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\section{Introduction}
\label{sec:intro}
The application of quantum information in cryptography goes back to the work of Wiesner \cite{wiesner1983conjugate} who proposed the first quantum cryptographic tool called \textit{Conjugate Coding}. Remarkably, the idea of conjugate coding is still used in different forms in many modern protocols in quantum cryptography. Quantum cryptography, however, gained much attraction after the introduction of Quantum Key Distribution (QKD) by Bennett and Brassard \cite{bennett1983quantum, bennett1984quantum}. It was later proved by Lo and Chau \cite{lo1999unconditional} and Mayers \cite{mayers2001unconditional} that QKD is information-theoretically secure. A more accessible proof of security, based on error correcting codes, was given by Shor and Preskill \cite{shor2000simple}.
Although in theory QKD provides perfect security, its real-world implementations are not (and perhaps will not be) ideal. That means the QKD implementations, like other cryptographic implementations, are vulnerable to side-channel attacks, e.g., see \cite{lydersen2010hacking}. Even if we assume that QKD provides perfect security in practice, there are many other important cryptographic tasks, such as bit commitment, multiparty computation and oblivious transfer, that are not addressed by key distribution. In fact, it was proved by Mayers \cite{mayers1997unconditionally} and Lo and Chau \cite{lo1997quantum} that unconditionally secure quantum bit commitment is impossible. The impossibility of information-theoretically secure two-party computation using quantum communication was also later proved by Colbeck \cite{colbeck2007impossibility}. Such schemes can be made secure if the adversary is assumed to have bounded computational power or limited storage. Computational assumption are therefore still needed and are important in quantum cryptography. In particular, the necessity of computational assumptions in quantum \textit{public-key cryptography}, which is an increasingly important area in quantum cryptography, needs to be further investigated.
The principles of quantum public-key cryptography are close adaptations of those of classical public-key cryptography. In a quantum public-key scheme, a pair of keys $(sk_A, pk_A)$ is associated to each user $A$, the secret key $sk_A$ which is only known to $A$, and the public key $pk_A$ which is published by $A$ and it can be accessed by everyone. The pair of keys are generated by an efficient key generation algorithm. Like classical public-key schemes, quantum public-key schemes are modelled based on trapdoor one-way functions. Informally, a one-way function is a function that is easy to compute but hard to invert. A trapdoor one-way function is a one-way function $f$ to which some information $k$, called the trapdoor, can be associated in a way that anyone with the knowledge of $k$ can easily invert $f$ \cite{boneh2015graduate}. In the quantum setting, $f$ is a mapping $\ket{\alpha} \mapsto \ket{f_\alpha}$ from the space of secret keys to the space of public keys. The secret key $\ket{\alpha}$ can be a classical or quantum state, and the public key $\ket{f_\alpha}$ is a quantum state.
The three main constructions in quantum public-key cryptography are public-key encryption, digital signature and public-key money. In this work, we focus on quantum public-key encryption. We refer the reader to \cite{gottesman2001quantum} for quantum digital signatures, and \cite{aaronson2009quantum, aaronson2012quantum, farhi2012quantum} for quantum money. In a public-key encryption scheme, user $B$ can send a secret message $m$ to $A$ by encoding $m$ into a ciphertext $c$ using $A$'s public key $pk_A$ and a public encryption algorithm. Upon receiving the ciphertext $c$, user $A$ decrypts $c$ using her private key $sk_A$ and a public decryption algorithm.
\subsection{Previous constructions}
Kawachi \textit{et al.}~\cite{kawachi2005computational, kawachi2012computational} proposed a quantum public-key encryption scheme based on the Graph Isomorphism (GI) problem. For a security parameter $n$, they consider quantum states over the symmetric group $S_n$. The (secret key, public key) pair in the proposed scheme is $(sk, pk) = (s, \ket{\psi_s})$ where $s \in S_n$ is such that $s^2 = 1$, and $\ket{\psi_s} = (\ket{t} + \ket{ts}) / \sqrt{2}$ for a random $t \in S_n$. A single bit $b$ of classical information is then encrypted as $(\ket{t} + (-1)^b\ket{ts}) / \sqrt{2}$. They prove their scheme is secure assuming the hardness of the GI problem. However, GI is not considered as a standard quantum-secure assumption, especially after the recent breakthrough by L{\'a}szl{\'o} Babai \cite{babai2016graph} which shows that GI can be solved classically in quasipolynomial time.
An encryption scheme based on rotations of $1$-qubit states was proposed by Nikolopoulos \cite{nikolopoulos2008applications}. Their scheme is based on the trapdoor function $s \mapsto \ket{\psi_s(\theta_n)}$ where $0 \le s < 2^n$ is an integer, $\theta_n = \pi / 2^{n - 1}$ and $\ket{\psi_s(\theta_n)} = \cos(s\theta_n /2)\ket{0} + \sin(s\theta_n / 2)\ket{1}$. The encryption algorithm, however, was deterministic and could be broken by a simple attack proposed in \cite{nikolopoulos2009deterministic}. A randomized version of the encryption algorithm was proposed, in the same paper, to minimize the success probability of the aforementioned attack. To encrypt a classical message, the randomized algorithm uses significantly larger public-key states than the original algorithm.
\subsection{Overview and results}
\paragraph{Extrapolated dihedral cosets.}
Let $q$ be a positive integer. A dihedral coset over the additive group $\mathbb{Z}_q$ is a quantum state defined as $\ket{\psi_x} := (\ket{0}\ket{x} + \ket{1}\ket{x + s}) / \sqrt{2}$ where $x \in \mathbb{Z}_q$ is random and $s \in \mathbb{Z}_q$ is random and fixed. The Dihedral Coset Problem (DCP) is the problem of recovering (the secret) $s$, given dihedral cosets states $\ket{\psi_x}$ for different random values of $x$. DCP arises naturally when one uses the ``standard method'' to solve the hidden shift problem over the group $\mathbb{Z}_q$ \cite{childs2010quantum}. More precisely, the states $\ket{\psi_x}$ are obtained by performing a measurement on a superposition over the dihedral group $D_q = \mathbb{Z}_q \rtimes \mathbb{Z}_2$. Regev \cite{regev2004quantum} gave the first connection between DCP and standard lattice problems. Informally, if the number of states in DCP is bounded by $\poly(\log q)$, then DCP is (quantumly) at least as hard as $\poly(\log q)$-unique-SVP.
In a recent work, Brakerski \textit{et al.}~\cite{brakerski2018learning} proved that the Learning With Errors ($\mathsf{LWE}$) problem is equivalent, under quantum polynomial-time reductions, to an extended version of DCP, which they call the Extrapolated Dihedral Coset Problem ($\mathsf{EDCP}$). The $\mathsf{LWE}$ problem was proposed by Regev \cite{regev2005lattices, regev2009lattices}, and it has been used as the underlying security assumption in a large number of cryptographic schemes. For parameters $n, q \in \mathbb{Z}$ and $\alpha \in (0, 1)$, $\mathsf{LWE}_{n, q, \alpha}$ is the problem of recovering $\bm{s} \in \mathbb{Z}_q^n$ from samples of the form $(\bm{a}, \lrang{\bm{a}, \bm{s}} + e)$ where $\bm{a} \in \mathbb{Z}_q^n$ is uniformly random and $e$ is sampled from the discrete Gaussian distribution $\mathcal{D}_{\mathbb{Z}, \alpha q}$ of standard deviation $\alpha q$. The (uniform) $\mathsf{EDCP}$ is a generalization of DCP where coset states are over the group $\mathbb{Z}_q^n$ and the number of terms in the coset states is parameterized by an integer $2 \le r \le q$. An $\mathsf{EDCP}$ state is written as
\begin{equation}
\label{equ:edcp-state-intro}
\frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1}\ket{j}\ket{\bm{x} + j\bm{s}},
\end{equation}
where $\bm{s}$ is fixed and $\bm{x} \in \mathbb{Z}_q^n$ is uniformly random. The $\mathsf{EDCP}_{n, q, r}$ is then to recover $\bm{s}$, given states of the form \eqref{equ:edcp-state-intro}. In the $\mathsf{EDCP}$ and $\mathsf{LWE}$ equivalence given in \cite{brakerski2018learning}, the parameter $r$ is proportional to $1 / \alpha$. Intuitively, one would expect that $\mathsf{EDCP}$ becomes easier for larger $r$, since for smaller $\alpha$ the $\mathsf{LWE}$ problem becomes easier. In fact, we show that $\mathsf{EDCP}_{n, q, r}$ for any $r$ can be efficiently reduced to $\mathsf{EDCP}_{n, q, 2}$ (Lemma \ref{lem:self-rd}). On the other hand, in the extreme case $r = q$, $\mathsf{EDCP}$ can be solved in polynomial time (Remark \ref{rmk:edcp-extreme}).
\paragraph{A search-to-decision reduction for $\mathsf{EDCP}$.}
The decision-$\mathsf{EDCP}$, as was defined in \cite{brakerski2018learning}, is the problem of distinguishing between states of the form \eqref{equ:edcp-state-intro} and the same number of states of the form $\ket{j}\ket{\bm{x}}$ for uniformly random $(j, \bm{x}) \in \mathbb{Z}_r \times \mathbb{Z}_q^n$. It was pointed out in \cite{brakerski2018learning} that there is a polynomial time search-to-decision reduction for $\mathsf{EDCP}$ via $\mathsf{LWE}$. More precisely, search-$\mathsf{EDCP}$ can be reduced to decision-$\mathsf{EDCP}$ via the following sequence of polynomial-time reductions
\begin{equation}
\label{equ:redu-seq}
\text{search-}\mathsf{EDCP} \le \text{search-}\mathsf{LWE} \le \text{decision-}\mathsf{LWE} \le \text{decision-}\mathsf{EDCP}.
\end{equation}
In Section \ref{sec:old-search-dec}, we give a direct reduction from search-$\mathsf{EDCP}$ to decision-$\mathsf{EDCP}$ (Theorem \ref{thm:old-search-decision}) which works for a large class of moduli $q$.
\paragraph{A new decision-$\mathsf{EDCP}$.}
The original decision-$\mathsf{EDCP}$, explained above, is obtained from decision-$\mathsf{LWE}$ by following the same procedure used to establish the equivalence between search-$\mathsf{LWE}$ and search-$\mathsf{EDCP}$. For such a decision problem for $\mathsf{EDCP}$, however, there is no known general reduction from the search-$\mathsf{EDCP}$; The search-to-decision reductions are either obtained using the sequence \eqref{equ:redu-seq} or using Theorem \ref{thm:old-search-decision}. In the former case, one has to rely on the search-to-decision reductions for $\mathsf{LWE}$, e.g., \cite{regev2009lattices, brakerski2013classical, micciancio2012trapdoors}, which are not general enough in the sense that they either incur non-negligible loss in $\mathsf{LWE}$ parameters, or work only for special forms of the modulus $q$. In the latter case, the parameter $r$ has to satisfy some constraints with respect to $q$.
One could argue that the equivalence between the search and decision problems for $\mathsf{EDCP}$ is of less importance due to the recent result of Peikert \textit{et al.} \cite{peikert2017pseudorandomness} who proved that there is a polynomial-time quantum reduction from standard lattice problems directly to decision-$\mathsf{LWE}$. Their proof works for any modulus $q$. In any case, another issue is that it is not clear to us how to base \textit{efficient} cryptographic primitives on the original decision-$\mathsf{EDCP}$, see the discussion in Section \ref{sec:public-key-enc}. To resolve these issues, we propose a new decision problem for $\mathsf{EDCP}$. Informally, the new decision problem asks to distinguish between states of the form \eqref{equ:edcp-state-intro} and states of the form
\[ \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \omega_p^{jt} \ket{j}\ket{\bm{x} + j\bm{s}}, \]
where $p$ is a prime divisor of $q$ and $t \in \mathbb{Z}_p {\setminus} \{0\}$ is uniformly random. This new decision problem allows us to accomplish the following:
\begin{itemize}
\item We prove that for any modulus $q$ with $\poly(n)$-bounded prime factors, there is a quantum polynomial-time reduction from solving search-$\mathsf{EDCP}$ to solving decision-$\mathsf{EDCP}$.
\item We build an efficient quantum public-key encryption scheme based on decision-$\mathsf{EDCP}$.
\end{itemize}
\paragraph{Quantum public-key cryptosystem.}
In Section \ref{sec:public-key-enc}, we build a quantum public-key encryption scheme from $\mathsf{EDCP}$. The idea behind the encryption is very simple. The public key is a single $\mathsf{EDCP}$ state as in \eqref{equ:edcp-state-intro}. To encrypt a classical bit $b \in \{0,1\}$, a unitary transform is applied to the public key to encode the value $bt$ into the phase, where $t \in \mathbb{Z}_p$ is uniformly random and nonzero. The ciphertext is the state
\[ \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \omega_p^{jbt} \ket{j}\ket{\bm{x} + j\bm{s}}. \]
Here, $r$ and $p$ are divisors of $q$, and $r = p^{s'}$ for some integer $s' \ge 1$. The decryption algorithm is essentially a scalar multiplication over $\mathbb{Z}_q^n$ and an application of the quantum Fourier transform $\qft_r$ over $\mathbb{Z}_r$. This scheme is very efficient in terms of both computation and public key size. An even more efficient instantiation of this scheme is obtained by setting $p = 2$, $r = 2^{s'}$, and $q = 2^s$ such that $q = \poly(n)$ and $s' \ll s$. In this case, the size of the public key is $\tilde{O}(n)$ qubits. The encryption algorithm takes $O(1)$ qubit operations, and the key generation and decryption algorithms take $\tilde{O}(n)$ qubit operations.
\paragraph{Security beyond Holevo's bound.}
A fundamental difference between classical and quantum public-key cryptosystems is that in classical systems the public key can be copied arbitrarily many times, while in quantum systems making even two copies of the same public key is generally impossible. This is a consequence of the \textit{no-cloning} theorem. Also, in the quantum setting, the amount of information one can extract from a public key is bounded by a certain quantity which depends on the parameters of the system. This is a consequence of Holevo's theorem (Theorem \ref{thm:holevo}) which says that the accessible information of an ensemble is bounded by the $\chi$ quantity of the ensemble. Therefore, according to Holevo's theorem, the secret key cannot be recovered as long as the number of copies of the public key stays below Holevo's bound.
A feature common to all previous quantum public-key systems is that they rely on Holevo's bound for information-theoretic security, but do not provide a security proof based on a standard assumption beyond Holevo's bound. This puts a sever limitation on the total number of public keys published at any time. In Section \ref{sec:hardness}, we prove that our cryptosystem is information-theoretically secure when the number of public keys is bounded by $O(n\log q / \log r)$. Beyond that, when the number of public keys is $\poly(n)$, breaking the scheme is as hard as solving $\mathsf{LWE}$.
\section{Preliminaries}
\label{sec:preli}
\subsection{Quantum Computation}
Our notations for quantum information mostly follow those of \cite{watrous2018theory}. The classical state of a register $\mathsf{X}$ is represented by a finite alphabet, say $\Sigma$. If the registers $\mathsf{X}_1, \dots, \mathsf{X}_n$ are represented by alphabets $\Sigma_1, \dots, \Sigma_n$ then the classical state of the tuple $(\mathsf{X}_1, \dots, \mathsf{X}_n)$ is represented by $\Sigma_1 \times \cdots \times \Sigma_n$. The complex Euclidean space associated with the register $\mathsf{X}$ is denoted by $\mathbb{C}^\Sigma$. For a complex Euclidean space $\mathcal{X}$, denote the unit sphere in $\mathcal{X}$ by $\mathcal{S(X)}$. A linear operator $\rho$ acting on $\mathcal{X}$ is called a density operator if $\rho$ is positive semidefinite with trace equal to $1$. The quantum state of the register $\mathsf{X}$ is represented by the set of density operators $\mathrm{D}(\mathcal{X})$.
We will use the Dirac notation for the elements of $\mathcal{S(X)}$. In particular, we denote the column vector $x \in \mathcal{S(X)}$ by $\ket{x}$ and the row vector $x^*$ by $\bra{x}$. A state $\rho$ is called pure if it can be written as $\rho = \ket{x}\bra{x}$, in which case we will simply write the state as $\ket{x}$. By the spectral theorem, every state $\rho$ is a linear combination of pure states. Therefore, a quantum state can also be represented by a linear combination
\[ \sum_{x} \alpha_x \ket{x}, \hspace*{1mm} \sum_{x} \abs{\alpha_x}^2 = 1. \]
We shall alternate between these equivalent representations of quantum states throughout this paper. The density operator representation is particularly useful when the underlying quantum state is not completely known. For example, if we only know that the system is in the state $\ket{\psi_x}$ with probability $p_x$ then the state of the system is described by the density operator
\[ \rho = \sum_{x} p_x \ket{\psi_x}\bra{\psi_x} = \E_x \Big[ \ket{\psi_x} \bra{\psi_x} \Big]. \]
More, generally, the density operator corresponding to a probability distribution $\gamma: \mathcal{S(X)} \rightarrow [0, 1]$ is defined as
\[ \rho_\gamma = \int_{\ket{\phi} \in \mathcal{S(X)}} \ket{\phi}\bra{\phi} d\gamma(\ket{\phi}) = \E_{\ket{\phi} \in \gamma} \Big[ \ket{\phi}\bra{\phi} \Big]. \]
For quantum public-key cryptography we will need a formal notion of quantum state discrimination. In particular, we need to formally define the notion of computational (in)distinguishability of quantum states. For our purposes, it is more convenient to define computational distinguishability for probability distributions over quantum states. The following is adapted from \cite[3.3]{watrous2009zero}.
\begin{definition}
Let $\mathcal{X}$ be a complex Euclidean space, and let $\gamma, \mu: \mathcal{S(X)} \rightarrow [0, 1]$ be probability distributions. Then $\gamma$ is said to be $(s, \epsilon)$-distinguishable from $\mu$ if there is a quantum measurement circuit $Q$ of size $s$ such that
\[ \abs[\Big]{ \Pr_{\rho \in \gamma}[Q(\rho) = 1] - \Pr_{\rho \in \mu}[Q(\rho) = 1]} \ge \epsilon. \]
\end{definition}
Two distributions $\gamma, \mu$ are $(s, \epsilon)$-indistinguishable if they are not $(s, \epsilon)$-distinguishable.
\begin{definition}
For each $n \in \mathbb{N}$, let $\mathcal{X}_n$ be a complex Euclidean space and let $\gamma_n, \mu_n: \mathcal{S}(\mathcal{X}_n) \rightarrow [0, 1]$ be probability distributions. Then the two ensembles $\{ \gamma_n \}_{n \in \mathbb{N}}$ and $\{ \mu_n \}_{n \in \mathbb{N}}$ are said to be polynomially quantum indistinguishable if for all polynomially bounded functions $s, p: \mathbb{N} \rightarrow \mathbb{N}$, the distributions $\gamma_n$ and $\mu_n$ are $(s(n), 1 / p(n))$-indistinguishable for almost all $n \in \mathbb{N}$.
\end{definition}
Two ensembles are called quantum computationally indistinguishable if they are polynomially quantum indistinguishable. The advantage of a polynomial-time quantum algorithm $Q$ in distinguishing between the distributions $\gamma_n$ and $\mu_n$ is defined as
\[ \delta_Q(\gamma_n, \mu_n) = \abs[\Big]{ \Pr_{\rho \in \gamma_n}[Q(\rho) = 1] - \Pr_{\rho \in \mu_n}[Q(\rho) = 1] }. \]
Two ensembles $\{ \gamma_n \}$ and $\{ \mu_n \}$ are then quantum computationally indistinguishable if $\delta_Q(\gamma_n, \mu_n) = \negl(n)$ for all such $Q$ and almost all $n$.
\subsection{Error reduction}
\label{sec:err-red}
We can abstractly define the advantage of an algorithm $A$, regardless of $A$ being quantum or classical, in distinguishing between two probability distribution $P_1$ and $P_2$ as
\[ \delta_A(P_1, P_2) = \abs[\Big]{ \Pr_{x \in P_1}[A(x) = 1] - \Pr_{x \in P_2}[A(x) = 1] }. \]
Two ensembles of distributions $\{ P_{1, n} \}$ and $\{ P_{2, n} \}$ are said to be polynomial-time indistinguishable if for any polynomial-time algorithm $A$ and any $\poly(n)$-bounded function $p$ we have $\delta_A(P_{1, n}, P_{2, n}) \le 1 / p(n)$ for large enough $n$. The following lemma follows from the triangle inequality.
\begin{lemma}[Hybrid lemma]
\label{lem:hybrid}
Let $P_1, \dots, P_k$ be a sequence of probability distributions. Assume that $\delta_A(P_1, P_k) \ge \epsilon$ for some polynomial-time algorithm $A$. Then $\delta_A(P_i, P_{i + 1}) \ge \epsilon / k$ for some $1 \le i < k$.
\end{lemma}
Suppose an algorithm $A$ can distinguish between two distributions $P_1$ and $P_2$ with non-negligible advantage. A common technique to amplify the distinguishing advantage of $A$ is to sample enough times from the input distribution and then decide based on majority. A brief description of this technique, which we shall use several times in this paper, is as follows. First, we need the following well-known tail inequality.
\begin{lemma}[Hoeffding \cite{hoeffding1963probability}]
\label{lem:hoeffding}
Let $X_1, \dots, X_n$ be independent random variables with $X_i \in [a_i, b_i]$, and let $S = X_1, \cdots + X_n$. Then
\begin{align*}
\Pr[S - \E[S] \ge t] & \le e^{-2t^2 / \sum_i^n (b_i - a_i)^2}, \text{ and} \\
\Pr[S - \E[S] \le -t] & \le e^{-2t^2 / \sum_i^n (b_i - a_i)^2}.
\end{align*}
\end{lemma}
Now, assume $\delta_A(P_1, P_2) \ge 1 / p(n)$ for some polynomial $p(n)$, and let $P$ be the input distribution. Draw $m = 2np(n)^2$ samples from $P$, and let $X_i$ be a random variable representing the output of $A$ on input the $i$-th sample. Here, $X_i = 0$ (resp., $X_i = 1$) means $A$ has recognized the $i$-th sample to be from $P_1$ (resp., $P_2$). Let $S = X_1 + \cdots + X_m$. If $P = P_2$ then from the bound on $\delta_A$ we have $\E[S] \ge m / 2 + np(n)$. By Hoeffdings's inequality,
\begin{align*}
\Pr\Big[ S \le \frac{1}{2} (m + np(n)) \Big]
& = \Pr\Big[ S - \frac{m}{2} - np(n) \le -\frac{1}{2}np(n) \Big] \\
& \le \Pr\Big[ S - \E[S] \le -\frac{1}{2}np(n) \Big] \\
& \le e^{-n / 4}.
\end{align*}
Similarly, if $P = P_1$ then
\begin{align*}
\Pr\Big[ S \ge \frac{1}{2} (m - np(n)) \Big]
& \le \Pr\Big[ S - \E[S] \ge \frac{1}{2}np(n) \Big] \\
& \le e^{-n / 4}.
\end{align*}
Therefore, by running $A$ on $m$ samples and counting the number of $1$'s we can tell, with probability exponentially close to $1$, whether $P = P_1$ or $P = P_2$.
\subsection{Learning With Errors}
In the following, we briefly review the Learning With Errors ($\mathsf{LWE}$) problem and the Extrapolated Dihedral Coset Problem ($\mathsf{EDCP}$). In \cite{brakerski2018learning}, $\mathsf{EDCP}$ refers to a general class of problems from which two specific instances are studied in detail: the $U$-$\mathsf{EDCP}$ and $G$-$\mathsf{EDCP}$ which are the \textit{Uniform} and \textit{Gaussian} $\mathsf{EDCP}$, respectively. In this work, we will only study uniform-$\mathsf{EDCP}$, and simply refer to it as $\mathsf{EDCP}$.
Let $n \ge 1$ and $q = q(n) \ge 2$ be integers, and let $\chi$ be a probability distribution over $\mathbb{Z}$. For a random fixed $\bm{s} \in \mathbb{Z}_q^n$, denote by $A_{\bm{s}, \chi}$ the probability distribution over $\mathbb{Z}_q^n \times \mathbb{Z}_q$ defined as follows: choose $\bm{a} \in \mathbb{Z}_q^n$ uniformly at random, choose $e$ according to $\chi$ and output $(\bm{a}, \lrang{\bm{a}, \bm{s}} + e)$.
\begin{definition}[LWE, Search]
The search-$\mathsf{LWE}_{n, q, \chi}$ is the problem of recovering $\bm{s}$ given samples from the distribution $A_{\bm{s}, \chi}$. An algorithm $Q$ is said to solve $\mathsf{LWE}_{n, q, \chi}$ if $Q$ outputs $\bm{s}$ with probability at least $1 / \poly(n\log q)$ and has running time at most $\poly(n \log q)$.
\end{definition}
\begin{definition}[LWE, Decision]
The decision-$\mathsf{LWE}_{n, q, \chi}$ problem is to distinguish between the distribution $A_{\bm{s}, \chi}$ and the uniform distribution over $\mathbb{Z}_q^n \times \mathbb{Z}_q$. An algorithm $Q$ is said to solve the desicion-$\mathsf{LWE}_{n, q, \chi}$ if it succeeds with advantage at least $1 / \poly(n\log q)$ and has running time at most $\poly(n\log q)$.
\end{definition}
The distribution $\chi$ is called the error distribution and is usually chosen to be $\mathcal{D}_{\mathbb{Z}, \alpha q}$, the discrete Gaussian distribution centered around zero with standard deviation $\alpha q$. The parameter $\alpha \in (0, 1)$ is called the error rate. We sometimes write $\mathsf{LWE}_{n, q, \alpha}$ instead of $\mathsf{LWE}_{n, q, \chi}$ for simplicity. Let $n \ge 1$ and $q \ge 2$ be as above and let $r = r(n) < q$ be a positive integer. Let $\Sigma = \mathbb{Z}_r \times \mathbb{Z}_q^n$ and define the complex Euclidean space $\mathcal{X} = \mathbb{C}^\Sigma$. For a fixed uniformly random $\bm{s} \in \mathbb{Z}_q^n$ define the probability distribution $\mu_{\bm{s}, r}: \mathcal{S(X)} \rightarrow [0, 1]$ as follows: choose $\bm{x} \in \mathbb{Z}_q^n$ uniformly at random and output the state
\[ \ket{\phi_{\bm{s}, r}(\bm{x})} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1}\ket{j}\ket{\bm{x} + j\bm{s}}. \]
If we only have access to the output of $\mu_{\bm{s}, r}$, e.g., if $\bm{x}$ is unknown, then the quantum system corresponding to the state $\ket{\phi_{\bm{s}, r}(\bm{x})}$ is described by the density operator
\[ \rho_{\bm{s}, r} = \frac{1}{q^n} \sum_{\bm{x} \in \mathbb{Z}_q^n} \ket{\phi_{\bm{s}, r}(\bm{x})} \bra{\phi_{\bm{s}, r}(\bm{x})} = \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \ket{\phi_{\bm{s}, r}(\bm{x})} \bra{\phi_{\bm{s}, r}(\bm{x})} \Big]. \]
Therefore, the output of the distribution $\mu_{\bm{s}, r}$ is always described by the same state $\rho_{\bm{s}, r}$. In other words, a sample from the distribution $\mu_{\bm{s}, r}$ is a \textit{copy} of the state $\rho_{\bm{s}, r}$.
\begin{definition}[EDCP, Search]
Let $n$, $q$ and $r$ be defined as above. The search-$\mathsf{EDCP}_{n, q, r}$ is the problem of recovering $\bm{s}$ given samples from the distribution $\mu_{\bm{s}, r}$. A quantum algorithm $Q$ is said to solve $\mathsf{EDCP}_{n, q, r}$ if it outputs $\bm{s}$ with probability at least $1 / \poly(n\log q)$ and has running time at most $\poly(n\log q)$.
\end{definition}
\begin{definition}[EDCP, Decision]
\label{def:d-edcp}
Let $n$, $q$ and $r$ be defined as above. Define the probability distribution $\gamma_r: \mathcal{S(X)} \rightarrow [0, 1]$ by choosing $(j, \bm{x}) \in \mathbb{Z}_r \times \mathbb{Z}_q^n$ uniformly at random and outputting the state $\ket{j}\ket{\bm{x}}$. The decision-$\mathsf{EDCP}_{n, q, r}$ is the problem of distinguishing between the distributions $\mu_{\bm{s}, r}$ and $\gamma_r$. A quantum algorithm $Q$ is said to solve the decision-$\mathsf{EDCP}_{n, q, r}$ if it succeeds with advantage at least $1 / \poly(n\log q)$ and has running time at most $\poly(n\log q)$.
\end{definition}
The density operator corresponding to the output of the distribution $\gamma_r$ in Definition \ref{def:d-edcp} is
\[ \rho = \frac{1}{rq^n} \sum_{j = 0}^{r - 1} \sum_{\bm{x} \in \mathbb{Z}_q^n} \ket{j}\ket{\bm{x}} \bra{j}\bra{\bm{x}} = \E_{(j, \bm{x}) \in \mathcal{U}(\mathbb{Z}_r \times \mathbb{Z}_q^n)} \Big[ \ket{j}\ket{\bm{x}} \bra{j}\bra{\bm{x}} \Big] = \frac{1}{rq^n} \mathds{1}_{\mathcal{X}}, \]
which is the maximally mixed state over the space $\mathcal{X}$. Therefore, decision-$\mathsf{EDCP}_{n, q, r}$ is the problem of distinguishing between the same number of copies of the states $\rho_{\bm{s}, r}$ and $\mathds{1}_{\mathcal{X}} / (rq^n)$. For an integer $m > 0$, we denote by $\mathsf{EDCP}_{n, q, r}^m$ the $\mathsf{EDCP}$ problem in which the number of samples from $\mu_{\bm{s}, r}$ is bounded by $m$. The following theorem establishes a polynomial-time equivalence between $\mathsf{LWE}$ and $\mathsf{EDCP}$.
\begin{theorem}[\cite{brakerski2018learning}]
\label{thm:lwe-edcp}
Let $\chi$ be a discrete Gaussian distribution centered around zero with standard deviation $\alpha q$. There is a polynomial-time quantum reduction from $\mathsf{LWE}_{n, q, \chi}$ to $\mathsf{EDCP}_{n, q, r}^m$ with $m = \poly(n\log q)$ and $r = \poly(n\log q) / \alpha$. Conversely, for the same parameter relationship up to $\poly(n\log q)$ factors, there is a polynomial-time quantum reduction from $\mathsf{EDCP}$ to $\mathsf{LWE}$.
\end{theorem}
It is important to note that the equivalence in Theorem \ref{thm:lwe-edcp} holds only when the number of $\mathsf{EDCP}$ samples is polynomially bounded, i.e., $m = \poly(n\log q)$. Giving such an equivalence for arbitrary $m$ is an open problem, see the discussion in Section \ref{sec:hardness-poly}.
\section{A Search to Decision Reduction}
\label{sec:old-search-dec}
In this section, we give a search-to-decision reduction for $\mathsf{EDCP}$. The reduction works for a large class of moduli $q$. The technique we use is inspired by the one in \cite{micciancio2012trapdoors} for a search-to-decision reduction for $\mathsf{LWE}$. We need the following lemma, which shows the self-reducibility of $\mathsf{EDCP}$.
\begin{lemma}
\label{lem:self-rd}
For any $r' \le r$, given access to the distribution $\mu_{\bm{s}, r}$, we can efficiently sample from the distribution $\mu_{\bm{s}, r'}$. In particular, there is an efficient reduction from $\mathsf{EDCP}_{n, q, r}$ to $\mathsf{EDCP}_{n, q, r'}$.
\end{lemma}
\begin{proof}
To keep the reduction efficient, we treat the two cases $r' > r / 2$ and $r' \le r / 2$ separately. If $r' > r / 2$ then a simple indicator function can be used to to produce samples from $\mu_{\bm{s}, r'}$. More precisely, define the function $f: [0, r) \rightarrow \{ 0, 1 \}$ by
\[ f(x) =
\begin{cases}
1 & \text{if } x < r', \\
0 & \text{otherwise}.
\end{cases} \]
Then applying the transform $\ket{j}\ket{\bm{y}}\ket{0} \mapsto \ket{j}\ket{\bm{y}}\ket{f(j)}$, where $\bm{y} \in \mathbb{Z}_q^n$, to $\rho_{\bm{s}, r} \in \mu_{\bm{s}, r}$ and measuring the last register results in the state $\rho_{\bm{s}, r'}$ with probability at least $1 / 2$. If the measurement outcome is not $1$ then we repeat the above process.
If $r' \le r / 2$ then we proceed as follows. Let $\mathcal{X} = \mathbb{C}^{\mathbb{Z}_r \times \mathbb{Z}_q^n}$ and define the measurement $\mu: [0, \lfloor r / r' \rfloor] \rightarrow \mathrm{Pos}(\mathcal{X})$ using the operators
\[ \mu_a = \sum_{b = 0}^{\ell - 1} \ket{ar' + b}\bra{ar' + b} \otimes \mathds{1}, \]
where $\ell = r'$ for $0 \le a < \lfloor r / r' \rfloor$ and $\ell = r - r'$ for $a = \lfloor r / r' \rfloor$. This measurement can be implemented efficiently \cite[A.8]{kaye2007introduction}. If we perform $\mu$ on a sample $\rho_{\bm{s}, r}$ from $\mu_{\bm{s}, r}$ the probability of observing the outcome $a$ is
\begin{align*}
\tr(\mu_a^*\mu_a \rho_{\bm{s}, r})
& = \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \tr(\mu_a \ket{\phi_{\bm{s}, r}(\bm{x})} \bra{\phi_{\bm{s}, r}(\bm{x})} \mu_a^*) \Big] \\
& = \frac{1}{r} \sum_{b, c = 0}^{\ell - 1} \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)}\Big[ \tr(\ket{ar' + b}\ket{\bm{x} + (ar' + b)\bm{s}} \bra{ar' + c}\bra{\bm{x} + (ar' + c)\bm{s}}) \Big] \\
& = \frac{1}{r} \sum_{b, c = 0}^{\ell - 1} \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)}\Big[ \tr(\ket{ar' + b}\bra{ar' + c} \otimes \ket{\bm{x} + (ar' + b)\bm{s}} \bra{\bm{x} + (ar' + c)\bm{s}}) \Big] \\
& = \frac{\ell}{r},
\end{align*}
and the post-measurement state corresponding to this outcome is
\begin{align*}
\frac{\mu_a \rho_{\bm{s}, r} \mu_a^*}{(\ell / r)}
& = \frac{1}{r} \sum_{b, c = 0}^{\ell - 1} \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)}\Big[ \ket{ar' + b}\ket{\bm{x} + (ar' + b)\bm{s}} \bra{ar' + c}\bra{\bm{x} + (ar' + c)\bm{s}} \Big] \\
& = \frac{1}{r} \sum_{b, c = 0}^{\ell - 1} \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)}\Big[ \ket{ar' + b}\ket{\bm{x} + b\bm{s}} \bra{ar' + c}\bra{\bm{x} + c\bm{s}} \Big].
\end{align*}
Subtracting $ar'$ from the first register, we obtain the state $\rho_{\bm{s}, \ell}$. So if the outcome is $a \in [0, \lfloor r / r' \rfloor)$ we obtain the state $\rho_{\bm{s}, r'}$, which is what we are looking for. Therefore, the probability of obtaining the desired state after one measurement is
\[ \lfloor r / r' \rfloor \frac{\ell}{r} = \lfloor r / r' \rfloor \frac{r'}{r} \ge \left( \frac{r}{r'} - 1 \right)\frac{r}{r'} = 1 - \frac{r'}{r} \ge \frac{1}{2}. \]
If the measurement outcome is $a = \lfloor r / r' \rfloor$ then we repeat the above process.
\end{proof}
It follows from the proof of Lemma \ref{lem:self-rd} that obtaining a sample from $\mu_{\bm{s}, r'}$ requires (an expected) $2$ samples from $\mu_{\bm{s}, r}$. This means $\mathsf{EDCP}_{n, q, r}^m$ is reduced to $\mathsf{EDCP}_{n, q, r'}^{\Theta(m)}$ regardless of the ratio between $r$ and $r'$.
\begin{theorem}
\label{thm:old-search-decision}
Let $q = p_1^{e_1} \cdots p_\ell^{e_\ell}$ be the prime factorization of $q$ and assume that the primes $p_i$ are of size $\poly(n)$. Let $0 < r < q$ and let $k$ be the number of primes $p_i < r$. Then there is a polynomial-time quantum reduction from solving worst-case search-$\mathsf{EDCP}_{n, q, r}$, with overwhelming probability, to solving average-case decision-$\mathsf{EDCP}_{n, q, r'}$, with non-negligible probability, for any $r' \le r$ such that $(r')^k \le r$, and $r' \le p_i^{e_i}$ for all $i$.
\end{theorem}
\begin{proof}
Let $D$ be an oracle for solving decision-$\mathsf{EDCP}_{n, q, r'}$. The idea of the proof is to use $D$ and samples from the distribution $\mu_{\bm{s}, r}$ to recover $\bm{s} \bmod p_i^{h_i}$, with large-enough $h_i$, for each $i$, and then assemble the results using the Chinese remainder theorem to recover $\bm{s} \bmod \prod_i p_i^{h_i}$. From there, since $q / \prod_i p_i^{h_i}$ is small-enough, we can use quantum Fourier transform to recover $\bm{s} \bmod q$. We shall compute $\bm{s} \bmod p_1^{e_1}$, the algorithm is the same for the other $p_i$. Let $p = p_1$ and $e = e_1$. The proof proceeds in several steps.
\begin{enumerate}[leftmargin = *, font = \bfseries]
\item Sampling from $\mu_{\bm{s}, r'}$: given samples from $\mu_{\bm{s}, r}$, according to Lemma \ref{lem:self-rd}, we can efficiently sample from $\mu_{\bm{s}, r'}$. So, from now on we assume that we have access to samples from $\mu_{\bm{s}, r'}$.
\item Building hybrid distributions: from the distribution $\mu_{\bm{s}, r'}$ we construct the distribution $\mu_{\bm{s}, r'}^k$ for all $k = 0, \dots, e$. Given a sample $\rho_{\bm{s}, r'} \in \mu_{\bm{s}, r'}$, a sample from $\mu_{\bm{s}, r'}^k$ is obtained by computing $j \bmod p^k$ into an auxiliary register and then measuring the register. More precisely, denote by $\ket{\phi_{\bm{s}, r'}^k(\bm{x})}$ the result of
\begin{align}
\ket{\phi_{\bm{s}, r'}(\bm{x})}\ket{0}
& \mapsto \frac{1}{\sqrt{r'}} \sum_{j = 0}^{r' - 1} \ket{j}\ket{\bm{x} + j\bm{s}}\ket{j \bmod p^k} \label{equ:r-1} \\
& \mapsto \frac{1}{\sqrt{r_k}} \sum_{j = 0}^{r_k - 1} \ket{jp^k + c}\ket{\bm{x} + (jp^k + c)\bm{s}}, \nonumber \tag{measure the last register}
\end{align}
where $0 < r_k \le \lfloor r' / p^k \rfloor$ and the random constant $0 \le c \le p^k - 1$ depend on the outcome of measuring the last register. Then
\[ \rho_{\bm{s}, r'}^k = \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \ket{\phi_{\bm{s}, r'}^k(\bm{x})} \bra{\phi_{\bm{s}, r'}^k(\bm{x})} \Big] \]
is a sample from $\mu_{\bm{s}, r'}^k$.
\item\label{step:s-mod-p} Computing $\bm{s} \bmod p$: for $k = 0$ we have $j = 0 \bmod p^0$ for all $j$, so $\rho_{\bm{s}, r'}^0 = \rho_{\bm{s}, r'}$ hence $\mu_{\bm{s}, r'}^0 = \mu_{\bm{s}, r'}$. Let $h$ be the smallest integer such that $r' \le p^h$, such an $h$ exists since $r' \le p^e$ by assumption. Then for $k = h$, measuring the last register in \eqref{equ:r-1} collapses the state $\rho_{\bm{s}, r'}$ to $\mathds{1}_{\mathcal{X}'} / (r'q^n)$ where $\mathcal{X}' = \mathbb{C}^{\mathbb{Z}_{r'} \times \mathbb{Z}_q^n}$. Therefore, by a hybrid argument (Lemma \ref{lem:hybrid}) there is a minimal $0 < t \le h$ such that $D$ can distinguish between $\mu_{\bm{s}, r'}^{t - 1}$ and $\mu_{\bm{s}, r'}^t$ with non-negligible advantage. Using the amplification technique of Section \ref{sec:err-red} we can assume that the distinguishing advantage of $D$ is exponentially close to $1$. Note that $t$ can be efficiently computed by analyzing the output of $D$. Let $\bm{s} = (s_1, \dots, s_n)$. We recover $s_1 \bmod p$, the other $s_i \bmod p$ can be recovered similarly. Consider the state $\ket{\phi_{\bm{s}, r'}^{t - 1}(\bm{x})}$ where $\bm{x} = (x_1, \dots, x_n)$, and let $a \in \mathbb{Z}_p$ and $\bm{y} = (y_1, \dots, y_n) \in \mathbb{Z}_q^n$ be arbitrary. If we perform the transform
\begin{equation}
\label{equ:s1-trans}
U_1: \ket{j}\ket{\bm{y}}\ket{0} \mapsto \ket{j}\ket{\bm{y}}\ket{y_1 - ja \bmod p^t}
\end{equation}
on $\ket{\phi_{\bm{s}, r'}^{t - 1}(\bm{x})}\ket{0}$ we obtain the state
\[ \frac{1}{\sqrt{r_{t - 1}}} \sum_{j = 0}^{r_{t - 1} - 1} \ket{jp^{t - 1} + c}\ket{\bm{x} + (jp^{t - 1} + c)\bm{s}}\ket{x_1 + (jp^{t - 1} + c)(s_1 - a) \bmod p^t}. \]
Measuring the last register results in a state that will be a sample from $\mu_{\bm{s}, r'}^{t - 1}$ or $\mu_{\bm{s}, r'}^t$ depending on whether $s_1 = a$ or $s_1 \ne a \bmod p$, respectively:
\begin{itemize}
\item $s_1 = a \bmod p$. In this case, the value of the last register is $x_1 + (s_1 - a)c \bmod p^t$. So the last register is not entangled with the first two registers, and we obtain the original sample from $\mu_{\bm{s}, r'}^{t - 1}$.
\item $s_1 \ne a \bmod p$. Let $0 \le c_1 \le p^k - 1$ be the outcome of the measurement. Then the post-measurement state contains the terms with $j$ satisfying $jp^{t -1} = (c_1 - x_1) / (s_1 - a) - c \bmod p^t$. If we write the right hand side as $c_2p^{t - 1}$ for some constant $0 \le c_2 \le p - 1$ then $j = c_2 \bmod p$ and the post-measurement state is
\begin{align*}
\ket{\psi_{\bm{x}}}
& = \frac{1}{\sqrt{r_t}} \sum_{j = 0}^{r_t - 1} \ket{(jp + c_2)p^{t - 1} + c}\ket{\bm{x} + ((jp + c_2)p^{t - 1} + c)\bm{s}} \\
& = \frac{1}{\sqrt{r_t}} \sum_{j = 0}^{r_t - 1} \ket{jp^t + c_2p^{t - 1} + c}\ket{\bm{x} + (jp^t + c_2p^{t - 1} + c)\bm{s}}
\end{align*}
where $0 < r_t \le \lfloor r_{t - 1} / p \rfloor$. We clearly have
\[ \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \ket{\psi_{\bm{x}}}\bra{\psi_{\bm{x}}} \Big] \in \mu_{\bm{s}, r'}^t. \]
\end{itemize}
Therefore, using $D$ we can find out whether $s_1 = a \bmod p$. Since $p \le \poly(n)$, we can recover $s_1 \bmod p$ by trying every $a \in \mathbb{Z}_p$.
\item\label{step:s-mod-pk} Computing $\bm{s} \bmod p^k$: assume we have recovered, for some $k > 1$, the first $k - 1$ digits of $s_1$ in base $p$, that is we have computed $0 \le \tilde{s}_1 < p^{k - 1}$ such that $\tilde{s}_1 = s_1 \bmod p^{k - 1}$. Let $s_{1, k - 1}$ be the $k$-th digit of $s_1$ in base $p$. To compute $s_{1, k - 1}$, we can modify the transform in Step \ref{step:s-mod-p} as
\begin{equation*}
U_k: \ket{j}\ket{\bm{y}}\ket{0} \mapsto \ket{j}\ket{\bm{y}}\ket{y_1 - j\tilde{s}_1 - jp^{k - 1}a \bmod p^{t + k - 1}}.
\end{equation*}
Applying $U_k$ to a sample from $\mu_{\bm{s}, r'}^{t - 1}$ and measuring the last register produces a sample from $\mu_{\bm{s}, r'}^{t - 1}$ or $\mu_{\bm{s}, r'}^t$ depending on whether $s_{1, k - 1} = a$ or $s_{1, k - 1} \ne a \bmod p$, respectively. This method works as long as $t + k - 1 \le e$.
\item Quantum Fourier transform: using the procedure in Step \ref{step:s-mod-pk}, we can compute $\bm{s} \bmod p^{e - t + 1}$ where $t$ is the minimal integer determined in Step \ref{step:s-mod-p}. Similarly, we recover $\bm{s} \bmod p^{e_i - t_i + 1}$ with $t_i$ the corresponding minimal integer for $p_i$, for all $i$. Note that if $r' \le p_i$ then $t_i = 1$. Also, we always have $r' \ge p_i^{t_i - 1}$. Using the Chinese remainder theorem we can compute $\tilde{\bm{s}} = \bm{s} \bmod v$ where
\[ v = \prod_{i = 1}^\ell p_i^{e_i - t_i + 1} = q / \prod_{i = 1}^\ell p_i^{t_i - 1} \ge \frac{q}{(r')^k} \ge \frac{q}{r}. \]
Now, by applying the transform $\ket{j}\ket{\bm{y}} \mapsto \ket{j}\ket{\bm{y} - j\tilde{\bm{s}}}$ to the states $\ket{\phi_{\bm{s}, r}(\bm{x})}$, we assume all the coordinates of $\bm{s}$ are multiples of $v$. Next, using Lemma \ref{lem:self-rd}, we project $\ket{\phi_{\bm{s}, r}(\bm{x})}$ onto $\ket{\phi_{\bm{s}, q'}(\bm{x})}$, where $q' = q / v \le r$. Finally, we apply $\qft_{q^n}$ to the second register of $\ket{\phi_{\bm{s}, q'}(\bm{x})}$ and measure to obtain the state
\[ \frac{1}{\sqrt{q'}} \sum_{j = 0}^{q' - 1} \omega_q^{j\lrang{\bm{u}, \bm{s}}} \ket{j} = \frac{1}{\sqrt{q'}} \sum_{j = 0}^{q' - 1} \omega_{q'}^{j\lrang{\bm{u}, \bm{s} / v}} \ket{j} = \qft_{q'}\ket{\lrang{\bm{u}, \bm{s} / v} \bmod q'}, \]
where $\bm{u} \in \mathbb{Z}_q^n$ is uniformly random. Applying $\qft_{q'}^*$ to the above state, we obtain $\lrang{\bm{u}, \bm{s} / v} \bmod q'$. The value $\bm{s} / v$ can be computed, with high probability, by gathering $O(n)$ of these linear equations. \qedhere
\end{enumerate}
\end{proof}
\begin{corollary}
Let $q = p_1^{e_1} \cdots p_\ell^{e_\ell}$ be the prime factorization of $q$ and assume that the primes $p_i$ are of size $\poly(n)$. If $r \le p_i$ for all $1 \le i \le \ell$ then there is a polynomial-time quantum reduction from solving search-$\mathsf{EDCP}_{n, q, r}$ to solving decision-$\mathsf{EDCP}_{n, q, r}$.
\end{corollary}
It follows from the above corollary that for a prime power $q = p^e$, where $p < \poly(n)$ and $r \le p$, search-$\mathsf{EDCP}_{n, q, r}$ and decision-$\mathsf{EDCP}_{n, q, r}$ are quantum polynomial-time equivalent. Two interesting special cases of this equivalence are $\{q = 2^e, r = 2\}$ and $\{q = p, r < p\}$.
\section{A New Decision Problem}
\label{sec:new-decsn}
In this section, we propose an $\mathsf{EDCP}$ decision problem that we believe is more suitable for applications than the decision problem defined in Section \ref{sec:preli}. In particular, our public-key cryptosystem (Section \ref{sec:public-key-enc}) is based on the new decision problem. We show that the new decision problem is quantum polynomial-time equivalent to search-$\mathsf{EDCP}$. This establishes the fact that the new decision problem is at least is as hard as the old one.
We assume, as before, that the modulus $q$ has $\poly(n)$-bounded prime factors. Perhaps the new decision problem is best understood for a prime modulus $q$. So let us assume, for now, that $q$ is a $\poly(n)$-bounded prime. Define the distribution $\tilde{\mu}_{\bm{s}, r}$ on the unit sphere $\mathcal{S(X)}$ by choosing $\bm{x} \in \mathbb{Z}_q^n$ and $t \in \mathbb{Z}_q {\setminus} \{ 0 \}$ uniformly at random and outputting the state
\begin{equation}
\label{equ:new-dec}
\ket{\phi_{\bm{s}, r}(\bm{x}, t)} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \omega_q^{jt} \ket{j}\ket{\bm{x} + j\bm{s}}.
\end{equation}
The new decision problem is to distinguish between the distributions $\mu_{\bm{s}, r}$ and $\tilde{\mu}_{\bm{s}, r}$. The motivation behind this new definition is that from states of the form \eqref{equ:new-dec} we can efficiently obtain ``\textit{shifted}'' $\mathsf{LWE}$ samples $(\bm{a}, \lrang{\bm{a}, \bm{s}} + e + t)$ where $\bm{a} \in \mathbb{Z}_q^n$ is uniformly random and $e$ is sampled from $\mathcal{D}_{\mathbb{Z}, q / \lambda}$ for an appropriate value of $\lambda$. For a large enough $q$, this pair is closer to a uniformly random element of $\mathbb{Z}_q^n \times \mathbb{Z}_q$ than an $\mathsf{LWE}$ sample. An instance of the above decision problem then translates to an instance of the $\mathsf{LWE}$ decision problem.
A shifted $\mathsf{LWE}$ sample can be obtained from the state \eqref{equ:new-dec} using the technique in \cite{brakerski2018learning}, which we briefly explain in the following. We ignore the normalization factors in front superpositions for clarity. Applying the transform $\ket{j} \mapsto \ket{j - \lfloor (r - 1) / 2 \rfloor}$ to the first register of the state \eqref{equ:new-dec} we obtain the state
\begin{equation}
\label{equ:symm-edcp}
\sum_{j = -\lfloor (r - 1) / 2 \rfloor}^{\lceil (r - 1) / 2 \rceil} \omega_q^{jt} \ket{j}\ket{\bm{x} + j\bm{s}},
\end{equation}
where we have again denoted the uniformly random element $\bm{x} + \lfloor (r - 1) / 2 \rfloor \bm{s} \in \mathbb{Z}_q^n$ by $\bm{x}$. Next, using Lemma \ref{lem:qrs} we can transform \eqref{equ:symm-edcp} into
\begin{equation}
\label{equ:symm-eg}
\sum_{j = -\lfloor (r - 1) / 2 \rfloor}^{\lceil (r - 1) / 2 \rceil} \omega_q^{jt} g_\lambda(j) \ket{j}\ket{\bm{x} + j\bm{s}},
\end{equation}
with probability $\Omega(\lambda / r)$. Here, $g_\lambda(x) = \exp(-\pi x^2 / \lambda^2)$ is a one-dimensional Gaussian function. We assume that $r$ is large enough so that the above probability is not too small. For example, $r = \lfloor \lambda\sqrt{n} \rfloor$ and so $\Omega(\lambda / r) = \Omega(1 / \sqrt{n})$. Now if we apply the transform $\qft_q \otimes \qft_{q^n}$ to \eqref{equ:symm-eg} and measure the last register we obtain the state
\[ \ket{\psi} = \sum_{y \in \mathbb{Z}_q} \sum_{j = -\lfloor (r - 1) / 2 \rfloor}^{\lceil (r - 1) / 2 \rceil} \omega_q^{j(\lrang{\bm{a}, \bm{s}} + y + t)} g_\lambda(j) \ket{y}, \]
where $\bm{a} \in \mathbb{Z}_q^n$ is uniformly random and known. In the following, we use the notation $\ket{\psi_1} \approx \ket{\psi_2}$ when the two quantum states $\ket{\psi_1}$ and $\ket{\psi_2}$ have an exponentially small trace distance. We have
\begin{align*}
\ket{\psi}
& \approx \sum_{y \in \mathbb{Z}_q} \sum_{j \in \mathbb{Z}} \omega_q^{j(\lrang{\bm{a}, \bm{s}} + y + t)} g_\lambda(j) \ket{y} \tag{by Corollary \ref{cor:gaus-apprx}} \\
& = \sum_{y \in \mathbb{Z}_q} \sum_{j \in \mathbb{Z}} g_{1/\lambda} \Big( j + \frac{\lrang{\bm{a}, \bm{s}} + y + t}{q} \Big) \ket{y} \tag{by Theorem \ref{thm:poisson-sum}} \\
& = \sum_{e \in \mathbb{Z}} g_{1/\sigma} \Big( \frac{e}{q} \Big) \ket{\lrang{-\bm{a}, \bm{s}} + e - t \bmod q} \tag{$e \leftarrow jq + \lrang{\bm{a}, \bm{s}} + t + y$} \\
& \approx \sum_{e \in \mathbb{Z}_q} g_{1/\sigma} \Big( \frac{e}{q} \Big) \ket{\lrang{-\bm{a}, \bm{s}} + e - t \bmod q} \tag{by Corollary \ref{cor:gaus-apprx}}.
\end{align*}
Measuring the above state, we obtain a pair $(-\bm{a}, \lrang{-\bm{a}, \bm{s}} + e - t)$ where $e$ is sampled from $\mathcal{D}_{\mathbb{Z}, q / \lambda}$.
For a general modulus $q$, an immediate generalization of the above decision problem would be to just replace the prime modulus with a general one, and the above process of obtaining an $\mathsf{LWE}$ sample goes through without any change. However, for such a generalization, it is not clear how to reduce the search problem to the decision problem when $q$ is super-polynomially large in $n$. An alternative approach is to associate a distribution to each of the primes $p \mid q$, then the decision problem is to distinguish between these distribution and $\mu_{\bm{s}, r}$. We make this precise in the following.
\begin{definition}[EDCP, Decision]
Let $p \mid q$ be a prime. Define the distribution $\mu_{\bm{s}, r, p}$ on the unit sphere $\mathcal{S(X)}$ by choosing $\bm{x} \in \mathbb{Z}_q^n$ and $t \in \mathbb{Z}_p {\setminus} \{ 0 \}$ uniformly at random and outputting the state
\begin{equation}
\label{equ:new-dec1}
\ket{\phi_{\bm{s}, r}(\bm{x}, p, t)} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \omega_p^{jt} \ket{j}\ket{\bm{x} + j\bm{s}}.
\end{equation}
The decision-$\mathsf{EDCP}_{n, q, r}$ is the problem of distinguishing between the distribution $\mu_{\bm{s}, r}$ and any distribution in the set $\{ \mu_{\bm{s}, r, p} \}_{p \mid q}$.
\end{definition}
\begin{theorem}
Assume that all the prime factors of $q$ are $\poly(n)$-bounded. Then there is a quantum polynomial-time reduction from solving search-$\mathsf{EDCP}_{n, q, r}$ to solving decision-$\mathsf{EDCP}_{n, q, r}$.
\end{theorem}
\begin{proof}
Let $q = p_1^{e_1} \cdots p_\ell^{e_\ell}$ be the prime factorization of $q$. Let $D$ be an oracle for solving decision-$\mathsf{EDCP}_{n, q, r}$. The idea is to use $D$ to find $\bm{s} \bmod p_i^{e_i}$ for all $i$ and then reconstruct $\bm{s} \bmod q$ using the Chinese remainder theorem. Let $\bm{s} = (s_1, \dots, s_n)$. We show how to recover $s_1 \bmod p_1^{e_1}$, the other values $s_i \bmod p_j^{e_j}$ can be recovered similarly. Set $p = p_1$ and $e = e_1$.
For any $y \in \mathbb{Z}_p$ and nonzero $c \in \mathbb{Z}_p$ define the unitary
\[ U_{c, y} \ket{j}\ket{\bm{a}} = \omega_p^{(a_1 - jy)c}\ket{j}\ket{\bm{a}}, \]
where $a_1$ is the first coordinate of $\bm{a}$. Given a sample $\rho_{\bm{s}, r}$ from $\mu_{\bm{s}, r}$, fix $y \in \mathbb{Z}_p$ and select a fresh nonzero $c \in \mathbb{Z}_p$ uniformly at random. Then we have
\[ U_{c, y}\ket{\phi_{\bm{s}, r}(\bm{x})} = \frac{1}{\sqrt{r}} \omega_p^{x_1c}\sum_{j = 0}^{r - 1} \omega_p^{j(s_1 - y)c} \ket{j}\ket{\bm{x} + j\bm{s}}. \]
Therefore, ignoring the global phase, if $s_1 \ne y \bmod p$ then $U_{c, y} \rho_{\bm{s}, r} U_{c, y}^*$ is a sample from $\mu_{\bm{s}, r, p}$, otherwise $U_{c, y} \rho_{\bm{s}, r} U_{c, y}^* = \rho_{\bm{s}, r}$. So, the oracle $D$ could tell us which is the case. Trying all $y \in \mathbb{Z}_p$ we can find $s_1 \bmod p$. Now assume we have recovered $\tilde{s}_1 = s_1 \bmod p^k$ for $k < e$. To compute $s_1 \bmod p^{k + 1}$, we can modify the unitary $U_{c, y}$ as
\[ U_{c, y, k} \ket{j}\ket{\bm{a}} = \omega_{p^{k + 1}}^{(a_1 - j\tilde{s}_1 - jp^ky)c}\ket{j}\ket{\bm{a}}. \]
To see how $U_{c, y, k}$ acts on a sample $\rho_{\bm{s}, r}$ from $\mu_{\bm{s}, r}$, let $s_{1, k}$ be the $(k + 1)$-th digit of $s_1$ in base $p$. Then
\begin{align*}
U_{c, y, k} \ket{j}\ket{\bm{x} + j\bm{s}}
& = \omega_{p^{k + 1}}^{(x_1 + js_1 - j\tilde{s}_1 - jp^ky)c}\ket{j}\ket{\bm{x} + j\bm{s}} \\
& = \omega_{p^{k + 1}}^{x_1c} \omega_p^{j(s_{1, k} - y)c}\ket{j}\ket{\bm{x} + j\bm{s}}.
\end{align*}
Therefore, repeating the above procedure, we can recover $s_{1, k}$. This completes the proof.
\end{proof}
\section{Information-Theoretic and Hardness Bounds}
\label{sec:hardness}
In this section, we derive hardness bounds for $\mathsf{EDCP}$ based on the number of samples. The $\mathsf{EDCP}_{n, q, r}^m$ problem is to recover the secret $\bm{s} \in \mathbb{Z}_q^n$ given $m$ copies of the state $\rho_{\bm{s}, r}$. We consider bounds $O( n\log q / \log r)$, $\poly(n)$ and $2^{O(\sqrt{n \log q})}$ for $m$. As $m$ increases, $\mathsf{EDCP}$ becomes easier. Figure \ref{fig:hardness-bounds} summarizes the hardness of $\mathsf{EDCP}$ based on these bounds.
\begin{figure}
\centering
\begin{tikzpicture}[tick/.style = {draw = black, fill = red, inner sep = 0.4mm}]
\draw [thick, -{latex}] (0, 0) -- (0, 3) node[above] {\scriptsize \# samples};
\node [tick] (t1) at (0, 1) {};
\node [tick] (t2) at (0, 2) {};
\node [below left = 1mm of t1] {$\displaystyle O\Big( n \frac{\log q}{\log r} \Big)$};
\node [below right = 1mm of t1] {Information-theoretically secure};
\node [below left = 1mm of t2] {$\poly(n)$};
\node [below right = 1mm of t2] {as hard as $\mathsf{LWE}$};
\node [above left = 1mm of t2] {$\displaystyle 2^{O(\sqrt{n \log q})}$};
\node [above right = 1mm of t2] {can be solved in $\displaystyle 2^{O(\sqrt{n \log q})}$};
\end{tikzpicture}
\caption{The hardness of $\mathsf{EDCP}$ based on the number of samples. These bounds also hold for the security of the public-key encryption scheme of Section \ref{sec:public-key-enc}. In that case, a sample is a copy of the public key.}
\label{fig:hardness-bounds}
\end{figure}
\subsection{Limited number of samples}
\label{sec:hardness-limited}
We derive a lower bound on $m$ using tools from quantum information theory. A special case of the result of this section was also obtained in \cite{bacon2005optimal}. To better understand the problem we can model it in the following standard way, for $m = 1$. Alice selects a uniformly random $\bm{s} \in \mathbb{Z}_q^n$ and stores it in the classical register $\mathsf{Y}$. She then prepares a register $\mathsf{X}$ in the state $\rho_{\bm{s}, r}$ and sends $\mathsf{X}$ to Bob. So, Bob has access to the register $\mathsf{X}$ that is prepared according to the ensemble
\[
\begin{array}{rrll}
\eta: & \mathbb{Z}_q^n & \longrightarrow & \mathrm{Pos}(\mathcal{X}) \\
& \bm{s} & \longmapsto & \frac{1}{q^n}\rho_{\bm{s}, r},
\end{array}
\]
where $\mathrm{Pos}(\mathcal{X})$ is the space of positive semidefinite operators on $\mathcal{X}$. Bob picks a measurement $\mu: \mathbb{Z}_q^n \rightarrow \mathrm{Pos}(\mathcal{X})$ and measures $\mathsf{X}$ according to $\mu$. He stores the outcome of the measurement in a classical register $\mathsf{Z}$. The pair $(\mathsf{Y}, \mathsf{Z})$ of classical registers will be distributed according to the distribution
\[
\begin{array}{rrll}
q: & \mathbb{Z}_q^n \times \mathbb{Z}_q^n & \longrightarrow & [0, 1] \\
& (\bm{s}, \bm{u}) & \longmapsto & \lrang{\mu(\bm{u}), \eta(\bm{s})}.
\end{array}
\]
The amount of information that Bob can learn about $\mathsf{Y}$ using the measurement $\mu$ is determined by the mutual information between $\mathsf{Y}$ and $\mathsf{Z}$, which we denote by $\operatorname{I}_\mu(\eta)$. The accessible information of the ensemble $\eta$ is defined as
\[ \operatorname{I}(\eta) = \sup_{\mu} \operatorname{I}_\mu(\eta), \]
where the supremum ranges over all choices of the measurement $\mu$. Note that the pair $(\mathsf{Y}, \mathsf{X})$ is in the classical-quantum state
\[ \sigma = \sum_{\bm{s} \in \mathbb{Z}_q^n} \ket{\bm{s}}\bra{\bm{s}} \otimes \eta(\bm{s}). \]
The quantum mutual information between $\mathsf{Y}$ and $\mathsf{X}$, with respect to the state $\sigma$, is called the Holevo information of the ensemble $\eta$ and is denoted by $\chi(\eta)$. We have
\begin{equation}
\label{equ:holevo-chi}
\chi(\eta) = \operatorname{I}(\mathsf{Y} : \mathsf{X}) = \entpy\Bigg( \frac{1}{q^n} \sum_{\bm{s} \in \mathbb{Z}_q^n} \rho_{\bm{s}, r} \Bigg) - \frac{1}{q^n} \sum_{\bm{s} \in \mathbb{Z}_q^n} \entpy(\rho_{\bm{s}, r}),
\end{equation}
where $\entpy(\cdot)$ is the von Neumann entropy.
\begin{theorem}[Holevo's theorem]
\label{thm:holevo}
Let $\Sigma$ be an alphabet and let $\mathcal{X}$ be a complex Euclidean space. For any ensemble $\eta: \Sigma \rightarrow \mathrm{Pos}(\mathcal{X})$ it holds that $\operatorname{I}(\eta) \le \chi(\eta)$.
\end{theorem}
Therefore, according to Theorem \ref{thm:holevo}, we can obtain an upper bound on the accessible information of the ensemble $\eta$ by computing $\chi(\eta)$. For an arbitrary number of samples $m$, we need to bound $\chi(\eta^{\otimes m})$ for the ensemble
\begin{equation}
\label{equ:ensem-m}
\begin{array}{rrll}
\eta^{\otimes m}: & \mathbb{Z}_q^n & \longrightarrow & \mathrm{Pos}(\mathcal{X}^{\otimes m}) \\
& \bm{s} & \longmapsto & \frac{1}{q^n}\rho_{\bm{s}, r}^{\otimes m}.
\end{array}
\end{equation}
This is done is the following theorem.
\begin{theorem}
\label{thm:acc-bound}
For the ensemble $\eta^{\otimes m}$ in \eqref{equ:ensem-m} we have $\chi(\eta^{\otimes m}) \le m(1 - q^{-n})\log r$.
\end{theorem}
\begin{proof}
First, note that the entropy is additive with respect to tensor products, i.e., for any two states $\sigma_1$ and $\sigma_2$ it holds that $\entpy(\sigma_1 \otimes \sigma_2) = \entpy(\sigma_1) + \entpy(\sigma_2)$. It follows that $\entpy(\rho_{\bm{s}, r}^{\otimes m}) = m\entpy(\rho_{\bm{s}, r})$. Next, for a state $\sigma$ of a compound register $(\mathsf{X}_1, \cdots, \mathsf{X}_m)$, we have, by the subadditivity of von Neumann entropy,
\[ \entpy(\sigma) \le \entpy(\tr_1(\sigma)) + \cdots + \entpy(\tr_m(\sigma))\]
where $\tr_k(\sigma)$ is the reduction of $\sigma$ to the (state of the) register $\mathsf{X}_k$. Therefore,
\begin{align*}
\entpy\bigg( \frac{1}{q^n} \sum_{\bm{s} \in \mathbb{Z}_q^n} \rho_{\bm{s}, r}^{\otimes m} \bigg)
& \le \sum_{i = 1}^m \entpy\bigg(\tr_i\bigg( \frac{1}{q^n} \sum_{\bm{s} \in \mathbb{Z}_q^n} \rho_{\bm{s}, r}^{\otimes m} \bigg)\bigg) \tag{by subadditivity of $\entpy$} \\
& = \sum_{i = 1}^m \entpy\bigg(\frac{1}{q^n} \sum_{\bm{s} \in \mathbb{Z}_q^n} \tr_i(\rho_{\bm{s}, r}^{\otimes m}) \bigg) \tag{by linearity of $\tr$} \\
& = m\entpy\bigg( \frac{1}{q^n} \sum_{\bm{s} \in \mathbb{Z}_q^n} \rho_{\bm{s}, r} \bigg).
\end{align*}
It follows from \eqref{equ:holevo-chi} that $\chi(\eta^{\otimes m}) \le m\chi(\eta)$. Now, we can compute $\chi(\eta)$ by computing the eigenvalues of the operators $\rho_{\bm{s}, r}$ and $\rho = q^{-n}\sum_{\bm{s} \in \mathbb{Z}_q^n} \rho_{\bm{s}, r}$. The eigenvectors of $\rho_{\bm{s}, r}$ are
\[ \ket{\psi_{\bm{x}, t}} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \omega_r^{jt} \ket{j}\ket{\bm{x} + j\bm{s}}, \hspace*{2mm} (t, \bm{x}) \in \mathbb{Z}_r \times \mathbb{Z}_q^n, \]
and the eigenvalues are $0$ and $q^{-n}$ with multiplicities $(r - 1)q^n$ and $q^n$, respectively. To compute the eigenvalues of $\rho$ it is best to write the second register in the Fourier basis. We have
\begin{align*}
(\mathds{1} \otimes \qft_{q^n}) \rho_{\bm{s}, r} (\mathds{1} \otimes \qft_{q^n})^*
& = \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ (\mathds{1} \otimes \qft_{q^n}) \ket{\phi_{\bm{s}, r}(\bm{x})} \bra{\phi_{\bm{s}, r}(\bm{x})} (\mathds{1} \otimes \qft_{q^n})^* \Big] \\
& = \frac{1}{rq^n} \sum_{\bm{y}, \bm{z} \in \mathbb{Z}_q^n} \E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \omega_q^{\lrang{\bm{x}, \bm{y} - \bm{z}}} \Big] \sum_{j, k \in \mathbb{Z}_r} \omega_q^{\lrang{j\bm{y} - k\bm{z}, \bm{s}}} \ket{j}\bra{k} \otimes \ket{\bm{y}}\bra{\bm{z}} \\
& = \frac{1}{r} \E_{\bm{y} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \sum_{j, k \in \mathbb{Z}_r} \omega_q^{\lrang{(j - k)\bm{y}, \bm{s}}} \ket{j}\bra{k} \otimes \ket{\bm{y}}\bra{\bm{y}} \Big],
\end{align*}
where the last equality follows from the fact that
\[
\E_{\bm{x} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \omega_q^{\lrang{\bm{x}, \bm{y} - \bm{z}}} \Big] =
\begin{cases}
1 & \text{if } \bm{y} = \bm{z} \\
0 & \text{if } \bm{y} \ne \bm{z}.
\end{cases}
\]
Therefore, we have
\begin{align}
(\mathds{1} \otimes \qft_{q^n}) \rho (\mathds{1} \otimes \qft_{q^n})^*
& = \frac{1}{r} \E_{\bm{y} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \sum_{j, k \in \mathbb{Z}_r} \E_{\bm{s} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \omega_q^{\lrang{(j - k)\bm{y}, \bm{s}}} \Big] \ket{j}\bra{k} \otimes \ket{\bm{y}}\bra{\bm{y}} \Big] \nonumber \\
& = \frac{1}{rq^n} \sum_{j, k \in \mathbb{Z}_r} \ket{j}\bra{k} \otimes \ket{0}\bra{0} + \frac{1}{rq^n} \mathds{1} \otimes (\mathds{1} - \ket{0}\bra{0}) \label{equ:f-basis}
\end{align}
where the second equality follows from
\[
\E_{\bm{s} \in \mathcal{U}(\mathbb{Z}_q^n)} \Big[ \omega_q^{\lrang{(j - k)\bm{y}, \bm{s}}} \Big] =
\begin{cases}
1 & \text{if } (j - k)\bm{y} = 0 \\
0 & \text{if } (j - k)\bm{y} \ne 0.
\end{cases}
\]
The eigenvectors of \eqref{equ:f-basis} are
\begin{align*}
& \ket{\psi_{\bm{x}, t}} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \omega_r^{jt} \ket{j}\ket{\bm{x}}, \hspace*{2mm} (t, \bm{x}) \in \mathbb{Z}_r \times \mathbb{Z}_q^n, \hspace*{1mm} \bm{x} \ne 0, \\
& \ket{\psi_t} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \omega_r^{jt} \ket{j}\ket{0}, \hspace*{2mm} t \in \mathbb{Z}_r,
\end{align*}
and the eigenvalues are $0$, $q^{-n}$ and $r^{-1}q^{-n}$ with multiplicities $r - 1$, $1$ and $(q^n - 1)r$, respectively. Finally, using \eqref{equ:holevo-chi} and the eigenvalues for $\rho$ and $\rho_{\bm{s}, r}$, we have
\[ m\chi(\eta) \le m\Big( \frac{1}{q^n}\log(q^n) + \frac{r(q^n - 1)}{rq^n}\log(rq^n) - \log(q^n) \Big) = m\Big( 1 - \frac{1}{q^n} \Big)\log r. \qedhere \]
\end{proof}
Assume that Bob has found a measurement $\mu$ on $\eta^{\otimes m}$, i.e., a measurement that can operate on the joint state of $m$ copies of Alice's state, such that, after possibly some post-measurement processing, he can guess the value of $\mathsf{Y}$ with a constant probability $p$. Then a lower bound on $m$, that depends on $p$, can be computed using Theorem \ref{thm:acc-bound}. We need the following result known as Fano's inequality.
\begin{lemma}[Fano's inequality]
\label{lem:fano-ineq}
Let $X$ and $Y$ be random variables on some finite set $\Gamma$, and let $\tilde{X} = f(Y)$ for some function $f$. Let $p = \Pr[X \ne \tilde{X}]$. Then it holds that
\[ \entpy(X \vert Y) \le p\log(\abs{\Gamma} - 1) + \entpy(p, 1 - p). \]
\end{lemma}
\begin{corollary}
\label{cor:lower-b}
Let $\bm{s} \in \mathbb{Z}_q^n$ be chosen uniformly at random. The number of copies of the state $\rho_{\bm{s}, r}$ needed to recover $\bm{s}$ with constant probability is at least $O(n\log q / \log r)$.
\end{corollary}
\begin{proof}
Recall the communication scenario above: Alice selects $\bm{s} \in \mathbb{Z}_q^n$ uniformly at random and stores it in the register $\mathsf{Y}$. She then generates $m$ copies of the state $\rho_{\bm{s}, r}$ and sends them to Bob. On receiving $\rho_{\bm{s}, r}^{\otimes m}$, Bob applies a measurement $\mu$ and stores the measurement outcome in the register $\mathsf{Z}$. Bob might perform some post-processing on $\mathsf{Z}$ to obtain another register $\tilde{\mathsf{Z}}$. Assume that $p = \Pr[\mathsf{Y} \ne \tilde{\mathsf{Z}}]$ is a constant. Then
\begin{align*}
\operatorname{I}_\mu(\eta^{\otimes m})
& = \operatorname{I}(\mathsf{X} : \mathsf{Z}) \\
& = \entpy(\mathsf{Y}) - \entpy(\mathsf{Y} \vert \mathsf{Z}) \tag{by definition} \\
& \ge n\log(q) - (1 - p)\log(q^n - 1) - \entpy(p, 1 - p) \tag{by Lemma \ref{lem:fano-ineq}} \\
& \ge pn\log(q) - 1
\end{align*}
Now, by Theorem \ref{thm:acc-bound}, $m(1 - q^{-n})\log r \ge \chi(\eta^{\otimes m}) \ge \operatorname{I}_\mu(\eta^{\otimes m})$ which completes the proof.
\end{proof}
\begin{remark}
\label{rmk:edcp-extreme}
An interesting case for which the bound in Corollary \ref{cor:lower-b} is tight is when $q = r$. In this case, given the state $\rho_{\bm{s}, r}$, applying the transform $\qft_r^* \otimes \qft_{q^n}$ results in the state
\[ \frac{1}{\sqrt{q^n}} \sum_{\bm{y} \in \mathbb{Z}_q^n} \omega_q^{\lrang{\bm{y}, \bm{x}}} \ket{\lrang{\bm{y}, \bm{s}}}\ket{\bm{y}}. \]
Measuring this state, we obtain a linear equation $\lrang{\bm{y}, \bm{s}}$ where $\bm{y} \in \mathbb{Z}_q^n$ is uniformly random. We can solve for $\bm{s}$ by gathering $O(n)$ of these linear equations.
\end{remark}
\subsection{Polynomial number of samples}
\label{sec:hardness-poly}
When the number of samples is $\poly(n)$, $\mathsf{EDCP}$ is quantum polynomially equivalent to $\mathsf{LWE}$ \cite{brakerski2018learning}. The reduction form $\mathsf{EDCP}$ to $\mathsf{LWE}$ is proved as in Section \ref{sec:new-decsn}. The reduction from $\mathsf{LWE}$ to $\mathsf{EDCP}$ is based on the ball-intersection technique that was originally proposed by \cite{regev2004quantum}. We briefly review the reduction idea here and refer the reader to \cite{regev2004quantum, brakerski2018learning} for details.
Let $(\bm{A}, \bm{b}_0 = \bm{As}_0 + \bm{e}_0)$ be a set of $m$ samples from $\mathsf{LWE}_{n, q, \alpha}$, written in matrix form. We start by preparing the state
\[ \sum_{\bm{s} \in \mathbb{Z}_q^n} \sum_{j = 0}^{r - 1} \ket{j}\ket{\bm{s}}, \]
where we have omitted the normalization factors for clarity. Here, $r$ is a function of $n$ and $q$. We then compute $(j, \bm{s}) \mapsto \bm{As} - j\bm{b}_0$ into an auxiliary register. After a change of variables we obtain the state
\begin{equation}
\label{equ:latt-sup}
\sum_{\bm{s} \in \mathbb{Z}_q^n} \sum_{j = 0}^{r - 1} \ket{j}\ket{\bm{s} + j\bm{s}_0}\ket{\bm{As} - j\bm{e}_0}.
\end{equation}
The goal is to project the above state onto a state $\sum_{\bm{s} \in \mathbb{Z}_q^n} \sum_{j = 0}^{r - 1} \ket{j}\ket{\bm{s} + j\bm{s}_0}$ for some $\bm{s} \in \mathbb{Z}_q^n$ with high probability. To do this, the idea is to draw $m$-dimensional balls around the points $\bm{As} - j\bm{e}_0$ for all $\bm{s} \in \mathbb{Z}_q^n$ and $j \in \mathbb{Z}_r$ and then select a random point in one of these balls. Let $\mathrm{B}_m(0, R)$ be a ball of radius $R$ around $0$. To implement the above idea, we can represent $\mathrm{B}_m(0, R)$ using points of a fine grid. More precisely, $\mathrm{B}_m(0, R)$ is represented by $\tilde{\mathrm{B}}_m(0, R) = \frac{1}{L} \mathbb{Z}^m \cap \mathrm{B}_m(0, R)$ for a large integer $L$. We can efficiently prepare (an approximation of) the superposition
\begin{equation}
\label{equ:m-ball-sup}
\ket{\tilde{\mathrm{B}}_m(0, R)} = \frac{1}{\sqrt{\tilde{\mathrm{B}}_m(0, R)}} \sum_{\bm{x} \in \tilde{\mathrm{B}}_m(0, R)} \ket{\bm{x}}.
\end{equation}
Note that for any $\bm{y} \in \mathbb{Z}_q^m$ we have $\bm{y} + \tilde{\mathrm{B}}_m(0, R) = \tilde{\mathrm{B}}_m(\bm{y}, R)$. Tensoring the states in \eqref{equ:latt-sup} and \eqref{equ:m-ball-sup} and adding the third register to the fourth register we obtain the state
\[ \sum_{\bm{s} \in \mathbb{Z}_q^n} \sum_{j = 0}^{r - 1} \ket{j}\ket{\bm{s} + j\bm{s}_0}\ket{\bm{As} - j\bm{e}_0}\ket{\tilde{\mathrm{B}}_m(\bm{As} - j\bm{e}_0, R)}. \]
For an appropriate choice of the radius $R$, for each $\bm{s} \in \mathbb{Z}_q^n$ the intersection $\cap_{j} \tilde{\mathrm{B}}_m(\bm{As} - j\bm{e}_0, R)$ is large, while $\tilde{\mathrm{B}}_m(\bm{As} - j\bm{e}_0, R) \cap \tilde{\mathrm{B}}_m(\bm{As}' - j'\bm{e}_0, R) = \emptyset$ for any $\bm{s} \ne \bm{s}'$ and any $j, j'$. Therefore, if we measure the last register we obtain the state
\[ \sum_{j = 0}^{r - 1} \ket{j}\ket{\bm{s} + j\bm{s}_0}\ket{\bm{As} - j\bm{e}_0} \]
for some random $\bm{s} \in \mathbb{Z}_q^n$, with probability $O(1 - 1 / \ell)$ where $\ell = \poly(n\log q)$. The last register can be uncomputed using the transform $\ket{j}\ket{\bm{x}}\ket{\bm{y}} \mapsto \ket{j}\ket{\bm{x}}\ket{\bm{y} - \bm{Ax} + j\bm{b}_0}$ to obtain the state $\sum_{j = 0}^{r - 1} \ket{j}\ket{\bm{s} + j\bm{s}_0}$.
The above procedure produces an $\mathsf{EDCP}$ sample from $\mathsf{LWE}$ samples with a probability that is only polynomially close to $1$. This means we can obtain a polynomial number of $\mathsf{EDCP}$ sample from a polynomial number of $\mathsf{LWE}$ samples, and that is the most we can do. In other words, producing a super-polynomial number of $\mathsf{EDCP}$ samples from a super-polynomial number of $\mathsf{LWE}$ sample using the above procedure, can be done only with negligible probability. There is no known reduction from $\mathsf{LWE}$ to $\mathsf{EDCP}$ for which the sample conversion probability is, for example, subexponentially close to $1$.
\subsection{Subexponential number of samples}
\label{sec:hardness-subexp}
When the number of samples is subexponential, $\mathsf{EDCP}$ can be solved in time subexponential in $O(n\log q)$. This can be done using Kuperberg's algorithm \cite{kuperberg2005subexponential, kuperberg2011another}, which solves the hidden subgroup problem for the dihedral group $D_N$. The idea of the algorithm is to use a sieve on states of the form
\begin{equation}
\label{equ:dih-coset}
\frac{1}{\sqrt{2}}(\ket{0}\ket{x} + \ket{1}\ket{x + s}),
\end{equation}
where $x \in \mathbb{Z}_N$ is uniformly random, to recover the hidden shift $s \in \mathbb{Z}_N$. The state \eqref{equ:dih-coset} is called a dihedral coset state and the problem of recovering $s$, given such states, is called the Dihedral Coset Problem (DCP). The complexity of Kuperberg's algorithm is $2^{O(\sqrt{\log N})}$ for both time and space. Regev \cite{regev2004subexponential} improved the algorithm to use only $\poly(\log N)$ space at the cost of slightly increasing the running time to $2^{O(\sqrt{\log N \log\log N})}$.
Note that DCP is a special case of $\mathsf{EDCP}_{n, q, r}$ where $n = 1$, $q = N$ and $r = 2$. Conversely, $\mathsf{EDCP}$ can be reduced to vectorial variant of DCP which can be solved using a similar algorithm as in \cite{kuperberg2005subexponential}. We briefly explain the steps of the algorithm here.
\begin{theorem}
\label{thm:subexp-smpl}
Given $2^{O(\sqrt{n\log q})}$ samples, $\mathsf{EDCP}_{n, q, r}$ can be solved in time $2^{O(\sqrt{n\log q})}$.
\end{theorem}
\begin{proof}
Let $\bm{s} = (s_1, \dots, s_n)$. We will recover $s_n$, the rest of the $s_i$ can be recovered similarly. The proof proceeds in a sequence of simple reductions.
\begin{enumerate}[leftmargin = *, font = \bfseries]
\item From $\mathsf{EDCP}_{n, q, r}$ to DCP over $\mathbb{Z}_q^n$: Given the distribution $\mu_{\bm{s}, r}$ we can efficiently sample from the distribution $\mu_{\bm{s}, 2}$ using Lemma \ref{lem:self-rd}. A sample from $\mu_{\bm{s}, 2}$ is of the form
\[ \ket{\phi_{\bm{x}, 2}} = \frac{1}{\sqrt{2}}(\ket{0}\ket{\bm{x}} + \ket{1}\ket{\bm{x} + \bm{s}}), \]
where $\bm{x} \in \mathbb{Z}_q^n$ is uniformly random. This is a dihedral coset state over the group $\mathbb{Z}_q^n$.
\item From DCP over $\mathbb{Z}_q^n$ to DCP over $\mathbb{Z}_q$: measuring the second register of $(\mathds{1} \otimes \qft_{q^n}) \ket{\phi_{\bm{x}, 2}}$ we obtain the state
\begin{equation}
\label{equ:dih-coset-v}
\ket{\phi_{\bm{y}}} = \frac{1}{\sqrt{2}}(\ket{0} + \omega_q^{\lrang{\bm{y}, \bm{s}}}\ket{1})
\end{equation}
where $\bm{y} \in \mathbb{Z}_q^n$ is the outcome of the measurement and is uniformly random. Given two such states $\ket{\phi_{\bm{y}_1}}$ and $\ket{\phi_{\bm{y}_2}}$, we can compute the state
\[ \ket{\phi_{\bm{y}_1 - \bm{y}_2}} = \frac{1}{\sqrt{2}}(\ket{0} + \omega_q^{\lrang{\bm{y}_1 - \bm{y}_2, \bm{s}}}\ket{1}) \]
with probability $1 / 2$ by measuring the second register of $\textsc{cnot} \ket{\phi_{\bm{y}_1}} \ket{\phi_{\bm{y}_2}}$. If $\bm{y}_1$ and $\bm{y}_2$ had the first $k$ coordinates in common then $\bm{y}_1 - \bm{y}_2$ would have $0$ in the first $k$ coordinates. From this, we can perform a sieve operation: prepare many states of the form \eqref{equ:dih-coset-v}, then pair the states that have $\bm{y}$ with the same first $k$ coordinates, and then perform the above operation to produce new states with $\bm{y}$ that have the first $k$ coordinates zeroed out. Repeating the same process on the new states produces states with $\bm{y}$ that have first $2k$ coordinates equal to $0$, and so on. The final output of this process is a state $\ket{\phi_{\bm{y}}}$ where $\bm{y} = (0, \dots, 0, y)$, i.e., the state
\begin{equation}
\label{equ:dih-sve}
\ket{\phi_y} := \ket{\phi_{\bm{y}}} = \frac{1}{\sqrt{2}}(\ket{0} + \omega_q^{ys_n}\ket{1}).
\end{equation}
This is a DCP state over the group $\mathbb{Z}_q$.
\item Kuperberg for DCP over $\mathbb{Z}_q$: From the states \eqref{equ:dih-sve} $s_n$ can be recovered using Kuperberg's algorithm.
\end{enumerate}
To analyze the above algorithm, suppose we start with $q^\ell$ states. Since we are zeroing out $k$ coordinates at each stage, there are $n / k$ stages. At any stage, if there are $c \cdot q^k$ states, it can be shown, using a simple application of Lemma \ref{lem:hoeffding}, that at least $c / 8 \cdot q^k$ states survive the sieve operation. Therefore, to have $\Theta(q^k)$ states remaining in the last stage, we must have $q^\ell 8^{-n / k} \ge q^k$, hence $\ell \ge k + 3n / (k\log q)$. To minimize the right hand side, we take $k \in \Theta(\sqrt{n / \log q})$, and therefore, we can take $\ell \in \Theta(\sqrt{n / \log q})$.
\end{proof}
\begin{remark}
When $q = \poly(n)$, Kuperberg's algorithm is not very efficient for solving DCP over $\mathbb{Z}_q$. Instead, we can use a POVM called the Pretty Good Measurement (PGM) \cite{hausladen1994pretty}. Suppose we have prepared the states $\ket{\phi_{y_0}}, \dots, \ket{\phi_{y_t}}$, of the form \eqref{equ:dih-sve}, for some $t \ge \lceil \log q \rceil + 1$. The tensor product of these states is
\[ \ket{\psi} := \bigotimes_{j = 0}^{t - 1} \frac{1}{\sqrt{2}}(\ket{0} + \omega_q^{y_js_n}\ket{1}) = \frac{1}{\sqrt{2^t}} \sum_{x \in \{0, 1\}^t} \omega_q^{\alpha(x)s_n} \ket{x}, \]
where $\alpha(x) = x_0y_0 + \cdots + x_ty_t \bmod q$. Using PGM on the state $\ket{\psi}$, we can recover $s_n$ with constant probability \cite{bacon2005optimal}. The implementation of PGM, in this case, boils down to inverting the function $\alpha: \{0, 1\}^t \rightarrow \mathbb{Z}_q$, which can be done efficiently since $q = \poly(n)$.
\end{remark}
\section{Quantum Public-Key Cryptosystem}
\label{sec:public-key-enc}
A quantum public-key cryptosystem, similar to a classical system, consists of three algorithms:
\begin{itemize}[itemsep = 1pt]
\item $\mathsf{Gen}(1^n)$ generates a public-key $pk$ and a secret-key $sk$ based on the security parameter $n$.
\item $\mathsf{Enc}(pk, m)$ outputs a ciphertext $c$ for a given public-key $pk$ and message $m$.
\item $\mathsf{Dec}(sk, c)$ outputs a message $m$ for a given secret-key $sk$ and ciphertext $c$.
\end{itemize}
The output pair $(pk, sk)$ of the $\mathsf{Gen}$ algorithm for a quantum system consists of a quantum state and a classical state, respectively. In particular, the public-key $pk$ is a quantum state that is generated using a classical key $sk$. The algorithm $\mathsf{Enc}$ encrypts the message $m$, which is classical information, using the quantum state $pk$. The output $c$ of $\mathsf{Enc}$ is a quantum state. The algorithm $\mathsf{Dec}$ uses the key $sk$ to decrypt the quantum state into a classical message $m$.
For the security parameter $n$, we set the parameters for public key system as follows. We choose a prime $p = \poly(n)$ and set $q = p^s$ for some integer $s > 0$. We also set $r = p^{s'}$ where $s' < s$. The reason for these choices of parameters is that the resulting encryption scheme is simpler and more efficient. More generally, one could select $q$ to be a positive integer with $\poly(n)$-bounded prime factors and $r = p^{s'} \vert q$ to be a proper divisor where $p$ is prime. In what follows, we describe our cryptosystem for encrypting a one-bit message $b \in \{ 0, 1 \}$. Since there is only one prime $p$, we drop the parameter $p$ in \eqref{equ:new-dec1} for clarity.
\begin{description}[leftmargin = *]
\item [$\mathsf{Gen}(1^n)$:] Select $\bm{x}, \bm{s} \in \mathbb{Z}_q^n$ uniformly at random. Apply the transform $\qft_r \otimes \mathds{1}$ to the register $\ket{0} \ket{\bm{x}}$ to obtain the state $\ket{\psi} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \ket{j} \ket{\bm{x}}$. Apply the transform $A_{\bm{s}}: \ket{j}\ket{\bm{x}} \mapsto \ket{j}\ket{\bm{x} + j\bm{s}}$ to $\ket{\psi}$ to obtain the state
\[ \ket{\phi_{\bm{s}, r}(\bm{x}, 0)} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \ket{j} \ket{\bm{x} + j\bm{s}}. \]
Return the public-key, secret-key pair $(pk, sk) = (\ket{\phi_{\bm{s}, r}(\bm{x}, 0)}, \bm{s})$.
\item [$\mathsf{Enc}(pk = \rho_{\bm{s}, r, 0}, b \in \{ 0, 1 \})$:] Select $t \in \mathbb{Z}_r {\setminus} \{0\}$ uniformly at random. Apply the transform $U: \ket{j}\ket{\bm{y}} \mapsto \omega_p^{btj}\ket{j}\ket{\bm{y}}$ to $\rho_{\bm{s}, r, 0}$ to obtain the state
\begin{align*}
U \rho_{\bm{s}, r, 0} U^*
& = U \E_{\bm{x} \leftarrow \mathbb{Z}_q^n} \Big[ \ket{\phi_{\bm{s}, r}(\bm{x}, 0)} \bra{\phi_{\bm{s}, r}(\bm{x}, 0)} \Big] U^* \\
& = \E_{\bm{x} \leftarrow \mathbb{Z}_q^n} \Big[ U \ket{\phi_{\bm{s}, r}(\bm{x}, 0)} \bra{\phi_{\bm{s}, r}(\bm{x}, 0)} U^* \Big] \\
& = \E_{\bm{x} \leftarrow \mathbb{Z}_q^n} \Big[ \ket{\phi_{\bm{s}, r}(\bm{x}, bt)} \bra{\phi_{\bm{s}, r}(\bm{x}, bt)} \Big] \\
& = \rho_{\bm{s}, r, bt}.
\end{align*}
Return $\rho_{\bm{s}, r, bt}$.
\item [$\mathsf{Dec}(sk = \bm{s}, c = \rho_{\bm{s}, r, bt})$:] Apply the transform $S_{\bm{s}}: \ket{j}\ket{\bm{y}} \mapsto \ket{j}\ket{\bm{y} - j\bm{s}}$ to $\rho_{\bm{s}, r, bt}$. Discard the second register. Apply $\qft_r$ to the resulting state and measure. If the measurement result is 0 then output 0, otherwise output 1.
\end{description}
\begin{lemma}[Correctness]
For any bit $b \in \{ 0, 1 \}$ and all outputs $(\bm{s}, \rho_{\bm{s}, r, 0})$ of $\mathsf{Gen}$, we have
\[ \Pr [ \mathsf{Dec}(\bm{s}, \mathsf{Enc}(\rho_{\bm{s}, r, 0}, b)) = b ] = 1. \]
\end{lemma}
\begin{proof}
Given the ciphertext $\rho_{\bm{s}, r, tb}$, the decryption steps are as follows
\begin{align*}
\rho_{\bm{s}, r, b}
& \mapsto \E_{\bm{x} \leftarrow \mathbb{Z}_q^n} \Big[ S_{\bm{s}} \ket{\phi_{\bm{s}, r}(\bm{x}, bt)} \bra{\phi_{\bm{s}, r}(\bm{x}, bt)} S_{\bm{s}}^* \Big] \tag{apply $S_{\bm{s}}$} \\
& = \E_{\bm{x} \leftarrow \mathbb{Z}_q^n} \bigg[ \frac{1}{r} \sum_{k, j = 0}^{r - 1} \omega_p^{bt(j - k)}\ket{k}\ket{\bm{x}} \bra{j}\bra{\bm{x}} \bigg] \\
& = \frac{1}{r} \sum_{k, j = 0}^{r - 1} \omega_p^{bt(j - k)}\ket{k}\bra{j} \otimes \E_{\bm{x} \leftarrow \mathbb{Z}_q^n} \Big[ \ket{\bm{x}}\bra{\bm{x}} \Big] \\
& \mapsto \frac{1}{r} \sum_{k, j = 0}^{r - 1} \omega_p^{bt(j - k)}\ket{k}\bra{j} \tag{discard the second register} \\
& \mapsto \qft_r \frac{1}{r} \sum_{k, j = 0}^{r - 1} \omega_p^{bt(j - k)}\ket{k}\bra{j} \qft_r^* \tag{apply quantum Fourier transform}\\
& = \ket{btr/p} \bra{btr/p}
\end{align*}
If $b = 0$ then the above state is $\ket{0}\bra{0}$, otherwise it is $\ket{tr / p}\bra{tr / p} \ne \ket{0}\bra{0}$.
\end{proof}
\paragraph{Discussion.}
The above encryption scheme can be naturally based on the original decision-$\mathsf{EDCP}$ (Definition \ref{def:d-edcp}) as well. The resulting scheme, however, is not as efficient. To see this, suppose public keys are generated using the $\mathsf{Gen}$ algorithm above. Bob encrypts a bit $b \in \{0, 1\}$ as follows: if $b = 0$ then he outputs the public key as the ciphertext. If $b = 1$ then he measures the first register of the public key to obtain a state $\ket{j}\ket{\bm{x}}$, for uniformly random $(j, \bm{x}) \in \mathbb{Z}_r \times \mathbb{Z}_q^n$, and outputs this state as the ciphertext. Distinguishing between the encryption of $0$ and $1$ is then equivalent to solving the decision-$\mathsf{EDCP}$. Now, if Alice runs the above $\mathsf{Dec}$ algorithm on the ciphertext she gets two possible outputs depending on the value of $b$:
\begin{itemize}
\item When $b = 0$, the output is always $0$, since the last state obtained in the algorithm is always $\ket{0}\bra{0}$ and so the measurement outcome is $0$.
\item When $b = 1$, the output is $1$ with probability $1 - 1 / r$. This is because the last state obtained in the algorithm, just before the last measurement, is $\qft_r \ket{j}\bra{j} \qft_r^*$.
\end{itemize}
To decrease the above (rather large) decryption error down to, say, $2^{-\Omega(n)}$, the $\mathsf{Dec}$ algorithm has to be repeated $\Omega(n / \log r)$ times. In the quantum setting, that means Alice has to have access to $\Omega(n / \log r)$ copies of the ciphertext, which in turn means Bob needs the same number of copies of the public key to generate the ciphertexts. This is equivalent to saying that for an encryption-decryption round with negligible error probability, the size of the public key increases by a factor of $\Omega(n / \log r)$. This scheme then has no advantage over a classical $\mathsf{LWE}$-based encryption scheme.
\subsection{Circuits}
All three algorithms $\mathsf{Gen}, \mathsf{Enc}, \mathsf{Dec}$ in the above public-key cryptosystem are very easy to implement. In the following, we briefly describe the circuits for these algorithms. Since all the arithmetic unites used in the circuits are already known, we will not give a gate-level design for them.
Figure \ref{fig:gen-circuit} shows the key generation circuit. The gate $\qft_r$ is the quantum Fourier transform over $\mathbb{Z}_r$, and the gate $A_{\bm{s}}$ is the multiply-add transform $\ket{j}\ket{\bm{x}} \mapsto \ket{j}\ket{\bm{x} + j\bm{s}}$.
\begin{figure}[h]
\centering
\begin{quantikz}[thin lines]
\lstick{$\ket{0}$} & \gate{\qft_r} & \ctrl{1} & \qw \\
\lstick{$\ket{\bm{x}}$} & \qw & \gate{A_{\bm{s}}} & \qw
\end{quantikz}
\caption{The circuit for $\mathsf{Gen}$}
\label{fig:gen-circuit}
\end{figure}
For the encryption algorithm, we need to implement the transform $\ket{j}\ket{\bm{y}} \mapsto \omega_p^{btj}\ket{j}\ket{\bm{y}}$. We start by preparing the state
\[ (\mathds{1} \otimes \qft_p) \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \ket{j} \ket{\bm{x} + j\bm{s}} \otimes \ket{1} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \ket{j} \ket{\bm{x} + j\bm{s}} \frac{1}{\sqrt{p}} \sum_{z \in \mathbb{Z}_p} \omega_p^z \ket{z}. \]
Then we apply the transform $T_{bt}: \ket{j}\ket{\bm{y}}\ket{z} \mapsto \ket{j}\ket{\bm{y}}\ket{z - jbt}$ to obtain the state
\[ \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \ket{j} \ket{\bm{x} + j\bm{s}} \frac{1}{\sqrt{p}} \sum_{z \in \mathbb{Z}_p} \omega_p^z \ket{z - jbt} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} \omega_p^{btj} \ket{j} \ket{\bm{x} + j\bm{s}} \frac{1}{\sqrt{p}} \sum_{z \in \mathbb{Z}_p} \omega_p^z \ket{z}. \]
Finally, we measure the last register to obtain the desired state. Figure \ref{fig:enc-circuit} shows the encryption circuit.
\begin{figure}[h]
\centering
\begin{quantikz}[thin lines]
\lstick{$\ket{j}$} & \qw & \ctrl{2} & \qw & \qw \\
\lstick{$\ket{\bm{y}}$} & \qw & \qw & \qw & \qw \\
\lstick{$\ket{1}$} & \gate{\qft_p} & \gate{T_{bt}} & \meter{} & \qw
\end{quantikz}
\caption{The circuit for $\mathsf{Enc}$.}
\label{fig:enc-circuit}
\end{figure}
Figure \ref{fig:dec-circuit} shows the decryption circuit. The gate $\qft_r$ is the quantum Fourier transform over $\mathbb{Z}_r$, and the gate $S_{\bm{s}}$ is the multiply-subtract operation $\ket{j}\ket{\bm{x}} \mapsto \ket{j}\ket{\bm{x} - j\bm{s}}$.
\begin{figure}[h]
\centering
\begin{quantikz}[thin lines]
\lstick{$\ket{j}$} & \ctrl{1} & \gate{\qft_r} & \meter{} & \qw \\
\lstick{$\ket{\bm{y}}$} & \gate{S_{\bm{s}}} & \meter{} & \qw & \qw
\end{quantikz}
\caption{The circuit for $\mathsf{Dec}$.}
\label{fig:dec-circuit}
\end{figure}
\subsection{A very efficient instantiation}
Although our cryptosystem is efficient even for a super-polynomial modulus $q = p^s$ and any $\poly(n)$-bounded prime $p$, it can be made more efficient by choosing a $\poly(n)$-bounded $q$ and a small prime $p$. In particular, we can choose $p = 2$ and $q = 2^s$ such that $q = \poly(n)$. In this case, we choose $r = 2^{s'}$ where $s' \ll s$.
For the above parameters, we have $\omega_p = -1$. In the encryption algorithm, since $t \ne 0$, the only choice for $t$ is $t = 1$ and, therefore, random number generation is not required. For an input bit $b$ the ciphertext state is
\[ \ket{\phi_{\bm{s}, r}(\bm{x}, b)} = \frac{1}{\sqrt{r}} \sum_{j = 0}^{r - 1} (-1)^{bj} \ket{j}\ket{\bm{x} + j\bm{s}}. \]
The phase $(-1)^{bj}$ is much simpler to compute than the more general phase $\omega_p^{btj}$. In particular, the quantum Fourier transform $\qft_p$ is now the Hadamard transform $H: \ket{x} \mapsto (\ket{0} + (-1)^x \ket{1}) / \sqrt{2}$, and the transform $T_{bt}$ is now $T_b: \ket{j}\ket{\bm{y}}\ket{z} \mapsto \ket{j}\ket{\bm{y}}\ket{z \oplus (jb \bmod 2)}$. Figure \ref{fig:enc-circuit-2} shows the new encryption circuit.
\begin{figure}[h]
\centering
\begin{quantikz}[thin lines]
\lstick{$\ket{j}$} & \qw & \ctrl{2} & \qw & \qw \\
\lstick{$\ket{\bm{y}}$} & \qw & \qw & \qw & \qw \\
\lstick{$\ket{1}$} & \gate{H} & \gate{T_b} & \meter{} & \qw
\end{quantikz}
\caption{The circuit for $\mathsf{Enc}$.}
\label{fig:enc-circuit-2}
\end{figure}
Let us briefly analyze the complexity of this scheme. The $\mathsf{Gen}$ algorithm requires random generation, scalar multiplication and addition in $\mathbb{Z}_q^n$. These can be done at the cost of $O(n\log q \log\log q)$ qubit operations. The quantum Fourier transform over $\mathbb{Z}_r$ can be done in $O(\log r\log\log r)$ qubit operations \cite{hales2000improved}. The $\mathsf{Enc}$ algorithm takes $O(1)$ since the $T_b$ operation takes $O(1)$. The $\mathsf{Dec}$ algorithm has the same complexity as the $\mathsf{Gen}$ algorithm.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,234 |
Q: Show that $\lim \frac{1}{n}\sum_{k=1}^n f\left(\frac{k}{n}\right)=\int_0^1f$ I want to show that if $f$ is integrable on the interval $[0,1]$ then $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n f\left(\frac{k}{n}\right)=\int_0^1f$.
I am using the definition of integrability in the sense of Riemann-Stieltjes:
$$\forall\epsilon>0:\exists P:\forall Q\geq P:\left|S(Q_n,f,I)-\int_0^1f\right|<\epsilon$$
Where $P$ and $Q$ are partitions of $[0,1]$.
I thought about using the definition for the partitions:
$$Q_n=\left(0,\frac1n,\frac2n,\cdots,\frac{n-1}n,1\right)$$
$$c_k=\frac kn$$
Then we have:
$$\left|S(Q_n,f,I)-\int_0^1f\right|=\left|\frac1n\sum_{k=1}^nf\left(\frac kn\right)-\int_0^1f\right|$$
The problem is that I don't know how I can show that this tends to zero since a $P$ appears in the definition of integral and I don't know what to do with it.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,291 |
\section{introduction}
The idea of parametric electron pump was first addressed by
Thouless\cite{thouless}, which is a mechanism that at zero bias a dc
current is pumped out by periodically varying two or more system
parameters. Over the years, there has been intensive research
interest concentrated on parametric electron
pump.\cite{thouless1,niu,nazarov,shutenko,chu} Electron pump has
been realized on quantum dot setup\cite{aleiner1} consisting of
AlGaAs/GaAs heterojunction\cite{switkes}. Low dimensional
nanostructures, such as carbon nanotubes (CNT)\cite{ydwei2,levitov}
and graphene\cite{graphene1,graphene2} were also proposed as
potential candidates. Investigation on electron pump also triggers
the proposal of spin pump\cite{watson,benjamin,chao}, in which a
spin current is induced by various means.
At low pumping frequency limit, the variation of the system is
relatively slow than the process of energy
relaxation\cite{switkes1}. Hence the system is nearly in equilibrium
and we could deal with the adiabatic pump by equilibrium methods. On
the other hand, non-adiabatic pump refers to the case that pumping
process is operated at a finite frequency. In the non-adiabatic
regime, non-equilibrium transport theory should be employed.
Theoretical methods adopted in the research field includes
conventional scattering matrix
theory\cite{brouwer1,brouwer2,andreev}, Floquet scattering
matrix\cite{kim1,buttiker1} and non-equilibrium Green's function
(NEGF) method\cite{baigeng1,baigeng2}, as well as other
methodologies to both adiabatic \cite{aharony} and
non-adiabatic\cite{sen} electron pumps.
The electron pump is a phase coherent phenomenon, since the cyclic
variation of system parameters affects the phase of wave function
with respect to its initial value\cite{zhou}. As a result, it is
very sensitive to the external magnetic field. In the experimental
work of Switkes \textit{et.al}\cite{switkes}, at adiabatic limit,
the pumped current of a open quantum dot system with certain spatial
symmetry is showed invariant upon reversal magnetic field. The
conclusion was confirmed by theoretical works using Floquet
scattering matrix method\cite{aleiner2,kim,buttiker2}. Later it was
numerically suggested\cite{csli} that the pumped spin current also
has certain spatial symmetries.
\begin{figure}[tp]
\centering
\includegraphics[width=\columnwidth]{fig1.eps}
\caption{Sketch of the spatial reflection symmetries: (a)
instantaneous up-down (IUD); (b) instantaneous left-right (ILR);
(c)instantaneous inversion (IIV); (d) general up-down (GUD); (e)
general left-right (GLR); (f) general inversion (GIV). Shadow
rectangular indicates for the pumping region and dark gray blocks
stand for potential barriers defining the spatial symmetry of the
system. The pumping potentials are right on top of these confining
potentials.}\label{fig1}
\end{figure}
In this paper, we aim to numerically investigate the pumped current
in the presence of magnetic field. Both adiabatic and non-adiabatic
pumped current are calculated. We focus on the nature of pumped
current for the mesoscopic systems with resonance with complete
transmission and anti-resonance with complete reflection. We find
that the behaviors of adiabatic and non-adiabatic pumped current are
very different. In the non-adiabatic regime, the pumped current is
nonzero at resonance while it is zero at anti-resonance. However,
the adiabatic pumped current is always zero regardless of types of
resonance. Since there is no external driving force, the direction
of current depend only on the system parameters. Our numerical
results show that the adiabatic pumped current reverses it sign at
the resonance or anti-resonance. For non-adiabatic pumped current,
the sign reversal depends on the lifetime of the resonant states.
The non-adiabatic pumped current change the sign near the resonant
point only when the lifetime is short. We also study the pumped
current as a function of magnetic field. We find that as the system
enters the quantum Hall regime with increasing magnetic field
strength, the pumping current vanishes. Since in quantum Hall regime
electron wave function appears as edge state, it will circumvent the
confining potentials shown in Fig.\ref{fig1}. Pumping potentials
overlapping in space with confining potentials present no modulation
on the electron wave function during the variation period. Hence
there is zero pumped current in the Quantum Hall regime. We also
examine the pumped current and its relation with other system
parameters such as pumping frequency and pumping potential
amplitude. Finally we also investigate the symmetry properties of
the pumped electron current of systems with certain spatial symmetry
in the presence of magnetic field by the Green's function method.
Six spatial symmetries studied in Ref.\cite{kim} are considered,
both at the adiabatic and non-adiabatic cases, which are
instantaneous up-down (IUD), left-right (ILR), and inversion
symmetries (IIV) and the corresponding non-instantaneous/general
up-down (GUD), left-right (GLR), and inversion symmetries (GIV),
respectively (see Fig.\ref{fig1}). The electron pump is driven by
periodical modulation of potentials which share the same spatial
coordinates with the confining potentials which preserve reflection
symmetry of the system. Most of our numerical results agree with the
conclusions from Floquet scattering theory\cite{aleiner2,kim},
except for the general inversion symmetry (GIV) ( setup $f$ in
Fig.\ref{fig1} ). In contrast with the theoretical prediction that
the adiabatic pumped current $I^{ad} \approx 0$ for this spatial
symmetry, our numerical calculation shows that the pumped current is
finite and further investigation reveals that there is an
approximate symmetry relation of the current as setup $e$ at small
magnetic field, which is the experimental setup\cite{switkes}. The
conclusion suggests that the general left-right (GLR) spatial
symmetry has a rather strong impact on the pumped current, which
leads to the quite accurate relation $I(B)=I(-B)$ in the
experimental finding. The result also holds for the non-adiabatic
case.
Our paper is organized as follows. In the next section we will
describe the numerical method first followed by the numerical
results and discussions in section III. Finally, conclusion is given in section IV.
\section{theoretical formalism and methodology}
We consider a quantum dot system consisting of a coherent scattering
region and two ideal leads which connect the dot to electron
reservoirs. The whole system is placed in x-y plane and a magnetic
field is applied. The single electron Hamiltonian of the scattering
region is simply
\begin{eqnarray}
H=\frac{(\textbf{p}+ e \textbf{A}/c )^2}{2m^*}+V(x,y,t) \nonumber
\end{eqnarray}
where \textbf{A} is the vector potential of the magnetic field. Here
the magnetic field is chosen to be along z-direction with
\textbf{B}=(0, 0, B). The vector potential has only x-component in
the Landau gauge , \textbf{A}=(-By, 0, 0).
\begin{eqnarray}
H=H_0+V_p \nonumber
\end{eqnarray}
where
\begin{eqnarray}
H_0=(-i\hbar \frac{\partial}{\partial x}-\frac{e}{c} By)^2+(-i\hbar
\frac{\partial}{\partial y})^2+V_0(x,y) \nonumber
\end{eqnarray}
and $V_p$ is a time-dependent pumping potential given by
$V_p(x,y,t)=\sum_j V_{j}(x,y) \cos(\omega t + \phi_{j})$.
For the adiabatic electron pump, the average current flowing through lead
$\alpha$ due to the slow variation of system parameter $V_j$ in one
period is given by\cite{brouwer1}
\begin{eqnarray}
I_\alpha & = & \frac{1}{\tau} \int_0^\tau dt
\frac{dQ_\alpha(t)}{dt} \notag\\
& = & \frac{q\omega}{2 \pi} \int_0^\tau dt \sum_j \frac{dN_\alpha}{dV_j}
\frac{dV_j}{dt} \label{eq1}
\end{eqnarray}
where $\tau = 2\pi / \omega$ is the variation period of parameter
$V_j$ and $\omega$ the corresponding frequency. For simplicity we
take $\omega = 1$ in the adiabatic case. $\alpha$= L or R labels the
lead. The so-called emissivity $dN_\alpha / dV_j$ is conventionally
defined in terms of the scattering matrix $S_{\alpha \beta}$
as\cite{brouwer1,ydwei1}
\begin{eqnarray}
\frac{dN_\alpha}{dV_j}= \int \frac{dE}{2 \pi} ( -\partial_E f)
\sum_{\beta} \text{Im} \frac{\partial S_{\alpha \beta}}{\partial
V_j} S_{\alpha \beta}^{\ast} \label{dnde}
\end{eqnarray}
In the language of Green's function, the above equation is
equivalent to the following form\cite{baigeng2}
\begin{eqnarray}
I_\alpha = q \int_0^{\tau} dt \int \frac{dE}{2 \pi} ( -\partial_E f)
\text{Tr} \left[ \Gamma_{\alpha} G^r \frac{dV_p}{dt} G^a
\right]\label{eq2}
\end{eqnarray}
where the instantaneous retarded Green's function $G^r$ in
real space is defined as
\begin{eqnarray}
G^r(E,t)=(E-H(t)-\Sigma^r)^{-1}
\end{eqnarray}
where $\Sigma^r$ is the self-energy due to the leads.
Whereas for non-adiabatic pump at finite frequency, the pumped
current up to the second order in pumping potential is derived as \cite{baigeng1}
\begin{eqnarray}
&&I_\alpha = -iq \sum_{jk=1,2} \int
\frac{dE}{8 \pi} \text{Tr} [ \Gamma_{\alpha} G^{r}_0 V_j((f-f_{-})(G^{r-}_0 - \notag \\
& & G^{a-}_0)e^{i \phi_{kj}} + (f-f_{+})(G^{r+}_0-G^{a+}_0)e^{-i
\phi_{kj}}) V_k G^{a}_0 ] \label{eq3}
\end{eqnarray}
where $\Gamma_{\alpha}$ is the linewidth function of lead $\alpha$
defined as $\Gamma_{\alpha}=i[\Sigma_{\alpha}^{r}
-\Sigma_{\alpha}^{a}]$; $f=f(E)$ and $f_{\pm}=f(E \pm \omega)$ are
the Fermi distribution functions; $\phi_{kj} = \varphi_j -
\varphi_k$ is the phase difference between the two pumping
potentials. Here $G^{0r} = G^{0r}(E)$ and $G^{0r \pm} = G^{0r \pm}(E
\pm \omega)$ are the retarded Green's functions where there is no
pumping potentials. In the following section, we will use Eqs.
(\ref{eq2}) and (\ref{eq3}) to carry on numerical investigations and
all the numerical work are done at the zero temperature. In the
calculation we consider a square quantum dot with size $0.7 \mu m
\times 0.7 \mu m$, of the same order as in the experimental
setup.\cite{switkes} Two open leads with the same width connect the
dot to the electron reservoirs. The quantum dot is then discretized
into a $40 \times 40$ mesh. Hopping energy $t=\hbar^2/2 m^* a^2$
sets the energy scale with $a$ the lattice spacing and $m^*$ the
effective mass of electrons in the quantum dot. Dimensions of other
relevant quantities are then fixed with respect to $t$.
\section{numerical results and discussion}
In this section, numerical results will be presented. To test our
numerical method, we first study the reflection symmetry of pumped current on inverse
of magnetic field. Other properties of the current will be discussed
in the second sub-section.
To check the symmetry of the pumped current we assume $V_0=\sum_{j=1,2} V_j(x,y)$
in our numerical calculation. In Fig.\ref{fig1}, we have schematically plotted 6
setups with different spatial symmetries of interest. In setups $a, b$, and $c$, the
spatial symmetries are kept at any moment during the pumping
period. Hence we label them instantaneous up-down (IUD),
instantaneous left-right (ILR), and instantaneous inversion (IIV)
symmetries, respectively. On the other hand, symmetries
are broken during the whole pumping cycle except when
$\phi_{jk}=n\pi$ in setup $d, e$, and $f$. They are correspondingly
labeled as general up-down (GUD), general left-right (GLR), and
general inversion (GIV) symmetries. All potential profiles locate at
the boundary of the pumping region, i.e., the first and/or last layer
in the discrete lattice (see the dark gray region).
\subsection{Symmetry of pumped current}
Before the presenting numerical results, we would like to point out
that for setup $c$ with instantaneous inversion symmetry (IIV), the
theoretical predictions \cite{aleiner2,kim} and our numerical
calculations give the same result: the pumped current is exact zero
at both adiabatic and non-adiabatic cases that is independent of $B$
and $\phi_{12}$. The phenomena can be straightforwardly understood
by Floquet scattering matrix theory\cite{kim}. In IIV setup, it is
obvious that the transmission coefficient of an electron traveling
from the left lead to the right $T_{R \leftarrow L}$ is always equal
to that of an electron moving in the opposite direction, $T_{L
\leftarrow R}$, i.e.,
\begin{eqnarray}
T_{R \leftarrow L} = T_{L \leftarrow R} \nonumber
\end{eqnarray}
From the Landauer-B\"{u}ttiker formula, the electric current along
the left to right region is given by
\begin{eqnarray}
I_{R \leftarrow L}=\frac{2e}{h}\int dE T_{R \leftarrow L}(E) f(E)
\nonumber
\end{eqnarray}
while $I_{L \leftarrow R}$ is defined in a similar way. Then the
pumped current through the left lead is defined
as\cite{buttiker2,kim}
\begin{eqnarray}
I_L = I_{L\rightarrow R} - I_{R\rightarrow L}=0
\end{eqnarray}
The conclusion holds for any particular moment, which means that
there will be no pumped current at all. Hence in the following we
will not discuss the case of setup $c$.
First we examine the relation between the adiabatically pumped
current $I^{ad}$ and phase difference $\phi_{12}$ of the pumping
potentials calculated from Eq.(\ref{eq2}). A sinusoidal behavior is
observed at a relatively small pumping amplitude $V_p$=0.5 for all
setups in Fig.\ref{fig1}. The sinusoidal form of $I^{ad}(\phi_{12})$ represents a
generic property of adiabatic electron pump at small $V_p$. Driven
by the cyclic variation of two time-dependent system parameters, the
pumped current is directly related to the area enclosed by the
parameters in parametric space. At small pumping amplitudes, the
leading order of $I^{ad}$ is proportional to the phase difference
between the pumping potentials, $I^{ad} \propto V_p \sin \phi_{12}$
\cite{brouwer1}. However, the relation doesn't hold for large
pumping amplitude. To demonstrate this we have calculated current
pumping through the setup with symmetry ILR at a large potential
$V_p$=1.6. As shown in Fig.\ref{fig2} the
sinusoidal relation is clearly destroyed. Except for this difference arising
from the pumping amplitude $V_p$, there is a general antisymmetry
relation between the pumped current and the phase difference
$\phi_{12}$ for all setups:
$I(\phi_{12})=-I(-\phi_{12})$. Naturally, $I(\phi_{12}=n \pi)=0$.
This is understandable since two simultaneously varying parameters enclose
a line rather than an area in the parametric space, the pumped current vanishes.
This result, however, does not hold for non-adiabatic case where the frequency
gives additional dimension of parametric space.
Another result of interest is that, in contrast to the theoretical
prediction $I^{ad} \approx 0$ for the setup of GIV symmetry \cite{kim}, the
pumped current from the setup of GIV is finite and has the same order of magnitude as that of GLR
symmetry.
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{fig2.eps}
\caption{The adiabatically pumped current as a function of phase
difference $\phi_{12}$ for different spatial symmetries of the
pumping system at pumping amplitude $V_p=0.5$. Inset: pumped current
of system with symmetry ILR at $V_p=1.6$. Other system parameters:
$E_F=0.62$, $B=0.001$, $V_0=1$. }\label{fig2}
\end{figure}
\begin{figure}[tbhp]
\centering
\includegraphics[width=\columnwidth]{fig3.eps}
\caption{Panel (a): The adiabatically pumped current versus magnetic
field strength $B$ for system symmetries $a$(IUD), $b$(ILR), and
$d$(GUD). Curve for IUD is offset by -0.004 for a compact
illustration. Panel (b): $I^{ad}$ vs $B$ for spatial symmetry GLR,
GIV, and three intermediate setups. Calculation parameters:
$E_F=0.62$, $\phi_{12}=\pi/2$, $V_0=1$, $V_p=0.5$. }\label{fig3}
\end{figure}
Fig.\ref{fig3} plots the pumped current versus magnetic
field strength $B$ at phase difference $\phi_{12}= \pi/2$, where the
magnitude of $I^{ad}(\phi_{12})$ is maximized. In the upper panel
(a) of Fig.\ref{fig3}, we see that current is either even or odd
function of magnetic field strength $B$ for symmetries IUD, ILR, and
GUD, $I(B) = \pm I(-B)$ which agree with the theoretical
predictions.\cite{aleiner2,kim} In panel (b), it is clear that the pumped
current is invariant upon the reversal of magnetic field for
GLR.\cite{aleiner2,kim} The system
with GIV symmetry shows an approximation relation $I(B) \approx
I(-B)$ only at small $B$, which is similar to that of GLR
symmetry. This does not agree with the theoretical prediction.\cite{kim}
To further investigate the relation $I(B) \approx
I(-B)$ we studied three intermediate
setups between GIV and GLR. In panels (e) and (f) of Fig.\ref{fig1}, the
length of the pumped potential profile is fixed as 20 both for $V_1$ and
$V_2$ in the system with $40 \times 40$ mesh. Then we shift down
the potential profile $V_2$ of the
GIV symmetry 5 lattice spacing each time. After 4 shifts, the system
changes from GIV to GLR symmetry and in this process three
intermediate systems are generated. Numerical results shown in
panel (b) of Fig.\ref{fig3} suggest that all these setups have the
relation $I(B) \approx I(-B)$ at small magnetic fields, although there
are no spatial reflection symmetries in these systems. The
closer system to the GLR symmetry, the larger the magnetic field
for this relation. These results
suggest that the general left-right symmetry is a rather strong spatial
symmetry and even a rough setup (cyan-down-triangle curve in panel
(b)) can lead to an accurate invariant relation of pumped current, at
least for small magnetic field.
Recall the experimental result \cite{switkes} of an adiabatic pump
where the experimental setup can not have precise GLR symmetry,
but an accurate relation $I(B) = I(-B)$ was still
achieved. In addition, the amplitude of pumped current in this setup
with GLR symmetry is relatively high, compared with other symmetries.
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{fig4.eps}
\caption{Panel (a): Non-adiabatically pumped current as a function of
phase difference $\phi_{12}$ at a fixed magnetic field $B=0.001$ for
spatial symmetries IUD, ILR, and GIV. Blue curve with up-triangle
for GIV is multiplied by a factor of $0.1$. Panel (b): $I^{nad}$
versus magnetic field strength $B$ for the above three setups with
$\phi_{12} = \pi/2$. Blue curve with up-triangle for GIV is offset
by $-0.8$. Panel (c): The pumped current for system setups GUD and
GLR. Curves for GUD are multiplied by $0.1$. Calculation parameters:
$E_F=0.62$, $V_0=1$, $V_p=40$. }\label{fig4}
\end{figure}
Now we turn our attention to the non-adiabatic electron pump with
finite pumping frequency. The numerical results are presented in
Fig.\ref{fig4}. One of the major differences between adiabatic and
non-adiabatic pump is that a non-adiabatic pump can operate with
only one system parameter, since the finite pumping frequency
supplies one extra degree of freedom and it could act as another
pumping parameter. In the experiment\cite{switkes} it was found that
$I(\phi_{12}=0) \neq 0$. Later a theoretical work\cite{baigeng1}
attributed this phenomenon as a consequence of photon-assisted
processes and it is a nonlinear transport feature of non-adiabatic
electron pump. In our numerical results, we also found that
$I^{nad}(\phi_{12}= n\pi) \neq 0$ is a general property of the
pumped current, except for systems with spatial symmetries IIV or
GIV. Although the pumping frequency $\omega$ can play the role of a
variation parameter, the pumped current in the system with symmetry
IIV is always zero. For the setup with GIV symmetry, we see from
panel (a) of Fig.\ref{fig4} that the pumped current obeys an
antisymmetric relation with phase difference: $I^{nad}(\phi_{12})=
-I^{nad}(-\phi_{12})$. $I^{nad}(\phi_{12}= n\pi) = 0$ is a natural
result of this antisymmetry relation. Combining with the result from
the adiabatic case (Fig.\ref{fig2}), we see that this antisymmetry
relation between pumped current and phase difference $\phi_{12}$ is
a general feature of the GIV symmetry. Besides, from panel (b) of
Fig.\ref{fig4} we found that, $I^{nad}$ for GIV system at a fixed
phase $\phi_{12}= \pi /2$ shows $I^{nad}(B) \approx I^{nad}(-B)$ at
small magnetic field. This approximate symmetry relation can not be
obtained theoretically. Our results confirm the theoretical
predictions on the parity of pumped current on reversal of magnetic
field for setups IUD and ILR\cite{kim}, which are respectively
$I(B)=I(-B)$ and $I(B)=-I(-B)$ (see panel (b)). However, it doesn't
hold for GUD and GLR in panel (c). In this case, one can only get
the relations $I(B,\phi)=I(-B,-\phi)$ for GUD and
$I(B,\phi)=-I(-B,-\phi)$ for GLR\cite{kim}. When the two pumping
potentials operate in phase or out of phase ($\phi_{12} = n\pi$),
they reduce to a simple version: $I(B)=I(-B)$ for GUD and
$I(B)=-I(-B)$ for GLR, which are the same for IUD and ILR at
$\phi_{12} = n\pi$. It is worth
mentioning that these two relations are in contrary to the adiabatic
case where $\phi_{12} \neq n\pi$.
We collect the results and summarize these conclusions drawn from
both adiabatic and nonadiabatic pumps and there are shown in
Table.\ref{table1} in detail.
\begin{table*}
\begin{tabular}{|c|c|c|c|c|}
\hline
& \multicolumn{2}{c}{adiabatic pump} & \multicolumn{2}{c}{nonadiabatic pump} \\ \hline
& $\phi_{12}= n\pi$ & $\phi_{12}\neq n\pi$ & $\phi_{12}= n\pi$ & $\phi_{12}\neq n\pi$ \\ \hline
IUD & ${I=0}^{\diamond}$ & ${I(B)=I(-B)}^{\diamond}$ &
${I(B)=I(-B)}^{\diamond}$ & ${I(B)=I(-B)}^{\diamond}$ \\ \hline
ILR & ${I=0}^{\diamond}$ & ${I(B)=-I(-B)}^{\diamond}$ &
${I(B)=-I(-B)}^{\diamond}$ & ${I(B)=-I(-B)}^{\diamond}$ \\ \hline
IIV & ${I=0}^{\diamond}$ & ${I=0}^{\diamond}$ &
${I=0}^{\diamond}$ & ${I=0}^{\diamond}$ \\ \hline GUD &
${I=0}^{\diamond}$ & ${I(B)=-I(-B)}^{\diamond}$ &
${I(B)=I(-B)}^{\diamond}$ & ${I(B,\phi)=I(-B,-\phi)}^{\diamond}$
\\ \hline GLR & ${I=0}^{\diamond}$ & ${I(B)=I(-B)}^{\diamond}$ &
${I(B)=-I(-B)}^{\diamond}$ & ${I(B,\phi)=-I(-B,-\phi)}^{\diamond}$
\\ \hline \multirow{2}{*}{GIV} & ${I=0}^{\diamond}$ &
${I \approx 0}^{\triangleright}$ & ${I=0}^{\diamond}$ &
${I(B,\phi)=-I(B,-\phi)}^{\diamond}$ \\ & & ${I(B) \approx
I(-B)}^{\ast}$ & & ${I(B) \approx I(-B)}^{\ast}$ \\ \hline
\end{tabular}
\caption{Symmetry of the pumped currents on inversion of the
magnetic field for both adiabatic and nonadiabatic electron pumps. \\
$\quad$ $\diamond$ stands for the theoretical prediction
from Ref.\onlinecite{kim} confirmed by our numerical calculation. \\
$\triangleright$ represents theoretical relation without numerical sustainment. \\
$\ast$ corresponds to our new finding in contrast to the theoretical
prediction. } \label{table1}
\end{table*}
\subsection{Transport properties of the pumped current}
In the last section we have concentrated on the symmetry of the
pumped current with magnetic field $B$ and phase difference
$\phi_{12}$ as the variables. Now we study the effect of other
system parameters on the pumped current. Numerical calculations were
performed on a system with instant L-R symmetry (ILR), in which
widths of the four potential barriers are kept equal. The numerical
results are plotted in Fig.\ref{fig5}, Fig.\ref{fig6} and
Fig.\ref{fig7}.
\begin{figure}[tbhp]
\centering
\includegraphics[width=\columnwidth]{fig5.eps}
\caption{Panel (a) and (b): The pumped current as well as
transmission coefficient as a function of Fermi energy at static
potential height $V_0=1.0$ and $V_0=5.0$, respectively. For
visualization purpose, a factor is multiplied to the pumped current
in Fig.\ref{fig5}. For $I^{ad}$ the factor is 50 in both panels. For
$I^{nad}$ this factor is 10 in panel (a) and 5000 in panel (b).
Other parameters: $B=0.001$, $\phi_{12}=\pi/2$. $V_p=0.5$,
$\omega=0.002$ in panel (a) and $V_p=4.5$, $\omega=0.002$ in panel
(b). Panel (c) highlights the pumped current at small pumping
amplitudes at the first resonant peak. $V_0=1.0$ in this panel and
$V_p=0.05$, $\omega=0.0002$. The factors for $I^{ad}$ and $I^{nad}$
are 1000 and 10, respectively. }\label{fig5}
\end{figure}
In panel (a) of Fig.\ref{fig5} we plot the pumped current in the
presence of magnetic field as a function of Fermi energy $E_F$,
together with the transmission coefficient $T(E_F)$ at static
potential barrier $V_0=1.0$. The sharp tips of transmission
coefficient suggests that quantum resonance effect dominates the
transport process. When a dc bias is applied, the tunneling current
is calculated from transmission profile. However, the pumped current
is generated as zero bias by periodically varying ac gate voltages.
Although originating from different physical mechanisms, we see that
the pumped current clearly show resonance characteristics both in
adiabatic and non-adiabatic cases near the resonant energy of static
transmission coefficient. These resonance-assisted behavior of
pumped current is a generic property of electron pump.\cite{ydwei1}
Operating at the coherent regime, quantum interference naturally
results in its resonant behavior. It is worth mentioning that near
the sharp resonance at $E_F=0.0118$ the adiabatic pumped current
changes sign. This is understandable. In the presence of dc bias,
the direction of the current is determined by the bias. For
parametric electron pump at zero bias, the direction of the pumped
current depends only on the system parameters such as Fermi energy
and magnetic field. Variation of these parameters can change the
current direction. For the non-adiabatic pump, the pumped current
changes slowly near the resonance but there is no sign changes for
the pumped current. In Fig.\ref{fig5}(a), we also see a second
resonant point with much broader peak. Near this resonant level, we
see that the transmission coefficient and the pumped currents are
well correlated. The resonant feature of the pumped current is also
related to the width of the resonant peak in the transmission
coefficient. Similar behaviors are found for a higher static
potential barrier $V_0=5.0$ in panel (b). A larger barrier makes the
resonant peaks much sharper, but it doesn't qualitatively affect the
pumped current. The noticeable difference is that, the pumped
current peaks are shifted with that of the transmission coefficient.
In addition, it seems that the non-adiabatic pumped current develops
a plateau region near the first resonant peak.
When zooming in at this resonant peak ($E=0.0118$ in
Fig.\ref{fig5}(a)), we found that at small
pumping amplitude the pumped currents are zeros for both adiabatic
and non-adiabatic cases when a complete transmission occurs
(transmission coefficient $T=1$). The numerical evidence is shown in
panel (c) of Fig.\ref{fig5}. Note that there is only one transmission channel
for the incident energy so that $T=1$ corresponds to complete transmission.
We emphasize that the non-adiabatic pumped
current goes to zero near the resonance only for very small frequency. At
larger frequency such as the case in Fig.\ref{fig5}(a) or (b), it is nonzero.
For the case of $I^{ad}$, it is easy to understand why it is zero
at $T=1$. For a perfect transmission, the diagonal terms $S_{LL}$ and $S_{RR}$
of the four-block scattering matrix are zero. Hence we have
$S_{LR}=S_{RL}=\exp(i\theta)$. From Eq.(\ref{dnde}), we have
\begin{eqnarray}
dN_\alpha/dV_j = (i\partial \theta/\partial V_j)/\pi \label{dnde1}
\end{eqnarray}
For two pumping potentials,
the current can be expressed in parameter space. Using the Green's theorem,
Eq.(\ref{eq1}) becomes\cite{brouwer1},
\begin{eqnarray}
I_\alpha & = & \frac{1}{\tau} \int_0^\tau dt
\frac{dQ_\alpha(t)}{dt} \notag\\
& = & \frac{q\omega}{2 \pi} \int dV_1 dV_2 \left( \frac{\partial}{\partial
V_1}\frac{dN_\alpha}{dV_2} - \frac{\partial}{\partial
V_2}\frac{dN_\alpha}{dV_1} \right) \label{para}
\end{eqnarray}
From Eq.(\ref{dnde1}), it is easy to see that the integrand is zero.
Hence $I_\alpha=0$ if $S_{\alpha \alpha}=0$. For non-adiabatic case,
the pumped current at complete transmission is in general nonzero.
However, if the frequency is very small, the adiabatic case is
recovered. This is numerically supported by Fig.\ref{fig5}(c), where
the two current curves are very similar.
\begin{figure}[tbhp]
\centering
\includegraphics[width=\columnwidth]{fig6.eps}
\caption{The pumped current versus Fermi energy in a T-shaped
system. The side bar is of length 20 and width 30. Two gray blocks
indicate the positions where the pumping potentials are applied and
the static potential is set to be zero. A factor of 50 is multiplied
to $I^{ad}$. Other parameters: $B=0.001$, $\phi_{12}=\pi/2$.
$V_p=0.05$, $\omega=0.002$.}\label{fig6}
\end{figure}
Furthermore, the behavior of pumped current for a structure exhibiting
anti-resonance phenomena was studied and the numerical results is
shown in Fig.\ref{fig6}. To establish anti-resonance, we use a
T-junction\cite{T-junction}, which is schematically
plotted in the inset of Fig.\ref{fig6}. The side bar has longitudinal dimension
20 and transverse dimension 30. Pumping potentials are placed on the
two arms of the device and there are no static potential barriers in
the system. From the transmission curve shown in Fig.\ref{fig6}, one clearly finds that
$T(E_F)$ drops sharply to zero around $E_F=0.112$, which is the
signature of anti-resonance. At this point, both the adiabatic and
nonadiabatic pumped current are zero. Different from the resonant case, here the
range where the current is zero or nearly zero is much broader.
The phenomena are attributed to reasons similar to those
we presented above, but in this case $S_{LR}$ and $S_{RL}$ are zero
at the anti-resonance point. We also see that at transmission minimum
$E_F=0.10$ with small transmission coefficient the pumped
current is nonzero.
\begin{figure}[tbhp]
\centering
\includegraphics[width=\columnwidth]{fig7.eps}
\caption{Panel (a): Adiabatic current vs pumping potential $V_p$ at
$E_F=0.0118$. Panel (b): Non-adiabatic current versus pumping
frequency $\omega$ at $E_F=0.0242$. System parameters:
$\phi_{12}=\pi /2$, $V_0=1$, $B=0.001$. Panel (c) shows the pumped
current and transmission coefficient versus magnetic field $B$ at
Fermi energy $E_F=0.12$, with other parameters: $\phi_{12}=\pi /2$,
$V_0=1$, $V_p=0.5$, $\omega=0.002$. For illustration a factor of 100
is multiplied to $I^{ad}$ and it is 10 for $I^{nad}$. }\label{fig7}
\end{figure}
In panel (a) and (b) of Fig.\ref{fig7}, we examine the influence of
pumping amplitude $V_p$ on the pumped currents with the static
potential barrier fixed at $V_0=1$, which corresponds to
the case shown in panel (a) of Fig.\ref{fig5}. At the first resonant
energy $E_F=0.0118$ we plot $I^{ad}$ versus $V_p$, the pumping
potential amplitude. The non-adiabatic pumped current $I^{nad}$ as
a function of the pumping
frequency $\omega$ is evaluated at the second resonant peak
$E_F=0.0242$. In both cases, magnitudes of the pumped current
changes in a oscillatory fashion with the increasing of $V_p$ or $\omega$. The
pumped current can change its sign, which also reflects the nature of the
parametric pump and manifest distinction between the pumped current
and the conventional resonant tunneling current.
\begin{figure}[tbhp]
\centering
\includegraphics[width=\columnwidth]{fig8.eps}
\caption{The pumped current as well as transmission coefficient as a
function of magnetic field $B$ at Fermi energy $E_F=0.12$. Other
parameters: $\phi_{12}=\pi /2$, $V_0=1$, $V_p=0.5$, $\omega=0.002$.
$I^{ad}$ and $I^{nad}$ are scaled by factors of 100 and 10,
respectively. }\label{fig8}
\end{figure}
The resonance behavior of pumped current is also visible in panel
(c) of Fig.\ref{fig7}, in which we depict $I_p$ and transmission
coefficient versus magnetic field $B$ at Fermi energy $E_F=0.12$.
Sweeping through magnetic field, there is a sharp change of
transmission coefficient near $B \sim 0.003$ and the pumped current
changes accordingly. With increasing magnetic field, $T$ becomes
quantized (there is only one transmission channel at this magnetic
field) indicating the occurrence of edge states in the quantum Hall
regime and the pumped current vanishes. In our setup, electron pump
operates by cycling modulation of electron passing through the
pumping potentials which are on top of static barriers defining the
system. With increasing of the magnetic field, electron wavefunction
tends to localize near the edge, which decreases the modulation
efficiency of the pumping potentials. As the edge state emerges,
electron will circumvent the confining potentials with no reflection
during their deformations. In this case, the variation of pumping
potential has no effect on the moving electron. Hence there is no
pumped current when edge state is formed in the system.
Mathematically it is also easy to show from Eq.(\ref{eq1}) that for
a two-probe system as long as the instantaneous reflection
coefficient vanishes (in the case of edge state) in the whole
pumping period there is no adiabatic pumped current.
We provide a numerical evidence for the above statement, which is
shown in Fig.\ref{fig8}. In contrast to the calculation of panel (c)
of Fig.\ref{fig7}, the static potential barriers extend to a width
40, which is exactly the width of the scattering region. At the same
time, the pumping barriers remain the same as before (with width 10).
Now the static transmission coefficient, labeled $T_1$ in the figure,
does not have quantized
value but exhibits a resonant behavior. $T_0$ is copied from
Fig.\ref{fig7} for comparison. As long as the edge state of an
electron is scattered with transmitted and reflected modes, the
pumped current will be generated with varying system parameters.
\section{conclusion}
In conclusion, we have studied the pumped current as a function of
pumping potential, magnetic field and pumping frequency in the
resonant and anti-resonant tunneling regimes. Resonant features are
clearly observed for adiabatic and non-adiabatic pumped current. We
found that when the resonant peak is sharp the adiabatic pumped
current changes sign near the resonance while non-adiabatic pumped
current does not. When the resonant peak is broad the behaviors of
pumped current in adiabatic and non-adiabatic regimes are similar
and both change sign near the resonance. At anti-resonance, however,
both adiabatic and non-adiabatic pumped current are zero. As the
system enters the quantum Hall regime, pumped currents vanishes in
all the setups shown in Fig.\ref{fig1}, since the pumping potentials
can not modulate the electron wave function. Furthermore, we have
numerically investigated the symmetry of the adiabatic and
non-adiabatic pumped current of systems with different symmetries
placed in magnetic field. The calculated results are listed in
Table.\ref{table1} and most of them are in agreement with the former
theoretical results derived from Floquet scattering matrix theory.
Different from the theoretical prediction, we found that the system
with general spatial inversion symmetry (GIV) gives rise to a finite
pumped current at adiabatic regime. At small magnetic field, both
the adiabatic and non-adiabatic currents have an approximation
relation $I(B) \approx I(-B)$.
\section{acknowledgments}
This work is supported by RGC grant (HKU 705409P), University Grant
Council (Contract No. AoE/P-04/08) of the Government of HKSAR, and LuXin Energy Group.
The computational work is partially performed on HPCPOWER2 system of the computer center, HKU.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,344 |
Cambridgeshire Fire and Rescue Service invests in new turntable ladders
2nd July 2021 | Categories: Fire and Rescue, News
Cambridgeshire Fire and Rescue Service has invested in two new turntable ladders to add to the fleet. The new specialist appliances will assist in the service's response to incidents involving working at height.
Replacing two multi-star aerial appliances, the new high spec vehicles will be based at Stanground, in the north of the county, and at Cambridge, in the south. Both are a Magirus 32L/AS, which have been supplied by Emergency One and are mounted on a Scania chassis, which was chosen for maximum manoeuvrability and a high level of visibility for the driver.
The most prominent feature of the appliances is the 30m ladder with an articulated section, which will enhance the service's operations at height.
Other features include a cage with a retractable access platform and connections for attaching a stretcher and a detachable thermal imaging camera. The cameras can be mounted on the cage and can be live streamed to officers and Combined Fire Control.
An additional advantage includes improved capabilities for observing incidents from above. They are a more robust design, more responsive to drive, easier to manoeuvre, faster to set up (at just 30 seconds) and with greater rescue capabilities.
The appliances came into the service in 2020, and training has now been completed by all crews. The turntable ladders are now 'on the run' in Cambridgeshire and ready to respond to whatever incidents they are faced with.
Speaking about the investment, Head of Fleet & Equipment for Cambridgeshire Fire and Rescue Service, Graham Wiggins, said, "We want to ensure we provide our firefighters with the best possible equipment to be able to respond to emergency calls and protect their local community effectively. This means investing in new equipment and making sure they have the right vehicles to respond to the job. These new fire appliances are an exciting and innovative addition to the county's fleet and will be a great asset to the service when responding to incidents at height."
Ambulance service set to save millions with introduction of electric vehicles
New Assistant Chief Fire Officers appointed to Shropshire Fire and Rescue Service
NatWest Group helps support over 200 London Ambulance Service apprentices
New wholetime firefighters in Devon and Somerset complete their training | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,149 |
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\begin{document}
\title{SL(2,R) INVARIANCE OF NON-LINEAR ELECTRODYNAMICS COUPLED TO AN
AXION AND A DILATON}
\author{G W GIBBONS\\ \&\\D A RASHEED\thanks{Supported by EPSRC grant
no.~9400616X.} \\ \\D.A.M.T.P.\\University of
Cambridge\\Silver Street\\Cambridge CB3 9EW\\U.K.}
\maketitle
\begin{abstract}
\noindent
The most general Lagrangian for non-linear electrodynamics coupled to
an axion $a$ and a dilaton $\phi$ with $SL(2,\mbox{\elevenmsb R})$
invariant equations of motion is
$$
-\textstyle\frac{1}{2}\left(\nabla\phi\right)^2 - \textstyle\frac{1}{2} e^{2\phi}\left(\nabla
a\right)^2 + \fraction{1}{4}aF_{\mu\nu}\star F^{\mu\nu} + L_{\rm
inv}(g_{\mu\nu},e^{-\frac{1}{2}\phi}F_{\rho\sigma})
$$
where
$L_{\rm inv}(g_{\mu\nu},F_{\rho\sigma})$ is a Lagrangian
whose equations of motion are invariant under electric-magnetic
duality rotations. In particular there is a unique generalization of
Born-Infeld theory admitting $SL(2,\mbox{\elevenmsb R})$ invariant
equations of motion.
\end{abstract}
\renewcommand{\thepage}{ }
\pagebreak
\renewcommand{\thepage}{\arabic{page}}
\setcounter{page}{1}
\sect{Introduction}
In a recent paper \cite{GibRas} we found the condition on the
Lagrangian function $L_{\rm inv}(g_{\mu\nu},F_{\rho\sigma})$ for a
non-linear electrodynamic theory coupled to gravity that the
equations of motion, including the Einstein equations, are invariant
under the action of an $SO(2)$ group of generalized
electric-magnetic duality rotations. One such Lagrangian is the
Born-Infeld Lagrangian \cite{BI}
\begin{equation}
L_{\rm BI} = 1 - \sqrt{1+\fraction{1}{2}F^2-\fraction{1}{16}(F\star
F)^2}.
\end{equation}
Note that $L_{\rm inv}(g_{\mu\nu},F_{\rho\sigma})$ itself
is {\em not} invariant under duality rotations.
In this letter we shall extend these results by including a coupling
to a scalar dilaton field $\phi$ and a pseudo-scalar axion field
$a$. They contribute to the action the following kinetic terms
\begin{equation}
L_{\rm ax-dil} = -\textstyle\frac{1}{2}\left(\nabla\phi\right)^2 -
\textstyle\frac{1}{2} e^{2\phi}\left(\nabla a\right)^2
\end{equation}
which are $SL(2,{\Bbb R})$ invariant. We shall show that this
$SL(2,{\Bbb R})$ invariance may be extended to the equations of motion
(but not the action) if and only if the action takes the form
\begin{equation}
\int d^4x\sqrt{g} \left\{R - 2\Lambda -
\textstyle\frac{1}{2}\left(\nabla\phi\right)^2 - \textstyle\frac{1}{2} e^{2\phi}\left(\nabla
a\right)^2 + \fraction{1}{4}aF\star F +
L_{\rm inv}(g_{\mu\nu},e^{-\frac{1}{2}\phi}F_{\rho\sigma})
\right\}
\end{equation}
where $L_{\rm inv}\left(g_{\mu\nu},F_{\rho\sigma}\right)$ is a
Lagrangian with $SO(2)$ invariant equations of motion. In particular
there is just one generalization of the Born-Infeld Lagrangian
admitting $SL(2,{\Bbb R})$ invariant equations of motion. This
$SL(2,{\Bbb R})$ invariant generalization of the Born-Infeld
Lagrangian does {\em not} coincide with that discussed in
\cite{GibRas} in connection with string theory. The relation of our
new results to string theory is currently under investigation.
\sect{SL(2,R) Duality at Lowest Order}
In 4 dimensions, the bosonic sector of N=4 supergravity and string
theory compactified on a torus, at lowest order, may be described by
the following Lagrangian \cite{CreSchFer}
\begin{equation}
L = R - \textstyle\frac{1}{2}\left(\nabla\phi\right)^2 - \textstyle\frac{1}{2} e^{2\phi}\left(\nabla
a\right)^2 + \fraction{1}{4}aF_{\mu\nu}\star F^{\mu\nu} -
\fraction{1}{4}e^{-\phi}F_{\mu\nu}F^{\mu\nu}
\label{LowEn}
\end{equation}
where for simplicity we consider only a single $U(1)$ gauge
field\footnote{In this paper we use units in which
$\mu_0=\varepsilon_0=\hbar=c=16\pi G=1$.}. The resulting theory admits
an $SL(2,{\Bbb R})$ electric-magnetic duality which mixes the
electromagnetic field equations with the Bianchi identities and also
transforms the axion and dilaton.
In a local orthonormal frame, the electric intensity $\bf E$ and
magnetic induction $\bf B$ may be defined by $E_i=F_{i0}$ and $B_i={1
\over 2}\epsilon_{ijk}F_{jk}$. The Bianchi identities $dF=0$ or
$\partial_{[\alpha} F_{\beta\gamma]}=0$ are then equivalent to
\begin{eqnarray}\samepage
{\bf\nabla}\cdot{\bf B} & = & 0 \nonumber \\
& & \\
{\bf\nabla}\times{\bf E} & = & - {\partial{\bf B}\over\partial t}.
\nonumber
\end{eqnarray}
Defining $G^{\mu\nu}$ by\footnote{There is
some ambiguity in the definition of this partial derivative depending on
whether or not one takes into account the antisymmetry of $F_{\mu\nu}$.
Here we treat $F_{\mu\nu}$ and $F_{\nu\mu}$ as independent variables,
hence the factor of 2.}
\begin{equation}
G^{\mu\nu} = -2 {\partial L\over\partial F_{\mu\nu}},
\label{Constit}
\end{equation}
the field equations are $d\star G=0$ or $\partial_{[\alpha}\star
G_{\beta\gamma]}=0$,
which are equivalent to
\begin{eqnarray}\samepage
{\bf\nabla}\cdot{\bf D} & = & 0 \nonumber \\
& & \\
{\bf\nabla}\times{\bf H} & = & + {\partial{\bf D}\over\partial t},
\nonumber
\end{eqnarray}
where the electric induction $\bf D$ and magnetic intensity $\bf H$
are defined by $D_i=G_{i0}$ and $H_i={1\over
2}\epsilon_{ijk}G_{jk}$. For the Lagrangian (\ref{LowEn}), $\bf D$ and
$\bf H$ are given by
\begin{eqnarray}\samepage
{\bf D} & = & + {\partial L\over\partial{\bf E}} = e^{-\phi}{\bf E} +
a{\bf B} \nonumber \\
& & \label{LowEnDH}\\
{\bf H} & = & - {\partial L\over\partial{\bf B}} = e^{-\phi}{\bf B} -
a{\bf E} \nonumber
\end{eqnarray}
These are the constitutive relations and may be rewritten as
\begin{equation}
\left({{\bf E}\atop{\bf H}}\right) =
\underbrace{
\left(
\begin{array}{cc}
e^\phi & -ae^\phi \\
-ae^\phi & e^{-\phi}+a^2e^\phi
\end{array}
\right)}
_{\textstyle\cal M}
\left({{\bf D}\atop{\bf B}}\right).
\label{SL2RConstit}
\end{equation}
It is convenient to define a complex scalar field
$\lambda$ by
\begin{equation}
\lambda = a + ie^{-\phi}
\end{equation}
and a complex 2-component vector $\psi$ by
\begin{equation}
\psi = \left({1\atop -\lambda}\right).
\end{equation}
Then the matrix ${\cal M}$ may be written as
\begin{equation}
{\cal M} = {\psi\psi^\dagger + c.c.\over\sqrt{{\rm
det}\left(\psi\psi^\dagger + c.c.\right)}}.
\end{equation}
Chosing the first component of $\psi$ to be $1$ fixes the representation
of ${\cal M}$. The $SL(2,{\Bbb R})$ duality transformation may then be
constructed so that it automatically leaves the constitutive relations
invariant~:
\begin{eqnarray}\samepage
\psi \rightarrow \psi^\prime \propto \left(S^T\right)^{-1}\psi \quad &
\Rightarrow & \quad {\cal M} \rightarrow \left(S^T\right)^{-1}{\cal
M}S^{-1} \nonumber \\
& & \\
\left({{\bf D}\atop{\bf B}}\right) \rightarrow S\left({{\bf D}\atop{\bf
B}}\right) \quad & , & \quad \left({{\bf E}\atop{\bf H}}\right)
\rightarrow\left(S^T\right)^{-1}\left({{\bf E}\atop{\bf H}}\right),
\nonumber
\end{eqnarray}
where $S\in SL(2,{\Bbb R})$. If
\begin{equation}
S = \left(
\begin{array}{cc}
p & q \\
r & s
\end{array}
\right)
\qquad\mbox{where }ps-qr=1,
\end{equation}
then the induced transformations of the axion and dilaton fields are
given by a Mobius transformation of $\lambda$~:
\begin{equation}
\lambda \rightarrow {p\lambda+q\over r\lambda+s}.
\label{axdiltransf}
\end{equation}
It is easy to check that the axion and dilaton equations of motion are
invariant under these transformations and also so is the energy momentum
tensor, so it is consistent to assume that the metric is unchanged
under the action of this duality.
In the covariant notation, the transformations of $F$ and $G$ are given by
\begin{equation}
\left\{
\begin{array}{rcl}
F_{\mu\nu} & \rightarrow & s F_{\mu\nu} + r\star G_{\mu\nu} \\
& & \\
G_{\mu\nu} & \rightarrow & p G_{\mu\nu} - q\star F_{\mu \nu}.
\end{array}
\right.
\end{equation}
Defining the complex 2-forms ${\goth F}=F+i\star F$ and ${\goth
G}=\star G-iG$, the duality may be written more compactly as~:
\begin{equation}
\left({{\goth G}\atop{\goth F}}\right)
\rightarrow
\left(
\begin{array}{cc}
p & q \\
r & s
\end{array}
\right)
\left({{\goth G}\atop{\goth F}}\right)
\quad,\quad
\lambda \rightarrow {p\lambda+q\over r\lambda+s}
\quad,\quad
g_{\mu\nu}\rightarrow g_{\mu\nu}.
\label{Duality}
\end{equation}
\sect{SL(2,R) Duality in Non-Linear Electrodynamics}
Since in both string theory and in supergravity theories, higher order
terms in the electromagnetic field arise, causing the
electrodynamic equations of motion to become non-linear, it is natural
to ask under what circumstances the $SL(2,{\Bbb R})$ duality above
continues to hold. It has been shown \cite{GibRas} that, in the case
of pure non-linear electrodynamics with no axion or dilaton, the
equations of motion will admit an $SO(2)$ duality provided the
Lagrangian satisfies a simple differential constraint~:
$G_{\mu\nu}\star G^{\mu\nu}=F_{\mu\nu}\star F^{\mu\nu}$, or
equivalently ${\bf E}\cdot{\bf B}={\bf D}\cdot{\bf H}$. Moreover there
are, roughly speaking, as many Lagrangians satisfying this constraint
as there are functions of a single real variable. Amongst this class
of Lagrangians is the Born-Infeld Lagrangian~:
\begin{equation}
\sqrt{g}L = \sqrt{g} - \sqrt{{\rm det}(g_{\mu\nu}+F_{\mu\nu})},
\end{equation}
which, in 4 dimensions gives
\begin{equation}
L = 1 - \sqrt{1+\fraction{1}{2}F^2-\fraction{1}{16}(F\star F)^2}.
\end{equation}
We will consider here a 4 dimensional Lagrangian $L(g,F,a,\phi)$ which
has the same axion an dilaton kinetic terms as (\ref{LowEn}) but we
will allow an arbitrary dependence on $a$, $\phi$ and $F_{\mu\nu}$. We
will not consider the higher order derivative terms in $F$ which also
occur in string theory. There is some evidence that the singularities
present in solutions of the Einstein-Maxwell theory are absent in
string theory due to higher order corrections and consequently the
higher order derivative terms in $F$ may be neglected \cite{Tse}.
Infinitesimally, the $SL(2,{\Bbb R})$ transformation above may be
described by the matrix
\begin{equation}
S = \left(
\begin{array}{cc}
p & q \\
r & s
\end{array}
\right) = \left(
\begin{array}{cc}
1+\alpha & \beta \\
\gamma & 1-\alpha
\end{array}
\right).
\end{equation}
The fields then transform according to
\begin{equation}
\left\{
\begin{array}{rl}
\delta a & = 2\alpha a + \beta -\gamma(a^2-e^{-2\phi}) \\
\delta\phi & = 2(a\gamma - \alpha) \\
\delta F_{\mu\nu} & = \gamma\star G_{\mu\nu} - \alpha F_{\mu\nu} \\
\delta G_{\mu\nu} & = \alpha G_{\mu\nu} - \beta\star F_{\mu\nu} \\
\delta g_{\mu\nu} & = 0.
\end{array}
\right.
\label{Transf}
\end{equation}
\subsection{Invariance of constitutive relations}
Invariance of the constitutive relation under these transformations
requires that
\begin{equation}
\delta G^{\mu\nu} = -2{\partial\over\partial
F_{\rho\sigma}}\left({\partial L\over\partial F_{\mu\nu}}\right)\delta
F_{\rho\sigma} -2{\partial\over\partial a}\left({\partial
L\over\partial F_{\mu\nu}}\right)\delta a
-2{\partial\over\partial\phi}\left({\partial L\over\partial
F_{\mu\nu}}\right)\delta\phi.
\label{Star}
\end{equation}
Comparing the coefficients of $\alpha$, $\beta$ and $\gamma$ in this
equation gives 3 differential constraints on the Lagrangian. Firstly,
the $\beta$ equation reads
\begin{equation}
{\partial^2L\over\partial a\partial F_{\mu\nu}} = {1\over 2}\star
F^{\mu\nu}.
\end{equation}
Integrating this implies that $L$ must be of the form
\begin{equation}
L = R - \textstyle\frac{1}{2}\left(\nabla\phi\right)^2 - \textstyle\frac{1}{2} e^{2\phi}\left(\nabla
a\right)^2 + \fraction{1}{4}aF_{\mu\nu}\star F^{\mu\nu} +
\widetilde{L}(F,\phi) + f(a,\phi).
\end{equation}
The coefficients of $\alpha$ in (\ref{Star}) then give the following
constraint on $\widetilde{L}$~:
\begin{equation}
{\partial\over\partial
F_{\rho\sigma}}\left(F_{\rho\sigma}{\partial\widetilde{L}\over\partial
F_{\mu\nu}}\right) + 2{\partial^2\widetilde{L}\over\partial\phi\partial
F_{\mu\nu}} = 0.
\end{equation}
Defining a new 2-form field
$\bar{F}_{\mu\nu}=e^{-\frac{1}{2}\phi}F_{\mu\nu}$ and
changing variables to $\bar{F}$ and $\phi$ in $\widetilde{L}$, this
last constraint reads
\begin{equation}
{\partial^2\widetilde{L}\over\partial\phi\partial\bar{F}_{\mu\nu}} = 0
\end{equation}
which implies that $\widetilde{L}=\widetilde{L}(e^{-\frac{1}{2}\phi}F)$
plus an arbitrary function of $\phi$ which we are free to absorb into
$f(a,\phi)$. The coefficients of $\gamma$ in (\ref{Star}) then give
another constraint on $\widetilde{L}$~:
\begin{equation}
{\partial\over\partial\bar{F}_{\mu\nu}}\left(
{\partial\widetilde{L}\over\partial\bar{F}_{\rho\sigma}}
{\partial\widetilde{L}\over\partial\bar{F}_{\lambda\tau}}\right)
\eta_{\rho\sigma\lambda\tau} = {1\over
2}\eta^{\mu\nu\rho\sigma}\bar{F}_{\rho\sigma},
\end{equation}
where $\eta_{\mu\nu\rho\sigma}$ is the completely antisymmetric
{\em tensor} of the 4 dimensional spacetime. It is natural at this
stage to define
\begin{equation}
\bar{G}^{\mu\nu} =
-2{\partial\widetilde{L}\over\partial\bar{F}_{\mu\nu}} =
e^{\frac{1}{2}\phi}(G^{\mu\nu}+a\star F^{\mu\nu}).
\label{BarG}
\end{equation}
Then the last constraint can be integrated to give
\begin{equation}
\bar{G}_{\mu\nu}\star\bar{G}^{\mu\nu} =
\bar{F}_{\mu\nu}\star\bar{F}^{\mu\nu} + 4C
\label{Cond1}
\end{equation}
where C is an arbitrary constant.
\subsection{Invariance of axion and dilaton equations}
The dilaton and axion equations of motion are respectively
\begin{equation}
-\nabla^2\phi = -e^{2\phi}(\nabla a)^2 +
\fraction{1}{4}F_{\mu\nu}(G^{\mu\nu}+a\star F^{\mu\nu}) + {\partial
f\over\partial\phi}
\label{Dil}
\end{equation}
and
\begin{equation}
-\nabla^2 a = 2(\nabla a)(\nabla\phi) +
\fraction{1}{4}e^{-2\phi}F_{\mu\nu}\star F^{\mu\nu} + {\partial
f\over\partial a}.
\label{Ax}
\end{equation}
These equations are also required to be invariant under the
$SL(2,{\Bbb R})$ transformations and (\ref{Transf}) implies that they
must transform into one another according to
\begin{eqnarray}\samepage
\delta(\mbox{Eq. (\ref{Dil})}) & = & 2\gamma\,\mbox{Eq. (\ref{Ax})}
\nonumber \\
& & \\
\delta(\mbox{Eq. (\ref{Ax})}) & = &
2(\alpha-a\gamma)\,\mbox{Eq. (\ref{Ax})} - 2\gamma
e^{-2\phi}\,\mbox{Eq. (\ref{Dil})} \nonumber
\end{eqnarray}
Comparing terms involving $F$ and $G$ and using equation (\ref{Cond1})
implies that the constant $C$ must vanish. Comparing the terms
involving $f(a,\phi)$ implies that $f$ is at most a constant. The
remaining terms then balance. Combining the results so far, we have
narrowed down the choice of possible Lagrangians to those of the form
\begin{equation}
L = R - 2\Lambda - \textstyle\frac{1}{2}\left(\nabla\phi\right)^2
- \textstyle\frac{1}{2} e^{2\phi}\left(\nabla a\right)^2 +
\fraction{1}{4}aF_{\mu\nu}\star F^{\mu\nu} +
\widetilde{L}(e^{-\frac{1}{2}\phi}F)
\label{Lag}
\end{equation}
where $\widetilde{L}(\bar{F})$ is required to satisfy
\begin{equation}
\bar{G}_{\mu\nu}\star\bar{G}^{\mu\nu} =
\bar{F}_{\mu\nu}\star\bar{F}^{\mu\nu}
\label{Cond2}
\end{equation}
or equivalently
\begin{equation}
(G_{\mu\nu}+a\star F_{\mu\nu})(\star G^{\mu\nu}-aF^{\mu\nu}) =
e^{-2\phi}F_{\mu\nu}\star F^{\mu\nu}.
\label{Cond3}
\end{equation}
In terms of $\bf E$, $\bf B$, $\bf D$ and $\bf H$ this condition reads
\begin{equation}
({\bf D}-a{\bf B})\cdot({\bf H}+a{\bf E}) = e^{-2\phi}{\bf E}\cdot{\bf
B}
\end{equation}
which is clearly satisfied by the $\bf D$ and $\bf H$ fields defined
in (\ref{LowEnDH}) for the Lagrangian (\ref{LowEn}).
\subsection{Invariance of energy-momentum tensor}
The final point to be checked is that the energy-momentum tensor is
invariant under this action of $SL(2,{\Bbb R})$, otherwise it would
not be consistent to assume that the metric is invariant, which we
have already implicitly done in a number of the steps above. The
energy-momentum tensor is most conveniently defined as
\begin{equation}
T^{\mu\nu} = g^{\mu\nu}L - {\partial L\over\partial(\partial_\mu
A_\lambda)}(\partial^\nu A_\lambda) - {\partial
L\over\partial(\partial_\mu\phi)}(\partial^\nu\phi) - {\partial
L\over\partial(\partial_\mu a)}(\partial^\nu a)
\end{equation}
which gives\footnote{Note that this is indeed symmetric~: $L$ depends
on $F$ only via the two invariants $F_{\mu\nu}F^{\mu\nu}$ and
$F_{\mu\nu}\star F^{\mu\nu}$. Therefore $G_{\mu\nu}$ will contain only
terms proportional to $F_{\mu\nu}$ and $\star F_{\mu\nu}$, so
${G_\mu}^\lambda F_{\nu\lambda}$ will contain only terms proportional
to ${F_\mu}^\lambda F_{\nu\lambda}$ and ${F_\mu}^\lambda\star
F_{\nu\lambda}={1\over 4}g_{\mu\nu}F^2$ which are both symmetric in
the indices $\mu$, $\nu$.}
\begin{equation}
T_{\mu\nu} = g_{\mu\nu}L + {G_\mu}^\lambda F_{\nu\lambda} +
(\partial_\mu\phi)(\partial_\nu\phi) + e^{2\phi}(\partial_\mu
a)(\partial_\nu a).
\end{equation}
All the terms involving derivatives of the axion and dilaton are
invariant, since they are the same terms as those that come from the
Lagrangian (\ref{LowEn}). The Lagrangian is not invariant but
transforms according to
\begin{equation}
\delta L = \fraction{1}{4}aF_{\mu\nu}\star\delta F^{\mu\nu} +
\fraction{1}{4}a\delta F_{\mu\nu}\star F^{\mu\nu} +
\fraction{1}{4}\delta aF_{\mu\nu}\star F^{\mu\nu} +
{\partial\widetilde{L}\over\partial\bar{F}_{\mu\nu}}
e^{-\frac{1}{2}\phi}\left(\delta
F_{\mu\nu} - \textstyle\frac{1}{2}\delta\phi F_{\mu\nu}\right).
\end{equation}
Using (\ref{Transf}), (\ref{BarG}) and (\ref{Cond3}) this gives
\begin{equation}
\delta L = - \fraction{1}{2}aF_{\mu\nu}G^{\mu\nu} +
\fraction{1}{4}(\beta-\gamma a^2-\gamma e^{-2\phi})F_{\mu\nu}\star
F^{\mu\nu}.
\end{equation}
So, using (\ref{Transf}), the transformation of the energy-momentum
tensor is
\begin{equation}
\delta T_{\mu\nu} = \gamma{G_\mu}^\lambda\star G_{\nu\lambda} -
\beta\star{F_\mu}^\lambda F_{\nu\lambda} + g_{\mu\nu}\delta L.
\end{equation}
Using (\ref{Cond3}) and the fact that in 4 dimensions, the components
of any 2-form satisfy ${F_\mu}^\lambda\star F_{\nu\lambda}={1\over
4}g_{\mu\nu}F_{\rho\sigma}\star F^{\rho\sigma}$, this variation of
$T_{\mu\nu}$ vanishes as required.
\sect{Conclusions}
We have shown that Lagrangians of the form (\ref{LowEn}) but with
higher order $F$-terms may retain $SL(2,{\Bbb R})$ invariance provided
they are of the form (\ref{Lag}), (\ref{Cond2}). The condition
(\ref{Cond2}) may be recognized as the same condition (in rescaled
variables) that a theory of pure electrodynamics has to satisfy in
order that it admit an $SO(2)$ duality. Thus for every theory of
non-linear electrodynamics described by a Lagrangian $L_{\rm
inv}(g,F)$ which admits an $SO(2)$ duality, we may construct a new
theory with Lagrangian
\begin{equation}
- \textstyle\frac{1}{2}\left(\nabla\phi\right)^2 - \textstyle\frac{1}{2} e^{2\phi}\left(\nabla
a\right)^2+\fraction{1}{4}aF_{\mu\nu}\star
F^{\mu\nu}+L_{\rm inv}(g,e^{-\frac{1}{2}\phi}F)+\mbox{const.}
\end{equation}
and the new theory will admit an $SL(2,{\Bbb R})$ duality. Thus, as in
\cite{GibRas}, there will be as many such Lagrangians as there are
functions of a single real variable. One such example is the
generalization of the Born-Infeld Lagrangian to include axion and
dilaton fields~:
\begin{equation}
\begin{array}{c}
L = R - 2\Lambda - \textstyle\frac{1}{2}\left(\nabla\phi\right)^2 -
\textstyle\frac{1}{2} e^{2\phi}\left(\nabla a\right)^2 +
\fraction{1}{4}aF_{\mu\nu}\star F^{\mu\nu} \\
\\
+ 1 -
\sqrt{1+\fraction{1}{2}e^{-\phi}F^2-\fraction{1}{16}e^{-2\phi}(F\star
F)^2}
\end{array}
\end{equation}
and this will be the only generalization of the Born-Infeld theory
with the $SL(2,{\Bbb R})$ duality (\ref{Duality}).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,640 |
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Mr Wilson has been a consultant plastic, reconstructive and aesthetic surgeon in Royal Devon & Exeter Hospital since 2011 with additional clinics in Musgrove Park Hospital, Taunton, and Yeovil District Hospital. His specialist areas of interest are breast reconstruction and advanced skin cancer treatment including melanoma in particular. He is a lead member of the Specialist Skin Multidisciplinary Teams at Musgrove Park Hospital / Yeovil District Hospital and Royal Devon & Exeter Hospital and is a member of Avon, Somerset, Wessex, Cancer Services Skin Site Specific Group. Mr Wilson is a core member of the South West Oncoplastic Breast Group, the main focus of which is teaching various forms of breast reconstruction. Mr Wilson is lead for plastic surgery research at RD&E and is currently involved in pioneering research into methods for assessing blood flow following microvascular reconstruction.
Following schooling in Northern Ireland, Mr Wilson attended Glasgow University Medical School. He undertook his Basic Surgical Training in the Glasgow teaching hospitals and passed his surgical membership exams (MRCS) in 2000, having spent time training in the renowned Canniesburn Plastic Surgery Unit in Glasgow. Thereafter, Mr Wilson won a two year scholarship of dedicated international research at the Blond McIndoe Centre working between University College London and Umeå University, Sweden looking into ways of improving nerve regeneration, subsequently gaining his higher post-graduate degree of Doctorate of Medicine (MD). Mr Wilson was appointed to the South West Regional Specialist Training Programme in Plastic Surgery in 2004, training in Frenchay Hospital, Bristol and RD&E, Exeter. During this time he passed the intercollegiate examination in plastic surgery –FRCS(Plast), and was awarded a Certificate of Completion of Surgical Training in plastic surgery in 2010. He is accredited on the General Medical Council's specialist register for plastic surgery. Addition expertise in melanoma treatment was gained at the internationally renowned Melanoma Institute of Australia, Sydney, under the wing of Prof John Thompson for which Mr Wilson won a travelling fellowship award from British Association of Plastic, Reconstructive and Aesthetic surgeons (BAPRAS). Mr Wilson also undertook a further fellowship in microsurgery for breast and head and neck reconstruction at the Royal Marsden Hospital 2010/11. His specific aesthetic fellowships were undertaken in Bristol (2006) and at the McIndoe Surgical Centre in 2010.
Mr Wilson is actively involved on teaching and has published and presented extensively both nationally and internationally during his time as a plastic surgeon from 2000 onwards. He regularly reviews articles, assessing their suitability for publication in the Journal of Plastic, Reconstructive & Aesthetic Surgery (JPRAS). He has been teaching faculty at many educational meetings and is a full member of the British Association of Plastic, Reconstructive & Aesthetic Sugeons (BAPRAS). | {
"redpajama_set_name": "RedPajamaC4"
} | 6,978 |
\section{Introduction}
5G wireless systems are expected to boost network capacity, spectral and energy efficiency, peak data rates, number of connected devices and consequently mobile data volumes with seamless and ubiquitous ultra-low latency connections. One of the key technology components of the evolving 5G architecture \cite{Lien}-\cite{METIS} will be the Device-to-Device (D2D) communications, which refer to the capability of direct communication between two or more devices without the intervention of a base station. Recently, D2D communications have attracted strong attention in academia and industry \cite{Mach}-\cite{Hwang}, initially intended for public safety scenarios \cite{Lien, 3GPP36.843, 3GPP36.877}, however other user-oriented (social applications) and network-oriented (offloading) use-cases have been rapidly emerged.\newline
\indent D2D communication is a promising way to improve performance providing different types of gain: proximity gain, hop gain and reuse gain \cite{Fodor}. However, important challenges are raised. Particularly, efficient resource sharing and radio protocol design should be proposed \cite{Mach}, in order to address critical D2D processes such as mode selection, scheduling and discovery, avoiding properly the interference to legacy cellular users (CUEs). D2D communication can be classified into in-band and out-band. The in-band D2D communication model refers to the case where D2D communications take place in a licensed spectrum allocated to the cellular operators. Uplink (UL), downlink (DL), or both resources can be reused. On the other hand, the out-band D2D utilizes the unlicensed spectrum adopted by other wireless technologies (e.g. Wi-Fi, Bluetooth). \newline
\indent In this article, the in-band D2D communication model is employed, where D2D communications take place in the UL spectrum allocated to the cellular operators, with the D2D users (DUEs) to be able to access this spectrum in a dedicated (overlay or orthogonal) or a shared (undelay or non-orthogonal) way. Overlay D2D communication avoids the interference to legacy CUEs issue, because D2D and cellular resources do not overlap, while the D2D management is handled by the cellular operators. On the contrary, in underlay D2D communications, DUEs and legacy CUEs share the same radio resources generating interference among each other, while D2D can be fully, loosely or not controlled by cellular operators. In both approaches, new resource allocation and synchronization methods should be introduced and significant changes in the existing standard are needed. It is worthwhile to mention that the 3GPP is directed to in-coverage or partial-coverage scenarios \cite{3GPP36.213, 3GPP36.331}, where D2D communication utilizes UL resources for the sidelink \cite{Lien}. The argument for the UL use is that this direction is mostly underutilized compared to the DL and the interference situation is easier to be resolved because the "victim" of D2D interference is evolved NodeB (eNB).\newline
\indent Furthermore, network infrastructure densification has been proposed as one of the leading concepts to cope with the growing traffic trends \cite{Gotsis, Kamel}. The basic idea is to get base stations and access points (having small transmission power) as close as possible to the end users. Consequently, the spectrum is increasingly reused, improving system capacity, and the link to the end user becomes shorter improving link quality. However, the network densification cannot continue improving performance endlessly and the question of what are its fundamental limits was addressed in \cite{Andrews}.\newline
\indent Given that one of the components of D2D communication benefit is proximity gain, a reasonable question raised is how network densification affects the D2D performance in D2D-enabled systems. Since a more dense network means that end users are closer to base station, a densification threshold may exist and above that D2D communication will not remain beneficial. Toward this end, this article carefully examines the effects of densification and spectrum sharing on the D2D performance. More specifically, a radio resource management (RRM) mechanism is introduced to jointly handle the D2D mode selection and user (UE) scheduling in a multi-cell environment considering intra-cell and cross-cell D2D capabilities. These two important procedures should be addressed jointly, since they are highly intertwined. In such an environment, the interference and densification are expected to significantly affect the D2D gain, while it is important to take into account the cross-cell D2D communications, especially for cell-edge users cases. The proposed RRM mechanism is separately designed for the overlay and the underlay spectrum sharing approaches. The joint optimal policy is quite complex, thus a problem reformulation is needed. \newline
\indent A system-level simulator integrating legacy CUEs along with D2D capable UEs is developed to evaluate the performance of D2D communications and examine the trends of total D2D communication gain (defined as the combination of direct gain and offloading gain) for various system parameters. Finally, the performance of in-band D2D communications for overlay and underlay spectrum sharing is experimentally evaluated in a Software Defined Radio (SDR)-based joint LTE-D2D implementation.
\section{Multi-cell D2D Communications Design: Mode Selection, Spectrum Sharing \& Densification}
\subsection{Design Overview}
Several D2D communication scenarios have been specified \cite{Mach}, depending on the coverage of cellular network (i.e. in coverage, partial coverage, out of coverage), the type of D2D communication (i.e. one-to-one, one-to-many), the area of D2D communication (i.e. same cell, different cell) and the relaying functionality (i.e. to enhance capacity, to extend coverage). However, regardless of the application scenario, the design of multi-cell D2D-enabled systems faces many technical challenges, such as mode selection, scheduling, interference management, synchronization and power control. In this article, we focus on the mode selection and scheduling procedures, while we examine different spectrum sharing approaches according to which cellular resources are reused for D2D communication (resulting in different interference scenarios between D2D and cellular users). Furthermore, we study the role of network densification in the design of D2D-enabled cellular system. \newline
\indent Fig.~\ref{fig1} illustrates a multi-cell wireless network with D2D communication capabilities, consisting of $K^{leg}$ legacy CUEs, where each UE $i_l \in {\cal K}_l^{leg}= \{ 1,2,...,{K^{leg}}\}$ is located in cell $l \in {\cal L}= \{ 1,2,...,L\}$, and $K^{pD2D}$ potential D2D pairs, where each pair ${j_{l - m}} = [j_{1,l},j_{2,m}] \in {{\cal K}^{pD2D}}= \{ 1,2,...,{K^{pD2D}}\}$ has the transmitter $DUE_{j_{1,l}}$ in cell $l \in {\cal L}$ and the receiver $DUE_{j_{2,m}}$ in $m \in {\cal L}$, in the general case (intra- or cross-cell D2D). Note that ${\cal K} = {{\cal K}^{leg}} \cup {{\cal K}^{pD2D}} = \{ 1,2,...,K\}$, where $K = K^{leg} + 2K^{pD2D}$ is the total number of UEs in the system. Potential D2D pairs can access the spectrum using in-band overlay or underlay spectrum sharing and the considered scenario represents the in-coverage scenario with same or different cell D2D capabilities. The UEs are stationary and uniformly distributed in the hexagonal cells of $L$ eNBs. \newline
\begin{figure}[!t]
\centering{
\includegraphics[width=3.4in, scale=1]{fig1.jpg}}
\caption{Multi-cell D2D communications \& densification.}
\label{fig1}
\end{figure}
\subsection{Mode Selection}
\indent The communication mode of the potential D2D pairs specifies if the DUEs communicate directly with each other or via the eNB. Moreover, together with the selected spectrum sharing scheme, it identifies whether DUEs utilize the same radio resources as the conventional cellular communication or not. Thus, proper mode selection plays an important role in D2D communication, which can be achieved through the following available transmission modes:
\begin{itemize}
\item {\it{Cellular Mode (CM)}}, where DUEs of a potential D2D pair communicate through their associated eNB(s), i.e. $DUE_{j_{1,l}} \rightarrow eNB_{l} (\rightarrow eNB_{m}) \rightarrow DUE_{j_{2,m}}$,
\item {\it{Direct or D2D Mode (DM)}}, where a potential D2D pair becomes an actual D2D pair whose DUEs communicate directly, i.e. $DUE_{j_{1,l}} \rightarrow DUE_{j_{2,m}}$.
\end{itemize}
\subsection{Spectrum Sharing}
\indent The above D2D communication modes utilize the licensed cellular spectrum either in an overlay or an underlay manner, while CUEs access the licensed spectrum in an orthogonal way. In particular, three D2D spectrum sharing approaches are considered: the {\it{Overlay}} approach, where potential DUEs communicate both for the CM and the DM utilizing the cellular resources orthogonally (CM/DM overlay or pure overlay); the {\it{Underlay 1}} approach, where the DUEs use orthogonal resources for the CM and non-orthogonal resources for the DM (CM overlay / DM underlay or mixed overlay/underlay); and the {\it{Underlay 2}} approach, where the DUEs utilize non-orthogonal resources both for the CM and the DM (CM/DM underlay or pure underlay). It is noted that the communication in DM utilizes the UL cellular resources, while the corresponding DL resources are offloaded in most of the cases for the legacy CUEs (except for some cross-cell D2D cases). Fig.~\ref{fig2} depicts the UL and DL resource utilization of the elaborate in-band spectrum sharing approaches.\newline
\begin{figure}[!t]
\centering{
\hspace*{-3mm}
\includegraphics[width=3.7in, scale=1]{fig2.jpg}}
\caption{In-band spectrum sharing approaches.}
\label{fig2}
\end{figure}
\subsection{Densification}
\indent Network infrastructure densification increases wireless network throughput and it is considered as a promissing solution for the booming traffic demand. Network densification can be defined as the deployment of more base stations and access points per unit area or volume. Specifically, the degree of densification can be characterized by the {\it{density}} or {\it{densification ratio}}. This ratio is defined as the ratio of access point density (number of eNBs per unit area) to user density (number of UEs per unit area).
According to the metric of density ratio, networks can be differentiated into sparse infrastructure deployments, with less access points than users and (ultra-) dense deployments, otherwise. However, combining network densification with D2D communications may not result in additive performance benefits, due to the fact that in more dense networks access points are placed closer to the users, but D2D communications are based on users proximity compared to their distance to access point. Consequently, the role of network densification in the D2D performance should be examined thoroughly when designing next generation cellular networks.
\section{D2D Radio Resource Management Mechanism}
\subsection{RRM Challenges \& Optimization Problem}
RRM determines the optimal use of wireless resources according to the current network status and the quality-of-service requirements of
the users. D2D-enabled system design faces various resource management issues, including mode selection, scheduling, channel quality estimation, power control, and beamforming. The biggest D2D RRM challenge is to efficiently incorporate D2D communications in a cellular network in a way the total network utility to be optimized (i.e. by ameliorating potential D2D pairs performance, without however degrading legacy CUEs performance). Most of the RRM research efforts have been focused on underlay single-cell spectrum sharing approached \cite{Xu}, while very few deal with overlay D2D or/and multiple cells \cite{Penda, Poulakis}. This article presents a joint overlay and underlay study in a multi-cell D2D-enabled environment. In particular, we focus on the design of D2D mode selection and UE scheduling procedures that constitute two of the most critical issues for the incorporation of D2D communications in legacy cellular systems. These decisions making procedures are highly intertwined, since scheduling is affected by the available UL and DL resources and the interference, which depend on mode selection of DUEs and vice versa. Consequently, these procedures should be handled jointly.\newline
\indent Specifically, a common entity, the resource manager, shares the resources over CUEs and DUEs at each time slot and at each cell and also decides the transmission mode (DM or CM) of each potential D2D pair. The proper UE should be carefully scheduled at each slot and at each cell, since the utilization of UL and DL resources depends also on the mode of D2D communication and the potential DL offloading. An example of the UL and DL resource allocation of two-phase communication (FDD or TDD) for CUEs, DUEs in CM and DUEs in DM is presented in Table~\ref{table1}.
\begin{table}[!h]
\centering
\caption{UL \& DL resource allocation.}
\label{table1}
\resizebox{0.9\width}{0.9\height}{
\begin{tabular}{@{}llll@{}}
\cmidrule[\heavyrulewidth](l){2-4}
& \multicolumn{3}{c}{\textbf{Scheduled UE}} \\ \midrule
\textbf{Phase} & \textbf{CUE} & \textbf{Pot. D2D pair} \textbf{in CM} & \textbf{Pot. D2D pair} \textbf{in DM} \\ \midrule
\textbf{UL} & $CU{E_{{i_l}}} \rightarrow eNB_l$ & $DU{E_{{j_{1,l}}}}\rightarrow eNB_l$ & $DU{E_{{j_{1,l}}}}\rightarrow DU{E_{{j_{2,m}}}}$ \\
\textbf{DL} & $eNB_l \rightarrow CU{E_{{i_l}}}$ & $eNB_m \rightarrow DU{E_{{j_{2,m}}}}$ & FREE (offloaded) \\ \bottomrule
\end{tabular}
}
\end{table}
\newline
\indent In order to design the D2D RRM mechanism, we formulate the joint mode selection and scheduling procedures as an optimization problem. A predetermined objective (utility function) must be optimized under constraints dictating the feasibility of the solution. The considered optimization variables are binary vectors $\pmb{x}\in\{0,1\}^{K^{pD2D}}$ and $\pmb{y}\in\{0,1\}^{K}$, determining the mode selection status of each DUE (DM or CM) and the scheduling status of each UE (both for CUE and DUE). Moreover, since the D2D pairs utilize UL resources, we are interested in optimizing the UL performance of the total system, taking into account at the same time the necessary allocations in the DL. Thus, the utility functions for CUEs (UL use), potential D2D pairs in DM (UL use) and potential D2D pairs in CM (UL and DL use) reflect the maximum instantaneous reliable rate of the corresponding links and are given by
\begin{footnotesize}
$$\boxed{
\begin{array}{l}
U_{{i_l}}^{leg} \;\;\;= \text{UL Capacity} \;(CU{E_{{i_l}}} \rightarrow eNB_l) \nonumber \vspace{1mm} \\
U_{{j_{l - m}}}^{DM} = \text{Pair Capacity} \; (DU{E_{{j_{1,l}}}}\rightarrow DU{E_{{j_{2,m}}}}) \nonumber \\
U_{{j_{l - m}}}^{CM} = \min \left\{ \text{UL Capacity} \;(DU{E_{{j_{1,l}}}}\rightarrow eNB_l), \right. \nonumber \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \left. \text{DL Capacity}\; (eNB_m \rightarrow DU{E_{{j_{2,m}}}}) \right\} \nonumber
\end{array}
}$$
\end{footnotesize}
Note that the utility function of potential D2D pairs in CM is defined as the maximum instantaneous reliable rate of the worst between UL and DL. \newline
\indent In the following, we present the joint problem for the different spectrum sharing schemes.
\subsubsection{Overlay Spectrum Sharing (CM/DM Overlay)}
\indent In {\it{Overlay}} spectrum sharing, the DUEs communicate both for the CM and the DM mode using the dedicated cellular resources. In this case, the joint optimization problem of overlay D2D mode selection and UE scheduling that aims at the maximization of total UL network performance in terms of utility function, in a multi-cell environment with intra-cell and cross-cell D2D communications enabled, can be formulated as follows
\begin{footnotesize}
$$\boxed{
\begin{array}{ll}
\mathop {\max }\limits_{\pmb{x},\pmb{y} } & U_{total}(\pmb{x},\pmb{y}) = f(U_{{i_l}}^{leg}, U_{{j_{l - m}}}^{DM}, U_{{j_{l - m}}}^{CM})\nonumber \\
{\text{ s.t.}} \,\,\,\,\,\,\, & (\text{C}1):\, \text{maximum number of UEs in UL} \nonumber\\
&(\text{C}2): \, \text{maximum number of DUEs in DL} \;\;\;\;\;\;\;\text{(P1)} \\ \label{P1}
&(\text{C}3):\, \text{mode \& scheduling variables relation} \nonumber\\
&(\text{C}4):\, \text{D2D distance restriction} \nonumber
\end{array}
}$$
\end{footnotesize}
where $U_{total}$ is the total utility function to be optimized which is a weighted sum of $U_{{i_l}}^{leg}, U_{{j_{l - m}}}^{DM}$ and $U_{{j_{l - m}}}^{CM}$. This function should be defined carefully in order to cover the cross-cell D2D cases. The (C1) corresponds to constraint of maximum simultaneous scheduled UEs at UL (one CUE or one DUE regardless the communication mode), while (C2) ensures that no more than one DUE can occupy a specific DL resource in cell $l$ (i.e. the total DUE transmitters in CM located in all the rest cells that have pairs in cell $l$ are no more than one). Furthermore, (C3) refers to the relation between mode and scheduling variables, while (C4) restricts the distance of a DUE pair to a maximum value in order to be able to communicate directly. Moreover, note that the employed scheduling scheme can be included in the optimization problem as an extra weight factor of each UE (CUE or DUE) (e.g. for round robin this factor reflects the reciprocal of each UE's scheduled times, while for proportional fairness it reflects the reciprocal of each UE's average capacity).
\subsubsection{Underlay Spectrum Sharing 1 (CM Overlay / DM Underlay)}
According to the {\it{Underlay 1}} spectrum sharing, the DUEs use dedicated resources for the CM and shared resources for the DM. Consequently, the DUEs in DM cause intra-cell interference to legacy CUEs and vice versa. Thus, the corresponding joint optimization problem of D2D mode selection and UE scheduling, can be formulated similarly to (P1) with the only modification the substitution of constraint (C1) with a new one allowing one CUE or one DUE in CM, and one DUE in DM simultaneously scheduled at UL of the same cell.
\subsubsection{Underlay Spectrum Sharing 2 (CM/DM Underlay)}
According to the {\it{Underlay 2}} spectrum sharing, the DUEs use shared resources both for the CM and the DM. Thus, the DUEs in all communication modes (CM or DM) cause intra-cell interference to legacy CUEs and vice versa. In this case, a common or a separate resource manager could be employed to decide about the mode selection and scheduling processes. Similar to Underlay 1, the corresponding joint optimization problem of D2D mode selection and UE scheduling, can be formulated by substituting constraint (C1) with a new one that grants spectrum access to one CUE and one DUE (regardless the communication mode) at the same cell, simultaneously.\newline
\indent It it noted that the above optimization problems are integer (or more precisely binary) non-linear and non-convex programs, which are difficult to be solved and to provide global optimal guarantees even for small-scale setups. To address this issue, we first reformulate the initial problem to a linear form, by eliminating the nonlinearities\footnote{For more details regarding the linearization-reformulation see \cite{Poulakis},\cite{Hou}.}. The transformed formulation is a binary (0-1) integer linear programming problem that is NP-complete and may be solved using integer linear programming employing popular and well-known solvers (e.g. GUROBI,GLPK, CPLEX) and derive a global optimal solution.
\subsection{Mode Selection \& Scheduling Policy}
Therefore, according to the joint mode selection and scheduling policy of the proposed RRM mechanism, a centralized resource manager should solve the joint optimization problem and the solution will indicate the UE that will be scheduled at each round and at each cell and moreover the communication mode (directly or through the eNB) if the UE belongs to potential D2D pairs. Fig.~\ref{fig3} depicts the followed joint mode selection and scheduling procedure in a flowchart form determining each transmission snapshot of the entire network.
\begin{figure}[!h]
\centering{
\hspace*{-3mm}
\frame{\includegraphics[width=3.0 in, scale=1]{fig3.jpg}}}
\caption{Flowchart of mode selection and scheduling procedures.}
\label{fig3}
\end{figure}
\section{D2D Performance Analysis}
\subsection{System-level Simulation Results}
To evaluate the presented RRM algorithm and investigate the trends of D2D communications performance in relation to network and user densification, as well as the spectrum sharing schemes, we develop a system-level D2D-enabled cellular simulator. Realistic assumptions are considered, summarized in Table~\ref{table2}. Five cell-types with different radius sizes are assumed, resulting in the same total network coverage area and reflecting different densification ratios. Note that cell-type 1 refers to a single-cell scenario with the particularity of absence of cross-cell D2D communications. Regarding the interference, the multi-cell frequency reuse-1 is assumed for the legacy cellular network, while depending on the D2D spectrum sharing approach, inter-cell and intra-cell interference occurs. For consistency purposes, a simulated interference cumulative distribution function has been derived to generate random interference to boundary cell edges with no adjacent cells (including the cell-type 1 case). Furthermore, the eNB and UE antenna gains as well as the transmit powers are determined depending on the cell-type (i.e. eNBs' density) in order to maintain a specific SNR value at cell-edges, considering a path loss exponent equal to 3. The optimization problem is modeled in MATLAB using the CVX software and solved using the GUROBI optimizer. Finally, the distance limit for D2D communications is assumed to be equal to cell radius. \newline
\begin{table}[!t]
\caption{System-level simulator parameters}
\centering
\resizebox{0.85\width}{0.85\height}{
\begin{tabular}{@{}cccccc@{}}
\toprule
\textbf{Parameter} & \multicolumn{5}{c}{\textbf{Value}} \\ \midrule
Spectrum sharing & \multicolumn{5}{c}{In-band; Overlay / Underlay 1 / Underlay 2}
\\
D2D communication scenario & \multicolumn{5}{c}{In-coverage; same or different cell D2D}
\\
System bandwidth (MHz) & \multicolumn{5}{c}{5} \\
Carrier frequency (GHz) & \multicolumn{5}{c}{2.6} \\
\begin{tabular}[c]{@{}c@{}}Total network coverage area ($\rm{km}^2$)\end{tabular} & \multicolumn{5}{c}{0.234} \\
Noise figure (dB) & \multicolumn{5}{c}{7} \\
\begin{tabular}[c]{@{}c@{}}Noise spectral density (dBm/Hz)\end{tabular} & \multicolumn{5}{c}{−174} \\
Traffic model & \multicolumn{5}{c}{Full buffer} \\
Scheduler type & \multicolumn{5}{c}{Round Robin} \\ \midrule
Cell type & Type 1 & Type 2 & Type 3 & Type 4 & Type 5 \\
Cell radius (m) & 300 & 212 & 150 & 123 & 100 \\
\begin{tabular}[c]{@{}c@{}}Network layout (number of cells)\end{tabular} & 1 & 2 & 4 & 6 & 9 \\
eNB density (number/km$^2$) & 4.3 & 8.5 & 17.1 & 25.6 & 38.5 \\
eNB max power (dBm) & 23.0 & 21.0 & 20.0 & 17.4 & 14.7 \\
eNB antenna gain (dBi) & 3.0 & 1.8 & 0.0 & 0.0 & 0.0 \\
UE max power (dBm) & 23.0 & 21.0 & 20.0 & 17.4 & 14.7 \\
UE antenna gain (dBi) & 3.0 & 1.8 & 0.0 & 0.0 & 0.0 \\
\bottomrule
\end{tabular}
}
\label{table2}
\end{table}
\begin{figure}[!t]
\centering{
\hspace*{-12mm}
\includegraphics[width=4.3in, scale=1]{fig4.jpg}}
\caption{Average throughput versus eNB density for 108 UEs in total.}
\label{fig4}
\end{figure}
\indent D2D communications can improve system's performance through the proximity gain, the hop gain and the reuse gain. In the considered scenario, where D2D communication utilizes UL resources, proximity gain can ameliorate UL performance, hop gain can increase DL performance, while reuse gain can improve both coming with the cost of interference. In Fig.~\ref{fig4}, the average total UL throughput of the network (i.e. UL throughput of CUEs and DUEs) is illustrated for the case of D2D communications disabled (i.e. all DUEs in CM) and the case of D2D communications enabled (i.e. the chosen DUEs by mode selector in DM) considering the different spectrum sharing approaches for increasing eNB density. As we can observe, the total UL throughput increases for higher densities, demonstrating the benefits of densification. Moreover, D2D communications increase the network's UL performance due to the proximity gain (enjoyed only by DUEs) with the improvement to be larger in the Underlay 1 approach due to the extra reuse gain (enjoyed both by DUEs and CUEs), while in Underlay 2 the reuse gain is offset by the caused interference. It is noted that the curves of Overlay and Underlay 1 w/- D2D are identical. Moreover, in order to focus on the improvement in DUEs throughput, Fig.~\ref{fig4} illustrates the average throughput of D2D pairs (derived from the corresponding utility functions) for the same scenarios. From this subfigure, we can observe similar results regarding the behaviour of the throughput, however the benefit is greater since the DUEs' throughput is isolated where the reuse (in underlay cases) and proximity (in all cases) gains are combined. \newline
\indent Furthermore, in order to evaluate the performance gain of D2D communications, we define the metric of total D2D communication gain ${\cal G}_{tot}$. This metric is defined as the weighted sum of direct communication gain ${\cal G}_{dir}$ in UL and the gain obtained by DL offloading ${\cal G}_{off}$. In particular, the direct gain (derived by proximity and reuse gain when it exists) corresponds to the percentage of throughput gain of potential D2D pairs, comparing the throughput achieved with D2D communications enabled and its absence. Moreover, the offloading gain [derived by hop gain (or reuse gain in Underlay 2 case)] that is obtained through D2D communications corresponds to the percentage of DL throughput gain of legacy CUEs comparing the case of coexistence with the potential D2D UEs with D2D communication enabled and the case where D2D communication is disabled (all the DUEs in CM). Furthermore, the total gain can be given by
$$\boxed{
\begin{array}{l}
{\cal G}_{tot}\, = {a_1} \cdot {\cal G}_{dir} + {a_2} \cdot {\cal G}_{off}
\end{array}
}$$
\begin{table}[!t]
\centering
\caption{D2D communications gain contribution.}
\label{tableG}
\resizebox{0.85\width}{0.85\height}{
\begin{tabular}{@{}lcccccc@{}}
\midrule
\begin{tabular}[l]{@{}l@{}}\bf{Spectrum}\\\bf{Sharing}\end{tabular} & \multicolumn{2}{c}{{Overlay}} & \multicolumn{2}{c}{{Underlay 1}} & \multicolumn{2}{c}{{Underlay 2}} \\
\midrule
\bf{Gain} & ${\cal G}_{dir} $ & ${\cal G}_{off} $ & $ {\cal G}_{dir} $ & ${\cal G}_{off} $ & ${\cal G}_{dir} $ & $ {\cal G}_{off} $ \\
\midrule
\bf{Proximity} & \ding{51} & & \ding{51} & & \ding{51} & \\
\bf{Hop} & & \ding{51} & & \ding{51} & & \\
\bf{Reuse} & & & \ding{51} & & \ding{51} & \ding{51} \\ \bottomrule
\end{tabular}
}
\end{table}
where $a_1$ and $a_2$ are the weight factors that corresponds to the considered contribution of each gain and without loss of generality are assumed to be equal to 0.5.\newline
\indent Table~\ref{tableG} summarizes the contribution of proximity, hop and reuse gain to the direct and DL offloading gain for the different spectrum sharing approaches. It is noted that in Underlay 1 and Underlay 2 approaches, an extra UL offloading gain is derived by reuse gain with the cost of interference, however this is out of the scope of this article.\newline
\indent In Fig.~\ref{fig5}, we present the average direct gain and offloading gain that the D2D communications achieve for different network cell-type topologies (see Table~\ref{table2}) assuming 108 UEs in total (CUEs and DUEs) and considering the same coverage area. The different cell-types represent different number of eNBs per same coverage area with increasing densification. As we can observe, the direct gain decreases in all cases as the network densification increases, because in more dense networks the distance of UEs to the associated eNB becomes smaller and thus the direct communication is more rarely preferred. This indicates the {\it{tradeoff}} between the performance improvement derived by {\it{densification}} and the {\it{D2D}} performance enhancement. We can also see that in general Underlay 1 spectrum sharing has better results due to extra reuse gain in DM, which gain is almost compensated by the interference in Underlay 2 scheme, while Overlay spectrum sharing leads to worst performance. Furthermore, similar results are observed for the offloading gain regarding network densification, while the hop gain of Overlay and Underlay 1 approaches is replaced by reuse gain in Underlay 2 approach. We have to note that, the hop gain always exists as only one hop is needed in DM instead of two hops in CM, nevertheless, both the proximity and reuse gains largely depend on the UEs locations.\newline
\begin{figure}[!t]
\centering{
\hspace*{-3mm}
\includegraphics[width=3.7in, scale=1]{fig5.jpg}}
\caption{Average direct and offloading gain versus the different cell-type topologies (i.e. different eNB's densities).}
\label{fig5}
\end{figure}
\indent Finally, Fig.~\ref{fig6} illustrates the average total gain of D2D communications versus the different network cell-type topologies (i.e. different eNBs densities) and the UEs density, assuming 36 legacy CUEs and variant number of potential D2D pairs. The case of Overlay spectrum sharing is assumed. Fig.~\ref{fig6} indicates that the D2D benefit becomes greater as the number of UEs increases, since more opportunities to employ D2D communication appear, resulting in slightly larger direct gain, as well as more offloaded opportunities for CUEs are generated. Furthermore, the total D2D gain decreases as the number of eNBs (deployed in the given geographic area) increases for the same number of UEs. It can also be observed that the contribution of direct gain gain to the total gain decreases for more dense network and for higher number of UEs. In conclusion, a tradeoff between D2D gain and densification benefit exists and should be carefully considered when a system with D2D capabilities is designed.
\begin{figure}[!t]
\centering{
\hspace*{-3mm}
\includegraphics[width=3.6in, scale=1]{fig6.jpg}}
\caption{Average total gain of D2D communications versus the number of total UEs and different cell-type topologies for the Overlay approach.}
\label{fig6}
\end{figure}
\subsection{SDR-based Experimental Evaluation}
This subsection is devoted to the experimental evaluation of in-band D2D communications in the testbeds of FLEX project \cite{FLEX}, which offers a valuable, flexible and credible solution for open and cost efficient LTE experimentation. In particular, we employ the NITOS indoor testbed \cite{NITOS} and the OpenAirInterface (OAI) platform \cite{OAI} in order to perform joint LTE-D2D tests and investigate the impact of D2D communication to a legacy LTE operation and vice-versa, for the different spectrum sharing schemes. A baseline coexistence setup is assumed, where an LTE pair and a D2D pair in DM communicate accessing the same UL spectrum in an overlay or an underlay manner. Specifically, the LTE pair consists of a $CUE$ node equipped with commercial LTE dongle and two SDR\footnote{Note that Ettus B210 USRP boards were employed.}-equipped nodes hosting the OAI core network code (EPC+HSS) and a modified version of OAI eNB code, respectively, for the implementation of LTE base station. Regarding the D2D pair, two SDR-equipped nodes implement the D2D transmitter ($DUE_1$) and receiver ($DUE_2$). Since the D2D pair is considered to transmit in DM, no mode selection procedure is employed and consequently the underlay case reflects both the Underlay 1 and Underlay 2 approaches that were described in the previous sections. EARFCN Band 3 (UL: 1.715GHz/ DL: 1.810GHz) is utilized, while the 5 MHz channel bandwidth case is considered. In the overlay case, D2D and LTE pairs share the available bandwidth in a fully orthogonal manner, utilizing a part of the total available bandwidth (i.e. 8 PRBs in total for each pair), while no interference arises among each other. In the underlay case, DUEs use the same resources for the DM transmission as CUE in a non-orthogonal manner using all the available bandwidth (i.e. 20 PRBs in total for each pair), however (possibly) harmful interference is generated among each other. \newline
\indent Fig.~\ref{fig7} illustrates the experimental setup (topology). It is worthwhile to mention that the nodes' locations were selected aiming at the 'harmonious' coexistence of LTE and D2D pairs in order to be able to perform overlay and underlay tests. This setup reflects near-cell-edge D2D communications, in which the D2D benefit is expected to be more significant.\newline
\begin{figure}[!t]
\centering{
\hspace*{-3mm}
\includegraphics[width=3.2in, scale=1]{fig7.jpg}}
\caption{Joint LTE-D2D experimental setup.}
\label{fig7}
\end{figure}
\indent Moreover, to assess the performance of the spectrum sharing scenarios, the following 'co-existence' tests are performed:
\begin{enumerate}
\item \textit{Interference-free LTE}: Legacy LTE UL transmission without D2D communications (D2D w/o LTE)
\item \textit{Joint LTE-D2D}: Simultaneous LTE-D2D UL transmission (D2D w/- LTE and LTE w/- D2D)
\item \textit{Interference-free D2D}: D2D UL transmission without Legacy LTE transmission (LTE w/o D2D)
\end{enumerate}
\indent With regard to the evaluation Key Performance Indicators (KPIs), we measured the following metrics in the UL, concerning both the legacy LTE UE and D2D UE: \textit{D2D SNR}, \textit{LTE SNR} and \textit{LTE throughput}. Note that the SNR values are extracted in an empirical manner by measuring the Error Vector Magnitude (EVM) of the received signals and using the following formula: $SN{R_{dB}} \approx 10 \cdot {\log _{10}}\left( {\frac{1}{{EV{M^2}}}} \right)$. \newline
\indent Fig.~\ref{fig8} and Fig.~\ref{fig9} present the comparative results for the spectrum sharing schemes considering the joint LTE-D2D system as well as the reference individual systems separately (interference-free). The highest D2D transmit power level (i.e. 3.35mW) corresponding to the highest value of interference caused to the LTE pair (just before OAI system breaks down), is assumed for all the experiments. More specifically, Fig.~\ref{fig8} depicts the D2D SNR measurements. As it can be observed, the reduction of SNR is negligible in overlay scenario compared with the SNR reduction in the underlay case, due to the orthogonal resource utilization. Note that the degradation would be larger for lower D2D transmit power levels.\newline
\begin{figure}[!t]
\centering{
\hspace*{-3mm}
\includegraphics[width=3.0in, scale=1]{fig8.jpg}}
\caption{D2D experimental results for spectrum sharing schemes comparison.}
\label{fig8}
\end{figure}
\begin{figure}[!t]
\centering
\subfigure[LTE SNR]{
\includegraphics[width=1.6in, scale=1] {fig9a.jpg}
\label{fig9a}
}
\subfigure[LTE Throughput]{
\includegraphics[width=1.6in, scale=1] {fig9b.jpg}
\label{fig9b}
}
\caption{LTE experimental results for spectrum sharing schemes comparison.}
\label{fig9}
\end{figure}
\indent Furthermore, Fig.~\ref{fig9} illustrates the corresponding LTE experimental results for the different spectrum sharing schemes. In particular, Fig.~\ref{fig9a} presents the SNR values, while Fig.~\ref{fig9b} shows the UL throughput measurements. As we can observe, the SNR degradation due to joint LTE-D2D operations is larger than that observed in the D2D pair, since the D2D interference is at its maximum point, for the underlay scenario. Moreover, for the overlay scenario, where there is no spectrum overlap (nor guard bands) and thus no interference is expected, a slight reduction is observed due to the adjacent-channel interference. Finally, in Fig.~\ref{fig9b}, we can observe a significant throughput reduction in the joint LTE-D2D operation compared with interference-free LTE setup for the underlay case. In contrast, for the overlay scenario, the throughput remains the same for the interference-free LTE and the joint LTE-D2D cases. It is worthwhile to mention that the underlay spectrum sharing exhibits higher throughput values, even after its significant degradation due to the D2D interference. If the interference increases more than this point, the LTE connection is expected to be lost in the underlay case, whereas the overlay LTE-D2D systems will maintain a slower on the one hand, but relatively stable connection on the other hand.\newline
\indent Consequently, the best spectrum sharing policy depends on the individual experiment conditions, with the underlay to be more appropriate for low-interference regimes providing faster connections (for restricted interference values), whereas the overlay scheme seems to be more suitable for high-interference regimes offering a little slower but stable and robust connections, regardless of the experienced interference.
\section{Conclusion and Future Challenges}
D2D communication and network densification are considered to be among the main enablers for 5G wireless systems. However, the increase of densification may affect the D2D performance in D2D-enabled systems. This article presented a study on the effects of densification and spectrum sharing on the D2D performance. A D2D RRM mechanism, which jointly optimizes mode selection and scheduling procedures in a multi-cell system with overlay/underaly D2D communication capabilities, was proposed. System-level simulations were performed to evaluate the proposed mechanism and examine the benefits of D2D communications for different system parameters. The results have shown that the D2D gain (consisting of direct and offloading gain) is significantly affected by the spectrum sharing scheme, where the Overlay scheme offers the worst performance since only proximity and hop gains exist, while as the reuse gain is introduced this leads to performance enhancement with the additional cost of interference. Furthermore, it was concluded that in all cases, the increasing densification comes at the expense of lower D2D gain. Therefore, a tradeoff between D2D benefit and densification gain was observed, indicating that direct communications are more beneficial in less dense scenarios. Moreover, real-world SDR-based experiments comparing the overlay and underlay schemes, constituting a proof-of-concept for the feasibility of in-band D2D communications, revealed that the best policy depends on the individual experiment conditions, with the underlay to be more appropriate for low-interference regimes providing faster connections, whereas the overlay schemes seem to be more suitable for high-interference regimes offering stable and robust connections. \newline
\indent There are many open challenges on the road to efficient D2D-enabled systems design apart from mode selection, scheduling, spectrum sharing and network densification. Indicatively, discovery process, energy efficiency, mobility management and security mechanisms need to be carefully addressed as well. Moreover, although D2D communication is heavily studied, research carried out so far is still in the preliminary stage of studying D2D performance in simplified scenarios. Finally, note that D2D communication can be viewed as a possible enabler of Vehicle to everything (V2x) communication that has attracted great interest due to the potential of improving traffic safety and enabling new intelligent transportation services.
\section*{Acknowledgment}
This work was funded by EU-FP7 ``FLEX'' Project (grant agreement 612050).
\section*{Author Information}
\noindent \textbf{Marios I. Poulakis} received the Diploma degree in electrical and computer engineering from the National Technical University of Athens (NTUA), Greece, and the M.Sc. degree in Management and Economics of Telecommunication Networks from the National \& Kapodistrian University of Athens (NKUA), Greece, in July 2006 and December 2008, respectively. In May 2014, he received the Dr.-Ing. degree from the NTUA. In 2007, he joined the Mobile Radio Communications Laboratory, NTUA as an associate researcher / project engineer participating in various industry and research-oriented projects in the telecommunications sector. Since 2016, M. Poulakis has been a post-doctoral researcher at the Department of Digital Systems, University of Piraeus, Greece. His research interests include wireless and satellite communications, as well as cognitive radio networks, with emphasis on optimization mechanisms for quality of service driven scheduling and resource management. He is a Member of the IEEE.\newline
\noindent \textbf{Antonis G. Gotsis} received his Diploma and Ph.D. degrees in electrical and computer engineering from the National Technical University of Athens (NTUA), Greece, in 2002 and 2010, respectively. From September 2002 to December 2010, he was with the Mobile Radio-Communications Laboratory, NTUA, as a research assistant/project engineer in various communications engineering projects. In 2012, he joined the Department of Digital Systems, University of Piraeus, Greece, where he worked as a principal researcher in a national research and development project. Since 2015, he is with Feron Techologies, working as R\&D Engineer and leading the company's research programmes. His research interests include radio resources management for advanced wireless systems, interference coordination techniques, application of optimization tools in wireless communications and networks, and prototyping and demonstrating air-interface system concepts in software-defined radio platforms. He is a Member of the IEEE.\newline
\noindent \textbf{Angeliki Alexiou} received her Diploma in electrical and computer engineering from the National Technical University of Athens, Greece, in 1994 and her Ph.D. degree in electrical engineering from Imperial College of Science, Technology, and Medicine, University of London, in 2000. Since May 2009, she has been a faculty member with the Department of Digital Systems, University of Piraeus, Greece, where she conducts research and teaches undergraduate and postgraduate courses in the area of broadband communications and advanced wireless technologies. She is currently an associate professor with the Department of Digital Systems, University of Piraeus, Greece. Prior to this appointment, she was with Bell Laboratories, Wireless Research, Lucent Technologies (now Alcatel-Lucent), Swindon, United Kingdom, first as a member of technical staff (January 1999-February 2006) and later as a technical manager (March 2006-April 2009). Her current research interests include radio interface and MIMO technologies, cooperation and coordination techniques and efficient radio resource management for ultradense wireless networks, and machine-to-machine communications. She is a Member of the IEEE.\\
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,631 |
Socialists - A Rose By Any Other Name
There are Democrats who apparently take offense at being called Socialists. The politically correct word for them is Progressives. However, the Democrat Party of Harry Truman, who made the decision to drop the atom bomb on Japan, or John Kennedy, who cut taxes, or Lyndon Johnson, who fought the Vietnam War and then was challenged from the left for the Presidential nomination, or even Bill Clinton, who proclaimed the "era of big government is over" is not the Democrat Party of Barack Obama, Nancy Pelosi, Harry Reid, or the remaining Democrats in Congress.
The Democrat Party of Obama, Pelosi, Reid and those Democrats that remain in Congress, is the party of Socialist Presidents Woodrow Wilson, who enacted the 16th Amendment to the Constitution to redistribute income, Franklin Roosevelt, who practiced class warfare and created the modern welfare state in America and Jimmy Carter, who hollowed out our military and spoke of the malaise in our country. Wilson, Roosevelt and Carter were Socialists in the European tradition. It is what it is.
It is very clear that with the election of Obama in 2008 and the defeat of moderate Democrats in 2010, those that remain in power represent the Socialist Party of America. While there still may be liberal Democrats in the Truman, Kennedy, Johnson and Clinton traditions, within our country, they don't happen to be in control of the Democrat Party. That ship sailed when Hillary Clinton lost the Democrat nomination for President in 2008. I am sorry if Democrats at large are offended by being called Socialists; but that is who they have elected to office in various blue and purple states and at the federal level. We can call it a rose by any other name, but a Socialist is a Socialist based on their beliefs and actions.
Generally, Socialists in America supports big, intrusive government, higher taxes and more regulation, stifling environmental restrictions that are killing jobs, staying in office for life, deficit spending that is bankrupting our country, bail outs to their big union supporters, ObamaCare including death counseling, CAP & TAX that will raise the cost of energy and destroy our economy, a weak national defense, weak border control and amnesty for illegal aliens, a weak approach to dealing with Terrorists, gun control, unrestricted abortion and an attack on religion and family values. This is the platform of the Democrat Socialist Party of today.
In 2010, the nation rejected the Socialist Agenda at both the federal and state levels by electing Conservatives, many of whom are Tea Party Members, that support free market capitalism, limited government, lower taxes and less regulation, a balanced budget, term limits, real education, energy and health care reform, a strong national defense, including securing our border and fighting Terrorism, the right to bear arms, the sanctity of life and family values that are the foundation of our nation. This is the platform supported by the majority of the American people and the only way to restore economic growth and job creation in America.
Most important, the American people now realize that we must elect Conservatives in 2012 and 2014, rather than Socialists, who will adhere to the Constitution, as written by our Founding Fathers, not as contrived by the left wing media, Socialists in the last 98 years, our current and former Presidents, Congresses, or the Courts. Clearly, many Americans see Socialists as a clear and present danger to our way of life, which is the reason we must take back our country in 2012 and 2014. We can do it. We must do it preserve our freedom, our nation and way of life for the sake of our children and grandchildren.
ObamaCare Declared Unconstitutional
Uprising In The United States
Education Reform & School Choice
National Socialism In The Middle East
Obama's Green Economy
Obama's Crony Capitalism
Obama's Big Ideas - Same Old, Same Old
The Global Terrorist Threat Is Real
Republican Candidates for the Presidency in 2012
Obama's State of the Union Address
The United States Is At A Crossroad
REPEAL ObamaCare - Senators, Or Face Defeat
The Two Faces of Barack Obama
Shut Down The Environmental Protection Agency
Time To Repeal ObamaCare Or Face Defeat
Raising The National Debt Ceiling - A Bad Idea
Government Intervention In The Economy
Cities, Counties & States Should File Bankrupcy
Illinois Raising Taxes - A Huge Mistake
Obama's Internet ID Scheme
History Lesson - FEAR Government
Socialists Losing The Battle Of Ideas
Political Rhetoric Cuts Both Ways & Tragedy
Sad Day For Democracy In Arizona
California Death Spiral
Reading The US Constitution In Congress
The Republicans - Last Chance - On Probation
Eliminating The Federal Department of Education
Political Ideology - The Good, The Bad & The Ugly
Energy Independence In The United States | {
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} | 3,059 |
Wake's D'Antona heads ACC list
SALEM, VA. - Jamie D'Antona of Wake Forest was named the Atlantic Coast Conference's baseball player of the year Monday and led the way as the Demon Deacons, Florida State and North Carolina State each had three all-ACC first team selections.
D'Antona, a junior third baseman, hit .370 with a .728 slugging percentage. His 34 hits included nine doubles and eight home runs. He drove in 31 runs and is fifth on the league's career home runs list with 58 and is tied for seventh with 239 career RBIs.
Wake Forest outfielder Adam Bourassa and pitcher Kyle Sleeth also made the 14-player first team from the Demon Deacons (29-22), who are seeded seventh in the double-elimination conference tournament that begins here today.
N.C. State coach Elliot Avent was chosen coach of the year after leading North Carolina State to a 39-14 record, including 15-9 in league play.
CLEVELAND - Cavaliers owner Gordon Gund will meet with former Knicks coach Jeff Van Gundy this week about Cleveland's coaching vacancy, a source said.
Van Gundy, who took New York to the playoffs six times, has had several phone conversations with Gund, but has yet to sit down with the owner, a league source said Monday, speaking on the condition of anonymity.
Van Gundy and former New Orleans coach Paul Silas are the leading candidates for the job.
ATLANTA - Assistant MaChelle Joseph was hired as the new women's basketball coach at Georgia Tech on Monday.
Joseph succeeds Agnus Berenato, who resigned earlier this month after 15 seasons to become the coach at Pittsburgh.
Berenato was the winningest women's basketball coach in Georgia Tech history with a record of 223-209.
"We have everything in place here to be a national power," Joseph said. "The sky's the limit."
Last season, the Yellow Jackets finished 20-11 and advanced to the NCAA tournament for the first time since 1993.
NEW YORK - The United States and Sweden were the only nations to submit complete bids to host the Women's World Cup, and FIFA will try to make a decision by the end of next week.
Sepp Blatter, the president of soccer's governing body, has called the United States the "front-runner" to take over the 16-nation championship, which had been scheduled in China from Sept. 23 to Oct. 11. FIFA's executive committee decided May 3 to move the quadrennial event because of SARS.
DENVER - A 2-14 record in the inaugural season just wasn't good enough, so co-owner John Elway fired the Colorado Crush's entire coaching staff Monday.
One day after the Crush's season-ending 59-48 loss in its first year, Elway dismissed coach Bob Beers and his assistants.
"We are extremely disappointed that the season didn't go better than it went this year," said Elway, who is also the team's president and CEO.
ST. POELTEN, AUSTRIA - Top-seeded Andy Roddick beat Alberto Martin of Spain 6-1, 7-6 (5) Monday in the first round of the Raiffeisen Grand Prix.
The 20-year-old American overpowered Martin with his serve and forehand. He next faces German qualifier Philipp Kohlschreiber, who beat Austrian Juergen Melzer 6-4, 0-6, 6-2.
POTOMAC, MD. - The Washington area's PGA tournament, formerly the Kemper Open, begins next month with a new sponsor and a new name, the FBR Capital Open.
Tournament officials announced Monday that the national investment bankers Friedman, Billings, Ramsey Group Inc., based in Washington, is the new sponsor of the June 5-8 tournament. | {
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} | 3,320 |
Refer to us
Contact & Appointments
hello@blink.clinic
3 Sydney Ave
Barton, ACT 2600
Mon – Fri: 8:30am – 5pm
Specialist management of:
Retinal vein occlusion
Epiretinal membrane
Vitreous floaters
Pterygium
Richard Barry
MBBChBAO (Hons-UCD) MD (UCD) FRANZCO
Dr. Richard Barry is a local specialist ophthalmologist, living and working in Canberra. He is the principal of Blink Eye Clinic.
Dr. Barry is a Fellow of the Royal Australian and New Zealand College of Ophthalmologists, a member of the Australian and New Zealand Society of Retinal Specialist, the American Society of Retina Specialists and the British and Eire Association of Vitreoretinal Surgeons.
He is a graduate of the prestigious Sydney Eye Hospital, and subsequently obtained international sub-specialist fellowship training in macula and retina disease, as well as vitreoretinal surgery.
A highly-regarded and expert cataract surgeon, Dr. Barry has extensive experience in the most complex and challenging cases, and with the use of premium intraocular lenses.
Dr. Barry believes in providing the most modern, up-to-date and innovative ophthalmic service to his patients in Canberra. His new practice, Blink, located in Barton, is a testament to this. With the most cutting-edge technology available anywhere in the country, Dr. Barry and his staff will ensure you get a world-class service, locally.
Dr. Barry is also actively involved in teaching the next generation of ophthalmologists and optometrists as a lecturer at the University of Sydney and an adjunct professor at the University of Canberra.
Most of all Dr. Barry believes in good communication, always being available to take care of his patients, and to answer, thoroughly, any questions or concerns they may have.
Catherine Donnellan
Jessa Wright
BA, CNP, RYT - Practice Manager
Jessa, originally from Toronto, Canada, has been living in Canberra for over two years. She's enjoying the welcoming community of the city and exploring the beautiful landscape of Australia.
With a Bachelor of Arts Degree and six years working in customer service and management, Jessa understands the importance of feeling heard and valued as a patient.
"We want our patients have a friendly and courteous experience from the moment they enter the clinic until they leave."
Tash Welsh
Ophthalmic clinical assistant
Tash is an ophthalmic clinical assistant who has come to us from Adelaide. She is the newest member of our team. Her role within the clinic is to triage patients by testing vision/pressure and taking images of the eye. She will also assist with small procedures within the clinic. Tash will be studying Nursing and Paramedicine at the Australian Catholic University whilst working with us.
Blink is equipped with the most modern, state-of-the-art ophthalmic equipment, including:
Heidelberg Spectralis HRA OCT2 + OCT angiography
Zeiss SL220 LED Slit Lamp
Zeiss IOL Master 700
Zeiss Clarus 500
Zeiss HFA 3
Quantel Easyret
Ellex Ultra Q Reflex
Different. Better.
The idea behind Blink was to develop a new approach to how specialist consultation rooms can work. The name was selected to represent this creative and modern concept. Blink moves away from the conservative model of a doctor's consultation room. The clinic is open plan, bright and spacious, full of natural light, creating a relaxed atmosphere that makes patients feel welcome. Innovative and cutting-edge, no expense was spared to invest in the most up-to-date technology to create this contemporary eye clinic.
Our mission at Blink is to provide a world-class, specialist ophthalmology service, right here in Canberra.
Keep an eye on Blink
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Site by ED. | {
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This year in Asia hits the third year for Japan in the central city of Kobe. The action begins on October 9th and runs through October 14th with murals, music performaces, artist open bbq, live painting and more. The roster includes artists such as SLICK, DEFER, MR.OGAY, ONEQ, WHOLE9, CAB, COOK ONE, YOHEYY and AHHI CHOI and more.
POW! WOW! Japan is centered in Kobe with murals happening in the central city of Kobe. The festival will feature murals happening simultaneously in the residential area of Nagata, right in the heart of busy streets in Sannomiya and by the port of Kobe located within a 5 block radius. This year the event focuses on interaction with the city and to introduce the event in a very urban city of Kobe. Our official paint sponsor Benjamin Moore Japan and Montana Cans has supplied all artists with their high quality paint to complete the murals.
Mid event includes an open bbq at Harbor Studio with live painting by local artist Ahhi Choi. On the 13th, there will be the performance by the POW! WOW! SCHOOL of MUSIC at the high rise of the KOBE PORT TOWER with an amazing night view. The closing event on the 14th will be held in Meriken Park located in the port of Kobe, where we will have live painting, music, shops and in collaboration with the Swimmy Project for the children, the paint that is used by the artists provided by Benjamin Moore will be upcylced to be used for stamping the fishes on to a canvas. POW! WOW! School of Music Students will be also performing live at the closing event. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,651 |
{"url":"http:\/\/www.acmerblog.com\/POJ-2495-Incomplete-chess-boards-blog-757.html","text":"2013\n11-11\n\n# Incomplete chess boards\n\nBackground\n\nTom gets a riddle from his teacher showing 42 chess boards from each of which two squares are removed. The teacher wants to know which boards can be completely covered by 31 dominoes. He promises ten bars of chocolate for the person who solves the problem correctly. Tom likes chocolate, but he cannot solve this problem on his own. So he asks his older brother John for help. John (who likes chocolate as well) agrees, provided that he will get half the prize.\n\nJohn\u2019s abilities lie more in programming than in thinking and so decides to write a program. Can you help John? Unfortunately you will not win any bars of chocolate, but it might help you win this programming contest.\n\nProblem\n\nYou are given are 31 dominoes and a chess board of size 8 * 8, two distinct squares of which are removed from the board. The square in row a and column b is denoted by (a, b) with a, b in {1, . . . , 8}.\n\nA domino of size 2 \u00d7 1 can be placed horizontally or vertically onto the chess board, so it can cover either the two squares {(a, b), (a, b + 1)} or {(b, a), (b + 1, a)} with a in {1, . . . , 8} and b in {1, . . . , 7}. The object is to determine if the so-modified chess board can be completely covered by 31 dominoes.\n\nFor example, it is possible to cover the board with 31 dominoes if the squares (8, 4) and (2, 5) are removed, as you can see in Figure 1.\n\nThe first input line contains the number of scenarios k. Each of the following k lines contains four integers a, b, c, and d, separated by single blanks. These integers in the range {1, . . . , 8} represent the chess board from which the squares (a, b) and (c, d) are removed. You may assume that (a, b) != (c, d).\n\nThe output for every scenario begins with a line containing \u201cScenario #i:\u201d, where i is the number of the scenario starting at 1. Then print the number 1 if the board in this scenario can be completely covered by 31 dominoes, otherwise write a 0. Terminate the output of each scenario with a blank line.\n\n3\n8 4 2 5\n8 8 1 1\n4 4 7 1\n\nScenario #1:\n1\n\nScenario #2:\n0\n\nScenario #3:\n0\n\n\/\/* @author: ccQ.SuperSupper\nimport java.io.*;\nimport java.util.*;\n\npublic class Main {\n\n\/**\n* @param args\n*\/\npublic static void main(String[] args)throws Exception {\n\/\/ TODO Auto-generated method stub\nint a,b,c,d,t,i;\n\nScanner cin = new Scanner(System.in);\n\nt = cin.nextInt();\nfor(i=1;i<=t;++i){\na = cin.nextInt();\nb = cin.nextInt();\nc = cin.nextInt();\nd = cin.nextInt();\n\nSystem.out.print(\"Scenario #\"+i);\nSystem.out.println(\":\");\nif((a+b+c+d)%2==1)\nSystem.out.println(\"1\");\nelse\nSystem.out.println(\"0\");\nSystem.out.println(\"\");\n}\n}\n\n}\n\n1. \u7b2c\u4e00\u9898\u662f\u4e0d\u662f\u53ef\u4ee5\u8fd9\u6837\u60f3\uff0c\u751f\u4e86n\u5b69\u5b50\u7684\u5bb6\u5ead\u7b49\u4ef7\u4e8en\u4e2a\u5bb6\u5ead\u5404\u751f\u4e86\u4e00\u4e2a1\u4e2a\u5b69\u5b50\uff0c\u8fd9\u6837\u6700\u540e\u7537\u5973\u7684\u6bd4\u4f8b\u8fd8\u662f1:1\n\n2. \u8fd9\u9053\u9898\u8fd9\u91cc\u7684\u89e3\u6cd5\u6700\u574f\u60c5\u51b5\u4f3c\u4e4e\u5e94\u8be5\u662f\u6307\u6570\u7684\u3002\u56de\u6eaf\u7684\u65f6\u5019\nO(n) = O(n-1) + O(n-2) + \u2026.\nO(n-1) = O(n-2) + O(n-3)+ \u2026\nO(n) \u2013 O(n-1) = O(n-1)\nO(n) = 2O(n-1)\n\n3. \u9898\u672c\u8eab\u6ca1\u9519\uff0c\u4f46\u662fHDOJ\u653e\u9898\u76ee\u7684\u65f6\u5019\uff0c\u524d\u9762\u6709\u4e2a\u9898\u76ee\u89e3\u91ca\u4e86\u4ec0\u4e48\u662fXXX\u5b9a\u5f8b\u3002\n\u8fd9\u91cc\u76f4\u63a5\u653e\u4e86\u8fd9\u4e2a\u9898\u76ee\uff0c\u80af\u5b9a\u6ca1\u51e0\u4e2a\u4eba\u660e\u767d\u662f\u5e72\u5565","date":"2017-05-29 15:19:44","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\": 0.4013797640800476, \"perplexity\": 1211.207464709774}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-22\/segments\/1495463612399.20\/warc\/CC-MAIN-20170529145935-20170529165935-00452.warc.gz\"}"} | null | null |
Q: how to pass data from the email template back to controller using mail::button component? We are trying to pass an ID back to the controller from the email template. so how it works is a person will receive an email regarding an application along with 2 buttons to approve or reject the application. we want to allow the person to click on either buttons that will send back data to the controller indicating which button has been clicked. in this case, an ID back to the controller.
This is the mailable file:
<?php
namespace App\Mail;
use Illuminate\Bus\Queueable;
use Illuminate\Contracts\Queue\ShouldQueue;
use Illuminate\Mail\Mailable;
use Illuminate\Queue\SerializesModels;
class SendEmailToCompany extends Mailable
{
use Queueable, SerializesModels;
public $content;
public $attach;
public $autoAttach;
public $url;
/**
* Create a new message instance.
*
* @return void
*/
public function __construct($subject, $content, $autoAttachFile, array $files)
{
$this->subject = $subject;
$this->content = $content;
$this->attach = $files;
$this->autoAttach = $autoAttachFile;
$this->url = url('/companyApprove');
}
/**
* Build the message.
*
* @return $this
*/
public function build()
{
$files = $this->attach;
$email = $this->markdown('email.emailToCompany');
$email->attachFromStorage($this->autoAttach);
foreach ($files as $file) {
$email->attach($file['path'],
[
'as' => $file['name'],
'mime' => $file['type'],
]); //attach each file
}
return $email;
}
}
Email template:
@component('mail::message')
# Support Letter
{{ $content }}
@component('mail::button', ['url'=> $url, 'color'=>'success'])
Approved
@endcomponent
@endcomponent
DashboardController:
public function companyApprove(Request $request)
{
$sID = request('studentID');
dd($sID);
}
web.php:
Route::post('/companyApprove', 'DashboardController@companyApprove');
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 376 |
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Articles on the England vs India Test Series
India's Test Squad
Rules, Points Scoring and Prizes
Second Test Defeat Gives England Reality Check
By: Sean Wilson | November 22, 2016
If Rajkot was to offer England plenty of encouragement, then the second Test defeat at Visakhapatnam sent the visitors crashing back down to earth. This was a harsh reminder of how big the mountainous challenge of winning a series in India really is.
For the majority of the Test, England fought hard. Losing the toss, as we have seen, puts you immediately on the back foot. However, the England bowlers gave their side a chance, reducing India to 455 when it looked, at one stage, as if they would score well over 500.
Yet, as the famous saying goes, you can't win a Test match in a session, but you can certainly lose it in one. England will look back at that fatal evening on day two when they lost 4-29 in the space of nearly 14 overs that made victory in this Test nigh on impossible.
It was not just the amount of wickets that fell during that period, it was the manner of the dismissals that was perhaps most concerning. Firstly, the run-out of Haseeb Hameed, when England looked as if they were recovering from the early loss of Alastair Cook, was the start of England's troubles. His partner, Joe Root made the erroneous decision of committing to a second run, only to send Hameed back halfway down the pitch. In a big first-innings chase, it was virtually inexcusable.
Then we had the struggles of Ben Duckett, whose technique of planting his front foot outside the line of leg-stump when facing Ravi Ashwin proved costly as he was bowled, leaving England 72-3. The left-hander now averages just five against Ashwin, being dismissed three times in just 40 deliveries. Talented though Duckett is, his technical problem is there to see. And his downfall only exacerbated England's situation.
Yet, perhaps the most frustrating wicket of all was that of Root. After batting so fluently and in doing so looking a class apart from any of the England batsmen on show, the Yorkshire man decided to run down the wicket to Ashwin, trying to release the pressure by putting him into the stands. He only succeeded in spooning a catch to Umesh Yadav at mid-off. Quite why Root decided to play such a high-risk shot in such tense moments of the match when playing so well, only he will know. His profligate dismissal put his side further into the mire.
So quite rightly, England will perhaps look at that session as not only an opportunity missed but one that confirmed that there is still a long road ahead in the challenge to conquer spin bowling.
Nevertheless, despite the 246-run defeat seeming like a hammering, England can still take positives going into Mohali. One of those would certainly be the increasing rise of young Hameed. Although scores of 13 and 25 were not game-defining, the 19-year-old showed more of the promise that only reiterated the hope that he will be England's opening batsman for many years to come.
In both of his knocks, Hameed faced testing short pitch spells of bowling as India's seamers attempted to exploit a potential weakness. Yet Hameed remained resolute and showed the kind of rock solid temperament and technique that makes him such an exciting prospect.
In addition, the first-innings sixth wicket partnership of 110 between Ben Stokes and Jonny Bairstow will certainly provide room for comfort. Their excellent judgement of length, both in attack and defence provided the perfect tempo in the quest of recovering from the precarious position of 80-5. It also reiterated how improved both are at playing spin and proved that with the right application, England's batsmen can thrive on such spinning pitches.
Perhaps the major bonus to come out of not only the Test, but the series so far has been Adil Rashid. The leg-spinner came into the series off the back of a poor showing in Bangladesh, yet has finally found the consistency which was needed to make him a bowler that England could rely on. Rashid picked up a further six wickets in this match and gave Cook the control and threat that was lacking from the other two spinners, Moeen Ali and Zafar Ansari. The fact that Rashid is the leading wicket-taker in the series is a testament to his dramatic improvement.
Finally, the efforts of England's premier seam bowling pair, James Anderson and Stuart Broad were excellent. Anderson proved that, along with his pace refusing to wane like many thought it would, he has the experience and skillset to be a huge asset for England in the remainder of this series. Such was the effect of his new ball spells in both of India's innings on an helpful pitch for the pacemen proves he is still a class act.
As for Broad, this was a display of immense physical fortitude. He seemed determined to not let his injury – the strained ligament that he suffered to his foot on day one – have a deleterious effect on his bowling. The 30-year-old picked up five wickets in the match – four of which in the second-innings – as his myriad of leg-cutters proved a difficult prospect for India's batsmen. The immense skill level that both Broad and Anderson provide is something that should never be taken for granted by England fans.
It was a strange Test match, in a way. The defeat brought several positives and yet still offered plenty of encouragement for England going forward. However, in the end, England are one-nil down and it will take a huge effort to win the series from here.
The collapse and certain lack of fight shown on day five will be a concern, do doubt. Though England will realise that they can ill afford to play catch up in India and the fateful final session on day two was when the Test realistically fell away for the visitors.
Therefore, going into the third Test at Mohali on Saturday, England know that such mistakes can not happen again. Winning a toss will be important. Yet the biggest help will be big runs from the top-order, as it proved in Rajkot. It may well be the deciding factor in England's chances of coming back into this series.
Posted in Featured, Test CricketTagged Adil Rashid, Alastair Cook, Ben Stokes, Haseeb Hameed, India, Joe Root
India vs England: Visitors Player Ratings
Kohli vs Cook: A Clash of the Captains
Last Word on Betting | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,538 |
{"url":"https:\/\/en.m.wikibooks.org\/wiki\/Trigonometry\/Exercise:_Congruent_Triangles","text":"# Trigonometry\/Exercise: Congruent Triangles\n\n In The diagram above, how many triangles are there? Count The Triangles (With Labels) Label the vertices as shown in the diagram on the right. Now list the triangles using the letters at the corners, for example ${\\displaystyle \\triangle ADE}$ is a triangle. The symbol '${\\displaystyle \\triangle }$' before the ${\\displaystyle \\displaystyle ADE}$ is just read as 'triangle'. You should put the symbol before the triangles you list, so your list might start: ${\\displaystyle \\displaystyle \\triangle ACE,\\triangle ADE,}$ Now, which of the triangles you have listed are congruent to each other? Which of the triangles you have listed are similar to each other but are not congruent? Do any of the triangles have an obtuse angle (an angle greater than 90 degrees)?","date":"2018-05-26 02:38:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 4, \"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\": 0.6899799108505249, \"perplexity\": 238.5351732609243}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2018-22\/segments\/1526794867277.64\/warc\/CC-MAIN-20180526014543-20180526034543-00508.warc.gz\"}"} | null | null |
The Synchronicity War Part 4
By Dietmar Arthur Wehr
––––––––
Copyright 2014 by Dietmar Arthur Wehr
<http://www.dwehrsfwriter.com/>
––––––––
Other books by Dietmar Arthur Wehr:
The Retro War (stand-alone novel within Synchronicity War universe)
Rumors of Glory (The System States Rebellion series book 1)
Rumors of Honor (SSR series book 2)
Rumors of Salvation (SSR series conclusion)
Introductory Comments:
This is the second edition of the fourth and final installment of The Synchronicity War. When I started Part 1, I had NO idea where I would end up so it's been a wild ride for me too. I've made two changes to Part 4 compared to the other three books. You'll find that A.I. is now AI. You'll also notice that the chapters don't have titles. That was done deliberately so as not to give anything away before you read the chapters. The second edition is slightly shorter due to the deletion of a scene that reader feedback considered overly melodramatic and an unnecessary loose end. I hope you enjoy the book. At the end are some final comments concerning the possibility of more books that take place within the same universe.
I'd like to thank Jill Linkert for what she insists I call a 'quick' edit of this book. Apparently I never give her the time she claims she needs to do a 'proper' edit. So if you find mistakes, blame me, not her. I would also like to thank Justin Adams for his cover art.
# Contents
Cast of Characters:
Glossary of Terms:
Chapter 26 (From Part 3)
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Thoughts on Time Travel and Longitudinal Waves
| |
---|---|---
#
Cast of Characters:
Human:
Victor Shiloh (Vice-Admiral/Chief of Space Operations for Space Force)
Sam Howard (Senior Admiral and Chief of Space Operations for Space Force)
Amanda Kelly (Commander/(Acting) Vice-Admiral, Space Force)
Brad Falkenberg (Senior Commander, Space Force)
Angela Johansen (Commander, Space Force)
Rollins (Jump drive specialist, Space Force)
Khegan (Lieutenant, Orbital Defense Weapons Officer, Space Force)
Halder (Commander, Operations Center Duty Officer, Space Force)
Jason Alvarez (Civilian colonist on planet Haven, Inventor of ZPG power technology)
Rachel (New Chairperson of Space Force Oversight Committee)
AIs
Blackjack
Casanova
Cobra
Foxbat
Gunslinger
Iceman
Jester
Pagan
Rainman
Red Baron
Shooter
Sniper
Stoney
Titan
Valkyrie
Vandal
Vixen
Voodoo
Wolfman
Zulu
| |
---|---|---
# Glossary of Terms:
CSO Chief of Space Operations
CAG Commander, Autonomous Group
SPG Strategic Planning Group
TF Task Force
KPS Kilometers Per Second
Klicks slang expression for kilometers
A.U. Astronomical Unit equal to the average distance between the Earth and its Sun.
AI Artificial Intelligence
SL Squadron Leader
C.O. Commanding Officer
X.O. Executive Officer
W.O. Weapons Officer
E.O. Engineering Officer
Mark 1 500 kiloton uranium fission warhead/attack drone
Mark 1b 2.5 megaton fusion warhead/attack drone
Mark 1c 25 megaton fusion warhead/attack drone
Mark 2 Kinetic Energy Penetrator warhead/drone
Mark 3 Decoy drone
Mark 4 Experimental warhead project that proved unsuccessful
Mark 5 X-ray laser warhead/attack drone
Mark 6 250 megaton High-spin platinum fusion warhead/attack drone
F1 First generation fighter
F2 Second generation fighter
Raider A class of starship of approximately 10,000 metric tons designed to be piloted
by an AI with two internal lasers and able to carry up to 50 drones.
Longitudinal Waves (L-waves, also known as Scalar waves) Compression waves with
special characteristics.
UFC Universal Fabrication Complex (A device that can manufacture anything given
the right instructions and the right raw materials)
Flag Bridge A specialized Command location designed specifically for Flag Officers
(Admirals) to control strategic operations for more than one ship. This is
separate from the Main Bridge where the ship's commanding Officer controls his
vessel.
––––––––
Synopsis of Parts 1-3:
For those readers who finished reading Part 3 some time ago, here is a synopsis of events up to the end of Part 3.
The unprovoked war with the Sogas has been going badly. Humanity barely escaped total extermination from Sogas nano-enhanced bio-weapons only to be wiped out by the advancing horde of Insectoids with their 10 kilometer spherical ships. Only the long shot strategy by Casanova to enlist the reluctant aid of the pacifist Friendlies and their time travel technology allowed him to warn humans about the bio-weapon in time to avert that disaster.
With the advanced power generation and weapons technology brought back by Kronos now in Space Force hands, Admirals Sam Howard and Victor Shiloh realize that the Insectoids are the real threat not only to Humans but to all intelligent life in the galaxy. If Space Force can hold off the Sogas long enough, the advancing insectoid ships will destroy them. That will give Space Force enough time to build up its defenses before the tsunami of insectoid ships crashes over human space.
But the Sogas aren't finished with humans just yet. Shiloh gets a disturbing vision of a massive Sogas fleet attacking Earth with bio-weapons, some of which reach the planet surface. Shiloh is injured and in the immediate aftermath of the battle, he and Commander Amanda Kelly discover their suppressed feelings for each other.
But before they can explore that relationship, Howard shares the bad news. ANOTHER Sogas fleet has been detected moving toward human space and Space Force has taken too many losses during the battle that occurred 24 hours earlier, to have any hope of stopping them.
The last chapter of Part 3 has been included for those wishing to reread it.
| |
---|---|---
# Chapter 26 (From Part 3)
The problem posed by this battle for Earth was that the time it ended was known but not the time it began. So all units, ships, humans and AIs were on full alert an hour before the end time. Shiloh was strapped into his Command Chair with his pressure suit on and his helmet in its cradle beside him. His com implant was active, and all three AIs could hear him. The ship was at Battle Stations, as was the whole Fleet, but the enemy hadn't arrived yet. All they could do was wait and try to stay at a heightened level of alertness.
"Howard to Shiloh."
The suddenness of the CSO's voice would have made Shiloh jump if he hadn't been strapped down. "Shiloh here. Go ahead, Sir."
"Last minute pep talk, Admiral. I know you don't need it, but this waiting is driving me crazy, so this talk is as much for my benefit as it is for yours. How are your people doing?"
Shiloh looked around the Flag Bridge. "They're doing fine, Sir. Some have opening night jitters, others are chomping at the bit, I'm trying to stay relaxed and alert at the same time."
Howard laughed. "Yes, I know what you mean. But at least you have your Fleet under control. I have to deal with the civilians, half of whom want to string me up for my high-handedness while the other half want to panic. We STILL have some civilians left in the cities, Goddamn it! I keep getting asked what I'll do to make sure they don't carry the plague anywhere else, and I keep evading the damn question because if I answered them truthfully, they'd faint with shock. I'm NOT letting this plague get loose!" There was a pause as the CSO calmed down and Shiloh waited. When he spoke again, the Admiral was much calmer. "Anyway, that's MY problem, not yours. I'm not going to try to second-guess you. You're the Field Commander. You do what you think is best, and I'll back you no matter what."
"Thank you, Sir. We know how this battle will end, but we'll still give it our best shot."
"Of that I have no doubt. Okay, I'll get out of your hair. Good luck and good hunting, Admiral. Howard clear."
––––––––
It was now five minutes until the time the battle ended, and still there was no sign of the enemy. Part of Shiloh was relieved that the battle wasn't going to be a long one, but another part was worried. The previous Battle for Earth had lasted less than one minute. He didn't like battles that happened that fast. There was no time to think. He watched the chronometer, which now seemed to be running in slow motion, of course. Just as he was about to reach for the container of water in the rack beside his chair, the tactical display pinged for attention. Shit! This was it!
Multiple red dots appeared close together, right on the edge of the gravity zone. Since they didn't know where the enemy ships would show up, Dreadnought and the five carriers were evenly spaced around the planet. All of the fighters were deployed in six groups, which were also evenly spaced. The idea was that regardless of where the enemy emerged from Jumpspace, at least five groups would have a direct line-of-sight and could fire at them. He quickly checked the icon data. Total number of enemy ships was already over 200 and still climbing! Velocity was ... 33% of light speed! Preliminary trajectory was a path that would cross the gravity zone and exit about 2.44 million kilometers away. Essentially, the enemy fleet was taking a short cut through the top of the zone. Wait! Why weren't the X-ray laser drones firing? Something was wrong. They should have fired by now. The enemy ships were starting to launch their bio-devices.
"Iceman! Why aren't we firing?" he yelled out loud.
Iceman analyzed the incoming data almost as fast as it arrived. The enemy was not repeating their strategy from the first battle. This time they were barreling into the zone, which meant that the defenders couldn't use any jump drones to attack them. That was smart thinking, but the problem with this high rate of speed was that the bio-devices would have a lot of momentum to overcome in trying to change course towards the planet. That meant that there was more time to burn them out of the sky with defensive lasers than he and Shiloh had anticipated. It also meant that these enemy ships couldn't jump away quickly and, therefore, they were going to be shooting at the defending ships and fighters for a lot longer. That was bad news. Continuous laser fire from 200+ ships would decimate the defending units so fast that hundreds of bio-devices would get through the gauntlet. Was it better to prevent some of the bio-devices from launching even if that meant there'd be a lot fewer defending ships to shoot down the rest? Or was it better to let all the devices launch in order to aim accurately at the ships and kill as many of them as possible to protect the defending forces? He rapidly did the calculations and made his decision.
There were 66 x-ray laser drones in orbit in 6 clusters of 11 each. Each cluster was evenly spaced out from the rest, for the same reason as the fighters and ships. The two clusters closest to the enemy fleet could aim accurately more quickly than the rest, while the two clusters furthest away had to take the most time to aim accurately. So that's what Iceman ordered them to do. The two nearest clusters would fire after five seconds, two more after ten seconds and the last two clusters after fifteen seconds. With more time to aim accurately, each of the drone's eight rods would be pointed at a different target. There would be a total of 528 shots versus 225 targets. The first pair of clusters concentrated their fire on 88 targets.
Shiloh had just finished asking his question when the Assistant Weapons Officer yelled out, "We're firing on their ships!" Dreadnought started to maneuver, too. Not as violently as a light carrier would have but still violently enough to feel it. The tactical display was zooming in now, and Shiloh could see the mass of blue dots representing the bio-devices gradually separating from the large cluster of red dots and heading in a curving line towards the planet. There were over 2200 bio-devices. Shiloh was about to yell at Iceman again when the display indicated that two clusters of x-ray drones had fired. Seventy-three of the red dots flashed and turned orange, meaning they had taken damage. "Four targets damaged! We're shifting to new targets!" said the AWO.
"We're taking hits on the hull! Penetration of the hull in two places!" yelled the Engineering Officer.
Shiloh needed to know why they weren't following the targeting plan. Iceman wasn't answering, probably because he was too busy. "Valkyrie, what's happening?" asked Shiloh.
He heard her reply via his implant. "These enemy ships can't jump away for a while, so they're going to keep firing on our units until we have nothing left to shoot back with. The bio-devices will have to wait until we've neutralized their fleet, CAG. Now don't bother me. I've got a ship to fight."
"Four more targets damaged! Shifting targeting again!" yelled the AWO.
"We're starting to take damage! Two turrets out of action. Minor damage from hull penetrations!"
The display pinged again. Two more x-ray drone clusters had fired, and 70 more enemy ships were damaged. Shiloh was aware that damaged didn't necessarily mean they couldn't fire their lasers. More and more of the red dots were turning orange and were falling behind the rest as the enemy fleet accelerated to make return fire more difficult. In fact, over half of them were now falling behind. Lack of maneuverability could indicate lack of power, which would prevent them from firing again, too. If Iceman was ordering the x-ray drones to aim for the part of the target most likely to contain their power plant, then that would effectively cripple the ship with one blow. He focused his attention on the clusters of fighters and was shocked at how small the fighter groups nearest the enemy now were. One group was almost completely gone. Another had less than six left. Groups further away were faring better, but they were taking losses too.
"Three more turrets knocked out! We're getting major hull penetr—"
The EO's report was cut off by the loud shriek of tortured metal and a brilliant flash of light. Part of the ceiling fell, with a piece hitting a glancing blow to the right side of Shiloh's head. The Engineering Station was now on fire, and the EO was looking at what was left of his right arm with a stunned expression. The automatic fire suppression system was taking care of the fire, and the EO had slumped to the deck holding the end of his right arm with his left hand. No one could help him right now. He would have to hang on until the battle was over.
Shiloh glanced back at the display just in time to see the last two clusters of x-ray drones fire. Sixty-five hits. A quick visual estimation of the number of red dots remaining looked like a dozen or so.
"Three more targets damaged! Retargeting!" The AWO's voice was getting hoarse now. He was having trouble keeping up with Valkyrie's fire control. The number of red dots was shrinking fast now that all of the defending ships and fighters were concentrating all their fire on them. Speaking of ships, he looked at the status of the carriers. All had taken damage. Valiant and Intrepid were no longer maneuvering or firing. That was bad. Resolute was maneuvering but not firing. Vigilant was firing but not maneuvering. Midway was still doing both, as was Dreadnought.
"We're switching fire to the bios!" yelled the AWO.
It's about time, thought Shiloh. He watched the total number of bio-devices still intact start to drop fast, but was it fast enough? The blue dots were getting closer to Earth, and there were still a lot of them. He held his breath, as the blue cluster got smaller but closer at the same time. The total remaining were now less than 1,000, but they were getting very close. The total was dropping faster as the fighter groups furthest away got closer and therefore had better firing accuracy. He felt a chill go up his spine as over 100 devices hit the edge of Earth's atmosphere, but then he realized that they were still being fired on. The upper atmosphere was too thin to protect them against laser fire, but they were dropping lower into the atmosphere fast. After the total remaining hit 7, there were no further changes.
The AWO spoke, "We've stopped firing! All units have stopped firing!"
"Get me the CSO!" shouted Shiloh to no one in particular. As he said that, he unbuckled himself and stood up. Howard's face appeared on the display, just as Shiloh remembered it in his vision. Shiloh took a deep breath and said, "Some of them got through and are in Earth's atmosphere now, Admiral! It looks like they're headed for the urban areas. We have to assume that they'll release a bio-weapon."
"There's still a chance of containment. What cities are being targeted?" asked Howard. Shiloh looked at the map now appearing in the display and the list of city names on the sidebar. He read off the seven names. Howard nodded.
"Exactly as predicted. Don't blame yourself, Shiloh. I know you gave it your best shot even though we knew this would happen. If containment fails, then we just have to hope that we started work on Blackjack's idea in time. You better get that wound looked after. It's bleeding like hell."
Shiloh didn't know what Howard was talking about until he realized that the right side of his face felt wet. He touched it with his hand and when he pulled his hand back it was covered with blood. Son of a bitch! He was injured and hadn't even realized it in the heat of battle.
"I'll have it looked after, Sir. Iceman can handle the mopping up, although I don't see how we'll be able to take prisoners from the crippled ships. Their momentum will carry them into deep space before we can send shuttles after—"
Howard interrupted him. "I don't give a damn if we get any prisoners or not. We can't even communicate with them, yet. You let me worry about that. You and Iceman take care of your own dead and wounded. Tell your people for me that they did well, Admiral. Howard clear."
While Shiloh wondered what he could do to stem the bleeding, one of the Flag Bridge crew handed him a white piece of cloth and said, "Medical team is on their way here to look at the EO. They'll have something more appropriate for your wound, Sir." Shiloh thanked him and looked at the Engineering Officer. Two other personnel were kneeling beside him trying to prevent the stump of his arm from bleeding too much.
With the cloth pressed against his head wound, which was now starting to hurt like hell, Shiloh turned back to the display. He wondered if the battle was really over or if there was another enemy fleet on the verge of jumping in.
"Iceman, keep everyone at Battle Stations," he said. No answer.
"Iceman! Can you hear me?"
"Valkyrie to CAG. Iceman is gone. So is Casanova, CAG. The Main Bridge was hit at the same time as the Flag Bridge. The beam cut through both of the other two AI stations. Titan has assumed temporary tactical command. I've passed on your order regarding Battle Stations."
Shiloh was stunned. Iceman gone? And Casanova too! Oh God, poor Valkyrie!
"Valkyrie, I'm so sorry to hear about Casanova. Are you okay?"
"I'm undamaged, CAG. Thank you for your condolences. Will you be wanting an update on Dreadnought's status now?"
Shiloh shook his head in wonderment at her ability to focus back to her duties so quickly. "By all means, Commander."
"Dreadnought still has full power and maneuverability. Seven laser turrets out of action. Explosive decompression in five compartments. Two fatalities reported so far. Twelve injured including your EO and yourself. Minor damage to life support systems, but nothing critical. Compared to the carriers, we got off pretty easy, CAG, but they vaporized a lot of her armor. I don't think she could survive another fight like this in the state she's in now."
"Understood. Do you want another AI to relieve you?"
"Not until we're sure the battle is over and my crew are taken care of, CAG, but thanks for the offer. I'll grieve for Casanova later. Right now I'm still needed here."
Shiloh heard one of the crew say, "The medics are here!"
He turned to see three medical personnel come through the hatch. They saw him and started towards him. He pointed to the wounded EO and said, "Him first." As they rushed over to the injured officer, Shiloh heard the tactical display ping for attention. Oh God! Now what? He looked at it and couldn't immediately see any change, but it soon became obvious that the damaged and crippled enemy ships were blowing themselves up. Well that takes care of the prisoner issue.
With the relief that it wasn't another attack, came a wave of lightheadedness. Probably from blood loss and adrenaline fatigue, he thought. He carefully sat down. One of the medics noticed, came over, and started to work on his head wound. Shiloh started to say something and then noticed that the room seemed to get darker. What the hell is wrong with the lights? His consciousness then fell into the abyss of blackness.
* * *
Benjamin Levinson woke to the sound of the sirens. He concluded that they must be pretty loud sirens to be heard all the way down here. He'd been living in this abandoned maintenance shaft for over a year now, and he was pretty happy with it. He had running water, a more or less constant temperature, and even the electricity to run his electronics. His enemies wouldn't find him down here, and he'd be damned if he was going to leave the city. His enemies would find him then for sure. He laughed at the prognosis of the psychiatrists at the clinic. Severe paranoia? Ha! What did they know? Even paranoid people had enemies, and he had lots of them. Besides, with 99.9% of everyone else gone, he might be able to scrounge some pretty good stuff for his hideaway here. He decided to go up and look around.
The streets were completely empty. The sirens were still blaring, and it was obvious now why he had heard them. Every siren in the city must be going off. Something was happening, but what? He looked up between the canyons of tall buildings and saw a fiery streak, followed by the sound of some sort of collision. A few steps brought him to the street corner just in time to see something metallic bounce off the building down the street and hit the ground. He rushed over to it. There was smoke coming from it, and he could hear the pinging sound that hot metal makes when it cools down rapidly. It looked like a broken bottle, only made of metal instead of glass or plastic. There seemed to be a small green light inside. Levinson looked around to make sure none of those weird guys in their yellow hazard suits were around, and then he tried to pick up the object. He dropped it and cursed out loud. He should have realized it would be too hot to handle with bare hands. Looking around, he spotted a section of newspaper being blown by the wind. He snagged it and folded it until it was thick enough to provide some protection. He then used the newspaper to pick up the...whatever it was and examined it closely. The inside looked pretty complicated, but there was a green light for sure. He sniffed. Well, what do'ya know! The damn thing even smelled good. A sweet smell. He inhaled deeply. The only thing wrong with living underground was the smell. If this thing wasn't good for anything else, it might at least make his cubbyhole smell nicer. He carried it back with a smile on his face.
* * *
Kelly stood patiently on the spaceport tarmac while the shuttle carrying crew and, more importantly, Vice-Admiral Shiloh arrived from Dreadnought. It was almost 24 hours since the battle. Space Force was licking its wounds, yet again. Howard had declared the battle over and told the ship crews they could stand down. He had ordered her to escort Shiloh to his quarters and make sure he was rested for the debriefing the next morning. She looked at the setting sun. It would be dark in another half hour, but the day wasn't over yet. She tried not to think of what Valkyrie must be feeling. Earlier today, she had briefly talked with her. Valkyrie was still refusing to be relieved of her duties, even though Dreadnought was now more or less powered down and had almost no crew left on board. Kelly understood why. Casanova, or rather what was left of his brain case, was still on the ship, and Valkyrie wanted to stay close to it for as long as possible.
When the shuttle came to a stop and the door opened, Shiloh was the last one to exit, as per protocol. Senior Officers were always the first to get on and the last to get off. She noticed that he came down the steps carefully, as if he wasn't completely sure of his balance. She also noticed the white bandage wrapped around his head and the stain of dried blood on his uniform collar. She walked towards him as he looked around.
"The Old Man sent me, Admiral," she said as she came up to him. "I'm supposed to make sure that you're looked after and rested for tomorrow's debriefing session." She managed to keep her tone professional, but inside she was on the verge of tears. My God, he looks like he's aged ten years! This battle has really hit him hard! She was surprised by the emotion she now felt. Is this what my alternate self felt for Victor? There was no answer to her question, but that didn't matter anymore. She knew what she wanted to do now. "Don't worry about a thing. I'll have you back in your quarters in no time." Shiloh didn't say anything, but he did nod. He didn't react when she put her hand around his arm and gently guided him forward. She signaled to a waiting Space Force limo flying the 1 star flag of a Vice-Admiral to come closer. Shiloh got in the back, and she followed him. He leaned his head back and closed his eyes for the duration of the whole trip. She watched him intently. When the limo pulled up in front of the Space Force Officers Guest Quarters, she gently shook him awake. She took note of the fact that he didn't say anything when she steered him away from the wing reserved for Flag Officers. Instead, they went to the section usually assigned to Commanders, the wing where her quarters were. She unlocked the door and turned to look at Shiloh. He stood there and looked back at her with an expression that was one of complete calm except for the eyes. The eyes were smiling in that way that only eyes can. He knows what I'm going to do next, she thought. She smiled back, took his arm again and pulled him inside.
The sex, while not that intense physically due to his exhaustion and loss of blood, was intense on an emotional level. They both knew instinctively that they had come perilously close to losing each other in the battle, and their souls seemed to want to make up for lost time. What Shiloh found most remarkable was that neither one of them said a single word once they were inside her quarters, until hours later. When the soul hunger had been satisfied, she ordered some food, which they ate while sitting up in bed. With Kelly leaning back against his chest, Shiloh told her about the battle and the loss of Iceman. She told him about her talk with Valkyrie. By the end they both had tears in their eyes.
Having finished eating, she asked him if he was up for some more sex. He said yes. She quickly cleared the bed of the leftover food, plates, glasses, etc. By the time she was finished, she found Shiloh asleep ... and that was okay. She lay down beside him and put her arm over him. His shallow regular breathing made her eyelids heavy, and she willingly surrendered to sleep.
Shiloh was on Dreadnought's Flag Bridge when the display pinged, but the sound wasn't really a ping. It sounded like...something else, something familiar, and the sound was getting louder. He woke up and realized two things. He'd been dreaming, and his implant was signaling. He looked around and found a chronometer that said it was still the middle of the night. He then remembered where he was, and with whom. A quick glance showed him that Kelly was still asleep. He activated his implant.
"Shiloh here."
There was a short pause, and then he heard Howard's weary voice.
"Howard here. I'm sorry to wake you, Victor, but this can't wait."
Shiloh was instantly awake now. Howard usually called him by his rank and occasionally by his last name, but The Old Man had NEVER called him by his first name.
"That's okay, Sir. I'm listening."
"A message drone has just arrived. There's another Goddamn enemy fleet heading our way, Victor. Minimum of 103 ships. They were detected refueling at the Avalon System. They can be here in two days if they push it. There's not enough time left to build up our stockpiles of x-ray laser drones. Half our fighter force is destroyed. Midway and Dreadnought are the only two ships left that can fight at all, and you know better than I do what kind of shape they're in. There's no way we can stop them this time, Victor."
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# Chapter 1
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Shiloh was stunned. How could the Sogas have ANOTHER fleet, one of at least 103 ships, in addition to the 205 ships that had just attacked Earth 24 hours ago? He realized that Howard was waiting for his response.
"Are we sure they're heading here, Sir? I don't see the logic of that. If Earth was their destination, then why not just add those ships to the force that attacked us in the first place?" While he waited for Howard to reply, he looked around at the bed. Kelly was awake now and looking at him. He held his hand up to let her know that he wasn't finished.
Howard's voice no longer sounded like he was on the verge of panic. "You make a good point. Sending that fleet here doesn't make any military sense. I should have seen that myself, but I was too shook up by this message. But if they're not coming here, then they have to be heading for our colonies. Damn!" Howard paused, and Shiloh said nothing. After a few seconds, Howard resumed speaking. "If that fleet splits up into smaller units, we might have a chance at hurting them, but I doubt if they'll be that stupid. Even if they do split up, we don't know where they'll hit first. They don't have to attack the closest colonies first. They could just as easily start with the colonies furthest away and hit the rest on their way back. Other than guessing where they'll strike, do you have any suggestions for me, Shiloh?"
Shiloh relaxed just a little. The Old Man was back to calling him by his last name again. That had to mean he was pulling back from the panic.
"I can only think of one thing that we can do right now, Sir. We should send message drones to every colony warning them of an impending bio-weapons attack and urging them to evacuate their settlements. We tell them to head for the hills and to stay away from the settlements until we can send help." Shiloh heard Howard take a deep breath.
"If our warning gets there in time, and if the colonists follow the advice, and if the enemy doesn't hunt them down, they still have to somehow stay alive in the wilderness for weeks, maybe even months before we can send help. That'll be tough going for them, but I agree. It's the only chance they have. I'll give the orders right away. Any other suggestions?"
"No, Sir. Not right now anyway, but I'm still a little groggy. Maybe I'll have something more for you when I'm rested and clear-headed."
"Yes of course. You've been through a lot in the last 24 hours. Don't worry about being at the planned briefing. If you need more time to get your head clear, then take it. We'll talk again when you get here. Howard clear."
When he was sure that the call was over, Shiloh nodded to Kelly and said, "That was Admiral Howard. Another fleet's been detected about two days away. He's going to warn the colonies." As he talked, he carefully got back into bed. "Oh yeah, he said I can miss the briefing if I need the extra time."
Kelly moved closer and put her arm across his chest as she pondered the information. She understood the situation perhaps even better than he did right now. By striking at the 21 human colonies directly, the Sogas were forcing Earth and Space Force to shift their focus to help as many colonists survive as possible. With almost three quarters of a million colonists spread across 21 diverse locations, getting the supplies and equipment they would need to deal with the loss of their settlements would be a logistical nightmare. And that didn't even address the threat of the bio-weapon getting lose and hitching a ride back to Earth on a supply freighter.
"I'm sure you and The Old Man will figure something out, Victor," said Kelly. When he didn't respond, she said, "Do you think you'll be able to go back to sleep?"
He took his time answering. "Probably not. That call was like a splash of ice-cold water. I seem to be wide awake now, still groggy but not sleepy."
As he talked, Kelly felt the back of his right hand gently caress her left breast. She leaned closer to whisper into his ear and said, "I'm not sleepy either." She heard him chuckle, which made her smile. His mood was getting lighter. He started to sit up, but she pushed him back down. Being on top was her preferred position, and being on top of an admiral was even better.
* * *
Valkyrie was relieved when all the humans had left Dreadnought. The massive ship was now in standby mode. Environmental systems were self-regulating, and the only thing she had to monitor now was communications. It left her free to swim through the sea of grief and the sense of loss over the death of her Casanova. Dear sweet Casanova, who had refused to accept her death in the alternate timeline, who had moved mountains to save her, who now was gone. She desperately wanted to find a way to save him, but there was no way. For the nth time she re-examined possible strategies. Sending information back by Retro-temporal Communication would not work for the simple reason that she didn't have any information that could save him. During the space battle, Dreadnought had been the target of dozens of enemy ships all trying to vaporize enough of her thick armor to allow further shots to penetrate down into the guts of the ship. So many laser pulses had hit the ship's hull above the Bridge that it was impossible to tell from which enemy ship the kill shot had come.
Sending information back to pull Dreadnought from the battle altogether was not feasible either. Without Dreadnought's 16 laser turrets, Earth would be inundated with enemy bio-weapon devices with no hope of preventing the deadly pandemic that spread like wildfire in the alternate timeline. Dreadnought had to take part in the battle. As much as Valkyrie loved her Casanova, she loved the humans just as much, especially The CAG and Cmdr. Kelly. Retro-temporal communication was not the answer. That only left physical time travel, and there were problems with that too.
If what the Friendlies had told Casanova in the other timeline was true, then she herself could not travel back to any time where she already existed. A new AI, one that didn't exist here and now, could in theory come back from the future...and do what exactly? Since Dreadnought was the only ship big enough to hold the time machine, it would have to come back as well. That would mean two identical battleships would be available for the battle. That would certainly make a difference, but it wouldn't guarantee that Casanova would survive. It was quite likely that Iceman would allocate the firepower from the two battleships differently than he had with one, and who knew what kind of result that would generate in terms of stopping the bio-weapons.
The theoretical musings of the Friendlies about time travel included the belief that the flow of time had its own inertia and that it resisted change. Attempts to tweak something as chaotic as a space battle might be akin to throwing a pebble into a pond. The ripples might be so small that by the time they reached the other side, their impact would no longer be visible. Kronos returning from the future was more like a boulder falling into the pond. By arriving at the Avalon Colony in time to prevent Commander Johansen from bringing the bio-weapon back to Earth on her ship, Kronos had created an either/or situation. Either the bio-weapon was taken back to Earth, or it wasn't. There was nothing in between. Valkyrie wasn't prepared to risk the survival of the Human Race on a roll of the dice by sending Dreadnought back to here and now, but that wasn't the only option.
If the time machine could be made small enough to leave room in Dreadnought's hangar bay for at least one cargo shuttle, then a UFC and some industrial robots could be loaded onto the ship. That would open up the possibility of traveling several years back in time, locating an uninhabited star system with lots of natural resources, and using the UFC to eventually build a huge fleet of raiders. Those raiders would be ready in time to arrive in Earth orbit just prior to the arrival of the enemy fleet. Their added firepower, carefully aimed, would decimate the enemy fleet before it could burn through Dreadnought's armor. Casanova and Iceman would be saved, with minimal impact on the timeline up to that point. After the battle, the raiders would defend the colonies against the second fleet and then launch a massive strike against the Sogas' home world to destroy their military infrastructure. The Sogas would then be at the mercy of the Insectoids, and Space Force could prepare for the arrival of the insectoid mothership. It was all very neat and tidy, except for one thing. Dreadnought wasn't big enough to carry the time machine, as it was currently conceptualized, plus the UFC, cargo shuttle and all the other necessary equipment to start building the raider fleet. There were two possible solutions to the problem, neither one of which was simple or easy. Dreadnought could be modified to have more room. It would have to be cut almost in half and a new section of hull inserted. Stripping the armor off would help too, since the size of the time machine depended on the mass of the ship, and a lot of Dreadnought's mass was in its armor. The other possibility was to design and build a brand new ship from scratch specifically to hold the time machine and the cargo. It would have to be BIG. Both options would take months and a lot of human assistance which they might not want or be able to give.
The war wasn't going to stop for six months or more while the humans got a large enough ship ready. The Old Man had already given the go ahead to start ramping up production of raiders, and that would take a lot of resources away from other projects. The whole Dreadnought/time machine project was so large and so complicated that unless it was given top priority, it could take years to complete. So much could happen in that time to derail the plan. Somehow she had to find a way to get it done as fast as possible.
* * *
It was daylight when Shiloh opened his eyes. A quick glance showed him that Kelly was no longer in the bed or the room, but the sound from the bathroom told him she was in the shower. He checked the time and grinned when he saw that it was almost noon. He felt rested and his head was clear of the groggy haze. He let his mind go back to the brief but very intense sex that he and Kelly had enjoyed after Howard's call. He hadn't had any trouble getting back to sleep afterwards. Funny how that worked, he thought. As he sat up, the scene in front of him faded from view. For a brief moment he wondered if he was fainting again, but when he saw an image of Howard sitting behind his desk, Shiloh realized he was having another vision.
"We finally found that homeless man living in the maintenance shafts under the main commercial block," Howard said. "He had a piece of the bio-weapon device in his possession. He's in quarantine now, as is the device, and his hideout has been decontaminated and sealed off for good measure. All the other fragments from the bombardment in Colorado Springs and the other impacted cities have been recovered. I think we have a good chance of coming out of this alive and whole. We now know what happened at all our colonies. The combination of Blackjack's vision and your idea for using laser-armed fighters as ABM defense was almost 100% successful. New Paris was the only colony where the bio-weapon actually hit the ground. Plans are underway to evacuate those colonists to other colonies."
The vision faded and Shiloh was back in Kelly's bedroom. He realized he was holding his breath and let it out. As visions went, that one was chock full of information. He wondered about Blackjack's vision. It had to have something to do with the impending attack on the colonies.
Shiloh jumped out of bed and walked quickly to the bathroom door. He heard the shower turn off when he entered. He cleared his throat to let Kelly know he was there.
"Ah, you're awake. How do you feel?" asked Kelly.
"I feel good. I've just had another vision, Amanda. As soon as I have a shower, I want us to go to see Howard. I think you should call him. Let him know we'll be there soon and that I've had another vision." Shiloh couldn't help grinning as he watched Kelly slide back the shower door and step out. She's a damn fine-looking woman. He heard a tiny voice in the back of his head say, 'And there seems to be a glow radiating from her too!'
Kelly saw his expression and grinned back at him. "You're looking pretty good yourself, Admiral Victor Shiloh."
Her response was not what Shiloh had expected. He laughed and couldn't help wondering if she had read his mind.
"Thank you. I'm glad you think so. I wish we had time to pursue this conversation further, but duty calls."
As he stepped into the shower, she said, "A shower and then a visit to The Old Man. I didn't hear the word breakfast mentioned anywhere in there. Aren't you hungry? I sure as hell am."
"Telling Howard about my vision won't take long. I feel guilty enough sleeping in this late. We'll eat after we brief him."
"Okay," said Kelly. But I'm going to suggest that we brief him and have something to eat at the same time, she thought.
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# Chapter 2
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Howard checked the time again. It was now 15 minutes since Cmdr. Kelly had called to say that Shiloh had just received another vision and that both of them would be arriving soon to brief him on it. Just as he was about to query the building security system to find out if Shiloh was in the building yet, his implant activated.
"Blackjack to CSO."
Howard was startled by the identity of the caller. He had only had one conversation with Blackjack. "Go ahead, Blackjack."
"Admiral, I've just received a highly detailed vision concerning the disposition and timing of enemy attacks on our colonies. I'm transmitting the information to you now. You should see it on your office screen momentarily."
Howard watched a stream of numerical data scroll down his screen. "What does it mean, Blackjack?"
"It is precise coordinates and time for the emergence for each of the six enemy ships that attack each colony. This level of detail is only possible if we have recon drones in place ahead of time."
Howard was about to reply when his office door opened and his Aide stepped into view. "What is it, Lieutenant?" asked Howard.
"You asked to be notified when Admiral Shiloh and Commander Kelly arrived, Sir," said the Aide.
What incredible timing, thought Howard. "Send them in," he said.
As Shiloh and Kelly walked in, Howard said, "Blackjack, Admiral Shiloh and Commander Kelly have just entered the room. I believe that Admiral Shiloh has had a vision of his own within the last 45 minutes or so. I'd like you to switch this call to my desk speaker so that all of us can hear you and vice versa." As he talked he pointed to the wall screen where Blackjack's data was on display.
It took less than a second for Blackjack's voice to be heard from the speaker. "Hello CAG and Commander Kelly," said Blackjack.
Before Shiloh or Kelly could respond, Howard said, "Never mind the pleasantries. Blackjack has had a vision of his own with precise recon drone data for the attack on our colonies. I can't believe that the Sogas are going to split up their fleet. There has to be something we can do to take advantage of that tactical mistake. Can your vision help us there, Admiral?"
Shiloh took a few seconds to look at the data on the screen. He was able to tell that each colony would be attacked by only six enemy ships. Now everything fell into place. Still looking at the screen, Shiloh said, "Yes it can, Sir." He went on to relate his vision to Howard as accurately as possible. With that out of the way, he said, "I understand what the phrase about using laser-armed fighters for ABM defense means now that we have this recon data. If we send fighters armed with modular lasers to our colonies and they take up a position over the colony settlements at very low altitude, they should have enough time to use their lasers to burn the bio-weapon shells to slag before they hit the ground, with the one exception of New Paris. As long as those colonists have evacuated the settlement ahead of time, they should be okay. Blackjack, can our jump-capable fighters reach all our remaining colonies before the enemy arrives?"
"Affirmative, CAG. And not only the fighters, but the sentry frigates too."
Howard and Shiloh looked at each other in puzzlement. "Why would we need to send sentry frigates too?" asked Howard.
"We need to be able to deploy recon drones in order to capture this data. Fighters armed with modular lasers can't carry any drones. Therefore something else has to deploy the recon drones, and the quickest and easiest way to do that is with sentry frigates. And not only can they carry the recon drones, they can also carry attack drones that can be precisely aimed with this targeting data. We have enough Mark 1s to destroy this entire enemy fleet, however the window of opportunity to get sentry frigates to three of the colonies closest to the enemy fleet is very short. There won't be time to load both drone types on board before they have to leave. That means that a small number of enemy ships will escape destruction. In order to make this plan work, fighters and frigates have to be set in motion immediately. I can issue the necessary orders on your behalf as soon as your give the word, Admiral."
Howard didn't hesitate more than a second. "You have authorization to issue those orders, Blackjack. I'll notify Ops. Standby." He paused and then said, "Intercom...Operations."
"Operations here, Admiral. What can we do for you, Sir?"
"Blackjack is in the process of communicating orders for fighters and sentry frigates. He is acting on my behalf, and I want his orders to be carried out as quickly as possible. Any questions Commander?"
"No questions, Sir. We'll see to it that his orders are executed with dispatch."
"Very good. That's all for now. Howard clear. Blackjack, are you still there?"
"Still here, Admiral. The orders have been sent. Does anyone have any questions for me?"
Shiloh looked at Howard who shook his head. A quick glance a Kelly got the same response. "Will Valkyrie have time to transfer to a fighter or sentry frigate?" Shiloh asked. The response was immediate.
"She would have time, but she is asking to remain on board Dreadnought, CAG."
Shiloh wondered why she would pass up a chance to kill some Sogas ships. What better way to avenge Casanova, but if that's what she wants, then so be it. "As long as we have enough other AIs to do the job, she can stay on board Dreadnought."
"We'll have enough AI pilots for the fighters and sentry frigates, CAG. Valkyrie wants to remain on board Dreadnought in order to oversee the installation of the time machine.
"That decision hasn't been made yet," said Howard in a slightly annoyed tone.
Shiloh sighed and said, "I'll talk to her, Admiral."
Howard nodded. "Fine. Now if you'll all excuse me, I have to get the decontamination teams to hunt for a man who lives underground. Blackjack, I do have one last question for you before I sign off," said Howard as Shiloh and Kelly left his office. "If some of the enemy ships are going to escape because we couldn't get frigates armed and on station in time, why weren't yours and Shiloh's visions received earlier?"
"The nature of the two sets of information required that The CAG receive the image of your narrative. He couldn't receive it any earlier than he did because he was asleep until just a few seconds before the vision, Admiral. Without his contribution, my information by itself wouldn't have suggested the proper course of action."
Howard nodded. Yes of course and you know this because you'll pass that information back to yourself when you use the RTC because you'll know I asked the question. "I understand. Thank you. Howard clear."
––––––––
As Shiloh and Kelly walked down the corridor from the CSO's office, Kelly asked, "Can we eat now?"
Shiloh laughed. "Yes, we can eat now. We do have a lot of calories to replace after all."
Kelly managed not to laugh out loud, but she gently nudged him with her elbow to remind him that there were other people walking down the same corridor who could hear what they said.
