text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Q: WCF service not working for iPhone I have a WCF service:
http://parentportal.technologyorg.com/ParentPortal.svc
which I have to access and get the data. but unfortunately I am getting some HTML code.
my code is as follows
-(IBAction)doit:(id)sender{
NSString *soapMessage = [NSString stringWithFormat:@"<SOAP-ENV:Envelope xmlns:SOAP-ENV=\"http://parentportal.technologyorg.com/ParentPortal.svc\"><SOAP-ENV:Body><GetDictionary></GetDictionary></SOAP-ENV:Body></SOAP-ENV:Envelope>"];
NSURL *url = [NSURL URLWithString:@"http://parentportal.technologyorg.com/ParentPortal.svc"];
NSMutableURLRequest *theRequest = [NSMutableURLRequest requestWithURL:url];
NSString *msgLength = [NSString stringWithFormat:@"%d", [soapMessage length]];
[theRequest addValue: @"text/xml; charset=utf-8" forHTTPHeaderField:@"Content-Type"];
[theRequest addValue: @"GetStudentTerms" forHTTPHeaderField:@"SOAPAction"];
[theRequest addValue: msgLength forHTTPHeaderField:@"Content-Length"];
[theRequest setHTTPMethod:@"POST"];
[theRequest setHTTPBody: [soapMessage dataUsingEncoding:NSUTF8StringEncoding]];
NSLog(@"%@",theRequest);
NSURLConnection *theConnection = [[NSURLConnection alloc] initWithRequest:theRequest delegate:self];
if (theConnection) {
webData = [NSMutableData data] ;
}
}
- (void)connection:(NSURLConnection *)connection didReceiveResponse:(NSURLResponse *)response {
NSLog(@"___didReceiveResponse");
NSHTTPURLResponse *httpResponse = (NSHTTPURLResponse *) response;
NSLog(@"Status code %d", [httpResponse statusCode]);
[webData setLength:0];
}
- (void)connection:(NSURLConnection *)connection didReceiveData:(NSData *)data {
NSLog(@"___didReceiveData");
[webData appendData:data];
NSLog(@"%@",webData);
}
- (void) connectionDidFinishLoading:(NSURLConnection *)connection {
NSLog(@"___connectionDidFinishLoading");
NSLog(@"DONE. Received Bytes: %d", [webData length]);
NSString *xmlData = [[NSString alloc]
initWithBytes:[webData mutableBytes]
length:[webData length]
encoding:NSUTF8StringEncoding];
}
I am not sure whats going wrong were if I NSLog it I am the output as some HTML code.
Can any one help me on this.
A: Try SudzC SOAP-based web service definition WSDL files
Code generation for Objective-C libraries for iPhone development.
Follow the steps below
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 592 |
package org.camunda.bpm.engine.rest.sub.runtime.impl;
import com.fasterxml.jackson.databind.ObjectMapper;
import org.camunda.bpm.engine.ProcessEngine;
import org.camunda.bpm.engine.impl.RuntimeServiceImpl;
import org.camunda.bpm.engine.rest.sub.impl.AbstractVariablesResource;
import org.camunda.bpm.engine.variable.VariableMap;
import org.camunda.bpm.engine.variable.value.TypedValue;
import java.util.List;
public class ExecutionVariablesResource extends AbstractVariablesResource {
private String resourceTypeName;
public ExecutionVariablesResource(ProcessEngine engine, String resourceId, boolean isProcessInstance, ObjectMapper objectMapper) {
super(engine, resourceId, objectMapper);
if (isProcessInstance) {
resourceTypeName = "process instance";
} else {
resourceTypeName = "execution";
}
}
protected String getResourceTypeName() {
return resourceTypeName;
}
protected void updateVariableEntities(VariableMap modifications, List<String> deletions) {
RuntimeServiceImpl runtimeService = (RuntimeServiceImpl) engine.getRuntimeService();
runtimeService.updateVariables(resourceId, modifications, deletions);
}
protected void removeVariableEntity(String variableKey) {
engine.getRuntimeService().removeVariable(resourceId, variableKey);
}
protected VariableMap getVariableEntities(boolean deserializeValues) {
return engine.getRuntimeService().getVariablesTyped(resourceId, deserializeValues);
}
protected TypedValue getVariableEntity(String variableKey, boolean deserializeValue) {
return engine.getRuntimeService().getVariableTyped(resourceId, variableKey, deserializeValue);
}
protected void setVariableEntity(String variableKey, TypedValue variableValue) {
engine.getRuntimeService().setVariable(resourceId, variableKey, variableValue);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,789 |
\section{Background}
We recall a classic theorem of Erd\H{o}s and Gallai~\cite{erdHos1959maximal}.
\begin{theorem}[Erd\H{o}s, Gallai~\cite{erdHos1959maximal}]
Let $n$, $k$ be positive integers and let $G$ be an $n$-vertex graph containing no path of $k$ edges, then
\[
e(G) \le \frac{(k-1)n}{2}.
\]
Equality is obtained if and only if $k$ divides $n$ and $G$ is the graph consisting of $n/k$ disjoint complete graphs of size $k$.
\end{theorem}
Erd\H{o}s and S\'os~\cite{erdos1984some} conjectured that the same bound would hold for any tree with $k$ edges. A proof of this conjecture for sufficiently large $k$ was announced in the 90's by Ajtai, Koml\'os, Simonovits and Szemer\'edi. We will consider a variant of this problem in the setting of hypergraphs and multi-hypergraphs. We obtain exact results for the case of large uniformity.
Given a hypergraph $\mathcal{H}$, we denote the vertex and edge sets of $\mathcal{H}$ by $V(\mathcal{H})$ and $E(\mathcal{H})$, respectively. We denote the number of vertices and hyperedges by $v(\mathcal{H}) = \abs{V(\mathcal{H})}$ and $e(\mathcal{H}) = \abs{E(\mathcal{H})} $. A hypergraph is said to be $r$-uniform if all of its hyperedges have size $r$. We now provide some definitions which we will need.
\begin{definition}
For a given uniformity $r$ and a fixed graph $G$, an $r$-uniform multi-hypergraph $\mathcal{H}$ is a \emph{Berge copy} of G, if there exists an injection $f_1: V(G) \to V(\mathcal{H})$ and a bijection $f_2:E(G) \to E(\mathcal{H})$, such that if $e = \{v_{1},v_{2}\}\in E(G)$, then $\{f_1(v_{1}),f_1(v_{2})\} \subseteq f_2(e)$. The set of Berge copies of $G$ is denoted by $\mathcal{B} G$. The sets $f_1(V(G))$ and $f_2(E(G))$ are called the \emph{defining} vertices and hyperedges, respectively.
\end{definition}
We recall the classical definition of the Tur\'an number of a family of hypergraphs.
\begin{definition}
The Tur\'an number of a family of $r$-uniform hypergraphs $\mathcal{F}$, denoted $\ex_r(n,\mathcal{F})$, is the maximum number of hyperedges in an $n$-vertex, $r$-uniform, simple-hypergraph which does not contain an isomorphic copy of $\mathcal{H}$, for all $\mathcal{H} \in \mathcal{F}$, as a sub-hypergraph.
\end{definition}
The same question may be asked for multi-hypergraphs, we denote the Tur\'an number for multi-hypergraphs by $\ex_r^{multi}(n,\mathcal{F})$.
\begin{remark}
If every hypergraph in $\mathcal{F}$ has at least $r+1$ vertices, then $\ex_r^{multi}(n,\mathcal{F})$ is infinite, since a hypergraph on $r$ vertices and multiple copies of the same hyperedge is $\mathcal{F}$-free.
\end{remark}
The classical theorem of Erd\H{o}s and Gallai was extended to Berge paths in $r$-uniform hypergraphs by Gy\H{o}ri, Katona and Lemons~\cite{gyorikatonalemons}.
\begin{theorem}[Gy\H{o}ri, Katona, Lemons \cite{gyorikatonalemons}] \label{gkl}
Let $n,k,r$ be positive integers and let $\mathcal{H}$ be an $r$-uniform hypergraph with no Berge path of length $k$. If $k>r+1>3$, we have
\begin{displaymath}
e(\mathcal{H}) \le \frac{n}{k} \binom{k}{r}.
\end{displaymath}
If $r \ge k>2$, we have
\begin{displaymath}
e(\mathcal{H}) \le \frac{n(k-1)}{r+1}.
\end{displaymath}
\end{theorem}
The remaining case when $k = r + 1$ was settled later by Davoodi, Gy\H{o}ri, Methuku and Tompkins~\cite{davoodi}, the Tur\'an number matches the upper bound of Theorem~\ref{gkl} in the $k>r+1$ case.
We now turn our attention to the case of trees in hypergraphs. The Tur\'an number of certain kinds of trees in $r$-uniform hypergraphs has long been a major topic of research. For example, there is a notoriously difficult conjecture of Kalai \cite{kalai} which is more general than the Erd\H{o}s-S\'os conjecture. The trees which Kalai considers are generalizations of the notion of tight paths in hypergraphs. In another direction, F\"uredi \cite{fur} investigated linear trees, constructed by adding $r-2$ new vertices to every edge in a (graph) tree. In this setting, he proved asymptotic results for all uniformities at least $4$. Whereas, the articles above considered classes of trees containing tight and linear paths, respectively, we will consider the setting of Berge trees.
In the range when $k > r$, a number of results on forbidding Berge trees were obtained by Gerbner, Methuku and Palmer in~\cite{bigk}. In particular they proved that if we assume the Erd\H{o}s-S\'os conjecture holds for a tree $T$ with $k$ edges and all of its sub-trees and also that $k>r+1$, we have $\ex_r(n,\mathcal{B} T) \le \frac{n}{k}\binom{k}{r}$ (a construction matching this bound when $k$ divides $n$ is given by $n/k$ disjoint copies of the complete $r$-uniform hypergraph on $k$ vertices). In the present paper, we will consider the range $r>k$, where we prove some exact results.
\section{Main Results}
Considering multi-hypergraphs, we prove the following.
\begin{theorem}
\label{Multi_tree_theorem}
Let $n,k,r$ be positive integers and let $T$ be a $k$-edge tree, then for all $r\geq (k-1)(k-2)$,
\begin{displaymath}
\ex_r^{multi}(n,\mathcal{B} T)\leq \frac{n(k-1)}{r}.
\end{displaymath}
If $r > (k-1)(k-2)$ and $T$ is not a star, equality holds if and only if $r$ divides $n$ and the extremal multi-hypergraph is $\frac{n}{r}$ disjoint hyperedges, each with multiplicity $k-1$. If $T$ is a star equality holds only for all $(k-1)$-regular multi-hypergraphs.
\end{theorem}
We conjecture that Theorem \ref{Multi_tree_theorem} holds for the following wider set of parameters.
\begin{conjecture}
\label{treeconj}
Let $n,k,r$ be positive integers and let $T$ be a $k$-edge tree, then for all $r \ge k+1$,
\begin{displaymath}
\ex_r^{multi}(n,\mathcal{B} T)\leq \frac{n(k-1)}{r}.
\end{displaymath}
For all trees $T$, where $T$ is not a star, equality holds if and only if $r$ divides $n$ and the extremal multi-hypergraph is $\frac{n}{r}$ disjoint hyperedges each with multiplicity $k-1$.
\end{conjecture}
The special case of Conjecture \ref{treeconj}, when the forbidden tree is a path, was settled by Gy\H{o}ri, Lemons, Salia and Zamora~\cite{new} (see the first corollary).
We now define a class of hypergraphs which we will need when we classify the extremal examples in our main result about simple hypergraphs, Theorem~\ref{Not_Star_Theorem}.
\begin{definition}
An $r$-uniform hypergraph $\mathcal{H}$ is \emph{two-sided} if $V(\mathcal{H})$ can be partitioned into a set $X$ and pairwise disjoint sets $A_i$, $i=1,2,\dots,t$ (also disjoint from $X$) of size $r-1$, such that every hyperedge is of the form $\{x\} \cup A_i$ for some $x \in X$. We say that a two-sided $r$-uniform hypergraph is $(a,b)$-regular if every vertex of $X$ has degree $a$ and every vertex of $\displaystyle\bigcup_{i=1}^{t}A_i$ has degree $b$.
\end{definition}
\begin{remark} A two-sided $r$-uniform hypergraph can also be viewed as a graph obtained by taking a bipartite graph $G$ with bipartite classes $X$ and $Y$, and ``blowing up" each vertex of $Y$ to a set of size $r-1$, and replacing each edge $\{x,y\}$ by the $r$-hyperedge containing $x$ together with the blown up set for $y$.
