experience compression or pain in the low back or SI joint, experiment with small shifts in form to alleviate the discomfort. Slightly changing elbow position—moving them forward or back, a little bit wider or somewhat closer together—can help. Also, focus fully on rolling the tops of your arms back and dropping your coccyx down while softening your belly—maybe sinking a little lower down through the upper body—as you pull with the forearms. Tone and reach back through the legs so there is a sense of the spine being pulled forward as the sitting bones are held back evenly in the opposite direction.
6. On the inhalation, lengthen the psoas muscle behind the belly, almost like you're breathing _through_ the muscle itself. Eventually the inhalation will pull the middle point of the pelvic floor up into the midline. On the exhalation, maintain, observe, and keep the tone that is building in the pelvic floor and throughout the body.
**P AÑCA VARIATION**
_Bhujaṅgāsana (Cobra Pose)_
1. After five rounds of ujjāyī breath in Sphinx Pose, take a big breath in. At the top of inhalation, lift and spread the elbows out to the sides. Use the pressure of the palms of your hands to pull yourself forward into Bhujaṅgāsana. You may also enter Bhujaṅgāsana directly from any variation of Catvāri.
2. Roll the tops of the arms back in space. The shoulder blades will automatically slide down the back, broadening and supporting a gentle lift to the heart from behind. Keep the coccyx dropped, the pubic bone and thighs on the floor, and the legs gently toned. Stay in Bhujaṅgāsana for several rounds of the breath or move immediately into Ṣaṭ, the sixth form.
**Ṣ AṬ**
The sixth form in Sūrya Namaskāra is Adho Mukha Śvānāsana (Downward Facing Dog Pose). In an Aṣṭāṅga Vinyāsa practice, practitioners spend more time in Downward Facing Dog Pose than in any other single pose. It is one to work on and refine endlessly.
1. At the crest of an inhalation in Upward Facing Dog Pose, release the palate to stimulate a wavelike, rolling movement that, on the exhalation, triggers the feet to pull back and flip over. This closes the hip joints and creates the perfect counterpose to Upward Facing Dog Pose.
2. Entering the pose, the shoulder blades must be fully protracted. As your weight shifts forward from the heels of thumbs to the roots of the index and middle fingers, the shoulder blades rotate back. Focusing on the collarbones, you may experience a sense of them rolling back and spreading.
3. The head gradually falls through the arms to complete the form. Keep the legs and arms activated and the fingers and toes spreading.
4. Hold the pose for at least five full rounds of the breath, gazing at a point on the floor between your feet or, eventually, at your navel. Gradually lengthen the arms to move the sitting bones farther back and to release and lengthen the psoas muscles; this allows you to find Mūlabandha and the central channel.
**Ṣ AṬ VARIATION**
_Puppy Pose_
1. From Pañca or one of its variations, exhale and push up onto the knees with the toes turned under.
2. Position the thighs vertical to the floor, walk the hands as far forward as possible, and place the forehead on the floor. (If your forehead doesn't reach the floor, put a block under the forehead as support.)
3. Come up onto the fingertips as you reach through the arms, cupping the palms away from the floor and protracting the shoulder blades. Pushing your fingertips down moves the middle of the armpit back and up, away from the floor. This is one way of learning the illusive shoulder form in full Downward Facing Dog Pose.
**Ṣ AṬ VARIATION**
_Old Dog Pose_
1. You may come directly into this variation by pushing up onto the knees when transitioning out of Pañca. Alternatively, you may practice the Puppy Pose first and work into the Old Dog.
2. Begin on all fours, with the arms and thighs vertical and the toes turned under. Move the hips back in space until the knees automatically begin to lift off the floor. You will experience a sensation of the buttocks expanding just before the knees automatically lift away from the floor. This is good because it makes the pelvic floor shrink.
3. Leave the knees bent deeply, and notice that your thumbs naturally ground to the floor. Keep the knees bent until you've moved the coccyx, pubic bone, and sitting bones as far back as possible.
4. This variation is particularly good if you have tight hamstrings that prevent you from deeply folding the groins closed in Downward Facing Dog Pose. Tight hamstrings may also inhibit the primary prāṇic action of the pubic bone moving up and back in the full form. Because the hamstrings are such strong muscles, when tight they pull down on the sitting bones, which prevents the pelvis from rotating around the femurs and keeps the coccyx tucked. All this can prevent the primary action of the pubic bone. In the final form of the pose, both actions are present: the pubic bone moves back and up, while the coccyx moves back and down. Too much action from the coccyx at the beginning of the pose may move the sacrum in the same direction as the coccyx, which will overpower the desired up and back motion of the pubic bone.
5. Stay in the pose for five breaths. Then, if appropriate, gradually straighten the legs into the full Downward Facing Dog Pose form and hold for five more rounds of breath.
**T RANSITIONING**
1. When practicing Puppy Pose, transition to the front of the mat and the next movement of Sūrya Namaskāra by lifting the hips and walking the feet forward until they are between the hands.
2. For Old Dog or the full form of Ṣaṭ, at the end of the fifth exhalation, bend the knees to prepare to jump forward. The end of the exhalation pulls down on the coccyx by toning and setting the PC muscle. That stability gives you the control to get your feet back to the front of the mat.
3. As you bend the knees, lift the head to look at the floor between the hands. Keeping the head away from the floor and behind the arms, walk, step, hop, or float forward, landing with the feet approximately between the hands at the front of the mat. However you get there is fine. Over time and as your strength and trust in your own movement increase, you will eventually be able to hop easily and land lightly.
**S APTA, AṢṬA, NAVA, AND SAMASTHITIḤ**
_Seven, Eight, Nine, and Samasthitiḥ_
1. Just as the feet land, extend the spine and inhale, looking up and lifting into Sapta (the seventh position, which is the same as Trīṇi).
2. Exhale and fold forward, engaging the legs and abdomen. Gaze down the line of the nose. This is Aṣṭa (the eighth position, which is the same as Dve).
3. Lifting the head, inhale as you stand up. Reach up and place the palms of your hands together above and in front of the head, gazing down the line of the nose at your thumbs. This is Nava (the ninth position, which is the same as Ekam).
4. Exhale, lengthen up through the midline, and return to Samasthitiḥ, allowing the arms to float back down to the sides.
5. Once you have become familiar with the movements and underlying structure within Sūrya Namaskāra A, you should practice it as a rhythmic, moving form, synchronizing the breath, gaze, and movement into a smooth and continuous pattern. All of the movements except Ṣaṭ (Downward Facing Dog Pose), from Ekam back to Samasthitiḥ, are held for only one breath, inhaling to enter the movement and exhaling to transition out of it. Ṣaṭ is held for five rounds of breath before transitioning forward into Sapta.
**S ŪRYA NAMASKĀRA B**
Sūrya Namaskāra B has seventeen forms and builds the heat before the full practice. Refer to the earlier descriptions of Sūrya Namaskāra A for more detail on some of these movements. (See full sequence of Sūrya Namaskāra A in Appendix 5.)
1. Stand in Samasthitiḥ with the hands together in prayer position.
2. Inhale into Ekam. As you inhale, bend the knees 90 degrees into a half squat and lift the arms over the head. Place the palms together and look along the nose at the thumbs. Keep the lower belly hollowed just above the pubic bone, inner knees touching, and the arms reaching up but in front of the head. This is Utkaṭāsana.
3. Exhale into Dve by straightening the legs, then coming up and folding over on the wave of the breath.
4. Inhale into Trīṇi, straightening the spine as you lift the head.
5. Exhale into Catvāri. Jump or step back into a high or low plank position (whichever is appropriate for you) with the corresponding gaze.
6. Inhale into Pañca. Pull yourself forward and roll the feet over into Upward Facing Dog Pose.
7. Exhale into Ṣaṭ, pushing back into Downward Facing Dog Pose.
8. As you complete the exhalation, bring the left heel onto the centerline of the mat, toes pointing out, and the right foot forward between the hands into a lunge position. Inhale into Sapta by lifting the arms and placing the palms together, keeping both feet well grounded. Bend the right knee until it is over the right ankle. Keep the left leg firm and straight and the left foot entirely grounded. Draw down the coccyx, pubic bone, and sitting bones, while opening the left groin. Gaze at the thumbs with the head behind the arms. This is Vīrabhadrāsana A (Warrior Pose A).
9. Exhale into Aṣṭa. Place the hands on the mat at shoulder-width distance. Step back with the right foot and lower the body into the form of Catvāri that is appropriate for you.
10. Inhale into Nava. Lift the heart, point the toes, and come up into Upward Facing Dog Pose.
11. Exhale into Daśa (the tenth movement). Pull back into Downward Facing Dog Pose on the out-breath, and at the end of the exhalation, bring the right heel in and step forward with the left foot between the hands in a lunge position.
12. Inhale into Ekādaśa (the eleventh movement). Lift the arms and bend the left knee into Vīrabhadrāsana A, the mirror image of step eight.
13. Exhale into Dvādaśa (the twelfth movement). Place the hands on the mat and step back into whichever plank position you did for Catvāri, with the elbows bent and drawn in toward your sides.
14. Inhale into Trayodaśa (the thirteenth movement) by pulling forward into Upward Facing Dog Pose.
15. Exhale into Caturdaśa (the fourteenth movement). Draw the hips back and wrap your shoulders out, engaging the serratus muscles so they protract easily as you move into Downward Facing Dog Pose. Remain in this position for five breaths.
16. Inhale into Pañcadaśa (the fifteenth movement). At the end of the fifth exhalation, bend the knees, look up, and hop forward, placing the feet in Samasthitiḥ position. Just as your feet land, inhale and lift the head.
17. Exhale into Ṣodaśa (the sixteenth movement) by folding forward and looking at the tip of the nose.
18. Inhale into Saptadaśa (the seventeenth movement). Bend the knees deeply and raise the arms into Utkaṭāsana.
19. Exhale as you press back up to standing tall in Samasthitiḥ.
Sūrya Namaskāra A and B should each be practiced a minimum of five times at the beginning of each practice. It's best to do them until a feeling of stability arises, and the breath seems to pervade your whole body. Sūrya Namaskāra is the great elixir among yoga practices; beginners and those who feel weak or emotionally distraught should do many repetitions! Sūrya Namaskāra builds the foundation for the rest of the series by creating coordination, endurance, and strength. Its importance cannot be overemphasized. Sūrya Namaskāra B helps to open the _granthis_ (energy blockages) around the sacrum. Remember to keep the shoulders away from the ears and the heart open throughout all the movements.
In a vinyāsa practice, poses are not done in isolation from one another. Instead, one unfolds into the next as a moving meditation, linked together by the breath and the gaze. They flow naturally, like waves within patterns of breath, movement, alignment, form, and mind. Ujjāyī breathing and a steady, nongrasping gaze, combined with a sense of bandha and mudrā, construct the foundation on which this integration of body and mind is built. To maintain this sense of a pattern of movement and form rather than independent poses, the entrance and exit to any pose are considered as important as the pose itself. Keeping the internal, meditative connection alive, we move between poses with Full or Half Vinyāsas or by returning for a full breath cycle to Samasthitiḥ.
FULL VINYĀSA AND HALF VINYĀSA
A Full Vinyāsa may be practiced between any and all poses, if time and energy permit. Typically, however, it is practiced during the second half of the standing sequence, and Half Vinyāsa is practiced between every pose (and eventually between the sides of most poses) after the standing sequence. Both forms increase strength and stability and serve to keep the practice moving as an interconnected series of movements while helping to reset the mind and nervous system to neutral between poses. Upward and Downward Facing Dog Poses and the rhythmic movements in and out of them are full, symmetrical expressions of the interlinking prāṇa-apāna patterns that help to process the residue of a pose and take you back into the middle path of breathing, bandha, mudrā, and dṛṣṭi.
A Full Vinyāsa follows the pattern of Sūrya Namaskāra A, returning to standing through the two Dog Poses and then starting again the Sūrya Namaskāra A pattern, and from Downward Facing Dog Pose moving into the next pose on the following inhalation. Half Vinyāsa is when you jump back into Catvāri between poses; move into Upward Facing Dog Pose on the next inhalation followed by Downward Facing Dog Pose on the exhalation; then glide into the next pose on the following inhalation.
TWO EXTRAORDINARY LINKING MOVEMENTS
To accommodate the cyclical return to the two Dog Poses, there are two other wonderful movements that defy easy categorization: Jumping Back and Cakrāsana (Wheel Pose). While you are learning the movements, or if you have physical limitations, these poses may feel (and be) out of reach; however, there are alternatives so that everyone can work on the same patterns of strength, counteraction, and movement and perhaps move into the full form one day. Remember, it is the patterns of Prāṇa and the thoughts that ride upon them that are of interest. The mindful, meditative step-by-counterstep process of intelligent vinyāsa is the actual practice. The achievement games that the mind constructs are to be observed by an unbiased intelligence.
**J UMPING BACK**
1. From a sitting position on the floor, begin by crossing the legs, drawing the thighs and knees in toward the chest, and placing the hands down at the sides while finishing a firm exhalation. This helps the apāna pattern bring the rectus abdominis muscle into its most contracted and toned form.
2. Lift the hips off the floor and inhale into the higher portion of the lift. As you lift and swing the legs back between the arms, move your chest and face forward and down.
3. Extend the legs to land in Catvāri once the chest and shoulders are down in correct Catvāri position.
4. If you can't perform this entire movement, merely lift the hips and feet off the floor as you balance for a breath resting on the hands. Then lower back down, place the hands in front of the knees, and step or hop back into Catvāri. Practicing the lift alone initiates the desired pattern in your body. Whether lifting up to jump back or simply lifting and stepping back, the movements should be done elegantly and on the wave of the breath. Straining and using yoga as a competitive ego sport is usually counterproductive in the long run.
**C AKRĀSANA**
_Wheel Pose_
Another useful and potentially wonderful vinyāsa linking movement is Cakrāsana, a backward roll. This pose can cause fear or trepidation because it's disorienting to roll backward, and it may seem that your neck is in danger. Actually with practice and some trust in the breath, Cakrāsana is accessible to most people.
1. Lie on your back in Tāḍāgī Mudrā (see page 27). Exhale deeply and firm the rectus abdominis muscles.
2. Before inhaling, begin lifting the legs. When they are about halfway up (at about 40 degrees from the floor), begin to inhale.
3. Lifting the pelvis off the floor, place the hands on the floor next to your ears. As your feet travel over the head, turn the eyes slightly upward in dṛṣṭi.
4. As the center of gravity moves between the hands, begin to push with them, almost as if you were pushing up into a backbend. Keep the legs straight, reaching them back toward the wall behind you as you move into a backward roll and wind up in Catvāri. The heart will approach the chin at the point of greatest flexion in the lower neck.
5. In the final stage of the roll, gaze down the line of the nose and roll symmetrically. Never pull the chin into the throat, and do not turn the head to either side.
6. Finish the pose in Catvāri, and then proceed through Half Vinyāsa to the next pose.
7. A qualified and patient teacher can be of tremendous help in learning this movement. It is important to resist doing the cheating method of rolling over one ear or the other. This is mechanically unsound and does not produce the symmetrical stretch that serves as a vinyāsa meditation.
_6_
Standing Poses
EVEN IF YOU DON'T DO MUCH YOGA, A CONSIDERABLE amount of your time is spent in standing poses: at the sink in the morning brushing your teeth, waiting in line for the bus, or strolling along the beach at sunset. When we begin practicing āsana, we discover that there are many "official" standing poses too, each with specific benefits and identifiable form. Those are the poses we will explore in this chapter.
As we begin to understand the underlying principles of form and alignment required to practice the āsanas as vinyāsa, we see how extraordinarily valuable standing poses are; they are titillating, grounding, and integrating. Most standing poses, such as Samasthitiḥ or Trikoṇāsana, are less dramatic than many other forms within our practice, and we may take them for granted or overlook their importance. For that very reason they are a practical and intelligent place to begin experiencing the internal forms that connect us from palate to perineum, allowing us to feel fully rooted to the earth through the practice. Well-executed standing poses, more than any other āsana form, make it possible for us to wake up and connect to what's happening before our eyes, even when we find ourselves in everyday "standing poses" like walking the dog.
The category of āsanas referred to as standing poses have many virtues. First, most can be done with little or no warm-up, so they may be practiced wherever you happen to be and whenever you feel like tapping into the benefits of the practice. Unlike most seated and balancing poses, backbends, or inversions, standing poses usually do not require a formal vinyāsa of poses to prepare and stretch the body beforehand or to get into them easily and safely.
Second, these poses connect us top to bottom, earth to sky. All standing poses naturally emphasize the feet and legs, and consequently they must involve the hips in a competent and integrated way; otherwise, the practitioner would fall over. With our feet fully awakened and intelligently rooting down, we build standing poses from the ground up. Bringing awareness to the arches, we then focus on the feeling of the leg muscles being drawn up from the feet (as if we were putting on a pair of stockings) as we experience a sense of connection to the earth and grounding down through the feet. This makes it easy to see that waking up the feet is immediately reflected in the pelvic floor. On a more subtle level, standing poses are stabilizing, not only in the literal sense of connecting us to the earth, but because the mechanical requirements they teach underscore the correct use of the pelvic floor, co-contractions of opposing muscle groups, and the obvious value of simultaneously working primary rotations and counterrotations. This deepens the impact of standing poses so we can experience and get the most benefit from the healthy and competent movements in and out of the forms.
All standing poses also require balance and thereby demand that we hone the skill of constantly adapting the forces and movements in the body to different structural plumb lines. On a subtle-body level, this is the function of Prāṇa (intelligence), and it quickly offers a visceral lesson on the wisdom of following the middle path. As such, standing poses, like deity visualization, can be a simple doorway into this internalized experience of the middle way.
Another good thing about standing poses is that, because there are so many of them, we can get a complete and balanced yoga practice using these poses alone. So if you ever find yourself in a tiny space for a long time—say, you get stuck in an elevator one day—you can do a full practice using very little floor space. Such a practice would include poses that fully express extension, flexion, and twisting as well as balancing and meditative form. Even the asymmetrical standing poses such as Parivṛtta Trikoṇāsana, because they are done on both sides, are conducive to meditation, bringing you back to a powerful feeling of symmetry and the invitation to drop into the midline of the body.
For the most part, standing poses are safer for many people than other poses. This is partly due to the fact that standing itself is a familiar form to anyone who ambulates through the world. Usually standing requires that we be awake and relate to the physical necessities of our environment. For this reason, during standing poses, we are less likely to obsess on and strain in any particular movement or position while ignoring the rest of the body (which is not uncommon when doing some other types of poses). Standing forms are simple, and beneath them all messages from the subtle body are constantly bringing our focus back to balancing, the familiar territory of not falling over. This is true even in more difficult standing poses, like Utthita Hasta Pādāṅguṣṭhāsana, where little opportunity is left for strain or distraction even though full expression of the pose may be beyond the practitioner's current reach. Because we are connecting lines of movement and Prāṇa rigorously and intelligently—and in most cases, more simply than in other forms—it is easy to recognize the necessary patterns of moving in sync with the breath in standing poses. They simply and directly invite us to feel and dissolve into integrative, full-body lines of movement and awareness.
This chapter groups together the poses traditionally practiced as the "standing sequence" in the Aṣṭāṅga Vinyāsa form, defining this family as poses that ground us into our legs and the earth. They connect us from top to bottom, from fingertips to toes, so when we wake up through the standing poses the mind no longer tends to separate out āsana and yoga practice as different from every other moment of our day. Standing poses bring with them the clarity, maturity, and competence required to be grounded physically and emotionally in the real world; as such, they are an antidote to neurotic patterns that may arise from attempting meditation, prāṇāyāma, or extreme yoga āsansas in an incorrect manner. Standing poses are of particular importance to practitioners who have difficulty focusing on what is arising in front of them and for those who are dull, distracted, distraught, or dysfunctional, as the benefits of well-executed standing poses can be a remedy to these sorts of problems.
An interesting point of controversy is the question of how far apart to place the feet in standing poses such as the Trikoṇāsana family or Pārśvottānāsana. A general rule is that you should place your feet apart approximately the length of one of your legs (measured from the top of the femur to the ankle). More important is that the feet should be placed at a distance that invites the pose to be "turned on" and that requires the feet and legs to wake up so you can experience the "juice" of the pose. You must discover the most healthy and beneficial distance for your specific circumstances. When your feet are at the optimal distance, a pose works your entire body—including the spine, head, neck, and shoulders—to make the pose a beautiful expression of integrated form.
Ultimately a yoga pose should not damage the joints, ligaments, tendons, or cartilage (or any other body system, for that matter). For example, you can create beneficial poses with the feet fairly close to each other (less than one leg length apart) in Trikoṇāsana or Vīrabhadrāsana, and this may be valuable or absolutely necessary if you have an infirmity, are older, or lack flexibility or strength. But if you are stronger, younger, more flexible, or more athletic, you can move the feet farther apart. Equally, a number of injuries can result from placing the feet _too_ far apart and exceeding the structural limits of the hip joints and the many groups of muscles that cross over these joints.
In the end, we must each understand the structural requirements of the poses and the breathing patterns necessary to integrate the form. We then practice with the intention of waking up both the body and the mind through each movement and micromovement as we transition in and out of poses and take on the full forms.
**S AMASTHITIḤ**
_Equal Standing_
Standing in Samasthitiḥ may seem relatively unchallenging, perhaps not even much of a yoga pose, since we stand around all the time anyway. Yet it is actually a very _advanced_ and important āsana. It is translated as "equal standing" because we stand rooted deeply into the earth while finding that point in time and space that falls directly along the plumb line and drops us into the present moment. Everything is equal in all directions: perfect efficiency in relation to gravity; a complete lack of bias from left to right, front to back, up to down; and when it's really tapped in, past to future. Samasthitiḥ is a strong yet meditative pose characterized by a clear, soft, open heart.
More than anything, it places the details and difficulties of other poses in the perspective of the present moment. Establishing a solid Samasthitiḥ sets the tenor for the entire practice, and as we return again and again to the form, the seed of connection to our present circumstances arises, inviting us to wake up!
1. Stand with the sides of the big toes touching. The heels can be together or slightly separated, depending on individual preference or limitations. Take a moment to breathe into the pose and settle the attention along the plumb line of the body, connecting the pose from the crown of the head through the core of the body and evenly down through the feet.
2. The weight drops immediately in front of the heels as the toes spread opening evenly and rooting down through the mounds of each toe into the earth. Draw the kneecaps up, to facilitate a feeling along the whole front surface of the body that it too is being drawn up. The back surface of the body is drawn down.
3. The heart area is lifted and spreads evenly. The front edges of the armpits lift and widen as the back edges drop down and broaden. Behind the kidneys, the floating ribs, the kidney wings, lift up with as much buoyancy as the heart.
4. The centers of the ears are over the centers of the shoulders, which are over the centers of the hip joints. The center of the perineum—the mūla point—ascends, which facilitates lifting the core of the body and making the crown of the head ascend.
5. Draw the lower belly, just above the pubic bone, slightly back to give you an awareness of Mūlabandha. The eyes remain steady, either slightly downcast or at the level of the horizon, resting on a point straight ahead.
6. Tune in to the body's central vertical axis, which runs from the center of the crown through the center of the perineum falling just in front of the heels. If you wish, fold the hands in Añjali Mudrā (prayer position) in front of the heart.
**P ĀDĀṄGUṢṬHĀSANA**
_Big Toe Pose_
Pādāṅguṣṭhāsana and Pādahastāsana (the pose that follows it) are usually practiced together, one immediately after the other, without a Half Vinyāsa between them. Of course, they may be practiced separately as well, but they complement and inform one another, both presenting an excellent opportunity for learning to passively rotate the pelvis around the tops of the femurs in a simple forward bend without straining either the lower back or hamstrings.
1. Stand with the feet hip-width apart and exhale to fully ground through the legs and feet. The distance between the legs should be just wide enough so that eventually, when the forward bend is quite deep, the head can fit between the upper shins.
2. Place hands on waist and on the inhalation, lift the front of the body without strain but with strength to feel an even upward pull through the core of the body, as if you were preparing to do a subtle backbend. Bring awareness to the pelvic floor, toning and dropping the coccyx down and forward as the pubic bone drops down and back.
3. On the exhalation, begin to fold forward keeping the spine straight and the legs active and awake so that the pelvis rotates around the top of the femurs. If your hamstrings are tight or injured, bend the knees slightly, but even while bent, your legs should remain vibrant and engaged, pushing down into the earth through the feet, especially the mounds of your big toes.
4. At the end of the exhalation, wrap the first two fingers of each hand around the big toes. Then inhale and lift the head to reset the pose, elongating the spine as if reaching through the crown of the head. (Remember you may bend your knees.)
5. On the exhalation, fold forward completely, pulling straight out (not up) along the line of the floor on the toes, bending your arms, and spreading your collarbones by reaching the elbows out to the sides. Allow the head, neck, and spine to hang as you fold forward, so there is a feeling of depth in the groins and the weight of the head encourages the spine to elongate. Pull with the arms, but do not put any strain on the neck, throat, or palate. Straighten the legs, if possible, and gaze at the tip of the nose.
6. After five breaths, at the end of an exhalation, tone the abdominal muscles and pelvic floor, bringing awareness to the sense of a heavy coccyx. On the inhalation, keep the awareness in the pelvic floor and coccyx as you straighten the arms. Still holding the toes, lift the head and reach out through the crown, straightening the spine and coming halfway up.
7. Place hands under feet, palms up, and move immediately into Pādahastāsana. Or, to exit the pose, follow step 4 in Pādahastāsana below.
**P ĀDAHASTĀSANA**
_Hand and Foot Pose_
Pādahastāsana requires more flexibility in the hamstrings and wrists than does Pādāṅguṣṭhāsana. It is acceptable to bend the knees in order to place the hands under the feet, but remember to keep the legs active and awake, which allows for a "micro-bend" (slight bend) that encourages the kneecaps to lift as the feet remain fully grounded.
1. After lifting the head in the last step of Pādāṅguṣṭhāsana, slide the hands, palms up, under the front of the feet until the toes touch your wrists. Bend the knees, if necessary, but keep the legs engaged. (To enter the pose without doing Pādānġuṣṭhāsana beforehand, follow steps 1 through 3 for Pādāṅguṣṭhāsana.)
2. Spread the toes and fold the hip joints closed as you exhale. Pull the hands forward toward the front of the mat, as if trying to pull them out from under the feet. Bend the arms and work the elbows out to the sides.
3. Gaze at the tip of the nose with an empty palate and smooth, complete breathing. Hold the pose for five breaths.
4. To exit, as you exhale, tone the abdominal muscles and PC muscle. At the end of the exhalation, keep the coccyx heavy (PC muscle engaged). Lift the head as you inhale and straighten the arms and spine, lifting halfway up. As you complete the inhalation, place the hands on the hips and push the skin there back as you exhale. On the following inhalation, return to standing with the spine straight and the palate released.
5. Hop the feet back together and take one round of breath in Samasthitiḥ before entering the next pose.
**T RIKOṆĀSANA**
_Triangle Pose_
Trikoṇāsana and its counterpose, Parivṛtta Trikoṇāsana, are usually practiced together, one immediately after the other, without returning to the front of the mat. Of course, they may be practiced separately as well, but they complement one another on many levels. For Trikoṇāsana, maintaining a straight spine while entering and exiting the pose is very important and is instrumental in cuing the legs to rotate and counter-rotate properly. This allows the pelvis to rotate passively around the tops of the femurs so the full benefits of the pose can be felt.
1. From Samasthitiḥ, on an exhalation, hop open to the right side, placing the feet about one leg length apart. Line the heel of the right foot up with the back of the left arch. Turn the right toes out toward the end of the mat, and turn the left foot in about 20 to 40 degrees. Make sure the right kneecap points in the same direction as the right foot, and engage both legs.
2. With the arms stretched out parallel to the floor, shoulder blades down, and torso facing the side of the mat, inhale and lift the quadriceps, kidney wings, and the center of the pelvic floor, as you scoop up in the cave of the sacrum. On an exhalation, keeping the PC muscle toned and the spine straight, tilt the pelvis to the right for the setup of the initial position.
3. The right leg will rotate slightly outward, and the left leg will rotate slightly inward—the primary rotations of this pose. Keep the spine straight as you reach out through the right arm.
4. Place the right hand in the appropriate position for your physical circumstances. The official placement is to hold the big toe of the right foot with the first two fingers of the right hand. If you are too stiff to manage this, placing the hand on the right ankle or shin, or a block positioned next to the outer edge of the foot can provide a starting point for working toward the full form. At this stage do not sacrifice the straightness in the spine in leaning over to take the big toe if you cannot reach it. Instead, rely on the rotations in the legs and the straightening spine to inform the pose and facilitate the rotation of the pelvis around the tops of the femurs to bring you naturally into the pose.
5. Once in the basic pose, introduce the counteractions in the legs, hips, and torso with the next in-breath. This inhalation wakes up the coccyx to bring it closer to your pubic bone. Ground the outer edge of your left foot while simultaneously reaching down into the earth through the mound of the right big toe. Keep the arches of both feet awake, and as you reach up to the ceiling through the left arm, imagine both arms extending evenly in opposite directions so there is space for the spine to continue to elongate.
6. Turn the head to gaze at the fingers of the left hand as you draw up the kneecaps and thighs on both legs. Make the buttocks firm and draw the shoulder blades down the back. Keep a micro-bend in the right knee and draw the back of your body toward the front of the body, starting with the sacrum; keep the PC muscle toned.
7. Hold the pose for five breaths. At the end of an exhale, keeping the legs engaged, the spine straight, and the pelvic floor toned, take a smooth inhale and reach up and out through the fingertips of the left hand to guide you back up to standing.
8. On the next exhalation, turn the feet to the other side; on the inhalation, repeat the pose to your left. Again, after five breaths, exit the pose on an inhalation. As you exhale, move into the counterpose (Parivṛtta Trikoṇāsana) or come back to Samasthitiḥ.
**P ARIVṚTTA TRIKOṆĀSANA**
_Revolving Triangle Pose_
This is the counterpose to Trikoṇāsana and is an example of a pose that is difficult to categorize: it is both a significant standing sequence pose and an important twist. To fully enter the pose, the spine should be elongated while reaching forward into the twist. This facilitates the proper actions and relationships of rotations and muscle tones in pelvis and hips. Once in the pose, watch and play with the interpenetration of actions within the pose; never locking the pelvis or hips helps to refine these relationships.
1. Standing in Samasthitiḥ, exhale, and hop out to the right side. As a counterpose, simply move into this pose after completing Trikoṇāsana on the second side. With the feet about one leg length apart, turn the right toes out toward the end of the mat and turn the left foot in 40 to 60 degrees. Turn the pelvis toward the right foot and draw up the fronts of the legs. Wake up the arches of both feet, and make sure not to lock or hyperextend the knees.
2. Angle the left foot forward enough to accommodate the rotation of the pelvis, which should be squaring (but not locked) toward the wall in front of your right foot. Be sure to keep the outer edge of the left heel and the inner edge of the right foot grounded. For extra stability, the feet can be positioned slightly wider apart—toward the sides of the mat—than they would fall if you were simply to turn directly from Trikoṇāsana. This added width can be helpful if you are very stiff, have hamstring injuries, or have difficulty maintaining balance.
3. On an inhalation, place the right hand on the right hip and reach up toward the ceiling through the left arm, protracting the left shoulder blade and reaching through the fingertips of the left hand.
4. On an exhalation, begin to fold forward at the hip joints. Elongate through the spine, as you reach forward and down through the left arm, allowing the left hand to come flat on the floor outside the right foot. If you cannot place your hand on the floor without curling your spine too much, position a block on the floor along the outer edge of the right foot and place your hand on the block.
5. On the next inhalation, reach up toward the ceiling through the right arm as you move the right hip back slightly and extend long through the crown of the head. Draw both shoulder blades down the back, opening the fronts of the armpits. Pay attention to your left shoulder so that the top of the humerus does not rotate forward, collapsing the heart. Spread the kidney wings and reach through the fingertips of the right hand to facilitate the twist and extension in the spine. Gaze softly at the thumb of the right hand. There should be no strain in the neck, jaw, or palate. Check to be sure the feet remain fully grounded, especially the outer edge of the left foot.
6. Hold the pose for five breaths, breathing smoothly and adjusting the pose with subtle actions and counteractions that will automatically bring you into the final form.
7. To exit, look down at the left hand at the end of an exhalation. With strong legs and a long spine, inhale and spin like a windmill to come up and out of the pose. Turn to the left and on the next exhalation enter the pose on the other side. Move smoothly and rhythmically into and out of this pose to receive maximum benefit from the form. Do not overanalyze the form or microadjustments you "think" you should make. Instead tune in to the feelings and sensations as they are arising and respond so that the pose comes to life with an open heart, strong legs, and a soft palate.
8. Hold the pose for five breaths. Exit smoothly by looking down on an exhalation and, with strong legs, spinning back up to standing. Hop or step back to Samasthitiḥ.
**P ĀRŚVAKOṆĀSANA**
_Side-Angle Pose_
This is an important pose for establishing a sense of strength in the legs and learning Uḍḍīyāna Bandha, which helps to lift the lower belly out of the pelvic basin. It is important, especially for those with knee injuries or who have had knee surgery, to drop the weight of the bent leg directly down through the heel so the force is not jammed forward in the knee joint.
1. From Samasthitiḥ, as you exhale, hop the feet out to the right with feet about one leg length apart. With the torso facing the side of the mat and arms reaching out to the sides at shoulder height, begin to lower into the pose. Ground evenly through the right heel as you spread the foot, and bend the knee until the right thigh is parallel to the ground. Keep the left leg long and firm and the outer edge of the left foot grounded.
2. Be certain to bring the right knee over the center of the right ankle. Keep the knee over the line connecting the two heels when entering and exiting the pose, and keep the upper body lifting toward the ceiling, with heavy sitting bones, as you first enter the pose.
3. Inhale and reach through the right arm to extend the right waist. On the exhalation, keeping the spine straight, tilt the pelvis to the right and place the fingertips (and eventually the flattened palm) of the right hand on the floor outside the right foot.
4. Keep both legs strong, grounding through the outer edge of the left foot, lifting the left thigh away from the floor while keeping the right knee precisely over the ankle.
5. On the inhale and from the hollow in front of the sacrum, generate a spiral extension up the front of the spine as you sweep the left arm up to point fingers toward the wall near your head. Turn the palm down, reaching through the fingertips and keeping the arm at the same angle as the left leg. Firm the buttocks to move your sacrum into your body. Curl the coccyx closer to the pubic bone and gaze at the palm of the left hand to stretch the psoas line.
6. Do not hunch the shoulders. Externally spiral and lift the left shoulder blade so that the flesh on the back of the neck moves down the back. Be sure that you have spiraled enough so there is no tension when you tilt your head back to look at the center of your palm. Reach as far as you can through the left fingertips, as if trying to touch infinity.
7. At the end of five breaths, root firmly into the floor through the legs and, maintaining strong legs, come back up to standing on the inhale. Immediately turn around to the second side and exhale as you bend the left leg to enter the pose on the other side. Gaze at the palm of the right hand and keep the inner right thigh lifting away from the floor.
**P ARIVṚTTA PĀRŚVAKOṆĀSANA**
_Twisted Side-Angle Pose_
As with Parivṛtta Trikoṇāsana, it is helpful to study this pose as both a standing pose and an important twist. In the traditional Aṣṭāṅga Vinyāsa practice, it is practiced as a counterpose immediately following Pārśvakoṇāsana without returning to the front of the mat, which is why it has been placed in this chapter. It is very important to flex the spine while entering Parivṛtta Pārśvakoṇāsana so as not to collapse into the front hip joint, which can eventually damage the hip and possibly contribute to femoroacetabular impingement.
1. If you are starting from Pārśvottānāsana, exit the pose with the left knee bent, come up to center on an inhale and, exhaling, rotate the right toes out, bending the right knee, and angling the left foot in about 45 to 60 degrees. If you are starting from Samasthitiḥ, hop out to the right so the feet are about one leg length apart. Rotate the feet and legs as already described.
2. With strong emphasis on the legs, on an inhale draw the spine long from deep in the low belly, reaching up. Then, fully exhaling reach forward with the left arm. Curling the spine, as if drawing the navel back to touch the spine, wrap the left upper arm around the outside of the right thigh and knee.
3. Place the fingertips, and eventually the flattened palm, of the left hand, with the fingers pointing in the same direction as the right toes, on the floor outside the right foot. Do not lose the engagement in the left leg. Keep reaching back and down through the outer edge of the left foot without letting the left thigh collapse down toward the mat. The foot may not fully ground at first, but that is not a major problem if the intention to keep it grounded is there.
4. Once the legs are firmly established, add the right arm into the pose. As you inhale, straighten the arm and sweep it around in an arc so it is stretched out above your head in the same line as the back leg. The palm should face down toward the floor. Keep the arm connected by using the serratus anterior muscle to protract the shoulder blade out and the humerus away from the ear as you reach in one long extended line from the outer edge of the left foot all the way out through the fingertips of the right hand.
5. If the top of the left humerus does not reach past the right knee as you rotate to enter the pose, you may substitute a modified version of the pose with hands together in Añjali Mudrā until you are flexible enough to do the full form. Do not abandon the aim to shift into the full form simply because it is easier to keep the hands in Añjali Mudrā. The pose will continue to deepen and offer new insights to the practice if you work toward the full form rather than becoming complacent and settling for a modified form.
6. Hold this position for five breaths. On an exhalation, look down at the left hand. Reestablish the strength and grounding in both legs and the pelvic floor. Inhaling, work the legs and spin the arm to come up, then rotate around to practice the pose on the second side.
PRASĀRITA PĀDOTTĀNĀSANA (FORWARD BEND WITH FEET SPREAD POSE) A, B, C, AND D
Within the Aṣṭānġa Vinyāsa system, all four forms of Prasārita Pādottānāsana are practiced together, flowing seamlessly from one to the next. They complement one another and build on the actions in the feet, legs, spine, and pelvic floor. When practiced together, they provide an excellent way to tap into the pelvic floor and Mūlabandha. As related poses, many of their alignment details are the same; these are described in detail for form A and then built on for the other forms. If practicing the different forms separately, follow the vinyāsas described in this section for entering and exiting the poses.
**P RASĀRITA PĀDOTTĀNĀSANA A**
1. Begin in Samasthitiḥ. On an exhalation, hop out to the right so that the feet are just over one leg length apart. Place the hands on the hips, and inhale fully to wake up the feet and legs, keeping the coccyx heavy. This allows you to establish a connection to the pelvic floor and tap into a sense of spaciousness in the hip joints as the crest of the inhalation lifts and spreads the core of the heart while elongating the spine.
2. On the exhalation, with the spine straight and the chin out, fold forward and place the palms of the hands on the floor between the feet. If necessary, bend the knees and rotate the feet inward (pigeon-toed) to reach the floor.
3. Keep the legs straight without locking the knees, and keep the feet evenly grounded. Especially as you enter the fold, watch carefully to keep the inner edges of the feet grounded. By reaching down through the mounds of the big toes and the inner edges of the heels first and _then_ grounding through the outer edges of the feet, the arches of both feet are invited to wake up. This is good.
4. Inhale and lift the head again, straightening the arms, lifting the chest, and reaching forward through the crown of the head to stretch the spine straight. On the next exhalation, contract the abdominal muscles as you fold forward, allowing the head to fall toward the floor. With the hands placed shoulder-width apart between the feet, press from the heels of the hands out through the fingertips to bring the head farther between the legs.
5. Keep the shoulders spreading and lifting away from the floor and the upper arms parallel as you gaze along the line of the nose. In all the forms, cultivate Uḍḍīyāna and Mūlabandhas by toning the PC muscle, connecting the ends of the breath and gazing at the tip of the nose. This will help you learn the true inner alignment of the poses. Hold the position for five breaths.
6. At the end of an exhalation, bring the awareness to the feet, legs, and center of the pelvic floor, while leaving the hands where they are. Inhale and lift the head, straightening the arms and spine so the back is approximately parallel to the floor.
7. Exhale and place the hands on the hips, pushing the skin of the outer hips back. As you inhale, straighten the spine to return to standing.
**P RASĀRITA PĀDOTTĀNĀSANA B**
1. Keeping the legs in a wide stance, exhale deeply to root down through the feet and legs. Inhaling, lift the arms straight out to the sides with a spin to encourage the feeling of the shoulder blades sliding down the back. Exhaling, place the hands on the hips, then inhale to lift the sternum and extend the spine as you use the first two fingers of each hand to apply gentle pressure to the lower belly at the area of the psoas buttons.
2. On the next exhalation, fold forward with the spine long, leaving the fingers in the low belly and allowing the elbows to reach evenly out to the sides. Keep the shoulders wide and pulled toward the hips, with the gaze steady along the line of the nose.
3. Soften the jaw and palate, and bring the awareness to the low belly (under your fingertips) and the connection of that part of the body to the pelvic floor. Hold the pose for five breaths, gazing along the line of the nose.
4. At the end of an exhalation, activate the legs, set the pelvic floor (holding the serpent's tail), and on the next inhalation, return to standing.
**P RASĀRITA PĀDOTTĀNĀSANA C**
1. Keeping the legs in a wide stance and the hands on the hips, exhale and ground through the legs and feet, making sure the arches of both feet are active. Before folding forward, inhale to straighten the arms out to the sides and roll the tops of the arms forward to facilitate clasping the hands behind the back; make the coccyx heavy to avoid collapsing the lumbar spine. The arms will bend slightly as you do this. Then roll the tops of the arms up, back, and out slightly as you straighten the arms.
2. On the exhalation, fold forward, dropping the clasped hands behind you and over your head toward the floor. The palms of the hands should face each other, and the shoulders should flow back and down. Don't worry if the hands don't touch the floor. Patience.
3. If you are flexible, you can try rotating the arms inward and flipping the hands over so the palms, still interlaced, are flat on the floor in order to make the pose more challenging. If you are new to the practice or tend to be stiff, you can hold a stick or strap between the hands with the thumbs pointing out to the sides to maximize the benefits from the pose.
4. After five breaths, exhale fully, then exit to standing on the next inhalation. Exhale and place the hands on the hips again.
**P RASĀRITA PĀDOTTĀNĀSANA D**
1. Keeping the legs in a wide stance and the hands on the hips, inhale lifting the chest and heart. Exhale and straighten the spine as you fold forward to grab the big toes with the middle and index fingers of each hand. Pull firmly with the arms—without hunching the shoulders or straining the neck—as you reach out to the sides through the elbows. Allow the head to hang. Ground the mounds of your big toes in opposition to the slight upward pull of the fingers and make sure not to hyperextend or lock the knees. If necessary, you may bend the knees in order to hold the toes. Gaze softly along the line of the nose.
2. Hold for five breaths. At the end of an exhalation, strengthen the legs and activate the feet while leaving the hands in place. Inhale and lift the head, straightening the arms and lifting the spine away from the floor.
3. Exhale as you place the hands on the hips, pushing the skin of the outer hips back. As you inhale, return to standing with a straight spine.
4. If necessary, shift the feet together slightly to make it safe for you to hop back to the front of the mat, landing in Samasthitiḥ on an exhalation.
**P ĀRŚVOTTĀNĀSANA**
_Forward Bend to the Side Pose_
This standing pose is an example of an important twisting action from the kidney to the knee, which activates a line of connection diagonally across the front of the body from the serratus anterior muscle on one side through the external oblique muscles on that side to the internal oblique muscles on the opposite side. This valuable twisting action is found in a number of other poses and is a fundamental expression of the apānic form.
1. Start in Samasthitiḥ. As you exhale, hop out to the right toward the back of the mat so that the feet are about one leg length apart. Turn the right toes to face the back of the mat, and angle the left foot in about 40 degrees, squaring the hips to the right. Do not lock the knees or hips.
2. Roll the tops of the arms forward as you bend your elbows to place the hands behind your back in Añjali Mudrā. Once the hands are in place, roll the tops of the arms back, sliding the shoulder blades broad and dropping them down the back. If you are stiff, you can make fists behind the back and during the pose push the fists together so the elbows lift and the collarbones spread.
3. On an inhalation, work both legs and, keeping the kneecaps lifted, also lift the core of the heart. The hands will eventually be at the level of the thoracic spine, immediately behind the heart, so as you inhale, they can help to facilitate a subtle backbend. Gaze down the line of the nose.
4. Be aware that arching back before the forward fold in this pose is an advanced move and is optional. It requires properly grounded feet, strong legs, and fully rotated hips.
5. Once fully extended on the inhalation, begin to fold forward toward the right leg on the next exhalation. Keep both legs activated and wrap the left kidney wing toward the right knee. Eventually the chin will come to the shin and you will be able to gaze at the nose. Until then, gaze at the right big toe as you fold and rest in the pose.
6. The inner edge of the right foot and the outer edge of the left heel stay grounded, especially when entering and exiting the pose. Keep the weight centered just in front of the heels. Notice the profound changes that occur within the pose with different distributions of weight in the feet. Lifting the elbows toward the ceiling can help to keep the upper body buoyant and the heart open.
7. Hold for five breaths. Inhaling with a heavy coccyx and using the heart and head to lengthen the torso, extend through the spine and return to standing with hands still in Añjali Mudrā behind the back.
8. Turn the feet to the other side and repeat the pose. After five breaths, return to standing on an inhalation. Turn the feet so they are parallel, release the hands from behind the back, and hop or step back to Samasthitiḥ at the front of the mat.
**U TTHITA HASTA PĀDĀṄGUṢṬHĀSANA**
_Standing Hand-to-Big-Toe Pose_
This pose appears early in the Aṣṭāṅga Vinyāsa sequencing of poses and is, for many beginning students, the first major roadblock to moving through the series too quickly. It is an asymmetrical, twisting pose that also requires balance. Pattabhi Jois would often laugh and say, "Why dancing?" as his students hopped around the room to avoid falling over in this pose. It requires patience and concentration; even if you feel there's no hope, it will improve through practice.
1. Begin in Samasthitiḥ. Grounding evenly through the left foot and leg, place the left hand on the waist. Lift and bend the right leg, rotating the femur slightly out to the side and taking the big toe firmly with the middle and index fingers of the right hand. Find a steady gazing point out in front of you on the floor along the line of the nose.
2. On an inhalation, press the right big toe against the fingers as you straighten the right leg and rotate the foot so it is reaching evenly out in front. Pull up on the toe with the arm slightly bent and a sense of the shoulder blades flowing down the back. At the same time, pull down through the right leg, as though trying to escape the grip of the right hand.
3. On the next exhalation, fold forward from the waist to put your chin on your shin. Wrap the left kidney wing toward the right knee, bowing to your foot and gazing gently along the line of the nose. Continue to pull up with the right arm as the heel pulls down toward the floor, facilitating the wrap and the bend. This complementary action between hand and foot actually makes the pose more stable.
4. Hold for five breaths, then straighten the spine and come out of the forward bend on an inhalation. On the next exhalation, open the right leg out to the side, dropping the outer right hip toward the floor and standing tall to resist puffing the left groin forward. Turn the head to gaze at a point on the horizon over the left shoulder. If you are a beginner, you may gaze at a point on the floor. Hold this position for five breaths.
5. On an inhale, draw the leg back to center, bow again for just one exhale, then stand tall on the next inhale. Lift the leg even higher and then release; while "fointing" (both pointing and flexing simultaneously) the foot, allow the leg to float for five breaths. Keep the heart area light and lifting up and your face free of tension. Hold for five breaths, then on an exhalation, return the right foot to the floor and do the pose on the left side. After practicing the pose on both sides, stand for one round of breath in Samasthitiḥ.
6. Beginners may use a wall for balance if necessary, placing the elbow of the arm that is not holding the leg near or barely touching the wall. This can overcome initial fear of the pose, but it can also become a habit. Better than using a wall is just holding the unengaged arm out to the side and possibly holding the pose for fewer breaths. Beware: using a wall does not force you to learn to balance. Remember, this pose could well be called Humble Āsana! It is no crime to lose your balance and drop the floating leg back to the floor. Many people get frustrated doing this pose and may notice that breathing and emotional intensity increase while practicing it. This is perfectly fine. Emotions can give needed energy to the practice.
**A RDHA BADDHA PADMOTTĀNĀSANA**
_Half-Bound Lotus Forward Bend Pose_
Ardha Baddha Padmottānāsana, though practiced early in an Aṣṭāṅga Vinyāsa practice, is actually quite advanced. Not only does it assume the practitioner can take Half Padmāsana, but he or she must also have enough flexibility to hold the Padmāsana foot and fold forward while balancing. However, for some, jumping into the deep end of a pool is the best way to learn to swim! Even if the full form is not accessible, modifications and internal forms make it easier to learn, and the actions and counteractions, as well as the concentration skills required for this pose, are excellent for everyone to practice and refine. Those with knee injuries or tight hips and knees should be cautious and practice only the first preparatory position, working gradually into the pose over many weeks or months (or perhaps never).
1. Stand in Samasthitiḥ. Find a sense of strength and balance in the left leg, making sure the knee is slightly bent (micro-bent) to give stability to the pose as you exhale to ground into the pose. Inhaling, lift the right foot, bend the right knee, and externally rotate the femur as you lift the leg to draw the right heel toward the top edge of the pubic bone. _Carefully_ press the knee down to point toward the floor.
2. Next, with an exhalation, reach around the back with the right arm to grab the right big toe. If you cannot reach the toe, don't worry. Use that hand to help you balance in the next phase.
3. Inhale and reach up with the left arm to stretch the psoas line and stabilize the pose, then bend forward to place the left hand or fingers on the floor as you exhale. Find a point on which your eyes can rest on the floor, and adjust the hips and knee as needed to fold comfortably and in order to square the hips and shoulders. Inhale and lift the head to straighten the spine, and then fold deeply into the full form, placing the left hand flat on the floor beside the left foot.
4. Hold the pose for five breaths. In this final position, gaze at the tip of the nose. If you are flexible, you can look to the left big toe.
5. At the end of an exhale, tone the pelvic floor, and on the next inhale, lift the head and straighten the left arm to come halfway up. At first, look between the eyebrows as you lift the spine to a horizontal position.
6. Exhale completely again, staying in place to ground fully through the left foot and leg—making sure to keep a micro-bend in the leg. On the next inhalation, come up to standing. Release the right foot and place it back on the floor.
7. Repeat the pose on the other side, then return to Samasthitiḥ.
**U TKAṬĀSANA**
_Difficult Pose_
Sometimes also translated as Horrible Pose, Utkaṭāsana is an excellent way to experience Uḍḍīyāna Bandha while learning to keep the legs strong, the pelvic floor awake, and the shoulders protracted while reaching up. Don't shy away from bending your knees _deeply_ to keep the pose alive, and remember to breathe fully—even as your arms are reaching enthusiastically up to the sky.
1. This pose is traditionally entered through a Full Vinyāsa. From Downward Facing Dog Pose, at the end of an exhalation, bend the knees, lift the head to look at the mat between the hands, and hop or step forward, inhaling just as your feet touch the floor. Bend the knees deeply as you land.
2. Keep the toes spreading, the heels down, and the inner knees touching as you drop down as if to sit in a chair. The coccyx should feel heavy. At the same time, draw back the lower belly just above the pubic bone. Do not tuck the sitting bones under! Draw them back and drop them down.
3. Keep the heart open, the fronts of the armpits wide, and the throat free of extraneous tension as you wrap the shoulders outward to lift the arms and reach up toward the ceiling. Reach up through the arms in front of the head with the palms touching, and do not crunch the neck. If putting the palms together is impossible, you may keep your hands slightly apart until you become more flexible. Do not strain your neck.
4. Look at the thumbs; because the hands are in front of the face, the line of the gaze is eventually _down_ along the line of the nose.
5. Hold the pose for five breaths, then on an inhalation, push through the feet and legs to return to standing. Drop the arms along the sides of the body as you come into Samasthitiḥ, or move directly into a Full Vinyāsa.
VĪRABHADRĀSANA (WARRIOR POSE) A AND B
Vīrabhadrāsana is a true expression of the union of opposites—the strength and vigor of a warrior and the centered, mellow feel of someone who is at peace within. That is the feeling we cultivate in this pose. By keeping the feet, legs, arms, and hands awake, space is created for the head, throat, torso, and pelvis to fully express stability. It is of particular importance to keep the gaze steady, the tongue soft, the palate released, and the breath free and easy. In the Aṣṭāṅga Vinyāsa form, Vīrabhadrāsana follows Utkaṭāsana with a Full Vinyāsa between, but of course it can be practiced by simply coming into it from Samasthitiḥ.
**V ĪRABHADRĀSANA A**
1. From Downward Facing Dog Pose, at the end of an exhalation, bend the knees and lift the head to look at the floor between the hands. Turn the left foot in about 20 to 45 degrees and step forward into a lunge with the right foot. Ground through both feet as you inhale and rotate the pelvis and upper body toward the front of the mat, reaching up overhead through both arms.
2. As you reach up through the arms do not strain the neck. Lift the sides of the torso and the fronts of the armpits as you stretch the arms over the head. Gaze at the thumbs and soften the palate.
3. Do not allow the right knee to drift to either side, and do not drop down so far that the knee goes past the ankle. Keep the left leg very strong as you descend, increasing the distance between the feet if you need more stretch or more space so the right thigh can be parallel to the floor.
4. Keep the left foot grounded by making certain that the outer edge of the heel stays touching or near the floor. To do this, wake up the arches of the foot and spread the toes. Lifting the back edge of the foot will eventually injure the knee, and it encourages a collapse into the hip, which can cause an injury to the hip joint. The upper body should feel light, as if it is floating up and out of the pelvis as the inner left thigh draws up away from the floor.
5. Contract the pelvic floor as you draw the coccyx and sitting bones down. Gaze at the thumbs and hold the pose for five breaths. On an inhalation straighten the right leg and, leaving the arms and gaze up, rotate the feet and torso to the left side and drop down, bending the left leg, to repeat the pose on the other side. After five breaths here, move directly into Vīrabhadrāsana B.
**V ĪRABHADRĀSANA B**
1. On an exhalation, lower the arms halfway so they are parallel to the floor with the palms down; at the same time, rotate the torso and pelvis to the right. Reach out through the fingertips and drop the shoulder blades broadly down the back. Keep drawing down the back surface of the body, while toning the pelvic floor. Keep the legs and feet strong and activated as you lift the front of the spine to prevent a sense of collapsing into the hip joints.
2. Ground the outer edge of the right heel and the inner edge of the left foot as you gaze out over the left shoulder at the middle finger of the left hand. Hold this position for five breaths. On an inhale, straighten the left knee, turn the right toes out toward the front of the mat, and angle the left foot in, then bend the right knee so you are in the pose on the other side. Gaze out over the right shoulder at the fingertips of the right hand, and hold the pose for five breaths.
3. Exit the pose by sliding the left heel back a couple of inches and twisting the pelvis around to face the right leg. You may reach up for one breath in Vīrabhadrāsana A or step directly back into the fourth position, Catvāri.
_7_
Forward Bends
WHEN WE CONSIDER THE PRIMARY SERIES, forward bending is the theme that comes to mind. Beyond those poses practiced as part of the opening sequence in the Sūrya Namaskāras and those found in the standing sequence—beginning with Paśchimottānāsana (Tuning Up the Back Pose) and ending with Ūrdhvā Mukha Paśchimottānāsana (Upward-Facing Forward Bend)—there are many forward folds. When practiced correctly, in sync with the internal cues to alignment and the patterns of Prāṇa, forward bends are incredibly grounding, therapeutic, calming, and integrating. If they are practiced aggressively or without attention to the internal forms that create healthy alignment, forward bends can contribute to shutting the heart area and a _tamasic_ (apathetic, depressed) mind state. For those with tight hamstrings or an unstable lumbar region, bending forward without the internal cues of alignment can cause physical discomfort and injury. One popular criticism of Aṣṭāṅga Vinyāsa yoga, and the Primary Series in particular, is that Aṣṭāṅga practitioners are grumpy (closed-hearted and tamasic) and prone to hamstring and low back injuries due to all the repetitive movements into forward bends. These generalizations are based on a fundamental misunderstanding of the proper mechanics that help practitioners experience the benefits of forward bends. It's true that repetitive misalignment can cause problems, but with proper form, even those with physical limitations can actually practice forward folds over and over with no problems whatsoever.
The first vinyāsa of forward bends is that in order to rotate the pelvis around the head of the femur, we begin by inhaling to enhance the prāṇa pattern, which holds the spine straight and opens the heart area. To do this, scoop back under the lower belly using Uḍḍīyāna Bandha to guarantee the participation of the pelvic floor. This subtle, internal vinyāsa is so frequently overlooked that it could be considered the "secret" first step of folding forward. Once this foundation is established and the spine is elongated to let the prāṇa pattern shine out, rotate the pelvis passively around the heads of the femurs, gradually reintroducing the apāna pattern. This allows us to get into the pose intelligently and to exactly the right degree for our particular circumstances. The hip flexor muscles, such as the psoas muscle, should not be involved in a well-executed forward bend. Rather the opposing muscles of hip extension, which include the hamstrings, glutes, and occasionally the deep hip (lateral rotator) muscles, must stay alert and online—slightly toned—to give control and reversibility to the movement. But the basic movement of the pelvis around the heads of the femurs is a function of keeping the spine straight and drawing breath, awareness, and attentiveness up and out of the hip joints, so as we fold forward, we have a sense of elongation in the spine and a passive movement of the hips, which creates space in the hip joints and spine.
The next stage of the vinyāsa for forward bends is when, having closed the hip joints in the primary movement, the tensions around the hip joints slightly reverse so that in the end of the exhale counteractions come into play. This closes the hip joints; signals the pelvic floor to tone; and stimulates the apāna pattern, which guarantees that the hip flexors do not tighten unnecessarily. Remember that the hip flexors are involved in the prāṇa pattern, and the hamstrings are involved in the apāna pattern.
Forward bends in general allow us to enhance the quality of the exhalation because we are folding over the front area of the body, including the abdominal organs and particularly the lungs. Exhaling smoothly and fully makes it easy to find the appropriate fold in the hip joints and to release tension in the palate. Consciously releasing tension in the palate is calming to the nervous system and is essential for forward bends, because as we fold, the space available for air to be drawn into the lungs is constricted, making it impossible for them to expand, which may trigger feelings of fear. Once fear is introduced into any pose, the potential for meditative, liberating movements decreases. In forward bends we release the palate to keep the heart open as we let go of inhalation patterns that have opened spatially in and around the body. This makes these poses profoundly relaxing, and their deep therapeutic and restorative potential is revealed.
Once we are in the pose, its internal dialectic between action and counteraction begins, and we make fine adjustments in terms of the tone of the pelvic floor—front to back and side to side. Thus, the question of whether prāṇa or apāna is leading the pose can be considered, knowing that—as in all poses—both ends of the breath must work in tandem. Sometimes the heart has to move farther forward, and the pubic bone has to stimulate the poses. At other times the kidney area has to expand, and the coccyx has to show the way out of the pose by offering a sense of dropping and pulling to a point of solidity in the pelvic floor.
The final step in the vinyāsa sequence for forward bends is when these two patterns simultaneously awaken in the awareness; when that unification is stable, we can let go of both patterns. Coming out of the pose, we complete an exhalation to stabilize the pelvic floor and then, keeping the sense of an awakened and toned pelvic floor in the forefront of the awareness, the inhalation facilitates a spooning out under the belly (Uḍḍīyāna) that initiates the exit from the pose. Ideally we can exit the depth of the pose on the inhalation without losing the integrity of the apāna pattern.
Forward bends can be either symmetrical, as with Pādāṅguṣṭhāsana (Big Toe Pose), or asymmetrical, as with Jānuśīrṣāsana (Head-to-Knee Pose). In forward bends in which the folded leg is close to the midline in adduction, as in Jānuśīrṣāsana, the range of motion is much more restricted than when the folded leg is abducted, or angled out away from the midline, as in Upaviṣṭha Koṇāsana. Because of their even movement, symmetrical forward folds have the qualities of simplicity and purity. Tensions along the spine and the pelvic floor are uniform from side to side, so they encourage the apāna pattern to flow. This can be very relaxing and grounding. Symmetrical forward bends allow us to observe and breathe into the pose as we adjust for inherent asymmetries in the body, making it easy to experience the beneficial aspects of this family of poses. When we practice forward bends as part of smooth, pleasant breath cycles, we feel the natural rhythms up and down the spine and out from the central channel, and we can experience these patterns communicating with each other across the pelvic floor and palate. Tapping into this deep level of awareness is easier in symmetrical forward bends. Once we have a taste for the sensations associated with the depth of awareness experienced through these poses, the same sensations are more easily found in asymmetrical forward bends as well, demonstrating that paying attention to subtle levels of awareness and form is how the poses deepen and become most beneficial.
In some symmetrical forward bends in which the legs are close to the midline of the body, such as Paśchimottānāsana, it is important to understand that the hip joint can not close as deeply as it can when the legs are abducted in poses like Prasārita Pādottānāsana, where each leg is extended more than 30 degrees past the midline. This is due to the length and angle of the neck of the femur, which can be quite dissimilar in different people. If the neck of the femur is angled severely or is short, it can hit the labrum or the edge of the acetabulum, especially in the transitions in and out of the pose. This can also happen if the feet are wide apart and the legs abducted when folding, as in Prasārita Pādottānāsana if we attempt to put the head on the floor. (In Prasārita Pādottānāsana, the problem is exacerbated particularly when exiting the pose, so if the feet are moved in toward the midline of the body when exiting, impingement can be avoided.) Extreme movements in forward bends with abducted legs can eventually cause injury to the joint, but with close attention to anatomy and mechanics, the poses are very safe and healthy.
When forward bends are practiced incorrectly or inattentively, disk and spinal problems (particularly in the lumbar region) as well as overstretched hamstrings can occur. Although sometimes these complications have to do with unique individual structure, more often they are aggravated because the legs and feet are not engaging properly (which they must in all forward bends, even if the knees are bent), the hamstrings are naturally short or tight, and/or the spine is not kept straight in the transitions into and out of the poses.
With regard to curving the spine as we fold or unfold, this is a pattern that may be habitual, as it is common in everyday activities; it's quite natural to look down at whatever has caught our interest as we begin to bend toward it. This automatically curls the spine, beginning with the cervical vertebrae, which almost magically tucks the pelvis under. The pattern can even approach a pure apānic coil, like a worm working its way out through the tip of the coccyx. Structurally the problem with folding forward in this way is that flexion of the spine automatically triggers the hamstrings and glutes to tone; this causes a posterior tilt in the pelvis, which may become locked in position. In either of these scenarios, the pelvis is not able to rotate passively around the heads of the femurs. To force a forward bend while the pelvis is locked, particularly if the legs and spine are not used intelligently, puts strain on the hamstrings and the lower back.
The role of the hamstrings in folding forward is also crucial. Depending on one's structure, hamstrings may be slightly longer or shorter—which gives more or less play before they reach their limit when bending forward with straight legs. Hamstrings can lengthen gradually through stretching, but they can also tighten due to certain activities such as running and biking or simply through lack of use. Because the hamstrings attach to the sitting bones, if they are tight and we fold forward with hyperextended or inactive legs, the sitting bones are pulled down and the pelvis rolls back along with them. So when we fold forward, especially if the spine is not straight, the low back can be compromised. If we push too hard under these circumstances, the hamstrings themselves can also be overstretched or strained, particularly at their attachments to the sitting bones.
Within the family of forward bends, there are two other general categories of poses. Those that are accessible and most beneficial when the spine is initially straight, especially during the entrance to and exit from the pose. Paśchimottānāsana, Baddha Koṇāsana, and Yoga Nidrāsana are examples of this form. There are also poses that are considered extreme forward bends, meaning the hip joints are deeply closed, but require extreme flexion of the spine initially. Poses such as Kūrmāsana and Dvi Pāda Śīrṣāsana fall into this latter category.
Forward bends that are not grounded in the full apāna pattern with its strong prāṇic complement can have the opposite effect of backbends and leave an emotional residue of depression or misery in the mind and body. When this happens, the area of the heart closes as a result of improper alignment; the internal breath is inhibited due to a lack of actions and counteractions in the pelvic floor, spine, and shoulders. It is not unusual for beginners to strain too much in forward bends, especially if they think there is something to accomplish. Touching the toes is a familiar goal we may struggle for, but putting too much effort toward that goal rather than tuning in to what's actually arising physically and mentally can cause the hamstrings to tighten even more, the result being that the "goal" is further away than it was at the beginning. Although we must push ourselves through our stuck patterns of resistance, if we exert too much grasping in any pose, it becomes increasingly difficult to observe the core of the body from the pelvic floor up through the head. When this happens in forward bends, and we tune out the feelings and sensations that are arising in the body, it is easy to undo the benefits of the pose. The sense of groundedness and of complete release disappear with each breath when we try too hard.
When practicing the seated forward bends as part of an Aṣṭāṅga Vinyāsa series, a Half Vinyāsa is taken between each pose; eventually, as strength increases and time permits, it is taken between each side of a given pose. For therapeutic reasons or if you have the desire (and the time) to do so, a Full Vinyāsa may also be practiced between seated poses. For this reason, this chapter gives instructions for entering and exiting the poses within a vinyāsa. When a pose is traditionally practiced without a Half Vinyāsa between it and the following pose, there is no mention of vinyāsa in the description.
PAŚCHIMOTTĀNĀSANA (TUNING UP THE BACK POSE) A, B, AND C
In the Primary Series, these three variations of Paśchimottānāsana are practiced in sequence, one immediately after the other, and each is held for five breaths. Though there are subtle differences between them, there are also some common internal and external forms. These commonalities are described for form A but apply to all three forms. Paśchimottānāsana is also included as part of the traditional finishing sequence.
**P AŚCHIMOTTĀNĀSANA A**
1. From Downward Facing Dog Pose, jump through to a seated position with the legs stretched out straight in front of you. Exhale to bring awareness to the sensation of the sitting bones dropping into the earth. From this reference point, the following inhalation establishes a strong internal connection to the central channel of the body that reaches from the center of the pelvic floor up through the crown of the head.
2. On an inhalation, reach the arms overhead to stretch the sides of the body and the psoas lines from the tips of the fingers down into the lower belly and the backs of the thighs.
3. As you exhale, fold forward to take the big toes with the middle and index fingers of each hand with the spine straightening. Extend through the spine and lift the belly up and over the tops of the thighs to allow the pelvis to rotate passively around the heads of the femurs.
4. If you cannot reach the toes, hold both ends of a strap, like a pair of reins that have been looped around the big toes, and/or bend the knees. It is far better to keep the alignment strong and active as you work to lengthen the hamstrings than to strain your back or hamstrings and collapse the heart area simply to reach the desired hand position. Patience is essential in all yoga āsanas, and especially so in Paśchimottānāsana.
5. Once you have the toes, activate the pose, moving in a sort of pulsating motion on the wave of the breath. On each inhalation, you may find that you come slightly out of the pose as you extend through the heart and the crown of the head, not trying to go down but reaching long and forward. On each exhalation, as you fold more deeply and drop into the pelvic floor, continue to pull back on the toes to encourage length in the spine and hamstrings.
6. Keep the legs firm and straight with a slight inward rotation. This is expressed by activating the feet, which stimulates a feeling of pushing forward through the roots of the big toes as you spread all of the toes and square the bottoms of the feet to the front in response to pulling back on the big toes.
7. Before folding forward completely, draw up the front surface of the torso as the groins and the inner tops of the legs drop down. This action is done on the inhalation as you extend the spine and spread the collarbones to keep the heart open. Gradually deepen the fold around the hip joints by bringing the torso up and over the legs. As a counteraction pull the back surface of the body down, grounding the sitting bones. Do not hunch the shoulders or strain the neck.
8. Gaze softly at the tip of the nose. Eventually, when you have folded very deeply and are flat against the legs (closed like a large book), gaze between the eyebrows, but never strain the neck to keep the dṛṣṭi you imagine you should be taking.
9. Once in the pose, hold for five breaths. At the end of the last exhalation, bring awareness to the pelvic floor and ground down through the sitting bones. Inhaling, extend long through the crown of the head and rise back to a seated position.
**P AŚCHIMOTTĀNĀSANA B**
1. Follow the same basic form as for Paśchimottānāsana A, but instead of reaching up with the arms to enter the pose, begin Paśchimottānāsana B just as you lift out of form A.
2. At the end of the inhalation that brought you out of the pose, switch the hand position to hold the feet with your hands. Curl your fingers around the sides of the feet placing thumbs on top of the first metatarsals (the top side of the mounds of your big toes). Push forward through the thumbs and spread the toes as you open the feet, and straighten the legs with a slight inward rotation.
3. Enter the full form on the wave of the breath (as in form A), inhaling one more time to straighten and then folding forward on the following exhale. Hold the pose for five breaths, gazing at the feet or along the line of the nose, depending on your flexibility.
4. If you cannot reach the feet, bend the knees and/or hold the ends of a strap looped around your feet for the pose. Even if you are doing one of these variations, you must still keep the legs and feet active and awake and work from a straightening spine. Just because the knees are bent is no reason for them to doze off!
5. After five breaths, exit the pose on an inhalation (as in form A).
**P AŚCHIMOTTĀNĀSANA C**
1. Follow the same basic form as for Paśchimottānāsana A, but instead of reaching up with the arms to enter the pose, begin Paśchimottānāsana C just as you lift out of form B.
2. At the end of the inhalation that brought you out of the pose, switch the hand position so you are holding one wrist with the opposite hand on the far side of your feet. In time with the inhale, pull back on the feet through the backs of your hands, then fold forward coming up and over the thighs to eventually place the ribs on the thighs. Bend the elbows out to the sides and push down slightly through the elbows toward the floor. Let the shoulder blades protract and move up to lengthen the arms. This stretches the latissimus dorsi, tracing the kidney wing pattern. Keep the upper trapezius muscles at the base of your neck soft, and release the tongue and palate.
3. After five breaths, exit the pose on an inhalation (as in form A).
**A RDHA BADDHA PADMA PAŚCHIMOTTĀNĀSANA**
_Half-Bound Lotus Forward Bend Pose_
Because Half Padmāsana is part of this pose, not all students are able to take this form; however, practicing variations that work with limitations is an excellent way to build confidence and hone the practice, even if Padmāsana never arrives! Those with knee injuries or tight hips should be cautious and may use a strap to hold the toes behind the back, use a block or blanket to elevate the pelvis, or practice only the first preparatory position for the duration of the pose. The ability to do the pose comes gradually as a result of all the other poses that _are_ possible for the practitioner.
1. Come to a seated position with the legs stretched straight out in front of you. Fold the left leg into Half Padmāsana, placing the foot as deeply into the lower belly and left groin as possible. While entering Half Padmāsana, the pelvis may tilt back slightly at first, but once in the fold, sit straight and turn the pelvis to bring the folded knee forward and down. The release of the hip and knee joint is facilitated by the Uḍḍīyāna Bandha action, which simultaneously lifts the front of the spine.
2. Keep the right leg awake, pressing the inner edge of the right foot forward through the mound of the big toe. There should be a twisting action in the hips, as you pull back the right sitting bone and bring the left leg forward. Press the folded knee down, creating an inward rotation in the femur.
3. Exhaling, reach behind the back with the left arm to grasp the left big toe. Inhaling, breathe up through the center of the body to straighten the spine and settle into the base of the pose in the pelvic floor. While holding the toe behind the back, do not hunch the shoulders toward the ears. If you cannot reach the toe, you may use a strap to loop around the toes. Alternatively, you can use the left hand to secure the folded leg. Proceed slowly if you are stiff! Some will need to only bring the heel of the folded leg back near the groins as in Jānuśīrṣāsana.
4. Still on the inhalation, reach up with the right arm, protracting that shoulder blade and stretching the psoas muscle on the right side. As you exhale, fold forward, reaching through the right arm and hand to hold the outer edge of the right foot. If you were unable to hold the toes in step 3 (above), instead of using a strap to hold the foot you may reach forward with both arms to grab the right foot with both hands. Once in the pose, inhale and roll the top of the left shoulder back and down, lifting the left elbow slightly to facilitate the action.
5. Hold the pose for five breaths. Begin to exit the pose on an inhalation, elongating the spine as you lift through the core of the heart. Once upright, release the left toe. Either switch legs to practice the pose on the other side, or preferably, cross the legs and do a Half Vinyāsa before folding the right leg into Half Padmāsana for the second side.
**T IRYANG MUKHA EKA PĀDA PAŚCHIMOTTĀNĀSANA**
_One Leg Reversed Forward Bend Pose_
This forward bend, found in the Primary Series, is an excellent example of the oblique line pattern of twisting—wrapping from the serratus anterior muscle on one side of the body to the obliques on the other. If folding the knee closed in Half Vīrāsana causes discomfort in the knee or hips, sitting on a blanket or block may help. To relieve discomfort in the knee, a neatly folded hand towel placed behind the knee may also be helpful, as it makes more space in the joint. You may also experiment with the distance between the knees—it is not mandatory that the thighs be completely parallel and/or touching. In fact, this is an extreme pose, so like all poses you should work slowly, bit by bit, working the edge of sensation but not pushing into or through pain. The pose should feel open, free, and comfortable.
1. Float through from Downward Facing Dog Pose to a seated position with the right leg folded in Vīrāsana position and the left leg stretched straight out in front. Inhaling reach up with both arms, and exhaling fold forward to take the left foot with both hands. The right arm needs to reach so that it brings the right kidney area forward and down toward the left knee. The hips twist gently as they begin to square toward the left leg.
2. If you have stiff or injured knees, sit on a cushion or block so you will not be tilted to the side. It is often best to support only the sitting bone on the straight leg side with the prop, allowing the other sitting bone to drop down slightly so you can maintain more mobility in the pelvis while entering and exiting the pose.
3. If you experience pain in the right knee joint, be sure to roll the calf muscle out to the side and tuck the skin of the outer thigh down toward the floor.
4. Draw back the left sitting bone and attempt to ground the right one. This action may be facilitated initially by using the left hand on the floor as an "outrigger" to push into the floor on the left side of the body and help position the hips correctly. Pushing the tops of the toes of the right foot into the floor, as if flexing the foot, can also facilitate this action.
5. Bring the right kidney wing area firmly forward and then down toward the left knee. Use the pull of the arms on the left foot to help deepen the fold and set the twist in the pose.
6. Keep the left leg awake. Draw the skin of the inner thigh of this leg and the skin of the outer thigh on your right leg down toward the floor. Hold the pose for five breaths. To exit, place the hands on the floor by your knees, and on an exhalation lean forward and lift up, then step or jump back, straightening the legs into Catvāri. From Catvāri, transition through Upward and Downward Facing Dog Poses, then return to a seated position to do the pose on the other side.
**K RAUÑCHĀSANA**
_Krauñcha's Pose_
This āsana is named after the sage Krauñcha, who is said to have split open a mountain pass in the Himalayas to let the Himalayan geese fly through that middle path. When doing the pose, the uplifted leg feels almost like a mountain you must navigate around, and as you reach forward, there is a sense of splitting the mountain, starting in the pelvic floor and continuing up through the central channel. The pose appears in the Intermediate Series immediately following Paśāsana and also represents an element of conclusion or summary to the Primary Series. Like so many of the forward bends within the Primary Series, Krauñchāsana is another fine example of the oblique pattern of twisting while folding forward.
1. From Downward Facing Dog Pose, at the end of an exhalation, float through to a seated position with the right leg folded back in Vīrāsana position and the left leg straight out in front of you.
2. On the inhalation, bend the left knee and clasp the left wrist with the right hand beyond the sole of the left foot. Through a sweeping action through the left foot, lift the left leg so that it is pointing toward the ceiling as you square the foot, reaching through the mound of the left big toe. To position the left leg correctly, allow a small external rotation in the femur, which involves bending the knee and dropping it out to the side while entering the pose. In this way, the head of the femur settles properly into place in the hip socket.
3. Straighten the left leg into the pose, emphasize a slight internal spin by reaching out through the big toe mound of the left foot. Resist the pull of the arms by pushing down toward the floor through the left heel. This wakes up the pose.
4. Lift the head as if to look at the big toe of the left foot, but gaze down the nose. Allow the shoulder blades to broaden and flow down the back; keep the tops of the shoulders away from the ears. Work the right foot as if to flex it; because the floor is in the way, the foot and muscles of the right leg will engage, stabilizing the base of the pose and signaling the right sitting bone to drop down toward the floor as the pelvic floor is activated. Keep the palate released and the tongue soft as you transition into the full form.
5. On an exhalation, bend the arms and draw the left leg closer to the torso, reaching up and forward through the upper body to place the chin on the shin as you wrap the right kidney area over toward the left knee.
6. Gaze up at the big toe and again activate the base of the pose. Hold this position for five breaths. On a final big inhalation, straighten the arms, still holding the foot, to return to the original position, then release the hands and jump back. Move through a Half Vinyāsa for the other side of the pose.
JĀNUŚĪRṢĀSANA (HEAD-TO-KNEE POSE) A, B, AND C
These three forms of Jānuśīrṣāsana are traditionally practiced one after another with a Half Vinyāsa between the individual forms. For more advanced students, a Half Vinyāsa is practiced between sides.
**J ĀNUŚĪRṢĀSANA A**
1. Come to a seated position with the legs stretched straight out in front of you. Fold the right knee closed and drop the knee out to the side with the sole of the foot near or touching the left thigh. Depending on your flexibility, the femurs should be at approximately a 90-degree angle. Pull back the left sitting bone as you press the right knee back and down with a slight inward rotation of the femur. This allows you to turn or square the hips toward the left leg as you enter the pose. The asymmetry of the action around the hip joints is the basic material of this family of poses.
2. Reach the arms over your head on an inhalation. As you exhale, fold forward and across the midline to hold the inner edge of the left foot with the right hand. Clasp the left wrist with the right hand.
3. The pelvis will naturally rotate to face the left leg. You may feel a slight lifting of the right sitting bone as you reach and lift up to fold, because the right leg has a primary inward rotation. As you deepen into exhalation and the fold, there is a counterrotation that drops that sitting bone back down toward the floor.
4. Wrapping the hands beyond the left foot and clasping the wrist takes time and flexibility, but working patiently and on the wave of the breath facilitates this action. Once in the pose, continue to unite deep internal actions and counteractions in the pelvic floor. Hold the pose for five breaths.
5. The counteraction of the left leg is an inward rotation with the inseam drawn down toward the floor. These leg actions will be facilitated if you keep the left leg awake and the left foot squared forward, especially through the root of the left big toe. As in Paśchimottānāsana and other forward bends, work with the breath to enter and deepen the pose. Using the breath, play with alternating primary and counterrotation patterns back and forth in the legs, feet, and around the hip joints. Their interplay can become even and refined as you learn the function of Mūlabandha.
6. At the end of an exhalation, ground the awareness into the pelvic floor. On the next inhalation, lift the torso and reach forward through the core of the heart as you straighten back to a seated position. Switch sides, or do a Half Vinyāsa before doing the pose on the other side.
**J ĀNUŚĪRṢĀSANA B**
1. Enter the pose as for Jānuśīrṣāsana A, beginning in a seated position with the legs stretched out straight in front.
2. Fold the right knee closed and sit up on the heel, placing it in front of the anus. Flex the foot (do not point the toes) so the inner edge of the right foot will eventually be visible along the inseam of the left thigh. Open the thighs to about an 65-degree angle (or to a lesser angle if you are stiff).
3. In this pose, you are sitting squarely on your heel so the sitting bones are approximately equidistant from the floor. Do not lean to one side!
4. On an inhalation, reach the arms over your head and, as in form A, fold forward to hold the sides of the left foot, or if you are more flexible, clasp the hands beyond the foot. Hold the pose for five breaths before transitioning to the other side, then do a Half Vinyāsa to Jānuśīrṣāsana C.
**J ĀNUŚĪRṢĀSANA C**
1. Enter the pose as for Jānuśīrṣāsana A and B, beginning in a seated position with the legs stretched out in front of you on the floor.
2. Fold the right knee closed and flex the foot. Bring the heel toward the lower belly and sit up straight. Turn the toes of the right foot down place them on the floor next to the inseam of the left leg. The right toes will eventually face squarely out to the right, and the sole of the foot will be perpendicular to the floor.
3. With the toes turned down this way, the right femur is encouraged to rotate inward as you start to lean forward into the pose. At this point, draw the right knee forward and eventually down to touch the floor, so the thighs form about a 45-degree angle. Be patient. If the knee does not easily reach the floor, do not push it. Remember to _flex the foot firmly_ when entering the pose.
4. As in the previous forms of Jānuśīrṣāsana, reach up through the arms on an inhalation, and as you exhale to fold forward, draw the kidney area on the right side forward and clasp the hands onto the sides of the foot or hold the wrists beyond the foot. Gaze along the line of the nose, eventually at the toes.
5. Hold the pose for five breaths. Exit on an inhalation, as you sit up with a straight spine. Switch legs via a Half Vinyāsa or simply by changing sides.
MARĪCHYĀSANA (MARĪCHI'S POSE) A AND B
Within the Primary Series, there are four forms of the Marīchyāsana poses, named after the sage Marīchi. The first and second forms (A and B) are forward bends, and the third and fourth (C and D) are twists. Although they are traditionally practiced as a continuous sequence (which is highly recommended), for the purpose of understanding the principles of the families of poses into which they fall, they are included in the forward bend section of this book.
**M ARĪCHYĀSANA A**
1. From Downward Facing Dog Pose, float through to a seated position, bending the right knee and placing the right foot flat on the floor with the left leg straight along in front of you. Place the right foot about one hand-width from and parallel to the left thigh.
2. The right sitting bone will be slightly off the floor, as if you were squatting. When working into the pose and as you hold it, keep that sitting bone as low as possible, but do not collapse back onto the bone forcing it to come to the floor; this compromises the psoas muscle on that side and makes the upward spiraling lift necessary to exit the pose impossible.
3. On an inhalation, reach up and forward with the right arm. On an exhalation, draw the right kidney area forward as you reach in the direction of the left foot, past the inner right thigh.
4. Lower the right shoulder as much as possible and wrap the right arm around the right leg, clasping your hands behind your back; use a strap if your hands won't reach. Once you have the bind, inhale and pull the clasped hands up the back, bending the right elbow slightly, which will help to keep the heart open. Exhaling, fold forward more deeply to put the chin on your left shin.
5. Gaze at the tip of your nose. If it is difficult to clasp the hands behind the back or to fold forward, gaze along the line of the nose and smooth out the ends of the breath. When the chin reaches the shin, gaze up toward the left big toe. Hold the pose for five breaths. On an inhalation, straighten up out of the fold and release the hands. Either switch sides or go through a Half Vinyāsa before returning to a seated position to do the pose on the other side.
**M ARĪCHYĀSANA B**
1. From a seated position with the legs stretched out in front, fold the left knee and place the heel near the upper pubic bone, entering Half Padmāsana. Slowly release the flexion of the Padmāsana foot, and draw the right foot back to place it flat on the floor just outside the right hip, as in Marīchyāsana A.
2. Press the left knee down toward the floor. On an inhalation, reach up with the right arm. On the exhalation, fold forward to reach the right arm out and wrap it around the right shin, clasping the hands behind the back.
3. If you cannot fold into Half Padmāsana, you may bend the left knee and place the heel in front of the right sitting bone. The right foot is then placed directly in front of the left ankle, with the right toes pointing forward. You may practice the pose in this form for a while and then gradually introduce the Half Padmāsana as flexibility increases. As with any variation, don't assume this is the final pose for you in this lifetime and become complacent in the form. Push yourself slowly, and gradually the full pose may be possible.
4. Once you are established in the preliminary seated position, inhale to reset the connection to the pelvic floor. On the next exhalation, begin to fold forward with the upper body, drawing the right kidney area toward the inside of the right leg, as in form A.
5. While folding forward, reach through the spine and the crown of the head rather than curling the spine to get the head to the floor. Eventually, your chin will rest comfortably on the floor. When you have reached the limit of your fold, lift the right elbow up and back, drawing the right shoulder slightly back and up behind you, and gaze down the nose. Once the chin comes to the floor, the gaze can be between the eyes. Hold the pose for five breaths. On an inhalation, sit up straight, unravel the legs or place the second leg into Full Padmāsana, and then jump back into Catvāri and complete the Half Vinyāsa to enter the second side (or simply switch sides).
**B ADDHA KOṆĀSANA A, B, AND C**
_Bound Angle Pose_
This pose is one that many people throughout the world assume for sitting in a casual manner or for work, so for some Baddha Koṇāsana is quite easy. In the West, this sitting position is less common. After many years of sitting on furniture, the hips become less flexible, and Baddha Koṇāsana can be challenging. But with patience and practice, the pelvis can become more vertical in this pose, and the hips will slowly open. It may take some time to work into the pose, and it helps to sit in this position as much as possible while doing simple activities. It may take many months of practice for the hip joints to relax enough for the legs to descend, but it's worth the wait.
1. From a seated position, bring the heels together about 3 inches in front of the groins. Sit up straight by rolling the pelvis to point the sitting bones straight down to the floor. If you feel that you are falling backward and/or if your knees are more than 6 inches above the floor, place a blanket under the sitting bones to position the pelvis perpendicular to the floor. Rolling the pelvis to vertical is initially more important than bringing your knees down.
2. For form A, have the soles of the feet together and clasp the big toes with the middle and index fingers of each hand. Sit straighter and straighter by first dropping the pubic bone toward the floor and then dropping the coccyx as well. Dropping the pubic bone makes the heart float up (prāṇa); dropping the coccyx makes the kidney area float up (apāna). Ultimately, both patterns are wide awake and active in Baddha Koṇāsana.
3. Soften the tongue and release the palate as you gaze at the tip of the nose, cultivating Uḍḍīyāna and Mūlabandhas. Keep the heart lifted as the knees press gently toward the floor. Stay here for five breaths.
4. To enter form B, open the feet like a book, so the soles of the feet are facing the ceiling. Hollow back the lower belly, scooping out the cave of the sacrum. On an exhalation, with the spine straight via the prāṇa pattern, slowly fold forward to place the belly in the cradle of the feet and the chin on the floor. Gaze between the eyebrows or down the nose to create the internal form of this pose. The gaze should encourage the spine to remain elongating and should create no tension or strain in the neck, face, or jaw. Experiment with the relative "weights" of the coccyx and pubic bone by inverting the feet, pulling and pushing their inner edges away from and into each other. Do not depress the heart. If you cannot place the belly on the feet, maintain the same pattern of a straight spine and an awakened pelvic floor, folding forward just as far as is appropriate for you. Over time, the pose will evolve and deepen. Remain in form B for five breaths.
5. To exit, ground the awareness in the sitting bones and imagine that water is running down to the floor through gutters in the edges of your thighs. On an inhalation, with the spine straight and long, return to an upright, seated position based in the pelvic floor.
6. For form C, on an exhalation curl the spine forward, tucking the crown of the head down toward the soles of the feet. Broaden the kidney wings and rest in the pose for five breaths. Exit on an inhale as for form B, then lift up and jump back for a Half Vinyāsa, or simply move into the next pose.
UPAVIṢṬHA KOṆĀSANA (SEATED ANGLE POSE) A AND B
At first glance, this pose seems similar to Prasārita Pādottānāsana with the legs stretched out to the side during a forward bend. But because Upaviṣṭha Koṇāsana is a seated pose, and the feet and legs are not stabilized by the force of gravity from the upper body, the actions and counteractions in the hips, legs, and spine are quite dissimilar. Within the Aṣṭāṅga Vinyāsa system, the following three related forms of this pose are part of the Primary Series and are generally practiced in one flowing sequence.
**U PAVIṢṬHA KOṆĀSANA A**
1. If you are entering the pose from a Half Vinyāsa, jump through to a seated position and open the legs immediately to about a 100- to 120-degree angle. Otherwise, simply sit and open the legs. If you are flexible, do not spread the legs farther than this.
2. If you are less flexible, the pelvis is not vertical, and it is difficult to sit up straight, elevate the pelvis by sitting on a blanket or block. You may also place the hands on the floor behind your back, fingers pointing forward, and then draw your whole spine in and up. Even in this variation, keep the legs active and alive.
3. On an inhalation, as you ground down through the sitting bones, lift the core of the heart and straighten the spine. As you exhale, keep the spine straight and fold forward to clasp the sides of the feet with the hands. If that is not possible, hold your big toes or simply place the hands on the floor out in front of you. You may need to decrease the angle between the legs in order to fold correctly or to hold the feet.
4. On the next inhalation, again straighten the spine and pull yourself forward as you straighten the legs completely, drawing the upper inner backs of the knees toward the floor. If the hands are on the floor in front you, gently pull back through the hands toward your thighs, which can help to elongate and to pull the spine forward. Press out through the heels and keep the skin rolling up the front of the torso and pulling down the back of the body.
5. Folding forward, place the belly and then the chin on the floor, maintaining the hollow of Uḍḍīyāna Bandha in the forward bend. Throughout the pose, continue to lengthen the legs through the heels and maintain a grounded sense in the sitting bones. Gaze between the eyebrows. If you cannot fold completely forward, simply breathe into your circumstances, keeping the spine straight and all of the actions and counteractions of the pose awake. In this way, little by little you may find you are much closer to the floor than you ever thought imaginable.
6. Hold the pose for five breaths. On an inhalation, straighten the spine, return to an upright position, and draw the legs together. Either cross the legs and hop back for a Half Vinyāsa, or simply sit up for the next pose.
**U PAVIṢṬHA KOṆĀSANA B**
1. From a seated position with the legs wide (as in form A), reach forward to clasp the big toes. As you exhale, curl the spine forward to puff out your kidney area. Inhale, bend the elbows slightly, and pull back to bounce the calf muscles off the floor and lift the feet into the air maintaining the wide spread of the legs. If this is impossible, try the bouncing action a time or two, then release the toes as you rock back to draw your legs up. Then reclasp your toes.
2. Balance on the back edges of the sitting bones, lifting the heart with the face parallel to the ceiling and gazing down along the line of the nose. Lengthen the inner edges of the legs and arms, lift the front edges of the armpits, and hold this form for five breaths.
3. To exit, on an exhalation drop the legs back down to the mat, cross the legs, and move through a Half Vinyāsa. Or simply take the next pose.
**S UPTA KOṆĀSANA**
_Reclining Angle Pose_
This version of Upaviṣṭha Koṇāsana begins by going into a wide-legged version of Halāsana (Plough Pose). As you rock up to a seated form, using the breath to control the movement, there is a brief pause before the legs drop gently to the floor. It takes practice, breath awareness, and a trust in the pattern of breath to do the pose correctly without slamming the heels down into the floor. To avoid the fear of hurting your heels while learning this pose, it can help to practice it on a thick carpet or, better yet, on a nice, thick lawn.
1. Lie on your back with the legs stretched out along the floor in Tāḍāgī Mudrā. At the end of an exhalation, empty of breath, begin to lift the legs, keeping them straight. When they have reached a 30-degree angle from the floor, inhale and continue lifting the hips to bring the feet over the head; place the toes on the floor. Reach the arms over your head as soon as the feet approach the floor, and clasp the big toes with the middle and index fingers of each hand. Spread the feet apart as for Upaviṣṭha Koṇāsana.
2. Gaze at the tip of the nose. Keep the legs straight, the pubic bone drawn up toward the ceiling, the throat soft, and the chin neutral (not pulled into the throat). This is an excellent position for Mūla and Uḍḍīyāna Bandha cultivation.
3. After five breaths in this position, you will rock up to Upaviṣṭha Koṇāsana B and then drop forward into Upaviṣṭha Koṇāsana A with the belly on the floor. To do this, first complete an exhale. Empty of breath and still holding the toes, begin to rock up toward the sitting bones, keeping the legs straight. Partway up, inhale and lift the heart to stop the movement. Do not inhale too soon, or you will not be able to rock up properly. Pause briefly, balancing on the back edges of the sitting bones, keeping the arms and legs straight with face lifted toward the ceiling and gazing gently down the nose. At the peak of the inhalation, in the gap where the breath begins to turn around, drop forward onto the calves, exhaling as you land. By lifting up away from the floor through the arms and keeping the legs straight as you drop forward, you can avoid dropping heavily onto your heels.
4. These actions assume a lot—that your hamstrings are long enough to flow easily through the movements, and that the actions and counteractions of the prāṇa and apāna patterns are deeply rooted in your nervous system. Practice and patience, along with a sense of humor, eventually make this pose possible.
5. At the end of the exhalation that brought you forward onto the floor, immediately sit back up, release the feet, and through a Half Vinyāsa move into the next pose.
**U BHAYA PĀDĀṄGUṢṬHĀSANA**
_Two Big Toes Pose_
Ubhaya Pādāṅguṣṭhāsana and Ūrdhvā Mukha Paśchimottānāsana are generally practiced in sequence, one immediately following the other. They are similar in that the setup for each is helpful in learning the correct shoulder action for Sarvānġāsana (Shoulderstand), and each places a strong emphasis on folding forward. Yet as similar as they are in form, holding the sides of the feet and the extreme fold of Ūrdhvā Mukha Paśchimottānāsana make it considerably more challenging for most practitioners.
1. Lie on the back in Tāḍāgī Mudrā. At the end of an exhalation, when you are empty of breath, begin to lift the legs. Inhale only when they have reached a 30-degree angle from the floor. Keep lifting the legs, beginning to exhale as the feet are straight up in the air and as the toes come near the floor above your head. Exhale fully and reach up to grab the big toes with the middle and index fingers of each hand.
2. Draw the pubic bone toward the ceiling to straighten the spine and legs. Stay in this position with legs awake, softening the palate and focusing on smoothing out the breath for five breaths. Gaze softly at the tip of the nose.
3. At the end of an exhale, empty of breath and still holding the toes, begin to rock up. Halfway up, inhale, and lift your heart to stop the momentum forward. Keep straightening up through the spine, and also maintain an even pull between the arms and legs so you can balance on the back edges of the sitting bones. Do not curl the spine and rest on the sacrum.
4. Drop the shoulder blades down the back and feel the upward flow of the front surface of the body. Smile softly to empty the palate. The face looks up to the sky, but the gaze follows the line of the nose to the toes. This facilitates a feeling of broadening the skin on the back of the neck and head as the tongue, jaw, and palate soften. Hold this position for five breaths.
5. To exit the pose, on an exhalation, release the toes, cross the legs, and draw the knees up. On the inhalation, lift up and swing back through to Catvāri, then move through Half Vinyāsa to the next pose.
**Ū RDHVĀ MUKHA PAŚCHIMOTTĀNĀSANA**
_Upward-Facing Forward Bend Pose_
Patience and an acceptance of things as they are is the key to this pose. It is far more difficult to rock up here than in the previous pose, but what's the rush?
1. Enter the pose in the same way as for Ubhaya Pādāṅguṣṭhāsana, lifting the legs up from Tāḍāgī Mudrā and placing the feet on the floor over the head while moving on the wave of the breath. Take hold of the sides of the feet for this form.
2. Rock up and establish the same balance and upward lift as in step 3 of Ubhaya Pādānġuṣṭhāsana with straight legs and, this time, holding on to the outsides of the feet. Once balanced, bend the arms partially and point the elbows out to the sides without hunching the shoulders.
3. Pull the legs toward a vertical position, and draw the chin toward the ankles. Do not drop the center of your heart. Gaze at the juncture of the big toes or between the eyebrows. Hold this position for five breaths. Then cross the legs and jump back for the Half Vinyāsa.
**N ĀVĀSANA**
_Boat Pose_
Nāvāsana is one pose that people love to rush through or skip. But it is an excellent way to build stamina; embody actions and counteractions in the pelvic floor, abdominal muscles, and back; and to build heat within the practice. It's a pose to relish—or at least pretend to enjoy. Remember, _any_ pose we want to avoid is most likely the one we should never skip as the mind is always looking to sabotage the practice.
1. Jump through from Downward Facing Dog Pose to land in a seated position with the legs lifted into the air at about a 45-degree angle from the floor. Or you may begin in a seated position and simply lift the legs. Be sure to balance on the back edges of the sitting bones in a way that makes you distinctly aware of the coccyx.
2. As you lift the legs, ground fully through the coccyx. Use the actions described in Chapter 3 for holding the tail of the serpent to stabilize the base of the body and give you a point from which to extend through the core.
3. The legs should be strong, with a sense of pushing forward through them. Reach forward through the hands to straighten the arms on either side of the legs in front of you with the palms facing; keep the arms parallel to the floor.
4. Keep the heart lifted, the fronts of the armpits high, and the shoulder blades low. The abdominal muscles are toned, but the muscles that run along the spine—the erector spinae muscles—do not engage firmly. This will keep your spirits up even through five repetitions.
5. Press the inner edges of the feet forward, straightening the legs and gazing at the juncture of the big toes. After five breaths, lean forward, bend the knees to cross the legs on the exhalation. Inhale as you lift the buttocks off the floor and rock forward to balance on your hands as if to jump back, but then sit back down. When lifting, keep the shoulder blades flowing down the back. Advanced students can press up into Adho Mukha Vṛkṣāsana (Downward Facing Pose or a Handstand) between each set instead of simply lifting the buttocks off the floor.
6. Repeat Nāvāsana and the lift five times. After the fifth repetition, inhale and swing back through, exhaling into Catvāri for Half Vinyāsa, or proceed to the next pose.
**A RDHA NĀVĀSANA**
_Half Boat Pose_
This version of Nāvāsana is good for training the apāna pattern, which can help eliminate lower back and SI joint pain.
1. Start in full Nāvāsana. On an exhalation, curl back so you are balanced on the sacrum, then lower straight legs until the heels are about 6 inches from the floor. Keep the head up, gazing at the toes.
2. At the same time, lower the shoulders to about 6 inches from the floor. Reach the arms along the sides of the body with hands resting on the sides of the thighs; the chin will be on or near the sternum.
3. Hold this pose for at least five breaths. It's easy to imagine the coccyx and sitting bones lengthening to become a long, strong dragon's tail. The abdominal muscles are strong and the PC muscle tones as the legs lengthen and reach out long.
4. At the end of an exhalation, lift the chest, straight legs, and feet, and inhale as you return to full Nāvāsana. This is like a straight-legged sit-up, so it takes focus and toning in the pelvic floor at the end of the initial exhalation. Cross the legs as for Nāvāsana and proceed through a Half Vinyāsa to carefully lengthen the abdominal muscles in Upward Facing Dog Pose.
**S UPTA HASTA PĀDĀṄGUṢṬHĀSANA**
_Reclining Hand-to-Big-Toe Pose_
Very similar to Utthita Hasta Pādāṅguṣṭhāsana, this pose doesn't require the same balance. Both forms allow a clear focus on the rotation of the femur in the hip socket and the actions and counteractions that facilitate this movement. It is also important to pay attention to the rotations of the pelvis in relation to the straightened leg as it is dropped over to the side.
1. Lie on the back in Tāḍāgī Mudrā. At the end of an exhalation, empty of breath, lift the straightened right leg and take hold of the big toe with the middle and index fingers of the right hand. If you can't reach the toe, loop a strap around your foot and hold the ends with your right hand. Bend the right arm, bowing it out to the side in a plane perpendicular to the end of the mat. As you begin to sit up, wrap the left kidney toward the right thigh and pull the chin to the shin.
2. Keep the left arm straight, with the palm along the top of the left thigh. The left leg is straight with the toes pointed and the heel pushing into the floor.
3. As you wrap the left kidney area toward the right knee, resist the pull of the arm through the leg by pushing the right heel toward the floor. Keep both shoulder blades flowing evenly down the back. (Advanced students can try lifting the back entirely off the floor.) Gaze along the line of the nose, and hold the position for five breaths.
4. Lower the head to the floor on an inhalation, then open the right leg out to the side on the exhalation. Keep the right arm slightly bowed as you lower the leg to the floor (use a strap to hold the foot if necessary, but still hold the strap with just the right hand). As the leg descends, the toes come toward the floor first, then the heel drops once you have reached the limit of your flexibility. This allows the proper rotation of the top of the femur in the hip joint. If the foot does not reach the floor, no problem. Simply breathe into the sensations along the midline of the body.
5. Once the right leg is in place, accentuate the reach out and down through the left leg to even out the hips. Do not lock the pelvis as you enter the pose; the safest vinyāsa for entering this second phase of the pose with leg out to the side is to allow some mobility (and possibly a lifting away from the floor) of the hip on the straight-leg side of the body.
6. Turn the head to the left and gaze at a point along the line of the floor. Keep the right shoulder rolled back toward the floor with the right buttock down as well. Continue to reach out and down through the left foot. The key is to move evenly from the area 2 inches below the navel.
7. Hold this position for five breaths. Using the strength of the arm and lower belly, inhale and draw the right leg back up to center. As you exhale, lift the chin toward the shin and bow to the leg again, for one breath. Inhale and drop the head back to the floor.
8. Exhaling, lower the right leg down next to the left leg on the floor and repeat the pose on the other side. To exit the left side and the full pose, return to Tāḍāgī Mudrā, then flow through Cakrāsana and complete Half Vinyāsa in preparation for the next pose.
**G ARBHA PIṆḌĀSANA**
_Embryo in the Womb Pose_
To those who are not familiar with yoga, particularly the Primary Series, Garbha Piṇḍāsana may appear to be the most bizarre of all poses. It _is_ quite unusual and can be challenging, but it's fun! The full form requires a safe Padmāsana and relies entirely on getting the correct vinyāsa of breath. Possibly one of the most beneficial aspects to this pose is that it is difficult, if not impossible, for the mind to wander, especially during the rocking phase of the pose. It is excellent training for learning to trust the intelligence of the breath while working through subtle layers of movement within the body.
1. From a straight, seated position, bring the feet into Padmāsana. Use the hands to draw the right foot back, bringing the heel toward the lower left side of the belly. Press the right knee toward the floor using the hip muscles. Next draw the left foot in over the right leg, bringing the heel toward the lower right side of the belly so you are in Padmāsana. Once in Padmāsana at first, flex the feet so the ankles do not collapse. If you cannot manage Padmāsana, then just cross your legs. Exhale to settle into the pose.
2. Exhaling, lift the knees away from the floor, folding at your hip joints. On the next few rounds of the breath thread the right hand through the gap on the right side between the upper thigh and the calf. It is helpful to keep the hand flat. Move at an angle approximately parallel to the left shin so that the right thumb grazes the left shin as you push the arm through. When the forearm is halfway through, turn the palm of the hand toward your face to complete the insertion. Next, thread the left hand through the gap in the left side of the body between the upper thigh and the calf. Push both arms through until the elbows are beyond the shins. Use water to lubricate your skin if the arms don't slide through easily. Hold your face with hands cupped beneath the chin, gazing at the tip of the nose for five breaths.
3. If you are unable to assume Padmāsana and have crossed your legs, fold at the hip joints to lift the knees and wrap the arms around the outsides of the knees. Clasp the hands together and draw the knees in toward your chest, then hold for five breaths.
4. On an exhalation, rock backward onto your upper back while maintaining a curled spine and keeping the hands tucked in toward the forehead. On the next inhalation, rock partway up, approximately to the sacrum, using the abdominal muscles to navigate a turn to the right. Repeat this rocking and turning nine times to make a complete circle. On the final inhalation, rock forward onto the hands and into Kukkuṭāsana (or with the hands on the floor outside the upper thighs if you are not in Padmāsana). Hold for five breaths with the face lifted and gazing down the line of the nose.
**K UKKUṬĀSANA**
_Cock Pose_
This arm balance is typically practiced immediately following Garbha Piṇḍāsana and relies fully on coordinating the wave of the breath with the pattern of movement within the form. You must exhale fully as you rock back to facilitate an inhalation that is grounded enough in both the core of the body and the pelvic floor to bring you up to balancing.
1. Enter the pose from step 4 of Garbha Piṇḍāsana; you are still in Padmāsana with the arms through your legs and the hands on the floor in front of the upper shins. On the inhalation and the final rock, continue to draw your chest and hips forward and up to balance on the hands in Kukkuṭāsana. The hands should be flat on the floor and well articulated, the arms straightening, and sitting bones lifted off the floor. Turn the head up, and focus the gaze slightly down along the line of the nose.
2. If when rocking you were unable to balance on the hands, sit back down, lift the knees as you exhale to rock forward. As the hands come to the floor work the abdominal muscles to lift the legs and buttocks off the floor. Inhale and look between the eyebrows when the center of gravity comes over the hands.
3. If you are not in Padmāsana after completing Garbha Piṇḍāsana, rock forward and lift the buttocks off the floor, balancing on the hands. Lift the face toward the ceiling while gazing down along the line of the nose for five breaths, then lower down to sit on the floor.
4. Remove the arms from between the legs. Move through a Half Vinyāsa either keeping the legs in Padmāsana or uncrossing the legs before jumping back.
**K ŪRMĀSANA**
_Turtle Pose_
The turtle figures prominently in Indian mythology as the support of the universe. Like other āsanas named after animals, it can be helpful in Kūrmāsana to imagine yourself as a turtle with the curve of the back and the arms around the body representing the turtle's shell. This pose requires considerable flexibility in the hips and spine. It is one that also demands patience and consistency to move into fully. Nonetheless, if you take it step-by-step, the form unfolds with time.
1. At the end of an exhale jump forward from Downward Facing Dog Pose so the feet land on the floor outside the arms, with the thighs beginning to wrap around the arms. Sit down.
2. Hook the inner knees as high up on the arms as possible, exhaling, and allow the feet to be out in front and just a little wider than the hips. Keeping the arms threaded under the legs, straighten the arms out and back on the floor with the palms facing down. Be careful not to place the thighs directly on the elbow joints, particularly if your elbows tend to hyperextend. Work the pose so there is no strain in any joints.
3. As you exhale, gradually push through the heels to extend the legs forward and out to the sides slightly until they are flat on the floor. The feet will move apart as this movement deepens, and the chest will automatically drop toward the floor.
4. When the belly is on the floor (or you have reached your limit of flexibility), gaze up between the eyebrows to maintain the internal lift of the heart. By straightening the legs as much as possible, the heels will eventually pop up off the floor automatically.
5. Hold this position for five breaths. On an inhalation, release the pose and return to a seated position. Release the arms and unwrap the legs, then move directly into Supta Kūrmāsana.
**S UPTA KŪRMĀSANA**
_Reclining Turtle Pose_
This pose represents a turtle withdrawing into its shell and into the calming flexion of the spine and a strong expression of the apāna pattern. It appears in the Primary Series and is an excellent preparatory pose for those working toward the Eka Pāda family of poses.
1. Reclining, with the legs wrapped around the upper arms and bent at 90-degree angles, bring the heels together and cross the right ankle over the left.