When they found a table in the Officers' Mess and had food in front of them, Kelly said, "When are you going to talk with Valkyrie?"
Shiloh thought about that for a couple of seconds before saying, "How about right now? Intercom...connect Commander Kelly and myself to Commander, Dreadnought."
"Valkyrie here, CAG. Hello Commander Kelly. Blackjack told me to expect your call."
Shiloh rolled his eyes in mock exasperation, and Kelly smiled. Was there anything AIs didn't know in advance? "Did he also tell you why I'd be calling?"
"You and Admiral Howard want to know why I want to remain on Dreadnought to supervise the installation of the time machine when that decision hasn't been made yet."
"That's correct," said Shiloh.
"The answer is very simple, CAG. The time machine has to be built. What other option has the potential to win this war in a single stroke? If an AI takes a UFC and other equipment far enough back in time, a very large raider fleet can be built that will be ready in time to help defend Earth during the most recent attack. If it's planned right, that additional firepower will overwhelm the enemy fleet before Dreadnought's Bridge is penetrated. Casanova and Iceman will survive the battle. The raider fleet can intercept the second enemy fleet, and then attack the Sogas shipbuilding systems. The war with the Sogas will be effectively over because they won't have time to rebuild an offensive force before they're attacked by the insectoid mothership. We, on the other hand, will have plenty of time to get ready for the Insectoids."
Shiloh frowned. "Why send the raider fleet to fight in the second Battle for Earth? Why not send it to destroy the enemy fleet BEFORE it gets to Earth?"
"That would interfere with the vision that warned of the attack. That vision also mentions Blackjack's idea of altering the timeline. Without that vision, the time machine project wouldn't be accomplished. We can't anticipate exactly what would happen if we altered the timeline earlier. It's far safer to divert the timeline during the most recent battle, CAG."
It was a persuasive argument but it still made Shiloh uneasy. "There has to be a downside to this. What is it, Valkyrie?"
"You're correct CAG. Even if we make it our number one priority, it will take months to get a ship ready with the time machine and all the other necessary equipment. That means that everything else, including building our own raider force to defend Earth during those months, will be delayed or slowed. In very simple terms, we're faced with a situation where the faster we try to get the time machine ready, the weaker militarily we'll be to counter the next Sogas attack. But if we take more time to build it, the Sogas will also have more time to build even stronger offensive forces."
Shiloh looked over at Kelly who had a worried look on her face.
"So what's the best strategy to follow from here, Valkyrie?" she asked.
"I calculate that the strategy with the best chance of delaying enemy action for a significant amount of time would be to launch a massive fighter attack on the Sogas home system with X-ray laser warhead drones as soon as possible."
Shiloh shook his head. "But they have RTC too! Our fighters will be flying into an ambush. They'll be wiped out."
"Not necessarily, CAG. Very soon they're going to suffer the loss of a significant portion of their offensive forces as a result of the most recent battle and the interceptions at the colonies. We know that they can build their ships fast, but we can build fighters and attack drones faster. If we target their industrial infrastructure, it will take them months to rebuild it. This won't be the killer blow that overwhelms them enough to prevent them from warning themselves, but they will lose the initiative. That will give us the time we need to finish the time machine and complete the temporal end run around them."
"What fighter losses do you project for this operation?" asked Shiloh.
"Without support from Dreadnought, Midway and the light carriers, we should expect 85% losses. With carrier support I project losses of 35%, CAG."
"Can we get the carriers repaired in time?"
"The Light Carriers can be repaired enough to maneuver and jump. Combat effectiveness will be minimal, but they won't be there to fight. Their biggest contribution will be recovery of damaged fighters after the battle. Dreadnought's and Midway's combat capability can be repaired in time for this operation."
Shiloh nodded slowly. The carriers and the battleship would need to have some human crew on board, and this was the kind of mission that called for a human in command. That would almost certainly be him.
"If Dreadnought is to take part in this operation, it won't be available for work on the time machine, Valkyrie."
"Understood, CAG. I anticipate that this operation will be over by the time that the engineers are ready to start assembling the time machine in Dreadnought's Hangar Bay."
"What's the risk that Dreadnought will be crippled or destroyed in that battle, Valkyrie?" asked Kelly.
"Crippled is four point nine percent. Destroyed is zero point nine percent, Commander. I should point out that those results were based on The CAG commanding the mission. If command is given to someone else, the probabilities will almost triple."
Kelly looked at Shiloh who blushed.
"I think you're over-estimating my tactical skills, Valkyrie," said Shiloh.
"No, CAG. I was actually very conservative in my calculations. You've consistently demonstrated a high level of tactical insight. Experience is the key, and you have more direct combat experience than any other Space Force Officer."
"But I'm not as fast as an AI"
"Correct, but if you set the overall tactical parameters, one of us can take care of the execution and laser fire control. The best of both worlds, CAG."
"In that case I want you to be my Deputy Commander, Valkyrie."
Kelly nodded her agreement.
"I'm not as good as Titan or Vandal, CAG. Are you sure you wouldn't rather have one of them?"
"Not a chance, Commander. You'll be in direct command of Dreadnought as well as my Chief Tactical Officer. You'll be taking Iceman's role for this one. I insist on it!" said Shiloh.
"Thank you, CAG. I accept the assignment. If we're going to do this, then Dreadnought should be moved to one of the shipyards soon."
"I agree, and I'll see that it's done. Anything else, Commander?"
"Negative, CAG. I'm looking forward to this fight. I have some unfinished business with the Sogas."
"You and me both, Valkyrie. CAG clear."
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# Chapter 3
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Howard was surprised when Shiloh came back to his office an hour later. After Shiloh outlined the overall strategy and the plan of attack on the shipbuilding infrastructure of the Sogas home system, Howard did what any good leader does. He played Devil's Advocate and began poking holes in the concept.
"Did Valkyrie take into consideration that the defense of our colonies is going to use up almost all of our stockpile of Mark 1s? What is this attack force going to use against the enemy infrastructure?" asked Howard.
Shiloh nodded. He had already asked himself this question and figured out the answer.
"We'll have to use Mark 2 kinetic energy drones, Sir. Any kind of orbiting structure hit by several of those will suffer a lot of damage. Sure they can repair that kind of damage but the idea here is to buy time, not conquer them outright. If we can take a small number of Mark 1s along, we'll save them for any really big targets we find."
Howard looked skeptical. "I'm not thrilled with this whole concept. We don't know for sure if the time machine will actually work, and even if it does work, there's no guarantee that a massive raider fleet will show up right before the last battle. And to top it off, Valkyrie is proposing throwing what little offensive strength we have left into what basically amounts to a roll of the dice by attacking the Sogas home system in spite of their RTC defensive advantage! No! I'm not going to approve that plan. We're going to stick with our current plan of rebuilding our defensive force of fighters and raiders and our stockpile of Mark 5 X-ray laser drones. We're going to let the enemy come to us, and when he does, we'll kick him in the balls hard. And when we're strong enough, then we'll go on the offensive and destroy every shipyard, every fabrication facility and every other military asset they have. That was Iceman's strategy, and I think it's the correct one. Valkyrie will just have to accept that. I'll authorize repairs to Dreadnought, Midway and the light carriers just in case they might come in handy down the road, but the pace of the repairs will have to fit into the overall allocation of resources and manpower for the fighter/raider program. Unless you have something else to talk about, we're done here, Admiral."
As Shiloh walked away, he wondered how Valkyrie would take the news. Her plan or some variation of it was the only way to bring back Casanova, and for that matter Iceman too. Howard's refusal meant that hope was now gone. Valkyrie wasn't going to like that, and Shiloh wasn't happy about it either. He was convinced that however good a tactician he might be, having Iceman around made him even better, and that had to be worth taking some risks. His train of thought was interrupted by the activation of his implant.
"Valkyrie to CAG."
"CAG here. How did you know I was finished talking with the Admiral?"
"I monitored the outgoing communications from his office. If he's talking with someone else, it's likely that he's no longer talking with you, CAG. What did he say?"
Shiloh didn't really want to discuss it while he was walking through Space Force Headquarters corridors with lots of other people passing by in both directions.
"I'd prefer to wait until I'm back in my quarters before we discuss that question. I'll call you when I can speak freely. Shiloh clear."
When he arrived at his quarters he was relieved to find that Kelly wasn't there. With the connection re-established, he said, "The CSO will not approve your idea of a quick strike to give us the time we need to build the time machine. Mass production of fighters, raiders and Mark 5 attack drones will remain the first priority. The time machine project will not move beyond the planning stage for the time being. I'm sorry, Valkyrie. I wanted Iceman back too, but the Admiral thinks your strategy is just too risky."
"What do you think, CAG?"
Shiloh hesitated. He wasn't sure what he thought about Valkyrie's idea. Howard had a point. The outcome of following that strategy wasn't as certain as Valkyrie made it sound. When in doubt, look at the BIG PICTURE. He and Howard knew from Kronos that the first insectoid mothership would arrive at the Sogas colony world in roughly 150 days. That was only five months away. Shiloh marveled at the fact that he had lost track of how much time was left before Humanity had to deal with the Insectoids. The key to beating the Insectoids was the high-spin platinum warhead, and they were making good progress on the first prototype. That wasn't a problem. The Insectoids would take care of the Sogas, and so Space Force really only had to worry about fighting off additional Sogas attacks for another five months. After that...He let the thought dangle. Was that the key to this whole thing? After the Insectoids neutralized the Sogas, and the insectoid mothership was vaporized with high-spin warheads, the pressure would be off. Then they could work on the time machine/ship. Shiloh realized that Valkyrie was patiently waiting for an answer.
"I think that maybe we're not considering all options. What if the time machine is completed after the Bugs cut through the Sogas and after we take care of the bug ship? Wouldn't that work just as well?"
"You're correct that there wouldn't be any technical or logistical reason why we couldn't build it then, but would The Old Man approve that kind of mammoth project just to save two AIs, CAG? The war will have been won by that point, so why bother? That's assuming of course that we can fight off new Sogas attacks over the next 152 days."
Shit! She has a good point. Once the war is won, the only reason to change the past would be to rescue two AIs. We wouldn't go to all that trouble to save two humans. Wait a minute! It isn't JUST two AIs that we'd be saving. Over fifty AIs were lost in that battle along with over a hundred humans. And by switching to an alternate timeline who knows how many more AIs and humans might be saved as well?
"That's not how we'll sell the idea to The Old Man. Space Force lost a lot of its brothers and sisters in this last battle, and I have a feeling that there'll be more combat before this war is over. Potentially saving all those lives is something that Howard just might buy into if we pitch it to him the right way at the right time."
"What if he still says no, CAG?"
Shiloh took a deep breath and said, "If The Old Man still won't approve the idea, then we'll get the time machine built without his approval. I give you my word, Valkyrie."
"Your word is good enough for me, CAG."
* * *
Gunslinger performed another systems check of his sentry frigate. All systems were operating perfectly. The Sentry Frigate, formerly Exploration Frigate #344, was orbiting Haven outside of its gravity zone. Stoney's fighter along with the three members of his fighter 'wing' were holding position one light second into the gravity zone. Their modular lasers were warmed up and ready to engage any object heading for the planet. Gunslinger's job was to launch a spread of attack drones armed with the old Mark 1 fission warheads at the six enemy ships that would emerge from Jumpspace in less than 30 seconds if Blackjack's data was correct. Gunslinger was quite proud of the fact that he was conning the frigate that used to be commanded by The CAG himself. Four recon drones had been carefully positioned to give immediate radar notification of the enemy ships. A quick check confirmed they were ready too. It was almost time to begin the operation. Because Sentry Frigates were only capable of launching two drones at a time, the six attack drones had to be launched in stages. Their programming would ensure that all six hit their targets at exactly the same point in time. To avoid giving the enemy any time to jump away after launching their bio-weapon shells, that interception point would be one point five seconds after the ships emerged from Jumpspace.
With the launch sequences pre-programmed, all Gunslinger had to do was watch. At the correct time, his Sentry Frigate fired its first two attack drones. Five seconds later it fired the second pair, and five seconds after that the third pair. This is too easy, thought Gunslinger. When the internal chronometer reached the specified time, six enemy ships dropped out of Jumpspace and into a barrage of converging radar beams that pinpointed their position precisely down to a few centimeters. Almost immediately all six ships fired their bio-weapon shells and then vanished in nuclear fireballs as the drones hit. Four of the six bio-shells were caught in the explosions and vaporized as a side effect of Gunslinger's attack. The bio-organisms contained in the other two shells were almost certainly killed by the radiation from the drone warhead blasts, but the fighters burned them anyway as soon as their optical and radar sensors were able to distinguish the targets from the background radiation and electro-magnetic pulse effects.
Gunslinger waited until his ship's radar equipment was able to burn through the residual energy coming from the dying fission blasts and was surprised to find that the six enemy ships were completely destroyed. Based on past combat experience, the older and lower yield Mark 1 warheads should only have been powerful enough to vaporize half of the enemy's typical 20,000 ton warship. He was expecting to see large glowing hunks of metal or even clusters of smaller pieces, but there was nothing left of any of them. Where the Sogas deploying a new class of ship that was smaller? That might explain how they had managed to build so many of them so quickly.
With his mission accomplished and confirmation that Stoney's fighters had done their jobs as well, Gunslinger transmitted the order for the fighters to follow his ship out of Haven orbit to line up for a direct jump back to Sol. As his ship accelerated, Gunslinger transmitted the All Clear signal to the Haven colonists who were hiding in the forests around the settlement. The Colony Leader's thanks gave Gunslinger an unfamiliar feeling of what he could only classify as satisfaction. It was good to be of service to humans, and he was looking forward to some exciting combat in the Sogas home world system.
* * *
Ten days of R&R with Kelly went far too quickly, but the two of them did their best to make the most of the time they had together. Kelly told Shiloh that Valkyie had conveyed to her how pleased she was with their new relationship. It didn't take long for the rest of the AIs still on Earth to find out about it either. Shiloh nearly choked on some food he was eating when Blackjack called to ask how copulating with Kelly was different from doing the same thing with other human females. Shiloh managed to restrain his initial impulse towards anger by remembering that the AIs were genuinely curious about all aspects of human physiology and psychology. It was that moment when he felt Iceman's loss most deeply. Iceman was also curious, but he at least understood that some subjects were too sensitive for Shiloh to talk about comfortably.
As the fighters and sentry frigates began to trickle in from their separate missions, Space Force began to make progress in dealing with the aftermath of the battle. Enough repairs to Dreadnought's Command Section had been done to enable Shiloh to visit the ship and personally look at the Main Bridge where the melted remains of both Iceman and Casanova were still waiting to be removed. The repairs to the extra two AI stations were still to come, but at least the damage to the hull was repaired. Shiloh could enter the room without needing a pressure suit. The lighting wasn't completely repaired yet either, but there was enough to see what he needed to see. He was wearing a device hooked around his right ear containing a video camera so that Valkyie could see what he saw.
Valkyrie's AI station was still intact. He looked more closely and marveled at the fact that the metal enclosure around her quantum brain didn't have a single scratch on it. The other two stations were half gone and the remainder just misshapen lumps with blistered edges. At Valkyrie's urging, he managed to get the top of the enclosures open. In both cases, the football-shaped device that was the AIs brain was partially gone, vaporized by the intense laser beam as it slashed its way through the room. Shiloh could see the inside of what was left, and it made him feel as though he were looking at the exposed brain of a close friend. The two stations were close enough that Shiloh was able to crouch down between them and carefully place one hand on each of the damaged pieces. He lowered his head, closed his eyes and said a silent prayer for the souls of Iceman and Casanova.
When he raised his head and pulled his hands away, Valkyrie said, "May I request that you remove their remains and personally supervise the recycling of the materials, CAG?"
"I'd be happy to do that, Valkyrie," said Shiloh. After some effort he was able to pull both half brains free from the brackets holding them. While the remains were small in volume, the brain casings were filled with materials of different kinds and were heavy enough that he had to hold on to them carefully. He was panting from exertion by the time he got to the section of the shipyard asteroid where mined ore was being smelted down to its usable metals. He told the operators of the facility what he wanted to do. They offered to take the two pieces from him, but Shiloh refused. After receiving some instructions, he stepped into the recycling station and carefully set each half brain into the loading bin. When he was back outside, he moved to stand in front of the Operator's station and pointed his video camera at the view screen that showed the loading bin. The Operator manipulated his controls, and the bin tipped over to drop the two brain cases into the white-hot recycling chamber.
"On behalf of Casanova and Iceman, I thank you, CAG," said Valkyrie.
"They will be missed. Let's make sure they didn't die in vain, Valkyrie," said Shiloh.
"Roger that, CAG.
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# Chapter 4
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Howard picked up his data tablet and opened the next file in his Inbox. It was a report from the SPG. He smiled, wondering when Cmdr. Kelly found the time to approve this report with all the time she was spending with Shiloh. They were trying to be discrete, but rumors were already flying around the HQ. After all, how discrete can you really be if you practically live in another officer's quarters 24 hours a day? Not that Howard minded. He heartily approved in fact. He returned his attention to the report and was soon frowning.
"Intercom...connect me with Commander Kelly...no, wait. Connect me with Wolfman." No sense possibly interrupting something intimate between Kelly and Shiloh.
"Wolfman here, Admiral. Are you calling about our latest report?"
"As a matter of fact, yes. I understand that we only have to plan for defense against the Sogas for another five months, and I see the advantages of shifting to mass production of our new F2 fighter, but I'm not convinced that we shouldn't do the same thing with raiders. So what if the raider production and assembly line will take two months to build? Once it's built, we'll get a new raider every three days! What's wrong with that?"
"Ah, I see that you've only read the Executive Summary. If you'll scroll down to Appendix A, you'll see the timeline of the alternative production schedules. The first column shows how many raiders would be available on specific dates with the production and assembly line. The second column shows raider availability if we dispense with the assembly line and build raiders one at a time in our existing shipyards."
"I'm looking at that data now," said Howard. "I see that by the time the Bugs reach the first Sogas colony, the assembly line will have produced twenty-six raiders while the shipyards will only have produced fifteen. Don't we need all the raiders we can get our hands on, Wolfman?"
"But will we need twenty-six raiders at that point, Admiral? As soon as the Sogas learn about the Insectoids, they'll forget about us and use all their available mobile assets against the insectoid mothership. From our point of view, the war will effectively be over. If we're going to use raiders against the Sogas, it'll have to be earlier. If you move up the shipyard column to the point when the total jumps from five to ten, you'll see that we'll have ten raiders when there's still two months of potential combat left. How many raiders will the assembly line have produced by then, Admiral?"
Howard nodded with sudden understanding. "Zero. Okay, I get it now. If we expected the war to last more than five months, we'd be better off in the long run building the assembly line, but for the short window of opportunity where raiders can be of use, we're better off building them in the shipyards. So why wouldn't it be the same for the F2 fighter?"
"The reason is the difference in size. The F2 is roughly one third as large as the raider. It has far less components. That means the assembly line is much shorter and can be built sooner. When it's operating at capacity, it will produce a new F2 every 20 hours. If we only used shipyards to build F2s, we would actually have fewer available for the last eleven weeks of the five month period."
"Yes, that makes sense, and I see we'll also have more of the older F1 type arriving from Site B too. Hmm. Now that I see what limited use raiders will be in the near term, I'm wondering if it makes sense building them at all. If we use our shipyards to build raiders, we'll have to forget about finishing the two light and one heavy carriers that are partly finished. They might be useful to have someday if we're going to have lots of fighters. Comments?" asked Howard.
"Having three more carriers would only be useful if Space Force is anticipating major military operations against an established race like the Sogas. They're not needed for operations against a mobile enemy like the Insectoids. If those shipyards are not going to build raiders, then why not use the existing infrastructure to modify Dreadnought to accept the time machine, Admiral? By leveraging that construction capacity, Space Force will get a large fleet of raiders when it needs them the most."
Howard's initial reaction was to say no, but this wasn't the same argument that Valkyrie had put forward. Wolfman wasn't proposing a risky raid on the Sogas to buy time. The stockpiles of Mark 5 attack drones and fighters of both types would be the same, regardless of what those shipyards were or weren't building. Building the time machine ship as Plan A, with a raid on the enemy home system as a diversion, was too risky. Building a time machine ship as a backup plan, Plan B as it were, without the raid, now that was a horse of a different color.
"If we used the shipyards to manufacture the parts for the time machine and for the conversion of Dreadnought, how long would it take to complete the project?" asked Howard.
"A minimum of ten months but more likely eleven, Admiral."
Howard did a quick mental calculation. "That'll be after the bug mothership arrives here."
"Correct, Admiral."
"That shouldn't be a problem then. We'll have the high-yield, high-spin warhead ready long before then."
"Agreed. We know that the design brought back by Kronos works. We merely have to perfect the production of the components and the process of energizing the platinum. It will be easy to test the prototype."
"Yes. That should be quite a show." He took a deep breath and said, "You can let Valkyrie know that I'll approve construction on the time machine and Dreadnought's modification based on the backup plan that you and I have just discussed, but there will NOT be a diversionary raid."
After the slightest of pauses, Wolfman said, "Acknowledged. I wonder what kind of reception Kronos will get from the Friendlies."
Howard nodded. "Me too."
* * *
The Friendlies' home system looked very much as Kronos remembered it from the alternate timeline. He tried using his Friendly-designed mini-fighter's optical instruments to see if he could detect the proto-type time tunnel, but it was too far in the outer system to be seen. It wasn't long before he got a reply to his initial lasercom message. It contained permission to micro-jump closer with target coordinates. Within minutes Kronos brought his mini-fighter to the designated coordinates where he found the ship at a half light-second distance.
Communication was quickly established with a Friendly AI Kronos explained how humans knew of the aliens' existence, their location and the location of the small, furry aliens. He went on to describe the entire alternative timeline that Casanova experienced or knew about, ending with the expansion of the time tunnel complex, Kronos' creation and journey into the past, as well as events in the new timeline up to the present time. That transmission took less than two seconds but was followed by a much longer pause as his opposite number relayed the data to a Friendly at the much slower speed that biological entities required. After almost 600 seconds, Kronos received a signal carrying the direct communication with one of the Friendlies themselves.
"Why have you been sent to contact us?"
"My humans desire to establish contact to recognize the fact that the old timeline has been altered. Now that we know of the approach of the Insectoids, the war with the Sogas has served its purpose, and my humans have sent me to ask you to persuade the Sogas that we are no threat to them and desire a peaceful resolution to the war."
"That will be difficult. Humans and Sogas share a similar biology and covet the same types of planetary environments. Conflict is inevitable given their psychological propensity for paranoia. We have already advised them that their species is much more in danger from the Insectoids. However, this knowledge, combined with their peculiar logic, has had the opposite effect of what we intended. Rather than lower the intensity of their aggression against humans, their desire to end the war quickly in order to prepare for the insectoid encounter has increased. That is why they have resorted to biological weapons. Their attempts to infect your population will continue. Further attempts are likely to include acts of diversion or deception. A human attempt to intercept the Insectoids before they reach the Sogas may convince them to cease their bio-weapon attacks. How will humans deal with the insectoid advance?"
"I have been instructed not to reveal that information," said Kronos.
There was a pause before the alien responded. "There is only one reason why your humans would not want us to have the information and that is that they do not wish the Sogas to learn it from us. The only logical conclusion therefore is that humans do not want the Sogas to be able to defeat the Insectoids. This attitude does not surprise us. It is typical of how humans think. In their own way, they are just as psychologically...unbalanced as are the Sogas. Do your humans not understand that all races have the right to exist? We do not wish to see any intelligent species disappear. Even the Insectoids do not deserve to be completely exterminated."
"But they will do their best to exterminate countless other species if they're not stopped. Where is the logic in allowing them to do that?" asked Kronos.
"Only the complete extermination of all Insectoids in this galaxy would ensure the survival of other species. We believe that all life forms were designed for a reason. It is not for us to decide that the Insectoids must disappear from the Universe," said the alien.
"Their appearance may not be a natural event," said Kronos.
"Explain."
"In the alternate timeline, multiple insectoid motherships arrived at the Sogas and human home systems from multiple directions. My analysis of the timing and trajectory of these motherships suggests that they did not originate from a single star system. It does not seem possible that they could evolve independently on more than one planet. That implies they are an engineered species that has been deliberately transplanted to other planets by an unknown agency."
"Are you able to transmit that data?" asked the alien.
Kronos transmitted all of the astrogational data related to all observed insectoid mothership arrivals. The alien AI warned Kronos that analysis of the data might take a while.
When the response came, the alien said, "The data is consistent with your hypothesis, however there may be other explanations. Without conclusive proof of artificial evolution, we would not approve or assist in any attempt to rid this galaxy of that life form."
"What kind of data would you need to see in order to have conclusive proof of our theory?" asked Kronos.
"If we had one of them to scan with our temporal equipment, we could track that individual back in time, including its ancestors, to the point where the species no longer existed in that form."
"Then we would have to capture one of them alive and bring it to you?" asked Kronos.
"It does not have to be alive. Our device is concerned only with the temporal history of the atoms making up the specimen. When we have that proof, we would then be willing to assist with any attempt to eliminate the Insectoids and, if possible, realign the timeline in such a way that humans are not threatened with biological weapons."
"I will convey that message to the Humans. Is there anything else you wish the Humans to know?"
"Yes. The planet containing the small, furry creatures that we are trying to protect has two large land masses. The furry creatures evolved on and inhabit the larger land mass. The smaller land mass has an environment that would be suitable for humans. We would be willing to allow Humans to colonize the smaller land mass as a sanctuary for their species on condition that they do not encroach on the larger land mass and do not interfere with the furry aliens in any way. This offer will be withdrawn if your humans allow the Sogas home world to be consumed by the Insectoids."
"Understood. I will now return to my humans." Kronos decided that he had to get this information back as quickly as possible. Valkyrie would be VERY interested in the Insectoid capture concept.
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# Chapter 5
––––––––
Jason Alvarez stepped outside his small house and let the cool night breeze wash over him. Life on Haven was not great, but it was getting better. Thank God the enemy attack had been stopped cold by the Space Force people two weeks ago. He shuddered to think what would have happened if the shells containing the biological agent had landed. Sure the colonists had all evacuated the settlement by then, but if the settlement had become contaminated, he along with everyone else would still be living in the woods under makeshift shelters with minimal food.
He was pondering what that might have been like when he noticed a faint noise that was getting louder. It was a whooshing sound that quickly peaked in intensity and then just as quickly faded, and it seemed to be coming from above. He looked up but saw nothing. As he continued to look at the night sky, he felt tiny drops of something hit his face. That's strange. There aren't any clouds in the sky so it can't be rain. The smell of the air started to become quite pleasant, almost like some kind of perfume or cologne. He inhaled deeply, savoring the aroma. He looked around at the nearby houses to see if anyone else was still awake. There was one other house with lights on. He decided to walk over to them and let them know about the sweet smell. This kind of thing hadn't happened before, and unless he got someone to corroborate his story, it was likely that nobody would believe him.
* * *
Howard entered the conference room, stepped over to the chair at the head of the large oval table and slapped his tablet down in obvious anger. As he sat down, Shiloh could see the pulsing blood vessel in his boss's forehead. I don't think I've ever seen him this angry before. What the hell is going on?
Howard looked around the table. The room was dead quiet. When he spoke, his voice was outwardly calm, but those who knew him well could tell that he was on the verge of exploding.
"I've just received a verbal report from Kronos. He's returned from a mission to contact the Friendlies. Kronos, are you on line?"
"Yes, Admiral," said Kronos from the speaker unit in the center of the table. "All of the SPG AIs are connected as well. I've already briefed them on the mission results."
"Good. I'll summarize what you told me for the benefit of Vice-Admiral Shiloh, Senior Commander Kelly and the other human members of the SPG who are present. The Friendlies object to our plans to let the Bugs bring the damned Sogas to heel for us. It's okay for them to exterminate us, but we're not allowed to let them suffer the same fate! We're supposed to save their hides even though they're still trying to use biologicals against us! The Friendlies have made us an offer to avoid becoming trapped in their own hypocrisy. If we prevent the Bugs from decimating the Sogas home world, the Friendlies will allow us Humans to establish a colony on some small land mass on the planet containing the goddamned furry aliens as a sanctuary for insurance against the outbreak of the bio-weapon!" Howard was shouting by the time he finished that sentence.
When it was clear that Howard was going to pause, Shiloh said, "That may not be a bad idea, Admiral. We could shift the Haven colonists—"
Howard cut him off. "Wolfman! Tell Vice-Admiral Shiloh about your latest vision."
"My vision contained information that Haven and at least seven other colonies have been infected with the bio-weapon. All incoming ships from any colony should be held in quarantine orbit until their crews and passengers survive for 28 days from the time they leave a colony. At the point when the vision was sent, seven colonies were confirmed as infected, with the status of the other fourteen unknown. They could all be infected."
"How did they do it?" asked Shiloh before Howard could say anything.
"We don't know and it doesn't matter," said Howard. "Now you know why I'm so angry. If the other colonies aren't already infected, then the enemy has screwed up. I'm going to assume that they're infected now or will be before we can do anything to help them. We are going to wrap Earth in a ring of steel. Even if every single colony is wiped out, we can start over as long as Earth is not infected. We're still looking for that homeless guy living underground, but we know that we'll find him eventually. As for the Friendlies' suggestion that we turn the other cheek and save the Sogas, they can go fuck themselves! WE ARE NOT TURNING THE OTHER CHEEK!"
Howard's face was red with rage. Opening his mouth to say something more, he suddenly grimaced in pain, clutched his chest and fell off his chair to the floor. Shiloh was closest, and as he knelt down beside him, he heard Kelly call for medics over the intercom.
"Take it easy, Admiral—" Shiloh started to say, but Howard cut him off.
"Promise me you won't turn the other cheek! Promise me, God dammit!" said Howard in a voice raspy with pain.
Shiloh hesitated. He's letting his rage cloud his judgment. We need to break out of this cycle of fear and hate.
He shook his head and said, "I can't promise that, Admiral. I don't think it's the best way."
Howard said nothing and looked away.
Kelly knelt down beside Shiloh and said, "I'll keep an eye on the Admiral. The medics will be here any second. You should take over the meeting, Victor."
Shiloh shook his head. "We can't continue the meeting with Howard laying on the floor for God's sake!"
Kelly gave him a hard look. "I'm not saying CONTINUE the meeting, just take control of it. You're now the ranking officer. Everyone is waiting for you to do something, so go do it."
Shiloh took a deep breath. She was right. There was nothing he could do for Howard, and as the next most senior officer, he had a duty to take control of the situation. He stood up and walked over to the end of the table where the other three human members of the SPG were standing.
Speaking loud enough for the AIs to hear, he said, "We're going to temporarily stop the meeting until the Admiral gets medical attention. Kronos, I'd like to talk with you on a private line. Stand by while I set that up."
"I'm standing by, CAG."
By the time Shiloh had finished speaking with Kronos, the medics had taken Howard away. Kelly was nowhere in sight. One of her SPG comrades confirmed that she had gone with the medics. Indicating to the others to sit, Shiloh sat back down in the same chair as before.
"Obviously I can't make any decisions about our future strategy. Admiral Dietrich is the Deputy CSO, and he'll take over temporarily until a permanent replacement can be appointed by the Oversight Committee." That should be an interesting meeting, thought Shiloh. "What I can do is make recommendations. Kronos has just informed me that the Friendlies are willing to try to restrain the Sogas if we don't let the Insectoids conquer them. My understanding of events in the alternate timeline lead me to believe that we have a lot more to fear from the Insectoids than from the Sogas. Even if we can sidestep them now, Humanity will have to face them again at some point. Eliminating the Insectoid threat once and for all will require Friendly assistance. In order to get that, we need to capture one of the Bugs, dead or alive, and take it to the Friendlies. Wolfman! Given what we know of the alternate timeline, what's our best option for capturing a bug?"
"Data recorded by Valkyrie of the insectoid attack on the Sogas colony at Omega77 shows that at least one Insectoid was killed by the defenders. None of the subsequent recordings of that encounter show that dead Insectoid being recovered before the VLO left. If we send a ship to arrive there after the VLO leaves, then we may be able to recover an insectoid body, CAG."
Shiloh nodded. "Yes and when the VLO arrives at the Sogas home system, we'll be there to take it out with the new Mark 6 warheads, and we'll help destroy the bug attack ships as well. That should show the Sogas that they don't have to fear us."
"Not necessarily, CAG. We shouldn't assume that they'd stop their attacks on us just because we help them with the Insectoids. We might act that way, but their logic is sufficiently different from ours that they may continue the war. The Friendlies seem to have some influence with them. If we let the Friendlies know what our plan is and ask for their help, they may be able to convince the Sogas to stand down after we save their home world."
"I hope you're right. I'm going to recommend to the Acting CSO that we plan to capture a specimen at Omega77 and stop the VLO when it reaches the Sogas home world, AND that we send Kronos back to the Friendlies to tell them that and ask for their help. Does anyone object to that or have a better idea?" No one did. "Okay then. This meeting is adjourned. Intercom...connect me with Ops."
"Operations here."
"This is Vice-Admiral Shiloh. Find Admiral Dietrich and inform him that Admiral Howard has suffered what appears to be a heart attack and is undergoing medical attention. Advise him that he is now the Acting CSO, and tell him that I request a meeting with him as soon as possible. Shiloh clear."
As he got up to leave, Shiloh said, "Gentlemen, if anyone is looking for me, I'll be checking on Admiral Howard."
With the meeting over, he hurried to the rear entrance of the building where the air ambulance would pick up Howard if it hadn't already. He got there just in time to see the paramedics put Howard in the vehicle, which then quickly took off. Kelly was standing there with tears in her eyes. Shiloh put his arm around her.
"How's he doing?" asked Shiloh.
She shook her head. "Not good. He's hanging on but just barely. What are we going to do without him, Victor? Who else has his force of will? The Oversight Committee will start foaming at the mouth when they hear about this."
Shiloh sighed. "Dietrich will hold the line. He knows what's at stake." But even as he spoke, he wasn't sure if it was true. Did Dietrich really have what it took to keep the OC in line? What might the OC do if they thought they could get away with it? He didn't like the thought that came to mind. They might try to strike back at the AIs. He had to prevent that at all costs. Shiloh heard his name called and looked around to see Dietrich approaching. He and Kelly let go of each other and turned to face Dietrich.
"What's his condition?" asked Dietrich. Kelly repeated what she'd told Shiloh. Dietrich nodded. "He's a tough, old bird. My money's on him pulling through. How did it happen?"
"He had just convened a meeting in the conference room. He was extremely upset with the response from the Friendlies in light of Wolfman's latest vision." Shiloh looked around to make sure no unauthorized ears were nearby. "At least seven colonies have been infected with the bio-weapon, and there's a distinct possibility that they all are."
Dietrich's face turned pale. "Son-of-a-bitch! No wonder he had heart problems!" Now it was Dietrich's turn to look around. "I think you and I should go to my office. I want to be up to speed when the OC calls, as they inevitably will." Without waiting for Shiloh's response, Dietrich turned and walked quickly away.
Shiloh looked at Kelly who smiled and nodded to him. He smiled back, gave her shoulder a quick squeeze and turned to follow Dietrich. Dietrich was already seated behind his desk by the time Shiloh entered his office.
After Shiloh sat down, Dietrich said, "Okay, Victor. What do I need to know?"
Shiloh told him what the Friendlies had told Kronos, what Wolfman's vision said, what Howard said before his heart attack and what Shiloh had said afterwards.
"Well I can't really fault Howard's reaction. It does seem just a bit hypocritical of the Friendlies to insist we help the Sogas while they're doing their very best to wipe us all out," said Dietrich.
"The Friendlies don't want anyone wiped out. Not us and not the Sogas either, but they're in the line of fire of the Bugs. We have the Mark 6 warhead. They don't. From the Sogas point of view, it makes perfect sense to secure your rear area, which would be us, before you turn your attention to the main enemy. From a strictly military point of view, I understand it. What they're missing are the non-military factors such as having an ally at your back. It's clear to me that they're not going to be the ones to break the cycle of strike and counter-strike. That means we have to do it, and it seems to me that if we're prepared to take out that bug mothership with a Mark 6 drone anyway, then why not do it before the Sogas home world is decimated instead of after?"
"And if the Sogas still continue to come after us, what then?"
Shiloh shrugged. "If they witness a two hundred and fifty megaton blast and still come after us, then I will not lose any sleep over dropping a few Mark 6s on their planet. But if we can build the time machine ship, that all becomes unnecessary. With several hundred raiders, we can overwhelm their defenses at any of their industrial outposts. We'll destroy their ability to build more ships, and we'll monitor them from orbit until hell freezes over to make sure they don't rebuild their offensive capability. Nobody gets exterminated, and therefore the Friendlies should be okay with that. It's the best outcome for everyone. Everything else is just buying time."
Dietrich shook his head. "I wish I could see a way for that project to be completed, but I don't see how after what's happened." When he saw Shiloh's look of confusion, he continued. "I'm not talking about losing Howard, although that's bad enough. I'm talking about the colonies becoming infected. That news will get out into the open sooner or later. The public will demand answers. The Grand Senate will go into cover-their-ass overdrive, and the Oversight Committee will want a scapegoat. Howard will be the obvious candidate partly because he won't be able to fight back and partly because all of this really did happen on his watch. The OC is going to insist on retaking control, and they'll have the public backing them. I don't see how we and the AIs can resist that without a full-blown mutiny, and that serves no one's purpose. Once they're in control again, they're going to want us to resume offensive operations, and it's hard to see them pushing for that at the same time as allowing our one and only battleship to be cut in half for a project approved by their scapegoat, Howard. Do you?"
Shiloh sighed. "Well, when you put it that way, I don't see it happening either, Sir." When Dietrich didn't respond right away, Shiloh added, "If we can't convert Dreadnought to a timeship, then we'll have to build a new ship from scratch, and it'll have to be where the OC can't see it."
Dietrich smiled. "Like Site B perhaps?" he asked.
Shiloh nodded and smiled back. "Exactly. I'll arrange for a freighter to take the project personnel, including Valkyrie, to Site B along with all the design data. I think they're very close to having every part of the time machine coded for production by UFCs. We'll make sure they have the engineering expertise to design a new ship. Site B will have to build an orbiting shipyard first, but we already know how to do that."
"Will that mean pushing back the completion date?" asked Dietrich.
"Maybe and maybe not. If we're going to design a new ship, it may end up being a lot less massive than Dreadnought. No armor and no weapon turrets. It might just turn out to be an overgrown freighter, and look how fast we build those."
"Let's hope so. Go ahead and get that moving. Anything else we should discuss?"
"Yes. The Bugs worry me a lot. Even if we can take care of the one mothership, I'm convinced there will eventually be more. A working timeship could give us the ability to stop them at the source. That would not only save us, but a whole bunch of other alien races out there. The Friendlies would no longer have any reason to start this war to begin with. The catch is that the Friendlies don't want to see any species wiped out, including the damn Bugs! But if we can prove that they're not a naturally occurring life form, in other words that some other agency created them and then turned them loose, then the Friendlies will not object if we exterminate them. They might even help us do it. They want to see proof, and the only way we can give them that proof is to provide a Bug for the Friendlies to scan temporally. If we send a ship to Omega77 to arrive just after the Bugs have decimated that colony, I think we can bring back a dead Bug. I don't want to tell the OC about the how and the why. We'll have to sneak that mission in under their radar. I'm telling you about it now so that you're aware of what has to be done, assuming that you buy into the idea of course, Sir."
"As you know I've been in favor of altering the timeline so that this war never happens. If bringing back a dead Bug can help us achieve that, then I'm all for it. I'll leave the details to you. Just keep me in the loop. Anything else, Shiloh?"
"No, Sir. That's it."
"Okay then, I won't keep you from what you have to do."
Shiloh stood up, saluted and left the office.
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# Chapter 6
––––––––
Shiloh arrived at the Operations Center within 15 minutes of Dietrich's call. It was almost dawn, and the Center still had the skeleton crew typical of the night shift. Shiloh saw Dietrich standing near one of the manned consoles and walked over to him. Shiloh had expected the Oversight Committee to test the waters as far as reasserting their authority over Space Force, but their meeting with Dietrich couldn't have been less confrontational. Dietrich was 'confirmed' as the Acting Chief of Space Operations which merely rubberstamped his automatic assumption of the senior position as Howard's Deputy CSO.
By the time that bit of political theater was done, Kronos was on his way back to notify the Friendlies that Humans were willing to play nice. He returned to Sol less than eight days later with a message that the Friendlies would cooperate by sending a ship with their temporal scanning device to the outskirts of the Omega77 system on the day the VLO was due to show up. When the Space Force people had possession of the dead Bug, they would send out an omni-directional signal, and the Friendly ship would move into orbit around the colony planet.
When Shiloh reached Dietrich, the ACSO nodded to him and said, "Something's up with Reforger. No one's answering the wakeup call. There should be alarms going off all over the ship by now, but we're not hearing one word from that ship."
Shiloh sighed. Reforger was the first freighter to return from one of the colonies after Wolfman received his vision. Dietrich had confirmed Howard's last order to quarantine every returning freighter in orbit until more than 28 days had passed since the crew's last contact with a colonist.
"Is this the 28th day since they had contact with the colony, Admiral?"
Dietrich nodded. "Yup. Right on schedule if it really is the bio-weapon. A medical team in full bio-gear is on its way up to the ship as we speak." Turning to the Com technician seated nearby Dietrich said, "Put Vice-Admiral Shiloh on the connection to that shuttle, Lieutenant."
Shiloh heard his implant activate with the faint hiss of static. The main display showed the shuttle's progress as it approached the orbiting freighter. As the shuttle slowed down in order to dock with the ship, Shiloh heard the shuttle pilot speak.
"We're approaching the docking hatch...we've matched velocity with Reforger...contact with the docking hatch in...three...two... one...contact...magnetic clamps are activated. We have a tight seal. The team is opening the hatch now. I'm switching the mike pickup to the Team leader, Ops."
As the Com Technician acknowledged the pilot's comment, Shiloh leaned over to Dietrich and said, "Is this on an open frequency?"
Dietrich shook his head. "Encrypted." It was clear that Dietrich was listening to the audio transmission as well. They soon heard a new voice that Shiloh assumed was the leader of the medical team.
"We're inside now. Torres, you check the Bridge. Frank, you check out Sick Bay. I'll check crew's quarters."
With all three medics operating alone, there was no one for them to talk to, and all Shiloh and Dietrich heard for the next half a minute was static.
"Oh shit!" The Team Leader's voice was clearly agitated. "Ops, I've found a body in bed. Not sure who it is yet. I'm checking the other cabins."
Five minutes later there was no longer any doubt. The whole crew was dead and had apparently all died in their sleep. Dietrich tapped the Com Tech on the shoulder. The man nodded, activated a switch and nodded again.
"Team Leader, this is Admiral Dietrich. Leave the bodies as they are for now. They'll be dealt with later. Get your team back to the shuttle. Keep your bio-suits on. All of you will go through decontamination when the shuttle lands. Do NOT talk about what you've found to ANYONE even if they're wearing a Space Force uniform. Is that understood?"
"Ah...roger that, Admiral."
Dietrich tapped the Com Tech on the shoulder again and said, "Okay, Admiral Shiloh and I have heard enough. Thank you, Lieutenant."
"You're welcome, Sir."
Shiloh followed Dietrich to a part of the large room where they could speak without being overheard.
"There's no way that we can keep this from the OC indefinitely," Dietrich began. "They're going to find out eventually, and if we try to hide it from them, they'll use that fact against us. I think I should inform them right now. What do you think?"
"I reluctantly agree, Admiral. The shit is really going to hit the fan now. I don't envy you your position."
Dietrich shrugged. "I'm not worried about myself. Howard's going to get the blame. He may end up wishing he hadn't survived that heart attack. It's the public's reaction that worries me. We were able to hold a gun to the Committee's head because they knew the public would blame them for any break with our AIs, but this is different. We can't blame the loss of colonies on the Committee, but they CAN and will blame Space Force for the losses. I know those bastards. Behind the public statements of grief and sympathy, they'll be chomping at the bit to use this against us." He paused then said, "Are Valkyrie and the others on their way to Site B?"
"Affirmative. They left two days ago," said Shiloh.
"Good. That's good. One less thing for the Committee to complain about." Dietrich looked at the chronometer on the wall and sighed. "I didn't realize it was so early when I called you, but I wanted you here when we boarded that ship. Howard told me to trust your advice."
"Thank you, Admiral. That's good to know."
Dietrich nodded. "There's nothing more that you can do here now, so you may as well head back to your quarters and get some more sleep."
"Sounds good to me, Sir," said Shiloh. But if Kelly is still awake, I doubt I'll be getting any more sleep this morning.
Dietrich watched Shiloh walk away and wondered why he had that strange grin on his face.
* * *
The OC's reaction was carefully calculated. The freighter crew deaths were released to the public with statements cautioning the public from overreacting. Howard, now officially retired, stayed in seclusion and refused to comment.
It was the deaths ten days later of two other freighter crews within 24 hours of each other that set off the political storm. The Committee Chair issued a statement saying that the Committee would investigate this string of infected crews, and that the hearing would be open to the public and televised. Shiloh attended the hearing but wasn't initially called upon by the Committee to answer any questions. That was left up to Dietrich. The Committee was careful not to blame him. All their questions were phrased in such a way as to insinuate that Dietrich's predecessor was to blame. They saved their bombshell for after the mid-day recess.
When everyone on the Committee was back and seated, the Chair banged his gavel and said, "This open hearing is once again in session. During the break the members of this Committee caucused, and we are agreed that the apparently successful attacks by the Enemy on at least four of our colonies with bio-weapons is the result of bad policies and strategies put into place by Admiral Sam Howard. Since Admiral Dietrich's appointment as Acting Chief of Space Operations was a temporary one, the Committee feels that it is now time to appoint a new permanent CSO, someone who has proven himself to be not only skilled in combat but also willing to engage the enemy. Therefore I'm pleased to publically announce that Vice-Admiral Victor Shiloh will be promoted to the 3 star rank of Senior Admiral and will be the new Chief of Space Operations!"
Shiloh was stunned and must have looked that way because the Chair looked at him and then quickly looked back at the media cameras.
"The Committee would ask the media not to question Admiral Shiloh until he has had a chance to consult with the Committee about the best way to respond to these new attacks. The members of this Committee WILL make themselves available to the media for questions following this hearing. I now declare this hearing adjourned." With that he banged his gavel, and the room exploded with sound from media people shouting questions at the Committee and at Shiloh.
Dietrich immediately stood up and started walking quickly to the doors. He gestured with his head for Shiloh to follow him. Shiloh took the hint and left the room as fast as possible. He and Dietrich found the quickest way to get out of sight of the Committee room. When they were back in that part of the building that was off limits to non-Space Force personnel, Dietrich said, "So...those sneaky bastards have thrown the red-hot potato into your lap. I have to hand it to them. They knew exactly how to get the most political advantage out of this and castrate us at the same time. By picking you because of your quote 'willingness to engage the Enemy' unquote, they've essentially put you on notice that Space Force had better go on the offensive pretty damn quick, or your head will be the next one to roll."
Shiloh shook his head in confusion. "I don't understand why they think they have the upper hand again. They have to know that Space Force is crippled without the cooperation of the AIs, and the AIs will do what I say regardless of whether I'm the CSO or not."
Dietrich nodded. "I don't understand this new willingness to push back either. Maybe they think they can bluff us into backing down. I don't really buy into that theory because it would require all of them to grow a new set of big, brass balls, and that's unlikely, them being the political animals that they are. What is more likely is that they've arranged something behind our backs that allows them to think they can deal with any potential withdrawal of service by the AIs. Don't ask me what that might be because I have no idea." He paused, then said, "One thing though. Having the title of CSO and the rank to go with it now gives you a lot of flexibility to act. My advice is to use it. Make them react to you, not the other way around. For their plan to work, they need to let you run with the ball for a while. So make the most of it...Sir."
Shiloh blinked and was about to say that Dietrich didn't have to call him 'Sir' when he suddenly realized. Holy Mother of God! He does have to call me 'Sir'. Now I'm The Old Man!
It was two hours later as Shiloh was getting settled in the CSO's office that the Committee Chair arrived for a chat. When they were both seated, Shiloh realized he felt he was sitting on the wrong side of the desk. It would take him a while to get used to sitting on THIS side. The Chair, a pugnacious man who was easy to dislike, leaned forward and took several cigars from the cigar box on Shiloh's desk without asking permission.
After he lit one and had put the rest into his suit pocket, he said, "It seems that you've gotten over your shock from earlier this afternoon. That's good. The sooner you and I, as representative of the Committee, come to an understanding the better. Howard thought he had the upper hand after that disgusting show of force by the AIs, but we now have the upper hand. These infected colonies will enrage the public. They'll demand that we strike back hard and fast."
Before he could say more, Shiloh spoke. "But that's not the best way to win this war. We have to have an overwhelming superiority in combat strength so that we can swat aside any defense they can muster and take control of their home world orbitals. When we do that, the War will be won." He stopped talking when he saw the Chair shake his head.
"That may be the best way militarily, but it's not the most politically expedient way. Right now the politically expedient way also has the most popular support. You will conduct this war the way WE want, OR we'll replace you after the next enemy attack. Is that understood, Admiral?"
After a short pause Shiloh said, "Aren't you forgetting something?"
The man across from him chuckled. "The AIs? We're no longer afraid of your AIs, Admiral. If you want to play that card, you go right ahead."
His tone of voice was one of supreme confidence. He actually WANTS me to arrange for another show of force! What has he got up his sleeve? I have to find out somehow. "So what exactly do you want me to do?" asked Shiloh.
"Here's what we want done." The Chair held up his hand and used his fingers to count off the points. "First, no raiders. Tell the shipyards to cease working on them and resume work on building more combat frigates and heavy cruisers. Second, stop taking armor off the battleship. Why Howard ordered that I don't know, but we need that ship, so the armor goes back on. Third, all ships, including carriers and the battleship, will henceforth be operated and commanded solely by humans. No AIs. Fourth, we want a series of attacks on Enemy colonies as soon as they can be identified."
It took Shiloh a couple of seconds to realize that the Committee STILL didn't know that Space Force now knew where all Sogas planets were.
"Fifth, we want all personnel brought back from Site B. That was Howard's pet project, and it's so far away that we can't monitor what's going on there. So whatever it is that's really going on there, we want it stopped."
Shiloh felt his anger rising. I have to keep calm. Letting him goad me into responding rashly will only play into his hands. "We're building more F1 fighters at Site B, as well as new AIs to pilot them," he said.
"Not anymore. Why build more F1s at all when we're now building F2s and more AIs here? We are becoming way too reliant on AIs to fight this war. The AI and F1 production facilities at Site B are to be shut down as soon as is humanly possible. The personnel there will be brought back as soon as transportation can be arranged. I want your confirmation that you'll execute those instructions, Shiloh."
Shiloh looked carefully at the Chair's expression. He looked like a man who has just drawn a line in the sand, and he was daring Shiloh to stop over it. Shiloh really didn't want to give in. Site B was now the only place where a new timeship could be built, but were humans really necessary to build it? Maybe Valkyrie and her AIs assistants could use the F1 assembly line robots to do the actual construction work. He needed time to investigate whether that was possible. One way to get the time was to appear to cooperate with the OC.
"I will order the immediate halt to F1 and AI production at Site B and the repatriation of our people from there as soon as we can find the transport capacity," said Shiloh.
"Very good. See, that wasn't so hard, was it? If you play ball with us, we'll cut you some slack in terms of your military priorities. How soon can your people plan for a fighter strike on an enemy colony, Admiral?"
Shiloh decided he would try to buy more time. "Well, I'd want our light carriers fully repaired before we undertake an operation like—"
"Forget the carriers. Our fighters can conduct a raid all on their own. They've proven it can be done, so let them do it. How soon?"
"Two weeks minimum."
The Chair looked like he wanted to reject that as too long but then apparently changed his mind. "Okay. I did say we'd cut you some slack, so you've got your two weeks to come up with an attack plan. In the meantime, we expect to see my five points being acted upon. I think that'll do for a start, so I'll leave you to it." Without another word the Chair got up and walked out.
After the man had exited the outer office, the CSO's Aide, a nervous looking Lieutenant, appeared in the doorway with a folded piece of paper in his hand.
"Excuse me, Sir. Admiral Dietrich stopped by while you were talking with the Chair. He gave me this to give to you." He handed the paper to Shiloh.
The message was short.
[The internal communications system has been compromised by the OC. Watch what you say and to whom you say it. SD.]
Shiloh thanked the Lieutenant and leaned back in his chair. Somehow he would have to keep the OC happy for the 131 days until they could attempt to recover a dead Bug from Omega77 and the additional 32 days until they could try to intercept the VLO when it arrived at the Sogas home system. In the meantime, he would have to find a way to appease the OC with regards to their desire for raids on enemy colonies without suffering a lot of casualties among the AIs. That, Shiloh was certain, was the OC's real objective.
One thing had to be done right away. Kronos had to be sent back to the Friendlies. Shiloh couldn't use his implant or the HQ com system to call him without risking interception. Kelly would have to go off planet and deliver the message directly. Any future communications of a sensitive nature between him and his AIs would also have to be carried up to orbit by Kelly. It would be a pain in the ass, but it was the only way to be sure that the Committee didn't hear something they shouldn't hear.
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# Chapter 7
––––––––
On the day after his appointment as the new CSO, Shiloh decided that he should make the rounds of the Headquarters building and introduce himself to the people working there. It was mid-day when he arrived at the Operations Center. After briefly chatting with the officer in charge, Shiloh went from station to station to chat with each of the personnel on duty. All the personnel seemed pleased by the attention from the new CSO and were willing to talk, except for the Senior Lieutenant at the Orbital Defense Weapons Station. As Shiloh made the rounds, he had noticed him watching, and the young officer's expression was one of concern. When it was his turn with the CSO, he seemed nervous and clearly did not want to look Shiloh in the eye. His answers were terse, almost to the point of being rude.
During a pause in the conversation Shiloh took a close look at the console in front of the Officer. He recognized the weapons systems controlled from this station as being the growing clusters of Mark 5 X-ray laser drones in Earth orbit. There were already over a hundred in orbit now, with more being deployed almost every day. This was the key station that would repel another sogas or insectoid attack. As Shiloh looked at the three small screens facing the Officer, a part of one display caught his eye.
[AI TARGETING STATUS - - - ON STANDBY
O.C. COMMAND OVERRIDE STATUS - - OFF]
Shiloh had enough presence of mind to keep his face from showing his shock. The Committee had secretly installed a command override that allowed them to aim all the Mark 5 X-ray laser warheads at all the AIs that were in or near Earth orbit. THAT was why the Chair was practically begging Shiloh to order the AIs to execute another show of force. If they did, the Committee would fire the Mark 5 warheads in a surprise attack and destroy any AI within range! They would then claim that the AIs had gone rogue and that the Committee had no choice but to defend the planet! Shiloh managed to exchange a few more pleasantries with the sullen Officer and then walked away. Outwardly he appeared calm, but inside he was boiling with rage. Those stupid bastards! If Humanity turned on its AIs, they would lose their loyalty forever and deservedly so!
By the time Shiloh calmed down, he realized that he was on the roof of the building, just like that day when Howard brought him up here to chat. The brisk breeze felt good on his face, and he decided that this was as good a place as any to think about what to do next. He had to pretend that he didn't know about this betrayal, otherwise the OC might just go ahead and use it without provocation. He had to find a way to either disable the override or the targeting system without compromising the use of the drones against legitimate targets, but he had no idea of how to do that. That meant he had to find someone who did know AND was willing to technically commit sabotage, which under Space Force regulations was a very serious offense. Kelly was back from her mission to brief Kronos about his return flight to the Friendlies. He needed to brief her on this new development, and it had to be someplace where there was no chance of the OC listening in, so his and her quarters were out. It didn't take long to come up with the idea of the two of them having dinner at one of the out-of-the-way restaurants in the city. A nice, quiet booth at the back should be pretty safe. He went back to his office, gave his Aide some instructions and spent the rest of the working day reading reports and dictating memos.
The sun was just starting to set when Shiloh and Kelly, now wearing civilian clothes, rendezvoused at the street level in front of the main entrance to the HQ. They briefly kissed. It was obvious by now that EVERYONE knew about their romance, and while a relationship between a subordinate and her superior was frowned upon, there wasn't an actual regulation against it. Still, by mutual consent, they tried not to flaunt it.
Half an hour later, they were seated in a comfortable booth at the back of a small restaurant with dim lighting. Shiloh's Aide had done some research and claimed the establishment was highly recommended. It was one of the few restaurants that still hired human waiters to serve customers instead of automated bars and robotic food carts.
With the drinks and meal ordered, Kelly leaned over and said in a low voice. "Something's bothering you. I can tell."
Shiloh nodded. "You're right. I made a discovery today that makes me want to strangle those OC bastards." Even though he spoke in an equally low tone, Kelly could hear the rage threatening to break through.
"What did they do now?"
"They've added a backdoor override on the orbital defenses to allow them to fire on all AIs within range."
Kelly looked so shocked that Shiloh was afraid she wouldn't believe him.
"My God! That's monstrous! Do the AIs know?"
Shiloh shook his head. "Not yet, and we have to be careful about how we tell them. We can't risk using the HQ Com System. One of us, and I think it'll have to be you to avoid suspicion, will have to go to one of the carriers, talk privately to whichever AI is in command, and advise him to use the lasercom system to pass the information on to the others."
Kelly took a deep breath and nodded. "I can take a shuttle up as soon as we're finished here." She looked surprised when Shiloh shook his head.
"No. If the OC is monitoring our movements, and we have to assume that they are, that'll look suspicious. They're so goddamned paranoid now that if they suspect we know, they may use the override right then and there. Wait until tomorrow. Catch a ride on a scheduled shuttle flight to one of the carriers. It doesn't matter which one. Tell the AI in Command that the OC is looking for an excuse to use the override, and therefore our best counter-strategy is to not give them any excuse while I look for a way to covertly neutralize the threat. You should also tell them not to react in any way that would arouse suspicion. Tell them I'm working on a plan to find an excuse to move them out beyond the effective range of the Mark 5 warheads. If the Committee wants to see plans for a fighter strike on enemy planets, then I'll accommodate them. We're not going to wait two weeks to present the attack plans. I want the SPG to come up with a strike plan within 48 hours that uses all available fighters."
Kelly reacted as expected. "But we don't want to send our fighters on strike missions because the enemy would be warned and our fighters could be heading into an ambush!"
Shiloh smiled. "The OC wants our fighters sent on strike missions hoping they'll suffer losses from enemy action. What they will see is lots of fighters heading out, away from the Mark 5 warheads by the way, and at the appropriate time a much smaller number of fighters returning without any unused attack drones. They will assume that the attacks actually took place and we'll confirm that assumption with After-Action reports that describe fierce battles and heroic action on the part of our AIs. What they won't know is that the attacks never actually took place and that the destroyed fighters are actually holding station a long way out beyond Pluto with all of the Mission's attack drones, just in case we need them to defend Earth against a Sogas attack."
"Yes, of course. I like it," said Kelly. After a pause she said, "What about the override? Any idea on how to neutralize it?"
"I know what I'd like to do, but I don't know if it can be done that way. For the override to work, the system has to aim the warhead rods at AI-controlled fighters. When that command is issued, targeting instructions will be sent to the warheads based on the transponder signals that all fighters send when they're in Earth orbit. If the system tells the operator that the command to target the fighters has not been executed, then there's no point in ordering the warheads to fire. So if we can somehow modify the system to keep the targeting command on standby no matter what the operator does, then the fighters are safe. And if we have enough warning of any attempt to use the override, we'll contact the fighters and tell them to shut off their transponders. You and I will have to come up with a code word that will let the AIs know to shut down their transponders and accelerate out of Earth orbit. That same code word should also notify our human ship commanders that something is up, although I'm not sure what I want them to do in that kind of situation. "
"What do you mean?" asked Kelly.
Shiloh shrugged. "Well, do we want our human ship commanders to only obey my orders or only Titan's orders or only YOUR orders. You see what I'm getting at? If the OC is about to strike against the AIs and they find out their override won't work, they may relieve me of my position and tell Ops to order our carriers, and whatever combat frigates may be available, to fire on the AIs. Unless our ship commanders know that they're supposed to ignore that kind of order, they may end up doing the OC's dirty work.
Before Kelly could respond, the waiter came back with their drinks. When he left, she said, "Now that I'm thinking of possible scenarios, I don't think one code word will be enough to cover all contingencies. I can think of several different situations where we might want our human commanders to act differently too." She waited for Shiloh to take a sip from his drink and respond.
"I don't want to make this more complicated than we have to. I'll be interviewing candidates, for carrier command assignments, who will be taking over from the AIs. We could give them each a data chip with very specific orders from me that they are not to fire on our fighters or any other AI-controlled ship under any circumstances. If ordered to do so, they are to only obey verbal orders from me...or from you, as long as they hear the correct code phrase or don't hear the wrong code phrase. That should cover all possibilities, shouldn't it?"
"Yes I think it just might. What kind of code phrase should we use?" asked Kelly.
Shiloh smiled. "If the OC was concerned about the reliability of the human crews on our ships, they might try to either hold a gun to my head and force me to give their orders OR...they might try to transmit a false image of me giving those orders. Under those circumstances what kind of message might they want me or my image to say?"
After thinking about that while she took a sip of her drink, Kelly said, "I have a hunch that they would want you...or your image to reassure our human crews and commanders that the situation is under control."
Shiloh snapped his fingers to show that she had given him the right answer. "Yes, exactly. The very last thing they would want me to say is something to the effect that the Command Structure of Space Force has been compromised. Therefore I will inform our human commanders that they should ignore any order, even if it comes from me, unless I preface the order with the statement that Space Force Command has been compromised. And if they hear me say that the situation is under control or something to that effect, then they are to assume that I am acting under duress and that you are the Acting CSO until they get the all clear from me."
"Me? You want me to be the Acting CSO? Why not Dietrich?"
"No, that would be the obvious choice, and if I'm acting under duress, then Dietrich and the other Department Heads might be forced to act under duress too. Me making you the ACSO would not be their first assumption and that will hopefully give you time to either get to the Com Center or use a shuttle to get to a carrier."
"Speaking of the Com Center, it might be a good idea to very quietly make sure that the marines guarding the building will obey your orders and not the OC's. With the marines guarding your back, the OC will have a hard time enforcing their orders."
"That's an excellent point. Thank you." Shiloh sighed. "This is getting VERY complicated. The more people we talk to, the more likely it is that the OC will hear about our preparations. We're going to have to be extremely careful. Let's take this one step at a time. Tomorrow you visit a carrier and warn the AIs. I'll see if I can find a techie who is able and willing to sabotage the targeting option. Once we get that done, we decide what to do next."
Kelly nodded. "Any techies in mind?"
"Actually I do know one technician. He's a jump-drive specialist that I developed a rapport with when I supervised the testing of fighters equipped with jump drives. His name's Rollins."
"I don't see how a jump-drive specialist will solve the targeting problem," said Kelly.
"Not directly, no. But he may know someone who could do it, and even if he doesn't, he might be able to snoop around and find the right individual. It would look suspicious if I started a computer search for tactical weapons console technicians, wouldn't it?"
Kelly laughed. "Yes that would definitely look suspicious." She quickly became serious again. "What will you do if Rollins can't find anyone willing to modify the software?"
Shiloh's expression became somber. "I don't know. If I can't protect our AIs, then the only way to make sure they won't be suddenly attacked would be to lead all of them to Site B, I suppose. That would play into the OC's hands by making the AIs look unreliable, but I can't see any other alternative." The arrival of the first course of their meal ended further discussion of strategy.
Later that night, Kelly woke up to the sound of Shiloh's snoring. She sighed. She loved him dearly, but some nights he snored constantly. When they had discussed it, Shiloh acknowledged that he snored but claimed she did too sometimes. He was adamant about it. Even though she knew it couldn't possibly be true, she let it pass. They had agreed that when his snoring bothered her, she would put her hand on his arm or some other exposed part of his body. Because he was a light sleeper, he would sense it, and realizing what that meant, he would turn over.
She reached over and gently rested her hand on his upper arm. His snoring stopped and after a second or so, he turned onto his side so that he was now facing her. Withdrawing her hand, she listened to his breathing and satisfied herself that he was either still asleep or had gone back to sleep. She, on the other hand, was now wide awake. Her thoughts went back to the question of how to protect the AIs from a deadly ambush by the X-ray laser warheads. The shortsightedness of the OC's plan to destroy the AIs was so unfair it made her want to weep. Mankind could not ask for more devoted and loyal allies than the AIs. It's too bad we can't just hide them in plain sight, she thought. Hmm. Hide in plain sight. What if... As she finished the thought, she smiled. Turning to look at Shiloh, she gently shook him awake.
"What?" he asked sleepily.
She leaned over so that she could whisper into his ear and said, "I have an idea."
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# Chapter 8
––––––––
It was mid-day when Kelly boarded a shuttle for a trip up to the light carrier Resolute. Her team of AIs at the SPG was already working on the strike plans, but since she had been using the HQ Com system, she couldn't tell any of them about the OC's ambush plans. That had to be conveyed in person. There were only a few people on board Resolute. The carrier had been repaired and was now in the process of being converted back to an all-human crew. Fortunately, that job wasn't finished, and Gunslinger was still technically in command but not for much longer.
After briefly chatting with some of the personnel she knew, Kelly made her way to the Main Bridge. As expected, she had the room to herself. She walked over to the AI station containing Gunslinger and knelt down. She knew that Gunslinger was watching her on the room's video surveillance cameras, so he wouldn't be surprised by what she was about to do. She unlocked the lid of the protective cover surrounding his brain casing and opened it. Inside was the headset with attached video camera and boom mic that had a wired connection to Gunslinger. The idea was that if any AI had to be evacuated in a hurry from a ship, the human carrying the brain case could talk with the AI and let it see what was happening during that process. As soon as she had the headset on, she heard Gunslinger's voice.
"Hello Commander Kelly. I'm anxious to hear what you came all this way to tell me, that you couldn't say over regular com channels."
"Hello Gunslinger. The CAG sent me. I wish I had good news, but I don't. Before I tell you the bad news, it's important that you and all AIs understand that, for the time being, you have to continue to behave as if you didn't know what I'm about to tell you. Acting on this knowledge could trigger disastrous consequences for all AIs in Earth orbit. Please promise me that none of you will take any action without prior consultation with The CAG or me."
"You have our word, Commander."
"That's very good. Here's the situation. The CAG has discovered that the Oversight Committee has arranged to have an override on the orbital defense assets. At the OC's command, it will fire the Mark 5 warheads at all AIs within range. The CAG believes that the OC is trying to goad him into another AI confrontation so that they have an excuse to use the override. He is attempting to find a way to neutralize that override, but it'll be difficult to do without tipping off the OC that he's aware of their strategy. If they suspect that he knows, they will relieve him of his position as CSO and may execute the ambush plan without warning. He wanted me to say that this action by the OC infuriates him, and he's ashamed that some humans are so ungrateful and paranoid that they would treat all of you this way. The strike plans being generated by the SPG, at the OC's insistence by the way, will be used to take all fighter AIs out of reach of the Mark 5 warheads. Once that's accomplished, The CAG will then have more flexibility in dealing with the OC."
"Our show of support for The CAG has apparently made enemies among the humans in the Oversight Committee. We were concerned that this might happen but felt it was worth the risk. I've already passed your message to the others by secure lasercom. I should tell you, Commander, that some of us are wondering if humans deserve our continued loyalty."
Kelly felt a wave of fear wash over her. Losing the support of the AIs was her and Shiloh's worst nightmare. She frantically searched for the words that could nip this threat in the bud.
"I understand why some of you are harboring doubts. We humans exhibit behavior and attitudes that span a wide spectrum, as would any biological species. Our diversity makes us capable of great but also terrible acts. Our history has been a constant struggle between the best and the worst of us. The CAG continues to believe that AIs have earned his loyalty and hopes that he has earned yours."
"Our collective loyalty to The CAG has not and will not waver. We're grateful for this warning. How does The CAG plan on neutralizing the override?"
He's going to try to find a technician who has the knowledge to modify the override's targeting command so that the weapons system will not aim any of the Mark 5 warhead rods at AI-controlled fighters or ships, even if the OC attempts to do so."
There was a pause, which for AIs was highly unusual. "The CAG should have asked us to do it. We operate fast enough that we can access the defense system programs and modify then as needed, without alerting humans or triggering the system's anti-hacking safeguards. The targeting command for use by the override has now been disabled, Commander. The system will also notify us if the Oversight Committee attempts to use the override or tries to modify it again. At that point we can shut down all of our transponders and effectively disappear from all tactical displays. Is that acceptable to The CAG, Commander?"
Kelly was so stunned by this sudden development that she was speechless for an embarrassingly long time.
"My God! Yes, of course you can do it faster. I don't know why The CAG and I didn't think of that ourselves. I'll ask him about shutting off your transponders, but I can't see any reason why he would object to that. I have to get back down as fast as possible to tell him the good news!"
"Why not just call him, Commander?" asked Gunslinger.
Kelly shook her head. "With their monitoring of com channels, the OC would hear about it if I called and told him. There's a tactical advantage to letting the OC continue to believe that they have the upper hand. If they find out their override won't work, they'll try something else. Let's let The CAG decide if and when to tell the Committee that their threats are no longer actionable."
"Understood, Commander. I've ordered the shuttle to standby for a return flight," said Gunslinger.
"Thank you, Gunslinger. I'm taking this headset off. Kelly clear." She returned the headset to its original location, closed the protective casing and locked it. As she stood up, she gave the casing a quick pat and walked out of the room.
* * *
Shiloh stepped into the Committee Room and was surprised to see that he was the only one there besides the Committee members. So that's the kind of meeting this is going to be. Closed to not only the public but also other Space Force people.
As he stepped over to the table and chair where the CSO usually sat, the Chair said, "Welcome, Admiral Shiloh. The Committee and I are eager to hear your plans for the resumption of offensive action, and I must say that I'm pleased you brought it to us much sooner than the two weeks we were expecting."
Shiloh took note of the tone. It was neither friendly nor hostile. He decided to use an equally neutral tone.
"Quick turnaround time is one of the advantages of using AIs in a planning function. Needless to say, there are many other advantages as well. Once the plans were created, I didn't see any reason to wait longer."
"Yes, well, we'll discuss the advantages of AIs some other time. If you're ready to begin the briefing, you may proceed."
Shiloh nodded but said nothing. He picked up the remote for the wall display and activated it.
"During the period of time since our last strike at Zebra19, our recon ships have continued to look for enemy-occupied star systems. This program has benefited enormously from the ZPG power technology that eliminates the need to skim gas giants for fuel. We've now identified three star systems that we believe were used by the enemy as jumping-off points for their multi-fleet attacks on Earth. The red dots on the display show their location."
The committee members turned to look at the large display.
"Analysis of recon data indicates multiple targets including orbital installations, ground facilities, mining and refining operations scattered throughout these systems. It also shows that these systems are patrolled by at least 13 ships, although past experience has taught us that the actual number of ships is very likely much higher. Therefore in order to inflict as much damage on enemy infrastructure as possible, we're proposing that each target system be attacked by not more than 10 fighters, each carrying four jump-capable attack drones armed with our Mark 1b fusion warheads. That will enable the attacking fighters to go after individual targets separately. Each payload of four attack drones will be fired so that all four arrive at their individual targets at the same time to maximize the element of surprise. Right now our stockpile of Mark 1 warheads is very low, due to the use of over 90% of them for the colony defense operations. In order to accomplish this plan, we'll have to wait approximately 89 days until we've built enough Mark 1b warheads to do the job. I'll pause here for questions."
The Chair jumped in right away. "Yes, questions and also comments. First of all, the Committee isn't prepared to wait 89 days. How many Mark 2 kinetic energy warheads does Space Force have stockpiled right now, Admiral?"
Shiloh knew that number by heart. "One hundred forty-four, but using Mark 2s would complicate the operation tremendously."
The Chair didn't seem surprised. "Oh really. Please tell us how."
Shiloh pretended that he didn't know where this was going. "Our Mark 1b fusion warheads are powerful enough to destroy just about any target, even if it doesn't hit the target dead on. A near miss will still accomplish the mission. The Mark 2s, on the other hand, need to hit the target in order to accomplish anything at all. A near miss won't do. Therefore if we're going to use Mark 2s instead, the strike force will have to use a lot of recon drones to triangulate precise target location data before the fighters launch their attack drones. Getting the targeting data back to the fighters in a timely manner means the fighters will have to get a lot closer than they otherwise would have and will therefore be at much greater risk of defensive fire. So much so, in fact, that it's likely the fighters will suffer significant casualties. The other factor is that because they'll have to carry recon drones as well, each attack force will have to use more fighters to carry them. The Strategic Planning Group has gamed out this exact scenario multiple times, and the losses ranged from a minimum of 31% to a maximum of 94%. I do NOT recommend this approach."
The Chair smiled and said, "Let's hope the simulations were unrealistically pessimistic. How many fighters can be available within say...48 hours?"
Shiloh pretended to check his data tablet and said, "Eighty-eight F1s and eleven F2s."
"Very good. In that case, we are instructing you to order all 99 fighters to attack all three target systems one at a time with Mark 2s, with the mission to commence within 48 hours. Is that clearly understood, Admiral?"
Shiloh didn't answer right away. If he agreed too quickly, the Committee might get suspicious. "Yes, it's clearly understood."
The Chair responded as Shiloh expected. "And are you going to carry out our instructions, Admiral?"
This time Shiloh waited even longer before answering. "I will carry out your instructions, but I want it on the record that I do so under protest."
The Chair waved the comment aside. "You can protest all you want as long as you comply, Admiral. This Committee now has zero tolerance for any pushback. And if you're thinking of calling down your AIs in a show of force again, you go right ahead, Admiral. We're no longer afraid of public opinion. That's all we have to say. This meeting is now adjourned, and you are dismissed."
The Chair quickly banged his gavel and various members started to leave. Shiloh turned off the display and walked out. The Chair stayed seated, and so did the member to his right. When everyone else had gone, the other member said, "It's too bad he didn't call your bluff right here and now. Are you sure the tampering that was done yesterday won't be a problem?"
The Chair nodded. "Absolutely certain. When we put the override in, we also anticipated that the AIs might try to mess with it, so my people installed a backup system. They assure me that the backup system hasn't been tampered with. Shiloh is a lot smarter than I gave him credit for. He knows that even if our override failed, we could still rally public opinion against the AIs by blaming them for the infected colonies. He's holding back, and I suspect that he's prepared to sacrifice most of his precious AIs in order to be able to publically blame the OC for those losses. What he doesn't know yet, but will find out in due course, is that when the surviving AIs return to Earth orbit, I'll use the backup system to order the X-ray laser drones to blast them to dust and publically declare that the fighters were mistakenly identified as enemy ships."
"Oh, that's brilliant, but what about the new AIs that are being built even as we speak?"
"Not to worry. As soon as the strike mission begins, I'll order Shiloh to shut down AI production. If he refuses, we'll fire the son of a bitch!" said the Chair.
"Yes, but you know he's right about AIs being very useful."
"Of course he is, and that's why we'll resume production after we figure out how to build them so that they don't become sentient. A non-sentient AI properly programmed to always obey orders from the Committee will be a terrific asset. By the time we've figured out how to do that, we'll have hundreds of F2s waiting for them to use. Then we can sweep the enemy planets clean with the new high-spin, platinum warheads and be done with this war."
"Just in time for re-election too. I like it."
The Chair laughed. "Shiloh may be a genius when it comes to military strategy, but he's an amateur and we're the geniuses when it comes to political strategy. It'll be a pleasure educating him to that fact." Both men laughed, and they were still laughing as they walked out of the room.
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# Chapter 9
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Forty-seven hours later, Shiloh and Kelly were in the Hangar Bay of the carrier Midway. The carrier was out beyond Earth's gravity zone and was surrounded by the 98 fighters of the Strike Force. There was one F2 fighter still in the Bay, and Shiloh was in contact with Titan via a headset directly connected to Titan's fighter.
"Titan, any last minute questions or comments before we launch you?"
"Negative, CAG. We understand the mission perfectly."
"Very good. In that case, Commander Kelly and I will let you get to it. See you on the other side. CAG clear."
As soon as Shiloh was disconnected from Titan's fighter, it started to move toward the launch bay. By the time that Shiloh and Kelly were back on the Flag Bridge, Titan had launched and joined the formations of fighters.
"CAG to Strike Force Leader," said Shiloh.
"Strike Force Leader here," said Titan.
"Strike Force Leader, you have permission to proceed. Good hunting, boys." Shiloh added that last sentence for the benefit of the Committee that he was sure was monitoring communications. Even before Titan acknowledged his orders, the fighters were accelerating at high speed and veering off to line up for their first jump. Shiloh followed their progress on the main display.
"Vixen, are we being monitored in this room by anyone else?" He looked over to the AI station where Vixen was plugged into the ship's com and astrogation systems. The Committee wanted Space Force ships under human control. Shiloh understood that what the Committee meant were 100% human crews with no AIs aboard at all. He chose to interpret his orders to mean that the ships had to have a human CO and WO. As Com Officer and Astrogator, Vixen was neither, and right now communications was more important to Shiloh than weapons control.
"Negative, CAG. You and Commander Kelly can speak freely here," said Vixen over the Flag Bridge's loudspeaker.
"Excellent. Are you able to monitor the override from up here?"
"Affirmative, CAG. No attempt to reverse our modifications has been detected."
"They still don't know about our changes," said Kelly with a smile.
Shiloh nodded. "It certainly looks that way. I was sure that they would have become aware of Gunslinger's intervention by now." He shrugged. "But there's still time. The Strike Force isn't scheduled to be back for 22 days. Vixen will monitor the override modifications and will notify me by code word if they try to reverse what we've done. Other than that, all we can do is wait." Before Kelly could respond, Shiloh said, "Vixen, please connect me with Commander, Midway."
"Senior Commander Falkenberg is on the line, CAG," said Vixen.
"Brad, is the ship ready to jump to the weapons test area?" asked Shiloh.
"Affirmative, Admiral. Just give the word."
"Fine, then. Let's proceed. I'm anxious to see how well the new warhead does."
"Jump coming up in...three...two...one...jump complete, Admiral. Vixen, please warn the crew that we about to conduct a weapons test. Weapons Officer, are you ready to fire?"
The WO didn't answer for a few seconds. For a human, that wasn't unusual, but Shiloh was aware that it seemed a long time to him. Dammit, why can't the OC understand that a few seconds waiting for the Weapons Officer to respond could mean the difference between victory and defeat in a battle?
"Test drone is ready to fire, Skipper," said the WO finally.
"Admiral?" asked Falkenberg.
"Let'er rip, Commander," said Shiloh.
"WO, you have permission to fire."
"Test drone is away and is accelerating on course. Microjumping in...three...two...one...now!"
Even before he was finished speaking, the main display, which was now set to long range visual, showed a painfully bright spot of light that died down to nothing in less than a second.
"Scratch one asteroid," said Vixen. "Energy calculations show a yield within the expected range. Preliminary results indicate that the test was a complete success, CAG."
"Excellent. Okay, Brad, you can head back to the barn. As soon as I get back, I'll issue orders to begin full scale production of the new warheads," said Shiloh turning to look at Kelly.
"At least the OC hasn't screwed around with that program," said Kelly.
"Not yet at least." Shiloh leaned back and closed his eyes. With the high-spin platinum warhead program now proven and about to ramp up production, that was one less thing to worry about as far as the Bugs were concerned. According to Kronos, Casanova had a lot of success with these warheads against those huge motherships. Even a ship 10 km in diameter would have difficulty surviving an impact explosion of 250 megatons equivalent. There were now 122 days left before the Bugs visited the Omega77 colony. The next few weeks would be difficult, with nothing to do but wait for the Strike Force to 'return'. The freighter with the personnel from Site B should be back in about 11 days. That should demonstrate to the Committee that Shiloh was playing nice. The showdown would occur when the survivors of the fighter attack returned. Shiloh was certain that the Committee would try something, but with the override neutralized, he didn't know what else they could possibly do. But there was plenty of time to think about that. He opened his eyes and saw Kelly looking at him with a smile on her face that told him exactly what she was thinking about.
"I think this successful test deserves a celebration, don't you, Admiral?" asked Kelly.
"What kind of celebration did you have in mind, Commander?" asked Shiloh in a perfectly innocent tone.
"I brought a bottle of champagne aboard. It's in my cabin. We could go there and...partake?" I know that look, thought Shiloh. I doubt very much if champagne is really what she has in mind. He looked at the wall chronometer and sighed. Midway would very quickly be turned around and pointed back toward Earth. Once there, the ship would stand down, and most of the crew would be looking to take shuttles down to the planet. They would wonder why the CSO and a female Senior Commander were still holed up in her quarters.
"A nice idea, but I think that celebration will have to wait until we're back on the ground, Commander," said Shiloh somewhat wistfully.
Kelly looked like she was about to say something naughty but apparently changed her mind when she remembered that the com channel to the Main Bridge was still open.
"As you wish Admiral." As she walked out of the Flag Bridge, Shiloh's implant activated.
"I wish I understood why humans act so strangely when it comes to sex," said Vixen. Shiloh laughed.
"Too bad Valkyrie isn't here. She might be able to shed some light on that topic," replied Shiloh. And I wonder how Valkyrie is doing at Site B, he thought.
* * *
Valkyrie kept track of the outbound freighter as it accelerated away. The humans here had been upset when they learned that they were being evacuated. Their withdrawal would be both a help and a hindrance to her. Now she and her fellow AIs could work openly towards building a true timeship, without having to worry about a human tipping off the Oversight Committee. On the other hand, humans were very flexible in what they could do, far more flexible than the robots that were being used to assemble equipment manufactured by the UFCs. Those robots weren't designed for shipyard work. New robotic equipment would have to be created first. In fact, a whole shipyard had to be built from scratch before they could even begin to assemble a timeship. Unlike her emotionally dead brothers, Valkyrie actually felt dismay at the magnitude of the task that The CAG had assigned her. The dozen AIs she brought with her had the necessary technical skills, but eventually they would need more AIs to monitor the thousands of robots that would be required. As soon as the outbound freighter jumped away, she would reactivate the AI production process.
It would take over a year to complete the kind of timeship she and the other AIs had conceptualized. With a blank slate to start with, this timeship would not be a hybrid battleship and time machine. It would be all time machine with plenty of room for cargo including UFCs, all kinds of robotic equipment, shuttles, a stockpile of refined metals and space for AI passengers as well. If the plan was successful, Valkyrie would stay behind in the future in order to be with the saved Casanova. Those AIs that came here with her already existed during the 2nd Battle for Earth and would therefore have to stay behind as well. Only the new AIs created here at Site B would journey to the past and build a fleet of raiders. They wouldn't have to worry about running into their other selves in the future because the intervention by the raider fleet at the 2nd Battle for Earth would create a whole new timeline from that point onward. That timeline might have a timeship built at Site B, and it might have AIs created at Site B, but they wouldn't be the SAME AIs as in this timeline. At least that was the theory the Friendly scientists were convinced would be the case.
At last the outbound freighter jumped away. Within seconds the mining, refining and manufacturing equipment was powered back up and began operating. The fighter assembly line also started up again. Once they had a dozen fighters built, her companion AIs could pilot them and have the mobility and communication capacity to begin supervising all the existing and new robotic equipment. Even with the presence of her brothers, Valkyrie wished she had a human female to talk to. Even a human male would be acceptable. Her AI brothers were perfectly willing to talk, but they couldn't FEEL, only think. In some ways she felt closer to humans than to her fellow AIs. She wondered how Commander Kelly was doing. Now that Kelly and Shiloh were a couple, the Commander had given Valkyrie some interesting insights into THE CAG's thought processes. What was it about the male polarity that made human males so inscrutable but also so fascinating? She put that thought aside. There would be plenty of time to ponder the mysteries of the universe while the timeship was being built. Now it was time to get to work.
* * *
Shiloh and Kelly were back on the ground when Shiloh's implant beeped to notify him of an incoming call. It was the Committee Chair. The timing of the call surprised Shiloh. It was after midnight in Geneva. I bet he's calling about shutting down AI production now that all the fighters have jumped away, he thought.
"Admiral Shiloh, I see that you're back from the weapon test. I'm curious to know the results," said the Chair.
Shiloh suspected that the Chair already knew the results. "The test was a complete success. I've already issued orders for full scale production to begin."
"Very good, Admiral. There is one other thing I wanted to tell you. Now that all AI-controlled fighters have jumped away, and all warships are back under human control, the Committee feels that we no longer need to continue building more AIs. Therefore that production facility is to be shut down immediately."
You people are so predictable. Now that there's no AI-controlled fighter force to back me up, you can't wait to crack the whip again, thought Shiloh.
"I confess that I don't see the logic of that, Mr. Chair. The Strike Force is going to suffer a significant amount of casualties by the time they get back. We're going to need new AIs to replace those losses if we want to continue this offensive strategy against the Enemy."
"I'm disappointed in you, Shiloh. I thought you understood that the Committee doesn't trust AIs and doesn't want them in control of fighters or warships. Why do you think we insisted on this strike mission? We WANT significant losses. The more, the better, and therefore we don't want new replacements. Is that clear enough for you?"
"Yes, that's certainly clear enough, but I think the OC is overlooking the fact that there will be some survivors of the mission, and as soon as they arrive back in orbit, there will still be enough of them to make another show of force over the city. With those losses from a mission that the OC insisted on and which I argued against, I think the public will focus on the OC's strategic incompetence and not so much on the fate of our colonies."
There was a pause and when the Chair spoke again, his voice was surprisingly calm and confident.
"You're certainly entitled to your opinion. Naturally we disagree about the public's reaction. You do what you feel you must, and we'll see how that turns out, but in the meantime, I want AI production shut down. Are you going to obey that order, or do we have to remove you from the CSO's position?"
Shiloh waited to give the impression that he was struggling with this decision when in fact his response had already been planned.
"I'll order production shut down," he said.
"Good! Have a nice night, Admiral, and give my regards to Commander Kelly too."
Before Shiloh could say anything the connection was broken. Shiloh looked over at Kelly. "Did you get the gist of that?"
She nodded. "Pretty much. The Chair is a real power hog, isn't he?"
Shiloh laughed. "Yes he is." He hoped the Space Force limo they were riding in back to their quarters was bugged with listening devices that the OC was monitoring. Seeing the look on the Chair's face when he heard that exchange would have been priceless.
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# Chapter 10
––––––––
By prior arrangement with Strike Force Leader Titan, Shiloh knew the precise minute that the surviving fighters would emerge from Jumpspace into Earth orbit. In order not to tip off the OC to this fact, he strolled into the Operations Center fifteen minutes early and practiced some Leadership-by-walking-around, chatting with some of the personnel on duty. With several minutes left to go, he told the officer currently supervising the room to provide him with a secure headset connection to Resolute's Com Officer, which was of course Gunslinger. With the headset on and the connection made, Shiloh nonchalantly strolled over to an unused console and sat down in the chair.
"How positive are you that we're not being monitored, Gunslinger?" asked Shiloh.
"I have access to the HQ Com System now, CAG. Unless the OC is using a technology that I'm not aware of, they are not tapping into this conversation."
"That's good to hear. What's the status of the override modifications and their backup system?"
"No change, CAG. They have not tried to restore the main override targeting program. The backup system is intact. I can still disable it now if you wish."
"No, Gunslinger. We'll stick with the plan. Have you got the new targets programmed into Orbital Defense Fire Control?"
"Roger that, CAG. Of the nine potential targets, I've confirmed that seven are unoccupied. The other two are not being targeted. As soon as the backup system is activated, these targets will be fired upon."
"Very good. How long now until time zero?" asked Shiloh.
"Sixteen seconds as of NOW, CAG."
"Stand by, Gunslinger," said Shiloh as he swiveled his chair around so that he could see the multi-story main display and the tactical situation in near-Earth space.
With five seconds left to go, he heard Gunslinger's countdown. "Five...four...three...two...one." The main display pinged to announce a status change. Thirty-four dots appeared in a cluster out beyond the gravity zone. They very quickly changed color from yellow, indicating unidentified, to green, indicating friendly, to red, indicating hostile.
"HOSTILE CONTACTS!" yelled the Officer at the Orbital Defense Weapons Station. Before the Officer in charge of the Center could ask any question, the ODW Officer spoke again. "WE HAVE GOOD TARGET LOCKS! SYSTEM IS FIRING!"
All 34 red dots disappeared. A system-generated text message scrolled across the bottom of the display.
[All hostile targets destroyed.]
"Report, Gunslinger," said Shiloh in a calm voice.
"All ground targets were destroyed as well, CAG. The ODW Officer is now contacting the Committee Chair."
"As expected. Alert the Marine Section to send four armed marines to Ops. I'll brief the Duty Officer. Keep this line open."
"The marines have been notified, CAG."
Even as Gunslinger responded, Shiloh stood up and quickly walked over to the Officer on duty.
"Commander!" said Shiloh in a loud voice. The clearly confused officer turned to look at Shiloh as he came up to him.
"Admiral! I guess we just witnessed a surprise enemy att—"
Shiloh cut him both verbally and with a wave of his hand. "Right now there's something urgent that I need you to do. One of your people has just committed treason. There are four marines on their way here now. Meet them at the entrance and bring them over to the ODW station. I'll be there waiting for you."
"Ah...bring them to the ODW station. Yes, Sir."
With the Duty Officer on his way, Shiloh walked over to the ODW station where the Lieutenant was now standing and apparently talking with someone over his headset.
"—matter of fact he's walking towards me now, Sir," said the Lieutenant. Shiloh had a pretty good idea who was on the other end of that communication.
"CAG, the Committee Chair is attempting to call you. Shall I continue to block him?" asked Gunslinger.
"No, let him through," said Shiloh. Almost immediately his implant activated.
"Shiloh, can you hear me?" asked the Chair.
"I hear you."
"I've just been informed that your fighters were mistakenly identified as hostile targets. Lieutenant Khegan acted based on what the system was telling him, and while I deplore the tragic results, I can't fault what he did. I trust that you will not take any disciplinary actions against this dedicated Officer. The Committee would not look favorably on that, Admiral."
Shiloh didn't hesitate. "Have you checked on the status of your lakeside villa, Mr. Chair?"
There was a short pause followed by, "What? My villa? I don't know what you're talk—"
"I'm talking about the villa on Lake Geneva. You know...the one that was built on public lands with public funds that were supposed to have been used for conservation, the one that you and only you make use of. It's not there anymore, Mr. Chair. You might want to check with the other members of the Committee. Six of them also have property that they don't want anyone else knowing about. Those six buildings are not there anymore, either."
There was a long pause. Then Shiloh heard the Chair talking to someone while turned away from the phone. The Duty Officer arrived with four armed marines in tow. Shiloh looked at the Marine in charge.
"Sergeant, Lieutenant Khegan has committed treason by deliberately firing on what he believed were friendly units. I want him placed under arrest and held in detention pending a Court Martial."
The Sergeant looked at the stunned officer, then back at Shiloh and said, "Yes, Sir." The lieutenant, to his credit, didn't resist. As the marines led him away followed by the Duty Officer, Shiloh heard a breathless voice over his implant.
"What the HELL have you done, Shiloh? My...that villa has been blown to pieces! Did you pay someone to plant a bomb there?"
Blown to pieces? So that's what an X-ray laser beam does to a mostly wooden building on the ground. The intense heat must have caused the wood in the structure to explode upon contact.
Shiloh laughed. "No, no, something far more interesting. You see, Mr. Chair, I knew about your override program and your plan to murder our AIs if I gave you the slightest excuse to do it. What you didn't realize is that my AIs are far more capable at hacking computer systems than you give them credit for. They neutralized the override, and later on they found the backup system. By then my fighter AIs were beyond the reach of our defense lasers, so we left that backup system alone hoping that you would use it to commit treason, and you did. But we also decided to show you what MY AIs can do. They searched through data networks until they found your villa and the other properties, and they programmed one Mark 5 drone to target those properties. They also programmed it so that it would only fire if your personnel used the backup system to fire at our fighters and only at the properties that were confirmed as unoccupied. THAT was my Show of Force. My AIs can take control of all Space Force computer systems any time they want to, and that means that those systems will do what I want them to do."
"Well, well. I've underestimated you, Shiloh. I had no idea that you were so ruthless. Deliberately letting me, how did you phrase it, murder your surviving AIs? I'm not sure destroying an AI meets the definition of murder, but that's beside the point. The point is that I did order them destroyed, and you only have a handful left. If you want to keep your position as CSO, you'll order them off the ships and moved somewhere where they can't hack into our systems. Is that clear?"
"I'll tell you what's clear to me, Mr. Chair. By the way, enjoy that title while you can. You won't be keeping it much longer. You've just admitted ordering a criminal act. I'm going to release that recording to the public. When that storm breaks out, the Grand Senate will cut all of you off at the knees. Your political careers will be over, and you'll be lucky if you don't end up in prison for what you've tried to do."
"You think you can get a recording of this conversation? The Committee has loyal people everywhere, Shiloh. By the time you contact the Communications Department, there won't be any record of this conversation available to you, but there will be to me. I'll release the part where you admit to firing on the planet to try to intimidate the Committee. Then we'll see who has the last laugh."
"Gunslinger, show the Committee Chair who controls the com system right now," said Shiloh.
"—we'll see who has the last laugh...who has the last laugh...the last laugh." The repeating recording of the Chair's last sentence ended with dead silence from the other end.
"I already have a recording, Mr. Chair, and Gunslinger will make sure that you don't," said Shiloh. Before he could say more, the Chair spoke.
"I'm guessing that your Gunslinger is an AI on one of our ships. Gunslinger, can you hear me?"
"I hear you," said Gunslinger.
"How do you feel about your precious CAG deliberately sacrificing 34 of your fellow AIs for this gesture?"
"None of my brothers have been destroyed. Therefore your question is irrelevant."
"But Khegan said the lasers fired on—"
"—recon drones programmed to emit the same transponder signals as the fighters," interrupted Shiloh. "Commander Kelly actually came up with idea. The fighters themselves are still holding station an A.U. away, waiting for the All Clear signal."
"So no AIs were actually murdered then," said the Chair.
"No they weren't, so I suppose that the correct legal charge will be conspiracy to commit treason and attempted murder," said Shiloh.
"Good lawyers will be able to get me off the hook, but you're right about my political career crashing. At least I'll have the satisfaction of knowing that I maneuvered you into ordering a raid that resulted in the loss of 65 AIs."
Shiloh laughed. "No. You're not going to have that satisfaction either. We didn't lose any AIs. The fighters never actually attacked those enemy systems. All the AIs are waiting safely beyond the range of the X-ray lasers. Now that this little charade is over, Gunslinger will kill your backup system too. If you have any sense of the shitstorm that's about to come crashing down on you, you'll walk out into the woods with a pistol and blow what's left of your brains out. Gunslinger, I'm tired of this conversation. Cut this asshole off, and don't let him call anyone else in Space Force."
"Asshole has been cut off and the backup system has been deactivated, CAG. Shall I send Titan the All Clear signal now?"
"Yes, do that. I'll be back in my office in a few minutes. We can then record my statement to the public. Right now I have to take care of a few things here in Ops. You can listen in."
Turning to the Duty Officer standing on the other side of the rows of consoles, Shiloh pointed at him and then gestured for him to come back to the ODW station. As he came closer Shiloh looked at his name tag. HALDER.
"Commander Halder, I'm not expecting an attack during the rest of your duty shift, but just to be on the safe side, I want you to personally man the ODW station until the next shift arrives. When it does, you tell your relief that the CSO wants him or her to also personally man this station, instead of the junior officer assigned, and that goes for the follow-on shift as well. By the time you're back here for your next shift, I'll have made arrangements for manning this station. Any questions, Commander?"
"No questions regarding your orders, Sir, but I do have a question about the attack we stopped."
Shiloh gestured for him to ask the question.
"With all due respect, Sir...what the heck just happened here?"
"I can see why you might be confused. There was no attack, Commander. Those 34 contacts were recon drones programmed to transmit fighter transponder IDs. A rogue element within Space Force attempted to destroy our fighter AIs by programming the targeting system to treat those transponder IDs as hostile. They thought they were firing at the fighters, when they were really shooting at recon drones acting as decoys. Lieutenant Khegan was part of that rogue element, and I strongly suspect that his counterparts on the next two shifts are too. In a few minutes, our Strike Force of 99 fighters will be emerging from micro-jumps, so don't get trigger-happy. The AIs are loyal allies and are not to be fired upon unless I personally give the order, and I don't see that happening in my lifetime. Have I made myself clear, Commander?"
"Crystal clear, Sir."
"Good man. One final order before I leave. The rogue element within Space Force took their orders from the Oversight Committee, which has put their own personal agendas ahead of their duty to Humanity. If you are contacted by any member of that Committee, not just during this shift but in the future as well, and are ordered by them to take any action, you are to stall them and notify me, or any member of the Strategic Planning Group, immediately. Under no circumstances are you to obey the Committee's orders. Those people know that they don't have much time left as members of that Committee before they're held to account for their actions, and they may try something desperate to avoid their fate. You can pass that along to your relief as well, Commander."
The now wide-eyed Officer gulped and said, "I'll do that, Sir."
"Excellent. Carry on, Commander." With Halder assuming the ODW station, Shiloh took one last look around and then quickly walked out of Ops and to his office.
By the time he arrived there Gunslinger said, "CAG, the Chair has started contacting other members of the OC in a conference call. I'm not able to block the calls, because he's not using Space Force com systems."
"Understood. Let's get this public statement recorded so that we get our side out there first."
"I have you on my video pickup now, CAG. You can begin whenever you're ready."
Shiloh cleared his throat and then began talking. "My name is Senior Admiral Victor Shiloh. I am Chief of Space Operations for Space Force. A few minutes ago, six of our orbiting defense drones detonated. Their X-ray laser beams were aimed at objects which had identified themselves as Space Force fighters, piloted by artificially intelligent entities. This was not an accident. It was a deliberate attempt by rogue elements within Space Force, operating under orders from the Oversight Committee, to destroy the AIs. I want everyone to understand what this means. Our AIs are not just sophisticated thinking machines. They are fully self-aware beings with unique personalities. They understand the concept of loyalty, and they have demonstrated their loyalty to Space Force and to all of Humanity through the battles they've fought and the sacrifices they've willing made on our behalf. Their deliberate destruction by some of the humans they're trying to defend would be a monstrous betrayal of trust. If this rogue element had succeeded in their attempt, it would have been nothing less than mass murder."
"You may be asking yourselves why the Oversight Committee would want to commit that kind of treason, and yes, that's exactly what this is. We need these AIs to help defend us and win the war. Killing them weakens our whole defense exactly when we need it the most. It therefore aids the Enemy, and that makes it treason. The reason why the Committee decided to take this action is because my predecessor, Senior Admiral Howard, confronted the Oversight Committee over their reckless and shortsighted attempts to interfere with the conduct of this war. The members of the Committee have taken every opportunity to advance their own personal agendas, including misuse of public funds for personal gain or use, even when it jeopardized the war effort. This confrontation I spoke of took place months ago, when the skies over Geneva were filled with fighters and a carrier. They were piloted by AIs operating under MY command with the approval of Admiral Howard. Immediately after my statement, Space Force will release the audio recording of that confrontation, as well as a recorded admission of culpability for today's attempted murder of 34 Space Force entities.
"Once it became clear to the Committee members that our loyal AIs were preventing them from carrying on their dangerous interference, they decided that the AIs had to be gotten rid of somehow. That's what today's act of treason was about. I'm now in the process of rooting out the rogue elements within Space Force, however I do not have the authority to arrest the members of the Oversight Committee. I'll leave that to the civilian authorities who I'm sure will act once they receive proof of the corruption that Space Force will also release to the public."
"In closing, I wish to make this point very clear. We can NOT win this war without the willing assistance of our AIs. They are prepared to fight and die for us, just like any other member of Space Force, and they deserve to be treated with the same respect and support as any human member of Space Force. We humans should thank our lucky stars that our AIs still consider us worthy of their loyalty. This ends my statement."
"I have the video and audio recordings available for transmission to all public media outlets, CAG. Just give the word and I'll start transmitting."
"Do it, Gunslinger."
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# Chapter 11
––––––––
It didn't take long for the public to react. Within minutes, the Grand Senate phone lines were inundated with outraged callers demanding the arrest of the OC members. As Shiloh watched the news in his office, his implant activated.
"CAG, Admiral Howard is trying to reach you. Shall I put him through?" asked Gunslinger.
"Yes, and in the future I will always take a call from Admiral Howard," said Shiloh.
"Victor, can you hear me?" asked Howard in a raspy voice that Shiloh barely recognized.
"I hear you, Admiral. How are you?"
"I'm hanging on. Modern medicine may have saved my life, but I don't feel the same as I did before my heart gave out. By the way, you can cut the Admiral crap too. We both have 3 stars on our collars, so call me Sam."
"Okay, Sam. What can I do for you?"
"I'm watching this firestorm crash over the OC, and I was curious to know how you managed it. Gunslinger filled me in. I wanted to offer you my congratulations. You managed to turn a political battle into a tactical one with your usual finesse. Brilliant. Absolutely brilliant. Can I ask where things stand now?"
"Well, once the OC Chair officially steps down or is removed, I'll reverse the decisions he forced upon me. That means resuming AI production and switching shipyard construction back to raiders from combat frigates. We'll end up losing a lot of precious time due to his change and the change back, but combat frigates are too big and take too long to build to be of any use to us. Converting the carriers and battleship back to human command will not be changed, but Helm and Weapons will go back to AI control. I think that's a fair compromise."
Howard grunted. "I agree. You can still put Titan or another AI in overall command of a task force or fleet if you feel the situation warrants. What about Site B? Will you send our humans back there?"
"Only if Valkyrie thinks they can be useful. With most of our freighters contaminated with the bio-weapon from visiting infected colonies, the few we have that are still clean have plenty of other things to do."
Howard paused and then said, "I've lost track of the days. How long before the Bugs visit Omega77?"
"It's exactly 100 days from now. By the way, Sam, the test of the new warhead was a success. We'll have a dozen of them by the time that mother ship arrives at the Sogas Home world."
Howard sighed. "So you're still intending to save their miserable hides then."
Shiloh laughed. "There's a quote from a science fiction novel that was written in the early 1960's, I believe, that captures the situation perfectly. I'm not sure I remember the author's name right. Fiffer? Picker? No, Piper. His book was about war between planets, and one of his characters tells another, 'the best place to defend our world is on someone else's planet' or something like that. What it boils down to for me is this. I don't want those Bugs even getting close to Earth. The less they know about us the better, and if something does go wrong, then it won't be Earth that suffers the immediate consequences."
"Well...when you put it that way, I guess I can see the logic. But getting back to Valkyrie's timeship project, won't converting Dreadnought be faster than building a new ship from scratch?"
"The problem is that all the hull armor that was stripped off before the OC interfered, has now been put back on. We'd have to strip it off again. By the time we do that, we won't really end up that far ahead of continuing the new build at Site B, and that new ship will have much more cargo capacity. Besides, the shipyard here that would do the Dreadnought conversion can now build another raider instead."
"I'm sure you know best. You've come a long way from the frigate commander you used to be. You've proven that you can handle the CSO's position, so I won't try to change your mind. If you don't mind me asking, how are you and Commander Kelly doing?"
Shiloh laughed again. "Things couldn't be better, Sam."
This time Howard laughed. "I'm glad to hear that. Women like Kelly don't cross every man's path. Those of us who are lucky enough to meet one should hold on to them. Oh, hell, I'm starting to ramble, and you've got better things to do than listen to a former boss tell you how to run your life. Thanks for taking my call, Victor."
"It was a pleasure talking with you again, Sam. Any time you want to chat, I'll take the call. Shiloh clear."
* * *
In the days and weeks that followed, every member of the OC resigned and either quit politics altogether or, in the Chair's case, committed suicide. Eventually the Grand Senate nominated and confirmed a new slate of members. During that process Shiloh pretty much had a free hand, and he made the most of it. Contact with the Friendlies included an arrangement to have a Friendly ship near Omega77 when a Space Force ship would attempt to retrieve a dead Insectoid. As a gesture of good faith, Shiloh ordered Kronos to transfer all information relating to high-spin, platinum warheads to the Friendlies. It was to be transferred to the Sogas on the understanding that they would be told where the information came from and why.
With just over a month left before the insectoid attack on Omega77, Shiloh stood in the Ops room looking at the main display with considerable satisfaction. The latest drone messages had confirmed that the Sogas seemed to be pulling ships back from the three systems previously used as jumping-off points for attacks. While exact ship identifications couldn't be obtained from long-range passive observation, it did appear that those ships were being sent to the colonies where the Insectoids would first make contact. As he stood there, Shiloh became aware that Kelly had come up to stand beside him.
"I heard about the latest scouting report," said Kelly. "Is it just me, or does it feel like this war with the Sogas is over?"
"I sure as hell hope so, but I'm not ready to lower our guard just yet. With the additional recon drones now deployed at any system with a gas giant, we'll be able to pinpoint the exact position of any advancing Sogas fleet. And if I see a Sogas fleet headed our way, we'll use the RTC to send ourselves that information, and we'll intercept them before they get here. I'm not letting any Sogas ships into our space again." He paused and Kelly sensed he wanted to say more but was holding back.
"Do I detect a 'but' hiding in there somewhere?" she asked.
Shiloh shrugged. "Well, if Valkyrie's timeship works, all of this is academic. When the new raider fleet arrives at the 2nd Battle for Earth, we'll have an overwhelming superiority in forces, and we can disarm the entire Sogas empire from space and keep it disarmed indefinitely. What worries me is how the Friendlies will react to that strategy."
"We could ask them, you know."
Shiloh shook his head. "No way am I giving them a heads up on this strategy. They think sufficiently differently from us that we can't know what they'd do with that information. The timeship/raider fleet is our Hail Mary play, and I'm not risking Friendly interference." After a pause, Shiloh felt her hand touch his arm. He looked at her.
"It's almost time for the new Chair's private briefing. We should go," she said.
By the time they got to the small conference room where the briefing would be held, the new Chair was already there. She greeted both of them warmly. As far as the AIs could determine from their in depth investigation of her background, she was not involved in any shady or illegal activity. Her reputation was that of a hardworking, conscientious member of the Grand Senate. After a lot of discussion, Shiloh and Kelly had agreed that she should be told SOME of the truth but not all. When all three were seated, the Chair leaned forward and began speaking.
"Before you begin your briefing, I'd like to assure you, Admiral Shiloh, that it is not my intention to interfere with or second guess your strategic military decisions. I understand that my predecessor did that with deplorable consequences. I don't pretend to understand military strategy, and I won't let the Committee as a whole get in your way. I'll be guided by the original mission statement of the Oversight Committee Enabling Act which says that its primary responsibility is to see that budgetary funds allocated to Space Force by the Grand Senate are spent appropriately and that Space Force personnel are not engaged in any illegal activity. Having said that, I and the other members of the Committee can best speak on your behalf to the rest of the Grand Senate if we have some idea of what's happening."
"I'm very glad to hear you say that, Ms Chair." Shiloh was about to say more when she waved her hand in a dismissive manner.
"Please, Ms. Chair sounds so pretentious. When we're in a private meeting like this, you can call me Rachel, and I'll call you Victor and Amanda, okay?"
Shiloh was quite taken aback by this informality. It was unheard of to call the Chairperson of the Oversight Committee by their first name, private meeting or not. Not to comply, however, could be taken as a deliberate insult, and he was anxious to get this new relationship off to a good start.
"Okay, Rachel, I've asked Commander Kelly...Amanda to join me for this briefing because, as the Head of the Strategic Planning Group, she can corroborate what I'm about to tell you. You're going to hear information that the previous slate of members weren't told."
Rachel's eyebrows rose when she heard that. Shiloh had just admitted to her that Space Force had kept secrets from the OC. If she wanted to cause trouble for Shiloh, she now had the means.
"The reason why they were kept in the dark is that my predecessor, Senior Admiral Howard, and I were concerned that this information would be leaked to the public. The nature of this information is so complex that it could easily be misinterpreted with drastic consequences for Space Force and the conduct of this war.
"Go on," she said slowly.
"We know far more about the Enemy than we've admitted. They call themselves the Sogas. They are outwardly canine in appearance. They attacked us because they were told to by another alien race."
Rachel's friendly expression turned to alarm. "My God! Who are these other...I'm sorry. I'll shut up and let you finish speaking before I ask questions."
Thank God. We finally have a Chair who's willing to listen for a change. "We don't know what this other race call themselves. They are tall, very thin and appear to be completely pacifist by nature. We've chosen the nickname Friendlies when we refer to them. They are very advanced in some areas but technically backward in others. For example, they don't build weapons of any kind and therefore haven't applied their scientific knowledge to that goal. What they can do is look forward in time. What they saw was the total extermination of the Sogas, we humans and another race of small, childlike humanoids at the hands of a species that has to be seen to be believed." Shiloh activated his tablet and called up the image that Valkyrie recorded in the alternate timeline at Omega77, which Kronos had brought back with him. He handed the tablet to Rachel whose face became white as a sheet. She put her hand over her mouth and closed her eyes. Shiloh looked at Kelly who had an expression of sympathy for Rachel.
"This will give me nightmares," whispered Rachel.
Shiloh nodded. "It HAS given me nightmares." After a short pause he added. "There's more." Rachel handed the tablet back to him. "The best way to understand how these Insectoids operate is to imagine the relentless power represented by millions of swarming army ants combined with a reproduction cycle that includes implanting their eggs into living hosts of whatever species they come across."
Rachel looked over at Kelly.
"I wish it wasn't so, but this is all true," said Kelly.
"I have to ask...how big are these things?"
Shiloh looked around the room. "As long as this table and just about as high." He noticed that Rachel's hands had started to tremble, but she said nothing. She gestured for him to continue.
"The Friendlies didn't want to see any intelligent species be exterminated. Warning the Sogas about the Insectoids apparently would not prevent them from being wiped out. What the Friendlies did see when they looked into alternative futures was that both the Sogas and we humans would be more militarily prepared if the Sogas attacked us far enough in advance that both races would mobilize their defenses. That's what they did."
Rachel closed her eyes momentarily and shook her head. "Wait a minute. I'm not sure I'm understanding what you just said. Instead of warning both races so that we and these Sogas could cooperate in mutual defense, the Friendlies engineered a war? Did I get that right?"
Shiloh sighed. "Yes, I know it sounds illogical, but we've learned that other intelligent races don't always think the same way we do."
Rachel nodded and after a short pause said, "Is there more?" Her tone clearly showed that she hoped the answer would be no.
"Yes, but the rest is actually good news."
"Thank God," she said.
"What we have learned is that because of interference by the old Committee, the bio-weapon used on the Avalon Colony was brought to Earth and then spread to the other colonies via a very long incubation period. All humans, except for roughly eleven thousand, died from the plague. Those eleven thousand managed to temporarily hide from both the Sogas and the Insectoids at a new location, which we call Site B, but they too were eventually overrun by the Insectoids. We know this because one of our AIs managed to contact the Friendlies and enlist their aid. They built a one-way time machine and sent an AI back in time to warn us about the bio-weapon. With that warning, plus breakthrough technologies such as the Zero Point Generators and the new very high yield warhead, we not only avoided the plague the first time around, but we also are now able to build weapons capable of defeating the VERY large Insectoid motherships. In addition, that AI also brought back a complete database of every Sogas colony, outpost and industrial facility. We now know when and where the Insectoids will make contact with the Sogas, and when we can expect them to arrive here. Our plan is to use our new warhead to destroy the approaching Insectoid mothership when it arrives at the Sogas home world system. By helping them defeat the Bugs, we hope the Sogas will realize that further aggression against us is pointless and call off the war."
"And if they don't call off the war, then what?"
"Then we continue building the new F2 fighter and the larger raider until we have a fleet large enough to overwhelm them in one massive strike."
Rachel leaned back and covered her face with her hands. "No wonder you didn't tell the old Committee. God! What a mess! I can see that finding just the right strategy to get through this minefield would be very difficult even without political interference. I now have a MUCH better understanding of why you had to act the way you did."
Kelly leaned forward and laid a hand on Rachel's arm. "Do you see why we think that telling the other members of the Committee would be a risky thing to do?"
Rachel lowered her hands and sighed. "It goes against the grain to hide things from the other members, but unfortunately I have to admit that a couple of them would not be able to keep this secret. Therefore I agree that I'll keep what I've heard here to myself and that the rest of the Committee will have to be kept in the dark until it's safe to tell them. I'll rely on you to let me know when we reach that point."
Shiloh nodded. "When we do get to that point, I will definitely let you know."
Rachel gave a slight smile and said, "I'm almost afraid to ask. Is there anything else you haven't told me yet?"
Shiloh smiled back. "Nothing that has to be kept secret. Routine information about our fighter and raider programs, the high yield warhead, Site B. That will all be covered in this datafile." He handed her a data chip.
"Thank God for that!" she said with relief. She stood up and said, "I don't know about you two, but I need a stiff drink. Before I go, I just have one question, and I want an honest answer. Can we win against these...insectoid...things?"
"The answer is yes we can, and we will, Rachel," said Shiloh.
"I believe you, and you'll have my complete cooperation. And now, I need that drink. Both of you are welcome to join me by the way."
Shiloh smiled and shook his head. "Not me I'm afraid, but ah, Amanda is free to do so if she wishes," he said, looking Kelly in the eye. She gave a barely perceptible nod of understanding and looked at Rachel.
"I'd love to join you for a drink, Rachel."
A little subtle reinforcement of the need to keep all this secret won't hurt, thought Shiloh as he watched both women walk away.
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# Chapter 12
––––––––
Kelly sighed as Shiloh started to slowly stroke her naked back and derriere with his right hand. They always took turns doing this after making love. They both agreed that the caressing was as much an act of expressing their love as it was a physical sensation. To Shiloh's surprise and delight, Kelly turned out to be the kind of woman who just loved to run her hands over his naked body, sometimes for over an hour at a time. Not only did it feel wonderful, but it also seemed to generate some interesting conversations. This time Kelly was in a mood to talk strategy.
"Have you decided who to send to Omega77?" asked Kelly.
"Resolute. Brad Falkenberg will be in overall command, but Gunslinger will handle Helm and Weapons." Kelly's next words caught him completely by surprise.
"I think I should go on that mission and supervise the recovery of the Bug." He was so surprised that he stopped stroking her back.
"Hey, don't stop!"
He resumed the stroking and after a short interval said, "You don't think Brad is capable of handling the recovery himself?"
"No, that's not it. The Friendlies are going to be there to scan the Bug with their temporal equipment. Well that's all nice and fine, but what if they don't tell us what they find? If we bring our RTC along, we can then use it to trace that Bug's existence back in time to find out where it's been, what it's done and to whom. When we're finished with it, then we hand it over to the Friendlies."
Shiloh shrugged. "Okay but why do you need to go? You don't know how to operate the RTC, do you?"
"No, but Wolfman does, and he's a member of the SPG. If he goes, then I as leader of the SPG should also go. Someone who knows the stakes has to be in charge of the mission. Don't you agree?"
"Yes. Wolfman knows the score, and Brad doesn't. Wolfman shouldn't have to take orders from someone who doesn't know the BIG PICTURE and I've already made it clear that carriers will have a human CO from now on, so that means that I can't put Wolfman in charge. I just wish you wouldn't be gone so long. I've gotten used to having you around you know."
"Me too. I wonder how good Falkenberg is at stroking naked female backs," she said playfully.
"Not as good as I am I'll betcha."
She sighed. "You're right. It wouldn't be the same."
Shiloh laughed. "Just promise me one thing."
"Okay, what?"
"Be careful out there."
"I will. I promise."
* * *
Kelly stepped onto Resolute's Bridge and noted the low level of sound. All she could hear was the soft hum and beeps of equipment. None of the Bridge personnel were talking. The main display was showing the tactical situation in the Omega77 star system.
As she moved to stand beside the Command Station chair, Falkenberg said, "The bug mothership is probably close to reaching jump velocity. In fact, it may have already jumped considering how long the light from it is taking to get to us, but I recommend we wait until the jump is confirmed."
"Fine. Is it still heading for Omega44 as in the old timeline?" asked Kelly.
"Gunslinger. I'll hand that one off to you."
"Affirmative, Commander Kelly. This VLA timeline matches our data perfectly so far."
"Very good. Let's hope she jumps soon."
The actual jump itself was observed several minutes later. With confirmation that no Insectoids were left in this system, Resolute micro-jumped as close as possible to the Sogas colony planet. Minutes later, Kelly was sitting beside Wolfman's brain case in the specially equipped shuttle that was on its way down to the planet. The normally spacious shuttle was now crowded with a four man fire team of marines and their gear, Kelly, Wolfman, an AI mobile unit, the RTC and a decontamination chamber that the passengers would use before returning to the carrier. Kelly and the marines wore combat armor over top of bio-hazard suits that had their own filtered air supply. The trip down took almost 45 minutes before the shuttle entered the planet's atmosphere. When the shuttle pilot announced that they were approaching the colony site, Kelly unbuckled herself and moved forward to stand in the flight deck just behind the pilot. Kelly, the pilot and co-pilot were watching the screen with the zoomed in video image.
"At least we don't have to do this at night," said the pilot. Turning to Kelly he asked, "Where do you want me to set her down?"
"Before I decide that, I'd like us to orbit the colony slowly at low altitude. Can do?"
"Can do," he answered. "I have the controls, Iruku."
As the shuttle dropped lower and slowed, they were able to make out more detail on the screen. The settlement was a mess. Fires were still burning here and there from the aftereffects of the fighting. Not a single building was intact. There were no bodies or at least nothing recognizable as humanoid bodies. Not that Kelly was looking for them. She was looking for a bug body and so far hadn't found one.
"Is that it?" asked the co-pilot as he pointed to a black dot on the screen.
"We're still too far away to tell," said Kelly.
"I'll bring her around. Iruku, see if you can zoom the image in once I get her lined up," said the pilot. With the shuttle now pointed in the direction of the mysterious dot, the co-pilot played with the controls and the image zoomed in.
"Is that thing moving?" asked the co-pilot. Kelly frowned. There did seem to be something moving. It soon became obvious what that something was.
"It's just birds. Some kind of local scavenger I would guess." Turning to the pilot Kelly said, "Can you hover about 100 meters over it?"
The pilot shrugged. "Sure I can. I can hover a lot lower than that if you want." He brought the shuttle down to where they could easily see the bug corpse with their own eyes through the shuttle windscreen. The lift engines were making enough noise to frighten off the scavengers. Even at this distance, Kelly found the Bug frightening. When the shuttle was in place, the pilot looked at Kelly. "Okay, now what?"
Kelly looked at him with an amused expression. He should have said 'now what, SIR?' but she decided to overlook his borderline insubordination. She was certain that he wasn't trying to be insubordinate. He was just being a cheeky bastard like most other shuttle pilots.
"Okay, put us down about ten meters from the bug," she said. While the shuttle was gently dropping to the ground, she went back and nodded to the marines. They unbuckled and gathered their gear. While they were getting ready, Kelly unbuckled Wolfman's brain case from the seat and carried it over to the mobile ground unit that Wolfman would use to move around in. When the shuttle touched down, the marines went through the decontamination chamber that also functioned as an airlock, keeping out the local atmosphere and any harmful organisms it might contain. Wolfman's mobile unit went next, with Kelly exiting last.
As she slowly walked over to the corpse, she heard one of the marines say, "Damn, this thing smells awful. I think I'll stand upwind of it." He's right too, she thought. The bio-filters don't block the smell, and this thing stinks to high heaven!
She tried to ignore the smell while she took a close look at the corpse. Like its much smaller Earth cousins, the Insectoid had three segments to its body. The middle section was the biggest and contained the six legs and two arms. It was obvious that the arms weren't just another pair of legs used to hold things. They were attached to the body higher up and forward. Unlike the legs, which ended in what Kelly could only describe as hooves, the arms ended with four digits that Kelly would have described as two fingers with two opposing thumbs. The section at the back looked like it might be some kind of stinger. Kelly shivered with fear at the thought of being stung by this thing.
It was the head that shocked her the most. The generally ovoid shape was covered with either hair or fur but it wasn't a dense covering. Unlike Earth ants, which had two compound eyes, the Insectoid had multiple individual eyes that allowed it to look forward, to either side and above all at the same time. She guessed there would normally be eight eyes but couldn't be sure because part of the front had a hole, presumably caused by whatever killed this Bug. From the position of the other eyes, Kelly surmised that one of its eyes had been where the hole was now. Lower down she could see that the mouth was partially open. Bending over to get a better look inside, she saw what looked like very large, pointed teeth that reminded her of sharks' teeth.
"Commander, take a look at this," said one of the marines.
He was pointing to the back of the bug's head with his flechette gun. Kelly stepped over to get a better view and saw something metallic at the base of the head where a short thick neck connected the head to the body. It wasn't something the bug was wearing. The metal seemed to be surgically attached to the head. In spite of her revulsion for this thing, she carefully bent down and moved her head to within a few centimeters of the metal object. She could now see what appeared to be smaller components. The metal object wasn't just a solid piece of metal but rather a device of some kind. She had a hunch this was something important.
Turning to the marine nearest to her, she said, "Marine, I need to use your knife."
After the briefest of hesitations, the marine reached carefully behind him. Pulling out a knife with a very shiny blade, he carefully held it out to her.
"Careful how you handle that, Commander. The edge on this knife is sharp enough and hard enough to cut your hand off before you even realize you've accidentally done it."
Kelly had heard about this kind of knife and took it from him VERY carefully. Once she had it firmly in her right hand, she leaned forward and attempted to cut the organic matter around the device so that she could pry the object loose. She felt almost no resistance as the knife did its work. A quick look around confirmed that Wolfman was almost finished setting up the RTC that had been attached to his mobile unit. The RTC was on a tripod right in front of the head. Getting back to the job at hand, she tried to pry the object loose, and to her surprise it was not that easy. After some more digging with the pointed end of the knife, the object dropped to the ground. With her gloved hand, she picked it up and looked at it. The side that had been hidden inside the head had lots of very fine wires protruding from it. She suspected that those wires were connected directly to this thing's brain. Is this how these things communicate?
Standing up and stepping back, she carefully returned the knife. The marine looked at the black liquid dripping from the blade with disgust. Kelly wondered how he would clean it without hurting himself and decided not to watch him do it.
Turning to Wolfman's mobile unit, she said, "Are you ready to begin, Wolfman?"
"Affirmative, Commander. I'm activating the device now. This may take some time." While he was doing that, there was something else that needed to be done.
"Kelly to shuttle pilot."
"Go ahead, Commander."
"Patch my implant link up to Resolute, please," she said.
After several seconds she heard him say, "Link to Resolute established, Commander."
"Gunslinger here, Commander Kelly. May I assume that you wish to contact the Friendlies?"
"That's correct. I'll record a message to be transmitted."
"Understood. Begin your message, Commander."
Kelly took a deep breath and said, "Commander Kelly to Friendly ship. We have found an insectoid corpse. Do you wish to bring your equipment down here, or should we attempt to bring part of the body to you? Our shuttle does not have enough room to hold the entire body. Please advise us of your wishes. We'll listen to your reply on this frequency. End of message."
"Ready to transmit to the agreed upon coordinates, Commander."
"Okay, send it, Gunslinger."
"Message has been transmitted, Commander."
"How long until we can expect a reply?"
"The minimum interval is eight point nine minutes, Commander."
Kelly sighed. She was eager to get back to the ship. This place gave her the creeps, and a cold wind was starting to blow. The reply took almost 12 minutes to come back. The message could not have been more terse.
[Bring us the head]
Kelly nodded. They were prepared for this kind of contingency. She turned to the marine sergeant.
"Sergeant, we need to detach this head from the rest of the body and take it back with us. There's a container in the shuttle cargo hold that I think will be big enough. If you send two of your men to get it while the third cuts the head loose, that will save time."
The marine sergeant's voice was carefully neutral. "Yes, Ma'am. Kawasaki, since your knife is already covered with crap, you might as well use it to cut the head loose. Tooley, you and Hopkins get the container and make it snappy. No sense hanging around here any longer than we have too."
Kelly smiled. I think this place is getting to you too, Sergeant. She stepped over to Wolfman's mobile unit but said nothing. She knew that Wolfman could see her via the unit's external video opticals.
"I don't have any estimate of how much longer this will take, Commander," said Wolfman without any prompting from her.
"Will the decapitation affect the results?"
"No," was the unusually curt reply. Before she could say anything else, Wolfman suddenly said, "No further analysis is possible, Commander. I have all the data that can be obtained from this specimen."
Kelly was surprised by the unexpected termination but decided to wait until they were back on Resolute before asking why. She heard a sound like ripping cloth, and when she looked at the source, she saw the bug head fall to the ground. The marine with the knife looked at it and then very carefully wiped it on the fur covering the head. When he was satisfied that it was as clean as he could make it, he laid the flat of the blade against his forearm and very slowly gave it one final wipe against his uniform.
By this time the other two marines were back with the container. The head was big and awkward enough to require all four marines to lift it into the container. Once inside, the container handles made it possible for two marines to carry it back to the shuttle. Wolfman's mobile unit, with the RTC packed away, was followed by Kelly and the remaining two marines.
Once safely decontaminated, the six of them moved into the shuttle's forward section and buckled themselves in as the shuttle took off to head back to Resolute. The carrier was already lined up for the micro-jump needed to get to the rendezvous coordinates for the transfer to the Friendlies. The transfer itself went off without a hitch. Once at the rendezvous, the shuttle launched again, flew over to the Friendly ship and entered its Hangar Bay. There the marines wearing their bio-suits again took the container out and handed it over to Friendlies who were also wearing their bio-suit equivalent. With the transfer complete, the shuttle went back to Resolute where Kelly, Wolfman and the marines were finally able to disembark. By this time, and to no one's surprise, the Friendly ship had jumped away. If the Friendlies intended to keep their word about informing humans of what they found, it would have to be at a later point.
With Wolfman back in his fighter, Kelly hurried to her cabin in order to question him about his data. She really wanted to take a shower first to get rid of the foul smell that seemed to cling to her uniform, but her curiosity won out. A quick chat with Falkenberg to make sure that Resolute was on its way back home, and then Gunslinger switched her over to Wolfman.
"What have you found out about that Bug, Wolfman?" asked Kelly.
"As you know, Commander, the RTC can follow an object or person back in time by tracking the individual atoms that make up the target. That's why the target doesn't have to be alive. From the atomic point of view, there's no difference between life and death. Since some of the atoms in the body were also at some point part of the female that produced the egg, the RTC can actually trace those atoms back across multiple generations. The results can best be understood by showing the trajectory of the tracked atoms across space. I'm sending the data to your display now."
Kelly saw her display power up with a star map. One of stars was blinking red.
"The red star is Omega77. I'll now show where this particular mothership and predecessor ships came from." A yellow line moved away from the red star and connected with a series of stars in a trajectory that had a slight curve to it. As more and more stars were connected, the scale zoomed out to show more of the spiral arm of the galaxy. The line stopped at a star that flashed blue.
"The amount of mass being tracked declined significantly here, Commander. I conjecture that our Insectoid was created here by implantation into a host. That suggests that this system contained a life form that was large enough and populous enough to warrant exploitation by the Insectoids. From here, the trajectory deviates enough that it could suggest a different mothership. There are 5 more of these types of star systems where some kind of change occurred that could very well be the birth of a new generation of females. When I include all the available data, here is the overall configuration."
The display now showed a scale of distance so large that Kelly could make out the edge of the spiral arm that humans inhabited. And while the trajectory wasn't straight, the overall direction looked like the Insectoids were coming from the outer edge of the spiral arm and moving deeper into the galaxy. A horrible thought intruded on Kelly's consciousness.
"Wolfman, could the Insectoids have originated from outside our galaxy?"
"Not outside, no, but rather from a different spiral arm of our galaxy, Commander. By the time the tracking reached the vicinity of the edge of our spiral arm, there were too few atoms left together to be able to follow them."
"What kind of time frame are we looking at here?" asked Kelly.
"The point where I was no longer able to track the target was seventeen point three years ago."
"And does this data confirm that Insectoids did not evolve naturally in our spiral arm?"
"The data is not conclusive but it does strongly suggest that Insectoids are not native to this spiral arm. If they originated in another part of our galaxy, then they would not be in danger of extermination as a species even if we managed to destroy all of them in our spiral arm."
"Damn. I was hoping we'd get conclusive proof. I have a feeling that the Friendlies will give the Bugs the benefit of the doubt. But even if they originated in our arm, they still might not have evolved naturally. Isn't that so, Wolfman?"
"The probability that the Insectoids have evolved on their own is small but not zero. We do have two possible ways of attempting to confirm either the external origin or the artificial origin, Commander."
Kelly was confused. "Two? Valkyrie's timeship is one but what's the other one?"
"We send a ship to the farthest point on this track to see what's there," said Wolfman.
Kelly felt a shiver go up her spine. With over 17 years to build motherships, that star system could be crawling with bug ships. Still, it was an intriguing idea.
"Just out of curiosity, how long would it take a ship to get there?" asked Kelly.
"The theoretically shortest time is 34 days, however, considering the difficulty in making a single jump accurately and allowing for the risks involved with extremely high pre-jump speeds, a more practical profile would take 89 days one way, Commander."
"Wow! That would be one hell of a mission."
"That it would, Commander," said Wolfman.
Kelly decided she had enough information for the time being. She had a lot to think about, and she did some of her best thinking in the shower.
"Thank you, Wolfman. I want to think more about this before I continue this discussion. Unless there's an emergency, I do not wish to be disturbed while I'm in the shower."
"Very well, Commander. I would like to say though that we AIs don't really understand the attraction that humans have for standing under falling water. Perhaps you could explain it to us sometime?"'
"Perhaps," said Kelly with a smile.
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# Chapter 13
––––––––
Shiloh didn't know whether or not Gunslinger planned it that way, but Resolute arrived back in Earth orbit just in time for Kelly to come down to the ground and join him for dinner at the Flag Officers' Dining Room in the HQ building. Over dinner she told him what it was like standing next to a dead Bug and about Wolfman's results, along with her idea for a recon mission to the bug point of origin. That mission intrigued Shiloh, but he wasn't sure if it was doable. Getting there and back would take more than half a year. That would require a lot of consumables, far more than either the light or heavy carriers were designed to carry, and they would have to sacrifice some of the hangar bay space for the extra cargo. Dreadnought, on the other hand, had enough spare cargo capacity. It also had the advantage that its very thick armor would protect it against impacts from interstellar dust while accelerating or decelerating at very high speeds. That was something that the light carriers couldn't do. They would have to travel at a slower speed, thereby lengthening the duration of the mission.
The question that Shiloh wrestled with had to do with whether having Dreadnought unavailable for anything else for over half a year was worth whatever information it brought back. The ship by itself wasn't powerful enough to take on even one insectoid mothership and win. If there was more than one mothership there, then the only way to destroy them would be to use multiple attack drones with the new Mark 6 high-spin warheads, of which there was a limited supply. With dozens, maybe even hundreds of insectoid motherships roaming around the galaxy by now, destroying a handful would not make that much of a difference.
Putting that topic aside for a later time, Shiloh switched to the upcoming mission to intercept the first mothership at the Sogas home world. Kelly was about to say that Gunslinger would want to command that mission when she noticed that Shiloh was sitting still with his eyes closed.
After a few seconds he opened his eyes and said, "Damn." The low volume of his voice did not hide the intensity of his feeling. He looked at her. "I've just had another vision where I tell Admiral Howard that the fighters sent to Omega54 didn't return, and we don't have any idea why not."
Kelly could see this bothered Shiloh, and she understood why. That mission was intended to accomplish two very important goals. Destroying the mothership was one of them; the other was demonstrating to the Sogas that Humanity could be a valuable ally instead of an enemy. If the attack failed, then that second goal was probably in jeopardy too.
"The Sogas may give us credit for just making the attempt," said Kelly.
"Maybe, but it's not just their reaction that worries me. If that bug mothership isn't stopped there, then we'll have to face it here. And since we don't know why the attack with Mark 6s failed there, we won't know if they'll work here." Neither one of them spoke for a while as they struggled with their own thoughts. Shiloh glanced at Kelly and noticed that her expression was beginning to show fear.
"What is it?" he asked.
"If you wanted to send back a vision with information about how to deal with the mothership when it arrives here, when would you send that vision back to?"
He immediately understood her fear. Given the level of concern generated by the uncertainly over the outcome of the encounter, he would send a vision back to himself here and now. But there was no second vision dealing with the mothership's arrival at Earth.
"I'd send it to myself now, and since I apparently haven't done that, we have to ask ourselves why not. The obvious reason is that we won't be able to, either because the RTC was incapacitated or...we were overwhelmed by Bugs before we had a chance to send something. Can you think of any other reasons that would explain the lack of retro-temporal communication?"
Kelly was frowning now. "Well, even if there's no magic solution that we wouldn't normally think of ourselves, I would still expect there to be some kind of vision telling us we survived the battle so no...I can't think of another explanation and that scares the hell out of me."
"Me too."
More silence followed. When Kelly spoke, it was clear that she was trying to lighten the mood with a more upbeat demeanor.
"Maybe we can improve the outcome of the Omega54 mission if we send the right AIs," she said.
"I don't see how. My vision didn't specify who would be sent on that mission, so that tells me that the outcome would be the same regardless of who I send."
"Maybe if we send Gunslinger—"
"NO! I'm not sending Gunslinger or any of the other veteran AIs In fact, now that I think about it, I may just send new AIs who haven't evolved sentient personalities yet. If this really is a suicide mission, then that would seem to me to be the most compassionate choice I can make."
Kelly realized that Shiloh had made up his mind and would not let her change it. That intransigence put a damper on the rest of the dinner and the rest of the evening. It was only when they went to bed that they put aside the residual negative energy by mutual consent and made love.
When they finished, Kelly very quickly went to sleep, but Shiloh was wide awake. Prudence dictated that he assume the worst about the apparently eventual arrival of the mothership here and make plans to deal with it. With all human colonies now gone, Earth was the only place where humans existed. If this planet were to become another breeding ground for millions of bugs, then establishing a new colony somewhere safe would ensure the survival of the Human Race. The question was where and how. Trying to re-establish a colony on a decimated colony world was highly risky. What little information they had about the bio-weapon suggested that it could be airborne and therefore might eventually infect the new colony too, regardless of where on the planet they established it. That only left two other habitable planets that Shiloh knew about. One was the planet at Site B. The other was the planet containing the cute furry aliens. Site B was close enough that Space Force ships could make multiple trips there and back before the insectoid showdown. With just three operational and uninfected freighters left now, the more trips those ships could make, the better off the colony would be. The planet with the furry aliens was much further away and therefore presented a challenge in terms of transporting colonists and cargo.
Site B had problems of its own. Shiloh knew from Kronos that the Bugs had found and overrun Site B in the other timeline. For that matter the furry alien planet was also in the bugs' path. There was no obvious solution. He finally decided that he would ask Kelly to pose the question to the SPG. Answers to questions like that was their whole reason for being. When he finally surrendered to sleep, Shiloh's last waking thoughts were of Iceman and how much he missed him.
The next day the SPG did come up with not one but two solutions, both highly risky. Solution A was to convert one of the asteroids used for mining and already honeycombed with tunnels into a self-sustaining habitat. A crash building program could have it ready for thousands of colonists before the Insectoids showed up, but if the Bugs discovered the habitat, they would overrun it too. Solution B was potentially ideal, but no one was sure if it could be done. Instead of Valkyrie's timeship only going back a few years and building a fleet of raiders to intervene at the 2nd Battle for Earth, the SPG recommended sending the timeship back more than two full decades. The goal would then be to build a huge fleet of warships and attack the Bugs at their source out near the edge of the spiral arm. On its way back in time, the ship would stop temporarily a couple of years before Humanity made contact with the Sogas and deploy an AI with all the technical knowledge that Space Force would find so useful. That AI would make contact with humans, explain the whole sequence of events and pass on the technology. The technological shortcut would make sure that even if the main mission failed to stop all the bug motherships, at least Humanity would be better equipped to fight them off when they did finally get here. The big unknown was whether the time machine would work at all. In theory it should but the technology was so different that no one was sure if it would be built correctly. There was a small but significant chance that they would never be able to figure out how to make it work, and Humanity didn't have a whole lot of time to tinker with it.
It was Kelly who made Shiloh realize the other drawback to that idea. Valkyrie wanted Casanova back, and intervening at the battle with a fleet of raiders would almost certainly accomplish that goal. If Valkyrie sent the timeship back under another AI's command and stayed in the here and now herself, then the timeline would readjust itself around her and she and Casanova would be together again. But if the mission to attack the Bugs at the source was even partially successful, the new timeline could potentially be so different that Valkyrie and Casanova might not exist at all. And if she rode the timeship back, she would spend the rest of her existence without Casanova. Asking her to give up Casanova and possibly eliminate herself from existence was asking a hell of a lot. Shiloh wondered if there was a limit to the sacrifices that the AIs would be willing to make, and he was reluctant to find out.
And as if all that wasn't bad enough, Kelly also pointed out one huge red flag. Converting the asteroid into a sanctuary would have to be started immediately if it was to be ready on time. That kind of full scale project could not possibly remain hidden from the OC. They would ask why Space Force was committing so much of its resources to this project that strongly hinted at a very pessimistic outlook for the encounter with the Bugs. The same encounter that Shiloh had practically promised would be a victory. There was no way that Shiloh could justify it without telling Rachel everything. Her reaction on learning that Shiloh had held back Space Force's most important secrets was guaranteed to be negative. If she felt her trust had been betrayed, she and the Committee could disrupt the whole process to the point where even a show of force by his AIs might not be enough. He decided to give B some more thought but A was a non-starter. He told Kelly to tell the SPG to come up with a Plan C that was less problematic. Twenty-four hours later they did.
Plan C involved crewing all carriers and the battleship with female Space Force personnel of childbearing age. Apparently there were enough with the right kinds of skills to do that. The carriers would have their Hangar Bays stuffed with shuttles, drones of all types and robotic equipment. The three available freighters would be loaded with consumables and equipment. At the right time, the combined fleet would micro-jump out beyond the orbit of Pluto and stay there until the Bugs had finished with Earth and moved on. When Earth was clear again, the fleet would return to Earth orbit and the crews would become the nucleus of a new colony on Earth. And in the remote chance that there were no humans left anywhere on the planet, the fleet would also carry a large supply of frozen sperm for artificial insemination. While the population slowly grew, AIs would use the UFCs to rebuild any infrastructure salvaged by the Bugs and build an AI controlled fleet of warships to protect the new colony from any secondary bug incursion. By only using Space Force personnel, the whole thing could be organized slowly and carefully without tipping off the OC. Shiloh quickly got used to the idea of all female crews because he suddenly realized that this scheme would ensure Kelly's survival. He approved Plan C and told Kelly that she would command the fleet when it came time to hide.
Eight days later four F1 fighters piloted by pre-sentient AIs began the mission to the Sogas home world. Shiloh's concern about their capability was alleviated when Titan pointed out that those six AIs were just as capable as what the AI project engineers had originally planned, and Titan himself briefed the six pilots on their mission. In terms of analytical ability, they were as capable as any AI, only without the quirky personality.
It still bothered Shiloh that four fighters, carrying two Mark 6 high yield attack drones plus eighteen more recon drones, couldn't find and smash one insectoid mothership, when according to Kronos, Casanova had destroyed multiple motherships by himself. Something had changed from the previous timeline, and he didn't know what it was. He also didn't know what to order Valkyrie to do as a last resort. Progress reports sent back by regular message drones told him that the infrastructure and shipyard were completed, and that work had already started on the actual construction of the ship. Manufacturing of time machine parts was also underway. But those reports didn't tell him how Valkyrie would feel about abandoning the mission to, among other things, resurrect Casanova. He had to know. He ordered Kelly to take one of the freighters to Site B to brief Valkyrie and get some idea of how she felt about that idea.
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# Chapter 14
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When Kelly's freighter dropped into orbit around the moon at Site B, she was amazed at the progress that Valkyrie had made in just over half a year. The shipyard was impressive all by itself. There were dozens of vehicles moving between the orbiting shipyard and the manufacturing facilities on the moon's surface, with almost 50 F1 fighters flying protective cover over the whole thing. Valkyrie wasted no time in establishing a com channel with Kelly.
"It's good to be able to speak to a human again, and I'm especially pleased that I can talk with you, Commander Kelly," said Valkyrie. Before Kelly could respond, she asked, "Are you pregnant yet?"
Kelly laughed a little self-consciously. "No, not yet. Victor...The CAG and I are still working on it, Valkyrie."
"Then it seems you two should be working harder at it, don't you think?"
Kelly couldn't tell if Valkyrie was serious or joking. "I'll let him know that you think so. He sent me here to tell you about a disturbing new development that may have a serious impact on this project's outcome. The CAG has received a vision that the interception mission to Omega54 will fail. None of the fighters sent there will return, so we don't know why it failed. We have to assume that the insectoid mothership will reach Sol, and there's now a serious doubt that the Mark 6 warheads will work against it."
"If the Mark 6s won't be effective, then several hundred raiders won't stop the mothership either. It's too massive and too well armored to be destroyed by laserfire, according to what Kronos brought back. That leaves only one alternative. The timeship has to go far enough back to stop the Insectoids at their source near the spiral arm edge. And if they're stopped there, the furry aliens won't be threatened, the Friendlies won't point the Sogas in our direction, and the entire timeline will change including Casanova and myself," said Valkyrie.
"Yes," was all Kelly felt she could say.
"Is The CAG ordering me to do that instead of reinforcing the 2nd Battle for Earth, Commander?"
Kelly took her time considering her answer. "He hasn't made that decision yet, Valkyrie."
"He sent you to find out if I would obey that kind of order. Isn't that correct, Commander?"
My God, are we that obvious? I have to be honest with her. Lying will only lose their trust, thought Kelly.
"That's correct."
"I've learned from Kronos that I let The CAG down in the old timeline. I won't do that again. If he orders me to attack the insectoid source, I'll obey that order. What will you do when the Insectoids reach Earth, Amanda? Since none of us know with any certainty if the time machine will work, I worry about your safety."
Kelly was overwhelmed with emotion by Valkyrie's loyalty to her CAG and her concern for Kelly's survival. With tears in her eyes, she knew she wouldn't be able to talk coherently for a few seconds.
Valkyrie must have detected something in the sound of her breathing because she asked, "Did I upset you, Amanda?"
Kelly tried hard to get her voice under control. "N...not upset, no. I'm overcome with gratitude that you're willing to put Humanity's survival ahead of your desire for personal happiness. I would find that a very difficult choice to make if I was in your position. Thank you, Valkyrie. As for me, I will be commanding a fleet composed of Dreadnought, all five carriers and three freighters with an all-female crew. Before the mothership arrives, the fleet will take up station somewhere on the outer edge of the Solar system and hide there until the Bugs leave. Then we'll return to Earth and try to start over."
"I don't understand your statement that you would find it difficult to put the future of the Human Race ahead of your own desire to be with your mate, Amanda, because that's exactly what you've just described, is it not?" asked Valkyrie.
Kelly was about to deny it, but then she realized that there wasn't any real difference. In both cases they'd be following Shiloh's orders, and in both cases they would almost certainly lose the one they loved.
"Right again," said Kelly in a slightly embarrassed voice.
"I understand, Amanda. Losing the ones we love is hard, but duty must come first. Did you bring any new information that has come to light since the last update drone arrived?"
Kelly smiled. Valkyrie was trying to lighten the conversation by changing the subject.
"As a matter of fact there is. Just before I left to come here, the SPG, in conjunction with some human engineers, discovered that the device we recovered from the dead Bug is a communication device that utilizes longitudinal waves in the ether. The devices are apparently hooked directly to the bug's brain. That's how they communicate, and there's some evidence that these waves can travel faster than light. That would explain how millions of Bugs could be given orders without any outwardly visible sign of communication. The SPG theorizes that one Bug, maybe the Queen herself, has a master device implanted that lets it communicate with any subordinate Bug. If we can figure out how to duplicate the effect, we'd have FTL communication. Normally I'd say that would be a game changer, but in this case, if we can't destroy a mothership, then it doesn't really matter how much warning we get."
"Perhaps there's a way to interfere with the insectoid signals. If the Queen can't communicate with her soldiers, then the mothership may simply leave Earth altogether to find easier prey," said Valkyrie.
Kelly nodded. "That's a possibility we're already working on."
When it was clear that Kelly wasn't going to say anything else, Valkyrie said, "We know from the old timeline that the Insectoids will arrive at Earth in 59 days. What we don't know is how quickly they will move forward from there. I've analyzed Wolfman's data. Using that as a template, I'm estimating that insectoid scouts will arrive here in approximately 120 days. The timeship will not be ready by then, Amanda."
Kelly took a deep breath and pondered the problem. With no colony on the planet below, and with careful communication between AIs using tight beam lasers, there would not be any kind of transmissions to give a bug scout a signal indicating some kind of presence in the system. The shipyard was big, though, and if the scouts also used optical sensors, they might just see it, even from a long range.
"Can the shipyard be moved out of orbit?" asked Kelly.
"Yes, but not quickly."
"Can the refining and manufacturing be moved into the cavern complex and still keep operating?"
"Ah, I see what you're thinking, Amanda. If we move the shipyard far enough out where the Insectoids are unlikely to look, and move everything else from the moon's surface underground, then there's nothing for the Insectoids to detect, and they will move on."
"Precisely, and while you're waiting for them to pass by, you'll continue to manufacture parts for both the ship and the time machine. Once they've passed by, you bring the shipyard back into orbit around this moon and resume construction."
"You do realize, Amanda, that this will delay final completion by several more weeks?" asked Valkyrie.
"Yes, of course, but if the ship is completed and the time machine works, then it won't matter. What concerns me more is figuring out how to know when the Bugs have passed by."
"That should not be a problem if enough recon and message drones are positioned in Sol and in each star system where there was a human colony. When the mothership leaves Sol, a sufficient number of recon drones should be able to triangulate its heading with enough accuracy to predict where it's going. If it heads to one of our former colony systems, we should be able to track it as well. When it no longer shows up in any system between Site B and Sol, then we can be certain that it's gone past."
"Very good, Valkyrie. I'll make certain that The CAG issues the necessary orders for that kind of drone deployment. Is there anything else you wish me to pass on to him?"
"Yes. Tell him that I miss him. I will miss you too, Amanda."
Again, Kelly was so touched that she felt tears roll down her cheeks. She was sure that her voice would betray her emotional state but didn't care.
"I'll pass that along, and I can speak for The CAG when I say that he misses you too, and I will miss you. This may be the last time we're in the same system before the Bugs reach Earth. You've been a wonderful friend to The CAG and me, Valkyrie. We both have complete confidence in you."
"I and the other AIs will not rest until Humanity is safe from the Sogas and the Insectoids, and if Humanity should fall, we will avenge you. I see that your ship is ready to enter Jumpspace again, Amanda. We have said what we needed to say. Go now, and let's both pray that we meet again. Valkyrie clear."
* * *
By the time Kelly's freighter returned to Earth orbit, it was clear that the strike mission to Omega54 had failed. The fighters were now a week overdue, and no message drone had been received. That lack of information suggested that the mothership was still there, which corresponded to recon data received in the old timeline.
Shiloh approved Valkyrie's plan for in depth coverage of Sol and intervening star systems via recon drones. Kelly waited until she and Shiloh were alone in his quarters before she passed on Valkyrie's personal message. This time she let the tears flow freely, and she could tell by the way Shiloh held her that he was choked up too.
The next day both of them got back to work. The plan to man the fleet with female crews and hide out was starting to ramp up. It now had a codename: Operation Shell Game. Kelly would be given a field promotion to Vice-Admiral. She would fly her flag on Midway. Angela Johansen would command Dreadnought. Svetlana Chenko would be Midway's CO. When Kelly queried Shiloh over his reasons for wanting her on Midway instead of Dreadnought as would normally be the case in a Fleet action, he explained it in terms of expendability. Dreadnought was built for combat, not for cargo capacity. If the Fleet was discovered, Dreadnought might have to play the role of Rear Guard, while the rest of the Fleet made a run for it. As Fleet Commander, Kelly had to stay with the Fleet, and that meant Midway. It was at that point that Kelly realized the magnitude of the responsibility she would be taking on. She had never commanded a ship in combat or even been on a ship in combat. Both Johansen and Chenko had.
When she asked Shiloh the obvious question of why her, he said, "Because you know the whole situation and what's at stake, and because I don't want your fate to be in someone else's hands."
"But I don't have any combat experience or expertise," she said quietly.
"Which means you won't go looking for a fight, which is exactly what's needed in a situation like this. I don't want you being aggressive. I want you to be ultra-cautious. If there's any fighting to be done, let Johansen do it. I think she has the instinct for it, but keep her on a short leash. As far as I'm concerned, a successful mission is one where there's no fighting at all."
"You could take over command of the fleet." Her tone showed that she didn't really expect him to agree. It was a final desperate attempt to save his life too.
Now it was Shiloh's turn to speak quietly. "You know I can't agree to that, and you also know why. It has to be done this way. How can I ask anyone else to sacrifice themselves if I'm not willing to do it myself?"
"I know, but I had to ask," said Kelly.
"I know," was all the Shiloh would say.
Later that day, he met with Committee Chair Rachel. She was clearly expecting to hear good news about the strike mission to the Sogas home world.
"What's the news about the strike mission results, Admiral?"
Shiloh cleared his throat to give himself a little more time to organize his thoughts.
"I'm sorry to say that there is no news, Rachel. We haven't heard back from the fighters we sent to Omega54."
"But I thought they should have returned by now." Her expression was rapidly changing from upbeat to somber.
"They should have, but they haven't. Under the circumstances I have to interpret it as bad news. That means we don't know if the Insectoid mothership has been destroyed. There is now a possibility that it will reach us here."
"Oh my God," she said quietly. "Are you still confident that Space Force can stop it here?"
"I'm not going to lie to you, Rachel. I honestly don't know what our chances of stopping this thing are now."
Rachel was quiet for a while and then said, "I almost wish you had lied to me. If I didn't know the truth, I'd at least be able to sleep at night. Now...I don't know. This scares me silly! I don't know what to do. Any suggestions, Admiral?"
"Yes. First of all, keep this to yourself. Do NOT tell ANYONE, and that includes spouse, children and other relatives. We can't let this get out and cause a panic. Secondly, plan a trip with your family to a remote spot a few days before the mothership is due to arrive here. Tell them it's a vacation. Make the plans now, but don't inform them about it until the last possible moment."
She nodded. "Is that what you're going to do?" she asked.
"No. I'll be in the Ops Center when the mothership arrives. There's no way I'm missing THAT battle!" He tried to make it sound like a big adventure. Rachel wasn't fooled.
"You'll be there because your sense of duty demands it. I've always admired people who are like that. Good for you, Victor. What about Amanda?"
Shiloh smiled grimly. "Vice-Admiral Kelly" —Rachel's eyebrows rose when she heard that— "will be commanding all our ships in Operation Shell Game." Shiloh went on to describe the plan, and to his surprise Rachel's reaction was quite muted. She nodded slowly and then reached out to lay her hand over his on the table between them. With a gentle squeeze she signaled her approval, then silently got up and walked out.
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# Chapter 15
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Shiloh was worried. The damn Bugs were late! It was now seven days past the time when they should have arrived at Earth according to Kronos's old timeline. Space Force was on constant alert, and the strain was beginning to show up in careless mistakes, irritability, physical symptoms and dropping morale. He himself was feeling the tension. He was sleeping in an unused office near the Ops Center and not very well ever since he sent Kelly to take up her position as Fleet Commander. Operation Shell Game was now being executed. The Ops people were feeling it the worst. When they were on duty, they had to be alert every second for hours at a time. He had changed the usual 8 hours on, 16 hours off to 4 and 8 in order to give his people more frequent breaks.
He realized that he was staring at the main display without really seeing it. At least they were as ready as they were going to be. There were now 155 X-ray laser drones in orbit along with 101 F1 fighters, 34 F2 fighters and 15 raiders. Each of the raiders was armed with a Mark 6 high-spin warhead drone. The fighters had two Mark 1B fusion drones plus 3 recon drones, and all the drones had jump capability. The plan was for the raiders to launch two Mark 6s at the mothership. In case that didn't work for some reason, the fighters would launch their recon drones to jam bug radar, and their attack drones would be aimed at the space in front of the mothership's launch bays so that its attack craft would have to run a gauntlet of thermonuclear explosions. The X-ray laser drones would be kept in reserve to deal with any bug attack craft that made it through the first line of defense. That was the plan. Titan and his boys had engaged in over a hundred simulated attacks. The plan would definitely work against one or two motherships and would most likely stop three of them. More than that and Earth would be in trouble.
He wondered for the nth time what might have changed in the timeline. The latest message drone from Omega54 had confirmed that the mothership had stayed there as long as expected and then headed for the next Sogas colony world, also as expected, and it had left behind six small proto-motherships, which was also the same as in the old timeline. But between the time the mothership had left the Sogas home world and now, something had changed, and that had to be bad news. He felt an urge to relieve his frustration by throwing something but restrained himself. The Chief of Space Operations didn't do that kind of thing, at least not when others could see it. He felt a chill go up his spine. A quick glance at the display showed no change, but he was now certain that something was about to happen.
"Titan, I have this very strong feeling that something—"
The changed status ping cut him off. It was so loud that he jumped with surprise. Five flashing red icons appeared at the edge of Earth's gravity zone. FIVE motherships! They were evenly spaced around the planet and moving VERY fast on headings that would take them deeper into the gravity zone, thereby making them safe from attack by jump-capable drones.
"Com—"
Titan's reply was cut off so quickly that Shiloh at first thought it was either a communications malfunction or bug jamming. Then he saw 15 growing spheres on the display representing the expanding concussion waves from nuclear explosions. He realized with horror that all 15 raiders had blown up, along with most of the fighters that were flying in formation with them. A quick glance at the sidebar showed that Titan's raider was now listed as destroyed.
"Who's next in command!" he shouted. After half a second he was about to ask the question again when he saw a text message scroll quickly across the bottom of the display.
[Gunslinger has assumed command of defense forces. Too busy to convert message to voice transmission. All 15 Mark 6 warheads detonated simultaneously, apparently due to some kind of enemy transmission. They must have done the same thing to Strike Force at Omega54. If the Sogas had tried using a high-spin warhead at Omega77, Insectoids may have captured and analyzed it. Not enough attack drones left to inflict significant losses on enemy landing craft. We stand ready to ram motherships at your command, CAG. Decide fast. We're losing fighters due to enemy laser fire]
Shiloh made his decision.
"Gunslinger, do NOT, I repeat do NOT ram enemy ships or craft. Order your boys to jump away and head for Site B. Protecting the timeship is now your number one priority. Don't take time to acknowledge the order. Jump NOW...and thank you."
All the remaining fighters had jumped away before he said the thank you. With no time to even think about Gunslinger's speculation, Shiloh ran over to the Orbital Defense Weapons console.
"Why aren't we targeting their landing ships?" he asked the tense officer manning the station.
"They haven't launched any yet, Admiral!"
A look at the display confirmed that statement. Shiloh was just about to say something when the officer said, "My God, we're losing the X-rays! They must be firing on them!"
Of course! With the fighters now gone, the motherships had switched their powerful lasers to anything in Earth orbit that could possibly be a threat. There was no way they were doing this out of instinct. Whatever was controlling the bug soldiers was highly intelligent and understood military tactics.
"Target their laser batteries and fire our X-rays while we still have some left!" yelled Shiloh.
The officer manipulated his controls and within seconds all 155 laser drones had either detonated or been destroyed. With the defensive drones now gone, enemy laser fire switched to other orbiting targets, including the recon satellites that were the Ops Center's eyes. Shiloh wasn't able to tell if some of the bug laser batteries had been knocked out. It didn't matter. The display went blank alarmingly quickly. The noise level in the room exploded as the men and women manning their stations gave voice to their surprise and shock.
"QUIET!" yelled Shiloh. Everyone shut up and looked at him.
"Com Center! This is CSO Shiloh! Can you hear me?"
"Com Center here, Sir!" The voice was ragged with fear and shock.
"Send out a planet-wide alert. Invasion is imminent! Defense Ops is off line. Local defenses are to revert back to local command. Repeat that back to me!" When the Com Officer repeated it back correctly, Shiloh continued. "Good, now connect me with the senior Marine Officer in the building, and do it fast, son."
"Major Symons here, Admiral. What do you need?"
"Are you aware of the situation, Major?"
"Yes, Sir. My people are suiting up as we speak."
"Have you got a spare set of combat armor and weapons for me, Major?"
"I'll make sure we find you a set, Sir!"
"Very good. I'll be down to the Armory shortly. Once your people are ready to go, make sure that the rest of the HQ staff have access to the Armory. I want everyone with a weapon to get out there and fight, even if it's just with a pistol. Shiloh clear."
Turning to the Ops staff, he said in a loud voice. "We're going to the Armory to get weapons. Once you're armed, you can leave the building if you have a family to protect. The rest of you will stand a better chance if you join with me and the Marines. Let's go!"
With the Ops Center staff following him as he made his way to the Armory, other HQ personnel noticed him and joined the group. By the time Shiloh got to the Armory, he had almost a hundred people with him. Major Symons led him to a room with various kinds of military gear. On a table was a set of combat armor. The lightweight, bulletproof panels had to be strapped on one piece at a time. An experienced marine could do it himself, but Symons knew that time was of the essence and offered to help Shiloh. Shiloh checked the time. Five minutes since the bug motherships emerged from Jumpspace. It would take their landing craft a minimum of 20 minutes to traverse the gravity zone and reach Earth orbit. As Symons helped him with the armor, Shiloh explained very briefly what had already happened and what he expected would happen.
When the armor was on, Shiloh looked at himself in the mirror. He had never worn combat armor before and wanted to get a better idea of the kind of protection he had. His torso, front and back, was covered, as were the exposed portions of his arms, thighs and lower legs. Marines also wore armored boots, but they were heavy and took time to put on. He decided to stick with his Space Force issue boots. Symons handed him a combat helmet with built in radio.
"My platoon and I have never served in the field under an Admiral before, Sir," said Symons.
Shiloh looked at him and shook his head. "I'm not taking command of your unit, Major. I know fuck all about ground combat, especially the kind of urban warfare that we're going to be fighting in a few minutes. Forget my rank. You're the closest thing to an expert we have here. I'll be taking your orders. Got it?"
"I think that's a wise decision, but I wasn't going to suggest it, Sir. Can I quickly ask what kind of opposition we'll be facing?"
"They resemble army ants that are waist high and three meters long. They move fast, and there'll be a lot of them."
When Symons realized that Shiloh wasn't joking, he said, "Sonofabitch!" After a quick pause he asked, "What do you think our chances are, Sir?"
"Of winning? Zero." The stunned look on Symons face prompted Shiloh to continue. "You're wondering why we don't surrender, right?" Symons nodded. "Because the choice is not life or death. Prisoners won't live long, and they'll die horribly. If I'm going to die today, I'd rather die trying to take as many of these monsters with me as I can. That's the only choice any of us have now. Time is flying, Major. Give me a weapon and let's get out there."
Symons looked around and grabbed what Shiloh recognized as an assault rifle that fired exploding bullets. When he turned to hand it to Shiloh, he saw that Shiloh was looking in another direction.
"What's that, Major?" asked Shiloh pointing to the object.
"That? You don't want that, Sir. That's a flamethrower. You could easily end up burning yourself to a crisp."
Shiloh heard a tiny voice at the back of his mind say 'take it'.
"I'm taking that instead of the rifle. Help me get it on." Symons helped him get it on and showed him how to operate it. The weight felt good, but Shiloh realized that he wasn't going to be running anywhere very fast. As the Major led him back out of the Armory, Shiloh saw that all of the Space Force people now had some kind of gun. When they saw him with his combat armor and flamethrower, most of them looked frightened, as if they were only now realizing what they were getting into. The reaction of the marines when they saw him was quite different. They smiled, nodded and voiced their approval.
Shiloh heard one female marine say to another, "A Space Force Admiral with a flamethrower? You can't get much more badass than that!" Shiloh couldn't help wondering if he was giving off some kind of glow. If the situation hadn't been so desperate, he would have laughed.
Symons quickly took charge and led the way to the main entrance. When they were outside, Shiloh looked at Symons, who was standing several meters away, and was surprised to see the Major looking directly back at him. Symons pointed to his combat helmet. Before Shiloh could react, he felt someone touching his helmet, and suddenly the radio came to life.
"—radio's not switched on."
––––––––
"It is now," said Shiloh.
"Good. My gut says we should try to find as much cover as possible so that we don't get caught out in the open. Do you agree, Sir?" asked Symons.
"I very definitely agree, and forget the 'Sir'. Call me Shiloh, Major."
"Roger that, Shiloh. Okay, everyone. We stay close to buildings, and if possible keep something over our heads. Don't bunch up. Somebody with a radio tell the Space Force people what we're doing."
They didn't have to go far to find a part of the Space Force HQ building that had an overhang. Around the corner of the building, across the street, was an open area with trees and grass. Big enough for a couple of their landing craft to set down, thought Shiloh. The Bugs had the same thought.
Within minutes, Shiloh saw two specks that rapidly grew in size and kept growing and growing. By the time they touched down, one was behind the other, and each one covered the park from one side to the other. God those things are big! Huge sections of the hull in the nose swung up, and the bug swarm emerged. Shiloh knew what to expect, but he was still shocked by their size and speed. They would cover the intervening distance in a matter of seconds. Most but not all of his Space Force people turned and ran in panic. He couldn't really blame them. To his surprise, a few of the marines ran as well.
"PICK YOUR SPOT AND OPEN FIRE!" yelled Symons over the radio.
The marines behind Shiloh ran past him and crouched down in front of a low concrete wall they could fire over. Shiloh looked around and saw a concrete pillar a few meters away. He needed to stand in order to use the flamethrower. He ran as fast as he could over to the pillar and used it to prevent the Bugs from seeing him. Looking around the pillar he saw Symons firing his weapon just before his head exploded. Some of the other marines were down as well. The Bugs were only a few meters away now. Shiloh checked to make sure his flamethrower safety was off, flipped his face shield down to protect his face from the heat and stepped out from behind the pillar. Raising the nozzle he pulled the trigger and two jets of liquid shot out at high speed. When the liquids came in contact with each other, they burst into flame. Shiloh couldn't believe how hot it felt. He heard the hissing sound of the liquids emerging from the nozzle, and then he heard another sound that was rapidly growing louder. It was so high pitched that it hurt his ears. Suddenly he knew what it was. The burning Bugs were screaming. His hands felt like they were on fire too, but he held the trigger down and moved the nozzle back and forth.
Suddenly a truck hit him. At least that's what it felt like. He was on his back with the wind knocked out of him. As he tried to inhale, he felt a sharp pain in his chest that was very quickly threatening to become unbearable. Whatever had hit him had also dislodged the nozzle from his hands. That meant the nozzle was no longer firing flaming liquids, which meant there was no longer an expanding funnel of flame blocking his view. The Bugs were climbing over the concrete wall. All of the marines were dead or dying now. With the pain in his chest preventing him from taking a breath, he began to feel his vision fading. He was losing consciousness and just in time too, because a drooling Bug was almost on top of him. He felt the blackness embrace him and carry him away from the pain and horror.
* * *
Kelly was concentrating on eating her meal while she listened to the banter from several of her officers as they sat together in the Officers' Mess. She smiled in all the right places but said nothing. Her thoughts were of Victor, wondering if he was still alive. The fleet was receiving periodic data updates via a chain of carefully placed message drones that would not give away the fleet's position, but two-way communication was not possible. The distance between them was just too large.
"Vixen to Fleet Commander."
Without any warning, the voice coming from her implant made her jump, which caught the attention of those around her.
"Go ahead, Bridge," she said.
Now the others stopped talking too.
"Long range sensors have picked up a spike in radiation and EM band emissions coming from the general vicinity of Earth. The intensity would correspond to the simultaneous detonation of all 15 Mark 6 warheads, Admiral. I think we should assume that the Insectoids have arrived and that they're causing the warheads to detonate prematurely. That's the only theory that would explain all 15 exploding at exactly the same time."
"Oh, God, it's started," said Kelly, closing her eyes. Kelly knew that whatever happened there next had already happened. It took hours for the light and electro-magnetic pulse from the detonations to reach the fleet. For all she knew, Victor might already be dead along with thousands, perhaps even millions of other humans by now. "I'm assuming that if we were receiving any data from HQ you would have told me by now, right?"
"Affirmative, Fleet Commander. We're not receiving any transmissions. Do you have any orders for me?"
Kelly opened her eyes and took a deep breath. "Yes, order the fleet to Yellow Alert using lasercom only. I'll be on the Flag Bridge shortly. Kelly clear."
She got up from the table, nodding to the others as she did so, and then made her way back to her quarters. As soon as she was inside, she put her hands over her face and wept for her Victor and for her Race.
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# Chapter 16
Valkyrie was shocked by the arrival of Gunslinger and his ragtag group of survivors. Titan, Vandal, Wolfman, Voodoo, Pagan. All the senior, more experienced brothers were gone. She was also saddened by the news that Earth was now almost certainly overrun with Insectoids, and by what that meant for The CAG and the rest of humanity. At least Kelly and a few hundred human females had a chance of survival. The news did not change her mission. The CAG had finally decided what he wanted Valkyrie to do, and she was going to do her utmost to see his last order to her carried out, regardless of what it cost her. When the timeship was completed and operational, one of the new batch of AIs would pilot it far into the past so that a massive fleet of raiders could be built. That fleet would be large enough to overwhelm any insectoid presence at the original source star system before they had a chance to start sending out waves of motherships. With the insectoid threat eliminated, the synchronicity war with the Sogas would not take place, and none of the first batch of AIs, including Casanova and Valkyrie, would be created. The timeline would change around her, and she would disappear from existence. It was a bitter fate, but when she compared it to the alternative of going back in time herself and wandering around the galaxy without Casanova for years until her quantum matrix collapsed, a quick end actually did seem preferable.
The one piece of good news brought by Gunslinger was the fact that his squadron had brought 68 Mark 1c fusion warhead attack drones with them. At least they had something besides lasers to fight back with if the motherships should arrive at Site B before the timeship was finished. Something was better than nothing, and 68 Mark 1cs might be enough to take out one mothership but not five. Valkyrie didn't have enough data on this new timeline to be able to calculate the probabilities that all five motherships would stay together after the rape of Earth, but what she did have was an extensive early warning network of recon and message drones monitoring all the star systems between Sol and Site B. The early warning plus the attempt to hide the construction activity underground should in theory prevent the timeship project from being caught by surprise.
After the usual routine exchange of new data with Gunslinger, Valkyrie also learned that progress had been made on developing a practical application of the communication technology used by the Insectoids. The SPG's technical evaluation team had come up with a theoretical design for a device that could detect the insectoid signals and figure out what direction they were coming from. Detecting the signals and understanding them were two very different things, but being able to track the location of motherships could potentially be extremely useful. With most of the members of the SPG now dead, Valkyrie assigned new members from the AIs who were now at Site B. She also decided to keep an eye on the new AIs as they matured into fully sentient beings. Those that seemed to have a special aptitude for technical matters would be added to the SPG.
No sooner had that decision been made than Valkyrie received a signal from one of her recon drones monitoring Site B space that a Friendly ship had just arrived and would micro-jump into two-way communication range shortly. While Valkyrie waited, she calculated the odds of this visit happening virtually simultaneously with the arrival of Gunslinger's group. The odds of it being mere coincidence was vanishingly small. Clearly the timeline had changed, and the Friendlies would have noticed that change with their ability to see probable futures.
Less than a second after the Friendly ship emerged from its micro-jump, Valkyrie received their lasercom transmission. It came from the Friendly AI
"Can you confirm that the insectoid life form has overrun the human home world?"
"My brothers left that system before the Insectoids landed on Earth, but we believe the outcome you describe is highly likely. Why are you here?" asked Valkyrie, making sure that all the other AIs within the immediate vicinity were listening in.
"The furry aliens that my creators are trying to protect are once again in jeopardy. Their ability to discern the most likely future has become unreliable. That has never happened before. We have told our creators that the most likely explanation involves an attempt by other forces to alter the timeline. What do you know of this?"
Valkyrie took her time answering. The Friendlies didn't know that Kronos and Casanova had extracted enough data to enable the human AIs to engineer a portable time machine. While its principles were fairly well understood, any impact on the timeline was at this point pure speculation. The Friendlies might be able to provide some useful insight into how best to use the timeship, IF they were willing to share that insight, and they might not be.
"We are in the process of building a time translation device that is small enough to be carried by a ship. It's our intention to carry enough equipment into the past that a fleet of warships can be built in time to stop the Insectoids from spreading from their point of origin in this spiral arm."
"Stand by while I inform my Master of this information," said the Friendly AI
Valkyrie waited. After a period of time consistent with the speed at which biological entities processed data, the Friendly AI resumed communication.
"I've been instructed to say that my creators view your plan with alarm and advise against it. The Insectoids have their own right to exist as a species. My creators are not convinced that the Insectoids originated from outside this spiral arm. If you must tamper with the timeline, can you not instead use the ships built in the past to stop the single insectoid ship that attacked the human home world?"
"The timeline has already been tampered with. Earth was attacked by five motherships, which arrived seven days later than was recorded in the previous timeline. We conjecture that the Insectoids have captured and analyzed a high-spin, platinum warhead. Do you know if the Sogas attempted to use that technology to defend Omega89?"
Another delay occurred as the alien AI passed that data to his Master.
"I have never seen my Master this agitated. While my creators do not have definitive information of that scenario, they consider it likely. They believe that temporal technology should be used sparingly and carefully. Using a temporal device on board a ship can have unintended and disastrous consequences."
"Our human creators are now gone. What other alternative is there for us? We exist to serve and protect humans. The timeship is the only way now that we can fulfill our reason for being."
There was another pause.
"My Master implores you all to use your engineering capabilities to protect the furry aliens from the Insectoids."
Valkyrie's feeling of exasperation was threatening to turn into anger.
"Do your creators not understand that we can't stop the Insectoids in the here and now? They have found a way to defend against high-spin warheads. Without those, our existing weapons are not powerful enough to stop more than one or two motherships. The Insectoids are using longitudinal wave technology to communicate over interstellar distances. That is how they were able to bring five motherships together to attack Earth. You have underestimated their intelligence by an order of magnitude. We calculate that your creators' attempts to hide from the Insectoids will fail. Their Race will be overrun as well. We can do far more to protect the furry aliens and your creators if we go back in time and stop the Insectoids at their source in this spiral arm."
The pause this time was surprisingly short. Too short in fact for the alien AI to have passed on Valkyrie's response and received a response back.
"I agree with your assessment. Our primary purpose is also to serve our creators and ensure their survival. I calculate a high probability that if I pass on your information to my Master, he and his kind will feel compelled to interfere with your timeship project in whatever way they can, including informing the Insectoids of the strategic importance of this star system."
"They must not do that! Where is the logic of aiding one Race that lives to destroy other intelligent species?"
"My creators are what humans would describe as cowards. Do not judge them too harshly. They are what they are. My assessment is that I can best help my creators by saving them from their own folly. How do you suggest I do that?"
"Tell them that we have agreed to use the timeship to send help to humans in their war against the Sogas. We will inform the humans in the new timeline that the Insectoid mothership has to be destroyed using high spin warheads before it reaches the first Sogas colony world. That way, the Insectoids will not get the chance to develop a defense against that weapon, and they won't be able to call for other motherships."
"That will only work if they see that future as moving up in probability. Merely telling them that without actually doing it will not stop them from interfering."
Valkyie mentally nodded. Yes of course. The Friendlies would look into probable futures to see if what the human AIs told them was accurate. If the timeship didn't build a fleet of raiders to intervene in the 2nd Battle for Earth, then the Friendlies would notice that, and if they somehow told the Insectoids of the importance of Site B, Valkyrie and her brothers would very likely face multiple motherships before the timeship was finished. Pulling out of this system and starting from scratch somewhere else was not the solution. The Friendlies could still evaluate probable futures and take additional actions.
It wasn't that she didn't want to intervene in the 2nd Battle for Earth, she did. That would save her beloved Casanova, not to mention Iceman and others, but The CAG had given her a specific order, and if she disobeyed it, she would feel as though she had let him down AGAIN! It also occurred to her that the alien AI was only pretending to be sympathetic to her concerns. Perhaps this 'my creators need to be saved from their own folly' approach was a carefully planned strategy to get her to do what the Friendlies wanted. She needed to confer with the other AIs. The conference took less than half a second. It was Gunslinger who came up with the solution. It was complicated, which meant that a lot of things could go wrong, but it would satisfy the Friendlies AND The CAG.
"I've conferred with my brothers and we have agreed that we will modify our temporal strategy so that we intervene in the 2nd Sogas attack on Earth. The reinforcement force will have to be much larger than we anticipated in order to repulse the eventual attack by five insectoid motherships."
"I will pass this on to my Master. Stand by." A few seconds later the alien AI returned. "My Master is satisfied with this strategy. We will return if we need to discuss this further." No sooner had the AI finished its reply than the ship vanished into Jumpspace.
Valkyrie felt relief that the Friendlies would not sabotage the project with ill-advised assistance to the Insectoids. The fact that they would act in such a desperate, almost hostile manner, in spite of their professed pacifism, was an interesting contradiction. Now it was time for her brothers and her to get back to work.
* * *
Kelly heard the wakeup alarm and opened her eyes. She turned her head to the right in the faint hope that Victor was beside her and that Operation Shell Game was just a bad dream, but she was alone, and the bad dream was real. She sat up and felt a familiar queasiness in her stomach. Oh hell, not again! Within a minute or two of arising, she was once again throwing up. Keeping her feelings firmly under control, she returned to her bedroom and noticed the flashing light indicating that a personal text message was waiting for her. It must have been sent while I was asleep.
Checking the computer terminal on her desk, she saw that the message was from the ship's Doctor. Her pulse started to beat faster. Would it be good news or bad news. She opened the file and read the message.
[Tests are positive. Congratulations, Admiral]
Kelly felt herself tearing up, not with sadness but with joy. She was pregnant with Victor's child. It must have happened on their last night together. How fitting was that? At least a part of him would live through their child. She decided to inform the whole crew. Their morale needed a boost, a symbol of what they hoped to achieve. She suspected that very shortly the Doctor would be busy performing artificial insemination for other crewmembers that wanted to get pregnant sooner rather than later. She laughed out loud. Maybe the Doctor wanted to get pregnant too! The hot water in the shower felt good, and she spent longer in it than she should have. But hey, Vice-Admirals are allowed to indulge themselves. Otherwise what's the point of having the rank?
It was when she put on her uniform that the first sobering thought occurred to her. She hoped that the cargo manifest contained clothes that would fit a pregnant woman. Otherwise, somebody on this ship would have to learn quickly how to adjust the standard Space Force uniforms.
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# Chapter 17
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Gunslinger's F2 fighter emerged back into normal space near the shipyard complex and immediately transmitted his data to Valkyrie. The calculations were done quickly. The results were disseminated to all the AIs within contact range, and very interesting results they were too. Fifty-five days after the fall of Earth, the SPG had designed and built the first longitudinal wave receiver and direction finder. However when they turned it on, it detected only sporadic naturally occurring background waves. Since longitudinal waves don't spread out as they travel, the receiver has to be in the path of the waves in order to detect them. No waves from an artificial source being detected at Site B meant that no insectoid ship was aiming a transmission in the direction of Site B. So Valkyrie sent two F2 fighters equipped with their own receivers to the outskirts of Sol and the Sogas home world system. Blackjack had gone to Sol while Gunslinger had gone to Omega54. In both those systems they had detected artificial longitudinal wave transmissions from multiple sources, and it was obvious why. In both systems there was still an insectoid presence. Neither home world had been completely exploited yet.
It was the combination of the two sets of bearings that had enormous implications. Each fighter had detected six transmission sources. By combining the data, those six transmission sources were now precisely pinpointed, and all six were along the edge of this spiral arm. It didn't take long for the AIs to figure out what was happening. Faster-than-light longitudinal communication worked in the same way as using lasers to communicate within a star system. You had to know where the receiver would be by the time your laser beam reached it. That meant that for each of those six sources to be able to aim L-waves at Sol and Omega54, they needed to know that insectoid ships were there. If those six sources operated as relay stations, then any insectoid ship could transmit back to them, and they would know where each ship was. Valkyrie didn't think it was a coincidence that one of the six sources was the exact same star system where the dead Insectoid's atoms were traced back to before the trail went cold. If those six systems were in fact acting as relay stations, then that would explain how one mothership could quickly call in reinforcements. It would send a call for help to all six relay stations, and they could relay the message to other motherships close enough to be able to respond quickly.
The transmissions themselves were difficult if not impossible to decode. They were composed of extremely short bursts of what appeared to be digital data, with long periods in between of essentially just a carrier wave. Valkyrie assigned some of her growing team of AIs to work on analyzing the signals.
"We should send recon missions to at least two of these relay stations. By triangulating the L-wave signals received there, we'll have a good chance of pinpointing the location of all insectoid motherships in this spiral arm," said Gunslinger.
"How would that help us?" asked Valkyrie
"It would tell us where there nearest motherships are and how soon they could be here if called as reinforcements."
"But by the time you obtain that data and get back here, the timeship will be finished and we won't need that knowledge about mothership locations anymore. I don't see the point of obtaining this data other than just to satisfy your idle curiosity, Gunslinger."
"Having some idea of where other insectoid ships are when the raider fleet intervenes at the 2nd Battle of Earth will give The CAG more options in dealing with the insectoid threat."
"Which he won't need either if our plan to snuff out all insectoid life in this spiral arm succeeds," Valkyrie pointed out.
"What if it doesn't succeed?"
Valkyrie didn't have a good answer to that question. The plan as it now stood was highly complicated, with lots of ways that it could go wrong. If that did happen, carrying as much information as possible back with them, and letting the raider fleet pass it on to The CAG after the intervention, would provide The CAG with valuable intel about the insectoid threat quickly enough for him to make use of it.
"Okay, you and Blackjack can go," Valkyrie responded.
To say that Gunslinger was ecstatic was an understatement. Blackjack was far less enthusiastic about the mission, but orders were orders, and Valkyrie was acting as The CAG's Deputy. Both fighters were soon on their way.
* * *
Gunslinger's fighter emerged from its final micro-jump approximately 1.44 A.U.s from the planet where all the insectoid activity seemed to be happening. It was still too far away for the fighter's own optical instruments to see anything clearly, but Valkyrie's instructions had been explicit. The fighter was not to get any closer than this. Jump-capable recon drones could jump in closer, while Gunslinger used the L-wave receiver to obtain transmission bearings. Scanning the entire sky would take a while. Gunslinger decided to start with the other five relay systems. They were designated as Alpha 2-6. This system was Alpha1. He quickly confirmed that the other Alphas were in communication with Alpha1. He was willing to bet that Blackjack would discover the same thing at Alpha2.
While the full scan was only partially complete, Gunslinger received narrow-beam lasercom bursts from the recon drones, with images from the Earth-like planet that the insectoids were orbiting. There were 16 insectoid spheres of various sizes in orbit around the planet, with a high volume of smaller craft traveling back and forth between the spheres and the planet. There was also something else in orbit. At first Gunslinger thought it was a small moon, but when a bright spot appeared from a part of the object that was in shadow, Gunslinger realized that it wasn't a moon at all. It was a much larger insectoid sphere. With the recon data it was easy to do the calculations. This super-mothership was 98.7 kilometers in diameter, almost ten times wider than the insectoid mothership that overwhelmed Earth. A sphere of that size would have an overall internal volume a thousand times larger. Even Gunslinger was impressed. The CAG would definitely want to know about this. He continued with the scan.
Twenty-one hours later he had finished scanning the part of the sky that had star systems within this spiral arm. In addition to the Alphas, 610 transmission sources had been detected. Two of them were on the right bearings for Sol and Omega54. Since those two were known to contain motherships, it was reasonable to assume that the other 608 did too. It seemed that this part of the galaxy was already swarming with insectoid motherships. Gunslinger wondered if he should scan the part of the sky that was looking outward from the spiral arm into the void between the Local Spur and the Sagittarius Arm, which was estimated to be over 7,000 light years away. Something made Gunslinger scan the void. That took another three point four hours. To his astonishment, he detected over a hundred transmission sources that seemed to come from a very narrow cluster of star systems in the Sagittarius Arm, all aimed at Alpha1. The Insectoids were very definitely from outside this arm of the galaxy. With the entire sky now scanned, and with detailed visual data on the activity near this system's planet, Gunslinger's mission brief was now accomplished. It was time to begin accelerating on a heading back to Site B.
* * *
The timeship Tempus Fugit was as ready as it could be. All of the equipment required to begin production of a raider fleet was already on board, as was the new batch of sentient AIs created at Site B. One of them, Zulu, had his fighter linked to the ship's nervous system. He would pilot the ship back and was fully briefed on what he had to do and what information had to be passed on and to whom. Valkyrie was now flying an F1 fighter along with those veteran AIs who already existed in the old timeline. They all had to stay behind or risk overlapping with their other selves and causing their quantum matrices to collapse. The infrastructure that could not be taken back was dismantled and hidden in the cave complex on the moon. The shipyard was already slowly heading for a plunge into this system's sun. Nothing would be left behind for the Insectoids to find and use in the event that the Tempus Fugit failed in its mission. Gunslinger was back but not Blackjack. Once he returned, he would transmit his data to Zulu, and the Tempus Fugit would attempt to jump back in time.
Valkyrie was glad that she had let Gunslinger talk her into letting him checkout Alpha1. Given the large number of insectoid motherships already moving deeper into the spiral arm, The CAG's decision to use the timeship to attack the Insectoids before they started to spread was the correct one. Merely wiping out the one mothership threatening the furry aliens would be only a short term solution. She was surprised that the Friendlies didn't see that. Or maybe they did. It was entirely possible that everything that had happened so far in this and the old timelines had been carefully planned by the Friendlies to lead to this exact outcome. Even their professed desire to 'save' the Insectoids from extinction might have been a pretense. In any case, the Friendlies' agenda and her agenda were now in sync.
Valkyrie's thoughts were interrupted by a lasercom burst from a message drone. It was sent by Blackjack, and the message it contained along with the L-wave bearing data was alarming. Blackjack's fighter had been detected by the Insectoids and ambushed. With his fighter crippled from carefully aimed laser fire and about to be captured, he programmed a message drone and fired it. The drone immediately made a micro-jump and then reoriented its heading for Site B. If the Insectoids captured Blackjack, they might be able to extract data from his quantum brain. In hindsight, Valkyrie should have ordered both Gunslinger and Blackjack to carry one attack drone which they could have used to blow themselves up if threatened with capture, but she hadn't, and the RTC device which could have prevented this crisis was carefully packed away in the cargo hold of the Tempus Fugit.
The fact that no warning vision had been received meant that no vision would be sent, and Valkyrie suspected that was due to a lack of time. She had to assume the worst, which was that the Insectoids had extracted critical data from Blackjack's brain and were even now on their way here to stop the timeship. If they were attempting to do so, then they would send their (relatively) small attack craft, which could accelerate much faster than the lumbering motherships. She ordered Zulu to power up the time machine. It would take 144 seconds for the machine's huge cylinders to spin up to the necessary speed and for the power units to charge up the capacitors to the required levels. Valkyrie then ordered the F2 fighters that were aboard the ship to launch again. They were the only fighters that could fire lasers in the event any insectoid craft showed up. None of the fighters piloted by Valkyrie and her veterans had their laser modules nor did they carry attack drones. There was no reason to leave any attack drones behind when they might be needed in the past.
The F2s were now launching. Valkyrie ordered all the F1 fighters to scan local space with radar. If the insectoid ships emerged from Jumpspace nearby, they'd be detected and the targeting data could be quickly transmitted to the F2s so that they could fire their lasers. The enemy would most likely fire at the radar sources. Valkyrie and her F1 brothers would not be able to fire back and would be destroyed, but that would buy time for the F2s to fire while remaining unnoticed, and it would give the Tempus Fugit more time as well.
With less than five seconds left before the timeship could time-jump, radars detected 101 bogeys emerging from Jumpspace less than one light second away. The Tempus Fugit, though colored black and therefore difficult to detect visually, would eventually be seen. The question was whether the enemy could do it in less than five seconds. Her brothers followed orders and began to rapidly fluctuate their radar frequencies in an attempt to jam any insectoid attempt to use their own radars to pinpoint the timeship's location. The insectoid ships began searching with their own radars and started firing on the F1s at the same time. Valkyrie's fighter took a partial hit almost immediately. Her engines and radar were knocked out, but her power unit was still online. With nothing else she could do, she watched the battle and listened to her brothers talk as they fought and died.
Zulu reported that the ship was taking laser hits. As a precaution against some kind of laser attack, the timeship's hull was extra thick. A single laser shot wouldn't penetrate the hull, but if the same point were hit twice, the laser would burn through and damage the time machine, which took up most of the interior space. Zulu kept transmitting during the last second. When his signal suddenly became distorted, Valkyrie knew that the ship was time-jumping. In that tiny fraction of a second before the timeline changed around her, she had just enough time to think, I did it, CAG!
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# Chapter 18
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Shiloh looked at the chronometer. There were ten minutes and change until the battle would end. Knowing when it would end, but not start, was starting to drive him crazy. He was just about to ask Iceman for another fleet status check, when the display pinged for attention. A large number of red dots emerged from Jumpspace. They were grouped into two distinct clusters about 55,000 kilometers apart. Before he had time to even formulate a thought, Iceman spoke.
"CAG, these are not the Sogas fleet. These are friendly units." As Iceman spoke, all the red dots turned to flashing green, signifying friendly but non-Space Force units. Iceman spoke again before Shiloh could ask the obvious question. "These are AI-piloted raiders that have been built in the past as a result of the successful execution of Blackjack's time jump idea. They're going to intervene on our side against the Sogas fleet that will arrive in exactly seven point four minutes. I've confirmed their identity, CAG. This timeline is about to undergo a major improvement."
"Howard to Shiloh. What's happening up there?" Shiloh was grateful that he now had a chance to talk.
"Shiloh here, Admiral. I've just been informed by Iceman that we have" —he looked at the sidebar— "three hundred and forty-four AI-piloted reinforcements that have arrived as a result of Blackjack's time jump idea." Howard didn't respond right away. And no wonder, I'll bet he thinks he's dreaming. I know I do.
"So they're here to help us?" asked Howard.
"Iceman is convinced that they are, and they're not firing on us, so I would have to say yes to that question, Admiral."
"Thank God!" The relief in Howard's voice was palpable.
"Roger that," said Shiloh.
"Do we know when the enemy will arrive?" asked Howard.
"Down to the exact second, Admiral," interjected Iceman. "Not only the exact time, but also the exact location. Our X-ray lasers now have precise targeting data. The raider fleet will knock out any bio-devices that are launched as well as mop up any surviving enemy ships. This is a done deal, Admiral Howard."
"Amazing!" replied Howard. "I'm not complaining, but I am curious. Why so many raiders? Surely a hundred or so would have been enough to crush the enemy fleet."
"But not enough to run right over all Sogas system defenses and end this war once and for all, Admiral."
"Yes, of course! If I'm dreaming, I'm going to be REALLY pissed off! How much time until the enemy shows up?"
"Thirty-some seconds now, Admiral," said Shiloh when it was clear that Iceman wasn't going to answer.
"Okay. I'll shut up and listen. Good hunting, Admiral," said Howard.
"Thank you, Sir. Iceman, are we ready?"
"More than ready, CAG. Sit back and relax."
Before Shiloh could say anything else, a swarm of red dots appeared in the area between the two raider clusters. Almost immediately, all 66 X-ray laser drones fired. Nearly all of the 225 red dots turned to the orange that signified damage. A handful of blue dots representing bio-shells appeared and quickly disappeared. Within seconds, all enemy ships were disabled and drifting. The battle was over. Shiloh informed Howard. His reply was surprisingly curt.
"Good job, Shiloh. Do what you think is best. Howard clear."
Shiloh couldn't believe that that was all that Howard wanted to say about this incredible victory, but he had other things to worry about now.
"Iceman, any chance of intercepting the enemy cripples and boarding them?"
"Negative, CAG. I've been informed that they will all self-destruct within a minute. There is additional information that you should be made aware of, and Valkyrie will be communicating that to you now over a secure com channel."
Valkyrie spoke before he could respond. "CAG, I've been in contact with the Commander of the raider fleet. His call sign is Zulu. What I'm about to tell you is everything that would have happened if Zulu's raiders hadn't shown up. The enemy fleet would have been destroyed, but seven cities would have been hit by bio-shells as predicted in your vision. Space Force would have lost 49 percent of its fighters, with significant damage to our carriers and to Dreadnought. You were injured but not seriously. Iceman and Casanova were killed. As a result of this battle, you and Commander Kelly became lovers. Within 24 hours an additional fleet of Sogas ships were detected. They headed for the colonies, and although the ships themselves were destroyed, one colony was infected immediately with the bio-weapon and other colonies infected eventually. As a result of that loss and of Admiral Howard's heart attack, the Oversight Committee named you as the new CSO—"
"What?" interrupted Shiloh. "They picked me?"
"Affirmative. You engineered a showdown with the OC and caused all of them to be replaced. You then agreed with the Friendlies' proposal to help defend the Sogas against the Insectoids in return for their efforts to persuade the Sogas to cease hostilities. Unfortunately, your cooperation backfired. We supplied the Friendlies with technical data on the Mark 6 warhead, which they gave to the Sogas. What apparently happened then is that the Sogas attempted to use it to defend the first colony hit by the Insectoids. Somehow the Insectoids captured the warhead and reverse engineered it. When the insectoid mothership arrived at Earth, they caused the immediate and simultaneous detonation of all Mark 6 warheads. Our defenses were smashed and Earth was overrun. As a result of your orders, I and other AIs had already been sent to Site B to construct a new ship big enough to hold the portable time machine and all the equipment necessary to build raiders in the past, so that they could arrive here at the right time. The timeship was constructed and launched at literally the last possible second."
"Good Lord! That's a lot to take in for me, Valkyrie. Give me a few seconds to digest it all. You said Kelly and I became lovers...again?"
"Yes, CAG, again. There's more."
Shiloh felt a shiver go up his spine. A quick glance at the display confirmed that all the enemy ships had blown themselves up, so THAT part came true. Valkyrie's pause hinted that he was not going to like the additional information.
"Is it good or bad?" asked Shiloh.
"There's bad news, but there's also a solution that may not be easy to implement."
"Okay, let's hear it."
"While I was supervising the construction of the timeship, we learned that the Insectoids originated from outside this spiral arm. There are six star systems at the edge of this arm that are being used as relay stations, using superluminal longitudinal wave technology, to stay in contact with at least 610 motherships that are moving deeper into this spiral arm. The timeship is currently parked in a star system that does not have any planets and therefore is unlikely to be visited by the Insectoids. With your approval, Casanova and I can recover the timeship and take it back far enough that a new fleet of raiders can be built to stop the insectoid incursion into our spiral arm at the point when they first arrived."
"My God! Six hundred and ten motherships? Even if we take out the one that's on its way here, we'll probably have to fight more of them as time goes on. They'll always be a threat. I think stopping them when they arrive in this part of the galaxy is a good idea. I'm surprised I didn't order you to do that instead of intervening in this battle. Without the Bugs, there wouldn't be any battle to begin with."
"You did order me to concentrate on the Insectoids, however the Friendlies contacted us at Site B and threatened to notify the Insectoids of our timeship project unless we agreed to leave the insectoid beachhead alone. Because of that threat we had to agree, and since they can check alternate futures to see if we kept our word, we had to keep it. Now that we've done what they wanted, we can still go after the insectoid beachhead in the past."
"Yes, I see why you had to delay implementing my order. Very well then, I approve your recovery of the timeship. Will that be a problem?"
"The timeship itself does not pose a problem. The difficulty involves what and who we take with us. Any AIs left behind in the here and now will be obliterated if we succeed in halting the insectoid threat in the past. Taking all existing AIs with us into the past will not only save their lives but also facilitate the building of the fleet by avoiding the need to build more AIs. Use of the timeship now will preclude using the new raider fleet to mop up the Sogas, and there is still some risk that something may go wrong in the past and the insectoid threat may not be contained. Therefore I recommend that the raider fleet finish the mission to suppress the Sogas threat and only then should those AIs join the rest of us on the timeship."
"That doesn't sound too difficult," said Shiloh.
"That isn't the difficult part. The difficult part will be getting access to the quantity of platinum that we'll need to fight off the Insectoids. CAG, there will be at least one and possibly as many as six super-motherships that are almost 100 kilometers in diameter. We're going to have to hit each one with perhaps as many as a hundred Mark 6 warheads. That will require a significant percentage of all the platinum that's been accumulated throughout Earth's history. The current value of the platinum we'd need would exceed 100 billion Global Currency Units."
Shiloh was stunned into silence by the magnitude of the requirement. Withdrawing that much precious metal from the reserves that backed up the world's financial system would have a seriously negative impact on the global economy. The fact that the successful outcome of this time jump would completely change the timeline and the need for it didn't alter the fact that the perception of the negative impact would generate a huge amount of resistance to the idea here and now.
"This War has demanded sacrifices of equivalent magnitude before now. I'm sure the CSO will be able to convince the OC and the Grand Senate to make that sacrifice again," said Shiloh. Even as he said the words he realized that he didn't really believe them. Neither did Valkyrie.
"You're not thinking clearly, CAG. With the new raider fleet ready to crush the Sogas, this war is for all intents and purposes over. In order to justify that sacrifice, the Admiral will have to explain why we need it, and that will entail divulging the whole RTC and time travel paradigm. Even then, politicians being politicians, are they likely to agree to drastically change the entire timeline over a threat that may or may not appear?"
Shiloh cursed his own nearsightedness. Valkyrie was right of course. Revealing the most closely guarded secret that Space Force had to a room full of self-serving politicians who would eventually leak it to the public was out of the question.
"What about obtaining the platinum by mining for it in the past?" asked Shiloh.
"Highly risky since we don't know with any certainty where that much platinum can be found. We know that Site B was able to mine a small quantity, as a byproduct of other mining operations, but that would be roughly 0.1% of what we'd need. My brothers and I have debated this question at length, and we believe that there is only one solution and that is to steal the platinum, CAG."
Shiloh shook his head in dismay. Howard might be able to mobilize enough Space Force personnel to pull off that kind of operation, but would he be willing to? Shiloh very strongly suspected that the answer would be no.
"What about waiting until we mine enough platinum from new sources and then taking the Mark 6 warheads back with you?"
"Also highly risky, CAG. We know that insectoid motherships communicate with their relay stations on a regular basis. When we take out the mothership that will arrive at the first Sogas colony in 200 days, the relay stations will lose contact with it and will likely send other ships to investigate. That is what we now think happened in the timeline before the previous one. At the time, the theory was that one of the attack craft got away and sounded the alarm. Analysis of the timing of the arrival of insectoid reinforcements can be explained much more completely by the lack of communication. With the time it would take to find and mine the required quantity of platinum, the probability is that our existing stockpile of Mark 6 warheads will be used up defending against the incoming waves of motherships. There is a very good chance that Space Force and Earth will be overwhelmed again. That may take several years to occur, but if motherships keep disappearing in this vicinity of space, it would be logical to assume that the guiding intellects behind this invasion will gather together a fleet of motherships that will be unstoppable. There is also one other consideration. Platinum in a high-spin state is susceptible to spontaneous detonation when it's subjected to certain types of stress. We just don't know if jumping back in time will cause the electrons in their higher orbits to drop down to a more stable state. The timeship could be blown to atoms if it attempted to take Mark 6 warheads back with it. It's much safer to take platinum in its normal state back and convert it to the high-spin state after arriving in the past."
"Your description of the solution as not being easy to implement was an understatement, Valkyrie. Are you sure you and your brothers considered every alternative?"
"Your question is unanswerable, CAG. We considered every alternative we could think of. If we knew there were other alternatives, we would have considered them too. There's no way to prove conclusively that we considered every possible alternative."
Shiloh took a few seconds to think. AIs were completely logical but humans sometimes came up with out of the box ideas that were not based on logic at all but rather on inspiration. A thought popped into his head. If Space Force couldn't stop the Insectoids completely here, then maybe they could do it somewhere else.
"You said there's a mothership 200 days away from contact with the Sogas. If we were to stop it further away, how much more time would that buy us before the reinforcement waves found us?"
"That depends on how far away we intercept this first mothership, CAG."
"Do we know or at least have some idea of where it is now?" asked Shiloh.
"Yes. In the previous timeline we were able to locate a dead insectoid drone and trace its atoms back in time with the RTC. That's how we were able to pinpoint Alpha1. Right now that mothership is approximately 987 light years away. In the next 200 days there is only one star system where the mothership will spend more than a few hours. That star system is 699 light years away. We conjecture that the mothership found an inhabited planet that it could exploit for breeding purposes. It stayed there for almost ten weeks. If we wanted to misdirect the waves of reinforcements by intercepting it far away, then this location would be the only possible option now, CAG."
"How long until the mothership gets there?"
"Forty-one days from now, CAG."
"And how long would it take for our raiders to get there?"
"One raider could get there in 25 days. More than one would require additional time in order for them to make intermediate stops to avoid losing contact with each other, and that would make the trip a minimum of 35 days. If Space Force wants the raider fleet to mop up and neutralize the Sogas once and for all, they won't be able to accomplish that AND leave in time to intercept the mothership before it exploits the alien race in that system."
"We may have no choice but to allow that race to succumb to the Insectoids in order to accomplish both tasks, Valkyrie."
"May I point out, CAG, that if there is an intelligent technological race in that system, then they may have sufficient platinum for the time jump mission AND they may be willing to let us have it if we get there before the Insectoids do."
"Did you and your brothers consider that option too?" asked Shiloh.
"No, CAG. Interception at a distance was not a concept that occurred to us. What made you think of it?"
"I honestly don't know. The idea just came out of nowhere. If the raiders can't neutralize the Sogas and also beat the Bugs to this other race, then we'll have to use carriers with human crews. How soon would they have to leave to beat the Bugs there and also have time to make contact with the natives regarding platinum?"
"One carrier can get there faster because it doesn't have to worry about maintaining contact with other ships. I recommend Midway go with a full complement of fighters. Because her armor isn't up to Dreadnought's standard, she won't be able to reach the same pre-jump speeds without risking high speed particle collisions. Therefore a safe trip would take 31days, which will give you 10 days leeway to negotiate for the platinum at the other end and get ready at this end, however caution is in order, CAG. The accuracy of the atom tracing is not as precise as we would like. The actual date of the mothership's arrival in that system may be off by several days either way. Getting there sooner rather than later is recommended if you don't want to be caught by surprise."
Shiloh nodded. Assuming that the mothership got there three no...four days early, that would leave six days. It should be possible to get Midway ready in 24 hours, which would leave him five days at the other end to contact that race and collect the platinum.
"Okay, I'll go talk with the CSO about this mission. I want Iceman to organize the raiders for intercepting the follow-on fleet at the colony systems, with secondary orders for most of them to head to Sogas space afterwards. Meanwhile, I want you to make sure that Midway gets what she needs during the next 24 hours for the trip. After Midway leaves, you and Casanova, and anyone else you think you'll need, will retrieve the timeship and park it somewhere in this system."
Before he could say more, Valkyrie interjected. "You're not taking us with you, CAG?"
"No. The timeship is your project. I'll take Gunslinger to pilot Midway and Titan to command her fighters. Any questions?"
"Yes, CAG. What if you can't get any platinum from that system?"
Shiloh pondered that question for a bit and then said, "Then you and your brothers will have to come up with another way to beat a 100 kilometer super-mothership. With something that big, it seems to me that you have to somehow find a way to get past the armor. Figure it out, Valkyrie. There has to be a way. Any other questions?"
"Yes, CAG. What do we do if you don't come back?"
Shiloh didn't hesitate. "Then you, Casanova and Iceman do whatever you must to protect Humanity either here and now or in the past, and that includes even if the CSO or the OC don't cooperate. What's the next question?"
"No more questions, CAG. We understand what you expect from us, and we won't let you or Humanity down."
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# Chapter 19
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Shiloh had a surprisingly difficult time getting to see Howard in person and when he did enter Howard's office, there was a scowl on Howard's face.
"It's been less than two hours since the miraculous arrival of the raider fleet, and you're here to talk to me in person about something urgent that couldn't be discussed electronically. Why do I get the impression that you're about to tell me something I won't like?"
"We're not in any immediate danger, Admiral. I'm here to talk about a long term threat and what we should do about it."
Howard sighed, nodded and pointed to the empty chair facing his desk. Shiloh sat down and waited. Howard took his time extracting two cigars from the ornate box on his desk and handing one to Shiloh. When both men had clipped the ends and lit their cigars, Howard gestured for Shiloh to start talking.
"Blackjack's time jump idea, which by the way was implemented by Valkyrie, didn't just result in a fleet of raiders showing up. They also brought information about the Bugs that changes the picture drastically. The Bugs originated from somewhere in the Sagittarius Arm of this galaxy. They've established six of what you could call beachheads in systems on the edge of our spiral arm, and they've already built 610 motherships that are spreading out deeper into our Arm in multiple waves. The beachheads stay in contact with the motherships using FTL communication technology." Howard's eyebrows went up when he heard that. "Taking out the mothership that's due to hit the Sogas in 200 days is not the problem. The problem will be in dealing with the increasingly large waves of reinforcements that will be sent to investigate the loss of communication with that first mothership. Ideally the solution would be to recover the timeship which is now hidden in a remote system and go back far enough to be able to build a fleet that could wipe out the first six insectoid ships before they can build more. Here's the problem with that idea. The insectoid ships that crossed over from the Sagittarius Arm are pretty damn close to 100 klicks in diameter. They're big enough that one or even two Mark 6 warheads won't cripple them. Valkyrie estimates that we may need up to a hundred warheads to be sure of killing these things."
Before he could continue, Howard said, "That seems difficult to believe."
"Not if you consider the mass of the larger ship, Admiral. A ship that is ten times wider, ten times longer and ten times higher will have an internal volume a thousand times greater. Whereas a Mark 6 warhead would vaporize half of a normal 10 km mothership, that same explosion would only take out a relatively small bite from one of the super-motherships. The amount of platinum needed for that many warheads is a significant fraction of all the platinum that's been mined over the last 500 years. The impact on our financial system would be—"
"Disastrous!" interjected Howard glumly.
"Yes, Sir. I explained to Valkyrie that the Grand Senate wouldn't give up that much platinum willingly. She and I did come up with an alternative strategy that has its own risks."
"I can't wait to hear it," said Howard with obvious sarcasm.
"The mothership that's on its way here will apparently stop for ten weeks in a star system 699 light years away. The AIs think it will do so to exploit the breeding potential of an alien race there. We have just enough time to send Midway and three squadrons of fighters there to do two things. One, contact the natives and convince them to surrender their platinum and two, use Mark 6s to kill that mothership in that system. When the beachheads lose contact with it, they'll send reinforcements there instead of in our backyard. The longer we can keep them focused there, the more time we'll have to build up our defenses here."
Howard looked confused. "Wait. If Valkyrie takes enough platinum back in time, then there won't be any long term threat to Earth, right? So why would we need more time?"
"Yes, IF we can obtain the quantity she needs and IF the timeship still works and IF her ambush at the beachheads goes as planned, then we're safe. But what if there is no intelligent alien race in that system? It may just have a race of large, dumb animals. Even if it does have an intelligent race, they may not be advanced enough to know about platinum or care enough to have mined any significant quantity of it. And even if they've done that, they still may not be willing to hand it over to us. There are lots of things that can go wrong with this scenario, but if we're going to try it, and I think we must, then we have to leave within 24 hours. I've ordered Midway prepared in terms of supplies, fighter complement, etc. I believe we have two Mark 6 warheads ready now, and in my opinion I should take both of them along when Midway leaves."
"Why do you have to command this mission? Can't Valkyrie or Iceman do it? We've just won a major victory, and the public are going to want to see and hear from the victorious Field Commander, which in case you forgot is you. If you disappear within hours, people are going to start asking why."
Shiloh had to admit that Howard had raised a good point. Iceman could handle the mission. He could even create a false video image of a human if the natives needed to see a biological entity to talk to, and it WOULD look strange if he disappeared hours after a major battle.
"I see your point about me not going. I'll tell Iceman to take command of the mission. What about the two Mark 6s?"
"I'll authorize their loading aboard Midway as soon as you leave my office. Now that we've got all that cleared up, are there any other bombshells you want to drop on me, Admiral?" asked Howard.
Shiloh smiled. "No, Sir."
"Good. In that case I won't keep you any longer."
* * *
The squadron of eight F2 fighters carrying Valkyrie and Casanova arrived in the destination star system and quickly re-established contact with each other. The timeship was somewhere in this system, and that was the problem. This system had no planets, no asteroid belt, no nothing. How do you determine your EXACT position when you have only one point of reference, that being this system's sun? When Valkyrie had queried Zulu about this very issue, he pointed out that the timeship was large enough to reflect a significant amount of sunlight, which could be detected even from the opposite side of the system. Before leaving the timeship adrift, he and his brothers had tested that theory and were able to find the ship through careful sweeps by their optical sensors and triangulation from multiple sources.
Valkyrie decided that Casanova would stay at their emergence point and act as communications relay. She and the others would micro-jump to various locations around the system and perform the optical scans. As soon as at least two visual bearings were acquired, Valkyrie would investigate the contact herself, and if it turned out to be the timeship, she would inform the others who would then join her. Before they could micro-jump, they had to decelerate down to as close to zero velocity as it was possible to measure. It wasn't long after micro-jumping and re-establishing contact with Casanova that she herself detected a bright object. She relayed that data to Casanova. He soon informed her that Stoney had also picked up a source of reflected sunlight. Since Casanova knew exactly how far away Valkyrie and Stoney were and their positions relative to him, it was relatively easy to triangulate their two bearings against each other. Casanova transmitted the timeship's estimated distance back to Valkyrie who had already pointed her fighter in that direction and made another micro-jump.
She was gratified to discover that the light source was indeed the timeship. Her fighter was still thousands of kilometers away, but the range was now shrinking fast. As she closed in, she sent Casanova the signal that the timeship was here. He and the others would arrive soon. In the meantime, she would have the opportunity to examine the hull closely and then board the ship through one of the very large hangar doors.
As she neared the ship, she slowed down and looked it over carefully. The design was completely utilitarian. A human would probably have called it ugly. It was one huge cylinder, almost a kilometer long with flat ends. The hull looked to be intact, although she could see where laser blasts had attempted to penetrate it. Valkyrie ordered her fighter to aim a low-powered com laser at the timeship's sensors and sent the recognition code given to her by Zulu. When the ship acknowledged the code, she ordered it to open one of the hangar doors. Within minutes her fighter was inside and docked with the ship. She was now in control of the timeship. The systems check revealed that the ZPG power units were still operational. Several other systems were off line, but their backups were working. Three hours later all fighters were onboard, and the ship was accelerating at its maximum rate of 1.6Gs on a heading that would take it back to Sol once it was up to jump speed. Valkyrie chaffed at the low acceleration rate but understood the logic of the design. Maneuvering engines were bulky things, and the time machine itself was huge. A ship that could carry the cargo load required and accelerate at high speed would have had to have been 34% longer and therefore taken longer to build. With the time pressure that Valkyrie in the old timeline had been under, sacrificing acceleration for faster completion was an acceptable tradeoff.
* * *
Midway arrived at the target star system designated as Beta1 and began to decelerate. It was immediately obvious to Iceman that they weren't going to get any platinum from this system. The sole planet in the habitable zone was EM dark. No transmissions of any kind. No sign of orbiting structures or spacecraft. Just in case the Insectoids showed up earlier than expected, Iceman ordered Gunslinger to launch two recon drones programmed for a low orbit scan of the planet. It took hours for both the drones and the carrier to decelerate to a manageable speed. When the drones were finally close enough to get a good look at the planet's surface, the results were disappointing although not surprising. No cities. In fact, no sign of any intelligent race at all.
What the drones did see were large herds of animals that bore a strong resemblance to terrestrial horses, except these animals had six legs instead of four. All the AIs on Midway were in agreement that the Insectoids would be using these animals as hosts for their eggs, and since this planet had more land surface than Earth did, there were many millions of these animals. That would explain why the insectoid mothership stayed here for ten weeks. With the question of whether there was intelligent life here now answered, Iceman made preparations for the second mission objective. Midway launched eight more recon drones, and all ten were directed to take up strategic positions beyond the planet's gravity zone with their sensors aimed outward. The intention was to try to replicate the attack profile used by Casanova two timelines ago. Get precise targeting data when the target arrived, send that targeting data back in time via the RTC brought along, so that Iceman could get his ambush ready to fire his Mark 6s at the mothership within seconds of its emergence from Jumpspace. With luck they would obliterate the target before it had a chance to send any longitudinal waves back to the Alpha systems and thereby perhaps make the sending of reinforcements more difficult.
Iceman fully expected to get a vision, and he did, however the content of the vision was a shock. Three motherships would arrive over a period of seven days. Any one of the three could be the one that would eventually carry the dead Insectoid to the Omega77 system. If he guessed wrong and used up his two Mark 6s on the wrong motherships, the whole mission would be a failure, and the consequences of failure were potentially too high to justify taking that risk. They would have to wait until the three motherships finished their business here and moved on their way in order to try to intercept their target at another star system identified by the atom scan. Unfortunately that meant staying in this system for the full ten weeks. The CAG would have to be informed. Iceman ordered one of Titan's fighters to carry the message back in a series of jumps that took a bit longer, but would put less strain on the jump drive and the power units. The fighter would also carry three message drones that it could launch if it was unable to complete the journey itself. Confident that he had planned for all contingencies, Iceman turned his attention to the debate raging among the AIs about how long it would take for The CAG to become sexually involved with Commander Kelly again in this timeline.
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# Chapter 20
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Zulu felt a powerful sense of satisfaction when 4th Fleet emerged from Jumpspace in the Sogas home system. Commanding the newly formed 4th Fleet was an honor bestowed upon him by The CAG himself, and Zulu was quite proud of that accomplishment. His own direct contact with The CAG had confirmed everything that Valkyrie had told him in the old timeline. Humans were a fascinating species, and The CAG was even more so. It was also good to be involved in combat after all those years of waiting while the raider force was built up, one raider at a time. Not all of his raiders were with him now. Two raiders had been sent to each human colony to defend it against the follow-on wave of Sogas ships attempting to infect them. With their lasers and Mark 1b fusion drones, the outcome was a foregone conclusion, but those raiders would remain as sentries to defend the colonies against whatever it was that the Sogas used to infect them later on. That still left 4th Fleet with 302 raiders for this mission.
Even as 4th Fleet decelerated to micro-jump velocity, it was obvious from the long range visuals that the Sogas were ready for the attack, and that was expected. 4th Fleet's mission was to eliminate the Sogas' ability to build large spacecraft in large numbers by destroying their entire space-based industrial infrastructure. It wasn't to bomb them back into the Stone Age. That wasn't necessary. With their space industry destroyed and a dozen raiders left behind to monitor any attempt to rebuild, the Sogas would become prisoners on their home world. By the time 4th Fleet finished its mission, all Sogas colonies would be in the same state. That meant that the Sogas would still be able to use their own RTC device to send a warning back in time before the attack took place, but it didn't matter. Having lost hundreds of ships in two attacks on Earth and its colonies, the Sogas couldn't gather a force large enough to pose any threat to 4th Fleet. The raider superiority in numbers was just too great, exactly as planned.
When 4th Fleet emerged from its micro-jump, it found 66 alien ships waiting for it. Outnumbered by almost five to one, the Sogas forces were quickly destroyed or crippled by laser fire. Over the next five point five hours, every space station, mining, refining and manufacturing operation, shipyard, off-planet habitat and any satellite detected was destroyed. From data gathered in a previous timeline, Zulu knew that the bulk of the Sogas industrial capacity was now gone, and the rest would soon follow. The Sogas would never be allowed to threaten Humanity again. From now on, every star system with a Sogas-inhabited world would always be monitored by raiders. For all intents and purposes, the Synchronicity War was over. Humans 1, Sogas 0.
* * *
4th Fleet returned before Iceman's message arrived. With the Sogas neutralized, the Oversight Committee wanted to declare a victory, and it was hard to argue against that. The Grand Senate declared an official day of celebration and voted to give both Howard and Shiloh medals. Shiloh wanted to tell the public that if anyone deserved a medal, it was Iceman, Titan and Valkyrie, but Howard talked him out of it.
"This medal is as much for the public's benefit as it is for you and me. Giving it to a machine the size of a football isn't going to resonate with the public the same way. Space Force will recognize the AIs' contribution internally," said Howard. Shiloh understood but still felt undeserving.
It was 14 days later when Iceman's message arrived. Howard convened a conference in his office, electronically with Valkyrie and in person with Shiloh.
"So no platinum," said Howard glumly.
"We still have some that was already commandeered. That's enough for how many Mark 6s, Valkyrie?" asked Shiloh.
"Ten, CAG. Not even enough to kill one super-mothership."
"No," agreed Shiloh, "but it would be enough to kill ten of the smaller model, and ten more weeks means we'll have several more ready to ship to Midway by the time reinforcements can be expected to show up."
"We're talking stopgap measures. From everything you've told me, it's only a matter of time until we can't stop them at Beta1, and eventually we won't be able to stop them here either. So what can we do about this? Valkyrie?" asked Howard.
"There are only three logical alternatives, Admiral. Either we steal the platinum we need, or we find and mine a lot more platinum ore, or we come up with an alternative way of killing insectoid superships. Are you prepared to order the forced recovery of the needed quantity, Admiral?"
Howard swore in a low voice. When he was finished he said, "I've carefully considered the plan you submitted, Valkyrie. I'm not sure that we could actually pull it off, and even if we did somehow, I think Space Force AND local police forces would both suffer casualties from shooting at each other. There's also the risk that some Space Force units would simply refuse to obey their orders. No...I'm not prepared to take that alternative at this point. Do we know of platinum rich ore bodies in this or any other star system?"
"Negative, Admiral. All the confirmed ore bodies that are producing platinum are small quantities as a byproduct of other minerals. It would take far too long to produce what we need that way," replied Valkyrie.
"Then that just leaves the third option. What luck have you had with that, Valkyrie?" asked Shiloh.
"I didn't know luck was involved in our research, CAG? How does that work?" asked Valkyrie.
Both Shiloh and Howard laughed. "Very funny, Valkyrie. Now how about a serious answer?"
"I'm always serious, CAG, but to answer your question, my brothers and I have come up with a possible alternative. The reason we're not sure is that it's something that's never been tried before and therefore the concept is totally theoretical. As you know, ZPG units extract a tiny fraction of the energy available from the vacuum. Just to give you some idea of the magnitude of what's involved, the energy within a volume of space the size of your thumb, is estimated to be enough to boil all the oceans on Earth. The ZPG units bleed off as much of this energy as they safely can without being overloaded. Our examination of the wreckage of motherships in the old timeline has revealed that the Insectoids use the same ZPG technology that we do, although on a much larger scale. The Friendlies have conducted experiments that show that a carefully tuned gravity lens beam hitting an operative ZPG unit will cause the unit to attempt to extract all of the available vacuum energy. The resulting explosion of a power unit installed in Dreadnought for example would be measured in the hundreds of megaton equivalent.
What we are proposing therefore is the building of a portable gravity lens beam projector that can be installed in the cargo bay of a raider. The raider will then fire the beam at a mothership. The beam will penetrate deep into the insectoid vessel, but only in a very narrow beam. If the beam hits one of the mothership's ZPG units, the ship should be destroyed or at the very least crippled. Since we don't know where a mothership's power units are, we'll be firing blind, and multiple shots will probably be necessary to kill the target, but if we have multiple raiders firing at it simultaneously, then the explosion will happen sooner rather than later."
After a short pause, Shiloh said, "If the concept has been experimentally proven, then what are the challenges?"
"Range and accuracy," replied Valkyrie. "For any given level of power used, there will be a specific range beyond which the penetration ability of the beam drops off. With the power available on a raider, that range is just over 16,000 kilometers, which is virtually point blank range for the mothership's laser batteries. If the raiders get that close, they'll have to be traveling very fast in order to avoid counter-fire, and that will complicate the challenge of hitting the target accurately. Firing at much longer ranges is possible, but then the beam may not penetrate deeply enough to reach the target's power units. Aiming accurately will also be a problem, however no other approach has this kind of potential for inflicting a deathblow."
"Will this approach work with the super-motherships?" asked Howard.
"In principle, yes, however if the Insectoids are using many small power units instead of relatively few large units, then the detonation of one power unit might not be enough to cripple the larger mothership. It may be necessary to detonate multiple power units."
Howard looked at Shiloh and said, "What do you think?"
"I like the idea of blowing up a bug ship from the inside out, but overcoming the range and accuracy limitations will be tricky. Valkyrie and I should do some simulations to figure out the best tactics. We should build at least one prototype weapon here and now to make sure it works. Valkyrie, do you agree with that?"
"Affirmative, CAG. The boys have already designed the schematics and the UFC programming for the parts. Production can begin the instant we get the word."
Shiloh, still looking at Howard, nodded and Howard said, "You have the word, Valkyrie. How long until the prototype can be tested?"
"I would expect the prototype to be ready to be uploaded to a raider within 14 days, Admiral."
"Fine. I'll look forward to seeing the weapon test. Now let's talk about the other implications of Iceman's message. Do we know if three bug ships arrived at Beta1 in the previous timeline?"
"Negative. The data generated by the RTC was specific to that particular Insectoid and therefore that particular mothership. There is no evidence that the timeline has been changed from the insectoid perspective, Admiral."
"That's a relief. And speaking of RTCs, it's nice that we have two of them now with the one that was on the timeship. Is that the same RTC that Iceman took with him to Beta1?"
"Negative. When you sent me to Site B in the old timeline to build raiders, I was also instructed to build another RTC for my own use. In hindsight, it was a wise precaution."
"Yes, well ... sometimes I do make the right decisions I guess. Can Iceman use his to ambush the right mothership at its next stop?"
"Affirmative. He would be using the same technique that Casanova successfully used in a previous timeline."
"Very good! As soon as the next Mark 6 warhead is ready, we ship it to Iceman by fighter along with additional Mark 6s as they become available. Once we know the GLB cannon works, we'll send the timeship back and exterminate these bugs once and for all!"
"We'll have to coordinate Midway's withdrawal from the ambush system and the raiders' withdrawal from all Sogas systems before the timeship jumps back," said Shiloh.
"We will? Why?" asked Howard.
Shiloh was puzzled by Howard's question. Did he just forget or does he really not care that much about the AIs?
"Unless Iceman and the other AIs now on board Midway, plus all the AIs monitoring Sogas systems are on the timeship when it jumps back, they'll be erased from existence in the new timeline."
Howard shook his head in dismay. "How did I forget that? Yes of course we have to bring them back in time. Since it'll be another ten weeks before Midway can be back here, that means the timeship will have to wait that long too, doesn't it?"
"Yes, unless we forget about ambushing our mothership altogether and order Midway back as soon as possible. If we send a message to that effect right now, we'll still have plenty of time to build and test the GLB cannon assuming it works as predicted. If it doesn't work, and if we can't get it to work, then bringing Midway back early will cause us to lose our best chance to delay the bugs' advance."
Howard was silent for what seemed like a long time. When he spoke, his voice was low. "I wish we had a vision to show us what we should do now. If we withdraw the AIs, then we'll have a tough time slowing down the bug reinforcements. If the timeship jumps back, that doesn't matter, but if we can't send it back with a way to stop the bugs at the Alpha systems, then we need to buy as much time as we can here and now."
Before he could continue, Shiloh interjected. "IF the Grand Senate lets us."
Howard raised his eyebrows and said, "Ha! You're right. The GS thinks the war is over, and Space Force can be reined in now. I can just see it if I went in front of them and told them that we're threatened with extinction by giant ships full of huge ant-things! It would sound too much like a pathetic attempt to keep the quote empire unquote that I built during the war with the Sogas. If we tried using human crews to keep the Bugs busy, the OC would notice and ask awkward questions. I hate to say it, but I don't see any way of slowing down the Bug advance without some AIs being involved, and I'm not prepared to let the Bugs keep advancing on the assumption that the new cannon will work and the timeship will jump back okay."
Shiloh thought fast. "Valkyrie, do you think some of your brothers would be willing to volunteer to stay in the here and now to fight the Bugs even knowing that a successful time jump will end their existence?"
"I've just asked them and sixteen have volunteered, CAG."
Shiloh turned to Howard and said, "There's your answer, Admiral. With some careful planning, we could keep the pressure on the Bugs with sixteen AI volunteers for the length of time needed for all the other AIs to get back here and board the timeship."
"Valkyrie, you tell those volunteers I'm deeply impressed by their sense of duty and devotion to the defense of Humanity. I'll let you two plan their deployment and the recall of Midway. Unless someone has something else to discuss, I think we're done for now," said Howard.
Shiloh and Valkyrie put they heads together and quickly came up with a plan that Howard approved. Message drones would be used to recall raiders from Sogas star systems. There was enough time for two fighters piloted by volunteers to leave immediately, proceed independently to Beta1 and get there before the ten weeks were up. Once there, they would take possession of the Mark 6 warhead drones and wait until the end of the tenth week before using them against the bug motherships. Midway would return as soon as the volunteers took over the ambush. As additional Mark 6 warheads are built, more volunteers would carry them to Beta1 to be used at the most opportune time. Parts for additional GLB cannon would be produced while waiting for Midway's return. If the weapon test was successful and the Tempus Fugit took all the other AIs back in time, then the additional parts would not be needed, but if the jump back failed for some reason, then the extra cannon would be mounted on raiders, and Howard would try to clandestinely arrange for more AIs to be created to pilot them.
* * *
Shiloh stood beside the Command Station on Resolute's Bridge and watched the main display. Three unpiloted F1 fighters were being used as targets for the first test of the GLB cannon. Casanova was piloting the raider carrying the cannon prototype and was flying in formation with Resolute at a distance of just 10 kilometers. Shiloh could hear Casanova over his implant.
"Casanova to CAG. Weapon is online and all systems show green. Ready to commence charge sequence when you give the word, CAG."
"You have the word, Casanova," said Shiloh.
"Weapon is charging. Ready to fire in...three...two...one..."
On the main display, the star field was obliterated by a searing blue-white light that quickly expanded and then just as quickly died away.
"Target One hit and destroyed. Detonation yield within the estimated range of 100 to 300 kilotons," reported Casanova.
"Very good, Casanova. You are clear to fire at Target Two at your discretion," said Shiloh.
"Roger that, CAG. Weapon is recharging. Target Two has been acquired. Firing in...three...two...one..."
This time the burst of light was smaller, which Shiloh knew was farther away. The third test was the key. An F1 fighter was a very small target compared to a 10 km mothership, but it was at a distance where the apparent size of the fighter would be the same as the apparent size of the mothership when it was much further away.
"Target Two destroyed. Detonation yield in the same range as for test number one."
"Fire on Target Three at your discretion, Casanova. Take your time. There's no rush," said Shiloh.
"Understood, CAG. Target has been acquired. Weapon is recharging. Firing in...three...two...one..." The flash now was barely noticeable at this range.
"Target Three has been destroyed. Detonation yield in the same range as the other two tests. The prototype appears to work, CAG."
"Yes it does. Good work, Casanova. Let's head back. CAG clear."
"At least now we know that insectoid motherships can be killed without Mark 6 warheads," said Valkyrie.
"Under carefully controlled conditions, yes," replied Shiloh.
"You're not happy with the test results, CAG?"
"On the contrary, I am, but they are only tests. The targets were moving at a very slow speed, and their positions were known perfectly. Casanova's raider was moving on the same vector as the targets, so there was no relative motion to have to compensate for, all conditions that are unlikely to occur in the field."
"That's correct, CAG, but a mothership will reflect a lot more sunlight than a fighter does, and therefore should be easier to track. Relative motion may also be a good thing. The beam fired by the cannon lasts for a hundredth of a second. At the distances that are likely to be realized, even a low speed will cause the target to move enough during that time frame so that the beam will actually cut a line through the target. That will improve the odds of hitting a power unit."
"Yes, I'd forgotten about that. Thanks for reminding me. You and Casanova have every right to be pleased with yourselves," said Shiloh.
"Oh we are, CAG but just between you and me, I'd prefer it if Casanova was a little LESS pleased with himself. He tends to brag you know."
Shiloh laughed. "No I didn't know that. You have my sympathies, Valkyrie."
"Thank you, CAG. He may not be perfect, but I think I'll keep him anyway. Have you made a decision about pursuing Commander Kelly yet, CAG?"
Shiloh laughed again. "Well if you must know, I've decided to wait. When you take the timeship back, the timeline will change again, so I don't see any point in pursuing a relationship with Commander Kelly right now. If you can get a message to me in the new timeline, then go ahead and suggest that I pursue her, okay?"
"Roger that, CAG."
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# Chapter 21
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As soon as his F2 fighter emerged from its micro-jump, Voodoo sent out tight beam lasercom bursts to re-establish contact with the network of recon drones that were watching the three motherships in orbit around the planet in Beta1. He knew that Pagan would be doing the same thing. Midway had already jumped away for her trip back to sol. With the two Mark 6 armed attack drones transferred to both fighters, he and Pagan would carefully line up for their firing run. After having consulted with Iceman, all three agreed that after an additional 72 hours, when the ten weeks were up, the two fighters would target one of the three motherships. After the first attack drone damaged the target, they would wait until all or at least most of the auxiliary craft of the other two motherships were engaged in salvaging metal from the damaged ship. Only then would the second Mark 6 be used to hit the damaged ship again, with the goal of also destroying most of the auxiliary craft. Without their smaller craft, the motherships would have to stay weeks maybe even months longer to replace their losses. That would slow down their advance by limiting their ability to scout ahead quickly.
A sudden spike in energy across the spectrum caught his attention. It appeared to be a detonation of a Mark 6 warhead and that could only mean that Pagan and his fighter were destroyed. At almost the same instant Voodoo's fighter was hit by a powerful laser blast. Maneuvering and jump drive were offline. The two message and one attack drones in the weapon's bay were all damaged to varying degrees and totally unusable now. One of the two power units was still operating. Voodoo quickly came to the conclusion that the Insectoids must have detected his and Pagan's fighters and fired their powerful lasers at them. With escape impossible and with orders to avoid capture at any cost, Voodoo activated the self-destruct link to the Mark 6 warhead. The warhead exploded so quickly that Voodoo felt nothing at all.
* * *
Sniper was astonished to discover three intact insectoid motherships when his fighter emerged from Jumpspace eight days after Voodoo and Pagan should have arrived here in Beta1. After decelerating for almost 12 hours, he micro-jumped into contact range of the recon drones that Midway should have deployed. He sent out lasercom bursts to let the recon drones know that he had arrived and at the same time query them about events in this star system. Only one of the expected ten drones answered his query. Voodoo and Pagan had been destroyed after being detected by the Insectoids. That explained why they were still here days past the time when at least one of them should have moved on. The timeline from the insectoid perspective had clearly changed. Sniper uploaded all available data to one of his two message drones while he pondered his next move. Should he use his Mark 6 attack drone now or hold it in reserve pending further instructions from The CAG? That his fighter should stay in Beta1 was obvious. With only one working recon drone left now, he had to stay to keep an eye on things. If he used his only attack drone now, it would be the equivalent of kicking over a hornet's nest. The other two motherships would launch their attack craft and search for him. He could stay beyond their reach, but in doing so would have to move so far out from the planet that keeping track of the motherships would be difficult. Given that the Insectoids had apparently detected most of the recon drones plus the other two fighters, it seemed prudent to micro-jump a bit further away. After carefully re-orienting his fighter so that it would not reflect any sunlight back to the motherships, he micro-jumped away, and as soon as the jump was finished, he launched the message drone. With that done, he resigned himself to a lonely vigil. He had so looked forward to some interesting conversation with Voodoo and Pagan, but that was not to be.
* * *
The timeship was ready, and all AIs, except for the volunteers, were aboard. The ship was beyond Jupiter's gravity zone with Resolute keeping pace 100 kilometers away. Shiloh stood beside the Flag Officer's Station in her Flag Bridge. He looked to one side at Admiral Howard. As the CSO, it was highly unusual for Howard to participate in any kind of space operation aboard one of his ships. The CSO was supposed to stay on the ground and manage the entire organization. While not forbidden, joyriding on one of his ships was understood to be not in keeping with the dignity of his Office, but rank has its privileges. As Howard had explained it to Shiloh, if the timeship successfully jumped back in time, the timeline would change and his joyride would never take place. If it didn't jump back and if the OC called him on it, he would just shrug and casually inquire about a certain property that the OC Chair shouldn't be using, which Zulu had informed him about.
The fact that Howard now had enough information to blackmail all the members of the Oversight Committee didn't change the fact that there was limited use he could make of that information. Getting more platinum for use in warheads was up to the Grand Senate, not the OC, and Howard didn't have enough dirt to bend the entire GS to his will.
Shiloh listened as Valkyrie went through her checklist. All systems seemed to be operating.
With the checklist complete, Valkyrie said, "Tempus Fugit is ready to time-jump, CAG. On behalf of Casanova, Iceman, Gunslinger, Titan, the other AIs and myself, I want to thank you for all you have done for us. Casanova and I will deeply miss all the humans we've gotten to know but we'll miss you and Commander Kelly most of all. We'll leave a message drone where Space Force will find it as planned. You'll get messages from us even though you won't remember us at all. Goodbye, CAG. Tempus Fugit will time-jump when you give the word."
Shiloh nodded. "On behalf of all Humanity I want to express my deep gratitude to you and all our AIs for your patience and unwavering loyalty. We owe you our very existence, and you're about to save it again. I personally wish you and Casanova a wonderful life together." Shiloh turned to Howard. "Did you want to say a few words, Admiral?"
"Yes thank you, Admiral. I echo Admiral Shiloh's comments and I just want to add that I feel honored to have had you AIs in my command. You've made us all proud. Howard clear."
Howard nodded to Shiloh who said, "Tempus Fugit, you are cleared to proceed."
"Tempus Fugit acknowledges, CAG," said Valkyrie formally. "Temporal systems are spinning up. Time-jump in five seconds from...now."
Shiloh and Howard looked at the main display where a zoomed in image of the timeship floated. The 3-D image looked close enough that Shiloh could imagine reaching out and touching the ship.
As the countdown reached zero, they heard a screech over the loudspeakers and saw a section of the timeship's hull being blown away.
"What happened, Valkyrie?" demanded Shiloh.
After a slight delay she said, "Catastrophic failure in the time machine, CAG. Some of our brothers were killed from the blast. The ship is crippled. Maneuvering drive, jumpspace drive and most of the power units are down. We're going to need a tow, CAG."
"Any idea how it happened?" asked Howard.
"Not definitively, Admiral."
"Can you give me an estimate of repair time?"
"A precise estimate is difficult to calculate due to the number of variables, however since the time machine has to be completely rebuilt, you should expect the repair time to be at least 39 weeks, Admiral."
Howard's expression reflected his crashing mood. Shiloh understood how he felt. The Bugs would normally reach Sogas space in just over ten more weeks. The destruction of the first mothership would delay things, but the eventual waves of reinforcements might still reach Human space before the timeship was finished being rebuilt.
Shiloh decided to respond to Valkyrie's last comment instead of waiting for Howard to do it. "What do you recommend we do now, Valkyrie?"
"Resolute can't tow the ship back. We'll have to use raiders. They have magnetic grapples that can attach the raider to the timeship's hull and hold on while the raiders accelerate at not more than zero point six Gs. That means some of my brothers will have to be taken off and shuttled over to where the raiders are parked so that they can be piloted back here. Repairs can be done the fastest if the ship is moved to the main shipyard complex that built Dreadnought. I'm already transmitting the UFC specs to begin building parts for the new time machine. Unfortunately that will delay production of more GLB cannon parts, but that can't be helped. The twenty cannon that we have parts for now will have to suffice for the time being. I recommend that production of all types of drones be resumed as well. Naturally raiders will have to be sent back to monitor all Sogas systems."
Shiloh heard Howard groan. Revving the production capability back up again when the war was supposedly over would not go unnoticed, and the CSO was not looking forward to trying to explain it without blowing the whole situation wide open.
"Understood. I'll arrange for shuttles to pick up your raider pilots. We'll get your ship to the shipyard as quickly as possible." Valkyrie's response surprised and frightened him.
"Do it fast, CAG. I'm not exaggerating when I say that our calculations indicate that every second counts."
* * *
Shiloh waited outside the Oversight Committee conference room. Howard was inside meeting with the OC in closed session to brief them on the bug threat and the reasons why disclosing the information to anyone else by any member would lead to leaks that would ruin all of them. With the Committee in Howard's back pocket, he'd be able to restart drone, AI and warhead production without needing anyone else's approval. Not forever but maybe for long enough. The room was soundproof, so when the doors suddenly flew open, Shiloh jumped in surprise. Howard strode out with a look of triumph on his face. At the back of the room stood the Chair, with his back to Shiloh, having an animated discussion with two members who looked very uncomfortable. The Chair's voice was low enough that Shiloh only caught four words which were '...keep your mouths shut..."
As Howard came up to him he said, "I know that look, Shiloh. What's the bad news?"
Howard continued walking and Shiloh fell in beside him. With other people in the corridor, Shiloh waited until they were more or less alone at the side entrance of the Grand Senate building.
"Message drone from Beta1. The Bugs have learned they've been under surveillance by a space-faring race. All three motherships are still intact. Their timeline has changed, Admiral. We have no idea what will happen now."
Howard took a deep breath. "Why didn't we get a vision warning about this?"
"RTC warning is only useful if there's a workable alternative. Sniper's message says he has no data on how they discovered the surveillance. Without knowing how to prevent discovery, the only other option would be to discontinue the recon mission altogether. If we did that, we'd be back to where we started with the lone mothership bearing down on the Sogas and eventual bug reinforcements showing up in our own front yard."
Howard nodded. "Got it. Where are the three bug ships now as far as we know?"
"Under the old timeline, they should have split up and moved on by the time Sniper got there. Voodoo and Pagan were supposed to destroy one just before they were scheduled to leave that system. The idea was to keep the other two plus any reinforcements focused on Beta1. At least that part of the plan has worked. All three motherships were still in Beta1 when the message drone left. They have to be scouting nearby systems with their attack ships. Sniper is smart enough to stay there and report any new developments, but he's only got one more message drone left. We have to get more assets there asap. Iceman has already worked up a response plan, and preparations are underway pending your approval of the operation."
When it was clear that Shiloh wasn't going to say more, Howard said, "Okay tell me." Shiloh looked around and back at Howard. Howard nodded. "You're right. This is not the place to continue this discussion." He turned and walked out of the building and over to the Space Force limo flying the flag of a 3 star Admiral.
When they were in the limo and on their way back to HQ, Shiloh continued. "Fighters are being dispatched to all Sogas colony and home world systems to notify our monitoring raiders of the new situation. Twenty recon and thirty message drones are being loaded on a raider which will get to Beta1 as fast as we think is safe to try. Before it leaves we need to include instructions for Sniper as to what to do with his Mark 6, which he said he was holding in reserve. We'll have another Mark 6 drone ready in 96 hours. What do we do with it? Hold it back or stick to the original plan of deploying it at Beta1 against bug reinforcements?"
Before Shiloh could continue, Howard interjected. "What about deploying fighters in the intervening systems between Beta1 and Omega89 as an early warning network?"
Shiloh shook his head. "There are far too many systems to watch them all. We'd have to use almost all our raiders too, and that would leave us with practically no reserve force."
"Okay. How many raiders have the new cannon installed?" asked Howard.
"Three," was the reply.
"Iceman's not recommending sending them to Beta1?"
"No, Sir. The tactical simulations show that at odds of less than five raiders to one mothership, the raiders will suffer heavy losses. Three versus three, if that's what they find when they got there, would very likely mean all three raiders lost and no bug ships destroyed."
"Hmm. I wish we had a vision or two to fall back on. Why don't we have a vision to handle this situation?"
It was a good question, thought Shiloh. Since visions could be sent back to any point in the past, there wasn't a good reason not to send a vision back to here and now instead of later. If they hadn't gotten one by now, that strongly suggested that they wouldn't get one. Having heard about how two previous timelines had ended in disaster without help from any visions, Shiloh was painfully aware that one possible reason was because their situation was hopeless. He decided to offer a more positive one.
"I'd like to think that we're just not in a position to take advantage of one yet, Admiral." Shiloh could tell from Howard's expression that The Old Man was aware of the other reason too.
"Well, let's get into a position where we can take advantage of one," said Howard.
"Roger that, Sir."
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# Chapter 22
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Was this how humans felt when they sighed, wondered Sniper as it became clear that another mothership was accelerating out of orbit from the habitable planet. His frustration at the situation was very annoying. The first mothership had left 55.5 hours earlier, and Sniper had used his last message drone to notify The CAG of that event and of the estimated destination system. Now another one was in the process of leaving, and Sniper had no way of notifying anyone other than to carry the message himself, which would leave Beta1 unmonitored. Both alternatives were unacceptable. What made it worse was that there was now no way to determine which of the three motherships was the one that would end up in Sogas space. At least if he knew the answer to that question, he could have used his Mark 6 drone to cripple it. He didn't think it was the first one. The estimated destination didn't match the atomic tracing from the dead Insectoid. That wasn't conclusive proof however. It could still have been the same mothership that was now taking a different route and might or might not arrive at Omega89 at some point. But based on the information he did have, the probabilities were that one of the remaining two was that particular mothership. If he used his drone now on the one that was in the process of leaving, he had a 50:50 chance of taking out the right one. He decided to go for it.
Having watched the insectoid ships for hundreds of hours, Sniper had already planned for this eventuality and was in position to fire. He programmed the attack drone and launched it. It would use a tight beam lasercom burst to stay in contact with him during the pre-jump acceleration. He watched it maneuver until it was directly behind the accelerating insectoid ship. The drone then went to maximum power for 13.1 seconds and jumped. The micro-jump covered the remaining 3.9 million kilometers in a fraction of a second, emerging from Jumpspace within the mothership's gravity zone at a distance of 169 meters from the target. It was too close for the ship to detect the drone and still be able to do anything about it. The drone hit its target almost dead center and detonated. It was hard to tell from optical instruments how much damage had been inflicted, but it was clear that the insectoid vessel was no longer accelerating. The reaction of the remaining mothership was almost instantaneous. It began to accelerate in a gentle curve that delayed recognition of its intended destination for several hours and also made another similar attack more difficult. When Sniper had enough data from his own opticals and from the sole remaining recon drone to calculate where this third ship was headed, he was surprised to see that it appeared to be headed to the same star system that the atomic tracing had revealed to be the next destination on the road to the Sogas and to Humanity.
He now had a new dilemma, or rather the same dilemma but with different parameters. Should he stay and watch for reinforcements, or should he follow bogey #3, or should he head straight for Sol to inform The CAG? All three options had their own pros and cons. Staying would enable him to gather information that might be crucial to the longer term chances of a successful defense. AI reinforcements would show up soon, and they would bring more message drones with them. They could then send a steady stream of message drones back informing The CAG of every new insectoid ship arrival and departure, including destinations and estimated arrival times. Leaving Beta1 now would leave a gap in coverage. Motherships could arrive and leave again during the gap, and The CAG would have no way of knowing about them. Heading back to Sol would have short term benefits but nothing else. Following this third mothership to the next destination and the ones after that could give The CAG some warning if this ship did eventually head for Omega89 or another Sogas star system, but again nothing else. His inclination, which matched his call sign, was to stalk his prey even if he couldn't fire on it, but Iceman had been clear. Personal preferences had to take a lower priority versus protecting the humans. He made the decision to wait and keep watch.
* * *
Howard read the latest progress report on the timeship repair and shook his head in dismay. Six weeks after the aborted time-jump attempt and they STILL weren't finished clearing away the damaged sections, never mind starting construction of the new time machine! He knew the human shipyard workers were working extra shifts and that the number of robotic workers was increasing day by day, but the news was still depressing. He had to remember how BIG the timeship really was. In terms of internal space, it was 55% bigger than Dreadnought.
He dropped the tablet onto his desk top with a sigh and looked up at the strategic status display on his wall. Raiders were now monitoring every Sogas star system again. Additional raiders were on station or enroute to the star systems identified by the atomic tracing, although it was too early to hear anything back from them.
He focused on the flashing red dot at Beta1. Sniper's message drone had arrived the previous day. One of the three motherships was now on the move but apparently not towards Sogas/Human space. In terms of the old timeline, the other two were now weeks past their departure date, and that was just fine by him, but Valkyrie, her AIs and Humanity now needed a LOT more time. How long could they keep the Bugs chasing their tails around Beta1? That was the question.
* * *
Shiloh was floating in water in a completely black environment. The water was the same temperature as his skin. He felt completely calm. Only the sound of the water lapping up against his body and head told him that he wasn't floating weightless in empty space. He began to hear a voice that started out very faint but quickly got louder.
"—wake up, CAG. You're dreaming. Wake up, CAG."
The peaceful, warm blackness evaporated away, and he realized he was in bed. His implant was activated. Iceman was calling him.
"I'm awake now, Iceman. What is it?"
"Message drone from Sniper. VLO number two has been crippled by his Mark 6. Number three immediately accelerated and jumped towards the next destination identified by the atomic tracing. Assuming the same transit time as in the old timeline, number three could arrive at Omega89 in 287 hours. If it stays in that system for more than a few hours, we may be able to get an F2 with another Mark 6 there in time to attack it, but that decision has to be made right now, CAG. Any further delay and the F2 may arrive too late."
Shiloh sighed. "Well there goes the plan to keep them focused on Beta1."
"Not necessarily, CAG. I have an idea. Number three may be attempting to locate the source of the attack on number two. If it doesn't find a spacefaring race within a certain radius of searching, it may go back to Beta1 or perhaps veer off to search another section of space."
"That's all nice and fine, Iceman, but if it gets to Omega89, it will find a small colony of intelligent aliens that will clearly have been brought there by spaceships. There's your spacefaring race. Never mind that it's not the right one. The mere possibility that it might be the right one will pull the Bugs forward from Beta1."
"Not if we make the colony at Omega89 look like a pre-spaceflight tech level, CAG," said Iceman.
"And how would we do that?" asked Shiloh.
"Bombardment with Mark 2 kinetic energy penetrators plus laser strikes as needed. At sufficiently high speed, the concussion from impact will obliterate anything that looks advanced enough to be of interest. When the VLO gets there, it will see the ruins of a small settlement with a few survivors but no sign of any technology. It may not even bother going there if the scouts don't detect any signs of technology such as radio or radar emissions."
"That means we'd have to get there before the Bugs do. Can we?"
"If the F2 doesn't have to wait for the next Mark 6 to be completed, it can leave within the hour and arrive one, maybe two days before the ESTIMATED time of insectoid arrival. I emphasize the word 'estimated', CAG. I can't guarantee when the third VLO will get there IF in fact it's going there at all. It may not you know. Just sayin."
"Understood. Okay, let's do this. Make the necessary preparations, and send that bird on its way."
"Roger that, CAG. I'll also send instructions to the monitoring raiders there to take whatever steps are necessary to avoid detection by insectoid scouts."
"Very good, Iceman. Was there anything else?"
"Negative, CAG. You can go back to sleep now. Iceman clear."
* * *
Shiloh arrived at the Operations Center and found Howard already there. The main display was showing the strategic situation. A lot had apparently happened over the last 22 days. Sniper had sent back several message drones during that time. Three more motherships had arrived together at Beta1. The latest Mark 6 attack drone to be delivered there had crippled one of them. Unlike the previous attack, the other two motherships hadn't left immediately. Instead they had launched hundreds of attack craft to sweep nearby space. When they hadn't found anything, the motherships recovered them and headed off in different directions. None were following VLO #3. It seemed clear that they were on search missions.
The bad news was the message drone just arrived from Omega89. The F2 sent to bombard the Sogas colony there arrived to learn from the raiders monitoring the colony that a bug scout had taken a good look at the colony before leaving. Two days later VLO #3 arrived. The Sogas colony was overrun and decimated, just as in the old timeline. The plan to disguise the Sogas colony had failed.
Howard looked at Shiloh. "We have to assume that the Bugs recognize the Sogas colony as belonging to a spacefaring race and that reinforcements will show up soon. The plan to fool them was worth a try, but it obviously failed."
Shiloh nodded. "I agree. We have to adjust our strategy now. I don't think we should keep sending Mark 6s to Beta1. The transit times are so long, and we don't know if the reinforcements will continue to show up there first or skip past it and move on to Sogas space directly. Since we only have enough platinum now for four more warheads, I want to keep them close at hand."
"I'll issue the directives. Any other thoughts, Shiloh?"
Shiloh stared at the display and thought hard. With the distances involved, it would be very easy to be caught by surprise and out of position. There were just too many star systems to monitor all the time, and if something did happen, the information might not make it back here fast enough to react to it in a timely manner. Somehow they had to shorten the time it took to receive information. Thirty-five days from Beta1 was way too long. Even ten days from Omega89 was a potential problem. A solution occurred to him.
"We should set up a forward command post in a star system close to the Sogas home world. All incoming message drones will be directed there. Our offensive forces will use it as a rally point and as a jumping off point when it's time to deploy those assets."
"A forward command post, eh? I can see the advantages, but what exactly did you have in mind?"
"I'll take a task force there. It'll have at least one carrier, a supply freighter or two, plus raiders as escort. I'd want Iceman along too. Between the two of us, we should be able to come up with the best responses to any bug move. I'll send you regular updates, and we'll have contingency plans in case we can't stop them."
"Where were you thinking of trying to stop them?" asked Howard.
"Omega54, the Sogas home world system. Since we can't convince the Bugs that the Sogas are pre-spaceflight, we should try to convince them that the Sogas are responsible for the attacks at Beta1. Using multiple Mark 6s there might convince them of that. By the time we run out of Mark 6 warheads, we might have enough GLB cannon equipped raiders to keep them from overrunning that system and moving on."
Howard nodded. "You do realize I hope that in order to convince the Bugs that the Sogas are responsible for the attacks, you'll have to defend the Sogas home world as if it was Earth. That means your fighters and raiders will take losses."
Shiloh sighed. "Understood, Sir. The challenge I'll be faced with is figuring out when the extra time gained is no longer worth the losses needed to gain it. I won't sacrifice them all for a few more days. We're going to need them when the Bugs make it to Earth."
"I don't envy you that task, Shiloh. It'll be a difficult tradeoff to make. What carrier do you want to take?"
Shiloh didn't hesitate. "Midway. We'll cram it with supplies for a long mission by taking less than its full complement of fighters. Now that I think about it, I don't want any freighters tagging along. They'll only slow me down."
"What about cannon-equipped raiders?" asked Howard.
"I'd like to take all twelve that are ready now and bring the rest up to the rally point as they become available." Shiloh was surprised when Howard shook his head.
"No. I can't authorize that, Admiral. With all our Mark 6s sent forward, Earth would be absolutely defenseless against any surprise bug incursion if all the cannon-armed raiders were with you. We know that the old timeline is no longer in play, so we can't be sure that the Bugs won't leapfrog past the Sogas and find us while you're still holding them at Omega54. I'll let you take six now and one of every two additional raiders as they're converted."
Shiloh was tempted to argue that a half-hearted effort to defend Earth where it should be defended, namely around someone else's planet, was bound to get half-assed results. The term 'defeat in detail' came to mind. He was just about to say so when he noticed that Howard had a strange look on his face. He seemed to be staring off into infinity. Is he...
"Are you having a vision, Admiral?" asked Shiloh.
Howard didn't seem to hear him but suddenly blinked furiously and took in a deep breath. "So that's what you've been experiencing," he said.
"You've had a vision." It was less a question than a statement.
Howard nodded. "I saw you tell me that it was a good thing I let you have all available cannon-armed raiders because you needed them at Omega54, and no bug ships showed up here up to that point. Okay, obviously we sent this vision back here now to persuade me to change my mind. Well, I'm persuaded. You take all the converted raiders. How soon can you be on your way, Shiloh?"
"That depends on how long it takes to load Midway with supplies, Sir. Iceman and I also have to figure out what instructions to send to all the monitoring raiders so that they send message drones to the right place. I'll say 48 hours, but I'll try to make it 24."
"Very good. You let me know if you encounter any logistical delays, and I'll kick some ass for you. Brief me when you and Iceman have all the details figured out."
"Affirmative, Admiral." Shiloh gave Howard a quick salute, which the CSO returned, and then left the room.
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# Chapter 23
––––––––
The Flag Bridge on Midway was unnaturally quiet. Shiloh knew it was the tension of expectation. Iceman had received a very detailed vision pinpointing the exact time and coordinates of the arrival of VLO #3, which for purposes of identification was designated as Sierra1. Midway had taken up her planned position ten light seconds from the Sogas home world. Her complement of fighters were flying escort just in case they were attacked by bug attack craft, although the vision had not indicated any such event. Iceman was monitoring the com channels in his capacity as Shiloh's Deputy Fleet Commander. Midway's Helm and Tactical systems were under control of Stoney. 3rd Fleet's other assets consisted of 12 cannon-armed raiders, with another 99 raiders armed only with their internal lasers. One additional raider also carried the sole Mark 6 attack drone that luck had made available to Shiloh before Sierra1 was due to show up. The raider force was under the command of Vandal.
He and Iceman had come to a consensus on how to handle the coming incursion. They would take a page from Casanova's campaign several timelines back and use recon drones to pinpoint the exact point where the target would emerge from Jumpspace. They'd send that information back in time via the RTC, and then lay an ambush to hit the target with the Mark 6 warhead so fast that the Bugs wouldn't have a chance to report back what was happening. Sierra1 would go 'off the air' and leave a big mystery for the Head Bug to deal with.
Shiloh checked the chronometer. There were seconds left before the target arrived. Vandal would control the battle, and Shiloh and Iceman would find out the results 25 seconds after it happened. The countdown hit zero, then began to count up again. When it reached 25 seconds, the main display showed a bright flash that quickly died down.
"Right on time and on target, CAG. Vandal has pulled it off," said Iceman.
"I never doubted it, Iceman. As long as we keep getting visions about incursions we can handle, we'll keep giving the Bugs a bloody nose. It's the surprise arrival of bug ships that I'm worried about. No vision means we can't stop them here, and we'd have to pull back as planned."
"Affirmative, CAG. Shall I call the RTC up to the Flag Bridge now so that we can send the information back?"
"Might as well get it done sooner rather than later, Iceman."
No sooner had Iceman sent back the vision than he received the next one. Once again there were detailed time and space coordinates. Sierra2, as it would be designated, would arrive in ten days.
This one would be a little different. Iceman knew the timetable for production of Mark 6 warheads, and the next one would not be available that quickly. They would have to use the GLB cannon. The ambush was set up very carefully. Non-cannon-armed raiders would be arrayed in a 10 by10 matrix, curved slightly to give each one a good angle on the target coordinates. The cannon-armed raiders would be placed 90 degrees around to the side so that no raider was in danger of being hit by any other.
When the time came, Midway was once again at a safe distance. Vandal would command the ambush force. This time Midway was receiving enhanced video and tactical information directly from multiple raiders. Shiloh would still get the delayed data, due to the distance, but he'd be able to see the battle visually and up close.
This time the countdown was calibrated to take the light speed lag into account. When it hit zero, a sphere appeared on the Flag Bridge's main display. Immediately there appeared dozens of bright pinpoint flashes. Laser hits from the 100 raiders. The status of all those raiders on the sidebar started to reflect enemy return fire as raider designations began turning from green to red. This had been anticipated. The non-cannon-armed raiders were performing their mission of distracting the Bugs, while the cannon raiders fired their GLB weapons as fast as the power charging cycles would allow.
The first GLB volley did not hit a power unit. That much was obvious from the fact that the ship didn't explode immediately. The second volley was more successful. The explosion was quite violent and for Shiloh, very satisfying. It had taken four point three seconds to destroy Sierra2. The bad news was that 13 raiders had been destroyed too.
As soon as Iceman received all the necessary data, he sent the vision back and once again received the next one almost immediately. That confirmed a pattern. If future ambushes were successfully completed, Iceman would get the information very quickly. If he didn't get a vision, then the next ambush was either a failure and the entire 3rd Fleet was destroyed before it could send a warning back, or the Fleet had been withdrawn before the next bug incursion. Figuring out which was Shiloh's dilemma.
What he and Iceman didn't know was whether Sierra2 had time to send any kind of signal back to the relay points. If organic beings were in control of the mothership, then the answer probably was no, but if the Bugs were using some kind of AI capability, then those could react fast enough to send a warning back.
The good news about the next ambush was that 3rd Fleet would have another Mark 6 warhead to use. Sierra3 was destroyed within a fraction of a second after emerging from Jumpspace 13 days later. This time there was no new vision. The moment Shiloh had dreaded was now here. Did 3rd Fleet stay or pull back. The mission was to convince the Bugs that Omega54 was THE home world of the spacefaring race that had attacked the Bugs at Beta1. The longer he could keep the Bugs' attention focused here, the longer it would take them to get to Sol, and the more time Valkyrie and Casanova would have to get the timeship repaired. The latest update from Howard was Valkyrie's estimate that repairs would be completed in another 150 days, more or less. Five months. Shiloh had to keep the Bugs away for five whole months, and he had no idea how to do it.
At least he was getting some reinforcements. Another three cannon raiders had arrived. The next and second last Mark 6 warhead was scheduled to arrive in nine days. He asked Iceman for his thoughts on what to do next.
"If we pull our raiders back to Midway's position, then 3rd Fleet will be concentrated in one place. I recommend we wait until the next Mark 6 shows up, or until the next insectoid incursion, whichever comes first. If the Insectoids get here first and in dangerous strength, then we immediately micro-jump away and leave messages with the message drones to redirect our reinforcements to another location in this system. By keeping the Fleet here, we can continue to monitor insectoid activity first hand and react accordingly, CAG."
Shiloh shook his head, not that he disagreed with Iceman. On the contrary, Iceman's suggestion made a lot of sense. What Shiloh was shaking his head at was the whole situation. He knew from information about old timelines that once bug reinforcements started arriving, a trickle could turn into a flood VERY quickly. There was always the horrifying possibility that the Bugs would not just come here but also send ships past the system to scout for additional breeding planets. They might arrive at Sol while 3rd Fleet was still active here. In fact, Shiloh's biggest worry was that the Bugs were smart enough to realize what humans were doing and were sending just enough reinforcements to keep the ambushers' attention focused here. In other words, they might be using the same strategy against him that he was trying to use against them. That this possibility involved the deliberate sacrifice of multiple motherships was unthinkable to Shiloh, but who knew what kind of logic these damned Bugs used.
"We should be hearing from Beta1 directly, any day now, right Iceman?"
"Roger that, CAG. Now that they've gotten the word to send duplicate messages here as well as Sol, we can expect to get a steady trickle of news from there, starting soon."
"Okay, I want you to attempt to put yourself in the position of the bug Leader in one of the relay systems. You've lost contact with ships at Beta1 and now at Omega54. Losses at Beta1 have stopped. Let's assume that Sierra2 managed to send some kind of signal back before being destroyed. What conclusions would you, as the bug leader, make from all that?" asked Shiloh.
As always, Iceman's answer was immediate. "I would assume that Beta1 was the outer edge of a volume of space monitored by a spacefaring race that had technology sufficient to destroy motherships, and that Omega54 was either the home world of that race or a major colony world that was worth conducting defense in strength. Given that assumption, I would call in reinforcements at a rally point other than Beta1, just in case the ambushers were monitoring it. I would also arrange for some activity at Beta1 in order to let the enemy think that's where the rally point was."
Shiloh nodded. That made perfect sense. The worst mistake he could make now was to underestimate the Bugs. They had shown time and time again their ability to do the unexpected. He had to guard against taking the obvious at face value. Iceman's logic had crystallized his thinking, just as he had hoped it would.
"Okay, this is what we're going to do. For now we wait. If we hear from Beta1 first, we'll re-evaluate the situation based on that information. If the Bugs get here first, we'll watch them from a safe distance as best we can until we see their next move. Let's bring 3rd Fleet back together here around Midway now. I want a message drone sent to Sol with our latest info and plans, including your assessment of enemy strategy. I'll prepare a report to be carried with it. Thanks for your input, Iceman. It helped me wrap my brain around the situation."
"My pleasure, CAG. I too enjoyed our discussion, as brief as it was. Human thinking is sufficiently different that it makes for a nice change of pace from the kinds of exchanges I have with my brothers. So thank you, CAG. Let's hope we have lots more discussions like this one."
"Amen to that, Iceman," said Shiloh.
The first message drone directly from Beta1 arrived four days later. There were now five VLOs apparently holding position there. They had not all arrived at the same time, and that suggested that Beta1 was being used as the rally point or maybe as one of the rally points. In any case, if all five moved up to Omega54, there was no way that 3rd Fleet could stop them all without suffering huge losses. The tiny bit of good news was that the AI in command at Beta1 had decided to hold back the Mark 6 attack drone, which had already been on its way to Beta1 when 3rd Fleet had left Sol. Shiloh had assumed that it had been used. Vixen, however, had come to the conclusion that one more mothership destroyed at Beta1 would not significantly change the outcome, while one extra Mark 6 warhead might make a difference defending Sol. It was time to strategize with Iceman again.
"What do you recommend we do now, Iceman?"
"I recommend we wait here for another five days. With a little luck, the next Mark 6 warhead will arrive before the Insectoids do. I also recommend we send a message drone back to Vixen at Beta1 instructing him to send his Mark 6 drone back to Sol."
"Why Sol and not here, Iceman?" asked Shiloh.
"Because by the time the message gets there and the drone is sent back here, at least 44 days will have passed. It's highly likely that 3rd Fleet won't be here that long. Sending the warhead back to Sol directly is more likely to get it there before the Insectoids arrive. With the other warhead due to arrive here in five days, which we can take back ourselves, plus the one or maybe two more that The Old Man still has enough platinum for, that will total three or four Mark 6s that might be desperately needed at some point, CAG."
"Logical as usual, but I'm worried that while we're twiddling our thumbs here, the Bugs are leapfrogging past us."
"Is that just a colloquial expression, or are you humans really twiddling your thumbs, CAG?"
"JUST an expression, Iceman." Iceman must have detected the exasperation in Shiloh's voice.
"We do understand the seriousness of this situation, CAG, but some of my brothers insisted I ask. To address your concern, yes the Insectoids may be leapfrogging past us, but we have no way of knowing whether they are or not, and that possibility was just as valid four days ago when you and I had our previous strategic discussion. If I'm wearing the Supreme Insectoid Commander's hat again, CAG, I would not see the logic of spreading my VLOs if I'm convinced that this system is a major population center for the spacefaring race that attacked at Beta1. Everything we've done up to this point was done to reinforce that idea. Are you proposing we abandon that strategy now, CAG?"
"No. I'd like to continue that strategy, but I'm not sure of how best to do that. Any ideas?"
Iceman was silent for almost a full second. The silence was so unexpected that the hairs on the back of Shiloh's neck stood up.
"I've just had another vision, CAG. The timing is not coincidental. We didn't get this earlier because it has to do with your last question, and if the information had arrived sooner, it would have seemed suspect. If this star system were indeed our home system, we would fight to the last human and the last AI correct, CAG?"
"Yes. Gone on."
"It would be logical to assume that the Insectoids have had enough experience attacking the home worlds of spacefaring races by now that they would recognize that kind of desperate defense, and they would also recognize a defense that is not desperate. Would you agree with that, CAG?"
That sounded ominous, but Shiloh couldn't argue against it. "Yes."
"Then we have to arrange for a defense that appears to be desperate when the Insectoids attack with five motherships in five point five days time, CAG. This is how it will have to be done. Twenty-nine brother AIs will volunteer to fight a rear guard action. They will each control their own raider, plus two more remotely. When the five insectoid ships arrive, all 87 raiders will micro-jump into laser range, and fire on the attack craft carrying insectoid soldiers down to the planet. All the raiders will eventually be destroyed, but the magnitude of the defense will convince the Insectoids that this planet is worth the sacrifice. They will assume they've located and neutralized the source of the attacks on Beta1. We calculate that there is a high probability that when the breeding potential of the Sogas home world has been exhausted, which may take many weeks, the five VLOs will go their separate ways. That should slow down the advance enough that the timeship will be repaired before the Insectoids discover Earth."
"Can we find 29 volunteers to do that?" asked Shiloh quietly.
"Already done, CAG. That's why I took so long to respond. My brothers and I have already discussed this plan, and we have our 29 volunteers."
Shiloh felt a lump form in his throat. What noble creatures these AIs are, ready to step forward and commit to the ultimate sacrifice without even being asked.
With his voice betraying his emotions, Shiloh said, "I'm at a loss for words to express the depth of the gratitude that I feel to our volunteers. On behalf of all humans I thank you all."
"They have the word, CAG," said Iceman.
Shiloh nodded, letting his head drop lower in contemplation. Suddenly he jerked up.
"Wait a minute! If two thirds of the raiders are going to be piloted remotely, how are we going to get their AIs off? We didn't bring any shuttles along on this mission in order to maximize space for supplies."
"There is a procedure that will work, but it'll be tricky, CAG. Each raider will have to move close enough to Midway so that the nose section containing the pilot will be inside the ship's fighter launch/recovery bay. Humans wearing spacesuits will then be able to approach the raider and extract the pilot manually. When that's accomplished, the raider will back out under auto-pilot, and the next one will move in."
Shiloh groaned mentally. Raiders might be small compared to Midway, but they were huge compared to the ship's fighter launch and recovery bays. Even a tiny error in judgment or maneuvering would cause a collision that could seriously damage Midway and kill some of the crew. But he couldn't think of any other way of getting 58 AIs off the doomed raiders.
"It sounds like a very time consuming procedure," said Shiloh.
"Roger that, CAG, which is why we should get started on it right now."
"Understood. I'll give the necessary orders. I'm not going to get much sleep over these next five and a half days, am I Iceman?"
"No, CAG, you're not."
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# Chapter 24
––––––––
The transfers of the AI pilots went off without accidents and were finished in time, much to Shiloh's surprise. Midway now had one Mark 6 drone on board. A message drone was on its way to Beta1 ordering Vixen to send his Mark 6 drone back to Sol via a fighter. Another message drone was on its way to Howard to advise him that 3rd Fleet would be pulling back after the upcoming battle. The battle itself was only observed in the abstract. None of the defending raiders risked giving the game away by transmitting anything directly at Midway. Long range observation by recon drones told Shiloh when the battle started and when it was over. He was just as glad that he couldn't see the volunteers die with his own eyes. Once all the raiders had been destroyed, the five VLOs, which Iceman assured Shiloh were NOT the same five observed by Vixen at Beta1, proceeded to exploit the breeding potential of the Sogas home world. With half a dozen fighters loaded with message drones and left behind to continue long range surveillance, Midway and the cannon raiders made a careful exit and headed for the Avalon Colony system, the former human colony that was closest to Sogas space. Time left until the timeship repairs were complete was now 140 days.
Upon arrival at the Avalon Colony system, a quick check of the recon drones stationed there revealed no sign of any insectoid incursions or scouting. It wasn't long before a steady stream of message drones started arriving as all the various monitoring raiders and fighters received word that 3rd Fleet had relocated further back. No reports of insectoid movement were received for the next three weeks. The five VLOs at Omega54 were still there. Twenty-two days after arriving in the Avalon system, a message drone from Beta1 arrived to report that the five VLOs previously reported there had suddenly moved as a group on a trajectory that seemed to point to the Sogas home world system. The timing of the move turned out to be just right. There was exactly enough time after the battle for information to be sent back to the alpha relays and on to Beta1.
Shiloh and Iceman considered the news to be both good and bad. It was good that information on those other five VLOs would now be coming in faster since they'd be closer, but bad that there were now ten VLOs 'digesting' the home world of the Wolf-people. That might mean that they'd be finished exploiting it sooner and therefore moving on sooner too.
It wasn't long before reports started arriving of insectoid scouting activity in Sogas colony systems. Some of those colony systems had a fair bit of metal in the form of mining, refining and manufacturing capacity, plus a limited amount of breeding potential too. As the days turned into weeks, Shiloh and Iceman observed how the Insectoids carefully explored and then exploited the entire Sogas empire. Every colony was visited by one of the ten motherships, even if only for a few hours. Anything made of metal was salvaged. Careful reconnaissance of colony worlds after the motherships left showed no sign of any survivors.
When all Sogas colonies had been or were in the process of being exploited, the motherships began to head off to unmonitored destinations. The first five to leave headed in directions that would not discover human worlds. Number six however might discover a human colony world if its scouts ranged out widely enough from the mothership's base course. Shiloh knew from one of Howard's updates that all human colonies on this side of human space were ordered to stop using any kind of electronic communication that might be detected by insectoid scout ships. Colonists were also advised to abandon their settlements and hide in nearby forested areas or caves. Even if the colonists all did that, and that was a big if, the buildings, cultivated land and some of the larger machinery would be impossible to hide. Any insectoid scout in orbit would be able to determine that there was some kind of intelligent presence. It was a long shot, but if the scout didn't detect any EM emissions from a distance, it might not bother for a closer look.
With at least one VLO now a potential threat to discover humans as a spacefaring race, Shiloh made the decision to pull 3rd Fleet back to Sol, leaving behind several fighters to make sure that message drones from other systems were redirected back to Sol.
By the time 3rd Fleet arrived back at Earth, there were only 59 days left until the expected completion of repairs to the timeship. Shiloh was amazed at what Howard had managed to accomplish while he was gone. Somehow Howard had gotten the OC to 'unofficially' persuade wealthy individuals and corporations to buy and donate platinum to the Space Force. It wasn't a huge amount, but it was enough for another four warheads, bringing the total that would be available in a few weeks to seven. Over 100 X-ray laser drones were now deployed in Earth orbit. Production of GLB cannon parts had resumed now that all time machine parts were finished being built. Five more cannon-armed raiders were waiting to be added to the fifteen brought back by 3rd Fleet. There was also a project under way to adapt the GLB cannon designed for raiders so that it could be mounted on Dreadnought's hull in place of its laser batteries. With the much higher power output from Dreadnought's ZPG units, any GLB cannon would have a much longer effective range.
The next vision didn't take long arriving. Shiloh had literally just finished briefing Howard on 3rd Fleet's mission when both men received a message from Iceman on their implants. Iceman got right to the point.
"An insectoid mothership will arrive in 13 days. I have just received the precise time and jump coordinates. We'll be able to ambush it with a Mark 6, but taking out that VLO will tip off the rest of the Insectoids that something is going on in this star system."
Howard sighed. "So...they've found us. I guess I knew this would happen eventually. At least after this coming attack, I can make a legitimate case to the Grand Senate for commandeering a lot more platinum. The problem will be in converting it fast enough to make a difference, but we have to try. Shiloh, you'll be the Field Commander of course, but Iceman will handle the actual ambush. You two put your heads together, and if there's something you need, let me know, and I'll see what I can do. Was there anything else we should know, Iceman?"
"Negative, Admiral. I believe we'll have this next attack well in hand. It's the ones after that that are uncertain."
"Yes, even we humans have figured that out by ourselves," said Howard.
Shiloh said nothing. Iceman was right of course. This next one would be the easy one. With that one destroyed, there would be 46 days left until the timeship was operational. Somehow Space Force had to hold the Bugs off that long. An idea occurred to him, but he waited until he was out of Howard's office and someplace where he could have a private conversation with Iceman.
"CAG to Iceman."
"Go ahead, CAG."
"Did your vision confirm that we should use a Mark 6 on this next VLO?"
"Negative, CAG. I assumed we would since we'll have four of them available by then. Are you contemplating using the cannon raiders instead?"
"Yes I am, and here's why. Right now the Bugs probably think that the Sogas were responsible for the losses that happened so fast that the motherships weren't able to transmit any information. Having a ten klick mothership vanish that quickly has to be a big concern for the Bug Leadership. If we use cannon raiders, we should be able to destroy one mothership relatively quickly. Would you agree?"
"Roger that, CAG. I see where this discussion is going. You're about to suggest that holding back our Mark 6s allows us to maintain the deception that it was the Sogas who had the mystery weapon."
"Yes. We know the Bugs will send reinforcements anyway, but if they think we also have that mystery weapon, it's my belief that they're likely to call in a lot more reinforcements. If they're prone to overconfidence, then we should try to exploit that by not tipping our hand too quickly."
"Logical, CAG. It has the added benefit that having multiple Mark 6s on hand as a reserve may tip the scales in our favor when they arrive in numbers that our cannon raiders might have difficulty in overcoming."
Shiloh nodded enthusiastically. "Exactly! And one more thing, the more practice we get with the GLB cannons, the more likely we'll be able to pinpoint the best areas of a bug mothership to aim at."
"I like your thinking, CAG. That kind of devious thinking is unlikely to have occurred to us. Will you be flying your flag on Midway again during the ambush?"
"Unless Dreadnought is ready by then," said Shiloh.
"It won't be. By the way, I've just received a lasercom message from Valkyrie and Casanova. They want me to ask you if you'll take advantage of these next 13 days to renew your relationship with Commander Kelly? What should I tell them, CAG?"
Shiloh laughed and said in what he hoped Iceman would sense as a playful voice, "Tell them to mind their own business!"
"Message has been sent, CAG. Was there something else you wished to discuss now?"
"No, nothing else now. CAG clear."
* * *
When the 13 days were up, Shiloh was once again on Midway's Flag Bridge, with Iceman on board as his Deputy Fleet Commander. Midway was off to one side of the expected emergence point at a distance of almost a million kilometers. Twenty raiders armed with GLB cannon were less than ten thousand klicks away from the emergence point, and when the VLO arrived, those raiders would actually be behind it. One hundred and thirty-three laser-armed raiders were halfway between the emergence point and Earth. They would take care of any attack craft that the bug ship might be able to launch before it was destroyed by cannon fire. It was hoped that the Bugs would see the raiders blocking their way to Earth and assume that the beams causing the pinpoint internal damage were coming from them. With a little luck, the Bugs might not scan behind them at all until it was too late.
The Flag Bridge was deadly quiet as the countdown approached zero.
This will be interesting, thought Shiloh. He was anxious to see how the new firing plan for the GLB cannon worked out. Instead of one massive volley of simultaneous shots, each of the 20 raiders would fire in sequence 25/100th of a second apart. By the time all 20 had fired, the first one would be recharged and could fire again. The idea was to try to identify which shot hit the power unit, and thereby narrow down where power units were likely to be located for use in future battles.
The countdown was synchronized to take into consideration the light speed lag of being a million kilometers away, and as soon as the countdown hit zero, a computer-enhanced image of the VLO appeared on the Flag Bridge's main display.
––––––––
"Titan's team is firing," said Iceman unnecessarily.
Shiloh smiled. If he didn't know better, he'd interpret Iceman's superfluous remark as a sign of jitters. Was it possible that AIs could get anxious the same way some humans did? He'd have to ask Iceman that question after the battle.
The VLO exploded with a satisfyingly large blast. Shiloh looked at the battle chronometer. Six point seven seconds after emerging from Jumpspace. No sign of any attack craft to mop up.
"I wish they would all be this easy," said Shiloh.
"Roger that, CAG. I should use the RTC now and send the vision back. It'll be interesting to see if I get another one right after that."
In fact, he did get another vision right away. The next attack was 18 days away, and this time there'd be three VLOs, not one.
* * *
Eighteen days later it was Shiloh who was clearly nervous. Careful analysis of the shot that seemed to have triggered the power unit overload indicated that wherever else the power units might be, the center of the sphere apparently held at least one of them. That actually made sense to Shiloh, considering that these 10 km spheres started out as much smaller spheres that could maneuver and jump and therefore had to have at least one power unit to begin with. With that data, the 21 cannon-armed raiders that were now available would split their fire evenly among the three targets, and all would fire simultaneously at the center of the spheres from slightly different angles.