\end{remark}
\begin{theorem}
\label{Not_Star_Theorem}
Let $n,k,r$ be positive integers and let $T$ be a $k$-edge tree which is not a star, then for all $r\geq k(k-2)$,
\begin{displaymath}
\ex_r(n,\mathcal{B} T)\leq \frac{n(k-1)}{r+1}.
\end{displaymath}
Equality holds if and only if $r+1$ divides $n$, and the extremal hypergraph is obtained from $\frac{n}{r+1}$ disjoint sets of size $r+1$, each containing $k-1$ hyperedges.
Unless $k$ is odd, and $T$ is the balanced double star, where the balanced double star is the tree obtain from and edge by adding $\frac{k-1}{2}$ incident edges to each of the ends of the edge, in which case equality holds if and only if $r+1$ divides $n$ and $\mathcal{H}$ is obtained from the disjoint union of sets of size $r+1$ containing $k-1$ hyperedges each and possibly a $(k-1,\frac{k-1}{2})$-regular two-sided $r$-uniform hypergraph (see Figure~\ref{Extremal}).
\end{theorem}
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale = 0.65,every node/.style={scale=.8}]
\draw (0,3);
\filldraw[blue,fill opacity=0.3] (2,0) arc (0:360: 2cm and 2cm);
\draw (0,0) node[above]{$r+1$ vertices};
\draw (0,0) node[below] {$k-1$ hyperedges};
\begin{scope}[xshift=5cm]
\filldraw[blue,fill opacity=0.3] (2,0) arc (0:360: 2cm and 2cm);
\draw (0,0) node[above]{$r+1$ vertices};
\draw (0,0) node[below] {$k-1$ hyperedges};
\end{scope}
\begin{scope}[xshift=10cm]
\filldraw[blue,fill opacity=0.3] (2,0) arc (0:360: 2cm and 2cm);
\draw (0,0) node[above]{$r+1$ vertices};
\draw (0,0) node[below] {$k-1$ hyperedges};
\end{scope}
\draw(0,-2.5);
\end{tikzpicture} \qquad
\begin{tikzpicture}[rotate=90,scale = 0.65,every node/.style={scale=.8}]
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,3.68) (.04,3.83) (2.85,2.25) (3.1,2) (2.70,1.75) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,2.18) (.04,2.33) (2.85,2.25) (3.1,2) (2.70,1.75) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,.68) (.04,.83) (2.85,2.25) (3.1,2) (2.70,1.75) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,-0.82) (.04,-0.6) (2.85,2.25) (3.1,2) (2.70,1.75) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,-3.68) (.04,-3.83) (2.85,-2.25) (3.1,-2) (2.70,-1.75) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,-2.18) (.04,-2.33) (2.85,-2.25) (3.1,-2) (2.70,-1.75) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,-.68) (.04,-.83) (2.85,-2.25) (3.1,-2) (2.70,-1.75) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,0.82) (.04,0.6) (2.85,-2.25) (3.1,-2) (2.70,-1.75) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,3.68) (.04,3.83) (2.85,.25) (3.1,0) (2.70,-.1) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,2.18) (.04,2.33) (2.85,.25) (3.1,0) (2.70,-.1) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,-3.68) (.04,-3.83) (2.85,-.25) (3.1,0) (2.70,.1) } ;
\filldraw[blue, fill opacity=0.10] plot [smooth cycle] coordinates { (0.02,-2.18) (.04,-2.33) (2.85,-.25) (3.1,0) (2.70,.1) } ;
\filldraw[red] (2.8,0) circle (5pt) (2.8,2) circle (5pt) (2.8,-2) circle (5pt);
\filldraw[green] (0,3.75)circle (3pt) (0,2.25)circle (3pt) (0,.75)circle (3pt) (0,-3.75)circle (3pt) (0,-2.25)circle (3pt) (0,-.75)circle (3pt);
\draw (4,0) node{$\abs{A_i} =r-1$, $d(A_i) =\frac{k-1}{2} $};
\draw (-.3,0) node[below]{$d(x)=k-1$};
\end{tikzpicture}
\caption{An extremal graph for Theorem~\ref{Not_Star_Theorem} is pictured. Any such graph can be obtained from disjoint copies of a sets of $r+1$ vertices with $k-1$ hyperedges and if $T$ is the balanced double star, possibly a $(k-1,\frac{k-1}{2})$-regular two-sided $r$-uniform hypergraph.}
\label{Extremal}
\end{figure}
\section{Proofs of the main results}
We start with some results about graphs.
\begin{definition} For a Graph $G$, we denote by $d(G)$ the average degree of $G$, that is $d(G) = \frac{2e(G)}{v(G)}.$
\end{definition}
\begin{lemma}\label{folklore} Any non-empty graph $G$ contains a subgraph $G'$ with minimum degree greater than $d(G)/2$.
\end{lemma}
The previous lemma is a well-known result in graph theory, which can be proved using the following lemma.
\begin{lemma}\label{average}
Let $G$ be a graph and $V' \subseteq V$, if $V'$ is incident with at most $\frac{d(G)}{2}\abs{V'}$ edges, then $d(G[V\setminus V']) \geq d(G)$.
\end{lemma}
\begin{proof}
Note that if $m$ is the number of edges incident with $V'$, then we have that \begin{displaymath} 2e(G[V\setminus V']) = 2e(G) - 2m \geq d(G)v(G) - d(G)\abs{V'} = d(G)(\abs{V}-\abs{V'}) = d(G)v(G[V\setminus V']). \qedhere \end{displaymath} \end{proof}
We are going to use the following fact about trees, before proving the next bound on the degrees of the vertices in clusters.
\begin{claim} \label{lowdegree} If $T$ is $k$-edge tree which is not a star, then there exists a vertex of $T$ which is not a leaf and has degree at most $\frac{k+1}{2}$.
\end{claim}
\begin{proof}
Let $T'$ be the tree obtained by $T$ by removing every leaf of $T$, since $T$ is not a star, $T'$ has at least two vertices, take any $v,w$ which are leaves in $T'$, and note that for each, every neighbor but one is a leaf, and also, since at most one the $k$ edges of $T$ is incident with both $u$ and $v$, we have that $d_T(u)+d_T(v)\leq k+1$.
And so, one of these vertices have the desired properties.
\end{proof}
Now we introduce two more definitions which we will need in the proofs.
\begin{definition}
Let $\mathcal{H}$ be a (multi-)hypergraph. A $(k-1)$-\emph{cluster} is a set of $k-1$ hyperedges of $\mathcal{H}$ that intersect in at least $k-1$ vertices. The intersection of the $k-1$ hyperedges is called the \emph{core} of the $(k-1)$-cluster. The union of the $k-1$ hyperedges is called the \emph{span} of the $(k-1)$-cluster.
\end{definition}
\begin{definition} Let $\mathcal{H} = (V,E)$ be a multi-hypergraph. A multi-hypergraph $\mathcal{H}' = (V',E')$ is called a
\emph{reduced sub-hypergraph} of $\mathcal{H}$ if $V' \subseteq V$
and there exists an injection $f:E' \to E$ such that $h \subseteq f(h)$ for all $h\in E'$.
For an edge $h\in E'$ we call $f(h)\in E$ its \emph{correspondent} edge in $\mathcal{H}$.
\end{definition}
In the following claims, we bound the degrees of the vertices in a $(k-1)$-cluster for a hypergraph which does not contain a copy of a Berge tree.
\begin{claim}\label{core}
Let $n,k,r$ be positive integers, with $r \ge k+1$, and let $T$ be a $k$-edge tree. If $\mathcal{H}$ is an $r$-uniform multi-hypergraph containing no Berge copy of $T$ and $S$ is a $(k-1)$-cluster in $\mathcal{H}$, then the vertices in the core of $S$ have degree exactly $k-1$. In particular, the core vertices of $S$ are only incident with the hyperedges of $S$.
\end{claim}
\begin{proof}
Let $C$ be the set of vertices in the core of $S$.
Suppose, by contradiction, there is a vertex $v$ in $C$ with degree at least $k$, and let $T'$ be a tree obtained from $T$ by removing any two leaves $x,y$. Suppose that the neighbors of these leaves are $x'$ and $y'$ respectively (it is possible that $x'=y'$).
Since $C$ has at least $k-1$ vertices and there are $k-1$ hyperedges containing all the vertices in $C$, we can greedily embed $T'$ in $C$ in such a way that $v$ takes the role of $x'$.
Suppose the vertex $u$ takes the role of $y'$ in this greedy embedding. We can complete the embedding of $T$ by using the last hyperedge of S and an unused vertex in it (one exists since $r\geq k+1$) to embed $y$.
Then since the degree of $v$ is at least $k$, we have a hyperedge available to embed $x$ as a unused vertex of this hyperedge.
Thus we have found a Berge copy of $T$ in $\mathcal{H}$, a contradiction.
\end{proof}
\begin{claim}\label{Y}
Let $n,k,r$ be positive integers, with $r \ge k+1$, and let $T$ be a $k$-edge tree which is not a star. If $\mathcal{H}$ is an $r$-uniform multi-hypergraph containing no Berge copy of $T$ and $S$ is a $(k-1)$-cluster of $\mathcal{H}$, then any vertex in the span of $S$ that is incident with a hyperedge not from $S$, has degree at most $\floor{\frac{k-1}{2}}$.
\end{claim}
\begin{proof}
Since $T$ is not a star, by Claim~\ref{lowdegree}, there is a vertex $x \in V(T)$ which is not a leaf and has degree $s$, $s\leq \floor{\frac{k+1}{2}}$, such that all but one of its neighbors is a leaf, let $y$ be the neighbor of $x$ which is not a leaf.
Suppose, by contradiction, there is a vertex $v$ in the span of $S$ which is incident with a hyperedge that is not in $S$ and $v$ has degree at least $\floor{\frac{k+1}{2}}$.
Let $C$ be the set of vertices in the core of $S$.
From Claim~\ref{core} we know that $v$ cannot be in $C$.
Pick $s$ hyperedges $h_1,h_2,\dots,h_s$ incident to $v$ in such a way that $h_1$ is not in $S$ and $h_2$ is in $S$.
Choose a vertex $w \in h_1$ not in $C$ (in fact, every vertex in $h_1$ is outside $C$ by Claim~\ref{core}) and $u \in h_2$ in $C$.
Choose further distinct vertices $v_3,v_4,\dots,v_s$ from the hyperedges $h_3,h_4,\dots,h_s$.
The vertex $v$ will be assigned to the vertex $x$ in the tree, and the vertex $u$ will be assigned to the vertex $y$ ($v_3,v_4,\dots,v_s$ will be assigned to the leaves adjacent to $x$).
Thus, using the hyperedges $h_1,h_2,\dots,h_s$ we can embed the vertex $x$ and all its neighbors in $T$ using at most $s-1$ hyperedges from $S$ and at most $s-1$ vertices from $C$ ($v$ and $w$ are not in $C$).
There are at least $(k-1)-(s-1)=k-s$ remaining vertices in $C$.
Each of these is contained in at least $k-s$ unused hyperedges of $S$.
Thus, the remaining $k-s$ vertices of the tree can be mapped to distinct vertices from $C$, and the remaining edges of the tree may be assigned to distinct unused hyperedges of $S$.
\end{proof}
\begin{remark}\label{disjoint}
Note that by Claim \ref{core} and Claim \ref{Y}, if $\mathcal{H}$ is a multi-hypergraph with uniformity $r \ge k+1$ that does not contain a Berge copy of a tree on $k$ edges which is not a star, then $(k-1)$-clusters of $\mathcal{H}$ are edge-disjoint.
\end{remark}
\begin{lemma}\label{cluster} Let $k$ be a positive integer and let $T$ be a $k$-edge tree which is not a star. Let $\mathcal{H}$ be a multi-hypergraph not necessarily uniform, not containing a Berge copy of $T$, and assume that each hyperedge in $\mathcal{H}$ has size at least $k+1$. If there exists a reduced sub-hypergraph $\mathcal{H}' = (V',E')$ of $\mathcal{H}$ such that $d_{\mathcal{H}'}(v) \geq k-1$ for each $v \in V'$ and $\abs{h} \geq k-1$ for each $h\in E'$, then $\mathcal{H}'$ contains a $(k-1)$-cluster.
Note that if $S$ is a $(k-1)$-cluster in $\mathcal{H}'$, then the correspondent edges of $S$ in $\mathcal{H}$ are a $(k-1)$-cluster.
\end{lemma}
\begin{proof}
Let $h_2 \in E'$.
We will show that every vertex in $h_2$ is contained in the same set of hyperedges in $E'$.
Let $v_1,v_2 \in h_2$, and suppose by contradiction that there exists a hyperedge $h_3$ incident to $v_2$ and not to $v_1$.