2. Rotate the arms inward. Drop the shoulders to reach the arms under the thighs and around the back. Clasp the hands behind the back, pulling them up the spine. Gaze at the tip of the nose, and hold this form for five breaths.
3. If you are a beginning student, you can work with the pose by simply bringing the heels together instead of crossing them. Then reach the arms under the thighs, reaching out and slightly back behind you (as far as flexibility allows) and if possible touch your head to the floor. If you are an advanced student, you can cross the ankles or shins behind the head or neck while sitting (as in Dvi Pāda Śīrṣāsana) and then lower forward to the floor. In this case, look between the eyebrows.
4. Hold the pose for five breaths. To exit either release the hands and use them to push yourself up to seated, lift up, unravel the legs, and come into Ṭiṭṭibhāsana, then move through Bakāsana into Catvāri, or simply release hands and ankles, sit up, and jump back into Catvāri. From Catvāri unroll into Pañca, and then roll back into Ṣaṭ.
**E KA PĀDA ŚĪRṢĀSANA**
_One Foot Behind the Head Pose_
For many Aṣṭāṅga Vinyāsa practitioners, the Eka Pāda family (which actually begins with Kūrmāsana in the Primary Series) is both daunting and strangely attractive—like those relatives who are eccentric and nobody outside the family can really tolerate or understand, but everyone in the family loves. These Eka Pāda poses can be considered forward bends in that they involve extreme flexion in both the hip joints and the spine.
It is important to work slowly and in stages to keep the lower back safe. Although for first-timers it can be a thrill to get the leg behind the head, blowing out a disk to do so is not worth it. Be patient and work only to your limit, without pain and with great integrity.
1. From Downward Facing Dog Pose on an exhale, hop forward to a seated position with the right thigh wrapped around the right arm. Or simply begin the pose from a seated position and wrap the leg around the arm. Roll back on the pelvis, bending the right knee to take the right foot in your arms, as if cradling a baby. Leave the left leg stretched out in front of you, but do not rigorously engage that leg or foot while working into the pose.
2. Lace the right arm under the right thigh, rolling the right calf toward the back of the body, and as you exhale work the leg as far up the arm as possible. On another exhale, take the right shin behind the neck, and position the foot near the left shoulder, toes pointing to the ceiling.
3. Keep the spine flexing and the outer right hip relaxed, with the abdominal muscles toned but not fully engaged. Deepen the position of the leg by working the right shoulder down and through beneath the calf. Twisting the upper body to the left and pulling with the right hand on the left thigh can facilitate this.
4. Once the leg is in position, inhale and sit up a little more straight, then straighten the left leg by reaching out through the mound of the left big toe to create an inward rotation in the leg. Fold the hands in front of the heart in Añjali Mudrā. Gaze softly along the line of the nose at a point on the floor, and hold the position for five breaths.
5. Release the mudrā, and on an exhalation, fold forward to place the chin on the left shin, stretching the arms out in front and clasping the left wrist with the right hand beyond the left foot. Gaze along the line of the nose or at the left big toe as you hold the position for five breaths.
6. On an inhalation, release the clasp of the hands and sit up. Exhale and with the right leg still behind the head, place the hands beside the hips. On the next inhalation, lift the hips up off the floor, pointing the left leg straight up with toes pointing to the ceiling and gazing at the left toes. On the following exhalation, swing back into Catvāri, then move through a Half Vinyāsa before doing the pose on the other side.
**C AKORĀSANA**
_Moonbeam-Drinking Bird Pose_
This pose strongly contracts the abdominal muscles and gives a sense of lengthening and complete action in the pelvic floor and coccyx. At the same time, it opens the heart so that it floats upward with the prāṇa opening fully in the presence of strong apāna. Cakorāsana is used as a transition when moving out of many poses in the Eka Pāda family and is also a separate pose in and of itself.
1. Follow steps 1 through 3 for Eka Pāda Śīrṣāsana, with the right leg behind the head. Bring the chest forward and draw the right leg down behind the lower neck so the neck can straighten and the head can roll back.
2. Place the hands on the floor next to the hips and exhale completely. While empty of breath, lift the hips up off the floor and reach up through the left leg to a vertical position, pointing the toes. With practice, the chin will touch the upper shin of the left leg. Gaze optimistically upward like the _cakora_ bird as it's approaching the moon for its nectar.
3. Hold the pose for five breaths. This position really works the rectus abdominis muscles and activates the apāna pattern based in the pelvic floor, while keeping the heart open.
4. On an exhalation, rock the pelvis forward and then back slightly using that momentum to help you swing back through into Catvāri, and do a Half Vinyāsa before practicing the pose on the other side.
**D VI PĀDA ŚĪRṢĀSANA**
_Two Feet Behind the Head Pose_
1. From Downward Facing Dog Pose at the end of an exhale, hop forward to land standing with bent legs, with the feet beyond the hands and the thighs wrapped around the arms. Sit gently, leaving the legs beyond the arms.
2. Sit up as in Eka Pāda Śīrṣāsana. Wrap the _left_ leg behind the head first. Once it is securely in place, draw the right leg back, hooking that ankle outside the left. Work the upper arms through the thighs and the legs as far down the neck and back as possible and draw the ankles and feet slightly apart.
3. Fold the hands in front of the heart in Añjali Mudrā, and gaze gently down to the floor along the line of the nose. Hold this position for five breaths.
4. Place the hands flat on the floor beside the hips. On an inhalation, lift straight up, balancing on the hands so your hips are off the floor. Hold for five breaths, still gazing ahead softly.
5. Unhook the feet and lift the hips away from the floor, pushing down through the arms and reaching out through the legs into Ṭiṭṭibhāsana (Small Water Bird Pose). Hold this for only one breath and then jump back (via Bakāsana, if possible) into Catvāri and do a Half Vinyāsa.
**Y OGA NIDRĀSANA**
_Yoga Sleeping Pose_
Very similar to Dvi Pāda Śīrṣāsana but in a prone position, some practitioners find this pose far more accessible than the upright form. Because the back is supported by the floor in Yoga Nidrāsana, there is far less risk of overpushing and injuring the spine or disks, so this is a good pose to experiment with if Dvi Pāda Śirṣāsana does not come easily and as preparation for that pose.
1. Lying on your back, bring both legs up and out toward the sides of the body, bending the knees and pointing the toes toward the floor.
2. As you exhale, wrap the left leg behind the head (as you did for Dvi Pāda Śirṣāsana) rolling the left calf back toward the floor and wrapping the left shoulder through the leg. Move the left leg and foot down the back with the foot as far over toward the right shoulder behind the neck as possible.
3. Next wrap the right leg behind the neck, moving the right shoulder through and positioning the right ankle outside the left, close to the floor. The ankles should overlap, and the toes should be pointed.
4. Wrap the arms around the sides and clasp them behind the back. Gaze gently down the line of the nose; make the heart float with the prāṇa pattern while releasing the jaw and palate. Hold the pose for five breaths.
5. To exit, release the arms and legs, straighten the legs, and roll through Cakrāsana into Catvāri and Half Vinyāsa.
_8_
Backbends
FOR SOME PRACTITIONERS, BACKBENDS ARE THE glory poses—coming from being stuck in the mud to rise up and stretch the rays of intelligence and enthusiasm out in all directions. For others, backbends are the nemesis; they're frightening, miserable, and to be avoided at all costs. From either of these disparate perspectives, it is quite easy (if not inevitable) to injure yourself practicing backbends, unless you pay attention to the internal actions and counteractions of the apāna and the prāṇa as you move in and out of the poses. Perhaps you get a free pass from either extreme view for a number of months or years before an injury occurs. However, if you repeatedly ignore the internal essence of the form—maybe because it feels kind of good to stretch the abdomen and fold deeply in the lower back; or maybe it seems safe to grit the teeth and push through pain with grimly determined muscle power—at some point, either during a particularly stiff practice or when the mind wanders one day, you are likely to experience an injury. Fortunately even at that juncture, it's not too late, and the internal forms are there to save you, presenting backbending benefits for your particular circumstance.
The odds of being injured in backbends without good alignment are higher than those in most other families of poses primarily because we find ourselves in Upward Facing Dog Pose (a rather advanced pose) many times during any given practice. Chapter 5 covers Upward Facing Dog Pose and variations that teach proper alignment because they are so central to every practice. So not only are there innumerable opportunities to drop deeply into the intricacies of the form, there are just as many chances to move by rote or in extreme ways and wind up with an injury. With practice though, almost everyone can learn to get into and out of backbends deeply and safely.
Backbends are the epitome of the prāṇa family, expressing a sense of extension and expansion up and out. Just as we know when we've been overemphasizing the apānic pattern without its beloved prāṇa present in forward bends—because we feel physically and emotionally shut down, defensive, and despairing—we also have an internal barometer that lets us know when we've gone too far with the expansive prāṇic pattern as well. Apāna gives a visceral sense of impermanence and interconnectedness to forms projected into the sense fields opened by the inhaling prāṇa pattern. When we ignore this grounding, stabilizing aspect of apāna in backbends, a sense of an asymmetrical, manic "hyperprāṇa" arises in which it is difficult to focus the mind, and an underlying sense of anxiety arises. We may be overwhelmed by waves of emotion or we may feel spacy or ungrounded in our day-to-day interactions with others. Unfortunately, because many of these manifestations of excessive prāṇa are intoxicating and ego building (rather than depressing like the overly apānic patterns of poorly balanced forward bends), they are easy to overlook. Consequently, we may not recognize overly prāṇic patterns, and even if we do, we may not see the need to adjust the practice to come back down to earth to an interdependent and healthy existence with other beings.
To bring balance into backbends, it is important to focus and refocus on the complementary opposite patterns of movement within the Prāṇa that fully engage us in the poses. In this way, the poses naturally unfold as deeply as is appropriate not only for our body, but also for our mental and emotional states on any given day. Because a central physiological aspect of backbends is that we literally "open the heart" to do the poses, taking care not to overdo this physical pattern when we are feeling emotionally unsettled or vulnerable is very important. Sometimes, even for those who can do extremely deep backbends, it is not advisable to go to the body's limit; it is better to maintain smooth and even breathing while taking the pose just a bit less deep.
For all backbends the underlying pattern is that as we inhale and consider the pose, we then ground fully on the exhalation to set the apānic pattern solidly in the pelvic floor as the base of the pose. This allows us to experience backbends as integrating poses that bring intelligence and protection to the lower back and groins. The apānic pattern, connecting to the coccyx while toning the pelvic floor and then spreading and lifting the kidney wings, grounds us so the prāṇic pattern can stimulate and passively stretch the hip flexors and groins. Once the complementary patterns are in play, backbends will manifest naturally and safely. They will deepen just the right amount as we do our part to allow the conversation between prāṇa and apāna to evolve into an affectionate embrace.
As novices we may think that backbends are just that—bending the back of the body, particularly the spine, backward. Though that is correct up to a point, the way to go deeply into backbends and really feel their benefits is to work up and through the body—to lengthen and stretch the _front_ of the body from the pelvic floor and to avoid compression of the vertebral facet joints in the back of the spine. Although the natural S shape of the spine does allow for easy extension in the areas of the lumbar and cervical spine, it soon becomes apparent from this internal perspective that extension of the whole spine is vital and more important to healthy backbends than is extreme extension of the spine at only a couple of spots. Moving from the deadened spine rather than the pelvic floor and full body usually causes pain and injury because we fold into the easily bendable areas—compressing the lumbar vertebrae, pressing the kidney wings closed at T12, and collapsing the head back on the atlas in the cervical region. When practicing backbends in a more integrated manner, we flow on the wave of the breath, lengthening top to bottom through the central channel and up the front of the body. We must also tone, extend, and open the structures in the front and the back of the body—the hip joints, the psoas muscles, the back of the diaphragm, the full thoracic and the lower cervical vertebrae—to ensure a spacious and stable backbend. Moving on the wave of the breath makes these individual actions natural and nonaggressive. They are difficult when not hooked together in a wave.
Most backbends involve a deep opening and extension of the front of each hip joint, often referred to as the groin. This extension is an apānic movement because actively, in its purest manifestation, it employs the abdominal muscles, hamstrings, and buttocks muscles. At a certain phase as we move into backbends, the muscles associated with the apāna family (hamstrings and gluteus muscles) cause a release of the hip flexors through reciprocal inhibition. This can be understood initially as an active tail-tucking pattern, which slows down extension of the spine and defines a stable root from which the prāṇa pattern can unfold symmetrically. Tail tucking, like everything else, can be overdone, causing external rotation of the femurs, narrowing of the pelvic floor, and ejection of the pubic bone from the narrowed pelvic floor. This is where Mūlabandha allows the essential counter-counteraction of the prāṇa to come into play, so there is a sensation of the coccyx lengthening and moving forward toward the pubic bone as the sacrum lifts up and in toward the lower belly.
Because it is so stimulating to open the chest, diaphragm, and throat, it is challenging to find, let alone move from, the pelvic floor. Yet awareness of the pelvic floor, the presence of the coccyx, and the opening of the kidney wings are the counteractions that deepen the poses and protect the SI joints, the lower spine, and the lower ribs so the expansive prāṇic patterns can unfold fully. If we become accustomed to being overly prāṇic in backbends, the extreme ungrounded version of this pattern can invade many other poses and ruin their contemplative effects on both body and mind.
In more extreme backbends, such as Ūrdhvā Dhanurāsana (Upward Bow Pose), it is also important to engage and position the shoulders correctly to support the bend in the upper lumbar and lower thoracic spine. Even though the backbending pattern is often perceived as most pronounced around the area of the heart (the part of the body that opposes and complements the thoracic area) and in the upper spine, the lower thoracic region can be compromised if the shoulders are misaligned. Learning the correct protraction of the shoulder blades and then the art of extending the upper spine from the heart area up through the head, without compressing the vertebrae and without breaking the rhythmical sequence of extension that is rooted in the pelvic floor, is essential while backbending. Proper shoulder alignment is also dependent on opening the kidney wings to get the belly to stretch and lengthen without losing the apānic tone into a sort of _psychic hernia_ that expands out from the line of the spine through an overstretched solar plexus or overly flaccid buttocks. This misalignment splays out the intelligence into a multiplicity of options and divides the subtle aspects of uniting prāṇa and apāna along the midline of the body.
In backbends, we must also pay close attention to the position of the head, the gaze, where in the body the pose is physically connected to the earth, and finally the relationship of the open palate to the pelvic floor. These more subtle aspects of the poses may be considered separately, but it is most effective in terms of supporting internal patterns of the form to study them in relationship to the body as a whole and through a sense of overall movement of the spine.
For most practitioners, the easiest place in the spine to fold backward is in the area of the upper cervical spine. If you tilt your head back to look up, you're doing a backbend in the cervical spine. Because looking up is so much a part of day-to-day life, we may not realize the impact that an unconscious folding of the uppermost cervical spine (the atlanto-occipital joint) has on a full backbend; it actually stops the flow of Prāṇa from top to bottom and tends to perpetuate an ungrounded feeling. The head and neck can either intelligently express a rhythmical extension from the heart through the crown, or the head can act as a heavy weight that simply falls back habitually, with the occiput dropping back onto the atlas. So as we move into backbends, if we can imagine the feeling of extension up through the core of the body from the chest (or better yet from the pelvic floor through the chest) and out through the crown of the head, we can begin to avoid dropping the head back unconsciously and overextending the cervical spine. The head is always the last component of the spine to roll back in healthy extension. This action of gradual extension from lower to higher in the cervical spine and full expression of the backbending form may be facilitated by dropping awareness deep into the center of the pelvic floor where we can feel the connection to the midline of the body. If we then imagine the central channel from top to bottom as smooth, luminous, and long, curving backward on the wave of the breath as if the breath were a fountain of movement up, back, up and over, then the backbend comes naturally. This feeling can be supported by tuning in to a sense of lifting and spreading the kidney wings and also paying attention to sensations in the skin of the throat and the back of the head; consciously softening and broadening and gently lifting all these areas of the body.
It is also important to cultivate a soft, downcast gaze as we move into backbends. When we look up, the posterior suboccipital muscles under the back of the skull are signaled to contract in direct response to the eye movement. This usually also translates into a contraction of the erector spinae muscles. If we look up as we move into backbends, both of these groups of muscles are likely to fire, automatically causing the head to fall back on the atlas. This stops extension in the lower cervical and upper thoracic areas. In addition, engaging the posterior suboccipitals has the powerful effect of triggering larger patterns of hyperextension throughout the body, such as closing the kidney area and releasing the pelvic floor. On the other hand, looking down as we enter backbends contracts the anterior suboccipital muscles, which tilt the head forward on the atlas and prevent it from falling back. So keeping the dṛṣṭi down along the line of the nose—without tension in the eyes—is integral to healthy backbends. It makes us extend the cervical spine from bottom to top after opening the upper chest and heart area.
A curious aspect of practicing backbends is that even though naturally flexible practitioners can extend and get into backbends with seemingly little effort, and we would expect them to benefit greatly from the poses, this is not always the case. Having arrived in the territory of the backbend, it still requires a great deal of internal and meditative focus to actually turn a gymnastic position into a fully functioning yoga pose and receive the benefits of that pose. Because those who are naturally flexible arrive more easily in the pose, they also run a greater risk than stiffer individuals of becoming intoxicated by the waves of expansive, prāṇic residue that may sweep through the body in the initial stages of the pose. So paradoxically, it is not unusual that stiffer practitioners taste the benefits of backbends first.
**Ś ALABHĀSANA A AND B**
_Locust Pose_
This is not a glory backbend—one that impresses friends and family members who don't think too much about yoga but love to watch you try extreme poses. Imagining yourself as a locust flying close to the ground can make the pose more enjoyable and also informs the alignment well. Śalabhāsana is an active backbend in the sense that it relies on muscles along the back of the spine and body (hamstrings and buttocks muscles) for its form, whereas in more flashy backbends, like Ūrdhvā Dhanurāsana, these muscles are eventually softened at different phases or throughout the pose. Forms A and B may be practiced separately; however, it is part of the traditional Intermediate Series to practice them together and also makes sense physically, as there is just a simple movement of the hands that allows the posture to deepen and that separates the two variations of this pose.
1. Lie on the belly with the forehead on the floor, the legs together, the toes pointed, and the arms stretched down along the side with the palms facing up. Lift the tops of the arms away from the mat about 2 or 3 inches, being careful not to pinch the shoulder blades together.
2. Exhale fully, then as you inhale, begin lifting the head, shoulders, chest, and legs away from the floor. Bend the elbows slightly, pushing the backs of the hands into the mat to aid in the lift.
3. Keep the legs engaged, the jaw soft, and the gaze steady as you reach back through the inner edges of the feet. Do not sickle the feet so that the toes touch and heels spread apart; keep the feet parallel or with the sides touching. It can be helpful to hold a block between the feet or ankles to prevent external rotation of the legs and to accentuate this foot action.
4. This is form A. Hold it for five breaths, gazing along the line of the nose with the head, chest, and legs lifted.
5. On an exhalation, place the palms of the hands on the floor next to the waist. As you inhale, push gently to lift the chest a little more. This is form B. Hold it for five breaths, then move the hands along the side of the body next to the heart, and on an exhalation, turn the toes under and push with the arms to pop up into Catvāri. Move through a Half Vinyāsa into the next pose.
**P ŪRVOTTANĀSANA**
_Stretching Up the Front Pose_
This is the perfect counterpose to Paśchimottānāsana and is the most obvious embodiment of the tail of the serpent. It wakes up the hamstrings and is very beneficial if the hamstrings are either tight or overstretched at their attachment points because it requires the belly of the muscles to engage. All backbends are actually a stretch of the front of the body, and no pose underscores this stretch more than Pūrvottanāsana.
1. Enter the pose from Downward Facing Dog Pose, jumping through to a sitting position on the end of an exhalation with the legs stretched straight out in front. Inhaling place the hands about 8 inches behind the buttocks, with the fingers pointing toward the front of the mat. Drop the chin down near the upper sternum, and roll the shoulders back and down. This is similar to Daṇḍāsana.
2. Bend the elbows as you exhale, and strongly curl the coccyx as you lengthen out through the legs. This movement is the epitome of the tail of the serpent movement. Lean back and start to lift the hips, still curling the coccyx as if to collect and hold the uniting of prāṇa and apāna. Imagine lengthening the coccyx all the way to the feet.
3. While inhaling, draw the sacrum up as you straighten the arms. Unroll the spine bottom to top, lifting the heart and gazing down the line of the nose. Remember that the head should be the last part to roll back.
4. Keep the legs engaged as you enter the pose, at first reaching through the balls of the toes and, as you reach the full lift, pointing the toes toward the floor. Keep the apānic grounding firm during the entire pose. Be sure to roll the shoulders back and down as you enter the pose, and keep a slight bend in the elbows while holding it. The sacrum should lift up and into the body as you keep the legs strong throughout the pose. It is also very important to keep the hands pressing evenly and intelligently into the mat, keeping the roots of the fingers—especially the index fingers—firmly rooted to the earth throughout the transitions and while in the pose.
5. Hold the pose for five breaths. Lower back to the floor on an exhalation, keeping the legs strong as you do so and prepare for the next pose.
**S UPTA VĪRĀSANA**
_Reclining Hero Pose_
It can be helpful in backbends to stretch the quadriceps. Within the Intermediate Series, Bhekāsana does just that. For those who are not ready for Bhekāsana or those who want an additional quadricep stretch, Vīrāsana and Supta Vīrāsana are excellent poses to explore as well, even though they are not officially part of the Aṣṭāṅga sequences. In all of the movements of this pose, move on the waves of the breath, and breathe fully and evenly throughout.
1. From a kneeling position, lower the pelvis to the floor between the lower legs, imagining the sitting bones plugging into the floor. As you position the body in this sitting position, use the hands to gently rotate the calves out to the sides and guide the skin of the outer thighs in and down toward the floor. If you experience discomfort in the knees or hips, support the sitting bones with a blanket or block. Place a blanket beneath the knees and ankles if the ankles are uncomfortable. You should feel no pain in the knees, ankles, or hips.
2. This is Vīrāsana and may be used as a form for sitting meditation or simply sitting during the course of the day. It is even and stable, and some practitioners find it easier to keep the pelvis vertical to the floor in Vīrāsana than in simple cross-legged forms. Because the legs are folded back, the quadriceps are naturally stretched and begin to lengthen.
3. To increase the stretch, after a few breaths, begin to lean the torso back, as if to lie down. If you are very flexible, this may be easy; if you are not, it may seem impossible. Find your limit and do not overstrain in this pose. It is most beneficial as a preparatory stretch and to aid digestion, so you may practice it after eating and before beginning a practice.
4. If it is difficult or impossible to lie back on the floor, arrange bolsters or blankets behind your body on which to lie. If lying back is easy, do so slowly, keeping the lower body toned but not gripped. If it is not comfortable to lie back, you may lean back placing the hands on the mat behind you with fingers pointing toward the front of the mat.
5. Once back as far as you can go (without overstraining), place the arms along the sides of the body. If you are using the hands behind you as support, simply keep them there. To experiment with the stretch, gently lift the pelvis off the floor and tuck the coccyx up toward the pubic bone. This will cause the femurs to rotate outward, and the knees may move apart slightly.
6. Next drop the pubic bone down toward the coccyx, tilting the pelvis down slightly, which will cause the knees to move closer together as the femurs rotate inward.
7. For a more extreme stretch if you are lying down, reach up along the floor over your head with both arms, spinning the arms so the palms face the midline of the body as the shoulder blades protract.
**B HEKĀSANA**
_Frog Pose_
It may be difficult to imagine ourselves as frogs until we find ourselves in Bhekāsana, simultaneously "spring-loaded" and "stuck in the mud," which provides useful perspective for evaluating other difficult situations that arise in life. Bhekāsana is practiced early in the Intermediate Series as a preparatory pose for the more intense backbend sequence. Lying on the belly with the legs in reverse Vīrāsana and the chest lifted, the quadriceps and psoas muscles are stretched, the arms and shoulders are positioned to engage the serratus anterior muscles, and the shoulder blades are positioned as support for the thoracic spine.
1. Lying on the belly lift the feet to bend the knees. Reach back and clasp the fronts of the feet, rotating the hands so the fingers point in the same direction as the toes. Position the upper arms approximately parallel to one another.
2. Exhale fully, then on an inhalation, lift the shoulders and chest as you push the feet down toward the floor outside the thighs. Keep the upper arms parallel and reach out through the sternum and crown of the head as you gaze down the nose. If possible, flatten the soles of the feet on the floor, but do not strain and injure the knees. The thighs may roll out to the sides slightly, but do not let them move too far apart.
3. Hold this position for five breaths, gazing down the line of the nose and breathing smoothly. On an exhalation, release the feet and place the hands flat on the floor next to the heart as you straighten the legs back down to the floor, with the feet flexed. Push into Catvāri on the next exhalation, move into Upward Facing Dog Pose on the next inhalation, then go through a Half Vinyāsa into the next pose.
4. If it is too difficult to clasp both feet at the same time, a preparatory version of this pose may be helpful. Lying on the belly, lift the upper body as if moving into the Sphinx Pose or Bhujaṇgāsana on the elbows. Place the left arm out in front of the body with the forearm pointing straight ahead and the elbow just beneath the left shoulder. Bend the right knee and clasp the top of the foot with the right hand; point the fingers forward if possible. Breathe into the pose, increasing the stretch in the leg and upper body on the inhalation and releasing it slightly on the exhalation. After five breaths on the left side, repeat this form on the other side. Then exit the pose as for full Bhekāsana.
**D HANURĀSANA**
_Bridge Pose_
This pose is delightful, with a sense of liberation and freedom—even for beginning practitioners. It is a traditional part of the Intermediate Series, but those who are not quite ready for the full series can experiment with it as a means of learning the internal actions and counteractions that inform all backbends.
1. Exhaling, lie down on the belly. Place the top of the forehead on the floor while gazing softly down toward the tip of the nose. Use the same position as described for the beginning of Śalabhāsana.
2. At the end of an exhalation, bring awareness to the coccyx, imagining it to be heavy and a ballast for the pelvic floor. Then, on an inhale lift the head, chest, and knees from the floor as you bend the knees and place the inner edges of the feet together. (This is technically a variation of Śalabhāsana.) Immediately clasp the ankles with the hands so that the feet are free to articulate. If you are flexible, you can clasp the legs with hands up higher on the shins toward the knees.
3. On an inhalation, lift the feet back and up, higher and higher as if you were a marionette and strings were drawing the inner edges straight up to the ceiling. Bend the elbows slightly to feel the small counterinward or secondary rotation of the upper arms. Maintain an awareness of the coccyx, imagining it as a tail. Cultivate the sense of an empty palate to lengthen the front of the spine rather than compressing the back of the spine. Gaze along the line of the nose.
4. After five breaths, come down on an exhalation. Continuing with that same exhalation, enter Catvāri, and then inhale into Upward Facing Dog Pose. Exhale into Downward Facing Dog Pose to digest, study, and enjoy the residue of the pose. If you are practicing the Intermediate Series, instead of going immediately into Upward and Downward Facing Dog poses, do Pārśva Dhanurāsana on both sides before exiting.
**P ĀRŚVA DHANURĀSANA**
_Side-Angle Bridge Pose_
Practice this pose immediately following Dhanurāsana, rocking from the Dhanurāsana over to the side and then returning to Dhanurāsana between sides. Of course this pose can be practiced on its own, but it is much easier and often more beneficial to practice after full Dhanurāsana.
1. Exhaling, lie down on your belly with the same alignment form as for Dhanurāsana.
2. If you are not working with Pārśva Dhanurāsana as part of the Intermediate Series, still incorporate some breaths in Dhanurāsana in preparation for this pose. As you inhale, lift the feet into full Dhanurāsana for five breaths. On an exhalation, rock over to the right, bringing the right arm and hip to the floor. Work the top of the arm back so you are resting as far to the front of the top of the humerus as possible. This action should feel good, and it helps to establish an openness in the heart.
3. Inhaling, activate the pose by bring the sacrum up and into the body as you feel the coccyx moving down and in toward the pubic bone. Engage the quadriceps and reach out and back through both legs, keeping the feet even and thighs parallel to gradually move the legs to a straighter form.
4. When the posture feels fully expressed, turn the head back symmetrically as if looking at the toes. Then with the lower neck in full extension, turn the head to look up. Do _not_ turn the head until the neck is first fully extended.
5. After five breaths, exhale and roll back onto the belly. As you inhale, lift back to full Dhanurāsana, then exhale and rock over to the left side and repeat the entire sequence of movements on that side.
6. After five breaths, exhale and roll back onto the belly. Inhale as you lift back to full Dhanurāsana, and hold for five breaths.
7. To exit the pose, lower the knees, release the feet, and straighten the legs down to the floor with the toes turned under. Place the hands in Catvāri position and immediately on an exhalation, lift your groins and belly into Catvāri. Inhale into Upward Facing Dog Pose, and exhale into Downward Facing Dog Pose. Again, take advantage of the open ears and the easy bandhas of this pose to digest the residue of Pārśva Dhanurāsana.
**Ū ṢṬRĀSANA**
_Camel Pose_
Though this pose is found in the Intermediate Series, it can be beneficial and informative—and even accessible—to less seasoned practitioners. It is very instructional in terms of backbending basics: strong legs with an internal spin, and a connection to the pelvic floor so the spine elongates while entering backbends. The instructions that follow include a few extra breaths from the traditional sequence. These extra breaths allow the body to settle into the lines of the pose and move on the wave of the breath rather than from a place of muscular engagement. Once the pose has been practiced repeatedly and you are used to it, you can attempt it without these extra breaths.
1. From Downward Facing Dog Pose, at the end of an exhale hop forward to a kneeling position with the knees about hip-width apart. Keep the lower legs parallel and the tops of the feet against the floor. Place the hands on the hips.
2. Exhale and drop the awareness into the base of the pose, grounding into the legs as they spiral down. Isometrically squeeze them together as you tone the PC muscle; drop the coccyx down and forward while keeping the internal action of the pubic bone moving down and back.
3. Inhaling, bring the awareness to subtle movements that straighten and elongate the spine along the plumb line of the body. Follow the wave of the breath as it arcs up through the central channel.
4. Gently push down on the hips with your hands and lift the heart, broaden the back of the body, and lengthen the spine from coccyx to crown. Keep the head lifted and gaze down the nose as you move into a subtle backbend. As in all backbends inhale with the gesture of lifting up and arching, and exhale with the sense of a heavy coccyx and dropping down through the legs into the earth.
5. Maintain the length in the spine and a sense of connection to the earth through the legs. Exhaling, place the hands on the ankles or heels (if you are stiff, flex the feet with the toes turned under, or place the hands on blocks placed outside the ankles). Still gazing down the nose and keeping the jaw soft, tilt the head back by extending the neck and reaching out through the crown of the head. Do not allow the back of the head to fall back on the atlas and interrupt the smooth, even flow of the arc from the knees to the crown of the head.
6. The position of the hands on your ankles directs the shoulder action. As you become more flexible in the spine and shoulders, experiment with turning the hands so the thumbs are along the outsides of the ankles. If you are less flexible, place the hands on your heels, and when you can hold the ankles, place the thumbs along the insides.
7. Hold the pose for five smooth, full breaths, gazing down the nose and softening the jaw and tongue as you release the palate as if to drink nectar. To exit, ground down into the earth through the knees on an exhalation. Inhale, keeping the legs toned and in place, and gently squeeze the thighs together as you return to kneeling, bringing the head up last. Exhaling, place your hands beside the knees and jump back into Catvāri for a Half Vinyāsa.
**L AGHU VAJRĀSANA**
_Light Thunderbolt Pose_
Ūṣṭrāsana and Laghu Vajrāsana are typically practiced in preparation for Kapotāsana. Laghu Vajrāsana is more difficult for many practitioners because the head is placed on the floor and the arms are kept straight. But moving slowly into the pose, working at it patiently over a period of weeks (or months), can eradicate—or at least tame—the fear, stiffness, and disorientation that are the usual obstacles to this āsana.
1. From Downward Facing Dog Pose, at the end of an exhale hop forward into a kneeling position with the thighs parallel and about hip-width apart. Exhaling, lean back to place the hands on the lower legs. Experiment with the position of the hands on the legs to fit with your proportions, level of practice, leg strength, and flexibility. The hands may be as far forward as the top of the calf and as far back as the ankles. Different hand positions change some dynamics of the pose, so adapt them to your circumstances.
2. Inhaling and keeping the legs strong with a sense of inward rotation, move the pelvis slightly forward to lift the sacrum in and up within the body, as you push down through the arms.
3. Keeping the arms and legs toned and strong, drop back until the head reaches the floor. Gaze down the nose as you descend and throughout the pose. Hold the position for five breaths, breathing smoothly and evenly.
4. At the end of an exhalation, engage and isometrically squeeze the legs together. From deep in the belly, on an inhalation, lift the pelvis to return the upper body to an upright position, bringing the head up last. Exhale into the legs and then, placing the hands on the floor next to the knees, hop back into Catvāri and move through a Half Vinyāsa.
**K APOTĀSANA**
_Pigeon Pose_
Imagining what it feels like to be a pigeon standing motionless on a city ledge with an enormous puffed chest and strong, straight legs can help as you move into Kapotāsana. It is a pose to work at slowly and with good form; for some practitioners, the full pose with the head resting on the feet may be a "next lifetime" event. Nonetheless, working the internal forms is accessible to everyone. Kapotāsana is a combination of flexibility and intelligent movement. Forcing the pose as if there is something to attain, or moving into a deep backbend by bending at the most flexible point in the spine, can have gratifying short-term results but usually ends in injury. Take time with this and other extreme poses.
1. Come to a kneeling position at the front of the mat with the legs about hip-width apart, the lower legs parallel, and the tops of the feet flat on the floor. Exhale to ground down through the legs and tone the pelvic floor.
2. Inhale, and begin to enter the pose as for Ūṣṭrāsana, extending through the core from the tip of the coccyx through the entire length of your upper body.
3. Inhaling, reach up through the torso, shoulders, and chest, then exhaling, lean back and begin to drop into a backbend with arms reaching up, back, and eventually toward the feet. Continue to follow in a smooth arc and extend through the spine from coccyx to crown. Gaze down the nose as you enter the pose. Soften the jaw, lift and broaden the kidney wings, and release the palate. Breathe smoothly.
4. Place the hands on the floor behind the toes or, eventually, on the heels. Bend the elbows and place the lower arms parallel to each other on the floor. Rest the top of the head on the soles of the feet as you continue gazing down the nose and breathing evenly. The smoother the breath in this pose—entering, exiting, and during—the more benefit you will get from the subtle internal actions.
5. After five breaths in the full form, move the hands toward the back of the mat, a few inches away from the toes and at shoulder width. Gradually straighten the arms, as if doing Ūrdhvā Dhanurāsana on your knees. Carefully protract the shoulder blades as the arms straighten. Continue to breathe smoothly, soften the tongue, and release the palate as you gaze down the nose. Hold this form for five breaths.
6. At the end of an exhalation, drop the awareness into the legs and pelvic floor by placing the hands on your hips. Inhaling, lift up out of the backbend to kneeling. Exhale, place your hands next to your knees, and jump back into Catvāri and a Half Vinyāsa.
**S UPTA VAJRĀSANA**
_Reclining Lightning Bolt Pose_
Though there are many ways to do this pose alone, it is helpful to have someone assisting to stabilize the legs. If you are practicing alone, you may try the advanced method by curling forward and then rocking back onto the elbows (place a blanket behind you where the elbows will fall), allowing the knees to lift away from the floor. Then extend back, lifting the heart as in Matsyāsana, the Fish Pose. To come up, reverse the process using a strong abdominal action to curl forward while still resting on the elbows. From there rock up to a seated position. If there is an assistant available and Padmāsana is not possible, sitting in Half Padmāsana or with the legs crossed is a viable option for completing the sequences (vinyāsas) of the pose.
1. From a seated position, come into Padmāsana with the left leg on top. Exhaling, reach behind the body leaning forward slightly to clasp the left big toe with the left hand. Inhale; on the next exhalation, again lean forward, wrap the right arm behind and under the left arm to grab the right big toe. Draw the shoulder blades down the back.
2. With an assistant to hold the thighs in and slightly down (allowing some room for movement in the lower body), inhale and lift the heart to form a backbend with deep lower neck extension. Exhale and drop the upper body back to place the top of the head on the floor as you gaze down the nose. Hold this position for five breaths. At the end of an exhalation, empty of breath, engage the pelvic floor and lower belly to initiate a lift in the upper body. Keep the head back and the chest puffed forward; return to a seated position on the inhalation.
3. Keep the hands on the feet, and with the following five breaths, drop back to touch the head to the floor on the exhalation and return to sitting on the following inhalation; leave the head back while gazing softly down the nose the entire time. On the fifth exhalation having dropped back, remain with the head on floor for five breaths (as in step 2), then return to sitting on an inhalation.
4. To exit, place the hands on the floor beside the legs, lift up, and jump back, unlacing the Padmāsana position to land in Catvāri. If you cannot do this lift yet, unlace the legs from Padmāsana and lift up, which continues to train the apānic pattern of curling and lifting. This will eventually lead to swinging and jumping back through into Catvāri.
**S ETU BANDHĀSANA**
_Bridge Pose_
Like Cakrāsana, Setu Bandhāsana can frighten the practitioner, who imagines the possibility of a broken neck for starters! But like in Cakrāsana, the neck is stable and safe when Setu Bandhāsana is practiced carefully and without excess tension in the neck and jaw. Using the legs intelligently and taking a certain amount of weight _forward_ into the legs as you roll back is essential. Also, it is important to move into the pose on the wave of the breath while gazing down the nose. When practiced in this manner, Setu Bandhāsana can actually have not only a comfortable but brilliant opening effect in the palate.
1. Lie on the back with the legs stretched straight out. Place the heels together with the feet turned out with medial edges of feet touching the mat. The toes should be pointing to the sides.
2. Using the hands under the buttocks, come up onto the elbows briefly and tilt the pelvis up to place the sitting bones on the floor as you would for Matsyāsana (the Fish Pose). As you do this, the knees will automatically bend out to the sides in line with the feet, and the feet will be drawn closer to the pelvis. Depending on the length of your legs, your feet should be approximately 12 to 30 inches from your buttocks.
3. Holding the outer thighs with the hands, inhale and lift the chest and draw back the head. Roll the shoulders back, lengthening the throat as you gaze down your nose. Use your elbows, so that as you exhale, you can bring the top of the head to the floor (eventually bring the center of the crown of the head to the floor) as in full Matsyāsana. Sink the sitting bones as you open the center of the heart. Breathe freely and continue to gaze down the nose. This position should feel wonderful in the front of the neck.
4. Set and activate the coccyx on an exhale to begin the lifting of the pelvis, pulling it slightly toward the feet. Then with an inhalation, press the soles of the feet toward the floor. Lift the hips and slowly roll from the crown of the head back toward the forehead. Keep the heart open, the throat released, and the eyes looking down the nose. Cross the arms across the chest. Straighten the legs and contract the buttocks firmly. Hold the pose for five breaths.
5. Exit, on an exhalation, using the strength in the legs to bring the hips back down to the floor into a Fish Pose. Use the elbows on the next exhalation to lift the head from the floor and to roll the spine onto the floor from the sacrum on up, so that the head is the last to come back to the floor. If you feel discomfort in the full, extended form, the form is incorrect. Carefully follow the coiling and countercoiling vinyāsa when entering or exiting this pose.
6. If you are a beginner, you may try a training position with your legs and/or arms when moving into the pose. For one version, bend the legs with the feet flat on the floor as in Setu Bandha Sarvāṅgāsana (Bridge Shoulderstand). Use the arms to position the head as for Matsyāsana (see Chapter 11), then straighten the arms along the sides and, inhaling, push through the legs to roll back on the head, gazing down the nose.
7. Alternatively, position the legs in the full form of Setu Bandhāsana with the arms along the sides of the body or reaching out to the sides as you initiate the roll. In this variation, you may eventually fold the arms over the chest, as in the full form, if the pose is strong and stable. Both variations use the arms as guides to draw the shoulders into their proper position and release the chest and throat. Note that the arms are rotated outward. If you have a neck injury or severe tension in the neck or shoulders, attempt only a Setu Bandhāsana variation.
**Ū RDHVĀ DHANURĀSANA**
_Upward Bow Pose_
Ūrdhvā Dhanurāsana is the pose most frequently associated with the classic backbend and rightly so. Many of us played around with backbends as kids, and it's a form that seems more "normal" than many other contorted yoga āsanas. It is also the perfect ground for learning the internal and physical forms common to all backbends—from shoulder movement and strength in the legs to activation of the pelvic floor, stretching of the psoas line, and riding the wave of the breath.
1. Lie on your back with the feet parallel to each other and drawn up near the outer edges of the buttocks. The feet should be slightly wider than hip-width apart and are kept parallel throughout the pose.
2. Place the hands on the floor by the ears, with the fingers pointing toward the shoulders. Exhale to establish a connection to the pelvic floor and Mūlabandha. On another exhalation, lift the sacrum off the floor, moving the ring of the pelvis (coccyx, pubic bone, and sitting bones) toward the knees, which are moving toward the front of the mat as the heels lift. This tones the hamstrings, and highlights the coccyx and the entire apānic pattern. Pause at the end of the exhalation on the crown of the head, with most of your weight remaining in the arms and legs and only a small amount on the head. Lift the heels and observe the feeling of protracting the shoulder blades while keeping the arms parallel to one another.
3. On the inhalation, lift the hips toward the ceiling, straightening the arms as you spread the lower back. Lengthen the belly and move the sacrum up and in toward the lower abdomen. Lower the heels to the floor if they are lifted. Take the head back completing the form only after everything else has rolled sequentially into place.
4. Spread the back surface of the body like a cobra's back. Relax the mouth and feel the skin on the back of the neck soften and then flow down the back as well as spreading up over the neck and forming a cobra hood over the head. Open, lift, and widen the fronts of the armpits and the shoulder blades, lengthening the psoas lines without limitation. Imagine that the coccyx is moving forward and slightly up as the pubic bone drops back and down as if to rest on the coccyx. Breathe freely.
5. The buttocks muscles may be contracted while initially entering the pose, but after a time and with practice, they can soften and release. Maintain a feeling of the sacrum moving in and up toward the navel as the coccyx moves away from the sacrum toward the pubic bone. Draw the backs of the thighs up into the legs to encourage Mūlabandha.
6. If you are a beginner and need motivation, gaze up between the eyebrows or at a point on the floor. When the pose is complete and easy, gaze down at the tip of or along the line of the nose.
7. Hold the full pose for at least five breaths. On an exhalation, lower carefully back down to the mat. To come down, first tuck the head, then place the shoulders down, unroll the spine flat on the floor, and finally inhale as you drop the sacrum to the floor. Rest briefly for one or two breaths and then repeat the pose. Each full Ūrdhvā Dhanurāsana in a series should become easier, more expressive, and more open.
8. After lowering to the floor on the final exhalation, placing the sacrum down last, rest in the residue of the pose for a few rounds of breath.
**D VI PĀDA VIPARĪTA DAṆḌĀSANA**
_Two-Footed Inverted Staff Pose_
This and the following two poses are extreme backbends found in the Advanced A Series, but they are also informative for less advanced practitioners working to refine subtle movements and extensions of the spine in all backbends.
1. Begin in Baddha Hasta Śīrṣāsana A (see Chapter 10). On an exhalation, carefully bring the feet down to the floor and come into a backbend. Inhaling, spread the kidney wings and wrap the shoulders as you ground through the elbows to lengthen through the crown of the head as you would in a well-aligned headstand.
2. Move the feet so that the legs can straighten. Be sure to move the sacrum in and up to keep the back comfortable. If possible, place the inner edges of the feet together on the floor and hold for five breaths, gazing softly along the line of the nose.
3. If the primary form of this pose is comfortable you may increase the stretch. On an inhalation, pushing the sacrum up and in, lift the head off the floor and walk the feet in closer to the hands. Keep the feet parallel to each other and shoulder-width apart. If you are flexible, hold the ankles. Keep the shoulder blades distinctly protracted the entire time.
4. To exit, in either form exhale and place the top of the head on the floor. Then with the kidney wings spread and the elbows grounded, rock and inhaling come back up into Headstand, then drop down into Catvāri and move through a Half Vinyāsa. If this is not possible, simply lower onto the back and roll out of the pose into Cakrāsana to take a Half Vinyāsa.
**E KA PĀDA VIPARĪTA DAṆḌĀSANA**
_One-Footed Inverted Staff Pose_
1. Traditionally you come directly into this pose following step 2 of Dvi Pāda Viparīta Daṇḍāsana. Alternatively, you can drop into it as for step 1 of that pose.
2. In either case, leave the left foot on the floor and take the right leg up toward the ceiling, pointing the toes. Keep the hips level and the lower back wide. Eventually the left leg will remain straight with the mound of the big toe firmly grounded. Hold this pose for five breaths.
3. Lower the right leg and lift the left, holding this side for five breaths as well. Lower the left leg and exit the pose as for Dvi Pāda Viparīta Daṇḍāsana.
4. If flexibility allows you may walk the feet in close to the hands. Center one foot and, inhaling, lift the other leg straight up to the ceiling, pointing the toes. Hold for five breaths. Once you have completed the second side, walk both feet out away from the head and exit the pose as for Dvi Pāda Viparīta Daṇḍāsana.
**E KA PĀDA RĀJA KAPOTĀSANA**
_One-Footed King Pigeon Pose_
Similar to Kapotāsana in feel, this more extreme version is an excellent method of exploring extension and flexion—without impingement—along the entire spine while working into backbends.
1. From Downward Facing Dog Pose, on an exhale jump forward bending the left leg to bring it forward and across the midline as you sit down, stretching the right leg out in back of you. Bring the right psoas button toward the left heel.
2. Pick up the right leg, and as you exhale, reach around with the right arm (palm up) and clasp the mound of your right big toe from the outside edge of the foot. On a deep, smooth inhalation, bring the right elbow and right kidney area forward and up as you drop the outer right hip toward the floor. This will rotate the right hand so the palm is turned down, and the chest will turn forward toward the front of the mat.
3. Lift the left arm and take the right foot (and eventually the right ankle) with both hands. Bring the right foot up, until eventually the sole meets the top of the head (or the heel may come to the eyebrows if you are extraordinarily flexible). Bring the elbows toward each other. Spread and lift the kidney wings as the heart continues to float up. Hold this for five breaths. On an exhalation, release the foot, place the hands down next to the hips, step back into Catvāri, and move through a Half Vinyāsa for the other side.
4. When first clasping the foot, resist the action of the arms by using the quadriceps to push back through the foot into the hand. This will further lengthen the front of the body and will eliminate compression in the lower back, helping to open the thoracic spine.
_9_
Twists
COMPENSATING FOR OR BALANCING ASYMMETRIES in the body and breath comprises much of the work of yoga āsana and prāṇāyāma. With practice, we may begin to notice and track these imbalances as they relate to all poses, but twisting poses easily expose asymmetries in the legs, hip joints, SI joints, spine, rib cage, and neck. Twists also reveal a dramatic interplay of actions and counteractions throughout the body. Indeed, the appropriate use of counteractions and counterrotations is the key to the alignment, form, and benefit of twisting poses.
The basic movement and breathing patterns entering, exiting, and during all twists is similar. Before you even begin, you exhale to tone the pelvic floor creating a sense of being grounded—usually through the feet or, in seated twists, through the sitting bones. This grounding contact with the foundation of the pose gives space within the body, mind, and nervous system to pause for an instant and then really feel the full movement pattern on the inhalation that follows. These first two breathing patterns are vital "setup steps" to the actual pose.
Exhaling establishes evenness, stability, and full lines of the apāna pattern of the twist, often using the abdominal wall. Inhaling brings in counteractions: extension of the torso and spine as well as a lifting of the heart and more awareness of the palate and pelvic floor connection. The inhalation wakes up the seeds in the nervous system of all the counterspins, rotations, pushes, and pulls for the primary actions of the particular twist. Having set up these two important patterns, we move into the actual twisting pose on the next exhalation. The following inhalation switches on or "lights up" the full pose and in theory pulls the central thread of the pelvic floor up into the body. This introduces all of the counteractions to the primary actions without the primary actions disappearing. Then they work together, churning out the nectar of integrated prāṇa and apāna.
In many twists, there are important counteractions to level the hips or minimize lateral flexion/extension (depending on the twist). These counteractions are an important means of identifying, balancing, and correcting bad postural habits. In the fine-tuning of twists, the side-to-side work across the midline is asymmetrical and creates rotations in opposite directions—like unscrewing two halves of a sphere or moving as if opposite ends of the body were opposing wheels in a system of gears that rotate in complementary yet opposite directions.
Most twisting occurs in the upper lumbar and lower thoracic sections of the spine (the junction of T12 and L1) and in the neck at the atlanto-axial joint (the junction of C1 and C2). There is very little twisting in the upper thoracic spine and the lower part of the neck. The SI joint may be injured by improperly aligned twisting, although micro-twisting in the SI joint _is_ allowed with good communication of complementary rotations side to side and through the pelvic floor.
The most common forms of twists are flexion twists, like Paśāsana (Noose Pose). Others, like Trikoṇāsana (Triangle Pose), are referred to as extension twists. Less common are spiraling extension twists, such as Parighāsana (Hinge Pose). We initiate movement into flexion twists on the exhalation, working the abdominal muscles to keep the lumbar spine slightly flexed and the coccyx curled. This is the apāna pattern. After initially getting into any flexion twist, the next inhalation brings in a slight amount of counterextension through the spine. This can be complemented on the following exhalation with a bit more flexion and on the next inhalation with additional extension.
As in flexion twists, extension twists are entered on the exhalation. The PC muscle is toned, the coccyx is awakened, and the spine is relatively straight when entering the pose. Extension twists allow for more length in the whole spine, particularly in the lumbar region and the neck. They do not involve as strong an initial compression of the abdominal muscles as in flexion twists, and they allow for more space between the vertebral bodies along the spine.
A flexion twist has more of a compressive effect on abdominal organs because it shortens the abdominal muscles, which may distort the positioning of the stomach, liver, and intestines in the abdominal cavity. Therefore it is easier and advisable to practice twists, especially flexion twists, when the bowels and bladder are not full. An extension twist has a more dramatic effect on the relationship of the diaphragm and the abdominal organs. The ribs and abdomen are less compressed and allow fuller motion in the diaphragm. It is usually easier to breathe in extension twists, and they are often pleasant to hold for longer periods of time or to use in more therapeutic contexts.
In all twists, there should be a feeling of dignity in the final pose—as though you are growing longer up through the center of the spine and out through the crown of the head. Whatever form the final twist takes, it is really important not to lock the pelvis and sitting bones in place either when entering the pose or when continuing to make subtle microadjustments while pausing in it. Also important is a strong awareness of the hips and pelvic floor. When you are first learning some new and "gnarly" twists, such as Marīchyāsana D, there may be a certain amount of strain getting into the pose, but this should be minimized by smoothing out the ends of the breath and not pushing or pulling yourself into the pose from an external perspective. It is sometimes difficult for beginners to breathe fully in extreme twists, so a feeling of anxiety may arise. In this case, it is important to soften the gaze and the tongue and sometimes even to back out of the twist a tiny bit in order to calm the mind. Grimacing, clenching the teeth, and grunting are usually methods of forcing a situation and ignoring an awareness of pain, but they don't help the pose unfold.
A general benefit of twists is that they can aid the "fire of digestion," helping the movement of the gastrointestinal track and possibly the function of the liver and stomach. This is mostly mechanics—the organs in the abdomen are moved around as you twist your torso, and this can help to get things moving! Another benefit of twisting is that you can distinctly feel the side-to-side imbalances around the hip joints, pelvic floor, abdominal wall, shoulders, and neck. Consequently, twisting can give you a real handle on how to work with and lessen these individual imbalances.
Commonsense contraindications for twists are herniated disks (especially those that have been recently herniated); a fractured spine; broken ribs; and swelling, such as with abdominal tumors. Spinal irregularities such as scoliosis require the practitioner to work slowly and carefully so as not to exacerbate the problem. Those with inguinal and other hernias should use caution with or avoid twisting in certain dimensions as well. It is also contraindicated to practice extreme twists during pregnancy (though open twists in which the belly is not compressed are usually alright). In all of these cases, or if you experience difficulty or pain when twisting, you should seek the advice of a highly experienced yoga teacher.
**M ARĪCHYĀSANA C**
This simple twist from the Primary Series is wonderful in that it can be done to some degree by almost everyone. It is easy to discover the main actions that underlie all twists and to really _feel_ counteractions deep in the body that are reminiscent of churning butter.
1. From Downward Facing Dog Pose, at the end of an exhale jump through to a seated position inhaling as you land. You may land with the right foot in place on the floor or with both legs stretched straight out in front. In the latter case, bend the right knee and place the right heel near the right hip and buttock, about a hand-width from the inner left thigh.
2. With the spine straight, place the right hand on the floor behind you and lean back to turn the belly to the right. This will initially turn the pelvis to the right, giving the feeling that the left leg is lengthening in an inward spiral.
3. Reach up with the left arm while inhaling, then exhale and wrap the arm around the outside of the right leg or simply hook the left elbow or hand on the outer right knee. If the arm is wrapped, reach behind the back with the right arm and clasp the left wrist.
4. The next inhalation "activates" the pose, bringing in all of the movement components and counteractions. The outer right hip and right sitting bone come back down toward the floor and slightly forward. This brings the pelvis somewhat back into a countertwist, which seems to shorten the left leg with a subtle outward rotation. Here it is possible to play with and listen to the dialogue between prāṇa and apāna while toning the pelvic floor.
5. Turn the head in the direction of the twist (over the right shoulder), and select an easy gazing point on the horizon. Notice transformations that occur in the body, mind, and breath over the course of five breaths.
6. To exit the pose, first exhale fully to deposit some tone in the pelvic floor. Inhale releasing the arms and turn the upper body toward the front. Exhale to settle the pelvis steadily onto the mat and square the body to the front, then proceed through the appropriate vinyāsa to repeat the same pattern on the other side.
**M ARĪCHYĀSANA D**
This is just like Marīchyāsana C, except that one of the legs is in Padmāsana. One small detail like that can make a seemingly accessible pose impossible! But even if the pose is difficult, don't give up. Work steadily and carefully just to the edge of your ability, and over time the pose will evolve.
1. From a seated position with the legs out in front of you, bend the left leg into a deep Half Padmāsana position with the heel near the psoas button area of the right lower belly and the ankle at the crease in the right groin.
2. Bend the right knee, placing the foot flat on the floor up toward the right sitting bone. Rotate the hips to the right as in Marīchyāsana C, pressing forward and down with the left knee.
3. Inhale as you lean back and slightly to the right to move the belly up out of the way and to the right. It is helpful on the next exhalation to rest the upper body on the right arm as you do this. On the following inhalation, reach up high with the left arm. Then with the top of the arm bone moving forward, exhale, flex the lumbar spine slightly as if lengthening the coccyx along the floor. Continue to wrap the arm as low around the right leg as possible. Reach behind the back with the right hand. Clasp the right wrist with the left hand and, if possible, hold the left ankle with the fingers of the right hand.
4. To activate the pose, inhale and pull in opposite directions with the arms so there is a sense of the heart spreading and lifting and a feeling of spiraling up through the crown of the head as you sink into the pose. Play with the counteraction, which brings the right sitting bone back down and forward toward the floor. Find a steady gazing point on the horizon, and keep the tongue silent and still.
5. Hold the twist for five breaths. At the end of an exhalation, ground into the base of the pose, and on the next inhalation, release the arms and turn the upper body to the front. Continue to release the arms, grounding into the pelvic floor on the exhale. After going through the appropriate vinyāsas, practice the posture on the second side.
6. If you cannot bind the arms, catch the right knee with the left elbow or hold the knee with the palm of the left hand. Press the elbow/ hand against the knee, and push back into the elbow/hand with the knee. You may also use a strap to connect the hands behind you. In form D, if Padmāsana is not possible, you may fold the left foot in front of the body and place it on the floor with the heel in front of the right sitting bone. You can then put the right foot in front of the left ankle to work the pose from there.
**P ARIGHĀSANA**
_Hinge Pose_
This twist from the end of the Intermediate Series allows a radical lateral spinning, spiraling movement in the lower spine and pelvis. It profoundly impacts the hip joints, SI joints, abdominal wall, and diaphragm.
1. Begin by floating through from Downward Facing Dog Pose or simply sitting and folding the right leg into Half Vīrāsana. Open the left leg out to the side so the thighs form a little more than a 90-degree angle. Sit with the pelvis and upper body turned out to face the folded leg.
2. Inhaling, reach up high with the left arm, releasing the left psoas line. Exhaling, reach straight out and forward with the left arm on the floor, palm up. Next slide the left arm and shoulder down along the floor to create the initial twist and lateral flexion.
3. Inhaling, reach up toward the ceiling with the right arm, then bend the elbow as you continue to stretch the arm and the right side in order to take the outer edge of the left foot with the right hand (or secure a connection between the hand and foot using a strap) with the elbow pointing up toward the ceiling. Then, if possible, bring the left hand to the left foot. Eventually the back of the head can rest on the left shin.
4. Gradually introduce a slight outward rotation to the right femur. This will bring the right sitting bone back toward the floor. Do this just the right amount to create Mūlabandha, remembering that the inward rotation of the right femur is the primary, leading action that takes your torso out over the straightened leg and that deeply and correctly lengthens the left hamstrings and adductors.
5. If you are flexible, the gazing point is the left big toe as the back of the head rests on the left shin. If you are stiffer, gaze along the line of the nose while smiling and breathing kindness into the whole-body patterns as they manifest. Hold the pose for five breaths.
6. Come out of the pose by releasing the hands on an exhalation and sitting up on an inhalation, then place the hands on the upper rims of the pelvis to ground the sitting bones equally for one breath. This provides a unique space for digesting the residue of this radical pose.
7. Move through a Half Vinyāsa to get back into the central axis, then float through to begin the pose on the other side.
**P AŚĀSANA**
_Noose Pose_
In Indian mythology, Gaṇeśa's vehicle was once Krauñcha (literally "crane"), a celestial musician who was cursed and turned into a huge mouse or rat. He was tamed when Gaṇeśa sat on him after throwing a noose around his neck. (Krauñcha really _wanted_ to be tamed, so he was delighted.) Paśāsana is the embodiment of this story; while squatting, the folded legs feel a like huge mouse or rat. Summoning the combination of forward bending with the action of an oblique stretch, we can throw our arms around and "lasso" our own legs, summarizing the Primary Series, to bring both the image of the mouse and the apānic residue from the series together.
1. From Downward Facing Dog Pose, at the end of an exhalation, hop to the front of the mat, landing with the feet together and sitting down in a squatting position. If it is difficult to squat with the heels flat on the floor, you may roll the mat slightly to provide support.
2. On an inhalation, lean back onto the left hand and reach up with the right arm as you turn the upper body to the left. Exhaling deeply, reach around the outside of the left thigh with the upper right arm, bending the elbow to bring the lower part of the arm near the left shin and up behind the back. Reach behind you with the left arm to clasp the hands together or hold the left wrist with the right hand.
3. Now turn on the juice in the pose. Inhaling, push down through the feet as if to stand up, but at the same time, keep a sense of the sitting bones being heavy, so you do not actually stand but instead activate the legs. Move the left sitting bone back down toward the left ankle to create a countertwisting of the pelvis.
4. Roll the top of the left arm back, sliding the shoulder blade down the back as the sternum lifts. Turn the head to the left, keeping the chin neutral, and gaze at a point on the wall over the left shoulder.
5. Pull the arms in opposite directions, as if to break the clasp of the hands, but keep them clasped so the pose comes to life. Pull the hands up the back as if to squeeze the rat. Keep the gaze steady and the breath smooth and even. Hold this for five breaths.
6. At the end of an exhalation, bring the awareness to the tone in the pelvic floor. Keeping this awareness, release the hands and return the upper body to center on the inhalation. Exhale to reawaken the feet and pelvic floor, then switch sides twisting to the right. After five breaths on the second side, exit the pose by setting the tone in the pelvic floor on an exhale before jumping back into Catvāri.
7. Two variations on full Paśāsana can be used when you are learning the pose or are using it therapeutically. If squatting is not possible, sit on the heels as in Vajrāsana. Inhale and reach up with the right arm to release the psoas line. Then exhaling deeply, reach across the thighs, bringing the right shoulder to the outer left knee and the right elbow to the floor near the left shin or ankle. Inhaling, reach up toward the ceiling with the left arm, then bend it at the elbow to drop the left hand behind the back to take the inner right thigh with the left fingers. This is a flexion twist. Work with the counteractions of pushing the right elbow into the left thigh and pulling the left hand back and up keeping the head away from the floor, extending and spiraling long through the spine.
8. If you are pregnant or for a variation that works for most, squat with the feet and knees about hip-width apart. Inhale and reach up with the right arm, then exhale and reach forward to wrap the right arm back around only the right leg. Clasp the hands behind the back, and work the pose intelligently as you would in the full form. Release after five breaths and repeat on the second side.
**B HARADVĀJĀSANA**
_Bharadhvāja's Pose_
This pose, named after the sage Bharadhvāja, appears in two similar forms within the Intermediate Series. The first Bharadvājāsana appears midway through the series, just before beginning poses with the legs behind the head. The second form appears at the very end of the series and is called Supta Ūrdhvā Pāda Vajrāsana.
1. From Downward Facing Dog Pose after an exit, slide through to a seated position and inhale with the left leg folded in Vīrāsana and the right leg stretched straight out in front. Angle the left knee out to the side and draw the right foot back toward the lower belly to put the right leg in Padmāsana as you exhale.
2. Spin the torso to the right, elongating the spine as you reach up through the left arm and fingertips, rotating the arm and protracting the shoulder blades. Exhaling, place the left palm on the floor under the right knee, with the fingers pointing toward the midline of the body. Wrap the right arm behind the back to clasp the right big toe or foot.
3. Activate the twist by engaging the arms, pushing down and forward through the left arm and hand and bending the right elbow to pull the right foot as the top of the right arm rolls back. Turn the head to the right, keeping the tops of the shoulders broad and dropping away from the ears as you find a steady gazing point on the horizon over the right shoulder.
4. Bring the awareness into the midline of the body and the sense of the sternum lifting, the kidney wings broadening and lifting, and the palate releasing. Experiment with the countertwist of the pelvis by bringing the left sitting bone back down toward the floor. The rotations of the femurs at the hip joints shift with different degrees of counteraction. Notice the effect of this on the pelvic floor. Hold this position for five breaths.
5. At the end of an exhalation, tone the pelvic floor, release the arms, and as you inhale, turn back to center. Cross the legs, place the hands next to the hips, lift up and jump back to Catvāri for a Half Vinyāsa. Then glide through and do the pose on the other side.
**S UPTA ŪRDHVĀ PĀDA VAJRĀSANA**
_Reclining Feet Up Thunderbolt Pose_
Similar in final form to Bharadvājāsana, the entrance to this pose is quite different as is the effect of the final form on the pelvic floor. It's an interesting exercise to practice the form with wide knees (Bharadvājāsana) and then immediately after with the knees close together (Supta Ūrdhvā Pāda Vajrāsana) to tune in to the subtle differences.
1. Lie on the back with the legs stretched out straight along the floor arms along the sides of the body, palms down as in Tāḍāgī Mudrā. At the end of an exhalation, begin to lift the legs and hips and at about a 30-degree lift from the floor inhale to continue the lift, until feet are straight overhead, then exhale to place the feet on the floor above your head as in Halāsana.
2. Still in Halāsana position, fold the right leg into Padmāsana and wrap the right arm behind the back to clasp the right big toe. Gaze down the line of your nose, and hold this position for five breaths.
3. On a well-defined exhalation, still in Halāsana position, fold the left leg into Vīrāsana, grabbing the left big toe or the side of the foot with your left hand. On the next inhalation, rock up to a seated position, keeping the hold on the feet as you roll up. The knees should be about 4 to 6 inches apart, and you should still be holding both feet with the hands.
4. Exhaling, release the left hand from the foot and place the left palm on the floor under the right knee, with the fingers pointing toward the midline of the body. Wrap the right arm behind your back to clasp the right big toe or foot.
5. Activate the twisting action by engaging the arms, pushing down and forward through the heel of the left hand and bending the right elbow to pull the right foot as the top of the right arm rolls back. It is similar to shooting a bow and arrow behind your back. Turn the head to the right, and gaze at a point on the horizon over the right shoulder. Hold this position for five breaths.
6. To exit, bring tone back to the pelvic floor, release the arms, and turn back to center. Place the hands next to the knees and jump back, or unravel the legs, cross them, lift up, and jump back to Catvāri for a Half Vinyāsa. Next jump through and lie down to repeat the pose on the other side.
**A RDHA MATSYENDRĀSANA**
_Half Spinal Twist Pose_
This extended spine twist appears in two forms within the Intermediate and Advanced A series. The half twist, in the Intermediate Series, is accessible and informative as an exploratory pose, even for those who do not practice Aṣṭāṅga Vinyāsa. It can teach us how, when breathing fully, the pose is refined as an endless subtle conversation between prāṇic and apānic patterns, and between our different theories and techniques. Close attentiveness to these conversations can give us insight into the nature of reality as well as our biases, beliefs, and emotional conditioning.
1. From Downward Facing Dog Pose, at the end of an exhalation, jump forward and inhale just as you land in a seated position. As you exhale, turn the pelvis to the right and bring the left heel back toward the sitting bones, pointing the left knee toward the front of the mat. Place the right foot on the floor just inside the left knee, with the toes pointing forward and the entire sole on the floor.
2. Inhaling, reach up with the left hand, rotating the arm and protracting the shoulder blade to stretch and open the psoas line. Exhaling completely, reach around the right leg, catching the upper part of the arm on the outer knee. Take the inner edge of the right foot with the left hand. If you are a beginner, you might need to exhale again to move fully into the twist. Wrap the right arm behind the body and clasp the inner left thigh.
3. Inhaling, turn on the pose by feeling the coccyx wake up to flow in toward the pubic bone. Bring the outer right hip and sitting bone back down and in toward the floor, moving them gently toward the front of the mat in a slight countertwist. Push the right leg back against the left arm and resist this push through the upper arm.
4. The arms stretch and pull in opposite directions both to open the heart and to extend the tail (as in holding the tail of a serpent). The back-and-forth conversation frees the head of any extraneous tension so that it floats and can inspire a feeling of nobility. (Any and all of the Internal Forms practices and visualizations can be used as we align the āsanas.) Find a steady gazing point over the right shoulder, and breathing smoothly, hold the pose for five breaths.
5. Before exiting, exhale to consolidate the actions, counteractions, thoughts, feelings, sensations, and the overall effect of the pose that can be sensed into at the root in the center of the pelvic floor. Then on an inhalation, release, and unwrap the form to return to a symmetrical seated position.
6. Cross the legs and place the hands beside the hips. Inhale as you lift up, and exhale back into Catvāri and complete a Half Vinyāsa to do the other side or simply switch sides.
**P ŪRṆA MATSYENDRĀSANA**
_Full Spinal Twist Pose_
This pose is similar to Ardha Matsyendrāsana, except the bottom leg is in Padmāsana rather than on the floor.
1. From Downward Facing Dog Pose, at the end of an exhalation, jump forward and inhale just as you land in a seated position. Inhale again as you draw the left leg into Padmāsana, dropping the knee to the floor and toward the middle of the mat with a slight internal spin of the left femur. Place the left heel down in the area of the right psoas button.
2. Place the right foot on the floor just inside the left knee, with the toes pointing toward the front of the mat and the entire sole on the floor.
3. Inhaling, reach up with the left hand, rotating the arm and protracting the shoulder blade to stretch and open the psoas line. Exhaling, curl the spine and slide the left arm outside the right thigh, wrapping the lower arm along the edge of the right shin so you can take the right foot at the inner edge of the mound of the big toe with the fingers of the left hand. Exhaling again, wrap the right arm behind the body and clasp the inner left thigh.
4. Inhaling, resist the hold with each hand so that the spine is invited to twist and extend as you lift the core of your heart.
5. Keeping the head and neck neutral, turn the head to the right and gaze softly at a point on the horizon beyond the right shoulder. Hold the pose for five breaths.
6. Exhale and bring the awareness to the toning of the pelvic floor. Maintaining that awareness, return to center on an inhalation.
7. Cross the legs and place the hands on the floor next to the hips. Inhale as you lift up, and exhale back into Catvāri for a Half Vinyāsa before repeating the pose on the other side.
_10_
Balancing Poses
IMAGINE YOU'RE STANDING IN SAMASTHITIḤ ON TOP of a flagpole, finding whatever it takes within your body and mind to keep you upright and stable—even when the wind picks up. That's the feeling we're after in all balancing poses, from simple standing poses like Samasthitiḥ with the feet together (which is challenging enough and could be considered the quintessential balancing pose) to more complicated poses such as Utthita Hasta Pādāṅguṣṭhāsana.
Balance is mostly unconscious and automatic, the result of a fine dialogue between opposing patterns of movement, alignment, and thought within the body and mind. Balancing embodies the awakening of a focused and refined intelligence. In successful balancing poses, the mind is open to a whole background of sensations, yet it is intensely focused on a seed pattern of sensation within the body that is a synthesis of kinesthetic relationships. It is never just smooth breathing, a steady gaze, or intelligent legs that are required for balance; it's always an open, focused, multifaceted, and adaptive conversation between many aspects of body and mind. If we concentrate too strongly on lower levels of technique, like a particular muscular line in the body, then balance becomes difficult and usually fails. If the mind is scattered, balancing is impossible! In balancing, the attention is focused, but it is difficult to say exactly on what the attention is focused.
In yoga, when we think of balancing poses, extreme forms such as a one-armed handstand may come to mind. Yet even these more demanding types of poses function in the same way as Samasthitiḥ, requiring deep and open focus. Of course, they usually also involve a specialized and intense awareness of particular patterns within the body, but their foundation is the same. Even in Samasthitiḥ, which could be considered the "beginner's" balancing pose, there is a constant flowing of synthesized techniques into the plumb line. When practiced in this way, Samasthitiḥ comes to life as vividly as any of the more intense balancing poses; it practically becomes an adoration of the plumb line by countless subsystems. Sri K. Pattabhi Jois used to say, "Correct Samasthitiḥ is very difficult." Perhaps this subtle level of awareness is what he was referring to.
The logical next step in understanding balancing poses is to turn Samasthitiḥ over into a headstand. The basic requirements are more de-manding, and the attention needed is more intense, as is the necessity of integrating the opposing actions of prāṇa and apāna for a successful and useful headstand. However, the basics of open-mindedness and clear focus are the same. As we increase the difficulty of the balance in even more complex poses, we work toward āsana in which legs veer off in opposite directions or in which we balance on just one limb; all of the same integrating principles and patterns apply, just with increasing difficulty. Good balance is the perfection of the whole mental process of finding a pattern on which to focus and then opening the background of that pattern. This ability is of particular significance in yoga.
**D AṆḌĀSANA**
_Staff Pose_
Daṇḍāsana seems almost like a "nonpose," one that is difficult to put into a family because it's just sitting, not doing much of anything. Looked at closely, however, it is revealed to be a pose in which we are doing a great deal and becomes the epitome of what is meant by joining together opposite patterns of form, breath, movement, and mind—a tribute to balancing on the plumb line.
1. From Downward Facing Dog Pose, at the end of an exhalation jump through to a seated position at the front of the mat. Stretch the legs straight out, with the feet together and the sides of the big toes touching. Square the feet and press the inner edges of the feet forward and out. Draw the upper, inner backs of the knees and the skin of the inner thighs toward the floor. The heels may come up off the floor slightly, but do not hyperextend the knees to achieve this.
2. Place the hands behind the back at the outer edges of the buttocks, with the fingers pointing toward the front of the mat. The elbows should not be locked; you may come up onto your fingertips to bend the elbows and rotate the tops of the arms up, back, and down while spreading the shoulder blades, as if shrugging.
3. On an inhalation, begin to place the chin in Jālandhara Bandha position (see Chapter 1), extending through the neck, spreading the skin on the back of the neck, and releasing the tongue and palate as you reach up through the center of the head. Exhale and nestle the chin down into the sternum.
4. Pull the sitting bones back and then ground them by imagining the kidney wings broadening and spreading up the back. Draw the lower belly, about two inches below the navel, into Uḍḍīyāna Bandha. Lift the front of the spine and broaden the back of the head, neck, and entire body as you hold the pose.
5. Lift the heart, as if that entire area is light and buoyant. It may be helpful to imagine that the chest is that of a swan, rounded and puffed, so the chin can settle easily into the core of the heart. Gaze along the line of the nose with a soft smile that opens the back of the palate. Hold this position for five breaths.
**H ANUMANĀSANA**
_Hanuman's Pose_
Like Daṇḍāsana, this classic split is a refined internalized pose that epitomizes the process of making tiny adjustments to stay balanced.
1. From Downward Facing Dog Pose, step the right leg forward between the arms and as you exhale slide the heel along the floor out in front of you, squaring the foot and reaching long through the leg. Breathe smoothly and over the course of one or several breaths, lower down so the right sitting bone rests on or close to the floor, reaching out and back through the left leg, with the toes pointed. Keep both legs activated. Do not collapse into the hip joints, and do not attempt to square the hips to the front.
2. Keep the upper body straight, the heart lifted, and the pelvic floor awake. If you cannot come into a full split, keep the hands along the side and support the weight, gradually, over several weeks or months, gently working the pose to eventually be able to sit fully down into it. When in the full position, place the hands on the hips for five breaths, gazing along the line of the nose.
3. Inhaling, reach the arms above the head, palms together, as in Ekam, and gaze at the thumbs for five breaths before releasing the hands down to the sides.
4. On an exhalation, fold forward over the right leg, and clasp the left wrist with the right hand beyond the sole of the foot, again holding for five breaths and gazing at the toes.
5. Inhaling, return to an upright position and fold the palms in front of the heart in Añjali Mudrā for five more breaths. Place the hands on the floor by the hips, lift up and step back into Catvāri, then move through a Half Vinyāsa before repeating the pose with all of the hand positions on the other side.
**B HUJAPĪḌĀSANA**
_Arm Squeezing Pose_
1. From Downward Facing Dog Pose, exhale and hop forward, landing with the feet on the floor in front of the shoulders and the legs wrapped around the outsides of the arms. Breathing smoothly, work the inner knees up as high as possible on the arms and, while bending the knees and leaning back slightly, inhale and lift the feet off the ground.
2. Cross the right ankle over the left and squeeze the heels in toward the sitting bones. Lift the core of the heart toward the ceiling, lift the chin slightly, and gaze between the eyebrows or at the horizon. Hold the pose for five breaths.
3. An alternative is to hook the inner knees around the upper arms and lean back to balance without crossing the feet. This is a way of working toward the full form of the pose and is still effective. Do not expect to do poses like this without a lot of practice!
4. On an exhalation, pull back the sitting bones and heels to curl as you slowly rock forward. Place the top of the head on the floor. Alternatively, you may bend the elbows deeply to place the chin on the floor. In this case, your spine is not as flexed, but the center of gravity is farther back. Both variations of this stage of the pose are beneficial. In both forms the hands and the crown of your head or your chin should form an equilateral triangle.
5. In this form pull the shoulders high and wide. Make the arms parallel to each other. Draw the feet off the floor, squeezing them up and in toward the sitting bones. Gaze at the tip of the nose. Hold for five breaths.
6. Exhale and, empty of breath, carefully lean back to lift the head. As the head ascends, inhale and gaze up between the eyebrows. Exit by separating the feet and straightening the legs out to the sides in Ṭiṭṭibhāsana (see pages 258–60) for one breath. On the exhalation, draw the knees back and straighten the arms in Bakāsana and jump back into Catvāri. You may also jump back into Catvāri directly from Ṭiṭṭibhāsana.
BAKĀSANA (CRANE POSE) A AND B
This pose is the embodiment of the apāna pattern. Like other poses named after animals, imagining yourself to be a crane with long, straight legs and a broad, round upper body can be helpful here. In the full form, the knees rest on the outer edges of the straightened arms, the feet are together and flexed, and the heels are drawn up toward the sitting bones.
**B AKĀSANA A**
1. From Downward Facing Dog Pose, hop forward at the end of an exhalation, then inhale as you land in a squatting position with the feet about 6 or 8 inches from the wrists.
2. Exhaling, lean forward over the hands as you place the knees on the outer edges of the upper arms. Drop the hips and push into the floor through the arms to connect them strongly to the sides of the body; puff out the area of the kidneys by toning the pelvic floor, curling the coccyx, and broadening the low back. Push your shoulders down away from the ears and let your feet begin to lift up away from the floor.
3. Inhaling, lift the head, heart, hips, and feet working the arms to eventually straighten them. Keep the feet flexed and square, with the inner edges touching as you squeeze the heels up toward the sitting bones. Squeeze the knees into the arms and simultaneously push out through the arms to resist the squeeze, drawing strength and awareness into the lower belly and pelvic floor.
4. Gaze along the line of the nose to a point out in front of you on the floor, and hold the pose for five breaths. Take a full inhalation, then on the next exhale jump back into Catvāri. Move through a Half Vinyāsa and on to form B.
**B AKĀSANA B**
1. From Downward Facing Dog Pose, hop forward on an exhalation as for Bakāsana A, but instead of landing in a squat, bend the knees midflight and land (on the arms) lightly with the knees on the outer edges of the upper arms. Having landed in the full form of the posture at the end of an exhalation, you can then inhale to pull the heels even closer to the sitting bones. Imagining yourself to be a small bird landing in a tree (the tree being your arms) and looking forward optimistically can help when learning to float into this form.
2. Once in the pose, follow steps 3 and 4 for form Bakāsana A, holding the pose for five breaths as for form A, before jumping back into Catvāri and moving through a Half Vinyāsa into the next pose.
**P IÑCHA MAYŪRĀSANA**
_Peacock Feather Pose_
This pose is excellent training for the important shoulder action of connecting to the serratus anterior muscles while protracting and stabilizing the shoulders. This action supports many poses, from Piñcha Mayūrāsana itself to Śirṣāsana and even Catvāri.
1. From Downward Facing Dog Pose, step forward and bend the elbows to bring your forearms down on the mat, with the elbows directly beneath the shoulders and the palms facing down. The upper arms should be parallel to each other as should the lower arms, so they form a square or box that can support the body evenly. Roll the shoulders back and around to activate the serratus anterior muscles to spread the kidney wings. Lift the hips away from the floor so you are supporting the pose with the arms and slowly step forward bringing the hips closer to the hands. The wrists, elbows, forearms, and the inner edges of the hands (the heel of the thumb and the index finger) root into the earth. Lowering the hips slightly and bending the knees can facilitate the stepping forward movement, especially while you are learning the pose.
2. As you move closer to the hands with the upper body, keep the 90-degree angle (or just a little past that) in the elbows. Lift the hips toward the ceiling and root again into the sense in the outer shoulder blades of protracting (wrapping around the sides of the body) as you ground thoroughly through the elbows and wrists. Feel the kidney wing power in the elbows.
3. Exhale to set the base of the pose, then inhale to step or kick up one leg at a time. Keep the serratus muscles and protraction of the shoulders engaged and pressed back, maintaining the 90-degree bend in the elbows. If the elbows bend less than 90 degrees, the posture is more difficult and unstable. Engage the legs, squeezing them together and reaching up toward the ceiling through the inner edges of the half-flexed feet. Feel the coccyx lifting up toward the ceiling, as the arms work to press against the floor. Imagine that you are holding the central axis between the legs as if it were a long, thin straw. Lift the head slightly to gaze at a point between the wrists or thumbs. Keep the tongue soft and the palate released. Hold the position for five breaths.
4. To exit, on an exhalation, step back down and move into Catvāri. Alternatively, after completing the breath cycle, inhale and quickly, with a small push upward, move the hands next to the heart as you snap down into Catvāri. This quick exit takes practice. Move through a Half Vinyāsa into the next pose.
**K ARANDAVĀSANA**
_Crane Pose_
This pose is not easy (which is a major understatement). For many practitioners, it is one that may not be part of the menu this go-round. But even if dropping into the full form and then lifting back up doesn't quite happen, the actions we work on to prepare for and eventually do these movements are very informative and integrating.
1. From Downward Facing Dog Pose, step forward and place the elbows on the floor and follow steps 1 through 3 as if to set up and move into Piñcha Mayūrāsana. Once balanced in the pose, you may take a few extra breaths or immediately move into Karandavāsana.
2. On an exhalation, cross the legs in Padmāsana. The key to this is that once the right leg is in Padmāsana position, extend that hip joint by reaching up toward the ceiling and slightly back through the right knee. This way, the folded right leg is not in the way of the left as it moves into Padmāsana.
3. With a firm and complete exhalation, curl the legs and lower torso to bring the tops of the shins in to rest on the middle back portion of the upper arms. This exhalation must be smooth and audible to go with the careful, firm contraction of the rectus abdominis and a strong sense of puffing the kidney area while strongly protracting the shoulders and maintaining a sense of pushing down through the elbows and wrists. At no point while in the pose should you release the abdominal contraction. If you do, you will not be able to return to Piñcha Mayūrāsana.
4. In the pose, push the shoulders back to minimize the flexion at the elbows. Do not allow the nose to go forward past the line between the thumbs. This rolling and pushing back of the shoulders makes it feel as if the heart is open, even though the spine (from the lower thoracic area down through the coccyx) feels as if it is coiled tightly. Strong apāna is held with strong prāṇa. Hold this position for five breaths.
5. To come out of the pose, exhale and redouble on the apānic coil the sense of curling deeply through the spine, dropping the coccyx and maintaining the tone in the abdominal muscles. Before you inhale, start bringing the legs, still in Padmāsana, back up toward the ceiling. Do not inhale until you clear the tight point halfway up. Then unfold the legs into Piñcha Mayūrāsana and exit as for Piñcha Mayūrāsana.
6. One way to train that may allow you to feel the pattern of the movement is to place the legs in Bakāsana position instead of Padmāsana as you bring them down and to then place the upper shins on the backs of the upper arms.
7. Another training trick is to sit in Padmāsana on the floor. Stand up on the knees and place the elbows on the floor just in front of the tops of the shins as if you were setting up for Piñcha Mayūrāsana. Exhale and pull the shins to the elbows and then up a few inches onto the backs of the upper arms. Notice the patterns that need to be strengthened and coordinated.
**M AYŪRĀSANA**
_Peacock Pose_
This pose requires that the engagement and use of the abdominal muscles be differentiated correctly to provide both a strong surface on which the elbows can rest and the ability to continue breathing while doing so. It can be considered an arm balance not only because we literally balance the body on the arms, but also because it requires strong core awareness and toning in coordination with a sense of release. In the Intermediate Series, Mayūrāsana comes after Karandavāsana, and there is a Full Vinyāsa—with Mayūrāsana hand placement—between the two poses.
1. From Samasthitiḥ, on an exhalation, hop the feet approximately hip-width apart, placing the hands on the hips. Inhaling, extend long through the spine and upper body. Exhale and fold forward to place the hands on the floor between the feet, with your fingers pointing toward the back of the mat and the outer sides of the hands almost touching. Inhale and straighten the arms, lifting the head and the heart while gazing along the line of the nose.
2. Exhale and jump back into a transitional version of Catvāri, with the elbows only slightly bent and the hands in this preliminary position. Flex the lower spine and slide forward on the feet to place the elbows close to one another and against the abdomen, slightly above and near the navel, keeping the legs strong and the feet flexed with the toes on the floor.
3. Keep the rectus abdominis muscle toned and the core of the body engaged as you move your center of gravity forward, reaching forward through the sternum and lifting the head to look at the floor just in front of the body. Remember this initial setup is done as a complete exhalation.
4. As you glide forward with the heart on an inhalation and reach the balance point, the feet will automatically lift off the floor. Hold this position for five breaths, then roll back slightly until the toes touch the floor.
5. Leaving the hands as they are inhale and move directly into Upward Facing Dog Pose with hands close together and fingers pointing toward the back of your mat. Exhaling, back into Downward Facing Dog Pose with hands reversed, and then jump forward to Sapta (seventh form of the Sun Salutation). Inhaling, release the hand position and stand up, placing the hands on the hips. Exhale and hop to bring the feet together for Samasthitiḥ.
ṬIṬṬIBHĀSANA (SMALL WATER BIRD POSE) A, B, AND C
Watching the small birds that fly between prime spots and then hop along on an open stretch of beach, moving side to side while pecking at the sand for delicacies that lie beneath the surface, is an image that can be helpful to keep in mind when you're working on this pose. When practicing Ṭiṭṭibhāsana in its various forms, there _is_ a sense of being that small creature with sea-soaked, sandy feet moving along the shore. The first form of the pose is an arm balance, which is used as a transition form in vinyāsas out of other arm balances such as Bhujapīḍāsana, while the second and third forms are extreme forward bends similar to balanced upright variations of Kūrmāsana.
**Ṭ IṬṬIBHĀSANA A**
1. From Downward Facing Dog Pose, at the end of an exhalation, jump forward and wrap the legs around the outside of your arms, keeping your feet flat on the floor and knees bent. Work the inner knees up the arms to the area of the deltoid muscles.
2. Finishing the exhalation, drop the sitting bones toward the floor and lean back, while reaching forward through the sternum and spreading the collarbones to lift up and balance on your hands. Finally inhale to reach out through the legs and straighten the arms.
3. Hold this position for five breaths. As you exhale, drop the head forward and puff out the kidneys. On an inhalation, bend the knees and draw them up the outer edges of the arms to rest for one breath in Bakāsana. Jump back into Catvāri on an exhalation and move through a Half Vinyāsa to the next pose.
**Ṭ IṬṬIBHĀSANA B, C, AND D**
These three forms of Ṭiṭṭibhāsana are practiced one immediately after the next so even though they are distinct forms, they are included in one continuous instruction because that is how they are best practiced.
1. Follow step 1 for Ṭiṭṭibhāsana A to position the feet flat on the floor beyond the arms. If you are a beginner, you should place your feet wider than in the previous form.
2. Exhaling and leaving the legs slightly bent, curl the spine and move the upper body more deeply through the groins into a strong forward bend. Work the upper arms, head, and shoulders through and behind the legs. Reach up to clasp the hands behind the back. Straighten the legs as much as possible, and lift the head slightly to look at a point on the floor just in front of the head. This is Ṭiṭṭibhāsana B and can be held for five breaths. Without exiting form B, move immediately into form C.
3. On the inhalation, lean to the left, putting more weight in the left leg. Exhaling, step forward with the right foot. Then lean to the right as you inhale, and step forward with the left foot as you exhale. Repeat this walking motion, forward and then back, for five breaths in each direction to complete Ṭiṭṭibhāsana C.
4. When you are back to the starting point at the front of the mat, immediately move into form D. Release the hands, but keep your arms wrapped behind the legs. Inhaling, move the feet closer together, with the toes pointing out to the side and the heels touching. Exhaling, work the upper arms even more deeply through the legs and lace the arms around the lower legs, clasping the hands in front of the ankles. Spread and push the elbows back to move the lumbar spine farther through the legs. Gaze softly along the line of the nose and hold this position for five breaths.
5. To exit the pose, release the hands and, exhaling, place them flat on the floor as you move the feet apart to lower the sitting bones toward the floor. Inhaling, lift and engage the legs and point your toes as you move into form A, then move through Bakāsana into Catvāri and a Half Vinyāsa. Jumping back into Catvāri directly without pausing briefly in Bakāsana is a matter of style, but it is also effective.
**V ĀSIṢṬHĀSANA**
_Vāsiṣṭha's Pose_
This advanced balancing pose is beautiful in its expression of the central axis and is named after Vāsiṣṭha, a legendary sage who carried a magical staff that is represented by the raised arm in this pose.
1. From a well-aligned Downward Facing Dog Pose, exhale and turn the feet so the toes point to your left. This will place the outer edge of the right foot on the mat. The left leg will rest on the right leg as you turn the belly to the left while swinging the left arm up off the floor. In the first position of this pose the right arm and outer right foot support the straightened torso. The left arm is straight up, and your head is turned so that you gaze at your left thumb.
2. On an exhalation, ground through the right arm and right foot to stabilize the pose. Inhale and straighten the left leg up toward the ceiling and take the left big toe with the middle and index fingers of the left hand. Keep an emphasis on the external rotation of the femur at the left hip joint. If lifting the left leg while straight is not possible, you may bend the leg in order to take the toe. In either case, press the left big toe firmly into the fingers holding it and lift the right outer thigh away from the floor.
3. Established with full complements of counteractions and counterrotations on the inhalation, this pose expands radiantly out. The head, heart, shoulder, and kidney wing actions spread both the front and the back of the torso. The unique patterns of force around the hip joints are organized by Mūlabandha.
4. After five breaths with the left leg up, exhale, release the toe, and bring the left leg down to join the right leg. Leave the left arm up and continue to gaze at the left thumb.
5. On the next exhalation, bring the left arm down to your side. Turn the torso toward the floor, spreading the feet apart slightly as you step onto the toes and place the left hand down so you are in a high plank position like Catvāri, but the arms are only slightly bent.
6. Rolling from the cave of the sacrum, drop the coccyx and, inhaling, unroll into Upward Facing Dog Pose. Then ride that same wave back into Downward Facing Dog Pose on the exhale. Repeat the pose on the other side.
**N AKRĀSANA**
_Crocodile Pose_
There is something both entertaining and confounding about this pose. It can be helpful to imagine yourself as a crocodile, with strong, short legs and a deft ability to hop forward and back on the floor.
1. From Downward Facing Dog Pose, lower down into Catvāri on an exhalation. Keep the core of the body and the abdominal muscles toned, the legs and arms strong, and the shoulders set properly for Catvāri.
2. As you inhale, hop up off the floor in the plank position, moving forward by working the hands and feet almost as if walking on all fours, then land as you exhale. This pattern is triggered by engaging the psoas muscles just as you are preparing to leave the floor, which pulls back the groins allowing you to hop. Repeat this five times, hopping forward.
3. Reverse the pattern in the hands and feet to hop backward five times, again lifting on the inhalation and landing on the exhalation. From Catvāri, move directly into Upward Facing Dog Pose on the inhalation, then a Half Vinyāsa and into the next pose.
**V ATAYANĀSANA**
_Air Vehicle Pose_
This interesting pose appears near the end of the Intermediate Series and, although it doesn't _look_ terribly difficult, it is surprisingly hard to balance in the final form.
1. Begin in Samasthitiḥ. Fold the right leg into Half Padmāsana and reach behind you with the right arm to clasp the right big toe.
2. Inhaling, reach up with the left arm, protracting the shoulder blade and stretching the psoas line on the left side of the body. At the peak of the inhalation, release the right foot and, exhaling, fold forward to put both hands on the floor beside the left foot.
3. Inhale and come to a modified Trīṇi position, with the right leg still in Half Padmāsana. On the next exhalation, jump back into Catvāri, then inhaling move through a one-legged Upward Facing Dog, and exhale into Downward Facing Dog Pose, with your right leg remaining in Half Padmāsana.
4. Next, on an exhalation, bend the left leg. Inhale as you jump forward, placing the right knee and left foot on the ground. The left heel should be near or touching the right knee, and the toes should point out to the side at about a 45- to 90-degree angle, depending on your flexibility.
5. Inhaling, begin squaring the hips toward the front and lift the upper body out of the pose, pushing down into the earth through the right knee and the left foot to maintain balance. If it is impossible to balance, you may move the left foot away from the right knee slightly until the pose is accessible.
6. Draw the arms to the front of the body. Bend your elbows and place the right upper arm on top of the left, then wrap the right lower arm around the left so the palms of the hands are together.
7. Inhaling, gradually lift the hands and arms straight up toward the ceiling, and gaze at the tips of the thumbs. Hold this for five breaths.
8. Exhale and release the arms, placing the hands on the floor by the knees with the right leg still in Padmāsana. Jump back into Catvāri on an exhalation, then keeping the Half Padmāsana form, move through a Half Vinyāsa to Downward Facing Dog Pose.
9. Take the right leg out of Half Padmāsana to place the foot in full Downward Facing Dog Pose position for one breath. Inhaling, draw the left leg up into Half Padmāsana, using your hand to help position the foot if necessary. Jump forward as in step 4, this time with the left knee on the ground and the right foot flat on the floor. Wrap the arms with the left one on top and inhaling reach up through the arms, gazing at the ceiling. Hold the pose for five breaths.
10. To exit the pose, work through a Half Vinyāsa to Downward Facing Dog Pose with the left leg in Half Padmāsana. Jump forward to land on the right foot and inhaling, lift the head and heart into a modified Trīṇi position. Reach behind the back, take the left foot with the left hand on the wave of the breath, then fold forward, exhaling. Inhaling, come back up to standing, and exhaling, bring the left foot down from Half Padmāsana into Samasthitiḥ.
**G OMUKHĀSANA**
_Calf Face Pose_
Gomukhāsana in the Aṣṭāṅga tradition is slightly different from the form practiced by many other traditions in that the feet are very close and one almost hovers to balance above the legs while sitting. Finding the plumb line and pelvic floor in this version makes the pose possible, and it is representational of the skills needed to find balance in any pose. There are two phases to this pose that affect the throat and lower neck at the home of the Udāna Vāyu. It grounds and integrates the intensity that can arise within the nervous system, emotions, and the physical body.
1. From Downward Facing Dog Pose, at the end of an exhalation hop through and land on your knees with the right thigh in front of the left, feet slightly out to the side. Sit down, and as you do so, move the lower legs and the sides of the feet so they are touching, if possible.
2. Drop the awareness into the sitting bones, making them heavy in order to stabilize the interplay of the pelvic floor muscles. Place the hands on the left knee, palms down, with the left hand under the right. Roll the shoulders back and down, spread the collarbones, and drop the chin into Jālandhara Bandha position.
3. Gaze softly inward with the eyes half open and downcast. Hold this position for five breaths.
4. Release the hands. Bend the left elbow and wrap the left arm up the back of your body with the back of the hand on your back. Reach up with the right arm, then bend the elbow to reach down the back with the palm facing the back and clasp the fingers or hands together. Reach up and slightly forward through the right elbow, which will protract the shoulder blade. Tilt the head back and gaze down the nose, holding this position for five breaths.
5. To exit the pose, release the arms, move through a Half Vinyāsa, and do the pose on the other side.
**D IKĀSANA**
_Compass Pose_
Dikāsana appears at the end of the Advanced A Series in the Aṣṭāṅga Vinyāsa system, and even though it isn't as showy as some of the others within that series, it is possibly one of the more difficult poses for many. It is a fine example of the subtleties of working into balancing poses by finding actions and counteractions, maintaining a strong yet soft wave pattern of the breath, and of incorporating the plumb line even when the body is not in a straight or upright form.
1. From Trīṇi, exhale and ground well through the left leg, toning the PC muscle. While inhaling, lift the right leg straight back and up so it is parallel to the floor. Lift the arms and bring them forward with palms of hands together if possible and head behind the arms, closer to the ceiling. Gaze at a point on the floor out in front, balancing for five breaths.
2. Be sure to stretch back through the inner edge of the right foot and use the pelvic floor and scooping up of the lower belly as a strong base for the pose. Do not collapse into the left hip, which can be avoided by micro-bending the left knee and fully articulating the left foot while following the breath. This keeps the hip joint comfortable and keeps the pelvis level.
3. After five breaths, on an inhale take the arms out to the sides angled slightly back behind you, but still parallel to the floor. The arms will be like wings of a bird in flight. Hold this form five more breaths. Then inhaling, reach overhead again for just one breath and on an exhale bring the leg and hands down to the floor coming into Dve position. On the next inhale repeat the pose on the second side.
MUKTA HASTA ŚĪRṢĀSANA (UNBOUND HAND HEADSTAND POSE)
The final sequence of poses within the Intermediate Series comprises seven headstands practiced one following the other, dropping down into Catvāri between poses. In the first three of these headstands, the hands are not clasped together, and in the remaining variations, they are (see Chapter 11).
**M UKTA HASTA ŚĪRṢĀSANA A, B, AND C**
1. From Downward Facing Dog Pose, on an exhalation, hop forward to place the head on the mat in front of the hands, with the hips raised and the feet on the floor. If you are a more advanced practitioner, you may position the hands for the variation being practiced and then float forward from Downward Facing Dog Pose, landing in the headstand.
2. For form A, place the palms of the hands flat on the floor with the fingers pointing toward the back of the mat, at a distance that allows the arms to be parallel to one another.
3. Exhale fully to establish the base of the pose as you protract the shoulder blades, drawing them up toward the waist and out slightly away from the ears and away from the floor. Inhaling, tone the legs and lift them together, keeping them straight, as you lift into Śīrṣāsana. Hold this position for five breaths.
4. At the end of the final exhalation, take one additional in-breath, moving the hands back into the original position as for a tripod headstand. Exhaling, drop down into full Catvāri, then immediately move into Upward Facing Dog Pose on the next inhale and Downward Facing Dog Pose on the exhale. Prepare for form B.
5. For Mukta Hasta Śīrṣāsana B, from Downward Facing Dog Pose, at the end of an exhalation, position the head as for form A, but place the arms straight out along the floor so you are looking at them, keeping the arms parallel with the palms facing up. Exhale to establish the base of the pose, then, keeping the arms firmly in place, lift the legs into Śīrṣāsana. Adjust the pressure in the backs of the hands, and make subtle adjustments in your shoulders and arms to find strength and balance. Hold this form for five breaths, then, as in step 4, move the hands back to a tripod position and drop into Catvāri and move through a Half Vinyāsa.
6. For Mukta Hasta Śīrṣāsana C, enter the pose as for form B, except place the palms down and reach the arms out to the sides of the body. Your hands may be angled slightly forward toward the end of the mat you are looking at in order to allow for easier balance. Lift the legs into Śīrṣāsana, keeping the arms straight and using subtle adjustments in the arms and hands to facilitate balance. Hold the posture for five breaths.
7. Exit the pose by placing the hands flat on the floor in front of your face on an inhalation. As in step 4, drop into Catvāri and move into a Half Vinyāsa ending in Downward Facing Dog Pose.
**B ADDHA HASTA ŚĪRṢĀSANA A, B, C, AND D**
In all of these variations of Śīrṣāsana, the hands are bound, touching each other or another part of the body.
1. For form A, from Downward Facing Dog Pose, on an exhalation, place the head on the mat in front of the arms. Position the arms in the classic Śīrṣāsana position with the hands clasped and cupped and the head placed in the nest of the hands.
2. Establish a strong base in the pose, then on an inhalation, lift the legs straight into Śīrṣāsana. Hold this pose for five breaths. As in step 4 for Mukta Hasta Śīrṣāsana, on an inhalation, move the hands so they are flat on the floor to prepare for Catvāri. Drop down on the exhalation and move through a Half Vinyāsa before coming into the next form.
3. For Baddha Hasta Śīrṣāsana B, position the arms in front of your face with the elbows bent and hands holding opposite elbows. Again, lift the legs into Śīrṣāsana on an inhalation. Hold the form for five breaths before dropping into Catvāri.
4. For form C, place the hands as if you were preparing for Piñcha Mayūrāsana, then position the head for a headstand and lift into Śīrṣāsana on an inhalation. Hold for five breaths before moving the hands to drop down into Catvāri.
5. For form D, place the arms up by the head, with the elbows bent and the fingers gently touching the upper trapezius muscles. Again, move in and out of the pose as for other variations of this sequence of headstands. For the final exit, after moving through a Half Vinyāsa, hop through to a seated position and the next pose.
**A DHO MUKHA VṚKṢĀSANA**
_Downward Facing Tree Pose_
Commonly referred to as Handstand, this pose can be practiced after backbends, as part of the Nāvāsana sequence, or simply on its own to build core strength and fine-tune proper shoulder alignment.
1. When you are first learning this pose, use a wall to kick up against. Lean over and place the hands shoulder-width apart about 1 to 2 feet from the base of the wall. Firmly protract the shoulder blades and confirm the grounding of the inner edges of the hands as in Downward Facing Dog Pose.
2. Exhale to set the pelvic floor. Kick up with one of legs as you inhale, leaving a slight bend in the elbows and maintaining hand and shoulder alignment as described in step 1. Your second leg will naturally follow the first, and when learning the pose you can place both heels on the wall, then begin to practice balancing without the wall.
3. At first keep the head fully behind the arms and gaze at a point on the floor just in front of the fingertips. Keep the legs stiff and awake using an inward rotation, and stretch up through the inner edges of the feet, keeping the buttocks soft.
4. The key to balancing is to be fluid with tiny back-and-forth adjustments in the belly, spine, hands, arms, legs, and (of course) pelvic floor. These constant adjustments are what makes this one of the finest yoga poses!
5. Once you have stability, you can enter the pose slowly with both legs straight and feet held together. Hold the pose for at least five breaths, then exhale to drop the feet slowly back down to the ground, placing them between the hands just in front of the face. Establishing and maintaining control entering and exiting the pose is where the juice in the pose is found. Work intelligently and slowly for lasting results. Another method of exiting the pose is to slowly bend the arms and bring the heart between the hands to exhale into Catvāri.
_11_
Finishing Poses
FINISHING POSES, MORE THAN ANY OTHER GROUP of poses, teach subtleties of the core of the body that lead to deep meditative states. We call them finishing poses because we generally do them at the end of an āsana practice, though under certain circumstances, they may be the entirety of the practice. Finishing poses include the whole family of poses associated with Sarvāṅgāsana (Shoulderstand) and a number of variations, which may be used therapeutically. The finishing poses also include Śirṣāsana (Headstand) and its variations as well as Matsyāsana (Fish Pose), which creates a deeply rooted set of counteractions and movements to balance the deep effects of both Shoulderstand and Headstand. There are also a number of extremely internal and meditative linking poses in the sequence, such as Balāsana (Child's Pose), Yoga Mudrā (Seal of Yoga), and Tāḍāgī Mudrā (Pond Mudra), as well as a long Śavāsana (Corpse Pose) that are included in the finishing postures.
A nice thing about this particular sequence is that it provides a good balance between the vibrant, precise state of mind that is required for any of the Aṣṭāṅga series, and the tranquil mood that allows the residue of an intense practice to be assimilated and transformative. We see that this balance makes the finishing poses important for everyone, but particularly for those with a rigorous Aṣṭāṅga Vinyāsa practice because the poses automatically induce a profound feeling of vairāgyam, or letting go. This freeing phase of the practice exposes the ritualistic and sometimes linear framework that the mind sometimes creates as a means of starting and maintaining a yoga practice. All too often, enthusiastic practitioners neglect or leave too little time for the finishing sequence as if fueled by a background fear that prevents them from entering the deeper meditative states of yoga in which there is a sense of delight in dissolving into the truth of impermanence. One of the obstacles in yoga is to circumvent the stillness and silence that proper practice of the finishing sequence necessitates. More rare in the Aṣṭāṅga Vinyāsa world is the opposite fault—that of being too attached to dissolving and being unable to create precise, radiant form in a true yoga āsana. However, some practitioners avoid the depth offered by the finishing poses, conceptualizing them as strictly therapeutic or restorative practices, which can create an imbalance toward an attachment to dissolution before there is actually anything to dissolve—in other words, before they have done the work. This can quickly become tamasic dullness in yoga. Anything within our yoga practice that creates avoidance of a vibrant, awakened, and precise state of mind should be avoided.
All of the finishing poses stretch, stimulate, and balance the myofascial sheaths of the head, chest, and neck. Thanks to this and their contemplative nature, they also make us keenly aware of multifaceted patterns of sensation in the physical structures of the body and, on a more subtle-body level, within patterns of Prāṇa that interpenetrate and connect the entire body and mind. When practicing the finishing poses, it is important to go slowly, progressing step by step, particularly because of the delicacy of the upper neck and structures of the head. The inversions require precise and correct placement of the shoulders, as well as positioning and alignment of the head in relation to the upper vertebrae of the neck so that Prāṇa moves freely throughout the core and into the periphery of the body. In this way, practicing the finishing poses creates a sense of liberation within the forms and underscores the necessity of navigating poses in an intelligent, grounded, and spontaneous manner. Even more than in all other families of poses, when working with the finishing sequence we must begin with an awareness and softening of the palate, mouth, face, and heart by using the technique of proper gazing. This way we can naturally make the correct and usually subtle adjustments to bring a pose into a pleasant, luminous balance.
The finishing poses require a highly refined sense of the entire structure and movement of the spine and the pelvic floor in order to bring them into a full form. One of the great benefits of experiencing these subtle relationships within the body is that the three bandhas will come naturally and easily when we practice finishing poses in a meditative manner. Of course, this not only has a palpable effect on our overall āsana practice and the day-to-day residue from that, but it also helps to deepen our prāṇāyāma practice.
Finishing poses are vital for all levels of practitioners. Even beginners who are physically not ready for some of the poses in the full sequence can and should practice appropriate variations. No matter the level or intensity of our practice, the finishing sequence will help us digest the residue and assimilate the physical benefits of the practice.
**P AŚCHIMOTTĀNĀSANA AND VARIATION**
_Tuning Up the Back Pose_
After finishing the full Ūrdhvā Dhanurāsana, folding the hip joints closed and elongating the spine forward provides the counterpose alignment that brings a sense of ease and balance.
1. Lying flat on the back with the knees bent, clasp the hands around the upper shins and, with the thighs apart, squeeze the knees in toward the armpits. Keep the sacrum on the floor to maintain the natural lumbar curve. Imagine that the pubic bone is heavy and dropping down toward the mat so that the PC muscle responds to hold the coccyx. Hold this for five breaths.
2. Next, draw the knees together and squeeze the legs into the chest. Resist the pull of the arms slightly through the legs so the spine is drawn into smooth, even traction. Again, hold for five breaths.
3. On an exhalation, extend the legs up toward the ceiling, then rock up to a seated position.
4. With the legs stretched straight out in front and the spine elongated, fold forward over the legs, taking your favorite hand position from the Paśchimottānāsana sequence (see pages 162–66 for full description). Hold for at least ten breaths. On an inhalation, sit up.
**S ARVĀṄGĀSANA**
_Shoulderstand_
This pose also contains Halāsana both in preparation for the full Shoulderstand and as part of the exit from the pose.
1. Enter Sarvāṅgāsana from Tāḍāgī Mudrā (see page 27). At the end of an exhalation, start lifting the straightened legs. Once they are about 30 degrees from the floor, begin to inhale, lifting the hips and buttocks and moving the feet up toward the ceiling.