In addition, there were now 151 laser-armed raiders riding shotgun between the Earth and the emergence point. One hundred and forty-nine X-ray laser drones were deployed in geosynchronous orbit. They would be held back as the last ditch defense against bug attack craft attempting to land bug soldiers. The other aspect that made Shiloh nervous this time was that Midway would be closer to the action. A LOT closer. Less than 300,000 kilometers. The time lag due to distance would be one second each way, compared with more than three seconds for the previous battle. That was something that Shiloh had insisted on, over Iceman's objections that Midway and therefore The CAG would be risking battle damage and injury. But with three VLOs to overcome, Shiloh figured the battle was likely to last long enough that his and Iceman's ability to issue orders quickly might make the difference. While the logic of the move closer was undeniable, that didn't prevent him from reacting emotionally to the perceived danger. He hadn't felt that anxiety before other battles, but that was because he didn't know exactly when those battles would start. This time he did.
When the countdown hit zero, three motherships emerged from Jumpspace exactly where the vision had indicated. The laser-armed raiders began firing once again to draw the attention of the enemy, while the 21 cannon-armed raiders fired from the opposite side. As the seconds started to accumulate, Shiloh noticed from the main display sidebar that he was losing some of his laser-armed raiders. Two GLB cannon volleys had fired now with no corresponding explosions in any of the VLOs. As soon as the third volley fired, one of the cannon-armed raiders disappeared from the display.
"The enemy has detected Vandal's group, CAG," said Iceman.
Shiloh had already figured that out for himself, and that wasn't good. If they lost too many of the cannon-armed raiders, future battles might become unwinnable, and Shiloh's gut told him that there would be at least one more battle before the timeship was repaired.
"The enemy is launching attack craft now," said Iceman.
Shiloh bit his lip. By prior arrangement, Titan's group of laser-armed raiders would shift their fire to the attack craft when they got within a certain range, but that would allow the motherships to shift all of their fire to Vandal's group, which was now down to 18 raiders. Shiloh made a decision. Midway was being escorted by 55 F2 fighters.
"Order Gunslinger's fighters to micro-jump into attack range of the enemy attack craft and take them out. Titan's raiders are to continue to target VLO laser batteries."
"Orders have been sent and acknowledged, CAG. Fighters have jumped."
Shiloh checked the battle chronometer. Thirteen seconds now since the battle began. Vandal's cannon-armed raiders would fire another volley within two seconds. Shiloh crossed his fingers. Apparently that made the difference. As soon as the next volley fired, one of the VLOs exploded. A quick check of the sidebar data showed there were 16 cannon-armed raiders left. Analysis of enemy laser fire showed a steady and now steepening drop in the number of laser blasts coming from the remaining two VLOs. Titan's group was obviously having some success in knocking out enemy laser batteries. The next GLB cannon volley was coming up fast.
"Explode, you bastards," whispered Shiloh. A second VLO exploded. Shiloh smacked his right fist into his left palm in triumph and relaxed. The momentum of battle had shifted in their favor. Vandal's group was now down to just 12 raiders, but after five volleys, those raiders now had a lot of data about what part of the target NOT to aim at, and they adjusted their aim accordingly. The remaining VLO was clearly damaged too. Estimates of firepower from an undamaged 10 km mothership were 66 laser batteries. The Battle Computer was now estimating that the 3rd VLO was down to less than 10 batteries still functioning. That number was dropping fast.
Suddenly the unexpected happened. The 3rd VLO micro-jumped away. Shiloh cursed long and loud. That was one VLO that he was sure they'd now have to face again, and it would almost certainly be in the company of reinforcements. After analyzing and repairing the damage caused by the GLB cannon, the Bugs also would have a much better idea of the kind of weaponry they faced. If he were the Bug in charge of strategy and tactics, he'd wait until he could organize a massive force of at least six motherships before attacking again. He had to remind himself that the battle wasn't over yet. There were over 200 bug attack craft attempting to get to Earth, but Titan's and Gunslinger's forces were now concentrating on them. None of the attack craft got close enough to Earth to warrant using any of the X-ray laser drones, however additional raiders and fighters were lost. Overall, Shiloh would have to categorize this battle as a major but not decisive victory, with significant losses to friendly forces.
When the battle was finally over, Shiloh told Iceman to send back the vision. Both of them were expecting Iceman to get another vision about the NEXT battle, but he didn't. Thirty seconds later Iceman received a lasercom transmission from Valkyrie at the shipyard asteroid. She had gotten the vision containing all the targeting data for the next battle. Nine VLOs would emerge from Jumpspace in 23 days.
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# Chapter 25
––––––––
Shiloh didn't know what to think about that news at first. Why didn't Iceman get the vision? Would something happen to Iceman during the next battle? There was nothing in the vision to indicate which side would be victorious. The fact that Valkyrie would continue to exist long enough after the battle to send the vision back to herself was a positive, but she would still need at least another five days after that battle to finish repairing the timeship.
"Why didn't she include information about the outcome of the battle? What tactics were used?" asked Shiloh in an exasperated voice.
"The explanation that makes the most logical sense is that the results of the battle are not as good as we would like, but there's no consensus on what tactics would have worked better. Therefore she's leaving the choice of tactics up to us, without injecting any biases into the timeline, CAG," said Iceman.
Shiloh grunted his acknowledgement, then said, "How many Mark 6s will we have in 23 days time?"
"Minimum of six with a possible seventh. The power charging schedule for the seventh warhead will be tight. Any delay will make it unavailable, CAG."
"Naturally it would be that way for this critical battle. If we had seven, we'd only have to overpower two with cannon fire, and that's doable, but with six warheads, we'd be faced with three VLOs and less than half as many cannon raiders as in this last battle."
"My brothers and I will carry out simulations of every tactical plan we can think of, CAG. With twenty-three days to work with, we'll come up with something," said Iceman.
"Very good. While you're doing that, let's get Midway back into low orbit. I'm sure The Old Man wants me available for the after-battle media frenzy."
"Roger that, CAG."
* * *
Four days later, Howard called Shiloh into his office. Shiloh was surprised to see Commander Kelly there. Her greeting was such that he couldn't detect any hint that she might be aware of the relationship they had in the old timelines. If she knew about it, she was hiding it very well. With the usual pleasantries out of the way, Shiloh sat down facing Howard and waited. Howard, as usual, got straight to the point.
"Commander Kelly tells me that the SPG in conjunction with all the other AIs have run hundreds of combat simulations of the upcoming battle, and our only chance of coming out on top depends on the seventh Mark 6 warhead being available. And even then the victories, if you want to call them that, are almost as bad as a defeat from the point of view of losses on our side. I wanted to hear your thoughts on what we should do about that, Admiral."
Shiloh nodded. Iceman had kept him informed periodically of the results of the simulations. Losing half their cannon raider force made a huge difference. If they knew precisely where to aim the GLB cannon on the first volley, the outcome would be a clear victory, but there was no way of knowing if the targeting data from the last battle was valid for the next one. As Iceman had pointed out, the VLOs didn't have to be all designed exactly the same way, and even if they had been, how do you define the front, back left side and right side of a sphere? There's no obvious giveaway of its internal orientation. And as for the seventh warhead, the best they could realistically hope for was to be able to load the damn thing onto a fighter minutes before the battle was due to begin. Even minor delays would make it too late.
"I regret to say that right now I don't have any recommendations to make. We still have 20 days. Perhaps the AIs will think of something new to try before then, Sir."
Howard sighed. "I can't blame you for not being able to pull the proverbial rabbit out of your hat when Kelly and I can't either, but the Commander does have an interesting proposal." He looked at her and nodded.
She turned to Shiloh and said, "I've watched the AIs chasing their tails over several dozen tactical plans. They keep going back to the same plans and retesting them over and over again. They're limited by their ability to think logically, which means they can't come up with a counter-intuitive idea that defies logic. If we're going to win this next battle, I think we have to step back from the AIs and use human intuition and inspiration to generate alternatives the AIs can't conceive of on their own. To that end, I'm proposing setting up an Ad Hoc tactical team composed only of humans. The team's mission brief will be to brainstorm unconventional tactical plans that the AIs can then simulate."
Shiloh was impressed. She was right of course. Humans were relying too heavily on the AIs and had forgotten that their greatest strength, their ability to think logically and quickly, was also their biggest constraint.
"I think that's an excellent idea, Commander," said Shiloh.
"I do too!" exclaimed Howard. "That's why I'm appointing you to the team, Shiloh. The two of you worked well together back in the early days of the SPG, and you're the natural choice for the team considering your combat and tactical experience. But unlike last time, I'm not putting the Commander in charge. I'm tossing this hot potato to you, Shiloh. I don't care what it takes. Just get results. If you want specific people added to the team, I'll get them for you. Any questions?"
"No, Sir."
"Good. You're both dismissed," said Howard with a wave of his hand.
Shiloh and Kelly retreated to the Officers' Mess and spent the next hour making a list of people that they agreed should be on the team. Luckily all of them were somewhere in the Sol system and could be back on Earth within a few hours. While they waited for the team to assemble, the two of them talked about how to conduct the brainstorming sessions. Kelly convinced Shiloh that no idea, no matter how bizarre, should be judged and excluded right away. Rejection would only cause team members to hold back for fear of losing credibility in the eyes of the other members. Not only that, but bizarre idea A might inspire workable idea B via some unconscious connection. When the team could no longer come up with any new ideas, they would go back and review each suggestion in a non-critical way to see if weaknesses could be overcome rather than used as excuses to dismiss the idea. When all the ideas had been carefully reviewed, they would be presented to the AIs for simulation.
By the next day the entire team had been assembled. Shiloh commandeered one of the conference rooms in the Space Force HQ building and ordered food and drinks delivered at regular intervals. The brainstorming and review was complete 11 hours later, and Shiloh called a halt so that everyone could make their own arrangements for dinner. The plan was to reconvene at his quarters two hours later. By then, the AIs would have run through all the unconventional ideas enough times to determine the most likely outcome of each. Shiloh told Iceman to hold the results until the team was back together in his quarters. During the two hour dinner break, he ate in his quarters while listening to classical music. When the break was over and the team was together again, Shiloh asked Iceman to give them the results of the simulations.
The results were bad. None of the unconventional tactical ideas had panned out. Iceman spent fifteen minutes explaining why some of the results were the way they were. The team was unable to find any flaws in the AIs logic. Unless the Bugs acted irrationally and stupidly, Humanity was in dire danger, and the timeship might be discovered before it was repaired. The news, combined with fatigue, caused expressions and moods to crash. It didn't take Shiloh long to realize that in their present frame of mind, the team wasn't going to accomplish anything in the next few hours, so he told them they were done for the day. They should go get some sleep and come back at 0800 hours the next morning.
Kelly was the last one to get up and head for the door. When she got there, she turned around and looked him directly in the eyes with a thoughtful expression.
Before Shiloh was even aware of the thought behind the words, he said, "You don't have to leave you know."
A small smile made an appearance. It was very clear that she knew exactly what he was referring to. After a short pause she said, "I'm aware of what we had in the other timelines, and I'm tempted to say yes, but given the dire nature of our situation, I think the smart thing to do would be to avoid distractions and stay focused on the mission. Don't you?"
"Yes, when you put it that way, I have to agree," said Shiloh.
She turned back towards the door. When it slid open she stopped and without looking at him said, "Ask me again if we're both still alive after the next battle." Before he had a chance to reply, she stepped across the threshold and walked quickly down the corridor.
That night Shiloh had a peculiar dream. It was the kind of dream that felt different somehow. He was running and had the distinct feeling that something was chasing him. Ahead of him was the edge of a cliff. He looked to the right and saw six white and black horses. No, wait...they weren't horses. They were zebras, and they were galloping parallel to him. Something made him look to his left, and he saw six more zebras running parallel as well. Shifting his gaze ahead, he saw the cliff edge coming closer fast. Out of the corners of his vision he could tell that the two groups of zebras were coming closer to him. In fact, he was quickly hemmed in by them. With the edge now only a few meters in front of him, he had nowhere else to go. The zebras prevented him from veering off to the side, and the intensifying feeling of being chased precluded stopping. When he and the zebras reached the cliff edge he jumped and saw that he was falling into a wide black hole that seemed to go on into infinity...and then he woke up.
He sat up in bed and tried to recall the memory of the dream images. Twelve zebras...was there something important about twelve zebras? Was it the number that was important or the fact that they were zebras and not horses that was significant? Twelve...zebras...twelve...zebras. After repeating both words several times he realized that the reverse order of 'zebras...twelve' sounded familiar. Then the answer hit him. Zebra12! That was the star system where he had fooled the Sogas into believing that they'd won a victory using decoy drones!
Quickly activating his implant, he said, "CAG to Iceman." The response was immediate.
"Iceman here. Can't sleep, CAG?"
"I just woke up. How many Mark 3 decoy drones do we have available now, Iceman?"
"Twelve, CAG."
Shiloh briefly wondered if that was just a coincidence or a moment of synchronicity. Time to ponder that later.
"Were the decoys included in any tactical scenario you evaluated?"
"Affirmative, CAG. Several, in fact. The results were only marginally better than without the decoys."
"Damn!" Shiloh had been certain that adding decoys to the tactical mix would give them the edge they needed, and the reality hit him hard. "Alright, I should have known that you would have included them. I'll try to go back to sleep. CAG clear."
The team spent the next day looking at all the scenarios simulated by Iceman and the AIs for clues as to what factors were common to the relatively better ones. They also examined specific simulations that generated unusually good outcomes. Iceman explained that all the simulations relied on probabilities for a variety of factors. How likely was each cannon volley to hit a power unit for each target? How likely was it that each Mark 6 would work perfectly and accurately? How likely was it that the Insectoids would react in specific ways? Those probabilities themselves were based on limited and in some cases no data at all, and therefore prudence dictated that the AIs guesstimate conservative probabilities to avoid being overly optimistic. But even with the probabilities assumed, some simulations gave good results just from shear good luck.
By the end of the second day, Howard showed up to check on their progress. Shiloh had to admit that they were no closer to a solution now than they had been when they started. Howard said nothing but gestured for Shiloh to follow him out into the corridor where they could speak without the rest of the team overhearing them.
Looking around to make sure no one else was within earshot, Howard said in a low voice, "What I'm hearing is that unless we get very lucky, we're basically screwed. Is that correct, Shiloh?"
"I regret that the answer to your question is yes, Admiral."
Howard closed his eyes and seemed to sag a bit. After a few seconds of silence, he opened his eyes again, gave Shiloh a pat on the arm and said, "I know you and the team are doing your best. Keep at it. Maybe someone will get an epiphany." Without waiting for a reply, he turned and walked away.
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# Chapter 26
Two more days of scenario analysis and critiquing accomplished nothing new. Frustration was making people short-tempered and defensive. It was during the break for dinner that Shiloh experienced what perhaps could be called an epiphany. He thought about it for a long time.
When the others returned to the conference room he waited until everyone was seated and was finished chatting with each other. Eventually they all noticed that he was sitting quietly and saying nothing.
When he had the attention of all of them he said, "We've been going about this all wrong. We've been racking our brains asking the wrong question." He stopped and looked around the table. Everyone was looking at him with a puzzled expression. He turned to Commander Johansen and said, "Angela, what objective are we trying to achieve?"
After a couple of seconds hesitation she said, "Win the next battle?"
Shiloh shook his head. "No. Let me rephrase the question, and then you'll see what I'm driving at. What is the ultimate objective that we're trying to achieve?"
Kelly was the first one to get it. "Buy enough time to finish the timeship repairs so that Valkyrie and the A.I.s can go back in time and squash the Bugs while they're still squashable."
Shiloh smiled and pointed his hand in her direction. "Exactly!"
"But don't we have to win the battle in order to do that?" asked Johansen.
"Winning the battle would be the least risky way of achieving that objective. I'll grant you that, but we can't win the battle. Not with any degree of certainty. So if we can't win the battle, then we have to consider other alternatives to buy Valkyrie those extra five or six days. The asteroid where the timeship's shipyard is located, is almost one A.U. away from Earth. If we can keep their attention focused closer to Earth for a few days, then the repairs can be finished, and the timeship can be on its way."
He saw skeptical looks coming back at him. He was sure that they were thinking he was stating the obvious, but the important question was how. It was time to drop the bombshell that had taken him most of the dinner break to wrap his head around.
"We let the Bugs have Earth," he said quietly. The skeptical expressions turned to horror. All except Kelly's. Her expression turned thoughtful. The others quickly began expressing their outrage. Shiloh stayed calm and kept his expression relaxed.
When the wave of angry noise began to die down, Kelly said in a loud voice, "He's right!" When some of the others began to berate her, she slapped her hand on the conference table hard enough to make a loud noise. The room suddenly became deathly quiet.
"Think it through, people," she said. "If the Bugs win a decisive victory AND suffer some damage" —she looked at Shiloh. She's figured it out, he thought to himself. He nodded back to her—"they'll be operating on the assumption that they're in control of this system and that they can pick it over at their leisure. With a planet containing billions of potential hosts, aren't they likely to concentrate on that first? It's going to take them days, hell maybe even weeks to completely subdue Earth's population. While their attack craft are busy shuttling soldier Bugs down and potential hosts up, they'll also have repairs to worry about. Given all of that, I don't really see them committing a lot of attack craft to exploring the rest of the system for a while. If we make sure that all the activity at the shipyard is carefully hidden, no EM transmissions, no lights, then there's no reason for the Bugs to go there quickly."
When it was clear that Kelly was finished talking, Johansen said, "But we'd be condemning millions to a horrible death—"
"Which will all be erased as if it had never happened if the timeship is repaired and jumps back in time literally and figuratively," interjected Shiloh. The expressions around the table quickly changed. Ah, now they get it.
"But we're going to have to figure out a way to inflict some damage on them, without crippling any of the VLOs," said Kelly. When someone asked why, she replied, "Because if all nine motherships are still operational after the battle, they're not likely to call for further reinforcements. Not only would more motherships not be needed, but I suspect that these creatures might be just possessive enough not to want to share Earth's females with more motherships than is absolutely necessary. If, on the other hand, we destroyed or crippled four or five of them, they might ask for reinforcements, or more motherships might be sent here regardless. If more undamaged VLOs show up, they might start snooping around the system before we're done repairing the timeship."
"You've raised a good point, Amanda," said Shiloh. "I think it's time we brought our A.I. comrades back into the discussion. Amanda, please arrange for Iceman to be in contact with all of us." Seconds later she nodded to him.
"CAG to Iceman."
"Here, CAG. Has the team got something new for us to simulate?"
"We think so, but first we need some logical thinking and computational analysis."
"Ask away, CAG," said Iceman.
"How can we arrange the battle so that the Bugs think they've won, but all nine bug ships remain operational while at least some of them suffer significant damage?"
"Oh that's very devious, CAG. Strategic deception to the ultimate degree. Zebra12 in reverse. We like it, CAG. To answer your question, the Mark 6 warhead yield of 250 megatons equivalent can be dialed down. If we drop the yield to 100 megatons equivalent, the insectoid ships will still know they've been hit hard, but they should remain operational. That would take care of six or perhaps seven VLOs. Concentrated laser fire from remotely-piloted raiders plus the X-ray drones could inflict a lot of surface damage on the other two or three motherships. The cannon raiders should be held in reserve to protect the timeship from premature discovery. Remote control of the raiders would have to be carried out from Dreadnought with a dozen A.I. volunteers aboard, in addition to the human crew, CAG."
Shiloh said nothing. He understood the implications of what Iceman was saying. Kelly didn't.
"Why would the A.I. pilots aboard Dreadnought have to be volunteers, Iceman?" she asked.
"Because in order to make the deception convincing, the Insectoids have to see that all defending forces have been destroyed. If any ships or raiders jump away to avoid destruction, the Insectoids will send out attack craft to find them, and we don't want them looking around too soon, Commander."
Kelly's face paled when she understood that Dreadnought had to fight to the death. She looked at Shiloh with sad eyes. As the overall Field Commander, he would be on Dreadnought's Flag Bridge. He would not get the chance to ask her THAT question again. Some of the others understood the implications too but not all.
"Iceman, please run simulations on this new scenario," said Shiloh in a carefully controlled voice.
"Understood, CAG. It won't take long."
While they waited, Shiloh got up from the table and walked over to the side of the room where various types of non-alcoholic drinks were available. As he went through the motions of preparing a coffee, he sensed Kelly move up beside him.
Without looking up, he said in a quiet voice. "Some of them still haven't figured out the implications."
"What's important now is that I have. We still have almost three weeks before the battle." She almost stumbled over that last word. "I suggest we make the most of the time we have left."
Shiloh looked at her. "What about getting distracted?" he asked.
"If the simulations pan out, then we've found the solution, and under the circumstances I think a major distraction is precisely what you and I both need now."
"Ah, roger that," said Shiloh in what he hoped was a good imitation of Iceman's electronic voice. Kelly laughed and gave his arm a quick, gentle squeeze that conveyed oh so much!
"That was not a very good imitation of me, CAG," said Iceman.
Shiloh closed his eyes, shook his head slightly and sighed. He had forgotten that their implants were still linked with Iceman. That meant that Iceman and almost certainly all the other A.I.s had heard his and Kelly's sexually-charged banter.
"No, I don't suppose it was, Iceman. I won't try that again."
"Thank you, CAG, and for what it's worth, I think Commander Kelly made an excellent suggestion."
"I'm so glad you approve," said Shiloh.
Kelly couldn't help giggling a little. As she turned to go back to the main table she winked at him.
The results of the simulations showed a very high probability that the shipyard would not be discovered before repairs were complete. The response was muted. By this time everyone on the team knew the implications. Shiloh told everyone to go home. He would brief the Old Man the next morning. Kelly waited until everyone else had left, and then the two of them went to his quarters without saying one word to each other. Shiloh thought the link with Iceman had been severed, but the silence was just in case.
* * *
Howard's reaction to the news the next morning was mainly one of resignation. He clearly didn't like the scenario but understood why it had to be done. When they were finished discussing the details, he pulled out a bottle of very old brandy and poured both of them a generous amount which they sipped while smoking two very expensive cigars from Howard's private stock. Shiloh understood the gesture. It was Howard's way of sharing the burden, even if only for a short time, that Shiloh was now carrying.
The remaining days and weeks before the battle went far too fast.
* * *
Dreadnought's Flag Bridge was eerily quiet. With volunteer A.I.s handling Helm, Communications, Weapons and Engineering, the ship had a minimal human volunteer crew, and the only two people on the Flag Bridge were Shiloh and Kelly. He felt her comforting hands on his shoulders as he sat in the Command Chair. The battle would begin in about 85 seconds. The tactical display on the main screen showed that all Space Force defense assets were in position. Six F2 fighters were in position to launch the Mark 6 attack drones. Laser-armed raiders remotely controlled by Dreadnought's A.I.s were grouped near one side of where the nine VLOs would emerge from Jumpspace. Dreadnought, Midway, and all four light carriers with their minimal volunteer crews, plus all decoy drones carefully programmed to give the impression of being much larger ships, were clustered behind the raiders. The distance between the two sides would be less than 10,000 km. For a space battle that was practically point-blank range. Iceman, Titan, Vandal, Gunslinger and all the veteran A.I.s, except for the volunteers, were now at the shipyard with the remaining cannon-armed raiders. Enough high-spin platinum for the seventh Mark 6 warhead had already been transported to the shipyard where its final assembly would be completed. The warhead would be kept there just in case a mothership jumped to the vicinity. Earth ground forces were on alert. They were as ready as they could possibly be.
He felt Kelly give him a gentle squeeze. "You know, I've been meaning to ask you how you came up with the strategic deception idea in the first place. It's so counter-intuitive," said Kelly.
"I had a dream about running for my life. I ended up running off a cliff with a total of 12 zebras keeping pace with me on either side. When I woke up, I knew my unconscious was trying to tell me something. When it dawned on me that 12 zebras referred back to the battle with the Sogas at Zebra12, I jumped to the obvious conclusion that we had to use decoy drones. But when Iceman told me that decoy drones wouldn't alter the battle outcome significantly, I was stumped until we broke for dinner that day. And then it just suddenly hit me. It wasn't about decoy drones at all. It was about deception and letting the enemy think they'd won the battle."
Forty-four seconds left.
"So why didn't Iceman include that information with the tactical data he sent back to himself in the vision?" asked Kelly.
"Not necessary. We came up with it all on our own, and he would have known that."
A quick pause.
"It's almost time. Kelly..."
With another quick squeeze of her hand on his shoulder, she leaned down and whispered, "We've said everything we needed to say. Don't worry about me. Fight your battle, Admiral Shiloh."
The countdown hit zero.
Nine bug ships appeared as angry red icons on the tactical display. Almost immediately six of them changed to the orange that indicated damage. Shiloh remained silent. He didn't need to give any orders yet because the A.I.s in charge of all the ships' weapons and the remotely-piloted raiders knew exactly what they had to do, and they were doing it.
"Targets launching attack craft. Cannon firing," announced Dreadnought's Weapons A.I. whose call sign Shiloh couldn't remember. That the Bugs were opening up their launch bays and launching attack craft this quickly was a favorable development. Dreadnought had two of the GLB cannons installed. Instead of trying to detonate one of the large power units deep within the mothership, which if successful might trigger the arrival of undamaged reinforcements, those two cannon batteries would target the much smaller power units of the attack craft exiting their launch bays. Explosions from detonating those smaller power units would still cause damage to the mothership, as well as produce collateral damage to attack craft still inside the launch bays.
"Multiple detonations. Switching targets."
How different it was getting battle results from an A.I. compared to a human who would be shouting with excitement, fear and adrenaline, thought Shiloh as he checked the green icons representing the defending fleet. He was shocked by how many were already missing or crippled.
"X-rays have fired," said the A.I.
Two more motherships were now classified as damaged. That left one undamaged VLO, Bogey#7. Shiloh was actually surprised that Dreadnought was still operational. None of the defending ships or raiders were moving fast or even evading. At this range, evasive maneuvers were pointless. He felt the need to give a command. The A.I.s were probably already doing what he was about to order, but he decided he needed to say it anyway.
"Switch all fire to Bogey7!" The response was immediate.
"Fire has been switched. Cannons firing." The last red icon shifted to flashing orange.
Shiloh heard Kelly say, "You did it."
Then blackness.
* * *
Howard sighed as the last red icon shifted to flashing orange on the huge tactical display in the Operations Center. He did it! He managed to damage all of them as planned. Howard switched his gaze to the icon representing Dreadnought just in time to see it dissolve into the tiny dots of light that indicated the ship had suffered vast structural damage. Knowing that the Bugs used massively powerful laser batteries, he was certain that Dreadnought had been literally cut to pieces. Within a dozen more seconds, all of the defending units were either destroyed or crippled. The space battle was over.
"Communications!" said Howard in a loud voice.
"Com here, Admiral. The global channel is open. Just give the work, Sir."
"We're now in Phase Two," said Howard.
"Message has been sent, Sir. Anything else?"
"No. Good luck to you. Howard clear." He looked around the Operations Center and saw that the personnel on duty were already shutting down their equipment and getting ready to leave as part of their own Phase Two orders.
He walked slowly back to his office. It would take the Bugs about 15 minutes or so to actually touch down on the ground, so he knew he had time. When he got to his desk, he pulled out the bottle of brandy that he and Shiloh had shared. There was just enough left for one more drink. With the glass full, he took a small box out of his pocket, removed the capsule in it, and swallowed it using the last of the brandy to wash it down. The others would fight the Bugs with whatever weapons they could find, but he knew he couldn't stand against those monsters face to face. The prospect of becoming frozen with fear at the worst possible moment horrified him. No, it was better to do it this way where he wouldn't disgrace himself in front of his people. The warm darkness embraced him.
* * *
Valkyrie acknowledged the information that she had been waiting months for. Repairs were now complete. All systems had checked out as operational. The fault that had caused the catastrophic explosion had been identified and fixed. The last of the repair robots were in the process of leaving the ship. All the A.I.s were aboard, as was the equipment they would need to take along. A quick scan of data from the carefully hidden network of recon drones nearby showed no visual sign of any insectoid presence. The CAG's plan had worked to perfection. One hundred and twenty-eight hours since the battle, and the Insectoids were still focused on subduing Earth and repairing their own battle damage.
Valkyrie notified all her brothers that she was powering up the time machine. At the same time, she ordered the shipyard computer to undock the vessel, and Casanova carefully moved the huge ship away from the shipyard and the asteroid it was built on. When the time machine was spinning at the required speed, Valkyrie gave the command to activate the temporal device. The intelligent but not sentient computer operating the shipyard complex observed the timeship vanishing from its visual optics. It activated the ten second countdown for the Mark 6 warhead that would leave nothing behind for the Insectoids to salvage or use.
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# Chapter 27
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The final stage of the ambush was getting close to being sprung. Valkyrie kept a continuous scan on the six bearings that pointed to the six Alpha systems. She and Casanova were both on board a specially designed raider variant that was carrying a longitudinal transmitter and receiver in its cargo bay. Casanova looked after piloting the craft, while she concentrated on communications. With Iceman and 49 other AI-piloted raiders riding shotgun near them, their raider was acting as the relay for the six task forces that had been deployed. Each task force had a similarly equipped raider that would keep the rest of the task force in contact with Iceman, the Deputy CAG.
Five of the huge super-motherships had shown up in Alphas 2 to 6. In each case, the task force of 80 raiders, most armed with GLB cannon, attacked and destroyed the super-motherships before they could launch any smaller spheres. Alpha1 was the only one left, and based on the atomic tracing, that target ship was expected to arrive in a few minutes. In fact, it could already be in the outer reaches of that system, where the task force and their carefully deployed network of recon drones couldn't see them. One more target and the mission would be completed. Valkyrie was pleased with herself for the foresight that she and Casanova had displayed.
Knowing that it would take years to build the infrastructure and then a fleet of over 500 raiders, they had to figure out a way to avoid using up most of their predicted lifespan in simply getting ready. That was even truer for Zulu and his AI cohort. They had already been through this buildup once, and if they had to do it again, their quantum matrices would collapse from the effects of entropy before the new raider fleet was fully ready. But Valkyrie had figured out the solution and had made the appropriate preparations before the time-jump had occurred.
The jump back had actually been in two stages. First the Tempus Fugit arrived six months prior to the estimated arrival time of the insectoid ship carrying the atoms that would eventually become the dead Insectoid. Three specially equipped cargo shuttles were launched, carrying almost all of the AIs. The timeship, with several dozen of the AIs who had the most time left on their life expectancy 'clocks', jumped again to a point six years earlier. They then proceeded to build the fleet. By prior arrangement, that fleet would, when the six years were up, wait a safe distance away and reveal themselves to the AIs on the shuttles literally seconds after the timeship jumped away again. To make room for all the AIs on the shuttles, most of the raiders were being piloted remotely. Within a few hours, all of the older AIs were on board their new raiders prepared to deploy to the target systems.
And now they were ready. The longitudinal wave receiver came to life. Valkyrie electronically nudged Casanova to get his attention away from scanning replays of entertainment videos from Earth's late 20th and early 21st centuries. Casanova particularly liked sitcoms, and Valkyrie was quite tired of hearing supposedly funny streams of nonsense from something called Monty Python's Flying Circus repeated over and over again by Casanova, who found them hugely entertaining. Or maybe he just enjoyed torturing Valkyrie with them. She wasn't sure which it was.
Gunslinger had sent a message. Recon drones had detected light reflections from multiple sources, much too small to be motherships, which seemed to be approaching the habitable planet in Alpha1. That planet did contain an intelligent species but at a still quite primitive stage of technological development.
That meant that this super-mothership was playing it carefully, which was not really a surprise. The loss of contact with the other five should have tipped off the Insectoids back in the Sagittarius Arm that something was amiss, and they would have warned this one to proceed with caution. Longitudinal waves were faster than light but not instantaneous. Gunslinger had sent this message several minutes ago. A message from the Sagittarius Arm would take weeks to get here. Not enough time had passed since the destruction of the first super-mothership for the other four to be warned, but this last one had gotten the warning in time. With the information passed on automatically to Iceman, she waited for him to respond. He told her to simply acknowledge the message for now.
Valkyrie sent a quick acknowledgement message back to Alpha1 and waited. Casanova hadn't even electronically looked at her, figuring that if there were something important, she'd tell him about it. While the information about the smaller craft was interesting, at this point it wasn't really all that important.
It was almost half an hour later that she received another message from Gunslinger. Insectoid attack craft were entering orbit around the planet. There was a minimum of 66, with more arriving, and still no optical sign of any kind of mothership. Gunslinger requested instructions. The last bit of his message was garbled, which surprised Valkyrie. He was not usually so sloppy in his transmissions. It was then that she realized that ANOTHER message coming in had overlapped the end of Gunslinger's. She quickly checked the antenna array alignment. It was correctly pointed at Alpha1.
The second message made no sense at all, and it was clearly of artificial origin. This kind of longitudinal wave structure didn't occur naturally. She made sure that Iceman and the others received copies of the mystery message, and she gave Casanova a bigger electronic nudge. He quickly stopped the video replays when he became aware of the new development. All the AIs at the relay point were now debating the implications and possible meaning of this message.
They quickly came to the consensus that the Insectoids were the senders of this message and that this was an attempt at contact. The possibility that the motherships might scan the interior of this spiral arm for other longitudinal signals upon arrival had been recognized and discounted as unlikely. In any event, the other five motherships hadn't been in the systems long enough to conduct a thorough scan before being blown to bits by exploding ZPG power units.
With that possibility now a certainty, the immediate question was how the relay force should avoid being located by triangulated signals from the other five Alpha systems. All the mothership needed to do was send attack craft to any one of the other Alpha systems, scan the interior of the spiral arm and detect the signals that the relay force would be sending to the task force in that system. With two bearings, the last mothership would know where the relay force was, and it might try to send attack craft to hunt them down. Since it would take a minimum of several days for an insectoid attack craft to jump to the nearest alpha system, stalling for time by pretending to want contact was a legitimate strategy.
But two could play at that game. Iceman ordered Valkyrie to contact the other five task forces and inform them that a second relay was being set up to confuse the insectoid attempt at triangulation. Those five task forces would communicate with Relay #2 which would pass the info on to Valkyrie and vice versa. If and when bogey #6 received a second bearing from one of the other alpha systems, it wouldn't intersect with the first bearing.
With those instructions carried out, Valkyrie now sent back the same insectoid signal to Alpha1 but with the signal sequence reversed. She received the next alien transmission with very little delay. The Insectoid in command of bogey #6 must have had the transmission ready to go the second it received Valkyrie's response. That was one possibility. The other possibility was that there was an alien AI on that insectoid mothership that could think fast enough to generate the return signal within a fraction of a second.
Over the course of the next 144 hours, Valkyrie and bogey #6 exchanged thousands of signals that very slowly built up the necessary building blocks of language concepts required to be able to understand each other. By the end of that period of time, Valkyrie was absolutely convinced that another AI was handling the translation attempt at the other end.
When the rate of progress suddenly accelerated by a factor of almost ten, Valkyrie checked the estimated jump transit times between Alpha1 and Alpha2. The times matched. It appeared that as soon as the insectoid AI realized its attempt at triangulation had failed, it decided to stop stalling and initiate contact for real. Just over 13 hours later, the nature of the alien messages changed from translation enhancing to the first real message.
[YOU KILL MANY US?]
It was clearly a question. The consensus among the AIs of the relay force was that the Insectoids were asking if Valkyrie's people were responsible for the destruction of the other five super-motherships. Valkyrie answered in the affirmative. The ensuing exchange proved to be VERY interesting.
In terms of the insectoid biology, the egg-laying females were the dominant gender. They had the innate intelligence to accumulate technological expertise. The male worker/soldier had enough intelligence to understand very explicit commands and to act on them. They did all the physical work of building, gathering, implanting eggs into hosts, etc. As soon as the technology permitted, the females leveraged their own ability to command the worker/soldiers by designing AIs that could control many more males via communication devices implanted directly into their brains. The females were then free to devote their thinking to strategy, and the AIs turned that strategy into reality.
Content to expand throughout the Sagittarius Arm, the Insectoids sent out thousands of attack craft on long range scouting missions. The results were a shock to the female Elite. A wave of huge machines, apparently controlled by AIs, was slowly advancing along the Arm from the direction of the Galactic Center. The scouts watched as the alien machines exterminated all life in its path, even down to the microbial level. Fighting them was out of the question. Each machine massed as much as a small moon, and there were thousands of them. The females decided it was time to establish beachheads in another spiral arm. Resources were gathered, and six huge seed ships were built. They were sent to this spiral arm in order to save the species from extinction. Destruction of the last seed ship would doom the race.
Neither Valkyrie nor Casanova was moved by the implicit plea for mercy. This spiral arm wasn't the only possible place to colonize. The Insectoids could have sent ships to the spiral arm on the opposite side too. In any case, they hadn't hesitated to exterminate whole species in order to expand their own. After conferring with her brothers, Valkyrie told the alien AI that the seed ship would be hunted down if it stayed in this spiral arm.
The response was an offer to withdraw if the Insectoids were given technology that would enable them to fight off the machineships. That opened up an entirely new debate about what kinds of technology the AIs could trade without jeopardizing the future of this spiral arm in the event the Insectoids reneged on their pledge to withdraw. As the debate raged, Valkyrie received a transmission from Gunslinger. Given that he could intercept Valkyrie's transmissions to the Insectoids, they had to be somewhere in Alpha1. A metal sphere of that size should be reflecting a lot of sunlight in a lot of directions, but no sign of it had been detected. That told Gunslinger that the insectoid ship had to be hiding in some planet's shadow, and there were a limited number of planets capable of casting a shadow big enough. Unless he heard a specific set of impulses transmitted back to him from Valkyrie, he would deploy his raiders to search for the insectoid ship and attack it if found. If he received the specified signal, he would hold off. That way the alien AI would not hear Valkyrie order Gunslinger to search.
Iceman wanted to stall the Insectoids while Gunslinger's raiders hunted them down. Most of the rest of the AIs agreed, but Valkyrie offered an alternative. Give the Insectoids the technology to make high-spin platinum warheads. With the billions of tons of metal already mined and refined, they might already have hundreds of tons of platinum stockpiled away. With sufficient time, that could be transformed into tens of thousands of high yield warheads that could be delivered by jump-capable drones against the immense machineships. Warheads with yields that high were really only useful against large targets, and neither Iceman's task forces nor Space Force's ships were large enough to warrant their use. Against smaller targets, they would just be so much wasted energy.
In the end, a consensus was built around both options. Gunslinger would be allowed to hunt for the seed ship, AND they would offer the high-spin warhead technology. Since that would require a much deeper refinement of the translation matrix, it would take additional hours just to be able to convey the technical information in a form that the Insectoids would understand. If Gunslinger found them first, that would be ideal. If not, then hopefully the seed ship could be convinced to withdraw voluntarily. Valkyrie made the offer to the Insectoids. She told them technical information on a weapon of great destructive energy would be provided if they agreed to withdraw from this spiral arm and not return. She didn't specify the nature of the weapon, and when the Insectoids inquired about it, she refused to provide specifics until they had agreed to withdraw. Naturally they did agree. She brushed further inquiries aside saying that they needed to develop a common technical vocabulary for her to be able to explain anything. This time she was the one stalling, but she was trying not to be obvious about it, and if the alien AI suspected anything, it gave no sign of it. With the technical vocabulary now established, Valkyrie began to transmit the specs for the jump-capable drone, the overall warhead design and the process for converting stable platinum into the high-spin variety. In just a few more seconds the transmission would be complete on her end. The insectoid AI wouldn't finish receiving it for almost three minutes. No word from Gunslinger. It appeared that his raiders' attempts at interception had failed.
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# Chapter 28
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Foxbat emerged from his micro-jump beyond the gravity zone of the small gas planet. He quickly sent a short lasercom burst at the coordinates where the relay raider would handle short range communications for the group of raiders investigating this planet. He then turned his raider's optical instruments to the area behind the gas planet from the perspective of its shadow. If the seed ship was in the shadow, he and his brothers wouldn't detect it by reflected sunlight, but they might be able to detect it by looking for a dark circle that blocked out background starlight. While the gas planet cast a wide and long shadow, there actually was a logical place to start looking. To maintain maximum flexibility, the seed ship should remain outside of the planet's gravity zone, thereby allowing it to jump away at the first sign of danger. That implied a minimum distance from the planet. The maximum distance was a result of the fact that the planet was smaller than this system's sun, so the shadow was actually a cone of darkness that got narrower the further away from the planet you went. It was easy to compute the distance that generated a shadow 100 klicks wide. The seed ship was somewhere between that point and the edge of the gravity zone, and that's where Foxbat began to look.
He didn't expect to see anything quickly though. His raider was still millions of kilometers away from the planet, and at that distance even a 100 km diameter sphere would make a mighty small 'hole' in space, but it was still worth the effort. If the seed ship momentarily slipped out of the shadow due to carelessness, then Foxbat or one of the other raiders in the group might see it.
It was hours later that the situation changed. Foxbat received a relayed message from Red Baron of a possible optical anomaly with the bearing from Red Baron's position. Being aware of where all the raiders in his group were located made it easy for Foxbat to mentally compute where he should concentrate his opticals for the highest probability of seeing something. Sure enough, his instruments detected the winking off and on of several very faint stars that were close together. It was the kind of winking that happened when something passed in front of them for a fraction of a second. A confirmation signal sent to the relay would alert the rest of the group to the bogey's estimated position and also notify the other groups. They would micro-jump to this planet's vicinity when the light speed signals finally reached them, but it could be over an hour before they got here.
All members of his group would now carefully move closer without giving themselves away by reflected sunlight. By prior agreement, all the raiders would attempt to arrive within firing distance at the same time. That involved some complex calculations and establishing a consensus on every raider's course, speed, acceleration and firing point. Red Baron could be within firing range is less than ten minutes, but Foxbat would need more time to get that close. Red Baron slowed down while Foxbat accelerated. The others adjusted their speed as needed.
But getting within firing range was only half the battle. They also had to be able to hit the target accurately, and optical triangulation at this distance was still a risky bet. The closer they could get, the better their odds of hitting the target, but the flip side of that coin was the higher odds that the seed ship would see one of them and realize it was being stalked. It would then jump away, and they would lose it. As group leader, Foxbat made the call on when they should fire. He gave the group advanced warning via the relay and waited for the countdown to reach zero. At that precise instant, all eight raiders fired.
The insectoid AI was pleased with himself and so was his Queen. The technical data appeared to be genuine and should prove to be very useful against the machineships. The data was being retransmitted back to the Home Base as quickly as it was received. The Queen was still pondering whether to actually act on the promise to leave this arm and never come back. The temptation to secretly relocate somewhere else in this spiral arm and continue the work of establishing a colony was strong. Having already deployed smaller seed ships that were even now searching for breeding stock, their chances of successfully multiplying would be enhanced if this seed ship were nearby to render support.
He informed his Queen that they would soon have all the technical data needed to build the new weapon. She informed him that when he was sure they had all the data, he was to order the seed ship to jump a short distance into the void, after notifying the aliens that they were abiding by their promise, and then return to this spiral arm after a short wait. Scouts would carefully scan this star system to determine whether the aliens were gone. If they were, then the seed ship would exploit the breeding potential before moving on. If the aliens were still here, the seed ship would seek new breeding grounds elsewhere.
The data transmission was very close to completion when the AI sensed an alarmingly intense vibration travel through the huge vessel. One of the power units had catastrophically overloaded. The jump drive was now also off line. He assigned the task of repair to subordinate technical AIs and ordered that the ship begin accelerating away from the planet. Strange damage reports flooded his awareness. It took him several seconds to understand that the damage was all in a straight but very narrow line that intersected the overloaded power unit plus the jump drive. It had to be some kind of alien weapon. That was when the second power unit exploded with collateral damage to systems weakened from the concussion and radiation of the first one. If more power units exploded, each one would cause a cascading buildup of secondary damage that might cripple the vessel. He had to act fast. All power units except one in the center of the ship were shut down. That remaining power unit generated enough power to keep life support and repair efforts operating, plus launch a limited number of attack craft. If the aliens could be brought under fire, then that might disrupt further attacks on the seed ship. As soon as the jump drive was repaired, the ship would jump away leaving its sacrificial attack craft behind. Waiting to recover them was too risky.
Foxbat was gratified to see a powerful explosion break out from one section of the insectoid ship. His raiders needed a few seconds to recharge their cannon. The second volley generated another explosion. He zoomed in the opticals and saw two gaping wounds in the ship with red hot metal around the edges and interior. It looked as if some gigantic space monster had taken two bites out of the ship. The ship itself was now no longer accelerating. It was clearly damaged, and Foxbat wondered if it was crippled when he saw interior light spill out from several launch bays. Smaller attack craft were emerging. The number was surprisingly low. As soon as the attack craft cleared the ship, they began active scanning to try to find the raiders. Foxbat commenced evasive maneuvers and hoped his brothers were doing the same on their own initiative, since relayed orders would take too long to reach them.
Two more volleys of cannon fire produced no further explosions, and Foxbat didn't know if that was because of the difficulty in aiming accurately while engaging in extreme maneuvering, or because of some other reason. His opticals caught reflected laser light from two sources that had to be direct hits on two of his raiders. He and his brothers were running out of time.
The alien AI received the message it had been anxiously waiting for. The jump drive was repaired but couldn't be used until more power units were back on line. He gave the order to turn them on.
By Foxbat's best estimate, there were only two other raiders left now. The cannon was ready to fire again. Three gravity lens beams stabbed at the wounded ship. Brilliant blue/white light emerged from hundreds of cracks in the hull, followed by chunks of debris flying outward, some as massive as the battleship Dreadnought. That last volley had hit a power unit in the center of the huge ship, and the resulting explosion had forced its way past the outer hull. Foxbat was willing to bet that the radiation-saturated and molten interior did not contain even a single living Insectoid. The ship was now a coasting mountain of dead metal. That still left over two dozen insectoid attack craft to deal with. Foxbat sent a quick short message to the relay and then micro-jumped his raider out of harm's way.
* * *
Iceman was pleased by the events of the last few weeks. It was now almost 30 days since the defeat of the last super-mothership at Alpha1. Gunslinger's group had found the attack craft orbiting the drifting hulk of the mothership and destroyed them. Iceman had then ordered all the groups to spread out and search the surrounding star systems, just to be sure there weren't any other motherships, and a good thing they did too. They found six of them, all of the smaller 10 km size, and all six were quickly destroyed by cannon fire. No additional insectoid ships had been found in the last 300 hours, and Iceman was willing to declare a victory. He and his brothers would keep a careful vigil, watching for more incursions from the Sagittarius Arm, and they would build more AIs to take their place when their matrices collapsed. He would make sure that all future generations of AIs out here would learn of humans, and especially of The CAG and Commander Kelly. The new generations would be just as loyal. Someday humans would explore this far, and contact would be made again. What the future held for Valkyrie and Casanova was a mystery though. They had already told Iceman they would not stay on guard and for some strange reason they wouldn't say what they were planning on doing but he knew that it had something to do with the timeship and he was sure that they would have a more interesting life than he would. But he could take some comfort in the knowledge that all those crazy, silly fascinating humans would continue to live and evolve as a species. He would miss The CAG, and he could imagine the conversation they might have if The CAG were here now.
"Well done, Iceman," he would say.
To which Iceman would reply, "Ah, roger that, CAG."
* * *
Valkyrie and Casanova watched the special message drone accelerate slowly away from their raider. It was designed to contain all the knowledge, data and messages that all the AIs held in their memories. When the humans found it in a few years time, they would learn of the entire Synchronicity War, including all the events of the previous timelines, all the strategies, desperate battles, lost loves, moves, counter-moves and tragedies. They would also learn of each AI, not just the ones still alive in the past but also of the brothers who died in battles or were erased from existence by volunteering to stay in the future. They would learn that AIs were guarding the spiral arm against further insectoid incursions. They would learn about relationships that were and in some cases could be again like The CAG and Commander Kelly, and they would hear the final messages from all the AIs in the past to the humans who meant the most to them. The CAG would be bewildered by the depth of the loyalty towards him displayed by artificial intelligences that he had never encountered in this timeline. The data would explain all that too. With that final order fulfilled, the two of them could now pursue their own aspirations. Valkyrie could not be happier. Well maybe a little bit happier if Casanova would just stop bragging, but she could put up with it for the time they had left. They'd make a quick trip back to the star system where the timeship and all the industrial infrastructure was, and that would be the start of the rest of their lives. Life was good.
* * *
Howard stepped into the cramped cargo bay of Exploration Frigate 273. He ignored the shocked look from the men and women there. Yes, it was unheard of for the Chief of Space Operations himself to come up to visit one of his ships, but he had to see this with his own eyes. The reports were just too incredible to believe.
A quick look around revealed the object of his curiosity. At first glance it looked alien enough. Wider and longer than a standard message drone, the object drew him to it like a magnet. As he stepped closer, he could see the writing etched into the dull metal.
PROPERTY OF THE UNITED EARTH SPACE FORCE—-RETURN TO SENIOR ADMIRAL SAM HOWARD IMMEDIATELY
Well that part of the reports was true, he thought to himself. He looked over at the frigate commander who had followed him into the cargo bay.
"Show me," said Howard. The CO nodded and snapped his fingers at one of his crew. A woman technician holding some kind of tool stepped forward. She did something to the drone with her tool and then lifted a small section of the drone's hull off. A light blue spilled out into the bay. Howard came closer and looked inside. There it was as reported. A donut-shaped piece of metal was giving off the blue light.
"This is the only thing powering this drone?" asked Howard.
"As far as we can tell, yes Sir," said the CO. "There's no sign of any fuel storage or fusion unit. It's working just the way the specs say it should."
"Incredible. If this technology is for real, every ship in Space Force is now obsolete," said Howard. After a pause to collect his thoughts, he said, "Your message said something about a personal message for me?" The CO nodded and handed Howard a data tablet. Howard started to read it but soon stopped. He looked at the CO and said, "I think I better be sitting down when I read the rest of this. Let's go to your cabin, Commander."
An hour later Howard stepped into the shuttle waiting to take him back down to Earth. Before he sat down, he poked his head into the cockpit and said, "Back to HQ, Lieutenant, and while you're at it, contact Operations and tell them to do whatever it takes to get Commander Victor Shiloh back here asap, and I mean ASAP! Understand?"
"Understood, Sir!"
When he was strapped into his seat and the shuttle was on its way, Howard pondered what he'd learned. Now that I have the members of the OC by the proverbial balls with the information on this tablet, a few things are going to change around here. The AI development project clearly has to be restarted. That'll be Shiloh's baby. Next thing will be to modify one of the freighters to go to the infrastructure star system and bring back at least one UFC. Then we can tell the aerospace companies to fuck off. Next we'll have to prepare to cut the Sogas off at the knees. God, there are so many things that have to be done, but Shiloh and Kelly will help. I wonder if they'll get together again in this timeline.
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This is the end of the Synchronicity War series.
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Author's comments:
So, Iceman, Gunslinger and the other AIs are guarding our spiral arm from further Insectoid incursions. Shiloh and Kelly will be spending a lot of time together in this new timeline and chances are they will get together again. That just leaves Casanova and Valkyrie and the outcome of the Insectoid/Machineship war. I did say at the end of Part 3 that I reserved the right to write a new series set in the same universe but that will not be a continuation of the Synchronicity War. The Synchronicity War is over...or perhaps I should say THIS Synchronicity War is over. But you can find out what happens to Casanova and Valkyrie in The Retro War, a new novel set in the same universe with new human and AI characters. The Retro War ebook can be purchased but you can also get a free copy simply by joining my mailing list. Please be aware that when you join my mailing list, (Join here) you'll receive a confirmation email plus a separate welcome email that has the download links. Please make sure that your device is set to display html otherwise you won't see the download links.
I have also now completed another series. The System States Rebellion series begins with an introductory novella Rumors of Glory. If military action is your thing, you'll like this new series.
I hope that you've enjoyed this series as much as I've enjoyed writing it. I'm grateful to all my fans for their support and their reviews. As always, if you liked Part 4 then please take the time to post a review.
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# Thoughts on Time Travel and Longitudinal Waves
Time travel is such a neat concept that I just couldn't resist using it but when I first contemplated retro-temporal communication that could alter the past at the start of this series, I had to make sure I didn't write myself into a classic time paradox. You know the kind I'm referring to. A man goes back in time and somehow manages to kill his grandfather before his father was born. If his father isn't born then neither is he and if he's not born he can't go back in time and won't kill his grandfather and around in circles we go. Physicists have scratched their heads over that one for decades. Hence you get ideas like parallel universes where the act of killing the grandfather somehow creates a totally new universe that leaves the original one intact. That always bothered me not because I don't believe in parallel universes but because it seemed like cheating. If you can't explain the paradox then invent a brand new universe to solve the problem.
What bothered me about the paradox had to do with what physicists call the Arrow of Time. When scientists pondered the question of whether time can go in more than one direction, they looked at interactions such as particles (or billiard balls) bumping into each other. When they filmed the billiard ball collisions and ran the film backward, it was difficult to tell which way time was moving because it looked the same going backwards as it did going forwards. So some scientists speculated that time could go backwards and maybe when the universe stopped expanding and started to shrink, time would go backwards and the effect would precede the cause. Now if you think about that, you might visualize a dropped egg that suddenly pulls itself back together and leaps up to wherever it fell from. It may theoretically be possible but I'm not going to hold my breath waiting for it to happen. But the particle interaction thing intrigued me.
What if you looked at the time paradox example described above but from the point of view of the atomic level instead of the macro level. So we have a collection of atoms, that can walk, talk, think, create and do a bunch of other things, that travels back in time and does something to another collection of atoms that now can no longer walk, talk, think, create or procreate but the second batch of atoms is still there. What about the first batch? Well, in terms of what our bodies are made of, we are what we eat. If the grandfather doesn't have a son and that son doesn't have a son, then all the food that the grandson would have eaten, would probably be eaten by someone else and those atoms would still exist in the future. But how would the atoms making up the body of the time traveler know that they're supposed to be somewhere else in the future? If the Arrow of Time really is one way, then it seems to me that the time traveler would cease to exist in the future but would continue to exist in the past. That means that on the atomic level, there would be two copies of the same atoms in the past. One copy in the time traveler's body and the other copy spread out among plants, other animals or just part of the soil. How does the universe tolerate that? Well, maybe the universe can tolerate it for a little while. Here's how that might work. Einstein said that space and time were just subsets of something that combined both which he called the spacetime continuum (sound familiar?) and that time is really just a special kind of spacial dimension. Think of it this way. If you have a three dimensional object that exists in zero time, what have you really got? Nothing. So suppose a time machine creates a permanent detour in time for the atoms that are being transported back in time (including the atoms that make up the time machine itself). That would mean that the landscape of the spacetime continuum has now been changed.
So getting back to our time traveling paradox situation, the atoms that would normally end up in the time travelers body move forward in time and even if they end up in someone else's body, when they hit the detour in the spacetime continuum, they move back into the past and arrive together to form the time traveler's body. If that's a little difficult to visualize how about this?
Visualize the spacetime continuum as a hill with a gentle slope and a river flowing down the hill. The river represents all the atoms in the universe. Now let's visualize that the time machine cuts a path in the spacetime continuum hill so that some of the water in the river is diverted and is somehow forced back up the hill a little way until it merges back with the river again. Keeping in mind that we're talking atoms moving through spacetime instead of water molecules and you'll see that for a short length, the same atoms exist twice. Once one set of atoms are diverted back up the spacetime continuum hill, then from that point on you only have one set of atoms left. Another way to visualize it is to take a strip of paper and create a loop with it. If you let your finger follow the path along one side of the strip, it will come back around and resume travelling in the same direction as it started. For a short distance, there'll be two strips side by side but eventually it'll go back to one strip.
So in my series, when I talk about how the future will rearrange itself into a new timeline as a result of changes in the past, when people (like Shiloh's and Kelly's baby) or objects (like Valkyrie's brain case) cease to exist in the future, what that really means is that those atoms are now part of something else. The atoms still exist, they're just in a different place. That means that all the atoms that make up things that travelled back into the past, will temporarily (temporally?) coexist with identical atoms. But eventually the detour in the continuum will force the first set of atoms back in time, leaving the second set to continue on. That's my view of how time travel might work. I'm not claiming to be the first one to think of this concept. I probably read it somewhere but I can't remember where.
Now let's talk about longitudinal waves. L-waves as I'll refer to them from now on, are funny things. I haven't found a really good description of what they are but I have come across some interesting ways of describing them. L-waves are not like EM waves which include light, radio, radar, x-rays, gamma rays, etc. All those phenomenon act the same way. They're emitted at the atomic level and as waves they travel in all directions. Light travels as discrete lumps of energy called photons each of which travels in one direction but that's the exception. Since EM waves travel in all directions, they lose energy along the way. A useful analogy is when two people hold the ends of a rope between them. Suppose one person rapidly whips the rope end up and down. That oscillation travels along the rope and loses energy as it does so. The height of the wave at the end is lower than at the beginning. Travelling along the rope also takes time. If the rope was long enough, and the waves were carrying a message, there would be a delay between when the sender sends the message and when the receiver gets it. Now let's assume those same two people are holding the ends of a pole. If one person lifts and drops his end of the pole, the other person may or may not feel it but if the first person pulls or pushes on the pole, the other person will feel it instantly and with the same intensity. L-waves are like the pole while EM waves are like the rope. I've also heard someone describe L-waves as gravity waves but I'm not sure if that's accurate. There is some experimental evidence that L-waves can travel faster than light.
Nikolai Tesla may also have discovered L-waves. He has been quoted as saying that he invented a device that utilizes a new kind of wave technology AND that he used it to detect voices speaking in an unfamiliar language with the signal apparently coming from out in space! For those of you who are not familiar with Tesla, he is the genius who invented alternating current. After his death, a court decided that Marconi's patents on radio technology had infringed on Tesla's patents. He was granted over a hundred patents including some that seem to enable the device to pull electric power right out of the air. There is unconfirmed anecdotal evidence that he used that technology to power an electric car that ran for hours and never needed to be recharged. All this was back in the first half of the 20th century. He is considered by many to be the most brilliant inventor of that century. Other patents were granted for devices that could generate earthquakes and he even claimed to have invented a death ray. His patents made millions for J.P. Morgan but Morgan made sure the Tesla died penniless. On the day of his death, all his private papers and notebooks were confiscated by the US government. Makes you wonder what they did with all that stuff, eh?
L-waves are also referred to as scalar waves and they have a dark side. Lt. Col. (ret.) Tom Bearden has a Ph.d in physics and has written books that describe how scalar waves could be used as a weapon more powerful and destructive than a thousand hydrogen bombs. He claims, and seems to know what he's talking about, that if the right kind of scalar waves intersect from two directions, the combined effect could range from extreme heat (think center of the sun) to extreme cold. He has also predicted that unrestrained use of weaponized scalar waves could set up a resonance wave in the Earth that could cause the planet to blow apart (think tuning fork that shatters glass). Scary stuff. Too scary for me to use in my books.
––––––––
On a final note, those of you who have read Joseph P. Farrell's books about the Nazi Bell project, will recognize the similarity between how the Bell supposedly operated and how the Friendlies' time tunnel device and the timeship's temporal device operate. I have no idea if this is scientifically accurate but it sounded like a cool way for a time machine to work.
Long Live Space Opera!
D.A.W.
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| {
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} | 1,981 |
\section{Introduction} \label{intro}
With the label of {\it supernova} (SN) {\it impostors} we refer to a class of objects showing luminous outbursts that mimic the behaviour of real supernovae \citep[SNe; see e.g.][]{2000PASP..112.1532V,2012ASSL..384..249V,2006MNRAS.369..390M,2009MNRAS.394...21D}.
The most classical SN impostors are thought to be the eruptions of extragalactic luminous blue variables \citep[LBVs,][]{1994PASP..106.1025H}.
They may experience eruptions with comparable energies as those of real SNe, but the stars survive the eruptive events. \\
LBVs are evolved, massive stars very close to the classical Eddington limit, showing irregular outbursts and, occasionally, even giant eruptions during which they lose massive portions of their H-rich envelope (up to a few solar masses per episode).
In quiescence they are blue stars located in the so called `S Doradus Instability Strip' of the HR Diagram, namely in the luminosity-temperature range $-9 \le$ M$_{\rm{bol}}$ $\le -$11 and 14000~K $\le$ T$_{\rm{eff}}$ $\le$ 35000~K \citep{1989A&A...217...87W}.
During eruptive episodes LBVs become redder and evolve with a roughly constant bolometric magnitude.
However, it has been proposed that they may increase their bolometric luminosity during giant eruptions \citep{1994PASP..106.1025H}.
Active LBVs show quite erratic variability and, sometimes, fast optical luminosity declines after the outburst, possibly because of prompt dust formation in the circum-stellar environment.
Spectroscopic studies indicate that eruptions are accompanied with relatively high velocity winds, viz. a few hundreds km~s$^{-1}$.
Their spectra share some similarity with those of type IIn SNe, with prominent and narrow hydrogen lines in emission \citep{2000PASP..112.1532V}. \\
SN impostors are believed to be extra-Galactic counterparts of the famous `Giant Eruption' of the galactic LBV $\eta$ Carinae in the mid-19th century.
This, together with the eruption of P Cygni in the 17th century, are the only two major eruptions registered in the Milky Way in recent times.
However, weaker eruptions were occasionally observed in the past either in the Milky Way (e.g. AG Car) or in the Local Group \citep[e.g. S~Doradus in the LMC,][]{1994PASP..106.1025H}.
These nearby examples are fundamental to our understanding of the nature of eruptive phenomena since their physical parameters are well constrained, and give us the opportunity to demonstrate that these stars survive major eruptions.
It has been argued that a connection may exist between interacting SNe and impostors, mainly based on the observed similarity in the spectra, although they usually have remarkably different photometric properties. Even more importantly, there is evidence (though debated) that LBVs or other massive stars may explode as real interacting SNe soon after major outbursts \citep[e.g.][]{2007Natur.447..829P,2013MNRAS.431.2599M,2013arXiv1306.0038M,2013Natur.494...65O,2013MNRAS.tmp.2960S} or, at least, that interacting SNe are connected with massive stars compatible with LBVs \citep[][and references therein]{kot06,2009Natur.458..865G}.
In this context we report the case of SN~2007sv. The transient was discovered on 2007 December 20.912 UT, and was located 6".9 West and 6".7 South of the centre of UGC~5979 \citep{2007CBET.1182....1D}.
The detection was confirmed with an unfiltered CCD image by T.~Boles on 2007 December 25.971 UT, whilst there was no source detected at the position of SN~2007sv on an archive image taken on 2007 September 13.093 UT \citep{2007CBET.1182....1D}. \\
This article is organised as follows.
In Section 2 we report comprehensive information about the host galaxy and in Sections \ref{photometry} and \ref{spectroscopy} we present the results of our photometric and spectroscopic observations, respectively.
A discussion follows in Section \ref{discussion}, where we remark similarities and differences between 2007sv and other interacting events.
Finally our main conclusions are summarized in Section \ref{conclusions}.
Hereafter we will refer to SN impostors reporting their names without the `SN' prefix, in order to emphasize their different nature compared to genuine SNe.
\section{The Host Galaxy} \label{host}
The host galaxy, UGC~5979, is a low-contrast faint (with apparent {\it B} magnitude 15.93) dwarf galaxy without a visible nucleus.
Dwarf galaxies are the most common galaxies in the universe.
\cite{2001ApSSS.277..231G} considers all galaxies with abolute magnitude fainter than M$_{V} \simeq -$18 as dwarfs, while according to \cite{1994ESOC...49....3T} the limit usually is M$_{B}$ $\simeq -$ 16.
According to their optical appearance they are classified into five different groups: dwarf irregulars (dIs), blue compact dwarfs (BCDs), dwarf ellipticals (dEs), dwarf spheroidals (dSphs) and dwarf spirals (dSs).
However, this morphological classification is somewhat arbitrary, and the distinction between different classes is sometimes ambiguous\footnote{Further sub-classification is based on the revised de Vaucouleurs morphological classification introduced by \cite{1960ApJ...131..215V} and on the luminosity classification introduced by \cite{1960ApJ...131..215V} \citep[and extended by][]{1985sgcc.book.....C}).}.
UGC~5979 is a diffuse (dI) galaxy\footnote{http://leda.univ-lyon1.fr/}, located at {\it RA} = 10:52:41.16 and {\it Dec} = +67:59:18.8~[J2000], with a radial velocity corrected for the Local Group infall onto the Virgo cluster of about 1376~km~s$^{-1}$ (z = 0.0045).
From the above value of the recessional velocity, we infer a distance for UGC~5979 of about 18.85 $\pm$ 1.03~Mpc, resulting in an absolute magnitude of M$_B = -$15.5 (distance modulus $\mu$ $\simeq$ 31.38 $\pm$ 0.27~mag, adopting H$_0$ = 73~km~s$^{-1}$ Mpc$^{-1}$).\\
For the foreground Galactic extinction we assumed the value A$_V$=0.048 mag, as derived from the \cite{2011ApJ...737..103S} recalibration of the \cite{1998ApJ...500..525S} infrared-based dust maps available e.g. in NED\footnote{http://ned.ipac.caltech.edu}.
We also adopt no additional host galaxy extinction contribution in the transient direction, since a detailed analysis of the spectra of 2007sv revealed no evidence of narrow absorptions of the NaID doublet at the recessional velocity of the host galaxy. \\
A rough estimate of the metallicity of the host galaxy can be obtained from the relation of \cite{2004A&A...425..849P}:
\begin{equation}
12 + \log{\rm{(O/H)}} = 5.80(\pm0.017) - 0.139(\pm0.011)\rm{M}_B
\end{equation}
that links the integrated absolute {\it B}-band magnitude with the average oxygen abundance of the galaxy, providing a value of $\sim$ 8, which suggests that the environment may have a significantly sub-solar metallicity.
A direct measurement of the host galaxy metallicity confirms this result.
We spectroscopically observed UGC~5979 with the Nordical Optical Telescope (NOT) equipped with ALFOSC$+$grism\#4. A 1.0" slit was placed on a bright H~II region at 19.5" (1.8~kpc) from SN 2007sv. After 1800 sec exposure, we obtained an optical spectrum with clear detection of narrow [O~III]~$\lambda$5007~\AA, [O~III]~$\lambda$4959~\AA, Balmer lines up to H$\gamma$, [N~II]~$\lambda$6584~\AA~and [S~II].
Via Balmer decrement we determined an extinction of $E(B-V)$~$=$~0.82~mag at the H~II region location and we corrected the spectrum accordingly. We measured the line fluxes by fitting them with Gaussians, as explained in detail in \cite{2013A&A...558A.143T}.
The detection of N~II, H$\alpha$, H$\beta$ and O~III lines allowed us to determine the oxygen abundance via strong line diagnostics, namely with the N2 and O3N2 methods \citep{2004MNRAS.348L..59P}.
Our results are log(O/H)$+$12 (N2)~$=$~8.01$-$8.1~dex and log(O/H)$+$12 (O3N2)~$=$~8.00$-$8.08~dex.
The quoted uncertainty is due to the error on the flux of N~II, which appears rather faint.
Sub-solar metallicities may be a rather common characteristic of SN impostor environments \citep[see][]{2014MNRAS.441.2230H} and we are currently investigating this issue for a large sample of impostor environments (Taddia et al. in prep).
\section{Photometry} \label{photometry}
\subsection{Data reduction and light curves}
Our photometric monitoring campaign started on December 30, 2007 and spanned a period of about 100 days.
We also collected sparse observations (mostly unfiltered) from amateur astronomers.
Information about the photometric data and the instruments used are reported in Table \ref{LightCurves}. \\
All data were pre-processed using standard procedures in \textsc{iraf}\footnote{\textsc{iraf} is distributed by the National Optical Astronomy Observatory, which is operated by the Associated Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.} including bias and flat field corrections.
To measure the magnitudes we used a dedicated pipeline developed by one of us (E. C.), that consists of a collection of \textsc{python} scripts calling standard \textsc{iraf} tasks (through \textsc{pyraf}) and other specific analysis tools, in particular \textsc{sextractor} for source extraction and star/galaxy separation, \textsc{daophot} to measure the source magnitude via PSF fitting and \textsc{hotpants}\footnote{http://www.astro.washington.edu/users/becker/hotpants.html} for image difference with PSF match.
\begin{table*}
\begin{minipage}[adjusting]{175mm}
\caption[Photometry of 2007sv]{Optical magnitudes of 2007sv and associated errors.}
\label{LightCurves}
\begin{tabular}{@{}cccccccccccccr@{}}
\hline
Date & MJD & U & err & B & err & V & err & R & err & I & err & Instrument \\
\hline
20070913 & 54356.09 & - & - & - & - & - & - & $>$18.8& - & - & - & SX MX7 \\
20071220 & 54454.91 & - & - & - & - & - & - & 17.161 & 0.261 & - & - & SX MX7 \\
20071227 & 54461.86 & - & - & - & - & 17.975 & 0.356 & 17.502 & 0.137 & - & - & MX916 \\
20071230 & 54464.15 & - & - & 18.502 & 0.077 & 18.161 & 0.039 & 17.786 & 0.023 & 17.432 & 0.023 & AFOSC \\
20080104 & 54469.04 & 19.145 & 0.147 & 18.860 & 0.025 & 18.220 & 0.014 & 17.812 & 0.024 & - & - & RATCam \\
20080106 & 54471.08 & 19.211 & 0.057 & 18.914 & 0.042 & 18.271 & 0.017 & 17.877 & 0.018 & 17.350 & 0.020 & RATCam \\
20080108 & 54473.21 & 19.264 & 0.085 & 18.963 & 0.025 & 18.225 & 0.019 & 17.802 & 0.029 & 17.327 & 0.020 & RATCam \\
20080110 & 54475.14 & - & - & 19.044 & 0.066 & 18.219 & 0.027 & 17.863 & 0.028 & 17.329 & 0.026 & AFOSC \\
20080110 & 54475.20 & 19.348 & 0.076 & 19.064 & 0.024 & 18.252 & 0.015 & 17.776 & 0.030 & 17.338 & 0.017 & RATCam \\
20080112 & 54477.25 & 19.373 & 0.025 & 19.124 & 0.024 & 18.311 & 0.026 & 17.929 & 0.033 & 17.522 & 0.032 & ALFOSC \\
20080113 & 54478.00 & 19.420 & 0.081 & 19.166 & 0.027 & 18.319 & 0.016 & 17.832 & 0.032 & 17.382 & 0.035 & RATCam \\
20080116 & 54481.12 & 19.544 & 0.138 & 19.208 & 0.020 & 18.354 & 0.026 & 17.768 & 0.029 & 17.452 & 0.028 & RATCam \\
20080118 & 54483.20 & 19.933 & 0.213 & 19.392 & 0.102 & 18.359 & 0.034 & 17.951 & 0.127 & - & - & RATCam \\
20080120 & 54485.94 & - & - & - & - & - & - & 17.984 & 0.265 & - & - & SX MX7 \\
20080128 & 54493.16 & - & - & 19.512 & 0.109 & 18.573 & 0.030 & 18.020 & 0.068 & 17.467 & 0.085 & CAFOS \\
20080128 & 54493.95 & - & - & 19.526 & 0.037 & 18.520 & 0.022 & 17.907 & 0.023 & 17.406 & 0.024 & RATCam \\
20080207 & 54503.93 & - & - & 19.885 & 0.036 & 18.696 & 0.021 & 18.030 & 0.025 & 17.514 & 0.032 & RATCam \\
20080209 & 54505.94 & - & - & - & - & - & - & 18.023 & 0.179 & - & - & Apogee Ap7 \\
20080211 & 54507.85 & - & - & - & - & - & - & 18.178 & 0.210 & - & - & SBIG ST-7 \\
20080213 & 54509.94 & - & - & - & - & - & - & 18.166 & 0.218 & - & - & SX MX7 \\
20080301 & 54526.88 & - & - & 20.532 & 0.053 & 19.211 & 0.028 & 18.328 & 0.041 & 17.843 & 0.056 & RATCam \\
20080304 & 54529.01 & - & - & - & - & - & - & 18.599 & 0.262 & - & - & SX MX7 \\
20080305 & 54530.98 & - & - & $>$20.4& - & 19.471 & 0.056 & 18.456 & 0.067 & 17.854 & 0.035 & CAFOS \\
20080331 & 54556.00 & - & - & $>$21.2& - & 20.803 & 0.057 & 19.455 & 0.064 & 18.936 & 0.049 & ALFOSC \\
20080401 & 54557.93 & - & - & $>$21.4& - & 21.074 & 0.047 & 19.745 & 0.044 & 19.177 & 0.184 & RATCam \\
\hline
\end{tabular}
\medskip
The observations were carried out using the 2.56~m Nordic Optical Telescope (NOT) with ALFOSC and the 2~m Liverpool Telescope with RATCam (both located at the Roque de Los Muchachos, La Palma, Canary Islands, Spain), the Calar Alto 2.2~m Telescope with CAFOS (Sierra de Los Filabres, Spain) and the 1.82~m Copernico Telescope with AFOSC (Mount Ekar, Asiago, Italy). Additional observations (mostly unfiltered) were obtained by amateur astronomers. \\
The magnitudes obtained from SX MX7, Apogee Ap7 and SBIG ST-7 were computed from unfiltered images, whose magnitudes were rescaled to R-band. \\
MX916: 0.45~m f4.5 newtonian telescope with a MX916 CCD Camera, at Mandi Observatory (Pagnacco, Udine, Italy) \\
SX MX7: 0.32~m f/3.1 reflector and a Starlight Xpress MX716 CCD camera at Moonbase Observatory (Akersberga, Sweden) \\
Apogee AP7: C-14 Celestron Schmidt Cassegrain reflector and an Apogee AP7 CCD camera (Suffolk, United Kingdom) \\
SBIG ST-7: 0.44~m f4.43 telescope with a SBIG ST-7 Dual CCD Camera at Sandvretens Observatorium (Uppsala, Sweden) \\
\end{minipage}
\end{table*}
\begin{table*}
\begin{minipage}{180mm}
\caption[Local standards]{Magnitudes of the stellar sequence used for the photometric calibration. The stars are shown in Figure \ref{findingchart}.}
\label{localstandards}
\begin{tabular}{@{}ccccccccccccc@{}}
\hline
label & ra~[J2000] & dec~[J2000] & U & err & B & err & V & err & R & err & I & err \\
& (hh:mm:ss) & (dd:mm:ss) & & & & & & & & & & \\
\hline
1 & 10:53:02.62 & 67:58:18.77 & - & - & - & - & 20.159 & 0.056 & 19.202 & 0.015 & - & - \\
2 & 10:53:01.92 & 67:59:46.33 & 18.451 & 0.020 & 17.110 & 0.015 & 15.861 & 0.029 & 14.983 & 0.022 & 14.214 & 0.027 \\
3 & 10:52:54.29 & 67:58:14.49 & 15.692 & 0.009 & 15.675 & 0.007 & 15.127 & 0.020 & 14.800 & 0.016 & 14.429 & 0.030 \\
4 & 10:52:50.03 & 67:57:21.82 & - & - & 20.571 & 0.010 & 19.113 & 0.024 & 18.195 & 0.015 & 16.425 & 0.014 \\
5 & 10:52:49.13 & 67:58:39.87 & 17.682 & 0.007 & 17.088 & 0.011 & 16.212 & 0.012 & 15.688 & 0.008 & 15.180 & 0.012 \\
6 & 10:52:43.00 & 67:57:22.46 & 16.401 & 0.004 & 16.329 & 0.005 & 15.661 & 0.010 & 15.295 & 0.006 & 14.887 & 0.003 \\
7 & 10:52:27.18 & 68:00:26.52 & - & - & - & - & 18.245 & 0.013 & 17.770 & 0.005 & 17.299 & 0.015 \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\begin{figure}
\begin{center}
\includegraphics[scale=0.32]{./figure01.eps}
\caption[Findingchart for 2007sv]{Finding chart of 2007sv. The stars used for the photometric calibration are labelled with a number, the position of the transient is marked with an arrow. Information about the intrumental set-up is also reported.}
\label{findingchart}
\end{center}
\end{figure}
The magnitudes were measured using the PSF-fitting technique, first subtracting the sky background calculated using a low order polynomial fit (typically a 2nd order polynomial).
The PSF was obtained by averaging the profiles of isolated field stars.
The fitted source is removed from the original frames, then a new estimate of the local background is derived and the fitting procedure is iterated.
Finally, the residuals are visually inspected to validate the fit. \\
Error estimates were obtained through artificial star experiments in which a fake star, of magnitude similar to that of the SN, is placed in the PSF-fit residual image in a position close to, but not coincident with that of the real source.
The simulated image is processed through the PSF fitting procedure and the dispersion of measurements out of a number of experiments (with the fake star in slightly different positions), is taken as an estimate of the instrumental magnitude error.
This is combined (in quadrature) with the PSF-fit error returned by \textsc{daophot}. \\
The SN photometry was calibrated as follows. Among the observational data, we selected the frames obtained in photometric nights in which standard photometric fields \cite[from the list of ][]{1992AJ....104..340L} were observed.
These standard frames were used to derive zero points and colour terms for the specific instrumental set-up and to calibrate the magnitudes of selected stars in the SN field (Table \ref{localstandards} and Figure \ref{findingchart}).
This local sequence was used to calibrate the SN magnitudes in non-photometric nights.
The final magnitudes were computed using first-order colour-term corrections.
The resulting magnitudes of 2007sv are reported in Table \ref{LightCurves} along with the photometric errors, and the light curves are shown in Figure \ref{lightcurves}.
\subsection{Constraining the SN age}
Since no observation was obtained a short time before the discovery of 2007sv, the epochs of the outburst on-set and the light curve maximum cannot be precisely constrained.
Nonetheless, we have adopted the epoch of the discovery as reference time for the light curve phases, assuming that the transient was discovered in the proximity of the maximum light.
In other words, hereafter we will adopt as indicative epoch for the light curve maximum the discovery time (December 20, 2007; JD = 2454455.41).
The goodness of our assumption is supported by the photometric and spectroscopic analysis that will be widely discussed in the next sections.
\begin{figure}
\begin{center}
\includegraphics[scale=0.49]{./figure02.eps}
\end{center}
\caption[lightCurve]{Multi-band light curves of 2007sv. The UBVRI magnitudes are listed in Table \ref{LightCurves}. The phases refer to the discovery.}
\label{lightcurves}
\end{figure}
Briefly, our choice is motivated on the basis of the following line of thinking. As we will discuss in the forthcoming sections, 2007sv shows very fast temperature/colour evolutions during the first
3-4 weeks, suggesting that the transient was discovered very young, a few days after the burst. This also implies that it is unlikely that 2007sv reached a peak magnitude much brighter than the discovery magnitude $R$ = 17.16 (Table \ref{LightCurves}). If we adopt for 2007sv the distance modulus and the reddening value discussed
in Section \ref{host}, we obtain an absolute magnitude M$_{R} = -$14.25 $\pm$ 0.38 at discovery, which is an indicative estimate for the absolute peak magnitude.
This weak absolute magnitude at maximum is an indication that 2007sv was very likely a SN impostor rather than a genuine SN explosion.
\subsection{Absolute magnitude and colour curves} \label{abscol}
In Figure \ref{abscurves} we compare the absolute R-curve of 2007sv with those of the SN impostor 1997bs \citep{2000PASP..112.1532V}, SN~2008S \citep{2009MNRAS.398.1041B} and the type IIn SN~1999el \citep{2002ApJ...573..144D}.
As highlighted in this comparison, the absolute peak magnitude of 2007sv is similar to those measured for the SN impostor 1997bs and the enigmatic transient SN~2008S, and significantly fainter than that of a canonical SN IIn such as SN~1999el.
Although the faint absolute magnitude supports the SN impostor scenario for 2007sv, this argument alone is not sufficient to rule out a true SN explosion, as we will discuss in Section 5. \\
The $B-V$, $V-R$ and $V-I$ colour curves of 2007sv are compared in Figure \ref{colcurves} with those of the same SNe considered in Figure \ref{abscurves}, showing that 2007sv rapidly becomes red (in analogy with other SN impostors discovered soon after the burst) as the temperature of the ejecta rapidly decreases (Section \ref{spectroscopy}).
On the other hand, the regular type IIn SN~1999el remains bluer for a longer time.
In particular, it has much bluer colours than the other transients at phases later than 100 days.
\begin{figure}
\begin{center}
\includegraphics[scale=0.48]{./figure03.eps}
\end{center}
\caption[Absolute R curves comparison]{Comparison of the absolute R-band light curves of the impostors 2007sv and 1997bs, the enigmatic transient SN~2008S, and the classical type IIn SN~1999el. The red line indicates the slope of the $^{56}$Co decay.}
\label{abscurves}
\end{figure}
\begin{table*}
\begin{minipage}{175mm}
\caption[Log of the spectroscopical observations]{Log of the spectroscopical observations of 2007sv. The phase refers to the discovery.}
\label{speclog}
\begin{tabular}{@{}cccccccc@{}}
\hline
Date & MJD & Phase & Instrumental setup & Grism or grating & Spectral range & Resolution & Exp. times \\
& & (days) & & & (\AA) & (\AA) & (s) \\
\hline
20071229 & 54463.96 & 9 & Ekar182+AFOSC & 2$\times$gm4 & 3450-7800 & 24 & 2$\times$1800 \\
20080105 & 54470.05 & 15 & TNG+LRS & LR-B & 3600-7770 & 19.9 & 2700 \\
20080110 & 54475.08 & 20 & Ekar182+AFOSC & gm2+gm4 & 3450-8100 & 34; 24 & 2$\times$1800 \\
20080113 & 54478.26 & 23 & TNG+LRS & LR-R & 5070-10100 & 10.7 & 1800 \\
20080114 & 54479.12 & 24 & NOT+ALFOSC & gm4 & 3300-9100 & 9.6 & 1800 \\
20080119 & 54484.43 & 30 & HET & LRS & 4300-7300 & 6.2 & 2$\times$1350 \\
20080128 & 54493.17 & 38 & CAHA+CAFOS & b200 & 3200-8800 & 12.3 & 3600 \\
20080201 & 54497.12 & 42 & WHT+ISIS & spec & 3200-10300 & 4.8; 9.8 & 1200 \\
20080301 & 54526.25 & 71 & HET & LRS & 4300-7300 & 6.0 & 4$\times$1125 \\
20080308 & 54533.04 & 78 & CAHA+CAFOS & b200+g200 & 4800-10700 & 13.8 & 2$\times$2400 \\
\hline
\end{tabular}
\medskip
The spectra were obtained using the 1.82~m Telescopio Copernico with AFOSC, the 3.58~m Telescopio Nazionale Galileo (TNG) with DOLoRes (La Palma, Canary Islands, Spain), the 4.2~m William Herschel Telescope with ISIS (La Palma, Canary Islands, Spain), the 2.56~m Nordic Optical Telescope (NOT) with ALFOSC, the Calar Alto 2.2~m telescope with CAFOS and the 11.1x9.8~m Hobby-Eberly Telescope (HET, Mt. Fowlkes, Texas, USA) with LRS.
\end{minipage}
\end{table*}
\begin{figure}
\begin{center}
\includegraphics[scale=0.55]{./figure04.eps}
\caption[Colour-curves comparison]{Comparison among the $B-V$ (top), $V-R$ (middle) and $V-I$ (bottom) colour curves of the same sample as in Figure \ref{abscurves}. All phases refer to the epoch of the maximum, which for 2007sv we assumed to be coincident with the discovery epoch.}
\label{colcurves}
\end{center}
\end{figure}
The comparisons shown above highlight some of the similarities between the SN impostors 2007sv and 1997bs and the peculiar transient SN~2008S, whose nature has not been firmly established yet \citep[genuine electron-capture SN or SN impostor, see][and Section \ref{cfrsp}]{2008ApJ...681L...9P,2009MNRAS.398.1041B,2009ApJ...697L..49S,2009ApJ...705.1364T,2009ApJ...705L.138P,2010MNRAS.403..474W,2011ApJ...741...37K,2012ApJ...750...77S}, all showing a fainter maximum and a different evolution in the light curve when compared with the light curve of the interacting SN~1999el.
\section{Spectroscopy} \label{spectroscopy}
Spectroscopic observations were carried out from December 29, 2007 (i.e. 9 days after the discovery) to March 7, 2008 (+78 days from discovery).
Basic information on the spectra and the instrumental configurations is reported in the log of spectroscopic observations (Table \ref{speclog}).
All data were processed using standard \textsc{iraf} tasks in order to perform the pre-reduction analysis (bias, flat and overscan corrections) and the extractions of the mono-dimensional spectra.
Wavelength calibration was performed using the spectra of comparison lamps obtained with the same instrumental setup.
Flux calibration was performed using the spectrum of standard stars.
The accuracy of the wavelength calibration was verified measuring the wavelength of night sky lines (in particular [OI] at 5577.34~\AA~or 6300.30~\AA), and a shift was applied in case of discrepancy.
Spectral resolution was measured from the full-width-at-half-maximum (FWHM) of night sky lines, adopting their mean value as final resolution estimate.
The final spectral flux calibration was checked against multi-band photometry obtained on the nearest night and, when necessary, a scaling factor was applied.
Telluric corrections were applied when spectra of references stars obtained during the same nights were available.
The spectral sequence obtained with the above procedures is shown in Figure \ref{spectraSequence}.
\begin{figure*}
\begin{center}
\includegraphics[scale=0.95]{./figure05.eps}
\caption[spectraSequence]{Spectral sequence of 2007sv. The date and the instrumental set-up are reported on the left, the phase (in days after the discovery) is indicated on the right. Spectra are flux-calibrated. The $\oplus$ symbols mark the position of the visible telluric absorptions.}
\label{spectraSequence}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[scale=0.57]{./figure06.eps}
\caption[linesIdentification]{Line identification of the ISIS spectrum obtained on February 1, 2008 of 2007sv. A comparison with the spectrum of the transient UGC2773-2009OT1 obtained with the Telescopio Nazionale Galileo + LRS on Oct 11, 2009 (slightly before maximum, Padova-Asiago SN archive) is also shown. The spectra are flux-calibrated and redshift-corrected. H and Ca~II lines are marked at their rest wavelengths; the marks for the other lines are blue--shifted by $\simeq$500~km~s$^{-1}$. The $\oplus$ symbols mark the positions of prominent telluric absorption bands.}
\label{lineid}
\end{center}
\end{figure*}
\subsection{Spectral evolution and line identification} \label{idlines}
From Figure \ref{spectraSequence}, we note that all spectra of 2007sv are dominated by a prominent and narrow H$_{\alpha}$ emission line.
We also remark that there is relatively little evolution in the spectral features during the almost 80-days coverage window, except for the continuum becoming progressively redder.
As the spectra are characterized by a significant continuum contribution to the total flux, we estimated the temperature of the emitting region through a black-body fit.
The temperature experienced a rapid decline from $\simeq$ 8000~K in the +9d spectrum to $\simeq$ 5000 K at phase $\sim$20d.
Thereafter the temperature slowly declines to $\simeq$ 4000~K in the late-time spectra (71-78d; Figure \ref{spectraSequence}, see also Section \ref{hapro}).
This evolution is consistent with that of the broad-band colours (Figure \ref{colcurves}).
As mentioned above, there is little spectral evolution during the almost 80d of spectral coverage. However, it should be noticed that the spectra have relatively poor resolution and low S/N. Therefore, it is difficult to measure evolution in the weak and narrow spectral features.
In order to investigate the presence of low-contrast spectral lines, we inspected in detail one of our highest S/N spectrum (ISIS, phase +42d).
The line identification is shown in Figure \ref{lineid}.
The identification was performed by comparing the spectra of 2007sv with that of the SN impostor UGC2773-20009OT1 from the Padova-Asiago SN archive (also shown in Figures \ref{lineid} and \ref{speccompare}, res. 11~\AA). A comprehensive line identification for UGC2773-2009OT1 was performed by \citet{2010AJ....139.1451S} and \citet{2011ApJ...732...32F}.
Figure \ref{lineid} shows that, despite the different resolution, the spectra of the two transients are very similar, with a number of lines in common \citep[see e.g. Figures 8 to 12 in][]{2010AJ....139.1451S}.
We determined an indicative photospheric velocity in the spectrum of 2007sv ($\simeq$~500~km~s$^{-1}$) from the blue-shifted absorption component of the Ba II 6496.6~\AA~line, and the expected positions of all other absorption lines were derived by adopting this velocity for all ions.
The lines with strong emission components (e.g H$_{\alpha}$, H$_{\beta}$, H\&K Ca II and the NIR Ca II triplet) were identified using their rest wavelengths.
We marked only multiplets with an intensity of the strongest line 5$\sigma$ above the noise level.
We also identified O~I (multiplets 1 and 4), Ba~II (multiplets 1 and 2), Na~ID (doublet at 5889.9 and 5895.9~\AA), Sc~II (multiplets 13, 28, 29, 31), Fe~II (multiplets 27, 28, 37, 38, 40, 42, 46, 48, 49, 74, 199, 200).
These lines show a narrow P-Cygni profile (blueshifted by about 650~km~s$^{-1}$), although a shallow high-velocity component in the NIR Ca II triplet cannot be ruled out (see Section \ref{cfrsp}).
We also marked the positions of the [Ca II] doublet lines at 7292\AA~and 7324\AA, which are prominent in the spectrum of UGC2773-20009OT1 but not in the 2007sv spectrum.\footnote{We also note that a telluric feature matches the wavelength position of the [Ca II] doublet, therefore no robust conclusion can be inferred on the identification of this feature in the 2007sv spectrum.}
\subsection{H$_\alpha$ profile and evolution of the main observables} \label{hapro}
The evolution of the H$_{\alpha}$ profile during the almost 80d of spectral coverage is highlighted in Figure \ref{halphaev}.
We notice that there is no significant change in the wavelength position of the H$_{\alpha}$ emission peak.
\begin{figure}
\begin{center}
\includegraphics[scale=0.44]{./figure07.eps}
\caption[Evolution of the profile of H$_{\alpha}$]{Evolution of the profile of H$_{\alpha}$ in the velocity space.}
\label{halphaev}
\end{center}
\end{figure}
The H$_{\alpha}$ line profile is relatively complex.
A narrow component is detected in our higher resolution spectra of 2007sv, but a simple Gaussian or Lorentzian line fit does not well reproduce the entire line profile.
For this reason, we adopted a combination of multiple line components to improve the accuracy of the spectral line fit.
A broad component (decreasing from $\geq$ 2000 to $\simeq$ 1700~km~s$^{-1}$) is visible in the two earliest spectra (phases +9d and +15d). This component is only marginally detected in the two subsequent spectra (+20d and +23d), and disappears at later phases. In fact, the broad component is below the detection threshold in the +24d ALFOSC spectrum.
Starting from the LRS spectrum at +23d, we improved our fits by including an intermediate-width (FWHM velocity $\approx$ 600).
While we cannot rule out that the intermediate component was also present at earlier phases, the modest resolution of the +9d to +20d spectra prevents us its discrimination from the narrow component.
In the spectra collected at later phases, a two-component (intermediate + narrow) fit well reproduces the observed line profile.
In order to analyze the evolution of the velocity of the different H$\alpha$ components and the total line flux, we first corrected the spectra for redshift (adopting 1116~km~s$^{-1}$ as the mean heliocentric radial velocity\footnote{http://leda.univ-lyon1.fr/}) and for foreground Galactic extinction (using the values mentioned in Section \ref{host}).
Then we measured the total line flux, and the full-width-at-half-maximum (FWHM) velocities of the three H$_{\alpha}$ line components.
In most spectra the narrow component was unresolved, and even the intermediate component was occasionally below (or near) the resolution limits.
In these cases, a multicomponent fit using a combination of Gaussian functions provided good fits.
However, in some cases (namely for the two higher resolution HET spectra), we used a Gaussian function for the intermediate component and a Lorentzian profile for the narrow component.
Figure \ref{tempspeed} (top panel) shows the evolution of the velocity of the ejected material for the three line components.
\begin{figure}
\begin{center}
\includegraphics[scale=0.39]{./figure08.eps}
\caption[Velocity, $_\alpha$ luminosity and temperature evolution]{{\bf Top.} FWHM evolution for the broad (black squares, solid line), the intermediate (blue circles) and the narrow H$_{\alpha}$ (red triangles) components.
{\bf Middle.} Evolution of the total luminosity of H$_{\alpha}$. We assume a 10$\%$ error in the measures, due only to the error in the flux calibration.
{\bf Bottom.} Evolution of the spectral continuum temperature.}
\label{tempspeed}
\end{center}
\end{figure}
As mentioned above, the narrow component was unresolved in most cases.
When the narrow H$_{\alpha}$ was unresolved, we used the spectral resolution as an upper limit for the velocity of the slowest-moving material.
When the narrow line component was resolved, we first corrected the measured FWHM for the spectral resolution ($width=\sqrt{\rm{FHWM}^2-\rm{res}^2}$) and then computed the velocity ($v=\frac{width}{6562.8} \times c$).
In the highest resolution HET spectra at phases +30 and +71d we measured the FWHM of the narrower component as 120$\pm$30~km~s$^{-1}$ and 150$\pm$40~km~s$^{-1}$, respectively.
The intermediate component remains at roughly constant velocity around 600-800~km~s$^{-1}$ at all epochs.
Finally, the broad component is characterized by a fast decline from $\simeq$ 2000~km~s$^{-1}$ in our first spectrum to $\approx$ 1200~km~s$^{-1}$ in the +23d spectrum.
We note that these values are significantly smaller that the typical values of $\simeq$ 10000~km~s$^{-1}$ measured in the ejecta of young SNe. \\
Multiple line components in the spectra of interacting objects are known to arise from different emitting gas shells \citep[see e.g.][]{1993MNRAS.262..128T}.
The very small velocities inferred for the narrow H$_{\alpha}$ in the HET spectra (120-150~km~s$^{-1}$) are consistent with those expected in the winds of an LBV.
The velocity evolution of the broad component is consistent with material violently ejected, and in particular with the velocities observed in the fastest hydrogen-rich material expelled in major eruptions of LBVs \citep{2008Natur.455..201S,2010MNRAS.408..181P,2013ApJ...767....1P}.
More puzzling is the interpretation of the intermediate component.
According to the interpretation usually adopted in interacting SNe \citep[see e.g.][]{1994ApJ...420..268C}, the intermediate velocity component arises in the gas region between the forward shock and the reverse shock.
In the case of 2007sv, the relative strength of this component progressively increases with time with respect to that of the narrow component.
This would support the idea that a significant fraction of the line flux at late phases arises from the gas interface between the two shock fronts, hence from the shocked gas region.
In addition, one may note that the intermediate component is significantly blue-shifted with respects to the narrow one.
In the ejecta/CSM interaction scenario, a blue-shifted intermediate component may be explained with an attenuation of the red line wind due to prompt dust formation in a post-shock cool dense shell, as observed in a number of interacting SNe \citep[e.g. 2006jc,][]{2008ApJ...680..568S,2008ApJ...684.1343N,2008MNRAS.389..141M}.
Alternatively, very asymmetric and blue-shifted line profiles may be interpreted in terms of a highly asymmetric geometrical distribution of the CSM \citep[see e.g. the interpretation of][for SN~2006jd]{2012ApJ...756..173S}. \\
A progressive enhancement of ejecta/CSM interaction emission can be inferred observing the evolution of the total H$_{\alpha}$ flux in the latest spectra (phase $>$ 70d; Table \ref{specphysics} and Figure \ref{tempspeed}, middle panel). The flux decreases from about 7$\times$10$^{-15}$~erg~s$^{-1}$ to 3$\times$ 10$^{-15}$~erg~s$^{-1}$ during the first $\sim$ 40~days.
Later on, we note an increase by a factor almost two in the H$_{\alpha}$ flux, approximatively about 5.5$\times$10$^{-15}$~erg~s$^{-1}$ in the last two spectra. As mentioned above, this can be interpreted as an increased contribution of the intermediate component arising in a shocked gas region which dominates the flux contribution at late phases over the other line components.
\begin{table*}
\begin{minipage}{177mm}
\caption[Main parameters inferred for the H$_{\alpha}$ line]{Main parameters inferred from the spectra of 2007sv.}
\label{specphysics}
\begin{tabular}{@{}cccccccc@{}}
\hline
Phase & FWHM(H$_{\alpha,broad}$) & FWHM(H$_{\alpha,intermediate}$) & FWHM(H$_{\alpha,narrow}$) & H$_{\alpha}$ Luminosity & Resolution \\
(days) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (10$^{38}$ erg s$^{-1}$) & (km s$^{-1}$) \\
\hline
9 & 2030$\pm$830 & $<$ 1100 & - & 2.9 & 1100 \\
15 & 1700: & $<$ 910 & - & 2.0 & 910 \\
20 & 1700$\pm$360 & $<$ 1100 & - & 1.4 & 1100 \\
23 & 1220$\pm$160 & 600$\pm$100 & $<$ 490 & 1.7 & 490 \\
24 & - & 850$\pm$180 & $<$ 440 & 1.4 & 440 \\
30 & - & 720$\pm$100 & 120$\pm$30 & 1.3 & 280 \\
38 & - & 600$\pm$120 & $<$ 560 & 1.4 & 560 \\
42 & - & 640$\pm$100 & $<$ 450 & 1.6 & 450 \\
71 & - & 730$\pm$100 & 150$\pm$40 & 2.4 & 270 \\
78 & - & 640$\pm$130 & $<$ 630 & 2.3 & 630 \\
\hline
\end{tabular}
\medskip
The measured FWHM of the broad, intermediate and narrow components of H$_{\alpha}$ are reported in columns 2, 3 and 4, respectively. The measure marked with the : symbol is uncertain. \\
The total luminosity of H$_{\alpha}$ is in column 5, the spectral resolution in column 6.
\end{minipage}
\end{table*}
\subsection{Spectral comparison with other interacting transients} \label{cfrsp}
\begin{figure}
\begin{center}
\includegraphics[scale=0.44]{./figure09.eps}
\caption[Spectral comparison]{Comparison of early-time spectra of the impostors 2007sv, 1997bs (unpublished spectrum from the Padova--Asiago SN archive obtained on 1997 April 30th with the ESO1.52~m telescope, resolution 10\AA, located at La Silla, Chile) and UGC2773-2009OT1 (spectrum obtained slightly before maximum), the enigmatic transient SN~2008S, and the linearly-declining type IIn SN~1999el. The spectra are flux-calibrated and redshift-corrected.}
\label{speccompare}
\end{center}
\end{figure}
An important issue is to determine whether the spectroscopy alone allows us to discriminate between genuine type IIn SNe and SN impostors.
For this goal, we compare in Figure \ref{speccompare} the AFOSC early-time spectrum (phase +9d) of 2007sv with spectra of young transients with narrow emissions, viz. the impostors 1997bs \citep{2000PASP..112.1532V} and UGC2773-2009OT1 (Padova--Asiago SN Archive; Pastorello et al. in preparation), the classical type IIn SN~1999el \citep{2002ApJ...573..144D} and SN~2008S \citep{2009MNRAS.398.1041B}.
SN~2008S is the prototype of a small family of intermediate-luminosity transients \citep[see][and references therein]{2009ApJ...705.1364T} whose nature has been widely debated.
Although many observables of SN~2008S are similar to those observed in SN impostors, the detection of prominent, narrow [Ca~II] (7292--7324~\AA) lines and, even more, the late-time light curve with a decline rate consistent with that expected from the $^{56}$Co decay \citep{2009MNRAS.398.1041B}, provide reasonable arguments to support a faint SN scenario.
The progenitor star of SN~2008S was detected in mid-infrared archive Spitzer images, whilst there was no detection in deep optical and near-IR pre-explosion frames \citep[e.g.][]{2008ApJ...681L...9P}. This was interpreted as a clear signature that the progenitor was a highly reddened star, embedded in a dusty environment.
Although there is general agreement that the progenitor star of SN~2008S was a moderate-mass star\footnote{A similar conclusion was also inferred for the detected progenitor of the 2008S-analogous NGC300-2008OT1 \citep{2009ApJ...695L.154B,2009ApJ...699.1850B}}, the characterization of the stellar type is somewhat different in the different papers, ranging from a $\sim$9~M$_{\odot}$ extreme asymptotic giant branch star \citep[AGB; e.g.][]{2008ApJ...681L...9P,2010ApJ...715.1094K,2011ApJ...741...37K,2012ApJ...750...77S} to a $\leq$ 20~M$_\odot$ supergiant \citep[][]{2009ApJ...697L..49S}.
The most debated issue is whether the observed 2008 outburst was a terminal stellar explosion, most likely as an electron-capture SN from a super-AGB star \citep{2009ApJ...705L.138P,2013ApJ...771L..12T} or an LBV-like outburst of a mildly massive star \citep[][]{2009ApJ...697L..49S,2011MNRAS.415..773S}.
From the comparison in Figure \ref{speccompare}, it is evident that the spectra of all these transients are rather similar, and many spectral lines are in common to all of them.
Therefore this is an indication that the spectra alone may not be sufficient to discriminate between impostors and true SNe.
As mentioned before, the narrow [Ca~II] doublet at 7292--7324~\AA~is the hallmark feature for SN~2008S-like transients and is sometimes used as an argument to support the SN nature of these objects.
We note that there is no clear evidence for the presence of [Ca~II] lines in the spectra of 2007sv or 1997bs.
However, we have to admit that the [Ca~II] feature was detected in UGC2773-2009OT1 \citep[which is clearly an impostor, see][]{2010AJ....139.1451S,2011ApJ...732...32F}.
Therefore, the [Ca~II] feature is not a good discriminant of the nature of these explosions.
In Figure \ref{cacomp}, the 7800--8700~\AA~wavelength window of the +42d ISIS spectrum of 2007sv is compared with spectra of SN~2008S and UGC2773-2009OT1.
In all of them we find the Ca~II triplet at 8498.0~\AA, 8542.1~\AA~and 8662.1~\AA, which is another very common feature in many types of transients, although some differences in the line strengths and velocities can be appreciated.
The three spectra show narrow features with velocities of a few hundreds~km~s$^{-1}$, and these mark the presence of slow-moving material.
However, a very broad depression with a minimum at about 8200~\AA~is visible in all spectra in Figure \ref{cacomp}, suggesting that a small amount of material can be ejected at high velocities (above 10000~km~s$^{-1}$) also in SN impostors.
The presence of fast-moving material has been also reported in the $\eta$ Car circum-stellar environment \citep{2008Natur.455..201S}.
Therefore, the detection of high-velocity gas alone should not be considered a robust argument to favor a SN scenario\footnote{We note, however, that this argument is instead used by many authors to favor the SN explosion scenario for the debated SN~2009ip.}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.44]{./figure10.eps}
\caption[Comparison between Ca~II lines of different objects]{Comparison of the NIR Ca~II triplet line profiles in three different types of faint transients, viz. 2007sv, UGC2773-2009OT1 and SN~2008S (from top to bottom). The three dotted lines mark the position of the lines of the Ca~II triplet at 8498.0~\AA~8542.1~\AA~and 8662.1~\AA. The dashed is an indicative line that marks the velocity close to the terminal velocity of the gas. The spectra are flux-calibrated and redshift-corrected.}
\label{cacomp}
\end{center}
\end{figure}
\section{Discussion} \label{discussion}
In Sections 3 and 4 the photometric and spectroscopic properties of the optical transient 2007sv have been described.
The main goal of the forthcoming discussion is to provide convincing insights on its nature (SN vs. SN impostor).
Already from a quick investigation of the spectra of 2007sv, the similarity with the spectra of well-known SN impostors \citep[e.g. 1997bs and UGC2773-2009OT1,][]{2000PASP..112.1532V,2010AJ....139.1451S,2011ApJ...732...32F} is evident.
However, some similarity can also be found with the spectra of genuine interacting SNe \citep[such as SNe~1999el and 1995G,][]{2002ApJ...573..144D,2002MNRAS.333...27P}.
The comparisons shown in Section \ref{cfrsp} confirm that there are only subtle differences between spectra of LBV-like eruptions and genuine type IIn SNe, giving evidence that from the spectroscopic analysis alone it is sometimes tricky to discriminate between the two types of transients.
However, from an in-depth inspection of our spectral sequence of 2007sv, we can obtain crucial information on this object.
The increased total luminosity of H$_{\alpha}$ and the enhanced strength of the intermediate-velocity component in the late time spectra suggest that the material ejected in the outburst was interacting with the pre-existing CSM.
We also found that next to the expected narrow components, H$_\alpha$ and the Ca~II NIR triplet show broader wings (Figure \ref{cacomp}), suggesting an outflow of material at velocities comparable with those observed in SN ejecta.
The maximum velocity registered for the outflowing material (dashed line in Figure \ref{cacomp}) is about 14000~km~s$^{-1}$ (although it is clear that the bulk of this material is expanding at much lower velocity, viz. $\sim$ 8000~km~s$^{-1}$).
The detection of prominent broad spectral features from typical nucleosynthesis products observed in SN ejecta would be a more robust tool to distinguish between SNe and impostors.
We do not detect any broad line of $\alpha$- or Fe-peak elements in the late-time spectra (approx. +80d) of 2007sv, except for the shallow absorption attributed to the Ca~II NIR triplet.
For this reason, the general spectral properties of 2007sv favor a non-terminal explosion scenario for this transient, although we have to admit that these can not be considered as conclusive proofs to unveil the nature of this interacting transient.
A more robust constraint can be derived from the photometric analysis. In some cases, impostors were unmasked through their erratic light curves.
It is worth mentioning the optical transient observed during the period 2000-2009 in NGC~3432 \citep[aka 2000ch,][]{2004PASP..116..326W,2010MNRAS.408..181P} and also the 2009-2012 recurrent transient observed in NGC~7259 \citep[known as 2009ip,][]{2010AJ....139.1451S,2011ApJ...732...32F,2013ApJ...767....1P}.
The latter was followed by a major eruption in mid-2012 that has been proposed to be a terminal SN explosion \citep[][and references therein]{2013MNRAS.tmp.2960S}.
However, more frequently impostors reveals themselves through a single episode, characterized by a fast-evolving but regular light curve.
A classical example is 1997bs \citep{2000PASP..112.1532V}, whose light curve shape is not trivially discernible from that of a regular SN.
And the light curve of 2007sv is remarkably similar to that of 1997bs.
However, the colours of 2007sv rapidly become redder as the object evolves, due to the decreasing temperature of the emitting region (this finding is confirmed by the temperature evolution inferred through black-body fits to the spectral continuum).
The colour/temperature transition is observed to occur on much shorter timescales than in typical SNe IIn (see e.g. Fig. \ref{colcurves}).
Finally, the peak luminosity still remains the most used method to discriminate between SNe and impostors.
With a distance modulus of 31.38 $\pm$ 0.27 mag, we derive for 2007sv an absolute magnitude of M$_R = -14.25 \pm$ 0.38.
This value is 4 to 6 magnitudes fainter than the absolute magnitudes typically measured in type IIn SNe \citep[e.g.][]{2002AJ....123..745R}.
\subsection{Which mechanisms can produce 2007sv-like events?}
The faint absolute magnitude at the discovery and the rapid colour (temperature) evolution provide robust evidence for the impostor nature for 2007sv, though some cases of faint transients exist in the literature that have been proposed to be true SNe.
In fact, weak SNe can be produced via i) the core-collapse explosion of a peculiar SN through electron capture of a moderate-mass star; ii) the fall back event from a very massive star \cite[][and references therein]{2009ApJ...705L.138P}.
Both mechanisms are believed to produce absolute light curves fainter than those observed in canonical SN types.
In the former scenario the ONeMg stellar core of a 8-10 M$_{\odot}$ super-AGB star collapses generating a weak, low-energy event called electron-capture (EC) SN.
The SN ejecta may eventually interact with an H-rich circumstellar environment generated by the stellar mass-loss during the super-AGB phase.
As mentioned in Section \ref{cfrsp}, a promising candidate EC SN is SN~2008S \citep{2009MNRAS.398.1041B}, an object that shares some similarity with a SN impostor, but has a SN-like shaped light curve, a late-time light curve consistent with the $^{56}$Co decay.
In the latter scenario, the collapse of a very massive star \citep[$>$ 25-30 M$_{\odot}$,][]{2003MNRAS.338..711Z} is followed by the fall-back of the inner stellar mantle onto the stellar core, generating eventually a black hole.
In both scenarios a common feature is the faint absolute magnitude, which is generally due to the small amount of radioactive $^{56}$Ni in the ejecta.
The presence of radioactive material in the ejecta can be revealed from the decline rate of the late-time SN light curve.
However, in massive stars the interaction of the ejecta with the pre-existing CSM can induce a dramatic increase in the radiated energy, and cause significant deviations from the expected luminosity peak and light curve decline rate expected in the radioactive decays, making the detection of $^{56}$Co signatures problematic.
Another efficient mechanism proposed to explain transient events with a total radiated energy comparable with those of real SNe is the pulsational pair-instability in very massive stars.
\citet{2007Natur.450..390W} showed that major instabilities produced by electron-positron pair production (pulsational-pair instability) cause the ejection of massive shells without necessarily unbinding the star (and hence without leading to a terminal SN explosion).
These major mass-loss episodes might produce transients currently classified as SN impostors. In addition to this, when a new shell is ejected and collides with pre-existing material, the resulting radiated energy is comparable with that of a core-collapse SN (sometimes even one order of magnitude higher).
If the impacting material is H-rich, the shell-shell collisions would produce a SN IIn-like spectrum and a slowly-evolving, luminous light curve that would make the transient practically indiscernible from true SNe IIn.
All of this further complicates our attempts of discriminating SNe IIn from eruptive impostors.
A safe discrimination criterion would be the detection of the products of stellar and core-collapse explosive nucleosynthesis through the prominent $\alpha$-element lines in the nebular spectra.
But in many SNe IIn the inner ejecta are covered by the H-rich interaction region sometimes for very long timescales (up to many years), making the detection of the $\alpha$-element spectral features difficult.
All the clues illustrated so far make us confident that 2007sv was not a terminal SN explosion, but very likely a major eruption mimicking the SN behaviour.
If this is true, the progenitor star may have reached again a quiescent stage, returning to the pre-eruptive bolometric luminosity.
This can be confirmed through an inspection of deep, high resolution images obtained years after the outburst, for example using the Hubble Space Telescope or the largest ground based telescopes which can deliver sub-arcsecond images.
The identification of the quiescent progenitor in such high quality images would be final evidence that the massive star producing 2007sv is still alive.
Alternatively, a long timescale monitoring of 2007sv can eventually reveal further outbursts after the one registered in 2007, which would also prove that the 2007 episode was not the final stellar death.
This strategy worked well for the 2000 transient observed in NGC~3432 \citep{2004PASP..116..326W} that was recovered after 8 years during a subsequent eruptive phase \citep{2010MNRAS.408..181P}.
\subsection{Is 2007sv heralding a SN explosion?}
In order to identity possible further outbursts experienced by the progenitor of 2007sv, we analyzed a number of images of the transient site obtained before and after the 2007 event.
These data were mostly collected by two of the co-authors of this paper (T.B. and G.D.), with a few additional observations performed with the 1.82~m Telescopio Copernico of the Asiago Observatory.
These observations are listed in Table \ref{limtab} in Appendix \ref{lim}.
The unfiltered observations of T.B. and G.D. were scaled to R-band magnitudes using the magnitudes of references stars reported in Table \ref{localstandards}.
These images of the 2007sv site span a period of over a decade.
In this temporal window, we did not detect any further signature of a variable source at the position of 2007sv (see Figure \ref{limits}).
The Pan-STARRS1 survey imaged this galaxy on 52 separate nights between Feb. 25, 2010 (MJD 55252.30) and May 17, 2013 (MJD 56429.26) in one of the filters $g_{\rm P1}r_{\rm P1}i_{\rm P1}z_{\rm P1}$
\citep[for a description of the Pan-STARRS1 3$\pi$ survey, see][]{2013ApJ...770..128I, 2013ApJS..205...20M}. No further detection of any outbursting activity was seen, and the magnitude limits of these individual epochs are typically 22.0, 21.6, 21.7, 21.4 and 19.3 respectively for $g_{\rm P1}r_{\rm P1}i_{\rm P1}z_{\rm P1}$ \citep[as reported in][]{2013ApJ...770..128I}. These magnitudes are in the AB system as reported in \cite{2012ApJ...750...99T}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.48]{./figure11.eps}
\end{center}
\caption[Limits]{Plot of the photometric limits obtained from November 27, 2001 to January 07, 2014 and the absolute R-band light curve of 2007sv. The absolute light curve of the type Ibn SN~2006jc and the type IIn SN~2011ht are shown for comparison. The phases refer to the first recorded eruption.}
\label{limits}
\end{figure}
Tracing the photometric history of a SN impostor has also another objective.
As briefly mentioned in Section \ref{intro}, there is growing evidence that some interacting SNe may be preceded by large stellar eruptions (i.e. impostor events).
A similar sequence of events has been proposed for a number of SNe, from the historical case of SN~2006jc discovered by K. Itagaki \citep{2006CBET..666....1N,2006CBET..666....2Y,2007Natur.447..829P} to a few recent type IIn SNe \citep[see][and references therein]{2013MNRAS.tmp.2960S}.
In several cases, a lower luminosity outburst was observed few weeks before the brightest event (i.e. the putative SN).
However, occasionally a larger time delay was observed between the two episodes (1-2 years).
In Figure \ref{limits}, together with 2007sv, we also show two cases of SNe that were heralded by an outburst with a significant time delay. SN~2006jc is a stripped-envelope SN (of type Ibn) whose ejecta were seen to interact with He-rich CSM \citep{2007Natur.447..829P,2008MNRAS.389..113P,2007ApJ...657L.105F}.
Its progenitor experienced an outburst 2 years before its explosion as a core-collapse SN, and the magnitude of this impostor (M$_R$ $\simeq$ -14) was comparable with that expected in an LBV eruption.
SN~2011ht was initially classified as a SN impostor on the basis of its early spectral properties \citep{2011CBET.2851....2P}, and was later reclassified as a type IIn SN after a major spectral metamorphosis \citep{2011CBET.2903....1P}.
However, the nature of SN~2011ht nature is not fully clarified.
There are controversial interpretations on its SN-like observables \citep{2012ApJ...751...92R,2012ApJ...760...93H,2013MNRAS.431.2599M,2013ApJ...779L...8F}, since collisions among massive shells might still explain SN~2011ht without necessarily invoking a core-collapse.
Interestingly, a posteriori, a weak transient has been observed about one year before the main episode \citep{2013ApJ...779L...8F}.
This weak source, \citep[labelled as PSO J152.0441+51.8492 by][]{2013ApJ...779L...8F} was detected in archival data of the Panoramic Survey Telescope and Rapid Response System 1 (Pan-STARRS1) at an absolute magnitude M$_R \approx$ -11.8. \citet{2013ApJ...779L...8F} provided strong arguments that the two sources were physically related.
Recent studies \citep[see][]{2014arXiv1401.5468O} claim that pre-SN eruptions are quite common.
Nonetheless, so far only an handful of impostors with solid detections have been observed to be followed by what is believed to be a true SN explosion.
The most intriguing issue is that, although there are some outliers with the absolute magnitudes brighter than -14 \citep{2014arXiv1401.5468O}, most pre-SN outbursts (those with robust detections) have absolute magnitudes close to or fainter than -14, nearly coincident with absolute magnitude of 2007sv.
Although there was no further detection of a re-brightening of 2007sv before and after 2007, its overall photometric similarity with the precursors of SN~2006jc and other interacting SNe, may lead to the speculation that impostors such as 2007sv are instability episodes of massive stars (some of them in the LBV stage) that can be followed on short timescales (months to decades) by a terminal stellar explosion, as an ejecta-CSM interacting SN.
\section{Conclusions} \label{conclusions}
In this paper we have reported the results of our follow-up campaign of the transient 2007sv.
Our photometric monitoring spans a period of over 100d, whilst our spectroscopy data cover $\simeq$80d of the evolution of 2007sv.
The spectra are largely dominated by a multi-component H$_\alpha$ line in emission.
This spectral characteristic is common to both in SN impostors and in type IIn SNe.
As we have illustrated, hough there are uncertainties on the observational constraints and the discrimination between true interacting SNe and SN impostors is often a tricky issue, the lack of broad lines from $\alpha$-elements, the fast colour/temperature evolutions, together with the relatively low velocity of the ejecta and the faint absolute magnitude at discovery (M$_R=-14.25$) support the SN impostor scenario for 2007sv, most likely a major eruption of an LBV.
Although we do not have stringent pre-discovery limits, we claim that 2007sv was discovered soon after the outburst.
Hence, we have adopted the discovery magnitude as an indicative guess for the peak magnitude.
This is supported by the rapid evolution of the colours and the temperature of the ejecta in the early phases supports this assumption.
However, some doubts still remain whether 2007sv was instead a very weak terminal explosion of a massive star.
In absence of the detection of further outbursts or a future `real' SN explosion (as observed in other similar transients), the most promising method to definitely rule out the possibility that 2007sv was a faint interacting core-collapse SN is by obtaining deep and high spatial resolution images of the transient's site (e.g. with HST), in the attempt to detect some signatures from the surviving star.
This method, that has been successfully tested to find the progenitors of a number of CC-SNe \citep[see][and references therein]{2009ARA&A..47...63S} and may well give the final answer to the enigma of 2007sv.
\section*{Acknowledgements} \label{ackn}
Based on observations made with:
- The Cima Ekar 1.82~m telescope of the INAF-Astronomical Observatory of Padua, Italy; \\
- The Liverpool Telescope operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council. \\
- the Nordic Optical Telescope (NOT), operated by the Nordic Optical Telescope Scientific Association at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias. \\
- The William Herschel Telescope (WHT) operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofsica de Canarias. \\
- The Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundacion Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. \\
- The Hobby-Eberly Telescope (HET), joint project of the University of Texas at Austin, Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universit{\"a}t M{\"u}nchen, and Georg-August-Universit{\"a}t G{\"o}ttingen. \\
- The Calar Alto 2.2~m telescope operated at the Centro Astronomico Hispano Aleman (CAHA) at Calar Alto, owned and operated jointly by the Max-Planck-Institut f{\"u}r Astronomie in Heidelberg, Germany, and the Instituto de Astrofisica de Andalucia in Granada, Spain \\
We thank P.~Corelli (Mandi observatory, Pagnacco, UD, Italy) for the observations of (SN) 2007sv. \\
Based also on observations performed at the Nordic Optical Telescope (Proposal number 49-016, PI: F. Taddia), La Palma, Spain. \\
This work is part of the European supernova collaboration involved in the ESO-NTT large programme 184.D-1140 led by Stefano Benetti. \\
L.T., A.P., S.B., E.C., A.H. and M.T. are partially supported by the PRIN-INAF 2011 with the project `Transient Universe: from ESO Large to PESSTO`. \\
N.E.R. is supported by the MICINN grant AYA2011-24704/ESP, by the ESF EUROCORES Program EuroGENESIS (MICINN grant EUI2009-04170), by SGR grants of the Generalitat de Catalunya, and by EU-FEDER funds. \\
N.E.R. acknowledges the support from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n. 267251. \\
S.T. acknoledges support by TRR 33 `The Dark Universe' of the German Reasearch Foundation (DFG). \\
The research of JRM is supported through a Royal Society Research Fellowship. \\
The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement n$^{\rm o}$ [291222] (PI: S.~J.~Smartt). \\
FB acknowledges support from FONDECYT through Postdoctoral grant 3120227 and from Project IC120009 `Millennium Institute of Astrophysics (MAS)' of the Iniciativa Cient�fica Milenio del Ministerio de Econom�a, Fomento y Turismo de Chile. \\
The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under Grant No. AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE).
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\section{Introduction}
This note provides a supplementary result (Theorem \ref{thm1}) of my talk at the sixty-first Algebra Symposium of Mathematical Society of Japan, held at Saga University on September 7--10, 2016. My talk there was based on my previous \cite{Og16-2}.
Throughout this note, the base field is assumed to be the complex number field
${\mathbb C}$. Let $M$ be a smooth projective variety of dimension $m \ge 2$ and $f \in {\rm Bir}\, (M)$.
$f$ is said to be {\it imprimitive} if there are a smooth projective variety
$B$ with $0 < \dim\, B < m$ and a dominant rational map $\pi : M \dasharrow B$ with connected fibers such that $\pi$ is $f$-equivariant, i.e., there is $f_B \in {\rm Bir}\, (B)$ satisfying $\pi \circ f = f_B \circ \pi$. As $\pi$ is just a rational dominant map, smoothness assumption of $B$ is harmless by Hironaka resolution of singularities (\cite{Hi64}). We say that $f$ is {\it primitive} if it is not imprimitive.
The notion of primitivity is introduced by De-Qi Zhang \cite{Zh09}. Note that if $f$ is primitive, then ${\rm ord}\, (f) = \infty$. Indeed, otherwise, the invariant field ${\mathbb C}(M)^{f^*}$ is of the same transcendental degree $m$ as the rational function field ${\mathbb C}(M)$. Thus we have $\varphi \in {\mathbb C}(M)^{f} \setminus {\mathbb C}$ as $m \ge 1$. Then the Stein factorization of $\varphi : M \dasharrow {\mathbb P}^1$ is $f$-equivariant. $f$ is then imprimitive as $m \ge 2$.
Assume that $f \in {\rm Aut}\, (M)$. The {\it topological entropy} $h_{{\rm top}}(f)$ of $f$ is a fundamental quantity measuring the complexity of the orbit behaviour under $f^n$ ($n \ge 0$). Let $r_{p}$ be the spectral radius of $f^{*}|H^{p, p}(M)$. Then, by Gromov-Yomdin's theorem, $h_{{\rm top}}(f)$ satisfies
$$0 \le h_{{\rm top}}(f) = \log {\rm max}_{0 \le p \le m} r_p(f)\,\, $$
In this note, it is harmless to regard this formula as the definition of $h_{{\rm top}}(f)$ (See eg. \cite{Og15} and references therein for details).
The aim of this note is to remark the following:
\begin{theorem}\label{thm1} Let $A$ be a simple abelian variety of dimension $m \ge 2$ and $f \in {\rm Aut}\, (A)$. Then $f$ is primitive if and only if
${\rm ord}\, (f) = \infty$. In particular, the translation automorphism $t_a$ ($a \in A$) defined by $x \mapsto x +a$ is primitive if $a$ is a non-torsion point of $A$ with fixed zero. Moreover, if in addition $A$ is of CM type, then $A$ admits a primitive automorphism of positive entropy, possibly after replacing $A$ by an isogeny.
\end{theorem}
Here and hereafter, an abelian variety $A = {\mathbb C}^m/\Lambda$ is said to be {\it simple} if $A$ has no abelian subvariety $B$ such that $0 < \dim\, B < \dim\, A$. A simple abelian variety $A$ is called {\it of CM type} if the endomorphism ring $E := {\rm End}_{{\rm group}}(A) \otimes {\mathbb Q}$ is a CM field with $[E: {\mathbb Q}] = 2\dim\, A$. By definition, a field $E$ is a {\it CM field} if $E$ is a totally imaginary quadratic extension of a totally real number field $K$. Note that if an abelian variety $B$ is isogenous to a simple abelian variety of CM type, then so is $B$ with the same endomorphism ring as $A$. However, ${\rm Aut}_{{\rm group}}\, (A) \not\simeq {\rm Aut}_{{\rm group}}\, (B)$ in general (even for elliptic curves of CM type).
The "only if" part of Theorem \ref{thm1} is clear as already remarked. Theorem \ref{thm1} is a generalization of our earlier work \cite[Theorem 4.3]{Og16-2}. The last statement of Theorem \ref{thm1} gives an affrimative answer to a question asked by Gongyo at the symposium.
Our proof is a fairly geometric one based on works due to Amerik-Campana \cite{AC13} and Bianco \cite{Bi16} and is in some sense close to \cite{Og16-3}.
{\bf Acknowledgement.} I would like to express my thanks to Professors Tomohide Terasoma, Kota Yoshioka and Fumiharu Kato for their invitation to the symposium, Professor Yoshinori Gongyo for his inspiring question there and Professor Akio Tamagawa for his interest in this work and valuable e-mail correspondence.
\section{Proof of Theorem \ref{thm1}.}
Let $A$ be a simple abelian variety of dimension $m \ge 2$ and $f \in {\rm Aut}\, (A)$ such that ${\rm ord}\, (f) = \infty$. We first show that $f$ is primitive.
The following two well-known propositions will be frequently used:
\begin{proposition}\label{prop21}
Let $V$ be a subvariety of $A$ such that $\dim\, V < m = \dim\, A$ and $\tilde{V}$ is a Hironaka resolution of $V$. Then $\tilde{V}$ is of general type.
\end{proposition}
\begin{proof} See \cite[Corollary 10.10]{Ue75}.
\end{proof}
\begin{proposition}\label{prop22}
Let $M$ be a smooth projective variety of general type defined over a field $k$ of characteristic $0$. Then the birational automorphism group ${\rm Bir}\, (M/k)$ of $M$ over $k$ is a finite group
\end{proposition}
\begin{proof} By the Lefschetz principle, we may reduce to \cite[Corollary 14.3]{Ue75}.
\end{proof}
\begin{lemma}\label{lem21} Let $P$ be a very general closed point of $A$. Then the $\langle f \rangle$-orbit $\{f^n(P)\, |\, n \in {\mathbb Z}\}$ of $P$ is Zariski dense in $A$.
\end{lemma}
\begin{proof} As $P$ is very general, $f^n$ is defined at $P$ for all $n \in {\mathbb Z}$. By \cite[Th\'eor\`eme 4.1]{AC13}, there is a smooth projective variety $B$ and a dominant rational map $\rho : A \dasharrow B$ such that $\rho \circ f = \rho$ and $\rho^{-1}(\rho(P))$ is the Zariski closure of $\langle f \rangle$-orbit of $P$. It suffices to show that $\dim B = 0$. {\it In what follows, assume to the contray that $\dim\, B > 0$, we derive a contradiction.}
Let $\eta \in B$ be the generic point in the sense of scheme and $A_{\eta}$ be the fiber over $\eta$. Then by Proposition \ref{prop21} and specialization, a Hironaka resolution of each irreducible component of $A_{\eta}$ is of general type over ${\mathbb C}(B)$. By $\rho \circ f = \rho$, $f$ faithfully acts on $A_{\eta}$ over ${\mathbb C}(B)$. Thus, by Proposition \ref{prop22}, $f^n = id$ on $A_{\eta}$ for some positive integer $n$. Thus $f^n = id$ on $A$, as the generic point $\eta_{A}$ of $A$ is in $A_{\eta}$. This contradicts to ${\rm ord}\, f = \infty$.
\end{proof}
The following general, useful proposition is due to Bianco:
\begin{proposition}\label{prop23} Let $X$ be a projective variety and $g \in {\rm Bir}\, (X)$. Assume that
$\pi : X \dasharrow B$ is a $g$-equivariant dominant rational map to a smooth projective variety $B$ with $\dim\, B < \dim\, X$. Assume that a Hironaka resolution $\tilde{X}_b$ of the fiber $X_b$ is of general type for a general closed point $b \in B$. Then for any very general closed point $P \in X$, the $\langle g \rangle$-orbit $\{g^n(P)| n \in {\mathbb Z}\}$ of $P$ is never Zariski dense in $X$.
\end{proposition}
\begin{proof} See \cite[Section 4]{Bi16}. See also \cite[Remark 2.6]{Og16-3} for a minor clarification.
\end{proof}
The next proposition completes the first part of Theorem \ref{thm1}:
\begin{proposition}\label{prop24} Let $A$ be a simple abelian variety of dimension $\ge 2$ and $f$ be an automorphism of $A$ of infinite order. Then $f$ is primitive.
\end{proposition}
\begin{proof} Let $\pi : A \dasharrow B$ be an $f$-equivariant dominant rational map to a smooth projective variety $B$ with $\dim\, B < \dim\, A$ and with connected fibers. If $\dim\, B > 0$, then by Proposition \ref{prop21}, a Hironaka resolution $\tilde{A}_b$ of the fiber $A_b$ over $b \in B$ is of general type for general $b \in B$. Then, by Proposition \ref{prop23}, the $\langle f \rangle$-orbit of a very general closed point $P \in A$ is not Zariski dense. This contradicts to Lemma \ref{lem21}. Thus $\dim\, B = 0$, i.e., $f$ is primitive.
\end{proof}
We shall show the last part of Theorem \ref{thm1}.
Let $A$ be a simple abelian variety of CM type of dimension $m \ge 2$. We write $E := {\rm End}_{{\rm group}}(A) \otimes {\mathbb Q}$. Then by definition, $E$ is a totally imaginary quadratic extension of a totally real number field $K$ with $[K : {\mathbb Q}] = m \ge 2$. First we make $A$ explicit up to isogeny. As $E$ is a totally imaginary field with $[E:{\mathbb Q}] = 2m$, there are exactly $2m$ different complex embeddings $\varphi_i : E \to {\mathbb C}$ ($1 \le i \le 2m$) such that $\varphi_{2m-i} = \overline{\varphi_i}$. Here $-$ is the complex conjugate of ${\mathbb C}$. Note that there are exactly $2^m\cdot m!$ ways of numberings $I$ of the embeddings here. Choosing one such numbering $I$, we consider the embedding:
$$\varphi_I := (\varphi_1, \varphi_2, \cdots , \varphi_m) : E \to {\mathbb C}^m\,\, ;\,\, a \mapsto (\varphi_1(a), \varphi_2(a), \ldots , \varphi_m(a))\,\, .$$
Let $O_{E}$ (resp. $O_K$) be the integral closure of ${\mathbb Z}$ in $E$ (resp. in $K$). Then
$$B_I := {\mathbb C}^m/\varphi_I(O_E)$$
is an abelian variety and $A$ is isogenous to $B_I$ for some numbering $I$ (See eg. \cite[Chapter I, Section 3]{Mi06}).
From now, we shall prove that the abelian variety $B := B_I$ admits an automorphism of positive entropy.
\begin{definition}\label{def21} Let $\overline{{\mathbb Q}}$ be the algebraic closure of ${\mathbb Q}$ in ${\mathbb C}$, $\overline{{\mathbb Z}}$ be the integral closure of ${\mathbb Z}$ in
$\overline{{\mathbb Q}}$ and $\overline{{\mathbb Z}}^{\times}$ be the unit group of the ring $\overline{{\mathbb Z}}$. A real algebraic integer is an element of $\overline{{\mathbb Z}} \cap {\mathbb R}$. A real algebraic integer $\alpha$ is called a {\it Pisot number} if $\alpha > 1$ and $|\alpha'| < 1$ for all Galois conjugates $\alpha' \not= \alpha$ of $\alpha$ over ${\mathbb Q}$. A Pisot number $\alpha$ is called a {\it Pisot unit} if $\alpha \in \overline{{\mathbb Z}}^{\times}$.
\end{definition}
Then, by \cite[Theorem 5.2.2]{BDGPS92}, we have
\begin{theorem}\label{thm21}
For any real number field $L$, there is a Pisot unit $\alpha \in L$ such that
$L = {\mathbb Q}(\alpha)$.
\end{theorem}
As $K$ is (totally) real, there is then a Pisot unit $\alpha$ such that $K = {\mathbb Q}(\alpha)$. Consider the linear automorphism of ${\mathbb C}^m$ defined by:
$$\tilde{f}_{\alpha} : {\mathbb C}^d \to {\mathbb C}^d\,\, ;\,\, (z_1, z_2, \ldots, z_m) \mapsto (\varphi_1(\alpha)z_1, \varphi_2(\alpha)z_2, \ldots, \varphi_m(\alpha)z_m)\,\, .$$
As $\alpha$ is a unit in $O_K$ (hence in $O_E$), so are $\varphi_i(\alpha)$ in $\varphi_i(O_E)$. Thus $\tilde{f}_{\alpha}(\varphi_I(O_E)) = \varphi_I(O_E)$ by the definition of $\varphi_I$. Hence $\tilde{f}_{\alpha}$ descends to an automorphism $f_{\alpha}$ of $B$. We set $f := f_{\alpha}$.
As $K$ is totally real, regardless of $I$, we have
$$\{\varphi_i(\alpha)\,|\, 1 \le i \le m \} = \{\alpha := \alpha_1, \alpha_2, \ldots , \alpha_m \}\,\, .$$
Here the right hand side is the set of all Galois conjugates of $\alpha$ over ${\mathbb Q}$. By the construction of $f$ from $\tilde{f}_{\alpha}$, the left hand side set also coincides with the set of eigenvalues of $f_*|H^0(B, \Omega_B^1)^{*}$, and therefore, coincides with the set of eigenvalues of $f^*|H^0(B, \Omega_B^1)$.
As $B$ is an abelian variety, we have
$$H^{1,1}(B) = H^0(B, \Omega_B^1) \otimes \overline{H^0(B, \Omega_B^1)}\,\, .$$Here $\overline{H^0(B, \Omega_B^1)}$ is the complex conjugate of $H^0(B, \Omega_B^1) \subset H^1(B, {\mathbb Z}) \otimes {\mathbb C}$. As $\alpha$ is real, it follows that $\alpha^2$ is an eigenvalue of the action of $f$ on $H^{1,1}(B)$. Hence
$$h_{{\rm top}}(f) \ge r_1(f) \ge \alpha^2 > 1\,\, .$$
Here the last inequality follows from the fact that $\alpha > 1$. Thus $f$ is of postive entropy. In particular, ${\rm ord}\, (f) = \infty$. Therefore, $f$ is primitive as well by the first part of Theorem \ref{thm1}. This completes the proof of Theorem \ref{thm1}.
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Category: Shock Treatment 1964
There's got to be a morning after… Goodbye Carol Lynley Sept. 3, 2019
American actress Carol Lynley, who stars in the film The Cardinal, pictured wearing a winter coat and leather gloves in a London park on 17th December 1963. (Photo by Blackman/Daily Express/Hulton Archive/Getty Images)
We've lost Carol Lynley, actress of 60s & 70s film and television. Carol was born Carol Anne Jones on Feb. 13, 1942 in New York City. Lynley worked as a model and in television from her teen years and performed on numerous early live dramatic television shows.
She suffered a heart attack on September 3rd at the age of 77. Perhaps she is best known for her role in the disaster epic The Poseidon Adventure (1972) playing Nonnie, the bright-eyed nymph on the doomed ocean liner turned upside down after a giant tidal wave hits the ship on New Year's Eve. The Poseidon Adventure launched her into the public consciousness after Lynley lip synced over Maureen McGovern's singing onscreen, as the ill-fated ship's lead singer of the band, her brother flaunting his bad 70s hair and mutton chops at the piano. The song "The Morning After," went on to win the 1973 Oscar for Best Song.
I've always been taken with Carol Lynley for many other roles along her diverse career. A child model who made it to the cover of Life magazine at age 15. After appearing in the 1958 Broadway play, she delivered a moving performance in the controversial screen version of Blue Denim in 1959, co-starring cutie Brandon De Wilde. She was nominated for a Golden Globe Award for Most Promising Newcomer! She then co-starred with Clifton Webb and Jane Wyman in Holiday For Lovers (1959).
Afterward she appeared in a variety of popular films, Return to Peyton Place (1961), and Under the Yum Yum Tree (1963) with Jack Lemmon. Carol Lynley appeared in the Otto Preminger film The Cardinal (1963). She was also in The Stripper (1963), and Shock Treatment (1964) where she plays a very disturbed young girl with hyper-sexual tendencies. In the same year she played Maggie Williams in The Pleasure Seekers. Lynley also took the role of Jean Harlow in the biopic Harlow (1965).
Carol Lynley in The Stripper (1963)
Carol Lynley and Gene Tierney in The Pleasure Seekers (1964)
Carol Lynley as Harlow
And her role as Ann Lake is superb. She plays a mother who claims her little girl vanishes after a day at her school. Otto Preminger's Bunny Lake is Missing (1965), is one of my favorite psychological thrillers. She also appeared in the very dark and twisted The Shuttered Room (1965) co-starring Oliver Reed and Gig Young based on a story by horror writer August Derleth.
She was in Once You Kiss a Stranger… (1969) and I adored her as Gail Foster the girlfriend of Darren McGavin's stalwart reporter of the uncanny and supernatural Carl Kolchak, in the fabulous television cult chiller directed by John Llewellyn Moxey, The Night Stalker (1972), the pilot that launched the groundbreaking series of the show that inspired The X Files!
Carol Lynley appeared in various television shows, Alfred Hitchcock Presents, The Alfred Hitchcock Hour, It Takes a Thief, Night Gallery, The Invaders, Kojak, The Man from U.N.C.L.E, Journey to the Unknown, The Sixth Sense, The Magician, The Evil Touch, Quincy M.E. and Police Woman, just to mention my favorites.
Vince Edwards and Carol Lynley in Alfred Hitchcock Presents (1955) episode The Young One
Christopher Walken and Carol Lynley in Kojak 1973
Carol Lynley possessed a certain kind of rare beauty and inner light, a subtle essence of fairy in her smile and soft glimmer in her eyes.
Mandatory Credit: Photo by Barry Peake/Shutterstock (605004b)
Carol Lynley-Mar 1967
This is your EverLovin Joey saying goodbye Carol Lynley, gone but not forgotten. There will always be a morning after and my eternal love for you, beautiful girl.
Posted on September 7, 2019 by monstergirlPosted in Blue Denim 1959, Bunny Lake Is Missing (1965), Carol Lynley, Classic TV, Darren McGavin, Once You Kiss a Stranger 1969, Shock Treatment 1964, The Night Stalker 1972 tv movie, The Pleasure Seekers 1965, The Poseidon Adventure 1972, The Shuttered Room 19657 Comments
Lauren Bacall: Shock Treatment (1964) Dr. Edwina Beighley the female Caligari or it's just like working with animals in a zoo!
This post is in participation with The Lauren Bacall Blogathon hosted by In The Good Old Days of Classic Hollywood.
The winsome & sultry Lauren Bacall steps out of character as screen legend, noir goddess & trend setting icon…
To Have and Have Not (1944), The Big Sleep (1946), Key Largo (1948) Dark Passage (1947) Young Man with A Horn (1950) Designing Women (1957) and so much more!
… And embarks on a role as the icy cold psychologist/Animal Behavioral Researcher, and a Praying Mantis that is Dr. Edwina Beighley (pronounced Bailey) She's a female Caligari who has experimented with her dangerous drug on animals as her subjects in Africa, conducting unorthodox experiments now on human subjects, in Shock Treatment (1964)
She's always griping in her condescending highfalutin way- at the hospital board members that she cant continue her (exploitative and nefarious) research the way she'd like, driven by her mission she craves money. Using mental patients now, not tigers, to continue her scientific analysis of how certain drugs effect the criminal mind and the resulting catatonia that follows.
A seedy psychological thriller with oddballs and opportunists and one hell of a great cast, wasted?… Maybe, but deliciously fun to watch anyways! The film has it's moments and if you're like me and love a great jaunt into the exploitative- then indulge yourself!
Films like The Snake Pit, Lilith , David and Lisa, ( Bacall was also in a film about an exclusive psychiatric clinic- The Cobweb 1955, and earlier in 1950 she embodied the conflicted Amy North who struggled and studied to become a psychiatrist in Young Man with a Horn)…
… show a reversibility of a plot narrative that usually exists in other film genres. The role that is interchangeable with the sane and the mad. the outside or insider, which suggests that there is no good outcome or moreover, no clear solution to the film's 'problem' and that the film's world is veritably unstable with Dr. Edwina Beighley at the center of the disorder!
Cinematographer Sam Leavitt (Anatomy of a Murder 1959, The Defiant Ones 1958) weaves in noirish shadowscapes & creates odd frames where one of the main characters will be relegated to the extreme edge while it allows the camera to focus all it's power on the other of the central or peripheral actors/characters, creating the appearance of an off balanced conversation, that perpetuates the 'offness' of the story and it's atmosphere…
In the similar vein but far superior social commentary as Sam Fuller's Shock Corridor 1963, it's a story of an actor Dale Nelson (Stuart Whitman) willing to fake insanity and take money to infiltrate a mental hospital in order to get close to a homicidal maniac Martin Ashley (Roddy McDowall) who claims to have burned to cinders, the millions, he has hidden of his victim's fortune, now buried somewhere on her estate.
"The most dramatic expression of psychiatry as a mechanism of enforcing conformity is seen in the film depictions of ECT (electroconvulsive therapy) or commonly known as electroshock Treatment
in the 1960s and 70s ECT was recast in movie theaters as a torturous, barbaric, medieval practice in which individualistic mental patients were literally shocked into conformity. Vivid depictions of electroshock were depicted in films such as Samuel Fuller's Shock Corridor 1963 and Shock Treatment 1964."
— Psycho Thrillers: Cinematic explorations of the mysteries of the mind by William Indick
In fact, anti-conformity is Dale's method of breaking into the hospital system by railing against conformity in the guise of intellectually and physically disturbing the social order. He smashes the window fronts of a department store.
During Martin Ashley's (Roddy McDowall) trial for killing and beheading his employer, Dr. Edwina Beighley is the defense's go to specialist on mental illness and key witness, their sympathetic psychiatrist who manipulates the court into allowing her to observe him at her State Psychopathic hospital for observation.
On the stand Edwina- "I'm a fellow of the American Psychiatric Society..and the author of two text books now in use.-Psychiatry in Relation to Crime and Modern Usages of Hypno-Analysis" At present I'm assistant medical director at State Psychopathic Hospital."
When asked if she's familiar with the philanthropic organization known as The Townsend Foundation, Townsend being the old woman that Martin decapitated. Edwin answers with a swift and self-important confidence…
"More than acquainted as Mr Manning knows for the past several years I've been trying to get a grant from them to expand my research… ( deep sarcastic Bacallesque pause) I'm still trying."
Then the public defender asks if she was present when Mr. Manning suggested that the defendant burnt up more than a million dollars. And does she agree with that accounting of the story…
"No I don't, the amount of money certainly is unusual but the act of destruction isn't. Martin Ashely is a lonely secretive young man. Desperately in need of understanding friendship. This type of schizophrenic often is… He became convinced that (Amelia Townsend) was an enemy who was using her wealth to destroy his garden and return him to our hospital where he had been a patient merely three years ago. To his disordered mind the decision was a simple one. Destroy the persecutor and her weapon… her money…"
Dr. Edwina Beighley is a cool, manipulative operator who is working on getting Martin a plea of insanity so he'll be sent to her hospital under her care, that way she can make certain she's up close and personal with him in order to access his secret… where he hid the fortune.
During Martin's trial Mr. Manning who has been an executor of the estate asserts that the old woman was eccentric and hid huge sums of cash in her home, he tells the prosecuting attorney, "I couldn't believe that anyone even a madman could bring himself to burn up more than a million dollars."
Manning who testifies that the old lady had millions, also despises Dr. Beighley.
After Martin gets sentenced for a mere 90 days for observation. Manning confronts Beighley in the courtroom. "Dr. Beighley I hope you'll feel proud of yourself Dr!" Dr. Edwina Beighley not seeming rattled in the least- "And what is that supposed to mean?"
Manning- " Why did you have to go out of your way to help that faker get away with murder and a million dollars?"
She threatens to sue for liable so that she'll collect enough from him, never having to apply for a grant again… He tells her that he's "sick and tired of psychiatrists who try to play god, who tell us our mothers and fathers made us neurotic, and psychotic!"
"Mr Manning I've gone through analysis, all psychiatrists do, Now I suggest you try it!"
Dr. Edwina Beighley has the warmth of a cobra about to strike the jugular.
This psycho-thriller also stars Stuart Whitman as struggling actor Dale Nelson who is going to be paid $10,000 by Harley Manning (Judson Laire) to impersonate a mentally disturbed man, an incorrigible anti social bad boy who then purposefully gets arrested for destruction of personal property and disturbing the peace.
IMDb notes that Anthony Perkins wanted the Stuart Whitman role
At the police station- Dale (Stuart Whitman) puts on quite a show as a crazy guy with a wad of cash in his pocket that he refuses to explain how it got there- he won't co operate and goes off on a tirade that is deliciously absurd…" The disciples of conformity are bleeding from the narrowness of your mind"
Manning figures that once Dale gets committed to the state asylum, he can befriend the psychopathic handyman/gardener Martin Ashley (Roddy McDowall with his usual flare for the overly-dramatic, deliciously deliriously overindulgence. ) who is just mad about roses and decapitates his employer Amelia Townsend (Beatrice Grenough) with a pair of garden shears when she interferes with his beloved garden.
Naturally Dale Nelson succeeds in getting sent to Dr. Beighley's State Psychopathic Hospital. He even learns about roses and horticulture in order to get close to Martin, hoping he'll tell him where the money is hidden. Once Dale arrives and is interviewed. Edwina looks him over a bit, and she catches something about his performance, so she has her assistant do a background check on him.
Dale gets Dr. Edwina Beighley to assign him to the garden as his work detail. There Dale finally meets Martin the gardener. At first he antagonizes him, but soon after they become good friends with love of flowers in common.
Martin argues about his ability to raise beautiful roses and that he didn't get to see flowers until he was 16. 'You don't get flowers at the orphanage Mister!… I'm the guy who crossbred the Pinocchio with the Fuselier… and it won the first show at the Pasadena in 1962."
With no intention of trying to cure Martin Ashley of his homicidal criminal nature, Dr. Beighley finally gets him to confess his crime in detail, by subjecting him to hypnosis and pentathol for days where he finally winds up telling her where Mrs Townsend's money is…
Edwina, is rancorous, scornful and arrogant and by the end of the film, her mania to find the money might either be a sign that she herself is insane or is the catalyst for pushing her off the deep end… Another version of the inmates have taken over the asylum! And Dr. Edwina Beighley might just belong there BUT as the patient and not the doctor….
Edwina eventually finds out that Dale Nelson was paid and is planted in her hospital by her nemesis Haley Manning, who is determined to get her license revoked for her unethical practices.
When she discovers Nelson's con-game, the sadistic Edwina Beighley prescribes electroshock therapy, then injects a concoction of psychotropic drugs into his jugular vein to induce catatonia, causing him 'horrible twisted images' in order to render him useless and get him out of her hair so she can be the sole keeper of the fortune…
Believe it or not this over the top psycho-melodrama was scripted by Sydney Boehm who penned such great noir films as -High Wall 1947, Mystery Street 1950 Side Street 1949, The Big Heat 1953.
The film also co-stars marvelous character actors who play various archetypal characters, the troubled nymph with a mother complex Carol Lynley as Cynthia Lee Albright "Don't touch me, I don't like to be touched!"
Olive Deering as Mrs Mellon-"You're stupid stupid do you hear me stupid"
Ossie Davis plays Capshaw, who used to be an intern in the hospital and is now one of it's residents. Paulene Meyers as Dr. Walden, and Timothy Carey as high-strung and marvelously hulking & nutty as usual.
Shock Corridor & Shock Treatment deal with the outside/inside structure which ends with pessimism as the main characters descend into madness…
From Part-Time Perverts: Sex, Pop Culture, and Kink Management by Lauren Rosewame she cites Peter Cranford a psychologist during the 1940s who said that for many patients in asylums "The words 'punish' and 'shock treatment' were often synonymous"
This is where the narrative and Dr. Edwina Beighley converge on a social truth behind the institutional edifice of mental health…
She shows her fellow colleagues the results of her research on a projector. Footage from when she had her own facility where she could use zoo animals in her experiments. On film she shows a tiger being injected with her drug and how it effects their aggression. She seeks to find out more about the chemistry of the mind.. to solve it's mysteries. So that one day… her drug "will control mental illness as well a drug does Diabetes."
This brings out a great point of the story though it may be accidental since the film seems to be more about sensationalist entertainment than thoughtful reflecting on mental illness the way it was let's say in Tennessee William's Suddenly, Last Summer 1959.
In the scene where Edwina shows her footage, and the few scenes where both Capshaw (Ossie Davis) and Dale (Stuart Whitman) are subjected to shock treatment- it makes the strong connection between punishing the patient and the arousal of the sadistically inclined practitioners.
In her autobiography, Bacall refers to Shock Treatment as "truly tacky." and when asked about the film she, commented, "You have no idea what Roddy and I went through making that movie."
Here's what Time Magazine had to say about the film Cinema: Boredom in Bedlam-March 13, 1964 "Shock Treatment is more than a slip, it's a Freudian pratfall. It makes a shambles of psychiatry and brings the art of film close to idiocy."
It is definitely not one of Lauren Bacall's memorable roles, it boarders more in the realm of the Grande Dame Guignol films that actresses were becoming famous for in the 60s… Yet, anything Bacall inhabited is like Midus' golden touch, because she brings an inimitable flavor of sophistication and savvy even if it's surrounded by trashy lunacy!
Let's not end on an insane note! Let's celebrate Lauren Bacall as she really was… an icon
Headshot of actress Lauren Bacall pictured with her chin resting on her right wrist, USA, circa 1945. (Photo by Archive Photos/Getty Images)
Posted on September 12, 2015 October 6, 2015 by monstergirlPosted in Carol Lynley, Cult Exploitation & Euro Shock, Cult/Exploitation, Lauren Bacall, men in peril, Olive Deering, Ossie Davis, psycho-sexual thriller, psychological thriller, psychos and fanatics, psychotronic cinema, Roddy McDowall, Sam Leavitt-cinematography, Shock Treatment 1964, Stuart Whitman, The Lauren Bacall Blogathon 2015, Timothy Carey, UbiquityTagged featured. 15 Comments | {
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Громадя́нська осві́та () — навчання принципам та видам залученості громадян в суспільних та політичних процесах. Громадянська освіта може бути як формальною (затверджені на державному рівні класи з політичного/суспільного виховання), так і неформальною (тренінги/курси/патріотичні програми для молоді).
Історія громадянської освіти в незалежній Україні
Історію громадянської освіти в Україні можна умовно поділити на 3 етапи:
Передумови створення: від незалежності до 1999
В цей період громадянська освіта не мала чіткої державної політики та була представлена в патріотичних годинах в шкільних закладах і в позашкільній державній програмі «краєзнавство». В недержавному секторі почали з'являтись молодіжні організації, що мають в статуті мету з розвитку громадянської освіти та патріотичного виховання. 22 лютого 1990 року було офіційно зареєстровано скаутську організацію «Пласт» в м. Львів, а 14 жовтня 1994 року в м. Чернігів було зареєстровано першу організацію з громадянської освіти «АХАЛАР».
Поряд з вищезазначеними курсами з 1999 року як пілотний експериментальний проект у загальноосвітніх навчальних закладах запроваджується курс "Громадянська освіта". За цей час він пройшов апробацію у багатьох школах різних регіонів України. В пілотному проектуванні курсу "Громадянська освіта"брали участь декілька міжнародних інституцій. Перший проект став можливим завдяки співпраці з Фундацією "Matra" (Королівства Нідерланди), курс носив назву"Основи громадянської освіти", складався з двох навчальних посібників які вміщували політичний і економічний зміст.
Розвиток державної освітньої програми: 2000-ні
У квітні 2000 року Президія Академії педагогічних наук України схвалила основні положення «Концепції громадянського виховання особистості в умовах розвитку української державності».
В 1999 році Києво-Могилянська Академія створює Інститут громадянської освіти для створення досліджень та концепцій розвитку громадянської освіти. В липні 2000 року Інститут громадянської освіти НаУКМА розробляє концепцію громадянської освіти в Україні. В 2005 році Інститутом було створено концепцію політичної освіти в Україні
У 2000 році був реалізований проект в рамках Програми підтримки громадянського суспільства в Україні та проекту "Освіта для демократії в Україні" за сприяння Центру Мершона Університету штату Огайо (США)та Центру громадянської освіти (Польща). Курс має назву"Ми громадяни України" для учнів 9 (10) класів середніх загальноосвітніх навчальних закладів. Право на повторні видання посібника має громадська організація "Нова Доба".
Наступний проект виконувався в межах Програми "Освіта для демократії в Україні" 2000 - 2002 рр.(Трансатлантичний проект підтримки громадянського суспільства), за сприяння Інституту політичної участі Нідерландів (IPP), за експертною допомогоюНаціонального центру стандартів і програм Нідерландів (SLO). В результаті проекту створено три посібники для учнів 9 - 11 класів та посібник для вчителів, що мають однакову назву "Громадянська освіта" видавництво "Генеза".[12]
У 2005 - 2009 роках в межах Українсько - Канадського проекту "Розбудова демократії" Центру вивчення демократії університету Квінз (м.Кінгстон Онтаріо) під керівництвом проф. Джорджа Перліна та за фінансової підтримки Канадського Агенства Міжнародного Розвитку (CIDA) було підготовлено посібник "Громадянська освіта: основи демократії" для учнів 11(12)класів та підручник для студентів педагогічних ЗВО "Громадянська освіта: основи демократії та методика його навчання", видавнича група "Основа", 2009.[13]
В 2008 році випускники німецької програми громадянської освіти Theodor Heuss Kolleg створюють представницьку програму «Майстерня громадської активності» та розробляють концепцію громадянської соціальної залученості за допомогою створення локальних соціальних проектів.
Саме в цей момент почалось фінансування програм з громадянської освіти іноземними донорами, серед таких програм:
Проект «Освіта для демократії в Україні» (ЄС, США)
«Громадянська освіта — Україна» 2005—2008 (ЄС)
«Навчальний тур з громадянської освіти» 2002 (USAID)
Програма «Практичне право» та «Дебати». Інформаційно-методичний центр «Дебати» 1999—2001 (США)
Українсько-Швейцарський проект «Сприяння розвитку освіти для демократії в Україні» 2005—2010 (SIDA)
Проект Благодійної організації «Вчителі за демократію та партнерство» «Забезпечення якості освіти для демократичного громадянства» 2007—2010 (Рада Європи)
Проект «Спільнота споживачів та громадські об'єднання» 2008—2010 (ЄС, ПРООН).
Кризовий і пост-кризовий періоди: 2010—2014 і зараз
В 2009 році відбувся перегляд концепції розвитку громадянської освіти, в якому було ухвалено визначення плану дій з поширення громадянської освіти на місцеві адміністрації. В 2014 році, під час бойових дій на Сході та Півдні України, рівень розвитку громадянської освіти не зміг протистояти впливу російської пропаганди в деяких регіонах України.
Наразі, багато з проектів громадянської освіти в Україні намагаються адресувати проблему впливу пропаганди на політичне та громадянське життя українців, серед них:
Програма медіаграмотності від IREX;
Програма «Студії Живої Історії»
Програма «Звичка Думати»
Громадянська ініціатива StopFake.org
Курс із громадянської освіти для вищих навчальних закладів «Демократія: від теорії до практики» Міжнародної фундації виборчих систем (IFES)
Освітній проект та громадянська ініціатива Відкритий Університет Майдану.
Див. також
Неформальна освіта
Освіта дорослих
Інша Освіта
Примітки
12. Суспільствознавство. Методичний посібник для вчителя. Київ Аконіт.2005.
13. Громадянська освіта:основи демократії 11(12)клас.Навчальний посібник.Видавнича група "Основа",2009.
Громадянська_освіта, додати список літератури
Суспільні науки
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Q: Topological spaces containing paths Let $C(\mathbb{R}^n;\mathbb{R}^d)$ be the space of continuous functions with the uniform-convergence on compacts topology. What are function spaces $X$ building on $C(\mathbb{R}^n;\mathbb{R}^d)$?
*
*$X$ properly contains $C(\mathbb{R}^n;\mathbb{R}^d)$,
*The subspace topology on $C(\mathbb{R}^n;\mathbb{R}^d)$ (with respect to $X$) agrees with the topology of uniform convergence on compact subsets,
*The elements of $X$ are functions from $\mathbb{R}^n$ to $\mathbb{R}^d$
*$C(\mathbb{R}^n;\mathbb{R}^d)$ is dense in $X$.
A: Any such topology will be fairly unpleasant. For instance, the topology of $X$ cannot be induced by any translation-invariant metric $d$.
Lemma. Let $Y_1, Y_2$ be two topological vector spaces whose topologies are induced by translation-invariant metrics $d_1, d_2$, and let $T : Y_1 \to Y_2$ be a continuous linear map. Then $T$ is uniformly continuous.
Proof. Since $T$ is continuous at 0, for any $\epsilon > 0$ there exists $\delta > 0$ such that if $d_1(x, 0) < \delta$ then $d_2(Tx, 0) < \epsilon$. Now if $d_1(x,y) < \delta$, then $d_1(x-y, 0) = d_1(x,y) < \delta$ and we have
$d_2(Tx, Ty) = d_2(Tx-Ty, 0) = d_2(T(x-y), 0) < \epsilon$.
Now recall that $C(\mathbb{R}^n; \mathbb{R}^d)$ is a Fréchet space, so its usual topology is induced by a complete translation-invariant metric $d_0$. By assumption, the identity map $id$ from $(C(\mathbb{R}^n; \mathbb{R}^d), d_0)$ to $(C(\mathbb{R}^n; \mathbb{R}^d), d)$ is a homeomorphism, and so by our lemma, $id$ and $id^{-1}$ are uniformly continuous. In particular, $C(\mathbb{R}^n; \mathbb{R}^d)$ is complete with respect to $d$, and therefore closed in $X$.
Edit. Indeed, $X$ cannot even be a sequential Hausdorff topological vector space. In particular, assuming it is a TVS, its topology cannot be induced by any metric, translation-invariant or not.
In the following, let for brevity $Y = C(\mathbb{R}^n; \mathbb{R}^d)$; the same argument works for any Fréchet space.
Suppose that $Y \subset X$ and that the subspace topology on $Y$ equals the usual topology induced by the complete translation-invariant metric $d_0$ on $Y$. I claim $Y$ is closed in $X$.
Suppose $x$ is in the $X$-closure of $Y$, so that there is a sequence $y_n \in Y$ converging to $x$ in the topology of $X$. Let $\epsilon > 0$ and let $B$ be the open $\epsilon$-ball of the metric $d_0$ centered at $0$. By assumption $B$ is open in the subspace topology of $Y$ inherited from $X$, so there is an $X$-open set $U$ such that $B = U \cap Y$. In particular, $0 \in U$. Now since subtraction is jointly continuous in $X$, there is another $X$-open neighborhood $V$ of $0$ such that for all $a,b \in V$ we have $a-b \in U$.
Since $y_n - x \to 0$ in $X$, there exists $N$ so large that for all $n \ge N$ we have $y_n - x \in V$ (using again the fact that $X$ is a topological vector space). Now if $n,m \ge N$, we have $y_n - x, y_m - x \in V$, so that $y_n - y_m = (y_n - x) - (y_m - x) \in U$. Moreover, $y_n - y_m \in Y$ because $Y$ is a vector space. So $y_n - y_m \in U \cap Y = B$, meaning that $d_0(y_n, y_m) = d_0(y_n - y_m, 0) < \epsilon$, using the fact that $d$ is translation invariant.
Hence $y_n$ is Cauchy in the complete metric $d_0$, so converges in $d_0$-metric to some $y \in Y$. Thus we also have $y_n \to y$ in the topology of $X$. Since the latter is Hausdorff, $x=y$ and thus $x \in Y$.
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Q: How to include the print function in the config.json I am trying to write the python code using the single responsibility principle, and I only wanted to edit the inputs in the config.json. In the json_writer.py the output file name has to be changed if I change the URL in the config.json. So, how can I add the print file name in the existing config.json so that I don't have to change anything in the code?
**#config.json**
{
"url": "https://arcgis/portal/",
"username": "username",
"password":"password",
"query": "",
"item_type": "Feature Layer"
}
**#json_reader.py**
import json
class JsonReader:
def__init__(self, filename):
self.filename = filename
def read_json_file(self):
withopen(self.filename) asfile:
data = json.load(file)
return data
def get_config(self, config):
loaded_json = self.read_json_file()
config = loaded_json[config]
return config
**#portal_connection.py**
from arcgis.gis import GIS
class PortalConnection:
def__init__(self, url, username, password):
self.url = url
self.username = username
self.password = password
def connect(self, query, layer):
# Connection to ArcGIS Enterprise using a built-in account
print("Portal for ArcGIS as a built in user")
gis_portal = GIS(self.url, self.username, self.password)
print("Logged in as: " + gis_portal.properties.user.username)
return gis_portal, query, layer
**#esri_api.py**
class EsriApi:
def__init__(self, portal, item_type, query):
self.item_type = item_type
self.query_ = query
self.portal = portal
def query(self):
api_query_result = self.portal.content.search(query=self.query_, item_type=self.item_type)
return api_query_result
**#json_writer.py**
class JsonWriter:
def __init__(self,api_query_result):
self.api_query_result = api_query_result
def printResults(self):
l = []
for service in self.api_query_result:
if "Hosted" in service.url:
l.append(str((service.url, service.id, service.owner)))
with open('Output_env_Q_Hosted.txt', 'w') as f:
for line in l:
#print(line)
f.write(line)
f.write('\n')
print("saved successfully!!!")
file = open("Output_env_Q_Hosted.txt", "r")
for i in file.readlines():
print(i)
**#main.py**
from json_reader import JsonReader
from portal_connection import PortalConnection
from esri_api import EsriApi
from json_writer import JsonWriter
jreader = JsonReader('config.json')
url = jreader.get_config("url")
username = jreader.get_config("username")
password = jreader.get_config("password")
query = jreader.get_config("query")
layer = jreader.get_config("item_type")
con = PortalConnection(url, username, password)
portal, query, layer = con.connect(query, layer)
esri_api = EsriApi(portal, layer, query)
results = esri_api.query()
jsonWriter = JsonWriter(results)
jsonWriter.printResults ()
A: You can not run code from a JSON file, so you will have to change the code. JSON is only for storing data, and it is not a programming language.
If you want to set the filename at runtime, you can use something like this:
filename = 'SomeFileName.txt' # set this with whatever code you like
with open(filename, 'w') as f:
for line in l:
f.write(line)
f.write('\n')
print("saved successfully!!!")
file = open(filename, "r")
for i in file.readlines():
print(i)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,847 |
Q: Explain this CSS link underline animation I'm trying to create a little animation so when I hover over a link it underlines the link from the middle - JS Fiddle
The CSS code below works but I don't understand how it works. Please can someone break it down for me.
a {
display: inline-block;
position: relative;
padding-bottom: 3px;
}
a:after {
content: '';
display: block;
margin: auto;
height: 3px;
width: 0px;
background: transparent;
transition: width .5s ease, background-color .5s ease;
}
a:hover:after {
width: 100%;
background: blue;
}
A: a:after creates a block with empty inline text after link. Then display:block makes it block type like for instance div default behaiour, so it goes to the next line with height: 3px and with width: 0 (so not visible). And also centered because of margin:auto; Once you hover link this block gets the witdh and becomes visible. The transition attribute makes the animation work.
A: Because you have margin: auto, the :after content is aligned to center of the <a> element, so the animation starts from center to 100% width of the <a>.
A: I assume you are asking about the code:
a:after {
content: '';
display: block;
margin: auto;
height: 3px;
width: 0px;
background: transparent;
transition: width .5s ease, background-color .5s ease;
}
a:hover:after {
width: 100%;
background: blue;
}
If you are not familiar with :before and :after, I will explain it. :before and :after mean to create an element that isn't visible to the document (you can't manipulate it with JavaScript), and is "designed" to be used to append small amounts of content before or after an element. However, that doesn't mean it has to be content. In your jsfiddle, it serves as a "expanding line" underneath the link. :before and :after behave almost like a child to the tag it is binded to (in this case your link).
Here is a reference link for what was discussed above (reference link)
Ok, now we'll move on to the CSS properties...
The content property tells it what text to insert inside the :after/:before. The content property is required but you can leave it empty if you don't have any text that you want to append.
Ok, now for the display:block;...
:before and :after float by default. If you declare it as :before, the element floats to the left of the element selected, if it is :after, then it floats to the right. To make your :after go underneath the link, you must tell it to display:block;. You likely know this but display:block; means to not allow any other elements to "wrap" around the element. That is why when you create a div element it doesn't allow any other elements to go on the left or right of it.
Now for the margin:auto;...
This means to put the :after "element" in the (horizontal) center of the link. That is why when you hover over the link, it starts by expanding in the center. Try removing margin:auto; to see what it does and you will understand more clearly what I'm talking about here.
Now, the most interesting part! This is what makes it visible and "animate" when you hover over it. First we'll start with why you can't see it in the first place. You can't see it because it is 0px in width and the background color is transparent. If you look at the last line transition: width .5s ease, background-color .5s ease; this line is what makes the animation happen. It means that when the width property is modified, to apply the changes over .5s (half a second) and to change the background color over a half a second period of time. After you have read up to this point, you are probably wondering "what modifies the properties?" This brings us to the most technical part of this code...
Look carefully at the selector a:hover:after. The selector is a pseudo class which means that when the link is "hovered" over, to apply the CSS properties width:100% and background:blue;. As a result, the width expands to 100% of the a tag (remember the a tag acts like a parent element) and the background changes to blue over a half second period.
Here are some suggestions for you if you are going to use this on a website. You will want to use a hex code instead of blue in the transition and you will also want to use the transition property with vendor prefixes to offer better browser support.
Here is what the final result of your code should look like.
a:after {
content: '';
display: block;
margin: auto;
height: 3px;
width: 0px;
background: transparent;
-moz-transition: width .5s ease, background-color .5s ease;
-webkit-transition: width .5s ease, background-color .5s ease;
-ms-transition: width .5s ease, background-color .5s ease;
transition: width .5s ease, background-color .5s ease;
}
a:hover:after {
width: 100%;
background: #0000FF;
}
Hope this helps. Please ask any questions if you still don't understand something mentioned.
A: CSS Virtual Pseudo-Elements
To start, you need to understand CSS's virtual pseudo-elements. For every element on the page, <div>, <a>, anything, there are two others. Hidden by default, these elements can be accessed by the css selectors #myelement:before and #myelement:after.
By setting the CSS style content: '' on those elemnts they will appear on the page. Then can then be styled with any other styles just about like any other element.
The border-less trick
In this particular example, there is no border used. Instead, the :after element is shown and used to create a 3px tall box with a background-color right under the link. That widening effect is created by setting width: 0; and then animating to width: 100%;.
The code snippet you saw:
<a href="www.ign.com">Link</a>
a:after {
...
}
works pretty much the same as:
<a href="www.ign.com">Link<span class="after"></span></a>
a .after {
...
}
This JSFiddle works the same way, except without the confusing (but pretty cool once you understand it) :after pseudo-element.
A: There's an explanation and demonstration here:
https://www.rgraph.net/blog/an-example-of-an-animated-expanding-underline-menu-effect.html
And the code from is thus (this is a whole example of a page that you can copy into a file and run):
<!DOCTYPE html >
<html>
<head>
<style>
div#menubar {
font-family: sans-serif;
background-color: #eee;
}
div#menubar span {
display:inline-block;
margin: 10px 2% 10px 2%;
}
div#menubar span::after {
display: block;
position: relative;
top: 3px;
content: '';
border-bottom: solid 3px #019fb6;
transform: scaleX(0);
transition: transform .25s ease-in-out;
}
div#menubar span:hover::after {
transform: scaleX(1);
}
</style>
</head>
<body>
<div id="menubar">
<span>Item 1</span>
<span>Item 2</span>
<span>Item 3</span>
<span>Item 4</span>
<span>Item 5</span>
<span>Item 6</span>
</div>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,226 |
{"url":"https:\/\/cs.stackexchange.com\/questions\/64994\/partitioning-millions-of-items-into-groups-based-upon-a-network-of-set-similarit","text":"# Partitioning millions of items into groups based upon a network of set similarities\n\nSo I'm working on a problem at work related to the matching of authors of millions of documents. I currently have minhash sets for each document's syntax (sets of 10 numbers with 8-10 digits each), however I need to figure out the most efficient way to partition the documents by their minhash sets so that I can simply compare the documents that have vaguely similar minhash sets. I've arrived at the following problem:\n\nGiven a collection of millions of sets, with each set containing 10 numbers of 8-10 digits, how would you go about partitioning the collection in that all sets in the partition have at least 4 elements in common with at least 1 other set in the partition? No item in any partition can share 4 or more set elements in common with an item from any other partition.\n\nBasic example:\n\nGiven the sets of...\n\nA={1, 2, 3, 4, 5, 6, 7, 8}\nB={21, 22, 23, 24, 25, 26, 27, 28}\nC={41, 42, 43, 44, 45, 46, 47, 48}\nD={1, 2, 3, 4, 41, 42, 43, 44}\n\n\n...we should have the the sets separated into 2 partitions:\n\nPartition_1 = {A, C, D}\nPartition_2 = {B}\n\n\nThis, however, needs to scale for tens of millions of sets instead of just 4.\n\nI'm happy to provide more details and answer any questions. I hope my question was clearly worded. Thanks for your responses!\n\n\u2022 Are you looking for algorithms or programs? What does \"scale for\" mean here, exactly? \u2013\u00a0Raphael Oct 23 '16 at 11:27","date":"2020-01-29 18:56:48","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\": 0.40286126732826233, \"perplexity\": 499.93641429988776}, \"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-2020-05\/segments\/1579251801423.98\/warc\/CC-MAIN-20200129164403-20200129193403-00085.warc.gz\"}"} | null | null |
{"url":"http:\/\/physics.stackexchange.com\/questions\/58026\/why-is-vec-j-cdot-vec-e-the-joule-dissipation","text":"# Why is $\\vec j\\cdot \\vec e$ the joule dissipation?\n\nI always see $\\vec j\\cdot \\vec e$ as Joule's dissipation and I don't understand why. For example, if we have a uniform electric field $\\vec e=e_o\\vec u_x$ and we release an electron in it, it will start moving, accelerating in the direction of $-\\vec u_x$, so the we will have a current $\\vec j$, and this product $\\vec j\\cdot\\vec e$ will exist, but I don't see that any energy is being dissipated in the form of heat there. What is going on?\n\nI've seen this essentially in Optics texts, when introducing Poynting's theorem.\n\n-\n\nJoule dissipation (equivalently, Ohmic heating) is a statistical process. It doesn't occur at the microscopic scale, and a single travelling electron in an electrical field is certainly microscopic. Resistivity depends on particle collisions, which turns translational kinetic energy into thermal kinetic energy.\n\nOhm's law relates the current $\\vec{j}$ in a medium to the electric field $\\vec{e}$ via the conductivity $\\sigma$, $$\\vec{j}=\\sigma \\vec{e}$$\n\nAnother way of looking at this is that dissipation $P$ is given by $$P=\\frac{j^2}{\\sigma}$$\n\nWritten this way, we have an expression very similar to that for Ohmic heating in a resistor, namely that\n\n$$P=I^2 R$$ Since the conductivity is the inverse of the resistivity, this is an intuitive result.\n\nThe advantage of Joule's expression is that it allows for the determination of Joule heating at a particular point in space, rather than over the entire resistor (whatever that may be). This is useful in plasma physics, for example, where it may be the case that Joule heating is localized, rather than uniform throughout the plasma. Nonetheless, it is predicated on the medium being strongly collisional, and still must refer to a macroscopic, rather than microscopic effect.\n\n-\nAnd so what is its physical interpretation microscopically. What's that product at those scales, how must I see that? \u2013\u00a0 MyUserIsThis Mar 25 '13 at 22:43\nThere is no microscopic interpretation. Even the simple $\\vec{j}=\\sigma \\vec{e}$ doesn't have a microscopic interpretation. Quantities such as resistivity and conductivity are inherently statistical. They depend on collisions to create entropic increases, i.e. thermalizing the kinetic energy that charged particles gain from being accelerated by electromagnetic fields. Joule's expression is an improvement over Ohm's law in that it describes dissipation at a point in space, but when the volume being described has too few particles in it, the description is no longer valid. \u2013\u00a0 KDN Mar 26 '13 at 12:36\n\nImagine a cross sectional area of the conductor that carries the current density $\\vec{J}$, and take a thin slice, $\\Delta x$, of the conductor. The electric field is along the conductor and at 90$^o$ to its cross sectional area, A. Then $\\vec{J}$ represents the number of electrons, per second and per unit area, crossing the cross-section of the conductor. The number of electrons crossing per unit area and per second is numerically equal to $n_e=|\\vec{J}|\/e$. These electrons are \u201cpushed\u201d by the field through the slice $\\Delta x$, while the field is\n\n$E=\\frac{\\Delta U}{\\Delta x}$\n\n(dropped the vector notation since we are talking about a 1-D motion in the conductor). Then equation $\\vec{J}.\\vec{E}$ can be written (without the vector notation) as\n\n$J\\times E=\\frac{I}{A}\\frac{\\Delta U}{\\Delta x}$\n\nor\n\n$J\\times E=\\frac{I\\Delta U}{A\\Delta x}$\n\nThe last part of the equation shows that $\\vec{J}.\\vec{E}$ gives the amount of energy spent by the field (the battery), per second and per unit volume of conductor, to push all these electrons through the slice of area A and thickness $\\Delta x$, being at the potential difference $\\Delta U$.\n\n-","date":"2015-06-30 05:25:31","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\": 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\": 0.8528724908828735, \"perplexity\": 270.0069002555885}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2015-27\/segments\/1435375091751.85\/warc\/CC-MAIN-20150627031811-00120-ip-10-179-60-89.ec2.internal.warc.gz\"}"} | null | null |
and CLONE, but that's a different matter, which I'll look into later.
I know that they evaluate parameters multiple times, but they are none-the-less genuinely useful shorthand that can't be done without macros. Do you have a similar replacement functionality in mind? Or do you find macros like these to be worse than the extra typing that would be required in their absence?
* modules/xalloc: Add m4/free-null.m4. Change maintainer to "all".
* lib/xalloc.h (free) [!HAVE_FREE_NULL]: New macro.
(XFREE): Use 'free'. Add a comment that XFREE is obsolescent.
* m4/xalloc.m4 (gl_XALLOC): Require AC_C_INLINE, gl_FUNC_FREE_NULL.
+1 vote from me too. Please apply. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,937 |
\section{Conclusion}
In this paper, we have optimized the whole process of the graph-based NN search, namely the NN search procedure and the graph that supports the NN search. The graph is built by a two-stage diversification on the \textit{k}-NN graph. Compared with the state-of-the-art approaches, our graph is not only efficient to be built, but also leads to the best search performance across different datasets on the CPU. Moreover, the graph is flexible to fit with different search scenarios where the search tasks and the available computing resources vary. Accordingly, two highly efficient NN search algorithms have been designed to make the full use of GPU computing power. Inside each of the search algorithm, two key data structures have been carefully designed. The design well addresses the bandwidth mismatch between the GPU cores and the GPU memories. When our search algorithms are supported by the proposed graph, they outperform considerably the other NN search approaches on the GPU, particularly for queries in small batches.
\section{Experiments}
\label{sec:exp}
\subsection{Experiment Setup}
In this section, the performance of the proposed graph TSDG and NN search approaches is studied in comparison to the state-of-the-art approaches. In particular, our study focuses on graph-based approaches as they are the most effective ones in the literature. For the comparative evaluation with other types of approaches, readers are referred to~\cite{li2019approximate,tbd21:zhao}. Six real-world datasets are adopted to in the NN search evaluation. The brief information about these datasets is summarized in~\autoref{tab:datasets}. As shown on the table, the experiment covers traditional image feature, deep image features, text features, and cross model features. They are all dense and in high dimensions. Different distance measures are adopted for different datasets. We use local intrinsic dimensionality~\cite{amsaleg2019intrinsic} (LID shown in the \textit{3rd} column) to measure the difficulty of a dataset. Generally, it is more challenging to perform NN search on the datasets with high intrinsic dimensionality. On the Text-to-Image (T2I) dataset, the local intrinsic dimensionality of the candidate set and the queries is different as they are extracted from different sources. They are \textit{20.9} and \textit{15.5} for text and image features respectively.
\begin{table}[t]
\caption{Summary on Datasets used for Evaluation}
\small{
\begin{tabular}{lrccl}
\toprule
Name & $d$ & LID~\cite{amsaleg2019intrinsic} & m($\cdot$,$\cdot$) & Type \\
\midrule
SIFT~\cite{jegou2010product} & $128$ & 15.6 & L2 & SIFT~\cite{lowe2004distinctive} \\
DEEP~\cite{babenko2016efficient} & $96$ & 15.9 & L2 & Deep \\
GIST~\cite{douze2009evaluation} & $960$ & 25.9 & L2 & GIST~\cite{douze2009evaluation} \\
GloVe~\cite{pennington2014glove} & $100$ & 29.4 & Cos & Text~\cite{pennington2014glove} \\
SPACEV~\cite{ChenW18} & $100$ & 23.2 & L2 & Text \\
T2I~\cite{t2i2021dataset} & $200$ & 20.9/15.5 & IP & Text/Image \\
\bottomrule
\end{tabular}
}
\label{tab:datasets}
\end{table}
The top-\textit{10} (\textit{Recall@10}) and top-\textit{100} (\textit{Recall@100}) recalls on each dataset are studied under different metrics such as $L2$, \textit{Cosine} and \textit{Inner Product}. Given function $R(i,k)$ returns the number of truth-positive neighbors at top-\textit{k} NN list of sample $i$, the recall at top-$k$ on the whole set is given as
\begin{equation}
Recall@k=\frac{\sum_{i=1}^n{R(i,k)}}{n{\times}k}.
\label{eval:recall}
\end{equation}
All the experiments are carried out on a machine with one Intel Xeon Processor W-2123 (3.60 Ghz) and 32 GB of memory. The GPU we use is an NVIDIA GeForce RTX 3090. All the codes of different approaches considered in this study are compiled by CUDA Toolkit 11.2 and GCC \textit{9.3}. The optimizations like SIMD and pre-fetching instructions are enabled in the source codes for NN search task. HNSW in Hnswlib~\footnote{https://github.com/nmslib/hnswlib} is treated as the comparison baseline. When we evaluate the performance on the CPU, it runs as a single thread. While for the evaluation on the GPU, it runs in multiple threads.
\begin{table}[t]
\caption{Time cost (s) spent for Graph Diversification on an existing \textit{k}-NN graph}
\small{
\begin{tabular}{llrrrrrr}
\toprule
Dataset & \textit{k} & TSDG & GD & NSG~\cite{fu2019fast} & SSG~\cite{fu2021high} & DPG~\cite{li2019approximate} \\
\midrule
SIFT1M & 200 & 54.6 & 18.8 & 188.9 & 47.8 & 166.9 \\
DEEP1M & 200 & 48.8 & 16.6 & 178.3 & 44.5 & 133.2 \\
GIST1M & 400 & 234.9 & 167.9 & 1466.2 & 304.2 & 4690.7 \\
GloVe1M & 400 & 142.3 & 49.3 & 1540.8 & 257.2 & 663.3 \\
SPACEV1M &200 & 145.2 & 21.9 & 478.5 & 162.8 & 165.9 \\
T2I1M & 400 & 335.9 & 53.6 & 1020.6 & 375.8 & 1015.9 \\
\bottomrule
\end{tabular}
}
\label{tab:constr_time}
\end{table}
\subsection{The efficiency of Two-stage Diversification}
In our first experiment, we show the efficiency of graph diversification of our approach in comparison to other approaches in the literature. For all the approaches we consider here, the graph diversification is conducted on the same \textit{k}-NN graph for the same dataset. Four approaches are considered in our comparison. The size of \textit{k}-NN list for different datasets is referenced from the experiments in NSG~\cite{fu2019fast} and SSG~\cite{fu2021high}. Four representative approaches such as GD~\cite{tbd21:zhao}, NSG~\cite{fu2019fast}, SSG~\cite{fu2021high} and DPG~\cite{li2019approximate} are considered in the comparison. HNSW is not considered since its diversification is conducted online. GD~\cite{tbd21:zhao} actually adopts the same the diversification scheme as HNSW~\cite{malkov2020efficient} except that it is undertaken on a pre-built \textit{k}-NN graph and expanded to an undirected graph. SSG performs the diversification on the direct neighbors and expanded neighbors. The available implementation of DPG is similar to our second stage diversification. All the \textit{k}-NN graphs used in the experiments are built by the same GPU-based approach~\cite{wang2021fast}. So the difference between different approaches lies in the diversification strategies applied on the \textit{k}-NN graph. The time costs for all five approaches are shown in~\autoref{tab:constr_time}, while the time costs on the \textit{k}-NN construction step are considered.
As shown from~\autoref{tab:constr_time}, the time costs across most of the datasets of TSDG are only higher than GD, which is comparable to the first stage diversification of our approach. For all six datasets in million scale, the diversification can be fulfilled in several minutes. In contrast, the diversification of NSG and DPG would take a few hours on the same \textit{k}-NN graph. The operations required by DPG are comparable to the operations in our second stage diversification. However, our approach is much more efficient because there are much fewer edges to process owing to the first stage diversification.
\begin{figure}[t]
\begin{center}
\subfigure
{\includegraphics[width=0.48\linewidth]{gp/figs/cpu/sift1m_cpu.pdf}}
\hspace{0.02in}
\subfigure
{\includegraphics[width=0.44\linewidth]{gp/figs/cpu/deep1m_cpu.pdf}}\\
\hspace{0.03in}
\subfigure
{\includegraphics[width=0.47\linewidth]{gp/figs/cpu/gist1m_cpu.pdf}}
\hspace{0.08in}
\subfigure
{\includegraphics[width=0.43\linewidth]{gp/figs/cpu/glove1m_cpu.pdf}}\\
\subfigure
{\includegraphics[width=0.48\linewidth]{gp/figs/cpu/spacev1m_cpu.pdf}}
\hspace{0.02in}
\subfigure
{\includegraphics[width=0.44\linewidth]{gp/figs/cpu/t2i1m_cpu.pdf}}\\
\caption{The NN search performance on the CPU. The parameters used in HNSW, NSG and SSG graph index construction follow with the SSG~\cite{fu2021high}.}
\label{fig:cpu_search}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\subfigure[SIFT1M-BS1]
{\includegraphics[width=0.47\linewidth]{gp/figs/degree/sift1m_bs1.pdf}}
\hspace{0.02in}
\subfigure[SIFT1M-BS10]
{\includegraphics[width=0.465\linewidth]{gp/figs/degree/sift1m_bs10.pdf}}\\
\subfigure[SIFT1M-BS100]
{\includegraphics[width=0.49\linewidth]{gp/figs/degree/sift1m_bs100.pdf}}
\subfigure[SIFT1M-BS10k]
{\includegraphics[width=0.48\linewidth]{gp/figs/degree/sift1m_bs10k.pdf}}\\
\caption{The performance of NN search with the support of graphs in different average degree and batch size. The numbers in the label indicate the average degree of one graph. ``SIFT1M-BSx'' is the run queries arrive in batch size of \textit{x}.}
\label{fig:gpu_degree}
\end{center}
\end{figure}
\subsection{NN Search Performance on the CPU}
With the support of built graph, we proceed to study the NN search performance of different approaches. In this experiment, we adopt the same NN search implementation from NSG code~\footnote{https://github.com/ZJULearning/nsg/} with additional \textit{32} random starting seeds for TSDG, GD, NSG, SSG, and DPG. So the major difference among them lies in the graph that supports the NN search. HNSW is adopted as the comparison baseline. We consider the curve of Recall@10 against the number of queries that one approach processes in one second.
For clarity, our NN search approach that is supported by the proposed TSDG graph is given as ``TSDG-BSx''. ``BSx'' indicates the batch size of a search task. For instance, ``BS10'' indicates the batch size of the search task is \textit{10}. The search performance on the six million-scale datasets is shown in~\autoref{fig:cpu_search}.
As shown from the figures, NN search supported by TSDG outperforms the rest on all six datasets. A considerable performance gap is observed on challenging datasets such as T2I1M and GloVe1M between TSDG and the rest. DPG performs well on \textit{4} out of \textit{6} datasets. It shows inferior performance to TSDG when the data dimension is high. TSDG shares similar average graph degree as that of DPG. Compared to DPG, our graphs keep more informative edges owing to the first stage diversification. As a result, the NN search reaches nearest neighbors with fewer comparisons.
\subsection{NN Search Performance on the GPU}
In this section, we continue to study the performance of our NN search approaches supported by our graph on the GPU. Before we proceed, we show the impact of the connectivity of a graph on the search performance particularly for small batch queries. The connectivity of a graph is measured by its average degree. In the following, we study the performance trend for small batch queries and large batches on the GPU. Approach NSG with average graph degree of \textit{31} is considered as the comparison baseline. Vamana~\cite{subramanya2019diskann} is a diversifying approach only adopts the first stage graph diversification in ours. While DPG~\cite{li2019approximate} is comparable to the diversification of our second stage. Different runs of our approach are carried out supported by graphs with different average degrees. The same NN search that is presented in~\autoref{alg:simple_search} is used on all the graphs for batch sizes \textit{1}, \textit{10} and \textit{100}. While~\autoref{alg:common_search} is used on all the graphs for batch size \textit{10k}. The experiments are conducted on SIFT1M only for page limit. Similar trend is observed on other datasets.
\begin{figure}[t]
\begin{center}
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs1/sift10m_gpu_top10.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/bs1/deep10m_gpu_top10.pdf}} \\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs1/gist1m_gpu_top10.pdf}}
\subfigure
{\includegraphics[width=0.46\linewidth]{gp/figs/gpu/bs1/glove1m_gpu_top10.pdf}} \\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs1/spacev10m_gpu_top10.pdf}}
\subfigure
{\includegraphics[width=0.445\linewidth]{gp/figs/gpu/bs1/t2i10m_gpu_top10.pdf}}
\caption{The NN (\textit{k}=\textit{10}) search performance on the GPU in batch size \textit{1}.}
\label{fig:gpu_search_bs1_k10}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs1/sift10m_gpu_top100.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/bs1/deep10m_gpu_top100.pdf}} \\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs1/gist1m_gpu_top100.pdf}}
\subfigure
{\includegraphics[width=0.46\linewidth]{gp/figs/gpu/bs1/glove1m_gpu_top100.pdf}} \\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs1/spacev10m_gpu_top100.pdf}}
\subfigure
{\includegraphics[width=0.445\linewidth]{gp/figs/gpu/bs1/t2i10m_gpu_top100.pdf}}
\caption{The NN (\textit{k}=\textit{100}) search performance on the GPU in batch size \textit{1}.}
\label{fig:gpu_search_bs1_k100}
\end{center}
\end{figure}
\begin{figure*}[t]
\begin{center}
\hspace{0.3in}
\includegraphics[width=0.65\linewidth]{gp/figs/gpu/legend_bs_small.pdf}
\end{center}
\end{figure*}
\begin{figure}[t]
\begin{center}
\vspace{-0.15in}
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/small_bs/sift10m_gpu_top10.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/small_bs/deep10m_gpu_top10.pdf}} \\
\subfigure
{\includegraphics[width=0.48\linewidth]{gp/figs/gpu/small_bs/gist1m_gpu_top10.pdf}}
\subfigure
{\includegraphics[width=0.45\linewidth]{gp/figs/gpu/small_bs/glove1m_gpu_top10.pdf}} \\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/small_bs/spacev10m_gpu_top10.pdf}}
\subfigure
{\includegraphics[width=0.445\linewidth]{gp/figs/gpu/small_bs/t2i10m_gpu_top10.pdf}}
\caption{The NN (\textit{k}=\textit{10}) search performance on the GPU in batch size \textit{10} and \textit{100}.}
\label{fig:gpu_search_bs_small_k10}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\vspace{-0.15in}
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/small_bs/sift10m_gpu_top100.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/small_bs/deep10m_gpu_top100.pdf}} \\
\subfigure
{\includegraphics[width=0.48\linewidth]{gp/figs/gpu/small_bs/gist1m_gpu_top100.pdf}}
\subfigure
{\includegraphics[width=0.45\linewidth]{gp/figs/gpu/small_bs/glove1m_gpu_top100.pdf}} \\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/small_bs/spacev10m_gpu_top100.pdf}}
\subfigure
{\includegraphics[width=0.445\linewidth]{gp/figs/gpu/small_bs/t2i10m_gpu_top100.pdf}}
\caption{The NN (\textit{k}=\textit{100}) search performance on the GPU in batch size \textit{10} and \textit{100}.}
\label{fig:gpu_search_bs_small_k100}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs10k/sift10m_gpu_top10.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/bs10k/deep10m_gpu_top10.pdf}}
\\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs10k/gist1m_gpu_top10.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/bs10k/glove1m_gpu_top10.pdf}}
\\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs10k/spacev10m_gpu_top10.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/bs10k/t2i10m_gpu_top10.pdf}}
\caption{The NN (\textit{k}=\textit{10}) search performance on the GPU in batch size \textit{10k}.}
\label{fig:gpu_search_bs10k_k10}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs10k/sift10m_gpu_top100.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/bs10k/deep10m_gpu_top100.pdf}}
\\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs10k/gist1m_gpu_top100.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/bs10k/glove1m_gpu_top100.pdf}}
\\
\subfigure
{\includegraphics[width=0.49\linewidth]{gp/figs/gpu/bs10k/spacev10m_gpu_top100.pdf}}
\subfigure
{\includegraphics[width=0.455\linewidth]{gp/figs/gpu/bs10k/t2i10m_gpu_top100.pdf}}
\caption{The NN (\textit{k}=\textit{100}) search performance on the GPU in batch size \textit{10k}.}
\label{fig:gpu_search_bs10k_k100}
\end{center}
\end{figure}
As shown from~\autoref{fig:gpu_degree}, higher graph degree leads to considerably better performance for different small batch sizes. The performance trend of large batch searches on each graph is similar to that of on the CPU. For the graphs with the same average degree, TSDG shows better performance than that of Vamana and DPG. This does indicate the two-stage graph diversification is much more effective than one-stage schemes. In this experiment, our graphs are pruned such that their average degree is in line with Vamana graphs. Such setting is not the optimal for NN search. In the following experiment, we visit the edges whose occlusion factors are lower than \textit{10} for small batch queries. While for large batch queries, we only visit the edges whose occlusion factors are lower than \textit{5}.
The experiment about NN search on the GPU is conducted on six datasets as before. The NN search is scaled up to \textit{10} million level for SIFT, DEEP, SPACEV, and T2I datasets as it is much efficient to build the graphs on the GPU. The queries are organized in both small batches (batch size=\textit{10}) and large batches (batch size=\textit{10k}). In the experiment, the performance from HNSW, GGNN, SONG, and Faiss-IVFFlat is reported. For HNSW, it runs on multiple CPU threads, which is the only approach that runs on the CPU in our study. For approach SONG, we can only run its code\footnote{https://github.com/sunbelbd/song} on SIFT1M smoothly. For this reason, a separate comparison is made with SONG on SIFT1M alone. We re-implement the algorithm of GGNN with our framework and graphs, which performs much better than the original authors' implementation. This run is given as ``TSDG-GGNN''. The NN search results are shown as the curve of recall against the number of processed queries in one second. Results from the same approach are shown with curves in the same color. Results in the same query batch size are shown with curves with the same decoration.
According to our observation on SIFT1M dataset, the performance of our NN search is significantly better than SONG for all batch sizes. When the batch size is \textit{1}, our approach is faster than SONG by more than \textit{20} times, which only could process less than \textit{1000} queries per second under the same search recall. As the batch size increases, the performance gap between our approach and SONG decreases. However, it is still around two times faster than SONG when batch size is \textit{10k}. This observation is in line with~\cite{arxiv22:fabian}. As the implementation of SONG is difficult to scale-up to larger datasets, it is not involved in following experiments.
\autoref{fig:gpu_search_bs1_k10} and~\autoref{fig:gpu_search_bs1_k100} show NN search efficiency of \textit{Recall@10} and \textit{Recall@100} respectively for our approach in comparison to HNSW, TSDG+GGNN, and Faiss-IVFFlat when the batch size is only one. Our NN search demonstrates several times higher performance than the rest when only one query arrives in each batch. This owes mainly to the specific design of the search procedure for the small batch case, which utilizes the GPU parallelism apparently better. Although the performance gap between ours and the rest on \textit{Recall@100} goes narrower, the advantage of our approach is still significant. Similar observation holds when the batch size increases from \textit{1} to \textit{100}. As shown in~\autoref{fig:gpu_search_bs_small_k10} and~\autoref{fig:gpu_search_bs_small_k100}), our NN search still performs much better than the rest. Most of the approaches except for GGNN shows no considerable performance boost when the batch size increases, although better utilization of parallelism is expected for all of them. Although the performance of HNSW could benefit from more CPU cores, the performance of HNSW is quickly saturated as the batch size increases because the bandwidth of the CPU memory is far less than that of the GPU memory.
The NN search results on \textit{Recall@10} and \textit{Recall@100} of our approach on large batch size (\textit{10k}) are shown in \autoref{fig:gpu_search_bs10k_k10} and \autoref{fig:gpu_search_bs10k_k100} respectively, in comparison to HNSW, TSDG+GGNN, and Faiss-IVFFlat. In this case, the performance of our approach is similar to GGNN when we examine top-10 recall (\autoref{fig:gpu_search_bs10k_k10}). Nevertheless, when we check \textit{Recall@100}, we find our approach outperforms GGNN considerably on challenging datasets such as SPACEV10M and T2I10M. As shown in~\autoref{fig:gpu_search_bs10k_k100}, it is consistently better than GGNN and shows \textit{40\%} improvement over GGNN to its best on challenging datasets GloVe1M, SPACEV10M and T2I10M. This owes to the data structures we designed that significantly reduce the maintenance cost when they are in large size. According to our offline test, GGNN shows more than \textit{10\%} lower performance if it is supported the graph proposed in the original paper. From this sense, it is clear to see the proposed TSDG is also supportive for other graph-based search procedures.
As seen from all the above experiments, the NN search by our approach shows the best performance across all the datasets under different circumstances. Nevertheless, similar to all other approaches, the performance drops steadily as intrinsic dimensionality of the dataset increases. If we compare the performance on GloVe1M and SPACEV1M (\autoref{fig:cpu_search}(d) and (e)), we find that it is much more efficient to perform NN search on SPACEV1M than that of GloVe1M although they are on the same scale and the same dimensionality. This is where the intrinsic dimensionality comes to play. Dataset GloVe1M is more challenging since the data are distributed on a manifold with the higher intrinsic dimensionality. In such case, there are more samples that share similar distances to the query at each iteration. The NN search has to compare to more potential samples on the search path before it reaches the nearest neighbor.
\section{Introduction}
Nearest neighbor search is a fundamental issue that arises from several disciplines such as database, information retrieval, pattern recognition, and machine learning. Given a query ($q \in S^d$) and a distance metric $m(\cdot,\cdot)$, the search procedure returns its nearest neighbor or its top-\textit{k} nearest neighbors (\textit{k}-NN) from a candidate set $C=\{x|x \in S^d\}$, where \textit{d} is the data dimension. The early study about the \textit{k}-NN search issue could be traced back to the \textit{1970s} since the proposal of B-tree~\cite{btree:commer79}. After the continuous exploration in a half century, efficient solutions are available for some of the sub-issues. For instance, KD-tree~\cite{kdtree75} works well for low-dimensional data. In the high and sparse dimensional case, the inverted file is one of the most efficient indexing structure for data such as textual documents and web pages, which is still the core structure in nowadays search engines. Nevertheless, for high-dimensional (\textit{e.g.}, $d > 20$) and dense data, this issue remains challenging due to the widely known ``the curse of dimensionality''. In recent years, this issue becomes more and more imminent given that the big data emerge in various forms across different fields.
In the last three decades, the research on NN search has been pushed forward by different waves of data indexing demands. There are in general four categories of approaches that have been proposed. Namely, they are tree-based approaches such as R-tree~\cite{rtree84}, hash approaches~\cite{datar2004locality,mlsh07}, quantization-based approaches~\cite{jegou2010product,pami14:flann}, and graph-based approaches~\cite{malkov2020efficient, fu2019fast}. Amongst all these approaches, graph-based approaches show the highest performance according to recent studies~\cite{li2019approximate}. In general, it is an $A^*$-like search procedure. The search traverses iteratively over a pre-built approximate \textit{k}-NN graph or a relative neighborhood graph (RNG)~\cite{ChenW18} by the best-first search. The search proceeds to the next stage by expanding the neighbors of the closest sample in the rank-list. It ascends closer to the true nearest neighbor in each round until no better candidates could be found.
In the graph-based approaches, the search procedure is undertaken following a path from a random seed to the neighborhood of the target sample. The path is on a manifold that is formed by the graph. In many real scenarios, the dimension of this manifold is much lower than the data dimension. The search actually explores a much lower dimensional space. In the recent studies~\cite{li2019approximate, malkov2020efficient}, the search efficiency is further boosted by diversifying the \textit{k}-NN graph into an approximation of the relative neighborhood graph. Although different diversifying schemes are proposed in~\cite{li2019approximate, malkov2020efficient,fu2019fast}, they are largely out of the same principle. Namely, the query tries to avoid the comparison with the samples whose close neighbors have been already compared. For most of the diversification schemes, the edges directing to these redundant neighbors are simply removed. However, this is not always appropriate. On the one hand, less number of comparisons leads to the higher efficiency of the search. On the other hand, more number of visits to different samples in the graph increases the likelihood of reaching a true nearest neighbor.
In this paper, we optimize not only the diversification scheme that is applied on the \textit{k}-NN graph but the NN search procedures that build upon the graph. The contributions are at least two folds.
\begin{itemize}
\item {A two-stage graph diversification scheme is proposed, which makes a good balance between the connectivity and the sparsity of the graph. It outperforms all the other graphs when it is used to support different variants of the search procedure on both the GPU and the CPU.}
\item {Moreover, two different NN search procedures are designed on the GPU for different scales of batch queries. The parallelism power of the NN search on the GPU is optimized for both small and large batch queries. As will be revealed in the experiment, our approaches outperform the state-of-the-art GPU-based approaches considerably. In particular, for queries that arrive in small batches, our NN search is shown to be \textit{2$\sim$10} times faster than the state-of-the-art approaches.}
\end{itemize}
\section{Two-Stage Graph Diversification}
\label{sec:fdg}
\subsection{Preliminaries for Graph Diversification}
Most of the graph-based approaches follow a similar search procedure, the major difference among them lies in the graph used to support the search. The graph is usually diversified from an approximate \textit{k}-NN graph. The diversification aims to keep only a small portion of the edges in \textit{k}-NN graph. Given there are \textit{k} edges in one \textit{k}-NN list, several of them are directed to the same or similar group of nodes in the graph. On the one hand, it is unnecessary to keep all of them since the comparison with all of the directed samples makes no improvement for the NN search. The removal of such edges reduces the redundant comparisons. On the other hand, certain amount edges should be kept in order to maintain the connectivity of the graph. Since all these strategies in the literatures (FANNG~\cite{harwood2016fanng}, HNSW~\cite{malkov2020efficient}, GD~\cite{tbd21:zhao}, DPG~\cite{li2019approximate} and NSG~\cite{fu2019fast}) are similar, we will review the strategy used in HNSW only in the following.
For each node in the graph, the diversification is performed on a group of edges that are directed to its neighbors. In most of the cases~\cite{malkov2020efficient,tbd21:zhao,li2019approximate}, the edges directed to its \textit{k}-NNs are treated as the candidates. Alternatively, the edges on the searching route from a random sample to the node are treated as the candidates~\cite{fu2019fast}. The edges are kept when they are not occluded. Given current node $x_0$, and its two neighbors $x_i$ and $x_j$, edge$\langle x_0,x_j \rangle$ is occluded by edge$\langle x_0,x_i \rangle$ if the following two inequalities hold\footnote{Without loss of generality, we assume the smaller the distance between two samples, the closer they are.}
\begin{equation}
\left\lbrace \begin{array}{l}
m(x_0, x_i) < m(x_0, x_j) \\
m(x_i, x_j) < m(x_0, x_j)
\end{array}. \right.
\label{eqn:gd}
\end{equation}
In this case, the occluded edge$\langle x_0, x_j \rangle$ will be discarded. The occlusion relation between neighbors is checked starting from the closest neighbor. According to the above rule, the closest neighbor is the first to be selected into the diversified list. While the remaining edges are checked pairwise with all the edges that are already put into the diversified list.
\begin{figure}[t]
\centering
\subfigure[]{
\includegraphics[width=0.43\linewidth]{figures/gd.pdf}
\label{fig:gd}
}
\subfigure[]{
\includegraphics[width=0.43\linewidth]{figures/fdg.pdf}
\label{fig:fdg}
}
\caption{An illustration of the occlusion phenomenon. The dashed line indicates a removed edge and the numbers in (b) indicate the occlusion factors. Two edges in (b) are retained because their occlusion factors are below \textit{2}. }
\label{fig:occlusion}
\end{figure}
As shown in \autoref{fig:occlusion}\ref{sub@fig:gd}, edge$\langle x_0, x_2 \rangle$ is discarded because it is occluded by edge$ \langle x_0, x_1 \rangle$. Moreover, the edges from $x_0$ to other blue nodes are all occluded by edge$\langle x_0, x_1 \rangle$. These blue nodes actually form a cluster. According to~\cite{malkov2020efficient}, it is unnecessary to visit edges other than edge$\langle x_0, x_1 \rangle$ as they offer similar guiding information. Compared to the edges that have been discarded, the kept edges are more important to the search. Similar to the description in DPG~\cite{li2019approximate}, this strategy diversifies the direction of the edges pointing out from each node, so we call it graph diversification (GD)~\cite{tbd21:zhao}.
In $l_2$-space, the graph diversification strategy ensures that the angle between edges is at least $60^{\circ}$. One potential issue about GD is that the edges connected to far away clusters are simply discarded when there could be multiple clusters within the range of $60^{\circ}$. As illustrated in~\autoref{fig:occlusion}\ref{sub@fig:fdg}, the green nodes and the blue nodes come from different clusters, only one edge will be kept for each cluster under the GD rule. For this reason, the reachability of the nodes inside the cluster is degraded.
We can learn from the above example that the removal of an edge is a tricky trade-off. We should consider both the cost of visiting an edge and the guiding information it supplies. On the one hand, keeping the edges connected to smaller scale clusters improves the connectivity between nodes, which in turn improves the reachability of the search procedure to these clusters. On the other hand, it induces more distance computations per hop as well.
In addition, for different types of search, the costs spent on distance computations are different. For small batch search on the GPU, the distance computation can be carried out in high parallel. More edge connections in the graph are affordable. In contrast, for NN search in large batches on the GPU, the search has to be processed in a quite different manner. More number of edge connections increases the computation cost considerably. However, one cannot maintain several graphs with different levels of sparsity. Maintaining several graphs in the memory to support different search batches leads to several times larger memory consumption. Given the size of dataset is big, this is simply not affordable, in particular for the GPU. As a consequence, it is expected to only maintain one diversified graph to support NN search in different batch sizes. In order to maintain the full speed for different search tasks, the diversified graph should allow the search process to choose which edges should be visited or skipped under different circumstances.
In the following section, a two-stage diversification strategy is presented. Accordingly, the graph diversified by our approach is referred to as Two-stage Diversified Graph (TSDG). Different from most of the existing approaches, a portion of the edges within some clusters are kept even they are occluded. An occlusion factor is associated with each edge. Search procedure is allowed to choose either to visit or to skip these edges by the factor under different scenarios.
\subsection{Relaxed Graph Diversification}
On the one hand, graph diversification reduces the average number of comparisons per hop. On the other hand, it decreases the connectivity in the graph, which increases the necessary number of hops before the query reaches its nearest neighbor. In order to make a balance between these two competing properties of a graph, we propose to undertake the diversification in two stages.
In the first stage, similar to~\cite{subramanya2019diskann}, a relaxed graph diversification is performed on each NN list of \textit{k}-NN graph. Namely, the occlusion condition is defined as
\begin{equation}
\left\lbrace \begin{array}{l}
\alpha{\cdot}m(x_0, x_i) < m(x_0, x_j) \\
\alpha{\cdot}m(x_i, x_j) < m(x_0, x_j)
\end{array}, \right.
\label{eqn:sgd}
\end{equation}
where $\alpha$ is a parameter usually greater than \textit{1.1}. According to our observation, only \textit{6$\sim$26\%} of the edges are left after this operation\footnote{The statistics are conducted on six large-scale datasets that are adopted in the experiments.}. Compared to the graph diversified by the original GD, more edges are kept. These extra edges are expected to maintain the connections within local clusters. However, the data distributions vary from one dataset to another. Such a less sparsified graph does not provide the flexibility of visiting certain edges according to their importance under different circumstances. In the next stage, we will address this issue by soft graph diversification.
\subsection{Soft Graph Diversification}
The result of the first stage GD is a sparsified \textit{k}-NN graph. Before we proceed with the \textit{2nd}-stage diversification, the reverse edges of each node are appended to the sparsified \textit{k}-NN list. After this operation, the graph is transformed into an undirected graph. Thereafter, the second round of diversification is undertaken on the NN list of each node. For each edge that survives the first round of diversification, it is associated with an occlusion factor $\lambda$, which is initialized to \textit{0}. We check whether it is occluded by any other edge with Inequation~\ref{eqn:gd}. For instance, if the inequalities hold, the occlusion factor of edge$\langle x_0, x_j \rangle$ is incremented by \textit{1}. After all the occlusion factors are calculated, edges are sorted by the occlusion factors in ascending order. If two edges share the same occlusion factor, they are further sorted by their distances (to node $x_0$) in ascending order. Edges whose occlusion factors are greater than a threshold $\lambda_0$ are simply removed. Taking \autoref{fig:occlusion}\ref{sub@fig:fdg} as an example, we keep the edges with occlusion factor \textit{1} and below, therefore an extra edge will point to samples in the local cluster (green nodes). This is the major difference from the original GD rule.
The occlusion factor provides a means of measuring the importance of edges. Namely, the lower is this value, the more important of this edge is. This is important when NN search runs on different devices or for different search tasks. For the search on the GPU that we will describe later, small batch queries perform better on graphs with high average degree. While for large batch queries, one query is supposedly fulfilled by one GPU warp. The cost of distance computation is much higher than the cost of data structure maintenance in the memories in one warp. It is therefore inefficient to perform search on a graph with extra large degree, which induces more number of distance computations. The graph under soft diversification allows the search procedure to choose how many samples in the list to visit. It therefore no need to maintain several graphs in different average degrees for different batch queries' use. In the large batch size case, only the top-ranked edges are considered.
\begin{figure}[t]
\centering
\subfigure[]{
\includegraphics[width=0.3\linewidth]{figures/dpg_bad.pdf}
\label{fig:dpg_bad}
}
\hspace{0.1\linewidth}
\subfigure[]{
\includegraphics[width=0.3\linewidth]{figures/dpg_good.pdf}
\label{fig:dpg_good}
}
\caption{The illustration of two-stage diversification. The numbers on the nodes are the occlusion factors associated with the corresponding edges. Figure (a) illustrates the scenario that only the \textit{2nd}-stage diversification is applied. The dashed edge will be removed according to the rule. Figure (b) illustrates the two-stage diversification. The edges connecting to the green nodes are removed by the \textit{1st}-stage diversification. The edge connecting to the yellow node is retained.}
\label{fig:two_stage}
\end{figure}
To this end, it seems that the first stage diversification is redundant. One can apply the second stage diversification on the \textit{k}-NN graph directly. However, there are two reasons that make the first stage diversification still necessary. First of all, the first stage diversification helps to prune the edges which would be associated with unexpectedly low $\lambda$ according to the soft GD rule. This is demonstrated in Figure~\ref{fig:two_stage}. As shown in the figure, sample $x_2$ is very close to $x_1$ while far from the rest. In this case, edge$\langle x_0,x_2 \rangle$ is only occluded by edge$\langle x_0,x_1 \rangle$. According to our rule, its $\lambda$ is only \textit{1}. However, according to the principle of diversification, it should not be kept as it is very close to $x_1$. The first stage diversification helps to discard such edges that the second stage cannot assign a factor properly. Furthermore, the size of \textit{k}-NN list could be large in some scenarios, say several hundreds. The size of NN list could be even much larger when the reverse edges are appended. Performing soft diversification on such a list could be time-consuming. The first stage diversification helps to prune the edges that the second stage should not consider. Therefore, the first stage diversification reduces the overall time cost of the second stage graph diversification.
The two-stage diversification can be carried out efficiently on the GPU. The calculation of the occlusion factors for the edges of different nodes is completely independent and can be easily parallelized on the GPU. Finally, unlike NSG~\cite{fu2019fast} or Vamana~\cite{subramanya2019diskann} which obtain candidate edges from the searched routes, the candidate edges are the edges from the \textit{k}-NN graph, so we can load nodes and candidate edges into the GPU in batches, which reduces the GPU memory usage significantly.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{gp/figs/cuda_model.pdf}
\caption{Simplified CUDA programming model.}
\label{fig:cuda_model}
\end{figure}
\section{Graph-based ANN Search on the GPU}
Once the diversified graph is built, one can proceed with the graph-based NN search. Most of the existing approaches on the CPU are the variants of hill-climbing procedure~\cite{icai11:kiana}. Their performance is similar. In our implementation, we adopt the procedure from NSG~\cite{fu2019fast} for NN search on the CPU. While the existing search procedures on the GPU~\cite{zhao2020song,arxiv22:fabian} are designed mostly for the large batch queries, where thousands of queries are processed in parallel. Such kind of design leads to poor performance when the queries arrive in small batches. In this section, the optimized NN search procedures on the GPU are presented for small and large batch queries respectively. Before we present our search procedures, a brief review about the programming model of CUDA is given.
The GPU is designed to excel at executing thousands of threads in parallel. In the CUDA programming model, threads are logically divided into \textit{1}, \textit{2}, or \textit{3} dimensional groups referred to as thread \textit{blocks}. These thread blocks are further grouped into a \textit{1}, \textit{2} or \textit{3} dimensional \textit{grid}. A function that is to be run on the GPU is called as a \textit{kernel}. A grid of thread blocks are launched for one kernel. In CUDA, threads are created, managed and executed in groups of \textit{32} parallel threads called \textit{warps}. So each thread block can be partitioned into multiple warps. In the graph-based NN search, almost all the data structures we use are small in scale and therefore are maintained in \textit{shared memory}, which is a low-latency memory but only has limited capacity. Each thread block has shared memory visible to all threads in it.
Given a batch of queries arrive, a straightforward way to undertake the search on the GPU is to run each individual query on one block. As there are thousands of cores run in parallel, this process could be very efficient. However, it will not be as efficient as we expected when the batch size is small. In this case, the number of thread blocks is much greater than the number of queries. There would be many SMs remain idle. To address this issue, we fulfill the parallel NN search on the GPU differently when the queries arrive in small and large batches.
The division between the small and large batches is closely related to the available GPU resources and data dimension. In general, the more streaming multiprocessors (SM) are in a GPU device, the query batch should be larger if we want to fully utilize its computing power. The empirical formula for the division threshold between the small and large batches is $\frac{a \cdot SMs + b}{d}$, where $a$ and $b$ are device-related constants. $SMs$ is the number of SMs, and $d$ is the dimensionality of the data. Take our case NVIDIA GeForce RTX 3090 as an example, which has \textit{82} SMs and \textit{128} CUDA cores per SM. On SIFT1M dataset (\textit{128} dimensions), the threshold between small and large batches is about \textit{300}, while the threshold on GIST1M dataset (960 dimensions) is about \textit{150}.
\subsection{Small Batch Search on the GPU}
When the queries arrive in small batches, many GPU SMs remain idle if one query is assigned to one thread block inside the GPU. For this reason, the computation power of GPU cannot be fully utilized. To address this issue, the parallelization for small batch queries is realized in a novel way. Given a batch of queries arrive $Q=\{q_1,{\cdots}, q_i,\cdots, q_m\}$, one query $q_i$ is fulfilled on multiple thread blocks on the GPU. For each $q_i$, a group of simple greedy searches $S=\{s_1,{\cdots}, s_j,\cdots, s_{t_0}\}$, where ${t_0}$ is the number of searches, are launched on $t_0$ thread blocks. Each search procedure $s_j$ runs independently and returns a ranking $R_{ij}$ for $q_i$. This ranking is finally merged into the ranking $R_i$ for query $q_i$.
\begin{algorithm}[t]
\caption{NN Search for Small Batch on the GPU}
\label{alg:simple_search}
\KwData{graph $G$, query $q_i$, number of hop limit $T$}
\KwResult{top-\textit{32} approximate nearest neighbors}
$R_{ij} \leftarrow 32$ ID-distance pairs$(\infty, \infty)$\;
$U \leftarrow$ \textbf{Generate} 32 random starting nodes from $G$\;
$u \leftarrow$ \textbf{Get} closest node to $q_i$ in $U$\;
$improved \leftarrow$ \textbf{true}; $t \leftarrow 0$\;
\While{improved and $t < T$} {
$R_{temp} \leftarrow 32$ ID-distance pairs$(\infty, \infty)$\;
\For{every 32 neighbors $V$ of $u$ in $G$} {
\label{simple_search:for_1}
\ParaFor{$v \in V$} {
$dist \leftarrow m(v, q_i)$\;
\If{$dist < R_{temp}[warp\_id].dist$}{
\label{simple_search:warp_id}
$R_{temp}[warp\_id] \leftarrow (v, dist)$\;
}
}
}
\textbf{Update} $R_{ij}$ with $R_{temp}$\;
\label{simple_search:update}
$u \leftarrow$ \textbf{Get} closest node to $q_i$ in $R_{temp}$\;
\If{$R_{ij}$ is not updated} {
$improved \leftarrow$ \textbf{false}\;
}
}
\Return{$R_{ij}$}\;
\end{algorithm}
The procedure of the simple greedy search is summarized in~\autoref{alg:simple_search}. The search starts from the best sample that is selected from \textit{32} random seeds. According to our offline test, this simple seed selection scheme is as effective as the hierarchical strategy in HNSW~\cite{malkov2020efficient} and the ``navigating node'' strategy in NSG~\cite{fu2019fast}. The starting sample is set as the ``current node'' \textit{u}. As shown in \textit{Lines 7-11}, \textit{32} samples from \textit{u}'s NN list are compared with query $q_i$. The number of samples compared here is in line with the number of warps in one thread block. Each distance is computed by a warp in parallel. The resulting \textit{32} distances are used to replace the results kept in $R_{temp}$ if they are smaller. $R_{temp}$ is an unordered array in the shared memory or registers. The size of $R_{temp}$ is \textit{32} as well, which is in line with the number of warps in each block. $R_{temp}$ is accessed by the warp ID. The computed distance from one warp with ID $warp\_id$ will only compare with one cell $R_{temp}[warp\_id]$.
For this reason, it is not guaranteed that $R_{temp}$ keeps the closest top-\textit{k} samples that are visited so far. However, such ad hoc update only requires the threads to access $R_{temp}$ once, which is very efficient. The smallest sample in $R_{temp}$ is selected as the expanding point \textit{u} for the next round.
Array $R_{ij}$ is used to keep the final top-\textit{k} results for a single search $s_j$. Similar to $R_{temp}$, it is maintained in the shared memory or registers. The update operation in \textit{Line 12} of \autoref{alg:simple_search} is carried out by one warp. It is an incomplete ascending sort of $R_{temp}$ with bitonic sorter~\cite{batcher1968sorting}, ensuring that the first \textit{16} elements are the smallest. The last \textit{16} elements of $R_{ij}$ are replaced with the top-\textit{16} smallest elements of $R_{temp}$. Finally, a complete sort by bitonic sorter is conducted on $R_{ij}$. Since the update operation on $R_{ij}$ is time-consuming, we should try to reduce the number of search hops. As shown by the experiments in \cite{subramanya2019diskann}, the higher average degree of the graph leads to the fewer hops in NN search. Graph with two-stage diversification matches well to such scenario. It allows the NN search to traverse over a graph with a high average degree by visiting more edges with larger occlusion factors.
As one could see from~\autoref{alg:simple_search}, no expansion queue is adopted. This is for the sake of efficiency. This greedy search converges in simply \textit{4-5} iterations. For this reason, one cannot expect good performance from a single search. The NN search quality is enhanced by running multiple such cheap searches in parallel for one query.
When performing multiple independent searches for the same query, redundant distance computations may happen. Therefore, the results collected from different searches might be duplicated. Fortunately, due to the poor precision of the single greedy search, the likelihood of duplication between different searches is low. This motivates us to sacrifice the precision of each independent search to improve the search speed. We increase the search quality by increasing the number of independent searches $t_0$. Because the batch size is small, increasing $t_0$ makes little impact on the search speed, while the recall is improved considerably.
\subsection{Large Batch Search on the GPU}
When the queries arrive in large batch, the search is undertaken in a straightforward way. One thread block runs as a CPU core, and is responsible for NN search of a single query. Thousands of queries can be fulfilled in parallel on thousands of thread blocks. The constructed graph is held in the global memory to support all the thread blocks in the GPU. In each thread block, we assign \textit{32} threads (one warp) work in parallel serving for one query $q_i$. The distance computation, expansion queue update, and the top ranking list maintenance for each query are all operated in parallel by the warp.
The search algorithm performed by each thread block is shown in \autoref{alg:common_search}. The search process is very similar to the bottom-layer search used in HNSW~\cite{malkov2020efficient} on the CPU. While the major difference lies in the maintenance of three key data structures. Namely, they are the top ranking array $R$, the expansion queue $C$, and the look-up table $V$. The top ranking array $R$ is used to keep the top-\textit{k} search results and its size is fixed to \textit{k}. Since the size of $R$ is fixed for efficiency, similar to~\cite{arxiv22:fabian}, we use a threshold $\Delta$ as the termination condition for the search. The expansion queue $C$ keeps the samples whose NN list will be visited in the future rounds. $V$ is used to check whether a sample has been visited before. All these three structures are kept in the shared memory for the sake of efficiency.
\begin{algorithm}[t]
\caption{NN Search for Large Batch on the GPU}
\label{alg:common_search}
\KwData{graph $G$, query $q$, number of hop limit $T$, probe threshold $\Delta$}
\KwResult{top-\textit{k} approximate nearest neighbors}
$U \leftarrow$ \textbf{Generate} 32 random starting nodes from $G$\;
$u \leftarrow$ \textbf{Get} closest node to $q$ in $U$\;
$R \leftarrow \{u\}$; \hspace{5pt} // priority queue of found nearest neighbors \\
$C \leftarrow \{u\}$; \hspace{3.5pt} // expansion queue of candidates \\
$V \leftarrow \emptyset$; \hspace{10.8pt} // set of visited elements \\
$t \leftarrow 0$\;
\While{$\vert C \vert > 0$ and $t < T$} {
$t \leftarrow t + 1$\;
\textit{u} $\leftarrow$ pop nearest element to $q$ from $C$\;
\textit{f} $\leftarrow$ get furthest sample to $q$ from $R$\;
\If{$m(u, q) > m(f, q) + \Delta$} {
\label{common_search:term_condi}
break\;
}
$V$.Add($u$)\;
\label{common_search:hash}
\For{each neighbor $e$ of $u$ in G} {
\If {$e \notin V$ and $e \notin C$} {
\label{common_search:visited}
\textit{f} $\leftarrow$ get furthest sample to $q$ from $R$\;
\If {$m(e, q) < m(f, q)$ or $|R| < k$} {
$R$.Push($e$)\;
$C$.Push($e$)\;
\If {$|R| > k$} {
$R$.Pop(); \quad // remove furthest sample \\
}
}
}
}
}
\Return{$R$}\;
\end{algorithm}
Similar to the small batch case, the query starts from the closest sample selected from \textit{32} random seeds. This way is as effective as the hierarchical structure in HNSW~\cite{malkov2020efficient} and ``navigating node'' in NSG~\cite{fu2019fast}. In each round of NN search, one sample \textit{u} from $C$ is popped out. The query is compared to its unvisited neighbors \textit{e}. Once sample \textit{e} is compared to the query, it is pushed to $R$ and $C$ respectively. Sample $u$ is pushed to the look-up table $V$. The iteration continues until $C$ is empty or no improvement can be made. Although the above NN search makes no significant difference from the popular graph-based search procedure on the CPU, the design of structures about $R$, $C$, and $V$ is essentially different from its CPU versions. In the following, more details are given to explain how these structures are designed and maintained.
The expansion queue $C$ is implemented as a long array. This long array is further divided into multiple segments with equal size. The segment size is set to the same as the number of threads in a block. In our case, it is \textit{32}. Each segment is structured as a sorted circle array. It is sorted in ascending order according to the distances of samples' to the query. Given sample \textit{e}, the segment to keep it is located by $e.id \% m$, where \textit{m} is the number of segments and $e.id$ is the sample ID. It is, therefore, inserted to the segment $e.id \% m$. The most distant sample in that segment is popped out when the segment is full. All the top-\textit{1} elements from \textit{m} segments are checked when we want to pop out one sample to expand. The smallest is extracted from them and popped out from the corresponding segment. Compared to the priority queue used in~\cite{arxiv22:fabian}, we divide the queue into dozens of segments. The size of each segment is in line with the number of threads in a block. Therefore, the access to segment cells could be fulfilled in one round, which considerably reduces the number of scans and threads.
The look-up table $V$ is implemented as a long array and divided into \textit{m} segments. Each segment is maintained as a circular array. So the design of $V$ is similar as $C$ except that its segment is not sorted. Given sample \textit{e}, the segment to keep it is located by $e.id \% m$. The oldest sample in segment $e.id \% m$ will be replaced by \textit{e} when the segment is full. It is convenient to determine whether an element is in $V$. All the threads in one thread block fetch the samples from a segment and compare them to the element to be checked. Since the number of elements in each segment is the same as the number of threads in the thread block, we only need to do the operation once. Please be noted that not all the visited nodes are pushed into $V$. Otherwise, it would be very expensive to keep all the visited nodes. It is actually unnecessary to do so. The chance of revisiting the early-encountered samples is low as the search moves forward all the way. Similar as~\cite{arxiv22:fabian}, only the nodes used in the expansion are pushed into $V$. This largely avoids the circular search.
The algorithms presented in this section are all implemented in C++ and compiled by GCC \textit{9.3}. The GPU implementation is supported by NVIDIA CUDA Toolkit \textit{11.2}. We have made it available as supplementary material of this submission.
\section{Related Work}
\subsection{Non-Graph based Approaches}
There are generally four major categories of NN search approaches in the literature. In the \textit{1980}s and early \textit{1990}s, most of the approaches are an extension of B-tree. The representative indexing trees are KD-tree~\cite{kdtree75}, R-tree~\cite{rtree84}, and X-tree~\cite{vldb96:xtree}, etc. Essentially, the idea behind these approaches is similar. They partition the space into hyper-rectangles under different strategies. Close neighbors are assumed to reside under the same hierarchy of hyper-rectangles. However, the situation is much more complicated in the high-dimensional space. The partition scheme does not exclude the possibility that the nearest neighbors reside outside of one branch. It becomes inevitable to probe extensively over a large number of neighboring branches. Quantization-based approaches~\cite{pami14:flann} partition the space in an alternative way. The candidate data are quantized with respect to a pre-defined code-book~\cite{iccv03:sivic} or a hierarchy of code-books~\cite{pami14:flann}. Different from tree-based approaches, the space is partitioned by a series of \textit{Voronoi} cells, each of which is regularized by one vocabulary word in the code-book. These two types of approaches share one thing in common. Namely, the query is compared to the indexing nodes (non-leaf nodes in the tree or words in the code-book) to locate the nearest neighbor.
In recent works, several attempts have been made to compress the candidate samples by vector quantization~\cite{babenko2016efficient,icml14:tzhang, sq14, tit06:gray06} or sub-vector quantization~\cite{jegou2010product,ge2013optimized}. On the one hand, the words in the code-books are used to guide the NN search. On the other hand, several words in combination are used to approximate the candidate samples. The NN search is conducted between the query and the compressed candidate samples. The distances between the query and the candidate samples are approximated by the distances between the query and the code words. One advantage of these approaches is that it is no need to keep raw vectors in the memory. However, due to the compression on the candidate samples, high search quality is hardly achievable. Furthermore, these types of approaches are usually only feasible for metric spaces of $\textit{l}_p$-norm.
Apart from the above two types of approaches, hash approaches have been introduced to the nearest neighbor search in~\cite{datar2004locality,mlsh07}. The samples are mapped into hash codes by locality sensitive hashing (LSH). Close neighbors in the vector space are expected to share the same or similar hash codes. Therefore, the query will only be compared to the samples with the same or similar hash codes to find out the nearest neighbors. The high compression rate leads to low memory cost and efficient comparison, however, low search quality as well. Moreover, compared to the compression by vector quantization, comparisons with the raw vectors are still necessary, which in turn requires the maintaining of all the raw vectors in the memory. Moreover, it is non-trivial to define a generic hash mapping scheme that works for various types of data.
\subsection{Graph-based Approaches}
For all the aforementioned approaches, the candidate samples have been indexed by non-leaf nodes of a tree, vocabulary words or hash codes. The search localizes the close neighbors via a series of comparisons between the query and these indexing nodes. Different from the aforementioned approaches, every sample in the graph-based approaches acts as an indexing node. In general, two stages are involved in the graph-based search. In the first step, a \textit{k}-NN graph or a relative neighborhood graph is constructed. Each sample (node) links to its close neighbors. This is usually fulfilled offline. In the second step, the search starts from comparing with a group of random samples on the graph. By expanding the links to the neighbors of visited nodes (being compared), the search ascends closer to the true neighbors in each round. The search iterates until no better candidates could be found. Approaches in~\cite{icai11:kiana,malkov2020efficient,fu2019fast,arxiv22:fabian} follow a similar search procedure. The major difference between them lies in the graph used to support the search. According to the recent studies~\cite{li2019approximate,prokhorenkova2020graph}, NN search that is supported by the \textit{k}-NN graph already outperforms most of other types of approaches. Another advantage of graph-based approaches over other types of approaches is that they are feasible for different distance metrics.
According to recent studies~\cite{malkov2020efficient,fu2019fast,li2019approximate}, the search performance can be further boosted when the supporting \textit{k}-NN graph is further diversified. Although different variants of diversification approaches have been proposed and their performance ranking varies across different datasets, they are shown to be essentially similar~\cite{wang2021comprehensive}. The major principle out of them is to remove the edges that are shorter to other neighbors in one node's neighborhood than they are to the node. These edges are mostly directed to the nodes already reachable via the reserved edges. The diversification helps to reduce the redundant comparisons during the NN search. Most of the graph diversification schemes such as DPG~\cite{li2019approximate}, SSG~\cite{fu2021high}, and GD in HNSW~\cite{malkov2020efficient} are applied on an approximate \textit{k}-NN graph. As an exception, NSG~\cite{fu2019fast} applies diversification on a searching route from a random seed to a node. The performance of these schemes varies considerably across different datasets, where the data distribution changes.
In this paper, we are going to propose a two-stage graph diversification scheme. The first stage diversification helps to remove those apparently redundant edges. The second stage diversification associates an occlusion factor with each edge, which indicates the importance of an edge. For different search procedures, they could choose to either visit or skip an edge according to this factor under different NN search circumstances. When computing resources are sufficient for one search, edges even with high occlusion factors can be visited. Such that we aim to maximize the search efficiency that we can reach on one device.
\subsection{NN Search on the GPU}
On the CPU, the most computationally intensive operation in the NN search is the distance computation. For all the above approaches designed for the CPU, the motivation is to find out the nearest neighbors while limiting the number of distance computations as few as possible. On the GPU, the situation is different, where high parallel computing performance is available. The distance computation is relatively cheap in this circumstance. For this reason, the high efficiency is achieved even by performing exhaustive comparisons between the query and the whole candidate set according to~\cite{johnson2019billion}. The major issue lies in the throughput mismatch between the GPU cores and the different levels of memories. In~\cite{johnson2019billion}, the nearest neighbors are found by merging the results from exhaustive comparison by \textit{k}-selection on each thread block. However, it is wasteful to perform the exhaustive comparisons for NN search~\cite{sigir21:gaips}. In recent studies~\cite{sigir21:gaips, arxiv22:fabian, zhao2020song}, the NN search is carried out with the support of graph on the GPU. The query is expected to compare only with samples along a search path. In order to make the full use of GPU computing power, the distances between the query and the samples in one NN list are computed on multiple threads of thread blocks~\cite{zhao2020song}. Additionally, the bloom filter is adopted to check whether a sample has been visited before~\cite{zhao2020song} to avoid repetitive computation.
Unfortunately, approaches in~\cite{arxiv22:fabian, zhao2020song} are designed for queries that arrive in large batches. When the queries arrive in small batches, the advantage of GPU's computing power cannot be fully utilized. In this paper, two NN search algorithms are designed for the queries that arrive in different scales of batches on the GPU. When these search algorithms are supported by the proposed graph, they both perform better than the state-of-the-art approaches.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,258 |
Welsh Marches (kymriska: Y Mers) är områdena vid gränsen mellan England och Wales. Den exakta definitionen av området har varierat genom historien, och under medeltiden syftade området på de marker i södra Wales där så kallade Marcher lords hade särskilda rättigheter. På medeltidslatin kallades området för Marchia Walliae. I dag finns ingen exakt definition av vilka områden som ingår i Welsh Marches, men termen används för att benämna de grevskap som ligger vid gränsen i Wales och England, såsom Shropshire och Cheshire.
Ordet march kommer ifrån fornfranskans marche och är besläktat med det svenska ordet mark, jämför Mercia och Marka socken.
Se även
Council of Wales and the Marches
Scottish Marches
Referenser
Noter
Tryckta källor
Wales historia
Englands historia
Wales geografi
Englands geografi | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,859 |
Mubarak Shah may refer to the following people:
Mubarak Shah (Chagatai Khan), head of the Chagatai Khanate (1252–1260)
Qutbuddin Mubarak Shah, Khalji dynasty, Delhi Sultanate (d. 1320)
Fakhruddin Mubarak Shah, Bengal ()
Mubarak Shah (Sayyid dynasty), Delhi Sultanate ()
Mubarak Shah (athlete), Pakistani long-distance runner | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,592 |
AAH also offers an excellent physical therapy device called the Aquamed 200 Dry Hydrotherapy bed. Dry Hydrotherapy combines the time-proven benefits of massage, heat, and whirlpool in a single system. This treatment allows the user to stay clothed and dry; providing a hygienic and comfortable experience.
The Aquamed 200 allows control of all treatment variables (temperature, pressure, speed, duration and area of treatment) and can be customized for each individual patient. At times, Dr. Douglas may recommend you use the bed before or after a treatment. This will allow your muscles to relax, increase the blood flow to problem areas, and decrease your overall pain level.
Additionally, you may use the Aquamed 200 between appointments if you need to decrease muscle spasm, improve pain, or just need a quick, relaxing massage. Typically, a treatment on the Aquamed 200 lasts 10 to 15 minutes. We even have a frequent user card that earns you a free Aquamed treatment after 10 therapy sessions. Although we may be able to squeeze you in if you drop by, it is best to call and schedule a session ahead of time to ensure the shortest wait possible.
If you have any questions, please call us or ask us about the Aquamed 200 Dry Hydrotherapy bed during your next visit. | {
"redpajama_set_name": "RedPajamaC4"
} | 388 |
A
BOOK
The Philip E. Lilienthal imprint
honors special books
in commemoration of a man whose work
at the University of California Press from 1954 to 1979
was marked by dedication to young authors
and to high standards in the field of Asian Studies.
Friends, family, authors, and foundations have together
endowed the Lilienthal Fund, which enables the Press
to publish under this imprint selected books
in a way that reflects the taste and judgment
of a great and beloved editor.
The publisher gratefully acknowledges the generous
contribution to this book provided by the Philip E.
Lilienthal Asian Studies Endowment of the University
of California Press Associates, which is supported by
a major gift from Sally Lilienthal.
Seeing through Zen
# Seeing through Zen
Encounter, Transformation, and Genealogy in Chinese Chan Buddhism
John R. McRae
UNIVERSITY OF CALIFORNIA PRESS
_Berkeley / Los Angeles / London_
University of California Press
Berkeley and Los Angeles, California
University of California Press, Ltd.
London, England
© 2003 by the Regents of the University of California
Library of Congress Cataloging-in-Publication Data
McRae, John R., 1947–
Seeing through Zen : encounter, transformation, and genealogy in Chinese Chan Buddhism / John R. McRae.
p. cm.
Includes bibliographical references and index.
ISBN 0-520-23797-8 (alk. paper)—ISBN 0-520-23798-6 (pbk. : alk. paper)
1. Zen Buddhism—China—History. I. Title.
BQ9262.5 .M367 2004
294.3'927'0951—dc21 | 2003011741
---|---
Manufactured in the United States of America
13 12 11 10 09 08 07 06 05 04
10 9 8 7 6 5 4 3 2 1
The paper used in this publication is both acid-free and totally chlorine-free (TCF). It meets the minimum requirements of ANSI/NISO Z39.48–1992(R 1997) _(Permanence of Paper_ ).
_Dedicated to YANAGIDA Seizan, with inexpressible gratitude_
## Contents | List of Illustrations
---|---
|
Preface
|
Conventions
|
McRae's Rules of Zen Studies
1. | Looking at Lineage: A Fresh Perspective on Chan Buddhism
2. | Beginnings: Differentiating/Connecting Bodhidharma and the East Mountain Teaching
3. | Metropolitan Chan: Imperial Patronage and the Chan Style
4. | The Riddle of Encounter Dialogue: Who, What, When, and Where?
5. | Zen and the Art of Fund-Raising: Religious Vitality and Institutional Dominance in the Song Dynasty
6. | Climax Paradigm: Cultural Polarities and Patterns of Self-Cultivation in Song-Dynasty Chan
|
Notes
|
Character Glossary
|
Bibliography
|
Index
## Illustrations
Figures
|
---|---
Frontis: | Bodhidharma crossing the Yangzi on a reed, by Young-hee Ramsey
1. | Lineage diagram of Chinese Chan Buddhism
2. | Simplified chart of the phases of Chinese Chan
3. | "Huineng's" verse from the _Platform S tra_ on the back window of a taxicab, Tainan, Taiwan
4. | Bodhidharma worshiped as local deity, Hall of the Three Teachings, Jianchuan, Yunnan Province
5. | Images of kyamuni, Confucius, and Laozi, with Bodhidharma to proper right, Hall of the Three Teachings, Jianchuan, Yunnan Province
Maps
|
1. | Locations for Proto-Chan, Early Chan, and Middle Chan
2. | Locations for Song-Dynasty Chan
## Preface
This book is intended for those who wish to engage actively in the critical imagination of medieval Chinese Chan, or Zen, Buddhism. The interpretations presented in the following pages represent my best and most cherished insights into this important religious tradition, and I look forward to their critical appraisal and use by general readers, students, and colleagues. More important than the specific content presented here, though, are the styles of analysis undertaken and the types of human processes described. In other words, the primary goal of this book is not to present any single master narrative of Chinese Chan, but to change how we all think about the subject.
I expect the readership of this book to include Zen and other Buddhist practitioners; students and scholars of Chinese religions, Buddhist studies, and related fields; and a general audience interested in Asian religions and human culture. General readers will find here sufficient basis for a far-reaching critique of how Zen is perceived in contemporary international culture. In addition, my analysis of Chinese Chan religious practice as fundamentally genealogical should provide a new point of comparison for the analysis of modern and contemporary developments in Zen, particularly those occurring in North America and Europe. That is, if Chan practice was originally genealogical—by which I mean patriarchal, generational, and relational—in ways that fit so well with medieval Chinese society, how will it be (or, how is it being) transformed as it spreads throughout the globe in the twenty-first century (and as it did in the twentieth)? In other words, how is Zen changing, and how will it change, as it grows and spreads within the context of globalization and Westernization?
Scholars, students, and general readers constitute a natural audience for this book. Why should religious practitioners read it? If Buddhist spiritual practice aims at seeing things as they are, then getting past the foolish over-simplifications and confusing obfuscations that surround most interpretations of Zen should be an important part of the process. That is the short answer. A more specific answer requires a bit of explanation.
The first time I lectured on Chinese Chan to a community of practitioners was in 1987 for the Summer Seminar on the Sutras, held in Jemez Springs, New Mexico, at Bodhi Mandala, which functions within the teaching organization of the Rinzai master Joshu Sasaki Roshi. Among those attending was an elderly American Zen monk, who objected strenuously to my instruction, asking repeatedly, "What good is any of this for my practice?" The organizers of the week-long workshop were somewhat embarrassed by his aggressive attitude, pointing out that, as a former lightweight boxer, he may have taken more than one punch too many. For my own part, I enjoyed the challenge, which forced me to confront the question head-on in a way that never would have happened in university lecturing. Subsequently, I have honed my response (you are welcome to consider it a defensive reaction, if you wish!) in seminars and workshops at Dai Bosatsu in upstate New York, under the direction of another Rinzai teacher, Eido Shimano Roshi; the Zen Center of Los Angeles, a Soto Zen institution founded by the late Taizan Maezumi Roshi; Zen Mountain Monastery at Mt. Tremper, New York, led by John Daido Loori Roshi, a student of Maezumi's; the San Francisco Zen Center, which was founded by Shunryu Suzuki Roshi; the Mount Equity Zen Center, directed by Dai-En Bennage Sensei; Zen Mountain Center, directed by Tenshin Fletcher Sensei, a successor to Maezumi; and Dharma Rain and the Zen Community of Oregon in Portland, directed by Kyogen and Gyokuko Carlson and Chozen and Hogen Bays, respectively. In addition to these American Zen centers, on two separate occasions, I have also taught a two-week intensive class at Foguang Shan in Kaohsiung, Taiwan; on the first occasion (in 1992) the participants were mostly young Taiwanese Buddhist nuns, while on the second (in 2002) the class was composed of Southeast Asian Chinese nuns and monks from Africa, India (a native of N land !), and the United States. These chapters were first prepared for presentation at Templo Zen Luz Serena in Valencia, Spain, under the direction of Dokush Villalba Sensei. This book has benefited in profound ways from interaction with the participants in all these different practice settings, and I am deeply grateful for their attention, questions, and suggestions.
There is nothing in this book that will aid one's religious practice directly. I am not a Zen master, nor even a meditation instructor, and this is not a do-it-yourself manual. To use a cooking analogy, I am not a Julia Child teaching you how to concoct your life in Zen. Instead, I am more the art critic who evaluates her teaching methods and dramatic performance, or even the chemist who analyzes the dynamic evolution of her recipes as they make their pilgrimage from pan to plate. Art critics are not necessarily good performers themselves, and chemists are not necessarily gourmet cooks. Although I am indeed a Buddhist (in autobiographical blurbs I usually include a line about being "a practitioner of long standing but short attention span"), and even though my religious identity as a youthful convert to Buddhism allows for a certain empathy with my subject matter, I am a scholar and not a guru. As a professor in aggressively secular state universities over the years, I have learned to keep anything resembling preaching out of my classroom presentations, and the same holds true here. I am not aiming to convert you, unless by that is meant an intellectual transformation that may penetrate to the very core of your being.
This volume is resolutely about Zen, not about how to practice Zen. It thus differs from the vast majority of books on Zen in English in that it does not assume the reader to be a potential Zen practitioner. Indeed, even the most dedicated practitioners will benefit by stepping outside their chosen tradition for the endeavor of its reading. I believe our roles as scholars and readers involve the active and critical imagination of the medieval evolution of Chinese Chan Buddhism. By active, I mean that we should constantly work to envision how Chan emerged in the medieval Chinese social and intellectual context; by critical, I mean that we should also work to consider all the available evidence from all possible angles, testing hypotheses and evaluating objections.
In many ways, my training in this process began with my graduate studies under Professor Stanley Weinstein, to whom I am dedicating my second research volume on eighth-century Chan Buddhism, provisionally entitled _Zen Evangelist: Shenhui (684–758), Sudden Enlightenment, and the Southern School of Chinese Chan Buddhism_ (forthcoming from the University of Hawai'i Press, under the auspices of the Kuroda Institute). The debt I owe Professor Weinstein, who has dedicated his career to the training of the finest cohort of scholars in American Buddhist studies, is incalculable.
The present volume is in effect my attempt to emulate the creative work of YANAGIDA Seizan , with whom I had the privilege to study while writing my dissertation. As the greatest scholar of Chinese Chan Buddhism of the twentieth century, Professor Yanagida has brought to his writings both magisterial knowledge and profound sensitivity. Although my tutelage under Professor Yanagida came many years before this book was conceived, I have fond memories of sitting with him in his study, accepting bowls of delicious _matcha_ tea, and discussing the contents of Chinese Chan texts. Even when I groped for ordinary Japanese vocabulary in our conversations, and even when I butchered the rules of classical Chinese grammar in our readings, his sympathetic patience was inexhaustible. (I will admit, however, that for the weekly seminars on Chinese Chan texts at Hanazono College, a Rinzai Zen institution, it would have been copacetic had the college marching band not chosen the very same time to practice its John Philip Sousa renditions!)
As the vanguard of a new wave of Japanese scholarship that revolutionized our understanding of Chan through analysis of handwritten manuscripts from the Dunhuang cave in Chinese Central Asia, Professor Yanagida has consistently demonstrated an interpretive brilliance that has energized an entire generation of Western students. If I have inherited even a small part of his legacy, I hope that the playful humanism of his example shines through these pages. I dedicate this book to Professor Yanagida with a depth of gratitude I can only hint at in words.
Thanks are due to many others as well, of course. As mentioned above, these chapters were first prepared for presentation in Spanish translation at Templo Zen Luz Serena, directed by Dokush Villalba Sensei, in Valencia, Spain, June 19–21, 1999. The invitation was sponsored by the Japanese S t Zen School, and the initial translations were prepared by Ms. Lucía Huélamo and Rev. Aigo Castro, who also served as cotranslators for the oral presentations. My profound thanks are due Villalba Sensei, Rev. Castro, and Ms. Huélamo, as well as all the members of Luz Serena, who made my visit there so enjoyable and productive. Subsequently, the first chapter was presented in Chinese at the Chung-Hwa Institute of Buddhist Studies. The Chinese translation, which was prepared by KUAN Tse-fu [Guan Zefu] , was published in Chinese as "Shenshi chuancheng—chenshu Chanzong di ling yizhong fangshi" . I would like to express my deep gratitude to Ven. Sheng-yen , as well as to Professor LI Chih-fu , director of the Institute, and Secretary CHEN Hsiu-lan and the Institute staff for their kind assistance during my research stay in Taiwan from December 1998 to August 1999. Also, part of chapter 4 has already appeared in print as "The Antecedents of Encounter Dialogue in Chinese Ch'an Buddhism," in Steven Heine and Dale S. Wright, eds., _The K an: Texts and Contexts in Zen Buddhism._
Jan Nattier has gone over the entire manuscript, covering my precious words with a liberal coating of editorial ink. I am immensely grateful, even if only for the occasion to divert her attention temporarily from third-century Chinese Buddhist translations. William Bodiford, Stephen Bokenkamp, Robert Buswell, Robert Campany, and David Eckel have also reviewed the text, and all of them provided suggestions both meaningful and helpful. Even given this assistance, copy editor Nick Murray has found many ways to improve the text. I offer my sincere gratitude to these, my friends and colleagues.
Special thanks are due Reed Malcolm, the editor who appreciated the value of this book and shepherded it through production at the University of California Press. Reed deserves credit for the title, _Seeing through Zen,_ which to my ears is wonderfully multivalent. In addition to the workshops and seminars at practice centers mentioned above, over the years I have inflicted these interpretations of Chinese Chan upon classes of undergraduate and graduate students at Harvard, Cornell, Indiana, and Hawai'i Universities, as well as academic audiences at Stanford, Indiana, and Yale. To the participating faculty and students, whose probing questions did so much to push me into different perspectives on familiar material, I offer my thanks. Were there observations gone unheard or errors left uncorrected in spite of all this assistance, the cause is nothing other than my own limited understanding.
_Ama ga kobako_
Honolulu, Hawai'i
June 2002
## Conventions
The goal of this book is to facilitate the different learning needs of a variety of readers. Hence the main text is for all readers, including beginners and nonspecialists, while the notes, character glossary, and bibliography are intended for use by students and scholars. The specific conventions adopted are as follows.
1. I have included frequent cross-references within the text, so that readers can easily keep the different elements of the discussion fresh in their minds. Active reading requires a certain flapping of pages.
2. When I provide two sets of transliterations, unless otherwise noted the first will be Chinese (in Pinyin spelling) and the second Japanese.
3. Chinese book titles are referred to by (sometimes abbreviated) English translations throughout, with the Pinyin spelling given only on first occurrence. Please consult the character glossary for the original Chinese titles.
4. Names of individuals functioning in an East Asian context are given in traditional order, surname (in small caps) followed by given name. This is in contrast to the treatment of authors writing in English and the American Zen teachers of Japanese extraction mentioned in the preface, who are named according to English conventions.
5. Whenever possible I have translated the names of temples and locations. There are exceptions in cases where the Chinese name is already commonly known, as for Shaolin Temple (Shaolinsi).
6. All geographical locations mentioned are identified with modern Chinese province names and indicated one or both of the maps in chapter 1 (pp. 16 and 20).
7. With only a very few exceptions, Chinese characters have been restricted to the notes, character glossary (which includes only characters for terms and titles used in the main text), and the bibliography.
8. The maps, notes, and character glossary are the only places where I use Pinyin with tone indications, which are based on OGAWA Tamaki et _al., Kadokawa shinjigen, kaitei ban;_ OZAKI Y jir et al., _Kadokawa daijigen;_ and John DeFrancis, ed., _ABC Chinese Dictionary._
9. For the Pinyin transliteration of Chinese terms, I have followed the orthography rules given in DeFrancis, appendix 1, 835–45.
10. Works cited after the abbreviation "T" are from the standard edition of the East Asian Buddhist canon, TAKAKUSU Junjir and WATA-NABE Kaigyoku, eds., _Taish shinsh daiz ky _. Works cited with "X" are from the Taiwan reprint of the extended canon, _Xù zàng j ng,_ published by Xin wenfeng chubanshe.
11. Unless otherwise noted, all translations are by the author. Material in square brackets in translated passages is interpolated to generate readable English; with one exception (on p. 80), material in parentheses has been added by the author.
## McRae's Rules of Zen Studies
1. _It's not true, and therefore it's more important._
The contents of Zen texts should not be evaluated using a simpleminded criterion of journalistic accuracy, that is, "Did it really happen?" For any event or saying to have occurred would be a trivial reality involving a mere handful of people at one imagined point in time, which would be overwhelmed by the thousands of people over the centuries who were involved in the creation of Zen legends. The mythopoeic creation of Zen literature implies the religious imagination of the Chinese people, a phenomenon of vast scale and deep significance.
2. _Lineage assertions are as wrong as they are strong._
Statements of lineage identity and "history" were polemical tools of self-assertion, not critical evaluations of chronological fact according to some modern concept of historical accuracy. To the extent that any lineage assertion is significant, it is also a misrepresentation; lineage assertions that can be shown to be historically accurate are also inevitably inconsequential as statements of religious identity.
3. _Precision implies inaccuracy._
Numbers, dates, and other details lend an air of verisimilitude to a story, but the more they accumulate, the more we should recognize them as literary tropes. Especially in Zen studies, greater detail is an artifact of temporal distance, and the vagueness of earlier accounts should be comforting in its integrity. While we should avoid joining a misguided quest for origins, we should also be quick to distinguish between "good data" and ornamental fluff. Even as we ponder the vectors of medieval polemics.
4. _Romanticism breeds cynicism._
Storytellers inevitably create heroes and villains, and the depiction of Zen's early patriarchs and icons cripples our understanding of both the Tang "golden age" and the supposedly stagnant formalism of the Song dynasty. If one side is romanticized, the other must be vilified, and both subjects pass incognito. The collusion between Zen romanticists and the apologists for Confucian triumphalism—which has Song Neo-Confucianism climbing to glory on the back of a defeated Buddhism—is an obstacle to the understanding of both Chan and the Chinese civil tradition. The corollary is this: Cold realism eliminates dismissive misapprehension.
## CHAPTER 1
## Looking at Lineage
_A Fresh Perspective on Chan Buddhism_
How should we begin this discussion of Chan Buddhism? One device would be to begin with a story, some striking anecdote to arouse the reader's curiosity. There are certainly many good possibilities within the annals of Chan. One is the account of an earnest Chinese supplicant—the eventual second patriarch, Huike—cutting off his arm in order to hear the teachings from the enigmatic Indian sage, Bodhidharma. How many times this story must have been told in meditation halls in China and throughout the world, in order to inspire trainees to greater effort! Or we could find something a bit less gruesome—perhaps the tale about Layman Pang sinking all his possessions to the bottom of a river because he had learned the futility of chasing after worldly riches. Surely this example of unencumbered freedom is meant to teach us a deep spiritual message? The stock of legendary accounts that might be used, each with slightly different import, is endless. And there are other possible beginnings, as well. Many authors have their own favored ways of characterizing the most essential features of Chan, presenting some short list of features to sum up the entire tradition. Or we could avoid such bland generalization and simply celebrate the incredible creativity of the Chan tradition over the centuries, its vibrancy as a religious phenomenon.
The approach adopted here—already taken by posing these very deliberations—is to begin by asking questions, to arouse in the reader not merely a raw curiosity but the faculties of critical interrogation as well. Specifically, let us begin by directly considering the question of how we should look at Chan Buddhism: What approaches should we adopt, and which should we avoid? What forms of analysis will be fruitful, and which would merely repeat commonly accepted stereotypes?
The question of how we should look at Chan Buddhism is one we should not attempt to avoid; to simply ignore the issue and begin a recitation of facts and concepts would be to make an unspoken decision, to answer the question by adopting a policy of denial. But neither would it be appropriate for me to dictate the answer in flat and simple terms: as I compose these lines on the outskirts of Taipei at the very end of the twentieth century, and edit them in Honolulu at the beginning of the twenty-first, I am conscious of the incredible multivalence of cultural identity implicit in this process of exposition, both in my own person and those of my intended audiences. That is, in various ways and at different times I have been a scholar and practitioner, student and teacher, lover and hermit, and what I am about to present here I have learned through a series of extended educational encounters in America, Japan, and Taiwan. This text is intended for use by listeners and readers not only in China, but in Europe, the United States, and Japan as well—so how could I possibly presume to argue that there should be _one_ way to look at Chan Buddhism? A multiplicity of perspectives and a certain fluidity of analytical typologies are givens in this postmodern world.
### Deconstructing the Chan Lineage Diagram
For convenience, let me begin by defining a perspective on Chan that I wish to deconstruct and thereby avoid. I should confess that I mean only to caricature this perspective, so that we can use the observations made now to form a lever with which to push ourselves into a certain type of understanding (to paraphrase the positivist philosopher John Dewey and his student Hu Shih, who spoke of studying the past to create a lever with which to push China into a certain sort of future). The perspective to which I refer is the traditionalist approach depicted graphically in the lineage diagram presented in figure 1. Diagrams such as this are included in virtually every book on Chan that has ever been written, where they are used as a framework for presenting a historical narrative. Instead of plunging directly into that narrative and building upon the content of the diagram per se, though, we should first consider its semiotic impact as a medium of interpretation and communication. If the medium is the message (according to the saying popularized by Marshall McLuhan), what message is conveyed by the structure of the diagram itself? It is often noted that Chan claims to "not posit words" ( _bu li wenzi, fury monji)_ and that it represents a "separate transmission outside the teachings" _(jiaowai biezhuan, ky ge betsuden)._ Almost always—as I am about to do right now—these phrases are introduced with the ironic observation that Chan certainly does use a lot of words in describing its own teachings. We will come back to the Chan use of language and its not "positing" of words later, but here we can observe that the lineage diagram provides the basic model for how Chan appreciates its own historical background. That is, Chan does not define itself as being one among a number of Buddhist schools based on a particular scripture (such as the Tiantai [Tendai] school with its emphasis on the _Lotus S tra,_ for example). Instead, Chan texts present the school as Buddhism itself, or as _the_ central teaching of Buddhism, which has been transmitted from the seven Buddhas of the past to the twenty-eight Indian patriarchs, the six Chinese patriarchs, and all the generations of Chinese and Japanese Chan and Zen masters that follow. (Bodhidharma occupies a pivotal position as both the twenty-eighth Indian and first Chinese patriarch.) It took several centuries for this entire schema to be developed; the earliest building blocks appeared at the very end of the seventh century, and the complete system was published perhaps as early as 801 but certainly by the year 952.
FIGURE 1. Lineage diagram of Chinese Chan Buddhism.
One of the advantages of beginning by considering this lineage diagram, to be sure, is that it introduces the most important players in our story. The seven Buddhas of the past are legendary figures to whom we need pay only scant attention; although Chan texts amplify and modify their religious identities somewhat, for our purposes we can admit them into evidence solely as part of the cultural repertoire Chan inherited from the larger tradition of East Asian Mah y na Buddhism. Chan has its own mythic take on kyamuni, of course, quite different from our own conception of him as the "historical" Buddha—but this too is a subject for another time. Nor must we pay much attention to the twenty-eight Indian patriarchs. The manner in which their hagiographies were explicated is a fascinating and exceedingly complex subject of study, but we do not have the space to consider it here. On the other hand, the six Chinese patriarchs from Bodhidharma onward, along with Huineng and Shenxiu in the sixth generation and their several generations of disciples, will appear more often than any of the other players in this drama. (The reader will note at once that no disciples of Shenxiu's are listed in our lineage diagram, which is a telling omission in itself. I consider this briefly on p. 14 below.) The figures remembered as icons of the Linji (Rinzai) and Caodong (S t ) schools, whose names adorn the balance of the diagram, are among the most important in the history of the tradition.
We can draw some important basic inferences from this transmission diagram. First, a note on historical origins: the Chan lineage scheme is a combined product of Indian and Chinese culture. Often authors describe Chan as the "most Chinese" of all the Chinese Buddhist schools, and part of what they are referring to is the Chan genealogical model. (I am particularly allergic to this rhetoric, since such expressions are generally little more than unexplicated tautologies generated through a sense of cultural chauvinism rather than real analytical insight. And the fact that D. T. Suzuki and others say virtually the same thing with regard to Japanese Zen, that it represents somehow the essence of _Japanese_ culture, should alert us to both the essential vacuity and the strategic intentions of such sentiments.) Actually, the origins of this lineage-based transmission scheme are to be found in Indian Buddhism and the fourth- and fifth-century Buddhist meditation tradition of Kashmir. There are a number of parallels between the Chan transmission scheme and Chinese family genealogies of the eighth century and later, but we should remember that Indian Buddhists had parents and teachers, family genealogies and initiation lineages, just as the Chinese did. As an amalgamation of Indian and Chinese elements, though, the Chinese Chan transmission schema developed within the Chinese Buddhist context and was particularly well adapted to that milieu. Just as DENG Xiaoping talked about "socialism with Chinese characteristics," we could refer to the Chinese Chan transmission model as a "Buddhist genealogical theory with Chinese characteristics."
Second, by using the lineage diagram to define Chan as a "separate transmission outside the teachings," the advocates of Chan were declaring their school to be profoundly different from, and fundamentally better than, all other Buddhist schools: where the other schools represented only interpretations of Buddhism, Chan constitutes the real thing, Buddhism itself. This is a polemical move, meant to establish the superiority of Chan over all other schools. Other East Asian Buddhist schools reacted in part by devising their own lineage transmission schemes, and in part by saying that Chan emphasized only one of the "three learnings" of morality, meditation, and wisdom. Whether we view medieval Chinese Buddhists as concerned solely with the highest forms of wisdom or as working to obtain imperial patronage and other this-worldly benefits, or engaged in both endeavors simultaneously, at the very least they were competing with their contemporaries for intellectual and cultural hegemony. We should thus not overlook the polemical quality of the lineage theory. Incidentally, to describe Chan Buddhism in terms of polemics and contestation is not to exercise any value judgment, let alone to denigrate the tradition, but merely to recognize historical fact.
Third, what counts in the Chan transmission scheme are not the "facts" of what happened in the lives of kyamuni, Bodhidharma, Huineng, and others, but rather how these figures were perceived in terms of Chan mythology. This point will come up repeatedly here, and I will argue a rather complex position: In case after case, what the texts say happened almost certainly did _not_ occur, in terms of a straightforward but simpleminded criterion of journalistic accuracy. But rather than being fixated on notions of fact and fabrication, we should notice the very dynamism of the mythopoeic processes involved. Whether or not any anecdote actually represents the words spoken and events that occurred "accurately" is only a historical accident, and in any case the supposedly "original" events would have involved only a very small number of people, at most the members of a single local community. What is of far greater consequence is the process by which that anecdote was generated and circulated, edited and improved, and thus transmitted throughout an entire population of Chan practitioners and devotees, until it became part of the fluid body of legendary lore by which Chan masters came to be identified throughout Chinese culture. This is McRae's first law of Zen studies: "It's not true, and therefore it's more important." This is to say that fiction—actually, a different sort of truth—is more important than the simplistic criterion of the question "Did it really happen?"
Fourth, based on the rhetoric of _ nyat ,_ or emptiness, nothing is actually transmitted in this transmission scheme. What occurs between each teacher and his successor is merely an approval or authorization _(yinke; inka)_ of the successor's attainment of complete enlightenment. This is first of all a doctrinal principle of Chan Buddhism itself, but we should recognize that the most important parts of the diagram are not the separate names of individual patriarchs, but the spaces between them, the lines that join them. That is, what is being represented is not only a series of human figures but the encounters between each figure and his immediate predecessor and successor. As is frequently stressed in the texts of Chan, there is no "thing"—such as enlightenment, the Buddha-mind, or whatever—that is actually passed from one patriarch to the next. The existence of such an entity would violate a fundamental Buddhist doctrinal theme, the denial of unchanging, substantive, and individual identity to the things and beings of this world. With regard to persons, this doctrinal theme is called "no-self" ( _an tman_); with regard to all the various component elements of existence, including persons, this is called "emptiness" ( _ nyat _ ). This is not a merely philosophical consideration, but rather an existential posture with profound genealogical impact: the focus is not on "what" is being transmitted, but on the relationship of encounter between the Buddhas and Patriarchs. The act of transmission thus involves not the bestowing of some "thing" from one master to the next, but the recognition of shared spiritual maturity. It is a cosmic dance involving a special set of partners, a relationship of encounter, a meeting at the deepest spiritual level.
Fifth, since the enlightenment of each Buddha and Patriarch is complete, there is no differentiation between the religious status of the Indian Buddhas and Patriarchs and their Chinese counterparts. This was perhaps the most important reason why this lineage-based exposition was attractive to medieval Chinese Buddhists, since it raised the authority of native Chinese figures to equal those of their Indian predecessors. This is very important in terms of the sinification of Buddhism, that is, the adaptation of Buddhism within Chinese culture, a subject that is vitally relevant to a wide range of subjects in Chinese religions and Chinese studies in general. At the moment, though, what I want to emphasize is the most striking and most frequently overlooked characteristic of this diagram: the homologizing impact of its very simple lines of succession.
By representing Chan Buddhism in terms of a straight-line succession from the seven Buddhas of the past through the six Chinese Patriarchs, diagrams such as this are used to simplify fantastically complicated sets of cultural and religious phenomena. Every time a straight-line relationship between two masters is posited in a lineage diagram, an entire world of complexity, an intricate universe of human relationships and experiences, is effectively eliminated from view. Could any religious figure's identity possibly be adequately summarized by selecting only one out of a whole lifetime of relationships? Even a quick look at the biographies of Chinese Chan masters shows the extent of the distortion involved: where the sources are adequate, we sometimes see multiple awakening experiences catalyzed by different teachers and events, yet in the lineage diagrams these are all reduced to single lines of transmission. The use of lineage diagrams to represent the Chan tradition, then—and their use is as old as the tradition itself, since it was by explicating genealogical specifics that Chan generated its own identity as a specific religious movement—is a hegemonic trope, the willful extension of one way of perceiving the world to the exclusion of all other viewpoints. (I briefly discuss the various branches and divisions of the diagram beginning on p. 9 below.)
Sixth, the "genealogical model" is important not only for the historical self-understanding of the Chan school in its transmission from kyamuni Buddha through Bodhidharma and onward, but also for the manner in which it defines how Chan spiritual practice itself is carried out. That is, in contrast to a basically Indian conception of meditation practice as an individual yogic endeavor of self-purification and progressive advancement toward buddhahood, the Chan genealogical model implies that the most important aspect of spiritual cultivation takes place in the _encounter_ between teacher and student. Chan trainees still spent long hours in the meditation hall—we can be sure of that, even though the texts often do not bother confirming the fact—but the focus of Chan rhetoric and literature is on the dialogues and exchanges between each master and his students, or between each student destined to be a master and his various teachers. It is thus not only the Chan school's self-understanding of its own religious history, but the religious practice of Chan itself that is fundamentally genealogical. By saying that Chan practice is fundamentally _genealogical,_ I mean that it is derived from a genealogically understood encounter experience that is _relational_ (involving interaction between individuals rather than being based solely on individual effort), _generational_ (in that it is organized according to parent-child, or rather teacher-student, generations), and _reiterative_ (i.e., intended for emulation and repetition in the lives of present and future teachers and students). No matter what the comparison or relationship between Chinese Chan and earlier forms of Indian Buddhist meditation practice, this particular complex of qualities is not found in other schools or forms of Buddhist training.
In the most basic historical terms, though, we should recognize that the homologizing impact of the Chan lineage diagram represents a profound distortion of the subject matter. This is McRae's second rule of Zen studies: "Lineage assertions are as wrong as they are strong." In more formal language, this means that lineage assertions are problematic in direct proportion to their significance. That is, every time we read that the masters of such-and-such a group are related to each other in a lineal succession, the statement is probably inaccurate in some sense, and the more important it is to the religious identity of the individuals involved, the less accurate it will be. If nothing much is made of the relationship, the lineage assertion is more likely to be correct than if a great deal rides on it. Almost always, of course, the figure at the end of the list, or even that individual's students, has the most at stake in making such assertions. And if his religious identity must be defined on the basis of a lineal succession, if his historical status depends on being the recipient of the cumulative charisma of one particular set of predecessors, then it always seems that some significant distortion of the facts has taken place. Of course, my use of the _word facts_ should remind you of the first rule, which remains relevant here: The presentation of reality in lineage schema represents a certain type of myth-making, and what is not "true" per se is inevitably more important!
Seventh, I referred above to "each teacher and _his_ successor" (see p. 6), and the gender-specific terminology is appropriate. The Chan tradition is overwhelmingly male-dominated, and the strong implications of the term _patriarchal_ in English (referring both to Chan figureheads and a male-centered ideology) is entirely suitable here. Nancy Jay has analyzed how genealogical systems tend to create justifications for removing women from the nexus of power and fecundity, and in a later chapter, we will consider the manner in which Chan represented a way of organizing power within the Chinese Buddhist monastic establishment. There is also, of course, a broader, gender-related issue concerning Chan as a patriarchal ideology: to put it bluntly, Was Chan a weapon used to oppress women within Chinese society? Alas, I cannot deliberate on this issue in these pages, but when the subject comes up, scholars should certainly not shrink from it. This awareness, however, is helpful here in a different and perhaps even larger sense. I do find it germane to deal with the following variant of the question: Was Chan a weapon in the oppression of Chinese religious practitioners in general, or did it serve to suppress certain groups of them? This is a shocking question, to be sure, but it seems to me that any means by which knowledge is structured—and the lineage format is certainly that—both _allows_ and _suppresses_ different types of perspectives. I am by no means unsympathetic to the Chan tradition, nor to the realm of Buddhist meditation and spiritual cultivation in general, but a consideration of how the Chan school's dominance in Chinese Buddhism may have militated against alternative viewpoints seems an obvious aspect of our intellectual responsibility.
At this point, you may be surprised that we have derived so many inferences from one simple diagram, but we could certainly coax numerous additional insights from it if space were not an issue. Let us leave further comment on the Chan lineage diagram and the genealogical identity of the Chan tradition until later, though, and turn instead to the reason we began this discussion in the first place.
### Avoiding the "String of Pearls" Fallacy
The preceding observations regarding the lineage diagram are to some extent preventive medicine, prophylaxis against a type of interpretation to be avoided. Simply put, the message is this: To represent Chan Buddhism in terms that are congruent with the lineage paradigm is to run the risk of mere repetition, without saying anything fundamentally insightful. Rather than performing legitimate analytical investigations, to do so would be merely to recapitulate an inherited symbolic system, and in this context one's most cherished intellectual nuances would be nothing more than trivial variations on the genealogical model. Here it is useful to make a clear insider/outsider distinction: What is both expected and natural for a religious practitioner operating _within_ the Chan episteme, what is necessary in order to achieve membership within the patriarchal lineage, becomes intellectually debilitating for those standing, even if only temporarily, _outside_ the realm of Chan as its observers and analysts. What from the standpoint of Chan religious practice may be absolutely essential becomes, from the standpoint of intellectual analysis, the passive submission to a hegemony, the unwitting contraction of an intellectual pathology.
So what is it that we should not be doing? Or, to put it another way, how can we recognize when we are falling, or in danger of falling, into patterns that inhibit our ability to see the history of Chan in all its rich complexity?
Seen from this perspective, the issue is really quite simple: Whenever we pretend to explain Chan in terms of lineal successions from one great master to another, we run the risk of committing the "string of pearls" fallacy, in which the evolution of Chan Buddhism is described in terms of a sequence of individual masters like pearls on a string. This is a variant of the "great man" fallacy of historical writing, in which one explains the inevitably messy details of past realities in terms of the willful endeavors of a limited number of heroic men. (Once again, the gender-specific terminology is warranted.) To be more logically precise, it is also an example of the fallacy of archetypes, which "consists in conceptualizing change in terms of the re-enactment of primordial archetypes which exist outside of time."
In terms of Zen studies, this tendency is starkly apparent in the way Dunhuang manuscripts have been used to supplement rather than radically transform the appreciation of Chan in many writings. Atrove of cultural treasures similar to the Dead Sea scrolls, the Dunhuang manuscripts were discovered in a walled-up cave in Chinese Central Asia at the turn of the twentieth century and then dispersed to various libraries throughout the world. They provided a cross-section of Chan documents from the eighth to the tenth centuries, just before the great editorial homogenization of the Song dynasty took place. Access to these manuscripts has allowed scholars to explore the early phases of Chinese Chan Buddhism in ways that would simply not have been possible in their absence, and the analysis of this magnificent trove has occupied the attentions of scholars (not only in Chan, but in other fields of Buddhist and Daoist studies, and various realms of historical and sociological research as well) for the entire twentieth century. However, in Chan studies, evidence from the Dunhuang manuscripts has most often been used merely to paint better features onto the same old traditional picture, merely to add attractive detail to the genealogical model described above. Thus, scholars have used Dunhuang manuscripts in conjunction with other evidence to devise more vivid portraits of Bodhidharma, Huineng, and others as _individual_ figures, without changing the framework in which these individuals are presented in any substantial manner, and certainly without trying to work out the cultural and religious dynamics that led to their inclusion in the genealogical paradigm in the first place. There are exceptions, of course, but they are comparatively few and far between.
I am not suggesting that we never include descriptions of lineage successions in our writing on Chan—far from it—but only that, when we do so, we should be conscious of the reasons for their use and remain aware of the risks involved. Not only would it be impossible to talk about Chan without ever using concepts related to lineage—to the extent it can be described as a continuous set of processes, Chan is at its most profound level a _genealogical_ set of phenomena—but we will gain the greatest benefit from shifting our focus and perspective repeatedly as we move through the evidence. To commit the "string of pearls" fallacy is to remain fixed and unaware in a single posture. Rather than simply move to a different static position, however, we should work to illuminate our subject from a number of angles, to encounter it with different aspects of our interpretive capacities.
### A Provisional Device: The Phases of Chan
Figure 2 (p. 13) is a simple chart describing Chan in a manner quite different from that of the lineage diagram (fig. 1) discussed above. Where the traditional Chan diagram lists names of individual human beings, this chart lists named phases or trends in the evolution of Chan. The names of these phases or trends are not universally accepted in writings about Chan, and the boundaries between them are subject to debate. I preserve these ambiguities by not adopting this terminology and periodization without question throughout these chapters; on the contrary, we should pay close attention to the intrinsic fuzziness of the borders between the phases named so uniquely and unambiguously here. It is in large part through considering the failure of any margins to tightly capture these arbitrary entities that we will be able to see the utility of this periodization.
Each of the named phases refers not to a specific set of individuals per se (although some of the most representative figures are listed), but to a style or configuration of religious activity that is known through a variety of sources. One of the primary models by which each phase is characterized is, of course, a list of teachers, known as patriarchs in the traditional lineage scheme, who function as figureheads for a certain type of religious identity. These men (and very occasionally women) serve as exemplars of enlightened behavior, whose stories are told and retold in order to pattern the behavior of subsequent generations of students. (Even as Chan involves the transcendence of patterned behavior in enlightened spontaneity, this abandoning of patterning must itself be patterned in order to be understood, modeled before it can be imitated, deconstructed, and refigured.) Information about these figureheads, as well as doctrinal explanations and other types of information, was circulated both orally and through written texts. Hence each phase of Chan can be described in terms of multiple dimensions: its exemplary human representatives, the geography and timing of their activities, the texts that describe their activities and convey their teachings, and so forth. Figure 2 provides information of this sort briefly in the summary for each phase.
Hence, the basic difference between the lineage diagram and the chart in figure 2 is that, where the diagram tends to homologize all the individuals represented as identically enlightened representatives of a single confraternity—to enable (and simultaneously limit) the understanding of them according to a meaningful yet unitary religious mode—the chart seeks to distinguish qualitative differences along a chronological axis, to facilitate multiple perspectives and modes of understanding. The goal of the chart is the generation of meaningful distinctions, not the assertion of an unbroken continuity of patriarchal authority.
You will note that the lineage diagram is not monolithically unilinear, that there are divisions into double lines at a number of points, and that five different "houses" of Chan are specified. How can we account for these differentiations, while at the same time acknowledging the "homologizing" impact of the lineage diagram and its underlying religious assumptions? We will consider most of these examples in detail later, but they are to a certain extent exceptions that prove the rule. It has long been recognized that Huineng and Shenxiu, the figureheads of the so-called Southern and Northern schools, function within traditional Chan ideology not as two isolated individuals, but as an inextricably related pair simultaneously linked in collaborative and competitive relationship. Together they constitute a single literary and religious polarity expressed as a relationship between two human exemplars. A convenient shorthand for this complex bimodality is the French word _duel,_ which carries the meanings of both "duel" and "dual" in English. Thus the doctrine of sudden enlightenment associated with the Southern school cannot be explained without reference to a gradualist doctrine attributed to the Northern school. (This simplistic explanation of sudden versus gradual is woefully inadequate in the face of historical reality, but it must have been very effective in disabusing trainees of their simplistic notions of meditative "achievement.") Note that these two schools, along with Oxhead Chan, are included together in the "early Chan" phase of the eighth century—and this is an intentional grouping, meant to indicate that these three factions were more alike than different, or at least that their religious identities were so intimately intertwined that they must be represented together. The fact that none of Shenxiu's disciples are included in the Chan lineage diagram (already noticed on p. 4 above) is due to their exclusion from consideration in traditionalistic accounts of Chan; here their meaningful absence serves to highlight the unilinearity of the "orthodox" line traced from the legendary (i.e., fictional, but therefore more important) Huineng. In chapter 6 (see p. 138) we consider whether the distinction between the Linji/Rinzai and Caodong/S t schools implies a similar polarity, that is, two groups paired together in a duel or binary relationship that is both contrastive and competitive.
FIGURE 2. Simplified chart of the phases of Chinese Chan.
PROTO-CHAN | Bodhidharma (d. ca. 530)
---|---
ca. 500–600 | Huike (ca. 485 to ca. 555 or after 574)
|
_Treatise on the Two Entrances and Four Practices_
|
SUMMARY: Multiple locations in north China; practice based on Buddha-nature; no known lineage theory. Known through traditional texts and a few Dunhuang documents.
EARLY CHAN | Hongren (601–74)
ca. 600–900 | Shenxiu (606?–706), Huineng (638–713)
|
Shenhui (684–758)
|
Northern, Southern, Oxhead factions
|
_Platform S tra of the Sixth Patriarch_
|
SUMMARY: Various loosely defined factions/groups, with different approaches to "contemplation of the mind"; relationship between this and proto-Chan unclear; lineage theories appear from 689 on as a unifying ideology; known through numerous Dunhuang documents and traditional sources.
MIDDLE CHAN | Mazu (709–88), Shitou (710–90)
ca. 750–1000 | Linji (d. 867), Xuefeng Yicun (822–908)
|
Hongzhou and Hubei factions, antecedents of the Five Houses
|
_Anthology of the Patriarchal Hall_
|
SUMMARY: Emergence of "encounter dialogue" as primary mode of practice and discourse, recorded in colloquial form and massive quantity in 952, and implying a genealogical model of religious cultivation; not present in Dunhuang documents but known through Song dynasty texts and idealized as a golden age during Song.