Enumerate the vertices of $T$ by $x_0,x_1,\dots,x_k$ in such a way that the graph induced by the vertices $x_0,x_1,\dots,x_i$ is connected for all $i$, $x_0$ is a leaf of $T$ and $x_0,x_1,x_2,x_3$ is a path of length 3 (such a path exists since $T$ is not a star). For each $i = 1,2,\dots,k$, the vertex $x_i$ is adjacent to exactly one vertex of smaller index, call the edge using $x_i$ and the vertex of smaller index $e_i$.
We can embed $T$ into $\mathcal{H}$ in the following way. First assign $v_1$ to $x_1$, $h_2$ to $\{x_1,x_2\}$, $v_2$ to $x_2$, $h_3$ to $\{x_2,x_3\}$ and any vertex in $v_3 \in h_3\setminus\{v_1,v_2\}$ to $x_3$. For $i=4,\dots,k$, suppose $e_i = \{x_i,x_{j_i}\}$.
Pick any hyperedge $h_i \in E'$ incident to $v_{j_i}$ and distinct from $h_2,h_3,\dots,h_{i-1}$ (such hyperedges exist since $d_{\mathcal{H}'}(v_{j_i}) \geq k-1$) and assign it to $e_i$. If $i\leq k-1$, pick any $v_i \in h_i\setminus\{v_1,v_2,\dots,v_{i-1}\}$, and if $i= k$, then let $\tilde{h}_k$ be the correspondent hyperedge of $h_k$ in $\mathcal{H}$.
As $\tilde{h}_k$ has size bigger than $k$, let $v_{k}$ be any vertex in $\tilde{h}_k\setminus\{v_1,v_2,\dots,v_{k-1}\}$.
This vertex $v_k$ is assigned to $x_k$.
Finally, since $v_1$ is incident with at least $k-1$ hyperedges distinct to $h_3$, there is a hyperedge $h_1$ incident to $v_1$ and distinct from the already chosen hyperedges.
Let $\tilde{h}_1$ be the correspondent hyperedge of $h_1$.
Take any vertex in $\tilde{h}_1$ which has not been assigned yet and assign it to $x_0$.
Thus, by replacing the edge $h_i$ with their correspondent hyperedges, we have found a Berge copy of $T$ in $\mathcal{H}$, a contradiction.
It follows that for any $v_1,v_2 \in h_2$, we have that $v_1$ and $v_2$ must be incident with the same set of hyperedges in $\mathcal{H}'$ (by assumption at least $k-1$), and so $\mathcal{H}'$ contains a $(k-1)$-cluster.
\end{proof}
Lemma \ref{cluster} says that if $\mathcal{H}$ does not contain a Berge copy of a tree and we are able to find a large enough reduced sub-hypergraph, then $\mathcal{H}$ must have at least one $(k-1)$-cluster. The main idea of the proofs of the main results is to show that if $\mathcal{H}$ has too many hyperedges and no Berge copy of a tree, then after removing all $(k-1)$-clusters, we would still be able to find a large enough reduced sub-hypergraph. This would imply that there is still another $(k-1)$-cluster in $\mathcal{H}$, a contradiction.
\begin{proof}[Proof of Theorem \ref{Multi_tree_theorem}]
Let $T$ be a $k$-edge tree, which is not a star.
Suppose that $\mathcal{H}$ is an $n$-vertex $r$-uniform hypergraph with at least $\frac{n(k-1)}{r}$ hyperedges such that $\mathcal{H}$ does not contain a Berge copy $T$, and let $G$ be the incidence bipartite graph of $\mathcal{H}$, i.e., the bipartite graph with color classes $V(\mathcal{H})$ and $E(\mathcal{H})$ where $v\in V(\mathcal{H})$ is adjacent to $h \in E(\mathcal{H})$ if and only if $v\in h$.
Since $\displaystyle e(\mathcal{H}) \ge \frac{n(k-1)}{r}$, we have
$\displaystyle
\frac{e(G)}{v(G)} = \frac{re(\mathcal{H})}{n + e(\mathcal{H})} = \frac{r}{\frac{n}{e(\mathcal{H})}+1} \ge \frac{r}{\frac{r}{k-1}+1} = \frac{r(k-1)}{r+k-1},$ and note that
\begin{displaymath}
\frac{r(k-1)}{r+k-1} \geq k - 2 \Leftrightarrow r(k-1) \geq (k - 2)(r+k-1) = r(k-1) + (k-2)(k-1) - r \Leftrightarrow r \geq (k-2)(k-1).
\end{displaymath}
Hence $d(G) = \frac{2e(G)}{v(G)}\geq 2\left(\frac{r(k-1)}{r+k-1}\right) \geq k-2$, since $r \geq (k-2)(k-1).$
Suppose $\mathcal{H}$ has $t$ distinct $(k-1)$-clusters $S_1,S_2,\dots,S_t$ (recall that by Remark~\ref{disjoint} $(k-1)$-clusters are edge-disjoint).
For each $S_i$, let $X_i$ be the set of vertices which are incident only with hyperedges of $S_i$, let $X = \bigcup_{i=1}^t X_i$ and let $Y$ be the set of vertices that are not in $X$ but are incident with at least one of the $(k-1)$-clusters.
Let $G_1$ be the induced subgraph of $G$ obtained by removing $X$, $Y$ and all $(k-1)$-cluster hyperedges from the vertex set of $G$. We will show that $d(G_1) \geq d(G)$ (provided $G_1$ is not the empty graph).
The number of edges removed in $G$ is $\sum_{v\in X} d_{\mathcal{H}}(v) + \sum_{v\in Y} d_{\mathcal{H}}(v)$. Since the degree of each $v\in X$ is at most $k-1$, we have that
$\displaystyle \left(\sum_{v\in X} d_{\mathcal{H}}(v)\right) \leq \abs{X}(k-1)$. Also $X$ is only incident with the $(k-1)$-cluster hyperedges, thus we also have the bound
$\displaystyle \left(\sum_{v\in X}d_{\mathcal{H}}(v)\right) \leq tr(k-1)$, and since the degree of each $v\in Y$ is at most $\frac{k-1}{2}$ (Claim~\ref{Y}), we have that $\displaystyle \left(\sum_{v\in Y} d_{\mathcal{H}}(v) \right) \leq \frac{(k-1)\abs{Y}}{2}$.
Therefore
\[\left(\sum_{v\in X}d_{\mathcal{H}}(v) + \sum_{v\in Y} d_{\mathcal{H}}(v) \right)(r+k-1)\]
\[= \left(\sum_{v\in X}d_{\mathcal{H}}(v)\right)r + \left(\sum_{v\in X}d_{\mathcal{H}}(v)\right)(k-1) + \left(\sum_{v\in Y} d_{\mathcal{H}}(v) \right)(r+k-1)\]
\[\leq \abs{X}r(k-1) + tr(k-1)^2 + \frac{(k-1)\abs{Y}}{2}(r+k-1) \leq r(k-1)(\abs{X} + t(k-1) + \abs{Y}),\]
where in the last inequality we used $\frac{r+k-1}{2} < r$.
Thus, equality can hold only if $Y = \emptyset$.
Rearranging we have
\begin{equation}
\label{g}
\left(\sum_{v\in X}d_{\mathcal{H}}(v) + \sum_{v\in Y} d_{\mathcal{H}}(v) \right)\le \frac{r(k-1)}{r+k-1}\left(\abs{X} + t(k-1) + \abs{Y}\right).
\end{equation}
The left-hand side of \eqref{g} is the number of removed edges, and the right-hand side is $d(G)/2$ times the number of removed vertices.
Therefore, by Lemma~\ref{average}, if $G_1$ is non-empty, we have that
\[d(G_1) \geq d(G) \geq 2(k-2).\]
Hence, by Lemma~\ref{folklore} there is a subgraph $G_2$ of $G_1$ with minimum degree at least $k-1$. Suppose that $G_2$ has bipartite classes $A \subseteq V(\mathcal{H})$ and $B \subseteq E(\mathcal{H})$, and define $\mathcal{H}'$ by taking the vertex set $V' = A$ and $E' = \{h\cap V': h \in B\}$.
The condition on the minimum degree of $G_2$ implies that every vertex of $\mathcal{H}'$ has degree at least $k-1$ and every hyperedge of $\mathcal{H}'$ has size at least $k-1$.
Then by Lemma \ref{cluster}, $\mathcal{H}'$ contains a $(k-1)$-cluster, but this $(k-1)$-cluster corresponds to a $(k-1)$-cluster in $\mathcal{H}$ contradicting the fact that we removed every $(k-1)$-cluster from $\mathcal{H}$.
So $\mathcal{H}$ must contain a Berge copy of $T$, unless $G_1$ is empty.
Note that, for $G_1$ to be empty it is necessary that $d(G) = 2\frac{r(k-1)}{r+k-1}$ and that equality holds in the inequality~\eqref{g}.
This can be possible only if $Y = \emptyset$ and
\[\abs{X} = \frac{1}{k-1}\sum_{v\in X} d_{\mathcal{H}}(v) = tr.\]
Since every $(k-1)$-cluster contains at least $r$ vertices, we have $\abs{X_i}\geq r$, and so
each $X_i$ must have size exactly $r$, hence $\mathcal{H}$ is the disjoint union of $t$ hyperedges each with multiplicity $k-1$.
Therefore the number of vertices would be a multiple of $r$ and $e(\mathcal{H}) = \frac{n(k-1)}{r}$.
Hence if $e(\mathcal{H}) \geq \frac{n(k-1)}{r}$, then $\mathcal{H}$ must contain a Berge copy of $T$, or $r|n$ and $\mathcal{H}$ is the disjoint union of $\frac{n}{r}$ hyperedges each with multiplicity $k-1$.
\end{proof}
\begin{remark}
For $r=(k-2)(k-1)$, the proof above also shows that if $e(\mathcal{H}) > \frac{n(k-1)}{r},$ then $\mathcal{H}$ must contain a Berge copy of $T$. However, the extremal construction does not follow from that proof. \end{remark}
\begin{proof}[Proof of Theorem \ref{Not_Star_Theorem}]
Let $T$ be a $k$-edge tree which is not a star. We may assume $k>3$, since otherwise $T$ is a path, and we already know the result for paths.
Let $\mathcal{H}$ be an $n$-vertex hypergraph with at least $\frac{n(k-1)}{r+1}$ hyperedges and $r \ge k(k-2)$. We will proceed by induction on the number of vertices $n$; the base cases $n \le r+1$ are trivial.
If there is a set $U$ of size $r+1$ which is incident with at most $k-1$ hyperedges,
put $V'=V\setminus U$ and let $n' = |V'|=n-r-1$.
By induction, $\mathcal{H}'$ the hypergraph induced by $V'$, has at most $\frac{n'(k-1)}{r+1}$ hyperedges and equality holds if $r+1|n'$ and $\mathcal{H}'$ is the disjoint union of cliques, unless $T$ is the balanced double star,
then it may contain a $(k-1,\frac{k-1}{2})$-regular two-sided hypergraph as described in the statement of the theorem.
Note that if one of the hyperedges incident with $U$ is incident with a vertex $v$, $v \in V'$, then $v$ has degree at least $\floor{\frac{k+1}{2}}$, and $v$ is in a $(k-1)$-cluster of $\mathcal{H}'$, thus we have a Berge copy of $T$ from Claim~\ref{Y}.
Hence, the $k-1$ hyperedges incident with $U$ are contained in the vertex set $U$ and $\mathcal{H}$ has the desired structure.
Similarly to the proof of Theorem~\ref{Multi_tree_theorem}, we have that
\[\displaystyle\frac{e(G)}{v(G)} = \displaystyle\frac{re(\mathcal{H})}{n+r(\mathcal{H})} = \frac{r}{\frac{n}{e(\mathcal{H})}+1} \ge \frac{r}{\frac{r+1}{k-1} + 1} = \frac{r(k-1)}{r+k},\] and note that
\begin{displaymath}
\frac{r(k-1)}{r+k} \geq k - 2 \Leftrightarrow r(k-1) \geq (k - 2)(r+k) = r(k-1) + (k-2)k - r \Leftrightarrow r \geq k(k-2).
\end{displaymath}
Hence $d(G) = \frac{2e(G)}{v(G)}\geq 2\left(\frac{r(k-1)}{r+k-1}\right) \geq k-2$, since $r \geq (k-2)(k-1).$
Suppose that $\mathcal{H}$ has $t$ distinct $(k-1)$-clusters $S_1,S_2,\dots, S_t$. Define the sets $X_1,\dots,X_t,X$ and $Y$ as in the proof of Theorem~\ref{Multi_tree_theorem}.