2. Follow the big toes with the eyes, but don't lift the chin, as, while exhaling, the feet disappear straight over the nose and eventually find their way to the floor above the head.
3. Walk or roll the shoulders back and under. Draw the elbows close to each other to make the upper arms parallel. Press the elbows into the floor and clasp the hands behind your back, reaching out and down through the arms and hands. Your toes may be flexed at this stage, pushing gently into the floor, which helps to keep the sitting bones lifting toward the ceiling and the spines traightening.
4. After a few rounds of breath, bend the elbows and work the hands up and into the back to draw the skin of the upper back toward the ceiling. Roll the shoulders even farther back into the floor; this should position the neck and chin correctly. Maintain the cervical curve in the neck, but keep the neck, chin, jaw, and tongue soft.
5. Draw the whole spine into the body. Make the legs firm and with an inhalation lift them, together or one at a time, flexing the feet slightly so the inner edges of the feet are reaching up to the ceiling. Keep both legs engaged and alive as you work more deeply into the pose, working the hands high up the back, as close to the upper shoulder blades and neck as possible. Stretch out through the heels before pointing the toes.
6. Gaze at the big toes. Keep the feet above the hips rather than allowing them to drift forward over the face.
7. Do not press the back of the neck into the floor or draw the chin in toward your throat! The large vertebra at the base of the neck (C7) is drawn up into the body. Your upper chest should expand, bringing the top edge of the sternum toward—and eventually to gently touch—the chin. Do _not_ actively draw the chin toward the sternum.
8. Remain in Sarvāṅgāsana for at least fifteen breaths, if comfortable.
9. If you experience discomfort during or after the pose and/or cannot keep the back of the neck off the floor, place a folded blanket(s) under the shoulders and arms. The shoulders and elbows will then be elevated on the firm, flat surface, while the head remains down at floor level. If using a blanket, be sure to have it folded neatly with the folded edge (not the raw or fringe edge) beneath the neck. The shoulders should be positioned so that when in the pose, C7 just grazes the edge of the folded blanket. A qualified teacher can demonstrate the details of this technique. There should be no discomfort or pain in any of the Finishing Poses.
**S ETU BANDHA SARVĀṄGĀSANA**
_Bridge Shoulderstand_
For beginning students or those with neck injuries, this variation of full Shoulderstand is a good alternative as part of the finishing sequence. It is also an excellent restorative pose for anyone, in or out of this sequence.
1. Lie on your back in Tāḍāgī Mudrā (see page 27) for five breaths. Draw the knees up and place the feet on the floor at the outer edges of the buttocks.
2. On an exhalation, lift the sacrum about 3 inches off the floor, working the knees forward toward the toes. On the next inhalation, engage the quadriceps and lift the pelvis off the floor, with the sacrum moving in and up.
3. Walk or roll the shoulders back and down, clasping the hands behind your back. The sternum will move toward the chin, but keep the chin and face neutral. Keep the legs engaged, as if squeezing a block between the thighs, and keep the feet parallel.
4. Gaze along the line of the nose, and hold the pose for at least ten breaths. Exit on an exhalation, releasing the hands to unroll the spine so that the sacrum is the last part to come down to the floor.
**H ALĀSANA**
_Plough Pose_
Halāsana is often practiced as a stage while working through Sarvāṅgāsana, but it can also be practiced on its own as a meditative and restorative form.
1. Lying on the back in Tāḍāgī Mudrā, gazing down the nose, exhale fully as you begin to lift the straightened legs away from the floor. When they are about 30 degrees from the floor, begin to inhale as the hip joints fold more deeply. Keep the head neutral, but gaze at the feet as they move over the head to the floor.
2. Clasping the hands behind your back, roll the shoulders completely back into the floor and make the spine straight, moving the sitting bones up toward the ceiling. You may bend the elbows at first, but work toward straightening the arms to further guarantee that the shoulders roll under.
3. Shift the softened gaze to the tip of the nose. The back of the neck remains off the floor, and the throat stays soft.
4. Straighten the spine, keeping the legs firm. This is an excellent opportunity for cultivating Uḍḍīyāna and Mūlabandhas. Point the toes only if you can keep the legs absolutely straight.
5. Remain here for ten breaths. To exit, continue into Karṇa Pīḍāsana or release the hands and, exhaling, roll down onto the back. Move into Catvāri via Cakorāsana.
**K ARṆA PĪḌĀSANA**
_Squeezing the Ears Pose_
This deeply meditative pose incorporates flexion of the spine into the shoulder structure that is used in Sarvāṅgāsana, an alignment that creates a feeling of safety within a strong form. Cupping the knees next to the ears focuses attention on _listening_ to the breath so that breathing becomes smooth, even, and deep.
1. Begin in Halāsana. Exhaling, bend the knees and lower them toward the floor next to the ears. Releasing the palate as you exhale will lengthen the entire spine in both directions. If the knees reach the floor, gently cup the ears with the inner knees. If they do not reach the floor, do not strain, but allow gravity and consistent practice to slowly deepen the pose until they can touch the ears. This may take a long time (years), but you should not strive too much for it.
2. Keep the hands clasped behind the back, and press the straight arms down into the floor. Gaze at the tip of the nose. Relax the ears. Soften and steady the eyes. Empty the palate as if softly smiling. Circulate the breath evenly through the entire field of perception as you hold the pose for ten breaths.
3. To exit, either roll down onto the back on an exhalation, or step back up into Sarvāṅgāsana to complete the full finishing sequence.
**Ū RDHVĀ PADMĀSANA**
_Upturned Lotus Pose_
This pose is an excellent opportunity to begin to feel movement in the pelvic floor at the ends of the breath. It is great training for Uḍḍīyāna and Mūlabandhas and also for keeping the PC muscle toned at will.
1. From Karṇa Pīḍāsana or Halāsana, place the arms back in Sarvāṅgāsana position behind the back with the elbows bent and the hands on the upper back. Inhaling, step back up into Sarvāṅgāsana.
2. Still in Sarvāṅgāsana, cross the legs into Padmāsana. Use one of the hands for assistance if needed to position the feet and legs. If you cannot yet do Padmāsana, simply cross the legs or place the soles of the feet together as in Baddha Koṇāsana.
3. Place the hands on the ends of the femur bones at the medial edges of the knees. Straighten the arms. Press the knees into the hands, while pushing back with the hands against your knees. Gaze softly along the line of the nose, holding the pose for five breaths. To exit move immediately into Piṇḍāsana on an exhalation.
**P IṆḌĀSANA**
_Embryo Pose_
This is an excellent counterpose for the spine after Sarvāṅgāsana and Halāsana, and it also serves as a threshold to awaken the spine for counteractions required for the subsequent cervical backbends in the Finishing Sequence.
1. From Ūrdhvā Padmāsana, on an exhalation fold at the hip joints to draw the knees down toward the fronts of the shoulders and, if possible, gently reach them to the floor, then inhale to set the pose.
2. Exhaling again, take the arms around the backs of the legs and clasp the hands, if possible. Make the body into an even ball, curling the spine and releasing the shoulders and neck completely. Leave the shoulder blades rolled back into the floor. Notice that this movement lessens the angle of flexion in your neck, preparing the body for its descent.
3. Gaze internally along the line of the nose with the eyes half closed. Stay in this position for ten breaths.
4. To exit, exhale while rolling down onto the back and placing the arms down along the sides of your body. Move immediately into Matsyāsana.
**M ATSYĀSANA**
_Fish Pose_
Due to the sense of a heavy coccyx, which is necessarily maintained by the legs being in Padmāsana (or crossed), this strong backbend opens the heart area and thoracic spine with less chance of excessive folding in the lumbar spine than with other deep backbends. Gazing down the nose keeps the neck properly extended and the pose integrated.
1. Lying on your back, keep the legs in Padmāsana or easily crossed. Take the outsides of the upper thighs with your hands. Ground the elbows, lift the head, and glance toward the navel as you complete an exhalation. Hold the seed of the exhalation in the center of the pelvic floor as you begin to inhale to unroll the spine into a backbend. Initially push the sacrum up and in, unroll up to the heart, and then up to extend the lower neck. Finally the head is allowed to rotate back at the very top of the inhalation. Exhale as you place your head down to the floor. Inhaling, activate the pose, release the palate, and gaze down the nose.
2. Use the elbows to adjust the shoulders, positioning the center of the crown of the head onto the floor. Take the elbows away from the floor and clasp the feet. Either pull back slightly through the hands or push with the hands and arms to pull or stretch the skin at the crown of the head in different ways. Notice the effects of these actions. Push down toward the floor through your knees. Give yourself time in this pose to position yourself comfortably and on the wave of the breath, inhaling as you push through the elbows and lift the heart area into a backbend and exhaling as the head is repositioned on the floor until it is in exactly the right, most comfortable place for your particular circumstances. If Padmāsana is not possible, leave the legs stretched out straight, and still work through the elbows to open the heart area into a backbend.
3. Continue to gaze down the nose as you drink nectar from the root of your palate. Silence the tongue, release the ears, and sink the sitting bones as the heart opens. Stay here for at least ten breaths rather than moving immediately into Uttāna Pādāsana. If you prefer to exit Matsyāsana, release the feet with the hands and place the elbows on the floor near the waist. Push through the elbows to lift the heart and then exhale, lift the head to look at the navel. Inhaling, lower the back onto the floor from bottom to top, placing the back of the head down last. Unravel the legs and then exit through Cakrāsana.
**U TTĀNA PĀDĀSANA**
_Upwardly Stretched Legs Pose_
This pose is reminiscent of an enthusiastic fish. Maintaining the form of Matsyāsana in the upper body, the legs are straightened and angled from the floor so that it you were a fish doing Uttāna Pādāsana, you'd be flying!
1. Move directly into Uttāna Pādāsana from Matsyāsana, leaving the top portion of the body in Matsyāsana.
2. With the legs still in Padmāsana (or crossed), flex the hip joints to lift the knees off the floor so the thighs are at about a 60-degree angle from the floor. Unravel the legs and straighten them into the air, reaching out through the feet. The legs act to ground the sitting bones even more.
3. Choose a comfortable angle for the legs. If you are a beginner, you might keep them at 45 degrees or higher, while more advanced practitioners could lower them closer to the floor.
4. Straighten the arms out in front of the chest, placing the palms together and keeping the arms at about the same angle as the legs.
5. Gaze down the line of the nose and stay in the pose for eight breaths. 6. To exit, on an exhalation, place the arms along the sides and bend the elbows, hands pointing toward the ceiling. Exhale deeply and push through the elbows to lift the chest and the head off the floor, curling the neck to look at the navel, then inhaling, lower the spine to the floor as you lift the legs to vertical. Place the hands up by the ears, fingers pointing toward the tops of the shoulders and roll back through Cakrāsana into Catvāri and on through a Half Vinyāsa.
**Ś ĪRṢĀSANA**
_Headstand_
One of the most important poses in any yoga practice is Śīrṣāsana. It takes time to learn the pose well, but even for beginners, it can offer the immediate benefit of changing their perspective. Over time, it becomes clear that this pose relies on proper shoulder alignment and arm strength, in addition to a trust in gravity. The pose is meditative, bringing form and focus into the plumb line, as well as connecting forces of movement and form out through both the feet and the crown of the head to invite awareness into subtleties that arise moment by moment. It is truly Samasthitiḥ upside down. Once stable in it, visualization of deities and internal forms reveals the central channel beautifully.
1. Create the foundation of the pose carefully. Place the outer edges of the lower arms on the floor and draw the elbows in so they are exactly shoulder-width apart with the forearms parallel. Interlace the fingers. Press the roots of the index fingers together and pull the roots of the little fingers slightly apart. Touch the tips of the thumbs; pull the wrists away from each other isometrically but keep them parallel and upright as they ground down through the ulnae. The palms should remain vertical (not cupped under). A few practitioners with proportionally short arms find that pressing the heels of the hands together allows the elbows to bear weight correctly in the pose.
2. Lift the hips and walk the feet in toward the shoulders to place the center of the crown of the head lightly down on the floor. The exact point that should be touching is an individual matter, and you should experiment with it intelligently so there is no strain created in the neck or head. It is important that the majority of the weight for the pose be held through the base of the arms—especially if you are first learning the pose. So you must work out the placement of the head (the angle and exactly where it should be within the nest of the hands) for your individual circumstances.
3. The heels of the thumbs should evenly touch the back of the head. Spread the shoulders as wide as possible, and lift them as far from the floor as possible. When the center of the crown of the head is down, you will be able to see the ceiling, just as you are able to see the floor in Samasthitiḥ when the head is level. Again, the Headstand is similar in form to Samasthitiḥ.
4. Distributing the weight within the broad frame of the arms and shoulders, walk the feet toward the face. Exhale and then, with the elbows grounding, inhale to draw the straightened legs up. Stretch the inner edges of the legs through the inner edges of the heels during the ascent. This will give you control and will give the elbows a greater share of the weight. If a straight-legged ascent is not possible, bend your knees and curl partway or completely up into the pose. Alternatively, you may lift the legs one at a time. If you are a beginner, you can do this pose against a wall until the balance becomes natural.
5. Hollow (draw back) the groins as in Samasthitiḥ; do not push out through them across the room. Stretch the inner heels as high as possible before pointing the toes. The legs are held together with an inward rotation in each leg. Throughout the pose, keep the inner edges of the feet together, constantly lengthening the inner seams of the legs toward the ceiling. Imagine holding a hollow straw between the legs as if holding the plumb line at the center of the pelvic floor, the mūla.
6. Widen the front edges of the armpits as if drawing the skin back and together around the backs of the shoulders. Pull the shoulder blades firmly down the back (when you're upside down, this means they are going up toward the ceiling), keeping the kidney area broad. The back of the neck should be long and strong but without tension. There should be no discomfort in the head or neck.
7. Press the sides of the hands and the edges of the forearms into the floor. Spread the weight wide through your whole body. It will feel light at the crown of your head as you push the plumb line into the floor using only your sense of dignity. The heart and throat feel spacious and there will be a sense of openness in the ears, and brightness at the root of the palate.
8. When you are stable and happy in the pose, gaze at a point straight out from the eyes or eventually gaze at the tip of the nose. It should be easy to keep the gaze strong, soft, and steady, and the breathing should be smooth and even. This is a great Mūlabandha pose. Hold for at least twenty-five breaths.
9. If you are an advanced student, you can finish by lowering the straight legs to a horizontal position with legs parallel to the floor. Reach out through the heels and inseams of the legs while gazing at the big toes for five breaths. Keep the lower belly hollowed back at the edges of the pubic bone and the shoulders stretched high and wide. Do not wrinkle the skin on the back of the neck as you lower into or hold this form.
10. Inhale and lift the legs back up into full Śīrṣāsana. Exhaling, activate the legs and push down through the frame of the arms, while shifting the weight toward your elbows without allowing the wrists to ascend. On an inhale push evenly down through elbows and wrists to effortlessly lift the head entirely off the floor. Do not do a backbend here. Rather, keep the legs vertical. Hold for ten breaths, then lower carefully back down into Śīrṣāsana for one breath before lowering the legs to the floor to rest in Balāsana.
11. If you are a beginner, you can finish by coming slowly down from Śīrṣāsana on an exhalation without lifting the head off the floor and resting in Balāsana.
**B ALĀSANA**
_Child's Pose_
It is very important to practice this deeply restorative pose after Śīrṣāsana to reestablish smooth circulation and absorb the residue from the pose. It is also an excellent pose to take when stressed, tired, or on one's moon cycle.
1. Sit on the mat with the legs folded back and the feet under the buttocks. The knees may be almost touching or as far as hip-width apart, depending on what feels best. Do not open the hips extremely wide for this pose. If you experience pain in the knees, place a folded blanket behind them for support.
2. Fold forward at the hip joints to place the forehead on the floor. If your head does not reach the floor, place a soft block or blanket to support the forehead in a position so as not to interfere with breathing.
3. Relax the arms. You can take various arm positions, each creating a slightly different effect. One or all may be practiced on any given day. The most common arm position is to reach up along the floor over the head with the elbows bent and the forearms resting on the floor. Or you may drape the arms along the sides of the body, with the palms facing up.
4. A third variation that is particularly good to practice after Śīrṣāsana is to rest the head in the hands and place the neck in a gentle traction. To do this form, with the arms reaching out in front, lift the head and bend the elbows so the hands easily reach the head. Walk the elbows forward on the mat, and place the hands along the sides of the head with the roots of the fingers along the temples, between the tops of the ears and the corners of the eyelids.
5. In this form, bring awareness to the spot on the head where you were in contact with the floor when in Śirṣāsana. Concentrate on the quality of the breath cycle, with a soft gaze toward the tip of the nose. Imagine that prāṇa and apāna have linked together deep in the navel and practice umbilical breathing to release all residue.
**P ADMĀSANA**
_Lotus Pose_
If you cannot grab the toes, hold the forearms behind your back and whether holding toes or not lower the shoulders. If you cannot cross the legs into Padmāsana, sit in a comfortable cross-legged position.
1. Sit and tilt the pelvis slightly back as you bend the right knee and use the hands to draw the right foot back, bringing the heel toward the lower left side of the belly in the psoas button area. Be careful with the knee and do not force the placement of the foot if there is knee discomfort. Press the right knee toward the floor using the hip muscles.
2. Bring the left foot in over the right toward the lower right side of the belly and the psoas button on that side. Once the feet are in place and the heels are touching or close to the lower abdomen, flex or turn the feet out so the ankles do not collapse and the bottoms of the feet begin to look out to the sides.
3. Fine-tune the pose so you have no discomfort in the knees or ankles. Do this first by softly turning the skin on the top of the ankle outward so the front of the tibia can rotate slightly toward the floor. Once in the pose, draw the knees closer together or keep some active outward rotation and flexion, pulling the knees slightly apart, to set the pose properly. Keeping the heels in contact with or close to the lower abdomen keeps the knees closed properly and comfortably. A good Padmāsana should naturally stimulate Uḍḍīyāna and Mūlabandhas. When beginning to learn the pose, do not grit your teeth in order to sit through any pain in the knees or ankles as if to train the body. This will only result in distracted mind and potential injury. Instead, you may try the posture for a few breaths only and then move into Half Padmāsana (below) for the remainder of your time in the pose.
4. Padmāsana should be used for meditation and prāṇāyāma only after it is very comfortable for a long period of sitting. Otherwise the knees or ankles could be damaged. You should be able to move the ankles and feet easily in a full, comfortable Padmāsana. If full Padmāsana is too difficult, you may sit in Half Padmāsana—with one foot up and the other leg folded under and supporting the Padmāsana leg and knee.
**B ADDHA PADMĀSANA**
_Bound Lotus Pose_
This is a meditative form of Padmāsana that encourages the kidney area to open and allows a deep meditative state to spontaneously arise upon the releasing of the binding of the arms.
1. Sit comfortably in Padmāsana, Half Padmāsana, or a simple cross-legged pose. Squeeze the knees closer together slightly more than you would if you were merely sitting. Both knees will eventually come to the floor when your flexibility has developed. However, do not sacrifice the symmetrical grounding of both sitting bones to bring the knees down.
2. Exhaling, wrap the left arm behind the back and take hold of the left big toe. After an inhalation, exhale strongly again to wrap the right arm around the back to grab the right big toe. Balance the shoulders as best you can, considering the asymmetry.
3. At the top of an inhalation, lift up the front of the spine and place the chin down on the heart in Jālandhara Bandha position for a few breaths to establish the Uḍḍīyāna and Mūlabandhas activity under your belly.
4. Lift the head and gradually roll it back so the face is parallel to the ceiling, but do not collapse the cervical spine. Look down the nose, empty the palate, and carefully stretch the front of the neck holding this form for at least three breaths. Move immediately into Yoga Mudrā.
**Y OGA MUDRĀ**
_Seal of Yoga Pose_
1. From the previous pose, keep the head up and back while looking down the nose. For a few breaths, release the palate as if to spread the front of the throat on the ceiling. Drop the front edges of the sitting bones deep into the earth using Mūlabandha. Imagine you are drinking the nectar of compassion from the root of the palate as each wave of inhalation crests.
2. Fold forward on an exhalation. Touch the chin to the floor, if possible. Otherwise, take the forehead down to the floor. You may place a block under the forehead if the head does not reach the floor.
3. Draw down the back surface of the body to bring the sitting bones toward the floor and to maintain the apānic base. Complete each exhalation while internally stretching the middle line from the heart through the crown of the head. You should feel the palate-perineum reflex in the full cycle of inhaling and exhaling.
4. If the chin is on the floor, gaze between the eyebrows and keep the front surface of the spine flowing upward in contrast to the strong downward sealing action of the sitting bones dropping and the PC muscle toning. If your forehead is on the floor or a block, softly gaze down the nose. Stay here for ten breaths and after a full exhalation then inhale to sit up straight. Release the hands to move into Padmāsana with Ujjāyī Prāṇāyāma for at least ten breaths.
**P ADMĀSANA WITH UJJĀYĪ PRĀṆĀYĀMA**
_Lotus Pose Variation_
This straight-arm form is held in response to the powerful residue left in the arms and shoulders from the previous pose. If you sit longer, the straightness of the arms and the exaggeration of the Jñāna Mudrā can be relaxed.
1. Sit in Padmāsana (or with your legs crossed). Place the hands in _Jñāna Mudrā_ (Seal of Wisdom) with index fingers and thumbs touching and the middle, ring, and little fingers extended and awake. Straighten the arms and place the hands on the knees turning the palms up and spreading the three outer fingers on each hand.
2. Lift and widen the fronts of the armpits as you pull down and widen the backs of the armpits. The arms should be lightly stiffened at this point. This helps to set the shoulders down into the correct position. The whole front of the spine seems to rise.
3. Draw the lower belly, 2 inches below the navel, back and tone the perineum as its center lifts.
4. Drop the chin to rest it on the upper edge of the sternum, as the collarbones lift and widen. This is Jālandhara Bandha position. It is similar in form to a swan resting its head on its chest. If your chin does not easily rest on the sternum, use a folded washcloth for support, placing it on the sternum and resting the chin on the cloth.
5. Practice ten rounds of full Ujjāyī Prāṇāyāma in this position. Gaze softly down toward the tip of the nose, and use only natural, pleasant retentions of a few seconds at the ends of the breaths. Listen carefully to the quality of the breath. Such breathing is extraordinarily pleasant. Do not strain! This pose may be practiced on its own when practicing a gentle or moon day practice, or as part of the finishing sequence in a more active practice, in which case move directly into Utplutiḥ after completing the final exhale of this pose.
**U TPLUTIḤ**
_Uprooting Pose_
1. Still in Padmāsana, place the hands on the ground next to the hips and lift the knees off the floor. On an inhalation, lift the buttocks off the floor for ten breaths. Turn the face up toward the sky, but gaze down along the line of the nose to release the back of the neck. It might take some practice to stay for ten breaths or even to lift at all. That is not a problem. The attempt to lift will contract the necessary muscles and produce the desired result, eventually leading to the ability to lift.
2. While lifting, keep the hands evenly rooted into the floor, the jaw released, and the tongue soft. Maintain a sense of the shoulder blades flowing down the back as you lift.
3. After completing the pose, exhale as you swing the legs, still in Padmāsana, back through the arms and unwrap them quickly to land in full Catvāri position. Or lower down to the floor on an exhalation and unwrap the legs into a simple cross-legged position. Inhale to lift up, and exhale as you swing back through into Catvāri and a Half Vinyāsa.
**Ś AVĀSANA**
_Corpse Pose_
Sri K. Pattabhi Jois used to say that Śavāsana is the most difficult pose. Many students thought he was kidding, but once you've been practicing for a while, this sentiment rings true. The essence of the pose is to embody complete balance in all directions but also to find equanimity between the state of being completely alert and that of being absolutely relaxed. In more advanced Śavāsana, one does not fall asleep, but a calm and removed, yet alert, open feeling pervades the body and mind. In Śavāsana, all of the residue within the body, mind, and nervous system has time to be assimilated. Depending on the intensity of the particular practice and nonpractice circumstances, it may be necessary to hold Śavāsana for anywhere from ten to twenty minutes until everything has settled and has been integrated properly.
1. Lie on the back as if in Samasthitiḥ. Lightly stiffen the arms and legs. Roll the shoulders back and down to the floor. Draw the lower tips of the shoulder blades up into the body as the kidney area falls back and widens. Lightly press the back of the head into the floor with the chin a hair's breadth lower than the eyebrows.
2. Gaze, with the eyes closed, down the line of the nose. Feel the seed of a smile to "empty" the palate, as the breathing pulls the Mūlabandha like a steady flame.
3. This is the formal Tāḍāgī Mudrā pose and should be practiced before dissolving into full Śavāsana. Carefully arrange the body so it is symmetrical. Remain for one to five minutes in this position, breathing smoothly. Allow the breath to fine-tune the subtle alignment of the body.
4. Now relax. Let the breath go. Leave everything alone and as it is in the present moment. The mouth releases. The hands and feet release. The eyes soften. The heart floats up, bright and empty. The palms of the hands and the soles of the feet soften. Let the ears relax into listening. The tongue is silent, letting everything be, just as it is.
_Acknowledgments_
WE WOULD LIKE TO THANK ALL THOSE WONDERFUL people who have inspired us and who have contributed to the production of this book. First, we thank our son Gabe who drew the concise anatomical drawings and then the beautiful illustrations that capture the feeling of the subtle internal practices described in the text.
We appreciate the hard work, enthusiasm, and remarkable patience of Sara Bercholz of Shambhala Publications in getting this project to finally take shape. Other members of the closely knit Shambhala team have also played instrumental roles in bringing the book to life. Beth Frankl, our editor, has given support, insight, and encouragement at just the right moments along the way. Julia Gaviria has worked tirelessly with the details of editing when the two of us have gone cross-eyed looking for correct spelling, diacritical markings, and general form. We have benefited from the fine aesthetic of Shambhala's art department and those who worked carefully with us, Lora Zorian, Hazel Bercholz, Jim Zaccaria, and the book's designer, Steve Dyer. The project literally took form because of our photographer Robert Muratore, whose work with light and composition was so quick, brilliant, and spontaneous that even a couple of aging yoga teachers found the three-day photo shoot easy.
We also thank the worldwide Aṣṭāṅga Vinyāsa yoga community—our friends who are remarkably kind and humorous, being so grounded in themselves and their tradition that the synthesizing of the many traditions and views presented here is a natural process. We salute the broader background of our lineage in the brilliant work of Sri T. Krishnamacarya, B. K. S. Iyengar, T. K. V. Deshikachar, and so many others. And we are continually thankful to our teachers and friends in many manifestations of the Buddhist tradition who keep us sharp by holding up a brilliant mirror to the many practices and philosophies of the yoga tradition.
APPENDIX 1
_Ancient Wisdom, Contemporary Circumstances_
WE FIND OURSELVES IN THIS DAY, THIS TIME, THIS VERY moment. If we're lucky, we awaken to the circumstance—the beauty of what is before us, an astonishing, open kaleidoscope of interconnected patterns of perception. Waking up may happen by chance, or we may decide it's worth the time and effort to lay the groundwork for it to occur. Through yoga, meditation, and prāṇāyāma, we practice with open hearts and inquisitive minds, and occasionally there's a moment of clarity or insight as we touch the foundation (primordial immediacy) of breath, dṛṣṭi, bandha, and mudrā. Sometimes it all falls into place easily, like a puzzle just waiting to be solved; at other times, the pieces of form, movement, mind, and attention seem to be parts of disparate stories jumbled together. Yet we still return to whatever arises and the practice with a strand of faith that there's more to it all than we will ever know. We understand that all we really _must_ do is show up.
When things are going your way, it's simple to show up, and that's good. But the test is when things aren't going so well. When life throws the unexpected, complex, or difficult circumstance in your path, what then? Can you practice if you have only limited time, if everything is in flux, if habitual patterns of mind and body seem to have taken over your entire existence, or if you're ill or injured? Of course! In fact it's in those times of complexity, transition, difficulty, and doubt when practice is paramount.
Remember that the mind is always looking for an excuse not to practice, because the more we practice, the more the mind itself (our ego structure) starts to dissolve. You tell yourself, "I'm too stiff / too sore / too tired." You think the practice just isn't for you, it's too difficult, you can't practice because you're an emotional wreck, you're too old or too sick. And when you've proven each of these to be untrue, you come up with more excuses because somehow the easiest solution for the wandering mind is not to practice. That makes sense. The mind's full-time job is to take in information, organize the data, make decisions, construct theories of "you" and guide you smoothly through time. For a responsible, intelligent mind, the most disastrous prospect is to give up its identity—to dissolve. The same rational mind that gets dragged into a practice eventually begins to notice that _after_ practicing, there's a feeling of safety, happiness, clarity, and a lack of tension. During practice, the mind forms a relationship with its complementary partner, the breath, and in this context, it gradually softens to trust process rather than conceptualization. Eventually, you see that the mind's razor-sharp intellect and ability to discriminate are necessary, but when you grasp ideas too tightly, these very same ideas are a hindrance.
These appendices offer support material for a daily practice. The traditional sequences for Primary and Intermediate Series, the names of dṛṣṭi points, Sanskrit counting, and alternative practices for when you run into complex situations that might provide the out your mind is looking for to abandon the practice.
APPENDIX 2
_Invocation_
**vande gurūṇāṁ caraṇāravinde**
**sandarśita svātma sukhāva bodhe**
**niḥśreyase jāṅgalikāyamāne**
**saṁsāra hālāhala moha śāntyai**
I bow to the two lotus feet of the [plurality of] gurus, which awaken insight into the happiness of pure being, which are the complete absorption into joy, the jungle physician, eliminating the delusion caused by the poison of samsāra [conditioned existence].
**ābāhu puruṣākāraṁ**
**śankha cakrāsi dhāriṇam**
**sahasra śirasaṁ śvetaṁ**
**praṇamāmi patañjalim**
I prostrate before the sage Patañjali who has thousands of radiant white heads [as the divine serpent, Ananta] and who has, as far as his arms, assumed a human form, holding a conch shell [divine sound], a wheel [a discus of light or time], and a sword [discrimination].
**Oṁ**
APPENDIX 3
_Sequencing_
_Primary and Intermediate Series_
When practicing within the Aṣṭāṅga Vinyāsa tradition, we begin the practice with five Sūrya Namaskāra As and Bs, followed by the standing sequence. Sometimes the standing sequence is shortened to allow time for the more difficult poses in some of the advanced series. After the standing sequence, poses in a particular series (Primary, Intermediate, and so on) are practiced. Then the finishing poses are performed to absorb the practice fully.
When learning the Primary Series, poses are added one at a time, tacked on to the end of one's usual practice. When moving into a new series, poses are added to the end of the practice until the next series is complete. Then the two series can be practiced independently of one another. Once the Intermediate Series is learned, one should still maintain the skills of the Primary Series. If one knows all six of the basic series, then all of them should still be practiced on a rotational basis. Pattabhi Jois would even divide the series into segments and combine different segments to create special skills within the student.
One should stay with the series as a method of discovering the inner principles (forms) upon which the series have been constructed. These inner principles carry the value, interest, and purpose of the practice.
**I NTRODUCTORY SEQUENCE**
Samasthitiḥ
Sūrya Namaskāra A and B (see Appendix 5)
_**Sanskṛit Counting**_
One: ekam
Two: dve
Three: trīṇi
Four: catvāri
Five: pañca
Six: ṣaṭ
Seven: sapta
Eight: aṣṭa
Nine: nava
Ten: daśa
Eleven: ekādaśa
Twelve: dvādaśa
Thirteen: trayodaśa
Fourteen: caturdaśa
Fifteen: pañcadaśa
Sixteen: ṣodaśa
Seventeen: saptadaśa
Eighteen: aṣṭādaśa
Nineteen: ekoṇaviṁśati
Twenty: viṁśati
_**Eight Traditional Dṛṣṭi or Gazing Points**_
_añguṣṭha_ (the middle of the thumb)
_bhrūmadhya_ (between the eyebrows)
_nāsāgra_ (the end of the nose)
_hastāgra_ (the tip of the hand)
_pārśva_ (the side—right or left)
_ūrdhvā_ (upward)
_nābhi cakra_ (the navel)
_pādayoragra_ (the tip of the foot)
**TWO ADDITIONAL GAZING POINTS**
_antara_ dṛṣṭi (internal)
_ātmā_ dṛṣṭi (pure awareness, free of construction of subject and object)
APPENDIX 4
_Illustrations of Mūlabandha, Kidney Wings, and Cobra Hood_
_Mūlabandha_ is associated with an astonishingly deep concentration of the mind, initially causing a strong drawing up of the center of the pelvic floor along the central axis under the plane of the navel. Its effect is felt profoundly throughout the body and eventually causes a wonderful release starting in the soft palate and going up and back into the crown of the head, balancing and grounding all tensions and sensations of the body around the central axis. All this gives the subjective experience that the body and its surroundings are vibrant and joyous.
When _kidney wings_ finally open to spread back, out, and up, they balance the natural buoyancy and spreading of the open heart. This makes the flame of pure attention just above the center of the pelvic floor replicate itself at stations up along the central axis until finally it unfolds in full form just above the crown of the head.
The _cobra hoods_ are represented in the half-human and half-serpent (Nāga) deity form of the sage Patañjali. Here the apāna pattern manifests as the grounding tail of infinity, which in turn makes endless numbers of effulgent cobra heads open their hoods to shelter the beloved prāṇa pattern of the heart.
APPENDIX 5
_Sūrya Namaskāra_ _A and B_
**SŪRYA NAMASKĀRA A**
**SŪRYA NAMASKĀRA B**
_Index_
_Note: Index entries from the print edition of this book have been included for use as search terms. They can be located by using the search feature of your e-book reader_.
abdominal muscles
abduction and adduction
abhiniveśa
abhyāsa
accuate line
acetabular notch
acromion
Adho Mukha Śvānāsana (Downward Facing Dog)
Adho Mukha Vṛkṣāsana (Downward Facing Tree Pose)
Adiśeṣa
Advanced A Series
agonist
ahiṃsā
Air Vehicle Pose. _See_ Vatayanāsana
ajñā cakra
alignment
balance and
cultivating inner
in forward bends
skin flow and
in twists
ujjāyī and
using imagery for
whole-body patterns of
amṛta
amṛta plavana
anal sphincter muscles, anatomy
Añjali Mudrā
antagonist
anterior cruciate ligament (ACL)
apāna
in backbends
in forward bends
in Mūlabandha
in twists
_See also under_ prāṇa
aparigraha
_Aparokṣānubhuti_
appendicular skeleton
apuṇya
arches of feet
Ardha Baddha Padma Paśchimottānāsana (Half-Bound Lotus Forward Bend Pose)
Ardha Baddha Padmottānāsana (Half-Bound Lotus Forward Bend Pose)
Ardha Matsyendrāsana (Half Spinal Twist Pose)
Ardha Nāvāsana (Half Boat Pose)
_Art of Happiness_ (Dali Lama XIV)
āsana
ahiṃsā in
dṛṣṭi and
internal forms and
Mūlabandha, importance of in
plumb line, importance of to
prāna and apāna in
Prāṇa and citta in
ujjāyī in
Aṣṭa
Aṣṭānġa tradition
Aṣṭānġa Vinyāsa
breath in
criticism of
Downward Facing Dog in
finishing poses, importance of to
standing sequence in
Sūrya Namaskāra, role of in
_See also_ Intermediate Series; Primary Series
asmitā
asteya
asymmetry
Avalokiteśvara
avidyā
axial skeleton
backbends
Baddha Hasta Śīrṣāsana (Bound Hand Headstand Pose)
Baddha Koṇāsana (Bound Angle Pose)
Baddha Padmāsana
Bakāsana A, B (Crane Pose)
balancing poses
Balāsana (Child's Pose)
Bali
bandhas. _See also individual bandhas_
Bharadvājāsana (Bharadhvāja's Pose)
Bhekāsana (Frog Pose)
Bhujanġāsana (Cobra Pose)
Bhujapīḍāsana (Arm Squeezing Pose)
biceps
biceps femoris
Big Toe Pose. _See_ Pādānġuṣṭhāsana
bindu dhāraṇa
Boat Pose. _See_ Nāvāsana
bodhisattva vow
Bound Angle Pose. _See_ Baddha Koṇāsana
Bound Hand Headstand Pose. _See_ Baddha Hasta Śīrṣāsana
Brahmā
Brahmā nāḍī
Brahmacarya
breath
in backbends
listening to
movement and
in Mūlabandha
preferences in
psoas buttons for training
in psoas line
in Reclining Angle Pose
retention
riding
in Samasthitiḥ
skeletal system and
in Sūrya Namaskāra
trusting
in twists
Bridge Poses. _See_ Dhanurāsana; Setu Bandhāsana
Bridge Shoulderstand. _See_ Setu Bandha Sarvānġāsana
Cakorāsana (Moonbeam-Drinking Bird Pose)
cakras
Cakrāsana (Wheel Pose)
Calf Face Pose. _See_ Gomukhāsana
Camel Pose. _See_ Uṣṭrāsana
Caturdaśa
Catvāri and variations
celibacy
central channel (suṣumnā nāḍī)
in backbends
bandhas and
cakras and
in Downward Facing Dog
Gaṇeśa and
in headstands
internal forms and
in Khecarī Mudrā
in kidney wings
in Mūlabandha
palate and
paramātman in
prāṇa and apāna in
in Samasthitiḥ
ujjāyī and
umbilical breathing and
cervical spine
Child's Pose. _See_ Balāsana
circulatory system
cit
citta
clavicle
cobra hood
Cobra Pose. _See_ Bhujanġāsana coccygeus
coccyx
apāna and
in backbends
in Downward Facing Dog
in holding serpent's tail
in Mūlabandha
pelvic floor and
sacrum and
in Samasthitiḥ
in twists
Cock Pose. _See_ Kukkuṭāsana
co-contraction
Compass Pose. _See_ Dikāsana
compassion
as amṛta
in boundless abodes
discriminating awareness and
ego and
ethics and
in Khecarī Mudrā
in Mūlabandha
in releasing palate
for suffering of others
visualization and
contemplative practices
Corpse Pose. _See_ Śavāsana
costal cartilage
counternutation
Crane Poses. _See_ Bakāsana; Karandavāsana
Crocodile Pose. _See_ Nakrāsana
Daṇḍāsana (Staff Pose)
Daśa
death
deep six
deity visualization
Dhanurāsana (Bridge Pose)
diaphragm
Difficult Pose. _See_ Utkaṭāsana
digestion
Dikāsana (Compass Pose)
discriminating awareness
divine sound (nāda)
Downward Facing Dog. _See_ Adho Mukha Śvānāsana
dṛṣṭi
in finishing poses
in Khecarī Mudrā
in psoas line
in Samasthitiḥ
traditional, list of
ujjāyī and
as view
_See also_ gaze
duḥkha
Dvādaśa
Dve
dveṣa
Dvi Pāda Śīrṣāsana (Two Feet Behind the Head Pose)
Dvi Pāda Viparīta Daṇḍāsana (Two-Footed Inverted Staff Pose)
eccentric contraction
Eka Pāda family
Eka Pāda Rāja Kapotāsana (One-Footed King Pigeon Pose)
Eka Pāda Śīrṣāsana (One Foot Behind the Head Pose)
Eka Pāda Viparīta Daṇḍāsana (One-Footed Inverted Staff Pose)
Ekādaśa
Ekam
embodiment
emotions
erector spinae
ethics
fascia
fear
feet, general placement of
femoroacetabular impingement
femurs
fibula
finishing poses
Fish Pose. _See_ Matsyāsana
Forward Bend to the Side Pose. _See_ Pārśvottānāsana
Forward Bend with Feet Spread Pose. _See_ Prasārita Pādottānāsana A, B, C, D
forward bends
four boundless abodes
foveal ligament
Full Vinyāsa
Gaṇeśa
Gaṇeśa Belly
Garbha Piṇḍāsana
Garuda
gaze. _See also_ dṛṣṭi
gemellus inferior and superior
glenoid fossa
gluteus maximus
Gomukhāsana (Calf Face Pose)
granthis (energy blockages)
Halāsana (Plough Pose)
Half Padmāsana
Half Vinyāsa
Half Vīrāsana
Half-Bound Lotus Forward Bend Pose. _See_ Ardha Baddha Padmottānāsana
hamstrings
in backbends
in forward bends
Hand and Foot Pose. _See_ Pādahastāsana
Handstand. _See_ Adho Mukha Vṛkṣāsana (Downward Facing Tree Pose)
Hanumanāsana (Hanuman's Pose)
hard palate
hatha yoga
_Haṭha Yoga Pradīpikā_
Headstand. _See_ Śirṣāsana
Head-to-Knee Pose, A, B, C. _See_ Jānuśīrṣāsana
heart
in backbends
in cobra hood
in forward bends
in Jālandhara Bandha
in nauli
prāṇa and
in psoas line
in Samasthitiḥ
scalenes and
in Utkaṭāsana
Hero Pose. _See_ Vīrāsana
Hinge Pose. _See_ Parighāsana
hip flexors
hips
in forward bends
in twists
holding serpent's tail
humerus
idā nāḍī (moon channel)
iliacus
iliococcygeus
iliopsoas muscle
ilium
imagination and imagery
in animal pose names
central channel and
Prāṇa and
in psoas line
impermanence
Indian mythology and iconography
infraspinatus
insertion of muscle
insight
integration
intention
interconnectedness
intercostal muscles, external and internal
interdependence
Intermediate Series
backbends in
balances in
headstands in
twists in
internal forms
alignment and
in backbends
defining
in forward bends
importance of
in headstand
importance of
as sādhanā bhāṣā
in standing poses
in Sūrya Namaskāra
invocations
iṣṭa devatā
Īśvara-praṇidhāna
Jālandhara Bandha
Jānuśīrṣāsana A, B, C (Head-to-Knee Pose)
Jihvā Bandha
Jñāna Mudrā
joints
acetabulofemoral
atlanto-axial
atlanto-occipital
ball-and-socket
glenohumeral
hinge
sternoclavicular
synovial hinge
_See also_ sacroilian (SI) joints
Jois, Sri K. Pattabhi
Jumping Back
Kapotāsana (Pigeon Pose)
Karandavāsana (Crane Pose)
Karṇa Pīḍāsana (Squeezing the Ears Pose)
Khecarī Mudrā
kidney wings
kindness
boundless
Prāṇa and
toward oneself
in yamas and niyamas
kleśas
knees
Krauñcha
Krauñchāsana
kriya practices
Kukkuṭāsana (Cock Pose)
Kuṇḍalinī
_Kūrma Purāṇa_
Kūrmāsana (Turtle Pose)
labrum
Laghu Vajrāsana (Light Thunderbolt Pose)
lateral collateral ligament (LCL)
life breath. _See_ Prāṇa
Light Thunderbolt Pose. _See_ Laghu Vajrāsana
linea alba
lineage
Locust Pose. _See_ Śalabhāsana A, B
lotus, symbolism of. _See also_ sahasrāra (thousand-petaled lotus)
Lotus Pose. _See_ Padmāsana
lumbar spine
maitrī
maṇipūra cakra. _See_ nābhi cakra
mantra
Marīchyāsana (Marīchi's Pose)
A, B
C, D
Matsyāsana (Fish Pose)
Mayūrāsana (Peacock Pose)
medial collateral ligament (MCL)
meditation (meditative states)
asana and
in cave of sacrum
embodiment and
finishing poses and
internal forms and
nāḍīs and
name and form in
in Padmāsana
in Sūrya Namaskāra
Vīrāsana for
viveka khyātiḥ in
meniscus
mind
altered states of
in balancing poses
discursive
dṛṣṭi and
mantra and
in mudrās
pelvis floor and
Prāṇa and
resetting
role of
symbols and
wandering
moon channel. _See_ idā nāḍī
muditā
mudrā. _See also_ _individual mudrās_
Mukta Hasta Śīrṣāsana A, B, C, (Unbound Hand Headstand Pose)
mūla
Mūlabandha
in backbends
in Downward Facing Dog
feet and pelvic floor in
in finishing poses
in forward bends
Gaṇeśa and
hip joints and
imagery for
importance of
other bandhas and
palate in
pelvic floor in
psoas and
sacrum and
in Samasthitiḥ
serpent's tail in
in Sun Salutation
in twists
Yoni Mudrā and
muscles, origin of
muscular system
myofascia
mystical experiences
nābhi cakra
nāḍīs (internal channels). _See also_ central channel (suṣumnā nāḍī)
nagas
Nakrāsana (Crocodile Pose)
nasal septum
nauli
Nava
Nāvāsana (Boat Pose)
nectar
nervous system
cobra hood and
impact of matrī on, integrating
nāḍīs and
palate release and
pattern sensations in
plumb line in
reciprocal inhibition in
resetting
twists, effect of on
whole-body patterns in
nirodha
niyamas
nonattachment
Noose Pose. _See_ Paśāsana
nutation
oblique muscles, internal and external
obturator externus and internus
Old Dog Pose
One Foot Behind the Head Pose. _See_ Eka Pāda Śīrṣāsana
One-Footed Inverted Staff Pose. _See_ Eka Pāda Viparīta Daṇḍāsana
One-Footed King Pigeon Pose. _See_ Eka Pāda Rāja Kapotāsana
One Leg Reversed Forward Bend Pose. _See_ Tiryang Mukha Eka Pāda Paśchimottānāsana
Pādahastāsana (Hand and Foot Pose)
Pādānġuṣṭhāsana (Big Toe Pose)
Padmāsana (Lotus Pose)
Padmāsana with Ujjāyī Prāṇāyāma (Lotus Pose Variation)
palate
in Jālandhara Bandha
in Mūlabandha
in nauli
opening of
releasing
root of
palate-perineum reflex
Pañca and variations
Pañcadaśa
paramātman
Parighāsana (Hinge Pose)
Parivṛtta Pārśvakoṇāsana (Twisted Side-Angle Pose)
Parivṛtta Trikoṇāsana (Revolving Triangle Pose)
Pārśva Dhanurāsana (Side-Angle Bridge Pose)
Pārśvakoṇāsana (Side-Angle Pose)
Pārśvottānāsana (Forward Bend to the Side Pose)
Paśāsana (Noose Pose)
Paśchimottānāsana A, B, C (Upward-Facing Forward Bend)
Paśchimottānāsana and variations (Tuning Up the Back Pose)
patella
patience
Peacock Feather Pose. _See_ Piñcha Mayūrāsana
pelvic diaphragm. _See_ pelvic floor
pelvic floor
apāna and
in backbends
balance in
feet reflecting
in finishing poses
in forward bends
in holding serpent's tail
in Mūlabandha
nauli and
palate and
psoas and
respiratory diaphragm and
sacrum and
in Samasthitiḥ
skeletal system and
in standing poses
in twists
ujjāyī and
visualizing
Yoni Mudrā and
pelvic nostril
pelvis
perception
pieriformis
Pigeon Pose. _See_ Kapotāsana
Piñcha Mayūrāsana (Peacock Feather Pose)
Piṇḍāsana (Embryo Pose)
piñgālā nāḍī (sun channel)
pituitary gland
Plough Pose. _See_ Halāsana
plumb line
in balancing poses
in headstands
scalenes and
in skeletal system
in standing poses
_See also_ central channel (suṣumnā nāḍī)
Pond Mudra. _See_ Tāḍāgī Mudrā
posterior cruciate ligament (PCL)
prāṇa
and apāna, integration of
in backbends
in forward bends
separation from apāna
Prāṇa
in backbends
in bandhas
in central channel
citta and
finishing poses and
in integrating body and mind
movement and
nāḍīs and
in psoas line
in releasing palate
seed sound for
skin and
during Yoni Mudrā
prāṇāyāma
Prasārita Pādottānāsana A, B, C, D
Primary Series
forward bends in
twists in
Proust, Marcel
psoas ( _psoas major_ )
awakening
in backbends
buttons
kidney wings and
line of
in Mini Uḍḍīyāna Bandha
stretch
in Ūrdhvā Dhanurāsana
pubic bone
pubic crest
pubic symphysis
pubococcygeus (PC) muscle
puṇya
Puppy Pose
Pūrṇa Matsyendrāsana (Full Spinal Twist Pose)
Pūrvottanāsana (Stretching Up the Front Pose)
quadratus femoris
quadratus lumborum (QL)
quadriceps
radius bone
rāga
range of motion
rasa
reciprocal inhibition
Reclining Angle Pose. _See_ Supta Koṇāsana
Reclining Feet Up Thunderbolt Pose. _See_ Supta Ūrdhva Pāda Vajrāsana
Reclining Hand-to-Big-Toe Pose. _See_ Supta Hasta Pādānġuṣṭhāsana
Reclining Lightning Bolt Pose. _See_ Supta Vajrāsana
Reclining Turtle Pose. _See_ Supta Kūrmāsana
rectus abdominis
rectus femoris
_Remembrance of Things Past_ (Proust)
resistance
respiratory system
Revolving Triangle Pose. _See_ Parivṛtta Trikoṇāsana
rhomboid muscle
rib cage
rotator cuff
SĀ-HAṀ mantra
sacroilian (SI) joints
sacrum
sādhanā bhāṣā (practice language)
sahasrāra (thousand-petaled lotus)
Śalabhāsana A, B (Locust Pose)
samāna vāyu
Samasthitiḥ
as quintessential balance
in Sūrya Namaskāra
ujjāyī in
saṃskāras
saṃtoṣa
Sanskrit counting
Sapta
Saptadaśa
sartorius muscle
Sarvānġāsana (Shoulderstand)
sattva
satyam
śauca
Śavāsana (Corpse Pose)
scalene muscles
scapula
Seated Angle Pose. _See_ Upaviṣṭha Koṇāsana A, B
semimembranosus muscle
semitendinosus muscle
serratus anterior
Setu Bandha Sarvānġāsana (Bridge Shoulderstand)
Setu Bandhāsana (Bridge Pose)
ṣaṭ and variations
ṣodaśa
shoulders
Shoulderstand. _See_ Sarvānġāsana
Side-Angle Bridge Pose. _See_ Pārśva Dhanurāsana
Side-Angle Pose. _See_ Pārśvakoṇāsana Śirṣāsana (Headstand)
sitting bone (ischial tuberosity)
in forward bends
in Mūlabandha
nutation and counternutation of
origin and insertion, examples of
in twists
in Utkaṭāsana
Śiva and Śakti
_Śiva Saṃhitā_
skeletal system
skin flow
Small Water Bird Pose. _See_ Ṭiṭṭibhāsana
soft palate
sphenoid sinus
Sphinx Pose
spine/spinal column
in backbends
in finishing poses
in forward bends
pelvic floor and
psoas and
respiratory diaphragm and
in twists
Squeezing the Ears Pose. _See_ Karṇa Pīḍāsana
Staff Pose. _See_ Daṇḍāsana
standing poses
sternocleidomastoid muscles
Stretching Up the Front Pose. _See_ Pūrvottanāsana
suboccipital muscles
subscapularis
subtle body
anatomy of
balance in
encountering
patterns in
Prāṇa and citta in
psoas in
purification of
working with
suffering. _See also_ duḥkha
sukha
sun channel. _See_ piñgālā nāḍī
Sun Salutation. _See_ Sūrya Namaskāra
supraspinatus
Supta Hasta Pādānġuṣṭhāsana (Reclining Hand-to-Big-Toe Pose)
Supta Koṇāsana (Reclining Angle Pose)
Supta Kūrmāsana (Reclining Turtle Pose)
Supta Ūrdhva Pāda Vajrāsana (Reclining Feet Up Thunderbolt Pose)
Supta Vajrāsana (Reclining Lightning Bolt Pose)
Supta Vīrāsana (Reclining Hero Pose)
Sūrya Namaskāra (Sun Salutation)
dṛṣṭi during
origin and insertion in
palate-perineum reflex in
transitioning in
Sūrya Namaskāra A
Sūrya Namaskāra B
suṣumnā nāḍī. _See_ central channel
svādhyāya (self-inquiry)
Tāḍāgī Mudrā (Pond Mudra)
tail tucking
tamas
tapas
Tenzin Gyatso, H. H. Dalai Lama XIV
teres minor
ṬHAṀ mantra
thighbone
thoracic spine
throat
tibia
Tiryang Mukha Eka Pāda Paśchimottānāsana (One Leg Reversed Forward Bend Pose)
Ṭiṭṭibhāsana (Small Water Bird Pose) A B, C, D
tongue
transversus abdominis
trapezius, upper
Trayodaśa
Triangle Pose. _See_ Trikoṇāsana
triceps
Trikoṇāsana (Triangle Pose)
Trīṇi
_Trivikrama_
trochanters, greater and lesser
trust
Turning Up the Back Pose. _See_ Paśchimottānāsana and variations
Turtle Pose. _See_ Kūrmāsana
Twisted Side-Angle Pose. _See_ Parivṛtta Pārśvakoṇāsana
twists
Two Big Toes Pose. _See_ Ubhaya Pādānġuṣṭhāsana
Two Feet Behind the Head Pose. _See_ Dvi Pāda Śīrṣāsana
Ubhaya Pādānġuṣṭhāsana (Two Big Toes Pose)
udāna
Udāna Vāyu
Uḍḍīyāna Bandha
in finishing poses
in forward bends
Mini
nauli in
psoas in
sacrum and
in Staff Pose
Tāḍāgī Mudrā and
in Utkaṭāsana
ujjāyī
Ujjāyī Prāṇāyāma
ulna
umbilical breathing
Unbound Hand Headstand Pose A, B, C. _See_ Mukta Hasta Śīrṣāsana
union of opposites
Upaniṣads
Upaviṣṭha Koṇāsana A, B (Seated Angle Pose)
upekṣā
Uprooting Pose. _See_ Utplutiḥ (Ūrdhvā Padmāsana)
Upward Bow Pose. _See_ Ūrdhvā Dhanurāsana
Upward Facing Dog. _See_ Ūrdhvā Mukha Śvānāsana
Upward-Facing Forward Bend. _See_ Paśchimottānāsana
Upward-Facing Forward Bend Pose. _See_ Ūrdhvā Mukha Paśchimottānāsana
Upwardly Stretched Legs Pose. _See_ Uttāna Pādāsana
Ūrdhvā Dhanurāsana (Upward Bow Pose)
Ūrdhvā Mukha Paśchimottānāsana (Upward-Facing Forward Bend Pose)
Ūrdhvā Mukha Śvānāsana (Upward Facing Dog)
Ūrdhvā Padmāsana (Upturned Lotus Pose)
Uṣṭrāsana (Camel Pose)
Utkaṭāsana (Difficult Pose)
Utplutiḥ (Uprooting Pose)
Uttāna Pādāsana (Upwardly Stretched Legs Pose)
Utthita Hasta Pādānġuṣṭhāsana (Standing Hand-to-Big-Toe Pose)
uvula
vairāgyam
Vāsiṣṭhāsana (Vāsiṣṭha's Pose)
Vatayanāsana (Air Vehicle Pose)
vinyāsa
cobra hood image and
complementary movement in
definitions of
deity visualization and
flow of poses in
Prāṇa and citta in
visualization in
Vīrabhadrāsana A, B (Warrior Pose)
Vīrāsana (Hero Pose)
Viṣṇu
visualizations. _See also_ deity visualization
viveka khyātiḥ
vṛtti
vyāna vāyu
Warrior Pose. _See_ Vīrabhadrāsana A, B
Wheel Pose. _See_ Cakrāsana
whole-body patterns
xiphoid process
yamas
Yoga Mudrā (Seal of Yoga)
Yoga Nidrāsana (Yoga Sleeping Pose)
Yoga Sūtra
Yoni Mudrā
_About the Authors_
**R ICHARD FREEMAN** has been a student of yoga since 1968. He has studied and taught a variety of yoga and contemplative traditions, which he synthesizes in his approach to the Aṣṭāṅga Vinyāsa methodology as taught by his principal teacher, the late Sri K. Pattabhi Jois of Mysore, India. He remains an avid student fascinated by the linking points between different traditions and cultures. He is the cofounder of the Yoga Workshop in Boulder, Colorado, and author of the _Mirror of Yoga_ (Shambhala).
**M ARY TAYLOR** began studying yoga in 1972, but it was not until 1988 and finding her primary teacher, Sri K. Pattabhi Jois, and the Aṣṭāṅga Vinyāsa system that she felt the profound and transformative impact that a dedicated and daily practice can have. She continues to study and practice, incorporating the residue that is produced on the mat into other aspects of her life, training as a professional chef, and applying a contemplative approach to caregiving within hospital settings through her teaching. She cofounded with Richard the Yoga Workshop and is the author of several books.
_For more information, please visit_ richardfreemanyoga.com.
ALSO BY RICHARD FREEMAN
_The Mirror of Yoga_
ALSO BY MARY TAYLOR
_What Are You Hungry For? Women, Food, and Spirituality_
_New Vegetarian Classics: Soups_
_New Vegetarian Classics: Entrees_
Sign up to receive news and special offers from Shambhala Publications.
Or visit us online to sign up at shambhala.com/eshambhala.
| {
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{"url":"https:\/\/en.wikipedia.org\/wiki\/RGB_color_space","text":"# RGB color spaces\n\n(Redirected from RGB color space)\n1931 CIE chromaticity diagram showing some RGB color spaces as defined by their chromaticity triangles.\n\nAn RGB color space is any additive color space based on the RGB color model.[1][2]\n\nRGB color spaces are commonly found describing the input signal to display devices such as television screens and computer monitors.\n\n## Definition\n\nRGB-Cube\n\nThe normal human eye contains three types of color-sensitive cone cells. Each cell is responsive to light of either long, medium, or short wavelengths, which we generally categorize as red, green, and blue. Taken together, the responses of these cone cells are called the Tristimulus values, and the combination of their responses is processed into the psychological effect of color vision.\n\nAn RGB color space is defined by:\n\nAn RGB color space uses primaries based on the RGB color model. Mixing of the three primaries in different proportions then creates the perception of colors other than the primaries. Applying Grassmann's law of light additivity, the range of colors that can be produced are those enclosed within the triangle on the chromaticity diagram defined using the primaries as vertices. The TRC and white point further define the possible colors, creating a volume of encodable colors enclosed within the 3D-triangle.[3]\n\nThe primary colors are usually specified in terms of their xyY chromaticity coordinates, though the u\u02b9,v\u02b9 coordinates from the UCS chromaticity diagram may be used. Both xyY and u\u02b9,v\u02b9 are derived from the CIE 1931 color space, a device independent space also known as XYZ which covers the full gamut of human-perceptible colors visible to the CIE 2\u00b0 standard observer.\n\n## Applications\n\nOne million colors in RGB space, visible in full-size image.\n\nRGB color spaces are well-suited to describing the electronic display of color, such as computer monitors and color television. These devices often reproduce colours using an array of red, green, and blue phosphors agitated by a cathode ray tube (CRT), or an array of red, green, and blue LCDs lit by a backlight, and are therefore naturally described by an additive color model with RGB primaries.\n\nEarly examples of RGB color spaces came with the adoption of the NTSC color television standard in 1953 across North America, followed by PAL and SECAM covering the rest of the world. These early RGB spaces were defined in part by the phosphor used by CRTs in use at the time, and the gamma of the electron beam. While these color spaces reproduced the intended colors using additive red, green, and blue primaries, the broadcast signal itself was encoded from RGB components to a composite signal such as YIQ, and decoded back by the receiver into RGB signals for display.\n\nHDTV uses the BT.709 color space, later repurposed for computer monitors as sRGB. Both use the same color primaries and white point, but different transfer functions, as HDTV is intended for a dark living room while sRGB is intended for a brighter office environment.[citation needed] The gamut of these spaces is limited, covering only 35.9% of the CIE 1931 gamut.[4] While this allows the use of a limited bit depth without causing color banding, and therefore reduces transmission bandwidth, it also prevents the encoding of deeply saturated colors that might be available in an alternate color spaces. Some RGB color spaces such as Adobe RGB and ProPhoto intended for the creation, rather than transmission, of images are designed with expanded gamuts to address this issue, however this does not mean the larger space has 'more colors\". The numerical quantity of colors is related to bit depth and not the size or shape of the gamut. A large space with a low bit depth can be detrimental to the gamut density and result in high ${\\displaystyle \\Delta E}$ errors[further explanation needed].\n\nMore recent color spaces such as Rec. 2020 for UHD-TVs define an extremely large gamut covering 63.3% of the CIE 1931 space.[5] This standard is not currently realisable with current LCD technology, and alternative architectures such as quantum dot[6] or OLED[7] based devices are currently in development.\n\n## RGB color space specifications\n\nRGB color spaces\nColor space Reference Standard Year White point Primaries Display Transfer function parameters\nRed Green Blue \u03b3 \u03b1 \u03b2 \u03b4 \u03b2\u03b4\nx\u0280 y\u0280 x\u0262 y\u0262 x\u0299 y\u0299 EOTF a + 1 K0\/\u03c6 = Et \u03c6 K0\nNTSC-J Based on NTSC(M) 1987 D93 0.63 0.34 0.31 0.595 0.155 0.07 2.5\nNTSC, MUSE SMPTE RP 145 (C), 170M, 240M 1987 D65 20\/9 1.1115 0.0057 4 0.0228\nApple RGB (Apple Computer) 0.625 0.28 1.8\nPAL \/ SECAM EBU 3213-E, BT.470\/601 (B\/G) 1970 0.64 0.33 0.29 0.60 0.15 0.06 2.8 14\/5\nsRGB IEC 61966-2-1 1996, 1999 0.30 2.2 12\/5 1.055 0.0031308 12.92 0.04045\nscRGB IEC 61966-2-2 2003\nHDTV ITU-R BT.709 1999 2.4 20\/9 1.099 0.004 4.5 0.018\nM.A.C. ITU-R BO.650-2[8] 1985 0.67 0.14 0.08 2.8\nNTSC-FCC ITU-R BT.470\/601 (M) 1953 C 2.5 11\/5\nPAL-M ITU-R BT.470-6[9] 1972 2.2\neciRGB ISO 22028-4 2008, 2012 D50 1.8 3 1.16 0.008856 9.033 0.08\nDCI-P3 SMPTE RP 431-2 2011 6300K 0.68 0.32 0.265 0.69 0.15 0.06 2.6 13\/5\nDisplay P3 SMPTE EG 432-1 2010 D65 ~2.2 12\/5 1.055 0.0031308 12.92 0.04045\nUHDTV ITU-R BT.2020, BT.2100 2012, 2016 0.708 0.292 0.170 0.797 0.131 0.046 2.4 1.0993 0.018054 4.5 0.081243\nWide Gamut (Adobe) D50 0.735 0.265 0.115 0.826 0.157 0.018 2.2 563\/256\nRIMM ISO 22028-3 2006, 2012 0.7347 0.2653 0.1596 0.8404 0.0366 0.0001 2.222 20\/9 1.099 0.0018 5.5 0.099\nProPhoto (ROMM) ISO 22028-2 2006, 2013 1.8 9\/5 1 0.001953125 16 0.031248\nCIE RGB CIE 1931 color space 1931 E 0.2738 0.7174 0.1666 0.0089\nCIE XYZ 1 0 0 1 0 0 1\n\nThe CIE 1931 color space standard defines both the CIE RGB space, which is an RGB color space with monochromatic primaries, and the CIE XYZ color space, which is functionally similar to a linear RGB color space, however the primaries are not physically realizable, thus are not described as red, green, and blue.\n\nM.A.C. is not to be confused with MacOS. Here, M.A.C.refers to Multiplexed Analogue Components.","date":"2023-01-28 21:35:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 1, \"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\": 0.4659821093082428, \"perplexity\": 3370.2520602436557}, \"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-2023-06\/segments\/1674764499654.54\/warc\/CC-MAIN-20230128184907-20230128214907-00064.warc.gz\"}"} | null | null |
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Masoom(8.4 rating) 1 Starrer Masoom released in 1983, this was an Drama movie and it received 8.4 average rating on IMDB. Cast By – Naseeruddin Shah, Shabana Azmi, Jugal Hansraj, Urmila Matondkar,… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,742 |
«Майя и Пайя» — советский фильм-сказка 1990 года режиссёра Гунара Пиесиса по мотивам одноименной пьесы-сказки Анны Бригадере.
Сюжет
Сказка о чудесных приключениях в подземном царстве двух сестёр: доброй, трудолюбивой и честной сироте Майе и избалованной, ленивой, завистливой и мстительной дочери свирепой мачехи Пайе. В подземном царстве их ждут всевозможные испытания, и в итоге каждая из них получает награду по заслугам.
В ролях
Элина Силиня — Майя
Иева Савича — Пайя
Лилита Озолиня — мачеха
Инга Айзбалте — Лайма
Эльза Радзиня — Лайма
Герман Тихонов — Варис
Эвалдс Валтерс — дед Вариса
Арвидс Озолиньш — Каупиньш
Петерис Лиепиньш — Самтцупере
Дзинтра Клетниеце — ведьма
Улдис Пуцитис — ''отец
Антра Лиедскалныня — Зиле
Литература
Айна Адермане — В ожидании чуда (О создаваемом худож. фильме Риж. киностудии «Майя и Пайя») // Ригас Балсс, 26 января 1990
Галина Фролова — «Майя и Пайя» (O создаваемом одноим. худож . фильме Риж . киностудии) // Кино, № 12, 1989. — с. 4-5
О. Шумяцкая — Майя и Пайя (Об одноим. худож . фильме Риж. киностудии. Режиссёр Гунар Писсис) // Киномеханик, № 6, 1991. — с. 36-37
Майя и Пайя // Домашняя синематека: отечественное кино 1918—1996. -М.: Дубль-Д, 1996. — 520 с. — с. 236
Фильмы СССР 1990 года
Фильмы-сказки СССР
Экранизации литературных сказок
Фильмы Рижской киностудии
Фильмы, выпущенные сразу на видеоносителях | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,825 |
Q: register type variable value is not updating Problem statement
Consider a parking lot with a single entry and exit gate. Two pairs of photo sensors (a,b) are used to monitor the activity of cars.
When an object is between the photo transmitter and the photo receiver, the light is blocked and the corresponding output is asserted to 1. By monitoring the events of two sensors, we can determine whether a car is entering or exiting or a pedestrian is passing through. For example, the following sequence indicates that a car enters the lot:
Initially, both sensors are unblocked (i.e., the a and b signals are "00").
Sensor a is blocked (i.e., the a and b signals are " 10").
Both sensors are blocked (i.e., the a and b signals are "1 1 ").
Sensor a is unblocked (i.e., the a and b signals are "01 ").
Both sensors becomes unblocked (i.e., the a and b signals are "00").
Design a parking lot occupancy counter as follows:
*
*Design an FSM with two input signals, a and b, and two output signals, enter and exit . The enter and exit signals assert one clock cycle when a car enters and one clock cycle when a car exits the lot, respectively.
*Derive the HDL code for the FSM.
The following code is used:
`timescale 1ns / 1ps
module assignment1(a,b,clk,reset,en,ex);
input a,b,clk,reset;
output en,ex;
reg en,ex;
reg [1:0] state,next_state;
parameter unBlocked=2'b00,aBlocked=2'b10,bothBlocked=2'b11,bBlocked=2'b01;
// State register block
//if(reset)
//state<=unBlocked;
//else
always@(posedge clk) begin
state<=next_state;
end
//
always @(state or a or b) begin
case (state)
unBlocked: if((a==1'b1) && (b==1'b0))
begin next_state=aBlocked; en=0; end
else next_state=unBlocked;
aBlocked:if((a==1'b1) && (b==1'b1))
begin next_state=bothBlocked; en=0; end
else next_state=unBlocked;
bothBlocked: if((a==1'b0) && (b==1'b1))
begin next_state=bBlocked; en=0; end
else next_state=unBlocked;
bBlocked: if((a==1'b0) && (b==1'b0))
begin next_state=unBlocked; en=1; end
else next_state=unBlocked;
endcase
end
endmodule
I am trying to create a moore machine when car enters. In next_state variable I am only getting 00 i.e unBlocked state...That is states are not updating according to the inputs.Here inputs are a,b...car enters into the parking only when(a,b) take values 00,10,11,01,00.can you please help in this?
A: It looks like the reset is not being used. This means your state is not being intialized, which means it is an 'x' or unknown value at the beginning of the simulation. This then causes the case statement to not match anything and this leads to no response to the inputs.
On general you should be vigilant about the basics like always resetting registers and also checking for any x's in simulation after reset is released. It will save you loads of time down the road
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,822 |
Q: What level of permissions to give per site What is best practice for departmental sites' permissions? Should certain users within each department have permission to create objects (pages, lists, etc.) or should the permissions only allow them to edit already existing objects?
From our experience, when creating lists, it might be necessary to create site columns and content types, so that adds to the training necessary for the end users if they're given permission to create lists.
Obviously, it has to be a balance between user empowerment and central administration to maintain security and best practices. What approach do you take on this? Thanks.
A: As far as departmental site is concerned each department has a seperate site. So its better to provide users with both create and edit permission. But with in that, if you need to restrict the permission create groups with unique permissions and add users to it.
A: To be honest, the best practice is to figure out what works best for your organisation. As you've noticed, there are a large number of factors involved - the cost of centralising administration, the expertise and training of the users, what permissions levels you need within a site, do you give different rights at the list level, and so on. It very much depends on what you're trying to do.
I tend to approach the problem by identifying the types of users within a site, and then the rights they need at each level (e.g. lists or site), and then try setting one up with those rights to see if there is anything that I've missed (e.g. needing read rights on lists used by lookup columns, etc.).
| {
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}\label{sec:intro}
\subsection{Eigenvalue problem, inverse spectral method}\label{ssec:evalpblm}
We consider the non-self-adjoint Zakharov--Shabat eigenvalue problem \cite{ZS}:
\begin{equation}\label{eq:zs}
\epsilon\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}} x}
\vec{w} =
\begin{bmatrix}
-\ensuremath{\mathrm{i}}\lambda & \psi_0 \\
-\psi_0^* & \ensuremath{\mathrm{i}}\lambda
\end{bmatrix}
\vec{w}\,.
\end{equation}
In equation \eqref{eq:zs}, we have written
\[
\vec{w}(x;\ensuremath{\lambda},\ensuremath{\epsilon})=\begin{pmatrix} w_1(x;\ensuremath{\lambda},\ensuremath{\epsilon}) \\ w_2(x;\ensuremath{\lambda},\ensuremath{\epsilon})\end{pmatrix}
\]
for the solution, $\lambda\in\mathbb{C}$ is a spectral parameter, and the function $\psi_0:\ensuremath{\mathbb{R}}\to\ensuremath{\mathbb{C}}$ is a known potential. We suppose, to begin our discussion, that $\psi_0$ is specified by real-valued amplitude and phase functions $A_0$ and $S_0$, so that
\[
\psi_0(x)=A_0(x)\exp\big(\ensuremath{\mathrm{i}} S_0(x)/\ensuremath{\epsilon}\big)\,.
\]
We identify precise assumptions on $A_0$, $S_0$ in \S\ref{ssec:potentials} below.
The parameter $\ensuremath{\epsilon}\in\ensuremath{\mathbb{R}}$ is assumed to be positive but small; this introduces the ``semiclassical'' scaling.
Our principal interest here is in those values of $\lambda\in\ensuremath{\mathbb{C}}$ for which \eqref{eq:zs} has a solution in $L^2(\ensuremath{\mathbb{R}})^2$; these values comprise the discrete spectrum---the eigenvalues of \eqref{eq:zs}.
Our motivation for analyzing \eqref{eq:zs} comes from its role in the theory of the initial-value problem for the cubic focusing nonlinear Schr\"odinger (NLS) equation
\begin{equation}\label{eq:nls}
\ensuremath{\mathrm{i}}\ensuremath{\epsilon}\frac{\ensuremath{\partial}\psi}{\ensuremath{\partial} t}+\frac{\ensuremath{\epsilon}^2}{2}\frac{\ensuremath{\partial}^2\psi}{\ensuremath{\partial} x^2}+|\psi|^2\psi=0\,.
\end{equation}
To emphasize the connection between \eqref{eq:zs} and \eqref{eq:nls}, we recall that Zakharov \& Shabat \cite{ZS} identified the linear spectral problem \eqref{eq:zs} as one half of the Lax pair for the NLS equation \eqref{eq:nls}. That is, the nonlinear equation \eqref{eq:nls} can be represented as the compatibility condition for two auxiliary linear problems---the Lax pair---and this structure allows one (in principle, at least) to construct solutions of the initial-value problem by the inverse spectral method (often called the inverse scattering transform). The initial step in this solution procedure is a spectral analysis of \eqref{eq:zs} in which the potential $\psi_0$ is taken to be the initial data for equation \eqref{eq:nls}, and the essential properties of the data for the initial-value problem are encoded in the spectral information---eigenvalues, norming constants, and reflection coefficient---associated with \eqref{eq:zs}. The temporal evolution is governed by properties of the other half of the Lax pair (details omitted here), is completely explicit, and takes place in the spectral domain. Finally, the solution at times $t>0$ is recovered by an inverse spectral transform; that is, the solution $\psi(x,t;\ensuremath{\epsilon})$ is recovered from the time-evolved scattering data. A detailed discussion of this process for \eqref{eq:nls} can be found, for example, in the monographs \cite{APT,FT}.
\begin{note}[Semiclassical scaling]
We note that the small parameter $\ensuremath{\epsilon}$ appearing in \eqref{eq:nls} is the same as that appearing in the eigenvalue problem \eqref{eq:zs} above. In the NLS equation, the real parameter $\ensuremath{\epsilon}$ is a measurement of the ratio of dispersive effects to nonlinear ones. Our experiments here are focused on \eqref{eq:zs}, but they are motivated by a desire to understand the limiting behavior of solutions of the initial-value problem for \eqref{eq:nls} with fixed data in the singular limit $\ensuremath{\epsilon}\downarrow 0$. This zero-dispersion limit problem is sometimes called the semiclassical limit for the focusing NLS equation; the origin of this descriptor is based on the quantum-mechanical interpretation of the linear terms in \eqref{eq:nls}.
\end{note}
We recall that, in the inverse-spectral framework, the eigenvalues of \eqref{eq:zs} correspond to solitons, and these special solutions are fundamental elements of the theory of \eqref{eq:nls}. The remarkable properties of these solutions are well known; see, e.g., \cite{APT}.
Thus, to summarize, given initial data for \eqref{eq:nls} or, equivalently, the potential in \eqref{eq:zs}, belonging to some reasonable class of functions (for example, $\mathscr{S}(\ensuremath{\mathbb{R}})$---the Schwartz class \cite{FT}), one would like to be able to effect a complete spectral analysis of \eqref{eq:zs} including, in particular, the location and multiplicity of the eigenvalues. Indeed, the eigenvalue locations have a direct impact on the dynamics and structure of the solution $\psi$ of \eqref{eq:nls}. Unfortunately, this is a challenge, and the general problem of rigorously extracting the requisite spectral information from \eqref{eq:zs} in the limit $\ensuremath{\epsilon}\downarrow0$ for a general potential $\psi_0$ remains largely open.
\subsection{Known results}\label{ssec:known}
Despite the challenges that remain for the spectral analysis of \eqref{eq:zs} in general, there are a couple of important results that provide valuable guidance. Our discussion below assumes familiarity with the basic machinery and vocabulary of the inverse spectral method; we refer the interested reader who lacks this familiarity to the appendix of \cite{LLV} for a short but accessible outline of the steps in the inverse spectral method.
The first, most basic result of interest is due to Satsuma \& Yajima \cite{SY}. They have shown that for a hyperbolic secant potential, i.e.,
\begin{equation}\label{eq:sydata}
\psi_0(x)=A{\rm sech} x\,,\quad A\in\ensuremath{\mathbb{R}}\,,
\end{equation}
the eigenvalue problem (with $\ensuremath{\epsilon}=1)$ is \emph{exactly solvable}. In particular, Satsuma \& Yajima showed how to transform the equation \eqref{eq:zs} with potential given by \eqref{eq:sydata} to the hypergeometric equation which they were able to solve explicitly in terms of hypergeometric functions. In fact, they were able to write down formulae, in terms of the Gamma function, for the entries $a(\lambda)$ and $b(\lambda)$ in the scattering matrix,
\begin{equation}
\mat{S}(\lambda)=\begin{bmatrix} a(\lambda)^* & b(\lambda)^* \\ -b(\lambda) & a(\lambda) \end{bmatrix}\,,
\end{equation}
which relates the Jost solutions of \eqref{eq:zs} normalized at each of the spatial infinities. Importantly, these quantities give rise to the transmission and reflection coefficients, $T(\lambda)=1/a(\lambda)$ and $R(\lambda)=b(\lambda)/a(\lambda)$, that are essential ingredients in the solution of \eqref{eq:nls} by the inverse-spectral method. We recall that zeros of the analytic continuation of the transmission coefficient $T$ to the upper half plane correspond to eigenvalues of \eqref{eq:zs}, and that $R$ is associated with continuous spectrum which, in this case, is confined to the real line.
Inspecting Satsuma \& Yajima's formula,
\begin{equation}
b(\lambda)=\frac{\ensuremath{\mathrm{i}} |\Gamma(\ensuremath{\mathrm{i}}\lambda+\frac{1}{2})|^2}{\Gamma(A)\Gamma(1-A)}=\ensuremath{\mathrm{i}}\frac{\sin(\pi A)}{\cosh(\pi\lambda)}\,,
\end{equation}
we see that when $A=N\in\ensuremath{\mathbb{N}}$, the reflection coefficient vanishes identically, and it turns out that the solution is a pure $N$-soliton, and the $N$ eigenvalues are also given explicitly; see \eqref{eq:syevals} below. Of particular interest is what happens when $N\to\infty$. As described by Lyng \& Miller \cite{LM} and in Note \ref{note:sy}, this problem is equivalent to a special case of the zero-dispersion limit problem for \eqref{eq:nls}.
\begin{note}[Non-zero phase]
Tovbis \& Venakides \cite{TV} have cleverly extended the above analysis to a one-parameter family of initial data of the form
\[
\psi_\mathrm{tv}(x)=A_\mathrm{tv}(x)\exp(\ensuremath{\mathrm{i}} S_\mathrm{tv}^\nu(x)/\ensuremath{\epsilon})\,,
\]
where
\[
A_\mathrm{tv}(x)=-{\rm sech} x\,,\quad \frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}} x} S_\mathrm{tv}^\nu(x)=-\nu\tanh x\,.
\]
This is a particularly important result as it provides a fundamental example in the case of nonzero phase. However, our focus here will be exclusively on real-valued, bell-shaped potentials like the hyperbolic secant considered by Satsuma \& Yajima. Thus, for the remainder of the paper, we confine our attention to the case $S_0\equiv0$.
\end{note}
The second, more general, guiding result is more recent and is due to Klaus \& Shaw \cite{KS,KS2}. Their result says that, roughly speaking, eigenvalues for bell-shaped or ``single-lobe'' potentials are confined to the imaginary axis. Moreover, the eigenvalues are simple, and their number is given in terms of the $L^1$ norm of the potential.
We give a precise statement of this result in Theorem \ref{thm:ks} below, and we use it to guide our numerical experiments. However, we note that it does not give detailed information about the precise locations of the eigenvalues.
\subsection{Semiclassical soliton ensembles}\label{ssec:sse}
We continue to focus on real-valued, bell shaped potentials. Given the dearth of detailed information about the eigenvalues of \eqref{eq:zs}, a standard procedure has been to replace the potential $\psi_0$ with an $\ensuremath{\epsilon}$-dependent reflectionless one, we shall denote it by $\psi_0^{(\ensuremath{\epsilon})}$, whose eigenvalues are known exactly and are believed to be good approximations to the true (but unknown) eigenvalues corresponding to $\psi_0$; see Figure \ref{fig:cosine} (b) for an example of such a reflectionless potential. The mechanics of this process are described in more detail below in \S\ref{sec:sse}. Briefly, Ercolani et al.\ \cite{EJLM} have shown how to formally approximate the eigenvalue locations by exploiting a remarkable feature---known from the very beginning \cite{ZS}---of the problem \eqref{eq:zs}. Namely, in the small-$\ensuremath{\epsilon}$ limit, the nonselfadjoint problem \eqref{eq:zs} can be written as a semiclassical self-adjoint Schr\"odinger operator with a nonselfadjoint and formally small but $\lambda$-dependent correction. Ignoring this correction, one can apply standard results about the Schr\"odinger operator to obtain approximate eigenvalue locations which then satisfy a Bohr--Sommerfeld type quantization condition; see \eqref{eq:bs} below.
These approximate eigenvalue locations were used in the monograph of Kamvissis et al.\ \cite{KMM} as the starting point for their asymptotic analysis. They neglected reflection, and they and used the approximate WKB eigenvalues in place of the unknown true eigenvalues. This process creates a \emph{semiclassical soliton ensemble}---a sequence of exact multisoliton solutions of \eqref{eq:nls}.
\begin{note}[The Satsuma--Yajima Ensemble]\label{note:sy}
In the special case that $\psi_0=A{\rm sech} x$, then with $\ensuremath{\epsilon}_N\stackrel{\mathrm{def}}{:=} A/N$, this process reproduces the family of exact $N$-soliton solutions given by Satsuma \& Yajima, and the limit $N\to\infty$ is evidently a special case of the semiclassical limit, and there is no error induced by the use of $\psi_0^{(\ensuremath{\epsilon})}$ in place of $\psi_0$. In general, however, this is not the case. This issue is addressed in \cite{LLV,LL_PLA13}.
\end{note}
Our experiment here is part of a larger program to quantify the effects of this uncontrolled modification of the initial data for general bell-shaped potentials.
We recall that the Whitham (or modulation) equations for \eqref{eq:nls} are \emph{elliptic}, and this feature of the problem confounds a common approach to similar problems which relies on local well-posedness of the hyperbolic Whitham system to permit an asymptotically vanishing perturbation of the eigenvalues. For example, Miller \cite{M} has rigorously shown that the WKB approximation at $t=0$ is asymptotically pointwise convergent, and more recent numerical experiments of Lee, Lyng, \& Vankova \cite{LLV} suggest convergence of the modified data to the true data in $L^2(\ensuremath{\mathbb{R}})$. However, it is not possible, on the basis of this information, to conclude that the solutions are close for any $t>0$.
For further discussion of this point, see, e.g., \cite{KMM,M,LLV}
Remarkably, though, the numerical computations of Lee et al.\ \cite{LLV} suggest that this convergence indeed persists for $t>0$; we view this as quite intriguing, given the extreme modulational instabilities known to be present in the semiclassical regime. Indeed, in a follow-up work, Lee \& Lyng \cite{LL_PLA13} examined the sensitivity of of the semiclassical limit to qualitatively similar perturbations of the data, and they found that modulational instabilities almost instantly detected small, analytic perturbations of the data. These results give strong, but indirect, evidence that the WKB (approximate) eigenvalues used to generate the SSE are quite close to the true eigenvalues.
Here, continuing and complementing these investigations, we numerically measure the difference directly in the spectral plane. Indeed, we aim to quantify a rate of convergence of the approximate eigenvalues to the true eigenvalues as $\ensuremath{\epsilon}\downarrow 0$; our experiments suggest convergence at a rate of $O(\ensuremath{\epsilon}^2)$ as $\ensuremath{\epsilon}\downarrow0$. The experiment is described in \S\ref{sec:gsse}. We view this numerical experiment as a preliminary step toward incorporating the WKB approximation into a rigorous analysis built on the framework of Kamvissis et al.\ \cite{KMM}. Assuming that the rate of convergence established here can be established rigorously (see the discussion in \S\ref{sec:discuss}), this would be a major step towards the development of a rigorous theory for the semiclassical limit of \eqref{eq:nls} that incorporates data beyond two special, exactly solvable, cases. Admittedly, the extension to analytic, bell-shaped, real data may seem at first glance to be a quite modest improvement, but this goal is at least a tractable target. The extension---mandated by the needs of applications---of the theory to more general (for example, non-analytic) data appears to be, for now, effectively out of reach.
\subsection{Plan}
In \S\ref{sec:frame} we begin by specifying the nature of the potentials $\psi_0$ that we will consider in \eqref{eq:zs}, and we describe a couple of important features of the eigenvalue problem. We give a careful statement of the spectral confinement results of Klaus \& Shaw \cite{KS,KS2}.
In \S\ref{sec:sse}, we outline the fundamental elements of the WKB approximation to the eigenvalues. In \S\ref{sec:nm}, we describe and validate the numerical method, and in \S\ref{sec:gsse}, we perform the main numerical experiment of the paper. As in previous work \cite{LLV,LL_PLA13}, our numerical experiment focuses on the case that $\psi_0(x)=\exp(-x^2)$. Finally, in \S\ref{sec:discuss}, we put the results in context and speculate about the implications of these calculations for the zero-dispersion limit problem.
\section{Framework: the Zakharov--Shabat eigenvalue problem}\label{sec:frame}
\subsection{Potentials}\label{ssec:potentials}
In this note, we restrict our attention to analytic Klaus--Shaw potentials . That is, we work in the framework of Kamvissis et al.\ \cite{KMM}, and we restrict our attention to potentials (initial data) of the form
\begin{equation}
\psi_0(x)=A_0(x)\,,
\label{eq:amplitude}
\end{equation}
where $A_0:\mathbb{R}\to(0,A]\subset\ensuremath{\mathbb{R}}$ is even, bell-shaped, and real analytic.
More precisely, $A_0$ is assumed to satisfy the assumptions detailed below.
\begin{assume}[Analytic Klaus--Shaw Potentials]\label{ass:pot}
The potential $A_0:\ensuremath{\mathbb{R}}\to\ensuremath{\mathbb{R}}$ is assumed to satisfy all of the following properties.
\begin{description}
\item[Decay] There exists $\alpha>0$ such that $|A_0(x)|=O(\ensuremath{\mathrm{e}}^{-\alpha|x|})$ as $x\to\pm\infty$.
\item[Evenness] $A_0$ is an even function, i.e., $A_0(x)=A_0(-x)$ for all $x\in\mathbb{R}$.
\item[Single Maximum] $A_0$ has a single genuine maximum at $x=0$, i.e., $A_0(0)=A$, $A_0'(0)=0$, $A_0''(0)<0$.
\item[Analyticity] $A_0$ is real analytic.
\end{description}
\end{assume}
\begin{note}
\noindent
\begin{enumerate}
\item[(a)] For the numerical calculations in this note, we shall restrict ourselves to the two concrete cases
\[
A_0(x)=A{\rm sech} x\,,\quad\text{and}\quad A_0(x)=\exp(-x^2)\,.
\]
Evidently, these choices fall into the category of Klaus--Shaw potentials described above. The sensitivity of the semiclassical limit problem for \eqref{eq:nls} to nonanalytic data has been investigated at the level of the partial differential equation by Clarke \& Miller \cite{CM}; at the spectral level, the sensitivity of the spectrum of \eqref{eq:zs} to nonanalytic perturbations of the potential was investigated by Bronski \cite{B}.
\item[(b)] The spectral confinement result of Klaus \& Shaw (cf. Theorem \ref{thm:ks} below) does not require such stringent restrictions on the potential. For example, in \cite{KS}, Klaus \& Shaw assume---in addition to the essential single-lobe requirement---that $A_0\in L^1(\ensuremath{\mathbb{R}})$ is nonnegative, bounded, and piecewise smooth. However, analyticity is important for the analysis of \cite{KMM}; this is due to the ellipticity of the Whitham equations. There are questions about the ``stability'' of the limit, even within the analytic class \cite{CM,LL_PLA13}. The other, apparently unneeded, conditions (e.g., $A_0''(0)<0$) are used in the analysis of \cite{KMM} to guarantee that the WKB formulae below are sufficiently well behaved.
\end{enumerate}
\end{note}
\subsection{About the eigenvalue problem: symmetry}\label{ssec:symmetry}
It is known that \eqref{eq:zs} is not self adjoint \cite{M_PD01}; thus, a priori, there is no restriction on where, in the complex plane, the spectrum may be.
We observe that the eigenvalue problem \eqref{eq:zs} can be recast as
\begin{equation}\label{eq:zsd}
\mathscr{L}^{(\ensuremath{\epsilon})}\vec{w}=\lambda\vec{w}\,,
\end{equation}
where $\mathscr{L}^{(\ensuremath{\epsilon})}$ is the non-self-adjoint Dirac operator defined by
\[
\mathscr{L}^{(\ensuremath{\epsilon})}\stackrel{\mathrm{def}}{:=}
\begin{bmatrix}
\ensuremath{\mathrm{i}}\ensuremath{\epsilon}\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}} x} & -\ensuremath{\mathrm{i}} A_0 \\
-\ensuremath{\mathrm{i}} A_0 & -\ensuremath{\mathrm{i}}\ensuremath{\epsilon}\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}} x}
\end{bmatrix}\,.
\]
It is a simple exercise to verify that if $\lambda$ is an eigenvalue of \eqref{eq:zs} with eigenfunction $\vec{w}=(w_1,w_2)^\ensuremath{\mathrm{t}}$, then so is $\lambda^*$ with eigenfunction $\tilde{\vec{w}}=(w^*_2,- w^*_1)^\ensuremath{\mathrm{t}}$. Thus, we will follow the established convention of considering and counting only eigenvalues $\lambda$ with $\Im\lambda>0$. Also, we note that if $A_0\in\mathscr{S}(\ensuremath{\mathbb{R}})$, then the $L^2(\ensuremath{\mathbb{R}})$ spectrum is comprised of the the continuous spectrum, which satisfies $\sigma_{\mathrm{cts}}(\mathscr{L}^{(\ensuremath{\epsilon})})=\ensuremath{\mathbb{R}}$, and a discrete set (possibly empty) of simple eigenvalues in the complex plane \cite{EJLM}.
\subsection{Klaus \& Shaw: spectral confinement}
In fact, for smooth Klaus--Shaw potentials, more is known. For each $\epsilon>0$ and for $\psi_0=A_0$ as described above, Klaus \& Shaw \cite{KS,KS2} have shown that the discrete spectrum of \eqref{eq:zs} is confined to the imaginary axis.
Indeed, for bell-shaped functions $A_0$ as described above, Klaus \& Shaw have shown the following.
\begin{theorem}[Klaus \& Shaw \cite{KS,KS2}]\label{thm:ks}
For $A_0$ as in Assumption~\ref{ass:pot}, eigenvalue problem \eqref{eq:zs} has precisely $N$ simple, purely imaginary eigenvalues with positive real parts where
\begin{equation}\label{eq:n}
N=\left\lfloor \frac{1}{2}+\frac{1}{\ensuremath{\epsilon}\pi}\norm{A_0}{L^1(\ensuremath{\mathbb{R}})}\right\rfloor\,,
\end{equation}
and $\lfloor h\rfloor$ is the integer part of $h$.
\end{theorem}
\section{Semiclassical soliton ensembles and the WKB approximation}\label{sec:sse}
We begin by recalling the basic formulae for the WKB eigenvalues of \eqref{eq:zs}; for more details see \cite{EJLM,KMM,LLV}. We define the density function for $\eta\in(0,\ensuremath{\mathrm{i}} A)$ via
\begin{equation}
\rho^0(\eta)\stackrel{\mathrm{def}}{:=}\frac{\eta}{\pi}\int_{x_-(\eta)}^{x_+(\eta)}\frac{\ensuremath{\mathrm{d}} x}{\sqrt{A_0(x)^2+\eta^2}}
=\frac{1}{\pi}\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}}\eta}\int_{x_-(\eta)}^{x_+(\eta)}\sqrt{A_0(x)^2+\eta^2}\,\ensuremath{\mathrm{d}} x\,,
\label{eq:rho0}
\end{equation}
where $x_\pm(\eta)$ are the two real turning points; see Figure \ref{fig:tp}.
\begin{figure}[ht]
\centering
\includegraphics[width=3in]{figs/turning_point.eps}
\caption{The turning points $x_\pm(\eta)$.}
\label{fig:tp}
\end{figure}
Using $\rho^0$ we next define
\begin{equation}\label{eq:theta0}
\theta^0(\lambda)\stackrel{\mathrm{def}}{:=}-\pi\int_\lambda^{\ensuremath{\mathrm{i}} A}\rho^0(\eta)\,\ensuremath{\mathrm{d}}\eta\,;
\end{equation}
the function $\theta^0$ gives a measure of the number of WKB eigenvalues on the imaginary axis between $\ensuremath{\lambda}$ and $\ensuremath{\mathrm{i}} A$.
To finish the process, we identify a sequence of distinguished values of $\ensuremath{\epsilon}$; we put
\begin{equation}
\epsilon_N\stackrel{\mathrm{def}}{:=}-\frac{1}{N}\int_0^{\ensuremath{\mathrm{i}} A}\rho^0(\eta)\,\ensuremath{\mathrm{d}}\eta \label{eq:hbarn}
=\frac{1}{\pi N}\int_{-\infty}^\infty A_0(x)\,\ensuremath{\mathrm{d}} x\,,\,\quad N=1,2,3,\ldots.
\end{equation}
Finally, the WKB eigenvalues $\lambda^\mathrm{wkb}_{N,k}$ are defined (there are $N$ of them for each $\epsilon_N$) by the formula
\begin{align}
-\int_{\lambda^\mathrm{wkb}_{N,k}}^{\ensuremath{\mathrm{i}} A}\rho^0(\eta)\,\ensuremath{\mathrm{d}} \eta &=\epsilon_N\left(k+\frac{1}{2}\right)=\frac{\theta^0(\lambda^\mathrm{wkb}_{N,k})}{\pi}\,,\quad k=0,\ldots, N-1\,. \label{eq:bs}
\end{align}
Therefore, writing $\lambda^\mathrm{wkb}_{N,k}=\ensuremath{\mathrm{i}} \tau^\mathrm{wkb}_{N,k}$ for $\tau^\mathrm{wkb}_{N,k}\in(0,A)\subset\mathbb{R}$, we obtain the WKB eigenvalues as solutions to the equations
\begin{equation}
\int_0^{x_+(\ensuremath{\mathrm{i}} t_{N,k})}\sqrt{A_0(x)^2-(\tau_{N,k}^{\mathrm{wkb}})^2}\,\ensuremath{\mathrm{d}} x=\frac{\pi\epsilon_N}{2}\left(k+\frac{1}{2}\right)\,,
\label{eq:wkbeval}
\end{equation}
for $k=0,1,\ldots,N-1$.
Specializing the above discussion to the case that the potential $A_0$ is given by
\begin{equation}
\psi_0(t)=A_0(x)=\ensuremath{\mathrm{e}}^{-x^2}\,,
\label{eq:gauss}
\end{equation}
we find, from \eqref{eq:hbarn},
\begin{equation}\label{eq:epsilon_N}
\epsilon_N=\frac{1}{\pi N}\int_{-\infty}^\infty\ensuremath{\mathrm{e}}^{-t^2}\,\ensuremath{\mathrm{d}} t =\frac{1}{\sqrt{\pi}N}\,,
\end{equation}
and, from formula \eqref{eq:wkbeval},
\begin{equation}\label{eq:gausswkb}
\int_0^{x_\sp(\ensuremath{\mathrm{i}} \tau_{N,k}^\mathrm{wkb})}\sqrt{\ensuremath{\mathrm{e}}^{-2x^2}-(\tau_{N,k}^\mathrm{wkb})^2}\,\ensuremath{\mathrm{d}} x=\frac{\sqrt{\pi}}{2N}\left(k+\frac{1}{2}\right),\quad k=0,1,2,\ldots N-1\,.
\end{equation}
In this case the turning points $x_{\ensuremath{{\scriptscriptstyle\pm}}}$ are given explicitly by
\begin{equation}\label{eq:gxplus}
x_{\ensuremath{{\scriptscriptstyle\pm}}}(\ensuremath{\mathrm{i}} \tau)=\pm\sqrt{-\ln \tau}\,,
\end{equation}
and equation \eqref{eq:wkbeval} can be rewritten as
\begin{equation}
\int_0^{\sqrt{-\ln \tau_{N,k}^\mathrm{wkb}}}\sqrt{\ensuremath{\mathrm{e}}^{-2x^2}-(\tau_{N,k}^\mathrm{wkb})^2}\,\ensuremath{\mathrm{d}} x=\frac{\sqrt{\pi}}{2 N}\left(k+\frac{1}{2}\right)\,,\quad
k=0,1,2,\ldots, N-1\,.
\label{eq:gwkbeval}
\end{equation}
This equation was solved to high precision by Lee et al.\ \cite{LLV}, and the 250-digit accuracy of the obtained solutions was verified using both \textsc{Mathematica} and \textsc{Maple} routines. We report the computed values in Appendix \ref{sec:wkbevals} below. Numerical experiments \cite{LLV,LL_PLA13} suggest, indirectly, that these values are quite close to the \emph{true} eigenvalues of \eqref{eq:zs} and that the distinct eigenvalues coalesce in the limit $\ensuremath{\epsilon}\downarrow 0$. In the next section we describe a numerical method aimed at accurately approximating the differences for a number of values of $\ensuremath{\epsilon}$ so that proximity of the WKB eigenvalues to the true eigenvalues can be measured directly and a rate of convergence, as $\ensuremath{\epsilon}\downarrow0$, can be estimated.
\section{Asymptotic analysis \& numerical method}\label{sec:nm}
\subsection{Evans-function analysis}\label{ssec:evans}
We shall numerically compute eigenvalues of the Zakharov--Shabat problem \eqref{eq:zs} by means of a complex shooting method originally proposed and implemented by Bronski \cite{B}. Here, we outline how this method approximates zeros of the Evans function (or transmission coefficient) associated with \eqref{eq:zs}. This connection may be useful; in \S\ref{sec:discuss} we describe some possible future projects related to \eqref{eq:zs} that exploit recent developments in methods for the numerical approximation of Evans functions (see, e.g., \cite{HLZ_ARMA09,HL,HSZ,HZ}).
First, we note that the eigenvalue problem \eqref{eq:zs}
can be reformulated as
\begin{equation}\label{eq:zs2}
\vec{w}'=\mat{A}(x;\ensuremath{\lambda},\ensuremath{\epsilon})\vec{w}\,,
\end{equation}
where
\begin{align}
\mat{A}(x;\ensuremath{\lambda},\ensuremath{\epsilon})& \stackrel{\mathrm{def}}{:=}
\begin{bmatrix}
-\ensuremath{\mathrm{i}}\ensuremath{\lambda}/\ensuremath{\epsilon} & A_0(x)/\ensuremath{\epsilon} \\
-A_0(x)/\ensuremath{\epsilon} & \ensuremath{\mathrm{i}}\ensuremath{\lambda}/\ensuremath{\epsilon}
\end{bmatrix} \\
&=\begin{bmatrix}
-\ensuremath{\mathrm{i}}\ensuremath{\lambda}/\ensuremath{\epsilon} & 0 \\
0 & \ensuremath{\mathrm{i}}\ensuremath{\lambda}/\ensuremath{\epsilon}
\end{bmatrix}+
\begin{bmatrix}
0 & A_0(x)/\ensuremath{\epsilon} \\
-A_0(x)/\ensuremath{\epsilon} & 0
\end{bmatrix}\, \nonumber\\
&\stackrel{\mathrm{def}}{=:}\mat{A}_\infty(\ensuremath{\lambda},\ensuremath{\epsilon})+\mat{B}(x;\ensuremath{\epsilon})\,.
\end{align}
We observe that, due to the Assumption \ref{ass:pot} on the potential $A_0$, we find that there exists $\alpha>0$ such that
\begin{equation}\label{eq:expdecay}
\norm{\mat{A}(x;\ensuremath{\lambda},\ensuremath{\epsilon})-\mat{A}_\infty(\ensuremath{\lambda};\ensuremath{\epsilon})}{}=\norm{\mat{B}(x;\ensuremath{\epsilon})}{}=O(\ensuremath{\mathrm{e}}^{-\alpha|x|})\quad
\text{as}\;\;x\to\pm\infty\,.
\end{equation}
Here, $\|\cdot\|$ is any matrix norm.
Evidently, the eigenvalues of the limiting matrix $\mat{A}_\infty$ are given by
\begin{equation}\label{eq:mu}
\mu_{\ensuremath{{\scriptscriptstyle -}}}(\ensuremath{\lambda},\ensuremath{\epsilon})=-\frac{\ensuremath{\mathrm{i}}\ensuremath{\lambda}}{\ensuremath{\epsilon}}\,,\quad
\mu_\sp(\ensuremath{\lambda},\ensuremath{\epsilon})=+\frac{\ensuremath{\mathrm{i}}\ensuremath{\lambda}}{\ensuremath{\epsilon}}\,,
\end{equation}
and the corresponding eigenvectors are
\(
\vec{v}_{\ensuremath{{\scriptscriptstyle -}}}=\vec{e}_1
\)
and
\(
\vec{v}_\sp=\vec{e}_2
\).
Under the assumption \eqref{eq:expdecay}, for fixed $\ensuremath{\epsilon}>0$ and for $\lambda\in\{z\in\ensuremath{\mathbb{C}}\,:\,\im z>0\}$, we find that there are solutions
\[
\vec{w}_{\ensuremath{{\scriptscriptstyle\pm}}}(x;\ensuremath{\lambda},\ensuremath{\epsilon})
\]
of \eqref{eq:zs2} which approach the decaying solutions $\vec{y}_{\ensuremath{{\scriptscriptstyle\pm}}}(x)=\exp(\mu_{\ensuremath{{\scriptscriptstyle\pm}}}(\lambda,\ensuremath{\epsilon}))\vec{v}_{\ensuremath{{\scriptscriptstyle\pm}}}$ of the limiting constant-coefficient system
\[
\vec{y}'=\mat{A}_\infty(\lambda,\ensuremath{\epsilon})\vec{y}
\]
as $x\to\pm\infty$. Then, up to a non-vanishing analytic factor, the Evans function, an analytic function on its natural domain, is given by
\begin{equation}\label{eq:evans}
D^{(\ensuremath{\epsilon})}(\ensuremath{\lambda})\stackrel{\mathrm{def}}{:=}\det\big(\vec{w}_\sp(x;\ensuremath{\lambda},\ensuremath{\epsilon}),\vec{w}_{\ensuremath{{\scriptscriptstyle -}}}(x;\ensuremath{\lambda},\ensuremath{\epsilon})\big)|_{x=0}\,.
\end{equation}
An immediate consequence of this definition is that $D^{(\ensuremath{\epsilon})}(\lambda)=0$ if and only if $\lambda$ is an eigenvalue. For, $D^{(\ensuremath{\epsilon})}$ detects a linear dependence between solutions of \eqref{eq:zs2} decaying at $\pm\infty$. As described below, Bronski's shooting method is based on approximating $\vec{w}_{\ensuremath{{\scriptscriptstyle -}}}$ and determining the values of $\lambda$ for which such linear dependence exists. For this determination, analyticity is used in an essential way.
\subsection{Numerical Methods}\label{sec:nm-Bronski}
As noted above, we adopt the shooting method of Bronski \cite{B96} to locate the eigenvalues for \eqref{eq:zs}. However, to make the discussion here self-contained, we give a thorough description of the procedure.
\begin{description}
\item[Step\#1 (Spatial Integration)] Conventionally, the process of solving the eigenvalue problem begins with integrating the differential equation \eqref{eq:zs} for fixed $\lambda$ with the initial conditions
\begin{equation}
\vec{w}(-L) = \begin{pmatrix}
1 \\
0
\end{pmatrix}
=\vec{v}_{\ensuremath{{\scriptscriptstyle -}}}
\end{equation}
to $x=+L$, where $L$ is chosen so that $\psi_{0}(\pm L)\approx 0$. However, direct numerical integration of this system suffers from large numerical errors due to the exponential growth of the mode corresponding to $\mu_{\ensuremath{{\scriptscriptstyle -}}}(\ensuremath{\lambda},\ensuremath{\epsilon})$ at large $L$ when $\im\lambda$ is large and $\ensuremath{\epsilon}$ is small. To eliminate this growth, for our numerical calculations, we define
\begin{equation}
\vec{w}(x)=\ensuremath{\mathrm{e}}^{\mu_{\ensuremath{{\scriptscriptstyle -}}}(\ensuremath{\lambda},\ensuremath{\epsilon})x}\vec{u}(x)
\end{equation}
and we integrate
\begin{equation}
\frac{\ensuremath{\mathrm{d}} \vec{u}}{\ensuremath{\mathrm{d}} x}=(\mat{A}-\mu_{\ensuremath{{\scriptscriptstyle -}}}\mat{I})\vec{u}
\end{equation}
from $x=-L$ to $x=+L$. The specified data at $x=-L$ is given by
\[
\vec{u}(-L)=\begin{pmatrix} \exp\big(-\mu_{\ensuremath{{\scriptscriptstyle -}}}(\lambda,\ensuremath{\epsilon})L\big) \\ 0 \end{pmatrix}
\]
corresponding to the choice $\vec{w}(-L)=(1,0)^\ensuremath{\mathrm{t}}$. We use a 6th-order Runge--Kutta scheme developed in \cite{Butcher64} as the integrator; we typically take $L=40$ and $\Delta x=0.002$.
\begin{note}[Quantitative Gap Lemma]
Using the known decay rate of $A_0$, it is possible to quantify the size of the initialization error that arises from truncating the domain of \eqref{eq:zs} and integrating from $x=-L$. For example, this kind of analysis has been done by Humpherys et al.\ \cite{HLZ_ARMA09}---using the ``quantitative gap lemma''---in their numerical approximation of the Evans function associated with viscous shock-layer solutions of the compressible Navier--Stokes equations. On the other hand, the true error is based on the values computed at $x=+L$, and our validation process (see \S\ref{ssec:validate} below) suggests that these errors are quite small. We therefore omit a detailed analysis of the initialization error.
\end{note}
\item[Step \#2 (Integrand Assembly)]
We write a generic complex number $\ensuremath{\lambda}$ in terms of its real and imaginary part as $\ensuremath{\lambda}=\gamma+\ensuremath{\mathrm{i}}\tau$, and we suppose that
\begin{equation}
\Gamma^{j}:[0,1]\to\ensuremath{\mathbb{C}}\,,\quad j=1,2,3,4\,;
\end{equation}
are the four sides of a rectangle in the complex plane. Here, $(j=1)\equiv\text{top}$, $(j=2)\equiv\text{right side}$, $(j=3)\equiv\text{bottom}$, and $(j=4)\equiv\text{left side}$. We adopt the following labeling convention for the grid points on $\Gamma^{j}$:
\begin{equation}\label{eq:lamb-nj}
\lambda_{n}^{j}=\gamma_n^{j}+\ensuremath{\mathrm{i}}\tau_n^{j}\,,\quad n=1,\ldots,M^j\,.