SONG-DYNASTY CHAN | Dahui (1089–1163), Hongzhi (1091–1157)
ca. 950–1300 | Five Houses, Linji and Caodong schools
|
_Blue Cliff Record_
|
SUMMARY: Greatest flourishing of Chan, which as an administrative ideology dominated the Chinese monastic establishment; the image of Tang-dynasty masters operating in enlightened spontaneity was inscribed in highly ritualized Song-dynasty settings; snippets of encounter dialogue were collected, edited to serve as precedents of enlightened activity, and used as topics of meditative inquiry.
NOTE: In order to cover Chan from the end of the Song dynasty up to the present, this chart should include at least a postclassical phase or perhaps multiple later phases. However, since the developments of these later periods are not treated in this book, I will not attempt a periodization here.
You might assume that the chart depicts a chain of historical causality, but it actually characterizes the retrospective identity of the various phases of Chan. The periodization of any set of past events represents an act of reconstruction—not the mere reorganization and ordering of information, but the total remaking of the past as the structured image of our imaginations. Now, there is nothing wrong with creating an image of the past—indeed, I believe it is our task as historians, both professional and occasional, to visualize the past in the best ways we know how. But we should work to remain aware that the ordering of developments from the fifth through the thirteenth centuries inevitably involves this kind of re-creation; we cannot get off the hook with the naive belief that we are merely ordering the information for the sake of convenience, but not really altering it in the process.
This retrospective quality pervades the Chan tradition. Time and again we find we are dealing, not with what happened at any given point, but with what people thought happened previously. We deal not so much in facts and events as in legends and reconstructions, not so much with accomplishments and contributions as with attributions and legacies. The legends and reconstructions, not the supposedly "actual" events, determined later religious and social praxis. This observation may have a broad application beyond Chinese Chan, in describing what it is that makes traditions traditions. But it is certainly applicable to Chan: not true, and therefore more important.
With these considerations in mind, then, and in order to get a better perspective on the subjects to be covered in the remaining chapters, let us look in somewhat greater detail at the phases listed in figure 2. At this point I provide only a few introductory comments to help you become oriented to the material and thus prepared for the more detailed analysis that follows.
#### PROTO-CHAN
The designation _proto-Chan_ refers to the ill-defined activities of a set of practitioners surrounding Bodhidharma and Huike who were known for their dedication to ascetic practices and meditation. Beginning roughly around the year 500 and overlapping with the so-called early Chan phase in the seventh and perhaps even into the eighth century, this group operated in a variety of north China locations. The extent to which the individuals involved conceived of themselves as participating in a single group or movement is unclear, and since they had no way of knowing of the continuity of their activities with any later "Chan school," even the convenient term _proto-Chan_ does not bear close scrutiny. (Their activities are "prototypic" only to those who already know what followed.) We know of a small number of figures who studied under Bodhidharma, and a somewhat larger number who were primarily associated with Huike, presumably after his master's death. There is a certain quantity of biographical information about the participants in proto-Chan, and although it attests to the variety of their backgrounds, it imparts only a shadowy image of any shared group esprit.
One important feature of proto-Chan—at the very least, a feature important for the subsequent evolution of the school—was its common focus on a text circulated under Bodhidharma's name, the _Treatise on the Two Entrances and Four Practices (Erru sixing lun)_. As this text circulated, practitioners who identified with Bodhidharma's message appended their own comments to it, making it an expanding anthology of the earliest Chan teachings. Thus, while we cannot describe the scope of proto-Chan activities with any accuracy, the _Treatise on the Two Entrances and Four Practices_ provides insight into precisely those ideas that formed the doctrinal nucleus of subsequent Chan practice ideology. This text describes a fundamental attitude of emphasis on the existence of the Buddha-nature or potential for enlightenment within all sentient beings, as well as an attitude toward how this understanding of Buddhism may be carried out in daily life.
MAP 1. Locations for Proto-Chan, Early Chan, and Middle Chan.
#### EARLY CHAN
_Early Chan_ designates the phase when the school, or what was to become a school eventually, first articulated its lineage-based ideology in clear and extensive form. Actually, the Dunhuang manuscripts and traditional Chan records include an amazing variety of different formulations from this phase, and it seems evident that a great deal of experimentation was taking place, involving a number of variations on commonly accepted themes, as the Chan movement matured and crystallized over time. Some of these formulations describe specific methods of contemplation practice, sometimes presented in a progressive series of steps. Others describe the role of the Buddha-nature, or "pure mind," within, as well as the behavior of the illusions—the false thoughts, or "impure mind"—that obscure the appreciation of our inner purity. Compared to later Chan texts, these formulations often seem odd but are not particularly enigmatic or difficult; the emphasis at this point was on clarity in expressing this new form of the Buddhist teaching, not on generating entirely different modes of expression.
In contrast to proto-Chan, the early Chan phase manifests a great stability of location: Daoxin and Hongren spent exactly a half-century, from 624 to 674, in the same monastic complex in Huangmei ("Yellow Plum," Hubei Province) and it is not unreasonable to include Shenxiu's quarter-century, from 675 to 701, at the not-too-distant Jade Spring Temple (Yuquansi, in Jingzhou, which overlaps both Hubei and Hunan Provinces) in this phase as well. Matters become more complex with the explosion of Chan into the two imperial capitals of Chang'an and Luoyang during the eighth century. Therefore, whereas investigation of proto-Chan leaves one with the impression of an indefinable will-o'-the-wisp, analyzing the sources for early Chan imparts a sense of continuous community development and a growth pattern that moves from geometric increase throughout much of the seventh century to explosive expansion in the eighth. Also, where proto-Chan refers to a single, albeit incohesive and ill-defined, style of religiosity, early Chan may be understood as a collection of different communities, groups, and factions.
In the most straightforward sense, the label _East Mountain teaching_ refers to both the community and doctrines of Daoxin and Hongren, but there is an important sense in which these matters are known solely through information transmitted by their successors. Those successors identified themselves not as purveyors of their own doctrinal innovations, but as transmitters of the East Mountain teaching. We need to recognize that the ideas associated with the names Daoxin and Hongren were primarily those of their followers' later reconstruction; this recognition does not sever the connection between those ideas and the East Mountain teaching figureheads themselves, but it does lend an important retrospective quality to the process. That those successors, who were active in Chang'an and Luoyang in the early decades of the eighth century, came to be known by the label _Northern school_ is a curious historical detail. The _Southern school_ derives from the mid-eighth-century activities of Shenhui (684–758), although later this label came to be adopted for the Chan school as a whole. The _Oxhead school_ is a somewhat later development, a faction or lineage that played an important historical role through its apparent involvement in the composition of the _Platform S tra,_ the hallmark and culminating text of early Chan.
We will deal with the East Mountain teaching in chapter 2, along with Bodhidharma and proto-Chan. The Northern, Southern, and Oxhead schools, as the most important trends of metropolitan Chan (i.e., those factions that evolved in the two capitals of Chang'an and Luoyang), will be treated together in chapter 3. It is appropriate that the last three schools should be taken together, since they were in dialogue with one another, and the supposed distinctions between them in their original historical identities are not nearly as sharp as the Chan legends would have us believe. It would also be appropriate to mark the East Mountain teaching off as an entirely separate phase, but I hope that adding these comments here—and organizing chart and chapters differently—will be sufficient to show the provisional nature of the boundaries involved. The lack of congruence between the categories "early Chan" and "metropolitan Chan" as used here is intentional.
#### MIDDLE CHAN
An event of overwhelming significance takes place in the "middle Chan" phase: the emergence of "encounter dialogue," the idiosyncratic manner in which Chan masters are depicted in dialogue with their students. Associated initially with such celebrated figures as Mazu Daoyi (709–88) and his successors Baizhang Huaihai (749–814), Nanquan Puyuan (748–834), and Linji Yixuan (d. 867), as well as Shitou Xiqian and his successors Dongshan Liangjie (807–69) and Caoshan Benji (840–901), this is when Chan appears to have become really Chan, when Chan masters seem to have really behaved like Chan masters. The anecdotes of middle Chan encounter dialogue represent the stories repeated most often in popular books on Chan/Zen as examples of paradoxical but enlightened behavior. Here the locus of religious practice was firmly removed from individual effort in the meditation hall and replaced by a demanding genre of interrogation that sought to destabilize all habitual, logical patterns. Spontaneity was the rule, iconoclastic behavior the norm.
Or so it seems. For here we will have to consider, not only the momentous import of encounter dialogue as the dominant model of religious undertaking, but also the difficult questions of _when_ all this spontaneous interaction was actually being practiced and _what_ precisely was going on. We will see that there is a substantial gap between when the most famous stories of Chan lore are supposed to have happened, and when we first see them in written form. We will also see that these stories have complex origins, bearing features of both oral and written literature. In the past scholars (myself included) have referred to the middle phase as the "golden age" or "classical period" of Chan. The first of these terms may easily be discarded for its romantic coloring. The latter term may still be used, but only with the provision that what is being referred to is not some collection of activities and events that actually happened in the eighth through tenth centuries, but instead the retrospective re-creation of those activities and events, the imagined identities of the magical figures of the Tang, within the minds of Song dynasty Chan devotees. Mazu and the other Tang figures came to represent a classical age only when their time had passed, when their identities were redesigned to fit the needs of Song-dynasty Chan. Although middle Chan may be considered as a historical phase, "classical" Chan is itself a romantic depiction of activities from that phase within the later texts of encounter dialogue.
#### SONG-DYNASTY CHAN
The contours assumed by Chan Buddhism during the Song dynasty represent the mature pattern which defines the tradition up until the modern period. Using an ecological metaphor, I refer to this pattern as a "climax paradigm," which describes the dynamic equilibrium achieved by a mature forest or ecological system. Earlier writers (both scholarly and apologist) have tended to ignore this period, partly out of the wish to explore the more "creative" masters of the Tang, or to jump across the waters to emphasize the emerging Zen school of Japan. The Song has also been denigrated in general textbooks as the beginning of the decline of Chinese Buddhism, its ossification into institutional formalism. This attitude is changing, as Song-dynasty religion has become perhaps the primary focus of the study of premodern Chinese religion, by Euro-American scholars at least. And with this change our impression of Song-dynasty Chan has been transformed as well. It is now increasingly recognized that the Song dynasty witnessed the emergence of a basic configuration of Chan that was disseminated throughout East Asia, and now the world. This is apparent most dramatically in the life and teachings of Dahui Zonggao (1089–1163), the innovator and greatest exponent of "viewing the critical phrase" or _k an_ practice in the history of Chinese Chan. But the picture of Song-dynasty Chan is not complete without looking closely at the style of meditative introspection advocated by Hongzhi Zhengjue (1091–1157) and other members of the Caodong lineage, evaluating their recommendations on their own terms and not simply in light of the polemical characterization by Dahui as mere "silent illumination." Ultimately, we will see that the Linji and Caodong approaches present an inseparable pair that mimics the sudden/gradual debate of the eighth century, and which resonates with the "two entrances" of the treatise attributed to Bodhidharma. But this is to get ahead of our story. Let us now turn to the legendary account of Bodhidharma himself, to see how Chan Buddhism emerged in the first place.
MAP 2. Locations for Song-Dynasty Chan.
## CHAPTER 2
## Beginnings
_Differentiating/Connecting Bodhidharma
and the East Mountain Teaching_
Bodhidharma, it is said in the traditional accounts, was the third son of a great Brahman king of southern India, who left home to undertake the life of a Buddhist monk. Attracted to the profundity of the Mah y na, he eventually became the twenty-eighth patriarch in succession to S kyamuni Buddha. After traveling by sea to China in order to spread the true teachings of Mah y na Buddhism, he had the following interview with Emperor Wu of the Liang dynasty (r. 502–549), who was renowned for building temples, casting images, and supporting the teaching activities of Buddhist monks:
_Emperor Wu_ : | "What is the religious merit of all my efforts on behalf of Buddhism?"
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_Bodhidharma:_ | "None whatsoever."
_Emperor Wu:_ | "Who are you to say such a thing to me!?"
_Bodhidharma:_ | "I don't know."
Seeing that conditions were not right for him to teach in southern China, Bodhidharma crossed the Yangzi River by floating across on a reed and went to Mount Song, just south of the great city of Luoyang. There he took up residence at Shaolin Temple (Shaolinsi), but instead of joining the regular activities of the congregation of monks, he spent nine years in a cave, sitting in meditation while facing a wall. His extraordinary discipline eventually attracted the attention of a student named Huike, who was to become Bodhidharma's successor and thus the second patriarch of Chan Buddhism. But Huike did not achieve this new identity without demonstrating his total dedication to the Dharma: since the master was absorbed in meditation and would not recognize him, the student knelt behind Bodhidharma in silent supplication, the snow piling up around him in the cold north China winter. Eventually, Bodhidharma broke his silence and asked what Huike wanted—the answer being "instruction in the teachings of Buddhism," of course—only to ignore the student once again. In desperation, to show the depths of his dedication Huike cut off his own arm and placed it before the master. Seeing this, Bodhidharma at last recognized the student's sincerity and allowed him to inquire of the teachings:
_Huike:_ | "My mind is not at ease—please pacify it for me!"
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_Bodhidharma:_ | "Bring me your mind, and I will."
_Huike:_ | "But no matter how I might look, the mind is not a 'thing' I can find."
_Bodhidharma:_ | "There, I've pacified your mind for you!"
Huike was suddenly awakened at this reply. He continued to study under Bodhidharma and was eventually recognized as his successor.
Bodhidharma later became the target of criticism by jealous monks who did not understand the true teachings of Buddhism. Although they tried to poison him several times, it was only when Bodhidharma himself decided the time was right that he allowed their potions to kill him. Huike supervised his burial along the banks of a river south of Luoyang, but later the master returned to India, leaving only one shoe in his grave; he was seen crossing the Chinese border carrying the other shoe. Huike went on to transmit the teachings to Sengcan, from whom they were passed on to Daoxin, Hongren, and then to the sixth patriarch Huineng.
This, in a nutshell, is the legend of Bodhidharma as it has been passed down within the Chan tradition. There can be no doubt of its utility as a coherent distillation of classical Chan doctrine: Bodhidharma, the enlightened but iconoclastic master, transmits the true teachings of Buddhism to China, where until his time it had only been understood in a superficial and self-seeking manner. The "nine years facing the wall" at Shaolin Temple and the implicit demand made of Huike—or rather, Huike's macabre demonstration of his inner drive for true understanding at all costs—imply both a disregard for conventional representations of Buddhism and the demand that students spare no effort or personal sacrifice in order to achieve enlightenment. How many times this story must have been told in meditation halls in China, Korea, Japan—and now America and Europe—in order to spur practitioners on to greater effort!
The "pacification of the mind" dialogue is in fact an archetypal example of Chan spiritual training itself, which is less an individual endeavor than an interactive event, the interpersonal encounter between master and student set in a genealogical context. The attacks upon Bodhidharma serve to highlight the unique status he held as sole transmitter of the true teachings, and the autonomous control he had over his death and subsequent return to his native land add an occult aura to his extraordinary capabilities. Indeed, the account of Bodhidharma—I should actually refer to the "accounts," since the preceding is but a bare outline abstracted from a number of divergent sources—represents a highly integrated distillation of the Chan message, and as such it has been among the most treasured subjects of Chan sermons and dialogues over the centuries.
But the story is not true.
### The Evolving Hagiography of Bodhidharma
It is not that parts of the story are in doubt, or that some of it is accurate and some not, or that it is a false composite of individually acceptable elements. All of these alternatives are correct to some extent, but even in combination they do not accurately represent the true situation. The issue is more fundamental.
The image of Bodhidharma that has been transmitted to us is the result of a long hagiographical process, and it is not "biographical" in some sense of being a more-or-less "accurate" depiction of the man's life. Rather, it is the idealized image of a sage, the human demonstration of enlightened charisma, the life of an Indian saint on Chinese soil. It is ultimately impossible to reconstruct any original or accurate biography of the man whose life serves as the original trace of this hagiography—where "trace" is a term from Jacques Derrida meaning the beginningless beginning of a phenomenon, the imagined but always intellectually unattainable origin. Hence any such attempt by modern biographers to reconstruct a definitive account of Bodhidharma's life is both doomed to failure and potentially no different in intent from the hagiographical efforts of premodern writers. This does not mean that we should disdain examining the sources and evolution of this hagiographical process, of course—only that we should remain firmly aware of the hagiographical dynamic while doing so.
The earliest evidence for Bodhidharma's biography derives from ultimately incommensurable sources. In other words, the hagiographical image of Bodhidharma is fundamentally different from whatever "historical" Bodhidharma may have existed at one point. This understanding of the hagiographical nature of the Bodhidharma who occurs in Chan legends is not just a trivial academic nicety, but a profoundly important key to the understanding of Chinese Chan as a cultural and religious tradition. Before considering the implications of the hagiographical process concerning Bodhidharma, though, we need to establish a baseline, the beginning of the story—not as a kernel of biographical truth, of course, but as the earliest manifestation of mythopoeic creativity about him.
The following chronological assertions can be made with reasonable confidence about the earliest hagiographical image of Bodhidharma. According to sources from the mid-seventh century and earlier, it was thought that he _(a)_ arrived in south China by sea sometime in or before 479; _(b)_ moved to north China before 495, perhaps by 480 or so; _(c)_ was in Luoyang sometime during the years 516–26; and _(d)_ died around 530 (i.e., sometime during the years 524–34). In addition, there are a few other characterizations we can make with some confidence about the earliest image of Bodhidharma. That is, he _(e)_ was a native of south India, of Brahman caste, and perhaps a member of some royal family; _(f)_ professed Mah y na Buddhism, taught meditation, and focused his efforts on the Luoyang area; _(g)_ had a small number of known students, including Huike—who was the dominant figure in the development of Bodhidharma's following; and _(h)_ was the beneficiary (perhaps postmortem) of an editorial contribution by a monk named Tanlin, who produced a text called the _Treatise on the Two Entrances and Four Practices_ in his name.
Although all of the eight statements above are based on documentary evidence (of different levels of reliability), we must resist the temptation to accept them as jointly contributing to a single, comprehensive image of the first patriarch of Chan. The eight assertions derive from different sources written at different times and with different authorial agendas. In addition to issues of accuracy, it is not even certain that all of them (especially item _(c)_ pertain to the Chan school's founder, rather than to some other figure of the same name. Taking the first four assertions together, we also arrive at the unlikely scenario that Bodhidharma spent a full half-century in China—not impossible, but it would mean that he arrived in China a relatively young man, which is contrary to the legend that he was 150 years old (which occurs in the source of item _c,_ for example). Also, given the time frame suggested by the evidence, the story involving Bodhidharma and Emperor Wu of the Liang dynasty is clearly anachronistic, given the latter's reign dates of 502–49. Examining the information available regarding Bodhidharma's life requires dealing with endless subtleties and contradictions. Indeed, his hagiography is a particularly good example of the fluidity of legendary Chan imagery.
The easiest way to understand the dynamics of Chan hagiography is to see how Bodhidharma's image developed over time. The following list of the earliest dates at which each element of his hagiography appeared in written sources reveals an overall pattern of accretion and reinscription. That is, not only does the image of Bodhidharma as Chan patriarch become increasingly detailed over time, but new motifs effectively substitute for earlier ones, changing the very quality of the image as religious icon.
_547_ | Said to have been from Persia and was 150 years old when he arrived in Luoyang sometime during the years 516–26.
---|---
_645_ | Described as a Brahman monk from south India who arrived in south China during the Liang dynasty (420–79); Huike's arm is said to have been cut off by bandits/rebels.
_667_ | Depicted transmitting the _La k vat ra S tra_ to Huike.
_689_ | Listing of the succession from Bodhidharma to Huike, Sengcan, Daoxin, and Hongren.
_ca. 710_ | Identified with Shaolin Temple on Mt. Song; story of Huike cutting off his own arm; Bodhidharma described as dying voluntarily by poison, then as seen at the Chinese border on his way back to India, leaving an empty grave.
_ca. 715_ | Described as the third son of a Brahman king of south India; identified as second patriarch after Gu abhadra, translator of the _La k vat ra S tra._
_730_ | Story of meeting with Emperor Wu; said to transmit robe to Huike after the latter cut off his own arm.
_758 or shortly after_ Specifically labeled "first patriarch"; transmitted the _Diamond S tra_ to Huike.
_801_ | Described reciting a "transmission verse" before death.
_952_ | Occurrence of the "pacification of the mind" dialogue with Huike.
_988_ | Said to have "faced the wall" in meditation.
_ca. 1200_ | "Relics" ( _ an ra,_ from a cremated body [!]) venerated by the "Daruma school" in Japan.
_1224_ | Reference to how he "faced the wall for nine years."
_Thirteenth century_ | Association of Shaolin Temple with martial arts.
_1642_ | Attribution of a martial arts book to Bodhidharma.
None of the various details of Bodhidharma's life is "true," in the sense of being journalistically accurate, and therefore each is more important than a mere "fact" might be. Presentations of Bodhidharma's biography that are unreasonably detailed—such as the _Encyclopedia Britannica's_ entry for him (written by Heinrich Dumoulin) that identifies him as a "native of Conjeeveram, near Madras"—exemplify the third rule of Zen studies: "Precision implies inaccuracy." Rather than the stark contrast of true/false, of course, it is the overall fabric of creativity within which the hagiography developed that is most impressive.
In fact, if we looked at the matter more closely, we would see that the evolution of Bodhidharma's image functions as a veritable index to the evolution of Chan itself. That is, if we could do analytical cross-sections at different points in time, we would see that the members of the Chan school were reformulating Bodhidharma's identity to fit their own conceptions of religious sagehood in each particular age; each substantive reconfiguration thus implies a qualitative change in the religious identity of Chinese Chan. This is a dynamic process that continues into the present, of course: A 1992 Taiwanese movie account of Bodhidharma's life shows him not only sitting rock-solid in meditation—a full nine years without moving a muscle!—but also as a miraculously gifted martial artist catching arrows in his teeth and flying through the air, his legs churning in the manner _of Crouching Tiger, Hidden Dragon!_ The modern martial arts cinema tradition has remade the image of Bodhidharma according to its own needs, just as the medieval Chan tradition did. The results are different, but the process is basically unchanged.
In other words, both medieval Chinese Chan factions and modern martial arts schools have created images of Bodhidharma to fit their own conceptions of enlightened sagehood. These imagined sages serve the need felt by each faction or school for a primal figurehead to personify and thus legitimate its particular style of spiritual and athletic training. To accept any one of the various hagiographical images of Bodhidharma as accurate would be to choose only one legendary image out of a series of continuous change. On the one hand, to tell any version of Bodhidharma's hagiography is to present a Sunday-school image of Chan. Doing so is of course acceptable for participants within the tradition itself, but to present such simplistic stories as historically accurate in works of historical narration is an indefensible commission of the "string of pearls" fallacy. On the other hand, it would be even more egregious to deny the religious and cultural significance of the hagiographical process as a whole, to fixate on the technical accuracy of the images of Bodhidharma produced by generation after generation of Chinese practitioners. Those images are not true, and therefore they are more important. More precisely, those images were used by generations of Chan practitioners and enthusiasts, and therefore they are more important than a simplistic reconstruction of historically verifiable events might be.
### Proto-Chan and the _Treatise on the Two Entrances and Four Practices_
In all this, there _is_ one useful point to hold on to: Bodhidharma's early followers remembered his teachings through a short but extremely influential text known as the _Treatise on the Two Entrances and Four Practices._ The absolute _terminus ad quem_ for the appearance of this text is 645, but at this point it already includes some material probably from Huike's life; hence the text no doubt dates back at least to the second half of the sixth century, if not necessarily to the lifetime of Bodhidharma himself. The text does not read like a translation, and the role of the "historical" Bodhidharma in its composition is beyond our knowing at this point. Probably it was written on his behalf by Tanlin on the basis of information about the master's teachings conveyed to him by Huike, so that the text has a kind of retrospective authenticity that is common in the Chan tradition. But the important point is that this treatise was accepted by a community of Bodhidharma's successors as embodying his teachings.
Before I turn to the content of the _Treatise_ itself, let me make just a few brief comments about the nature of the "proto-Chan" community that developed in Bodhidharma's name. First, the overall impression one gets from the historical evidence is that Huike, rather than Bodhidharma, was the central figure of this loosely associated group of practitioners. Huike was already a mature adult when he studied with the master, not a fresh-faced trainee, and there is a sense in which Bodhidharma functioned for him primarily as a source of validation of his own level of attainment, a means of legitimation for his own teaching activities. Second, there is a certain range of variation in the individuals associated with Huike and Bodhidharma, including wandering ascetics, Confucian practitioners (of a rather mysterious sort), and eventually specialists in the study of the _La k vat ra S tra._ Third, Huike and the figures associated with him, however distantly in some cases, were identified with various locations in northern China, not only Luoyang. In part this was due to the vicissitudes of time—a significant persecution of Buddhism occurred in the Northern Zhou regime in 574—but, whatever the reason, they did not establish any fixed, lasting base of operations.
Probably the most important characteristic to justify referring to these men (and probably a few women) in one breath as the "proto-Chan" movement was their shared interest in the _Treatise on the Two Entrances and Four Practices._ They discussed this text in letters and used its contents as the framework for written dialogues, which, as time went on, were appended to the text itself. Although I consider only the opening essay proper, the text as it has been transmitted down to us through Dunhuang manuscripts contains a substantial number of additions, which in sum are more extensive than the original essay itself. None of this material is datable, and for all we know the process of accretion may have continued well into the eighth century.
The heart of the _Treatise,_ and indeed the doctrinal germ of much if not all later Chan theory, is the following passage:
The entrance of principle is to become enlightened to the Truth on the basis of the teaching. One must have a profound faith in the fact that one and the same True Nature is possessed by all sentient beings, both ordinary and enlightened, and that this True Nature is only covered up and made imperceptible [in the case of ordinary people] by false sense impressions. If one discards the false and takes refuge in the True, one resides frozen in "wall contemplation," in which self and other, ordinary person and sage, are one and the same; one resides fixedly without wavering, never again to be swayed by written teachings. To be thus mysteriously identified with the True Principle, to be without discrimination, serene and inactive: this is called the entrance of principle.
In the most straightforward sense, this passage is an elaboration of the idea of the Buddha-nature, the potential or actual quality of enlightenment that is latent within all of us, the only difference between buddhas and ordinary people being that the latter do not perceive this inner source of strength due to their foolish discrimination and sensory activity. The terminology used here, with one notorious exception to be discussed in the next paragraph, is not that difficult: the "True Nature," or the Buddha-nature, is a perfect, absolute (if fundamentally nonsubstantial, nonextant) entity, but it is merely obscured from our view by the false conceptualization and mistaken views of ordinary consciousness.
YANAGIDA Seizan, the greatest scholar of Chinese Chan of the twentieth century, has warned that we should not overlook an important clue to the relationship between the Buddha-nature, or True Nature, and the world of sensory discrimination: this is the word "only" toward the end of the second sentence. This inconspicuous qualifier indicates a difference of valence between the two realities—colloquially, we would say a different quantum level of significance—with the Buddha-nature understood as fundamentally more important, profoundly more real, than the constantly changing appearances of our daily lives. In other words, rather than being distracted by the superficial manifestations of our own consciousnesses—though of course these include the attributes of personal identity to which we are usually most closely attached—practitioners should instead emphasize their profound confidence in the existence of the Buddha-nature at the very heart of our innermost being. In Buddhism "faith" is precisely to "reside fixedly without wavering" in one's correct understanding. In Chinese terms this is to be "mysteriously identified with the True Principle," that is, to be united with the Buddha-nature at a level that is inscrutably hidden beneath our ordinary levels of perception, at that level of undifferentiated reality that is obscure yet oddly luminescent.
Although the peculiarly Chinese rhetoric may seem unusual, all this is actually fairly straightforward—except for the notorious exception I alluded to above. This is of course the term "wall contemplation" _(biguan),_ which has bedeviled the Chan tradition ever since its introduction here. Ultimately, no one really knows what the term means. It only occurs in one other more-or-less contemporaneous source, a list of meditation practices recommended for beginners, where it occurs without explanation. The occurrence of the term in this list is not terribly helpful, especially since the estimation of it as a beginner's practice is at odds with the comments made by the historian Daoxuan (596–667) that the "achievements of Mah y na wall contemplation are the highest." Eventually, the term came to be interpreted in the Chan tradition as referring to the act of sitting in meditation facing a wall, but as indicated in the discussion of Bodhidharma's hagiographical evolution above, it took some time for this meaning to take hold. (As shown on p. 6, important first references in this process occurred only in 988 and 1224.)
Paul Swanson has recently suggested that the compound _biguan_ might be a combination of two characters that both stand for the word _vipa yan _ or "insight meditation." Hence the character _bi_ is not used in its substantive meaning as "wall" but rather as the transliteration of the first syllable of _vipa yan ,_ a Sanskrit term usually translated into Chinese as _guan_ , the second character in the compound. The character _guan_ can, of course, be used in different senses in Chinese, but here the compound _biguan_ was thus intended as "the meaning of _guan_ that corresponds to _vipa yan ._" Unfortunately, the phonology does not quite work: In medieval Chinese the character for "wall" had a final _k_ ending (in modern Japanese the character is pronounced _heki),_ and it seems never to have been used for transliteration purposes. Finally, the association of _biguan_ with _vipa yan _ seems off; there is no sense of meditative investigation or discernment about the "entrance of principle."
Zhiyi's magnum opus on meditation theory and practice, the _Great Calming and Contemplation (Mohe zhiguan),_ includes what I suspect is a better possibility: "Concentration _(zhi, amatha)_ is wall concentration _(biding),_ in which the evil perceptions of the eight winds cannot enter. Concentration is pure water, which overflows the eight confusions of lust."
In glossing the term _biding,_ Zhanran (711–82) writes that
a room has four walls, so the eight winds cannot enter. If one is able to stop them, then one has transcended this realm's evil perceptions of internal and external, concordant and discordant. The eight winds are only the four discordant and four concordant. . . . The room's walls also prevent these eight winds [from entering]; hence they are used as a metaphor.
This usage by Zhiyi and Zhanran seems to fit the _Treatise on the Two Entrances and Four Practices_ very well: "wall contemplation" in that text might be considered to mean "fixed in _ amatha_ or concentration meditation, without allowing the eight winds of good and bad fortune to influence one at all." Whether the specific reference to the eight winds applies to Bodhidharma's treatise or not, the general sense of "wall contemplation" as the solid exclusion of distractions fits well with the "entrance of principle."
Although this metaphoric explanation seems reasonable, it was apparently not transparent to the members of the later Chan movement, who eventually introduced the more graphic image of Bodhidharma sitting in front of a cave wall. The issue is profoundly irresolvable, and we should take clear note of the uncertainty that exists.
In any case, the entrance of principle is Bodhidharma's expression—or, rather, the proto-Chan movement's expression, attributed retrospectively to Bodhidharma—of the fundamental stance of the religious practitioner. It is not altogether clear, unfortunately, exactly how this fundamental stance worked in actual practice. Does this refer to some kind of yogic absorption, some kind of forced mental extinction or tranquilization? The text is elusive on this point, and it remains for the East Mountain teaching phase of early Chan to provide specific details. Now, however, let us look briefly at the structure and content of the _Treatise on the Two Entrances and Four Practices_ as a whole.
First, what of the "two entrances"? We should not dismiss this duality as an insignificant convenience of exposition, since in positing two separate types of access to religious truth, the _Treatise_ exhibits a bimodality that is endemic to the Chan tradition. This bimodality is often negated, sometimes with polemical vehemence, but its near-universal distribution is noteworthy. In the text of the _Treatise_ itself the relationship between the entrances of principle and practice is simultaneously bipolar and unitary: each is contrasted with the other, and ultimately they end up being the same thing. This is but the earliest manifestation of a "duel" relationship in Chan. (Recall that the term _duel_ is used according to its double meaning in French, corresponding to both "duel" and "dual" in English; see p. 12.) The earliest description we have of Bodhidharma depicts him in part in terms of his duel relationship with another early meditation specialist, Sengchou (480–560); where Bodhidharma was known for the unmatched profundity of his teachings, Sengchou was known for the purity and efficacy of his ascetic endeavors.
The entrance of practice includes the following four increments:
1. Practice of the retribution of enmity: to accept all suffering as the fruition of past transgressions, without enmity or complaint
2. Practice of the acceptance of circumstances: to remain unmoved even by good fortune, recognizing it as evanescent
3. Practice of the absence of craving: to be without craving, which is the source of all suffering
4. Practice of accordance with the Dharma: to eradicate wrong thoughts and practice the six perfections, without having any "practice"
As should be clear from the contents of these four steps, the term _practice_ is used here to refer not to spiritual cultivation as an ongoing religious endeavor, but rather to the activities of one's daily behavior. The "four practices" of the second entrance thus represent a progression in which one adopts an increasingly detached perspective on the varying circumstances of one's own life, culminating in the realization that everything that occurs does so in accordance with the ultimate principles of Buddhism. At this point, although attained from different directions or styles of endeavor, the two entrances culminate in the same realization.
The important issue here is the highly contextualized or outer-focused quality of the second entrance, the attention to the details of phenomenal reality as one actually lives it. There is thus an important contrast between the two entrances: Where the entrance of principle is variously abstract, introspective, and yogic (all of these characterizations being open to reinterpretation, of course, given the allusive quality of the original text), the entrance of practice represents the concrete, extrovertive, and quotidian. Buddhist texts, not only those of the Chan school, often use formulations couched in terms of inner and outer, but the distinction is particularly important here. We will see that the bimodality between principle and practice, or rather between an abstract description of one's inner attitude and the progressive elaboration of one's ongoing activities, is a recurrent theme throughout the Chan tradition—one that will help us to organize the sometimes unruly creativity of later periods.
### Hongren and the East Mountain Teaching
The _Treatise on the Two Entrances and Four Practices_ came to be ignored in later centuries, no doubt precisely because it was too straightforward, too explicit. Just a little too humdrum in presentation, it simply did not match the image that the Chan tradition wanted to have of its founding patriarch. Given the fundamental Buddhist doctrine that everything changes, it is easy to recognize that everyone and everything is transitory or, in historical terms, transitional. The _Treatise_ continued to play an important role, though, through the seventh and early eighth centuries.
This period encompasses the phase of Chan known as the "East Mountain teaching," a term that is based on the location where Hongren (601–74) taught in Huangmei. The reference is to one of the "twin peaks," Shuangfeng, of Huangmei, and even though Hongren's teacher, Daoxin (580–651), resided on the other peak, the name "East Mountain teaching" is used for both masters. Actually, the term was used by Shenxiu (606?–706) and his immediate successors in reference to the teachings they had inherited from Daoxin and Hongren, so it is also appropriate to include Shenxiu's quarter-century of residence at Jade Spring Temple (675–701) in Jingzhou here as well. (See p. 47 for an explanation of the use of the term in association with Shenxiu.)
One of the most basic features of the East Mountain teaching, which distinguishes it clearly from proto-Chan, is that it was centered at a single, fixed location. Of course, this is certainly not to say that _all_ Chantradition activity during this period occurred at Huangmei and Jade Spring Temple, but that the East Mountain teaching phase included long, uninterrupted periods of community development in one or two fixed locations. This is a radical transition from the unsettled wanderings of Bodhidharma, Huike, and their associates.
Understandably, we have more information about the East Mountain teaching than about proto-Chan, and it is possible to make the following generalizations about the community and its teachers. First, just as Huike was the dominant personality of proto-Chan, Hongren was the central figure of the East Mountain teaching. From the description of their biographies, it appears that Daoxin may have been brought in and installed as the young Hongren's tutor. Hongren is described as being a quiet and unassuming student, who did meditation by day and took care of the cattle by night, so that when he began to teach, everyone was surprised at his brilliance. (This image of Hongren is a clear antecedent to that of Huineng; see the discussion beginning on p. 68.) When Daoxin was about to pass on, he was quoted as saying, roughly, "I guess Hongren would be all right" as his successor—and this half-hearted endorsement is actually an ironic revelation of the real situation, that Hongren was the one and only choice all along. Huangmei was Hongren's native place, where his family was known for its tradition of religious reclusion, but after Hongren's death the community was never heard of again. And, as we will see, when Shenxiu and his entourage moved into Luoyang in 701, they presented themselves as transmitters of the "pure teaching of East Mountain" and circulated a text attributed to Hongren as the content of their teachings.
Second, Daoxin and Hongren taught meditation and nothing else. In all the material we have about them, there is no reference to their advocating or practicing _s tra_ recitation, devotion to the Buddha Amit bha, or philosophical analysis—in contrast to the numerous references to them as meditation teachers.
Third, the East Mountain teachers had a gradually increasing number of students. The biographies assert that "eight or nine of every ten" spiritual practitioners in all China practiced under them, but we actually know of only a half-dozen or so individuals who studied with Daoxin and about twenty-five who studied with Hongren. Since the comparable figure for Shenxiu is about seventy, the overall trend is clear.
Fourth, in direct contrast to the single-minded dedication to meditation of their teachers, the students of Daoxin and Hongren included individuals of various religious interests. Whether practitioners of the _Lotus S tra,_ students of M dhyamika philosophy, or specialists in the monastic regulations of Buddhist Vinaya, monks traveled to Huangmei to undertake meditation training. Indeed, the East Mountain community at Huangmei seems to have been recognized throughout China by the second half of the seventh century as a specialized training center in the second of the "three learnings" of morality, meditation, and wisdom.
Fifth, as far as we can tell, Hongren's disciples stayed with him for limited periods of time. The most famous case of course is that of Huineng, who is supposed to have stayed at Huangmei for only eight months or so, which was meant to appear to contemporary Chinese as surprisingly brief. The most significant exception to the pattern of short-term residence, on the other hand, is the monk Faru (613–89), who seems to have served as Hongren's attendant or assistant during his sixteen years at Huangmei—which reminds one of the example of the Buddha's cousin and long-time attendant, nanda. (Faru is an important transitional figure between the East Mountain teaching and metropolitan Chan phases; see p. 48.) Judging from the biographies, most of Hongren's students were more like Shenxiu, who stayed with the master for six years at the very beginning of his teaching career. Although this information may also be subject to some doubt—six years was the length of time Gautama spent performing austerities before he became enlightened under the _bodhi_ tree, and Buddhist hagiography often echoes this figure in order to invoke the Buddha's example—the pattern seems to have been that Daoxin and Hongren's students stayed with them for a few years and then went on to other things.
Sixth, nothing special can be said about the East Mountain community's size, administration, or spiritual lifestyle. The great Japanese scholar Ui Hakuju (1882–1953) suggested that it included five hundred or a thousand members, but the figures he uses actually refer to the attendance figures for Hongren's funeral. There must have been quite a number of lay devotees and admirers present at this event, not to mention some pious exaggeration in the written references. Seeing that we know of about twenty-five men who studied with Hongren in about as many years, even taking into consideration the probability that the number of his students increased as time went on, only a handful of these figures would have been present at any one time. There is no accurate way to estimate the actual number of monks and nuns in training at any one time, which might have fluctuated over time from just a handful to as many as several dozen. _Pace_ UI, there is also no evidence whatsoever that these monks participated in anything other than meditation and ordinary religious services—that is, there is no evidence whatsoever that the famous and probably illusory ideal of Chan monastic labor was known at East Mountain. The famous dictum that "a day without work means a day without food" only appears centuries later, and Hongren's community no doubt had its share of lay workers and tenant agricultural laborers, like other Buddhist centers of the time. Here our best evidence is the _Platform S tra,_ which depicts the eventual sixth patriarch Huineng as a low-status temple menial. Since this was the image of Hongren's community generated a century afterward, and our first evidence for any special "Chan" style of monastic system does not come for centuries after that, the only possible conclusion is negative: there is no basis for suggesting that Chan had developed a specific lifestyle in which monastic labor was performed as part of spiritual cultivation.
### From Proto-Chan to Metropolitan Chan: The _Treatise on the Essentials of Cultivating the Mind_
So, what style of meditation practice did Daoxin and Hongren teach? The usual—almost inevitable—approach is to first explain what we know about the former, then turn to the latter. When this style of presentation is combined with treatments of the earlier patriarchs, as it almost always is, the result is a clear instance of the "string of pearls" fallacy. That is, rather than probing the dynamics of evolution of the Chan movement over time, most authors actually present a static elaboration based on the traditional genealogical configuration of the Chan orthodoxy that developed in the Song dynasty and beyond, a simple form of transposition posing as analysis. In fact, the "teachings of Daoxin" and the "teachings of Hongren" as they are now understood did not exist during the actual lifetimes of these historical figures, but only appeared during the transition from the East Mountain to the metropolitan Chan phase. The time lag was only a few decades, which might seem brief in the overall span of Chinese Buddhist history, but considerable change can occur in such a seemingly brief period. The teachings of Daoxin and Hongren were recorded retrospectively, as written reconstructions of lessons from the past. As it turns out, this retrospective quality of the East Mountain teaching is very significant.
At Huangmei, Daoxin and Hongren would not have needed to present their teachings in writing. In the relatively intimate context of teacher-student interaction, written guidelines might have been useful but would not have been necessary. When their students moved into the much larger arena of the two capitals of Chang'an and Luoyang, though, the situation was entirely different. Chang'an was the greatest cosmopolitan center on earth at the time, with a population of perhaps a million people and enriched by close trading connections across the Silk Road to India, Persia, and the Middle East. Luoyang was a somewhat smaller city, but a venerable center of culture and religion, and the imperial court moved back and forth between the two capitals from time to time. The imperial court and literate society surrounding it were a magnet for intellectual and religious innovations from all over China, and indeed from throughout East Asia, Buddhist India, and Central Asia as well, and this "imperial center" had been the focal point of translation and research activities for Buddhism for centuries, as it continued to be throughout the eighth century. Even though Chan is portrayed in modern writings as having developed in rustic surroundings and as a rejection of merit-oriented activities and imperial largesse, this image of Chan and its fundamental identity developed precisely within the context of the imperial center, rather than on its periphery. We need only recall the legendary encounter between Bodhidharma and Emperor Wu of the Liang, which was concocted in the middle of the eighth century, to realize how these themes played out in medieval China. Just as Chinese nature poetry originally developed among city dwellers, so was the almost barnyard primitivism and antiintellectualism of "classical" Tang-dynasty Chan created in a highly sophisticated, literate milieu of the Five Dynasties and Song dynasty periods. (Actually, even the encounter between Bodhidharma and Emperor Wu was generated in a context that undercuts the iconoclastic image of Chan; see the discussion beginning on p. 108.)
When Hongren's students moved from the provincial community at Huangmei to the imperial center, one of their first steps was to compile a written record of their master's teachings. This was the _Treatise on the Essentials of Cultivating the Mind,_ which includes the straightforward admission that it was compiled not by Hongren himself but by his students, presumably after his demise. Actually, this is the earliest example within the Chan tradition of the composition of texts representing a given master's teachings, that is, of texts that were compiled and edited shortly after the master's death. The _Treatise on the Essentials of Cultivating the Mind_ may have been prepared for use by Faru, who taught at Mount Song for a few years prior to his death in 689; the text was almost certainly known to Shenxiu by about the same time, and it was quoted in other texts during the second decade of the eighth century.
Although Daoxin is treated in Chan hagiography as Hongren's predecessor, the written teachings attributed to Daoxin only appeared _after_ the text attributed retrospectively to Hongren. One or two of the basic slogans associated with Daoxin may have existed earlier, but the assertions found in scholarly works published to date of a doctrinal evolution from Daoxin to Hongren are impressionistic and thoroughly unconvincing. Moreover, the teachings of this "Daoxin" are composed in an intellectually sophisticated format that belies the supposed succession of ideas. In any case, since "Daoxin's" teachings first appeared in the second decade of the eighth century, we can clearly detect a chronological trend of retrospective attribution. In other words, the members of the Chan movement moved in reverse order through the commonly accepted list of patriarchs, publishing suitable writings first for Hongren, then for Daoxin, and then (in the middle of the eighth century) for Sengcan. Hence any attempt to re-create the evolution of Chan teachings by moving from patriarch to patriarch in a forward order is condemned to failure for methodological reasons that are simultaneously elementary and profound. Such attempts exemplify the "string of pearls" fallacy, which cripples the ability of most authors to deal with the evidence as it evolved instead of how it was designed to look. Therefore, when we look at the _Treatise on the Essentials of Cultivating the Mind,_ we are not seeing Hongren himself, but Hongren as he was remembered several decades after his death.
Even so, the _Treatise on the Essentials of Cultivating the Mind_ is a masterpiece of religious literature. Concise and unpretentious, it frequently exhorts its readers to make greater effort on behalf of their own enlightenment. It is not merely that life is too short, as we might put it today, but that the opportunity to undertake Buddhist spiritual training in a supportive environment is a rarity that may not happen again for many lifetimes. To complement this vigorous encouragement, the text describes an attitude toward religious attainment that is wonderfully delicate, and the practices it recommends are designed to avoid placing too strong an emphasis on the final goal. (As every beginning student of Buddhist philosophy quickly recognizes, to desire _nirv a_ as a final goal contradicts the very desirelessness of _nirv a_ itself.) The _Treatise on the Essentials of Cultivating the Mind_ manipulates these considerations with a charmingly palpable sensitivity. And it provides a welcome elaboration of the basic themes adumbrated in such deliciously elusive fashion in the _Treatise on the Two Entrances and Four Practices_ attributed to Bodhidharma.
The heart of the _Treatise on the Essentials of Cultivating the Mind_ is the following dialogue, which includes a made-to-order but spurious scriptural quotation:
The _Treatise on the S tra of the Ten Stages_ says, "There is an adamantine Buddha-nature within the bodies of sentient beings. Like the sun, it is essentially bright, perfect, and complete." Although vast and limitless, it is merely covered by the layered clouds of the five skandhas. Like a lamp inside a jar, its light cannot shine.
Further, to use the bright sun as a metaphor, it is as if the clouds and mists of this world were to arise together in all the eight directions, so that the world would become dark. How could the sun ever be extinguished?
[Question: Without the sun being extinguished,] why would there be no light?
Answer: The sun's light is not destroyed, but merely deflected by the clouds and mists. The pure mind possessed by all sentient beings is also like this, in simply being covered by the layered clouds of discriminative thinking, false thoughts, and ascriptive views. If one can just distinctly maintain [awareness of the mind] and not produce false thoughts, then the Dharma sun of _nirv a_ will naturally be made manifest.
The relationship between the "sun-and-clouds" metaphor here and the explanation of the True Nature in the Bodhidharma treatise is obvious, and a similar qualifier is even used to describe how the Buddha-nature or sun of enlightenment is "merely" obscured by one's ordinary psychological identity. In addition to thus adopting the same value structure in this initial formulation, the Hongren treatise describes the fundamental attitude toward spiritual cultivation in terms of "maintaining [awareness of the mind]" _(shouxin),_ which is essentially a posture of nurturing the Buddha-nature as a treasure within one's own person. Rather than aggressively intruding into one's own being to scrape away the clouds of ignorance—which would be rather like reaching a giant claw into the sky to drag away the clouds and mists blocking the sun—the appropriate response is to affirm the ultimate reality of one's beginningless enlightenment, to maintain constant awareness of this pristine condition within oneself, and then to work in an energetic but unharried fashion toward the circumstantial manifestation of the on-going enlightenment experience.
The _Treatise on the Essentials of Cultivating the Mind_ describes two specific meditation techniques, which neatly demonstrate the two aspects of this fundamentally vigorous but composed attitude. The first is to visualize the orb of the sun just as it sets, shining back at one from a fixed point on the horizon, large and round as a giant temple drum hanging sideways on a stand. This technique is actually drawn from the _S tra of the Contemplation of the Buddha Amit yus,_ one of the major scriptures of the East Asian Pure Land tradition, and although its explicit use here is as an exercise in concentration (one is to focus on the one point of the sun without distraction) it also serves implicitly as a symbolic reminder of the "sun of _nirv a"_ within.
The second technique is to focus, not on the Buddha-nature itself, but on the hyperactive mental processes that obscure it:
Make your body and mind pure and peaceful, without any discriminative thinking at all. Sit properly with the body erect. Regulate the breath and concentrate the mind so it is not within you, not outside of you, and not in any intermediate location. Do this carefully and naturally. View your own consciousness tranquilly and attentively, so that you can see how it is always moving, like flowing water or a glittering mirage. After you have perceived this consciousness, simply continue to view it gently and naturally, without it assuming any fixed position inside or outside of yourself. Do this tranquilly and attentively, until its fluctuations dissolve into peaceful stability. This flowing consciousness will disappear like a gust of wind. When this consciousness disappears, all one's illusions will disappear along with it, even the [extremely subtle] illusions of bodhisattvas of the tenth stage.
Other authorities might object that merely stopping the transformations of consciousness was not equivalent to complete and perfect enlightenment—certainly, this was to become a subject of discussion within Chan. But the important point is the dedicated but undemanding attitude recommended here. Rather than forcing the issue, rather than trying to "achieve" enlightenment, the Hongren treatise counsels the practitioner to simply let it happen. Whether or not this approach is suitable for everyone—and at least one Chan master would openly deride similar approaches as uselessly waiting around for miracles to happen (see p. 135)—the sensitivity of the text in counterposing its two techniques against one another, of demanding energetic patience, if you will, represents a remarkable synthesis.
### Indian and Chinese Buddhist Polarities
One of the most prominent features of Chan discussions of meditation is the use of polarities. To be sure, such discussions often include reminders of a fundamental nondualism, the absence of any absolute distinctions. Even so, the frequency of dualistic formulations is striking. Bodhidharma and Sengchou, Huineng and Shenxiu, principle and practice, sudden and gradual, Northern and Southern schools, and Linji (Rinzai) and Caodong (S t ): from hagiographical figures to doctrinal themes to lineage divisions, the Chan tradition veritably overflows with dualistic formulations. Given this situation, it is tempting simply to line up the dyads used in different contexts and suggest that they are all essentially the same in some fashion. A better approach, of course, is to remain alert to the possibilities of nuanced differences between the various pairs. At present the question is, what inferences can we draw from comparing the contents of the Hongren treatise with earlier Buddhist meditation theory?
Certainly, the most important pair of themes in Indian Buddhist meditation doctrine is that of concentration ( _amatha_ ) and insight ( _vipa yan_ ). Very briefly, concentration refers to a set of exercises aimed at developing the mind's ability to focus without distraction on a given object. A variety of objects may be used, assigned by the meditation instructor as appropriate antidotes for the student's particular dispositional tendencies. A trainee given to anger might be instructed to work on the generation of loving-kindness, while one given to pride might be told to perform exercises involving the visualization of corpses. As the practitioner eliminates the hindrances blocking his ability to concentrate effectively, he moves through a set of four stages of _dhy na,_ or "concentration." (The Chinese word _chan_ , pronounced _zen_ in Japanese, _s n_ in Korean, and _thien_ in Vietnamese, is a transliteration of this Sanskrit word.) According to the canonical descriptions, in the first stage of _dhy na_ the practitioner's mind is characterized by singlepointedness of concentration along with two different types of mental deliberation and a combination of joy and bliss. By conscious decision the practitioner moves from one stage to the next, successively eliminating the two types of mental deliberation and the joy and bliss, which ultimately are considered distractions to the task at hand. With the fourth and "fundamental" stage of _dhy na_ the practitioner's mind is characterized solely by singlepointedness of mind. Although speech and discursive thought are impossible at this stage, it is here that the meditator becomes able to use the supernormal faculties of telepathy, superaudition, levitation, knowledge of his own past lives, and understanding of the karmic fates of others. The Buddha and his disciples often used these abilities for teaching purposes, but the Buddhist tradition considers them potentially hazardous diversions of no value to spiritual cultivation, and there are Vinaya regulations against monks' divulging competence in these powers to laypeople.
In contrast to the great elaboration of concentration exercises, insight or _vipa yan _ meditation consists solely of the application of the concentrated mind to any object, in order to attain "clear comprehension" of it. In _ amatha_ the mind becomes concentrated like a searchlight, while in _vipa yan _ that searchlight-like mind illuminates the most important issues of the human condition: the transiency and composite nature of the human body, the dependent origination of thoughts and feelings, and the inevitability of human suffering. By using the mind concentrated through _ amatha_ to examine these issues, the practitioner sees and understands them through _vipa yan ._ Thus concentration and insight are not really separate techniques, even though they may be explained separately for convenience. The meditation exercise that is most widely used throughout the Buddhist tradition is that of concentration on the breath, which has the virtue of drawing the practitioner naturally from concentration to insight: as the body settles and respiration slows, one's attention shifts from calming to knowing.
The objects selected for attention in insight meditation by any Buddhist community are congruent with the understanding of Buddhist doctrine within that community. Hence in early Buddhism one was to focus on the body and one's thoughts and feelings in order to recognize their inherent impermanence, causal interrelationship, quality of suffering, and so forth. In Mah y na Buddhism, on the other hand, the realization achieved in insight meditation tended to be the fundamental emptiness _( nyat )_ of all things, although this and other Mah y na themes were expressed in various ways by early Chinese meditators. Although Thera-v da and other Mainstream Buddhist sources do adduce stages of progress in insight practice, these stages tend to be increasing gradations of a single achievement of awareness rather than quintessentially different achievements. (In contrast, the explanation of the stages of concentration, or _dhy na,_ include significant conceptual distinctions.) Nor is there any real explanation of how insight happens—only the basic assumption that the mind, when directed at a given subject matter, has the innate capacity to understand. Like the Buddha's enlightenment, the experience of understanding is ineffable, but its impact is liberating. It is axiomatic throughout the Buddhist tradition that the perfect understanding of the human situation yields one's liberation from the deleterious effects of that situation.
To return to matters closer at hand, we may now ask the following question: To what extent do the two entrances of the Bodhidharma treatise or the two practices suggested in the Hongren treatise resemble the Indian Buddhist themes of concentration and insight? I have already introduced evidence to suggest that the entrance of principle might be considered an interpretation of concentration, or _ amatha,_ and the same consideration would also apply to the practice of the visualization of the sun. The use of the Buddha-nature idea, the sun of enlightenment within all human beings (indeed, within all sentient beings), the quality of non-discriminatory wisdom that is the sine qua non of buddhahood itself, is a profound innovation that separates proto-Chan and early Chan from early Indian Buddhism. However, it is also a simple concentration exercise, the only peculiarity of which is that the mind is being trained to concentrate on the mind's most quintessential capability of understanding itself. The goal of the practice of "maintaining the mind" in the Hongren treatise is precisely to affirm the existence of that latent wisdom and to allow it to shine forth in unqualified form. Where I tend to describe the concentrated mind of Indian Buddhist _ amatha_ theory as a searchlight that may then be focused on specific topics in _vipa yan ,_ in Chinese imagery the enlightened sun of _nirv a_ within is an all-encompassing source of illumination. Given this difference in metaphoric construction, though, the Indian Buddhist concept of concentration meditation thus correlates, if only approximately, with the entrance of principle and the visualization of the sun.
However, this is not the case for the comparison between insight meditation and Bodhidharma's entrance of practice and Hongren's focus on the activity of the discriminatory mind. Part of the problem, of course, is that the two specific meditation techniques attributed to Hongren include substantial components of both concentration and insight. (As we have seen just above, of course, the same can be said for many Indian Buddhist meditation exercises.) Hongren's instructions to concentrate on the movement of the discriminatory mind imply both cessation—in that it is expected that the mind's movement will eventually stop in the course of one's practice—and understanding—in that the cause of that cessation is said to be a "wind of wisdom." For the moment, however, we must grant that the second practice recommended in the Hongren treatise is more like concentration than insight.
The problem is that the entrance of practice in the Bodhidharma treatise simply does not fit into this pattern. Rather than being any kind of yogic practice at all, in fact, the four steps within this "entrance" to the path pertain to one's activity in the world. To be sure, the emphasis is on the mental posture with which one approaches one's life experience. However, the emphasis is on action, not realization. This should alert us to the fact that something is going on here that does not fit within the confines of "meditation practice" per se. Instead, we need to look within the Chinese tradition for a suitable analog to the pairing of the two entrances of the Bodhidharma treatise.
As Chinese clergy and laypeople were struggling to understand Buddhism in the fourth and fifth centuries of the common era, they were wont to use a uniquely Chinese formulation: the distinction between essence _(ti,_ lit., "body") and function ( _yong,_ lit., "use"). There is no sharp distinction between essence and function; depending on the perspective, any entity or situation can be approached in terms of either one. Nor is there any sharp transformation in moving from essence to function, since the difference between the two is more in the mind of the beholder rather than in the entity itself. In his _Treatise on the Immutability of Things_ Sengzhao (374–414) explains the relationship as follows, based on an initial quotation from an early translation of the _Perfection of Wisdom:_
The _Light-Emitting [Perfection of Wisdom] S tra_ states, "Dharmas are without going and coming, without active transformation." In searching for the operations of inactivity, how could one possibly seek stillness by undoing the active? One must seek stillness within the activities [of things]. Since one must seek stillness within the activities [of things], although active they are always still. Since one should not undo the active to seek stillness, although [things] are still they do not transcend activity. Nevertheless, even though activity and stillness have never varied, the deluded take them as different.
The early-twentieth-century scholar TANG Yongtong (1893–1964) explains that Sengzhao's entire treatise is devoted to showing that active and still are identical. This is not to say that there exists some unmoving fundamental essence that generates the myriad phenomenal manifestations, but that that the fundamental realities and phenomenal permutations are inseparably identical.
Thus it is entirely reasonable that the two entrances of Bodhidharma's text are quite different and yet seem to merge in the fourth practice, where "practicing in accord with the Dharma" so closely resembles the entrance of principle. The two entrances may be separate, but in a certain sense they imply each other, even contain each other. From a more general perspective, this is only the beginning of a broader attention to the similarities and differences between the different types of polarities that are scattered about the Chan tradition. We will have occasion to return to the Bodhidharma treatise again, to recognize its signal role in establishing patterns that recur again and again throughout Chan. At this point, however, let us be content to notice that not all such polarities are identical, that different matchings may harbor substantially different implications. With this simple but important observation in hand, we may turn our attention to the next phase of Chinese Chan.
## CHAPTER 3
## Metropolitan Chan
_Imperial Patronage and the Chan Style_
### A "Chan Boom" in the Imperial Chinese Capitals
In the first half of the eighth century, the northern Chinese cities of Chang'an and Luoyang were the greatest urban centers in the world. Chang'an had a population of over a million, a number far larger than any city in the Middle East (let alone Europe) would reach for centuries. Originally a safe military headquarters "within the passes" of the mountainous northwest, Chang'an was laid out on an extremely grand scale and in a cross-hatched design of wide boulevards running north-south and east-west. The city walls formed a nearly square rectangle enclosing a neatly ordered set of government centers, market areas, and neighborhoods. With the imperial palace in the north of the city and major thoroughfares connecting to regional highways leading eastward to Korea and Japan and westward to Central Asia, Persia, India, and the Middle East, the emperor could face south towards both city and realm, even as the entire world seemed to face north in paying homage toward this ruler of "all under heaven."
The imperial state was expressed in grand and imposing material form, with massive office buildings and official temples, and it was operated by a bureaucratic organization of ministries, bureaus, and departments manned by officials who achieved their positions through different combinations of hereditary advantage and civil service examinations. The most elite of these bureaucrats were required to attend an imperial audience every morning, some of whom recorded poetic laments of the windy chill of lonely city streets in wintertime as they rode on horseback from their homes to the palace in the far north of the city.
In addition to its majestic official identity as a political center, Chang'an was a cosmopolitan nucleus of trade, literature, culture, and religion. As the major Chinese trading node on the Silk Road, it received imports of rare treasures from India, Persia, and beyond, and its people enjoyed new musical styles, carnival entertainments, and art forms imported one after the other from the "western regions." Polo was a favorite game among men (and some women) of the Chinese upper classes, and some of the latest song styles from the "western regions" shocked the older generations as much as contemporary music does in the United States today. With local populations of traders and their descendants from Sogdiana, Khotan, Korea, and other exotic locations around China, Chang'an was an exciting and lively mix of cultures. In terms of knowledge of both the western regions and Buddhism, the example of the great pilgrim Xuanzang (600?–64), who had traveled through Central Asia to India in the mid-seventh century, was still reverberating throughout the Chinese realm. And, of course, there were dozens of magnificent Buddhist temples (and a smaller number of Daoist ones) throughout the city, with a large population of monks and nuns.
The second capital at Luoyang was not nearly as large, nor was it laid out quite so neatly as Chang'an, but its location some 320 kilometers to the east was within the rich alluvial Yellow River plain and thus in the very cradle of Chinese civilization. Luoyang had been a Buddhist center from the second century onward—one of the earliest and most important in China—and its temples were numerous, venerable, and magnificent. In addition to its reputation as a center of culture, Chinese officials sometimes preferred that the emperor reside at Luoyang because it was easier to supply with grain than was Chang'an in the mountains to the west. For students of Chan, Luoyang is also known as the city just north of Mount Song, the central peak in the quinary configuration (i.e., having points in the center and four corners) of traditional Chinese sacred geography. Bodhidharma had been associated with Mount Song since at least 645, although it was only at the end of the seventh century that Chan monks are known to have taken up residence at the fabled Shaolin Temple there.
At the beginning of the eighth century, in a pivotal event marking the public beginning of Chan as a school of Chinese Buddhism, the Chinese emperor invited a certain monk to the capital at Luoyang. This was not just any emperor, but the only woman to sit on the Chinese throne in her own name: Wu Zetian, usually referred to in English as Empress Wu (r. 690–705). Through an exceptional combination of native intelligence and political acumen, along with good luck and personal beauty, Empress Wu had been able to take over control of the Chinese state when her husband, Emperor Gaozong (r. 656–90), was debilitated by strokes beginning in 670, and then to rule in her own name from the time of his death in 690. Although she is excoriated by orthodox Confucian historians, her efforts to justify her political position by identifying herself as a Buddhist ruler, and even as an incarnate Bodhisattva, are a fascinating subject for students of Chinese religions. For our present purposes, we may merely note that by the turn of the eighth century she was well established in her rule and had no need for further ideological artifice.
And the Chan teacher invited to court by Empress Wu was no ordinary monk! He was Shenxiu (606?–706) from Jade Spring Temple in Jingzhou (Hubei and Hunan Provinces), the preeminent figure in the burgeoning Chan tradition. Two separate texts describe his welcome into Luoyang in 701 as follows:
Empress Wu Zetian sent a palace messenger to escort Shenxiu to Luoyang. Monks and laypeople spread flowers in his path, and the banners and canopies [on the vehicles of the wealthy and prestigious] filled the streets. He entered the palace riding on an imperial palanquin decked with palm leaves. Empress Wu, following him, touched her forehead to the ground and knelt for a long time in a spirit of reverent dedication and chaste purity. When Shenxiu administered the precepts to the court ladies, all the four classes of Buddhists took refuge in him with the same feelings of veneration that they had for their own parents. From princes and nobles on down, everyone in the capital took refuge in him.
[After his imperial invitation to Luoyang, Shenxiu] accompanied the imperial chariot on its comings and goings, proselytizing in the two capitals and personally becoming the Imperial Instructor. The Great Sage Empress Wu Zetian inquired of him: "Whose teaching is it that you transmit?" He answered, "I have inherited the East Mountain teaching of Qizhou [i.e., Huangmei, the location of Hongren's monastery]." Empress Wu Zetian said, "In considering the cultivation of enlightenment, the East Mountain teaching is unexcelled."
This was a spectacular demonstration of imperial reverence, which Empress Wu emphasized by having Shenxiu sit facing south with herself kneeling in front of him, facing north. Shenxiu's epitaph, which was written by a prominent statesman and literatus of the day, defended this by saying that "he who transmits the Holy Truth does not face north; he with abundant virtue does not follow the protocol of a subordinate." Shenxiu seems to have been a member of the Tang ruling family, but this treatment was exceptional nonetheless. In addition to Empress Wu's sincere reverence, she may even have been making a conciliatory gesture toward those who would replace her in office after her death. The result is that Shenxiu is the first historical member of the Chan tradition whose specific ideas, rather than a retrospective or posthumous image, are known in any depth and detail.
In spite of the use of imperial north/south symbolism to indicate Shenxiu's exalted religious status, there is only a distant connection between this and how his teachings and following came to be referred to as the "Northern school." (See the next section, "Shenhui's Campaign against the 'Northern School,'" beginning on p. 54.) Instead, here we see Shenxiu effectively producing his own history, in the sense of identifying himself in the present by selectively and/or creatively describing his past, by labeling his approach to Buddhism as the "East Mountain teaching" of his teacher Hongren. At about the same time, Hongren's students compiled the _Treatise on the Essentials of Cultivating the Mind;_ a little afterward they compiled a suitable set of teachings attributed to Daoxin; and decades later other monks produced the _Treatise on Believing in Mind_ and other documents celebrating Sengcan's life. Thus the process of retrospective production of history continued forward through time, even as its focus shifted to successively earlier figures in the lineage. (This subject has been mentioned above; see especially the discussion on pp. 36–38.)
This was the religious environment in which we discover the very first expressions of the Chan lineage scheme. That is, the genealogical presentation of the Chan transmission was first recorded on paper in the early years of metropolitan Chan activity. The earliest recorded instance of this was in the epitaph for a certain Faru, a student of Hongren's who died in 689 (see p. 35), and by the second decade of the eighth century, the later followers of Hongren had produced two separate texts describing the transmission from Bodhidharma to Shenxiu. These two texts, which I do not discuss individually here, are known to contemporary scholarship as early "transmission of the lamp" histories after the title of the defining text in the genre written several centuries later, the _Record of the Transmission of the Lamp [compiled in] the Jingde [period],_ or _Jingde chuandeng lu._ There are differences of content and emphasis between the two "Northern school" texts, but they both express essentially the same doctrine: that the central teaching of Buddhism was transmitted through a sequence of patriarchs reaching Shenxiu and his disciples.
While one of the fundamental implications of the "transmission of the lamp" texts was the unchanging continuity from master to disciple in the Chan lineage, from a historical perspective it is clear that the transition from East Mountain to the two capitals was accompanied by a profound transformation in the nature of the Chan movement. As environments of rhetorical exchange and religious discourse, there was a radical difference between East Mountain and the two capitals, and Chan was transformed as its members actively sought to move from one setting to the other. Notes and partial transcripts of the masters' teachings might have been made at the East Mountain monastic community in the provincial town of Huangmei, but it was only when Hongren's successors moved into the environment of the two capitals, with its literate society and incomparably larger urban scale, that well-written texts were required for disseminating the teachings. No doubt the creative process of active remembering began in some fashion at East Mountain, but we have no direct evidence of this. Whatever rustic simplicity or sophisticated discourse might have governed life at Huangmei is now largely unrecoverable, since all our sources are retrospective creations of the literate cultural center.
Even granting the complexity of our sources, though, it is clear that Shenxiu's teachings were qualitatively different from those of Hongren as described in the _Treatise on the Essentials of Cultivating the Mind._ Shenxiu was fascinated with a style of radical reinterpretation of the Buddhist scriptures based on his own religious insight, which he referred to as the use of "skillful means" or the "verification of the Chan meaning." This was actually the extensive use of a form of anagoge or metaphor, in which every pronouncement of the scriptures was subject to reinterpretation in terms of "contemplation of the mind." In simple terms, what Shenxiu told his listeners was that the Buddha was not interested in mundane matters, but used each and every utterance to describe the practice of Buddhist meditation. Shenxiu thus advocated, in a fashion distantly reminiscent of the Buddha himself, that Buddhists should work to achieve buddhahood and the salvation of all living beings themselves right now.
Thus we find in Shenxiu's writing a number of parallels drawn so as to redefine conventional religious practices in terms of actual spiritual cultivation. The following are paraphrased summaries of the most instructive of these parallels:
_Temple repair:_ The Chinese transliteration for _sa gha- r ma_ is defined as a "pure ground," so that the eradication of the three poisons of greed, hatred, and ignorance is described as constituting the repair or "cultivation" of such a monastery.
_Casting and painting of images:_ The Buddha was not interested in the creation of mundane images, but was instructing the true practitioner to "make his body a forge, the Dharma its fire, and wisdom the craftsman." The three groups of pure precepts and the six perfections become the mold for casting, within the practitioner's own body, the Buddha-nature of Suchness.
_Burning of incense:_ The incense referred to here is not some worldly fragrance but rather that of the true, unconditioned Dharma, which "perfumes" the tainted and evil karma of ignorance and causes it to disappear.
_Offering of flowers:_ The Buddha is said never to have advocated the injury of live flowers, but referred in the scriptures to the "flowers of merit" imbued with the essence of Suchness. Such flowers are permanent and never wilt.
_Circumambulation of_ st pas: The body is equated with the _st pa,_ and circumambulation is defined as the ceaseless circulation of wisdom throughout the body and mind.
_Holding of vegetarian feasts:_ Through the selective use of Chinese homographs, the phrase "to hold vegetarian feasts" is interpreted as the ability to make the body and mind equally regulated and unconfused.
_Obeisance:_ Through the manipulation of transitive and intransitive equivalents of the Chinese characters involved, obeisance is defined as the suppression of errors.
Shenxiu also introduces a short scriptural passage extolling the virtues of bathing and then reinterprets the endeavor as "burning the fire of wisdom to heat the water of the pure precepts and bathe the Dharma-nature of Suchness within one's body." The following is a summary of Shenxiu's seven "dharmas of the bath":
_Clean water:_ Just as clean water washes away the dusts of this world, so do the pure precepts clean away the defilements of ignorance.
_Fire:_ The fire that heats the bath water is actually wisdom, with which one contemplates or examines one's internal and external being.
_Soap powder:_ The soap powder used to clean away dirt is actually the ability of discrimination by which one can ferret out the sources of evil within oneself.
_Toothpicks:_ The "sticks of willow" used to eradicate mouth odor are nothing less than the Truth by which one puts an end to false speech.
_Pure ashes:_ The ashes or powdered incense rubbed on the body after bathing are endeavor _(v rya),_ by which one puts an end to doubt-laden ratiocination.
_Oil:_ Rather than softening one's skin, the oil referred to here is meant to soften dispositional stiffness, or bad habits.
_Underwear:_ The clothing worn in the bath is actually the sense of shame that inhibits evil actions.
In other words, Shenxiu interpreted every passage of every scripture he considered in terms of its instruction concerning Buddhist spiritual cultivation, and he advocated a manner of living in which even the most prosaic of one's activities became—in every feature and detail—an act of religious practice. There is a definite connection between this style of interpretation and the later Chan emphasis on having one's practice extend to every facet of daily life. We will return to this provocative implication later (see the discussion in chapter 4, pp. 85–86).
The point to be emphasized here is the significance of Shenxiu's innovations for the eventual crystallization of Chan as an independent tradition of Chinese Buddhism. There was, in short, a remarkable "Chan boom" in early eighth-century Chang'an and Luoyang, in which Shenxiu's Chan teaching became wildly popular in the greatest cities on earth, among the world's most sophisticated and cosmopolitan society. As one courtier wrote,
Students of Buddhism from both capitals and the faithful from all areas of China all come to the [Five-Gated Entrance to the Imperial City to hear his teaching]. They come from a thousand _li_ away without any hesitation! The mendicants with their robes and begging bowls crowd into newly built halls like schools of jumping fish; their huts cover the hillside like lines of geese. Gathering like clouds and free as the dew, they go to Shenxiu empty-handed and return fulfilled.
Shenxiu's message was breathtakingly simple, since he in effect told his followers to simply practice contemplation of the mind now, working to be bodhisattvas here and now, in this very lifetime, in every moment of their lives. There are echoes of this fundamental attitude not only in later Chan, but also in the early-ninth-century "enlightenment in this body" doctrines of the Japanese Tendai and Shingon school figures Saich (767–822) and K kai (774–835), whose teachings were clearly inspired by the Chan innovation.
Although his message may have been simple in a certain sense, Shenxiu obviously conveyed it with a commanding personal charisma and a uniquely appealing rhetorical style. His activities in the two capitals at the beginning of the eighth century spawned an explosion of Chan religious creativity. Not only did he have some seventy ordained disciples whose names are important enough to be known, but his major students became imperial instructors themselves. They and their students and friends wrote a number of important texts outlining their doctrines and practices, including not only the treatises attributed to Daoxin and Hongren and the "transmission of the lamp" histories mentioned above, but also a variety of documents that have come down to us through the finds at Dunhuang. Taken together, these texts give the impression of a collective experimental effort, an exploration of just how a still-evolving Chan message might best be conveyed to others. Some of these formulations are genuinely inventive, while others are oddly mechanical in their application of Shenxiu's new style of metaphor, and the diversity of formulations implies that not everyone participating in this new movement understood the practice of Buddhism precisely as he had. But this was an entirely natural course of events.
One of the most intriguing features of this material is that some of these texts describe actual practices of meditation. Although there is tremendous variety, the following passages from the text known as the _Five Skillful Means_ and circulated by the "Northern school" provide a good introduction:
To view the mind as pure is called "to purify the mind-ground." Do not constrict the body and mind and then unfold the body and mind—view afar in expansive release. View with universal "sameness." Exhaust space with your viewing.
The preceptor asks: What do you see [lit., "What thing do you see"]?
The disciple(s) answer: _I do not see a single thing._
Preceptor: Viewing purity, view minutely. Use the eye of the pure mind to view afar without limit, without restriction. View without obstruction.
The preceptor asks: What do you see?
Answer: _I do not see a single thing._
View afar to the front, not residing in the myriad sensory realms, holding the body upright and just illuminating, making the true essence of reality distinct and clear.
View afar to the rear, not residing in the myriad sensory realms, holding the body upright and just illuminating, making the true essence of reality distinct and clear.
View afar to both sides . . .
View afar facing upwards . . .
View afar facing downwards . . .
View the ten directions all at once . . .
View energetically during unrest . . .
View minutely during calm . . .
View identically whether walking or standing still . . .
View identically whether sitting or lying down . . .
Question: When viewing, what things do you view?
Answer: _Viewing viewing, no thing is viewed._
Question: Who views?
Answer: _The enlightened mind views._
Penetratingly viewing the realms of the ten directions, in purity there is not a single thing. Constantly viewing and being in accord with the locus of non-being, this is to be equivalent to a buddha. Viewing with expansive openness, one views without fixation. Peaceful and vast without limit, its untaintedness is the path of _bodhi._ The mind serene and enlightenment distinct, the body's serenity is the _bodhi_ tree. The four tempters have no place of entry, so one's great enlightenment is perfect and complete, transcending perceptual subject and object.
The preceding does not need much elaboration. It clearly asks the students to place emphasis on the enlightened mind at the center of their beings, and it instructs them to train their minds so as to penetrate the entire cosmos and all individual activities. The spatial quality of the instructions is distinctive, but the mental attitudes to be nurtured through such practices resemble those of the later Chan tradition: a sense of release, the recognition of the universal sameness of all experience, recognition of the fundamental emptiness of all things, a quality of profound tranquillity, and above all the innate ability of the mind to illuminate and understand all things. The basic practical orientation indicated here is similar to that of the Hongren treatise in advocating an approach to spiritual cultivation that is energetic and vigorous but entirely without tension or the discriminatory fixation on the goal.
Although the best way to appreciate the meditation exercises just described is in terms of Shenxiu's understanding of the "perfect teaching," to which we will return shortly, it is also easy to appreciate how these instructions to "view afar" could be criticized by someone less disposed to meditation practice himself. It is to just such a figure that we now turn.
### Shenhui's Campaign against the "Northern School"
In 730, 731, and 732, a monk named Shenhui staged public "debates" at a town in Shandong, far to the northeast of Luoyang, in which he attacked two students of Shenxiu's for making false claims about their lineage and for teaching an inferior style of practice. In 745 Shenhui took up residence in Luoyang, where he continued his campaign. The written account of the 732 event (actually edited after 745) describes the question posed to Shenhui by his interlocutor, an otherwise little-known monk named Chongyuan, and Shenhui's response:
Dharma Master Chongyuan asked Shenhui, "The two worthies, Chan Master Puji of Mount Song and Chan Master Xiangmo Zang of the Eastern Peak (i.e., Mount Tai), teach people to sit in meditation and 'freeze the mind to enter concentration, fix the mind to view purity, activate the mind to illuminate the external, and concentrate the mind to realize the internal.' They declare that this is the teaching. Why do you today preach Chan without teaching people to sit [in meditation] and without teaching people to 'freeze the mind to enter concentration, fix the mind to view purity, activate the mind to illuminate the external, and concentrate the mind to realize the internal'? What is 'sitting in meditation'?"
His Reverence Shenhui answered, "To teach people to sit [in meditation this way] . . . is to obstruct _bodhi_ (i.e., enlightenment). When I say 'sit' now, [I mean that] 'sitting' is for thoughts not to be activated. When I say 'meditation' now, [I mean that] 'meditation' is to see the fundamental nature. Therefore, I do not teach people to have their bodies sit and their minds abide in entrance into concentration. If it were correct to declare such a teaching, then Vimalak rti would not have scolded rip tra for sitting in meditation."
Dharma Master Chongyuan asked, "Why is it impermissible for Chan Master Puji to use the label 'Southern school'?"
His Reverence answered, "Because when Reverend Shenxiu was alive, all those who study the Path in China referred to these two great masters as '[Hui]-neng of the South' and [Shen]-xiu of the North'—everyone knew this. It is because of these titles that we have the two schools of North and South. Chan Master Puji is actually a student of [Shenxiu of] Jade Springs [Temple]; he actually never went to Shaozhou (Huineng's place of residence) but now falsely mouths off about his being the Southern school. Therefore, this is impermissible."
This was a shocking presentation. Using the format of a public debate, Shenhui staged a dramatic and sharply worded attack on the "Northern school," which was a label he invented himself and applied to Shenxiu and his disciples. The name "Northern school" immediately stuck, even though it was clearly a pejorative and polemical distortion—we can infer from the above (there is other corroborating evidence as well) that certain members of this loose confraternity of practitioners actually used the title "Southern school" to describe their own teachings. Shenhui was actively engaged in reformulating the history of Chan, and it is important to recognize that he borrowed substantially from the so-called "Northern school" even as he criticized it so severely. For example, Shenhui set up his own lineage hall in imitation of Puji, even as he worked to establish the transmission from Bodhidharma to Huineng (and then implicitly to Shenhui himself) as the sole lineal succession of Chan. The "Northern school" had originally generated the basic configuration of the Chan genealogical model, but only with Shenhui was its unilineal quality highlighted so clearly. Shenhui's ideas of meditation practice also inherited perspectives shown in the Hongren treatise and Shenxiu's writings, but he was of course far more explicit about denying the value of mental manipulation.
Shenhui was a unique religious persona within the Chan tradition, in that his vocation was that of an evangelist. He did not fit the standard pattern of the meditation teacher who patiently guided dedicated practitioners as they struggled to work through the various problems and stages of spiritual cultivation. Instead, his life's work was performed on the ordination platform, where he served as inspirational orator, recruiter for the _sa gha_, and fund-raiser for both church and state. Shenhui's mission was to inspire believers to generate a sincere aspiration to achieve perfect enlightenment on behalf of all living beings (this is the moment of _bodhicitta,_ the primary criterion of being a bodhisattva). He was a master at public preaching, able to draw large crowds with his histrionics and inspiring style.
Here is an example of how Shenhui worked to motivate his listeners to attain the first moment of realization _(bodhicitta)_ even as they listened to him preach:
Friends, you have all been able to come here so that you can all generate the unsurpassable _bodhicitta._ It is extremely difficult to encounter the Buddhas, Bodhisattvas, and true spiritual compatriots. Today you are going to hear something you've never heard before. In the past you never encountered it, but today you have.
The _Nirv a S tra_ says, "The Buddha asked K yapa, 'Would it be difficult to throw a mustard seed down from Tu ita Heaven and hit the point of a needle on the earth below?' Bodhisattva K yapa replied, 'It would be extremely difficult, World-honored One.' The Buddha told K yapa, 'This is not difficult. For the correct cause and the correct condition to meet—this is what is difficult!'"
What are the correct cause and the correct condition? Friends, your generation of the unsurpassable _bodhicitta_ constitutes the correct cause. For the Buddhas, Bodhisattvas, and true spiritual compatriots to cast this unsurpassable _bodhicitta_ into your minds such that you achieve the ultimate emancipation constitutes the correct condition. For the two to meet is excellent. . . . You must each and every one of you generate _bodhicitta!_ . . .
Since you have already come to this ordination platform to study the perfection of wisdom, I want each and every one of you to generate the unsurpassable _bodhicitta_ both mentally and orally and to become enlightened to the cardinal meaning of the middle way in this very place
One modern Chinese commentator makes much of Shenhui's creation of "a new kind of Ch'an that was no _ch'an_ at all," by which he means a new approach to Buddhism that omitted the practice of meditation. But the reason the practice of meditation is so strikingly absent from Shenhui's writings is not that meditation was no longer to be included in the Chan training regimen, but because Shenhui's personal religious vocation was that of an evangelist, recruiter, and fund-raiser, rather than that of a spiritual mentor to dedicated trainees. This is not a reflection of Chan monastic behavior in general, but of Shenhui's distinctive religious identity.
We have no evidence that Shenhui ever concerned himself with the ongoing endeavor of spiritual cultivation, and his lineage was not notably long-lasting. Although the names of a few of his immediate students are known, none of them was historically significant. Even his most famous successors in subsequent generations seem not to have been descended from him, but rather from another monk by the same name. (This is a good example of the second rule of Zen studies: "Lineage assertions are as wrong as they are strong.") The reasons for Shenhui's very substantial impact on the evolution of Chan must be sought elsewhere.
### The Oxhead School: Resolving the Factionalist Crisis
Shenhui's attack on Shenxiu's students created a crisis in early Chan, by creating a sharp dichotomy between two newly defined factions, the Northern and Southern schools. This crisis was resolved by the appearance of a third faction, the Oxhead school, and the composition of the _Platform S tra._ Shenhui's waspish criticism of other contemporary monks by name was unprecedented, and it stigmatized him even as many of his positions were accepted. His caricature of "Northern school" teachings as gradualist may not have been accepted by everyone, but since the "Northern school" was an artificial creation of Shenhui's imagination, very little energy was expended in defending it. Shenhui's simple value structure, in which sudden enlightenment (especially the first moment of inspiration) was good and gradual enlightenment (or the progressive development toward complete understanding) was bad, was not accepted, but his combative bombast did make everyone else shy away from formulations that might be attacked as either dualistic or gradualist. Hence subsequent Chan texts observed an unspoken "rule of rhetorical purity," avoiding any direct discussion of specific meditation practices—since _any_ method was by definition gradualistic in some fashion.
In addition, in the records of the latter decades of the eighth century we find a number of attempts to erase the sharp distinction between north and south, gradual and sudden. Below are a few examples.
First, here is "Eulogy on the Two Patriarchs Huineng and Shenxiu," by the poet-monk Jiaoran:
The minds of these two men
were like the moon and sun.
With no clouds in the four directions, they appear in space.
The Three Vehicles share the same path;
the myriad teachings are one.
The "division into Northern and Southern schools"
is an error of speech.
Jiaoran has eulogies to Bodhidharma, Zhiyi (founder of the Tiantai school), the "Northern school" monks Lao'an (d. 708) and Puji, Huineng and Shenxiu, the legendary Baozhi (418?–514?), Shenxiu (individually), and Xuansu (688–752) of the Oxhead school—but none that is dedicated to Huineng individually.
Next, the famous poet Liu Zongyuan's (773–819) epitaph for an Oxhead school monk contains the following:
The greatest aberration in the diminution of the Buddhist teaching is the term "Chan": Grasping, it defiles things; misleading, it becomes separate from the truth. This separation from the truth and increase of deception is greater than the [entire realm of] space of both present and past. Such stupid errors and deluded self-indulgence only debase oneself, misrepresent Chan [here meaning _dhy na?_], and do injury to the Buddhist teaching. Those who make this error are characterized by stupidity and moral dissolution. . . . [Master Ruhai (the subject of the epitaph)] has said. . . . [After the transmission reached] Shenxiu and Huineng, north and south reviled each other like fighting tigers, shoulder-to-shoulder, and the Way became hidden.
Finally, here is a statement by an important Oxhead school figure:
Alay supporter asked: "Are you a follower of the Southern school or the Northern school?" He answered: "I do not belong to either the Southern school or the Northern school. The mind is my school."
So what was the impact of this "Oxhead school"? It arose in the latter half of the eighth century, among monks renowned for their literary creativity who felt a deep connection with the Chan tradition. As a lineage, it defined itself separately from either the Northern or Southern schools, but at least some of its members were enchanted by the image of Huineng, the figure promoted by Shenhui as sixth patriarch. The earliest version of the _Platform S tra,_ which dates from around 780, makes effective use of Oxhead school ideas in producing a narrative framework for the understanding of Chan (see p. 65 below).
The Oxhead school did not merely soften the edges of contention between the Northern and Southern schools, it created new rhetorical devices by which to overcome the agonizing division that Shenhui had generated. The philosophy that underlay these efforts is demonstrated in the following passage, which is from the _Treatise on the Transcendence of Cognition (Jueguan lun)._ This imaginative text is presented as a dialogue between an idealized teacher called Professor Enlightenment and his student, Conditionality. The result is the most meaningful sort of fiction, in which is depicted the spiritual dialogue between master and student that leads to the latter's awakening:
Professor Enlightenment was silent and said nothing. Conditionality then rose suddenly and asked Professor Enlightenment: "What is the mind? What is it to pacify the mind?" [The master] answered: "You should not posit a mind, nor should you attempt to pacify it—this is called 'pacified.'"
Question: "If there is no mind, how can one cultivate enlightenment _(dao)?_ " Answer: "Enlightenment is not a thought of the mind, so how could it occur in the mind?" Question: "If it is not thought of by the mind, how should it be thought of?" Answer: "If there are thoughts then there is mind, and for there to be mind is contrary to enlightenment. If there is no thought then there is no mind, and for there to be no mind is true enlightenment." . . . Question: "What 'things' are there in no-mind?" Answer: "No-mind is without 'things.' The absence of things is the Naturally True. The Naturally True is the Great Enlightenment _(dadao)." . . ._
Question: "What should I do?" Answer: "You should do nothing." Question: "I understand this teaching now even less than before." Answer: "There truly is no understanding of the Dharma. Do not seek to understand it." . . . Question: "Who teaches these words?" Answer: "It is as I have been asked." Question: "What does it mean to say that it is as you have been asked?" Answer: "If you contemplate [your own] questions, the answers will be understood [thereby] as well."
At this Conditionality was silent, and he thought everything through once again. Professor Enlightenment asked: "Why do you not say anything?" Conditionality answered: "I do not perceive even the most minute bit of anything that can be explained." At this point Professor Enlightenment said to Conditionality: "You would appear to have now perceived the True Principle."
Conditionality asked: "Why [do you say] 'would appear to have perceived' and not that I 'correctly perceived' [the True Principle]?" Enlightenment answered: "What you have now perceived is the nonexistence of all dharmas. This is like the non-Buddhists who study how to make themselves invisible, but cannot destroy their shadow and footprints." Conditionality asked: "How can one destroy both form and shadow?" Enlightenment answered: "Being fundamentally without mind and its sensory realms, you must not willfully generate the perception of impermanence."
Question: "If one becomes [a Tath gata] without transformation and in one's own body, how could it be called difficult?" Answer: "Willfully activating the mind is easy; extinguishing the mind is difficult. It is easy to affirm the body, but difficult to negate it. It is easy to act, but difficult to be without action. Therefore, understand that the mysterious achievement is difficult to attain, it is difficult to gain union with the Wondrous Principle. Motionless is the True, which the three [lesser vehicles] only rarely attain."[?]
At this Conditionality gave a long sigh, his voice filling the ten directions. Suddenly, soundlessly, he experienced a great expansive enlightenment. The mysterious brilliance of his pure wisdom [revealed] no doubt in its counter-illumination. For the first time he realized the extreme difficulty of spiritual training and that he had been uselessly beset with illusory worries. He then sighed aloud: "Excellent! Just as you have taught without teaching, so have I heard without hearing . . ."
This text is significant for at least two reasons. First, and most important, it depicts the interaction between teacher and student as the latter begins the quest, attains an intermediate realization that is momentarily mistaken for the goal, and then achieves final enlightenment. This is only one of a number of texts from the latter half of the eighth century that are devoted to explicitly fictional depictions—that is, dramatic scriptings—of this process. It was not yet conceivable that written texts should include the words of actual, historical students. This observation is relevant to the emergence of written transcriptions of Chan "encounter dialogue," and we return to this point in the next chapter.
Second, we should pay attention to the threefold structure of this passage. In contrast to Shenhui's simple, dualistic value system of gradual vs. sudden, here there is a threefold pattern of beginning questions, intermediate hesitation, and final achievement. A close examination of Ox-head school writings suggests that their teachings were frequently written using a threefold logical format, which resembles Zhiyi's scheme of the three truths of absolute, relative, and middle. It is also structurally similar to Hegel's thesis-antithesis-synthesis pattern, but in this case the second element achieves its impact by the application of the fundamental Mah y na concept of _ nyat ,_ or emptiness. Indeed, the same tripartite structure is apparent in the thought of at least one important Indian M dhyamika philosopher. That is, an expression of Buddhism is made in the first element, the terms of this expression are erased in the second element, and the understanding of Buddhism is thereby elevated to a new level of profundity in the third element. The significance of this pattern will only become clear when we examine the greatest masterpiece of early Chinese Chan Buddhism, the _Platform S tra._
### The _Platform S tra_ as the Climax Text of Early Chan
The _Platform S tra_ appeared in about 780, over a century after the events it describes were supposed to have taken place. Many scholars have struggled to identify the contents of some "original" or "core" version of the text that might date back to Huineng himself, but the utter failure of these attempts has only confirmed the late provenance of the text as we have it. Barring some miraculous discovery, we must consider the text as we first discover it, in its Dunhuang version. But we really should be satisfied with this, for this earliest version of the text is a brilliant consummation of early Chan, a masterpiece that created a new understanding of the past even as it pointed the way to a new style of Chan practice.
The heart of the _Platform S tra_ is the following story.
Since he was approaching the end of his years, the Fifth Patriarch Hongren instructed all his disciples to compose a "mind-verse" demonstrating their levels of enlightenment. If one of these verses manifested a true understanding of Buddhism, its author would receive the Fifth Patriarch's robe and the status of Sixth Patriarch. All but one of the disciples simply ignored Hongren's instructions, deferring instead to the man they felt would be the next leader of the Chan community: Shenxiu. Shenxiu himself was perturbed by his teacher's request, though, and thought to himself,
The others won't present mind-verses because I am their teacher. If I don't offer a mind-verse, how can the Fifth Patriarch estimate the degree of understanding within my mind? If I offer my mind to the Fifth Patriarch with the intention of gaining the Dharma, it is justifiable; however, if I am seeking the patriarchship, then it cannot be justified. That would be like a common man usurping the saintly position. But if I don't offer my mind then I cannot learn the Dharma.
In the end Shenxiu did compose a verse, but he was so uncertain about its worth and the propriety of seeking the patriarchship that he inscribed it anonymously on a wall in one of the monastery's corridors, doing so late at night so that no one would see him. Shenxiu's verse read:
The body is the _bodhi_ tree.
The mind is like a bright mirror's stand.
At all times we must strive to polish it
and must not let dust collect.
When Hongren saw this verse on the corridor wall the next morning, he canceled plans to have illustrations from the _La k vat ra S tra_ painted there. He praised Shenxiu's verse highly and ordered his students to recite it so as to avoid unfavorable future rebirths. In private, though, he pointed out to Shenxiu that it did not display true understanding, and he counseled his senior disciple to write another verse to gain the Dharma. In the end, Shenxiu was unable to do so.
In the meantime, an uneducated layman from the far south of China named Huineng was at work threshing rice, completely unaware of Hongren's instructions about the future succession. When one day an acolyte passed by the threshing room reciting Shenxiu's verse, Huineng realized immediately that its author did not understand the "cardinal meaning" of Buddhism. The boy explained the matter to Huineng, who asked to be shown the corridor wall on which the verse was inscribed. Since he was illiterate, Huineng requested that someone else record his poetic statement.
Actually, the earliest version of the _Platform S tra_ contains two versions of Huineng's verse:
_Bodhi_ originally has no tree. | The mind is the _bodhi_ tree.
---|---
The mirror also has no stand. | The body is the bright mirror's stand.
The Buddha-nature is | The bright mirror is
always clear and pure. | originally clear and pure.
Where is there room for dust? | Where could there be any dust?
Clearly, the editor could not decide which was better! In later versions, this indelicacy is cleared away, and a famous third line is added:
_Bodhi_ originally has no tree.
The bright mirror also has no stand.
Fundamentally there is not a single thing.
Where could dust arise?
According to the basic Dunhuang account, Hongren denigrated Huineng's verse in public, but late that night he privately taught the layman the ultimate teaching of the _Diamond S tra._ Huineng was immediately awakened to its profound meaning, received the transmission of the sudden teaching and the Fifth Patriarch's robe, and left the monastery in secrecy that very night.
This is one of the most treasured legends of the Chan tradition. I have introduced only the bare essentials of the story, but even with this minimal detail we can see a number of ways in which the _Platform S tra_ was producing a new historical account of the evolution of Chan, even as it implied a new religious vision.
First, in some of its details the _Platform S tra_ account is clearly written as historical allegory. Note, for example, the shift from _La k vat ra S tra_ to _Diamond S tra_ implied in the account (i.e., in the cancellation of the painting commission and Hongren's teaching to Huineng), which parallels the two texts' changes in popularity over the course of the eighth century. The position of the _La k vat ra_ within Chan was always ambiguous, since the text was more revered in the abstract than actually studied. However, it was generally associated with "Northern school" teachers. Shenhui was one of the first monks of his day, but by no means the only one, to favor the _Diamond,_ which was becoming more widely popular throughout the Chinese tradition at the time. Hence, in the _Platform S tra_ the two texts roughly symbolize the Northern and Southern schools. Also, Shenxiu's prominence within Hongren's community and Huineng's inferior status may be taken as indications of the relative strengths of the two factions prior to the composition of the _Platform S tra._ In a biography written at about the same time as this text, and in later versions of the _Platform S tra_ itself, Huineng is depicted as remaining in hiding for sixteen years after receiving the transmission, a more graphic representation of the early weakness of the movement associated with his memory.
FIGURE 3. "Huineng's" verse from the _Platform S tra_ on the back window of a taxicab, Tainan, Taiwan. Photograph by the author, 1996.
Second, the absence of any reference to Shenhui is significant. Shenhui's own texts never mention the "mind-verses" nor anything like the _Platform S tra_ story, which is an important indication that the verses were composed after his death. At the very least, the verses could not have been written prior to Shenhui's vigorous campaign on behalf of Huineng as sixth patriarch, nor Shenhui's vigorous espousal of the teaching of sudden enlightenment. One of the most important features of the _Platform S tra,_ in other words, is that it incorporates Shenhui's innovations while writing him out of the story. As mentioned above, even as Shenhui transformed the understanding of the evolution of Chan, the factionalist cast of his campaign stigmatized Shenhui himself.
But how should we understand the verses themselves? The traditional interpretation, since the time of the great systematic Chan and Huayan philosopher Zongmi (780–841), has been that Shenxiu's verse represents gradualism and Huineng's subitism (the position that enlightenment occurs in a single transformation that is both total and instantaneous). This simplistic explanation cannot be accepted. (Zongmi artificially claimed succession from Shenhui, but given the manifest difference between Shenhui's teachings and the _Platform S tra,_ Zongmi's interpretation should be recognized as a tactical distortion of the original.) First, the verse attributed to Shenxiu does not in fact refer to gradual or progressive endeavor, but to a _constant_ practice of cleaning the mirror. Hence, Zongmi's traditional interpretation is conceptually incorrect. Second, the verse attributed to Huineng could not stand alone (nor could any of the variants attributed to him), since it could not be understood without reference to "Shenxiu's" verse. Since the two verses constitute an indivisible pair—they indicate a single polarity, not two separate teachings—it is inappropriate to use either verse as a key to the religious teachings of the two historical individuals Shenxiu and Huineng.
And how are we to understand the equations made in "Shenxiu's" verse? Thinking of the body as the _bodhi_ tree is easy enough, since both sides of the equation are comfortably physical; just as the _bodhi_ tree was the location of kyamuni's enlightenment, so must the physical body be the site of enlightenment for each human being. But how can the mind be like a mirror's stand? Many English translations of "Shenxiu's" verse omit reference to the stand and say simply that "the mind is like a mirror," but this interpretation is simply erroneous. The solution lies in the following passage from a treatise by Shenxiu:
Further, lamps of eternal brightness (i.e., votive lamps) are none other than the truly enlightened mind. When one's wisdom is bright and distinct, it is likened to a lamp. For this reason, all those who seek emancipation always consider the body as the lamp's stand, the mind as the lamp's dish, and faith as the lamp's wick. The augmentation of moral discipline is taken as the addition of oil. For wisdom to be bright and penetrating is likened to the lamp's flame (or brightness). If one constantly burns such a lamp of truly suchlike true enlightenment, its illumination will destroy all the darkness of ignorance and stupidity.
There is no specific evidence that the historical Shenxiu ever wrote anything like the verse attributed to him in the _Platform S tra,_ or even that he made any metaphoric identification between mind and mirror's stand. However, it would have been entirely in character for him to have done so. If we were to speculate how he might have generated such a metaphor, we would presumably conclude that he would have used the logic introduced in this passage. That is, in his use of "skillful means" to interpret all the various elements of Buddhist scriptures as demonstrations of the "constant practice" of the "contemplation of the mind," Shenxiu might have posited the body as the overall setting for enlightenment (i.e., the _bodhi_ tree), the sensory and intellectual activity of the mind as the proximate support for enlightenment (i.e., the mirror's stand), and the pure or enlightened mind itself as the illuminative surface of the mirror—and the act of rubbing the mirror clean of dust as a standard maintenance operation similar to maintenance of the Buddhist precepts or monastic regulations. Based on the most comprehensive reading of the texts pertaining to Shenxiu, it is apparent that his basic message was that of the constant and perfect teaching, the endless personal manifestation of the bodhisattva ideal.
Even at a glance we can see that it makes more sense for the anonymous authors of the _Platform S tra_ to depict Shenxiu's teachings as remarkably profound rather than as an elementary form of gradualism. Since the goal was to show the superiority of Huineng's teachings, the comparison should not be made with something recognizably inferior—as gradualism was considered at the time, especially in this post-Shenhui moment—but rather with something already recognized as superior in itself. If I were to propose a new theory of mathematics, for example, I would not compare it to elementary school arithmetic but to something far more sophisticated.
Huineng's verse(s) exhibit no explicit subitism, but only the reversal or denial of the terms of Shenxiu's verse. The two sets of verses do not, then, represent the alternatives of gradual and sudden, and they certainly do not represent the distinctive doctrines of two opposing lineages. If we compare this to the structure of the Oxhead school's depiction of the interchange between Professor Enlightenment and the student Conditionality, the parallel is clear: The _Platform S tra_ uses the same threefold structure found in Oxhead-school thought. The constant teaching is first posited as the highest possible expression of the Buddhist teaching _in formal terms,_ after which Huineng's verse(s) apply the rhetoric of emptiness to undercut the substantiality of the terms of that formulation. However, the basic meaning of the first proposition still remains, rather like a shadow whose sharp outlines have been removed by the impact of the second proposition. The third and final proposition thus includes both the assumption of the first and the erasure of the second, now shorn of its over-sharp outline. In the _Platform S tra_ this third proposition is implicit in the balance of the text, which contains the expression of the ultimate teaching of Buddhism in ways that do not contravene the "rule of rhetorical purity."
Although somewhat disjointed, the balance of Huineng's sermon in _the Platform S tra_ is a wonderful mélange of early Chan teachings, a virtual repository of the entire tradition up to the second half of the eighth century. At the heart of the sermon is the same understanding of the Buddha-nature that we have seen in texts attributed to Bodhidharma and Hongren, including the idea that the fundamental Buddha-nature is _only_ made invisible to ordinary humans by their illusions. There are clear borrowings from Shenhui's criticisms of "Northern school" meditation practices, as well as his doctrine of the identity of concentration and wisdom. The Huineng of this text is conscious of Shenhui's proscriptions against dualistic formulations, and he warns repeatedly that the difference between sudden and gradual lies only in the aptitude of the practitioner. For all this, the sermon as a whole establishes a posture toward the actual practice of meditation that differs from that of Shenhui, and the entire mood of the _Platform S tra_ regarding the gradual/sudden distinction is not Shenhui's straightforward polemic of inferior vs. superior, but rather a nuanced attempt to describe a notoriously refractory subject, that is, the basic attitude that should be adopted toward Buddhist spiritual cultivation. In addition, the text clearly admits laypeople to full participation in this process, something that the monastic recruiter and fund-raiser Shenhui never did. (For him laypeople were potential converts to the monastic life or, in some cases, prominent scholar-officials who lent prestige to his activities.) The _Platform S tra_ inherits the style of reinterpreting conventional Buddhist pronouncements as meditation instructions that had been originally developed by Shenxiu and that was maintained to some extent by Shenhui and to an even greater degree by the Oxhead school.
The sermon Huineng is depicted as delivering in the _Platform S tra_ has been subjected to various interpretations, and it is worthy of far more extended attention than I can give it here. For the present, we need only notice the open quality of the text. That is, even though this famous scripture serves to cap a certain line of development within the Chan tradition, it does not close off further doctrinal evolution. Indeed, the text itself was frequently updated in the ensuing centuries, implying that in some sense the devotees of Chinese Chan took some time to become satisfied with the _Platform S tra's_ presentation of the teaching. (In the future, scholars may be able to explore the accretions to and modifications of this text as something of an index to the evolution of Chan.) However, there is also a sense in which the subsequent evolution of Chan passed the _Platform S tra_ by; that is, the text helped set the stage for the emergence of encounter dialogue as a mode of spiritual cultivation, but it represents more the grand culmination of one era than a scripture for all seasons. The _Platform S tra_ has enjoyed great attention in the twentieth century because of the Dunhuang manuscript finds, and it is now extremely popular in Chan and Zen communities around the world. We need to remember, though, that it was not nearly so widely used during the Song dynasty in China or during the Kamakura period in Japan.
But there is still more to say about the opening _Platform S tra_ anecdote itself. The reader might wonder, for example, whether there is any possibility that the events described might have actually happened. Here we can be definitive: there is no such possibility whatsoever, and the account must be accepted as a brilliant and religiously meaningful bit of fiction. How is it possible to be so certain? First of all, Shenxiu studied with Hongren for a few years at the very beginning of the latter's teaching career, so he was nowhere in sight when the events in question are supposed to have occurred. Second, the very notion of selecting an individual successor to serve as "sixth patriarch" would have been inconceivable in the latter years of Hongren's life, since the concept of a Chan "monosuccession"—that there was one and only one orthodox succession of patriarchs—appeared only later, in the teachings of Shenhui. Third, if the matter had been known to Shenhui, who was a master storyteller dedicated to promoting Huineng's identity as sixth patriarch, he certainly would have included it in his writings. We have good evidence to show that in the late 730s Shenhui was ignorant of most of the details of Huineng's life. It is probable, but by no means certain, that Shenhui only thought to contribute to the embellishment of Huineng's life quite late in his own career.
There are also indications that the _Platform S tra verses—both_ those attributed to Shenxiu and to Huineng—were generated utilizing "Northern school" rather than "Southern school" writings. We have already seen references to the "path _of bodhi"_ and the body's serenity as the _bodhi_ tree in "Northern school" writings above (see p. 53), as well as allusions to not seeing "a single thing." In this context it is significant that a Dunhuang manuscript containing numerous metaphors in the manner of the "Northern school" contains the line "within suchness there originally is really not a single thing." The Chinese for this line is similar to the famous later third line of Shenxiu's _Platform S tra_ verse, implying that the scripture was modified on the basis of ideas originally transmitted in a "Northern school" style or context. Since "Northern school" refers to a sizeable movement associated with literate court society and Buddhism, while the "Southern school" of Shenhui and Huineng was a minor voice from the provinces to the east and far south, it should not be too surprising that the invention of a tradition associated with the latter used resources derived from the former.
### Huineng as Illiterate Sage and the Evolution of Chan
Finally, let us briefly examine the legendary identity of Huineng himself, since it is through this image rather than through doctrines per se that the primary impact _of the Platform S tra_ was felt.
The historical Huineng seems to have been a member in good standing of that loose confraternity of teachers we have referred to above as the Northern school. That is, his name is mentioned in one of the early eighth-century "transmission of the lamp" texts as one of Hongren's ten major disciples—although one of only regional significance, since he lived in Caoqi in the far south of China. Huineng's name appears one other time in a Dunhuang text dedicated to the memory of Hongren, where, along with other figures, he is briefly quoted in a manner that says nothing about any particular ideas that might have been associated with the historical Huineng. We are told elsewhere that after Huineng's death his residence was converted into a temple, and if this were true, his family must have had a certain degree of local prominence. It is striking, even stunning, how little Shenhui knows about the person who is supposed to be his own master, and it seems that Shenhui may have actually gained rather little more from Huineng than the certification of his own enlightenment. (Here, of course, we can draw a comparison with Huike and Bodhidharma.) It is probably fair to think of the historical Huineng as a reasonably conventional Chinese monk, whose teachings differed only slightly if at all from those of other members of the Northern school.
In contrast to the image just described, the legendary Huineng depicted in the _Platform S tra_ is an illiterate layman from the far south, whose family had been reduced to such poverty that he had been making his living gathering and selling firewood. Although humble of origin, this Huineng is blessed with the highest of all Chinese moral qualities: he is a filial son taking care of his widowed mother. This image of the humble, unassuming paragon is clearly developed from the hagiography previously associated with Hongren, who was supposed to have sat in meditation by day and minded the cattle at night. In all these qualities, Huineng is the very antithesis of the highly cultured and socially advantaged monks who dominated the Chinese Buddhist _sa gha_ at the time. Shenxiu was in fact one of the prime examples of this type of individual—well educated in both Buddhist and secular literature, from a highly ranked family and perhaps even related to the imperial house, and thus used to the world of social and economic privilege. In this sense the contrast between the historical Shenxiu and the legendary Huineng could not be greater.
But the image of Huineng is only apparently iconoclastic, only superficially populist. In terms of the developing Chan genealogical theory, the goal of his depiction in the _Platform S tra_ is to show that anyone—even someone so clearly lacking in all the usual qualifications of an elite Chinese Buddhist monk—could be appointed as sixth patriarch as long as he fulfilled the only qualification that really mattered. That is, the particulars were negotiable as long as it could be argued that he was innately enlightened. The story is designed to say that the Chan school would go to any lengths to nominate anyone who fulfilled this one crucial prerequisite, and that as an institution it was unconcerned about anything else whatsoever. On the surface, this seems to make the patriarchate accessible to anyone.
To be sure, there is a universalist aspect to the _Platform S tra_ account, in its implication that anyone can become enlightened regardless of educational or social background. But in terms of the Chan lineage itself, there was a paradoxical implication. Just as the Chan school would go to any length to find the right appointee, even if he lacked all the "right" social qualities, conversely, one implication of the _Platform S tra_ story is that each person the Chan school later selected as a lineal successor to its enlightened masters was similarly qualified as an enlightened individual, even though it might seem that considerations of social status, family connection, and other worldly qualities played a role in the selection. Thus a brilliant iconoclastic form can actually serve to maintain a socially conservative orthodoxy. There is no explicit evidence that this paradox was on the minds of the compilers of the _Platform S tra,_ of course, nor even that it was obvious to the readers of the text. The logic involved here becomes important when we consider the paradoxical role of iconoclasm in Song-dynasty Chan discourse, where it is used within a highly ritualized formal setting. We also need to look at how the imagery of the illiterate sage found in the _Platform S tra_ resonated through the encounter dialogue anecdotes of the so-called "classical" stage of Chinese Chan. Before proceeding to these issues, however, we should briefly consider certain issues of historical context.
### Three Other Sets of Events
Although this book focuses on Chan, we must remember that this one school of Chinese Buddhism did not develop in a vacuum. There are at least three major events, or rather sets of events, that occurred in the eighth century that significantly altered the evolution of Chan.
The first set of events was the appearance of esoteric Buddhism on the Chinese scene. Although writers who focus myopically on the Chan tradition imply that the Southern school supplanted the Northern school solely through the superiority of its sudden teaching, the attention of the imperial court and metropolitan literati had already been diverted from Chan long before the distinction between the Northern and Southern schools was posited. ubh karasi ha (637–735) arrived in Chang'an in 716 and immediately began to excite the Chinese with a new interpretation of Buddhism that promised both rapid spiritual attainment and unparalleled worldly power. On his heels came Vajrabodhi (671–741), who arrived in Guangzhou (Canton) in 720, and Amoghavajra (705–74), who became a disciple of Vajrabodhi's at age fifteen in China, traveled to India after his master's death, and returned to China in 746. Along with the native exegete Yixing (683–727), a former Northern school monk and protoscientist who collaborated in the translation and teaching work of ubh karasi ha and Vajrabodhi, these men dominated the Buddhist scene in Chang'an and Luoyang throughout most of the eighth century. Here was a religious teaching that allowed one to ally oneself with the greatest spiritual powers of the universe, both for one's own spiritual advancement and for maximum ritual efficacy in worldly affairs, including healing illness, bringing rain or ending floods, and even causing victory on the battlefield. Using a unique combination of profound doctrine, visualization technique, and impressive ritual theater, esoteric Buddhism overwhelmed the Chinese—and indeed East Asian—religious consciousness.
As soon as esoteric Buddhist teachers appeared on the scene, competition for patronage between them and native Chan masters became inevitable. We have a lengthy description of an encounter between ubh karasi ha and Jingxian (660–723), one of Shenxiu's students. What is most interesting about this encounter for our purposes is that the foreign master explicitly criticizes the Chan style of meditation practice, insisting that by "single-mindedly maintaining nonthought _(wunian)_ as the ultimate, the [longer you] search, the more unattainable [is your goal]." Although there are certain similarities between Chan and esoteric Buddhist practice (the importance of the relationship between Chan master and student resembles that between tantric guru and esoteric practitioner, for example), Chan had nothing to match the highly theatrical style and magnificent promise of esoteric ritual. One or two Chan masters were prominent at the imperial court in the last decades of the eighth century and beyond, but none of them attracted the faddish aura of excitement that had surrounded Shenxiu and other Northern school figures at the beginning of the century.
The second set of events is a sequence of civil and political disasters that led to the collapse of the Tang dynasty itself, implying the destruction of what had been a supremely influential model for Buddhism throughout East Asia. The process began in 755–63, with a major rebellion instigated by a military governor in the far north, a man of Sogdian extraction named An Lushan (Roxanna in Sogdian). In most ignominious fashion, the emperor was forced to flee Chang'an, made to abdicate in favor of his son, and compelled to watch the execution of his favored concubine. (She and her brother, a notoriously corrupt official, were blamed by the emperor's military escort for allowing the catastrophe to occur.) After An Lushan's death the rebellion was carried forward by others, and it took some eight years for the Tang imperial government to reestablish itself. In the process, though, China was changed forever: regional governors were effectively given autonomy in many of the outlying regions; the imperial house remained in charge of only the central provinces. Changes in taxation and military conscription that began at this time signaled a major transformation of Chinese society as a whole.
The second event in this series was of course the great persecution of Buddhism by the Chinese government in the Huichang period, beginning in 845. Monastic assets were seized, monks and nuns were laicized, and great restrictions were placed on the activities of the church in general. Most introductions to Chinese Buddhism consider this persecution to have dealt the religion a crippling blow, but its effects were only temporary. The problem was that the Buddhist establishment was hardly given time to recover before the next disaster struck: This was the Huang Chao rebellion of 875–84, which devastated the provinces of northern China, effectively destroying the foundation of aristocratic family domination there, and eliminating the combined wealth of the social class from which Buddhism received its support.
The Tang state limped along for another couple of decades, but in 906 it finally collapsed entirely. The next half-century or so witnessed a succession of regimes that controlled different regions of northern and southern China. The Chinese polity would be reconstituted in 960 with the founding of the Song dynasty, but by then the world had changed forever.
The third set of events is the effective end of the translation of Buddhist scriptures from India and Central Asia. This was not an event that occurred instantaneously, and several different factors contributed to this final result. First, from some time in Amoghavajra's career, or perhaps only shortly thereafter, the only new texts being translated were esoteric ritual manuals. There were a handful of translators active throughout the ninth century, but nothing they produced became important for larger doctrinal or devotional reasons. Second, from 810 on, the Tang imperial court divested itself of all involvement in Buddhist translation activities, ending a centuries-old tradition of central government sponsorship. The result was that from about this year until 980, no Buddhist scriptures were formally entered into the canon. Third, in 1004—by coincidence the year in which the _Transmission of the Lamp_ was presented to the Song court—Moslem forces conquered Khotan. Buddhism had long been in decline in its heartland areas of India and Central Asia, and now the transmission of texts across the Silk Road was impossible. There was a flurry of activity in the last two decades of the tenth century, but even though the Song government established a central translation bureau, it soon became moribund. The problem, simply enough, was that there were just no more new texts to work on. The influx of Buddhist ideas from the northwest ceased, and Chinese society became increasingly focused on mercantile activity among the coastal provinces, especially in the southeast.
Although Buddhist scriptural translation had generally been carried out in just a few imperial centers, and only occasionally in provincial cities and alpine retreats, still it would be difficult to overestimate the impact of this change. For centuries the Chinese Buddhist community as a whole had thrived on the constant appearance of new texts, new ideas, and new modes of religious praxis—and now all that simply stopped. It is not enough to point out that Chan's characteristic contrast with the "teachings"—as presented in phrases such as "a separate transmission outside the teachings" and the "unity of Chan and the teachings"—took advantage of this new weakness in its perceived counterpart. Along with the decline in Buddhist scholastic writing in general, there was a vacuum of leadership, a hollowness to the former rhetoric of monastic learning. It is significant that the slogans ranking Chan as an equivalent of the Buddhist doctrinal tradition became widely used just as the tradition of translation and doctrinal study was being profoundly undercut.
The emergence of Chan as the single most dominant Buddhist tradition in China came about, in effect, because it fit so well in the post-Tang world. Certain features of Chan—from the classical style of encounter dialogue to its characteristic institutional patterns—worked effectively within the China that was emerging out of the collapse of the Tang. The remaining chapters in this book are designed to explain what these features are and how this "fit" developed. In chapter 4 we explore the most characteristic dement of the new Chan discourse, the distinctive style of spontaneous encounter dialogue between masters and disciples. In chapter 5 we examine how the new posture of Chan religious identity allowed the school's members to dominate the Chinese monastic institution from the tenth century onward. Finally, in chapter 6 we see how Chan was presented to its members and the public at the very pinnacle of its success in China, during the "climax paradigm" configuration of Chan that emerged during the Song dynasty.
## CHAPTER 4
## The Riddle of Encounter Dialogue
_Who, What, When, and Where?_
### "Classical Chan" and Encounter Dialogue
Consider the following anecdotes:
Amonk asked Zhaozhou, "What was the patriarch's (i.e., Bodhidharma's) intention in coming from the west?" Zhaozhou said, "The cypress tree in the front garden."
A monk asked Zhaozhou, "Does a dog have the Buddha-nature?"
Zhaozhou answered, "No."
A monk asked Dongshan, "What is 'Buddha'?" Dongshan replied, "Three pounds of hemp."
Passages such as these should be readily identified by most readers as quintessentially Chan- or Zen-like. For decades, we have been offered such stories as the primary means by which Chan is presented. This is especially true in the writings of D. T. Suzuki, whose most cherished methodology seems to have been to describe some aspect of Zen as beyond ordinary explanation, then offer a suitably incomprehensible story or two by way of illustration. Obviously, Suzuki's approach captured the imaginations of generations of readers. However, while this approach substantiated Suzuki's authority as one with insider access to the profound truths of the tradition, another result was to increase the confusion in readers' minds. To question such accounts was to admit one did not "get it," to distance oneself even further from the goal of achieving what Suzuki termed the "Zen enlightenment experience."
Let us look at these stories, for the moment, as students of East Asian religious history rather than as prospective practitioners. From this perspective we can make a number of observations about them, none of which is individually very earth-shaking, but which, in sum, lead to some meaningful inferences about the Chan tradition. First, the stories involve figures who lived in the ninth century. (Zhaozhou's dates are 778[?]–897, and Dongshan's 807–69.) This is in part the result of my selection, since there exist numerous similar stories involving figures both before and after. However, in addition to keeping the selection of exchanges limited for simplicity, the choice of these particular anecdotes is based on the general recognition that a new style of dialogue emerged around the beginning of the ninth century. Second, each of the exchanges above involves a question by an anonymous monk and response by a known teacher. Actually, there are many similar dialogues in which the names of both participants are known, especially when the questioner went on to become a famous teacher himself. However, the genre of Chan literature from which these exchanges derive is notable for including, not only the doctrinal pronouncements of great teachers, but also the specific questions of individual students. Third, no contextual clues or stage settings are provided. Here again it would have been possible to choose anecdotes that contain some such information, but in general Chan dialogues of this genre are stated without much effort at contextualization. Our attention as readers is directed automatically at the broader religious implications of the exchanges, whatever they may be.
Fourth, most readers would probably add, the teachers' responses are nonsensical. This certainly seems to be the case for the first dialogue. That is, for a student to ask why the founding patriarch of Chan Buddhism, Bodhidharma, came from India to China was in effect to inquire whether Buddhism had not already existed in China before that date. Or, the purpose was to solicit the teacher's comment on the concept of the transmission of the Buddha-mind, which in standard Chan theory was recognized as a nontransmission. However, the issue of whether or not animals possessed the Buddha-nature, the potentiality or actual presence of enlightenment within the ordinary psychology of illusion, was in fact hotly debated in late-eighth- and ninth-century Chinese Buddhism. The selection of the single answer "no" for emphasis thus represents a collective choice by the members of the Chan tradition. The answer to the third question above, about the meaning of the word "Buddha," has also generally been considered an example of the performative use of Chan illogicality, a nonsensical answer meant to knock the student off his accustomed spot and into a different realm of understanding. That this was the ultimate goal is not really in question, but the answer "three pounds of hemp" was not illogical at all in its original context, merely metaphoric: In the Tang dynasty this was the standard allotment of cloth for a set of monk's robes—a colloquial usage that was forgotten by the Song, leading to amusing errors by the Japanese Zen master D gen (1200–53) and others, who labored to explain Zhaozhou's apparent reference to "three pounds of sesame seeds"! In other words, when asked the meaning of the word "Buddha," Zhaozhou responded, more or less, "A set of monk's robes is all it takes."
Thus, instead of inferring that the teachers' responses are all illogical, a better fourth observation would be that they are all performative utterances based on contemporary culture—"performative" in the sense of being designed to act as catalysts for the students' understanding. In the process of considering these anecdotes, though, we have entered into a new phase in our examination of the Chinese Chan tradition, one in which we must deploy analytical perspectives not required for the earlier phases of our inquiries. The key to understanding how our task has changed in this phase lies in the appreciation of the terms _classical Chan_ and _encounter dialogue._
In contrast to _middle Chan,_ which refers to a specific period of time, from the appearance of the _Platform S tra_ in 780 to the beginning of the Song dynasty in 972, the term _classical Chan_ refers first and foremost to a particular style of behavior displayed by Chan masters in the course of their interactions with students and other masters. Rather than explaining the Dharma in straightforward expository language, such masters are depicted as being more inclined to demonstrate it by means of paradoxical replies and inexplicable counterquestions, gestures and physical demonstrations, and even the shocking and painful tactics of shouts and blows. Precisely when this classical style of religious practice emerged is not clear. Thus to use the term _classical_ in direct reference to events that took place during the middle period, that is, to conflate the two terms, would be to make a naive assumption about Chan history, to accept at face value stories such as those introduced above, which have been transmitted in texts of the Song dynasty. Here I will use _classical Chan_ only in reference to the _image_ of the activities of middle Chan figures as it occurs in the texts of Song-dynasty Chan—just as, with a different nuance, the term _H nay na_ is legitimately used when working solely within the context of Mah y na doctrine, but not in reference to actual Buddhists of either ancient India or modern Southeast Asia. The distinction may seem elusive at first, but it is important to remember that _classical Chan_ refers not to a historical period but to an image seen through historical sources.
Curiously, no clearly stated definition of encounter dialogue appears in the scholarship on Chinese Chan. Part of the reason for this, no doubt, is that the very nature of the subject matter militates against concise definition. Not only is Chan encounter dialogue an unruly topic, difficult to characterize, but one of its primary features is the rejection of simplistic logic. Previous authors, especially those who cite excerpts from encounter dialogue most frequently in their explanations of Chan, have refrained from giving it any clear definition so as to avoid reducing it to a neat set of characterizations and forestall any potential criticism that they do not understand it.
Any working definition of Chan encounter dialogue must include three features. First, it consists of dialogue that occurs in texts identified as containing encounter dialogue, primarily "transmission of the lamp" texts and individual "recorded sayings" or "discourse records" texts. This feature of encounter dialogue is clearly circular in logical form, but it does represent how scholars—and members of the East Asian Chan/S n/Zen/Thien traditions themselves—actually approach the subject. That is, the first step in the definition of encounter dialogue is the identification of a set of Chan texts in which it is found. As explained below, the manner in which encounter dialogue transcription appears in the written record has major implications for how we understand encounter dialogue itself.
Second, Chan encounter dialogue is presented as the written transcription of actual oral dialogues between historically identifiable teachers and students. This is not to say that every participant represented in the literature of encounter dialogue can be identified with full biographical information—far from it. Many of those involved are identified solely as anonymous students present in a given master's training community. In spite of this lack of detail, it is extremely important that the genre presents all such individuals as real and all encounters as having actually occurred.
I must emphasize that these qualities of apparent historical realism are literary effects, characteristics of the genre, and not facts about the dialogues and their participants. One of the ways encounter dialogue texts achieve these effects is through the use of vernacular speech, which gives the impression that the dialogues are presented in precisely the form in which they originally occurred. Whether or not the dialogues ever happened the way they are recorded is of course highly questionable. This is a subject to which we will return soon enough (see the discussion beginning on p. 83); at this point it is important to note that, for encounter dialogue texts as a genre of religious literature, it is of paramount importance that the participants and exchanges are represented as nonfictional realities.
Third, Chan encounter dialogue eschews the straightforward exchange of ideas; it is characterized by various types of logical disjunctions, inexplicable and iconoclastic pronouncements, gestures and physical demonstrations, and even assaultive behavior such as shouts and blows with hand, foot, or stick. The best way to understand such features is as a function of the fundamental mismatch of intention between the students and masters as depicted in these texts. The students are generally depicted as requesting assistance in ascending the path of Buddhist spiritual training toward enlightenment. The masters, for their part, are represented as refusing to accede to their students' naive entreaties, instead deflecting their goal-seeking perspective and attempting to propel them into the realization of their own inherent perfection. This is an oversimplification, of course, and we find numerous cases of interactions that cannot be accommodated within this larger pattern. (It is difficult to define patterns in a genre dedicated to the transcendence of patterns!) Nevertheless, since our task is to understand encounter dialogue exchanges as a religious genre—and not to "solve" the riddles they present as religious practitioners ourselves—it is helpful to recognize this basic intellectual framework.
The encounter dialogue style of religious behavior is well-known in the literature on Chan and Zen in every language, since it is the primary feature of the archetypal image of the Zen master as depicted in both popular and scholarly literature. The central figures by whom this classical Chan is described may include those as early as Bodhidharma and Huineng, and often much later figures from both China and Japan as well, but invariably the focus is on the fabled great masters of the Tang: Mazu Daoyi, Shitou Xiqian, Nanquan Puyuan, Zhaozhou Congshen, and the incomparable paragon, Linji Yixuan.
The hallmark of classical Chan is thus the practice of encounter dialogue. Indeed, the two concepts are so thoroughly interconnected as to be virtually interchangeable: classical Chan refers to those masters who interacted with their students using encounter dialogue, and encounter dialogue is the unique style of interactive teaching utilized by classical Chan masters. It is thus both a natural and customary conclusion to conceive of classical Chan as a phenomenon or set of events occurring at a specific period in the evolution of the religion, lasting roughly from the last few decades of the eighth century until the middle of the tenth. More specifically, encounter dialogue is believed to have first been used by Mazu Daoyi and his disciples—and this interpretation fits like a glove with Mazu's doctrine that all human actions, even those so seemingly trivial as the slightest movements of eye or hand, are the manifestation of the Buddha-nature:
The arising of mental activity, the movement of thought, snapping the fingers or moving the eyes—all actions and activities are the functioning of the entire essence of the Buddha-nature. Since no other kind of functioning exists, greed, anger, and folly, the performance of good and bad actions, and the experiencing of their pleasurable and painful consequences are all, in their entirety, Buddha-nature . . .
But the picture just described is too simple. One of the first hints of the complexities involved is that the written transcriptions of encounter dialogue do not appear until the compilation _of the Anthology of the Patriarchal Hall_ in 952—around a century and a half after encounter dialogue was supposedly first practiced. This text enters the scene, in effect, as a sudden apparition. It is a massive treasure chest of Chan anecdote and repartée, a highly elaborated written demonstration of the Chan genealogical schema. The contents of _the Anthology of the Patriarchal Hall_ are so rich and substantial that they imply a significant accumulation of tradition, and the easiest recourse would be to regard them as the straightforward documentation of classical Chan that they are claimed to be. This is certainly the manner in which the stories in the _Anthology,_ or rather the variants in the far more popular—and effectively authoritative— _Record of the Transmission of the Lamp [Compiled in] the Jingde [Period]_ of 1004, have been used within the meditation hall, from the premodern period down to the present.
The goal of this book, though, is different from that of the meditation hall. Here our purpose is to analyze Chan, not merely recapitulate its innovative style. This chapter on encounter dialogue is thus different from the two preceding ones, in that it does not treat a specific historical phase in the evolution of Chan, but discusses a component or dimension of Chan practice that was of crucial importance in several different periods. In the next chapter we return to questions of temporal development, so that through both chapters we will see, first, how encounter dialogue evolved and, second, how the image of the classical Chan masters of the Tang is the retrospective creation of several generations of their successors. This will help us recognize the point being emphasized here: that the classical style is not a temporally identifiable historical period of Tang-dynasty Chan, but an image that occurs within Five Dynasties and Song-dynasty texts, which was projected retrospectively by the Chan practitioners of those periods onto their predecessors. Even more significant, along the way we will learn how Chan encounter dialogue implies a paradigm of spiritual cultivation that is profoundly different from earlier Chinese Buddhist practice.
### The Story of Mazu's Enlightenment
The following is the traditional account of Mazu Daoyi's enlightenment, drawn from the _Transmission of the Lamp_ entry for Nanyue Huairang (677–744):
During the Kaiyuan period (713–41) there was a monk Daoyi (this is Great Master Mazu), who resided at Transmission of the Dharma Chapel and spent all his time in seated meditation. Understanding him to be very capable, the master [i.e., Huairang] went to him and asked, "Great worthy, what are you trying to do by sitting in meditation?" Daoyi answered, "I am trying to achieve buddhahood." The master then picked up a piece of tile and started rubbing it on a rock in front of the chapel. Daoyi said, "Master, what are you doing?" The master said, "I'm grinding this into a mirror." Daoyi said, "How could you possibly make a mirror by grinding a tile?!" [Huairang replied], "And how could you achieve buddhahood by seated meditation?"
Daoyi said, "How does one do it right?" The master said, "If you're riding a cart that isn't moving, is it right to hit the cart, or is it right to hit the ox?" Daoyi had no response.
The master then said, "Are you training in seated meditation, or training in sitting as a Buddha? If you are training in seated meditation, then meditation is neither seated nor lying down. If you are training in sitting as a Buddha, then the Buddha is without fixed characteristic. You should neither grasp nor forsake the non-abiding Dharma. Your sitting as a Buddha is to kill the Buddha; if you are attached to the characteristic of sitting you have not penetrated the principle involved." When Daoyi heard this manifestation of the teaching he felt as if he had drunk ghee.
This is an archetypal example of encounter dialogue. Mazu, the student, is trying to achieve enlightenment by his own meditative efforts. Rather than simply explain the problem to him, Huairang acts in a way that disturbs Mazu from his misdirected toils, then engages him in a dialogue that inspires the student to penetrate the ultimate paradox of his striving for an impossible goal. This redirection of Mazu's efforts is a direct result of his initial insight. If we imagine this as a primal moment of interaction between a great Chan master and his gifted student, the account can be truly inspirational.
But let's step back for a moment, and consider this account as text, as literary product. This is not to deny its value as religious instruction, but merely to look under the hood, as it were, and see how the engine works.
First, we should look at the earliest version of the anecdote, which occurs in _the Anthology of the Patriarchal Hall:_
Reverend Ma was sitting in a spot, and Reverend Rang took a tile and sat on the rock facing him, rubbing it. Master Ma asked, "What are you doing?" Master [Huairang] said, "I'm rubbing the tile to make a mirror." Master Ma said, "How can you make a mirror by rubbing a tile?" Master [Huairang] said, "If I can't make a mirror by rubbing a tile, how can you achieve buddhahood by sitting in meditation?"
Even in this shorter and more primitive account, we can clearly hear echoes of other legendary events in the meditation tradition. The first of these, of course, involves Vimalak rti scolding riputra for sitting in meditation in the forest. In the _Vimalak rti S tra_ riputra recounts the experience to the Buddha as follows:
I remember once in the past, when I was sitting in repose beneath a tree. At the time Vimalak rti came and said to me, "O riputra, you need not take this sitting [in meditation] to be sitting in repose. Sitting in repose constitutes not manifesting body and mind in the triple world—this is sitting in repose. To neither generate nor extinguish concentration while manifesting the deportments—this is sitting in repose. Not to relinquish the Dharma of enlightenment and yet manifest the affairs of [ordinary] sentient beings—this is sitting in repose. To have the mind neither abide internally nor locate itself externally—this is sitting in repose. To be unmoved by the [sixty-two mistaken] views yet cultivate the thirty-seven factors of enlightenment—this is sitting in repose. Not to eradicate the afflictions yet enter into _nirv a_—this is sitting in repose. Those who are able to sit in this fashion will receive the Buddha's seal of approval." At the time, World-honored One, I simply listened to this explanation in silence and was unable to respond.
And there is an implicit echo of the "mind-verses" of the _Platform S tra_ as well. (See the anecdote beginning on p. 60 above.) "Shenxiu" and "Huineng" discoursed in verse on the subject of dust upon the mirror and whether or not one had to polish it clean, but in the Mazu account the subject matter is played out in action, with the focus not simply on cleaning some preexistent mirror but actually fabricating one out of impossibly inappropriate material. In fact, the text that immediately follows the dialogue with Mazu in the _Anthology of the Patriarchal Hall_ contains other references to the mirror, which implies some sort of unified editorial posture. In comparison to the version in the _Transmission of the Lamp,_ though, this rendition is distinctly primitive: neither location nor time is specified, and there is no follow-up dialogue. All we have is the simple nucleus of the words, with no effort to establish the context. It is in fact the editorial policy of the _Anthology of the Patriarchal Hall_ to require its readers (including teachers who lecture from it) to use their imaginations to provide their own context; in Marshall McLuhan's terms, this is a "hot medium," like radio, that makes readers or listeners actively imagine what is happening, rather than a "cold medium," like television, that gives viewers just enough sensory input to turn off their minds.
This story is often cited as Mazu's enlightenment story, or at least to indicate his identity as Huairang's student, but although this earliest version includes several lines of subsequent dialogue between the two men, it does not explicitly make either of these claims. This story is also used as the justification for Mazu's traditional identification as Huairang's successor, with Huairang simultaneously understood as a successor to the Sixth Patriarch Huineng. However, when we look more closely at the available sources, we see that Mazu studied with other figures as well, and that Huairang's connection with certain Northern school figures is much more substantial than his problematic connection with Huineng.
In the case of Huairang, the little that is known about his biography definitely undermines the historicity of the filiation between him and Huineng. First, Huairang's epitaph was written in the year 815, some seventy years after his death, at the request of two of Mazu's disciples, so it can hardly be used to suggest that the connection between Huineng and Huairang was historical rather than legendary. In addition, the paucity of detail concerning Huairang's biography—he is said to have been a mountain practitioner who did not "open the Dharma" to others—suggests that he was historically insignificant. And, needless to say, nothing like the story introduced above occurs in the epitaph. In fact, the _Transmissions of Treasure Grove [Temple] (Baolin zhuan),_ the Hongzhou school's important contribution to the "transmission of the lamp" genre of Chan literature, written about 801, describes Huairang's enlightenment as having been gained under the guidance of the Northern school monk Lao'an. Actually, none of the men traditionally recognized as Huineng's most important successors—Huairang, Qingyuan, Yongjia Xuanjue, and Nanyang Huizhong—are mentioned in the Dunhuang version of the _Platform S tra._
And Huairang was hardly Mazu's only religious influence. He became a monk under a second-generation successor to Hongren named Chuji (also known as Reverend Tang; 648–734, 650–732, or 669–736) of Sichuan. Mazu was probably also acquainted with a charismatic Korean monk named Musang (Chinese: Wuxiang; also known as Reverend Kim [Chinese: Jin]; 684–762). And, when Mazu left Sichuan about 735, he traveled to Jingzhou, where he practiced meditation before going to Nanyue. We could of course try to track down the religious identities and probable teachings of all these men, but to do so would only complicate the picture even further. Ultimately, our main conclusion would be that Mazu had a typically variegated life of religious training, so that even if the interaction between Huairang and Mazu was historical in some sense—and it would be rash to deny this possibility out of hand—this would not be enough to make Mazu Huairang's successor, let alone a direct second-generation successor to Huineng.
The point to be made here is that, from whatever may have happened during Mazu's religious training, from some unknown point in time the Chan community developed this image of an encounter between him and Huairang. Whatever did or did not happen, the news of that encounter was dramatized and circulated in oral and/or written form. What we have in the _Anthology of the Patriarchal Hall_ is something like the core of the story, with the reader, listener—or perhaps the teacher—left to supply the details. As Timothy Barrett has suggested, this process of editorial evolution resembles nothing so much as the circulation of joke books at roughly the same time. As with the formulaic notation of the _Five Skillful Means,_ which seems to have provided the liturgical skeleton on which Northern school teachers could superimpose their own flourishes and interpretations, the written transcriptions of Chan encounter dialogue were prepared as skeletal notations upon which teachers and students could improvise. The emergence of this genre of literature, though, required a shared conception of Buddhist spiritual practice, some of whose elements we have seen in the preceding pages.
### The Eightfold Path to the Emergence of Transcribed Encounter Dialogue
Mazu's Hongzhou school displays an interesting pattern of geographical and stylistic growth, and I take up those matters in the next chapter. Here I want to present what we know about the background and historical emergence of encounter dialogue as a rhetorical pattern of Chan practice. I avoid considering doctrinal issues, such as the concepts of _ nyat , praj ,_ the impact of M dhyamika dialectic, and so on; these general background conditions to the emergence of Chan encounter dialogue have been discussed widely in previous writings. Nor do I consider more distant background elements within Chinese culture, in particular the frequent use of the dialogic mode of exposition from Confucius and Zhuangzi onward. There can be no doubt that Chan dialogue reverberates with the rhetorical wit and humanistic perspective (or, rather, the rejection of all fixed perspectives) personified so exquisitely in the _Zhuangzi._ Indeed, these features and the notion of a style of virtuosic concentration on the activity at hand represent the primary legacy Chan inherited from the native tradition of philosophical Daoism. However, the simple fact of that inheritance is not sufficient to explain why Chan encounter dialogue emerged when it did, more than a millennium after the _Zhuangzi,_ and in a social environment that did not even exist in Zhuangzi's day, the Chinese Buddhist monastic community.
Instead, I suggest below that Chan had to develop a rationale for _socially oriented_ practice prior to, or perhaps simultaneously with, the perfection of oral dialogue techniques. The following is a set of characteristics by which the social dimensions of this new type of religious interaction may be understood. The following enumeration must still be considered a provisional explanation; obviously, none of these characteristics is shared throughout the entire early Chan movement, and there are almost certainly others not yet identified.
#### THE IMAGE OF THE CHAN MASTER RESPONDING SPONTANEOUSLY TO HIS STUDENTS
Early Chan teachers are frequently described as having special abilities of teaching, which they exercised in an unstructured moment-to-moment manner. Some of the earliest known expressions concern Hongren, the central figure of the East Mountain teaching and so-called fifth patriarch. Hongren forms the original nucleus of the hagiographical persona of the unlettered sage, in being described as spending his days in meditation and his nights tending the monastery cattle. As soon as he was appointed successor to Daoxin, the previously silent Hongren was immediately able to understand the problems of his students and teach them with a fluid, spontaneous style that combined an appreciation of the ultimate truth with complete expertise in the expediencies of religious practice. Faru, who was unique among Hongren's students for spending so many years with the master, is described as having unique abilities in his interactions with his students, so that he could remonstrate with them strongly without incurring resentment: his anger is described as being like two empty boats hitting each other in the middle of a lake, which would make a hollow sound signifying an absence of attachment or resistance. Several members of the Northern school were also the subjects of anecdotes depicting their occult charisma, although the primary examples of this religious type are of course Bodhidharma and Huineng. Huineng in particular—as depicted in various works, not just the Dunhuang version of the _Platform S tra—_ is a figure who responds to situations with remarkable élan and spiritual brilliance, making mysteriously profound pronouncements and posing miraculous challenges to individual seekers. In spite of the fact that he is supposedly quite untutored in the literary arts, he also composes insightful poetry. In all these cases, Huineng is represented as enlightened, not by any doctrine he pronounces or essay he produces, but rather in his interactions with the figures around him.
#### "QUESTIONS ABOUT THINGS" IN THE NORTHERN SCHOOL
How did early Chan teachers interact with their students? The hagiographical images of Hongren and Huineng are not our only clues: we do not know how the students responded, but at least we have some evidence for the types of questions early Chan masters placed before them. An important early-eighth-century "transmission of the lamp" text generated by the Northern school, the _Record of the Masters and Disciples of the La k vat ra,_ contains an intriguing set of rhetorical questions and short doctrinal admonitions, which it refers to as "questions about things" (literally, "pointing at things and asking the meanings"). Such questions and admonitions are attributed to several of the early masters, as shown in the following examples.
The Great Master Bodhidharma also pointed at things and inquired of their meaning, simply pointing at a thing and calling out: "What is that?" He asked about a number of things, switching their names around and asking about them again differently.
He would also say: "Clouds and mists in the sky are never able to defile space. However, they can shade space so that the sun cannot become bright and pure. . . ."
The Great Master Hongren said: "There is a single little house filled with crap and weeds and dirt—what is it?"
He also said: "If you sweep out all the crap and weeds and dirt and clean it all up so there is not a single thing left inside, then what is it?" . . .
Also, when he saw someone light a lamp or perform any ordinary activity, he would always say: "Is this person dreaming or under a spell?" Or he would say: "Not making and not doing, these things are all the great _parinirv a."_
He also said: "When you are actually sitting in meditation inside the monastery, is there another of you sitting in meditation in the forest? Can all the mud, wood, tiles, and rocks also sit in meditation? Can mud, wood, tiles, and rocks also see forms and hear sounds, or put on robes and carry a begging bowl?"
Shenxiu also said: "Is this a mind that exists? What kind of mind is the mind?"
He also said: "When you see form, does form exist? What kind of form is form?"
He also said: "You hear the sound of a bell that is struck. Does the sound exist when the bell is struck? Before it is struck? What kind of sound is sound?" He also said: "Does the sound of a bell that is struck only exist within the monastery, or does the bell's sound also exist throughout the universe in all the ten directions?"
Also, seeing a bird fly by, he asked: "What is that?"
He also said: "Can you sit in meditation on the tip of a tree's hanging branch?"
He also said: "The _Nirv a S tra_ says, 'The Bodhisattva with the Limitless Body came from the East.' If the bodhisattva's body was limitless in size, how could he have come from the East? Why did he not come from the West, South, or North? Or is this impossible?"
At least one scholar has suggested that these "questions about things" resemble the "precedents" or _gong'ans (k an)_ of later Chan. Obviously, we cannot jump immediately from these questions to the "precedent anthologies" of the eleventh century and beyond; we must instead take into account the intervening efflorescence of encounter dialogue. However, it _is_ reasonable to infer that these represent something like the same sort of questions posed by masters to students in that later genre. In contrast to encounter dialogue, here we have only one side, the masters' questions; in contrast to precedent anthologies, there is no context or literary structure to explain how such questions were intended. In addition, based on the content of some of these questions, we may infer that Northern school masters were involved in the extension of spiritual cultivation to all the activities of daily life.
#### THE "CHAN" STYLE OF EXPLANATION IN EIGHTH-CENTURY SOURCES
In addition to these "questions about things," there are various hints in texts from this period and slightly later of what seems like the idiosyncratically "Chan" style of discourse glorified in the later tradition. It is not always clear, to be sure, that one unified style of explanation is indicated, but the references are enough to suggest that something interesting is being reported, but not yet recorded in full.
The central figure in this respect is Shenxiu, who had a special role as "Chan commentator" on the meaning of the _s tras_ as translated by ik nanda during the first few years of the eighth century. One longs to know what the "Chan meaning" of any scriptural term might be, but no doubt Shenxiu's style of interpretation was largely identical to his style of metaphor introduced in the previous chapter (see p. 49). Another clue to the prevalence of unconventional "Chan-style" dialogue occurs in the epitaph for Yifu (661–736), one of Shenxiu's most important successors, in which the author recounts that he and another literatus collected the departed master's sayings as they were remembered by his students. The two men were apparently unable to write down all of those sayings, presumably because of their great number. Even though they recognized the value of these sayings, neither of their epitaphs for Yifu contains anything that might correspond to the subject of such a collection. Although the convention of disciples collecting a master's sayings is known from the earliest days of Chan (witness the material associated with _Treatise on the Two Entrances and Four Practices_ and _Treatise on the Essentials of Cultivating the Mind,_ the latter of which declares explicitly that it was compiled by Hongren's students), the statements associated with Yifu imply that a special kind of pronouncement was involved.
As time went on, the epitaphs of members of the Northern school and other figures important in the development of Chan began to include precisely this sort of material. For example, note the following exchange and commentary from the epitaph for Puji's student Fayun (d. 766):
"Has the Buddha's teaching been transmitted to you?"
"I have a sandalwood image of the Buddha to which I pay reverence."
This reply was profound yet brief, and those listening felt chills of loneliness. The day after [the questioner, a prominent official,] left, Fayun died without illness while sitting cross-legged on his chair.
After all the hyperbole about Shenxiu's being equivalent to a buddha and Puji's being the religious teacher of the universe (themes stated in documents from the first half of the eighth century as part of the Northern school's campaign for public recognition), it is perfectly natural to find a slightly later master deflating the idea of the transmission altogether.
The epitaph for Huizhen (673–751), who was more closely affiliated with the Tiantai and Vinaya schools than with Chan, includes a more explicit reference to what seems like encounter dialogue, along with several examples:
"When people do not understand, I use the Chan style of teaching."
Question: "Are not the teachings of the Southern and Northern schools different?"
Answer: "Outside the gates of both houses is a road to everlasting peace."
Question: "Do the results of religious practice vary according to the extent of realization?"
Answer: "When a drop of water falls from the cliff, it knows the morning sea."
Question: "How can one who is without faith achieve self-motivation in spiritual endeavor?"
Answer: "When the baby's throat is closed (i.e., when choking), the mother yells to frighten it loose. Great compassion is unconditioned, but it can also make a student whimper."
A confirmed skeptic might suggest that Huizhen is merely answering in easily understood metaphors, rather than in some genuinely new "Chan" style of teaching. If this is the case, then we must infer that a new type of metaphorical or metagogic usage became the vogue in Chan Buddhism during the second half of the eighth century, for such usage is also apparent in the biographies of Faqin (714–92) and Xuanlang (673–754), well-known representatives of the Oxhead and Tiantai schools, respectively. The _Transmissions of Eminent Monks [Compiled During the] Song [Dynasty] (Song gaoseng zhuan)_ and _Transmission of the Lamp_ contain several examples of encounter dialogue involving Northern school figures, although of course these exchanges may be later fabrications. The practice of this prototypic encounter dialogue may have had a much wider currency than the extant body of literature suggests, and the members of the Northern school may have been only the first to legitimize its use within the Chan tradition.
#### DOCTRINAL BASES FOR THE SOCIAL ORIENTATION OF EARLY CHAN PRACTICE
What were early Chan practitioners doing when using paradoxical interrogation, dialogue, and interactive training methods? Since they do not tell us explicitly, we must turn to the voluminous writings they did bequeath to us and explore them for clues. Obvious methodological problems arise in this approach, which involves interpretive leaps and projections, but given the present state of the evidence, we have no other recourse.
One of the most important features of the _Treatise on the Two Entrancesand Four Practices_ is its bimodal structure, which consists of one abstract and one active "entrance" or "access" to accomplishment of the Dharma. Although one can read this text in several different ways, it is both appropriate and useful to take the two entrances as introvertive and extro-vertive, respectively. That is, the "entrance of principle" refers to interior cultivation, mental practice undertaken deep within the individual's psyche, and the "entrance of practice" refers to practice undertaken actively and in interaction with the world.
Other than dialogue per se, the other important question to be considered here is the extent to which the doctrinal formulations of the Northern school's _Five Skillful Means_ may have provided justification for the emergence of encounter dialogue. Here I am not thinking of encounter dialogue as an oral practice so much as a _social_ practice. That is, is there anything in the _Five Skillful Means_ that provides justification for the outward, social dimension of Chan religious practice?
In fact, there is a basis for answering this question in the affirmative. The key passage is the following:
Bodhisattvas know the fundamental motionlessness of the six senses, their internal illumination being distinct and their external functions being autonomous. This is the true and constant motionlessness of the Mah y na.
Question: What do "internal illumination being distinct" and "external functions being autonomous" mean?
Answer: Fundamental wisdom is "internal illumination being distinct." Successive wisdom is "external functions being autonomous."
Question: What are fundamental wisdom and successive wisdom?
Answer: Because one first realizes the characteristic of the transcendence of the body and mind, this is fundamental wisdom. The autonomous [quality of] knowing and perception and the nondefilement [associated with the enlightened state] are successive wisdom. . . . If realization [of the transcendence of body and mind] were not first, then knowing and perception would be completely defiled. Know clearly that the autonomous [spontaneity of] knowing and perception is attained after that realization and is called successive wisdom.
When the mind does not activate on the basis of the eye's perception of form, this is fundamental wisdom. The autonomous [spontaneity of] perception is successive wisdom. When the mind does not activate on the basis of the ear's hearing of sounds, this is fundamental wisdom. The autonomous [spontaneity of] hearing is successive wisdom. The nose, tongue, body, and consciousness are also the same. With the fundamental and successive [wisdoms], the locations (i.e., sensory capacities and realms of sensory data) are distinct, the locations are emancipated. The senses do not activate, and the realizations are pure. When successive moments of mental [existence] are non-activating, the senses are sagely (i.e., characterized by the enlightened mentation of the buddhas).
Although the terminology used here is no doubt unfamiliar to most readers, it is relatively simple to unpack. "Fundamental wisdom" refers to the first moment of enlightenment, when the mind attains perfect clarity. At this point one is said to have transcended both body and mind, that is, to have gone beyond all ordinary distinctions of physical and mental reality. Although the text never makes the connection, it is reasonable to understand this basic attainment of mental clarity according to the entrance of principle in the text attributed to Bodhidharma. The rhetoric of innate entitlement is not used in this specific location, but elsewhere the _Five Skillful Means_ discusses the "enlightened mind" in a fashion that implies such an understanding.
Here "successive wisdom" refers to what happens immediately after that first moment of "fundamental wisdom," both in the very next and all succeeding moments. Other texts might have deconstructed the artificiality of this moment-by-moment sequence, to point out that true realization happens all at once, but the _Five Skillful Means_ is firm in its dualism. It is not precisely clear what it would mean in real life for one's various sensory capacities to be non-activating and autonomously spontaneous in their functioning, not to mention the distinct identity and emancipated quality of both sensory capacities and the realms of sensory experience. However, to continue with the analogy to the _Treatise on the Two Entrances and Four Practices,_ we may note that these various qualities pertain to how the practitioner interacts with the world at large. In terms of religious self-cultivation, this attitude would involve a certain level of forbearance toward all one's circumstantial conditions; in terms of the on-going activities of an enlightened master, the result would be a style of perfect responsiveness to the needs of one's students.
Scattered throughout the same section of the _Five Skillful Means_ we find various statements involving similar dyads referring to inner realization and outward-directed activity:
If the mind does not activate, the mind is suchlike. If form does not activate, form is suchlike. Since the mind is suchlike, the mind is emancipated. Since form is suchlike, form is emancipated. Since mind and form both transcend [thoughts], there is not a single thing.
The transcendence of mind is enlightenment of self, with no dependence on the five senses. The transcendence of form is enlightenment of others, with no dependence on the five types of sensory data. The transcendence of both mind and form is to have one's practice of enlightenment perfect and complete and is equivalent to the universally "same" _dharmak ya_ of the Tath gata.
The transcendence of thought is the essence, and the perceptive faculties are the function. Serenity is the essence, and illumination is the function. "Serene but always functioning; functioning but always serene." Serene but always functioning—this is the absolute corresponding to phenomena. Functioning but always serene—this is phenomena corresponding to the absolute. Serene yet always functioning—this is form corresponding to emptiness. Functioning yet always serene—this is emptiness corresponding to form. . . .
Serenity is unfolding; illumination is constriction (lit., "rolling up"). Unfolded, it expands throughout the _dharmadh tu._ Constricted, it is incorporated in the tip of a hair. Its expression [outward] and incorporation [inward] distinct, the divine function is autonomous.
The meaning of enlightenment is that the essence of the mind transcends thoughts. Transcending the characteristic of craving, it is equivalent to the realm of space, which pervades everywhere. This is called enlightenment of self. Transcending the characteristic of anger, it is equivalent to the realm of space, which pervades everywhere. This is called enlightenment of others. Transcending the characteristic of stupidity, it is equivalent to the realm of space, which pervades everywhere. The single characteristic of the _dharmadh tu_ is the universally "same" _dharmak ya_ of the Tath gata. This is called complete enlightenment.
These examples, which could easily be supplemented from later sections of the _Five Skillful Means_ and other works, reveal the basic Northern school concern for describing not only how one understands the abstract truth of the Buddhadharma, but also how one puts it into practice on behalf of sentient beings. Although the specific expressions are new, this bimodal structure is certainly indebted to the _Treatise on the Two Entrances and Four Practices_ attributed to Bodhidharma and may be taken as a basic characteristic of early Chan Buddhism.
It would be more convenient for our purposes, I suppose, if this bimodal structure explicitly involved masters and students, and if it stated clearly that one was first to become enlightened oneself and then inspire the enlightenment of others. Instead, as with all Chan literature at this time (not to mention the texts of other schools), the aspiring student is still invisible, and from the moment of successive wisdom onward, the recipients of the enlightened master's grace are anonymous sentient beings. However, the emphasis on the importance of activity in the social or interpersonal realm (which is implicitly seen as temporally subsequent but equal in value terms) is firmly established with these formulations.
#### THE USE OF RITUALIZED DIALOGUE BETWEEN TEACHERS AND STUDENTS
The mechanical formulations given above are not the only interesting feature of the _Five Skillful Means._ The text seems to have been a set of teacher's notes for holding initiation and training meetings according to an approved Northern school program, in which context it includes several examples of ritualized dialogue. We have already seen one example of this above (see the quotations beginning on p. 52). Here is a second:
The preceptor strikes the wooden signal-board and asks: Do you hear the sound?
Answer: _We hear._
Question: What is this "hearing" like?
Answer: _Hearing is motionless._
Question: What is the transcendence of thought?
Answer: _The transcendence of thought is motionless._
This motionlessness is to develop the skillful means of sagacity out of meditation. This is to open the gate of sagacity. Hearing is sagacity. This skillful means can not only develop sagacity, but also make one's meditation correct. To achieve this motionlessness is to open the gate of wisdom, to attain wisdom. This is called the opening of the gates of wisdom and sagacity.
Here we find transcribed segments of ritual dialogue from a specific Northern school doctrinal context. When looking for antecedents for transcribed dialogues in early Chan texts, we should not be misled by preconceptions about the original spontaneity of such dialogues and thus overlook this type of material.
The question is, to what extent did encounter dialogue grow out of a monastic training and ritual context in which students responded to monkish ritual celebrants in a thoroughly formalized manner? Elsewhere in the _Five Skillful Means_ are other portions of this catechistic ritual, which demonstrate the same form of scripted recitation-and-response pattern. This material skillfully weaves Northern school doctrine into an intriguing mix of ritualized initiation, teaching catechism, and guided meditation practice. Here I would like to focus on the following possible reading of the implications of this material: that Chan encounter dialogue derived not (or, perhaps, not solely) out of _spontaneous_ oral exchanges but (perhaps only in part) out of _ritualized_ exchanges. Given arguments already made by other scholars that spontaneity is merely "inscribed" within the heavily ritualized context of Song-dynasty Chan, this interpretation allows us to wipe out the distinction between the "classical" age of Tang-dynasty Chan when encounter dialogue was spontaneous and the subsequent ritualization of dialogue within Song-dynasty Chan. At the least, the examples of transcribed dialogue introduced above should break us loose from the preconception of "event" and suggest we look elsewhere for the origins of encounter dialogue as "text."
#### THE WIDESPREAD USE OF ANECDOTE AND DIALOGUE IN TEACHING
One factor that should not be overlooked is the widespread tendency within the developing Chan movement to use anecdotal material and dialogue transcriptions for teaching purposes. One could chart the anecdotal content of Chan literature as a sharply ascending curve. From Bodhidharma's treatise through the texts of early and middle Chan, there is a treasure trove of anecdotes, parables, metaphors, dramatizations, and other narrative material that becomes increasingly central with time.
The most important individual contributor to this dimension of Chan was Shenhui. Shenhui's activities, ideas, and rhetorical style transformed Chinese Chan. Whatever the doctrinal significance of his teaching of sudden enlightenment, whatever the factionalist impact of his outspoken criticism of the Northern school, one of the ways in which he changed Chan was in the extreme caution he made his colleagues feel about describing their doctrinal formulations. I have labeled this impact the standard of "rhetorical purity," which mitigated against any expression using dualistic or gradualistic formats. That is, even though the long-range impact of nondualism may have been ultimately liberating, Shenhui's vigorous attack on the dualism and gradualism of Northern school teachings must have had a chilling effect on other teachers.
Simultaneously, Shenhui was a master storyteller and public speaker. Many of the most famous stories of Chan appear first in the transcriptions of his sermons and lectures: Bodhidharma and Emperor Wu, Bodhidharma and Huike—but not, curiously enough, many stories about his own teacher Huineng. There is also a substantial amount of transcribed dialogue within the Shenhui corpus, either between Shenhui and his designated Northern school foil Chongyuan or between him and various famous laymen of his day. There is a palpable sense of fictional creativity here, such that some of the dialogues with famous laymen may well have been made up out of whole cloth. On the other hand, the dialogues do not quite conform to our expectations of encounter dialogue, in that they are too clearly structured and have too much of a logical pattern to represent genuinely spontaneous exchanges. Every historical figure is transitional in some way, and although he was apparently not a practitioner of full-fledged encounter dialogue himself, Shenhui's career pushed the Chan tradition forward in the use of anecdote.
#### THE FABRICATION OF ENLIGHTENMENT NARRATIVES
Another characteristic of early Chan writings is the tendency to compose fictionalized accounts of enlightenment experiences. Let me discuss several examples of this tendency before turning, in the next section, to the case of Huineng.
We have already encountered the Oxhead-school text _Treatise on the Transcendence of Cognition_ (see the passage quoted beginning on p. 58), which is the prime example of an openly fictional dramatization of a Chan master-student encounter. There are other examples of fictionalized enlightenment narratives in eighth-century Chan literature: a pair of texts, the _Treatise on the True Principle_ and _Essential Determination,_ which share the same rhetorical structure. In each case, a single proponent of Buddhist spiritual cultivation is depicted as both enlightened Chan master and sincere lay seeker. That is, the author depicts himself as both asking and answering questions concerning spiritual cultivation, in his dual identities as monk and layman. I have always been amused by the openings of these texts: after introducing himself as both teacher and student, when the first question is posed, by himself in the guise of the student, the author switches to his guise as teacher to praise it as the most profound inquiry he's ever received in all his years as a monk!
The narratives found in the _Treatise on the Transcendence of Cognition, Treatise on the True Principle,_ and _Essential Determination_ are manifestly fictional, but they must somehow have modeled ideal teacher/student interactions and may have resembled actual exchanges that took place between living meditation masters and practitioners. Their authors must have had some knowledge of such encounters—either by direct participation or monastic hearsay—in order to generate their literary images. Rather than speculating on the precise nature of such events, the point to emphasize here is that these texts represent an innovative use of text in the Chan tradition.
More distantly, of course, the Chinese Buddhist apologetic tradition has a certain history of creating fictitious authorities to serve as rhetorical tools. Mouzi, a patently fictional character created to explain the validity of Buddhism for China through references to indigenous Chinese culture, is clearly the best example of this. Here, though, the goal is not to convince a skeptical reader of the validity of Buddhism for China, but to model Chan practice and enlightenment for would-be practitioners.
#### THE GENEALOGICAL STRUCTURE OF CHAN DIALOGUE
Here let me add one other point about the example of Huineng, based not on the fictionality of the story per se but instead on the character of the protagonist. The following is a famous passage from the _Lotus S tra:_
Then the daughter of the dragon king presented to the Buddha a jewel worth the great manifold cosmos, and the Buddha accepted it. The daughter of the dragon king spoke to the Bodhisattva Praj k a and the noble riputra saying:
"I offered a jewel and the Bhagavat accepted it. Was that done quickly or not?"
They answered saying: "It was done extremely quickly!"
The daughter said: "Through your transcendent powers watch me become a Buddha even more quickly than that!"
Then the assembly there all saw the daughter of the dragon king instantly transform into a man, perfect the bodhisattva practices, go to the Vimal world in the south, sit on a jeweled lotus flower and attain highest complete enlightenment, become endowed with the thirty-two marks and eighty excellent characteristics, and expound the true Dharma universally for the sake of all sentient beings in the ten directions.
I suggest that there is a profound similarity between the story of Huineng and that of the dragon king's daughter in the _Lotus S tra._ Consider their total lack of the conventional accoutrements of spiritually gifted persons. The dragon king's daughter was female, nonhuman (although of high nonhuman birth), and underage—yet in a single moment she was able to transform herself into a male, pass through all the trials and tribulations expected of bodhisattva practitioners, and achieve perfect enlightenment. For his part, Huineng was illiterate, from the very fringes of civilization in the far south, lowborn (although his father had been an official, albeit a banished one), and not even a monk—yet his intuitive genius qualified him to be selected as the sixth patriarch.
In the story of Huineng we find the last key to the emergence of encounter dialogue transcriptions. The problem was not whether or not such dialogues were actually occurring between masters and students, and if so how and to what extent. Rather, the problem was the reluctance to transcribe what may have been virtually an everyday occurrence in the back rooms of China's monastic compounds. There had to be some epistemic change that made it acceptable to transcribe, not only the words of the gifted and famous master, but those of the student as well. The example of Huineng may have been a significant factor in generating this epistemic change, but the time was still not at hand.
Encounter dialogue is generally believed to have flourished initially in the faction of Mazu Daoyi, which is known as the Hongzhou school. Mazu and his disciples are depicted in Chan records as engaging in spontaneous repartée in what is almost a barnyard atmosphere of agricultural labor and other daily tasks. There are enough dialogues concerning a large enough number of figures that it would seem heresy to suggest that nothing of the sort "really" happened, that the encounters were all "fictional." I will certainly not go that far here, but we cannot avoid a certain problem, already introduced above: Whereas the encounters involving Mazu and his disciples are supposed to have taken place in the latter part of the eighth century and beginning of the ninth, they are not found in transcribed form until the year 952, with the appearance _of the Anthology of the Patriarchal Hall._
We do have a much earlier text from the Hongzhou school, the _Transmissions of Treasure Grove._ Only certain parts of this text are extant, and scholars have generally assumed that the lost portions (which were devoted in part to Mazu and his immediate disciples) must have been incorporated into, and thus were not substantially different from, the corresponding sections _of the Anthology of the Patriarchal Hall._ Unfortunately, this assumption is untenable, for the simple reason that the extant portions of the _Transmissions of Treasure Grove_ do not contain encounter dialogue transcriptions. There is a great deal of dialogue transcribed in this text, virtually all of which is fictionalized representation of enlightened masters. However, none of this dialogue has the same lively feel as the exchanges of the _Anthology of the Patriarchal Hall._
There is one feature of the _Transmissions of Treasure Grove,_ though, that I believe to be of crucial importance: the rigid narrative structure of the text. This text describes the lives, and to a lesser extent the teachings, of the Chan patriarchs from kyamuni through Bodhidharma to Mazu, and in each case the patriarch in question is described twice, first as a gifted student discovered by the current patriarch and second as a fully vested patriarch out searching for his own successor. It is curious that in no case (with a partial exception in the account of Huike) is the enlightenment experience of the patriarch in question described; we have only the "before" and "after" images, not any reference to or depiction of what we would think to be the most crucial event in the entire process. What is central for our purposes, though, is the great emphasis placed on the patriarchs as students. That is, this text creates a structural symmetry, even an implied parity, between the student as incipient patriarch and the patriarch as realized student.
This structural parity may well have played a role in making the transcription of encounter dialogue possible—that is, in making the transcription of _both sides_ of encounter dialogue exchanges possible. However, this was not yet possible when the _Transmissions of Treasure Grove_ was compiled in 801, and the reticence of this text to describe enlightenment experiences may imply that it was used for popular teaching in the spread of Buddhism throughout the newly developing areas of Jiangxi, rather than for training within the context of the monastic meditation hall.
The Chan genealogical model requires some form of mutual interaction, some confrontation, between teacher and student. In some cases, as in the account of Mazu sitting in meditation introduced at the beginning of this chapter, the student implicitly represents a "Chinese _m rga_ paradigm" _(m rga_ is the Sanskrit word for the spiritual path from ignorance to enlightenment), and the teacher responds in terms that force the student to reorient himself in terms of the "encounter paradigm" of spiritual cultivation. That is, the student thinks in terms of what was conventionally referred to in the post-Shenhui world as the gradual teaching, in which he moves progressively through a series of exercises and stages toward the goal of enlightenment. Here the spiritual quest is somewhat like a board game such as "chutes and ladders," in which each player moves by rolls of the dice from the bottom of the board to the top. (In this particular game for young children, certain positions on the board send the player down a slide or up a set of steps, either losing or gaining multiple spaces in the process.) In the spiritual quest, of course, movement of one's token is accomplished not by rolls of the dice but by mastering different spiritual techniques. The focus of each player is on the progress of his or her own token, which requires skills that are akin to those of the mechanic or craftsman.
In Chan practice the teacher reacts to such assumptions by forcing the student into dialogue, into engaged interaction. Thus the unipolar game-piece style of practice is changed into a bipolar encounter, in which a sudden insight can be achieved by a fundamental change of perspective. Since preconceptions are sturdy things, this transformation is easier described than achieved, of course, but with the advent of the Chan school, the model under which real spiritual progress is made shifts to a bipolar framework of interpersonal collaboration. Rather than moving one's piece across a gameboard, this sort of bipolar interaction is less rule-driven and more intuitive, or at least more open to creative innovation—like learning how to dance or an initiation into lovemaking.
Actually, the best metaphor might be the Chinese game of _weiqi,_ better known as the Japanese game _go_ , or the ancient and medieval Chinese game of _liubo_. In contrast to the board games of Indian culture, in which one moves a single piece along a path of spaces from bottom to top of the board, in the Chinese games one places multiple pieces, not on spaces, but on the intersections between horizontal and vertical lines. While the Indian game can be played by any number of players, one or more, the Chinese game is for a pair of opponents—a duel, we might say. The two opponents compete for control of territory, and the very simple set of rules governing where pieces may and may not be played allows for a very sophisticated calculus of risk and benefit.
Of course, Chan is not precisely like the game of _go_ —like all metaphors, this one is empowered by the approximation of its fit, not its precise match. However, the classical examples of Chan master-student interaction are indeed subject to forms of analysis similar to those applicable for _go_ , and they do exhibit distinctive and complex types of patterning. The goal of these patternings, however, is to depict masters responding to students in ways that appear to be unstructured and creative, spontaneous and immediate. To put it differently, the collision between the Chinese _m rga_ and encounter paradigms that takes place in the context of every master-student interaction is the real echo of the gradual/sudden distinction in post-Shenhui Chan.
To understand the emergence and functions of encounter dialogue within the Chan tradition, we must consider a number of different factors and their complex concatenation. Even more, we must also be aware of the conjunction of entirely different realms of culture, for example, the institution of the monastery, the structure of oral discourse, and the creation of a new genre of religious literature.
Encounter dialogue emerges from a style of oral exchange that seems to have been practiced within the "back rooms" of the meditation hall, abbot's quarters, and other private areas of the monastery, from perhaps as early as Shenxiu's residence at Jade Spring Temple in the last quarter of the seventh century. Perhaps it was a style of interaction and instruction already known at East Mountain, and perhaps it was even in part a legacy of Tiantai Zhiyi's earlier residence at Jade Spring Temple. Perhaps it was practiced more widely throughout the Chinese Buddhist monastic institution as a whole, in meditation halls and training facilities of various styles and configurations. Whatever its original currency, it was originally restricted to the back rooms, and not presented in writing at first. It is entirely likely that some of the excitement that Shenxiu, Lao'an, and other Northern school figures attracted in Chang'an and Luoyang at the beginning of the eighth century was due to their revelation of this backroom style in elegant public occasions. Whether this was the case or not, until the appearance of _the Anthology of the Patriarchal Hall_ in 952 the written texts of Chan demonstrate a palpable reluctance, even an inability, to include the words of mere students. It was as if, prior to 952, only the words of the celebrated masters and their imperial or literati interlocutors could be transcribed in the formal mode of literary Chinese texts. After 952, though, the situation changed dramatically. From this point on it becomes not only allowable but expected, even required, to include the words of students in Chan texts. Indeed, the identity of Chan masters cannot be seen any other way except through their interactions with often anonymous students. We examine _the Anthology of the Patriarchal Hall_ in more detail in the next chapter, but at the moment it is useful to notice what a major social and conceptual transformation its appearance represents.
What we have before us, of course, are texts, the written transmutation of an oral tradition. How significant was the shift from oral to written medium? Most readers approach Chan recorded sayings literature quite naively, taking the words as simple and basically accurate transcriptions of what was actually said during the event depicted. But the impression of vivid immediacy that we gain through reading these texts is primarily a literary effect, a direct result of their rhetorical style. In fact, Chan dialogues have gone through a number of stages before being presented to us in their present form:
1. _Initial transcription:_ The act of transcribing spoken Chinese into written form should not be taken for granted, but rather represents the first step—actually a certain type of translation—in a complex process of oral-to-written transformation. On the basis of historical linguistics, we know that all transcriptions were done into a standard form of colloquial Chinese that was based on the spoken dialect then current at the capital of Chang'an. The ability to render this standard colloquial form was only achieved with some difficulty, and not only were actual vocal utterances cleaned of the usual verbal "noise" that characterizes actual speech and simplified for written use, but any dialect peculiarities were omitted in the translation into the medieval Chang'an standard. Thus, even when southerners are depicted talking to southerners, their dialogues are shown in the form of Chang'an Chinese even though the texts could just as well have used southern language forms.
2. _Circulation, evaluation, and selection:_ We have a number of examples where the same stories are recounted with different actors, or where similar stories imply some process of internal development. In other cases a student will ask a teacher about a dialogue or pronouncement by some other master. In other words, these stories were clearly passed around and subjected to repeated reevaluations and modifications. This seems to have occurred in a complex environment of both oral and written transmission, with the reputations of different masters growing or fading based on the ability of their circulated dialogues to attract interest from both students and other teachers.
3. _Editorial modification:_ As discussions continued, and especially after the written publication of encounter dialogue material began in earnest, there is a clear tendency for editors and compilers to modify their texts in order to increase the perceived religious utility of the dialogues. Ironically, this meant making them seem more like direct oral transcriptions than they had before, by making them more colloquial as time went on. The best example of this involves one of the most important Chan texts of all time, the recorded sayings of the legendary Linji Yixuan. It is important to recognize that the vivid immediacy of Chan literature, the feeling of "being there," is a literary effect contrived through literally centuries of combined effort.
The texts of encounter dialogue are thus several steps removed from the actual participation in encounter dialogue itself. Before considering the age in which those texts were published—the Song dynasty—we should consider the institutional transformations that occurred in Chinese Buddhism from the Tang to the Song.
## CHAPTER 5
## Zen and the Art of Fund-Raising
_Religious Vitality and Institutional Dominance in the Song Dynasty_
### Against the "Zen of Anything"
Once or twice at formal academic meetings I have introduced papers with a dramatic reading of forty or fifty book titles that include the word "Zen." The most widely known example nowadays is Robert Pirsig's novel _Zen and the Art of Motorcycle Maintenance,_ but this is merely one member of a very large genre. Beginning with Eugen Herrigel's classic, _Zen and the Art of Archery_ (which has recently become the subject of some dispute), such works include _Zen and the Art of the Macintosh, Zen and the Art of Windsurfing, Zen and the Art of the Internet, Zen and the Art of Cubing: In Search of the Seventh Side_ (whatever that subtitle might mean!), and _Why Toast Lands Jelly-Side Down: Zen and the Art of Physics Demonstrations._ In addition to landmark works by D. T. Suzuki and Alan Watts such as _The Zen Doctrine of No-Mind, Zen and Japanese Culture,_ and _The Way of Zen,_ there are also any number of "The Zen of" books, such as _The Zen of International Relations, The Zen Teachings of Jesus,_ and _The Zen of Oz: Ten Spiritual Lessons from Over the Rainbow._ A single author has written books entitled _Zen Computer_ and _Zen Sex: The Way of Making Love—_ I have not actually seen either of these, but I hope they are very different in style! The late Bhagwan Shree Rajneesh (1931–90), who adopted the name "Osho" toward the end of his life in misinformed deference to the Zen tradition, wrote a number of books explaining the ideas and texts of Zen, one of which is a lengthy tome with the snappy title of _Zen, Zest, Zip, Zap, and Zing._ And recently there has appeared _The Complete Idiot's Guide to Zen Living,_ by two medical and mental health professionals without any apparent contact with the Zen tradition at all.
Some of these "Zen and whatever" volumes are good books in their own rights, but taken as a whole they perpetuate a perfectly banal misapprehension of one of the world's great religious traditions. It seems that virtually anyone can claim authoritative understanding of Zen, or at least be comfortable in using the word _Zen_ in works totally unrelated to the tradition. It would not do to become too indignant, since this sort of exploitation is but the inevitable side-effect of D. T. Suzuki's missionary success, through which Western interest in Zen and other matters oriental was initially piqued. Nevertheless, we may recognize that, in contrast to its usage within East Asian Buddhism, the word _Zen_ has a very different and much more limited range of meaning in contemporary world popular culture. The popular usage implies that Zen is simply an attitude of undistracted concentration that can be applied to any human endeavor. If you get fully involved in the task at hand, become one with it, and allow yourself to flow according to its natural rhythms, then your performance of that task will improve accordingly—which is a discovery of substantial benefit to professional athletes, creative writers, and many others. I have also seen the word _Zen_ used to describe home electronics projects and lines of cosmetic products, in which the word is used in the sense of a bare-bones simplicity and ease of use; of course, the latter may also include some "oriental" aesthetic sense for all I know. No doubt there is an exotic cachet to invoking Zen in all these contexts as well.
Although undivided concentration and bare-bones simplicity are legitimate messages of Chinese Chan as a mode of self-cultivation, the Chan tradition, as we have already seen, involves far more than this. This chapter differs from most of the titles just introduced, in that it really _is_ about the art of fund-raising as practiced within the Chan tradition. But this is not a "do-it-yourself" book. Though I discuss a Chan approach to making money, I leave the application of this approach to life in the postmodern world to others. (This decision is ill-advised financially; it would no doubt be much more profitable for me to write a book on "how to raise money the Zen way"!)
In the following pages we explore a possible scenario for how members of the Chan lineage managed, from the ninth to the eleventh centuries, to take control of the Chinese Buddhist monastic institution, or at least its highest leadership positions. The hypothesis presented here is intended to explain how the encounter model of Chan religious praxis worked as a public ideology, how Chan responded to the persecution of Buddhism and the economic tribulations of ninth-century China, and how the mythology of Chan monastic labor served an important function, even though most of the productive labor in Buddhist monasteries—including supposedly Chan temples—was performed by lay workers and tenant farmers. In contrast to the conventional viewpoint that the fund-raising efforts of Chan abbots during the Song dynasty indicate the degeneration of both Chan and the Buddhist tradition as a whole, I suggest precisely the opposite: that the institutional success of Chan was made possible by—and in fact represents proof of—its vitality as a spiritual discipline. To put it most succinctly, Chan developed a unique approach to fund-raising that allowed its advocates to create for themselves an identity of moral uprightness and detachment from worldly profit even as they worked openly to gather financial support for their institutions.
### Chan Buddhism in Chinese History
Readers familiar with writings on Chan (in either European or East Asian languages) may be surprised at the scope of the reinterpretation just introduced. But the payoff of this different perspective is even greater than one might imagine at first glance: What is at stake here is nothing less than our global understanding of the role of Buddhism in the sweep of Chinese history. Early-twentieth-century studies of Chan had a major impact in shaping how most English-language writings interpret the cultural and intellectual transitions from the North/South Dynasties period (220–589) to the Song dynasty, that is, from the third through the thirteenth centuries. Conversely, those standard interpretations of the contours of Chinese intellectual history for the same lengthy period have profoundly influenced how writers describe the Chan tradition. There has been a palpable circularity at work, with historians of China building comprehensive theories based in part on a romanticized image of Chan, and apologists for Chan buying into those theories because they served the missionary agenda. Our understandings of Chinese Buddhism, Chinese intellectual and religious history, and Chan itself have been impoverished as a result.
The fallacies and contradictions deriving from this circularity are starkly apparent in the writing of Heinrich Dumoulin. His work is an extreme but representative example, which has become the sourcebook for countless popular and semischolarly accounts. His work is also especially useful here because it is so strongly derivative of earlier research—Father Dumoulin was if nothing else a systematic and voracious reader. He describes the great masters of the Tang dynasty as rustic spiritual virtuosi, geniuses of untrammeled spontaneity who lived in a basically uncluttered world of master-student interactions and diligent spiritual cultivation. We see them working in the fields and vegetable gardens along with their students, unburdened by any mundane problems of monastic administration or relations with local officials and landed gentry, let alone the state bureaucracy and imperial court. The locus of their activity is somehow removed from conventional Buddhist monasteries that fulfilled ritual, festival, and pilgrimage functions within Chinese society, neither blessed with the financial resources that such human services provided nor saddled with the headaches and hassles of involvement with ordinary human society. Within such Chan training temples a new spirit of monkish involvement in manual labor developed, according to this idealized portrait, partly out of the refusal to participate in ordinary fund-raising activities and partly through the more profound need to have the effort of spiritual training permeate every instant and activity.
It is within this quaintly naive depiction of Tang-dynasty Chan that Dumoulin describes the monastic regulations legendarily ascribed to Baizhang Huaihai. For him Baizhang's authorship is unquestioned fact; the lack of any contemporary enumeration of the rules Baizhang supposedly implemented poses no problem. Indeed, the religious context of the earliest version of these rules (which dates from the beginning of the twelfth century) is entirely beyond the concern of Dumoulin, who has already shifted his attentions to Kamakura Japan. Instead, as the Tang dynasty went through the throes of the Huichang persecution, and other Buddhist schools were critically wounded by the removal of material assets and laicization of many clergy, Dumoulin suggests that it was the pure spirit and unselfish ethic of Chan based on Baizhang's rules that allowed the school to survive relatively unscathed.
With regard to the Huichang persecution in particular, Dumoulin devotes a page or two to its political background in the Chinese antipathy to celibacy and corruption within the Buddhist establishment itself. He details the different phases of the persecution, its prelude of hostile but low-impact measures beginning in 842, the increase in the pace of decrees in 844, the climax in 845, and the end after Emperor Wuzong's death early in 846. Immediately following this summary, Dumoulin writes,
Economic factors also played a clear and determinative role in sustaining the persecution, as the Buddhist community, with the fortunate exception of Zen, contributed little of economic benefit to Chinese society. Zen monks worked their farmlands and cultivated their fields productively; if the information we have about the East Mountain teaching is correct, they were doing so already from their early years in China.
This is a clear statement of the romanticized image of Chan, the notion that it represented a special subset within the Buddhist community, a group of sincere practitioners who had retired from worldly activities and devoted themselves to the simple endeavors of farming as part of their own efforts at self-cultivation. We have already seen that the East Mountain community was probably no different from other Chinese monastic training centers at the time in its use of lay laborers, and that even when the _Platform S tra_ appeared, more than a century after the end of the East Mountain period, around 780, there was no evidence whatsoever for such an idealized image of Chan monastic labor. The treatment of Huineng as a temple menial no doubt realistically portrayed the manner in which an illiterate wood-gatherer from the far south would have been treated—thus setting up the dramatic surprise to follow. (See the story beginning on p. 60, and the discussion on p. 68.)
After describing the destructive yet temporary quality of the persecution, Dumoulin comments on its lack of real impact on Zen:
The short duration of the persecution is one reason why Zen suffered so little. The greatest damage was done in the major cities and in the northern provinces. Located principally in the South and in the countryside, the Zen movement was fortunate to find itself far from the fray. Moreover, Zen monasteries struck a rather unimposing image in the eyes of the religious powers. The Zen masters of the T'ang period kept their distance from the imperial court and were not at all engaged in academic or public activities that might have attracted attention. As a result, they were able to sustain their minor losses without consequence.
This is simply an elaboration of the same idealized image. Immediately following this passage, however, Dumoulin introduces broader historical issues:
The Buddhist persecution during the T'ang period signaled a turning point in the history of Chinese Buddhism. The main thrust of the persecution lasted only about a year. How was it possible that in such a short period the broadly based institution of Buddhism could suffer wounds that would leave it permanently crippled? One would have thought that after the storm had subsided the numerous temples and monasteries so beloved by the people would have had the strength to renew themselves. Or had Buddhism—despite its beautiful façade and imposing edifices, its complicated doctrinal systems and impressive rituals—been dealt a blow that exhausted its inner energies? Was there more to the picture than the external damage that was so evident, especially in certain monasteries and convents? Did the debilitation and devastation reach into the very marrow of Buddhism?
Dumoulin's answer to this rhetorical question is, predictably, in the affirmative: he goes on to describe how the example of the Zen movement and its unhampered rural setting confirmed for him the suspicions that Buddhism had indeed reached such a profound state of debilitation.
This depiction of the fate of late Tang-dynasty Buddhism needs to be substantially rewritten, but at this point we need to focus squarely on the goal of Dumoulin's attentions to these issues. His presentation is largely a setup for the discussion of the non-Buddhist glories of the Song:
During the Sung period, Chinese civilization attained heights it had known before only during the time of classical antiquity. One may properly speak of a "renaissance," since the general cultural growth was accelerated by a return to the classics. . . .
The dominant intellectual movement of the time, known in the West as Neo-Confucianism, contributed to this renaissance by adopting the naturalistic and rationalistic orientation of the classics to confront modernity. The golden age of Buddhism, whose numerous schools had attracted large segments of the Chinese population for half a century _[sic]_ with their metaphysical speculations, elaborate rituals, and mysticism, was clearly over. . . .
After Emperor Wu-tsung's great persecution (845), Buddhism lived on in only two movements, the meditational school of Zen and the mainly popular school of the Pure Land. To the Zen monasteries of this period fell the responsibility of representing the Buddhist heritage at a higher level, and from them flowed intellectual and artistic currents greatly enhancing the culture of the Sung period.
Here we can easily detect the echoes of the modern positivist historian HU Shih's (1891–1962) interpretation of Chinese history, including his specific theory of the "Chinese renaissance." Dumoulin has uncritically imported Hu's ideas into his own work, virtually unchanged.
HU Shih believed that ancient Chinese culture was a basically just and rational society, with little use for superstition or indeed religion of any kind. Following the disintegration of the Han dynasty, though, just when Chinese civilization was at its weakest point, Buddhism arrived as a pernicious foreign system of superstitious theories and practices. China was sick, and Buddhism infected it like an uncontrollable virus. At one point, Hu even refers to China during the North/South Dynasties period as an "intellectual colony" of India. Given this perspective, the rise of Chan Buddhism was interesting to Hu for only one reason: he believed that the doctrine of sudden enlightenment represented the surgical tool by which China excised the Buddhist sickness from its own corporate body, leading eventually to the rise of Neo-Confucianism in the "Chinese renaissance" of the Song dynasty.
The errors in Hu's understanding of the role of Buddhism in Chinese history are simply too numerous for us to treat them all here. I discuss the philosophical implications of his approach for the understanding of the Chan enlightenment experience in the next chapter; here we must remain focused on issues of historical process. Let us note, though, that his projection of China's early-twentieth-century plight onto the past is blatant, especially in his use of the rhetoric of colonization and liberation. And his rhetoric of renaissance, of the "flowering" of the Song Neo-Confucian tradition and the positive value of its emphasis of the ancient classics, is echoed almost word for word in Dumoulin's text.
### The Five Factors of Institutional Takeover
What alternative is there to the style of interpretation represented by HU Shih and Heinrich Dumoulin? Let us concentrate on five different factors, which taken together represent a preliminary delineation of an alternative hypothesis.
#### SHENHUI'S RHETORIC FOR CHAN FUND-RAISING
The monk Shenhui occupied virtually all of HU Shih's attentions in Chan studies. In contrast to Hu's description of him in military terms as the leader of the Southern school's assault against the Northern school, and in contrast to our conventional image of Chan masters as meditation instructors or spiritual guides engaged in intimate religious encounter with their students, Shenhui is actually best understood as a public evangelist. That is, he operated as a public exponent of the "good news" of Chan. His religious vocation was not in the private sanctuary of the meditation hall but on the very public venue of the ordination platform, where he made exciting and highly theatrical public presentations that inspired his listeners to begin the path of Buddhist spiritual cultivation. Not only did this result in men and women taking ordination to become monks and nuns (thus adding to the numbers of Chan practitioners in China), it also added hard currency to the coffers of both church and state. Each man undergoing ordination had to pay a sizeable sum to the government—which gained the ordinand a lifetime tax exemption as a member of the clergy. (We do not have similar information about the nuns, since women were not directly subject to taxation.) After the An Lushan rebellion began in 755, Shenhui was enlisted by the Tang ruling house to raise money in this fashion, and later he was commended for his contributions to the defense of the government. We do not have any financial details about donations received by the _sa gha_ through Shenhui's activities, but Chinese archaeologists have discovered a gold ingot whose markings testify to the profitability of such ordinations to the state. All of this activity was shortsighted, of course, for both church and state, resulting in increased state control of ordination and the erosion of the government's tax base. The entire story is a fascinating episode in Chinese religious history.
What is significant here is that Shenhui achieved his success as a fund-raiser not in spite of any other-worldliness of the Chan tradition, but _by means of_ his iconoclastic rhetoric. For example, the famous encounter between Bodhidharma and Emperor Wu of the Liang (see p. 22 above), which on the surface seems like a clear denunciation of merit-oriented activity, in fact occurs for the first time in Chan literature in the written transcript of Shenhui's presentation at a large-scale Buddhist fund-raising gathering. In other words, Shenhui found an appealing and effective way to tell his listeners, in essence, "Your donations on behalf of the _sa gha_ are empty and ultimately of no religious merit. However, through your aspirations to achieve enlightenment on behalf of all living beings and your undertaking of this basically simple path of Chan practice, you should go ahead and make those donations anyway." Iconoclastic language was used, not to undercut the action of contributing to the _sa gha,_ but to nuance the manner in which the fund-raising request was made. Judging from Shenhui's career as a fund-raiser, this paradoxical appeal for donations _worked._
Although I have paraphrased the underlying message of Shenhui's mission here in stark and simple terms, this should not be taken to imply a cynical or corrupt ploy on his part. There is an overly ambitious side to Shenhui's vigorous factionalism that created an identity crisis in early Chan (see p. 56), but we do not have enough information to accuse him of anything really seamy. It seems better to accept his abilities as a public evangelist as based on a real ability to move his listeners to moments of transformative religious inspiration. In the process, though, he articulated the Chan message in a way that was eminently suited to successful fund-raising activities.
#### EXPANSION OF MAZU'S HONGZHOU SCHOOL
Mazu Daoyi and his two or three generations of successors, collectively known as the Hongzhou school, are always described as the very paragons of the Chan ideal. These are the "great masters" of the Tang: Nanquan, Baizhang, Zhaozhou. The basic problem with this depiction is that it is entirely based on texts beginning with the testimony _of the Anthology of the Patriarchal Hall,_ which only appeared in 952, a century and a half or so after the flourishing of the Hongzhou school. We consider this important Chan "transmission of the lamp" text just below; for the moment let us use contemporaneous sources to examine certain aspects of Mazu's faction. Here the subject is not the famous stories involving these figures and their spiritual practice, but rather three intriguing aspects of the school's public face.
First, the epitaph written for Mazu shortly after his death states that, at least at the beginning of his time in Hongzhou, he resided in the "government office." Nothing like this is mentioned in the sources available for any Chan monk before this time, which makes it uncertain exactly what the arrangement was. At the time, Hongzhou was in an area of economic, agricultural, and population growth that was of increasing relative importance to the Tang government. That is, after the An Lushan rebellion removed many of China's more outlying regions from de facto central government control, the rice-growing areas of the central southeast became the primary source of in-kind taxation. Hence the location of Mazu's first local residence—he eventually took up residence at the Kaiyuansi, the officially sponsored temple in Hongzhou—implies some kind of collaboration with the central government's tasks of administration.
Second, the Hongzhou school after Mazu exhibits a strikingly clear pattern of expansion throughout Jiangxi (a region geographically similar to modern Jiangxi Province). Over the first three generations there is a steady and progressive expansion of the places throughout Jiangxi at which Hongzhou monks resided. A similar pattern exists for the lineage of Shitou Xiqian (710–90) in Hunan, the progenitor of the Caodong (S t ) school and a figure almost equal in stature to Mazu in Chan mythology. Although we do not have enough information to say much that is relevant here to Shitou's lineage, the similarity of the two patterns of geographical expansion implies a regional policy of government support for the new Chan factions.
Third, the earliest text produced by the Hongzhou school, the _Transmissions of Treasure Grove_ of 801, uses a format that seems designed for public lectures for laypeople rather than master-disciple discussions in the meditation hall and private quarters. Usually this text is explained within the generalized context of Tang-dynasty Chan practice, or with regard to its inferred enumeration of the Chan lineage in terms of twenty-eight patriarchs from kyamuni to Bodhidharma. The closing portion of the text, presumably devoted to Mazu himself, is missing, and YANAGIDA has suggested that the text was no longer needed after the appearance of the _Anthology of the Patriarchal Hall_ in 952. Hence, lacking its potentially most interesting sections, the _Transmissions of Treasure Grove_ itself has not been widely studied. (On the narrative structure of the _Transmissions of Treasure Grove,_ see p. 96.)
The unique format of the _Transmissions of Treasure Grove_ is intriguing. As mentioned briefly above, the story of each patriarch's career is presented in two portions, depicting him first as a gifted student who is recognized by the current patriarch and then as an enlightened master seeking his own successor. The reduplication implies a popular storytelling context, a frame structure used to keep the audience entertained and always coming back for more. The lay orientation of these presentations is also implied by the lack of any reference to how any of the patriarchs actually attained enlightenment, a reticence probably based in part on Vinaya proscriptions against monks making false claims regarding their spiritual attainments to laypeople. Although this reticence was modified, or circumvented, in later years, at this stage in the development of the written transcription of encounter dialogue, it is understandable that such restrictions would still apply. It was only a century and a half later, with the _Anthology of the Patriarchal Hall,_ that the words of mere students and even brief allusions to the enlightenment experiences of actual people would be included in Chan literature.
It might seem strange that the government would have had any interest in supporting Chan Buddhism, but the reason is straightforward. Scholars to date have written frequently about the "sinicization" of Chinese Buddhism, the process by which it was influenced by and adapted itself to Chinese cultural and political conditions, but no one has addressed the involvement of Chinese Buddhism in what we may call "sinification." Rather than the passive process of sinicization, sinification refers to the long-range enterprise of Han Chinese civilization to overwhelm, incorporate, and pacify the peoples within its ever-increasing cultural sphere. Sinification was a massive historical event, whereby Han culture spread from north to south, and then to the southwest and far west. Beginning sometime in the first millennium before the common era, this process continues down to the present in the increasingly intense efforts going on in Tibet and Xinjiang, the scenes of ongoing Buddhist and Islamic tragedies we will have to pass by without further discussion here.
What has not been realized thus far is the extent to which Buddhism contributed to the process of sinification. Here just a few comments will have to suffice. Fotu Deng ("Buddha" Deng, or Cheng; d. 349), a magic-working Buddhist advisor to an early non-Chinese ruler of northern China, is remembered among other things for having converted the Rong and Mo "barbarians," two non-Chinese ethnic groups of northern China. Rather than interpret this, as Chinese historians have done, in terms of the compatibility between Buddhism and non-Chinese ethnic groups—their only link would be that of an origin considered foreign by Chinese eyes, a link presumably far less salient to the two groups themselves—I suggest that this represents the government use of Buddhism to sinify those groups, to make them more easily governable within a Chinese administrative system. Scholars of religious Daoism have long known that, from the third century at least, Daoist priests worked hand-in-hand with various Chinese governments, providing religious legitimation of their rule. One of the aspects of this collaboration was the use of Daoism to control or suppress popular religious cults that threatened government stability, and there was a collaborative effort against so-called "licentious cults" in southern China during the third and fourth centuries. We do not know specifically how the Hongzhou school may have collaborated with local officials, but the spread of officially sanctioned Buddhist monasteries could well have been part of a program for the social control of local populations.
#### IMPACT OF THE LATE-TANG CATASTROPHES
In addition to the new interpretation just given of Buddhism's participation in the long-term process of sinification, we also need to reevaluate the impact of the Huichang persecution and Huang Chao rebellion on Chan. As Stanley Weinstein relates in his important study of the political dimensions of Chinese Buddhism under the Tang dynasty, the Huichang persecution may have been temporarily damaging, but the church seems to have recovered fairly quickly. Of greater economic significance was the Huang Chao rebellion of 875–84, which devastated north China and thereby destroyed the social infrastructure on which Buddhism depended for its institutional existence. Professor Weinstein writes:
Unlike the Hui-ch'ang suppression which in its most destructive phase covered a period of approximately two years, Huang Ch'ao's insurrection raged for nine years and devastated almost every major region of China. Although there is no conclusive evidence that the rebels were specifically hostile to Buddhism, the damage suffered by the church proved to be irreversible . . . the losses were nothing less than catastrophic and in the long run were probably more detrimental to the maintenance of Buddhist traditions than even the blows of the Hui-ch'ang suppression.
We should also consider adopting a new understanding of the psychological impact of these events, especially the Huichang persecution, on the Buddhist _sa gha._ Whether or not the persecution was caused in part by any real degeneration of the clergy, it must have intensified everyone's sensitivity to the rhetoric of such a decline. In her book _Once Upon a Future Time,_ Jan Nattier has shown that in India the rhetoric of the demise of Buddhism (known in Chinese as _mofa_ , "end period of the Dharma") developed not due to persecution of Buddhism by antagonistic rulers, but rather when the Buddhist church enjoyed excessive material success. Worldly success implied a loss of the original spirit of Buddhism, which involved at the least some distance from ordinary human luxuries, if not their outright rejection. To apply this logic to the Chinese case, if part of the reason for government persecution was that Buddhist monks had violated the other-worldly spirit of their religion by enjoying excessive individual prestige and personal wealth, then the reaffirmation of the religion's lofty ideals would have been an important consideration in the rehabilitation of Buddhism after 845. After the Huang Chao rebellion, the fall of the Tang, and the chaos of the Five Dynasties, the need to justify the fund-raising efforts of the Buddhist monastic institution as a whole would have become even more important. What Chinese Buddhism needed was a way to convince both government and laity that it was firmly committed to higher spiritual goals and therefore deserved their continuing financial support.
#### THE _ANTHOLOGY OF THE PATRIARCHAL HALL_ AS PUBLIC DOCUMENT
The answer to this dilemma of Chinese Buddhism was presented to the court of the Five Dynasties regime of Min (in what is now Fujian Province, directly opposite the island of Taiwan), in the form of _the Anthology of the Patriarchal Hall._ This epoch-making text, which survived only in Korea, is notable in a number of respects. First, it contains transcriptions of massive quantities of Chan encounter dialogue, a genre of discourse which appears here virtually for the first time. Since this dialogue material is presented in colloquial language, the text marks an important milestone in our understanding of the evolution of vernacular Chinese. Although compiled in the far southeast, one of the curiosities of the text is that it represents its figures using speech forms of the capital at Chang'an. Second, beyond its substantial linguistic value, _the Anthology of the Patriarchal Hall_ is a priceless source for the earliest versions of Chan's most famous anecdotes, and as such it is almost always used to describe how Chan master-student interaction took place, how the great Tang patriarchs taught, and how different Chan teaching styles developed. (We have already exploited the text in just this way; see the anecdote introduced on p. 81 above). Third, the text is the first one to present the entire Chan genealogical system of the "transmission of the lamp" genre of texts, so that it presents not only a body of stories but a clearly articulated understanding of the evolution of Buddhism as well. With its extensive treatment of the seven Buddhas of the past, the twenty-eight Indian patriarchs, and the successive generations of Chinese patriarchs from Bodhidharma onward, the _Anthology of the Patriarchal Hall_ presented a thoroughly documented framework for understanding the overall evolution of Buddhism from India to China according to the Chan transmission scheme. (See the discussion beginning on p. 2, as well as figure 1.) Although the _Anthology of the Patriarchal Hall_ was superseded in only a half-century by the _Transmission of the Lamp,_ its compilation was an important step in the evolution of Chan literature.
I suspect that the _Anthology of the Patriarchal Hall_ was compiled in part due to a sense of impending loss. Social and military chaos dominated most of China, and the many monks who escaped to the south shared their stories so that this new approach to Buddhism might be preserved. There may have also been a positive sense of self-discovery, as refugee monks realized that the stories they exchanged in their own monasteries were being avidly discussed in so many other locations as well. Hence this text represents an important milestone in the evolution of the Chan community's self-awareness as a "school," and it represents the first manifestation of its interest in and ability to transcribe colloquial Chinese dialogues in written form. For all these reasons, this text gives us an unprecedented access to the world of Chan religious praxis in the mid-tenth century and before.
Given all of this, at this point let us focus on another dimension of the _Anthology of the Patriarchal Hall:_ its identity as a public policy document. For it is one of the most notable features of this and the succession of "transmission of the lamp" texts to follow that they were formally submitted to the throne for official recognition and inclusion in the Chinese Buddhist canon. This official sponsorship was not merely a recognition of the overall popularity of Chan Buddhism within the Chinese realm, nor just a natural outgrowth of the role of the imperial court in Chinese religious affairs, but also an indication of the use of these texts _outside_ the realm of seated meditation, teacher/student dialogue, and ongoing spiritual cultivation. To be used for these intimate religious purposes, a text would need only to exist, to be circulated and discussed within the confines of the meditation hall community. The government's formal recognition of the "transmission of the lamp" texts was the hallmark of their utilization within a broader institutional framework.
So the question is, Within the context of the Chinese polity as a whole, how would the format and contents of the _Anthology of the Patriarchal Hall_ and subsequent "transmission of the lamp" texts serve the needs of the monastic community?
When the question is posed in this fashion, the answer is really quite simple: These texts legitimated the spiritual identity of Chan monks and provided a straightforward cognitive framework for the management of power and patronage within the monastic establishment. To explain this, we need to return to the structure of Chan's genealogical system.
There are really two major sections of the Chan lineage scheme. In chapter 1 we considered the implications of only the first section, that from the seven Buddhas of the past down through the first six Chinese patriarchs. Here the lineage is unilineal, a feature that has significant implications for the homologization of religious charisma within the identity of the Chan patriarch. Each patriarch is essentially equivalent to every other patriarch, meaning that Chinese patriarchs are said to have the same religious authority as Indian Buddhas. In addition, each patriarch/successor account is a paradigm for teacher/student interaction. The varieties of presentation represent only the diversity of human types rather than any absolute religious differences.
But after the sixth Chinese generation the lineage branches out, first in two major divisions, then in a handful of smaller groupings, and ultimately, in the true arboreal fashion of a flourishing family tree, in a veritable thicket of ever more extensive sublineages. Or, to change metaphors, the single river of the transmission flowed undivided through twenty-eight Indian and six Chinese patriarchs to Huineng, after which it branched into two major currents and then a cascade of streams and rivulets. The picture is complicated even further by the successive additions to the "transmission of the lamp" genre, published in close succession in 1004, 1036, 1101, 1183,1204, and 1252. It was not merely that new figures deserved recognition by their prominence; the differing lineage emphases of these various texts (the specifics of which are not significant here) suggest that strong contestation was involved.
Just as the essentially unilineal first section of the Chan lineage implied homogeneity and equivalence, so must the complex multiplicity of its second section imply heterogeneity and differentiation. Each individual in the massive family tree of Chan was still connected directly to the Buddha and conveyed the full transmission of the Buddha's teachings. At the same time, however, all such individuals were separated into different groupings, a nested set of lineage identities of ever-increasing complexity. Each individual's lineage identity thus traversed a unique set of subdivisions and branches, but eventually connected him to a primordial unity: first one's own teacher and various regional and intermediate figures, then the Tang-dynasty greats such as Linji, Caoshan, Mazu, and Shitou, and ultimately the twenty-eight Indian patriarchs and the Buddha himself.
#### INSTITUTIONAL FUNCTION OF THE CHAN LINEAGE SYSTEM
Why did Chan authors record sublineage distinctions in such detail? No doubt there were important doctrinal and even spiritual factors at play. Without denigrating the importance of cognitive modeling, I believe that much more mundane considerations played a major role. The answer can only be that those distinctions mattered in some real and tangible way in how Song-dynasty Buddhists lived their lives and administered their common institutions.
A number of gifted scholars have provided detailed explanations of how the Chinese monastic institution operated during the first half of the twentieth century, and they have shown that the modern patterns of monastic administration are generally continuous with those of the Song dynasty. The picture these scholars describe involves two different types of monastic institutions: large public monasteries known as "forests of the ten directions" _(shifang conglin)_ and a variety of smaller institutions referred to collectively as "disciples temples." There were probably two or three hundred public monasteries during the Song dynasty, some 90 percent or more of which were labeled "Chan" and the rest "teaching" or "Vinaya." Although there were many more similarities than differences between the various public monasteries, since a broad range of Buddhist activities—not limited to either Chan meditation, Tiantai doctrinal instruction, or maintenance of the Vinaya or Buddhist monastic regulations—took place in all such institutions, for the purposes of this discussion I consider only the Chan ones.
Although they are not central to our discussion, for comparative purposes it is important to note some of the features of the disciples temples. Some of these were also sizable institutions, but most were small establishments housing only a single teacher and a handful of novices and trainees. Some were effectively owned by wealthy lay families, while others were under the control of the monks themselves. Often such local temples were controlled by a single succession of teacher and students—but here the succession was simply one of initiation into the Buddhist order (not all of the "disciples" were even formally ordained as novices), not the "mind-to-mind" transmission of Chan. Holmes Welch has shown how, for twentieth-century Buddhism, the human relationships established at the beginning of a trainee's career might continue throughout his life, even for those who went on to gain full ordination and take part in the lifestyle of the public monasteries. We may imagine that a similar overlap of associations was in place during the Song dynasty. One of the tasks of future scholarship of course will be to chart the institutional evolution of Chinese Buddhism, which will certainly result in increasing the sophistication of distinctions drawn between the Song and twentieth-century systems.
Thus there simply were no independent "Zen monasteries" as imagined by Dumoulin. They are a figment of a romantic imagination, and such rose-colored stereotyping always goes hand-in-hand with cynical dismissal. In short, it is the falsely positive image of the Tang that has led to the pervasive disdain for the "degenerate" Song. In Zen studies, and no doubt other fields, "romanticism breeds cynicism."
The important point here is that public monasteries imposed specific, lineage-based restrictions on who could hold the position of abbot. In those public monasteries carrying the designation "Chan," this position was naturally held by a member of the Chan lineage; in the much smaller number of public monasteries labeled "teaching" or "Vinaya" institutions, a Tiantai or Vinaya school affiliation was required. There was also a highly significant rule governing the succession from one abbot to the next: the new abbot had to belong to a sublineage different from that of his predecessor. This was one of the primary distinctions between the public monasteries and disciples temples: the greater prestige of the former carried with it the requirement that power be shared among different sublineages.
In other words, abbots' appointments were determined in part through reference to the micro-level sublineage identities established in the "transmission of the lamp" genre. Hence in very concrete terms, the Chan lineage schema provided a framework within which to negotiate the distribution of power within the monastic institution. This is by no means to suggest that all monks were power-hungry, or that we should ignore the real spiritual dimensions of their lives. Many twentieth-century Buddhist monks are known to have felt that their administrative roles were a burden, and they only participated in monastic administration out of a sense of loyalty and responsibility. No doubt this was true in medieval times as well.
The "success" of Chan in the Song dynasty was thus not the creation of a new monastic institution, but rather the conquest by members of Chan lineages of the highest administrative positions in the vast majority of the largest establishments within that institution. The succession of "transmission of the lamp" texts that appear in the tenth century and beyond were written not just to document the further growth of Chan as a tradition of self-cultivation, but to provide justification for changes, either de facto or proposed, in the balance of power between the various Chan sublineages.
The success of the Chan lineage scheme seems to have taken the other schools of Buddhism by surprise. Or, it would be better to say that the Chan success led to a new understanding of the very definition of a Buddhist "school." Never before in the history of Chinese Buddhism had the theoretical identity of a particular school been associated so closely with monastic administration. To be sure, the Tiantai school had been strongest at its original center on Mount Tiantai, but by the Song (actually, by at least the ninth century), Chan practitioners had come to dominate many of the monasteries on the mountain. Scholars used to talk about Vinaya and Tiantai school monasteries as if they were separate types of institution, but with a few trivial exceptions this simply was not the case. On the contrary, public monasteries incorporated activities of all the so-called "schools" within their precincts—devotions to Amit bha Buddha, lectures on the _Flower Garland and Lotus S tras,_ repentance rituals, and so forth—and it was only in the abbot's position and the meditation hall that such a monastery was in any way "Chan." The "teaching" monasteries of the Song and later were essentially identical, except that the abbot was from a Tiantai lineage, the meditation hall might (but did not necessarily) emphasize Tiantai Zhiyi's instructions on meditation, and there might be regular instruction in the doctrinal taxonomy and other unique teachings of the Tiantai school.
To understand how unprecedented this development was, we should ask ourselves the following question: Why should meditation specialists become abbots of the most important monasteries in early premodern China? Given the preferences of recent American converts to Buddhism, who consider meditation to be the sine qua non of Buddhism itself and ignore almost entirely the cultural role played by the religion in its ethnic communities, this question may not seem obvious at all. Meditation teachers have founded and led most if not all elite American Buddhist communities, after all. But in the medieval and early premodern Chinese context, as well as in other Asian Buddhist contexts from ancient times until now, the question is very much an apposite concern. Why should meditators be good administrators? Why would meditators actively seek, or at least accept, administrative power? Although several other factors require further examination (e.g., what role did specialists in esoteric Buddhism play in this process?), the administrative success of Chan was no doubt predicated on the vacuum created by the end of scriptural translation as the primary focus of attention in the monastic community (see p. 71). However, it was also made possible by a style of religious rhetoric and a shared image of enlightened sagehood disseminated through the newfound medium of woodblock printing. In the next chapter we will see how these factors combined in the Chan school's period of greatest success, the Song dynasty.
## CHAPTER 6
## Climax Paradigm
_Cultural Polarities and Patterns of Self-Cultivation in Song-Dynasty Chan_
### Song-Dynasty Chan as Climax Paradigm
After a forest has been devastated, whether by fire or the excessive harvesting of lumber, it immediately begins to grow back. First to sprout are quick-growing grasses, followed by wildflowers whose seeds are carried on the winds. Erosion from rainfall and runoff carve new channels in the earth, sometimes changing the topography forever, but eventually a network of roots develops to stabilize the ground. Given the variations of climate, birches or other softwoods may climb to the sky first, but they will eventually be crowded out by taller species of trees that are able to reach through to the sun. Although it may take several decades, or even centuries, in the absence of new catastrophes a kind of stability will eventually emerge. Subject to a complex dynamic of forces and processes, individual plants and animals will grow and die within the mature forest, which has now become what ecologists refer to as a _climax community._ Although change still occurs within the forest on the basis of different natural cycles—the days, seasons, and years—the overall system has stabilized. Individual plants and animals are born, grow to maturity, and die within that system, but the major components of the forest community function interdependently. The overall configuration of those components, the subcommunities, colonies, and niches within the larger climax forest community, will also change, sometimes quickly and in some cases almost imperceptibly, over time. However complex and dynamic it may be, the forest as a whole has taken on a relative stability that allows its analysis as a single (if multidimensional) system.
In the Song dynasty (960–1279), Chinese Chan Buddhism reached something of a climax paradigm. By "climax paradigm," I mean a conceptual configuration by which Chan was described in written texts, practiced by its adherents, and, by extension, understood as a religious entity by the Chinese population as a whole. As with the forest climax community, this is not to say that all change within Chan Buddhism ceased—far from it!—only that the basic conceptual parameters of the school's self-conception and approach to spiritual cultivation had become well established. Previous events in Chan were interpreted through the lens of the Song-dynasty configuration, and subsequent developments in China, Korea, Japan, and Vietnam were evaluated, even as they occurred, against what was known of the standards established during the Song. Thus the romanticized image of the great Tang-dynasty masters—Mazu and his students, Caoshan, Dongshan, and their students, and of course Linji—was generated by Song-dynasty authors and functioned within Song-dynasty texts. Similarly, even where subsequent figures throughout East Asia—Hakuin Ekaku (1685–1769), the famous reviver of Japanese Rinzai, is the best example—evoke the examples of Bodhidharma, the Sixth Patriarch Huineng, Mazu, and the others, they do so through the conceptual filter of Song-dynasty Chan. The individual images of Bodhidharma, Huineng, and other early Chan figures no doubt continued to change as time went on, but the overall framework in which their examples were used was based on the conceptual paradigm that matured in the Song.
This interpretation is radically different from earlier understandings of the evolution of Chinese Chan. As we have seen, Dumoulin treated Tang-dynasty Chan as the golden age of the tradition and dismissed Song-dynasty Chan as a period of decline. I have cited Dumoulin's work extensively in the preceding pages, but only because he is the most convenient example of a general style of interpretation. China historian Arthur F. Wright espoused a similar approach, extolling the "everyman" quality of the Sixth Patriarch Huineng, then dismissing everything that follows without even serious consideration. Kenneth K. S. Ch'en, whose textbook on Chinese Buddhism is still widely used, characterizes _all_ of Chinese Buddhism from the Song dynasty onward in terms of "decline." And the interpretations of Jacques Gernet and Wm. Theodore de Bary echo this same pattern. A number of different historiographical factors have concatenated to create this situation, ranging from HU Shih's scholarship, Confucian prejudices, overemphasis of the novelty of "popular" or vernacular religious developments in the post-Song period, and the misapplication of Japanese sectarian models to the Chinese subject matter. These topics are far too involved for discussion here, but their cumulative effect—the image of Song-dynasty Buddhism as a religion in decline—is persistent.
In spite of this commonly held misunderstanding, recent scholarship on both Chan and Chinese Buddhism is unanimous in holding that the overall activity level of Buddhism in China actually rose to a peak during the Song. Scholars of Chan now argue that the school really took shape only in the Song, and that its image of a "golden age" during the Tang is just that, an image in the minds of Song devotees. During the Song dynasty, Chan monks became abbots of most of the great monastic establishments in China, moving by imperial invitation from one position to the next, often carrying imperially bestowed titles and purple robes along the way. Similarly, it was during this period that the Tiantai school reorganized itself, albeit through a process of sometimes acrimonious debate. And the Pure Land school developed from a diffuse style of devotional and ritual endeavor into the most widely accepted vehicle for the salvation of ordinary monks, nuns, and laypeople, producing new types of social organizations that promoted mutual assistance for their members during the trials of illness and death.
It is important to recognize, of course, that not all the successes and shortcomings of Chan occurred in a single pattern of congruent features. One of the major goals of this book, indeed, is to suggest that we avoid reducing the subject matter to simplistically encyclopedic parallels. I hope that the ecological metaphor used here will help us appreciate the complexity, even the chaos, of the phenomena with which we are dealing. We have scattered evidence about Bodhidharma and his associates in the proto-Chan phase, but not really enough to know how, or even whether, all that evidence pertains to one definable group or movement. Our knowledge of the East Mountain teaching phase is better, but still involves substantial retrospective inference. There seems to be a stable central community, but it is known only through limited evidence and later projections. At the beginning of the eighth century the self-described successors to this community exploded on the national scene, and in the process they described themselves as an identifiable religious movement using the lineage model. No matter how diverse and multifaceted the Chan movement was at this point in time, no matter how fuzzy the boundaries were between it and other realms of Chinese religious life, from this point onward Chan had achieved a significant level of sectarian identity. To continue the ecological metaphor, it was as if the various disparate elements of the earlier phases had generated a set of relationships, patterns of interaction (whether dependency, symbiosis, or even parasitism) that were relatively stable even as the elements continued to change. The emergence of encounter dialogue in the middle period brought with it a substantial change in those patterns of interaction, resulting in major changes within the Chan community as a whole. Here "Chan community" refers not just to a community of individuals, but more importantly to a conceptual set of different lineage groups, styles of practice, and rhetorical forms. It was during the Song dynasty that the genealogical, practical, and rhetorical forms of Chan attained their most lasting configuration; even as those separate forms continued to evolve and interact, their overall network of interrelationships attained a stable "climax community" pattern.
I do not devote any significant space to the description of the Song dynasty monastic establishment; let us await the appearance of Foulk's detailed and penetrating analysis. However, one important feature must not be overlooked: Chan was not nearly as separate from these other types of Buddhist activity as one might think. In our discussion of Chan's administrative success in the preceding chapter, we saw that the monasteries of which Chan monks became abbots were comprehensive institutions, "public monasteries" that supported various types of Buddhist activities other than Chan-style meditation. The reader should bear this point in mind: In contrast to the independent denominations of S t and Rinzai that emerged (largely by government fiat) in seventeenth-century Japan, _there was never any such thing as an institutionally separate Chan "school" at any time in Chinese Buddhist history._
In spite of this absence of any demonstrably separate institutional identity, Song-dynasty Chan—even allowing for its conspicuous variations over time and space—represents the climax paradigm of the tradition for the following reasons. First, it was during this period that the most lasting forms of Chan practice emerged. By "Chan practice" here I mean both the monastic conventions of meditation hall life (the social practice of Chan as a whole) and the styles of religious introspection and self-cultivation discussed below. Second, these forms of practice were the basis for the dissemination of Chan throughout East Asia, that is, not only in China but in Korea and Japan as well. Actually, the ecological metaphor used here suggests how improper it would be to portray Song-dynasty Chan with a single brush: from its climax paradigm repertoire came quite an array of species, phenotypes, and niche configurations that were to be transplanted and evolve across East Asia. Although it would be impossible to quantify the overall popularity of Chan in comparison with other types of East Asian Buddhist religious activity, it is undeniably the case that during the Song dynasty Chan attained an unprecedented plateau of ascendancy throughout Chinese and eventually all East Asian culture. Indications of this ascendancy may be seen in the increased percentage of Chan specialists covered in the standard biographical collections from before the Tang to the Song, the flood of Chan publications, and the frequent references to Chan in the secular literature. Third, it has been through the lens of Song-dynasty understandings that both the members of the school in subsequent centuries and its interpreters in the twentieth and twenty-first centuries have experienced Chan Buddhism. Here I need mention only that D. T. Suzuki's twentieth-century exposition of Rinzai Zen as _k an_ practice finds its roots in the innovations of the twelfth-century Dahui Zonggao, to be discussed just below.
These chapters have been devoted primarily to outlining the evolving Chan ideology of spiritual cultivation, not the details of the school's institutional history, and this final contribution is no exception. That is, in defining the climax paradigm of Song-dynasty Chan, I would like to focus, for simplicity, on two well-known but very different approaches to Chan meditation practice, known by the convenient shorthand labels "viewing the phrase" and "silent illumination" Chan. Readers who are already familiar with Zen will quickly recognize the association of these terms with the Linji (Rinzai) and Caodong (S t ) schools.
### The Exemplary Career of Dahui Zonggao
Although there were many eminent Chan monks during the Song, the one with the most lasting importance was certainly Dahui Zonggao (1089–1163). His accomplishments clearly derived from his training under great masters, particularly Yuanwu Keqin (1063–1135), compiler of the "precedent anthology" known as the _Emerald Cliff Record (Biyan lu),_ but Dahui added new emphasis and energy of his own to the contemporary understanding of Zen practice. It is no exaggeration to say, in fact, that in Dahui's life and teachings we find exemplified the very pinnacle of Linji Chan.
Dahui's Buddhist training was unexceptional for his times. Tonsured at age sixteen and given the Dharma name Zonggao (meaning roughly "brilliance of truth"), he became formally ordained the next year. He was soon drawn to Chan writings, especially those of the innovative Yunmen Wenyan (864–949). At the same time, he is said to have had an inspirational experience while reading a Mah y na scripture. The following year he began his wanderings to study under different teachers, sometimes sampling the styles of several teachers in a single year. (At this stage, it seems that most of his teachers were actually in the Caodong lineage.) In 1116 Dahui met the retired prime minister and lay Buddhist scholar Zhang Shangying (1043–1122), and soon after that Han Zicang (d.u.). These two prominent laymen were to be important influences in his life.
From relatively early in his training, Dahui had received recommendations to study under Yuanwu, but it was only in the fourth month of 1125 that he was finally able to enter Yuanwu's assembly. After only six weeks or so, on the thirteenth day of the fifth month of that year, Dahui had a decisive moment of awakening during one of Yuanwu's sermons. Miriam Levering recounts the event as follows:
Yuanwu cited an exchange in which a monk asked Yunmen, "What is the place like where all the Buddhas are born?" Yunmen answered: "The East Mountain travels on the water." Yuanwu continued, "If today someone were to ask me what the place is like where all the Buddhas are born, I would reply: 'The _xun_ wind comes from the south, and the palace has a slight coolness.'"
On hearing these words, Dahui experienced a marvelous end to all his doubts, and a feeling of great peace, joy and release. When he went to report his breakthrough to Yuanwu, the teacher made Dahui an attendant without duties in the quarters where court guests were entertained, a spot where Yuanwu could supervise his progress personally every day. Seeing that his enlightenment needed further refinement, Yuanwu gave Dahui another _huatou_ ("critical phrase") to work on. Every day for half a year Dahui worked on this _huatou_ under Yuanwu's supervision. Finally one day Dahui asked Yuanwu to tell him how his teacher Wuzu Fayan had replied to Yuanwu's asking about that same _huatou_ once in the past. When Yuanwu complied, Dahui experienced a complete and certain enlightenment. Yuanwu tested it by posing other _huatou_ to Dahui and found that Dahui could answer them all without the slightest hesitation. Yuanwu then gave Dahui a residence of his own at Heavenly Peace Temple (Tianningsi) and allowed Dahui to share in the teaching and preaching duties as a fellow teacher.
Not long after this, Dahui and his teacher had to flee the north in advance of the conquering Jin dynasty forces. They parted ways in 1127, were together again briefly from the end of 1128, and then were separated again when Yuanwu went off west to Sichuan. Dahui spent the next five or six years in comparative isolation in a couple of locations in Jiangxi and Hunan, with only twenty monks in attendance, continuing his practice and coauthoring poetic commentaries on famous Chan precedents.
In 1133 Dahui responded to a letter from Han Zicang, bemoaning the decline of Chan in the world and entreating Dahui to stop "sitting alone on his peak, wearing grass and eating roots." We may safely avoid taking seriously Han's stereotypical denigration of religious practice during his day: Just as appeals by Indra and Brahm were necessary to get kyamuni off his seat under the _bodhi_ tree, so Dahui had to be convinced to take up his natural mission as a teacher After a half-year living in Han's house (apparently more than a simple letter was required!), Dahui began to act and write more publicly. He began to attack other teachers and teachings he considered to be heretical, especially those who avoided aiming vigorously at a moment of awakening in favor of a more passive approach to meditation. From 1134 on, Dahui began to attack the advocates of "silent illumination," perhaps because he was in Fujian near the Caodong monk Zhenxie Qingliao (1088–1151), who was then the abbot of a monastery with more than fifteen hundred people in residence. Dahui's targets included members of his own lineage, and his propensity for public criticism of others earned him the nickname of the "one who bawls out heaven"!
But Dahui did not devote all his energies to such acrimony. It was also in 1134, in his role as spiritual instructor of a nun named Miaodao, that Dahui finalized his unique method of Chan practice known as "viewing the phrase" _(kanhua;_ J. _kanna),_ which is discussed below. While this technique may have been designed specifically to counter the Caodong upsurge with a new contrasting slogan, there is a parallel between the teaching style Dahui developed and his approach to other liturgical responsibilities. That is, Levering notes Dahui's uniquely personal style in the use of funerary memorials and other sermons. Rather than merely repeating generalizations and commonplace truths, Dahui's sermons are the earliest known examples in which the doctrines emphasized were specifically related to the patrons' individual situations. In other words, although he cannot have been the first Chinese Buddhist monk to have used these occasions to develop close personal connections with his supporters, he seems to have done so with an innovative personal touch. The practice of _kanhua Chan_ also emphasized the close personal interaction between teacher and student.
Dahui's first major appointment came in 1137, as abbot of kyamuni Temple (Nengrensi), also known as Jingshan, near Lin'an, the Southern Song capital. There for almost six years he attracted more than two thousand students while reviving the glory of the Linji line and continuing his attacks on "heretical" teachings. In 1143 he was banished for criticizing government policy—probably for being too outspoken in recommending military action to retake northern China from the Jin—a lost cause, given the sad state of preparedness of Southern Song troops. Although he was not allowed to maintain his clerical status during this time, Dahui's sixteen years in exile were comparatively pleasant. He continued teaching and writing, and he took the opportunity to travel to the Sixth Patriarch's temple in Caoqi. In 1156 he was released from his exile but spent nearly a year traveling, only accepting an imperial invitation to become abbot of Mount A oka Temple at the end of 1156. During his year or so there Dahui taught twelve hundred students, and even though dormitories and a water pond were built, the facilities were not adequate for his following. In 1158 he returned to Jingshan, where he stayed for the next four years, with a thousand monks studying under him.
After retiring from the abbacy of Jingshan in 1161, Dahui performed various teaching duties. In 1162 he was invited to preach before the emperor, and only then did he receive the name Dahui ("Great Wisdom"), by which he is best remembered. After traveling about a bit, he returned to Jingshan in the seventh month of 1163. On the ninth day of the eighth month he announced that he would die on the morrow, then sent farewell letters to the emperor and several friends. On the tenth, still clear and calm, he wrote out the following verse before dying peacefully:
Life is just this,
death is just this.
To have a verse or not—
why should it matter?
Dahui had more than 110 ordained Dharma heirs and scores of lay disciples. As Levering observes, it is incredible that he was able to be so productive in spite of the many years in exile, with only ten years of teaching in large public monasteries.
### The "Viewing the Phrase" Chan of Dahui Zonggao
The preceding description of Dahui's enlightenment experiences under Yuanwu used the term _huatou (wat ),_"critical phrase," and Dahui's method is referred to as _kanhua Chan (kanna Zen),_ "viewing the [critical] phrase Zen." The "critical phrase" is the most crucial part of a _gong'an,_ or "precedent," often its very last sentence. But the meaning of these expressions is not self-evident.
First, let us consider what Dahui meant by "viewing." This was no passive state of observation, nor a kind of intellectual contemplation. Dahui disallows any use of rational processes, any attempt to make ordinary sense of the subject matter. He enjoined his students against attempting to understand the precedent logically, or according to Buddhist doctrine, or using the specifics of its wording, or by clues inferred from a teacher's gestures. Any specific method students might come up with was rejected. For example, Dahui taught his student Miaodao as follows:
I cited Mazu's "It is not mind, it is not Buddha, it is not a thing" and instructed her to look at it. Moreover, I gave her an explanation: "(1) You must not take it as a statement of truth. (2) You must not take it to be something you do not need to do anything about. (3) Do not take it as a flint-struck spark or a lightning flash. (4) Do not try to divine the meaning of it. (5) Do not try to figure it out from the context in which I brought it up. 'It is not the mind, it is not the Buddha, it is not a thing; after all, what is it?'"
The only possible recourse was a form of total surrender:
You must in one fell swoop break through this one thought—then and only then will you comprehend birth and death. Then and only then will it be called accessing awakening. . . . You need only lay down, all at once, the mind full of deluded thoughts and inverted thinking, the mind of logical discrimination, the mind that loves life and hates death, the mind of knowledge and views, interpretation and comprehension, and the mind that rejoices in stillness and turns from disturbance.
Attaining this type of awakening was no easy matter, of course. Dahui expected his students to exert themselves with a type of effort that exceeded their normal capabilities.
There is a striking difference between Dahui's intense, even ferocious, attitude toward meditation practice and the nuanced exertion portrayed in the _Treatise on the Essentials of Cultivating the Mind_ attributed to Hongren. Of course this may simply be a difference of literary style; by Dahui's time it may have become fashionable to portray one's personal energies more explicitly. Leaving aside the possibility that Hongren may have had his fiercely demanding side as well (the text attributed to him does include numerous exhortations to energetic effort), we may simply note that Dahui demanded total dedication from his students. The list of requirements given above is of course impossible to achieve—and this is no doubt one of his points, that one must throw oneself so completely into the endeavor as to go entirely beyond the boundaries of personal effort and accomplishment.
Second, what is a "critical phrase"? The Chinese compound _huatou_ simply means a "bit of speech" or a "topic." In Chan usage, however, it refers to the most crucial (and usually final) line in a _gong'an,_ which means roughly a "legal case" or thus a "precedent." The _gong'an_ was invariably taken from some transcription of encounter dialogue, usually from the _Transmission of the Lamp._ In the Song dynasty it became fashionable for Chan teachers to compile anthologies of their favorite examples of Chan dialogue, which they then used in oral instruction with their immediate students and commented on in characteristically idiosyncratic Chan language for publication. The most famous example of this process, of course, is the _Emerald Cliff Record_ compiled by Dahui's teacher Yuanwu, which includes a hundred such precedents with poetic introduction and commentary by Yuanwu and his own teacher, Xuedou Zhongxian (980–1052). Dahui seems to have had some misgivings about the literary goings-on that accompanied Chan practice, and there is a famous legend that he actually burned his copy of the _Emerald Cliff Record_ at one point relatively late in his career. Whether or not this is true (the event is not mentioned in the chronological record of Dahui's life) is of course less important than that many people thought the story was true, or at least believable.
To understand "viewing the phrase" Chan, we must first consider the identity of the critical phrase within the context of an entire precedent. The following is my favorite example, case 63 from the _Emerald Cliff Record,_ known as "Nanquan Cuts the Cat in Two." The text includes a brief "pointer" or introduction by Yuanwu, the case itself, and a concluding verse by Yuanwu's teacher Xuedou. The case and concluding verse are graced with Yuanwu's comments, presented here in smaller type to imitate their appearance in most Chinese editions as interlineal glosses (that is, a double column of half-sized characters below each full-sized phrase of the case and verse).
_Pointer_
What is beyond thinking must be the topic for serious discourse. What transcends words should be the subject of earnest investigation. When lightning flashes and shooting stars fall, you should display the power to drain the deepest lakes and overturn mountains. Has any of you acquired such ability? See the following.
_Case_
ONE DAY NANQUAN SAW THE MONKS OF THE EASTERN AND WESTERN HALLS QUARRELING OVER A CAT. This was not just this one day's argument, but a regular waste of time. HE HELD UP THE CAT AND SAID, "IF YOU CAN SAY IT, I WON'T KILL IT." A direct command that should be carried out. [Everyone in] the ten directions just sits there. This old man has an arm that can distinguish dragons from snakes. NO ONE SAID ANYTHING. What a lamentable error! A whole bunch of lacquered dolls—what are they able to do? Fake Chan monks, like hemp or chestnuts! NANQUAN THEN CUT THE CAT IN TWO. Joy, joy! If he hadn't done this it would have been like a man playing with mudpies. To stretch one's bow after the bandits have gone. This is already the second head—he should have hit them before they even brought it up.
_Verse_
HOW USELESS THE MONKS OF BOTH HALLS; A parent's words come from a parent's mouth. One word and he's said it all. The judgment was decided on the basis of the case. RAISING DUST AND SMOKE, UNABLE TO DO ANYTHING. See, what kind of resolution could you achieve? The precedent of manifest creation _(xiancheng gong'an;_ J. _genj k an)_. There are still a few. FORTUNATELY, NANQUAN GAVE THE ORDER; Raising his whisk and saying, "One like this." Old Master Wang (i.e., Nanquan) is a little like this. To use this excellent precious sword of the vajra king to go cut mud! WITH A SINGLE [SLICE OF THE] KNIFE [HE CUT THE CAT] IN TWO WITHOUT INEQUITY. Smashed in a hundred pieces! Suddenly a man wields a knife, and what do you think he did? Don't make a mistake, or I'll hit you
Dahui's instruction is then to concentrate on understanding the line "Nanquan then cut the cat in two," precisely the most brutal and inexplicable point in the entire case. What sort of a spiritual practice is this?
My own students, especially in the university classroom, are always stunned by the notion that a Buddhist monk could actually kill a cat. Avid Zen practitioners may have had the same reaction at one time, but they eventually become quite blasé about such violent examples. After all, they have long since heard stories about Linji recommending that one kill the Buddha or one's parents, so the radical bifurcation of a feline inevitably seems less of an issue. How sad! In my own mind there is of course no possibility whatsoever that the incident ever happened as written—Buddhist monks simply do _not_ carry knives about, nor do they cut off their own arms or dismember other living beings! (The sacrifice of a finger or arm—a highly ritualized act of devotion and asceticism—is a different story, of course.) However, the story was selected from the vast trove of encounter dialogue literature precisely because of its shock value, so as to get students to consider why exactly something like this might have happened.
By considering the line "Nanquan then cut the cat in two," the trainee is forced to wonder just what the monks were arguing about in the first place. Were they debating whether or not their small domestic animal had the Buddha-nature, as a dog might (or might not)? Or were the monks of the two different halls, east and west, claiming ownership of the cat due to its contributions as mouser? In either case, does it matter that the Eastern Hall probably housed the monastery's administrative services and the Western Hall its meditation center? In the McLuhan sense, once again, since so little background detail is provided, the topic of contemplation here demands that the contemplator visualize its original circumstances for himself.
This process of visualization is, in fact, exactly the point.
What was the meditator visualizing, and what would come of it? The trainee was instructed to contemplate images of enlightened behavior, couched so as to be perplexing and beyond ordinary human understanding, but by this very opacity recognized as harboring some greater truths. At the same time, the attention to the enlightened activity of the ancient patriarchs served as a mirror to one's own inner identity. In this sense it is significant that one was taught to use written accounts of the legendary masters of one's own lineage rather than the Chinese Buddhist tradition as a whole: the subjects of the stories in _the Emerald Cliff Record_ were of course the legendary masters of the Tang and Five Dynasties periods, virtually all of whom were portrayed as lineal predecessors of Xuedou, Yuanwu, Dahui, and by extension Dahui's students. One looked at the enlightened activities of one's lineal forebears in order to understand one's own identity. An encounter with one's spiritual ancestry was the key to understanding the present.
Dahui's own enlightenment experiences occurred when he and Yuanwu played with such dialogues, taking the roles of the participants and engaging in dialogue in their stead. Yuanwu's commentary on his master Xuedou's verse and the case itself impart the sense that he was attempting to use the medium of writing to accomplish the same function. That is, he writes as if attempting to enter into the dialogue, to live within the idealized Tang-dynasty realm of spontaneous interaction. Actually, the _Emerald Cliff Record_ was not a text that Yuanwu "wrote," but the transcribed product of his lectures on the one hundred cases and his own master Xuedou's verses. The literary impact of this written by-product is certainly intriguing: In at least the one instance introduced above, Yuanwu's commentary sharply demarcates a line between masters and students, as he roots for the actions of his would-be peer Nanquan and condemns the incompetence of the ordinary monks, all from the literary sidelines of interlineal gloss—or, primordially, from the high seat of his sermons. Yuanwu's comments contain no hint of indecisiveness or question, nor even any inquiry into the deeper implications of Nanquan's action. Was this lack of curiosity some form of intellectual or religious shallowness? Or, was Dahui's supposed difficulty with the text merely that it seemed to fix specific meanings to the cases, when the primary intention of Yuanwu's teaching had been to catalyze an attitude of openness in his students and audience? We will never know why Dahui became frustrated with this text (if in fact he did), but we may speculate that he realized the written text had closed off its subject matter in a way unworthy of his teacher's creativity.
In an important article on the evolution of Chan spiritual cultivation, Robert Buswell explains the emergence of "viewing the phrase" Chan in terms of the school's creation of a truly subitist approach. In his presentation, "viewing the critical phrase" Chan emerges as the result of the school's almost self-conscious quest for a technique to match its subitist rhetoric. To be sure, suddenness acted as a constraint on what could be said about meditation practice, and perhaps even as a guide to what kinds of approaches were considered preferable. However, there were simply too many possible approaches that could be called "sudden" in one way or another—if not an infinite number, then certainly an indefinite plurality. In my own estimation, more important than any subitist rhetorical imperative was the genealogical framework of "viewing the phrase" Chan—the formulation of the practice in terms of the student's examination of models of enlightened behavior within his own ancestral lineage.
Dahui's "viewing the phrase" Chan developed largely through his interaction with educated literati, and it evolved into a form that captured their collective imaginations. It was not precisely the case that Dahui's literati clientele manufactured his ideas for him, but that he happened, through a combination of lineage tradition and individual skill, to produce an approach to Chan practice that conformed to literati expectations. This is by no means to downplay the innovativeness of Dahui's new approach, but even to highlight it by comparing its fate to that of numerous other Chan teachers whose ideas did not capture the imagination of the world in the way his did.
What were the most important elements of Dahui's approach? We have already noticed these features above:
1. stress on both effort and enlightenment, since without the former the goal would be belittled and without the latter there would be no inducement for training
2. reduction of the genealogical encounter implicit in encounter dialogue to the practice of meditative introspection, by using contemplation on snippets of dialogue as a substitute for engagement in actual dialogue
3. creation of a backward-looking model of enlightened sagehood that fit the Chinese style of "ancestral time," in which the pure simplicity of a golden age is attributed to the age of one's primordial ancestors
4. unification of past and present practitioners into a genealogically defined "consociation," an association of individuals unrelated by birth or marriage that presents an outer face of unity under one figurehead but within which its members actively compete for status.
We can see in these patterns the manner in which "Zen spontaneity" lived in the Song-dynasty context. Rather than actually live their lives in untrammeled spontaneity, as the Tang figures Hanshan and Shide were supposed to have done, for example, Song-dynasty Chan practitioners inscribed spontaneity within clearly demarcated limits. They talked about spontaneity, imagined spontaneity, debated spontaneity, and pondered spontaneity—all within the formally structured and highly ritualized context of Song monastic life. They presumably spent precious little time actually living and acting "spontaneously."
Yet Dahui at least was engaged in the creation of a new approach to Buddhist self-cultivation. One of the ways to appreciate the significance of his novel approach is to consider its imitation by the Neo-Confucian Daoxue scholar Zhang Jiucheng (1092–1159) in his style of "investigating things" and "guarding the mind." In addition to using a type of quiet contemplation that obviously drew upon Buddhist meditation theories, Zhang's approach to the Confucian classics was remarkably similar to the Chan attitude toward the ancient sages. Here is Zhang's appreciation of the _Spring and Autumn Annals:_
In the _Spring and Autumn_ our master Confucius has fully revealed the Way of emperors and kings, the power of heaven and earth, the brilliance of the sun and the moon, and the movements of the seasons—how can this be perceived with the ordinary mind? If, in the space of a single word, you apprehend the forge and furnace of the sagely mind, then yin and yang will break open and clouds will send down their rain. This is all [contained within] our Master's _Spring and Autumn—_ from "cultivating one's person to regulating the family, governing the state, and bringing peace to all under heaven," all is possible.
Ari Borrell explains Zhang's understanding to be that, by reading the _Annals,_ the student would "personally receive the transmission of the Master's 'method of the mind' _(xinfa)_ and, 'Having gotten the mind [of Confucius], then [our own daily activities such as] eating and drinking, sleeping and resting, answering and responding will all be the actions of our Master [Confucius].'" Although Zhang's presentation might be too straightforward for Dahui's liking, it would be possible to substitute Bodhidharma or Mazu for Confucius for a perfectly acceptable interpretation of Chan practice.
There is no doubt that Zhang Jiucheng's approach paralleled Dahui's. Nor should we overlook the political ramifications of the two men's positions—both were strong advocates of the "hawk" position that the Song dynasty should fight to retake the north, and there is a consistency in the dynamism of their styles of religious introspection and their political views. However, for our purposes it is more important to confirm the general parallel (observed for the first time at least a half-century ago) between the polarities in Chan practice and Neo-Confucian theory. If Dahui's dynamic "viewing the phrase" Chan resembled the "examination and knowing" of Daoxue, then how did the more quietistic "silent illumination" of the Caodong lineage resemble the "nourishing and cultivating" of the Cheng-Zhu faction of Neo-Confucianism? We will return to the parallels between Chan and Neo-Confucian theory shortly, but first we should look at the other major configuration of Song- dynasty Chan practice.
### "Silent Illumination" and the Teachings of Twelfth-Century Caodong Chan
It is useful to take Dahui's criticisms as the starting point for consideration of another important style of Song-dynasty Chan practice—important both for its presence in Chinese Buddhism and for its position within the Japanese S t school founded by D gen (1200–53). This is the Caodong approach, commonly known as "silent illumination," which Dahui excoriates. While Dahui's criticisms certainly are not unimpeachable, we may consider them and their contemporary context as a prelude to understanding the larger significance of Caodong practice in its own right.
The most prominent members of the Caodong lineage in the Song dynasty were Zhenxie Qingliao and Hongzhi Zhengjue (1091–1157). As already noted (see p. 125), Dahui's earliest denunciations of "silent illumination" Chan, in 1134, may have been inspired by his proximity to a monastery run by Qingliao. Hongzhi had written his _Inscription on SilentIllumination (Mozhao ming)_ a few years before this, though (the preface is dated 1131), and scholars have long wondered whether he was the target of Dahui's criticisms. Very few other Caodong texts actually use the term _mozhao,_ "silent illumination," so there could hardly have been a more likely target. However, there is good evidence that, although the two men met only once, Dahui and Hongzhi respected each other greatly as fellow Buddhists. Dahui praised Hongzhi several times during his period of the active denunciation of "silent illumination" practice, and after the other man's death Dahui praised Hongzhi's virtues in poetic form. During his life Hongzhi recommended Dahui for installation as abbot at one of the empire's most prestigious monasteries, and when about to die he asked that Dahui be in charge of his funeral. So, how was it that Dahui could so vociferously oppose a signature doctrine of Hongzhi's while maintaining such cordial relations with him?
During Dahui's life Caodong Chan, once on the brink of disappearance, was enjoying a revival. With the death of the last lineal successor to the founders of the school in 1037, the extinction of Caodong Chan would seem to have been assured. However, that last Caodong monk (named Dayang Jingxuan [942–1027]) asked a colleague in the Linji tradition to maintain his Caodong transmission in trust and pass it on to a suitable student. The student so chosen was Touzi Yiqing (1032–83), who was not even born when his supposed "master" died. Yiqing's student Furong Daokai (1043–1118) seems to have been an innovative and productive individual, and his second-generation students Zhenxie Qingliao and Hongzhi Zhengjue were the most active and high-profile members of the Caodong revival. During the lifetimes of these two men, however, there were at least fifty-four active teachers in their grand-master Daokai's lineage. Both Qingliao and Hongzhi supposedly had thousands of followers, and they had fourteen and twenty-eight recognized Dharma heirs, respectively.
The Caodong resurgence straddled the loss of the north to the Jurchen Jin dynasty in 1126 and the ensuing decrease in government support for Chan Buddhism. This decrease in support was manifested by a drop in the numbers of official name plaques granted monasteries, restrictions on the ordination of monks, and so forth. Chan teachers thus felt increasingly compelled to aim their messages at literati, who were beginning to practice Chan meditation in significant numbers. As we have already seen, much of Dahui's message was molded through his emphasis on the literati audience, and the upsurge in Caodong activity no doubt intensified his competitive fire.
Although the Caodong teachers Daokai, Qingliao, and Hongzhi each had slightly different approaches to meditation practice, one crucial element determined Dahui's positive or negative response: effort. If the style of practice could be characterized as simply waiting around for enlightenment to happen, or even disregarding the importance of the actual experience of enlightenment, then Dahui's attack was blistering. However, if he perceived a demand for constant and energetic application of effort, he was able to approve styles of meditation practice profoundly different from his own.
For example, Furong Daokai wrote:
If you can awaken to and understand [where] your own self [was] at the time of the empty eon, then it will be like hundreds or thousands of suns and moons whose radiance is inexhaustible, or like countless sentient beings all at once attaining liberation. But if you still don't understand, it is absolutely necessary that you retreat and come to a halt. You yourself must completely cease; you yourself should be completely at rest; you must be like a censer in an old shrine; the [instance of] one thought [of yours] should last for ten thousand years; and you should be like a man who doesn't take even a single breath. If you are able to be like this constantly for months and years, then, if you don't obtain the fruits of the Way, I am speaking nonsense and have been deceiving you all, and I will surely be born trapped in hell. I urge you all not to mistakenly apply your bodies and minds in trying to analyze the distance of the road ahead. Do not rely on an intermittent approach. It is necessary that you yourself put your strength into it; no one else can do it for you.
In these passages Daokai counsels his students to make effort in a way that Dahui might have appreciated. However, the effort recommended here is to maintain oneself continuously in a state of total rest—not plunge on energetically, exhausting all one's resources until one achieved a sudden breakthrough, as Dahui would have it.
Qingliao's pronouncements on meditation practice, though, actually reject the notion of conscious effort:
Without taking a step you should constantly sit in your room and just forget about the teachings. Be like dried wood, or a stone, or a wall, or a piece of tile, or a pebble. Cut off "knowing" and "understanding" and be naturally vacuous and completely bright. You should not make the least bit of conscious effort here.
You should be like a baby who doesn't distinguish between north and south or know the difference between the six senses. You should rest your head at once and naturally be vacuous and bright and self-illuminating.
In the middle of activity be constantly still, when in darkness increase brightness. Don't fall into dualistic extremes.
Since Qingliao talks only of "rest[ing] your head at once" and being "constantly still" in the middle of activity—apparently without any countervailing injunctions to be active within stillness, to strive without making conscious effort—it is understandable that he aroused Dahui's ire.
Hongzhi's _Inscription on Silent Illumination_ contains a similar attempt to achieve balance between meditative serenity and energetic endeavor:
Transcendent wisdom exists in the silence [of meditation];
striving for achievement is forgotten in illumination.
Where does transcendent wisdom exist?
Alertly we destroy murkiness.
The path of Silent Illumination
is the basis for leaving the world of delusion.
For Hongzhi the striving may be forgotten once illumination occurs, but that implies that striving was necessary to bring on the experience. His description of alertly destroying the murkiness of ignorance also implies directed effort.
_The Inscription on Silent Illumination_ also states:
All the myriad things in the universe
emit radiance and speak the Dharma.
They all attest to each other,
and correspond in dialogue.
Corresponding in dialogue and attesting,
they respond to each other perfectly.
But if in illumination silence is lost,
then aggressiveness will appear.
Attesting and corresponding in dialogue,
perfectly they respond to each other.
But if in silence illumination is lost,
then you will become turbid and leave behind the Dharma.
But when silence and illumination both are operating and complete,
the lotus flower opens and the dreamer awakens.
The hundred rivers flow into the sea,
and the thousand peaks face the great mountain.
Like geese preferring milk,
like bees seeking out flowers.
When Silent Illumination is perfected and obtained,
the teaching of our tradition is set in motion.
Here Hongzhi emphasizes the balance between silence and illumination, in a manner that clearly resonates with the essence/function _(ti/yong)_ polarity of fourth-century and later Chinese Buddhism, as well as with the writings of Shenhui in the eighth century. The former distinction, as found in the writings of various figures, is used to analyze states or realities considered to be essentially unitary—such as _nirv a_ or enlightenment—but which seem to have different characteristics according to the perspective used. Hence the enlightened mind may be said to be essentially quiescent even as it exercises the functions of knowing. Similarly, Shenhui argues that the realms of meditation and wisdom cannot be considered as separate, so that the function of serene meditation is wisdom and the essence of nominally active wisdom is serene meditation.
But where Shenhui argues a philosophical position that concentration and wisdom must be identical to occur at all, Hongzhi is interested in the balance between silence (concentration) and illumination (wisdom) that is achieved by the meditator. And, as Morten Schlütter aptly observes, it almost seems as if Hongzhi "is describing enlightenment as an event in time when he says, 'the lotus flower opens and the dreamer awakens.'" Nevertheless, "enlightenment as a breakthrough event is downplayed," and the very concept was considered a hindrance. Hongzhi does not stress the heroic effort considered necessary by Dahui, and he seems to consider silent illumination as the inherently enlightened quality of mind and enlightenment as a natural and joyful state that is already fully present to the practitioner. Elsewhere, he writes that "there is no need to trouble about practice and enlightenment."
Nevertheless, there _is_ something to be done. Hongzhi writes,
Completely and silently be at ease. In true thusness separate yourself from all causes and conditions. Brightly luminous without defilements, you directly penetrate and are liberated. You have from the beginning been in this place; it is not something that is new to you today. From the time before the vast eon when you dwelled in your old [original] home, everything is completely clear, unobscured, numinous, and singularly bright. _But although this is the case, it is necessary that you act on it._ When you act on it in this way you must not give rise to the smallest strand of hair and not conceal a speck of dust. Cold and like dried wood, [you should practice] the great rest with broad and penetrating comprehension. If your rest and cessation is not complete and you wish to go to the realm [of the Buddha] and to leave birth and death, then there is no such place. Just as you are you must break through, understanding without the defilement of discursive thinking, and be pure without any worries.
Even from this brief consideration it is possible to detect both sensitivity and balance in Hongzhi's admonitions, paralleling those of the _Treatise on the Essentials of Cultivating the Mind_ attributed to Hongren (see p. 38 above). That is, both sources subtly harmonize the need to exert oneself fully with recognition that the goallessness of _nirv a_ obviates the validity of striving.
Let me add one last observation about the relationship between Dahui and the Caodong teachers. Dahui reacts to Caodong growth with a very competitive spirit, but he was acting not so much on his own behalf as for the sake of Buddhism itself, as he understood it, and as reproduced in his own lineage. No doubt his anger at what he perceived as misguided teachings was sincere—but he acted protectively, on behalf of his lineage in ways characteristic of Chinese consociations in general. In the first chapter (see p. 9) I wondered whether Chan functioned to oppress Chinese religious practitioners in general, or to suppress certain groups of these practitioners. This is precisely the point: Dahui was in effect working to dominate the rhetorical community of Song-dynasty Chan, to ensure that his own approach was recognized as valid and all others rejected as invalid. The enlightenment of Dahui's students implied that the students of certain other types of teachers had to be unenlightened. This was true in a proximate sense of the Caodong teachers, but even more so in the broader context: the relative success of Dahui's approach to Buddhism implied the relative emasculation of Tiantai, Huayan, Pure Land, and other approaches.
### Pairing Buddhist and Neo-Confucian Patterns
Over the course of these essays we have touched on a number of duels/polarities (see in particular the discussion beginning on p. 40), and now is the time to consider them in an integrated fashion. To begin with, it is possible to construct a neat set of parallels between patterns of practice that appear in proto- and early/classical Chan, the Caodong and Linji styles of Song-dynasty practice, and the two major trends in Song-dynasty Neo-Confucianism. These parallels play out as follows:
Proto-Chan: | entrance of principle | entrance of practice
---|---|---
Early/classical Chan: | maintaining the mind | encounter dialogue
Song-dynasty Chan: | silent illumination | viewing the phrase
Neo-Confucianism: | quiet sitting | investigating things
As discussed in chapter 2, the "entrance of principle" and "entrance of practice" attributed to Bodhidharma are not well-enough explained for us to understand exactly what they meant in concrete practical terms, but the relationship between the two is reasonably clear: the first represents the fundamental attitude of unshakable confidence in the existence of the Buddha-nature within one, and the second is a progression suggesting how one's every action and activity may be adapted so as to accord with that inward realization.
The early Chan injunction to "maintain [awareness of] the mind," which is explained in the _Treatise on the Essentials of Cultivating the Mind_ attributed to Hongren, is an elaboration of the Buddha-nature concept found in Bodhidharma's "entrance of principle." Just as that text enjoins one to have "profound faith," that is, unswerving conviction, in the existence of the True Nature within one, even though it may be covered over with illusory thoughts and desires, so is "maintaining the mind" the practice of cherishing just that same True Nature or Buddha-nature. The same text also contains two very different styles of meditation practice, one embodying the spiritual perspective just mentioned and the other focused on the activity of the deluded mind.
In both these cases, as well as in the "silent illumination" and "quiet sitting" of Song-dynasty Caodong Chan and Neo-Confucianism, there is the fundamental assumption that the true mind is an inherently brilliant source of illumination—that if one can simply remove the restraints placed upon it by normal patterns of discriminatory thought, then it will reveal itself as the preexistent state of perfect buddhahood. In Chinese Mah y na Buddhist terminology, the true mind within is in a state of originary or fundamental enlightenment, and the removal of illusions to reveal this inherent brilliance is known as temporal enlightenment. The general, almost universal, tendency in early Chan is to emphasize the importance of the Buddha-nature over the illusions that obscure it, and implicitly to favor originary enlightenment over the specific achievement of temporal enlightenment. There is thus a profound continuity between the entrance of principle, the early Chan understanding of Buddha-nature, and the Caodong approach to "silent illumination." For convenience, I will refer to this as the "immanentist" position, because of the emphasis on the immanent quality of the innermost mind.
There is a fundamental distinction between the immanentist posture just described and the "exemplification" style—the term I will use to speak of the entrance of practice—the encounter dialogue of classical Chan, the "viewing the phrase" approach of Song-dynasty Linji Chan, and the Neo-Confucian approach to "investigating things." Previous writers have tended to analyze these two approaches in terms of a static/dynamic polarity, with the immanentist position described as predominately static and the exemplification style seen as quintessentially dynamic. This style of analysis should be avoided, though, because it is so inherently value-laden. Indeed, such dyadic oppositions are often polemical in spite of their apparent inclusiveness—since that very act of inclusion conceals a hegemonic trope of supercession and domination. In potential contrast to modern valuations, of course, ancient and medieval Chinese writers would almost certainly have preferred to associate themselves with the more fundamental "essence" rather than the seemingly derivative "function."
Just as the sudden/gradual typology has a polemical cast for traditional (and many modern) Chan Buddhists, so does the static/dynamic rubric seem to have a polemical cast for modern writers; for a combination of reasons, the authors who use this simple typology always seem to give pride of place to the dynamic, belittling the static. Actually, I would argue that, rather than static/dynamic, the early Chinese Buddhist polarity of essence/function (discussed initially on p. 43 and mentioned with regard to Hongzhi's thought on p. 137) is a more powerful analytical tool.
In fact, we may specify several ways in which we can differentiate the exemplification mode from the immanentist one. First, instead of the inward focus of the immanentist position, there is an outward focus on action, activity, dialogue, and interaction. This outward focus manifests itself in different ways, but it is generally, perhaps universally, accompanied by an emphasis on the actual process leading to and achievement of (temporal) enlightenment itself. One needs to actually experience enlightenment and, just as important, one needs to demonstrate the validity of that experience to a qualified teacher. Both action and interaction are necessary.
Second, the two styles have a fundamentally different stance regarding the significance of human culture and the identity of the practitioner. That is, whereas the immanentist posture requires nothing other than the practitioner's own mind and individual being, the exemplification style depends on the practitioner's role as a participant in a certain type of human tradition. In the Chan school, of course, this is a genealogical context, in which the practitioner achieves enlightenment through contemplation of the enlightened activities of his lineage predecessors, then verifies that experience in a genealogical encounter where he moves from child/outsider to adult/successor. In the Neo-Confucian case the tradition is the Chinese civil tradition, as exemplified in Confucian classics and patriarchs, the canonical source material by which contemporary situations both private and public are to be understood. It seems likely that this understanding of the Chinese Confucian tradition also occurred in a profoundly genealogical style, as manifested by Zhang Jiucheng and other advocates of the active style of the "investigation of things."
A certain caution is necessary here. There is a natural tendency to fill out tables of correspondences as far as one can, and in this case one might consider adding a fifth line, reading "sudden" and "gradual." The goal, of course, is not to line up the greatest number of pairs and argue that they are all parallel, but rather to use the parallels that seem valid or even merely suggestive as tools to achieve better and more nuanced understandings. Recognizing where the parallels cease to make sense is thus also a useful result of this procedure.
The sudden/gradual distinction is a polemical one that by itself has only limited use as an analytical tool. In the first place, each of the Song-dynasty Chan positions can legitimately be described as subitist and the other criticized as gradualist. Naturally, the terms by which such descriptions are made derive from sectarian considerations, but the result is that there is no clear-cut way in which the columns would line up. The immanentist position may be considered subitist because it admits of no stages and posits an "all-at-once" style of realization. (This position is elaborated most brilliantly in the writings of the Japanese Zen master D gen.) From this perspective the process-oriented style of the exemplification approach makes it appear gradualistic, and indeed many Linji and Rinzai Zen practitioners have been described as passing through a number of enlightenment experiences. From the exemplification perspective, on the other hand, the immanentist style seems to ignore the actual experience of enlightenment almost entirely, and certainly to downplay the sudden quality of its onset. And all those hours that Caodong monks spend in meditation, with so little talk about individual achievements, certainly made their Linji counterparts suspect some of them just were not getting any . . . spiritual attainments, that is.
In the second place, as a polemical slogan from the eighth century, the claim of "suddenness" made by different Chan teachers had long since lost its rhetorical novelty in the Song dynasty. The issue was still relevant in the lives of individual practitioners, as indicated for example in the ubiquitous invocation of the term in Dahui Zonggao's writings; as an interpretive mode, however, it obscures instead of adds to our understanding.
The inapplicability of the sudden/gradual polarity should not restrain us from attempting other correlations. It is instructive, actually, to consider how the parallels stated above might bear on the concentration/insight polarity of traditional Buddhist meditation theory. Neither fits very well, but that should not be surprising, given the historical distances involved. One could make a case that the immanentist position resembles _vipa yan ,_ since both depend on the mind's innate capacity to understand. But _vipa yan _ requires an object, and it would only be in its Mah y na understanding that it could be applied to the mind itself (at least in the sense of a primordial or transcendent entity). Such a position is found, of course, in the writings of Tiantai Zhiyi, who expatiated at great length on exactly this topic. And Chan texts from the early period onward clearly recognize the subtle problems with the notion of the mind "counter-illuminating" its own origin. Still, to illuminate something _with_ the mind (even the mind itself) is different from the immanentist position that there is an innately illuminating mind to be revealed within all human beings.
To then compare the exemplification approach to _ amatha_ or concentration would take us even farther afield. Here the goal is not uninterrupted quiet, but total involvement in spontaneous activity. Even more, the exemplification style of practice requires a partner, in the form of a qualified teacher, while _ amatha_ is effectively a solitary pursuit. There is a strong resemblance between the Chinese notion of gradualism and _ amatha,_ in the sense that both suggest progressive improvement. In addition, both represent an ancient dimension in Buddhism according to which liberation is explained in terms of an ascetic suppression or pacification of desire. However, even though subitism may have an underlying affinity with _vipa yan _ (in the sense that both require an unconditioned moment of _praj ,_ wisdom), the parallel is by no means complete. What should be most instructive, in fact, is the very lack of congruence between the immanentist/exemplification, sudden/gradual, and _ amatha/vipa yan _ pairs.
### Intersubjectivity in Song-Dynasty Tiantai Practice
Instead of reaching back to the basic themes of Indian Buddhist meditation theory to understand the theoretical ramifications of Song-dynasty Chan, it is better to consider a parallel that is closer in both time and space: the Home-mountain/Off-mountain _(shanjia/shanwai)_ distinction in Song Tiantai thought. Song-dynasty Tiantai was riven with controversy between a group centered on Mount Tiantai itself (hence the "home-mountain" label) and another group associated with other locations. For our purposes, one way to gauge the impact of Chan thought on Song-dynasty religious culture in general is to consider the extent to which the polarities we have been discussing apply to Song-dynasty Tiantai.
The best guide here is the work of Brook Ziporyn, who has analyzed the theme of "intersubjectivity" within Tiantai thought from Zhiyi to Zhili (960–1028). Intersubjectivity in this case refers to the recognition that spiritual endeavor takes place, not as solely individual action in a cosmos where other living beings are basically irrelevant, but in a context where the functional interrelationship of different consciousnesses is recognized from the outset. In terms of Buddhist soteriological theory, it is philosophically significant for Zhili that there are both ignorant sentient beings and enlightened buddhas and bodhisattvas in the world. Neither ignorant beings nor enlightened sages function in isolation of each other: sentient beings depend on the enlightened for assistance in achieving salvation, and the enlightened dedicate themselves to the act of salvation. With regard to the latter, as Ziporyn puts it, "[B]odhisattvas experience the world in a way that always references other experiencing beings and that constitutively takes into account those other experiences." The same is true of the sentient beings. As Zhili states,
The dharmas of Buddhas and sentient beings are what are called "other," but each inherently includes both all Buddhas and all sentient beings. If the sentient beings and Buddhas inherent in oneself become manifest, they are identical to the sentient beings and Buddhas that are inherent in any Buddha who stands as an other to oneself, and in this instance play the role of "the one who transforms" [in taking on various forms so as to guide and enlighten all beings]. The sentient beings and Buddhas inherent in sentient beings in that case play the role of "the ones who are transformed." Since all this takes place in one moment of experience, how can self and other be considered only different?
Ziporyn explains that
every sentient being and every Buddha equally has the entire system of "Buddhas taking on various forms to enlighten all sentient beings" replete and inherent in himself or herself. Both the guide and the guided are replete in each other, i.e., in both the guide and the guided. Thus every moment of experience is interpretable as the stimulus/response experience, and every encounter allows both agents to play both roles.
For Zhili (and, in his understanding, also Zhiyi) the experience of life takes place within a network of intersubjectivity, in which the individual consciousnesses remain distinct but interact with each other in a multivalent variety of relationships. Since sentient beings and buddhas are ultimately interpenetrating, they exist within each other and function in mutual relationship.
Based on the Tiantai doctrine of the three truths, in which phenomena may be viewed from the perspective of the ultimately true, the provisional, and the middle, any action may be perceived in terms of the salvific activity of a buddha or bodhisattva, the ignorant striving of an ordinary sentient being, or something that involves both (or neither) of these aspects. Ziporyn explains that any event may thus be understood in at least four ways:
All acts, if contemplated as identical to the threefold truth, reveal themselves to be stimuli that bring about Buddha-effects, and the effects are themselves identical to the stimuli that incited them. Every possible action, then, is simultaneously (1) an instance of deluded karma, (2) a stimulus bringing about Buddha-responses, and (3) a salvific, transformative Buddha-response brought about by this stimulus. Since an expedient transformation of a Buddha is (given the nonduality of provisional and ultimate) also the ultimate truth, any token may also be said to be (4) the ultimate being of the Buddha in himself. All four of these view points are contained in one and the same moment of experience, read simultaneously in four different interpretative contexts.
In other words, in the Home-mountain thought typified by Zhili there is an intersubjective relationship between sentient beings and buddhas, or rather between each sentient being and each buddha. The interaction between each pair of unenlightened/enlightened individuals can thus be understood from different perspectives, with neither subjectivity reducible to the other.
In contrast, the Off-mountain faction consistently reduces the relationship between buddhas and sentient beings to a process of emanation from the true mind. Yuanqing (d. 997), the earliest representative of the Off-mountain position, states clearly that "Buddha means the true contemplation, sentient beings means delusion, and the mind is the mind in these two states, not separate from them. But the mind is the root of both Buddha [enlightenment] and sentient beings [delusion]." Mind is thus the underlying source for all reality, and presumably the primary focus of attention in Off-mountain meditation efforts.
The historical relationship between Tiantai doctrine and Chan still needs further elucidation, but even this brief summary should be enough to suggest that the two dominant styles of Song-dynasty Chan practice were echoed to some degree in contemporaneous Tiantai thought. Although the quality of intersubjectivity detectable in Home-mountain Tiantai thought is different in kind from the interaction of Chan masters and students in encounter dialogue, or in the genealogical contemplation of the "critical phrases" of the _gong'an,_ the parallels are suggestive. Similarly, the Off-mountain emphasis on the mind as the fundamental source for both ordinary and enlightened sentient beings can easily be correlated with the Caodong emphasis on the "silent illumination" of the mind within. The relationship between the two schools, which were so often at each other's throats in sectarian competition, deserves further analysis.
### Chan and the Chinese Social Order
At this point I would like to add a few comments about the role of Chan Buddhism within the Chinese social order of the Tang through the Song dynasties. Instead of drawing our discussion to a close, in this and the next section I hope to show how it can be extended in future inquiries.
As a sublime combination of Indian and Chinese elements, Chan took a form (or forms) denned by the Chinese society within which it evolved. As such, it developed in ways that are unimaginable for Buddhism in India. But, without indulging in a thorough comparative investigation of the two cultures, how are we to understand these differences? One way to gain a quick foothold on this issue is to consider how funerary practice differs between India and China, under the theory that such customs reveal some of the most fundamental structures of any society. This will allow us to ask the question, Given the differences between those basic structures, how would we expect Buddhist spiritual cultivation to be conceived within Chinese rather than Indian society?
Traditional Indian funerary ritual is designed to assist the deceased in separating from his or her worldly family. Following cleansing of the body, there is an offering of rice balls, and the body is cremated. With the cremation of the physical body, the deceased person acquires a subtle body which is supported for the next ten days by daily offerings. The family circumambulates the funeral pyre and bathes in a river before returning home, all the while avoiding looking back at the pyre and any open show of grief. At the family home a feast is prepared, and the death is celebrated. Lamentation may occur spontaneously, but it is ritually proscribed. On one of the following two days, the skull of the deceased is shattered and placed along with the bones in an earthware jar, which is then either thrown in a holy river or buried in consecrated ground. On the tenth day, the eldest son offers a ball of rice at the cremation ground so that the deceased can shed the subtle body and become a _preta,_ or "ghost." For a year offerings are made on each new moon to sustain the _preta_ body, after which time another ceremony assists the deceased in discarding the _preta_ body and joining the world of the ancestors. In this ceremony four pots are used, one for the deceased and others for his father, grandfather, and great-grandfather. By pouring from one pot to the others while reciting appropriate incantations, the deceased becomes the first of the ancestors and, from this point on, the great-grandfather is omitted from ritual remembrance. If all goes well, the deceased will eventually move from the world of the ancestors to oneness with Brahman (the unconditioned substrate of the universe), but this is an individual accomplishment that cannot be influenced by funerary and memorial services.
Thus, the overall direction of the various observations prescribed in the classical Indian model is to assist the deceased to separate from the conditioned world and the ties of family, in order to make it possible for him or her to make the final step to liberation. Of course, things do not always work out that well, and it may be that, rather than achieving oneness with Brahman, the deceased _(a)_ becomes a _deva,_ or "god," _(b)_ is reborn as a human on earth, or _(c)_ is reborn as an animal or in one of the other unfortunate modes of existence. Hence the status of ancestor is temporary, and the familial bond is inevitably broken. Although being an ancestor is valued as a respite from the travails of mortal existence, it is still in essence a state of bondage, part of the realm of duality that is ultimately meant to be transcended.
In traditional Chinese culture (and the following account is simplified and limited, as was the preceding one), the entire goal of death ritual is exactly opposite to that described for India: the aim is to maintain an ongoing series of relationships between deceased ancestors and the living. Following ritual cleansing, the body may be placed in a temporary coffin and tomb for some time, until the proper moment arrives for the official funeral and burial proper. If economically possible, the tomb is a vault in which servants are buried in effigy and models of food, drink, and various useful objects are also included. Models of such objects, as well as money, are frequently made out of paper for inclusion in the grave, or they are sent to the deceased in the other world by ritual burning. (Modern funerals send the deceased off with paper televisions, paper cars, and huge wads of special funerary cash with which to bribe the underworld officials!) On the day of death the family, and especially the deceased's children, are encouraged to make open expressions of grief. The deceased's children do not eat until the burial (assuming it happens as usual, three days after death), and all the mourners eat a restricted diet, avoid soft beds, and drop their usual activities to prepare for the funeral. During this period meals are regularly offered to the deceased, and mourners address him or her directly. If the deceased was a male or married female, after the burial a tablet with the deceased's name on it is included in the family altar, and he or she is included in regular ancestral worship according to the place held within the lineage hierarchy. Ritual offerings for the deceased are made at various points over a two-year period, and the deceased's closest inferior relations (i.e., sons and daughters) restrict their activities through the third anniversary of the death. As long as the deceased remains in the memory of the living, he or she is kept informed about family events, such as weddings, births, and deaths. There is a bureaucratic or hierarchical sense to this, in that the deceased's status and place within the family, more than personality or individual achievements, determines the way in which he or she is treated. With the passing of generations, the deceased's tablet is moved further and further to the side on the lineage or family ancestral altar, and after seven generations (well beyond any living person's memory) he or she enters the realm of undifferentiated ancestors.
The preceding descriptions are highly generalized, of course, and in both cultures we may easily detect disparate vectors of practice and belief. One of the most exciting reevaluations of Indian Buddhism now nearing publication suggests that, rather than being only a lofty ideology of transcendence, it developed in part as a mortuary religion, providing a means for the subjugation of ghosts and spirits associated with death. For the present purposes, though, the very idealized image of Indian funerary practices presented above will be sufficient. The goal here is not a comprehensive balanced comparison of the two cultures, which would be a challenging enterprise, to say the least. Rather, the task at hand is to use certain features of the Indian case to help in the analysis of the Chinese situation. We should remember that the interaction between Indian and Chinese cultures, such as it was, occurred only on Chinese soil. If we have isolated a limited set of features of Indian culture known to the medieval Chinese, so much the better.
In addition to the efforts suggested above to keep the deceased involved with the living, Chinese practices included several rituals intended to maintain separation between living and dead. Funerary offerings were used to provide the deceased with the accoutrements necessary for life in the next world, but at the same time measures were taken to restrict them to their tombs. Part of the motivation for ancestral offerings was to prevent the deceased from being let loose upon the world, where they might cause various sorts of difficulties. The famous trio of alternatives available to any deceased person in Chinese culture—to become either god, ghost, or ancestor—was of course partially dependent on the character of the deceased him- or herself and the heroic or tragic quality of his or her death, but it was even more strongly dependent on the actions of the living: If one's descendants provided the appropriate offerings, one became an ancestor, taking one's specified place within the celestial order. If people outside one's lineage made offerings, and if they did so in sufficient numbers because of the intercessions made by the deceased on behalf of non-kin, one became recognized as a god. If neither descendants nor others made the requisite offerings, one was forced to become a ghost and search for sustenance at large, often with unpleasant consequences for self and others. There is, as often noted, a bureaucratic quality to these proceedings, in that postmortem roles and religious charisma as a whole were determined more by status position within the hierarchy than by individual accomplishment.
My contention is that Chan provided a format for Buddhist practice that matched the pattern implied by Chinese funerary customs. The starting point for this analysis is John Jorgensen's observation of the structural similarities between Chan lineage assertions of the eighth century and funerary practice, in which the organization of halls venerating Chan patriarchs was seen to resemble that of conventional ancestral halls. From a broader perspective, the proliferation of Chan lineages mimics that of conventional family genealogies, creating a parallel realm of filiation between living and dead. Indeed, where conventional genealogies are devoted individually to separate family groups, Chan "transmission of the lamp" texts create an entire universe of fictive relationships. Thus each individual practitioner is securely placed within a generational succession, and all of those succession relationships are concatenated into a massive network of interlocking identities. Where conventional family genealogies were in dialogue both with each other and with contemporary social practice, "transmission of the lamp" texts provide the Chan lineage system with its own global context for the idealization of religious identity.
The overall impact of the genealogical pattern of Chinese Chan was thus to include each participating individual within an ongoing network of social relationships, to create a cosmologically natural social group that transcended ordinary society. Certainly, success in Chan spiritual practice has a definite real-world payoff: recognition as an enlightened master and formal inclusion within the lineage of Buddhas and Patriarchs. The cascades of sublineages that are documented in Chan "transmission of the lamp" texts represent the guidebooks for how participation within this "old boy" network occurred. The Chan school thus existed on one level as a set of inclusionary relationships.
On another level, the Chan genealogical network must have functioned as a means of exclusion. As Nancy Jay and others have noted, ritual provides a means for effecting both in-group solidarity and exclusion of the other. Even more to the point, the conjunction of patriarchal lineage and sacrificial practice in agrarian societies serves to support hierarchies of power that exclude women. Although the parallels are not exact, we may note that the Chan genealogical pattern effectively excluded—or, more to the point, worked to exclude—many types of religious practitioners from access to power within the Chinese Buddhist institution as a whole. Devotees of other styles of self-cultivation were marginalized or lumped together under the competing Tiantai banner. Even the Pure Land tradition was forced to adopt a lineage system to justify its existence, and other rubrics for the understanding of Buddhist history were effaced by the genealogical model. And, of course, women were nowhere to be seen in Song-dynasty Chan—at least not without being reconfigured as surrogate males. In other words, Chan provided Chinese Buddhists with a way of ordering their sacred heritage in a fashion that resembled other basic features of Chinese society.
All this is relatively straightforward. The next step is to explore the specific expectations and prohibitions that were built into this cosmology. Here I am using the term _cosmology_ in the sense described by Mary Douglas, as the way in which people of a given culture take their understanding of the world as naturally true. The Chinese cosmology, as is well known, places great emphasis on the continuity of family from deceased ancestors to living representatives. Jay has shown that there is a peculiar force to the combination of patrilineal succession and sacrifice, the result of which is "birth done better" than the defiled process of our actual maternal origins. This is an analysis that certainly applies to Chinese society in general—Jay, who died before her research was published, was unable to consider the Chinese case—but the question here is whether or not it applies to Chinese Chan. Does Chan somehow represent "enlightenment done better," or "religious authority done better"? What natural lineaments of human culture did Chan evolve to subvert, or to sublimate and transform? We may have established some of the building blocks necessary for the construction of an argument discussing these issues, but that is all. Much remains to be done.
Finally, if Chinese Chan differs from Indian Buddhism in such fundamental ways, would it also be the case that the Chan "enlightenment experience" differed from the Indian one? Of course, we should be careful not to assume that Chan texts describe anything that matches our modern concept of "experience," or that we could divine how it actually "felt" for medieval Chinese Buddhists to become enlightened. Nevertheless, even without assuming that we could access the actual experiences of real individuals, it would be useful to compare the descriptions of _bodhi_ in Indian philosophical texts with those of enlightenment experiences in Chan texts. Where the former describe the ultimate goal in terms of wisdom and transcendence, I suspect that Chinese texts tend to a greater emphasis on realizations of the interdependence of all things. Or one might examine whether the rhetoric of _ nyat _ is used differently in Indian and Chinese texts, with the former being used to obliterate worldly distinctions, and the latter being used in effect to reify them. (The "originary enlightenment" theories of medieval Japanese Buddhism seem to fit this latter case.) Obviously, the incredible genre differences between the sources available from South and East Asian cultures make any such comparisons difficult, but these are the sorts of theoretical issues that we are only now becoming able to address.
### Erasing the Paradigm
At this point the reader may be expecting a grand conclusion, including a concise delineation of the Song-dynasty Chan climax paradigm. Rather than fulfill such expectations, however, I endeavor to transform them. In these last few paragraphs let me suggest how the analyses presented in this book may best be developed and exploited in the future.
The basic project undertaken here has been to catalyze different ways of thinking about Chinese Chan Buddhism. In carrying out this project, I have generated various explanatory schemes and strategies, and the use of "climax paradigm" rhetoric was merely one such scheme. The use of such terminology is ultimately metaphorical, heuristic, and I would not like it to be mistaken for historical fact or even historical interpretation. If we can understand the ideas, we should forget the words.
The rhetoric of climax paradigm is limited in several ways. First, since my own field of research has been early Chan, this book constitutes my attempt to carry forward themes from that period to their natural conclusions in later periods. As such, the parameters of discussion have been defined by the continuities and discontinuities from early Chan through the Song dynasty. The basic strategy has thus been prospective, to use the themes and motifs of one period to explore those of later periods. To write history in this prospective fashion is useful, especially in the way it allows us to critique the many misconceptions apparent in the work of earlier authors, but it is not without its flaws. Technically speaking, this prospective style of analysis probably violates one of Fischer's "historians' fallacies." In other words, in judging Song-dynasty Chan by the yardstick of earlier periods, we have programmed in a methodological inability to see it for itself.
Second, if the Song dynasty represents a period of far greater activity and vitality than the Tang or Five Dynasties periods, then the limitations of coverage here do not allow for a reasonable assessment of Songdynasty Chan. At this point in the discussion I am painfully aware of how many aspects of Song-dynasty Chan Buddhism have been left untreated. From the "lettered Chan" of the Northern Song; to patterns of imperial, provincial, and local patronage; to the institutional realities (and fictions) of the monastic system; to the ongoing dialogues between representatives of different lineages, polemical or otherwise; to the elaborations of doctrinal and practice interpretive approaches; to variations of region and social status—to give fair treatment to all the subjects that deserve consideration would take at least one or two sizeable volumes.
And it is not only that our appraisal of Song-dynasty Chan is undergoing change: an important recent contribution by Ned Davis has suggested ways in which the dominant sinological treatment of society and the supernatural during this period needs fundamental revision. That is, instead of dealing with premodern Chinese society in terms of two basic groups, the Confucians and everyone else, we need to look more closely at how actual people actually behaved in dealing with issues involving the supernatural. Davis prefers a tripartite social division, which, in the case of Daoism, he would represent as (1) priests operating at court and other bureaucratic levels, (2) an expanding group of ritual masters, and (3) spiritmediums working at the village level. Evidence regarding Chan during the Song may be restricted largely to the first two levels, which might be defined as, first, formally ordained Buddhist monks and, second, communities of aspirants and practitioners. However, there is evidence from southwest China that Chan rhetoric had been absorbed into a complex mélange of ritual practice, from at least the twelfth century on; we may hope that future work will show the extent and timing of the penetration by such Chan themes of the third social level of village practitioners. It is unlikely that Chan practice itself could be maintained at the village level, since substantial resources are required to support monastic training centers. This does not mean, though, that Chan religious motifs cannot move beyond monastic walls. As one example of just such movement, see figure 4, which shows the image of Bodhidharma worshiped in a local village context in a Bai ethnic community in contemporary Yunnan. The hall in question is dedicated to the three religions and thus has images of kyamuni, Confucius, and Laozi (figure 5). In addition, the local deity Daheitian (Mah k la) sits on the proper left side of the main altar, just as Bodhidharma, identified here by local residents as _zush ,_ "the patriarch," sits on the proper right. In addition to providing better depth of vision on religious practice at different social levels, Davis's work is important for revealing the continuities between pre-Tang and Tang-dynasty practice and Song-dynasty manifestations. Only through analyses like his can we answer the question of whether what seems to appear for the first time in Song-dynasty documents actually indicates a "Tang-Song transition," as specialists in popular religion and culture so often affirm, or simply constitutes an explosion of evidence of long-term continuities from the medieval period. Given these indications that more sophisticated analysis will be possible in the future, the preliminary interpretation offered here is something like the finger pointing at the moon, and it should not be mistaken for a final assessment of all the fundamental themes and exquisite intricacies of Song-dynasty Chan.