We are going to remove all vertices and hyperedges of these $(k-1)$-clusters as in the previous proof, and we will denote the incidence bipartite graph of $\mathcal{H}$ by $G$.
By $G_1$ we will denote the incidence bipartite graph of the hypergraph $\mathcal{H}'$, obtained from $\mathcal{H}$ after removing the $(k-1)$-clusters.
If $\abs{X_i} \geq r+1$ for some $i$, then by taking $U\subseteq X_i$ of size $r+1$, we would have that $U$ is incident with at most $k-1$ hyperedges, and we would be done by induction.
Hence we assume that $\abs{X_i}\leq r$.
For each $i$, with $\abs{X_i} = r$, we have \[\sum_{v\in X_i} d_{\mathcal{H}}(v) \leq (r-1)(k-1) +1 = \abs{X_i}(k-1)-(k-2),\] since any hyperedge is incident with at most $r-1$ vertices from $X_i$, with the possible exception of at most one hyperedge ($X_i$, if $X_i \in E(\mathcal{H}))$.
For each $i$, with $\abs{X_i} \leq r-1$, we have \[\sum_{v\in X_i} d_{\mathcal{H}}(v) \leq \abs{X_i}(k-1) \leq (r-1)(k-1).\]
Let $a$ be the number of $X_i$, $1 \le i \le t$, with the size $r$. Then we have the following inequalities
\begin{equation}
\label{aaaa}
\displaystyle\sum_{v\in X}d_{\mathcal{H}}(v) = \sum_{\substack{{\abs{X_i}= r} \\ v\in X_i}} d_{\mathcal{H}}(v) + \sum_{\substack{{\abs{X_i} < r} \\ v\in X_i}} d_{\mathcal{H}}(v) \leq t(r-1)(k-1) + a,
\end{equation}
and
\begin{equation}
\label{bbbb}
\sum_{v\in X}d_{\mathcal{H}}(v) \leq \sum_{\substack{{\abs{X_i}= r} \\ v\in X_i}} (\abs{X_i})(k-1)-(k-2)) + \sum_{\substack{{\abs{X_i}< r} \\ v\in X_i}}\abs{X_i}(k-1) =\abs{X}(k-1) - a(k-2).
\end{equation}
We also have
\begin{equation}
\label{zyx}
tr(k-1) \leq \sum_{v\in X}d_{\mathcal{H}}(v) + \sum_{v \in Y} d_{\mathcal{H}}(v) \leq t(r-1)(k-1) + a + \frac{k-1}{2}\abs{Y},
\end{equation}
where in the first inequality follows from the fact the set the edges of $t(k-1)$ hyperedges of the $t$ cluster are incident only with the set $X\cup Y$, and the second inequality follows directly from Claim~\ref{Y} together with the fact that $d_{\mathcal{H}}(v) \leq k-1$ by definition.
Rearranging \eqref{zyx} yields
\begin{equation}
\label{gh}
\displaystyle t(k-1) \leq a + \frac{\abs{Y}(k-1)}{2}.
\end{equation}
The following three bounds come from multiplying inequality \eqref{bbbb} and \eqref{aaaa} by $r$ and $k$, respectively, and the bound from Claim \ref{Y} by $k+r$.
\begin{equation}
\label{ab}
\left(\sum_{v\in X}d_{\mathcal{H}}(v)\right)r \leq \abs{X}r(k-1) - ar(k-2).
\end{equation}
\begin{equation}
\label{cd}
\left(\sum_{v\in X}d_{\mathcal{H}}(v)\right)k \leq t(r-1)k(k-1) + ak.
\end{equation}
\begin{equation}
\label{ef}
\left(\sum_{v\in Y} d_{\mathcal{H}}(v)\right)(k+r) \leq \frac{\abs{Y}(k-1)}{2}(k+r).
\end{equation}
Now we bound the number of deleted hyperedges times $r+k$.
From \eqref{ab}, \eqref{cd}, \eqref{ef} and then \eqref{gh}, it follows that
$$\left(\sum_{v\in X}d_{\mathcal{H}}(v) + \sum_{v\in Y} d_{\mathcal{H}}(v)\right)(k+r) \leq \abs{X}r(k-1) - ar(k-2) + t(r-1)k(k-1) + ak + \frac{\abs{Y}(k-1)}{2}(k+r)$$
$$= \abs{X}r(k-1) - ar(k-2) + tr(k-1)^2 + t(k-1)(r-k) + ak + \frac{\abs{Y}(k-1)}{2}(k+r)$$
$$\leq \abs{X}r(k-1) - ar(k-2) + tr(k-1)^2 + (r-k)\left(a + \frac{\abs{Y}(k-1)}{2}\right) + ak + \frac{\abs{Y}(k-1)}{2}(k+r)$$
$$= \abs{X}r(k-1) - ar(k-3) + tr(k-1)^2 + \abs{Y}(k-1)r = r(k-1)(\abs{X}+\abs{Y}+t(k-1)) - ar(k-3)$$ $$\leq r(k-1)(\abs{X}+\abs{Y}+t(k-1)).$$
Rearranging we have
\begin{equation}
\label{f}
\left(\sum_{v\in X}d_{\mathcal{H}}(v) + \sum_{v\in Y} d_{\mathcal{H}}(v) \right)\le \frac{r(k-1)}{r+k}\left(\abs{X} + t(k-1) + \abs{Y}\right).
\end{equation}
The left-hand side of \eqref{f} is the number of removed edges, and the right-hand side of \eqref{f} is $d(G)/2$ times the number of removed vertices.
Hence, by Lemma~\ref{average} if $G_1$ is nonempty, we have that
\[d(G_1) \geq d(G) \geq 2(k-2).\]
Thus, by Lemma~\ref{folklore} we can find a subgraph $G_2$ of $G_1$ with minimum degree at least $k-1$.
Suppose that $G_2$ has bipartite classes $A \subseteq V$ and $B \subseteq E(\mathcal{H})$, define $\mathcal{H}'$ by taking the vertex set $V' = A$ and hyperedge set $E' = \{h\cap V': h \in B\}$.
The condition on the minimum degree of $G_2$ implies that every vertex of $\mathcal{H}$ has minimum degree at least $k-1$, and every hyperedge of $\mathcal{H}'$ has size at least $k-1$.
Then by Lemma~\ref{cluster}, $\mathcal{H}'$ contains a $(k-1)$-cluster, which
contradicts that we have removed all $(k-1)$-clusters in $\mathcal{H}$.
For $G_1$ to be empty it is necessary that $d(G) = 2\frac{r(k-1)}{r+k}$, and for \eqref{f} to hold with equality, we must have that $e(\mathcal{H}) = \frac{n(k-1)}{r+1}$.
To obtain equality in $\eqref{f}$,
it is necessary that $a=0$ (since $k>3$) and that every hyperedge contains one of the $X_i$.
It then follows that $\abs{X} = t(r-1)$, and by \eqref{gh}, $\abs{Y} = 2t$.
By \eqref{ef}, for every $v\in Y$, we have $d_{\mathcal{H}}(v) = \frac{k-1}{2}$, so $n = t(r+1)$. Then $\mathcal{H}$ is a disjoint union of sets of $r+1$ vertices with $k-1$ hyperedges, and a hypergraph
constructed from the
classes $A=\{X_1,X_2\dots,X_t\}$ and $B = Y$, where $\{y,X_i\}$ is an edge if $X_i \cup \{y\}$ is a hyperedge of $\mathcal{H}$.
Note that $2t = 2\abs{A} = \abs{B}$, the degree of every vertex in $B$ is $\frac{k-1}{2}$ and every vertex of $A$ has degree $k-1$; that is, $\mathcal{H}$ is a $(k-1,\frac{k-1}{2})$-regular two-sided hypergraph.
However, this is only possible if $k$ is odd, and it is simple to check that this construction contains a Berge copy of every $k$-edge tree which is not a balanced $k$-edge double star or the $k$-edge star.
\end{proof}
\section*{Acknowledgments}
The research of the first three authors was partially supported by the National Research, Development and Innovation Office NKFIH, grants K116769, K117879 and K126853. The research of the second author is partially supported by Shota Rustaveli National Science Foundation of Georgia SRNSFG, grant number DI-18-118. The research of the third author is supported by the Institute for Basic Science (IBS-R029-C1).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,518 |
HDT's Adroit® manipulator arms are unique. They closely match the dexterity, strength, and speed of a human arm. Designed to work closely with people – using force sensing in every joint and impedance control to create active compliance, Adroit arms can work in commercial settings alongside people, without the robot being locked up in a caged work cell.
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"redpajama_set_name": "RedPajamaC4"
} | 856 |
Jerome Thompson, American (1814-1886) Jerome Thompson came from a family of painters. His father, Cephas, was a successful itinerant portrait painter and Jerome trained himself in the studio in Middleborough, Massachusetts as a portraitist in spite of his father's opposition. The elder Thompson encouraged his oldest son, Cephas Giovanni, but thought his younger son should become a farmer. Perhaps because of severe opposition, Jerome became the best known and best regarded of the Thompson painters. At age seventeen, he left home to follow a career as an itinerant portraitist. After four years, he settled in New York City where he had considerable success. By 1844 he had changed his directory listing from "portrait painter" to "artist". This change coincided with his entry into genre painting. In a letter to the American Art Union dated September 5, 1848, Thompson wrote: "I herewith send a picture for sale – it is a composition called going to the 'Squires to be Married'" and for which I ask $150 – this is the second time I have ever offered a picture to the American Art Union." He exhibited extensively in many venues including the American Academy of Fine Arts, the National Academy of Design, Pennsylvania Academy of Fine Arts, and the Brooklyn Art Association. The painting that launched his career was "Pic Nic", first exhibited at the national Academy of Design in 1850, and again at the Pennsylvania Academy the next year. Genre paintings by Thompson are in the collections of the Museum of Fine Arts, Boston, the Fine Arts Museum of San Francisco, the Berkshire Museum, the Brooklyn Museum, the Metropolitan Museum of Art, the Cincinnati Art Museum and the Evansville Museum of Arts.
The Country Parson Disturbed at Breakfast by a Couple Wishing to be Married. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,556 |
Gjon Delhusa (AFI: [ˈd͡ʒon ˈdɛlhusɒ]) es un cantante, compositor y letrista húngaro de origen albanés, primo del futbolista húngaro Kálmán Ihász. Representó a Hungría en el Festival de la Canción de Eurovisión 1996.
Biografía
Delhusa nació el 9 de agosto de 1953, en Budapest. Su madre, Erzsébet Ihász, era tía de Kálmán Ihász. Sus abuelos paternos eran de Grecia y Albania y por el lado materno tiene orígenes germano-húngaros. Pasó su infancia en Zugló, queriendo en ese entonces ser mecánico de autos.
Comenzó su carrera de música ligera a una edad muy temprana. Tenía solo 18 años cuando, en 1971, interpretó la canción Hegyek lánya de Attila y Károly Kőszeghy en el Táncdalfesztiválon, un concurso de música organizado por Magyar Televízió, la televisión pública de Hungría. Gracias a esto, Delhusa se hizo conocido de inmediato.
El año 1974 fue muy exitoso para él, ya que todos sus discos (cuatro sencillos) lanzados antes de ese año se convirtieron en discos de oro. vendiendo 500.000 copias.
Ese mismo año obtuvo el Gran Premio del Festival Internacional de la Canción de Dresde, en la República Democrática Alemana. Firmó un contrato de grabación con el sello alemán Amiga y lanzó su primer single, Mein Erste Mädchen, cantado en alemán, vendiendo 500.000 copias. En 1975 también fue elegido cantante del año de la RDA. Estos premios también fueron de gran importancia, dado que él y la cantante Katin Kovács, son los únicos cantantes húngaros que ganaron estos premios.
En 1977, fue el cantante principal del conjunto Bergendy durante unas semanas, dando un concierto en la gala del Festival Metronome '77. En 1979, se lanzó su primer álbum importante, Delhusa Gjon. Hasta 1980, participó en los principales festivales de Europa del Este, en los que obtuvo premios.
Ha escrito canciones para Goldie Ens y Goombay Dance Band, entre otros artistas. En 1980 firmó un contrato por tres años con la compañía discográfica de Alemania Occidental TELDEC, y en 1983 le ofrecieron otro contrato por tres años, pero con la condición de que se estableciera en Alemania.
En 1988, participó en un concurso de canciones en Cuba, donde fue el mejor compositor.
En 1989 se edita en casa el disco Csavargó a cargo del sello Rákóczi, que resultó ser un éxito. Se convirtió en disco de oro en pocos días, y dos años después ya era disco de platino, lo que significó 100.000 copias vendidas.