\end{equation}
Evidently, the corner points carry multiple labels. That is, for example,
\[
\lambda_1^{1}=\lambda_M^{4}
\]
and so on. At this point, we use the labels
\begin{equation}
\begin{split}
\vec{w} = \begin{pmatrix} w_{1} \\ w_{2} \end{pmatrix},\quad
\vec{u} = \begin{pmatrix} u_{1} \\ u_{2} \end{pmatrix},
\end{split}
\end{equation}
and we are interested in finding zeros of $w_{1}(L;\lambda,\ensuremath{\epsilon})$. To this end, we compute for the selected $\ensuremath{\lambda}$-values the quantity
\begin{equation}\label{eq:integrand}
f(\ensuremath{\lambda}_n^{j},\ensuremath{\epsilon})=\frac{u_1'(L;\ensuremath{\lambda}_n^{j},\ensuremath{\epsilon})}{u_1(L;\ensuremath{\lambda}_n^{j},\ensuremath{\epsilon})}\,.
\end{equation}
For notational convenience, we denote the numerator and denominator of \eqref{eq:integrand} by $u_1'(\ensuremath{\lambda}_n^j)$ and $u_1(\ensuremath{\lambda}_n^j)$ respectively.
To evaluate \eqref{eq:integrand}, we need to approximate the derivative with respect to $\lambda$ in the numerator. Instead of using the finite difference approximation, to obtain a third-order approximation of the derivative $u_1'(\ensuremath{\lambda}_n^{j})$, after computing $u_1(\ensuremath{\lambda}_n^{j})$, we use the cubic spline (with not-a-knot endpoint condition) to interpolate $u_1(\ensuremath{\lambda}_n^{j})$ at $\lambda_{n}^{j}$, knowing that the coefficient of the spline function gives the derivative $u_1'(\ensuremath{\lambda}_n^{j})$ at $\lambda_{n}^{j}$.
\item[Step \#3 (Moment Calculations)]
Suppose that $\Omega\subset\ensuremath{\mathbb{C}}$ is a simply connected domain. If $\Gamma$ is a simple closed curve in $\Omega$ and if $f$ is holomorphic on $\Omega$ with zeros
\(
\ensuremath{\lambda}_1\,,\ldots,\ensuremath{\lambda}_N
\)
inside $\Gamma$, then the $p$th moment of $f$ about $z_0$ is given by
\begin{equation}\label{eq:moment}
M_p(z_0)=\frac{1}{2\pi\ensuremath{\mathrm{i}}}\oint_\Gamma\frac{(\zeta-z_0)^pf'(\zeta)}{f(\zeta)}\,\ensuremath{\mathrm{d}} \zeta\,,
\end{equation}
and
\[
M_p(z_0)=\sum_{k=1}^N(\ensuremath{\lambda}_k-z_0)^p\,.
\]
Thus, $M_0(0)$ returns the number of zeros inside the contour $\Gamma$, and $M_1(0)=\ensuremath{\lambda}_1+\cdots+\ensuremath{\lambda}_N$.
Therefore, provided that there is only one zero inside $\Gamma$, the first moment about zero returns its location. We may thus find eigenvalue locations by approximating integrals of the form \eqref{eq:moment}.
In our first numerical calculation, for the case $A_{0}(x)=A{\rm sech}(x)$---for which the eigenvalues are already known, we take $\Gamma$ to be a rectangle placed to enclose a solitary eigenvalue. For the principal experiment, we center each of the rectangles $\Gamma_k$ at the approximate eigenvalue location given by the solution of \eqref{eq:gwkbeval}. Now suppose that the four corners of a rectangular contour $\Gamma_{k}$ in the complex plane are labeled as shown in Figure \ref{fig:gamma}.
\begin{figure}[ht]
\centering
\includegraphics[width=3.5cm]{figs/gamma.eps}
\caption{The rectangular contour $\Gamma_{k}$.}
\label{fig:gamma}
\end{figure}
Thus, using the definition (\ref{eq:lamb-nj}) and the integrand (\ref{eq:integrand}), the two moments along the rectangular contour $\Gamma_{k}$ can be expressed as
\begin{subequations}\label{eq:moments}
\begin{multline}
n_{k}=\frac{1}{2\pi\ensuremath{\mathrm{i}}}\left[\int_{a}^{b}\frac{u_1'(\ensuremath{\lambda}^{1})}{u_1(\ensuremath{\lambda}^{1})}\,\ensuremath{\mathrm{d}}\gamma^{1}
+\ensuremath{\mathrm{i}}\int_{b}^{c}\frac{u_1'(\ensuremath{\lambda}^{2})}{u_1(\ensuremath{\lambda}^{2})}\,\ensuremath{\mathrm{d}}\tau^{2} \right.
\left.+\int_{c}^{d}\frac{u_1'(\ensuremath{\lambda}^{3})}{u_1(\ensuremath{\lambda}^{3})}\,\ensuremath{\mathrm{d}}\gamma^{3}
+\ensuremath{\mathrm{i}}\int_{d}^{a}\frac{u_1'(\ensuremath{\lambda}^{4})}{u_1(\ensuremath{\lambda}^{4})}\,\ensuremath{\mathrm{d}}\tau^{4} \right]\,,
\end{multline}
and
\begin{multline}
\ell_{k}=\frac{1}{2\pi\ensuremath{\mathrm{i}}}\left[\int_{a}^{b}\frac{\ensuremath{\lambda}^{1} u_1'(\ensuremath{\lambda}^{1})}{u_1(\ensuremath{\lambda}^{1})}d\gamma^{1} +\ensuremath{\mathrm{i}}\int_{b}^{c}\frac{\ensuremath{\lambda}^{2}u_1'(\ensuremath{\lambda}^{2})}{u_1(\ensuremath{\lambda}^{2})}d\tau^{2}\right.
\\
\left.+ \int_{c}^{d}\frac{\ensuremath{\lambda}^{3}u_1'(\ensuremath{\lambda}^{3})}{u_1(\ensuremath{\lambda}^{3})}d\gamma^{3} + \ensuremath{\mathrm{i}}\int_{d}^{a}\frac{\ensuremath{\lambda}^{4}u_{1}'(\ensuremath{\lambda}^{4})}{u_1(\ensuremath{\lambda}^{4})}d\tau^{4} \right].
\end{multline}
\end{subequations}
As described above, when $n_{k}\approx1$, the corresponding value of $\ell_{k}$ gives the approximate location of the eigenvalue enclosed by the contour $\Gamma_{k}$. For our numerical calculation, each integral in \eqref{eq:moments} is evaluated by the 6th-order Newton-Cotes integration formula (also referred to as Weddle's rule) \cite{DA08}.
\begin{note}
The superscripts in \eqref{eq:moments} refer to the labeling scheme described in \eqref{eq:lamb-nj} and should not be confused with exponents appearing in the moment formula \eqref{eq:moment}. Thus, $n_k$ is the zeroth moment about zero and $\ell_k$ is the first moment about zero.
\end{note}
\end{description}
\subsection{Validation: the Satsuma--Yajima ensemble}\label{ssec:validate}
To test the methodology, we look at the case
\[
A_0(x)=A{\rm sech} x\,,
\]
for which the eigenvalues are known exactly \cite{SY}.
Indeed, following the notation of \cite{LM}, the $N$ eigenvalues of \eqref{eq:zs} are given by
\begin{equation}\label{eq:syevals}
\ensuremath{\lambda}_{N,k}^{\mathrm{sy}}\stackrel{\mathrm{def}}{:=}\ensuremath{\mathrm{i}} A-\ensuremath{\mathrm{i}}\left(k+\frac{1}{2}\right)\ensuremath{\epsilon}_N\,,
\quad k=0,1,\ldots, N-1\,,
\end{equation}
and we recall that, in this case,
\[
\ensuremath{\epsilon}_N\stackrel{\mathrm{def}}{:=} A/N\,.
\]
We are thus considering a ``quantized'' sequence of values of $\ensuremath{\epsilon}$ which tends to zero as $N\to\infty$.
Following the algorithm described in \S\ref{sec:nm-Bronski}, we compute the eigenvalues for the cases $N=5, 10, 15$, and $20$. The $k^{th}$ eigenvalue approximation for the case of $N$ is denoted by $\lambda_{N,k}^\mathrm{app}=\ensuremath{\mathrm{i}}\tau_{N,k}^\mathrm{app}$. We define the maximum error for each $N$ to be
\begin{equation}
e_{N}\stackrel{\mathrm{def}}{:=} \max_{k} | \lambda_{N,k}^\mathrm{app} - \ensuremath{\lambda}_{N,k}^{\mathrm{sy}}|.
\end{equation}
\begin{table}[ht]
\begin{centering}
\caption{Validation: the Satsuma--Yajima ensemble}
\label{tab:syvalidate}
\begin{tabular}{c|ccccccc}
$N$ & $e_{N}$ & at $\lambda_{N, k}$ & $dx$ & $d\gamma$ & $d\tau$ & $\overline{ab}$ & $\overline{bc}$\\ \hline
5&6.7174E-10&$\lambda_{5,4}$&80/40,000&0.2/192&0.4/192&0.2&0.4\\ \hline
10&2.2438E-09&$\lambda_{10, 9}$&80/40,000&0.2/192&0.2/192&0.2&0.2\\ \hline
15&9.3286E-09&$\lambda_{15, 14}$&80/40,000&0.2/192&0.133/192&0.2&0.133\\ \hline
20&3.0100E-08&$\lambda_{20, 19}$&80/40,000&0.2/192&0.1/192&0.2&0.1\\
\end{tabular}
\end{centering}
\end{table}
In Table \ref{tab:syvalidate} we list $e_{N}$ and the location at which the maximum error occurs. We also list the mesh size $dx$ used for the eigenvalue problem, and the mesh size $d\gamma$ and $d\tau$ used for the moment calculations. We discover that the largest errors always occur at $(N-1)^{th}$ eigenvalue, closest to the real-axis. This agrees with Bronski's finding \cite{B96} that the numerical method suffers near $\sigma_{\mathrm{cts}}(\mathscr{L}^{(\ensuremath{\epsilon})})=\ensuremath{\mathbb{R}}$. Indeed, as noted by Bronski, the boundary conditions reverse roles in the lower half plane, and the method is not expected to be reliable close to the real line. We also see that the error increases when $\ensuremath{\epsilon}$ decreases. For the case of the smallest $\ensuremath{\epsilon}$ ($N=20$), we were able to control the error to the order of $10^{-8}$.
\section{The Gaussian Case}\label{sec:gsse}
\subsection{Experiment}
We now present the results of the principal calculation of the paper. We consider the Zakharov--Shabat problem \eqref{eq:zs} with Gaussian potential
\begin{equation}
\psi_0(x)=\exp(-x^2)\,.
\end{equation}
In Step \#1 of the algorithm, the domain of calculation is $-40\le x\le 40$ ($L=40$), and the step size of the integration is $dx=80/40,000$. To test whether our computational results are numerically convergent, we use a finer mesh size $dx=80/80,000$ to compute the eigenvalue problem for the case of $N=15$. We find that the difference between computed eigenvalue location is of the order of $10^{-11}$ between the two different mesh sizes. In Step \#3, with the SSE eigenvalue located at the center of the rectangle, the length of the top and the bottom side of each rectangle is $\overline{ab} =\overline{cd}= 0.2$, while the left and the right side of the rectangle is $\overline{bc} =\overline{da}= 0.0815$. A total number of $193$ grid points are evaluated at each side of the rectangle ($M=193$ in equation \eqref{eq:lamb-nj}). This gives that $d\gamma \approx \Delta\gamma= 0.2/192$ and $d\tau \approx\Delta\tau=0.0815/192$ in Eq. (\ref{eq:moments}).
We denote the difference (in absolute value) between $\lambda_{N,k}^\mathrm{app}$ and the $k^{th}$ WKB eigenvalue $\lambda_{N,k}^\mathrm{wkb}$ by
\begin{equation}
D_{k}^{N} =|\lambda_{N,k}^\mathrm{app}-\lambda_{N,k}^\mathrm{wkb}|,
\end{equation}
where $\lambda_{N,k}^\mathrm{wkb}=\ensuremath{\mathrm{i}}\tau_{N,k}^\mathrm{wkb}$ are the WKB eigenvalues computed in \cite{LLV}. The computed values for $\tau_{N,k}^\mathrm{wkb}$ and $\tau_{N,k}^\mathrm{app}$ for $N=10,\ldots, 22$ are recorded below in Appendix \ref{sec:wkbevals}.
\subsection{Least squares fit, rate of decay}
We perform a least squares fit of the data in terms of
\begin{equation}\label{eq:D_least}
D_{*}^{N}=C_*\cdot N^{\alpha_*}\,,
\end{equation}
for some constants $C$ and $\alpha$. If we take the logarithmic function to the above equation, then the least squares fit is reduced to a linear least square fit in the $\log-\log$ space. Now for each $N=10,\cdots, 20$, we have data for $k=0,\cdots, N-1$, and for $N=21$ and 22, we have data for the largest ($k=0$) eigenvalue. Therefore we have total 167 data points. Figure \ref{fig:least}(a) shows the least square fit for all 167 data points. The triangles are the computed differences. For example, there are 10 eigenvalues for $N=10$, and hence there are 10 computed differences. The solid line is the computed least square curve, which indicates the overall trend of decay of $D^{N}$ versus $N$. The rate of decay is $\alpha=-2.00848$. We remark that the largest difference for each case of $N$ always occurs at the eigenvalue closest to the real axis ($k=N-1$), whereas the smallest difference occurs for the largest eigenvalue.
Another way to monitor the rate of decay for $D^{N}$ versus $N$ is to compute the difference of the largest eigenvalue for each $N$. That is, we compute
\begin{equation}
D_{0}^{N} = |\lambda_{N,0}^\mathrm{app}-\lambda_{N,0}^\mathrm{wkb}|,\quad N=10,\cdots, 22.
\end{equation}
Figure \ref{fig:least}(b) is the least square fit for this collection of 13 data points. It shows that the rate of decay is $\alpha=-2.0135$. We are thus led to propose the following conjecture.
\begin{figure}[ht]
\centering
(a) \includegraphics[width=4.5in]{figs/all-data-LS.eps} \\
(b) \includegraphics[width=4.5in]{figs/top_1.eps}
\caption{(a) Least square fit for 167 data points, for which $N=10,\cdots, 20$ with $k=0,\ldots, N-1$, and $N=21, 22$ with $k=0$. (b) Least square fit for $k=0$, $N=10,\ldots, 22$.}
\label{fig:least}
\end{figure}
\begin{conjecture}\label{conj:eps2}
The WKB eigenvalues satisfy
\begin{equation}
|\ensuremath{\lambda}_{N,k}^\mathrm{wkb}-\ensuremath{\lambda}_{N,k}|=O(N^{-2})\quad\text{as}\;N\to\infty\,.
\end{equation}
Here, $\lambda_{N,k}^\mathrm{wkb}$ are the WKB eigenvalues given by \eqref{eq:bs} while $\lambda_{N,k}$ denote the true eigenvalues of \eqref{eq:zs} corresponding to the quantized values $\ensuremath{\epsilon}_N$.
\end{conjecture}
This conjecture agrees with formal calculations of Miller \cite{M_PD01}; see the concluding discussion in \S\ref{ssec:prove}.
\section{Discussion}\label{sec:discuss}
\subsection{Future directions}
One natural extension of this numerical experiment would be to try to use similar methods to examine the ``cosine-perturbed'' potentials used by Lee \& Lyng \cite{LL_PLA13} in their recent study of the stability of the semiclassical limit. They considered potentials of the form
\begin{equation}\label{eq:cosine}
\tilde{\psi}_0^{(\ensuremath{\epsilon})}=0.3\cos\left(\frac{x}{0.54\ensuremath{\epsilon}}\right)\exp(-x^2)\,,\quad\ensuremath{\epsilon}>0\,;
\end{equation}
these were chosen to mimic the potentials $\psi_0^{(\ensuremath{\epsilon})}$ that arise due to the use of the WKB eigenvalues. Figure \ref{fig:cosine} shows the close resemblance of of a member of the family of potentials in \eqref{eq:cosine} and the corresponding potential $\psi_0^{(\ensuremath{\epsilon})}$. Lee \& Lyng found that, despite the superficial similarity between these two data, numerical simulations of the temporal evolution under the equation \eqref{eq:nls} appear to be extremely sensitive to the differences between the two potentials. That is, the differences appeared to almost instantaneously trigger the acute modulational instabilities known to be a feature of \eqref{eq:nls} in the small-$\ensuremath{\epsilon}$ regime.
One possible explanation is that the spectrum of \eqref{eq:zs} is quite sensitive to the variations between perturbations of this kind. We observe (see Figure \ref{fig:cosine}) that the potentials in \eqref{eq:cosine} are \emph{not} single-lobe Klaus--Shaw potentials, and thus the spectrum need not be confined to the imaginary axis. Thus, we propose to revisit the spectral instability calculations of Bronski \cite{B}; his numerical results suggested that the eigenvalue problem with real potential is stable when subjected to nonanalytic perturbations. However, the focus there on analyticity is misleading; his nonanalytic perturbation was of Klaus--Shaw type. The cosine-perturbed potential provides an interesting example of an analytic but multiple-lobed potential.
In addition, this proposed numerical experiment provides an opportunity to develop and test the numerical techniques for Evans-function calculations aimed at detecting eigenvalues in exponentially asymptotic systems of the basic form
\[
\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}} x}\vec{w}=\mat{A}(x;\lambda,\ensuremath{\epsilon})\vec{w}\,.
\]
Here, we have adopted the complex shooting method of Bronski \cite{B96}, but one intriguing possibility is to adopt some of the techniques from the Evans function community. A focus of this community has been on large systems (see, e.g., \cite{HSZ,HZ}), but preliminary work by Humpherys \& Lytle \cite{HL} on tracking eigenvalues by continuation is quite intriguing. Their continuation method would make it straightforward to follow eigenvalue branches as the parameter $\ensuremath{\epsilon}$ varies, and the oscillatory nature of the potential in \eqref{eq:cosine} provides a challenging test case for the developing numerical method. The results of this experiment might give some additional insight into the spectral origins of the modulational instability in \eqref{eq:nls}.
\begin{figure}[ht]
\centering
\begin{tabular}{cc}
(a)\includegraphics[width=2.8in]{figs/N15_cosine_modified_initial.eps} &
(b)\includegraphics[width=2.8in]{figs/N15_SSE_modified_initial.eps}
\end{tabular}
\caption{(a) The cosine perturbation given in \eqref{eq:cosine}. (b) The reconstruction of the initial data $\psi_0^{(\ensuremath{\epsilon})}$ using the WKB eigenvalues for $N=15$. Figures taken from \cite{LL_PLA13}.}
\label{fig:cosine}
\end{figure}
\subsection{Proving the conjecture and implications for the semiclassical limit }\label{ssec:prove}
A second natural direction for future work would be to seek a rigorous proof of the $O(\ensuremath{\epsilon}^2)$ decay of the WKB eigenvalues to the true eigenvalues. Indeed, we believe that such a proof is highly likely to be an essential ingredient in the development of a complete theory for the semiclassical limit for \eqref{eq:nls} that is based on semiclassical soliton ensembles. Given that a completely rigorous theory is restricted to special, exactly solvable potentials, the extension to more general (bell-shaped, analytic) real data is a clearly worthwhile goal.
Miller has given a possible roadmap for finding such a proof in the concluding discussion of his paper \cite{M_PD01}; in this paper he introduces a certain complexified WKB method for analyzing the spectrum of \eqref{eq:zs}. Although the analysis is formal, Miller's method is able to reproduce the $\mathsf{Y}$-shaped configurations of eigenvalues that Bronski \cite{B96} observed for potentials with a nontrivial phase $S_0$. As a starting point, Miller suggests a change of variables that transforms \eqref{eq:zs} to a Weber equation plus a small correction, and he speculates about the kind of tools from Kato's perturbation theory for linear operators \cite{Kato} that will be necessary to deal with the two-parameter family of linear operators that results from this plan of attack.
This program has not, to our knowledge, been carried out completely, but it seems to be a natural starting point. We believe that the new numerical evidence presented here provides additional impetus for pursuing this line of analysis.
Finally, assuming the conjecture has been proved, an important next step will be to incorporate these error estimates into the asymptotic analysis of the semiclassical limit problem for \eqref{eq:nls}, as in \cite{KMM,LM}. However, we recall that a crucial step in this analysis is the ``sweeping away of the poles'' in a meromorphic Riemann--Hilbert problem (RHP). That is, one makes a change of variables which exchanges a meromorphic RHP for a sectionally holomorphic one. But, this change of variables is predicated on knowing the precise locations of the soliton eigenvalues. If the WKB approximations are used instead, this process will leave behind phantoms of the residues at these poles; the rate of decay in the conjecture provides a means of quantifying how quickly these phantoms disappear in the limit $\ensuremath{\epsilon}\downarrow0$.
\section*{Acknowledgement}
Research of YK and GL was supported in part by the National Science Foundation under grant number DMS-0845127.
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{"url":"http:\/\/us.metamath.org\/mpegif\/mmtheorems74.html","text":"Home Metamath Proof ExplorerTheorem List (p. 74 of 322) < Previous\u00a0\u00a0Next > Browser slow? Try the Unicode version.\n\n Color key: Metamath Proof Explorer (1-21498) Hilbert Space Explorer (21499-23021) Users' Mathboxes (23022-32154)\n\nTheorem List for Metamath Proof Explorer - 7301-7400 \u00a0 *Has distinct variable group(s)\nTypeLabelDescription\nStatement\n\nTheoremwdomimag\u00a07301 A set is weakly dominant over its image under any function. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)\n*\n\nTheoremunxpwdom2\u00a07302 Lemma for unxpwdom\u00a07303. (Contributed by Mario Carneiro, 15-May-2015.)\n*\n\nTheoremunxpwdom\u00a07303 If a cross product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)\n*\n\nTheoremharwdom\u00a07304 The Hartogs function is weakly dominated by . This follows from a more precise analysis of the bound used in hartogs\u00a07259 to prove that har is a set. (Contributed by Mario Carneiro, 15-May-2015.)\nhar *\n\nTheoremixpiunwdom\u00a07305* Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg\u00a06846 this shows that and have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.)\n*\n\n2.5\u00a0\u00a0ZF Set Theory - add the Axiom of Regularity\n\n2.5.1\u00a0\u00a0Introduce the Axiom of Regularity\n\nAxiomax-reg\u00a07306* Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg\u00a07309) that every non-empty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv\u00a07311). A stronger version that works for proper classes is proved as zfregs\u00a07414. (Contributed by NM, 14-Aug-1993.)\n\nTheoremaxreg2\u00a07307* Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)\n\nTheoremzfregcl\u00a07308* The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.)\n\nTheoremzfreg\u00a07309* The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the \"weak form.\" There is also a \"strong form,\" not requiring that be a set, that can be proved with more difficulty (see zfregs\u00a07414). (Contributed by NM, 26-Nov-1995.)\n\nTheoremzfreg2\u00a07310* The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg\u00a07309) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480. (Contributed by NM, 17-Sep-2003.)\n\nTheoremelirrv\u00a07311 The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr\u00a07316 and efrirr\u00a04374, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)\n\nTheoremelirr\u00a07312 No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)\n\nTheoremsucprcreg\u00a07313 A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.)\n\nTheoremruv\u00a07314 The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)\n\nTheoremruALT\u00a07315 Alternate proof of Russell's Paradox ru\u00a02990, simplified using (indirectly) the Axiom of Regularity ax-reg\u00a07306. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)\n\nTheoremzfregfr\u00a07316 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)\n\nTheoremen2lp\u00a07317 No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.)\n\nTheorempreleq\u00a07318 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)\n\nTheoremopthreg\u00a07319 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg\u00a07306 (via the preleq\u00a07318 step). See df-op\u00a03649 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)\n\nTheoremsuc11reg\u00a07320 The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)\n\nTheoremdford2\u00a07321* Assuming ax-reg\u00a07306, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)\n\n2.5.2\u00a0\u00a0Axiom of Infinity equivalents\n\nTheoreminf0\u00a07322* Our Axiom of Infinity derived from existence of omega. The proof shows that the especially contrived class \" \" exists, is a subset of its union, and contains a given set (and thus is non-empty). Thus it provides an example demonstrating that a set exists with the necessary properties demanded by ax-inf\u00a07339. (Contributed by NM, 15-Oct-1996.)\n\nTheoreminf1\u00a07323 Variation of Axiom of Infinity (using zfinf\u00a07340 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)\n\nTheoreminf2\u00a07324* Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf\u00a07340 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)\n\nTheoreminf3lema\u00a07325* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. (Contributed by NM, 28-Oct-1996.)\n\nTheoreminf3lemb\u00a07326* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. (Contributed by NM, 28-Oct-1996.)\n\nTheoreminf3lemc\u00a07327* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. (Contributed by NM, 28-Oct-1996.)\n\nTheoreminf3lemd\u00a07328* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. (Contributed by NM, 28-Oct-1996.)\n\nTheoreminf3lem1\u00a07329* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. (Contributed by NM, 28-Oct-1996.)\n\nTheoreminf3lem2\u00a07330* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. (Contributed by NM, 28-Oct-1996.)\n\nTheoreminf3lem3\u00a07331* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg\u00a07309. (Contributed by NM, 29-Oct-1996.)\n\nTheoreminf3lem4\u00a07332* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. (Contributed by NM, 29-Oct-1996.)\n\nTheoreminf3lem5\u00a07333* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. (Contributed by NM, 29-Oct-1996.)\n\nTheoreminf3lem6\u00a07334* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. (Contributed by NM, 29-Oct-1996.)\n\nTheoreminf3lem7\u00a07335* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3\u00a07336 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex\u00a05751. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)\n\nTheoreminf3\u00a07336 Our Axiom of Infinity ax-inf\u00a07339 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2\u00a07324, and the conclusion is the version of the Axiom of Infinity shown as Axiom 7 in [TakeutiZaring] p. 43. (Other standard versions are proved later as axinf2\u00a07341 and zfinf2\u00a07343.) The main proof is provided by inf3lema\u00a07325 through inf3lem7\u00a07335, and this final piece eliminates the auxiliary hypothesis of inf3lem7\u00a07335. This proof is due to Ian Sutherland, Richard Heck, and Norman Megill and was posted on Usenet as shown below. Although the result is not new, the authors were unable to find a published proof.\n (As posted to sci.logic on 30-Oct-1996, with annotations added.)\n\nTheorem: The statement \"There exists a non-empty set that is a subset\nof its union\" implies the Axiom of Infinity.\n\nProof: Let X be a nonempty set which is a subset of its union; the\nlatter\nproperty is equivalent to saying that for any y in X, there exists a z\nin X\nsuch that y is in z.\n\nDefine by finite recursion a function F:omega-->(power X) such that\nF_0 = 0 (See inf3lemb\u00a07326.)\nF_n+1 = {y<X | y^X subset F_n} (See inf3lemc\u00a07327.)\nNote: ^ means intersect, < means \\in (\"element of\").\n(Finite recursion as typically done requires the existence of omega;\nto avoid this we can just use transfinite recursion restricted to omega.\nF is a class-term that is not necessarily a set at this point.)\n\nLemma 1. F_n subset F_n+1. (See inf3lem1\u00a07329.)\nProof: By induction: F_0 subset F_1. If y < F_n+1, then y^X subset\nF_n,\nso if F_n subset F_n+1, then y^X subset F_n+1, so y < F_n+2.\n\nLemma 2. F_n =\/= X. (See inf3lem2\u00a07330.)\nProof: By induction: F_0 =\/= X because X is not empty. Assume F_n =\/=\nX.\nThen there is a y in X that is not in F_n. By definition of X, there is\na\nz in X that contains y. Suppose F_n+1 = X. Then z is in F_n+1, and z^X\ncontains y, so z^X is not a subset of F_n, contrary to the definition of\nF_n+1.\n\nLemma 3. F_n =\/= F_n+1. (See inf3lem3\u00a07331.)\nProof: Using the identity y^X subset F_n <-> y^(X-F_n) = 0, we have\nF_n+1 = {y<X | y^(X-F_n) = 0}. Let q = {y<X-F_n | y^(X-F_n) = 0}.\nThen q subset F_n+1. Since X-F_n is not empty by Lemma 2 and q is the\nset of \\in-minimal elements of X-F_n, by Foundation q is not empty, so q\nand therefore F_n+1 have an element not in F_n.\n\nLemma 4. F_n proper_subset F_n+1. (See inf3lem4\u00a07332.)\nProof: Lemmas 1 and 3.\n\nLemma 5. F_m proper_subset F_n, m < n. (See inf3lem5\u00a07333.)\nProof: Fix m and use induction on n > m. Basis: F_m proper_subset\nF_m+1\nby Lemma 4. Induction: Assume F_m proper_subset F_n. Then since F_n\nproper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper\nsubset.\n\nBy Lemma 5, F_m =\/= F_n for m =\/= n, so F is 1-1. (See inf3lem6\u00a07334.)\nThus the inverse of F is a function with range omega and domain a subset\nof power X, so omega exists by Replacement. (See inf3lem7\u00a07335.)\nQ.E.D.\n\n(Contributed by NM, 29-Oct-1996.)\n\nTheoreminfeq5i\u00a07337 Half of infeq5\u00a07338. (Contributed by Mario Carneiro, 16-Nov-2014.)\n\nTheoreminfeq5\u00a07338 The statement \"there exists a set that is a proper subset of its union\" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex\u00a07344.) The left-hand side provides us with a very short way to express of the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)\n\n2.6\u00a0\u00a0ZF Set Theory - add the Axiom of Infinity\n\n2.6.1\u00a0\u00a0Introduce the Axiom of Infinity\n\nAxiomax-inf\u00a07339* Axiom of Infinity. An axiom of Zermelo-Fraenkel set theory. This axiom is the gateway to \"Cantor's paradise\" (an expression coined by Hilbert). It asserts that given a starting set , an infinite set built from it exists. Although our version is apparently not given in the literature, it is similar to, but slightly shorter than, the Axiom of Infinity in [FreydScedrov] p. 283 (see inf1\u00a07323 and inf2\u00a07324). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2\u00a07343 and omex\u00a07344 and are based on the (nontrivial) proof of inf3\u00a07336. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2\u00a07342. Theorem inf0\u00a07322 shows the reverse derivation of our axiom from a standard one. Theorem inf5\u00a07346 shows a very short way to state this axiom.\n\nThe standard version of Infinity ax-inf2\u00a07342 requires this axiom along with Regularity ax-reg\u00a07306 for its derivation (as theorem axinf2\u00a07341 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2\u00a07342 instead of this one. The derivation of this axiom from ax-inf2\u00a07342 is shown by theorem axinf\u00a07345.\n\nProofs should normally use the standard version ax-inf2\u00a07342 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)\n\nTheoremzfinf\u00a07340* Axiom of Infinity expressed with fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)\n\nTheoremaxinf2\u00a07341* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf\u00a07339 and Regularity ax-reg\u00a07306.\n\nThis theorem should not be referenced in any proof. Instead, use ax-inf2\u00a07342 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.)\n\nAxiomax-inf2\u00a07342* A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2\u00a07343 shows it converted to abbreviations. This axiom was derived as theorem axinf2\u00a07341 above, using our version of Infinity ax-inf\u00a07339 and the Axiom of Regularity ax-reg\u00a07306. We will reference ax-inf2\u00a07342 instead of axinf2\u00a07341 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf\u00a07339 from ax-inf2\u00a07342 is shown by theorem axinf\u00a07345. (Contributed by NM, 30-Aug-1993.)\n\nTheoremzfinf2\u00a07343* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2\u00a07342 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)\n\n2.6.2\u00a0\u00a0Existence of omega (the set of natural numbers)\n\nTheoremomex\u00a07344 The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it, as shown by the reverse derivation inf0\u00a07322.\n\nA finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial ; this would lead to by omon\u00a04667 and (the universe of all sets) by fineqv\u00a07078. The finitist could still develop natural number, integer, and rational number arithmetic but would be denied the real numbers (as well as much of the rest of mathematics). In deference to the finitist, much of our development is done, when possible, without invoking the Axiom of Infinity; an example is Peano's axioms peano1\u00a04675 through peano5\u00a04679 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)\n\nTheoremaxinf\u00a07345* The first version of the Axiom of Infinity ax-inf\u00a07339 proved from the second version ax-inf2\u00a07342. Note that we didn't use ax-reg\u00a07306, unlike the other direction axinf2\u00a07341. (Contributed by NM, 24-Apr-2009.)\n\nTheoreminf5\u00a07346 The statement \"there exists a set that is a proper subset of its union\" is equivalent to the Axiom of Infinity (see theorem infeq5\u00a07338). This provides us with a very compact way to express of the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.)\n\nTheoremomelon\u00a07347 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)\n\nTheoremdfom3\u00a07348* The class of natural numbers omega can be defined as the smallest \"inductive set,\" which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. (Contributed by NM, 6-Aug-1994.)\n\nTheoremelom3\u00a07349* A simplification of elom\u00a04659 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)\n\nTheoremdfom4\u00a07350* A simplification of df-om\u00a04657 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)\n\nTheoremdfom5\u00a07351 is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)\n\nTheoremoancom\u00a07352 Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.)\n\nTheoremisfinite\u00a07353 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. The Axiom of Infinity is used for the forward implication. (Contributed by FL, 16-Apr-2011.)\n\nTheoremnnsdom\u00a07354 A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.)\n\nTheoremomenps\u00a07355 Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.)\n\nTheoremomensuc\u00a07356 The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)\n\nTheoreminfdifsn\u00a07357 Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)\n\nTheoreminfdiffi\u00a07358 Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)\n\nTheoremunbnn3\u00a07359* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn\u00a07113 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)\n\nTheoremnoinfep\u00a07360* Using the Axiom of Regularity in the form zfregfr\u00a07316, show that there are no infinite descending -chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)\n\nTheoremnoinfepOLD\u00a07361* Using the Axiom of Regularity in the form zfregfr\u00a07316, show that there are no infinite descending -chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)\n\n2.6.3\u00a0\u00a0Cantor normal form\n\nSyntaxccnf\u00a07362 Extend class notation with the Cantor normal form function.\nCNF\n\nDefinitiondf-cnf\u00a07363* Define the Cantor normal form function, which takes as input a finitely supported function from to and outputs the corresponding member of the ordinal exponential . The content of the original Cantor Normal Form theorem is that for this function is a bijection onto for any ordinal (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to ). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2\u00a07397 of this function in terms of df-oi\u00a07225. (Contributed by Mario Carneiro, 25-May-2015.)\nCNF OrdIso seq\ud835\udf14\n\nTheoremcantnffval\u00a07364* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)\nCNF OrdIso seq\ud835\udf14\n\nTheoremcantnfdm\u00a07365* The domain of the Cantor normal form function (in later lemmas we will use CNF to abbreviate \"the set of finitely supported functions from to \"). (Contributed by Mario Carneiro, 25-May-2015.)\nCNF\n\nTheoremcantnfvalf\u00a07366* Lemma for cantnf\u00a07395. The function appearing in cantnfval\u00a07369 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)\nseq\ud835\udf14\n\nTheoremcantnfs\u00a07367 Elementhood in the set of finitely supported functions from to . (Contributed by Mario Carneiro, 25-May-2015.)\nCNF\n\nTheoremcantnfcl\u00a07368 Basic properties of the order isomorphism used later. The support of an is a finite subset of , so it is well-ordered by and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso\n\nTheoremcantnfval\u00a07369* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 seq\ud835\udf14 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnfval2\u00a07370* Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 seq\ud835\udf14 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF seq\ud835\udf14\n\nTheoremcantnfsuc\u00a07371* The value of the recursive function at a successor. (Contributed by Mario Carneiro, 25-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 seq\ud835\udf14\n\nTheoremcantnfle\u00a07372* A lower bound on the CNF function. Since CNF is defined as the sum of over all in the support of , it is larger than any of these terms (and all other terms are zero so we can extend the statement to all instead of just those in the support). (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 seq\ud835\udf14 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnflt\u00a07373* An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent where is larger than any exponent which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 seq\ud835\udf14\n\nTheoremcantnflt2\u00a07374 An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnff\u00a07375 The CNF function is a function from finitely supported functions from to , to the ordinal exponential . (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnf0\u00a07376 The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnfreslem\u00a07377* The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.)\nCNF\n\nTheoremcantnfrescl\u00a07378* A function is finitely supported from to iff the extended function is finitely supported from to . (Contributed by Mario Carneiro, 25-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnfres\u00a07379* The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF CNF\n\nTheoremcantnfp1lem1\u00a07380* Lemma for cantnfp1\u00a07383. (Contributed by Mario Carneiro, 20-Jun-2015.)\nCNF\n\nTheoremcantnfp1lem2\u00a07381* Lemma for cantnfp1\u00a07383. (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso\n\nTheoremcantnfp1lem3\u00a07382* Lemma for cantnfp1\u00a07383. (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 seq\ud835\udf14 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 seq\ud835\udf14 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF CNF\n\nTheoremcantnfp1\u00a07383* If is created by adding a single term to , where is larger than any element of the support of , then is also a finitely supported function and it is assigned the value where is the value of . (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF CNF\n\nTheoremoemapso\u00a07384* The relation is a strict order on (a corollary of wemapso2\u00a07267). (Contributed by Mario Carneiro, 28-May-2015.)\nCNF\n\nTheoremoemapval\u00a07385* Value of the relation . (Contributed by Mario Carneiro, 28-May-2015.)\nCNF\n\nTheoremoemapvali\u00a07386* If , then there is some witnessing this, but we can say more and in fact there is a definable expression that also witnesses . (Contributed by Mario Carneiro, 25-May-2015.)\nCNF\n\nTheoremcantnflem1a\u00a07387* Lemma for cantnf\u00a07395. (Contributed by Mario Carneiro, 4-Jun-2015.)\nCNF\n\nTheoremcantnflem1b\u00a07388* Lemma for cantnf\u00a07395. (Contributed by Mario Carneiro, 4-Jun-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso\n\nTheoremcantnflem1c\u00a07389* Lemma for cantnf\u00a07395. (Contributed by Mario Carneiro, 4-Jun-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso\n\nTheoremcantnflem1d\u00a07390* Lemma for cantnf\u00a07395. (Contributed by Mario Carneiro, 4-Jun-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 seq\ud835\udf14 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnflem1\u00a07391* Lemma for cantnf\u00a07395. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct are -related as or , and WLOG assuming that , we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 seq\ud835\udf14 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF CNF\n\nTheoremcantnflem2\u00a07392* Lemma for cantnf\u00a07395. (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnflem3\u00a07393* Lemma for cantnf\u00a07395. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than has a normal form, we can use oeeu\u00a06601 to factor into the form where and (and a fortiori ). Then since , has a normal form, and by appending the term using cantnfp1\u00a07383 we get a normal form for . (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnflem4\u00a07394* Lemma for cantnf\u00a07395. Complete the induction step of cantnflem3\u00a07393. (Contributed by Mario Carneiro, 25-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremcantnf\u00a07395* The Cantor Normal Form theorem. The function CNF , which maps a finitely supported function from to to the sum over all indexes such that is nonzero, is an order isomorphism from the ordering of finitely supported functions to the set under the natural order. Setting and letting be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres\u00a07379, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremoemapwe\u00a07396* The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternative definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso\n\nTheoremcantnffval2\u00a07397* An alternative definition of df-cnf\u00a07363 which relies on cantnf\u00a07395. (Note that although the use of seems self-referential, one can use cantnfdm\u00a07365 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF OrdIso\n\nTheoremcantnff1o\u00a07398 Simplify the isomorphism of cantnf\u00a07395 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)\nCNF \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CNF\n\nTheoremmapfien\u00a07399* A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.)\n\nTheoremwemapwe\u00a07400* Construct lexicographic order on a function space based on a reverse well-ordering of the indexes and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.)\nOrdIso \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OrdIso\n\nPage List\nJump to page: Contents\u00a0 1\u00a01-100 2\u00a0101-200 3\u00a0201-300 4\u00a0301-400 5\u00a0401-500 6\u00a0501-600 7\u00a0601-700 8\u00a0701-800 9\u00a0801-900 10\u00a0901-1000 11\u00a01001-1100 12\u00a01101-1200 13\u00a01201-1300 14\u00a01301-1400 15\u00a01401-1500 16\u00a01501-1600 17\u00a01601-1700 18\u00a01701-1800 19\u00a01801-1900 20\u00a01901-2000 21\u00a02001-2100 22\u00a02101-2200 23\u00a02201-2300 24\u00a02301-2400 25\u00a02401-2500 26\u00a02501-2600 27\u00a02601-2700 28\u00a02701-2800 29\u00a02801-2900 30\u00a02901-3000 31\u00a03001-3100 32\u00a03101-3200 33\u00a03201-3300 34\u00a03301-3400 35\u00a03401-3500 36\u00a03501-3600 37\u00a03601-3700 38\u00a03701-3800 39\u00a03801-3900 40\u00a03901-4000 41\u00a04001-4100 42\u00a04101-4200 43\u00a04201-4300 44\u00a04301-4400 45\u00a04401-4500 46\u00a04501-4600 47\u00a04601-4700 48\u00a04701-4800 49\u00a04801-4900 50\u00a04901-5000 51\u00a05001-5100 52\u00a05101-5200 53\u00a05201-5300 54\u00a05301-5400 55\u00a05401-5500 56\u00a05501-5600 57\u00a05601-5700 58\u00a05701-5800 59\u00a05801-5900 60\u00a05901-6000 61\u00a06001-6100 62\u00a06101-6200 63\u00a06201-6300 64\u00a06301-6400 65\u00a06401-6500 66\u00a06501-6600 67\u00a06601-6700 68\u00a06701-6800 69\u00a06801-6900 70\u00a06901-7000 71\u00a07001-7100 72\u00a07101-7200 73\u00a07201-7300 74\u00a07301-7400 75\u00a07401-7500 76\u00a07501-7600 77\u00a07601-7700 78\u00a07701-7800 79\u00a07801-7900 80\u00a07901-8000 81\u00a08001-8100 82\u00a08101-8200 83\u00a08201-8300 84\u00a08301-8400 85\u00a08401-8500 86\u00a08501-8600 87\u00a08601-8700 88\u00a08701-8800 89\u00a08801-8900 90\u00a08901-9000 91\u00a09001-9100 92\u00a09101-9200 93\u00a09201-9300 94\u00a09301-9400 95\u00a09401-9500 96\u00a09501-9600 97\u00a09601-9700 98\u00a09701-9800 99\u00a09801-9900 100\u00a09901-10000 101\u00a010001-10100 102\u00a010101-10200 103\u00a010201-10300 104\u00a010301-10400 105\u00a010401-10500 106\u00a010501-10600 107\u00a010601-10700 108\u00a010701-10800 109\u00a010801-10900 110\u00a010901-11000 111\u00a011001-11100 112\u00a011101-11200 113\u00a011201-11300 114\u00a011301-11400 115\u00a011401-11500 116\u00a011501-11600 117\u00a011601-11700 118\u00a011701-11800 119\u00a011801-11900 120\u00a011901-12000 121\u00a012001-12100 122\u00a012101-12200 123\u00a012201-12300 124\u00a012301-12400 125\u00a012401-12500 126\u00a012501-12600 127\u00a012601-12700 128\u00a012701-12800 129\u00a012801-12900 130\u00a012901-13000 131\u00a013001-13100 132\u00a013101-13200 133\u00a013201-13300 134\u00a013301-13400 135\u00a013401-13500 136\u00a013501-13600 137\u00a013601-13700 138\u00a013701-13800 139\u00a013801-13900 140\u00a013901-14000 141\u00a014001-14100 142\u00a014101-14200 143\u00a014201-14300 144\u00a014301-14400 145\u00a014401-14500 146\u00a014501-14600 147\u00a014601-14700 148\u00a014701-14800 149\u00a014801-14900 150\u00a014901-15000 151\u00a015001-15100 152\u00a015101-15200 153\u00a015201-15300 154\u00a015301-15400 155\u00a015401-15500 156\u00a015501-15600 157\u00a015601-15700 158\u00a015701-15800 159\u00a015801-15900 160\u00a015901-16000 161\u00a016001-16100 162\u00a016101-16200 163\u00a016201-16300 164\u00a016301-16400 165\u00a016401-16500 166\u00a016501-16600 167\u00a016601-16700 168\u00a016701-16800 169\u00a016801-16900 170\u00a016901-17000 171\u00a017001-17100 172\u00a017101-17200 173\u00a017201-17300 174\u00a017301-17400 175\u00a017401-17500 176\u00a017501-17600 177\u00a017601-17700 178\u00a017701-17800 179\u00a017801-17900 180\u00a017901-18000 181\u00a018001-18100 182\u00a018101-18200 183\u00a018201-18300 184\u00a018301-18400 185\u00a018401-18500 186\u00a018501-18600 187\u00a018601-18700 188\u00a018701-18800 189\u00a018801-18900 190\u00a018901-19000 191\u00a019001-19100 192\u00a019101-19200 193\u00a019201-19300 194\u00a019301-19400 195\u00a019401-19500 196\u00a019501-19600 197\u00a019601-19700 198\u00a019701-19800 199\u00a019801-19900 200\u00a019901-20000 201\u00a020001-20100 202\u00a020101-20200 203\u00a020201-20300 204\u00a020301-20400 205\u00a020401-20500 206\u00a020501-20600 207\u00a020601-20700 208\u00a020701-20800 209\u00a020801-20900 210\u00a020901-21000 211\u00a021001-21100 212\u00a021101-21200 213\u00a021201-21300 214\u00a021301-21400 215\u00a021401-21500 216\u00a021501-21600 217\u00a021601-21700 218\u00a021701-21800 219\u00a021801-21900 220\u00a021901-22000 221\u00a022001-22100 222\u00a022101-22200 223\u00a022201-22300 224\u00a022301-22400 225\u00a022401-22500 226\u00a022501-22600 227\u00a022601-22700 228\u00a022701-22800 229\u00a022801-22900 230\u00a022901-23000 231\u00a023001-23100 232\u00a023101-23200 233\u00a023201-23300 234\u00a023301-23400 235\u00a023401-23500 236\u00a023501-23600 237\u00a023601-23700 238\u00a023701-23800 239\u00a023801-23900 240\u00a023901-24000 241\u00a024001-24100 242\u00a024101-24200 243\u00a024201-24300 244\u00a024301-24400 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292\u00a029101-29200 293\u00a029201-29300 294\u00a029301-29400 295\u00a029401-29500 296\u00a029501-29600 297\u00a029601-29700 298\u00a029701-29800 299\u00a029801-29900 300\u00a029901-30000 301\u00a030001-30100 302\u00a030101-30200 303\u00a030201-30300 304\u00a030301-30400 305\u00a030401-30500 306\u00a030501-30600 307\u00a030601-30700 308\u00a030701-30800 309\u00a030801-30900 310\u00a030901-31000 311\u00a031001-31100 312\u00a031101-31200 313\u00a031201-31300 314\u00a031301-31400 315\u00a031401-31500 316\u00a031501-31600 317\u00a031601-31700 318\u00a031701-31800 319\u00a031801-31900 320\u00a031901-32000 321\u00a032001-32100 322\u00a032101-32154\n Copyright terms: Public domain < Previous\u00a0\u00a0Next >","date":"2017-10-18 01:54:42","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, 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대방광불화엄경은 줄여서 '화엄경'이라고 부르기도 하며, 부처와 중생이 둘이 아니라 하나라는 것을 중심사상으로 하고 있다. 화엄종의 근본경전으로 법화경과 함께 한국 불교사상 확립에 중요한 영향을 끼친 경전이다.
이 책은 닥종이에 찍은 목판본으로 마치 병풍처럼 펼쳐서 볼 수 있는 절첩장 형태이며, 접었을 때의 크기는 세로 31.6㎝, 가로 12.3㎝이다. 검푸른 빛의 표지에는 금색으로 제목이 써 있고 끝에 '정(貞)'자가 있어 당나라 삼장반야(三臟般若)가 한문으로 번역한 정원본임을 알 수 있다.
화엄경 정원본(貞元本) 40권 중 권 제31에 해당하는 이 책은 고려 숙종때에 간행한 해인사고려각판(국보 제206호)에서 찍어낸 것으로, 간행시기는 13세기에서 14세기로 추정된다.
Avatamsaka Sutra is one of the canonical scriptures of Mahayana Buddhism; the central idea expressed in this sutra is the unity of Buddha and sentient beings. As the most fundamental text for Korean Buddhism, this sutra has had seminal influence on Korean Buddhist philosophy along with the Saddharmapundarika Sutra (Lotus Sutra). Printed on mulberry paper using woodblocks, this book measures 12.3 cm wide and 30.5 cm long when folded. The title of this book is written in gold on the dark blue cover, and the Chinese character "貞" indicates that this book is the Zhenyuan Version translated by Banruo of the Tang Dynasty. It was printed with the Printing Woodblocks of Miscellaneous Buddhist Scriptures in Haeinsa Temple, Hapcheon (National Treasure No. 206), and the production date of said woodblocks is believed to be around the 13th ~ 14th century. | {
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Alfonso Quiroz Cuarón (Ciudad Jiménez, Chihuahua, 8 de febrero de 1910 - Ciudad de México, 16 de noviembre de 1978) es considerado el padre de la criminología mexicana.
Semblanza biográfica
Fue hijo de Francisco Quiroz y de Refugio Cuarón. A los 15 años de edad, pierde a su padre, víctima de un asesinato en las oficinas del ferrocarril, en la estación de Tampico. Lejos de cambiar su vida, nace en él el interés de averiguar el por qué de las conductas homicidas en los hombres.
En 1929, llega a la Ciudad de México, desempeñándose como ayudante del Juzgado IV Correccional, donde formó parte del Consejo Supremo de Defensa y Prevención Social, y al año siguiente ingresa como practicante en el Servicio Médico Forense.
En 1939, se convierte en el primer criminólogo graduado en la Universidad Nacional Autónoma de México, y obtuvo la jefatura de la Sección Médico Psicológica del Centro de Observación del Tribunal de Menores. Fue discípulo del gran psiquiatra forense José Gómez Robleda, quien estuvo a cargo del Neuropsiquiátrico General de La Castañeda. Sus informes sobre perfiles criminales tuvieron repercusión en todo el mundo.
En 1932, en la cárcel de Lecumberri, junto con los doctores Matilde Carrillo, Benjamín Argüelles y González Enríquez se dispusieron a realizar los primeros estudios científicos sobre las personalidades atípicas de los reclusos, y logró clasificarlas. Quiroz sostiene entonces que la política criminológica debe considerar no solo la reclusión, sino también la rehabilitación, haciendo prevención en los aspectos sociales, económicos y psicológicos. Años más tarde, no solo logra la búsqueda de recuperación de los presos, sino que propone la edificación de diferentes unidades penales en la Ciudad de México, con lo cual desapareció el reclusorio Lecumberri.
Para él, las ciencias criminalísticas y las criminológicas, al unirse, se enriquecen una a la otra complementándose, porque los conocimientos técnicos de la primera perfeccionan los conocimientos de la segunda, y logran una síntesis para la explicación de las conductas antisociales, lo que ayuda a establecer medidas preventivas.
Aportaciones
Ramón Mercader
Uno de los casos más sonados en los que intervino es el de Ramón Mercader (conocido con el alias de Jaques Mornard), el asesino de León Trotski. Gracias a sus pericias, se logró obtener la identificación plena del criminal.
"Goyo" Cárdenas
Gregorio Cárdenas, asesino de mujeres, las enterraba en el patio de su casa, y fue también parte de su investigación en 1943. Este criminal no llegó a ser sentenciado, ya que durante su reclusión en la cárcel de Lecumberri cursó estudios de Derecho, lo que le valió poder defenderse y asesorar a otros reclusos en sus correspondientes procesos penales. Finalmente, Cárdenas fue puesto en libertad.
Enrico Sampietro
El famoso falsificador Enrico Sampietro fue finalmente atrapado por la policía gracias a la intervención del doctor Quiroz Cuarón, en el año 1948.
Higinio Sobera de la Flor
La personalidad de otro asesino de mujeres: Higinio Sobera de la Flor también fue investigada por él.
Los restos del tlatoani Cuauhtémoc
En 1952, coordina los estudios para establecer la autenticidad de los restos del último emperador mexica, Cuauhtémoc, encontrados por la arqueóloga Eulalia Guzmán.
República Dominicana
En 1965, la Organización de la Naciones Unidas (ONU) lo comisiona en la República Dominicana para realizar estudios comportamentales de los soldados estadounidenses que habían invadido el país.
Reconocimientos
A la fecha de este artículo, una de las seis comunidades que atienden a los adolescentes que entran en conflicto con la ley en la Ciudad de México, lleva su nombre. Es la Comunidad Especializada para Adolescentes "Dr. Alfonso Quiroz Cuarón".
Bibliografía
Un estrangulador de mujeres.
El tipo sumario - En colaboración con el criminólogo mexicano José Gómez Robleda. Criminalia, año XVII. México, 1951.
Higinio Sobera de la Flor - Criminalia, año XX, México, 1954
El asesino de León Trotsky y su peligrosidad - Revista de Criminalística de Cuba. La Habana, 1956, reeditado en Etudes Internationales de Psycho-Socieologie Criminelle. París, 1957.
El costo social del delito - En colaboración con Raúl Quiroz. Botas, México, 1970.
Medicina Forense. - Porrúa. México, 1976.
Referencias
Nacidos en Jiménez
Criminólogos de México
Fallecidos en Ciudad de México | {
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Europium(II)-hydroxid ist eine chemische Verbindung des Europiums aus der Gruppe der Hydroxide. Es liegt in Form seines Hydrates Eu(OH)2·H2O vor.
Gewinnung und Darstellung
Europium(II)-hydroxid entsteht bei der Zersetzung von Europium(II,III)-oxid an feuchter Luft neben Europium(III)-hydroxid. Direkt entsteht es bei der Reaktion von Europium mit Wasser in Natronlauge.
Eigenschaften
Europium(II)-hydroxid-hydrat ist ein leuchtend gelber Feststoff, der sich langsame in Europium(III)-hydroxid zersetzt. Dies erfolgt auch unter Schutzgas. Er besitzt eine orthorhombische Kristallstruktur mit der und den Gitterparametern a = 670,1 pm, b = 619,7 pm, c = 365,2 pm.
Einzelnachweise
Europiumverbindung
Hydroxid | {
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Q: Refactor my C# code - Switch statement I have the following code which I am are currently using .... Basically, this method assigns the correct boolean flag (TRUE/FALSE) for each Task. As more and more tasks need to be added .. I can see that the switch statement will have to grow to cater for every task.
There has to be an easier way ... to keep the method small.
Code: (forget naming convention, it has been changed for posting)
public ClassStructure.User AssignTaskStatusToUser(ClassStructure.User,
List<ClassStructure.Tasks> TaskStatus)
{
foreach (ClassStructure.Tasks data in TaskStatus)
{
string Task_CallID = data.Task_Call_ID;
switch (Task_CallID)
{
case ClassStructure.Tasks_CallIDs_Strings.TASK1:
User.TASK1 = data.Task_Flag;
break;
case ClassStructure.Tasks_CallIDs_Strings.TASK2:
User.TASK2 = data.Task_Flag;
break;
case ClassStructure.Tasks_CallIDs_Strings.TASK3:
User.TASK3 = data.Task_Flag;
break;
}
}
return User;
}
ClassStructure.Tasks_CallIDs_Strings = String Representation of the Tasks
data.Task_Flag = boolean
User.TASKX = boolean
Any feedback is welcome. I am sure there is an easy solution.
A: For a lot of values like these, I would use a map something like this:
Dictionary<ClassStructure.Tasks_CallIDs_Strings, Task_Flag>
and retrieve values by mapping the CallIDs strings.
Edit:
As everyone can now see, the real problem of refactoring this example lies in refactoring User.TASKX. Making it a list should suffice - as it could then be indexed by the same string ClassStructure.Tasks_CallIDs_Strings
A: Oh... Reconsider your naming scheme.
public delegate void TaskAssigner(User user, bool taskFlag)
IDictionary<string, TaskAssigner> taskAssigners = new Dictionary<string, TaskAssigner>();
...
taskAssigners.Add(ClassStructure.Tasks_CallIDs_Strings.TASK1, (u, t) => u.TASK1 = t;);
taskAssigners.Add(ClassStructure.Tasks_CallIDs_Strings.TASK2, (u, t) => u.TASK2 = t;);
...
foreach(ClassStructure.Tasks data in TaskStatus)
taskAssigners[data.Task_Call_ID](user, data.Task_Flag);
A: I was thinking something like this - but maybe I missed the point of what it is all for?
public class User
{
private Dictionary<string,Task> tasks;
internal Dictionary<string,Task> Tasks
{
get { return tasks; }
set { tasks = value; }
}
internal void AddTask(Task task)
{
tasks.Add(task.Task_Call_ID,task);
}
internal void AddTasks(List<Task> task)
{
foreach(Task task in Tasks)
{
tasks.Add(task.Task_Call_ID,task);
}
}
}
The Task class could have properties that allowed you to pass a function pointer (to the function that actually executes a task) if you needed that kind of flexibility - and you could add other methods like ExecuteTasks to User as well...
A: Could you have an array/list of tasks instead and use Task_CallID as an index into that?
e.g.
User.Tasks[Task_CallID] = data.Task_Flag;
If you must have them all as members there are other options:
*
*Maintain a mapping from Task_Call_ID to PropertyInfo reference and use that to set the correct property
*Use reflection to find the property based on the number bit (X) and set that property
Both of these are reflection based and a bit nasty.
A: Why not make a Users Tasks structured as a list:
User Class
public List<ClassStructure.Tasks> Tasks {
get; set;
}
Your Method becomes:
public void AssignTasks(User user, List<ClassStructure.Tasks> TaskStatus)
{
user.Tasks.AddRange(TaskStatus)
}
Which is to say that you don't need the method at all.
Your accessor then becomes running Find on a user's Tasks and checking the Tasks flag.
A: Dictionary is a great alternative for this. However, when a switch/case gets very complex look at using the strategy pattern (not for your scenario though).
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\section{Introduction}\label{sec:introduction}
With the rise of the big data phenomenon (and in particular, the rise of social networking), graph mining has become necessary to analyze the diverse relationships between objects and data, and to understand the complex structures of the underlying graph.
Analyzing such complex systems is crucial in understanding how a social network forms, or in making predictions about future network behavior.
However, graph mining is sensitive to the topological structures of a network. For example, in social network analysis, graph mining can leverage topology information to identify communities in a network, but does not have the ability to determine if the topology contains noise (or other errors). One way to avoid this is using multilayer network \cite{boccaletti2014structure}.
Multilayer network is a group of networks which depict multiple relationships between nodes, where each layer in the group represents a particular type of relationship \cite{de2013mathematical,boccaletti2014structure,loe2015comparison}.
Take the AUCS dataset \cite{kim2015community} as an example. The multiple layers represent five different relationship types between 61 employees of a university department: (i) coworking, (ii) having
lunch together, (iii) facebook friendship, (iv) spending leisure time together, and (v) coauthorship. Figure \ref{intro-layers} shows the distances among these different layers, calculated by \cite{de2015structural}, where the distance between layers is the distance value of the root node of the smallest subtree containing both layers. From Figure \ref{intro-layers}, we observe a number of key interactions: for example we observe that being coworkers and having lunch together (representing professional interactions) is strongly related through the low layer distance, and spending lesiure time together is likewise close to being connected on facebook (representing social interactions).
In general, by considering multiple relationship types, we argue that multilayer network inherently reflects essential interactions between nodes in a manner that is robust to the noise presents in any individual relationship type.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{Figures/intro-layers}
\caption{Dendrogram of layers for the AUCS dataset.}
\label{intro-layers}
\end{figure}
Nowadays, graph mining methods on multilayer network typically concentrate on different network granularities \cite{de2017community}, depending on the task. For example, on node/edge granularity, \cite{heaney2014multiplex} and \cite{hu2014conditions} focus on developing suitable centrality measures \cite{costenbader2003stability} like cross-layer degree centrality for multilayer network \cite{brodka2011degree,brodka2012analysis}. Brodka et.al \cite{kazienko2010individual,brodka2010method} proposed multilayered local clustering coefficient (MLCC) and cross-layer clustering coefficient (CLCC) to depict cluster coefficient \cite{zhou2005maximal} of a node in a multilayer network.
In contrast, the cluster level is often used for community detection \cite{boccaletti2014structure,de2015identifying,loe2015comparison,domenico2015identifying,chen2016multilayer,jeub2017local,deford2017spectral}, and the layer level used to analyze the interactions between different layer types \cite{de2015structural,benson2016higher}.
In this paper, we made an attempt to propose three novel graph mining methods for multilayer network by combining analysis at all levels of granularity that is suitable for a range of tasks. We achieve this via embedding the nodes of the multilayer network into a vector space. This vector space can be interpreted as a hidden metric space \cite{boguna2009navigability} for the multilayer network that naturally defines concepts of similarity between nodes, and captures many of the aspects of the original multilayer network, facilitating vector-based representation to a variety of machine learning algorithms to solve a range of tasks.
In standard network analysis, many such network embedding methods have been developed \cite{chen2017fast}, such as DeepWalk \cite{perozzi2014deepwalk}, LINE \cite{tang2015line} and node2vec \cite{grover2016node2vec}. These methods are all based upon generating samples of random walks over an input graph, with the properties of the random walk varying by method.
However, these methods are built on top of a single graph, and to the best of our knowledge, the graph embedding method for multilayer graph has not been rigorously explored.
Hence, we propose a generic multilayer graph embedding framework, which applies to any graph embedding method developed for single-layer graphs. In particular, we introduce three principled methods (``Network Aggregation,'' ``Results Aggregation'' and ``Network Co-analysis'') for extending graph embedding to multilayer networks:
\begin{itemize}
\item \textbf{Network Aggregation}: The assumption for this method is all edges from different layers are equal \cite{boccaletti2014structure,loe2015comparison}. Based on this assumption, this method aggregates all networks into a single (weighted) network (where multiple edges between nodes are not allowed) and then applies existing graph embedding algorithms to analysis the merged graph. Note that this merged graph no longer distinguishes between relationship types, as many edges for node pairs that share multiple edges in the multilayer network are not retained.
\item \textbf{Results Aggregation}: The assumption for this method is different layers have totally different kinds of edges \cite{berlingerio2013abacus}, which implies that even if two nodes have edges in different layers, these edges are totally different and cannot be merged. Based on this assumption, this method applies graph embedding to each layer separately, then merges the vector spaces together to define a vector space of the multilayer network.
\item \textbf{Layer Co-analysis}: Unlike the first two methods which only consider inter-layer edges (network aggregation) or intra-layer edges (results aggregation), this method leverages interactions among different layers to allow for traversal between layers, and to retain the structure of each individual layer. Specifically, we introduce a new random walk method that is capable of traversing layers, thereby encoding important interactions between nodes and layers. The methodology of random walk in a multilayer network is a general extension of single-layer embedding methods. For example, node2vec and DeepWalk are both state-of-the-art single-layer graph embedding methods. As node2vec added to DeepWalk the ability to control the homophily and structural equivalence properties of random walk samples, we enable the ability for random walk samples to traverse multiple layers. In particular, we control the degree to which this occurs through the addition of the $r\in[0,1]$ parameter, which determines the probability that the next step of a random walk will traverse to a different layer in the multilayer network.
\end{itemize}
In Section \ref{method}, we gives detailed information on three approaches for performing embedding on multilayer networks. In Section \ref{eva}, we use experimental results to demonstrate that our methods for multilayer network embedding can improve upon the results of node2vec for a link prediction task. In Section \ref{sec:related}, we outline related work in single network embedding, and conclude in Section \ref{sec:conclusion} with discussion and future work.
\section{Methods} \label{method}
Given a multilayer network $MN=(V,L,A)$ with vertex set $V$, layer set $L$, and multilayer edge set $A$ (where $A\subseteq \left\{ (x,y,l)|x,y\in V,l\in L \right\}$ and we denote the existence of an edge between $x$ and $y$ in layer $l$ by $a^l_{xy}=1$), our task in vertex embedding is to learn a mapping function $f:V \to \mathbb{R}^d$ where $d$ is the chosen dimension of the vector space. We are effectively looking to find a function $f$ that represents the features of a vertex from the multilayer network. Figure \ref{med-overview} shows the three proposed architectures for projecting a multilayer network into a vector space. The following subsections provide details on the implementation and structure of each architecture, as well as the corresponding strategy for constructing the function $f$.
\begin{figure}[!t]
\centering
\subfigure[Architecture of Network Aggregation]{
\label{intro-NA}
\includegraphics[width=\linewidth]{Figures/med-overview-NA}
}
\subfigure[Architecture of Results Aggregation]{
\label{intro-RA}
\includegraphics[width=\linewidth]{Figures/med-overview-RA}
}
\subfigure[Architecture of Layer Co-analysis]{
\label{intro-LC}
\includegraphics[width=\linewidth]{Figures/med-overview-LC}
}
\caption{The architecture of three methods
to do multilayer networks embedding.}
\label{med-overview}
\end{figure}
\subsection{Network Aggregation}\label{subsec:NA}
Network aggregation is the baseline of the proposed methods for multilayer network embedding, where the layers of the multilayer network are merged to obtain a single network, and the regular node2vec method is applied to the merged graph. Algorithm \ref{Alg:NA} represents the network aggregation process.
In this method, we have a network aggregation function $g: MN\to G$, i.e. a function that takes a multilayer MN and outputs a merged network $G=(V,E)$ that shares the vertex set $V$ with the multilayer network, but has an edge set $E=\left\{ { (i,j)|\sum _{ l\in L }^{ }{ a^{ l }_{ ij }\geq 1 } } \right\} $ that disallows multi-edges. The mapping function $f$ is defined by training on the merged network $G$ using node2vec.
Note that by combining all layers together in this method, we have lost the details of each layer and also have no way to leverage the interactions between layers when the function $f$ is learned.
\begin{algorithm}[!t]
\caption{Network Aggregation Algorithm}
\label{Alg:NA}
\KwIn{Multilayer Network $MN$}
Initialize $G = (V,\emptyset)$\;
\For{all $i,j \in V$}
{
\For{all $l\in L$}
{
\If{$a^l_{ij}=1$}{
// add an edge for $G$\;
$G$ $\leftarrow$ $G$ $\cup$ (i,j)\;
break}
}
}
$f \leftarrow node2vec(G)$\;
\end{algorithm}
\subsection{Results Aggregation}
Here we project each network layer of the multilayer network into a separate vector space, and concatenate the resulting vector spaces. We define each layer graph as $G_l=(V,E_l)$ where $E_{ l }=\left\{ {(x,y)|a^l_{xy}=1} \right\}$ for $l\in L$ and learn $|L|$ functions $f_l: V\to \mathbb{R}^{d'}$ that are combined to form the map $f$ as $f=f_1||f_2||...||f_{|L|}$, where $||$ denotes the concatenation operator s.t. $f: V \to \mathbb{R}^{d'|L|}$.
Unlike the network aggregation method, the dimension $d=d'|L|$ of the resulting vector space scales with the number of layers in the layer set $L$. Though this method can in theory preserves the structure of each layer, it is also unable to leverage interactions between layers when learning the mapping function. Algorithm \ref{Alg:RA} outlines the result aggregation process.
\begin{algorithm}[!t]
\caption{Results Aggregation Algorithm}
\label{Alg:RA}
\KwIn{Multilayer Network $MN$}
initialize $f$ as empty\;
\For{ each $layer$ $l\in L$}
{
Initialize $G_l = (V,\emptyset)$\;
\For{all $i,j \in V$}
{
\If{$a^l_{ij}=1$}{
// add an edge for $G_l$\;
$G_l$ $\leftarrow$ $G_l$ $\cup$ (i,j)\;
}
}
$f_l\leftarrow node2vec(G_l)$\;
$f\leftarrow f||f_l$
}
\end{algorithm}
\subsection{Layer Co-analysis}
To overcome the fact that neither network aggregation nor results aggregation can leverage interactions between layers, we adopt layer co-analysis for the construction of a vector space that is cognizant of interactions between layers as well as preserving the structure of each layer.
Consider random walks over the multilayer networks depicted in Figure \ref{med-interplay}, where dotted black lines represent correspondence between the nodes of each layer, and bold lines represent edges in the network. If we use result aggregation in Figure \ref{subfig-path}, there is no path from node $A$ to node $C$, as it is not possible to traverse between layers. Therefore the vector space cannot learn to associate these nodes. In contrast, if we run network aggregation on Figure \ref{subfig-multiedge}, we ignore the multi-edge between nodes $A'$ and $B'$ (represented by the bold red lines). These issues motivate the need for a method that can traverse the path (represented by the dotted red lines) between layers from $A$ to $C$, and that can retain the information implied by the multi-edge.
\begin{figure}[!t]
\centering
\subfigure[Path retention]{
\label{subfig-path}
\includegraphics[width=0.46\linewidth]{Figures/med-path-vertical}
}
\subfigure[Multi-edge retention]{
\label{subfig-multiedge}
\includegraphics[width=0.46\linewidth]{Figures/med-multiedge-vertical}
}
\caption{Random walks on multilayer networks.}
\label{med-interplay}
\end{figure}
As LINE improved upon the uniform random walks of DeepWalk with weighted 2nd order walks, and node2vec introduced the parameters $p$ and $q$ to control the local and global biases of the sample random walks, we enable random walks to traverse between layers of a multilayer network, and introduce the parameter $r$ to control this tendency.
Let $i_{|L|}$ represent the number of distinct layers connected to node $i$ $\forall i\in V$ of the multilayer network $MN$. That is, we can represent $i_{|L|}$ as in Equation \ref{eq:connected_layers}.
\begin{equation}
i_{ |L| }=\sum _{ l=1 }^{ |L| }{ I\left[ \left( \sum _{ (i,j,l)\in E }^{ }{ I[a^{ l }_{ ij }=1] } \right) \geq 1 \right] }
\label{eq:connected_layers}
\end{equation}
Where $I$ denotes the indicator function. Then we introduce a 2nd order random walk with parameters $p,q,r$. If the random walk previously traversed edge $(z,x,l')$ and the current node $x$ only has edges on layer $l'$ (i.e. $x_{|L|}=1$), then we step according to the node2vec strategy (here, with binary edges), i.e. $P\left( t_{ i }=(x,y,l)|t_{ i-1 }={ (z,x,l') } \right)\propto\alpha_{pq}(z,x,l)$, where $\alpha_{pq}(z,x,l)$ is a multilayer modification to the node2vec $p$, $q$ factors as follows:
\begin{align}
\alpha _{ pq }(z,x,l)=\begin{cases} 1/p & if\quad d^l_{ zx}=0 \\ 1 & if\quad d^l_{ zx }=1 \\ 1/q & if\quad d^l_{zx}=2 \end{cases}
\end{align}
where $d^l_{zx}$ is the shortest path between nodes $z$ and $x$ in layer $l$ of the multilayer graph (where nodes $z,x$ may be the same node).
Otherwise, if $x_{|L|}>1$, the random walk stays on the current layer $l'$ with probability $r$, and moves along the edge of another layer $l$ with probability $1-r$. That is, the random walk traversal probability for $x_{|L|}>1$ is given by Equation \ref{eq:traversal}.
\begin{align}
&P\left( t_{ i }=(x,y,l)|t_{ i-1 }={ (z,x,l') } \right) \nonumber\\
\propto &\begin{cases} \alpha _{ pq }(z,x,l)r & if\quad l=l' \\ \frac { \alpha _{ pq }(z,x,l) }{ x_{ |L| }-1 } (1-r) & otherwise \end{cases}
\label{eq:traversal}
\end{align}
Pseudocode for this method is given in Algorithm \ref{Alg:Co}, where $node2vecSGD$ refers to running stochastic gradient descent on the node2vec loglikelihood \cite{grover2016node2vec} with the multilayer random walks taking the place of the standard node2vec walks. In this paper, we only consider binary edges (support for weighted multilayer networks could be incorporated trivially through multiplying the edge traversal probabilities by the normalized edge weights).
Note that in this algorithm, the $r$ variable represents how important we view the relationships between layers to be in comparison to the interactions between nodes of the same layer.
For $r \rightarrow 0$, random walks will always traverse to different layers of the multilayer network when possible, whereas $r \rightarrow 1$ will restrict each random walk to stay on the layer in which it was initialized.
\begin{algorithm}[!t]
\caption{Layer Co-analysis Algorithm}
\label{Alg:Co}
\KwIn{Multilayer Network $MN$,$r,\alpha_{pq}$,$num\_walks$,$walk\_length$}
Initialize $walk\_list$ to empty\;
\For{$nw\_iter$ from 1 to $num\_walks$}
{
Initialize current edge $(i,j,l)\leftarrow (i_0,j_0,l_0)$ uniformly at random\;
\For{$wl\_iter$ from 1 to $walk\_length$}
{
$walk\_list[nw\_iter][wl\_iter]\leftarrow i$\;
with probability $r$, choose $next\_layer=l$, otherwise choose $next\_layer=l'$ uniformly at random for some layer $l'$ incident to $j$\;
set current edge $(i,j,l)\leftarrow(j,i',next\_layer)$ proportional to $\alpha_{pq}(j,i',next\_layer)$ for some $i'$ incident to $j$ through $next\_layer$\;
}
}
$f\leftarrow node2vecSGD(walk\_list)$
\end{algorithm}
\section{Evaluation} \label{eva}
Here we choose five real-world multilayer datasets, comparing the performance of our three multilayer network embedding methods on the link prediction task. In this paper, we set $p=q=r=0.5$, with 10 random walks of 80 steps initialized from each node (i.e. we have $10|V|$ random walks in total for each graph), and compare against two standard methods of link prediction \cite{zhou2009predicting}. All experiments in this paper were conducted locally on CPU using a Mac Book Pro with an Intel Core i7 2.5GHz processor and 16GB of 1600MHz RAM.
Though this limits the size of our experiments in this preliminary work, our method naturally inherits the runtime scalability of node2vec.
\subsection{Datasets}
Table \ref{tab-datasets} shows the size of the five datasets, with layer information as follows:
\begin{table}[!t]
\centering
\caption{Information for five datasets.}
\scalebox{0.85}{
\begin{tabular}{|c|c|c|c|c|}
\hline
Datasets & \# Layers & \# Nodes & \# Edges & \begin{tabular}[c]{@{}c@{}}labels (\# of corresponding nodes)\end{tabular} \\ \hline
AUCS & 5 & 61 & 353 & none \\ \hline
Terrorists & 4 & 78 & 623 & none \\ \hline
Students & 3 & 185 & 311 & none \\ \hline
VC & 3 & 29 & 250 & \begin{tabular}[c]{@{}c@{}}Boys (12)\\ Girls (17)\end{tabular} \\ \hline
LN & 4 & 191 & 511 & \begin{tabular}[c]{@{}c@{}}Leskovec's collaborator (87)\\ Ng's collaborator (104)\end{tabular} \\ \hline
\end{tabular}
}
\label{tab-datasets}
\end{table}
\begin{itemize}
\item AUCS \cite{kim2015community} (AUCS): The multiple layers represent five different relationship types
between 61 employees of a university department: (i) coworking, (ii) having
lunch together, (iii) Facebook friendship, (iv) onine
friendship (having fun together), and (v) coauthorship.
\item Terrorist network \cite{roberts2011roberts} (Terrorists): Each layer represents known interactions and ties between terrorists in the Noordin Top Terrorist dataset. These ties cover four different relationship types:
(i) communication, (ii) financial, (iii) operation, and (iv) trust.
\item Student cooperation \cite{fire2012predicting} (Students): Each layer represents a type of cooperation or coordination
between 185 students of Ben-Gurion University: (i) Computer Network, which represents students who finished their papers on the same machine. (ii) Partner's Network, which represents joint work on a submission. (iii) Time Network, which indicates if students submitted papers in the same epoch.
\item Vickers Chan 7th grader social dataset \cite{vickers1981representing} (VC): Each layer represents an aspect of interaction between students in a class who were asked the following questions: (i) Who do you get on with in the class? (ii) Who are your best friends in the class? (iii) Who would you prefer to work with?
\item Leskovec-Ng collaboration dataset \cite{chen2017multilayer,zhang2014name,saha2015name}\footnote{The dataset can be downloaded from https://sites.google.com/site/pinyuchenpage/datasets} (LN): The coauthorship networks of Jure Leskovec and Andrew Ng from 1995 to 2014. A four layer multilayer graph is defined by partitioning the coauthorship networks into 5-year intervals. For each layer, there is an edge between two researchers if they coauthored at least one paper in the corresponding 5-year interval. In addition, each researcher is labeled as ``Leskovec's collaborator'' or ``Ng's collaborator'' depending upon collaboration frequency.
\end{itemize}
\subsection{Link Prediction Evaluation}
Here we perform the link prediction task with respect to the merged graph $G=(V,E)$ as defined in Section \ref{subsec:NA}.
This is because the purpose of our proposed three methods is to leverage multilayer relationships of the datasets to inform the node embedding, this embedding does not inherently associate nodes (or edges between them) with particular layers of the multilayer network.
We split each edge set $E$ into a training subset $E^T$ and a test subset $E^{P}$, where $E=E^{T} \cup E^{P}, E^{T} \cap E^{P} = \emptyset$. In this paper, we randomly chose $10\%$ of the edges from each $E$ as $E^P$.
For a multilayer network trained on $E^{T}$, we use our proposed embedding methods to find the corresponding vector spaces. We then calculate the distances of node pairs corresponding to $E^{P}$, and reorder these distances into an ascending list.
In order to predict links in the sampled multilayer network, we treat the first $10\%$ node pairs in $list$ have edges. Comparing these predicted edges with the true edges in $E^{P}$, we can evaluate the outcome of our methods.
Here, we introduce accuracy rate (see Equation \ref{eq-accuracy}) and F1-score (see Equation \ref{eq-f1}) to do the evaluation.
First of all, accuracy rate indicates the number of edges $C$ that has been corrected estimated in $E^{P}$.
\begin{equation}
\label{eq-accuracy}
Accuracy = \frac{C}{|E^P|}
\end{equation}
Second, as F1-score is the harmonic mean of the precision and recall values for each layer, Equation \ref{eq-f1} evaluates the average F1-score of each layer. Here, $F-measure_{l}=\frac{2\cdot PREC_l \cdot RECALL_{l}}{PREC_l + RECALL_{l}}$, $PREC_l$ and $RECALL_l$ are the precision and recall values for each layer $l$. Larger F1-score means better prediction performance.
\begin{equation}
\label{eq-f1}
F_l(\{ C_l \}_{l=1}^{L}, \{ C'_l \}_{l=1}^{L'}) = \frac{1}{|L|} \sum F-measure_l
\end{equation}
In addition, we introduce two famous local link prediction methods \cite{zhou2009predicting} ``Common Neighbor Similarity'' and ``Jaccard Similarity'' on merged network as the comparison methods. Where Common Neighbor Similarity is defined in Equation \ref{eq-CN}, and Jaccard Similarity is defined in Equation \ref{eq-Ja}. Where $\Gamma(X)$, $\Gamma(Y)$ stands for the neighborhoods of the node $X,Y \in V$ in the merged graph.
\begin{equation}
\label{eq-CN}
CN_{xy} = |\Gamma(X) \cap \Gamma(Y)|
\end{equation}
\begin{equation}
\label{eq-Ja}
Jaccard_{xy} = \frac{|\Gamma(X) \cap \Gamma(Y)|}{|\Gamma(X) \cup \Gamma(Y)|}
\end{equation}
Table \ref{tab-lp} shows the accuracy and F1-Score results for different datasets. In addition, we use bold text to indicate the best performance for each datasets. From the table we can tell that except LN datasets, our methods can achieve higher accuracy and F1-score for the rest of the datasets.
We use Figure \ref{intro-layers} and Figure \ref{eva-layer} to show the layer distance for each corresponding dataset.
In addition, we use Figure \ref{eva-topology} to show topology of each layer and the corresponding merged layers of three multilayer networks. As VC and LN have label information, we use different shape and color to demonstrate nodes with different labels.
Combine all these evaluations, we give a detailed analysis for each datasets.
\begin{itemize}
\item For AUCS, Terrorists and Students datasets: As AUCS dataset represents the interactions activities among employees, Terrorists dataset shows how terrorists work together, and Students dataset indicates the collaboration among students. It is clear that different layers have strong influence with each other. Which means if we want to predict edges information among two nodes, the important interactions of these two nodes among different layers should be considered. What's more, take the topology of layers of Terrorists dataset in Figure \ref{eva-terrorist-graphs} as an example, there are strong interactions among four layers, although layer 2 (Financial) has a small number of nodes, but layer co-analysis can recover necessarily information by random walk on different layers. Hence, it is reasonable and important to consider the interactions for different layers.
\item For VC dataset: As different questions indicate different problems, these different layers have relatively weak connections. For example, we cannot argue that a person gets on with (Q1) are all of his/her best friends, while the best friends of a person (Q2) are the same group of people that the person wants to work with (Q3). As shown in Figure \ref{eva-VC-graphs}, the first question is too general, which causes to create lots of noises (unnecessary edges) and therefore cannot indicate the true relationships among these nodes. If we combine these layers together or put more concentration on interactions among layers, then neither network aggregation nor layer co-analysis can reveal true information instead of just introducing more noises into the analysis.
\item For LN dataset: this dataset is a temporal dataset across 20 years that people joins or leaves the Leskovec's group or Andrew Ng's group. As shown in Figure \ref{eva-LN-graphs}, different layers in different time do not show any interactions. For example, in the first layer (LN\_1995\_1999), there is no blue nodes, and for the second (LN\_2000\_2004) and third layer (LN\_2005\_2010), two groups are expanded by themselves. what's more, as the time span of the multilayer network is too large, this particular feature indicates the fact that the interaction among these layers is not the key reason to form the topology in each corresponding layer. What's more, as there are a 5 years span between layers, the noises in these layers are the major reason why our methods cannot function well. Instead, the original Jaccard method is the best. Because this method only cares about the average number of shared neighbors for two nodes. So the noise has the least affected for such method.
\end{itemize}
\begin{table*}[!t]
\centering
\caption{Accuracy and F1-Score for Different Methods}
\begin{tabular}{|c|c|c||c|c|c|}
\hline
\multirow{3}{*}{Datasets} & \multicolumn{5}{c|}{Accuracy / F1-Score for Different Methods} \\ \cline{2-6}
& \multicolumn{2}{c||}{Regular Link Prediction Methods}
& \multicolumn{3}{c|}{Our Methods} \\ \cline{2-6}
& Common Neighbor & Jaccard Similarity & Network Aggregation & Results Aggregation & Networks Co-analysis \\ \hline
AUCS & 0.029 / 0.056 & 0.051 / 0.097 & 0.184 / 0.311 & 0.092 / 0.168 & \textbf{0.207} / \textbf{0.343} \\ \hline
Terrorists & 0.012 / 0.024 & 0.016 / 0.032 & 0.229 / 0.373 & 0.090 / 0.166 & \textbf{0.347} / \textbf{0.515} \\ \hline
Students & 0.015 / 0.030 & 0.138 / 0.243 & 0.139 / 0.225 & 0.063 / 0.119 & \textbf{0.127} / \textbf{0.2444} \\ \hline
VC & 0.049 / 0.093 & 0.098 / 0.178 & 0.400 / 0.571 & \textbf{0.650} / \textbf{0.788} & 0.550 / 0.710 \\ \hline
LN & 0.027 / 0.053 & \textbf{0.206} / \textbf{0.342} & 0.103 / 0.187 & 0.070 / 0.130 & 0.083 / 0.153 \\ \hline
\end{tabular}
\label{tab-lp}
\end{table*}
\begin{figure*}[!t]
\centering
\includegraphics[width=\linewidth]{Figures/eva-layers}
\caption{Layers Distance for the four datasets.}
\label{eva-layer}
\end{figure*}
\begin{figure*}[!t]
\centering
\subfigure[Four Layers of Terrorists Dataset]{
\label{eva-terrorist-graphs}
\includegraphics[width=\linewidth]{Figures/eva-terrorist-graphs}
}
\subfigure[Three Layers of VC Dataset]{
\label{eva-VC-graphs}
\includegraphics[width=\linewidth]{Figures/eva-VC-graphs}
}
\subfigure[Four Layers of LN Dataset]{
\label{eva-LN-graphs}
\includegraphics[width=\linewidth]{Figures/eva-LN-graphs}
}
\caption{Topology of three multilayer networks.}
\label{eva-topology}
\end{figure*}
\section{Related Work on Random Walk based Network Embedding}\label{sec:related}
In this section, we review related work on standard network embedding, that is, embedding methods proposed for single-layer graphs.
With the development of unsupervised feature learning techniques \cite{bengio2013representation}, deep learning methods proved successful in natural language processing tasks through neural language models. These models have been used to capture the semantic and syntactic structures of human language \cite{collobert2008unified}, and even logical analogies \cite{mikolov2013linguistic}, by embedding words as vectors.
As a graph can be interpreted as a kind of language (by treating random walks as the equivalent of sentences),
DeepWalk \cite{perozzi2014deepwalk} introduced such methods into network analysis, allowing for the projection of network nodes into a vector space. To solve the scalability problem of this method when applied to real world information networks (which often contain millions of nodes), LINE \cite{tang2015line} was developed. LINE extended the uniform random walks of DeepWalk to 1st and 2nd order weighted random walks, and it can project a network with millions of vertices and billions of edges into a vector space in a few hours.
However, both methods have limitations.
As DeepWalk uses uniform random walks for searching, it cannot provide control over the explored neighborhoods. In contrast, LINE proposes a breadth-first strategy to sample nodes and optimize the likelihood independently over 1-hop and 2-hop neighbors, but it has no flexibility in exploring nodes at future depths. In order to deal with both of these limitations, node2vec \cite{grover2016node2vec} provides a flexible and controllable strategy for exploring network neighborhoods through the parameters $p$ and $q$. From a practical standpoint, node2vec is scalable and robust to perturbations. Of course, none of these methods can deal with random walk samples that intelligently consider traversals between layers of multilayer networks. One of our methods (layer co-analysis) is therefore a natural progression of the literature in extending the capabilities of the random walk samples.
\section{Discussion and Future Work}\label{sec:conclusion}
In this paper, we demonstrate three different methods to project multilayer network into a representative vector space. The first method (Network Aggregation) aggregates all layers together to construct a merged network, and use standard network embedding method to project a multilayer network into a vector space.
The second method (Results Aggregation) uses standard network embedding to obtain a vector space for each corresponding layer, and then combines these vector spaces together to construct a new vector space for the multilayer network.
At last, as the first two methods do not leverage the important interactions between layers, we introduce a layer co-analysis method which leverage interactions among layers. In layer co-analysis, we use $r$ to constrain the behavior of the walk, where the greater the $r$, the greater the chance of the random walk to stay in the same layer. On the contrary, the smaller the $r$, the greater the possibility of random walk to choose different layers.
In the evaluation part, we compare the accuracy and F1-score for five datasets, by comparing to regular link prediction methods, we have proved that our method do have the ability to project a multilayer network into the suitable vector space.
To the best of our knowledge, since this paper is a first-line research for principled graph embedding a multilayer network into a vector space, our experimental results suggest some future work and new challenges along this line:
(i) From evaluation aspect, as we only use link prediction as the evaluation in this paper, the performance on multi-label classification is worth exploring. In addition, as our methods can be simply applied to layers with weighted edges and weighted interactions, we will test the performance on weighted multilayer network.
(ii) From algorithm perspective, the proposed co-layer analysis method involves an additional layer transition probability $r$ for multilayer network embedding. In the future work, we will further discuss how to automatically learn $r$ by analyzing layer distance for a multilayer network.
(iii) From the data type perspective, as attributed graphs have been widely introduced in big data analysis, we will continue to discuss the possibility to project an attributed multilayer graphs into a proper vector space by taking into the node/edge's properties.
\newpage
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,476 |
Q: Having an ERROR: java.lang.ClassNotFoundException I'm new to coding. Currently, I am using Appium Java Eclipse.
Below is the error log.
Nov 29, 2019 6:37:48 PM io.appium.java_client.remote.AppiumCommandExecutor$1 lambda$0
INFO: Detected dialect: W3C
Exception in thread "main" java.lang.NoClassDefFoundError: org/apache/commons/lang3/StringUtils
at io.appium.java_client.internal.ElementMap.getElementClass(ElementMap.java:77)
at io.appium.java_client.internal.JsonToMobileElementConverter.newRemoteWebElement(JsonToMobileElementConverter.java:67)
at org.openqa.selenium.remote.internal.JsonToWebElementConverter.apply(JsonToWebElementConverter.java:55)
at io.appium.java_client.internal.JsonToMobileElementConverter.apply(JsonToMobileElementConverter.java:61)
at org.openqa.selenium.remote.RemoteWebDriver.execute(RemoteWebDriver.java:561)
at io.appium.java_client.DefaultGenericMobileDriver.execute(DefaultGenericMobileDriver.java:41)
at io.appium.java_client.AppiumDriver.execute(AppiumDriver.java:1)
at io.appium.java_client.android.AndroidDriver.execute(AndroidDriver.java:1)
at org.openqa.selenium.remote.RemoteWebDriver.findElement(RemoteWebDriver.java:323)
at io.appium.java_client.DefaultGenericMobileDriver.findElement(DefaultGenericMobileDriver.java:61)
at io.appium.java_client.AppiumDriver.findElement(AppiumDriver.java:1)
at io.appium.java_client.android.AndroidDriver.findElement(AndroidDriver.java:1)
at org.openqa.selenium.remote.RemoteWebDriver.findElementByXPath(RemoteWebDriver.java:428)
at io.appium.java_client.DefaultGenericMobileDriver.findElementByXPath(DefaultGenericMobileDriver.java:151)
at io.appium.java_client.AppiumDriver.findElementByXPath(AppiumDriver.java:1)
at io.appium.java_client.android.AndroidDriver.findElementByXPath(AndroidDriver.java:1)
at DemoActualAutomation.main(DemoActualAutomation.java:15)
Caused by: **java.lang.ClassNotFoundException**: org.apache.commons.lang3.StringUtils
at java.net.URLClassLoader.findClass(Unknown Source)
at java.lang.ClassLoader.loadClass(Unknown Source)
at sun.misc.Launcher$AppClassLoader.loadClass(Unknown Source)
at java.lang.ClassLoader.loadClass(Unknown Source)
DemoActualAutomation.class:
import java.net.MalformedURLException;
import java.util.concurrent.TimeUnit;
import io.appium.java_client.android.AndroidDriver;
import io.appium.java_client.android.AndroidElement;
public class DemoActualAutomation extends AppiumDemo {
public static void main(String[] args) throws MalformedURLException {
// TODO Auto-generated method stub
AndroidDriver<AndroidElement> driver = Capabilities();
driver.manage().timeouts().implicitlyWait(10, TimeUnit.SECONDS);
driver.findElementByXPath("//android.widget.TextView[@text='Preference']").click();
}
}
Am I missing something? Please help. Thank you!!
A: *
*From the https://commons.apache.org/proper/commons-lang/download_lang.cgi
Download the commons-lang3-3.9-bin.zip
*Extract locally
*In Eclipse Go to your Project Properties
*On the Java Build Path > Libraries Tab use the Add External JARs.
*Add the Commons-lang3-3.9.jar from the extracted folder.
A: Download the commons-lang3-3.9-bin.zip
In Eclipse, Go to your Project >Rt. Click > Build Path >Configure Build Path >Add External Jars >Select all 4 jars > Apply and Close.
ur code will run...
A: org/apache/commons/lang3/StringUtils indicates you are using commons-lang3. You should add commons-lang3-xxx.jar to classpath. commons-lang-2.6 doesn't have lang3 packages.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,546 |
"use strict";
const MongoClient = require("mongodb").MongoClient
const URI = process.env.MONGODB_URI;
function withDB(callback) {
return function(...args) {
return new Promise((resolve, reject) => {
MongoClient.connect(URI, function(err, db) {
if (err) {
reject(err);
return;
}
callback(db, ...args).then(result => {
db.close();
resolve(result);
}, result => {
db.close();
reject(result);
});
});
});
}
}
export const getConfigForPath = withDB(async function(db, path) {
let collection = db.collection("paths");
path = path.slice(0);
while (path.length > 0) {
let config = await collection.findOne({ path });
if (config) {
return config;
}
path.pop();
}
return null;
});
export const setConfigForPath = withDB(async function(db, path, config) {
let collection = db.collection("paths");
if (config) {
let doc = {
path,
...config,
};
await collection.findOneAndReplace({ path }, doc, { upsert: true });
}
else {
await collection.findOneAndDelete({ path });
}
});
export const getConfigForPathPrefix = withDB(async function(db, prefix) {
let collection = db.collection("paths");
let query = {};
prefix.forEach((p, i) => { query[`path.${i}`] = p });
return await collection.find(query).toArray();
});
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\section{Introduction}
Magnetic van der Waals compounds have recently attracted significant attention both regarding new fundamental physical phenomena that they demonstrate and also in view of their potential for applications in next-generation spintronic devices \cite{Huang2017,Gong2017,Otrokov2019,Gong2019,Yang21}. One interesting subclass of these compounds is the family of transition metal (TM) tiophosphates $M_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$ where $M$ stands for a TM ion \cite{Brec86,Grasso02}. The spins associated with the $M$ ions are arranged in the crystallographic $ab$-plane on a two-dimensional (2D) honeycomb spin lattice (Fig.~\ref{fig:structure}). By virtue of the weak van der Waals coupling between the planes along the $c$-axis the honeycomb spin lattice often retains its quasi-2D character even in bulk crystals which thus provide an excellent platform for the realization and studies of the fundamental models of 2D magnets \cite{Pokrovsky90}. Depending on the choice of the $M$ ion different Hamiltonians can be realized in the $M_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$ family. In the case of $M$\,=\,Mn the system is a Heisenberg antiferromagnet \cite{Joy92,Wildes98} whereas the antiferromagnetism is of the Ising type for $M$\,=\,Fe \cite{Joy92,Lancon16,Selter21}.
The situation is more complex in the case of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$. It orders antiferromagnetically (AFM) at a N\'eel temperature $T_{\rm N} = 155$\,K due to the residual interplane interactions and below $T_{\rm N}$ it demonstrates not only a significant 'easy'-plane anisotropy but also an anisotropic magnetic response within the $ab$-plane enabling to classify this material alternatively as anisotropic Heisenberg or the {\it XXZ} antiferromagnet \cite{Joy92,Wildes15,Lancon18,Selter21}. While in the above cited works the magnetic properties of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ were studied in quite detail by static magnetometry and also by elastic and inelastic neutron scattering (INS), addressing the spin dynamics by local spin probes received much less attention. Recently a detailed nuclear magnetic resonance (NMR) study of single crystals of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ demonstrated the sensitivity of the $^{31}$P nuclear spin probe to quasi-2D correlations \cite{Dioguardi20}. However, a direct addressing the electronic spin system of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ with electron spin resonance (ESR) spectroscopy has not been attempted so far whereas this method was recently successfully applied for studies of magnetic van der Waals topological insulators providing interesting insights onto the relationship between the spin dynamics and the electronic properties \cite{Otrokov2019,Vidal2019,Alfonsov2021,Alfonsov2021b}.
Here we present the results of a detailed high-frequency/high-field ESR (HF-ESR) spectroscopic study of well-characterized single crystalline samples of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ carried out in a broad range of excitation frequencies, magnetic fields and temperatures. Three temperature regimes could be identified. At $T>T_{\rm N}$ the HF-ESR parameters are almost independent of the direction of the applied field. The $g$-factors are slightly anisotropic characteristic of an 'easy'-plain anisotropy of the Ni$^{2+}$ ions. Entering the AFM ordered state yielded a strongly anisotropic response indicating anisotropic spin fluctuations at the HF-ESR frequencies which persist at temperatures far below $T_{\rm N}$. Finally, at the base temperature of 4\,K the frequency {\it versus} field dependence of the previously unobserved spin-wave mode was measured and its excitation energy was quantified.
These findings shed new light on the previous INS, NMR and computational studies of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ and
call for the development of the comprehensive theoretical model of spin excitations in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$.
\begin{figure}[ht]
\centering
\includegraphics[clip,width=0.9\columnwidth]{structure_NPS_3.jpg}
\caption{Structure of the individual Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ layer: Ni$^{2+}$ ions (turquoise spheres) are octahedrally coordinated by six S ligands (yellow spheres). The NiS$_6$ octahedra make a honeycomb network in the $ab$-plane with the $C_3$ symmetry axis of the octahedra normal to the $ab$-plane ($c^*$-axis). The P-P dumbbell occupies the void of each honeycomb. The honeycomb layers are stacked along the $c$-axis and form the crystallographic structure of the monoclinic symmetry with space group $C2/m$ \cite{Klingen1973,Ouvrard1985}. }
\label{fig:structure}
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics[clip,width=1.9\columnwidth]{spectra_paramag.pdf}
\caption{Temperature dependence of the HF-ESR signal at a fixed excitation frequency $\nu\approx 329$\,GHz for ${\bf H}\parallel {\bf c^\ast}$ (a) and ${\bf H}\perp {\bf c^\ast}$ (b). (c) Left vertical scale: $\nu$ {\it versus} $H_{\rm res}$ dependence at $T = 250$\,K for ${\bf H}\parallel {\bf c^\ast}$ (blue squares) and ${\bf H}\perp {\bf c^\ast}$ (red circles). Solid lines are the fit according to the resonance condition $h\nu = g\mu_{\rm B}\mu_0H_{\rm res}$. Right vertical scale: HF-ESR signals at selected frequencies. }
\label{fig:spectra_paramag}
\end{figure*}
\section{Experimental details}
\label{experimental}
High-quality single crystals of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ were grown by the chemical vapor transport technique with iodine as the transport agent. Details of the growth and physical characterization of the crystals studied in this work are described in Ref.~\cite{Selter21}.
HF-ESR measurements were carried out with two home-made setups. For measurements at frequencies in a range 30--330\,GHz a vector network analyzer (PNA-X from Keysight Technologies) was employed for generation and detection of the microwave radiation. For measurements at higher frequencies up to 500\,GHz a modular Amplifier/Multiplier Chain (AMC from Virginia Diodes, Inc.) was used at the generation side in combination with a hot electron InSb bolometer (QMC Instruments) at the detection side. Samples were placed into a probe head operational in the transmission mode in the Faraday configuration. The probe head was inserted into a $^4$He variable temperature unit of a 16\,T superconducting magnet system (Oxford Instruments). For angular dependent measurements a piezoelectric step-motor-based sample holder was mounted inside the probe head \cite{Fuchs17}. The ESR spectra were recorded in the field-sweep mode at selected constant microwave frequencies by continuously sweeping the field from 0 to 16\,T and then back to 0\,T.
\section{Experimental results}
Typical HF-ESR spectra of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ at various temperatures for the magnetic field ${\bf H}$ applied parallel and perpendicular to the $c^\ast$-axis, and the frequency $\nu$ versus resonance field $H_{\rm res}$ dependence are shown in Fig.~\ref{fig:spectra_paramag}(a), (b) and (c), respectively. Here, the $c^\ast$-axis is defined as the normal to the $ab$-planes (Fig.~\ref{fig:structure}). The $\nu(H_{\rm res})$ dependence at $T = 250$\,K $> T_{\rm N} = 158$\,K \cite{Selter21} can be accurately fitted to the simple paramagnetic resonance condition $h\nu = g\mu_{\rm B}\mu_0H_{\rm res}$ yielding the slightly anisotropic $g$-factor values $g_{\parallel} = 2.16 \pm 0.02$ and $g_{\perp} = 2.18 \pm 0.02$ for ${\bf H}\parallel {\bf c^\ast}$ and ${\bf H}\perp {\bf c^\ast}$, respectively. Here $h$, $\mu_{\rm B}$ and $\mu_0$ are the Plank constant, Bohr magneton and vacuum permeability, respectively.
The ESR response of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ in the paramagnetic state above $T_{\rm N}$ is practically isotropic regarding both the linewidth $\Delta H$ and $H_{\rm res}$ (Fig.~\ref{fig:linewidth_shift_T}). The relative shift of the signal with respect to its position at room temperature $\delta H_{\rm res} = [H_{\rm res}(T) - H_{\rm res}(300\,K)]/ H_{\rm res}(300\,K)$ is $T$-independent within the experimental uncertainty down to $T_{\rm N}$ and is nearly zero while $\Delta H$ begins to increase below 180\,K for both magnetic field geometries. Remarkably, at the vicinity of $T_{\rm N}$ the signal for ${\bf H}\parallel {\bf c^\ast}$ rapidly wipes out [Fig.~\ref{fig:spectra_paramag}(a)] and cannot be detected at lower temperatures in the available frequency range. In contrast, the signal for ${\bf H}\perp {\bf c^\ast}$ continuously broadens upon entering the magnetically ordered state and strongly shifts to lower fields [Fig.~\ref{fig:spectra_paramag}(b) and Fig.~\ref{fig:linewidth_shift_T}].
Measurements of the HF-ESR signal at $T < T_{\rm N}$ and at different orientations of the applied magnetic field within the $ab$-plane revealed a significant angular dependence of its position with the 180$^\circ$ periodicity which follows a simple sine law $H_{\rm res} = A + B\sin(\alpha)$ (Fig.~\ref{fig:angular_dep}). Here, $\alpha$ is the angle between ${\bf H}$ and the $b$-axis. The anisotropy of the resonance field with $H_{\rm res}(\alpha = 0) < H_{\rm res}(\alpha = 90^\circ)$ is in agreement with the anisotropy of the static magnetization $M_{\rm b} > M_{\rm a}$ \cite{Wildes15,Selter21,Mperiodicity} since for a stronger magnetized direction less external field is needed to reach the resonance condition. The dependence $H_{\rm res}(\alpha)$ at $T = 4$\,K, shown in Fig.~\ref{fig:angular_dep} for comparison, could be obtained only in the limited range of angles since the signal at this low temperature could be measured only at much higher frequencies due to the opening of the AFM excitation gap (see below) and thus for many orientations the resonance field was out of the available field range.
\begin{figure}[ht]
\centering
\includegraphics[clip,width= \columnwidth]{linewidth_shift_T.pdf}
\caption{Temperature dependence of the linewidth (main panel) and of the relative shift of the HF-ESR signal (inset) at two close frequencies of 308.3\,GHz for ${\bf H}\parallel {\bf c^\ast}$ (a) and of 305.8\,GHz for ${\bf H}\perp {\bf c^\ast}$. }
\label{fig:linewidth_shift_T}
\end{figure}
Below $T\sim 70$\,K the HF-ESR signal for ${\bf H}\perp {\bf c^\ast}$ is not observable anymore in the available frequency range presumably due to its significant broadening. However, the signal recovers at the base temperature of 4\,K where its $\nu(H_{\rm res})$ dependence, referred hereafter as the AFM branch, can be measured with confidence (Fig.~\ref{fig:angular_AFM_branch_4K}).
In contrast to the paramagnetic state where the position of the signal follows the linear resonance condition $h\nu = g\mu_{\rm B}\mu_0H_{\rm res}$ [Fig.~\ref{fig:spectra_paramag}(c)], the AFM branch is shifted upwards significantly and gets apparently nonlinear. The "flattening" of this branch with lowering the frequency is reflected in the broadening of the signal because in the employed HF-ESR setup the spectra are recorded at a given fixed frequency by sweeping the magnetic field. Finally, the signal cannot be detected at $\nu < 350$\,GHz suggesting the presence of an energy gap for the AFM excitations.
\begin{figure}[ht]
\centering
\includegraphics[clip,width= 0.9\columnwidth]{angular_dep.pdf}
\caption{Dependence of the resonance field $H_{\rm res}$ on the angle $\alpha$ which the field ${\bf H}$ applied in the $ab$-plane makes with the $b$-axis at $T = 130$\,K and $\nu = 323$\,GHz (circles), and at $T = 4$\,K and $\nu = 485$\,GHz (triangles). Dashed lines are fits to the function $H_{\rm res} = A + B\sin(\alpha)$. }
\label{fig:angular_dep}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[clip,width= \columnwidth]{AFM_branch_4K_ab.pdf}
\caption{Left vertical scale: $\nu$ {\it versus} $H_{\rm res}$ dependence of the HF-ESR signal at $T = 4$\,K for ${\bf H}\perp {\bf c^\ast}$ (circles). Solid line depicts the fit of the data to the function $\nu =h^{-1}[(g_{\perp}\mu_{\rm B}\mu_0H_{\rm res})^2+\Delta^2]^{1/2}$. Dashed line corresponds to the paramagnetic resonance condition $\nu = h^{-1}g_{\perp}\mu_{\rm B}\mu_0H_{\rm res}$. Right vertical scale: HF-ESR signals at selected frequencies.
}
\label{fig:angular_AFM_branch_4K}
\end{figure}
\section{Discussion}
\subsection{Paramagnetic state}
The obtained $g$-values $g_{\parallel} = 2.16 \pm 0.02$ and $g_{\perp} = 2.18 \pm 0.02$ [Fig.~\ref{fig:spectra_paramag}(c)] fall into the common range of the $g$-factors found for the Ni$^{2+}$ (3$d^6$, $S = 1$) ions in the octahedral ligand coordination \cite{AbragamBleaney}. A small anisotropy with $g_\perp > g_\parallel$ indicates a splitting of the $S = 1$ spin triplet into a singlet $\mid0\rangle$ and a doublet $\mid\pm 1\rangle$ due to the second-order spin-orbit coupling effect in the presence of the trigonal distortion of the ligand octahedra along the $c^*$-axis which can be parameterized using Hamiltonian $\mathcal{H}=D[S_z^2+(1/3)S(S+1)]$. Here $D$ is the single-ion anisotropy (SIA) parameter.