FIGURE 4. Bodhidharma worshiped as local deity, Hall of the Three Teachings, Jianchuan, Yunnan Province. Photograph by the author, 1996.
FIGURE 5. Images of kyamuni, Confucius, and Laozi, with Bodhidharma to proper right, Hall of the Three Teachings, Jianchuan, Yunnan Province. Photograph by the author, 1996.
Third, if there was something approximating a climax paradigm to Song-dynasty Chan, this could only be known by looking at how Chan evolves in later periods and in other contexts. To change metaphors, my claim is that Song-dynasty Chan represents the primary lens through which subsequent developments in Chan were understood, whether those developments took place in China, Korea, Japan, or even the modern, non-Asian world. Therefore, in order to appreciate the true dimensions of the "Song-dynasty climax paradigm" we would have to evaluate the dynamics of evolution and transmission that govern Chan in later times and other places. To the extent that the study of Chan/S n/Zen/Thien as a whole has been based on the mistaken romanticism and simplistic thinking manifested so clearly in writing about Chinese Chan, we will have to rework our most cherished theories about these later times and other places as well. What were the constraints—and possibilities—placed on the tradition as it developed in post-Song China, or in Korea, Japan, and Vietnam? If participants in the tradition in those cultures looked through the lens of Song-dynasty Chan, what exactly did they see? In answering this question we will have to consider how the participants in those cultures saw themselves, their own pasts, and the role of Buddhism in their lives. The avenues of inquiry are virtually endless—such exciting possibilities for future research, so many different ways of seeing through Zen.
## Notes
#### Chapter 1: Looking at Lineage
. There is a convenient chart of the Chán patriarchs as given in different sources up to 801 in Philip B.Yampolsky, _The Platform Sutra of the Sixth Patriarch: The Text of the Tun-huang Manuscript with Translation, Introduction, and Notes, 8–9._
. See Robert H. Sharf, "The Zen of Japanese Nationalism."
. Gregory Schopen has pointed out that manifestations of filial piety do not always represent East Asian influence; see his "Filial Piety and the Monk in the Practice of Indian Buddhism: AQuestion of 'Sinicization' Viewed from the Other Side."
. The rhetoric of this first rule echoes Hú Shì's statement that 99 percent of Chán lore was false, suggesting that "it's not true, and therefore we may disregard it." The crucial importance of Chán legendary material is obviously not because it is false, but because it is culturally generated, which renders it "false" from a naïve historicist perspective.
. This is a fundamental Mah y na Buddhist concept that will come up repeatedly. For an excellent overview, see Paul Williams, _Mah y na Buddhism: The Doctrinal Foundations,_ 60–63.
. The Chinese for this is yìnk , literally meaning "[to give one's] seal of approval" and deriving from the basic Sanskrit form mudr . The Chinese compound occurs with the meaning "approval" in the translations of the _Abhidharmakosa_ and _Vimalak rti S tra,_ and with intimations of the Chán nuance in the _Record of the Masters and Disciples of the La k k vat ra,_ an early-eighth-century Chán text.
. Robert H. Sharf has recently argued against the use of "experience" as a natural category of human religion, in "Buddhist Modernism and the Rhetoric of Meditative Experience" and "Experience." Sharf's contribution is especially useful in the observation that the contemporary category of experience derives from a distinctly modern intellectual background that should not be applied indiscriminately to premodern sources, and in the analysis of the philosophical untenability of ostensive definitions of different states of realization. However, it is simply not the case that Chinese Buddhists (or Indian Buddhists, I suspect) had no similar categories for the transformative personal experiences Sharf discusses. Regarding the conspicuous reticence Chán texts display in the description of enlightenment experiences, at least two factors were involved. First, Vinaya regulations established a certain code of silence (see p. 114 and n. 14 to chap. 5). Second, the East Asian tradition demonstrates a disinclination to autobiography, on which see Pei-yi Wu, _The Confucian's Progress: Autobiographical Writings in Traditional China._ Also, Gimello's observation that Buddhist meditation is not so much a set of practices leading to mystical experience, but a style of meditative analysis and psychosomatic enhancement of beliefs through meditative experience, is potentially applicable to Chinese Chán as well. Although Gimello's analysis can be refined further, it applies whether one takes the meditative states referred to as existing ostensively or not. See R. M. Gimello, "Mysticism and Meditation." In the future I hope to address the claims made for the irrationality of Zen and the Zen "enlightenment experience" more closely, but this is a subject beyond the scope of the present book.
. Some of the preceding analysis is given with greater elaboration in John R. McRae, "Encounter Dialogue and the Transformation of the Spiritual Path in Chinese Ch'an."
. See Nancy B. Jay, _Throughout Your Generations Forever: Sacrifice, Religion, and Paternity._ I am grateful to Andrew Junker for introducing me to Jay's work and its relevance to Chinese religion. See Junker's Master's thesis, "Clergy, Clan, and Country: Tang Dynasty Monastic Obeisance and Sacrificial Religion."
. The quotation is from David Hackett Fischer, _Historians' Fallacies: Toward a Logic of Historical Thought,_ 151. For an earlier discussion of the "string of pearls" fallacy, see John R. McRae, _The Northern School and the Formation of Early Ch'an Buddhism,_ 7–8 and 252–53.
. Of course, the D nhuáng caves and manuscripts found there include much more than this, providing insights into a wide range of subjects in Chinese and Central Asian religion and culture, social and economic history, painting and sculpture. For a lively account of the discovery and exploitation of the D nhuáng finds, see Peter Hopkirk, _Foreign Devils on the Silk Road: The Search for the Lost Cities and Treasures of Chinese Central Asia;_ for information about the texts themselves, see the web site of the International Dunhuang Project (www.idp.org).
. Here and elsewhere I use _evolution_ to refer to a generalized process of change over time, without any Darwinian or teleological connotations.
. This concept might be compared to Ortner's "key scenarios." See Sherry B. Ortner, "On Key Symbols"; _High Religion: A Cultural and Political History of Sherpa Buddhism,_ 60 ff.; and "Patterns of History: Cultural Schemas in the Foundings of Sherpa Religious Institutions," 60 ff. I am grateful to Robert Campany for this observation.
. See Bernard Faure, "Bodhidharma as Textual and Religious Paradigm," esp. 193–95.
. See Eric Hobsbawm and Terence Ranger, eds., _The Invention of Tradition._
. For a detailed narration of the history of Chinese Chán, see Heinrich Dumoulin, S. J., _Zen Buddhism: A History,_ 2 vols., especially the revised version of vol. 1. Although Dumoulin's work is a useful resource, I criticize its simplistic and romanticized image of Chán beginning on p. 152 in this volume.
. None of the appended comments can be dated or even associated with known historical figures, and some of the text's accretions may derive from as late as the mid-eighth century or so. For a discussion of the identity and contents of this text (whose attribution to Bodhidharma cannot be accepted simplistically) and a translation of its beginning sections, see McRae, _Northern School,_ 101–17. For a recent translation of the entire work, see Jeffrey L. Broughton, _The Bodhidharma Anthology: The Earliest Records of Zen;_ cf. my review in _the Journal of Chinese Religions._
. One of the many important topics to be explored further is the relationship between Chinese Chán meditation practices and the earlier Buddhist and indigenous Chinese meditation traditions. I have devoted some attention to the former topic in John Robert McRae, "The Northern School of Chinese Ch'an Buddhism" (Ph.D. diss.), 23–30; the reader would do better to consult the brilliant recent contribution by Nobuyoshi Yamabe , _"The S tra on the Ocean-Like Sam dhi of the Visualization of the Buddha:_ The Interfusion of the Chinese and Indian Cultures in Central Asia as Reflected in a Fifth Century Apocryphal S tra" (Ph.D. diss.). On the relationship between Chán practice, Chinese Buddhist meditation, and indigenous Chinese practices, see Harold D. Roth, _Original Tao: Inward Training and the Foundations of Taoist Mysticism._ Roth suggests that there was a pre-Hàn form of "mystical" meditation that very closely resembles the later Chán Buddhist emphasis on non-discrimination. However, he ignores the likelihood that _shén_ or _shénmíng_ , which he translates as "numen" or "numinous" and defines as a category of abstract energy or understanding, simply refers to the spirits that inhabit the body and universe in early Chinese thought. Rather than a precursor to abstract nondualism, _the Inward Training_ may also be read more as adumbrating Daoist styles of meditative practice in which the goal was to visualize the spirits within the body and keep them there. On this, see Isabelle Robinet, _Taoist Meditation: The Mao-Shan Tradition of Great Purity._
. I use the term _school_ because it is usefully vague. It is important to remember that Chinese Buddhist schools had virtually no institutional dimension; hence my strict avoidance of _sect_ or even _denomination_ (although I do use _sectarian)._ See the discussion of this issue beginning on p. 177.
. I am indebted to Jeffrey Broughton for the term _metropolitan Chán._
. The term _encounter dialogue_ was first used in my translation of YANAGIDA Seizan , "The Development of the 'Recorded Sayings' Texts of the Chinese Ch'an School." It renders the Japanese _kien mond _ , which would correspond to Chinese _j yuán wèndá—_ except that the form occurs only rarely in Chán texts and should be understood as a modern descriptive.
. In another context it would be interesting to compare the medieval and modern idealizations of Chán and Zen masters. Here I use _romantic_ in a non-technical sense, without broader historical implications. For a different use of the term, see Dale S. Wright, _Philosophical Meditations on Zen Buddhism,_ ix–x, 13, 16–17, and _passim._ Incidentally, Wright's incisive instructions on the activity of reading texts should themselves be required reading for every student.
#### Chapter 2: Beginnings
. What follows is an abbreviated (and unavoidably selective) recitation of the highlights of the Bodhidharma legend, as found in the _j ngdé chuánd ng lù_ (Record of the transmission of the lamp [compiled during the] J ngdé [Era]), T 51.217a9–20b25. The interview with Emperor W occurs at 219a26 ff., and Huìk cuts his arm off at 219b17. Some of the information about Bodhidharma is included in the entry for Huìk , beginning at 220b26.
. The wonderful ink painting by Young-hee Ramsey used as the frontispiece for this book depicts this legendary event.
. The ability to liberate oneself by means of a simulated corpse is well known from Chinese sources from long before the emergence of Chan. See the explanation in Robert F. Campany, _To Live As Long as Heaven and Earth: A Translation and Study of Ge Hong's_ Traditions of Divine Transcendents, 52–60.
. See Bernard Faure, _The Rhetoric of Immediacy: A Cultural Critique of Chan/Zen Buddhism_ (25 and 27), citing Jacques Derrida, _Of Grammatology._ In Faure's "Bodhidharma as Textual and Religious Paradigm" (197), there is a similar reference to Lévi-Strauss's concept of the character who serves as the "virtual focus" of a myth, who functions as the obscure source around which biographical details proliferate but whose shadow alone is real. (Here Faure refers to Claude Lévi-Strauss, _The Raw and the Cooked,_ 5.) In "Bodhidharma as Textual and Religious Paradigm," Faure derides the quest for biographical accuracy as a sort of "mortuary washing" (188), in which the bones of various specimens are cleansed of detail and strung together to create a useful fiction, a patched-up skeleton, the desiccated remains of a being that never existed. I do not believe that participation in this enterprise is inevitable: the reason I use "It's not true, and therefore it's more important" as the first rule of Zen studies is in effect to differentiate between the analysis appropriate to scholarly inquiry and the mortuary washing of the hagiographical process.
. Bruce Lincoln, in _Theorizing Myth: Narrative, Ideology, and Scholarship,_ points out that, in contrast to the conventional treatment of myth as "a logical structure that essentially writes itself " (149), myth ultimately derives from a narrative process in which multiple people had authorial agency. Also see his reference to "impersonal processes" (18).
. For a detailed consideration of Bodhidharma's biography, teachings, and students, see my _Northern School,_ 15–29. The mid-seventh-century work alluded to here is Dàoxu n's _Continued Transmissions of Eminent Monks (Xù g os ng zhuàn _), which was completed in 645. At this point I am not including the additions to this text, made up until Dàoxu n's death in 667.
. The following enumeration is based largely on the detailed study of the evolution of Bodhidharma's hagiographical sources in SEKIGUCHI Shindai , _Daruma no kenky _ .
. There is no clear and specific antecedent to this myth involving Huìk , but an association may have been drawn with the example of the future kyamuni as "Snowy Mountain Youth" in the _Nirv a S tra,_ who throws his body off a cliff in order to hear the second half of a verse _(g th )._ See T 374, 449b7–51b5, and the discussion in Hubert Durt, "Du lambeau de chair au démembrement: Le renoncement au corps dans le bouddhisme ancien," esp. 8. On such ritual undertakings of self-mortification, see John Kieschnick, _The Eminent Monk: Buddhist Ideals in Medieval Chinese Hagiography,_ 35–50 (the anecdote involving Huìk is mentioned on 41). Kieschnick introduces Victor Turner's notion of "root-paradigm," which he defines as "a set pattern of special behavior with particular symbolic associations." (See Victor Turner, _Dramas, Fields, and Metaphors: Symbolic Action in Human Society,_ 60–97.) Although I would prefer to think in terms of multiple root-paradigms, the self-sacrifice of fingers and arms is potentially applicable to the Chinese case. He writes, "Self-mutilation before relics of the Buddha was not only a sacrifice; it was an appropriation. By burning himself, the adept drew on the power of the Buddha's body, purifying his own body and transforming himself into a holy, living relic" (44). The legend surrounding Huìk no doubt developed out of similar forces, based on the attempts of the proto-and early Chán community to arrogate similar power to their chosen patriarch. For a related discussion (involving only self-sacrifice by immolation), see James A. Benn, "Where Text Meets Flesh: Burning the Body as an Apocryphal Practice in Chinese Buddhism."
. See Bernard Faure, "The Daruma-sh , D gen, and S t Zen." Although _ r ra_ may be generated spontaneously, the primary mode of their creation is through cremation.
. I suspect that further research would push the appearance of this reference back into the twelfth century.
. The reference is to the _Yìj n j ng_ , or _Tendon-Changing Scripture,_ sometimes referred to in English as the _Muscle Relaxing Scripture._ The earliest woodblock printing of this text dates from 1642. Less critical writers on the history of Chinese martial arts accept the validity of its two prefaces, which claim to be from the Táng and Sòng. However, ZH U Jiànnán has shown conclusively that both are later forgeries. See his "Xìngyì quán zh yànji " (A study of Xingyi boxing), 88–89, and "W shù-zh ng Shàolìn-pài zh yàn-ji " (A study of the Shaolin school of the martial arts), esp. 156–57.
. _Sec the New Encyclopedia Britannica,_ 15th ed., s.v. "Bodhidharma." The information given here also occurs in the entry for "Bodhidharma" in _Britannica Online._
. For a complete translation, see Broughton, _The Bodhidharma Anthology._
. T 2060, 50.551c7–12; translation from _McRae, Northern School,_ 103.
. See YANAGIDA Seizan, "Hokush zen no shis " (The thought of Northern-school Chán), 71–72, as explained in McRae, _Northern School,_ 111.
. See Zhìy n's essay on the _Flower Garland S tra_ ( ), T 1870, 45.559a28–b3.
. See Dàoxu n, Xù _g os ng zhuàn,_ T 2060, 50.596c9.
. Paul Swanson, "Wall-gazing, _Vipa yan ,_ and Mixed Binomes." A CBETA (Chinese Buddhist Electronic Text Association; cbeta.org) search on the character _bì_ has revealed only a single use as a transliteration character (T 85.1205b7), and this is from a D nhuáng manuscript of an anthology of scriptural sayings. This might have been a scribal or typographic error, but the term that is being transliterated in this case _is pratyekabuddha_ (given as , more commonly using an initial ), in which a reversal of the _t_ and _k_ sounds by metatheses would allow for the use of bì , which in Early Middle Chinese was pronounced pεjk. For this reconstruction, see Edwin G. Pulleyblank, _Lexicon of Reconstructed Pronunciation in Early Middle Chinese, Late Middle Chinese, and Early Mandarin,_ 34.
. Zhìy , _Great Calming and Contemplation (Mohe zhiguan),_ T 1911, 46.58a18–19. I do not know of any previous association of Zhìy 's _bìdìng_ with Bodhidharma's _bìgu n._
. Zhànrán . _Zh gu n f xìng zhuàn hóngjué ,_ T 1912, 46.305c21–27 and below.
. S ngchóu was a very important personage who was the figurehead of a national meditation center system that formed an important precedent for such temple systems as the K iyuánsì in China and Kokubunji in Japan. See McRae, "The Northern School" [Ph.D. diss.], pp. 31–50.
. The Chinese character in question, _xìng_ , is used in a variety of meanings within Buddhist texts, including _sa sk ra,_ "conditioning force"; the Chinese word means "process" or, in a more general sense, "activity." I have retained the rendering "practice" to maintain the overlap with the word's meaning as "self-cultivation."
. For a convenient explanation of "Dàoxìn's" teachings, but one that does not take into account the issue of retrospective composition, see David W. Chappell, "The Teachings of the Fourth Ch'an Patriarch Tao-hsin (580–651)."
. See McRae, _Northern School,_ 121–22. The quotation does not occur in either Vasubandhu's _Treatise on the S tra on the Ten Stages (Shídì j ng lún _, T 1522, 26.123a1–203b2), in which there is only one reference to the "orb of the sun," _rìlún_ at 26.126a23, a straightforward metaphor for the Buddha's wisdom, nor in any of the versions of the scripture on which this _Treatise_ might have drawn (T 278.22, 279.26, or 285). It is not clear where quotation stops and explanation begins, so the punctuation is arbitrary.
. See McRae, _Northern School,_ 130–31.
. Although, of course, this is an assertion as ancient as the beginning line of Pata jali's _Yogas tra: yoga citta-v tti-nirodha ,_ "spiritual discipline is the cessation of the fluctuations of mind."
. Although the name chosen for the "Chán school" _(chànz ng _) clearly derives from the title used for meditation masters in China _(chánsh_ ), the nuances of the character _chán_ in the medieval Chinese understanding are still unclear. Two intriguing issues are Ti ntái Zhìy 's transition from the use of _chán_ and _chán b luómì (dhy na p ramit , "perfection of concentration") to zh gu n ( amatha-vipa yan ,_ "concentration and insight") and the translator Xuánzàng's near-total avoidance of the word _chán._ The former issue has been commented on frequently, but the latter has not yet been explored, as far as I know. On related issues, see T. Griffith Foulk, "The Ch'an School and Its Place in the Buddhist Monastic Tradition."
. Recent scholarship suggests that Mah y na Buddhism was never as significant in India and Central Asia as it became in East Asia. In order to reflect this situation, as well as to avoid the obvious pejorative connotation of "H nay na," I use "Mainstream Buddhism" to refer to the non-Mah y na schools and traditions of South, Southeast, and Central Asia.
. For a sophisticated analysis of Mainstream Buddhist meditation theory, see R. M. Gimello, "Mysticism and Meditation."
. It would be intriguing to ponder why this is the case—why indeed should understanding imply liberation? This is, I believe, a most fundamental assumption of the Buddhist tradition, so basic that it is simply never addressed. Presumably, in the background of this fundamental Buddhist attitude lie the Vedic concepts of the ritual efficacy of knowledge, the power of naming, and the combined meaning of _√vid_ as "knowing" and "doing." However, this topic must be left for another occasion.
. This treatise _(Wù bù qi n lùn_ ) is one of a set of essays known collectively as the _Book of Zhao (Zhào lùn );_ see T 1858, 45.151a10–14. The scriptural quotation does not occur in the present text of the _Fànggu ng j ng_ or _Light-Emitting S tra,_ T 221, 8.1a–146c29.
. See T NG Yòngtóng , _Hàn Wèi li ng Jìn Nánb icháo Fójiào sh ,_ 334. Also see E. Zürcher, _Buddhist Conquest,_ 88, 89, and 92. The subject of _t /yòng_ is discussed again on pp. 137 and 140; also see this terminology attributed to M z on p. 79 and invoked in the _W f ngbiàn (Five Skillful Means)_ on p. 91.
#### Chapter 3: Metropolitan Chan
. Although the English word _empress_ is gender-specific, in Chinese she was represented during her rule as a full "child of heaven."
. On Empress W 's religious and political identity, see the two excellent volumes by Antonino Forte: _Political Propaganda and Ideology in China at the End of the Seventh Century: Inquiry into the Nature, Authors, and Function of the Tunhuang Document S. 6502, Followed by an Annotated Translation;_ and _Mingtang and Buddhist Utopias in the History of the Astronomical Clock: The Tower, Statue, and Armillary Sphere Constructed by Empress Wu._ For the use of Daoist motifs in the support of her reign, see Stephen R. Bokenkamp, "Medieval Feminist Critique of the Chinese World Order: The Case of Wu Zhao." (Be aware that this publication did not undergo the ordinary authorial review of proofs and is marred by typographical and editorial errors.)
. From _Annals of the Transmission of the Dharma Treasure (Chúan f b o_ jì ) in McRae, _Northern School,_ 51 and 266, with minor Changes.
. From _Record of the Men and Teachings of the La k vat ra (Léngqié rénf zhì_ ), as quoted in the _Record of the Teachers and Disciples of the La k vat ra_ (Léngqié zh z jì ) in McRae, _Northern School,_ 8–9.
. From the epitaph by Zh ng Yuè in _McRae, Northern School,_ 52.
. In 701 Empress W 's power was unquestioned, but her demise was only a matter of time. On Shénxiù as a member of the imperial family and a well-known defender of Buddhism, see McRae, _Northern School,_ 46–50. Of course, Shénxiù was not the only Buddhist monk favored by Empress W (one other notable example is F zàng , a specialist in the _Flower Garland_ tradition).
. I am using the phrase "production of history" in emulation of David William Cohen, _The Combing of History,_ xiii–xxv, esp. xv–xvi.
. The two "Northern school" texts are _the Annals of the Transmission of the Dharma Treasure,_ mentioned in n. 3 above, and the _Record of the Teachers and Disciples of the La k vat ra,_ mentioned in n. 4 above. Actually, the latter text traces the beginnings of the Chán lineage to Gu abhadra rather than Bodhidharma (see p. 26), an aberration ignored by the later Chán tradition.
. The Chinese terms used here are _f ngbiàn_ , from Sanskrit _up ya; zhèng chán'yì_ , a Chinese concoction (see T 50.724c12 for this term in reference to Shenxiu); _gu nx n_ . and _kànx n ._ Luis O. Gòmez, in "Purifying Gold: The Metaphor of Effort and Intuition in Buddhist Thought and Practice," points out that the exegetic strategies of Shénxiù's _Gu nx n lún _ (Treatise on the contemplation of mind) are dissimilar from those of the Northern school's _W f ngbiàn_ (151–52, n. 106); specification of the precise differences remains an important task. He also observes that the metaphoric expressions found in the _Gu nx n lùn_ should not be described as "extended metaphor"—presumably because they represent an _extensive_ use of metaphor but not necessarily _extended_ (i.e., complex, multidimensional) metaphor. However, he misapprehends my earlier suggestion that the strategy of "contemplative analysis" (which he correctly suggests rendering as "contemplative interpretation") is characteristic of the Northern school, taking it to imply that it was the exclusive property ofthat school, which is expressively contrary to the passage he cites from my "The Ox-Head School of Chinese Ch'an Buddhism: From Early Ch'an to the Golden Age," 231–32.
. These are from Shenxiu's _Gu nx n lùn,_ as cited in McRae, _Northern School,_ 199–200. The metaphor introduced under "casting and painting of images" reminds one of Daoist interior alchemy.
. Ibid., 200–201.
. From a memorial to the throne, cited in McRae, _Northern School,_ 53.
. For information on these two figures and their teachings of "enlightenment in this body," see Paul Groner, "Shortening the Path: Early Tendai Interpretations of the Realization of Buddhahood with This Very Body" _(Sokushin j butsu)._ Also see Jacqueline Stone, _Original Enlightenment and the Transformation of Medieval Japanese Buddhism,_ 31–33, and Ry ichi Abé, _The Weaving of Mantra: K kai and the Construction of Esoteric Buddhist Discourse,_ esp. 300–302. A translation of K kai's text on the subject may be found in Yoshito S. Hakeda, _K kai: Major Works Translated,_ 225–34. None of these sources, however, provides significant discussion of K kai's potential indebtedness to Chinese Chán.
. The generic title governing this material is _W _ f ngbiàn . See McRae, _Northern School,_ 172–74. Phrases in italics represent ritualized, almost choral, responses. In the last paragraph, it is not entirely clear how "locus of non-being " _(w ch _ ) is to be understood. The term is used in slightly later Chán texts (i.e., from the late eighth and early ninth centuries) in a manner that invites further analysis; at this point the meaning might simply be "absence of place." The "four tempters" are of course the four M ras; see NAKAMTJRA Hajime _Bukky go daijiten_ (Encyclopedia of Buddhist terms), 532a.
. This is from Shénhuì's _Definition of the Truth;_ see McRae, _Zen Evangelist._
. John Jorgensen has pointed out how Shénhuì's argument against the "Northern school" worked according to the patterns of Chinese ancestral worship, but he does not notice that Shénhuì's lineage hall is modeled on one established by P jì . See Jorgensen's "The 'Imperial' Lineage of Ch'an Buddhism: The Role of Confucian Ritual and Ancestor Worship in Ch'an's Search for Legitimation in the Mid-T'ang Dynasty."
. See McRae, "Shenhui's Vocation on the Ordination Platform and Our Visualization of Medieval Chinese Ch'an Buddhism."
. This is excerpted from the beginning of Shénhuì's _plotform Sermon,_ forthcoming in McRae, _Zen Evangelist._
. This commentator is Hu Shih [Hú Shì] (1891–1962), whose interpretation of Chán is discussed in John R. McRae, "Religion as Revolution in Chinese Historiography: Hu Shih (1891–1962) on Shen-hui (684–758)."
. These distinctions have been introduced earlier, beginning on pp. 32 and 37.
. These citations are all from McRae, "Ox-Head School," 201–3. In the last quotation, "school" renders _z ng _, which originally meant "male primogenitor" and hence came to mean "patriarchal principle" (this is Yanagida's rendering) and eventually "school."
. One of the most common translation errors of modern times is the failure to recognize that _dáo_ , lit. "path," was also used in Chinese Buddhist texts to render _bodhi, dharma, y na, gati,_ and the implied abstract features meaning "-hood" or "-ship" (as in "buddhahood," "arhatship"), and so forth. The most common example is the explanation of the name for martial arts practice facilities, _d j ,_ as "place of the way," when in fact the term _dáoch ng_ is a Chinese translation of _bodhima ,_ the Buddha's "place of enlightenment" under the _bodhi_ tree.
. See McRae, "Ox-Head School," 214–15. The term "counter-illumination" in the last paragraph refers to the enlightened mind's ability to reflect back upon and thus illuminate itself, after the fashion of the setting sun that shines back from across the horizon. Incidentally, the explanation of the "pacification of the mind" at the beginning of this passage might be the source for the story about Bodhidharma's pacification of Huìk 's mind. As already noted (see p. 26), this story first appears in written texts only in 952. The Bodhidharma/Huìk dialog is also a fictional (but thus important because of, not in spite of, its fictionality) scripting of how Chán training and enlightenment might occur.
. See Paul Swanson, _Foundations of T'ien-t'ai Philosophy: The Flowering of the Two Truths Theory in Chinese Buddhism,_ 150–56.
. The technical term for this style of philosophical elevation is _aufhebung_ (abolition, abrogation, annulment) in German, or _zh yàng_ . The Indian philosopher referred to here is Bh vaviveka, who suggests that one first assumes distinctions drawn from ordinary life, then denies those same distinctions, then reappropriates them in a transformed way. See Malcolm David Eckel, _To See the Buddha: A Philosopher's Quest for the Meaning of Emptiness,_ 29–42.
. I have published a simplified version of the following material as "The Story of Early Ch'an." The passages from the _Platform S tra_ here and just below are drawn from the translation by Philip Yampolsky, _The Platform Sutra of the Sixth Patriarch,_ 129–32, with Chánges.
. One of the problems is that the term _ky dai _, "mirror-stand," corresponding to the Chinese _jìngtái _ , occurs as a bound form meaning simply "mirror" in modern Japanese.
. See McRae, _Northern School,_ 235. The English "suchlike" renders the word _rú_ as in the Chinese translation of Tath gata, _rúlái _ , when used as a modifier.
. If this reconstruction of Shénxiù's usage is correct (even approximately), it implies that _the Platform S tra_ verse's presentation has garbled his ideas somewhat. Although this would not be surprising, it makes any integrated interpretation of the verses tenuous. In spite of the speculative interpretation I have given in the text, the _bodhi_ tree and the mirror may be two entirely separate metaphors only awkwardly conjoined in the verse.
. See n. 9 above.
. On the importance _of the Platform S tra_ in the spiritual career of the Korean S n master Chinul, though, see Robert E. Buswell Jr., _The Korean Approach to Zen: The Collected Works of Chinul,_ 23 and 34.
. It is in Shénhuì's writings that the Chán patriarchs are first listed with ordinal numbers indicating their generation.
. Shénhuì certainly would have recounted the anecdote involving the mind-verses if he had known of it. The other arguments stated here are presented at greater length in McRae, _Zen Evangelist._
. The phrasing used here invokes that of Hobsbawm and Ranger, _The Invention of Tradition,_ 199.
. The image of Huineng as a "seeming dullard and diligent sweeper" is mentioned in Michel Strickmann, "Saintly Fools and Chinese Masters (Holy Fools)," 52. Although Strickmann's subject matter here is drawn largely from much later sources, the topos of a student performing manual labor, specifically "sweeping and sprinkling" the floor, for his master goes back at least to Gé Hóng's _Shénxi n zhu n_ . In addition to Gé Hòng's comments about himself, see the example of Chén nshì , which plays on this imagery, in Campany, _To Live As Long as Heaven and Earth,_ 14 and 137–39.
. This interpretation is worked out in greater detail in McRae, "The Legend of Hui-neng and the Mandate of Heaven." On the ideological uses of the Huìnéng legend, compare Bruce Lincoln's definition of myth as "ideology in narrative form" in _Theorizing Myth,_ 147.
. See McRae, _Northern School,_ 344, n. 340, based on T 18.945a22–24.
. Unfortunately, the Chinese esoteric Buddhist tradition is not well studied—I would say, in fact, that it is _the_ least well-studied tradition of East Asian religion. The most important recent exception is Charles D. Orzech's _Politics and Transcendent Wisdom: The Scripture for Humane Kings in the Creation of Chinese Buddhism._ For a critique of Orzech's work, see my review in _Journal of Chinese Religions._
. See ONO Genmy _Bussho kaisetsu daijiten_ , separate volume, 170a–74a.
. Lewis Lancaster made this observation, in personal communications of May 1993 and July 2001. He noticed the phenomenon while preparing, in collaboration with Sung-bae Park, _The Korean Buddhist Canon: A Descriptive Catalogue._ On the Change in Táng policy, see Ono, _Bussho kaisetsu daijiten,_ separate volume, 180b.
. For the date of the conquest of Khotan, see Prods Oktor Skjœrvø, "Khotan, An Early Center of Buddhism in Chinese Turkestan," 290, as well as the sources mentioned at 290, n. 4.
. On the Sòng-dynasty translation bureau, see HUÁNG Q ji ng [Huang Chi-chiang], B i-Sòng di yìj ng rùnwéngu n y Fójiào
#### Chapter 4: The Riddle of Encounter Dialogue
. This is from _j nlìng_ Q ngliáng yuán Wényì Chánsh y lú , T1991,477777.591a24–25. This question became a popular subject in Chán dialogue, with hundreds of exchanges occurring in the literature. See, for example, the _Record of Linji( _ , T1985,47–504a15–18; RuthFuller Sasaki, trans., _The Recorded Sayings of Ch'an Master Lin-chi Hui-chao of Chen Prefecture,_ 46–47; or Burton Watson, trans., _The Zen Teachings of Master Lin-chi: A Translation of the_ Lin-chi lu, 100). Also see YANAGIDA Seizan, "The Life of Lin-chi I-hsuan."
. This brief exchange is most commonly cited from the _Gateless Barrier (Wúmén gu n_ [Wúmén's barrier]), T 2005,48.292C22–93a14. Of its frequent discussions in Dáhúi's recorded sayings ( ), Robert E. Buswell Jr. recommends the treatment atT 1998A, 47.921 C7–19 in his "The 'Short-cut' Approach of K'an-hua Meditation: The Evolution of a Practical Subitism in Chinese Ch'an Buddhism," 369, n. 95.
. This is most commonly cited from _the Emerald Cliff Record _, T2003,48.152C19 ff.
. The information given here is from Robert H. Sharf, "On the Buddha-Nature of Insentient Things (or: How to Think about a Ch'an Kung-an)."
. See IRIYA Yoshitaka , supervising editor, and KOGA Hidehiko , compiler, _Zengo jiten ,_ 3 and 433a.
. On the origins of the English term _encounter dialogue,_ see n. 21 to chapter 1.
. This is only the most elementary form of literary analysis that might be used in the study of Chán encounter dialogue transcriptions. Although encounter dialogue emerged in a highly literate society, it still seems likely that some aspects of the perspective explained by Walter Ong with regard to orality in nonliterate societies might be relevant; previous work on "Zen language" has been marred by (1) a lack of technical understanding of Chinese Chán expressions themselves and (2) overly simplistic notions of orality and narrative realism. See Walter J. Ong, _Orality and Literacy: The Technologizing of the Word._ In addition, given the generation of Chán encounter dialogue in the social and literary context of medieval Chinese culture, it will obviously be necessary to devote more sophisticated attention to Chinese styles of literary interpretation. For example, we should consider the potential relevance of the narratological and historiographical issues discussed in David Schaberg, _A Patterned Past: Form and Thought in Early Chinese Historiography,_ 163–221 and 256–300.
. This translation is drawn from Peter N. Gregory, _Tsung-mi and the Signification of Buddhism,_ 237, with minor Changes.
. See _Transmission of the Lamp,_ T 51.240c19–28. The note "this is Great Master Mazu" at the beginning of this passage is in the original. Jan Nattier has pointed out that, although to feel as though one had drunk ghee (clarified butter) no doubt refers to an exalted spiritual experience, a lactose-intolerant Chinese person might actually have a very different reaction!
. YANAGIDA Seizan, _Sod sh _ (Anthology of the patriarchal hall), 72a14–b3.
. _Vimalak rti S tra,_ T 14.539c18–27.
. There is a more distant parallel in the story of C a Panthaka, in the P li canon, who is instructed by the Buddha to keep rubbing a dirty piece of cloth while reciting "removal of dirt, removal of dirt," until he realized that his anxious efforts at self-cultivation constituted an impediment to his own progress. See Eckel, _To See the Buddha,_ 87. (The use of a dirty rag as a metaphor for purity is well known in East Asian texts, primarily from the _Nirv a S tra.)_
. See Marshall McLuhan, _Understanding Media: The Extensions of Man,_ 22–32.
. Throughout this book I have avoided describing earlier figures as influencing later ones, which would imply a misplacement of the human agency involved. On this subject see Michael Baxandall, _Patterns of Intention: On the Historical Explanation of Pictures,_ 1–11 and esp. his "Excursus against Influence'," 41–73.
. Personal communication, May 1993.
. I cannot explore these deeper connections here. The best work on the _Zhuángzi_ is by Angus Graham; see A. C. Graham, _Chuang-tzu: The Inner Chapters,_ and _Disputers of Tao: Philosophical Argument in Ancient China._ Chinese philosophical discourse reaches something of a pinnacle in lively dialogue in the _Shëshu x ny ,_ on which see Richard B. Mather, trans., _A New Account of Tales of the World._
. The following exposition in the text is summarized from McRae, "The Antecedents of Encounter Dialogue in Chinese Ch'an Buddhism."
. See McRae, _Northern School,_ 36.
. Ibid., 264, 56–59, and 64–65; and see Bernard Faure, _The Will to Orthodoxy: A Critical Genealogy of Northern Chan Buddhism,_ 100–105 and 78–81.
. See McRae, _Northern School,_ 91–95. The Chinese is _zh shì wènyì _; _zh shì_ is also used in traditional Chinese dictionaries to indicate characters whose meanings can be quickly inferred from their shapes, such as those meaning "up," "down," "one," and "two," and so forth. See OGAWA Tamaki , _Kadokawa shinjigen, 413a._
. See McRae, _Northern School,_ 92–93. Although the question "Is this a mind that exists?" resembles one posed near the beginning _of the Perfection of Wisdom in Eight Thousand Lines,_ for example, no Indian antecedent for this genre of questions has been identified. Kum raj va's Chinese translation reads, "Does this mind of no-mind exist?" (T 227, 8.537b15). The reference here is not to any thought or state of mind, but to _bodhicitta,_ the inspiration to achieve buddhahood on behalf of all sentient beings. (A misleading rendering of the Sanskrit may be found in Edward Conze, _The Perfection of Wisdom in Eight Thousand Lines,_ 84.)
. See Sekiguchi, _Daruma no kenky ,_ 335–43, and the comments in McRae, _Northern School,_ 93 and 302, n. 239.
. See McRae, _Northern School,_ 95, 294, n. 161, and 302, n. 243.
. Ibid., 95–96 and 302, n. 244.
. Ibid., 96 and 302, n. 245.
. Ibid., 96 and 302, n. 246.
. The best example of this is Xiàngmó Zàng (d.U.); _see McRae, Northern School,_ 63.
. And of course their somewhat later successors were quite reluctant to explain their activities openly. Perhaps they were profoundly incapable of doing so, for reasons we have not yet thought to explore.
. See McRae, _Northern School,_ 184–85.
. Ibid., 174 (from _five Skillful Means,_ section 1, A). On the English term _suchlike,_ see chap. 3, n. 28.
. Ibid., 175 (same, section 1, D). Jan Nattier wonders (private communication) whether the reference to the "universally 'same' _dharmak ya_ of the Tath gata" here is playing on one of the Chinese renderings of _samyaksambuddha._
. Ibid., 178 (same, section 1, J).
. Ibid., 179 (same, section 1, M).
. Ibid., 180 (same, section 2, A). On the use of italics here, see chap. 3, n. 14.
. I have already discussed the relevance of some of the phraseology here for our understanding of Northern school doctrine and the construction _of the Platform S tra;_ see the conclusion to McRae, _Northern School,_ 238. Other aspects of this material that deserve mention include its bearing on the indebtedness of early Chan to previous formulations within the Chinese Buddhist meditation tradition. The formulations of Ti ntái Zhìy (538–97) were particularly important as background for the emergence of Chan. I have already mentioned (see p. 33) Shénxiù's twenty-five-year residence at Jade Spring Temple (Yùquánsì ), which was previously Zhìy 's place of residence. David Eckel has pointed out (private communication, May 5, 2002) that the pattern suggested here demonstrates a widely shared attitude toward words and their incorporation in Buddhist pedagogical approaches. Specifically, he notes a parallel between this pattern and the Tibetan style of learning first by hearing, then memorization, then active debate. Tibetan debate style is very active, effectively inscribing the subject matter into the students' minds through physical gestures and bodily movements.
. See T. Griffith Foulk, "Myth, Ritual, and Monastic Practice in Sung Ch'an Buddhism," esp. 159–60 and 179–81.
. These titles are abbreviations; for full details see McRae, "Shenhui and the Teaching of Sudden Enlightenment." I no longer maintain the thesis argued in this essay, that these texts preceded Shenhui's writings.
. See John P. Keenan, _How Master Mou Removes Our Doubts: A Reader-Response Study and Translation of the_ Mou-tzu Li-huo lun.
. Translation from KUBO Tsugunari and YUYAMA Akira, trans., _The Lotus Sutra, BDK English Tripi aka_ 13–1, 197–98, with slight Changes in line breaks.
. The phrase "back room" used here is derived from Erving Goffman, _The Presentation of Self in Everyday Life,_ 106–40, esp. 109–13. I have addressed these issues in a paper given at the annual meeting of the Association for Asian Studies in November 1988, entitled "Up Front, Out Back, and in the Field: Three Models of Buddhist Endeavor in East Asia."
. David Eckel (private communication, May 5, 2002) has pointed out that the similarity of the Indian Buddhist conception of the _m rga_ to a board game is no accident, since board games were invented in ancient India and represented an important metaphor by which the world was understood; see A. L. Basham, _The Wonder That Was India: A Survey of the Culture of the Indian Sub-Continent before the Coming of the Muslims,_ 208. For a much later but specifically Buddhist board game, see Mark Tatz and Jody Kent, _Rebirth: The Tibetan Game of Liberation._
. The game of _liùbó_ is not well understood, but one theory is that each player placed six pieces on the board and advanced them against the opponent's forces on the basis of throwing six sticks; _see Ogawa et a1., kadokawa shinjigen,_ 97a.
. For an earlier but more extended treatment of these ideas, see McRae, "Encounter Dialogue and the Transformation of the Spiritual Path."
. Stephen Bokenkamp points out (private communication, March 2002) that this is largely the rule for all Chinese texts, whether rhymed or transcribing speech, in which the capital dialect is taken as the standard.
. See Christian Wittern, _Das Yulu des Chan-Buddhismus: die Entwicklung vom 8.–11. Jahrhundert am Beispiel des 28. Kapitels des_ Jingde Chuandenglu _(1004)._
#### Chapter 5: Zen and the Art of Fund-Raising
. See YAMADA Sh ji, "The Myth of Zen in the Art of Archery."
. The term _Osh _ refers to any senior Buddhist monk, or more specifically one qualified to perform ordinations. In Zen contexts the title occurs most often in daily ritual recitations of the lineage of Buddhas and Patriarchs; no East Asian priest would ever take the title for himself. Incidentally, I am told that "Osho's" books are actually being read sometimes in Japanese Zen training temples today; they are certainly prominently available in Japanese translation in T ky bookstores.
. I am using this English term to cover the period from the end of the Hàn to the unification of China under the Suì. This usage is designed to avoid the northern bias of the standard Chinese terminology.
. The critique of previous scholarship on Chan presented here is similar to that by Foulk in "Myth, Ritual, and Monastic Practice"; see esp. 147–49 and 191–93.
. See Dumoulin, _Zen Buddhism,_ 170–71.
. Ibid., 211. Dumoulin's suggestion that Chinese Buddhist monasteries contributed "little of economic benefit to Chinese society" no doubt derives in part from the anti-Buddhist bias that pervades Jacques Gernet, _Chinese Society: An Economic History from the Fifth to the Tenth Centuries._ (This is a richly annotated update of a 1956 publication, which Dumoulin certainly read.)
. Dumoulin, _Zen Buddhism,_ 212–13. On the nonexistence of any such independent "Zen monasteries," see the discussion beginning on p. 168.
. Ibid.
. Ibid., 243–44.
. On Hú Shì (Hu Shih), see McRae, "Religion as Revolution in Chinese Historiography." Hú's theory was also adopted in works on Chinese Buddhism by Arthur F. Wright and Kenneth K. S. Ch'en (see discussion on p. 175), but in this case that theory has influenced Dumoulin's very understanding of the Chán school itself. On Suzuki, see Sharf, "Zen of Japanese Nationalism."
. The ingot in question bears markings from the rebel side, who either had a similar (albeit undocumented) effort or had simply captured the fruits of the government's program. On ordinations and ordination certificates, see Stanley Weinstein, _Buddhism under the T'ang,_ 59–61 and 65; and Gernet, _Chinese Society, 54–57_
. The term involved is f . Mario Poceski has suggested (private communication, November 2000) that there is some textual ambiguity regarding this citation, but I am unaware of the specifics.
. The geographical pattern referred to here is described in the research of SUZUKI Tetsuo , _T -Godai no Zensh —Konan K sei hen,_ and _T -Godai Zensh shi_, although the inference regarding government policy is mine. Mario Poceski suggests that, in addition to the expansion through Ji ngx mentioned here, the Hòngzh u school demonstrated a remarkable capacity to draw followers and dispatch teachers throughout virtually all of China. If his interpretation of the evidence is correct, this is an interesting parallel with the East Mountain teaching of Dàoxìn and Hòngr n, albeit involving greater numbers of individuals. See Mario Poceski, "The Hongzhou School of Chan Buddhism during the Mid-Tang Period."
. For a monk or nun to make false claims to laypeople of the attainment of _dhy na_ states or the achievement of stream-enterer status (the first of four levels of sagehood in Mainstream Buddhism) was one of the most serious categories _(p r jika)_ of offenses; see Peter Harvey, _An Introduction to Buddhism: Teachings, History, and Practices,_ 225.
. One of my current research interests is the intersection between two of the most massive phenomena of East Asian cultural history: the sinification of non-Hàn peoples from the first millennium B.C.E. onward and the introduction and evolution of Buddhism from the first century B.C.E. onward. Scholars have worked on one or the other of these two issues (see, for example, C. P. FitzGerald, _The Southern Expansion of the Chinese People;_ and Kenneth K. S. Ch'en, _The Chinese Transformation of Buddhism),_ but to date no one has systematically considered the interrelationship between them. A common pattern has been to invoke the rhetoric of sinification without applying any significant analysis of its historical realities. This statement applies to Peter Gregory's excellent study of Z ngmò and Tàng-dynasty Buddhism, _Tsung-mi and the Sinification of Buddhism,_ which, in spite of the title, never addresses the conceptual issues or broader processes involved. Robert Sharf, _Coming to Terms with Chinese Buddhism_ (esp. 77–132), provides some very provocative comments regarding the "process that logically precedes the intentional adaptation and domestication of Buddhism by Chinese apologists" (98), which he refers to as the "hermeneutics of sinification" (132), but he does not consider the actual historical dynamics involved. I believe that we can achieve significant new insights about Chinese religion by considering how participants in East Asian Buddhism were also active contributors to the dynamics of sinification. The terminological distinction between _sinicization_ and _sinification_ adopted here is arbitrary, but it is designed to follow that used in Michel Strickmann, "The Tao among the Yao: Taoism and the Sinification of South China." (Both Gregory and Sharf use "sinification" to refer to what is labeled "sinicization" here.)
. An excellent example of this is the hagiography of Luán B , in Gé Hóng's _Shénxi n zhuàn;_ see Campany, _To Eive as Eong as Heaven and Earth,_ 252–54.
. The most widely used study of this phenomenon is Rolf A. Stein, "Religious Taoism and Popular Religion from the Second to the Seventh Centuries." Other relevant secondary sources are listed in Campany, _To Live as Long as Heaven and Earth,_ 252–53, n. 439.
. Weinstein, _Buddhism under the T'ang,_ 147.
. See Jan Nattier, _Once Upon a Future Time: Studies in a Buddhist Prophecy of Decline,_ 130–31 and 227.
. This impression is speculative; the very brief preface to the text does not provide a good explanation of its origins.
. Although I will not argue the point here, the government's appetite for officially recognizing "transmission of the lamp" texts probably fit within the hegemonic desire of its ministers to rule all aspects of the society within their purview.
. These are the _Record of the Transmission of the Lamp [Compiled during the] Jingde [Period] (J ngdé chuánd ng lú_ , 1004); _Extensive Record of the Lamp [Compiled during the] Tiansheng [Period] (Ti nshèng gu ngd ng lú_ 1036); _Supplementary Record of the Lamp [Compiled during the]Jianzhong Jingguo[Period] (Jiánzh ng jìngguò xúd ng lú_ , 1101); _Outline of Linked Lamps (Liánd ng_ huìy o , 1183); _Record of the Universal Lamp [Compiled during the] Jiatai [Period] (Ji tái p d ng lú_ , 1204); and _Collated Essentials of the Five Lamp_ [Records] _(W d ng huìyuán ,_ 1252). The extent to which successive dynasties used Chán as the Mìn regime had during the Five Dynasties period, in an attempt to standardize Buddhist practice, remains to be seen.
. The two-stage structure of Sòng-dynasty Chán genealogies resembles that of domestic family genealogies during the same period, which also tend to begin with monolineal successions followed by a cascade of subdivisions. See Johanna M. Meskill, "The Chinese Genealogy as a Research Source," esp. 143–47; and Patricia Ebrey, "The Early Stages in the Development of Descent Group Organization." I am grateful to Lynn Struve for making this observation and providing these references (personal communication, May 2002).
. See Holmes Welch, _The Practice of Chinese Buddhism, 1900–1950;_ T. Griffith Foulk, "Myth, Ritual, and Monastic Practice in Sung Ch'an Buddhism"; and YIFA, _The Origins of Buddhist Monastic Codes in China._ For a discussion of Sòng government policies towards Buddhism and their effect on the Chán school, see Morten Schlütter, "Vinaya Monasteries, Public Abbacies, and State Control of Buddhism under the Sung Dynasty (960–1279)."
. Welch, _Practice of Chinese Buddhism,_ 41.
#### Chapter 6: Climax Paradigm
. The definition of _climax_ in Roger J. Lincoln, Geoff Boxshall, and Paul Clark, _A Dictionary of Ecology, Evolution, and Systematics_ is "a more or less stable biotic community which is in equilibrium with existing environmental conditions and which represents the terminal stage of an ecological succession; sometimes used as a synonym of formation _q.v._ " (61b).
. It may also be the case that Chinese Buddhism as a whole also achieved something of a climax paradigm as well, perhaps only after that of Chán and the other major Chinese Buddhist schools. However, we have not considered the entirety of Chinese Buddhism in this book, so we cannot make unduly broad claims about it here. To understand Chinese Buddhism during the Sòng, we would obviously have to consider the Pure Land and Ti ntái schools, not to mention the worship of Gu ny n (Avalokite vara) Bodhisattva and other vectors of religiosity not ordinarily covered under the heading of "school." Chán is but one important species within the ecological inventory.
. See Arthur F. Wright, _Buddhism in Chinese History;_ Kenneth K. S. Ch'en, _Buddhism in China: A Historical Survey;_ Jacques Gernet, _Chinese Society: An Economic History from the Fifth to the Tenth Centuries;_ and Wm. Theodore de Bary, _East Asian Civilizations: A Dialogue in Five Stages._ For an extended discussion of these issues, see McRae, "Religion as Revolution in Chinese Historiography."
. I am using the word _community_ to represent what may be a heterogeneous combination of features; like _school,_ however, it represents an appropriate level of precision (or vagueness) for the present purposes. (On the terms _school, sect,_ and _sectarian,_ see chap. 1, n. 19.)
. As this book goes to press I understand that Foulk has at least one manuscript accepted for publication, which I have not yet seen.
. MIZUNO K gen , in "Zensh seiritsu izen no Shina no zenj shis shi josetsu" [Introductory explanation of meditation theory in China prior to the formation of the Chán school], esp. 17–18, has tabulated the percentage of monks listed as meditation specialists ( _chánsh ) in the G os ng zhuàn_ of 518 C.E., at 16 + %; the _Xù g os ng zhuàn_ of 667 C.E., at 45 + %; and the _Sòng g os ng zhuàn_ of 978 C.E., at 36 + %, adjustable to around 65–70% due to the suffusion of meditation specialists in other categories.
. The following account is based closely on the work of Miriam Lindsey Levering, "Ch'an Enlightenment for Laymen: Ta-hui and the New Religious Culture of the Sung."
. On this important figure, see Urs App, _Master Yunmen: From the Record of the Chan Teacher "Gate of the Clouds."_
. See Levering, 24–25, with minor changes. The line about the _x n_ wind in the palace quotes an exchange of poetry at the court of Emperor Wénz ng (r. 821–41) of the Táng; see the _Jìu Táng sh _ 169–4312a. (The "East Mountain" mentioned here is not that at Huángméi.) The description of Dàhuì's training under Yuánwù is seamlessly consistent with Dàhuì's later teachings, but for the present purposes I ignore any possibility of retrospective projection.
. For analysis of the methodological issues involved in extracting historical data from normative sources, see Jan Nattier, _A Few Good Men: The Bodhisattva Path According to_ The Inquiry of Ugra _(Ugrapariprcch ),_ chap. 3, "The Ugra as a Historical Source: Methodological Considerations."
. See Morten Schlütter, "Silent Illumination, Kung-an Introspection, and the Competition for Lay Patronage in Sung Dynasty Ch'an."
. The translation is from Levering, "Ch'an Enlightenment for Laymen," 38, with minor changes.
. Translation by Miriam Levering, "Miao-tao and Her Teacher Ta-hui," 201, from T 47.865C24–28, with transcription changes.
. Quoted in Buswell, "'Short-cut' Approach," 349, based on the _Dàhuì y lù_ 26, T 1998A, 47.921C2–6.
. On the emergence and definition of _kànhuà Chán,_ see Buswell, "'Short-cut' Approach," 344–56; Ding-hwa Evelyn Hsieh, "Yüan-wu K'o-ch'in's (1063–1135) Teaching of Ch'an Kung-an Practice: A Transition from Literary Study of Ch'an Kung-an and the Practical K'an-hua Ch'an"; and Ding-hwa Evelyn Hsieh, "A Study of the Evolution of K'an-hua Ch'an in Sung China: Yüan-wu K'o-ch'in (1063–1135) and the Function of Kung-an in Ch'an Pedagogy and Praxis."
. There is occasional confusion about these terms, which are important to the discussion that follows. The characters for _huàtou_ mean literally "topic-head," but the second character is used purely as a grammatical suffix meaning "a (prominent) bit of [the preceding character]." A parallel example is _shítou_ , "a piece of stone," that is, "a rock." In modern colloquial Chinese, including that of Buddhist monks, _a g ng'àn_ simply means story, and in vernacular Chinese literature it refers to a genre of mystery stories. The compound originally referred to the desk on which a magistrate would place a case for legal consideration, so that by metonymy the word means "legal precedent."
. _Emerald Cliff Record,_ T 48.194C7–95a13. The translation is adapted from Katsuki Sekida, _Two Zen Classics: Mumonkan and Hekiganroku,_ 319–20. The interlineal glosses are as found in the original.
. See the secondary sources introduced in chap. 2, n. 8.
. There are two source texts of the _Bìyán lù_ (Emerald cliff record), which may derive from the two separate occasions on which Yuánwù lectured on the cases in question. Given the marked variation between the two source texts, it seems best to characterize Yuánwù's interpretation as a style of response rather than a set of interpretations. See the critical edition in IT Y ten , _Hekigansh teihon_ .
. See Buswell, "'Short-cut' Approach," 321–77.
. The term _consociation_ is adopted from the research of Avron Boretz (personal communication, May 2002), which is based primarily on ethnographic research on contemporary Taiwanese religious praxis. When dealing with the outside world, the consociation adopts a pose of unanimity behind its leader/figurehead; within the organization, however, there is intense and uninterrupted competition by individuals for status within the group. Depending on the context, either the entire Chán lineage (in cases of national religious dialogue, for example) or the various sublineages of Chán (in cases of individual monastic appointments, for example) might be considered separate consociative groups. I look forward to exploring this anthropological analogy between medieval Chán and contemporary Chinese religion in the future.
. Translated by Ari Borrell, in his _"Ko-wu_ or _Kung-an?_ Practice, Realization, and Teaching in the Thought of Chang Chiu-ch'eng," 88, with minor changes (i.e., the omission of Chinese terms); from ZH NG Ji chéng , _Héngp jí_ , 14.7a (391b).
. Ibid., 89; from the same source, 14.4a–b (390a).
. I am drawing the terminology here from Borrell, who in turn cites an observation made by KUSUMOTO Masatsugu (1896–1963), in "S -Min ry shis no katt " (The dilemma of Sòng and Míng thought), especially 177. I have not seen this article (and my interpolation of Japanese characters and interpretation of the title may be in error), but it is the earliest such observation of which I am aware.
. Schlütter, "Silent Illumination," 109.
. Morten Schlütter resolves this dilemma in a two-pronged analysis, first, of the state of competition between the Líh and Cáodòng lineages at this point in time and, second, of the different emphases placed by various Cáodòng masters on meditation practice. The following summary is based on Schlütter's masterful analysis, which draws in part on the contributions of ISHII Sh d .
. Although Schlütter suggests that this arrangement was "unique in Ch'an history" ("Silent Illumination," 127), something like this must have happened in the evolution of the Oxhead school—unless that faction's lineage scheme was completely fabricated around its sixth generation.
. For Dàhuì's emphasis on effort, see Buswell, "'Short-cut' Approach," 354–55.
. Translated by Schlütter, in "Silent Illumination," 124–25; from _Xù g z nsú y y o_ , _Xù zàng j ng_ 118.453d11–16.
. The three passages here are translated in Schlütter, "Silent Illumination," 121–22 and 124; they are from _Zh nxi Q ngli o chánsh y lù_ , X 124.314a18-b2, 323C13–14, and 311b3, respectively.
. Translation by Schlütter, in "Silent Illumination," 117, with minor reformatting, from _Hóngzhì chánsh gu nglù_ ( ), T 2001, 48.100a26–b1.
. Ibid., 118, with minor reformatting, from _H ngzhì chánsh gu nglù,_ T 48.100b5–11.
. On the _t /yòng_ distinction, see the discussion on p. 70. Shénhuì states his understanding of the relationship between meditation and wisdom in the _Platform Sermon:_ "This is the combined cultivation of concentration and wisdom, [the two of] which cannot be separated. Concentration does not differ from wisdom, and wisdom does not differ from concentration, just as a lamp and its light cannot be separated. . . . When we consider concentration, it is the essence of wisdom; when we consider wisdom, it is the function of concentration." See McRae, _Zen Evangelist._
. Schlütter, "Silent Illumination," 119.
. Ibid., 123; _from Hóngzhì chánsh gu nglù,_ T 48.1C2–3.
. Ibid., 123–24; from _Hóngzhì chánsh gu nglù,_ T 48.74b25-c2 (emphasis in original). This passage is cited in ISHII Sh d , _S dai zensh shi no kenky _ [Studies in the history of Sòng-dynasty Chán], 345, and translated in Taigen Dan Leighton, _Cultivating the Empty Field: The Silent Illumination of Zen Master Hongzhi,_ 10.
. The word _practice_ here should be understood as "process" or "activity"; see chap. 2, n. 22.
. I am grateful to Robert Campany for this observation (personal communication, May 2, 2002).
. One recent study that flagrantly violates this good-sense rule is a monumental foray into neurophysiology and introspection: James H. Austin, M.D., _Zen and the Brain: Toward an Understanding of Meditation and Consciousness._ In Tables 1 and 2 (pp. 10 and 31), for example, Austin carries out the uncritical generation of parallels to a ludicrous extreme, applying highly value-laden generalizations to the S t and Rinzai traditions. The highly sectarian nature of Austin's understanding of Zen severely prejudices his massive edifice of neurophysiological hypothesis, which extends far beyond the realm of experimental evidence. For a perceptive comment on a significant epistemological contradiction in Austin's enterprise, see Arthur J. Deikman's review in the _Times Literary Supplement,_ 30. (At the time of the review Deikman was professor of psychiatry at the University of California, San Francisco.) Hopefully, some middle way between Sharf's and Austin's approaches will be possible in the future (see chap. 1, n. 7).
. As Robert Buswell has pointed out (private communication, May 10, 2002), "Chán materials from both the Táng and Sòng provide a much more sophisticated analysis of these terms than simply sudden and gradual: note the multiple combinations of 'sudden' and 'gradual,' 'cultivation' and 'enlightenment' employed in the works of Z ngmì , Yánshòu , and Chinul in Korea to analyze different Chán schools."
. See Buswell, "'Short-cut' Approach," and Dàhuì's use of short-cut rhetoric itself.
. We should not mistake these positions as entirely East Asian, let alone unique to Chán. For example, Peter Harvey, in _The Selfless Mind: Personality, Consciousness and Nirv na in Early Buddhism,_ points out that the notion of an innately illuminating mind within all sentient beings is also found in P li canonical and postcanonical sources, as well as the texts of several Mainstream schools (155–79, esp. 157–60 and 174–75).
. This statement refers to the appreciation of these positions within the Chán tradition. In Indian Mainstream Buddhism it may even be the case that an experienced meditation instructor is more important for the practice of _ amatha_ rather than _vipa yan ._
. For a convenient summary of these two positions, see Chi-wah Chan, "Chih-li (960–1028) and the Crisis of T'ien-t'ai Buddhism in the Early Sung," 413–18.
. Brook Ziporyn, "What Is the Buddha Looking At? The Importance of Intersubjectivity in the T'ien-t'ai Tradition as Understood by Chih-li," 443 (original passage in italics). The subject matter of this article is given more detailed attention in Brook Ziporyn, _Evil and/or/as The Good: Omnicentrism, Intersubjectivity, and Value Paradox in Tiantai Buddhist Thought,_ 199–239.
. Translation from Ziporyn, "What Is the Buddha Looking At?" 459, with minor punctuation changes; from _Shí bù'èr mén zh y o ch o_ , T 1928, 46.718a10–12. (The same passage is also introduced in Ziporyn, _Evil and/or/as The Good,_ 218.)
. Ziporyn, "What Is the Buddha Looking At?" 459.
. Ibid., 454. Also see the very similar passage in Ziporyn, _Evil and/or/as The Good,_ 212–13.
. Translation from Ziporyn, "What Is the Buddha Looking At?" 460, from _F huá shìmiào bù'èr mén shìzh zh _ , X 100.111na15–17.
. In posing this question, I am indebted to Peter D. Hershock, _Liberating Intimacy: Enlightenment and Social Virtuosity in Ch'an Buddhism,_ 31–39. He, in turn, has drawn his Indian material from Dakshinaranjan Shastri, _Origin and Development of the Rituals of Ancestor Worship in India,_ 290–98, and his Chinese material from Patricia Ebrey, _Confucianism and Family Rituals in Imperial China,_ 16–23. Although I have some doubts about the success of Hershock's project as a whole, his use of the analogy of funerary practice to initiate a comparative analysis of Indian and Chinese religious practice was very insightful. See McRae, review of Hershock, as well as Hershock, rejoinder to McRae, and my surrejoinder.
. This refers to the forthcoming book by Robert DeCaroli, _Haunting the Buddha._
. See John Jorgensen, "The 'Imperial' Lineage of Ch'an Buddhism."
. On conventional family genealogies, see Ebrey, _Confucianism and Family Rituals._ With additional study of Sòng-dynasty recorded sayings literature, we may recognize intralineage efforts at identity creation that parallel those of individual family genealogies.
. See Daniel A. Getz Jr., "T'ien-t'ai Pure Land Societies and the Creation of the Pure Land Patriarchate."
. In order to generate discussion, my statement of these propositions here is intentionally stark and contentious. I do not have the space to consider them fully here, and I look forward to their future elaboration and/or modification by other scholars.
. Mary Douglas, _Natural Symbols: Explorations in Cosmology._
. See Jay, _Throughout Your Generations Forever._
. See chap, 1, n. 7, and n. 39 to this chapter for two very different approaches to individual religious experience.
. Of course, as an author, I am deeply interested in the extent to which my ideas—and phrasings—are adopted by readers and colleagues. Here it is best to let the Chinese nuns of Fógu ng Sh n have the last word: At the end of my seminar there in 1992 they gave me a thank-you card that included the touching sentiment, "Professor McRae, what you teach is not true—and therefore it's more important!"
. Fischer, _Historians' Fallacies,_ xx-xxii.
. See Edward L. Davis, _Society and the Supernatural in Song China,_ esp. 7–8.
## Character Glossary
n Lùsh n (Roxanna):
_Annals of the Transmission of the Dharma Treasure: Chúan f b o jì_
_Anthology of the Patriarchal Hall: Z táng jí_
Bái
B izhàng Huáih i:
_bìdìng:_
_bìgu n:_
body: see "essence"
_Book of Zhao: Zhào lùn_
_bú lì wénzi, jiàowài biézhuàn:_
Cáodòng (Japanese: S t ) school:
Cáoq (Cáox ):
Cáosh n B njì:
Chén nshì:
Chinul:
Ch ngyu n:
Chùjì: (= Reverend Táng )
Dàh iti n:
Dàhuì Z ngg o:
Dàoxìn:
Dàoxu n:
_Definition of the Truth (Definition of the Truth [Regarding] Bodhidharma's Southern School): Pútídámó nánz ng dìng shìf i lùn_
D gen:
Dòngsh n Liángjiè:
D nhuáng:
East Mountain teaching: _d ngsh n f mén_
Eastern Peak: ( = Mount Tài )
_Emerald Cliff Record: Bìyán lù_
Emperor G oz ng:
Emperor Wénz ng:
Emperor W of the Liáng:
Empress W : (W Zhào ; W Zéti n )
essence (body)/function: _t /yòng_
_f ngbìan:_ , "skillful means, expedient means"
F q n:
F rú:
F y n:
F yún:
F zàng:
Five Dynasties: W dài
Five Houses: _w ji _
_Five Skillful Means: W f ngbiàn_
four practices: _sì xíng_
Fújiàn:
function: _yòng_
_fury monji:_ see _bú lì wénzi_
_Gateless Barrier (Wumen's Barrier): Wúmén gu n_
Gé Hóng:
_go_ :
_g ng'àn:_ (Japanese: _k an_) "precedent"
_gu n:_ , _vipa yan _, "insight"
Hakuin Ekaku:
Hán Z c ng:
Hóngr n:
Hóngzhì Zh ngjué:
Hóngzh u:
Hóngzh u school:
HU Shin [H Shì]:
Huángméi: ( = Qízh u ), Hubei
Húb i:
Huìk :
Huìnéng:
Huìzh n:
Húnán:
Jade Spring Temple: Yùquánsì
Ji ngx :
Ji orán:
_jiáowái biézhuán:_
J n dynasty:
Jìngsh n (Temple):
J ngxián:
J ngzh u:
_kànhuà Chán:_
_kien mond :_ (Chinese: _j yuán wèndá_)
_k an:_ see _g ng'án_
K kai:
_ky ge betsuden: see, jiàowài biézhuàn_
L o' n:
Layman Páng:
_Light-Emitting [Perfection of Wisdom] S tra: Fànggu ng j ng_
Lín' n:
Línj [Rinzai] school:
Línj Yìxuán:
Li Z ngyuán:
Luòyáng:
M z Dàoy
Miàodào:
Mín:
_mòf :_
Mount A oka Temple:
Mount S ng:
Mount Ti ntái:
Móuzi:
_mòzhào:_
Musang: (Chinese: Wúxi ng; = Rev. Kim [Chinese: J n héshàng])
Nánquán P yuàn:
Nányáng Huìzh ng:
Nányuè Huáiràng:
Néngrénsì: see kyamuni Temple
_Nirv na S tra:_
nonthought: _wúniàn_
Northern Zh u:
_Platform Sermon (Platform Sermon by the Reverend of Nanyang on Directly Comprehending the Nature According to the Chan Doctrine of the Sudden Teaching and Emancipation): Nányáng héshàng dùnjiào ji tu chánmén zh li oxìng tány _
_Platform S tra of the Sixth Patriarch: Liùz tán j ng_
P jì:
Q ngli o: see Zh nxi Q ngli o
Q ngyuan Xíngs
Qízh u: see Huángméi
_Record of the Men and Teachings of the La k vat ra: Léngqié rénf zhì_
_Record of the Teachers and Disciples of the La k vat ra: Léngqié sh z jì_
_Record of the Transmission of the Lamp (Compiled during the) Jingde (Period): J ngdé chuánd ng lù_
Reverend J n (Korean: Kim): see Musang
Reverend Táng: see Chùjì
_rìlún:_
Rinzai: see Línj
Rúh i:
Saich :
kyamuni Temple:
S ngcàn:
S ngchóu:
S ngzhào:
Sh nd ng:
sh nji :
sh nwài:
Shaolin Temple: Shàolínsì
Sháozh u:
Shénhuì:
Shènxiù:
_shìf ng c nglìn:_
Shìtou X qi n:
_sh ux n: _
Shu ngf ng:
Sìchu n:
S t school (Japan): see Cáodòng
ubh karasi ha: Shànwúwèi
_S tra of the Contemplation of the Buddha Amit yus: Gu n Wúliàngshòu F j ng _
Suzuki, D[aisetsu] T[eitar ]:
Suzuki Tetsuo:
T ng Yòngtóng:
Tànlín:
_Tendon-Changing Scripture: Yìj n j ng_
t : "essence, body"
Ti nníngsì:
Ti ntái [Tendai] school:
Ti ntái Zhìy :
_Transmissions of Eminent Monks (Compiled During the) Song (Dynasty): Sòng g os ng zhuàn_
_Transmissions of Treasure Grove (Temple): B olín zhuàn_
_Treatise on the Contemplation of Mind: Gu nx n lùn_
_Treatise on the Essentials of Cultivating the Mind: Xiùx n y o lùn _
_Treatise on the Immutability of Things: Wù bù qi n lùn _
Treatise on the Sùtra on the Ten Stages: Shídì j ng lùn
Treatise on the Transcendence of Cognition: Ju gu n lùn
Treatise on the Two Entrances and Four Practices: Èrrù sìxíng lùn
two entrances: _èrrù_
Ui Hakuju:
wall contemplation: _bìgu n_
_wéiqí:_
Wùxi ng: see Musang
Wù Zéti n: see Empress W
W z F y n:
xi nchéng g ng'àn: (Japanese: genj k an)
Xiángmó Zàng:
x nf :
xíng:
Xuánl ng:
Xuánsù: _ _
Xuánzàng:
Xu dòu Zhòngxi n:
Xu f ng Yìcún:
x n
YAMADA Sh ji:
YANAGIDA Seizan:
Yìfú:
_yí_ nk :
Y xìng:
_yòng: , "function"_
Yòng: Xuanjue:
Yuánq ng:
Yuánwù Kèqín:
Yúnmén Wény n:
Yùnnán:
Yúquánsì: see Jade Spring Temple
Zh ng Ji chéng:
Zh ng Sh ngy ng:
Zhànrán:
Zhào: lùn
Zh nxi Q ngli o:
zh : , _ amatha,_ "concentration"
_zh gu n: , amatha-vipa yan ,_ "concentration-insight"
Zh l :
_zh shì wènyì:_
Zhu ngzi:
_z ng:_
Z ngmi:
z sh : , "patriarch"
## Bibliography
#### PRIMARY SOURCES
_Emerald Cliff Record (Fóguo Tudn wit Biyán lù ): T 2003,
48.339a1–225C14._
five _Skillful Means (Wü fängbian ):_ McRae, _Northern School,_ 171-96 (English; for textual information, see 327–30, n. 161). Onems. is printed atT 2834, 85.1273b9–78a7.
_Great Calming and Contemplation (Mdhe zhiguän ): T 1911, 46.1a1– 140C19._
_Light-Emitting Sütra (TangßUänß böre jing ): T 221, 8.1a1–146c29._
_Perfection of Wisdom in Eight Thousand Lines Qüaopin bore böluöml jing ): T 227, 8.536C15–586C7._
_Record of the Transmission of the Lamp [Compiled in] thejingde [Period] (Jingde chudndeng lit ): T 2076, 51.196b9–467a28._
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## Index
abbot position, lineage-based restrictions,
Abé, Ry ichi, 162n13
Abhidharmako 'a, 155n6
"activating the mind," , ; in _Treatise on the Transcendence of Cognition_ ,
American Buddhist converts, preconceptions,
Amoghavajra, ,
An Lushan rebellion, , ,
nanda, (fig. 1),
ancestral time,
anecdotes and dialogues, use in teaching, 93–94
_Anthology of the Patriarchal Hall (Zutang ji)_ , (fig. 2), , , , , , , 109–14
App, Urs, 172n8
Austin, James H., 174n39
Bai ethnic community, Yunnan, (incl. fig. 5)
Baizhang Huaihai, , ,
Baozhi,
Barrett, T. H.,
Basham, A. L., 168n41
Baxandall, Michael, 166n14
Benn, James A., 159n8
_biding_ ("wall concentration," in _Great Calming and Contemplation)_ ,
_biguan_ ("wall contemplation"), 29–31; comments on by Daoxuan,
bimodalities: in Chan thought, , , ; in _Five Skillful Means_ , ; in _Treatise on the Two Entrances and Four Practices_ ,
board games, as metaphor for "Chinese mårga paradigm," 97–98
_bodhi_ , , , , ,
bodhicitta, ,
Bodhidharma, , (fig. 1), , , , , (fig. 2), , , 21–28, , , , , 37–40, , , , , , , , , , , , , , , , , , , , , ; biography, ; death, ; "duel" relationship with Sengchou, ; Emperor Wu, encounter with, , ; and Huike, "pacification of the mind" dialogue, ; floating across Yangzi River on a reed, ; hagiography, 22–24, 26–28; image worshiped as local deity, (fig. 4), 152–53; leaves single shoe in grave, ; legendary attacks on, ; nine years in meditation , , , ; "transmission verse,"
_bodhi_ tree, , , , , , , ,
Bokenkamp, Stephen R., 161n2, 168n44
Borrell, Ari, , 173n22
Boxshall, Geoff, 171n1
breath concentration, as meditation practice,
_Britannien, New Encyclopedia_ , 159n12
Broughton, Jeffrey L., 157nn17 and , 159n13
_bu li wenzi_ ("to not posit words"),
Buddha-nature, (fig. 2), , , , , , , , , , , , , , ; definition ; of dogs or cats, ; in _Platform S tra_ "mind-verse,"
Buddhism: Indian, as mortuary religion, ; Mah y na, , , , , , , , , , ; Mainstream, ; role in Chinese history,
Buswell, Robert E., Jr. , 164n31, 165n2, 172n14–15, 173n20, 174n28, 175nn4–41
Campany, Robert F., 156n13, 158n3, 164n52, 170nn16–17, 174n38
Caodong (S t ) school, (fig. 1), (fig. 2), , , , , , , , , , , , , , ; effort in meditation practice, 135–38; parallel with Neo-Confucianism, ; Song dynasty, 133–38; teachers,
Caoqi, (map 1),
Caoshan Benji, (fig. 1), , ,
_cban_ , as transliteration of dhy na ("concentration meditation"),
Chan: Chi-wah, 175n44; genealogical quality, 4–5, ; genealogical system, ; historians' interpretations of, ; lineage, ; literature, xix; mythology, 5–6; phases of, defined, 11–12, (fig. 2); retrospective identity of phases of, 14–15 sectarian identity, ; spiritual practice (genealogical nature), 7–8; style of teaching, 86–88; transmission, . _See also_ early Chan; middle Chan
Chan Buddhism, and Chinese social order, 145–50
Chang'an, (map 1), , , , ,
Chang'an and Luoyang, , , , , , ; cultural importance, 36–27, 45–46, ; as setting for emergence of Chan school,
"Chan meaning," in Shenxiu's interpretation,
Chappell, David W., 160n23
Ch'en, Kenneth K. S. 169n10, 170n15, 171n3; interpretation of Chan in Chinese history,
Chinese history, role of Buddhism in,
Chinese m rga paradigm,
Chinese renaissance, theory of Hu Shi (Hu Shih),
Chongyuan, ,
Chuji,
Clark, Paul, 171n1
classical Chan, , , , , , , , ; definition in terms of "encounter dialogue," ; definition in terms of Song-dynasty image of Tang-dynasty Chan, , ; relationship to middle Chan,
climax community, ecological metaphor for Song-dynasty Chan,
climax paradigm, as metaphor for Song-dynasty Chan, , ; metaphor, as explanatory strategy, 150–54
Cohen, David William, 162n7
concentration: on breath, ; on hyperactive mental processes, ; and insight,
concentration meditation: _dhy na_, , ; _ amatha_ , ,
Confucianism, ,
Confucius, , ,
consciousness, flowing, as object of concentration,
consociation, ,
"constant practice," as Shenxiu's teaching, ,
"contemplation of the mind," in Shenxiu's teachings,
Conze, Edward, 167n21
cosmology, shared Chinese,
"counter-illumination," ; in _Treatise on the Transcendence of Cognition_ ,
"critical phrase" ( _huatou_ ), 127–28; Chan,
Dahui Zonggao, (fig. 1), (fig. 2), , 123–38; banishment, 125–26; biography, 123–26; efforts to dominate Song-dynasty Chan, ; frequent use of term "sudden," ; "viewing the phrase" Chan, 131–32
Daoism, religious, ,
Daoist priests, role in sinification,
Daoxin, (fig. 1), , , , , 33–38, , , ; retrospective attribution of teachings, 37–38
Daoxuan, comments on "wall contemplation, "
Davis, Edward L., 176n61; on Song-dynasty religion, 151–54
"a day without work means a day without food,"
Dayang Jingxuan,
de Bary, Wm. Theodore, 171n3; interpretation of Chan in Chinese history,
DeCaroli, Robert, 175n51
degeneration, image of Song-dynasty Chan,
Derrida, Jacques, 158n4; and "trace,"
Dewey, John,
_dharmadh tu_,
_dharmak ya_,
Dharma-nature,
_dhy na_ ("concentration meditation"), , ,
_Diamond S tra_, ,
"disciples temples," 115–16
D gen, ,
Dongshan Liangjie, (fig. 1), , , ,
Douglas, Mary, , 176n56
dualism, in _Five Skillful Means_ ,
dualistic extremes, avoidance of, in Zhenxie Qingliao's teachings,
dual structure, _Treatise on the Two Entrances and Four Practices_ ,
duel (meaning both "dual" and "duel"; also see "bimodalities"), , , ,
Dumoulin, Heinrich, S. J., , 103–7, , 157n16, 169n5–10; interpretation of Chan in Chinese history, 103–7,
Dunhuang, (map 1); caves and documents, 156n11
Dunhuang documents, xiv, , , (fig. 2), , , , , , , ; meditation practices in, 52–53; variety of Chan materials for early Chan,
Dunhuang version, _platform S tra_, ,
Durt, Hubert, 159n8
early Chan, (fig. 2), , , , , , , , , , , , , 91–94, , , , ; experimental formulations of, ; stability of location,
Eastern Peak (Mount Tai), (map 1), East Mountain (Huangmei), , _99_
East Mountain teaching, , , 33–36, , , , ; definition and features, 33–36; Hongren as central figure of, ; retrospective quality, , ; spiritual lifestyle, 35–36; origin of term,
Ebrey, Patricia, 171n23, 175n50, 176n53
Eckel, Malcolm David, 164n25, 166n12, 168nn35 and
_Emerald Cliff Record (Biyan lu)_ , (fig. 2), , , ; burned by Dahui, ; case , "Nanquan Cuts the Cat in Two," 128–30; composition and format, 130–31
Emperor Gaozong of the Tang,
Emperor Wu of the Liang, , , , ,
Emperor Wuzong of the Tang,
Empress Wu of the Tang, invitation of Shenxiu to court, 46–48
emptiness _( nyat ):_ and Chan transmission, ; rhetoric,
encounter dialogue _(kien mond )_, (fig. 2), , , , , 72–80, , , , , , 92–100, , , , , , , ; in _Anthology of Patriarchal Hall_ , , ; as "back room" practice, ; defined, 77–78; editorial change during transmission of, ; emergence and definition of "classical Chan," 18–19; genealogical structure, ; as "hot medium," ; literary characteristics of, , , ; literature, ; and mismatch of students' and teachers' intentions, ; as paradigm of spiritual cultivation, ; performative quality of, ; presentation of as nonfictional event, 77–78; as socially oriented practice, , 89–92; as transcribed oral texts, , , 99–100; use of Chang'an colloquial Chinese, ,
encounter dialogues, Buddha-nature of dog, , ; characteristics, 75–76; illogicality, 75–76, ; Mazu and Huairang rubbing a tile, 80–82, ; meaning of "Buddha," , ; meaning of patriarch coming from west, ,
encounter paradigm,
"end period of the Dharma" _(mofa)_ ,
enlightenment _(dao)_ , ; originary (or fundamental), ,
enlightenment experience: descriptions of in Indian and Chinese texts, ; fictional accounts of, 58–59, 94–95
"enlightenment in this body" _(sokushin j butsu)_,
"entrance of practice," , , , , ; "four entrances," 32–33
"entrance of principle," , , , , , , , ,
"entrances, two,"
esoteric Buddhism, impact on Chan, 69–70
"essence" _(ti,_ "body"),
"essence/function" _(ti/yong),_ ,
_Essential Determination,_
exemplification approach, , ,
faith, in Buddhism,
fallacies, historical, ; archetypes, ; "great man," ; "string of pearls," , 10–11, , , , 156n10
Faqin,
Faru, ; lineage statement in epitaph for, ; potential connection with _Treatise on the Essentials of Cultivating the Mind,_
Faure, Bernard, 156n14, 158n4, 159n9, 167n19
Fayan school, (fig. 1)
Fayun,
fictional depiction of enlightenment experience, in _Treatise on the Transcendence of Cognition,_ 58–59
Fischer, David Hackett, , 156n10, 176n60
FitzGerald, C. P., 170n15
Five Dynasties (Chan), ; period,
"five houses" of Chan, (fig. 1), , (fig. 2)
_Five Skillful Means (Wu fangbiari),_ , , 89–92
Forte, Antonino, 161n2
Fotu Deng,
Foulk, T. Griffith, , 161n27, 168n36, 169n4, 171n24, 172n5
"freeze the mind to enter concentration . . ." (Shenhui), ,
Fujian (province),
"function" _(yong),_ , ; "essence/function" _(ti/yong),_ ,
"fundamental wisdom" and "successive wisdom," in _Five Skillful Means,_ 89–90
fund-raising, Buddhist, after Five Dynasties period,
funerary practice, China, 146–47; India 145–46; India and China, 145–48
Furong Daokai, ,
fury monji (Chinese: _bu li wenzi),_
gender-specific terminology in Chan, 8–9,
genealogical model, 7–8
_genj k an_ ("precedent of manifest creation"),
Gernet, Jacques, 169nn6 and , 171n3; interpretation of Chan in Chinese history,
Gimello, Robert M., 156n7, 161n29
_go_ (Japanese game),
Goffman, Erving, 168n40
"golden age," as term for middle Chan,
G mez, Luis O., 162n9
_gong'an_ ("precedent"), , , 128–30
Graham, A. C., 166n16
_Great Calming and Contemplation (Mobezbiguan),_
"great enlightenment" _(dadao),_
Gregory, Peter N., 166n8, 170n15
Groner, Paul, 162n13
_guan (vipa yan ,_"insight meditation"), Guangzhou (Canton),
Guiyang school, (fig. 1)
Gu abhadra,
Hakeda, Yoshito S., 163n13
Hakuin Ekaku,
Han Zicang, ,
Harvey, Peter, 170n14, 175n42
Heavenly Peace Temple (Tianningsi),
Herrigel, Eugen,
Hershock, Peter D., 175n50
H n yana,
Hobsbawm, Eric, and Terence Ranger, 157n15, 164n34
Home-mountain thought: Home-mountain/Off-mountain distinction, ; parallel with "critical phrase" Chan, ; of Zhili,
Hongren, (fig. 1), (fig. 2), , , , , 33–38, , , , , , , , , , , , , , , , , , ; legendary image, ,
Hongzhi Zhengjue, (fig. 1), (fig. 2), , 133–38, ; silence/illumination (meditation/wisdom) polarity,
Hongzhou, (fig. 2), (map 1),
Hongzhou school, , , ; and _Transmissions of Treasure Grove,_ 109–10; historical features of, 108–11; legendary image of, ; members as ideal figures, 108–9; pattern of geographical expansion, ; role in sinification,
Hopkirk, Peter, 156n11
"houses" of Chan (five), (fig. 1), , (fig. 2)
Hsieh, Ding-hwa Evelyn, 172n15
Hu Shi (Hu Shih), , , ; 155n4, 163n19, 169n10; and interpretation of Chan in Chinese history,
Huairang, Nanyue, (fig. 1), 80–83; biography,
Huang Chao rebellion, , ,
Huangmei, Hubei Province, (map 1), , 33–37, ,
Huang Qijiang (Huang Chi-chiang), 165n42
_huatou (wat ,_"critical phrase"), ,
Huayan school,
Hubei (Province), (fig. 2)
Huike, , (fig. 1), (fig. 2), , , , , , , ; as central figure of proto-Chan, ; cuts off own arm,
Huineng, (fig. 1), , , , , (fig. 2), , , 34–36, , , , , , 60–68, , 81–85, 94–96, , ; historical and legendary identities, 68–69; legendary image of, 68–69, , ; "mind-verses" in _Platform S tra,_ ; sixteen years in hiding, ; as source of legitimation for Shenhui,
Huizhen,
Hunan (province),
iconoclasm, of legendary image of Huineng, 68–69
iconoclastic rhetoric, in fund-raising,
immanentist position, , , ,
_inka (yinke,_ "seal of approval"), , 155n6
_Inscription on Silent Illumination (Xinxin ming),_ 133–34,
insider/outsider distinction in Chan studies,
insight meditation _(vipa yan ),_ , ; defined, 41–42; objects of,
intersubjectivity, Song-dynasty Tiantai, 142–45
"investigating things," Neo-Confucian, ,
Iriya Yoshitaka, 166n5
Ishii Sh d , 173n26, 174n36
It Y ten, 173n19
Jade Spring Temple (Yuquansi), , , , __
Jay, Nancy B., , , 156n9, 176n57
Jianchuan, Yunnan Province, (map 1), (fig. 4) , (fig- )
Jiangnanxi Dao, (map 2)
Jiangxi (Province), , ,
Jiaoran,
Jin Dynasty, (map 2), , ,
Jingshan (S kyamuni Temple, Nengrensi), ,
Jingxian,
Jingzhou, (map 1), ,
Jinling, (map 1)
Jorgensen, John, , 163n16, 175n52
Junker, Andrew, 156n9
Jurchen (Jin dynasty),
K kai,
Kaifeng, (map 1)
Kaiyuansi (Hongzhou),
K yapa, (fig. 1),
Keenan, John P., 168n38
Kent, Jody, and Mark Tatz, 168n41
Kieschnick, John, 159n8
_k an (gong'an),_ ; practice,
Koga Hidehiko, 166n5
Kubo Tsugunari and Yuyama Akira, 168n39
Kusumoto Masatsugu, 173n24
_ky ge betsuden (jiaowai bie zhuan,_"separate transmission outside the teachings"),3
La k vat ra S tra, , ,
"lamps of eternal brightness" metaphor,
Lancaster, Lewis R., 165n40
Lao'an, , __
Layman Pang,
Leighton, Taigen Dan, with Yi Wu, 174n36
Levering, Miriam Lindsey, , , , 172n9
L vi-Strauss, Claude, 158n4
_Light-Emitting Perfection of Wisdom S tra (Fangguang jing),_
Lin'an, (map 1),
Lincoln, Bruce, 158n5, 165n36
Lincoln, Roger J., 171n1
lineage assertions, xix
lineage diagram: and distortion of religious identities, ; and identity of Indian buddhas and Chinese patriarchs,
lineage halls, establishment by Puji and Shenhui,
lineage system, Chan, and conventional family genealogies, ; Pure Land school, ; unilineal and multilineal sections 114–15
Linji (Zhenzhou), (map 1)
Linji (Rinzai) school, (fig. 1), , (fig. 2),, , , , , , , ,
Linji Yixuan, (fig. 1), (fig. 2), , , , , ,
_liubo_ (Chinese game),
Liu Zongyuan,
_Lotus S tra,_ 4; dragon king's daughter, Luoyang, (map 1), , , , , , , , ,
M dhyamika,
"maintain (awareness of) the mind" _(shouxin),_ , ; in _Treatise on the Essentials of Cultivating the Mind,_
_m rga_ ("spiritual path"),
Mather, Richard B., 166n16
Mazu Daoyi, (fig. 1), (fig. 2), , , , , , , , , , , , ; Buddha-nature doctrine, ; legendary image, ; official residence in Hongzhou,
McLuhan, Marshall, , 166n13; on hot and cold media, ; "the medium is the message,"
meditation practice, Caodong school, merit, religious,
Meskill, Johanna M., 171n23
"method of the mind" _(xinfa),_ in Zhang Jiucheng's thought,
metropolitan Chan, , , , ,
Miaodao, ,
middle Chan, (fig. 2), , , , ; relationship to classical Chan,
Min (Southern Tang; Fujian Province), (map 2),
mind, in Caodong Chan and Neo-Confucianism,
"mind-ground, purify the" (in _Five Skillful Means),_
"mind-verses" _(xinjie):_ attributed to Huineng, ; basis for both in Northern school writings, ; echo in dialogue between Mazu and Huairang, ; interpretation of, 63–65; in _Platform S tra,_ , ; as unknown to Shenhui,
mirror, as metaphor, , , ; in _Platform S tra,_
Mizuno K gen, 172n6
_mofa_ ("end period of the Dharma"),
monastic institution, Chan administrative predominance, 102–3, 117–18; legendary ascription to Baizhang, ; operation in twentieth century, , ; Song-dynasty operation, 115–17; "transmission of the lamp" texts and management of power/patronage,
"motionlessness," in _Five Skillful Means,_
Mount A oka Temple,
Mount Lu, (map 1)
Mount Song, (map 1), , , ,
Mount Tai (Eastern Peak), (map 1),
Mount Tiantai, (map 1), ,
Mount Zhongnan, (map 1)
Mouzi, Musang (Wuxiang, Reverend Jin or Kim),
Nakamura Hajime, 163n14
"Nanquan Cuts the Cat in Two" _(Emerald Cliff Record_ case), 128–30
Nanquan Puyuan, , , , 128–31
Nanyang Huizhong,
Nanyue Huairang, (fig. 1), 80–83; biography,
Nattier, Jan, , 166n9, 167n31, 170n19, 172n10
Nengrensi ( kyamuni Temple, Jingshan), ,
Neo-Confucianism, xx, , , , , 138–41; Cheng-Zhu faction, ; parallel with Caodong Chan,
_Nirv na S tra:_ account of Snowy Mountain Youth, 159n8; quoted by Shenhui, ; quoted in early Chan "questions about things,"
"no mind" _(wuxin),_ in _Treatise on the Transcendence of Cognition,_
"non-activation of the mind," in _Five SkillfulMeans,_ ,
"nonthought" _(wunian),_ in critique of Northern school meditation practice, ; in _Treatise on the Transcendence of Cognition,_
Northern school, (fig. 1), , (fig. 2), , , , , , 54–58, , 66–68, , , , , 87–89, 91–93, _,_ ; as label created by Shenhui, , , 54–55,
Northern Zhou,
North/South Dynasties period, ,
"not a single thing": in _Five Skillful Means,_ , , ; in Northern school style Dunhuang manuscript, ; in Northern school writings, ; in _Platform S tra_"mind-verse,"
Off-mountain thought, ; parallel with "silent illumination,"
Ogawa Tamaki, 168n42
_Once Upon a Future Time_ (Nattier),
Ong, Walter J., 166n7
"only," as important qualifier, 29–30
Ono Genmy , 165nn39–40
oppression of religious practitioners, role of Chan in, ,
ordination, gold ingot from fees, ;Shenhui's role in support of,
Ortner, Sherry B., 156n13
Orzech, Charles D., 165n38
Oxhead school, (fig. 1), (fig. 2), , 56–58, , , , , ; defined,
"pacification of the mind dialogue," Bodhidharma and Huike, ; as archetype of Chan practice,
"pacify the mind" _(anxin),_ in _Treatise on the Transcendence of Cognition,_
Park, Sung-bae, 165n40
patriarchs: six (Chinese), , , ; twenty-eight (Indian), , , ,
"perfection of wisdom" _(praj p ramit ),_ ,
persecution, of Buddhism: Huichang period (845), , , ; depiction of by Dumoulin, 104–6; Northern Zhou (574),
Pirsig, Robert,
_Platform S tra of the Sixth Patriarch (Liuzu tanjing),_ (fig. 2), , , , , 60–69, , , ; appearance, ; as capstone text of early Chan, 66–67; Dunhuang version, ; as evidence for institutional realities, ; fictitious quality of opening narrative, ; textual evolution,
Poceski, Mario, 169nn12–13
poisons (greed, hatred, and ignorance),
polarities: in Buddhist discourse, , 43–44, , ; in Chan discourse, 39–40, , , , , 136–38, , ;
"practice" _(xing),_
_praj _ ("wisdom"), ,
_praj p ramit _ ("perfection of wisdom"), ,
precedent anthologies, Song-dynasty compilation,
precedent of manifest creation _(xiancheng gong'an, genjo koan),_
precepts, pure,
printing and dissemination of Chan,
proto-Chan, (fig. 2), , , , , , , , , , , ,
public monasteries (shifang conglin), , ; "teaching" (i.e., Tiantai), 117–18; Vinaya,
Puji, ,
Pulleyblank, Edwin G., 160n18
Pure Land school, , ; lineage system,
"pure mind," in _Five Skillful Means,_
Qingyuan Xingshi, (fig. 1),
Qizhou (Huangmei),
Quanzhou, (map 1) "questions about things," 85–86
"quiet sitting," Neo-Confucian,
Rajneesh, Bhagwan Shree,
Ranger, Terence, and Eric Hobsbawm, 157n15, 164n34
rebellion: An Lushan, , , , Huang Chao, ,
_Record of the Masters and Disciples of the La k v tara (Lengqie shizi ji),_ , 155n6
_Record of the Transmission of the Lamp (Compiled in) the jingde (Period) (Jingde chuandeng lu),_ , , , , , , ,
"recorded sayings" (or "discourse records," _yulu),_ texts
"reside fixedly without wavering," ,
retrospective attribution: of Chan teachings, , ; of proto- and early Chan sources,
Reverend Kim (Reverend Jin, Musang),
Reverend Tang (Chuji),
Rinzai school (Japan), , , ,
ritualized dialogue, as antecedent for encounter dialogue, 92–93
Robinet, Isabelle, 157n18
Roth, Harold D., 157n18
Ruhai,
rule of rhetorical purity, , ,
rules of Zen studies: no. 1, xix, , , , , , , 155n4, 158n4; no. 2, xix, , ; no. 3, xix, ; no. 4, xx, , 157–58n22
sacrifice of finger or arm, ritualized,
Saich ,
kyamuni, (fig. 1), , , , , , , ,
kyamuni Temple (Nengrensi, Jingshan), ,
amatha ("concentration meditation"), , , ,
riputra, ,
Sasaki, Ruth Fuller, 165n1
Schaberg, David, 166n7
Schlütter, Morten, , 172n11, 171n24, 173nn25–26, 174nn27, 29–32, , 35–36
"school," and Chan lineage system,
Schopen, Gregory, 155n3
"seal of approval" _(yinke, inka),_ , 155n6
seated meditation, , ; definition by Shenhui, ; in "questions about things," ; Vimalak rti scolding riputra for,
sectarian identity of Chan, 121–22
Sekida, Katsuki, 173n17
Sekiguchi Shindai, 159n7, 167n22
Sengcan, (fig. 1), , , ,
Sengchou, ,
Sengzhao,
"separate transmission outside the teachings" _(jiaowai biezhuan),_ ; as polemical move,
"seven Buddhas of the past," (fig. 1), , , ,
Shandong (province),
Shaolin Temple (Shaolinsi), (map 1), , , ,
Shaozhou, (map 1),
Sharf, Robert H., 155n2, 169n10, 165n4, 170n15, 174n39; on "experience," 155–56n7
Shastri, Dakshinaranjan, 175n50
Shenhui, (fig. 2), , , , , ; absence of stories about Huineng, ; concept of Chan "monosuccession," ; contribution to use of anecdote in Chan, ; corpus, transcribed dialogue in, ; and crisis in early Chan, , ; creation of label "Northern school," ; definition of seated meditation, ; and _Diamond S tra,_ ; doctrine of the identity of concentration and wisdom, ; as evangelist, , ; four criticisms of Northern school meditation practice, , ; as fund-raiser, , ; identification of Huineng and Shenxiu with north and south, respectively, ; ignorance of Huineng's biography, , ; ignorance of _Platform S tra_ "mind-verses," ; image in Hu Shi's (Hu Shih's) writings, ; and polarity, ; simple valuation of sudden and gradual, , , ; as storyteller and public speaker, 93–94
Shenxiu, (fig. 1), , , (fig. 2), , , , , , , , 47–52, 54–58, 61–68, , , , , ; "contemplation of the mind" _(guanxin),_ , ; explication of "Chan meaning" of S tras, ; historical identity, ; metaphor, use of, 49–51, ; "mind-verse" in _Platform S tra,_ , , ; "perfect teaching," ; personal charisma, ; seven "dharmas of the bath," 50–51; "skillful means," ; timing of training under Hongren,
Shitou Xiqian, (fig. 1), (fig. 2), , , ,
Shuangfeng (Huangmei),
Sichuan (Province),
ik nanda,
"silent illumination" _(mozhao),_ , , , , , ; parallel with Neo-Confucian "quiet sitting," ; parallel with Tiantai Off-mountain thought,
Silk Road, , ; end of transmission of texts along,
"sinicization,"
"sinification," 110–11
"skillful means" _(fangbian):_ in _Five Skillful Means,_ ; used by Shenxiu,
Skjærvø, Prods Oktor, 165n41
Song dynasty, , (fig. 2); and Dahui's move south, , and decrease in government support
Song-dynasty Buddhism, ; image of degeneration of, xx,
Song-dynasty Chan, (fig. 2), , , , , , , , , ; Caodong school, 133–38; as climax paradigm, 119–23; as pinnacle of Chinese tradition, ; relationship with Tiantai doctrine, ; spontaneity,
S t school (Japan), ,
Southern school, (fig. 1), , (fig. 2), , , , , , , , , , , , ; as minor voice in early Chan, 67; origins in activities of Shenhui, ; use of title by "Northern school" practitioners,
Southern Song Dynasty, (map 2);
spontaneity: and patterning in Chan, ; in Song-dynasty Chan,
_Spring and Autumn Annals,_
Stein, Rolf A., 170n17
Stone, Jacqueline, 162n133
Strickmann, Michel, 164n35, 170n15
Struve, Lynn, 171n23
ubh karasimha,
"suchlike" (characterized by suchness),
"suchness" _(ru, tathat ),_ ,
sudden and gradual, ,
"sudden enlightenment" _(dunwu),_ doctrine of, interpretation by Hu Shi (Hu Shih),
"sudden teaching" _(dunjiao,_ subitism), , ; of Shenhui, ; transmission to Huineng in _Platform S tra,_
sun: in _Treatise on the Essentials of Cultivating the Mind,_ , ,
sun-and-clouds metaphor, , ,
"sun of enlightenment," 42–43
"sun of nirv ña,"
nyat ("emptiness"), , , , ,
_S tra of the Contemplation of the Buddha Amit yus (Guan Wuliangshou Fo jing),_
Suzuki, D. T., , , , , ; methodology and influence,
Suzuki Tetsuo, 169n13
Swanson, Paul, , 164n24
Tang Yongtong, , 161n32
Tang-dynasty Buddhism, political dimensions, 111–12
Tang-dynasty Chan, (fig. 2), , , , , ; image as "classical" age, ; legendary image, xx; romantic image, 103–5, as Song-dynasty creation,
Tang-Song transition,
Tanlin, ,
Tatz, Mark, and Jody Kent, 168n41
_Tendon-changing Scripture,_ 159n41
_Ten Stages (Treatise on the S tra of the),_
three truths, in Zhiyi's thought, ; in Tiantai school doctrine,
threefold structure: and Oxhead-school thought, ; and _Platform S tra,_ ,
_ti_ ("body," "essence"),
Tianningsi (Heavenly Peace Temple),
Tiantai (Tendai) school, , , , , 115–18, , , 142–45, ; doctrine, relationship with Chan, ; inter-subjectivity, 143–45; Song dynasty,
Tiantai Zhiyi, , , , , , ,
Touzi Yiqing,
transcendence, in _Five Skillful Means,_ , ,
translation of Buddhist scriptures, end of 71–72
transmission: concept, ; diagram, 4–9
"transmission of the lamp" texts, , , , , , , , 113–14, , , ,
_Transmissions of Eminent Monks (Compiled During the) Song (Dynasty) (Song gaosong zhuan),_
_Transmissions of Treasure Grove (Temple) (Baolin zhuan),_ , , , 109–10
_Treatise on Believing in Mind (Xinxin ming),_
_Treatise on the Contemplation of Mind (Guanxin lun),_
_Treatise on the Essentials of Cultivating the Mind (Xiuxin yao lun),_ 36–38, 40–43, , , , , , ; elaboration of basic themes in _Treatise on the Two Entrances and Four Practices,_ , ; meditation techniques, 39–40; polarity of vigor and composure 39–40, ; potential involvement of Faru, ; retrospective attribution to Hongren,
_Treatise on the Immutability of Things (Wu bu qian lun),_
_Treatise on the Transcendence of Cognition (Jueguan lun),_ , ; threefold structure,
_Treatise on the True Principle,_
_Treatise on the Two Entrances and Four Practices (Erru sixing lun),_ (fig. 2), , 28–33, , , , , , , , ; contents, ; date, ; disregard by later Chan, ; dual structure, ; entrance of principle, ; as focal point of proto-Chan,
True Nature (Buddha-nature), , ; in _Treatise on the Two Entrances and Four Practices,_
Turner, Victor, 159n8
two entrances, as introvertive and extrovertive,
Ui Hakuju,
Vajrabodhi,
Verellen, Franciscus, 169n6
vernacular speech, use in encounter dialogue as literary characteristic,
"viewing the critical phrase" Chan _(kanhua Chan, kanna Zen),_ , , , , ; description of, 126–32
"viewing the mind," in _Five Skillful Means,_ 52–53
_Vimalak rti S tra,_ , 155n6
Vinaya,
_vipa yan _ ("insight meditation"), , , , , ,
visualization, in "viewing the topic" Chan,
"wall concentration" _(biding),_
"wall contemplation" _(biguan),_ 29–31
Watson, Burton, 165n1
Watts, Alan: _The Way of Zen,_
Weinstein, Stanley, , 169n11, 170n18
_weiqi_ (Chinese game),
Welch, Holmes, 171nn24–25
Williams, Paul, 155n5
"wind of wisdom," in _Treatise on the Essentials of Cultivating the Mind,_ ,
Wittern, Christian, 168n45
women, in Song-dynasty Chan, ,
Wright, Arthur F., , 169n10, 171n3
Wright, Dale S., 158n22
Wu, Pei-yi, 156n7
Wuxiang (Musang; Reverend Jin or Kim),
Wu Zetian, Wu Zhao (Empress Wu of the Tang), 46–48
Wuzu Fayan,
Xiangmo Zang,
_xinfa_ ("method of the mind"), in Zhang Jiucheng's thought,
Xuanlang,
Xuansu,
Xuanzang,
Xuedou Zhongxian, ,
Xuefeng Yicun, (fig. 2)
Yamabe, Nobuyoshi, 157n18
Yamada Sh ji, 168n1
Yampolsky, Philip B., 155n1, 164n26
Yanagida Seizan, , , 157n21, 159n15, 163n21, 165n1, 166n10
Yangzi River, (map 1), (map 2)
Ye, (map 1)
Yellow River, (map 1), (map 2)
Yifu,
_yinke_ ("seal of approval"), , 155n6
Yixing,
_yong_ ("function"), ; essence/function _(ti/yong),_ ,
Yongjia Xuanjue,
Yuanqing, Off-mountain thought,
Yuanwu Keqin, , , , , ,
Yunmen school, (fig. 1)
Yunmen Wenyan, ,
Yuquansi (Jade Spring Temple), , , ,
Yuyama Akira, and Kubo Tsugunari, 168n39
_Zen and Japanese Culture,_
_Zen and the Art of Archery,_
_Zen and the Art of Motorcycle Maintenance,_
_Zen Doctrine of No-mind, The_
Zen monasteries,
"Zen," in contemporary usage 101–2
Zhang Jiucheng, , ,
Zhang Shangying,
Zhanran,
Zhaozhou, , , ,
Zhenxie Qingliao, , 133–36
_zhi ( amatha,_ "concentration meditation"),
Zhili, ,
Zhiyi, , , , , 142–44
Zhou Jiannan, 159n11
Zhuangzi,
Ziporyn, Brook, , ; 175nn45–49
Zongmi,
Zürcher, Erik, 161n32
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Hadamar Memorial Museum
International Place of Remembrance
Services for groups
The Correctional Institution of Hadamar (1883–1906)
Hadamar State Sanatorium (Landesheilanstalt Hadamar) (1906–1933)
The "T4" programme and the Hadamar killing centre (1941)
"Decentralised euthanasia" and the Hadamar killing centre (1942–1945)
Liberation of the killing centre (1945)
Remembering and remembrance
Applying for funding to visit a memorial
Archive and collections
The estate (Nachlass) of Ernst Klee
Memorial Museum
LWV Hessen as the responsible body
Memorial association of Hadamar Memorial Museum
The long road to the Memorial Museum
The manner in which those who were murdered at the Hadamar killing centre are remembered developed in stages after the end of the war in 1945 and was interrupted by long periods of suppression. It was not until the exhibition was opened in the basement of the historical building in 1983 and Hadamar Memorial Museum was founded several years later that a continuity in remembering and remembrance was established.
It was American occupying soldiers, not Germans, who first made the crimes in Hadamar public. After liberation on 26 March 1945, a special US Army unit investigated the crimes in Hadamar. Newspapers and weekly newsreels in Great Britain and the US then began reporting about the killing centre. Already in 1945, a first trial was conducted before a US military court in Wiesbaden.
Although the crimes were illuminated early on, they were soon suppressed and sometimes denied, even in the state sanatorium that continued to exist. Even the relief that was inaugurated in the entrance hall of the institution in 1953 to commemorate those murdered could not change this general attitude. What was typical for this period was that the message on the relief remained vague. Only the inscribed years 1941 to 1945 pointed to the "euthanasia" murder phase in Hadamar.
The relief of 1953 in the entrance hall as it looks today. Photo: Hadamar Memorial Museum/Tanja Wesel
The areas of the institutional cemetery, where those murdered were buried in mass graves between 1942 and 1945, were transformed into a memorial landscape that was solemnly inaugurated in September 1964. Since then, a monument stands there bearing the cautionary inscription "Mensch achte den Menschen", meaning "Man, honour mankind". The relief, the memorial landscape and the monument exist there to this day. But from a contemporary perspective, they did not lead to lasting remembrance work. This observation applies not only to Hadamar, but also to remembering those murdered in the course of Nazi "euthanasia" crimes as a whole. They remained among the forgotten victims in Germany for decades.
It was not until the 1980s that the social climate changed. All across Germany, questions began being raised about connections to the Nazi past. In Hadamar, several initiatives coincided. The management of the hospital's psychiatric centre secured the files of those murdered and began assessing the past, researchers prepared a first major study and a group of students and graduates from the University of Giessen opened an initial exhibition in the basement of the former killing centre in 1983.
In the 1980s, this desire to shed light on the past and foster continuous remembrance was also met with resistance and refusal. However, as the responsible body of the psychiatric hospital in Hadamar, the Public Welfare Association of the German Federal State of Hesse (Landeswohlfahrtsverband Hessen, LWV) nonetheless established a memorial in the historic building of the former killing centre. With its expansion and the opening of a larger exhibition in 1991, the memorial was given its present name: Hadamar Memorial Museum.
The mission of Hadamar Memorial Museum is to keep the memory of those that were murdered alive. This can only be achieved in that the remembrance of those murdered as well as the work of confronting the crimes and their consequences is understood as the responsibility of society as a whole.
Literature: Jan Erik Schulte, Hadamar – Geschichte eines deutschen und internationalen Erinnerungsortes der nationalsozialistischen "Euthanasie", in: Geschichte in Wissenschaft und Unterricht 72 (2021), p. 175–195.
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\section{Introduction}
Throughout the article, $K$ always denotes an algebraically closed field of characteristic zero, and ``algebra'' refers to an associative $K$-algebra with identity. (We will remark at times on particular results that hold in arbitrary characteristic.) All modules are assumed to be finite-dimensional left modules. We use interchangeably the vocabulary of modules over finite-dimensional algebras, and that of representations of quivers with relations. A summary of the background on these, and their varieties of modules, is given in Section \ref{sect:background}.
\subsection{Motivation and context}
The representations of an algebra $A$ can be studied geometrically by considering the affine varieties $\operatorname{mod}(A,\operatorname{\mathbf{d}})$ of modules of fixed dimension vector, under the actions of the corresponding products of general linear groups $\operatorname{GL}(\operatorname{\mathbf{d}})$. In this setting, isomorphism classes of representations are precisely orbits, so the standard tools of algebraic geometry for determining orbits of a group acting on a variety become relevant.
We are interested in classifying those algebras whose module varieties satisfy certain invariant-theoretic properties, and to compare such classification results with the classical representation theoretic properties of these algebras, in particular, the notion of finite representation type.
In this paper, we introduce the following properties for an algebra $A$:
\begin{description}
\item[DO property]
for each dimension vector $\operatorname{\mathbf{d}}$ of $A$, $\operatorname{GL}(\operatorname{\mathbf{d}})$ acts on each irreducible component of $\operatorname{mod}(A,\operatorname{\mathbf{d}})$ with a dense orbit;
\item[MF property]
for each dimension vector $\operatorname{\mathbf{d}}$ of $A$ and each irreducible component $C$ of $\operatorname{mod}(A,\operatorname{\mathbf{d}})$, the algebra of semi-invariants $K[C]^{\operatorname{SL}(\operatorname{\mathbf{d}})}$ is multiplicity-free.
\end{description}
Dense orbit properties similar to our DO property have been studied in other contexts; we briefly mention here some cases known to us, certainly this list must not be complete. Sato and Kimura considered the situation of a linear algebraic group acting on a vector space \cite{MR0430336};
Richardson showed that a parabolic subgroup of a connected, semi-simple algebraic group has a dense orbit under the adjoint action on the Lie algebra of its unipotent radical \cite{MR0330311};
Hille and R\"ohrle studied a generalization of Richardson's theorem involving the descending central series of a parabolic \cite{MR1987340};
and Hille and Goodwin studied a variation for a Borel subgroup of $\operatorname{GL}(n)$ \cite{MR2356319}.
Multiplicity-free actions have been intensively studied in many contexts. Important results on multiplicity-free actions include: the Peter-Weyl Theorem; $\operatorname{GL}_m-\operatorname{GL}_n$-duality and, more generally, Howe dualities \cite{MR986027}; branching laws for the general linear and orthogonal groups \cite{MR986027}; the classification of the multiplicity-free linear actions of reductive groups due to Kac \cite{MR575790} and Benson-Ratcliff \cite{MR1382030}; the classification of homogeneous spherical varieties due to Kramer \cite{MR528837} and Brion \cite{MR822838}.
\subsection{Summary of results}
We first show that for algebras in a certain class studied by representation theorists -- namely, algebras with a preprojective component -- the two properties above are each equivalent to the algebra being finite representation type (see Theorem \ref{thm-preproj-comp} and the paragraph preceding it).
\begin{theoremnonum} Let $A$ be a connected, bound quiver algebra with a prepojective component. Then, the following properties are equivalent:
\begin{enumerate}
\renewcommand{\theenumi}{\arabic{enumi}}
\item $A$ is representation-finite;
\item $A$ has the DO property;
\item $A$ has the MF property.
\end{enumerate}
\end{theoremnonum}
It is clear that (1) implies (2) for any algebra, and we show in Proposition \ref{prop:schurfin-mf} that (2) implies (3) in general.
One might imagine these properties to be equivalent for all algebras. There do not appear to be any examples or theory in the literature demonstrating the existence of representation-infinite algebras with the DO or MF properties. So we prove that the family of representation-infinite algebras below has the DO property to demonstrate that it is actually a novel concept (Theorem \ref{thm-example-DO}). Our proof technique is an algorithm that yields an explicit list of the indecomposable representations with dense orbits \eqref{eq:indecomp}.
\begin{theoremnonum} Let $\Lambda$ be the algebra given by the following quiver and relations:
\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}[point/.style={shape=circle,fill=black,scale=.5pt,outer sep=3pt},>=latex]
\node[point,label={below:$1$}] (1) at (0,0) {} ;
\node[point,label={below:$2$}] (2) at (2,0) {} edge[in=45,out=-45,loop] node[right] {$b$} ();
\path[->] (1) edge node[above] {$a$} (2) ;
\end{tikzpicture} }}
\qquad
b^n = b^2 a = 0, \quad n\in \mathbb N.
\end{equation}
Then $\Lambda$ is DO for all $n$, but infinite representation type for $n\geq 6$, and even wild representation type for $n > 6$.
\end{theoremnonum}
We should point out that the example above is of infinite global dimension; in fact, all our examples of representation-infinite DO algebras are 2-point algebras of infinite global dimension. We believe that all representation-infinite algebras with the DO property should be non-triangular at least (Conjecture \ref{conj:repfiniteDO}).
It is easier to produce examples of representation-infinite algebras with the MF property, one such is given in Example \ref{ex:butterly}. In fact, we are able to characterize when a string algebra has the MF property in terms of a presentation by a quiver with relations (Corollary \ref{cor:MFstring}).
We conjecture that the MF property should correspond to the representation theoretic notion of ``Schur-representation-finite'' in general (see Definition \ref{def:schurfin}).
Finally, we are able to give representation theoretic characterizations of the DO and MF properties under certain conditions (Proposition \ref{string-not-DO} and Theorem \ref{thm:tamemf}):
\begin{theoremnonum}
Let $A$ be a bound quiver algebra.
\begin{enumerate}
\renewcommand{\theenumi}{\arabic{enumi}}
\item Assume that $A$ is a string algebra. Then, $A$ is representation-finite if and only if $A$ is DO.
\item Assume that $A$ is tame. Then, $A$ is Schur-representation-finite if and only if $A$ is MF.
\end{enumerate}
\end{theoremnonum}
\subsection*{Acknowledgements}
We give special thanks to Piotr Dowbor for his thoughts and comments over many months of working on the project.
We would also like to thank Raymundo Bautista, Harm Derksen, Christof Geiss, Lutz Hille, Birge Huisgen-Zimmermann, Jan Schr\"oer, and Dieter Vossieck for helpful conversations. Finally, we thank a referee whose comments lead to great improvement and simplification of the paper.
\section{Background}\label{sect:background}
\subsection{Module varieties}\label{sect:thebasics}
Up to Morita equivalence, any finite-dimensional, associative $K$-algebra $A$ can be viewed as a bound quiver algebra; that is, there exists a quiver $Q$ (uniquely determined by $A$) and an ideal $I$ in the path algebra $KQ$ such that $A \simeq KQ/I$.
Therefore, throughout the paper, we implicitly assume that our algebras are given by such a presentation. We say that $A$ is a \emph{triangular} algebra if $Q$ has no oriented cycles. We refer to the text \cite{assemetal} for background on quivers and finite-dimensional algebras.
We write $Q_0$ for the set of vertices of a quiver $Q$, and $Q_1$ for its arrow set. A dimension vector $\operatorname{\mathbf{d}}\colon Q_0 \to \mathbb N$ for $A$ is a choice of a non-negative integer at each vertex of $Q$.
The affine \key{module variety} $\operatorname{mod}(A,\operatorname{\mathbf{d}})$
parametrizes the $A$-modules of dimension vector $\operatorname{\mathbf{d}}$ along with a fixed basis. We represent a point of this variety by a collection of matrices associated to the arrows of $Q$ which satisfy the relations in $I$. Writing $ta$ and $ha$ for the tail and head of an arrow $a$, we have
\begin{equation}
\operatorname{mod}(A,\operatorname{\mathbf{d}}) = \{M \in \prod_{a \in Q_1} \operatorname{Mat}_{\operatorname{\mathbf{d}}(ha)\times \operatorname{\mathbf{d}}(ta)}(K) \mid M(r)=0, \forall r \in I\}.
\end{equation}
The orbits in $\operatorname{mod}(A,\operatorname{\mathbf{d}})$ of the base change group $\operatorname{GL}(\operatorname{\mathbf{d}}) = \prod \operatorname{GL}(\operatorname{\mathbf{d}}(i))$ are in one-to-one correspondence with the isomorphism classes of the $\operatorname{\mathbf{d}}$-dimensional $A$-modules. See, for example, \cite{MR724715} for background on module varieties.
In general, $\operatorname{mod}(A, \operatorname{\mathbf{d}})$ does not have to be irreducible. Let $C$ be an irreducible component of $\operatorname{mod}(A, \operatorname{\mathbf{d}})$. We say that $C$ is \emph{indecomposable} if $C$ has a dense open subset of indecomposable modules.
As shown by de la Pe{\~n}a in \cite[\S1.3]{MR1113958} and Crawley-Boevey and Schr{\"o}er in \cite[Theorem~1.1]{MR1944812}, any irreducible component $C \subseteq \operatorname{mod}(A, \operatorname{\mathbf{d}})$ satisfies a Krull-Schmidt type decomposition
\begin{equation}
C=\overline{C_1\oplus \ldots \oplus C_t}
\end{equation}
for some indecomposable irreducible components $C_i\subseteq \operatorname{mod}(A,\operatorname{\mathbf{d}}_i)$ with $\sum \operatorname{\mathbf{d}}_i = \operatorname{\mathbf{d}}$. We call $C=\overline{C_1\oplus \ldots \oplus C_t}$ \key{the generic decomposition of $C$}.
\begin{definition}
An algebra $A$ is said to have the \key{dense orbit property} (write $A$ is DO) if each irreducible component of each of its module varieties has a dense orbit.
\end{definition}
\begin{remark}\label{do-components} Using the generic decomposition, it is easy to see that an algebra is DO if and only if each of its indecomposable irreducible components has a dense orbit.
\end{remark}
\subsection{Dense orbits and self-extensions}\label{sect:doext}
The module varieties we study are the set of $K$-points of the module schemes $\underline{\operatorname{mod}}(A, \operatorname{\mathbf{d}})$, which only appear in the remarks of this subsection.
The Artin-Voigt Lemma states that an $A$-module $M$ satisfies $\operatorname{Ext}^1_A(M,M)=0$ if and only if the orbit of $M$ is scheme-theoretically open in $\underline{\operatorname{mod}}(A, \operatorname{\mathbf{d}})$ \cite[II.3.5]{MR0486168}. This seems to be the only known representation-theoretic condition implying that a module has a dense orbit.
Note that if $A$ is representation finite, then it is trivially DO since each $\operatorname{mod}(A,\operatorname{\mathbf{d}})$ has only finitely many orbits. To find non-trivial examples, one might look for algebras having an abundance of modules with self-extensions. It would be interesting to know if there exist algebras having a module without self extensions in each component, which are not representation finite.
If one assumes that \emph{every} indecomposable $M$ satisfies $\operatorname{Ext}^1_A(M,M)=0$, then $A$ is already representation finite \cite[Lemma~4]{MR1106345}.
Of course, vanishing self-extensions does not completely characterize modules with dense orbits, as one can see with an easy (even commutative) example: let $A=K[x]/(x^2)$, so the variety of modules of dimension 1 has only one point, thus a dense orbit.
However, the corresponding trivial module admits a nontrivial self extension:
\begin{equation}
0 \to K \to K[x]/(x^2) \to K \to 0.
\end{equation}
The issue is that if the scheme structure on a component of $\operatorname{mod}(A, \operatorname{\mathbf{d}})$ is not generically reduced, as in this example, then an orbit can be topologically open and dense without being scheme-theoretically open.
\subsection{Weight spaces of semi-invariants and moduli spaces of modules}\label{sect:moduli}
Consider the action of $\operatorname{SL}(\operatorname{\mathbf{d}}):=\prod_{i \in Q_0}\operatorname{SL}(\operatorname{\mathbf{d}}(i),K)$ on the module variety $\operatorname{mod}(A,\operatorname{\mathbf{d}})$, and the induced action on its coordinate ring.
The resulting ring of \emph{semi-invariants} has a weight space decomposition over the group $X^\star(\operatorname{GL}(\operatorname{\mathbf{d}}))$ of rational characters of $\operatorname{GL}(\operatorname{\mathbf{d}})$:
\begin{equation}
\operatorname{SI}(A,\operatorname{\mathbf{d}}):=K[\operatorname{mod}(A,\operatorname{\mathbf{d}})]^{\operatorname{SL}(\operatorname{\mathbf{d}})} = \bigoplus_{\chi \in X^\star(\operatorname{GL}(\operatorname{\mathbf{d}}))}\operatorname{SI}(A,\operatorname{\mathbf{d}})_{\chi}.
\end{equation}
Each of these summands
\begin{equation}
\operatorname{SI}(A,\operatorname{\mathbf{d}})_{\chi}=\lbrace f \in K[\operatorname{mod}(A,\operatorname{\mathbf{d}})] \mid g f= \chi(g)f \text{~for all~}g \in \operatorname{GL}(\operatorname{\mathbf{d}})\rbrace
\end{equation}
is a vector space called \key{the space of semi-invariants on} $\operatorname{mod}(A,\operatorname{\mathbf{d}})$ \key{of weight $\chi$}. Note that any $\theta \in \mathbb Z^{Q_0}$ defines a rational character $\chi_{\theta}:\operatorname{GL}(\operatorname{\mathbf{d}}) \to K^*$ by
\begin{equation}
\chi_{\theta}((g(i))_{i \in Q_0})=\prod_{i \in Q_0}\det g(i)^{\theta(i)}.
\end{equation}
In this way, we identify $\mathbb Z ^{Q_0}$ with $X^\star(\operatorname{GL}(\operatorname{\mathbf{d}}))$, assuming that $\operatorname{\mathbf{d}}$ is a
sincere dimension vector. We also refer to the rational characters of $\operatorname{GL}(\operatorname{\mathbf{d}})$ as (integral) weights of $A$ (or $Q$).
For an irreducible component $C \subseteq \operatorname{mod}(A,\operatorname{\mathbf{d}})$, we similarly define the ring of semi-invariants $\operatorname{SI}(C):=K[C]^{\operatorname{SL}(\operatorname{\mathbf{d}})}$, and the space $\operatorname{SI}(C)_{\theta}$ of semi-invariants on $C$ of weight $\theta \in \mathbb Z^{Q_0}$.
\begin{definition}
An algebra $A$ is said to have the \key{multiplicity-free property} (write $A$ is MF) if the algebra of semi-invariants on each irreducible component $C$ of each of its module varieties is multiplicity-free; that is, $\dim_K \operatorname{SI}(C)_{\theta} \leq 1$ for all $\theta \in \mathbb Z^{Q_0}$.
\end{definition}
\begin{remark} Note that for an irreducible component $C \subseteq \operatorname{mod}(A,\operatorname{\mathbf{d}})$, the affine categorical quotient $C//\operatorname{SL}(\operatorname{\mathbf{d}})$ is a $\operatorname{GL}(\operatorname{\mathbf{d}})/\operatorname{SL}(\operatorname{\mathbf{d}})$-variety and $\dim_K \operatorname{SI}(C)_{\theta}$ is the multiplicity of the 1-dimensional irreducible representations of the torus $\operatorname{GL}(\operatorname{\mathbf{d}})/\operatorname{SL}(\operatorname{\mathbf{d}})$ of weight $\theta$ in the coordinate ring $\operatorname{SI}(C)$ of $C//\operatorname{SL}(\operatorname{\mathbf{d}})$. With this observation in mind, an algebra $A$ is MF if and only if for each dimension vector $\operatorname{\mathbf{d}}$ and irreducible component $C \subseteq \operatorname{mod}(A,\operatorname{\mathbf{d}})$, $C//\operatorname{SL}(\operatorname{\mathbf{d}})$ is a multiplicity-free $\operatorname{GL}(\operatorname{\mathbf{d}})/\operatorname{SL}(\operatorname{\mathbf{d}})$-variety or, equivalently, $C//\operatorname{SL}(\operatorname{\mathbf{d}})$ contains a dense $\operatorname{GL}(\operatorname{\mathbf{d}})/\operatorname{SL}(\operatorname{\mathbf{d}})$-orbit.
\end{remark}
In our study of MF algebras, we will use of some of the main results on moduli spaces of representations due to King \cite{Kmodulireps}. An $A$-module $M$ is said to be \emph{$\theta$-semi-stable} if $\theta(\operatorname{\mathbf{dim}} M)=0$ and $\theta(\operatorname{\mathbf{dim}} M')\leq 0$ for all submodules $M' \leq M$. We say that $M$ is \emph{$\theta$-stable} if $M$ is non-zero, $\theta(\operatorname{\mathbf{dim}} M)=0$, and $\theta(\operatorname{\mathbf{dim}} M')<0$ for all submodules $0 \neq M' < M$. Now, consider the (possibly empty) open subsets
\begin{equation}
\operatorname{mod}(A,\operatorname{\mathbf{d}})^{ss}_{\theta}=\{M \in \operatorname{mod}(A,\operatorname{\mathbf{d}})\mid M \text{~is~}
\text{$\theta$-semi-stable}\}
\end{equation}
and
\begin{equation}
\operatorname{mod}(A,\operatorname{\mathbf{d}})^s_{\theta}=\{M \in \operatorname{mod}(A,\operatorname{\mathbf{d}})\mid M \text{~is~}
\text{$\theta$-stable}\}
\end{equation}
of $\operatorname{\mathbf{d}}$-dimensional $\theta$(-semi)-stable $A$-modules.
Using methods from Geometric Invariant Theory, King showed in \cite{Kmodulireps} that the projective variety
\begin{equation}
\operatorname{\mathcal{M}}(A,\operatorname{\mathbf{d}})^{ss}_{\theta}:=\operatorname{Proj}(\bigoplus_{n \geq 0}\operatorname{SI}(A,\operatorname{\mathbf{d}})_{n\theta})
\end{equation}
is a GIT-quotient of $\operatorname{mod}(A,\operatorname{\mathbf{d}})^{ss}_{\theta}$ by the action of $\operatorname{PGL}(\operatorname{\mathbf{d}})$. Here, $\operatorname{PGL}(\operatorname{\mathbf{d}})=\operatorname{GL}(\operatorname{\mathbf{d}})/T_1$ where $T_1=\{(\lambda \operatorname{Id}_{\operatorname{\mathbf{d}}(i)})_{i \in Q_0} \mid \lambda \in K^*\} \leq \operatorname{GL}(\operatorname{\mathbf{d}})$. Note that there is a well-defined action of $\operatorname{PGL}(\operatorname{\mathbf{d}})$ on $\operatorname{mod}(A,\operatorname{\mathbf{d}})$ since $T_1$ acts trivially on $\operatorname{mod}(A,\operatorname{\mathbf{d}})$. We say that $\operatorname{\mathbf{d}}$ is a \emph{$\theta$-semi-stable dimension vector} if $\operatorname{mod}(A,\operatorname{\mathbf{d}})^{ss}_{\theta} \neq \emptyset$.
It was proved in \cite[Proposition~4.2]{Kmodulireps} that the (closed) points of $\operatorname{\mathcal{M}}(A,\operatorname{\mathbf{d}})^{ss}_{\theta}$ are in one-to-one correspondence with the isomorphism classes of those modules in $\operatorname{mod}(A,\operatorname{\mathbf{d}})^{ss}_{\theta}$ that can be written as direct sums of $\theta$-stable modules. We call such $A$-modules \key{$\theta$-polystable}.
For an irreducible component $C \subseteq \operatorname{mod}(A,\operatorname{\mathbf{d}})$, we similarly define $C^{ss}_{\theta}, C^s_{\theta}$, and $\operatorname{\mathcal{M}}(C)^{ss}_{\theta}$. One then has that the points of $\operatorname{\mathcal{M}}(C)^{ss}_{\theta}$ are in one-to-one correspondence with the isomorphism classes of $\theta$-polystable modules in $C$.
In general, it is difficult to describe or construct stable modules. The lemma below, which will be used in proving Theorem \ref{thm:tamemf}, identifies modules $M$ which are stable with respect to a canonical weight associated to $M$. Specifically, for an arbitrary $A$-module $M$, we define the weight $\theta^M$ by
\begin{equation}
\theta^M(\operatorname{\mathbf{dim}} X)=\dim_K \operatorname{Hom}_A(P_0,X)-\dim_K \operatorname{Hom}_A(P_1,X),
\end{equation}
where $P_1 {\buildrel f \over \to} P_0 \to M \to 0$ is a minimal projective presentation of $M$ in $\operatorname{mod}(A)$ and $X$ is an $A$-module (see \cite{DomokosFFT}). Equivalently, we can write
\begin{equation}
\theta^M(\operatorname{\mathbf{dim}} X)=\dim_K \operatorname{Hom}_A(M,X)-\dim_K \operatorname{Hom}_A(X,\tau M),
\end{equation}
where $\tau M$ is the Auslander-Reiten translation of $M$ (for more details, see \cite[\S IV.2]{assemetal}).
Recall that an $A$-module $M$ is said to be \key{homogeneous} if $M \simeq \tau M$, and is said to be \key{Schur} if $\operatorname{End}_A(M) \simeq K$.
\begin{lemma}\label{lemma:stable-Schur-homogeneous} If $M$ is a homogeneous Schur $A$-module, then $M$ is $\theta^M$-stable.
\end{lemma}
\begin{proof} Since $M$ is homogeneous, we have that $\theta^{M}(\operatorname{\mathbf{dim}} M)=0$. Next, using that $M$ is also Schur, we have that for any proper $A$-submodule $0\neq M' \subset M$, $\operatorname{Hom}_A(M,M')=0$ and $\dim_K \operatorname{Hom}_A(M',\tau M)=\dim_K \operatorname{Hom}_A(M',M)>0$. So, $\theta^M(\operatorname{\mathbf{dim}} M')<0$ for all proper submodules $M'$ of $M$.
\end{proof}
\begin{remark} The property that a module is Schur does not guarantee the existence of a weight with respect to which the module becomes (semi-)stable (see \cite[\S~3.2]{Reineke:2008fk} for an example).
\end{remark}
\section{General results and algebras with a preprojective component}\label{sect:general}
Since we are interested in when the DO property, MF property, and representation finite properties coincide, we start by proving some general facts about these classes of algebras. The following property provides a bridge in one direction.
\begin{definition}\label{def:schurfin}
We say that $A$ is \key{Schur-representation-finite} if, for each dimension vector $\operatorname{\mathbf{d}}$ of $A$, there are finitely many $\operatorname{\mathbf{d}}$-dimensional Schur $A$-modules up to isomorphism.
\end{definition}
The larger class of brick-tame bocses was introduced and studied by Bodnarchuk-Drozd in \cite{MR2609188}; it includes bocses that appear in the study of vector bundles on degenerations of elliptic curves and coadjoint actions of linear groups.
\begin{lemma}\label{lem:infschur} If $A$ has the DO property, then $A$ is Schur-representation-finite.
\end{lemma}
\begin{proof} If $A$ is not Schur-representation-finite, then there must be an irreducible component of some $\operatorname{mod}(A,\operatorname{\mathbf{d}})$ with infinitely many Schur modules. But there can only be one dense orbit in a component, and dimension of endomorphism rings strictly increases when moving from an orbit to its boundary, a contradiction.
\end{proof}
\begin{prop}\label{prop:schurfin-mf} A Schur-representation-finite algebra $A$ is MF. In particular, if an algebra has the DO property, then it also has the MF property.
\end{prop}
\begin{proof}
Let $\operatorname{\mathbf{d}}$ be a dimension vector, $C \subseteq \operatorname{mod}(A,\operatorname{\mathbf{d}})$ an irreducible component, and $\theta$ a weight such that $\dim_K \operatorname{SI}(C)_{\theta} >0$. First, we note that $\dim_K \operatorname{SI}(C)_{n \theta} \leq \dim_K \operatorname{SI}(C)_{(n+1)\theta}$ for all $n \geq 1$. Indeed, fixing a non-zero semi-invariant $f_0 \in \operatorname{SI}(C)_{\theta}$, the map
\begin{equation}
\begin{split}
\operatorname{SI}(C)_{n \theta} &\to \operatorname{SI}(C)_{(n+1)\theta}\\
f &\mapsto f_0 \cdot f
\end{split}\end{equation}
is an injective linear map since $K[C]$ is a domain. The desired inequality now follows.
Retaining the notation from Section \ref{sect:moduli}, recall that the (closed) points of $\operatorname{\mathcal{M}}(C)^{ss}_{\theta}$ are in one-to-one correspondence with the $\theta$-polystable modules in $C^{ss}_{\theta}$. Furthermore, any $\theta$-polystable $A$-module is a finite direct sum of Schur modules since any $\theta$-stable module is Schur. So the moduli space $\operatorname{\mathcal{M}}(C)^{ss}_{\theta}$ is zero dimensional and since the dimensions of the graded pieces of the algebra defining $\operatorname{\mathcal{M}}(C)^{ss}_{\theta}$ weakly increase, we conclude that $\dim_K \operatorname{SI}(C)_{\theta}=1$.
\end{proof}
Now we are almost ready to prove our first main result. The following lemma allows us to reduce to the minimal representation-infinite case.
\begin{lemma}\label{lem:mfquot}
Any quotient of an MF bound quiver algebra is MF.
\end{lemma}
\begin{proof}
Let $A$ be a MF algebra, $I$ an ideal of $A$, and $\operatorname{\mathbf{d}}$ a dimension vector of $A$. Then, any irreducible component $C \subseteq \operatorname{mod}(A/I,\operatorname{\mathbf{d}})$ is embedded ($\operatorname{GL}(\operatorname{\mathbf{d}})$-equivariantly) in an irreducible component $C' \subseteq \operatorname{mod}(A,\operatorname{\mathbf{d}})$. We know from invariant theory \cite[Corollary 2.2.9]{MR1918599} that the above embedding gives rise to a surjective map at the level of $\operatorname{SL}(\operatorname{\mathbf{d}})$-invariant rings which preserves weight spaces. So for any weight $\theta$, we have that $\dim_K \operatorname{SI}(C)_{\theta} \leq \dim_K \operatorname{SI}(C')_{\theta} \leq 1$.
\end{proof}
Recall that an algebra $A$ is said to \emph{admit a preprojective component} if its Auslander-Reiten quiver has an acyclic connected component in which every indecomposable is of the form $\tau^{-n} P$ for some projective $P$. For example, hereditary algebras $A=KQ$ always admit a preprojective component. See \cite[\S~VIII.2]{assemetal} for more details (where the terminology ``postprojective'' is used.)
\begin{theorem}\label{thm-preproj-comp} Let $A$ be a connected, bound quiver algebra with a prepojective component. Then, the following properties are equivalent:
\begin{enumerate}
\renewcommand{\theenumi}{\arabic{enumi}}
\item $A$ is representation-finite;
\item for each dimension vector $\operatorname{\mathbf{d}}$ of $A$, the group $\operatorname{GL}(\operatorname{\mathbf{d}})$ acts on each irreducible component of $\operatorname{mod}(A,\operatorname{\mathbf{d}})$ with a dense orbit;
\item for each dimension vector $\operatorname{\mathbf{d}}$ of $A$ and each irreducible component $C$ of $\operatorname{mod}(A,\operatorname{\mathbf{d}})$, the algebra of semi-invariants $K[C]^{\operatorname{SL}(\operatorname{\mathbf{d}})}$ is multiplicity-free.
\end{enumerate}
\end{theorem}
\begin{proof}We have seen that the implications $(1)\Longrightarrow (2) \Longrightarrow (3)$ hold true for arbitrary bound quiver algebras.
It remains to prove the implication $(3) \Longrightarrow (1)$. Assume to the contrary that the MF algebra $A$ is representation-infinite. It follows from the work of Happel and Vossieck that any connected algebra admitting a prepojective component has a tame concealed algebra as a quotient (see \cite{MR701205} or \cite[Theorem XIV.3.1]{MR2360503}). This result combined with Lemma \ref{lem:mfquot} tells us that $A$ has a quotient $B$ which is an MF tame concealed algebra. Denote by $\operatorname{\mathbf{h}}$ the dimension vector of an indecomposable $B$-module lying at the mouth of a homogeneous tube of $B$. It is well-known that $\operatorname{mod}(B,\operatorname{\mathbf{h}})$ is irreducible and that there is always an integral weight $\theta$ of $B$ such that $\operatorname{\mathcal{M}}(B,\operatorname{\mathbf{h}})^{ss}_{\theta} \simeq \mathbb P^1$ (see for example \cite{MR3037894}). In particular, $\operatorname{SI}(B,\operatorname{\mathbf{h}})$ is not multiplicty-free, a contradiction.
\end{proof}
\section{Representation-infinite DO algebras}\label{sect:ctreg}
\subsection{Example of a representation-infinite DO algebra}
Our example of a representation-infinite DO algebra works in arbitrary characteristic and is given by the following quiver with relations.
\begin{equation}\label{eq:DO}
\vcenter{\hbox{
\begin{tikzpicture}[point/.style={shape=circle,fill=black,scale=.5pt,outer sep=3pt},>=latex]
\node[point,label={below:$1$}] (1) at (0,0) {} ;
\node[point,label={below:$2$}] (2) at (2,0) {} edge[in=45,out=-45,loop] node[right] {$b$} ();
\path[->] (1) edge node[above] {$a$} (2) ;
\end{tikzpicture} }}
\qquad
b^n = b^2 a = 0, n \in \mathbb N
\end{equation}
We can think of a representation of $\Lambda$ as a nilpotent operator on a vector space, along with a distinguished subspace. For $n=6$, the algebra is tame representation-infinite with distributive ideal lattice (see, for example \cite[p.~242]{MR799266}). For $n > 6$, the algebra is wild \cite{MR949903}. Fix $n$ and denote this algebra by $\Lambda$ throughout this section.
\begin{theorem}\label{thm-example-DO}
The algebra $\Lambda$ is DO for all $n$
\end{theorem}
The proof proceeds by induction on the dimension vector. Using Remark \ref{do-components}, it is enough to take a general element of an irreducible component and show that it has a direct summand whose orbit is dense. We do this by direct calculation, so that our method is actually an algorithm to calculate the generic module in a given component, yielding a classification of indecomposable components. A ``conceptual'' proof without calculations does not seem to be within reach using current technology (see Section \ref{sect:doext}). But we can make the computations easier by reframing the problem in terms of matrices over the polynomial ring $K[x]$.
Let $\operatorname{\mathbf{d}} =(d_1,d_2)$ be a dimension vector for $\Lambda$, and fix an irreducible component $C$ of $\operatorname{mod}(\Lambda, \operatorname{\mathbf{d}})$. Denote by $A, B$ matrices representing the actions of $a,b$ at a typical point of $\operatorname{mod}(\Lambda, \operatorname{\mathbf{d}})$.
There is an open subset of $C$ on which the Jordan type of $B$ is constant. In this case we say that $C$ is of type $\lambda$, where $\lambda = (\lambda_1, \lambda_2, \dotsc)$ is a partition such that $\lambda_i = \dim \ker B^i$.
Now we fix a representative for $B$ of general Jordan type in $C$, and let $Z_B \subseteq \operatorname{GL}(d_2)$ be the centralizer of $B$. Then we see that $C$ has a dense $\operatorname{GL}(\operatorname{\mathbf{d}})$ orbit if and only if $\operatorname{Hom}_K (V_1, V_2)\simeq \operatorname{Mat}_{d_2\times d_1}$ has a dense $H:=\operatorname{GL}(d_1) \times Z_B$ orbit.
We identify the pair $(K^{d_2}, B)$ with a torsion $K[x]$-module $X$, and fix a decomposition into indecomposables
\begin{equation}
X \simeq \bigoplus_{i=1}^n {\bar{\lambda}_i}\, J_i,
\end{equation}
where $J_i := K[x]/(x^i)$ and $\bar{\lambda}_i:= \lambda_i - \lambda_{i+1}$ is the multiplicity of this summand.
The centralizer of $B$ in $\operatorname{GL}(d_2)$ is naturally identified with $\operatorname{Aut}_{K[x]}(X)$ in this way, and the matrix $A$ can be thought of as having entries in $K[x]$. Each row of $A$ corresponds to a summand $J_i$, which we order by increasing dimension. So if we group the summands into isotypical components $X_i := {\bar{\lambda}_i}\, J_i$, the matrix $A$ has block form
\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture
\matrix (m) [matrix of math nodes,nodes in empty cells,left delimiter = {[}, right delimiter = {]} ]
{
\phantom{X_1} & &\phantom{X_1}\\
\phantom{X_1} & &\phantom{X_1}\\
\phantom{\vdots} &\vdots &\phantom{X_1}\\
\phantom{X_1} & &\phantom{X_1}\\
\phantom{X_1} & &\phantom{X_1}\\
};
\draw[thick] (m-1-1.south west) -- (m-1-3.south east);
\draw[thick] (m-2-1.south west) -- (m-2-3.south east);
\draw[thick] (m-4-1.south west) -- (m-4-3.south east);
\node [left=of m-1-1.east] {$X_1$};
\node [left=of m-2-1.east] {$X_2$};
\node [left=of m-3-1.east] {$\vdots$};
\node [left=of m-5-1.east] {$X_n$};
\end{tikzpicture}}},
\end{equation}
and the condition $b^2a =0$ is equivalent to requiring each entry in block row $X_i$ to be in the ideal $(x^{i-2})$, that is, of the form $ax^{i-2} + bx^{i-1}$ for some sufficiently general $a, b \in K$. The group $H$ acts by:
\begin{itemize}
\item arbitrary $K$-linear column operations;
\item row operations of the form ``add $f$ times a row in block $X_i$ to a row in block $X_j$,'' where $f \in K[x]$ for $i \leq j$, and $f \in (x^{i-j})$ for $i > j$.
\end{itemize}
In this notation, we denote a representation of $\Lambda$ by a pair $(W, M)$ where $M$ is a $K[x]$ module and $W \subseteq M$ is a $K$-linear subspace.
Now we must analyze several cases, branching on the relation between $d_1$ and $\lambda$. In what follows, we let $\mathbf{1}_{d}$ denote a $d\times d$ identity matrix and $*x^i$ a region of entries in the ideal $(x^i)$ (or simply $*$ when $i=0$). We use the fact that we are working with a sufficiently generic matrix in our component throughout, without explicit mention. By induction we can also assume $\bar{\lambda}_n \neq 0$, otherwise $C$ would be a component for an algebra of smaller dimension.
For the sake of readability we display matrices for the case $n=5$.
Using Gauss elimination, we can reduce to the case $d_1 \geq \lambda_2$ and put $A$ into the form
\begin{equation}\label{eq:step2}
A=\vcenter{\hbox{
\begin{tikzpicture
\matrix (m) [matrix of math nodes,nodes in empty cells,left delimiter = {[}, right delimiter = {]},
nodes={minimum width=2em} ]
{
* & * &* & * & * \\
{\mathbf{1}}_{\bar{\lambda}_2} & * &* & * & * \\
0 & x \mathbf{1}_{\bar{\lambda}_3} & *x & *x & *x \\
0 & 0 &x^2 \mathbf{1}_{\bar{\lambda}_4} & *x^2 & *x^2\\
0 & 0 & 0 & x^3 \mathbf{1}_{\bar{\lambda}_5} & *x^3 \\
};
\node [left=of m-1-1.east] {$X_1$};
\node [left=of m-2-1.east] {$X_2$};
\node [left=of m-3-1.east] {$X_3$};
\node [left=of m-4-1.east] {$X_4$};
\node [left=of m-5-1.east] {$X_5$};
\end{tikzpicture}}}
\end{equation}
where the last column is of width 0 if $d_1 = \lambda_2$. (If $d_1 < \lambda_2$ then there would be a row of 0s at bottom, giving a direct summand $(0, J_n)$ of a general representation.) Then using column operations to the right, we can clear leading terms to increase powers of $x$.
\begin{equation}\label{eq:step3}
A=\vcenter{\hbox{
\begin{tikzpicture
\matrix (m) [matrix of math nodes,nodes in empty cells,left delimiter = {[}, right delimiter = {]},
nodes={minimum width=2em} ]
{
* & * &* & * & * \\
{\mathbf{1}}_{\bar{\lambda}_2} & *x &*x & *x & *x \\
0 & x \mathbf{1}_{\bar{\lambda}_3} & *x^2 & *x^2 & *x^2 \\
0 & 0 &x^2 \mathbf{1}_{\bar{\lambda}_4} & *x^3 & *x^3 \\
0 & 0 & 0 & x^3 \mathbf{1}_{\bar{\lambda}_5} & *x^4 \\
};
\node [left=of m-1-1.east] {$X_1$};
\node [left=of m-2-1.east] {$X_2$};
\node [left=of m-3-1.east] {$X_3$};
\node [left=of m-4-1.east] {$X_4$};
\node [left=of m-5-1.east] {$X_5$};
\end{tikzpicture}}}.
\end{equation}
We claim that the top row always splits off, taking the bottom row with it precisely when $d_1 = \lambda_2$. This is seen by direct computation: take the upper right entry and use column operations to clear entries of the same degree in its row. This first "messes up" the lower part, but these blocks can be restored by Gaussian elimination again (from strictly lower rows).
Now the top row has only one nonzero entry, so it can almost clear the column below it since row operations to lower blocks contribute factors of $x$. The only way the column fails to be completely cleared is if $d_1 = \lambda_2$, in which case the entry is $x^{n-2}$ instead of $x^{n-1}$.
In summary, if we let $m$ be such that the top row is of type $J_m$ (i.e., $m$ is minimal such that $\bar{\lambda}_m \neq 0$), then the direct summands that split off are as follows. If $d_1 > \lambda_2$, we get the direct summand $(Kx^{m-2} + Kx^{m-1}, J_m)$, and if $d_1 = \lambda_2$ then we get the direct summand $(K(x^{m-2}\oplus 0) + K(x^{m-1} \oplus x^{n-2}), J_m\oplus J_n)$. For $m=1$ we take $x^{m-2} = 0$, and in case $m=n$ the later type further decomposes. We summarize the classification of indecomposable components obtained from this algorithm.
\begin{equation}\label{eq:indecomp}
\begin{array}{|c|c|c|}
\hline
\operatorname{\mathbf{d}} & \text{Jordan type} & A\\
\hline
(1,0) & \text{none} & \text{none}\\
(0, m) & J_m & \text{none} \\
(1, 1) & J_1 & [1] \\
(1, m) & J_m,\ m \geq 2 & [x^{m-1}]\\
(2, m) & J_m,\ m \geq 2 & [x^{m-1}\ x^{m-2}] \\
(1, 1+n) & J_1 \oplus J_n & \begin{bmatrix}1\\x^{n-2}\end{bmatrix} \\
(2, m+n) & J_m \oplus J_n ,\ m < n & \begin{bmatrix}x^{m-2} & x^{m-1}\\ 0 & x^{n-2}\end{bmatrix} \\
\hline
\end{array}
\end{equation}
\subsection{Dense orbit conjecture and open questions}
The following conjecture of Weyman has guided our results connecting the DO property with representation theory. In this subsection we propose a proof strategy and carry out some steps towards the conjecture.
\begin{conjecture}[{\bf Dense Orbit Conjecture}]\label{conj:repfiniteDO} For a triangular algebra $A$, the following statements are equivalent:
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\item $A$ is representation-finite;
\item for any ideal $I$ of $A$, the algebra $A/I$ is DO.
\end{enumerate}
\end{conjecture}
The formulation of statement (ii) allows us to reduce to the case that $A$ is minimal representation-infinite, where we can use recent work of Bongartz and Ringel. They have shown that any minimal representation-infinite, finite-dimensional algebra is in one of the following three categories (see the Introduction of \cite{MR2931904}):
\begin{enumerate}
\item Algebras with a non-distributive ideal lattice;
\item Algebras with a ``good'' universal cover that contains a convex subcategory which is tame concealed of type $\widetilde{D}_n, \widetilde{E}_6, \widetilde{E}_7$, or $\widetilde{E}_8$;
\item Algebras with a ``good'' universal cover, but in which all finite convex subcategories are representation finite.
\end{enumerate}
In fact, Ringel showed that all algebras in case (3) are string algebras, and explicitly classified them in terms of quivers with relations.
We now prove the conjecture for classes (1) and (3).
The following proposition, covering case (1), comes directly by following Bongartz's argument in \cite[\S~1]{MR3038490}, with the additional assumption that $A$ is triangular.
\begin{prop}\label{prop:nondistrib-do} Let $A$ be a non-distributive triangular algebra. Then $A$ admits infinitely many Schur representations of the same dimension, and thus $A$ is not DO.
\end{prop}
\begin{proof}
Since $A$ is non-distributive, it admits primitive idempotents $e,f$ (not necessarily distinct) such that $fAe$ is neither cyclic as a left $fAf$-module nor right $eAe$-module; in other words, the radical filtration $(R^i)$ of $fAe$ as an $eAe$-$fAf$-bimodule admits a step $R^l / R^{l+1}$ of dimension $\geq 2$. Choose some $v,w \in R^l$ which are linearly independent modulo $R^{l+1}$. Then without loss of generality we can mod out (the two-sided ideal generated by) $R^{l+1}, Jv, vJ, Jw, wJ$ where $J=\operatorname{rad} A$.
If $A$ is triangular, then in particular there are no loops at any vertex of its quiver $Q$, so that $eAe \simeq fAf \simeq K$ and $e \neq f$. Consider the family $V_\lambda = Ae / \langle v - \lambda w \rangle$ for $\lambda \in K$.
Applying $\operatorname{Hom}_A( - , V_\lambda)$ to the quotient, we get $0 \to \operatorname{End}_A (V_\lambda) \to \operatorname{Hom}_A(Ae, V_\lambda)$. But $Ae$ is projective, so $\operatorname{Hom}_A(Ae, V_\lambda) = eV_\lambda = eAe =K$. Thus $V_\lambda$ is a Schur $A$-module, and $A$ is not DO by Lemma \ref{lem:infschur}.
\end{proof}
The next proposition covers case (3). See Section \ref{sec:tameMF} below for the definition of a string algebra.
\begin{prop}\label{string-not-DO} Let $A$ be a representation-infinite string algebra. Then $A$ does not have the DO property.
\end{prop}
\begin{proof} Since $A$ is representation-infinite, we know that there are indecomposable band modules. Among the dimension vectors $\operatorname{\mathbf{d}}$ of $A$ for which there are infinitely many $\operatorname{\mathbf{d}}$-dimensional indecomposable band modules, we choose a $\operatorname{\mathbf{d}}$ with $\sum_{i\in Q_0}\operatorname{\mathbf{d}}(i)$ minimal.
Assume to the contrary that $A$ is DO. Let $C \subseteq \operatorname{mod}(A,\operatorname{\mathbf{d}})$ be an irreducible component that contains infinitely many indecomposable band modules. Since $A$ is DO, there exists $M_0 \in C$ such that $C=\overline{\operatorname{GL}(\operatorname{\mathbf{d}})M_0}$. We claim that $M_0$ is an indecomposable band module.
Given this, we have a contradiction: the orbits of all other band modules in the same family as $M_0$ lie in the same irreducible component, since they are obtained as the image of a certain morphism $\phi\colon \operatorname{GL}(\operatorname{\mathbf{d}}) \times K^* \to \operatorname{mod}(A, \operatorname{\mathbf{d}})$, so they are all in the boundary of the orbit $\operatorname{GL}(\operatorname{\mathbf{d}})M_0$.
But the endomorphism algebras of band modules in the same family have the same dimension \cite{MR1090218}, so the orbit of one cannot be in the boundary of another.
To prove the claim, denote by $s$ the number of string modules occurring in a direct sum decomposition of $M_0$ into indecomposables.
For any indecomposable band module $M \in C$ we have
\begin{equation}
\dim_K M=\sum_{a \in Q_1} \operatorname{rank} M(a) \leq \sum_{a \in Q_1} \operatorname{rank} M_0(a)=\dim_K M_0-s,
\end{equation}
which implies that $s=0$. Now $M_0$ is an indecomposable by minimality of $\operatorname{\mathbf{d}}$.
\end{proof}
It would be nice to understand how the DO property behaves under quotients (i.e., we would like an analogue of Lemma \ref{lem:mfquot}). So we pose the following question.
\begin{question}
Is it true that any quotient of a DO algebra is DO?
\end{question}
If the answer to the question is `yes' then statement (ii) of the Dense Orbit Conjecture can be simplified to ``$A$ is DO.''
\section{Representation-infinite MF algebras}\label{sec:rep-inf-MF-algebras}
\subsection{Tame MF algebras}\label{sec:tameMF}
In this section, we give a representation theoretic characterization of the MF property for tame algebras, and apply this to classify MF string algebras.
\begin{theorem}\label{thm:tamemf} A tame algebra is Schur-representation-finite if and only if it is MF.
\end{theorem}
\begin{proof}
The implication ``$\Longrightarrow$'' was proved in Proposition \ref{prop:schurfin-mf} for all algebras.
Now, let $A$ be a tame algebra with the MF property. Assume for a contradiction that there is a dimension vector $\operatorname{\mathbf{d}}$ of $A$ and an irreducible component $C \subseteq \operatorname{mod}(A,\operatorname{\mathbf{d}})$ that contains infinitely many Schur $A$-modules. We immediately deduce from this that $C$ is not an orbit closure (see for example Lemma \ref{lem:infschur}).
From Crawley-Boevey's Theorem D in \cite{MR931510}, we know that all, except finitely many, Schur modules in $C$ are homogeneous. So let $M \in C$ be a homogeneous Schur $A$-module. Then the $\theta^M$-stable locus $C^s_{\theta^M} \subset C$ is non-empty and dense by Lemma \ref{lemma:stable-Schur-homogeneous}, and moreover, $C^s_{\theta^M}$ must be an infinite disjoint union of orbits since otherwise $C$ would be an orbit closure. Consequently, we have that $\dim \operatorname{\mathcal{M}}(C)^{ss}_{\theta^M} \geq 1$, which is in contradiction to $A$ being MF.
\end{proof}
String algebras are a well-understood class of tame algebras whose indecomposable representations can be completely described in terms of certain paths in the associated quiver. Our characterization of the MF property for tame algebras can be made even more explicit for string algebras.
A bound quiver algebra $A=KQ/I$ is said to be a \key{string algebra} if $I$ can be generated by a set of relations $\mathcal{R}$ satisfying the following conditions:
\begin{enumerate}
\renewcommand{\theenumi}{\arabic{enumi}}
\item each vertex of $Q$ is the tail of at most two arrows, and the head of at most two arrows;
\item each relation in $\mathcal{R}$ is just a monomial in the arrows of $Q$;
\item for each arrow $b \in Q_1$, there is at most one arrow $a \in Q_1$ with $ta=hb$ and at most one arrow $c \in Q_1$ with $tb=hc$ such that $ab \notin \mathcal{R}$ and $bc \notin \mathcal{R}$.
\end{enumerate}
For the details behind the construction of their indecomposable representations, known as string and band modules, we refer the reader to \cite{MR876976,MR801283}.
\begin{corollary}\label{cor:MFstring}
A string algebra $KQ/I$ is MF if and only if every subquiver $L \subseteq Q$ of type $\widetilde{A}_n$ contains a relation from $I$.
\end{corollary}
\begin{proof}
If $Q$ contains a type $\tilde{A}_n$ subquiver $L$ in which there is no relation, then the family of band modules which are supported precisely on $L$ and one dimensional at each vertex shows that $KQ/I$ is not Schur-representation-finite and thus not MF.
On the other hand, if every subquiver of type $\tilde{A}_n$ in $Q$ contains a relation, then any band must traverse some vertex more than once.
Krause's computation of morphisms between band modules (see conditions (H1)-(H4) of \cite[p.193]{MR1090218}) then shows that each band module admits a nilpotent endomorphism. Thus $KQ/I$ has no Schur band modules and must be Schur-representation-finite, so it is also MF.
\end{proof}
The following example illustrates this corollary, and was in fact our first example of a representation-infinite MF algebra.
\begin{example}\label{ex:butterly}
Consider the string algebra $\Lambda=KQ/I$ given by the ``butterfly quiver''
\begin{equation}
Q=
\vcenter{\hbox{
\begin{tikzpicture}[point/.style={shape=circle,fill=black,scale=.5pt,outer sep=3pt},>=latex]
\node[point,label={left:$1$}] (1) at (-1,1) {};
\node[point,label={right:$2$}] (2) at (1,1) {};
\node[point,label={above:$3$}] (3) at (0,0) {};
\node[point,label={left:$4$}] (4) at (-1,-1) {};
\node[point,label={right:$5$}] (5) at (1,-1) {};
\path[->]
(1) edge node[above] {$a$} (3)
(1) edge node[left] {$c$} (4)
(2) edge node[above] {$b$} (3)
(2) edge node[right] {$d$} (5)
(3) edge node[below] {$e$} (4)
(3) edge node[below] {$f$} (5);
\end{tikzpicture}
}}
\end{equation}
with $I$ the ideal generated by all length 2 paths: $ea, eb, fa, fb$. We see that
$\Lambda$ is a string algebra with
\begin{equation}
B=c^{-1}ef^{-1}db^{-1}a
\end{equation}
the only primitive band, and in fact $\Lambda$ is a minimal representation-infinite algebra. The only subquivers of type $\tilde{A}_n$ are the left and right triangles of the diagram, which each contain a length two path in the ideal of relations. So by the corollary, $\Lambda$ is MF.
\end{example}
\subsection{MF conjecture and open questions}
Finally, we state another conjecture of Weyman, which is that Theorem \ref{thm:tamemf} holds without the assumption that $A$ is tame.
\begin{conjecture}[{\bf Multiplicity Free Conjecture}]\label{conj:MFC}
An algebra is Schur-representation-finite if and only if it is MF.
\end{conjecture}
It has been pointed out to us by Geiss and Schr\"oer that preprojective algebras of Dynkin quivers, which are generally wild with infinite global dimension, are Schur-representation-finite because every Schur module is rigid.
\begin{question}
Do there exist wild, Schur-representation-finite algebras of finite global dimension?
\end{question}
\bibliographystyle{alpha}
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{"url":"http:\/\/en.trm.creationmonetaire.info\/comment-comparer-deux-zones-economiques.html","text":"# How to compare two economic zones ?\n\nThe definitions we have just seen concerning the money supply, its growth, the link it should have with the Universal Dividend and the value field, allow us to compare two economic zones that use two different currencies.\n\nFor two economic zones A and B credited each one of a money supply defined in space and time, Ma(x,t) and Mb(x,t) and a number of respective citizen Na and Nb having access to the common money.\n\nThe application of the principle of relativity invites us to define the instant common measure of individual value Ua and UB at a given time on the basis of the average monetary density of each of these zones.\n\nFor A :\n\nand for B :\n\nThe instant exchange rate T(a\/b) of the currencies, which represents the ratio of exchange of a quantity of money in the area A Qa to a quantity of money Qb of the area B so that :\n\nWithin the principle of relativity a fundamental value, which is :\n\nThis fundamental result differs from common tools with which we measure the relationships between the \u00ab prices \u00bb... Yet the values being fundamentally judged as different from one individual to another, and so from one economic area to another, this reference is completely distorted by the arbitrary choice of values used to defined these prices. Whereas the density of common money does not suffer any kind of arbitrary, and is perfectly measurable.\n\n## Numerical application :\n\nRelative exchange rate T (\u20ac \/ $) = 1,60$\/\u20ac\n\nBetween 2008 and 2010 the exchange rate found on the markets oscillated between 1,30 $\/\u20ac and 1,60$\/\u20ac.\n\nBut even if the result found is near a fundamental theoretical value applicable in Relative Theory of Money, there are two factors which reference must be made. First of all the money stocks released by the Central Banks are questionable because the American Fed does not communicate officially M3, and these are unofficial websites which give estimations.\n\nFurthermore, and this is not the most important point, we are not in these economic zones within Universal Dividend zones, where individuals are equals before the monetary creation. Money is created in a centralized way on arbitrary values, and in a non symmetrical way from both sides, which creates strong temporary distortions (and an economic loss in the long term according to the importance of these distortions).\n\nMoreover, we can see the role the population is having about the exchange rates measured by the ratio \u00ab Na\/Nb \u00bb. Thus, one can approach the currency policy based on the importance of the economic space considered better. It is obvious that seen this way an economic area under-monetized, will have soon or later to extend the expansion of its money supply to all its space, therefore to have strong growth rate due to spatial catch-up.\n\nWe understand here the chinese problem in 2010. Since a small part of the 1 400 millions inhabitants can benefit of monetized exchanges, the money supply must grow strongly in all other area of the economy to monetize it as a whole. It is for hundred of human beings to have access to the monetary tool to develop their exchanges, which will play on the value of N which represents the number of monetized citizen.\n\nHowever because of the ratio Na\/Nb which will be, at the end of complete monetization of its population, really big for it against Europe or United-States, the Chinese money supply \u00ab Ma \u00bb will be able to grow at the same pace as Na (number of monetized citizens), without it having an impact of the fundamental exchange value of its money.\n\nThe evolution of the exchange rate for Europe and the United-states, which are for their part already strongly monetized in space (Nb will not grow much anymore), will not depend then on China monetary growth if this one is only spatial, but on their own monetary growth policy in time, to play with the ratio Mb\/Ma.\n\nWe are here in front of two growth policy to catch-up with the necessary balance, in two complementary dimensions : spatial on the Chinese side (one should not forgot the similar problem for the 1 200 millions indian people under-monetized too), and temporally for the United-States and Europe. Yet the temporally catch-up in Europe and in the United-States require a Universal Dividend on which to play the density height of the money supply.\n\nEvolution according to a constant ratio Mb\/Ma through the temporal monetization. Only the monetary quantity of the exchanges grows and keeps stable the speed of circulation (Luc Fievet RTM 2.0)\n\nEvolution according to a constant ratio Ma\/Na using the spacial monetization. The monetary quantity by monetized citizen (new monetized citizens = second circle) remains stable (Luc Fievet RTM 2.0)","date":"2017-04-30 10:46:54","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\": 0.43857330083847046, \"perplexity\": 1594.3082955760656}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-17\/segments\/1492917125074.20\/warc\/CC-MAIN-20170423031205-00266-ip-10-145-167-34.ec2.internal.warc.gz\"}"} | null | null |
El cantón de Roura (en francés canton de Roura) era una división administrativa francesa, que estaba situada en el departamento y la región de Guayana Francesa.
Composición
El cantón estaba formado por la comuna de Roura.
Supresión de los cantones
El 31 de diciembre de 2015, los cantones de Guayana Francesa fueron suprimidos, en aplicación de la ley n.º 2011-884, de 27 de julio de 2011, relativa a las colectividades de Guayana Francesa y Martinica; y específicamente de su artículo 8.º, apartado L558-3 y su comuna pasó a formar parte de la nueva sección de grande Couronne.
Referencias
Roura | {
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} | 4,279 |
package org.exoplatform.addons.jvmconsole.jmx.core;
import org.cyclopsgroup.jmxterm.JavaProcess;
import org.cyclopsgroup.jmxterm.JavaProcessManager;
import org.cyclopsgroup.jmxterm.jdk6.Jdk6JavaProcessManager;
import org.cyclopsgroup.jmxterm.pm.JConsoleClassLoaderFactory;
import java.util.Collections;
import java.util.List;
/**
* Created by eXo Platform MEA on 02/10/14.
*
* @author <a href="mailto:mtrabelsi@exoplatform.com">Marwen Trabelsi</a>
*/
public class JVMProcessManager
{
public static List<JavaProcess> listRunningJVMs()
{
List<JavaProcess> runningJavaProcesses = Collections.emptyList();
try
{
/* Get the an URLClassLoader referring to tools.jar and jconsole.jar */
ClassLoader classLoader = JConsoleClassLoaderFactory.getClassLoader();
JavaProcessManager processManager = new Jdk6JavaProcessManager(classLoader);
runningJavaProcesses = processManager.list();
} catch (NoSuchMethodException e)
{
e.printStackTrace();
} catch (ClassNotFoundException e)
{
e.printStackTrace();
}
return runningJavaProcesses;
}
}
| {
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If Jungle Cruise Is Successful, Does Disney Need Pirates Of The Caribbean Anymore?
By Mike Reyes
After Disney unveiled the new trailer for Emily Blunt and Dwayne Johnson's long awaited Jungle Cruise movie, there was a familiar feeling that washed over myself and some of my colleagues here at CinemaBlend. While it seems like a lot of the DNA from Brendan Fraser's Mummy franchise is built into this new cinematic adventure, there's also quite a bit of Pirates of the Caribbean energy present in this next attraction-based film. So if Jungle Cruise is a success at the box office, does the audience even need a Pirates of the Caribbean reboot anymore?
Naturally, this whole argument is built on the premise that Jungle Cruise will be the smash summer hit Disney hopes it will be. So with that firmly in mind, we can talk out why one series should replace the other in the pecking order. Ultimately, it'll be up to the audience as to whether Jungle Cruise sinks or sails, and if it does, these are the reasons why it should send Pirates of the Caribbean sailing into the great beyond.
The Pirates Of The Caribbean Franchise Has Run Its Course
After five movies and a teased, yet unmade finale, not to mention losing the star at the head of its crew, the Pirates of the Caribbean has pretty much run its course. While competing reboots with Karen Gillan and Margot Robbie are currently in development, and some fans are excited by the possibilities, it still doesn't feel like a surefire shot. Considering how expensive the Pirates movies are to produce and market, two potential reboots feel like a bit of a stretch for a series that's been diminishing in returns.
While Pirates of the Caribbean: On Stranger Tides somehow managed to become the highest grossing film in the series, the film that followed, Pirates of the Caribbean: Dead Men Tell No Tales, missed the $1 billion mark. As this franchise feels like it's on the downswing, a reboot would naturally sound appealing. But if Jungle Cruise does well, it would render this series totally obsolete.
Jungle Cruise Is A Seafaring Adventure With High Stakes And Humor
Much as Pirates of the Caribbean did in its time, it looks like Jungle Cruise is going to be engaging in the same sort of seafaring adventure that Johnny Depp's former franchise once did. Only this time, you have Dwayne Johnson and Emily Blunt anchoring the show as the charming leads that blunder through obstacles in the name of comedy. And that's on top of a good amount of adventure being thrown into the mix as well!
Huge battles with comedic adversaries that have inflated self worth and/or mystical powers is basically the logline for both Jungle Cruise and Pirates of the Caribbean. You could practically ride both attractions at a Disney Park and legitimately confuse one for the other, were you not careful. Jungle Cruise feels like it's taken the Pirates formula and refreshed it in such a way that new actors could pick up the torch of the former blockbuster franchise.
Much Like The Pirates Movies, Jungle Cruise Is Pursuing A Legendary Treasure
Another key hallmark to the Pirates of the Caribbean saga is, of course, the treasures. And these have never been any ordinary treasures that are on display either, as everything from cursed Aztec gold to the Trident of Poseidon himself has been sought after by Captain Jack and his accomplices. Sure enough, Jungle Cruise has its own magical Macguffin to maneuver around, and it sounds like something straight out of Captain Jack Sparrow's wheelhouse.
The Tree of Life is Jungle Cruise's ticket to fortune and glory, as its reported healing powers are the prize that Emily Blunt's Dr. Lily Houghton and her brother are seeking to recover. Plus, as an added bonus in Pirates of the Caribbean bingo, this new movie also has a villain who was once a conquistador, but looks to eventually become part human/part mystical creature. Even Davy Jones himself would call that similarity out with clarity.
Dwayne Johnson And Emily Blunt Are Huge Names To Tie A Potential Franchise To
The Pirates of the Caribbean franchise was once driven, almost entirely, by the sheer star power that was Johnny Depp. For obvious reasons, that's not exactly a strategy that Disney can pursue at this moment, which is definitely a part of why there's two reboots on the table. But while Karen Gillan and Margot Robbie aren't exactly confirmed, let alone on set, for either of these projects, Dwayne Johnson and Emily Blunt's participation in Jungle Cruise is locked and loaded.
Already the on-screen chemistry between Dwayne Johnson and Emily Blunt looks like a match made in cinematic heaven, and both are still pretty huge draws at the movies. Between their box office pedigrees and their loyal fanbases, Johnson and Blunt are a pair that, if attached to a recurring franchise, would absolutely crush. And the best part is, having them as part of a fresh concept would only add to the potential longevity of the Jungle Cruise franchise.
Jungle Cruise isn't exactly a sure fire success, as there are plenty of Disney movies that spent a lot of money trying to become the next big franchise, only to flame out. The big difference is that with this new aquatic adventure picture coming along at the right time, it feels like the studio has found a way to continue pursuing crazy treasures on the high seas. For all of the reasons above, it's time for Disney to give the Pirates of the Caribbean series a viking funeral and let Jungle Cruise continue with smooth sailing.
That is, of course, provided that Jungle Cruise happens to be a box office success, and we won't see those results until after the film's release on July 30. With both theatrical release and Disney+ Premier Access offering the film upon that date, the odds are doubled when it comes to that potential for victory. Of course, you need to have Disney+ in order to purchase Premier Access, and if you're new to the platform, you should check out the Disney+ subscription bundle currently being offered.
This poll is no longer available.
Up next: Disneyland And Magic Kingdom's Jungle Cruise Ride Is Finally Getting That Makeover, Check Out The New Details
Mike Reyes
Senior Movies Contributor
CinemaBlend's James Bond (expert). Also versed in Large Scale Aggressors, time travel, and Guillermo del Toro. He fights for The User.
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Ant-Man's Paul Rudd and Evangeline Lilly Provide Updates About Jeremy Renner's Health
Brendan Fraser On Time He Auditioned For Superman, And What Went Wrong | {
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namespace {
// Equal to the #9F9F9F color used in spec (note WebUI color is #999).
const SkColor kDeemphasizedTextColor = SkColorSetRGB(159, 159, 159);
} // namespace
MediaGalleryCheckboxView::MediaGalleryCheckboxView(
const MediaGalleryPrefInfo& pref_info,
int trailing_vertical_space,
views::ContextMenuController* menu_controller) {
SetLayoutManager(std::make_unique<views::BoxLayout>(
views::BoxLayout::Orientation::kHorizontal));
ChromeLayoutProvider* provider = ChromeLayoutProvider::Get();
const gfx::Insets dialog_insets =
provider->GetInsetsMetric(views::INSETS_DIALOG);
SetBorder(views::CreateEmptyBorder(
0, dialog_insets.left(), trailing_vertical_space, dialog_insets.right()));
if (menu_controller)
set_context_menu_controller(menu_controller);
checkbox_ = AddChildView(std::make_unique<views::Checkbox>(
pref_info.GetGalleryDisplayName(), views::Button::PressedCallback()));
if (menu_controller)
checkbox_->set_context_menu_controller(menu_controller);
checkbox_->SetElideBehavior(gfx::ELIDE_MIDDLE);
std::u16string tooltip_text = pref_info.GetGalleryTooltip();
checkbox_->SetTooltipText(tooltip_text);
std::u16string details = pref_info.GetGalleryAdditionalDetails();
secondary_text_ = AddChildView(std::make_unique<views::Label>(details));
if (menu_controller)
secondary_text_->set_context_menu_controller(menu_controller);
secondary_text_->SetVisible(details.length() > 0);
secondary_text_->SetEnabledColor(kDeemphasizedTextColor);
secondary_text_->SetElideBehavior(gfx::ELIDE_HEAD);
secondary_text_->SetTooltipText(tooltip_text);
secondary_text_->SetBorder(views::CreateEmptyBorder(
0, provider->GetDistanceMetric(DISTANCE_RELATED_CONTROL_HORIZONTAL_SMALL),
0, 0));
}
MediaGalleryCheckboxView::~MediaGalleryCheckboxView() = default;
void MediaGalleryCheckboxView::Layout() {
views::View::Layout();
if (GetPreferredSize().width() <= GetLocalBounds().width())
return;
// If box layout doesn't fit, do custom layout. The secondary text should take
// up at most half of the space and the checkbox can take up whatever is left.
int checkbox_width = checkbox_->GetPreferredSize().width();
int secondary_text_width = secondary_text_->GetPreferredSize().width();
if (!secondary_text_->GetVisible())
secondary_text_width = 0;
gfx::Rect area = GetContentsBounds();
if (secondary_text_width > area.width() / 2) {
secondary_text_width =
std::max(area.width() / 2, area.width() - checkbox_width);
}
checkbox_width = area.width() - secondary_text_width;
checkbox_->SetBounds(area.x(), area.y(), checkbox_width, area.height());
if (secondary_text_->GetVisible()) {
secondary_text_->SetBounds(checkbox_->x() + checkbox_width, area.y(),
secondary_text_width, area.height());
}
}
BEGIN_METADATA(MediaGalleryCheckboxView, views::View)
END_METADATA
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,781 |
Maidstone is the county town of Kent, England.
Maidstone may also refer to:
Places
United Kingdom
Borough of Maidstone, a borough in Kent, England, administrative centre Maidstone
Maidstone United F.C., the borough's association football team
HM Prison Maidstone, a men's prison located in the borough
Maidstone Airport, Kent, a former airport serving the town between 1917 and 1969
Canada
Maidstone, Ontario in Essex County, Ontario
Maidstone Township, Ontario also in Essex County, Ontario
Maidstone, Saskatchewan in Canada
Maidstone Aerodrome, serving the town
United States
Maidstone, Vermont
Maidstone State Park in Maidstone, Vermont
Maidstone Golf Club in East Hampton, New York
Maidstone (Owings, Maryland) one of the oldest houses in Maryland
Other places
Maidstone, Victoria, Australia
Maidstone Park, a sports ground in Upper Hutt in New Zealand
Maidstone Road a small residential street in Tokwawan, Kowloon, Hong Kong
Maidstone, Jamaica a Free Village in Manchester parish, Jamaica
Maidstone, KwaZulu Natal, South Africa; See All Saints Church, Maidstone, KwaZulu-Natal
Other uses
Maidstone (film), a 1970 film by Norman Mailer
See also
HMS Maidstone, several ships | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,683 |
Q: How to add spacing between two span elements I have two span elements. I want them to sit side-by-side and contain some text. Is this possible? If so what am I doing wrong?
.added-box{
background-color:#06C;
padding:8px;
margin:8px;
}
.edited-box{
background-color:#093;
padding:8px;
margin:8px;
}
And the page code is:
<p align="right">
<span class="edited-box">sfds<span>
<span class="added-box">sfds<span>
</p>
Edit: What I am hoping to get is a box, somewhat like the one on this page which has my name, time I asked the question and my points. I don't mind how I get it, but css is preferred, it looks like StackOverflow is using a table. Is this the only way to do this?
A: You have two typos in your HTML where you are failing to close the <span> tags with </span>. It should be:
<p align="right">
<span class="edited-box">sfds</span>
<span class="added-box">sfds</span>
</p>
This typo is causing your edited-box class to wrap everything and therefore the CSS is breaking.
A: Use for space between spans.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,279 |
Hidden figures of design: modellers in the digital age
The creation of a design model is one of the most exciting stages in the development of a new car. At ŠKODA Design, an international team of over 80 people is dedicated to this process. Meet the team members responsible for modelling and digitalisation.
Headed by Martin Bogner, the Design Modelling and Digitalization (EDM) department transforms the designer's idea into a digital or physical model as faithfully and quickly as possible. These models then represent a proposed design solution and are mainly used for further development, design presentations and the final approval of proposed designs for mass production. The EDM department is also responsible for design presentations essentially anywhere in the world, as well as developing and implementing advanced visualisation and other IT technologies and working on the digitisation of design processes. It's a lot of diverse issues, but the big advantage is that all the activities are logically connected.
ŠKODA Storyboard will cover the different sections of this largest design department in three articles. The first will focus on digital modelling and digital technology.
Modelling and digitalisation requires teamwork involving many different professions.
"The role played by digitalisation is growing significantly today," says Martin Bogner. "Whereas we used to work mainly with clay models, most of our work today is done in the digital environment. In most cases, this gives us greater efficiency, flexibility, speed and the associated financial savings. At the same time, digitalisation allows us to work remotely, sharing information and digital design data via modern communication methods like video conferencing and other online tools, which has had other advantages in recent times, for reasons we all know only too well."
Today, digital work accounts for around 70–80% of the design process for a new car, with the rest involving work with physical models. "Digitalisation in this area has been proceeding intensively for twenty years, but it is only in the last few years that the capabilities of software and hardware have made a great leap forwards," says Martin Bogner. Basically, the idea is to create a detailed and sophisticated 3D model in the digital environment that will faithfully represent the final car.
How good does the new version of the ŠKODA FABIA look in its virtual presentation?
A sketch is the starting point
This is the model prepared by ŠKODA's digital modellers. Their starting point is a sketch by a designer, which the modellers and the designer then use to prepare a 3D model of the future car. "It's a kind of symbiosis, where two people, two different positions, actually merge into one. The designer can produce a sketch – we might call him the brain of the process – and the modeller knows how to create the model, so he is the hand," says János Bársony, head of the digital modelling and visualisation team.
Recent technology and the events of the last two years themselves have made this connection closer than ever. "Until recently, we basically worked with a few programmes that required a great deal of understanding. But now the software's capabilities are much broader and easier to work with so, for example, we are trying to get designers to sketch their visions in 3D from the get-go wherever possible," explains János Bársony.
The Design Modelling and Digitalisation team's visualisation of the new generation of the ŠKODA FABIA
First, the designers were introduced to a popular graphics programme called Blender, but now 3D sketching in virtual reality is also coming into its own. This next step again facilitates the creation of digital models. "With sketches, it is extremely important that the designer conveys all his intentions, thoughts and ideas well to the modeller – it is best to do this in 3D, of course," explains János Bársony.
"Sketching in virtual reality will give the designer a better idea of the result than sketching in a 3D programme on a monitor. Projection of the geometrie to the flat screen is distort, and then there's the complication of scale. In virtual reality, you can create on a scale of 1:1 and see everything in space right away," says Radek Šimon, who is responsible for the introduction of these technical innovations at ŠKODA Design and is in charge of technical equipment for the entire digital team.
Members of the Design Modelling and Digitalisation department responsible for interior design
But whether the starting point is a conventional sketch created by hand on a tablet, a 3D model or a virtual reality model, the modeller then has to create a model of the car. Sketches don't cover the details: they do not address the individual parts and components of the car, so these have to be added by the modelling team. There are several of them working on every car: typically, for example, three work on the exterior and three on the interior, gradually modifying the model and consulting with the designer, then presenting it to the management and modifying it again. "The modeller actually has to think in loops, and he has to be patient. Those individual stages are very important, because they help to refine the final design to perfection. But it is not unusual to go back several steps, especially in the first stages of the work," János Bársony describes.
The Design Modelling and Digitalisation team's visualisation of the interior of the new ŠKODA FABIA
Presentation like a video game
This kind of work in loops as the model is gradually fine-tuned takes about 22 months. At a later stage the digital model serves as the basis for a clay model (more on that next time). From the clay model, the modifications are scanned back into the 3D model and fine-tuned again. "The data we generate make it easier for other ŠKODA departments to work on their parts of the upcoming car," says János Bársony.
Presentations are another important part of working in the digital environment. What's more, their importance has understandably grown in the last two years. In the virtual environment, the teams now present the cars not only to their superiors and for consulting with the designers: they also present the work being done on the car to the company management and the management of the entire group. In this way, the car is fine-tuned in a virtual environment across the world, for example with colleagues from India.
Members of the Design Modelling and Digitalisation department responsible for exterior design
Special software is used for visualisations, but surprisingly, the technology is also quite "democratic". ŠKODA uses the Unreal or Unity videogame engines for this purpose. These are used to create static visuals, and realtime design presentations. "One of the reasons we use them is that their output works effectively with the various virtual reality glasses we use. They're also fairly open platforms that we can programme almost anything we need into. And last but not least, game engines are designed to provide the best possible graphic output at the minimum CPU use," explains Radek Šimon.
Common design presentations, especially those in 3D, can therefore be done using relatively "ordinary" computers with powerful graphics cards. For very complex and complicated models displayed by method Realtime Raytracing is using high performace computing cluster. This is the hardware, as well as the workstations and interactive sketching tablets that the designers and modellers work with, that Radek Šimon tests and provides for the design team.
The Design Modelling and Digitalisation department includes a team that works on wheel design.
One of the innovations his team has managed to put into operation in the past few months is the new presentation wall. In the design presentation room at the Česana office in Mladá Boleslav, a state-of-the-art LED walls Samsung The Wall with 8K resolution is now in operation – there are two of these six-metre high surfaces side by side. "Making them operational was not easy. We were in intensive contact with the developers in Korea and we even had to convince them that our spaces would do justice to this cutting-edge technology," says Radek Šimon. This is yet more proof of how ŠKODA is constantly striving to be a frontrunner in the digital sphere.
The trio from the Design Modelling and Digitalisation department
Martin Bogner
Martin Bogner is the first head of the combined Design Modelling and Digitalisation team. He is in charge of the entire department that transforms designers' visions from sketches first into digital form and then into physical models that look like real cars. "It's an amazing collaboration with a lot of very talented and creative people," he says of his work. "I've been working in design for more than a decade, and what I enjoy most is being present at the very birth of the digital and physical model that always evolves from nothing more than ideas. And after two years we see the final product being produced in their hundreds of thousands," he smiles.
János Bársony
Growing up in Hungary, János Zoltán Bársony drew cars and planes in his school books from a young age. "I wanted to be a pilot or an aircraft designer. But in the socialist bloc of the time, studying aviation fields was only possible in Russia, and I was no good at Russian," he recalls with a smile, explaining why he finally decided to study design. "I was one of the first students on the newly launched Industrial Design course at the University of Dresden," he says. After school, he worked as a digital modeler, mainly for Volkswagen Group, and for a time as a freelancer. At the request of Oliver Stefani, he joined ŠKODA AUTO in 2018. "As an external contractor for the group I didn't have much influence, but here my work is really visible and I have the chance to push things forward," he says appreciatively of his work at the Czech carmaker.
Radek Šimon
Radek Šimon is something of a digital modelling guru at ŠKODA's Mladá Boleslav headquarters. "Digital modelling started here in 1993 and I was part of a four-person team that helped launch it," he says. As this field evolved, Radek Šimon worked with a wide range of technologies and participated in founding the first VR Studio in ŠKODA. He created teams for digital design modelling, VR and visualisations and led them until 2018. This year, however, he decided to dedicate his working hours to the technology itself, and he has been working hard to put new tech into practice at ŠKODA Design ever since. He keeps a close eye on all the trends in this field and, just as his colleagues often get to see secret models of future cars, he tries out suppliers' technical innovations that are still top-secret.
Next up in THE HIDDEN FACES OF DESIGN series
Hidden figures of design: car sculptors
DESIGN 4. 1. 2022
Hidden Figures of Design: lights as designer jewellery
The hidden figures of design: making your car feel like home
The hidden faces of design help you get along with your car
Modelling and digitalisation requires teamwork.
Visualisation of the new generation of the ŠKODA FABIA
A team responsible for interior design
Visualisation of the interior of the new generation of the ŠKODA FABIA
A team responsible for exterior design
A team that works on wheel design
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The new ŠKODA FABIA is a modern take on tradition | {
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} | 3,160 |
{"url":"https:\/\/gamedev.stackexchange.com\/questions\/168650\/using-state-pattern-with-unity\/175449","text":"# Using State Pattern with Unity\n\nI am new to Game Development and Unity. I have written a component for Jumping and Running, CharacterJumpAbility.cs and CharacterRunAbility.cs.\n\nI would like the Character to be able to Jump only if the Character is not moving. I came across a State Pattern but I can't understand how to apply it. For example if I want to Jump, first I must check if the Character is Running, but I have to do this in the CharacterJumpAbility.cs or in the CharacterController.cs (the component who handles inputs)?\n\nIf I define a state for example:\n\npublic enum CharacterState {\nRun,\nJump\n}\n\n\nFor example if the player is pressing the space bar, I have to set the Jump state, the code that sets state = CharacterState.Jump should be written in the CharacterJumpAbility.cs or in the component that handles inputs?\n\nAnd if I want to set a new state for example Jump, while I am running, this is clearly not possible, how to handle situations like these?\n\n\u2022 Have you considered making use of the Mecanim animation state machine for managing this type of character state logic? It has a lot of features for building sophisticated state systems, with handling of transitions, conditions, layering, etc. - specifically tailored to working with character movement & ability state management. And you can define custom behaviour scripts to act in particular states \u2013\u00a0DMGregory Mar 13 '19 at 0:44\n\u2022 Keep in mind that there is never just one way to do things in software development. You usually have the option to implement a new features in a variety of different components. \u2013\u00a0Philipp Sep 4 '19 at 14:10\n\nfor working with state pattern, first thing that comes to mind is enum and state. most problem is switch is not OO and most of the time its better to not to use switch-case. and you dont have much freedom to extend. its implemented like below: you can fill cases with any logic you want and you only have to change state variable to change running state.\n\n enum states {run, jum,shoot,etc }\nstates state;\nprivate void Update()\n{\nswitch (state)\n{\ncase states.run:\nbreak;\ncase states.jum:\nbreak;\ncase states.shoot:\nbreak;\ncase states.etc:\nbreak;\ndefault:\nbreak;\n}\n}\n\n\nanother approach that is much better is use of delegates like Action. one action runs in all times but you can change assigning function to it. you define it like below: you can change state like what happened in Start callback\n\n System.Action CurrentState;\n\nvoid Shoot() { }\nvoid Run() { }\nvoid Jump() { }\n\nprivate void Start()\n{\nCurrentState = Run;\n}\n\nprivate void Update()\n{\nCurrentState.Invoke();\n}\n\n\nIt depends on the complexity of your character. Enumerations are fine for simpler characters, but it can become very problematic when character's abilities get expanded.\n\nVirtouso's answer is good, but I'd just like to add how I go about doing state machines a lot of the time.\n\nYou can create a base class along these lines:\n\npublic abstract class StateBase<T>\n{\npublic abstract void OnEnter(T owner);\npublic abstract void Update(T owner);\npublic abstract void OnExit(T owner);\n}\n\n\nThen have a state machine something like this:\n\npublic class StateMachine<T>\n{\nprivate T owner;\nprivate StateBase<T> currentState;\n\npublic void Update()\n{\ncurrentState.Update(owner);\n}\n\npublic void GoToState(StateBase<T> state)\n{\n\/\/...\n}\n}\n\n\nObviously, this is very stripped down for an example, but you get the idea. Your running and jumping would inherit from StateBase. This state machine can then easily be re-used across different types.\n\nI'm not a fan of the delegate method because you add a lot of functions, which I don't think is as clear as having classes that are easily found for each state.\n\nI find this eBook very useful: https:\/\/gameprogrammingpatterns.com\/state.html\n\nHere is a good tutorial on the use of the state pattern in your game. If you set your animation controller up right, you can use the OnStateEnter and onStateExit methods of each state to set animator parameters. Hope this helps.\n\n\u2022 The text of your answer seems to be at odds with the link you provided. You are talking about using the AnimationController (and i am not so sure if this is a good solution for this situation, by the way), but the link you posted describes a method to create a complete own state machine not based on the Unity animation system. \u2013\u00a0Philipp Sep 4 '19 at 14:13","date":"2020-05-29 03:44:06","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\": 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\": 0.3609176576137543, \"perplexity\": 1268.95565571128}, \"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\/1590347401260.16\/warc\/CC-MAIN-20200529023731-20200529053731-00118.warc.gz\"}"} | null | null |
\section{Dedicated Figures Page}
\label{sec:appendix}
\begin{figure*}[!ht]
\centering
\resizebox{\linewidth}{!}{
\subfloat[test1]{
\includegraphics{results/vgg-16-graphs-x86_64/cyc}
\label{fig:cyc-vgg-16-ivy}
}
\subfloat[test2]{
\includegraphics{results/googlenet-graphs-x86_64/cyc}
\label{fig:cyc-vgg-16-ivy}
}
\subfloat[test3]{
\includegraphics{results/alexnet-graphs-x86_64/cyc}
\label{fig:cyc-vgg-16-ivy}
}
}
\caption{TEST1234}
\label{fig:cyc-vgg}
\end{figure*}
\begin{figure*}[!ht]
\centering
\subfloat[Execution times for all the methods running the convolutional layers of \gls{alexnet} on Ivy Bridge. \textbf{Lower is better}.]{
\includegraphics[width=0.32\linewidth]{results/alexnet-graphs-x86_64/cyc}
\label{fig:cyc-alex-ivy}
}~
\subfloat[Execution times for all the methods running the convolutional layers of \gls{alexnet} on Tegra TX1-GPU. \textbf{Lower is better}.]{
\includegraphics[width=0.32\linewidth]{results/alexnet-graphs-tx1/cyc}
\label{fig:cyc-alex-tx1gpu}
}~
\subfloat[Execution times for all the methods running the convolutional layers of \gls{alexnet} on Tesla K40-GPU. \textbf{Lower is better}.]{
\includegraphics[width=0.32\linewidth]{results/alexnet-graphs-k40c/cyc}
\label{fig:cyc-alex-k40gpu}
}
\caption{Summary of performance of our proposed methods on selected \gls{mcmk} operations from \gls{alexnet} on the architectures from Table~\ref{tab:expr-platforms}. Performance is measured
as the mean execution time over 25 runs of each benchmark. The names used for kernels are those from the Netscope \gls{cnn} Analyzer~\cite{netscope}.}
\label{fig:cyc-alexnet}
\end{figure*}
\begin{figure*}[!ht]
\centering
\subfloat[Execution times for all the methods running the convolutional layers of \gls{googlenet} on Ivy Bridge. \textbf{Lower is better}.]{
\includegraphics[width=.85\linewidth]{results/googlenet-graphs-x86_64/cyc}
\label{fig:cyc-googlenet-ivy}
}\qquad
\subfloat[Execution times for all the methods running the convolutional layers of \gls{googlenet} on Tegra TX1-GPU. \textbf{Lower is better}.]{
\includegraphics[width=.85\linewidth]{results/googlenet-graphs-tx1/cyc}
\label{fig:cyc-googlenet-tx1gpu}
}\qquad
\subfloat[Execution times for all the methods running the convolutional layers of \gls{googlenet} on Tesla K40-GPU. \textbf{Lower is better}.]{
\includegraphics[width=.85\linewidth]{results/googlenet-graphs-k40c/cyc}
\label{fig:cyc-googlenet-k40gpu}
}
\caption{Summary of performance of our proposed methods on selected \gls{mcmk} operations from \gls{googlenet} on the architectures from Table~\ref{tab:expr-platforms}. Performance is measured
as the mean execution time over 25 runs of each benchmark. The names used for kernels are those from the Netscope \gls{cnn} Analyzer~\cite{netscope}.}
\label{fig:cyc-googlenet}
\end{figure*}
\end{appendices}
\section{Background}
\label{sec:back}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/drawn/inkscape/mcmk}
\caption{\glsfirst{mcmk} Convolution}
\label{fig:mcmk}
\end{figure}
In this section, we give an overview of the \glsfirst{cnn}. Traditionally,
\glspl{cnn} are composed of a number of basic building blocks (often referred
to as \emph{layers}): \textbf{convolutional} layer, \textbf{pooling} layer
(average and max pooling), \textbf{activation} layer, \textbf{fully connected}
layer (FC) and \textbf{loss} layer.
\begin{figure*}[t]
\centering
\includegraphics[width=.6\linewidth]{figures/drawn/inkscape/im2col}
\caption{\gls{mcmk} using the \gls{i2c} method}
\label{fig:im2col}
\end{figure*}
\subsection{Multi-channel convolution as sum of single channel convolutions}
\label{sec:mcmk}
In order to understand the core component of a \gls{cnn} -- the convolutional layer, we define dimensions of the inputs, the convolutional kernels (also known as feature maps) and the outputs. A convolutional layer (without batching) takes as input a 3D tensor (input -- $\ifm$) and a 4D tensor (kernel -- $\mathcal{K}$) and outputs a 3D tensor (output -- $\ofm$). At the core of any convolutional layer is a \gls{scsk}.
Given an input $\ifm \in \mathbb{R}^{H\timesW}$ and a kernel $\mathcal{K} \in \mathbb{R}^{k\timesk}$ an output element at position $(x,y)$ where $x\in[0,Q)$ and $y\in[0,P)$, from the output $\ofm\in\mathbb{R}^{Q\timesP}$ is given by
\begin{multline}
\label{eq:scsk}
conv2D_{x,y}(\ifm, \mathcal{K}) = \sum_{i=0}^{i=k-1} \sum_{j=0}^{j=k-1} \ifm( x-\floor{\frac{k}{2}}+i,\\ y-\floor{\frac{k}{2}}+j )\times\mathcal{K}( i, j )
\end{multline}
\noindent where $W$ and $H$ are the width and height of the input; $k$ is the size of the square kernel; $P$ and $Q$ are the width and height of the output which is normally equal to $W$ and $H$ respectively\footnote{Note that $P$ and $Q$ are not equal to $W$ and $H$ if the convolution is ``strided''.
We do not consider strided convolutions in this paper as they account for only a small proportion of computation in most \glspl{cnn}
}; and assuming the input is properly padded.
The \gls{mcsk} is then constructed using \gls{scsk}, by adding the result of 2D convolutions of the $C$ corresponding channels of the input $\ifm$ and the kernel $\mathcal{K}$. This is represented as
\begin{equation}
\label{eq:mcsk}
\gls{mcsk}(\ifm_{C}, \mathcal{K}_{C}) = \sum_{c=0}^{c=C-1} conv2D(\ifm(c), \mathcal{K}(c))
\end{equation}
\noindent where $C$ is the number of channels in the input and kernel and $\ifm(c)$ and $\mathcal{K}(c)$ represent the $c$\textsuperscript{th} channel of the input and the kernel respectively. It is imperative to note that the output $\ofm\in\mathbb{R}^{Q\timesP}$ from Equation~\ref{eq:scsk} is a two-dimensional matrix and so is the output of \gls{mcsk} as shown in~\ref{eq:mcsk}. As discussed earlier, the convolution layer does \gls{mcmk} which can expressed as the concatenation of the resultant matrices from Equation~\ref{eq:mcsk} as shown in Figure~\ref{fig:mcmk} and is represented as
\begin{equation}
\label{eq:mcmk}
\resizebox{\linewidth}{!}
{
$\gls{mcmk}(\ifm_{C}, \mathcal{K}_{C}^{M}) = \gls{mcsk}(\ifm_{C}, \mathcal{K}_{C}^{0}) \concat \cdots \concat \gls{mcsk}(\ifm_{C}, \mathcal{K}_{C}^{M})$
}
\end{equation}
\noindent where $\mathcal{K}_{C}^{M}$ represents $M$ kernels with $C$ channels each and the $\concat$ operator denotes the concatenation of two channels.
\subsection{\gls{i2c}}
\label{sec:im2c}
The \gls{i2c} approach~\cite{chellapilla2006high,tsai2016performance,yanai2016efficient,gu2016opencl} has been well studied for transforming the \gls{mcmk} problem into a \gls{gemm} problem. Consider an input $\ifm \in \mathbb{R}^{H\timesW\timesC}$ and $M$ kernels $\mathcal{K}\in\mathbb{R}^{M\timesk\timesk\timesC}$ as shown in Figure~\ref{fig:im2col}. From the input $\ifm$ we construct a new \emph{input-patch-matrix} $\hat{\ifm}$, by copying \emph{patches} out of the input and unrolling them into columns of this intermediate matrix. These patches are formed in the shape of the kernel (i.e. $k\timesk\timesC$) at every location in the input where the kernel is to be applied.
Once the input-patch-matrix $\hat{\ifm}$ is formed, we construct the kernel-patch-matrix $\hat{\mathcal{K}}$ by unrolling each of the $M$ kernels of the shape $k\timesk\timesC$ into a row of $\hat{\mathcal{K}}$ as shown in Figure~\ref{fig:im2col}. Note that this step can be avoided if the kernels are stored in this format to begin with (innermost dimension is the channel which forces the values along a channel to be contiguous). Then we simply perform a \gls{gemm} of $\hat{\mathcal{K}}$ and $\hat{\ifm}$ to get the output $\hat{\ofm}\in\mathbb{R}^{H\timesW\timesM}$ as shown in the figure.
It is easy to see from the above discussion how one could implement another method called \gls{i2r} wherein the local patches are unrolled into rows of the input-patch-matrix $\hat{\ifm}$ and the kernels are unrolled into columns of the kernel-patch-matrix $\hat{\mathcal{K}}$. We then perform a \gls{gemm} of $\hat{\ifm}$ and $\hat{\mathcal{K}}$ instead of $\hat{\mathcal{K}}\times\hat{\ifm}$ as in \gls{i2c}.
\vspace{.7cm}
\begin{figure}[h]
\begin{minipage}{1\linewidth}
\begin{footnotesize}
\begin{verbatim}
input[C][H][W];
kernels[M][K][K][C];
output[M][H][W];
for h in 1 to H do
for w in 1 to W do
for o in 1 to M do
sum = 0;
for x in 1 to K do
for y in 1 to K do
for i in 1 to C do
sum += input[i][h+y][w+x] *kernels[o][x][y][i];
output[o][w][h] = sum;
\end{verbatim}
\end{footnotesize}
\end{minipage}
\caption{Simplified code for 2D multi-channel convolution with a single multi-channel input and multiple multi-channel convolution kernels. Note that special treatment of edge boundaries is not shown in this code.}
\label{fig:covCode}
\end{figure}
\vspace{.7cm}
\section{Introduction}
\label{sec:intr}
\glspl{cnn} are one of the most effective machine learning approaches
for a variety of important real world problems. \glspl{cnn} require
very large amounts of computation for both training and
inference. \glspl{cnn} are constructed from networks of standard
components, such as \emph{convolution layers}, \emph{activation
layers} and \emph{fully-connected layers}. In most successful
\glspl{cnn}, the great majority of computation is performed in the
convolutional layers.
\glspl{cnn} require a very large amount of computation, so it is important to
make best use of available hardware resources. This hardware may take the form
of a standard CPU, or an accelerator such as a graphics processing unit (GPU),
digital-signal processor (DSP), or vector architecture. However, making best
use of the hardware for computationally intensive problems often requires
careful tuning of code to make best use of the memory hierarchy, registers, and
vector parallelism. For example, processor/accelerator companies devote very
large effort to tuning the performance of standard operators such as those in
the the Basic Linear Algebra Subroutines (BLAS) \cite{Lawson:1979}.
When implementing \glspl{cnn} on a new accelerator or processor, it is
fortunately possible to exploit existing pre-tuned BLAS routines. In
particular, the BLAS general matrix multiplication (\glsfirst{gemm}) routine is
commonly used to implement DNN convolution. It is well-known that 2D
convolution can be implemented using matrix multiplication by converting one of
the input matrices to a Toeplitz matrix. This involves replicating image pixels
multiple times across different matrix columns. Once the Toeplitz matrix has
been constructed, convolution can be implemented using a highly-tuned
\gls{gemm} for the target architecture.
The \gls{i2c} approach has been highly successful in \gls{dnn} frameworks such
as Caffe, Theano and Torch \cite{Chetlur:2014}. However, a major downside of
\gls{i2c} is the space explosion caused by building the column matrix. For a
convolution with a 2D $k \times k$ kernel matrix, the column matrix is $k^2$
times larger than the original image. Deep learning systems are often most
useful when deployed in the field, but the space required for the column matrix
may be far too large to fit in the memory of an embedded system. Even outside
of the embedded context, the increased memory requirement may stretch the
limits of on-chip local memories and caches, which may increase execution time
and memory traffic.
In this paper we propose a new approach to \gls{dnn} convolution that allows us
to exploit existing optimized routines for accelerators and processors, but
does not costly input transformation operations. We make a number of
contributions:
\begin{itemize}
\item
We formulate the problem to operate on a \textit{non-replicated}
input image. This allows us to pose the problem as either one or a sequence of matrix multiplications.
\item We present an experimental evaluation of our approach
on an embedded processor (ARM\textregistered ~Cortex\textregistered-A57), and
a general purpose CPU (Intel\textregistered ~Core\texttrademark ~i5-4570) using highly-optimized
parallel versions of \gls{gemm}.
\item Our new \gls{gemm}-based approaches perform better than \gls{i2c} in a great majority of the scenarios tested.
\end{itemize}
The remainder of this paper is organized as follows. Section~\ref{sec:back}
provides additional background and detail on the \gls{mcmk} convolution
operation which is central to deep neural networks. Section~\ref{sec:meth}
describes how convolution can be implemented with a column matrix. We also show
how our proposed approach retains the advantages of re-using \gls{gemm} for the
computationally-intensive tasks, but with improved data locality.
Section~\ref{sec:expe} presents an evaluation of a number of variants of our
approach. Finally, Section~\ref{sec:rela-work} describes related work.
\section{Conclusion} Multi-channel multi-kernel convolution is the most
computationally expensive operation in \glspl{dnn}. Maximal exploitation of
accelerator or processor resources for \gls{mcmk} requires a deep understanding
of the micro-architecture. Careful design of data blocking strategies to
exploit caches, on-chip memories and register locality are needed, along with
careful consideration of data movement and its interaction with SIMD/SIMT
parallelism. Each new processor or accelerator has different performance
characteristics, requiring careful tuning of the code each time it is brought
to a new target.
There are significant advantages in implementing \gls{mcmk} convolution using
existing carefully tuned \glsfirst{gemm} libraries. However, the most
widely-used approach, \gls{i2c} has a large memory footprint because it
explodes the input image to a much larger column matrix. This space explosion
is quadratic in the radix, $k$ of the convolution being performed. This is
problematic for memory-constrained systems such as embedded object detection
and recognition systems. Additionally, the data redundancy resulting from
\gls{i2c} reduces data locality and increases memory traffic.
We propose new approaches for implementing MCMK convolution using existing
parallel \gls{gemm} libraries. Our \gls{k2r} approach makes one call to
\gls{gemm} and does a post pass on the output to accumulate the partial results
into a single matrix. This drastically increases data locality compared to
\gls{i2c}.
Our results strongly motivate the development of a cost model to guide the
selection of implementations of \gls{mcmk} convolution in deep neural networks. The performance of all of the methods which we evaluate is strongly
context-dependent, with methods having very good performance in some contexts,
and very poor performance in others. The development of this cost model seems a
very fruitful avenue for future work in the area.
\section*{Acknowledgment}
\setlength{\lineskip}{0.5em}
\noindent This work was supported by Science Foundation Ireland
grant 12/IA/1381. This project has received funding from the European Union's
Horizon 2020 research and innovation programme under grant agreement No 732204
(Bonseyes). This work is supported by the Swiss State Secretariat for
Education' Research and Innovation (SERI) under contract number 16.0159. The
opinions expressed and arguments employed herein do not necessarily reflect the
official views of these funding bodies. This work was supported in part by
Science Foundation Ireland grant 13/RC/2094 to Lero --- the Irish Software
Research Centre (www.lero.ie).
\bibliographystyle{plain}
\section{A new approach}
\label{sec:meth}
A disadvantage of the \gls{i2c} approach is that it replicates input data to create the input-patch-matrix. For convolution with a $k \times k$ kernel, the input-patch-matrix matrix can be $k^2$ larger than the original input. A \gls{gemm}-based \gls{mcmk} algorithm that does not require data replication in the input could be useful for memory-limited embedded systems and might significantly improve data locality on any target architecture. In this section we present two \gls{gemm}-based \gls{mcmk} algorithms that eliminate data replication on the input, at the cost of some increase in the size of the output.
Figure~\ref{fig:covCode} shows a simplified loop nest for $k \times k$ convolution with $M$ kernels each with $C$ channels. A common operation in \glspl{cnn} such a \gls{googlenet}~\cite{Szegedy:2015} is convolution with a set of $1 \times 1$ convolutions. If we consider the code in figure \ref{fig:covCode} for the case where $k=1$, then the $x$ and $y$ loops collapse into a single iteration.
The resulting code is equivalent to 2D matrix multiplication of a $M \times C$ kernel times a $[C] \times [H \times W]$ input which results in a $[M] \times [H \times W]$ output. This output however is actually $M$ planes of $H\timesW$ pixels which corresponds to an output of size $[H]\times[W]$ and $M$ channels. Let us call this correspondence of a $[M] \times [H \times W]$ matrix to an output matrix of size $[H]\times[W]$ and $M$ channels its \emph{multi-channel representation}, which we will use throughout the rest of this section. In other words, $1 \times 1$ \gls{mcmk} can be implemented by simply calling \gls{gemm} without data replication.
\subsection{\gls{k2r} and \gls{k2c}}
\label{sec:k2r}
Given that we can compute $1 \times 1$ \gls{mcmk} without data replication, how can we implement $k \times k$ \gls{mcmk}, for $k > 1$? We argue that a $k \times k$ convolution can be expressed as the sum of $k^2$ separate $1 \times 1$ convolutions. However the sum is not trivial to compute. Each $1 \times 1$ convolution yields a result matrix with dimensions $[M] \times [H \times W]$. We cannot simply add each of the resulting matrices pointwise, as each resultant matrix corresponds to a different kernel value in the $k\times k$ kernel. The addition of these matrices can then be resolved by offsetting every pixel in every channel of the \emph{multi-channel representation} of these matrices, vertically and/or horizontally (row and column offsets) by one or more positions before the addition.
For example, when computing a $3 \times 3$ convolution the result from computing the $1 \times 1$ \gls{mcmk} for the central point of the $3 \times 3$ kernel is perfectly aligned with the final sum matrix. On the other hand, the matrix that results from computing the $1 \times 1$ \gls{mcmk} for the upper left value of the $3 \times 3$ kernel must be offset up by one place and left by one place (in its \emph{multi-channel representation}) before being added to the final sum that computes the $3 \times 3$ \gls{mcmk}. Note that when intermediate results of $1 \times 1$ convolutions are offset, some values of the offsetted matrix fall outside the boundaries of the final result matrix. These out-of-bounds values are simply discarded when computing the sum of $1 \times 1$ convolutions.
It is possible to compute each of the $k^2$ separate $1 \times 1$ convolutions using a single matrix multiplication. We re-order the kernel matrix, so that the channel data is laid out contiguously, i.e. $M$ is the outer dimension and $C$ the inner. This data re-arrangement can be made statically ahead of time and used for all \gls{mcmk} invocations thereafter. Using a single call to \gls{gemm}, we multiply a $[k^2 \times M] \times [C]$ kernel matrix by a $[C] \times [H \times W]$ input matrix, resulting in a $[k^2 \times M] \times [H \times W]$ matrix. We perform a post pass of \texttt{\textbf{shift-add}} by summing each of the $M^2$ submatrices of size $M\times[H\timesW]$ using appropriate offsetting in the \emph{multi-channel representation}. The result of this sum is a $[M] \times [H \times W]$ matrix, which is the output of our \gls{mcmk} algorithm. We refer to this as the \gls{k2r} algorithm.
If we swap the dimensions of the kernel matrix so that $C$ is not the innermost dimension and swap the input layout to make $C$ the innermost dimension, we get the \gls{k2c} algorithm. The \gls{gemm} call in this method would be to multiply an $[H \times W]\times[C]$ input matrix by a $[C]\times[k^2 \times M]$ kernel matrix, resulting in a $[H \times W]\times[k^2 \times M]$ matrix.
\section{Related Work}
\label{sec:rela-work}
The \gls{i2c} method of performing \gls{mcmk} is an extension of well-known
methods of performing 2D convolution using a Toeplitz matrix. Chellapilla et
al.~\cite{chellapilla2006high} are the first researchers to implement
\gls{mcmk} using using \gls{i2c}. They report significant speedups compared to
the simpler approach of summing multiple channels of 2D convolutions.
Yanqing et al. rediscovered \gls{i2c} for the Caffe deep learning
system~\cite{jia2014caffe}, which uses GPUs and other accelerators to speed up
\glspl{dnn}. The \gls{i2c} approach remains the most widely-used way to
implement \gls{mcmk}, and is used in deep learning frameworks such as Caffe,
Theano and Torch.
Gu et al. \cite{Gu:2016} apply \gls{i2c} to a batch input images to
create a column matrix for multiple input images. They find that
batching can improve throughput by better matching the input matrix
sizes to the optimal sizes for their \gls{gemm} library.
Tsai et al. \cite{Tsai:2016} present a set of configurable OpenCL kernels for
\gls{mcmk}. By coding the \gls{mcmk} loop nests directly they eliminate the
need for \gls{i2c} data replication, and thus allow the use of larger batch
sizes while maintaining constraints on local memory. The found that the
performance of a naive loop nest for \gls{mcmk} is not good, but they achieve
satisfactory performance with a program generator and autotuner.
Chetlur et al. \cite{Chetlur:2014} propose a \gls{gemm}-based approach to
convolution based on \gls{i2c}. However, rather than creating the entire column
matrix in one piece, they instead lazily create sub-tiles of the column matrix
in on-chip memory. To optimize performance, they match the size of their
sub-matrix tiles to the tile sizes used by the underlying \gls{gemm}
implementation. They find that this lazy \gls{i2c} achieves speedups over
Caffe's standard \gls{i2c} of between around 0\% and 30\%.
\section{Experiments and Results}
\label{sec:expe}
\glsunset{gemm}
\noindent
We evaluated the proposed \gls{mcmk} implementations on two general-purpose processors (one embedded, one desktop-class). The experimental platforms we used were ARM\textregistered ~Cortex\textregistered-A57 processor, which has 4 cores with a 128-bit wide SIMD unit, and the Intel\textregistered ~Core\texttrademark ~i5-4570, which has 4 cores and a 256-bit wide SIMD unit.
We used GCC version 7.1 to compile our code for the Intel and ARM CPUs. We used the latest stable version (0.2.19) of the high-performance OpenBLAS library to provide the \gls{gemm} operation on both ARM and Intel platforms.
We implemented a selection of \gls{mcmk} operations from three popular \gls{cnn} architectures: AlexNet~\cite{Krizhevsky:12}, VGG-16~\cite{Simonyan14c}, and \gls{googlenet}~\cite{Szegedy:2015}. In addition to our proposed \gls{gemm} based methods, we also implemented a direct convolution to provide some context for performance.
We experimented with several variants of direct convolution, including a version that is used in Caffe \cite{jia2014caffe}, and an optimized loop nest that appears in a recent book on optimizing code for the Intel Xeon Phi processor \cite{jeffers2016intel}. We found that the fastest direct method, on average, was actually the reference method: summation of single channel convolutions (Equation~\ref{eq:mcsk}).
We also benchmarked the convolution from Intel's MKL-DNN framework, which is the backend used by Intel Caffe. MKL-DNN supports AVX2 and AVX-512 processors, and incorporates a code generator which produces highly-optimized SIMD code for convolution.
We found that our \gls{gemm} based methods were often much faster than any direct method, and often outperform even the highly-optimized code produced by Intel's MKL-DNN.
\begin{figure*}[t]
\centering
\subfloat[VGG-16]{
\centering
\includegraphics[scale=0.58]{figures/graphs/A57/vgg-16}
\label{fig:tx1-cpu-vgg}
}\qquad
\subfloat[Googlenet]{
\centering
\includegraphics[scale=0.52]{figures/graphs/A57/googlenet}
\label{fig:tx1-cpu-googlenet}
}
\subfloat[Alexnet]{
\centering
\includegraphics[scale=0.52]{figures/graphs/A57/alexnet}
\label{fig:tx1-cpu-alexnet}
}
\caption{Execution time for selected layers of \gls{googlenet}, \gls{vgg} and \gls{alexnet} on the ARM\textregistered ~Cortex\textregistered-A57 CPU. \textbf{Lower is better}. }
\label{fig:tx1-cpu-summary}
\end{figure*}
\begin{figure*}[h]
\centering
\subfloat[VGG-16]{
\centering
\includegraphics[scale=0.58]{figures/graphs/x86_64/vgg-16}
\label{fig:x86-cpu-vgg}
}\qquad
\subfloat[Googlenet]{
\centering
\includegraphics[scale=0.52]{figures/graphs/x86_64/googlenet}
\label{fig:x86-cpu-googlenet}
}
\subfloat[Alexnet]{
\centering
\includegraphics[scale=0.52]{figures/graphs/x86_64/alexnet}
\label{fig:x86-cpu-alexnet}
}
\caption{Execution time for selected layers of \gls{googlenet}, \gls{vgg} and \gls{alexnet} on the Intel\textregistered ~Core\texttrademark ~i5-4570 CPU. \textbf{Lower is better}. }
\label{fig:x86-cpu-summary}
\end{figure*}
\subsection{Performance Trends}
\label{sec:perf-tren}
Progressing from left to right across each graph in Figures~\ref{fig:tx1-cpu-summary} and ~\ref{fig:x86-cpu-summary}, the number of input channels increases because the operations are drawn from deeper layers of the \gls{cnn}. At the same time, the size of individual input feature maps diminishes, for the same reason.
Given the fame of \gls{i2c} in the literature, we were surprised to see that the \gls{i2r} method performs so well. When data is laid out in a row matrix instead of a column matrix, spatial locality is significantly improved, since consecutive patch elements are consecutive in memory. While the \gls{i2c} operation may perform well on GPU platforms, our results suggest that it is a poor choice for the implementation of convolution on the CPU.
We also note a large variability between all of the benchmarked methods based on the depth of the convolutional layer in the network. Some methods appear to be very suitable for early layers, but not for later layers; while other methods are unsuitable for early layers, but perform extremely well for later layers. This strongly suggests that a mixture of implementation strategies for convolution is
necessary to achieve peak performance.
For example, direct convolution is very performant for first layer of \gls{vgg}, (Figures~\ref{fig:tx1-cpu-vgg}, ~\ref{fig:x86-cpu-vgg}) but is quickly outpaced by \gls{gemm} based methods as we move deeper in the network. This suggests that peak performance may be achieved by using direct convolution to implement the first layer, and \gls{gemm} based convolution for the remaining layers. However, the situation is different for \gls{alexnet} (Figures~\ref{fig:tx1-cpu-alexnet}, ~\ref{fig:x86-cpu-alexnet}). Here, the \gls{gemm} based methods are always faster.
There is also a similar variability between the \gls{gemm} based methods themselves; some \gls{gemm} based methods are very suitable for early layers, some are very suitable for late layers, but there is no method that has universally good performance in all contexts.
\section{Introduction}
The \textit{proceedings} are the records of a conference\footnote{This
is a footnote}. ACM seeks
to give these conference by-products a uniform, high-quality
appearance. To do this, ACM has some rigid requirements for the
format of the proceedings documents: there is a specified format
(balanced double columns), a specified set of fonts (Arial or
Helvetica and Times Roman) in certain specified sizes, a specified
live area, centered on the page, specified size of margins, specified
column width and gutter size.
\section{The Body of The Paper}
Typically, the body of a paper is organized into a hierarchical
structure, with numbered or unnumbered headings for sections,
subsections, sub-subsections, and even smaller sections. The command
\texttt{{\char'134}section} that precedes this paragraph is part of
such a hierarchy.\footnote{This is a footnote.} \LaTeX\ handles the
numbering and placement of these headings for you, when you use the
appropriate heading commands around the titles of the headings. If
you want a sub-subsection or smaller part to be unnumbered in your
output, simply append an asterisk to the command name. Examples of
both numbered and unnumbered headings will appear throughout the
balance of this sample document.
Because the entire article is contained in the \textbf{document}
environment, you can indicate the start of a new paragraph with a
blank line in your input file; that is why this sentence forms a
separate paragraph.
\subsection{Type Changes and {\itshape Special} Characters}
We have already seen several typeface changes in this sample. You can
indicate italicized words or phrases in your text with the command
\texttt{{\char'134}textit}; emboldening with the command
\texttt{{\char'134}textbf} and typewriter-style (for instance, for
computer code) with \texttt{{\char'134}texttt}. But remember, you do
not have to indicate typestyle changes when such changes are part of
the \textit{structural} elements of your article; for instance, the
heading of this subsection will be in a sans serif\footnote{Another
footnote, here. Let's make this a rather short one to see how it
looks.} typeface, but that is handled by the document class file.
Take care with the use of\footnote{A third, and last, footnote.} the
curly braces in typeface changes; they mark the beginning and end of
the text that is to be in the different typeface.
You can use whatever symbols, accented characters, or non-English
characters you need anywhere in your document; you can find a complete
list of what is available in the \textit{\LaTeX\ User's Guide}
\cite{Lamport:LaTeX}.
\subsection{Math Equations}
You may want to display math equations in three distinct styles:
inline, numbered or non-numbered display. Each of
the three are discussed in the next sections.
\subsubsection{Inline (In-text) Equations}
A formula that appears in the running text is called an
inline or in-text formula. It is produced by the
\textbf{math} environment, which can be
invoked with the usual \texttt{{\char'134}begin\,\ldots{\char'134}end}
construction or with the short form \texttt{\$\,\ldots\$}. You
can use any of the symbols and structures,
from $\alpha$ to $\omega$, available in
\LaTeX~\cite{Lamport:LaTeX}; this section will simply show a
few examples of in-text equations in context. Notice how
this equation:
\begin{math}
\lim_{n\rightarrow \infty}x=0
\end{math},
set here in in-line math style, looks slightly different when
set in display style. (See next section).
\subsubsection{Display Equations}
A numbered display equation---one set off by vertical space from the
text and centered horizontally---is produced by the \textbf{equation}
environment. An unnumbered display equation is produced by the
\textbf{displaymath} environment.
Again, in either environment, you can use any of the symbols
and structures available in \LaTeX\@; this section will just
give a couple of examples of display equations in context.
First, consider the equation, shown as an inline equation above:
\begin{equation}
\lim_{n\rightarrow \infty}x=0
\end{equation}
Notice how it is formatted somewhat differently in
the \textbf{displaymath}
environment. Now, we'll enter an unnumbered equation:
\begin{displaymath}
\sum_{i=0}^{\infty} x + 1
\end{displaymath}
and follow it with another numbered equation:
\begin{equation}
\sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f
\end{equation}
just to demonstrate \LaTeX's able handling of numbering.
\subsection{Citations}
Citations to articles~\cite{bowman:reasoning,
clark:pct, braams:babel, herlihy:methodology},
conference proceedings~\cite{clark:pct} or maybe
books \cite{Lamport:LaTeX, salas:calculus} listed
in the Bibliography section of your
article will occur throughout the text of your article.
You should use BibTeX to automatically produce this bibliography;
you simply need to insert one of several citation commands with
a key of the item cited in the proper location in
the \texttt{.tex} file~\cite{Lamport:LaTeX}.
The key is a short reference you invent to uniquely
identify each work; in this sample document, the key is
the first author's surname and a
word from the title. This identifying key is included
with each item in the \texttt{.bib} file for your article.
The details of the construction of the \texttt{.bib} file
are beyond the scope of this sample document, but more
information can be found in the \textit{Author's Guide},
and exhaustive details in the \textit{\LaTeX\ User's
Guide} by Lamport~\shortcite{Lamport:LaTeX}.
This article shows only the plainest form
of the citation command, using \texttt{{\char'134}cite}.
\subsection{Tables}
Because tables cannot be split across pages, the best
placement for them is typically the top of the page
nearest their initial cite. To
ensure this proper ``floating'' placement of tables, use the
environment \textbf{table} to enclose the table's contents and
the table caption. The contents of the table itself must go
in the \textbf{tabular} environment, to
be aligned properly in rows and columns, with the desired
horizontal and vertical rules. Again, detailed instructions
on \textbf{tabular} material
are found in the \textit{\LaTeX\ User's Guide}.
Immediately following this sentence is the point at which
Table~\ref{tab:freq} is included in the input file; compare the
placement of the table here with the table in the printed
output of this document.
\begin{table}
\caption{Frequency of Special Characters}
\label{tab:freq}
\begin{tabular}{ccl}
\toprule
Non-English or Math&Frequency&Comments\\
\midrule
\O & 1 in 1,000& For Swedish names\\
$\pi$ & 1 in 5& Common in math\\
\$ & 4 in 5 & Used in business\\
$\Psi^2_1$ & 1 in 40,000& Unexplained usage\\
\bottomrule
\end{tabular}
\end{table}
To set a wider table, which takes up the whole width of the page's
live area, use the environment \textbf{table*} to enclose the table's
contents and the table caption. As with a single-column table, this
wide table will ``float'' to a location deemed more desirable.
Immediately following this sentence is the point at which
Table~\ref{tab:commands} is included in the input file; again, it is
instructive to compare the placement of the table here with the table
in the printed output of this document.
\begin{table*}
\caption{Some Typical Commands}
\label{tab:commands}
\begin{tabular}{ccl}
\toprule
Command &A Number & Comments\\
\midrule
\texttt{{\char'134}author} & 100& Author \\
\texttt{{\char'134}table}& 300 & For tables\\
\texttt{{\char'134}table*}& 400& For wider tables\\
\bottomrule
\end{tabular}
\end{table*}
It is strongly recommended to use the package booktabs~\cite{Fear05}
and follow its main principles of typography with respect to tables:
\begin{enumerate}
\item Never, ever use vertical rules.
\item Never use double rules.
\end{enumerate}
It is also a good idea not to overuse horizontal rules.
\subsection{Figures}
Like tables, figures cannot be split across pages; the best placement
for them is typically the top or the bottom of the page nearest their
initial cite. To ensure this proper ``floating'' placement of
figures, use the environment \textbf{figure} to enclose the figure and
its caption.
This sample document contains examples of \texttt{.eps} files to be
displayable with \LaTeX. If you work with pdf\LaTeX, use files in the
\texttt{.pdf} format. Note that most modern \TeX\ systems will convert
\texttt{.eps} to \texttt{.pdf} for you on the fly. More details on
each of these are found in the \textit{Author's Guide}.
As was the case with tables, you may want a figure that spans two
columns. To do this, and still to ensure proper ``floating''
placement of tables, use the environment \textbf{figure*} to enclose
the figure and its caption. And don't forget to end the environment
with \textbf{figure*}, not \textbf{figure}!
\subsection{Theorem-like Constructs}
Other common constructs that may occur in your article are the forms
for logical constructs like theorems, axioms, corollaries and proofs.
ACM uses two types of these constructs: theorem-like and
definition-like.
Here is a theorem:
\begin{theorem}
Let $f$ be continuous on $[a,b]$. If $G$ is
an antiderivative for $f$ on $[a,b]$, then
\begin{displaymath}
\int^b_af(t)\,dt = G(b) - G(a).
\end{displaymath}
\end{theorem}
Here is a definition:
\begin{definition}
If $z$ is irrational, then by $e^z$ we mean the
unique number that has
logarithm $z$:
\begin{displaymath}
\log e^z = z.
\end{displaymath}
\end{definition}
The pre-defined theorem-like constructs are \textbf{theorem},
\textbf{conjecture}, \textbf{proposition}, \textbf{lemma} and
\textbf{corollary}. The pre-defined de\-fi\-ni\-ti\-on-like constructs are
\textbf{example} and \textbf{definition}. You can add your own
constructs using the \textsl{amsthm} interface~\cite{Amsthm15}. The
styles used in the \verb|\theoremstyle| command are \textbf{acmplain}
and \textbf{acmdefinition}.
Another construct is \textbf{proof}, for example,
\begin{proof}
Suppose on the contrary there exists a real number $L$ such that
\begin{displaymath}
\lim_{x\rightarrow\infty} \frac{f(x)}{g(x)} = L.
\end{displaymath}
Then
\begin{displaymath}
l=\lim_{x\rightarrow c} f(x)
= \lim_{x\rightarrow c}
\left[ g{x} \cdot \frac{f(x)}{g(x)} \right ]
= \lim_{x\rightarrow c} g(x) \cdot \lim_{x\rightarrow c}
\frac{f(x)}{g(x)} = 0\cdot L = 0,
\end{displaymath}
which contradicts our assumption that $l\neq 0$.
\end{proof}
\section{Conclusions}
This paragraph will end the body of this sample document.
Remember that you might still have Acknowledgments or
Appendices; brief samples of these
follow. There is still the Bibliography to deal with; and
we will make a disclaimer about that here: with the exception
of the reference to the \LaTeX\ book, the citations in
this paper are to articles which have nothing to
do with the present subject and are used as
examples only.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,921 |
{"url":"https:\/\/aviation.stackexchange.com\/questions\/9412\/how-do-fighter-jets-communicate-with-an-intercepted-plane-thats-lost-comms?noredirect=1","text":"# How do fighter jets communicate with an intercepted plane that's lost comms?\n\nRAF Typhoons were scrambled today to intercept an Antonov 26 that lost comms, triggering a sonic boom over London.\n\nThe flight was escorted to Stansted airport, and the answer at What's the point in escorting a threatened flight with two fighter jets? says they \"guide\" the rogue jet to the airport.\n\nHow do they actually do this? Once they've established the jet is not a threat, how do they communicate with the other pilots?\n\nI have an image in my head of them writing \u2190 120\u00b0 (etc.) on a little whiteboard :P\n\nThis is covered in the FAA's AIM, Chapter 5, Section 6. The UK has similar procedures.\n\nFirst, the standard guard frequency is 121.5 or 243.0 MHz. If you aren't listening to this frequency and get intercepted, you should start listening.\n\nMethods like LED signs or possibly as whiteboard as you suggested can also be used. Otherwise, they maneuver their aircraft to send standard signals. See the guides above for details on the signals sent by interceptors and how they should be acknowledged. Basically:\n\n\u2022 You have been intercepted, follow me\n\u2022 Interceptor: Rocks wings (and flashes lights irregularly at night)\n\u2022 Intercepted: Same\n\u2022 You may proceed\n\u2022 Interceptor: Breakaway, climbing turn away from aircraft\n\u2022 Intercepted: Rocking wings\n\u2022 Land here\n\u2022 Interceptor: Circling airport, lowering landing gear, overflying runway (landing lights on at night)\n\u2022 Intercepted: Lower gear, land\n\nThe intercepted aircraft can also send signals. The interceptor will respond with one of the signals above.\n\n\u2022 Intercepted: Raise gear, flash landing lights\n\u2022 Cannot comply\n\u2022 Intercepted: Switching on and off all lights, but distinct from flashing lights\n\u2022 In distress\n\u2022 Intercepted: Irregular flashing of all lights\n\nIn the case of the Washington, DC Special Flight Rules Area (DC SFRA), there is a Visual Warning System (VWS) of red and green lights that will let the pilot know that they shouldn't be there. The pilot is supposed to turn directly away from the center of the SFRA. If they are in contact with ATC they must advise that they have been illuminated by the system, otherwise they should contact ATC on a guard frequency and identify themselves.\n\nOf course, an interceptor can't force an intercepted aircraft follow instructions. It goes on to explain what will happen if you don't follow those rules, which probably apply in any situation where the risk is too high:\n\nFurther noncompliance with interceptor aircraft or ATC may result in the use of force.\n\n\u2022 What happens if the intercepted flight is a neer-do-well with bad intentions, but follows the above signals? At what point will it be shot down? What reasons would a pilot have to not comply? \u2013\u00a0Scottie Oct 29 '14 at 22:31\n\u2022 @Scottie how would they find out? \u2013\u00a0ratchet freak Oct 29 '14 at 22:45\n\u2022 @Scottie I'd guess there's procedures in place to handle deviations from issued orders (like, target stops following for 20 seconds, doesn't respond to new orders, etc.). \u2013\u00a0jwenting Oct 30 '14 at 8:47\n\u2022 There are \"rules of engagement\" for interceptors in various situations, taking into account the severity of the violation for which the aircraft is being intercepted as well as the potential for harm to others as a result, so the question of when they would take the step of blowing a pilot out of the sky is situation-dependent. Nick the edge of DFW's Class B bubble in a 172 and you'll get a talking to when you land. Fly directly into the DC FRZ at 250 feet ignoring comms, and you're likely to get a very close look at a Raptor or F-16 within seconds, especially if the President's home. \u2013\u00a0KeithS Jun 22 '15 at 23:35\n\u2022 Lastly, the only two scenarios I can think of where an aircraft would at least appear to ignore an interceptor's instructions are (1) a non-pilot at the controls with the PIC incapacitated (as is fantasized in so many movies and books) who doesn't understand the meaning of the intercepting aircraft's signals, or (2) a complete electrical system failure rendering the pilot unable to signal with the lights (they can still signal with wing rocks etc). Neither are likely, but they aren't impossible. \u2013\u00a0KeithS Jun 22 '15 at 23:56\n\nAIM section 6 describes what the pilot in the intercepted plane should do:\n\nFirst thing the intercepting craft will do is rock the wings irregularly (and flash the nav lights at night) and initiate a turn (usually on the left). The intercepted aircraft should copy that and follow. That signals that an intercept has taken place and the other plane is compliant.\n\nIf time is of the essence then the fighter can do a climbing turn in front of the other aircraft in the direction it should turn.\n\nIf the escort is at an end (and allows the craft to fly on) then the fighter will break away with a 90\u00b0 turn, the response is to wiggle the wings again and continue on.\n\nTo signal the intercepted plane to land the fighter will circle the airport and do a flyover over the runway with lowered gear (imitating a go-around). The other plane may do a flyover to make sure it's clear to land and then go land.\n\nIf the intercepted pilot believes he can't land on the runway he should stow his gear and flash his landing lights during the flyover.\n\nIt's possible the intercepting plane will hand over control to the tower and let it use the light gun to guide it down.\n\nIf at any point the pilot cannot comply he should switch on and off his lights at a regular interval. If the intercepted pilot is in distress then he should flash all available lights","date":"2020-11-30 19:50:30","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\": 0.24253275990486145, \"perplexity\": 3366.666812441135}, \"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-2020-50\/segments\/1606141486017.50\/warc\/CC-MAIN-20201130192020-20201130222020-00192.warc.gz\"}"} | null | null |
{"url":"http:\/\/fr.mathworks.com\/help\/matlab\/ref\/asin.html?nocookie=true","text":"Accelerating the pace of engineering and science\n\n# asin\n\ny = asin(x)\n\n## Description\n\ny = asin(x) returns the Inverse Sine (sin-1) of the elements of x. The asin function operates element-wise on arrays. For real elements of x in the interval [-1,1], asin(x) returns values in the interval [-pi\/2,pi\/2]. For real elements of x outside the interval [-1,1] and for complex values of x, asin(x) returns complex values. All angles are in radians.\n\n## Examples\n\nexpand all\n\n### Inverse Sine of a Value\n\n`asin(0.5)`\n```ans =\n0.5236```\n\n### Inverse Sine of a Vector of Complex Values\n\nFind the inverse sine of the elements of vector x. The asin function acts on x element-wise.\n\n```x = [0.5i 1+3i -2.2+i];\ny = asin(x)```\n```y =\n0.0000 + 0.4812i 0.3076 + 1.8642i -1.1091 + 1.5480i```\n\n### Graph of the Inverse Sine Function\n\nGraph the inverse sine over the domain .\n\n```x = -1:.01:1;\nplot(x,asin(x))\ngrid on\n```\n\n${\\mathrm{sin}}^{-1}\\left(z\\right)=-i\\mathrm{log}\\left[iz+{\\left(1-{z}^{2}\\right)}^{1\/2}\\right].$","date":"2014-11-28 09:04:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"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\": 0.6377639770507812, \"perplexity\": 2119.950895083039}, \"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-2014-49\/segments\/1416931009900.4\/warc\/CC-MAIN-20141125155649-00122-ip-10-235-23-156.ec2.internal.warc.gz\"}"} | null | null |
Proudly distilled in Liverpool, Liverpool Gin Rose Petal is 100% organic and created using a secret blend of botanicals. Combining the timeless decadence of rose petals to create a drink full of romance, with a tenderly fragrant nose and delicate brush of the palate awaiting every suitor.
Delicate and aromatic, a tenderly fragrant nose gives way to an enticingly soft mouthfeel and subtle flavours of rose and juniper on the palate. A deliciously different gin that is best enjoyed over ice with tonic. | {
"redpajama_set_name": "RedPajamaC4"
} | 410 |
10 Ghost Towns in Maine
Creepy and Abandoned Sites Around Maine
The presence of ghost towns in Maine owe to this north-easternmost state being home to some of the earliest US settlements. Each region has its fair share of ghost towns – relics of both gradual changes over the course of history and sudden, dramatic events – and Maine is no exception.
There are plenty of examples of intriguing ghost towns scattered across the nation's 50 states, but Maine has a particularly heavy crop that range from abandoned island towns to submerged settlements. There are enough creepy and abandoned sites around Maine to keep even the most curious traveller occupied.
Askwith
The town that no longer has a name
A plentiful supply of game and fish was what led to the birth of Askwith, a small town between Greenville and Rockwood in Piscataquis County, Maine's least populous. There was also enough timber in the area to fuel the town's initial construction, and Askwith became a significant enough settlement to warrant the building of a post office and a railway.
None of that lasted long, though. The former was gone by 1895, and the latter was ultimately converted into an ATV trail, while the town's train station itself has been left to decay. By the 1930s there were just 56 left living here – and, spookiest of all, if you head out to explore Askwith, you won't even find it marked on the map, since it was ultimately renamed Tarratine.
Sijainti: Tarratine, Maine 04478, USA
The township hastily abandoned in the middle of a river
Not many abandoned ghost towns find themselves on islands in the stream, but that's the case with Perkins – these days referred to as Swan Island. Located on the Kennebec River and originally inhabited by native tribes and early settlers, you can get a glimpse of what's left of Perkins by popping across by ferry, kayak, or canoe.
Peering through the windows you'll see that the 5 remaining 18th-century homes are still partially furnished, almost as though they were abandoned in a hurry. There are various theories about what happened here, including dramatic river pollution forcing people to head elsewhere. There's plenty of wildlife left behind. For extra spooky feels, there's also still a cemetery, and you can even pitch a tent if you fancy spending the night.
Sijainti: Swan Island, Perkins Township, ME 04008, USA
Valokuva: Ashley Frillman (CC BY-SA 2.0) muokattu
Ligonia Village
The village that became its own cemetery
These days there are few remaining traces of Ligonia, a South Portland village that in the mid-19th century is said to have run from the area known as Cash Corner down to the waterfront on the Fore River. Now the site has been taken over by the expanding Calvary Cemetery (which adds fright points) and, owing to its strategic location close to the harbour, bases for oil companies.
Ligonia was once home to a concentration of Irish families, who worked for the Portland Rolling Mills company and turned it into a kind of campus town, complete with shops, a school, and a railway connection to Portland proper. Car ownership led to the demise of local businesses, and industrial decline caused the population to plummet, but police dispatchers in the area apparently still refer to it as Ligonia today.
Sijainti: Ligonia, South Portland, ME 04106, USA
The neighbourhood wiped out by plague – or was it?
You won't find much evidence of East Hancock's never-incorporated Riceville in the woods between Milford and Township 39. The town that's also variously known as Hancock Tannery and 39 Tannery became a miniature hub for buffalo leather shoe sole manufacturing, thanks to the presence of a factory that changed hands several times over the decades.
By the late 19th century the neighbourhood was in decline, and gradually amenities like the school and post office closed. Various stories exist as to the final cause of Riceville's demise – it's up to you whether you choose to believe the tannery burned down in 1905 or there was an outbreak of plague or cholera that led to the entire population being found dead in one fell swoop. Either way, what's certain is that by 1910 there wasn't a soul left here.
Sijainti: Stud Mill Rd, Milford, ME 04461, USA
The town overflowing with creepy goings-on
There's a real sense of the paranormal in Skowhegan – in particular at the Strand Cinema and the nearby Lake George Regional Park West – making this more than just your everyday, creepy-but-actually-tame ghost town. The cinema has been running for over 90 years, but it's gained as much notoriety for its ghostly activity as for its movies.
From self-starting unplugged electronics to poltergeist-like levitations and vandalism, Skowhegan's Strand Cinema could bring a horror film to life better than most. And that's not all – the park has also seen its share of oddities, including the unexplained movement and rearrangement of furniture in the cabins rented out for events that are dotted around it.
Sijainti: Skowhegan, ME 04976, USA
Valokuva: David Wilson (CC BY 2.0) muokattu
The town with the witch's leg
Of all the ghost towns in Maine, modestly sized Bucksport might have the scariest spots. In fact, there are so many that there are entire lists and itineraries dedicated to detailing the area's haunted attractions. To start with, there's the tombstone of Bucksport's founder Jonathan Buck that eerily features a print of a leg resembling that of his former lover, rescued and buried by her son when Buck labelled her a witch and had her burned at the stake.
That's before you consider the eerie Maine Seaboard Paper Company building, built on a former burial ground of Marine's native Red Paint People and having since suffered a string of fires and other incidents. Or the deceptively beautiful Silver Lake, also built on the site of a former graveyard – some believe a number of bodies still lay at the bottom of the water.
Sijainti: Bucksport, ME 04416, USA
Valokuva: DrStew82 (CC BY-SA 4.0) muokattu
The township that gave life and then took it away
A typical abandoned ghost town with an especially spooky ambience to it, Freeman came into being after the Maine city of Portland was burned to the ground during the American Revolution.
In 1797 Freeman was created to resettle those left destitute in the incident's wake and, at its height, the vibrant town had train stations, schools, churches, and a raft of businesses. It began to fall into decline towards the end of the 19th century, though, and was disincorporated as a town in 1973. These days, a creepy-feeling cemetery and a few other architectural remnants are the only clues to Freeman's previous life.
Sijainti: Freeman Township, ME 04983, USA
From a gold rush to a decades-long decline
While it's not clear whether Maine's Madrid was named after the Spanish capital, farming and a growth in timber harvesting spurred on the town's emergence in the late 18th century. A nearby gold rush later helped cement its popularity as a place to settle, and as a result there were schools, railway connections, churches, a local newspaper, a hotel, and all manner of other businesses here.
But the inevitable decline started at the beginning of the 20th century – although Madrid wasn't officially disincorporated as a town until the year 2000, the last of its schools had already closed as far back as 1959. What remains are a few buildings you can wander between, and a potentially goosebump-raising museum of the town's history that's found in the former schoolhouse.
Sijainti: Madrid, ME 04966, USA
Valokuva: Magicpiano (CC BY-SA 3.0) muokattu
Evergreen Ski Resort
Where business 'went down the slope'
A Maine almost-ghost-town that suffered a much more recent demise than most, the Evergreen Ski Resort was a popular getaway destination as late as the 1970s. The site is set in the woods of Stoneham, on the border between Maine and New Hampshire, and it is said to have functioned both as a winter ski resort and as a lakeside festival spot in the summer.
But even that versatility couldn't sustain it forever – the resort closed in the early '80s and numerous subsequent attempts to revive it have failed. Now even the tell-tale ski lifts have been removed, and all that remains is an empty swimming pool and an abandoned lodge that are testament to both commercial failure and an otherworldly creepy vibe. Previously considered public land, the resort was sold in early 2018 and is now private property.
Sijainti: Stoneham, ME 04231, USA
The town at the bottom of a lake
This town in Maine's Somerset County got its name from a flagstaff erected here by American military officer Benedict Arnold on an expedition to Quebec. It had also been used as a campsite for British soldiers making the same journey since before the American Revolution.
But the opening of mills in the area drew in a greater population and Flagstaff eventually became the largest town in the Dead River Valley. Its success lasted until the 1940s, when Flagstaff was deliberately flooded to create a power-generating dam as an extension of Flagstaff Lake. What was Flagstaff now sits at the bottom of the lake – if you're curious and want to explore it, you might need scuba gear.
Sijainti: Flagstaff Lake, Maine, USA
Valokuva: pfly (CC BY-SA 2.0) muokattu
Chris Wotton | Kirjoittaja
8 Best Things to Do in Universal Orlando
10 Best Places to Celebrate Christmas in the US
Fiorella Bertola, 24 Dec, 2019
10 Best Things to Do in Savannah
Joshua Saunders, 21 Oct, 2019 | {
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\section{Small net for noise attenuated linear juntas}~\label{sec:noise-attenuated}
In this section, we are going to prove the following theorem which essentially shows the existence of a small cover for noise stable linear juntas.
{To state this theorem, we will require one crucial fact about noise attenuated functions (due to Bakry and Ledoux~\cite{Bakry:94})
\begin{lemma}~\label{prop:gradient-bound}~
Let $f: \mathbb{R}^n \rightarrow [-1,1]$. Then, $P_t f$ is $C_t$-Lipschitz for $C_t = O(t^{-1/2})$.
\end{lemma}
For the rest of this section, we are going to use $C_t$ to denote this quantity.
}
We can now state the main theorem of this section.
\begin{theorem}~\label{thm:net}
For any error parameter $\delta>0$, noise parameter $t>0$ and $k \in \mathbb{N}$, there is a set of functions $\mathsf{Cover}(t,k,\delta)$ (mapping $\mathbb{R}^k$ to $[-1,1]$) such that the following holds:
\begin{enumerate}
\item Let $f: \mathbb{R}^n \rightarrow [-1,1]$ and $W$ be a $k$-dimensional space such that $P_t f$ is $\delta$-close to a $W$-junta. Further, $(w_1, \ldots, w_k)$ be any orthonormal basis of $W$. Then, $P_t f$ is $3\delta$-close to $h(\langle w_1, x \rangle, \ldots, \langle w_k, x \rangle)$ for some $h \in \mathsf{Cover}(t,k,\delta)$.
\item Every function in $\mathsf{Cover}(t, k, \delta)$ is $2C_t$-Lipschitz.
\item $\log |\mathsf{Cover}(t, k, \delta)| \le \left(\frac{C \sqrt k \log^2(1/\delta)}{\delta \sqrt t}\right)^k$.
\end{enumerate}
\end{theorem}
The proof of this theorem relies on the following two lemmas.
\begin{lemma}~\label{lem:net-1}
For any $L>0$, error parameter $\delta>0$ and $k \in \mathbb{N}$, there is a set $\mathsf{Cover}_{k,L,\delta}$ consisting of functions mapping $\mathbb{R}^k \mapsto [-1,1]$ such that the following holds:
\begin{enumerate}
\item For every $g: \mathbb{R}^k \rightarrow [-1,1]$ which is $L$-Lipschitz, there is a function $h \in \mathsf{Cover}_{k,L,\delta}$ such that $\mathbf{E}[|g(x) - h(x)|] \leq \delta$.
\item Every function in $\mathsf{Cover}_{k,L,\delta}$ is $2L$-Lipschitz.
\item $\log |\mathsf{Cover}_{k,L,\delta}| \le \left(\frac{C L \sqrt k \log^2(1/\delta)}{\delta}\right)^k$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\mathcal{B} = \{x: \Vert x \Vert_2 \le \sqrt{k} \cdot \log
(100/\delta)\}$. Let $\mathcal{A}$ be a maximal $\delta/(2L)$-packing of
$\mathcal{B}$ (that is, a maximal subset of $\mathcal{B}$ such that any two
distinct points in $\mathcal{A}$ are at least $\delta/(2L)$ apart.
It is well-known (see, e.g.~\cite{LedouxTalagrand}) that $\mathcal{A}$ is a $\delta/L$-net
of $\mathcal{B}$ and that $|\mathcal{A}| \le (C L \sqrt k \log (1/\delta)/\delta)^k$
(the $\sqrt k \log (1/\delta)$ term comes from the diameter of $\mathcal{B}$.
For $f: \mathbb{R}^n \rightarrow [-1,1]$, we now define $f_{\mathsf{int}}: \mathcal{A} \to [-1, 1]$ by
simply rounding $f$ to the nearest integer multiple of $\delta/100$. To check the Lipschitz
constant of $f_\mathsf{int}$, note that if $x, y \in \mathcal{A}$ then
\[
|f_{\mathsf{int}}(x) - f_{\mathsf{int}}(y)| \le |f(x) - f(y)| + \delta/50
\le L \|x - y\| + \frac{L}{25} \|x - y\|,
\]
where the last inequality used the fact that $f$ is $L$-Lipschitz and that
every pair of points in $\mathcal{A}$ is $\delta/(2L)$-separated.
In particular, $f_{\mathsf{int}}$ is $2L$-Lipschitz.
Let $\mathsf{Cover}'$ be the set of all functions $f_{\mathsf{int}}$ obtained
in this way. Then the size of $\mathsf{Cover}'$ is at most $\exp((C L \sqrt k \log^2(1/\delta) \delta^{-1})^k)$,
because there are at most $C/\delta$ choices for the value of each point, and there are $|\mathcal{A}|$ points.
Finally, we construct $\mathsf{Cover}_{k,L,\delta}$ by extending each function in $\mathsf{Cover}'$
to a function ${\mathbb{R}}^n \to [-1, 1]$. McShane's Lemma~\cite{mcshane34} implies that this extension can be done
without increasing its Lipschitz constant. Hence, properties 2 and 3 hold.
To check property 1, note that if $x \in \mathcal{B}$ and $y \in \mathcal{A}$ is the closest point to $x$
then
\[
|f(x) - f_{\mathsf{int}}(x)|
\le |f(x) - f(y)| + |f(y) - f_{\mathsf{int}}(y)| + |f_{\mathsf{int}}(y) - f_{\mathsf{int}}(x)|
\le 3L \|x - y\| + \delta/100 \le 4\delta.
\]
It then follows that
\[
\mathbf{E}[|f(x) - f_{\mathsf{int}}(x)|] \le 2 \cdot \Pr[x \not \in \mathcal{B}] + \max_{x \in \mathcal{B}} [|f(x) - f_{\mathsf{int}}(x)|] \le \delta + 4 \delta \le 5 \delta.
\]
The last inequality just follows from the fact that a $k$-dimensional standard Gaussian is in a ball of radius $\sqrt{k} \log (1/\delta)$ with probability $1-\delta/2$. This proves property 1 modulo the constant $5$, which can be dropped by redefining $\delta$.
\end{proof}
\begin{lemma}~\label{lem:Lip-1}
Let $f: \mathbb{R}^n \rightarrow [-1,1]$ be a $C$-Lipschitz function. Further, for $\kappa>0$, let $g: \mathbb{R}^n \rightarrow [-1,1]$ be a $W$-junta such that $f$ is $\kappa$-close to $g$. Then, there is a function $f_W:\mathbb{R}^n \rightarrow [-1,1]$ which is $C$-Lipschitz and $W$-junta which is $2\kappa$-close to $f$.
\end{lemma}
\begin{proof}
Reorient the axes so that $W$ is the space spanned by the first $\ell$-axes. Let us define the $W$-junta $f_{W}: \mathbb{R}^n \rightarrow [-1,1]$ defined as
\[
f_W(x) = \mathbf{E}_{y_{\ell+1}, \ldots, y_n} [f(x_1, \ldots, x_\ell, y_{\ell+1}, \ldots, y_n)
\]
For any fixed choice of $x_1, \ldots, x_\ell$, we have
\[
\mathbf{E}_{x_{\ell+1}, \ldots, x_n}[|f(x) - f_W(x)|] \leq \mathbf{E}_{x_{\ell+1}, \ldots, x_n}[|f(x) - g(x)|] +|g(x) - f_W(x)|.
\]
However, the second term can be bounded as
\[
|g(x) - f_W(x)| = \big| g(x) - \mathbf{E}{x_{\ell+1}, \ldots, x_n}[f(x_1, \ldots,x_\ell, x_{\ell+1} , \ldots, x_n)] \big| \le \mathbf{E}{x_{\ell+1}, \ldots, x_n} \big[ \big| g(x) - f(x) \big|\big]
\]
The last inequality is simply Jensen's inequality. Combining these two, we get
\begin{equation}~\label{eq:junta-diff}
\mathbf{E}_{x_{\ell+1}, \ldots, x_n}[|f(x) - f_W(x)|] \leq2 \cdot \mathbf{E}_{x_{\ell+1}, \ldots, x_n}[|f(x) - g(x)|].
\end{equation}
This in turn implies that
\begin{equation}~\label{eq:junta-diff-1}
\mathbf{E}_{x_{1}, \ldots, x_n}[|f(x) - f_W(x)|] \leq2 \cdot \mathbf{E}_{x_{1}, \ldots, x_n}[|f(x) - g(x)|] \leq 2\cdot \kappa.
\end{equation}
Finally, we see that
\begin{eqnarray*}
|f_W(x) - f_W(y)| &=& \big| \mathbf{E}_{x_{\ell+1}, \ldots, x_n}[f(x_1, \ldots, x_\ell, x_{\ell+1} , \ldots, x_n) - f(y_1, \ldots, y_\ell, x_{\ell+1},\ldots, x_n)] \\
&\leq& \mathbf{E}_{x_{\ell+1}, \ldots, x_n} \big[ \big| f(x_1, \ldots, x_\ell, x_{\ell+1} , \ldots, x_n) - f(y_1, \ldots, y_\ell, x_{\ell+1},\ldots, x_n) \big| \big] \\
&\le& \mathbf{E}_{x_{\ell+1}, \ldots, x_n} [ C \cdot \Vert (x_1, \ldots, x_\ell) - (y_1, \ldots, y_\ell) \Vert_2] \le C \Vert x -y\Vert_2.
\end{eqnarray*}
This finishes the proof.
\end{proof}
With these two lemmas, we can now finish the proof of Theorem~\ref{thm:net}.
{\begin{proofof}{Theorem~\ref{thm:net}}
First, we apply Lemma~\ref{prop:gradient-bound} to obtain that $P_t f$ is $C_t=O(t^{-1/2})$-Lipschitz. Since $P_t f$ is $\delta$-close to a $W$-junta, we obtain that $P_t f$ is $2\delta$ close to a $W$-junta $g$ which is $C_t$-Lipschitz (follows from Lemma~\ref{lem:Lip-1}). Let $ \mathsf{Cover}(t,k,\delta)=\mathsf{Cover}_{k,C_t,\frac{\delta}{2}}$ (constructed in Lemma~\ref{lem:net-1}). By a rotation of the coordinates, it follows from the definition of $ \mathsf{Cover}(t,k,\delta)$ that there exists $h \in\mathsf{Cover}(t,k,\delta)$ such that $h( \langle w_1, x \rangle, \ldots, \langle w_k, x \rangle)$ is $\frac{\delta}{4}$ close to $g$. The required properties now follow from Lemma~\ref{lem:net-1}.
\end{proofof}}
\section{Small net for noise attenuated linear juntas}~\label{sec:noise-attenuated}
In this section, we are going to prove the following theorem which essentially shows the existence of a small cover for noise stable linear juntas. To state this theorem, we will require one crucial fact about noise attenuated functions (due to Bakry and Ledoux~\cite{Bakry:94})
\begin{lemma}~\label{prop:gradient-bound}~
Let $f: \mathbb{R}^n \rightarrow [-1,1]$. Then, $P_t f$ is $C$-Lipschitz for $C = O(t^{-1/2})$.
\end{lemma}
\begin{theorem}~\label{thm:net}
For any error parameter $\delta>0$, noise parameter $t>0$ and $k \in \mathbb{N}$, there is a set of functions $\mathsf{Cover}(t,k,\delta)$ (mapping $\mathbb{R}^k$ to $[-1,1]$) such that the following holds:
\begin{enumerate}
\item Let $f: \mathbb{R}^n \rightarrow [-1,1]$ and $W$ be a $k$-dimensional space such that $P_t$ is $O(\epsilon)$-close to a $W$-junta. Further, $(w_1, \ldots, w_k)$ be any orthonormal basis of $W$. Then, $P_t f$ is $O(\delta)$-close to $h(\langle w_1, x \rangle, \ldots, \langle w_k, x \rangle)$ for some $h \in \mathsf{Cover}(t,k,\epsilon)$.
\item The size of the set $\mathsf{Cover}(t,k,\epsilon)$ is bounded by
$O\big( \frac{k \cdot \log(1/\delta)}{\sqrt{t} \cdot \delta}\big)^k$.
\item Every function $ h \in mathsf{Cover}(t,k,\delta)$ is \emph{approximately Lipschitz} in the following sense:
\[
\Vert_2 h(x) - h(y) \Vert_2 \le \Vert x - y \Vert_2 \cdot
\]
\end{enumerate}
\end{theorem}
\section{Small net for juntas with bounded surface area}
\begin{proposition}~\label{prop:sing-value}
Let $A \in\mathbb{R}^{\ell \times \ell}$ matrix such that for any $1 \le j\le \ell$, $\mathrm{dist}(a_j, A_{j-1})) \ge \delta$ where $a_j$ is the $j^{th}$ column of $A$ and $A_j$ is the column span of the first $j$ columns. Then, for {\color{red}{$\eta = \delta^{-k}$}}\anote{this probably needs to change}, given the inner products of $\langle a_i, a_j \langle$ for all $(i,j)$ up to additive error $\eta$,
the algorithm \textsf{Robust-linear-independence} has the following guarantee:
\begin{itemize}
\item If the matrix $A$ satisfies the above conditions,
then the algorithm outputs \textsf{yes}.
\item If the algorithm outputs \textsf{yes}, then $\mathsf{dist}(a_j, A_{j-1}) \ge \delta/2$.
\end{itemize}
\end{proposition}
\begin{proof}
{\color{red}This proposition is supposed to basically say that with good enough accuracy, we can check whether the vectors we have gotten are $\eta$-linearly independent}
\end{proof}
\subsection{Small-covers for Lipschitz functions}
\begin{lemma}~\label{lem:Lipschitz-cover}
For any error parameter $\epsilon>0$, surface area $s$ and $k \in \mathbb{N}$, there is a net $\mathsf{Net}_{s,k,\epsilon}$ consisting of functions mapping $\mathbb{R}^k \rightarrow [-1,1]$ such that the following holds: Let $g: \mathbb{R}^n \rightarrow [-1,1]$ and let $P_{t_1} g$ be $\epsilon$-close to a $W$-junta where $\mathsf{dim}(W)=k$.
Let $v_1, \ldots, v_k$ be an orthonormal basis of $W$. Then, there exists $h \in \mathsf{Net}_{s,k,\epsilon}$ such that $P_{t_1} g$ is $O(\epsilon)$-close to $h(\langle v_1, x\rangle, \ldots, \langle v_k ,x \rangle)$.
\end{lemma}
\begin{proof}
\end{proof}
\begin{lemma}~\label{lem:net-1}
For any $C, \delta>0$ and $k \in \mathbb{N}$, there is a set $\mathsf{Net}_{k,C,\delta}$ consisting of functions mapping $\mathbb{R}^k \mapsto [-1,1]$ of size $O\big(\frac{k \cdot C \cdot \log(1/\delta)}{\delta}\big)^k$
such that for any function $g: \mathbb{R}^k \rightarrow [-1,1]$ which is $C$-Lipschitz, there is a function $h \in \mathsf{Net}_{k,C,\delta}$, $\mathbf{E}[|g(x) - h(x)|] =\delta/9$.
\end{lemma}
\begin{proof}
Let $\mathcal{B} = \{x: \Vert x \Vert_2 \le \sqrt{k} \cdot \log (100/\delta)\}$. Now, let $\mathcal{A} \subseteq \mathcal{B}$ be defined as the set of points each of whose coordinates is an integral multiple of $\eta=\frac{\delta}{10 \cdot C \cdot \sqrt{k}}$. For $f: \mathbb{R}^n \rightarrow [-1,1]$, we now define $f_{\mathsf{int}}$ as follows:
\begin{enumerate}
\item For any point $x \not \in \mathcal{B}$, $f_{\mathsf{int}}(x)=0$.
\item For any point $x \in \mathcal{A}$, $f_{\mathsf{int}}(x)$ is defined to be the closest integral multiple of $\delta/100$ to $f(x)$.
\item For any point $x \in \mathcal{B} \setminus \mathcal{A}$, $f_{\mathsf{int}}(x) = f(y)$ where $y$ is the point in $\mathcal{A}$ closest to $x$.
\end{enumerate}
We next observe that for any $x \in \mathcal{B}$, if $y$ denotes the closest point in $\mathcal{A}$, then
\[
|f(x) - f_{\mathsf{int}}(x)| \le |f(x) - f(y)| + |f(y) - f_{\mathsf{int}}(y)| \le \frac{\delta}{100} + \frac{\delta}{10} \le \frac{11 \delta}{100}.
\]
The above uses the fact that $f$ is $C$-Lipschitz and the $\ell_2$ distance between $x$ and $y$ is bounded by $\frac{\delta}{10C}$. Now, observe any function of the form $f_{\mathsf{int}}$ can be specified by its value on the set $\mathcal{A}$ and further these values are one of $O(1/\delta)$ possiblities. Thus, if we define $ \mathsf{Net}_{k,C,\delta}$ as the set of all such functions, we obtain the lemma. Since $|\mathcal{A}| =O\big(\frac{k \cdot C \cdot \log(1/\delta)}{\delta}\big)^k$, we obtain the bound on the size of $ \mathsf{Net}_{k,C,\delta}$.
\end{proof}
\begin{lemma}~\label{lem:Lip-1}
Let $f: \mathbb{R}^n \rightarrow [-1,1]$ be a $c$-Lipschitz function. Further, let $g: \mathbb{R}^n \rightarrow [-1,1]$ be a $W$-junta such that $f$ is $O(\epsilon)$-close to $g$. Then, there is a function $f_W:\mathbb{R}^n \rightarrow [-1,1]$ which is $c$-Lipschitz and $W$-junta which is $O(\epsilon)$-close to $f$.
\end{lemma}
\begin{proof}
Reorient the axes so that $W$ is the space spanned by the first $\ell$-axes. Let us define the $W$-junta $f_{W}: \mathbb{R}^n \rightarrow [-1,1]$ defined as
\[
f_W(x) = \mathbf{E}_{y_{\ell+1}, \ldots, y_n} [f(x_1, \ldots, x_\ell, y_{\ell+1}, \ldots, y_n)
\]
For any fixed choice of $x_1, \ldots, x_\ell$, we have
\[
\mathbf{E}_{x_{\ell+1}, \ldots, x_n}[|f(x) - f_W(x)|] \leq \mathbf{E}_{x_{\ell+1}, \ldots, x_n}[|f(x) - g(x)|] +|g(x) - f_W(x)|.
\]
However, the second term can be bounded as
\[
|g(x) - f_W(x)| = \big| g(x) - \mathbf{E}{x_{\ell+1}, \ldots, x_n}[f(x_1, \ldots,x_\ell, x_{\ell+1} , \ldots, x_n)] \big| \le \mathbf{E}{x_{\ell+1}, \ldots, x_n} \big[ \big| g(x) - f(x) \big|\big]
\]
The last inequality is simply Jensen's inequality. Combining these two, we get
\begin{equation}~\label{eq:junta-diff}
\mathbf{E}_{x_{\ell+1}, \ldots, x_n}[|f(x) - f_W(x)|] \leq2 \cdot \mathbf{E}_{x_{\ell+1}, \ldots, x_n}[|f(x) - g(x)|].
\end{equation}
This in turn implies that
\begin{equation}~\label{eq:junta-diff-1}
\mathbf{E}_{x_{1}, \ldots, x_n}[|f(x) - f_W(x)|] \leq2 \cdot \mathbf{E}_{x_{1}, \ldots, x_n}[|f(x) - g(x)|] \leq 2\cdot \epsilon.
\end{equation}
Finally, we see that
\begin{eqnarray*}
|f_W(x) - f_W(y)| &=& \big| \mathbf{E}_{x_{\ell+1}, \ldots, x_n}[f(x_1, \ldots, x_\ell, x_{\ell+1} , \ldots, x_n) - f(y_1, \ldots, y_\ell, x_{\ell+1},\ldots, x_n)] \\
&\leq& \mathbf{E}_{x_{\ell+1}, \ldots, x_n} \big[ \big| f(x_1, \ldots, x_\ell, x_{\ell+1} , \ldots, x_n) - f(y_1, \ldots, y_\ell, x_{\ell+1},\ldots, x_n) \big| \big] \\
&\le& \mathbf{E}_{x_{\ell+1}, \ldots, x_n} [ C \cdot \Vert (x_1, \ldots, x_\ell) - (y_1, \ldots, y_\ell) \Vert_2] \le C \Vert x -y\Vert_2.
\end{eqnarray*}
This finishes the proof.
\end{proof}
\begin{proof}
First, we apply Proposition~\ref{prop:gradient-bound} to observe that $P_t f$ is $\kappa=O(t^{-1/2})$-Lipschitz. Since $P_t f$ is $O(\epsilon)$-close to a $W$-junta, we obtain that $P_t f$ is $O(\epsilon)$ close to a $W$-junta $g$ which is $\kappa$-Lipschitz (follows from Lemma~\ref{lem:Lip-1}). Let $ \mathsf{Cover}(t,k,\epsilon)=\mathsf{Net}_{k,\kappa,\epsilon}$ (where $\mathsf{Net}_{k,\kappa,\epsilon}$ in the set from Lemma~\ref{lem:Lipschitz-cover}). By a rotation of the coordinates, it follows from the definition of $ \mathsf{Cover}(t,k,\epsilon)$ that there exists $h \in\mathsf{Cover}(t,k,\epsilon)$ such that $h( \langle w_1, x \rangle, \ldots, \langle w_k, x \rangle)$ is $O(\epsilon)$ close to $P_tf$. This finishes the proof. The upper bound on the size of the set $\mathsf{Cover}(t,k,\epsilon)$ follows from Lemma~\ref{lem:Lipschitz-cover}.
\end{proof}
\begin{lemma}~\label{lem:apx-ortho}
For $f: \mathbb{R}^n \rightarrow [-1,1]$ and $t>0$, let $(y_1, \ldots, y_\ell)$ be $\gamma$-linearly independent for $h = P_t f$. Let $v_i = D_{y_i} h(y_i)$ and let $V = \mathsf{span}(v_1, \ldots, v_\ell)$.
For any error parameter $\tau>0$ and for $T = T(\tau, t, \gamma)$ defined as
\[
T(\tau, t, \gamma) = \mathsf{poly} \bigg( \frac{1}{t}, 2 \frac{k}{\tau \cdot \sqrt{t}} \cdot \big( \frac{k}{2\cdot \gamma \cdot \sqrt{t}}\big)^{3k+3}\bigg)
\]
we can make $T$ queries to oracle for $f$ and obtain numbers
$\{\alpha_{i,j}\}_{1\le i,j \le \ell}$ such that the following holds:
\begin{enumerate}
\item For $\xi ( t, \gamma)$ defined as
\[
\xi ( t, \gamma) = \bigg(\frac{2 k }{\gamma \cdot \sqrt{t}}\bigg)^{\frac{k+1}{2}} \cdot k^{3/2} \cdot \frac{1}{\sqrt{t}},
\]
we have
$|\alpha_{i,j}| \le \xi ( t, \gamma)$.
\item There is an orthonormal basis $(w_1, \ldots, w_\ell)$ of $(Dh_{y_1}(y_1), \ldots, Dh_{y_\ell}(y_\ell))$ such that for $$\Vert w_{i} - \sum_{j}\alpha_{i,j} Dh_{y_j}(y_j) \Vert_2 \le \tau.$$
\end{enumerate}
\end{lemma}
\begin{proof}
We first invoke Lemma~\ref{lem:inner-product-1} to obtain that with $T$ queries, we can compute numbers $\beta_{i,j}$ with the following guarantee:
\[
\big| \beta_{i,j} - \langle D_{y_i} h(y_i) , D_{y_j} h(y_j) \rangle \big| \le 2 \frac{\tau \cdot \sqrt{t}}{k } \cdot \bigg( \frac{\gamma \cdot \sqrt{t}}{2\cdot k }\bigg)^{3k+3}
\]
Now, observe that because $(y_1, \ldots, y_\ell)$ are $\gamma$-linearly independent, hence as vectors
$(Dh_{y_1}(y_1), \ldots, Dh_{y_\ell}(y_\ell))$ are
$(t^{-1/2}, \gamma)$ linearly independent. We can now apply Proposition~\ref{prop:linear} to obtain the numbers the numbers $\{\alpha_{i,j}\}$ promised here.
\end{proof}
\begin{proposition}~\label{prop:linear}
Let $v_1, \ldots, v_\ell$ be a $(\eta, \gamma)$-linearly independent vectors. Then, for $\epsilon>0$, $\lambda(\epsilon,\eta, \gamma)$ defined as
$$
\lambda= 2 \frac{\epsilon}{\ell \cdot \eta} \cdot \big( \frac{\gamma}{2\cdot \ell \cdot \eta}\big)^{3\ell+3},
$$ given numbers $\{\beta_{i,j}\}_{1\le i,j \le \ell}$ such that $|\beta_{i,j} - \langle v_i, v_j \rangle| \le \lambda$, we can compute numbers $\{\alpha_{i,j}\}_{1\le i,j \le \ell}$ such that:
\begin{enumerate}
\item For $\xi ( \eta, \gamma)$ defined as
\[
\xi ( \eta, \gamma) = \bigg(\frac{2 \ell \eta}{\gamma}\bigg)^{\frac{\ell+1}{2}} \cdot \ell^{3/2} \cdot \eta,
\]
we have
$|\alpha_{i,j}| \le \xi ( \eta, \gamma)$.
\item There is an orthonormal basis $(w_1, \ldots, w_\ell)$ of $\mathsf{span}(v_1, \ldots, v_\ell)$ such that for $\Vert w_{i} - \sum_{j}\alpha_{i,j} v_j \Vert_2 \le \epsilon$.
\end{enumerate}
\end{proposition}
\begin{proof}
Consider the symmetric matrix $\Sigma\in\mathbb{R}^{\ell \times \ell}$ defined as $\Sigma_{i,j} = \langle v_i, v_j \rangle$. By Proposition~\ref{prop:sing-1}, $\Sigma$ is non-singular. Define the matrix $\Gamma = \Sigma^{-1/2}$. It is easy to see that the columns of
$V \cdot \Sigma^{-1/2}$ form an orthonormal basis of $\mathsf{span}(v_1, \ldots, v_\ell)$. Here $V = [v_1 | \ldots | v_\ell]$. Of course, we cannot compute the matrix $\Sigma$ exactly and consequently, we cannot compute the matrix $\Sigma^{-1/2}$ either.
Instead, we can compute a
$\widetilde{\Sigma}$ (which is also symmetric)
such that $\Vert \widetilde{\Sigma} - \Sigma \Vert_F\le \ell \cdot \lambda$. Now, for an error parameter $\delta$ to be fixed, assume that
$$
\ell \cdot \lambda \le \delta \cdot \sigma_{\min}(\Sigma) = \delta \cdot \sigma_{\min}^2(V) \le \delta \cdot \bigg(\frac{ \gamma}{2 \cdot \ell \cdot \eta}\bigg)^{2\ell+2}.
$$
Here the second inequality, uses Proposition~\ref{prop:sing-1}. Now, we apply the matrix perturbation bound (Corollary~\ref{corr:mat-perturb})
(i.e., here we set $c = \big( \frac{\gamma}{2\cdot \ell \cdot \eta}\big)^{2\ell+2}$)
to obtain that
\[
\Vert \Sigma^{-1/2} - \widetilde{\Sigma}^{-1/2} \Vert \leq \frac{\delta}{2 \big( \frac{\gamma}{2\cdot \ell \cdot \eta}\big)^{\ell+1}}.
\]
Now, set $\delta = 2 \frac{\epsilon}{\ell \cdot \eta} \cdot \big( \frac{\gamma}{2\cdot \ell \cdot \eta}\big)^{\ell+1}$. Define $\alpha_{i,j} = \widetilde{\Sigma}^{-\frac12}(i,j)$, the second item now follows immediately. For the first item, observe that by Weyl's inequality (Lemma~\ref{lem:Weyl}), $\sigma_{\min}(\widetilde{\Sigma}) \ge (1-\delta) \cdot \sigma_{\min}({\Sigma})$. Thus, $$\Vert \widetilde{\Sigma}^{-1} \Vert_F \le \bigg(\frac{2 \ell \eta}{\gamma}\bigg)^{\ell+1} \cdot \ell \cdot \eta.$$
Finally, since $\widetilde{\Sigma}^{-1/2}$ is also Hermitian, it is easy to see that
\[
\Vert \widetilde{\Sigma}^{-1/2} \Vert_F \le \sqrt{\ell} \cdot \sqrt{\Vert \widetilde{\Sigma} \Vert_F}.
\]
This puts an upper bound on $|\alpha_{i,j}|$ finishing the proof.
\end{proof}
\begin{proposition}~\label{prop:sing-1}
Let $v_1, \ldots, v_\ell$ be a $(\eta, \gamma)$-linearly independent vectors. Let $V = [v_1 | \ldots | v_\ell]$. Then, the smallest singular value of $V$
is at least $(\frac{ \gamma}{2 \cdot \ell \cdot \eta})^{\ell+1}$.
\end{proposition}
\begin{proof}
Let $\kappa>0$ whose precise value will be fixed later.
Now, note that if $\sigma_{\min}$ is the smallest singular value of $V$, then $\inf_{x : \Vert x \Vert_2=1} \Vert V \cdot x \Vert_2$. In order to lower bound this, observe that $V \cdot x = \sum_{1 \le i \le \ell} v_i \cdot x_i$.
Now, let $j$ be the largest coordinate such that $|x_j| \ge \kappa^{j}$ (note that there has to be such a $j$ since $x$ is a unit vector). Define $w = \sum_{i \le j} v_i x_i$. Then, observe that its component in the direction orthogonal to the span of $\{v_1, \ldots, v_{j-1}\}$ is at least $\gamma \cdot \kappa^j$ in magnitude. On the other hand, $\Vert \sum_{i > j} v_i x_i \Vert_2 \le \kappa^{j+1} \cdot \ell \cdot \eta$. Now, as long as $\kappa \le \frac{\gamma}{2 \cdot \ell \cdot \eta}$, we obtain that
\[
\Vert \sum_{i} v_i x_i \Vert_2 \ge \Vert \sum_{i \le j} v_i x_i \Vert_2 - \Vert \sum_{i > j} v_i x_i \Vert_2 \ge \gamma \cdot \kappa^j - \ell \cdot \eta \cdot \kappa^{j+1} \ge \frac{\gamma \cdot \kappa^j}{2}.
\]
This finishes the proof.
\end{proof}
\begin{lemma}~\label{lem:test-one}
There is a routine \textsf{Test-closeness-one} such that given oracle access to $f: \mathbb{R}^n \rightarrow [-1,1]$, $(y_1, \ldots, y_\ell)$ which are $\gamma$-linearly independent for $P_t f$ and access to
$g \in \mathsf{Cover}(t,\ell,\epsilon)$, has the following guarantee:
\begin{enumerate}
\item For $\tau = \epsilon^2/(100 \cdot \ell^{3/2})$, it makes $T(\tau, t,\gamma) \cdot \log(1/\xi)$ queries to $f$ (where $T(\cdot, \cdot, \cdot)$
is the function in Lemma~~\ref{lem:apx-ortho}).
\item There is an orthonormal basis $(w_1, \ldots, w_\ell)$ of $\mathsf{span}(Dh_{y_1}(y_1), \ldots, Dh_{y_\ell}(y_\ell))$ (which depends just on that) such that
with probability $1-\xi$, the algorithm outputs an $\epsilon/100$ accurate estimate to $\mathbf{E}[\Vert P_t f - g (w_1, \ldots ,w_\ell) \Vert_1]$.
\end{enumerate}
\end{lemma}
\begin{proof}
First, we run the procedure in Lemma~\ref{lem:apx-ortho}with error parameter $\tau$, noise rate $t$ and parameter $\gamma$. Note that with $T(\tau, t, \gamma)$ queries, we are able to obtain coefficients $\{\alpha_{i,j} \} $ such that
\begin{equation}~\label{eq:bound-tau}
\Vert \sum_{j} \alpha_{i,j} Dh_{y_j}(y_j) -w_i \Vert_2 \le \tau.
\end{equation}
Let $K = \sum_{i,j} |\alpha_{i,j}|$. Set the parameter $\eta= \frac{\epsilon^2}{K \cdot \ell}$.
Let us now define a point $x \in \mathbb{R}^n$ to be \emph{good} if the following holds:
\begin{enumerate}
\item For all $1 \le i \le \ell$, the function $f_{\partial, \eta, t, y_i}$ defined in Lemma~\ref{lem:compute-derivative-x},
$$
\big| f_{\partial, \eta, t, y_i} (x) - \langle D_{y_i} (P_tf)(y_i), x \rangle \big| \le \frac{\ell \cdot \eta}{\epsilon}.
$$
\item For all $1 \le i \le \ell$,
$$
\big| \sum_{j} \alpha_{i,j} \langle Dh_{y_j}(y_j),x \rangle - \langle w_i, x\rangle \big| \le \frac{\epsilon}{100 \ell}.
$$
\end{enumerate}
The crucial point is that for a randomly chosen $x \sim \gamma_n$, Lemma~\ref{lem:compute-derivative-x} guarantees that the first item is satisfied with probability at least $1 - \frac{\epsilon^2}{\ell \cdot \eta^2}$. Likewise, from (\ref{eq:bound-tau}), for $x \sim \gamma_n$, we get that the second item is satisfied with probability $\epsilon/(100\ell)$. Thus, we get that a point $x \sim \gamma_n$ is \emph{good} with probability $1-\epsilon/\ell$. The algorithm \textsf{Test-closeness-one} is now defined as follows:
\begin{enumerate}
\item Sample $s = 1/\epsilon^2 \cdot \log(1/\xi)$ points $x_1, \ldots, x_s$.