Delhusa tocó durante un corto tiempo con András Markó , un ex miembro de la banda Gemini, y luego se mudó a la RDA, donde trabajó durante varios años. En 1989, recibió el premio EMeRTon de la radio húngara.
En 1996, recibió la Cruz de Oro del Gobierno de Albania por su álbum Mediterranean. En 1997, se lanzó un CD en Curazao con las canciones ganadoras del Festival Cubano de 1988.
En 2002, alcanzó el millón de copias vendidas. Sus canciones más exitosas incluyen una canción escrita en memoria de su amigo de la infancia, Csaba Kesjár (Gyertyák a síron), un piloto de carreras que murió joven.
El 15 de marzo de 2005, recibió la Cruz de Oro al Mérito de la República de Hungría, y en abril de ese mismo año, fue distinguido con la Orden de Vitéz, como Miembro de la Orden de los Caballeros en la Iglesia de los Héroes de Dunaújváros.
Eurovisión
En 1996, Delhusa fue seleccionado a través de una final nacional para representar a Hungría en el Festival de la Canción de Eurovisión, celebrado ese año en Oslo, con la canción Fortuna, pero fue eliminado en la preselección de audio. Esta ronda pre-clasificatoria, que no fue transmitida de ninguna manera y cuyas votaciones completas tampoco se hicieron públicas, generó gran controversia, ya que junto con Hungría, Alemania, uno de los principales contribuyentes financieros del concurso, también fue eliminada, por lo que los organizadores se enfrentaron con muchos menos medios económicos de los esperados. Debido a esto, este peculiar sistema de preselección se eliminó después de ese año.
Premios y reconocimientos
1974 – Gran Premio del Festival de Dresde
1975 – Cantante del año en la RDA
1988 – Premio al Mejor Compositor del Concurso de la Canción Cubana
1989 – Premio EMeRTon
1996 – Cruz de Oro del Gobierno de Albania
2000 – Comunidad Gitana - Premio Cantante del Año
2005 – Cruz de Oro al Mérito de la República de Hungría
2005 – Miembro de la Orden de los Caballeros en la Iglesia de los Héroes de Dunaújváros - Orden de Vitéz
Discografía
1979 I.
1979 Flieg mit dem Wind
1983 Matina
1986 Delhusa Gjon
1989 Csavargó – aranylemez
1990 Átviszlek az éjszakán – aranylemez
1991 Mindenem a farmerem – aranylemez
1992 Tévelygő angyal
1993 Túl vakmerő – aranylemez
1994 Delhusa líra – platinalemez
1995 Mediterrán – aranylemez
1996 Rossz pénz
1997 Indián nyár
1999 Szerenád – aranylemez
2000 Líra kettő
2001 Kicsordul majd a szívem – aranylemez
2002 Szívek tengerén
2003 Bulizós slágerek – aranylemez
2004 Szeress még
2007 Best Of
2009 Fény
Referencias
Enlaces externos
Sitio web oficial de Gjon Delhusa
Ficha en music.eng
Allmusic.hu
Delhusa Gjon már nem csajozik annyit! ("¡Delhusa Gjon ya no coquetea tanto!") Nice.hu, 7 de diciembre de 2009.
Representantes de Hungría en Eurovisión
Participantes en Eurovisión 1996
Cantantes de Hungría
Cantantes en húngaro
Nacidos en Budapest | {
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<?php
namespace app\models;
use Yii;
use yii\base\Model;
use yii\data\ActiveDataProvider;
use app\models\Empresa;
/**
* EmpresaSearch represents the model behind the search form about `app\models\Empresa`.
*/
class EmpresaSearch extends Empresa
{
/**
* @inheritdoc
*/
public function rules()
{
return [
[['id', 'estado', 'cidade'], 'integer'],
[['razao_social', 'cnpj', 'inscricao_estadual', 'inscricao_municipal', 'logradouro', 'numero', 'cep', 'bairro', 'telefone1', 'telefone2', 'email', 'site', 'logo', 'responsavel_id'], 'safe'],
];
}
/**
* @inheritdoc
*/
public function scenarios()
{
// bypass scenarios() implementation in the parent class
return Model::scenarios();
}
/**
* Creates data provider instance with search query applied
*
* @param array $params
*
* @return ActiveDataProvider
*/
public function search($params)
{
$query = Empresa::find();
// add conditions that should always apply here
$dataProvider = new ActiveDataProvider([
'query' => $query,
]);
$this->load($params);
if (!$this->validate()) {
// uncomment the following line if you do not want to return any records when validation fails
// $query->where('0=1');
return $dataProvider;
}
// grid filtering conditions
$query->andFilterWhere([
'id' => $this->id,
'estado' => $this->estado,
'cidade' => $this->cidade,
]);
$query->andFilterWhere(['like', 'razao_social', $this->razao_social])
->andFilterWhere(['like', 'cnpj', $this->cnpj])
->andFilterWhere(['like', 'inscricao_estadual', $this->inscricao_estadual])
->andFilterWhere(['like', 'inscricao_municipal', $this->inscricao_municipal])
->andFilterWhere(['like', 'logradouro', $this->logradouro])
->andFilterWhere(['like', 'numero', $this->numero])
->andFilterWhere(['like', 'cep', $this->cep])
->andFilterWhere(['like', 'bairro', $this->bairro])
->andFilterWhere(['like', 'telefone1', $this->telefone1])
->andFilterWhere(['like', 'telefone2', $this->telefone2])
->andFilterWhere(['like', 'email', $this->email])
->andFilterWhere(['like', 'site', $this->site])
->andFilterWhere(['like', 'logo', $this->logo])
->andFilterWhere(['like', 'responsavel_id', $this->responsavel_id]);
return $dataProvider;
}
}
| {
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Q: Inverse of Braiding in Monoidal Category is Braiding In Mac Lane's Categories for the Working Mathematician in section $\mathrm{XI}.1$ about symmetric monoidal categories we find that a braided monoidal category $M$ is a monoidal category $(M,\oplus,e,\alpha,\lambda,\rho)$ where in addition to $\alpha, \lambda, \rho$ we have a natural equivalence $$\gamma_{a,b}:a\oplus b\cong b\oplus a$$ which must satisfy the following commutative diagrams:
$\qquad\qquad$
$\gamma$ is then called a braiding.
Now Mac Lane writes that the left hexagon for $\gamma$ implies the right hexagon for $\gamma^{-1}$, concluding that $\gamma^{-1}$ is a braiding as well
I don't really see how this works. Let's write $\beta$ for $\gamma^{-1}$, so that $\beta_{ab}=\gamma_{ba}^{-1}$. To turn the left diagram into (the shape of) the right, I replace the upper $\gamma$ by the inverse $\beta$ while the lower $\gamma$'s are replaced by the inverses of their opposites (instead of $\gamma_{bc}$ we have the inverse $\beta_{bc}$ of $\gamma_{cb}$. Now, the inner hexagon commutes, but I don't see why the outer hexagon should commute.
On the other hand, the outer hexagon commutes if and only if the following diagram commutes:
A: I don't have my copy of Cats Work at hand to check that those diagrams were correctly transcribed, but those are not the correct hexagon diagrams for a braiding. (They would be okay for a symmetry, where $\gamma_{ab}^{-1} = \gamma_{ba}$ by one of the axioms.) Edit: How do I know they're incorrect? Because they fail on the groupoid of braids!
The easy way to remember the correct diagrams is to pretend for a moment that the underlying monoidal category is strict monoidal (so that in particular the associativity constraints $\alpha$ are identities). On account of the coherence theorem for monoidal categories, this is no real loss of generality. In this case the axioms are (suppressing the $\otimes$ symbols)
$$(abc \stackrel{\gamma_{a,bc}}{\to} bca) = (abc \stackrel{\gamma_{a,b}c}{\to} bac \stackrel{b\gamma_{a,c}}{\to} bca),$$
$$(abc \stackrel{\gamma_{ab,c}}{\to} cab) = (abc \stackrel{a\gamma_{b,c}}{\to} acb \stackrel{\gamma_{a,c}b}{\to} cab),$$
which have very easily digested interpretations as braids. It's pretty clear in this simplified case that by taking inverses on the first condition, one gets the second condition applied to the transformation $\gamma^{-1}$, and vice-versa. And, it's not hard to reinsert associativities to get the corresponding correct hexagon identities, and check the claim there directly.
Now that I've taken a closer look at your post and see that you've posted images, I guess those were correctly "transcribed". This is a genuine error in the text in that case. Assuming that, what I think must have happened is this: the first edition of Cats Work was from the early 70's, well before braided monoidal categories were introduced, but certainly with material there on symmetric monoidal categories. As said before, the displayed images are fine for that case. Now, there are in fact several ways to present the hexagon identities for the symmetric monoidal case; with the "right" presentation, one can define a braiding just by retaining the hexagons and by dropping the axiom $\gamma_{a,b}\gamma_{b,a} = 1_{ba}$. But this won't work if you choose the "wrong" one. Mac Lane, who wrote the second edition when he was well into his 80's, might not have noticed the "wrong" one was used, and all this slipped through the cracks of the editorial process.