In the case of isolated Ni$^{2+}$ ions the initial (zero-field) splitting of the spin levels due to a finite value of $D$ should yield two ESR peaks corresponding to the transitions $|+1\rangle\!\!\leftrightarrows | 0\rangle$ and $|-1\rangle\!\!\leftrightarrows |0\rangle$ separated in field by $2| D|/g\mu_{\rm B}\mu_0$ \cite{AbragamBleaney}. Here $D = E_{|\pm 1\rangle} - E_{|0\rangle} $ is the energy difference of the zero-field split levels. However, for the interacting paramagnetic Ni centers in concentrated magnets such as Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$, the two signals merge into a single line at the mid resonance field due to the exchange narrowing effect \cite{Smirnov2008}. This complication prevents a direct quantification of the magnitude of $D$ from the distance between the two fine-structure-split ESR lines. Nevertheless, the sign and an order of magnitude of the SIA in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ can be estimated considering the relation between $D$ and $g$-factors in the form $D = \lambda (g_{\parallel} - g_{\perp})/2$ \cite{AbragamBleaney}. Here $\lambda$ is the spin-orbit coupling constant. Taking the Ni$^{2+}$ free ion value of $\lambda \sim -300$\,cm$^{-1} = -87$\,meV \cite{Ballhausen1962} and the experimentally determined $g$-factors one obtains $D\sim 0.7$\,meV. One should note, however, that the magnitude of $D$ might be overestimated since in a covalent solid such as Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ $|\lambda|$ could be smaller than in the ionic limit.
The positive sign of $D$ implies that the $c^*$-axis should be the 'hard' axis for the Ni spins in agreement with the density functional theory (DFT) result~\cite{Olsen2021}. Multireference configuration-interaction calculations are even in the quantitative agreement with the HF-ESR estimate \cite{Dioguardi20}. $D >0$ is compatible with the experimentally determined almost in-plane zigzag spin structure in the AFM ordered state of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ with the spins aligned along the $a$-axis~\cite{Wildes15}. This secondary, in-plane 'easy' axis may possibly arise due to the spin-orbit coupling (SOC) driven anisotropy of the Heisenberg superexchange in a combination with dipole-dipole interactions \cite{Kim2021}. Indeed, such an 'easy'-plane structure is found to be most stable in the DFT\,+\,$U$\,+\,SOC calculations~\cite{Koo2021}. Here, $U$ is the on-site electron repulsion energy. The positive $D$ would also naturally explain the suppression of the AFM order in the monolayer of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ \cite{Kim2019} because according to the Mermin-Wagner theorem a magnetic out-of-plane 'easy' axis is required to stabilize magnetic order in a 2D spin lattice at $T>0$ \cite{commentKim2021}. At odds with these findings, the 'easy'-axis type of SIA was derived from the analysis of the INS data~\cite{Lancon18}.
It is remarkable that both the resonance field $H_{\rm res}$ and the linewidth $\Delta H$ of the HF-ESR signal in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ remain nearly isotropic and in particular $T$-independent by cooling the sample down to temperatures close to $T_{\rm N}$ (Fig.~\ref{fig:linewidth_shift_T}). Usually, in the quasi-2D spin systems the ESR line broadening and shift occur at $T> T_{\rm N}$ due to the growth of the in-plane spin-spin correlations resulting in a development of slowly fluctuating short-range order \cite{Benner1990}. Specifically, a distribution of the local fields and shortening of the spin-spin relaxation time due to the slowing down of the spin fluctuations increase the ESR linewidth.
The onset of the short-range ordered regime is typically associated with the broad maximum in the static magnetic susceptibility $\chi(T)$ which occurs in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ at $T_{\chi_{\rm max}} \sim 260$\,K \cite{Selter21}. It manifested in an anomalous breakdown of the scaling between the $^{31}$P NMR shift and static $\chi$ but, interestingly, the NMR linewidth and the relaxation rate $1/T_1$ remained practically unchanged down to $T_{\rm N}$ \cite{Dioguardi20}. This suggests that by approaching $T_{\rm N}$ from above the electron spin dynamics in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ still remains much faster than the NMR time window of the order of 10--100\,$\mu$s. On a much shorter, nano- to picosecond time scale of HF-ESR the slowing down of the spin dynamics becomes visible in the increasing of $\Delta H$ only below 180\,K, i.e. just at $T \lesssim 1.1T_{\rm N}$. Such a short temperature interval above $T_{\rm N}$ for the manifestation of the low-D dynamic spin correlations suggests that from the viewpoint of local spin probes, such as NMR and particularly HF-ESR, Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ can be considered rather as a quasi-3D than a quasi-2D spin system, i.e. the interlayer coupling should be sufficiently strong. Indeed, a relatively high AFM ordering temperature is maintained only in bulk single crystals and is suppressed towards the pure 2D limit which implies the significance of the coupling in the third dimension for the stabilization of magnetic order in a quasi-3D system with the 'easy'-plane magnetic anisotropy~\cite{Kim2019}.
\subsection{Transition to the AFM ordered state}
For ${\bf H}\perp {\bf c^\ast}$, the shift of the HF-ESR signal at $T<T_{\rm N}$ from its paramagnetic position can be ascribed to the opening of the energy gap for spin excitations arising due to magnetic anisotropy \cite{Turov}. Concomitantly, $\Delta H$ continue to increase strongly without tendency to saturate which indicates significant spin fluctuations also in the AFM long-range ordered state of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$. In this field geometry both in-plane and out-of-plane fluctuations should contribute to the line broadening while for ${\bf H}\parallel {\bf c^\ast}$ the in-plane fluctuations make a dominant contribution (see, e.g., Supplemental Material in Ref.~\cite{Alfonsov2021} for details). One can conjecture that a wipeout of the signal for the out-of-plane orientation of ${\bf H}$ at $T<T_{\rm N}$ might be a consequence of the sharp boosting of the in-plane spin fluctuations upon establishment of the long-range magnetic order. Eventually below $T\sim 70$\,K the signal gets unobservable also for the other field geometry indicating a strong spectral density of the spin fluctuations at HF-ESR frequencies persisting in the AFM ordered state of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ far below~$T_{\rm N}$.
\subsection{Magnon gap}
Eventually spin fluctuations apparently cease at a low temperature of 4\,K enabling to detect the HF-ESR signal in the ${\bf H}\perp {\bf c^\ast}$ configuration. Considering its specific frequency versus field dependence (Fig.~\ref{fig:angular_AFM_branch_4K}) one can ascribe this resonance mode with confidence to the lowest in energy uniform ($q = 0$ wave vector) spin wave (magnon) excitation in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ (note that owing to the long wavelength of the applied microwave radiation it is usually not possible in an ESR experiment to excite the modes with a finite momentum transfer).
The data can be reasonably well fitted with the function describing the AFM branch for the 'hard' direction of a two-sublattice collinear antiferromagnet $\nu =h^{-1}[(g_{\perp}\mu_{\rm B}\mu_0H_{\rm res})^2+\Delta^2]^{1/2}$ \cite{Turov} with the in-plane $g$-factor $g_\perp = 2.18$ obtained from the measurements in the paramagnetic state [Fig.~\ref{fig:spectra_paramag}(c)]. The fit yields the excitation gap $\Delta = 365$\,GHz (1.51\,meV). The choice of the fit function appears reasonable since the $\nu(H_{\rm res})$ dependence was measured for the direction of the applied field close to the $b$-axis which is the 'hard' in-plane direction of the AFM zigzag spin structure of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ \cite{Lancon18}.
Observation of such a low-energy mode with an excitation gap in the zero magnetic field limit amounting to only $\Delta \approx 1.5$\,meV sheds new light into the spectrum of magnetic excitations in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$. The energy spectrum of the spin waves was studied recently by INS on single crystals of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ in Ref.~\cite{Lancon18}. It was concluded from the analysis of the magnon dispersions that the minimum excitation gap lies at the center of the 2D Brillouin zone ($\Gamma$-point) and amounts to $E_{\Gamma}\sim 7$\,meV. It followed from the same analysis that the "competing" gap of a similar magnitude $E_{C}$ should be present at the Brillouin zone corner ($C$-point). It was argued that the closeness of the magnitudes of these two gaps could give rise to the instability of the magnetic structure of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$. It appears now that the low-energy mode with the magnitude $\Delta \ll E_{\Gamma}, E_{C}$ found by HF-ESR was overlooked in the INS experiment possibly due to the problems with the energy resolution at small scattering vectors $q \approx 0$ where strong elastic scattering dominates the total response. Therefore accounting for the "new" mode in the excitation spectrum calls for the reanalysis of the exchange and anisotropy constants estimated from the INS data in Ref.~\cite{Lancon18}.
Furthermore, the occurrence of the low-energy magnon excitation with the gap $\Delta \approx 1.5$\,meV (18\,K) much smaller than $E_{\Gamma}, E_{C} \sim 7$\,meV (81\,K) may explain the puzzling observation of the power-law dependence of the $^{31}$P NMR relaxation rate $1/T_1 \propto T^5$ down to temperatures significantly smaller than 80\,K instead of the expected gap-like-activated behavior \cite{Dioguardi20}. This kind of $\propto T^5$ dependence is typical for the relaxation via the three-magnon scattering process and holds at temperatures higher that the magnon gap \cite{Beeman1968}. The low-lying excitation observed by HF-ESR is thus likely to provide an additional relaxation channel for nuclear spins at $T< E_{\Gamma}, E_{C} \sim 80$\,K.
\section{Conclusions}
In summary, we have performed a detailed HF-ESR spectroscopic study of the single-crystalline samples of the van der Waals compound Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$, a member of the family of TM tiophosphates hosting
by the virtue of the layered crystal structure a stack of 2D honeycomb spin planes weakly coupled in the third dimension. From the analysis of the $g$-tensor in the paramagnetic state above the AFM ordering temperature $T_{\rm N} = 158$\,K we determined the positive sign of the SIA constant $D$ of the Ni$^{2+}$ ions and estimated its upper limit to be $D \lesssim 0.7$\,meV.
This result supports computational predictions of an 'easy'-plane type of SIA and of the energetic stability of the in-plane order of Ni spins in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ \cite{Dioguardi20,Olsen2021,Koo2021} in agreement with the experimentally determined spin structure~\cite{Wildes15} but at variance with the conclusion of the INS work in Ref.~\cite{Lancon18} proposing the 'easy'-axis type of SIA.
A critical broadening of the HF-ESR signal usually observed in quasi-2D magnets far above $T_{\rm N}$ due to the development of the slowly fluctuating 2D short-range order is found in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ only in the vicinity of $T_{\rm N}$. This suggests a significant interlayer coupling which could be responsible for the high 3D AFM ordering temperature of this compound.
A nearly isotropic HF-ESR response in the paramagnetic state of Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ gets strongly anisotropic in the ordered state indicating strong and anisotropic spin fluctuations at HF-ESR frequencies.
The fluctuations cease at low temperatures enabling to measure the $\nu(H_{\rm res})$ dependence of the in-plane magnon excitation branch which has a small energy gap $\Delta \approx 1.5$\,meV in the zero field limit. The occurrence of this low-energy spin wave excitation not observed in the INS study \cite{Lancon18} explains the unexpected 3-magnon-assisted $^{31}$P NMR relaxation process at low temperatures \cite{Dioguardi20}. Altogether our results call for the revisiting of the analysis of the spin wave excitations in Ni$_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$\ where a sizable interlayer magnetic coupling might be considered as well and should stimulate fundamental theoretical understanding of magnetic excitations also in other van der Waals TM tiophosphates.
\section{ACKNOWLEDGMENTS}
The authors would like to thank L. Hozoi for helpful discussions.
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through grants No. KA1694/12-1 and AS523/4-1, and within the Collaborative Research Center SFB 1143 ``Correlated Magnetism – From Frustration to Topology'' (project-id 247310070) and the Dresden-Würzburg Cluster of Excellence (EXC 2147) `` ct.qmat - Complexity and Topology in Quantum Matter'' (project-id 39085490). K. M. acknowledges the Hallwachs–Röntgen Postdoc Program of ct.qmat for financial support.
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An editor's cut of a motion picture is made by the film editor on their own, or working with the film director. The editor tapes together the first cut of the film, the "editor's cut", arranging the separate takes into a coherent story according to the plan communicated by the director. The editor's version of the film will often be as much as two hours beyond the final running time of the film. Working from the editor's cut, decisions then need to be made, usually together with other creative staff, to improve continuity, balance the story, trim or delete scenes, etc.
A version supposedly nearer to the director's original creative vision is sometimes marketed as a director's cut. These special-market versions of a movie DVD are more expensive than the regular edition, as they are usually longer than the theatre version, and have extra discs often including "making of ... " documentaries, out-take collections, extended interviews with cast and crew, etc.
Film and video terminology | {
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\section{Introduction}
In real-world applications, such as spam filtering \citep{drucker1999support} and medical diagnosing \citep{huang2010bayesian}, the loss of misclassifying a positive instance and negative instance can be rather different. For instance, in medical diagnosing, misdiagnosing a patient as healthy is more dangerous than misclassifying healthy person as sick. Meanwhile, in reality, it is often very difficult to define an accurate cost for these two kinds of errors \citep{liu2010learning,zhou2016large}. In such situations, it is more desirable to keep the classifier working under a small tolerance of false-positive rate (FPR) $\tau$, i.e., only allow the classifier to misclassify no larger than $\tau$ percent of negative instances. Traditional classifiers trained by maximizing classification accuracy or AUC are not suitable due to mismatched goal.
In the literature, classification under constrained false-positive rate is known as Neyman-Pearson (NP) Classification problem \citep{scott2005neyman,lehmann2006testing,rigollet2011neyman}, and existing approaches can be roughly grouped into several categories. One common approach is to use \emph{cost-sensitive learning}, which assigns different costs for different classes, and representatives include cost-sensitive SVM \citep{osuna1997support,davenport2006controlling,davenport2010tuning}, cost-interval SVM \citep{liu2010learning} and cost-sensitive boosting \citep{masnadi2007asymmetric,masnadi2011cost}. Though effective and efficient in handling different misclassification costs, it is usually difficult to find the appropriate misclassification cost for the specific FPR tolerance.
Another group of methods formulates this problem as a constrained optimization problem, which has the FPR tolerance as an explicit constraint \citep{mozer2002prodding,gasso2011batch,Mahdavi2013}. These methods often need to find the saddle point of the Lagrange function, leading to a time-consuming alternate optimization. Moreover, a surrogate loss is often used to simplify the optimization problem, possibly making the tolerance constraint not satisfied in practice.
The third line of research is scoring-thresholding methods, which train a scoring function first, then find a threshold to meet the target FPR tolerance \citep{drucker1999support}. In practice, the scoring function can be trained by either class conditional density estimation~\citep{tong2013plug} or bipartite ranking~\citep{narasimhan2013relationship}. However, computing density estimation itself is another difficult problem. Also most bipartite ranking methods are less scalable with super-linear training complexity.
Additionally, there are methods paying special attention to the positive class. For example, asymmetric SVM \citep{wu2008asymmetric} maximizes the margin between negative samples and the core of positive samples, one-class SVM \citep{ben2001support} finds the smallest ball to enclose positive samples. However, they do not incorporate the FPR tolerance into the learning procedure either.
In this paper, we address the tolerance constrained learning problem by proposing \emph{$\tau$-False Positive Learning} ($\tau$-FPL). Specifically, $\tau$-FPL is a scoring-thresholding method. In the scoring stage, we explicitly learn a ranking function which optimizes the probability of ranking any positive instance above the centroid of the worst $\tau$ percent of negative instances. Whereafter, it is shown that, with the help of our newly proposed Euclidean projection algorithm, this ranking problem can be solved in linear time under the projected gradient framework. It is worth noting that the Euclidean projection problem is a generalization of a large family of projection problems, and our proposed linear-time algorithm based on bisection and divide-and-conquer is one to three orders faster than existing state-of-the-art methods.
In the thresholding stage, we devise an out-of-bootstrap thresholding method to transform aforementioned ranking function into a low false-positive classifier. This method is much less prone to overfitting compared to existing thresholding methods. Theoretical analysis and experimental results show that the proposed method achieves superior performance over existing approaches.
\section{From Constrained Optimization to Ranking}
In this section, we show that the FPR tolerance problem can be transformed into a ranking problem, and then formulate a convex ranking loss which is tighter than existing relaxation approaches.
Let $\mathcal{X} =\{\bm{x} \mid \bm{x} \in \mathbb{R}^d: ||\bm{x}|| \leq 1 \}$ be the instance space for some norm $||\cdot||$ and $\mathcal{Y} = \{-1, +1\}$ be the label set, $\mathcal{S} = \mathcal{S}_+ \cup \mathcal{S}_-$ be a set of training instances, where $\mathcal{S}_+ = \{ \bm{x}_i^+ \in \mathcal{X}\}_{i=1}^m$ and $\mathcal{S}_- = \{ \bm{x}_j^- \in \mathcal{X}\}_{j=1}^n$ contains $m$ and $n$ instances independently sampled from distributions $\mathbb{P}^+$ and $\mathbb{P}^-$, respectively. Let $0 \leq \tau \ll 1$ be the maximum tolerance of false-positive rate.
Consider the following Neyman-Pearson classification problem, which aims at minimizing the false negative rate of classifier under the constraint of false positive rate
\begin{equation}
\begin{split}
\min_{f, b} ~ ~ \mathbb{P}_{\bm{x}^+ \sim \mathbb{P}^+}\left(f(\bm{x}^+\right) \leq b)~~~~\textrm{\small s.t.} ~ ~ \mathbb{P}_{\bm{x}^- \sim \mathbb{P}^-}\left(f(\bm{x}^-) > b\right) \leq \tau,
\end{split}
\end{equation}
where $f:\mathcal{X} \to \mathbb{R}$ is a scoring function and $b \in \mathbb{R}$ is a threshold. With finite training instances, the corresponding \emph{empirical risk minimization} problem is
\begin{equation}\label{empirical plm}
\begin{split}
\min_{f, b} ~&~ {\cal L}_{emp}(f, b) = \frac{1}{m} \sum \nolimits_{i=1}^m\mathbb{I}\left(f(\bm{x_i^+}) \leq b\right) \\\\
\textrm{\small s.t.} ~&~ \quad \frac{1}{n} \sum \nolimits_{j=1}^n\mathbb{I}\left(f(\bm{x_j^-}) > b\right) \leq \tau,
\end{split}
\end{equation}
where $\mathbb{I}(u)$ is the indicator function.
The empirical version of the optimization problem is difficult to handle due to the presence of non-continuous constraint; we introduce an equivalent form below.
\begin{prop}\label{tcp-rp}
Define $f(\bm{x}_{[j]}^-)$ be the $j$-th largest value in multiset $\{f(\bm{x}_i) \mid \bm{x}_i \in \mathcal{S}_-\}$ and $\lfloor.\rfloor$ be the floor function. The constrained optimization problem (\ref{empirical plm}) share the same optimal solution $f^*$ and optimal objective value with the following ranking problem
\begin{equation}\label{origin rank plm}
\min_{f}\frac{1}{m}\sum_{i=1}^m \mathbb{I}\left(f(\bm{x}_i^+) - f(\bm{x}_{[\lfloor \tau n \rfloor + 1]}^-) \leq 0\right).
\end{equation}
\end{prop}
\begin{proof}
For a fixed $f$, it is clear that the constraint in (\ref{empirical plm}) is equivalent to $b \geq f(\bm{x}_{[\lfloor \tau n \rfloor + 1]}^-)$. Since the objective in (\ref{empirical plm}) is a non-decreasing function of $b$, its minimum is achieved at $b = f(\bm{x}_{[\lfloor \tau n \rfloor + 1]}^-)$. From this, we can transform the original problem (\ref{empirical plm}) into its equivalent form (\ref{origin rank plm}) by substitution.
\end{proof}
Proposition \ref{tcp-rp} reveals the connection between constrained optimization \eqref{empirical plm} and ranking problem \eqref{origin rank plm}. Intuitively, problem \eqref{origin rank plm} makes comparsion between each positve sample and the $(\lfloor \tau n \rfloor + 1)$-th largest negative sample. Here we give further explanation about its form: it is equivalent to the maximization of the partial-AUC near the risk area.
\begin{figure}[ht]
\centering
\includegraphics[width=0.4\linewidth]{partial-AUC-big.pdf}
\caption{Neyman-Pearson Classification is equivalent to a partial-AUC optimization near the specified FPR.}
\label{Fig.0}
\end{figure}
Although it seems that partial-AUC optimization considers fewer samples than the full AUC, optimizing (\ref{origin rank plm}) is intractable even if we replace $\mathbb{I}(\cdot)$ by a convex loss, since the operation $[\cdot]$ is non-convex when $\lfloor\tau n\rfloor > 0$. Indeed, Theorem \ref{np-hard} further shows that whatever $\tau$ is chosen, even for some weak hypothesis of $f$, the corresponding optimization problem is NP-hard.
\begin{mydef}\label{def:surr}
A \emph{surrogate function} of ~$\mathbb{I}(u \leq 0)$ is a contionuous and non-increasing function $L:\mathbb{R} \to \mathbb{R}$ satisfies that (i) $\forall u \leq 0$, $L(u) \geq L(0) \geq 1$; (ii) $\forall u > 0$, $0 \leq L(u) < L(0)$; (iii) $L(u) \rightarrow 0$ as $u \rightarrow +\infty$.
\end{mydef}
\begin{thm}\label{np-hard}
For fixed $\lambda > 0$, $0 < \tau < 1$ and surrogate function $L(\cdot)$, optimization problem $\tau$-$OPT_{L}^{\lambda}$
\begin{equation*}
\min_{f \in \mathcal{H}^d}\frac{1}{m}\sum_{i=1}^m L\left(f(\bm{x}_i^+) - f(\bm{x}_{[\lfloor \tau n \rfloor + 1]}^-)\right)
\end{equation*}
with hyperplane hypothesis set $\mathcal{H}^d = \{f(\bm{x}) = \bm{w}^{\top}\bm{x} \mid \bm{w} \in \mathbb{R}^d, ||\bm{w}|| \leq \lambda\}$ is NP-hard.
\end{thm}
This intractability motivates us to consider the following upper bound approximation of (\ref{origin rank plm}):
\begin{equation}\label{new rank plm}
\min_{f}\frac{1}{m}\sum_{i=1}^m \mathbb{I}\left(f(\bm{x}_i^+) - \frac{1}{\lfloor \tau n \rfloor + 1}\sum_{i = 1}^{\lfloor \tau n \rfloor + 1} f(\bm{x}_{[i]}^-) \leq 0\right)
\end{equation}
which prefers scores on positive examples to exceed the mean of scores on the worst $\tau$-proportion of negative examples. If $\tau = 0$, optimization (\ref{new rank plm}) is equivalent to the original problem (\ref{origin rank plm}). In general cases, equality also could hold when both the scores of the worst $\tau$-proportion negative examples are the same.
The advantages of considering ranking problem (\ref{new rank plm}) include (details given later):
\begin{itemize}
\item For appropriate hypothesis set of $f$, replacing $\mathbb{I}(\cdot)$ by its convex surrogate will produce a tight convex upper bound of the original minimization problem (\ref{origin rank plm}), in contrast to the cost-sensitive classification which may only offer an insecure lower bound. This upper bound is also tighter than the convex relaxation of the original constrained minimization problem \eqref{empirical plm} (see Proposition \ref{prop:upper})
\item By designing efficient learning algorithm, we can find the global optimal solution of this ranking problem in \emph{linear time}, which makes it well suited for large-scale scenarios, and outperforms most of the traditional ranking algorithms;
\item Explicitly takes $\tau$ into consideration and no additional cost hyper-parameters required;
\item The generalization performance of the optimal solution can be theoretically established
\end{itemize}
\subsection{Comparison to Alternative Methods}
Before introducing our algorithm, we briefly review some related approaches to FPR constrained learning, and show that our approach can be seen as a \emph{tighter upper bound}, meanwhile maintain linear training complexity.
\textbf{Cost sensitive Learning}~One alternative approach to eliminating the constraint in \eqref{empirical plm} is to approximate it by introducing asymmetric costs for different types of error into classification learning framework:
\begin{equation*}
\min_{f, b} {\cal L}_{emp}^C(f, b) = \frac{1}{m} \sum_{i=1}^m\mathbb{I}\left(f(\bm{x_i^+}) \leq b\right) + \frac{C}{n} \sum_{j=1}^n\mathbb{I}\left(f(\bm{x_j^-}) > b\right),
\end{equation*}
where $C \geq 0$ is a hyper-parameter that punishes the gain of false positive instance. Although reasonable for handling different misclassification costs, we point out that methods under this framework indeed \emph{minimize a lower bound} of problem \eqref{empirical plm}. This can be verified by formulating \eqref{empirical plm} into unconstrained form ${\cal L}_{emp}'(f, b)$
\begin{eqnarray*}
& &\!\!\!\!{\cal L}_{emp}'(f, b) \\
&\triangleq&\!\!\!\! \max_{\lambda \geq 0} \frac{1}{m} \sum_{i=1}^m\mathbb{I}\left(f(\bm{x_i^+}) \leq b\right) \!+\! \lambda\left( \frac{1}{n} \sum_{j=1}^n\mathbb{I}\left(f(\bm{x_j^-}) > b\right) - \tau \right) \\
&\geq&\!\!\!\! \frac{1}{m} \sum_{i=1}^m\mathbb{I}\left(f(\bm{x_i^+}) \leq b\right) + C\left( \frac{1}{n} \sum_{j=1}^n\mathbb{I}\left(f(\bm{x_j^-}) > b\right) - \tau\right)\\
&=&\!\!\!\! {\cal L}_{emp}^C(f, b) - C\tau.
\end{eqnarray*}
Thus, for a fixed $C$, minimizing ${\cal L}_{emp}^C$ is equivalent to minimize a lower bound of ${\cal L}_{emp}$. In other words, cost-sensitive learning methods are \emph{insecure} in this setting: mismatched $C$ can easily violate the constraint on FPR or make excessive requirement
, and the optimal $C$ for specified $\tau$ is only known after solving the original constrained problem, which can be proved NP-hard.
In practice, one also has to make a continuous approximation for the constraint in \eqref{empirical plm} for tractable cost-sensitive training, this multi-level approximation makes the relationship between the new problem and original unclear, and blocks any straightforward theoretical justification.
\textbf{Constrained Optimization}~~The main difficulty of employing Lagrangian based method to solve problem \eqref{empirical plm} lies in the fact that both the constraint and objective are non-continuous. In order to make the problem tractable while satisfying FPR constraint strictly, the standard solution approach relies on replacing $\mathbb{I}(\cdot)$ with its convex surrogate function (see Definition \ref{def:surr}) to obtain a convex constrained optimization problem \footnote{For precisely approximating the indicator function in constraint, choosing ``bounded'' loss such as sigmoid or ramp loss \citep{collobert2006trading,gasso2011batch} etc. seems appealing. However, bounded functions always bring about a non-convex property, and corresponding problems are usually NP-hard \citep{yu2012polynomial}. These difficulties limit both efficient training and effective theoretical guarantee of the learned model.}. Interestingly, Proposition \ref{prop:upper} shows that whichever surrogate function and hypothesis set are chosen, the resulting constrained optimization problem is a \emph{weaker upper bound} than that of our approach.
\begin{prop}\label{prop:upper}
For any non-empty hypothesis set $\mathcal{H}$ and $f \in \mathcal{H}$, convex surrogate function $L(\cdot)$, let
\begin{eqnarray*}
\bar{R}_0 \!\!&=&\!\!\! \frac{1}{m}\sum \nolimits_{i=1}^m \mathbb{I}\left(f(\bm{x}_i^+) - f(\bm{x}_{[\lfloor \tau n \rfloor + 1]}^-) \leq 0\right)\\
\bar{R}_1 \!\!&=&\!\!\! \frac{1}{m} \sum \nolimits_{i=1}^m L\left(f(\bm{x}_i^+) - \frac{1}{\lfloor \tau n \rfloor + 1}\sum_{j=1}^{\lfloor \tau n \rfloor + 1} f(\bm{x}_{[j]}^-)\right)\\
\bar{R}_2 \!\!&=&\!\!\! \min_{b \in \mathbb{R}} \frac{1}{m} \sum_{i=1}^m L\left(f(\bm{x}_i^+) - b\right)~\textrm{s.t.}~\frac{1}{n} \sum_{j=1}^n L\left(b - f(\bm{x}_j^-)\right) \leq \tau,
\end{eqnarray*}
we have
\begin{equation*
\bar{R}_0 \leq \bar{R}_1 \leq \bar{R}_2.
\end{equation*}
\end{prop}
Thus, in general, there is an exact gap between our risk $\bar{R}_1$ and convex constrained optimization risk $\bar{R}_2$. In practice one may prefer a tighter approximation since it represents the original objective better. Moreover, our Theorem \ref{bound p} also achieves a tighter bound on the generalization error rate, by considering empirical risk $\bar{R}_1$.
\textbf{Ranking-Thresholding}~Traditional bipartite ranking methods usually have super-linear training complexity. Compared with them, the main advantage of our algorithm comes from its linear-time conplexity in each iteration without any convergence rate (please refer to Table \ref{tabcomp}). We named our algorithm $\tau$-FPL, and give a detailed description of how it works in the next sections.
\section{Tolerance-Constrained False Positive Learning}
Based on previous discussion, our method will be divided into two stages, namely \emph{scoring} and \emph{thresholding}. In scoring, a function $f(\cdot)$ is learnt to maximize the probability of giving higher a score to positive instances than the centroid of top $\tau$ percent of the negative instances. In thresholding, a suitable threshold $b$ will be chosen, and the final prediction of an instance $\bm{x}$ can be obtained by
\begin{equation}
y = \mathrm{sgn}\left(f(\bm{x}) - b\right) \ .
\end{equation}
\subsection{Tolerance-Constrained Ranking}
In (\ref{new rank plm}), we consider linear scoring function $f(\bm{x}) = \bm{w}^{\top}\bm{x}$, where $\bm{w} \in \mathbb{R}^d$ is the weight vector to be learned, $\text{L}_2$ regularization and replace $\mathbb{I}(u < 0)$ by some of its convex surrogate $l(\cdot )$. Kernel methods can be used for nonlinear ranking functions. As a result, the learning problem is
\begin{equation}\label{learning problem}
\min \limits_{\bm{w}} \frac{1}{m}\sum \nolimits_{i=1}^m l\left(\bm{w}^{\top}\bm{x}_i^+ - \frac{1}{k}\sum \nolimits_{j=1}^k\bm{w}^{\top}\bm{x}_{[j]}^-\right) + \frac{R}{2}\|\bm{w}\|^2,
\end{equation}
where $R > 0$ is the regularization parameter, and $k = \lfloor \tau n \rfloor + 1$.
Directly minimizing (\ref{learning problem}) can be challenging due to the $[\cdot]$ operator, we address it by considering its dual,
\begin{thm}\label{dual form}
Define $\bm{X}^+=[\bm{x}_1^+, ..., \bm{x}_m^+]^{\top}$ and $\bm{X}^-=[\bm{x}_1^-, ..., \bm{x}_m^-]^{\top}$ be the matrix containing positive and negative instances in their rows respectively, the dual problem of (\ref{learning problem}) can be written by
\begin{eqnarray}\label{dual problem}
\min \limits_{(\bm{\alpha}, \bm{\beta} \in \Gamma_k)}\! g(\bm{\alpha}, \bm{\beta}) \!\! = \!\!\frac{1}{2mR}||\bm{\alpha}^{\top}\bm{X}^+ \!-\! \bm{\beta}^{\top}\bm{X}^-||^2 + \sum_{i=1}^m l^*(-\alpha_i),
\end{eqnarray}
where $\bm{\alpha}$ and $\bm{\beta}$ are dual variables, $l^*(\cdot)$ is the convex conjugate of $ l(\cdot)$, and the domain $\Gamma_k$ is defined as
\begin{eqnarray}\label{dual set}
\nonumber \Gamma_k = \left\{\bm{\alpha} \in \mathbb{R}_+^m, \bm{\beta} \in \mathbb{R}_+^n \mid \sum_{i=1}^m \alpha_i = \sum_{j=1}^n \beta_j; \beta_j \leq \frac{1}{k}\sum_{i=1}^m \alpha_i, \forall j\right\}.
\end{eqnarray}
Let $\bm{\alpha}^*$ and $\bm{\beta}^*$ be the optimal solution of (\ref{dual problem}), the optimal
\begin{equation}
\bm{w}^* = (mR)^{-1}(\bm{\alpha}^{*\top}\bm{X}^+ - \bm{\beta}^{*\top}\bm{X}^-)^{\top}.
\end{equation}
\end{thm}
According to Theorem 1, learning scoring function $f$ is equivalent to learning the dual variables $\bm{\alpha}$ and $\bm{\beta}$ by solving problem \eqref{dual problem}. Its optimization naturally falls into the area of projected gradient method. Here we choose $l(u)=[1-u]_+^2$ where $[x]_+ \triangleq \max\{x, 0\}$ due to its simplicity of conjugation. The key steps are summarized in Algorithm \ref{alg:t-FPL}. At each iteration, we first update solution by the gradient of the objective function $g(\bm{\alpha}, \bm{\beta})$, then project the dual solution onto the feasible set $\Gamma_k$. In the sequel, we will show that this projection problem can be efficiently solved in {\it linear time}. In practice, since $g(\cdot)$ is smooth, we also leverage Nesterov's method to further accelerate the convergence of our algorithm. Nesterov's method \citep{Nesterov03} achieves $O(1/T^2)$ convergence rate for smooth objective function, where $T$ is the number of iterations.
\begin{algorithm}[htb!]
\caption{$\tau$-FPL Ranking} \label{alg:t-FPL}
\begin{algorithmic}[1]
\REQUIRE $X^+ \in \mathbb{R}^{m \times d}, X^- \in \mathbb{R}^{n \times d}$, maximal FPR tolerance $\tau$, regularization parameter $R$, stopping condition $\epsilon$
\STATE
Randomly initialize $\bm{\alpha_0}$ and $\bm{\beta_0}$\\
\STATE Set counter: $t \leftarrow 0$\\
\WHILE{$t = 0$ \textbf{or} $|g(\bm{\alpha_{t}}, \bm{\beta_{t}}) - g(\bm{\alpha_{t-1}}, \bm{\beta_{t-1}})| > \epsilon$}
\STATE Compute gradient of $g(\cdot)$ at point $(\bm{\alpha_t}, \bm{\beta_t})$\\
\STATE Compute $\bm{\alpha_{t+1}'}, \bm{\beta_{t+1}'}$ by gradient descent;\\
\STATE Project $\bm{\alpha_{t+1}'}, \bm{\beta_{t+1}'}$ onto the feasible set $\Gamma_k$:
\begin{equation*}
(\bm{\alpha_{t+1}}, \bm{\beta_{t+1}}) \leftarrow \Pi_{\Gamma_k}(\bm{\alpha_{t+1}'}, \bm{\beta_{t+1}'})
\end{equation*}
\STATE Update counter: $t \leftarrow t+1$;
\ENDWHILE
\STATE Return $\bm{w} \leftarrow (mR)^{-1}(\bm{\alpha_{t}}^{\top}X^+ - \bm{\beta_{t}}^{\top}X^-)^{\top}$
\end{algorithmic}
\end{algorithm}
\subsection{Linear Time Projection onto the Top-k Simplex}\label{proj sec}
One of our main technical results is a \emph{linear time} projection algorithm onto $\Gamma_k$, even in the case of $k$ is close to $n$. For clear notations, we reformulate the projection problem as
\begin{equation}\label{proj plm}
\begin{split}
\min_{\bm{\alpha} \geq 0, \bm{\beta} \geq 0} ~ & ~ \frac{1}{2}||\bm{\alpha} - \bm{\alpha}^0||^2 + \frac{1}{2}||\bm{\beta} - \bm{\beta}^0||^2\\
\text{\small s.t.}~ & ~\sum \nolimits_{i=1}^m \alpha_i = \sum \nolimits_{j=1}^n \beta_j, \quad \beta_j \leq \frac{1}{k}\sum \nolimits_{i=1}^m \alpha_i, \forall j.
\end{split}
\end{equation}
It should be noted that, many Euclidean projection problems studied in the literature can be seen as a special case of this problem. If the term $\sum_{i=1}^m \alpha_i$ is fixed, or replaced by a constant upper bound $C$, we obtain a well studied case of \emph{continuous quadratic knapsack problem} (CQKP)
\begin{align*}
\min_{\bm{\beta}} ~~ ||\bm{\beta} - \bm{\beta}^0||^2 \quad
\textrm{\small s.t.} ~\sum \nolimits_{i=1}^n \beta_i \ \leq\ C, 0\leq \beta_i \leq C_1 \ ,
\end{align*}
where $C_1 = C/k$. Several efficient methods based on median-selecting or variable fixing techniques are available \citep{patriksson2008survey}. On the other hand, if $k=1$, all upper bounded constraints are automatically satisfied and can be omitted. Such special case has been studied, for example, in \citep{liu2009efficient} and \citep{li2014top}, both of which achieve $O(n)$ complexity.
Unfortunately, none of those above methods can be directly applied to solving the generalized case (\ref{proj plm}), due to its property of unfixed upper-bound constraint on $\beta$ when $k >1$. To our knowledge, the only attempt to address the problem of unfixed upper bound is \citep{lapin2015top}. They solve a similar (but simpler) problem
\[
\min_{\beta} ~~ ||\bm{\beta} - \bm{\beta}^0||^2 ~~ \textrm{\small s.t.}~~ 0 \leq \beta_j \leq \frac{1}{k}\sum \nolimits_{i=1}^n \beta_i
\]
based on sorting and exhaustive search and their method achieves a runtime complexity $O(n\log(n) + kn)$, which is super-linear and even quadratic when $k$ and $n$ are linearly dependent. By contrast, our proposed method can be applied to both of the aforementioned special cases with minor changes and remains $O(n)$ complexity. The notable characteristic of our method is the efficient combination of bisection and divide-and-conquer: the former offers the guarantee of worst complexity, and the latter significantly reduces the large constant factor of bisection method.
We first introduce the following theorem, which gives a detailed description of the solution for (\ref{proj plm}).
\begin{thm}\label{sol of proj}
$(\bm{\alpha}^*\in \mathbb{R}^m, \bm{\beta}^*\in \mathbb{R}^n)$ is the optimal solution of (\ref{proj plm}) if and only if there exist dual variables $C^* \geq 0$, $\lambda^*$, $\mu^* \in \mathbb{R}$ satisfy the following system of linear constraints:
\begin{eqnarray}
C^*&=&\sum \nolimits_{i=1}^m [\alpha_i^0 - \lambda^*]_+ \label{def C}\\
C^*&=&\sum\nolimits_{j=1}^n \min\{[\beta_j^0 - \mu^*]_+, C^*/k\}\label{def mu}\\
0&=&\lambda^* + \mu^* + \frac{1}{k} \sum\nolimits_{j=1}^n\left[ \beta_j^0 - \mu^* - C^*/k \right]_+ \label{3 var}
\end{eqnarray}
and $\alpha_i^* = [\alpha_i^0 - \lambda^*]_+ \label{def alpha}$, $\beta_j^* = \min\{[\beta_j^0 - \mu^*]_+, {C^*}/{k}\} \label{def beta}$.
\end{thm}
Based on Theorem \ref{sol of proj}, the projection problem can be solved by finding the value of three dual variables $C$, $\lambda$ and $\mu$ that satisfy the above linear system. Here we first propose a basic bisection method which guarantees the worst time complexity. Similar method has also been used in \citep{liu2009efficient}. For brevity, we denote $\alpha_{[i]}^0$ and $\beta_{[i]}^0$ the $i$-largest dimension in $\bm{\alpha^0}$ and $\bm{\beta^0}$ respectively, and define function $C(\lambda)$, $\mu(C)$, $\delta(C)$ and $f(\lambda)$ as follows\footnote{Indeed, for some $C$, $\mu(C)$ is not one-valued and thus need more precise definition. Here we omit it for brevity, and leave details in section \ref{redifine_muc}.}:
\begin{eqnarray}
C(\lambda) &=& \sum \nolimits_{i=1}^m [\alpha_i^0 - \lambda]_+ \label{f:C}\\
\mu(C) &=& \mu \text{ satisfies } (\ref{def mu})\label{f:mu}\\
\delta(C) &=& \mu(C) + {C}/{k} \label{f:delta}\\
f(\lambda) &=& k\lambda + k\mu(C(\lambda)) + \sum \nolimits_{j=1}^n[\beta_j^0 - \delta(C(\lambda))]_+. \label{f:f}
\end{eqnarray}
The main idea of leveraging bisection to solve the system in theorem \ref{sol of proj} is to find the root of $f(\lambda) = 0$. In order to make bisection work, we need three conditions: $f$ should be continuous; the root of $f$ can be efficiently bracketed in a interval; and the value of $f$ at the two endpoints of this interval have opposite signs. Fortunately, based on the following three lemmas, all of those requirements can be satisfied.
\begin{lemma}\label{c=0}
(Zero case) \quad $(\bm{0}^m, \bm{0}^n)$ is an optimal solution of (\ref{proj plm}) if and only if $k\alpha_{[1]}^0 + \sum_{j=1}^k \beta_{[j]}^0 \leq 0$.
\end{lemma}
\begin{lemma}\label{bound}
(Bracketing $\lambda^*$)\quad If $C^* > 0$, $\lambda^* \in (-\beta_{[1]}^0,\ \alpha_{[1]}^0)$.
\end{lemma}
\begin{lemma}\label{inc-dec}
(Monotonicity and convexity)
\begin{enumerate}
\item $C(\lambda)$ is convex, continuous and strictly decreasing in $(-\infty$, $\alpha_{[1]}^0)$; \label{c1}
\item $\mu(C)$ is continuous, monotonically decreasing in $(0, +\infty)$; \label{c2}
\item $\delta(C)$ is continuous, strictly increasing in $(0, +\infty)$; \label{c3}
\item $f(\lambda)$ is continuous, strictly increasing in $(-\infty, \alpha_{[1]}^0)$. \label{c4}
\end{enumerate}
Furthermore, we can define the inverse function of $C(\lambda)$ as $\lambda(C)$, and rewrite $f(\lambda)$ as:
\begin{equation}
f(\lambda(C)) = k\lambda(C) + k\mu(C) + \sum \nolimits_{j=1}^n[\beta_j^0 - \delta(C)]_+,
\end{equation}
it is a convex function of $C$, strictly decreasing in $(0, +\infty)$.
\end{lemma}
Lemma \ref{c=0} deals with the special case of $C^*=0$. Lemma \ref{bound} and \ref{inc-dec} jointly ensure that bisection works when $C^* > 0$; Lemma \ref{bound} bounds $\lambda^*$; Lemma \ref{inc-dec} shows that $f$ is continuous, and since it is also strictly increasing, the value of $f$ at two endpoints must has opposite sign.
\textbf{Basic method: bisection \& leverage convexity } We start from select current $\lambda$ in the range $(-\beta_{[1]}^0,\ \alpha_{[1]}^0) \triangleq [l, u]$. Then compute corresponding $C$ by (\ref{f:C}) in $O(m)$, and use the current $C$ to compute $\mu$ by (\ref{f:mu}). Computing $\mu$ can be completed in $O(n)$ by a well-designed median-selecting algorithm \citep{kiwiel2007linear}. With the current (i.e. updated) $C$, $\lambda$ and $\mu$ in hand, we can evaluate the sign of $f(\lambda)$ in $O(n)$ and determine the new bound of $\lambda$. In addition, the special case of $C=0$ can be checked using Lemma \ref{c=0} in $O(m + n)$ by a linear-time k-largest element selecting algorithm \citep{kiwiel2005floyd}. Since the bound of $\lambda$ is irrelevant to $m$ and $n$, the number of iteration for finding $\lambda^*$ is $\log(\frac{u - l}{\epsilon})$, where $\epsilon$ is the maximum tolerance of the error. Thus, the worst runtime of this algorithm is $O(m +n)$. Furthermore, we also leverage the convexity of $f(\lambda(C))$ and $C(\lambda)$ to further improve this algorithm, please refer to \citep{liu2009efficient} for more details about related techniques.
Although bisection solves the projections in linear time, it may lead to a slow convergence rate. We further improve the runtime complexity by reducing the constant factor $\log(\frac{u - l}{\epsilon})$. This technique benefits from exploiting the monotonicity of both functions $C(\lambda)$, $\mu(C)$, $\delta(C)$ and $f(\lambda)$, which have been stated in Lemma \ref{inc-dec}. Notice that, our method can also be used for finding the root of arbitary piecewise linear and monotone function, without the requirement of convexity
\begin{algorithm}[htb!]
\caption{Linear-time Projection on Top-k simplex} \label{alg:proj}
\begin{algorithmic}[1]
\REQUIRE $\bm{\alpha^0} \in \mathbb{R}^m, \bm{\beta^0} \in \mathbb{R}^n$, maximal accuracy $\epsilon$
\STATE Calculate initial uncertainly intervals for $\lambda$, $C$, $\delta$ and $\mu$;
\STATE Initialize breakpoint caches for $C(\lambda)$, $\mu(C)$, $\delta(C)$, $f(\lambda)$:
\begin{equation*}
Cache_C \leftarrow \{\alpha_i^0 \mid \forall i\},~~ Cache_{\mu}, Cache_{\delta}, Cache_f \leftarrow \{\beta_j^0 \mid \forall j\}
\end{equation*}
\STATE Initialize partial sums of $C(\lambda)$, $\mu(C)$, $\delta(C)$, $f(\lambda)$ with zero;
\STATE Set $t \leftarrow 0$ and $\lambda_0 \leftarrow (\alpha_{[1]}^0 - \beta_{[1]}^0)/2$;
\WHILE{$t = 0$ \textbf{or} $|\lambda_{t} - \lambda_{t-1}| > \epsilon$}
\STATE Calculate $C_t$, $\mu_t$, $\delta_t$, $f_t$(by leveraging corresponding caches and partial sums);
\STATE Prune caches and update partial sums;
\STATE Shrink intervals of $\lambda$, $C$, $\delta$ and $\mu$ based on sign of $f(\lambda_t)$;
\STATE $t \leftarrow t + 1$;
\STATE Set $\lambda_t$ as midpoint of current new interval
\ENDWHILE
\STATE Return $\lambda^* \leftarrow \lambda_t$, $\mu^* \leftarrow \mu_t$, $C^* \leftarrow C_t$
\end{algorithmic}
\end{algorithm}
\textbf{Improved method: endpoints Divide \& Conquer } Lemma \ref{inc-dec} reveals an important chain monotonicity between the dual variables, which can used to improve the performance of our baseline method. The key steps are summarized in Algorithm \ref{alg:proj}. Denote the value of a variable $z$ in iteration $t$ as $z_t$. For instance, if $\lambda_t > \lambda_{t-1}$, from emma \ref{inc-dec} we have $C_t < C_{t-1}$, $\mu_t > \mu_{t-1}$ and $\delta_t < \delta_{t - 1}$.
This implies that we can set uncertainty intervals for both $\lambda$, $C$, $\mu$ and $\delta$. As the interval of $\lambda$ shrinking, lengths of these four intervals can be reduced simultaneously. On the other hand, notice that $C(\lambda)$ is indeed piecewise linear function (at most $m + 1$ segments), the computation of its value only contains a comparison between $\lambda_t$ and all of the $\alpha_i^0$s. By keeping a cache of $\alpha_i^0$s and discard those elements which are out of the current bound of $\lambda$ in advance, in each iteration we can reduce the expected comparison counts by half. A more complex but similar procedure can also be applied for computing $\mu(C)$, $\delta(C)$, and $f(\lambda)$, because both of these functions are piecewise linear and the main cost is the comparison with $O(m+n)$ endpoints. As a result, for approximately linear function and evenly distributed breakpoints, if the first iteration of bisection costs $\gamma (m+n)$ time, the overall runtime of the projection algorithm will be $\gamma (m+n) + \gamma (m+n)/2 +... \leq 2\gamma (m+n)$, which is much less than the original bisection algorithm whose runtime is $\log(\frac{u - l}{\epsilon})\gamma (m+n)$.
\subsection{Convergence and Computational Complexity}
Following immediately from the convergence result of Nesterov's method, we have:
\begin{thm}\label{conv rate}
Let $\bm{\alpha}_T$ and $\bm{\beta}_T$ be the output from the $\tau$-FPL algorithm after T iterations, then $g(\bm{\alpha}_T,\bm{\beta}_T) \leq \min g(\bm{\alpha}, \bm{\beta}) + \epsilon$, where $T \geq O(1/\sqrt{\epsilon})$.
\end{thm}
\begin{table}[htb
\centering
{
\begin{tabular}{lcc}
\hline
\textbf{Algorithm} & \textbf{Training} & \textbf{Validation} \\
\hline\hline
$\tau$-FPL &$O(d(m+n)/T^2)$ & Linear\\\\
TopPush &$O(d(m+n)/T^2)$ & Linear \\\\
CS-SVM &$O(d(m+n)/T)$ & Quadratic \\\\
$\text{SVM}_{\text{tight}}^{\text{pAUC}}$ &$O((m\log m+ n\log n + d(m+n))/T)$ & Linear\\\\
Bipartite & $O((d(m+n) + (m+n)\log(m+n))/T)$ &\multirow{2}{*}{Linear}\\
Ranking & $\sim O(dmn + mn\log(mn)/\sqrt T)$ & \multirow{2}{*}{}\\
\hline
\end{tabular}
\caption{Complexity comparison with SOTA approaches}
\label{tabcomp}}
\end{table}
Finally, the computational cost of each iteration is dominated by the gradient evaluation and the projection step. Since the complexity of projection step is $O(m + n)$ and the cost of computing the gradient is $O(d(m + n))$, combining with Theorem \ref{conv rate} we have that: to find an $\epsilon$-suboptimal solution, the total computational complexity of $\tau$-FPL is $O(d(m+n)/\sqrt{\epsilon})$. Table \ref{tabcomp} compares the computational complexity of $\tau$-FPL with that of some state-of-the-art methods. The order of validation complexity corresponds to the number of hyper-parameters. From this, it is easy to see that $\tau$-FPL is asymptotically more efficient.
\subsection{Out-of-Bootstrap Thresholding}\label{OOB}
In the thresholding stage, the task is to identify the boundary between the positive instances and $(1-\tau)$ percent of the negative instances. Though thresholding on the training set is commonly used in \citep{joachims1996probabilistic,davenport2010tuning,scheirer2013toward}, it may result in overfitting. Hence, we propose an out-of-bootstrap method to find a more accurate and stable threshold. At each time, we randomly split the training set into two sets $\mathcal{S}_1$ and $\mathcal{S}_2$, and then train on $\mathcal{S}_1$ as well as the select threshold on $\mathcal{S}_2$. The procedure can be running multiple rounds to make use of all the training data. Once the process is completed, we can obtain the final threshold by averaging. On the other hand, the final scoring function can be obtained by two ways: learn a scoring function using the full set of training data, or gather the weights learned in each previous round and average them. This method combines both the advantages of out-of-bootstrap and soft-thresholding techniques: accurate error estimation and reduced variance with little sacrifice on the bias, thus fits the setting of thresholding near the risk area.
\section{Theoretical Guarantees}\label{sec:theory}
Now we develop the theoretical guarantee for the scoring function, which bounds the probability of giving any positive instances higher score than $1-\tau$ proportion of negative instances. To this end, we first define $h(\bm{x}, f)$, the probability for any negative instance to be ranked above $x$ using $f$, i.e. $h(x, f) = \mathbb{E}_{x^- \sim \mathbb{P}^-}[\mathbb{I}(f(\bm{x}) \leq f(\bm{x}^-))]$, and then measure the quality of $f$ by $P(f, \tau) = \mathbb{P}_{\bm{x}^+ \sim \mathbb{P}^+}(h(\bm{x}^+, f) \geq \tau)$, which is the probability of giving any positive instances lower score than $\tau$ percent of negative instances. The following theorem bounds $P(f, \tau)$ by the empirical loss $L_{\bar{k}}$.
\begin{thm}\label{bound p}
Given training data $\mathcal{S}$ consisting of $m$ independent instances from distribution $\mathbb{P}^+$ and $n$ independent instances from distribution $\mathbb{P}^-$, let $f^*$ be the optimal solution to the problem (\ref{learning problem}). Assume $m \geq 12$ and $n \gg s$. We have, for proper $R$ and any $k \leq n$, with a probability at least $1-2e^{-s}$,
\begin{equation}\label{p bound eq}
P(f^*, \tau) \leq L_{\bar{k}}+ O\left(\sqrt{(s + \log(m)/m)}\right),
\end{equation}
where $\tau = k/n + O\left(\sqrt{\log m/n}\right)$, and $L_{\bar{k}} = \frac{1}{m} \sum_{i=1}^m l\left(f^*(\bm{x}_i^+) - \frac{1}{k}\sum_{j=1}^k f^*(\bm{x}_{[j]}^-)\right)$.
\end{thm}
Theorem \ref{bound p} implies that if $L_{\bar{k}}$ is upper bounded by $O(\log(m)/m))$, the probability of ranking any positive samples below $\tau$ percent of negative samples is also bounded by $O(\log(m)/m))$. If $m$ is approaching infinity, $P(f^*, \tau)$ would be close to 0, which means in that case, we can almost ensure that by thresholding at a suitable point, the true-positive rate will get close to 1. Moreover, we observe that $m$ and $n$ play different roles in this bound. For instance, it is well known that the largest absolute value of Gaussian random instances grows in $\log(n)$. Thus we believe that the growth of $n$ only slightly affects both the largest and the centroid of top-proportion scores of negatives samples. This leads to a conclusion that increasing $n$ only slightly raise $L_{\bar{k}}$, but significant reduce the margin between target $\tau$ and $k/n$. On the other hand, increasing $m$ will reduce upper bound of $P$, thus increasing the chance of finding positive instances at the top. In sum, $n$ and $m$ control $\tau$ and $P$ respectively.\\
\section{Experiment Results}\label{sec:experiment}
\subsection{Effectiveness of the Linear-time Projection}
We first demonstrate the effectiveness of our projection algorithm. Following the settings of \citep{liu2009efficient}, we randomly sample 1000 samples from the normal distribution ${\cal N}(0,1)$ and solve the projection problem. The comparing method is \emph{ibis} \citep{liu2009efficient}, an improved bisection algorithm which also makes use of the convexity and monotonicity. All experiments are running on an Intel Core i5 Processor. As shown in Fig.\ref{Fig.lable}, thanks to the efficient reduction of the constant factor, our method outperforms \emph{ibis} by saving almost $75\%$ of the running time in the limit case.
We also solve the projection problem proposed in \citep{lapin2015top} by using a simplified version of our method, and compare it with the method presented in \citep{lapin2015top} (PTkC), whose complexity is $O(n\log(n) + kn)$. As one can observe from Fig.\ref{Fig.sub.2}, our method is linear in complexity regarding with $n$ and does not suffer from the growth of $k$. In the limit case (both large $k$ and $n$), it is more than three-order of magnitude faster than the competitors.
\begin{figure}[tb!]
\centering
\subfigure[Run time against the method ibis (log-log).]{
\label{Fig.sub.1}
\includegraphics[width=0.45\linewidth]{ibis2.jpg}}
\subfigure[Run time against PTkC (log-log).]{
\label{Fig.sub.2}
\includegraphics[width=0.45\linewidth]{nips152.jpg}}
\caption{Running time against two peer projection methods.}
\label{Fig.lable}
\end{figure}
\subsection{Ranking Performance}\label{RP}
\begin{table*}[htb!]\small\centering\resizebox{1\textwidth}{!}{\begin{tabular}{c|cc|ccccc|ccccc|cccccc}\hline\multirow{2}{*}{\textbf{Dataset}} & \multicolumn{2}{c|}{\textbf{heart}} & \multicolumn{5}{c|}{\textbf{spambase}} & \multicolumn{5}{c|}{\textbf{real-sim}} & \multicolumn{6}{c}{\textbf{w8a}} \\ & \multicolumn{2}{c|}{120/150,d:13} & \multicolumn{5}{c|}{1813/2788,d:57} & \multicolumn{5}{c|}{22238/50071,d:20958} & \multicolumn{6}{c}{2933/62767,d:300}\\ \hline$\tau(\%)$ & 5 & 10 & 0.1 & 0.5 & 1 & 5 & 10 & 0.01 & 0.1 & 1 & 5 & 10 & 0.05 & 0.1 & 0.5 & 1 & 5 & 10 \\\hline\hline\textbf{CS-SVM}& .526 & .691 & .109 & .302 & $\bm{.487}$ & .811 & .920 & .376 & $\bm{.748}$ & .921 & .972 & .990 & .501 & .520 & .649 & .695 & .828 & .885 \\\textbf{TopPush}& $\bm{.541}$ & .711 & .112 & .303 & .484 & .774 & .845 & $\bm{.391}$ & .747 & .920 & .968 & .983 & $\bm{.508}$ & $\bm{.551}$ & .627 & .656 & .761 & .842 \\\textbf{$\text{SVM}_{\text{tight}}^{\text{pAUC}}$}& .509 & .728 & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A \\\textbf{$\tau$-Rank}&$\bm{.541}$&$\bm{ .740}$&$\bm{ .112}$&$\bm{ .305}$&$ .460$&$\bm{ .842}$&$\bm{ .929}$&$\bm{ .391}$&$.747$&$ .920$&$\bm{ .975}$&$\bm{ .991}$&$\bm{ .508 }$&$\bm{ .551 }$&$.645$&$\bm{ .710 }$&$\bm{ .832 }$&$\bm{ .894}$\\\textbf{2$\tau$-Rank}&$\bm{.547}$&$\bm{ .734}$&$\bm{ .112}$&$\bm{ .311}$&$ .477$&$\bm{ .862}$&$\bm{ .936}$&$\bm{ .391}$&$.747$&$\bm{ .922}$&$\bm{ .978}$&$\bm{ .992}$&$\bm{ .508 }$&$.549$&$\bm{ .675 }$&$\bm{ .739 }$&$\bm{ .841 }$&$\bm{ .902}$\\\hline\end{tabular}}\caption{Ranking performance by different values of the tolerance $\tau$. The number of positive/negative instances and feature dimensions (`d') is shown together with the name of each dataset. The best results are shown in bold. 'N/A's denote the experiments that require more than one week for training.}\label{tab:real_exp_rank}\end{table*}
Next, we validate the ranking performance of our $\tau$-FPL method, i.e. scoring and sorting test samples, and then evaluate the proportion of positive samples ranked above $1- \tau$ proportion of negative samples. Considering ranking performance independently can avoid the practical problem of mismatching the constraint in (\ref{empirical plm}) on testing set, and always offer us the optimal threshold.
Specifically, we choose (\ref{origin rank plm}) as evaluation and validation criterion. Compared methods include cost-sensitive SVM (CS-SVM) \citep{osuna1997support}, which has been shown a lower bound approximation of (\ref{origin rank plm}); TopPush \citep{li2014top} ranking, which focus on optimizing the absolute top of the ranking list, also a special case of our model ($\tau = 0$); $\text{SVM}_{\text{tight}}^{\text{pAUC}}$ \citep{Narasimhan2013ASS}, a more general method which designed for optimizing arbitrary partial-AUC. We test two version of our algorithms: $\tau$-Rank and $2\tau$-Rank, which correspond to the different choice of $\tau$ in learning scheme. Intuitively, enlarge $\tau$ in training phase can be seen as a top-down approximation---from upper bound to the original objective (\ref{empirical plm}). On the other hand, the reason for choosing $2\tau$ is that, roughly speaking, the average score of the top $2\tau$ proportion of negative samples may close to the score of $\tau n$-th largest negative sample.
\textbf{Settings}. We evaluate the performance on publicly benchmark datasets with different domains and various sizes\footnote{\url{https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/binary}}. For small scale datasets($\leq 10,000$ instances), 30 times stratified hold-out tests are carried out, with $2/3$ data as train set and $1/3$ data as test set. For large datasets, we instead run 10 rounds. In each round, hyper-parameters are chosen by 5-fold cross validation from grid, and the search scope is extended if the optimal is at the boundary.
\textbf{Results}. Table \ref{tab:real_exp_rank} reports the experimental results. We note that at most cases, our proposed method outperforms other peer methods. It confirms the theoretical analysis that our methods can extract the capacity of the model better. For TopPush, it is highly-competitive in the case of extremely small $\tau$, but gradually lose its advantage as $\tau$ increase. The algorithm of $\text{SVM}_{\text{tight}}^{\text{pAUC}}$ is based on cutting-plane methods with exponential number of constraints, similar technologies are also used in many other ranking or structured prediction methods, e.g. Structured SVM \citep{tsochantaridis2005large}. The time complexity of this kind of methods is $O((m+n)\log(m+n))$, and we found that even for thousands of training samples, it is hard to finish experiments in allowed time.
\subsection{Overall Classification Accuracy}
\begin{table*}[htb!]\small
\centering
\resizebox{1\textwidth}{!}
{
\begin{tabular}{c| c| cccc |c ccc
\hline
\textbf{Dataset(+/-)} & \textbf{$\tau$(\%)} & \textbf{BS-SVM} & \textbf{CS-LR} & \textbf{CS-SVM} & \textbf{CS-SVM-OOB} & \textbf{$\tau$-FPL} & \textbf{$2\tau$-FPL}\\
\hline
\hline
\textbf{heart}&5&$(.069, .675), .713$&$(.035, .394), .606$&$(.027, .327), .673$&$(.058, .553), .609$&$(.053, .582), \bm{.468}$&$(.055, .584), \bm{.514}$\\
120/150,d:13&10&$(.121, .774), .435$&$(.058, .615), .385$&$(.078, .666), .334$&$(.088, .682), .318$&$(.086, .686), \bm{.314}$&$(.080, .679), \bm{.317}$\\
\hline
\textbf{breast-cancer}&1&$(.015, .964), .559$&$(.007, .884), \bm{.116}$&$(.006, .870), .130$&$(.014, .955), .451$&$(.013, .955), .324$&$(.011, .949), .192$\\
239/444&5&$(.063, .978), .276$&$(.013, .965), .035$&$(.017, .965), .034$&$(.046, .974), \bm{.026}$&$(.041, .976), \bm{.025}$&$(.045, .974), \bm{.026}$\\
d:10&10&$(.113, .985), .142$&$(.035, .970), .030$&$(.044, .973), .027$&$(.095, .981), .020$&$(.098, .982), \bm{.018}$&$(.094, .982), \bm{.018}$\\
\hline
\textbf{spambase}&0.5&$(.008, .426), 1.220$&$(.007, .011), 1.362$&$(.002, .109), .891$&$(.005, .275), .790$&$(.005, .278), \bm{.722}$&$(.004, .268), \bm{.732}$\\
1813/2788&1&$(.013, .583), .748$&$(.007, .011), .989$&$(.004, .256), .744$&$(.009, .418), .582$&$(.008, .416), .584$&$(.008, .440), \bm{.560}$\\
d:57&5&$(.054, .895), .192$&$(.007, .011), .989$&$(.020, .667), .333$&$(.047, .793), .207$&$(.041, .822), \bm{.178}$&$(.046, .845), \bm{.155}$\\
&10&$(.103, .941), .087$&$(.007, .011), .989$&$(.051, .716), .284$&$(.090, .902), .099$&$(.087, .925), \bm{.075}$&$(.090, .928), \bm{.072}$\\
\hline
\textbf{real-sim}&0.01&$(.002, .813), 22.376$&$(.001, .207), 7.939$&$(.000, .209), .791$&$(.000, .268), .833$&$(.000, .270), \bm{.730}$&$(.000, .270), \bm{.730}$\\
22238/50071&0.1&$(.008, .919), 7.09$&$(.001, .207), .826$&$(.001, .700), .428$&$(.001, .584), .416$&$(.001, .585), \bm{.415}$&$(.001, .585), \bm{.415}$\\
d:20958&0.5&$(.023, .966), 3.680$&$(.001, .207), .794$&$(.001, .755), .245$&$(.003, .810), .190$&$(.003, .829), \bm{.174}$&$(.003, .827), \bm{.181}$\\
&1&$(.036, .978), 2.570$&$(.001, .207), .794$&$(.007, .880), .121$&$(.007, .875), .125$&$(.007, .894), \bm{.115}$&$(.006, .891), \bm{.109}$\\
&5&$(.094, .994), .878$&$(.078, .994), .575$&$(.029, .931), .139$&$(.039, .965), .035$&$(.041, .972), \bm{.028}$&$(.044, .974), \bm{.028}$\\
&10&$(.133, 0.997), .336$&$(.078, .994), \bm{.007}$&$(.069, .993), .007$&$(.099, .986), .019$&$(.092, .991), .009$&$(.094, .991), .009$\\
\hline
\textbf{w8a}&0.05&$(.001, .525), .966$&$(.000, .101), .900$&$(.000, .420), .580$&$(.000, .438), \bm{.562}$&$(.000, .428), .572$&$(.000, .428), .572$\\
1933/62767&0.1&$(.001, .585), .710$&$(.000, .119), .881$&$(.000, .447), .553$&$(.001, .493), .507$&$(.001, .495), \bm{.505}$&$(.001, .499), \bm{.501}$\\
d:123&0.5&$(.006, .710), .437$&$(.000, .119), .881$&$(.002, .595), .405$&$(.003, .634), .366$&$(.003, .654), \bm{.347}$&$(.003, .667), \bm{.333}$\\
&1&$(.011, .749), .341$&$(.014, .696), .715$&$(.006, .642), .358$&$(.006, .695), .305$&$(.006, .702), \bm{.298}$&$(.007, .726), \bm{.274}$\\
&5&$(.048, .823), .177$&$(.014, .696), .305$&$(.013, .701), .299$&$(.046, .805), .195$&$(.033, .818), .182$&$(.036, .827), \bm{.173}$\\
&10&$(.049, .823), .177$&$(.014, .696), .305$&$(.013, .701), .299$&$(.053, .814), .186$&$(.042, .833), \bm{.167}$&$(.038, .826), \bm{.174}$\\\hline
\end{tabular}}
\caption{[(mean false positive rate, mean true positive rate), NP-score] on real-world datasets by different values of the tolerance $\tau$. In the leftmost column, the number of positive/negative instances and feature dimensions (`d') in each dataset. For each dataset, the best results are shown in bold.}
\label{tab:real_exp}
\end{table*}
In this section we compare the performance of different models by jointly learning the scoring function and threshold in training phase, i.e. output a classifier. To evaluate a classifier under the maximum tolerance, we use \emph{Neyman-Pearson score} (NP-score) \citep{scott2007performance}. The NP-score is defined by
$\frac{1}{\tau} \max\{fpr, \tau\} - tpr$ where $fpr$ and $tpr$ are false-positive rate and true-positive rate of the classifier respectively, and $\tau$ is the maximum tolerance. This measure punishes classifiers whose false-positive rates exceed $\tau$, and the punishment becomes higher as $\tau \to 0$.
\textbf{Settings}. We use the similar setting for classification as for ranking experiments, i.e., for small scale datasets, 30 times stratified hold-out tests are carried out; for large datasets, we instead run 10 rounds. Comparison baselines include: Cost-Sensitive Logistic Regression (CS-LR) which choose a surrogate function that different from CS-SVM; Bias-Shifting Support Vector Machine (BS-SVM), which first training a standard SVM and then tuning threshold to meet specified false-positive rate; cost-sensitive SVM (CS-SVM). For complete comparison, we also construct a CS-SVM by our out-of-bootstrap thresholding (CS-SVM-OOB), to eliminate possible performance gains comes from different thresholding method, and focus on the training algorithm itself. For all of comparing methods, the hyper-parameters are selected by 5-fold cross-validation with grid search, aims at minimizing the NP-score, and the search scope is extended when the optimal value is at the boundary. For our $\tau$-FPL, in the ranking stage the regularization parameter $R$ is selected to minimize (\ref{origin rank plm}) , and then the threshold is chosen to minimize NP-score. We test two variants of our algorithms: $\tau$-FPL and $2\tau$-FPL, which corresponding different choice of $\tau$ in learning scheme. As mentioned previously, enlarge $\tau$ can be seen as a top-down approximation towards the original objective.
\textbf{Results}. The NP-score results are given in Table \ref{tab:real_exp}. First, we note that both our methods can achieve the best performance in most of the tests, compared to various comparing methods. Moreover, it is clear that even using the same method to select the threshold, the performance of cost sensitive method is still limited. Another observation is that both of the three algorithms which using out-of-bootstrap thresholding can efficiently control the false positive rate under the constraint. Moreover, $\tau$-FPLs are more stable than other algorithms, which we believe benefits from the accurate splitting of the positive-negative instances and stable thresholding techniques.
\subsection{Scalability}
\begin{figure}[tb]
\centering
\includegraphics[width=0.8\linewidth]{scalability2.pdf}
\caption{Training time of $\tau$-FPL versus training data size for different $\tau$ (log-log).}
\label{Fig.scale}
\end{figure}
We study how $\tau$-FPL scales to a different number of training examples by using the largest dataset real-sim. In order to simulate the limit situation, we construct six datasets with different data size, by up-sampling original dataset. The sampling ratio is $\{1,2,2^2,...2^5\}$, thus results in six datasets with data size from 72309 to 2313888. We running $\tau$-FPL ranking algorithm on these datasets with different $\tau$ and optimal $R$ (chosen by cross-validation), and report corresponding training time. Up-sampling technology ensures that, for a fixed $\tau$, all the six datasets share the same optimal regularization parameter $R$. Thus the unique variable can be fixed as data size. Figure \ref{Fig.scale} shows the log-log plot for the training time of $\tau$-FPL versus the size of training data, where different lines correspond to different $\tau$. It is clear that the training time of $\tau$-FPL is indeed linear dependent in the number of training data. This is consistent with our theoretical analysis and also demonstrate the scalability of $\tau$-FPL.
\section{Proofs and Technical Details}
In this section, we give all the detailed proofs missing from the main text, along with ancillary remarks and comments.
\textbf{Notation.} In the following, we define $z_{[i]}$ be the $i$-th largest dimension of a vector $\bm{z} = (z_1,...,z_N) \in \mathbb{R}^N$, define $\alpha_{[0]}^0 = \beta_{[0]}^0 = \infty$, $\alpha_{[n+1]}^0 = \beta_{[n+1]}^0 =-\infty$, define $B_i^j$ as the range $(\beta_{[j]}^0, \beta_{[i]}^0]$ and $B_i$ as the abbreviation of $B_i^{i+1}$.
\subsection{Proof of Theorem \ref{np-hard}: NP-Hardness of $\tau$-$OPT_{L}^{\lambda}$}
Our proof is based on a turing-reduction of the following Maximum Agreement Problem (MAP) to $\tau$-$OPT_{L}^{\lambda}$.
\begin{myplm}
Define $\mathcal{H'} = \{f(\bm{x}) = \bm{w}^{\top}\bm{x} - b \mid \bm{w} \in \mathbb{R}^p, b \in R\}$. We say $f \in \mathcal{H'}$ and point $(\bm{x}, y) \in R^p \times \{+1, -1\}$ reach an \emph{agreement} if $yf(\bm{x}) > 0$. Given data set $D = \{(\bm{x}_1, y_1),...(\bm{x}_N, y_N) \}$ contains $N$ samples, find a $f \in \mathcal{H'}$ with the maximum number of agreements reached on $D$.
\end{myplm}
It is clear that MAP is in fact equivalent to the problem of binary classification using 0-1 loss and hyperplane hypothesis set. In general, both solving MAP or approximately maximizing agreements to within some constant factor (418/415) are NP-hard \citep{ben2003difficulty}. Now we introduce the decision problem version of MAP (DP-MAP).
\begin{myplm2}
Given data set $D = \{(\bm{x}_1, y_1),...(\bm{x}_N, y_N) \}$ and any $0 \leq k \leq N$, whether there exists some $f \in \mathcal{H'}$ reaches at least $k$ agreements on $D$ ? If yes, output one of such $f$.
\end{myplm2}
If we have an oracle algorithm $O$ of DP-MAP, we can solve MAP by applying bisection search on $k$ and take the maximum value of $k$ that $O$ output \emph{Yes} as solution. The overall number of calling $O$ is at most $\log(N)$. This naive reduction shows that DP-MAP is also NP-hard. More precisely, it is a NP-Complete problem.
Now we consider to solve DP-MAP by $\tau$-$OPT_{L}^{\lambda}$. This means that, for fixed $0 < \tau < 1$, $\lambda > 0$ and surrogate $L(\cdot)$, we have an oracle algorithm of $\tau$-$OPT_{L}^{\lambda}$, denoted by $O$. We need to design an algorithm so that for all $0 \leq k \leq N$ and data set $D$ with $N$ $p$-dimensional points, we can determine if there exists a hyperplane reaches at least $k$ agreements on $D$ by polynomial calls to $O$.
\textbf{Case-1: $N - \lfloor \tau N\rfloor < k \leq N$}.~~In this case, we construct the following dataset $D_O$ as input to $O$:
\begin{eqnarray*}
m &=& 1\\
n &=& N + A\\
d &=& p + 1\\
\bm{x}^+ &=& \bm{0}^{d}\\
\bm{x}_i^- &=& (-y_i\bm{x}_i^\top, -y_i),~~i = 1,...,N\\
\bm{x}_{N + j}^- &=& \bm{0}^d,~~j= 1,...,A
\end{eqnarray*}
Here, $A \in \mathbb{N}$ is the smallest non-negative integer that satisfies
\begin{equation}\label{g(A)}
g(A) \triangleq \lfloor\tau(N + A)\rfloor - A = N - k
\end{equation}
We give some properties about $g(A)$.
\begin{lemma}
(Properties of $g: \mathbb{N} \rightarrow \mathbb{Z}$)
\begin{enumerate}[itemindent=1em]
\item $0 \leq g(A) - g(A + 1) \leq 1$;
\item $g(A) \leq 0$ when $A \geq \frac{\tau}{1 - \tau}N$;
\item For any integer $T \in [0, g(0)]$, there exist $A = \mathcal{O}(N)$ such that $g(A) = T$.
\end{enumerate}
\end{lemma}
Both of them are easy to verify, so we omit the details. Combine these properties and the fact $ 0 \leq N - k < g(0)$, we know that there must exist some $A \in [0, \lceil\frac{\tau}{1 - \tau}N\rceil]$ satisfies \eqref{g(A)}, and thus the size of dataset we constructed above is linearly related to $N$ and $p$. Now we introduce the following lemma.
\begin{lemma}\label{keq}
There exists hyperplane reaches at least $k$ agreements iff the minimum value that $O$ found is less than $L(0)$.
\end{lemma}
\begin{proof}
On the one hand, if there exists a hyperplane $f_0(\bm{x}) = \bm{w}_0^{\top}\bm{x} + b_0 \in \mathcal{H'}$ reaches at least $k$ agreements on $D$, we know $|\mathcal{T}| \leq N - k$ where $\mathcal{T} = \{t \mid (\bm{w}_0^{\top}, b_0)(y_t\bm{x_t}^{\top}, y_t)^{\top} \leq 0\}$. Define $\bm{w_1} = \frac{\lambda(\bm{w}_0^{\top}, b_0)^{\top}}{||(\bm{w}_0^{\top}, b_0)^{\top}||}$. Now $\bm{w}_1 \in \mathcal{H}^d$ and at most $N - k$ different values of $t \in \{1,...,N\}$ satisfies
\begin{equation}\label{kkk}
\bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_t^- \leq 0
\end{equation}
Note that for any $j = N + 1,...,N + A$ we have $\bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_j^- = 0$, combine these two observations we have: at most $N - k + A$ different values $t \in \{1,...,N+A\}$ satisfies \eqref{kkk}. Thus,
\begin{eqnarray*}
&& \bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_{[N- k + A + 1]}^- > 0\\
\Rightarrow && L(\bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_{[N- k + A + 1]}^-) < L(0)\\
\Rightarrow && L(\bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_{[\lfloor\tau(N + A)\rfloor + 1]}^-) < L(0)\\
\Rightarrow && \min \limits_{\bm{w} \in \mathcal{H}^d} L(\bm{w}^T\bm{x}^+ - \bm{w}^T\bm{x}_{[\lfloor \tau n\rfloor + 1]}^-) < L(0)
\end{eqnarray*}
The LHS of the last inequality is indeed the minimum value that $O$ output.
On the other hand, it is obvious that the above proof is easily reversible, this completes the proof of lemma.
\end{proof}
By Lemma \ref{keq}, we can determine if there exists a hyperplane reaches at least $k$ agreements by calling $O$ once. If the output minimum value is less than $L(0)$, the hyperplane that $O$ learned is exactly corresponds to the hyperplane that reaches enough agreements on $D$, otherwise there is no such hyperplane. We thus complete the reduction.
\textbf{Case-2: $0 \leq k \leq N - \lfloor \tau N\rfloor$}.~The dataset we used here as input to $O$ is
\begin{eqnarray*}
m &=& 1\\
n &=& N + A\\
d &=& p + 2\\
\bm{x}^+ &=& \bm{0}^{d}\\
\bm{x}_i^- &=& (-y_i\bm{x}_i^\top, -y_i, 0)^\top,~~i = 1,...,N\\
\bm{x}_{N + j}^- &=& (\bm{0}^{p+1\top}, -1)^\top,~~j= 1,...,A
\end{eqnarray*}
$A \in \mathbb{N}$ is the smallest non-negative integer that satisfies
\begin{equation}\label{h(A)}
h(A) \triangleq \lfloor\tau(N + A)\rfloor = N - k
\end{equation}
\begin{lemma}\label{keqq}
(Properties of $h: \mathbb{N} \rightarrow \mathbb{N^+}$)
\begin{enumerate}[itemindent=1em]
\item $0 \leq h(A + 1) - h(A) \leq 1$;
\item $h(A) > N$ when $A \geq \frac{1 - \tau}{\tau}N + \frac{1}{\tau}$;
\item For any integer $T \in [\lfloor \tau N\rfloor, N]$, there exist $A = \mathcal{O}(N)$ such that $h(A) = T$.
\end{enumerate}
\end{lemma}
Combine Lemma \ref{keqq} and the fact $\lceil \tau N \rceil \leq N - k \leq N$ we know that the size of dataset constructed above is linearly related to $N$ and $p$. Now the claim in Lemma \ref{keq} is also true in this case, we give a proof sketch below.
\begin{proof}
We follow the same definitions of $f_0$ and $\mathcal{T}$ as in the proof of Case-1. Define $\bm{w_1} = \frac{\lambda(\bm{w}_0^{\top}, b_0, 1)^{\top}}{||(\bm{w}_0^{\top}, b_0, 1)^{\top}||}$. Now $\bm{w}_1 \in \mathcal{H}^d$ and we have
\begin{eqnarray*}
\bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_i^- &=& (\bm{w}_0^{\top}, b_0)(y_i\bm{x_i}^{\top}, y_i)^{\top},~~~\forall i = 1,...,N\\
\bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_j^- &=& 1,~~~\forall j = N+1,...N+A
\end{eqnarray*}
Thus, \eqref{kkk} holds for at most $N - k$ different values of $t$ in $\{1,...,N+A\}$. This implies
\begin{eqnarray*}
&&\bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_{[N- k + 1]}^- > 0\\
\Rightarrow && L(\bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_{[N- k + 1]}^-) < L(0)\\
\Rightarrow && L(\bm{w}_1^T\bm{x}^+ - \bm{w}_1^T\bm{x}_{[\lfloor\tau(N + A)\rfloor + 1]}^-) < L(0)\\
\Rightarrow && \min \limits_{\bm{w} \in \mathcal{H}^d} L(\bm{w}^T\bm{x}^+ - \bm{w}^T\bm{x}_{[\lfloor \tau n\rfloor + 1] }^-) < L(0)
\end{eqnarray*}
Above proof can be easily reversed for another direction, we omit the details.
\end{proof}
Combine both Case-1 and Case-2, we complete the reduction from DP-MAP to $\tau$-$OPT_{L}^{\lambda}$. Since DP-MAP is NP-Complete, we conclude that $\tau$-$OPT_{L}^{\lambda}$ must be NP-hard.\\
\textbf{Remark 1}.~It is clear that above reduction can not be used for $\tau = 0$. Indeed, for convex surrogate $L(\cdot)$ the objective of $0$-$OPT_{L}^{\lambda}$ is convex and its global minimum can be efficiently obtained, see \citep{li2014top} for more details.\\
\textbf{Remark 2}.~In problem definition and above proof we implicitly suppose that the minimum of $\tau$-$OPT_{L}^{\lambda}$ can be attained. In general cases, by considering the decision problem of $\tau$-$OPT_{L}^{\lambda}$ one can complete a very similar reduction and we omit details for simplicity here.
\subsection{Proof of Proposition \ref{prop:upper}}
\begin{comment}
\begin{prop}
For any non-empty hypothesis set $\mathcal{H}$ and $f \in \mathcal{H}$, non-constant convex surrogate function $L(\cdot)$, let
\begin{eqnarray*}
\bar{R}_0 \!\!&=&\!\!\! \frac{1}{m}\sum \nolimits_{i=1}^m \mathbb{I}\left(f(\bm{x}_i^+) - f(\bm{x}_{[\lceil \tau n \rceil]}^-) < 0\right)\\
\bar{R}_1 \!\!&=&\!\!\! \frac{1}{m} \sum \nolimits_{i=1}^m L(f(\bm{x}_i^+) - \frac{1}{\lceil \tau n \rfloor}\sum_{j=1}^{\lceil \tau n \rceil} f(\bm{x}_{[j]}^-))\\
\bar{R}_2 \!\!&=&\!\!\! \min_{b \in \mathbb{R}} \frac{1}{m} \sum_{i=1}^m L(f(\bm{x}_i^+) - b)\quad\textrm{s.t.}~\frac{1}{n} \sum_{j=1}^n L(b - f(\bm{x}_j^-)) \leq \tau
\end{eqnarray*}
we have
\begin{equation*
\bar{R}_0 \leq \bar{R}_1 \leq \bar{R}_2.
\end{equation*}
The second equality holds iff both of the following three conditions are satisfied: (i) the scores of the worst $\tau$-proportion negative examples are the same; (ii) loss of the remaining $(1 - \tau)$-proportion negative samples are 0; (iii) $\tau n \in \mathbb{N}$.
\end{prop}
\end{comment}
$\bar{R}_0 \leq \bar{R}_1$ can be easily obtained by combining the relationship between mean and maximum value and the definition that $L(\cdot)$ is an upper bound of $\mathbb{I}(\cdot)$. We now prove $\bar{R}_1 \leq \bar{R}_2$. Define $k = \lfloor\tau n \rfloor + 1$, we have
\begin{eqnarray*}
&&kL(0) \geq k > \tau n \\
&\geq& \sum_{j=1}^n L(b - f(\bm{x}_j^-))~\text{(Constraint on FPR)}\\
&\geq& \sum_{j=1}^k L(b - f(\bm{x}_{[j]}^-))~\text{(Nonnegativity of } L\text{)}\\
&\geq& kL(\frac{1}{k}\sum_{j=1}^k (b - f(\bm{x}_{[j]}^-)))~\text{(Jensen's Inequality)}\\
&=& kL(b - \frac{1}{k}\sum_{j=1}^k f(\bm{x}_{[j]}^-))\\
&\Leftrightarrow& b > \frac{1}{k}\sum_{j=1}^k f(\bm{x}_{[j]}^-)~\text{(Monotonicity of } L\text{)}
\end{eqnarray*}
Thus
\begin{eqnarray*}
\bar{R}_2 & = &\min_{b \in \mathbb{R}} \frac{1}{m} \sum_{i=1}^m L(f(\bm{x}_i^+) - b)\\
&\geq& \frac{1}{m} \sum \nolimits_{i=1}^m L(f(\bm{x}_i^+) - \frac{1}{\lfloor \tau n \rfloor + 1}\sum_{j=1}^{\lfloor \tau n \rfloor + 1} f(\bm{x}_{[j]}^-))\\
&=& \bar{R}_1.
\end{eqnarray*}
\subsection{Proof of Theorem \ref{dual form}}
Since truncated quadratic loss is non-increasing and differentiable, it can be rewritten in its convex conjugate form, that is
\begin{equation}
l(z) = \max \limits_{\alpha \leq 0}\ \{ \alpha z - l_*(\alpha) \}
\end{equation}
where $l_*(\alpha)$ is the convex conjugate of $l$. Based on this, we can rewrite the problem (\ref{learning problem}) as
\begin{eqnarray}\label{plm:1}
\nonumber \min \limits_{\bm{w}} \max_{\alpha \leq 0} \sum_{i=1}^m \alpha_i(\bm{w}^{\top}\bm{x}_i^+ - \frac{1}{k}\sum_{j=1}^k \bm{w}^{\top}\bm{x}_{[j]}^-) - \sum_{i=1}^m l_*(\alpha_i) + \frac{mR}{2}||\bm{w}||^2
\end{eqnarray}
where $\bm{\alpha} = (\alpha_1,..., \alpha_m)^{\top}$ is the dual variable.\\
On the other hand, it is easy to verify that, for $\bm{t} = (t_1,..., t_n)^{\top} \in \mathbb{R}^n$,
\begin{equation*}
\sum_{i=1}^k t_{[k]} = \max \limits_{\bm{p} \in \Omega}\ \bm{p}^{\top}\bm{t}
\end{equation*}
with $\Omega = \{\bm{p} \mid \bm{0} \leq \bm{p} \leq \bm{1}, \bm{1_n}^{\top}\bm{p} = k\}$. By substituting this into (\ref{plm:1}), the problem becomes
\begin{eqnarray*}
\nonumber \min \limits_{\bm{w}} \max \limits_{\bm{\alpha} \leq \bm{0}^m, \bm{p} \in \Omega} \sum_{i=1}^m \alpha_i(\bm{w}^{\top}\bm{x}_i^+ - \frac{1}{k}\sum_{j=1}^n p_j\bm{w}^{\top}\bm{x}_j^-) -
\sum_{i=1}^m l_*(\alpha_i) + \frac{mR}{2}||\bm{w}||^2
\end{eqnarray*}
Now, define $\beta_j = \frac{1}{k} p_j\sum_{i=1}^m \alpha_i$, above problem becomes
\begin{eqnarray*}
\nonumber \min \limits_{\bm{w}} \max \limits_{\bm{\alpha} \leq \bm{0}^m, \bm{\beta} \leq \bm{0}^n} && \sum_{i=1}^m \alpha_i\bm{w}^{\top}\bm{x}_i^+ - \sum_{j=1}^n \beta_j \bm{w}^{\top}\bm{x}_j^- - \sum_{i=1}^m l_*(\alpha_i) + \frac{mR}{2}||\bm{w}||^2\\
s.t. ~&&\sum_{i=1}^m \alpha_i = \sum_{j=1}^n \beta_j, \beta_j \geq \frac{1}{k}\sum_{i=1}^m \alpha_i
\end{eqnarray*}
Notice that this replacement is able to keep the two problems equivalent.\\
Since the objective above is convex in $\bm{w}$, and jointly concave in $\bm{\alpha}$ and $\bm{\beta}$, also its feasible domain is convex; hence it satisfies the strong max-min property \citep{boyd2004convex}, the min and max can be swapped. After swapping, we first consider the inner minimization subproblem over $\bm{w}$, that is
\begin{equation*}
\min \limits_{\bm{w}} \sum_{i=1}^m \alpha_i \bm{w}^{\top}\bm{x}_i^+ - \sum_{j=1}^n\beta_j \bm{w}^{\top}\bm{x}_j^- + \frac{mR}{2}||\bm{w}||^2
\end{equation*}
Here we omit items which does not depend on $\bm{w}$. This is an unconstrained quadratic programming problem, whose solution is $\bm{w}^* = -\frac{1}{mR}(\bm{\alpha}^{\top}X^+ - \bm{\beta}^{\top}\bm{X}^-)^{\top}$, and the minimum value is given as
\begin{equation}
-\frac{1}{2mR}||\bm{\alpha}^{\top}\bm{X}^+ - \bm{\beta}^{\top}\bm{X}^-||^{\top}
\end{equation}
Then, we consider the maximization over $\bm{\alpha}$ and $\bm{\beta}$. By replacing them with $-\bm{\alpha}$ and $-\bm{\beta}$, we can obtain the conclusion of Theorem \ref{dual form}. \qed
\subsection{Proof of Theorem \ref{sol of proj}}
Let $\lambda, \mu, u_i \geq 0, v_i \geq 0, \omega_i \geq 0$ be dual variables, and let $C = \sum_{i=1}^m \alpha_i^0 = \sum_{j=1}^n \beta_j^0$. Then the Lagrangian function of (\ref{proj plm}) can be written as
\begin{eqnarray*}
\mathcal{L} &=& \frac{1}{2}||\bm{\alpha} - \bm{\alpha_0}||^2 + \frac{1}{2}||\bm{\beta} - \bm{\beta_0}||^2 + \lambda(\sum_{i=1}^m \alpha_i^0 - C) + \\
& & \mu(\sum_{j=1}^n \beta_j^0 - C) - \sum_{i=1}^m u_i\alpha_i^0 - \sum_{j=1}^n v_i\beta_j^0 + \sum_{j=1}^n(\beta_j^0 - \frac{C}{k}).
\end{eqnarray*}
The KKT conditions are
\begin{eqnarray}
\frac{\partial \mathcal{L}}{\partial \alpha_i^*} &=& \alpha_i^* - \alpha_i^0 + \lambda - u_i = 0 \label{alpha1}\\
\frac{\partial \mathcal{L}}{\partial \beta_j^*} &=& \beta_j^* - \beta_j^0 + \mu - v_j + \omega_j = 0 \label{eq:beta1} \\
\frac{\partial \mathcal{L}}{\partial C} &=& -\lambda - \mu - \frac{1}{k}\sum_{j=1}^n \omega_j = 0 \label{eq:3var}\\
0 &=& u_i\alpha_i^* \label{eq:alpha2}\\
0 &=& v_j\beta_j^* \label{eq:beta2}\\
0 &=& \omega_j(\beta_j^* - \frac{C}{k})\label{eq:beta3} \\
C &=& \sum_{i=1}^m \alpha_i^* \label{eq:alphaC}\\
C&=& \sum_{j=1}^n \beta_j^*\label{eq:betaC}\\
\bm{0} &\leq& \bm{\alpha}^*, \bm{\beta}^*, \bm{u}, \bm{v}, \bm{\omega}
\end{eqnarray}
Consider $\beta_i^*$. By (\ref{eq:beta1}) and (\ref{eq:beta2}), we know that if $v_j = 0$, then $\beta_j^* = \beta_j^0 - \mu - \omega_j \geq 0$; else if $v_j > 0$, then $0 = \beta_j^* > \beta_j^0 - \mu - \omega_j$. This implies that $\beta_j^* = [\beta_j^0 - \mu - \omega_j]_+$. Following similar analysis we can show that $\alpha_i^* = [\alpha_i^0 - \lambda]_+$.
Further, by (\ref{eq:beta3}), we have that if $\omega_i = 0$, then $\beta_j^* = [\beta_j^0 - \mu]_+ \leq \frac{C}{k}$; else if $\omega_i > 0$, then $\frac{C}{k} = \beta_j^* < [\beta_j^0 - \mu]_+$. Thus $\beta_j^* = \min\{[\beta_j^0 - \mu]_+, \frac{C}{k}\}$. \\\\
Substituting the expression of $\beta_j^*$ and $\alpha_i^*$ into (\ref{eq:alphaC}) and (\ref{eq:betaC}), we have that both constraints (\ref{def C}), (\ref{def mu}) and the closed form solution of $\bm{\alpha}$ and $\bm{\beta}$ holds. Now we verify (\ref{3 var}).
\textbf{Case 1.} First consider $C >0$, we have that if $\omega_j = 0$, then $\frac{C}{k} \geq \beta_j^* = \beta_j^0 - \mu$; else if $\omega > 0$, then $0 < \frac{C}{k} = \beta_j^* = [\beta_j^0 - \mu - \omega_j]_+$ and thus $\beta_j^0 - \mu - \omega_j = \frac{C}{k}$. To sum up we know that $\omega_j = [\beta_j^0 - \mu - \frac{C}{k}]_+$, and thus by (\ref{eq:3var}), (\ref{3 var}) holds.
\textbf{Case 2.} Now Suppose $C = 0$. If this is the case, according to (\ref{eq:alphaC}) and (\ref{eq:betaC}), we have $\bm{\alpha}^* = \bm{\beta}^* = \bm{0}$. By a trivial discussion, above KKT conditions can be reduced to:
\begin{eqnarray}
\lambda &\geq& \alpha_{[1]}^0\\
0 &\geq& \lambda + \mu + \frac{1}{k} \sum_{j=1}^n [\beta_j^0 - \mu]_+ \label{simp KKT}
\end{eqnarray}
Notice that there is no any upper bounded constraint on $\lambda$. Thus, if $\lambda$ and $\mu$ satisfy the simplified KKT condition, by choosing a large enough $\lambda' \geq \lambda$, both (\ref{simp KKT}) and (\ref{3 var}) hold, and optimal solution is still zero vector. This completes the proof of necessity.
At last, notice that KKT condition is the necessary and sufficient conditions in the case of convex problem, and the above transformations are reversible, we complete the proof of sufficiency. \qed
\subsection{Proof of Lemma \ref{c=0}}
First suppose $(\bm{\alpha}^*, \bm{\beta}^*) = (\bm{0}^m, \bm{0}^n)$. Denote corresponding dual variables by $C^*, \lambda^*$ and $\mu^*$. First, we have
\begin{equation}\label{cond of lambda}
\bm{\alpha}^* = \bm{0}^m \Leftrightarrow \lambda^* \geq \alpha_{[1]}^0
\end{equation}
Moreover, $(\bm{\alpha}^*, \bm{\beta}^*) = (\bm{0}^m, \bm{0}^n)$ implies $C^*=0$, and equality (\ref{def mu}) is automatically holds for arbitrary $\mu \in \mathbb{R}$. Thus the unique constraint on $\mu^*$ is (\ref{3 var}), i.e.
\begin{equation}\label{111}
k\lambda^* + k\mu^* + \sum_{j =1}^n [\beta_j^0 - \mu^*]_+ = 0.
\end{equation}
Consider $f(\mu) = k\mu + \sum_{j =1}^n [\beta_j^0 - \mu]_+$. It is easy to verify that $f(\mu)$ is continuous and piecewise-linear on $\mathbb{R}$. Moreover, if $\mu \in B_t$, we can write $f(\mu)$ as
\begin{eqnarray}
f(\mu) = \sum_{i=1}^t \beta_{[i]}^0 + (k - t)\mu
\end{eqnarray}
Thus, $f(\mu)$ is strictly decreasing in $B_{k+1}^{n+1}$, strictly increasing in $B_0^k$, and the minimum is achieved in $B_k$, which is $\sum_{i=1}^k \beta_{[i]}^0$. Combine this conclusion with (\ref{cond of lambda}) and (\ref{111}), we have
\begin{eqnarray*}\label{abc}
k\alpha_{[1]}^0 + \sum_{j=1}^k \beta_{[i]}^0 &=& \min \limits_{\lambda^*, \mu} \{ k\lambda^* + k\mu + \sum_{j =1}^n [\beta_j^0 - \mu]_+\} \\
&\leq& k\lambda^* + k\mu^* + \sum_{j =1}^n [\beta_j^0 - \mu^*]_+ \\
&=& 0.
\end{eqnarray*}
This proves the necessity. \\
On the other hand, if $k\alpha_{[1]}^0 + \sum_{j=1}^k \beta_{[j]}^o \leq 0$, by setting $C = 0, \mu = \beta_{[k]}^0$ and $\lambda = -\frac{1}{k} \sum_{j =1}^k \beta_j^0$, one can check that all the optimal conditions in theorem \ref{sol of proj} can be satisfied, and corresponding $(\bm{\alpha}^*, \bm{\beta}^*) = (\bm{0}, \bm{0})$. This completes the proof.$\qed$
\subsection{Redefine $\mu(C)$ }\label{redifine_muc}
As we mentioned in footnote, we need to redefine function $\mu(C)$ to ensure its one-valued property. We begin by introducing following lemma.
\begin{lemma}\label{sol of mu}
Consider (\ref{def mu}). Denote $\ C_0 = k(\beta_{[k]}^0 - \beta_{[k+1]}^0)$.
\begin{enumerate}
\item If $\ C > C_0$, then there exists a unique $\mu$ satisfying (\ref{def mu}).
\item If $\ 0 < C \leq C_0$, then arbitrary $\mu \in [\beta_{[k+1]}^0, \beta_{[k]}^0 - \frac{C}{k}]$ can satisfy (\ref{def mu}).
\end{enumerate}
\end{lemma}
\begin{proof}
For a fixed $C > 0$, consider
\begin{equation}
g(\mu) = \sum_{j =1}^n \min \{[\beta_j^0 - \mu]_+, \frac{C}{k}\} - C
\end{equation}
Obviously it is a continuous and decreasing function, and the range is $[-C$,$\infty)$. By intermediate value theorem, $g=0$ must have a root. Another fact is that $g$ is piecewise-linear and has at most $2n$ breakpoints: $\{\beta_i^0, \beta_i^0 - \frac{C}{k}\}_{i=1}^n$. Thus if the root of $g$ is not unique for some $C$, all of these roots must be located on a continuous segment with slope 0. Otherwise, if there exists a root only located on segment whose slope is not 0, we can ensure no other root exists.
Let $\mu'$ be a root of $g$, we first show that $\mu' \in (\beta_{[k+1]}^0 - \frac{C}{k}, \beta_{[k]}^0)$. Notice that
\begin{equation*}
\begin{split}
&g(\beta_{[k]}^0) = \sum_{j =k}^{n} 0 + \sum_{j =1}^{k-1} \min \{[\beta_{[j]}^0 - \beta_{[k]}^0]_+, \frac{C}{k}\} - C \leq \sum_{j=1}^{k-1} \frac{C}{k} - C = -\frac{C}{k} < 0,\\
&g(\beta_{[k+1]}^0 - \frac{C}{k}) \geq \sum_{j =1}^{k+1} \min \{[\beta_{[j]}^0 - \beta_{[k+1]}^0 + \frac{C}{k}]_+, \frac{C}{k}\} - C \geq \sum_{j=1}^{k+1} \frac{C}{k} - C = \frac{C}{k} > 0.
\end{split}
\end{equation*}
Thus above claim holds by the intermediate value theorem and the monotonicity of $g$.\\
Suppose $\mu' \in B_i$ and $\mu' + \frac{C}{k} \in B_j$, we have $j \leq k \leq i$, and $g$ can be written as
\begin{equation}\label{aaaa}
0 = g(\mu') = \sum_{t = j+1}^i \beta_{[t]}^0 - (i-j)\mu' - (1-\frac{j}{k})C
\end{equation}
Now we can prove the claims in Lemma \ref{sol of mu}.
\begin{description}
\item[Case 1.] If $C > C_0$, we know that $\beta_{[k+1]}^0 > \beta_{[k]}^0 - \frac{C}{k}$, thus $B_k$ and $B_k - \frac{C}{k}$ are disjoint. This means $i \neq j$ (once $i=j$, we must get $i=k=j$, and thus there exists $\mu'$ belongs to $B_k$ and $B_k - \frac{C}{k}$ simultaneously, that is a contraction), and by (\ref{aaaa}) we know the slope which $\mu'$ lies on is not 0, thus $\mu'$ is the unique root of $g$.
\item[Case 2.]If $C \leq C_0$, we know that $\beta_{[k+1]}^0 \leq \beta_{[k]}^0 - \frac{C}{k} < \beta_{[k]}^0$, and thus for all $\mu' \in [\beta_{[k+1]}^0, \beta_{[k]}^0 - \frac{C}{k}]$, $i=k=j$ and (\ref{aaaa}) is a identities. This means that $\mu' = [\beta_{[k+1]}^0, \beta_{[k]}^0 - \frac{C}{k}]$.
\end{description}
This completes the proof.
\end{proof}
\noindent \textbf{Remark.} Note that in fact in the case of $C > C_0$, we can get a stronger bound on $\mu'$: $\mu' \in (\beta_{[k]}^0 - \frac{C}{k}, \beta_{[k+1]}^0)$ and thus $j < k < i$. This is based on the fact that
\begin{eqnarray*}
g(\beta_{[k+1]}^0) &=& \sum_{j =k+1}^{n} 0 + (\beta_{[k]}^0 - \beta_{[k+1]}^0) + \sum_{j =1}^{k-1} \min \{[\beta_{[j]}^0 - \beta_{[k]}^0]_+, \frac{C}{k}\} - C
\\&<& \frac{C}{k} + \sum_{j=1}^{k-1} \frac{C}{k} - C = 0,\\
g(\beta_{[k]}^0 - \frac{C}{k}) &\geq& \min \{[\beta_{[k+1]}^0 - \beta_{[k]}^0+\frac{C}{k}]_+, \frac{C}{k}\} + \sum_{j =1}^{k} \min \{[\beta_{[j]}^0 - \beta_{[k]}^0 + \frac{C}{k}]_+, \frac{C}{k}\} - C \\
&>& 0 + \sum_{j=1}^{k} \frac{C}{k} - C = 0. \qed
\end{eqnarray*}
Based on Lemma \ref{sol of mu}, we can redefine $\mu(C)$ as follow.
\begin{eqnarray}
\mu(C)&=&\left\{\begin{array}{ll}
\beta_{[k+1]}^0 & C \in (0, C_0)\\\\
\mu \ satisfies\ (\ref{def mu}) & C \in [C_0, +\infty]
\end{array}\right. \label{fmu}
\end{eqnarray}
This function is one-valued, and both of our discussions below are based on this new formulation.
\subsection{Proof of Lemma \ref{bound}}
When $C^* > 0$, it is obvious that $\lambda^* < \alpha_{[1]}^0$. Now we consider the claim of lower bound.
On the one hand, if $\mu^* + \frac{C^*}{k} > \beta_{[1]}^0$, we have that for all $j \leq n$, $\beta_j^0 \leq \beta_{[1]}^0 < \mu^* + \frac{C^*}{k}$. Thus
\begin{eqnarray*}
&&0 = f(C^*) = k\lambda^* +k\mu^* + \sum_{j=1}^n[\beta_j^0 - \mu^* - \frac{C^*}{k}]_+ = k(\lambda^* +\mu^*)
\end{eqnarray*}
and then
\begin{eqnarray*}
0 < C^* = \sum_{j=1}^n \min\{[\beta_j^0 - \mu^*]_+, \frac{C^*}{k}\} = \sum_{j=1}^n [\beta_j^0 + \lambda^*]_+
\end{eqnarray*}
The last equality implies that $\lambda^* > -\beta_{[1]}^0$.
On the other hand, consider the situation of $\mu^* + \frac{C^*}{k} \leq \beta_{[1]}^0$. According to the proof of Lemma \ref{sol of mu}, we know that $\mu^* + \frac{C^*}{k} > \beta_{[k+1]}^0$, thus
\begin{eqnarray*}
0 = f(C^*) &=& k\lambda^* +k\mu^* + \sum_{j=1}^n[\beta_j^0 - \mu^* - \frac{C^*}{k}]_+\\
&=& k\lambda^* +k\mu^* + \sum_{j=1}^k[\beta_j^0 - \mu^* - \frac{C^*}{k}]_+\\
&\leq& k\lambda^* +k\mu^* + k[\beta_{[1]}^0 - \mu^* - \frac{C^*}{k}]_+\\
&=& k\lambda^* +k\mu^* + k(\beta_{[1]}^0 - \mu^* - \frac{C^*}{k})\\
&=& k\lambda^* + k\beta_{[1]}^0 - C^*\\
&<& k\lambda^* + k\beta_{[1]}^0
\end{eqnarray*}
This also means $\lambda^* > -\beta_{[1]}^0$. \qed
\subsection{Proof of Lemma \ref{inc-dec}}
Our proof is based on a detailed analysis of each function's sub-gradient. The correctness of claim \ref{c1} is verified in \citep{liu2009efficient}, so we only focus on claim \ref{c2}, \ref{c3}, \ref{c4}. Due to space limitation, we only detail the key points.
Consider $\mu(C)$. First, according to Lemma \ref{sol of mu}, we know that $\mu(C)$ is well defined in $[0, \infty)$. Now let us claim that $\mu(C)$ is continuous. It is not difficult to check that $\mu(C)$ is piecewise-linear, and continuous at both of these breakpoints. Thus, $\mu(C)$ is continuous. In order to verify that $\mu(C)$ is decreasing with $C$, we only need to show that the slope of \emph{any} segment of $\mu$ is less than 0.
Like the proof of Lemma \ref{sol of mu}, suppose that $\mu(C) \in B_i$ and $\mu(C) + \frac{C}{k} \in B_j$, we have $j \leq k \leq i$, and obtain a linear relation between $C$ and $\mu$:
\begin{eqnarray*}\label{tttt}
(i-j)\mu = \sum_{t = j+1}^i \beta_{[t]}^0 - (1-\frac{j}{k})C
\end{eqnarray*}
Thus, if $C > C_0$, according to the remark of Lemma \ref{sol of mu} we know that $i > k > j$. and the slope of $\mu$ is $-\frac{k-j}{k(i-j)} < 0$. Else in the case of $C \leq C_0$, in terms of the definition of $\mu$ we know corresponding slope is 0. In conclusion, we can ensure that $\mu(C)$ is strictly decreasing in $[C_0, +\infty)$, and decreasing in $(0, +\infty)$.
Similar analysis shows that $\delta(C)$ is also piecewise-linear, has at most $O(n)$ breakpoints, and the slope of each segment is $\frac{i - k}{k(i-j)}(i \neq j)$, which is strictly large than 0. In the case of $C \leq C_0$, the slope is $\frac{1}{k} > 0$, leads to the conclusion of $\delta$ is strictly increasing in $(0, +\infty)$.
At last, consider $f(\lambda)$. Because both $\mu$ and $-\tau$ are decreasing with $\lambda$, it is obviously to see that $f$ is strictly decreasing in $(-\infty, +\infty)$.
Now we prove the convexity of $f(C)$. We prove that both $\lambda(C)$ and $T(C) = k\mu + \sum_{j=1}^n [\beta_j^0 - \delta]_+$ are convex function. The convexity of $\lambda$ is guaranteed in \citep{liu2009efficient}. Now we only discuss the convexity of $T(C)$. If $C \geq C_0$, reuse the definition of $i$ and $j$ above, and define $x = i - k >0$, $y = k - j > 0$, one can verify that, the sub-gradient of $f(C)$ is
\begin{eqnarray*}
f' = \frac{xy}{k(x+y)} - 1 = \frac{1}{\frac{k}{x} + \frac{k}{y}} - 1 > -1
\end{eqnarray*}
Following the conclusions that $\mu$ is decreasing with $C$, and $\delta = \mu + \frac{C}{k}$ is increasing with $C$, we know that both $x$ and $y$ is increasing with $C$. Thus $f'$ is increasing with $C$, which means that $f$ is convex in $[C_0, +\infty)$.