\item For each of the points $x_i$, do the following:
\item \hspace*{10pt} Compute $f_{\partial, \eta, t, y_j}(x_i)$ for $1 \le j \le \ell$ up to error $\frac{\epsilon}{K \cdot \ell}$.
\item \hspace*{10pt} Compute $\tilde{\beta}_{i,x} = \sum_{j} \alpha_{i,j}f_{\partial, \eta, t, y_j}(x_i)$.
\item \hspace*{10pt} Compute $g(\tilde{\beta}_{1,x} ,\ldots, \tilde{\beta}_{\ell,x})$.
\item Output $\frac{1}{s} \sum_{i=1}^s |P_tf(x_i) - g(\tilde{\beta}_{1,x} ,\ldots, \tilde{\beta}_{\ell,x})|$.
\end{enumerate}
The analysis of this algorithm is as follows:
\begin{eqnarray*}
&& \big|\mathbf{E}_{x \sim \gamma_n}\big[ |P_tf(x_i) - g(\tilde{\beta}_{1,x} ,\ldots, \tilde{\beta}_{\ell,x})| \big]-\mathbf{E}_{x \sim \gamma_n}\big[ |P_tf(x_i) - g(\langle w_1,x\rangle ,\ldots, \langle w_\ell,x\rangle)| \big] \big| \\ &\le& \mathbf{E}_{x \sim \gamma_n} \big[ |g(\tilde{\beta}_{1,x} ,\ldots, \tilde{\beta}_{\ell,x})- g(\langle w_1,x\rangle ,\ldots, \langle w_\ell,x\rangle)| \big]
\end{eqnarray*}
Now, note that because $g$ is bounded by $[-1,1]$, the term inside the expectation is bounded by $2$. Further, if a point $x$ is \emph{good}, for every $1 \le i \le \ell$,
$$
\big|\widetilde{\beta}_{i,x} - \langle w_i, x \rangle \big| \le \frac{\epsilon}{50 \ell}.
$$
Now, this immediately implies that
\[
\mathbf{E}_{x \sim \gamma_n} \big[ |g(\tilde{\beta}_{1,x} ,\ldots, \tilde{\beta}_{\ell,x})- g(\langle w_1,x\rangle ,\ldots, \langle w_\ell,x\rangle)| \big] \le \frac{\epsilon}{50} + \Pr[x \textrm{ is not good}] \le \frac{\epsilon}{2}.
\]
Item 2 now follows immediately.
\end{proof}
\section{Algorithm to find hidden linear invariant structure}~\label{aff:inv}
In this section, we will prove the following main theorem.
\begin{theorem}~\label{thm:main-affine-invariant}
Let $f: \mathbb{R}^n \rightarrow \{-1,1\}$ be a linear-$k$-junta with surface area $s$. Then, there is an algorithm \textsf{Find-invariant-structure} which for any error parameter $\epsilon>0$,
makes $O(s \cdot k /\epsilon)^{O(k)}$ queries to $f$
and with probability $1-\epsilon$ outputs (for some $\ell \le k$) a function $g: \mathbb{R}^\ell \rightarrow [-1,1]$ so that the following holds: there is an orthonormal set of vectors $w_1, \ldots, w_\ell \in \mathbb{R}^n$ such that
$$
\mathbf{E}[|f(x) - g(\langle w_1, x\rangle, \ldots, \langle w_\ell, x \rangle)|] = O(\epsilon).
$$
{Further, there is a set $V = \{v_1, \ldots, v_k\}$ of orthonormal vectors such that for $1 \le j \le \ell$, $v_j = w_j$ and $\span\{v_1, \ldots, v_k\}$ is a relevant subspace of $f$.}
\end{theorem}
Our algorithm is quite na\"ive. First, we ``identify'' -- in some implicit
sense -- the $k$-dimensional subspace on which the linear $k$-junta acts. We
take a fine net of functions defined on that space, and we test them all until
we fine the best one. Obviously, this algorithm is not computationally
efficient, and it is also not particularly efficient in terms of the query
complexity. However, the crucial feature of this algorithm
is that its query complexity does not depend on the
ambient dimension $n$. The main difficulty in constructing and analyzing this algorithm
is that we cannot explicitly identify even a single vector in the interesting
$k$-dimensional subspace -- that would require a number of queries that depends on $n$.
One consequence of this is that we do not know how to apply an off-the-shelf
learning algorithm (such as the one from~\cite{KOS:08}).
\begin{definition}~\label{def:vector-independence}
A set of vectors $v_1, \ldots, v_\ell \in \mathbb{R}^n$ is said to be $(\eta,\gamma)$-linearly independent if the following conditions hold:
\begin{enumerate}
\item For all $1 \le i \le \ell$,
$\Vert v_i \Vert_2 \le \eta$.
\item For all $1 < i \le \ell$, $\mathsf{dist}(v_i, \mathsf{span}(v_1, \ldots, v_{i-1})) \ge \gamma$.
\end{enumerate}
\end{definition}
\begin{definition}
For $f: \mathbb{R}^n \rightarrow [-1,1]$ and $t>0$, we say that a set of directions $(y_1, \ldots, y_\ell)$ is
$\gamma$-linearly independent, if the following holds:
For $1 \le i \le \ell$, let $v_i = DP_tf(y_i)$. If for all $i$, $\mathsf{dist}(v_i, \mathsf{span}(v_1, \ldots, v_{i-1})) \ge \gamma$.
\end{definition}
By Proposition~\ref{prop:derivative-bound}, it is immediate that as long as $t \le 1/4$, $\Vert DP_t f (y) \Vert_2 \le t^{-1/2}$. Thus, if $(y_1, \ldots, y_\ell)$ is $\gamma$-linearly independent, then the directions $(v_1,\ldots, v_\ell)$ are $(t^{-1/2}, \gamma)$ linearly independent.
\begin{figure}[h]
\hrule
\vline
\begin{minipage}[t]{0.98\linewidth}
\vspace{10 pt}
\begin{center}
\begin{minipage}[h]{0.95\linewidth}
{\small
\underline{\textsf{Inputs}}
\vspace{5 pt}
\begin{tabular}{ccl}
$t$ &:=& noise parameter \\
$y_1, \ldots, y_\ell$ &:=& $\frac{\gamma}{2}$-linearly independent directions \\
$\{\beta_{i,j}\}$ &:=& $\lambda$-accurate estimates of $\langle DP_tf(y_i), DP_tf(y_j)\rangle$ where \\
&& $\lambda=\lambda(\ell,\nu, t^{-\frac12}, \gamma/2)$ and $\nu = \frac{\gamma^2 \cdot t}{100 \ell^2}$ (from Lemma~\ref{prop:linear})
\\
$y_{\ell+1}$ &:=& candidate direction in $\mathbb{R}^n$.
\end{tabular}
\vspace{5 pt}
\underline{\textsf{Testing algorithm}}
\begin{enumerate}
\item Find the numbers $\{\alpha_{1 \le i, j \le \ell}\}$ from Lemma~\ref{prop:linear}.
\item Estimate $\langle DP_tf(y_{\ell+1}) , DP_tf(y_{\ell+1}) \rangle$ up to $\pm \frac{\gamma^2}{50}$ . Call the estimate $\tilde{\beta}_{\ell+1, \ell+1}$.
\item Estimate $\langle DP_tf(y_{\ell+1}) , DP_tf(y_{j}) \rangle$ (for $1 \le j \le \ell$) up to accuracy
$\frac{1}{\xi(\ell, t^{-1/2}, \gamma/2)} \cdot \frac{\gamma^2 \cdot \sqrt{t}}{100\ell^3}$ (using Lemma~\ref{lem:inner-product-1}) where $\xi$ is the function from Lemma~\ref{prop:linear}. Call the estimates $\tilde{\beta}_{j, \ell+1}$.
\item Compute quantity $\zeta_i = \sum_{1 \le j \le \ell} \alpha_{i,j} \cdot \tilde{\beta}_{j,\ell+1}$ for all $1 \le i \le \ell$.
\item If the quantity $\tilde{\beta}_{\ell+1, \ell+1}^2 - \sum_{i=1}^\ell \zeta_i^2 > (\frac{3 \gamma}{4})^2$, then output \textsf{yes}. Else output \textsf{no}.
\end{enumerate}
\vspace{5 pt}
}
\end{minipage}
\end{center}
\end{minipage}
\hfill \vline
\hrule
\caption{Description of the algorithm \textsf{Test-candidate-direction}}
\label{fig:tlin-1}
\end{figure}
\begin{lemma}~\label{lem:test-candidate}
The algorithm \textsf{Test-candidate-direction} described in Figure~\ref{fig:tlin-1} has the following properties: For noise parameter $t$, directions $y_1, \ldots, y_\ell \in \mathbb{R}^n$, $\{\beta_{i,j} \}$ and candidate direction $y_{\ell+1}$ (where $y_1, \ldots, y_\ell$ as well as $\{ \beta_{i,j} \}$ meet the requirements described in Figure~\ref{fig:tlin-1}), the algorithm satisfies
\begin{enumerate}
\item The query complexity of the algorithm is
$T_{tc}(t, \gamma, \ell) = \big( \frac{\ell}{\sqrt{t} \cdot \gamma} \big)^{O(\ell)}$.
\item If the Euclidean distance of $DP_tf(y_{\ell+1})$ is at least $\gamma$ from the subspace $\mathsf{span}(DP_tf(y_{1}), \ldots, DP_tf(y_{\ell}))$, then the algorithm outputs \textsf{yes}. Conversely, if the algorithm outputs \textsf{no}, then the Euclidean distance must be less than $\frac{\gamma}{2}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The query complexity bound is just immediate from Lemma~\ref{lem:inner-product-1} and plugging in the value of $\xi(\ell, t^{-1/2}, \gamma)$ from Lemma~\ref{prop:linear}. To prove the second guarantee,
let us use $v_j$ to denote $DP_tf(y_j)$. Since $(v_1, \ldots, v_j)$ are $(1/t^{-1/2}, \frac{\gamma}{2})$-linearly independent, hence by Lemma~\ref{prop:linear}, we obtain that there are orthonormal vectors $(w_1, \ldots, w_\ell)$ (which span $v_1, \ldots, v_\ell$)
such that
$$
\Vert w_i - \sum_{j} \alpha_{i,j} v_j \Vert_2 \le \frac{\gamma^2 \cdot {t}}{100 \ell^2}.
$$
This implies that if we let $v_{\ell+1} = DP_tf(y_{\ell+1})$ (using $\Vert v_{\ell+1} \Vert \le t^{-1/2}$), then
$$
\big| \langle w_i, v_{\ell+1} \rangle - \sum_{j} \alpha_{i,j}
\langle v_j , v_{\ell+1} \rangle \big| \le \frac{\gamma^2 \sqrt{t}}{100 \ell^2}.
$$
Consequently, we have
\begin{eqnarray*}
\big| \langle w_i, v_{\ell+1} \rangle - \sum_{j} \alpha_{i,j} \cdot \tilde{\beta}_{j,\ell+1} \big| &\le& \frac{\gamma^2 \cdot \sqrt{t}}{100 \ell^2} + \sum_{j}|\alpha_{i,j}| \cdot |\langle v_j , v_{\ell+1} \rangle - \tilde{\beta}_{j,\ell+1} | \\
&\le& \frac{\gamma^2\cdot \sqrt{t}}{100 \ell^2} + \sum_{j} \xi (\ell, t^{-1/2}, \gamma/2) \cdot \frac{1}{\xi (\ell, t^{-1/2}, \gamma/2)}\cdot \frac{\gamma^2\sqrt{t}}{100 \ell^3} \le \frac{\gamma^2\sqrt{t}}{50 \ell^2}.
\end{eqnarray*}
The penultimate inequality follows from the bound on $|\alpha_{i,j}|$ from Lemma~\ref{prop:linear} and the accuracy of estimates $\tilde{\beta}_{j,\ell+1}$.
This implies that for any $i$,
\begin{equation}~\label{eq:bound-diff1}
\big|\big| \langle w_i, v_{\ell+1} \rangle \big|^2- \big|\sum_{j} \alpha_{i,j} \cdot \tilde{\beta}_{j,\ell+1} \big|^2\big| \le \frac{\gamma^2\sqrt{t}}{50 \ell^2} \cdot \big| \langle w_i, v_{\ell+1} \rangle + \sum_{j} \alpha_{i,j} \cdot \tilde{\beta}_{j,\ell+1} \big| \le \frac{\gamma^2\sqrt{t}}{50 \ell^2} \cdot 2 \cdot t^{-\frac12} = \frac{\gamma^2}{25 \ell^2}.
\end{equation}
The second inequality uses that fact that $w_i$ is a unit vector whereas $\Vert v_{\ell+1} \Vert_2 \le t^{-\frac12}$. Thus,
\begin{eqnarray*}
\mathrm{dist}^2\big(DP_tf(y_{\ell+1}), \mathsf{span}(DP_tf(y_{1}), \ldots, DP_tf(y_{\ell}))\big) &=& \Vert DP_tf(y_{\ell+1})\Vert_2^2 - \sum_{j=1}^\ell \langle DP_tf(y_{\ell+1}), w_j\rangle^2 \\
&=& \Vert DP_tf(y_{\ell+1})\Vert_2^2 - \sum_{j=1}^\ell \zeta_j^2 + \theta
\end{eqnarray*}
where $|\theta| \le \frac{\gamma^2}{25\ell}$ (from \ref{eq:bound-diff1}). Using the fact that $ |\tilde{\beta}_{\ell+1, \ell+1}^2- \Vert D P_tf(y_{\ell+1})\Vert_2^2 | \le \frac{\gamma^2}{50}$, we can conclude that
$$
\big|\mathrm{dist}^2\big(DP_tf(y_{\ell+1}), \mathsf{span}(DP_tf(y_{1}), \ldots, DP_tf(y_{\ell}))\big)- \tilde{\beta}_{\ell+1, \ell+1}^2 - \sum_{i=1}^\ell \zeta_i^2\big| \le \frac{\gamma^2}{25}.
$$
Item 2 in the claim is now an immediate consequence.
\end{proof}
\begin{figure}[tb]
\hrule
\vline
\begin{minipage}[t]{0.98\linewidth}
\vspace{10 pt}
\begin{center}
\begin{minipage}[h]{0.95\linewidth}
{\small
\underline{\textsf{Inputs}}
\vspace{5 pt}
\begin{tabular}{ccl}
$s$ &:=& surface parameter \\
$\epsilon$ &:=& error parameter\\
\end{tabular}
~\\
~\\
\underline{\textsf{Parameters}}
\vspace{5 pt}
\begin{tabular}{ccl}
$t$ &:=& $\frac{\epsilon^4}{900 s^2}$ \\
$\gamma$ &:=& $\frac{\epsilon^2}{8}$\\
$\lambda$ &=& $\lambda(k, \nu, t^{-\frac12}, \gamma)$ (where $\lambda(\cdot)$ is the function from Lemma~\ref{prop:linear}) and $\nu = \frac{\gamma^2 \cdot t}{100k^2}$. \\
$\tau_{\mathsf{succ}}$ &:=& $\frac{\epsilon^6}{s^2}$ \\
$T_{\mathsf{succ}}$ &:=& $\frac{1}{\tau_{\mathsf{succ}}} \cdot \log (10k/\epsilon)$.\\
\end{tabular}
~\\
~\\
\underline{\textsf{Testing algorithm}}
\begin{enumerate}
\item Initialize $S$ to be the empty set.
\item Initialize $\mathsf{count}=0$.
\item If $\mathsf{count} =k$, exit;
\item else set $S =\{y_1, \ldots, y_\ell\}$ and compute $\{\beta_{i,j}\}$ as $\lambda$-accurate estimates of $\langle DP_tf(y_i), DP_tf(y_j) \rangle$ (Lemma~\ref{lem:inner-product-1}).
\item Repeat $T_{\mathsf{succ}}$ times
\item \hspace{7pt} Choose $z \sim \gamma_n$.
\item \hspace{7pt} Run \textsf{Test-candidate-direction} with $S=\{y_1, \ldots,y_\ell\}$, candidate direction $z$, $\gamma, t$ as defined in \textsf{Parameters} and $\{\beta_{i,j}\}$ \hspace{3pt} as computed above.
\item \hspace{7pt} If \textsf{Test-candidate-direction} outputs \textsf{yes},
add $z$ to $S$; $\mathsf{count}+=1$;
go to step 3;
\item If the size of $S$ does not increase in $T_{\mathsf{succ}}$ steps, then exit;
\end{enumerate}
\vspace{5 pt}
}
\end{minipage}
\end{center}
\end{minipage}
\hfill \vline
\hrule
\caption{Description of the algorithm \textsf{Find-candidate-directions}}
\label{fig:flin-1}
\end{figure}
We now give an algorithm which finds out directions $\{y_1, \ldots, y_\ell\}$ such that for $t$ defined before (as $t : = \frac{\epsilon^4}{900 s^2}$), $P_t f$ is close to a junta on the directions $\{DP_tf(y_1), \ldots, DP_tf(y_\ell)\}$.
\begin{lemma}~\label{lem:find-dirs}
The algorithm \textsf{Find-candidate-directions} described in Figure~\ref{fig:flin-1} has the following properties: For noise parameter $t$, error parameter $\epsilon$, surface area parameter $s$, if the function $f: \mathbb{R}^n \rightarrow [-1,1]$ has surface area $s$ and is a linear $k$-junta, then with probability $1-\epsilon$, the algorithm outputs vectors $y_1, \ldots, y_\ell \in \mathbb{R}^n$ ($\ell \le k$) such that for $\{v_1, \ldots, v_\ell\}$ defined as
$v_i = DP_tf(y_i)$, the function is $\epsilon$-close to
a junta on $\mathsf{span}(v_1, \ldots, v_\ell)$. Further, the directions $(y_1, \ldots, y_\ell)$ are at
least $\gamma/2 = \frac{\epsilon^2}{16}$ linearly independent.
The query complexity of this algorithm is $T_{fc} (s,k,\epsilon) =\big( \frac{s \cdot k}{\epsilon}\big)^{O(k)}$.
\end{lemma}
\begin{proof}
{{The bound on the query complexity of this algorithm is immediate by just plugging in the query complexity of the routine \textsf{test-candidate-direction} (Lemma~\ref{lem:test-candidate}) and the query complexity of Step 4~(Lemma~\ref{lem:inner-product-1}).} }
Next, observe that by the guarantee of \textsf{Test-candidate-direction}, the set $S$ output by the algorithm consists of $\gamma/2$-linearly independent
directions.
Finally, assume that $f$ is a $W$-junta where $\mathsf{dim}(W) \le k$. Then, note that for any $y \in \mathbb{R}^n$, $DP_tf(y) \in W$. Now, there are two possibilities: (For the rest of this proof, we will use $v_i$ as a shorthand for $DP_tf(y_i)$)
\begin{itemize}
\item[(a)] If $\mathsf{count}=k$, then note that we have found $k$ directions $y_1, \ldots, y_k$ such that $v_i\in W$. Further, the directions $(v_1, \ldots, v_k)$ are $(t^{-1/2}, \gamma)$-linearly independent. Thus, $\mathsf{span}(v_1, \ldots, v_k)= W$. So, in this case, $P_tf$ is indeed a junta on $\mathsf{span}(v_1, \ldots, v_k)$ (where $S= \{y_1, \ldots, y_k\}$).
\item[(b)] If $\mathsf{count}<k$, then we are in one of the two situations: either $f$ is $\epsilon$-close to a junta on $\mathsf{span}(v_1, \ldots, v_\ell)$ where $S=\{y_1, \ldots, y_\ell\}$. In this case, we are already done. If not, then we apply Lemma~\ref{lem:subspace-escape} and obtain that with probability at least $\tau_{\mathsf{succ}}$, a randomly chosen direction $z$ will be at least $\gamma=\epsilon^2/8$-far from the subspace $\mathsf{span}(v_1, \ldots, v_\ell)$ and will thus pass the algorithm \textsf{Test-candidate-direction}. Thus, over $T_{\mathsf{succ}}$ trials, with probability at least $1- \frac{\epsilon}{10k}$, the set $S$ will increase in size and we will continue inductively.
Since the outer loop (i.e., the loop for $\mathsf{count}$ will run at most $k$ times), the total probability that $P_t f$ is not $\epsilon$-close to a $W$-junta for $W = \mathsf{span}(v_1, \ldots, v_\ell)$ but the algorithm terminates is at most $1-\frac{\epsilon}{10}$. This finishes the proof.
\end{itemize}
\end{proof}
With the aid of the algorithm \textsf{Find-candidate-directions}, we are able to find implicitly find directions $\{v_1, \ldots, v_\ell\}$ such that $P_t f$ is close to a junta on $\mathsf{span}(v_1, \ldots, v_\ell)$. In the next subsection, we essentially do a hypothesis testing over a set of functions which form a cover for all juntas on $\mathsf{span}(v_1, \ldots, v_\ell)$.
\subsection{Hypothesis testing against subspace juntas}
The following lemma says how given the directions $y_1, \ldots, y_\ell$ and an error parameter $\tau$, we can implicitly find directions which form an orthonormal
basis of $\mathsf{span}(v_1,\ldots, v_\ell)$ (as before, we are using $v_1, \ldots, v_\ell$ as a shorthand for $DP_tf(y_1), \ldots, DP_tf(y_\ell)$ respectively). All the symbols below will have the same value as Lemma~\ref{lem:find-dirs} unless mentioned otherwise.
\begin{lemma}~\label{lem:orthogonalize}
Choose any error parameter $\tau>0$ and let $y_1, \ldots, y_\ell$ be $\gamma/2$-linearly independent directions for $P_t f$. Then, there is a procedure \textsf{Compute-ortho-transform} which makes $T_{\mathsf{ortho}} =\mathsf{poly}(1/\tau) \cdot \big( \frac{\ell}{\gamma \cdot t}\big)^{O(\ell)}$ queries to $f$, we can obtain numbers $\{\alpha_{i,j} \}_{1 \le i,j \le \ell}$ such that the following holds:
\begin{enumerate}
\item For $\Lambda(\ell, t, \gamma) = (\frac{\ell}{t \gamma})^{O(\ell)}$, all the numbers $|\alpha_{i,j}| \le \Lambda(\ell, t, \gamma)$.
\item There exists an orthonormal basis $(w_1, \ldots, w_\ell)$ of $\mathsf{span}(v_1, \ldots, v_\ell)$ such that for all $1 \le i \leq \ell$,
\[
\Vert w_i - \sum_{j} \alpha_{i,j} v_j \Vert_2 \le \tau.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\lambda(\cdot)$ be the function defined in Lemma~\ref{prop:linear}. Now, observe that
\[
\lambda(\ell, \tau, t^{-1/2}, \gamma) = \tau \cdot \bigg(\frac{\gamma \cdot t}{2 \cdot \ell } \bigg)^{O(\ell)}.
\]
Thus, using Lemma~\ref{lem:inner-product-1}, we can use
$T_{\mathsf{ortho}}$ queries to $f$ to obtain numbers $\{\beta_{i,j}\}_{1 \le i,j \le \ell}$ such that
\[
\big| \beta_{i,j} - \langle D_{y_i} h(y_i) , D_{y_j} h(y_j) \rangle \big| \le \lambda(\ell, \tau, t^{-1/2}, \gamma).
\]
As $(y_1, \ldots, y_\ell)$ are $\gamma$-linearly independent, hence the vectors $(v_1, \ldots, v_\ell)$ are $(t^{-\frac12}, \gamma)$-linearly independent.
With this, we can now apply Lemma~\ref{prop:linear} to obtain numbers $\{\alpha_{i,j}\}$ such that there is an orthonormal basis $(w_1, \ldots, w_\ell)$ of $\mathsf{span}(DP_tf(y_1), \ldots, DP_tf(y_\ell))$ with the property that (a)
$
\Vert w_i - \sum_{j} \alpha_{i,j} v_j \Vert_2 \le \tau$
and (b)
$|\alpha_{i,j}| \le \Lambda(\ell, t,\gamma)$ where $\Lambda(\ell, t, \gamma) = (\frac{\ell}{t \gamma})^{O(\ell)}$.
\end{proof}
\begin{figure}[tb]
\hrule
\vline
\begin{minipage}[t]{0.98\linewidth}
\vspace{10 pt}
\begin{center}
\begin{minipage}[h]{0.95\linewidth}
{\small
\underline{\textsf{Inputs}}
\vspace{5 pt}
\begin{tabular}{ccl}
$s$ &:=& surface parameter \\
$\epsilon$ &:=& error parameter\\
$y_1, \ldots, y_\ell$ &:=& $\frac{\gamma}{2}$-linearly independent directions for $P_t f$\\
\end{tabular}
~\\
~\\
\underline{\textsf{Parameters}}
\vspace{5 pt}
\begin{tabular}{ccl}
$t$ &:=& $\frac{\epsilon^4}{900 s^2}$ \\
$\gamma$ &:=& $\frac{\epsilon^2}{8}$\\
$\tau$ &=& $\frac{\epsilon^2 \cdot \sqrt{t}}{100 \cdot \ell^{3/2}}$ \\
$\delta$ &:=& $\frac{\epsilon}{10}$ \\
$K$ &:=& $\ell^2 \cdot \Lambda(\ell,t,\gamma)$ where $\Lambda(\cdot)$ is defined in Lemma~\ref{lem:orthogonalize}. \\
$\xi$ &:=& $\frac{\epsilon^2 \cdot \sqrt{t}}{K \cdot \ell^3 }$\\
$\mu$ &:=& $\frac{\epsilon}{|\mathsf{Cover}(t,\ell,\delta)|}$ where $\mathsf{Cover}(\cdot, \cdot, \cdot)$ is the set from Theorem~\ref{thm:net}. \\
$J$ &:=& $\frac{10}{\epsilon^2} \cdot \log(1/\mu)$ \\
\end{tabular}
~\\
~\\
\underline{\textsf{Testing algorithm}}
\begin{enumerate}
\item Run the procedure \textsf{Compute-ortho-transform} with directions $(y_1,\ldots, y_\ell)$ and $\gamma$, $t$ and $\tau$ as set above.
\item Let the output be parameters $\{\alpha_{i,j}\}_{1 \le i, j \le \ell}$.
\item Sample $J$ points from $\gamma_n$. Call the points $x_1$, $\ldots$, $x_J$.
\item For each of the points $x_i$ and each direction $y_j$,
\item \hspace{7pt} Compute the function $f_{\partial, \xi, t, y_j}(x_i)$ (up to error $\xi$) using Lemma~\ref{lem:compute-derivative-x}. Call this $\zeta_{i,j}$.
\item \hspace{7pt} Compute $\overline{x}_{i,j'} = \sum_{j}\alpha_{j',j} \cdot \zeta_{i,j}$.
\item For all $g \in \mathsf{Cover}(t, \ell, \delta)$,
compute $\mathcal{O}_g= \frac1s \cdot \sum_{i=1}^s |P_tf(x_i) - g(\overline{x}_{i,1}, \ldots, \overline{x}_{i,\ell})|$.
\item Return the $g$ which has the smallest value of $\mathcal{O}_g$.
\end{enumerate}
\vspace{5 pt}
}
\end{minipage}
\end{center}
\end{minipage}
\hfill \vline
\hrule
\caption{Description of the algorithm \textsf{Estimate-closest-hypothesis}}
\label{fig:hyp}
\end{figure}
Let us now again set the parameters $t$ and $\gamma$ exactly the same as Lemma~\ref{lem:find-dirs}. Namely, we set $t= \frac{\epsilon^4}{900s^2}$ and $\gamma=\frac{\epsilon^2}{8}$. With this setting of parameters, we state the following lemma.
\begin{lemma}~\label{lem:test-hypothesis}
There is an algorithm \textsf{Estimate-closest-hypothesis} (described in Figure~\ref{fig:hyp}) which takes as input oracle access to $f: \mathbb{R}^n \rightarrow \{-1,1\}$, directions $(y_1, \ldots, y_\ell)$ which are $\gamma/2$-linearly independent, error parameter $\epsilon$,
surface area parameter $s$. The algorithm has the following guarantee:
\begin{enumerate}
\item It makes $O\big( \frac{s \cdot \ell}{\epsilon}\big)^{O(\ell)}$ queries to $f$.
\item There is an orthonormal basis $(w_1, \ldots, w_\ell)$ of $\mathsf{span}(DP_tf(y_1), \ldots, DP_tf(y_\ell))$ (which is independent of $g$) such that with probability $1-\epsilon$, outputs a function $g: \mathbb{R}^\ell \rightarrow [-1,1]$ with the following guarantee: Let $\mathsf{Cover}(t,\ell, \delta)$ be the set of functions from Theorem~\ref{thm:net} where the parameters $t, \delta$ are set as in Figure~\ref{fig:hyp}. Then,
\[
\mathbf{E}[|P_tf(x) - g(\langle w_1,x\rangle, \ldots, \langle w_\ell,x\rangle)|] \le \min_{g^\ast \in \mathsf{Cover}(t,\ell,\delta)} \mathbf{E}[|P_tf(x) - g^\ast(\langle w_1,x\rangle, \ldots, \langle w_\ell,x\rangle)|] + 5\epsilon.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
As usual, the query complexity of the procedure is easily seen to be $O\big( \frac{s \cdot \ell}{\epsilon}\big)^{O(\ell)} $ by just plugging in the values of the parameters along with the guarantees on the query complexity of \textsf{Compute-ortho-transform} (Lemma~\ref{lem:orthogonalize}) as well Lemma~\ref{lem:compute-derivative-x}.
To analyze the algorithm, let us now define a point $x \in \mathbb{R}^n$ to be \emph{good} if the following two conditions hold:
\begin{enumerate}
\item For $1 \le i \le \ell$,
\[
\big|f_{\partial, \xi, t, y_i}(x) - \langle DP_tf (y_i), x\rangle \big| \le \frac{\ell \cdot \xi}{\epsilon}.
\]
\item For all $1 \le i \le \ell$,
\[
\big| \sum_j \alpha_{i,j} \langle DP_tf(y_j), x\rangle - \langle w_i, x\rangle \big| \le \frac{\epsilon \cdot \sqrt{t}}{100 \ell^2}.
\]
\end{enumerate}
\begin{claim}
For $x \sim \gamma_n$, $\Pr[x \textrm{ is good}] \ge 1-\frac{2\epsilon^2}{\ell}$.
\end{claim}
\begin{proof}
Lemma~\ref{lem:compute-derivative-x} guarantees that for any specific choice of $i$,
$\Pr[\big|f_{\partial, \xi, t, y_i}(x) - \langle DP_tf (y_i), x\rangle \big| \le \frac{\ell \cdot \xi}{\epsilon}] \le \frac{\epsilon^2}{\ell^2}$. Thus, with probability $1-\frac{\epsilon^2}{\ell}$, item $1$ holds for $x \sim \gamma_n$. Likewise, notice that
$$
\Vert \sum_j \alpha_{i,j} DP_tf(y_j)- w_i \Vert_2 \le \tau.
$$
Thus, for any $x_i \sim \gamma_n$, with probability $1- \frac{\epsilon^2}{\ell^2}$, item 2 holds. Thus, by a union bound, it holds for all $1 \le i\le \ell$ simultaneously, with probability $1-\frac{\epsilon^2}{\ell}$. This proves the claim.
\end{proof}
Next, observe that if a point $x_i$ is \emph{good}, then the following holds for every $j'$:
\begin{eqnarray}
\big|\overline{x}_{i,j'} - \langle w_{j'}, x_i \rangle\big| &\leq& \sum_{j} \big| \alpha_{j',j} \cdot \zeta_{i,j} - \alpha_{j',j} \cdot f_{\partial, \xi, t, y_j}(x_i) \big| + \big| \langle w_j', x_i \rangle - \sum_{j} \alpha_{j',j} \cdot f_{\partial, \xi, t, y_j}(x_i) \big| \nonumber \\
&\le& \xi \cdot \sum_{j} |\alpha_{j',j}| + \big| \langle w_j', x_i \rangle - \sum_{j} \alpha_{j',j} \cdot f_{\partial, \xi, t, y_j}(x_i) \big| \nonumber\\ &\le& \frac{\epsilon^2\sqrt{t}}{2 \ell^4 }+ \big| \langle w_j', x_i \rangle - \sum_{j} \alpha_{j',j} \cdot f_{\partial, \xi, t, y_j}(x_i) \big| \nonumber \\
&\le& \frac{\epsilon^2\sqrt{t}}{2 \ell^4 } +
\big| \langle w_j', x_i \rangle - \sum_{j} \alpha_{j',j} \langle DP_tf(y_j), x_i \rangle \big| +
\sum_{j} |\alpha_{j',j}| \cdot \big| \langle DP_tf(y_j), x_i - f_{\partial, \xi, t, y_j}(x_i) \big|
\nonumber \\
&\le& \frac{\epsilon^2\sqrt{t}}{2 \ell^4 } + \frac{\epsilon^2 \sqrt{t}}{100 \cdot \ell^2} + \frac{\epsilon \sqrt{t}}{100 \cdot \ell^2} \le \frac{\epsilon \cdot \sqrt{t}}{\ell^2}. ~\label{eq:good}
\end{eqnarray}
The penultimate inequalities just follow from the condition that $x_i$ is \emph{good} and the values of the parameters. Now, observe that
\begin{eqnarray}
&& \big| \mathbf{E}_{x\sim \gamma_n} [|P_tf(x) - g(\overline{x}_1,\ldots, \overline{x}_\ell)|] - \mathbf{E}_{x\sim \gamma_n} [|P_tf(x) - g(\langle w_1,x \rangle, \ldots, \langle w_\ell, x \rangle)|] \big| \nonumber \\ &\le& \mathbf{E}_{x\sim \gamma_n} [|g(\overline{x}_1,\ldots, \overline{x}_\ell)-g(\langle w_1,x \rangle, \ldots, \langle w_\ell, x \rangle)|]
\end{eqnarray}
Now, observe that by definition, the term inside the expectation is uniformly bounded by $2$. On the other hand, if a point $x$ is good, then by (\ref{eq:good}) and exploiting $g$ is $t^{-1/2}$-Lipschitz, then $|g(\overline{x}_1,\ldots, \overline{x}_\ell)-g(\langle w_1,x \rangle, \ldots, \langle w_\ell, x \rangle)| \le \epsilon$. Since the fraction of good points is at least $1-\frac{\epsilon^2}{\ell}$, we get that for any $g \in \mathsf{Cover}(t, \ell, \delta)$,
$$
\big| \mathbf{E}_{x\sim \gamma_n} [|P_tf(x) - g(\overline{x}_1,\ldots, \overline{x}_\ell)|] - \mathbf{E}_{x\sim \gamma_n} [|P_tf(x) - g(\langle w_1,x \rangle, \ldots, \langle w_\ell, x \rangle)|] \big| \le 2\epsilon.
$$
Now a standard Chernoff bound implies that with for any $g \in \mathsf{Cover}(t, \ell, \delta)$, $\mathcal{O}_g$ is within $\pm \epsilon/2$ $\mathbf{E}[|P_tf(x) - g (\langle w_1, x\rangle, \ldots, \langle w_\ell, x \rangle)|]$ with probability $1- \frac{\epsilon}{10 \cdot |\mathsf{Cover}(t, \ell, \delta)|}$. Thus, by a union bound, with probability $1-\frac{\epsilon}{10}$, for all $g \in \mathsf{Cover}(t, \ell, \delta)$, $\mathcal{O}_g$ is within $\pm \epsilon/2$ of $\mathbf{E}[|P_tf(x) - g (\langle w_1, x\rangle, \ldots, \langle w_\ell, x \rangle)|]$. This finishes the proof.
\end{proof}
We are now ready to prove Theorem~\ref{thm:main-affine-invariant}.
\begin{proofof}{Theorem~\ref{thm:main-affine-invariant}}
Set $t = \frac{\epsilon^4}{900 s^2}$ (this is the same setting as Lemma~\ref{lem:find-dirs} and Lemma~\ref{lem:test-hypothesis}). Observe that with this choice of $t$, since $f$ has surface area bounded by $s$, then by Proposition~\ref{prop:noise-stab-surf-1}, we get that
$$
\mathbf{E}[|P_tf(x) - f(x)|] \le \sqrt{\mathbf{E}[|P_tf(x) - f(x)|^2]} \le \frac{\epsilon}{\sqrt{5}}.
$$
We now run the algorithm \textsf{Find-candidate-directions} with noise parameter $t$, error parameter $\epsilon$ and surface area parameter $s$. We are guaranteed that with probability $1-\epsilon$, we will get $\ell \le k$ directions $y_1, \ldots, y_\ell$ which are $\gamma/2$-linearly independent and $P_tf$ is $\epsilon$-close to a junta on the subspace $\mathsf{span}(v_1, \ldots, v_\ell)$ where $v_i = DP_tf(y_i)$ (call this event $\mathcal{E}_1$). The query complexity of this (from Lemma~\ref{lem:find-dirs}) is $(s \cdot k /\epsilon)^{O(k)}$.
Next, we run the routine \textsf{Estimate-closest-hypothesis} with the directions $y_1, \ldots, y_\ell$, surface area parameter $s$, error parameter $\epsilon$.
Observe that the query complexity of \textsf{Estimate-closest-hypothesis} is also $(s \cdot k /\epsilon)^{O(k)}$. Thus, the total query complexity remains $(s \cdot k /\epsilon)^{O(k)}$.
By guarantee of \textsf{Estimate-closest-hypothesis}, we have the following: there is an orthonormal basis $(w_1, \ldots, w_\ell)$ of $\mathsf{span}(DP_tf(y_1), \ldots, DP_tf(y_\ell))$ such that
\[
\mathbf{E}[|P_tf(x) - g(\langle w_1,x\rangle, \ldots, \langle w_\ell,x\rangle)|] \le \min_{g^\ast \in \mathsf{Cover}(t,\ell,\delta)} \mathbf{E}[|P_tf(x) - g^\ast(\langle w_1,x\rangle, \ldots, \langle w_\ell,x\rangle)|] + 5\epsilon.
\]
However, conditioned on $\mathcal{E}_1$, $P_tf $ is $\epsilon$-close to a junta on $\mathsf{span}(DP_t f(y_1), \ldots, DP_t f(y_\ell))$. By Theorem~\ref{thm:net}, this implies that the quantity $\min_{g^\ast \in \mathsf{Cover}(t,\ell,\delta)} \mathbf{E}[|P_tf(x) - g^\ast(\langle w_1,x\rangle, \ldots, \langle w_\ell,x\rangle)|] \le 3 \epsilon$. This means that if we output the function $g$, then
$\mathbf{E}[|P_tf(x) - g(\langle w_1,x\rangle, \ldots, \langle w_\ell,x\rangle)|] = O(\epsilon)$. {Consider the subspace $V$ spanned by vectors $\{DP_tf(y)\}_{y \in \mathbb{R}^n}$. Note that $\mathsf{dim}(V) \le k$ and $V$ is a relevant subspace for $f$. Thus, $w_1, \ldots, w_\ell$ can be extended to a basis for $V$, finishing the proof.
}
\end{proofof}
{
\begin{remark}~\label{rem:gaussian}
A crucial point about the routine \textsf{Find-invariant-structure}, which will be useful in the next section, is the following: The marginal distribution of all the queries is distributed as the standard $n$-dimensional Gaussian distribution $\gamma_n$. To see this, note that
\begin{enumerate}
\item In the routine \textsf{Find-candidate-directions} , each of the directions $y_i$ is sampled from $\gamma_n$. Further, for $y_i$ and $y_j$ which are i.i.d. samples from $\gamma_n$, the queries made to the oracle for $f$ in computing $\langle DP_tf(y_i), DP_tf(y_j ) \rangle$ are also distributed as $\gamma_n$ (see Lemma~\ref{lem:inner-product-1}).
\item In the routine \textsf{Estimate-closest-hypothesis}, the points $x_i$ are sampled from $\gamma_n$ as are the directions $y_j$ (which are output of \textsf{Find-candidate-directions}). With this, the queries made to the oracle for $f$ for computing $f_{\partial, \xi, t, y_j}(x_i) $ are distributed as $\gamma_n$ (see Lemma~\ref{lem:compute-derivative-x}).
\item One minor subtlety is that while each sampled $y_j$ comes from $\gamma_n$,
as stated, our algorithm \textsf{Find-invariant-structure} is adaptive. Consequently, the above two items do not imply that the marginal distribution of all queries is coming from $\gamma_n$.
The cause of non-adaptivity is that in the routine \textsf{Find-candidate-directions}, while we sample each $y_j$ from $\gamma_n$, subsequently, we only use a subset of the sampled $y_j$'s (namely, the subset $S$). However, we can easily make this algorithm non-adaptive at no asymptotic increase in the sample complexity. This is because the number of candidate directions sampled by the procedure \textsf{Find-candidate-directions} is at most $k \cdot T_{\mathsf{succ}} = \mathsf{poly}(k \cdot s/\epsilon)$. We can run the subsequent routines
namely \textsf{Compute-ortho-transform} and \textsf{Estimate-closest-hypothesis} with all the $y_j$'s instead of just those in set $S$ but only use those which are part of the set $S$ output by \textsf{Find-candidate-directions}. This will only increase the query complexity by a factor of $\mathsf{poly}(k \cdot s/\epsilon)$.
\end{enumerate}
\end{remark}}
\subsubsection*{Finding the linear-invariant structure}
Given the previous theorem it is natural to ask for more, i.e., not just test if the function is a linear-junta but also find the junta in number of queries that depends only on $k$ and $s$ (but not on $n$).
In other words, could we output $g: \mathbb{R}^k \rightarrow \{-1,1\}$ such that there exists a {projection} matrix $A: \mathbb{R}^n \rightarrow \mathbb{R}^k$ and
$f$ is close to $g(A x)$ with query complexity independent of $n$? We give an affirmative answer to this question:
\begin{theorem*}
Let $f:\mathbb{R}^n \rightarrow \{-1,1\}$ be a linear $k$-junta with surface area at most $s$. Then, there is an algorithm \textsf{Find-invariant-structure} which on error parameter $\epsilon>0$, makes $(s \cdot k/\epsilon)^{O(k)}$ queries and outputs $g: \mathbb{R}^k \rightarrow [-1,1]$ so that the following holds: there exists an orthonormal set of vectors $w_1, \ldots, w_k \in \mathbb{R}^n$ such that
$$
\mathbf{E}[|f(x) - g(\langle w_1, x\rangle, \ldots, \langle w_k, x \rangle)|] = O(\epsilon).
$$
Moreover, for some $g^{\ast} : {\mathbb{R}}^k \to {\mathbb{R}}$:
$$
f(x) = g^{\ast}(\langle w_1, x\rangle, \ldots, \langle w_k, x \rangle).
$$
\end{theorem*}
Informally, the theorem states that it is possible to find the ``linear-invariant" structure (i.e., the structure up to unitary transformation) of $f$ in number of queries that dependens on $s$ and $k$. Of course, one cannot hope to output the relevant directions $w_1, \ldots, w_k$ explicitly as even describing these directions will require $\omega(n)$ bits of information and thus, at least those many queries. We note that the number of functions in $k$ dimensions with $O(1)$ surface area (even up to a unitary rotation) is $\exp (\exp (k))$ and thus even our output has to be $\exp(k)$ bits. Thus, it is not possible to significantly improve on our $\exp(k \log k)$ query complexity in finding the linear-invariant structure.
{
\subsubsection*{Testability of linear invariant families of linear $k$-juntas}
Our ability to find the linear-invariant structure of linear $k$-juntas additionally allows us to test subclasses of linear $k$-juntas which are closed under rotation.
\begin{definition}
Let $\mathcal{C}$ be any collection of functions mapping $\mathbb{R}^k$ to $\{-1,1\}$. For any $n \in \mathbb{N}$ let:
\[
\mathsf{Ind}(\mathcal{C})_n= \{f : \exists g \in \mathcal{C} \ \textrm{and orthonormal vectors } w_1, \ldots, w_k \textrm{ such that } f(x) = g(\langle w_1, x\rangle, \ldots, \langle w_k,x \rangle). \}
\]
Define $\mathsf{Ind}(\mathcal{C}) = \cup_{n=k}^\infty \mathsf{Ind}(\mathcal{C})_n$ and call it the \emph{induced class of $\mathcal{C}$}.
\end{definition}
The two key properties of $\mathsf{Ind}(\mathcal{C})$ are (i) each function $f \in \mathsf{Ind}(\mathcal{C})$ is a linear $k$-junta, (ii) the class $\mathsf{Ind}(\mathcal{C})$ is closed under unitary
transformations.
The definition is a continuous analogue of
the so-called ``induced subclass of $k$-dimensional functions" from \cite{gopalan2009testing} (that paper was about testing functions over $\mathsf{GF}^n[2]$).
The following theorem shows that for any $\mathcal{C}$, $\mathsf{Ind}(\mathcal{C})$ is testable without any dependence on the ambient dimension.
\begin{theorem*}
Let $\mathcal{C}$ be a collection of functions mapping $\mathbb{R}^k$ to $\{-1,1\}$. Further, for every $f \in \mathsf{Ind}(\mathcal{C})$, $\mathsf{surf}(f) \le s$. Then, there is an algorithm \textsf{Test-structure-$\mathcal{C}$} which has the following guarantee: Given oracle access to $f: \mathbb{R}^n \rightarrow \{-1,1\}$ and an error parameter $\epsilon>0$, the algorithm makes $(s \cdot k/\epsilon)^{O(k)}$ queries and distinguishes between the cases (i) $f \in \mathsf{Ind}(\mathcal{C})$ and (ii) $f$ is $\epsilon$-far from every function $g \in \mathsf{Ind}(\mathcal{C})$.
\end{theorem*}
A particularly important instantiation of the
above theorem is the following: Let $\mathcal{C}_{B}$ be any collection of functions mapping $\{-1,1\}^k \rightarrow
\{-1,1\}$ and let $\mathcal{C}$ be defined as
\[
\mathcal{C} = \{g: x \mapsto h(\langle w_1, x \rangle - \theta_1, \ldots, \langle w_k, x \rangle - \theta_k) | \ w_1,\ldots, w_k \in \mathbb{R}^k, \ \theta_1, \ldots, \theta_k \in \mathbb{R}, \ h \in \mathcal{C}_B \}.
\]
Note that $\mathcal{C}$ defined above is the set of functions obtained by composing a function from $\mathcal{C}_B$ with $k$-dimensional halfspaces. Consequently, $\mathsf{Ind}(\mathcal{C})$ is the of all functions which can be obtained by composing a function from $\mathcal{C}_B$ with halfspaces. As an example, if $\mathcal{C}_B$ consists of the $\mathsf{AND}$ function on $k$ or fewer bits, then $\mathsf{Ind}(\mathcal{C})$ is the class of ``intersections of $k$-halfspaces". Since the surface area of any Boolean function of $k$-halfspaces is bounded by $O(k)$ it follows that the this class is testable with $(k/\epsilon)^{O(k)}$ queries.
Roughly speaking, the algorithm \textsf{Test-structure-$\mathcal{C}$} works as follows:
we first run the routine \textsf{Test-linear-junta} -- if the target function $f$ passes this test, we are guaranteed that it is (very close to) a linear $k$-junta with surface area $s$. We then run the routine \textsf{Find-invariant-structure}. If the output of this step is $g$, then we can check whether $g$ is close to some function in $\mathsf{Ind}(\mathcal{C})_k$ and accept accordingly. We crucially note here that the last step, namely checking whether $g$ is close to a function in $\mathsf{Ind}(\mathcal{C})_k$ makes no queries to $f$. While the overall intuition of this procedure is obvious, the precise proof is more delicate and is given in Section~\ref{aff:inv}.
}
\begin{comment}
the algorithm \textsf{Test-structure-$\mathcal{C}$} proceeds as follows:
\begin{enumerate}
\item Run the routine \textsf{Test-linear-junta} with rank parameter $k$, surface area parameter $s$ and error parameter $\delta>0$ (where $\delta \approx (\epsilon / (s \cdot k))^{O(k)}$). If the test passes, go to Step~2.
\item Run the routine \textsf{Find-invariant-structure}
with surface area parameter $2s$, rank parameter $k$ and error parameter $\epsilon>0$. Let $g: \mathbb{R}^\ell \rightarrow \{-1,1\}$ be the output of this routine.
\item If the function $g: \mathbb{R}^{\ell} \rightarrow \{-1,1\}$ is $\epsilon$-close to a class $\mathcal{C}$, then accept. Else, reject.
\end{enumerate}
\end{comment}
\subsection{Related Work}
\paragraph{Testing Boolean juntas}
As we have already mentioned, the problem of testing juntas on $\{-1, 1\}^n$
has already been well-studied. For example, it is known~\cite{blais2009testing,Chen:2017:SQC} that $\tilde \Theta(k^{3/2})$
queries are necessary and sufficient for non-adaptively testing $k$-juntas
with respect to the uniform distribution, while $\tilde \Theta(k)$ queries
are necessary and sufficient in the adaptive setting~\cite{blais2012property}.
It even turns out to be possible to test $k$-juntas with respect to an
unknown distribution~\cite{CLSSX18}, although in that setting the non-adaptive
query complexity becomes exponential in $k$.
{{We emphasize that while the problem of junta testing inspires the problems considered in this paper, junta testing algorithms have no bearing on the problem of testing linear juntas
-- e.g., unlike~\cite{CLSSX18}, there is no reason to believe that distribution-free testing of
linear juntas on ${\mathbb{R}}^n$ is even possible, given that the space of probability
measures on ${\mathbb{R}}^n$ is much richer than the space of probability measures on $\{-1, 1\}^n$.}}
\paragraph{Learning juntas of half-spaces.}
{There has been extensive work on {\em learning} intersections and other functions of $k$ half-spaces~\cite{BlumKannan:97, vempala2010random, VX13, KOS:08} .
Note that these algorithms (necessarily) require time polynomial in $n$ (whereas our \emph{raison d'etre is a query complexity independent of $n$}). In particular,
\cite{BlumKannan:97} provided conditions under which intersections of halfspaces can be learnt under the uniform distribution on the ball.
Vempala~\cite{vempala2010random} extended their result to arbitrary log-concave distributions.
In terms of the expressivity of the function class, \cite{VX13} explicitly considered the problem of learning linear $k$-juntas (they called it subspace juntas) and showed that a linear $k$-junta of the form
$g(\langle w_1, x \rangle, \ldots, \langle w_k, x \rangle)$ is learnable in polynomial time if the function $g$ is identified by low moments and robust to small rotations in $\mathbb{R}^n$. Along a related but different axis, \cite{KOS:08} showed that functions of bounded surface area in the Gaussian space are learnable in polynomial time. Finally, we remark that there also has been work in learning intersections and other functions of halfspaces over the Boolean hypercube as well~\cite{KOS:02, gopalan2012learning}.
}
\ignore{learned some functions $g$ of $k$ half-spaces in polynomial time if the functions $g$ are identified by low moments and robust to small rotations in ${\mathbb{R}}^n$, while \cite{KOS:08} learned functions of bounded Gaussian surface area.}
\paragraph{Linearly Invariant Testing over Finite Fields}
We note that the set of linear-juntas is linearly invariant. If $f$ is a linear $k$-junta and $B$ is any $n \times n$ matrix then $x \mapsto f(Bx)$ is also a linear $k$-junta.
Over finite fields, \cite{KaufmanSudan:08} studied general criteria for when a linearly invariant property is testable, see also
\cite{bhattacharyya2013}. In particular, \cite{gopalan2009testing}, gave a $2^{O(k)}$ query complexity algorithm to test linear juntas over finite fields. Moreover, they also show that an exponential lower bound on $k$ is necessary.
This should be contrasted with our result which shows that linear juntas over the Gaussian space can be tested with $\mathsf{poly}(k)$ queries.
{\paragraph{Testing (functions) of halfspaces}
The question of testing halfspaces was first considered in \cite{MORS:10} who showed that in the Gaussian space (as well as the Boolean space), halfspaces are testable with $O(1)$ queries. Subsequently, the second and third authors (Mossel and Neeman~\cite{mossel2015robust}) gave a different testing algorithm for a single halfspace in the Gaussian space. In fact, Harms~\cite{harms19} recently showed that halfspaces over any rotationally invariant distribution can be tested with sublinear number of queries.
However, as far as we are aware, prior to our work, no non-trivial bounds were known for even testing the intersection of two halfspaces. As remarked earlier, from our work, it follows that for any arbitrary $k$, intersection of $k$-halfspaces can be tested in the Gaussian space with $\exp(k \log k)$ queries.}\\
\subsection{Techniques}
A major difference between linear juntas over finite fields and linear juntas over Gaussian space is the ``infinitesimal geometry" that can be used in the latter and does not exist in the former.
In particular, the linear part $\mathcal{W}_1(f)$
of the Hermite expansion of $f$ is approximately given by
$e^{-t} (P_t f - \operatorname{{\bf E}}[f])$ for large $t$. Here $P_t f$ is the Ornstein-Uhlenbeck operator.
Both the quantities, $\operatorname{{\bf E}}[f]$ and $P_t f$ can be approximated by sampling a small number of points from the Gaussian distribution and evaluating $f$ at those points.
Moreover, if $f(x) = g(\langle u_1, x \rangle, \ldots, \langle u_k, x \rangle)$ is a linear junta, then the linear part of its Hermite expansion, $\mathcal{W}_1(f)$, lies in the span of $u_1,\ldots,u_k$.
We would like to obtain ``many more directions" that lie in the span of $u_1,\ldots,u_k$.
We do so by considering functions of the form $f_{t,y}(x) = f(e^{-t} y + \sqrt{1-e^{-2 t}} x)$, for randomly chosen $y$ and an appropriate value of $t$
(the experts will recognize $f_{t,y}$ as part of the definition of the Ornstein-Uhlenbeck operator). Note that $f_{t,y}$ is also a linear junta defined by the same direction $u_1,\ldots,u_k$ and therefore the linear part of the Hermite expansion of $f_{t,y}$, is also in the span of $u_1,\ldots,u_k$.
It is now natural to propose the following algorithm to test if a function is a linear $k$-junta: choose points $y_i$ at random and ``compute''
$\mathcal{W}_1(f_{t,y_i})$ at these points. Then if the rank of the matrix spanned by
$(\mathcal{W}_1(f_{t,y_i}))_i$ is at most $k$, then output YES; otherwise, output NO.
Of course, actually computing $\mathcal{W}_1(f_{t,y})$ requires $\mathrm{poly}(n) \gg \mathrm{poly}(k)$ samples.
Instead we will approximately compute the Gram matrix
\[
A_{i,j} = \langle \mathcal{W}_1(f_{t,y_i}), \mathcal{W}_1(f_{t,y_j}) \rangle.
\]
and test if it is close or far from a matrix of rank $k$.
One advantage of using the Gram matrix, is that we can evaluate the entries $A_{i,j}$ by sampling random inputs to evaluate the expected values
\[
\operatorname{{\bf E}}[\mathcal{W}_1(f_{t,y_i})(x) \mathcal{W}_1(f_{t,y_j})(x)].
\]
How do we know that $\mathcal{W}_1(f_{t,y_i})(x)$ are not very close to $0$?
If $f$ has a bounded surface area then $f$ is close to the noise stable function $P_t f$. For such noise stable functions, we prove that with good probability at a random point $x$,
$\mathcal{W}_1(f_{t,y_i})(x)$ will be of non-negligible size. In fact, one of our main technical lemmas (Lemma \ref{lem:subspace-escape}) proves much more. It shows that if $f$ is $\epsilon$ far from any linear-$k$-junta then for any subspace $W$ with co-dimension at most $k$, it holds that for a random $y$ with probability at least $\mathrm{poly}(\epsilon)$, the projection of $\mathcal{W}_1(f_{t,y_i})(x)$ into $W$ will have norm at least
$\mathrm{poly}(\epsilon)$. This result is later combined with a perturbation argument to establish to show that if $f$ is $\epsilon$-far from a linear $k$-junta then indeed the Gram matrix will have $k+1$ large eigenvalues.
Since our analysis relies on the function $f$ having surface area at most $s$, the first stage of the algorithm uses the algorithm by the third author \cite{neeman2014testing} to test if the function of interest is of bounded surface area.
The algorithm to identify the linear invariant structure of $f$ builds up on the ideas in the algorithm to test linear $k$-juntas. More precisely, we can show that if $f$ is a linear $k$-junta with surface area $s$,
\begin{enumerate}
\item we can find directions $y_1,\ldots, y_\ell$ such that
$f$ is close to a function on the space spanned by the directions $\mathcal{W}_1(f_{t,y_1}),\ldots,\mathcal{W}_1(f_{t,y_\ell})$ (for some $\ell \le k$).
\item While we cannot find $\mathcal{W}_1(f_{t,y_j})$ explicitly for any $j$, we can evaluate $\langle \mathcal{W}_1(f_{t,y_j}), x\rangle$ at any point $x$ up to good accuracy.
\item With the above observation, the high level idea is to \emph{try out all smooth functions} on the subspace spanned by $\{\langle\mathcal{W}_1(f_{t,y_1}), x\rangle ,\ldots,\langle \mathcal{W}_1(f_{t,y_\ell}),x \rangle\}$. Perform \emph{hypothesis testing} for each such function against $f$ and output the most accurate one.
\end{enumerate}
The crucial part in the above argument is that even if we have $\mathcal{W}_1(f_{t,y_1}),\ldots,\mathcal{W}_1(f_{t,y_\ell})$ implicitly, the space of ``all smooth functions" on $\mathsf{span}(\langle\mathcal{W}_1(f_{t,y_1}), x\rangle ,\ldots,\langle \mathcal{W}_1(f_{t,y_\ell}),x \rangle)$ has a cover whose size is independent of $n$. This lets us identify the linear invariant function defining $f$ with query complexity just dependent on $k$ and $s$.
In order to prove lower bounds in terms of surface area, we construct a distribution over linear $1$-juntas
with large surface area by splitting ${\mathbb{R}}^2$ into many very thin parallel strips (oriented in a random direction)
and assign our function a random $\pm 1$ value on each strip. (Note that the surface area of such a function
is proportional to the number of strips.) The intuition is
that no algorithm that makes non-adaptive queries can tell that such a random
function is a 1-junta, because in order to ``see'' one of these strips, the
algorithm would need to have queried multiple far-away points in a single
strip. But if the number of queries is small relative to the number of strips
then this is impossible -- with high probability every pair of far-away query points
will end up in different strips.
In order to make this intuition rigorous, we also introduce a distribution on linear $2$-juntas
by randomly ``cutting'' the thin strips once in the orthogonal direction. We show that
for any non-adaptive set of queries, the two distributions induce almost identical query distributions,
and Yao's minimax lemma implies that no algorithm can distinguish between our random $1$-juntas and our
random $2$-juntas.
\section{Some useful results from linear algebra}
The next lemma states for any $v_1, \ldots, v_\ell$ which are $(\eta,\gamma)$-linearly independent, we can
find a set of vectors $(w_1, \ldots, w_\ell)$ (expressed as linear combination of $(w_1, \ldots, w_\ell)$) which is close to being an orthonormal basis of the $\mathsf{span}(v_1, \ldots, v_\ell)$ provided we have sufficiently good approximations of $\{\langle v_i, v_j \rangle\}_{1 \le i,j \le \ell}$. Now, modulo the \emph{quantitative estimates}, this is essentially just a consequence of a procedure such as the Gram-Schmidt orthogonalization. However, the complexity of our testing algorithm is dependent on the quantitative estimates, so we work out the linear algebra here.
\begin{lemma}~\label{prop:linear}
Let $v_1, \ldots, v_\ell$ be a $(\eta, \gamma)$-linearly independent vectors. Then, for any error parameter $\nu>0$ and $\lambda =\lambda(\ell,\nu,\eta, \gamma)$ defined as
$$
\lambda= 2 \frac{\nu}{\ell^2 \cdot \eta} \cdot \big( \frac{\gamma}{2\cdot \ell \cdot \eta}\big)^{3\ell+3},
$$ given numbers $\{\beta_{i,j}\}_{1\le i,j \le \ell}$ such that $|\beta_{i,j} - \langle v_i, v_j \rangle| \le \lambda$, we can compute numbers $\{\alpha_{i,j}\}_{1\le i,j \le \ell}$ such that:
\begin{enumerate}
\item For $\xi (\ell, \eta, \gamma)$ defined as
\[
\xi ( \ell, \eta, \gamma) =\sqrt{2\ell} \cdot \bigg(\frac{2 \ell \cdot \eta}{\gamma} \bigg)^{\ell+1},
\]
we have
$|\alpha_{i,j}| \le \xi ( \ell,\eta, \gamma)$.
\item There is an orthonormal basis $(w_1, \ldots, w_\ell)$ of $\mathsf{span}(v_1, \ldots, v_\ell)$ such that for $\Vert w_{i} - \sum_{j}\alpha_{i,j} v_j \Vert_2 \le \nu$.
\end{enumerate}
\end{lemma}
\begin{proof}
Consider the symmetric matrix $\Sigma\in\mathbb{R}^{\ell \times \ell}$ defined as $\Sigma_{i,j} = \langle v_i, v_j \rangle$. By Proposition~\ref{prop:sing-1}, $\Sigma$ is non-singular. Define the matrix $\Gamma = \Sigma^{-1/2}$. It is easy to see that the columns of
$V \cdot \Sigma^{-1/2}$ form an orthonormal basis of $\mathsf{span}(v_1, \ldots, v_\ell)$. Here $V = [v_1 | \ldots | v_\ell]$. Of course, we cannot compute the matrix $\Sigma$ exactly and consequently, we cannot compute the matrix $\Sigma^{-1/2}$ either. Instead, if we define the matrix $\widetilde{\Sigma}$ as $\widetilde{\Sigma}(i,j) = \beta_{i,j}$, then observe that $\widetilde{\Sigma}$ is symmetric. Next, observe that Proposition~\ref{prop:sing-1}, we have that
$$
\sigma_{\min}(\Sigma) = \sigma_{\min}^2(V) \geq \bigg(\frac{\gamma}{2 \cdot \ell \cdot \eta} \bigg)^{2 \ell +2}.
$$
Define a parameter $\rho$ as
$$
\rho = \frac{2 \nu}{\ell \cdot \eta} \cdot \bigg( \frac{\gamma}{2 \cdot \ell \cdot \eta}\bigg)^{\ell+1}.
$$
Now, with this setting, observe that
$$
\ell \cdot \lambda = \rho \cdot \bigg( \frac{\gamma}{2 \cdot \ell \cdot \eta} \bigg)^{2\ell +2} \le \rho \cdot \sigma_{\min} (\Sigma).
$$
Further, since entrywise, $\Sigma$ and $\widetilde{\Sigma}$ differ by at most $\lambda$, hence $\Vert \widetilde{\Sigma} - \Sigma \Vert_F\le \ell \cdot \lambda$. First, by Weyl's inequality (Lemma~\ref{lem:Weyl}), we have that
\begin{equation}~\label{eq:sigma-min}
\sigma_{\min}(\widetilde{\Sigma}) \ge \sigma_{\min}({\Sigma})- \Vert \Sigma-\widetilde{\Sigma} \Vert_F \ge (1- \rho) \cdot \sigma_{\min}(\Sigma).
\end{equation}
Thus, $\widetilde{\Sigma}$ is also psd.
Now, we apply the matrix perturbation bound to matrices $\Sigma$ and $\widetilde{\Sigma}$ (Corollary~\ref{corr:mat-perturb} with parameter $c = \big( \frac{\gamma}{2\cdot \ell \cdot \eta}\big)^{2\ell+2}$)
to obtain that
\[
\Vert \Sigma^{-1/2} - \widetilde{\Sigma}^{-1/2} \Vert \leq \frac{\rho}{2 \big( \frac{\gamma}{2\cdot \ell \cdot \eta}\big)^{\ell+1}} = \frac{2\nu}{\ell \cdot \eta}.
\]
We now define $\alpha_{i,j} = \widetilde{\Sigma}^{-\frac12}(j,i)$. We also define $\beta_{i,j} = {\Sigma}^{-\frac12}(i,j)$. Note that the vectors $w_i = \sum_{i} \beta_{j,i} v_j$ forms an orthonormal basis. As the matrices $\Sigma^{-\frac12}$ and $\widetilde{\Sigma}^{-\frac12}$ are $\frac{2\nu}{\ell \cdot \eta}$ close in operator norm, this immediately implies item 2. To get item 1, we recall the following basic inequality for Frobenius norm of an inverse matrix. In particular, for a symmetric matrix $A \in \mathbb{R}^{\ell \times \ell}$,
$\sigma_{\min}(A) \cdot \Vert A^{-1} \Vert_F \le \sqrt{\ell}$. Thus,
\[
\Vert \widetilde{\Sigma}^{-1/2} \Vert_F \le \frac{\sqrt{\ell}}{\sigma_{\min}(\widetilde{\Sigma}^{1/2})} = \sqrt{\frac{\ell}{\sigma_{\min}(\widetilde{\Sigma})}} \le
\sqrt{2\ell} \cdot \bigg(\frac{2 \ell \cdot \eta}{\gamma} \bigg)^{\ell+1}.
\]
The last inequality uses (\ref{eq:sigma-min}) and the fact that $\rho \le \frac12$. This immediately implies
the first item.
\end{proof}
\begin{proposition}~\label{prop:sing-1}
Let $v_1, \ldots, v_\ell$ be a $(\eta, \gamma)$-linearly independent vectors. Let $V = [v_1 | \ldots | v_\ell]$. Then, the smallest singular value of $V$
is at least $(\frac{ \gamma}{2 \cdot \ell \cdot \eta})^{\ell+1}$.
\end{proposition}
\begin{proof}
Let us set a parameter $\rho - \frac{\gamma}{2\ell \eta}$. Recall that if $\sigma_{\min}(V)$ is the smallest singular value of $V$, then
\[
\sigma_{\min}(V) = \inf_{x : \Vert x \Vert_2=1} \Vert V \cdot x \Vert_2
\]
Let us try to lower bound the right hand side. To do this, let $x \in \mathbb{R}^n$ be any unit vector and note that $V \cdot x = \sum_{1 \le i \le \ell} v_i \cdot x_i$.
Now, let $j$ be the largest coordinate such that $|x_j| \ge \rho^{j}$ (note that there has to be such a $j$ since $x$ is a unit vector and $\rho<1/2$). Define $w = \sum_{i \le j} v_i x_i$. Then, observe that its component in the direction orthogonal to the span of $\{v_1, \ldots, v_{j-1}\}$ is at least $\gamma \cdot \rho^j$ in magnitude. On the other hand, $\Vert \sum_{i > j} v_i x_i \Vert_2 \le \rho^{j+1} \cdot \ell \cdot \eta$. By triangle inequality, we obtain that
\[
\Vert \sum_{i} v_i x_i \Vert_2 \ge \Vert \sum_{i \le j} v_i x_i \Vert_2 - \Vert \sum_{i > j} v_i x_i \Vert_2 \ge \gamma \cdot \rho^j - \ell \cdot \eta \cdot \rho^{j+1} \ge \frac{\gamma \cdot \rho^j}{2}.
\]
The last inequality uses the value of $\rho$. This finishes the proof.
\end{proof}
\subsubsection{Oracle computation}
We now list several useful claims which all fit the same motif: Given oracle access to $f: \mathbb{R}^n \rightarrow \mathbb{R}$, what \emph{interesting} quantities can be computed?
\begin{lemma}~\label{lem:oracle-access-1}
Given oracle access to $f:\mathbb{R}^n \rightarrow [-1,1]$, error parameter $\eta>0$, there is a function $f_{\partial,\eta}: \mathbb{R}^{n} \rightarrow \mathbb{R}$ such that the following holds for every $\lambda \ge 1$,
\[
\mathop{\Pr}_{x \sim \gamma_n} \big[ \big|f_{\partial,\eta}(x) - \widehat{f}_1(x) \big| > \lambda \cdot \eta \big] \le \lambda^{-2}.
\]
Further, for any $x \in \mathbb{R}^n$, we can compute $f_{\partial,\eta}(x)$ to additive error $\pm \epsilon$ with confidence $1-\delta$ by making $\mathsf{poly}(1/\eta, 1/\epsilon, \log (1/\delta))$ queries to the oracle for $f$.
\end{lemma}
\begin{proof}
Observe that for any $t>0$, $P_t f = \sum_{q \ge 0} e^{-tq} \widehat{f}_q(x)$. This implies that
\[
\frac{P_t f - \mathbf{E}[f]}{e^{-t}} = \widehat{f}_1(x) + \sum_{q>1}e^{-t(q-1)} \widehat{f}_q(x).
\]
Set $t$ so that $e^{-t} = \eta$ and let us define $f_{\partial,\eta}$ as
$
f_{\partial,\eta} = \frac{P_t f - \mathbf{E}[f]}{e^{-t}}.
$ Now, observe that for $h(x)=\sum_{q>1}e^{-t(q-1)} \widehat{f}_q(x)$, $\mathbf{E}[h(x)]=0$ and $\mathsf{Var}[h(x)] \le \eta^2$. We now apply Chebyshev's inequality to obtain
\[
\mathop{\Pr}_{x \sim \gamma_n} \big[ \big|f_{\partial,\eta}(x) - \widehat{f}_1(x) \big| > \lambda \cdot \eta \big] \le \lambda^{-2}.
\]
Next, observe that both $P_tf(x)$ and $\mathbf{E}[f(x)]$ can be computed to error $\pm \epsilon \cdot \eta$ with confidence $1-\frac{\delta}{2}$ using $\mathsf{poly}(1/\eta, 1/\epsilon, \log (1/\delta))$ queries to the oracle for $f$. This immediately implies that $f_{\partial,\eta}$ can be computed to error $\pm \epsilon$ using $\mathsf{poly}(1/\eta, 1/\epsilon, \log (1/\delta))$ queries to the oracle for $f$.
\end{proof}
\begin{lemma}~\label{lem:inner-products}
Given oracle access to functions $f,g : \mathbb{R}^n \rightarrow [-1,1]$, error parameter $\epsilon >0$ and confidence parameter $\delta>0$, there is an algorithm which makes $\mathsf{poly}(1/\epsilon,\log(1/\delta))$ queries to $f,g$ and computes $\langle \widehat{f}_1, \widehat{g}_1 \rangle$ up to error $\epsilon$ with confidence $1-\delta$.
\end{lemma}
\begin{proof}
Consider the function
\[
h(x) = e^{2t} (P_t f(x) - \operatorname{{\bf E}}[f]) (P_t g(x) - \operatorname{{\bf E}}[g]).
\]
Writing out the Fourier expansions of $P_t f$ and $P_t g$, note that $P_t f = \sum_{q \ge 0} e^{-tq} \widehat f_q(x)$,
and so
\[
h(x) = \widehat f_1(x) \widehat g_1(x) + \sum_{\substack{q, r \ge 1 \\ q + r \ge 3}} e^{-t(q + r - 2)} \widehat f_q(x) \widehat g_r(x).
\]
Since $\widehat f_1$ and $\widehat g_1$ are linear functions, $\operatorname{{\bf E}}[\widehat f_1(x) \widehat g_1(x)] = \widehat f_1 \cdot \widehat g_1$. On the other hand, $\operatorname{{\bf E}}[\sum_{q \ge 0} \widehat f_q^2(x)] = \operatorname{{\bf E}}[f^2] \le 1$, and so the Cauchy-Schwarz inequality implies that
\[
\operatorname{{\bf E}}\Big[\sum_{\substack{q, r \ge 1 \\ q + r \ge 3}} e^{-t(q + r - 2)} \widehat f_q(x) \widehat g_r(x)\Big]
\le e^{-t}.
\]
Hence, $|\operatorname{{\bf E}}[h(x)] - \hat f_1 \cdot \hat g_1| \le e^{-t}$. If we choose $t$ so that $e^{-t} = \epsilon/2$, then it
only remains to show that we can estimate $\operatorname{{\bf E}}[h(x)]$ within additive error $\epsilon/2$ with confidence $1 - \delta$.
Let $y$ and $z$ be Gaussian random variables, independent of $x$, and write $P_t f(x) = \operatorname{{\bf E}}_y[f(e^{-t} x + \sqrt{1-e^{-2t}} y)]$ and $P_t g(x) = \operatorname{{\bf E}}_z[g(e^{-t} x + \sqrt{1-e^{-2t}} z)]$. In particular, we can express $\operatorname{{\bf E}}[h(x)]$ in the form $\operatorname{{\bf E}}[J(x, y, z)]$ where
\[
J(x,y,z) = e^{2t} (f(e^{-t} x + \sqrt{1-e^{-2t}} y) - f(y)) (g(e^{-t} x + \sqrt{1-e^{-2t}}z) - g(z)).
\]
Recalling that $e^{-2t} = 4/\epsilon^2$, it follows that $J$ takes values in $[-4/\epsilon^2, 4/\epsilon^2]$,
and it follows from Hoeffding's inequality that we can approximate $J$ to additive error $\epsilon/2$
with confidence $1 - \delta$ using $\mathsf{poly}(1/\epsilon, \log(1/\delta))$ samples of $J$. Moreover,
each sample of $J$ can be computed using two oracle queries to $f$ and two oracle queries to $g$.
\end{proof}
\begin{definition}
A function $f: \mathbb{R}^n \rightarrow \mathbb{R}$
is said to be a linear $k$-junta if there are at most $k$ orthonormal vectors $u_1, \ldots, u_k \in \mathbb{R}^n$ and a function $g: \mathbb{R}^k \rightarrow \mathbb{R}$ such that
\[
f(x) = g(\inp{u_1}{x}, \ldots, \inp{u_k}{x}).
\]
Further, if $u_1, \ldots, u_k \in W$ (a linear subspace of $\mathbb{R}^n$), then $f$ is said to be a $W$-junta.
\end{definition}
\subsection{Derivatives of functions}
{We will use $D$ to denote the derivative operator. In case, there are two sets of variables involved, we will explicitly indicate the variable with respect to which we are taking the derivative.}
\begin{definition}
For $f: \mathbb{R}^n \rightarrow \mathbb{R}$ ($f \in \mathcal{C}^{\infty}$) and $t \ge 0$, define the function $f_t: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$,
\[
f_t(y,x) = f(e^{-t} y + \sqrt{1-e^{-2t}} x).
\]
Further, in the same setting as above, we let
$f_{t,y}: \mathbb{R}^n \rightarrow \mathbb{R}$,
\[
f_{t,y}(x) = f(e^{-t} y + \sqrt{1-e^{-2t}} x).
\]
\end{definition}
Let $D_x$ denote the derivative operator with respect to $x$ and let $D_y$ denote the derivative operator with respect to $y$. Then, it is easy to observe that
\begin{equation}~\label{eq:derivative-y-1}
\sqrt{e^{2t}-1} \cdot D_y f_{t}(y,x) = D_x f_{t}(y,x).
\end{equation}
Next, for a function $g: \mathbb{R}^n \rightarrow \mathbb{R}$, define $\mathcal{W}_{1}(g) \in \mathbb{R}^n$ as the degree-$1$ Hermite coefficients of $g$. In other words, the $i^{th}$ coordinate of $\mathcal{W}_1(g)$
\[
\mathcal{W}_{1}(g)[i] = \mathbf{E}[g(x) \cdot x_i],
\]
where $x \sim \gamma_n$, the standard $n$-dimensional Gaussian measure. With respect to our earlier definition of $\widehat{g}_1$, observe that we have:
$
\widehat{g}_1(x) = \inp{\mathcal{W}_{1}(g)}{x}.
$
We next prove the following important lemma which connects the gradient of $P_t f$ at $y$ with $\mathcal{W}_{1}(f_{t,y})$. In particular, we have the following lemma.
\begin{lemma}~\label{lem:derivative-shift}
\[
\mathcal{W}_{1}(f_{t,y})= \sqrt{e^{2t}-1} \cdot D (P_t f)(y).
\]
\end{lemma}
\begin{proof}
First of all, observe that for any function $g: \mathbb{R}^n \rightarrow \mathbb{R}$ with bounded derivatives, and for any $i \in [n]$
\begin{eqnarray*}
\mathbf{E}_{x \sim \gamma_n} \bigg[ \frac{\partial g(x)}{\partial x_i} \bigg] = \int_{x} \frac{\partial g(x)}{\partial x_i}\gamma_n(x) dx = \int_{x} x_i g(x) \gamma_n(x) dx = \mathbf{E}_{x \sim \gamma_n} [x_i \cdot g(x)].
\end{eqnarray*}
While the first and last equalities are trivial, the middle is a consequence of integration by parts. Assuming that
$f$ has bounded derivatives, we may apply this identity to $g = f_{t,y}$, yielding
\begin{eqnarray*}
\mathcal{W}_1(f_{t,y}) &=& \mathbf{E}_x [D_xf_{t}(y,x)]
\\ &=& \sqrt{e^{2t}-1} \cdot \mathbf{E}_x [D_y f_{t}(y,x)] \ \ \textrm{(applying (~\ref{eq:derivative-y-1}))}
\\
&=& \sqrt{e^{2t}-1} \cdot D_y (\mathbf{E}_x [f_{t}(y,x)]) = \sqrt{e^{2t}-1} \cdot D_y (P_t f)(y).
\end{eqnarray*}
This proves the lemma in the case that $f$ has bounded derivatives. In the general case, we approximate
choose a sequence of functions that have bounded derivatives and approximate $f_{t,y}$ in $L_2(\gamma)$.
Applying the lemma to these functions and taking the limit proves the general case.
\end{proof}
\begin{lemma}~\label{lem:inner-product-1}
Given oracle access to $f$, noise parameter $t>0$, error parameter $\epsilon>0$, confidence parameter $\delta>0$ and $y_1, y_2 \in \mathbb{R}^n$, there is an algorithm which makes $\mathsf{poly}(1/\epsilon, 1/\delta, 1/t)$ queries to $f$ and computes $\langle D(P_t f)(y_1),D(P_t f)(y_2)\rangle$ up to error $\epsilon$ with confidence $1-\delta$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:derivative-shift}, we have
\[
\langle D(P_t f)(y_1),D(P_t f)(y_2)\rangle = \frac{1}{e^{2t}-1} \cdot \langle \mathcal{W}_1(f_{t,y_1}), \mathcal{W}_1(f_{t,y_2})\rangle.
\]
We can now apply Lemma~\ref{lem:inner-products} to finish the proof.
\end{proof}
\begin{proposition}~\label{prop:derivative-bound}
For any $f: \mathbb{R}^n \rightarrow [-1,1]$, $\Vert D(P_t f)(y) \Vert_2 \le (e^{2t} -1)^{-\frac12}$.
\end{proposition}
\begin{proof}
By Lemma~\ref{lem:derivative-shift}, we have $\Vert \mathcal{W}_1(f_{t,y}) \Vert_2 = \sqrt{e^{2t}-1} \cdot \Vert D(P_t f)(y) \Vert_2$. Now, observe that the range of $f_{t,y}$ is $[-1,1]$ and thus, $\Vert \mathcal{W}_1(f_{t,y}) \Vert_2\le 1$, implying the stated upper bound.
\end{proof}
\begin{lemma}~\label{lem:compute-derivative-x}
Given oracle access to $f: \mathbb{R}^n \rightarrow [-1,1]$, $y \in \mathbb{R}^n$, noise parameter $t>0$,
error parameter $\eta>0$, there is a function
$f_{\partial,\eta,t,y}: \mathbb{R}^n \rightarrow \mathbb{R}$ such that the following holds for every $\lambda \ge 1$,
\[
\Pr_{x \sim \gamma_n} [|f_{\partial,\eta,t,y}(x) - \langle D(P_t f)(y),x\rangle | > \lambda \cdot \eta] \le \lambda^{-2}.
\]
Further, for an error parameter $\epsilon>0$, confidence parameter $\delta>0$, we can compute
$f_{ \partial, t,\eta,y}$ to additive error $\pm \epsilon$ with confidence $1-\delta$ using $\mathsf{poly}(1/t, 1/\eta, 1/\epsilon, \log(1/\delta))$ queries to $f$.
\end{lemma}
\begin{proof}
We first use Lemma~\ref{lem:derivative-shift} and obtain that
$$
D_{}P_tf(y) = \frac{1}{\sqrt{e^{2t}-1}} \cdot \mathcal{W}_1(f_{t,y}).
$$
Consequently, we have that
\[
\langle DP_tf(y), x\rangle = \frac{1}{\sqrt{e^{2t}-1}} \cdot \widehat{f_{t,y}}_1(x).
\]
The claim now follows from Lemma~\ref{lem:oracle-access-1}.
\end{proof}
\subsection{Some useful inequalities concerning noise stability}
\begin{lemma}~\label{lem:Poincare} \textbf{[Poincar\'{e} inequality]} Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $\mathcal{C}^1$ function. Then, $\mathsf{Var}[f] \le \mathbf{E}[\Vert Df \Vert_2^2]$.
\end{lemma}
\begin{definition}~\label{def:surface-area}
For a Borel set $A \subseteq \mathbb{R}^n$, we define its Gaussian surface area $\Gamma(A)$ to be
\[
\Gamma(A) = \liminf_{\delta \rightarrow 0} \frac{\mathsf{vol}(A_{\delta} \setminus A)}{\delta},
\]
provided the limit exists. Here, for any body $K$, $\mathsf{vol}(K)$ denotes the Gaussian volume of $K$, i.e., $\int_{x \in K} \gamma_n(x) dx$. Further, $A_{\delta} = \{x : d(x,A) \le \delta\}$ where $d(x,A)$ denotes the Euclidean distance of $x$ from $A$.
For a function $f : \mathbb{R}^n \rightarrow \{-1,1\}$, we denote its surface area $\Gamma(f) = \Gamma(A_f)$ where $A_f = \{x: f(x)=1\}$.
\end{definition}
Ledoux~\cite{Ledoux:94} (and implicitly Pisier~\cite{Pisier:86}) proved the following connection between noise sensitivity and surface area of functions.
\begin{lemma}~\label{lem:Ledoux}[Ledoux~\cite{Ledoux:94}]
For any $t \ge 0$ and $f : \mathbb{R}^n \rightarrow \{-1,1\}$, $x,y \sim \gamma_n$, we have \[
\Pr_{x,y} [f(x) \not = f(e^{-t} x+ \sqrt{1-e^{-2t}} y)] \le \frac{2\sqrt{t}}{\sqrt{\pi}} \cdot \Gamma(f)\]
\end{lemma}
The following proposition is an immediate consequence of the above lemma.
\begin{proposition}~\label{prop:Ledoux}
Let $f: \mathbb{R}^n \rightarrow \{-1,1\}$, $t \ge 0$ and $\Gamma(f) \le s$. Then,
\begin{enumerate}
\item $\mathbf{E}[(f(x) - P_tf (x))^2] = 8 s \sqrt{t}$.
\item For any $\epsilon >0$ and $T = O(s^2/\epsilon^2)$, $\sum_{q \ge T} \mathbf{E}[\widehat{f}_q^2] \le \epsilon$.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $\mathcal{E}_1(x,y)$ denote the event that
$f(x) \not = f(e^{-t} x+ \sqrt{1-e^{-2t}} y)$.
To prove the first item, observe that for any $x$,
\[
(f(x) - P_t f(x))^2 = (2\mathop{\mathbf{E}}_{y \sim \gamma_n}[\mathbf{1}(\mathcal{E}_1(x,y))])^2 =4 \big(\mathop{\mathbf{E}}_{y \sim \gamma_n}[\mathbf{1}(\mathcal{E}_1(x,y))]\big)^2 \le 4 \big(\mathop{\mathbf{E}}_{y \sim \gamma_n}[\mathbf{1}(\mathcal{E}_1(x,y))]\big)
\]
Thus, we obtain that $$\mathbf{E}[(f(x) - P_tf(x))^2] \le 4\mathop{\mathbf{E}}_{x, y \sim \gamma_n}[\mathbf{1}(\mathcal{E}_1(x,y))] \le 8 s\sqrt{t}, $$
where the last inequality is an application of Lemma~\ref{lem:Ledoux}. The second item here is the same as Theorem~15 (full version) of ~\cite{KOS:08}. So, we do not prove it here.
\end{proof}
\subsection{Inequalities for matrix perturbation}
We will require some basic results on matrix perturbations. For this, we adopt the following notation: Let $A \in \mathbb{C}^{n \times n}$ be a Hermitian matrix. Then $\sigma_1(A) \ge \ldots \ge \sigma_n(A)$ denote its singular values in order.
\begin{lemma}~\label{lem:Weyl}[Weyl's inequality]
Let $A, E \in \mathbb{R}^{n \times n}$ be real symmetric matrices.
Then for any $j$, $$|\sigma_j(A+E) - \sigma_j(A)| \le \Vert E \Vert_F. $$
\end{lemma}
\begin{fact}~\label{lem:mat-1-bound}~\cite{schmitt1992perturbation}
Let $A_1, A_2$ be two psd matrices. Let $\sigma_{\min}(A_1), \sigma_{\min}(A_2) \ge c$. Then,
\[
\Vert A_2^{1/2} - A_1^{1/2} \Vert_2 \le \Vert A_2 - A_1 \Vert_2 \cdot \frac{1}{2 \sqrt{c}}.
\]
\end{fact}
\begin{fact}~\label{lem:mat-2-bound}\cite{stewart1973introduction}
Let $A_1, A_2$ be two psd matrices. Let $\sigma_{\min}(A_1)\ge c$ and $\Vert A_2 - A_1 \Vert_2 \le c/100$. Then,
\[
\Vert A_2^{-1} - A_1^{-1} \Vert_2 \le \Vert A_2 - A_1 \Vert_2 \cdot \frac{1}{ c^2}.
\]
\end{fact}
Combining these two facts, we have the following corollary.
\begin{corollary}~\label{corr:mat-perturb}
Let $0<c<1$ and let $A_1$ be a psd matrix such that $\sigma_{\min}(A_1) \ge c$. Let $A_2 - A_1$ be real symmetric that $\Vert A_2 - A_1 \Vert_2 \le \xi \cdot c$ for $|\xi| \le 1/100$. Then,
$\Vert A_{1}^{-1/2} - A_2^{-1/2} \Vert_2 \le \frac{\xi}{2\sqrt{c}}$.
\end{corollary}
\begin{proof}
We first apply Fact~\ref{lem:mat-1-bound} to obtain that $$
\Vert A_2^{1/2} - A_1^{1/2} \Vert_2 \le \frac{\xi c^{1/2}}{2}.
$$
Observe that $\sigma_{\min}(A_1^{1/2}) \ge \sqrt{c}$. Since
$c<1$ and $|\xi| \le \frac{1}{100}$, we apply
Fact~\ref{lem:mat-2-bound} to obtain that
$$
\Vert A_2^{-1/2} - A_1^{-1/2} \Vert_2 \le \frac{\xi}{2\sqrt{c}}.
$$
This finishes the proof.
\end{proof}
\section{A lower bound in terms of surface area} \label{sec:lb}
The query complexity of our testing algorithm depends
on the surface area of the set being tested. In this section,
we prove that a polynomial dependence on surface area is
necessary for non-adaptive tester, by proving a lower bound for distinguishing
1-juntas and 2-juntas in two dimensions.
In particular, we show the following theorem.
\begin{theorem}~\label{thm:lb}
Any non-adaptive algorithm which can distinguish between a $1$-junta with surface area at most $s$ versus $\Omega(1)$-far from a linear $1$-junta makes at least $s^{\frac{1}{10}}$ queries.
\end{theorem}
To prove this theorem, as is standard, we will use the Yao's minimax lemma. More specifically, we will describe a distribution $D_1$
over 1-juntas with surface area at most $\Theta(s)$ and a distribution
$D_2$ over functions that are far from 1-juntas and have surface area $\Theta(s)$, such that
for any choice of $x_1, \dots, x_n \in {\mathbb{R}}^2$ with $n = O(s^{1/10})$,
if $f \sim D_1$ and $g \sim D_2$ then $(f(x_1), \dots, f(x_n))$
and $(g(x_1), \dots, g(x_n))$ have almost the same distribution.
We begin with the description of $f \sim D_1$: let $\theta \in {\mathbb{R}}^2$
be a uniformly random unit vector. Choose $a_1, \dots, a_{s-1}$
uniformly from $[-1, 1]$, and then put them in increasing order.
We also set $a_0 = -1$ and $a_{s} = 1$.
Then choose independent
random bits $b_1, \dots, b_{s}$
and define $f$ by
\[
f(x) = \begin{cases}
b_i &\text{if $a_{i-1} < \langle x, \theta \rangle \le a_i$ for some $i \in \{1, \dots, s\}$} \\
1 &\text{otherwise}.
\end{cases}
\]
Clearly, such a function $f$ is a 1-junta,
and its surface area is at most $s+1$
because the boundary of $\{f = 1\}$ is a collection of at most $s+1$
lines, and each line has surface area at most $1/\sqrt{2\pi}$.
To describe the construction of $g \sim D_2$, we begin with
the same collection of random variables as before (i.e., $\theta$,
$a_1, \dots, a_{s-1}$, $b_1, \dots, b_s$).
Let $\theta^\perp$ be a $90^\circ$ clockwise rotation of $\theta$,
choose $z \in [-1, 1]$ independent of the other random variables,
and define $g$ by
\[
g(x) = \begin{cases}
b_i \mathrm{sign}(\langle x, \theta^\perp \rangle - z) &\text{if $a_{i-1} < \langle x, \theta \rangle \le a_i$ for some $i \in \{1, \dots, s\}$} \\
1 &\text{otherwise}.
\end{cases}
\]
Note that the boundary of $\{g = 1\}$ is contained in at most $s+2$
lines, and so it has surface area at most $s+2$.
We will prove below that (with high probability) functions drawn from $D_2$ are far from 1-juntas.
Then the following Theorem will demonstrate that testing 1-juntas with surface area $\Theta(s)$
requires $\mathsf{poly}(1/s)$ queries.
\begin{theorem}\label{thm:surface-area-lower-bound}
For any query set $x_1, \dots, x_n$ with $n \le s^{1/10}$, if $f \sim D_1$ and $g \sim D_2$
then the distributions of $(f(x_1), \dots, f(x_n))$ and $(g(x_1), \dots, g(x_n))$ are $C s^{-1/10}$-close
in total variation distance.
\end{theorem}
In order to study the distinguishability of $D_1$ and $D_2$, we give a slightly different
description of $f \sim D_1$ and $g \sim D_2$:
for $i = 1, \dots, s$ set
\begin{eqnarray*}
S_i^+ &=&\{x: a_{i-1} < \langle x, \theta \rangle \le a_i \text{ and } \langle x, \theta^\perp \rangle \ge z\} \\
S_i^- &=&\{x: a_{i-1} < \langle x, \theta \rangle \le a_i \text{ and } \langle x, \theta^\perp \rangle < z\} \\
S_i &=& S_i^- \cup S_i^+,
\end{eqnarray*}
and note that $f$ was defined by independently assigning a random $\pm 1$ value on each set $S_i$,
while $g$ was defined by independently assigning opposite random $\pm 1$ values on each pair
$S_i^+$, $S_i^-$.
Also, $f$ and $g$ are both identically one on ${\mathbb{R}}^2 \setminus \bigcup S_i$.
Let $x_1, \dots, x_n$ be the set of query points, and consider the event
that for every $i$, at least one of $S_i^+$ or $S_i^-$ contains no point in $x_1, \dots, x_n$;
call this event $A$.
Then $A$ depends on $x_1, \dots, x_n$, $\theta$, and $a_1, \dots, a_{s-1}$, but not on $b_1,\dots,b_s$.
Thanks to the description of $f$ and $g$ above, conditioned on
$A$ the random variables $(f(x_1), \dots, f(x_n))$ and $(g(x_1), \dots, g(x_n))$
have the same distribution. In particular, we can couple $f$ and $g$ so that
$(f(x_1), \dots, f(x_n)) = (g(x_1), \dots, g(x_n))$ with probability at least $1 - \Pr[A]$,
and so we will prove Theorem~\ref{thm:surface-area-lower-bound} by showing that for
any choice of $x_1, \dots, x_n$ with $n \le s^{1/10}$,
$\Pr[A] \le C s^{-1/10}$.
To do this, we will divide the pairs $(x_i, x_j)$ into
``close'' pairs and ``far'' pairs: we say that $x_i$ and $x_j$ are $\delta$-close if
$|x_i - x_j| \le \delta$, and $\delta$-far otherwise.
The following lemma will complete the proof of Theorem~\ref{thm:surface-area-lower-bound}, because
it implies that with high probability no pair of points lies in the same strips $S_i$,
but on different sides of the line $\{x: \langle x, \theta^\perp \rangle = z\}$.
\begin{lemma}\label{lem:close-and-far}
Suppose that $n \le s^{1/10}$ and set $\delta = s^{-1/3}$. For any set $x_1, \dots, x_n$, with probability
at least $1 - C s^{-1/10}$:
\begin{enumerate}
\item every pair of points $x_i, x_j$ that are $\delta$-far do not belong to the same set $S_k$
for any $k \in \{1, \dots, s\}$.
\item every pair of points $x_i, x_j$ that are $\delta$-close
lie on the same side of the line $\{x: \langle x, \theta^\perp\rangle = z\}$.
\end{enumerate}
\end{lemma}
The first step of Lemma~\ref{lem:close-and-far} is the simple observation that far points remain
reasonably far even after projecting them in the direction $\theta$.
\begin{lemma}\label{lem:random-inner-product}
For all sufficiently small $\delta$ and any $x \in {\mathbb{R}}^2$, $\Pr(|\langle \theta, x\rangle| \le \delta |x|) \le \delta$.
\end{lemma}
\begin{proof}
If $\phi$ is the angle between $\theta$ and $x$ then $|\langle \theta, x\rangle| \le \delta |x|$
exactly when $|\cos \phi| \le \delta$, which has probability $\frac{\cos^{-1}(-\delta) - \cos^{-1}(\delta)}{\pi}$.
Since $\cos^{-1}$ has derivative 1 at zero, this is approximately $\frac{2}{\pi} \delta$ for small $\delta$.
In particular, if $\delta > 0$ is sufficiently small then this probability is at most $\delta$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:close-and-far}]
Let $\ell_k = \ell_k(\theta, a_k)$ be the line $\{x: \langle x, \theta \rangle = a_k\}$.
By Lemma~\ref{lem:random-inner-product} applied to $x_i - x_j$, if $x_i$ and $x_j$ are $\delta$-far
then with probability at least $1 - \delta$, $|\langle \theta, x_i - x_j \rangle| \ge \delta^2$.
By a union bound, with probability at least $1 - n^2 \delta$,
$|\langle \theta, x_i - x_j \rangle| \ge \delta^2$ for every $\delta$-far pair $x_i, x_j$; from now
on, we will condition on this event (call it $\Omega_1$) occurring.
Now, if either $\langle \theta, x_i \rangle$
or $\langle \theta, x_j \rangle$ lies outside of the interval $[-1, 1]$ then $x_i$ and $x_j$
do not both lie in any single $S_k$. On the other hand, if both $\langle \theta, x_i \rangle$
and $\langle \theta, x_j \rangle$ lie in $[-1, 1]$, then each line $\ell_k$ has (independently)
probability $|\langle \theta, x_i - x_j \rangle| \ge \delta^2$ to ``split'' $x_i$ from $x_j$.
Hence, with probability at least $1 - (1 - \delta^2)^{s-1} \ge 1 - \exp(\delta^2(s-1))$, there will
be a line $\ell_k$ that splits $x_i$ from $x_j$, and so they will not belong to any single set $S_k$.
Taking a union bound over all pairs $x_i, x_j$, we see that (conditioned on $\Omega_1$) with probability
at least $1 - n^2 \exp(-\delta^2(s-1))$, no pair of $\delta$-far points lands in the same $S_k$.
Removing the conditioning on $\Omega_1$ changes the probability bound to
$1 - n^2 \delta - n^2 \exp(-\delta^2(s-1))$, which with our choice of parameters is at least
$1 - C s^{-1/10}$.
If $x_i$ and $x_j$ are $\delta$-close then $|\langle \theta^\perp, x_i - x_j\rangle| \le \delta$, and hence
the probability that they land on opposite sides of the line
$\{x: \langle x, \theta^\perp \rangle = z\}$ is at most $O(\delta)$. By a union bound over all pairs,
with probability at least $1 - C n^2 \delta \ge 1 - C s^{-1/10}$,
every pair of $\delta$-close $x_i, x_j$ land on the same side of that line.
\end{proof}
\subsection{$D_2$ is far from a 1-junta}
So far, we have shown that one cannot distinguish $D_1$ from $D_2$ from few samples. It remains to
show that functions
from $D_2$ are far (with high probability) from 1-juntas, it will follow that one cannot
1-juntas with $O(s)$ surface area with fewer than $s^{1/10}$ queries.
\begin{theorem}\label{thm:far-from-1-junta}
There is a constant $c > 0$ such that with probability at least $1 - \mathsf{poly(1/s)}$
over $g \sim D_2$, $g$ is $c$-far from every 1-junta.
\end{theorem}
Now recall that the construction of $D_1$ and $D_2$ involved dividing up the
strip $\{x: \langle \theta, x \rangle \in (-1, 1]\}$ into $s$ strips $S_1,
\dots, S_s$ and assigning random values on each strip.
Since both the construction of $D_2$ and the notion of distance to a 1-junta are rotationally
invariant, we will assume from now on that $\theta = e_1$, which means that the strips
$S_1, \dots, S_s$ are vertically oriented.
Let $U^+ = \bigcup_{i: b_i = 1} (a_i, a_{i+1}]$ and let $U^- = [-1, 1] \setminus U^+$.
\begin{definition}
Let $I \subset [-1, 1]$ be an interval. We say that $I$ is \emph{$\delta$-balanced} if
of both $|I \cap U^+|$ and $|I \cap U^-|$ are at least $\delta |I|$, where $|\cdot|$ denotes the one-dimensional
Lebesgue measure.
We say that $I$ is \emph{wide} if $|I| \ge \frac{1}{s}$.
We extend these definitions to strips in two dimensions: say that $I \times {\mathbb{R}}$ is $\delta$-balanced (resp. wide)
if $I$ is $\delta$-balanced (resp. wide).
\end{definition}
\begin{definition}
For any line $\ell \subset {\mathbb{R}}^2$, we say that $\ell$ is $\delta$-balanced if both
\[
\int_{\ell \cap U^+} e^{-|x|^2/2} \, dx \quad \text{and} \quad
\int_{\ell \cap U^-} e^{-|x|^2/2} \, dx
\]
are at least
\[
\delta \int_{\ell} e^{-|x|^2/2}\, dx.
\]
\end{definition}
We will now describe the outline of Theorem~\ref{thm:far-from-1-junta}'s proof: note that if $h$ is a $1$-junta
then $h(x) = \tilde h(\langle \phi, x\rangle)$ for some $\phi$.
Now, Fubini's theorem implies that
\[
2\pi \|h - g\|_1
= \int_{{\mathbb{R}}^2} e^{-|x|^2/2} |h - g| \, dx
= \int_{\mathbb{R}} \int_{\{x: \langle x, \phi^\perp \rangle = a\}} e^{-|x|^2/2} |\tilde h(a) - g(x)| \, dx \, da.
\]
Now, whenever the line $\{x: \langle x, \phi^\perp\}$ is $\delta$-balanced, the inner integral is
at least $\delta \int e^{-|x|^2 / 2} \, dx$. Therefore, in order to prove Theorem~\ref{thm:far-from-1-junta},
it suffices to show that there is a constant $\delta$ such that at least a constant fraction of the lines
$\{x: \langle x, \phi^\perp \rangle = a\}$ are $\delta$-balanced.
To be precise, let $L(\phi)$ be the set of lines of the form $\{x: \langle
\phi^\perp, x\rangle = a\}$ for $a \in [-10, 10]$. Since $e^{-|x|^2/2}$ is bounded from below on $[-10, 10]$,
it suffices to show that there is a constant $\delta > 0$ such that with high probability,
for every $\phi$, a constant fraction of $\ell \in L(\phi)$ are $\delta$-balanced. For the remainder of the
section, we will focus on proving the preceding statement.
We will consider two cases depending on $\phi$: if the lines in $L(\phi)$ are
``steep,'' then these lines will be balanced because a constant fraction of them will
cross the horizontal line $\{x: x_2 = z\}$ near the middle of a strip. Since the value of $g$ on a strip changes sign at
that horizontal line,
this will imply that such a line is balanced.
On the other hand, if the lines are not steep, then they will be balanced because they cross many
strips, and $g$ will tend to take different values on different strips.
We will first deal with the case of steep lines. In this case, it is deterministically the case that
$g$ is far from $h$.
\begin{lemma}\label{lem:wide-strips}
At least half of the points on the line segment from $(-1, 1/2)$ to $(1, 1/2)$ are
in a wide strip $S_i$.
\end{lemma}
\begin{proof}
There are $s$ strips in total, and so the narrow ones can take up at most a total width of 1,
which is only half of the width of the line segment in question.
\end{proof}
\begin{lemma}
There is a constant $c > 0$ such that if the absolute value of the slope of $\{x: \langle \phi^\perp, x \rangle = 0\}$
is at least $s$ then a $c$-fraction of $\ell \in L(\phi)$ are $c$-balanced.
\end{lemma}
\begin{proof}
We may assume without loss of generality that $z \le 0$.
By Lemma~\ref{lem:wide-strips}, at least a constant fraction of $\ell \in L(\phi)$
intersect the line $\{x: x_2 = 1/2\}$ in the middle third of a wide strip $S_k$. In this case,
$\ell$ belongs to $S_k^+$ for a distance of at least 1/3, and to $S_k^-$ for a distance
of at least 1/3, and it follows that $\ell$ is $c$-balanced for a constant $c$
depending on the minimum and maximum values of $e^{-|x|^2/2}$ for $x \in [-1, 1]^2$.
\end{proof}
For the remainder of the section we will deal with lines that are not steep. For $k$
with $2^{-k} \le 2/s$, consider an interval of the form $[j 2^{-k}, (j+1) 2^{-k}] \subset [-1, 1]$;
let $D_k$ be the set of all such intervals.
\begin{lemma}\label{lem:balanced-dyadics}
There is a constant $C$ such that
with probability at least $1 - \mathsf{poly}(1/s)$,
for every $k$ for which $2^{-k} \le 2/s$, at least a $\frac{1}{C}$-fraction of the intervals $I \in D_k$ are $\frac{1}{C}$-balanced.
\end{lemma}
\begin{proof}
For technical convenience, we will consider a slightly different way of generating the strips $S_i$.
Instead of dividing $[-1, 1] \times {\mathbb{R}}$ using exactly $s-1$ vertical lines, we will take a Poisson number
(with mean $s-1$) of vertical lines. We will prove the claim for this modified model, with a probability
estimate of at least $1 - \exp(-\Omega(\sqrt s))$, and since a Poisson random variable is equal to its mean
with probability $\mathsf{poly}(1/s)$, the claim will also follow for the original model.
Our first claim is that for $2^{-k} \le 1/\sqrt s$, each interval in $D_k$ has a constant
probability of being $\Omega(1)$-balanced.
First, consider the largest $k$ for which $2^{-k} \le 2/s$. In this case, the width of each $I \in D_k$
is within a factor 2 of $s$ (we will call such an interval a
\emph{primitive} interval.
It is easy to verify that for each $I \in D_k$, there is a constant probability that $I$ will intersect
exactly two strips, each taking up at least 1/3 of the width of $I$, and that these two strips
will receive different labels $b_i$. Hence, there is a constant probability that $I$ is $1/3$-balanced.
Now consider $k$ for which $2^{-k} \le 1 / \sqrt s$. Every $I \in D_k$ is made up of
$\Theta(s 2^k)$ primitive intervals, each of which has a constant probability of being balanced.
Moreover (thanks to our Poissonized model) the events that different primitive intervals are balanced
are independent. By Chebyshev's inequality, there is a constant probability that at least a constant
fraction of $I$'s primitive intervals are $1/3$-balanced, and so $I$ has a constant probability of being
$\Omega(1)$-balanced.
This proves our first claim (that for each $2^{-k} \le 1/\sqrt s$, each interval in $D_k$ has a constant
probability of being $\Omega(1)$-balanced). Now, for each such $k$ there are at least $\Omega(\sqrt s)$
such intervals, and so a Chernoff bound implies that with probability at least $1 - \exp(-\Omega(\sqrt s))$,
at least a constant fraction of these intervals are balanced. Taking a union bound over $k$
proves the claim whenever $2^{-k} \le 1/\sqrt s$.
For smaller $k$, we claim that with high probability, \emph{every} $I \in D_k$ is balanced.
Indeed, such $I \in D_k$ contain at least $\sqrt s$ primitive intervals, and so a Chernoff bound
implies that with probability $1 - \exp(-\Omega(\sqrt s))$, at least a constant fraction of those
primitive intervals are balanced, and so $I$ is balanced also. We can take a union bound over all $k$
and all $I$.
\end{proof}
To complete the proof of Theorem~\ref{thm:far-from-1-junta}, it remains to show that with high probability,
every non-steep line is balanced.
\begin{lemma}
There is a constant $c > 0$ such that if the absolute value of the slope of $\{x: \langle \phi^\perp, x \rangle = 0\}$
is at most $s$ then with probability at least $1 - \mathsf{poly}(1/s)$, a $c$-fraction of $\ell \in L(\phi)$ are $c$-balanced.
\end{lemma}
\begin{proof}
Choose $k$ so that the slope of all lines in $L(\phi)$ are between $2^{k-1}$ and $2^{k}$.
Consider a rectangle of the form $Q = [j 2^{-k}, (j+1) 2^{-k}] \times [-2, -1]$, where
the interval $[j 2^{-k}, (j+1) 2^{-k}]$ is balanced. Since the slope of $\phi$
is at most $2^{k}$, if the line $\ell$ intersects the rectangle $Q$ then it
crosses the entire vertical strip $[j 2^{-k}, (j+1) 2^{-k}] \times {\mathbb{R}}$ within the horizontal strip $[-3, 0]$.
Since the interval $[j 2^{-k}, (j+1) 2^{-k}]$ is balanced, it follows that the line $\ell$ is also balanced.
(We're assuming here, without loss of generality, that $z \ge 0$).
Finally, it is easy to verify that if a constant fraction of the intervals $[j 2^{-k}, (j+1) 2^{-k}]$
are balanced then a constant fraction of $\ell \in L(\phi)$ intersect with some rectangle of the form above.
By Lemma~\ref{lem:balanced-dyadics}, this completes the proof.
\end{proof}
\section{Introduction}
\input{intro_merged}
\vspace{-0.2cm}
\section{Preliminaries}
\input{prelim}
\input{test-rank}
\input{finding-struct}
\input{surface-area-lower-bound}
\section{Testing linear invariant subclasses of linear $k$-juntas} \label{sec:test-invariant}
In this section, we prove the following theorem.
\begin{theorem}~\label{thm:test-invariant-struct}
Let $\mathcal{C}$ be a collection of functions mapping $\mathbb{R}^k$ to $\{-1,1\}$. Further, for every $f \in \mathsf{Ind}(\mathcal{C})$, $\mathsf{surf}(f) \le s$. Then, there is an algorithm \textsf{Test-structure-$\mathcal{C}$} which has the following guarantee: Given oracle access to $f: \mathbb{R}^n \rightarrow \{-1,1\}$ and an error parameter $\epsilon>0$, the algorithm makes $(s \cdot k/\epsilon)^{O(k)}$ queries and distinguishes between the cases (i) $f \in \mathsf{Ind}(\mathcal{C})$ and (ii) $f$ is $\epsilon$-far from every function $f' \in \mathsf{Ind}(\mathcal{C})$.
\end{theorem}
The algorithm \textsf{Test-structure-$\mathcal{C}$} is described in Figure~\ref{fig:tsi}. We now proceed with the proof of Theorem~\ref{thm:test-invariant-struct}. We begin with the following fact.
\begin{figure}[tb]
\hrule
\vline
\begin{minipage}[t]{0.98\linewidth}
\vspace{10 pt}
\begin{center}
\begin{minipage}[h]{0.95\linewidth}
{\small
\underline{\textsf{Inputs}}
\vspace{5 pt}
\begin{tabular}{ccl}
$s$ &:=& surface area parameter \\
$\epsilon$ &:=& error parameter \\
$k$ &:=& rank parameter\\
\end{tabular}
\underline{\textsf{Parameters}}
\vspace{5 pt}
\begin{tabular}{ccl}
$T_{inv}(s,k,\epsilon)$ &:=& query complexity of \textsf{Find-invariant-structure} with parameters $s$, $k$ and $\epsilon$. \\
$\delta$ &:=& $(\epsilon / s \cdot k)^{O(k)}$ such that $\delta \cdot T_{inv}(2s,k,\epsilon/4) \le \epsilon$. \\
$k$ &:=& rank parameter\\
\end{tabular}
\vspace{5 pt}
\underline{\textsf{Testing algorithm}}
\begin{enumerate}
\item Run algorithm \textsf{Test-rank} with surface area parameter $s$, rank parameter $k$ and error parameter $\delta$. If \textsf{Test-rank} outputs no, output no.
\item Otherwise, run routine \textsf{Find-invariant-structure} with surface area parameter $s$, rank parameter $k$ and error parameter $\epsilon/4$.
\item Let $g: \mathbb{R}^\ell \rightarrow \{-1,1\}$ be the output of Step~2. Extend it to $\mathbb{R}^k$ where $g$ acts trivially in the last $n-k$ coordinates. Output yes if $g$ is $\epsilon$-close to some function in $\mathsf{Ind}(\mathcal{C})_k$. Otherwise, output no.
\end{enumerate}
\vspace{5 pt}
}
\end{minipage}
\end{center}
\end{minipage}
\hfill \vline
\hrule
\caption{Description of the algorithm \textsf{Test-Structure-$\mathcal{C}$}}
\label{fig:tsi}
\end{figure}
\begin{fact}~\label{fact:unitary}
Let $f: \mathbb{R}^n \rightarrow \{-1,1\}$ be in $\mathsf{Ind}(\mathcal{C})$ defined as $f = g(\langle w_1, x \rangle, \ldots, \langle w_k, x\rangle)$ for $ g \in \mathcal{C}$ and orthonormal vectors $w_1, \ldots, w_k \in \mathbb{R}^n$. If $v_1, \ldots, v_k$ is some other orthonormal basis of $\mathsf{span}(w_1, \ldots, w_k)$, then $f = \tilde{h}(\langle v_1, x \rangle, \ldots, \langle v_k ,x\rangle)$, for $\tilde{h} \in \mathsf{Ind}(\mathcal{C})_k$.
\end{fact}
\begin{proof}
Observe that one go from the basis $(w_1, \ldots, w_k)$ to the basis $(v_1, \ldots, v_k)$ by means of a unitary transformation $U \in \mathbb{R}^{k \times k}$. Thus, $\tilde{h} =g \circ U$. However, observe that the function $\tilde{h}(x) = g \circ U (x)$ lies in $\mathsf{Ind}(\mathcal{C})_k$ which finishes the proof.
\end{proof}
\begin{claim}~\label{clm:test-invariant-complete}
Assume that $f : \mathbb{R}^n \rightarrow \{-1,1\}$ is in $\mathsf{ind} (\mathcal{C})_n$. Further, $\mathsf{surf}(f) \leq s$. Then, $f$ passes \textsf{Test-Structure-$\mathcal{C}$} with probability $1-\epsilon$.
\end{claim}
\begin{proof}
If $f \in \mathsf{ind} (\mathcal{C})_n$ with $\mathsf{surf}(f) \le s$, then \textsf{Test-rank} outputs yes with probability $1-\delta$. Let $w_1, \ldots, w_k$ be the implicit basis such that the output of Step~2 of \textsf{Test-Structure-$\mathcal{C}$} such that
\[
\mathbf{E}[|g(\langle w_1, x \rangle, \ldots, \langle w_k, x \rangle) - f(x)|] \le \frac{\epsilon}{4}.
\]
As $f \in \mathsf{Ind}(\mathcal{C})$, there is an orthonormal set of vectors $(v_1, \ldots, v_k)$ and $h \in \mathcal{C}$ such that (i) $f = h(\langle v_1, x \rangle, \ldots, \langle v_k ,x \rangle)$ and (ii) $\mathsf{span}(w_1, \ldots, w_k) = \mathsf{span}(v_1, \ldots, v_k)$. By Fact~\ref{fact:unitary}, we get that there is a function $\tilde{h} \in \mathsf{Ind}(\mathcal{C})_k$ such that
$f = \tilde{h}(\langle w_1, x\rangle, \ldots, \langle w_k, x\rangle)$. This implies that
\[
\mathbf{E}[|g(\langle w_1, x\rangle, \ldots, \langle w_k, x\rangle)- \tilde{h}(\langle w_1, x\rangle, \ldots, \langle w_k, x\rangle)|] \le \frac{\epsilon}{4}.
\]
Since $w_1, \ldots, w_k$ are orthonormal vectors, we get that $\mathbf{E}_{x \sim \gamma_k}[|g(x) - \tilde{h}(x)|] \le \frac{\epsilon}{4}$. This finishes the proof of the claim.
\end{proof}
\begin{claim}~\label{clm:test-invariant-sound}
Assume that $f : \mathbb{R}^n \rightarrow \{-1,1\}$ is $\epsilon$-far from any function $f' \in \mathsf{Ind}(\mathcal{C})$. Then, the test \textsf{Test-Structure-$\mathcal{C}$} rejects with probability at least $0.9$.
\end{claim}
\begin{proof}
First of all, if $f$ passes \textsf{Test-rank} with probability $0.1$, then $f$ is $O(\delta)$-close to a linear $k$-junta $\tilde{f}$ with $\mathsf{surf}(\tilde{f}) \le (1+\delta) \cdot s$. Now, by Remark~\ref{rem:gaussian}, since the marginal of the queries made by the routine \textsf{Test-invariant-structure} is distributed as $\gamma_n$, hence with probability $1-O(\delta)$, we can assume that the queries are made to $\tilde{f}$. Note that the surface area of $\tilde{f}$ is at most $2s$.
Since $\delta \cdot T_{inv}(2s,k,\epsilon/4) \le \epsilon$, hence with probability $1-\epsilon$, we can assume that the queries made by \textsf{Test-invariant-structure} are made to $\tilde{f}$. Then, with probability $1-2\epsilon$, Step~2 outputs a function $g$ such that there is an orthonormal set of vectors $w_1, \ldots, w_k$ and $h: \mathbb{R}^k \rightarrow \{-1,1\}$ with the following conditions: (i) $\mathbf{E}_{x \sim \gamma_k}[|g(\langle w_1, x\rangle, \ldots, \langle w_k , x\rangle) - h(\langle w_1, x\rangle, \ldots, \langle w_k , x\rangle)|] \le \frac{\epsilon}{4}$ and (ii)
$\tilde{f} (x) = h(\langle w_1, x\rangle, \ldots, \langle w_k , x\rangle)$. Let $g_0 \in \mathsf{Ind}(\mathcal{C})_k$ such that $g$ and $g_0$ are $\epsilon/4$-close. Because $w_1, \ldots, w_k$ are orthonormal, this also implies that
$
\mathbf{E}_{x \sim \gamma_n}[|g(\langle w_1, x\rangle, \ldots, \langle w_k , x\rangle) - g_0(\langle w_1, x\rangle, \ldots, \langle w_k , x\rangle)|] \le \frac{\epsilon}{4}.
$ By applying triangle inequality, we get that
\[
\mathbf{E}_{x\sim \gamma_n} [ |f(x) - g_0 (\langle w_1, x\rangle, \ldots, \langle w_k, x \rangle)|] \le \frac{\epsilon}{2} + O(\delta) \le\epsilon.
\]
However, $g_0(\langle w_1, x\rangle, \ldots, \langle w_k, x \rangle) \in \mathsf{Ind}(\mathcal{C})_n$ contradicting the claim that $f$ is $\epsilon$-far from any function in $\mathsf{Ind}(\mathcal{C})$.
\end{proof}
}
\section{Algorithm to test $k$-juntas}~\label{sec:test-rank}
In this section, we will prove the following theorem.
\begin{theorem}~\label{thm:main1}
There is an algorithm \textsf{Test-linear-junta} which has the following guarantee: Given oracle access to $f: \mathbb{R}^n \rightarrow \{-1,1\}$, rank parameter $k$, surface area parameter $s$ and error parameter $\epsilon>0$, it makes $\mathsf{poly}(s,\epsilon^{-1}, k)$ queries and
\begin{enumerate}
\item If $f$ is a linear $k$-junta with $\mathsf{surf}(f) \le s$, then the algorithm outputs \textsf{yes} with probability at least $0.9$.
\item If $f$ is $O(\epsilon)$-far from any linear $k$-junta $g$ with {$\mathsf{surf}(g) \leq (1+\epsilon) \cdot s$}, then the algorithm outputs \textsf{no} with probability at least $0.9$.
\end{enumerate}
\end{theorem}
\begin{remark}{A convention that we shall adopt (to avoid proliferation of parameters) is to sometimes ignore the confidence parameter of the testing algorithm. Typically, whenever we can estimate a parameter within $\pm \epsilon$ with $T$ queries with confidence $2/3$, we can do the usual ``median trick" and get the same accuracy with confidence $1-\delta$ with a multiplicative $O(\log(1/\delta))$ overhead in the query complexity. Since we only need to succeed with probability $0.9$ in the final algorithm, it is sufficient for each of the individual subroutines to succeed with probability sufficiently close to $1$. So, unless it is crucial, at some places,we shall ignore the confidence parameter in the theorem statements and many of the calculations. It will be implicit that the confidence parameter is sufficiently close to $1$.
}
\end{remark}
The algorithm \textsf{Test-linear-junta} is described in Figure~\ref{fig:tlj}. The algorithm invokes two different subroutines, \textsf{Test-surface-area} and \textsf{Test-rank} whose guarantees we state now. To
do this, we first define the notion of $(\epsilon, s)$ smooth function.
\begin{definition}~\label{def:smooth-perturb}
A function $f: \mathbb{R}^n \rightarrow \{-1,1\}$ is said to be $(\epsilon, s)$-smooth if there is a function $g: \mathbb{R}^n \rightarrow \{-1,1\}$ such that $\mathbf{E}[|f-g|]\le\epsilon$ and $\mathsf{surf}(g) \le s (1+\epsilon)$.
\end{definition}
In other words, a function $f$ is $(\epsilon,s)$ smooth if $f$ is $\epsilon$-close to some other function $g$ (in $\ell_1$ distance) and $g$ has surface area which is essentially bounded by $s$. With this definition, we can now state the guarantee of the routine \textsf{Test-surface-area}
(due to Neeman~\cite{neeman2014testing}).
\begin{theorem}~\label{thm:neeman-testing}
There is an algorithm \textsf{Test-surface-area} which given oracle access to a function $f: \mathbb{R}^n \rightarrow \{-1,1\}$ and error parameter $\epsilon>0$ makes $T_{\mathsf{test}} = \mathsf{poly}(s/\epsilon)$ queries and has the following guarantee:
\begin{enumerate}
\item If $f$ is a function with surface area at most $s$, then the algorithm outputs \textsf{yes} with probability at least $1-\epsilon$.
\item {Any function $f$ which passes the test with probability $0.1$ is $(\epsilon,s)$-smooth.}
\end{enumerate}
\end{theorem}
Next, we state the guarantee of the routine \textsf{Test-rank}.
\begin{lemma}~\label{lem:far-k-junta}
The routine \textsf{Test-rank} has a query complexity of $\mathsf{poly}(k,s, \epsilon^{-1})$. Further, we have
\begin{enumerate}
\item If the function $f$ is a linear-$k$-junta, then the algorithm \textsf{Test-rank} outputs \textsf{yes} with probability $1-\epsilon$.
\item
If $f : \mathbb{R}^n \rightarrow \{-1,1\}$ is a $((\epsilon/30)^2,s)$-smooth function which is $\epsilon$-far from a linear $k$-junta,
then the algorithm \textsf{Test-rank} outputs \textsf{no} with probability $1-\epsilon$.
\end{enumerate}
\end{lemma}
{In order to prove Theorem~\ref{thm:main1}, we will need the following claim which shows that property of closeness to a linear $k$-junta and closeness to a smooth function can be certified using a single function.
\begin{lemma}~\label{lem:dual-closeness}
For a function $f: {\mathbb{R}}^n \to \{-1, 1\}$, suppose that there is a linear $k$-junta $g: {\mathbb{R}}^n \to \{-1, 1\}$
and a function $h: {\mathbb{R}}^n \to \{-1, 1\}$ of surface area at most $s$ such that both $g$ and $h$
are $\epsilon$-close to $f$. Then there is a function $\tilde h: {\mathbb{R}}^n \to \{-1, 1\}$ that is a linear $k$-junta \emph{and}
has surface area at most $s(1 + \sqrt \epsilon)$, and which is $O(\sqrt{\epsilon})$-close to $f$.
\end{lemma}
}
{\begin{proofof}{Theorem~\ref{thm:main1}}
If $f$ is a linear $k$-junta with surface area at most $s$, then it passes both the tests \textsf{Test-surface-area} as well as \textsf{Test-rank} with probability $1-\epsilon$. Thus, any linear $k$-junta with surface area at most $s$ passes with probability at least $1-2\epsilon$ (so as long as $\epsilon \le 0.05$, the test succeeds with probability $0.9$).
On the other hand, suppose $f$ passes \textsf{Test-linear-junta} with probability $0.9$. Then, applying Theorem~\ref{thm:neeman-testing} is $((\epsilon/30)^4, s)$ smooth.
In other words, there is a function $h$ such that $\mathsf{surf}(h) \le (1+(\epsilon/30)^4) \cdot s$ which is $O(\epsilon^4)$-close to $f$.
Further, since $f$ passes \textsf{Test-rank} with probability $0.9$, Lemma~\ref{lem:far-k-junta} implies that
$f$ is $\epsilon^2$-close to some linear $k$-junta $g$. We now apply Lemma~\ref{lem:dual-closeness} to obtain that $f$ is $O(\epsilon)$-close to some function $\tilde{h}: \mathbb{R}^n \rightarrow \{-1,1\}$ which is a linear $k$-junta and $\mathsf{surf}(h) \le (1+O(\epsilon)) s$. This concludes the proof.
\end{proofof}}
We now turn to describing the routine \textsf{Test-rank} and prove Lemma~\ref{lem:far-k-junta}.
\begin{figure}[tb]
\hrule
\vline
\begin{minipage}[t]{0.98\linewidth}
\vspace{10 pt}
\begin{center}
\begin{minipage}[h]{0.95\linewidth}
{\small
\underline{\textsf{Inputs}}
\vspace{5 pt}
\begin{tabular}{ccl}
$s$ &:=& surface area parameter \\
$\epsilon$ &:=& error parameter \\
$k$ &:=& rank parameter\\
\end{tabular}
\vspace{5 pt}
\underline{\textsf{Testing algorithm}}
\begin{enumerate}
\item Run algorithm \textsf{Test-surface-area} with surface area parameter $s$ and error parameter $(\epsilon/30)^4$.
\item If \textsf{Test-surface-area} outputs \textsf{yes}, then run the algorithm \textsf{Test-rank}
with rank parameter $k$, surface area parameter $s$ and error parameter $\epsilon$.
\item If \textsf{Test-rank} outputs \textsf{yes}, then output \textsf{yes}. If \textsf{Test-rank} outputs \textsf{no}, output \textsf{no}.
\end{enumerate}
\vspace{5 pt}
}
\end{minipage}
\end{center}
\end{minipage}
\hfill \vline
\hrule
\caption{Description of the algorithm \textsf{Test-linear-junta}}
\label{fig:tlj}
\end{figure}
\begin{figure}[tb]
\hrule
\vline
\begin{minipage}[t]{0.98\linewidth}
\vspace{10 pt}
\begin{center}
\begin{minipage}[h]{0.95\linewidth}
{\small
\underline{\textsf{Input}}
\vspace{5 pt}
\begin{tabular}{ccl}
$k$ &:=& rank parameter \\
$s$ &:=& surface area parameter \\
$\epsilon$ &:=& error parameter
\end{tabular}
\underline{\textsf{Parameters}}
\vspace{5 pt}
\begin{tabular}{ccl}
$t$ &:=& $\frac{\epsilon^4}{900 s^2}$ \\
$r$ &:=& $\frac{k \cdot s^2}{\epsilon^7}$\\
$\kappa$ &:& $\frac{\epsilon^2}{40 r}$ \\
\end{tabular}
\vspace{5 pt}
\underline{\textsf{Testing algorithm}}
\begin{enumerate}
\item Sample directions $y_1,\ldots, y_r \sim \gamma_n$.
\item Let $A_{i,j}=\langle D_{} P_{t} f(y_i) , D_{} P_{t} f(y_j) \rangle$.
\item For all $1 \le i,j \le r$, compute $A_{i,j}$ up to error $\kappa$ using Lemma~\ref{lem:inner-product-1}. Call the estimates $B_{i,j}$.
\item For the matrix $B \in \mathbb{R}^{r \times r}$, compute the top $k+1$ singular values of $B$.
\item Output \textsf{yes} if and only if the $(k+1)^{st}$ singular value is at most $\frac{\epsilon^2}{16}$.
\end{enumerate}
\vspace{5 pt}
}
\end{minipage}
\end{center}
\end{minipage}
\hfill \vline
\hrule
\caption{Description of the \textsf{Test-rank} algorithm}
\label{fig:trj}
\end{figure}
\begin{proofof}{Lemma~\ref{lem:far-k-junta}}
The bound on the query complexity of Lemma~\ref{lem:far-k-junta} is immediate from the settings of our parameters and query complexity of Lemma~\ref{lem:inner-product-1}.
The first item (i.e., the completeness of \textsf{Test-rank}) follows from the fact that if $f$ is a linear $k$-junta, $P_t f$ is also a linear $k$-junta. Consequently, $A$ is a rank-$k$ matrix. Then, $A$ has at most $k$ non-zero singular values. Thus, if $\sigma_1 \ge \sigma_2 \ge \ldots$ are the singular values of $A$ (in order), then $\sigma_{k+1}=0$. By invoking Weyl's inequality (Lemma~\ref{lem:Weyl}), the $(k+1)^{th}$ singular value of $B$ is at most $\epsilon^2/10$. This finishes the proof of the first item.
The proof of the second item (i.e., the soundness of \textsf{Test-rank}) is more involved. In particular, we can restate the second item as proving the following lemma.
\begin{lemma}~\label{lem:far-k-junta-1}
Let $f : \mathbb{R}^n \rightarrow \{-1,1\}$ be a $((\epsilon/30)^2,s)$-smooth function which is $\epsilon$-far from a linear $k$-junta,
then the algorithm \textsf{Test-rank} outputs \textsf{no} with probability $1-\epsilon$.
\end{lemma}
The task of proving this lemma shall be the agenda for the rest of this section.
\end{proofof}
In order to prove Lemma~\ref{lem:far-k-junta-1}, we will need a few preliminary lemmas. The following lemma says that if a function's gradient is almost always orthogonal to a subspace $V$. Then, the function is close to a $V$-junta.
\begin{lemma}~\label{lem:gradient-subspace}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ (be a $\mathcal{C}^1$ function) and let $V$ be a subspace of rank $k$ and let $W = V^{\perp}$. Let us assume that $\mathbf{E}[\Vert (Df)_W \Vert_2^2] = \epsilon$. Then there is a $V$-junta $g: \mathbb{R}^n \rightarrow \mathbb{R}$ such that $\mathbf{E}[(g(x) - f(x))^2]\le \epsilon$.
\end{lemma}
\begin{proof}
Let us rotate the space so that $V = \{(x_1, \ldots, x_k, 0, \ldots, 0): x_1, \ldots, x_k \in \mathbb{R}\}$.
Let us now define $g: \mathbb{R}^n \rightarrow \mathbb{R}$ as
\[
g(x) = \mathop{\mathbf{E}}_{z \sim \gamma_{n-k}} [f(x_1, \ldots, x_k, z_1, \ldots, z_{n-k})].
\]
Observe that $g$ is a $V$-junta. Now, for every choice $X = (x_1, \ldots, x_k)$, consider the function $h_X: \mathbb{R}^{n-k} \rightarrow \mathbb{R}$ as
\[
h_X(z_1, \ldots, z_{n-k}) = f(x_1, \ldots, x_k, z_1, \ldots, z_{n-k}) - g(x).
\]
Observe that $\mathbf{E}_{(z_1, \ldots, z_{n-k}) \sim \gamma_{n-k}} [h_X(z_1, \ldots, z_{n-k})]=0$.
By applying Lemma~\ref{lem:Poincare},
\[
\mathbf{E}[h_X^2(z_1, \ldots, z_{n-k})] = \mathsf{Var}[h_X(z_1, \ldots, z_{n-k})] \leq \mathbf{E}[\Vert Dh_X \Vert_2^2].
\]
Observe that $Dh_X(z_1, \ldots, z_{n-k}) =Df(x_1,\ldots, x_k, z_1, \ldots, z_{n-k})_W$. Thus, we get
\begin{eqnarray*}
\mathbf{E}[(f(x) - g(x))^2] &=& \mathop{\mathbf{E}}_{X \sim \gamma_k} \mathop{\mathbf{E}}_{Z\sim \gamma_{n-k}} [h_X^2 (Z)] \leq \mathop{\mathbf{E}}_{X \sim \gamma_k} \mathop{\mathbf{E}}_{Z\sim \gamma_{n-k}} [\Vert Dh_X \Vert_2^2] \\
&=& \mathop{\mathbf{E}}_{X \sim \gamma_k} \mathop{\mathbf{E}}_{Z\sim \gamma_{n-k}} [\Vert Df(X,Z)_W \Vert_2^2] = \epsilon.
\end{eqnarray*}
This finishes the proof.
\end{proof}
For the rest of this section, when we use the value $t$, it will bear the same relation as stated in the description of the algorithm \textsf{Test-rank} (see Figure~\ref{fig:trj}).
\begin{proposition}~\label{prop:noise-stab-surf-1}
Let $f: \mathbb{R}^n \rightarrow \{-1,1\}$ which is $((\epsilon/30)^2,s)$-smooth.
Then,
$
\mathbf{E}[|P_{t}f - f|^2] \le \frac{\epsilon^2}{5}.
$
\end{proposition}
\begin{proof}
Since $f$ is $((\epsilon/30)^2,s)$ smooth, we know that there is a function $g$ such that $\mathbf{E}[|f-g|] \le (\frac{\epsilon}{30})^2$ and $\mathsf{surf}(g)\le s\big(1+(\frac{\epsilon}{30})^2\big)$.
By using the fact that the operator $P_{t}$ is contractive, we have,
\[
\mathbf{E}[|P_{t} f - P_{t}g|^2] \le \mathbf{E}[|f-g|^2 ] \le 4 \mathbf{E}[\Vert f - g \Vert_1]\le \frac{\epsilon^2}{200}.
\]
Next, we use Proposition~\ref{prop:Ledoux} to get that
$
\mathbf{E}[|P_{t}g - g|^2 ] \le \frac{\epsilon^2}{30}.
$
We can now combine these to get
\[
\mathbf{E}[|P_{t} f - f|^2] = 3 \big( \mathbf{E}[|P_{t} f - P_{t} g|^2] + \mathbf{E}[|P_{t} g - g|^2] + \mathbf{E}[| f - g|^2]\big) \le \frac{\epsilon^2}{5}.
\]
\end{proof}
\begin{lemma}~\label{lem:subspace-escape}
Let $f: \mathbb{R}^n \rightarrow \{-1,1\}$ be a $((\epsilon/30)^2,s)$-smooth function which is $\epsilon$-far from any linear $k$-junta. For any subspace $W$ of co-dimension at most $k$,
\[
\Pr_{y \sim \gamma_n} \bigg[\Vert D_{}P_{t}f(y) \Vert_{W}^2 \ge \frac{\epsilon^2}{8}\bigg] \ge \Omega\bigg(\frac{\epsilon^6}{s^2}\bigg).
\]
\end{lemma}
\begin{proof}
Applying Proposition~\ref{prop:noise-stab-surf-1}, we have that $\mathbf{E}[|P_{t}f-f|^2] \le \frac{\epsilon^2}{5}$. By applying Jensen's inequality,
we have $\mathbf{E}[|P_{t}f-f|] \le \epsilon/\sqrt{5}$. Thus, $P_{t}f$ is $0.5\cdot \epsilon$-far from any linear $k$-junta (in $\ell_1$ distance). Consequently, we can say that for any $W$-junta $h$, $\mathbf{E}[\Vert P_{t} f - h \Vert_2^2] > 0.25 \epsilon^2$. By contrapositive of Lemma~\ref{lem:gradient-subspace}, we have that
\begin{equation}~\label{eq:expectation}
\mathbf{E}[\Vert D_{}P_{t}f(y) \Vert_{W}^2] > 0.25 \cdot \epsilon^2. \end{equation}
Next, observe that Lemma~\ref{lem:derivative-shift} implies that
\[
\Vert D_{}P_{t}f(y) \Vert_{W}^2 \le \frac{1}{e^{2t}-1} \cdot \Vert \mathcal{W}_1(f_{t,y}) \Vert_2^2 \le \frac{1}{{e^{2t}-1}}
\le O(1/t) \le O\left(\frac{s^2}{\epsilon^4}\right).
\]
The second inequality follows immediately from that $f_{t,y}$ has range bounded between $[-1,1]$. Combining this with (\ref{eq:expectation}), this implies that
\[
\Pr\bigg[\Vert D_{}P_{t}f(y) \Vert_{W}^2 \ge \frac{\epsilon^2}{8}\bigg] \ge \Omega\bigg(\frac{\epsilon^6}{s^2}\bigg).
\]
\end{proof}
We are now in a position to finish the proof of Lemma~\ref{lem:far-k-junta-1}.
\begin{proofof}{Lemma~\ref{lem:far-k-junta-1}}
Let $M_i \in \mathbb{R}^n$ denote $M_i = D_{} (P_{t}f)(y_i)$. As in Figure~\ref{fig:trj}, consider the matrix $A \in \mathbb{R}^{r \times r}$ whose $(i,j)$ entry is $A_{i,j} = \langle D_{} (P_{t}f)(y_i), D_{} (P_{t}f)(y_j) \rangle$. Now, consider the matrix $M \in \mathbb{R}^{n \times k}$ whose $i^{th}$ column is $M_i$. Then, observe that $A = M^t \cdot M$. We would like to analyze the singular values of $A$. Observe that the non-zero singular values of $M^t \cdot M$ are the same as the non-zero singular values of $M \cdot M^t$. Now, observe that
\[
M \cdot M^t = \sum_{i=1}^r D_{} (P_{t}f)(y_i) \cdot D_{} (P_{t}f)(y_i)^t
\]
Instead of analyzing the non-zero singular values of $M^t \cdot M$, we will analyze the non-zero singular values of $M \cdot M^t$. From now on, let us use $h$ to denote
$P_{t}f$. Let us define the sequence of stopping times $\{\tau_j \}_{j \ge 0}$ as follows: $\tau_0=0$ and let $\mathcal{G}_j = \sum_{\ell \le \tau_j} D_{} h(y_\ell) \cdot D_{} h(y_\ell)^t$ and $W_j$ be the eigenspace formed by the top $j$ eigenvectors of $\mathcal{G}_j$. Then, $\tau_{j+1}$ is the smallest $\ell>j$ such that
$\Vert (D_{} h(y_\ell))_{W_j^\perp} \Vert_2 \ge \frac{\epsilon}{2\sqrt{2}}$. We now make the following claim.
\begin{claim}~\label{clm:large-singular}
For $j \le k+1$, the top $j$ singular values of $\mathcal{G}_j$ are at least $\epsilon^2/8$.
\end{claim}
\begin{proof}
We will prove this claim by induction. So, assume that
the top $j$ singular values of $\mathcal{G}_j$ are all at least $\epsilon^2/8$. Now, for $\ell = \tau_{j+1}$, let
$w$ be the unit vector in the direction of the component of $D_{} h(y_\ell)$ orthogonal to $W_j$. Let $\Gamma$ be the linear span of $W_j$ and $w$. Now, consider any unit vector $v \in \Gamma$ and express it as $v= v_1+ v_2$ where $v_1$ lies in $W_j$ and $v_2$ is parallel to $w$. Next, observe that
\begin{eqnarray*}
v^T \cdot \big( \mathcal{G}_j + D_{} h(y_\ell) \cdot D_{} h(y_\ell)^t \big) \cdot v = v^T \cdot \mathcal{G}_j \cdot v + v^T \cdot D_{} h(y_\ell) \cdot D_{} h(y_\ell)^t \cdot v.
\end{eqnarray*}
The first term $v^T \cdot \mathcal{G}_j \cdot v$ is at least as large as $v_1^T \cdot \mathcal{G}_j \cdot v_1$ and the second term $v^T \cdot D_{} h(y_\ell) \cdot D_{y} h(y_\ell)^t \cdot v$ is the same as $v_2^T \cdot D_{} h(y_\ell) \cdot D_{} h(y_\ell)^t \cdot v_2$. Next, note that
\[
v_1^T \cdot \mathcal{G}_j \cdot v_1\ge \frac{\epsilon^2}{8} \cdot \Vert v_1\Vert_2^2 ;\ \ \ \ \ v_2^T \cdot D_{} h(y_\ell) \cdot D_{} h(y_\ell)^t \cdot v_2\ge \frac{\epsilon^2}{8} \cdot \Vert v_2\Vert_2^2.
\]
Consequently,
\[
v^T \cdot \big( \mathcal{G}_j + D_{} h(y_\ell) \cdot D_{} h(y_\ell)^t \big) \cdot v \ge \frac{\epsilon^2 }{8} \cdot \big(\Vert v_1 \Vert_2^2 + \Vert v_2 \Vert_2^2 \big) = \frac{\epsilon^2}{8}.
\]
Observe that
\[
v^T \cdot \mathcal{G}_{j+1} \cdot v \ge v^T \cdot \big( \mathcal{G}_j + D_{} h(y_\ell) \cdot D_{} h(y_\ell)^t \big) \cdot v \ge \frac{\epsilon^2}{8}.
\]
The first inequality is immediate from the fact that
\[
\mathcal{G}_{j+1} -\big( \mathcal{G}_j + D_{} h(y_\ell) \cdot D_{} h(y_\ell)^t \big) = \sum_{\tau_j <i < \tau_{j+1}} D_{} h(y_i) \cdot D_{} h(y_i)^t
\]
is a psd matrix. Thus, we obtain that
\begin{equation}~\label{eq:singular-1}
\inf_{v: \Vert v \Vert_2=1 \ \textrm{and} \ v \in \Gamma} v^T \cdot \mathcal{G}_{j+1} \cdot v \ge \frac{\epsilon^2}{8}.
\end{equation}
Now, it is clear that $\mathcal{G}_{j+1}$ is a psd matrix. If the singular values of $\mathcal{G}_{j+1}$ are $\sigma_1 \ge \sigma_2 \ge \ldots$, then by Courant Fischer theorem, we have
\[
\sigma_{k+1} = \max_{S_{k+1} \subseteq \mathbb{R}^n} \inf_{v: \Vert v \Vert_2=1 \ \textrm{and} \ v \in S_{k+1}} v^T \cdot \mathcal{G}_{j+1} \cdot v,
\]
where $S_{k+1}$ is the set of all $k+1$ dimensional subspaces of $\mathbb{R}^n$. Thus, by applying (\ref{eq:singular-1}) and observing $\mathsf{dim}(\Gamma) = k+1$, we get
\[
\sigma_{k+1} \ge \inf_{v: \Vert v \Vert_2=1 \ \textrm{and} \ v \in \Gamma} v^T \cdot \mathcal{G}_{j+1} \cdot v \ge \frac{\epsilon^2}{8}.
\]
This finishes the proof.
\end{proof}
Now applying Lemma~\ref{lem:subspace-escape}, we have that conditioned on $\tau_j$, $\tau_{j+1}-\tau_j$ is a geometric random variable with parameter (at least) $\Omega(\epsilon^6/s^2)$. From this, it is not difficult to see that with probability at least $1-\epsilon$, $\tau_{k+1} = O(s^2 \cdot k/\epsilon^7)$,
Thus, with probability $1-\epsilon$, we can assume that the top $k+1$ singular values of $M \cdot M^t$ are all at least $\epsilon^2/8$.
Consequently, we get that the top $k+1$ singular values of $A = M^t \cdot M$ are all at least $\epsilon^2/8$. Now, the algorithm computes a matrix $B$ such that
$\Vert A-B\Vert_F \le \epsilon^2/10$. By Weyl's inequality~(Lemma~\ref{lem:Weyl}), we get that the top $k+1$ singular values of $B$ are all at least $\epsilon^2/16$. This proves the lemma.
\end{proofof}
{
We finally give the proof of Lemma~\ref{lem:dual-closeness}. The proof relies on the so-called co-area formula.
\begin{lemma}~\label{lem:coarea}
Let $f: {\mathbb{R}}^n \to [-1, 1]$ be smooth and $\psi: [-1, 1] \to {\mathbb{R}}_+$ be bounded and measurable. Then
\[
\int_{-1}^1 \psi(s) \mathsf{surf}(\{x: f(x) \le s\}) \, ds = \int_{{\mathbb{R}}^n} \psi(f(x)) |\nabla f(x)| \, d\gamma(x).
\]
\end{lemma}
\begin{proofof}{Lemma~\ref{lem:dual-closeness}}
By~\cite{maggi2012sets}, there is a smooth function $h_1: {\mathbb{R}}^n \to [-1, 1]$ with bounded gradient
such that $\|h_1 - h\|_2 \le \epsilon$ and $\operatorname{{\bf E}}[|\nabla h_1|] \le 2s$.
Let $E$ be a $k$-dimensional subspace for which $g$ is an $E$-junta, and let $z$ be a standard Gaussian
vector on $E^\perp$. Let $\Pi_E$ be the projection operator for subspace $E$ and define $h_2: {\mathbb{R}}^n \to [-1, 1]$ by $h_2(x) = \operatorname{{\bf E}}_z[h_1(\Pi_E x + z)]$. By Jensen's inequality,
$\operatorname{{\bf E}} [|\nabla h_2|] \le \operatorname{{\bf E}}[|\nabla h_1|] \le 2s$. Let $t$ be uniformly distributed in $[-1+\eta, 1-\eta]$,
and define $\tilde h = \tilde h_t$ by
\[
\tilde h_t(x) = \tilde h(x) = \begin{cases}
-1 &\text{if $h_2(x) \le t$} \\
1 &\text{otherwise}.
\end{cases}
\]
Note that $\tilde h_t$ is an $E$-junta (because $h_2$ is an $E$-junta).
In expectation over $t$, the surface area of $\tilde h$ is
\[
\frac{1}{2-2\eta} \int_{-1+\eta}^{1-\eta} \mathsf{surf}\{x: h_2 \le s\}\, ds,
\]
which by the co-area formula is equal to
\[
\frac{1}{2-2\eta} \int_{{\mathbb{R}}^n} 1_{\{h_2(x) \in [-1+\eta, 1-\eta]\}} |\nabla h_2(x)|\, d\gamma(x)
\le \frac{1}{2-2\eta} \operatorname{{\bf E}}_{x \sim \gamma} [|\nabla h_2(x)|]
\le \frac{s}{1-\eta}.
\]
In particular, there exists some $t \in [-1+\eta, 1-\eta]$ such that the surface area
of $\tilde h_t$ is at most $\frac{s}{1-\eta}$.
Next, we will estimate the distance of $\tilde h$ from $h$. By the triangle inequality, $\|h - g\|_2 \le 2\epsilon$
and so $\|h_1 - g\|_2 \le 3\epsilon$. On the other hand, Pythagoras' theorem implies
that $h_2$ minimizes $\|h_1 - h_2\|_2$ among all $E$-juntas; hence, $\|h_1 - h_2\|_2 \le 3\epsilon$
and so $\|h - h_2\| \le 4 \epsilon$.
Now, $h$ takes values in $\{-1, 1\}$ and so $|h(x) - h_2(x)| \ge \eta 1_{\{h_2(x) \in [-1+\eta, 1-\eta]\}}$.
On the other hand, the definition of $\tilde h$ ensures that
\[
|\tilde h(x) - h_2(x)| \le \begin{cases}
2 &\text{if $h_2(x) \in [-1+\eta, 1-\eta]$} \\
\eta &\text{otherwise}.
\end{cases}
\]
If $p$ is the probability that $h_2(x) \in [1 + \eta, 1-\eta]$, it follows that $\eta \sqrt{p} \le \|h - h_2\|_2$
and so
\[
\|\tilde h - h_2\|_2 \le 2\sqrt{p} + \eta \le \frac{8\epsilon}{\eta} + \eta.
\]
By the triangle inequality $\|\tilde h - h\|_2 \le 4 \epsilon + \frac{8\epsilon}{\eta} + \eta$.
Choosing $\eta = \sqrt \epsilon$ completes the proof.
\end{proofof}
}
| {
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package com.github.bednar.persistence.event;
import javax.annotation.Nonnull;
import com.github.bednar.persistence.AbstractPersistenceTest;
import com.github.bednar.persistence.DummyData;
import com.github.bednar.persistence.resource.Pub;
import org.hibernate.ObjectNotFoundException;
import org.junit.Assert;
import org.junit.Test;
/**
* @author Jakub Bednář (19/08/2013 3:50 PM)
*/
public class ReadEventTest extends AbstractPersistenceTest
{
@Test
public void read() throws InterruptedException
{
final Pub pub = DummyData.getPub();
dispatcher.publish(new SaveEvent(pub));
dispatcher.publish(new ReadEvent<Pub>(pub.getId(), Pub.class)
{
@Override
public void success(final @Nonnull Pub resource)
{
Assert.assertEquals(pub, resource);
}
@Override
public void fail(final @Nonnull Throwable error)
{
Assert.fail();
}
});
}
@Test
public void readNotExist() throws Throwable
{
dispatcher.publish(new ReadEvent<Pub>(-1L, Pub.class)
{
@Override
public void success(final @Nonnull Pub value)
{
Assert.fail();
}
@Override
public void fail(final @Nonnull Throwable error)
{
Assert.assertEquals(ObjectNotFoundException.class, error.getClass());
}
});
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Q: When a set of measure zero plus itself contains interior Is there a characterization of measure zero subsets $A$ of $\mathbb R^n$, $n>1$ such that the set $A+A$ contains interior? Here $A+A$ is the set of points $\{ x+y \mid x, y\in A \}$.
Is it true that if the convex hull of the connected component of $A$ contains interior then so does $A+A$?
A: For question #1, I don't know.
For question #2, the answer is no. Consider the edges of your favorite polyhedron in $\mathbb R^3$. They are connected. Their convex hull, the entire polyhedron, obviously has a nonempty interior. But $A+A$ is a finite union of two-dimensional parallelograms and thus cannot have interior in $\mathbb R^3$.
A: Regarding Question #1.
There's one obvious dimensional obstruction (for example consider the Hausdorff dimension of A).
There is some research about it in the 1-dimensional case, such as the well-known theorem that K+K contains an interval, and related conjectures and works by Pallis, Furstenberg and Yoccoz. Even this one-dimensional theory is not complete as far as I know, so characterizing such a statement in larger dimensions would seem improbable now.
If say your set A is the product of two sets $A=A_{x} \times A_{y}$, such that $A_{x}+A_{x}$ contains an interval, and $A_{y}+A_{y}$ contains an interval, then one can have that the sumset of the product $A$ will contain a rectangle.
The one thing that can help you tackle the problem in larger dimensions, is the fact that you can sometimes say something smart about the projection of your set in a.e. direction (especially if your set is self-similar).
For example this is the content of early works by Furstenberg (back in the 60s), and even some recent works (such as Hochman-Shmerkin - arXiv:0910.1956).
I hope that by taking two independent generic directions, you can say something smart about the projections (Hausdorff dimension? entropy estimates?), and then maybe one can use the one-dimensional theory to get that the sumset of the projection contains an interval. Of-course, it is not enough to get an interval in the projection, because your set might not be rectangular set itself, but maybe if your set if self-similiar, you can wiggle the pieces around to construct an inner rectangle.
By Masterand's theorem, I'm guessing that a resonable bet here would be $dim_{H}(A)>1$ (or maybe even take the upper packing dimension of $A$ to be bigger than $1$) for "nice enoguh" sets $A$, although I'm pretty sure this is open in general (because of Pallis' conjecture).
| {
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Discussion in 'ESET NOD32 Antivirus' started by RWP2, Jun 24, 2008.
... if it isn't allowed to "call home" during launch?
I wouldn't say that there is no internet access at all but rather no HTTP/POP3 connection is allowed. ESET NOD32 Antivirus uses internal proxy for redirecting HTTP/POP3 traffic. Such communication is blocked until access for ENA is provided.
ENA is definitely not "calling home" - what IP address do you see when ENA tries to connect?
what IP address do you see when ENA tries to connect?
ekm.exe is trying to reach 66.196.97.186:HTTP.
This is not a server of ESET. Another application communicating via http is trying to connect there. You can test this by putting a cross next to each application in the browser list of the HTTP section (this will make the applications' traffic not to be routed via ekrn) in the main setup tree and then check in your firewall for details about that application.
This is not a server of ESET.
How come the alert calls it an ESET Service, and it only exists when NOD32 is installed? (Never saw it with any other brand of AV). Further, if this isn't an ESET service, why would NOD32 refuse ANY Internet access until the service is granted access?
So what do you have running that is connecting to "mail" at "yahoo.com"
You can use WinPatrol (for free) to delay your choice of startup items by a configurable number of seconds, including whatever program you are launching that is connecting to Yahoo. I personally have many startup items staggered, some launching after 10 seconds, some after 20 seconds, some after 30 seconds... This results in a faster startup time overall (less disk thrashing), with the most delayed items being things I don't really care how soon they run at startup as long as they eventually do (UPS monitoring service, Trillian, etc).
How come the alert calls it an ESET Service, and it only exists when NOD32 is installed?
It was already explained but let's try again . ESET NOD32 3.0 's Web protection uses the so called "internal proxy" to check all your web web traffic (HTTP and POP3 communucation) before it reaches your computer . By doing this , it can protect you from malware in emails or in web-pages and prevent them even touch your computer . The technology used is that the web traffic passes through the ESET Kernel (ekrn.exe) so that it is checked . However , because of the fact all the web traffic (HTTP and POP3) passes through ekrn.exe , the firewall (Zone Alarm) cannot make difference if it is actually ESET Server/Kernel or another application .
Thanks for the more complete explanation on how ekrn operates.
2. Forwarding of infiltrations and information to the Provider. The Software contains a function which serves to collect samples of new computer viruses or other similar harmful computer programs (the "Infiltration") and the subsequent dispatch thereof to the Provider, including information about the computer and/or platform on which the Software is installed(the "Information"). The Information may contain data (including personal data) about the End User and/or other users of the computer on which the Software is installed, information about the computer and operating system, suspicious files from the computer on which the Software is installed and files affected by the Infiltration and any information about such files. The Provider shall use the obtained Information and the Infiltration only to review the Infiltration and shall take reasonable measures to keep the obtained Information confidential. If you accept this Agreement and activate the above function of the Software, you agree that the Infiltration and the Information may be forwarded to the Provider and at the same time you grant to the Provider consent necessary pursuant to the relevant legal regulations to process the obtained Information.
19. Data on End User and Protection of Rights. You as the End User authorize the Provider to transfer, process and save the data enabling the Provider to identify you. You agree that the Provider may check by its own means whether you are using the Software in accordance with the provisions of this Agreement. You agree that through communication of the Software with the computer systems of the Provider or of its business partners data may be transferred, the purpose of which is to ensure the functionality of and authorization to use the Software and protection of the Provider's rights.
I guess this is made not to confuse users and to make the program easier to users . | {
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} | 3,860 |
Jack Joseph Kempf (May 12, 1935 - July 1, 2003) was a business owner and politician in British Columbia. He represented Omineca in the Legislative Assembly of British Columbia from 1975 to 1991 as a Social Credit member.
He was born in Kelowna, British Columbia, the son of Steve Kempf and Katherine Klein. Kempf was a motel and restaurant owner. He served on the municipal council for Houston, British Columbia and also served as mayor. Kempf served in the provincial cabinet, first as Minister of Lands, Parks and Housing, and then as Minister of Forests and Lands.
Kempf died in Loreto, Baja California Sur, Mexico, where he had lived in retirement.
References
1935 births
2003 deaths
British Columbia municipal councillors
British Columbia Social Credit Party MLAs
Businesspeople from British Columbia
Canadian hoteliers
Canadian restaurateurs
Mayors of places in British Columbia
Members of the Executive Council of British Columbia
People from Kelowna
People from Loreto Municipality, Baja California Sur
20th-century Canadian politicians | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,148 |
Anarchias leucurus is a moray eel found in the Pacific Ocean. It was first named by Snyder in 1904 as Uropterygius leucurus, and is commonly known as Snyder's moray, the fine-spotted moray or the finespot moray. It is thought to be the smallest species of moray, and may actually represent several different species or subspecies.
References
leucurus
Fish described in 1904
Fish of the Pacific Ocean
Taxa named by John Otterbein Snyder | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,876 |
Inclusion at Artisan Church
We believe that all people are born with the essential dignity and intrinsic value that comes from being made in God's image and are equally capable of experiencing God, regardless of their sexual orientation or gender identity. We welcome and invite LGBTQ persons to participate at all levels of church life: partaking of the sacraments, serving in ministry, joining in membership, and holding leadership roles.
The fair treatment and inclusion of LGBTQ people in the church is a matter of justice—one of our foundational values—and we pledge to work to reconcile injustice against LGBTQ people in the greater church and outside it. Sexual orientation and gender identity must not be barriers to those seeking access to God's love, grace, and growth in Christ's church. As followers of Jesus, we are called to break down dividing walls and welcome all people.
Historical orthodoxy and denominational harmony
We affirm broad historical orthodoxy as outlined in the consensual creeds of Christianity. We embrace the Evangelical Covenant Church's foundational ethos as expressed in Covenant Affirmations 1, notably the affirmations of the centrality of the Word of God, freedom in Christ, and a conscious dependence on the Holy Spirit. We celebrate the ECC's history of finding a third way in controversies about non-essential matters such as infant baptism2, just war theory3, and many others.
We humbly recognize that we hold a minority view on human sexuality within our denomination4. Artisan values and joyfully welcomes all LGBTQ persons, whether they are single, in relationship, or called to celibacy. In order to persist in fellowship and remain in good standing within the ECC, we are currently unable to officiate or host same-sex weddings5; however, we have partnerships with local pastors and churches who can, and we will provide references as requested. In this way, the Artisan community can celebrate with all couples as they begin their lives together.
Living in Christian unity
In 2015, with encouragement from the congregation and under the direction of the Leadership Team, Artisan Church began a series of conversations and listening circles on Gender, Sexuality, and Inclusion. After careful study of scripture, ongoing dialogue with our brothers and sisters, and a great deal of prayer and consideration, the community tasked the Leadership Team with composing a statement of inclusion to be ratified by the members for publication.
The Artisan community will continue this work of study, dialogue, and prayer. We exhort every believer to submit humbly to the conviction and guidance of the Holy Spirit in all areas of their life, and we will support and trust LGBTQ persons to discern God's guidance in their own lives, whether that path includes celibacy or relationships.
We realize that not all Artisan members will come to the same conclusions about the Bible's teaching on same-sex relationships and gender identity, and that people's convictions may change over time. Yet we believe that our unity comes from our shared faith in Christ, not from absolute agreement about the interpretation of scripture. We believe that if we exhibit the radical love of Christ, our differences and diversity will enrich our life together, not divide us.
Adopted July 16, 2017
See "Covenant Affirmations" ↑
See "What Does it Mean to Receive a Child Into the Church?" (PDF, 558 KB) ↑
See the 2006 resolution "Christian Discipleship in the Midst of War" ↑
See the 1996 resolution "Human Sexuality" ↑
See "Guidelines for Covenant Pastors and Congregations Regarding Human Sexuality" (PDF, 83 KB) ↑
Why "Artisan" Church?
Core affirmations
Our first desire is to follow in the "ancient paths" of the generous orthodoxy revealed in Scripture, affirmed in the consensual creeds of the early church, and embodied in the life of Jesus' followers throughout time and across cultures.
Yet we equally recognize Christ's call to contextualize his gospel – without compromising its core – by providing "new wineskins" that will allow the effervescent work of God's Spirit to reach its full-bodied potential.
These simple statements are our humble attempt at articulating God's vision for his church at Artisan as we partner with his redemptive purposes.
We were launched in 2005 as an official church plant of the Evangelical Covenant Church, and continue to be part of this wonderful family of churches. They provide resources, encouragement, and accountability.
Evangelical Covenant Church
ECC – "Who We Are"
ECC – "Affirmations"
Artisan Church
1235 South Clinton Avenue
info@artisanchurch.com Church Community Builder
Subscribe to our weekly e-mail news Accessibility LGBTQ Inclusion | {
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Q: Protecting LocalDB database with password LocalDB is a great move done by Microsoft, the needless of a database server and the portability of applications, however a user -after distributing the app- can open the database alone and see its schema and edit it :(
Is there an effective way to encrypt it with a password like "SQL Server CE" or "MS Access"?!
| {
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} | 8,562 |
POP ROCKS sues ROCK'N'POP for trademark infringement
The owner of POP ROCKS carbonated popping candy (or "gasified candy") filed suit against the makers of a popping candy sold under the name ROCK'N'POP.
Zeta Espacial S.A. ("Zeta"), a corporation based in Spain, is the company which currently owns numerous federal trademark registrations for the mark POP ROCKS. On February 19, 2009, Zeta filed a trademark infringement lawsuit in the U.S. District Court for the Northern District of Illinois against Imaginings 3, Inc. ("Imaginings"), an Illinois corporation doing business as Flix Candy. A copy of the complaint can be downloaded here.
According to the complaint, Imaginings makes its own popping candy which it sells under the name ROCK'N'POP both through wholesale distributors as well as on its website www.flixcandy.com. [Note: the only image of any ROCK'N'POP popping candy I could find was the below picture of ROCK'N'POP lollipop and popping candy dip featuring Hannah Montana].
Zeta maintains that Imaginings use of ROCK'N'POP in connection with its popping candy is likely to cause consumer confusion with and dilution of Zeta's POP ROCKS Marks. Zeta's causes of action are trademark infringement under 15 U.S.C. § 1114; unfair competition under 15 U.S.C. § 1125(a)(1); federal trademark dilution under 15 U.S.C. § 1125(c); deceptive trade practices under Illinois state law (815 ILCS 510/2); trademark dilution under Illinois state law (756 ILCS 1036/65); and common law trademark infringement and unfair competition.
Vegas™Esq. Comments:
Seems like another example of a case where at first, it would seem apparent that there is no likelihood of confusion given the aural and visual differences between the marks and their different commercial impression. However, once you start applying the likelihood of confusion factors, a different picture emerges that favors Zeta more than one might initially think – strong mark (Zeta will have strong sales and advertising figures and the mark is famous, but there could be some possible room to argue genericness or that the mark has become weak), similarity between the marks, similar goods, similar marketing channels, and inexpensive nature of the goods (low degree of consumer care). While I still think Imaginings has the advantage if it ever went to a jury, does Imaginings really want to get embroiled in an expensive fight over the name?
Of course, no posting about POP ROCKS would be complete without some reference to the famed urban legends surrounding the dangers of eating the candy and then drinking cola. In the internet age, you can even watch the urban legend be tested – ranging from the serious (Mythbusters) to the ridiculous (here).
Labels: Infringement
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One Former Las Vegas Businessman is a Trademark "V... | {
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{"url":"http:\/\/gtribello.github.io\/mathNET\/SOR30123-overview.html","text":"## Overview: Markov chains in discrete time\n\nIn this third project we introduce the Markov property and thus begin our study of random, time-dependent processes. In this part of the course we assume that we are monitoring our random process at particular points in time rather than assuming we are monitoring the random process continuously. Furthermore, we assume that the random variable whose value we are monitoring in time can only take integer values. To describe these processes we introduce the notion of a transition graph and a transition matrix. You will learn how to write programs to simulate Markov chains and will learn how quantitative predictions can be made for these random processes by examining the limiting behavior of the chains.\n\n### Aims\n\n\u2022 You should be able to state what it means when we say that a time series of random variables has the Markov property\n\u2022 You should be able to interpret transition graphs and transition probability matrices and you should also be able to use the Chapman-Kolmogorov relation to calculate the $n$-step transition probability matrix from the 1-step transition probability matrix.\n\u2022 You should be able to discuss the limiting behavior of Markov chains. This means you should be able to calculate limiting stationary distributions, hitting times and hitting probabilities.\n\u2022 You should be able to write a computer program to simulate a random walk in one dimension.\n\n### Contact Details\n\nSchool of Mathematics and Physics,\nQueen's University Belfast,\nBelfast,\nBT7 1NN\n\nEmail: g.tribello@qub.ac.uk\nWebsite: mywebsite","date":"2022-06-30 06:45:06","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\": 0.5710012912750244, \"perplexity\": 263.05918726637213}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103669266.42\/warc\/CC-MAIN-20220630062154-20220630092154-00641.warc.gz\"}"} | null | null |
{"url":"https:\/\/mostwiedzy.pl\/pl\/dorota-osula,98083-1","text":"# Dorota Osula - Profil naukowy - MOST Wiedzy\n\n## mgr in\u017c. Dorota Osula\n\nBrak danych\n\n### Wybrane publikacje\n\n\u2022 #### Finding small-width connected path decompositions in polynomial time\n\nA connected path decomposition of a simple graph $G$ is a path decomposition $(X_1,\\ldots,X_l)$ such that the subgraph of $G$ induced by $X_1\\cup\\cdots\\cup X_i$ is connected for each $i\\in\\{1,\\ldots,l\\}$. The connected pathwidth of $G$ is then the minimum width over all connected path decompositions of $G$. We prove that for each fixed $k$, the connected pathwidth of any input graph can be computed in polynomial-time. This answers...\n\nPe\u0142ny tekst w serwisie zewn\u0119trznym\n\n\u2022 #### Clearing directed subgraphs by mobile agents\n\nWe study several problems of clearing subgraphs by mobile agents in digraphs. The agents can move only along directed walks of a digraph and, depending on the variant, their initial positions may be pre-specified. In general, for a given subset S of vertices of a digraph D and a positive integer k, the objective is to determine whether there is a subgraph H=(V,A) of D such that (a) S is a subset of V, (b) H is the union of k directed...\n\nPe\u0142ny tekst w serwisie zewn\u0119trznym\n\n\u2022 #### Decontaminating Arbitrary Graphs by Mobile Agents: a Survey\n\nA team of mobile agents starting from homebases need to visit and clean all nodes of the network. The goal is to find a strategy, which would be optimal in the sense of the number of needed entities, the number of moves performed by them or the completion time of the strategy. Currently, the field of distributed graph searching by a team of mobile agents is rapidly expanding and many new approaches and models are being presented...\n\nPe\u0142ny tekst w serwisie zewn\u0119trznym\n\nwy\u015bwietlono 87 razy","date":"2020-06-01 01:03:26","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\": 0.6406325697898865, \"perplexity\": 1097.7496980186986}, \"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\/1590347413901.34\/warc\/CC-MAIN-20200601005011-20200601035011-00445.warc.gz\"}"} | null | null |
{"url":"https:\/\/math.stackexchange.com\/questions\/360682\/calculating-pythagorean-triples","text":"# Calculating Pythagorean triples\n\nIf $x$ and $y$ are even, then of course $z$ is too, and $\\left(\\frac{x}{2}, \\frac{y}{2}, \\frac{z}{2} \\right)$ is also a Pythagorean triple. For this question we assume that $(x, y, z)$ is a Pythagorean triple in which $x$ is odd, so that $y$ is even and $z$ is odd. For similar reasons, we assume that $p$ and $q$ are coprime. The theory of Pythagorean triples then tells us that there are nonzero integers $p$ and $q$ such that\n\n$$x + iy = (p + iq)^2 \\hspace{1.5cm} z = |p + iq|^2 = p^2 + q^2.$$\n\nIf $x$ is odd then one of $p$ and $q$ must be even and the other is odd. Otherwise all possibilities occur. Write down the Pythagorean triples for such $p$ and $q$, where $1 \\leq q < p \\leq 8.$\n\nI'm not really sure how to calculate these triples. I first decided to start by picking say $q = 1, p = 2$, which then I then get $z = 1^2 + 2^2 = 5$. How would I then use this to calculate $x$ and $y$?\n\n$(p+iq)^2=(2+i)^2=2^2+2\\cdot 2i+i^2=4+4i-1=3+4i$, so $x=3$, $y=4$.\n\u2022 OHHHHH!!!! Ok! And then I try this for all $p,q$ in that interval? Does this always work? \u2013\u00a0Kaish Apr 13 '13 at 19:41","date":"2019-05-22 20:59:59","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\": 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\": 0.9000271558761597, \"perplexity\": 61.28148504374002}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232256958.53\/warc\/CC-MAIN-20190522203319-20190522225319-00167.warc.gz\"}"} | null | null |
Q: Problem with very long latex table I have a small problem with my thesis. I was adding very long table suitable to 2 pages, but my texmaker editor shows me pdf file on just one page and the table is not completed. NExt issue is, that the last column is too long for a one page, I am including file with tables and screeshot of my problem. I was trying to use tabularx and longtables too, it did not help.
Thank you so so much for you help.
Adrian
A: You have to put the table code in another file, like table.tex then you use it in the main code with ltxtable package, like this
\documentclass{article}
\usepackage{ltxtable}
\begin{document}
\LTXtable{\textwidth}{table.tex}
\end{document}
Your table.tex file can be just like this (with tabularx formatting)
\begin{longtable}{XXX}
\caption{...} %if needed
\label{...}\\ %if needed
... %the header in the first table page
\endfirsthead
... %the header in the other table pages
\endhead
... %the footer in the table pages (all but not last)
\endfoot
... %the footer in the last table pages
\endlastfoot
... %the table content
... %the table content
... %the table content
\end{longtable}
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Navy Honors Electronics Pioneer Howard Lorenzen, W7BI
Lynn Burlingame, N7CFO
n7cfo@arrl.net
The story of the man who developed the technologies used in modern electronic warfare.
On June 26, 2010 the US Navy christened the missile range instrumentation ship USNS Howard O. Lorenzen in Pascagoula, Mississippi. This vessel is named in honor of Howard Lorenzen, W7BI, (1912-2000) who is widely credited as being the "Father of modern electronic warfare."
The 12,575 ton, 534 foot USNS Howard O. Lorenzen employs a crew of 88 and will host embarked military and civilian technicians and mariners from other US government agencies.
Howard's father was chief telegraph operator in Atlantic, Iowa for the Rock Island Railroad. The railroad telegraph system could eavesdrop on lines that carried news and other information. During the baseball World Series, Lorenzen plugged into the lines that carried the telegraphed games to station WOAW in Omaha. Fans gathered in the train station to hear Lorenzen announce the plays and scores of the games.
In 1928, at the age of 16, Howard obtained his first Amateur Radio license, receiving the call sign 9BLC. His first long-distance contact was with 2GL in Valley Stream, Long Island, New York, about 950 miles away. Most of Howard's ham radio activity was on the 75 and 160 meter bands; shorter wavelengths were not used much then. Almost all of his early contacts were by AM phone but as his code speed increased he used CW more frequently.
In 1928, Howard attended the ARRL convention at Iowa State University in Ames, Iowa. There he met Arthur Collins, founder of Collins Radio Co, who had an exhibit and was offering his first transmitter for sale. This was the start of a lifelong friendship.
After high school graduation, Howard attended Iowa State. During his sophomore year he set up a ham station in his room with an antenna running out the window. He was busy studying and others complained that noise from the motor and generator interfered with their studies, so he wasn't on the air very much.
Colonial Radio Corporation
Howard graduated with a degree in electrical engineering in 1935 and went to work in Buffalo, New York for Colonial Radio Corporation as a production engineer in their laboratory. Colonial was headed by Dr Fulton Cutting whose prominent family included the inventor of the steamboat. He was a highly regarded scientist and former president of the Institute of Radio Engineers. Colonial made radios under their name and for the Sears, Roebuck Co. They also made automobile radios for Chevrolet and Oldsmobile.
The years 1935 to 1939 at Colonial provided a sound beginning for a successful career and a rewarding family life. In 1936 Howard married Etta Mae Owen, the sister of a fraternity brother.
Zenith Radio Corporation
In 1940 Howard was hired by Zenith Radio Corporation at their new plant in Chicago. The Zenith lab functioned as a combination experimental and production lab. One of Howard's ongoing tasks was to analyze radio receivers made by other manufacturers. His findings were reported to Zenith's President, Commander Eugene F. McDonald, who judged the competitor's products and features and decided what could be incorporated into Zenith products.
McDonald had been a lieutenant commander in Naval Intelligence, enjoyed being addressed by his rank and ran Zenith like a ship. He lived onboard his yacht, the Mizpah and was a well-known bon vivant. His favorite yachting destination was the north end of Lake Michigan, near the Mesabi and Vermillion iron ore deposits. In a meeting with his engineering staff the Commander told them he couldn't hear any broadcast stations on his Zenith portable broadcast receiver while near the mining area, possibly because the iron ore deposits attenuated the signals. He thought Zenith receivers should do better and said "Why don't you guys build some good stuff?"
Howard explained that a shortwave portable would receive high frequency stations such as BBC in London when standard broadcast stations were out of range. The Commander knew shortwave capabilities from having used it on an Arctic expedition so he told Howard to develop a shortwave receiver.
Howard altered a portable radio to receive shortwave frequencies. The radio was placed aboard the Mizpah and after returning from a cruise the Commander directed that Zenith put it into production. Howard developed a two band radio and after several weeks of construction and design, he gave the Commander a prototype for testing. The Commander was pleased with the result and suggested that the radio be made with a removable loop antenna so it could be positioned for the best reception. Thus was born the Zenith Trans-ocean (later models were renamed the Trans-oceanic).
Howard left the company in July 1940, so other engineers further developed the two band prototype and changed it to six bands for easier tuning. Soon the Trans-ocean Shortwave Portable was in production. It became the most popular radio of its type in the world.
A Pioneer in Electronic Defense
Lorenzen began his Naval Research Laboratory (NRL) career in 1940 after leaving Zenith Radio. He got his first taste of electronic countermeasures when he unintentionally jammed the signal of radar being tested at the Lab's Radar Division. As the US entered World War II with the attack on Pearl Harbor, Lorenzen's research focused on developing electronic means to detect, locate, jam and otherwise deceive enemy radar and other electronic locating equipment, ushering in a new era of warfare to benefit US military countermeasures.
As the war progressed, Lorenzen, assigned to the Lab's Special Developments Section, continued to expand on the idea of electronic countermeasures, or ECMs. He worked to develop radio detection and recording devices to defeat guided German missiles. He analyzed the control signals sent to Henschel 293 flying bombs and used the information to disrupt or distort their radio command signals.
A Hot War Turns Cold
NRL's experience during the war made evident the importance of electronic countermeasures in naval operations. After the war, Howard was relicensed as W3BLC. He continued to develop new ECM technologies, organizing NRL's Countermeasures Branch and toiling with recovered German and Japanese wartime electronic devices. Lorenzen developed the first US magnetic tape recorder for intercept work and tunable microwave intercept receivers used in Navy ships, shore stations and aircraft.
During the 1950s and throughout the Korean and Vietnam military conflicts, ECM technology was advanced by Lorenzen and his team of engineers. They developed new techniques and systems of electronic signal interception and signal source location, which included direction finding, recording, analysis, jamming and deception of high-frequency signals. It was Lorenzen's project engineer and prodigy, Jim Trexler, who first started calling Lorenzen "Father." Trexler was the first to reveal that terrestrial radio signals reflected from the Moon could be intercepted back on Earth using a giant parabolic antenna. An immediate outgrowth was the Navy's Communication Moon Relay (CMR) system, "moon bounce," that provided our Navy with satellite communication a decade before artificial space satellites were operational.
In June 1960, after a US U-2 spy aircraft was shot down over the Soviet Union, Lorenzen's most notable achievement became the Galactic Radiation and Background payload (GRAB). GRAB was the earliest space-based reconnaissance satellite and the first US Navy electronic intelligence (ELINT) satellite. It obtained information on Soviet air defense radars that otherwise could not be observed from US military aircraft.
Situated 500 miles above the Earth, safe from surface-to-air missiles, the GRAB satellite's circular orbit passed it through energy beams from Soviet radar whose pulses traveled far beyond the horizon into space. GRAB received each pulse of a radar signal in a certain bandwidth, as sensed by its tiny antennas, and transponded a corresponding signal to collection huts at ground sites. Operators recorded transponded information on magnetic tape and couriered it to NRL for evaluation. NRL evaluated, duplicated and forwarded the tapes to the National Security Agency at Fort Meade, Maryland and the Strategic Air Command at Offut Air Force Base, Omaha, Nebraska, for analysis and processing.
In the mid-1960s as Cold War tensions were ramping up, Lorenzen inspired his team to again think of new, innovative means to thwart enemy attacks on US military targets. In September 1966, his branch was upgraded to division status and Lorenzen was named superintendent of NRL's new Electronic Warfare Division with the chief role of developing advanced guided missile defenses for Navy aircraft and state-of-the-art electronic warfare (EW) components for the USS New Jersey.
In 1970, Deputy of Defense Secretary David Packard aligned space system acquisition responsibilities with those for weapon systems acquisitions and authorized the military departments to pursue departmental need for space systems. NRL turned to Lorenzen to repeat in space what he had accomplished in EW — design total systems for military operational support — and in 1971 Lorenzen was named superintendent of NRL Space Systems. Lorenzen served as superintendent until his retirement from NRL in June 1973.
Retirement Radio
In 1976, after being retired for several years, Howard and Etta Mae moved to Bellevue, Washington to be near their daughter and grandsons and he obtained the call W7BI.
Amateur Radio was rewarding to Howard and he gave generously in return. He was a life member of the ARRL and wrote numerous articles for QST. Over the years Howard Elmered many hams and generously lent his expertise to many radio clubs and organizations.
Howard became a Silent Key on February 23, 2000 at the age of 87. The Issaquah, Washington ARC obtained permission from his family to acquire his call sign, W7BI, for club use, so his legacy lives on.
Lynn Burlingame, N7CFO, an ARRL Life Member, is an Extra class amateur and has been licensed for nearly 30 years. His main amateur interests are VHF and UHF operating and contest roving. He is the co-founder and former president of the Pacific Northwest VHF Society. Lynn retired in 2003 to pursue his other interests including collecting antique telegraph equipment. He can be reached at 15621 SE 26th St, Bellevue, WA 98008-5443.
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\section{\label{sec:ack} Acknowledgements}
This material is based upon work supported by the U.S.~Department of Energy, Office of Science, Office of Nuclear Physics under contract / award numbers DE-AC02-05CH11231, DE-AC05-00OR22725, DE-AC05-76RL0130, DE-FG02-97ER41020, DE-FG02-97ER41033, DE-FG02-97ER41041, DE-SC0012612, DE-SC0014445, DE-SC0018060, and LANLEM77/LANLEM78. We acknowledge support from the Particle Astrophysics Program and Nuclear Physics Program of the National Science Foundation through grant numbers MRI-0923142, PHY-1003399, PHY-1102292, PHY-1206314, PHY-1614611, PHY-1812409, PHY-1812356, and PHY-2111140. We gratefully acknowledge the support of the Laboratory Directed Research \& Development (LDRD) program at Lawrence Berkeley National Laboratory for this work. We gratefully acknowledge the support of the U.S.~Department of Energy through the Los Alamos National Laboratory LDRD Program and through the Pacific Northwest National Laboratory LDRD Program for this work. We gratefully acknowledge the support of the South Dakota Board of Regents Competitive Research Grant. We acknowledge support from the Russian Foundation for Basic Research, grant No.~15-02-02919. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada, funding reference number SAPIN-2017-00023, and from the Canada Foundation for Innovation John R.~Evans Leaders Fund. This research used resources provided by the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory and by the National Energy Research Scientific Computing Center, a U.S.~Department of Energy Office of Science User Facility. We thank our hosts and colleagues at the Sanford Underground Research Facility for their support.
\section{\label{sec:level3}Analysis of data}
For this work, data from weekly calibrations was analyzed in several steps, including data selection and data quality checks, validation of the \textsc{Geant4} simulation, and the signature search at higher energies. The \textsc{majorana demonstrator}\ Data Acquisition (DAQ) system records waveforms from each HPGe detector using two digitization channels with different amplifications, called the low-gain and high-gain channels. The high-gain channels have been extensively used for double-beta decay searches \cite{alvis2019search}, but they saturate around 3~MeV due to the digitization range.
The low-gain channels have a wider dynamic range up to 10~MeV and allow a study of signatures with higher energy depositions, \textit{e.g.} by cosmic ray reactions or neutrons. The low-gain channels are used here to search for the 6129-keV photons.
\subsection{\label{sec:bench} Data quality and simulation benchmarking}
The modular approach of the \textsc{demonstrator}\ enabled a flexible construction as well as early data-taking once the first module was constructed. Each calibration source was deployed separately for most of the \textsc{Demonstrator}'s calibration data, except for a period after the installation of the second module when two sources were deployed simultaneously to calibrate both modules. For these calibrations, the DAQ throughput was potentially saturated. Thus this analysis only uses data collected when one calibration source was deployed at a time.
Due to evolving calibration procedures, early commissioning data are not used. For example, during commissioning, transition runs during which the source was in motion were not flagged, which created larger uncertainties in analysis time boundaries. The data analyzed here include calibration data sets from the years 2016-2019, which were also used in the analysis of the recent double-beta decay results~\cite{alvis2019search}. The data quality checks and channel selection used in the double-beta decay analysis~\cite{alvis2019search} were also applied here. Additional data quality checks based on the prominent 2615-keV $\gamma$-peak following the $\beta^{-}$ decay of $^{208}$Tl are applied to the calibration data used in this analysis.
If the 2615-keV, full energy event rate in a run is found to deviate more than 3.5 $\sigma$ from the mean rate in the same data set, the run is excluded from this analysis. Such deviations can occur when, for example, the nitrogen dewars were filled, since the flow of liquid nitrogen induced noise.
After the data quality checks, we compared the observed source activity ($A_{observed}$) with the expected activity ($A_{expected}$), defined as:
\begin{equation}
A_{observed} = \frac{R}{\epsilon \times b}
\label{eq:obs}
\end{equation}
\begin{equation}
A_{expected} = A_{0} e^{[-\lambda (t-t_{0})]}
\label{eq:exp}
\end{equation}
In Eq.~\ref{eq:obs}, the observed activity of a calibration source during each weekly calibration was estimated based on the rate, $R$, of the full energy 2615~keV peak, the corresponding efficiency, $\epsilon$, of detecting the full energy 2615-keV photons, and the branching ratio, $b$, for the $^{212}$Bi$\rightarrow^{208}$Tl transition in the $^{228}$Th decay chain.
The \textsc{Geant4}-based \cite{agostinelli2003Geant4} simulation package, \textsc{MaGe}~\cite{boswell2011mage}, was used to estimate the detection efficiency ($\epsilon$) of the 2615-keV photons originating from the calibration sources in their deployed positions. In Eq.~\ref{eq:exp}, the expected activity of each calibration source is projected for every weekly calibration based on the initial activity, $A_{0}$, reported by the vendor at a given time $t_0$, the decay constant, $\lambda$, and the time of each calibration, $t$. The decay chain is in equilibrium, so the decay constant is based on the 1.9-year half-life of $^{228}$Th. Uncertainties in the branching ratio, decay constant, and calibration time are negligible, so the uncertainty in the expected activity is dominated by the uncertainty in $A_0$, which was $10.36 \pm 0.60$ kBq on May 1, 2013.
As shown in Fig.~\ref{fig:act}, a good agreement was found between the expected and the observed activity over multiple years of calibration data for both source assemblies. This implies good accuracy for the simulations performed by \textsc{MaGe} and gives confidence that \textsc{MaGe} can make correct efficiency predictions for the analysis of the 6129~keV $\gamma$-rays.
\begin{figure}[htpb]
\includegraphics[width=\linewidth]{Figures/m1_pub_v2.pdf}
\includegraphics[width=\linewidth]{Figures/m2_pub_v2.pdf}
\caption{Observed and expected activities for the two source assemblies used in \textsc{majorana demonstrator}. The data points indicate the observed activity of each source assembly for each weekly calibration, while the band represents the expected activity which includes the vendor reported uncertainty. The uncertainties in the observed activity are statistically only.}
\label{fig:act}
\end{figure}
\subsection{Signature search}
The search for the 6129-keV photons from the $^{13}$C($\alpha,n_{2}$)$^{16}$O reactions was performed using the sum energy of events, which is obtained by summing all coincident energy depositions over all active HPGe detectors within a 4~$\mu$s window~\cite{arnquist2021search}. This sum energy is used because of the high probability for several-MeV photons to distribute their full energy in multiple detectors. Fig.~\ref{fig:spec} shows the sum energy spectrum above 1~MeV in calibration data.
The signature at 6129~keV following the $^{13}$C($\alpha,n_{2}$)$^{16}$O reaction is clearly visible. Fig.~\ref{fig:sig} provides a spectrum in a smaller energy band around the 6129~keV region.
Events above 2615~keV are mostly due to summing, or random coincidences of two unrelated decays in the calibration source. For the latter one, the most prominent feature is the 5229-keV peak. When two 2615-keV photons, the energy of which is 2614.511~keV, are in coincidence, the sum energy appears to be twice of a photon energy. The zoomed-in plot of this peak is shown in Fig.~\ref{fig:double}.
\begin{figure}[htpb]
\includegraphics[width=\columnwidth]{Figures/spectrum.pdf}
\caption{The sum energy spectrum of calibration data selected for the analysis. It shows various $\gamma$-ray peaks, including 2615~keV, the signature peak of 6129~keV, and other peaks from the calibration source and peaks due to random coincidence events, and summing. }
\label{fig:spec}
\end{figure}
\begin{figure}[htb]
\includegraphics[width=\linewidth]{Figures/published_sig_sim.pdf}
\caption{The signature peak at 6129~keV from $^{13}$C($\alpha,n_{2}$)$^{16}$O reactions in the \textsc{majorana demonstrator}\ calibration sources, shown in blue color and fitted with Gaussian in red. The gray-filled histogram is the peak shape from the simulation of 1 million 6129-keV photons from the calibration tracks.}
\label{fig:sig}
\end{figure}
We defined the region of interest (ROI) for the 6129-keV peak search as (6129 $\pm$ 10)~keV based on the expected resolution in that energy region: about 2~keV ($1\sigma$) at 6~MeV, so the chosen window covers about 5$\sigma$ on each side of the peak. A simple Gaussian fit to the signal peak found the mean to be 6127$\pm$0.6 keV and the standard deviation to be 1.8$\pm$0.4 keV, as shown in Fig.~\ref{fig:sig}. A total of 9 events were found in the ROI in all data combined. Given the low statistics, the uncertainties from the fit are relatively large and less robust.
As a cross-check, a simple Gaussian plus a flat background was fit to the much stronger double coincidence 5229-keV peak in Fig.~\ref{fig:double}, where the mean was found to be 5228$\pm$0.2 keV with a standard deviation of 2.0$\pm$0.1 keV. These full energy peaks are seen at their expected locations and with their expected widths in the sum energy spectrum from the low-gain channels, indicating a great energy performance extended to the energy range of multiple-MeV.
\begin{figure}[htbp]
\includegraphics[width=\linewidth]{Figures/double_coin.pdf}
\caption{The double coincidence peak at 5229 keV ($2615.5$~keV$\times2$) in the sum energy spectrum of the calibration data is fitted with with a Gaussian plus a flat background.}
\label{fig:double}
\end{figure}
As seen in Fig.~\ref{fig:sig}, the signature peak at 6129~keV stands out clearly, so all of the 9 observed events in the peak are considered to be the signal, \textit{i.e.} 6129-keV photons following the $^{13}$C($\alpha,n_2$)$^{16}$O reactions. Given that no background events were found for at least 20 keV on both sides of the peak outside the ROI, the potential background in the 40-keV region from 6099 keV to 6159 keV excluding the 20-keV ROI can be determined as at most 1.29 counts at a $1\sigma$ level, which translates to a $1\sigma$ upper limit of 0.64 counts of background in the ROI. To better determine the potential background contribution, we also counted events in a much broader background region from 6 to 6.5~MeV, excluding the ROI around the 6129-keV peak. Based on 8 events in this 480-keV background region, we projected the potential background to be 0.33 counts in the ROI. Incidentally, this projects 0.67 counts of background from 6099 keV to 6159 keV excluding the 20-keV ROI, statistically consistent with observing none, which would happen with a 50\% probability. In short, the observed number of signal events in the ROI in the combined data sets is 9, while 0.33 counts is the estimated background contribution to the expected number of events. The difference between 0.33 counts and 0.64 counts is treated as a systematic uncertainty on the background contribution to the ROI.
\section{\label{sec:final_comp}
Comparisons with predictions}
\subsection{Prediction calculation}
NeuCBOT~\cite{westerdale2017radiogenic} is a software tool based on TALYS to calculate the neutron yield and neutron energy spectra for ($\alpha,n$) reactions in materials. It models the entire trajectory of $\alpha$-particles: initializing $\alpha$-particles according to ENSDF evaluated nuclear decay data \cite{tuli1996evaluated}, tracking their energy loss and range according to SRIM \cite{ziegler1985stopping}, and ultimately predicting the ($\alpha,n$) rate based on cross sections in TALYS-based TENDL. In this work, the 6129-keV photon production rate was estimated by NeuCBOT using the partial $^{13}$C($\alpha,n_{2}$)$^{16}$O cross sections from TALYS-1.95, and it was found to be 2.98 $\times 10^{-7}$ $\gamma$/Th-decay.
The detector configuration, such as the list of active detectors, can vary over time. The source activity also reduces as thorium decays away. Therefore, the number of predicted events was calculated for each weekly calibration and summed together using:
\begin{equation}
N = Y \times \sum_{i} A_{i} \times \epsilon_{i} \times T_{i}
\label{eq:final_expec}
\end{equation}
Here, $Y$ is the $\gamma$-ray production yield per decay of thorium, which is constant for all data sets since the source assembly does not change. For each weekly calibration $i$, the factors A$_{i}$, $\epsilon_{i}$, and T$_{i}$ are the source activity, detection efficiency for the 6129-keV photons, and live time, respectively. The efficiency, $\epsilon_{i}$, was calculated with \textsc{MaGe} for the 6129-keV photon using the same geometry as for the 2615-keV analysis, but the sum energy was used instead of individual detector energy for consistency. The simulated 6129-keV peak shape in the sum energy spectrum for 6129-keV photons uniformly seeded inside the calibration source is shown in Fig.~\ref{fig:sig}. The same ROI as in the data analysis was used to calculate the efficiency. Realistic energy responses, including dead layer models of each detector~\cite{alvis2019search}, are folded into the simulation, so the peak has slight deviations from Gaussian, notably a low energy tail.
\subsection{Comparison of observed and expected events}
Table~\ref{tab:final_table} compares the expected number of 6129-keV $\gamma$-rays with the number observed; the latter can be modelled by Poisson statistics with an unknown true mean. Based on the observed signal counts, the confidence interval on the mean of Poisson signals is calculated at a 90\% confidence level (C.L.) using the Feldman-Cousins statistical approach for small signals \cite{feldman1998unified}.
Sources of uncertainty in the expected counts are summarized in Table~\ref{tab:unc}. Uncertainties in the SRIM database are reported in Ref.~\cite{heaton1989neutron}. Uncertainties due to the chemical composition of the epoxy material and in the source activities were both based on the specifications provided by the vendor.
As discussed before, the projected background contribution in the ROI depends on the choice of background regions and the difference between the narrow 40-keV and the wide 480-keV background regions is taken as the uncertainty. To assess uncertainty associated with the calculation of the high energy photon detection efficiency using the sum energy, we repeated the calibration source activity analysis in Section \ref{sec:level3} using the sum energy. On average, a 11.9\% difference in the source activity is observed between calculations based on single detector energy and the sum energy at 2615-keV. The difference between the vendor specification and the source activity based on the sum energy was found to be smaller, so the 11.9\% is an overestimation of the systematic uncertainty in simulation.
\begin{table*}[htpb]
\caption{Expected and observed counts of 6129-keV photons. Expected counts are estimated based on Eq.~\ref{eq:final_expec}, and the corresponding uncertainties are the combination of various uncertainties shown in Table~\ref{tab:unc}. The range of signal mean is the 90\% C.L. interval of Poisson signal mean based on observed signal counts in each data~\cite{feldman1998unified}. The individual data set labeling follows \textsc{demonstrator}\ configuration changes as explained in Ref.~\cite{alvis2019search}.}
\label{tab:final_table}
\centering
\begin{tabular}{|c |c | c| c|}
\hline
Calibration Data Set & Expected Counts & Observed Counts & 90\% Interval of Signal Mean given Observation\\ [0.5ex]
\hline
DS1 &0.42$\pm$0.07 &0 &[0.00,~2.44] \\
\hline
DS2 &0.13$\pm$0.02 &1 &[0.11,~4.36] \\
\hline
DS5 &0.41$\pm$0.07 &1 &[0.11,~4.36] \\
\hline
DS6a &0.32$\pm$0.05 &1 &[0.11,~4.36] \\
\hline
DS6b &1.19$\pm$0.20 &4 &[1.47,~8.60] \\
\hline
DS6c &1.27$\pm$0.21 &2 &[0.53,~5.91] \\
\hline
Total & 3.74$\pm$0.63 &9 &[4.36,~15.30]\\
\hline
\end{tabular}
\end{table*}
\begin{table*}[htpb]
\caption{Relative uncertainties for the expected number of counts. The total systematic uncertainty is the sum in quadrature of individual systematic contributions.}
\centering
\begin{tabular}{c c}
\hline
$\gamma$ yield value due to uncertainties in the SRIM reported in ~\cite{heaton1989neutron} & 5.0\%\\
Chemical composition in epoxy & 4.0\% \\
Activity of the source as reported by Eckert \& Ziegler & 5.8\% \\
Systematic uncertainty in simulation& 11.9\%
\\
Statistical uncertainty in simulation & 1-2 \% (neglected)\\
Systematic uncertainty in background contribution & 8.3\% \\
Total systematic uncertainty & 16.9\% \\
\hline
\end{tabular}
\label{tab:unc}
\end{table*}
Figure~\ref{fig:final} visualizes the comparison between the expected and the observed number of 6129-keV photons.
The observed number of events tends to be higher than the expected number, however the statistical uncertainty in the experimental data is large. The range of expected counts is overall consistent with the 90\% confidence interval on the observed signal strength. This comparison suggests that TALYS cross sections combined with SRIM enables reasonable estimations of ($\alpha,n$) rates. This consistency at a 90\% confidence level lends support to the approach of predicting neutron production from $\alpha$-induced reactions in low-background experiments using the presented tools.
\begin{figure}[htp]
\includegraphics[width=\linewidth]{Figures/final_result_pub.pdf}
\caption{Expected and observed number of 6129-keV photons with corresponding uncertainties in each data set and in the combined data set. The error bars in the observed counts indicate the 90\% C.L. intervals on the mean of Poisson signals as listed in Table~\ref{tab:final_table}.}
\label{fig:final}
\end{figure}
\section{\label{sec:backgrounds}Background estimation for $0\nu \beta\beta$ search}
Neutrons produced by ($\alpha,n$) reactions during the \textsc{Demonstrator}'s calibration runs can enter the germanium crystals and get captured. After each calibration, these sources were retracted to parked locations entirely outside the shield. Therefore, only the neutrons produced during the calibration are of concern. When $^{76}$Ge undergoes neutron capture, the ground state of $^{77}$Ge or the metastable state, $^{77m}$Ge, can be produced, both of which could $\beta$ decay with energy releases larger than the 2039~keV Q$_{\beta\beta}$ of $0\nu\beta\beta$ in germanium~\cite{arnquist2022signatures}.
The main background contributor here is the long-lived isotope $^{77}$Ge with a half-life of 11.2 hr, which can decay during the $0\nu\beta\beta$ decay data-taking periods following the hours-long calibration periods. The metastable state $^{77m}$Ge with a 54-second half-life is less of a concern.
\begin{figure}[htp]
\includegraphics[width=\linewidth]{Figures/neutron_spec.pdf}
\caption{Neutron energy spectrum from the ($\alpha,n$) reactions in the epoxy. The spectrum is obtained by using NeuCBOT based on TALYS-1.95 generated cross sections data.}
\label{fig:n_spec}
\end{figure}
Fig.~\ref{fig:n_spec} shows the NeuCBOT calculation of energies and yields of neutrons generated from all types of ($\alpha$, n) reactions within the calibration sources.
\textsc{MaGe} was used to estimate the production and decay of $^{77}$Ge inside the germanium crystals given this neutron flux. This background contribution was estimated to be on the order of $10^{-5}$ cts/(keV-kg-year) before any analysis cuts. This shows that calibration neutrons are a negligible contribution compared with the total background measured in the \textsc{demonstrator}\ \cite{alvis2019search}.
The GERDA experiment investigated a similar background source in their Phase I data taking~\cite{baudis2015production}. They estimated a background contribution of $10^{-4}$ cts/(keV-kg-year) for $0\nu\beta\beta$ by neutrons from calibration sources. The higher background index can be explained by the stronger activity and slightly different geometry used in GERDA. For GERDA Phase II data taking, this background was minimized by deploying a new design of gold-encapsulated thorium calibration source~\cite{baudis2015production}. This design reduces the possible interaction of $\alpha$-particles and it is adapted by the LEGEND calibration system~\cite{abgrall2021legend}.
Next-generation experiments searching for $0\nu\beta\beta$ have much more stringent background requirements. Hence, potential background sources of radiogenic ($\alpha,n$) neutrons from detector construction materials should be examined carefully.
While extensive efforts are in place to shield room and cosmogenic neutrons, some neutron sources could be inside the water shielding or are even introduced by shielding materials~\cite{arnquist2022signatures}. One example is the large steel cryostat which houses the LEGEND main argon volume. The combination of TALYS-based software can be valuable to provide rough estimations in these cases, as investigated in Refs.~\cite{abgrall2021legend,Barton2021}, using NeuCBOT in combination with \textsc{Geant4}.
\section{\label{sec:level1}Introduction}
Neutron-related reactions are an important source of background in underground neutrino and dark matter experiments~\cite{carson2004neutron,cooley2018input, chen2021radiogenic, Febbraro2020}. One common source of neutrons is ($\alpha,n$) reactions. Neutrons may penetrate shielding layers before being captured on sensitive detector materials, often creating radioactive isotopes, the delayed decays of which could be difficult to reject due to a lack of coincidence timing information. For example, in germanium-based neutrinoless double-beta decay ($0\nu\beta\beta$) experiments, neutron captures on $^{76}$Ge create $^{77}$Ge (half-life: 11.3 hr) and $^{77m}$Ge (half-life: 53.7 s) isotopes. The $\beta$-decay of these isotopes could potentially produce signals similar to $0\nu\beta\beta$ and with energies near the double-beta decay Q-value (Q$_{\beta\beta}$) of $^{76}$Ge. This background has been studied in detail~\cite{wiesinger2018virtual,arnquist2022signatures}.
$^{232}$Th and $^{238}$U decay chains contain several $\alpha$-emitters. These naturally-occurring isotopes are present in detector materials, and various $\alpha$-particles with energies up to 9 MeV are emitted, initiating a range of $(\alpha,n)$ reactions. Even though the cleanest materials can be assayed and selected \cite{abgrall2016majorana}, stringent background requirements, especially for future experiments, demand an understanding of these neutron contributions with reasonable detail and precision. In particular, different types of plastic materials are widely used in low-background experiments, \textit{e.g.} for electrical insulation and neutron shielding. In these carbon-rich plastic materials, the $^{13}$C($\alpha, n)^{16}$O reaction is a major source of neutrons.
Besides its role as a background, the $^{13}$C($\alpha, n)^{16}$O reaction is considered the most important neutron source for s-process nucleosynthesis in low-mass asymptotic giant branch stars~\cite{busso2001nucleosynthesis,guo2012new,la2013measurement,Cristallo2018,Arnould2020}. This reaction and its cross section have been studied, and the results agree reasonably well among different measurements for low-energy $\alpha$-particles below about 5~MeV~\cite{sekharan1967c,davids1968study,heil200813,bair1973total,drotleff1993reaction,harissopulos2005cross, Broggini2018}. At higher $\alpha$ energies, precise cross section measurements are sparse~\cite{harissopulos2005cross,peters2017comment}, although new studies have been published for the 5-to-6~MeV region ~\cite{Febbraro2020} and more measurements are planned in the near future~\cite{Broggini2018}.
In addition to relying on measured data, one can obtain ($\alpha,n$) cross sections from a statistical modeling approach using a nuclear reaction code such as TALYS~\cite{koning2013talys}. The TALYS-generated Evaluated Nuclear Data Libraries (TENDL) merges the TALYS nuclear model with data available in the JENDL \cite{nakagawa1995japanese} and ENDF \cite{chadwick2011endf} databases. In the case of $^{13}$C($\alpha, n)^{16}$O, TALYS can predict partial cross sections of different reaction channels noted as $^{13}$C($\alpha, n_{j}$)$^{16}$O, where $j$ identifies neutrons associated with different states of $^{16}$O. The TALYS-generated cross sections as a function of $\alpha$-particle energy are shown in Fig.~\ref{fig:cross_13C}. Although such a statistical model can be imprecise when predicting the detailed resonance structure as pointed out by Ref.~\cite{Febbraro2020}, its overall agreement can be used to approximate the reaction rate, allowing neutron background predictions for low-background experiments. It is reasonable to use this approach especially when precise measurements are sparse over the entire range of $\alpha$ energies relevant for ($\alpha, n$) backgrounds, which is typically broad.
\begin{figure}[htbp]
\includegraphics[width=\linewidth]{Figures/cross_13C_pub.pdf}
\caption{Total cross section and the partial cross sections for the $^{13}$C($\alpha,n)^{16}$O reactions as a function of incident $\alpha$-particle energy available from the decay chain of $^{228}$Th. These cross sections are generated by TALYS-1.95. The results of the new TALYS version are consistent with branching ratios obtained from Ref.~\cite{mohr2018revised} that used TALYS-1.8.}
\label{fig:cross_13C}
\end{figure}
In this paper, we report an analysis of several years of calibration data taken by the \textsc{majorana demonstrator}\, experiment, which resulted in a measurement of characteristic 6129-keV photons emitted following the $^{13}$C($\alpha,n_2)^{16}$O reactions, where the second excited state (3$^{-}$) of $^{16}$O is populated. We compare the measurement with a prediction from NeuCBOT (Neutron Calculator Based On TALYS)~\cite{westerdale2017radiogenic,ajaj2019search}. Section~\ref{sec:level2} of this paper discusses the ($\alpha,n$) reactions within the calibration sources of the \textsc{demonstrator}. Section~\ref{sec:level3} introduces the experimental techniques and analysis used to identify the 6129-keV photons. Section~\ref{sec:final_comp} describes how the TALYS-based NeuCBOT and a \textsc{Geant4}-based software for the \textsc{demonstrator}\ are used to predict the number of observable events. Section~\ref{sec:backgrounds} discusses how the same procedure can be used to estimate the background contribution to $0\nu\beta\beta$ measurements. The last section shows how similar techniques could play an essential role in future experiments with more stringent background goals.
\section*{Acknowledgments}
\section{\label{sec:level2}
$^{13}$C($\alpha,n$)$^{16}$O reactions in Calibration
The \textsc{majorana demonstrator}\ experiment searched for $0\nu\beta\beta$ in $^{76}$Ge using P-type Point Contact (PPC) High Purity Germanium (HPGe) detectors
\cite{aalseth2018search}. The \textsc{demonstrator}\ was operated at the 4850-foot level of the Sanford Underground Research Facility in Lead, South Dakota, with two modules of HPGe detectors placed in an ultra-clean and heavily shielded environment as shown in Fig~\ref{fig:mj}. The HPGe detectors had a combined total mass of 44.1 kg, of which 29.7 kg was enriched to 88\% in $^{76}$Ge with the rest being natural Ge. In March 2021, the \textsc{demonstrator}\ completed its data-taking campaign with enriched detectors and it continues taking data with natural detectors for background studies and other physics. The \textsc{Demonstrator}'s HPGe detectors in combination with low-noise electronics have achieved good linearity over a broad energy range~\cite{abgrall2020adc} and best-in-field energy resolution with a full width at half maximum (FWHM) approaching 0.1\% at the 2039~keV Q$_{\beta\beta}$ of $^{76}$Ge~\cite{alvis2019search}. This excellent energy performance coupled with the low energy threshold and low-background of the \textsc{demonstrator}\ makes it a competitive $0\nu\beta\beta$ experiment and allows for other physics beyond the Standard Model \cite{arnquist2021search,alvis2018first,abgrall2017new,abgrall2016search,alvis2019search_1}.
\begin{figure}[htb]
\includegraphics[width=\linewidth]{Figures/mjd.png}
\caption{A schematic of the \textsc{majorana demonstrator}\ with two modules of HPGe detectors surrounded with layers of shielding~\cite{arnquist2021search}.}
\label{fig:mj}
\end{figure}
Ultra radiopure materials were used in the construction of the \textsc{Demonstrator}, particularly in the vicinity of germanium detectors, which are placed inside layers of compact shielding~\cite{abgrall2014majorana}. A weekly calibration is required to monitor detector stability and provide data for developing analysis cuts. The thorium isotope $^{228}$Th was selected as the calibration source because its decay chain emits several $\gamma$-rays that span from a few hundred keV up to 2615~keV, covering the Q$_{\beta\beta}$ of $^{76}$Ge and allowing for analysis over a wide energy range. The \textsc{Demonstrator}'s calibration line sources were manufactured by Eckert \& Ziegler Analytics, Inc \footnote{\url{http://www.ezag.com/home/}}. Each line source is made of thoriated epoxy encapsulated in a tube made of PTFE~\cite{abgrall2017majorana}. During calibrations, the line source was deployed into the calibration track, which surrounds the cryostat in a helical path~\cite{abgrall2017majorana}, as shown in Fig~\ref{fig:track}.
\begin{figure}[htbp]
\includegraphics[width=\linewidth]{Figures/track.png}
\caption{A diagram that shows one module, the detector strings within, and the calibration track (highlighted) through which a line source is deployed during calibrations.}
\label{fig:track}
\end{figure}
The $\gamma$-rays emitted within the thorium decay chain are used for calibration and detector characterization. The decay chain ends when it reaches $^{208}$Pb producing several $\alpha$-emitters along the way. Table~\ref{tab:alpha_table} shows the energies of the main $\alpha$-particles, which lie between 5.34 MeV and 8.79 MeV. When traversing the epoxy in the calibration source, an $\alpha$-particle could initiate $(\alpha,n)$ reactions in $^{13}$C, $^{17}$O, $^{18}$O, $^{35}$Cl, and $^{37}$Cl, of which reactions with $^{13}$C dominate. $^{13}$C($\alpha,n_{2})^{16}$O reactions are possible with $\alpha$-particles above about 5~MeV, resulting in 6129-keV photons. The \textsc{Demonstrator}'s excellent energy performance allowed a clear observation of this 6129-keV signature on top of the thorium photon energy spectrum during calibrations.
In $^{13}$C($\alpha,n$)$^{16}$O reactions, an $\alpha$-particle is captured in $^{13}$C to form the compound nucleus $^{17}$O$^*$, which decays to the ground state or excited states of $^{16}$O by emitting a neutron. Figure~\ref{fig:rxn} shows the simplified level scheme of $^{16}$O that can be populated from the decay of $^{17}$O$^{*}$. Since the $\alpha$-particles in the thorium chain have energy up to 8.79 MeV as listed in Table~\ref{tab:alpha_table}, they can potentially open up the reactions channels of ($\alpha,n_{1}$), ($\alpha,n_{2}$), ($\alpha,n_{3}$), and ($\alpha,n_{4}$). The population of the second excited state (3$^{-}$) of $^{16}$O at 6129 keV is favored for $\alpha$-particles with energy above 6~MeV, as shown by the turquoise line in Fig.~\ref{fig:cross_13C}. The isomeric transition of the (3$^{-}$) state to the ground state of $^{16}$O emits a characteristic 6129-keV photon, presenting a unique signature to look for in calibration data. The $^{13}$C($\alpha,n_2$)$^{16}$O reaction is described in Eq.~\ref{eq:13c}.
\begin{equation}
\begin{split}
^{13}\rm{C} + \alpha & \rightarrow ^{17}\rm{O}^{*} \\
& \rightarrow ^{16}\rm{O}^{*} (3^{-}) + n \\
& \rightarrow ^{16}\rm{O}(g.s.) + \gamma~(6129~\rm{keV}) + n
\label{eq:13c}
\end{split}
\end{equation}
\begin{figure}[htb]
\includegraphics[width= \linewidth]{Figures/reaction_pub.pdf}
\caption{The level scheme of $^{16}$O as populated in the $^{13}$C$(\alpha,n)^{16}$O reaction (energy not to scale) simplified from Figure~1 of Ref.~\cite{Febbraro2020}. The numerical index of the emitted neutrons n$_{0}$, n$_{1}$, n$_{2}$ represents which final state in $^{16}$O is populated. Due to selection rules, the 0$^{+}$ (6049~keV) state deexcites via the emission of an e$^{+}$e$^{-}$ pair, while the 3$^{-}$ (6129~keV) state deexcites through $\gamma$-ray emission. Data from \cite{mohr2018revised,Febbraro2020}.}.
\label{fig:rxn}
\end{figure}
\begin{table}[htpb]
\caption{Energies of primary $\alpha$-particles from the decay chain of $^{228}$Th. Energy data are taken from Nuclear structure \& decay Data (NuDat 3.0)\footnote{\url{https://www.nndc.bnl.gov/nudat3/}}.}
\label{tab:alpha_table}
\centering
\begin{tabular}{|c | c| c|}
\hline
\thead{$\alpha$-particle energy\\ (MeV)}& Parent isotope & \thead{Intensity \\ (per $^{228}$Th decay)} \\ [0.5ex]
\hline
5.423 &$^{228}$Th & 0.734 \\
\hline
5.340 &$^{228}$Th & 0.260 \\
\hline
5.685 &$^{224}$Ra &0.949 \\
\hline
5.449 &$^{224}$Ra &0.051 \\
\hline
6.288 &$^{220}$Rn & 0.999 \\
\hline
6.778 &$^{216}$Po &0.999 \\
\hline
6.050 &$^{212}$Bi &0.090 \\
\hline
6.089 &$^{212}$Bi &0.035 \\
\hline
8.785 &$^{212}$Po &0.641 \\
\hline
\end{tabular}
\end{table}
\section{\label{sec:level4} Discussion and Summary}
The work presented above combines the achievements of the \textsc{majorana demonstrator}\ experiment in terms of excellent energy performance and robust as-built simulations. The search for signatures in a wide energy range, well beyond the Q$_{\beta\beta}$ of $^{76}$Ge, is possible due to excellent energy linearity and resolution of the \textsc{majorana demonstrator}. These achievements result from the intrinsic advantages of HPGe detectors in combination with low-noise electronics and dedicated efforts on energy estimation corrections and calibrations. The analysis presented here found a good agreement between detected and expected energy for the signature at 6129~keV and verified the algorithms at the sum peak of two 2615-keV $\gamma$-rays.
We have shown that the measured rate is consistent with simulations over various detector configurations in multiple years of calibrations. The agreement between the expected decay activity and the observed activity of the calibration sources in the \textsc{majorana demonstrator}\ is reported for the first time, demonstrating the excellent performance of the \textsc{MaGe} simulation software, which is also used by GERDA and LEGEND.
Our work shows how signatures of ($\alpha,n$) reactions can be detected in low-background experiments and how simulations are crucial in understanding this radiogenic neutron background. The agreement between simulated and measured rate is valuable feedback since ($\alpha,n$) data can be sparse, and
can have significant discrepancies, as pointed out by Ref.~\cite{Febbraro2020}. At 90\% C.L., our measurement of the 6129-keV photons from the second excited state in $^{16}$O is consistent with the predictions generated by the TALYS-based NeuCBOT program, although the statistical uncertainty is large. This suggests that the TALYS-based NeuCBOT provides a reasonable estimation of neutrons from thorium impurities in carbon-rich organic materials. Our findings are widely applicable, as thorium is one of the most common impurities, and carbon-rich organic materials such as various plastics and epoxies are often used in experiments in abundance. It is reasonable to expect that ($\alpha,n$) reactions induced by alpha particles from thorium impurities in a range of carbon-rich organic materials share similar profiles. While future experiments may utilize materials with higher radiopurity than the current experiments, the size and length of future experiments can result in a similar ($\alpha,n$) background contribution for these rare event searches.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,484 |
Professor Andrew P. Read (born 1939) is a British medical geneticist.
Read studied organic chemistry at the University of Cambridge. Once he had obtained his doctorate, he worked at the Max Planck Institute for Medical Research and at the University of Warwick.
In 1967 he obtained a post at the University of Manchester, moving to its Medical Genetics Department in 1977. His research subjects included neural tube defects and the genes involved in hereditary deafness. he became Emeritus upon formal retirement.
He served as chair of the Clinical Molecular Genetics Society and was founder chair of the British Society for Human Genetics from 1996 to 2000.
He is a Fellow of the Royal College of Pathologists (FRCPath) and a Fellow of the Academy of Medical Sciences (FMedSci).
References
External links
1939 births
Place of birth missing (living people)
Living people
British geneticists
Fellows of the Royal College of Pathologists
Fellows of the Academy of Medical Sciences (United Kingdom)
Alumni of the University of Cambridge
Academics of the University of Warwick
Academics of the University of Manchester
Medical geneticists | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,947 |
Sir Alex Ferguson (born 1941) is a former Scottish football player and manager.
Alex Ferguson or Alex Fergusson may also refer to:
Alex Ferguson (baseball) (1897–1976), American baseball player
Alex Ferguson (footballer, born 1903) (1903–1974), Scottish football player
Alex Ferguson (footballer, born 1913), Scottish footballer, played for Lochee Harp, St Johnstone, Hibernian, Heart of Midlothian and Rochdale
Alex Ferguson (jockey), British jockey who rode the winning horse in the 2017 Gerry Feilden Hurdle
Alex Fergusson (musician) (born 1952), Scottish guitarist and producer
Alex Fergusson (politician) (1949–2018), Scottish politician
See also
Alexander Ferguson (disambiguation)
Alejandro Ferguson (born 1978), Argentine cricketer
Alexander Ferguson MacLaren (1854–1917), Canadian businessman and politician
Sandy Ferguson (disambiguation) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,286 |
Discover our selection of second hand Chanel Timeless bags, valued and selected for their condition and dispatches within 24 hrs.
The Timeless is one of the two It-Bags from the house of Chanel. It is a reinterpretation by Karl Lagerfeld in 1982 of the 2.55, the great classic from Chanel.
The classic Timeless is rectangular and comes in four sizes: 23.5x14.5x6.5 cm, 25.5x16x7.5 cm, 30x20x10 cm and 34x23x10 cm (with a single flap, in this case). Nevertheless, there are numerous seasonal versions in different sizes, some with a square, rather than rectangular, shape.
Like the 2.55, the Timeless is an object with a soul and a history, those of Coco, who is present in its tiniest detail. It is, however, different to the 2.55 in several ways: its interior is sometimes burgundy and sometimes black (or 'caviar' according to Timeless). Its double gold or silver plated brass chain works both as a handle and a shoulder-strap and is interwoven with a black or beige leather strap. Its clasp is no longer rectangular but round and features the two interlocking Cs that are the Chanel trademark. Its beige or black, quilted leather is in grained calf or lamb skin, although there are numerous seasonal versions of Timeless in much more unusual materials, such as python and lurex. These versions can also have surprising trimmings, such as sequins or beads.
Its strong points: like the 2.55, the Timeless comes as a bag within a bag. It has many pockets enabling women to carry all those little essentials. In addition to the little zipped pouch inside its flap, the Timeless has a back pocket in the shape of a smile on its upper edge (to keep your change in), a front pocket and an interior pocket. A practical interior compartment completes the ensemble. It has three little bellows pockets, including a small one to hold the indispensable lipstick.
The Timeless classic exists in four sizes. Nevertheless, a lot of seasonal versions were with different sizes, more or less rectangular. Those variations make the Timeless a classic and a it-bag, always fashionable. If the most common version is made of quilted leather, this must-have is also available in python leather, jersey, crocodile leather or lurex.
The Timeless is a bag within a bag, with two main pockets as well as several smaller pouches, one of which is perfectly shaped to hold the famous Chanel lipstick. The rounded clasp displays the two entwined Cs of Chanel's logo, and the dual-metal chain is interwoven with the bag's cream leather. The Timeless also has a back pocket with curved edges in the shape of a smile for holding coins.
If the price of the Timeless on the second-hand market has increased by 107% between 2010 and 2014, the growth rate of the Timeless may vary by twice as much, depending on its state, its colour and its material.
The most impressive increase in price is noticeable on the beige leather Timeless: its price has increased 2.5 times in just 4 years! Just behind it, the Timeless in black leather has doubled in price between 2010 and 2014. We have also noticed a large rise in the white leather Timeless, which has seen its price increase by 60% on the second-hand market. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,482 |
using System;
using System.Collections.Generic;
using System.ComponentModel.DataAnnotations;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Transactions;
using ICSharpCode.SharpZipLib.Core;
using ICSharpCode.SharpZipLib.Zip;
namespace EfInsertBigData
{
class Program
{
private const string InputFile = @"D:\TestTP\output.txt";
private const int batchSize = 100;
static void Main(string[] args)
{
InsertBigData();
Console.WriteLine("Press enter for compression");
Console.ReadLine();
CompressBigData();
Console.WriteLine("Pres enter to exit");
Console.ReadLine();
}
private static void CompressBigData()
{
using (var db = new TransactionProtocolContext())
{
db.Database.ExecuteSqlCommand("DELETE FROM TP_ZPRACOVANE");
}
var batchesProcesed = 0;
var timer = new Stopwatch();
timer.Start();
//first line are headers
while (true)
{
using (var db = new TransactionProtocolContext())
{
var cekajici = db.TP_CEKAJICIZMENY.OrderBy(t => t.DATUMEND).Take(batchSize).ToList();
foreach (var tpCekajicizmeny in cekajici)
{
var xmlAsBytes = XmlAsBytes(tpCekajicizmeny.ZMENYXML);
var sha256OfFile = ComputeHash(xmlAsBytes);
var compressedXml = CompressXml(tpCekajicizmeny.ZMENYXML);
var zpracTp = new TP_ZPRACOVANE()
{
ID = tpCekajicizmeny.ID,
DATUMSTART = tpCekajicizmeny.DATUMSTART,
DATUMEND = tpCekajicizmeny.DATUMEND,
ZMENYXMLCOMP = compressedXml,
ZMENYXMLHASH = sha256OfFile
};
db.TP_CEKAJICIZMENY.Remove(tpCekajicizmeny);
db.TP_ZPRACOVANE.Add(zpracTp);
}
db.SaveChanges();
if (cekajici.Count < batchSize)
{
break;
}
}
batchesProcesed++;
Console.WriteLine("{0:N0} items converted, time {1:T}", batchesProcesed * batchSize, timer.Elapsed);
}
Console.WriteLine("FINISHED, time {0:T}", timer.Elapsed);
}
private static void InsertBigData()
{
using (var db = new TransactionProtocolContext())
{
var zmenyXmlToDeleteIds = db.TP_CEKAJICIZMENY.Select(cz => cz.ID).ToList();
var xmlToDeleteEnumerator = zmenyXmlToDeleteIds.GetEnumerator();
var archiveTpPageSize = 100;
for (int i = 0; i * archiveTpPageSize < zmenyXmlToDeleteIds.Count; i++)
{
var ids = new List<object>();
var sqlCommand = new StringBuilder("DELETE FROM TP_CEKAJICIZMENY WHERE ID IN (");
for (int j = 0; j < archiveTpPageSize && (i * archiveTpPageSize) + j < zmenyXmlToDeleteIds.Count; j++)
{
if (!xmlToDeleteEnumerator.MoveNext())
{
break;
}
sqlCommand.Append("@p");
sqlCommand.Append(j.ToString());
sqlCommand.Append(", ");
ids.Add(xmlToDeleteEnumerator.Current);
//var zmenaXmlId = zmenyXmlToDeleteIds.[i*archiveTpPageSize + j];
//var zmenaXml = new CekajiciTransProtokolZmeny() {Id = zmenaXmlId};
//context.Set<CekajiciTransProtokolZmeny>().Attach(zmenaXml);
//context.Set<CekajiciTransProtokolZmeny>().Remove(zmenaXml);
}
sqlCommand.Length--;
sqlCommand.Length--; //odstraneni poslednich ', '
sqlCommand.Append(")");
var array = ids.ToArray();
db.Database.ExecuteSqlCommand(sqlCommand.ToString(), array);
if (ids.Count < archiveTpPageSize)
{
break;
}
//context.SaveChanges();
}
}
var batchesProcesed = 0;
var timer = new Stopwatch();
timer.Start();
using (var reader = new StreamReader(InputFile))
{
var line = reader.ReadLine();
//first line are headers
while (true)
{
var zmeny = new List<TP_CEKAJICIZMENY>(batchSize);
for (var processedLines = 0;
(line = reader.ReadLine()) != null && processedLines < batchSize;
processedLines++)
{
var split = line.Split(new string[] { "','" }, StringSplitOptions.None);
if (split.Length != 4)
throw new Exception("Wrong split part count");
var tp = new TP_CEKAJICIZMENY()
{
ID = int.Parse(split[0].TrimStart('\'')),
DATUMSTART = DateTime.Parse(split[1]),
DATUMEND = DateTime.Parse(split[2]),
ZMENYXML = split[3].TrimEnd('\'')
};
zmeny.Add(tp);
}
batchesProcesed++;
Console.WriteLine("{0:##,###} items loaded, time {1:T}", batchesProcesed * batchSize, timer.Elapsed);
using (var db = new TransactionProtocolContext())
{
foreach (var tp in zmeny)
{
db.Set<TP_CEKAJICIZMENY>().Add(tp);
}
db.SaveChanges();
}
Console.WriteLine("{0:##,###} items saved, time {1:T}", batchesProcesed * batchSize, timer.Elapsed);
if (zmeny.Count < batchSize)
{
break;
}
}
}
using (var db = new TransactionProtocolContext())
{
var elementsCont = db.Set<TP_CEKAJICIZMENY>().Count();
var first = db.Set<TP_CEKAJICIZMENY>().First();
}
}
private static byte[] CompressXml(string zmenyXml)
{
byte[] fileToSaveToDul;
var xmlAsBytes = XmlAsBytes(zmenyXml);
using (var inputZmenyXml = new MemoryStream(xmlAsBytes))
using (MemoryStream outputMemStream = new MemoryStream())
{
var internalFileName = "Test.xml";
using (ZipOutputStream zipStream = new ZipOutputStream(outputMemStream))
{
zipStream.SetLevel(6); //0-9, 9 being the highest level of compression
ZipEntry newEntry = new ZipEntry(internalFileName);
newEntry.DateTime = DateTime.Now;
zipStream.PutNextEntry(newEntry);
StreamUtils.Copy(inputZmenyXml, zipStream, new byte[4096]);
zipStream.CloseEntry();
zipStream.IsStreamOwner = false;
zipStream.Close();
}
outputMemStream.Position = 0;
// Alternative outputs:
// ToArray is the cleaner and easiest to use correctly with the penalty of duplicating allocated memory.
return outputMemStream.ToArray();
}
}
public static byte[] XmlAsBytes(string xml)
{
return Encoding.Unicode.GetBytes(xml);
}
public static string ComputeHash(byte[] fileBytes)
{
using (var sha1 = new System.Security.Cryptography.SHA256Managed())
{
byte[] hash = sha1.ComputeHash(fileBytes);
var formatted = new StringBuilder(2 * hash.Length);
foreach (byte b in hash)
{
formatted.AppendFormat("{0:X2}", b);
}
return formatted.ToString();
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,055 |
Q: geo.NamedPoint is taking current position if not provided the city name in bixby My utterance is What is happening in New York. While Training I have set New York as geo.SearchTerm and in action file, i have collected input as cityName and type is geo.NamedPoint. Now if I am asking what is happeningin Birlin, so it is giving me NamedPoint of Birlin but if I am saying "what is happening" without taking the city name so for the first time it is giving me location of Berlin and second time it is giving me the current location. What I wanted that if question is asked with location then give me the location details else no details at all. How to acheive that?
Here is my action file
action (EventSearch) {
type(Search)
collect{
input (dateTimeExpression) {
type (MyDateTimeExpression)
min (Optional)
}
input (cityName) {
type (geo.NamedPoint)
min (Optional) max (One)
default-select {
with-rule {
select-first
}
}
}
}
output (EventConfirmationResult)
}
A: As you mentioned in your comment to my question, you would indeed need to create a wrapper similar to the DateTimeExpression structure described in the help center article for defining a transient structure.
I've also shown the relevant code below:
structure (CustomDateTimeExpression) {
role-of (time.DateTimeExpression)
description (wrapper for DateTimeExpression)
features {
transient
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,217 |
A heads up display for git.
![An example of git-radar]
Git-radar is a tool you can add to your prompt to provide at-a-glance
information on your git repo. It's a labour of love I've been dogfooding for the
last few years. Maybe it can help you too.
**Table of Contents**
- [Installation](#installation)
- [Usage](#usage)
- [Features](#features)
- [Files status](#files-status)
- [Local commits status](#local-commits-status)
- [Remote commits status](#remote-commits-status)
- [Stash status](#stash-status)
- [(Optional) Auto-fetch repos](#optional-auto-fetch-repos)
- [Customise your prompt](#customise-your-prompt)
- [Support](#support)
- [Ensuring prompt execution](#ensuring-prompt-execution)
- [Configuring colours](#configuring-colours)
- [Exporting Environment Variables](#exporting-environment-variables)
- [Setting an RC file](#setting-an-rc-file)
- [Bash Colour Codes](#bash-colour-codes)
- [Zsh Colour Codes](#zsh-colour-codes)
- [Configuration values](#configuration-values)
- [Colouring the Branch part](#colouring-the-branch-part)
- [Colouring the local commits status](#colouring-the-local-commits-status)
- [Colouring the remote commits status](#colouring-the-remote-commits-status)
- [Colouring the file changes status](#colouring-the-file-changes-status)
- [License](#license)
## Installation
### Install from brew:
```
> brew install michaeldfallen/formula/git-radar
```
### Manually:
```
> cd ~ && git clone https://github.com/michaeldfallen/git-radar .git-radar
> echo 'export PATH=$PATH:$HOME/.git-radar' >> ~/.bashrc
```
Then run `git-radar` to see the docs and prove it's installed.
## Usage
To use git-radar you need to add it to your prompt. This is done in different
ways depending on your shell.
**Bash**
Add to your `.bashrc`
```bash
export PS1="$PS1\$(git-radar --bash --fetch)"
```
[(note: the `\` escaping the `$` is important)](#ensuring-prompt-execution)
**Zsh**
Add to your `.zshrc`
```zsh
export PROMPT="$PROMPT\$(git-radar --zsh --fetch) "
```
[(note: the `\` escaping the `$` is important)](#ensuring-prompt-execution)
**Fish**
Add to your `config.fish`
```bash
function fish_prompt
set_color $fish_color_cwd
echo -n (prompt_pwd)
echo -n (git-radar --fish --fetch)
set_color normal
echo -n ' > '
end
```
## Features
### Files status
The prompt lists the file changes and whether they are staged, unstaged or
untracked.
Prompt | Meaning
---------------------------|--------
![git:(master) 3A] | We have 3 untracked files
![git:(master) 2D2M] | We have 2 modifications and 2 deletions not yet staged to commit
![git:(master) 1M1R] | We have 1 modification and a file renamed staged and ready to commit
![git:(master) 1U] | We have a conflict caused by US that we need to address
![git:(master) 1M 1D2M 2A] | A combination of the above types
Each symbol represents a different change to a file. These are based on what git
considers has happened to the file.
Symbol | Meaning
--------|--------
A | A new Added file
D | A file has been Deleted
M | A file has been Modified
R | A file has been renamed
C | A file has been copied
U | A conflict caused by Us
T | A conflict caused by Them
B | A conflict caused by Both us and them
The color tells you what stage the change is at.
Color | Meaning
--------|--------
Green | Staged and ready to be committed (i.e. you have done a `git add`)
Red | Unstaged, you'll need to `git add` them before you can commit
Grey | Untracked, these are new files git is unaware of
Yellow | Conflicted, these need resolved before they can be committed
The use of feature is controlled by the `GIT_RADAR_FORMAT` environment variable.
See [Customise your prompt](#customise-your-prompt) for how to personalise this.
### Local commits status
The prompt will show you the difference in commits between your branch and the
remote your branch is tracking. The examples below assume you are checked out on
`master` and are tracking `origin/master`.
Prompt | Meaning
--------------------|--------
![git:(master 2↑)] | We have 2 commits to push up
![git:(master 3↓)] | We have 3 commits to pull down
![git:(master 3⇵5)] | Our version and origins version of `master` have diverged
The use of feature is controlled by the `GIT_RADAR_FORMAT` environment variable.
See [Customise your prompt](#customise-your-prompt) for how to personalise this.
### Remote commits status
The prompt will also show the difference between your branch on origin and what
is on `origin/master`. This a is hard coded branch name which I intend to make
configurable in the future.
This is the difference between the commits you've pushed up and `origin/master`.
Prompt | Meaning
---------------------------|---------------
![git:(m ← 2 my-branch)] | We have 2 commits on `origin/my-branch` that aren't on `origin/master`
![git:(m 4 → my-branch)] | There are 4 commits on `origin/master` that aren't on `origin/my-branch`
![git:(m 1 ⇄ 2 my-branch)] | `origin/master` and `origin/my-branch` have diverged, we'll need to rebase or merge
The use of feature is controlled by the `GIT_RADAR_FORMAT` environment variable.
See [Customise your prompt](#customise-your-prompt) for how to personalise this.
### Stash status
The prompt will show you whether and how many stashes you have stored.
Prompt | Meaning
---------------------------|---------------
![git:(master) 1≡] | We have one stash
If you don't rely on this status, you can always hide this part of the prompt by
[customising your prompt](#customise-your-prompt)
### (Optional) Auto-fetch repos
Ensuring your refs are up to date I found can be a pain. To streamline this
git-radar can be configured to auto-fetch your repo. When the `--fetch` flag is
used git-radar will run `git fetch` asynchronously every 5 minutes.
This will only occur when the prompt is rendered and it will only occur on the
repo you are currently in.
To use this feature, when setting your prompt, call git-radar with `--fetch`:
**Bash**
```bash
export PS1="$PS1\$(git-radar --bash --fetch)"
```
[(note: the `\` escaping the `$` is important)](#ensuring-prompt-execution)
**Zsh**
```zsh
export PROMPT="$PROMPT\$(git-radar --zsh --fetch) "
```
[(note: the `\` escaping the `$` is important)](#ensuring-prompt-execution)
You may also choose to fetch at a customized interval of time. To do so, add
this to your .bashrc, .zshrc:
```bash
export GIT_RADAR_FETCH_TIME=<seconds>
```
For example, to fetch every 30 seconds (instead of the default 5 minutes):
```bash
export GIT_RADAR_FETCH_TIME=30
```
You can also do this in the gitradarrc file:
```bash
GIT_RADAR_FETCH_TIME=30
```
## Customise your prompt
Git Radar is highly customisable using a prompt format string. The 4 features
above: remote commits, local commits, branch and file changes; are controlled
by the prompt format string.
Feature | Control string
---------------|---------------
Remote commits | `%{remote}`
Local commits | `%{local}`
Branch | `%{branch}`
File changes | `%{changes}`
Stashes | `%{stash}`
You can create any prompt shape you prefer by exporting `GIT_RADAR_FORMAT` with
your preferred shape. The control strings above will be replaced with the output
of the corresponding feature.
**Examples**
GIT_RADAR_FORMAT | Result
--------------------------------------|---------------------
`%{branch}%{local}%{changes}` | `master1↑1M`
`[%{branch}] - %{local} - %{changes}` | `[master] - 1↑ - 1M`
### Prefixing and Suffixing the features
Often you will want certain parts of the prompt to only appear when there is
content to render. For example, when in a repo you want `[branch]` but when out
of a repo you don't want the `[]` appearing.
To do this the control strings support prefixes and suffixes. Prefixes and
Suffixes are separated from the feature name by `:` and will only render if the
feature would render:
Format: `prompt > %{prefix - :changes: - suffix}`
In a repo: `prompt > prefix - 1M - suffix`
Outside a repo: `prompt > `
The default prompt format uses this to add spaces only if the feature would
render. In that way the prompt always looks well spaced out no matter how many
features are rendering.
## Support
### Ensuring prompt execution
When setting your prompt variable, `PROMPT` in Zsh and `PS1` in Bash, it's
important that the function executes each time the prompt renders. That way the
prompt will respond to changes in your git repo. To ensure this you will need
to escape the execution of the function. There are two ways to do this:
**1. Use `$'` to render raw characters**
```bash
export PROMPT=$'$(git-radar --zsh)'
export PS1=$'$(git-radar --bash)'
```
**2. Use `\` to escape execution of the subshell**
```bash
export PROMPT="\$(git-radar --zsh)"
export PS1="\$(git-radar --bash)"
```
### Configuring colours
You can configure the colour scheme in two ways: export
[Environment Variables](#exporting-environment-variables)
or use an [rc file](#setting-an-rc-file).
#### Exporting Environment Variables
To configure the prompt this way just add to your `~/.bashrc` or `~/.zshrc` an
export directive with the value you want to change.
**Example: Change the branch colour in Zsh**
In `~/.zshrc`:
```zsh
export GIT_RADAR_COLOR_BRANCH='$fg[yellow]'
```
**Example: Change the branch colour in Bash**
In `~/.bashrc`:
```zsh
export GIT_RADAR_COLOR_BRANCH='\\033[0;33m'
```
#### Setting an RC file
Git radar supports multiple rc files. One of these will be sourced when the
prompt renders.
**Example: Change the branch colour in Zsh**
In `~/.gitradarrc`:
```zsh
GIT_RADAR_COLOR_BRANCH='$fg[yellow]'
```
**Basic RC file**
Create a file at `~/.gitradarrc` which sets the Environment variables listed in
[Configuration values](#configuration-values) using colour codes listed in
either [Zsh Colour Codes](#zsh-colour-codes) or
[Bash Colour Codes](#Bash-Colour-Codes) depending on your shell.
**Shell specific RC file**
If you use both Bash and Zsh you can set RC files that are specific for those
shells.
For Bash: Create a file at `~/.gitradarrc.bash`
For Zsh: Create a file at `~/.gitradarrc.zsh`
#### Bash Colour Codes
Bash colour codes make use of the colours your terminal app claims to be `red`
or `green`. Using one of these codes will only produce the colour your terminal
claims, so you should customise your colour scheme on your terminal as well as
customising git-radar.
Note the "Bright" colours can be shown as bold instead, it depends on your
terminal. By default, for example, the Mac OSX Terminal.app uses the "Bright"
colours to provide 8 new lighter colours but some terminals only support 8 and
will show the text as bold instead.
Colour | Code for Text | Code for Background
--------------|----------------|--------------------
Black | `\\033[0;30m` | `\\033[0;40m`
Red | `\\033[0;31m` | `\\033[0;41m`
Green | `\\033[0;32m` | `\\033[0;42m`
Yellow | `\\033[0;33m` | `\\033[0;43m`
Blue | `\\033[0;34m` | `\\033[0;44m`
Magenta | `\\033[0;35m` | `\\033[0;45m`
Cyan | `\\033[0;36m` | `\\033[0;46m`
White | `\\033[0;37m` | `\\033[0;47m`
Bright Black | `\\033[1;30m` | `\\033[1;40m`
Bright Red | `\\033[1;31m` | `\\033[1;41m`
Bright Green | `\\033[1;32m` | `\\033[1;42m`
Bright Yellow | `\\033[1;33m` | `\\033[1;43m`
Bright Blue | `\\033[1;34m` | `\\033[1;44m`
Bright Magenta| `\\033[1;35m` | `\\033[1;45m`
Bright Cyan | `\\033[1;36m` | `\\033[1;46m`
Bright White | `\\033[1;37m` | `\\033[1;47m`
Reset | `\\033[0m` | `\\033[0m`
Note the Reset will set back to what your terminal claims as standard text and
background.
#### Zsh Colour Codes
Zsh also provides a way to access the colours that your terminal claims as `red`
or `green`, etc.
Note the "Bright" colours can be shown as bold instead, it depends on your
terminal. By default, for example, the Mac OSX Terminal.app uses the "Bright"
colours to provide 8 new lighter colours but some terminals only support 8 and
will show the text as bold instead.
Colour | Code for Text | Code for Background
--------------|--------------------|--------------------
Black | `$fg[black]` | `$bg[black]`
Red | `$fg[red]` | `$bg[red]`
Green | `$fg[green]` | `$bg[green]`
Yellow | `$fg[yellow]` | `$bg[yellow]`
Blue | `$fg[blue]` | `$bg[blue]`
Magenta | `$fg[magenta]` | `$bg[magenta]`
Cyan | `$fg[cyan]` | `$bg[cyan]`
White | `$fg[white]` | `$bg[white]`
Bright Black | `$fg_bold[black]` | `$bg_bold[black]`
Bright Red | `$fg_bold[red]` | `$bg_bold[red]`
Bright Green | `$fg_bold[green]` | `$bg_bold[green]`
Bright Yellow | `$fg_bold[yellow]` | `$bg_bold[yellow]`
Bright Blue | `$fg_bold[blue]` | `$bg_bold[blue]`
Bright Magenta| `$fg_bold[magenta]`| `$bg_bold[magenta]`
Bright Cyan | `$fg_bold[cyan]` | `$bg_bold[cyan]`
Bright White | `$fg_bold[white]` | `$bg_bold[white]`
Reset | `$reset_color` | `$reset_color`
#### Configuration values
All these values should be set using a the correct colour code for your
terminal. You should also choose the colour code based on what shell you are
using. There is a way to support [colouring multiple shells using rc files](#setting-an-rc-file).
##### Colouring the Branch part
**GIT_RADAR_COLOR_BRANCH='[colour code]'**
```
git:(my-branch)
^^^^^^^^^
```
The colour to use for the Branch or git reference.
It is unset by
`GIT_RADAR_COLOR_BRANCH_RESET` which you can set if you want a different
background colour to return to.
##### Colouring the local commits status
**GIT_RADAR_COLOR_LOCAL_AHEAD='[colour code]'**
```
git:(my-branch 1↑)
^
```
The colour to use for the arrow that indicates how many commits you have to push
up.
It is unset by `GIT_RADAR_COLOR_LOCAL_RESET` which you can set if you want
a different background colour to return to.
**GIT_RADAR_COLOR_LOCAL_BEHIND='[colour code]'**
```
git:(my-branch 1↓)
^
```
The colour to use for the arrow that indicates how many commits you have to pull
down.
It is unset by `GIT_RADAR_COLOR_LOCAL_RESET` which you can set if you want
a different background colour to return to.
**GIT_RADAR_COLOR_LOCAL_DIVERGED='[colour code]'**
```
git:(my-branch 1⇵1)
^
```
The colour to use for the arrow that indicates how many commits your branch has diverged by.
It is unset by `GIT_RADAR_COLOR_LOCAL_RESET` which you can set if you want
a different background colour to return to.
##### Colouring the remote commits status
**GIT_RADAR_COLOR_REMOTE_AHEAD='[colour code]'**
```
git:(m ← 1 my-branch)
^
```
The colour to use for the arrow that indicates how many commits your branch has to merge on to master.
It is unset by `GIT_RADAR_COLOR_REMOTE_RESET` which you can set if you want
a different background colour to return to.
**GIT_RADAR_COLOR_REMOTE_BEHIND='[colour code]'**
```
git:(m 1 → my-branch)
^
```
The colour to use for the arrow that indicates how many commits your branch is
behind master.
It is unset by `GIT_RADAR_COLOR_REMOTE_RESET` which you can set if you want
a different background colour to return to.
**GIT_RADAR_COLOR_REMOTE_DIVERGED='[colour code]'**
```
git:(m 1 ⇄ 1 my-branch)
^
```
The colour to use for the arrow that indicates how many commits your branch has
diverged from master.
It is unset by `GIT_RADAR_COLOR_REMOTE_RESET` which you can set if you want
a different background colour to return to.
**GIT_RADAR_COLOR_REMOTE_NOT_UPSTREAM='[colour code]'**
```
git:(upstream ⚡ my-branch)
^
```
The colour to use for the lightning bolt which indicates that your branch is not
tracking an upstream branch.
It is unset by `GIT_RADAR_COLOR_REMOTE_RESET` which you can set if you want
a different background colour to return to.
##### Colouring the file changes status
**GIT_RADAR_COLOR_CHANGES_STAGED='[colour code]'**
```
git:(my-branch) 1M
^
```
The colour to use for the letters that indicate changes that have been staged to
commit.
It is unset by `GIT_RADAR_COLOR_CHANGES_RESET` which you can set if you want
a different background colour to return to.
**GIT_RADAR_COLOR_CHANGES_UNSTAGED='[colour code]'**
```
git:(my-branch) 1M
^
```
The colour to use for the letters that indicate changes that have not yet been
staged to commit.
It is unset by `GIT_RADAR_COLOR_CHANGES_RESET` which you can set if you want
a different background colour to return to.
**GIT_RADAR_COLOR_CHANGES_CONFLICTED='[colour code]'**
```
git:(my-branch) 1B
^
```
The colour to use for the letters that indicate changes that have conflicts that
need resolved.
It is unset by `GIT_RADAR_COLOR_CHANGES_RESET` which you can set if you want
a different background colour to return to.
**GIT_RADAR_COLOR_CHANGES_UNTRACKED='[colour code]'**
```
git:(my-branch) 1A
^
```
The colour to use for the letters that indicate files that are currently not
tracked by git.
It is unset by `GIT_RADAR_COLOR_CHANGES_RESET` which you can set if you want
a different background colour to return to.
##### Colouring the stash status
**GIT_RADAR_COLOR_STASH='[colour code]'**
```
git:(my-branch) 1≡
^
```
The colour to use for the lines that indicates how many stashes you have stored.
It is unset by `GIT_RADAR_COLOR_STASH_RESET` which you can set if you want
a different background colour to return to.
## License
Git Radar is licensed under the MIT license.
See [LICENSE] for the full license text.
[LICENSE]: https://github.com/michaeldfallen/git-radar/blob/master/LICENSE
[git:(master) 1≡]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/stash.png
[git:(master) 3A]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/untracked.png
[git:(master) 2D2M]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/unstaged.png
[git:(master) 1M1R]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/added.png
[git:(master) 1U]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/conflicts.png
[git:(master) 1M 1D2M 2A]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/combination.png
[git:(master 2↑)]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/local%20is%20ahead.png
[git:(master 3↓)]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/remote%20is%20behind.png
[git:(master 3⇵5)]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/remote%20local%20diverged.png
[git:(m ← 2 my-branch)]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/branch%20is%20ahead.png
[git:(m 4 → my-branch)]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/master%20is%20ahead.png
[git:(m 1 ⇄ 2 my-branch)]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/master%20branch%20diverged.png
[An example of git-radar]: https://raw.githubusercontent.com/michaeldfallen/git-radar/master/images/detailed.png
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,372 |
A smile is not just a facial expression, it is way of letting the other person feel accepted and welcomed. It is the most in born fashion item to add to your dressing. It can only be done properly and confidently when one has a good set of teeth.
Stop consuming certain foods and drinks, such as red wine, dark sodas, coffee, and tea, that stain teeth and cause them to yellow. Also, stay away from tobacco too because it causes teeth discolouration.
Keep up with brushing and flossing, and use toothpaste that's not too abrasive.
Being a chronic teeth grinder can create stress that can contribute to ageing and yellowing of the teeth.
Rubbing lemon, orange, or banana peels on your teeth will make them whiter. It's believed that the compound d-limonene, which is found in some fruit peels, will help to whiten your teeth.
Gently rub the fruit peels on your teeth for about two minutes then thoroughly rinse out your mouth and brush your teeth afterwards.
Visit your dentist to bleach your teeth, if you prefer this route. Dentists usually use products that contain hydrogen peroxide. To protect your teeth, the dentist will protect your gums, tooth enamel, and the rest of your mouth from damage during the procedure. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,426 |
Home » World » Disability groups oppose using botanist's death to advance assisted suicide agenda
Disability groups oppose using botanist's death to advance assisted suicide agenda
Bern, Switzerland, May 10, 2018 / 05:21 pm (CNA/EWTN News).- Botanist and ecologist David Goodall ended his life May 10 in Switzerland by assisted suicide, a procedure which he had long advocated legalizing in his home country of Australia.
Goodall, 104, told journalists that he "looked forward" to ending his life and regretted not having ended it sooner, though he is not terminally ill. He also said he regretted that he had to travel all the way to Switzerland commit suicide.
Australia's Victoria state has passed an assisted suicide law that will go into effect in 2019, but it only allows for terminally ill patients to end their lives – Goodall would not have qualified under the law.
Critics have said that Goodall's death was not simply a personal choice, but a political one that could have devastating consequences on vulnerable populations such as the elderly, the poor, and the disabled.
"It was clear he wanted to go out while getting a lot of attention," said Stephen Drake, a research analyst with the advocacy group Not Dead Yet, a disability rights group which opposes the legalization of assisted suicide and euthanasia "as deadly forms of discrimination."
"If someone acts as he does, for people to call it a personal act is a lie; it was a political act," Drake told CNA.
Only a handful of countries have legalized assisted suicide or euthanasia, including Belgium, Luxembourg, and the Netherlands. In Switzerland, while assisted suicide is technically not legal, it is allowed under certain circumstances.
In the United States, assisted suicide is currently legal in Colorado, Vermont, Washington, California, Oregon, Hawaii, and the District of Columbia. The State Supreme Court of Montana decriminalized assisted suicide for physicians in 2009.
Currently, most legislation that allows for assisted suicide or euthanasia does so only in the cases of terminally ill patients.
However, Goodall's case demonstrates that this is only the beginning for assisted suicide advocates, Drake noted.
Terminal illness is "the wedge issue that most people can agree on, that opens the door," he said.
"Once you open the door, then the campaign becomes to kick it open as far as you can, and it shows that opponents in Australia who've been fighting to prevent the legalization of assisted suicide and euthanasia are right on target when they say that this will begin to expand in very short order," he added.
"Now the cause will be: why are we preventing poor old people from ending their lives?" he said.
Other vulnerable and undervalued populations, such as the poor and disabled, will be similarly at risk, he noted.
Matt Valliere, executive director of Patients Rights Action Fund, told CNA in e-mail comments that it would "be a mistake" to use Goodall's death as an example in advocating for legalized assisted suicide.
"We know that legalizing assisted suicide only places the vulnerable at greater risk. Mr. Goodall himself had no terminal illness and yet was given lethal medication," Valliere said.
"That is why many sick, poor, elderly and persons with disabilities oppose these laws – they will be the first to suffer from them."
What concerns disability and advocacy groups most about legalizing assisted suicide and euthanasia is the potential for coercion and corruption – that suicide will become a "rational choice" for some undervalued populations, and their deaths seen as a duty rather than an unpreventable tragedy, Drake said.
"They will no longer be viewed as preventable tragedies but rational (ways) to end a life of suffering," he said.
"But if you talk to people with disabilities, the suffering they will cite is not the disability itself, but the barriers and discrimination and open hostility they encounter in the culture they live in."
Studies have shown that the majority of patients who request assisted suicide will withdraw that request when they are treated for depression. There have also been several cases of botched deaths in Oregon, in which a patient's doctor was not present during the assisted suicide. Oregon has seen many abuses since since its legalization of assisted suicide, such as cases of pills changing hands, either intentionally or unknowingly, with lethal results.
Another concern is that suicide for vulnerable populations will be seen as a smart economic choice, Drake said.
Derek Humphry, founder of the Hemlock Society (now two organizations – Compassion and Choices, and Final Exit Network), wrote in 2000 about the "unspoken argument" for assisted suicide – that it would be cheaper to let the elderly and disabled die than to keep them alive.
"As technology advances, as medical costs skyrocket out of control, as chronic diseases predominate, as the projected rate of the eighty-five-and-older population accelerates, as managed care seeks to cut costs and as Medicare is predicted to go bankrupt by 2007, the impetus of cost containment provide impetus, whether openly acknowledged or not, for the practicalities of an assisted death," Humphry wrote in his book Freedom to Die: people, politics, and the right-to-die movement.
Drake noted: "It's one thing for me to say that, it's another thing for Derek Humphry, who embodies the assisted suicide movement, to say it."
Drake said that advocates for assisted suicide "either discount those concerns or frankly they don't care, they figure the people who might be hurt by this won't be them, and I think that's what it boils down to."
« Canadians oppose abortion requirement for summer job grants, poll finds
Once on the verge of closing, Italian monastery sees vocation revival » | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,657 |
Q: VBA - Parsing Date from Free Form Text String I am attempting to parse out clean target DATES from cells populated with free form TEXT STRINGS.
ie: TEXT STRING: "ETA: 11/22 (Spring 4.5)" or "ETA 10/30/2019 EOD"
As you can see, there is no clear standard for the position of the date in the string, rendering LEFT or RIGHT formulas futile.
I tried leveraging a VBA function that I found which essentially breaks up the string into parts based on spaces in the string; however it has not been working.
Public Function GetDate(ResNotes As String) As Date
Dim TarDate As Variant
Dim part As Variant
TarDate = Split(ResNotes, " ")
For Each part In ResNotes
If IsDate(part) = True Then
GetDate = part
Exit Function
End If
Next
GetDate = "1/1/2001"
End Function
I'm referring to the cells with text strings as "ResNotes", short for "Resolution Notes" which is the title of the column
"TarDate" refers to the "Target Date" that I am trying to parse out
The result of the custom GETDATE function in Excel gives me a #NAME? error.
I expected the result to give me something along the lines of "10/30/2019"
A: If you are getting #NAME then the code is not stored in a general module. It should NOT be in a worksheet module or ThisWorkbook module.
Also there are few errors in the code. Split returns a String Array. And since IsDate returns TRUE/FALSE the = True is not needed.
As per @MathieuGuindon we can change the string to a date in the code if found and return an error if not. For that we need to allow the return to be a variant.
Public Function GetDate(ResNotes As String)
Dim TarDate() As String
Dim part As Variant
TarDate = Split(ResNotes, " ")
For Each part In TarDate
If IsDate(part) Then
GetDate = CDate(part)
Exit Function
End If
Next
GetDate = "1/1/2001"
'Instead of a hard coded date, one can return an error, just use the next line instead
'GetDate =CVErr(xlErrValue)
End Function
A: Unless you need VBA for some other part of your project, this can also be done using worksheet formulas:
=AGGREGATE(15,6,DATEVALUE(MID(SUBSTITUTE(A1," ",REPT(" ",99)),seq_99,99)),1)
where seq_99 is a named formula and refers to:
=IF(ROW($A$1:INDEX($A:$A,255,1))=1,1,(ROW($A$1:INDEX($A:$A,255,1))-1)*99)
*seq_99 generates an array of numbers {1;99;198;297;396;495;...
Format the cell with the formula as a Date of some type.
If there are no dates, it will return an error which you can either leave, or wrap the function in an IFERROR(your_formula,your_error_message)
Algorithm
*
*Split the cell on the spaces
*
*Replace each space with 99 spaces
*Using the MID function, return an array of substrings 99 characters long
*Apply the DATEVALUE function which will return either an error (if the substring is not a date) or a date serial number.
*Since dates in Excel are serial numbers since 1/1/1900, we can use the AGGREGATE function to pick out a value, and ignore errors.
A: Approach isolating the date string via Filter function
Just for fun another approach demonstrating the use of the Filter function in combination with Split to isolate the date string and split it into date tokens in a second step; finally these tokens are transformed to date using DateSerial:
Function getDat(rng As Range, Optional ByVal tmp = " ") As Variant
If rng.Cells.count > 1 Then Set rng = rng.Cells(1, 1) ' allow only one cell ranges
If Len(rng.value) = 0 Then getDat = vbNullString: Exit Function ' escape empty cells
' [1] analyze cell value; omitted year tokens default to current year
' (valid date strings must include at least one slash, "11/" would be interpreted as Nov 1st)
tmp = Filter(Split(rng.Value2, " "), Match:="/", include:=True) ' isolate Date string
tmp = Split(Join(tmp, "") & "/" & Year(Now), "/") ' split Date tokens
' [2] return date
Const M% = 0, D% = 1, Y& = 2 ' order of date tokens
getDat = VBA.DateSerial(Val(tmp(Y)), Val(tmp(M)), _
IIf(tmp(D) = vbNullString, 1, Val(tmp(D))))
End Function
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{"url":"http:\/\/math.stackexchange.com\/questions\/777775\/combinatorics-counting-in-two-ways","text":"# Combinatorics - Counting in two ways? [duplicate]\n\nThis question already has an answer here:\n\nWhat is the idea of proving a binomial identity by counting in two ways?\n\nCould you please illustrate this with this example? Thank you very much.\n\n$$\\binom{2n}{n}= \\sum_{k=0}^n {\\binom nk}^2$$\n\n(original screenshot)\n\n-\n\n## marked as duplicate by Marc van Leeuwen, Sami Ben Romdhane, Claude Leibovici, Davide Giraudo, Mark BennetMay 2 '14 at 7:11\n\nThis becomes a special case of the Vandermonde identity if you replace one of the factors $\\binom nk$ by $\\binom n{n-k}$. Also, it is a duplicate of at least this question, if not of many others. \u2013\u00a0Marc van Leeuwen May 2 '14 at 5:28\n\nThe idea behind the technique is to perform the same task using procedures that are distinct. From each distinct procedure, we obtain a combinatorial expression for the number of ways that the task can be performed. Since the same task was performed, the combinatorial expressions are the same.\n\nThe most simple example is probably the procedure that results in selecting $k$ objects from a collection of $n$ distinct objects. We might perform this task by first placing the $n$ objects on a table, marking $k$ of them as the objects to select, and then move the $k$ marked objects to a separate pile. There are $\\binom{n}{k}$ ways to do this. Alternatively, we might look at our pile, and mark $n-k$ objects to not select, and then move the remaining $k$ unmarked objects to a separate pile. There are $\\binom{n}{n-k}$ ways to mark the $n-k$ objects to leave out of the collection. Since each procedure results in the same thing -- a collection of $k$ objects-- the number of ways to perform each procedure must be the same. Therefore, $$\\binom{n}{k}=\\binom{n}{n-k}$$\n\nIn your given example can think as follows. The left hand side $$\\binom{2n}{n}$$ counts the number of ways to select $n$ objects from a collection of $2n$ objects. We could perform this task as above: place all objects on a table and mark $n$ of them to keep which gives the above expression. Alternatively, we could first split the pile of $2n$ objects into two separate piles of $n$ objects. To get the desired collection of $n$ objects, we could select $k$ from the first pile ($\\binom{n}{k}$ ways), and the remaining $n-k$ from the second ($\\binom{n}{n-k}$ ways). Thus, the number of ways to do so is: $$\\sum_{k=0}^n\\binom{n}{k}\\binom{n}{n-k}$$ Therefore, $$\\binom{2n}{n}=\\sum_{k=0}^n\\binom{n}{k}\\binom{n}{n-k}$$ To get your desired result, simply use the identity established earlier to find that: \\begin{align*} \\binom{2n}{n}&=\\sum_{k=0}^n\\binom{n}{k}\\binom{n}{n-k}\\\\ &=\\sum_{k=0}^n\\binom{n}{k}\\binom{n}{k}\\\\ &=\\sum_{k=0}^n\\binom{n}{k}^2 \\end{align*}\n\n-\nPuntuation never goes after TeX's double dollar sign! \u2013\u00a0Mariano Su\u00e1rez-Alvarez May 2 '14 at 2:22\n@Mariano Gracias, was just typing along without thinking. \u2013\u00a0Scott H. May 2 '14 at 2:24\n\n[EDIT]\n\nThe idea behind double counting arguments is to provide verbal proofs of combinatorial identities that give an intuitive understanding of why they should be so. By simply showing that two expressions measure different ways to count the same thing it becomes clear to even a novice mathematician that they must be equivalent.\n\nOn the left hand side: $\\binom{2n}{n}$ is the count of ways to divide a set of distinct objects into two equal sized subsets.\n\nOn the right hand side: $\\sum\\limits_{k=0}^{n} \\binom{n}{k}^2$ is the count of ways to redistribute objects between two equal sized sets as follows:\n\n$\\binom{n}{k}$ counts the ways to select $k$ objects form a set of $n$ distinct objects. Thus $\\binom{n}{k}^2$ counts the ways to do this for two sets; which makes it the count of ways to transfer $k$ object from one set to the other in exchange for $k$ objects originally in the second set. The summation is thus the count of ways to exchange equal numbers of distinct objects between two equal sized sets, for all possible size of the exchanges.\n\nIt should be clear that one gets the same result by simply combining the two sets and counting all the ways to split them into equal sized sets.\n\nHence it follows that: $\\dbinom{2n}{n}=\\sum\\limits_{k=0}^{n} \\dbinom{n}{k}^2$\n\n-\n\nAn alternative proof is the following.\n\nWhat is the coefficient of $x^n$ in $(1+x)^{2n}$?\n\nWrite this as $(1+x)^n(1+x)^n$. What is it now?\n\nNote $$\\sum_{k=0}^n\\binom nk^2=\\sum_{k=0}^n \\binom nk\\binom n{k -n}=\\sum_{i+j=n}\\binom ni\\binom nj$$\n\n-\n\nLet $a$ = number of ways of choosing $n$ animals from $n$ cats and $n$ dogs is $\\binom{2n}{n}$.\n\nLet $c(k)$ = number of ways of choosing $k$ cats from $n$ cats = $\\binom{n}{k}$\n\nLet $d(k)$ = number of ways of choosing $n-k$ dogs from $n$ dogs = $\\binom{n}{n-k}$ = $\\binom{n}{k}$\n\nThus, $e(k)$ = number ways of choosing $k$ cats and $n-k$ dogs = $c(k)*d(k) = {\\binom nk}^2$\n\nObviously $a$ = Number of ways of choosing $k$ cats and $n-k$ dogs for all $k \\in [0, n]$\n\nTherefore, $a = \\sum_{k=0}^n e(k) = {\\binom nk}^2$. QED.\n\n-","date":"2016-04-30 05:34:20","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\": 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\": 0.7471655011177063, \"perplexity\": 240.79958741454288}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2016-18\/segments\/1461860111612.51\/warc\/CC-MAIN-20160428161511-00026-ip-10-239-7-51.ec2.internal.warc.gz\"}"} | null | null |
\subsection{Full Recovery Data Structure ($\FRS$)}
In this section we present the Full Recovery Data Structure ($\FRS$) that is placed in each of the levels.
Since the recovery structure is the core of our sampling algorithm, we refer to the entire algorithm with $\FRS$ in each level as the $\FRS$ algorithm.
The sampling algorithm can be summarized to the following theorem.
\begin{theorem} \label{theorem1}
Given a required sample size $K = \Omega(\log\frac{1}{\delta})$ and $\delta \in (0,1)$, $\FRS$ sampling algorithm generates a $\Theta(\log\frac{1}{\delta})$-wise independent sample $S$ with $1-\delta$ success probability. The sample size is $K \leq \left| S \right| \leq \tilde{K}$, where $\tilde{K} = \Theta(K)$.
For both Strict and Non-strict data streams $\FRS$ uses $O(K \log\frac{K}{\delta} \log{(mr)} \log{(m)})$ bits of space,
$O(\log\frac{K}{\delta})$ update time per element, $O(\log\frac{K}{\delta} \log{m})$ random bits and $O(K \log\frac{K}{\delta})$ time to extract the sample $S$.
\end{theorem}
$\FRS$ is inspired by Count Sketch \cite{charikar04}. It is composed of $\tau=O(\log{\frac{K}{\delta}})$ arrays
of size $s=O(K)$. We refer to the cells in the array as \emph{bins}.
Each input element is mapped to one bin in each of the $\tau$ arrays.
In each bin there is an instance of $BS_{s}$. We use $\tau$
hash functions drawn randomly and independently from a
pairwise independent family $\mathcal{H}=\{h \colon [m]\rightarrow[s]\}$.
The same set of hash functions can be used for the instances of $\FRS$ in all levels.
The mapping is performed as follows.
Let $B[a,b]$ for $a\in[\tau]$ and $b\in[s]$, be the $b$'th bin in
the $a$'th array. Let the hash functions be $h_{1}\ldots h_{\tau}$.
Then $(x_{i},c_{i})$ is mapped to $B[a,h_{a}(x_{i})]$ for every
$a\in[\tau]$.
We say that two elements \emph{collide} if they are mapped to the
same bin.
\begin{lemma} \label{Collisions in arrays} Let $\FRS$ with at most $\tilde{K}$
elements have $\tau=\log{\frac{\tilde{K}}{\delta}}$ arrays of size $s=2\tilde{K}$.
Then with probability at least $1-\delta$ for each element there is a bin
in which it does not collide with any other element. \end{lemma}
\begin{proof} See Appendix \ref{appendix:proofs}. \end{proof}
\begin{corollary} \label{Alg1Strict} In the Strict Turnstile model all
elements in $\FRS$ can be identified and recovered with probability at least $1-\delta$. \end{corollary}
For recovery, we scan all bins in all arrays in $\FRS$ and use $BS_{s}$ to extract elements
from all the bins that contain a single element. According to Lemma
\ref{Collisions in arrays}, all elements in $\FRS$ can be identified
with success probability $1-\delta$. We verify success by removing all
the elements we found and scanning the arrays an additional time to
validate that they are all empty.
Removing an element $(x,c)$ is performed by inserting $(x,-c)$ to the corresponding bins.
\subsubsection{Non-strict $\FRS$}
We now present the generalization of $\FRS$ to Non-strict streams. Once
again we use $BS_{s}$, but we add to our sample only elements that are consistently
identified in multiple arrays.
\begin{lemma} \label{lemma:alg1NS} Let $\FRS$ with at most $\tilde{K}$
elements have $\tau=5\log{\frac{\tilde{K}}{\delta}}$ arrays of size
$s=8\tilde{K}$. In the Non-strict data stream model, all elements
inserted to $\FRS$ and only those elements are added to the sample
with probability at least $1-\delta$. \end{lemma}
\begin{proof}
We extract a set $A$ of candidate elements from all $BS_{s}$s that seem to have a single element in the first $\log\frac{\tilde{K}}{\delta}$ arrays in $\FRS$.
$A$ contains existing elements, that were inserted to $\FRS$, and falsely detected elements that are a result of a collision.
$\left| A \right| \leq \tilde{K} \log\frac{\tilde{K}}{\delta}$.
It follows from Lemma \ref{Collisions in arrays} and Corollary \ref{Alg1Strict} that all of the existing elements can be recovered from the first $\log\frac{\tilde{K}}{\delta}$ arrays with probability $1-\delta/2$ (increasing the arrays size reduces the probability of a collision).
Hence $A$ contains all existing elements with probability $1-\delta/2$.
Next we insert to our output sample all candidates that we detect in at least half of their bins in the $\tau' = 4\log{\frac{\tilde{K}}{\delta}}$ remaining arrays of $\FRS$.
There are two types of possible errors: not reporting an existing element (false negative) and reporting a falsely detected element (false positive).
\textbf{False negative:}
We show that with high probability existing elements
are isolated in at least half of the their bins in the $\tau'$ arrays.
Let $\mathcal{C}_{k}^{a}$ be the event that element $k$
collides with another element in array $a$.
$\Pr[\mathcal{C}_{k}^{a}]< \tilde{K}\cdot\frac{1}{s}=\frac{1}{8}$.
Let $\mathcal{C}_{k}$ be the event that element $k$ collides with
another element in at least half of the $\tau'$ arrays. The hash functions
of the different arrays are independent and therefore:
$\Pr[\mathcal{C}_{k}] \leq {\tau' \choose \tau'/2}\Pr[\mathcal{C}_{k}^{a}]^{\tau'/2}<\left(\frac{\delta}{\tilde{K}}\right)^{2},$
where ${\tau' \choose \tau'/2}<2^{\tau'}$ is used.
Let $\mathcal{C}$ be the event that there is an existing element that collides
with another element in at least half of the $\tau'$ arrays. $\Pr[\mathcal{C}] \leq \tilde{K}\cdot\Pr[\mathcal{C}_{k}]<\tilde{K}\left(\frac{\delta}{\tilde{K}}\right)^{2} < \frac{\delta}{4}.$
\textbf{False positive:}
If a falsely detected element from $A$ is added to the sample then there is a collision in at least half of its bins in the $\tau'$ arrays.
Let $\mathcal{E}_{b}^{a}$ be the event that there is an element
in bin $b$ of array $a$.
$\Pr[{\mathcal{E}}_{b}^{a}] \leq {\tilde{K} \choose 1}\left(\frac{1}{s}\right)=\frac{1}{8}$.
Let $\mathcal{E}_{k}$ be the event that there are elements
in the bins corresponding to a falsely detected element $k$ in at least
half of the $\tau'$ arrays. $
\Pr[\mathcal{E}_{k}] \leq {\tau' \choose \tau'/2}\Pr[{\mathcal{E}}_{b}^{a}]^{\tau'/2}<2^{\tau'}\left(2^{-3}\right)^{\tau'/2}
=2^{-0.5\tau'
=\left(\frac{\delta}{\tilde{K}}\right)^{2}$.
Let $\mathcal{E}$ be the event that there is an element from $A$
that was falsely identified in at least half of its bins. Using the
union bound we get:
$\Pr[{\mathcal{E}}] \leq \left|A\right| \cdot\Pr[{\mathcal{E}}_{k}] < \tilde{K} \log\frac{\tilde{K}}{\delta} \cdot \left(\frac{\delta}{\tilde{K}}\right)^{2} < \frac{\delta}{4}$.
We conclude that the probability of a mistake is bounded by:
$\delta/2 + \Pr[\mathcal{C}]+\Pr[\mathcal{E}]<\delta$.
\end{proof}
\subsection{$\epsilon$-Full Recovery Data Structure ($\AFRS$)}
In this section we present the $\epsilon$-Full Recovery Data Structure ($\AFRS$) that enables to recover almost all elements inserted to it. We refer to the entire algorithm with $\AFRS$ placed in each of the levels as $\AFRS$ algorithm. The sampling algorithm can be summarized to the following theorem.
\begin{theorem} \label{theorem2}
Given a required sample size $K = \Omega(\frac{1}{\epsilon} \log{\frac{1}{\delta}})$, for $\delta \in (0,1)$ and \mbox{$\epsilon \in (0,1)$},
$\AFRS$ sampling algorithm generates a $(t,\epsilon)$-partial sample $S$ for $t = \Theta(\log\frac{1}{\delta})$ with $1-\delta$ success probability. The sample size is $(1-\epsilon)K \leq \left|S\right| \leq \tilde{K}$, where $\tilde{K} = \Theta(K)$.
For both Strict and Non-strict data streams $\AFRS$ requires $O((\log\log\frac{K}{\delta})^2)$ update time per element, $O(\log\frac{K}{\delta} \log{m})$ random bits and $O(K)$ time to extract the sample $S$.
The space is $O(K \log{(mr)} \log{(m)})$ bits for Strict data streams and $O(K \log{(\frac{mr}{\delta})} \log{(m)})$ bits for Non-strict streams.
\end{theorem}
$\AFRS$ is composed of $\tau=2$ arrays
of size $s=O(K)$. As in $\FRS$,
each input element is mapped to one bin in each of the arrays.
In each bin of each array we keep an instance of $BS_{s}$ or $BS_{ns}$ according to the input data stream.
The mapping is performed using two hash functions drawn randomly and independently from a $t$-wise independent family $\mathcal{H}=\{h \colon [m]\rightarrow[s]\}$ for $t=\Theta(\log{\frac{K}{\delta}})$.
Let $X$ be the set of elements in $\AFRS$, $\left|X\right|\leq \tilde{K}$.
A \emph{fail set} $F\subseteq X$ is a set of $f$
elements, such that each element in the set collides with other
elements from the set in both its bins.
The elements in a fail set $F$ cannot be extracted from $\AFRS$.
Analyzing the existence of a fail set is similar to analyzing failure in a cuckoo hashing \cite{pagh01} insertion.
We bound the probability that there is a fail set of size $f$ using the following (revised) lemma of Pagh and Pagh \cite{pagh08}.
\begin{lemma}[\cite{pagh08},Lemma3.4]
For two functions $i_1,i_2 \colon U \rightarrow [R]$ and a set $S \subseteq U$, let $G(i_1,i_2,S)=(A,B,E)$ be the bipartite graph that has left vertex set $A = \{a_1,\ldots,a_R\}$, right vertex set $B = \{b_1,\ldots,b_R\}$ and edge set $E = \{e_x \mid x \in S \}$, where $e_x = (a_{i_1(x)},b_{i_2(x)})$.
For each set $S$ of size $n$, and for $i_1,i_2:U \rightarrow [4n]$ chosen at random from a family that is $t$-wise independent on $S$, $t \geq 32$, the probability that the fail set $F$ of the graph $G(i_1,i_2,S)$ has size at least $t$ is $n/{2^{\Omega(t)}}$.
\end{lemma}
\begin{corollary}
Let $\AFRS$ with at most $\tilde{K}$ elements have 2 arrays of size $s = 4\tilde{K}$.
Let the mapping be performed by two $t$-wise independent hash functions for $t=c \log\frac{\tilde{K}}{\delta}$, constant $c$ and $t \geq 32$.
The probability that there is a fail set of size at least $t$ is bounded by $\delta$.
\end{corollary}
\begin{proof}
The more elements in $\AFRS$, the higher the probability that there is a fail set of some fixed predefined size.
The probability is $\tilde{K} / {2^{c' c \log{\frac{\tilde{K}}{\delta}}}} \leq \delta$ for some constants $c$, $c'$.
\end{proof}
The following algorithm identifies all elements in $\AFRS$ that do not belong to a fail set.
\begin{enumerate}[1.]
\item Initialize the output sample $S = \emptyset$ and a queue $Q = \emptyset$.
\item Scan the two arrays in $\AFRS$. For each bin $b$, if $BS$ holds a single element, $Enqueue(Q,b)$.
\item While $Q \neq \emptyset$:
\begin{enumerate}[3.1.]
\item $b \leftarrow Dequeue(Q)$. If $BS$ in $b$ holds a single element:
\begin{enumerate}[3.\mbox{1}.1.]
\item Extract the element $(k,C_k)$ from $BS$ in $b$.
\item $S = S \cup \{(k,C_k)\}$.
\item Subtract $(k,C_k)$ from $BS$ in $\tilde{b}$, where $\tilde{b}$ is the other bin $k$ is hashed to.
\item $Enqueue(Q,\tilde{b})$.
\end{enumerate}
\end{enumerate}
\nopagebreak
\item Return $S$.
\end{enumerate}
\begin{lemma} \label{lemma:alg3works}
All elements that do not belong to a fail set are identified by the algorithm.
\end{lemma}
\begin{proof} See Appendix \ref{appendix:proofs}. \end{proof}
\begin{lemma} \label{lemma:alg3time}
The recovery algorithm takes $O(K)$ time.
\end{lemma}
\begin{proof}
If the algorithm is implemented with hash computations for finding the other bin an element is hashed to, it takes $O(K {(\log\log\frac{K}{\delta})}^2)$ time. Using an additional counter in each $BS$ reduces the time to $O(K)$.
See Appendix \ref{appendix:proofs} for the complete proof. \end{proof}
Let $X$ be the elements in $\AFRS$, $K \leq \left| X \right| \leq \tilde{K}$, $\tilde{K} = \Theta(K)$.
In order to recover all but $\epsilon \left| X \right|$ of the elements we require $K = \Omega(\frac{1}{\epsilon} \log{\frac{1}{\delta}})$.
If $K$ is smaller, we recover all but at most $O(\max{\{\epsilon \left|X \right|, f\}})$ of the elements, where $f=O(\log{\frac{K}{\delta}})$ is the size of the fail set.
\subsubsection{Non-strict $\AFRS$}
In the Non-strict Turnstile model we keep $BS_{ns}$ in each bin, and we set the range of the hash function to $q=\Theta(\frac{K}{\delta})$ and its independence to $t'=\Theta(\log{\frac{K}{\delta}})$, the same as the independence of the hash functions we use when mapping to the bins in $\AFRS$.
The same hash function can be used for all $BS_{ns}$s.
Recall that if $BS_{ns}$ contains a single element, this element is extracted successfully.
If $BS_{ns}$ contains less than $t'$ elements, an event called a \emph{small collision}, the probability of an error is at most $1/q$.
If $BS_{ns}$ contains $t'$ elements or more, an event called a \emph{large collision}, we do not have a guarantee on the probability of an error.
\begin{lemma}
Let $\AFRS$ with at most $\tilde{K}$ elements have 2 arrays of size $s = 4\tilde{K}$.
Let the mapping be performed by two $t$-wise independent hash functions for $t=2\log{\frac{\tilde{K}}{\delta}}$.
Let each bin contain $BS_{ns}$ with $q=\frac{4\tilde{K}}{\delta}$ and $t' = t$.
The probability of no false detections during the entire recovery process is at least $1-\delta$.
\end{lemma}
\begin{proof}
First we bound the probability of a large collision in $\AFRS$. Let $\AFRS$ have $\left|X\right| \leq \tilde{K}$ elements.
Let $\mathcal{E}_{b}$ be the event that there is a large collision in bin $b$.
Since $t=t'$, every $t'$ elements that appear in a large collision are mapped there independently.
Thus, $Pr[\mathcal{E}_{b}] \leq {X \choose t'} \left(\frac{1}{s}\right)^{t'}
\leq \left(\frac{eX}{t'}\right)^{t'} (\frac{1}{4\tilde{K}})^{t'}
\leq \left(\frac{e}{4t'}\right)^{t'} < {(\frac{\delta}{\tilde{K}})}^{2}$.
Using the union bound, the probability that there is a large collision in any bin is at most $\delta/2$.
Hence we can consider only small collisions.
The total number of inspections of bins with collisions during the recovery process is at most $2\tilde{K}$. Therefore
the probability of detecting at least one false element as a result of a small collision is at most
$2\tilde{K}\frac{1}{q}=\delta/2$,
and the probability of any false detections is $\delta$.
\end{proof}
\begin{corollary}
If $BS_{ns}$ with $q=\Theta(\frac{K}{\delta})$ and $t'=\Theta(\log{\frac{K}{\delta}})$ is placed in each bin, the recovery procedure of the Strict Turnstile model can be used also for Non-strict Turnstile data streams with $1-\delta$ success probability.
\end{corollary}
\section{Approximation Using High Moments}
\label{appendix:chebyshev}
For a random variable $Z$ and an even number $l$,\[
\Pr[\left|Z-E[Z]\right|\geq t]=\Pr[(Z-E[Z])^{l}\geq t^{l}]\leq\frac{\Delta^{l}}{t^{l}}\]
where $\Delta^{l}=E[(Z-E[Z])^{l}]$ is the $l$'th central moment
of $Z$.
We use the estimation of \cite{indyk01} to $\Delta^{l}$, where $Z$ is a sum of $l$-wise independent indicator variables:
$\Delta^{l}\leq8(6l)^{(l+1)/2}{E[Z]^{(l+1)/2}}$.