| {
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{"url":"http:\/\/www.solutioninn.com\/consider-again-the-banks-customer-loan-decision-problem-in-problem","text":"# Question\n\nConsider again the bank\u00e2\u20ac\u2122s customer loan decision problem in Problem 51. Suppose now that the bank\u00e2\u20ac\u2122s utility function of profit x (in dollars) is U(x) = 1 \u00e2\u20ac\u201c e-x\/500000. Find the strategy that maximizes the bank\u00e2\u20ac\u2122s expected utility in this case.\nHow does this optimal strategy compare to the optimal decision with an EMV criterion? Explain any difference between the two optimal strategies.\n\nSales0\nViews57\nComments0\n\u2022 CreatedApril 01, 2015\n\u2022 Files Included\nPost your question\n5000","date":"2016-10-27 19:40:01","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8468565940856934, \"perplexity\": 4701.263754490736}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988721387.11\/warc\/CC-MAIN-20161020183841-00265-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
{"url":"http:\/\/ieeexplore.ieee.org\/xpls\/icp.jsp?reload=true&arnumber=6594891","text":"We are currently experiencing intermittent issues impacting performance. We apologize for the inconvenience.\nBy Topic\n\n\u2022 Abstract\n\nSECTION I\n\n## INTRODUCTION\n\nWHISPERING gallery mode (WGM) based resonators have been used widely for biosensor applications [1], [2], [3], [4]. In most cases, the presence of the analyte changes the effective refractive index of the resonator, which in turn changes its resonance wavelength. As such resonance wavelength shifts can be monitored very precisely, a very high sensitivity can be achieved, right down to single-molecule levels [2].\n\nFor most cases, two key factors that determine the overall sensing performance are the Q factor of the resonator and mode overlap, which is defined to be the fraction of the optical energy within the region affected by the analyte. Increasing the Q-factor increases the precision of the resonance peak determination. However, while Q-factors that exceed $10^{6}$ or more have been reported [1], [2], such high values are difficult to achieve using standard fabrication processes, and can also result in limited dynamic range and sensitivity to environmental noise. Increasing the mode overlap improves the overall sensitivity and detection limit regardless of such vagaries of fabrication or sensing processes. However, the need to confine the optical mode in the resonator has led most WGM based biosensors to rely on evanescent fields for sensing, which can reduce the practically achievable mode overlap.\n\nOf the many structures that have been proposed to solve this problem of low mode overlap [4], [5], [6], [7], the slot structure, which consists of a low-refractive index region sandwiched between regions of high refractive index, has attracted a particular attention. The presence of slot not only opens up the access to the interior of optical mode, where the optical field is the highest, but can further enhance the optical field inside the slot due to the continuity requirement for the D-field [8], making it ideal for applications such as non-linear optical devices and biosensing [9], [10]. Furthermore, high Q-factors are possible with a slot structure due to its all-dielectric construction. Recently, we have demonstrated fabrication of microdisks with a 40-nm thin horizontal air slot with an intrinsic Q-factor of 34,000 [11]. Consequently, such a structure can provide more than a ten-fold increase in sensitivity over conventional, evanescent-field based WGM biosensors [3], [4]. However, the presence of air-slot complicates the optical properties of the resonator in a non-intuitive way. Thus, while the potential for highly sensitive biosensing using horizontal slot resonator has been demonstrated, the limits of its performance and the optimum design for biosensor have not yet been investigated.\n\nIn this letter, we report on the results of numerically investigating the effects of slot geometry on the performance of such horizontal slot microdisk resonators based biosensors. We find that the mode overlap is the limiting factor for the performance. Based on calculated results, we suggest that slot resonators offer advantage over conventional, evanescent-field based resonators primarily for the surface sensing that detects the modification of the sensing surface by the analyte, typically in units of nm\/nm (change in resonant wavelength per analyte thickness) or nm\/pg (change in resonant wavelength per analyte mass). This requires strong optical field near the surface. At the same time, possible mode mixing needs to be avoided by proper design of the resonator structure. We further propose an optimized triple-slot structure to double the possible detector sensitivity.\n\nSECTION II\n\n## THEORY\n\nWhile there have been previous reports on optimization of waveguide structures with vertical slots [12], [13], such results cannot be applied directly to horizontal air slot resonator structures. In addition to the obvious difference in the structure, the physical quantities that affect the sensitivity are different as well. Using variation methods, the change in the effective index of a waveguide upon a small change in the refractive index can be given by [14]. TeX Source $$\\Delta n_{eff}={{\\int_{M}{d\\nu\\varepsilon E^{2}}}\\over{\\int_{\\infty}{d\\nu\\varepsilon E^{2}}}}\\times n_{g}(n_{2}^{2}-n_{1}^{2})\/2n_{1}^{2}\\eqno{\\hbox{(1)}}$$ where M in the integral in the numerator denotes the region whose index has changed, and ${\\rm n}_{{\\rm g}}$ is group index. ${\\rm n}_{1}$ and ${\\rm n}_{2}$ are the refractive indices of region M before and after the change, respectively. Note that the $\\Delta {\\rm n}_{\\rm eff}$ is directly proportional to the group index. Thus, the effects of dispersion such as the slow light can greatly affect the performance of waveguide-based biosensors such as Mach-Zehnder interferometers [15] or photonic crystals. However, in case of resonators, the resonance wavelength is a function of the refractive index, and we must consider the effect of dispersion. As a result, group index does not appear in resonator peak shift, and we obtain TeX Source $$\\Delta\\lambda=Overlap\\times{{\\lambda (n_{2}^{2}-n_{1}^{2})}\\over{2n_{1}^{2}}},\\;Overlap={{\\int_{M}{d\\nu\\varepsilon E^{2}}}\\over{\\int_{\\infty}{d\\nu\\varepsilon E^{2}}}}\\eqno{\\hbox{(2)}}$$ In fact, (2) is equivalent to the expression $\\Delta\\omega=-\\omega\\left\\langle E\\right\\vert\\!\\Delta\\varepsilon\\!\\left\\vert E\\right\\rangle\/2\\!\\left\\langle E\\right\\vert\\varepsilon\\!\\left\\vert E\\right\\rangle$ that was reported in [16], since $\\Delta\\omega\/\\omega=\\;-\\Delta\\lambda\/\\lambda$ and $\\varepsilon={\\rm n}_{1}^{2}$ at region of M.\n\nIn short, for a resonator-based biosensor, the only external factor that determines the shift in the resonance peak is the mode overlap, defined to be the fraction of optical energy in the region containing the analytes, as the difference between ${\\rm n}_{2}$ and ${\\rm n}_{1}$ is determined by sensing material. Thus we conclude that for whispering gallery mode based biosensors, the mode overlap optimization equals to sensor optimization. Henceforth, we will focus on design rules for optimizing the overlap.\n\nSECTION III\n\n## SIMULATION\n\nThe optical modes of horizontal slot disk structures were simulated using finite difference time-domain (FDTD) method. In all cases, a horizontal slot microdisk that consists of two 8 $\\mu{\\rm m}$ diameter Si disks, one on top of the other separated by a gap that forms the slot, and suspended in a low index matrix of either air or water, was used unless otherwise specified. Si was chosen for its high refractive index compatibility with standard microfabrication techniques, and possible ultra-high Q-factors [17]. The diameter was fixed at 8 $\\mu{\\rm m}$ to be small enough for efficient simulations, but large enough for low radiation losses. For optimization, the thickness of the slot was varied from 10 to 100 nm, while the thickness of the Si disks was varied from 100 to 200 nm. A minimum slot thickness of 10 nm was chosen to be larger than typical protein molecules [18]. The Si disks were chosen to be thick enough for reasonable optical confinements, but thin enough that for the TM modes only the zeroth order vertical modes are supported. A schematic description of the structure is given below in Fig.\u00a01. Clearly, such a suspended structure is not physically realizable. However, we have previously shown, both theoretically and experimentally via evanescent coupling using tapered fibers, that a pedestal type resonator with a concentric ${\\rm SiO}_{2}$ spacer layer between the two disks can accurately mimic such a suspended structure [11].\n\nFig.\u00a01. Simulation structure. The diameter was fixed at 8 $\\mu{\\rm m}$. Thickness of the Si disks was varied from 100 to 200 nm and thickness of the slot was varied from 10 to 100 nm. Disks are surrounded by air or water. Perfectly matched layer is used for absorbing boundary condition.\n\nFig.\u00a02(a) and (b) show the slot-mode overlap, defined to be the fraction of the optical energy in the slot region, of a disk in air and water, respectively. Fig. 2(c) and (d) show the homogeneous-mode overlap, defined to be the fraction of the optical energy in the entire environment surrounding the resonator, of a disk in air and water, respectively. For accurate comparisons between different resonators, we calculated every ${\\rm TM}_{1,0}$ (first-order radial and zeroth order vertical TM mode) slot-modes in the 1510 to 1570 nm range for a given resonator structure, and used the largest overlap value. Shaded regions indicate regions where the Q-factor became too low (below 700) to be reliable for calculation of mode overlap, and were discarded from further investigations. We note that a Q-factor of only 330 was shown to be sufficient for biosensing using slot-type ring resonators [19]. Thus, low Q-factors are not expected to be a problem for any mode we investigate here. The values of Q-factors, corresponding propagation loss, and effective refractive indices of the modes that were investigated ranged $10^{3}\\hbox{--}10^{7}$, 0.05\u2013200 db\/cm, and 1.2\u20132.5, respectively.\n\nFig.\u00a02. (a) and (b): Slot-mode overlap in air and water. Slot-mode overlap remains similar for a wide range of combination of disk and slot thicknesses except for the presence of periodic dips. (c) and (d): Homogeneous-mode overlap in air and water. Homogeneous-mode overlap decreases monotonically with increasing Si disk thickness.\n\nSomewhat surprisingly, Fig. 2(a) and (b) show that, except for the presence of periodic dips that will be discussed later, the slot-mode overlap remains largely the same for a wide range of combination of disk and slot thicknesses. Still, the largest slot-mode overlap is obtained when the Si disks are relatively thick. This is due to pulling of the evanescent fields from above and below the microdisk into the disk by the increased effective refractive index. But if the Si disks are too thick, then the fraction of the optical mode confined inside the disks becomes too large, reducing the slot-mode overlap. Conversely, this pulling-in of the evanescent field reduces the homogeneous-mode overlap such that it decreases monotonically with increasing Si disk thickness. A similar phenomenon has been reported for waveguide sensors as well [12].\n\nThe largest values for the slot-mode overlap in air and water are 30.4 and 32.3%, respectively. However, they are smaller than the largest values obtained for homogeneous-mode overlap, which were 60.9 and 68.1%. It is smaller than even the homogeneous-mode overlap calculated for a conventional, 200 nm thick Si disk resonator in water, which is 47.8% for a mode with the same azimuthal number, similar resonant frequency, and Q-factor. This indicates that for homogeneous sensing methods which probe the entire environment, a slot disk resonator offers only a little advantage in sensitivity, and none if only the slot region is used for sensing, over a conventional disk resonator.\n\nOn the other hand, the slot structure provides a significant advantage for surface sensing methods, which probe only the nm-thin layer at the surface. As shown in Fig. 3(a) and (b), the energy density in the slot, obtained by dividing the slot overlap by the slot thickness, increases monotonically with decreasing the slot thickness (except for the presence of shallow, periodic dips). As a result, a slot resonator with 10 nm thick slot and Si disks that are thicker than 150 nm will have as much as 19.5% of its optical mode within the 5 nm thin layer at the slot surface. In contrast, a conventional, 200 nm thick Si disk resonator in water will have only 1.87% of its optical mode within the 5 nm thin layer at the surface, which would result in a surface sensitivity that is 10 times weaker.\n\nFig.\u00a03. Energy density in the slot in air (a) and water (b), obtained by dividing the slot overlap by the slot thickness. Energy density increases monotonically with decreasing the slot thickness except for the presence of periodic dips.\n\nWe now investigate the cause of periodic dips in the overlap. To do so, we calculate the slot-thickness dependence of resonance wavelength of other modes whose resonance frequency lies near that of the ${\\rm TM}_{1,0}$, and has the same azimuthal number. The Si disk thickness was set to be 200 nm, and the resonator is taken to be immersed in water. As shown in Fig.\u00a04, we see that at the slot thickness of 42.5 nm, where the dip in the slot-mode overlap occurs as shown in Fig.\u00a02(b), we have anti-crossing of the ${\\rm TM}_{1,0}$ mode with the ${\\rm TE}_{1,1}$ mode, indicating mode-mixing. As TE modes contain more of the E-field inside the dielectric, such mixing reduces both the slot- and homogeneous-overlap, producing the observed dips.\n\nFig.\u00a04. Slot-thickness dependence of resonant wavelength WGM modes whose resonance frequency lies near that of the fundamental ${\\rm TM}_{1,0}$, and has the same azimuthal number. The Si disk thickness was set to be 200 nm, and the resonator is taken to be immersed in water.\n\nHowever, no such mixing is observed at the slot thickness of 33 nm, where the ${\\rm TM}_{1,0}$ mode is degenerate with ${\\rm TE}_{3,0}$ mode. This difference is ascribed to the symmetries of the field distribution of ${\\rm TM}_{1,0}$, ${\\rm TE}_{1,1}$, and ${\\rm TE}_{3,0}$ modes. As shown in Fig.\u00a05, ${\\rm TM}_{1,0}$ and ${\\rm TE}_{1,1}$ modes have the same symmetry of axial and radial field in the vertical direction even when far away from the crossing. Consequently, ${\\rm TE}_{1,1}$ mode mixes readily with the ${\\rm TM}_{1,0}$ mode when the k-vector matching condition is satisfied. On the other hand, ${\\rm TE}_{3,0}$ mode has a completely different symmetry, and thus does not mix with the even when the k-vector is matched. Such similar symmetry exists between ${\\rm TM}_{1,0}$ and ${\\rm TE}_{2,1}$ mode as well, leading to mode mixing and formation of the second dip at slot thickness of 85 nm for 200 nm thick Si slot disks in the water.\n\nFig.\u00a05. Axial and radial field distribution of ${\\rm TM}_{1,0}$, ${\\rm TE}_{1,1}$ and ${\\rm TE}_{3,0}$ modes. Note that the ${\\rm TM}_{1,0}$ and ${\\rm TE}_{1,1}$ modes have the same symmetry of axial and radial field distribution in vertical direction.\n\nPractically, this means that when designing a slot disk resonator for bio-sensing applications, care should be taken to avoid the conditions under which such mode mixing can occur. One way to reduce the possibility of such mode mixing is lowering of the refractive index contrast, as such reduction in index contrast increases the spacing between the dips. This is already demonstrated in Fig. 2(a) and (b) that show 3 dips for Si slot disks in air, but only 2 for Si disks in water. However, that reducing the refractive index contrast will lead to reduction of the optical confinement in the slot and consequent reduction in the sensitivity of the resonator.\n\nSECTION IV\n\n## SIMULATION OF MULTI SLOT DISK\n\nFinally, we note that by adding more slots to the disk, we can increase the sensing surfaces inside the slot that can utilize the confined optical field. Previous investigations into multiple slot ring resonators have demonstrated that presence of additional slot does not significantly affect either the coupling or the surface scattering losses [20]. On the other hand, doing so in effect increases the slot thickness, which reduces the optical field density. Fig.\u00a06(a) shows the calculated enhancement of surface sensitivity of double and triple-slot structure over a single-slot structure, as a function of separation between the slots. In this simulation, the slot thickness was fixed at 10 nm which gave the best single-slot performance and total Si thickness are fixed at 280 nm and 370 nm for disks in air and water. Fig.\u00a06(b) shows typical ${\\rm E}^{2}$ distribution of double and triple-slot disks. We find that increasing the number of slots does help. Interestingly, the maximum enhancement is obtained not when the slots are close together in the middle, where the E-field intensity is expected to be the highest, but when the slots are separated by a well-defined optimum. The effect of additional slot is larger when the resonator is in water. The maximum value of enhancement observed is 2.0 for a triple slot in water, and 1.6 for triple slots in the air. In fact, for a resonator in water, a simple double-slot can increase the surface sensitivity by a factor of 1.6.\n\nFig.\u00a06. (a) Enhancement of surface sensitivity of 10 nm double and triple-slot structure over a single-slot structure, as a function of slot interval. The maximum value of enhancement observed is 2.0 for a triple slot in water. (b) ${\\rm E}^{2}$ distribution of double- and triple-slot disks.\nSECTION V\n\n## CONCLUSION\n\nIn conclusion, we have investigated sensitivity of slot disks through the overlap. For homogeneous sensing, a slot structure provides little advantage over conventional, non-slotted disk resonator with similar Q factor. On the other hand, the slot structure provides a significant advantage for surface sensing methods, with the surface sensitivity increasing continuously with decreasing slot thickness for an order-of magnitude enhancement over a conventional, non-slotted disk. Possible mode mixing needs to be avoided by proper design of the resonator structure. Finally, adapting multislots we further enhance the surface sensitivity. Maximum value of enhancement observed is 2.0 for a triple slot in water.\n\n## Footnotes\n\nThis work was supported in part by the National Research Foundation of Korea Grant Funded by the Korea Government under Grant 2010-0029255 and in part by the Basic Science Research Program under Grant 2009-0087691.\n\nS. C. Eom is with the Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea.\n\nJ. H. Shin is with the Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea, and also with the Graduate School of Nanoscience and Technology, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail: jhs@kaist.ac.kr).\n\nColor versions of one or more of the figures in this paper are available online at http:\/\/ieeexplore.ieee.org.\n\n## References\n\nNo Data Available\n\n## Cited By\n\nNo Data Available\n\nNone\n\n## Multimedia\n\nNo Data Available\nThis paper appears in:\nNo Data Available\nIssue Date:\nNo Data Available\nOn page(s):\nNo Data Available\nISSN:\nNone\nINSPEC Accession Number:\nNone\nDigital Object Identifier:\nNone\nDate of Current Version:\nNo Data Available\nDate of Original Publication:\nNo Data Available","date":"2015-02-28 19:35:18","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\": 0.624580979347229, \"perplexity\": 986.6957739562315}, \"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-2015-11\/segments\/1424936462035.86\/warc\/CC-MAIN-20150226074102-00067-ip-10-28-5-156.ec2.internal.warc.gz\"}"} | null | null |
Redunca è un genere di mammiferi artiodattili della famiglia Bovidae noti come redunche o antilopi dei canneti, presenti in Africa.