On the other hand, if $C \leq C_0$, one can easily check that $T(C) = -C$. Thus this moment $f' = -1$, which is larger than the sub-gradient of $f$ in the case of $C > C_0$. Thus $f'$ is increasing in $(0, +\infty)$, and $f$ is convex in $(0, +\infty)$. \qed
\subsection{Proof of Theorem \ref{bound p}}
For the convenience of analysis, we consider the constrained version of the optimization problem (\ref{learning problem}), that is
\begin{equation}
\min \limits_{w \in W} L_{\bar{k}} = \frac{1}{m} \sum_{i=1}^m l(\bm{w}^{\top}\bm{x_i^+} - \frac{1}{k}\sum_{j=1}^k\bm{w}^{\top}\bm{x_{[j]}^-})
\end{equation}
where $W=\{\bm{w} \in \mathbb{R}^d \mid ||\bm{w}|| \leq \rho\}$ is a domain and $\rho > 0$ specifies the size of the domain that plays similar role as the regularization parameter $R$.\\
First, we denote $G$ as the Lipschitz constant of the truncated quadratic loss $l(z)$ on the domain $[-2\rho, 2\rho]$, and define the following two functions based on $l(z)$, i.e.
\begin{eqnarray}
h_l(\bm{x}, \bm{w}) &=& \mathbb{E}_{\bm{x}- \thicksim P-}[l(\bm{w}^{\top}\bm{x} - \bm{w}^{\top}\bm{x}^-)],\\
P_l(\bm{w}, \tau) &=& \mathbb{P}_{\bm{x}^+ \thicksim P^+}(h_l(\bm{x}_i^+ \bm{w}) \geq \tau)
\end{eqnarray}
The lemma below relates the empirical counterpart of $P_l$ with the loss $L_{\bar{k}}$.
\begin{lemma}\label{god lemma}
With a probability at least $1-e^{-s}$, for any $\bm{w} \in W$, we have
\begin{equation}
\frac{1}{m}\sum_{i=1}^m \mathbb{I} (h_l(\bm{x}_i^+,\bm{w}) \geq \delta) \leq L_{\bar{k}},\\
\end{equation}
where
\begin{eqnarray}\label{delta}
\nonumber \delta = \frac{4G(\rho +1)}{\sqrt{n}} + \frac{5\rho(s+\log(m))}{3n} + 2G\rho\sqrt{\frac{2(s + \log(m))}{n}} + \frac{2G\rho(k-1)}{n}.
\end{eqnarray}
\end{lemma}
\begin{proof}
For any $\bm{w} \in W$, we define two instance sets by splitting $\mathcal{S}_+$, that is
\begin{eqnarray*}
A(\bm{w}) &=& \{\bm{x}_i^+ \mid \bm{w}^{\top}\bm{x}_i^+ > \frac{1}{k} \sum_{i=1}^k \bm{w}^{\top}\bm{x}_{[i]}^- + 1\}\\
B(\bm{w}) &=& \{\bm{x}_i^+ \mid \bm{w}^{\top}\bm{x}_i^+ \leq \frac{1}{k} \sum_{i=1}^k \bm{w}^{\top}\bm{x}_{[i]}^- + 1\}
\end{eqnarray*}
For $\bm{x}_i^+ \in A(W)$, we define
\begin{equation*}
||P -P_n||_W = \mathop{sup} \limits_{||\bm{w}||\leq\rho}|h_l(\bm{x}_i^+, \bm{w}) - \frac{1}{n}\sum_{j=1}^n l(\bm{w}^{\top}\bm{x}_i^+ - \bm{w}^{\top}\bm{x}_j^-)|
\end{equation*}
Using the Talagrand's inequality and in particular its variant (specifically, Bousquet bound) with improved constants derived in \citep{bousquet2002bennett}, we have, with probability at least $1-e^{-s}$,
\begin{eqnarray}\label{god ineq1}
\nonumber ||P -P_n||_W \leq \mathbb{E}||P-P_n||_W + \frac{2s\rho}{3n} + \sqrt{\frac{2s}{n}(2\mathbb{E}||P -P_n||_W + \sigma_P^2(W))}.
\end{eqnarray}
We now bound each item on the right hand side of (\ref{god ineq1}). First, we bound $\mathbb{E}||P -P_n||_W$ as
\begin{eqnarray*}\label{god ineq2}
\mathbb{E}||P -P_n||_W &=& \frac{2}{n}\mathbb{E}[\mathop{sup} \limits_{||\bm{w}||\leq \rho}\sum_{j=1}^n\sigma_j l(\bm{w}^{\top}(\bm{x}_i^+ - \bm{x}_j^-))]\\
&\leq& \frac{4G}{n}\mathbb{E}[\mathop{sup} \limits_{||\bm{w}||\leq \rho}\sum_{j=1}^n\sigma_j (\bm{w}^{\top}(\bm{x}_i^+ - \bm{x}_j^-))] \\
&\leq& \frac{4G\rho}{\sqrt{n}}
\end{eqnarray*}
where $\sigma_j$'s are Rademacher random variables, the first inequality utilizes the contraction property of Rademacher complexity, and the last follows from Cauchy-Schwarz inequality and Jensen's inequality. Next, we bound $\sigma_P^2(W)$, that is,
\begin{equation*}\label{god ineq3}
\sigma_P^2(W) = \mathop{sup} \limits_{||\bm{w}||\leq \rho} h_l^2(\bm{x}, \bm{w}) \leq 4G^2\rho^2
\end{equation*}
By putting these bounds into (\ref{god ineq1}), we have
\begin{eqnarray*}
||P -P_n||_W &\leq& \frac{4G\rho}{\sqrt{n}} + \frac{2s\rho}{3n} + \sqrt{\frac{2s}{n}(4G^2\rho^2 + \frac{8G\rho}{\sqrt{n}})}\\
&\leq& \frac{4G(\rho + 1)}{\sqrt{n}} + \frac{5s\rho}{3n} + 2G\rho\sqrt{\frac{2s}{n}}
\end{eqnarray*}
Notice that for any $x_i^+ \in A(W)$, there are at most $k-1$ negative instances have higher score than it, we thus have
\begin{equation*}
\sum_{j=1}^n l(\bm{w}^{\top}\bm{x}_i^+ - \bm{w}^{\top}\bm{x}_j^-) \leq 2G\rho(k-1)
\end{equation*}
Consequently, by the definition of $||P -P_n||_W$ we have, with probability $1 - e^{-s}$,
\begin{eqnarray*}
|h_l(\bm{x}_i^+, \bm{w})| \leq ||P -P_n||_W + \frac{1}{n}\sum_{j=1}^n l(\bm{w}^{\top}\bm{x}_i^+ - \bm{w}^{\top}\bm{x}_j^-)\\
\leq \frac{4G(\rho + 1)}{\sqrt{n}} + \frac{5s\rho}{3n} + 2G\rho\sqrt{\frac{2s}{n}} + 2G\rho\frac{k-1}{n}
\end{eqnarray*}
Using the union bound over all $\bm{x}_i^+$'s, we thus have, with probability $1-e^{-s}$,
\begin{equation}
\sum_{\bm{x}_i^+ \in A(\bm{w})} \mathbb{I}(h_l(\bm{x}_i^+, \bm{w}) \geq \delta) = 0
\end{equation}
where $\delta$ is in (\ref{delta}). Hence, we can complete the proof by $|B(\bm{w})| \leq mL_{\bar{k}}$.
\end{proof}
Based on Lemma \ref{god lemma}, we are at the position to prove Theorem \ref{bound p}. Let $S(W, \epsilon)$ be a proper $\epsilon$-net of $W$ and $N(\rho, \epsilon)$ be the corresponding covering number. According to standard result, we have
\begin{equation}
\log N(\rho, \epsilon) \leq d\log(\frac{9\rho}{\epsilon}).
\end{equation}
By using concentration inequality and union bound over $\bm{w}' \in S(W, \epsilon)$, we have, with probability at least $1-e^{-s}$,
\begin{eqnarray}\label{god god ineq}
\nonumber \mathop{sup} \limits_{\bm{w}' \in S(W, \epsilon)} P_l(\bm{w}', \delta) - \frac{1}{m}\sum_{i=1}^m\mathbb{I}(h_l(\bm{x}_i^+, \bm{w}') \geq \delta)
\leq \sqrt{\frac{2(s+d\log(9\rho/\epsilon))}{m}}
\end{eqnarray}
Let $\bm{d} = \bm{x}^+ - \bm{x}^-$ and $\epsilon = \frac{1}{2\sqrt{m}}$. For $\bm{w}^* \in W$, there exists $\bm{w}' \in S(W, \epsilon)$ such that $||\bm{w}' - \bm{w}^*|| \leq \epsilon$, it holds that
\begin{eqnarray*}
\mathbb{I}(\bm{w}^{*\top}\bm{d} \leq 0) = \mathbb{I}(\bm{w'}^{\top}\bm{d} \leq (\bm{w}' - \bm{w}^*)^{\top}\bm{d})
\leq \mathbb{I}(\bm{w}'^{\top}\bm{d} \leq \frac{1}{\sqrt{m}}) \leq 2l(\bm{w}'^{\top}\bm{d}).
\end{eqnarray*}
where the last step is based on the fact that $l(.)$ is decreasing and $l(1/\sqrt{m}) \geq 1/2$ if $m \geq 12$.
We thus have $h_b(\bm{x}^+, \bm{w}^*) \leq 2h_l(\bm{x}^+, \bm{w}')$ and therefore $P_b(\bm{x}^*, \delta) \leq P_l(\bm{x}', \delta/2)$.\\
As a consequence, from (\ref{god god ineq}), Lemma \ref{god lemma} and the fact
\begin{eqnarray*}
L_{\bar{k}}(\bm{w}') \leq L_{\bar{k}}(\bm{w}^*) + \frac{G\rho}{\sqrt{m}}
\end{eqnarray*}
We have, with probability at least $1-2e^{-s}$
\begin{eqnarray*}
P_b(\bm{w}^*, \delta) &\leq& L_{\bar{k}}(\bm{w}^*) + \frac{G\rho}{\sqrt{m}} + \sqrt{\frac{2s+2d\log(9\rho)+d\log m}{m}},
\end{eqnarray*}
where $\delta$ is as defined in (\ref{delta}), and the conclusion follows by hiding constants.\qed
\section{Conclusion}\label{sec:conclusion}
In this paper, we focus on learning binary classifier under the specified tolerance $\tau$. To this end, we have proposed a novel ranking method which directly optimizes the probability of ranking positive samples above $1-\tau$ percent of negative samples. The ranking optimization is then efficiently solved using projected gradient method with the proposed linear time projection. Moreover, an out-of-bootstrap thresholding is applied to transform the learned ranking model into a classifier with a low false-positive rate. We demonstrate the superiority of our method using both theoretical analysis and extensive experiments on several benchmark datasets.
\acks{This research was mainly done when the first author was an intern at iDST of Alibaba. This work is supported by the National Natural Science Foundation of China (NSFC) (61702186, 61672236, 61602176, 61672231), the NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Information (U1609220), the Key Program of Shanghai Science and Technology Commission (15JC1401700) and the Joint Research Grant Proposal for Overseas Chinese Scholars (61628203). }
\vskip 0.2in
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,731 |
#include "nameNormalizerConverter.h"
#include <qrutils/nameNormalizer.h>
using namespace generatorBase::converters;
using namespace qReal;
QString NameNormalizerConverter::convert(const QString &data) const
{
return utils::NameNormalizer::normalizeStrongly(data, false);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,580 |
\section{Introduction}
\label{sec:intro}
Extreme events arise in many situations of practical interest \cite{albeverio2006extreme}. The most common ones are the extremes witnessed in geophysical phenomenon such as the earthquakes, cyclones and drought. These represent extreme events recorded in a univariate time series. However, networks have emerged as a standard modeling paradigm to describe such complex systems \cite{vespignani2012modelling,barabasi2012network}. For instance, traffic jams, large scale power blackouts, call drops in cellular phone networks, overloaded web servers are a result of large fluctuations beyond the capacity of the system to service the requests. These events take place, respectively, on a
network of roads, power distribution network, cellular network and on a network of routers.
In general, any large excursion of a dynamical variable from its typical behaviour can be termed
an extreme event. Though such events typically have low probability
of occurrence, they are regarded as significant due to the social and financial losses suffered on
account of most of these extreme events.
While the classical extreme value theory \cite{gumbel1958statistics}, formulated a century ago, deals with extreme events primarily
in univariate time series, the study of extreme events on complex networks has attracted attention
only since the last decade \cite{chen2013discovery,chen2014controlling,mahecha2017detecting}. In the latter case, a typical setting is that of a topology of a complex network, a system of nodes connected through edges \cite{albert2002statistical}. In order to discuss events and extreme events, a dynamical process needs to be additionally defined on the networks. If we then let $x(t)$ represent the flux of any dynamical variable on a node of a network, then an event is deemed to be an extreme event if $x(t) > x_{\text{th}}$, where $x_{\text{th}}$ is a suitable threshold that indicates, for instance,
the processing capacity of a node. An early work along this direction used non-interacting random walks as a dynamical
process \cite{kishore2011extreme}. By assuming $x_{\text{th}}$ to be proportional to the typical random walker flux on a node,
it was demonstrated that the probability that an extreme event occurs on that node depends on the
degree of the node in question. In particular, quite contrary to intuition, it was observed that,
on an average, hubs have low probability for occurrence of extreme events, while small degree nodes
have higher likelihood for extreme events \cite{kishore2011extreme}. Over the years, this proposition was observed in
many different settings.
For instance, this feature been theoretically reported in a network of coupled
chaotic maps \cite{Moitra_sinha_2019}, in stochastic processes such as Brownian motion in
an external potential \cite{Amritkar_Ma_Hu_2018}, and in biased random walks on networks \cite{kishore2012extreme}.
However, it appears that in the case of interactions among agents, the extreme events created
by the agents can possibly have a different profile for the probability for the occurrence of
extreme events \cite{chen2015extreme, gupta2021extreme}. The formulation introduced in Ref. \cite{kishore2011extreme}
has been used to study network failures as well \cite{mizutaka2013structural, mizutaka2014network}.
In general, the focus of these works has been on extreme events, specifically on the nodes of a network.
A natural extension of these works is to consider extreme events occurring at the interface where nodes interact - the edges of the network. Recently, using random walks as flows on a network, extreme events
occurring on edges and their properties were studied \cite{kumar2020extreme}.
Let $i$ and $j$ be two nodes connected by an edge $e_{ij}$. It was shown that
extreme events on nodes $i$ and $j$, and those on the edge $e_{ij}$ connecting the nodes
are nearly uncorrelated and that they should be regarded as events which do not influence one another \cite{kumar2020extreme}.
Hence results for the occurrence of extreme events on nodes do not characterise those taking place on the
edges. Indeed, most of our personal experience in the case of road traffic tells us that traffic jams on
junctions and at any other point on the roads are not always correlated. Furthermore, with unbiased walkers occurrence probability of extreme events on edges depends only on the gross network parameters such as
total number of edges and not on the local details in the vicinity of the edge. This feature differs appreciably from what we know about extreme events on nodes. Motivated by these considerations we study extreme events on the edges using the more broad dynamics of {\it biased} random walkers on complex networks. The standard random walk provides
a simple and yet non-trivial setting to explore this question. In the rest of this paper, a general formalism
is developed to study biased random walks on networks, and it is applied to the degree biased
random walks on scale-free networks, and also to other centrality measure based biases such as
eigenvector centrality, betweenness centrality and page-rank centrality. It is also applied to a
real-life planar street network. In all the cases studied here, extreme events arise from inherent fluctuations in the number of walkers, and provides insights into how inherent fluctuations, as opposed to external shocks, give rise to extreme events.
We describe the formalism in sections \ref{BRW_intro}, \ref{BRW_nodes}, \ref{BRW_edges}. In section \ref{DBRW} we apply it to a planar network and subsequently to several different types of biased random walkers. Finally, in section \ref{stability} we discuss the robustness of edges against occurrence of extreme events.
\section{Biased Random Walks on networks} \label{BRW_intro}
A connected, undirected network with $N$ nodes and $E$ edges is considered. Its connectivity
structure is described by an adjacency matrix ${\mathbf A}$ whose elements
$a_{ij}$ are either $1$ or $0$ depending on whether an edge connects nodes $i$ and $j$ or not.
On this network structure, diffusion of $W$ independent random
walkers is considered whose dynamics is biased by any network property that can be associated with the node, such as the degree. This is represented by a single-step transition probability $\pi_{ij}$ for a walker to go from node $i$ to node $j$, while satisfying the normalization condition $\sum_j \pi_{ij}=1$ for all $i$. In this work, we study a
class of biased random walks defined by the one-step transition probability
\begin{equation}
\pi_{ij}=\frac{a_{ij}f_{j}}{\underset{m}{\sum}a_{im}f_{m}}, \;\;\;\;i,j=1,2, \dots, N.
\label{transition prob matrix}.
\end{equation}
The biasing function $f_{j}$ here, could represent any structural or non-structural property of the network as long as it is time-independent and is associated with a node. The structural properties could simply be the degree of a node, the betweenness centrality, it could be any amalgam of structural features of the node and its local neighbourhood, it could be the ground-truth community the node is externally assigned, and so much more. This can be generalized to the case of multiplex
networks as well \cite{battiston2016efficient}. In the case of unbiased random walk $f_{j}=1$, for all $j$, indicating that the probability for a walker to jump to any of its neighbouring nodes is a constant.
Let $P_{ij}(t)$ denote the probability that a walker at node $i$ starting at time $t=0$
reaches node $j$ after $t$ discrete jumps. The time evolution of probability is
given by the master equation
\begin{equation}
P_{ij}(t)=\sum_{k=1}^{N} P_{ik}(t-1) ~ \pi_{kj}.
\end{equation}
By allowing only time independent biasing functions we are assured transition probabilities $\pi_{ij}$ that are stationary and that any node in the network is accessible from any other node via a finite number of steps. It then follows from Perron-Frobenius theorem \cite{ninio1976simple} that for any such biasing function $f_{j}$, the stationary probability distribution $p_{i}$ exists. This time-independent occupation probability on node $i$ turns out to be \cite{noh2004random}:
\begin{equation}
p_{i}=\frac{f_{i} \sum_j a_{ij} f_{j} }{\underset{k}{\sum} f_{k} \left( \underset{j}{\sum} a_{kj}f_{j} \right) },
\label{occprob1}
\end{equation}
in which the normalization condition $\sum_{j}p_{j}=1$ holds. Note that the occupation probability $p_i$
does not restrict the form of the biasing function or the network structure at this stage, thereby making it sufficiently general.
\section{Extreme Events on nodes} \label{BRW_nodes}
The occupation probability in Eq. \ref{occprob1} can be used to
evaluate extreme event probabilities on nodes. The probability of
finding $w>1$ non-interacting walkers on the node labelled $i$ is $p_{i}^w$ while the rest of the $W-w$ walkers are distributed among the other nodes of the network. The distribution of number of walkers on node $i$ is then a binomial distribution given by \cite{kishore2011extreme}
\begin{equation}
\mathcal{P}_i(w)=\binom{W}{w} ~ p_{i}^w ~ \left(1-p_{i} \right)^{W-w}.
\label{TraffDistNode}
\end{equation}
Intuitively, an extreme event represents a pronounced deviation from the mean flux.
The flux $w$ represents events in our framework and it is deemed to be an extreme
event if $w \geq q_i$, where $q_i$ represents a suitable threshold.
Then, using Eq. \ref{TraffDistNode}, the EE probability on $i$-th node can be
defined as
\begin{equation}
\mathcal{P}_{EE}(q_{i}) = \sum_{w=q_{i}}^{W} \mathcal{P}_i(w)\label{PEEN}.
\end{equation}
The threshold $q_{i}$ is chosen to be proportional to the natural variability
of the flux passing through the node. As was done earlier in Ref. \cite{kishore2012extreme}, in this work
too the threshold for identifying EE is taken to be
\begin{equation}
q_{i}=\langle w_{i}\rangle + m \sigma_{i},
\label{PEEN threshold}
\end{equation}
where $m \geq 0$. The average and the variance of the flux passing through $i$-th node are given as
\begin{subequations}
\begin{align}
\langle w_{i}\rangle & =Wp_{i}, \label{TraffMean} \\
\sigma_{i}^{2} & = W p_{i} \left(1-p_{i} \right). \label{TraffVar}
\end{align}
\end{subequations}
Since these quantities depend only on the biasing function and the local structural information about immediate neighbours of nodes, these results will hold good
for any network irrespective of its degree distribution.
This notion of EE is sufficiently general to encompass systems that accommodate congestion, {\it e.g.}, airplane/vehicular traffic in a network of airports/streets.
\section{Extreme Events on Edges} \label{BRW_edges}
\label{EEE}
While extreme events on nodes have been studied earlier, there has not been much work about EE on the edges of a network. As was recently shown in Ref. \cite{kumar2020extreme}, the occurrence of EE on nodes and
its edges are nearly uncorrelated. Hence, it is not possible to infer about EE on edges/nodes based on the
information about EE on nodes/edges. Physically, this is intuitive because once an EE takes place on a node, the walkers get scattered into a large number of edges and might not necessarily lead to an EE on the edges. These events on the edges will depend on the flux through edges and the carrying capacity
of the edges. For instance, in the water supply network, the reservoirs might have ample capacity to hold the water, though its transport channels might have a smaller \textit{load}-bearing capacity. A microprocessor has a network of logic gates, where the information is carried as electrical signals. In all these cases, the conduits (edges) can only support a finite range of fluctuations \cite{barabasi2004fluctuations}, beyond which it should be designated as an extreme event. In most cases, the occurrence of EE on the edges can lead to disasters. Therefore it is imperative that EE on edges be studied on its own right.
\subsection{Load and Flux Distribution on edges}
Firstly, we seek quantifiable definitions of the situations described above, {\it i.e.}, for load and flux.
Consider an edge $e_{ij}$ and the nodes it connects labelled $i$ and $j$. Suppose there are $w_{i}$ walkers
on node $i$ and $w_{j}$ walkers at node $j$. At the $t$-th time step, let $l_{i}$ out of the $w_{i}$ walkers jump into node $j$, and $l_{j}$ out of the $w_{j}$ walkers jump to node $j$. The load $l_{ij}$ and flux $f_{ij}$ of
the edge $e_{ij}$ at time $t$ is then defined as
\begin{equation}
l_{ij}(t)=l_{i}(t)+l_{j}(t) \,\,\,\,\,\,\,\,\,\,\,\,\, f_{ij}(t)=l_{i}(t)-l_{j}(t)\label{fload}
\end{equation}
Since the load $l_{ij}$ is stochastic, a natural question of interest is its
probability distribution for a particular edge $e_{ij}$. The details of obtaining
this probability distribution of load for a given edge $e_{ij}$ is given in the
appendix \ref{AppB}. The normalized probability distribution comes out to be
\begin{equation}
P_L(l_{ij})=\binom{W}{l_{ij}} ~ (2p_{i} \pi_{ij})^{l_{ij}} ~ (1-2p_{i} \pi_{ij})^{W-l_{ij}}.
\label{prdistload}
\end{equation}
Comparing this with Eq. \ref{TraffDistNode}, an effective occupation probability on edge $e_{ij}$
can be identified as $p_{ij}^{\text{eff}} = 2 p_{i} \pi_{ij}$. Physically, this corresponds to the product of the probability of a walker to be on node $i$ and probability for that walker to move into node $j$. Therefore, the distribution of load on an edge is dependent on the
product of the biasing functions of the two nodes involved in the
edge. It is dependent on the local network structure and not on the global network parameters.
The mean load and the variance, respectively, on the edge $e_{ij}$ are
\begin{subequations}
\begin{align}
\langle l_{ij}\rangle & = 2Wp_{i} \pi_{ij} = W p_{ij}^{\text{eff}} = \frac{2Wa_{ij}f_{i}f_{j}}{{\sum_{l}}({\sum_{k}}(a_{lk}f_{k})f_{l}},
\label{loadmean}\\
\sigma_{ij}^{2} & = W p_{ij}^{\text{eff}} ( 1 - p_{ij}^{\text{eff}} )
\label{loadstdev}
\end{align}
\end{subequations}
In the limit that $ p_{ij}^{\text{eff}} \ll 1$, then it is easy to see that
$\sigma_{ij}^2 \approx \sqrt{\langle l_{ij} \rangle}$, a result known in the case
of unbiased random walks on networks.
For the probability that a node $i$ has $l_i$ walkers, we make a similar calculation done in Appendix \ref{AppB} to get
\begin{equation}
\mathcal{L}(l_{i};\,i)=\binom{W}{l_{i}} ~ (p_{i} \pi_{ij})^{l_{i}} ~ (1-p_{i} \pi_{ij})^{W-l_{i}}
\label{eq:li}
\end{equation}
The flux distribution {\it i.e.}, the probability that the flux equals $f_{ij}$ on an edge $e_{ij}$ is then product of probabilities of getting different values $l_i$ and $l_j$ constrained by Eq. \ref{fload}. We sum over all such possibilities to get the flux distribution on an edge to be
\begin{equation}
P_F(f_{ij})=\sum_{m=0}^{W-f_{ij}} \mathcal{L}(l_{i}=m+f_{ij};\,i) ~ \mathcal{L}(l_{j}=m;\,j).
\label{flux_dist}
\end{equation}
Using the load (Eq. \ref{prdistload}) and flux distributions (Eq. \ref{flux_dist}), we can now compute the extreme events on edges for a variety of networks.
\subsection{Probability of EE on edges for load}
Similar to the definition of EE on nodes, if the load on an edge at any time $t$ exceeds a defined threshold $q_{ij}$ such that $l_{ij}(t) \geq q_{ij}$, then it indicates the occurrence of load EE on edge $e_{ij}$.
As in the case of nodal extreme events, $q_{ij}$ is taken to be
proportional to the natural variability of the load on the edge. Thus,
\begin{equation}
q_{ij}=\langle l_{ij}\rangle+m\sigma_{ij}\label{thresh-1},
\end{equation}
where $m\geq0$, and the mean load $\langle l_{ij}\rangle$ and standard
deviation $\sigma_{ij}$ are given by Eqs. \ref{loadmean} and \ref{loadstdev} respectively which inform us that $q_{ij}$ is a function of $p_{ij}^{\text{eff}}$. Hence, the probability for load EE on an edge $e_{ij}$ also depends on $p_{ij}^{\text{eff}}$. Then by using Eq. \ref{prdistload} and Eq. \ref{loadmean} we obtain for probability of
load EE on an edge $e_{ij}$
\begin{equation}
Q_L \left( \langle l_{ij} \rangle \right) = \sum_{l=q_{ij}}^{W} P_L(l).
\label{eenode-1}
\end{equation}
The probability for the occurrence of flux EE, $Q_F \left( \langle f_{ij} \rangle \right)$,
on an edge is quite similar to Eq. \ref{eenode-1}, with $P_L(l)$ replaced by flux
distribution $P_F(f)$ from Eq. \ref{flux_dist}. In general, the EE probabilities on
edge $e_{ij}$ depend only on the mean load or flux, which in turn carries information
about local network structure and strength of bias in random walk dynamics.
Furthermore, $\langle l_{ij}\rangle$ is characteristic of an edge irrespective of the
network topology, and allows us to compare EE probabilities for different biases
and even between networks on a common footing. This accessible quantity is a candidate for an observable when studying real life systems.
We are currently in a position to take our pick of biased random walk dynamics and a network structure and obtain information on the EE probabilities on edges and nodes for congestion-type systems. In the rest of the paper, this framework is tested and compared against simulation results for different random walk dynamics and network topologies.
\section{Biased Random Walks}
\label{DBRW}
\begin{figure}
\includegraphics{Figures_Latest/IISERP_osmnx.png}
\caption{The 1.5 km\protect\textsuperscript{2} planar street network around IISER Pune
with 1010 nodes and 1306 edges. The nodes in the two-dimensional plane denote street intersections and the edges represent streets.}
\label{IISERP}
\end{figure}
\subsection{Degree Bias}
In this section, we shall consider one of the most applicable biases - the degree biased random walk \cite{fronczak2009biased, wang2006traffic, zlatic2010topologically}. Its biasing function is of the form
\begin{equation}
f_{i}=k_{i}^{\alpha},
\label{eq:degbias}
\end{equation}
where $k_i=\sum_j a_{ij}$ is the degree of $i$-th node. The exponent
$\alpha$ is the parameter that determines the strength of bias imparted to the walkers.
Unbiased random walk corresponds to $\alpha=0$, whereas for $\alpha > 0$ the walkers
are biased towards hubs (nodes with high degrees), and for $\alpha < 0$ they are preferentially biased towards small degree nodes. $|\alpha|$ determines the strength of the bias towards or away from the hubs. On networks, biased random walks of this type had been studied earlier to enumerate the characteristic time scales, namely, the recurrence time
to the starting point, cover time, and time taken to visit a node for the first time \cite{goldhirsch1987biased,lee2014estimating}. The
parameter $\alpha$ can be tuned so as to minimize these time scales. In general, this results
in smaller time scales with biased random walks in comparison to the unbiased random walks \cite{bonaventura2014characteristic}. The local navigation rules for the biased walks are determined by the transition probability (see Eq. \ref{transition prob matrix})
\begin{equation}
\pi_{ij}=\frac{a_{ij} ~ k_{j}^{\alpha}}{\sum_{m}a_{im} ~ k_{m}^{\alpha}}.
\label{eq:dbrw trans}
\end{equation}
The stationary occupation probability follows from Eq. \ref{occprob1}, and is
\begin{equation}
p_{i}=\frac{k_{i}^{\alpha} ~ \underset{m}{\sum}a_{im}k_{m}^{\alpha}}{\underset{l}{\sum} ~ [k_{l}^{\alpha}(\underset{m}{\sum}a_{lm}k_{m}^{\alpha})]}.
\label{DBRW pstat}
\end{equation}
The index in each of the summations runs over all nodes.
The results are first obtained for a (planar) street network \cite{boeing2017osmnx} shown in Fig. \ref{IISERP}. The planarity of the network was confirmed using Kuratowski's theorem \cite{kuratowski1930probleme}.
Fig. \ref{F+L} displays the probabiity distribution for load and flux obtained from
biased random walk simulations for an arbitrary edge $e_{0,11}$ connecting nodes labelled 0 and 11.
For the purposes of this simulation, $W=2500$ non-interacting walkers executed a
degree biased random walk (DBRW) on the network in Fig.\ref{IISERP} for 100,000 time steps.
The results shown in Fig. \ref{F+L} represent an average over 15 realisations with a fixed
network, but with varying initial positions of the walker for each realisation.
Evidently, the simulation results are in good agreement with the analytical results in Eqs. \ref{eq:li} and \ref{flux_dist}. These results differ substantially from the corresponding distributions for
unbiased random walks, in which case {\it every} edge on the network has the identical probability
distribution for load and flux independent of network topology \cite{kumar2020extreme}.
In the present case of biased random walks, the distribution of flux or load depends on nodal
properties. Futher, as the load distributions reveal, for $\alpha >0$ since walkers preferentially
move towards hubs, edges with smaller mean loads do not attract walkers. On the other hand, if
strongly biased towards small degree nodes, it is possible for walkers to avoid edges which
connect fairly high degree nodes. Hence, the higher probability of null mean loads as seen in Fig. \ref{F+L} for $\alpha <0$. Thus, by
tuning the bias, an edge can be made to either attract or repel walkers.
The results for $\alpha=0$ (unbiased random walk) agree exactly with the results of Ref. \cite{kumar2020extreme}. However, unlike the case of unbiased random walk where every edge has the same probability distribution, edges with different mean loads have distributions according to Eq. \ref{prdistload}.
\begin{figure}
\includegraphics{Figures_Latest/F+L.png}
\caption{The probability distributions of (a) load and (b) flux for a randomly chosen edge $e_{0,11}$ connecting
nodes 0 and 11 with degrees 100 and 44, respectively. The different curves indicate the distributions obtained by executing DBRW for varying values of bias strength $\alpha\in\{-2,-1,0,1,2\}$. Each execution had 2500 walkers running for 100,000 time steps on the network in Fig. \ref{IISERP} and was repeated 15 times with varying initial positions of random walkers.}
\label{F+L}
\end{figure}
Next, the probability for the occurrence of load EE, $Q_L \left( \langle l_{ij} \rangle \right)$
on edge $e_{ij}$ is considered. The theoretical result is obtained from
Eq. \ref{eenode-1} and is juxtaposed with simulation results for
$Q_L \left( \langle l_{ij} \rangle \right)$ in Fig. \ref{PEEL}. It is displayed for all the edges for several threshold values indexed by $m$ at a fixed value of bias ($\alpha=1$). An excellent agreement is observed between the analytical results and the simulations.
In an average sense, the EE probability is higher for edges with lower values of
$\langle l_{ij} \rangle$ in comparison to edges with larger $\langle l_{ij} \rangle$.
This effect is more pronounced for higher EE thresholds. As $m \to 0$,
the mean load itself becomes the threshold value effectively leading to most events designated
as extreme. In this case, EE probability does not vary much with $\langle l_{ij} \rangle$. Clearly, this quantity varies with $\langle l_{ij} \rangle$ but does not vary much as $m$ is varied. Hence, for a fixed value of threshold indexed
by $m$, EE probability has been averaged over different number of extreme events. It was also found that the density of edges for each value of $\langle l_{ij}\rangle$ differed, but for a given value of $\langle l_{ij}\rangle$, the relative density remained the same for all values of $m$. This is not a peculiarity of biased random walks but is reflective of the inhomogeneity of the network structure.
\begin{figure}
\includegraphics{Figures_Latest/IISERE.png}
\caption{Probability of load EE vs mean load on edge for different thresholds of extremeness: $m\epsilon[1,5]$. 2500 walkers performed a DBRW with $\alpha=1$ for 100,000 time steps on the network mentioned in Fig.\ref{IISERP} }
\label{PEEL}
\end{figure}
Now that we have an understanding of the effect of the choice of threshold we proceed to analyse how changing the bias strength affects the system. Fig. \ref{PEEL_2sigma} shows $Q_L \left( \langle l_{ij} \rangle \right)$ plotted against the mean load $\langle l_{ij}\rangle$ for varying values of $\alpha$
at fixed EE threshold of $m=2$. We have run it on a scale-free network with 2000 nodes and 7984 edges. On this network, 15968 non-interacting walkers execute degree biased random walk for 75000 time steps (note that these are the parameters and network used for the rest of the graphs, unless mentioned otherwise). An excellent agreement between analytical and numerical simulations can be seen. Note that for the
unbiased random walk dynamics ($\alpha=0$) shown in Fig. \ref{PEEL_2sigma}(c) all the edges have the
same mean load and hence identical EE occurrence probability, in agreement with the earlier
result \cite{kumar2020extreme}. The effects of biasing is visible in the figure. When biased towards
hubs with $\alpha >0$, more walkers moving towards the hubs leads to EE on edges which connect two hubs. At the other end of the spectrum, when $\alpha < 0$, walkers preferentially move towards small degree nodes and consequently fewer or no EE takes place on edges connecting hubs. The discontinuity in $Q_L \left( \langle l_{ij} \rangle \right)$ arises due to the discreteness of walkers while the threshold $q_{ij}$ in Eq. \ref{eenode-1} is a real number.
In general, considerable variability in the EE probability can be observed. As Fig. \ref{PEEL_2sigma}(e)
shows for $\alpha =-2$, for mean load $\langle l \rangle < 1$,
EE probability varies by
4 orders in magnitude. For mean loads of $\langle l \rangle > 1$ (yellow background), the variability
is about one order of magnitude. Similar wide range of variabilities are observed for other values of
biases shown in Fig. \ref{PEEL_2sigma}. The exception is the case of $\alpha=0$ corresponding to
{\it unbiased} random walk, for which the mean load is same on all the edges of the network and
consequently the EE probability is identical on all the edges \cite{kumar2020extreme}. The variability
in EE probability becomes more pronounced for $\alpha < 0$. This is a new feature not encountered in the
EE probabilities on nodes or edges of networks on which unbiased random
walks are executed \cite{kishore2011extreme,kishore2012extreme}. It would imply the following:
on the edges characterised by {\it large} mean loads, the extreme events are approximately equiprobable irrespective of its local network structure. On edges with {\it small} mean loads, this probability varies by large orders of magnitude. Hence, two edges whose mean loads do not appreciably differ from one another can have very different EE probabilities. So as far as EE are concerned, biased walker dynamics on edges leads to the network displaying an approximately logical (not necessarily physical) phase separation in to two parts -- one in which EE have only small variations about an average value, and the other in which
large excursions about the mean are observed. This will have implications for strategies that try to harness extreme events. Finally, it must be emphasised that this variability is not due to insufficient averaging but an inherent feature of biased dynamics in networked system.
\subsection{Other Biases}
\begin{figure}
\includegraphics{Figures_Latest/DBRW_biases.png}
\caption{Log-log plot of the probabilities of edge EE for load vs mean edge load for different values of bias parameters (a) $\alpha=2$, (b) $\alpha=1$, (c) $\alpha=0$, (d) $\alpha=-1$, (e) $\alpha=-2$ and (f) $\alpha=-4$. The red squares represent analytical results and the green circles indicate numerical simulations. All runs have $m=2$, and were carried out for 75,000 time steps each. The network used was a scale-free network with 2000 nodes and 7984 edges with 15968 non-interacting biased random walkers.}
\label{PEEL_2sigma}
\end{figure}
In this subsection, we switch to a scale-free network \cite{barabasi1999emergence} and apply different kinds of biased walks based on network centrality measures. A scale-free network with 2000 nodes and 7984 edges is created. On this network, 15968 non-interacting walkers executed random walk dynamics for $75000$ time steps. The walkers are randomly distributed on the nodes of the network initially. Simulation data is recorded after discarding the first $10^3$ time steps on account of transient behaviour.
Since centrality measures provide different ways to rank the importance of nodes in a network, they can also be used as biasing functions. We chose four measures - Betweenness, Eigenvector, Closeness and PageRank centralities \cite{brandes2001faster,bonacich1987power,freeman1978centrality,page1999pagerank} and repeated the procedure followed in section \ref{DBRW}. The Betweenness centrality $B_v$ of a node labelled $v$ is given by
\begin{equation}
B_v = \sum_{a,b} \frac{\sigma_{a,b}(v)}{\sigma_{a,b}},
\end{equation}
where $\sigma_{a,b}$ is the total number of shortest paths from node $a$ to $b$, while $\sigma_{a,b}(v)$ is the number of such paths passing through node $v$. The summation is over all pairs of nodes $(a,b)$ on the network such that $a \ne b \ne v$. It represents the degree to which nodes stand between each other. The eigenvector centrality defines the importance of a node in terms of the neighbourhood of each node.
Let $\mathbf{A}$ be the adjacency matrix, then the Eigenvector centrality
is given by the left eigenvector $\mathbf{x}$ corresponding to the largest eigenvalue $\lambda$:
\begin{equation}
\lambda \mathbf{x} = \mathbf{x} \, \mathbf{A}.
\end{equation}
The $v$-th element of $\mathbf{x}$ (denoted by $X_v$) gives the centrality measure for the node $v$ on a network. The closeness centrality is the average length of the shortest path between the node and all other nodes in the graph. If $d(v,u)$ represents the length of shortest path between nodes
$v$ and $u$, then closeness centrality is defined as
\begin{equation}
C_v = \frac{N}{\sum_u d(v,u)}.
\end{equation}
Thus, if $C(v)$ is large, then it implies that most $d(v,u)$ are small, and hence node $v$ is "closer" to other nodes on the network. For the present purposes, the biasing function for the $v$-th node is taken to be
\begin{equation}
f_v \propto \Phi_v^{\alpha},
\end{equation}
where $\Phi_v=B_v,\, X_v$, $C_v$ or PageRank centrality \cite{page1999pagerank}. The transition probability and the occupation probability can be obtained by using $f_v$ in Eqs. \ref{transition prob matrix} and \ref{occprob1} respectively. From these, the load and flux distributions as well as their extreme event probabilities can be computed as was done earlier for degree biased walks.
In Fig. \ref{otherbiases}(b-e), the load EE probabilities on edge $e_{ij}$ is plotted for the four different biasing functions. For comparison, the degree bias ($k^\alpha$) is shown in Fig. \ref{otherbiases}(a). In all the cases, the same strength parameter $\alpha=1$ and the EE threshold is set to $q_{ij}=\langle l_{ij}\rangle+2\sigma_{ij}$. Even though each of these centrality measures
capture a different facet of the notion of importance for a node, the EE probabilities on edges of the network do not differ significantly irrespective of which biasing function was applied. This can be attributed to correlations found between the centrality measures \cite{oldham2019consistency,valente2008correlated}.
\begin{figure}
\includegraphics{Figures_Latest/BRW_biases.png}
\caption{Log-log plot of probabilities of edge EE for load against mean edge load using various biasing functions from (a) Degree, (b) Betweenness, (c) Closeness, (d) PageRank, and (e) Eigenvector centralities. The red squares represent analytical results and the green circles indicate numerical simulations. All runs have $m=2$, $\alpha$=1 and were carried out for 75,000 time steps each on a scale-free network with 2000 nodes and 7984 edges with 15968 non-interacting biased random walkers.}
\label{otherbiases}
\end{figure}
As seen in the case of degree biased random walks, in Fig. \ref{otherbiases} too, a large variability
in EE probability is observed for edges whose mean load is approximately in the range $\langle l \rangle < 1$,
as compared to the edges with
$\langle l \rangle > 1$ (yellow background). Thus, irrespective of the actual biased walk algorithm employed, pronounced
variability in EE probability for some of the edges appears to be a robust feature.
For a given biased dynamics on a network, let
$\Delta l = \langle l \rangle_{\text max} - \langle l \rangle_{\text min}$
represent difference between the maximum and minimum value of mean loads on its edges. It can be observed
that $\Delta l$ is a function of applied bias. As seen in Figs. \ref{PEEL_2sigma} and \ref{otherbiases}, $\Delta l$ is largest
for betweenness centrality based walk ($ = 8.542 \times 10\textsuperscript{2}$), and smallest for closeness centrality based walk ($ = 2.726 \times 10\textsuperscript{0}$). For unbiased
random walks, $\Delta l = 0$.
This doesn't preclude the possibility that the same edge under different biases can have different EE probabilities. Further, since the EE thresholds are defined based on these measures, it is evident that what might be extreme in one setting
would theoretically not be so in another. This implies that the choice of bias and its strength are crucial factors
in determining extreme event occurrence rates.
\section{Correlations between Extreme Events} \label{corr_EE
As reported in \cite{kumar2020extreme}, the unbiased random walk process on networks is uncorrelated, the extreme events on nodes are almost temporally uncorrelated with that on the edges. Yet, the extreme events arising from the
{\it biased} random walks can display significant time-lagged correlations, {\it i.e.}, extreme events
occurring on nodes can be correlated with the extreme events on the neighbouring nodes or edges after
a time lag $d$. Let us denote by $\mathcal{N}(i)$ the set of nodes that are connected to node $i$.
In this case, two
possible scenarios exist, ({\it i}) node-node correlation: EE occurs at time $t$ on a node $i$, and occurs
at time $t+d$ on node $j$ such that $j \in \mathcal{N}(i)$, ({\it ii}) edge-node correlation: EE occurs at
time $t$ on an edge $e_{ij}$, and occurs at time $t+d$ on $i$-th or $j$-th node connected
to this edge. To quantify these correlations, we coarse-grain
the time series of walkers as follows. Let $w(t),\, t=1,2,3,\dots,75000$ represent the time varying walker flux through an edge or node.
This series is modified as follows. If an EE occurs at time $t=\tau$, then $w(\tau)=1$, else $w(\tau)=0$. This coarse-grained time series is used to calculate the (normalized) time lagged cross correlation (TLCC) function \cite{gubner2006probability} between any two time series $x(t)$ and $y(t)$.
It is defined as
\begin{equation}
r_{d}=\frac{\sum_{t=1}^{T-d} (x(t)-\langle x\rangle) ~ (y(t+d)-\langle y\rangle)}{\sqrt{\sum_{t=1}^{T-d}(x(t)-\langle x\rangle)^{2}} ~ \sqrt{\sum_{t=1}^{T-d}(y(t+d)-\langle y\rangle)^{2}} } .
\end{equation}
Here $r_d \in [-1,1]$ and indicates if the two time series are correlated or anti-correlated at time lag $d$. We calculate node-node and edge-node correlation $r_d$ for the EE time series $w(t)$ for all pairs of neighbors in the network with DBRW dynamics for lags $d \in [-10,10]$. The results are shown in Fig. \ref{nodenode} are for five
node pairs. The five node pairs are representative of the different behaviour found, they are labelled with degrees of the nodes the edges connect. Each colour represents the same node pair across all subplots. As is evident in
Fig. \ref{nodenode}(c), the node-node correlations for unbiased walkers ($\alpha=0$) are barely significant for any time lag $d$. On the other hand, as the bias strength increases, {\it i.e}, $|\alpha| > 0$,
strong correlations emerge among connected pairs of nodes.
\begin{figure}
\includegraphics{Figures_Latest/node-node.png}
\caption{Node-Node correlations of EE time series for five representative node pairs labelled by their degrees [$k_i,k_j$] in the legend. Subplots compare this over different strengths of the degree bias (a) $\alpha=2$, (b) $\alpha=1$, (c) $\alpha=1$, (d) $\alpha=-1$, (e) $\alpha=-2$ and (f) $\alpha=-4$ against lag $d$. }
\label{nodenode}
\end{figure}
From Fig. \ref{nodenode}, we see that the node pairs display strong correlations for lags $d=-1$ and $d=1$.
This means that if an EE occurred in a node its neighboring node is likely to have experienced one in the previous time step and/or experience one in the next step. The likelihood of this happening increases as $\alpha$ gets more negative. Furthermore, there exist node pairs (such as the one labelled [$k_i,k_j$]=[5,4] in Fig. \ref{nodenode}(d-f)) whose correlations do not decay rapidly with lag $d$. This means that EE among neighbors can remain correlated for long times. This might be useful for predicting such events. As the bias gets more negative the fraction of pairs with a nontrivial amount of correlation ($\>0.4$ by convention) increases. This fraction depends on degree distribution of the network and for this scale-free network it is $<10\%$ until $\alpha=-2$. The strength of this lagged correlation depends on
the product of the degrees of the nodes the edge $e_{ij}$ connects, $k_ik_j$. If $k_ik_j$ is large, the
lagged correlation will be small since the EE tend to get "scattered" away in to the many neighbours at the next
time step. If $k_ik_j$ is small, the limited local neighbourhood of poorly connected nodes could result in a fraction of the negatively biased walkers to be dispersed again to the same node.
Next we investigate the lagged correlation between the EE for load on an edge $e_{ij}$ and the EE
occurring on one of the nodes it connects to, say, node $i$. In Fig. \ref{loadnode}, lagged correlation $r_d$
is shown for DBRW dynamics for varying bias strength $\alpha$. In general, edge-node correlations are
far weaker than the node-node correlations. For a fixed $\alpha$, the strongest cross correlation
is seen at time lags of $d=-1$ and $d=1$. It implies that if there is a load EE on an edge then one of
the nodes connecting it is likely to encounter an EE either one time step before and/or one time step later.
This effect gets more pronounced as $\alpha$ gets more negative. In this case, walkers preferentially
explore small degree nodes which have fewer edges connected to them. Hence, once an extreme event takes place
in one such node, the walkers have no option but to transit through one of its edges creating an extreme event
on the edge as well. Due to this, walkers continually circulate locally among the few small degree nodes
leading to extremely slow decay of correlations as seen in Fig. \ref{loadnode}.
As $\alpha$ gets more negative, larger fraction of edges display such slow decay of correlations with lag $d$.
On the other hand, for $\alpha > 0$, the walkers preferentially explore hubs on the
network. If an extreme event takes place in one of the hubs, the walkers are scaterred away through a large
number of edges connected to it. Hence, it might not exceed the threshold required to designate it as an
extreme event. Hence lagged correlations are not pronounced in the case of $\alpha >0$ as observed in
Fig. \ref{loadnode}(a-b).
\begin{figure}
\includegraphics{Figures_Latest/edge-node.png}
\caption{Edge-Node correlations between EE time series for five representative edges that were taken and labelled by the degrees of the nodes [$k_i,k_j$] they connected. Subplots compare this over different strengths of the degree bias with bias strength (a) $\alpha=2$, (b) $\alpha=1$, (c) $\alpha=1$, (d) $\alpha=-1$, (e) $\alpha=-2$ and (f) $\alpha=-4$ against lag $d$. The peak values are observed for lag = -1 and 0.}
\label{loadnode}
\end{figure}
\section{Network stability under edge extreme events} \label{stability}
Using the extremes related results obtained above, we are now in a position to discuss the robustness of a network in the context of extreme events faced by edges. To study this, we adopt two broad classes of node or edge deletion strategies:
({\it i}) targeted node/edge deletion based on decreasing order of EE occurrence probability,
({\it ii}) random edge/node deletion.
To illustrate the effects of these deletion strategies, we monitor the size of the relative
giant component $S(t)=\frac{N_{\text{max}}(t)}{N}$ in the network as a function of time. It is the
ratio of nodes in the largest connected component $N_{\text{max}}$ to the total number of nodes $N$
{\it initially} in a network. If all the nodes are reachable from any other node, $S=\frac{N_{\text{max}}}{N}=1$.
As we delete nodes/edges $S(t)$ decreases with time.
In Fig. \ref{delete}(a), the simulation results for a variety of biased random walks are considered
and for each one the targeted and random node deletion strategies are shown. In most cases, even if
30-40\% of the nodes are removed in a targeted manner, the giant component in the network falls to
less than 50\%. In the case of degree biased random walks, even as 50\% of nodes are targeted and
removed, giant component size becomes vanishingly small. However, if the nodes are randomly
removed, a reasonably sized giant component in the network survives even until about
80\% of the nodes are deleted. It is well-known that scale-free network is resilient to
random attacks but not to targeted attacks \cite{cohen2001breakdown}. The difference here is that the same viewpoint is reinforced even if the node deletion strategy is based on occurrence of extreme events.
A somewhat surprising behaviour is seen in Fig. \ref{delete}(b), in which edges deletion results
are shown for both targeted and random removal strategies. Unlike the node deletion case,
a significant fraction of the giant component survives well until about 80\% of the edges
are removed. Indeed, until 30\% of edges are deleted, the size of giant component does not
decrease appreciably. Thus, as far as the survival of network structure is concerned, node and
edge deletion processes are not equivalent. If $q$\% of nodes
are deleted, the corresponding giant component size is $S^n_q$. Similarly, if $q$\% of edges are
removed, let the giant component size be denoted by $S^{e}_q$. Generally, we observe that $S^{e}_q > S^n_q$
for nearly all $q$. Physically, this effect arises because if a node is deleted, typically
many edges becomes dysfunctional for transport of randowm walkers and this tends to quickly
reduce the size of giant component. On the other hand, if an edge is deleted, most other parts
of the network continues to remain functional for transport. Thus, deletion of a node and an
edge lead to different outcomes for network resilience.
It might also be pointed out that, unlike node deletion strategies, there is virtually
no difference between random and targeted edge deletion for the same reason that scale-free
networks are more resilient to edge deletions than node deletions.
\begin{figure}
\includegraphics{Figures_Latest/removalstrategy.png}
\caption{Size of the relative giant component tracked as (a) nodes are removed in decreasing order of their node EE probability ${P}_{EE}(q_{i})$ (b) edges are removed in decreasing order of $Q_L \left( \langle l_{ij} \rangle \right)$. In both cases an EE threshold of $m=2$ was used. In (b), curves for $\alpha=-1$ and $-2$ lie on top of each other as do the curves for $\alpha= 1$ and $2$.}
\label{delete}
\end{figure}
\section{Conclusions}
In summary, we have proposed a general formalism to study extreme events taking place on the edges, rather than on the nodes of a complex network. In this work, we considered the dynamics of biased random walkers on the network, and extreme events on the edges based on the deviation of flux of walkers above a defined threshold. Unlike the case of nodes, in the edge $e_{ij}$ of the network, walkers can go from node $i$ to node $j$ and vice versa. Hence, it was necessary to distinguish between two distinct dynamical quantities -- the load, which is the total number of walkers traversing the edge. Secondly, the flux denoting the difference between the number of walkers going $i$ to $j$ or in the reverse direction. Thus, enabling the consideration of extreme events using both of these dynamical observables. In this work,
we have applied the proposed formalism to study the occurrence of extreme events in load and flux on the edges. It must be pointed out that extreme events in this work represent one possible outcome due to intrinsic fluctuations in the number of biased random walkers, and is not triggered by external sources.
We obtained analytical and numerical estimates for the occurrence probabilities of extreme events on the edges of a network. The occurrence probability of extreme events on edges depends on the local network structure around the edge of interest. Within the framework of biased random walkers, it is shown that the edges can be characterised by the mean loads (or mean fluxes).
Then, $e_{ij} = e_{ij}(\langle l \rangle)$, where $\langle l \rangle$ denotes mean load. For the edges with low mean loads ($\langle l \rangle < 1$), it is shown that the variability of the extreme event probability is higher compared to edges with higher mean loads ($\langle l \rangle > 1$). Strongly biased random walks tend to increase the variability for $\langle l \rangle < 1$, and suppress the variability at $\langle l \rangle > 1$. We show these results for varying strengths of the degree biased random walk on scale-free networks. We also show similar trends are realised for other biases -- Betweenness, Closeness, Eigenvector, and PageRank centrality based biases. In addition, these are also demonstrated for a real-life planar network.
Further, we studied how the extreme events on nodes and edges are related. Earlier, it was shown that these two classes of extreme events are uncorrelated for unbiased random walkers on networks. However, while using biased random walkers a non-trivial correlation is found to exist. This implies that
an extreme event on a node tends to influence the occurrence or non-occurrence of an extreme event on an edge connected to it. We quantify such correlations for various strengths of the degree bias.
Finally, we also studied the vulnerability of the network as a whole to the occurrence of extreme events
on the edges. We removing nodes/edges in decreasing order of the probability of their extreme event occurrence. For instance, the edge with the highest probability will be removed first, and then the node with next highest probability will be removed and so on. After each removal, the size of
the largest connected component of the network is computed. It is shown that the largest connected component stays intact almost until 50\% of the edges are removed due to extreme events. This is in contrast to the case of node removal based on similar criteria outlined above.
In which case, the size of the giant component falls almost by 50\% when half of the nodes are
removed. Thus, networks can be expected to maintain their resilience much more in the face of
extreme events on the edges, than if it were to happen on the nodes.
It would be interesting to study extreme events on real-life network with measured loads and fluxes and compare them to patterns observed in synthetic networks. An absorbing exploration would be to map community structure \cite{zlatic2010topologically} but when supplied with the EE data. Keeping track of how EE cascades through the network using higher order versions of the correlations observed among the nodes and edges is another avenue of research. Using measures that quantify the local neighbourhood of nodes involved in node-node and edge-node correlations instead of $k_ikj$ can provide better insight about strong correlations observed. Developing a more accommodating definition of EE can lead to modeling more than congestion-type systems. Using time evolving properties of nodes, edges (and if possible even the walkers) can lead to interesting generalisations of the theory.
Trying out other biasing functions, especially non-structural ones whose values are bestowed externally could lead to interesting trends. One possible extension to this framework is to get equations analytically for time-dependent biasing functions as well. In general, dynamics on the edges have long been neglected and these results, we hope, might lead to more work on extreme events on the edges of networks.
\begin{acknowledgments}
GG thanks Aanjaneya Kumar for many discussions during this work. GG acknowledges support from IISER Pune and INSPIRE Scholarship for Higher Education (SHE). MSS acknowledges the support from MATRICS grant of SERB, Govt of India.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 274 |
package com.crossoutxtrem;
import android.app.Activity;
import android.app.AlertDialog;
import android.content.Context;
import android.content.DialogInterface;
import android.content.Intent;
import android.content.SharedPreferences;
import android.content.DialogInterface.OnClickListener;
import android.content.SharedPreferences.Editor;
import android.graphics.Color;
import android.os.Bundle;
import android.view.View;
import android.widget.LinearLayout;
import android.widget.ToggleButton;
public class TPOptions extends Activity implements OnClickListener
{
private ToggleButton tb1;
private ToggleButton tb2;
private ToggleButton tb3;
private ColorPicker p1Picker;
private ColorPicker p2Picker;
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.tpoptions);
SharedPreferences prefs = this.getSharedPreferences("TriangleOfCircles", Context.MODE_PRIVATE);
int first = prefs.getInt("tpfirst", 0);
tb1 = (ToggleButton) findViewById(R.id.toggleButton1);
tb2 = (ToggleButton) findViewById(R.id.toggleButton2);
tb3 = (ToggleButton) findViewById(R.id.toggleButton5);
if (first==0)
tb1.setChecked(true);
else if (first==1)
tb2.setChecked(true);
else
tb3.setChecked(true);
p1Picker = new ColorPicker(this, prefs.getInt("p1Color", Color.RED));
p2Picker = new ColorPicker(this, prefs.getInt("p2Color", Color.BLUE));
((LinearLayout) findViewById(R.id.linearLayout9)).addView(p1Picker);
((LinearLayout) findViewById(R.id.linearLayout10)).addView(p2Picker);
}
public void moveClicked(View view)
{
int id = view.getId();
if (id==tb1.getId())
{
if (tb1.isChecked())
{
tb2.setChecked(false);
tb3.setChecked(false);
}
else
{
tb1.setChecked(true);
}
}
else if (id==tb2.getId())
{
if (tb2.isChecked())
{
tb1.setChecked(false);
tb3.setChecked(false);
}
else
{
tb2.setChecked(true);
}
}
else
{
if (tb3.isChecked())
{
tb2.setChecked(false);
tb1.setChecked(false);
}
else
{
tb3.setChecked(true);
}
}
}
public void backClicked(View view)
{
finish();
}
public void startClicked(View view)
{
if (p1Picker.getColor()==p2Picker.getColor())
{
AlertDialog messageBox = new AlertDialog.Builder(this).create();
messageBox.setTitle("Invalid");
messageBox.setMessage("The color for player 1 and the color for player 2 must be different.");
messageBox.setButton("Ok", this);
messageBox.show();
return;
}
int move;
if (tb1.isChecked())
move = 0;
else if (tb2.isChecked())
move = 1;
else
move = 2;
TwoPlayerGame.whoGoesFirst = move;
TwoPlayerGame.p1Color = p1Picker.getColor();
TwoPlayerGame.p2Color = p2Picker.getColor();
Editor editor = this.getSharedPreferences("TriangleOfCircles", Context.MODE_PRIVATE).edit();
editor.putInt("tpfirst", move);
editor.putInt("p1Color", TwoPlayerGame.p1Color);
editor.putInt("p2Color", TwoPlayerGame.p2Color);
editor.commit();
Intent spIntent = new Intent(view.getContext(), TwoPlayerGame.class);
startActivity(spIntent);
finish();
}
public void onClick(DialogInterface dialog, int which)
{
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,351 |
@class StacManClient;
@interface StacManErrorMethods : NSObject
-(id)initWithClient:(StacManClient*)client;
//getAll(String filter, Integer page, Integer pagesize)
-(StacManResponse*)getAllWithFilter:(NSString*)filter page:(int)page pageSize:(int)pageSize delegate:(NSObject<StacManDelegate>*)del;
//simulate(int id, String filter)
-(StacManResponse*)simulateWithId:(int)_id filter:(NSString*)filter delegate:(NSObject<StacManDelegate>*)del;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,535 |
{"url":"https:\/\/itectec.com\/ubuntu\/ubuntu-how-to-find-the-pid-of-the-process-which-has-deleted-a-file\/","text":"# Ubuntu \u2013 How to find the pid of the process which has deleted a file\n\nprocessrm\n\nI am working on a project related to VM migration. Sometimes the VM image will disappear and I just want to know who the culprit is. I tried strace on suspicious processes but to no avail.\n\nsudo auditctl -w \/path\/to\/somefile -p wra\n\nausearch -f \/path\/to\/somefile -i","date":"2021-10-19 06:14:46","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\": 0.1876010298728943, \"perplexity\": 4054.6038884075133}, \"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-2021-43\/segments\/1634323585242.44\/warc\/CC-MAIN-20211019043325-20211019073325-00542.warc.gz\"}"} | null | null |
Longview Consulting, Inc.
Edward Moses is the President of Longview Consulting, specializing in advising high technology companies in technical, systems engineering, and management issues. His experience includes being:
President of the Giant Magellan Telescope Organization responsible for designing and building the world's largest aperture visible and infrared telescope.
Director of the UC Berkeley Center for Fusion Energy Science and Applications
Principal Associate Director at Lawrence Livermore National Laboratory (LLNL) for the National Ignition
Facility (NIF) and Photon Science Directorate. As leader of the NIF Directorate, he was responsible for completing construction and activation of the NIF, the largest laser facility in the world, and transforming it into an international user facility
Director of the National Ignition Campaign to study High Energy Density science, including laboratory astrophysics and laboratory planetary physics, studying materials under extreme temperature and pressure conditions, strategic missions, and fusion energy.
Responsible for the development of high-average-power lasers, short-pulse lasers and high-average-power gamma ray sources.
Program Manager in Physics Directorate for the Peregrine Program to develop and license advanced computational techniques for the radiation treatment of cancer
Director in Physics Directorate for Program Development to develop a strategic vision and detailed program plans for the spectrum of activities in space technology, material science, nuclear physics, high-energy physics and high-energy-density science.
Executive Vice President of Advanced Technology Application's, Inc. to develop advanced concepts and product development plans for technology products and systems for private industry and government programs.
Education: Ph.D. in electrical engineering from Cornell University.
Memberships: Member of National Academy of Engineering, a Fellow of SPIE, a Fellow of AAAS, and a Member of the California Council on Science and Technology
Awards: Project Management Institute award for 2010 Project of the Year, Thomas Jefferson Award for Educational Outreach, Edward Teller Medal for "leadership in the development and completion of the National Ignition Facility", Fusion Power Associates Leadership Award, The Memorial D. S. Rozhdestvensky Medal for Lifetime Contributions to Lasers and Optical Sciences, NNSA Defense Programs Award of Excellence for Significant Contribution to Stockpile Stewardship Program, and the R&D 100 Award for the PEREGRINE Project
Publications: Numerous publications, presentations and briefings on physics, computational physics, lasers and optics, technology, systems integration and systems engineering, inertial fusion energy.
Patents: in lasers, optics, fusion energy and radiation health physics.
Reviews: Managed and participated in reviews of other major projects for University of California and National Science Foundation and other international identities.
Educational Outreach: Large number of talks to school audiences, summer students, university audiences, American Junior Academy of Sciences, other national labs and many civic organizations on laboratory science and technology and engineering and numerous presentations on project and program management to all levels of government and private industry. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,562 |
{"url":"https:\/\/community.jmp.com\/t5\/Discussions\/How-to-check-the-aliasing-term-in-DOE-custom-design\/m-p\/55753","text":"Choose Language Hide Translation Bar\nHighlighted\nLevel II\n\n## How to check the aliasing term in DOE custom design\n\nHi Anyone,\n\nHow do we check what is the confounding factors during our initial custom design setup?\n\nRgrds\n\nIrfan\n\n7 REPLIES 7\nHighlighted\nStaff\n\n## Re: How to check the aliasing term in DOE custom design\n\nCustom design will seldom give you a design with perfect aliasing. So the report will use correlations to show you the degree of aliasing that is present.\n\nWhen creating the design, notice just below the model window is the \"Alias Terms\" part of the report. Open that up and put in the terms for which you wish to check for aliasing. For example, if the model has only main effects and you wish to see if they are aliased with two-factor interactions, then put the two-factor interactions in the Alias Terms window.\n\nNow, when you generate the design, under Design Evaluation, you will see the Alias Matrix as an option. Opening that up will show you the correlations between your model terms and the alias terms specified earlier. A value of 1 would show complete aliasing. Any non-zero term shows \"partial\" aliasing. Graphically, you can see this on the color map on correlations report.\n\nDan Obermiller\nHighlighted\nLevel II\n\nThanks Dan\n\nHighlighted\nLevel II\n\n## Re: How to check the aliasing term in DOE custom design\n\nHi Anyone,\n\nAble to explain to me what does the 'Temperature is biased by 0.333 times the true value of the grind*time? Does it mean that the parameter estimates for temperature is 1\/3 value of the grind*time estimates?\n\nRgrds\n\nIrfan\n\nHighlighted\nSuper User\n\n## Re: How to check the aliasing term in DOE custom design\n\nIrfan,\n\nThis links walks through an example of the effects of confounding (also called aliasing or partial aliasing)\n\nhttps:\/\/en.wikipedia.org\/wiki\/Fractional_factorial_design\n\nFor another example, I have attached 4-factor fractional factorial. The screenshot below displays the alias matrix. The response Y was created from the formula\n\n``20 + 1*:X1 + 2*:X2 + 3*:X3 + 4*:X4 + 0.5*:X12 + 1*:X34 + 5*:X13 -5*:X24 +5*:X14 +10*:X23 + 1*Random Normal()``\n\nSince X12 (simpler notation) is confounded with X34, and only X12 is in the model, it's coefficient should be approx. 1.5. And since X13 and X24 are correlated the coefficient of X13 will be approx. 0, and the coefficient of X14 will be approx. 15.\u00a0 This is why checking the evaluating the design which includes checking the alias structure is an important step in designing an experiment. If there is an expected important interaction, the design should be chosen so it has no confounding with other 2 factor interactions, or small partial interactions with multiple effects.\n\nGeorgia\n\nHighlighted\nLevel IV\n\n## Re: How to check the aliasing term in DOE custom design\n\nThe correlation of 0.333 means that the estimated effect of Temperature is increased by the effect of the gind*time-interaction multiplied by 1\/3. So basically you are estimating in the analysis\n\n[Temperature] =\u00a0Temperature + 1\/3*Grind*Time\n\nThus if there is a strong interaction effect it will inflate your main effect estimator.\n\nYou can see that by simulating some simple data. For demonstration purposes I will use a Plackett-Burrman screening design with the following correlation structure:\n\nAs you can see the main effect X1 has a correlation of 0.333 with the X2*X3-interaction. I generated the design and added some simulated data:\n\nAs you can see, the true effect of X1 = 10, the true effect of the interaction X2*X3 = 18. If we now fit a main effects model you get the following results:\n\nThe estimated effect of X1 = 16, which is the true effect of X1 - which is 10 - and it is biased by 1\/3*18 (the effect of X2*X3).\n\nHope this helps.\n\nSebastian\n\nHighlighted\nLevel II\n\n## Re: How to check the aliasing term in DOE custom design\n\nHi Sebastian,\n\nDoes it mean that if I have a correlation of -1 for X2*X3 with X1 then it might cancel out the estimated effect of X1?\n\nRgrds\n\nIrfan\n\nHighlighted\nLevel IV\n\n## Re: How to check the aliasing term in DOE custom design\n\nYou are correct. This might as well happen if you find a correlation of 1 and a negative effect of X2*X3 and a positive effect of X1.","date":"2020-05-25 17:41:28","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\": 0.801267683506012, \"perplexity\": 1767.0025407212283}, \"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-2020-24\/segments\/1590347389309.17\/warc\/CC-MAIN-20200525161346-20200525191346-00588.warc.gz\"}"} | null | null |
import os
import unittest
from gae_ext_runtime import testutil
RUNTIME_DEF_ROOT = os.path.dirname(os.path.dirname(__file__))
DOCKERFILE_TEXT = '''\
# This Dockerfile for a Ruby application was generated by gcloud.
# The base Dockerfile installs:
# * A number of packages needed by the Ruby runtime and by gems
# commonly used in Ruby web apps (such as libsqlite3)
# * A recent version of NodeJS
# * A recent version of the standard Ruby runtime to use by default
# * The bundler gem
FROM gcr.io/google-appengine/ruby:{base_image_tag}
# If your application requires a specific ruby version (compatible with rbenv),
# set it here. Leave blank to use the currently recommended default.
ARG REQUESTED_RUBY_VERSION="{ruby_version}"
# Install any requested ruby if not already preinstalled by the base image.
# Tries installing a prebuilt package first, then falls back to a source build.
RUN if test -n "$REQUESTED_RUBY_VERSION" -a \\
! -x /rbenv/versions/$REQUESTED_RUBY_VERSION/bin/ruby; then \\
(apt-get update -y \\
&& apt-get install -y -q gcp-ruby-$REQUESTED_RUBY_VERSION) \\
|| (cd /rbenv/plugins/ruby-build \\
&& git pull \\
&& rbenv install -s $REQUESTED_RUBY_VERSION) \\
&& rbenv global $REQUESTED_RUBY_VERSION \\
&& gem install -q --no-rdoc --no-ri bundler --version $BUNDLER_VERSION \\
&& apt-get clean \\
&& rm -f /var/lib/apt/lists/*_*; \\
fi
ENV RBENV_VERSION=${{REQUESTED_RUBY_VERSION:-$RBENV_VERSION}}
# Copy the application files.
COPY . /app/
# Install required gems if Gemfile.lock is present.
RUN if test -f Gemfile.lock; then \\
bundle install --deployment --without="development test" \\
&& rbenv rehash; \\
fi
# Temporary. Will be moved to base image later.
ENV RACK_ENV=production \\
RAILS_ENV=production \\
RAILS_SERVE_STATIC_FILES=true
# Run asset pipeline if we're in a Rails app.
RUN if test -d app/assets -a -f config/application.rb; then \\
bundle exec rake assets:precompile || true; \\
fi
# BUG: Reset entrypoint to override base image.
ENTRYPOINT []
# Start application on port $PORT.
CMD {entrypoint}
'''
class RuntimeTestCase(testutil.TestBase):
"""Tests for the Ruby external runtime fingerprinter."""
def file_contents(self, filename):
"""Reads the contents of the file from the tempdir.
Args:
filename: (str) filename to be joined with tempdir prefix.
Returns:
File contents.
"""
with open(self.full_path(filename)) as f:
return f.read()
def stub_response(self, response):
"""Stubs the console response from the user.
Args:
response: (str) stubbed response.
Returns:
A function to reset the stubbed functions to their original
implementations.
"""
can_prompt = self.exec_env.CanPrompt
prompt_response = self.exec_env.PromptResponse
def unstub():
self.exec_env.CanPrompt = can_prompt
self.exec_env.PromptResponse = prompt_response
self.exec_env.CanPrompt = lambda: True
self.exec_env.PromptResponse = lambda prompt: response
return unstub
def setUp(self):
self.runtime_def_root = RUNTIME_DEF_ROOT
super(RuntimeTestCase, self).setUp()
def test_generate_without_ruby_files(self):
self.write_file('index.html', 'index')
self.generate_configs()
self.assertFalse(os.path.exists(self.full_path('app.yaml')))
self.assertFalse(os.path.exists(self.full_path('Dockerfile')))
self.assertFalse(os.path.exists(self.full_path('.dockerignore')))
def test_generate_without_ruby_files_no_write(self):
"""Tests generate_config_data does nothing if no ruby files."""
self.write_file('index.html', 'index')
self.assertIsNone(self.generate_config_data())
self.assertFalse(os.path.exists(self.full_path('app.yaml')))
def test_generate_with_ruby_files(self):
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
unstub = self.stub_response('bundle exec rackup -p $PORT -E deployment')
self.generate_configs()
unstub()
app_yaml = self.file_contents('app.yaml')
self.assertIn('runtime: ruby\n', app_yaml)
self.assertIn('env: flex\n', app_yaml)
self.assertIn('entrypoint: bundle exec rackup -p $PORT -E deployment\n',
app_yaml)
self.assertFalse(os.path.exists(self.full_path('Dockerfile')))
self.assertFalse(os.path.exists(self.full_path('.dockerignore')))
def test_generate_with_ruby_files_no_write(self):
"""Tests generate_config_data with basic Ruby files.
Tests that app.yaml is written with correct contents given entrypoint
response, and that Dockerfile and .dockerignore not written to disk.
"""
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
unstub = self.stub_response('bundle exec rackup -p $PORT -E deployment')
cfg_files = self.generate_config_data()
unstub()
app_yaml = self.file_contents('app.yaml')
self.assertIn('runtime: ruby\n', app_yaml)
self.assertIn('env: flex\n', app_yaml)
self.assertIn('entrypoint: bundle exec rackup -p $PORT -E deployment\n',
app_yaml)
self.assertNotIn('Dockerfile', [f.filename for f in cfg_files])
self.assertNotIn('.dockerignore', [f.filename for f in cfg_files])
def test_generate_with_deploy(self):
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
self.write_file('.ruby-version', 'rbx-3.9')
unstub = self.stub_response('bundle exec rackup -p $PORT -E deployment')
self.generate_configs(deploy=True)
unstub()
dockerfile = self.file_contents('Dockerfile')
self.assertEqual(
dockerfile,
DOCKERFILE_TEXT.format(
ruby_version='rbx-3.9',
entrypoint='bundle exec rackup -p $PORT -E deployment'))
dockerignore = self.file_contents('.dockerignore')
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
def test_generate_with_deploy_no_write(self):
"""Tests generate_config_data with deploy=True.
Tests that .dockerignore and Dockerfile contents are correct
based on contents of app.
"""
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
self.write_file('.ruby-version', 'rbx-3.9')
unstub = self.stub_response('bundle exec rackup -p $PORT -E deployment')
cfg_files = self.generate_config_data(deploy=True)
unstub()
self.assert_genfile_exists_with_contents(
cfg_files,
'Dockerfile',
DOCKERFILE_TEXT.format(
ruby_version='rbx-3.9',
entrypoint='bundle exec rackup -p $PORT -E deployment'))
self.assertIn('.dockerignore', [f.filename for f in cfg_files])
dockerignore = [f.contents for f in cfg_files if
f.filename == '.dockerignore'][0]
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
def test_generate_with_custom(self):
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
unstub = self.stub_response('bundle exec rackup -p $PORT -E deployment')
self.generate_configs(custom=True)
unstub()
app_yaml = self.file_contents('app.yaml')
self.assertIn('runtime: custom\n', app_yaml)
self.assertIn('env: flex\n', app_yaml)
self.assertIn('entrypoint: bundle exec rackup -p $PORT -E deployment\n',
app_yaml)
dockerfile = self.file_contents('Dockerfile')
self.assertEqual(
dockerfile,
DOCKERFILE_TEXT.format(
ruby_version='',
entrypoint='bundle exec rackup -p $PORT -E deployment'))
dockerignore = self.file_contents('.dockerignore')
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
def test_generate_with_custom_no_write(self):
"""Tests generate_config_data with custom=True.
Tests that app.yaml is written with correct parameters and
Dockerfile, .dockerignore contents are correctly returned by method.
"""
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
unstub = self.stub_response('bundle exec rackup -p $PORT -E deployment')
cfg_files = self.generate_config_data(custom=True)
unstub()
app_yaml = self.file_contents('app.yaml')
self.assertIn('runtime: custom\n', app_yaml)
self.assertIn('env: flex\n', app_yaml)
self.assertIn('entrypoint: bundle exec rackup -p $PORT -E deployment\n',
app_yaml)
self.assert_genfile_exists_with_contents(
cfg_files,
'Dockerfile',
DOCKERFILE_TEXT.format(
ruby_version='',
entrypoint='bundle exec rackup -p $PORT -E deployment'))
self.assertIn('.dockerignore', [f.filename for f in cfg_files])
dockerignore = [f.contents for f in cfg_files if
f.filename == '.dockerignore'][0]
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
def test_generate_with_existing_appinfo(self):
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
appinfo = testutil.AppInfoFake(
entrypoint='bundle exec ruby index.rb $PORT',
runtime='ruby',
vm=True)
self.generate_configs(appinfo=appinfo, deploy=True)
self.assertFalse(os.path.exists(self.full_path('app.yaml')))
dockerfile = self.file_contents('Dockerfile')
self.assertEqual(
dockerfile,
DOCKERFILE_TEXT.format(
ruby_version='',
entrypoint='bundle exec ruby index.rb $PORT'))
dockerignore = self.file_contents('.dockerignore')
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
def test_generate_with_existing_appinfo_no_write(self):
"""Tests generate_config_data with passed appinfo."""
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
appinfo = testutil.AppInfoFake(
entrypoint='bundle exec ruby index.rb $PORT',
runtime='ruby',
vm=True)
cfg_files = self.generate_config_data(appinfo=appinfo, deploy=True)
self.assertFalse(os.path.exists(self.full_path('app.yaml')))
self.assert_genfile_exists_with_contents(
cfg_files,
'Dockerfile',
DOCKERFILE_TEXT.format(
ruby_version='',
entrypoint='bundle exec ruby index.rb $PORT'))
self.assertIn('.dockerignore', [f.filename for f in cfg_files])
dockerignore = [f.contents for f in cfg_files if
f.filename == '.dockerignore'][0]
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
def test_generate_with_ruby_version(self):
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
self.write_file('.ruby-version', '2.3.1\n')
appinfo = testutil.AppInfoFake(
entrypoint='bundle exec ruby index.rb $PORT',
runtime='ruby',
vm=True)
self.generate_configs(appinfo=appinfo, deploy=True)
self.assertFalse(os.path.exists(self.full_path('app.yaml')))
dockerfile = self.file_contents('Dockerfile')
self.assertEqual(
dockerfile,
DOCKERFILE_TEXT.format(
ruby_version='2.3.1',
entrypoint='bundle exec ruby index.rb $PORT'))
dockerignore = self.file_contents('.dockerignore')
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
def test_generate_with_ruby_version_no_write(self):
"""Tests generate_config_data with .ruby-version file."""
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
self.write_file('config.ru', 'run Index.app')
self.write_file('.ruby-version', '2.3.1\n')
appinfo = testutil.AppInfoFake(
entrypoint='bundle exec ruby index.rb $PORT',
runtime='ruby',
vm=True)
cfg_files = self.generate_config_data(appinfo=appinfo, deploy=True)
self.assertFalse(os.path.exists(self.full_path('app.yaml')))
self.assert_genfile_exists_with_contents(
cfg_files,
'Dockerfile',
DOCKERFILE_TEXT.format(
ruby_version='2.3.1',
entrypoint='bundle exec ruby index.rb $PORT'))
self.assertIn('.dockerignore', [f.filename for f in cfg_files])
dockerignore = [f.contents for f in cfg_files if
f.filename == '.dockerignore'][0]
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
def test_generate_with_prompt(self):
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
unstub = self.stub_response('bundle exec ruby index.rb $PORT')
self.generate_configs(deploy=True)
unstub()
dockerfile = self.file_contents('Dockerfile')
self.assertEqual(
dockerfile,
DOCKERFILE_TEXT.format(
ruby_version='',
entrypoint='bundle exec ruby index.rb $PORT'))
dockerignore = self.file_contents('.dockerignore')
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
def test_generate_with_prompt_no_write(self):
"""Tests generate_config_data with entrypoint given by prompt."""
self.write_file('index.rb', 'class Index; end')
self.write_file('Gemfile', 'source "https://rubygems.org"')
unstub = self.stub_response('bundle exec ruby index.rb $PORT')
cfg_files = self.generate_config_data(deploy=True)
unstub()
self.assert_genfile_exists_with_contents(
cfg_files,
'Dockerfile',
DOCKERFILE_TEXT.format(
ruby_version='',
entrypoint='bundle exec ruby index.rb $PORT'))
self.assertIn('.dockerignore', [f.filename for f in cfg_files])
dockerignore = [f.contents for f in cfg_files if
f.filename == '.dockerignore'][0]
self.assertIn('.dockerignore\n', dockerignore)
self.assertIn('Dockerfile\n', dockerignore)
self.assertIn('.git\n', dockerignore)
self.assertIn('.hg\n', dockerignore)
self.assertIn('.svn\n', dockerignore)
if __name__ == '__main__':
unittest.main()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,399 |
Home Sports Happy Harvick breaks winless drought at New Hampshire
Happy Harvick breaks winless drought at New Hampshire
Posted By: Trussville Tribune staffon: July 22, 2019 In: Sports
From The Trussville Tribune staff reports
NASCAR's Monster Energy NASCAR Cup Series (MENCS) and NASCAR Xfinity Series (NXS) took on New Hampshire Motor Speedway in Loudon, New Hampshire, while the ARCA Menards Series was at Iowa Speedway in Newton, Iowa, this past weekend. Below is a recap of the events.
Monster Energy NASCAR Cup Series
It was an intense battle between Kevin Harvick and Denny Hamlin on the final lap of Sunday's Monster Energy NASCAR Cup Series (MENCS) race at New Hampshire. Coming off the final turn, the No. 4 of Harvick made contact with the No. 11 of Hamlin, which slowed Hamlin's machine and catapulted Harvick to victory, breaking a 21-race winless streak and giving Harvick his first win of the season.
It was Harvick's second straight at the Magic Mile and fourth overall.
Hamlin crossed the line second, .210 seconds behind.
Erik Jones took home third, with Ryan Blaney and Matt DiBenedetto (second top five of the season) completing the top five.
The rest of the top 10 included: Martin Truex Jr., Ryan Newman, Kyle Busch, Joey Logano and pole winner Brad Keselowski.
Harvick, along with Hamlin, Truex, Kyle, Logano, Keselowski, Kurt Busch, Alex Bowman and Chase Elliott have clinched spots in the MENCS Playoffs with wins this season.
Talladega Superspeedway's second MENCS race of 2019 is set for Sunday, Oct. 13, with the running of the 1000Bulbs.com 500, and will be the second race in the Round of 12 of the playoffs.
Christopher Bell, who led 186 of 200 laps, waxed the field in the NASCAR Xfinity Series (NXS) event Saturday afternoon at New Hampshire. It was Bell's second triumph at the 1.058-mile track (second start), claiming his fifth victory of the season.
Runner-up and pole-sitter Cole Custer crossed the line 4.068 seconds behind.
Justin Allgaier followed in third, with Tyler Reddick in fourth and Cup Series regular Paul Menard in fifth.
With wins this season, Bell, Custer, Reddick and Michael Annett have secured spots in the NXS Playoffs.
ARCA Menards Series
General Tire Pole Award winner Chandler Smith dominated the ARCA Menards Series event at Iowa Speedway on Friday night, leading 140 of the 150-lap event and claiming his fourth victory of the season.
The margin of victory was 1.515 seconds ahead of second-place finisher Christian Eckes.
Michael Self, Carson Hocevar and Ty Gibbs rounded out the top five.
Bret Holmes (Munford, Ala.) finished eighth and sits second in the point standings, 90 points behind leader Self.
Tune-In TV/Radio Coverage
The MENCS will compete at Pocono Raceway, Sunday, July 28 at 2 p.m. (CDT). NBCSN will televise the event. MRN and SiriusXM will provide radio coverage.
The NXS will be at Iowa Speedway, Saturday, July 27 at 4 p.m. (CDT). NBCSN will televise the race, while MRN and SiriusXM will provide radio coverage.
The NASCAR Gander Outdoors Truck Series will return to action at Pocono Raceway on Saturday, July 27 at noon (CDT). FOX will televise the event. MRN and Sirius XM will provide radio coverage.
The ARCA Menards Series will also race at Pocono Raceway on Friday, July 26 at 3 p.m. (CDT). FOX Sports 1 will televise the event.
Tags: Happy HarvickNASCARnew hampshire
Revamped OxyContin was supposed to reduce abuse, but has it?
10 arrested for shoplifting in Trussville
Hewitt-Trussville wrestling falls to Huntsville in first round of 2020 AHSAA State Dual Championships
Hewitt-Trussville boys and girls' bowling teams book tickets to 2020 AHSAA State Championship | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,692 |
This year, two diocesan high schools made the top 10 in the Iowa AP Index, which measures student participation in advanced placement courses. Prince of Peace Catholic School in Clinton placed fifth while Regina Catholic Education Center in Iowa City placed sixth.
AP Environmental Science students from Prince of Peace Catholic School in Clinton walk through the prairie at Syracuse Wildlife Area in Calamus, Iowa.
All high schools in the Diocese of Davenport currently offer advanced placement classes with an option of taking an exam at the conclusion of the course. High scores on AP tests can result in college credit, depending on the college's requirements. This can help cut down on college costs down the road.
Generally, AP exams are optional. Regina and Prince of Peace weave them into the curriculum. Regina offers a weighted course grade to all students who take the AP exam at the end of a course. Prince of Peace uses the AP exam as the second-semester exam for most of its course offerings.
"We want our students to have every opportunity to get that college credit," said Nancy Peart, principal of Prince of Peace. The Clinton school requires all students to take at least one AP class.
Zoulek said an average three-credit college course costs between $550 and $1,200, sometimes more. The cost to take an AP exam is $93.
Catholic school alumni have reaped the benefits of taking AP classes and getting college credit based on their exam scores. David Rudolph, a 2012 Regina graduate, said taking AP exams and receiving college credit allowed him the freedom to take more elective courses at the University of Oklahoma, as well as study abroad. Mia Boldt, a 2014 Regina graduate, was able to save money at the University of Michigan by being able to get the college credit while still in high school. Another Regina graduate, who recently returned to the school to speak to students, said she was able to double major and still be on track to graduate in four years.
Zoulek said AP courses look great on a transcript and can help set a student apart from other college applicants, even if their exam scores did not qualify for college credit.
Katharine Atkinson, AP Environmental Science teacher at Prince of Peace Catholic School in Clinton, participated in a summer internship with the Department of Natural Resources last year, learning about prairie and wetland habitat management. She is engaging her students in a Pollinator Prairie restoration project at the Syracuse Wildlife Area in Calamus, Iowa, about 30 miles from the school. The students have cleared rocks, cement blocks and logs out of the site and planted Little Bluestem seeds hand-harvested from a nearby grass prairie area. Over the next decade, the AP Environmental Science classes will monitor the Pollinator Prairie planting, documenting what grows. Atkinson said, "We will have lists of desired plants seeded, as well as invasive plants that we hope will not thrive. As a beautiful flowering prairie develops, students should build a personal connection to caring for the Earth and fulfilling the pope's mandate" to care for the earth. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,105 |
\section{Introduction}
Nuclear multifragmentation is an important phenomenon, the study of which can reveal reaction mechanism in heavy ion collisions at intermediate and high energies \cite{Moretto,Gross1,Jacob,Cole,Mallik102}. Central collision fragmentation reactions around fermi energy domain are extensively used for producing neutron rich isotopes and for studying nuclear liquid gas phase transition.\\
Different theoretical models have been already developed for throwing light on the nuclear multifragmentation reaction and for explaining relevant experimental data. The models are mainly classified into two categories: (i) Dynamical models \cite{Dasgupta,Ono,Hartnack} and (ii) Statistical models \cite{Das1,Bondorf1,Gross2,Randrup,Brohm,Raduta,Lacroix}. The dynamical models are based on more microscopic calculations where the time evolution of projectile and target nucleons are studied. In statistical models the clusterization technique is nicely incorporated but the disadvantage of statistical model is that the calculation are started by assuming some initial conditions (like temperature, excitation energy, freeze-out volume, fragmenting source size etc.). These condition are either parameterized or obtained from some experimental observables.\\
In this work we develop a hybrid model for explaining multifragmentation reaction around fermi energy domain. We treat central collision only. Initially the excitation of the colliding system is calculated by using dynamical Boltzmann-Uehling-Uhlenbeck (BUU) approach \cite{Dasgupta,Mallik9} with proper consideration of pre-equilibrium
emission. Then the fragmentation of this excited system is calculated by Canonical Thermodynamical model (CTM) \cite{Das1}. The decay of excited fragments, which are produced in multifragmentation stage is calculated by an evaporation model \cite{Mallik1} based on Weisskopf theory. Different observables like charge distribution, largest cluster distribution etc. are calculated by using this hybrid model for $^{129}$Xe+$^{119}$Sn reaction at different projectile energies and compared with experimental data \cite{Hudan} The idea of setting the initial conditions for a statistical model from a dynamical model is of course not new; see for example Barz et al \cite{Barz}. In many Statistical Model of Multifragmentation (SMM), the initial conditions are fixed by some measured data. In our hybrid model the initial conditions for the thermodynamical model are set up almost entirely by the transport model calculation.\\
The concept of temperature is quite familiar in heavy ion collision and it is a better observable (compared to energy) for studying liquid gas phase transition. One standard way of extracting temperatures is the Albergo formula \cite{Albergo}, where temperature is calculated from the measured isotopic yields (i.e. cold fragments). Another common technique for obtaining temperature is to measure the kinetic energy spectra of emitted particles. But in this method, the effect of sequential decay from higher energy states, Fermi motion, pre-equilibrium emission etc complicate the scenario of temperature measurement. Our hybrid model calculation is an alternative method for deducing the freeze-out temperature and it bypasses all such problems.\\
\begin{figure*}
\includegraphics[width=16cm,keepaspectratio=true,clip]{Fig1.eps}
\caption{(Color online) Evolution of test particles at (a) 0 fm/c, (b) 75 fm/c and (c) 200 fm/c in center of mass frame for 45 MeV/nucleon $^{129}$Xe on $^{119}$Sn reaction.}
\label{fig1}
\end{figure*}
\section{Basics of the dynamical model}
The hybrid model consist of three different stages: (i) Initial condition determination by dynamical BUU calculation, (ii) fragmentation by canonical thermodynamical model and (iii) decay of excited fragments by evaporation model.\\\
We start our calculation when two nuclei in their respective ground states approach each other with specified velocities. The mean field potential energy
density is taken from Lenk-Pandharipande \cite{Lenk}:
\begin{equation}
v(\rho(\vec{r}))=\frac{A}{2}\rho^2(\vec{r})+
\frac{B}{\sigma+1}\rho^{\sigma+1}(\vec{r}) +\frac{c\rho_0^{1/3}}{2}
\frac{\rho(\vec{r})}{\rho_0}\nabla_r^2[\frac{\rho(\vec{r})}{\rho_0}]
\end{equation}
where first two term represents zero range Skyrme interaction and the derivative term does not affect nuclear matter properties but in a finite system it produces quite realistic diffuse surfaces and liquid drop binding energies. This can be achived for $A=$-2230.0 MeV $fm^3, B$=2577.85 MeV $fm^{7/2}, \sigma=$7/6, $\rho_0=0.16$ and $c$=-6.5 MeV$fm^{5/2}$ \cite{Lenk}. We first construct Thomas-Fermi solutions for ground states \cite{Lee}. The Thomas-Fermi phase space distribution will then be modeled by choosing test particles with appropriate positions and momenta using Monte Carlo. Each nucleon is represented by 100 test particles ($N_{test}=100$). We begin Boltzmann-Uehling-Uhlenbeck (BUU) model calculation to get the excitation of the fragmenting system.
In the center of mass frame the test particles of the projectile and the target nuclei (in their Thomas-Fermi ground state) are boosted towards each other. The test particles move in a mean-field $U(\rho(\vec{r}))$ (generated by the potential energy density of Eq.(1)) and will occasionally suffer two-body collisions when two of them pass close to each other and the collision is not blocked by Pauli principle. The mean-field propagation is done using the lattice Hamiltonian method which conserves energy and momentum very accurately \cite{Lenk}. Two body collisions are calculated as in Appendix B of ref. \cite{Dasgupta}, except that pion channels are closed, as there will not be any pion production in this energy region. Positions and momenta of the test particles are updated after each time steps ($\Delta t$) by the equations
\begin{eqnarray}
\frac{d\vec{p}_i}{dt}&=&-\nabla_r U(\rho(\vec{r}_i),t)\nonumber\\
\frac{d\vec{r}_i}{dt}&=&\vec{v}_i\nonumber\\
&& i=1,2,.....,(A_p+A_t)N_{test}
\end{eqnarray}
\section{Excitation Energy Determination}
We can calculate the excitation energy ($E^*$) from
projectile beam energy ($E_{beam}$) by direct kinematics by assuming that the projectile and the
target fuse together. In that case the excitation energy is
$E^*=A_pE_{beam}/(A_p+A_t)$ where $A_p$ and $A_t$ are projectile and target
masses respectively. This value is too high as a measure of the excitation
energy of the system which multifragments. Pre-equilibrium particles
which are not part of the multifragmenting system carry off a significant
part of the energy.
\begin{figure}[b]
\includegraphics[width=3.0in,height=3.0in,clip]{Fig2.eps}
\caption{Variation of energy of the central dense region (containing $80\%$ of total test particles) with time obtained from dynamical BUU calculation for $^{129}$Xe on $^{119}$Sn reaction at 45 MeV/nucleon.}
\label{fig2}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth,height=2.5in,clip]{Fig3.eps}
\caption{Left Panel indicates the variation of excitation energy per nucleon with projectile beam energy per nucleon obtained from dynamical BUU model. The Canonical Thermodynamical Model (CTM) can calculate average excitation energy per nucleon for a given freeze-out temperature, mass number and charge. Therefore to know the required freeze-out temperature corresponding to each excitation (obtained from BUU calculation) CTM is used. The variation of freeze-out temperature with projectile beam energy is shown in the right panel.}
\label{fig3}
\end{figure}
To get a better measure of excitation of the fragmenting system we need to do a BUU calculation where the pre-equilibrium particles can be identified and can be taken out to calculate excitation energy per nucleon. We exemplify our method with central collision reactions $^{129}$Xe+$^{119}$Sn at projectile beam energy $45$ MeV/nucleon. Initially the center of $^{129}$Xe and $^{119}$Sn are kept at (100fm, 100fm, 90fm) and (100fm, 100fm, 110fm) respectively and and they are boosted towards each other along $z$ direction. Fig. 1 shows the test particles at t=0 fm/c (when the nuclei are separate), 75 fm/c (the time when violent collisions occur) and 200 fm/c (almost all collisions are completed). From the figure it is clear that for t=200 fm/c some test particles are far distant from the central dense region. These fit the category of pre-equilibrium emission. In different multifragmentation experiments, it is observed that after pre-equilibrium emission, $75\%$ to $80\%$ of the total mass creates the fragmenting system \cite{Xu,Frankland,Verde}. Hence we choose the test particles which create $80\%$ of the total mass (i.e. $A_0=198$) from the most central dense region. Knowing the momenta of selected test particles the kinetic energy is calculated and from the positions of these selected test particles the potential energy is calculated by using Eq. 1. By adding kinetic and potential energy the energy of the fragmenting system is obtained. Fig. 2 shows the variation of excited state energy of the central dense region (i.e. $80\%$ of the total test particles) with time. Here total energy is always constant but as time progresses, pre-equilibrium particles having high kinetic energy, are escaping from the central dense region, therefore the energy of the central dense region is decreasing. It is clear that after t=100 fm/c, the energy becomes independent of time. Hence, we can stop BUU calculation at any time after t=100 fm/c and consider the corresponding energy as excited state energy. To get the excitation we need to know the ground state of the fragmenting system. For this we use the Thomas Fermi method for a spherical nucleus of mass $A=198$ ($80\%$ of $^{129}$Xe+$^{119}$Sn mass). Subtracting ground state energy from the calculated energy above the excitation energy is obtained.
\begin{figure}[t]
\includegraphics[width=\columnwidth,keepaspectratio=true,clip]{Fig4.eps}
\caption{(Color online) Theoretical charge distribution (red dotted lines) for $^{129}$Xe
on $^{119}$Sn reaction at (a) 32 MeV/nucleon (b) 39 MeV/nucleon
(c) 45 MeV/nucleon and (d) 50 MeV/nucleon. The experimental data are shown by
black squares.}
\label{fig4}
\end{figure}
\section{Computations with the statistical model: Extraction of Temperature}
We have described above how from BUU we extract the mass, charge and the excitation energy of the fragmenting system. Our next task is to obtain the freeze-out temperature. The canonical thermodynamic model (CTM) \cite{Das1} can be used to calculate average excitation per nucleon for a given temperature, charge and mass. Getting an excitation energy for a given temperature, mass and charge is described in detail in \cite{Das1}. We do the exploration for each beam energy. We will not repeat the formulae of CTM here but just mention that apart from neutrons and protons the following composites are included in CTM breakup. We include deuteron,triton, $^3He$, $^4He$ and for heavier nuclei we include a ridge along the line of stability. The composites that follow from CTM will further decay by evaporation. The details of how we do it can be found in \cite{Mallik1}.
\section{Results}
We have done calculations for the same $^{129}$Xe+$^{119}$Sn pair for projectile beam energies 32, 39, 45 and 50 MeV/nucleon. In each case, we have stopped the time evolution at $t=200$ fm/c, and selected $80\%$ of the total mass from central dense region for calculating the excited state energy. Then subtracting the ground state energy the excitation is obtained. The variation of calculated excitation energy with projectile beam energy is shown in the left diagram of Fig. 3. From this excitation energy we find out the corresponding freeze-out temperature. Thus the freeze-out temperature for a given beam energy is obtained. This is plotted on the right side of Fig.3.\\
\begin{figure}[h!]
\includegraphics[width=\columnwidth,keepaspectratio=true,clip]{Fig5.eps}
\caption{(Color online) Theoretical largest cluster probability distribution (red dotted lines) for $^{129}$Xe on $^{119}$Sn reaction at (a) 32 MeV/nucleon (b) 39 MeV/nucleon (c) 45 MeV/nucleon and (d) 50 MeV/nucleon. The experimental data are shown by black squares.}
\label{fig5}
\end{figure}
To check the accuracy of our model, we have compared the theoretical results with experimental data. Fig. 4 shows the comparison of charge distribution at projectile beam energies 32, 39, 45 and 50 MeV/nucleon. With the increase of energy (i.e. increase of temperature), fragmentation is more, therefore multiplicities of higher fragments gradually decrease. Fig. 5 represents the largest cluster probability distribution at different energies. Since with the increase of energy breaking increases, the peak of the largest cluster probability distribution shifts towards the lower atomic number side and the width of the distribution gradually decreases. The variation of average charge of largest cluster $\langle Z_{Largest}\rangle$ with projectile beam energy is shown in Fig. 6. In each case nice agreement between theoretical result and experimental data is obtained.\\
\begin{figure}[t]
\includegraphics[width=2.5in,height=2.5in,clip]{Fig6.eps}
\caption{(Color online) Variation of average size of largest cluster with projectile beam energy obtained from hybrid model calculation (red dotted lines) for $^{129}$Xe on $^{119}$Sn reaction. The experimental data are shown by black squares.}
\label{fig6}
\end{figure}
\section{Discussion}
In this work we do a BUU calculation to get the excitation energy of the multifragmenting system produced in the central collision reactions around Fermi energy domain, then do a CTM exploration to locate the temperature which will give this excitation. Once this temperature is fixed CTM is used to fit available experimental data. The agreement with data is pleasing. This work complements the work we did where we fitted the data obtained from
the decay of projectile like fragments at energies in the limiting fragmentation region. There \cite{PLF} we fitted the data using CTM with an assumed temperature profile first and later \cite{Mallik9} we showed that the temperature profile is obtainable from BUU calculations. Our present calculations are not prohibitively computer intensive. One virtue of these calculations is that equation 1 leads to reasonable values of binding energy of finite nuclei (even in Thomas-Fermi approximation) and realistic diffuse surface without having to supplement the zero range Skyrme interaction with a finite range interaction.
Vlasov propagation for large nuclei when finite range interaction is present is very computer intensive. The other pleasing aspect is that the lattice Hamiltonian method \cite{Lenk} gives remarkable accuracy in total energy and total momentum conservation in these calculations.\\
For fragmenting system, we adopted the value of $80\%$ of total mass from the experimental papers quoted in our paper. But our results (see Fig. 6) show that this was a reasonable choice. We show a plot of $\langle Z_{Largest}\rangle$ which agrees fairly well with data. Now $\langle Z_{Largest}\rangle$ depends upon the size of the fragmenting system as well as the temperature of the fragmenting system. The larger the fragmenting system, the larger is the $\langle Z_{Largest}\rangle$. The higher the temperature, the smaller is $\langle Z_{Largest}\rangle$. Now the temperature also depends upon what percentage of nucleons are left out as pre-equilibrium particles. The value $80\%$ we choose gives a combination of temperature and fragmenting mass that seems to be just about right. One could do a detailed best "fit" but this was not attempted.\\
What we presented in this work did not involve any radial flow. One reason is that the collision energy being only about 50 MeV/nucleon, the initial compression is small so any radial flow must also be small. The best signature for radial flow will be in the velocity distribution but we are only calculating multiplicity distribution. Neither CTM nor SMM can incorporate radial flow easily but in Lattice gas model, where flow is easily incorporated. it was found that even for significant radial flow, multiplicity distributions are hardly affected \cite{Das_LCG}.
\section{Acknowledgements}
This work was supported in part by Natural Sciences and Engineering Research Council of Canada.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,575 |
Q: problems with downloading Winehq on Ubuntu 16.04 I was following the instructions on the https://wiki.winehq.org/Ubuntu for winehq installation. When I executed
sudo apt-add-repository 'deb https://dl.winehq.org/wine-builds/ubuntu/ xenial main'
a blank line appeared with no response and it brought me back to the input line. I thought maybe it worked and then I executed sudo apt update and it said
Malformed entry 6 in list file /etc/apt/sources.list (component).
Can anyone help?
deb-src https://archive.Ubuntu.com/Ubuntu xenial-security main restricted universe multiverse
Also if I may add I am running a chroot
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,175 |
echo "127.0.0.1 localhost localhost.localdomain localhost4 localhost4.localdomain4" > /etc/hosts
echo "::1 localhost localhost.localdomain localhost6 localhost6.localdomain6" >> /etc/hosts
echo "127.0.0.1 $HOSTNAME" >> /etc/hosts
# folder permissions
chmod -R 777 /files
chmod -R 777 /scripts
# set timezone
ln -s -f /usr/share/zoneinfo/${TIME_ZONE} /etc/localtime
# disable SELinux
setenforce disabled
if [ -f /etc/selinux/config ]; then
sed -i -E 's:SELINUX=enforcing:SELINUX=disabled:g' /etc/selinux/config
fi
# update YUM
yum clean all
yum update -y
yum install -y openssl
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,669 |
Q: One dimensional heat equation using Fourier Transform for temperature distribution One dimensional heat equation is
$$ \dfrac{\partial u}{\partial t} = k \dfrac{\partial^2 u}{\partial x^2} $$
and initial condition is
$$ u(x,t) = f(x) $$
I'm trying to find the temperature distribution by using Fourier transform for $ -\infty < x < \infty $
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 576 |
Compatibility Requires iOS 4.0 or later.
By watching our high quality video lessons, learn the fundamentals of professional and olympic boxing from our USA Boxing certified professional coaches. You will learn the 10 punches of boxing, proper stance, footwork, blocking, head movement, slipping, weaving, bobbing, and much more.
Our world-class coaches have trained champion professional boxers and UFC fighters from around the world. As a bonus, we've included a week long workout program complete with a ring timer and music player that'll get you into fighting shape in no time. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,914 |
Q: Solving functional differential equation $f'(x)=2f(2x)-f(x)$
Show that there is at least a nonzero function $f$, differentiable on $[0,+\infty)$, satisfying $$f'(x)=2f(2x)-f(x) \qquad \forall x>0 $$
$$M_n:=\int_{0}^{\infty}x^nf(x)dx<\infty \qquad \forall n\in
\mathbb{N} $$
My best idea so far is to assume that the solution is a power series, i.e.
$$ f(x)=\sum_{n=0}^{\infty}a_nx^n\qquad \forall x>0$$
Then the equation becomes
$$ \sum_{n=0}^{\infty}na_nx^{n-1}=2\sum_{n=0}^{\infty}a_n2^nx^n-\sum_{n=0}^{\infty}a_nx^n$$
equating all the coefficients of the same degree I get
$$na_n=(2^{n}-1)a_{n-1}\qquad \forall n\geq 1$$
So setting $a_0=1$, I get
$$a_{n}=\frac{1}{n!}\prod_{k=1}^{n}(2^k-1) \qquad \forall n$$
But does the power series actually converge? Using Hadamard's formula, and that $2^{k}-1\geq 2^{k-1}$,
$$ |a_n|^{1/n}\geq\frac{1}{(n!)^{1/n}}\left[2^{n(n-1)/2}\right]^{1/n}\sim\frac{e}{n(2\pi n)^{1/2n}}2^{(n-1)/2}\to \infty$$
so the radius of converge of the series is $0$, so it doesn't actually define a solution on $[0,+\infty)$.
A: Well, it looks like your best idea (which by itself was not bad at all) fails precisely for the reasons you so comprehensively described. So what shall we do?
Guess what?
We turn it the other way around. Let's look for a function in the form of
$$f(x)=\sum_{n=0}^\infty a_ne^{-2^nx},\quad x\geqslant0$$
You might wonder what's with $2^n\cdot x$ up there in the exponent. Well, it's simple: I was going to write $e^{-nx}$ when I realized we won't be needing most of those terms, so let it be this way.
Now, the equation becomes
$$-\sum_{n=0}^{\infty}2^na_ne^{-2^nx}=2\sum_{n=0}^{\infty}a_ne^{-2^{n+1}x}-\sum_{n=0}^{\infty}a_ne^{-2^nx}$$
or
$$\sum_{n=0}^{\infty}2^na_ne^{-2^nx} = -2\sum_{n=1}^{\infty}a_{n-1}e^{-2^nx}+\sum_{n=0}^{\infty}a_ne^{-2^nx}$$
which gives
$$(2^n-1)a_n=-2a_{n-1},\quad n\geqslant1$$
So continuing in your footsteps, I set $a_0=1$ and get
$$a_n=\frac{2^n}{\prod_{k=1}^{n}(2^k-1)}, \quad n\geqslant1$$
which makes my series converge pretty fast, for the same reason why your series fails to do so.
Now that's our solution.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,656 |
\section{Introduction}
In 1988 Cipolla wrote an essay entitled The fundamental laws of human stupidity \cite{cipolla88}. The structure of this essay consisted in several
chapters with some of them intended for the introduction and discussion of each of the five fundamental laws that according to Cipolla rule the
human stupidity.
As the concept of stupidity can be ambiguous it is important to properly frame the meaning we embrace here. A stupid person is someone given to
unintelligent decisions or acts but here we consider those acts within a social context. Stupidity should not be understood as the opposite
of intelligence. In fact, according to some of the ideas of Cipolla, none of us gets rid of or will never get rid of a brief moment of stupidity.