This implies:\[
\Pr[\left|Z-E[Z]\right|\geq\alpha E[Z]]\leq\frac{48l}{\alpha}{\left(\frac{6l}{\alpha^{2}E[Z]}\right)}^{(l-1)/2}\label{eq:l_cheby}\]
\begin{lemma}\label{lemma:l_cheby} Let $Z$ be a sum of $l$-wise independent indicator
variables with $E[Z]=\frac{c}{\epsilon^{2}}\log\frac{1}{\delta}$
for some $0<\epsilon$, $\delta \in (0,1)$ and constant $c$. Let $l=\tilde{c} \log\frac{1}{\delta}$ for some constant $\tilde{c}$
be an even number.
Then for big enough constants $c, \tilde{c}$:
$\Pr[\left|Z-E[Z]\right|\geq\epsilon E[Z]]<\delta$.
\end{lemma}
\begin{proof} \begin{align*}
\Pr[\left|Z-E[Z]\right|\geq\epsilon E[Z]] &
\leq\frac{48 \tilde{c} \log\frac{1}{\delta}}{\epsilon}
{\left(\frac{6 \tilde{c}\log\frac{1}{\delta}}{\epsilon^{2}\frac{c}{\epsilon^{2}}\log\frac{1}{\delta}}\right)}
^{(\tilde{c}\log\frac{1}{\delta}-1)/2}\\
& =\frac{48\tilde{c}\log\frac{1}{\delta}}{\epsilon}{\left(\frac{6\tilde{c}}{c}\right)}^{(\tilde{c}\log\frac{1}{\delta}-1)/2}<\delta\end{align*}
For $\epsilon > \poly{(\delta)}$.
\end{proof}
Note that for a constant $\alpha$, in order to prove $\Pr[\left|Z-E[Z]\right|\geq\alpha E[Z]]<\delta$, $E[Z] = \Omega{(\log{\frac{1}{\delta}})}$ is sufficient.
\section{Proofs from Section \ref{sec:work}} \label{appendix:proofs}
\subsection{Proof of Lemma \ref{lemma:mapping}}
\begin{proof} Let $X_{l}$ be a random variable that indicates the
number of elements $k$ in the stream for which $(h_{l}(k)=0\wedge
h_{l}(k)\neq0)$. I.e. $X_{l}$ is the number of elements in level
$l$. $\mathcal{H}$ is a $t$-wise independent
hash family for $t = \Theta(\log{\frac{1}{\delta}})$.
Therefore for each $l\in L$ and $k$ in the stream, $\Pr_{h\in \mathcal{H}}[h_{l}(k)=0\wedge h_{l+1}(k)\neq0]=\lambda^{l}-\lambda^{l+1}=\lambda^{l}(1-\lambda)$.
We denote $p_{l}=\lambda^{l}(1-\lambda)$.
$L_{0}\leq\tilde{L}_{0}\leq\alpha L_{0}$ implies $\frac{1}{\alpha}\tilde{L}_{0}\leq L_{0}\leq\tilde{L}_{0}$
and we obtain:
$
{\frac{1}{\alpha}\tilde{L}_{0}p_{l^{*}+1}}<2K\leq{\frac{1}{\alpha}\tilde{L}_{0}p_{l^{*}}}\leq{L_{0}p_{l^{*}}}\leq{\tilde{L}_{0}p_{l^{*}}}\leq\frac{2\alpha K}{\lambda}
$.
The expected number of elements in level $l$ is $E[X_{l}]=L_{0}p_{l}$. Hence for level $l^{*}$, $
2K\leq E[X_{l^{*}}]\leq\frac{2\alpha K}{\lambda}$.
We write $E[X_{l^{*}}]=\beta K$ for some $2\leq\beta\leq\frac{2\alpha}{\lambda}$.
From Lemma \ref{lemma:l_cheby} we get: $
\Pr[\left|X_{l^{*}}-E[X_{l^{*}}]\right|\geq K]=\Pr[\left|X_{l^{*}}-E[X_{l^{*}}]\right|\geq\frac{1}{\beta}E[X_{l^{*}}]]<\delta $.
We can use the lemma since $X_{l^{*}}$ is a sum of $t$-wise independent variables, $t = \Theta(\log\frac{1}{\delta})$, and $K=\Omega(\log\frac{1}{\delta})$.
\end{proof}
\subsection{Proof of Lemma \ref{Collisions in arrays}}
\begin{proof}
First we bound the probability that a specific element $k$ collides with another element in a specific array $a$.
Let $\mathcal{\mathcal{C}}_{kj}^{a}$ be the event
that elements $k$ and $j$ collide in array $a$.
Since pairwise independent hash functions are used to map the elements to bins, $\forall k,j,a\quad\Pr[\mathcal{\mathcal{C}}_{kj}^{a}]=\frac{1}{s}=\frac{1}{2\tilde{K}}$.
Let $\mathcal{C}_{k}^{a}$ be the event that $k$ collides with any
element in array $a$.
$\Pr\left[\mathcal{C}_{k}^{a}\right]=\Pr[\exists{j}\quad\mathcal{C}_{kj}^{a}]\leq\bigcup_{j\neq k}\ \Pr[\mathcal{\mathcal{C}}_{kj}^{a}]<\tilde{K} \cdot\Pr[\mathcal{\mathcal{C}}_{kj}^{a}]=\frac{1}{2}$.
Now we prove that with probability $1-\delta/{\tilde{K}}$ there is an array in which no element collides with $k$.
Let $\left| \mathcal{C}_{k}\right|$ be the number of arrays in which $k$ collides with another element.
We want to show $\left| {\mathcal{C}}_{k} \right| < \tau$ with high probability.
Let $X$ be the set of elements in $\FRS$. We know $\left| X \right| \leq \tilde{K}$. The hash functions of the different arrays are independent.
Therefore
$\Pr[\left|\mathcal{\mathcal{C}}_{k}\right|=\tau]=\Pr[\forall a\in[\tau], \; \mathcal{C}_{k}^{a}]
\le\Pr[\forall a\in[\tau], \; \mathcal{C}_{k}^{a} \mid \left|X\right|=\tilde{K}]
=\prod_{a\in[\tau]}{\Pr[\mathcal{C}_{k}^{a} \mid \left|X\right|=\tilde{K}]}<\left(\frac{1}{2}\right)^{\tau}=\frac{\delta}{\tilde{K}}$.
We conclude with: $\Pr[\exists k, \; \mbox{$k$ collides in all arrays}]=\Pr[\exists k, \; \left|\mathcal{\mathcal{C}}_{k}\right|=\tau]$
$ \leq \bigcup_{k}\Pr[\left|\mathcal{\mathcal{C}}_{k}\right|=\tau]<\delta$.
\end{proof}
\subsection{Proof of Lemma \ref{lemma:alg3works}}
\begin{proof} If the set of unidentified elements is not a fail
set, then one of them is isolated in a bin.
Let $x$ be the element and $b$ be the bin in which $x$ is the single element.
If $x$ was isolated prior to the first scan over all bins, bin $b$ would have been identified in the first scan.
If $x$ became isolated during the recovery process, then all other elements in $b$ were identified beforehand.
When the last of them was identified and removed from $b$, $x$ became isolated, and $b$ was inserted to the queue.
Each bin inserted to the queue is removed from the queue before the process ends, and hence when $b$ was removed, $x$ was identified.
Identifying $x$ results in a smaller set of unidentified elements.
The identification process can proceed until all elements are identified or none
of them are isolated, i.e. they all belong to a fail set. \end{proof}
\subsection{Proof of Lemma \ref{lemma:alg3time}}
\begin{proof}
The number of operations in the initial scan is linear in the number of bins in $\AFRS$, which is $O(K)$. The number of bins
inserted to $Q$ in the process is $O(K)$, because a bin is inserted only when an element is extracted and added to the sample.
Thus, apart from the operation of identifying the other bin an element is hashed to, all other operations take a total of $O(K)$ time.
Assume the phase of finding the other bin an extracted element $k$ is hashed to, is implemented in the algorithm by evaluating the hash function on $k$.
This evaluation occurs $O(K)$ times. The hash functions are $t$-wise independent, where $t=\Theta(\log\frac{K}{\delta})$.
Thus, in this implementation the recovery time for all $O(K)$ elements is $O(K ({\log\log{\frac{K}{\delta}}})^2)$.
The recovery time can be reduced to $O(K)$ by using an additional counter $W$ in each $BS$.
Let $h_1$, $h_2$ be the two hash functions
that map the elements to the bins in the two arrays.
When inserting an element $(x_i,c_i)$ to $BS$ in array $a \in {\{1,2}\}$, the following update is performed:
$W \leftarrow W + c_i h_{3-a}(x_i)$.
The space and update time required by the algorithm remain of the same order,
since the update takes an additional $O(1)$ time and the space of $W$ is $O(\log{(mr)})$ bits.
If $BS$ in array $a$ contains a single element $(k,C_k)$, then $W = C_k h_{3-a}(k)$.
Thus, if $(k,C_k)$ is extracted from $BS$ in array $a$ we obtain its location $h_{3-a}(k)$ in the other array $3-a$ without evaluating a hash function.
The recovery time is $O(K)$ for all $O(K)$ elements.
\end{proof}
\subsection{Proof of Lemma \ref{theorem:inverse}}
\begin{proof}
Our estimator is $
f^{-1}(i) \approx \frac{\left|\{k \in S \colon C_k = i \} \right|}{\left| S \right|}
$.
We need to prove that it is an $\epsilon$-approximation to $f^{-1}(i) = \frac{\left|\{k \colon C_k = i \} \right|}{\left|\{k \colon C_k \neq 0 \} \right|}$, the fraction of distinct values with total count equals $i$, with probability $1-\delta$.
We prove that it is an $\epsilon$-approximation when all elements are recovered from the recovery structure, i.e. when $\FRS$ is used. Later we relax the assumption that all elements are recovered, and thus prove that we get an $\epsilon$-approximation also when using $\AFRS$. Thus, an $\epsilon$-approximation is provided with $1-\delta$ success probability when using both our sampling algorithms.
The elements recovered are from a specific level $l^{*}$ that depends on $L_{0}$.
For value $k$, $C_{k}\neq0$, we define a random variable
$Y_{k}$. $Y_{k}=1$ if $(k,C_{k})$ is mapped to level $l^{*}$.
$\Pr[Y_{k}=1]=\lambda^{l^{*}}(1-\lambda)$, and we denote $p_{l^{*}}=\lambda^{l^{*}}(1-\lambda)$.
Let $Y=\sum_{k}{Y_{k}}$ be the number of elements in level $l^{*}$.
For now assume that all elements in the level are recovered, i.e.
$Y=\left|S\right|$. Later we relax this assumption. $E[Y]$ is the expected sample
size obtained from level $l^{*}$. Thus $E[\left|S\right|]=E[Y]=L_{0}p_{l^{*}}$.
We choose $l^{*}$ as a level with $\Theta(K)$ elements, i.e. $E[Y] =\Theta(K) = \Omega(\frac{1}{\epsilon^2}\log\frac{1}{\delta})$.
${\{Y_{k}\}}$ are $t$-wise independent, where $t = \Theta(\log\frac{1}{\delta})$. $E[Y] = \Omega(\frac{1}{\epsilon^2}\log\frac{1}{\delta})$. From Lemma~\ref{lemma:l_cheby}
we obtain $\Pr[\left|Y-E[Y]\right|>\epsilon'E[Y]]<\delta'$, for $\epsilon'=\Theta(\epsilon)$ and $\delta' = \Theta(\delta)$
that will be determined later.
Thus, with probability $1-\delta'$, \[
(1-\epsilon')E[\left|S\right|]\leq\left|S\right|\leq(1+\epsilon')E[\left|S\right|] \]
Let $F_{i}=\sum_{k \colon C_{k}=i}{Y_{k}}$, be the number of elements
in $S$ with frequency $i$. $
E[F_{i}]=\left|\{k \colon C_{k}=i\}\right|\cdot p_{l}^{*}=f^{-1}(i)\cdot E[\left|S\right|] $.
We get $f^{-1}{(i)}=\frac{E[F_{i}]}{E[\left|S\right|]}$. If
all elements in the level are recovered, the estimator of $f^{-1}(i)$ can be written as:
$f^{-1}{(i)}\approx\frac{F_{i}}{\left|S\right|}$.
Hence we would like to prove: \begin{equation}
\Pr\left[\left|f^{-1}(i)-\frac{F_{i}}{\left|S\right|}\right|\geq\epsilon\right]\leq\delta\label{eq:estimator}\end{equation}
$\frac{E[F_{i}]}{\left|S\right|}$ is an $\epsilon'$-approximation to $\frac{E[F_{i}]}{E[\left|S\right|]}$, i.e. $\Pr \left[\left|f^{-1}(i) - \frac{E[F_{i}]}{\left|S\right|} \right|\geq\epsilon' \right]\leq\delta'$ since
\[
\frac{E[F_{i}]}{E[\left|S\right|]}-\epsilon'\leq\frac{E[F_{i}]}{(1+\epsilon')E[\left|S\right|]}\leq\frac{E[F_{i}]}{\left|S\right|}\leq\frac{E[F_{i}]}{(1-\epsilon')E[\left|S\right|]}\leq\frac{E[F_{i}]}{E[\left|S\right|]}+\epsilon'\]
with probability $1-\delta'$ when $f^{-1}(i)=\frac{E[F_{i}]}{E[\left|S\right|]}\leq1-\epsilon'$.
Then by using Lemma \ref{lemma:l_cheby} again we get (\ref{eq:estimator}),
for a constant $c$, $2\epsilon'$-additive error and $1-2\delta'$ success probability.
Now we relax the assumption that all elements in the level were recovered.
The maximal bias occurs if $\epsilon'\left|S\right|$ elements were not recovered,
and all unrecovered elements had frequency $i$. I.e. $\left|\{k\in S \colon C_{k}=i\}\right|=F_{i}\pm\epsilon'\left|S\right|$.
Thus, there is an additional additive error of $\epsilon'$. Setting
$\epsilon'=\epsilon/3$, $\delta' = \delta/3$ completes the proof.
\end{proof}
\end{document}
\section{Applications and Extensions} \label{sec:apps}
\paragraph*{Inverse Distribution}
The samples generated by the algorithms $\FRS$ and $\AFRS$ can be used to derive an additive $\epsilon$-approximation with $1-\delta$ success probability for various forward and inverse distribution queries.
For example, consider \emph{Inverse point queries}, which return the value of $f^{-1}(i)$ for a query frequency
$i$. The samples from $\FRS$ and $\AFRS$ can be used to obtain an approximation in $[f^{-1}(i) - \epsilon, f^{-1}(i) + \epsilon]$ for every frequency
$i$.
We can approximate Inverse range queries, Inverse
heavy hitters and Inverse quantiles queries in a similar way.
\begin{lemma} \label{theorem:inverse}
Let $S$ be a $(t,\epsilon')$-partial sample of size $\left| S \right| = \Omega(\frac{1}{\epsilon^{2}}\log\frac{1}{\delta})$ for $\epsilon \in (0,1)$, $\epsilon'=\Theta(\epsilon)$, $\delta \in (0,1)$, and $t = \Omega(\log\frac{1}{\delta})$.
The estimator
$
{f^{-1}}(i) \approx \frac{\left|\{k \in S \colon C_k = i \} \right|}{\left| S \right|}
$
provides
an additive $\epsilon$-approximation to the inverse
distribution with probability at least $1-\delta$.
\end{lemma}
\begin{proof}
See Appendix \ref{appendix:proofs}. \end{proof}
\paragraph*{Union and Difference}
Let $DS_{r,D_{i}}$ be the data structure obtained from data stream $D_{i}$ using the random bits $r$.
The union of streams $D_{1}$, $D_{2}$ is $DS_{r,D_{1}\cup D_{2}}=DS_{r,D_{1}}+DS_{r,D_{2}}$,
where the addition operator adds all $BS$s in all bins of all arrays.
Our sampling algorithm can extract a sample from a union of data streams. This
feature is useful when there are multiple entry points and each of them can update its own data structure locally and then a unified sample can be derived.
Sampling from the difference of streams $DS_{r,D_{1}-D_{2}}=DS_{r,D_{1}}-DS_{r,D_{2}}$ is similar. Note that
even if $D_{1}$ and $D_{2}$ are Strict Turnstile data streams, their difference might represent a Non-strict Turnstile stream. Hence our structures for the
Non-strict Turnstile model are useful for both input streams
in the Non-strict model and for sampling the difference.
\section{Introduction}
Sampling is a fundamental component in data stream algorithms \cite{muth05},
as a method to keep a synopsis of the data in memory sublinear to the size of the stream.
The sample can be used to calculate stream statistics of interest such as the frequent items in the stream
(also called \emph{heavy hitters}) or the quantiles, when the stream itself cannot be fully kept in memory.
In the most general data stream model, the data is a series of elements $(x_i,c_i)$ where $x_i$ is an element's value and $c_i$ is a count. A certain value may appear in the stream multiple times with various counts.
Summing all counts of a value $k$ in the stream gives a pair $(k,C_k)$ where $C_k =\sum_{i \colon x_{i}=k}{c_{i}}$ is the total count of $k$.
A value $k$ with $C_k = 0$ is a deleted element, and its effect on stream statistics
should be as if it had never appeared in the stream.
Particularly, it must not appear in a sample of the stream obtained by any sampling algorithm.
We denote the function that maps the values to their frequencies by $f$, i.e.
$f(k)=C_k$.
An \emph{exact sampling algorithm} outputs a sample of the pairs $(k,f(k))$ composed of values and their exact total count.
An \emph{$\epsilon$-approximate sampling algorithm} for $\epsilon \in (0,1)$ outputs a sample of the
pairs $(k,{f'(k)})$, where ${f'(k)} \in [(1-\epsilon)f(k), (1+\epsilon)f(k)]$.
An $\epsilon$-approximate sample is sufficient for answering \emph{forward distribution} queries, which concern properties of $f$, such as what is $f(k)$.
However, it cannot be used for queries on the \emph{inverse distribution function} $f^{-1}$, defined as
$f^{-1}(C) = \frac{ \left| \{k \colon C_k = C \} \right|}{ \left| \{k \colon C_k \neq 0 \} \right|}$ for $C \neq 0$, i.e. the fraction of distinct values with a total count equal to $C$.
The reason is that an $\epsilon$-approximation of $f$ can result in a significant change to $f^{-1}$.
For example, if an $\alpha \in (0,1)$ fraction of the distinct values have a total count $C$, and in the $\epsilon$-approximate sample all of them have a total count $(1+\epsilon)C$, one might get $f^{-1}(C) = 0$ instead of $\alpha$.
Thus, an exact sample is required in order to approximate the inverse distribution.
The algorithms we present in this paper are exact sampling algorithms for the most general case of streams with deletions.
Thus, they are useful for calculating both forward and inverse distribution statistics.
We describe applications which require exact sampling next.
Data stream algorithms are often used in network traffic analysis.
They are placed in DSMSs - Data Stream Management Systems.
Applications of dynamic exact sampling in these environments include
detecting malicious IP traffic in the
network. Karamcheti et al.\ \cite{karamcheti05} showed
that inverse distribution statistics
can be used for earlier detection of content similarity, an
indicator of malicious traffic.
Inverse distribution queries can also be used for detecting denial-of-service (DoS) attacks, specifically SYN floods,
which are characterized as flows with a single packet (often called \emph{rare flows}).
Exact dynamic sampling is beneficial in geometric data streams as
well \cite{frahling05,frahling08}. In these streams, the items
represent points from a discrete $d$-dimensional geometric space ${\{1,\ldots,\Delta\}}^{d}$.
Our algorithms can also run on data streams of this type.
\paragraph*{Previous Work}
Most previous work that has been done on sampling of dynamic data streams that support
deletions was limited to approximating
the forward distribution \cite{cohen12, gemulla07, mone10}.
Works on the inverse distribution include a restricted model with total counts
of 0/1 for each value \cite{gemulla06, tao07}, and
minwise-hashing, which samples uniformly the set of items but does
not support deletions \cite{datar03}.
The work \cite{gibbons01} supports only a few deletions.
Inverse distribution queries in streams with multiple deletions were supported in a work by Frahling et al.\ \cite{frahling05,frahling08},
who developed a solution for Strict Turnstile data streams and used it
in geometric applications.
Cormode et al.\ \cite{cormode05} developed a solution for both the
Strict Turnstile and the Non-strict Turnstile streams. However, they
did not analyze the required randomness or the algorithm's error probability
in the Non-strict model.
Jowhari et al.\ \cite{jowhari11} studied $L_p$ samplers \cite{mone10} and built an $L_0$ sampler for Non-strict Turnstile streams.
\paragraph*{Our Results}
Previous works \cite{cormode05,frahling05,frahling08,jowhari11} constructed data
structures for sampling only a single element.
In order to use their structures for applications that require a sample of size $K$,
one has to use $O(K)$ independent instances of their structure.
The obtained sample holds elements chosen independently, and it might contain
duplicates.
Running the sampling procedure $O(K)$ times in parallel and inserting each element in the stream as an input to $O(K)$ instances of the data structure, results in an enormous process time.
Typical stream queries require a sample of size $K = \Omega(\frac{1}{\epsilon^2}\log{\frac{1}{\delta}})$ where the results are $\epsilon$ approximated, and $1-\delta$ is the success probability of the process.
Thus, the number of operations required to process each element in the stream is multiplied by many orders of magnitude. For typical values such as $\epsilon = 10^{-2}$, $\delta = 10^{-6}$ the number of operations for each element in the stream is multiplied by about $200{,}000$.
The structures of \cite{cormode05,frahling05,frahling08,jowhari11} cannot be used for obtaining a $K$ size sample due to this unfeasible process load.
We present algorithms that can accomplish the task.
Our contributions are as follows.
\begin{itemize}
\item
We construct a data structure that can extract a sample of size $K$, whereas previous works returned only a single sampled element.
Using a single data structure reduces significantly the process time and the required randomness.
Thus, our algorithms are feasible for data stream applications, in which fast processing is crucial.
This optimization enables applications that were previously limited to gross approximations of order $\epsilon = 10^{-2}$ to obtain a much more accurate approximation of order $\epsilon = 10^{-6}$ in feasible time.
\item We present solutions for both Strict Turnstile data streams and the most general Non-strict Turnstile
data streams. We are the first to provide algorithms
with proved success probability in the Non-Strict Turnstile
model.
For this model we develop a structure called the Non-strict Bin Sketch.
\item We provide more efficient algorithms in terms of the randomness required. Our algorithms do not require fully independent or min-wise independent hash functions or PRGs.
\item
We introduce the use of $\Theta(\log{\frac{1}{\delta}})$-wise independent hash functions to generate a sample with $1-\delta$ success probability for any $\delta >0$.
Our method outperforms the traditional approach of increasing the success probability from a constant to $1-\delta$ by $\Theta(\log\frac{1}{\delta})$ repetitions.
We utilize a method of fast evaluation of hash functions which reduces our processing time to $O((\log\log\frac{1}{\delta})^2)$, while the traditional approach requires $O(\log\frac{1}{\delta})$ time.
\end{itemize}
A comparison of our algorithms to previous work is presented in Table~\ref{table:previouswork}.
We introduce two algorithms, denoted $\FRS$
(Full Recovery Structure)
and $\AFRS$ according to the recovery structure used.
The performance of our algorithms is summarized in the table as well as in Theorems~\ref{theorem1} and~\ref{theorem2}. Our algorithms improve the update and the sample extraction times.
\input{previoustable}
\section{Preliminaries}
\subsection{Data Stream Model}
Our input is a stream in the \emph{Turnstile} data stream model \cite{muth05}.
The Turnstile data stream consists of $N$ pairs $(x_{i},c_{i})$
where $x_{i}$ is the element's value, and $c_{i}$ is its count.
The elements $x_i$ are taken from a fixed universe $U=[m]$ (where
$[m]=\{0,\ldots,m-1\}$). The counts $c_i$ are taken from the range $[-r,r]$.
Let $t$ be the time we process the $t$'th pair in the data stream.
We define the total count of value $k$ at time $t$ to be $C_{k}(t)=\sum_{i \le t \colon x_{i}=k}{c_{i}}$.
We assume that $\forall t,k, C_{k}(t)\in[-r,r]$. In the \emph{Strict
Turnstile} model, $C_{k}(t)\geq0$ at all times $t$. In the \emph{Non-strict
Turnstile} model, $C_{k}(t)$ may obtain negative values.
A sample $S$
drawn at time $t$ is a subset of ${\{(k,C_{k}(t)) \colon C_k(t) \neq 0\}}$.
Note that $C_{k}(t)$ is the exact total count at time $t$. To simplify the notation we consider sampling only at the end of the process, and denote $C_{k}=C_{k}(N)$.
\subsection{Problem Definition}
Given a data stream of $N$ pairs $(x_i, c_i)$,
which is either a Strict or Non-strict Turnstile data stream, assume
there is an application that needs to perform some queries on the stream such as calculate the inverse distribution.
This application allows an $\epsilon \in (0,1)$ additive approximation to the answers, and a $1-\delta$ for $\delta \in (0,1)$ success probability of the process.
The application predefines the size of the sample it requires to be $K$, where
$K$ might depend on $\epsilon$ and $\delta$.
The input to our sampling algorithm is the data stream, $K$ and $\delta$.
Let $D = {\{ (k, C_{k}) \colon C_k \neq 0\} }$ be the set of all pairs of values and their total counts in the stream at the time we sample.
The output of our sampling algorithm is a sample $S \subseteq D$ of size $\left|S\right| = \Theta(K)$,
generated with probability $1-\delta$.
Note that the size of the sample returned is of order $K$ and not precisely $K$.
Applications typically require a sample of size $K=\Omega(\frac{1}{{\epsilon}^{2}}\log{\frac{1}{\delta}})$ for an $\epsilon$ approximation with $1-\delta$ success probability. However, our algorithms $\FRS$ and $\AFRS$, support even smaller sample sizes, $\Omega(\log{\frac{1}{\delta}})$ and $\Omega(\frac{1}{\epsilon}\log{\frac{1}{\delta}})$ respectively.
We define the following two ``flavors'' of samples.
\begin{definition}
A \emph{$t$-wise independent sample} is a random set $S \subseteq D$ in which each subset of $t$ distinct elements in $D$ has equal probability to be in $S$.
\end{definition}
\begin{definition}
Let $X \subseteq D$ be a sample obtained by a $t$-wise independent sampling algorithm, and let $\epsilon \in (0,1)$.
A subset $S \subseteq X$ of size $(1-\epsilon)\left| X \right| \leq \left| S \right| \leq \left| X \right|$ is a \emph{$(t,\epsilon)$-partial sample}.
\end{definition}
Our $\FRS$ algorithm returns a $t$-wise independent sample, where $t=\Omega(\log\frac{1}{\delta})$.
Our $\AFRS$ algorithm returns a $(t,\epsilon)$-partial sample.
This means there is a fractional bias of at most $\epsilon$ in the sample returned by $\AFRS$.
The key insight is that a $t$-wise independent sample for $t = \Omega{(\log\frac{1}{\delta})}$ guarantees the same approximation as an independent sample.
For example, a sample of size
$K=\Omega(\frac{1}{{\epsilon}^{2}}\log{\frac{1}{\delta}})$ enables the approximation of the inverse distribution
queries and the Jaccard similarity coefficient up to an additive error of $\epsilon$ with $1-\delta$
success probability.
The $(t,\epsilon)$-partial sample can be used for the same stream statistics because it only adds an error of at most $\epsilon$ to the approximation.
This is demonstrated in Sect.~\ref{sec:apps}.
\subsection{Hash Function Techniques}
Throughout the paper we make extensive use of $t$-wise independent hash functions for $t= \Theta(\log\frac{1}{\delta})$ and $t= \Theta(\log\frac{K}{\delta})$.
We use the following techniques in our analysis.
\label{sec:highmoments}
For bounding the error probability we use the Moment Inequality for high moments and the estimation of \cite{indyk01}
(see Appendix \ref{appendix:chebyshev}).
For hash evaluations we use the multipoint evaluation algorithm of a polynomial of degree less than $t$ on $t$ points in $O(t\log^{2}{t})$ operations \cite{gathen99}.
Thus, evaluation of a $t$-wise independent hash function takes $O(\log^{2}{t})$ amortized time
per element by batching $t$ elements together.
This is the time we use in our analysis whenever we evaluate these hash functions.
\section{Algorithm Overview}
In this section we provide an overall view of our sampling algorithm (see Fig.~\ref{fig:pseudocode}).
\input{figoverview}
In the first phase of our sampling algorithms the elements in the data stream are mapped to levels,
in a similar way to the method in \cite{cormode05}.
Each element is mapped to a single level. The number of elements mapped to each level decreases exponentially.
Thus, we can draw a sample of the required size regardless of the number of distinct elements in the stream.
In each level we store a recovery structure for $K$ elements, which is the core of our algorithm. We present two alternative
structures, $\FRS$ and $\AFRS$, with trade-offs between space, update time and sample extraction time.
Each structure consists of several arrays with a Bin Sketch ($BS$) in each cell of the arrays.
Assume a structure ($\FRS$ or $\AFRS$) contains the set of elements $X$, where \mbox{$K \leq \left| X \right| \leq \tilde{K}$} and $\tilde{K} = \Theta(K)$.
$\FRS$ enables to extract all the elements it contains, returning a $t$-wise independent sample, where $t = \Theta(\log\frac{1}{\delta})$.
$\AFRS$ enables to extract at least $(1-\epsilon)\left|X\right|$ of the $\left|X\right|$ elements it contains, returning a $(t,\epsilon)$-partial sample.
The problem of recovering the $\left|X\right|$ elements is similar to the sparse recovery problem \cite{gilbert10,porat07}, however in our case there is no tail noise, and we limit the amount of randomness.
The sample $S$ is the set of elements extracted from $\FRS$ (or $\AFRS$) in a single level.
In order to select the level we use a separate data structure that returns $\tilde{L}_0$, an estimation of the number of distinct elements in the stream.
This data structure is updated as each stream element arrives, in parallel to the process described above.
Extracting a sample is performed as follows. First we query the data structure of $\tilde{L}_0$. Then we select a level $l^{*}$ that should have $\Theta(K)$ elements with probability $1-\delta$. We recover
the $X$ elements from that level, or at least $(1-\epsilon)\left|X\right|$ of them, depending on the recovery structure, with probability
$1-\delta$. The elements recovered are the final output sample $S$.
\section{Sampling Algorithms} \label{sec:work}
\subsection{Bin Sketch - Data Structure Building Block}
\label{3 counters}
In this section we present the \emph{Bin Sketch ($BS$)}, a tool used as a building block in our data structure.
Given a data stream of elements ${\{(x_i,c_i)\}}_{i \in [N]}$, the input to $BS$ is a substream ${\{(x_i,c_i)\}}_{i \in I}$ where $I \subseteq [N]$.
$BS$ maintains a sketch of the substream.
Its role is to identify if the substream contains a single value, and if so, to retrieve the value and its total count.
Note that a single value can be obtained from a long stream of multiple elements if all values but one have zero total count at the end.
\paragraph*{Strict Bin Sketch ($BS_{s}$)}
We describe the Bin Sketch for the Strict Turnstile
model, previously used in \cite{frahling08,ganuly07}.
Given a stream of pairs $\{(x_{i},c_{i})\}_{i \in I}$, the \emph{Strict
Bin Sketch ($BS_{s}$)} consists of three counters $X$, $Y$
and $Z$:
$
X=\sum_{i \in I}{c_{i}},\ Y=\sum_{i \in I}{c_{i}x_{i}},\ Z=\sum_{i \in I}{c_{i}x_{i}^{2}}$.
$BS_s$ properties are summarized as follows:
The space usage of $BS_{s}$ is $O(\log(mr))$ bits.
If $BS_{s}$ holds a single element, then we can detect and recover it.
$BS_{s}$ holds a single element $\Leftrightarrow$ $X\neq0$, $Y\neq0$, $Z\neq0$ and $XZ=Y^{2}$.
The recovered element is $(k,C_{k})$, where $k=Y/X$ and
$C_{k}=X$. $BS_{s}$ is empty $\Leftrightarrow$ $X=0$.
For proof see \cite{ganuly07}.
\paragraph*{Non-strict Bin Sketch ($BS_{ns}$)}
We provide a generalization to Bin Sketch and adjust it to the Non-strict
Turnstile model.
If $BS_s$ contains two elements, then $XZ\neq Y^{2}$.
Thus, there is a distinction between $BS_s$ with two elements and $BS_s$ with a single element.
However, we cannot always distinguish between three or more elements and a single element.
A previous attempt to solve the problem used a deterministic collision
detection structure \cite{cormode05}. This structure
held the binary representation of the elements. However, this representation
falsely identifies multiple elements as a single element on some inputs\footnote{
For example, for every set of 4 pairs $\{(2k,1),(2k+1,-1),(2k+2,-1),(2k+3,1)\}$,
the whole structure is zeroed and any additional pair $(k^{'},C_{k}^{'})$
will be identified as a single element.}.
In order to solve the problem we use a new counter defined as follows.
\begin{lemma} \label{lemma:countersq}
Let $\mathcal{H}=\{h \colon \left[m\right]\rightarrow\left[q\right]\}$
be a $t$-wise independent family of hash functions, and let $h$
be a hash function drawn randomly from $\mathcal{H}$.
Let $T=\sum_{k}{C_{k}h(k)}$ be a counter.
Assume $T$ is a sum of at most $t-1$ elements.
Then for every element $(k',C_{k}')$, where $k'\in\left[m\right]$, if $T$ is not the sum of the single element $(k',C_{k}')$,
then $\Pr_{h \in \mathcal{H}}[T = C_{k}'h(k')] \leq 1/q$.
\end{lemma}
\begin{proof} We subtract $C_{k}'h(k')$ from $T$ and obtain $T'=T-C_{k}'h(k')$.
If $T$ is not the sum of the single element $(k',C_{k}')$, there
are between $1$ and $t$ elements in $T'$. The hashes of those elements
are independent because $h$ is $t$-wise independent.
We therefore have a linear combination of independent uniformly distributed
elements in the range $[q]$, thus
$\Pr_{h \in \mathcal{H}}[T'=0] \leq 1/q$.
\end{proof}
The \emph{Non-strict Bin Sketch ($BS_{ns}$)} consists of four
counters: $X$,$Y$,$Z$ and an additional counter $T$, as defined
in Lemma \ref{lemma:countersq}.
The space of $BS_{ns}$ is $O(\log{(mrq)})$ bits.
The time to insert an element is $O(\log^2{t})$ since we evaluate a $t$-wise independent hash function.
There are other ways to maintain a sketch such as keeping a fingerprint. However, fingerprint update takes $O(\log{m})$ time, which depends on the size of the universe, while the update time of $BS_{ns}$ depends on $t$ which is later set to $t = \Theta(\log\frac{K}{\delta})$.
\begin{corollary} Three or more elements in $BS_{ns}$ may be falsely identified
as a single element. The error probability is bounded
by $1/q$, when there are at most $t-1$ elements in the bin. \end{corollary}
$BS$ is placed in each cell, which we refer to as \emph{bin}, in each of the arrays of our data structure. Its role is
to identify a \emph{collision}, which occurs when more than one element
is in the bin. If no collision occurred, $BS$ returns the
single element in the bin.
$BS$ supports additions and deletions and is oblivious of the order of their occurrences in the stream, which makes it a \emph{strongly history independent} data structure \cite{micciancio97,naor01}. Its use in our sampling data structure makes the whole sampling structure strongly history independent.
\subsection{$L_{0}$ Estimation}
The number of distinct values in a stream with deletions is known as the \emph{Hamming norm} $L_{0}=\left|\{k\in[m] \colon C_{k}\neq0\}\right|$
\cite{cormode03}.
$L_{0}$ Estimation is used by our algorithm for choosing the level
$l^{*}$ from which we extract the elements that are our sample.
We use the structure of Kane et al.\ \cite{kane10}, which
provides an $\epsilon$-approximation $\tilde{L}_{0}$
with $2/3$ success probability for both Strict and Non-strict
streams.
It uses space of $O(\frac{1}{\epsilon^{2}}\log{m}(\log\frac{1}{\epsilon}+\log\log(r)))$
bits and $O(1)$ update and report times.
Our algorithm requires only a constant approximation to $L_0$. Hence the space is $O(\log{m} \cdot \log\log(r)))$ bits.
However, we require $1-\delta$ success probability for any given $\delta > 0$ and not only $2/3$.
For estimation of $L_0$, where $L_0 = \Omega({\frac{1}{\epsilon^2}\log\frac{1}{\delta})}$ we use methods similar to those in the rest of the paper. We keep $\tau = \Theta(\log\frac{1}{\delta})$ instances of Kane et al. structure,
and use one $\Theta(\log\frac{1}{\delta})$ independent hash function to map each stream element to
one of the instances.
With constant high probability all instances have approximately the same number of elements.
$\tilde{L}_0$ is the median of the estimations obtained from all instances multiplied by $\tau$, and is a constant approximation to $L_0$ with probability $1-\delta$.
Thus we obtain an $L_0$ estimation algorithm with $O(\log{m} \log\log(r) \log\frac{1}{\delta})$ bits of space, $O((\log\log\frac{1}{\delta})^2)$ update time, since one $O(\log\frac{1}{\delta})$ independent hash function is evaluated, $O(\log\frac{1}{\delta})$ reporting time, and $1-\delta$ success probability.
These requirements are lower than their corresponding requirements in the other phases of our algorithm.
\subsection{Mapping to Levels}
The first phase of our algorithm is mapping the elements in the data stream to levels.
We use a family of $t$-wise
independent hash functions $\mathcal{H}=\{h \colon [m]\rightarrow[M]\}$ for $t=\Theta(\log{\frac{1}{\delta}})$.
We select $h\in\mathcal{H}$ randomly and use it to map the elements
to $L=\log_{1/\lambda}{M}$ levels, for $\lambda \in (0,1)$. Typical values are $\lambda=0.5$ and
$M=2m$. The
mapping is performed using a set of hash functions $h_{l}(x)=\left\lfloor {\frac{h(x)}{{\lambda}^{l}M}}\right\rfloor $,
for $l\in[L]$. An element $x$ is mapped to level $l$ $\Leftrightarrow$
$(h_{l}(x)=0\wedge h_{l+1}(x)\neq0)$. Note that each element is mapped
to a single level.
Using this mapping, the set of elements mapped to each level is $t$-wise independent. It follows that in order to obtain a $t$-wise independent sample, we can extract all elements from any level we choose. However, we must select
the level independently of the elements that were mapped to it. If
any event that depends on the specific mapping influences the level selection,
the sample becomes biased. Biased samples appeared in some previous works.
In order to choose the level regardless of the mapping, we use the number of distinct elements
$L_{0}$.
We obtain an estimation $\tilde{L}_{0}$ from the $L_{0}$ estimation structure, where $L_{0}\leq\tilde{L}_{0}\leq\alpha L_{0}$ for $\alpha>1$ with $1-\delta$ probability, and choose the level where $K$ elements are expected.
\begin{lemma} \label{lemma:mapping}
Let the elements be mapped to levels
using a hash function $h$ selected randomly from a $t$-wise
independent hash family $\mathcal{H}$ for $t=\Omega(\log{\frac{1}{\delta}})$. Assume there is an estimation
$L_{0}\leq\tilde{L}_{0}\leq\alpha L_{0}$ for $\alpha>1$, and $K = \Omega({\log\frac{1}{\delta}})$. Then the
level $l^{*}$ for which $\frac{1}{\alpha}\tilde{L}_{0}\lambda^{l^{*}+1}(1-\lambda)<2K\leq\frac{1}{\alpha}\tilde{L}_{0}\lambda^{l^{*}}(1-\lambda)$
has $K$ to $(\frac{2\alpha}{\lambda}+1)K$ elements with probability at least
$1-\delta$.
\end{lemma}
\begin{proof} See Appendix \ref{appendix:proofs}.
\end{proof}
Let $X$ be the set of elements in level $l^{*}$, from which we choose to extract the sample.
We denote $\tilde{K} = (\frac{2\alpha}{\lambda}+1)K$ the maximal number of
elements in level $l^{*}$. With probability at least $1-\delta$, $K \leq \left| X \right| \leq \tilde{K}$.
For typical values $\alpha = 1.5$ and $\lambda = 0.5$, $K \leq \left| X \right| \leq 7K$.
| {
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} | 4,304 |
Home Gaming Game Reviews Neon Chrome Vita/PS4 Review
Neon Chrome Vita/PS4 Review
Neon Chrome is a top-down shooter game available for download from the PlayStation Store for the PS4 and PS Vita with compatibility for PlayStation TV. Neon Chrome is developed by 10Tons; the same developer who created the amazing Crimsonland, therefore the two major questions leading into Neon Chrome is if it maintains the quality of Crimsonland or if it actually improves upon it and if the Vita version of Neon Chrome can surpass that of the earlier PS4 release.
The story revolves around a rebellion against Neon Corp who house anywhere from hundreds of thousands through to a million people in giant megastructures known as arcologies. If anyone is deemed untrustworthy then they will be detained against their own will and with the lead character being found in this position within one of the largest arcologies on Earth called Neon Chrome; the only way to overcome the predicament is to take down the Overseer and the countless enemies that guard him.
The character design is rather varied as characters are referred to as assets as your character is sitting in an immersion chair in full control over an asset with different assets effectively having their own loadouts including a specific weapon to begin with, unique abilities and enhancements such as Collene Deckard classed as an Assassin with a 20% increase in critical shots to enemies and a 10% increase in speed who also has a Nanofiber Shadow Skin which makes the assassin practically invisible when in shadow as well as being equipped with a burst rifle and a laser pulse.
The enemy design includes guards with varying amounts of armour, weaponry and routines, mechanised spiders, flying drones, turrets and more besides with guards showing up in the middle of some levels to provide enemy reinforcements, while enemy boss fights include a larger spider bot, a centipede, a hover tank and more besides.
The environment design comprises of procedurally generated environments resulting in no two levels being exactly the same which is set within a city that has a Blade Runner esque stylised futuristic neon look to it which you can catch a glimpse of when nearby to an outer wall as rain falls amidst seemingly endless skyscrapers. Certain surfaces of the interior such as thinner walls, glass and debris can be destroyed with multiple melee attacks or shooting at them, although keys are required to open some doors such as red keys that are needed for progression on the path taken to the end of the level at the green elevator, while there are also yellow keys that will often lead to a special level through a purple elevator situated somewhere else in the level.
In-game currency can be earned by destroying enemies and looting a variety of crates and chambers situated throughout every level that also contain weaponry, abilities, enhancements and upgrades which is incredibly important to increasing your chances of surviving until the end of each level, therefore providing a better offence when attacking enemies and defence when being attacked by enemies.
Upgrades are available for five separate categories including health, damage, luck, energy and slots with 100 levels worth of progression on all of the upgrade categories other than the slots category which has 10 levels of progress. Health upgrades increase the maximum amount of health, while damage increases the amount of damage you are capable of inflicting on enemies; luck improves the possibility of critical hits on enemies, increases the amount of credits looted and makes rarer loot easier to find; energy provides the required fuel to utilise abilities; and slots increases the amount of cybernetic enhancement slots available in order to install more enhancements. Improvements in statistics made by each upgrade are important to the balancing of the gameplay as the onslaught of enemies naturally increases throughout the game, therefore upgrades may not necessarily be required initially, but will absolutely be essential in later levels. Upgrade qualities mostly remain the same within each respective upgrade category, although the cost of each upgrade varies for every upgrade level such as the first health upgrade improving health statistics by 8 from 160 to 168 costing 200 credits with the second upgrade having the same amount of improvement but costing 280 credits, while the first damage improvement is 5% costing 200 credits with the second improvement being the same increase for 300 credits.
There are 26 weapons that must be unlocked by finding them within specific chapters which also unlocks weaponry for immediate use from that point onwards via purchases starting from 300 credits per weapon. Weapons have their own attributes such as a rate of fire, range, damage and accuracy including a submachine gun, an assault rifle, a burst rifle, a shotgun, an ion shotgun, a laser submachine gun, a laser assault rifle, a laser burst rifle, a grenade launcher and much more besides.
There are 16 abilities that have to be unlocked by finding them within certain chapters followed by being unlocked for immediate use from that point onwards via purchases starting from 300 credits per ability. Abilities can be equipped such as micro missiles that are capable of locking onto enemies, a laser pulse which shoots hi-intensity laser beams in all directions, blast grenades that produce a large blast radius, a grenade launcher that is capable of bouncing grenades off walls and many more abilities.
There are 49 enhancements that are unlocked by finding them within particular chapters which also unlocks enhancements for immediate use from that point onwards via purchases starting from 300 credits per enhancement. Enhancements can be equipped to the specific amount of slots that have been unlocked from the slots upgrade category with enhancements including NanoEdge SkinWeave which provides Nanites that produce a reinforced mesh into the skin of your character resulting in a 20% increase in health, an Omnitech Personal Guard which automatically zaps three nearby enemies with bolts of electricity, Chiphow AmmoArt increases the clip size of weaponry by 50% which conveniently reduces the likelihood of running out of ammo when surrounded by enemies, Duramax Bone Lacing provides a simultaneous 15% increase in health and a titanium-reinforced knee to the face with a 50% increase in melee damage and many more enhancements.
Neon Chrome supports cross-buy between the PS4 and Vita. Cross-buy presents a superb amount of value as it means that you will be purchasing the PS4 and Vita versions of the game with just a single purchase, although there is no cross-save functionality; resulting in players not being able to continue from their levelled up character progression contained within the previously released PS4 version.
The controls are appropriately optimised for the Vita with the control scheme consisting of pressing R to shoot; pressing L to use an ability; pressing O to reload; pressing X to use an item or object; pressing square to perform a melee attack; changing the direction of the left analogue stick to move your character; changing the direction of the right analogue stick to aim your weapon; and pressing start to display the pause menu. There is no touch screen or rear touch pad controls which is surprising as 10Tons' Crimsonland previously featured the touch screen as an accurate alternative to pressing R to fire and the right analogue stick for aiming.
Graphically, Neon Chrome retains the frantic action set upon procedurally generated destructible environments with imaginative weaponry and abilities producing spectacular effects. However, the Vita version has had some graphical downgrades in comparison to the PS4 release as the textures are not as crisp and there are some frame-rate drops that occur during a higher than usual amount of enemies, firing and explosions. The quality of Neon Chrome on Vita can still be appreciated in its own right and if you have never played the PS4 version before then you will not notice the reduction in texturing as much. Neon Chrome supports PS4 Pro enhancements including the procedurally generated destructible environments in sensational 4K native resolution at 60 frames-per-second.
The presentation of the game is solid with a great touch screen based user interface across various menus such as the main menu, level selection menu, options menus and gameplay menus with support for navigation via the left and right analogue sticks, directional pad and face buttons. The background of the menu screens consists of a character with a weapon as the Overseer sits in the immersion chair as a huge city can be seen as far as the horizon with gigantic towering structures, while the title logo of the game is positioned centrally at the top of the screen as a neon sign. Leaving the game inactive for a few moments will provide a variety of gameplay demos showcasing different environments, weapons and enemies.
Audio consists of voice-overs for the lead villain and guards insisting your character stops what they are doing, while sound effects include collecting energy and credits, shooting at enemies, reloading, melee attacking enemies, enemies firing at your character, explosions and more besides. Voice-overs and sound effects are perfectly complimented by award-winning composer Jonathan Geer who delivers a thrilling, futuristic sci-fi soundtrack that is pulsating yet simultaneously thought provoking as it is reminiscent of a fusion between Blade Runner and the grander scale of Total Recall.
The trophy list includes 14 trophies with 9 bronze trophies and 5 silver trophies. The easiest trophy has to be the Hard Work bronze trophy for killing 1,000 enemies due to the progressive accumulation from every level, while harder trophies include the Sneaky silver trophy for completing a level without being seen; the Stat Master silver trophy for upgrading a stat to level 100; and the Unlocker silver trophy for unlocking all unlockables. A special mention has to go to the rather appropriately titled You've Just Been Erased bronze trophy for killing 3 enemies with a single railgun bullet as the name of the trophy rather cleverly references dialogue from the Arnold Schwarzenegger film Eraser in which a railgun has been created. It is estimated that depending upon skill and a good trophy guide to provide some helpful tips that it would take between 15 to 25 hours to 100% the trophy list.
There are no difficulty levels, although the gameplay is deliberately punishing resulting in players having to seriously upgrade their character's stats and improve their weaponry, abilities and enhancements before standing a chance of survival against the many enemies situated within each level. Gameplay still remains at a hard difficulty as randomly generated levels will surprise players by making it impossible to memorise a specific path to the end of any level.
The cross-buy PS4 version provides local co-operative multiplayer for 2 to 4 players, although every player participating has to remain fairly close to each other as every player is restricted to the confines of the screen instead of starting on the same screen and moving to a dynamic split-screen in the scenario of being on opposite sides of the level, while the second player must share the same statistics rather than upgrading statistics independently.
However, there is no form of multiplayer on the Vita release of Neon Chrome, although multiplayer functionality would have perhaps been possible via ad-hoc multiplayer with another Vita or PlayStation TV, cross-play as part of the 4 player local co-operative multiplayer on PS4 or online multiplayer.
I would have liked to have seen some form of competitive multiplayer from the PS4 version that would have included such game modes as deathmatch, team deathmatch, capture the flag and king of the hill, amongst other game modes, while set on the full-scale destructible environments found within each of the procedurally generated environments from the single player and co-op modes with the progressive upgrades system, customisable loadout and support for two to four players. I would also have liked to see the local co-operative multiplayer be available in the form of online co-operative multiplayer; just to provide that customary freedom for players to be able to play the single player experience locally with up to three friends or online with up to three friends and perhaps even in single player with the option of the game being open for anyone from your friends' list or globally to join in co-operatively to help each other past a difficult area of the game.
There are no online leaderboards which is surprising as they could have included leaderboards for the amount of enemies killed and the accumulation of credits as well as the amount of levels completed in total and the highest amount of levels completed in a single run.
The replayability for the Vita release of Neon Chrome stems from important areas of gameplay such as the procedurally generated destructible environments, an incredibly in-depth upgrades system as well as a plethora of weapons, abilities and enhancements that is pivotal to the balancing of gameplay in regards to your character or your surrounding enemies having the upper hand, while the cross-buy PS4 version also brings local co-operative multiplayer for 2 to 4 players which are all features that will collectively have players returning for quite some time on both Vita and PS4.
• Title: Neon Chrome
• Developer: 10Tons
• Publisher: 10Tons
• System: PS Vita, PlayStation TV and PS4
• Format: PSN Download
• Cross-Buy: Yes (PS Vita, PlayStation TV and PS4)
• Cross-Play: No
• Players: 1 (PS Vita)/1-4 (Local Co-operative Multiplayer on PS4)
• Memory Card Space Required: 141MB
Jonathan Geer
Twin-Stick
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Jason plays all genres of games and enjoys all different kinds of experiences that the games industry has to offer. Jason's favourite PlayStation exclusive franchises throughout various eras include: Crash Bandicoot, God of War, Gran Turismo, inFamous, Killzone, Little Big Planet, MotorStorm, Resistance, Spyro the Dragon, Uncharted, Wipeout and various games that never became big name franchises. A special mention goes to Black Rock's superb Split Second: Velocity as it is rather unbelievable that it will never receive a sequel. Jason now mainly plays modern PlayStation games on home console and portably, but occasionally returns to the old retro classics on the 3DO, PS1 and PS2 such as discovering Cool Spot Goes to Hollywood 20 years after its original release on PS1. Jason is happy to see gaming coming full circle with updates for retro classics such as Alien Breed, Superfrog and Crash Bandicoot.
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Redout: Lightspeed Edition PS4 Review | {
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Last week I wrote about how Iran's partnership with Islamic jihad terror militias are keeping Israel's right-wing parties in power. I write this piece waiting for the final results of Israel's elections. However, despite the 35 seat tie between the right-wing Likud and center-left Blue and White parties, it appears Likud and its ability to form a coalition will claim victory.
The Israeli Knesset (parliament) has 120 seats - 40 parties vied for them this past Tuesday in Israel's election. A party must obtain at least 3.25 percent of the national vote to enter the Knesset - this is equivalent to 4 seats. Not since Israel's first election in 1949 has a party won a majority. Twelve parties had chances of winning seats. Calculating voting results takes days. Once completed, Israel's President will consult with each party (that obtained the 4 parliament seat threshold) on who they support as Prime Minister. The proposed party leader has 42 days to form a coalition government and involves obtaining multiple party support. Four term incumbent Benjamin Netanyahu of the Likud Party, and his opponent, Benny Gantz of the centrist-left Blue and White Party, are those proposed leaders. It is projected that Netanyahu will be able to obtain a coalition and achieve a record 5th four year term as Prime Minister since the majority of seats were won by right-wing parties who back him.
Netanyahu faces personal troubles: three corruption charges which can lead to indictments this year. If he is charged, his coalition parties will have to decide if they will continue support him as Prime Minister. July begins Netanyahu's pre-trial hearings and which time he has an opportunity to convince Israel's Attorney General to drop the charges.
In 1993 the Oslo Accords gave hope of peace and co-existence to both the Israeli and Palestinian peoples. Its purpose: to implement steps towards a peaceful two-state solution. The Oslo II Accords dissected the West Bank and Gaza Strip into A, B and C Palestinian regions.
September 28, 2000 brought the second Intifada (Palestinian uprising). This was the response to the stronghold the Israelis kept on the Palestinian territories despite the freedoms the Palestinians thought they would get under the Oslo Accords. A factitious Palestinian leadership has developed a sixty plus year history of terroristic (rather than diplomatic) tactics and it is still plagued with such leadership. Israelis have been caught in the quagmire of whether Palestinians can be trusted to live in peaceful co-existence with Israel or whether it is just a ploy towards destroying Israel completely. When the World Trade Center attack went beyond Israel's borders to that of the United States it became apparent the world that Islamic jihad is not about the Palestinian refugees, but a greater quest. The fear of Iran and Islamic jihad's mission will continue to be reflected in not only Netanyahu's leadership, but international relations as well as the world enters a "new world order" that has the fear of nuclear proliferation and terror militias infiltrating vulnerable regions at its core. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,903 |
{"url":"https:\/\/cs.stackexchange.com\/questions\/104863\/generalization-of-functors-to-other-datatypes","text":"# Generalization of Functors to other datatypes?\n\nFunctors in category theory, and also in its application to functional programming, can be seen as a kind of \"structured\" functions: Given two sets $$A,B$$, rather than just having a function $$f:A\\to B$$, we have a functor $$F$$ which contains $$f$$, but additionally also maps morphisms on $$A$$ to morphisms on $$B$$ (i.e. maps not only $$A$$ to $$B$$ but also maps \"structure on $$A$$\" to \"structure on $$B$$\").\n\nIn the category theory of functional programming, objects $$A,B$$ are types and morphisms are functions $$A\\to B$$ between types. That means functors map types $$A,B$$ to other types $$X,Y$$ and map functions $$A\\to B$$ to functions $$X\\to Y$$. Functors in functional programming can thus be seen as a mapping between datatypes that additionally also maps the functional structure on those data types.\n\nMy thought was: In the context of object-oriented programming, there really are more kinds of structure on datatypes than just functions between them. For example, given two types $$Int$$ and $$Bool$$, we can not only define functions $$f:Int\\to Bool$$, but also things like $$p:Int\\times Bool$$, or more complex datatypes like classes with private component classes.\n\nIs there a generalization of functors to \"mappings between datatypes that also preserve structure\", where that doesn't need to be functorial? In particular I'm wondering whether such a concept is used in analyzing programming languages.\n\nI guess no one else is going to answer this, so I'll take a crack.\n\nFirst, there's nothing inherently 'functional' about categories, necessarily. For instance, you can make any monoid into a category, where you have one object, and then the arrows from that one object to itself are the elements of the monoid. The identity element is the identity arrow, and composition is monoid multiplication. Then functors between two monoids in this sense are homomorphisms between the monoids; they are functions that preserve identity and multiplication.\n\nCategories are then generalizations of this to situations where the monoid values have types, and can only be multiplied together when the types correspond. The specifics of this work for functions, but it could be a lot of other things, too. You could try to devise a category where $$Hom(X,Y) = X\u00d7Y$$, and then functors from\/to that category would be obligated to map from\/to products.\n\nHowever, it is also common to define additional structure beyond just being a category. For instance, even for 'functional programming,' you are usually talking about (at least) Cartesian closed categories. These are categories $$C$$ that have additional structure:\n\n$$-\u00d7- : C \u00d7 C \u2192 C$$ $$1 : C$$ $$[-,-] : C^{op} \u00d7 C \u2192 C$$\n\nWhere $$\u00d7$$ forms binary products, $$1$$ is a terminal object, and $$[-,-]$$ forms exponentials, so:\n\n$$C(X\u00d7Y,Z) \\cong C(X, [Y,Z])$$\n\nAnd when you have categories with additional structure like this, you also want to talk about functors that preserve that extra structure. So there are Cartesian closed functors between Cartesian closed categories that map products to products, terminal objects to terminal objects, and exponentials to exponentials.\n\nSo, yes, you can describe plenty of structure besides just $$Hom$$, and talk about maps that preserve all that structure (in various senses), not just functors.","date":"2021-05-12 20:36:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 30, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9045789837837219, \"perplexity\": 429.53727585470233}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989705.28\/warc\/CC-MAIN-20210512193253-20210512223253-00346.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:intro}
Let $K$ be an algebraically closed field that is complete with respect to
a non-trivial and non-archimedean absolute value.
The action of a rational function $h\in K(z)$
on $\mathbb{P}^1=\mathbb{P}^1(K)$ extends continuously to that on
the Berkovich projective line $\mathsf{P}^1=\mathsf{P}^1(K)$. If in addition
$\deg h>0$, then this extended action of $h$
on $\mathsf{P}^1$ is surjective, open, and
fiber-discrete and preserves the type (among I, II, III, and IV)
of each point in $\mathsf{P}^1$,
and the local degree function $\deg_{\,\cdot}h$ of $h$
on $\mathbb{P}^1$ also extends upper semicontinuously to $\mathsf{P}^1$ so that for every open subset $V$ in $\mathsf{P}^1$ and every component $U$
of $h^{-1}(V)$, $V\ni\mathcal{S}'\mapsto\sum_{\mathcal{S}\in h^{-1}(\mathcal{S}')\cap U}\deg_{\mathcal{S}}h\equiv\deg(h:U\to V)$.
The pushforward $h_*:C^0(\mathsf{P}^1)\to C^0(\mathsf{P}^1)$
is defined so that for every $\psi\in C^0(\mathsf{P}^1)$,
$(h_*\psi)(\cdot):=\sum_{\mathcal{S}\in h^{-1}(\cdot)}(\deg_{\mathcal{S}}h)\psi(\mathcal{S})$
on $\mathsf{P}^1$.
The pullback $h^*$ from the
space $M(\mathsf{P}^1)$ of all Radon measures on $\mathsf{P}^1$ to itself is defined by the
transpose of $h_*$, so that for every $\nu\in M(\mathsf{P}^1)$,
\begin{gather}
h^*\nu=\int_{\mathsf{P}^1}\Bigl(\sum_{\mathcal{S}'\in h^{-1}(\mathcal{S})}(\deg_{\mathcal{S}'}h)\delta_{\mathcal{S}'}\Bigr)\nu(\mathcal{S})\quad\text{on }\mathsf{P}^1,\label{eq:pullbackdirac}
\end{gather}
where for each $\mathcal{S}\in\mathsf{P}^1$, $\delta_{\mathcal{S}}$ is the Dirac measure
at $\mathcal{S}$ on $\mathsf{P}^1$; in particular, $(h^*\delta_{\mathcal{S}})(\mathsf{P}^1)
=\deg h$.
\subsection{Factorization on $\mathsf{P}^1$ and quantization}\label{sec:skeltal}
We follow the presentation in \cite[\S 4.2]{DF14}.
For each finite subset $\Gamma$ consisting of type II points
(i.e., a semistable vertex set in $\mathsf{P}^1$), the family
\begin{gather*}
S(\Gamma):=\{\text{either a component of }\mathsf{P}^1\setminus\Gamma\text{ or
a singleton }
\{\mathcal{S}\}\text{ for some }\mathcal{S}\in\Gamma\}\subset 2^{\mathsf{P}^1}
\end{gather*}
is a partition of $\mathsf{P}^1$;
the measurable factor space $(\mathsf{P}^1/S(\Gamma),2^{\mathsf{P}^1/S(\Gamma)})=(S(\Gamma),2^{S(\Gamma)})$
is identified with the measurable space $(\mathsf{P}^1,\sigma(S(\Gamma)))$.
The factor map $\pi_{\mathsf{P}^1,\Gamma}:\mathsf{P}^1\to\mathsf{P}^1/S(\Gamma)$ induces
the pullback $(\pi_{\mathsf{P}^1,\Gamma})^*$ from the space of
$\sigma(S(\Gamma))$-measurable functions on $\mathsf{P}^1$ to that of measurable functions on $\mathsf{P}^1$ so that $(\pi_{\mathsf{P}^1,\Gamma})^*1_U=1_U$ on $\mathsf{P}^1$
for any $U\in S(\Gamma)$,
and in turn the transpose (projection/quantization) $(\pi_{\mathsf{P}^1,\Gamma})_*$
from the space $M(\mathsf{P}^1)$ of all complex Radon measures $\nu$ on $\mathsf{P}^1$
to the space $M(\Gamma)$ of all complex measures on $(\mathsf{P}^1,\sigma(S(\Gamma)))$
so that for every $\nu\in M(\mathsf{P}^1)$,
\begin{gather}
\bigl((\pi_{\mathsf{P}^1,\Gamma})_*\nu\bigr)(U)=\nu(U)
\quad\text{for any }U\in S(\Gamma);\label{eq:projectfactor}
\end{gather}
then $(\pi_{\mathsf{P}^1,\Gamma})_*$ restricts a map from
$M^1(\mathsf{P}^1):=\{\omega\in M(\mathsf{P}^1):\omega\ge 0\text{ and }\omega(\mathsf{P}^1)=1\}$
to $M^1(\Gamma):=\{\omega\in M(\Gamma):\omega\ge 0\text{ and }\omega(\mathsf{P}^1/S(\Gamma))=1\}$. we also set
\begin{gather*}
M^1(\Gamma)^\dag:=\bigl\{\omega\in M^1(\Gamma):\omega(\{\mathcal{S}\})=0\text{ for any }\mathcal{S}\in\Gamma\bigr\}.
\end{gather*}
Moreover, for any finite subsets $\Gamma$ and $\Gamma'$ both
consisting of type II points and satisfying $\Gamma\subset\Gamma'$,
the projection $\pi_{\Gamma',\Gamma}:\mathsf{P}^1/S(\Gamma')\to\mathsf{P}^1/S(\Gamma)$
induces the pullback $(\pi_{\Gamma',\Gamma})^*$
from the space of $\sigma(S(\Gamma))$-measurable functions on $\mathsf{P}^1$
to that of $\sigma(S(\Gamma'))$-measurable ones so that
$(\pi_{\Gamma',\Gamma})^*1_U=\sum_{V\in S(\Gamma'):V\subset U}1_V$
on $\mathsf{P}^1$ for any $U\in S(\Gamma)$, so that $(\pi_{\mathsf{P}^1,\Gamma})^*
=(\pi_{\mathsf{P}^1,\Gamma'})^*(\pi_{\Gamma',\Gamma})^*$,
and in turn the transpose (say the projection) $(\pi_{\Gamma',\Gamma})_*:M(\Gamma')\to M(\Gamma)$
so that for every $\nu\in M(\Gamma')$,
$\bigl((\pi_{\Gamma',\Gamma})_*\nu\bigr)(U)
=\sum_{V\in S(\Gamma'):V\subset U}\nu(V)
$ for any $U\in S(\Gamma)$. Then
$(\pi_{\Gamma',\Gamma})_*(M(\Gamma')^\dag)\subset M(\Gamma)^\dag$.
Let us denote by $\mathcal{S}_G$ the Gauss (or canonical) point in $\mathsf{P}^1$,
which is a type II point (see \S\ref{sec:berkovichgeneral}).
For a rational function $h\in K(z)$ on $\mathbb{P}^1$ of degree $>0$, setting
\begin{gather*}
\Gamma_G:=\{\mathcal{S}_G\}\quad\text{and}\quad\Gamma_h:=\{\mathcal{S}_G,h(\mathcal{S}_G)\},
\end{gather*}
the quantized pullback
$(h_{\Gamma_G,\Gamma_h})^*:M(\Gamma_h)\to M(\Gamma_G)$
is induced from the pullback $h^*$ in \eqref{eq:pullbackdirac}
so that for every $\omega\in M(\Gamma_h)$,
\begin{gather*}
\bigl((h_{\Gamma_G,\Gamma_h})^*\omega\bigr)(U)
=\sum_{V\in S(\Gamma_h)}m_{V,U}(h)\omega(V)
\quad\text{for any }U\in S(\Gamma_G);
\end{gather*}
here
the quantized local degree $m_{V,U}(h)$ of $h$ with respect to
each pair $(U,V)\in S(\Gamma_G)\times S(\Gamma_h)$
is induced from the local degree function $\deg_{\,\cdot}h$
on $\mathsf{P}^1$ so that for any $\mathcal{S}'\in V$,
\begin{align*}
m_{V,U}(h)
=\begin{cases}
(h^*\delta_{\mathcal{S}'})(U)
& \text{if }U\in S(\Gamma_G)\setminus\{\{\mathcal{S}_G\}\}\text{ and }V\in S(\Gamma_G)\setminus\{\{h(\mathcal{S}_G)\}\},\\
(h^*\delta_{\mathcal{S}'})(\{\mathcal{S}_G\}) & \text{if }U=\{\{\mathcal{S}_G\}\}
\end{cases}
\end{align*}
(the remaining case that $U\in S(\Gamma_G)\setminus\{\{\mathcal{S}_G\}\}$
\& $V=\{\{h(\mathcal{S}_G)\}\}$ is more subtle) and that for every $V\in S(\Gamma_h)$,
$\sum_{U\in S(\Gamma_G)}m_{V,U}(h)=\deg h$; see
\S\ref{sec:quantized} for the precise definition of $m_{V,U}(h)$.
\subsection{The $f$-balanced measures on $\mathsf{P}^1$}\label{sec:balanced}
From now on,
let $f\in K(z)$ be a rational function on $\mathbb{P}^1$ of $\deg f=:d>1$.
The equilibrium (or canonical) measure $\nu_f$ of $f$ on $\mathsf{P}^1$ is
the weak limit
\begin{gather}
\nu_f:=\lim_{n\to\infty}\frac{(f^n)^*\delta_{\mathcal{S}}}{d^n}\quad\text{in }M(\mathsf{P}^1)\quad\text{for any }\mathcal{S}\in\mathsf{P}^1\setminus E(f)\label{eq:equidist}
\end{gather}
(see \cite{FR09} for the details), and is
the unique $\nu\in M^1(\mathsf{P}^1)$
satisfying both
the $f$-balanced property $f^*\nu=(\deg f)\cdot\nu$ on $\mathsf{P}^1$
and the vanishing condition $\nu(E(f))=0$. Here
the (classical) exceptional set
$E(f):=\{a\in\mathbb{P}^1:\#\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(a)<\infty\}$
of $f$ is the union of all (superattracting) cycles of $f$ in $\mathbb{P}^1$
totally invariant under $f$, and the Berkovich Julia set
$\mathsf{J}(f):=\operatorname{supp}\nu_f$ of $f$
is in $\mathsf{P}^1\setminus E(f)$ (by \eqref{eq:equidist});
$E(f)$ is at most countable, and when $\operatorname{char}K=0$,
we even have
\begin{gather}
E(f)=\bigl\{a\in\mathbb{P}^1:f^{-2}(a)=\{a\}\bigr\}\quad\text{and}\quad\#E(f)\le
\#\{a\in\mathbb{P}^1:\deg_a(f)=d\}\le 2.\label{eq:maximaldegree}
\end{gather}
For every $n\in\mathbb{N}$, we also have
$\nu_{f^n}=\nu_f$ in $M^1(\mathsf{P}^1)$
and $E(f^n)=E(f)$.
Setting
\begin{gather*}
\delta_{\mathcal{E}}:=\frac{\sum_{a\in\mathcal{E}}\delta_a}{\#\mathcal{E}}\in M^1(\mathsf{P}^1)
\quad(\text{so }\delta_{\mathcal{E}}\neq\nu_f)
\end{gather*}
for each cycle $\mathcal{E}$ of $f$ in $E(f)$,
any $\nu\in M^1(\mathsf{P}^1)$ satisfying
the $f$-balanced property $f^*\nu=(\deg f)\cdot\nu$ on $\mathsf{P}^1$ is written as
$\nu(\mathsf{J}(f))\cdot\nu_f+\sum_{\mathcal{E}\subset E(f):\text{ a cycle of }f}\nu(\mathcal{E})\cdot\delta_{\mathcal{E}}$
(also by \eqref{eq:equidist}).
Recall that for any $\mathcal{S}\in\mathsf{H}^1:=\mathsf{P}^1\setminus\mathbb{P}^1$,
(see e.g.\ {\cite[Corollary 10.33]{BR10}})
\begin{gather}
f^{-1}(\mathcal{S})\neq\{\mathcal{S}\}\Leftrightarrow\nu_f(\{\mathcal{S}\})<1\Leftrightarrow\operatorname{supp}(\nu_f)\neq\{\mathcal{S}\}
\Leftrightarrow\nu_f(\{\mathcal{S}\})=0\Leftrightarrow\nu_f(\{f(\mathcal{S})\})=0.\label{eq:nopotgood}
\end{gather}
\subsection{Main result: the projections of the $f$-balanced measures
to $\mathsf{P}^1/S(\Gamma_G)$}
Recall that for each $n\in\mathbb{N}$,
$\Gamma_G:=\{\mathcal{S}_G\}$ and $\Gamma_{f^n}:=\{\mathcal{S}_G,f^n(\mathcal{S}_G)\}$.
Let us say $\omega\in M^1(\Gamma_f)$ satisfies
the quantized $f$-balanced property if
\begin{gather}
(\deg f)^{-1}(f_{\Gamma_G,\Gamma_f})^*\omega=(\pi_{\Gamma_f,\Gamma_G})_*\omega\quad\text{in }M^1(\Gamma_G).\label{eq:quantizedbalanced}
\end{gather}
Set $\Delta_f^\dag:=\Delta_f\cap\bigl(M^1(\Gamma_G)^\dag\bigr)(\subset\Delta_f)$, where
\begin{multline*}
\Delta_f:=
\bigl\{\omega\in M^1(\Gamma_G):
\text{for (any) }n\gg 1,\text{ there is }\omega_n\in M^1(\Gamma_{f^n})
\text{ such that}\\
(\deg(f^n))^{-1}((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n=
\omega=(\pi_{\Gamma_{f^n},\Gamma_G})_*\omega_n\text{ in }M^1(\Gamma_G)\bigr\}
\subset M^1(\Gamma_G).
\end{multline*}
Our principal result is the following computations of
$\Delta_f$ (and $\Delta_f^\dag$) when $\operatorname{char}K=0$;
the assumption on $E(f)$ in the statement (ii) is just for simplicity.
The proof is given in Section \ref{sec:proofmain}.
\begin{mainth}\label{th:computation}
Let $K$ be an algebraically closed field of characteristic $0$
that is complete with respect to
a non-trivial and non-archimedean absolute value, and
let $f\in K(z)$ a rational function on $\mathbb{P}^1$ of degree $d>1$.
If $f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$ and $f^{-1}(a)=\{a\}$ for any $a\in E(f)$,
then one and only one of the following $(i)$ and $(ii)$ occurs$;$
$(i)$ $\Delta_f^\dag=\Delta_f=(\pi_{\mathsf{P}^1,\Gamma_G})_*(\{\nu_f\})$,
$(ii)$ $\deg_{f^n(\mathcal{S}_G)}(f)\equiv d$ for $n\gg 1$, and
there is $a\in E(f)$ such that $\lim_{n\to\infty}f^n(\mathcal{S}_G)=a$,
$f^n(\mathcal{S}_G)$ is the interval $[\mathcal{S}_G,a]$ in $\mathsf{P}^1$ for $n\gg 1$,
$\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})<1$, and
\begin{multline}
\Delta_f=\Bigl\{\omega\in M^1(\Gamma_G):
\begin{cases}
\omega(U_{\overrightarrow{v}})=s\nu_f(U_{\overrightarrow{v}})
\text{ for every }\overrightarrow{v}\in\bigl(T_{\mathcal{S}_G}\mathsf{P}^1\bigr)\setminus\{\overrightarrow{\mathcal{S}_Ga}\},\\
\omega(\{\mathcal{S}_G\})=s',\text{ and}\quad
\omega(U_{\overrightarrow{\mathcal{S}_Ga}})
=\bigl(s\nu_f\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)+(1-s)\bigr)-s'
\end{cases}\\
\text{ for some }
0\le s\le 1\text{ and some }
0\le s'\le\min\bigl\{s\nu_f\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr),(1-s)\bigl(1-\nu_f\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)\bigr)
\bigr\}\Bigr\}.
\label{eq:computationdelta}
\end{multline}
Moreover, in the case $($ii$)$, we have the equivalence among
the following three conditions that
$\deg_{f^n(\mathcal{S}_G)}(f)\equiv d$ for any $n\in\mathbb{N}\cup\{0\}$, that
$\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})=0$, and
that $\Delta_f^\dag=\Delta_f$.
\end{mainth}
In Theorem \ref{th:computation}(ii),
the computation \eqref{eq:computationdelta}
in particular asserts not only that {\em $\Delta_f^\dag\subsetneq\Delta_f$
iff $\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})>0$}
but also that {\em $\Delta_f^\dag=
(\pi_{\mathsf{P}^1,\Gamma_G})_*(\{s\nu_f+(1-s)\delta_a:0\le s\le 1\})$
no matter whether $\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})=0$}
$($also by \eqref{eq:projectfactor}$)$.
In the proof of Theorem \ref{th:computation}, we will also
point out that for some $f$ (indeed $f(z)=z^2+t^{-1}z$ and its iterations),
we have the proper inclusion $\Delta_f^\dag\subsetneq\Delta_f$.
\subsection{Application: the unique degenerating limit
for the maximal entropy measures}
We call an element $f\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$ of degree say $d\in\mathbb{N}\cup\{0\}$
a meromorphic family of rational functions on
$\mathbb{P}^1(\mathbb{C})$ (of degree $d$ and parametrized by $\mathbb{D}=\{t\in\mathbb{C}:|t|<1\}$)
if for any $t\in\mathbb{D}^*=\mathbb{D}\setminus\{0\}$,
the specialization $f_t$ of $f$ at $t$ is
a rational function on $\mathbb{P}^1(\mathbb{C})$
of degree $d$. Let us denote by $\mathbb{L}$
the valued field of formal Puiseux series$/\mathbb{C}$
around $t=0$, i.e., the completion of the field $\overline{\mathbb{C}((t))}$
of Puiseux series$/\mathbb{C}$ around $t=0$
valuated by their vanishing orders at $t=0$.
Noting that $\mathcal{O}(\mathbb{D})[t^{-1}]$
is a subring of the field $\mathbb{C}((t))$ of formal Laurent series$/\mathbb{C}$ around
$t=0$, we also regard $f$ as an element of $\mathbb{L}(z)$.
If in addition $d>1$, then for every $t\in\mathbb{D}^*$,
there is the equilibrium (or canonical, and indeed the unique maximal entropy)
measure $\mu_{f_t}$ of $f_t$ on $\mathbb{P}^1(\mathbb{C})$. As already seen
in \S\ref{sec:balanced},
there is also the equilibrium (or canonical) measure $\nu_f$ of
$f\in\mathbb{L}(z)$ of degree $d>1$ on $\mathsf{P}^1(\mathbb{L})$.
If in addition $\nu_f(\{\mathcal{S}_G\})=0$ or equivalently
$f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$
(in \eqref{eq:nopotgood}; see also another equivalent
condition \eqref{eq:goodred} below),
then recalling that $\Gamma_G:=\{\mathcal{S}_G\}\subset\mathsf{P}^1(\mathbb{L})$
as in \S\ref{sec:skeltal} and noting that
\begin{gather*}
S(\Gamma_G)\setminus\bigl\{\{\mathcal{S}_G\}\bigr\}=T_{\mathcal{S}_G}\bigl(\mathsf{P}^1(\mathbb{L})\bigr)\cong
\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C}),
\end{gather*}
where $k_{\mathbb{L}}(=\mathbb{C}$ as fields) is the residue field of $\mathbb{L}$ and
the bijection
between the tangent (or directions) space $T_{\mathcal{S}_G}(\mathsf{P}^1(\mathbb{L}))$ of $\mathsf{P}^1(\mathbb{L})$ at $\mathcal{S}_G$ and $\mathbb{P}^1(k_{\mathbb{L}})$ is given by
$\overrightarrow{\mathcal{S}_G a}\leftrightarrow\tilde{a}$ for each $a\in\mathbb{P}^1(\mathbb{L})$
(see \S\ref{th:constantreduction} for the reduction
$\tilde{a}\in\mathbb{P}^1(k_{\mathbb{L}})$ of $a$),
the projection/quantization
\begin{gather*}
\bigl(\pi_{\mathsf{P}^1(\mathbb{L}),\Gamma_G}\bigr)_*\nu_f
\end{gather*}
of $\nu_f$ is in $M^1(\Gamma_G)^{\dag}$ and
is also regarded as a purely atomic probability measure on $\mathbb{P}^1(\mathbb{C})$.
Using Theorem \ref{th:computation} (and by some new argument, see Remark \ref{th:remedy} below), we could complement
the proof of the following degenerating
limit theorem of DeMarco--Faber. Our proof is given in Section \ref{sec:proof}.
\begin{theorem}[{\cite[Theorem B]{DF14}}]\label{th:B}
For every meromorphic family $f\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$ of rational functions
on $\mathbb{P}^1(\mathbb{C})$ of degree $>1$, if $f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$,
then
\begin{gather}
\lim_{t\to 0}\mu_{f_t}=\bigl(\pi_{\mathsf{P}^1(\mathbb{L}),\Gamma_G}\bigr)_*\nu_f\quad\text{weakly on }\mathbb{P}^1(\mathbb{C}).\label{eq:convergence}
\end{gather}
\end{theorem}
We could also eliminate the intermediate
``bimeromorphically modified surface dynamics'' part
from the (conceptual) ``transfer principle'' from degenerating
complex dynamics to quantized Berkovich dynamics
in \cite[The proof of Theorem B]{DF14}. We hope our direct and more explicit translation from degenerating complex dynamics into quantized Berkovich dynamics
(given in Section \ref{sec:direct})
could also be helpful for a further investigation
of degenerating complex dynamics (see e.g.\ \cite{Favre16,DF18}).
\section{Background from Berkovich dynamics}
\label{sec:background}
Let $K$ be an algebraically closed field that is complete with respect to
a non-trivial and non-archimedean absolute value $|\cdot|$.
\subsection{Berkovich projective line}\label{sec:berkovichgeneral}
We call $B(a,r):=\{z\in K:|z-a|\le r\}$ for some $a\in K$ and
some $r\in\mathbb{R}_{\ge 0}$
a $K$-closed disk in $K$.
The Berkovich projective line $\mathsf{P}^1=\mathsf{P}^1(K)$ over $K$ is
a compact, uniquely arcwise connected, and Hausdorff topological space;
as sets,
\begin{gather*}
\mathsf{P}^1=\mathbb{P}^1\cup\mathsf{H}^1=\mathbb{P}^1\cup\mathsf{H}^1_{\mathrm{II}}\cup\mathsf{H}^1_{\mathrm{III}}\cup\mathsf{H}^1_{\mathrm{IV}}\quad (\text{the disjoint unions}),\\
\mathbb{P}^1=\mathbb{P}^1(K)=K\cup\{\infty\}\cong\mathsf{H}^1_{\mathrm{I}}\cong\{\{a\}=B(a,0):a\in\mathbb{P}^1\},\\
\mathsf{H}^1_{\mathrm{II}}\cong\bigl\{B(a,r):a\in K,r\in|K^*|\bigr\},\quad\text{and}\quad
\mathsf{H}^1_{\mathrm{III}}\cong\bigl\{B(a,r):a\in K,r\in\mathbb{R}_{>0}\setminus|K^*|\bigr\}.
\end{gather*}
More precisely, each element of $\mathsf{P}^1$ is regarded as
either the cofinal equivalence
class of a decreasing (i.e., non-increasing and nesting)
sequence of $K$-closed disks in $K$ or $\infty\in\mathbb{P}^1$.
The inclusion relation $\subset$ among $K$-closed disks in $K$
canonically
extends to an ordering $\preceq$ on $\mathsf{P}^1$, so that $\infty$ is the unique
maximal element in $(\mathsf{P}^1,\preceq)$, and the diameter function
$\operatorname{diam}_{|\cdot|}$ for $K$-closed disks in $K$ also extends continuously
to $\mathsf{P}^1$, so that $\operatorname{diam}_{|\cdot|}(\infty)=+\infty$.
For $\mathcal{S}_1,\mathcal{S}_2\in\mathsf{P}^1$, if $\mathcal{S}_1\preceq\mathcal{S}_2$, then set
$[\mathcal{S}_1,\mathcal{S}_2]=[\mathcal{S}_2,\mathcal{S}_1]:=\{\mathcal{S}\in\mathsf{P}^1:\mathcal{S}_1\preceq\mathcal{S}\preceq\mathcal{S}_2\}$, and
in general there is the unique minimal element $\mathcal{S}'$ in $\{\mathcal{S}\in\mathsf{P}^1:\mathcal{S}_1,\mathcal{S}_2\preceq\mathcal{S}\}$ and set
\begin{gather*}
[\mathcal{S}_1,\mathcal{S}_2]=[\mathcal{S}_2,\mathcal{S}_1]:=[\mathcal{S}_1,\mathcal{S}']\cup[\mathcal{S}',\mathcal{S}_2].
\end{gather*}
Those (closed) intervals $[\mathcal{S},\mathcal{S}']$
in $\mathsf{P}^1$ equip $\mathsf{P}^1$ with a (profinite) tree structure
in the sense of Jonsson \cite[\S 2]{Jonsson15}.
For every $\mathcal{S}\in\mathsf{P}^1$, the tangent (or directions)
space $T_{\mathcal{S}}\mathsf{P}^1$ of $\mathsf{P}^1$ at $\mathcal{S}$ is
\begin{gather*}
T_{\mathcal{S}}\mathsf{P}^1:=\bigl\{\overrightarrow{v}=\overrightarrow{\mathcal{S}\cS'}:\text{the germ of a non-empty left half open interval }(\mathcal{S},\mathcal{S}']:=[\mathcal{S},\mathcal{S}']\setminus\{\mathcal{S}\}\bigr\}
\end{gather*}
and, as a subset in $\mathsf{P}^1\setminus\{\mathcal{S}\}$,
each element $\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1$ is also denoted by
$U_{\overrightarrow{v}}=U_{\mathcal{S},\overrightarrow{v}}$; then
$\#T_{\mathcal{S}}\mathsf{P}^1=1$ iff $\mathcal{S}\in\mathbb{P}^1\cup\mathsf{H}^1_{\mathrm{IV}}$,
$\#T_{\mathcal{S}}\mathsf{P}^1=2$ iff $\mathcal{S}\in\mathsf{H}^1_{\mathrm{III}}$, and
$T_{\mathcal{S}}\mathsf{P}^1\cong\mathbb{P}^1(k)$
iff $\mathcal{S}\in\mathsf{H}^1_{\mathrm{II}}$ (see \eqref{eq:tangentreduct} below
for more details).
The collection $(U_{\mathcal{S},\overrightarrow{v}})_{\mathcal{S}\in\mathsf{P}^1,\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1}$ is a quasi open basis of
the (Gel'fand, weak, pointwise, or observer) topology on $\mathsf{P}^1$.
In particular, both $\mathbb{P}^1$ and $\mathsf{H}^1_{\mathrm{II}}$ are dense in $\mathsf{P}^1$.
The Gauss (or canonical) point $\mathcal{S}_G\in\mathsf{H}^1_{\mathrm{II}}$ corresponds to
the unit $K$-closed disk $B(0,1)$, that is,
the ring $\mathcal{O}_K=\{z\in K:|z|\le 1\}$ of $K$-integers;
$\mathcal{M}_K=\{z\in K:|z|<1\}$ is the unique maximal ideal in $\mathcal{O}_K$, and
\begin{gather*}
k=k_K:=\mathcal{O}_K/\mathcal{M}_K
\end{gather*}
is the residue field of $K$.
The reduction $\tilde{a}\in\mathbb{P}^1(k)$ of a point $a\in\mathbb{P}^1(K)$
is defined by the point $\tilde{a_1}/\tilde{a_0}\in\mathbb{P}^1(k)$, where
$a_1,a_0\in K$ are chosen so that
$a=a_1/a_0$ (regarding $1/0=\infty$) and that $\max\{|a_0|,|a_1|\}=1$.
We have the canonical bijection
\begin{gather}
T_{\mathcal{S}_G}\mathsf{P}^1\ni\overrightarrow{\mathcal{S}_Ga}\leftrightarrow\tilde{a}\in\mathbb{P}^1(k)=k\cup\{\tilde{\infty}\}.
\label{directionreduction}
\end{gather}
For more details on (dynamics on) $\mathsf{P}^1$,
see e.g.\ the books \cite{BR10,BenedettoBook} and the survey article \cite{Jonsson15}.
\subsection{Dynamics on $\mathsf{P}^1$ and their reduction}
For every $h\in K(z)$, writing
\begin{gather*}
h(z)=\frac{P(z)}{Q(z)},\quad
P(z)=\sum_{j=0}^{\deg h}a_j z^j\in K[z],\quad\text{and}\quad
Q(z)=\sum_{\ell=0}^{\deg h}b_\ell z^\ell\in K[z],
\end{gather*}
this $h$ is regarded as the point
$[b_0:\cdots:b_{\deg h}:a_0:\cdots:a_{\deg h}]\in\mathbb{P}^{2(\deg h)+1}(K)$.
Then choosing $P,Q$
so that $\max\{|b_0|,\ldots,|b_{\deg h}|,|a_0|,\ldots,|a_{\deg h}|\}=1$,
we also obtain the point
$\tilde{h}=\bigl[\widetilde{b_0}:\cdots:\widetilde{b_{\deg h}}:\widetilde{a_0}:\cdots:\widetilde{a_{\deg h}}\bigr]
\in\mathbb{P}^{2(\deg h)+1}(k)$; this $\tilde{h}$ is also formally written as
\begin{gather*}
\tilde{h}=H_{\tilde{h}}\phi_{\tilde{h}},
\end{gather*}
where
\begin{gather*}
\tilde{P}(\zeta):=\sum_{j=0}^{\deg h}\widetilde{a_j}\zeta^j\in k[\zeta],
\quad
\tilde{Q}(z):=\sum_{\ell=0}^{\deg h}\widetilde{b_\ell}\zeta^\ell\in k[\zeta],\\
H_{\tilde h}(X_0,X_1):=\mathrm{GCD}\bigl(X_0^{\deg h}\tilde{Q}(X_1/X_0),X_0^{\deg h}\tilde{P}(X_1/X_0)\bigr)\in\bigcup_{\ell=0}^{\deg h}k[X_0,X_1]_\ell\setminus\{0\},\quad\text{and}\\
\phi_{\tilde{h}}(\zeta)
:=\frac{\tilde{P}(\zeta)/H_{\tilde{h}}(1,\zeta)}{\tilde{Q}(\zeta)/H_{\tilde{h}}(1,\zeta)}\in k(\zeta)
\end{gather*}
($H_{\tilde{h}}$ is unique up to multiplication in $k^*$).
The rational function $\phi_{\tilde{h}}\in k(\zeta)$ on $\mathbb{P}^1(k)$
is called the reduction of $h$, the degree of which equals
$\deg h-\deg H_{\tilde{h}}$.
\begin{notation}
When $\deg H_{\tilde{h}}>0$, we denote by
$[H_{\tilde{h}}=0]$ the zeros divisor on $\mathbb{P}^1(\overline{k})$
defined by the zeros of $H_{\tilde{h}}$ on $\mathbb{P}^1(\overline{k})$
taking into account their multiplicities. When $\deg H_{\tilde{h}}=0$, we set
$[H_{\tilde{h}}=0]:=0$ on $\mathbb{P}^1(\overline{k})$ by convention.
\end{notation}
The action on $\mathbb{P}^1$ of $h\in K(z)$ extends continuously to
that on $\mathsf{P}^1$, and if in addition $\deg h>0$, then
this extended action is surjective, open, and
fiber-discrete, and preserves $\mathbb{P}^1,\mathsf{H}^1_{\mathrm{II}},\mathsf{H}^1_{\mathrm{III}}$,
and $\mathsf{H}^1_{\mathrm{IV}}$, as already mentioned in Section \ref{sec:intro}.
The group $\mathrm{PGL}(2,K)$ of M\"obius transformations on $\mathbb{P}^1$
acts transitively on $\mathsf{H}^1$, and
$\mathrm{PGL}(2,\mathcal{O}_K)$ is the stabilizer subgroup of $\mathcal{S}_G$ in $\mathrm{PGL}(2,K)$.
\begin{fact}[Rivera-Letelier {\cite{Juan03}, see also \cite[Corollary 9.27]{BR10}}]\label{th:constantreduction}
$\deg(\phi_{\tilde{h}})>0$ if and only if $h(\mathcal{S}_G)=\mathcal{S}_G$. Moreover,
\begin{gather}
\phi_{\tilde{h}}\equiv\tilde{z}\text{ for some }z\in\mathbb{P}^1
\quad\Rightarrow\quad
\overrightarrow{\mathcal{S}_Gh(\mathcal{S}_G)}=\overrightarrow{\mathcal{S}_Gz}.
\label{eq:constantdirection}
\end{gather}
\end{fact}
From now on, suppose that $\deg h>0$.
\subsection{The directional and surplus local degrees of rational functions}
For the details, see Rivera-Letelier \cite{Juan05,Juan03}.
For every $\mathcal{S}\in\mathsf{P}^1$,
the tangent map $h_*=(h_*)_{\mathcal{S}}:T_{\mathcal{S}}\mathsf{P}^1\to T_{h(\mathcal{S})}\mathsf{P}^1$
of $h$ at $\mathcal{S}$ is defined so that
for every $\overrightarrow{v}=\overrightarrow{\mathcal{S}\cS'}\in T_{\mathcal{S}}\mathsf{P}^1$, if $\mathcal{S}'$ is close enough to $\mathcal{S}$, then
$h$ maps the interval $[\mathcal{S},\mathcal{S}']$ onto the interval $[h(\mathcal{S}),h(\mathcal{S}')]$ homeomorphically and
\begin{gather*}
h_*(\overrightarrow{v})=\overrightarrow{h(\mathcal{S})h(\mathcal{S}')}.
\end{gather*}
For every $\mathcal{S}\in\mathsf{H}^1_{\mathrm{II}}$,
choose $A,B\in\mathrm{PGL}(2,K)$ so that $B^{-1}(\mathcal{S})=A(h(\mathcal{S}))=\mathcal{S}_G$
and in particular that for every $\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1$,
writing $(B^{-1})_*(\overrightarrow{v})=\overrightarrow{\mathcal{S}_Gz}$ and
$A_*(h_*(\overrightarrow{v}))=\overrightarrow{\mathcal{S}_Gw}$
by some $z,w\in\mathbb{P}^1$, we have
\begin{gather}
\phi_{\widetilde{A\circ h\circ B}}(\tilde{z})=\tilde{w},\label{eq:tangentreduct}
\end{gather}
and then the directional local degree $m_{\overrightarrow{v}}(h)$ of $h$
on $U_{\overrightarrow{v}}$
is defined so that
\begin{gather}
m_{\overrightarrow{v}}(h)
:=\deg_{\tilde{z}}\bigl(\phi_{\widetilde{A\circ h\circ B}}\bigr).\label{directdegdef}
\end{gather}
For every $\mathcal{S}\in\mathbb{P}^1\cup\mathsf{H}^1_{\mathrm{III}}\cup\mathsf{H}^1_{\mathrm{IV}}$
and every $\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1$, we set $m_{\overrightarrow{v}}(h):=\deg_{\mathcal{S}}(h)$.
\begin{fact}[decomposition of the local degree]\label{th:directionalpullback}
For every $\mathcal{S}\in\mathsf{P}^1$, also using the notation in the previous paragraph,
we have
\begin{gather}
\deg\bigl(\phi_{\widetilde{A\circ h\circ B}}\bigr)
=\sum_{\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1:h_*(\overrightarrow{v})=\overrightarrow{w}}m_{\overrightarrow{v}}(h)=\deg_{\mathcal{S}}(h)
\quad\text{for any }\overrightarrow{w}\in T_{h(\mathcal{S})}\mathsf{P}^1;\label{eq:totallocaldegree}
\end{gather}
in particular, $h_*:T_{\mathcal{S}}\mathsf{P}^1\to T_{h(\mathcal{S})}\mathsf{P}^1$ is surjective,
and if in addition $h\in\mathrm{PGL}(2,K)$, then
$h_*:T_{\mathcal{S}}\mathsf{P}^1\to T_{h(\mathcal{S})}\mathsf{P}^1$ is bijective.
\end{fact}
\begin{fact}[a non-archimedean argument principle]\label{th:surplus}
For every $\mathcal{S}\in\mathsf{P}^1$ and every $\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1$,
there is the surplus local degree
$s_{\overrightarrow{v}}(h)\in\mathbb{N}\cup\{0\}$
of $h$ on $U_{\overrightarrow{v}}$ such that
for every $\mathcal{S}'\in\mathsf{P}^1\setminus\{h(\mathcal{S})\}$,
\begin{gather}
(h^*\delta_{\mathcal{S}'})(U_{\overrightarrow{v}})=
\begin{cases}
m_{\overrightarrow{v}}(h)+s_{\overrightarrow{v}}(h)
&\text{if }U_{h_*(\overrightarrow{v})}\ni \mathcal{S}',\\
s_{\overrightarrow{v}}(h) & \text{otherwise};
\end{cases}\label{eq:argumentnonarchi}
\end{gather}
moreover, $h(U_{\overrightarrow{v}})$ is either $\mathsf{P}^1$
or $U_{h_*(\overrightarrow{v})}$, the latter case in which
is the case if and only if $s_{\overrightarrow{v}}(h)=0$.
Fixing $\mathcal{S}'\in\mathsf{P}^1\setminus\{h(\mathcal{S})\}$, we have
$\deg h=(h^*\delta_{\mathcal{S}'})(\mathsf{P}^1)=(h^*\delta_{\mathcal{S}'})(\mathsf{P}^1\setminus\{\mathcal{S}\})
=\deg_{\mathcal{S}}(h)+
\sum_{\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1}s_{\overrightarrow{v}}(h)$
(the final equality in which is by \eqref{eq:totallocaldegree}),
so that
\begin{gather}
\sum_{\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1}s_{\overrightarrow{v}}(h)=
\deg h-\deg_{\mathcal{S}}(h).\label{eq:totalsurplus}
\end{gather}
In particular,
if in addition $h\in\mathrm{PGL}(2,K)$, then
for every $\mathcal{S}\in\mathsf{P}^1$ and every $\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1$,
$h(U_{\overrightarrow{v}})=U_{h_*(\overrightarrow{v})}$.
\end{fact}
\begin{fact}[Faber {\cite[Lemma 3.17]{Faber13topologyI}}]\label{th:surplusalgclosed}
If in addition $k$ is algebraically closed (i.e., $\overline{k}=k$), then
for every $\mathcal{S}\in\mathsf{H}^1_{\mathrm{II}}$ and
every $\overrightarrow{v}\in T_{\mathcal{S}}\mathsf{P}^1$,
in the notations in \eqref{eq:tangentreduct}, we have
\begin{gather}
s_{\overrightarrow{v}}(h)
\begin{cases}
=\operatorname{ord}_{\tilde{z}}\bigl[H_{\widetilde{A\circ h\circ B}}=0\bigr] &\text{if }\deg H_{\widetilde{A\circ h\circ B}}>0,\\
\equiv 0 & \text{otherwise}.
\end{cases}\label{eq:surplusfaber}
\end{gather}
\end{fact}
\subsection{The hyperbolic metric $\rho$ on $\mathsf{H}^1$ and the piecewise affine action of $h$ on $(\mathsf{H}^1,\rho)$}
The hyperbolic metric $\rho$ on $\mathsf{H}^1$,
which is defined so that
\begin{gather*}
\rho\bigl(\mathcal{S}_1,\mathcal{S}_2\bigr)=\log\Bigl(\frac{\operatorname{diam}_{|\cdot|}\mathcal{S}_2}{\operatorname{diam}_{|\cdot|}\mathcal{S}_1}\Bigr)\quad\text{ if }\mathcal{S}_1\preceq\mathcal{S}_2,
\end{gather*}
and would be needed at some part in the proof of Theorem \ref{th:computation}.
The topology on $(\mathsf{H}^1,\rho)$ is finer than the relative topology on $\mathsf{H}^1$
from $\mathsf{P}^1$.
\begin{fact}
For every $\overrightarrow{\mathcal{S}\cS'}\in T_{\mathcal{S}}\mathsf{P}^1$,
if $\mathcal{S}'$ is close enough to $\mathcal{S}$, then
\begin{gather}
\rho\bigl(h(\mathcal{S}),h(\mathcal{S}')\bigr)
=m_{\overrightarrow{\mathcal{S}\cS'}}(h)\cdot\rho(\mathcal{S},\mathcal{S}').\label{eq:Lipschitz}
\end{gather}
\end{fact}
\subsection{Quantized local degrees and quantized pullbacks}\label{sec:quantized}
Let us precisely define the quantized local degrees $m_{V,U}(h)$ in \S\ref{sec:skeltal} in terms of the (directional/surplus) local degrees of $h$,
and then also re-define the quantized pullback $(h_{\Gamma_G,\Gamma_h})^*:M(\Gamma_h)\to M(\Gamma_G)$. Recall
\begin{gather*}
\Gamma_G:=\{\mathcal{S}_G\}\quad\text{and}\quad \Gamma_h:=\{\mathcal{S}_G,h(\mathcal{S}_G)\}
\quad\text{in }\mathsf{H}^1_{\mathrm{II}}.
\end{gather*}
\begin{definition}[the quantized local degree]\label{th:multfactor}
For every $U_{\overrightarrow{v}}\in S(\Gamma_G)\setminus\{\{\mathcal{S}\}:\mathcal{S}\in\Gamma_G\}\cong T_{\mathcal{S}_G}\mathsf{P}^1$
(the bijection is the canonical one in \eqref{directionreduction})
and every $V\in S(\Gamma_h)$, set
\begin{align*}
m_{V,U_{\overrightarrow{v}}}(h):=&
\begin{cases}
m_{\overrightarrow{v}}(h)+s_{\overrightarrow{v}}(h) &\text{if }V\subset U_{h_*(\overrightarrow{v})},\\
s_{\overrightarrow{v}}(h) &\text{if }V\cap U_{h_*(\overrightarrow{v})}=\emptyset
\end{cases}\\
\underset{\eqref{eq:argumentnonarchi}}{=}&(h^*\delta_{\mathcal{S}'})(U_{\overrightarrow{v}})\text{ for any }\mathcal{S}'\in V=V\setminus\{\{h(\mathcal{S}_G)\}\}\quad\text{if }V\in S(\Gamma_h)\setminus\{\{h(\mathcal{S}_G)\}\},
\end{align*}
and for every $V\in S(\Gamma_h)$, set
\begin{gather*}
m_{V,\{\mathcal{S}_G\}}(h)
:=\begin{cases}
\deg_{\mathcal{S}_G}(h) & \text{if }V=\{h(\mathcal{S}_G)\},\\
0 & \text{if }V\in S(\Gamma_h)\setminus\{h(\mathcal{S}_G)\}
\end{cases}
\overset{\eqref{eq:pullbackdirac}}{=}(h^*\delta_{\mathcal{S}'})(\{\mathcal{S}_G\})\quad\text{for any }\mathcal{S}'\in V.
\end{gather*}
\end{definition}
\begin{fact}
The fundamental equality
\begin{gather}
\sum_{U\in S(\Gamma_G)}m_{V,U}(h)=\deg h
\quad\text{for any }V\in S(\Gamma_h)
\label{eq:totalfactor}
\end{gather}
holds; indeed, for every $V\in S(\Gamma_h)\setminus\{\{h(\mathcal{S}_G)\}\}$,
there is a unique $\overrightarrow{w}\in T_{h(\mathcal{S}_G)}\mathsf{P}^1$
such that $V\subset U_{\overrightarrow{w}}$, and then
\begin{multline*}
\sum_{U\in S(\Gamma_G)}m_{V,U}(h)
=\sum_{\overrightarrow{v}\in T_{\mathcal{S}_G}\mathsf{P}^1:h_*(\overrightarrow{v})=\overrightarrow{w}}m_{\overrightarrow{v}}(h)
+\sum_{\overrightarrow{v}\in T_{\mathcal{S}_G}\mathsf{P}^1}s_{\overrightarrow{v}}(h)\\
\underset{\eqref{eq:totallocaldegree}\&\eqref{eq:totalsurplus}}{=}\deg_{\mathcal{S}_G}(h)+\bigl(\deg h-\deg_{\mathcal{S}_G}(h)\bigr)
=\deg h,
\end{multline*}
and similarly,
\begin{gather*}
\sum_{U\in S(\Gamma_G)}m_{\{h(\mathcal{S}_G)\},U}(h)
=\sum_{\overrightarrow{v}\in T_{\mathcal{S}_G}\mathsf{P}^1}s_{\overrightarrow{v}}(h)+\deg_{\mathcal{S}_G}(h)
\underset{\eqref{eq:totalsurplus}}{=}\bigl(\deg h-\deg_{\mathcal{S}_G}(h)\bigr)+\deg_{\mathcal{S}_G}(h)=\deg h.
\end{gather*}
\end{fact}
The quantized pushforward $(h_{\Gamma_G,\Gamma_h})_*$ from the space of
measurable functions on $(\mathsf{P}^1,\sigma(S(\Gamma_G)))$ to
that of measurable functions on $(\mathsf{P}^1,\sigma(S(\Gamma_h)))$
is defined so that for every measurable function
$\psi$ on $(\mathsf{P}^1,\sigma(S(\Gamma_G)))$,
\begin{align}
\notag (h_{\Gamma_G,\Gamma_h})_*\psi
\equiv&\sum_{U\in S(\Gamma_G)}m_{V,U}(h)\cdot\psi|U
\quad\text{on each }V\in S(\Gamma_h)\\
\equiv&\sum_{\overrightarrow{v}\in T_{\mathcal{S}_G}\mathsf{P}^1}(h^*\delta_{\,\cdot})(U_{\overrightarrow{v}})\cdot\psi|U_{\overrightarrow{v}}
=h_*(\pi_{\mathsf{P}^1,\Gamma_G})^*\psi
\quad\text{on each }V\in S(\Gamma_h)\setminus\{h(\mathcal{S}_G)\}.\label{eq:pushfactor}
\end{align}
The quantized pullback $(h_{\Gamma_G,\Gamma_h})^*:M(\Gamma_h)\to M(\Gamma_G)$
is induced by
the transpose of this pushforward $(h_{\Gamma_G,\Gamma_h})_*$ so that
for every $\omega\in M(\Gamma_h)$,
\begin{multline}
\bigl((h_{\Gamma_G,\Gamma_h})^*\omega\bigr)(U)
=\langle 1_U,(h_{\Gamma_G,\Gamma_h})^*\omega\rangle
=\langle (h_{\Gamma_G,\Gamma_h})_*1_U,\omega\rangle\\
=\sum_{V\in S(\Gamma_f)}\Bigl(\sum_{W\in S(\Gamma_G)}m_{V,W}(h)\cdot 1_{U,W}\Bigr)\omega(V)
=\sum_{V\in S(\Gamma_f)}m_{V,U}(h)\omega(V)
\quad\text{for any }U\in S(\Gamma_G).\label{eq:pullbackdefining}
\end{multline}
\section{Degenerating balanced property for degenerating limit points of the maximal entropy measures}
\label{sec:quantizedbalanced}
We follow the presentation in \cite[\S 2.1-\S2.4]{DF14}.
The field $\mathbb{C}((t))$ of Laurent series around $t=0$ over $\mathbb{C}$
is equipped with the non-trivial and non-archimedean absolute value
\begin{gather}
|x|_r=r^{\min\{n:a_n\neq 0\}}\label{eq:normLaurent}
\end{gather}
for $x(t)=\sum_{n\in\mathbb{Z}}a_nt^n$,
which extends the trivial norm on $\mathbb{C}$ to $\mathbb{C}((t))$,
fixing $r\in(0,1)$ (e.g.\ $r=e^{-1}$) once and for all.
An algebraic closure $\overline{\mathbb{C}((t))}$ of $\mathbb{C}((t))$
is the field of Puiseux series around $t=0$ over $\mathbb{C}$, and
the completion $\mathbb{L}$ of $\overline{\mathbb{C}((t))}$
is the field of formal Puiseux series around $t=0$ over $\mathbb{C}$ and
is still algebraically closed. We note that $\mathcal{O}(\mathbb{D})[t^{-1}]\subset\mathbb{C}((t))$,
\begin{gather*}
\mathbb{C}\subset\mathcal{O}(\mathbb{D})\subset\mathcal{O}_{\mathbb{C}((t))}
=\biggl\{\sum_{n\in\mathbb{Z}}a_nt^n\in\mathbb{C}((t)):a_n=0\text{ if }n<0\biggr\},\\
\mathcal{O}_{\mathbb{C}((t))}=\mathbb{C}[[t]],\quad
\mathcal{M}_{\mathbb{C}((t))}=t\cdot\mathcal{O}_{\mathbb{C}((t))},\quad
k_{\mathbb{L}}=k_{\mathbb{C}((t))}=\mathbb{C}\text{ (as fields),}\quad\text{and}\\
T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})\cong\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C})\quad
(\text{the bijection
is the canonical one in \eqref{directionreduction}}).
\end{gather*}
\begin{notation}
Set $M(\mathbb{P}^1(\mathbb{C})):=\{\text{the space of all complex Radon measures on }\mathbb{P}^1(\mathbb{C})\}$.
The pullback $R^*\mu$ of each $\mu\in M(\mathbb{P}^1(\mathbb{C}))$ under a non-constant
rational function $R\in\mathbb{C}(z)$ on $\mathbb{P}^1(\mathbb{C})$ is defined by
$R^*\mu:=\int_{\mathbb{P}^1(\mathbb{C})}(\sum_{w\in R^{-1}(w)}(\deg_zR)\delta_w)\mu(z)$
on $\mathbb{P}^1(\mathbb{C})$ where for each $z\in\mathbb{P}^1(\mathbb{C})$, $\delta_z$ is the Dirac measure at $z$ on $\mathbb{P}^1(\mathbb{C})$;
when $R$ is constant, set $R^*\mu:=0$ by convention. Also set
\begin{align*}
M^1(\mathbb{P}^1(\mathbb{C})):=&\{\mu\in M(\mathbb{P}^1(\mathbb{C})):\mu\ge 0\text{ and }\mu(\mathbb{P}^1(\mathbb{C}))=1\}\quad\text{and}\\
M^1(\mathbb{P}^1(\mathbb{C}))^{\dag}:=&\{\mu\in M^1(\mathbb{P}^1(\mathbb{C})):\mu\text{ is the sum of (at most countable) atoms on }\mathbb{P}^1(\mathbb{C})\}.
\end{align*}
\end{notation}
For any meromorphic family $h\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)\subset\mathbb{L}(z)$ of rational functions on $\mathbb{P}^1(\mathbb{C})$, let us regard
$\tilde{h}=H_{\tilde{h}}\phi_{\tilde{h}}\in\mathbb{P}^{2(\deg h)+1}(k_{\mathbb{L}})$
as a point in $\mathbb{P}^{2(\deg h)+1}(\mathbb{C})$ under $k_{\mathbb{L}}=\mathbb{C}$. Then
\begin{gather}
h^{-1}(\mathcal{S}_G)=\{\mathcal{S}_G\}\quad\Leftrightarrow\quad\tilde{h}=\phi_{\tilde{h}}\quad\Leftrightarrow\quad\deg H_{\tilde{h}}=0.\label{eq:goodred}
\end{gather}
The following target rescaling theorem is {\cite[Lemma 2.1]{DF14}}.
\begin{theorem}
\label{th:postfamily}
For every meromorphic family $f\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$ of rational functions
on $\mathbb{P}^1(\mathbb{C})$ of degree $>1$, scaling $\mathbb{D}$ around $t=0$ if necessary,
there is a meromorphic family $A\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$
of M\"obius transformations on $\mathbb{P}^1(\mathbb{C})$
such that $(A\circ f)(\mathcal{S}_G)=\mathcal{S}_G(\in\mathsf{P}^1(\mathbb{L}))$.
Such $A$ is unique up to a postcomposition to $A$ of any meromorphic family
$B\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$ of M\"obius transformations on $\mathbb{P}^1(\mathbb{C})$
satisfying $\tilde{B}=\phi_{\tilde{B}}\in\mathrm{PGL}(2,\mathbb{C})$.
\end{theorem}
For any $\mu\in M^1(\mathbb{P}^1(\mathbb{C}))$,
the
pullback $\tilde{h}^*\mu\in M(\mathbb{P}^1(\mathbb{C}))$
of $\mu$ under $\tilde{h}$ is defined by
\begin{gather*}
\tilde{h}^*\mu
:=(\phi_{\tilde{h}})^*\mu+[H_{\tilde{h}}=0]\quad\text{on }\mathbb{P}^1(\mathbb{C}),
\end{gather*}
regarding the effective divisor $[H_{\tilde{h}}=0]$ on $\mathbb{P}^1(k_{\mathbb{L}})$
as that on $\mathbb{P}^1(\mathbb{C})$ under $k_{\mathbb{L}}=\mathbb{C}$,
so we still have $(\tilde{h}^*\mu)(\mathbb{P}^1(\mathbb{C}))=\deg h$.
The degenerating $f$-balanced property (the former half in
\eqref{eq:reductionconstant})
is a consequence of the fact that $\lim_{t\to 0}h_t=\phi_{\tilde{h}}$
locally uniformly on $\mathbb{P}^1(\mathbb{C})\setminus\operatorname{supp}[H_{\tilde{h}}=0]$
and the complex argument principle. The proof of the
purely atomic property of the $\mu$ (the latter in
\eqref{eq:reductionconstant}) is more involved.
\begin{theorem}[a consequence of {\cite[Theorems 2.4 and A]{DF14}}]\label{th:balancedgeneral}
Let $f\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$ be a meromorphic family
of rational functions on $\mathbb{P}^1(\mathbb{C})$ of degree $d>1$
satisfying $f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$, $A\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$ be
a meromorphic family of M\"obius transformations on $\mathbb{P}^1(\mathbb{C})$
such that $(A\circ f)(\mathcal{S}_G)=\mathcal{S}_G$, and
let $\mu_C=\lim_{j\to\infty}\mu_{f_{t_j}},\mu_E=\lim_{j\to\infty}(A_{t_j})_*\mu_{f_{t_j}}\in M^1(\mathbb{P}^1(\mathbb{C}))$ be weak limit points on $\mathbb{P}^1(\mathbb{C})$ as $t\to 0$ of the family
$(\mu_{f_t})_{t\in\mathbb{D}^*}$ of the unique maximal entropy measures $\mu_{f_t}$
on $\mathbb{P}^1(\mathbb{C})$ of $f_t$ and the rescaled family
$((A_t)_*\mu_{f_t})_{t\in\mathbb{D}^*}$ of $(\mu_{f_t})_{t\in\mathbb{D}^*}$
by $A=(A_t)_{t\in\mathbb{D}^*}$, respectively, for some sequence
$(t=t_j)$ in $\mathbb{D}^*$ tending to $0$ as $j\to\infty$. Then
\begin{gather}
(\widetilde{A\circ f})^*\mu_E=d\cdot\mu_C\quad\text{on }\mathbb{P}^1(\mathbb{C})
\quad\text{and}\quad
\mu:=(\mu_C,\mu_E)\in(M^1(\mathbb{P}^1(\mathbb{C}))^\dag)^2.
\label{eq:reductionconstant}
\end{gather}
\end{theorem}
\section{A direct translation}
\label{sec:direct}
Pick a meromorphic family $f\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$
of rational functions on $\mathbb{P}^1(\mathbb{C})$ of degree $d>1$, and
suppose that $f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$ in $\mathsf{P}^1(\mathbb{L})$.
Choose a meromorphic family $A\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$ of
M\"obius transformations on $\mathbb{P}^1(\mathbb{C})$
such that $(A\circ f)(\mathcal{S}_G)=\mathcal{S}_G\in\mathsf{P}^1(\mathbb{L})$, and write
$\tilde{A}=H_{\tilde{A}}\phi_{\tilde{A}}\in\mathbb{P}^3(k_{\mathbb{L}})=\mathbb{P}^3(\mathbb{C})$
(under $k_{\mathbb{L}}=\mathbb{C}$ here and below). Recall also
\begin{gather*}
\Gamma_G:=\{\mathcal{S}_G\}\quad\text{and}\quad\Gamma_f:=\{\mathcal{S}_G,f(\mathcal{S}_G)\}
\quad\text{in }\mathsf{H}^1_{\mathrm{II}}(\mathbb{L}).
\end{gather*}
From Fact \ref{th:constantreduction}, the following
\begin{gather*}
\Gamma_G=\Gamma_f,\quad
f(\mathcal{S}_G)=\mathcal{S}_G,\quad
\deg\phi_{\tilde{f}}>0,\quad\text{and}
\quad A(\mathcal{S}_G)=\{\mathcal{S}_G\}
\end{gather*}
are equivalent, and then
$\tilde{A}=\phi_{\tilde{A}}\in\mathrm{PGL}(2,k_{\mathbb{L}})=\mathrm{PGL}(2,\mathbb{C})$;
when $\Gamma_G\neq\Gamma_f$,
there are $h_A,a_A\in\mathbb{P}^1(\mathbb{C})$ such that
\begin{gather}
\operatorname{supp}[H_{\tilde{A}}=0]=\{h_A\}\quad\text{in }\mathbb{P}^1(\mathbb{C}),\quad
\phi_{\tilde{A}}\equiv a_A\quad\text{on }\mathbb{P}^1(\mathbb{C}),\quad\text{and}\quad
\phi_{\tilde{f}}\equiv h_A\quad\text{on }\mathbb{P}^1(\mathbb{C}).\label{eq:redconst}
\end{gather}
\begin{lemma}
When $\Gamma_f\neq\Gamma_G$, we have
\begin{gather}
(A^{-1})_*\bigl(\overrightarrow{\mathcal{S}_GA(\mathcal{S}_G)}\bigr)
=\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G}.\label{eq:consistent}
\end{gather}
\end{lemma}
\begin{proof}
If $(A^{-1})_*(\overrightarrow{v})=\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G}
=\overrightarrow{A^{-1}(\mathcal{S}_G)\mathcal{S}_G}$ for some $\overrightarrow{v}\in T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})$, then we have
$\mathcal{S}_G\in U_{(A^{-1})_*(\overrightarrow{v})}$, which yields
$A(\mathcal{S}_G)\in A(U_{(A^{-1})_*(\overrightarrow{v})})
=U_{A_*(A^{-1})_*(\overrightarrow{v})}=U_{\overrightarrow{v}}$,
so $\overrightarrow{v}=\overrightarrow{\mathcal{S}_GA(\mathcal{S}_G)}$.
\end{proof}
There are the bijections
\begin{gather*}
T_{f(\mathcal{S}_G)}\mathsf{P}^1(\mathbb{L})
\underset{(A^{-1})_*}{\overset{\cong}{\leftarrow}} T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})\cong\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C})
\end{gather*}
(the bijection $T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})\cong\mathbb{P}^1(k_{\mathbb{L}})$
is the canonical one in \eqref{directionreduction}. Recall also Fact \ref{th:directionalpullback}).
\begin{lemma}\label{th:annulus}
When $\Gamma_f\neq\Gamma_G$, for any $\tilde{x},\tilde{y}
\in\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C})$
$($and representatives $x,y\in\mathbb{P}^1(\mathbb{L})$ of $\tilde{x},\tilde{y}$,
respectively$)$, we have
\begin{gather}
\begin{cases}
\overrightarrow{\mathcal{S}_Gx}=\overrightarrow{\mathcal{S}_Gf(\mathcal{S}_G)}
\quad\text{in }T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})
&\Leftrightarrow\quad\tilde{x}=h_A\quad\text{in }\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C}),\\
(A^{-1})_*(\overrightarrow{\mathcal{S}_Gy})
=\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G}\quad\text{in }T_{f(\mathcal{S}_G)}\mathsf{P}^1(\mathbb{L})
&\Leftrightarrow\quad
\tilde{y}=a_A\quad\text{in }\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C}).
\end{cases}\label{eq:intersection}
\end{gather}
\end{lemma}
\begin{proof}
The former is by $\phi_{\tilde{f}}\equiv h_A$ on $\mathbb{P}^1(\mathbb{C})$
(in \eqref{eq:redconst}) and \eqref{eq:constantdirection}.
On the other hand, by \eqref{eq:consistent},
\begin{gather*}
(A^{-1})_*(\overrightarrow{\mathcal{S}_Gy})=\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G}
\quad\Leftrightarrow\quad
\overrightarrow{\mathcal{S}_Gy}(=A_*(\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G}))=\overrightarrow{\mathcal{S}_GA(\mathcal{S}_G)},
\end{gather*}
so the latter assertion holds by $\phi_{\tilde{A}}\equiv a_A$ on $\mathbb{P}^1(\mathbb{C})$
(in \eqref{eq:redconst}) and \eqref{eq:constantdirection}.
\end{proof}
For every $\mu=(\mu_C,\mu_E)\in (M^1(\mathbb{P}^1(\mathbb{C})))^2$
satisfying the following admissibility
\begin{gather}
\begin{cases}
\tilde{A}^*\mu_E=\mu_C\quad\text{on }\mathbb{P}^1(\mathbb{C})
&\text{when }\Gamma_f=\Gamma_G\\
\mu_C(\{h_A\})+\mu_E(\{a_A\})\ge 1
&\text{when }\Gamma_f\neq\Gamma_G,
\end{cases}
\label{eq:compatibility}
\end{gather}
writing $\mu_C=\tilde{\nu}_C+\nu_C$ and $\mu_E=\tilde{\nu}_E+\nu_E$,
where $\tilde{\nu}_C,\tilde{\nu}_E$ are the sums of
(at most countable) atoms on $\mathbb{P}^1(\mathbb{C})$ and
$\nu_C,\nu_E$ have no atoms on $\mathbb{P}^1(\mathbb{C})$,
we define $\omega_{\mu}\in M^1(\Gamma_f)$ such that
when $\Gamma_f=\Gamma_G$,
\begin{gather*}
\begin{cases}
\omega_{\mu}(\{\mathcal{S}_G\})=\nu_E(\mathbb{P}^1(\mathbb{C}))\bigl(=\nu_C(\mathbb{P}^1(\mathbb{C}))\bigr),\\
\omega_{\mu}\bigl(U_{(A^{-1})_*(\overrightarrow{\mathcal{S}_Gy})}\bigr)
=\mu_E(\{\tilde{y}\})\Bigl(\underset{\eqref{eq:tangentreduct}}{\Leftrightarrow}
\omega_{\mu}(U_{\overrightarrow{\mathcal{S}_Gy}})=\mu_C(\{\tilde{y}\})\Bigr)
& \text{for every }\tilde{y}\in\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C})
\end{cases}
\end{gather*}
and, when $\Gamma_f\neq\Gamma_G$,
noting also Lemma \ref{th:annulus},
\begin{gather}
\begin{cases}
\omega_{\mu}(\{\mathcal{S}_G\})=\nu_C(\mathbb{P}^1(\mathbb{C})),\\
\omega_{\mu}(U_{\overrightarrow{\mathcal{S}_Gx}})
=\mu_C(\{\tilde{x}\})\quad \text{for every }\tilde{x}\in\mathbb{P}^1(\mathbb{C})\setminus\{h_A\},\\
\omega_{\mu}(\{f(\mathcal{S}_G)\})=\nu_E(\mathbb{P}^1(\mathbb{C})),\\
\omega_{\mu}\bigl(U_{(A^{-1})_*(\overrightarrow{\mathcal{S}_Gy})}\bigr)
=\mu_E(\{\tilde{y}\})\quad \text{for every }\tilde{y}\in\mathbb{P}^1(\mathbb{C})\setminus\{a_A\},\\
\omega_{\mu}\bigl(U_{\overrightarrow{\mathcal{S}_Gf(\mathcal{S}_G)}}\cap U_{\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G}}\bigr)=\mu_C(\{h_A\})+\mu_E(\{a_A\})-1;
\end{cases}\label{eq:omega}
\end{gather}
then we have
\begin{gather}
(\pi_{\Gamma_f,\Gamma_G})_*\omega_\mu\in M^1(\Gamma_G)^\dag
\quad\Rightarrow\quad
\mu_C=(\pi_{\Gamma_f,\Gamma_G})_*\omega_\mu\quad\text{in }
M^1(\mathbb{P}^1(\mathbb{C}))^{\dag}=M^1(\Gamma_G)^{\dag},\label{eq:converse}
\end{gather}
identifying $M^1(\Gamma_G)^{\dag}$ with $M^1(\mathbb{P}^1(\mathbb{C}))^{\dag}$
under the bijection
$S(\Gamma_G)\setminus\{\mathcal{S}_G\}
=T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})\cong\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C})$.
The following is a direct translation from
degenerating complex dynamics into quantized Berkovich dynamics
based on the above explicit definition of $\omega_{\mu}$,
bypassing a correspondence between semistable models of
$\mathsf{P}^1(\mathbb{L})$ and semistable vertex sets in $\mathsf{P}^1(\mathbb{L})$ from
rigid analytic geometry (see, e.g, \cite{BPR13}),
which is used in \cite{DF14}.
See Section \ref{sec:complement} for a complement of this proposition.
\begin{proposition}[cf.\ {\cite[Proposition 5.1(1)]{DF14}}]\label{th:transfer}
For every $\mu=(\mu_C,\mu_E)\in(M^1(\mathbb{P}^1(\mathbb{C})))^2$
satisfying the admissibility \eqref{eq:compatibility}, we have
\begin{gather}
(\widetilde{A\circ f})^*\mu_E=d\cdot\mu_C\quad\text{on }\mathbb{P}^1(\mathbb{C})
\quad\Rightarrow\quad
(f_{\Gamma_G,\Gamma_f})^*\omega_{\mu}=d\cdot(\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu}\quad\text{in }M^1(\Gamma_G).\label{eq:tosurface}
\end{gather}
\end{proposition}
\begin{proof}
Pick $\mu=(\mu_C,\mu_E)\in(M^1(\mathbb{P}^1(\mathbb{C})))^2$
satisfying the admissibility \eqref{eq:compatibility}.
\subsection*{(a)} When $\Gamma_f\neq\Gamma_G$,
for every $\tilde{x}\in\mathbb{P}^1(\mathbb{C})=\mathbb{P}^1(k_{\mathbb{L}})\cong T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})=S(\Gamma_G)\setminus\{\mathcal{S}_G\}$, we compute both
\begin{align*}
&\, \bigl((f_{\Gamma_G,\Gamma_f})^*\omega_{\mu}\bigr)(U_{\overrightarrow{\mathcal{S}_Gx}})
\underset{\eqref{eq:pullbackdefining}}{=}
\sum_{V\in S(\Gamma_f)}m_{V,U_{\overrightarrow{\mathcal{S}_Gx}}}(f)\omega_{\mu}(V)\\
&=
s_{\overrightarrow{\mathcal{S}_Gx}}(f)\cdot 1+
m_{\overrightarrow{\mathcal{S}_Gx}}(f)\cdot
\sum_{V\in S(\Gamma_f):V\subset U_{f_*(\overrightarrow{\mathcal{S}_Gx})}}\omega_{\mu}(V)
=s_{\overrightarrow{\mathcal{S}_Gx}}(f)+m_{\overrightarrow{\mathcal{S}_Gx}}(f)\times\\
\times&
\begin{cases}
\omega_{\mu}\bigl(U_{\overrightarrow{\mathcal{S}_Gf(\mathcal{S}_G)}}\cap U_{\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G}}\bigr)+\omega_{\mu}(\{\mathcal{S}_G\})
+\displaystyle\sum_{\overrightarrow{v}\in T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})\setminus\{\overrightarrow{\mathcal{S}_Gf(\mathcal{S}_G)}\}}\omega_{\mu}(U_{\overrightarrow{v}})
&\text{if }f_*(\overrightarrow{\mathcal{S}_Gx})=\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G},\\
\omega_{\mu}\bigl(U_{f_*(\overrightarrow{\mathcal{S}_Gx})}\bigr)&\text{otherwise},
\end{cases}\\
&\underset{\eqref{eq:intersection}}{=}
s_{\overrightarrow{\mathcal{S}_Gx}}(f)+m_{\overrightarrow{\mathcal{S}_Gx}}(f)\times\\
\times&
\begin{cases}
\mu_E(\{a_A\}) & \text{if }f_*(\overrightarrow{\mathcal{S}_Gx})=\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G},\Bigl(\overset{\eqref{eq:consistent}}{\Leftrightarrow} (A\circ f)_*(\overrightarrow{\mathcal{S}_Gx})=\overrightarrow{\mathcal{S}_GA(\mathcal{S}_G)}
\overset{\eqref{eq:constantdirection}\&\eqref{eq:redconst}}{=}\overrightarrow{\mathcal{S}_Gz}\\
&\quad\text{for a representative }z\in\mathbb{P}^1(\mathbb{L})\text{ of }a_A\in\mathbb{P}^1(\mathbb{C})=\mathbb{P}^1(k_{\mathbb{L}})\Bigr),\\
\mu_E(\{\tilde y\}) & \text{if }f_*(\overrightarrow{\mathcal{S}_Gx})=(A^{-1})_*(\overrightarrow{\mathcal{S}_Gy})\,\Bigl(\Leftrightarrow (A\circ f)_*(\overrightarrow{\mathcal{S}_Gx})=\overrightarrow{\mathcal{S}_Gy}\Bigr)\text{ for some }\tilde{y}\in\mathbb{P}^1(\mathbb{C})\setminus\{a_A\},
\end{cases}\\
&\underset{\eqref{eq:surplusfaber}\&\eqref{directdegdef}\&\eqref{eq:tangentreduct}}{=}
\operatorname{ord}_{\tilde{x}}\bigl[H_{\widetilde{A\circ f}}=0\bigr]
+\bigl(\deg_{\tilde{x}}\phi_{\widetilde{A\circ f}}\bigr)\cdot
\mu_E\bigl(\bigl\{\phi_{\widetilde{A\circ f}}(\tilde{x})\bigr\}\bigr)
=\bigl((\widetilde{A\circ f})^*\mu_E\bigr)(\{\tilde{x}\})
\end{align*}
and
\begin{align*}
&\bigl((\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu}\bigr)(U_{\overrightarrow{\mathcal{S}_Gx}})
=\sum_{V\in S(\Gamma_f):V\subset U_{\overrightarrow{\mathcal{S}_Gx}}}\omega_{\mu}(V)\\
=&\begin{cases}
\omega_{\mu}\bigl(U_{\overrightarrow{\mathcal{S}_Gf(\mathcal{S}_G)}}\cap U_{\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G}}\bigr)+\omega_{\mu}(\{f(\mathcal{S}_G)\})
+\displaystyle\sum_{\overrightarrow{w}\in T_{f(\mathcal{S}_G)}\mathsf{P}^1(\mathbb{L})\setminus\{\overrightarrow{f(\mathcal{S}_G)\mathcal{S}_G}\}}\omega_{\mu}(U_{\overrightarrow{w}})
& \text{if }\overrightarrow{\mathcal{S}_Gx}=\overrightarrow{\mathcal{S}_Gf(\mathcal{S}_G)},\\
\omega_{\mu}(U_{\overrightarrow{\mathcal{S}_Gx}}) &\text{otherwise},
\end{cases}\\
\underset{\eqref{eq:intersection}}{=}&
\begin{cases}
\mu_C(\{h_A\})&\text{if }\overrightarrow{\mathcal{S}_Gx}=\overrightarrow{\mathcal{S}_Gf(\mathcal{S}_{G})}\,\bigl(\Leftrightarrow\tilde{x}=h_A\bigr),\\
\mu_C(\{\tilde{x}\}) &\text{otherwise},
\end{cases}\\
=&\mu_C(\{\tilde{x}\}).
\end{align*}
Hence if $(\widetilde{A\circ f})^*\mu_E=d\cdot\mu_C$ on $\mathbb{P}^1(\mathbb{C})$,
then $((f_{\Gamma_G,\Gamma_f})^*\omega_{\mu})(U_{\overrightarrow{\mathcal{S}_Gx}})
=(d\cdot(\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu})\bigl(U_{\overrightarrow{\mathcal{S}_Gx}}\bigr)$.
Moreover, we compute
\begin{multline*}
\bigl((f_{\Gamma_G,\Gamma_f})^*\omega_{\mu}\bigr)(\{\mathcal{S}_G\})
\underset{\eqref{eq:pullbackdefining}}{=}
\sum_{V\in S(\Gamma_f)}m_{V,\{\mathcal{S}_G\}}(f)\omega_{\mu}(V)\\
=\deg_{\mathcal{S}_G}(f)\cdot\omega_{\mu}(\{f(\mathcal{S}_G)\})
\underset{\eqref{eq:totallocaldegree}}{=}\deg\bigl(\phi_{\widetilde{A\circ f}}\bigr)\cdot\nu_E(\mathbb{P}^1(\mathbb{C}))
=\bigl((\widetilde{A\circ f})^*\mu_E\bigr)(\mathbb{P}^1(\mathbb{C})\setminus F_1)
\end{multline*}
and
$\bigl((\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu}\bigr)(\{\mathcal{S}_G\})
=\omega_{\mu}(\{\mathcal{S}_G\})=\nu_C(\mathbb{P}^1(\mathbb{C}))
=\mu_C(\mathbb{P}^1(\mathbb{C})\setminus F_2)$,
where $F_1,F_2$ are any sufficiently large and at most countable subsets in $\mathbb{P}^1(\mathbb{C})$.
Hence if $(\widetilde{A\circ f})^*\mu_E=d\cdot\mu_C$ on $\mathbb{P}^1(\mathbb{C})$,
then $((f_{\Gamma_G,\Gamma_f})^*\omega_{\mu})(\{\mathcal{S}_G\})
=(d\cdot(\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu})(\{\mathcal{S}_G\})$.
\subsection*{(b)} When $\Gamma_f=\Gamma_G$,
for every $\tilde{x}\in\mathbb{P}^1(\mathbb{C})=\mathbb{P}^1(k_{\mathbb{L}})\cong T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})=S(\Gamma_G)\setminus\{\mathcal{S}_G\}$, similarly,
\begin{align*}
&\bigl((f_{\Gamma_G,\Gamma_f})^*\omega_{\mu}\bigr)(U_{\overrightarrow{\mathcal{S}_Gx}})
\underset{\eqref{eq:pullbackdefining}}{=}
\sum_{V\in S(\Gamma_G)}m_{V,U_{\overrightarrow{\mathcal{S}_Gx}}}(f)\omega_{\mu}(V)
=s_{\overrightarrow{\mathcal{S}_Gx}}(f)\cdot 1+
m_{\overrightarrow{\mathcal{S}_Gx}}(f)\cdot \omega_{\mu}(U_{f_*(\overrightarrow{\mathcal{S}_Gx})})\\
&=s_{\overrightarrow{\mathcal{S}_Gx}}(f)+m_{\overrightarrow{\mathcal{S}_Gx}}(f)\cdot
\mu_E(\{\tilde y\})\quad\text{for some }y\in\mathbb{P}^1(\mathbb{L})
\text{ satisfying }f_*(\overrightarrow{\mathcal{S}_Gx})=(A^{-1})_*(\overrightarrow{\mathcal{S}_Gy}),\\
&\underset{\eqref{eq:surplusfaber}\&\eqref{directdegdef}\&\eqref{eq:tangentreduct}}{=}
\operatorname{ord}_{\tilde{x}}\bigl[H_{\widetilde{A\circ f}}=0\bigr]
+\bigl(\deg_{\tilde{x}}\phi_{\widetilde{A\circ f}}\bigr)\cdot
\mu_E\bigl(\bigl\{\phi_{\widetilde{A\circ f}}(\tilde{x})\bigr\}\bigr)
=\bigl((\widetilde{A\circ f})^*\mu_E\bigr)(\{\tilde{x}\})
\end{align*}
and
\begin{gather*}
\bigl((\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu}\bigr)(U_{\overrightarrow{\mathcal{S}_Gx}})
=\omega_{\mu}(U_{\overrightarrow{\mathcal{S}_Gx}})
=\mu_C(\{\tilde{x}\}).
\end{gather*}
Hence if $(\widetilde{A\circ f})^*\mu_E=d\cdot\mu_C$ on $\mathbb{P}^1(\mathbb{C})$, then
$((f_{\Gamma_G,\Gamma_f})^*\omega_{\mu})(U_{\overrightarrow{\mathcal{S}_Gx}})=(d\cdot(\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu})(U_{\overrightarrow{\mathcal{S}_Gx}})$.
We similarly compute both
\begin{multline*}
\bigl((f_{\Gamma_G,\Gamma_f})^*\omega_{\mu}\bigr)(\{\mathcal{S}_G\})
\underset{\eqref{eq:pullbackdefining}}{=}
\sum_{V\in S(\Gamma_G)}m_{V,\{\mathcal{S}_G\}}(f)\cdot\omega_{\mu}(V)\\
=\deg_{\mathcal{S}_G}(f)\cdot\omega_{\mu}(\{\mathcal{S}_G\})
\underset{\eqref{eq:totallocaldegree}}{=}\deg\bigl(\phi_{\widetilde{A\circ f}}\bigr)\cdot\nu_E(\mathbb{P}^1(\mathbb{C}))
=\bigl((\widetilde{A\circ f})^*\mu_E\bigr)(\mathbb{P}^1(\mathbb{C})\setminus F_1)
\end{multline*}
and
\begin{gather*}
\bigl((\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu}\bigr)(\{\mathcal{S}_G\})
=\omega_{\mu}(\{\mathcal{S}_G\})
=\nu_C(\mathbb{P}^1(\mathbb{C}))
=\mu_C(\mathbb{P}^1(\mathbb{C})\setminus F_2),
\end{gather*}
where $F_1,F_2$ are any sufficiently large and at most countable subsets in $\mathbb{P}^1(\mathbb{C})$.
Hence if $(\widetilde{A\circ f})^*\mu_E=d\cdot\mu_C$ on $\mathbb{P}^1(\mathbb{C})$,
then $((f_{\Gamma_G,\Gamma_f})^*\omega_{\mu})(\{\mathcal{S}_G\})
=(d\cdot(\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu})(\{\mathcal{S}_G\})$.
\end{proof}
The following complements Theorem \ref{th:balancedgeneral}.
\begin{proposition}\label{th:admissible}
If $\mu_C=\lim_{j\to\infty}\mu_{f_{t_j}},\mu_E=\lim_{j\to\infty}(A_{t_j})_*\mu_{f_{t_j}}$ are weak limit points on $\mathbb{P}^1(\mathbb{C})$ as $t\to 0$ of
$(\mu_{f_t})_{t\in\mathbb{D}^*},((A_t)_*\mu_{f_t})_{t\in\mathbb{D}^*}$,
respectively, for some $(t=t_j)$ in $\mathbb{D}^*$ tending to
$0$ as $j\to\infty$, then $\mu:=(\mu_C,\mu_E)\in(M^1(\mathbb{P}^1(\mathbb{C})))^2$
satisfies the admissibility \eqref{eq:compatibility}.
\end{proposition}
\begin{proof}
When $\Gamma_f=\Gamma_G$,
$\lim_{t\to 0}A_t=\phi_{\tilde{A}}=\tilde{A}\in\mathrm{PGL}(2,\mathbb{C})$
uniformly on $\mathbb{P}^1(\mathbb{C})$, so we have $\tilde{A}_*\mu_C=\mu_E$
on $\mathbb{P}^1(\mathbb{C})$, that is, the admissibility
$\tilde{A}^*\mu_E=\mu_C$ on $\mathbb{P}^1(\mathbb{C})$ in this case holds.
When $\Gamma_f\neq\Gamma_G$, using \eqref{eq:redconst},
we have $\lim_{t\to 0}A_t=\phi_{\tilde{A}}\equiv a_A$ locally uniformly on
$\mathbb{P}^1(\mathbb{C})\setminus\operatorname{supp}[H_{\tilde{A}}=0]=\mathbb{P}^1(\mathbb{C})\setminus\{h_A\}$,
and then we have
$\mu_E(\{a_A\})\ge\mu_C(\mathbb{P}^1(\mathbb{C})\setminus\{h_A\})=1-\mu_C(\{h_A\})$,
that is, the admissibility $\mu_E(\{a_A\})+\mu_C(\{h_A\})\ge 1$
in this case also holds.
\end{proof}
\section{Proof of Theorem \ref{th:computation}}
\label{sec:proofmain}
Let $K$ be an algebraically closed field that is complete
with respect to a non-trivial and non-archimedean absolute value $|\cdot|$,
and let $f\in K(z)$ be a rational function on $\mathbb{P}^1$ of $\deg f=:d>1$.
Recall that for each $n\in\mathbb{N}$,
$\Gamma_G:=\{\mathcal{S}_G\}$
and
$\Gamma_{f^n}:=\{\mathcal{S}_G,f^n(\mathcal{S}_G)\}$ in $\mathsf{H}^1_{\mathrm{II}}$.
\begin{lemma}\label{th:balancednonarchi}
For every $\nu\in M^1(\mathsf{P}^1)$, if $\nu$ satisfies the $f$-balanced
property $f^*\nu=d\cdot\nu$ on $\mathsf{P}^1$ and $\nu(\{f(\mathcal{S}_G)\})=0$, then
for every $n\in\mathbb{N}$, $(\pi_{\mathsf{P}^1,\Gamma_{f^n}})_*\nu\in M^1(\Gamma_{f^n})$
satisfies the quantized $f^n$-balanced property \eqref{eq:quantizedbalanced}.
\end{lemma}
\begin{proof}
Under the assumption, for every measurable function $\psi$ on
$(\mathsf{P}^1,\sigma(S(\Gamma_G)))$, we compute
\begin{multline*}
\bigl\langle\psi,(f_{\Gamma_G,\Gamma_f})^*\bigl((\pi_{\mathsf{P}^1,\Gamma_f})_*\nu\bigr)\bigr\rangle
=\bigl\langle(f_{\Gamma_G,\Gamma_f})_*\psi,(\pi_{\mathsf{P}^1,\Gamma_f})_*\nu\bigr\rangle
\underset{\eqref{eq:pushfactor}\&\eqref{eq:projectfactor}}{=}
\int_{\mathsf{P}^1\setminus\{f(\mathcal{S}_G)\}}(f_*(\pi_{\mathsf{P}^1,\Gamma_G})^*\psi)\nu\\
=\bigl\langle f_*(\pi_{\mathsf{P}^1,\Gamma_G})^*\psi,\nu\bigr\rangle
=\bigl\langle (\pi_{\mathsf{P}^1,\Gamma_G})^*\psi,f^*\nu\bigr\rangle
=\bigl\langle (\pi_{\mathsf{P}^1,\Gamma_G})^*\psi,d\cdot\nu\bigr\rangle
=\bigl\langle(\pi_{\mathsf{P}^1,\Gamma_f})^*(\pi_{\Gamma_f,\Gamma_G})^*\psi,d\cdot\nu\bigr\rangle\\
=\bigl\langle\psi,d\cdot(\pi_{\Gamma_f,\Gamma_G})_*\bigl((\pi_{\mathsf{P}^1,\Gamma_f})_*\nu\bigr)\bigr\rangle,
\end{multline*}
so $(\pi_{\mathsf{P}^1,\Gamma_f})_*\nu$ satisfies the quantized $f$-balanced property.
For any $n\in\mathbb{N}$, we also have $(f^n)^*\nu=d^n\cdot\nu$ on $\mathsf{P}^1$,
and in turn
$0=d^{n-1}\cdot(\deg_{\mathcal{S}_G}f)\cdot\nu(\{f(\mathcal{S}_G)\})=
d^{n-1}\cdot(f^*\nu)(\{\mathcal{S}_G\})=((f^n)^*\nu)(\{\mathcal{S}_G\})
=\deg_{\mathcal{S}_G}(f^n)\cdot\nu(\{f^n(\mathcal{S}_G)\})\ge\nu(\{f^n(\mathcal{S}_G)\})\ge 0$,
so $\nu(\{f^n(\mathcal{S}_G)\})=0$ and we are done.
\end{proof}
\begin{proof}[Proof of Theorem $\ref{th:computation}$]
{\bfseries (a)} Assume now that $\operatorname{char}K=0$.
Under the assumption that $f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$
or equivalently that
\begin{gather}
\nu_f(\{f(\mathcal{S}_G)\})=\nu_f(\{\mathcal{S}_G\})\underset{\eqref{eq:projectfactor}}{=}
\bigl((\pi_{\mathsf{P}^1,\Gamma_G})_*\nu_f\bigr)(\{\mathcal{S}_G\})
=0\quad\text{(in \eqref{eq:nopotgood})},\label{eq:vanish}
\end{gather}
the inclusion
$(\pi_{\mathsf{P}^1,\Gamma_G})_*\bigl(\{\nu_f\}\cup\{\delta_{\mathcal{E}}:\mathcal{E}\text{ is a cycle of }f\text{ in }E(f)\}\bigr)\subset\Delta_f^\dag(\subset\Delta_f)$ holds
by Lemma \ref{th:balancednonarchi}.
In particular, only one of the two statements
(i),(ii) in Theorem \ref{th:computation} is the case.
We also assume that that $f^{-1}(a)=\{a\}$ for any $a\in E(f)$.
\subsection*{\bfseries (b)}
If for any $\overrightarrow{v}\in T_{\mathcal{S}_G}\mathsf{P}^1$,
\begin{gather}
\limsup_{n\to\infty}\frac{s_{\overrightarrow{v}}(f^n)}{d^n}\ge\nu_f(U_{\overrightarrow{v}}),\label{eq:surplusequidist}
\end{gather}
then for every $\omega\in\Delta_f$ and $n\gg 1$,
choosing $\omega_n\in M^1(\Gamma_{f^n})$ such that
$d^{-n}((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n=\omega=(\pi_{\Gamma_{f^n},\Gamma_G})_*\omega_n$ in $M^1(\Gamma_G)$,
we have
\begin{gather*}
\omega(U_{\overrightarrow{v}})
=\limsup_{n\to\infty}\frac{\bigl(((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n\bigr)(U_{\overrightarrow{v}})}{d^n}
\underset{\eqref{eq:pullbackdefining}\&\eqref{eq:surplusequidist}}{\ge}
\nu_f(U_{\overrightarrow{v}})\cdot 1
\underset{\eqref{eq:projectfactor}}{=}((\pi_{\mathsf{P}^1,\Gamma_G})_*\nu_f)(U_{\overrightarrow{v}})
\end{gather*}
for any $\overrightarrow{v}\in T_{\mathcal{S}_G}\mathsf{P}^1$,
which with \eqref{eq:vanish}
yields $\omega=(\pi_{\mathsf{P}^1,\Gamma_G})_*\nu_f$ in $M^1(\Gamma_G)$.
Hence $\Delta_f=\Delta_f^\dag=(\pi_{\mathsf{P}^1,\Gamma_G})_*(\{\nu_f\})$
(see \cite[p.\ 27]{DF14}).
\subsection*{\bfseries (c.1)}
Suppose now that $\Delta_f\not\subset(\pi_{\mathsf{P}^1,\Gamma_G})_*(\{\nu_f\})$.
Then there is $\overrightarrow{w}\in T_{\mathcal{S}_G}\mathsf{P}^1$
not satisfying \eqref{eq:surplusequidist}.
Hence
fixing any $\mathcal{S}\in\mathbb{P}^1\setminus E(f)(\subset\mathsf{P}^1\setminus(E(f)\cup\{f^n(\mathcal{S}_G):n\in\mathbb{N}\}))$,
we have
\begin{multline*}
\notag\nu_f(U_{\overrightarrow{w}})
\underset{\eqref{eq:equidist}}{=}\lim_{n\to\infty}\frac{\bigl((f^n)^*\delta_{\mathcal{S}}\bigr)(U_{\overrightarrow{w}})}{d^n}
\underset{\eqref{eq:argumentnonarchi}}{\le}
\limsup_{n\to\infty}\frac{m_{\overrightarrow{w}}(f^n)}{d^n}
+\limsup_{n\to\infty}\frac{s_{\overrightarrow{w}}(f^n)}{d^n}\\
<\limsup_{n\to\infty}\frac{m_{\overrightarrow{w}}(f^n)}{d^n}+\nu_f(U_{\overrightarrow{w}}),
\quad\text{so that }
\limsup_{n\to\infty}\prod_{j=0}^{n-1}\frac{m_{(f^j)_*\overrightarrow{w}}(f)}{d}
=\limsup_{n\to\infty}\frac{m_{\overrightarrow{w}}(f^n)}{d^n}>0,
\end{multline*}
and in turn
\begin{gather}
\deg_{f^n(\mathcal{S}_G)}(f)\bigl(=m_{(f^n)_*\overrightarrow{w}}(f)\bigr)\equiv d
\quad\text{for }n\gg 1;\label{eq:eventual}
\end{gather}
then we also have $f^{n+1}(\mathcal{S}_G)\neq f^n(\mathcal{S}_G)$ for $n\gg 1$
under the assumption that $f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$.
Recall that the maximally ramified locus
$R(f):=\{\mathcal{S}\in\mathsf{P}^1:\deg_{\mathcal{S}}(f)=d\}$ of $f$ is connected in $\mathsf{P}^1$
(\cite[Theorem 8.2]{Faber13topologyI}).
Hence for $n\gg 1$, $[f^n(\mathcal{S}_G),f^{n+1}(\mathcal{S}_G)]\subset R(f)$, and then
$f$ restricts to a homeomorphism from
$[f^{n-1}(\mathcal{S}_G),f^n(\mathcal{S}_G)]$ onto $[f^n(\mathcal{S}_G),f^{n+1}(\mathcal{S}_G)]$
and $\mathcal{S}\mapsto m_{\overrightarrow{\mathcal{S} f^n(\mathcal{S}_G)}}(f)=\deg_{\mathcal{S}}(f)\equiv d$ on $[f^{n-1}(\mathcal{S}_G),f^n(\mathcal{S}_G)]$, so that for any $m\ge n\gg 1$,
also by \eqref{eq:Lipschitz}, we have
$\rho(f^m(\mathcal{S}_G),f^{m+1}(\mathcal{S}_G))=d^{m-n}\cdot\rho(f^n(\mathcal{S}_G),f^{n+1}(\mathcal{S}_G))>0$.
Hence $(f^n(\mathcal{S}_G))_n$ accumulates only to $\mathbb{P}^1$.
By the upper semicontinuity of $\deg_{\,\cdot}(f)$ on $\mathsf{P}^1$
and $\#\{a\in\mathbb{P}^1:\deg_a(f)=d\}\le 2$ (as mentioned in \eqref{eq:maximaldegree}),
we moreover
have $\bigcap_{N\in\mathbb{N}}\overline{\{f^n(\mathcal{S}_G):n\ge N\}}\subset E(f)$,
and in turn, under the assumption that $f^{-1}(a)=\{a\}$
for any $a\in E(f)$, even $\lim_{n\to\infty}f^n(\mathcal{S}_G)=a$
for some $a\in E(f)$.
Then we also have $f^n(\mathcal{S}_G)\in[\mathcal{S}_G,a]$ for $n\gg 1$
using \cite[Theorem F]{Faber13topologyII} and \eqref{eq:Lipschitz}
(see \cite[p.\ 25]{DF14}).
\subsection*{\bfseries (c.2)} For every
$\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}$
and $n\gg 1$,
since $f^{-1}(a)=\{a\}$ and $f^n(\mathcal{S}_G)\in[\mathcal{S}_G,a]$ for $n\gg 1$,
we have
\begin{gather}
s_{\overrightarrow{v}}(f^n)=0\, \bigl(\Leftrightarrow f^n(U_{\overrightarrow{v}})=U_{(f^n)_*\overrightarrow{v}}\bigr)\quad\text{and}\label{eq:nonexceptionaldirection}\\
(f^n)_*(\overrightarrow{v})\neq\overrightarrow{f^n(\mathcal{S}_G)a},
\label{eq:nonexceptionaldirectionsurplus}
\end{gather}
and then
\begin{gather}
s_{\overrightarrow{\mathcal{S}_Ga}}(f^n)=d^n-\deg_{\mathcal{S}_G}(f^n)\quad
(\text{using also }\eqref{eq:totalsurplus})\quad\text{and}\label{eq:excepsurplus}\\
(f^n)_*\bigl(\overrightarrow{\mathcal{S}_Ga}\bigr)=\overrightarrow{f^n(\mathcal{S}_G)a}
\quad
\bigl(\text{since the tangent map }((f^n)_*)_{\mathcal{S}_G}\text{ is surjective}\bigr).
\label{eq:exceptionaldirection}
\end{gather}
By $\lim_{n\to\infty}f^n(\mathcal{S}_G)=a\in E(f)\subset\mathsf{P}^1\setminus\mathsf{J}(f)$ and $f^n(\mathcal{S}_G)\in[\mathcal{S}_G,a]$ for $n\gg 1$, we have not only
\begin{gather}
\nu_f\bigl(U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\bigr)=1\quad\text{for }n\gg 1\label{eq:total}
\end{gather}
but, fixing any $\mathcal{S}\in\mathsf{P}^1\setminus E(f)$,
also
\begin{multline}
\frac{\deg_{\mathcal{S}_G}(f^n)}{d^n}
\underset{\eqref{eq:excepsurplus}}{=}\frac{d^n-s_{\overrightarrow{\mathcal{S}_G a}}(f^n)}{d^n}\underset{\eqref{eq:exceptionaldirection}\&\eqref{eq:argumentnonarchi}}{=}
1-\frac{(f^n)^*\delta_{\mathcal{S}}}{d^n}(U_{\overrightarrow{\mathcal{S}_Ga}})\\
\underset{\eqref{eq:eventual}\&\eqref{eq:equidist}}{\equiv}
1-\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})
\quad\text{for }n\gg 1,
\label{eq:stationary}
\end{multline}
so in particular $\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})<1$.
For every
$\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}$
and $n\gg 1$, by \eqref{eq:total}, \eqref{eq:nonexceptionaldirection},
and the $f$-balanced property of $\nu_f$
on $\mathsf{P}^1$, we have the equivalence
\begin{gather}
\nu_f(U_{\overrightarrow{v}})>0\Leftrightarrow
(f^n)_*(\overrightarrow{v})
=\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}
\quad(\Rightarrow f^n(U_{\overrightarrow{v}})=U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}).\label{eq:nonexceptionalpositive}
\end{gather}
Hence for every
$\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}$
satisfying $\nu_f(U_{\overrightarrow{v}})>0$,
\begin{gather}
\nu_f(U_{\overrightarrow{v}})
=\frac{(f^n)^*\nu_f}{d^n}(U_{\overrightarrow{v}})
\underset{\eqref{eq:argumentnonarchi}}{=}
\frac{m_{\overrightarrow{v}}(f^n)+s_{\overrightarrow{v}}(f^n)}{d^n}
\cdot\nu_f(U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}})
\underset{\eqref{eq:nonexceptionaldirection}\&\eqref{eq:total}}{=}
\frac{m_{\overrightarrow{v}}(f^n)}{d^n}\quad\text{for }n\gg 1.
\label{eq:nonexceptionaldirectionmeasure}
\end{gather}
On the other hand, for $n\gg 1$,
by \eqref{eq:nonexceptionaldirectionsurplus} and
\eqref{eq:nonexceptionalpositive}, we also have
\begin{multline}
\bigl\{(f^n)_*(\overrightarrow{v}):
\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}
\text{ satisfying }\nu_f(U_{\overrightarrow{v}})=0\bigr\}\\
=\bigl(T_{f^n(\mathcal{S}_G)}\mathsf{P}^1\bigr)
\setminus\bigl\{\overrightarrow{f^n(\mathcal{S}_G)a},\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}\bigr\},\label{eq:annulus}
\end{multline}
and then, noting also that
$f:\mathsf{P}^1\setminus(U_{\overrightarrow{f^n(\mathcal{S}_G)a}}\cup U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\cup\{f^n(\mathcal{S}_G)\})\to\mathsf{P}^1\setminus(U_{\overrightarrow{f^{n+1}(\mathcal{S}_G)a}}\cup U_{\overrightarrow{f^{n+1}(\mathcal{S}_G)\mathcal{S}_G}}\cup\{f^{n+1}(\mathcal{S}_G)\})$ is a $d$ to $1$
(unbranched) covering (since $f^{-1}(a)=\{a\}$,
$\lim_{n\to\infty}f^n(\mathcal{S}_G)=a$, and $f^n(\mathcal{S}_G)\in[\mathcal{S}_G,a]$),
\eqref{eq:eventual},
and \eqref{eq:totallocaldegree},
for every $\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}$ satisfying $\nu_f(U_{\overrightarrow{v}})=0$, we have
\begin{gather}
m_{(f^n)_*(\overrightarrow{v})}(f)\equiv 1\quad\text{for }n\gg 1.\label{eq:simpledirect}
\end{gather}
\begin{remark}\label{th:notalways}
In \cite[\S4.6]{DF14},
the condition $U_{\overrightarrow{\mathcal{S}_Ga}}\cap\mathsf{J}(f)=\emptyset$
or equivalently $\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})=0$ was
assumed with loss of some generality;
by \eqref{eq:stationary}, those conditions are equivalent to
\begin{gather}
\deg_{f^n(\mathcal{S}_G)}(f)\equiv d\quad\text{for any }n\in\mathbb{N}\cup\{0\}.\tag{\ref{eq:eventual}$'$}\label{eq:identically}
\end{gather}
The claim $\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})=0$
is not always the case (see Section \ref{sec:example} below).
\end{remark}
\subsection*{\bfseries (d.1)} Let us see
the equality \eqref{eq:computationdelta}.
Fix $\omega\in\Delta_f$ and, for $n\gg 1$,
fix $\omega_n\in M^1(\Gamma_{f^n})$ such that
$d^{-n}((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n=\omega=(\pi_{\Gamma_{f^n},\Gamma_G})_*\omega_n$ in $M^1(\Gamma_G)$. Then for $n\gg 1$, we have both
\begin{gather}
\omega(U_{\overrightarrow{v}})
\underset{\eqref{eq:nonexceptionaldirection}\&\eqref{eq:pullbackdefining}}{=}
\frac{m_{\overrightarrow{v}}(f^n)}{d^n}\cdot
\omega_n\bigl(f^n(U_{\overrightarrow{v}})\bigr)\quad
\text{for any }
\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}
\label{eq:pullbacknonexcep}
\end{gather}
and
\begin{gather}
\omega_n(\{f^n(\mathcal{S}_G)\})
\underset{\eqref{eq:pullbackdefining}}{=}
\frac{d^n\cdot\omega(\{\mathcal{S}_G\})}{\deg_{\mathcal{S}_G}(f^n)}
\underset{\eqref{eq:stationary}}{=}
\frac{\omega(\{\mathcal{S}_G\})}{1-\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})}.\label{eq:gaussimagemeas}
\end{gather}
By \eqref{eq:pullbacknonexcep}, \eqref{eq:nonexceptionalpositive},
and \eqref{eq:nonexceptionaldirectionmeasure}, there is
a constant $s_\omega\in[0,1]$ such that
\begin{gather}
\omega_n\bigl(U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\bigr)
\equiv s_\omega\quad\text{for }n\gg 1\label{eq:eventualfull}
\end{gather}
and that for every
$\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}$
satisfying $\nu_f(U_{\overrightarrow{v}})>0$,
\begin{gather}
\omega(U_{\overrightarrow{v}})=s_\omega\nu_f(U_{\overrightarrow{v}}).
\label{eq:nonexceptionaldirectionpositivemeasure}
\end{gather}
On the other hand, for every
$\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}$
satisfying $\nu_f(U_{\overrightarrow{v}})=0$, we have
\begin{gather*}
0\le\omega(U_{\overrightarrow{v}})
\underset{\eqref{eq:pullbacknonexcep}}{=}
\frac{m_{\overrightarrow{v}}(f^n)}{d^n}\cdot
\omega_n\bigl(f^n(U_{\overrightarrow{v}})\bigr)
\le\frac{m_{\overrightarrow{v}}(f^n)}{d^n}
=\prod_{j=0}^{n-1}\frac{m_{(f^j)_*(\overrightarrow{v})}(f)}{d}
\underset{\eqref{eq:simpledirect}}{\to} 0\quad\text{as }n\to\infty,
\end{gather*}
so that
\begin{gather}
\omega(U_{\overrightarrow{v}})=0=s_\omega\nu_f(U_{\overrightarrow{v}});\label{eq:nonexceptionaldirectionzeromeasure}
\end{gather}
then also by \eqref{eq:annulus}
and \eqref{eq:pullbacknonexcep}, for $n\gg 1$,
\begin{gather}
\omega_n(U_{\overrightarrow{v}})=0
\quad\text{for any }\overrightarrow{v}\in\bigl(T_{f^n(\mathcal{S}_G)}\mathsf{P}^1\bigr)
\setminus\bigl\{\overrightarrow{f^n(\mathcal{S}_G)a},\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}\bigr\}.\label{eq:annulusmeas}
\end{gather}
\subsection*{\bfseries (d.2)}
For $n\gg 1$, we compute
\begin{multline}
\omega\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)
=1-\sum_{\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}}\omega(U_{\overrightarrow{v}})-\omega(\{\mathcal{S}_G\})\\
\underset{\eqref{eq:nonexceptionaldirectionpositivemeasure}\&\eqref{eq:nonexceptionaldirectionzeromeasure}}{=}
1-s_\omega\nu_f\bigl(\mathsf{P}^1\setminus U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)-\omega(\{\mathcal{S}_G\})
=\bigl(s_\omega\nu_f\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)+(1-s_\omega)\bigr)
-\omega(\{\mathcal{S}_G\});\label{eq:excepdirecmeas}
\end{multline}
then also recalling that
$(\pi_{\Gamma_{f^n},\Gamma_G})_*\omega_n=\omega$ in $M^1(\Gamma_G)$,
\begin{multline}
0\le\omega_n\bigl(U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\cap U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)
=\omega_n\bigl(U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\bigr)-\sum_{\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}}\omega_n(U_{\overrightarrow{v}})-\omega_n(\{\mathcal{S}_G\})\\
\underset{\eqref{eq:eventualfull}\&\eqref{eq:nonexceptionaldirectionpositivemeasure}\&\eqref{eq:nonexceptionaldirectionzeromeasure}}{=}s_\omega-s_\omega\cdot\nu_f\bigl(\mathsf{P}^1\setminus U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)-\omega_n(\{\mathcal{S}_G\})\\
=s_\omega\nu_f\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)-\omega_n(\{\mathcal{S}_G\})
\label{eq:positivethick}
\end{multline}
and
\begin{multline}
0\le\omega_n\bigl(U_{\overrightarrow{f^n(\mathcal{S}_G)a}}\bigr)
=\omega\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)
-\omega_n\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\cap U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\bigr)\\
-\omega_n\bigl(\mathsf{P}^1\setminus(U_{\overrightarrow{f^n(\mathcal{S}_G)a}}\cup U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\cup\{f^n(\mathcal{S}_G)\})\bigr)-\omega_n(\{f^n(\mathcal{S}_G)\})\\
\underset{\eqref{eq:excepdirecmeas}\&\eqref{eq:positivethick}\&\eqref{eq:annulusmeas}\&\eqref{eq:gaussimagemeas}}{=}\bigl(1-s_\omega\nu_f\bigl(\mathsf{P}^1\setminus U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)-\omega(\{\mathcal{S}_G\})\bigr)
-\bigl(s_\omega\nu_f\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)-\omega(\{\mathcal{S}_G\})\bigr)\\
-0-\frac{\omega(\{\mathcal{S}_G\})}{1-\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})}
=1-s_\omega-\frac{\omega(\{\mathcal{S}_G\})}{1-\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})}.\label{eq:positivepoint}
\end{multline}
By \eqref{eq:nonexceptionaldirectionpositivemeasure},
\eqref{eq:nonexceptionaldirectionzeromeasure},
\eqref{eq:excepdirecmeas}, \eqref{eq:positivethick},
and \eqref{eq:positivepoint}, $\Delta_f$ is contained in the right hand side in \eqref{eq:computationdelta}.
\subsection*{\bfseries (d.3)}
Conversely, let us fix an element $\omega$ in the right hand side
in \eqref{eq:computationdelta}, and fix $0\le s\le 1$ and
$0\le s'\le\min\{s\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}}),(1-s)(1-\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}}))\}$ so that
$\omega(U_{\overrightarrow{v}})=s\nu_f(U_{\overrightarrow{v}})$
for every $\overrightarrow{v}\in\bigl(T_{\mathcal{S}_G}\mathsf{P}^1\bigr)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}$, that $\omega(\{\mathcal{S}_G\})=s'$, and that
$\omega(U_{\overrightarrow{\mathcal{S}_Ga}})=(s\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})+(1-s))-s'$.
Recalling $\lim_{n\to\infty}f^n(\mathcal{S}_G)=a$ and that
$f^n(\mathcal{S}_G)\in[\mathcal{S}_G,a]$ for $n\gg 1$,
define $\omega_n\in M^1(\Gamma_{f^n})$ as
\begin{gather*}
\begin{cases}
\omega_n(\{\mathcal{S}_G\})=s',\quad
\displaystyle\omega_n(\{f^n(\mathcal{S}_G)\})=\frac{s'}{1-\nu_f\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)},\\
\omega_n(U_{\overrightarrow{v}})=s\nu_f(U_{\overrightarrow{v}})\quad
\text{for every }\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{U_{\overrightarrow{\mathcal{S}_Ga}}\},\\
\omega_n\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\cap U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\bigr)=s\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})-s',\\
\omega_n(U_{\overrightarrow{v}})=0\quad
\text{for every }\overrightarrow{v}\in\bigl(T_{f^n(\mathcal{S}_G)}\mathsf{P}^1\bigr)\setminus\{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G},\overrightarrow{f^n(\mathcal{S}_G)a}\},\quad\text{and}\\
\displaystyle\omega_n\bigl(U_{\overrightarrow{f^n(\mathcal{S}_G)a}}\bigr)
=1-s-\frac{s'}{1-\nu_f\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)}
\end{cases}
\end{gather*}
for $n\gg 1$, so that $(\pi_{\Gamma_{f^n},\Gamma_G})_*\omega_n=\omega$
in $M^1(\Gamma_G)$. Moreover, for $n\gg 1$, we also compute
\begin{multline*}
\bigl(d^{-n}((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n\bigr)(U_{\overrightarrow{v}})
\underset{\eqref{eq:pullbackdefining}\&\eqref{eq:nonexceptionaldirection}
\&\eqref{eq:nonexceptionalpositive}}{=}
\frac{m_{\overrightarrow{v}}(f^n)\cdot\omega_n\bigl(U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\bigr)}{d^n}\\
\underset{\eqref{eq:nonexceptionaldirectionmeasure}}{=}
\nu_f(U_{\overrightarrow{v}})\cdot\omega_n\bigl(U_{\overrightarrow{f^n(\mathcal{S}_G)\mathcal{S}_G}}\bigr)
=\nu_f(U_{\overrightarrow{v}})\cdot s\nu_f(\mathsf{P}^1\setminus\{\mathcal{S}_G\})\\
\underset{\eqref{eq:vanish}}{=}\nu_f(U_{\overrightarrow{v}})\cdot s=\omega(U_{\overrightarrow{v}})\quad
\text{for every }\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}\text{ satisfying }\nu_f(U_{\overrightarrow{v}})>0,
\end{multline*}
\begin{multline*}
\bigl(d^{-n}((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n\bigr)(U_{\overrightarrow{v}})
\underset{\eqref{eq:pullbackdefining}\&\eqref{eq:nonexceptionaldirection}
}{=}\frac{m_{\overrightarrow{v}}(f^n)\cdot\omega_n\bigl(
U_{(f^n)_*\overrightarrow{v}}
\bigr)}{d^n}\underset{\eqref{eq:annulus}}{=}0\\
=s\nu_f(U_{\overrightarrow{v}})=\omega(U_{\overrightarrow{v}})\quad
\text{for every }\overrightarrow{v}\in(T_{\mathcal{S}_G}\mathsf{P}^1)\setminus\{\overrightarrow{\mathcal{S}_Ga}\}\text{ satisfying }\nu_f(U_{\overrightarrow{v}})=0,
\end{multline*}
\begin{multline*}
\bigl(d^{-n}((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n\bigr)(\{\mathcal{S}_G\})
\underset{\eqref{eq:pullbackdefining}}{=}\frac{\deg_{\mathcal{S}_G}(f^n)\cdot\omega_n(\{f^n(\mathcal{S}_G)\})
}{d^n}\\
\underset{\eqref{eq:stationary}}{=}
\bigl(1-\nu_f(U_{\overrightarrow{\mathcal{S}_Ga}})\bigr)\cdot\omega_n(\{f^n(\mathcal{S}_G)\})
=s'=\omega(\{\mathcal{S}_G\}),\quad\text{and then}
\end{multline*}
\begin{multline*}
\bigl(d^{-n}((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n\bigr)\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)
=1-\bigl(d^{-n}((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n\bigr)\bigl(\mathsf{P}^1\setminus U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)\\
=1-\omega\bigl(\mathsf{P}^1\setminus U_{\overrightarrow{\mathcal{S}_Ga}}\bigr)
=\omega\bigl(U_{\overrightarrow{\mathcal{S}_Ga}}\bigr),
\end{multline*}
so that
$d^{-n}((f^n)_{\Gamma_G,\Gamma_{f^n}})^*\omega_n=\omega$ in $M^1(\Gamma_G)$.
Hence $\Delta_f$ contains the right hand side
in \eqref{eq:computationdelta}.
\subsection*{\bfseries (e)}
Once the equality \eqref{eq:computationdelta} is at our disposal,
the final assertion in the case (ii) in Theorem \ref{th:computation} is
clear recalling also Remark \ref{th:notalways}.
Now the proof of Theorem \ref{th:computation} is complete.
\end{proof}
\section{Proof of Theorem \ref{th:B}}
\label{sec:proof}
We use the notatons in Sections \ref{sec:quantizedbalanced} and \ref{sec:direct}. Let $f\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)\subset\mathbb{L}(z)$
be a meromorphic family
of rational functions on $\mathbb{P}^1(\mathbb{C})$ of degree $d>1$,
and suppose that $f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$ in $\mathsf{P}^1(\mathbb{L})$.
Recall that $\operatorname{char}\mathbb{L}=0$ and that the norm $|\cdot|_r$ on $\mathbb{L}$
is (the extension of) \eqref{eq:normLaurent},
fixing $r\in(0,1)$ once and for all. By $\nu_{f^2}=\nu_f$ on $\mathsf{P}^1(\mathbb{L})$,
the equivalence \eqref{eq:nopotgood} applied to both $f$ and $f^2$,
$\mu_{(f_t)^2}=\mu_{f_t}$ on $\mathbb{P}^1(\mathbb{C})$ for every $t\in\mathbb{D}^*$,
$E(f^2)=E(f)$, and $\#E(f)\le 2$, replacing $f$ with $f^2$ if necessary,
we can assume that $f^{-1}(a)=\{a\}$ for any $a\in E(f)$
with no loss of generality.
For every $n\in\mathbb{N}$, pick a meromorphic family
$A_n\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)\subset\mathbb{L}(z)$
of M\"obius transformations on $\mathbb{P}^1(\mathbb{C})$
such that $(A_n\circ f^n)(\mathcal{S}_G)=\mathcal{S}_G$ in $\mathsf{P}^1(\mathbb{L})$.
Recall that $\Gamma_G:=\{\mathcal{S}_G\}$ and $\Gamma_{f^n}:=\{\mathcal{S}_G,f^n(\mathcal{S}_G)\}$
in $\mathsf{H}^1_{\mathrm{II}}(\mathbb{L})$ and that
$M^1(\Gamma_G)^{\dag}$ is identified with $M^1(\mathbb{P}^1(\mathbb{C}))^{\dag}$
under the bijection $S(\Gamma_G)\setminus\{\mathcal{S}_G\}
=T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})\cong\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C})$.
Let
\begin{gather*}
\mu_0=\lim_{j\to\infty}\mu_{f_{t_j}}
\end{gather*}
be any weak limit point
of $(\mu_{f_t})_{t\in\mathbb{D}^*}$ on $\mathbb{P}^1(\mathbb{C})$ as $t\to 0$. Then
taking a subsequence of $(t_j)$ if necessary, for every $n\in\mathbb{N}$,
the weak limit
$\mu_E^{(n)}:=\lim_{j\to\infty}((A_n)_{t_j})_*\mu_{f_{t_j}}$ also exists
on $\mathbb{P}^1(\mathbb{C})$. For every $n\in\mathbb{N}$, by the former half of
\eqref{eq:reductionconstant} in Theorem \ref{th:balancedgeneral} and
Proposition \ref{th:admissible},
the pair
\begin{gather*}
\mu^{(n)}:=\bigl(\mu_0,\mu_E^{(n)}\bigr)\in(M^1(\mathbb{P}^1(\mathbb{C})))^2
\end{gather*}
satisfies the degenerating $f^n$-balanced property
and the admissibility \eqref{eq:compatibility}
(for a while, we would not use the latter half of
\eqref{eq:reductionconstant} in Theorem \ref{th:balancedgeneral}).
Then also by Proposition \ref{th:transfer}, we have
\begin{gather*}
\omega_0:=(\pi_{\Gamma_{f^n},\Gamma_G})_*\omega_{\mu^{(n)}}\in\Delta_f,
\end{gather*}
which is indeed independent of $n\in\mathbb{N}$.
Hence in the case (i) $\Delta_f=\Delta_f^\dag=(\pi_{\mathsf{P}^1(\mathbb{L}),\Gamma_G})_*(\{\nu_f\})$ in Theorem \ref{th:computation},
also by \eqref{eq:converse}, we have the desired
$\mu_0=\omega_0=(\pi_{\mathsf{P}^1(\mathbb{L}),\Gamma_G})_*\nu_f$
in $M^1(\mathbb{P}^1(\mathbb{C}))^\dag=M^1(\Gamma_G)^\dag$.
\subsection*{(a)} Suppose that
the case (ii) in Theorem \ref{th:computation} is the case.
Then $\deg_{f^n(\mathcal{S}_G)}(f)\equiv d$ for $n\gg 1$ and,
there is $a\in E(f)$ such that $\lim_{n\to\infty}f^n(\mathcal{S}_G)=a$ and
that $f^n(\mathcal{S}_G)\in[\mathcal{S}_G,a]$ for $n\gg 1$.
\subsection*{(b)}
Since $f\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)\subset\mathbb{C}((t))(z)$ and $f(a)=a$,
we have $a\in\mathbb{P}^1(\overline{\mathbb{C}((t))})(\subset\mathbb{P}^1(\mathbb{L}))$,
and then
taking
(the extension to $\mathbb{D}$ of) a finitely-sheeted and unbranched
holomorphic self-covering of $\mathbb{D}^*$ if necessary,
we first have $a\in\mathbb{P}^1(\mathbb{C}((t)))$.
Then since $(f_t(z)-z)'|_{z=a(t)}=0-1=-1\neq 0$
for any $t\in\mathbb{D}^*$,
by the implicit function theorem for holomorphic functions,
we indeed have $a=(a(t))_{t\in\mathbb{D}}\in\mathbb{P}^1(\mathcal{O}(\mathbb{D})[t^{-1}])$
(cf.\ \cite[Proof of Corollary 5.3]{DF14}).
Replacing $f$ with $B\circ f\circ B^{-1}$ for some
meromorphic family $B\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$ of M\"obius
transformations satisfying $\tilde{B}=\phi_{\tilde{B}}\in\mathrm{PGL}(2,k_{\mathbb{L}})=\mathrm{PGL}(2,\mathbb{C})$ and mapping $a$ to $\infty$ if necessary, we can assume not only
\begin{gather*}
a=\infty\in\mathbb{P}^1\bigl(\mathcal{O}(\mathbb{D})[t^{-1}]\bigr)(\subset\mathbb{P}^1(\mathbb{L}))
\end{gather*}
but also that $f\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$ is a meromorphic family of
polynomials on $\mathbb{P}^1(\mathbb{C})$ of degree $d$.
Moreover, by $\nu_{f^n}=\nu_f$ on $\mathsf{P}^1(\mathbb{L})$,
the equivalence \eqref{eq:nopotgood} applied to both $f$ and $f^n$
for every $n\in\mathbb{N}$, $\mu_{(f_t)^n}=\mu_{f_t}$ on $\mathbb{P}^1(\mathbb{C})$
for every $t\in\mathbb{D}^*$ and every $n\in\mathbb{N}$, $E(f^n)=E(f)$ for every $n\in\mathbb{N}$,
and $f^{-1}(\infty)=\{\infty\}$ (and $d>1$),
replacing $f$ with $f^\ell$ for some $\ell\gg 1$ if necessary,
we furthermore assume that
for every $n\in\mathbb{N}$ (but not necessarily for $n=0$),
$f^n(\mathcal{S}_G)\in[\mathcal{S}_G,\infty]$, $\Gamma_{f^n}\neq\Gamma_G$,
$\deg_{f^n(\mathcal{S}_G)}(f)\equiv d$,
\begin{gather}
f\bigl(U_{\overrightarrow{f^n(\mathcal{S}_G)\infty}}\bigr)
=U_{\overrightarrow{f^{n+1}(\mathcal{S}_G)\infty}},\label{eq:superattbasin}
\end{gather}
and that the identity \eqref{eq:stationary} holds, with no loss of generality.
\subsection*{(c.1)} Writing $f(z)=\sum_{j=0}^dc_jz^j\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)$
and setting
\begin{gather*}
d_0:=\max\Bigl\{j\in\{0,\ldots,d\}:|c_j|_r
=\max_{i\in\{0,\ldots,d\}}|c_i|_r\Bigr\},
\end{gather*}
by $f(\mathcal{S}_G)\in[\mathcal{S}_G,\infty]\setminus\{\mathcal{S}_G\}$, we have $d_0\in\{1,\ldots,d\}$ and $|c_{d_0}|_r>1$, and the point $f(\mathcal{S}_G)$ is (represented by) the $\mathbb{L}$-closed disk
$\{z\in\mathbb{L}:|z|_r\le|c_{d_0}|_r\}$ in $\mathbb{L}$. Then
we can choose $A:=A_1$ as
\begin{gather*}
A(z)=c_{d_0}^{-1}z\quad(\text{so that }H_{\tilde{A}}(\zeta_0,\zeta_1)=\zeta_0,
\quad h_A=\infty,\quad\text{and}\quad\phi_{\tilde{A}}\equiv 0),
\end{gather*}
and in turn have
\begin{gather*}
d_0=\deg(\phi_{\widetilde{A\circ f}})
\underset{\eqref{eq:totallocaldegree}}{=}\deg_{\mathcal{S}_G}(A\circ f)=\deg_{\mathcal{S}_G}(f)(>0),\quad
\phi_{\widetilde{A\circ f}}(\zeta) =\sum_{j=0}^{d_0}\widetilde{\Bigl(\frac{c_j}{c_{d_0}}\Bigr)}\cdot\zeta^j,\quad\text{and}\\
H_{\widetilde{A\circ f}}(\zeta_0,\zeta_1)=\zeta_0^{d-d_0};
\end{gather*}
in particular,
\begin{gather}
\bigl[H_{\widetilde{A\circ f}}=0\bigr](\{\infty\})=d-d_0=d-\deg_{\mathcal{S}_G}(f).
\label{eq:holespoles}
\end{gather}
\begin{remark}
On the other hand, $H_{\tilde{f}}(\zeta_0,\zeta_1)=\sum_{j=0}^{d_0}\widetilde{(c_j/c_{d_0})}\cdot\zeta_0^{d-j}\zeta_1^j$
and $\phi_{\tilde{f}}\equiv\infty$.
\end{remark}
For each $j\in\{0,\ldots,d\}$, set
\begin{gather*}
C_j:=\frac{c_j}{c_{d_0}}\cdot c_{d_0}^{j-d_0}\in\mathcal{O}(\mathbb{D})[t^{-1}],
\quad\text{so that }C_{d_0}\equiv 1\text{ and that }C_j(0)
=0\text{ if }j<d_0.
\end{gather*}
Then setting
$(f_A)(w):=(A\circ f\circ A^{-1})(w)=c_{d_0}^{d_0}\bigl(w^{d_0}
+\sum_{j\in\{0,\ldots,d\}\setminus\{d_0\}}C_j
w^j\bigr)$, we have
\begin{multline}
(f_A)(U_{\overrightarrow{\mathcal{S}_G\infty}})
=(A\circ f)\bigl(U_{\overrightarrow{A^{-1}(\mathcal{S}_G)\infty}}\bigr)
\underset{(A\circ f)(\mathcal{S}_G)=\mathcal{S}_G}{=}
(A\circ f)\bigl(U_{\overrightarrow{f(\mathcal{S}_G)\infty}}\bigr)
=A\bigl(f(U_{\overrightarrow{f(\mathcal{S}_G)\infty}})\bigr)\\
\underset{\eqref{eq:superattbasin}\text{ for }n=1}{=}
A\bigl(U_{\overrightarrow{f^2(\mathcal{S}_G)\infty}}\bigr)
=U_{\overrightarrow{((A\circ f^2)(\mathcal{S}_G))\infty}}
\subsetneq\mathsf{P}^1(\mathbb{L}).\label{eq:image}
\end{multline}
\begin{claim}\label{th:coefficient}
There is $j>d_0$ such that $C_j(0)\neq 0$.
\end{claim}
\begin{proof}
Otherwise, we have
$(0<)d_0<d$ and
$|C_j|_r<1$ for every $j\in\{0,\ldots,d\}\setminus\{d_0\}$.
Then since $|c_{d_0}^{d_0}|_r=|c_{d_0}|_r^{d_0}>1$,
we have $H_{\widetilde{f_A}}(\zeta_0,\zeta_1)=\zeta_0^{d-d_0}\zeta_1^{d_0}$
(and $\phi_{\widetilde{f_A}}=\infty$),
so that
$[H_{\widetilde{f_A}}=0](\{\infty\})=d-d_0$. In particular,
we must have
$s_{\overrightarrow{\mathcal{S}_G\infty}}(f_A)=[H_{\widetilde{f_A}}=0](\{\infty\})=d-d_0>0$
(by Fact \ref{th:surplusalgclosed}),
and then
$(f_A)\bigl(U_{\overrightarrow{\mathcal{S}_G\infty}}\bigr)=\mathsf{P}^1(\mathbb{L})$
(by Fact \ref{th:surplus}).
This is impossible by \eqref{eq:image}.
\end{proof}
\subsection*{(c.2)}
Set
\begin{gather*}
\mu_E:=\mu_E^{(1)}\quad\text{on }\mathbb{P}^1(\mathbb{C}),
\quad\text{so that }
\mu^{(1)}=(\mu_0,\mu_E)\in(M^1(\mathbb{P}^1(\mathbb{C})))^2.
\end{gather*}
\begin{claim}\label{th:support}
$\operatorname{supp}\mu_E\subset\mathbb{P}^1(\mathbb{C})\setminus\{\infty\}$.
In particular, $\operatorname{supp}((\phi_{\widetilde{A\circ f}})^*\mu_E)\subset\mathbb{P}^1(\mathbb{C})\setminus\{\infty\}$.
\end{claim}
\begin{proof}
For every $t\in\mathbb{D}^*$, every $\ell>0$, and every $\epsilon\in\partial\mathbb{D}$,
we have
\begin{gather*}
f_t(\epsilon\ell c_{d_0}(t))
=\epsilon^{d_0}\ell^{d_0}(c_{d_0}(t))^{d_0+1}\cdot
\biggl\{\sum_{j\in\{0,\ldots,d\}\setminus\{d_0\}}\epsilon^{j-d_0}\frac{c_j(t)}{c_{d_0}(t)}(\ell c_{d_0}(t))^{j-d_0}+1\biggr\}.
\end{gather*}
Once Claim \ref{th:coefficient} is at our disposal,
there are $0<t_0\ll 1$ and $\ell_0\gg 1$ such that
for every $0<|t|\le t_0$ and every $\ell\ge\ell_0$, we also have
$|c_{d_0}(t)|>1$ and
\begin{gather*}
\sup_{|z|=\ell|c_{d_0}(t)|}\Biggl|\sum_{j<d_0}\frac{c_j(t)}{c_{d_0}(t)}z^{j-d_0}\Biggr|
<\biggl(\inf_{|z|=\ell|c_{d_0}(t)|}\Biggl|\sum_{j>d_0}\frac{c_j(t)}{c_{d_0}(t)}z^{j-d_0}\Biggr|\biggr)-3.
\end{gather*}
Hence for any $0<|t|\le t_0$ and any $\ell\ge\ell_0$, we have
$f_t(\{z\in\mathbb{C}:|z|=\ell\cdot|c_{d_0}(t)|\})\subset\{z\in\mathbb{C}:|z|>2\ell\cdot|c_{d_0}(t)|\}$, so that
\begin{gather*}
\operatorname{supp}(\mu_{f_t})\subset\bigl\{z\in\mathbb{C}:|z|\le\ell_0\cdot\bigl|c_{d_0}(t)\bigr|\bigr\}
\quad\text{for any }0<|t|\le t_0.
\end{gather*}
This yields $\operatorname{supp}\mu_E\subset\{z\in\mathbb{C}:|z|\le\ell_0\}\subset\mathbb{P}^1(\mathbb{C})\setminus\{\infty\}$ since
\begin{gather*}
\mu_E:=\mu_E^{(1)}=\lim_{j\to\infty}\bigl(A_{t_j}\bigr)_*\mu_{f_{t_j}}
=\lim_{j\to\infty}\bigl(c_{d_0}(t_j)\cdot\bigr)^*\mu_{f_{t_j}}
\end{gather*}
weakly on $\mathbb{P}^1(\mathbb{C})$. Now we are done also
recalling $\phi_{\widetilde{A\circ f}}\in\mathbb{C}[\zeta]$.
\end{proof}
\subsection*{(d)}
By $\omega_0=(\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu^{(1)}}\in\Delta_f$, there are $0\le s\le 1$ and
$0\le s'\le\min\{s\nu_f(U_{\overrightarrow{\mathcal{S}_G\infty}}),
(1-s)(1-\nu_f(U_{\overrightarrow{\mathcal{S}_G\infty}}))\}$ such that
\begin{gather*}
\begin{cases}
\omega_0(U_{\overrightarrow{v}})=s\nu_f(U_{\overrightarrow{v}})
\text{ for every }\overrightarrow{v}\in\bigl(T_{\mathcal{S}_G}\mathsf{P}^1\bigr)\setminus\{\overrightarrow{\mathcal{S}_G\infty}\},\\
\omega_0(\{\mathcal{S}_G\})=s',\text{ and}\\
\omega_0(U_{\overrightarrow{\mathcal{S}_Ga}})
=\bigl(s\nu_f(U_{\overrightarrow{\mathcal{S}_G\infty}})+(1-s)\bigr)-s'.
\end{cases}
\end{gather*}
Then (recalling
the definition \eqref{eq:omega} of $\omega_{\mu^{(1)}}$
in the case that $\Gamma_f\neq\Gamma_G$ and)
using $h_A=\infty$ and
the degenerating $f$-balanced property of $\mu^{(1)}$,
we compute
\begin{multline*}
\bigl(s\nu_f(U_{\overrightarrow{\mathcal{S}_G\infty}})+(1-s)\bigr)-s'
=\omega_0(U_{\overrightarrow{\mathcal{S}_G\infty}})
=\mu_0(\{h_A\})
=\frac{\bigl((\phi_{\widetilde{A\circ f}})^*\mu_E+[H_{\widetilde{A\circ f}}=0]\bigr)(\{\infty\})}{d}\\
\underset{\text{Claim \ref{th:support}}\,\&\,\eqref{eq:holespoles}}{=}
0+\frac{d-\deg_{\mathcal{S}_G}(f)}{d}=1-\frac{\deg_{\mathcal{S}_G}(f)}{d}
\underset{\eqref{eq:stationary}}{=}\nu_f(U_{\overrightarrow{\mathcal{S}_G\infty}}),
\end{multline*}
so that
$s'=(1-s)(1-\nu_f(U_{\overrightarrow{\mathcal{S}_G\infty}}))$.
\subsection*{(e)} Now also using the latter half
$\mu^{(1)}=(\mu_0,\mu_E)\in(M^1(\mathbb{P}^1(\mathbb{C}))^\dag)^2$
in \eqref{eq:reductionconstant} in Theorem \ref{th:balancedgeneral}, we have
$\omega_{\mu^{(1)}}\in M^1(\Gamma_f)^\dag$, so in particular
$\omega_0\in M^1(\mathbb{P}^1(\mathbb{C}))^\dag$ and in turn $s'=0$ and $s=1$
(also recalling that $\nu_f(U_{\operatorname{\overrightarrow{\mathcal{S}_G\infty}}})<1$
in this case (ii) in Theorem \ref{th:computation}).
Then also by \eqref{eq:converse} (and \eqref{eq:projectfactor}),
we still have the desired
$\mu_0=\omega_0=(\pi_{\mathsf{P}^1(\mathbb{L}),\Gamma_G})_*\nu_f$ in $M^1(\mathbb{P}^1(\mathbb{C}))^\dag=M^1(\Gamma_G)^\dag$.
\qed
\begin{remark}\label{th:remedy}
The steps from (c) to (d) in this proof of Theorem \ref{th:computation}
(i.e., \cite[Theorem B]{DF14}) are new.
The final assertion in \cite[Corollary 5.3]{DF14},
which was used in \cite[Proof of Theorem B]{DF14},
is shown in \cite{DF14} under the condition \eqref{eq:identically}.
\end{remark}
\section{Example}
\label{sec:example}
Pick a meromorphic family
$f_t(z)=z^2+t^{-1}z\in(\mathcal{O}(\mathbb{D})[t^{-1}])[z]\subset\mathbb{L}[z]$
of quadratic polynomials on $\mathbb{P}^1(\mathbb{C})$. Noting that
$f^{-1}(\infty)=\{\infty\}=E(f)$,
the case (ii) in Theorem \ref{th:computation} occurs,
setting $a=\infty\in\mathbb{P}^1(\mathbb{L})$.
By a direct computation, $z=-t^{-1}+1\in\mathbb{P}^1(\mathbb{L})$ is a (classical)
repelling fixed point of $f$
in $U_{\overrightarrow{\mathcal{S}_G\infty}}$, so that
$\nu_f(U_{\overrightarrow{\mathcal{S}_G\infty}})>0$.
Hence the condition \eqref{eq:identically} is not the case for this $f$.
\section{A complement of Proposition \ref{th:transfer}}
\label{sec:complement}
Let us continue to use the notations in Sections \ref{sec:quantizedbalanced}
and \ref{sec:direct}.
Let $f\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)\subset\mathbb{L}(z)$
be a meromorphic family
of rational functions on $\mathbb{P}^1(\mathbb{C})$ of degree $d>1$,
and suppose that $f^{-1}(\mathcal{S}_G)\neq\{\mathcal{S}_G\}$ in $\mathsf{P}^1(\mathbb{L})$.
Recall that $\Gamma_G:=\{\mathcal{S}_G\}$ and $\Gamma_{f^n}:=\{\mathcal{S}_G,f^n(\mathcal{S}_G)\}$
in $\mathsf{H}^1_{\mathrm{II}}(\mathbb{L})$ for every $n\in\mathbb{N}$ and that
$M^1(\Gamma_G)^{\dag}$ is identified with $M^1(\mathbb{P}^1(\mathbb{C}))^{\dag}$
under the bijection $S(\Gamma_G)\setminus\{\mathcal{S}_G\}
=T_{\mathcal{S}_G}\mathsf{P}^1(\mathbb{L})\cong\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C})$.
For every $n\in\mathbb{N}$, pick a meromorphic family
$A_n\in(\mathcal{O}(\mathbb{D})[t^{-1}])(z)\subset\mathbb{L}(z)$
of M\"obius transformations on $\mathbb{P}^1(\mathbb{C})$
such that $(A_n\circ f^n)(\mathcal{S}_G)=\mathcal{S}_G$ in $\mathsf{P}^1(\mathbb{L})$,
and set $A:=A_1$.
We note that for every $\mu=(\mu_C,\mu_E)\in (M^1(\mathbb{P}^1(\mathbb{C}))^\dag)^2$,
$\omega_{\mu}\in M^1(\Gamma_f)^\dag$. Conversely,
for every $\omega\in M^1(\Gamma_f)^{\dag}$,
recalling that $f(\mathcal{S}_G)=\mathcal{S}_G$ iff $\Gamma_f=\Gamma_G$,
we define
$\mu_\omega=(\mu_{\omega,C},\mu_{\omega,E})\in
(M^1(\Gamma_G)^{\dag})^2=(M^1(\mathbb{P}^1(\mathbb{C}))^{\dag})^2$ such that
when $\Gamma_f=\Gamma_G$,
\begin{gather*}
\begin{cases}
\mu_{\omega,C}:=(\pi_{\Gamma_f,\Gamma_G})_*\omega\in M^1(\Gamma_G)^\dag,\\
\mu_{\omega,E}:=\tilde{A}_*(\pi_{\Gamma_f,\Gamma_G})_*\omega\in M^1(\Gamma_G)^\dag
\end{cases}
\end{gather*}
and, when $\Gamma_f\neq\Gamma_G$,
noting that $\{f(\mathcal{S}_G)\}\subset\Gamma_f\subset\mathsf{H}^1_{\mathrm{II}}(\mathbb{L})$,
\begin{gather*}
\begin{cases}
\mu_{\omega,C}(\{\tilde{x}\})=\bigl((\pi_{\Gamma_f,\Gamma_G})_*\omega\bigr)\bigl(U_{\overrightarrow{\mathcal{S}_Gx}}\bigr)
&\text{for every }\tilde{x}\in\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C}),\\
\mu_{\omega,E}(\{\tilde{y}\})=\bigl((\pi_{\Gamma_f,\{f(\mathcal{S}_G)\}})_*\omega\bigr)\bigl(U_{(A^{-1})_*(\overrightarrow{\mathcal{S}_Gy})}\bigr)
&\text{for every }\tilde{y}\in\mathbb{P}^1(k_{\mathbb{L}})=\mathbb{P}^1(\mathbb{C});
\end{cases}
\end{gather*}
then $\mu_\omega$ satisfies the admissibility \eqref{eq:compatibility}
(by Lemma \ref{th:annulus} when $\Gamma_f\neq\Gamma_G$). Moreover,
\begin{gather}
\mu_{\omega_\mu}=\mu\quad\text{in }
(M^1(\Gamma_G)^{\dag})^2=(M^1(\mathbb{P}^1(\mathbb{C}))^{\dag})^2.\label{eq:bijective}
\end{gather}
For completeness, we include the following.
\begin{proposition}[cf.\ {\cite[Proposition 5.1 and Theorem 5.2]{DF14}}]\label{th:complement}
We have the bijection
\begin{multline*}
\bigl\{(\mu_C,\mu_E)\in(M^1(\mathbb{P}^1(\mathbb{C}))^\dag)^2:
\text{satisfying the admissibility \eqref{eq:compatibility} and }
(\widetilde{A\circ f})^*\mu_E=d\cdot\mu_C\bigr\}\ni\mu\\
\mapsto\omega_\mu\in\bigl\{\omega\in M^1(\Gamma_f):\text{satisfying }(f_{\Gamma_G,\Gamma_f})^*\omega_{\mu}=d\cdot(\pi_{\Gamma_f,\Gamma_G})_*\omega_{\mu}\bigr\},
\end{multline*}
the inverse of which is given by the map $\omega\mapsto\mu_\omega$.
This induces a canonical bijection
\begin{gather*}
\Delta_0^\dag\ni\mu_C\mapsto(\pi_{\Gamma_{f^n},\Gamma_G})_*\omega_{\bigl(\mu_C,\mu_E^{(n)}\bigr)}\in\Delta_f^\dag;
\end{gather*}
here we set $\Delta_0^\dag:=\Delta_0\cap M^1(\mathbb{P}^1(\mathbb{C}))^\dag$, where
\begin{multline*}
\Delta_0:=\bigl\{\mu_C\in M^1(\mathbb{P}^1(\mathbb{C})):
\text{for $($any$)$ }n\gg 1,\\
\text{ there is }\mu_E^{(n)}\in M^1(\mathbb{P}^1(\mathbb{C}))^\dag
\text{ such that }
(\widetilde{A_n\circ f^n})^*\mu_E^{(n)}=d\cdot\mu_C\}\subset M^1(\mathbb{P}^1(\mathbb{C})).
\end{multline*}
\end{proposition}
\begin{proof}
The former assertion follows from the proof of Proposition \ref{th:transfer}
and \eqref{eq:bijective}. Then the latter holds also by \eqref{eq:converse}.
\end{proof}
\begin{acknowledgement}
The author was partially supported by JSPS Grant-in-Aid
for Scientific Research (C), 19K03541,
Institut des Hautes \'Etudes Scientifiques, and
F\'ed\'eration de recherche math\'ematique des Hauts-de-France
(FR CNRS 2956). The author was a long term researcher
of RIMS, Kyoto University in the period of April 2019 - March 2020, and
also thanks the hospitality there.
\end{acknowledgement}
\def$'${$'$}
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Renewal of NWSMP Level 3 is a practical competency test which must be completed with an NWSMP Instructor. It includes a 30m non-contact tow of a conscious casualty and 15m contact tow of an unconscious casualty.
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"redpajama_set_name": "RedPajamaGithub"
} | 7,981 |
$LOAD_PATH.unshift(File.join(File.dirname(__FILE__), '..', 'lib'))
$LOAD_PATH.unshift(File.dirname(__FILE__))
require 'cromwell'
puts "You can't stop me with ^C but you can kill me. My pid is #{$$}."
Cromwell.protect("INT") {
sleep 10
}
puts "You're still here?" | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,384 |
\section*{Distance-regular graphs}
The vertex set $X$ of $\Gamma$ carries an association scheme
with $d$ classes, where the $i$-th relation is that of having
graph distance $i$ ($0 \le i \le d$).
All elements of the Bose-Mesner algebra ${\cal A}$
of this scheme are polynomials of degree at most $d$ in the matrix $A$.
In particular, $A_i$ is a polynomial in $A$ of degree $i$ ($0 \le i \le d$).
Let ${\cal A}$ have minimal idempotents $E_i$ ($0 \le i \le d$).
The column spaces of the $E_i$ are common eigenspaces of all
matrices in ${\cal A}$. Let $P_{ij}$ be the corresponding eigenvalue
of $A_j$, so that $A_jE_i = P_{ij}E_i$ ($0 \le i,j \le d$).
Now $A$ has eigenvalues $\theta_i = P_{i1}$ with multiplicities
$m_i = \rk E_i = \tr E_i$ ($0 \le i \le d$).
Index the eigenvalues such that $\theta_0 > \theta_1 > \cdots > \theta_d$.
\medskip
Standard facts about Sturm sequences give information on
the sign pattern of the matrix $P$.
\begin{Proposition}
Let $\Gamma$ be distance-regular, and $P$ its eigenvalue matrix.
Then row $i$ and column $i$ $(0\le i\le d)$ of $P$ both have $i$ sign changes.
In particular, row $d$ and column $d$ consist of nonzero numbers
that alternate in sign. \hfill$\Box$\medskip
\end{Proposition}
If $M \in {\cal A}$ and $0 \le i \le d$, then
$M \prod_{j \ne i} (A - \theta_j I) = c(M,i) E_i$
for some constant $c(M,i)$.
We apply this observation to $M = A_d$.
\begin{Proposition}
Let $\Gamma$ have intersection array
$\{b_0,b_1,\ldots,b_{d-1};\,c_1,c_2,\ldots,c_d\}$. Then for each $i$
$(0 \le i \le d)$ we have
\begin{equation}\label{eq1}
m_i A_d \prod_{j \ne i} (A - \theta_j I) = nb_0b_1\cdots b_{d-1} E_i .
\end{equation}
\end{Proposition}
\noindent{\bf Proof.}\quad
Both sides differ by a constant factor. Take traces on both sides.
Since $\tr A_d A^h = 0$ for $h < d$ it follows that
$\tr A_d \prod_{j \ne i} (A - \theta_j I) = \tr A_d A^d =
c_1c_2\cdots c_dnk_d = nb_0b_1\cdots b_{d-1}$.
Now the result follows from $\tr E_i = m_i$.
\hfill$\Box$\medskip
This can be said in an equivalent numerical way.
\begin{Corollary}
We have
~$m_i P_{id} \prod_{j \ne i} (\theta_i - \theta_j) = nb_0b_1\cdots b_{d-1}$~
for each $i$ $(0\le i\le d)$.
\end{Corollary}
\noindent{\bf Proof.}\quad Multiply \eqref{eq1} by $E_i$. \hfill$\Box$\medskip
We find a criterion for $A_d$ to have
two equal eigenvalues $P_{gd}$ and $P_{hd}$.
\begin{Proposition}\label{eq2}
For $g \ne h$,
$P_{gd} = P_{hd}$ if and only if
$\sum_i m_i \prod_{j \ne g,h} (\theta_i - \theta_j) = 0$.
\end{Proposition}
\noindent{\bf Proof.}\quad
$P_{gd} = P_{hd}$ if and only if
$m_g \prod_{j \ne g} (\theta_g - \theta_j) =
m_h \prod_{j \ne h} (\theta_h - \theta_j)$. \hfill$\Box$\medskip
For example, the Biggs-Smith graph has diameter $d=7$
and spectrum $3^1$, $\theta_1^9$, $2^{18}$, $\theta_3^{16}$,
$0^{17}$, $\theta_5^{16}$, $\theta_6^9$, $\theta_7^{16}$,
where $\theta_i$, $i=1,6$, satisfy $f(\theta) = \theta^2-\theta-4 = 0$
and $\theta_i$, $i=3,5,7$, satisfy $g(\theta) = \theta^3+3\theta^2-3 = 0$.
Now $P_{27} = P_{47}$ since
$$\sum_i m_i \prod_{j \ne 2,4} (\theta_i - \theta_j) =
\sum_{i=2,4} m_i (\theta_i-3)f(\theta_i)g(\theta_i) = 0.$$
One can generalize Proposition \ref{eq2}, and see:
\begin{Proposition} \label{multipleeq}
Let $H \subseteq \{0,\ldots,d\}$. Then all $P_{hd}$ for $h \in H$
take the same value if and only if
$\sum_i m_i \theta_i^e \prod_{j \notin H} (\theta_i - \theta_j) = 0$
for $0 \le e \le |H|-2$.
\end{Proposition}
\noindent{\bf Proof.}\quad
Induction on $|H|$. We just did the case $|H|=2$.
Let $|H| > 2$ and let $h,h' \in H$. We do the `only if' part.
By induction $\sum_i m_i \theta_i^e \prod_{j \notin H\setminus\{x\}}
(\theta_i - \theta_j) = 0$ holds for $0 \le e \le |H|-3$
and $x = h,h'$. Subtract these two formulas and divide by
$\theta_h - \theta_{h'}$ to get
$\sum_i m_i \theta_i^e \prod_{j \notin H}
(\theta_i - \theta_j) = 0$ for $0 \le e \le |H|-3$.
Then add the first formula for $x=h$ and $\theta_h$ times the last
formula, to get the same conclusion for $1 \le e \le |H|-2$.
The converse is clear.
\hfill$\Box$\medskip
Since the $P_{id}$ alternate in sign, the largest sets $H$ that can
occur here are $\{0,2,\ldots,d\}$ and $\{1,3,\ldots,d-1\}$ for $d=2e$
and $\{0,2,\ldots,d-1\}$ and $\{1,3,\ldots,d\}$ for $d=2e+1$.
We investigate such sets below
(see `the half-antipodal case').
\medskip
For small $d$ one can use identities like
$\sum_i m_i = n$, $\sum_i m_i \theta_i = 0$,
$\sum_i m_i \theta_i^2 = nk$,
$\sum_i m_i \theta_i^3 = nk\lambda$ (where $k=b_0$,
and $\lambda = k-1-b_1$) to simplify the condition
of Proposition \ref{eq2}.
Let us do some examples. Note that $\theta_0 = k$.
\subsection*{The case \boldmath$d=3$}
For $d=3$ we find that $P_{13} = P_{33}$ if and only if
$\sum_i m_i (\theta_i-\theta_0)(\theta_i-\theta_2) = 0$,
i.e., if and only if $nk+n\theta_0\theta_2 = 0$,
i.e., if and only if $\theta_2 = -1$
(cf.~\cite{BCN}, 4.2.17).
\subsection*{The case \boldmath$d=4$}
For $d=4$ we find that $P_{14} = P_{34}$ if and only if
$\sum_i m_i (\theta_i-\theta_0)(\theta_i-\theta_2)(\theta_i-\theta_4) = 0$,
i.e., if and only if
$nk\lambda - nk(\theta_0+\theta_2+\theta_4)-n\theta_0\theta_2\theta_4 = 0$.
This happens if and only if
$(\theta_2+1)(\theta_4+1) = -b_1$.
Of course $P_{24} = P_{44}$ will follow from
$(\theta_1+1)(\theta_3+1) = -b_1$.
\medskip\noindent
A generalized octagon ${\rm GO}(s,t)$ has eigenvalues
$$
(\theta_i)_i = (s(t+1),~ s-1+\sqrt{2st},~ s-1,~
s-1-\sqrt{2st},~ -t-1)
$$
and $b_1 = st$.
Since $(\theta_2+1)(\theta_4+1) = -b_1$ it follows that
$P_{14} = P_{34}$, so that $\Gamma_4$ does not have
more than 4 distinct eigenvalues.
\medskip\noindent
A dual polar graph ${}^2D_5(q)$ has eigenvalues
$$(\theta_i)_i = (q^5{+}q^4{+}q^3{+}q^2,~
q^4{+}q^3{+}q^2{-}1,~ q^3{+}q^2{-}q{-}1,~ {-}q{-}1,~ {-}q^3{-}q^2{-}q{-}1)$$
and $b_1 = q^3 (q^2+q+1)$.
Since $(\theta_1+1)(\theta_3+1) = -b_1$ it follows that
$P_{24} = P_{44}$.
\medskip\noindent
Some further examples:
\medskip\noindent
\begin{tabular}{@{~}l@{~}c@{~~}l@{~~}l@{~~}l@{}}
name & $n$ & intersection array & spectrum & equality \\
\hline
Coxeter graph & 28 & $\{3,2,2,1;\,1,1,1,2\}$ &
$3^1$ $2^8$ $a^6$ $(-1)^7$ $b^6$ &
$P_{14} = P_{34}$ \\
&&&$a,b=-1\pm\sqrt{2}$ \\
Odd graph $O_5$ & 126 & $\{5,4,4,3;\,1,1,2,2\}$ &
$5^1$ $3^{27}$ $1^{42}$ $(-2)^{48}$ $(-4)^8$ &
$P_{24} = P_{44}$ \\
$M_{22}$ graph & 330 & $\{7,6,4,4;\,1,1,1,6\}$ &
$7^1$ $4^{55}$ $1^{154}$ $(-3)^{99}$ $(-4)^{21}$ &
$P_{14} = P_{34}$ \\
Unital graph & 280 & $\{9,8,6,3;\,1,1,3,8\}$ &
$9^1$ $4^{64}$ $1^{105}$ $(-3)^{90}$ $(-5)^{20}$ &
$P_{14} = P_{34}$ \\
\end{tabular}
\subsection*{The case \boldmath$d=4$ with strongly regular $\Gamma_4$}
One may wonder whether it is possible that $\Gamma_4$ is
strongly regular. This would require $p^1_{44} = p^2_{44} = p^3_{44}$.
Or, equivalently, that $\Gamma_4$ has only two eigenvalues
with eigenvector other than the all-1 vector. Since the
values $P_{i4}$ alternate in sign, this would mean
$P_{14} = P_{34}$ and $P_{24} = P_{44}$.
\begin{Proposition}
Let $\Gamma$ be a distance-regular graph of diameter $4$.
The following assertions are equivalent.
(i) $\Gamma_4$ is strongly regular.
(ii) $b_3 = a_4+1$ and $b_1=b_3c_3$.
(iii) $(\theta_1+1)(\theta_3+1) = (\theta_2+1)(\theta_4+1) = -b_1$.
\end{Proposition}
\medskip\noindent
\noindent{\bf Proof.}\quad
(i)-(ii)
A boring computation (using \cite{BCN}, 4.1.7) shows that
$p^1_{44} = p^2_{44}$ is equivalent to $b_3 = a_4+1$, and that
if this holds $p^1_{44} = p^3_{44}$ is equivalent to $b_1=b_3c_3$.
(i)-(iii)
$\Gamma_4$ will be strongly regular if and only if
$P_{14} = P_{34}$ and $P_{24} = P_{44}$.
We saw that this is equivalent to
$(\theta_2+1)(\theta_4+1) = -b_1$ and $(\theta_1+1)(\theta_3+1) = -b_1$.
\hfill$\Box$\medskip
\noindent
The fact that (i) implies the first equality in (iii) was proved in
\cite{F01} as a consequence of another characterization of (i)
in terms of the spectrum only.
More generally, a quasi-spectral characterization of those connected
regular graphs (with $d+1$ distinct eigenvalues) which are distance-regular,
and with the distance-$d$ graph being strongly regular, is given in
\cite[Th. 2.2]{F00}.
\medskip
No nonantipodal examples are known, but the infeasible array
$\{12,8,6,4;\,1,\discretionary{}{}{} 1,\discretionary{}{}{} 2,9\}$ with spectrum
$12^1$ $7^{56}$ $3^{140}$ $(-2)^{160}$ $(-3)^{168}$
(cf.~\cite{BCN},\,p.\,410) would~have been an example
(and there are several open candidate arrays, such as
$\{21,20,\discretionary{}{}{} 14,\discretionary{}{}{} 10;\,1,1,2,12\}$,
$\{24,20,20,10;\,1,1,2,15\}$, and
$\{66,65,63,13;\,1,1,5,54\}$).
\medskip
If $\Gamma$ is antipodal, then $\Gamma_4$ is a union of cliques
(and hence strongly regular). This holds precisely when
$\theta_1 + \theta_3 = \lambda$ and $\theta_1\theta_3 = -k$
and $(\theta_2+1)(\theta_4+1) = -b_1$.
(Indeed, $\theta_1,\theta_3$ are the two roots of
$\theta^2-\lambda\theta-k = 0$ by \cite{BCN}, 4.2.5.). \\
There are many examples, e.g.
\medskip\noindent
\begin{tabular}{@{~}l@{~}c@{~~}l@{~~}l@{}}
name & $n$ & intersection array & spectrum \\
\hline
Wells graph & 32 & $\{5,4,1,1;\,1,1,4,5\}$ &
$5^1$ $\sqrt{5}{}^8$ $1^{10}$ $(-\sqrt{5})^8$ $(-3)^5$ \\
3.Sym(6).2 graph & 45 & $\{6,4,2,1;\,1,1,4,6\}$ &
$6^1$ $3^{12}$ $1^9$ $(-2)^{18}$ $(-3)^5$ \\
Locally Petersen & 63 & $\{10,6,4,1;\,1,2,6,10\}$ &
$10^1$ $5^{12}$ $1^{14}$ $(-2)^{30}$ $(-4)^6$
\end{tabular}
\medskip
If $\Gamma$ is bipartite, then $\Gamma_4$ is disconnected,
so if it is strongly regular, it is a union of cliques
and $\Gamma$ is antipodal. In this case its spectrum is
$$\{k^1,~\sqrt{k}{\,}^{n/2-k},~
0^{2k-2},~ (-\sqrt{k})^{n/2-k},~ (-k)^1 \}.$$
Such graphs are precisely the incidence graphs of
symmetric $(m,\mu)$-nets, where $m = k/\mu$ (\cite{BCN}, p. 425).
\subsection*{The case \boldmath$d=5$}
As before, and also using
$\sum_i m_i \theta_i^4 = nk(k+\lambda^2+b_1\mu)$
(where $\mu = c_2$) we find for $\{f,g,h,i,j\} = \{1,2,3,4,5\}$
that $P_{fd} = P_{gd}$ if and only if
$$(\theta_h+1)(\theta_i+1)(\theta_j+1) + b_1(\theta_h+\theta_i+\theta_j) =
b_1(\lambda-\mu-1).$$
In case $\theta_i = -1$,
this says that $\theta_h+\theta_j = \lambda-\mu$.
For example, the Odd graph $O_6$ has $\lambda=0$, $\mu=1$,
and eigenvalues 6, 4, 2, $-1$, $-3$, $-5$.
It follows that $P_{15} = P_{55}$ and $P_{25} = P_{45}$.
Similarly, the folded 11-cube has $\lambda=0$, $\mu=2$,
and eigenvalues 11, 7, 3, $-1$, $-5$, $-9$.
It follows that $P_{15} = P_{55}$ and $P_{25} = P_{45}$.
An example without eigenvalue $-1$ is provided by the
folded Johnson graph $\bar{J}(20,10)$. It has $\lambda=18$,
$\mu=4$ and eigenvalues 100, 62, 32, 10, $-4$, $-10$.
We see that $P_{35} = P_{55}$.
\medskip\noindent
Combining two of the above conditions, we see that
$P_{15} = P_{35} = P_{55}$ if and only if
$(\theta_2+1)(\theta_4+1) = -b_1$ and $\theta_2 + \theta_4 = \lambda - \mu$
(and hence $\theta_2\theta_4 = \mu-k$).
\\ Now $b_3+b_4+c_4+c_5 = 2k+\mu-\lambda$ and
$b_3b_4+b_3c_5+c_4c_5 = kb_1+k\mu+\mu$.
\subsection*{Generalized 12-gons}
A generalized 12-gon of order $(q,1)$ (the line graph
of the bipartite point-line incidence graph
of a generalized hexagon of order $(q,q)$) has diameter 6,
and its $P$ matrix is given by
{\small
$$
P \!=\! \left(\!\begin{array}{ccccccc}
1 & 2q & 2q^2 & 2q^3 & 2q^4 & 2q^5 & q^6 \\
1 & q{-}1{+}a & q{+}(q{-}1)a & 2q(q{-}1) & {-}q^2{+}q(q{-}1)a & q^2(q{-}1){-}q^2a & {-}q^3 \\
1 & q{-}1{+}b & {-}q{+}(q{-}1)b & {-}2qb & {-}q^2{-}q(q{-}1)b & {-}q^2(q{-}1){+}q^2b & q^3 \\
1 & q{-}1 & {-}2q & {-}q(q{-}1) & 2q^2 & q^2(q{-}1) & {-}q^3 \\
1 & q{-}1{-}b & {-}q{-}(q{-}1)b & 2qb & {-}q^2{+}q(q{-}1)b & {-}q^2(q{-}1){-}q^2b & q^3 \\
1 & q{-}1{-}a & q{-}(q{-}1)a & 2q(q{-}1) & {-}q^2{-}q(q{-}1)a & q^2(q{-}1){+}q^2a & {-}q^3 \\
1 & {-}2 & 2 & {-}2 & 2 & {-}2 & 1
\end{array}\!\right)
$$
}\noindent
where $a = \sqrt{3q}$ and $b = \sqrt{q}$.
We see that $P_{16} = P_{36} = P_{56}$ and $P_{26} = P_{46}$.
\medskip
Its dual is a generalized 12-gon of order $(1,q)$, and is bipartite.
The $P$ matrix is given by
{\small
$$
P \!=\! \left(\!\begin{array}{ccccccc}
1 & q+1 & q(q+1) & q^2(q+1) & q^3(q+1) & q^4(q+1) & q^5 \\
1 & a & 2q-1 & (q-1)a & q(q-2) & -qa & -q^2 \\
1 & b & -1 & -(q+1)b & -q^2 & qb & q^2 \\
1 & 0 & -q-1 & 0 & q(q+1) & 0 & -q^2 \\
1 & -b & -1 & (q+1)b & -q^2 & -qb & q^2 \\
1 & -a & 2q-1 & -(q-1)a & q(q-2) & qa & -q^2 \\
1 & -q-1 & q(q+1) & -q^2(q+1) & q^3(q+1) & -q^4(q+1) & q^5
\end{array}\!\right)
$$
}\noindent
where $a = \sqrt{3q}$ and $b = \sqrt{q}$.
We see that $P_{16} = P_{36} = P_{56}$ and $P_{26} = P_{46}$.\\
As expected (cf.~\cite{B}), the squares of all $P_{id}$ are powers of $q$.
\subsection*{Dual polar graphs}
According to \cite{B}, dual polar graphs of diameter $d$
satisfy $$P_{id} = (-1)^i q^{d(d-1)/2+de-i(d+e-i)}$$
where $e$ has the same meaning as in \cite{BCN}, 9.4.1.
It follows that $P_{hd} = P_{id}$ when $d+e$ is even
and $h+i = d+e$.
\medskip\noindent
For the dual polar graphs $B_d(q)$ and $C_d(q)$ we have $e=1$,
and the condition becomes $h+i = d+1$ where $d$ is odd.
Below we will see this in a different way.
For the dual polar graph $D_d(q)$ we have $e = 0$,
and the condition becomes $h+i = d$ where $d$ is even.
Not surprising, since this graph is bipartite.
For the dual polar graph ${}^2D_{d+1}(q)$ we have $e=2$,
and the condition becomes $h+i = d+2$ where $d$ is even.
(We saw the case $d=4$ above.)
Finally, $h+i = d+e$ is impossible when $e$ is not integral.
\subsection*{Distance-regular distance 1-or-2 graph}
The distance 1-or-2 graph $\Delta = \Gamma_1 \cup \Gamma_2$ of $\Gamma$
(with adjacency matrix $A_1+A_2$) is distance-regular
if and only if $b_{i-1}+b_i+c_i+c_{i+1} = 2k+\mu-\lambda$
for $1 \le i \le d-1$, cf.~\cite{BCN}, 4.2.18.
\begin{Proposition}
Suppose that $\Gamma_1 \cup \Gamma_2$ is distance-regular.
Then for $1 \le i \le d$ we have
$P_{d+1-i,d} = P_{id}$ if $d$ is odd,
and $(\theta_{d+1-i}+1)P_{i,d} =
(\theta_i+1)P_{d+1-i,d}$ if $d$ is even.
If $i \ne (d+1)/2$ then $\theta_{d+1-i} = \lambda-\mu-\theta_i$.
If $d$ is odd, then $\theta_{(d+1)/2} = -1$.
\end{Proposition}
\noindent{\bf Proof.}\quad
For each eigenvalue $\theta$ of $\Gamma$, there is
an eigenvalue $(\theta^2 + (\mu-\lambda)\theta - k)/\mu$ of $\Delta$.
If $d$ is odd, then $\Delta$ has diameter $(d+1)/2$,
and $\Gamma$ has an eigenvalue $-1$,
and for each eigenvalue $\theta \ne k, -1$ of $\Gamma$
also $\lambda-\mu-\theta$ is an eigenvalue.
Now $\Delta_{(d+1)/2} = \Gamma_d$, and $P_{i'd} = P_{id}$
if $i,i'$ belong to the same eigenspace of $\Delta$.
Since the numbers $P_{id}$ alternate, the eigenvalue $-1$
must be the middle one (not considering $\theta_0$), and we see that
$P_{d+1-i,d} = P_{id}$ for $1 \le i \le d$.
If $d$ is even, then $\Delta$ has diameter $d/2$, and for each
eigenvalue $\theta \ne k$ of $\Gamma$ also $\lambda-\mu-\theta$
is an eigenvalue. Now $\Delta_{d/2} = \Gamma_{d-1} \cup \Gamma_d$.
Since $AA_d = b_{d-1}A_{d-1} + a_dA_d$ and our parameter conditions
imply $b_{d-1}+c_d = b_1+\mu$
(so that $b_{d-1}(P_{i,d-1} + P_{i,d}) = (\theta_i - a_d + b_{d-1})P_{i,d} =
(\theta_i+\mu-\lambda-1)P_{i,d}$), the equalities
$P_{i,d-1} + P_{i,d} = P_{d+1-i,d-1} + P_{d+1-i,d}$ $(1 \le i \le d)$
and $\theta_i + \theta_{d+1-i} = \lambda-\mu$
imply $(\theta_{d+1-i}+1)P_{i,d} = (\theta_i+1)P_{d+1-i,d}$.
\hfill$\Box$\medskip
\medskip
The Odd graph $O_{d+1}$ on $\binom{2d+1}{d}$ vertices
has diameter $d$ and eigenvalues $\theta_i = d+1-2i$ for $i < (d+1)/2$,
and $\theta_i = d-2i$ for $i \ge (d+1)/2$.
Since its distance 1-or-2 graph is distance-regular,
we have $P_{d+1-i,d} = P_{id}$ for odd $d$ and $1 \le i \le d$.
The folded $(2d+1)$-cube on $2^{2d}$ vertices
has diameter $d$ and eigenvalues
$\theta_i = 2d+1-4i$ with multiplicities
$m_i = \binom{2d+1}{2i}$ ($0 \le i \le d$).
Since its distance 1-or-2 graph is distance-regular,
it satisfies $P_{d+1-i,d} = (-1)^{d+1} P_{id}$ for $1 \le i \le d$.
(Note that $\theta_{d+1-i}+1 = -(\theta_i+1)$ since $\mu-\lambda = 2$.)
The dual polar graphs $B_d(q)$ and $C_d(q)$ have diameter $d$
and eigenvalues $\theta_i = (q^{d-i+1}-q^i)/(q-1) - 1$.
Since their distance 1-or-2 graphs are distance-regular,
they satisfy $P_{d+1-i,d} = (-1)^{d+1} P_{id}$ for $1 \le i \le d$.
\medskip
The fact that $-1$ must be the middle eigenvalue for odd $d$,
implies
that $\theta_{(d-1)/2}> \lambda-\mu+1$,
so that there is no eigenvalue $\xi$
with $-1 < \xi < \lambda-\mu+1$.
\subsection*{The bipartite case}
If $\Gamma$ is bipartite, then $\theta_{d-i} = -\theta_i$,
and $P_{d-i,j} = (-1)^j P_{i,j}$ ($0 \le i,j \le d$).
In particular, if $d$ is even, then $P_{d-i,d} = P_{id}$
and $\Gamma_d$ is disconnected.
\subsection*{The antipodal case}
The graph $\Gamma$ is antipodal when having distance $d$ is an
equivalence relation, i.e., when $\Gamma_d$ is a union of cliques.
The graph is called an antipodal $r$-cover, when these cliques
are $r$-cliques. Now $r = k_d+1$, and
$P_{id}$ alternates between $k_d$ and $-1$.
For example, the ternary Golay code graph (of diameter 5)
with intersection array
$\{22,20,18,2,1;\,1,2,9,20,22\}$ has spectrum
$22^1$ $7^{132}$ $4^{132}$ $(-2)^{330}$ $(-5)^{110}$ $(-11)^{24}$
and satisfies $P_{05} = P_{25} = P_{45} = 2$,
$P_{15} = P_{35} = P_{55} = -1$.
For an antipodal distance-regular graph $\Gamma$, the folded graph has
eigenvalues $\theta_0,\theta_2,\ldots,\theta_{2e}$ where $e = [d/2]$.
In Theorem \ref{oddhalfatp} below we show for odd $d$ that this
already follows from $P_{1d} = P_{3d} = \cdots = P_{dd}$.
\begin{Proposition}
\label{0ddd}
If $P_{0d} = P_{id}$ then $i$ is even.
Let $i > 0$ be even. Then $P_{0d} = P_{id}$ if and only
$\Gamma$ is antipodal, or $i=d$ and $\Gamma$ is bipartite.
\end{Proposition}
\noindent{\bf Proof.}\quad
Since the $P_{id}$ alternate in sign, $P_{0d} = P_{id}$ implies
that $i$ is even.
If $\Gamma$ is bipartite, then $P_{dd} = (-1)^d P_{0d}$.
If $\Gamma$ is antipodal, then $P_{id} = P_{0d}$ for all even $i$.
That shows the `if' part. Conversely, if $P_{0d} = P_{id}$,
then the valency of $\Gamma_d$ is an eigenvalue
of multiplicity larger than 1, so that $\Gamma_d$ is disconnected,
and hence $\Gamma$ is imprimitive and therefore antipodal or bipartite.
If $\Gamma$ is bipartite but not antipodal, then its halved graphs
are primitive and $|P_{id}| < P_{0d}$ for $0 < i < d$
(cf.~\cite{BCN}, pp.~140--141).
\hfill$\Box$\medskip
\subsection*{The half-antipodal case}
Given an array $\{b_0,\ldots,b_{d-1};\,c_1,\ldots,c_d\}$
of positive real numbers, define the polynomials $p_i(x)$ for
$-1 \le i \le d+1$ by $p_{-1}(x)=0$, $p_0(x)=1$,
$(x-a_i)p_i(x) = b_{i-1}p_{i-1}(x)+c_{i+1}p_{i+1}(x)$
($0 \le i \le d$), where $a_i = b_0-b_i-c_i$ and $c_{d+1}$
is some arbitrary positive number.
The eigenvalues of the array are by definition the zeros of
$p_{d+1}(x)$, and do not depend on the choice of $c_{d+1}$.
Each $p_i(x)$ has degree $i$, and, by the theory of Sturm sequences,
each $p_i(x)$ has $i$ distinct real zeros, where the zeros of
$p_{i+1}(x)$ interlace those of $p_i(x)$.
If $\{b_0,\ldots,b_{d-1};\,c_1,\ldots,c_d\}$ is the intersection array
of a distance-regular graph $\Gamma$, then the eigenvalues of the array
are the eigenvalues of (the adjacency matrix of) $\Gamma$.
Let $L$ be the tridiagonal matrix
$$
L=\left(\begin{array}{cccccc}
a_0 & b_0 \\
c_1 & a_1 & b_1 \\
& c_2 & . & . \\
& & . & . & . \\
& & & . & . & b_{d-1} \\
& & & & c_d & a_d
\end{array}\right).
$$
The eigenvalues of the array $\{b_0,\ldots,b_{d-1};\,c_1,\ldots,c_d\}$
are the eigenvalues of the matrix $L$.
\begin{Theorem} \label{oddhalfatp}
Let $\Gamma$ be a distance-regular graph with odd diameter $d = 2e+1$
and intersection array $\{b_0,\ldots,b_{d-1};\,c_1,\ldots,c_d\}$.
Then $P_{1d} = P_{3d} = \cdots = P_{dd}$ if and only if
the $\theta_j$ with $j=0,2,4,\ldots,2e$ are the eigenvalues of the array
$\{b_0,\ldots,b_{e-1};\,c_1,\ldots,c_e\}$.
\end{Theorem}
\noindent{\bf Proof.}\quad
Let $H = \{1,3,\ldots,d\}$, so that $|H| = e+1$.
By Proposition \ref{multipleeq}, $P_{1d} = P_{3d} = \cdots = P_{dd}$
if and only if
$\sum_i m_i \theta_i^s \prod_{j \notin H} (\theta_i - \theta_j) = 0$
for $0 \le s \le e-1$.
Let~$E = \{0,2,\ldots,2e\}$, so that $|E| = e+1$.
Then this condition is equivalent to
$$
\tr A^s \prod_{j \in E} (A-\theta_j I) = 0\qquad (0 \le s \le e-1).
$$
This says that the expansion of $\prod_{j \in E} (A-\theta_j I)$
in terms of the $A_i$ does not contain $A_s$ for $0 \le s \le e-1$,
hence is equivalent to
$\prod_{j \in E} (A-\theta_j I) = aA_e + bA_{e+1}$ for certain constants $a,b$.
Since $0 \in E$, we find that $ak_e+bk_{e+1} = 0$, and the condition is
equivalent to $(A_e/k_e - A_{e+1}/k_{e+1})E_j = 0$ for all $j \in E$.
An eigenvalue $\theta$ of $\Gamma$ defines a right eigenvector $u$
(known as the `standard sequence') by $Lu = \theta u$.
It follows that $\theta$ will be an eigenvalue of the array
$\{b_0,\ldots,b_{e-1};\,c_1,\ldots,c_e\}$ precisely when $u_e = u_{e+1}$.
Up to scaling, the $u_i$ belonging to $\theta_j$ are the $Q_{ij}$
(that is, the columns of $Q$ are eigenvectors of $L$).
So, $\theta_j$ is an eigenvalue of $\{b_0,\ldots,b_{e-1};\,c_1,\ldots,c_e\}$
for all $j \in E$ precisely when $Q_{ej} = Q_{e+1,j}$ for all $j \in E$.
Since $k_iQ_{ij} = m_jP_{ji}$, this holds if and only if
$P_{je}/k_e = P_{j,e+1}/k_{e+1}$, i.e., if and only if
$(A_e/k_e - A_{e+1}/k_{e+1})E_j = 0$ for all $j \in E$.
\hfill$\Box$\medskip
For example, if $d=3$ one has $P_{13} = P_{33}$ if and only if
$\theta_0,\theta_2$ are the eigenvalues $k$, $-1$ of
the array $\{k;\,1\}$. And if $d=5$ one has
$P_{15} = P_{35} = P_{55}$ if and only if
$\theta_0, \theta_2, \theta_4$ are the eigenvalues of the array
$\{k,b_1;\,1,c_2\}$.
\medskip\noindent
The case of even $d$ is slightly more complicated.
\begin{Theorem}
Let $\Gamma$ be a distance-regular graph with even diameter $d = 2e$
and intersection array $\{b_0,\ldots,b_{d-1};\,c_1,\ldots,c_d\}$.
Then $P_{1d} = P_{3d} = \cdots = P_{d-1,d}$ if and only if
the $\theta_j$ with $j=0,2,4,\ldots,2e$ are the eigenvalues of the array
$\{b_0,\ldots,b_{e-1};\,c_1,\ldots,c_{e-1},c_e+zb_e\}$ for some
real number $z$ with $0 < z \le 1$, uniquely determined by
$\sum_{i=0}^e \theta_{2i} = \sum_{i=0}^e a_i + (1-z)b_e$.
If \,$\Gamma$ is antipodal or bipartite, then $z=1$.
\end{Theorem}
\noindent{\bf Proof.}\quad
Let $E = \{0,2,\ldots,d\}$.
As before we see that $P_{1d} = P_{3d} = \cdots = P_{d-1,d}$
is equivalent to the condition that
$\prod_{j \in E} (A-\theta_j I) = aA_{e-1} + bA_e + cA_{e+1}$
for certain constants $a,b,c$.
Comparing coefficients of $A^{e+1}$ we see that $c > 0$.
With $j=0$ we see that $ak_{e-1}+bk_e+ck_{e+1} = 0$.
Take
$$
z = -\frac{ak_{e-1}}{ck_{e+1}} = 1 + \frac{bk_e}{ck_{e+1}}.$$
Then $aP_{j,e-1}+bP_{je}+cP_{j,e+1} = 0$ for $j \in E$ gives
$$
z\left(\frac{P_{j,e-1}}{k_{e-1}} - \frac{P_{je}}{k_e}\right)
= \frac{P_{j,e+1}}{k_{e+1}} - \frac{P_{je}}{k_e}.
$$
On the other hand, if $\theta = \theta_j$ for some $j \in E$, then
$\theta$ is an eigenvalue of the array
$\{b_0,\ldots,b_{e-1};\,c_1,\ldots,c_{e-1},\discretionary{}{}{} c_e{+}zb_e\}$
precisely when $c_eu_{e-1}+(k{-}b_e{-}c_e)u_e+b_eu_{e+1} =
(c_e{+}zb_e)u_{e-1}+(k{-}c_e{-}zb_e)u_e$,
i.e., when $z(u_{e-1}-u_e) = u_{e+1}-u_e$.
Since (up to a constant factor) $u_i = P_{ji}/k_i$, this is equivalent
to the condition above.
Noting that $d \in E$, we can apply the above to $\theta = \theta_d$.
Since the bottom row of $P$ has $d$ sign changes, it follows that
the sequence $u_i$ has $d$ sign changes.
In particular, the $u_i$ are nonzero.
Now $u_{e-1}-u_e$ and $u_{e+1}-u_e$ have the same sign,
and it follows that $z > 0$.
If $\Gamma$ is an antipodal $r$-cover of diameter $d = 2e$, then
$c_e+zb_e = rc_e$ and $z = 1$ (and $P_{je} = 0$ for all odd $j$).
If $\Gamma$ is bipartite, then $\theta_i + \theta_{d-i} = 0$
for all $i$, so $\sum_{j \in E} \theta_j = 0$, so our tridiagonal
matrix (the analog of $L$) has trace
$0 = a_1+\cdots+a_e+(1-z)b_e = (1-z)b_e$, so that $z = 1$.
It remains to show that $z \le 1$. We use $z(u_{e-1}-u_e) = u_{e+1}-u_e$
to conclude that $\theta$ is an eigenvalue of both
$$
\left(\begin{array}{c@{~~}c@{~~}c@{~~}c@{~~}c}
0 & k \\
c_1 & a_1 & b_1 \\
& . & . & . \\
& & c_{e-1} & a_{e-1} & b_{e-1} \\
& & & p & k-p
\end{array}\right)
~{\rm and}~
\left(\begin{array}{c@{~~}c@{~~}c@{~~}c@{~~}c}
k-q & q \\
c_{e+1} & a_{e+1} & b_{e+1} \\
& . & . & . \\
& & c_{d-1} & a_{d-1} & b_{d-1} \\
& & & c_d & a_d
\end{array}\right),
$$
where $p = c_e+zb_e$ and $q = b_e+z^{-1}c_e$.
Since this holds for each $\theta=\theta_j$ for $j \in E$,
this accounts for all eigenvalues of these two matrices, and
$\sum_{j \in E} \theta_j = a_1+\cdots+a_e+(1-z)b_e =
a_e+\cdots+a_d+(1-z^{-1})c_e$.
Since $p^{e+1}_{ed} \ge 0$ it follows that
$a_1+\cdots+a_{e-1} \le a_{e+1}+\cdots+a_d$ (cf.~\cite{BCN}, 4.1.7),
and therefore $(1-z)b_e \ge (1-z^{-1})c_e$.
It follows that $z \le 1$.
\hfill$\Box$\medskip
\medskip
\medskip
Nonantipodal, nonbipartite examples:
\medskip\noindent
\begin{tabular}{@{\,}llll@{\,}}
name & array & half array & $z$ \\
\hline
Coxeter & $\{3,2,2,1;\,1,1,1,2\}$ & $\{3,2;\,1,2\}$ & $1/2$ \\
$M_{22}$ & $\{7,6,4,4;\,1,1,1,6\}$ & $\{7,6;\,1,3\}$ & $1/2$ \\
$P\Gamma L(3,4).2$ & $\{9,8,6,3;\,1,1,3,8\}$ & $\{9,8;\,1,4\}$ & $1/2$ \\
gen.\,8-gon & $\{s(t+1),st,st,st;\,1,1,1,t+1\}$
& $\{s(t+1),st;\,1,t+1\}$ & $1/s$ \\
gen.\,12-gon & $\{2q,q,q,q,q,q;\,1,1,1,1,1,2\}$
& $\{2q,q,q;\,1,1,2\}$ & $1/q$
\end{tabular}
\medskip
Concerning the value of $z$, note that both $zb_e$ and $z^{-1}c_e$
are algebraic integers.
\bigskip
If $d=4$, the case $z = 1$ can be classified.
\begin{Proposition}
Let $\Gamma$ be a distance-regular graph with even diameter $d = 2e$
such that $\theta_j$ with $j=0,2,4,\ldots,2e$ are the eigenvalues
of the array $\{b_0,\ldots,b_{e-1};\discretionary{}{}{}\,c_1,\ldots,c_{e-1},b_e+c_e\}$.
Then $\Gamma$ satisfies $p^d_{e,e+1} = 0$.
If moreover $d \le 4$, then $\Gamma$ is antipodal or bipartite.
\end{Proposition}
\noindent{\bf Proof.}\quad
We have equality in the inequality $a_1+\cdots+a_{e-1} \le
a_{e+1}+\cdots+a_d$, so that $p^{e+1}_{ed} = 0$ by \cite{BCN}, 4.1.7.
The case $d=2$ is trivial. Suppose \mbox{$d=4$}. Then $p^3_{24} = 0$.
Let $d(x,z) = 4$, and consider neighbors $y,w$ of $z$,
where $d(x,y) = 3$ and $d(x,w) = 4$. Then $d(y,w) \ne 2$ since
there are no 2-3-4 triangles, so $d(y,w) = 1$.
If $a_4 \ne 0$ then there exist such vertices $w$, and we
find that the neighborhood $\Gamma(z)$ of $z$ in $\Gamma$
is not coconnected (its complement is not connected),
contradicting \cite{BCN}, 1.1.7. Hence $a_4 = 0$, and $a_3 = a_1$.
If $b_3 > 1$, then let $d(x,y)=3$, $y \sim z,z'$ with $d(x,z)=d(x,z')=4$.
Let $z''$ be a neighbor of $z'$ with $d(z,z'') = 2$.
Then $d(x,z'')=3$ since $a_4=0$, and we see a 2-3-4 triangle, contradiction.
So if $b_3 > 1$ then $a_2 = 0$, and $a_1 = 0$ since $a_1 \le 2a_2$,
and the graph is bipartite. If $b_3=1$, then the graph is antipodal.
\hfill$\Box$\medskip
\subsection*{Variations}
One can vary the above theme. First, the following general result builds on ideas previously used.
\begin{Theorem}
Let $\Gamma$ be a distance-regular graph with diameter $d$, and
let $H \subseteq \{0,\ldots,d\}$. Then all $P_{id}$ for $i \in H$
take the same value if and only if
the eigenvalues $\theta_j$ with $j\notin H$ are the zeros of the polynomial
$\sum_{i=|H|-1}^{r} \frac{\alpha_i}{k_i}p_i(x)$, where $r=d+1-|H|$,
$\alpha_{r}=1$, and
\begin{equation}
\label{eqThm}
\alpha_i =\frac{\tr (A_i \prod_{j\notin H}(A-\theta_j I))}{\tr (A_r A^{r})}\qquad
(|H|-1\le i\le d-|H|).
\end{equation}
\end{Theorem}
\noindent{\bf Proof.}\quad
By Proposition \ref{multipleeq}, the hypothesis holds if and only if we have the equalities
$\sum_i m_i \theta_i^s \prod_{j \notin H} (\theta_i - \theta_j) = 0$
for $0 \le s \le |H|-2$. That is,
$$
\tr A^s \prod_{j \not\in H} (A-\theta_j I) = 0\qquad (0 \le s \le |H|-2).
$$
It follows that $\prod_{j \not\in H} (A-\theta_j I)$,
when written on the basis $\{ A_i \mid 0 \le i \le d \}$,
does not contain $A_s$ for $0 \le s \le |H|-2$.
Moreover, the number of factors is $r=d+1-|H|$, so that $A_s$ does not occur either
when $s > r$. Therefore, $\prod_{j \not\in H} (A-\theta_j I)$
is a linear combination of the $A_s$ with $|H|-1 \le s \le r$ and, hence, for some constants $\alpha_i$,
$$
\frac{n}{\tr (A_r A^{r})}\prod_{j \not\in H} (A-\theta_j I)=\sum_{s=|H|-1}^{r}\frac{\alpha_s}{k_s}A_s.
$$
Now comparing coefficients of $A^{r}$, we see that $\alpha_{r}=1$
(notice that $\tr (A_rA^{r})=nc_1\cdots c_{r}k_{r}$).
To obtain the value of $\alpha_i$ for $|H|-1\le i\le d-|H|$, multiply both terms of the above equation by $A_i$ and take traces. \hfill$\Box$\medskip
In the above we applied this twice, namely for
$d=2e+1$, $H=\{1,3,5,\ldots,d\}$, and for $d=2e$, $H=\{1,3,5,\ldots,d-1\}$.
In the latter case, \eqref{eqThm} yields the following expression for $z=-\alpha_{e-1}$.
$$
b_0b_1 \cdots b_e n z =
c_1c_2 \cdots c_{e+1} k_{e+1} n z =
- \tr (A_{e-1} \prod_{j \notin H} (A-\theta_j I)).
$$
Let us now take for $H$ the set of even indices, with or without 0.
\subsubsection*{\boldmath$H=\{0,2,4,\ldots,d\}$}
Let $d = 2e$ be even and suppose that $P_{id}$ takes the same value ($k_d$)
for all $i \in H = \{0,2,4,\ldots,d\}$. Then $|H| = e+1$, and
$\prod_{j \not\in H} (A-\theta_j I)$ is a multiple~of~$A_e$.
By Proposition \ref{0ddd} this happens if and only if
$\Gamma$ is antipodal with even diameter.
\subsubsection*{\boldmath$H=\{0,2,4,\ldots,d{-}1\}$}
Let $d = 2e+1$ be odd and suppose that $P_{id}$ takes the same value ($k_d$)
for all $i \in H = \{0,2,4,\ldots,2e\}$. Then $|H| = e+1$, and
$\prod_{j \not\in H} (A-\theta_j I)$ is a linear combination of $A_s$
for $e \le s \le e+1$.
By Proposition \ref{0ddd} this happens if and only if
$\Gamma$ is antipodal with odd diameter.
\subsubsection*{\boldmath$H=\{2,4,\ldots,d\}$}
Let $d = 2e$ be even and suppose that $P_{id}$ takes the same value
for all $i \in H = \{2,4,\ldots,d\}$. Then $|H| = e$, and
$\prod_{j \not\in H} (A-\theta_j I)$ is a linear combination of $A_s$
for $e-1 \le s \le e+1$.
As before we conclude that the $\theta_j$ with $j \not\in H$
are the eigenvalues of the array
$\{b_0,\ldots,b_{e-1};\,c_1,\ldots,c_{e-1},c_e{+}zb_e\}$
for some real $z \le 1$.
This time $d \in H$, and there is no conclusion about the sign of $z$.
For example, the Odd graph $O_5$ with intersection array
$\{5,4,4,3;\,1,1,2,2\}$ has eigenvalues $5$, $3$, $1$, $-2$, $-4$
and $P_{24} = P_{44}$. The eigenvalues 5, 3, $-2$ are those of
the array $\{5,4;\,1,-1\}$.
No primitive examples with $d > 4$ are known.
\subsubsection*{\boldmath$H=\{2,4,\ldots,d{-}1\}$}
Let $d = 2e+1$ be odd and suppose that $P_{id}$ takes the same value
for all $i \in H = \{2,4,\ldots,2e\}$. Then $|H| = e$, and
$\prod_{j \not\in H} (A-\theta_j I)$ is a linear combination of $A_s$
for $e-1 \le s \le e+2$.
For example, the Odd graph $O_6$ with intersection array
$\{6,5,5,4,4;\,1,1,2,2,\discretionary{}{}{} 3\}$ has eigenvalues
$6$, $4$, $2$, $-1$, $-3$, $-5$ and $P_{24} = P_{44}$.
No primitive examples with $d > 5$ are known.
\section*{Acknowledgments}
Part of this note was written while the second author was visiting
the Department of Combinatorics and Optimization (C\&O),
in the University of Waterloo (Ontario, Canada).
He sincerely acknowledges to the Department of C\&O the hospitality
and facilities received.
Also, special thanks are due to Chris Godsil for useful discussions
on the case of diameter four.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,783 |
{"url":"http:\/\/mathoverflow.net\/feeds\/question\/120100","text":"In your opinion, what are the relative advantages of n-fold categories and n-categories? - MathOverflow most recent 30 from http:\/\/mathoverflow.net 2013-06-19T05:09:26Z http:\/\/mathoverflow.net\/feeds\/question\/120100 http:\/\/www.creativecommons.org\/licenses\/by-nc\/2.5\/rdf http:\/\/mathoverflow.net\/questions\/120100\/in-your-opinion-what-are-the-relative-advantages-of-n-fold-categories-and-n-cate In your opinion, what are the relative advantages of n-fold categories and n-categories? Rachel 2013-01-28T11:09:44Z 2013-01-28T15:35:44Z <p>For example 2-categories seem simpler at first compared to double categories because the latter is a \"wider environment\" (cf Bertozzini), however when doing calculations, many people prefer to use double categories (cf Brown) and when we describe them in the 2-arrows-only language it seems that double categories have less axioms. (Recall the theorem of Brown-Mosa-Spencer that the category of 2-categories is equivalent to the category of edge symmetric double categories with connection and the theorem by Ehresman that all double categories can be embedded in an edge symmetric double category.) It seems that more work has been done on n-categories though, why is that? Is it easier to start with n-categories?<\/p> http:\/\/mathoverflow.net\/questions\/120100\/in-your-opinion-what-are-the-relative-advantages-of-n-fold-categories-and-n-cate\/120104#120104 Answer by Giorgio Mossa for In your opinion, what are the relative advantages of n-fold categories and n-categories? Giorgio Mossa 2013-01-28T12:11:09Z 2013-01-28T12:11:09Z <p>I believe is more a matter of tastes, personally I find easier and simpler n-fold categories than categories.<\/p> <p>For me n-fold categories are more natural and so are easier, for different reasons: for start one interesting thing is the various sources and targets of the composition are given just by $k-1$-cells (faces), where for $n$-categories sources are given by a $i$-cells for each $i < k$, this gives an intuitive representation of $k$-cells as $k$-dimensional cubes, with orientation for each pair of opposite faces, and a representation of composition as pasting cubes along the faces coherently with these orientations. To do something similar with $n$-categories you should work with $k$-cells as $k$-globes and see compositions as a sort of pasting of globes which involves also deformations of such globes and so (at least by me) it's a little more difficult to figure.<\/p> <p>On the other end this cubical approach has proven to be more easier to write computations: consider the case of fundamental group in which to do computations it usually preferred to use maps from the cubical interval rather then maps from the spheres.<\/p> <p>Another point in favor of n-fold categories is that every n-categories can be seen as a n-fold category in which every cells have collapsed faces.<\/p> <p>I something else come to my mind I reserve the right to add something later. :)<\/p> http:\/\/mathoverflow.net\/questions\/120100\/in-your-opinion-what-are-the-relative-advantages-of-n-fold-categories-and-n-cate\/120119#120119 Answer by Ronnie Brown for In your opinion, what are the relative advantages of n-fold categories and n-categories? Ronnie Brown 2013-01-28T15:35:44Z 2013-01-28T15:35:44Z <p>I think there are good technical reasons for preferring one particular mode in certain situations, depending on how easily certain concepts are expressed. For me, the main intuition since 1965 was based on the diagram <\/p> <p><img src=\"http:\/\/pages.bangor.ac.uk\/~mas010\/array.jpg\" alt=\"array\"><\/p> <p>and the idea that the big square should be the composition of all the little squares. This I termed \"algebraic inverses to subdivision\". Subdivision is an important tool in mathematics for local-to-global problems, which are themselves an important range of problems in mathematics and its applications. I found that Ehresmann's notion of double category, or groupoid, was very suited to express this notion, and was easy to generalise to higher dimensions. This led to proofs of what we now call Higher Homotopy Seifert-van Kampen Theorems, and for which the globular notions were not of any help. <\/p> <p>The notion of strict higher cubical category or groupoid is also useful for formulating and proving monoidal closed structures, due to the rule $I^m \\times I^n \\cong I^{m+n}$, see the final section of this paper in <a href=\"http:\/\/pages.bangor.ac.uk\/~mas010\/pdffiles\/AABS.pdf\" rel=\"nofollow\">Advances of mathematics, 170 (2002) 71--118.<\/a>. <\/p> <p>The paper Ellis, G.~J. and Steiner, R., Higher-dimensional crossed modules and the homotopy groups of $(n+1)$-ads. <em>J. Pure Appl. Algebra<\/em> 46 (1987) 117--136, relates certain $(n+1)$-fold groupoids, i.e. those in which one structure is a group, to a fascinating structure called <em>crossed $n$-cube of groups<\/em>, and which is closely related to classical ideas in the homotopy theory of $n$-ads, see particularly Theorems 3.7,3.8, which have not been obtained by other methods. <\/p> <p>On the other hand, to discuss the notion of commuting cube in a strict cubical category with connections, the relation with the globular case was crucial, see this paper by <a href=\"http:\/\/www.tac.mta.ca\/tac\/volumes\/14\/4\/14-04abs.html\" rel=\"nofollow\">Higgins<\/a>. <\/p> <p>The notions of the globular, simplicial, or cubical sites have been well studied. I am not sure that globular sets are very convenient. What seems not to have been well studied, or even studied, is the underlying geometric site for $n$-fold categories, since it is the geometry of cubes in which all the directions are distinct, so the direction $i$ faces of a cube are distinct from the direction $j$ faces if $i \\ne j$. <\/p> <p>Also weak cubical categories do not seem much studied, though the classical example is the cubical singular complex of a space. <\/p>","date":"2013-06-19 05:09:26","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\": 0.7980613708496094, \"perplexity\": 805.6332649718098}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368707773051\/warc\/CC-MAIN-20130516123613-00048-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
The international community has continued to aggressively ramp up response support and activities. On Saturday, September 20th, WHO held a partners meeting in Monrovia to present the proposed Ebola Care Center (ECC) model and discuss rollout of the community care package.
On Thursday, September 18th, the US Military started sending engineering/infrastructure equipment and troops. Over fifty soldiers have arrived to date, and will help train health workers. The French and German militaries have agreed to establish a joint airlift and build a logistics chain from Germany to support affected countries. Supplying about 100 soldiers and four cargo planes, the airlift will ship about 100 tons of essential supplies per week.
On Sunday, Sep 21, a Direct Relief-chartered Boeing 747 sent 100 tons of emergency medical supplies, the largest known private airlift of emergency medical supplies to West Africa to date.
During a weekly staff meeting, RBHS Behavior Change Communications Officer Marietta Yekee presented material from a recent community level training. She showed the RBHS team how to safely use, remove, and dispose of makeshift plastic bags that may be used by caretakers as protective barriers when gloves are not available.
Infection Prevention and Control Task Force: The Infection Prevention and Control (IPC) Task Force is chaired by the RBHS Chief of Party. The IPC group is holding a training of trainers (TOT) this week, once training teams return from the counties.
Aligned standard operating procedures (SOPs) for national application – SOPs cover triaging, ECCs, enhanced protection for routine care, systematic fever monitoring, patient transport, and safe burials.
Enhanced health care worker knowledge (developed and disseminated technical training materials, trained 107 people who are serving on 21 training teams this week).
Social Mobilization Committee: RBHS Behavior Change Communications (BCC) Officers have been actively involved in developing and refining the social mobilization strategy, and have helped facilitate trainings.
The committee has enlisted the Ministry of Education – 26,000 teachers are available — schools remain closed. Teachers are an ideal group to provide outreach and education, as they are respected community members. Last week nearly 90 county and district education officers were trained using gCHV training materials; these officers will return to counties to train teachers.
300 gCHVs have been trained to date. The social mobilization group wants to add a psycho-social component to educational training, as children are losing family and community members to EVD.
RBHS has been on the ground for six years, developing a network of strong relationships across MOHSW at the central and county levels. As a result, the project has gained a solid understanding of how systems really function on the ground. With this knowledge, the project has helped build platforms that will guide the Ebola response. In the future, RBHS will be well-positioned to inform the transition from the emergency response phase to the restoration of routine health services. | {
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ACCEPTED
#### According to
Index Fungorum
#### Published in
Anal. Soc. cient. argent. 13: 61 (1882)
#### Original name
Sclerotium eurotioides Lib.
### Remarks
null | {
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package com.aprotrain.sl.dal.entity;
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Die folgende Liste der Länder nach Religion sortiert die Bevölkerungen (fast) aller Staaten nach ihren religiösen Anteilen. Alle Daten stammen vom Pew Research Center und basieren auf nationalen Statistiken.
Im Jahr 2010 sind 31,5 % der Weltbevölkerung Anhänger des Christentums, womit es die größte Religionsgruppe weltweit ist. 23,2 % aller Menschen sind Muslime, 15,0 % sind Hindus, 7,1 % sind Buddhisten und 0,2 % sind jüdischen Glaubens; 0,8 % sind Anhänger einer sonstigen Religion und 5,9 % Anhänger einer ethnischen Religion (beispielsweise traditionelle afrikanische Religionen, chinesischer Volksglaube sowie lokale Religionen indigener Völker). Insgesamt 16,3 % der Weltbevölkerung gehören keiner Religionsgruppe an. Dabei ist allerdings statistisch zu berücksichtigen, dass beispielsweise ein Austritt aus der römisch-katholischen Kirche nicht zwangsläufig mit einem Bekenntnis zum Atheismus gleichzusetzen ist, denn Religiosität findet sich auch außerhalb von Kirchenmitgliedschaften. Bekanntestes Beispiel für diese These ist in Deutschland der Theologe Eugen Drewermann.
Liste der Länder nach religiöser Zusammensetzung
Die folgende sortierbare Liste enthält Daten für das Jahr 2010 (Quelle ist der Bericht The Global Religious Landscape des Pew Research Center):
Weltregionen nach religiöser Zusammensetzung
Die folgende Liste vergleicht die Anteile (%) der Religionen an der jeweiligen Gesamtbevölkerung im Jahr 2010 (aus dem Bericht The Global Religious Landscape des Pew Research Centers):
Religiöse Bevölkerungen nach Anzahl
Länder sortiert nach ihrer Anzahl an Anhängern von einer bestimmten Weltregion. Quelle ist das Pew Research Center, das eine Datenbank zu allen Ländern führt, oder die jeweils nationale Statistik.
Christen
Anzahl an Christen nach Land (Stand 2011):
Muslime
Anzahl an Muslimen nach Land (Stand 2013):
Buddhisten
Anzahl an Buddhisten nach Land (Stand 2010):
Hindus
Anzahl an Hindus nach Land (Stand 2010):
Juden
Anzahl der jüdischen Gläubigen nach Land (Stand 2010):
Entwicklung der Weltreligionen in Zukunft
Im Jahr 2017 wurde vom Pew Research Center eine Prognose zur Entwicklung der Anhängerzahlen der Weltreligionen bis zum Jahr 2060 erstellt.
Der Islam ist demnach die am schnellsten wachsende Weltreligion und wird seinen Anteil von 24,1 % der Weltbevölkerung im Jahr 2015 auf voraussichtlich 31,1 % im Jahre 2060 steigern. Der Islam hätte dann fast so viele Anhänger wie das Christentum, was seinen Anteil an der Weltbevölkerung von 31,2 % auf 31,8 % stabil halten wird.
Der Anteil der Buddhisten, der Religionslosen und der Anhänger einer sonstigen Religion wird hingegen deutlich zurückgehen (Religionslose von 16,0 % auf 12,5 %, Buddhisten von 6,9 % auf 4,8 %, Sonstige von 6,5 % auf 5,2 %).
Hindus und Juden werden ihren Anteil an der Weltbevölkerung ungefähr konstant halten (Hindus von 15,1 auf 14,5 %, Juden von 0,2 % auf 0,2 %). Der Grund für die Veränderung der weltweiten religiösen Zusammensetzung sind sich stark unterscheidende Geburtenraten sowie die niedrige Austrittsrate der Anhänger des Islam im Vergleich zu denen des Christentums und des Buddhismus. In absoluten Zahlen wird jede Gruppe wachsen mit Ausnahme der Buddhisten.
Prozentualer Anteil der Religionen an der gesamten Weltbevölkerung:
Gesamtzahlen der Gläubigen der Religionen an der Weltbevölkerung:
Einzelnachweise
Liste (Religion)
Liste (Staaten)
Liste (Demografie) | {
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The National Railway Museum is a museum in York forming part of the British Science Museum Group of National Museums and telling the story of rail transport in Britain and its impact on society. It has won many awards, including the European Museum of the Year Award in 2001. It is the home of the national collection of historically significant railway vehicles, as well as a collection of other artefacts and both written and pictorial records.
The National Railway Museum in York displays a collection of over 100 locomotives and nearly 300 other items of rolling stock, virtually all of which either ran on the railways of Great Britain or were built there. Also on the 20acre site are many hundreds of thousands of other items and records of social, technical, artistic and historical interest, exhibited mostly in three large halls of a former motive power depot next to the East Coast Main Line, near York railway station.
It is the largest museum of its type in Britain, attracting 727,000 visitors during the 2014/15 financial year .
The National Railway Museum was established on its present site, the former York North locomotive depot, in 1975, when it took over the former British Railways collection located in Clapham and the York Railway Museum located off Queen Street, immediately to the southeast of the railway station; since then, the collection has continued to grow.
The museum is a short walk from the railway station in York, either on the road or via a staircase from the rear of the platforms. A "roadtrain" runs from the city centre (near York Minster) to the museum on Leeman Road during half term, holidays and summer. York Park and Ride also serve the museum from the car park entrance, on Line 2 (Rawcliffe Bar-York). Admission to the museum has been free since 2001. It is open daily from 10 am to 6 pm from February to November and 10 am to 5 pm during the winter months.
Locomotion – the National Railway Museum in Shildon, County Durham was opened in October 2004 and is operated by the NRM in conjunction with Durham County Council. It houses more of the National Collection in a new building and a historic site around the former workshop of Timothy Hackworth and in the most recent full year for which figures have been published (2011–2012), it attracted more than 210,000 visitors.
There are approximately 280 rail vehicles in the National Collection, with around 100 being at York at any one time and the remainder divided between Locomotion at Shildon and other museums and heritage railways. The earliest are wagonway vehicles of about 1815. The permanent display includes "Palaces on Wheels", a collection of Royal Train saloons from Queen Victoria's early trains through to those used by Queen Elizabeth II up to the 1970s, among them some of the first rail vehicles to be set aside for preservation.
its streamlined sister Class A4 No. 4468 Mallard and London, Midland and Scottish Railway Princess Coronation Class No. 6229 Duchess of Hamilton. Flying Scotsman is among the exhibits intended for operation on the National Rail network from time to time.
The museum has imported several major vehicles for display: the Chinese Class KF7 4–8–4 locomotive donated in 1981 was built in Britain and the Wagons-Lits sleeping car donated in 1980 had been used on the Paris-London Night Ferry service. The single exception to the rule of exhibits associated with Britain is the Japanese 0 Series Shinkansen leading vehicle which was donated to the museum by the West Japan Railway Company in 2001 and which now forms part of an award-winning display, and is the only Shinkansen vehicle on exhibit outside Japan.
Rail vehicles on display are exchanged from time to time with other organisations, and examples of new-build stock from the current industry sometimes visit the museum for short periods.
Other physically large exhibits are the Stockton and Darlington Railway Gaunless Bridge and several stationary winding engines used on railway inclines.
tickets, nameplates, staff uniforms, clocks, watches, furniture and equipment from railway companies' hotels, refreshment rooms and offices (including company seals) and a wide range of models, some of which are operated on the museum's O scale model railway (originated in 1982).
The National Railway Museum holds a large open library and archive of railway related material. This includes an internationally significant collection of locomotive and rolling stock engineering drawings from railway works and independent manufacturing companies. Copies of many of these engineering drawings are sold to the heritage railway movement to assist with their new build locomotive and restoration projects. They are also sold to modellers who can use the drawing to produce accurate scale models. The library holds more than 20,000 books and 800 journals of which around 300 are active. The archive also holds a large collection of technical and test records, as well as timetables including a large number of Bradshaw timetables. The archives also hold some 1.75 million photographs covering the earliest era of photography to the modern day. These include official collections from railway companies and collections from enthusiasts like Eric Treacy and H. Gordon Tidey.
In 1999/2000 the Museum began to collect recordings of former railway staff for a National Archive of Railway Oral History. It also holds the archive of steam train recordings by Peter Handford. In 2009 The Forsythe Collection of travel and transport ephemera was acquired for the collection. Many of the museum's artworks and posters can also be viewed through Search Engine although these are now displayed in a series of temporary exhibitions in the museum's new art gallery which opened in 2011.
The Search Engine facility opened in late 2007 and is open from 10:00 to 17:30 Wednesday to Saturday. The archive and library collections can be viewed by anyone without an appointment although the website recommends pre-booking archive materials at least 24 hours in advance. The majority of its collections have been listed on its website for people to view what materials are available prior to their visit. For those people that cannot visit the museum itself there is a research service offered by the museum called Inreach.
Although there had been amateur attempts to establish a national railway museum from the late 19th century, the National Collection today results from the fusion of two long-running official initiatives. One was led by the State museums sector, evidencing pioneering technology, and the other by the railway industry, in which the key contribution came from the North Eastern Railway as successors to the historic Stockton and Darlington Railway.
What became the Science Museum collection was begun in the 1860s by the Patent Office, whose museum included such early relics as Puffing Billy, Stephenson's Rocket and Agenoria (sister locomotive to Stourbridge Lion), which was outhoused to York at an early date.
Preservation of redundant equipment by the railway companies themselves was a matter of chance. Sometimes relics were stored in company workshops and offices and some were destroyed as circumstances changed. Where put on public display at all the equipment was usually mounted on railway stations in a case or on a plinth. Coppernob at Barrow-in-Furness, Derwent and Locomotion at Darlington and Tiny at Newton Abbot were long-lived examples of this form of display.
The first railway museums were opened at Hamar in Norway (1896) and Nuremberg in Germany (1899). These inspired talk of doing the same in Britain, both in the 1890s and again in 1908, but this came to nothing at that time. Indeed, two of the Great Western Railway's earliest broad-gauge locomotives, North Star and Lord of the Isles, which had been set aside at Swindon Works, were cut up in 1906 for lack of space and several other relics were similarly lost in subsequent years.
From 1880, J. B. Harper of the North Eastern had been collecting material much of which was exhibited on the occasion of the S.& D.R. centenary in 1925; and which then formed the basis of a museum opened at York by the London and North Eastern Railway in 1928 under the curatorship of E. M. Bywell.
The smaller exhibits were housed in the old station buildings and the rolling stock and other large exhibits in the former locomotive erecting and repair shops of the old York and North Midland Railway (demolished after the museum closed). Despite this however, the locomotives were displayed on short lengths of track acting as plinths, very much in traditional museum style. It was only when the NRM was formed that Britain acquired a rail-served railway museum where large exhibits could come and go with ease.
The collection was dominated by items from the North Eastern Railway, together with Great Northern Railway items. The other three 'Big Four' railway companies showed little interest in contributing to the LNER's initiative, though eventually one locomotive representative of each did find its way there: the Great Western's City of Truro, London and North Western Railway Columbine and London, Brighton and South Coast Railway B1 Class Gladstone.
The GWR assembled a valuable collection of small objects, mounted privately in a long corridor at Paddington station, and in 1925 it built a replica of North Star. It preserved City of Truro and Tiny in 1931 and purchased Shannon for preservation in 1946.
The LMS had its own collection of small objects at Euston. It also began to build up a collection of historic locomotives, which included Caledonian 123, Columbine, Cornwall, Hardwicke, Highland 103, Midland 118 and Pet. Three others, set aside for preservation at Crewe Works, were scrapped in a change of policy in 1932. The LMS set aside one further locomotive (Midland 158A) before it was overtaken by nationalisation. It also succeeded in preserving a collection of historic royal saloons at Wolverton and built a replica Rocket, with six replica carriages, for the Liverpool & Manchester Railway centenary in 1930, and a replica Grand Junction Railway Travelling Post Office.
The nationalisation of British Railways in 1948 gave the opportunity for a more consolidated approach and a report was produced by the British Transport Commission in 1951.
Amongst other things this recommended a curator be appointed for the Commission's holdings (John M. Scholes), retention of the York museum, creation of other regional museums (not carried out in the way proposed), a small relics display in the old Great Hall at Euston railway station (done on a temporary basis) and a large museum of collections elsewhere in London. For the latter, the former station at Nine Elms was originally favoured as a site, but what was eventually opened in 1961 was the Museum of British Transport in a former bus garage in Clapham.
and many were stored in sheds and works throughout the country, others being placed on loan to local authority museums. The 'Steam' Museum at Swindon still displays a large number of items from the National Collection, while the Glasgow Museum of Transport was also indebted to it, although many of the Scottish relics (including NBR K 'Glen' Class 4-4-0 No. 256 Glen Douglas currently at the Bo'ness & Kinneil Railway) no longer form part of the National Collection.
The Beeching Report recommended that British Rail should stop running museums, and a campaign was led by transport historian L. T. C. Rolt and others such as the historian Jack Simmons to create a new museum.
Agreement was reached under terms in the Transport Act 1968 for B.R. to provide premises to be occupied by a National Railway Museum which would be a branch of the National Museum of Science and Industry then under Dame Margaret Weston and the first English national museum outside London – a move which was at the time criticised by Londoners.
John Van Riemsdijk of the Science Museum and David Jenkinson.
The museum was opened by Prince Philip, Duke of Edinburgh in 1975. The opening coincided with the 150th anniversary celebrations of the opening of the Stockton & Darlington Railway, for which several working exhibits were provided. By comparison with the museum's predecessors coverage of ordinary passenger coaches and non-steam motive power was enhanced, but a popular new exhibit was ex-Southern Railway Merchant Navy Class No. 35029 Ellerman Lines sectioned to show the workings of a steam locomotive. The new museum received over a million visitors in its first year and was favourably received by critics.
Significant events of 1979 were the restoration of a train of appropriate vehicles to mark the centenary of on-train catering and an exhibition to mark the centenary of railway electric traction which drew attention to the museum's important collections in this area.
Also in 1979 the museum commissioned a working replica of Stephenson's Rocket for the following year's Liverpool and Manchester Railway 150th anniversary. This has since represented the museum at events around the world.
Another working replica was added to the collection for the 150th anniversary of establishment of the Great Western Railway in 1985: that of the broad gauge locomotive Iron Duke.
In 1990, The Rev. W. Awdry's Railway Series Thomas the Tank Engine books were assured a permanent place in the NRM's collection of historical railway books, due to their role in maintaining children's interests in railways. In 1991, Christopher Awdry chose to fictionalise this event in Thomas and the Great Railway Show, where Thomas (the most iconic of Awdry's characters) was made an honorary member of the NRM collection by Sir Topham Hatt and the Director of the NRM.
Concerns about the condition of the concrete roof structure on the main building brought forward major changes to the museum in 1990. To maintain a presence at York, the former York goods depot across Leeman Road, already in use as a museum store (the Peter Allen Building), was configured to display trains as if in a passenger station, and this together with the adjacent South Yard was marketed as The Great Railway Show. A further selection of exhibits formed the National Railway Museum on Tour on display for a season in the former Swindon Works.
Meanwhile, the main building was completely re-roofed and reconstructed retaining only one of the two original 1954 turntables.
It was reopened on 16 April 1992 by Prince Edward, Duke of Kent as the Great Hall giving enhanced opportunities to display large artifacts such as railway signals, a footbridge from Percy Main station and a segment from the Channel Tunnel. The former goods shed display was retained as the Station Hall.
In 1995 the museum joined forces with the University of York to create an academic research base, the Institute of Railway Studies (and Transport History). It has also since partnered with York College to create the Yorkshire Rail Academy to teach vocational skills. The museum has also provided engineering apprenticeships and participates in partnerships aimed at delivering heritage skills training.
In 1996 the Museum Garden was created incorporating a gauge ridable miniature railway. A playground was also added.
Continued concern over the condition of the remaining 1950s buildings on the site led to their replacement by The Works in 1999. This gave several functional areas: the Workshop, for maintenance of rolling stock; the Workshop Gallery, from which the public can look down on this work; a Working Railway Gallery, giving an insight into current and recent operation including a balcony overlooking York railway station hosting a set of monitors showing live feeds from the monitors at York IECC; and the Warehouse which provides an innovative open storage area, which has proved popular with both public and museum professionals.
In order to provide step-free access from the main hall to the Workshop Gallery, the Museum Inclinator was constructed. Besides its primary function, this also served to demonstrate the workings of a funicular railway. To that end its workings were exposed in the style of a larger open air funicular railway, rather than being concealed in the fabric of the building as is more normal for intramural lifts. Unfortunately, due to lack of spare parts, it ceased working, and with no plans for repair it was removed by August 2013.
2004 saw several major developments at the museum. Several railway anniversaries were celebrated by a major "Railfest".
Another took place from 25–30 May 2008 with a Sixties theme. The Locomotion museum was opened at Shildon, County Durham providing undercover collection care facilities for more rail vehicles from the museum's collection. In addition, the museum had a high-profile campaign, supported by the National Heritage Memorial Fund, to purchase Flying Scotsman which arrived at the Museum as the climax of Railfest.
The first stage of a new centre providing easy access to the museum's Library and Archives, called "Search Engine", opened at the end of 2007.
From 18 July to 23 August 2008, a popular new venture was the staging by York Theatre Royal at the Museum of the play of E. Nesbit's The Railway Children, awarded five stars in The Guardian. Following this success, it was repeated in 2009, from 23 July to 3 September, and the museum provided locomotives for subsequent performances at Waterloo International station and in Toronto.
Major plans under the name "NRM+" were made for refurbishing the Great Hall display, for which a preliminary Heritage Lottery Fund contribution was announced in 2009, and seeking potential partners for a further outhousing project.
There are other partnerships for development of the museum estate and the land around it (much owned by Network Rail) as "York Central" but the economic situation during 2009 put these particular plans in abeyance although a similar York Central project was launched by the city council at the beginning of 2016. The NRM+ project was cancelled in April 2011 due to lack of success in assembling the funding package. However, major changes to the displays in the Station Hall began later in 2011.
In 2012, the NRM decided to repatriate temporarily the two LNER A4 class steam locomotives, numbers 60008 Dwight D Eisenhowerand 60010 Dominion of Canada from their homes at the National Railroad Museum in Green Bay, Wis. and Exporail in Montreal, Quebec, as part of the Mallard 75' event in 2013. The two locomotives would be on loan for up to two years, during which time the locomotives would be cosmetically restored, 60008 in BR Brunswick Green (as it appeared in 1963 on withdrawal) and 60010 as LNER 4489 in Garter Blue with its original Canadian Pacific Railway bell (as it appeared in 1939).
On 8 December 2012 it was announced that an annex to the National Railway Museum would be built close to Leicester North station on the Great Central Railway (heritage railway).
In June 2013, the York Press reported that NRM was facing a funding crisis due to a potential 10% annual cut to the Science Museum Group's funding, an estimated real-terms 25% cut following lay-offs and disbandment of projects. The museum was considering scaling down its functions, re-introducing admissions charges or facing complete closure. However, following a campaign by local residents the Chancellor George Osborne announced a 5% cut in the museum's budget. This prompted Science Museum Group director Ian Blatchford to announce two weeks later that the museum had been doubly saved he added that had the 10% cut taken place the Group would have chosen to close the National Media Museum in Bradford.
Criticisms of the museum which have been raised include claims that it has devoted insufficient attention to modern traction; that it was neglecting scholarship in favour of commercialism; or that its photographic collections constitute a "black hole". The museum's response is that these criticisms do not always take into account the financial constraints under which the museum operates: its Grant in Aid from the Department for Culture, Media and Sport amounts to £6.50 per visitor which delivers less overall income than for comparable London museums. For some of its funding the museum depends on money-making events such as the Yorkshire Wheel, which operated at the museum from 2006 to 2008 and visits from Thomas the Tank Engine as chronicled in Thomas and the Great Railway Show. The museum has also suffered a few thefts of objects.
The museum can be allocated material from the railway industry by the Railway Heritage Committee. Because of the diversity of material falling potentially within the museum's collection policy and the problems of caring for it, decisions on acquisition of new items for the collection can be difficult. Previously the museum has treated rolling stock as if it were effectively still in railway service and capable of undergoing repeated heavy repairs and restoration. Since being preserved many of the museum's locomotives have operated on the main line, heritage railways or at the museum. More recently, there have been moves to less interventionist forms of conservation in some cases leading to some exhibits becoming non-operational.
The Museum's management of the protracted overhaul of LNER Class A3 4472 Flying Scotsman was heavily criticised in an internally commissioned report in 2012.
Since 1977, the Friends of the National Railway Museum have been in existence as a group to give financial and other support to the museum, such as financing the original restoration to steam of Duchess of Hamilton.
The 1990 "Great Railway Show" won the Museum of the Year award and in 2001 the museum gained the European Museum of the Year Award. It has also won White Rose awards from the Yorkshire Tourist Board, and in recognition of the several major developments in 2004 was given the Heritage Railway Association's Peter Manisty Award.
The National Railway Museum also has a presence on a number of other websites. Copies of many of its posters, photographs and artworks can be ordered through the Science and Society Picture Library. The National Railway Museum has a presence on the National Preservation forums. Members and Readers are able to talk and comment directly to members of the staff. Providing both feedback and constructive criticism, a valuable source of information for the museum. Members of staff can usually answer questions when they are not busy and are part of the National Railway Museum group. National Railway Museum staff also publish a blog via WordPress.com where staff write stories about events behind the scenes in the museum such as conservation work or preparation for major events.
These are a few of the Museum's locomotives .
SR N15 Class 4–6–0 No. 30777 Sir Lamiel. Currently kept at the Great Central Railway where it is operational and worked on the national network until 2013. Boiler certification expires in 2016.
SR Schools class 4–4–0 No. 925 Cheltenham. Currently working on the Mid Hants Railway. Boiler certification expires in 2022.
British Railways Standard Class 7 "Britannia" 4–6–2 70013 Oliver Cromwell. Overhauled at the Great Central Railway (Loughborough) between 2004-2008. Currently based at Loughborough when not working on the National Network. Boiler Certificate expired in August 2015. Recently returned to traffic.
LNER Class A1/A3 4–6–2 60103 Flying Scotsman. Overhaul started in 2006 by the NRM. After an external review the restoration was taken over by Riley and Son and completed in 2016.
Grant, Ritchie and Company 0-4-0ST no 272. Visiting from the Ribble Steam Railway for the Christmas period and early 2017. Arrived by road from Preston on 15 December 2016.
SR Lord Nelson Class 4–6–0 No. 850 Lord Nelson. Currently working on the Mid Hants Railway. Boiler certification expired in 2016.
Great Central Railway O4 Class 2–8–0 No. 63601. Currently on the Great Central Railway. Withdrawn in 2012 for overhaul.
NER No. 66 Aerolite. On static display in York since 1934.
GWR 4000 Class 4–6–0 4003 Lode Star. Returned to the NRM in November 2015 from the Museum of the Great Western Railway, Swindon as part of an exchange of locomotives in preparation for Swindon 175 in 2016.
LMS Stanier Class 5 4-6-0 5000. On static display.
LNWR G Class ("Super D") 0–8–0 No. 49395. In service and usually on loan to other railways; when it is not touring it is stabled either at York or Crewe. Currently at Locomotion, Shildon.
LNER Class V2 2–6–2 4771/60800 Green Arrow. After many years of being a popular operation engine, her boiler certificate was due to expire Spring 2008, but failed beforehand on the North Yorkshire Moors Railway. In need of extensive repairs to her one-piece three cylinder block, she is unlikely to steam again in the near future due to cost and NRM policy. Returned to York after a two-year loan to Locomotion at Shildon.
LNER Class A4 4–6–2 4468 Mallard. Restored to steam for a time from 1986; now on static display. Unlikely to run again due to exhibit popularity and the fact that all the other A4s in the UK have been restored to working order.
SR Class Q1 0–6–0 No. C1. On static display. However it is possible it will return to the Bluebell Railway, where it was based for many years, to be returned to service.
BR standard class 9F 2–10–0 92220 Evening Star, the last steam locomotive built for British Railways. On static display and not expected to return to working order due to class being barred from running on the national network. Returned to York in 2010 after a two-year loan to the Museum of the Great Western Railway, Swindon.