Tassonomia
Comprende tre specie:
Note
Altri progetti
Collegamenti esterni
Bovidi | {
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\section{Introduction}
Foams and related physical systems (like emulsions, biological tissues, or polycrystals) are ubiquitous, and serve as a paradigm for a wide range of physical phenomena and mathematical problems \cite{WeaireBook1,WeaireBook2,Morgan01,Morgan02}.
One of them deals with the general topological and geometrical properties of cellular materials \cite{Quilliet,Quilliet2,Delannay,Hilgenfeldt1,Hilgenfeldt2,Kafer,Hilgenfeldt3,Kraynik}.
In this connection, Quilliet \textit{et al.} \cite{Quilliet,Quilliet2} studied recently the topological features of two-dimensional (2D) soap froths under slow oscillatory shear. Such macroscopic strain induces rearrangements within the foam, and the number of sides of every cell evolves in time through local topological changes ($T1$ events) \cite{WeaireBook1,Cox1,Durand1}. Nevertheless, Quilliet \textit{et al.} reported the existence of an equilibrium state after few cycles, characterized by a stationary probability distribution of the number of sides per cell (topological disorder). They also showed that the width of this distribution of sides is strongly correlated to the distribution of bubble sizes (geometrical disorder) within the foam.
These results suggest that the macroscopic state of a homogeneously sheared foam can be adequately described using the ideas and formalism of statistical thermodynamics.
Indeed, the pioneering work of Edwards on granular matter \cite{Edwards} has shown how the powerful arsenal of statistical physics can be extended to athermal systems. This method has proven its applicability to other fields as well \cite{divers}.
Because there is no thermal averaging due to Brownian motion, this approach requires the presence of a macroscopic agitation (analogue to an effective temperature) allowing the system to explore its entire phase space.
Various attempts have been made in the past to describe the geometrical and topological properties
of 2D foams using the concepts of statistical thermodynamics \cite{Schliecker1,Schliecker2,Rivier1,Rivier2,Fortes3,Iglesias}.
However, these former theoretical approaches
rely on strong assumptions: either they use an ad-hoc interaction potential between bubbles \cite{Schliecker1,Schliecker2}, involve (rather than deduce) empirical laws correlating size and side distributions \cite{Rivier1,Rivier2,Fortes3}, or ignore some geometrical constraints (for instance, only the mean bubble area is specified, not the individual bubble areas \cite{Rivier1,Rivier2,Iglesias}).
Some of these models are based on the maximum entropy (information theory) formalism \cite{Rivier1,Rivier2}, which has been subject to controversy \cite{Chiu}. Other models invoke minimisation of the energy \cite{Fortes3}, or a combination of both principles \cite{Iglesias} to describe the state of a foam. However, it has been established \cite{Weaire1, Graner1, Vaz1, Jiang, Kraynik} that different arrangements (topologies) of a large number of bubbles do not really affect the energy (see discussion below).
Furthermore, none of these models can account for the correlations between topological and geometrical disorders reported by Quilliet \textit{et al.} \cite{Quilliet,Quilliet2}.
In this letter we set up a framework for describing the equilibrium state of a two-dimensional foam, basing our development on analogies with conventional statistical mechanics.
As for other athermal systems \cite{Edwards, divers}, we show that the energy is not relevant to describe the macroscopic state of a foam. Instead, a more appropriate state variable is introduced: the total cell curvature.
We establish the function of state which is minimized for a finite cluster of bubbles at equilibrium. This thermodynamic potential function differs from entropy or energy used in previous theories.
The formalism developed here allows to derive an analytical expression for the distribution of the number of sides.
We show that the semi-empirical expression conjectured by
other authors \cite{Schliecker1,Schliecker2,Sherrington1,Sherrington2} is recovered, in a certain limit.
Finally, the bubble size-topology correlations deduced from the present theory are investigated, and compared with experimental data.
\section{Physical, geometrical, and topological constraints}
Consider a given set of $N_{B}$ bubbles with prescribed areas $\lbrace A_{i} \rbrace$ (we focus on time scales much shorter than those typical of bubble coarsening and coalescence, so the bubbles preserve their integrity and size\footnote{Rigorously, the area and pressure of a cell vary from one configuration to another; only the number of molecules of gas it contains remains unchanged. We consider that the area fluctuations are small so that each bubble can be identified by its area.}).
A 2D foam is a partition of the plane without gaps or overlaps, and its structure must obey certain constraints\cite{WeaireBook1} (one considers the dry foam limit where liquid volume fraction is negligible).
The physical constraints follow from the mechanical equilibrium throughout the system:
first, the balance of film tensions at every vertex implies that the edges, or Plateau borders, are three-connected making angles of $120^{\circ}$ with each other (Plateau's laws).
Then, the balance of gas pressures in adjacent bubbles implies that every edge is an arc of circle, whose algebraic curvature $\kappa_{ij}$ is proportional to the pressure difference between the two adjacent bubbles $i$ and $j$ (Laplace's law):
\begin{equation}
\kappa_{ij}=-\kappa_{ji}=\dfrac{P_j-P_i}{\gamma},
\label{Laplace}
\end{equation}
where $\gamma$ is the film tension,
and $P_i$ and $P_j$ are the pressures in bubble $i$ and $j$, respectively (by convention, $\kappa_{ij} \geq 0$ when the center of curvature is outside the cell $i$, i.e.: when $P_j \geq P_i$).
As a consequence, the algebraic curvatures of the three edges that meet at the same vertex must add to zero:
\begin{equation}
\kappa_{ij}+\kappa_{jk}+\kappa_{ki}=0.
\label{Adjacent Laplace}
\end{equation}
The foam must also satisfy topological and geometrical requirements:
apart from the constraint of prescribed areas, its structure must obey Euler's rule, which relates the number of bubbles $N_B$, with those of Plateau borders $N_{Pb}$, and vertices $N_v$: $N_B+N_v-N_{Pb}=c$, where $c$ is the Euler-Poincar\'{e} characteristic ($c=0$ for a torus, $c=2$ for an infinite plane).
This rule, combined with Plateau's laws, immediately gives:
\begin{equation}
N_{Pb}=3(N_{B}-c).
\label{Euler-Plateau}
\end{equation}
Finally, the Gauss-Bonnet theorem applied to a $n$-sided cell yields $\sum_{j=1}^n l_{ij} \kappa_{ij}=\pi (n-6)/3$, where $l_{ij}$ denotes the length of the edge common to bubbles $i$ and $j$.
\section{Microcanonical ensemble $\left(N_{B},\kappa_{tot},N_s \right)$}
In statistical mechanics, the most fundamental entry is certainly via the microcanonical ensemble. Suppose that the foam is agitated slowly as
compared to the characteristic relaxation time after a $T1$ process \cite{WeaireBook1,Cox1,Durand1} (one considers only perturbations that preserve the area of every bubble, and so the total foam area). Then, the deformation of the foam is quasistatic: its structure evolves through configurations which always satisfy the physical, geometrical and topological constraints stated above. Such configurations will be referred to as \textit{accessible microstates}.
By analogy with the fundamental postulate of statistical mechanics \cite{Books}, we hypothesize that all the accessible microstates of a given set of bubbles filling the 2D space have equal probability, under a slow macroscopic agitation.
Obviously, a periodic or infinite foam fills the 2D space. It must be pointed out that the postulate applies to a free cluster too, provided that the surrounding air is included as a supplementary bubble. Then, this ``extra bubble" must also be taken into account in the counting of the number of accessible microstates. In the following, an infinite or periodic foam, or a free cluster plus the surrounding air shall be referred to as an \textit{unbounded foam}. Conversely, a free finite cluster, a cluster within a larger foam, or a foam enclosed in a container shall be referred to as a \textit{bounded foam}. Figure \ref{boundary_conditions} summarizes the different kinds of boundaries that exist for a 2D foam.
\begin{figure}[htb]
\begin{center}
\begin{overpic}[width=7.8cm]{boundary_conditions.eps}
\put(-6,55){\textbf{(a)}}
\put(50,55){\textbf{(b)}}
\put(-6,2){\textbf{(c)}}
\put(50,2){\textbf{(d)}}
\end{overpic}
\caption{Various situations for the boundary of a 2D foam. \textbf{(a):} infinite or periodic foam. \textbf{(b):} free cluster (figure taken from \cite{Graner1}). \textbf{(c):} cluster of bubbles within a larger foam. \textbf{(d):} foam enclosed in a container. Situation \textbf{(a)} is referred to as an unbounded foam, while situations \textbf{(c)} and \textbf{(d)} are referred to as bounded foams. Situation \textbf{(b)} is either an unbounded or bounded foam, depending on whether or not the surrounding air is included as a supplementary bubble.}
\label{boundary_conditions}%
\end{center}
\end{figure}
Rigorously, the accessible microstates do not all correspond to the same total surface energy. It is tempting, by analogy with the statistical mechanics of a gas, to restrict the equiprobability hypothesis to the microstates of equal energy.
This refinement appears to be unnecessary: previous studies have shown that different arrangements of a large number of bubbles of given areas do not affect the energy very much\footnote{Actually, there is also some uncertainty on the value of the energy of an isolated volume of gas \cite{Books}.}: in their computational studies of 2D foams under shear, Jiang \textit{et al.} \cite{Jiang} reported energy fluctuations less than $2\%$. Graner \textit{et al.} \cite{Graner1} obtained the same results, both numerically and experimentally: the energy values of the different metastable states of a 2D foam lie within $2\%$ of the ``ground state" value. Kraynik \textit{et al.} \cite{Kraynik} conducted similar studies on 3D foams, and reported fluctuations below $4\%$ for the surface energy of every bubble.
These observations are also consistent with the so-called ``equation of state" of a foam \cite{Graner1,Ross,Aref,Fortes2}, which relates surface energy $E$, areas $A_i$, and pressures $P_i$ within a cluster of bubbles:
$ E=\sum_{i}\left( P_{i}-P_{0}\right) A_{i} $, where $P_{0}$ is the external pressure. In the limit of fixed bubble pressures, the total surface energy of such a cluster would be conserved.
Hence, to a first approximation, one can reasonably assume that the energy of a large cluster does not depend on its configuration, but is directly determined from the number of bubbles $N_B$ and the area distribution $p(A)$.
On a macroscopic scale, the state of the foam should be described by a limited number of independent variables (besides the number of bubbles and the size distribution). These quantities must be ``constants of motion" \cite{Books}, i.e: they must keep constant values throughout the dynamics of the system (here, a slowly agitated foam). An important observation is that one can define two independent quantities which are conserved for an unbounded foam.