The association of stupidity to a source of collective troubles and nuisance and to the origin of social scourges has been manifested through
history in several nowadays popular quotes . Among them it is worth citing a phrase credited to A. Dumas: "One thing that humbles me
deeply is to see that human genius has its limits while human stupidity does not" \cite{dumas}. B. Russell, in his essay \textit{The triumph of
stupidity} wrote: "The fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt"
\cite{russ}.
Cipolla starts by preventing us about the silent danger of stupidity in the first law. There he affirms that detecting a stupid person is a hard
task. This law states that always and inevitably everyone underestimates the number of stupid individuals in circulation.
While it could be tempting to associate stupidity with lack of education or training, Cipolla affirms that the probability that a certain
person be stupid is independent of any other characteristic of that person. This is the content of the second law. This somehow suggests that there
is a natural stupidity, which resists any academic training.
But again, we are here interested in an operational definition of stupidity, the one that is dangerous for others, an idea that can be summarized
by a quote taken from one of M. Atwood novels, \cite{atwood}: "Stupidity is the same as evil if you judge by the results."
This idea is expressed in Cipolla's third law: A stupid person is a person who causes losses to another person or to a group of persons while himself
deriving no gain and even possibly incurring losses.
This law also suggest the definition of three other phenotypes that complement the stupid group (S). These three groups, according to Cipolla, are
the intelligent people (I), whose actions benefit both themselves and others, the bandits (B), who benefits themselves at the expense of others, and
finally the helpless or unaware people (U), whose actions enrich others at their own expense.
Stupid people are dangerous and damaging because their behavior is hard to understand and predict from a rational point of view. The bandit's
actions, while producing some damage, obey a predictable pattern of rationality. The possibility to foresee the behavior of a bandit can help an
individual to build up defenses against it. On the contrary, when facing a stupid person this is impossible.
So, while most of the time the evil has a clear face and is easily identifiable, stupidity is not. This biased evaluation is what is considered in
the forth law: Non-stupid people always underestimate the damaging power of stupid individuals. In particular non-stupid people constantly forget
that at all times and places and under any circumstances to deal and/or associate with stupid people always turns out to be a costly mistake.
The effect of stupidity and the difficulty to recognize it is what leads us to the fifth law: A stupid person is the most dangerous type of person.
Cipolla characterized the four groups in terms of two parameters; the own gains or losses
$p$, and the gains or losses that an individual inflicts on others, $q$. The payoff resulting from the interaction between two persons can be
definedin terms of these quantities associated to the identification of the participants with one of the four defined groups.
These four groups can then be characterized by the range of values adopted by $p$ and $q$ as follows :
\begin{itemize}
\item[S] : $p_s \leq 0$ y $q_s < 0$
\item[U]: $p_u \leq 0$ y $q_u\geq 0$
\item[I]: $p_i > 0$ y $q_e \geq 0$
\item[B]: $p_b > 0$ y $q_b < 0$
\end{itemize}
In \cite{joel} we presented a four strategy game based on this four groups. The payoff of the strategies was defined by the values of $p$ and $q$
as presented in Table \ref{tabla1}, that indicates which is the payoff of the strategy at the file when competing with the strategy at the column
\begin{table}[h]
\centering
\begin{tabular}{c| c| c|c |c}
&{\bf S} & {\bf U} &{\bf I} & {\bf B}\\
\hline
{\bf S} & $p_s +q_s$ & $p_s+q_u$& $p_s+q_i$ & $p_s+q_b$ \\
\hline
{\bf U} & $p_u +q_s$ & $p_u +q_u$& $p_u +q_i$ & $p_u +q_b$ \\
\hline
{\bf I} & $p_i +q_s$ & $p_i +q_u$& $p_i +q_i$ & $p_i +q_b$ \\
\hline
{\bf B} & $p_b +q_s$ & $p_b +q_u$ & $p_b +q_i$ & $p_b +q_b$
\end{tabular}
\caption{Payoff Table} \label{tabla1}
\end{table}
The results shown in \cite{joel} supported the validity of the law enunciated by Cipolla that contained some appreciations about the dangerousness
of the presence of stupid people. In this work, the evaluation of the effect of the actions of this group was done by calculating
the total wealth of the population in the steady state of the strategy profile resulting from an evolutionary game and an imitation dynamics.
The presence of stupid people not only undermined the total wealth but also promoted the inhibition of cooperative behaviors represented by
intelligent and unaware people. In that work we could not find a critical value for the fraction of stupids that could divide the behavior of the
system into two different regimes. According to the the first law it is not possible to know the number of stupid individuals in a
population, so it might be interesting to find different regimes for different fraction of stupid people.
In the present work we have adopted a different approach, interpreting Cipolla' s idea in a different way.
We will consider a three strategy game, where the participating strategies will be (I), (B) and (U) and we will let any individual to occasionally
behave as a stupid person with a given probability. This probability is the parameter that will govern the amount of stupid people at each time.
This means that being stupid will not be a permanent state by an occasional state accounting for the possibility that at any time any individual can
behave stupidly. In the following sections we present a more thorough description of the the model and the numerical results.
\section{The model}
As mentioned before, we are going to consider an evolutionary game, with four strategies though one of them, the (S), differs from the
others in the
sense that it is not durable and it can not be imitated. Any individual can be stupid at any time and during one time step with probability $\rho_s$ .
While stupidity will not be a permanent condition on this version of the game, we still need to define the payoff of each of the three strategies
when playing between them and when ocassionally confronting with a stupid person. We also need to define the payoff that
an eventual stupid player may receive.
We introduce then a 4x4 payoff matrix
\[A=\begin{pmatrix}
p_s +q_s & p_s+q_u& p_s+q_i & p_s+q_b\\
p_u +q_s & p_u +q_u& p_u +q_i & p_u +q_b \\
p_i +q_s & p_i +q_u& p_i +q_i & p_i +q_b\\
p_b +q_s & p_b +q_u & p_b +q_i & p_b +q_b
\end{pmatrix}\]
To compare the effects of this new model with those obtained in \cite{joel}, we will adopt the values in Table \ref{tabla2}.
\begin{table}[h]
\centering
\begin{tabular}{| c| c|c |c|c|c|c|c|}
\hline
$x_i$ &$x_b$ &$x_d$ &$x_e$&$y_i$&$y_b$&$y_d$&$y_e$\\
\hline
1 &[1.1,2] &[-2,-1] &[-2,-1]&1&-1&1&-1\\
\hline
\end{tabular}
\caption{Chosen values for the payoff matrix} \label{tabla2}
\end{table}
The only Nash equilibrium of this game is strategy (B). As shown in \cite{joel}, considering the chosen values for the payoffs, the sub game in which
only the strategies (I) and (B) participate constitutes a Prisoner's Dilemma (PD) or a Donation Game \cite{sigm10} .
Despite that there is only one Nash equilibrium and thus an evolutionary game ruled by the replicator mean field equations will have only one steady
state, when considering a spatially extended game with players located on top a network, the steady state can show a different steady configuration
\cite{dur,kup3,yukov,pavlogiannis}.
In order to compare the results obtained here with those previously shown in \cite{joel} we locate the players on top of the networks used in that
work . We consider a particular family of regular networks, i.e. with all the node having the same degree, but with a tunable degree
of disorder. These networks. described in \cite{kup2} present a topology that varies according to a disorder parameter $\pi_d$. This parameter is
responsible for the change in the value of the clustering coefficient $C$ of the network.
This dependence is shown if Fig. (\ref{clust})
\begin{figure}[h]
\includegraphics[width=.8\textwidth]{clust.eps}
\caption{Mean clustering coefficient of the used networks as a function of the degree of disorder}
\label{clust}
\end{figure}
We will adopt a simple evolutionary dynamics for the strategies considering a deterministic imitation. In each round a selected player plays with all
its neighbors. In turn, these neighbors do the same with their own. After that selected player analyses its performance or earnings
and compares them with that of its neighbors. Then, it adopts the strategy of the player with the highest gain, that eventually can be its own one.
In case of tie the choice is decided at random. This update dynamics is the simplest one, representing a deterministic imitation and closely linked
to the replicator dynamics \cite{hofb}.
Players will play according to their chosen strategy, that can be (I), (B) or (U) but at any time, any player can adopt the strategy (S) with
probability $\rho_s$. This election will not be permanent, will last only one time step (there is always a probability $\rho_s^n$ of
adopting the (S) strategy during $n$ consecutive steps) and after that, the player will adopt the original behavior or change it to imitate the
neighbor with the highest payoff. We recall that the (S)
strategy can not be imitated, but this is not an issue as under any circumstance a player adopting the (S) strategy will obtain the higher payoff.
The fact that there is probability $\rho_s$ of adopting strategy (S) means that at each time step there is a mean effective population of $\rho_s N$
stupids, where $N$ is the total population size.
\section{Results}
In our simulations we considered networks of $N=10^5$ nodes and the system evolved untill reaching a steady state. The degree of disorder $\pi_d$ and
the probability of adopting the (S) strategy at each
time step $\rho_s$ were chosen as parameters.
The results are shown in Fig. (\ref{figura1})
We observe that the topology of the network has a minor effect on the evolution of the strategy profile of the population. On the contrary
the probability of adopting the (S) strategy plays a crucial role. The system shows the existence of a critical value of $\rho_s$ separating the
evolution of the system towards two differently regimes. For low values of $\rho_s$ the dominating strategy is (I) while for higher values, (B)
dominates.
This is clearly reflected in the left panel of Fig (\ref{figura1}) showing the fraction of (I) in the steady state and also in the right panel
showing the mean gain of the population.
Clearly,
the prevalence of (B) attempts against the wealth of the population, and this prevalence is promoted by the sporadic appearance of the (S) behavior.
The interesting additional feature observe in this work is the existence of two well defined ranges, and a critical $\rho_s$ values more defined when
the network is more ordered.
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.8\linewidth]{int.eps}
\label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.8\linewidth]{gain.eps}
\label{fig:sub2}
\end{subfigure}
\caption{This plots shows the fraction of (I) individuals $\rho_i$ (left) and the main gain of the population in one round $<\epsilon>$ (right) as a
function of $\rho_s$. All the curves correspond to a the steady state}
\label{figura1}
\end{figure}
The presence of a sharp transition between a cooperative to a defective steady state was not observed in \cite{joel}.
To understand the relevance of this result we refer to previous results involving evolutionary games. There are several examples of the
effect of locating the players on networks with different topologies \cite{dur,kup3,kup2,now0,now3,sza2,now1,kup1,roc,ass,ift,vuk2,gan}.
These works show that the evolutionary behaviour of the strategies might be affected by the underlying topology of links
between players, sometime promoting cooperative states even when the Nash equilibrium is the defective strategy.
In these examples, the topology of the network is the factor responsible for different regimes. Here we show that while the topology of the networks
plays a non negligible role, the most important parameter is the probability of a player to adopt the (S) strategy.
The results show that as $\rho_i$ increases there is a transition from a scenario where (I) is prevalent to another one where most of the population
behaves as (B). The transition is sharper for highly ordered network and turns smoother as $\pi_d$ increases. Also, the critical value at which this
transition occurs moves to right with increasing $\pi_d$. This fact is shown in Fig, (\ref{icrit})
\begin{figure}[!tbp]
\includegraphics[width=.8\textwidth]{icrit.eps}
\caption{This plot shows the value of $\rho_s$ at which $\rho_i=0.5$ as a function of $\pi_d$. The curve is a spline for helping visualization.}
\label{icrit}
\end{figure}
\section{Conclusions}
As we stated in the introduction, the work by Cipolla should be understood in a cartoonish way. Nevertheless, it implies certain facts that deserve
to be taken into consideration. It is in this spirit that we analyzed the work by Cipolla and the results that can be obtained by translating these
ideas into a
mathematical model. Lets start by the first law. It affirms that we always underestimate the number of stupid individuals in
circulation. Inspired by this statement we wander whether despite the density of stupids is impossible to calculate there is a critical density
separating two different scenarios or not. We mean by this to evaluate the possibility that only by reaching a threshold density, the group of stupid
people can inflict a considerable harm to the entire population.
For that we proposed a model in which any individual is susceptible of behaving stupidly at any time and with a certain probability.
Our results show the existence of a critical probability $\rho_s \approx 0.35$. The transition from the cooperative regime to the defective one is
only sharp for slightly disordered networks, turning smoother as the disorder increases. Also, the increasing disorder produces a displacement of
$\rho_s$ to higher values. The curves show that while for low values of $\rho_s$ the disorder attempts against (I), the situation is reversed for
higher degrees of disorder. These results agree with those obtained in \cite{joel} for the case when the fraction of (I) individuals remains constant
along the whole simulation.
Another interesting feature is the increase in the fraction of (I) for higher values of $\rho_s$. This phenomenon can be attributed to a screening
effect played by the (S) population. The (I) players can not be affected and tempted by the presence of (B) ones and then they survive. This effect
has been also found in \cite{joel}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,937 |
Q: Django local dev server hangs with chrome after form submission. Works in firefox Ok this is a strange one.
Here is what I do to reproduce the problem (windows 10/python 3.7.1/django 2.5.5):
*
*Create a new virtual env with virtualenv-wrapper and 'mkvirtualenv' command
*Install Django in the virtual env using 'pip install django'
*Create a new Django project
*Migrate for first time to default sqlite database
*createsuperuser
*Run dev server
*Access 127.0.0.1:8000/admin via chrome
*Log in with superuser credentials
I see a http post in the dev server console window. It looks like this:
(default-users) C:\Users\kmfae\Documents\test\django-default-users>python manage.py runserver
Watching for file changes with StatReloader
Performing system checks...
System check identified no issues (0 silenced).
September 05, 2019 - 19:13:22
Django version 2.2.5, using settings 'defusers_project.settings'
Starting development server at http://127.0.0.1:8000/
Quit the server with CTRL-BREAK.
[05/Sep/2019 19:13:40] "GET /admin HTTP/1.1" 301 0
[05/Sep/2019 19:13:40] "GET /admin/ HTTP/1.1" 302 0
[05/Sep/2019 19:13:40] "GET /admin/login/?next=/admin/ HTTP/1.1" 200 1819
[05/Sep/2019 19:13:40] "GET /static/admin/css/login.css HTTP/1.1" 200 1233
[05/Sep/2019 19:13:40] "GET /static/admin/css/base.css HTTP/1.1" 200 16378
[05/Sep/2019 19:13:40] "GET /static/admin/css/responsive.css HTTP/1.1" 200 17944
[05/Sep/2019 19:13:40] "GET /static/admin/css/fonts.css HTTP/1.1" 200 423
[05/Sep/2019 19:13:40] "GET /static/admin/fonts/Roboto-Regular-webfont.woff HTTP/1.1" 200 85876
[05/Sep/2019 19:13:40] "GET /static/admin/fonts/Roboto-Light-webfont.woff HTTP/1.1" 200 85692
[05/Sep/2019 19:13:47] "POST /admin/login/?next=/admin/ HTTP/1.1" 302 0
But in the chrome browser I don't get redirected to the admin site. It sits there forever with the spinning icon as if it's loading a page. At least most of the time it does that. Rarely it actually does work as it should.
The same exact process works every time in firefox. Here's what the dev server looks like when I use firefox. Notice it gets past the initial http post where chrome gets hung:
(default-users) C:\Users\kmfae\Documents\test\django-default-users>python manage.py runserver
Watching for file changes with StatReloader
Performing system checks...
System check identified no issues (0 silenced).
September 05, 2019 - 19:06:37
Django version 2.2.5, using settings 'defusers_project.settings'
Starting development server at http://127.0.0.1:8000/
Quit the server with CTRL-BREAK.
[05/Sep/2019 19:06:45] "GET /admin/ HTTP/1.1" 302 0
[05/Sep/2019 19:06:45] "GET /admin/login/?next=/admin/ HTTP/1.1" 200 1819
[05/Sep/2019 19:06:52] "POST /admin/login/?next=/admin/ HTTP/1.1" 302 0
[05/Sep/2019 19:06:52] "GET /admin/ HTTP/1.1" 200 3042
Anyone have ideas what's going on???
A: You have two solutions:
*
*Upgrade Python to version 3.8, remove and rebuild the virtual environment.
or
*
*Downgrade Django from 3.0 to 2.2 version.
pip install "django>=2.2,<3"
Both solutions fix this problem. Execute python manage.py runserver
and logged into admin panel again.
Seems to be a problem with Python 3.7 and Django 3.0.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,030 |
Sviňucha černá je možná jedním z nejohroženějších malých kytovců při pobřeží Jižní Ameriky, ale je plachá a snadno se dá přehlédnout, takže se o ní ví jen málo. Její areál sahá od Ohňové země daleko na sever, na tichomořské straně až do severního Peru.
Potrava
Živí se rybami, hlavonožci a krilem nebo jinými korýši.
Váha
U dospělých je to asi 40–70 kg. Porodní váha není známa.
Určovací znaky
dozadu nakloněná hřbetní ploutev
zavalité a mohutné tělo
čelo má velmi ploché
málo čeří vodu
normálně žije v malých skupinách
Reference
Externí odkazy
Sviňucha černá na IUCN
Sviňuchovití | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,372 |
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"redpajama_set_name": "RedPajamaArXiv"
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Invercargill Golf Club golf professional Scott Riordan on a green in Otatara.
MEN'S health is the driving force for a new golfing event being held in conjunction with Movember.
As well as the annual moustache-growing challenge to raise awareness about men's health, the Movember Foundation charity is running a series of golf tournaments, the Mo Masters Series, throughout New Zealand.
It started in Auckland in September and will run to March, culminating in the championship finals in Queenstown. The Invercargill tournament will be held at the Invercargill Golf Club in Otatara on December 3.
New Zealand country manager for the Movember Foundation Robert Dunne said the Mo Masters was aimed at getting men active and talking with their friends over a round of golf, hopefully with men's health at the forefront of that.
"Sometimes people may not want to grow a moustache, so this is another fantastic way to support Movember.
"When it comes to their health, too many men don't talk, don't take action and die too young. There is no better platform than golf to get Kiwi men physically active, socially connected and talking about their health," Mr Dunne said.
Movember Foundation is a men's health charity which funds more than 1200 projects in 21 countries worldwide focusing on prostate cancer, testicular cancer, mental health and suicide prevention.
Invercargill Golf Club golf professional Scott Riordan said the club was happy to support the event.
The Mo Master series is a three-person Ambrose competition over 18 holes.
Each tournament includes a complimentary barbecue on-course, and beverages in the clubrooms, along with some competitive on-course activities such as long drive, closest to the pin, pro challenge and the hype hole with prizes valued up to $3000.
The winning team at each tournament will be invited to play in the finals weekend at the Millbrook Resort, near Arrowtown, to determine the overall Mo Master champion team. | {
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Q: How to properly register primitives and nulls in polymorphic serialization? I need to set up a serialization/deserialization mechanism for a polymorphic class hierarchy that also includes primitives and nulls. There are container classes containing collections with polymorphic objects, primitives, and nulls. And, the subclasses for these objects are spread across modules (therefore sealed is not an option).
I have been reading through the kotlinx.serialization polymorphism docs trying to come up with a solution. I've been able to make some incremental progress by working through that tutorial but I seem to still be hitting a wall when I try to put everything together.
The code I am posting here is a minimal example that brings together everything I need. If I can get this example to work, that should cover everything I need for my real project. This example does run without error but introduces some unnecessary readability and efficiency issues.
All classes in my custom class hierarchy are serializable data classes. The outermost container object that needs to be serialized/deserialized is a map wrapper. This map has keys which are each an instance of one of these data classes. And the values of this map can be primitives, nulls, or instances of one of my data classes. I think my main challenge here is to include those primitives and nulls in my polymorphic serialization in a clean way.
The goal of my code below is to represent this problem in the simplest way possible and to serialize and deserialize one container object successfully.
There are two main issues in the code:
*
*I've had to replace null with FakeNull. Without this, I get null cannot be cast to non-null type kotlin.Any. This will reduce the readability and simplicity of my code and I suspect it could decrease efficiency as well.
*I've had to add StringClassSerializer and DoubleClassSerializer and wrapper classes. I would also need to add serializers like these for every primitive class. If I don't register these primitives as subclasses of Any, I get Class 'String' is not registered for polymorphic serialization in the scope of 'Any'.. And if I try to register them with their default serializers (like subclass(String::class, String.serializer())) I get Serializer for String of kind STRING cannot be serialized polymorphically with class discriminator.. The problem with using serializers like StringClassSerializer and wrappers like StringWrapper is that it removes the efficiency and readability benefits of using primitives.
The json comes out looking like:
{"type":"MapContainer","map":[{"type":"SubA","data":1.0},{"type":"StringWrapper","s":"valueA"},{"type":"SubB","data":2.0},{"type":"DoubleWrapper","d":2.0},{"type":"SubB","data":3.0},{"type":"SubA","data":1.0},{"type":"SubB","data":4.0},{"type":"matt.play.FakeNull"}]}
I don't like the way this looks. I want the nulls to simply be null and the primitives to simply be primitives.
import kotlinx.serialization.KSerializer
import kotlinx.serialization.PolymorphicSerializer
import kotlinx.serialization.SerialName
import kotlinx.serialization.Serializable
import kotlinx.serialization.descriptors.buildClassSerialDescriptor
import kotlinx.serialization.encoding.Decoder
import kotlinx.serialization.encoding.Encoder
import kotlinx.serialization.json.Json
import kotlinx.serialization.modules.SerializersModule
import kotlinx.serialization.modules.polymorphic
import kotlinx.serialization.modules.subclass
import kotlin.collections.set
@Serializable
abstract class SuperClass
@Serializable
@SerialName("SubA")
data class SubA(val data: Double): SuperClass()
@Serializable
@SerialName("SubB")
data class SubB(val data: Double): SuperClass()
@Serializable
@SerialName("MapContainer")
data class MapContainer<K: SuperClass, V>(val map: Map<K, V>): Map<K, V> by map
@Serializable
@SerialName("StringWrapper")
data class StringWrapper(val s: String)
@Serializable
@SerialName("DoubleWrapper")
data class DoubleWrapper(val d: Double)
object StringClassSerializer: KSerializer<String> {
override val descriptor = buildClassSerialDescriptor("string")
override fun deserialize(decoder: Decoder) = decoder.decodeSerializableValue(StringWrapper.serializer()).s
override fun serialize(encoder: Encoder, value: String) =
encoder.encodeSerializableValue(StringWrapper.serializer(), StringWrapper(value))
}
object DoubleClassSerializer: KSerializer<Double> {
override val descriptor = buildClassSerialDescriptor("double")
override fun deserialize(decoder: Decoder) = decoder.decodeSerializableValue(DoubleWrapper.serializer()).d
override fun serialize(encoder: Encoder, value: Double) =
encoder.encodeSerializableValue(DoubleWrapper.serializer(), DoubleWrapper(value))
}
@Serializable
object FakeNull
fun main() {
val theMap = mutableMapOf<SuperClass, Any?>()
theMap[SubA(1.0)] = "valueA"
theMap[SubB(2.0)] = 2.0
theMap[SubB(3.0)] = SubA(1.0)
theMap[SubB(4.0)] = FakeNull /*wish I could make this just `null`*/
val theMapContainer = MapContainer(theMap)
val format = Json {
allowStructuredMapKeys = true
ignoreUnknownKeys = true
serializersModule = SerializersModule {
polymorphic(SuperClass::class) {
subclass(SubA::class)
subclass(SubB::class)
}
polymorphic(Any::class) {
/*I wish I could remove all of this primitive wrapper stuff*/
default {
when (it) {
StringWrapper::class.simpleName -> StringClassSerializer
DoubleWrapper::class.simpleName -> DoubleClassSerializer
else -> throw RuntimeException("unknown type: ${it}?")
}
}
subclass(String::class, StringClassSerializer)
subclass(Double::class, DoubleClassSerializer)
subclass(SubA::class)
subclass(SubB::class)
subclass(FakeNull::class)
}
polymorphic(
MapContainer::class, MapContainer::class, actualSerializer = MapContainer.serializer(
PolymorphicSerializer(SuperClass::class),
PolymorphicSerializer(Any::class)
) as KSerializer<MapContainer<*, *>>
)
}
}
val encoded = format.encodeToString(PolymorphicSerializer(MapContainer::class), theMapContainer)
println("\n\n${encoded}\n\n")
val decoded = format.decodeFromString(PolymorphicSerializer(MapContainer::class), encoded)
if (theMapContainer != decoded) {
throw RuntimeException("the decoded object is not the same as the original")
} else {
println("success")
}
}
A: Primitives (such as strings, numbers, and enums) by default are serialized as JSON primitives (e.g., "answer" or 42), not JSON objects ({ ... }). This is why they don't support polymorphic serialization; there is no "space" to place the type information in (the class discriminator).
There is no JSON object to place the class discriminator in, e.g., {"type": "fully.qualified.Name"} by default.
But, kotlinx serialization does allow you to write custom serializers, which allows you to work around this. I wrote a custom serializer for enums since I wanted to register enums as concrete types in polymophic serialization. It sounds like you should be able to do something similar. (Disclosure: I only read your problem description in detail; not your ongoing attempts/solution.)
A serializer which supports registering [Enum]s as subclasses in polymorphic serialization when class discriminators are used.
When class discriminators are used, an enum is not encoded as a structure which the class discriminator can be added to.
An exception is thrown when initializing [Json]: " "Serializer for of kind ENUM cannot be serialized polymorphically with class discriminator."
This serializer encodes the enum as a structure with a single value holding the enum value.
Use this serializer to register the enum in the serializers module, e.g.:
subclass( <enum>::class, PolymorphicEnumSerializer( <enum>.serializer() )
This custom serializer can possibly be generalized to any primitive type and thus support your use case.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,983 |
\section{Sensorimotor Control Across Weathers}\label{sec:method}
In this section, we introduce a computational framework for transferring knowledge of ground truth labels from one weather condition to multiple different scenarios without any semantic labels and additional human labeling effort. Figure~\ref{fig:highlevel} gives a high-level overview of the framework.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{images/TLwithoutSemantics.pdf}
\caption{This figure gives a high level overview of the 3 steps for transferring knowledge between two domains $D_0$ and $D_1$ for the purpose of sensorimotor control. Ground truth steering data for only a limited number of images from domain $D_0$ is available. Details of the framework are provided in Section \ref{sec:method}.}
\label{fig:highlevel}
\end{figure}
\subsection{Teacher End-to-End Training}
In this step, the teacher model is trained end-to-end in a supervised manner to predict the steering command of the vehicle from the raw RGB images generated by the camera placed at the front of the ego-vehicle. The training data is collected by an expert driver only once for that particular weather scenario. We refer to the images recorded under the weather condition under which this data was collected as belonging to domain $D_0$. Note that the teacher model is itself divided into a Feature Extraction Module (FEM), $F_0$ and a control module, $C_0$. The raw image (belonging to $D_0$) is passed through $F_0$ to retrieve a lower-dimensional feature representation. This feature representation is in turn fed to the $C_0$ which predicts the steering angle. A depiction of the model is shown in Figure~\ref{fig:end2end}. The FEM, $F_0$ is a sequential combination of 4 units where each unit comprises a convolutional, pooling, and activation layer. The output of unit 4 is flattened to a size of 800, which is in turn fed as an input to the module, $C_0$. The control module, $C_0$ is a series of fully connected layers and outputs the steering command.
\noindent{\textbf{Auxiliary network.}} It might be the case that the amount of images with labels is limited or the model is too large for the task at hand. Hence, the model may tend to overfit. Therefore, during training, to mitigate the effect of overfitting, $F_0$ additionally uses auxiliary networks connected to its intermediate layers~\cite{SzegedyCVPR2015}. Each of the auxiliary networks has a control module, $C_0$ with shared weights. The projection layers, $P_1$, $P_2$ and $P_3$ project the feature maps of the intermediate layers to the dimension of $C_0$ \emph{i.e.} 800. The overall output of the teacher model is the weighted sum of the outputs of the auxiliary networks. The loss is also described by a weighted combination of the individual losses of the 4 auxiliary networks. The loss for each of the control modules is the mean squared error (MSE) between the ground truth label provided by the expert and that predicted by $C_0$. The overall loss is a weighted sum of the losses from each of the 4 control modules.
\begin{align*}
\mathcal{L} &=\sum_{i=1}^{4} \alpha_i \cdot \mathcal{L}_{i},
&\text{s.t.} \sum_{i=1}^{4} \alpha_i = 1,
\end{align*}
where $\alpha_i$, and $\mathcal{L}_i$ are the weighting and the error for the auxiliary network, obtained from the intermediate unit $i$ of the FEM $F_0$. The error functions are calculated as follows:
\begin{equation*}
\mathcal{L}_i = \frac{1}{N}\sum_{j=1}^{N}(y_j - O_{ij})^2,
\end{equation*}
where $y_j$ is the ground truth steering angle obtained from the expert driver for a sample $j$ and $N$ denotes the number of total samples. $O_{ij}$ is the output of the control module corresponding to the $i$th auxiliary network for the $j$th sample.
The weights $\alpha_i$ are themselves learned by a separate weight network. The auxiliary network that has the greatest contribution towards the overall output would also have the highest relative weight. This is important in case of limited data, wherein not all layers may be essential to train the model. In such a case the weights of the shallower auxiliary networks would be higher in comparison to the deeper auxiliary networks. Hence, a significant contribution towards the overall prediction would come from these shallow layers, thereby making the deeper layers effectively dormant. An extreme case would be when the labeled data is so small that even the first layer is enough to give a correct model prediction. In such a case, only $\alpha_1 = 1$ and all other $\alpha_i = 0, \ \text{for}\ i = 2,3,4$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{images/WeightedEnd2EndTraining.pdf}
\caption{The figure depicts the general architecture of the model comprised of the FEM and the auxiliary control modules.}
\label{fig:end2end}
\end{figure}
\subsection{Knowledge Transfer}
As described in step 2 of Figure~\ref{fig:highlevel}, knowledge of ground truth labels from domain $D_0$ is transferred to domain $D_1$ using a teacher-student architecture. The output of the auxiliary networks acts as the teacher to provide supervised information to train the student.
We use the FEM, $F_0$ and control module, $C_0$ (combined, referred to as teacher) trained on images belonging to domain $D_0$, for which ground-truth steering labels are available, to transfer knowledge to a different combination of FEM, $F_1$ and control module, $C_1$ (referred to as student) for domain $D_1$, for which we have access to only unlabeled images. The subsequent procedure is detailed in the following steps:
\begin{enumerate}
\item Image $I_0$ belonging to domain $D_0$ is passed through an image-translation-network to generate image $I_1$ belonging to domain $D_1$ in a manner that only the semantic information is preserved but the weather condition is modified. \cite{ZhuICCV2017, HuangECCV2018, MinjunECCV2018} describe methods for training a translation network in an unsupervised manner using generative adversarial networks (GANs). We use \cite{ZhuICCV2017} for our experiments. A positive implication of using these networks is that they preserve the semantics of the scene and hence the steering angle label would also be the same.
\item \noindent{\bf Hard loss:} If $I_0$ happens to have a ground truth (\emph{hard}) label then the weights of the student network are updated with these labels and the loss is referred to as the \emph{hard loss}. \noindent{\bf Soft loss:} Otherwise, a forward pass can also be done by passing $I_0$ through the teacher. Meanwhile, the corresponding image $I_1$ is passed through the student network. The output of the teacher can then used as a soft target for updating the weights of the student via the soft loss. The overall loss is the weighted average of the soft and hard losses. The weights indicate the relative importance given to the soft targets in relation to the ground truth labels.
\end{enumerate}
Note that the student network can be fed not only images from domain $D_1$ but rather multiple domains including domain $D_0$. Hence, the student network would not only be capable of predicting the steering for multiple domains but would act as a regularizer for better generalization (See P1 in Section~\ref{sec:discussion}).
\subsection{Substitution}
This refers to step 3 described in Figure \ref{fig:highlevel}. At inference time, the teacher network can be substituted with the student network to predict the correct steering command on images from all domains which the student encountered during training.
\section{Related Work}\label{sec:related_work}
Vision-based autonomous driving approaches have been studied extensively in an academic and industrial setting~\cite{JanaiArXiv2017}. A plenty of real world~\cite{GeigerCVPR2012,CordtsCVPR2016,XuCVPR2017} as well as synthetic~\cite{RosCVPR2016,GaidonCVPR2016,RichterICCV2017,RichterECCV2016} datasets for autonomous driving research have become available. In recent years, neural network approaches have significantly advanced the state-of-the-art in computer vision tasks. Especially, end-to-end learning for sensorimotor control has recently gained a lot of interest in the vision and robotics community. In this context, different approaches to autonomous driving are studied: modular pipelines~\cite{ThrunJFR2006}, imitation learning~\cite{PomerleauNIPS1989}, conditional imitation learning~\cite{Codevilla2018ICRA}, and direct perception~\cite{ChenICCV2015}.
\noindent{\textbf{Embodied agent evaluation. }}Most available datasets~\cite{GeigerCVPR2012, CordtsCVPR2016} cannot be used for evaluating online driving performance due to their static nature. The evaluation of driving models on realistic data is challenging and often not feasible. Therefore, a lot of interest has emerged in building photo-realistic simulators~\cite{MullerIJCV2018, ShahFSR2017, DosovitskiyCoRL2017} to analyze those models. However, despite having access to simulation engines, there is currently no universally accepted benchmark to evaluate vision-based control agents. Therefore, our experimental setup is a step towards a field where it is still not quite established how to evaluate and measure the performance of the models~\cite{AndersonArXiv2018, CodevillaECCV2018}.
\noindent{\textbf{Unpaired image-to-image translation networks. }}Unsupervised image-to-image translation techniques are rapidly making progress in generating high-fidelity images across various domains~\cite{ZhuICCV2017, LiuNIPS2017, HuangECCV2018, MinjunECCV2018}. Our framework is agnostic to any particular method. Hence, continual improvements in these networks can be easily integrated into our framework by replacing a previous network.
\noindent{\textbf{Transfer learning via semantic modularity. }}Several works used semantic labels of the scene as an
intermediate representation for transferring knowledge between domains. In the context of autonomous driving, the authors of~\cite{MullerCoRL2018} proposed to map the driving policy utilizing semantic segmentation to a local trajectory plan to be able to transfer between simulation and real-world data. Furthermore, for making a reinforcement model trained in a virtual environment workable in the real world, the authors of~\cite{YouBMVC2017} utilize the intermediate semantic representation as well to translate virtual to real images. However, there is still little work on generalizing driving models across weathers. The work by~\cite{WenzelCoRL2018} showed how to transfer knowledge between different weather conditions using a semantic map of the scene. In contrast, in this paper, we demonstrate the possibility of transferring the knowledge between weathers even without semantic labels.
\noindent{\textbf{Knowledge distillation. }}Originally, knowledge distillation~\cite{HintonArXiv2015} was used for network compression (student network is smaller than the teacher while maintaining the accuracy). However, the authors of~\cite{YangArXiv2018} focus on extracting knowledge from a trained (teacher) network and guide another (student) network in an individual training process. Furthermore,~\cite{ShenArXiv2016} used a slightly modified version of knowledge distillation for the task of pedestrian detection. In this work, we use a teacher-student architecture, but rather to leverage unlabeled data for sensorimotor control.
\section{Introduction}\label{sec:introduction}
The ubiquity of a tremendous amount of processing power in contemporary computing units has proliferated the usage of deep learning-based approaches in control applications. In particular, supervised deep learning methods have made great strides in sensorimotor control, whether it be for autonomous driving~\cite{BojarskiArXiv2016}, robot perception~\cite{KaufmannCoRL2018}, or manipulation tasks~\cite{LevineJMLR2016, NairICRA2017, ZhangICRA2018}. However, the performance of such models is heavily dependent on the availability of ground truth labels. To have the best generalization capability, one should annotate data for all possible scenarios. Nonetheless, obtaining labels of high quality is a tedious, time consuming, and error-prone process.
We propose to instead utilize the information available for one domain and transfer it to a different one without human supervision as shown in Figure~\ref{fig:intro_fig}. This is particularly helpful for many robotic applications wherein a robotic system trained in one environment should generalize across different environments without human intervention. For example in simultaneous localization and mapping (SLAM), it is very important that the algorithm is robust to different lighting conditions~\cite{NewmanCVPR2017}.
In the context of autonomous driving, transferring knowledge from simulation to the real world or between different weather conditions is of high relevance. Recently,~\cite{MullerCoRL2018, YouBMVC2017, WenzelCoRL2018} have attempted to tackle these problems by dividing the task of vehicle control into different modules, where each module specialized in extracting features from a particular domain. In these works, semantic labels are used as an intermediate representation for transferring knowledge between different domains. However, obtaining these semantic labels requires human effort which is time-consuming, expensive, and error-prone \cite{WenzelCoRL2018}. In this work, we instead propose to use a teacher-student learning-based approach to generalize sensorimotor control across weather conditions without the need for extra annotations, \emph{e.g.}, semantic segmentation labels.
\begin{figure}
\centering
\includegraphics[width=0.75\linewidth]{images/intro_fig.pdf}
\caption{Teacher-student training for generalizing sensorimotor control across weather conditions. \textbf{Top:} The teacher network, trained on ground truth data collected on sunny weather is capable of predicting the correct steering angle when tested on this condition. \textbf{Middle:} However, the teacher fails to predict the correct steering when tested on an input image from a different domain (rainy weather). \textbf{Bottom:} With our proposed framework, the student network trained with supervised information from the teacher network is capable of predicting the correct steering for the rainy weather. This is done without any additional ground truth labels or semantic information.}
\label{fig:intro_fig}
\end{figure}
To this end, we make the following contributions:
\begin{itemize}
\item We demonstrate how knowledge of ground truth data for steering angles can be transferred from one weather scenario to multiple different weather conditions. This is achieved without the additional requirement of having semantic labels. We make use of an image-to-image translation network to transfer the images between different domains while preserving information necessary for taking a driving decision.
\item We show how the proposed method can also utilize images without ground truth steering commands to train the models using a teacher-student framework. The teacher provides relevant supervised information regarding the unlabeled images to train the features of the student. Hence, we can eliminate the need for an expert driver for data collection across diverse conditions.
\item If the sample data with ground truth labels is limited, then the teacher and student models may tend to overfit. To overcome this, we propose using weighted auxiliary networks connected to the intermediate layers of these models. During inference, the model size can be reduced by eliminating auxiliary layers with low weights without reducing accuracy.
\end{itemize}
In the following sections, we first review related work. We then present the details of our method, followed by an analysis of our model's performance. Finally, we discuss various parts of our model.
\section{Experiments}\label{sec:experiments}
In this section, we evaluate our approach on the CARLA simulator~\cite{DosovitskiyCoRL2017} version 0.8.2. It provides a total of 15 different weather conditions (labeled from 0 to 14) for two towns, \emph{Town1} and \emph{Town2}, respectively.
\subsection{Evaluation Metrics}
Finding appropriate evaluation metrics is rather challenging for navigation and driving tasks. There is no unique way to quantify these tasks. The authors of~\cite{AndersonArXiv2018} discuss different problem statements for embodied navigation and present based on these discussions evaluation metrics for some standard scenarios. In~\cite{CodevillaECCV2018}, a more extensive study on evaluation metrics for vision-based driving models is carried out. In particular, they analyzed the difference between online and offline evaluation metrics for driving tasks. The preliminary results showed that driving models can have similar mean squared error (MSE) but drastically different driving performance. As a result of this, it is not straight forward to trivially link offline to online performance due to a low correlation between them. Nevertheless, the authors of \cite{CodevillaECCV2018} found that among offline metrics not requiring additional parameters, the mean absolute error between the driving commands and that predicted ones yields the highest correlation with online driving performance.
In addition to using this offline metric, we evaluate the online performance of the models when executing multiple and diverse turnings around corners, since it is a much more challenging task in comparison with simply moving in a straight line. The online performance is tested on the CARLA simulator across all the 15 weather conditions. For each weather condition, we evaluate the models for multiple different turns. In all experiments, the starting positions of the vehicle is just before the curve. The duration of the turn is fixed to 120 frames because it covers the entire curvature of the turn. We report the percentage of time the car remains within the driving lane as a measure of success.
\subsection{Dataset}
For collecting ground truth training data, we navigate through the city using the autopilot mode. To demonstrate the superiority of our method, we collect a limited sample size of 6500 images for weather condition 0 of which only 3200 are labeled with ground truth steering commands. Using our proposed method we aim to transfer knowledge to the remaining 14 weather scenarios. Also, note that none of the 6500 images have any semantic labels.
The 3200 sample images with ground truth data are only available for \emph{Town2}, whereas all the offline and online evaluations are performed on \emph{Town1}. To focus the attention on the effectiveness of our approach and preserve compatibility with prior work~\cite{BojarskiArXiv2016, XuCVPR2017, CodevillaECCV2018}, the models are trained to predict the steering angle of the car while keeping the throttle fixed. The steering angles in CARLA are normalized to values between -1 and 1. The corresponding degrees for these normalized values depends on the vehicle being used. The default vehicle which we use for our experiments has a maximum steering angle of \SI{70}{\degree}.
\subsection{Models}\label{subsec:models}
The offline and online performance of the models described in this section are given in Figure~\ref{fig:l1errorplot} and Table~\ref{tab:turns}, respectively. Figure~\ref{fig:l1errorplot} shows the plot of the mean absolute error between the actual steering command and that predicted by all of the models. Table~\ref{tab:turns} contains the percentage for which the ego-vehicle remains within the driving lane while making turning maneuvers executed by the models across the 15 weather scenarios.
\noindent{\bf Oracle: Steering labels for all weathers. }Here we have assumed that we have access to the ground truth steering commands across all the 15 different weather conditions for \emph{Town1}. Since we are also evaluating the models on \emph{Town1} across all the weather conditions, we find in both the offline and online evaluation metrics that this model achieves the highest accuracy and hence it could serve as an upper bound for evaluating the other models along with our approach.
\noindent{\bf Model~\cite{WenzelCoRL2018}: Steering and semantic labels for weather 0. }Here we adopt the approach of~\cite{WenzelCoRL2018}, wherein the semantic labels of the images are additionally available for the 3200 labeled samples on weather 0. This additional information is used to first train what we refer to as the feature extraction module (FEM) in a supervised manner. The FEM module, in this case, is trained as an encoder-decoder architecture. The encoder encodes the input image into a lower-dimensional latent vector, while the decoder reconstructs the semantic map of the image from the latent vector. The latent vector is then used to train the control module from the ground truth steering labels. The FEM and control modules are hence trained independently and without any auxiliary networks. This FEM trained on the semantics of weather 0 is used as a teacher to train the student which is capable of producing the semantics of all the other 14 weather conditions. The authors of~\cite{WenzelCoRL2018} used the method of~\cite{ZhuICCV2017} and provide 10 separate networks for translating from weather 0 to weathers 2, 3, 4, 6, 8, 9, 10, 11, 12, and 13, respectively. The translated images for each of the 10 weather conditions along with weather 0 are fed in equal proportion to train the student. We would particularly like to evaluate our method which does not have access to any semantic labels against this model. In addition to this, we also evaluate the performance of this method on the model provided by the paper, which was trained with more than 30000 samples from both \emph{Town1} and \emph{Town2}. The performance of this model on \emph{Town1} is far superior since it was trained on much greater data and also had access to ground truth data from \emph{Town1}.
\noindent{\bf Teacher: Steering angles for weather 0. }This model is trained using only the available labeled data for weather 0 in an end-to-end manner. This model has a poor performance for the unseen weather conditions, particularly for conditions 3-14, which are considerably different in visual appearance compared to weather 0. Nevertheless, despite the poor performance this model can be used as a teacher to train the student for predicting the correct steering angles for weather conditions 1-14 for which no ground truth data exists. This approach is described in the next model. Also, note that the unlabeled data remains unutilized here.
\noindent{\bf Ours: Steering angles for weather 0. }This model is trained using the method described in Section~\ref{sec:method}, wherein knowledge is transferred from the teacher network trained on images and ground truth steering commands from weather 0 to the student network which is capable of handling images from all weathers 0-14. For a fair comparison against the model trained with semantic labels (Model~\cite{WenzelCoRL2018}, described earlier) we use the same data and generative models to translate even the unlabeled images to weathers 2, 3, 4, 6, 8, 9, 10, 11, 12, and 13, respectively. These generated images can then be fed to the student model for predicting the correct steering angles for all the 15 weather conditions.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{images/comparison.pdf}
\caption{This plot shows the mean absolute error between the actual steering angle and that predicted by the 5 different models (see subsection \ref{subsec:models}) on data collected across the 15 different weather conditions on \emph{Town1}. Lower is better.}
\label{fig:l1errorplot}
\end{figure}
\begin{table*}
\begin{center}
\resizebox{\linewidth}{!}{
\begin{tabular}{|l||l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l||l|}
\hline
& & \multicolumn{16}{c|}{Weather Conditions}\\
\cline{3-18}
Method & Trained on & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & overall \\
\hline\hline
Oracle & \emph{Town1} & 99.79 & 99.90 & 100 & 97.40 & 98.96 & 99.27 & 98.13 & 98.85 & 98.27 & 99.90 & 99.27 & 96.35 & 93.85 & 93.96 & 96.35 & 98.02\\
Model~\cite{WenzelCoRL2018} & \emph{Town1\&2} & 99.06 & 93.44 & 98.85 & 98.75 & 97.92 & 98.23 & 97.60 & 96.56 & 91.15 & 96.04 & 97.29 & 95.00 & 94.69 & 82.08 & 95.41 & 95.47 \\
Model~\cite{WenzelCoRL2018} & \emph{Town2} & 68.33 & 67.71 & 50.00 & 71.77 & 67.40 & 64.38 & 63.85 & 63.65 & 61.88 & 71.35 & 51.35 & 67.50 & 58.33 & 61.67 & 66.98 & 63.74\\
Teacher & \emph{Town2} & 92.19 & 92.40 & 82.12 & 44.38 & 51.77 & 73.65 & 32.50 & 61.56 & 49.48 & 80.10 & 60.63 & 48.54 & 35.20 & 34.27 & 50.52 & 59.29 \\
Ours & \emph{Town2} & 93.96 & 95.21 & 81.25 & 99.90 & 100 & 94.17 & 90.42 & 79.69 & 77.19 & 86.77 & 84.58 & 65.63 & 68.54 & 58.44 & 80.73 & 83.77 \\
\hline\hline
Ours (Auxiliary network 1) & \emph{Town2} & 93.96 & 93.44 & 80.73 & 92.40 & 100 & 99.69 & 90.42 & 80.10 & 77.19 & 87.50 & 91.98 & 67.40 & 66.25 & 57.29 & 81.15 & 83.97\\
\hline
\end{tabular}
}
\end{center}
\caption{This table shows the percentage for which the ego-vehicle remains within the driving lane while executing a turn for the models across the 15 different weather scenarios on \emph{Town1}. Higher is better.}
\label{tab:turns}
\end{table*}
\section{Conclusion}\label{sec:conclusion}
In this work, we showed how a teacher-student learning-based approach can leverage limited labeled data for transferring knowledge between multiple different domains. Our approach, specifically designed to work for sensorimotor control tasks, learns to accurately predict the steering angle under a wide range of conditions. Experimental results showed the effectiveness of the proposed method, even without having access to semantic labels as an intermediate representation between weather conditions. This framework may be extendable to other application areas for which a certain domain has ground truth data and shares a common characteristic with other domains for which no labels are available.
\section{Discussion}\label{sec:discussion}
In this section, we discuss some critical insights on the experimental observations we obtained while evaluating the models. Here are some points we found worthwhile to provide some commentary based on the results provided in Figure~\ref{fig:l1errorplot} and Table~\ref{tab:turns}.
\noindent{\bf P1 - Better regularization:} It is interesting to observe that the teacher model, trained only on the available 3200 labeled samples from \emph{Town2} on weather 0 has a worse offline performance for \emph{Town1} on weather 0 in comparison to our method. This seems to imply that our approach which has been trained on multiple kinds of weather has better generalization capabilities and can even outperform its teacher when evaluated in a different town. Hence, an additional positive consequence of training the student with generated images from multiple diverse domains is that it acts as a regularizer tending to prevent overfitting to one specific domain.
\noindent{\bf P2 - Semantic inconsistency:} Note that Model~\cite{WenzelCoRL2018} which in addition to having the same data and labels as our approach has also access to ground truth semantic labels. Yet, its performance is significantly poor. Upon investigation, we found that due to the limited number of semantic labels, the FEM trained as an encoder-decoder architecture seemed to be overfitting to the available data. Hence, when tested on unseen environments, the semantic segmentation output of the module breaks. The latent vector representing these broken semantics is then fed to the control module, which is incapable of predicting the correct steering command. Figure~\ref{fig:semantics} shows some sample images with the corresponding semantic segmentation outputs which are considerably different from the true semantics of the scene.
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{images/semantics.pdf}
\caption{This plot shows three sample images (column 1) with the corresponding semantic segmentation output by the model (column 2) for 3 different weathers. The segmentation produced by the model does not reflect the actual semantic characteristics of the scene (column 3).}
\label{fig:semantics}
\end{figure}
\noindent{\bf P3 - Modular training constraints:} Furthermore, the modular approach of Model~\cite{WenzelCoRL2018} wherein the FEM and control module are trained independently as opposed to an end-to-end model served to be a bottleneck in being able to learn the features universally. Also, an assumption to train the control module well is that the FEM would work perfectly well, which is not the case. Hence, the overall error of the modular pipeline would be an accumulation of the errors of the independent FEMs and control modules. We found that if we also shift the training of our approach to a modular one then performance deteriorates. This can be done in our approach by updating only the weights of the FEM of the student from the output features of the FEM of the teacher.
\noindent{\bf P4 - Auxiliary weights:} To prevent overfitting of the models, trained on limited data we used a weighted sum of the outputs of the auxiliary layers. The weights themselves were learned as part of the training. Once training of our student model was complete, we found that more than \SI{97}{\percent} of the weight was held by the first auxiliary network. This seemed to imply that only the first unit of the FEM is enough for predicting the steering command. Hence the remaining unit layers are not providing any additional information for the model. So we evaluated our model based on the output of the first auxiliary network rather than on the weighted sum of the 4 auxiliary networks. The online evaluation of this approach is given in Table~\ref{tab:turns} against the row labeled \emph{Ours (Auxiliary network 1)}. It is interesting to note that this approach is comparable in its performance with the original one. Therefore, at test time we can prune the network to a smaller size by making predictions only based on the first auxiliary network and removing the remaining 3 auxiliary networks. This would result in less computation and faster inference.
\noindent{\bf P5 - Online vs. offline evaluation:} Figure~\ref{fig:auxillaryUnit1} shows an offline evaluation of the two variations of our method described in the previous point across the 15 weather conditions. Note that apart from weather 0, 1, and 2, the two curves are indistinguishable from one another. However, the online evaluation results do not correspond with this observation. For weathers 3, 5, 7, and 9-14 the online performance is different despite having the same offline metric. This confirms the intuition presented in~\cite{AndersonArXiv2018} and the problems associated with evaluating embodied agents in offline scenarios. The topic of finding a correlation between offline evaluation metrics and online performance has therefore recently started to receive positive traction. It is therefore important to come up with a universal metric for evaluating various algorithms across the same benchmark. Due to the non-existence of such benchmarks, we created our own for the evaluation of the different approaches.
\begin{figure}
\centering
\includegraphics[width=0.85\linewidth]{images/comparison_aux.pdf}
\caption{This plot shows the mean absolute error between the ground truth steering label and that predicted by the two models. The \textcolor{new_blue}{\textbf{blue}} curve is the weighted sum of all the 4 auxiliary networks of our model. The \textcolor{new_orange}{\textbf{orange}} line depicts the output of only the first auxiliary network of our model.}
\label{fig:auxillaryUnit1}
\end{figure}
\noindent{\bf P6 - Activation maps:} To understand the behavior of the model, which also works with only the first auxiliary network, we took the sum of the activation maps of the first unit of the FEM of the student and displayed it as a heatmap as shown in Figure~\ref{fig:stud_filter} for a sample of 2 images. We see that the activation maps are most prominent in regions where there are lane markings, sidewalks, cars, or barriers. Knowing these cues seems to be enough for the network to take an appropriate driving decision in most of the cases. Therefore, the higher-level features determined by the preliminary layers of the model are already enough to detect these objects of interest.
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{images/stud_filter.pdf}
\caption{This figure shows the sum of the activation maps displayed as a heatmap of the first unit of the FEM of the student model for a sample taken from 2 different weather conditions. The activation maps are more prominent in regions where there are lane markings, sidewalks boundaries, other vehicles, or barriers.}
\label{fig:stud_filter}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,600 |
\section*{Methods}
\subsection*{Sample preparation method.}
Experiments were performed on 200 nm thick $\alpha$\nh GeTe(111)\ films grown by molecular beam epitaxy on BaF$_2$(111) substrates~\cite{BaF2_1,BaF2_2}. A protective amorphous Te-cap used to avoid surface oxidation and degradation was removed in the ultrahigh vacuum chamber by annealing the thin film for 30 min at 250\degC. The surface quality checked by LEED shows the 1$\times$1 diffraction pattern of the $\alpha$\nh GeTe(111)\ surface seen in Fig.~1c.
\subsection*{ARPES and spin-resolved ARPES}
The VUV-ARPES/SARPES experiments were performed at the COPHEE end-station of the SIS beamline ~\cite{Hoesch_JESRP}, Swiss Light Source (SLS), using $p$-polarized photons in the energy range 20--25 eV, and an Omicron EA 125 hemispherical energy analyzer equipped with two orthogonally mounted classic Mott detectors. The whole set-up allows the simultaneous measurements of all three spatial components of the spin-polarization vector for
each point of the band structure. The geometry of the experimental set-up is schematically shown in Fig.~1e. The spin-resolved MDCs were measured by rotating the sample azimuth $\phi$ such that $\overline{\mathrm{M}\Gamma\mathrm{M}}$\ or $\overline{\mathrm{K}\Gamma\mathrm{K}}$\ were aligned perpendicular to the scattering plane. Direct and immediate visualization of the spin splitting as a function of electron momentum was measured by exploiting the manipulator tilt angular degrees of freedom [see $\tau$ in Fig.~1(e)]. The spin\nh integrated Fermi surfaces were measured with the channeltron detectors of the SARPES set-up by \hbox{on-the-fly} sweeping the manipulator tilt angle ($\tau$) for selected polar angles $\theta$ as seen in the experimental setup in Fig.~1(e). The angular and resolution-broadened energy range in spin-resolved mode were 1.5\dg and 60 meV, respectively. In spin-integrated mode the resolutions were set to 0.5\dg and 20 meV. To enhance the bulk sensitivity compared to VUV-ARPES, experiments were also performed in the soft-X-ray energy range at the ADRESS beamline~\cite{Strocov_ADRESS}. The experimental results in the main text were reproduced on three samples with individual annealing preparation over different experimental runs. All data were taken at 20 K.
\subsection*{First principle calculations}
The ab-initio calculations are based on density functional theory (DFT)
as implemented within the multiple scattering theory (SPRKKR)~\cite{SPR13,Minar_RPP}. SOC has been naturally included by use of a fully relativistic four-component scheme. As s first step of our investigations we
performed self-consistent calculations (SCF) for 3D bulk as well as 2D semi-infinite surface of $\alpha$\nh GeTe(111)\ within the screened KKR formalism~\cite{Minar_RPP}. The corresponding ground state band structures are presented in terms of Bloch spectral functions (BSF). The converged SCF potentials served as an input quantities for our spectroscopic investigations. The ARPES calculations were performed in the framework of the fully relativistic one-step model of photoemission~\cite{JBraun_1SM,Ebert_book} in its spin-density matrix formulation~\cite{JBraun_Rashba}, which accounts properly for the complete spin-polarization vector, in particular for Rashba systems
like GeTe. Together with a realistic model for the surface barrier potential, one-step calculations based on a semi-infinite half-space configuration were decisive to substantiate the $\alpha$\nh GeTe(111)\ surface-generated spectral features on both qualitative and quantitative levels.
\subsection*{Acknowledgments}Constructive discussions with S. Picozzi, R. Calarco and R. Bertacco are gratefully acknowledged. This work was supported by the Swiss National Science Foundation Project No. PP00P2\_144742\/1. We acknowledge the financial support from German funding agencies DFG (SPP1666) and the German ministry BMBF (05K13WMA) is also gratefully acknowledged (H.V.,.H.E.,J.B. and J.M.). J.M. acknowledges the CENTEM project, Reg.No. CZ.1.05/2.1.00/03.0088, co-funded by the ERDF as part of the Ministry of Education, Youth and Sports OP RDI program. D.K. and V.H. acknowledge the financial support of Czech Science Foundation (project 14-08124S)
\endgroup
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 782 |
Why macrolides are a must for CAP
There is now overwhelming evidence to add macrolides to beta-lactams in empiric treatment of community acquired pneumonia (CAP), an Australian...
By Michael Woodhead
There is now overwhelming evidence to add macrolides to beta-lactams in empiric treatment of community acquired pneumonia (CAP), an Australian respiratory physician says. Despite controversy over the use of macrolides in CAP in recent years, most trials show major reductions in mortality, especially when they are used early, according to Professor Grant Waterer (pictured), Medical ... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 668 |
Q: How can I keep selected the previous value when updating a form? I am using laravel collctive form. I want to update only the quantity field of the following form keeping the blood_id field readonly. When I submit the form I am not getting blood_id value.
How can i solve it?
{!! Form::model($bloodBank, ['route' => ['bloodBanks.update', $bloodBank->id], 'method' => 'put']) !!}
<div class="row">
<div class="col-lg-8">
<div class="form-group">
{!! Form::label('blood_id', 'Blood Group', ['class' => 'form-control-label']);!!}
{!! Form::select('blood_id', $bloods , null , ['placeholder' => 'Choose Blood Group',"class"=>"form-control",'disabled' => true]) !!}
</div>
</div>
</div>
<div class="row">
<div class="col-lg-8">
<div class="form-group">
{!! Form::label('quantity','Quantity', ['class' => 'form-control-label']);!!}
{!! Form::number("quantity",null, ["class"=>"form-control form-control-label",'min'=>'0']) !!}
<span class="validation-error">{{ $errors->first("quantity") }}</span>
</div>
</div><!-- col-12 -->
</div>
<button class="btn btn-info">Update </button>
{!! Form::close() !!}
A: For disabled fields, you may want to add hidden fields which won't display on the rendered page BUT will be included in the request object. E.g.
{{ Form::hidden('blood_id', $bloods) }}
This is in addition to the displayed field you already have which is disabled.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,960 |
Carl Bechem GmbH — найстаріше мастильно виробниче підприємство Європи, засноване Карлом Бехемом 1834 року у м.Хаген, Північний Рейн — Вестфалія, Німеччина.
Концерн Carl Bechem GmbH створює високоякісні мастила, які знаходять своє застосування у автомобільній і харчовій промисловості, для гірничої промисловості (важкозаймисті гідравлічні рідини для підземних робіт) і підприємств сфери металообробки.
Історія
За більш ніж 175 років успішної роботи, компанія змогла зарекомендувати себе як надійний партнер таких підприємств як: Nestle, Pepsi, Bosch, Volkswagen, GM Group, Європейського Космічного Агентства та багатьох інших. Вся продукція концерну сертифікована міжнародними організаціями ISO, TÜV і IATF; в Україні - УкрСЕПРО і МЕПР. Крім того, група товарів для харчової та фармацевтичної промисловості має відповідні допуски NSF H1, H2 і Halal. Продукти для гірничо-видобувного сектору ринку дозволені до застосування в українській гірничій промисловості і відповідають всім міжнародним вимогам якості і безпеки.
Завод у Німеччині розташований за адресою Weststraße 120, Hagen, Germany
Напрямики діяльності
Мастила - промислові мастила з високоефективними комплексами добавок для редукторів, ланцюгів та ін.
Мастильно-охолодні рідини - водозмішуючі та неводосмішуючі рідини для обробки металів різанням, шліфуванням ...
Змазки - високоефективні змазки для тривалих строків та екстремальних режимів експлуатації
Захист від корозії - антикорозійні засоби для тривалої і ефективного захисту при транспортуванні та зберіганні
Пасти - протизадирні мастильні пасти для різьбових з'єднань, складання, підробітки
Очисні засоби - спеціальні рідини, призначені для очищення поверхонь
Антифрикційні лаки - сухі мастильні покриття, як альтернативне рішення, замість традиційного змащення і олив
Мастильні матеріали у спреях
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Carl Bechem GmbH
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Підприємства Північного Рейну-Вестфалії
Підприємства, засновані 1834 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,638 |
namespace llvm {
template <typename T, unsigned BitNum, typename BitVectorTy, bool isSigned>
class PackedVectorBase;
// This won't be necessary if we can specialize members without specializing
// the parent template.
template <typename T, unsigned BitNum, typename BitVectorTy>
class PackedVectorBase<T, BitNum, BitVectorTy, false> {
protected:
static T getValue(const BitVectorTy &Bits, unsigned Idx) {
T val = T();
for (unsigned i = 0; i != BitNum; ++i)
val = T(val | ((Bits[(Idx << (BitNum-1)) + i] ? 1UL : 0UL) << i));
return val;
}
static void setValue(BitVectorTy &Bits, unsigned Idx, T val) {
assert((val >> BitNum) == 0 && "value is too big");
for (unsigned i = 0; i != BitNum; ++i)
Bits[(Idx << (BitNum-1)) + i] = val & (T(1) << i);
}
};
template <typename T, unsigned BitNum, typename BitVectorTy>
class PackedVectorBase<T, BitNum, BitVectorTy, true> {
protected:
static T getValue(const BitVectorTy &Bits, unsigned Idx) {
T val = T();
for (unsigned i = 0; i != BitNum-1; ++i)
val = T(val | ((Bits[(Idx << (BitNum-1)) + i] ? 1UL : 0UL) << i));
if (Bits[(Idx << (BitNum-1)) + BitNum-1])
val = ~val;
return val;
}
static void setValue(BitVectorTy &Bits, unsigned Idx, T val) {
if (val < 0) {
val = ~val;
Bits.set((Idx << (BitNum-1)) + BitNum-1);
}
assert((val >> (BitNum-1)) == 0 && "value is too big");
for (unsigned i = 0; i != BitNum-1; ++i)
Bits[(Idx << (BitNum-1)) + i] = val & (T(1) << i);
}
};
/// \brief Store a vector of values using a specific number of bits for each
/// value. Both signed and unsigned types can be used, e.g
/// @code
/// PackedVector<signed, 2> vec;
/// @endcode
/// will create a vector accepting values -2, -1, 0, 1. Any other value will hit
/// an assertion.
template <typename T, unsigned BitNum, typename BitVectorTy = BitVector>
class PackedVector : public PackedVectorBase<T, BitNum, BitVectorTy,
std::numeric_limits<T>::is_signed> {
BitVectorTy Bits;
typedef PackedVectorBase<T, BitNum, BitVectorTy,
std::numeric_limits<T>::is_signed> base;
public:
class reference {
PackedVector &Vec;
const unsigned Idx;
reference(); // Undefined
public:
reference(PackedVector &vec, unsigned idx) : Vec(vec), Idx(idx) { }
reference &operator=(T val) {
Vec.setValue(Vec.Bits, Idx, val);
return *this;
}
operator T() const {
return Vec.getValue(Vec.Bits, Idx);
}
};
PackedVector() { }
explicit PackedVector(unsigned size) : Bits(size << (BitNum-1)) { }
bool empty() const { return Bits.empty(); }
unsigned size() const { return Bits.size() >> (BitNum-1); }
void clear() { Bits.clear(); }
void resize(unsigned N) { Bits.resize(N << (BitNum-1)); }
void reserve(unsigned N) { Bits.reserve(N << (BitNum-1)); }
PackedVector &reset() {
Bits.reset();
return *this;
}
void push_back(T val) {
resize(size()+1);
(*this)[size()-1] = val;
}
reference operator[](unsigned Idx) {
return reference(*this, Idx);
}
T operator[](unsigned Idx) const {
return base::getValue(Bits, Idx);
}
bool operator==(const PackedVector &RHS) const {
return Bits == RHS.Bits;
}
bool operator!=(const PackedVector &RHS) const {
return Bits != RHS.Bits;
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const PackedVector &operator=(const PackedVector &RHS) {
Bits = RHS.Bits;
return *this;
}
PackedVector &operator|=(const PackedVector &RHS) {
Bits |= RHS.Bits;
return *this;
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void swap(PackedVector &RHS) {
Bits.swap(RHS.Bits);
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// Leave BitNum=0 undefined.
template <typename T>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,758 |
President Obama said Sunday that counterterrorism efforts in Yemen are ongoing despite President Abd-Rabbu Mansour Hadi's ouster by Houthi rebels.
"Our top priority is to make sure our people on the ground are safe," Obama said during comments in New Delhi with Indian Prime Minister Narendra Modi.
"It is a dangerous country and dangerous part of the world," he added.
Obama said he has seen reports that those counterterrorism efforts have been suspended.
"This is one more sequence in what has been an ongoing turbulent process inside Yemen," he added.
"Yemen has never been a perfect democracy or an island of stability," Obama said. | {
"redpajama_set_name": "RedPajamaC4"
} | 932 |
\section{Introduction}\label{intro}
If $X$ is a reduced set of $n$ points in $\pr2$, the {\em fat point subscheme $Z = mX\subset\pr2$}
is the $(m-1)$-st infinitesimal neighborhood of $X$. Thus $mX$ is the subscheme defined by the
symbolic power $I(X)^{(m)}\subset R=k[\pr2]$ (that is, by the saturation of the ideal $I(X)^m$
with respect to the ideal generated by the coordinate variables in the ring $k[\pr2]$).
The question motivating this paper is: What
are the Hilbert functions of such subschemes of $\pr2$?
There have been two main approaches to this question,
and one goal of this paper is to demonstrate them in various situations.
The two approaches are exemplified by the papers \cite{GMS} and \cite{GHM}.
The approach of \cite{GMS} is to identify constraints that Hilbert functions
must satisfy and then for each function satisfying those constraints to try to find a
specific subscheme having that function as its Hilbert function.
A complete classification of all Hilbert functions of reduced 0-dimensional
subschemes of projective space was given in \cite{GMR} using essentially this approach.
The paper \cite{GMS} then uses \cite{GMR} as the starting point for
classifying Hilbert functions for subschemes of the form $Z = 2X\subset\pr2$
with $X$ reduced and 0-dimensional.
This approach is most effective when the class of possible functions
is fairly limited, hence the restriction in \cite{GMS} to the case $m=2$.
This approach has the advantage of providing explicit results often without needing
detailed information about the disposition of the points, but it has the disadvantage
of not providing a complete dictionary of which point sets give which Hilbert function.
The approach of \cite{GHM} is to use the geometry of the surface
$Y$ obtained by blowing up the points of the support of $Z$ to
obtain information about the Hilbert function of $Z$.
This approach is most effective when the geometry of $Y$ is well-understood,
hence the restriction in \cite{GHM} to the case $n\leq 8$.
Given points $p_i$ and non-negative integers $m_i$,
the subscheme defined by the ideal $\cap_{i=1}^n(I(p_i)^{m_i})$
is also called a fat point subscheme, and is denoted
$m_1p_1+\cdots+m_np_n$.
The advantage of the second approach, as implemented in \cite{GHM},
is that it provided complete results for all fat point subschemes
$Z=m_1p_1+\cdots+m_np_n$ with $n\leq8$,
together with a complete determination of which $Z$ give
the same Hilbert function, but it had the cost of needing a lengthy
analysis of the geometry of $Y$, and gives only recursive
determinations of the Hilbert functions. However, for $n\leq8$ and
$k=2$ there are only finitely many cases, so a complete list of
the Hilbert functions which occur can be given. See \cite{GHM}
for this list.
The first case left open by \cite{GHM} is $n=9$ points of $\pr2$.
It should, in principle, be possible to carry out the necessary analysis
to obtain a complete recursive classification of Hilbert functions and
corresponding points sets for $n=9$, but whereas for $n\leq 8$ there are only finitely
many classes of sets of $n$ points, there will certainly be infinitely many when $n=9$
(related to the fact that there can be infinitely many prime divisors on $Y$ of
negative self-intersection, and to the fact that effective nef divisors $F$ can occur with
$h^1(Y, {\mathcal O}_Y(F))>0$). Thus a complete classification in this case
using the methods of \cite{GHM} will be a substantial effort, which we leave
for future research (not necessarily by us).
Instead, in this paper we will focus on some special cases.
We devote \textsection\ref{9dblpts} to demonstrating the first approach by obtaining a complete answer in
the case of $n=9$ and $m=2$. This also shows how one could recover the result
for $n=8$ and $m=2$ obtained in \cite{GHM} using the methods of \cite{GMS}.
The rest of the paper is devoted to demonstrating both methods
for the case of $n$ points of multiplicity $m$ on cubics, under somewhat
different hypotheses chosen to play to the strengths of each method.
The Philosophy of the First Way is to use known facts about Hilbert functions to say things about what Hilbert functions are possible. The Philosophy of the Second Way is to use known facts about cohomology of blown up surfaces to say things about what dimensions of linear systems are possible. Sections
\S\ref{ptsoncubics} (using the First Way) and \S\ref{appII} (using the Second Way)
illustrate how we can attack the same problem and obtain overlapping and sometimes complementary
results, but using dramatically different ways to do so.
So, given points on a plane cubic,
for the First Way we will assume the cubic is irreducible, that $m=2$
and, in some cases, that $n$ is not too small.
Our main results here are Theorem \ref{smooth cubic} and Theorem \ref{singular cubic}.
For the Second Way
we will make no restrictions on $m$ nor assume the cubic is irreducible but
we will assume the points are smooth points of the cubic and
we will assume that the points are evenly distributed (meaning essentially that
no component contains too many of the points).
Under these two assumptions we give a complete determination of
all possible Hilbert functions in Theorem \ref{method2cor1}.
Using the same techniques we will, in Remark \ref{method2cor2},
also recover the Hilbert functions
for $X$ and $2X$ when $X$ is a reduced set of points contained in
a reduced, irreducible singular cubic curve in case the singular
point of the curve is one of the points of $X$.
We now discuss both methods in somewhat more detail.
For the first approach we will follow \cite{GHM} and \cite{GMS}
and sometimes work with the first difference, $\Delta h_{2X}$, of the Hilbert function $h_{2X}$
rather than with $h_{2X}$ directly, since for our purposes $\Delta h_{2X}$
is easier to work with, but we regard $\Delta h_{2X}$ as just an equivalent formulation
of the Hilbert function and so for simplicity we will refer to it as the Hilbert function.
The first approach can be summarized as follows. We start by listing
all Hilbert functions $\Delta h_X$ for reduced sets $X$ of $n$ points, using \cite{GMR},
and then we analyze each case in turn using $h_X$ to
constrain the behavior of $h_{2X}$. For example, in some extreme cases
the form of $\Delta h_X$ forces many of the points of $X$ to lie on a line;
knowing this can be very useful in determining $h_{2X}$.
Our analysis uses the following tools:
(a) a crude bound on the regularity of $I(2X)$, giving an upper bound for the
last degree in which $\Delta h_{2X}$ can be non-zero;
(b) B\'ezout considerations giving the values of $\Delta h_{2X}$ in most degrees;
(c) the fact that the sum of the values of $\Delta h_{2X}$ is 27; and
(d) a theorem of Davis \cite{davis} giving geometric consequences for certain
behavior of the function $\Delta h_{2X}$. The idea is that we know the value
of the Hilbert function for most degrees by (a), (b) and (c), and we can
exhaustively list the possibilities for the remaining degrees.
Then we use (d) to rule out many of these. Finally, for the cases that
remain, we try to construct examples of them (and in the situations
studied in this paper, we succeed).
For the second approach we study $h_Z$ for an arbitrary fat point subscheme
$Z=m_1p_1+\cdots +m_np_n\subset\pr2$ using the geometry
of the surface $Y$, where $\pi:Y\to\pr2$ is the morphism
obtained by blowing up the points $p_i$. This depends on
the well known fact that $\dim I(Z)_t=h^0(Y, {\mathcal O}_Y(F))$
where $F=tL-m_1E_1-\cdots-m_nE_n$, ${\mathcal O}_Y(L)=\pi^*{\mathcal O}_{\pr2}(1)$
and $E_i=\pi^{-1}(p_i)$. The fundamental fact here is
the theorem of Riemann-Roch:
\addtocounter{thm}{1}
\begin{equation}\label{RRoch}
h^0(Y, {\mathcal O}_Y(F))-h^1(Y, {\mathcal O}_Y(F))+
h^2(Y, {\mathcal O}_Y(F))=\frac{F^2-K_Y\cdot F}{2}+1=\binom{t+2}{2}-\sum_i\binom{m_i+1}{2}.
\end{equation}
To see the relevance of \eqref{RRoch}, note that $K_Y=-3L+E_1+\cdots+E_n$, so we have by duality that
$h^2(Y, {\mathcal O}_Y(F))=h^0(Y, {\mathcal O}_Y(K_Y-F))$ and
thus $h^2(Y, {\mathcal O}_Y(F))=0$ if $t<0$. Now, since we are interested in the values of Hilbert functions when
$t\geq0$, we have
$h_Z(t) = \dim(R_t)-\dim(I(Z)_t)=\binom{t+2}{2}-h^0(Y, {\mathcal O}_Y(F))$ which using \eqref{RRoch} becomes
\addtocounter{thm}{1}
\begin{equation}\label{RRoch2}
h_Z(t) =\sum_i\binom{m_i+1}{2}-h^1(Y, {\mathcal O}_Y(F)).
\end{equation}
This second approach, as applied in \cite{GHM},
depended on knowing two things: the set $\hbox{Neg}(Y)$
of all prime divisors $C$ on $Y$ with $C^2<0$ and on knowing
$h^0(Y,{\mathcal O}_Y(F))$ for every divisor $F$
for which we have $F\cdot C\geq0$ for all $C\in \hbox{Neg}(Y)$.
Given $\hbox{Neg}(Y)$,
one can in principle reduce the problem of computing
$h^0(Y, {\mathcal O}_Y(F))$ for an arbitrary divisor $F$
to the case that $F\cdot C\geq0$ for all $C\in \hbox{Neg}(Y)$.
If $n\geq 2$ and $F\cdot C\geq 0$ for all $C\in \hbox{Neg}(Y)$,
then $h^2(Y, {\mathcal O}_Y(F))=0$, so from Riemann-Roch
we have only $h^0(Y, {\mathcal O}_Y(F))\geq 1+(F^2-K_Y\cdot F)/2$.
When $n\leq8$ or the points $p_i$ lie on a conic (possibly singular),
this inequality is always an equality,
but for $n\geq9$ points not contained in a conic it needn't be,
so more information in general is needed.
Similarly, in case $n\leq8$ or the points $p_i$ lie on a conic (possibly singular),
it turns out, in fact, that $\hbox{Neg}(Y)$ is a finite set, but this also can fail
for $n\geq9$ points not contained in a conic.
As a consequence, given $\hbox{Neg}(Y)$
one can determine $h_Z$
for any fat point subscheme $Z=m_1p_1+\cdots +m_np_n\subset\pr2$
if either $n\leq8$ or the points $p_i$ lie on a conic.
This raises the question of what sets $\hbox{Neg}(Y)$
occur under these assumptions. We answered this question
in \cite{GHM}. There are only finitely many possibilities
and \cite{GHM} gives a complete list.
When $n\geq 9$ and the points $p_i$
do not lie on a conic then not only can $\hbox{Neg}(Y)$ fail to be finite
but $h^1(Y, {\mathcal O}_Y(F))$ need not vanish, even if
$F\cdot C\geq 0$ for all $C\in \hbox{Neg}(Y)$ and even if $F$ is effective.
Assuming that the points $p_i$ lie on a cubic curve does not
eliminate either difficulty, but it does mean that
$-K_Y$ is effective (whether the cubic is irreducible or not),
and thus the results of \cite{anticanSurf}
can be applied to the problem of computing $h^0(Y, {\mathcal O}_Y(F))$.
In case $-K_Y$ is effective, it is known what kinds of classes
can be elements of $\hbox{Neg}(Y)$, but no one has yet classified precisely which sets $\hbox{Neg}(Y)$ arise for $n
\geq 9$ (doing this for $n=7, 8$ was the new contribution in \cite{GHM}). On the other hand,
even without this complete classification, partial results can still be obtained
using the second approach, as we will show here
using information about the geometry of $Y$ developed in \cite{anticanSurf}.
\section{Approach I: Nine Double Points}\label{9dblpts}
It is natural to ask what can be said for fat point schemes $Z$ supported at
$r>8$ points. As observed in \cite[Remark 2.2]{GHM}, there are infinitely
many configuration types of $r>8$ points, so we will restrict our attention
to subschemes $2Z=2(p_1+\cdots+p_r)$ of $\pr2$. Since we are now restricting the multiplicities of the points to be at most 2,
it is not necessary to make an exhaustive list of the configuration types -- indeed, we will point out situations where different configurations exist but nevertheless do not give different Hilbert functions. Instead, in this situation we can bring to bear
the methods developed in \cite{GMS}, and to demonstrate additional methods
which can be used. We will determine all Hilbert functions
that occur for double point subschemes $2Z=2(p_1+\cdots+p_9)$ of $\pr2$, for every Hilbert function occurring as the
Hilbert function of a simple point subscheme $Z=p_1+\cdots+p_9$.
\begin{defn}\rm
Let $Z$ be a zero-dimensional subscheme of $\pr{n}$ with Hilbert function $h_Z$. The {\em difference function} of $Z$ is the first difference of the Hilbert function of $Z$, $\Delta h_Z (t) = h_Z(t) - h_Z(t-1)$. (This is sometimes also called the {\em $h$-vector} of $Z$, and sometimes the {\em Castelnuovo function} of $Z$.)
\end{defn}
The Hilbert function and its difference function clearly give equivalent information and it is
primarily because of the simpler bookkeeping allowed by the first difference that we use it.
Notice that $\Delta h_Z$ is the Hilbert function of any Artinian reduction of $R/I_Z$ by a linear form.
One problem raised in \cite{GMS} is the existence and determination of maximal and minimal
Hilbert functions. In the current context, this means that we fix an underlying Hilbert function $\underline{h}$ that exists for some set of 9 points in $\pr2$, and letting $X$ move in the irreducible flat family of all sets of points with Hilbert function $\underline{h}$, we ask whether there is a maximal and a minimal Hilbert function for the corresponding schemes $Z=2X$. It was shown in \cite{GMS} that there {\em does} exist a maximal such Hilbert function, denoted ${\underline h}^{max}$ (for any number of points). The proof in \cite{GMS} is nonconstructive, and \cite{GMS}
determines ${\underline h}^{max}$ in only a few special cases.
The paper \cite{GMS} also raises the question of whether
${\underline h}^{min}$ always exists; i.e., whether there exists an $X'$
such that $h_{2X}$ is at least as big in every degree as $h_{2X'}$ for
every $X$ with $h_X=h_{X'}$. This question remains open.
A useful tool is the following lemma. This lemma, and generalizations of it, are well-known.
For a very short proof of the statement given here see \cite[Lemma 2.18]{GMS}.
\begin{lem} \label{reg lemma}
Let $X$ be a reduced set of points in $\pr2$ with regularity $r+1$. Then the regularity of $I_{2X}$ is bounded by $\hbox{\rm reg}(I_{2X}) \leq 2 \cdot \hbox{\rm reg}(I_X) = 2r+2$.
\end{lem}
We will also use the following result of Davis \cite{davis}. It is a special case of a more general phenomenon \cite{BGM} related to maximal growth of the first difference of the Hilbert function.
\begin{thm} \label{davis thm}
Let $X \subset \pr2$ be a zero-dimensional subscheme, and assume that $\Delta h_X (t) = \linebreak \Delta h_X(t+1) = d$ for some $t,d$. Then the degree $t$ and the degree $t+1$ components of $I_X$ have a GCD, $F$, of degree $d$. Furthermore, the subscheme $W_1$ of $X$ lying on the curve defined by $F$ (i.e.\ $I_{W_1}$ is the saturation of the ideal $(I_X,F)$) has Hilbert function whose first difference is given by the truncation
\[
\Delta h_{W_1} (s) = \min \{ \Delta h_X(s), d \}.
\]
Furthermore, the Hilbert function of the points $W_2$ not on $F$ (defined by $I_{W_2} = I_X : (F)$) has first difference given by the (shifted) part {\em above} the truncation:
\[
\Delta h_{W_2}(s) = \max \{ \Delta h_X (s+d) -d , 0 \}.
\]
\end{thm}
We will see precisely the possibilities that occur for the first infinitesimal neighborhood of nine points, and we will see that there is in each case a maximum and minimum
Hilbert function. All together, there occur
eight Hilbert functions for schemes $X=p_1+\cdots+p_9$.
We give their difference functions, and the possible Hilbert functions that occur for double point schemes
$2X$, in the following theorem.
\begin{thm}\label{9 pts}
The following table lists all possibilities for the difference function for nine double points, in terms of the difference function of the underlying nine points. In particular, for each ${\underline h}$, both ${\underline h}^{max}$ and ${\underline h}^{min}$ exist, and we indicate by ``max'' or ``min" the function that achieves ${\underline h}^{max}$ or ${\underline h}^{min}$, respectively, for each $\underline{h}$. Of course when we have ``max = min," the Hilbert function of $2X$ is uniquely determined by that of $X$.
\addtocounter{thm}{1}
\begin{equation} \label{hfs of nine points}
\begin{array}{l|l|lccccccccccccccccccccccccccccccccc}
\hbox{\rm difference function of $X$} & \hbox{\rm possible difference functions of $2X$} & \hbox{\rm max/min} \\ \hline
\verb! 1 1 1 1 1 1 1 1 1! & \verb! 1 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1! & \hbox{\rm max = min} \\ \hline
\verb! 1 2 1 1 1 1 1 1! & \verb! 1 2 3 4 2 2 2 2 2 1 1 1 1 1 1 1! & \hbox{\rm max = min} \\ \hline
\verb! 1 2 2 1 1 1 1! & \verb! 1 2 3 4 4 3 2 2 1 1 1 1 1 1! & \hbox{\rm max = min} \\ \hline
\verb! 1 2 2 2 1 1! & \verb! 1 2 3 4 4 4 3 2 1 1 1 1! & \hbox{\rm max = min} \\ \hline
\verb! 1 2 2 2 2! & \verb! 1 2 3 4 4 4 4 2 2 1! & \hbox{\rm max} \\
& \verb! 1 2 3 4 4 4 3 2 2 2! & \hbox{\rm min} \\ \hline
\verb! 1 2 3 1 1 1! & \verb! 1 2 3 4 5 5 2 1 1 1 1 1! & \hbox{\rm max = min} \\ \hline
\verb! 1 2 3 2 1! & \verb! 1 2 3 4 5 6 4 2! & \hbox{\rm max} \\
& \verb! 1 2 3 4 5 6 3 2 1! \\
& \verb! 1 2 3 4 5 6 3 1 1 1! \\
& \verb! 1 2 3 4 5 6 2 2 1 1! & \hbox{\rm min}
\\ \hline
\verb! 1 2 3 3! & \verb! 1 2 3 4 5 6 6! & \hbox{\rm max} \\
& \verb! 1 2 3 4 5 6 5 1! \\
& \verb! 1 2 3 4 5 6 4 2! \\
& \verb! 1 2 3 4 5 6 3 3! \\
& \verb! 1 2 3 4 5 5 4 3! & \hbox{\rm min}
\end{array}
\end{equation}
\end{thm}
\begin{proof}
One has to ``integrate" the difference functions in order to verify the claims about ${\underline h}^{max}$ or ${\underline h}^{min}$. We leave this to the reader. The fact that the eight Hilbert functions listed above for $X$ give a complete list is standard, and we omit the proof.
\bigskip
\noindent \underline{Case 1}: \verb! 1 1 1 1 1 1 1 1 1!. If $X$ has this difference function then $X$ must be a set of 9 collinear points in $\pr2$. Such a set of points is necessarily a complete intersection, so it is easy to check that the difference function for $2X$ is the one claimed. (Even the minimal free resolution is well-known.)
\bigskip
\noindent \underline{Case 2}: \verb! 1 2 1 1 1 1 1 1!. If $X$ has this difference function then $X$ must consist of 8 points on a line and one point off the line (it follows from Theorem \ref{davis thm}). It is not hard to check, using B\'ezout arguments, that then $2X$ has the claimed difference function.
\bigskip
\noindent \underline{Case 3}: \verb! 1 2 2 1 1 1 1!. If $X$ has this difference function then $X$ must consist of seven points on a line, say $\lambda_1$, and two points off the line (again using Theorem \ref{davis thm}). Let $Q_1, Q_2$ be these latter points. We will see that the Hilbert function is independent of whether $Q_1$ and $Q_2$ are collinear with one of the seven other points or not. Note first that $2X$ contains a subscheme of degree 14 lying on a line. Hence the regularity is $\geq 14$, so the difference function ends in degree $\geq 13$.
Let $L_1$ be a linear form defining $\lambda_1$ and let $L_2$ be a linear form defining the line joining $Q_1$ and $Q_2$. Using B\'ezout's theorem, it is clear that there is no form of degree $\leq 3$ vanishing on $2X$. Furthermore, $L_1^2 L_2^2$ is the only form (up to scalar multiples) of degree 4 vanishing on $2X$. Now, in degree 5 we have that $L_1^2$ is a common factor for all forms in the ideal of $2X$. Hence $ (I_{2Q_1 + 2Q_2})_3 \cong (I_{2X})_5 $, where the isomorphism is obtained by multiplying by $L_1^2$. But $2Q_1 + 2Q_2$ imposes independent conditions on forms of degree 3, so these isomorphic components have dimension $10 - 6 = 4$.
The calculations above give the claimed difference function up to degree 5. But the sum of the terms of the difference function has to equal 27 ($= \deg 2X$), and the terms past degree 5 must be non-increasing and positive and non-zero through degree 13. Using also Lemma \ref{reg lemma} (which implies that the difference function must be zero no later than degree 14), this is enough to force the claimed difference function.
\bigskip
\noindent \underline{Case 4}: \verb! 1 2 2 2 1 1!. By Theorem \ref{davis thm}, $X$ must consist of six points, $X_1$, on a line, $\lambda_1$, and three collinear points, $X_2$, on another line, $\lambda_2$. The intersection of $\lambda_1$ and $\lambda_2$ may or may not be a point of $X_1$; it is not a point of $X_2$. We will see, as in Case 3, that this combinatorial distinction does not affect the Hilbert function of $2X$. Pictorially we have the following two possibilities:
\begin{picture}(200,90)
\put (80,18){\line (1,0){105}}
\put (65,17){$\scriptstyle \lambda_1$}
\put (92,15){$\bullet$}
\put (107,15){$\bullet$}
\put (122,15){$\bullet$}
\put (137,15){$\bullet$}
\put (152,15){$\bullet$}
\put (167,15){$\bullet$}
\put (80,38){\line (1,0){90}}
\put (65,37){$\scriptstyle \lambda_2$}
\put (92,35){$\bullet$}
\put (107,35){$\bullet$}
\put (122,35){$\bullet$}
\put (104,46){$\scriptstyle X_2$}
\put (129,4){$\scriptstyle X_1$}
\put (280,18){\line (1,0){105}}
\put (265,17){$\scriptstyle \lambda_1$}
\put (292,15){$\bullet$}
\put (307,15){$\bullet$}
\put (322,15){$\bullet$}
\put (337,15){$\bullet$}
\put (352,15){$\bullet$}
\put (367,15){$\bullet$}
\put (322,66){\line(1,-1){60}}
\put (330,53){$\bullet$}
\put (340,43){$\bullet$}
\put (350,33){$\bullet$}
\put (307,66){$\scriptstyle \lambda_2$}
\put (350,52){$\scriptstyle X_2$}
\put (329,4){$\scriptstyle X_1$}
\end{picture}
Combining Lemma \ref{reg lemma} with the fact that $2X$ contains a subscheme of degree 12 on a line, we get that the difference function of $2X$ ends in degree exactly 11. Using B\'ezout it is not hard to check that
\[
\begin{array}{lll}
h^0({\mathcal I}_{2X}) = h^0({\mathcal I}_{2X}(1)) = h^0({\mathcal I}_{2X}(2)) = h^0({\mathcal I}_{2X}(3)) = 0 \\
h^0({\mathcal I}_{2X}(4)) = 1 \\
h^0({\mathcal I}_{2X}(5)) = h^0({\mathcal I}_{2X_2}(3)) = h^0({\mathcal I}_{X_2}(2)) = 3 \\
h^0({\mathcal I}_{2X}(6)) = h^0({\mathcal I}_{2X_2}(4)) = h^0({\mathcal I}_{X_2}(3)) = 7.
\end{array}
\]
This means that the difference function of $2X$ begins \verb!1 2 3 4 4 4 3 ...! and arguing as in Case 3 gives the result.
\bigskip
\noindent \underline{Case 5}: \verb! 1 2 2 2 2!. This case corresponds to nine points on a reduced conic curve. There are three possibilities. If the conic is smooth then the nine points are arbitrary. If the conic consists of two lines then this case takes the form of five points on one line and four points on the other line. Here we can have (a) none of the nine points is the point of intersection of the two lines, or (b) one of the five points is the point of intersection. All of these cases have been studied in \cite{GHM}, and we omit the details.
\bigskip
\noindent \underline{Case 6}: \verb! 1 2 3 1 1 1!. Now $X$ consists of six point on a line plus three non-collinear points off the line. It is easy to check, using the same methods, that there is only one possibility for the Hilbert function of $2X$, independent of whether the line through two of the non-collinear points meets one of the six collinear points or not. We omit the details.
\bigskip
\noindent \underline{Case 7}: \verb! 1 2 3 2 1!. By Lemma \ref{reg lemma}, the difference function for $2X$ ends in degree $\leq 9$ and the entries again add up to 27. Furthermore, it is not hard to see that $X$ has at most 5 points on a line, and $X$ has at most one set of 5 collinear points.
The first main step in the proof is the following assertion:
\medskip
\begin{claim} \label{old claim 7.1}
$h^0({\mathcal I}_{2X}(5)) = 0$.
\end{claim}
Note that this implies that $h^0({\mathcal I}_{2X}(t)) = 0$ for $t \leq 5$. Suppose that there is a curve $F$, of degree 5 containing $2X$. There are several possibilities. By abuse of notation we will denote by $F$ also a form defining this curve.
\begin{itemize}
\item $F$ is reduced. Then $F$ has to contain 9 singular points, which form the points of $X$ (and hence have the difference function \verb!1 2 3 2 1!). This can happen in one of two ways:
\begin{itemize}
\item $F$ consists of the union of five lines, and $X$ consists of nine of the resulting ten double points. But from B\'ezout we note that the 10 double points do not lie on a cubic curve (since each of the five lines would have to be a component of the cubic), so the ten points have difference function \verb!1 2 3 4!, and hence $X$ cannot have difference function \verb!1 2 3 2 1!.
\item $F$ consists of the union of three lines and a smooth conic, and $X$ consists of all nine resulting double points. Now the three lines have to be components of any cubic containing $X$, so there is a unique such cubic, and again $X$ does not have difference function \verb!1 2 3 2 1!.
\end{itemize}
\item $F$ has a double conic. Then all the singular points of $F$ must lie on this conic. But, $X$ does not lie on a conic, so this is impossible.
\item $F$ has a double line, i.e. $F = L^2G$, $\deg G = 3$. Then $G$ contians at most 3 singular points of $F$. This forces the remaining 6 singular points to be on the line, contradicting the fact that at most 5 points of $X$ can lie on a line.
\end{itemize}
\noindent This concludes the proof of Claim \ref{old claim 7.1}.
\medskip
Thanks to Claim \ref{old claim 7.1}, we now know that the difference function for $2X$ has the form
\[
\verb! 1 2 3 4 5 6! \qed \qed \qed \qed
\]
where the last four spaces correspond to entries that are $\geq 0$ and add up to $27 - 21 = 6$.
Now notice that there is an irreducible flat family of subschemes of degree 9 with difference function \verb!1 2 3 2 1! \cite{ellingsrud}, and the general such is a complete intersection of two cubics. The difference function for the corresponding scheme $2X$ is easily checked to be \verb!1 2 3 4 5 6 4 2!. It follows that not only does this difference function exist, but in fact it corresponds to ${\underline h}^{max}$. (See also \cite[Remark 7.4]{GMS}.)
In particular, \verb!1 2 3 4 5 6 6! and \verb!1 2 3 4 5 6 5 1! do not occur. The following, then, are the remaining possibilities for the difference function of $2X$:
\begin{enumerate}
\item \verb! 1 2 3 4 5 6 4 2!
\item \verb! 1 2 3 4 5 6 4 1 1!
\item \verb! 1 2 3 4 5 6 3 3!
\item \verb! 1 2 3 4 5 6 3 2 1!
\item \verb! 1 2 3 4 5 6 3 1 1 1!
\item \verb! 1 2 3 4 5 6 2 2 2!
\item \verb! 1 2 3 4 5 6 2 2 1 1!
\end{enumerate}
\noindent For each of these we will either give a specific example (that the reader can verify directly, either by hand or on a computer program) or a proof of non-existence.
\begin{enumerate}
\item \verb! 1 2 3 4 5 6 4 2! . As we saw above, this occurs when $X$ is the complete intersection of two cubics, and this corresponds to ${\underline h}^{max}$.
\item \verb! 1 2 3 4 5 6 4 1 1!. This does not exist. Indeed, this difference function forces the existence of a line $\lambda$ that contains a subscheme of $2X$ of degree 9, which is impossible. (Any such subscheme must have even degree.)
\item \verb! 1 2 3 4 5 6 3 3!. This does not exist in our context. Note that it {\em does} exist when $X$ has difference function \verb!1 2 3 3!, as we will verify below.
To see that this does not exist, note that by Theorem \ref{davis thm}, the \verb!3 3! at the end forces the existence of a cubic curve $C$ that cuts out from $2X$ a subscheme $W$ of degree 21 with difference function \verb!1 2 3 3 3 3 3 3!. Observe that if $P$ is a point of $X$ which is a smooth point of $C$, then $C$ cuts out a non-reduced point of degree 2 at $P$. If $P$ is a point of $X$ which is a singular point of $C$, then $C$ contains the fat point $2P$ (which has degree 3).
Note also that our $h$-vector does not permit the existence of a subscheme of degree more
than 8 on a line.
Suppose first that $C$ is reduced. Since we only have the nine points of $X$ to work with, it is not hard to check, using the above observation, that the only way that $C$ can cut out from $2X$ a subscheme of degree 21 is if $X$ has the following configuration:
\begin{picture}(200,90)
\put (130,0){\line (1,1){65}}
\put (240,0){\line (-1,1){65}}
\put (130,15){\line(1,0){110}}
\put (142,12){$\bullet$}
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\put (170,40){$\bullet$}
\put (182,52){$\bullet$}
\put (194,40){$\bullet$}
\put (207,27){$\bullet$}
\end{picture}
\bigskip
\noindent But this uses all nine points, and its support lies on a {\em unique} cubic, contradicting the fact that $X$ has difference function \verb!1 2 3 2 1!. This configuration provides one of the correct difference functions for \verb!1 2 3 3! below.
Now suppose that $C$ is not reduced. Without loss of generality, $C$ has a double line. The difference function for $X$ would, in principle, allow five points of $X$ to lie on a line, but because the hypothetical difference function for the subscheme $W$ ends in degree 7, in fact at most four points of $X$ can lie on a line. So the double line contains at most four fat points of $2X$, which have degree 12. In order for $C$ to cut out a subscheme of degree 21, then, we must have a reduced line that cuts out an additional subscheme of degree at least 9. This forces at least five points of $X$ to be collinear, which again is impossible.
\item \verb! 1 2 3 4 5 6 3 2 1! . This difference function does exist. It occurs when $X$ is the union of one point and the complete intersection of a conic and a general quartic curve.
\item \verb! 1 2 3 4 5 6 3 1 1 1! . This difference function does exist. It occurs when $X$ is the union of five general points on a line, three general points on a second line, and one additional general point off both lines.
\item \verb! 1 2 3 4 5 6 2 2 2!. This difference function does not exist. Indeed, suppose that it did exist. Because of the \verb!2 2 2!, there must be a curve $C$ of degree 2 that cuts out on $2X$ a subscheme $W$ of degree 17 having difference function \verb!1 2 2 2 2 2 2 2 2!.
First note that $X$ cannot contain five points on a line (and hence a subscheme of $W$ of degree at least 10) since the hypothetical difference function ends in degree 8. Now consider cases.
\begin{enumerate}
\item $C$ is smooth: then it cannot cut out a subscheme of odd degree.
\item $C$ is reduced and reducible: then we cannot obtain the desired subscheme $W$ of degree 17 unless $X$ contains 5 points on a line, in which case $W$ contains a subscheme of degree at least 10 on that line.
\item $C$ non-reduced: then we cannot have a subscheme of degree 17 supported on that line.
\end{enumerate}
\item \verb! 1 2 3 4 5 6 2 2 1 1!. This difference function does exist. It occurs when $X$ has the following configuration:
\begin{picture}(200,90)
\put (133,3){\line (1,1){65}}
\put (243,-3){\line (-1,1){71}}
\put (145,15){$\bullet$}
\put (182,0){$\bullet$}
\put (219,15){$\bullet$}
\put (157,27){$\bullet$}
\put (170,40){$\bullet$}
\put (182,52){$\bullet$}
\put (194,40){$\bullet$}
\put (207,27){$\bullet$}
\put (231,3){$\bullet$}
\end{picture}
\end{enumerate}
\bigskip
\noindent \underline{Case 8}: \verb! 1 2 3 3!. This is the difference function for a general set of nine points in $\pr2$. We know (from \cite{anticanSurf}, for example) that the ``generic" difference function for nine general double points is \verb!1 2 3 4 5 6 6!. Hence this occurs and corresponds to the maximum possible Hilbert function. Clearly all other possibilities will end in degree $\geq 7$. On the other hand, Lemma \ref{reg lemma} guarantees that all other examples end in degree $\leq 7$. Note that again, $X$ can have at most four points on a line.
\medskip
\underline{Claim 8.1}: $h^0 ({\mathcal I}_{2X}(5)) \leq 1$.
Notice that as a consequence of this claim we also obtain $h^0({\mathcal I}_{2X}(4)) = 0$. Keeping in mind that it is possible that $h^0({\mathcal I}_{2X}(5)) = 0$ (e.g. the generic case), we will assume that $h^0({\mathcal I}_{2X}(5)) \neq 0$ and deduce that then it must be $=1$. So let $C$ be a curve of degree 5 containing the scheme $2X$. As before (Claim \ref{old claim 7.1}) there are a few possibilities.
\begin{itemize}
\item If $C$ is reduced then since it must have nine double points, it must consist of either the union of five lines, no three through a point, or the union of three lines and a smooth conic, with no three components meeting in a point. By B\'ezout, each component of $C$ is then a fixed component of the linear system $|(I_{2X})_5|$, so the claim follows.
\item If $C$ contains a double line then at most four (fat) points of $2X$ lie on this line, so we must have a cubic curve that contains the remaining five double points. Consider the support, $X_1$, of these five double points. The points of $X_1$ are not collinear, and they do not have four collinear points since $X$ lies on only one cubic. With these restrictions, clearly there is no cubic curve double at such a set of five points.
\item If $C$ contains a double conic (smooth or not), this conic contains at most seven points of $X$, because of the Hilbert function of $X$. Hence $C$ must have a line that contains two double points, which is impossible.
\end{itemize}
This concludes the proof of Claim 8.1.
\medskip
It follows that the possibilities for the difference function of $2X$ are the following:
\begin{enumerate}
\item \verb! 1 2 3 4 5 6 6!
\item \verb! 1 2 3 4 5 6 5 1!
\item \verb! 1 2 3 4 5 6 4 2!
\item \verb! 1 2 3 4 5 6 3 3!
\item \verb! 1 2 3 4 5 5 5 2!
\item \verb! 1 2 3 4 5 5 4 3!
\end{enumerate}
\noindent As before, we examine these each in turn.
\begin{enumerate}
\item \verb! 1 2 3 4 5 6 6!. We have seen that this occurs generically.
\bigskip
\item \verb! 1 2 3 4 5 6 5 1!. This exists, for instance from the following configuration:
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\noindent (That is, seven points on a conic, three points on a line, with one point in common.)
\bigskip
\item \verb! 1 2 3 4 5 6 4 2!. This exists, for instance from the following configuration:
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\end{picture}
\item \verb! 1 2 3 4 5 6 3 3!. This exists, for instance from the configuration mentioned earlier:
\begin{picture}(200,90)
\put (130,0){\line (1,1){65}}
\put (240,0){\line (-1,1){65}}
\put (130,15){\line(1,0){110}}
\put (142,12){$\bullet$}
\put (170,12){$\bullet$}
\put (194,12){$\bullet$}
\put (222,12){$\bullet$}
\put (157,27){$\bullet$}
\put (170,40){$\bullet$}
\put (182,52){$\bullet$}
\put (194,40){$\bullet$}
\put (207,27){$\bullet$}
\end{picture}
\bigskip
\item \verb! 1 2 3 4 5 5 5 2! We claim that this does not exist. The key is that such a double point scheme, $2X$, would have to lie on a unique quintic curve, say $C$. To see that this is impossible, the argument is very similar to that of Claim \ref{old claim 7.1}, but with a small difference. One checks as before that $C$ must consist either of five lines or the union of three lines and a conic, and in both cases we must have that no three components share a common point. In the first case, $X$ consists of nine of the ten double points of $C$ (it does not matter which nine), and in the second case $X$ consists of all nine double points of $C$. But in both of these cases one can check geometrically or on a computer that $h^0({\mathcal I}_{2X} (6)) = 4$, while the hypothetical difference function would require this dimension to be 3.
\bigskip
\item \verb! 1 2 3 4 5 5 4 3! This exists, and can be achieved by the configuration mentioned above: it is supported on nine of the ten intersection points of five general lines in $\pr2$.
\end{enumerate}
\end{proof}
\section{Approach I: Points on Cubics}\label{ptsoncubics}
For this section we will always let $C \subset \pr2$ be an irreducible cubic curve
defined by a polynomial $F$ of degree 3. Let $X$ be a
reduced set of $n = 3t + \delta$ points on $C$, where
$0 \leq \delta \leq 2$. Let $Z= 2X$ be the double point scheme
in $\pr2$ supported on $X$.
The object of this section is to describe the possible Hilbert functions of
$X$ and of the corresponding $Z$. In some instances we assume that $t$
is ``big enough" (with mild bounds), and in one instance (Theorem \ref{smooth cubic}(b))
we assume that the points are not too special and that $C$ is smooth.
\bigskip
\begin{prop} \label{ci case} Assume that $\delta = 0$, $t \geq 3$, and the Hilbert function of $X$ has first difference
\addtocounter{thm}{1}
\begin{equation} \label{hf of 1 2 3 ... 3 2 1}
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & \dots & t -1 & t & t+1 & t+2 \\
\hline
\Delta h_X & 1 & 2 & 3 & \dots & 3 & 2 & 1 & 0
\end{array}
\end{equation}
(where the values between $2$ and $t-1$, if any, are all 3). Then $X$ is a complete intersection with ideal $(F,G)$, where
$\deg F = 3$ and $\deg G = t$. Furthermore, if $C$ is singular
then the singular point is not a point of $X$. Assume that $t>3$,
so that $t+3 > 6$ and $2t > t+3$. Then we have the first difference of the Hilbert function of $Z$ is
\medskip
\begin{itemize}
\item[($t=3$)] $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 4 \ 2 \ 0$;
\item[($t=4$)] $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 6 \ 5 \ 3 \ 1 \ 0$;
\item[($t=5$)] $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 6 \ 6 \ 5 \ 4 \ 2 \ 1 \ 0$;
\item[($t \geq 6$)]
{\small
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & 5 & 6 & \dots & t +2 & t+3 & t+4 & t+5 & \dots & 2t-1 & 2t & 2t+1 & 2t+2 \\
\hline
\Delta h_Z & 1 & 2 & 3 & 4 & 5 & 6 & 6 & \dots & 6 & 5 & 4 & 3 & \dots & 3 & 2 & 1 & 0
\end{array}
\]
}
\end{itemize}
\end{prop}
\begin{proof}
We first show that $X$ must be a complete intersection. From the
Hilbert function (\ref{hf of 1 2 3 ... 3 2 1}), it is clear that $F$ is a
factor of every form in $I_X$ up to degree $t-1$, and that in fact
it generates the ideal up to this point. In degree $t$ there is
exactly one new form, $G$, in the ideal, and since $F$ is
irreducible, $F$ and $G$ have no common factor. But $(F,G)$
is a saturated ideal that is contained in $I_X$ and defines a
zero-dimensional scheme of the same degree as $X$, hence $I_X = (F,G)$.
Since $X$ is a complete intersection, if $C$ is singular and
$P \in X$ is the singular point of $C$, then $X$ must be
non-reduced at $P$, contradicting our assumption.
Now, it is a simple (and standard) argument that $I_Z = (F^2,FG,G^2)$,
and one can verify the claimed Hilbert function of $R/I_Z$,
for instance by using the fact that $(F,G)$ is directly linked
to the ideal of $Z$ by the complete intersection $(F^2,G^2)$, and using the formula for the behavior of Hilbert functions under linkage \cite{DGO} (see also \cite{migliore}). We omit the details.
\end{proof}
\bigskip
Because the form $F$ of least degree is irreducible, the Hilbert function
of $X$ has first difference that is strictly decreasing from the first
degree where it has value $< 3$ until it reaches 0. Having
proved Proposition \ref{ci case}, we can now assume without
loss of generality that the Hilbert function of $X$ has first difference
\addtocounter{thm}{1}
\begin{equation} \label{hf of 1 2 3 ... 3}
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & \dots & t & t+1 & t+2 \\
\hline
\Delta h_X & 1 & 2 & 3 & 3 & \dots & 3 & \delta & 0
\end{array}
\end{equation}
where $0 \leq \delta \leq 2$.
\begin{thm} \label{smooth cubic}
Assume that either $C$ is smooth, or else that no point of $X$ is
the singular point of $C$. Assume further that $t > 5-\delta$.
Then the Hilbert function of the double point scheme $Z$ supported on $X$ is
{\small
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & 5 & \dots & t+3 & t+4 & t+5 & \dots & 2t+\delta-1 & 2t+\delta \\
\hline
\Delta h_Z & 1 & 2 & 3 & 4 & 5 & 6 & \dots & 6 & 3+\delta & 3 & \dots & 3 & ??