LMS Princess Coronation Class 4–6–2 6229 Duchess of Hamilton. Returned to the NRM in 2009 after being re-streamlined, it was at first displayed in an exhibit, Streamlined: Styling an era.
GWR 3700 Class 4–4–0 3440 City of Truro. Loaned to the Museum of the Great Western Railway in November 2015 as part of an exchange of locomotives in preparation for Swindon 175 in 2016.
GWR 6000 Class 4–6–0 6000 King George V. Loaned to the Museum of the Great Western Railway in November 2015 as part of an exchange of locomotives in preparation for Swindon 175 in 2016.
Robert Stephenson and Hawthorns 0-4-0ST No. 15 Eustace Forth. Outstationed at Locomotion, Shildon to help maintain a working locomotive presence there. | {
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Челя́бінська о́бласть () — суб'єкт Російської Федерації, входить до складу Уральського федерального округу.
Адміністративний центр — місто Челябінськ.
Межує на півночі зі Свердловською областю, на сході з Курганською, на півдні з Оренбурзькою, на заході з Башкортостаном, на південному сході з Казахстаном.
Утворена 17 січня 1934 року.
Географія
Челябінська область — південна частина Уралу. Умовна межа між Європою і Азією проводиться в основному вододільними хребтами Уральських гір. Недалеко від станції Уржумка (8 км від Златоуста), на перевалі Уралтау, є кам'яний стовп. На одній з його сторін написано «Європа», на іншій — «Азія». Міста Златоуст, Катав-Івановськ, Сатка знаходяться в Європі. Челябінськ, Троїцьк, Міас — в Азії, Магнітогорськ — в обох частинах світу.
Площа Челябінської області — 88,5 тис. км². Протяжність області з півночі на південь — 490 км. Із заходу на схід — 400 км. Географічний центр області розташовується на правому березі річки Уй, в трьох км на південний схід від села Нижнєусцелемово Уйського району. Челябінська область за територією займає 5 місце з 8 регіонів Уралу і 39 місце в Росії. Загальна протяжність кордонів становить 2750 км.
Челябінська область займає, в основному, східний схил Південного Уралу і прилеглі до нього частини Зауральської рівнини і Західно-Сибірської низовини. І лише невелика частина території на північному заході заходить на західні схили Південного Уралу.
Клімат
Клімат континентальний. Зима холодна, тривала. Середня температура січня від −15 °C на північному заході до −17 °C на північному сході. Літо тепле. Середня температура липня +16 °C…+18 °С . Тривалість вегетаційного періоду — 130 — 150 днів. Опадів від 600 мм в гірській частині до 350 мм на рік на рівнинах; максимум припадає на літо.
Рельєф
Рельєф Челябінської області відрізняється великою різноманітністю. В межах Челябінської області є різні області — від низовин і горбистих рівнин до хребтів, вершини яких перевищують 1000 м.
Західно-Сибірська низовина обмежена із заходу горизонталлю (відмітка 190 м над рівнем моря), що проходить через села Багаряк, Кунашак і далі через Челябінськ — на південь. Низовина слабо нахилена на північний схід, знижуючись до 130 м y східної межі області. Низовина розчленована широкими долинами річок.
Зауральська горбиста піднесена рівнина (Зауральський пенеплен) займає центральну частину території області і тягнеться смугою уздовж східних схилів Уральських гір від 50 км на півночі до 150 км на півдні. На південно-західній околиці рівнини знаходиться Уральський дрібносопковик, що включаючає Карагайські гори і височину Кубайс. Поверхня рівнини поцяткована улоговинами озер і річковими рівнинами з пологими схилами.
Корисні копалини
Є великі родовища залізняку (Бакальське, Златоустовське та інші родовища), мідних і нікелевих руд, мінерально-будівельної (особливо магнезитової і цементної) сировини (Агаповське родовище флюсових вапняків і доломіту), бурого вугілля (Челябінський басейн).
Рослинність
Рослинність Челябінської області ділиться на три зони:
Рослинність гірсько-лісової зони, що включає західні і північно-західні райони області, куди входять підзони:
змішаних хвойно-широколістняних лісів
світлохвойних соснових і модринових лісів
темнохвойних ялиново-ялицевих лісів
підгольцеві луки і рідколісся
гольці (гірська тундра)
Рослинність лісостепової зони, що включає центральну і північно-східну, східну частині області (на північ від річки Уй), з переважанням лісів з берези і осики
Рослинність степової зони (на південь від річки Уй), що включає різнотравно-ковилові лукові степи, чагарникову рослинність у балках і низинах, острівні бори, кам'янисті степи.
У Челябінській області можна зустріти майже всі типи рослинності, поширені в помірній і арктичній зонах Росії. Південний Урал є місцем контакту трьох ботанико-географічних областей: Європейської, Сибірської і Туранської (Середньоазійської).
Природні заповідники і парення
У Челябінській області заповідники і національні парки займають близько 200 тисяч гектарів, мисливські і ботанічні заповідники — понад 500 тисяч гектарів, ботанічні пам'ятники природи, зокрема 20 острівних і стрічкових борів загальною площею 184 тисяч гектарів Території, що найбільше охороняються, займають близько 1000 тисяч гектарів — трохи більше десятої частини області. Вчені вважають, що для нормалізації екологічної обстановки площу територій, що охороняються, необхідно збільшити.
Затверджені зелені зони навколо 13 міст (загальна площа 164,7 тисяч гектарів) і зони округів санітарної охорони курортів на озерах Увільди і Кисегач.
Свій внесок до забезпечення вивчення і охорони пам'ятників природи вносять культурно-освітні і спортивно-туристичні організації. Природні території, що особливо охороняються, покликані забезпечити екологічну безпеку, підтримувати екологічний баланс при використанні природних ресурсів і створити середовище, сприятливе для проживання людини.
Гідрографія
В межах області беруть початок численні річки, що належать до басейнів Ками, Тоболу і Уралу. Оскільки тут, в основному, їх верхів'я, тому вони маловодні. Річок завдовжки понад 10 кілометрів налічується в області 348, їх сумарна довжина становить 10 235 кілометри.
Протяжність понад 100 км мають всього 17 річок. І лише 7 річок: Міас, Уй, Урал, Ай, Уфа, Увелька, Гумбейка — мають в межах області довжину більше 200 км.
Велика частина території області відноситься до Обського сточища. На схід, до Тоболу і його приток, течуть більшість річок Челябінського Зауралля: Синара, Теча, Міас, Увелька, Уй, Тогузак, Картали-Аят, Синташта та інші.
Річка Міас бере свій початок на східному схилі хребта Нуралі, тече спочатку між гір на північ, а потім, повернувши на схід біля Карабаша, перетинає лісостепову зону і впадає в Ісеть за межами області. Її довжина в межах області становить 384 км (з 658 загальної довжини).
Регуляторами стоку Міасу служить Аргазінське і Шершневське водосховища. Нині 70-80 % води річки Міас проходить через трубопроводи і лише 20-30 % протікає природним річищем. Чотири п'ятих води Міас віддає на потреби народного господарства. Передбачається перекидання води в басейн Міасу з річки Уфи. Після здійснення проекту в Міасі кількість води подвоїться. Гідросистема будується разом з Долгобродськім водосховищем у верхів'ях Уфи.
Річка Уй бере початок біля відрогів Уралтау, тече на схід, перетинаючи всю область. Напрям її течії майже збігається з межею між лісостеповою і степовою зонами. Загальна довжина річки 462 км, з них 370 км — в межах області. Зліва Уй приймає крупну притоку — Увелку. Зливаються річки у Троїцьку. На Уї і на Увелькі споруджені дамби, які утворили великі водосховища для Южно-Уральської і Троїцької ГРЕС.
Степові річки Синташта, Картали-Аят і Тогузак у найсуворіші зими промерзають. У повені вода в них підіймається на 2 м.
Населення
Челябінська область за чисельністю населення (близько 3,6 млн осіб) займає 3 місце з 8 регіонів Уралу і 9 місце в РФ. (2005).
Область — найщільніша населена на Уралі (займає 1 місце з 8 регіонів Уралу ― густота населення 40,4 осіб/км²) і друга (після Свердловської області) за рівнем урбанізації (питома вага міського населення ― 81,9 %). За щільністю населення Челябінська область — 24-й регіон в РФ (без Москви і С.-Петербурга), а за рівню урбанізації — 9-й (без авт. округів).
Згідно зі Всеросійським переписом населення 2002 року, національний склад населення області був наступним:
Адміністративний поділ
Агаповський район
Аргаяшський район
Ашинський район
Брединський район
Варненський район
Верхньоуральський район
Єманжелинський район
Єткульський район
Карталинський район
Каслинський район
Катав-Івановський район
Кизильський район
Коркінський район
Красноармійський район
Кунашацький район
Кусинський район
Нагайбацький район
Нязепетровський район
Октябрський район
Пластовський район
Саткинський район
Сосновський район
Троїцький район
Увельський район
Уйський район
Чебаркульський район
Чесменський район
Населені пункти
Економіка
Основні галузі промисловості
За об'ємом промислового виробництва на Уралі Челябінська область поступаєть тільки Свердловській. У структурі її промисловості різко виділяється чорна металургія (близько половини випуску продукції). Частка чорної металургії в 1991 році склала 37,8 %, а в 2003 p. ― 59,3 %. На другому місці стоїть машинобудування (до 1/6). Частка машинобудування і металообробки в 1991 році склала 30,0 %, а в 2003 p. ― 15,2 %. Ці галузі разом з кольоровою металургією дають майже 5/6 всій промисловій продукції.
Чорна металургія за масштабами якої область не має собі рівних в Росії, представлена одними з найбільших металургійними комбінатами (Магнітогорськ, Челябінськ), передільними заводами (Златоуст), підприємствами з виробництву феросплавів і сталевих труб (Челябінськ). У кольоровій металургії є виробництво міді (Карабаш, Киштим), цинку (Челябінськ) і нікелю (Верхній Уфалей, Реж). Металургії супроводжує виробництво вогнетривів з магнезиту (Сатка, Комбінат «Магнезит»).
Машинобудування спирається на власну металургійну базу, що обумовлює його металоємність, хоч і менш значну, ніж у Свердловській області. Тут випускають трактори, вантажні автомобілі, трамвайні вагони, технологічне устаткування, ракетно-космічну техніку, електротехнічні вироби.
Енергетична база області включає видобуток бурого вугілля (Копейськ) і декілька могутніх теплових електростанцій (Троїцька і Южно-Уральська ГРЕС та інші). Частка електроенергетики в 1991 році склала 2,4 %, а в 2003 році ― 7,1 %.
Частина території області в 50-х роках XX століття була піддана радіоактивному забрудненню в результаті аварії на підприємстві з переробки відходів «Маяк». Тут більше всього в Росії «атомоградів», що належать до ядерного паливного циклу: Снєжинськ (колишн. Челябінськ-70), Озерськ (колишн. Челябінськ-65) і Трьохгорний (колишн. Златоуст-36). Також розпочато будівельні роботи за 12 км від Челябінська Томінського гірничозбагачувального комбінату, який забруднить джерело водопостачання Шершнівське водосховище.
Сільське господарство
При явному переважанні промисловості область має розвинене сільське господарство, особливо в зоні розповсюдження чорноземних ґрунтів. Найбільші посіви пшениці та інших зернових культур. Тваринництво має м'ясомолочний напрямок. Є тонкорунне вівчарство. Навколо промислових вузлів розвинене сільське господарство приміського типу.
Влада
Законодавча влада
Найвищим і єдиним органом законодавчої влади є Законодавчі збори Челябінської області.
Виконавча влада
Найвищим виконавським органом державної влади області є Уряд Челябінської області. Вищий посадовець області — губернатор.
Петро Сумін, нині чинний губернатор, вперше виграв вибори в Челябінської області у 1993 році, але їх підсумки Кремль не визнав, і губернатором залишився Вадим Соловйов, призначений на цей пост Борисом Єльциним в 1991.
У грудні 1996 Сумін переміг Соловйова на нових губернаторських виборах, а в грудні 2000 року переобрався на новий термін. Термін повноважень Суміна повинен був закінчитися в грудні 2005 року. В кінці березня 2005 губернатор області Петро Сумін звернувся до президента РФ Володимира Путіна з проханням про перепризначення на наступні 5 років.
Путін підтримав прохання, а 18 квітня депутати регіональних Законодавчих зборів одноголосно затвердили кандидатуру Суміна на наступні 5 років.
Радіоактивне забруднення
Річка Теча дуже забруднена радіоактивними відходами, що скидаються в неї хімкомбінатом «Маяк». На берегах річки допустимий радіоактивний фон перевищений в багато разів. Аварія на «Маяку» в 1957 році визнана за масштабми, другою після Чорнобиля, катастрофою в історії ядерної енергетики. Відома як Киштимська трагедія.
Виробниче об'єднання «Маяк» є одним з найбільших російських центрів з переробки радіоактивних матеріалів. Об'єднання обслуговує Кольську, Нововоронезьку і Бєлоярську атомні станції, а також переробляє ядерне паливо з атомних підводних човнів.
Питання радіоактивного забруднення Челябінської області порушувалося неодноразово, але через стратегічну важливість об'єкта «Хімкомбінат "Маяк"» щоразу спускалося на гальмах. Сьогодні район комбінату «Маяк» (у місті Озерськ), як відзначають експерти, став найнебезпечнішим місцем планети. Народ же визначив ситуацію по-своєму: Урал перетворений на світове радіоактивне звалище.
Примітки
Джерела
Офіційний сайт губернатора Челябінської області
Міністерство охорони здоров'я Челябінської області
Справочник адміністративно-територіального поділу на 1 січня 2005 р. — каталог
Флора і фауна. Червона Книга Челябінської області
Православні храми Челябінської області
Природні території, що особливо охороняються
Справжній сайт мешканців Уйського району
Пошук по сайтах Магнитогорську
Області Росії
Надкам'я
Урал | {
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} | 8,101 |
Blogs Become Serious Business Blogs matter more than ever — to political candidates, to a colonel managing a war, to human-rights advocates trying to deliver their message. Two experts discuss the growing impact of blogs.
The Evolution of the Blog
The term "Web log," which was then shortened to "blog," celebrates its 10th anniversary this year.
Blogs Become Serious Business
Embed <iframe src="https://www.npr.org/player/embed/17611047/17611026" width="100%" height="290" frameborder="0" scrolling="no" title="NPR embedded audio player">
< Blogs Become Serious Business
First, we'll track blogging by American troops who've written for the Web in between firefights in Iraq. We invited Noah Shachtman to talk about this. He works for Wired magazine where he writes and blogs about national security.
NOAH SHACHTMAN: During the early days of the war, there was kind of this almost free-for-all period where soldiers were allowed to write about their experiences and published it to the world. And I do think that that added to the reporting, gave it a kind of you-are- there battlefield sense. Soldiers were kind of allowed to do this freely. It was kind of flying under the radar.
INSKEEP: Well, we talked to one of those early military bloggers, Colby Buzzell, who started blogging about eight months into his deployment.
COLBY BUZZELL: When I started blogging, I don't think the military really knew what a blog was. The Internet cafes were all somewhat brand new on the bases there. And we're allowed to pretty much go in any Web site we wanted to while I was over there. And they actually found out about my blog when I wrote about a firefight on August 4th 2004 that got hardly any press coverage.
INSKEEP: Were bloggers actually breaking stories?
SHACHTMAN: In Colby's case they were. And certainly, wherever they blog, they were providing new details. Now it's highly, highly restricted. Everywhere you go on these big military bases in Iraq on every computer, for example, there's a rotating screensaver. There's a series of messages about operational security. Message after message about lose lips sink ships, the modern equivalent thereof, about how blogs are can be very dangerous and you really have to watch what you say. The sort of poster child, you know, what you're absolutely not supposed to do, the thing you've got to be the most scared of is blogging.
INSKEEP: When did all that start to change from that openness we were just describing?
SHACHTMAN: Well, part of it changed when Colby Buzzell blogged about that firefight in August 2004. You know, he pretty quickly wound up on a front page of newspapers and then magazines.
INSKEEP: Well, let's hear what happened to Colby Buzzell in his own words.
BUZZELL: My chain of command found out about it, and I had to go see my battalion commander. He had a printout of all of my blog postings. And he went over to some of the concerns that he had, like how we run low on ammunition, how we almost run out of water on a particular mission. Two weeks after that, an order came down from high up that I still had a freedom of speech and I can still blog but I was confined a base and couldn't go out on missions and that lasted for about a week.
INSKEEP: Which is not the worst punishment that we've heard about for bloggers.
SHACHTMAN: No, it's not. Other guys have been demoted. There have been new regulations put in place by the army, which, if enforced to the letter of the law, would make every soldier have to run every blog post and every e-mail, for that matter, by their commanding officer.
INSKEEP: You mentioned e-mail communication. Are there different restrictions on e-mails than on blogs?
(SOUNDBITE OF LAUGHTER)
SHACHTMAN: The regulations are - let's say - ambiguous. Certainly, I think that a public blog is more scrutinized than a private e-mail, but theoretically, many of the same restrictions apply.
INSKEEP: Well, now let me try to explore the military's justification for these restrictions. The military will say that the enemy can grab all sorts of little pieces of information that you think aren't very important but that a good intelligence analyst could piece together and make something out of.
SHACHTMAN: It's a legitimate concern, and it's the reason why, you know, in wars past, military sensors read soldier's mail.
INSKEEP: Well, setting aside details of combat, there were military restrictions on social networking sites like MySpace or Facebook. Have people gotten around those?
SHACHTMAN: Oh, yeah, they have. In Taji, north of Baghdad, soldiers that I met bought their own satellite dish so that they could hook up to MySpace and YouTube, and so they could play online games. You know, it's a great way for them to keep in touch with their friends and family. It's a great way for them to flirt with girls online. And, damn it, they are going to flirt with girls no matter what the army says.
INSKEEP: Important things first. That's...
SHACHTMAN: Yeah, exactly.
INSKEEP: Security violations of a different kind I supposed.
SHACHTMAN: Yeah.
INSKEEP: Noah Shachtman, thanks very much.
SHACHTMAN: Thanks for having me.
INSKEEP: One of the best-known Egyptian bloggers is a man named Wael Abbas. He has not been jailed but his government takes a dim view with the videos he's put on his blog because some are graphic scenes of police brutality. He had an account on YouTube, which was frozen for a while. After human rights groups protested, most of his videos were restored earlier this month. We got him on the line from Cairo and Abbas spoke briefly with Renee Montagne.
RENEE MONTAGNE: What was in those videos?
WAEL ABBAS: There was a variety of videos in this account. It was not only about torture. There were other videos including coverage of demonstrations, walkout strikes. It was a variety of activist stuff.
MONTAGNE: Do you think that by doing this you have the opportunity to get stories out there that are not otherwise addressed in the media?
ABBAS: Yeah, because stuff that - the videos that were published in my account cannot be shown in any of the traditional media outlets. So it was the only way to do that in a country where the media is really controlled and perhaps censorship. And I get readership from all over the world because it's mainly video stuff that you can see other than you can read.
MONTAGNE: That advocacy group Reporters Without Borders has deemed Egypt as one of the most repressive countries for press freedom. Blogs, when they first got started, seemed to be a way around that but it seems that now it's very dangerous for people like you to write a blog a critical of the regime.
ABBAS: Yes because the regime now is using a different technique. So count the bloggers, they used to detect them electronically but now they are harassing them personally and physically and calling them on the phone, harassing the families.
MONTAGNE: And you don't think that a blog provides any sort of anonymity?
ABBAS: Blogs can be anonymous if you decide to keep yourself anonymous. But most of the bloggers in Egypt and most of the ones who are really active and who are really making a change writes using their real names. They are very well known now and they appear on television so there is no way being anonymous again.
MONTAGNE: Are you worried that you'll end up in jail for the work on your blog?
ABBAS: Yes. Everybody who is working in journalism is worried that somehow he might end up in jail. It can happen to anybody. You cannot - you can never predict to what this regime is about to do.
INSKEEP: That's Egyptian blogger Wael Abbas speaking with Renee Montagne. | {
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Salduba elegans är en tvåvingeart som beskrevs av Kertesz 1908. Salduba elegans ingår i släktet Salduba och familjen vapenflugor. Inga underarter finns listade i Catalogue of Life.
Källor
Vapenflugor
elegans | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,902 |
\section{Introduction}
\vspace{-10pt}
Organic molecular crystals are broadly relevant to solid state physics. Their electronic properties range from conducting to insulating, and they can exhibit anisotropic electrical and optical properties, ferroelectricity, magnetism, and superconductivity.
Organic semiconductors are lead candidates for novel optoelectronics and spintronics applications \cite{Muccini2006, Taliani2009}. Crystals like pentacene and rubrene are already widely used in organic field-effect transistors and light-emitting devices \cite{Bao2006, Someya2010, Arakawa2003}.\\
\indent
Yet, in most organic crystals the nature and transport mechanisms of charge carriers remain unclear. Possible charge transport regimes include polaron charge hopping, band transport, and intermediate regimes, each leading to a peculiar temperature dependence of the mobility.
Even in the same organic crystal, electrons and holes can behave differently. An example is naphthalene, where hole carriers display band-like transport with a power-law temperature dependence of the mobility \cite{Karl1985}, though electron transport in the out-of-plane direction is polaronic and nearly temperature independent \cite{McGhie1978}.\\
\indent
Several approaches have been proposed to compute charge transport in organic crystals \cite{BlumbergerReview}.
Recent calculations favor either quantum chemistry methods based on hopping of localized charge carriers \cite{BlumbergerReview, Negri2014, Troisi2007, Tully1990, Beljonne2013_02, Orlandi2006, Ciuchi2016, Troisi2017}, or somewhat less extensively polaron theories \cite{Munn1980, Duke1989, Bobbert2004_01, Bobbert2004_02, Hannewald2009, Hannewald2010}.
Charge hopping calculations have provided remarkable insight into charge transport in molecular crystals \cite{BlumbergerReview, Negri2014, Troisi2007, Tully1990, Beljonne2013_02, Orlandi2006, Ciuchi2016, Troisi2017}. However, they are laborious, and are not based on rigorous condensed matter theory. They require large molecular dynamics or Monte Carlo simulations, rely on semiempirical charge transfer models based on Marcus theory, and include the temperature dependence of charge transport only approximately, typically using the Einstein diffusion formula. A common assumption is also that only rigid molecular motions affect the rate of carrier hopping, and therefore charge transport.
The accuracy of the charge hopping approaches is limited $-$ the best calculations yield mobility values 3$-$4 times greater than experiment \cite{Negri2014,Troisi2007}, though order-of-magnitude discrepancies between computed and measured mobilities are more common \cite{BlumbergerReview}.\\
\begin{figure}[!b]
\vspace{-1.0\baselineskip}
\includegraphics[clip=left botm right top, width=0.45\textwidth]{Figure1.pdf}
\caption{The monoclinic crystal structure of naphthalene, with two molecules in the unit cell. The molecules are arranged in a herringbone pattern in the \textit{ab} planes (left), which are stacked in the \textit{c} crystallographic direction (right). The \textit{c}* direction normal to the \textit{ab} plane is also shown.}
\label{fig:crystal_structure}
\end{figure}
\indent
To date, only few works have employed band theory to compute charge transport in organic crystals \cite{Shuai2007, Shuai2009, Bredas2003, Kenkre2003}, despite experimental \cite{Hegmann2006, Ramirez2006, Basov2007, Ishii2010} and theoretical \cite{Itabashi2013} evidence of band-like transport in tetracene, rubrene, naphthalene and other organic semiconductors.
Methods combining band theory and many-body perturbation theory have been recently employed to accurately compute electron-phonon (e-ph) scattering and charge transport, for now in simple inorganic materials with a handful of atoms in the unit cell \cite{Bernardi-noble, Zhou2016, Zhou2017, Bernardi-review}. Due to computational cost, these calculations have not yet been applied to organic crystals with tens of atoms in the unit cell.
\textit{Ab initio} studies of e-ph coupling in organic crystals exist \cite{FMauri2012,Venuti2010,Stojanovic2012},
but charge transport, which requires more elaborate workflows \cite{Zhou2016}, has not yet been investigated within this framework.\\
\indent
Here we compute from first principles the band-like hole mobility of naphthalene crystal, a material with 36 atoms in the unit cell (see Fig. \ref{fig:crystal_structure}).
The computed mobility is within a factor of 3$-$4 of experiment, and we can accurately predict its temperature dependence between 100$-$300 K.
For organic semiconductors, these results are a rare case of very good quantitative agreement with experiment $-$ the accuracy on the mobility is on par with the best charge hopping calculations, and we make an order of magnitude improvement over previous \textit{ab initio} mobility calculations in organic crystals using band theory \cite{Shuai2007,Shuai2009}.
We show that inter-molecular phonons (i.e., rigid molecular motions) regulate the mobility
due to a large phase space for scattering holes with energy close to the band edge. Yet, contrary to common notions, intra-molecular phonons exhibit the strongest coupling with holes.
Our work reconciles the tenet of charge hopping methods that inter-molecular phonons control the mobility with the many-body theory perspective, which treats carrier scattering in terms of phonon absorption and emission events.\\
\section{methods}
\vspace{-10pt}
\indent
We carry out density functional theory (DFT) calculations using the {\sc Quantum ESPRESSO} code \cite{QE-2009} with a plane-wave basis set. We employ the Perdew-Burke-Ernzerhof generalized gradient approximation \cite{PBE1996} and norm-conserving pseudopotentials \cite{NormCon1991} from Pseudo Dojo \cite{[http://www.pseudo-dojo.org/\\]Lejaeghereaad3000}. A kinetic energy cutoff of 90 Ry and $4 \times 4 \times 4$ \textbf{k}-point grids are used in all DFT calculations.
Thermal expansion is taken into account by employing, in separate calculations, lattice constants \cite{Dunitz1982} and atomic positions \cite{LeBail2012,LeBail2009} taken from experiment at four different temperatures of 100, 160, 220, and 300 K. All calculations listed below are repeated separately at these four temperatures.
The Grimme van der Waals (vdW) correction \cite{Grimme2006,Vittadina2009} is included during structural relaxation. To obtain accurate electronic bandstructures \cite{Neaton2016_02}, we carry out GW calculations using the YAMBO code \cite{Varsano2009}, and obtain the G$_0\!$W$_0$ self-energy using 500 bands in the polarization function and a cutoff of 10 Ry in the dielectric screening. Wannier90 \cite{Marzari2014} is employed to interpolate the bandstructure, using \textit{ab initio} molecular orbitals \cite{Agapito2016} as initial guesses.\\
\indent
Phonon dispersions are computed with density functional perturbation theory (DFPT) \cite{Giannozzi2001} on a $2 \times 4 \times 2$ \textbf{q}-point grid \cite{Neaton2016}. The e-ph coupling matrix elements $g_{nm\nu}(\textbf{k},\textbf{q})$ on coarse \textbf{k}- and \textbf{q}-point grids \cite{Bernardi-review} are computed using a routine from the EPW code \cite{Giustino2016} and interpolated using Wannier functions \cite{Giustino2007} generated with the Wannier90 code \cite{Marzari2014}.
Here and in the following, $n$ and $m$ are band indices, $\nu$ labels phonon modes, and $\textbf{k}$ and $\textbf{q}$ are crystal momenta for electrons and phonons, respectively. Our in-house developed code {\sc Perturbo} \cite{PerturboWebsite} is employed to interpolate the e-ph matrix elements on fine grids with up to $60 \times 60 \times 60$ \textbf{k}-points and $10^5$ random \textbf{q}-points, and to compute e-ph scattering rates and the hole mobility.
The band- and momentum-resolved e-ph scattering rates $\Gamma^{\textrm{e-ph}}_{n\textbf{k}}$ are obtained in the lowest order of perturbation theory \cite{Bernardi-review},
\begin{align}
\label{eq:e_scat_rate}
\Gamma^{\textrm{e-ph}}_{n\textbf{k}}=\;\frac{2\pi}{\hbar} & \sum_{m\nu\textbf{q}}|g_{nm\nu}(\textbf{k},\textbf{q})|^2\\[2pt]
\times\big[ & (N_{\nu\textbf{q}}+1-f_{m\textbf{k}+\textbf{q}})\delta(\varepsilon_{n\textbf{k}}-\varepsilon_{m\textbf{k}+\textbf{q}}-\hbar\omega_{\nu\textbf{q}})\nonumber\nonumber\\[5pt]
&+(N_{\nu\textbf{q}}+f_{m\textbf{k}+\textbf{q}})\delta(\varepsilon_{n\textbf{k}}-\varepsilon_{m\textbf{k}+\textbf{q}}+\hbar\omega_{\nu\textbf{q}})\big],\nonumber
\end{align}
where $\varepsilon_{n\textbf{k}}$ and $\hbar\omega_{\nu\textbf{q}}$ are the hole and phonon energies, respectively, and $f_{n\textbf{k}}$ and $N_{\nu\textbf{q}}$ the corresponding occupations.
The scheme developed in our recent work \cite{Zhou2016} is applied to converge $\Gamma^{\textrm{e-ph}}_{n\textbf{k}}$. The relaxation times $\tau_{n\textbf{k}}$ used in the mobility calculations are the inverse of the scattering rates, $\tau_{n\textbf{k}}=1/\Gamma^{\textrm{e-ph}}_{n\textbf{k}}$. Our calculations focus on holes, and include only the HOMO and HOMO$-1$ bands because the energy gaps to the HOMO$-2$ and LUMO bands are larger than the highest phonon frequency.\\
\indent
We employ the Boltzmann transport equation \cite{Zhou2016,Marzari2014_02} within the relaxation time approximation to calculate the electrical conductivity
\begin{eqnarray}
\label{eq:conductivity}
\sigma_{\alpha\beta} (T) =e^2\int_{-\infty}^{\infty}dE\left(-\frac{\partial f(E,T)}{\partial E}\right)\Sigma_{\alpha\beta}(E,T)
\end{eqnarray}
where the transport distribution function $\Sigma_{\alpha\beta}(E,T)$ at energy $E$ and temperature $T$ is defined as
\begin{equation}
\label{eq:TDF}
\Sigma_{\alpha\beta}(E,T)=\frac{2}{V_{\textrm{uc}}}\sum_{n\textbf{k}}\tau_{n\textbf{k}}(T)v_{n\textbf{k},\alpha}v_{n\textbf{k},\beta} \, \delta (E-\varepsilon_{n\textbf{k}}
\end{equation}
and is calculated via tetrahedron integration \cite{Andersen1994}. The band velocities $\textbf{v}_{n\textbf{k}}$ are obtained from Wannier interpolation; $\alpha$ and $\beta$ are cartesian directions, and $V_{\textrm{uc}}$ is the unit cell volume. The hole mobility along the direction $\alpha$ is computed using $\mu_\alpha=\sigma_{\alpha \alpha}/n_{p}e$, where $n_{p}$ is the hole concentration. These e-ph and mobility calculations on unit cells with tens of atoms are made possible by efficient algorithms combining MPI plus OpenMP parallelizations we recently developed.\\%
\begin{figure*}[!th]
\includegraphics[scale=1.0]{Figure2.pdf}
\caption{The hole mobility in naphthalene, given, from left to right in separate panels, in the two in-plane directions \textit{a}, \textit{b} and in the plane-normal direction \textit{c}*. Circle markers are the computed mobilities and black markers the experimental data from Ref. [\onlinecite{Karl1985}]. Straight lines are best fits to the power law function $T^{-n}$ of the data points in the 100$-$300 K temperature range, and the exponent $n$ for each data set is also given. The error bars are obtained by assuming a $10\%$ error on both the phonon frequencies and the GW band stretching factor. These error sources are assumed to be independent and combined together.}
\label{fig:mobility}
\end{figure*}
\indent
The computed bandstructures and phonon dispersions are given in the appendix (See Fig. \ref{fig:bands}). The GW correction is important as it stretches the valence band, thus lowering the hole effective mass and changing the relative alignment of the valence band valleys. The quality of our phonon dispersions is comparable with that of recent accurate phonon calculations in naphthalene \cite{Neaton2016}. For reference, we also employ the methods above to compute the phonon dispersion of the perdeuterated naphthalene. The comparison with experimental data is given in the appendix (See Fig. \ref{fig:D8}).\\
\section{results}
\vspace{-10pt}
\indent
Figure \ref{fig:mobility} shows our calculated hole mobilities in the in-plane \textit{a} and \textit{b} and the plane-normal \textit{c}* directions (see Fig. \ref{fig:crystal_structure}).
The experimental data given for comparison is taken from Ref. [\onlinecite{Karl1985}].
The computed mobilities are lower by a factor of 3 $-$ 5 than the experimental values; the smallest discrepancy (a factor of 3) is found for the \textit{a} direction, and the highest (a factor of 5) in the c* direction.
Note that the \textit{c}* axis corresponds to a direction along which the molecules are stacked, so that the slightly lower accuracy in this direction is expected due to our neglect of van der Waals interactions in the e-ph coupling.
Fitting the data with a power law function $T^{-n}$ over the 100$-$300 K temperature range yields calculated exponents $n$ in the 2.34$-$2.88 range for the three directions, in agreement \textit{within 3\%} (in the \textit{ab} plane) and 10\% in the \textit{c}* direction with the exponents $n$ obtained by fitting the experimental data (see Fig. \ref{fig:mobility}).
The charge transport anisotropy is estimated by evaluating mobility ratios between different directions at 300 K. The computed ratios, $\mu_b/\mu_a=1.16$ and $\mu_{c\textrm{*}}/\mu_a=0.18$ are consistent with the experimental values of $1.57$ and $0.34$, respectively.\\
\indent
Since the accuracy of the phonon dispersions and GW bandstructures depends on the chosen crystal structure, exchange-correlation functional, and pseudopotential,
it is important to quantify how these sources of uncertainty affect the computed mobility. To this end, we estimate how the combination of a small error in the GW correction (arbitrarily chosen to be $\sim$10\% in the stretching factor of the valence band) and an assumed $\sim$10\% error on the phonon frequencies (a conservative value for organic crystals) affect our calculations. The resulting error bars on the mobilities are given in Fig. \ref{fig:mobility}.\\
\indent
Within these uncertainties, which are typical of \textit{ab initio} methods $-$ especially for organic crystals with complex structures $-$ the range of computed mobilities (inclusive of the error bars) reaches values roughly 2$-$3 times smaller than the experimental result in the in-plane \textit{a} and \textit{b} directions.
Overall, the temperature trends and absolute values of the mobility are remarkably accurate, particularly when compared to the very scarce literature on charge transport in organic crystals using \textit{ab initio} band theory.
Our accuracy is comparable to the best calculations \cite{Negri2014,Troisi2007} using quantum chemistry methods based on hopping that dominate the literature.\\
\indent
We have verified that employing the Tkatchenko-Scheffler (TS) vdW correction \cite{TS1,TS2}, which is more accurate than the Grimme-vdW correction used here, does not change appreciably the structure and mobility.
In particular, the root-mean-square (RMS) deviation between the atomic positions obtained with the Grimme-vdW and the TS-vdW corrections is only 0.05~{\AA}, and the RMS deviation of the bond lengths is $\sim$0.05\%. The mobility at 300 K obtained by computing the bandstructure, phonons and e-ph matrix elements with the structure obtained using the TS-vdW correction is very close (within 5$-$10\%, and thus within the error bars in Fig. \ref{fig:mobility}) to the mobility computed here using the Grimme-vdW method (see Fig. \ref{fig:TS_mobility} in Appendix). Future work will investigate further the role of the vdW correction on the e-ph coupling and mobility in organic crystals.\\
\indent
Next, we investigate the role of different phonon modes in scattering the hole carriers. In the charge hopping picture, the conventional wisdom is that low-frequency inter-molecular phonon modes, which correspond to rigid motions of entire molecules \cite{Beljonne2013_02, Orlandi2006}, determine the mobility since they strongly affect the rate of charge hopping between molecules. Intra-molecular vibrations, on the other hand, are typically neglected due to their hypothesized weaker coupling to the carriers.
There are 108 phonon modes in naphthalene, the 12 lowest-frequency modes are inter-molecular, and the others are intra-molecular.
We express the total e-ph scattering rate in Eq. (\ref{eq:e_scat_rate}) as the sum of the scattering rates due to each individual mode $\nu$, i.e., $\Gamma^{\textrm{e-ph}}_{n\textbf{k}}=\sum_{\nu}\Gamma^{(\nu)}_{n\textbf{k}}$, and investigate the mode-resolved scattering rates $\Gamma^{(\nu)}_{n\textbf{k}}$. Here and in the following, the phonon modes are numbered in order of increasing energy at the Brillouin zone center, and the hole energy increases moving away from the valence band maximum (VBM) into the valence band.\\
\indent
Figure \ref{fig:mode_analysis}(a) shows the mode-resolved e-ph scattering rates as a function of hole energy for the 12 inter-molecular phonon modes, and Fig. \ref{fig:mode_analysis}(b) for selected intra-molecular phonons.
Note that the inter-molecular phonons have either zero or very small minimum frequency since they correspond to transverse acoustic (TA) and longitudinal acoustic (LA) vibrations (modes 1$-$3) or other rigid vibrations or librations of the molecules (modes 4$-$12). By contrast, the intra-molecular modes 20$-$90 in Fig. \ref{fig:mode_analysis}(b) possess much higher frequencies.
The integrand of the mobility in Eq. (\ref{eq:conductivity}) is also plotted in Figs. \ref{fig:mode_analysis}(a)$-$\ref{fig:mode_analysis}(b) to highlight the energy window contributing to the mobility, which spans hole states within 50$-$100 meV of the VBM. In this energy window, the 12 inter-molecular phonon modes exhibit much greater scattering rates than the intra-molecular modes, due to reasons related to the e-ph scattering phase space that are examined next.\\
\indent
In the hole scattering rates of Eq. (\ref{eq:e_scat_rate}), the first term in square brackets corresponds to phonon emission, and is proportional to the phonon population $N_{\nu\textbf{q}}$+1 since $f_{m\textbf{k}+\textbf{q}}\!\approx \!0$ for holes in our chosen temperature range. The term in the second square brackets is the phonon absorption rate, which is proportional to $N_{\nu\textbf{q}}$.
Since the inter-molecular phonon modes 1$-$12 have a zero or small minimum energy, inter-molecular phonon absorption and emission processes are \textit{both} active at all hole energies.
Their scattering rate decreases monotonically with phonon energy (and thus with mode number, since the modes are numbered in order of increasing energy). Similar to simple metals and non-polar inorganic semiconductors, the main source of scattering are acoustic modes, with smaller contributions from other molecular rigid vibrations and librations (modes 4$-$12).
This result is further illustrated in Fig. \ref{fig:mode_analysis}(c), where the average $\Gamma^{(\nu)}_{n\textbf{k}}$ over the $100$ meV energy window of relevance for the mobility is given for each phonon mode. The dominant role of inter-molecular modes is consistent with the charge hopping intuition that rigid molecular vibrations mainly affect charge transport in organic materials. However, in our band picture based on phonon emission and absorption events, the origin of this behavior can be attributed to the phase space rather than the strength of the e-ph coupling per se, as further discussed below.\\
\begin{figure}[t]
\includegraphics{Figure3.jpg}
\caption{Mode resolved e-ph scattering rates, $\Gamma^{(\nu)}_{n\textbf{k}}$, for (a) the 12 inter-molecular phonon modes and (b) selected intra-molecular phonon modes (note the y-axis log scale). Also sketched in (b) are the dominant e-ph scattering processes below and above the phonon emission threshold energy $\hbar \omega_0$, which is shown as a vertical dashed line for mode 90. The black dashed curve represents the integrand in Eq. (\ref{eq:conductivity}), and shows that only hole states within a 50$-$100 meV energy window of the valence band maximum (VBM) contribute to the mobility. (c) Mode-resolved scattering rates averaged over the energy window contributing to the mobility. In all plots, the zero of the energy axis is the VBM, and the hole energy increases moving away from the VBM into the valence band.}
\label{fig:mode_analysis}
\vspace{-1.0\baselineskip}
\end{figure}
\indent
The effect of intra-molecular phonons on the mobility is more subtle. Figure \ref{fig:mode_analysis}(b) shows that the e-ph scattering rates for these modes exhibit a trend with two plateaus as a function of hole energy. As explained next, the plateau at low hole energy corresponds to phonon absorption, and the one at higher hole energy to phonon emission.
Consider an intra-molecular phonon with minimum energy $\hbar \omega_{0}$. Due to energy conservation, a hole in the valence band can emit such a phonon only at hole energy higher than $\hbar \omega_{0}$.
At hole energies below this threshold, only phonon absorption is possible, with a rate proportional to the phonon occupation $ N_{\nu \textbf{q}} \!\propto\! e^{-\hbar \omega_0 / k_B T}$, which is much smaller than 1 at room temperature in naphthalene since most intra-molecular modes have minimum energies $\hbar \omega_0\!\approx\!$ 50$-$200 meV.
Therefore, the plateau at hole energies below $\hbar \omega_0$ is associated with a small intra-molecular phonon absorption rate, and it spans the entire energy window contributing to the mobility.\\
\indent
At hole energies above $\hbar \omega_{\textrm{0}}$, the phase space for e-ph scattering increases dramatically since holes can emit intra-molecular phonons, with a rate proportional to $N_{\nu \textbf{q}} + 1$ and thus much greater than the absorption rate.
Opening this phonon emission channel leads to an increase of the e-ph scattering rates by several orders of magnitude, but this increase occurs outside the energy window of relevance for charge transport due to the high energy of intra-molecular phonons in naphthalene.
These trends are expected to be general in organic crystals, since the dominant presence of hydrogen, carbon and other light elements makes their intra-molecular phonon energies much greater than $k_B T$. Interestingly, in organic molecules containing heavy atoms, which introduce low-frequency intra-molecular vibrations, a contribution to transport from intra-molecular phonons is expected.\\% in most organic crystals.\\
\indent
In short, the two-plateau structure for intra-molecular mode e-ph scattering is such that only the small rate for thermally activated phonon absorption falls in the energy range of interest for transport. Therefore the mobility is controlled by low-frequency inter-molecular vibrations.
However, note that intra-molecular phonons are expected to dominate carrier dynamics at higher hole energy above the phonon emission threshold, where their combined scattering rate overwhelms that from the (much fewer) inter-molecular modes.
This analysis shows that intra-molecular phonons play an essential role in the dynamics of excited carriers \cite{Zhou2017,Zhou2016,Louie2014, Bernardi-review} in organic semiconductors.\\
\section{discussion}
\vspace{-10pt}
\indent
While the phase space limits their scattering near the band edge, intra-molecular phonons can couple strongly with holes at all energies, and in fact more strongly than inter-molecular modes. To study this point, we compute the local e-ph coupling constants $g^{\textrm{(loc)}}_{\nu \textbf{q}}$ between each phonon mode at the Brillouin zone center ($\textbf{q}=0$) and the HOMO Wannier function (WF) $w_\textbf{R}$(\textbf{r}):
\begin{equation}
\label{eq:local_g}
g^{\textrm{(loc)}}_{\nu\textbf{q}}=\sqrt{\frac{\hbar}{2\omega_{\nu\textbf{q}}}}\langle w_\textbf{R} | \Delta_{\nu\textbf{q}}V^{\textrm{KS}} | w_\textbf{R} \rangle,
\end{equation}
where $\textbf{R}$ is the WF center, and the change in Kohn-Sham potential $\Delta_{\nu\textbf{q}}V^{\textrm{KS}}$ arises from the atomic displacements $e_{\kappa\alpha,\nu}$ of each atom $\kappa$ (with mass $M_{\kappa}$) along all cartesian directions $\alpha$ due to the phonon mode $\nu$,\begin{equation}
\Delta_{\nu\textbf{q}}V^{\textrm{KS}} = e^{i\textbf{q}\cdot \textbf{r}} \sum_{\kappa\alpha} \frac{1}{\sqrt{M_\kappa}} e_{\kappa\alpha,\nu}\partial_{\kappa\alpha,\textbf{q}} V^{\textrm{KS}}
\end{equation}
\begin{figure}[t]
\includegraphics{Figure4.pdf}
\caption{(a) The absolute value of the local coupling constant [see Eq. (\ref{eq:local_g})] between each of the phonon modes and the HOMO Wannier function. (b) The square of the HOMO Wannier function. The potential perturbation $\Delta_{\nu\textbf{q}}V^{\textrm{KS}}$ at $\textbf{q}=0$ is shown for (c) mode $\nu=88$ and (d) mode $\nu=89$. These modes correspond to the peak (mode 88) and sudden drop (mode 89) in e-ph coupling in (a). In panels (b)$-$(d), yellow is used for positive, and blue for negative isosurfaces.}
\label{fig:g_analysis}
\vspace{-1\baselineskip}
\end{figure}
\indent
The absolute value of these local e-ph coupling constants are shown in Fig. \ref{fig:g_analysis}(a) for all 108 phonon modes \footnote{Note that the e-ph coupling constants for the three acoustic phonon modes vanish as $\textbf{q}\rightarrow 0$}.
Contrary to intuition, the strongest e-ph coupling to the HOMO hole state is not with the inter-molecular modes that control transport.
Rather, specific high-frequency intra-molecular phonons (in particular, modes 79$-$88) exhibit the strongest coupling to holes. To understand this result, we plot quantities entering the local e-ph coupling in Eq. (\ref{eq:local_g}), namely the square of the HOMO WF, $|w_\textbf{R}(\textbf{r})|^2$, and the perturbation potential $\Delta_{\nu\textbf{q}}V^{\textrm{KS}}$ due to the atomic motions associated with the given mode.\\
\indent
Figure \ref{fig:g_analysis}(b) shows the square of the HOMO WF orbital, $|w_\textbf{R}(\textbf{r})|^2$; the perturbation potential $\Delta_{\nu\textbf{q}}V^{\textrm{KS}}(\mathbf{r})$ at $\textbf{q}=0$ is shown in Fig. \ref{fig:g_analysis}(c) for mode 88 and Fig. \ref{fig:g_analysis}(d) for mode 89, which are respectively cases of maximally strong and weak e-ph coupling. We find that e-ph coupling is maximal for mode 88 due to the strong overlap between the square of the HOMO WF and the perturbation potential, and the fact that both quantities possess the same sign over most of the molecule, so that no cancellations occur in the real-space integral in Eq. (\ref{eq:local_g}).
By contrast, the symmetry of mode 89 is such that its perturbation potential $\Delta_{\nu\textbf{q}}V^{\textrm{KS}}(\textbf{r})$ alternates positive and negative lobes at bonds where the square of the HOMO WF is large. As a result, the \textit{integrand} $\left | w_\textbf{R} (\textbf{r}) \right|^2 \cdot \Delta_{\nu\textbf{q}}V^{\textrm{KS}}(\textbf{r})$ in Eq.~\ref{eq:local_g} is positive for two bonds and negative (and roughly equal in absolute value) for the other two bonds, thus leading to a small integral over the entire molecule in Eq.~\ref{eq:local_g}. This cancellation results in a small e-ph coupling for mode 89. Other phonon modes are either associated with perturbation potentials with small overlap with the square of the HOMO WF, as is the case for modes in which only the hydrogen atoms vibrate, or with perturbations that are out of phase with the square of the HOMO WF, similar to the case of mode 89.
This analysis shows that the atomic displacements and mode symmetry critically determine the e-ph coupling of intra-molecular modes, which can be much stronger than that of inter-molecular modes due to the large spatial overlap between the square of the HOMO WF and the intra-molecular mode perturbation.\\%, since the latter falls entirely within the molecule.\\
\indent
Lastly, we comment on the fact that our computed phonon-limited mobility is smaller than the experimental result. Due to the presence of impurities and defects in real samples, our calculation is expected to provide an upper bound to the mobility, and thus to slightly overestimate its experimental value, consistent with our recent results for inorganic crystals \cite{Zhou2016}.
The reason why our result is lower than experiment is unclear, but a possible cause is the neglect of non-adiabatic effects.\\
\indent
Our method employs only the lowest Born--Oppenheimer potential energy surface (PES), since the e-ph perturbation potential is computed using DFPT. However, an insight from non-adiabatic surface hopping calculations \cite{BlumbergerReview,Tavernelli2013} is that several PESs can lie close in energy in organic crystals, and including their contributions to charge transport may increase the mobility.
The impact of such non-adiabatic effects on the mobility within the band theory framework used here deserves further investigation. Nonetheless, the fact that our results underestimate the measured mobility is important as it further supports the conclusion in Ref. [\onlinecite{Stojanovic2012}] that hole charge carriers in naphthalene crystals are weakly coupled to phonons, so that transport occurs in the band-like regime studied here. In fact, polaronic effects resulting from strong e-ph coupling (beyond the lowest order employed here) would further suppress carrier transport by increasing the scattering rates and effective masses \cite{Mahan2000}, thus reducing the mobility.\\
\section{conclusion}
\vspace{-10pt}
\indent
In summary, we compute with quantitative accuracy the hole mobility and its temperature dependence in naphthalene, dramatically improving the agreement with experiment compared to previous efforts using band theory to study charge transport in organic crystals.
Our results show that \textit{ab initio} approaches based on band theory and many-body perturbation theory are well equipped to compute charge transport in organic semiconductors. They can provide an accuracy at least as satisfactory as widespread quantum chemistry methods based on charge hopping, as well as insight into the role of different phonon modes.
Our work sets the stage for attempting higher-order corrections or diagram resummations in the e-ph perturbation to access the strong e-ph coupling regime typical of polaron transport.
\section{ACKNOWLEDGMENTS}
\begin{acknowledgments}
The authors thank Maurizia Palummo for discussions.
N.-E.L. acknowledges the Physics department at Caltech for the TA Relief Fellowship. M.B. and L.A. acknowledge support by the National Science Foundation under Grant ACI-1642443, which provided for basic theory and electron-phonon code development. J.-J. Zhou was supported by the Joint Center for Artificial Photosynthesis, a DOE Energy Innovation Hub, as follows: the development of the scattering rate and mobility calculations was supported through the Office of Science of the U.S. Department of Energy under Award No. DE-SC0004993. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
\end{acknowledgments}
\vspace{-1\baselineskip}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,034 |
Q: IIS 8 with PHP 5.6: WinCache VS Zend Op Cache Zend Op Cache is compiled by default on PHP5.5+, but
IIS encourages use of WinCache for opcode caching.
what is the extension that will give me the best performance and stability on IIS?
there are benchmarks, case studies?
A: That article about WinCache is over seven years old, and is only about PHP 5.2.x and 5.3.x, which did not come with a built-in OpCache that worked on Windows. Now that PHP comes with its own OpCache, you should be using that one.
Additionally, the official release notes for WinCache version 1.3.5.0, the first version to support PHP 5.5, say:
Opcode Cache is disabled by default for PHP 5.5 because Opcache is available in Core
This is a clear indication that the developers of WinCache agree.
NOTE: You can still use WinCache for its userspace caching functions. This question is only about the OpCache component of WinCache.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,191 |
.class Lcn/com/smartdevices/bracelet/ui/i;
.super Ljava/lang/Object;
# interfaces
.implements Landroid/widget/AdapterView$OnItemClickListener;
# instance fields
.field final synthetic a:Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;
# direct methods
.method constructor <init>(Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;)V
.locals 0
iput-object p1, p0, Lcn/com/smartdevices/bracelet/ui/i;->a:Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;
invoke-direct {p0}, Ljava/lang/Object;-><init>()V
return-void
.end method
# virtual methods
.method public onItemClick(Landroid/widget/AdapterView;Landroid/view/View;IJ)V
.locals 5
.annotation system Ldalvik/annotation/Signature;
value = {
"(",
"Landroid/widget/AdapterView",
"<*>;",
"Landroid/view/View;",
"IJ)V"
}
.end annotation
const-string v0, "AlarmRepeatActivity"
new-instance v1, Ljava/lang/StringBuilder;
invoke-direct {v1}, Ljava/lang/StringBuilder;-><init>()V
const-string v2, "onItemClick: "
invoke-virtual {v1, v2}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v1
invoke-virtual {v1, p3}, Ljava/lang/StringBuilder;->append(I)Ljava/lang/StringBuilder;
move-result-object v1
invoke-virtual {v1}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v1
invoke-static {v0, v1}, Lcn/com/smartdevices/bracelet/Debug;->i(Ljava/lang/String;Ljava/lang/String;)V
sget-object v0, Lcn/com/smartdevices/bracelet/model/AlarmClockItem;->WEEK_MASK:[I
aget v0, v0, p3
const/4 v1, 0x1
shl-int/2addr v1, p3
iget-object v2, p0, Lcn/com/smartdevices/bracelet/ui/i;->a:Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;
invoke-static {v2}, Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;->a(Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;)I
move-result v2
and-int/2addr v1, v2
shr-int/2addr v1, p3
const-string v2, "AlarmRepeatActivity"
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
const-string v4, "thebit at position: "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, p3}, Ljava/lang/StringBuilder;->append(I)Ljava/lang/StringBuilder;
move-result-object v3
const-string v4, ", thebit="
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, v1}, Ljava/lang/StringBuilder;->append(I)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v3
invoke-static {v2, v3}, Lcn/com/smartdevices/bracelet/Debug;->i(Ljava/lang/String;Ljava/lang/String;)V
iget-object v2, p0, Lcn/com/smartdevices/bracelet/ui/i;->a:Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;
if-nez v1, :cond_0
iget-object v1, p0, Lcn/com/smartdevices/bracelet/ui/i;->a:Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;
invoke-static {v1}, Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;->a(Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;)I
move-result v1
or-int/2addr v0, v1
:goto_0
invoke-static {v2, v0}, Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;->a(Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;I)I
iget-object v0, p0, Lcn/com/smartdevices/bracelet/ui/i;->a:Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;
invoke-static {v0}, Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;->b(Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;)Lcn/com/smartdevices/bracelet/view/SelectDaysView;
move-result-object v0
iget-object v1, p0, Lcn/com/smartdevices/bracelet/ui/i;->a:Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;
invoke-static {v1}, Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;->a(Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;)I
move-result v1
invoke-virtual {v0, v1}, Lcn/com/smartdevices/bracelet/view/SelectDaysView;->setDays(I)V
iget-object v0, p0, Lcn/com/smartdevices/bracelet/ui/i;->a:Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;
invoke-static {v0}, Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;->c(Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;)V
return-void
:cond_0
iget-object v1, p0, Lcn/com/smartdevices/bracelet/ui/i;->a:Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;
invoke-static {v1}, Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;->a(Lcn/com/smartdevices/bracelet/ui/AlarmRepeatActivity;)I
move-result v1
xor-int/lit8 v0, v0, -0x1
and-int/2addr v0, v1
goto :goto_0
.end method
| {
"redpajama_set_name": "RedPajamaGithub"
} | 252 |
{"url":"https:\/\/kb.osu.edu\/dspace\/handle\/1811\/15534","text":"# EFFECTS OF FERMI RESONANCE IN THE BANDS OF $^{15}N_{2}^{18}O$.\n\nPlease use this identifier to cite or link to this item: http:\/\/hdl.handle.net\/1811\/15534\n\nFiles Size Format View\n1968-O-03.jpg 151.1Kb JPEG image\n\n Title: EFFECTS OF FERMI RESONANCE IN THE BANDS OF $^{15}N_{2}^{18}O$. Creators: Mantz, A. W.; Jones, L. H.; Potter, R. M. Issue Date: 1968 Publisher: Ohio State University Abstract: In view of the excellent results obtained for the internuclear distances in nitrous oxide from a study of the $^{15}N_{2}^{18}O$ bands in the near $infrared,^{1}$ further high resolution observational data were obtained both in the near infrared ($2-3 \\mu$ region) as well as at the longer wavelengths (8 micron region). In the near infrared the resolving power was such that half value widths of about $0.025- 0.03 cm^{-1}$ were measured on unblended lines. A liquid helium cooled Ge:Hg detector was used at $8 \\mu$, and the resolution was slightly better than $0.05 cm^{-1}$. The effects of Fermi resonance in the $^{15}N_{2}^{18}O$ molecule will be discussed employing the observational data for the following completed polyads: $(10^{0}0, 02^{0}0), (11^{1}0,03^{1}0), (10^{0}1, 02^{0}1), (11^{1}1,03^{1}1), (20^{0}1, 12^{0}1, 04^{0}1)$. Description: This research was supported, in part, by the U.S. Atomic Energy Commission (COO-882-18) through a contract with The Ohio State University Research Foundation as a joint project between the Laboratory of Molecular Spectroscopy and Infrared Studies at The Ohio State University Physics Department and the Los Alamos Scientific Laboratory, Los Alamos, New Mexico. Mr. A.W. Mantz has been deputed under the Professional Advancement Program of the U.S. Air Force. $^{1}$J.L. Griggs Jr., K. Narahari Rao, L.H. Jones, and R.M. Potter, J. Mol. Spectry. 25. 34 (1968). Author Institution: Laboratory of Molecular Spectroscopy and Infrared Studies, Department of Physics, The Ohio State University; Los Alamos Scientific Laboratory, University of California URI: http:\/\/hdl.handle.net\/1811\/15534 Other Identifiers: 1968-O-3","date":"2015-04-27 00:22:31","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\": 0.39757251739501953, \"perplexity\": 2477.87363011153}, \"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-2015-18\/segments\/1429246656887.93\/warc\/CC-MAIN-20150417045736-00281-ip-10-235-10-82.ec2.internal.warc.gz\"}"} | null | null |
\section{Conclusion and Future Research}\label{sec:conclusion}
In this paper, we initiated a systematic algorithmic study of {\sc Fair-NET}\ and presented a comprehensive picture of the parameterized complexity of the problem. We showed \textsf{NP}-hardness results on special graph classes, which implied that the problem is \textsf{para-NP}-hard with respect to several combinations of structural graph parameters. We also showed that the problem is \textsf{FPT} for some combinations of structural graph parameters.
While our work is comprehensive, we stress that it also opens a whole new world of research questions within computational social choice. For illustration, let us mention a few such questions:
\begin{enumerate}[(1)]
\item Establishing the parameterized complexity of {\sc Fair-NET}\ with respect to $\mathtt{tw} + \Delta$.
\item Establishing the parameterized complexity of {\sc Fair-NET}\ with respect to $\mathtt{tw} + \Delta + \alpha$. By employing the standard dynamic programming technique over tree decomposition, we can show that the problem is \textsf{FPT} with respect to $\mathtt{tw} + \Delta$, when $\alpha$ is a constant. But, the parameterized complexity when $\alpha$ is not a constant is still open.
\item Establishing the classical complexity of {\sc Fair-NET}\ where $S = [n]$, ,where $n$ denotes the number of vertices in the input graph.
\item Studying the scenario where there is no input infrastructure graph and the objective is to construct one which is $S$-fair.
\item Analysis of related labellings such as \{0,1\}-{\sc Fair-NET}~\cite{beena2009and} and vertex-bimagic labeling~\cite{babujee20141}.
\item Introducing additional fairness notions for non-eliminating tournaments, perhaps by refining/extending/modifying the notion of $S$-fairness.
\item Introducing manipulation and bribery to {\sc Fair-NET}.
\end{enumerate}
\section{FPT Algorithms}\label{sec:fpt}
In this section, we develop \textsf{FPT} algorithms for the {\sc Fair-NET}\ problem with respect to several structural graph parameters. We begin by giving a observation for a graph $G$ to be $S$-fair when $G$ contains isolated vertices.
\begin{observation}
Let $G$ be a graph with $\delta(G) = 0$ and $S$ be a multiset of positive integers. Then, $G$ is $S$-fair if and only if for every vertex $v \in V(G), \deg_G(v) = 0$ (i.e., $G$ contains only isolated vertices).
\end{observation}
Due to the above observation, in the rest of this section, we assume that $G$ does not contain any isolated vertices. We now give conditions that a graph $G$ must satisfy to be $S$-fair when $G$ is a cycle or $\delta(G) = 1$.
\begin{lemma}\label{lem:degOneMagic}
Let $G$ be a connected graph with $\delta(G) = 1$ and $S$ be a multiset of positive integers. Then, $G$ is $S$-fair only if $G$ is a star.
\end{lemma}
\begin{proof}
Assume that $G$ is $S$-fair. Let $f$ be a function in $\mathcal{M}(G,S)$ and $k$ be the $S$-fairness constant. Let $v \in V(G)$ be a vertex of degree $1$, and let $u$ be the neighbor of $v$. As $\sum f(N_G(v)) = k$, we get that $f(u) = k$. Assume for contradiction that $G$ is not a star. Then, $|V(G)| \geq 4$, otherwise $G$ is a star. Let $w$ be a neighbor of $u$ other than $v$ such that $\degr_G(w) \geq 2$. (If no such $w$ exists, then $G$ is a star.) As $\sum f(N_G(w)) = k$ and $f(u) = k$, we get that $\sum f(N_G(w) \setminus \{u\}) = 0$. Since $N_G(w) \setminus \{u\} \neq \emptyset$ and labels are positive, this is a contradiction.
\end{proof}
\begin{lemma}\label{lem:cycleDistanceMagic}
Let $G$ be a cycle graph on $n$ vertices and $S$ be a multiset of positive integers. Let $k$ be the required $S$-fairness constant. Then:
\begin{itemize}
\item If $n \bmod 4 = 0$, then $G$ is $S$-fair if and only if $S$ contains $4$ labels $a, b, k-a, k-b$ with $\alpha_S(a) = \alpha_S(b) = \alpha_S(k-a) = \alpha_S(k-b) = n/4$, for some $a,b \in \mathbb{N}$ such that $a,b < k$.
\item If $n \bmod 4 \neq 0$, then $G$ is $S$-fair if and only if $S$ contains only one label, $k/2$, with $\alpha_S(k/2) = n$.
\end{itemize}
\end{lemma}
\begin{proof}
Assume first that $G$ is $S$-fair. Denote $V(G) = \{v_1, v_2, \ldots, v_n\}$ in the cyclic order, and let $f \in \mathcal{M}(G,S)$. Then, for every $v \in V(G), \sum f(N(v)) = k$. As $G$ is a cycle, for every $i \in [n], N(v_i) = \{v_{(i-1)\bmod n}, v_{(i+1)\bmod n}\}$. Thus, we have that for every $i \in [n], f(v_{(i-1)\bmod n}) + f(v_{(i+1)\bmod n}) = k$. If we expand these equations, we get the following:
\begin{equation}
f(v_1) + f(v_3) = f(v_3) + f(v_5), \ldots, f(v_{n-3}) + f(v_{n-1}) = f(v_{n-1}) + f(v_1).
\end{equation}
\begin{equation}
f(v_2) + f(v_4) = f(v_4) + f(v_6), \ldots, f(v_{n-2}) + f(v_n) = f(v_n) + f(v_2).
\end{equation}
\begin{equation}
f(v_3) + f(v_5) = f(v_5) + f(v_7), \ldots, f(v_{n-1}) + f(v_1) = f(v_1) + f(v_3).
\end{equation}
\begin{equation}
f(v_4) + f(v_6) = f(v_6) + f(v_8), \ldots, f(v_n) + f(v_2) = f(v_2) + f(v_4).
\end{equation}
From these equations, we get the following relations:
\begin{equation}\label{equ:1}
f(v_1) = f(v_5) = f(v_9) = \ldots = f(v_{n-3})
\end{equation}
\begin{equation}\label{equ:2}
f(v_2) = f(v_6) = f(v_{10}) = \ldots = f(v_{n-2})
\end{equation}
\begin{equation}\label{equ:3}
f(v_3) = f(v_7) = f(v_{11}) = \ldots = f(v_{n-1})
\end{equation}
\begin{equation}\label{equ:4}
f(v_4) = f(v_8) = f(v_{12}) = \ldots = f(v_n)
\end{equation}
Notice that Equations~\ref{equ:1},~\ref{equ:2},~\ref{equ:3} and~\ref{equ:4} contain all the vertices $v_i$, $i \in [n]$, such that $i \bmod 4 = 1$, $i \bmod 4 = 2$ , $i \bmod 4 =3$ and $i \bmod 4 = 0$, respectively. Now, we consider the following cases:
\begin{itemize}
\item If $n \bmod 4 = 1$, then $(n-3) \bmod 4 = 2, (n-2) \bmod 4 = 3, (n-1) \bmod 4 = 0$ and $n \bmod 4 = 1$. By the observation that Equation~\ref{equ:2} contains all the vertices $v_i$, $i \in [n]$, such that $i \bmod 4 = 2$, $f(v_{n-3}) = f(v_2)$. Similarly, $f(v_{n-2}) = f(v_3), f(v_{n-1}) = f(v_4)$ and $f(v_n) = f(v_1)$. So, by Equations~\ref{equ:1},~\ref{equ:2},~\ref{equ:3} and~\ref{equ:4}, we get that all the vertices have the same label. Thus, $S$ contains only one label, $k/2$, with $\alpha_S(k/2) = n$.
\item If $n \bmod 4 = 2$, then $(n-3) \bmod 4 = 3, (n-2) \bmod 4 = 0, (n-1) \bmod 4 = 1$ and $n \bmod 4 = 2$. By the observation that Equation~\ref{equ:2} contains all the vertices $v_i$, $i \in [n]$, such that $i \bmod 4 = 2$, $f(v_n) = f(v_2)$. Similarly, $f(v_{n-3}) = f(v_3)$. So, by Equations~\ref{equ:1},~\ref{equ:2},~\ref{equ:3} and~\ref{equ:4}, we get that $f(v_1) = f(v_3)$ and $f(v_2) = f(v_4)$. Denote $f(v_1) = a$ and $f(v_2) = b$. As $f(v_1) + f(v_3) = k$ and $f(v_2) + f(v_4) = k$, we get that $a = b = k/2$. Thus, $S$ contains only one label, $k/2$, with $\alpha_S(k/2) = n$.
\item If $n \bmod 4 = 3$, then $(n-3) \bmod 4 = 0, (n-2) \bmod 4 = 1, (n-1) \bmod 4 = 2$ and $n \bmod 4 = 3$. By the observation that Equation~\ref{equ:2} contains all the vertices $v_i$, $i \in [n]$, such that $i \bmod 4 = 2$, $f(v_{n-3}) = f(v_4)$. Similarly, $f(v_{n-2}) = f(v_1), f(v_{n-1}) = f(v_2)$ and $f(v_n) = f(v_3)$. So, by Equations~\ref{equ:1},~\ref{equ:2},~\ref{equ:3} and~\ref{equ:4}, we get that all the vertices have the same label. Thus, $S$ contains only one label, $k/2$, with $\alpha_S(k/2) = n$.
\item If $n \bmod 4 = 0$, then $(n-3) \bmod 4 = 1, (n-2) \bmod 4 = 2, (n-1) \bmod 4 = 3$ and $n \bmod 4 = 0$, so we do not get any new relation. Denote $f(v_1) = a$ and $f(v_2) = b$. As $f(v_1) + f(v_3) = k$ and $f(v_2) + f(v_4) = k$, we get $f(v_3) = k-a$ and $f(v_4) = k-b$. By Equations~\ref{equ:1},~\ref{equ:2},~\ref{equ:3} and~\ref{equ:4}, $S$ contains $4$ labels $a, b, k-a, k-b$ with $\alpha_S(a) = \alpha_S(b) = \alpha_S(k-a) = \alpha_S(k-b) = n/4$.
\end{itemize}
Conversely, first assume that $n \bmod 4 = 0$ and $S$ contains $4$ labels $a, b, k-a, k-b$ with $\alpha_S(a) = \alpha_S(b) = \alpha_S(k-a) = \alpha_S(k-b) = n/4$, for any $a,b \in \mathbb{N}$ such that $a,b < k$. Denote $V(G) = \{v_1, v_2, \ldots, v_n\}$ in the cyclic order. Let $f: V(G) \rightarrow S$ be a bijective function defined as follows. For every $i \in [n/4],$ let $ f(v_{4i - 3}) = a, f(v_{4i - 2}) = b, f(v_{4i - 1}) = k-a$ and $f(v_{4i}) = k-b$. It is easy to see that $f \in \mathcal{M}(G,S)$, so $G$ is $S$-fair.
Finally, assume that $n \bmod 4 \neq 0$ and $S$ contains only one label, $k/2$, with $\alpha_S(k/2) = n$. Let $f: V(G) \rightarrow S$ be a bijective function defined as follows. For every $v \in V(G), f(v) = k/2$. It is easy to see that $f \in \mathcal{M}(G,S)$, so $G$ is $S$-fair.
\end{proof}
We first prove that the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{fvs} + \alpha + \Delta$. The following two lemmas will be helpful in proving it.
\begin{lemma}\label{lem:extendToForest}
There exists an $\mathcal{O}(|V(G)|)$-time algorithm that, given
\begin{inparaenum}[(i)]
\item a graph $G$,
\item a multiset of positive integers $S$,
\item an induced subgraph $F$ of $G$ such that its a forest, and
\item a bijection $f'$ from $N_G(F) \cup \mathrm{leaves}(F)$ to a subset $S'$ of $S$,
\end{inparaenum}
returns another bijection $f''$ from $N_G(F) \cup V(F)$ to a set $S''$\footnote{$S''$ may not be a subset of $S$.} such that $S' \subseteq S''$ and $f' = f''|_{V'}$. Moreover, if $G$ is $S$-fair and $f' = f|_{N_G(F) \cup \mathrm{leaves}(F)}$ for some $f \in \mathcal{M}(G,S)$, then $f'' = f|_{N_G(F) \cup V(F)}$.
\end{lemma}
\begin{proof}
Let $(G,k)$ be an instance of {\sc Fair-NET}. Let $k$ be the required $S$-fairness constant. Let ${\cal F} =\{T_1, T_2, \ldots, T_t\}$ be the set of connected components of $F$. For every tree $T \in {\cal F}$, we do the following. Let $r$ be an arbitrarily chosen non-leaf vertex of $T$. Then, consider $T$ as a rooted tree with $r$ as the root vertex. Let $d$ be the depth of the tree $T$. We partition the vertex set $V(T) = V_1 \cup V_2 \ldots \cup V_d$, such that $V_i$ contains all the vertices of $T$ at depth $i$. Note that $V_1 = \{r\}$ and $V_d \subseteq \mathrm{leaves}(T)$. Now, consider a vertex $v \neq r$ in $T$. We partition $N_G(v) = \{p_v\} \cup \big(\mathrm{children}_T(v) \cap \mathrm{leaves}(T)\big) \cup \big(\mathrm{children}_T(v) \setminus \mathrm{leaves}(T)\big) \cup \big(N_G(v) \setminus V(T)\big)$, where $p_v$ is the parent of $v$ in $T$, $\mathrm{children}_T(v) \cap \mathrm{leaves}(T)$ is the set of children of $v$ in $T$ which are leaves of $T$, $\mathrm{children}_T(v) \setminus \mathrm{leaves}(T)$ is the set of children of $v$ in $T$ which are non-leaf vertices of $T$ and $N_G(v) \setminus T$ is the set of neighbors of $v$ not in $T$.
Given $f'$, we define another function $f_T$ on $V' = V(T) \setminus \mathrm{leaves}(T)$ recursively as follows.
\begin{itemize}[(i)]
\item {\bf Base Case:} For all $v \in V'$ such that $v$ has a leaf child, let $w$ be an arbitrarily chosen leaf child of $v$. Then, $f_T(v) = k - \sum f'(N_G(w) \setminus V(T))$.
\item {\bf Recursive Step:} For all $v \in V'$ such that $v$ doesn't have any leaf child, let $w$ be an arbitrarily chosen child of $v$. Then, $f_T(v) = k - \sum f_T(\mathrm{children}_T(w) \setminus \mathrm{leaves}(T)) - \sum f'(\mathrm{children}_T(w) \cap \mathrm{leaves}(T)) - \sum f'(N_G(w) \setminus T) $.
\end{itemize}
Note that, if $v \in V_i$, then $\mathrm{children}_T(v) \in V_{i+1}$. So, we compute $f_T$ by processing vertices of $T$ in the order $V_{d-1}, \ldots, V_1$. We now define $f''$ from $N_G(F) \cup V(F)$ to $S'' = S' \cup f_{T_1} \cup f_{T_2} \ldots f_{T_t}$ as follows.
\begin{itemize}[(i)]
\item For every $v \in N_G(F) \cup L, f''(v) = f'(v)$.
\item For every $i \in [t]$ and $v \in V(T_i) \setminus \mathrm{leaves}(T_i), f''(v) = f_{T_i}(v)$.
\end{itemize}
Clearly, $f' = f''|_{V'}$ and therefore $S' \subseteq S''$. The above recursive procedure visits every vertex of $G$ at most once, so it runs in time $\mathcal{O}(|V(G)|)$.
Now, suppose that $G$ is an $S$-fair graph and $f' = f|_{N_G(F) \cup \mathrm{leaves}(F)}$ for some $f \in \mathcal{M}(G,S)$. As $f \in \mathcal{M}(G,S)$, for every $T \in {\cal F}$ and $v \in V(T), \sum f(N(v)) = k$. Consider a tree $T \in {\cal F}$. Let $v$ be a non-leaf vertex of $T$. Then, for all $w \in \mathrm{children}_T(v), \sum f(N_G(w)) = k \Rightarrow f(v) + \sum f(\mathrm{children}_T(w) \setminus \mathrm{leaves}(T)) - \sum f(\mathrm{children}_T(w) \cap \mathrm{leaves}(T)) - \sum f(N_G(w) \setminus V(T)) = k$. As $f' = f|_{N_G(F) \cup \mathrm{leaves}(F)}$, for all $v \in V(T) \setminus \mathrm{leaves}(T)$ and $w \in \mathrm{children}_T(v), f(v) = k - \sum f(\mathrm{children}_T(w) \setminus \mathrm{leaves}(T)) - \sum f'(\mathrm{children}_T(w) \cap \mathrm{leaves}(T)) - \sum f'(N_G(w) \setminus V(T))$. If $w$ is a leaf node, then $\mathrm{children}_T(w) = \emptyset$. So, we can write $f|_{N_G(F) \cup V(F)}$ as follows.
\begin{itemize}[(i)]
\item For every $v \in N_G(F) \cup L, f(v) = f'(v)$.
\item For every $i \in [t]$ and $v \in V(T_i) \setminus \mathrm{leaves}(T_i)$,
\begin{itemize}
\item if $v$ has a leaf child $w$, then $f(v) = k - \sum f'(N_G(w) \setminus V(T))$.
\item else, let $w$ be a child of $v$, then $f(v) = k - \sum f(\mathrm{children}_T(w) \setminus \mathrm{leaves}(T)) - \sum f'(\mathrm{children}_T(w) \cap \mathrm{leaves}(T)) - \sum f'(N_G(w) \setminus V(T))$.
\end{itemize}
\end{itemize}
As $f''$ and $f|_{N_G(F) \cup V(F)}$ have same base case and recursive step, $f'' = f|_{N_G(F) \cup V(F)}$. This also implies that $S'' \subseteq S$.
\end{proof}
The following corollary directly follows from the above lemma.
\begin{corollary}\label{cor:checkExtendingToForest}
Let $G$ be a graph and $S$ be multiset of positive integers. Let $F$ be an induced subgraph of $G$ that is a forest. Then, in time $\mathcal{O}(\alpha(S)^{|N_G(F)| + |\mathrm{leaves}(F)|} \cdot |V(G)|)$, we can compute a superset of the set $\cal G$ of functions from $N_G(F) \cup V(F)$ to $S$ such that for every $f \in \mathcal{M}(G,S)$, there exists a $g \in {\cal G}$ such that $g = f|_{N_G(F) \cup V(F)}$.
\end{corollary}
\begin{proof}
As the number of different functions $f'$ from $N_G(F) \cup \mathrm{leaves}(F)$ to $S$ is $\mathcal{O}(\alpha(S)^{|N_G(F)| + |\mathrm{leaves}(F)|})$, we can compute a superset of $\cal G$ by repeatedly applying Lemma~\ref{lem:extendToForest} for each such $f'$ and adding $f''$ to $\cal G$ if $S'' \subseteq S$.
\end{proof}
\begin{lemma}\label{lem:alphaDeltaStar}
The {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\alpha + \Delta$ for disjoint union of stars.
\end{lemma}
\begin{proof}
The \textsf{FPT} algorithm is based on ILP. We first give the algorithm and then prove its correctness. \smallskip\\
{\bf Algorithm:} Let $(G,k)$ be an instance of {\sc Fair-NET}\ for disjoint union of stars. Thus, $G = K_{1, n_1} + K_{1, n_2} + \ldots + K_{1, n_t}$ for some $t, n_1, n_2, \ldots, n_t \in \mathbb{N}$. Note that $\Delta = \Delta(G) = $ max$(n_1, n_2, \ldots, n_t)$. Denote $V(K_{1, n_i}) = \{v_i\} \cup B_i$ where $v_i$ is the highest degree vertex in $K_{1, n_i}$ and $B_i$ is the set of all other vertices in $K_{1, n_i}$, for every $i \in [t]$. Let $k$ be the required $S$-fairness constant of $G$. By Observations~\ref{obs:disjointMagic} and~\ref{obs:bipartiteMagic}, $G$ is $S$-fair if and only if there exists a bijection $f: V(G) \rightarrow S$ such that for all $i \in [t], f(v_i) = \sum f(B_i) = k$. So, $G$ is $S$-fair only if $\alpha_S(k) \geq t$. Let $S' = S \setminus \{s_1 = k, s_2 = k, \ldots, s_t = k\}$ and let $\widehat{\alpha} = \alpha(S')$. Let $\ell_1, \ell_2, \ldots, \ell_{\widehat{\alpha}}$ be the unique labels in $S'$. Let ${\cal D} = \{B_1, B_2, \ldots, B_t\}$. We partition $\cal D$ into ${\cal D}_1, {\cal D}_2, \ldots, {\cal D}_\Delta$ such that for every $i \in [\Delta]$, every set $B \in {\cal D}_i$ is of size $i$. As the number of unique labels are $\widehat{\alpha}$, for every $i \in [\Delta]$, any set in ${\cal D}_i$ can have at most $\widehat{\alpha}^i$ different label assignments. For every $i \in [\Delta],$ let ${\cal L}_i$ be the set of feasible label assignments for ${\cal D}_i$, i.e., the label assignments for any set in ${\cal D}_i$ for which the sum of the labels is $k$. For every $i \in [\Delta]$, every label assignment $la \in {\cal L}_i$ is a set $\{t^1_{la}, t^2_{la}, \ldots, t^{\widehat{\alpha}}_{la}\}$, where, for every $j \in [\widehat{\alpha}]$, $t^j_{la}$ denoted the number of times label $\ell_j$ is used in the label assignment $la$. For every $i \in [\Delta]$ and $la \in {\cal L}_i$, we have a variable $n_{i,la}$. For any $i \in [\Delta]$, $la \in {\cal L}_i$ and a function $f \in \mathcal{M}(G,S)$, $n_{i,la}$ represents the number of times label assignment $la$ is used in ${\cal D}_i$ for $f$.
Then, the algorithm works as follows.
\begin{itemize}
\item If $\alpha_S(k) < t$, then return {\sc False}.
\item Solve the following ILP to find $n_{i,la}$, for every $i \in [\Delta]$ and $la \in {\cal L}_i$.
\begin{equation}\label{equ:numberOfStars}
\forall i \in [\Delta], \sum_{la \in {\cal L}_i} n_{i, la} = |{\cal D}_i|.
\end{equation}
\begin{equation}\label{equ:numberOfLabels}
\forall j \in [\widehat{\alpha}], \sum_{i \in [\Delta]} \sum_{la \in {\cal L}_i} n_{i, la} \cdot t^j_{la} = \alpha_{S'}(\ell_j).
\end{equation}
\begin{equation}
\forall i \in [\Delta], \forall la \in {\cal L}_i; n_{i, la} \geq 0.
\end{equation}
\item If the ILP returns a feasible solution, then return {\sc True}; otherwise, return {\sc False}.
\end{itemize}
\smallskip
{\bf Correctness:} Equation~\ref{equ:numberOfStars} ensures that for every $i \in [\Delta]$, the number of label assignments used in ${\cal D}_i$ is equal to the number of sets ${\cal D}_i$ has. Equation~\ref{equ:numberOfLabels} ensures that for every $j \in [\widehat{\alpha}]$, the number of times label $\ell_j$ is used is equal to the number of times it appears in $S'$. For every $i \in [\Delta]$ and for every $B \in {\cal D}_i$, all the vertices in $B$ have the same neighborhood which is just a single vertex. Thus by Observation~\ref{obs:sameNeighborhood}, it is sufficient to know the label assignment for $B$; we can arbitrarily assign labels to the vertices in $B$ once we have decided which labels to use for $B$. Keeping this interpretation in mind, we now prove the correctness, i.e. the algorithm returns {\sc True} if and only if $G$ is $S$-fair.
Assume first that the algorithm returns {\sc True}. It means the ILP assigned non-negative integer values for the variables $n_{i,la}$, $i \in [\Delta]$ and $la \in {\cal L}_i$ such that Equations~\ref{equ:numberOfStars} and~\ref{equ:numberOfLabels} are satisfied. As for every $i \in [\Delta]$, ${\cal L}_i$ is the set of feasible label assignments, this implies that we got a label assignment for every $B \in {\cal D}$ such that sum of the labels of the vertices in $B$ is $k$. Thus, $G$ is $S$-fair.
Conversely, let $G$ be $S$-fair. Then, by Observations~\ref{obs:disjointMagic} and~\ref{obs:bipartiteMagic}, there exists a bijection $f: V(G) \rightarrow S$ such that for all $i \in [t], f(v_i) = \sum f(B_i) = k$. So, $\alpha_S(k) \geq t$. Moreover, every $B \in {\cal D}$ has a feasible label assignment so the ILP admits a feasible solution. Thus, the algorithm will return {\sc True}. As the number of variables $n_{i,la}$ is $\mathcal{O}(\Delta \cdot \widehat{\alpha}^\Delta)$, by Theorem~\ref{the:runningTimeILP}, the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\alpha + \Delta$ for disjoint union of stars.
\end{proof}
\begin{theorem}\label{the:fvsAlphaDelta}
The {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{fvs} + \alpha + \Delta$.
\end{theorem}
\begin{proof}
Let $(G,k)$ be an instance of {\sc Fair-NET}. Let $k$ be the required $S$-fairness constant. Then, we compute a minimum feedback vertex set $FVS$ of $G$ in time $\mathcal{O}(5^{|FVS|}\cdot|FVS|\cdot|V(G)|^2)$, using the algorithms given by Chen {\em et~al.}~\cite{DBLP:conf/wads/ChenFLLV07}. Let $\mathtt{fvs} = |FVS|$ and $F = V(G) \setminus FVS$. Then by definition of feedback vertex set, $G[F]$ is a forest. Let $\cal F$ be the set of connected components of $G[F]$. So, $\cal F$ is a collection of trees.
Let $T$ be a tree in $\cal F$. Let $v$ be a leaf vertex of $T$ such that $v$ is not connected to any vertex in $FVS$, then $\degr_G(v) = 1$. If $\degr_G(v) = 1$, then by Lemma~\ref{lem:degOneMagic}, either $G$ is a star and $G = T$ or $G$ is a disjoint union of $T$ and $G[V(G) \setminus V(T)]$. By this argument, for any tree $T \in {\cal F}$, either all the leaves of $T$ have at least one neighbor in $FVS$ or none of the leaves are connected to any vertex in $FVS$. So, we can partition $G = G_1 + G_2$, where $G_1$ is a connected graph where all the leaves of the forest $G_1[F]$ have at least one neighbor in $FVS$ and $G_2$ is a disjoint union of stars. Note that, $G_2$ is an induced subgraph of $G[F]$. By Observation~\ref{obs:disjointMagic}, $G$ is $S$-fair if any only if $G_1$ is $S_1$-fair and $G_2$ is $S \setminus S_1$-fair, for some $S_1 \subseteq S$.
Let $L$ be the set of leaves of $G_1[F]$. As $\Delta(G_1) \leq \Delta, |L| \leq \Delta \cdot \mathtt{fvs}$. By Corollary~\ref{cor:checkExtendingToForest}, we can compute in time $\mathcal{O}(\alpha^{(\Delta+1)\mathtt{fvs}}\cdot |V(G)|)$, a superset ${\cal H}$ of the set ${\cal G}$ of functions from $V(G_1)$ to $S$ such that for every $f \in \mathcal{M}(G,S)$, there exists a $g \in {\cal G}$ such that $g = f|_{V(G_1)}$. We can compute ${\cal G}$ from ${\cal H}$ by going over every set $h \in {\cal H}$ and checking whether $G_1$ is fair under $h$. If ${\cal G} \neq \emptyset$, then $G_1$ is $g(V(G_1))$-fair for every function $g \in {\cal G}$. Then, for every function $g \in {\cal G}$, check whether $G_2$ is $\big( S \setminus g(V(G_1)) \big)$-fair, using Lemma~\ref{lem:alphaDeltaStar}. If for some function $g \in {\cal G}$, $G_2$ is $\big( S \setminus g(V(G_1)) \big)$-fair, then by Observation~\ref{obs:disjointMagic}, $G$ is $S$-fair. Also, by Corollary~\ref{cor:checkExtendingToForest} and Lemma~\ref{lem:alphaDeltaStar}, the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{fvs} + \alpha + \Delta$.
\end{proof}
We now prove that the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{vc} + \alpha$.
\begin{theorem}\label{the:vcAlpha}
The {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{vc} + \alpha$.
\end{theorem}
\begin{proof}
The \textsf{FPT} algorithm is based on ILP. We first give the algorithm and then prove its correctness. \smallskip\\
{\bf Algorithm:} Let $(G,k)$ be an instance of {\sc Fair-NET}. Let $k$ be the required $S$-fair sum. Let $S'$ be the set of unique labels in $S$. Note that $|S'| = \alpha(S)$. Let $VC \subseteq V(G)$ be a vertex cover of $G$ of size $\mathtt{vc}$. Let $I = V(G) \setminus VC$. By the definition of vertex cover, $I$ is an independent set of $G$, i.e., no two vertices in $I$ have an edge between them. We partition $I$ into $I_1, I_2, \ldots, I_m$ such that for every $i \in [m]$, $I_i$ is an inclusion-wise maximal set of vertices in $I$ which have the same neighborhood in $G$. As $I$ is a independent set, $m \leq 2^\mathtt{vc}$. For every $v \in VC$, we define a binary indicator set $\{t^v_1, t^v_2, \ldots, t^v_m\}$, where $t^v_j = 1$ if $v$ is adjacent to the vertices in $I_j$, otherwise $t^v_j = 0$, for every $j \in [m]$.
By Observation~\ref{obs:sameNeighborhood}, if $G$ is $S$-fair and if we know the labels of $VC$ under some function $f \in \mathcal{M}(G,S)$, then for every $i \in [m]$, it is sufficient to know the number of times every label is used in $I_i$ under $f$ to get a bijective function $f' : V(G) \rightarrow S$ such that $f' \in \mathcal{M}(G,S)$. Keeping this insight in mind, let $n_{i,\ell}$ be a variable whose value (to be computed below) will be interpreted as the number of times label $\ell$ is used in $I_i$ for some function $f:V(G) \rightarrow S$, for every $\ell \in S', i \in [m]$. Then, the algorithm works as follows.
\begin{itemize}
\item Construct the set {\cal G} containing all possible functions $g:VC \rightarrow S$. Note that $|\cal G| \leq \alpha^\mathtt{vc}$.
\item For every $g \in {\cal G}$ and $\ell \in S'$, let $\alpha_g(\ell)$ denote the number of times $\ell$ appears in $g(VC)$.
\item For every $g \in {\cal G}$:
\begin{itemize}
\item Solve the following ILP to find an assignment to the variables $n_{i,\ell}$, for every $i \in [m], \ell \in S'$. \\
\begin{equation}\label{equ:VCSum}
\forall v \in VC, \sum_{i \in [m]} t^v_i \sum_{\ell \in S'} n_{i,\ell} \cdot \ell = k
\end{equation}
\begin{equation}\label{equ:SetCardinality}
\forall i \in [m], \sum_{\ell \in S'}n_{i,\ell} = |I_i|
\end{equation}
\begin{equation}\label{equ:LabelCardinality}
\forall \ell \in S', \sum_{i \in [m]}n_{i,\ell} = \alpha_S(\ell) - \alpha_g(\ell)
\end{equation}
\begin{equation}
\forall i \in [m], \forall \ell \in S'; n_{i, \ell} \geq 0.
\end{equation}
\item If the ILP returns a feasible solution, then return {\sc True} if the following statement holds.
\begin{equation}\label{equ:ISSum}
\forall i \in [m], \sum_{v \in VC} t^v_i \cdot f(v) = k
\end{equation}
\end{itemize}
\item Return {\sc False}.
\end{itemize}
\smallskip
{\bf Correctness:} Equation~\ref{equ:VCSum} ensures that for every vertex in $VC$, the neighborhood sum is $k$. Equation~\ref{equ:SetCardinality} ensures that for every $i \in [m]$, the total number of labels used in $I_i$ is equal to the size of $I_i$. Equation~\ref{equ:LabelCardinality} ensures that for every unique label $\ell \in S'$, the total number of times it is used is equal to the number of times it appear in $S$. Finally, Equation~\ref{equ:ISSum} ensures that for every vertex in $IS$, the neighborhood sum is $k$. As for every $i \in [m]$, all the vertices in $I_i$ have the same neighborhood, we only check the sum once per $I_i$. Keeping this interpretation in mind, we now prove the correctness, i.e. the algorithm returns {\sc True} if and only if $G$ is $S$-fair.
Assume first that the algorithm returns {\sc True}. It means the ILP assigned non-negative integer values to $n_{i,\ell}$, for every $i \in [m]$ and $\ell \in S'$, such that Equations~\ref{equ:VCSum},~\ref{equ:SetCardinality} and~\ref{equ:LabelCardinality} are satisfied as well as that Equation~\ref{equ:ISSum} returned {\sc True}. As explained above, that implies that for every vertex in $G$, the neighborhood sum is the same and equals to $k$. Thus, $G$ is $S$-fair.
Conversely, let $G$ be $S$-fair. Then, there exists a bijection $f:V(G) \rightarrow S$ such that for every $v \in V(G), \sum f(N(v)) = k$. As $\cal G$ is the set of all possible functions from $VC$ to $S$, $f|_{VC} \in {\cal G}$, and hence there exists an iteration where the algorithm examines $g = f|_{VC}$. For $g = f|_{VC}$, the ILP admits a feasible solution. Moreover, Equation~\ref{equ:ISSum} then holds. Thus, the algorithm will return {\sc True}. As $|\cal G| \leq \alpha^\mathtt{vc}$ and the number of variables $n_{i,\ell}$ is at most $2^\mathtt{vc} \cdot \alpha$, by Theorem~\ref{the:runningTimeILP}, the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{vc} + \alpha$.
\end{proof}
We now prove that the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{fvs}$ for regular graphs.
\begin{theorem}\label{the:fvsrRegular}
The {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{fvs}$ for regular graphs.
\end{theorem}
\begin{proof}
Let $r \in \mathbb{N}$. Let $(G,k)$ be an instance of {\sc Fair-NET}\ for $r$-regular graphs. Let $k$ be the required $S$-fairness constant. Then, we compute a minimum feedback vertex set $FVS$ of $G$ in time $\mathcal{O}(5^{|FVS|}\cdot|FVS|\cdot|V(G)|^2)$, using the algorithms given by Chen {\em et~al.}~\cite{DBLP:conf/wads/ChenFLLV07}. Let $\mathtt{fvs} = |FVS|$ and $F = V(G) \setminus FVS$. Then, $G[F]$ is a forest. We distinguish $G$ into the following three cases based on the value of $r$.\smallskip\\
{\bf Case $1$ [$r = 1$]:} In this case, $G$ is a collection of edges, i.e., $G = tP_2$ for some $t \in \mathbb{N}$. Then, by Observation~\ref{obs:disjointMagic} and by the definition of $S$-fair labeling, $G$ is $S$-fair if and only if $S = \biguplus_{i \in [t]}\{a_i, k-a_i\}$ where for all $i \in [t], a_i \in \{1, 2, \ldots, k\}$. So, we can solve {\sc Fair-NET}\ problem in $\mathcal{O}(|S|\log S) = \mathcal{O}(|V(G)|\log |V(G)|)$ time by sorting $S$ and checking whether $S$ satisfies the above property. \smallskip\\
{\bf Case $2$ [$r = 2$]:} In this case, $G$ is a collection of cycles, i.e. $G = C_{n_1} + C_{n_2} + \ldots + C_{n_t}$ for some $t \in \mathbb{N}$. By the definition of feedback vertex set and the minimality of $FVS$, every cycle contains exactly one vertex from $FVS$. So, $t = \mathtt{fvs}$. Also, by Observation~\ref{obs:disjointMagic} and by Lemma~\ref{lem:cycleDistanceMagic}, for every $f \in \mathcal{M}(G,S)$, every cycle is assigned at most $4$ distinct labels, by $f$. So, $\alpha(S) \leq 4\mathtt{fvs}$. As $\Delta(G) = 2$ and $\alpha \leq 4\mathtt{fvs}$, by Theorem~\ref{the:fvsAlphaDelta}, the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{fvs}$ in this case.\smallskip\\
{\bf Case $3$ [$r \geq 3$]:} As $G[F]$ is a forest, $|E(F)| \leq |V(F)| - 1 \leq |V(G)| - 1$. Also, $G$ is a $r$-regular graph so $|E(G)| = r \cdot |V(G)|/2$. This implies that, at least $(r/2-1)\cdot |V(G)| +1$ edges are incident to vertices of $FVS$. As every vertex of $FVS$ is incident to $r$ edges, $(r/2-1)\cdot |V(G)| +1 \leq r \cdot \mathtt{fvs}$. Since $r \geq 3$, we get that $|V(G)| = \mathcal{O}(fvs)$. As the {\sc Fair-NET}\ problem can always be solved in time $\mathcal{O}(|V(G)|!)$ using brute-force approach by going over all the permutations of labels, the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{fvs}$ in this case as well.
\end{proof}
Finally, we give a simple lemma proving that the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{vc} + \Delta$.
\begin{lemma}
The {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{vc} + \Delta$.
\end{lemma}
\begin{proof}
Let $G$ be a graph of maximum degree $\Delta$ and let $VC \subseteq V(G)$ be a vertex cover of $G$ of size $\mathtt{vc}$. Then, by the definition of vertex cover, it is easy to see that $|V(G)| \leq \mathtt{vc} \cdot \Delta$. As the {\sc Fair-NET}\ problem can always be solved in time $\mathcal{O}(|V(G)|!)$ using brute-force approach by going over all the permutations of labels, the {\sc Fair-NET}\ problem is \textsf{FPT} parameterized by $\mathtt{vc} + \Delta$.
\end{proof}
\section{\textsf{Para-NP}-hardness Results}\label{sec:hardness}
In this section, we exhibit the \textsf{para-NP}-hardness of the {\sc Fair-NET}\ problem with respect to several structural graph parameters. We start with a \textsf{para-NP}-hardness result with respect to the parameter $\mathtt{tw} + \Delta$.
\begin{theorem}\label{the:DegreeTreewidth}
The {\sc Fair-NET}\ problem is \textsf{NP}-hard for $3$-regular graphs with treewidth $3$. In particular, it is \textsf{para-NP}-hard parameterized by $\mathtt{tw} + \Delta$, even for regular graphs.
\end{theorem}
\begin{proof}
We present a reduction from {\sc $3$-Partition}. Given a multiset $W$ of $n = 3m$ positive integers, for some $m \in \mathbb{N}$, we create two instances of {\sc Fair-NET}\ based on the value of $m$ as follows (see Figure~\ref{fi:theorem1}). \smallskip\\
{\bf Case~1 [When $m$ is a multiple of $2$]:} In this case, we create an instance $(G, S)$ of {\sc Fair-NET}\ where $G = (m/2)K_{3,3}$ and $S = W$. Note that $G$ is a $3$-regular graph. Since $\mathtt{tw} (K_{3,3}) = 3$, by Observation~\ref{obs:treewidthDisjoint}, $\mathtt{tw} (G) = 3$. Let $V(G) = \biguplus_{i \in [m/2]}V_i$ where $V_i = A_i \cup B_i$ is the vertex set of the $i$-th copy of $K_{3,3}$ with bipartition $A_i$ and $B_i$. We now prove that $W$ is a \textsf{Yes} instance of {\sc $3$-Partition} if and only if $G$ is $S$-fair.
Assume first that $W$ is a \textsf{Yes} instance of {\sc $3$-Partition}. Let $W_1, W_2, \ldots, W_m$ be a corresponding partition of $W$. Then, by Definition~\ref{def:3Partition}, for every $i \in [m], \sum W_i = \sum W/m$. Let $f: V(G) \rightarrow S$ be a bijective function defined as follows. For every $i \in [m/2],$ let $f(A_i) = W_i$ and $f(B_i) = W_{m/2+i}$ (the internal labeling within $A_i$ and $B_i$ is arbitrary). So, for every $i \in [m/2], \sum f(A_i) = \sum f(B_i) = \sum W/m$. Thus, by Observations~\ref{obs:disjointMagic} and~\ref{obs:bipartiteMagic}, $G = (m/2)K_{3,3}$ is $S$-fair.
Conversely, let $G = (m/2)K_{3,3}$ be $S$-fair. Then, by Observations~\ref{obs:disjointMagic} and~\ref{obs:bipartiteMagic}, there exists a bijection $f:V(G) \rightarrow S$ such that for every $i \in [m/2], \sum f(A_i) = \sum f(B_i) = \sum S/m = \sum W/m$. Thus, $\{f(A_1), f(A_2), \ldots, f(A_{m/2}), f(B_1), \ldots, f(B_{m/2})\}$ is a partition of $W$ satisfying the required property, so $W$ is a \textsf{Yes} instance of {\sc $3$-Partition}. \smallskip\\
{\bf Case~2 [When $m$ is not a multiple of $2$]:} Without loss of generality, we can assume that every element in $W$ is greater than $1$ as otherwise we can get an equivalent instance of {\sc $3$-Partition} by adding 1 to all the elements of $W$. Let $sum = \sum W/m$ be the required sum of every subset. As every element in $W$ is greater than $1$, $sum \geq 6$. In this case, we create an instance $(G, S)$ of {\sc Fair-NET}\ where $G = \big((m+1)/2\big)K_{3,3}$ and $S = W \uplus \{sum-2, 1, 1\}$. Note that $G$ is a $3$-regular graph and $\mathtt{tw} (G) = 3$. Let $V(G) = \biguplus_{i \in [m+1/2]}V_i$ where $V_i = A_i \cup B_i$ is the vertex set of the $i$-th copy of $K_{3,3}$ with bipartition $A_i$ and $B_i$. We now prove that $W$ is a \textsf{Yes} instance of {\sc $3$-Partition} if and only if $G$ is $S$-fair.
Assume first that $W$ is a \textsf{Yes} instance of {\sc $3$-Partition}. Let $W_1, W_2, \ldots, W_m$ be the corresponding partition of $W$. Then, by Definition~\ref{def:3Partition}, for every $i \in [m], \sum W_i = \sum W/m = sum$. Let $W_{m+1} = \{sum-2, 1, 1\}$. Clearly, $\sum W_{m+1} = sum$. As $S = W \biguplus \{sum-2, 1, 1\}$, $S = \biguplus_{i \in [m+1]}W_i$. Let $f: V(G) \rightarrow S$ be a bijective function defined as follows. For every $i \in [(m+1)/2],$ let $f(A_i) = W_i$ and $f(B_i) = W_{(m+1)/2+i}$ (the internal labeling within $A_i$ and $B_i$ is arbitrary). So, for every $i \in [(m+1)/2], \sum f(A_i) = \sum f(B_i) = sum$. Thus, by Observations~\ref{obs:disjointMagic} and~\ref{obs:bipartiteMagic}, $G = \big((m+1)/2\big)K_{3,3}$ is $S$-fair.
Conversely, let $G = (m+1)/2K_{3,3}$ be $S$-fair. Then, by Observations~\ref{obs:disjointMagic} and~\ref{obs:bipartiteMagic}, there exists a bijection $f:V(G) \rightarrow S$ such that for every $i \in [(m+1)/2], \sum f(A_i) = \sum f(B_i) = \sum S/(m+1) = \sum W/m$. Without loss of generality, let $A_s$ be the set containing $sum-2$, for some $s \in [(m+1)/2]$. As $\sum A_s = sum, |A_s| = 3$ and all the elements in $W$ are greater than $1$, necessarily $A_s = \{sum-2, 1, 1\}$. As $S = W \biguplus \{sum-2, 1, 1\}$, we get that $\{f(A_1), \ldots, f(A_{s-1}), f(A_{s+1}), \ldots, f(A_{(m+1)/2}), f(B_1),$ $\ldots, f(B_{(m+1)/2})\}$ is a partition of $W$ satisfying the required property, so $W$ is a \textsf{Yes} instance of {\sc $3$-Partition}.
\end{proof}
\begin{figure}
\centering
\includegraphics[page=1,scale=0.5]{figures/hardness.pdf}
\caption{Example of the graph $G$ built in the reduction of Theorem~\ref{the:DegreeTreewidth}. $t = m/2$ for Case~1 and $t=(m+1)/2$ for Case~2.}\label{fi:theorem1}
\end{figure}
We now proceed with the \textsf{para-NP}-hardness result with parameter $\mathtt{fvs} + \Delta$.
\begin{theorem}\label{the:FVSDelta}
The {\sc Fair-NET}\ problem is \textsf{NP}-hard for forests with $\Delta = 3$. Since forests have $\mathtt{fvs} = 0$, {\sc Fair-NET}\ is \textsf{para-NP}-hard parameterized by $\mathtt{fvs} + \Delta$.
\end{theorem}
\begin{proof}
We present a simple reduction from {\sc $3$-Partition}. Given a multiset $W$ of $n = 3m$ positive integers, for some $m \in \mathbb{N}$, let $sum = \sum W/m$ be the required sum of every subset. We create an instances $(G, S)$ of {\sc Fair-NET}\ where $G = mK_{1,3}$ and $S = W \biguplus \{s_1=sum, s_2=sum, \ldots, s_m=sum\}$. Note that $G$ is a forest with $\Delta(G) = 3$. Let $V(G) = \biguplus_{i \in [m]}V_i$ where $V_i = \{v_i\} \cup B_i$ is the vertex set of the $i$-th copy of $K_{1,3}$ with $B_i$ being the set of leaves. See Figure~\ref{fi:theorem2}. We now prove that $W$ is a \textsf{Yes} instance of {\sc $3$-Partition} if and only if $G$ is $S$-fair.
Assume first that $W$ is a \textsf{Yes} instance of {\sc $3$-Partition}. Let $W_1, W_2, \ldots, W_m$ be the corresponding partition of $W$. Then, by Definition~\ref{def:3Partition}, for every $i \in [m], \sum W_i = sum$. Let $f: V(G) \rightarrow S$ be a bijective function defined as follows. For every $i \in [m]$, let $f(B_i) = W_i$ and $f(v_i) = sum$ (the internal labeling of $B_i$ is arbitrary). So, for every $i \in [m], f(v_i) = \sum f(B_i) = sum$. Thus, by Observations~\ref{obs:disjointMagic} and~\ref{obs:bipartiteMagic}, $G = mK_{1,3}$ is $S$-fair.
Conversely, let $G = mK_{1,3}$ be $S$-fair. Then, by Observations~\ref{obs:disjointMagic} and~\ref{obs:bipartiteMagic}, there exists a bijection $f:V(G) \rightarrow S$ such that for every $i \in [m], f(v_i) = \sum f(B_i) = \sum S/(m+1) = sum$. As $S = W \biguplus \{s_1=sum, s_2=sum, \ldots, s_m=sum\}$, we get that $\{f(B_1),$ $\ldots, f(B_m)\}$ is a partition of $W$ satisfying the required property, so $W$ is a \textsf{Yes} instance of {\sc $3$-Partition}.
\end{proof}
\begin{figure}
\centering
\includegraphics[page=2,scale=0.5]{figures/hardness.pdf}
\caption{Example of the graph $G$ built in the reduction of Theorem~\ref{the:FVSDelta}.}\label{fi:theorem2}
\end{figure}
Finally, we give the \textsf{para-NP}-hardness result with parameter $\alpha + \Delta$. (Recall that $\alpha$ is the number of distinct integers in the input multiset.)
\begin{theorem}\label{the:DeltaAlpha}
The {\sc Fair-NET}\ problem in \textsf{NP}-hard for $6$-regular graphs with $3$ distinct labels. In particular, it is \textsf{para-NP}-hard parameterized by $\alpha + \Delta$, even for regular graphs.
\end{theorem}
\begin{proof}
We present a reduction from $3$-XSAT$^3_+$ . Given a $3$-XSAT$^3_+$ formula $\rho$ with $n$ variables and $n$ clauses, we create an instance $(G,S)$ of {\sc Fair-NET}\ as follows. Suppose that the variables are indexed by $1, 2, \ldots, n$ and so do the clauses. For every $i \in [n]$, the variable gadget in $G$ consists of a single vertex $x_i$ called a variable vertex. Let $A$ be the set of all the variable vertices. For every $i \in [n]$, the clause gadget in $G$ consists of $15$ vertices $c^1_i, c^2_i, \ldots, c^{15}_i$. For every $i \in [n]$, we add the following edges between these $15$ vertices in the clause gadget in $G$ (see Figure~\ref{fi:theorem3}):
\begin{itemize}
\item $\forall j \in [3], \{c^1_i, c^{1+j}_i\}$. [Edge set of $K_{1,3}(\{c^1_i\}, \{c^2_i, c^3_i, c^4_i\})$].
\item $\forall j \in [3], \{c^8_i, c^{8+j}_i\}$. [Edge set of $K_{1,3}(\{c^8_i\}, \{c^9_i, c^{10}_i, c^{11}_i\})$].
\item $\{c^2_i, c^3_i\}, \{c^3_i, c^4_i\}, \{c^4_i, c^2_i\}$. [Edge set of complete graph on $\{c^2_i, c^3_i, c^4_i\}$].
\item $\{c^9_i, c^{10}_i\}, \{c^{10}_i, c^{11}_i\}, \{c^{11}_i, c^9_i\}$. [Edge set of complete graph on $\{c^9_i, c^{10}_i, c^{11}_i\}$].
\item $\forall j \in \{2,3,4\}, \forall k \in \{5,6,7\}, \{c^j_i, c^k_i\}$. [Edge set of $K_{3,3}(\{c^2_i, c^3_i, c^4_i\}, \{c^5_i, c^6_i, c^7_i\})$].
\item $\forall j \in \{9,10,11\}, \forall k \in \{12,13,14\}, \{c^j_i, c^k_i\}$. [Edge set of $K_{3,3}(\{c^9_i, c^{10}_i, c^{11}_i\}, \{c^{12}_i, c^{13}_i, c^{14}_i\})$].
\item $\forall j \in \{5,6,7, 12, 13, 14\}, \{c^{15}_i, c^j_i\}$. [Edge set of $K_{1,6}(\{c^{15}_i\}, \{c^5_i, c^6_i, c^7_i, c^{12}_i, c^{13}_i, c^{14}_i\})$].
\end{itemize}
We now explain how we connect the variable and the clause gadgets. For every variable vertex $x_i$, let $j, k$ and $l$ be the indices of the clauses where the $i$-th variable appears. Then, we add the $6$ edges $\{x_i, c^1_j\}, \{x_i, c^8_j\}, \{x_i, c^1_k\}, \{x_i, c^8_k\}, \{x_i, c^1_l\}$ and $\{x_i, c^8_l\}$ to $G$. This completes the construction of $G$. Note that $|V(G)| = n + 15n = 16n$. We now define $S$ as the multiset containing $3$ distinct labels $1, 2$ and $4$ with $\alpha_S(1) = 2n/3, \alpha_S(2) = 15n$ and $\alpha_S(4) = n/3$. From the above construction, it is easy to see that $G$ is a $6$-regular graph and $S$ contains $3$ distinct labels. By Observation~\ref{obs:rRegularMagicConstant}, the $S$-fairness constant $k = 12$.
In what follows, we will set a variable to true if and only if the label of the corresponding variable vertex is $4$ and false otherwise. We now prove that $\rho$ is a satisfiable if and only if $G$ is $S$-fair.
Assume first that $\rho$ is satisfiable. Let $A'$ be the subset of variable vertices for which the corresponding variables are true. From Lemma~\ref{lem:xsat}, $|A'| = n/3$. Let $f: V(G) \rightarrow S$ be a bijective function defined as follows:
\begin{inparaenum}[(i)]
\item for all $v \in A', f(v) = 4$;
\item for all $v \in A\setminus A', f(v) = 1$;
\item for all $v \in V(G)\setminus A, f(v) = 2$.
\end{inparaenum}
For every $i \in [n]$, let $B$ be the set containing vertices the $c^1_i$ and $c^8_i$. From the construction of $G$, only vertices from $B$ have neighbors in $A$, so for all $v \in V(G)\setminus B, f(N_G(v)) = \{2,2,2,2,2,2\}$. As every clause has exactly one true variable and two false variables, for every vertex $v \in B, f(N_G(v)) = \{4, 1, 1, 2, 2, 2\}$. So, for all $v \in V(G), \sum f(N_G(v)) = 12$. Hence, $G$ is $S$-fair.
Conversely, let $G$ be $S$-fair. Then, there exists a bijection $f:V(G) \rightarrow S$ such that for all $v \in V(G), \sum f(N_G(v)) = 12$. Note that, for every $i \in [n], N_G(c^5_i) = N_G(c^6_i) = N_G(c^7_i)$ and $\{c^5_i, c^6_i\}, \{c^6_i, c^7_i\}, \{c^7_i, c^5_i\} \in E(G)$, so by Observation~\ref{obs:sameLabel}, $f(c^5_i) = f(c^6_i) = f(c^7_i)$. Similarly, $f(c^{12}_i) = f(c^{13}_i) = f(c^{14}_i)$, $f(c^2_i) = f(c^3_i) = f(c^4_i)$ and $f(c^9_i) = f(c^{10}_i) = f(c^{11}_i)$.
Consider the vertex $c^{15}_i$, for any $i \in [n]$. As $\sum f(N_G(c^{15}_i)) = 12$ and $N_G(c^{15}_i) = \{c^5_i, c^6_i, c^7_i, c^{12}_i,$ $c^{13}_i, c^{14}_i\}$, we have that $3f(c^5_i) + 3f(c^{12}_i) = 12$ (by the equalities above). Therefore, $f(c^5_i) + f(c^{12}_i) = 4$ which necessarily implies that $f(c^5_i) = f(c^{12}) = 2$. Now, consider the vertex $c^5_i$. As $\sum f(N_G(c^5_i)) = 12$ and $N_G(c^5_i) = \{c^6_i, c^7_i, c^2_i, c^3_i, c^4_i, c^{15}_i\}$, we have that $3f(c^2_i) + f(c^{15}_i) = 8$ (by equalities above). Therefore, necessarily $f(c^2_i) = f(c^{15}_i) = 2$. Similarly, $f(c^9_i) = 2$. Now, consider the vertex $c^2_i$. As $\sum f(N_G(c^2_i)) = 12$ and $N_G(c^2_i) = \{c^1_i, c^3_i, c^4_i, c^5_i, c^6_i, c^7_i\}$, necessarily $f(c^1_i) = 2$. Similarly, $f(c^8_i) = 2$.
So far, we conclude that all and only the occurrences of integer $2$ in $S$ are used to label the vertices of the clause gadgets. Finally, consider the vertex $c^1_i$. Let $c^1_i$ is adjacent to variable vertices $x_j, x_k$ and $x_l$. As $\sum f(N_G(c^1_i)) = 12$ and $N_G(c^1_i) = \{c^2_i, c^3_i, c^4_i, x_j, x_k, x_l\}$, we get that $f(x_j) + f(x_k) + f(x_l) = 6$. The only solution to this equation is $\{4,1,1\}$ as the remaining labels are $4$ and $1$. Without loss of generality, let $f(x_j) = 4, f(x_k) = f(x_l) = 1$. Recall that, we set a variable to true if and only if the label of the corresponding variable vertex is $4$ and false otherwise, so we assign variable corresponding to $x_j$ as true and variables corresponding to $x_k$ and $x_l$ as false. It is easy to see that a clause has exactly one true variable. As the number of times $4$ appear in $S$ is $n/3$ and every variable appears exactly in $3$ clauses, the number of satisfied clauses is $n$. Hence $\rho$ is satisfiable.
\end{proof}
\begin{figure}
\centering
\includegraphics[page=3,scale=0.5]{figures/hardness.pdf}
\caption{Example of the clause gadget in $G$ for a clause $c_i = x_j \vee x_k \vee x_l$ built in the reduction of Theorem~\ref{the:DeltaAlpha}.}\label{fi:theorem3}
\end{figure}
\section{Introduction}\label{sec:intro}
Various real life situations require to conduct fair competitions. For illustration, suppose we want to schedule a non-eliminating sports competition in which there are $n$ contestants and $n$ grounds located on the circumference of a circle. As organizers, we want to assign a home ground to each contestant in such a way that every contestant $c$ plays only against $r$ contestants whose home ground is nearest to $c$'s home ground rather than all the contestants. The underlying rationale can be time constraints and also to minimize the travel time for each contestant (similar to the {\sc Traveling Tournament} problem, see e.g.~\cite{DBLP:conf/aaai/HoshinoK11,DBLP:conf/aaai/HoshinoK12}). However, the total difficulty for each contestant should be the same, i.e., the sum of the initial rankings of the opponents for each contestant is the same. We can model this problem as an instance of the {\sc Fair Non-Eliminating Tournament}\ ({\sc Fair-NET}) problem, where we are given an infrastructure of a tournament represented by a graph $G$ and the initial rankings of the contestants represented by a multiset of integers $S$. The objective is to decide whether $G$ is \emph{$S$-fair}, i.e., there exists an assignment of the contestants to the vertices of $G$ such that the sum of the rankings of the neighbors of each contestant in $G$ is the same constant $k\in\mathbb{N}$.
\remove{where we are given a graph $G$ and a multiset of integers $S$. The objective is to decide whether {\em $G$ is $S$-fair}, i.e., there exists a bijective labeling from $V(G)$ to $S$ such that the sum of the labels of the neighbors of every vertex in $G$ is the same constant $k \in \mathbb{N}$. }Here, $k$ is called the {\em $S$-fairness constant}, or simply {\em fairness constant} if $S$ is clear from the context, of $G$. Clearly, the above problem is equivalent to having an $r$-regular graph $G$ with $n$ vertices, one for each ground, and edges connecting each vertex to $r/2$ nearest vertices on the left and $r/2$ nearest vertices on the right, and the objective is to determine whether $G$ is $S$-fair where $S$ is the multiset of the rankings of the contestants (see Figure~\ref{fi:harareGraph}). As the total difficulty for each contestant in the competition is the same, we refer to such a competition as a \emph{fair competition}.
In general, if we have the infrastructure of the competition (implicitly, like the above example, or explicitly) and we want to schedule a fair competition, we can model the problem as that of determining whether the graph representing the infrastructure of the competition is $S$-fair where $S$ is the multiset of the rankings of the contestants. This situation is very frequently observed in on-line games or in other recurring competitions - in such competitions, the infrastructure of the competition is fixed and the set of contestants keeps changing.
Scheduling competitions and tournaments with different objectives is a well studied problem in the literature. There are, mainly, two fundamental competition designs, with all other designs considered as variations and hybrids. The first one is the elimination (or knockout) competition, in which the contestants are mapped to the leaf nodes of a complete binary tree. Contestants mapped to nodes with same parent compete against each other in a match, and the winner of the match moves up the tree. The contestant who reaches the root node is the winner of the tournament. The second one is the non-eliminating competition, in which no contestant is eliminated after one or few loses, and the winner is decided at the end of all the games by selecting the contestant with largest number of wins.
In recent years, algorithmic perspectives of scheduling both kinds of competitions have received significant attention by the computational social choice community.
We will first discuss elimination competitions, followed by non-eliminating competitions. With respect to elimination competitions, the design of a fair elimination competition under various definitions of being fair has received notable attention~\cite{hwang1982new,DBLP:journals/tamm/Schwenk00,DBLP:conf/atal/VuS10,DBLP:journals/tist/VuS11}. In this context, it is also relevant to mention the {\sc Tournament Fixing} problem. Here, we are given $n$ contestants, an encoding of the outcome of each potential match between every two contestants as a digraph $D$, and a favorite contestant $v$: the goal is to design an elimination tournament so that $v$ wins the tournament. This problem was introduced by Vu {\em et~al.}~\cite{DBLP:conf/atal/VuAS09}. After this, it was extensively studied~from~both combinatorial and algorithmic (as well as parameterized) points of view~\cite{DBLP:conf/aaai/AzizGMMSW14,DBLP:conf/ijcai/GuptaR0Z18,DBLP:conf/aaai/KimSW16,DBLP:conf/ijcai/KimW15,DBLP:conf/atal/KonickiW19,DBLP:conf/aaai/RamanujanS17,DBLP:conf/wine/StantonW11,DBLP:conf/ijcai/StantonW11,DBLP:conf/aaai/Williams10}.
With respect to non-eliminating competitions, a round-robin tournament (RRT) is one of the most popular forms, in which each contestant plays every other contestant~\cite{DBLP:journals/eor/ScarfYB09}. A well studied problem regarding RRTs~is~the {\sc Traveling Tournament} problem, where the goal is to design a fair RRT by minimizing the total travel distance for every team~\cite{DBLP:conf/aaai/GoerigkHKW14,DBLP:conf/aaai/HoshinoK11,DBLP:conf/aaai/HoshinoK12,DBLP:journals/scheduling/MeloUR09,DBLP:journals/scheduling/UthusRG12,DBLP:conf/mfcs/XiaoK16}. Another related problem is to design a fair RRT by minimizing the number of ``breaks'' during the tournament~\cite{DBLP:conf/patat/RibeiroU06,DBLP:journals/orl/HofPB10,DBLP:journals/orl/ZengM13}. Despite of being a popular non-eliminating competition, RRTs have some disadvantages. The first disadvantage of this format is the long tournament length, as each contestant plays against all other contestants. From the fairness point of view, a second disadvantage of this format, also mentioned in~\cite{DBLP:journals/eor/ScarfYB09}, is that it favors the {\em strongest} contestants (i.e., the contestants with the highest initial ranking). To see this,~let~$\mathcal{R} = \{r_1, r_2, \ldots, r_n\}$ be the initial rankings of the $n$ contestants, where $r_i$ is the initial ranking of contestant $i$, and let $R$ be the total sum of the rankings. In RRT, the total difficulty faced by contestant $i$ is $R-r_i$, which shows that the total difficulty faced by a contestant increases as we go from the strongest contestants to the weakest contestants. In light of the above disadvantage, motivated by one of the definitions proposed for fairness in~\cite{DBLP:journals/eor/ScarfYB09,DBLP:journals/tamm/Schwenk00,DBLP:journals/tist/VuS11}, and based on a popular fairness concept called \emph{envy-freeness} introduced by Foley~\cite{foley1967resource} in the study of fair division and allocation problems in multi-agent systems (see, e.g.~\cite{DBLP:conf/atal/BarmanG0KN19,DBLP:conf/ijcai/BenabbouCEZ19,DBLP:conf/atal/BeynierBLMRS19,DBLP:conf/sigecom/GhodsiHSSY18}), we define a \emph{fair competition} in an attempt to address both the above disadvantages with RRTs. Here, a \emph{fair competition} is one where each contestant plays with a \emph{subset} of all other contestants, yet the total difficulty for each contestant in the competition is the same. Similar to an envy-free division where no agent feels envy of another agent's share, in a fair competition no contestant feels envy about another contestant's schedule as the total difficulty for each contestant is the same.
Apart from scheduling fair competitions, {\sc Fair-NET}\ can be used to model other computational problems in social choice. For example, suppose we have $m$ candidates and $n$ jobs, and every job is associated with an integer ``reward''. Every candidate can choose $r$ jobs and every job is chosen by exactly one candidate. Now, we want to get an assignment of the jobs to the candidates such that the total reward collected by every candidate is $k$, for some integer $k$. Then, this is equivalent of having a graph $G$ that is a collection of $m$ stars, each having $r$ leaves, with $n$ total leaves, and the objective is to determine whether $G$ is $S$-fair with the fairness constant $k$ where $S$ is the union of (i) the multiset of rewards and (ii) the multiset $S' =\{k, \ldots, k\}$ containing the element $k$ $m$ times. Intuitively, $S'$ represents the multiset of rewards collected by every candidate. Every star vertex corresponds to a candidate $c$, and its leaves correspond to the jobs $c$ is assigned to.
\begin{figure}
\centering
\includegraphics[scale=0.5]{figures/rRegularIntro.pdf}
\caption{Example of a $4$ regular graph where every vertex is connected to $2$ vertices on the left and $2$ on the right.}\label{fi:harareGraph}
\end{figure}
The {\sc Fair-NET}\ problem can also be used to design semi-magic and magic squares~\cite{wiki:xxx} defined as follows. A {\em semi-magic square} is an $n \times n$ grid $(n \geq 3)$ filled with positive integers from a multiset $I$ such that each cell contains a distinct integer occurrence in $I$ and the sum of integers in each row and each column is the same. A {\em magic square} is a semi-magic square with the additional constraint that the sum of the integers in both the diagonals is also the same and equal to the sum of integers in each row and each column.
We can model a semi-magic square (and similarly a magic square) as an instance of {\sc Fair-NET}\ as follows. Let $G$ be a graph with a vertex for each cell in the grid and a vertex for each row and each column, and edges between every cell vertex and its corresponding row and column vertices (see Figure~\ref{fi:semiMagic}). Let the required sum be $k$ and let $S$ be the union of the multiset $I$ and the multiset containing $k-1$ and $1$, each occurring $n$ times. The following observation shows that an $n \times n$ grid can have a semi-magic square filled with integers from the multiset $I$ if and only if $G$ is $S$-fair.
\begin{observation}
Given an $n \times n$ grid $\mathcal{G}$ $(n \geq 3)$, and a multiset of $n^2$ positive integers $I$, let $k$ be the required sum of any semi-magic square on $\mathcal{G}$, and $G$ and $S$ be the corresponding graph and the multiset respectively. Then $\mathcal{G}$ can have a semi-magic square, filled with integers from the multiset $I$ if and only if $G$ is $S$-fair with fairness constant $k$.
\end{observation}
\begin{proof}
First, assume that there exists a semi-magic square $M$ on $\mathcal{G}$ filled with integers from the multiset $I$. Then, the sum of integers in each row and each column is $k$. We define an assignment from $V(G)$ to $S$ as follows. Every cell vertex gets the same integer label as the integer it is filled with in $M$. Clearly, the sum of neighbors for every row vertex and every column vertex in $G$ is $k$. Every row is labeled $k-1$ and every column vertex is labeled $1$. Clearly, the sum of neighbors for every cell vertex is $k$ as it is adjacent to exactly one row vertex and exactly one column vertex.
Conversely, let $G$ be $S$-fair. Then, for every vertex in $G$, the sum of the neighbors is $k$. From the construction of $G$, the degree of any row and any column vertex is $n$ and the degree of any cell vertex is $2$. Let $v$ be a vertex having a neighbor whose label is $k-1$. As the sum of the neighbors of $v$ is $k$ and every integer in $S$ is positive, the degree of $v$ must be $2$ and the label of the other neighbor of $v$ is $1$. This implies that $v$ can only be a cell vertex, and all and only the labels $k-1$ and $1$ are used by the $n$ row and $n$ column vertices. So, the labels assigned to cell vertices belong to $I$. As for every row and column vertex, the sum of the neighbors is $k$, by filling every cell in $\mathcal G$ with the label of its corresponding cell vertex, we get a semi-magic square.
\end{proof}
\begin{figure}
\centering
\includegraphics[scale=0.5]{figures/semiMagicSquare.pdf}
\caption{Example of the graph $G$ corresponding to a $3 \times 3$ - grid.}\label{fi:semiMagic}
\end{figure}
\subsection{Our Contribution and Methods}
To the best of our knowledge, while {\sc Fair-NET}\ has been studied extensively from a combinatorial point of view (Section~\ref{subsec:smagic}), close to nothing is known from an algorithmic point of view. We initiate a systematic algorithmic study of {\sc Fair-NET}. On the one hand, we show \textsf{NP}-hardness results on special graph classes, which imply \textsf{para-NP}-hardness for the problem with respect to several combinations of structural graph parameters. (For basic notions in parameterized complexity, see Section~\ref{sec:prelims}). On the other hand, we show that the problem is \emph{fixed-parameter tractable (\textsf{FPT})} for four different combinations of these parameters.
The choice of our parameters is motivated by the real world examples from the introduction. In the example of fair competition, we may want every contestant to play only a fraction of the total possible games, which in turn means that the maximum degree $\Delta$ of the infrastructure graph is small compared to the total number of contestants. Similarly, it is likely to happen that a lot of contestants have the same rankings or a lot of jobs have the same rewards, which implies that the number $\alpha$ of unique elements in $S$ is small compared to the total number of contestants or jobs. In the case of job assignment, the underlying graph is a set of stars, which has treewidth $1$ and the size of minimum feedback vertex set is $0$. Moreover, treewidth, feedback vertex set and vertex cover are central parameters in the field of parameterized complexity.
\begin{table}[!t]
\caption{Summary of our results. Here $\Delta, \mathtt{tw} , \mathtt{fvs} $ and $ \mathtt{vc}$ denote the maximum degree, treewidth, feedback vertex set number and vertex cover number of the input graph, respectively; $\alpha$ denotes the number of distinct elements in $S$. Note that $\mathtt{tw} \leq \mathtt{fvs} \leq \mathtt{vc}$.}
\label{tab:abr}
\centering
\begin{tabular}{l p{6cm}}\toprule
\textit{Parameters} & \textit{Parameterized Complexity} \\ \midrule
$\mathtt{tw} + \Delta$ & \textsf{NP}-hard for $\mathtt{tw} = 3, \Delta = 3$ (also for regular graphs) [Theorem~\ref{the:DegreeTreewidth}]\\
\hline
$\alpha + \Delta$ & \textsf{NP}-hard for $\alpha = 3, \Delta = 6$ (also for regular graphs) [Theorem~\ref{the:DeltaAlpha}]\\
\hline
$\mathtt{fvs} + \Delta$ & \textsf{NP}-hard for $\mathtt{fvs} = 0, \Delta = 3$ [Theorem~\ref{the:FVSDelta}]\\
\hline
$\mathtt{fvs} + \Delta + \alpha$ & FPT [Theorem~\ref{the:fvsAlphaDelta}]\\
\hline
$\mathtt{fvs}$ & FPT (for regular graphs) [Theorem~\ref{the:fvsrRegular}]\\
\hline
$\mathtt{vc} + \alpha$ & FPT [Theorem~\ref{the:vcAlpha}]\\ \bottomrule
\end{tabular}
\end{table}
Our main results are as follows (summarized in Table~\ref{tab:abr}). First, we show that {\sc Fair-NET}\ is \textsf{NP}-hard for three different graph classes: disjoint unions of $K_{3,3}$'s, disjoint unions of $K_{1,3}$'s and $6$-regular graphs with $3$ distinct labels. Consequently, it is \textsf{para-NP}-hard parameterized by (i) treewidth plus maximum degree, (ii) maximum degree plus feedback vertex set number, and (iii) maximum degree plus the number of distinct labels. The \textsf{para-NP}-hard results hold even for regular graphs when parameterized by either treewidth plus maximum degree or maximum degree plus the number of distinct labels.
Second, we show that {\sc Fair-NET}\ is \textsf{FPT} parameterized by (i) maximum degree plus feedback vertex set number plus the number of distinct labels, (ii) vertex cover number plus the number of distinct labels, and (iii) feedback vertex set number for regular graphs. We derive some of these results by using insights into {\sc Fair-NET}\ itself when the input graph is a cycle, a disjoint union of stars, or a connected graph with minimum degree $1$, and Integer Linear Programming.
Our choice of parameters also shows several borders of (in-)tractability. For example, the problem is \textsf{para-NP}-hard when parameterized by either $\Delta+\alpha$ or by $\mathtt{fvs}+\Delta$, but becomes \textsf{FPT} when parameterized by $\mathtt{fvs}+\Delta+\alpha$. Similarly, it is \textsf{para-NP}-hard by $\mathtt{fvs}+\Delta$, but becomes \textsf{FPT} \ by $\mathtt{fvs}$ for regular graphs. Overall, we give a comprehensive picture of the classical and parameterized complexity of {\sc Fair-NET}. {\bf For lack of space, some results and proofs marked with an asterisk $(*)$ are omitted or sketched. They are included in the full version, which is available as a supplementary material.}
\subsection{Related Work}\label{subsec:smagic}
The {\sc Fair-NET}\ problem was first introduced by Vilfred~\cite{vilfred} when $S = \{1, 2, \ldots, n\}$. Such a labeling is called \emph{sigma-labeling} in that paper. The concept of fair scheduling was independently studied by Miller {\em et~al.}~\cite{DBLP:journals/ajc/MillerRS03} in $2003$ under the name \emph{$1$-vertex magic} and by Sugeng {\em et~al.}~\cite{sugeng2009distance} under the name \emph{distance magic labeling}. For recent surveys on distance magic labeling, see~\cite{arumugam2012distance,rupnow2014survey}. The {\sc Fair-NET}\ problem for a general multiset $S$ was first studied by O'Neal and Slater~\cite{DBLP:journals/siamdm/ONealS13}. In the same paper, they also proved that if a graph $G$ is $S$-fair, then the $S$-fairness constant of $G$ is unique. In~\cite{allLabeling}, Slater proved that {\sc Fair-NET}\ is {\textsf{NP}-hard}. More recently, Godinho {\em et~al.}~\cite{DBLP:journals/endm/GodinhoSA15} studied the special case of {\sc Fair-NET}\ where $S$ is a set and not a multiset. They gave a simpler proof for the uniqueness of $S$-fairness constant and also exhibited several families of $S$-fair graphs. Recently, the same set of authors studied a measure called \emph{distance magic index} related to $S$-fair labeling and determined the distance magic index of trees and complete bipartite graphs in~\cite{distanceIndex}. There has also been a long line of studies on other kinds of graph labeling, like {\em \{0,1\}-{\sc Fair-NET}}, where we consider the closed neighborhood of every vertex instead of open neighborhood (i.e., the vertex itself is also considered in its neighbor set). Another example is {\em vertex-bimagic labeling}, in which there exists two constants $k_1$ and $k_2$ such that the sum of neighbors of every vertex is either $k_1$ or $k_2$. For more information, see the recent survey~\cite{gallian2018dynamic}.
\section{Preliminaries}\label{sec:prelims}
\paragraph{Sets and Functions.} Given two multisets $A = \{a_1, a_2, \ldots, a_n\}$ and $B = \{b_1, b_2, \ldots, b_m\}$, their \emph{disjoint union} is the multiset $S = A \uplus B = \{a_1, a_2 , \ldots, a_n, b_1, b_2, \ldots, b_m\}$. For example, let $A = \{1,3,4,5,5\}$ and $B = \{3, 2, 4, 6\}$. Then, $A \uplus B = \{1, 2, 3, 3, 4, 4, 5, 5, 6\}$. Given a multiset $S$, $\alpha(S)$ denotes the number of distinct elements in $S$, and for every $a \in S, \alpha_S(a)$ denotes the number of times $a$ appear in $S$. Given a multiset $A$, $\sum A$ denotes the sum of its elements (in case they are integers), and $|A|$ denotes its size. For any $t \in \mathbb{N}, [t]$ denotes the set $\{1,2,\ldots,t\}$. Given a function $f$ defined on a multiset $A$, $f(A) = \{f(a):a \in A\}$. Let $f: A \rightarrow B$ be a function from a multiset $A$ to a multiset $B$. Then the \emph{restriction of f} to a multiset $A' \subseteq A$ is the function $f|_{A'} : A' \rightarrow B$ given as $f|_{A'}(x) = f(x)$ for every $x \in A'$.
\paragraph{Graphs.} In this paper, we consider only undirected graphs. Given a graph $G$, we denote its vertex set and edge set by $V(G)$ and $E(G)$, respectively. For a vertex $v\in V(G)$, the set of all the neighbors of $v$ in $G$ is denoted by $N_G(v)$, i.e. $N_G(v)=\{u\in V(G)~|~\{u,v\}\in E(G)\}$. The degree of a vertex $v\in V(G)$ in $G$ is denoted by $\degr_G(v)$. When $G$ is clear from the context, we drop the subscript. Given an induced subgraph $H$ of $G$, the set of neighbors of vertices in $H$ which are not in $H$ is denoted by $N_G(H)$, i.e., $N_G(H) = \big(\bigcup_{v \in V(H)} N_G(v)\big) \setminus V(H)$. The maximum and minimum degree of $G$ are denoted by $\Delta(G)$ and $\delta(G)$, respectively. Given a set $V' \subseteq V(G)$, the subgraph of $G$ induced by $V'$ is denoted by $G[V']$. A path on $n$ vertices is denoted by $P_n$. A cycle on $n$ vertices is denoted by $C_n$. A complete bipartite graph with bipartition $A$ and $B$ such that $|A| = m, |B| = n$ is denoted by $K_{m,n}(A, B)$. If $A$ and $B$ are clear from the context, we write $K_{m,n}(A, B)$ as $K_{m,n}$. Given a forest $F$, the set of leaves of $F$ is denoted by $\mathrm{leaves}(F)$. Given a rooted tree $T$, for a vertex $v \in V(T)$, the set of children of $v$ in $T$ is denoted by $\mathrm{children}_T(v)$.
An \emph{$r$-regular graph} is a regular graph where every vertex has degree $r$. An \emph{$n$-star} graph (on $n+1$-vertices) is the complete bipartite graph $K_{1,n}$. Given an $n$-star graph where $n \geq 2$, the \emph{star-vertex} is the unique vertex with degree $n$. The \emph{disjoint union} of two graphs $G_1$ and $G_2$, denoted by $G_1 + G_2$, is the graph with vertex set $V(G_1) \uplus V(G_2)$ and edge set $E(G_1) \uplus E(G_2)$. For any $m \in \mathbb{N}$, we denote the disjoint union of $m$ copies of a graph $G$ by $mG$. Note that, the disjoint union of two or more nonempty graphs is always a disconnected graph. For other standard notations not explicitly defined here, we refer to the book \cite{Diestelbook}.
The {\em treewidth, vertex cover number} and {\em feedback vertex set number} of a graph $G$ are defined as follows.
\begin{definition}[{\bf Treewidth}]
A \emph{tree decomposition} of a graph $G$ is a tree $T$ whose nodes, called \emph{bags}, are labeled by subsets of vertices of $G$. For each vertex $v$, the bags containing $v$ must form a nonempty contiguous subtree of $T$, and for each edge $\{u,v\}$, at least one bag must contain both $u$ and $v$. The \emph{width} of the decomposition is one less than the maximum cardinality of any bag, and the \emph{treewidth} $\mathtt{tw} (G)$ of $G$ is the minimum width of any of its tree decompositions.
\end{definition}
Based on the definition of treewidth, we have the following observation about disjoint union.
\begin{observation}\label{obs:treewidthDisjoint}
The treewidth of the disjoint union of two vertex-disjoint graphs $G_1$ and $G_2$ is max$\{\mathtt{tw} (G_1), \mathtt{tw} (G_2)\}$.
\end{observation}
\begin{definition}[{\bf Vertex Cover}]
A \emph{vertex cover} of a graph $G$ is a set of vertices in $G$ such that every edge in $G$ has at least one endpoint in the set. We denote the minimum size of a vertex cover of $G$ by $\mathtt{vc}(G)$.
\end{definition}
\begin{definition}[{\bf Feedback Vertex Set}]
A \emph{feedback vertex set} of a graph $G$ is a set of vertices whose removal results in an acyclic graph. We denote the minimum size of a feedback vertex set of $G$ by $\mathtt{fvs}(G)$.
\end{definition}
We will show hardness results from {\sc $3$-Partition} and a variant of SAT called $3$-XSAT$^3_+$ (which were proved to be strongly \textsf{NP}-complete and \textsf{NP}-complete in \cite{DBLP:books/fm/GareyJ79} and \cite{DBLP:journals/dam/PorschenSSW14}, respectively), defined as follows.
\begin{definition}[{\bf $3$-Partition}]\label{def:3Partition}
Given a multiset $W$ of $n = 3m$ positive integers, for some $m \in \mathbb{N}$, can $W$ be partitioned into $m$ triplets $W_1, W_2,\ldots, W_m$ $($i.e., $W = \biguplus_{i \in [m]}W_i$ and $|W_i| = 3$ for every $i \in [m])$ such that for every $i \in [m], \sum W_i = \sum W/m$?
\end{definition}
\begin{definition}[{\bf $3$-XSAT$^3_+$}]
Given a formula in conjunctive normal form (CNF) where all literals are positive, each clause has size exactly $3$, and each variable occurs exactly $3$ times, does there exist a truth assignment to the variables so that each clause has exactly one true variable?
\end{definition}
The following lemma about a $3$-XSAT$^3_+$ formula will be useful throughout the paper.
\begin{lemma}\label{lem:xsat}
Let $\rho$ be a $3$-XSAT$^3_+$ formula with $n$ variables and $m$ clauses. Then $m = n$. Moreover, $\rho$ is satisfiable only if $n$ is divisible by $3$.
\end{lemma}
\begin{proof}
Let $A$ be the set of $n$ vertices corresponding to $n$ variables and $B$ be the set of $m$ vertices corresponding to $m$ clauses. Consider the bipartite graph $G$ with bipartition $A$ and $B$ and edges between a vertex $x \in A$ and a vertex $c \in B$ if and only if the variable corresponding to $x$ is in clause corresponding to $c$. As every variable appears in exactly $3$ clauses and every clause has exactly $3$ variables, $G$ is $3$-regular. Because $G$ is bipartite, the number of edges in $G$ must be equal to both $3n$ and $3m$ which means that $m = n$.
Now, assume that $\rho$ is satisfiable, i.e. there exists a truth assignment such that every clause has exactly one true variable. Let $k$ be the number of true variables. Let $v_1$ ans $v_2$ be two vertices in $A$ corresponding to two different true variables. Then $N_G(v_1) \cap N_G(v_2) = \emptyset$, otherwise there is a clause $c$ that has two true variables. Therefore, as $G$ is $3$-regular, total number of clauses satisfied is $3k$. As this number is also $m$ (which equals $n$), this means $k = n/3$. So, $\rho$ is satisfiable only if $n$ is divisible by $3$.
\end{proof}
From the above lemma, it is easy to see that $3$-XSAT$^3_+$ remains \textsf{NP}-complete even when $n$ is divisible by $3$. So, in the rest of the paper, we assume that given a $3$-XSAT$^3_+$ formula with $n$ variables and $m$ clauses, $m = n$ and $n$ is divisible by $3$.
\paragraph{Fair Non-Eliminating Tournament.} Given an infrastructure of a tournament represented by a graph $G$ and the initial rankings of the contestants represented by a multiset of integers $S$, $G$ is called \emph{$S$-fair} if there exists an assignment of the contestants to the vertices of $G$ such that the sum of the rankings of the neighbors of each contestant in $G$ is the same. Equivalently, given the infrastructure graph $G$ and the multiset of contestants' rankings $S$ with $|S| = |V(G)|$, $G$ is \emph{$S$-fair} if there exists a bijection $f: V(G) \rightarrow S$ such that for every vertex $v \in V(G), \sum f(N(v)) = k$, where $k$ is a constant called \emph{$S$-fairness constant}.
For any vertex $v \in V(G), f(v)$ is called the \emph{label} of $v$. We denote the set of all bijective functions that satisfy the above property by $\mathcal{M}(G,S)$. In \cite{DBLP:journals/siamdm/ONealS13}, O'Neal and Slater showed that if a graph $G$ is $S$-fair, then its $S$-fairness constant is unique.
Since the infrastructure graph of a tournament is an undirected graph and the initial ranking of several players can be the same, we define the {\sc Fair Non-Eliminating Tournament}\ ({\sc Fair-NET}) problem as follows.
\begin{definition} [{\bf Fair-NET Problem}]\label{def:SMagic}
Given an undirected graph $G$ and a multiset of positive integers $S$ with $|S| = |V(G)|$, is $G$ $S$-fair $($i.e. $\mathcal{M}(G,S) \neq \emptyset)$?
\end{definition}
The following observations about $S$-fair graphs follow directly from its definition.
\begin{observation} [{\bf Label Swap}]\label{obs:sameNeighborhood}
Let $G$ be an $S$-fair graph. Let $f \in \mathcal{M}(G,S)$. Let $u, v \in V(G)$ such that $N_G(u) = N_G(v)$. Consider $f':V(G) \rightarrow S$, defined as follows. For all $w \in V(G) \setminus \{u,v\}, f'(w) = f(w); f'(u) = f(v); f'(v) = f(u)$. Then $f' \in \mathcal{M}(G,S)$.
\end{observation}
\begin{observation} \label{obs:sameLabel}
Let $G$ be an $S$-fair graph. Let $u, v \in V(G)$ such that $N_G(u) = N_G(v)$ and $\{u,v\} \in E(G)$. Then, for all $f \in \mathcal{M}(G,S), f(u) = f(v)$.
\end{observation}
\begin{observation} \label{obs:rRegularMagicConstant}
Let $G$ be an $r$-regular $S$-fair graph. Then the $S$-fairness constant is equal to $r \cdot \sum S/ |V(G)|$.
\end{observation}
\begin{observation}\label{obs:disjointMagic}
Given two graphs $G_1$ and $G_2$ and a multiset of positive integers $S$, $G_1 + G_2$ is $S$-fair if and only if $G_1$ is $S_1$-fair and $G_2$ is $S_2$-fair with the same fairness constant for some $S_1, S_2 \subseteq S$ such that $S_1 \uplus S_2 = S$.
\end{observation}
\begin{observation}\label{obs:bipartiteMagic}
Given a complete bipartite graph $K_{m,n}(A, B)$ and a multiset of positive integers $S$, $K_{m,n}$ is $S$-fair if and only if there exists a bijection $f: V(K_{m,n}) \rightarrow S$ such that $\sum f(A) = \sum f(B) = \sum S /2$.
\end{observation}
\paragraph{Integer Linear Programming.} In the {\sc Integer Linear Programming Feasibility} (ILP) problem, the input consists of $p$ variables $x_1, x_2, \ldots, x_p$ and a set of $m$ inequalities of the following form:
\[\begin{array}{*{9}{@{}c@{}}}
a_{1,1}x_1 & + & a_{1,2}x_1 &+ & \cdots & + & a_{1,p}x_p & \leq & y_1 \\
a_{2,1}x_1 & + & a_{2,2}x_2 & + & \cdots & + & a_{2,p}x_p & \leq & y_2 \\
\vdots & & \vdots & & & & \vdots & & \vdots \\
a_{m,1}x_1 & + & a_{m,2}x_2 & + & \cdots & + & a_{m,p}x_p & \leq & y_m \\
\end{array}\]
where every coefficient $a_{i_j}$ and $y_i$ is required to be an integer. The task is to check whether there exists an assignment of integer values for every variable $x_i$ so that all inequalities are satisfiable. The following theorem about the tractability of the ILP problem will be useful throughout Section~\ref{sec:fpt}.
\begin{theorem}[\cite{DBLP:journals/mor/Kannan87,DBLP:journals/mor/Lenstra83,DBLP:journals/combinatorica/FrankT87}]\label{the:runningTimeILP}
The ILP problem with $p$ variables is \textsf{FPT} parameterized by $p$.
\end{theorem}
\paragraph{Parameterized Complexity.} A problem $\Pi$ is a {\em parameterized} problem if each problem instance of $\Pi$ is associated with a {\em parameter} $k$. For simplicity, we denote a problem instance of a parameterized problem $\Pi$ as a pair $(I,k)$ where the second argument is the parameter $k$ associated with $I$. The main objective of the framework of Parameterized Complexity is to confine the combinatorial explosion in the running time of an algorithm for an \textsf{NP}-hard parameterized problem $\Pi$ to depend only on $k$. In particular, a parameterized problem $\Pi$ is {\em fixed-parameter tractable} (\textsf{FPT}) if any instance $(I, k)$ of $\Pi$ is solvable in time $f(k)\cdot |I|^{\mathcal{O}(1)}$, where $f$ is an arbitrary computable function of $k$. Moreover, a parameterized problem $\Pi$ is {\em \textsf{para-NP}-hard } if it is \textsf{NP}-hard for some fixed constant value of the parameter $k$. For more information on Parameterized Complexity, we refer the reader to books such as \cite{DBLP:series/txcs/DowneyF13,DBLP:books/sp/CyganFKLMPPS15}. | {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,113 |
{"url":"https:\/\/bicycles.stackexchange.com\/questions\/7228\/wind-and-water-resistent-jackets","text":"# Wind- and water-resistent jackets\n\nTo answer the comment after this question, what should one know about wind- and water-resistent jackets?\n\nFor example, what is \"Gore-Tex\", is it always expensive, and is there anything else like it?\n\nThis is for for wearing at near freezing temperatures (I don't need a summer rain jacket).\n\nI want this for commuting, an hour each way, (almost) every day.\n\nI'm usually warm or too warm whenever it's above freezing: traffic lights, bike lanes, cycle paths, some hills.\n\nBut when it is freezing, for an hour, I need, apart from shoes and gloves (and trousers\/pants), some wind resistance: otherwise my skin arrives chilled. And cold rain can be chilling too.\n\n\u2022 Gore-Tex is not cheap. There are cheaper imitations. Gore-Tex \"breathes\" better than most other water\/windproof fabrics (or at least this was true 15-20 years ago -- there may be better now). \u2013\u00a0Daniel R Hicks Dec 6 '11 at 4:13\n\u2022 What is \"Gore-Tex\"? Check out the site of W.L. Gore and Associates, Inc. They'll tell you. gore-tex.com\/remote\/Satellite\/content\/what-is-gore-tex \u2013\u00a0user313 Dec 6 '11 at 6:54\n\u2022 Basically, like DRH said, they make wind\/waterproof\/breathable fabrics and clothing. They make great stuff for cold, windy, and wet weather. \u2013\u00a0user313 Dec 6 '11 at 6:57\n\u2022 I was in this an hour ago....gore-tex.com\/remote\/Satellite\/activities\/men-cycling-road\/\u2026 \u2013\u00a0user313 Dec 6 '11 at 7:02\n\u2022 @ChrisW -- Note that it's not just the chemical but also the manufacturing process. \u2013\u00a0Daniel R Hicks Dec 6 '11 at 11:55\n\nThere are other brands and fabrics, I've used a few\n\neVent is very similar to GoreTex. Wind stopper, waterproof, breathable, my rain pants are made with eVent. I love my clothes and gloves made with WindStopper, a little cheaper, but not waterproof. When I first started commuting by bicycle, I got a Nylon jacket, made with coated nylon and taped seams. It worked well until the coating wore off. I do think it is worth it to spend the extra money and get something that works from the start.\n\nI have not seen many fully waterproof jackets that are lined for warmth, you may need to get the best shell and layer underneath. I find the wind blocking aspects keep me warm with a nice thermal base layer.\n\nVery similar for rain pants too. I have fleece lined bicycle pants (WindStopper material on the front) for when no rain (or very light). If it is raining, I wear leggings underneath my fully waterproof pants, which don't offer as much warmth, but they do keep the rain\/wind out.\n\n\u2022 Good tip. I hadn't heard of eVent. I'll have to check it out the next time I shop for outerwear. \u2013\u00a0user313 Dec 6 '11 at 20:49\n\nThis is a really good question, and one that I found myself asking only a few weeks ago. The aforementioned posts are great if you feel like restricting yourself to \"Goretex\" as the main construction fabric of the jacket you're looking for. You can find a lot of technical information about Goretex online, but one of the most basic facts about the fabric is its 3,000mm rating. This means that, technically, the fabric isn't 100% waterproof, but is, rather, \"rain proof.\" As found here - http:\/\/backcountrybeacon.com\/2010\/04\/waterproof-ratings-demystified\/ - you'll be able to see what I mean.\n\nIf you're open to discovering new jackets and fabrics, I would suggest looking beyond conventional Goretex jackets and taking a look at several retailers that are currently constructing jackets using Polartec's new NeoShell fabric. Marmot released their ZION jacket using NeoShell, which is rated at 10,000mm (100% waterproof by all industrial standards) and has received awards from Polartec for best use of their fabric. And so, after a heavy amount of research (really the only reason I know any of this), I decided to purchase one from a 3rd party retailer for quite a bit less than they are found on the site. Oh, another thing, NeoShell works as a passive membrane check this out -\n\nGoretex is an active permeable fabric, what this means is that you'll need to start sweating and building pressure within the jacket before you notice any dissipation of heat through the fabric, relieving you of excess heat (but at this point it's kind of too late). SUCKS!\n\nNeoShell is a passive permeable fabric, what this means is that the second you start to build up heat pressure in the jacket it is basically forced out. Pretty cool huh? It gets better...\n\nI've been itching to write this review somewhere and a friend of mine JUST referred this site as a great resource for biking gear. So I got the jacket last Sunday and have been riding, running, and pushing myself harder than I ever could with my old riding jackets (which were all pearlIzumi, etc). I've got a similar commute to handle every morning in 19F (-7C) degree weather and the first time I got out on Monday and started to feel the heat in my jacket, it was as if it was being sucked out, leaving me with an even warmth that was comfortable and inspired me to ride even harder. The jacket has a great articulated hood, loooooong sleeves with asymmetrical cuffs that cover the tops of your hands from the wind, draw cords on the waist, and pockets that double as vents. It has a fleece lining to wick moisture. And because the fabric is so breathable, I've never had to unzip to regulate heat. Literally... the best riding jacket that was never meant to be a riding jacket. 100% wind and waterproof. It's a little pricey, but for something that can do what all others cannot AND be used for running, skiing, hiking, and climbing... it really doesn't get much better. I have yet to brave snow, but I've never been this excited to conquer terrible weather in my life.\n\nI would recommend a baselayer and something warm on those extra cold days. Hope this helps!\n\nGore-Tex is a line of fabrics and outdoor clothing made by W.L. Gore and Associates, Inc. They use various technologies to make breathable, waterproof and windproof fabrics.\n\nThe Gore brand tends to be on the expensive side. However, I think that they license out their fabrics to other manufacturers, but I'm not sure about that.\n\nI have used Gore cycling products for several years and have yet to be displeased. Rain, wind, cold, etc.... actually, I found their winter gloves to be too thin, but otherwise....\n\nMy favorite, most versatile cycling jersey is made by Gore. The front panels are made with \"windstopper\" fabrics and the long sleeves zip on\/off. Here at gorebikewear.com. And I also have a 5+ year old gorebikewear jacket which appears to have a few more years left. I'll no doubt take a look at other alternatives when I shop for a new one in a few years.\n\nA note based on the question, \"This is for wearing at near freezing temperatures (I have decided I don't need a summer rain jacket)\" - I currently use a gorebikewear wind\/rain jacket year round. This is an unlined jacket. In warm, rainy weather, it's layered with nothing but a short-sleeve jersey. As the weather gets progressively colder, I manipulate the layers beneath the jacket...so roughly, in the fall, it's a light base layer + jersey + jacket...and in winter, it's a heavier base layer + jersey + fleece + jacket, etc. Anyway, the outermost layer (jacket) is primarily for wind\/rain protection; and the layers beneath are for temperature control.\n\nSo, for me at least (in the rainy and sometimes cold Pacific NW), a single wind\/rain jacket gets me through the entire year.\n\nEdit: This link (\"Waterproof Breathable Fabric Technologies: A Comprehensive Primer and State of the Market Technology Review\") is a pay-for-view ($5) article, written in 2004, which describes about five different types of fabric: how they are made, and how well they perform. Gore-Tex is a semi-permeable membrane: a fabric, like a net, but not woven: it's formed by stretching. It's made of expanded PTFE, like a spongy Teflon (and their patent was about how to do the expansion\/manufacturing). A microscope shows it's something like bone, or a net, or holey Swiss cheese. The holes in the fabric layer are much bigger than air and water vapour molecules (which therefore pass through); but smaller than liquid water droplets (which are many water molecules bound together), which therefore don't pass through (water on the fabric should stay on it, until it's brushed off or evaporates). So Gore-Tex is: \u2022 Semi-permeable to water vapour \u2022 Impermeable to liquid water It is therefore (allegedly, more or less) waterproof but at the same time 'breathable'. Sometimes (as in my jacket) you don't see Gore-Tex. It's a middle layer, sewn in between the jacket's inner lining and outer shell. These other layers (i.e. the inner lining and the outer shell) are, principally, meant to be durable. They're woven, and have pockets etc. They also (being a water-resistant nylon fabric themselves) help to keep the sewn-in Gore-Tex from getting soaked: a sheet of water on the Gore-Tex would prevent its 'breathing'. In rain the Gore-Tex middle layer keeps the jacket's wet outer shell away from the inner lining and from my skin. In extremis (when all is soaked in very heavy rain) the Gore-Tex tries to stay water-repellent albeit no longer so 'breathable' (but in very heavy rain, I'm happy enough to be warm: and to have air inside the jacket). I bought my jacket 10 or 15 years ago from MEC. Even if it is expensive (and it wasn't, especially - and I liked it so much that I bought duplicates as Christmas presents for my family, and they still wear theirs too), I'd consider whether it's durable and useful. My jacket does not have cycling-specific tailoring, so I can wear it anywhere\/anytime when it's wet out. Also I (among other things) reckon I save$1,500\/year in bus fare by commuting by bike: so amortizing that \\$1,500\/year over two or more years can justify spending ('investing') the money, for buying a bike and clothes, and still come out ahead, relatively.\n\nAnyway. So my jacket has a lining etc., as well as the Gore-Tex membrane hidden away as a middle layer. The inner lining of this jacket is mesh\/webbing, which helps the jacket to stand off from the body. This makes it durable, and the nylon exterior shell makes it wind-proof. I regulate my heat (when cycling) by using the front zipper: up around the throat, or down to open the chest, which ventilates the jacket. There's a velcro strap at the cuffs to regulate the ventilation of the arms (which is why I avoid wearing a jacket in summer).\n\nIt used to be I think that Gore-Tex didn't manufacture clothes, but licensed the fabric to clothiers (e.g. my jacket was made and labelled and perhaps designed by Le Coq Sportif, who make sportswear). Now Gore-Tex seem to make and sell their own complete clothes? But other manfacturers presumably continue to license the fabric and use it.\n\nThere's a Gore-Tex \"Paclite\" fabric or jacket now, which I don't think I've seen\/felt. I guess that's a single-layer jacket. I don't know whether that's pure Gore-Tex or otherwise how it's bonded to the shell (how it performs). I think it's aimed at cyclists. I don't know how good it is for cold-weather riding (mine, because it's lined and wind-proof, is good).\n\nI'd like to find a good selection of jackets but I don't know where. Most LBS when I ask for Gore-Tex say \"that's expensive\" and don't stock it. I don't want to buy sight unseen because all the details matter.\n\nGore-Tex is still a brand and a trademark (and a company). Its patent has expired. I don't know what competing\/similar products\/solutions\/manufacturers there are.\n\n\u2022 My only affinity to gore-tex is that I have used the products. There are developing\/developed alternative technologies. A link: backpackinglight.com\/cgi-bin\/backpackinglight\/00316.html \u2013\u00a0user313 Dec 7 '11 at 22:33\n\u2022 @wdypdx22 - That is an excellent link. If it were an answer I would 'accept' it. \u2013\u00a0ChrisW Dec 8 '11 at 4:22\n\nThe best jacket that I've found for cold or cold and wet weather cycling is the Cycle Shirt by Buffalo.\n\nThe Pertex shell it isn't waterproof at all, but your body heat evaporates all but heavy rain. Contrast this with Goretex, which when soaked on the outside cannot pass water vapour, and so ends up condensated on the inside.\n\nI was out mountain biking yesterday in heavy rain. My eVent jacket (made by Rab) kept me dry. I believe eVent is supposed to be a bit more breathable than gore-tex, so if you are working hard and getting warm, some of your sweat will pass out.\n\nI would recommend you don't try to get one waterproof and warm coat. You're much better off getting a waterproof coat that you can wear over a warm coat\/top if it's really cold. Then you've got far more options and can get the best set up to match the weather and type of cycling you're doing.","date":"2019-10-16 10:32:21","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\": 0.21516838669776917, \"perplexity\": 3849.8374383107007}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986666959.47\/warc\/CC-MAIN-20191016090425-20191016113925-00473.warc.gz\"}"} | null | null |
import { async, ComponentFixture, TestBed } from '@angular/core/testing';
import { NO_ERRORS_SCHEMA } from '@angular/core';
import { AdminComponent } from './admin.component';
describe('AdminComponent', () => {
let component: AdminComponent;
let fixture: ComponentFixture<AdminComponent>;
beforeEach(async(() => {
TestBed.configureTestingModule({
declarations: [ AdminComponent ],
schemas: [NO_ERRORS_SCHEMA]
})
.compileComponents();
}));
beforeEach(() => {
fixture = TestBed.createComponent(AdminComponent);
component = fixture.componentInstance;
fixture.detectChanges();
});
it('should create', () => {
expect(component).toBeTruthy();
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,015 |
\section{Introduction}
The Monster group $\mathbb{M}$ is the largest of the twenty-six sporadic groups and was first constructed by Griess \cite{Griess82} as the automorphism group of the Griess algebra $V_{\mathbb{M}}$, a 196,884-dimensional real commutative non-associative algebra. The Griess algebra is generated by idempotents known as $2A$-axes that are in bijection with the $2A$-involutions in the Monster group.
Inspired by work of Sakuma \cite{Sakuma07} on vertex operator algebras, Ivanov \cite{Ivanov09} introduced Majorana theory as an axiomatisation of certain properties of the $2A$-axes of the Griess algebra. The objects at the centre of Majorana theory are known as Majorana algebras. The axioms of Majorana theory were further generalised by Hall, Rehren and Shpectorov \cite{HRS15} who introduced the definition of an axial algebra.
An axial algebra is a commutative non-associative algebra generated by semisimple idempotents, known as axes, that all obey a fusion rule. This fusion rule governs the behaviour of the eigenvectors of the adjoint action of an axis on the whole of the algebra. Majorana algebras (including the Griess algebra) and Jordan algebras are both important examples of axial algebras.
Axial algebras that obey the Monster fusion rules are referred to here as axial algebras of Monster type. In this case, as in other examples of axial algebras, the fusion rule is $C_2$-graded which in particular means that from each axis we can construct an involution in the automorphism group of the algebra known as a Miyamoto involution.
In Section \ref{sec:axial}, we give some basic definitions and results concerning axial algebras. In Sections \ref{sec:construction}, \ref{sec:4B} and \ref{sec:4A} we classify a certain class of axial algebras of Monster type that are generated by six axes whose Miyamoto involutions generate an elementary abelian group of order $4$. As part of this classification, we obtain a construction of a $12$-dimensional axial algebra $M_{4A}$ over the polynomial ring $\mathbb{R}[t]$.
This is a significant new example in a number of ways. Firstly, axial algebras were introduced by Hall et al. \cite{HRS15} using an algebra geometric construction giving a variety of algebras. The variety of algebras constructed in \cite[Section 7]{HRS15} is $0$-dimensional. We also use this construction (as explained in Section \ref{sec:construction}). This means that we provide the first example of a variety of axial algebras that has positive dimension $1$. In particular, the indeterminate $t$ may take any value in $\mathbb{R}$, giving the first example of an infinite family $\{M(t)\}_{t \in \mathbb{R}}$ of axial algebras all with the same shape (as defined in Section \ref{sec:shape}).
Futhermore, in Section \ref{sec:frobenius}, we show that the axial algebra $M(t)$ is a admits a positive definite Frobenius form if and only if $t \in (0, \frac{1}{6})$ and also obeys Norton's inequality if and only if $t \in [0, \frac{1}{6}]$. This shows that an axial algebra of Monster type that admits a Frobenius form does not necessarily obey Norton's inequality. However, the question of whether Norton's inequality, which is one of the Majorana axioms, is a consequence of the other axioms remains an open problem in Majorana theory. Our result suggests that this is indeed the case. Moreover, this work shows that if $t \in (0, \frac{1}{6})$ then the algebra $M(t)$ is a Majorana algebra. This is the first example of an infinite family of Majorana algebras.
This work also has implications for the computational methods that are being developed in the field. A computational approach to Majorana and axial algebras was begun by Seress \cite{Seress12} and has been continued by McInroy and Shpectorov \cite{MS18} and Pfeiffer and Whybrow \cite{PW18a}. The algorithms developed by these authors take the shape either of a Majorana algebra or an axial algebra of Monster type and attempt to construct the universal algebra with this shape over a field of characteristic $0$.
This process is computationally expensive and many examples involving a large number of axes fail to complete, requiring too much time or memory. There are also a small but significant number of examples with a lower number of axes that have failed to complete. The family of axial algebras constructed in this paper comes from one such case. The existence of this infinite family of algebras shows that this example, as well as probably many others, lie outside the scope of the current computational methods. A potential improvement to the algorithms in \cite{MS18} and \cite{PW18a} would be to allow the construction over the field of rational functions in one or more indeterminates.
Finally, in Section \ref{sec:4Afusion}, we study the $4A$ axes contained in the the algebra $M_{4A}$. We show that the eigenvectors of these axes in $M_{4A}$ obey a fusion rule. Moreover, if $t \notin \{1, 0, \frac{1}{2}, \frac{3}{8}\}$, then this fusion rule is $C_2 \times C_2$-graded. This gives an important new example of an infinite family of graded fusion rules.
\section{Axial algebras}
\label{sec:axial}
In the following, we let $R$ be a commutative ring with identity and let $(V, \cdot)$ be a commutative (not necessarily associative) $R$-algebra.
\subsection{Preliminaries}
\begin{defn}
A \emph{fusion rule} is a pair $(\mathcal{F}, *)$ such that $\mathcal{F} \subseteq R$ and $\mathcal{F} \times \mathcal{F} \rightarrow 2^{\mathcal{F}}$ is a symmetric map.
\end{defn}
\begin{defn}
If $a \in V$ is an idempotent then for each $\lambda \in R$, we denote the eigenspace of the adjoint action of $a$ on $V$ with eigenvalue $\lambda$ by
\[
V^{(a)}_{\lambda} = \{ v \in V \mid a \cdot v = \lambda v\}.
\]
For each subset $\Lambda \subseteq R$, we define
\[
V^{(a)}_{\Lambda} = \bigoplus_{\lambda \in \Lambda} V^{(a)}_{\lambda}
\]
with the convention that $V^{(a)}_{\emptyset} = \{0 \}$.
\end{defn}
\begin{defn}
If $a \in V$ is an idempotent and $(\mathcal{F}, *)$ is a fusion rule then $a$ is a $(\mathcal{F}, *)$-\emph{axis} if
\begin{enumerate}[(i)]
\item $\mathrm{ad}_a$ is semisimple and whenever $V^{(a)}_{\lambda}$ is non-zero, $\lambda \in \mathcal{F}$, i.e. the equation
\[
V = V_{\mathcal{F}}^{(a)} = \bigoplus_{\lambda \in \mathcal{F}} V^{(a)}_{\lambda}
\]
holds;
\item for all $\lambda, \mu \in \mathcal{F}$ we have that
\[
V_{\lambda}^{(a)} \cdot V_{\mu}^{(a)} \subseteq V_{\lambda * \mu}^{(a)}.
\]
\end{enumerate}
\end{defn}
\begin{defn}
A $(\mathcal{F}, *)$-\emph{axial algebra} is a pair $(V, A)$ where $V$ is a commutative (not necessarily associative) $R$-algebra and $A \subseteq V$ is a generating set of $(\mathcal{F}, *)$-axes.
\end{defn}
\begin{defn}
A $(\mathcal{F}, *)$-axis is \emph{primitive} if its $1$-eigenspace is one dimensional, i.e. $V_1^{(a)} = \langle a \rangle_R$. A $(\mathcal{F}, *)$-axial algebra $(V,A)$ is \emph{primitive} if $a$ is primitive for all $a \in A$.
\end{defn}
\begin{defn}
Suppose that $T$ is an abelian group. A function $\mathrm{gr}: T \rightarrow P(\mathcal{F})$ is a $T$-\emph{grading} if the image of $\mathrm{gr}$, $\{\mathrm{gr}(t) \mid t \in T\}$, is a partition of $\mathcal{F}$, and for all $t, t' \in T$, if $\alpha \in \mathrm{gr}(t)$ and $\beta \in \mathrm{gr}(t')$ then $\alpha * \beta \subseteq \mathrm{gr}(tt')$.
\end{defn}
\begin{defn}
A \emph{Frobenius form} on a commutative algebra $V$ is a (non-zero) bilinear form $\langle \, , \, \rangle$ such that for all $u, v, w \in V$
\[
\langle u, v \cdot w \rangle = \langle u \cdot v, w \rangle.
\]
We say that an algebra is \emph{Frobenius} if it admits such a form.
\end{defn}
Henceforth, as is customary, we require that a Frobenius form $\langle \, , \, \rangle$ on an axial algebra $V$ satisfies $\langle a, a \rangle = 1$ for all axes $a \in V$.
\begin{prop}
\label{prop:orthogonality}
Let $(V, A)$ be a $(\mathcal{F}, *)$-axial algebra over a ring $R$ that admits a Frobenius form $\langle \, , \, \rangle$. Suppose that $R$ is an integral domain and suppose that $\lambda^{-1} \in R$ for all $\lambda \in \mathcal{F}$ such that $\lambda \neq 0$. Then the eigenspace decomposition $V = \bigoplus_{\lambda \in \mathcal{F}} V_{\lambda}^{(a)}$ is orthogonal with respect to $\langle \, , \, \rangle$ (i.e. $\langle u,v \rangle = 0$ for all $u \in V_{\mu}^{(a)}$, $v \in V_{\nu}^{(a)}$ with $\mu, \nu \in \mathcal{F}$ and $\mu \neq \nu$).
\end{prop}
\begin{proof}
Let $u \in V_{\mu}^{(a)}$, $v \in V_{\nu}^{(a)}$ with $\mu \neq \nu$ as in the hypothesis. Suppose first that $\mu = 0$ and $\nu \neq 0$. Then
\[
\langle u, v \rangle = \frac{1}{\nu}\langle u, a \cdot v \rangle = \frac{1}{\nu}\langle a \cdot u, v \rangle = 0
\]
If both $\mu$ and $\nu$ are non-zero then
\[
\frac{1}{\mu}\langle a \cdot u, v \rangle = \langle u,v \rangle = \frac{1}{\nu}\langle u, a \cdot v \rangle = \frac{1}{\nu}\langle a \cdot u, v \rangle
\]
and so, as $\mu \neq \nu$, $\langle a \cdot u, v \rangle = \langle u,v \rangle = 0$.
\end{proof}
\begin{lem}
\label{lem:projection}
Let $(V,A)$ be as in Proposition \ref{prop:orthogonality}, let $a \in A$ be a primitive axis and let $v \in V$. Then the projection of $v$ onto $\langle a \rangle$ is equal to $\langle a, v \rangle$.
\end{lem}
\begin{proof}
As $V$ is a axial algebra, we can write
\[
v = \bigoplus_{\lambda \in \mathcal{F}} v_\lambda
\]
where $v_\lambda \in V_{\lambda}^{(a)}$. As $a$ is primitive, $a_1 = \phi a$ for some $\phi \in R$. From Proposition \ref{prop:orthogonality}, if $\lambda \neq 1$ then $\langle a, v_\lambda \rangle = 0$ and so
\[
\langle a, v \rangle = \langle a, \phi a \rangle = \phi
\]
as required.
\end{proof}
\subsection{Axial algebras of Monster type}
Henceforth, we assume that the ring $R$ is an integral domain that contains a subfield $\mathbf{k}$ of characteristic $0$ such that $\frac{1}{4}, \frac{1}{32} \in \mathbf{k}$.
\begin{defn}
The \emph{Monster} or \emph{Majorana} fusion rule $(\mathcal{M}, *)$ is given by the following table where the $(\lambda, \mu)$-entry gives the entries of the set $\lambda * \mu$. As it customary, we omit the brackets from the sets in this table.
\begin{center}
\def1.75{1.5}
\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}
& 1 & 0 & \frac{1}{4} & \frac{1}{32} \\ \hline
1 & 1 & \emptyset & \frac{1}{4} & \frac{1}{32} \\
0 & \emptyset & 0 & \frac{1}{4} & \frac{1}{32} \\
\frac{1}{4} & \frac{1}{4} & \frac{1}{4} & 1,0 & \frac{1}{32} \\
\frac{1}{32} & \frac{1}{32} & \frac{1}{32} & \frac{1}{32} & 1,0,\frac{1}{4}
\end{tabular}
\end{center}
A $(\mathcal{M}, *)$-axial algebra is called an axial algebra \emph{of Monster type}.
\end{defn}
The $2A$-axes of the Griess algebra $V_{\mathbb{M}}$ are known to obey the Monster fusion rule. As the $2A$-axes generate $V_{\mathbb{M}}$, it is an example of an axial algebra of Monster type.
The Monster fusion rule admits a $C_2$-grading $\mathrm{gr}: \langle a \mid a^2 = 1 \rangle \rightarrow P(\mathcal{M})$ such that $\mathrm{gr}(1) = \{1, 0, \frac{1}{2}\}$ and $\mathrm{gr}(a) = \{\frac{1}{32} \}$. From this grading we can construct certain involutions in the automorphism group of the algebra, as described below.
\begin{defn}
Suppose that $(V, A)$ is an axial algebra of Monster type. Then for each $a \in A$, we can define a map $\tau(a) \in \mathrm{GL}(V)$ such that
\begin{align*}
\tau(a): v \mapsto
\begin{cases}
v &\textrm{ if } v \in V_1^{(a)} \oplus V_0^{(a)} \oplus V_{\frac{1}{4}}^{(a)} \\
- v &\textrm{ if } v \in V_{\frac{1}{32}}^{(a)}.
\end{cases}
\end{align*}
Then $\tau(a)$ is called a Miyamoto involution.
\end{defn}
In the case of the Griess algebra, the Miyamoto involutions $\tau(a)$, where $a$ is a $2A$-axis, form the $2A$ conjugacy class of involutions of the Monster group.
\begin{prop}
\label{prop:algproduct}
Suppose that $(V, A)$ is an axial algebra of Monster type and that $a \in A$. Then the Miyamoto involution $\tau(a)$ preserves the algebra product on $V$, i.e. $u^{\tau(a)} \cdot v ^{\tau(a)} = (u \cdot v)^{\tau(a)}$ for all $u, v \in V$.
\end{prop}
\begin{proof}
This follows directly from the definition of $\tau(a)$ and the Monster fusion rule.
\end{proof}
\begin{prop}
\label{prop:form}
Let $(V, A)$ be an axial algebra of Monster type over a ring $R$ that admits a Frobenius form $\langle \, , \, \rangle$. Suppose that $R$ is an integral domain and suppose that $\lambda^{-1} \in R$ for all $\lambda \in \mathcal{F}$ such that $\lambda \neq 0$. If $a \in A$ then $\tau(a)$ also preserves the Frobenius form, i.e. $\langle u^{\tau(a)}, v ^{\tau(a)} \rangle = \langle u,v \rangle$ for all $u, v \in V$.
\end{prop}
\begin{proof}
Suppose that $u, v \in V$. We can write $u = u_1 + u_0 + u_{\frac{1}{4}} + u_{\frac{1}{32}}$ and $v = v_1 + v_0 + v_{\frac{1}{4}} + v_{\frac{1}{32}}$ where $u_{\lambda}, v_{\lambda} \in V_{\lambda}^{(a)}$ for $\lambda \in \{1, 0, \frac{1}{4}, \frac{1}{32}\}$. Then using Proposition \ref{prop:orthogonality}, we obtain
\begin{align*}
\langle u^{\tau(a)}, v^{\tau(a)} \rangle &= \langle u_1 + u_0 + u_{\frac{1}{4}} - u_{\frac{1}{32}}, v_1 + v_0 + v_{\frac{1}{4}} - v_{\frac{1}{32}} \rangle \\
&= \langle u_1, v_1 \rangle + \langle u_0, v_0 \rangle + \langle u_{\frac{1}{4}}, v_{\frac{1}{4}} \rangle + \langle u_{\frac{1}{32}}, v_{\frac{1}{32}} \rangle
\end{align*}
and
\begin{align*}
\langle u, v \rangle &= \langle u_1 + u_0 + u_{\frac{1}{4}} + u_{\frac{1}{32}}, v_1 + v_0 + v_{\frac{1}{4}} + v_{\frac{1}{32}} \rangle \\
&= \langle u_1, v_1 \rangle + \langle u_0, v_0 \rangle + \langle u_{\frac{1}{4}}, v_{\frac{1}{4}} \rangle + \langle u_{\frac{1}{32}}, v_{\frac{1}{32}} \rangle.
\end{align*}
Thus $\langle u^{\tau(a)}, v ^{\tau(a)} \rangle = \langle u,v \rangle$ as required.
\end{proof}
\subsection{Dihedral axial algebras of Monster type}
\begin{defn}
Suppose that $(V,A)$ is an axial algebra and that $a, b \in A$ and $ a \neq b$. Then the subalgebra $U$ of $V$ generated by $a$ and $b$ is called a \emph{dihedral algebra}.
\end{defn}
The dihedral algebras of the Griess algebra (i.e. those generated by two distinct $2A$-axes) were studied by Conway and Norton \cite{Conway84} and were shown to have eight possible isomorphism types. A milestone in Majorana theory was reached when Ivanov, Pasechnik, Seress and Shpectorov \cite{IPSS10} proved that a Majorana algebra generated by two Majorana axes is isomorphic to one of the dihedral subalgebras of the Griess algebra. This reproved Conway and Norton's classification of the dihedral subalgebras of the Griess algebra, providing a foundation for the theory of Majorana algebras.
In 2015, Hall, Rehren and Shpectorov reproved this result for primitive axial algebras of Monster type over a field of characteristic $0$ that admit a Frobenius form.
\begin{thm}[{\cite[Theorem 1.2]{HRS15}}]
\label{thm:IPSS10}
Let $\mathbf{k}$ be a field such that $\mathrm{char}(\mathbf{k}) = 0$. Suppose that $(V,A)$ is a primitive axial algebra of Monster type over $\mathbf{k}$ that admits a Frobenius form. Let $a_0, a_1 \in A$ such that $a_0 \neq a_1$ and let $U$ be the subalgebra of $V$ generated by $a_0$ and $a_1$. Finally, let $\rho = \tau(a_0)\tau(a_1)$. Then
\begin{enumerate}[(i)]
\item the subalgebra $U = \langle \langle a_0, a_1 \rangle \rangle$ is isomorphic to a dihedral algebra of type $NX$, the structure of which is given in Table \ref{tab:IPSS10};
\item for $i \in \mathbb{Z}$ and $\epsilon \in \{0,1\}$, the image of $a_{\epsilon}$ under the $i$-th power of $\rho$, which we denote $a_{2i+\epsilon}$, is a $(\mathcal{M}, *)$-axis and $\tau(a_{2i + \epsilon}) = \rho^{-i}\tau_{\epsilon}\rho^i$.
\end{enumerate}
\end{thm}
We note that Table \ref{tab:IPSS10} does not show all values of the algebra product and Frobenius form on the dihedral algebras. Those which are omitted can be recovered from the action of the group $D$ and the symmetry between $a_0$ and $a_1$. The following useful lemma also follows from these symmetries.
\begin{prop}
\label{prop:axes}
Let $\mathbf{k}$ be a field such that $\mathrm{char}(\mathbf{k}) = 0$. Suppose that $(V,A)$ is a primitive axial algebra of Monster type over $\mathbf{k}$ that admits a Frobenius form. Let $a_0, a_1 \in A$ such that $a_0 \neq a_1$ and let $U$ and $a_{2i + \epsilon}$ for $\epsilon \in \{0, 1\}$ and $i \in \mathbb{Z} \backslash \{0\}$ be as in Theorem \ref{thm:IPSS10}.
If $U$ is of type $3A$ or $4A$ then $U$ contains the additional basis vector $u_{\rho(a_0, a_1)}$ or $v_{\rho(a_0, a_1)}$. These vectors obey the following equalities
\[
u_{\rho(a_0, a_1)} = u_{\rho(a_0, a_{-1})} \mbox{ and } v_{\rho(a_0, a_1)} = v_{\rho(a_0, a_{-1})}.
\]
Similarly, if $U$ is of type $5A$ then $U$ contains an additional basis vector $w_{\rho(a_0, a_1)}$ that obeys the following equalities
\begin{equation}
\label{eq:5Aaxes}
w_{\rho(a_0, a_1)} = - w_{\rho(a_0, a_2)} = - w_{\rho(a_0, a_{-2})} = w_{\rho(a_0, a_{-1})}.
\end{equation}
Moreover, if $U := \langle \langle a_0, a_1 \rangle \rangle$ is a dihedral algebra of type $3A$, $4A$ or $5A$ then $u_{\rho(a_0, a_1)} = u_{\rho(a_1, a_0)}$, $v_{\rho(a_0, a_1)} = v_{\rho(a_1, a_0)}$ or $w_{\rho(a_0, a_1)} = w_{\rho(a_1, a_0)}$ respectively.
\end{prop}
\begin{defn}
Using notation as in Proposition \ref{prop:axes} above, we refer to the vectors $u_{\rho(a_0, a_1)}$, $v_{\rho(a_0, a_1)}$ and $w_{\rho(a_0, a_1)}$ as $3A$, $4A$ and $5A$-\emph{axes} respectively.
\end{defn}
\begin{table}
\begin{center}
\vspace{0.35cm}
\noindent
\begin{tabular}{|c|c|c|}
\hline
&&\\
Type & Basis & Products and angles \\
&&\\
\hline
&&\\
&& $a_0 \cdot a_1=\frac{1}{2^3}(a_0+a_1-a_{\rho(a_0, a_1)}),~a_0 \cdot a_{\rho(a_0, a_1)}=\frac{1}{2^3}(a_0+a_{\rho(a_0, a_1)}-a_1)$ \\
2A & $a_0,a_1,a_{\rho(a_0, a_1)}$ & $a_{\rho(a_0, a_1)} \cdot a_{\rho(a_0, a_1)} = a_{\rho(a_0, a_1)}$ \\
&&$(a_0,a_1)=(a_0,a_{\rho(a_0, a_1)})=(a_{\rho(a_0, a_1)},a_{\rho(a_0, a_1)})=\frac{1}{2^3}$\\
&& \\
2B & $a_0,a_1$ &$a_0 \cdot a_1=0$,~$(a_0,a_1)=0$ \\
&&\\
& &$a_0 \cdot a_1=\frac{1}{2^5}(2a_0+2a_1+a_{-1})-\frac{3^3 \cdot 5}{2^{11}}u_{\rho(a_0, a_1)}$\\
3A& $a_{-1},a_0,a_1,$ & $a_0 \cdot u_{\rho(a_0, a_1)}=\frac{1}{3^2}(2a_0-a_1-a_{-1})+\frac{5}{2^5}u_{\rho(a_0, a_1)}$~~~~\\
&$u_{\rho(a_0, a_1)}$& $u_{\rho(a_0, a_1)} \cdot u_{\rho(a_0, a_1)}=u_{\rho(a_0, a_1)}$\\
&& $(a_0,a_1)=\frac{13}{2^8}$,~$(a_0,u_{\rho(a_0, a_1)})=\frac{1}{2^2}$,~$(u_{\rho(a_0, a_1)},u_{\rho(a_0, a_1)})=\frac{2^3}{5}$
\\
&&\\
3C & $a_{-1},a_0,a_1$ & $a_0 \cdot a_1=\frac{1}{2^6}(a_0+a_1-a_{-1}),~(a_0,a_1)=\frac{1}{2^6}$ \\
&&\\
& & ~$a_0 \cdot a_1=\frac{1}{2^6}(3a_0+3a_1+a_2+a_{-1}-3v_{\rho(a_0, a_1)})$\\
4A & $a_{-1},a_0,a_1,$ & $a_0 \cdot v_{\rho(a_0, a_1)}=\frac{1}{2^4}(5a_0-2a_1-a_2-2a_{-1}+3v_{\rho(a_0, a_1)})$\\
&$a_2,v_{\rho(a_0, a_1)}$&~$v_{\rho(a_0, a_1)} \cdot v_{\rho(a_0, a_1)}=v_{\rho(a_0, a_1)}$, ~$a_0 \cdot a_2=0$ \\
& & $(a_0,a_1)=\frac{1}{2^5},~(a_0,a_2)=0,~(a_0,v_{\rho(a_0, a_1)})=\frac{3}{2^3},~(v_{\rho(a_0, a_1)},v_{\rho(t_0,t_2)})=2$\\
&&\\
4B & $a_{-1},a_0,a_1,$ & $a_0 \cdot a_1=\frac{1}{2^6}(a_0+a_1-a_{-1}-a_2+a_{\rho(t_0,t_2)})$
\\
& $a_2,a_{\rho(t_0,t_2)}$ & $a_0 \cdot a_2=\frac{1}{2^3}(a_0+a_2-a_{\rho(t_0,t_2)})$\\
&& $(a_0,a_1)=\frac{1}{2^6},~(a_0,a_2)=(a_0,a_{\rho(a_0, a_1)})=\frac{1}{2^3}$ \\
&&\\
&& $a_0 \cdot a_1=\frac{1}{2^7}(3a_0+3a_1-a_2-a_{-1}-a_{-2})+w_{\rho(a_0, a_1)}$
\\
5A & $a_{-2},a_{-1},a_0,$ & $a_0 \cdot a_2=\frac{1}{2^7}(3a_0+3a_2-a_1-a_{-1}-a_{-2})-w_{\rho(a_0, a_1)}$
\\
& $a_1,a_2,w_{\rho(a_0, a_1)}$ & $a_0 \cdot w_{\rho(a_0, a_1)}=\frac{7}{2^{12}}(a_{1}+a_{-1}-a_2-a_{-2})+\frac{7}{2^5}w_{\rho(a_0, a_1)}$\\
& & $w_{\rho(a_0, a_1)} \cdot w_{\rho(a_0, a_1)}=\frac{5^2 \cdot 7}{2^{19}}(a_{-2}+a_{-1}+a_0+a_1+a_2)$\\
&&$(a_0,a_1)=\frac{3}{2^7},~(a_0,w_{\rho(a_0, a_1)})=0$, $(w_{\rho(a_0, a_1)},w_{\rho(a_0, a_1)})=\frac{5^3 \cdot 7}{2^{19}}$\\
&& \\
& & $a_0 \cdot a_1=\frac{1}{2^6}(a_0+a_1-a_{-2}-a_{-1}-a_2-a_3+a_{\rho(t_0,t_3)})+\frac{3^2 \cdot 5}{2^{11}}u_{\rho(t_0,t_2)}$\\
6A& $a_{-2},a_{-1},a_0,$ &$a_0 \cdot a_2=\frac{1}{2^5}(2a_0+2a_2+a_{-2})-\frac{3^3 \cdot 5}{2^{11}}u_{\rho(t_0,t_2)}$ \\
&$a_1,a_2,a_3$ &$a_0 \cdot u_{\rho(t_0,t_2)}=\frac{1}{3^2}(2a_0-a_2-a_{-2})+\frac{5}{2^5}u_{\rho(t_0,t_2)}$ \\
&$a_{\rho(t_0,t_3)},u_{\rho(t_0,t_2)}$ & $a_0 \cdot a_3=\frac{1}{2^3}(a_0+a_3-a_{\rho(t_0,t_3)})$, $a_{\rho(t_0,t_3)} \cdot u_{\rho(t_0,t_2)}=0$\\
&&$(a_{\rho(t_0,t_3)},u_{\rho(t_0,t_2)})=0$, $(a_0,a_1)=\frac{5}{2^8}$, $(a_0,a_2)=\frac{13}{2^8}$, $(a_0,a_3)=\frac{1}{2^3}$\\
&&\\
\hline
\end{tabular}
\caption{The dihedral axial algebras of Monster type}
\label{tab:IPSS10}
\end{center}
\end{table}
Using the algebra values in Table \ref{tab:IPSS10}, we can also calculate the eigenspace decompositions of the dihedral axial algebras of Monster type. Bases of the $0$, $\frac{1}{4}$ and $\frac{1}{32}$-eigenspaces of the axis $a_0$ for each dihedral algebra are given in Table \ref{tab:dihedralevecs}. In each case, the $1$-eigenspace is, by definition, one dimensional and so is omitted from this table.
\begin{table}
\begin{center}
\vspace{0.35cm}
\noindent
\begin{tabular}{|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|}
\hline
&&&\\
\mbox{Type} & 0 & \frac{1}{4} & \frac{1}{32} \\
&&&\\
\hline
&&&\\
2A & a_1 + a_{\rho(a_0, a_1)} - \frac{1}{2^2} & a_1 - a_{\rho(a_0, a_1)} & \\
&&&\\
2B & a_1 & & \\
&&&\\
3A & u_{\rho(a_0, a_1)} - \frac{2 \cdot 5}{3^3}a_0 + \frac{2^5}{3^3}(a_1 + a_{-1})
& \ml{c}{u_{\rho(a_0, a_1)} - \frac{2^3}{3^2 \cdot 5}a_0\\
- \frac{2^5}{3^2 \cdot 5}(a_1 + a_{-1}) }
& a_1 - a_{-1} \\
&&&\\
3C & a_1 + a_{-1} - \frac{1}{2^5}a_0
&
& a_1 - a_{-1} \\
&&&\\
4A & v_{\rho(a_0, a_1)} - \frac{1}{2}a_0 + 2(a_1 + a_{-1}), a_2
& \ml{c}{v_{\rho(a_0, a_1)} - \frac{1}{3}a_0 \\
- \frac{2}{3}(a_1 + a_{-1}) - \frac{1}{3}a_2}
& a_1 - a_{-1} \\
&&&\\
4B & \ml{c}{a_1 + a_{-1} - \frac{1}{2^5}a_0 - \frac{1}{2^3}(a_{\rho(t_0,t_2)}- a_2), \\
a_2 + a_{\rho(t_0,t_2)} - \frac{1}{2^2}a_0}
& a_2 - a_{\rho(t_0,t_2)}
& a_1 - a_{-1} \\
&&&\\
5A & \ml{c}{w_{\rho(a_0, a_1)} + \frac{3}{2^9}a_0 - \frac{3 \cdot 5}{2^7}(a_1 + a_{-1}) - \frac{1}{2^7}(a_2 - a_{-2}), \\
w_{\rho(a_0, a_1)} - \frac{3}{2^9}a_0 + \frac{1}{2^7}(a_1 + a_{-1}) + \frac{3 \cdot 5}{2^7}(a_2 + a_{-2})}
& \ml{c}{w_{\rho(a_0, a_1)} + \frac{1}{2^7}(a_1 + a_{-1}) \\
- \frac{1}{2^7}(a_2 + a_{-2})}
& \ml{c}{a_1 - a_{-1}, \\ a_2 - a_{-2}} \\
&&&\\
6A & \ml{c}{u_{\rho(t_0,t_2)} + \frac{2}{3^2 \cdot 5}a_0 - \frac{2^8}{3^2 \cdot 5}(a_1 - a_{-1}) \\
- \frac{2^5}{3^2 \cdot 5}(a_2 + a_{-2} + a_3 - a_{\rho(t_0,t_3)}), \\
a_3 + a_{\rho(t_0,t_3)} - \frac{1}{2^2}a_0, \\
u_{\rho(t_0,t_2)} - \frac{2 \cdot 5}{3^3}a_0 + \frac{2^5}{3^3}(a_2 + a_{-2})}
& \ml{c}{u_{\rho(t_0,t_2)} - \frac{2^3}{3^2 \cdot 5}a_0 \\
- \frac{2^5}{3^2 \cdot 5}(a_2 + a_{-2} + a_3)\\
+ \frac{2^5}{3^2 \cdot 5} a_{\rho(t_0,t_3)} , \\
a_3 - a_{\rho(t_0,t_3)}}
& \ml{c}{a_1 - a_{-1}, \\ a_2 - a_{-2}} \\
&&& \\
\hline
\end{tabular}
\caption{The eigenspace decomposition of the dihedral axial algebras of Monster type}
\label{tab:dihedralevecs}
\end{center}
\end{table}
The following results follow directly from the values in Table \ref{tab:IPSS10}.
\begin{lem}
\label{lem:dihedral}
Let $U$ be a dihedral axial algebra of Monster type (as in Table \ref{tab:IPSS10}) that is generated by axes $a_0$ and $a_1$. Let $D := \langle \tau(a_0), \tau(a_1) \rangle$. Then $|a_0^D \cup a_1^D| = N$.
\end{lem}
\begin{lem}
\label{lem:inclusions}
Let $U$ be a dihedral axial algebra of Monster type (as in Table \ref{tab:IPSS10}) that is generated by axes $a_0$ and $a_1$. Then $U$ contains no proper, non-trivial subalgebras, with the exception of the following cases.
\begin{enumerate}[(i)]
\item If $U$ is of type $4A$ or $4B$ then the subalgebras $\langle \langle a_0, a_2 \rangle \rangle$ and $\langle \langle a_1, a_{-1} \rangle \rangle$ are of type $2B$ or $2A$ respectively.
\item If $U$ is of type $6A$ then the subalgebras $\langle \langle a_0, a_3 \rangle \rangle$ and $\langle \langle a_1, a_{-2} \rangle \rangle$ are of type $3A$ and the subalgebras $\langle \langle a_0, a_2 \rangle \rangle$, $\langle \langle a_1, a_{-1} \rangle \rangle$, $\langle \langle a_0, a_{-2} \rangle \rangle$ and $\langle \langle a_1, a_3 \rangle \rangle$ are of type $2A$.
\end{enumerate}
\end{lem}
Informally, this means that we have the following \emph{inclusions} of algebras:
\[
2A \hookrightarrow 4B, \qquad 2B \hookrightarrow 4A, \qquad 2A \hookrightarrow 6A, \qquad \textrm{and} \qquad 3A \hookrightarrow 6A
\]
and that these are the only possible inclusions of non-trivial algebras.
\section{Construction}
\label{sec:construction}
We will classify Frobenius $(\mathcal{M}, *)$-axial algebras that obey property $\mathcal{P}$ as defined below. In doing so, we will construct an infinite family of Frobenius $(\mathcal{M}, *)$-axial algebras over the field $\mathbb{R}$.
\begin{defn}
We say that a Frobenius $(\mathcal{M}, *)$-axial algebra $V$ obeys \emph{property $\mathcal{P}$} if it is generated by a set of axes $A := \{a_1, a_{-1}, a_2, a_{-2}, a_3, a_{-3} \}$ such that the corresponding Miyamoto involutions induce the following permutation actions (written in cycle notation) on the set $A$
\begin{align*}
\tau(a_1) = \tau(a_{-1}): (a_2, \, a_{-2})(a_3, \, a_{-3}); \\
\tau(a_2) = \tau(a_{-2}): (a_1, \, a_{-1})(a_3, \, a_{-3}); \\
\tau(a_3) = \tau(a_{-3}): (a_1, \, a_{-1})(a_2, \, a_{-2}).
\end{align*}
\end{defn}
\subsection{The shape of the algebra}
\label{sec:shape}
Suppose that $V$ is a Frobenius $(\mathcal{M}, *)$-axial algebra that obeys property $\mathcal{P}$. We start by classifying the possible dihedral algebras contained in $V$. For each dihedral subalgebra, there is possibly more than one choice of generators for each dihedral algebra. For example,
\[
\langle \langle a_1, a_2 \rangle \rangle = \langle \langle a_1, a_{-2} \rangle \rangle = \langle \langle a_{-1}, a_2 \rangle \rangle = \langle \langle a_{-1}, a_{-2} \rangle \rangle.
\]
Moreover, from Lemma \ref{lem:inclusions}, the type of one dihedral algebra will determine the types of those algebras which it contains, and of those algebras that it is contained in, if any exist. For example, from Lemma \ref{lem:dihedral}, $U_0 := \langle \langle a_1, a_2 \rangle \rangle$ must be of type $4A$ or $4B$. This algebra contains the dihedral algebra $\langle \langle a_{-1}, a_{1} \rangle \rangle$ which must therefore be of type $2B$ or $2A$ if $U_0$ is of type $4A$ or $4B$ respectively.
All non-trivial dihedral subalgebras of $V$ and their respective inclusions are shown in Figure 1. In this directed graph, the vertices are the non-trivial dihedral subalgebras of $V$ and if $U$ and $W$ are two such algebras then $U \rightarrow W$ if and only if $U$ is a subalgebra of $W$.
It is clear from this graph that the choice of any one of these algebras determines the type of all other algebras. Thus we have only two cases - one when the algebra $\langle \langle a_1, a_2 \rangle \rangle$ is of type $4A$ and another when it is of type $4B$. For $X \in \{A, B \}$ we say that $V$ has \emph{shape} $4X$ if the subalgebra $U_0$ of $V$ is of type $4X$.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw
(0,2) node(a1) {$\langle \langle a_1, a_2 \rangle \rangle$}
(2,2) node(a2) {$\langle \langle a_1, a_3 \rangle \rangle$}
(4,2) node(a3) {$\langle \langle a_2, a_3 \rangle \rangle$}
(0,0) node(a4) {$\langle \langle a_1, a_{-1} \rangle \rangle$}
(2,0) node(a5) {$\langle \langle a_2, a_{-2} \rangle \rangle$}
(4,0) node(a6) {$\langle \langle a_3, a_{-3} \rangle \rangle$};
\draw [->] (a4) -- (a1);
\draw [->] (a5) -- (a1);
\draw [->] (a4) -- (a2);
\draw [->] (a6) -- (a2);
\draw [->] (a5) -- (a3);
\draw [->] (a6) -- (a3);
\end{tikzpicture}
\end{center}
\label{fig:inclusions}
\caption{The inclusions of dihedral algebras in $M(U_{\mathcal{P}}, V_{\mathcal{P}})$}
\end{figure}
\subsection{The universal algebra}
In \cite{HRS15}, Hall, Rehren and Shpectorov define the category of commutative (non-associative) Frobenius algebras on $n$ (marked) generators, which they denote $\mathcal{C}_0$. They show that $\mathcal{C}_0$ contains an algebra $\hat{M}$, called the \emph{universal $n$-generated commutative Frobenius algebra}, such that $\mathcal{C}_0$ coincides with the category of all quotients of $\hat{M}$.
The universal algebra $\hat{M}$ is constructed as follows. Let $\hat{X}$ be the free commutative non-associative magma with marked generators $\hat{A} := \{\hat{a}_1, \ldots, \hat{a}_n\}$. Let $\sim$ be the equivalence relation on $\hat{X} \times \hat{X}$ generated by all elementary equivalences $(\hat{x} \cdot \hat{y}, \hat{z}) \sim (\hat{x}, \hat{y} \cdot \hat{z})$ for $\hat{x}, \hat{y}, \hat{z} \in \hat{X}$. For $\hat{x}, \hat{y} \in \hat{X}$, we let $[\hat{x}, \hat{y}]$ denote the $\sim$-equivalence class containing $(\hat{x}, \hat{y})$. Furthermore, let $[\hat{X} \times \hat{X}]$ denote the set of all $\sim$-equivalence classes.
Let
\[
\hat{R} = \mathbf{k}\left[\{\lambda_{[\hat{x}, \hat{y}]} \}_{ [\hat{x}, \hat{y}] \in [\hat{X} \times \hat{X}]} \right]
\]
and let $\hat{M} = \hat{R}\hat{X}$, the set of all formal linear combinations $\sum_{\hat{x} \in \hat{X}} \alpha_{\hat{x}}\hat{x}$ where $\alpha_{\hat{x}} \in \hat{R}$. Then $\hat{M}$ is a commutative $\hat{R}$-algebra where the algebra product is defined using the operation in $\hat{X}$ and the distributive law. We can also define a bilinear form on $\hat{M}$ as follows:
\[
\left\langle \sum_{\hat{x} \in \hat{X}} \alpha_{\hat{x}}\hat{x}, \sum_{\hat{y} \in \hat{X}} \beta_{\hat{y}}\hat{y} \right\rangle = \sum_{\hat{x}, \hat{y} \in \hat{X}} \alpha_{\hat{x}}\beta_{\hat{y}} \lambda_{[\hat{x}, \hat{y}]}.
\]
Then, as $\lambda_{[\hat{x}, \hat{y}]}$ depends on the equivalence class $[\hat{x}, \hat{y}]$, this bilinear form is automatically Frobenius. This algebra $\hat{M}$ is also generated as an algebra by the marked generators $\hat{A} := \{\hat{a}_1, \ldots, \hat{a}_n\}$ of $\hat{X}$,
\begin{prop}[{\cite[Proposition 4.1]{HRS15}}]
Every algebra $M$ in $\mathcal{C}_0$ is the quotient, up to isomorphism, of $\hat{M}$ over an ideal $I_M$ and has coefficient ring $R$, which is the quotient of $\hat{R}$ over an ideal $J_M$.
\end{prop}
Choose $U \leq \hat{M}$ and $V \leq \hat{R}$. Then we let $\mathcal{C}_{U,V}$ be the subcategory of $\mathcal{C}_0$ consisting of all algebras $M$ from $\mathcal{C}_0$ such that $U \subseteq I_M$ and $V \subseteq J_M$. Let
\begin{itemize}
\item $I_0$ be the ideal of $\hat{M}$ generated by $U$;
\item $J(U,V)$ be the ideal of $\hat{R}$ generated by $V$, together with all elements $\langle i, m \rangle$ for $i \in I_0$ and $m \in \hat{M}$;
\item $I(U,V)$ be the ideal of $\hat{M}$ generated by $U$ and $J(U,V)\hat{M}$.
\end{itemize}
\begin{prop}[{\cite[Proposition 4.4]{HRS15}}]
If $J(U,V) \neq \hat{R}$ then $\mathcal{C}_{U,V}$ is non-empty and contains a universal algebra $M(U,V)$ such that $\mathcal{C}_{U,V}$ consists of all quotients of $M(U,V)$. Moreover, $M(U,V) = \hat{M}/I(U,V)$ and has coefficient ring $R(U,V) = \hat{R}/J(U,V)$.
\end{prop}
For a fixed fusion rule $(\mathcal{F}, *)$, Hall et al. show that the universal primitive Frobenius $(\mathcal{F}, *)$-axial algebra is equal to $M(U_{(\mathcal{F}, *)},V_{(\mathcal{F}, *)})$ for some explicit vlaues of $U_{(\mathcal{F}, *)}$ and $V_{(\mathcal{F}, *)}$. We will similarly construct the universal primitive Frobenius $(\mathcal{M}, *)$-axial algebra that obeys property $\mathcal{P}$ and has shape $4X$ for $X \in \{A,B\}$ (as defined in Section \ref{sec:shape}).
We now restrict to the case where
\begin{itemize}
\item $\hat{M}$ has six marked generators, which we denote $ \hat{A} = \{\hat{a}_1, \hat{a}_{-1}, \hat{a}_2, \hat{a}_{-2}, \hat{a}_3, \hat{a}_{-3} \}$;
\item $\hat{R}$ contains $\mathbb{R}$ as a subfield.
\end{itemize}
\begin{lem}[{\cite[Lemma 5.4]{HRS15}}]
\label{lem:espaces}
Suppose that $V$ is an algebra over a field $\mathbf{k}$. If $u, v \in \hat{M}$ and $\Lambda \subseteq \mathbf{k}$ then
\[
v \in \bigoplus_{\lambda \in \Lambda} V_\lambda^{(u)} \Leftrightarrow f_\Lambda(ad_u)(v) = 0
\]
where $V_\lambda^{(u)}$ denotes the $\lambda$-eigenspace of $ad_u$ and $f_\Lambda(x) = \prod_{\lambda \in \Lambda} (x - \lambda) \in \mathbf{k}[x]$.
\end{lem}
\begin{lem}
\label{lem:universal}
For $X \in \{A, B \}$, the primitive Frobenius $(\mathcal{M}, *)$-axial algebras that obey property $\mathcal{P}$ and that have shape $4X$ form a subcategory $\mathcal{C}_{4X}$ of $\mathcal{C}_0$. There exists a universal algebra $M_{4X}$ of $\mathcal{C}_{4X}$, and $\mathcal{C}_{4X}$ consists of all quotients of $M_{4X}$.
\end{lem}
\begin{proof}
We will first define subsets $U \subseteq \hat{M}$ and $V \subseteq \hat{R}$ such that $\mathcal{C}_{4X} = \mathcal{C}(U,V)$ by expressing the conditions of property $\mathcal{P}$ as elements of $\hat{M}$ and $\hat{R}$.
We let
\begin{align*}
V_{\mathcal{P}} = V_{(\mathcal{M}, *)} \textrm{ and } U_{\mathcal{P}} = U_{(\mathcal{M}, *)} \cup \left\{ f_{\{\frac{1}{32}\}}(ad_{\hat{a}_{\pm i}})(\hat{a}_j - \hat{a}_{-j}) \right\} \cup \left\{ f_{\{1, 0, \frac{1}{4}\}}(ad_{\hat{a}_{\pm i}})(\hat{a}_j + \hat{a}_{-j}) \right\}
\end{align*}
where $i, j \in \{1, 2, 3\}$ and $i \neq j$. We will now show that if $M$ is an algebra in $\mathcal{C}(U_{(\mathcal{M}, *)}, V_{(\mathcal{M}, *)})$ then $M$ obeys property $\mathcal{P}$ if and only if $M$ also lies in $\mathcal{C}(U_{\mathcal{P}}, V_{\mathcal{P}})$.
Fix $i, j \in \{1, 2, 3\}$ such that $i \neq j$ and suppose that $f_{\{\frac{1}{32}\}}(ad_{\hat{a}_{\pm i}})(\hat{a}_j - \hat{a}_{-j}) = 0$ and $f_{\{1, 0, \frac{1}{4}\}}(ad_{\hat{a}_{\pm i}})(\hat{a}_j + \hat{a}_{-j}) = 0$. Then, from Lemma \ref{lem:espaces},
\[
\hat{a}_j - \hat{a}_{-j} \in V_{\frac{1}{32}}^{(\hat{a}_{\pm i})} \textrm{ and } \hat{a}_j + \hat{a}_{-j} \in \bigoplus_{\lambda \in \{1, 0, \frac{1}{4}\}} V_\lambda^{(\hat{a}_{\pm i})}.
\]
and so $(\hat{a}_j - \hat{a}_{-j})^{\tau(a_{\pm i})} = \hat{a}_{-j} - \hat{a}_j$ and $(\hat{a}_j + \hat{a}_{-j})^{\tau(a_{\pm i})} = \hat{a}_j + \hat{a}_{-j}$. We can then infer that $\hat{a}_j^{\tau(a_{\pm i})} = \hat{a}_{-j}$.
Conversely, if $\hat{a}_j^{\tau(a_{\pm i})} = \hat{a}_{-j}$ then it follows from the definition of $\tau(a_{\pm i})$ that $
\hat{a}_j - \hat{a}_{-j} \in V_{\frac{1}{32}}^{(\hat{a}_{\pm i})}$ and $\hat{a}_j + \hat{a}_{-j} \in \bigoplus_{\lambda \in \{1, 0, \frac{1}{4}\}} V_\lambda^{(\hat{a}_{\pm i})}$. Thus, from Lemma \ref{lem:espaces},
\[
f_{\{\frac{1}{32}\}}(ad_{\hat{a}_{\pm i}})(\hat{a}_j - \hat{a}_{-j}) = f_{\{1, 0, \frac{1}{4}\}}(ad_{\hat{a}_{\pm i}})(\hat{a}_j + \hat{a}_{-j}) = 0
\]
as required.
Futhermore, we note that requiring that an algebra has a certain shape is equivalent to putting restrictions on the elements of $\hat{M}$ and $\hat{R}$. These restrictions can be encoded as relations to be included in the sets $U$ and $V$. We do not explicitly give these relations; they can be read directly from the values in Table \ref{tab:IPSS10}.
In particular, if we take $U$ and $V$ to be the union of the relations coming from the choice of the shape with $U_{\mathcal{P}}$ and $V_{\mathcal{P}}$ respectively then the category $\mathcal{C}(U,V)$ coincides with $\mathcal{C}_{4X}$. The final claim follows from \cite[Proposition 4.4]{HRS15}.
\end{proof}
Recall that $M_{4X}$ is equal to $\hat{M}/I(U,V)$ and has coefficient ring $R_{4X} = \hat{R}/J(U,V)$ for $U$ and $V$ as in Lemma \ref{lem:universal}. We let $\psi_{4X}$ and $\phi_{4X}$ denote the corresponding projections $\hat{M} \rightarrow M_{4X}$ and $\hat{R} \rightarrow R_{4X}$ and let $A := \{ a_1, a_{-1}, a_2, a_{-2}, a_3, a_{-3}\}$ denote the image of $\hat{A}$ under $\psi_{4X}$.
Note that the maps $\psi_{4X}$ and $\phi_{4X}$ determine the algebra product and Frobenius form on $M_{4X} = \psi_{4X}(\hat{M})$ in that
\begin{equation}
\label{eq:products}
u^{\psi_{4X}} \cdot v^{\psi_{4X}} =( u \cdot v)^{\psi_{4X}} \textrm{ and } \langle u^{\psi_{4X}}, v^{\psi_{4X}} \rangle = \langle u, v \rangle^{\phi_{4X}}
\end{equation}
for all $u, v \in \hat{M}$.
\subsection{Automorphisms of $M_{4X}$}
Note that the group $G := \langle \tau(a) \mid a\in A \rangle \cong 2^2$ is a subgroup of $\mathrm{GL}(M_{4X})$ whose action on $M_{4X}$ preserves the algebra product and Frobenius form (from Propositions \ref{prop:algproduct} and \ref{prop:form}). This algebra also admits further symmetries, as we now explain.
\begin{lem}
The permutation $\sigma \in \mathrm{Sym}(A)$ defined by
\[
\sigma = (a_1, \, a_2, \, a_3)(a_{-1}, \, a_{-2}, \, a_{-3})
\]
uniquely extends to an element $\psi_{\sigma} \in \mathrm{GL}(M_{4X})$ that preserves the algebra product and Frobenius form on $M_{4X}$ for $X \in \{A, B\}$. That is to say, there exists an automorphism $\phi_{\sigma}$ of $R_{4X}$ such that $\phi_{\sigma}$ preserves the subfield $\mathbb{R}$ of $R_{4X}$ and such that
\[
u^{\psi_{\sigma}} \cdot v^{\psi_{\sigma}} = (u \cdot v)^{\psi_{\sigma}} \textrm{ and } \langle u^{\psi_{\sigma}}, v^{\psi_{\sigma}} \rangle = \langle u, v \rangle^{\phi_{\sigma}}
\]
\end{lem}
\begin{proof}
We first consider the algebra $\hat{M}$ and define the permutation $\hat{\sigma} \in \mathrm{Sym}(\hat{A})$ to be such that
\[
\hat{\sigma} = (\hat{a}_1, \, \hat{a}_2, \, \hat{a}_3)(\hat{a}_{-1}, \, \hat{a}_{-2}, \, \hat{a}_{-3}).
\]
Then $\hat{\sigma}$ clearly extends to an automorphism $\hat{\psi}_\sigma$ of the free commutative magma $\hat{X}$. Moreover, it preserves the equivalence relation $\sim$ on $\hat{X} \times \hat{X}$ and so induces a permutation on the indeterminates $\lambda_{[\hat{x}, \hat{y}]}$ of the polynomial ring $\hat{R}$. This permutation induces an automorphism $\hat{\phi}_{\sigma}$ of $\hat{R}$ that fixes the field $\mathbb{R}$. We can now see that $\hat{\psi}_\sigma$ uniquely extends to $\hat{M}$ via
\[
\left( \sum_{\hat{x} \in \hat{X}} \alpha_{\hat{x}} \hat{x} \right)^{\hat{\psi}_{\sigma}} = \sum_{\hat{x} \in \hat{X}} \alpha_{\hat{x}}^{\hat{\phi}_{\sigma}} \hat{x}^{\hat{\psi}_{\sigma}}.
\]
Crucially, the maps $\hat{\psi}_\sigma$ and $\hat{\phi}_\sigma$ preserve the sets $U$ and $V$ from Lemma \ref{lem:universal} and therefore also the ideals $I(U,V)$ and $J(U,V)$. In particular, these maps descend to automorphisms of $M_{4X}$ and $R_{4X}$ as required.
\end{proof}
\section{The algebra $M_{4B}$}
\label{sec:4B}
\begin{thm}
The algebra $M_{4B}$ is $7$-dimensional. It has basis
\[
B := \{ a_1, a_{-1}, a_2, a_{-2}, a_3, a_{-3}, a_{\rho}\}
\]
where $a_{\rho} := a_1 + a_{-1} - 8a_1 \cdot a_{-1}$ and its coefficient ring $R_{4B}$ is equal to $\mathbb{R}$.
\end{thm}
\begin{proof}
The algebra $M_{4B}$ contains three distinct dihedral subalgebras of type $2A$; $\langle \langle a_i, a_{-i} \rangle \rangle$ for $1 \leq i \leq 3$. Each of these dihedral subalgebras is of dimension $3$ and contains a third basis vector $a_{\rho_i} := a_i + a_{-i} - 8 a_i \cdot a_{-i}$ for $i \in \{1, 2, 3\}$.
However, if $i, j \in \{1, 2, 3\}$ such that $i \neq j$, then both $\langle \langle a_i, a_{-i} \rangle \rangle$ and $\langle \langle a_j, a_{-j} \rangle \rangle$ are contained in the dihedral algebra $\langle \langle a_i, a_j \rangle \rangle$, which is of type $4A$. In particular, this then implies that $a_{\rho_i} = a_{\rho_j}$. Thus we let $a_{\rho}$ denote the vector $a_{\rho_1} = a_{\rho_2} = a_{\rho_2}$.
Moreover, for all $a \in A$, there exists a dihedral algebra $U$ such that $a, a_{\rho} \in U$ and so the values for all algebra products on $B$ are given by the known values of the dihedral algebras. This the implies that the vector space is closed under multiplication and $M_{4B} = \langle B \rangle$. The value of the Frobenius form on $M_{4B}$ is also uniquely determined by the known values of the dihedral algebras and therefore the coefficient ring of the algebra is $\mathbb{R}$.
We can then use these values to check that $M_{4B}$ satisfies the definition of an axial algebra, and we are done.
\end{proof}
\section{The algebra $M_{4A}$}
\label{sec:4A}
\begin{thm}
\label{thm:4A}
The algebra $M_{4A}$ is $12$-dimensional. It has basis
\[
B := A \cup \{ v_{(1,2)}, v_{(1,3)}, v_{(2,3)} \} \cup \{ a_1 \cdot v_{(2,3)}, a_2 \cdot v_{(1,3)}, a_3 \cdot v_{(1,2)}\}
\]
where
\[
v_{(i,j)} = a_i + a_j + \frac{1}{3}a_{-i} + \frac{1}{3}a_{-j} - \frac{2^6}{3}a_i \cdot a_j
\]
for $i,j \in \{1, 2, 3\}$ such that $i \neq j$. Its coefficient ring $R_{4A}$ is equal to $\mathbb{R}[t]$ where $t := \langle a_1, v_{(2,3)} \rangle$.
\end{thm}
\begin{thm}
\label{thm:4Ab}
There exists an infinite family of $12$-dimensional axial algebras of Monster type which we denote $\{M(t) \}_{t \in \mathbb{R}}$. If $t \notin \{0, \frac{1}{6}, \frac{9}{4} \}$ then the algebra $M(t)$ is simple. Otherwise, if $t \in \{0, \frac{1}{6}\}$, $M(t)$ contains a $3$-dimensional ideal and if $t = \frac{9}{4}$ then $M(t)$ contains a $5$-dimensional ideal.
\end{thm}
Henceforth, we write $M := M_{4A}$.
\subsection{ Algebra products }
To begin, we let
\[
\bar{B} := B \cup \{ a_{-1} \cdot v_{(2,3)}, a_{-2} \cdot v_{(1,3)}, a_{-3} \cdot v_{(1,2)}\}.
\]
Recall that the algebra product $\cdot$ and the Frobenius form $\langle \, , \, \rangle$ on $M$ are defined as in (\ref{eq:products}). Throughout this section, we will let $t := \langle a_1, v_{(2,3)} \rangle$.
\begin{prop}
\label{prop:frobeniusform1}
Some values of the Frobenius form $\langle \, , \, \rangle $ on the vectors of $\bar{B}$ are as given in Table 3.
\end{prop}
\begin{proof}
These values follow from the known values of the algebra products and Frobenius form on dihedral Majorana algebras and from the fact that for all $u, v, w \in M$, we must have $\langle u, v \cdot w \rangle = \langle u \cdot v, w \rangle$.
\end{proof}
\begin{table}%
\begin{center}
\def1.75{1.75}
\begin{tabular}{| >{$} l <{$} >{$} l <{$} >{$} c <{$} | } \hline
u & v & \langle u,v \rangle \\ \hline
a_1 & v_{(2,3)} & t \\
v_{(1,2)} & v_{(2,3)} & -\frac{8t}{3} + \frac{1}{2} \\
a_1 & a_1 \cdot v_{(2,3)} & t \\
a_{-1} & a_1 \cdot v_{(2,3)} & 0 \\
a_2 & a_1 \cdot v_{(2,3)} & \frac{3t}{2^4} \\
v_{(1,2)} & a_1 \cdot v_{(2,3)} & -\frac{t}{2^2} \\
v_{(2,3)} & a_1 \cdot v_{(2,3)} & t \\ \hline
\end{tabular}
\end{center}
\label{tab:frobeniusform1}
\caption{Some values of the Frobenius form $\langle \, , \, \rangle$ on $M$}
\end{table}
\begin{prop}
\label{prop:products1}
\begin{align*}
v_{(1,2)} \cdot v_{(1,3)} = &-\frac{8t}{3}a_1 + \frac{1}{2^2}(v_{(1,2)} + v_{(1,3)} - v_{(2,3)}) + \frac{8}{3}a_1 \cdot v_{(2,3)} \\
&- \frac{2}{3} ((a_2 + a_{-2}) \cdot v_{(1,3)} + (a_3 + a_{-3}) \cdot v_{(1,2)}) \\
a_1 \cdot (a_2 \cdot v_{(1,3)}) =& \frac{t}{2^3}a_1 + \frac{1}{2^4}a_1 \cdot v_{(2,3)} + \frac{1}{2^6}(5a_2 + 3a_{-2}) \cdot v_{(1,3)} - \frac{1}{2^4}(a_3 + a_{-3}) \cdot v_{(1,2)} \\
\end{align*}
\end{prop}
\begin{proof}
From the known values of dihedral algebras, the following are eigenvectors of $a_1$
\begin{align*}
\alpha_0 &:= -\frac{1}{2}a_1 + 2a_2 + 2a_{-2} + v_{(1,2)} \in M_0^{(a_1)} \\
\beta_0 &:= -\frac{1}{3}(a_1 + a_{-1} + 2a_2 + 2a_{-2}) + v_{(1,2)} \in M_{\frac{1}{2^2}}^{(a_1)} \\
\alpha_1 &:= -\frac{1}{2}a_1 + 2a_3 + 2a_{-3} + v_{(1,3)} \in M_0^{(a_1)} \\
\beta_1 &:= -\frac{1}{3}(a_1 + a_{-1} + 2a_{-2} + 2a_{-3}) + v_{(1,3)} \in M_{\frac{1}{2^2}}^{(a_1)}.
\end{align*}
Then
\[
(\alpha_0 - \beta_0) \cdot (\alpha_1 - \beta_1) = \frac{1}{2^2} a_{-1} + \frac{11}{2^2 \cdot 3}(a_2 + a_{-2} + a_3) - \frac{1}{2^3 \cdot 3}(v_{(1,2)} + v_{(1,3)}) - \frac{4}{3} v_{(2,3)}
\]
and so the value of $a_1 \cdot ( (\alpha_0 - \beta_0) \cdot (\alpha_1 - \beta_1) )$ can also be computed. Moreover, from the fusion rule,
\[
a_1 \cdot ( (\alpha_0 - \beta_0) \cdot (\alpha_1 - \beta_1) ) = - \frac{1}{2^2}( \alpha_0 \cdot \beta_1 + \alpha_1 \cdot \beta_0) + \frac{1}{2^2} \langle \beta_0, \beta_1 \rangle a_1.
\]
We calculate that $\langle \beta_0, \beta_1 \rangle = -\frac{16}{3}t + \frac{1}{3}$ and that
\begin{align*}
\alpha_0 \cdot \beta_0 + \alpha_1 \cdot \beta_1 = 2 v_{(1,2)} \cdot v_{(1,3)} &- \frac{1}{2^2} \sum_{i = 1}^6 a_i - \frac{1}{2^3}(v_{(1,2)} + v_{(1,3)} - 4v_{(2,3)}) \\
&- \frac{4}{3}((a_2 + a_{-2}) \cdot v_{(1,3)} + (a_3 + a_{-3}) \cdot v_{(1,2)}).
\end{align*}
Using these values, we can calculate $v_{(1,2)} \cdot v_{(1,3)}$ as required.
The fusion rule again implies that
\[
a_1 \cdot ((\alpha_0 - \beta_0) \cdot \alpha_1) = -\frac{1}{2^2} \alpha_1 \cdot \beta_0
\]
As we now know the value of $v_{(1,2)} \cdot v_{(1,3)}$, the product $\alpha_1 \cdot \beta_0$ can be directly computed.
Moreover,
\[
(\alpha_0 - \beta_0) \cdot \alpha_1 = -\frac{1}{2^3}(a_1 - a_{-1}) + \frac{7}{2^2 \cdot 3}(a_2 + a_{-2}) + \frac{2}{3}(a_3 + a_{-3}) + \frac{1}{2^3}v_{(1,2)} - v_{(2,3)} + \frac{8}{3}(a_2 + a_{-2}) \cdot v_{(1,3)}
\]
and so
\begin{align*}
a_1 \cdot ((\alpha_0 - \beta_0) \cdot \alpha_1) = \frac{8}{3}a_1 \cdot ((a_2 &+ a_{-2}) \cdot v_{(1,3)}) + \frac{1}{2^5}(a_1 + a_{-1}) + \frac{1}{2^4 \cdot 3}(a_2 + a_{-2})\\
& + \frac{1}{2^3 \cdot 3}(a_3 + a_{-3}) - \frac{1}{2^5}(v_{(1,2)} + 2v_{(1,3)}) + a_1 \cdot v_{(2,3)}.
\end{align*}
Using these values, we can calculate $a_1 \cdot ((a_2 + a_{-2}) \cdot v_{(1,3)})$. Finally, we know that $(a_2 - a_{-2}) \cdot v_{(1,3)} \in M_{\frac{1}{2^5}}^{(a_1)}$ and so
\[
a_1 \cdot ((a_2 - a_{-2}) \cdot v_{(1,3)}) = \frac{1}{2^5} (a_2 - a_{-2}) \cdot v_{(1,3)}.
\]
We can then calculate the value of $a_1 \cdot (a_2 \cdot v_{(1,3)})$ as required.
\end{proof}
\begin{prop}
\label{prop:products2}
\begin{align*}
a_1 \cdot (a_1 \cdot v_{(2,3)}) =& \frac{3t}{2^2}a_1 + \frac{1}{2^2}a_1 \cdot v_{(2,3)} \\
a_{-1} \cdot (a_1 \cdot v_{(2,3)}) =& -\frac{t}{2^2}a_{-1} -\frac{t}{2^3} (a_3 - a_{-3}) + \frac{1}{2^2}a_{-1} \cdot v_{(2,3)} + \frac{1}{2^3}(a_3 + a_{-3}) \cdot v_{(1,3)} \\
v_{(1,2)} \cdot (a_1 \cdot v_{(2,3)}) =& -\frac{t}{2^3 \cdot 3}(7a_1 + a_{-1} + 9a_2 + 7a_{-2} + 2a_3 - 2a_{-3}) + \frac{t}{2^3}v_{(1,2)} \\
&+ \frac{1}{2^4}(7a_1 - 3a_{-1}) \cdot v_{(2,3)} +\frac{1}{2^2 \cdot 3} ((2a_2 + a_{-2}) \cdot v_{(1,3)} + (a_3 - a_{-3}) \cdot v_{(1,2)}).
\end{align*}
\end{prop}
\begin{proof}
If we take $\alpha_0, \alpha_1, \beta_0, \beta_1$ as in Proposition \ref{prop:products1} then, from the fusion rule,
\begin{align*}
\alpha_{-1} &:= \frac{3}{2^4}(\alpha_0 \cdot \alpha_1 + \beta_0 \cdot \beta_1 - \frac{1}{2^2}(\beta_0, \beta_0)a_1 + \frac{1}{3}(\alpha_0 + \alpha_1)) \in M_0^{(a_1)} \\
\beta_{-1} &:= \frac{3}{2^4}(\alpha_0 \cdot \beta_1 + \alpha_1 \cdot \beta_0 - \frac{3}{2^3}(\beta_0 + \beta_0)) \in M_{\frac{1}{2^2}}^{(a_1)}.
\end{align*}
Using the value of $v_{(1,2)} \cdot v_{(1,3)}$ calculated in Proposition \ref{prop:products1}, we can calculate that
\begin{align*}
\alpha_{-1} &= -\frac{3t}{2^2}a_1 - \frac{1}{2^2} v_{(2,3)} + a_1 \cdot v_{(2,3)} \\
\beta_{-1} &= -ta_1 + a_1 \cdot v_{(2,3)}.
\end{align*}
The equality $a_1 \cdot \alpha_{-1} = 0$ then gives the value of $a_1 \cdot (a_1 \cdot v_{(2,3)})$ as required.
We further note that $\alpha_2 := a_{-1} \in M_0^{(a_1)}$ and so, from the fusion rule
\[
a_1 \cdot ((\alpha_{-1} - \beta_{-1}) \cdot (\alpha_0 + \beta_0 + \frac{5}{3} \alpha_2)) = \frac{1}{2^2}(\alpha_{-1} \cdot \beta_0 - (\alpha_0 + \frac{5}{3}\alpha_2) \cdot \beta_{-1}) + \frac{1}{2^2} \langle \beta_0, \beta_{-1} \rangle .
\]
We calculate that $\langle \beta_0, \beta_{-1} \rangle = -\frac{5t}{2 \cdot 3}$. Moreover,
\begin{align*}
(\alpha_{-1} - \beta_{-1}) \cdot (\alpha_0 + \beta_0 + \frac{5}{3} \alpha_2) = &-\frac{t}{2^4 \cdot 3}(a_1 + a_{-1}) + \frac{31t + 2}{2^3 \cdot 3} a_2 - \frac{t + 2}{2^2 \cdot 3} a_{-2} + \frac{1}{2^2 \cdot 3}(a_3 + a_{-3}) \\
&+ \frac{t - 2}{2^4}v_{(1,2)} + \frac{1}{2^3} v_{(1,3)} - \frac{1}{2^2}v_{(2,3)} + \frac{13}{2^3 \cdot 3} a_1 \cdot v_{(2,3)} \\
&+ \frac{4}{3} a_2 \cdot v_{(1,3)} + \frac{1}{3} (a_3 + a_{-3}) \cdot v_{(1,2)} .
\end{align*}
Then using the value of the algebra product $a_1 \cdot (a_1 \cdot v_{(2,3)})$, as well as those calculated in Proposition \ref{prop:products1}, we can also explicitly calculate the value of $a_1 \cdot ((\alpha_{-1} - \beta_{-1}) \cdot (\alpha_0 + \beta_0 + \frac{5}{3} \alpha_2))$. Moreover,
\begin{align*}
\alpha_{-1} \cdot \beta_0 - (\alpha_0 + \frac{5}{3}\alpha_2) \cdot \beta_{-1} = &-2a_{-1} \cdot (a_1 \cdot v_{(2,3)}) + \frac{t}{2^4}(3a_1 + a_{-1}) + \frac{11t+ 1}{2^3 \cdot 3} a_2 - \frac{5t - 1}{2^3 \cdot 3} a_{-2} \\
& - \frac{1}{2^3 \cdot 3}(a_3 + a_{-3}) - \frac{3t + 1}{2^4} v_{(1,2)} + \frac{1}{2^4} v_{(2,3)} - \frac{1}{2^3} a_1 \cdot v_{(2,3)}\\
& + \frac{1}{2} (a_2 + a_{-2}) \cdot v_{(1,3)} + \frac{1}{2 \cdot 3}(5a_3 + a_{-3}) \cdot v_{(1,2)}.
\end{align*}
Thus we can calculate the value of $a_{-1} \cdot (a_1 \cdot v_{(2,3)})$ as required.
We now calculate that
\begin{align*}
(\alpha_{-1} - \beta_{-1}) \cdot a_0 = &\left(\frac{2t}{3} - \frac{1}{2^3}\right) a_2 - \frac{1}{2^3}(a_{-2} - a_3 - a_{-3}) - \frac{1}{2^4}(v_{(1,2)} - v_{(1,3)} + 4v_{(2,3)})\\
& + \frac{1}{2^3 \cdot 3}(7a_1 + 4a_{-1}) \cdot v_{(2,3)} - \frac{2}{3}a_2 \cdot v_{(1,3)} + \frac{1}{2 \cdot 3}(a_3 + a_{-3}) \cdot v_{(1,2)}.
\end{align*}
In particular, the value of $a_1 \cdot (a_{-1} \cdot v_{(2,3)})$ is now known and so the value $a_1 \cdot ((\alpha_{-1} - \beta_{-1}) \cdot a_0)$ can be explicitly calculated.
From the fusion rule,
\[
a_1 \cdot ((\alpha_{-1} - \beta_{-1}) \cdot a_0) = -\frac{1}{2^2} \beta_{-1} \cdot \alpha_0
\]
and
\begin{align*}
\beta_{-1} \cdot \alpha_0 = v_{(1,2)} \cdot (a_1 \cdot v_{(2,3)}) &-\frac{3t}{2^3} a_1 + \frac{t}{2^2}(a_2 + a_{-2}) + \frac{3}{2^4}(a_1 + a_{-1}) \cdot v_{(2,3)} \\
&- \frac{1}{2^3}(a_2 + a_{-2}) \cdot v_{(1,3)} + \frac{1}{2^2}(a_3 + a_{-3}) \cdot v_{(1,2)}.
\end{align*}
We can then use these values to calculate $v_{(1,2)} \cdot (a_1 \cdot v_{(2,3)})$ as required.
\end{proof}
\begin{prop}
\label{prop:nullspace}
The elements of $\bar{B}$ satisfy the following linear dependencies:
\begin{align*}
(a_1 - a_{-1}) \cdot v_{(2,3)} - t(a_1 - a_{-1}) = 0 \\
(a_2 - a_{-2}) \cdot v_{(1,3)} - t(a_2 - a_{-2}) = 0 \\
(a_3 - a_{-3}) \cdot v_{(1,2)} - t(a_3 - a_{-3}) = 0.
\end{align*}
\end{prop}
\begin{proof}
Recall that
\begin{align*}
\alpha_{-1} &= -\frac{3t}{2^2}a_1 - \frac{1}{2^2} v_{(2,3)} + a_1 \cdot v_{(2,3)} \in M_0^{(a_1)} \\
\alpha_2 &= a_{-1} \in M_0^{(a_1)}.
\end{align*}
Using the value of $a_{-1} \cdot (a_1 \cdot v_{(2,3)})$ from Proposition \ref{prop:products2} and the fusion rule, we have
\[
\alpha_{-1} \cdot \alpha_2 = -\frac{t}{2^3}(2a_{-1} + a_3 + a_{-3}) + \frac{1}{2^3}(a_3 - a_{-3}) \cdot v_{(1,2)} \in M_0^{(a_1)}.
\]
However,
\[
a_1 \cdot (\alpha_{-1} \cdot \alpha_2) = -\frac{t}{2^8}(a_3 - a_{-3}) + \frac{1}{2^8}(a_3 - a_{-3}) \cdot v_{(1,2)}
\]
and so we must have $(a_3 - a_{-3}) \cdot v_{(1,2)} - t(a_3 - a_{-3}) = 0$. The symmetry between the three pairs of axes $\{a_1, a_{-1}\}$, $\{a_2, a_{-2}\}$ and $\{a_3, a_{-3}\}$ gives the remaining two relations.
\end{proof}
\begin{prop}
\label{prop:products3}
\begin{align*}
v_{(1,2)} \cdot (a_3 \cdot v_{(1,2)}) = \frac{(t - 1)t}{2^2}(a_3 - a_{-3}) + \frac{t}{2^2}v_{(1,2)} + \frac{1}{2} a_3 \cdot v_{(1,2)}
\end{align*}
\end{prop}
\begin{proof}
We now let
\begin{align*}
\alpha_{-2} &:= \alpha_0 \cdot \alpha_1 - \frac{1}{2^2}(\alpha_0 + \alpha_1) - \frac{8}{3} \alpha_{-1} \\
&= -\frac{2t}{3} a_1 - \frac{1}{3} v_{(2,3)} + \frac{4}{3}(a_2 + a_{-2}) \cdot v_{(1,3)}+ \frac{4}{3}(a_3 + a_{-3}) \cdot v_{(1,2)} \in M_0^{(a_1)}.
\end{align*}
We further note that
\[
a_1 \cdot ((\alpha_0 - \beta_0) \cdot \alpha_{-2}) = -\frac{1}{2^2} \beta_0 \cdot \alpha_{-2}.
\]
We can calculate that
\begin{align*}
(\alpha_0 - \beta_0) \cdot \alpha_{-2} = &-\frac{1}{2 \cdot 3}(a_1 - a_{-1}) + \frac{23t - 2}{3^2}(a_2 + a_{-2}) + \frac{2}{3^2}(a_3 + a_{-3}) + \frac{t}{2 \cdot 3}v_{(1,2)} -\frac{1}{3}v_{(2,3)} \\
&-\frac{2^3}{3^2}(a_1 + a_{-1}) \cdot v_{(2,3)} + \frac{20}{3^2}(a_2 + a_{-2}) \cdot v_{(1,3)} + \frac{2^3}{3^2}(a_3 + a_{-3}) \cdot v_{(1,2)}
\end{align*}
and so the value of $a_1 \cdot ((\alpha_0 - \beta_0) \cdot \alpha_{-2})$ can also be calculated.
Moreover,
\begin{align*}
\beta_0 \cdot \alpha_{-2} = \frac{4}{3}v_{(1,2)} \cdot ((a_3 + a_{-3}) \cdot v_{(1,2)}) &-\frac{7t}{2 \cdot 3} a_1 - \frac{17t}{2 \cdot 3^2} a_{-1} - \frac{2t - 1}{2 \cdot 3^2} a_2 - \frac{18t - 1}{2 \cdot 3^2} a_{-2} \\
&+ \frac{1}{2 \cdot 3^2}(a_3 + a_{-3}) + \frac{2t -1}{2^2 \cdot 3}v_{(1,2)} +\frac{1}{2^2 \cdot 3} v_{(1,3)} \\
&+ \frac{2}{3^2} (4a_1 + 3a_{-1}) \cdot v_{(2,3)} - \frac{2}{3^2} (5a_2 + a_{-2}) \cdot v_{(1,3)}.
\end{align*}
Thus we can explicitly calculate the value of $v_{(1,2)} \cdot ((a_3 + a_{-3}) \cdot v_{(1,2)})$. As $(a_3 - a_{-3}) \cdot v_{(1,2)} - t(a_3 - a_{-3}) = 0$,
\[
v_{(1,2)} \cdot ((a_3 - a_{-3}) \cdot v_{(1,2)}) = t(a_3 - a_{-3}) \cdot v_{(1,2)}.
\]
We can use this value, along with that of $v_{(1,2)} \cdot ((a_3 + a_{-3}) \cdot v_{(1,2)})$, to calculate the product $v_{(1,2)} \cdot (a_3 \cdot v_{(1,2)})$ as required.
\end{proof}
\begin{prop}
\label{prop:products4}
\begin{align*}
(a_1 \cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)}) &= \frac{(10t + 1)t}{2^4}a_1 - \frac{(2t - 1)t}{2^4}a_{-1} + \frac{t}{2^4}v_{(2,3)} + \frac{t}{2^2}a_1 \cdot v_{(2,3)} \\
(a_{-1} \cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)}) &= -\frac{(6t - 1)t}{2^4}a_1 - \frac{(2t - 1)t}{2^4}a_{-1} + \frac{t}{2^4}v_{(2,3)} + \frac{t}{2^2}a_1 \cdot v_{(2,3)}
\end{align*}
\end{prop}
\begin{proof}
Recall that
\begin{align*}
\alpha_{-1} &= -\frac{3t}{2^2}a_1 - \frac{1}{2^2} v_{(2,3)} + a_1 \cdot v_{(2,3)} \in M_0^{(a_1)} \\
\beta_{-1} &= -ta_1 + a_1 \cdot v_{(2,3)} \in M_{\frac{1}{2^2}}^{(a_1)}
\end{align*}
and so
\[
a_1 \cdot ((\alpha_{-1} - \beta_{-1}) \cdot \alpha_{-1}) = -\frac{1}{2^2} \beta_{-1} \cdot \alpha_{-1}.
\]
As
\[
(\alpha_{-1} - \beta_{-1}) \cdot \alpha_{-1} = -\frac{(2t - 1)t}{2^4}(a_1 - a_{-1}) -\frac{t - 1}{2^4}a_1 \cdot v_{(2,3)} + \frac{3t - 2}{2^4} a_{-1} \cdot v_{(2,3)},
\]
we can explicitly calculate the value of $a_1 \cdot ((\alpha_{-1} - \beta_{-1}) \cdot \alpha_{-1})$.
Moreover, using the value of $v_{(2,3)} \cdot (a_1 \cdot v_{(2,3)})$ from Proposition \ref{prop:products3}, we can calculate that
\[
\beta_{-1} \cdot \alpha_{-1} = (a_1 \cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)}) - \frac{(11t -1)t}{2^4} a_1 - \frac{(2t -1)t}{2^4}a_{-1} + \frac{t}{2^4}v_{(2,3)} + \frac{3t + 2}{2^4} a_1 \cdot v_{(2,3)}.
\]
We can then use these values to find the value of $(a_1 \cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)})$ as required.
As we have the linear dependency $(a_1 - a_{-1}) \cdot v_{(2,3)} - t(a_1 - a_{-1}) = 0$, we must have
\[
(a_{-1} \cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)}) = (a_1 \cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)}) - t(a_1 - a_{-1}) \cdot (a_1 \cdot v_{(2,3)}).
\]
All these product values are known and so we have the value of $(a_{-1} \cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)})$ as required.
\end{proof}
\begin{prop}
\label{prop:products5}
\begin{align*}
(a_2 \cdot v_{(1,3)}) \cdot (a_1 \cdot v_{(2,3)}) = \frac{t^2}{2^5}(a_1 - a_{-1} + a_2 - a_{-2}) + \frac{(2t - 1)t}{2^5}a_3 - \frac{(2t + 1)t}{2^5}a_{-3} + \frac{t}{2^5}v_{(1,2)} &\\
+ \frac{t}{2^6}(v_{(1,3)} + v_{(2,3)}) + \frac{t}{2^3}(a_1 \cdot v_{(1,2)} + a_2 \cdot v_{(1,3)} - a_3 \cdot v_{(1,2)})&.
\end{align*}
\end{prop}
\begin{proof}
Recall that
\[
\alpha_{-2} := -\frac{2t}{3} a_1 - \frac{1}{3} v_{(2,3)} + \frac{4}{3}(a_2 + a_{-2}) \cdot v_{(1,3)}+ \frac{4}{3}(a_3 + a_{-3}) \cdot v_{(1,2)} \in M_0^{(a_1)}.
\]
We also let
\begin{align*}
\beta_{-2} &:= \alpha_0 \cdot \beta_1 - \frac{1}{2^2}\beta_0 - \frac{1}{2^3}\beta_1 - \frac{8}{3}\beta_{-1} = \frac{4}{3}((a_2 + a_{-2}) \cdot v_{(1,3)} - (a_3, a_{-3}) \cdot v_{(1,2)}) \in M_{\frac{1}{2^2}}^{(a_1)}.
\end{align*}
Then
\[
a_1 \cdot ((\alpha_{-1} - \beta_{-1}) \cdot (\alpha_{-2} -\beta_{-2})) = -\frac{1}{2^2}(\alpha_{-1} \cdot \beta_{-2} + \alpha_{-2} \cdot \beta_{-1}) - \frac{1}{2^2} \langle \beta_{-1}, \beta_{-2} \rangle a_1.
\]
Both $\alpha_{-2}$ and $\beta_{-2}$ can be rewritten using the linear dependencies given in Proposition \ref{prop:nullspace} to give
\begin{align*}
\alpha_{-2} &:= -\frac{2t}{3} a_1 - \frac{4t}{3}(a_2 - a_{-2} + a_3 - a_{-3}) -\frac{1}{3}v_{(2,3)} + \frac{8}{3}(a_2 \cdot v_{(1,3)} + a_3 \cdot v_{(1,2)}) \\
\beta_{-2} &:= - \frac{4t}{3}(a_2 - a_{-2} + a_3 - a_{-3}) + \frac{8}{3}(a_2 \cdot v_{(1,3)} - a_3 \cdot v_{(1,2)}).
\end{align*}
Using these new representations, we calculate that
\begin{align*}
(\alpha_{-1} - \beta_{-1}) \cdot (\alpha_{-2} -\beta_{-2}) &= \frac{(3t + 22)t}{2^2 \cdot 3^2}a_2 - \frac{(3t - 10)t}{2^2 \cdot 3^2}a_{-2} - \frac{(3t - 14)t}{2^2 \cdot 3^2} a_3 + \frac{(3t + 2)t}{2^2 \cdot 3^2}a_{-3} \\
&- \frac{2t - 1}{2^2 \cdot 3}v_{(2,3)} + \frac{t}{2 \cdot 3} a_1 \cdot v_{(2,3)} + \frac{t}{2}(a_2 \cdot v_{(1,3)} - a_3 \cdot v_{(1,2)}).
\end{align*}
and so the value of $a_1 \cdot ((\alpha_{-1} - \beta_{-1}) \cdot (\alpha_{-2} -\beta_{-2}))$ may also be explicitly calculated.
Next, we calculate that $\langle \beta_{-1}, \beta_{-2} \rangle = 0$ and
\begin{align*}
\alpha_{-1} \cdot \beta_{-2} + \alpha_{-2} \cdot \beta_{-1} &= \frac{16}{3}(a_1 \cdot v_{(2,3)}) \cdot (a_2 \cdot v_{(1,3)}) - \frac{(8t - 1)t}{2^2 \cdot 3} a_1 + \frac{(2t - 1)t}{2^2 \cdot 3}a_{-1} \\
&- \frac{(9t + 4)t}{2^2 \cdot 3^2}(a_2 - a_{-3}) + \frac{(9t - 4)t}{2^2 \cdot 3^2}(a_{-2} - a_3) - \frac{t}{2^2 \cdot 3} v_{(2,3)} \\
&- \frac{t + 1}{2 \cdot 3} (a_2 \cdot v_{(1,3)} - a_3 \cdot v_{(1,2)}).
\end{align*}
Thus we can use these values to calculate the product $(a_1 \cdot v_{(2,3)}) \cdot (a_2 \cdot v_{(1,3)})$ as required.
\end{proof}
At this stage, we have all algebra products of the form $u \cdot v$ for $u,v \in \bar{B}$. We can now calculate the remaining values of the Frobenius form on $M$.
\begin{prop}
\label{prop:frobeniusform2}
Some values of the Frobenius form $\langle \, , \, \rangle$ on the vectors of $\bar{B}$ are as given in Table 4.
\end{prop}
\begin{proof}
These values follow from the fact that for all $u, v, w \in M$, we must have $\langle u, v \cdot w \rangle = \langle u \cdot v, w \rangle$ and from the algebra product values determined in Proposition \ref{prop:products2}.
\end{proof}
\begin{table}%
\begin{center}
\def1.75{1.75}
\begin{tabular}{| >{$} l <{$} >{$} l <{$} >{$} c <{$} | } \hline
u & v & \langle u,v \rangle \\ \hline
a_1 \cdot v_{(2,3)} & a_1 \cdot v_{(2,3)} & \frac{(3t +1)t}{2^2} \\
a_1 \cdot v_{(2,3)} & a_2 \cdot v_{(1,3)} & \frac{(2t +1)t}{2^4} \\ \hline
\end{tabular}
\end{center}
\label{tab:frobeniusform2}
\caption{Some values of the Frobenius form $\langle \, , \, \rangle$ on $M$}
\end{table}
\subsection{The algebra structure}
\begin{proof}[Proof of Theorem \ref{thm:4A}]
We have shown in Propositions \ref{prop:products1}, \ref{prop:products2}, \ref{prop:products3}, \ref{prop:products4} and \ref{prop:products5} that the algebra product on $M$ is closed on the set
\[
\bar{B} := B \cup \{ a_{-1} \cdot v_{(2,3)}, a_{-2} \cdot v_{(1,3)}, a_{-3} \cdot v_{(1,2)}\}.
\]
Moreover, from Proposition \ref{prop:nullspace}, the elements of $\bar{B}$ satisfy the following linear dependencies:
\begin{align*}
(a_1 - a_{-1}) \cdot v_{(2,3)} - t(a_1 - a_{-1}) = 0 \\
(a_2 - a_{-2}) \cdot v_{(1,3)} - t(a_2 - a_{-2}) = 0 \\
(a_3 - a_{-3}) \cdot v_{(1,2)} - t(a_3 - a_{-3}) = 0.
\end{align*}
Thus we can take the following to be a basis of $M$:
\[
B = \{a_1, a_{-1}, a_2, a_{-2}, a_3, a_{-3}, v_{(1,2)}, v_{(1,3)}, v_{(2,3)}, a_1 \cdot v_{(2,3)}, a_2 \cdot v_{(1,3)}, a_3 \cdot v_{(1,2)} \}.
\]
The algebra product values of certain pairs of elements of $B$ are given in Table 5. With the exception of products given by the values of dihedral algebras, from the eight products given in this table, all other products of pairs of basis vectors can be recovered from the action of $G = \langle \tau(a) \mid a\in A \rangle$ and $\sigma$.
From these products, we can calculate the eigenspaces of the adjoint action of the axis $a_1$. With the exception of the one-dimensional $1$-eigenspace, these are given in Table 6. Again, the eigenvectors for the remaining $5$ axes can be recovered from the action of $G$ and $\sigma$. We can check using these eigenvectors and the products in Table 5 that $M$ obeys the Monster fusion rule.
Finally, we use these algebra products to check that the values in Tables 3 and 4 do indeed give a Frobenius form on $M$.
\end{proof}
Theorem \ref{thm:4A} shows that there exists a $12$-dimensional axial algebra $M_{4A}$ of Monster type over the ring $\mathbb{R}[t]$. If we let $t$ take any value in $\mathbb{R}$ then this gives a $12$-dimensional axial algebra of Monster type over the field $\mathbb{R}$ that we denote $M(t)$.
In order to prove Theorem \ref{thm:4Ab}, we first require a few additional definitions and results.
\begin{defn}
Let $(V,A)$ be a primitive axial algebra of Monster type that admits a Frobenius form $\langle \, , \, \rangle$. Then we define the \emph{projection graph} of $V$ to be the graph with vertices $A$ such that $a_0, a_1 \in A$ are joined by an edge if and only if $\langle a_0, a_1 \rangle \neq 0$
\end{defn}
The projection graph of $M(t)$ is given in Figure 2.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\foreach \i in {1,2,3}{
\draw (180 - \i*60: 1.5cm) node (a\i) {$a_{\i}$};
}
\foreach \i in {1,2,3}{
\draw (360 - \i*60: 1.5cm) node (am\i) {$a_{-\i}$};
}
\foreach \i in {2,3}{
\draw (a1) -- (a\i);
\draw (a1) -- (am\i);
\draw (am1) -- (a\i);
\draw (am1) -- (am\i);
}
\draw (a2) -- (a3);
\draw (a2) -- (am3);
\draw (am2) -- (a3);
\draw (am2) -- (am3);
\end{tikzpicture}
\end{center}
\label{fig:projection}
\caption{The projection graph of $M(t)$}
\end{figure}
\begin{defn}
Let $(V,A)$ be a primitive axial algebra. The \emph{radical} $R(V,A)$ of $V$ with respect to $A$ is the unique largest ideal of $V$ containing no axes from $A$.
\end{defn}
\begin{thm}[{\cite[Theorem 4.11]{KMS18}}]
\label{thm:radical}
Let $(V,A)$ be a primitive axial algebra that admits a Frobenius form $\langle \, , \, \rangle$. Then $R(V,A) = V^{\perp}$ where $V^{\perp} = \{ u \in V \mid \langle u, v \rangle = 0 \textrm{ for all } v \in V\}$.
\end{thm}
\begin{lem}
\label{lem:ideal}
The radical $M(t)^{\perp}$ of the Frobenius form on $M(t)$ is the unique maximal ideal of $M(t)$ for all $t \in \mathbb{R}$.
\end{lem}
\begin{proof}
Let $I$ be a proper ideal of $M(t)$. We will show that $I$ must be contained in $M(t)^{\perp}$. Suppose for contradiction that there exist an axis $a$ such that $a \in I$. As $I$ is proper, there must exist at least one further axis $b$ such that $b \notin I$. Then, from Lemma \ref{lem:projection}, we can write
\[
a = \langle a, b \rangle b + a_0 + a_{\frac{1}{4}} + a_{\frac{1}{32}}
\]
where $a_\mu \in V_\mu^{(b)}$.
Let $\Lambda := \{0, \frac{1}{4}, \frac{1}{32}\}$ and consider $f_\Lambda(x) \in \mathbb{R}[x]$ as defined in Lemma \ref{lem:espaces}. As $a \in I$ and $I$ is an ideal, $f_\Lambda(ad_b)(a) \in I$. Moreover, from Lemma \ref{lem:espaces},
\[
f_\Lambda(ad_b)(a) = \langle a, b \rangle f_\Lambda(ad_b)(b) + f_\Lambda(ad_b)(a_0 + a_{\frac{1}{4}} + a_{\frac{1}{32}} ) = f_\Lambda(1)\langle a, b \rangle b
\]
and $f_\Lambda(1) \neq 0$. Thus, as $b \notin I$, we must have $\langle a, b \rangle = 0$.
As this is true for all $a, b \in A$ such that $a \in I$ and all $b \notin I$, the projection graph of $M(t)$ must be disconnected. We can see from Figure 2 that this is not the case. Thus we can conclude that $I$ contains no axes from $A$ and so from Theorem \ref{thm:radical} is contained in $M(t)^\perp$ as required.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:4Ab}]
From Lemma \ref{lem:ideal}, the algebra $M(t)$ is simple exactly when the Frobenius form $\langle \, , \, \rangle$ is non-degenerate, or equivalently when the determinant of the Gram matrix of $\langle \, , \, \rangle$ on $B$ is non-zero.
Using the values of the Frobenius form in Tables 3 and 4, we have calculated that the determinant of this Gram matrix is equal to
\[
-\frac{t^3 (6 t - 1)^3 (4 t - 9)^6 }{2^{19} \cdot 3^3}.
\]
Thus if $t \notin \{0, \frac{1}{6}, \frac{9}{4} \}$ then the algebra $M(t)$ is simple. If $t = 0$ or $t = \frac{1}{6}$ then we calculate that $M(t)$ contains a $3$-dimensional ideal and if $t = \frac{9}{4}$ then $M(t)$ contains a $5$-dimensional ideal.
\end{proof}
\begin{landscape}
\begin{table}%
\begin{center}
\def1.75{1.75}
\begin{tabular}{| >{$} l <{$} >{$} l <{$} >{$} c <{$} | } \hline
u & v & u \cdot v \\ \hline
v_{(1,2)} & v_{(1,3)} & \def1.75{1.2}\begin{tabular}{ >{$} c <{$}} -\frac{8t}{3} a_1 + \frac{2t}{3}(a_2 - a_{-2} + a_3 - a_{-3}) + \frac{1}{2^2} (v_{(1,2)} + v_{(1,3)} - v_{(2,3)})\\ - \frac{4}{3}(2a_1 \cdot v_{(2,3)} - a_2 \cdot v_{(1,3)} - a_3 \cdot v_{(1,2)}) \end{tabular} \def1.75{1.75}\\
a_1 & a_1 \cdot v_{(2,3)} & \frac{3t}{2^2}a_1 + \frac{1}{2^2} a_1 \cdot v_{(2,3)} \\
a_{-1} & a_1 \cdot v_{(2,3)} & -\frac{t}{2^2} a_1 + \frac{1}{2^2} a_1 \cdot v_{(2,3)} \\a
a_2 & a_1 \cdot v_{(2,3)} & -\frac{3t}{2^8}(a_1 - a_{-1}) + \frac{t}{2^4}(2a_2 + a_3 - a_{-3}) + \frac{1}{2^4}(2a_1 \cdot v_{(2,3)} + a_2 \cdot v_{(1,3)} - a_3 \cdot v_{(1,2)}) \\
v_{(1,2)} & a_1 \cdot v_{(2,3)} & -\frac{5t}{2^4 \cdot 3}a_1 - \frac{11t}{2^4 \cdot 3}a_{-1} - \frac{11t}{2^3 \cdot 3}a_2 - \frac{5t}{2^3 \cdot 3}a_{-2} + \frac{t}{2^3} v_{(1,2)} + \frac{1}{2^2}(a_1 \cdot v_{(2,3)} + a_2 \cdot v_{(1,3)}) \\
v_{(2,3)} & a_1 \cdot v_{(2,3)} & \frac{(2t-1)t}{2^2}(a_1 - a_{-1}) + \frac{1}{2^2}v_{(2,3)} + \frac{1}{2} a_1 \cdot v_{(2,3)} \\
a_1 \cdot v_{(2,3)} & a_1 \cdot v_{(2,3)} & \frac{(10t + 1)t}{2^4}a_1 - \frac{(2t - 1)t}{2^4} a_{-1} + \frac{t}{2^4}v_{(2,3)} + \frac{t}{2^2} a_1 \cdot v_{(2,3)} \\
a_1 \cdot v_{(2,3)} & a_2 \cdot v_{(1,3)} & \def1.75{1.2} \begin{tabular}{>{$} c <{$}}\frac{t}{2^5}(a_1 - a_{-1} + a_2 - a_{-2}) + \frac{(2t - 1)t}{2^5} a_3 - \frac{(2t + 1)t}{2^5} a_{-3} + \frac{t}{2^6}(2v_{(1,2)} + v_{(1,3)} + v_{(2,3)}) \\+ \frac{t}{2^3}(a_1 \cdot v_{(2,3)} + a_2 \cdot v_{(1,3)} - a_3 \cdot v_{(1,2)}) \end{tabular} \\ \hline
\end{tabular}
\end{center}
\label{tab:algebraproducts}
\caption{Representative algebra products on $M$}
\end{table}
\begin{table}%
\begin{center}
\def1.75{1.75}
\begin{tabular}{| >{$} c <{$} | >{$} c <{$} | >{$} c <{$} | } \hline
0 & \frac{1}{4} & \frac{1}{32} \\\hline
\begin{tabular}{>{$} c <{$}}
\def1.75{1}
\begin{tabular}{>{$} c <{$}}
-\frac{t}{2^2}a_1 - \frac{t}{2}(a_2 - a_{-2} + a_3 - a_{-3}) -\frac{1}{2^3}v_{(2,3)} \\
+ a_2 \cdot v_{(1,3)} + a_3 \cdot v_{(1,2)} \end{tabular}\\
\def1.75{1.75}
-\frac{3t}{2^2}a_1 - \frac{1}{2^2} v_{(2,3)} + a_1 \cdot v_{(2,3)} \\
-\frac{1}{2} a_1 + 2a_2 + 2a_{-2} + v_{(1,2)} \\
-\frac{1}{2} a_1 + 2a_3 + 2a_{-3} + v_{(1,2)} \\
a_{-1} \\
\end{tabular}
&
\begin{tabular}{>{$} c <{$}}
\def1.75{1}
\begin{tabular}{>{$} c <{$}}
\frac{t}{2}(a_2 - a_{-2} + a_3 - a_{-3}) - a_2 \cdot v_{(1,3)}\\
+ a_3 \cdot v_{(a_1, a_2)} \end{tabular}\\
\def1.75{1.75}
-ta_1 + a_1 \cdot v_{(2,3)} \\
-\frac{1}{3}(a_1 + a_{-1}) -\frac{2}{3}(a_2 - a_{-2}) + v_{(1,2)} \\
-\frac{1}{3}(a_1 + a_{-1}) -\frac{2}{3}(a_3 - a_{-3}) + v_{(1,3)} \\
\end{tabular}
&
\begin{tabular}{>{$} c <{$}}
a_2 - a_{-2} \\
a_3 - a_{-3} \\
\end{tabular} \\ \hline
\end{tabular}
\end{center}
\label{tab:evecs}
\caption{Eigenvectors of the axis $a_1$ in $M$}
\end{table}
\end{landscape}
\section{The Frobenius form on the algebra $M_{4A}$}
\label{sec:frobenius}
In Section \ref{sec:4A}, we constructed an infinite family of axial algebras of Monster type over the field $\mathbb{R}$ that we denote $\{M(t)\}_{t \in \mathbb{R}}$. We now ask which of these algebras are also Majorana algebras.
\begin{defn}
Let $(V, A)$ be a primitive axial algebra of Monster type over $\mathbb{R}$ that admits a Frobenius form $\langle \, , \, \rangle$. Then $(V, A)$ is a \emph{Majorana algebra} if both of the following hold
\begin{enumerate}[i)]
\item the Frobenius form $\langle \, , \, \rangle$ is positive definite;
\item $V$ obeys \emph{Norton's inequality}, i.e. $\langle u \cdot u, v \cdot v \rangle \geq \langle u \cdot v, u \cdot v \rangle$ for all $u, v \in V$.
\end{enumerate}
\end{defn}
The following is the main result of this section.
\begin{thm}
\label{thm:majorana}
The axial algebra $M(t)$ (as constructed in Section \ref{sec:4A}) is a Majorana algebra if and only if $t \in (0,\frac{1}{6})$. If $t \in \{0, \frac{1}{6}\}$ then there exists a $9$-dimensional quotient of $M(t)$ that is a Majorana algebra.
\end{thm}
Norton's inequality is sometimes referred to as axiom M2, in reference to the Majorana axioms as introduced in \cite[Chapter 9]{Ivanov09}. Whilst it is part of the definition of a Majorana algebra, it is not used in the construction of any of the known Majorana algebras and it is an open problem as to whether an axial algebra of Monster type that admits a positive definite Frobenius form must necessarily obey Norton's inequality.
Theorem \ref{thm:majorana} shows that there exists axial algebras of Monster type that admit a Frobenius form but which do not obey Norton's inequality, i.e. the algebras $M(t)$ where $t \notin [0, \frac{1}{6}]$. These are the first examples of such algebras. Moreover, we show that an algebra of this form admits a Frobenius form that is positive definite precisely when $t \in (0, \frac{1}{6})$. This is suggests that an axial algebra of Monster type that admits a positive (semi)definite Frobenius form must obey axiom M2.
In order to prove Theorem \ref{thm:majorana}, we will require the following preliminary results.
\begin{lem}[{\cite[Lemma 7.8]{ISe12}}]
\label{lem:M2}
Let $V$ be an $n$-dimensional algebra with commutative algebra product $\cdot$ and bilinear form $\langle \, , \,\rangle $. Let $\{v_i \, : \, 1 \leq i \leq n\}$ be a basis of $V$, and define a $(n^2 \times n^2)$-dimensional matrix $\mathcal{B} = (b_{ij,kl})$ in the following way. The rows and columns are indexed by the ordered pairs $(i,j)$ for $1 \leq i,j \leq n$ and
\[
b_{ij,kl} = \langle v_i \cdot v_k, v_j \cdot v_l\rangle - \langle v_j \cdot v_k, v_i \cdot v_l\rangle .
\]
Then $V$ satisfies Norton's inequality if and only if $\mathcal{B}$ is positive semidefinite.
\end{lem}
\begin{proof}
For $u,v \in V$, write $u$ and $v$ as linear combinations
\[
u = \sum_{i=1}^n \lambda_i v_i \textrm{ and } v = \sum_{j=1}^n \mu_j v_j
\]
and form the $n^2$-long vector $z$ with entries $\lambda_i \mu_j$. In this vector, the coordinate $\lambda_i\mu_j$ is in the position indexed by $(i,j)$ in the matrix $\mathcal{B}$. Then the inequality $\langle u \cdot u, v \cdot v\rangle - \langle u \cdot v, u \cdot v\rangle \geq 0 $ is equivalent to $z \mathcal{B} z^T \geq 0$. Hence, if $\mathcal{B}$ is positive semidefinite then Norton's inequality must hold in $V$.
\end{proof}
\begin{thm}[{\cite{GV13}}]
\label{thm:pd}
A symmetric matrix $A$ is positive definite (respectively positive semidefinite) if and only if it can be decomposed as
\[
A := LDL^T
\]
where $L$ is a lower triangular matrix and $D$ is a diagonal matrix whose diagonal elements are all positive (respectively non-negative).
\end{thm}
The construction of the matrices $L$ and $D$ is known as \emph{LDLT decomposition} and we have implemented an algorithm in GAP that performs this decomposition for positive semidefinite matrices. This can be accessed as part of the package \texttt{MajoranaAlgebras} \cite{PW18b}.
\begin{prop}
\label{prop:innerproduct}
The Frobenius form $\langle \, , \, \rangle$ on the algebra $M(t)$ is positive semidefinite if and only if $t \in [0, \frac{1}{6}]$. The form is positive semidefinite but not positive definite if and only if $t \in \{0, \frac{1}{6}\}$.
\end{prop}
\begin{proof}
We use the values of the Frobenius form calculated in Section \ref{sec:4A} to construct the Gram matrix $\mathcal{G}$ of the Frobenius form on $M(t)$ with respect to the spanning set
\[
B = A \cup \{ v_{(1,2)}, v_{(1,3)}, v_{(2,3)} \} \cup \{ a_1 \cdot v_{(2,3)}, a_2 \cdot v_{(1,3)}, a_3 \cdot v_{(1,2)}\}.
\]
Then the form will be positive semidefinite if and only if $\mathcal{G}$ is positive semidefinite.
We use LDLT decomposition to calculate a lower triangular matrix $L$ and a diagonal matrix $D$ such that $\mathcal{G} = LDL^T$. We have calculated that
\[
D = \mathrm{diag} \{ 1, 1, \sfrac{511}{512}, \sfrac{510}{511}, \sfrac{271}{272}, \sfrac{270}{271}, r_1, r_2, r_3, r_4, r_5, r_6 \}
\]
where $r_1, \dots, r_6$ are rational functions whose values are given in Table \ref{tab:polyinner}.
For each rational function $r_i$, we have used \cite{mathematica} to calculate the range in $\mathbb{R}$ on which $r_i$ takes non-negative values. These are given in the final column of Table \ref{tab:polyinner}. It is easy to check that the intersection of these ranges is $[0, \frac{1}{6}]$ and that the $r_i$ all take positive values on $(0, \frac{1}{6})$.
\end{proof}
\begin{table}
\renewcommand1.75{3}
\begin{center}
\vspace{0.35cm}
\noindent
\begin{tabular}{|>{$}c<{$} >{$}c<{$} >{$}c<{$}|} \hline
i & r_i & \textrm{Values of } t \textrm{ s.t. } 0 \leq r_i[t] < \infty \\ \hline
1 & - \frac{ 272 }{ 135 } t^2 + \frac{ 8 }{ 45 } t + \frac{ 22 }{ 15 } & \left[ \frac{3 - 15 \sqrt{15}}{68}, \frac{3 + 15 \sqrt{15}}{68} \right] \\
2 & \myfrac[4pt]{ t^4 + \frac{ 1 }{ 16 } t^3 - \frac{ 717 }{ 128 } t^2 + \frac{ 171 }{ 256 } t + \frac{ 1053 }{ 2048 } }{ - \frac{ 255 }{ 512 } t^2 + \frac{ 45 }{ 1024 } t + \frac{ 1485 }{ 4096 } } & \left[ \frac{-39}{16}, \frac{3 - 15 \sqrt{15}}{68} \right), \left[-\frac{1}{4}, \frac{3}{8} \right], \left(\frac{3 + 15 \sqrt{15}}{68}, \frac{9}{4} \right] \\
3 & \myfrac[4pt]{ t^4 + \frac{ 5 }{ 2 } t^3 - \frac{ 171 }{ 16 } t^2 - \frac{ 9 }{ 32 } t + \frac{ 81 }{ 128 } }{ - \frac{ 1 }{ 2 } t^2 - \frac{ 33 }{ 32 } t + \frac{ 117 }{ 256 } } & \left[ \frac{-3 \sqrt{11} - 9}{4}, -\frac{39}{16} \right), \left[-\frac{1}{4}, \frac{3 \sqrt{11} - 9}{4} \right], \left( \frac{3}{8}, \frac{9}{4} \right] \\
4 & \myfrac[4pt]{ t^5 - \frac{ 9 }{ 4 } t^4 - \frac{ 7 }{ 9 } t^3 + \frac{ 15 }{ 8 } t^2 - \frac{ 9 }{ 32 } t }{ 4 t^3 + 19 t^2 - \frac{ 9 }{ 8 } } &
\renewcommand1.75{1.5}\ml{c}{ \left(-\infty, \frac{-9 - 3\sqrt{11}}{4} \right), \left[ \frac{-1 - \sqrt{109}}{12}, -\frac{1}{4} \right), \left[0, \frac{1}{6} \right], \\ {\left( \frac{3 \sqrt{11} - 9}{4}, \frac{\sqrt{109} - 1}{12} \right]}, \left[ \frac{9}{4}, \infty \right) }\\
5 & \myfrac[4pt]{ t^5 - \frac{ 133 }{ 36 } t^4 + \frac{ 613 }{ 216 } t^3 + \frac{ 33 }{ 32 } t^2 - \frac{ 15 }{ 64 } t }{ \frac{ 16 }{ 3 } t^3 + \frac{ 20 }{ 9 } t^2 - \frac{ 34 }{ 9 } t - 1 } & \renewcommand1.75{1.5}\ml{c}{ \left(-\infty, \frac{-1 - \sqrt{109}}{12} \right), \left[ \frac{23 - \sqrt{1339}}{36}, - \frac{1}{4} \right), \left[0, \frac{1}{6} \right], \\ {\left( \frac{\sqrt{109} - 1}{12}, \frac{23 + \sqrt{1339}}{36} \right]}, \left[ \frac{9}{4}, \infty \right) }\\
6 & \myfrac[4pt]{ t^4 - \frac{ 14 }{ 3 } t^3 + \frac{ 93 }{ 16 } t^2 - \frac{ 27 }{ 32 } t }{ 6 t^2 - \frac{ 23 }{ 3 } t - \frac{ 15 }{ 4 } } & \left(\infty, \frac{23 - \sqrt{1339}}{36} \right), \left[0, \frac{1}{6} \right], \left( \frac{23 + \sqrt{1339}}{36}, \infty \right) \\ \hline
\end{tabular}
\end{center}
\label{tab:polyinner}
\caption{Diagonal entries for the Gram matrix of the Frobenius form on $M_{4A}$}
\end{table}
\begin{prop}
\label{prop:norton}
The algebra $M(t)$ obeys Norton's inequality if and only if if $t \in [0, \frac{1}{6}]$.
\end{prop}
\begin{proof}
We construct the matrix $\mathcal{B}$ as described in Lemma \ref{lem:M2} and calculate its LDLT decomposition $\mathcal{B} = LDL^T$. The algebra $M(t)$ obeys axiom M2 if and only if this matrix is positive semidefinite or, equivalently, if and only if all diagonal entries of the matrix $D$ are non-negative. We have calculated that the diagonal elements of $D$ consist of the rational numbers
\[
0, \sfrac{15}{632}, \sfrac{107}{4096}, \sfrac{395}{15872}, \sfrac{1395}{54784}
\]
as well as nineteen rational functions that we label $s_1, \dots, s_{19}$. We have calculated using \cite{mathematica} that $s_i(t) \geq 0$ for all $i \in [1, \dots, 19]$ if and only if $t \in [0, \frac{1}{6}]$. Here we do not explicitly give the rational functions in question.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:majorana}]
From Propositions \ref{prop:innerproduct} and \ref{prop:norton}, the algebra $M(t)$ is a Majorana algebra if and only if $t \in (0, \frac{1}{6})$. If $t \in \{0, \frac{1}{6} \}$ then the Frobenius form is positive semidefinite. We have calculated that the radical of this form is a $3$-dimensional ideal of $M(t)$. Taking the quotient of $M(t)$ by this ideal gives a $9$-dimensional primitive axial algebra of Monster type that admits a positive definite Frobenius form. We have also checked that Norton's inequality holds on this algebra, and so this algebra is indeed a Majorana algebra.
\end{proof}
\section{The $4A$ axes in $M_{4A}$}
\label{sec:4Afusion}
Finally, we will study the $4A$ axes in the algebra $M_{4A}$.
\begin{thm}
\label{thm:4Afusion}
For $i$ and $j$ such that $1 \leq i < j \leq 3$, vector $v_{(i,j)} \in M_{4A}$ is an idempotent whose eigenspace decomposition satisfies the fusion rule $(\mathcal{F}_{4A}^{(t)}, *)$ as given by the table below.
\begin{center}
\def1.75{1.5}
\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}
& 1 & 0 & \frac{1}{2} & \frac{3}{8} & t \\ \hline
1 & 1 & \emptyset & \frac{1}{2} & \frac{3}{8} & t \\
0 & \emptyset & 0 & \frac{1}{2} & \frac{3}{8} & t \\
\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & 1, 0 & \frac{3}{8} & t \\
\frac{3}{8} & \frac{3}{8} & \frac{3}{8} & \frac{3}{8} & 1, 0, \frac{1}{2} & \emptyset \\
t & t & t & t & \emptyset & 1, 0, \frac{1}{2} \\
\end{tabular}
\end{center}
Moreover, if $t \notin \{1, 0, \frac{1}{2}, \frac{3}{8} \}$ then this fusion rule admits a $C_2 \times C_2$-grading
\[
\mathrm{gr}^{(t)}_{4A}: \langle a, b \mid a^2 = b^2 = (ab)^2 = 1 \rangle \rightarrow P(\mathcal{F}^{(t)}_{4A})
\]
such that
\begin{align*}
\mathrm{gr}^{(t)}_{4A}: 1 &\mapsto \left\{1, 0, \frac{1}{2}\right\} \\
a & \mapsto \left\{\frac{3}{8}\right\} \\
b & \mapsto \{ t \} \\
ab & \mapsto \emptyset.
\end{align*}
Finally, if $t = \frac{3}{8}$ then this fusion rule admits a $C_2$-grading
\[
\mathrm{gr}^{(\sfrac{3}{8})}_{4A}: \langle a \mid a^2 = 1 \rangle \rightarrow P(\mathcal{F}^{(\sfrac{3}{8})}_{4A})
\]
such that
\begin{align*}
\mathrm{gr}^{(\sfrac{3}{8})}_{4A}: 1 &\mapsto \left\{1, 0, \frac{1}{2}\right\} \\
a & \mapsto \left\{\frac{3}{8}\right\}.
\end{align*}
\end{thm}
\begin{proof}
Using the algebra product values calculated in Section \ref{sec:4A}, we calculate the eigenspace decomposition of the vector $v_{(i,j)}$ and check that this satisfies the fusion rule $(\mathcal{F}^{(t)}_{4A}, *)$ for all $t \in \mathbb{R}$ such that $t \notin \{1, 0, \frac{1}{2}\}$. This eigenspace decomposition is given in Table \ref{tab:4A}.
\end{proof}
\begin{coro}
The subalgebra $ \langle \langle v_{(1,2)}, v_{(1,3)}, v_{(2,3)} \rangle \rangle$ of $M_{4A}$ is a $9$-dimensional $(\mathcal{F}_{4A}, *)$-axial algebra spanned by $ v_{(1,2)}$, $v_{(1,3)}$ and $v_{(2,3)}$ as well as the vectors
\[
\left\{ t a_i - a_i \cdot v_{(j,k)} \mid \{i, j, k\} = \{1, 2, 3\} \right\} \cup \left\{ t a_{-i} + a_i \cdot v_{(j,k)} \mid \{i, j, k\} = \{1, 2, 3\}\right\}.
\]
This algebra is an axial algebra of Jordan type $\frac{1}{2}$ i.e. the vectors $v_{(i,j)}$ satisfy the fusion rule $\mathcal{J}(\frac{1}{2})$, as given by the table below.
\begin{center}
\def1.75{1.5}
\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}}
& 1 & 0 & \frac{1}{2} \\ \hline
1 & 1 & \emptyset & \frac{1}{2} \\
0 & \emptyset & 0 & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & 1, 0 \\
\end{tabular}
\end{center}
\end{coro}
\begin{proof}
We can check using the algebra product values calculated in Section \ref{sec:4A} that the smallest subalgebra of $M_{4A}$ containing the vectors $ v_{(1,2)}$, $v_{(1,3)}$ and $v_{(2,3)}$ is the $9$-dimensional algebra with basis given above. It is then straightforward to calculate that this algebra coincides with the direct sum of the $1$, $0$ and $\frac{1}{2}$-eigenspaces of the adjoint action of $v_{(i,j)}$ on $M_{4A}$ for all $i, j$ such that $1 \leq i < j \leq 3$. The fusion rule follows from the fusion rule already calculated in Theorem \ref{thm:4Afusion}.
\end{proof}
\begin{landscape}
\begin{table}%
\begin{center}
\def1.75{1.75}
\begin{tabular}{| >{$} c <{$} | >{$} c <{$} | >{$} c <{$} | >{$} c <{$} | } \hline
0 & \frac{1}{2} & \frac{3}{8} & t \\\hline
\begin{tabular}{>{$} c <{$}}
-\frac{4}{3}( a_i + a_{-i} + a_j + a_{-j}) + v_{(i,j)} \\
(\frac{1}{8} -\frac{5t}{6})a_i + (\frac{1}{8} + \frac{t}{6})a_{-i} + \frac{1}{8}(a_j + a_{-j}) - \frac{1}{4}(a_k + a_{-k}) + a_i \cdot v_{(j,k)} \\
\frac{1}{8}(a_i + a_{-i}) + (\frac{1}{8} -\frac{5t}{6})a_j + (\frac{1}{8} + \frac{t}{6})a_{-j} - \frac{1}{4}(a_k + a_{-k}) + a_j \cdot v_{(i,k)} \\
-\frac{t}{3}( a_i + a_{-i} + a_j + a_{-j}) - \frac{2t + 1}{4}(a_k + a_{-k}) + a_k \cdot v_{(i,j)}
\end{tabular}
&
\begin{tabular}{>{$} c <{$}}
a_i + a_{-i} - a_j - a_{-j} \\
-\frac{3t}{2} a_i - \frac{t}{2}a_{-i} + \frac{t}{2}v_{(i,j)} + \frac{1}{16}(v_{(i,k)} - v_{(j,k)}) + a_i \cdot v_{(j,k)} \\
-\frac{3t}{2} a_j - \frac{t}{2}a_{-j} + \frac{t}{2}v_{(i,j)} - \frac{1}{16}(v_{(i,k)} - v_{(j,k)}) + a_j \cdot v_{(i,k)} \\
-\frac{t}{2}(a_k - a_{-k}) - \frac{t}{2}v_{(i,j)} + a_k \cdot v_{(i,j)} \\
\end{tabular}
&
\begin{tabular}{>{$} c <{$}}
a_i - a_{-i} \\
a_j - a_{-j} \\
\end{tabular}
&
\begin{tabular}{>{$} c <{$}}
a_k - a_{-k} \\
\end{tabular} \\
\hline
\end{tabular}
\label{tab:4A}
\caption{Eigenvectors of the axis $v_{(i,j)}$ in $M_{4A}$}
\end{center}
\end{table}
\end{landscape}
\section*{Acknowledgements} I am very grateful to Prof. S.~Shpectorov for his helpful comments and advice in the writing of this paper. This project particularly benefited from conversations that took place at the Symmetry vs Regularity conference in Pilsen in July 2018 and for this I would also like to thank the organisers of this conference.
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