\end{array}
\]
}
For the behavior in degree $\geq 2t+\delta$, we have the following conclusions.
\begin{itemize}
\item[(a)] If $\delta = 1$ or $\delta = 2$ then $\Delta h_Z (2t+\delta) = 3-\delta$ and $\Delta h_Z(k) = 0$ for $k > 2t+\delta$.
\item[(b)] If $\delta = 0$, there are two possible Hilbert functions, these being determined by
\begin{itemize}
\item[i.] $\Delta h_Z(2t) = 3$ and $\Delta h_Z(k) = 0$ for $k > 2t$, and
\item[ii.] $\Delta h_Z(2t) = 2, \Delta h_Z(2t+1) = 1, \Delta h_Z(2t+2) = 0$.
\end{itemize}
\noindent Moreover, if the points $p_i$ are sufficiently general and $C$ is smooth, then the Hilbert function is the first of these two.
\end{itemize}
\end{thm}
\begin{proof}
A complete analysis of all cases with $\delta = 0$, where $C$ is a
reduced cubic and the points $p_i$ either are arbitrary smooth points
of $C$ or they are completely arbitrary and $C$ is also irreducible,
is given in the next section using the Second Way. The interested
reader can complete the current proof to those cases using the
techniques of this section, as a further comparison of the methods.
First note that the condition $t > 5-\delta$ implies $2t+\delta > t+5$.
We proceed via a number of claims.
\bigskip
\noindent {\bf Claim 1:} {\em For $\ell < 2t+\delta$, $(I_Z)_\ell$ has
the cubic form $F$ as a common factor (i.e.\ $C$ is part of the base locus).}
Suppose that $G \in (I_Z)_\ell$ does not have $F$ as a factor.
Then at each point of $X$, the intersection multiplicity of $F$ and
$G$ is at least 2 since $G$ is double at each point. Hence by
B\'ezout's theorem, $3\ell \geq 2n = 2(3t+\delta) = 6t+2\delta$.
Hence $\ell \geq 2t + \frac{2}{3} \delta$, and the claim follows.
\bigskip
\noindent {\bf Claim 2:} {\em For $\ell \leq t+3$, $(I_Z)_\ell$ has $F^2$ as a common factor.}
By Claim 1, since $F$ is not double at any point of $X$, for $\ell < 2t+\delta$ we have an isomorphism
\addtocounter{thm}{1}
\begin{equation} \label{isom of ideals}
(I_X)_{\ell -3} \cong (I_Z)_\ell
\end{equation}
where the isomorphism is given by multiplication by $F$. But from (\ref{hf of 1 2 3 ... 3}),
we see that $F$ is a common factor for $(I_X)_k$ for all $k \leq t$.
Hence $(I_Z)_\ell$ has $F^2$ as a factor whenever $\ell -3 \leq t$, as claimed.
This verifies the claimed first difference of the Hilbert function up to degree $t+3$.
Note that the Hilbert function, in degree $t+3$, has value equal to
\[
1 + 2 + 3 + 4 + 5 + 6 \cdot [(t+3) - 4] = 6t+9.
\]
We now compute the value in degree $t+4 < 2t$. Using the isomorphism (\ref{isom of ideals}), we have
\[
\begin{array}{rcl}
h_Z(t+4) & = & \binom{t+6}{2} - h^0({\mathcal I}_Z(t+4)) \\ \\
& = & \binom{t+6}{2} - h^0({\mathcal I}_X (t+1)) \\ \\
& = & \binom{t+6}{2} - \left [ \binom{t+3}{2} - h_X(t+1) \right ] \\ \\
& = & \binom{t+6}{2} - \left [ \binom{t+3}{2} - (3t+\delta) \right ] \\ \\
& = & 6t + 12 + \delta
\end{array}
\]
Then we easily see that $\Delta h_Z(t+4) = 3+\delta$ as claimed.
Next we compute the value in degree $t+5$. We have
$2t + \delta > t+5$, so we can use Claim 1. Then a similar computation gives
\[
h_Z(t+5) = 6t+15+\delta.
\]
From this we immediately confirm $\Delta h_Z(t+5) = 3$.
Since $F$ is a common factor in all components $< 2t+\delta$, and
since $\Delta h_Z$ takes the value 3 already in degree $t+5$, it
repeats this value until $F$ is no longer a common factor. In
particular, it takes the value 3 up to degree $2t+\delta -1$.
We now have to see what happens past degree $2t+\delta -1$.
Note that using our above calculations, it follows that
\[
\begin{array}{rcl}
h_Z(2t+\delta-1) & = & 6t+12+\delta + 3[2t+\delta-1 - (t+4)] \\ \\
& = & 3 ( 3t+ \delta) - 3 + \delta.
\end{array}
\]
Since $\deg Z = 3(3t+\delta)$, we have reached the multiplicity
minus $(3-\delta)$. We consider these cases separately.
When $\delta = 1$ or $\delta = 2$, we are adding only 2 or 1,
respectively, and since the first difference of the Hilbert function
cannot be flat at this point, $\Delta h_Z$ must be as claimed in (a). This completes (a).
Since the sum of the values of $\Delta h_Z$ up to degree $2t-1$ is $9t-3$, this observation that $\Delta h_Z$ cannot be flat at this point also proves that the possibilities listed in (b) are the only ones possible.
If $\delta = 0$, though, $\Delta h_Z$ can either end $\dots 3,3,0$ or
$\dots 3,2,1$. We now consider these two possibilities. The former
means that also in degree $2t+\delta = 2t$, all forms in $I_Z$ have
$F$ as a factor. The latter means that there is a form, $G$, of degree
$2t+\delta = 2t$ in $I_Z$ that does not have $F$ as a factor, and
hence $(F, G)$ is a regular sequence (since $F$ is irreducible).
Suppose that the latter holds. Note that the complete intersection
defined by $(F,G)$ has degree $3 \cdot 2t = 6t = 2n$. As in Claim 1,
$G$ cuts out on $C$ a divisor of degree at least $2n$, so in fact $G$
cuts out exactly the divisor $2X$ on $C$. So $X$ itself is not a
complete intersection (since it has the Hilbert function given by
(\ref{hf of 1 2 3 ... 3})), but the divisor $2X$ (as a subscheme of
$\pr2$) is a complete intersection, namely of type $(3,2t)$.
Note that $2X$, which is curvilinear, is not the same as $Z$.
Now suppose that $C$ is smooth. We know that then two effective
divisors of the same degree are linearly equivalent if and only if
they have the same sum in the group of $C$. The condition described
in the previous paragraph implies that the sum of the points of $X$ is a
2-torsion point in the group of $C$ but is not zero. Since there are at most three 2-torsion points in the group of $C$,
for general choices we have a contradiction, and so such a $G$ cannot exist (in general),
and we have proved the assertion about the general choice of the points.
Finally, we show that the Hilbert function ii.\ of (b) also occurs. We begin with four general lines,
$\lambda_1, \lambda_2,\lambda_3,\lambda_4 \subset \pr2$
and let $P_1,P_2, P_3, P_4, P_5, P_6$ be the six points of pairwise
intersection of these lines. Let $G_1$ be the form defining the union
of these four lines. Let $X_1 = \bigcup_{1 \leq i \leq 6} P_i$. Notice
that $X_1$ does not lie on any conic, since by B\'ezout any conic
containing $X_1$ has to contain all four lines $\lambda_1,\dots,\lambda_4$,
hence must have $G_1$ as a factor. Hence Hilbert function of $X_1$ has first difference $(1,2,3)$, and$X_1$ is not a complete intersection.
Let $C$ be a general cubic curve containing $X_1$, and let $F$ be the
defining polynomial of $C$. $C$ is smooth. Notice that the degree of
the complete intersection of $F$ and $G_1$ is 12, and this complete
intersection is at least double at each $P_i$, so in fact it is exactly
double at each $P_i$. In particular, there is no additional multiplicity
at any of the $P_i$ coming from tangency. As a divisor on $C$, note
that $X_1$ is not cut out by any conic, since it is not a complete intersection.
However, the divisor $2X_1$ is cut out by a quartic, namely $G_1$.
Now let $X$ be the union of $X_1$ with a general hypersurface section,
$W_1$, of $C$ cut out by a curve of degree $t-2$. Note that $W_1$ is a
complete intersection defined by $(F,H)$ for some form $H$ of degree
$t-2$. We first claim that $X$ is not a complete intersection. Indeed,
suppose that $X$ were a complete intersection defined by $(F,H')$ for
some $H'$ of degree $t$. Then $I_X$ links $W_1$ to $X_1$. But $W_1$
and $X$ are both complete intersections sharing a generator, so by
liaison theory the residual is also a complete intersection. But we
have seen that $X_1$ is not a complete intersection. Contradiction.
In particular, $\Delta h_X$ is given by (\ref{hf of 1 2 3 ... 3}).
Now let $Z$ be the fat point scheme supported on $X$, and consider
the form $G_1 H^2$. This has degree $2t$, and cuts out the divisor
$2X$ on $C$. Even more, $G_1 H^2$ is an element of $I_Z$ in
degree $2t$ that does not have $F$ as a factor. As we saw above,
this gives a value $\Delta h_Z(2t) = 2$ and $\Delta h_Z(2t+1) = 1$
as desired. This completes the proof of Theorem~\ref{smooth cubic}.
\end{proof}
Now we wish to explore the possibilities when $C$ is singular and one
point, $P$, of $X$ is the singular point of $C$. The arguments are very
similar, and we will primarily highlight the differences. The main
observation is that $C$ is already double at $P$ so we have to focus
on the remaining $n-1$ points.
\begin{lem} \label{ci poss}
Assume that $C$ is singular, that $P \in X \subset C$ is the singular
point of $C$, and that $n \geq 5$. Then $X$ is not a complete intersection.
\end{lem}
\begin{proof}
More precisely, we will show that if $P \in X \subset C$ with $X$ a
complete intersection, and if $P$ is the singular point of $C$, then
$X$ has one of the following types: $CI(1,1),\ CI(1,2),\ CI(2,2)$.
First note that if $X$ is a complete intersection defined by forms $(F,G)$,
where $F$ is the defining polynomial for $C$, then $X$ has multiplicity
$\geq 2$ at $P$, so $X$ is not reduced. Hence we have to determine
all the possibilities for reduced complete intersections on $C$ that do
not use $F$ as a minimal generator. The listed possibilities are clear:
one point, two points, four points, and these all exist even including
$P$ as one of the points. Using the irreducibility of $F$, it is not hard to
show that these are the only possibilities, and we omit the details.
\end{proof}
\begin{thm} \label{singular cubic}
Assume that $C$ is an irreducible singular cubic with singular point $P$,
and assume that $P \in X$, where $X$ is a reduced set of $3t + \delta$
points of $C$, with $0 \leq \delta \leq 2$. Assume further that $t > 3$.
Then the Hilbert function of the double point scheme $Z$ supported on $X$ is as follows.
\begin{enumerate}
\item If $\delta = 0$ then
{\small
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & 5 & \dots & t+2 & t+3 & t+4 & t+5 & \dots & 2t & 2t+1 & 2t+2 \\
\hline
\Delta h_Z & 1 & 2 & 3 & 4 & 5 & 6 & \dots & 6 & 5 & 3 & 3 & \dots & 3 & 1 & 0
\end{array}
\]
}
\item If $\delta = 1$ then either
{\small
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & 5 & \dots & t+2 & t+3 & t+4 & t+5 & \dots & 2t & 2t+1 & 2t+2 \\
\hline
\Delta h_Z & 1 & 2 & 3 & 4 & 5 & 6 & \dots & 6 & 6 & 3 & 3 & \dots & 3 & 3 & 0
\end{array}
\]
}
or
{\small
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & 5 & \dots & t+2 & t+3 & t+4 & t+5 & \dots & 2t & 2t+1 & 2t+2 \\
\hline
\Delta h_Z & 1 & 2 & 3 & 4 & 5 & 6 & \dots & 6 & 5 & 4 & 3 & \dots & 3 & 3 & 0
\end{array}
\]
}
\item If $\delta = 2$ then
{\small
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & 5 & \dots & t+2 & t+3 & t+4 & t+5 & \dots & 2t & 2t+1 & 2t+2 & 2t+3 \\
\hline
\Delta h_Z & 1 & 2 & 3 & 4 & 5 & 6 & \dots & 6 & 6 & 4 & 3 & \dots & 3 & 3 & 2 & 0
\end{array}
\]
}
\end{enumerate}
\end{thm}
\begin{proof}
The bound $t > 3$ is simply to ensure that in each case, some value of the Hilbert
function $\Delta h_Z$ takes the value 3. For instance, in the case $\delta = 0$,
we have $2t > t+3$. As a consequence of Lemma \ref{ci poss}, when
$n = 3t + \delta \geq 5$ the Hilbert function of $X$ must have first difference
\addtocounter{thm}{1}
\begin{equation} \label{hf of X on sing}
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & \dots & t & t+1 & t+2 \\
\hline
\Delta h_X & 1 & 2 & 3 & 3 & \dots & 3 & \delta & 0
\end{array}
\end{equation}
In analogy with Theorem \ref{smooth cubic}, we first have
\medskip
\noindent {\bf Claim 1:} {\em Assume that
\[
\ell \leq
\left \{
\begin{array}{ll}
2t & \hbox{if $\delta = 0$} \\ \\
2t+1 & \hbox{if $\delta = 1,2$}
\end{array}
\right.
\]
Then $F$ is a common factor of $(I_Z)_\ell$.}
The proof is the same as that of Claim 1 in Theorem \ref{smooth cubic},
except that the intersection multiplicity of $F$ and $G$ at $P$ is now at least 4.
\noindent {\bf Claim 2:} {\em For $\ell \leq t+2$, $(I_Z)_\ell$ has $F^2$ as a common factor. Furthermore,
\bigskip
\begin{itemize}
\item If $\delta = 0$ then $F^2$ is {\em not} a common factor of $(I_Z)_{t+3}$.
\item If $\delta = 2$ then $F^2$ is a common factor of $(I_Z)_{t+3}$.
\item If $\delta =1$ then $F^2$ may or may not be a common factor of $(I_Z)_{t+3}$ (examples exist for either option).
\end{itemize}
}
The proof of Claim 2 hinges on the possible Hilbert functions for $X-\{P\}$.
In particular, we show that $(I_{X-\{P\}})_{t-1}$ always has $F$ as a common
factor, and the differences in the three cases rest with the possibilities for $(I_{X-\{P\}})_t$,
which we get by comparing to those for $I_X$, obtained using Lemma \ref{ci poss}.
\begin{itemize}
\item If $\delta = 0$ then $X$ has Hilbert function with first difference
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & \dots & t & t+1 \\
\hline
\Delta h_X & 1 & 2 & 3 & 3 & \dots & 3 & 0
\end{array}
\]
so clearly the only possibility for $\Delta h_{X-\{P\}}$ is
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & \dots & t-1 & t & t+1 \\
\hline
\Delta h_{X-\{P\}} & 1 & 2 & 3 & 3 & \dots & 3 & 2 & 0
\end{array}
\]
Hence there is a form $G$ of degree $t$ vanishing on $X-\{P\}$ but not
containing $F$ as a factor, so $FG \in (I_Z)_{t+3}$ does not have $F^2$ as a factor.\\
\item If $\delta = 2$ then $X$ has Hilbert function with first difference
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & \dots & t-1 & t & t+1 & t+2 \\
\hline
\Delta h_{X-\{P\}} & 1 & 2 & 3 & 3 & \dots & 3 & 3 & 2 & 0
\end{array}
\]
\noindent so $\Delta h_{X-\{P\}}$ is
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & \dots & t-1 & t & t+1 & t+2 \\
\hline
\Delta h_{X-\{P\}} & 1 & 2 & 3 & 3 & \dots & 3 & 3 & 1 & 0
\end{array}
\]
\end{itemize}
\noindent We know that $(I_Z)_\ell \cong (I_{X - \{P\}})_{\ell -3}$ for $\ell$ satisfying
the bounds of Claim 1, and as a result of the above observations we know
when $(I_{X - \{P\}})_{\ell -3}$ is forced to have $F$ as a common factor, so the claim follows.
\item If $\delta = 1$ then $X$ has Hilbert function with first difference
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & \dots & t & t+1 & t+2 \\
\hline
\Delta h_X & 1 & 2 & 3 & 3 & \dots & 3 & 1 & 0
\end{array}
\]
\noindent so $\Delta h_{X-\{P\}}$ is either
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & \dots & t-1 & t & t+1 \\
\hline
\Delta h_{X-\{P\}} & 1 & 2 & 3 & 3 & \dots & 3 & 3 & 0
\end{array}
\]
or
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & \dots & t-1 & t & t+1 & t+2 \\
\hline
\Delta h_{X-\{P\}} & 1 & 2 & 3 & 3 & \dots & 3 & 2 & 1 & 0
\end{array}
\]
\noindent Since we have removed $P$, the remaining points could
be a complete intersection, so $F^2$ is a common factor of $(I_Z)_{t+3}$ if and only
if the points of $X - \{P\}$ are not a complete intersection of a curve of degree $t$ with $F$.
This completes the proof of Claim 2.
The rest of the proof is very similar to that of Theorem \ref{smooth cubic} and we omit the details.
\end{proof}
\section{Approach II: Points on Cubics}\label{appII}
Let $Z = m_1p_1 + \cdots + m_n p_n \subset \mathbb P^2$,
where the points $p_1,\dots,p_n$ are distinct and arbitrary.
When $n<9$, a complete determination of $h_Z$
is given in \cite{GHM}, but the case of $n\geq9$ remains of interest.
Giving a complete determination of $h_Z$
for all $n\geq9$ arbitrary distinct points $p_1, \ldots , p_n$
would involve solving some extremely hard
open problems.
For example, it is even an open problem to determine $h_Z$ for $n>9$
when the points $p_1, \ldots , p_n$ are general and $m_1=\cdots =m_n$.
So here, as in \textsection\ref{ptsoncubics},
we consider the case of $n\geq9$ points $p_i$ in special cases. These cases include those
considered in \textsection\ref{ptsoncubics}. We recover and in some cases extend the
results of \textsection\ref{ptsoncubics}, but the methods we use here are different.
To start, let $p_1,\cdots,p_n$ be $n\geq 9$ distinct points on a reduced plane
cubic $C$. If $C$ is not irreducible, we assume further that all the points
are smooth points of $C$. If $D$ is a component of $C$, let $n_D$ be the number of
these points on $D$. We will say that the points are {\em evenly
distributed\/} if $n_D=n(\deg(D))/3)$ for every reduced
irreducible component $D$ of $C$. Note that for $n$ points to be
evenly distributed, it is necessary either that 3 divide $n$ or that $C$
be irreducible.
We will use some facts about surfaces obtained by blowing up points in the plane,
in particular we'll make use of the intersection form on such surfaces, which we now briefly recall.
Given distinct points $p_1,\ldots,p_n\in\pr2$, let $\pi : Y\to {\bf P}^2$
be the morphism obtained by blowing up the points $p_i$. The divisor class group ${\rm Cl}(Y)$
of divisors modulo linear equivalence
is a free abelian group with basis $[L],[E_1],\ldots,[E_n]$, where
$L$ is the pullback to $Y$ of a general line, and $E_i=\pi^{-1}(p_i)$.
There is a bilinear form, called the {\em intersection form}, defined on the group of divisors,
which descends to ${\rm Cl}(Y)$. It is uniquely determined by the fact
that $L$, $E_1$, $\ldots$, $E_n$ are orthogonal with respect to the intersection form,
with $L\cdot L=L^2=1$ and $E_i^2=-1$ for $i=1,\ldots,n$. For two distinct, reduced, irreducible curves
$C_1$ and $C_2$ on $Y$, $C_1\cdot C_2$ is just the number of points of intersection
of the two curves, counted with multiplicity.
We recall that a divisor $F$ is {\em nef}
if $F\cdot C\geq 0$ for every effective divisor $C$. A useful
criterion for nefness is that if $F$ is an effective divisor such
that $F\cdot C\geq0$ for every component $C$ of $F$, then $F$ is
nef.
In preparation for stating Theorem \ref{method2cor1}, our main result in this section, we
set some additional notation. Let $Z=m(p_1+\cdots+p_n)$. In
degrees $t$ such that $3t=mn$,
the value of $h_Z(t)$ is influenced by torsion in the group $\Pic(C)$.
Our formula for $h_Z$ as given in Theorem \ref{method2cor1}
accounts for this influence via an integer-valued function we will denote by $s$.
In fact, $s$ depends on the points $p_i$, on $m$ and on $t$, but for a fixed
set of points $p_i$ it is convenient to mostly suppress the dependence on the points
and denote $s$ as $s(t,n,m)$, where the parameter $n$ is a reminder of the dependence on the $n$ points. To define $s(t,n,m)$,
let $L$ be a general line in the plane and fix evenly distributed smooth points
$p_1,\ldots,p_n$ of a reduced cubic $C$. Since Theorem \ref{method2cor1} applies only for
$n\geq9$ and we need $s(t,n,m)$ only when $t\geq nm/3$, we define $s(t,n,m)$ only for $n\geq9$
when $t\geq nm/3$:
\begin{itemize}
\item[(1)] If $t>nm/3$, we set $s(t,n,m)=0$.
\item[(2)] If $n=9$ and $t=3m$, let $\lambda$ be the order (possibly infinite) of
${\mathcal O}_C(3L)\otimes {\mathcal O}_C(-p_1-\cdots-p_9)$ in $\Pic(C)$.
We then set $s(t,n,m)=\lfloor m/\lambda\rfloor$.
\item[(3)] If $n>9$ and $t=nm/3$, we set $s(t,n,m)=1$ if
${\mathcal O}_C(tL)\otimes {\mathcal O}_C(-mp_1-\cdots-mp_n)={\mathcal O}_C$
in $\Pic(C)$, and we set $s(t,n,m)=0$ otherwise.
\end{itemize}
The value of $s(t,n,m)$ depends on whether
${\mathcal O}_C(tL)\otimes {\mathcal O}_C(-mp_1-\cdots-mp_n)$ is trivial.
Note triviality of this line bundle is equivalent to the divisor $mp_1+\cdots+mp_n$ on $C$
being the intersection of $C$ with a curve $H$,
necessarily of degree $t=mn/3$. Of course it can happen that
${\mathcal O}_C(tL)\otimes {\mathcal O}_C(-p_1-\cdots-p_n)$ is non-trivial even though
${\mathcal O}_C(tmL)\otimes {\mathcal O}_C(-mp_1-\cdots-mp_n)$ is trivial. For example,
if $p_1, p_2$ and $p_3$ are flexes on $C$ but not collinear, then
${\mathcal O}_C(L)\otimes {\mathcal O}_C(-p_1-p_2-p_3)$ is not trivial, but
${\mathcal O}_C(3L)\otimes {\mathcal O}_C(-3p_1-3p_2-3p_3)$ is trivial, and $H$ in this case
is the union of the lines tangent to $C$ at the points $p_1$, $p_2$ and $p_3$.
When $C$ is a smooth cubic curve, triviality of
${\mathcal O}_C(tL)\otimes {\mathcal O}_C(-mp_1-\cdots-mp_n)$
is equivalent to the sum $mp_1+\cdots+mp_n$ being trivial in the group law on the cubic
(with respect to a flex being taken as the identity element). (The divisor $X_1$ given in the proof of part (c) of Theorem \ref{smooth cubic} gives another example, and shows that this issue arose also with the first approach.)
\begin{rem}
When $n=9$, the values of $\lambda$ that can occur depend on
the torsion in $\Pic(C)$, and this depends on $C$ and on the characteristic
of the ground field; see Remark \ref{torsionRem}.
Thus knowing something about $C$ tells us something about
what Hilbert functions can occur for points on $C$, but the Hilbert
functions themselves depend only on $\lambda$, and
already for a smooth irreducible non-supersingular cubic $C$,
there is torsion of all orders.
\end{rem}
\begin{thm}\label{method2cor1}
Let $X =p_1+\cdots+p_n$ be a set of $n\geq 9$
evenly distributed smooth points on a reduced plane cubic $C$. Let $Z = mX$.
The value $h_Z(t) = \hbox{dim} (k[{\bf P}^2]/(I(Z)))_t$
of the Hilbert function in degree $t$ is:
\begin{itemize}
\item[(i)] $\binom{t+2}{2}$ if $t<3m$;
\item[(ii)] $n\binom{m+1}{2}-s(t,n,m)$ if $t \geq nm/3$; and
\item[(iii)] $\binom{t+2}{2}-\binom{t-3r+2}{2}+n\binom{m-r+1}{2}-s(t-3r,n,m-r)$ if $n>9$ and
$3m\leq t< mn/3$, where $r=\lceil (mn-3t)/(n-9)\rceil$.
\end{itemize}
\end{thm}
\begin{proof} This result is a corollary of the main result of
\cite{anticanSurf}.
Let $F=tL-mE_1-\cdots-mE_n$, where $\pi : Y\to {\bf P}^2$
is the morphism obtained by blowing up the points $p_i$,
$L$ is the pullback to $Y$ of a general line, and $E_i=\pi^{-1}(p_i)$.
Let $C'\subset Y$ be the proper transform of $C$ with respect to $\pi$.
Since the points $p_i$ blown up are smooth points on $C$, we see
$[C']=[3L-E_1-\cdots-E_n]$ (and hence $C'$ is an anticanonical divisor). Moreover,
each component of $C'$ is the proper transform
$D'$ of a component $D$ of $C$, and each of the components of $C$ (and hence of $C'$) is
reduced. (To see this note that $n_D>0$ for each component $D$ of $C$
since the points $p_i$ are evenly distributed, but the number of points $p_i$ which lie on $D$
is $n_D$ and all of the points $p_i$ are smooth points of $C$, so each component of $C$ has a smooth point
and hence must be reduced.)
In addition, the following statements are equivalent:
\begin{itemize}
\item[(a)] $F\cdot D'\geq 0$ for every irreducible component $D'$ of $C'$;
\item[(b)] $F\cdot C'\geq 0$; and
\item[(c)] $F\cdot D'\geq 0$ for some irreducible component $D'$ of $C'$.
\end{itemize}
Clearly, (a) implies (b), and (b) implies (c). We now show that (c) implies (a).
If $C'$ has only one component, then (c) and (a) are
trivially equivalent, so suppose $D_1'$ and $D_2'$ are distinct components of $C'$.
In order to show that $F\cdot D_1'\geq0$ implies $F\cdot D_2'\geq0$,
we will use the assumption that the points $p_i$ are evenly distributed smooth points of $C$.
Let $D_j=\pi(D_j')$, so $D_j'$ is the proper transform of $D_j$.
Because the points are evenly distributed, we have
$n_{D_j}=n(\deg(D_j))/3$. Thus $n_{D_j}$ of the $n$ points $p_i$
lie on $D_j$. Because the points are smooth points of $C$, we have
$[D_j']=[\deg(D_j)L-\sum_{p_i\in D_j}E_i]$, where the sum involves $n_{D_j}$ terms.
Thus $F\cdot D_j'\geq 0$ can be rewritten as $t\deg(D_j)-mn_{D_j}\geq0$.
Substituting $n(\deg(D_j))/3$ for $n_{D_j}$ gives $t\deg(D_j)-mn(\deg(D_j))/3\geq0$
which is equivalent to $3t-mn\geq0$, which is itself just $F\cdot C'\geq0$.
Thus $F\cdot D_1'\geq0$ and $F\cdot D_2'\geq0$ are both equivalent to $F\cdot C'\geq0$,
and hence $F\cdot D_1'\geq0$ if and only if $F\cdot D_2'\geq0$. This shows (c) implies (a).
We now show that $h^0(Y,{\mathcal O}_Y(F))=0$
if and only if $t<3m$. For $t\geq 3m$, we have ${\mathcal O}_Y(F)
={\mathcal O}_Y((t-3m)L+mC')$, and hence $h^0(Y,{\mathcal O}_Y(F))>0$.
If, however, $t<3m$, then $3t<9m\leq nm$ so
$F\cdot C'<0$, and hence, as we saw above,
$F\cdot D'< 0$ for each component $D'$ of $C'$, in which case
each component $D'$ of $C'$ is a fixed component of $|F|$ so
$h^0(Y, {\mathcal O}_Y(F))=h^0(Y, {\mathcal O}_Y(F-C'))
=h^0(Y, {\mathcal O}_Y((t-3)L-(m-1)E_1-\cdots-(m-1)E_n))$.
But $t-3<3(m-1)$, so, by the same argument, we can again
subtract off $C'$ without changing $h^0$. Continuing
in this way we eventually obtain $h^0(Y, {\mathcal O}_Y(F))=
h^0(Y, {\mathcal O}_Y((t-3m)L))=h^0(\pr2, {\mathcal O}_{\pr2}(t-3m))$, but
$h^0(\pr2, {\mathcal O}_{\pr2}(t-3m))=0$ since $t-3m<0$. Thus
$h_{Z}(t)=\binom{t+2}{2}$ for $t<3m$, which proves (i).
Next consider (ii). If $t\geq nm/3$, i.e., if $F\cdot C'\geq0$, then as we saw above
$F\cdot D'\geq0$ for every component $D'$ of $C'$.
But as we also saw above, $(t-3m)L+mC'\in |F|$, hence $F$ is nef.
If $t>nm/3$ (in which case $s(t,n,m)=0$), then $F\cdot C'>0$, so
by \cite[Theorem III.1(a,b)]{anticanSurf}, $h^1(Y,{\mathcal O}_Y(F))=0$.
Thus \eqref{RRoch2} gives $h_{Z}(t)=n\binom{m+1}{2}=n\binom{m+1}{2}-s(t,n,m)$ as claimed.
We are left with the case that $t=nm/3$.
Suppose $t=nm/3$ and $n=9$. Thus $F=mC'$ and
$F\cdot C'=0$ (because $n=9$ and $t=3m$), so $(C')^2=0$.
By duality we have $h^2(Y, {\mathcal O}_Y(mC'))=h^0(Y, {\mathcal O}_Y(-(m+1)C'))=0$,
so by Riemann-Roch we have
$h^0(Y, {\mathcal O}_Y(F))-h^1(Y, {\mathcal O}_Y(F))=1+(F^2+C'\cdot F)/2=1$.
Since $F$ is nef, so is $iC'$ for all $i\geq0$. Since $F\cdot C'=0$, either
$|F|$ has an element disjoint from $C'$ or $F$ and $C'$ share a common component.
If $|F|$ has an element disjoint from $C'$, then ${\mathcal O}_{C'}(F)$ is trivial, so
$h^0(C', {\mathcal O}_{C'}(F))=1$ since $C$ (and hence $C'$) is connected and
reduced.
Suppose $F$ and $C'$ share a common component. Then $C'$ is in the base
locus of $|F|$ by \cite[Corollary III.2]{anticanSurf}, and hence
$h^0(Y, {\mathcal O}_Y(F))=h^0(Y, {\mathcal O}_Y(F-C'))$.
Let $\phi$ be the least $i>0$ (possibly infinite) such that
$C'$ is not in the base locus of $|iC'|$. Then
we have $h^0(Y, {\mathcal O}_Y(jC'))=h^0(Y, {\mathcal O}_Y((j+1)C'))$
for $0\leq j<\phi-1$, so by induction (using the base case
$h^0(Y, {\mathcal O}_Y)=1$ and the fact $h^0(Y, {\mathcal O}_Y(jC'))-h^1(Y, {\mathcal O}_Y(jC'))=1$)
we have $h^0(Y, {\mathcal O}_Y(jC'))=1$ and $h^1(Y, {\mathcal O}_Y(jC'))=0$ for all $0\leq j<\phi$.
It follows that
$$0\to {\mathcal O}_Y((s-1)C')\to {\mathcal O}_Y(sC')\to {\mathcal O}_{C'}(sC')\to 0 \eqno(\star)$$
is exact on global sections for $1\leq s\leq \phi$, and that $h^0(C', {\mathcal O}_{C'}(sC'))=0$
and $h^1(C', {\mathcal O}_{C'}(sC'))=0$ for $0<s<\phi$. Thus ${\mathcal O}_{C'}(sC')$
is nontrivial for $0<s<\phi$. Since for all $m$, $|mC'|$ either has an element disjoint from $C'$ or
$C'$ is in the base locus of $|mC'|$, we see that ${\mathcal O}_{C'}(\phi C')$ is trivial,
and hence $\phi$ is the order of ${\mathcal O}_{C'}(C')$ in $\Pic(C')$. But since the points
$p_i$ blown up are smooth points of $C$, the morphism $\pi:Y\to\pr2$ induces an isomorphism
$C\to C'$, and under this isomorphism,
${\mathcal O}_C(3L)\otimes {\mathcal O}_C(-p_1-\cdots-p_n)$
corresponds to ${\mathcal O}_{C'}(C')$, so we see $\phi=\lambda$.
It follows that $h^0(C', {\mathcal O}_{C'}(sC'))=h^1(C', {\mathcal O}_{C'}(sC'))$
for all $s\geq0$, and these are both 1 if ${\mathcal O}_{C'}(sC')$ is trivial
(i.e., if $s$ is a multiple of $\lambda$) and they are 0 otherwise.
We now claim that $(\star)$ is exact on globals sections for all $s\geq 1$.
It is enough to show this
when $s$ is a multiple of $\lambda$, because otherwise, as we noted above,
$h^0(C', {\mathcal O}_{C'}(sC'))=0$ and hence
$(\star)$ is automatically exact on global sections.
But $|\lambda C'|$ (and hence also $|i\lambda C'|$ for all $i\geq1$) has an element disjoint from $C'$, so
$H^0(Y, {\mathcal O}_Y(i\lambda C'))\to H^0(C', {\mathcal O}_{C'}(i\lambda C'))$ is onto,
which shows that $(\star)$ is exact on global sections when $s$ is a multiple of $\lambda$.
It follows that $(\star)$
is also exact on $h^1$'s (since as above $h^2(Y, {\mathcal O}_Y(iC'))=0$
for all $i\geq0$), and hence that
$h^1(Y, {\mathcal O}_Y(mC'))=h^1(Y, {\mathcal O}_Y)+
\sum_{1\leq i\leq m}h^1(C', {\mathcal O}_{C'}(iC'))$.
Now $h^1(Y, {\mathcal O}_Y)=h^1(\pr2, {\mathcal O}_{\pr2})=0$
and $h^1(C', {\mathcal O}_{C'}(iC'))$ is 1 if and only if $i$ is a multiple of $\lambda$
and it is 0 otherwise. Thus $h^1(Y, {\mathcal O}_Y(mC'))$ is the number of summands
$h^1(C', {\mathcal O}_{C'}(iC'))$ for which $i$ is a multiple of $\lambda$; i.e.,
$h^1(Y, {\mathcal O}_Y(mC'))=\lfloor m/\lambda\rfloor$, which is just $s(t,n,m)$.
This implies that $h_{Z}(t)=n\binom{m+1}{2}-s(t,n,m)$, as claimed.
If $t=nm/3$ but $n>9$, then $F^2>0$ so by \cite[Theorem III.1(c)]{anticanSurf}
either ${\mathcal O}_{C'}(F)$ is trivial (in which case $s(t,n,m)=1$)
and $h^1(Y,{\mathcal O}_Y(F))=1$ (and hence
$h_{Z}(t)=n\binom{m+1}{2}-1=n\binom{m+1}{2}-s(t,n,m)$),
or $C'$ is in the base locus of $|F|$. If $C'$ is in the base locus,
then by \cite[Theorem III.1(d)]{anticanSurf} and the fact that $F^2>0$ we have
${\mathcal O}_{C'}(F)$ is not trivial (in which case $s(t,n,m)=0$)
and $h^1(Y,{\mathcal O}_Y(F))=0$, and hence
$h_{Z}(t)=n\binom{m+1}{2}-s(t,n,m)$, as claimed.
Now consider case (iii); i.e., $3m\leq t<nm/3$ and $n>9$. Then
$F\cdot D'< 0$ for each component $D'$ of $C'$ (since
the points are evenly distributed), in which case
$h^0(Y, {\mathcal O}_Y(F))=h^0(Y, {\mathcal O}_Y(F-C'))
=h^0(Y, {\mathcal O}_Y((t-3)L-(m-1)E_1-\cdots-(m-1)E_n))$.
If $t-3 < n(m-1)/3$, we can subtract $C'$ off again.
This continues until we have subtracted $C'$ off $r=\lceil (mn-3t)/(n-9)\rceil$ times,
at which point we have that $F-rC'$ is nef and effective
and $h^0(Y, {\mathcal O}_Y(F))=h^0(Y, {\mathcal O}_Y(F-rC'))$.
Applying (ii) to $F-rC'$ gives
$\binom{t-3r+2}{2}-h^0(Y, {\mathcal O}_Y(F-rC'))=h_{(m-r)Z}(t-3r)=n\binom{m-r+1}{2}-s(t-3r,n,m-r)$
or $h^0(Y, {\mathcal O}_Y(F-rC'))=\binom{t-3r+2}{2}-(n\binom{m-r+1}{2}-s(t-3r,n,m-r))$.
Substituting this in for $h^0(Y, {\mathcal O}_Y(F))$ in
$h_{Z}(t)=\binom{t+2}{2}-h^0(Y, {\mathcal O}_Y(F))$ gives (iii).
\end{proof}
\begin{rem}
We can now write down all possible Hilbert functions for
$n\geq 9$ points of multiplicity $m$ for each possible choice of Hilbert function
for the reduced scheme given by the points, if the points are smooth points of a reduced cubic curve
and evenly distributed. Suppose $X=p_1+\cdots+p_n$ and $m=1$.
If $3$ does not divide $n$, or it does but
$s(n/3,n,1)=0$, then the difference function
for the Hilbert function of $X$ is the same as given in \eqref{hf of 1 2 3 ... 3}, but
if $3$ divides $n$ and $s(n/3,n,1)=1$, then $X$ is a complete intersection
and the difference function
for the Hilbert function of $X$ is the same as given in \eqref{hf of 1 2 3 ... 3 2 1}.
We now compare our results for $Z = 2X=2(p_1+\cdots+p_n)$ with those obtained in Proposition \ref{ci case} and Theorem \ref{smooth cubic}, and we explicitly list those cases skipped there (because there we assumed $n=3t$ with $t\geq3$ in Proposition \ref{ci case} and $n=2t+\delta$ with
$t>5-\delta$ in Theorem \ref{smooth cubic}).
Say $n\equiv 1\mod 3$. Then
the difference function for the Hilbert function is:
$n=10$: 1 2 3 4 5 6 6 3 0
$n=13$: 1 2 3 4 5 6 6 6 4 2 0, and for
$n=10+3x$ for $x>1$: the result is the same as given in Theorem \ref{smooth cubic}(a).
Next, say $n\equiv 2\mod 3$. Then the difference function for the Hilbert function is:
$n=11$: 1 2 3 4 5 6 6 5 1 0, and for
$n=11+3x$ for $x>0$: the result is the same as given in Theorem \ref{smooth cubic}(a).
If $n=3x$, there are two possibilities.
If $s(2x,n,2)=0$ for the given points (i.e.,
the divisor $2p_1+\cdots+2p_n$ on $C$ is not cut out by a curve of degree
$2x$, or equivalently
${\mathcal O}_C(2xL-2E_1-\cdots-2E_n)$ is not trivial),
then the difference function for the Hilbert function is:
$n=9$: 1 2 3 4 5 6 6 0, and
$n=3x$: 1 2 3 4 5 6 $\ldots$ 6 3 $\ldots$ 3 0
for $x\geq4$,
where the number of 6's is $x-1$
and the number of trailing 3's is $x-3$.
For $x>5$, this is the same as the result given in Theorem \ref{smooth cubic}(b).
If $s(2x,n,2)=1$ for the given points (i.e.,
the divisor $2p_1+\cdots+2p_n$ on $C$ is cut out by a curve of degree
$2x$, or equivalently ${\mathcal O}_C(2xL-2E_1-\cdots-2E_n)$ is trivial),
but $s(n/3,n,1)=0$
(so $p_1+\cdots+p_n$ is not cut out by a curve of degree $x$, which is
equivalent to saying that ${\mathcal O}_C(xL-E_1-\cdots-E_n)$ is not trivial),
then the difference function for the Hilbert function is:
$n=9$: 1 2 3 4 5 6 5 1 0,
$n=12$: 1 2 3 4 5 6 6 6 2 1 0, and
$n=3x$: 1 2 3 4 5 6 $\ldots$ 6 3 $\ldots$ 3 2 1 0
for $x>4$,
where the number of 6's is $x-1$
and the number of trailing 3's is $x-4$.
For $x>5$, this is the same as the result given in Theorem \ref{smooth cubic}(c).
Now say $n=3x$ and $s(x,n,1)=1$. In this case, $X$ is the complete intersection of
$C$ and a form of degree $t$, and the difference function for the Hilbert function of $2X$ is:
$n=9$: 1 2 3 4 5 6 4 2 0
$n=12$: 1 2 3 4 5 6 6 5 3 1 0 and
$n=3x$ for $x>4$: the result is the same as given in Proposition \ref{ci case}.
\end{rem}
\begin{rem}\label{torsionRem}
The possible values of the Hilbert functions as given in Theorem \ref
{method2cor1} depend partly on what torsion occurs in $\Pic(C)$, and
this in turn is affected by the characteristic of $k$.
When $C$ is smooth,
see \cite[Example IV.4.8.1]{refHr} for a discussion of the torsion.
When $C$ is reduced but not smooth, the torsion is easy to understand since
it is all contained in the identity component $\Pic^0(C)$ of $\Pic(C)$,
whose group structure is isomorphic either
to the additive or multiplicative groups of the ground field.
(See for example \cite[Proposition 5.2]{refHL}, which states a result for
curves of so-called canonical type. But for any reduced cubic $C$, one can always find a set of 9 evenly distributed
smooth points of $C$,
and the proper transform $C'$ with respect to blowing those points up
is a {\em curve of canonical type}, meaning that $C'\cdot D=K_X\cdot D=0$
for every component $D$ of $C'$. Since the points blown up are smooth on $C$,
$C$ and $C'$ are isomorphic and thus so are $\Pic(C)$ and $\Pic(C')$, hence the conclusion of
\cite[Proposition 5.2]{refHL} applies to $C$, even though $C$ is not itself of canonical type.)
When $C$ is reduced and
irreducible but singular, for example, the result is that
$\Pic^0(C)$ is the additive
group of the ground field when $C$ is cuspidal and it is the
multiplicative
group of the field when $C$ is nodal \cite[Exercise II.6.9]{refHr}.
In particular, if $C$ is an irreducible cuspidal cubic curve over
a field of characteristic zero, then $\Pic^0(C)$ is torsion free, so
$h_
{2X}$ cannot be $(1,2,3,4,5,6,6,6,2,1)$; indeed, this follows, after
a simple calculation, because if ${\mathcal O}_{C}(2xL-2p_1-
\cdots-2p_n)$ is trivial, then so is ${\mathcal O}_{C}(xL-p_1-\cdots-
p_n)$. On the other hand, ${\mathcal O}_{C}(xL-p_1-\cdots-p_n)$ can
be nontrivial even if ${\mathcal O}_{C}(2xL-2p_1-\cdots-2p_n)$ is
trivial
if the characteristic is 2 or if the singular point is a node but the
characteristic is not 2, since in those cases $\Pic(C)$ has elements of
order 2.
\end{rem}
\begin{rem} \label{method2cor2}
We can also use the method of proof of Theorem \ref{method2cor1}
to recover the result of Theorem \ref{singular cubic}
for the Hilbert function of $mX=m(p_1+\cdots+p_n)$ for $n\geq9$ points on a
reduced, irreducible cubic curve $C$ where $p_1$, say, is the singular
point and $m$ is 1 or 2. As is now clear, the approach of Theorem \ref{method2cor1}
is to determine $h^0(Y, {\mathcal O}_Y(tL-mE_1-\cdots-mE_n))$ for all $t$,
and then translate this into the Hilbert function or the difference function
for $mX$.
This translation is purely mechanical and the resulting Hilbert functions
in the case that $n\geq 12$ are
already given in Theorem \ref{singular cubic} (we leave writing down the Hilbert functions
for $9\leq n\leq 11$ using the results that follow as an exercise for the reader).
Thus it is the calculation of
$h^0(Y, {\mathcal O}_Y(tL-mE_1-\cdots-mE_n))$
that is of most interest, and it is on this that we now focus.
Let $Y$ be the blow up of the points, let $C'$ be the proper
transform of $C$, and let $F_t=tL-E_1-\cdots-E_n$ and $G_t=tL-2(E_1+\cdots+E_n)$,
where we denote by $L$ both a general line in the plane and its pullback to $Y$.
Up to linear equivalence, note that $C'=3L-2E_1-E_2-\cdots-E_n$.
The goal here is to compute the values of $h^0(Y, {\mathcal O}_{Y}(F_t))$ and
$h^0(Y, {\mathcal O}_Y(G_t))$. For $t<3$, B\'ezout tells us that $h^0(X,{\mathcal O}_Y(F_t))=0$,
since $F_t\cdot C'<0$ (hence $h^0(Y, {\mathcal O}_Y(F_t))=h^0(Y, {\mathcal O}_Y(F_t-C'))$
and $(F_t-C')\cdot L<0$ (hence $h^0(Y, {\mathcal O}_Y(F_t-C'))=0$). If $t\geq3$, then certainly
$h^0(Y,{\mathcal O}_Y(F_t))>0$, since $F_t=(t-3)L+C'+E_1$. We consider three cases, according to whether
$F_t\cdot C'<0$, $F_t\cdot C'>0$ or $F_t\cdot C'=0$.
If $0<F_t\cdot C'=3t-2-(n-1)$ (i.e., if $3\leq t<(n+1)/3$),
then $h^0(Y,{\mathcal O}_Y(F_t))=h^0(Y,{\mathcal O}_Y(F_t-C'))=
h^0(Y,{\mathcal O}_Y((t-3)L+E_1))=h^0(Y,{\mathcal O}_Y((t-3)L))=
h^0(\pr2,{\mathcal O}_{\pr2}((t-3)L))=\binom{t-3+2}{2}$,
since $F_t-C'=(t-3)L+E_1$. If $F_t\cdot C'>0$ (i.e., $t>(n+1)/3$), then
$h^0(Y,{\mathcal O}_Y(F_t))=\binom{t+2}{2}-n$ (since $F_t$, meeting both
components of $-K_Y=C'+E_1$ positively, is nef and hence
$h^1(Y,{\mathcal O}_Y(F_t))=0$ by \cite[Theorem III.1(a,b)]{anticanSurf}).
This leaves the case that $t = (n+1)/3$. This means that $
{\mathcal O}_{C'}(F_t)$ has degree 0. Consider the exact sequence
$$
0\to {\mathcal O}_Y((t-3)L+E_1)\to {\mathcal O}_Y(F_t)\to {\mathcal
O}_{C'}(F_t)\to 0.
$$
By an analogous argument to the one used to show $h^1(Y,{\mathcal O}
_Y(F_t))=0$ when $t > (n+1)/3$, we obtain that $h^1({\mathcal O}_Y
((t-3)L+E_1)) = 0$. But $C'$ is a smooth rational curve, so also
the third sheaf in the sequence has vanishing first cohomology.
Thus we obtain $h^1(Y, O_Y(F_t))=0$, hence the points impose
independent conditions. It follows that $h^0(Y, O_Y(F_t))=\binom
{t +2}{2}-n$ also for $t = (n+1)/3$.
We thus have: $h^0(Y,{\mathcal O}_Y(F_t))=0$ for $0\leq t <3$;
$h^0(Y,{\mathcal O}_Y(F_t))=\binom{t-1}{2}$ for $3\leq t <(n+1)/3$;
and $h^0(Y,{\mathcal O}_Y(F_t))=\binom{t+2}{2}-n$ for $t\geq (n+1)/3$.
A similar analysis works for $2X$. There are now four ranges of degrees.
The first range is $t<6$, in which case $h^0(Y,{\mathcal O}_Y(G_t))=0$ by B\'ezout, arguing as above.
For $t\geq6$, we have $h^0(Y,{\mathcal O}_Y(G_t))>0$, since up to linear equivalence
we have $G_t=(t-6)L+2(C'+E_1)$.
The second range is now $6\leq t< (n+8)/3$; in this case $2C'$ is, by B\'ezout,
a fixed component of $|G_t|$, so $h^0(Y,{\mathcal O}_Y(G_t))=
h^0(Y,{\mathcal O}_Y((t-6)L+2E_1))=\binom{t-4}{2}$.
The third range is $(n+8)/3\leq t <(2/3)(n+1)$, for which $C'$ is a fixed component of $|G_t|$
(and $G_t-C'=(t-6)L+C'+2E_1$ is nef)
so $h^0(Y,{\mathcal O}_Y(G_t))=h^0(Y,{\mathcal O}_Y(G_t-C'))$
and we know $h^0(Y,{\mathcal O}_Y(G_t-C'))$ by Theorem \ref{method2cor1}(ii) if $n>9$,
while $h^1(Y,{\mathcal O}_Y(G_t-C'))=0$ by \cite[Theorem III.1(a,b)]{anticanSurf}) if $n=9$,
so again we know $h^0(Y,{\mathcal O}_Y(G_t-C'))$.
The last range is $t\geq (2/3)(n+1)$, in which case $G_t$ is nef.
If $t>(2/3)(n+1)$, then $G_t$ meets $-K_Y$ positively,
so $h^1(Y,{\mathcal O}_Y(G_t))=0$ \cite[Theorem III.1(a,b)]{anticanSurf}),
and $h^0(Y,{\mathcal O}_Y(G_t))=\binom{t+2}{2}-3n$.
We are left with the case that $t=(2/3)(n+1)$. Consider the exact sequence
$$0\to {\mathcal O}_Y((t-3)L-E_2-\cdots-E_n)\to {\mathcal O}_Y(G_t)\to {\mathcal O}_{C'}(G_t)\to 0.$$
Since $G_t\cdot C'\geq0$ and $C'$ is smooth and rational, we have
$h^1(C',{\mathcal O}_{C'}(G_t))=0$, and since ${\mathcal O}_Y((t-3)L-E_2-\cdots-E_n)=
{\mathcal O}_Y((t-6)L+C'+2E_1)$ and $(t-6)L+C'+2E_1$ is nef (as observed above)
with $(G_t-C')\cdot C'>0$, we have $h^1(Y,{\mathcal O}_Y(G_t-C'))=0$ \cite[Theorem III.1(a,b)]{anticanSurf})
and hence $h^1(Y,{\mathcal O}_Y(G_t))=0$, so in fact $h^0(Y,{\mathcal O}_Y(G_t))=\binom{t+2}{2}-3n$.
\end{rem}
\begin{rem}
Here we comment on what is left to do if one wants to recover the results of
section 2 using the methods of section 4.
So consider $n=9$ points on a given cubic $C$ (but note that there may be more
than one cubic through the points), either all of multiplicity 1 or all of multiplicity 2.
The case that the points are evenly distributed smooth points of $C$
is done above, as is the case that the curve $C$ is reduced and irreducible. The case that the
points all lie on a conic follows from the known result for configuration types
of points on a conic \cite{GHM}. What's left is that the points do not all lie on
any given conic (and hence $C$ is reduced) and either: one or more of the points
is not a smooth point of $C$ and $C$ is not irreducible, or the points are
not distributed evenly (and hence again $C$ is not irreducible).
The four reducible cubics that arise are: a conic and a line tangent
to the conic; a conic and a transverse line; three lines passing through
a point; and three lines with no point common to all three. Each of these cases
leads to a number of cases depending on how the points
are placed (such as how many are on each component and whether one or more
is a singular point of the cubic, but also depending on the group law of the cubic).
Analyzing these cases would give a complete result of the Hilbert functions
of the form $h_X$ and $h_{2X}$ for a reduced scheme $X$ consisting of 9 distinct
points of the plane.
\end{rem}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 932 |
was a textile worker and memoirist during the Meiji Era in Japan, daughter of samurai from Matsushiro, Shinano Province. She is known for writing a memoir called the "Tomioka Diary" (Tomioka Nikki) in which she chronicled her life among the female workers in the Tomioka silk mill.
She was among the daughters of samurai who were recruited in 1873, the sixth year of the Meiji period, from across the nation for practical training over a period of two to three years in the silk production process. They later became trainers in silk manufacture in their own prefectures.
Upon leaving Tomioka in 1874, the government awarded Wada and her colleagues medals and the special title "Women Spinners' Victory Battalion". She became a trainer at the Saijō Village Silk Reeling Factory (later Rokkōsha mill) in Matsushiro, Nagano Prefecture.
Wada married an army officer, who later died in 1913 from wounds he sustained in the Russo-Japanese war. She began composing the Tomioka Nikki while living at the Furukawa Mine company house at the Ashio Copper Mine, where her son was a manager. It was published after her death by him, leading some historians to doubt its authenticity.
See also
The Tomioka Silk Mill and Related Industrial Heritage (UNESCO World Heritage Site nomination)
References
Further reading
External links
Tomioka Nikki
1857 births
1929 deaths
Economic history of Japan
Japanese women writers
Textile workers
Silk production
19th-century Japanese women | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,219 |
Q: Vertical align an absolute div I have a div, it's has no declared height and is positioned absolute.
How can I vertically align it?
<div class="valign"><a href="#"><img src="whatever.jpg"></a></div>
.valign{
position: absolute;
left: 10px;
}
No flex box please.
A: .valign {
position: absolute;
top: 50%;
left: 10px;
-webkit-transform: translateY(-50%);
-moz-transform: translateY(-50%);
-ms-transform: translateY(-50%);
transform: translateY(-50%);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,833 |
Leah Culver
🌊 Cofounder and CTO of @breaker - the best app for listening to podcasts.
Leah Culver is the co-founder & CTO of Breaker - the best app for listening to podcasts. Leah has been building startups and products for over 10 years. She helped launch an early version and competitor to Twitter called Pownce with Kevin Rose and Daniel Burka in 2007. In 2011, she was working on a B2B chatting and instant messaging application that was accepted into Y Combinator, and has been also part of companies like Sincerely and Dropbox. Since 2016, she's been focused on building Breaker, a new podcast app centred around discovery that was also part of Y Combinator earlier this year. Leah joins us to share her story, how she got into startups, what it's been like building and working at startups in Silicon Valley for over 10 years, what it's been like been like building Breaker, what it was like going through Y Combinator a second time, and much more.
Learn more about how Leah got into tech & startups
Learn more about what it was like starting Breaker
Learn more about what it was like being a part of Y Combinator for a second time
Leah Culver is the co-founder & CTO of Breaker - the best app for listening to podcasts.
Leah has been building startups and products for over 10 years. She helped launch an early version and competitor to Twitter called Pownce with Kevin Rose and Daniel Burka in 2007. In 2011, she was working on a B2B chatting and instant messaging application that was accepted into Y Combinator, and has been also part of companies like Sincerely and Dropbox.
Since 2016, she's been focused on building Breaker, a new podcast app centred around discovery that was also part of Y Combinator earlier this year.
Leah joins us to share her story, how she got into startups, what it's been like building and working at startups in Silicon Valley for over 10 years, what it's been like been like building Breaker, what it was like going through Y Combinator a second time, and much more.
"How do you build something that people want to use? I don't think there's a clear answer, other than trial and error."
Can you give us a quick overview of some of the startups you were a part of or some of the products you worked on that really set the stage for you becoming a founder and launching Breaker?
Can you start by telling us more about what Breaker is all about (for those who might not know) and what motivated you to help launch it?
Why podcasts? What is it about the medium that excites you the most?
How did you approach building the first version of Breaker?
So Breaker was also part of Y Combinator. What was that entire experience like for you and Eric?
Given your experience as both a developer and a Product Manager, what's your approach to building products (in a startup setting)? Any insights to share with other founders?
How did you approach growing Breaker in the early days? What were some of the most successful tactics or channels?
What have been some of the biggest challenges or learning points along this journey?
Do you have any recommendations on some great content that you've come across lately?
Do you have any personal mottos or an outlook that you live by and would share with others?
Leah Culver on Twitter
Leah Culver on the web
Breaker on Twitter
Breaker on the web
Buglife
Product Development Cycle Fundamentals
YC Startup School | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,720 |
'use strict';
// What should I cache?
var urlsToCache = [
'/',
'index.html',
'manifest.json',
// js assets
'js/main.js',
'js/services/NetworkConnectivityService.js',
'js/components/CardList.js',
'js/components/SideNavBar.js',
'js/components/ToDoCard.js',
'js/indexedDB/IndexedDBLayer.js',
'js/indexedDB/TodoListDB.js',
'js/web-workers/db.js',
'js/vendor/cards.js',
'js/vendor/vanillatoasts.js',
'js/util/setDefaultTODOs.js',
// css assets
'style/css/main.css',
'style/vendor/vanillatoasts.css',
// img assets
'imgs/offline-rex.jpg',
'imgs/icon/favicon-96x96.png',
'imgs/icon/favicon-144x144.png',
'imgs/icon/favicon-196x196.png',
'imgs/plus.svg'
];
// used for logging
const logTextColor = "black";
const logBackgroundColor = "#FFD700";
// Cache name - should be changed whenever significant changes are made
var version = 1;
var getISODate = () => {
return new Date().toISOString().split('T')[0];
};
var getCacheName = () => {
return `${getISODate()}-static-${version}`;
};
// just checking is cache the most recent
var isLatestCacheName = (key) => {
return key == getCacheName();
};
// deletes all caches that are out of date / version
var deleteOldCaches = function () {
console.log('%c Service workers: ', 'color:' + logTextColor + '; background-color: ' + logBackgroundColor, 'Checking for old caches.');
var getOldCacheKeys = function () {
return caches.keys().then(function (keys) {
return keys.filter(function (key) {
return !isLatestCacheName(key);
});
});
};
return getOldCacheKeys().then(function (oldCacheKeys) {
return Promise.all(oldCacheKeys.map(function (key) {
return caches.delete(key);
}));
});
};
function updateCache() {
return caches.open(getCacheName())
.then(function (cache) {
return cache.addAll(urlsToCache);
});
};
// If all the files are successfully cached, then the service worker is installed.
self.addEventListener('install', function (event) {
// Perform install steps
console.log('%c Service workers: ', 'color:' + logTextColor + '; background-color: ' + logBackgroundColor, '2. Installed, all the files are downloaded and cached successfully. Activating...');
event.waitUntil(updateCache());
});
// After the service worker is installed, it is then activated, meaning it can start controlling what the user gets!
self.addEventListener('activate', function (event) {
console.log('%c Service workers: ', 'color:' + logTextColor + '; background-color: ' + logBackgroundColor, '3. Activated. Now is a good moment if we want to manage old caches.');
deleteOldCaches();
});
var deleteAllCache = function () {
console.log('%c Service workers: ', 'color:' + logTextColor + '; background-color: ' + logBackgroundColor, 'Deleting all cache :(');
var cacheWhitelist = `this should never be a cache name!!! this thing cost me way too much time!!`;
caches.keys().then(function (keyList) {
return Promise.all(keyList.map(function (key) {
if (cacheWhitelist.indexOf(key) === -1) {
return caches.delete(key);
}
}));
});
}
var doesRequestAcceptHtml = function (request) {
return request.headers.get('Accept')
.split(',')
.some(function (type) {
return type === 'text/html';
});
};
self.addEventListener('fetch', function (event) {
console.log('%c Service workers: ', 'color:' + logTextColor + '; background-color: ' + logBackgroundColor, 'Fetch intercepted: ', event.request.url);
event.respondWith(
caches.match(event.request)
.then(function (response) {
// Cache hit - return response
if (response) {
console.log('%c Service workers: ', 'color:' + logTextColor + '; background-color: ' + logBackgroundColor, 'Cache hit! No need for the server to work here :)');
return response;
}
// IMPORTANT: Clone the request. A request is a stream and
// can only be consumed once. Since we are consuming this
// once by cache and once by the browser for fetch, we need
// to clone the response
var fetchRequest = event.request.clone();
return fetch(fetchRequest).then(
function (response) {
// Check if we received a valid response
if (!response || response.status !== 200 || response.type !== 'basic' || event.request.method) {
return response;
}
// IMPORTANT: Clone the response. A response is a stream
// and because we want the browser to consume the response
// as well as the cache consuming the response, we need
// to clone it so we have 2 stream.
var responseToCache = response.clone();
caches.open(getCacheName).then(function (cache) {
cache.put(event.request, responseToCache);
});
return response;
}
);
})
);
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,297 |
Q: Open workbook, count a range and write count into another workbook I have a macro in a Excel workbook and I am performing count operation from different files and update it in the file which has macro.
Private Sub count_Click()
Dim MyCount(1 To 3) As Long
Dim myData As Workbook
Dim Rng As Range
Set myData = Workbooks.Open("C:\Users\xyz\Desktop\cas\Book3.xlsx") ' selecting a workbook'
With myData.Worksheets("sheet1")
Set Rng = Intersect(.Columns(1), .UsedRange)
MyCount(1) = WorksheetFunction.CountA(Rng)
End With
With myData.Worksheets("sheet1")
Set Rng = Intersect(.Columns(5), .UsedRange)
MyCount(2) = WorksheetFunction.CountA(Rng)
End With
'MsgBox "count is " & MyCount(1)
Set myData = Workbooks.Open("C:\Users\xyz\Desktop\cas\Book2.xlsm")
' the workbook where I want the values to be transferred'
With myData.Worksheets("sheet1").Range("a1")
.Offset(RowCount, 0) = MyCount(1)
End With
With myData.Worksheets("sheet1").Range("a2")
.Offset(RowCount, 0) = MyCount(2)
End With
End Sub
I have tried opening the workbook and updating the values, but the values are not displayed.
It is probably because the instance of the workbook is open and it is unable to update.
A: If you think it may be open read only you can try:
'MsgBox "count is " & MyCount(1)
Set myData = Workbooks.Open("C:\Users\xyz\Desktop\cas\Book2.xlsm")
If myData.ReadOnly Then
MsgBox "File is Read-only"
exit sub
Else
MsgBox "File is not read-only"
End If
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,021 |
// Copyright 2014 The Chromium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
// Include test fixture.
GEN_INCLUDE(['net_internals_test.js']);
// Anonymous namespace
(function() {
// Path to the page containing iframe. Iframe is used to load sdch-related
// content from the different origin. Otherwise favicon requests for the main
// page domain would spoil SDCH blacklists counters making test behavior hardly
// predicatble.
var BASE_PATH = 'files/sdch/base-page.html?iframe_url=';
/**
* Checks the display on the SDCH tab against the information it should be
* displaying.
* @param {object} sdchInfo Results from a sdch manager info query.
*/
function checkDisplay(sdchInfo) {
expectEquals(sdchInfo.sdch_enabled,
$(SdchView.SDCH_ENABLED_SPAN_ID).innerText === 'true');
NetInternalsTest.checkTbodyRows(SdchView.BLACKLIST_TBODY_ID,
sdchInfo.blacklisted.length);
NetInternalsTest.checkTbodyRows(SdchView.DICTIONARIES_TBODY_ID,
sdchInfo.dictionaries.length);
// Rather than check the exact string in every position, just make sure every
// entry does not have 'undefined' anywhere and certain entries are not empty,
// which should find a fair number of potential output errors.
for (var row = 0; row < sdchInfo.blacklisted.length; ++row) {
for (var column = 0; column < 3; ++column) {
var text = NetInternalsTest.getTbodyText(
SdchView.BLACKLIST_TBODY_ID, row, column);
expectNotEquals(text, '');
expectFalse(/undefined/i.test(text));
}
}
for (var row = 0; row < sdchInfo.dictionaries.length; ++row) {
for (var column = 0; column < 6; ++column) {
var text = NetInternalsTest.getTbodyText(
SdchView.DICTIONARIES_TBODY_ID, row, column);
expectFalse(/undefined/i.test(text));
if (column === 0) {
// At least Domain cell should not be empty.
expectNotEquals(text, '');
}
}
}
}
/**
* A Task that loads provided page and waits for the SDCH dictionary to be
* downloaded. The page response headers should provide Get-Dictionary header.
* @extends {NetInternalsTest.Task}
*/
function LoadSdchDictionaryTask() {
NetInternalsTest.Task.call(this);
}
LoadSdchDictionaryTask.prototype = {
__proto__: NetInternalsTest.Task.prototype,
/**
* Navigates to the page and starts waiting to receive the results from
* the browser process.
*/
start: function(url) {
g_browser.addSdchInfoObserver(this, false)
NetInternalsTest.switchToView('sdch');
// 127.0.0.1 is not allowed to be an SDCH domain, use test domain.
url = url.replace('127.0.0.1', 'testdomain.com');
this.url_ = url;
chrome.send('loadPage', [url]);
},
/**
* Callback from the BrowserBridge. Checks if |sdchInfo| has the SDCH
* dictionary info for the dictionary the page has advertised. If so,
* validates it and completes the task. If not, continues running.
* @param {object} sdchInfo Results of a SDCH manager info query.
*/
onSdchInfoChanged: function(sdchInfo) {
if (this.isDone())
return;
checkDisplay(sdchInfo);
if (sdchInfo.dictionaries.length > 0) {
var testDict = sdchInfo.dictionaries.filter(function(dictionary) {
return dictionary.domain === 'sub.testdomain.com';
});
if (testDict.length === 0)
return;
expectEquals(1, testDict.length);
var dict = testDict[0];
expectEquals('/', dict.path);
expectTrue(dict.url.indexOf('/files/sdch/dict') !== -1);
var tableId = SdchView.DICTIONARIES_TBODY_ID;
var domain = NetInternalsTest.getTbodyText(tableId, 0, 0);
var path = NetInternalsTest.getTbodyText(tableId, 0, 1);
var url = NetInternalsTest.getTbodyText(tableId, 0, 5);
expectEquals(dict.domain, domain);
expectEquals(dict.path, path);
expectEquals(dict.url, url);
this.onTaskDone(this.url_);
}
}
};
/**
* A Task that loads provided page and waits for its domain to appear in SDCH
* blacklist with the specified reason.
* @param {string} reason Blacklist reason we're waiting for.
* @extends {NetInternalsTest.Task}
*/
function LoadPageWithDecodeErrorTask(reason) {
NetInternalsTest.Task.call(this);
this.reason_ = reason;
}
LoadPageWithDecodeErrorTask.prototype = {
__proto__: NetInternalsTest.Task.prototype,
/**
* Navigates to the page and starts waiting to receive the results from
* the browser process.
*/
start: function(url) {
g_browser.addSdchInfoObserver(this, false)
NetInternalsTest.switchToView('sdch');
// 127.0.0.1 is not allowed to be an SDCH domain, so we need another one.
url = url.replace('127.0.0.1', 'testdomain.com');
chrome.send('loadPage', [url]);
},
/**
* Callback from the BrowserBridge. Checks if |sdchInfo.blacklisted| contains
* the test domain with the reason specified on creation. If so, validates it
* and completes the task. If not, continues running.
* @param {object} sdchInfo Results of SDCH manager info query.
*/
onSdchInfoChanged: function(sdchInfo) {
if (this.isDone())
return;
checkDisplay(sdchInfo);
if (sdchInfo.blacklisted.length > 0) {
var testDomains = sdchInfo.blacklisted.filter(function(entry) {
return entry.domain === 'sub.testdomain.com';
});
if (testDomains.length === 0)
return;
expectEquals(1, testDomains.length);
var entry = testDomains[0];
expectEquals(this.reason_, sdchProblemCodeToString(entry.reason));
var tableId = SdchView.BLACKLIST_TBODY_ID;
var domain = NetInternalsTest.getTbodyText(tableId, 0, 0);
var reason = NetInternalsTest.getTbodyText(tableId, 0, 1);
expectEquals(entry.domain, domain);
expectEquals(this.reason_, reason);
this.onTaskDone();
}
}
};
/**
* Load a page, which results in downloading a SDCH dictionary. Make sure its
* data is displayed.
*/
TEST_F('NetInternalsTest', 'netInternalsSdchViewFetchDictionary', function() {
var taskQueue = new NetInternalsTest.TaskQueue(true);
taskQueue.addTask(
new NetInternalsTest.GetTestServerURLTask(
BASE_PATH + encodeURI('/files/sdch/page.html')));
taskQueue.addTask(new LoadSdchDictionaryTask());
taskQueue.run();
});
/**
* Load a page, get the dictionary for it, and get decoding error to see
* the blacklist in action.
*/
TEST_F('NetInternalsTest', 'netInternalsSdchViewBlacklistMeta', function() {
var taskQueue = new NetInternalsTest.TaskQueue(true);
taskQueue.addTask(
new NetInternalsTest.GetTestServerURLTask(
BASE_PATH + encodeURI('/files/sdch/page.html')));
taskQueue.addTask(new LoadSdchDictionaryTask());
taskQueue.addTask(
new NetInternalsTest.GetTestServerURLTask(
BASE_PATH + encodeURI('/files/sdch/non-html')));
taskQueue.addTask(
new LoadPageWithDecodeErrorTask('META_REFRESH_UNSUPPORTED'));
taskQueue.run();
});
/**
* Load a page, which is said to be SDCH-encoded, though we don't expect it.
*/
TEST_F('NetInternalsTest', 'netInternalsSdchViewBlacklistNonSdch', function() {
var taskQueue = new NetInternalsTest.TaskQueue(true);
taskQueue.addTask(
new NetInternalsTest.GetTestServerURLTask(
BASE_PATH + encodeURI('/files/sdch/non-sdch.html')));
taskQueue.addTask(
new LoadPageWithDecodeErrorTask('PASSING_THROUGH_NON_SDCH'));
taskQueue.run();
});
})(); // Anonymous namespace
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,976 |
Areas responsible for speech and language have been mapped to two different brain regions. This image shows a 3D-printed reconstruction of the white matter pathway connecting these two areas (here shown from the left) which is called the arcuate fasciculus.
The dimensions of the 3D model are 9 cm x 7.5 cm x 10 cm.
To create this model, a map of the white matter pathway was generated using a type of magnetic resonance imaging (MRI) called diffusion imaging, which tracks the movement of water molecules. The data from the scan was used to create a 3D model made from clear resin, using 3D-printing technology. The 3D model was then illuminated using different coloured lights to create this image.
Stephanie and Ahmad work together as research scientists at Natbrainlab at King's College London. Stephanie studies how language processes recover after brain damage, and Ahmad focuses on the brain networks of the human visual system. For this project, they collaborated with Alfonso, who is an MRI specialist also working at King's College London. Alfonso provides technical engineering support to research projects that use MRI, including diffusion tractography, as used for this image. Find out more. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,388 |
Q: ArrayObject does not work with end() in PHP 7.4 On migrating to PHP 7.4 I have to deal with a different behavior of some array functions like reset(), current() or end() concerning ArrayObject. The following example produces different outputs:
<?php
$array = new \ArrayObject(["a", "b"]);
$item = end($array);
var_dump($item);
$array = ["a", "b"];
$item = end($array);
var_dump($item);
With php 7.4 the output is:
bool(false)
string(1) "b"
On PHP versions before 7.4 the output is the following:
string(1) "b"
string(1) "b"
A end($array->getArrayCopy()) produces a notice, but might be a workaround if used with a variable.
Is there a way to emulate the behavior of end() with an ArrayObject or ArrayIterator? The ArrayObject could be very big, an iteration to the end might not be the best solution.
A: From PHP 7.4 array methods don't operate on internal array, but on ArrayObject itself. I summarized two solutions for that.
1. Getting internal array of object.
$array = new \ArrayObject(["a", "b"]);
$item = end($array->getArrayCopy());
2. Creating Facade of ArrayObject and adding custom method end() to upgraded class.
A: You can make the arrayobject an array to get the keys then use end on the keys to get the last key.
$array = new \ArrayObject(["a", "b"]);
$keys = array_keys((array)$array);
$end_key = end($keys);
var_dump($array[$end_key]);
It's not a pretty solution but it works.
I suggest you make it a function so that you can call it when needed.
https://3v4l.org/HTGYn
As a function:
function end_object($array){
$keys = array_keys((array)$array);
$end_key = end($keys);
return $array[$end_key];
}
$array = new \ArrayObject(["a", "b"]);
$item = end_object($array);
var_dump($item);
A: A slightly faster approach without casting or using an iterator would be to not use the constructor in the first place, and instead use append method which will create an array itself and you can use end on that array later
$array = new \ArrayObject();
$array->append(["a", "b"]);
$item = end($array[count($array) - 1]);
var_dump($item);
count($array) - 1 in case you append another array later, we make sure that $item is always the last element in the last appended array.
A: Came up with this approach.
Should perform better than any of the above.
*
*No casting to another variable using memory and CPU
*Only one internal iterator.
*Only one new variable, and it's a reference to the item.
class MyClass extends ArrayObject {
public function last() {
foreach ($this as $entity) {}
return $entity ?? null;
}
}
$array = new \MyClass(["a", "b", "c"]);
$item = $array->last();
var_dump($item);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,976 |
Q: Xquery 1.0 - average over elements with subelements So I have an XML file that has elements with numbers in them, and also some sub elements with numbers in them. Something like:
<data>
<points>
<score>80</score>
<score>90</score>
<score>10</score>
<score>13</score>
</points>
<favor>50</favor>
<ranked>
<rank>50</rank>
<rank>10</rank>
</ranked>
</data>
I want to compute the average across all these elements that have numbers including sub elements. So, I want a query that could produce:
(80+90+10+13+50+50+10) / 7 = 43.285714286
A: You can use expression //*[not(*)] to select leaf elements (element that doesn't have child element), anywhere in the XML document :
let $elements := //*[not(*)]
return sum($elements) div count($elements)
xpathtester demo
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 263 |
Reforma administracyjna Norwegii – uchwalona 8 czerwca 2017 roku reforma podziału administracyjnego Norwegii. W jej wyniku zredukowana została liczba okręgów oraz gmin.
Okręgi
W wyniku reformy liczba okręgów została zmniejszona z 19 do 11. Część zmian weszła w życie 1 stycznia 2018 (połączenie Sør-Trøndelag i Nord-Trøndelag), większość 1 stycznia 2020.
Gminy
Spośród 428 istniejących gmin zlikwidowano 119 i w ich miejsce utworzono 47 nowych. Tym samym liczba gmin w kraju od 1 stycznia 2020 wynosi 356.
Odbiór
Wśród niektórych mieszkańców Norwegii reforma budzi kontrowersje. Połączenie Troms i Finnmarku spowodowało sprzeciw mieszkańców tego drugiego. W maju 2018 przeprowadzono w tym regionie niewiążące referendum, w którym 87% mieszkańców wypowiedziało się przeciw połączeniu okręgów (przy frekwencji 58%).
Przypisy
Podział administracyjny Norwegii | {
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Why is there a scaling problem with the newer version?
Cloud with a call out box.
How to remove bates numbering from bulk documents?
When Will Nitro Pro Support Flash?
"thumb through book" functionalty in Single Page-mode! | {
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Das Arrondissement Neufchâteau ist eine Verwaltungseinheit des französischen Départements Vosges innerhalb der Region Grand Est (bis Ende 2015 Lothringen). Verwaltungssitz (Unterpräfektur) ist Neufchâteau.
Im Arrondissement liegen fünf Wahlkreise (Kantone) und 175 Gemeinden.
Wahlkreise
Kanton Darney (mit 56 von 81 Gemeinden)
Kanton Le Val-d'Ajol (mit 2 von 21 Gemeinden)
Kanton Mirecourt (mit 26 von 56 Gemeinden)
Kanton Neufchâteau
Kanton Vittel (mit 44 von 45 Gemeinden)
Gemeinden
Die Gemeinden des Arrondissements Neufchâteau sind:
Ehemalige Gemeinden seit der landesweiten Neuordnung der Kantone
bis 2016:
Rocourt
Neuordnung der Arrondissements 2019
Durch die Neuordnung der Arrondissements im Jahr 2019 wurden die 32 Gemeinden Ambacourt, Baudricourt, Biécourt, Blémerey, Boulaincourt, Chauffecourt, Chef-Haut, Dombasle-en-Xaintois, Domvallier, Frenelle-la-Grande, Frenelle-la-Petite, Hymont, Juvaincourt, Madecourt, Mattaincourt, Mazirot, Mirecourt, Oëlleville, Pierrefitte, Poussay, Puzieux, Ramecourt, Rancourt, Remicourt, Repel, Rouvres-en-Xaintois, Saint-Prancher, Thiraucourt, Totainville, Valleroy-aux-Saules, Villers und Vroville aus dem Arrondissement Neufchâteau dem Arrondissement Épinal zugewiesen.
Dafür wechselten aus dem Arrondissement Épinal die zwei Gemeinden Grandrupt-de-Bains und Vioménil zum Arrondissement Neufchâteau.
Département Vosges
Neufchateau | {
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Biografia
Di ascendenza armena, secondo di tre figli, Margo (1926-1994) e Flora (1931-1962). Kevorkian è noto per aver praticato il suicidio assistito su 129 malati terminali, e per aver praticato l'eutanasia sul 130º paziente. Kevorkian, medico patologo, dopo essersi diplomato con lode all'età di 17 anni, si è laureato nel 1952 alla Medical School dell'Università del Michigan, ad Ann Arbor.
Nel 1980 Kevorkian ha scritto una serie di articoli in una rivista tedesca per la Medicina e una legge sull'etica dell'eutanasia. Come lui stesso affermò frequentemente, non era un sostenitore del suicidio assistito, ma un sostenitore di chi, nel pieno delle sue facoltà mentali, ha diritto di scegliere, in condizioni di patologia grave e incurabile, se continuare a vivere oppure no. Questa sua teoria, che gli valse il nomignolo di Dottor Morte, lo portò ad essere considerato alla stregua di un serial killer appartenente al ramo degli Angeli della Morte.
Venne quindi condannato a 25 anni di reclusione per omicidio di secondo grado. Kevorkian ha scontato parte della condanna - a partire dal 1999 - nel carcere di Lakeland, nel Michigan. Rilasciato nel 2007, condannato però alla libertà vigilata per 3 anni, si candidò come indipendente alla Camera, sebbene la condanna gli precludesse la partecipazione alle Commissioni. Da tempo malato ai reni e al cuore, nel maggio del 2011 venne ricoverato all'ospedale William Beaumont di Royal Oak, dove morì il 3 giugno 2011 a causa di una trombosi polmonare.
Carriera musicale
Kevorkian era un musicista e compositore jazz. La fondazione di autodeterminazione EXIT con sede a Ginevra ha incaricato il direttore d'orchestra David Woodard nel 1999 di preparare le opere per organo di Kevorkian per ensemble di fiati.
Influenza culturale
Nel 2010 è stato girato un film intitolato You Don't Know Jack - Il dottor morte sulla sua vita, interpretato da Al Pacino.
Nel 1999 Kurt Vonnegut scrisse Dio la benedica, dottor Kevorkian (God Bless you, Dr. Kevorkian), un libro in cui l'autore si fa assistere dal medico per fare dei "viaggi con ritorno" in paradiso per intervistare personaggi famosi deceduti.
La copertina dell'album Paegan Terrorism Tactics della band sludge metal Acid Bath raffigura un dipinto di Kevorkian.
La band heavy metal Anvil ha dedicato al medico la canzone Doctor Kevorkian nell'album del 1996 Plugged In Permanent.
La band industrial EBM Kevorkian Death Cycle è considerata crociata per il diritto alla morte, in omaggio a questo medico.
Note
Voci correlate
Exit Italia
World Federation of Right to Die Societies
Altri progetti
Collegamenti esterni
Nati in Michigan | {
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For tampa bay a year exhibit 10 lightning opened body
By | December 3, 2019 | 0
They are beautiful, but not decorative.
Allen finished with 22 points, playing in his fourth game of the season after right ankle surgery.
Both of their wins came against the Cardinals and, weirdly, both games were won by a score of 31 in overtime.
At times, the 899's too-taut suspension and racy powerband make it feel like I'm riding the wrong bike.
More than 3 former NHL players are members of the NHLAA.
This led to development of catalytic cracking .
But Palmieri forced overtime by scoring on a rebound with 7 seconds to play.
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You can stay better connected to back-seat passengers with the newly offered CabinWatch setup and CabinTalk .
The 22-year-old played college football at Weber State, a school that's not necessarily renowned for its football prowess.
The fire brigade said it had carried out over 400 operations as well as laying on extra boats as water ambulances.
Valtteri Bottas, Mercedes 2.
Every time I come home, it's 'When are you coming home to the Knicks?' MSG is a special place, man.
Rarely do you see him on the ground.
The Raiders ranked fifth in the NFL in rushing defense and seventh in the league in takeaways .
The Raiders ranked among the NFL's elite in offense, defense and special teams under Gruden's leadership in 2000, as the Silver and Black advanced to the AFC Championship Game for the first time since 1990.
The narrative surrounding the Hypermotard is that it's a nefarious hooligan, practically a danger to itself and others-like the kid in your grade school class who was banned from using scissors after one too many accidental gorings.
— Copyright 2019 by STATS LLC and Associated Press.
James leads all Eastern Conference players with 808 votes, followed https://www.footballjerseyscheapwholesale.com Iverson .
Club Retailer of the Year Award is https://www.baixargratisapp.com based on a combination of fan interest, vendor feedback, sales, creative marketing, support of league-wide initiatives and unique stadium shopping experiences:
The 90th MLB All-Star Game, at Progressive Field in Cleveland, will air nationally Fox on July 9; in Canada by Rogers Sportsnet and RDS; and worldwide by partners in more than 180 countries;
Now comes a fourth-round pick making his debut for a Cincinnati team that's already looking to next year and is willing to turn the most important position over to someone who hasn't even worked with the starters until this week;
The Cavs have outscored foes 464 with Ilgauskas on the court and have been outscored 789 when he's on the bench.
That is something that separates us from everybody else – our competitive attitude.
He also says the engine and frame have matching numbers.
One of only 70 players in history to play in 100 career regular season games for the Redskins as of the conclusion of the 2018 season…
Gregorius took a curtain call while fans chanted his name after the slam, the 12th in New York's postseason history.
We're 1 for 3 on the power play and getting booed, he said.
The Preds' offensive-minded defenseman had gone 14 games without a goal before finally finding the back of the net Tuesday.
Ford tied his career high with 24 and Michael Redd added 23 to help the Milwaukee Bucks beat James and the Cleveland Cavaliers 111 on Saturday night.
NFL football center Max Unger grew up on his family's McCandless Ranch, a 7000 acre working cattle ranch that starts on the slopes of Mauna Loa stretches down to Hookena Beach.
A return from the starting right guard would bode well for the Colts' offense.
The 13's represent the second-most in NBA history, tied with Steph Curry.
• Serena Williams will star in https://www.moulindathie.com multi-media campaign to support the launch of Serena Williams Jewelry with global diamond manufacturer K.P.
Traction control can be turned off-a recommended step when venturing off the beaten paths.
Both trims are more or less just appearance packages, but we're especially into the look of the Bobber Twenty, which is meant to evoke the original 1920 G-20 Scout, a model later made famous by land speed racer Burt Munro — subject of the film The World's Fastest Indian. | {
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{"url":"https:\/\/zbmath.org\/?q=0984.13001","text":"# zbMATH \u2014 the first resource for mathematics\n\nMaximal elements of support. (English) Zbl\u00a00984.13001\nLet $$R$$ be a commutative ring with 1. In this paper the author claims that the Jacobson radical of an $$R$$-module $$M$$ is tightly related to the support of $$M$$, where the support of $$M$$ is the set of prime ideals $$P$$ such that the localization of $$M$$ at $$P$$ is not zero. In particular, if the module is finitely generated and injective, the set of zero divisors of the module is equal to the union of the maximal elements of the support of the module.\nReviewer:\u00a0K.Koh (Raleigh)\n##### MSC:\n 13A10 Radical theory on commutative rings (MSC2000) 13C11 Injective and flat modules and ideals in commutative rings 13E15 Commutative rings and modules of finite generation or presentation; number of generators\n##### Keywords:\nsupport of a module; Jacobson radical\nFull Text:","date":"2021-04-23 08:57:01","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\": 0.3528051972389221, \"perplexity\": 242.05361434669535}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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\/1618039568689.89\/warc\/CC-MAIN-20210423070953-20210423100953-00568.warc.gz\"}"} | null | null |
Герб Торонто, Онтаріо, Канада, був розроблений Робертом Ваттом, головним герольдом Канади на той час, для міста Торонто після його об'єднання в 1998 році. Герб був наданий Канадійською геральдичною владою 11 січня 1999 р.
Опис
Герб можна описати так: Узолотому полі злиті синій стовп і глава. Клейнод: на кольоровому вінку, випущеному мурована корона, обтяжена червоним серцем між двома срібними трояндами, у клейноді на трав'янистій зеленій горі орел з піднятими крилами і золотим озброєнням. Щитотримачами є: зправа сидячий бобер з червоним коміром на якому золотий шестикутник обтяжений зеленим ясеновим листом; зліва - бурий ведмідьз червоним коміром на якому золотий шестикутник обтяжений квіткою колумбії. Обидва щитотримачі розміщені на трав'янистій горі над срібно-лазуровими смугами, над яким три хвилеподібних срібні потоки в стовп, кожен з яких обтяжений іншим лазурним потоком; над синьою хвилястою базою розміщений сувій з девізом "Різноманітність Наша сила" між двома кленовими червоними листками.
Офіційний опис герба:
Щит: золотий з блакитним костуром.
Клейнод: над золото-синім буралетом золота мурована корона, обтяжена червоним серцем між двома срібними трояндами, над яким зелений курган, на якому орел з піднятими крилами.
Щитотримачі: на зеленому кургані, що піднімається над срібно-лазуровими смугами та у супроводі обтяженими синім трьома срібними хвилястими смугами, справа бобер у червоному нашийнику із золотою шестикутною підвіскою з кленовим листом та ведмедем зліва у червоному нашийнику із золотою шестикутною підвіскою з синьою квіткою колумбіни.
Девіз: Різноманітність Наша сила.
Герб, зображений на щиті, виконаний таким чином, що представляє дві вежі мерії Торонто та велику літеру Т, як показано на зображенні гербів. Три хвилясті потоки під щитом представляють три річки Торонто: Гамбер, Дон і Руж. Хвиляста "основа" представляє озеро Онтаріо.
Колишні герби
Мережа офісів мера Торонто включає герби або геральдичні символи всіх попередніх муніципалітетів, включаючи муніципалітет столичного Торонто.
Герби нинішнього уряду міста Торонто, колишнього міста Скарборо та колишнього міста Йорк зареєстровані в Державному реєстрі гербів, прапорів та бейджів Канади Канадійського геральдичного управління. Колишнє місто Йорк було єдиним колишнім муніципалітетом у столичному Торонто, який мав девіз латинською мовою, тоді як девізи інших муніципалітетів були англійською мовою.
Торонто
Колишнє місто Торонто мало герб до об'єднання в 1998 році. Щит складався з чотирьох чвертей, розділених білим хрестом, обтяжений червоним кленовим листом. Перша чверть була червоною і обтяжена трьома золотими левами як натяк на герб Англії, друга чверть була синьою з білою стилізованою трояндою, щоб натякати на Йорк, третя чверть була синьою із бсрібною шестернею, а четверта чверть містила золотий пароплав у червоному полі, щоб показати важливість озера та водних шляхів у місті та навколо нього. Клейнод: бобер на вершині золотої мурованої корони; корона символізувала Форт-Йорк. Щитотримачами були воїн корінного народу Канади (який, ймовірно, представляв місцевих міссісогів) з луком (ліворуч від глядача) та алегорією Британії тризубом та щитом, намальованим Юніон Джеком (праворуч від глядача). Девізом було «Промисловість, інтелект, доброчесність».
У попередній версії герба на місці білої троянди був показаний бобер, а замість зубчатого колеса - сніп пшениці. Крім того, щитотримачем від перших націй на попередньому гербі був вождь, що тримав сокиру, і обидва щитотримачі були звернені прямо навпроти один одного.
Геральдичний бейдж HMCS Торонто має клейнод колишнього міста Торонто.
Східний Йорк
Гербовий знак району Східного Йорку був розроблений Гаррі Фолксом, жителем Лесайда. Він був обраний округом в 1967 році і складався з таких елементів:
Щит: бобер між двома кленовими листками над Білою трояндою Йорків;
Клейнод: бульдог;
Девіз: район Іст-Йорк.
Етобікок
Місто Етобіко отримало герб від Управління канадської інтелектуальної власності 16 листопада 1977 року Геральдичні елементи були такі:
Щит: У золотому полі на зеленій горі згруповання з чотирьох гілочок вільх.
Клейнод: На зелено-золотому буралеті канадська мурована корона, увінчана шісттьма кленовими листками.
Щитотримачі: Зправа корінний індіанець на зігнутому коліні, що тримає в правиці лук, а зліва фігура трапера на зігнутому коліні, що представляє Етьєна Бруля, що тримає в лівиці мушкет.
На срібній стрічці чорний напис "ETOBICOKE TRADITION PROGRESS.
Північний Йорк
Печатку в Північному Йорку створив архітектор Торонто Мюррей Браун (1885-1958), якому також було доручено в 1922-1923 роках спроектувати першу муніципальну будівлю тодішнього містечка. Місто Норт-Йорк, підзаконний акт 103, прийнятий 23 грудня 1923 року, визначав печатку як "Щит, що показує сніп пшениці та терези, увінчаний бобром на кроні, та облямівку кленового листя з правого та лівого боку, весь оточений словами "ПProgress With Economy". Колишній герб муніципалітету складався з:
Щит: бобер на короні, поверх снопа пшениці та терезів справедливості.
Значок: по три кленові листки по обидва боки щита, все в межах кола.
Девіз: Progress with Economy.
Скарборо
Герб міста Скарборо був наданий в 1996 році Канадською геральдичною владою, і офіційний герб гербу був таким:
Щит: У золоті синя квітка колумбії, у синій главі золоте напівсонце.
Клейнод: Червоний кленовий лист над золотою короною із чотирми колосками (видно одну і дві півполовини), що чергується з чотирма жорнами (два видимі).
Щитотримачі: Два золоті олені із синім озброєннями у комірах із плетінок срібла, синього і червоного кольору, що стоять на горі Скарборо-Блафсів, що підноситься над синьо-срібними водами озера Онтаріо.
Девіз: HOME ABOVE THE BLUFFS.
Попередній герб колишнього району мав щит у лавровому вінку. На цьому щиті були розташовані такі елементи, по чвертях:
герб провінції Онтаріо;
сніп пшениці;
зубчасті колеса і завод;
вигляд на Скарборо-Блафс.
Йорк
Герб міста Йорк був наданий в 1993 році Канадською геральдичною владою, і офіційний герб був таким:
Щит: У зеленому полі срібний хвилястий стовп, обтяжений синьою смугою та золота гілочка троянди із срібною квіткою.
Клейнод: срібний гірський голуб, що піднімається з настінної корони над срібно-зеленим буралетом.
Щитотримачі: На трав'янистому кургані, золотий бобер із залено-срібно-синьою стрічкою на шиї та сидячий лев з подібною стрічкою.
Девіз: E SINGULIS COMMUNITAS (лат. "Від окремих людей, спільнота").
Бейдж: зелений шестикутник, обтяжений золотою гілочкою троянди та срібною квіткою.
Девіз міста був на латині, і це був єдиний колишній муніципалітет з девізом на цій мові, тоді як інші мали девізи англійською.
Примітки
Зовнішні посилання
Сторінка Герба на вебсайті міста Торонто
Сторінка символів міста на вебсайті міста Торонто
Культура Торонто
Торонто | {
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La maison Ernest-Cormier est une résidence située au 1418, avenue des Pins Ouest à Montréal que l'architecte Ernest Cormier a fait construire en 1930-31 pour son propre usage. Elle a été classée monument historique en 1974. L'ancien premier ministre du Canada Pierre Elliott Trudeau en fut le propriétaire de 1980 jusqu'à sa mort en 2000.
Description
L'architecte montréalais Ernest Cormier réalise une résidence aux lignes sobres qui intègre un agencement de dégradés et de retraits, tout en mettant l'accent sur la verticalité de la composition. La grande fenêtre verticale en façade, surmontée de motifs floraux stylisés, traduit bien cet esprit. De plus, l'utilisation du béton armé dans un bâtiment résidentiel est innovatrice pour l'époque.
La construction de la maison suit de près celle du pavillon principal de l'Université de Montréal, du même architecte et dont la tour apparaît en miniature dans les mains d'une muse, au-dessus de l'entrée. Ce serait la dernière grande maison bourgeoise à être construite dans le Mille carré doré.
La demeure fut classée monument historique le en vertu de la Loi sur les biens culturels du Québec. Les meubles de la résidence ont aussi été classé œuvre d'art à la même date. Il s'agit de l'un des trois biens du patrimoine moderne, avec le mausolée des Évêques-de-Trois-Rivières et Habitat 67 à avoir été classé durant la vie de son concepteur. La résidence a été restaurée en 1982-83.
L'ancien premier ministre du Canada Pierre Elliott Trudeau en fut le propriétaire de 1980 jusqu'à sa mort en 2000.
Notes et références
Annexes
Liens externes
Site web de la Ville de Montréal
Bibliographie
Architecture Art déco au Canada
Lieu patrimonial de Montréal
Immeuble patrimonial classé du Québec
Pierre Trudeau
Maison à Montréal
Ville-Marie (arrondissement)
Lieu historique national au Québec
Lieu historique national en 2018
Bâtiment de Ernest Cormier | {
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Established by friends, colleagues and co-workers in memory of Bruce C. McDonald, (Law '63).
Bruce Carr McDonald was born and raised as the youngest of a family of five in Roland, a small town named after his great-uncle in rural Manitoba. He obtained his B.A. from the University of Manitoba, his LLB from Queen's University (where he was a gold medalist), and his Masters and Doctorate degrees in Law from the University of Michigan (Ann Arbour). He was a member of the Law Faculty of Queen's from 1964 to 1970 before joining the Toronto office of Lang Michener where he was a senior partner.
Bruce McDonald had expertise in a wide variety of areas of law, but his great love was competition law. He taught evidence, torts, and competition law while he was at Queen's and later taught competition law at York University. On his death in 1991, was recognized as one of the leading authorities in competition law.
Bruce McDonald was always very passionate about education and the law and he was proud of his connection to Queen's University. Upon his death, his family was approached by many friends and colleagues who wanted to contribute something in his honour. This award was an obvious choice. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,809 |
\section{Introduction}
In a recent series of papers\cite{BRMC} detailed Montecarlo studies
have been produced for Drell-Yan
$\mu^+\mu^-$ experiments at moderate
center of mass energies and dilepton masses ($S$ $\sim$ 30-400 GeV$^2$,
$M$ $<$ 12 GeV/c$^2$). These experiments would measure
unpolarized, single and double spin asymmetries
and include proposals and plans at GSI\cite{panda,assia,pax},
at RHIC\cite{rhic2}, and at COMPASS\cite{compass-hadron}.
For the following discussion the relevant variables are
the squared center of mass energy $S$, the parton longitudinal
fractions $x_1$ and $x_2$, the dilepton mass $M$, and
$\tau$ $\equiv$ $M^2/S$ $=$ $x_1x_2$.
At magnitude level
\begin{equation}
{{d\sigma} \over {dx_1dx_2}}\
\sim\ N {{f(x_1) f(x_2)} \over {S \tau}}.
\end{equation}
\noindent
For the cases considered here
$N$ can be 1 nbarn or smaller. The product $f(x_1)f(x_2)$
is a symbolic way to represent a bilinear combination of partonic
distribution
functions, that decreases fastly towards zero for $x_1$, $x_2$
or $\tau$ $>>$ 0.1.
According to the above equation we have
small overall event rates,
fastly decreasing at increasing $\tau$;
However, all transverse spin related
measurements require one or both of the longitudinal momentum
fractions in the valence region, corresponding to average $\tau$
values that cannot be too small.
In a recent publication\cite{SSVY} the cross sections for
Drell-Yan
$\mu^+\mu^-$ production in the kinematical ranges relevant for the
quoted experiments has been reconsidered,
with resummation of soft gluon emissions near the partonic
threshold $\tau$ $=$ 1,
rising the cross sections in a $\tau-$dependent way.
The effect is predicted to be huge for $\tau$ $\sim$ 1,
but present anyway, and sometimes strong, also for $\tau$
$<<$ 1.
If compared to the numbers by \cite{BRMC}, the enhancement
factors
can range from 2 to over 10 in regions that are relevant for
the quoted experimental proposals at $S$ $<$ 300 GeV$^2$ (for $S$
$=$ 300-500 GeV$^2$ the cross sections are fixed by
experimental data, see e.g.\cite{conway,anass}).
If the predictions of ref.\cite{SSVY} are respected,
several plans
for quantities to be measured at GSI could be reconsidered.
This makes an experimental confirmation
of the prediction, if possible, an urgent matter. Apart for
competing predictions, it would be useful to
establish the $\tau$-dependence of the cross sections as precisely
as possible.
In our opinion this confirmation is
possible within a few years with the COMPASS hadronic beam
facility\cite{compass-hadron}, using
a pion beam at 50 GeV/c.
The measurement is rather tricky.
The predicted cross section enhancements
can arrive to a factor 2, but at relatively large $\tau$,
where events are few. When the
cross sections are integrated over a large mass range (equivalently,
$\tau$ range) these enhancements are much less evident.
Here a Montecarlo simulation is organized to study the proposed
measurement at COMPASS, and also the case of the
PANDA experiment\cite{panda} at GSI, where the effect is expected
to be especially strong. The Montecarlo
apparatus is the same as described in ref.\cite{BRMC}, so all the related
details can be found in that group of references.
K-factors
(i.e. overall $\tau-$dependent factors renormalizing the cross sections)
have been rewritten
so that cross section values follow alternatively the
behavior predicted in refs.\cite{BRMC} and \cite{SSVY}.
\section{The simulation}
For COMPASS e consider (negative) pion-nucleon Drell-Yan with
$S$ $=$ 100 GeV$^2$. We assume that the
measurement is performed on a fixed target with $Z/A$ $=$ 0.5.
We produce 50,000 simulated events for 4 $<$ M $<$ 9 GeV/c$^2$,
$x_F$ $>$ 0 and 1 $<$ $P_T$ $<$ 3 GeV/c.
This means about 35,000 events in the
lower mass bin 4-5 GeV/c$^2$.
The cut $x_F$ $>$ 0 is related with the strong possibility that fixed
target experiments
have reduced acceptance for negative $x_F$ (see e.g. the event
scatter plot in ref.\cite{conway}).
The $P_T$ cutoffs are
related with the selection of Drell-Yan events in strict sense
(see the discussions in ref.\cite{BRMC}).
We perform the simulation both for the case of ``normal'' cross
sections, and with the
enhancement factors shown in ref.\cite{SSVY}. So we will speak
of ``enhanced'' and ``non enhanced'' data sets.
Defining $\Delta M$ as the overall mass range, we
divide it into 5 equal subranges, and compare the simulated
event population of the larger mass bins with the population of the
lowest mass one. Since the populations of the lower mass bins are
practically equal in the enhanced and non enhanced cases
(about 35,000 events), we may limit ourselves to studying the
populations of the upper mass bins. The underying
necessary assumption is that these populations are extracted from
well normalized samples of 50,000 events.
For each case (enhanced and non-enhanced
cross sections)
the simulation is repeated ten times, so that for each
mass bin we may calculate the average number of events and its
fluctuation. The results are shown in
fig.1. In figs. 2 and 3
we show the laboratory angle and energy distributions, useful
for the discussion on the feasibility of the measurement.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{compass.eps}
\caption{
COMPASS: Event mass distribution for enhanced (triangles)
and non-enhanced (circles) cross sections;
\label{compass}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{theta_compass.eps}
\caption{
COMPASS: Upper panel:
Laboratory polar angle distribution for (mixed) positive
and negative muons for pair invariant mass in the range 4-5 GeV/c$^2$.
Lower panel: the same for mass in the range 7-8 GeV7c$^2$.
\label{theta_compass}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{energy_compass.eps}
\caption{
COMPASS: Upper panel:
Laboratory energy distribution for (mixed) positive
and negative muons for pair invariant mass in the range 4-5 GeV/c$^2$.
Lower panel: the same for mass in the range 7-8 GeV7c$^2$.
\label{energy_compass}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{panda.eps}
\caption{
PANDA case: event mass distribution for enhanced (triangles)
and non-enhanced (circles) cross sections; The large error bars
in the second and third point are related with uncertainties
in the prediction of the $J/\psi$ production rate.
\label{panda}}
\end{figure}
Fig.1 shows a
delicate but interesting situation. In the last three mass
ranges (6-7, 7-8, 8-9 GeV/c$^2$) the average event numbers are,
in the enhanced vs non-enhanced cases:
3300$\pm$60 vs 2443$\pm$57
734$\pm$20 vs 432$\pm$19
97$\pm$9 vs 43$\pm$8
The statistical fluctuation is small enough to separate clearly the
two possible outcomes in each mass bin. The very last
mass bin contains a too small event number, so we forget about it and
focus the rest of our analysis on the mass bin 7-8 GeV/c$^2$.
The conclusions are valid for the mass bin 6-7 GeV/c$^2$ as well.
The errors in the evaluation of the relative populations of the
higher mass bin (7-8 GeV/c$^2$) to the
lower mass bins (4-5 GeV/c$^2$)
may arise from two sources especially:
1) The background of random coincidence events with large mass,
that in the
case of a pion beam is due to coincidence between a
beam halo muon with large energy and a collision-originated
muon with small energy.
The estimation of the fraction of ``wrong'' pairs
in real experiments is normally based
on the examination of like-charge pairs. According to the analysis
in ref.\cite{conway}($S$ $\approx$ 470 GeV$^2$), and
ref.\cite{anass}($S$ $\approx$ 250 GeV$^2$), this noise is not a
problem
since (i) the background falls as steeply as the true data frequency
at increasing masses, (i) it remains at least one order of magnitude
below data.
2) The precision by
which the relative acceptances for large and small mass events
is estimated. This is potentially a problem if muons
associated with large and small mass
events distribute very differently
in the laboratory frame phase space.
Section 4 of ref.\cite{anass} is devoted to a
detailed study of the acceptance, for a
$\pi^-$-nucleus Drell-Yan experiment that presents
some analogies
with the COMPASS case. They show
a flat efficiency in dilepton-track reconstruction for mass up
to 6.8 GeV/c$^2$. The case of larger masses is not shown.
In fig.2 we show
the distribution of single muon polar angles in the
(fixed target) laboratory reference frame.
These muons belong to simulated Drell-Yan
events in the above Compass conditions
(without distinguishing between positive and negative muons).
The upper panel reports the distribution of 10,000 events belonging to
the dilepton mass region
4-5 GeV/c$^2$, the lower panel an equal number of events belonging
to the mass region 7-8 GeV/c$^2$.
Despite there are differences, the two distributions are similar enough
to exclude problems related with differential angular acceptance.
In fig.3 we show the corresponding distributions for the muon
laboratory energy. In this case there are important differences,
but the major point is that both distributions assume
non-negligible values in a large fraction of
all the available energy phase space.
An examination of the $\mu^+\mu^-$ correlations shows that for each event
$E_+ + E_-$ $\approx$ constant (34 $\pm$ 7 GeV in
the lower mass range, 45 $\pm$ 4 GeV in the higher mass range)
with a strong level of energy asymmetry
$A_E$ $\equiv$
$2 \vert E_+ - E_- \vert / ( E_+ + E_- )$. In both cases
$A_E$ follows an approximate distribution $1 + 1.4 (A_E)^2$
meaning that most events concentrate towards larger energy asymmetry.
The above analysis means that identifying a dilepton pair requires,
at small as at large masses,
a good and well understood acceptance level throughout all the energy range,
since the typical pair includes both a low-energy and a high-energy
muon. A lack in this sense can decrease seriously the acceptance,
but does not necesarily discriminate pairs
with different masses.
To better estimates the potential errors introduced by ignorance
about energy acceptance, we simulate a really
extreme situation: we suppose that the acceptance for single muons
with energy $<$ 20 GeV
is reduced by 50 \% and one is not aware of this. In other words, one is
convinced that the apparatus acceptance is $\approx$ 1 at all energies,
but this is true for $E$ $>$ 20 GeV only.
In this case the fraction of silently lost pairs is
49 \% in the low mass range, 46 \% in the high mass range.
Things are worse in the opposite case (the acceptance is
reduced to 50 \% for energy $>$ 20 GeV, and one is not aware of this).
Now the fraction of lost pairs is
40 \% in the low mass case, 51 \% in the high mass case.
So, the lack of knowledge about true acceptance would lead to an
error 20 \% in the estimation of the relative population of the
two mass bins.
The proposed examples are really pessimistic ones, since normally
acceptance is reconstructed with far larger precision than this.
Despite this, also in these situations the event ratio reconstruction
is precise enough to estimate the searched enhancement factor.
For PANDA (GSI) we
consider $\bar{p}p$ Drell-Yan with
$S$ $=$ 30 GeV$^2$, in the mass range 2-4.5 GeV$^2$.
The other cuts are the same as for the COMPASS case.
To avoid influence by the $J/\psi$ region on the event normalization,
we normalize the collected data set
with the requirement: 40,000 events
in the mass range 2-2.5 GeV/c$^2$.
For masses between 2.5 and 3.5 GeV/c$^2$
we base the simulation
on Drell-Yan data by ref.\cite{biino} at large $x_F$,
where gluon-gluon $J/\psi$ production is partially
suppressed as it would happen at PANDA.
This still leaves room for a large uncertainty factor in the
$J/\psi$ production rates (see fig.4).
We assume the threshold enhancement to be
the same for the $J/\psi$ peak and for the background.
As evident from fig.4, for masses over 3.5 GeV/c$^2$ event numbers
increase by one order and
there is no doubt on
the benefit PANDA would receive from the enhancement.
To summarize, the
threshold enhancement can be verified
with satisfactory precision
in the COMPASS apparatus for pion beams.
The enhancement would be much more striking
at PANDA (GSI),
with large increases in the counting rates at masses
$>$ 3.5 GeV/c$^2$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,921 |
public class DefendWithShield implements DefenseBehavior {
public String defend() {
return "Shield defend!";
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,100 |
Nicolae Bălcescu è un comune della Romania di 1.581 abitanti, ubicato del distretto di Călărași, nella regione storica della Muntenia.
Il comune è formato dall'unione di 3 villaggi: Fântâna Doamnei, Nicolae Bălcescu, Paicu.
Altri progetti
Collegamenti esterni
Comuni del distretto di Călărași (Romania) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,910 |
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