The first one is the \textit{total number of sides} $N_s=\sum_{i}n_{i}$, where $n_{i}$ is the number of sides of bubble $i$. One has $N_s=2N_{Pb}$ for an unbounded foam ($N_s$, and not $N_{Pb}$, is an extensive and fluctuating variable for a finite cluster). Thus, according to Eq. (\ref{Euler-Plateau}), $N_s=6(N_B-c)$.
The second one is the \textit{total cell curvature} $\kappa_{tot}= \sum_{i}\kappa_{i}$, where $\kappa_{i}$ is the \textit{cell curvature} of bubble $i$, defined as:
\begin{equation}
\kappa_{i}=\sum_{j \in \mathcal{N}(i)} \kappa_{ij},
\label{cell_curvature}
\end{equation}
$\mathcal{N}(i)$ denoting the neighbouring cells of bubble $i$.
Obviously, $\kappa_{tot}$ is additive.
Since $\kappa_{ij}=-\kappa_{ji}$, the terms cancel in pairs in the double sum, yielding $\kappa_{tot}=0$ for an unbounded foam.
The property $\kappa_{ij}=-\kappa_{ji}$ correlates the two adjacent bubbles that share a common edge. It is also important to note that -- regardless of this mathematical property -- the constraint $\kappa_{tot}=0$ is also imposed by the curvature sum rule (\ref{Adjacent Laplace}), as it is illustrated on Fig. \ref{sumrule}. This rule, which correlates the three adjacent bubbles that share a common vertex, has a physical origin (Laplace's law).
Although it is unclear whether $N_s$ and $\kappa_{tot}$ are the only constants of motion for an unbounded foam, the constraints $N_s=6(N_B-c)$ and $\kappa_{tot}=0$ must be taken into account in the statistical description of such a system. Surprisingly enough, the constraint on the total curvature has always been ignored in previous theoretical models \cite{Schliecker1,Schliecker2,Rivier1,Rivier2,Fortes3,Iglesias}.
\begin{figure}[htb]
\centering
\begin{overpic}[width=5cm]{curvature_sum_rule.eps}
\end{overpic}
\caption{Illustration of the equivalence (for an unbounded foam) between the sum (over all vertices) $\sum_\alpha S_\alpha$ and the sum (over all cells) $\kappa_{tot}=\sum_i \kappa_i$, where $S_\alpha$ denotes the sum -- turning clockwise -- of the algebraic curvatures $\kappa_{ij}+\kappa_{jk}+\kappa_{ki}$ between the three bubbles $i$, $j$, $k$ that share the same vertex $\alpha$, while $\kappa_i$ denotes the cell curvature of bubble $i$ (see Eq. (\ref{cell_curvature})). Each term $\kappa_{ij}$ in the first sum is represented by an arrow pointing from $i$ to $j$. The equivalence between the two sums is immediate: the cell curvature $\kappa_i$ of cell $i$ is represented by the collection of arrows coming out of that cell. For an unbounded foam at mechanical equilibrium, the curvature sum rule (\ref{Adjacent Laplace}) yields $S_\alpha=0$ at every vertex $\alpha$, and thus $\kappa_{tot}=0$. \label{sumrule}}
\end{figure}
In the light of this discussion, we may restate the postulate as:
all the accessible states of a given set of $N_B$ incompressible cells corresponding to the same values of $N_{s}$ and $\kappa_{tot}$ are equally probable, under macroscopic agitation. By analogy with thermal physics, the \textit{microcanonical ensemble} $\left(N_{B},\kappa_{tot},N_s \right)$ refers to a large number of copies of a foam (with given bubble size distribution p(A)) whose parameters $N_{B}$, $\kappa_{tot}$, and $N_s$ are fixed.
\section{Thermodynamic limit}
For a bounded foam, $\kappa_{tot}$ reduces to the sum of the side curvatures along its boundary. Both $N_s$ and $\kappa_{tot}$ are fluctuating variables, although their values are usually restricted within a certain range. For instance, for a free cluster, $N_s<2N_{Pb}$ and $\kappa_{tot}<0$ since the pressure in any bubble of the cluster is higher than the external pressure \cite{Vaz3}.
For a cluster within a larger foam, $N_s<2N_{Pb}$, but $\kappa_{tot}$ can be positive or negative. For a foam enclosed in a container, $\kappa_{tot}=0$ (the container walls can be regarded as zero curvature sides), but $N_s$ fluctuates.
One can argue whether or not the fluctuations of $N_s$ and $\kappa_{tot}$ become negligible in the thermodynamic limit ($N_B \rightarrow \infty$, total surface area $A_{tot} \rightarrow \infty$, but $\langle A \rangle=A_{tot}/N_B$ remains finite).
The average number of sides of a large cluster ($N_B \gg 1$) scales as $\langle N_{s}\rangle \sim N_B$, while the total curvature, equal to the sum of the side curvatures along the cluster boundary, scales as $\langle \kappa_{tot} \rangle \sim \sqrt{N_B/\langle A \rangle}$.
The relative standard deviations of such quantities scale as $N_B^{-1/2}$ \cite{Books}. Thus, the fluctuations of $N_s$ and $\kappa_{tot}$ become vanishingly small as $N_B \rightarrow \infty$. Moreover, their thermodynamic limit values are known: $\left\langle N_s \right\rangle / N_B \rightarrow 6$, and $\left\langle \kappa_{tot} \right\rangle / N_B \rightarrow 0$.
Note however that $\langle N^{*}_{s}\rangle/ N^{*}_B \sim {N^{*}_B}^0$ converges much faster than $\langle \kappa_{tot}^{*} \rangle /N^{*}_B \sim {N^{*}_B}^{-1/2}$.
\section{Idealized foam}
In order to obtain the probability distribution of the number of sides for a cell of given size, we need to enumerate the microstates which correspond to the same macroscopic state. Ideally, a microstate is specified by the curvature, length, position and orientation of each Plateau border.
However, this description is too cumbersome to handle. Moreover, even for a given foam topology, the number of different accessible microstates depend in a non-trivial way on boundary conditions (e.g.: free, periodic, or enclosed cluster) and bubble size distribution \cite{Herdtle, Fortes1, Vaz2, Graner1, Weaire2}.
\begin{figure}[htb]
\begin{center}
\begin{overpic}[width=7.8cm]{dessin5.eps}
\put(17,39){$i$}
\put(24,39){$j$}
\put(26,32){$k$}
\put(65,36){$i$}
\put(88,30){$j$}
\put(73,14){$k$}
\put(-5,1){\textbf{(a)}}
\put(47,1){\textbf{(b)}}
\end{overpic}
\caption{\textbf{(a):} Real two-dimensional foam under shear. \textbf{(b):} Idealization of the foam (mean-field approximation): adjacent bubbles $i$, $j$ and $k$ are now regular cells surrounded by an effective foam and disconnected from each other.\label{2dfoams}}
\end{center}
\end{figure}
We shall use a simplified microscopic description of the foam:
we consider that for each particular bubble $i$, the rest of the foam can be replaced
on average by a ``mean field" of identical bubbles. Thus, the bubble $i$ is a regular cell with identical curved sides joining at each vertex with angles of $120^\circ$, as sketched on Fig. \ref{2dfoams}b.
The cell curvature of a $n$-sided regular cell with area $A$ is \cite{Iglesias,Graner1}:
\begin{equation}
\kappa\left(A,n \right) =\dfrac{\pi}{3}\dfrac{n \left(n-6 \right)}{e(n)\sqrt{A}},
\end{equation}
where $e(n)=\frac{\pi \mid n-6 \mid}{3 \sqrt{n}} \left( \frac{\pi}{n} - \frac{\pi}{6} - \frac{\sin (\pi/n - \pi/6) \sin(\pi/6)}{\sin(\pi/n)} \right)^{-1/2}$ is the elongation of the cell (ratio of perimeter to square-root of area). $e(n)$ is a slowly decreasing function, lying between $e(3)\simeq 3.74$ and $e(\infty)\simeq 3.71$ \cite{Graner1}.
It can be noticed that while the (surface) energy of such a bubble ($\sim e(n)\sqrt{A}$) is almost independent of $n$, its cell curvature $\kappa\left(A,n \right)$ increases rapidly with $n$, and is not upper-bounded.
In this idealized foam description, each bubble is ``disconnected" from the others.
By construction, all the physical, geometrical and topological constraints are satisfied, except the curvature sum rule (\ref{Adjacent Laplace}) and the Euler-Plateau relation (\ref{Euler-Plateau}), which both involve adjacency of the bubbles.
Consistent with the discussion above, the \textit{accessible} microstates of a foam tiling the entire plane are those which satisfy $N_s=6(N_B-c)$ and $\kappa_{tot}=0$ (the number of bubbles is large enough so $N_s$ and $\kappa_{tot}$ can be treated as continuous variables).
A microstate of the idealized foam is specified by the numbers of sides $\left\lbrace n_i \right\rbrace $ of all the bubbles.
Hence, the number of accessible microstates is obtained by enumerating the distributions of the $N_s=6(N_B-c)$ sides over the $N_{B}$ bubbles which satisfy the constraint $\kappa_{tot}=0$. This number is not easy to evaluate, and a grand-canonical description shall be more appropriate.
\section{Grand-canonical ensemble $\left(N_{B},\beta^{-1},\mu \right)$}
Consider then a sample of $N^{*}_B$ bubbles in the unbounded foam (the asterisk denotes the variables of this grand-canonical ensemble). This sample can be a cluster of bubbles, or a collection of isolated bubbles. Let $p^{*}(A)$ be the distribution of bubble size within this sample.
Although $N^{*}_B$ is fixed, this system can exchange sides and curvature with the rest of the foam, through $T1$ events. Hence, the total number of sides $N^{*}_s$ and the total curvature $\kappa^{*}_{tot}$ are now internal variables free to fluctuate for this system\footnote{$N^{*}_s$ and $\kappa^{*}_{tot}$ are independent variables: there are different ways of distributing $N^{*}_s$ sides over $N^{*}_B$ bubbles; each of these distributions corresponds to a different value of $\kappa^{*}_{tot}$.}.
We assume that possible other variables required to describe the macroscopic state remain fixed. Surface energy in particular does not fluctuate, since it depends only on $N^{*}_B$ and the distribution $p^{*}(A)$, and not on the specific configuration of the system.
Suppose that the rest of the foam is large in comparison with the system, so that it constitutes a reservoir of sides and curvature.
Using the formalism of conventional statistical mechanics \cite{Books}, it comes that the probability for the system to be in the microstate $\left( n_1,n_2,\ldots,n_{N^{*}_{B}} \right)$
is proportional to $e^{-\beta \kappa^{*}_{tot} +\mu N^{*}_s}$,
with $\kappa^{*}_{tot}=\sum_{i=1}^{N^{*}_B} \kappa(A_i,n_i)$
and $N^{*}_s=\sum_{i=1}^{N^{*}_B} n_{i}$.
$\beta^{-1}$ and $\mu$ (rigorously, $\mu \beta^{-1}$) denote respectively the ``temperature" of the reservoir of curvature, and the ``chemical potential" of the reservoir of sides. A large number of copies of a foam whose parameters $\left(N_{B},\beta^{-1},\mu \right)$ are fixed shall be referred to as a \textit{grand-canonical ensemble}.
As noticed before, the cell curvature of a bubble is not upper-bounded as $n \rightarrow \infty$, what ensures that the temperature $\beta^{-1}$ is always positive \cite{Books}.
Finally, the probability for a given cell of size $A$ to have $n$ sides is:
\begin{equation}
p_{A}(n)=\chi^{-1}(A) e^{-\beta \kappa(A,n) +\mu n},
\label{distribution1}
\end{equation}
where $\chi(A)= \sum_{n \geq 3} e^{-\beta \kappa(A,n) +\mu n}$ denotes the partition function of the cell.
The average total cell curvature and average number of sides are, respectively,
$\langle \kappa^{*}_{tot}\rangle=-\partial \ln \Xi / \partial \beta$
and $\left\langle N^{*}_{s}\right\rangle=\partial \ln \Xi / \partial \mu$,
with $\ln \Xi=N^{*}_B \int_{0}^{\infty}p^{*}(A) \ln \chi(A) dA$. $\Xi$ is the partition function of the system.
Using the formalism of conventional statistical mechanics \cite{Books}, one concludes that the thermodynamic equilibrium of a foam in contact with a reservoir of curvature and sides coincides with the minimum of the thermodynamic potential $\Phi=-\beta^{-1}\ln \Xi$.
The analogy between the statistical descriptions of an ideal gas and a 2D foam is summarized in Table \ref{tableau}.
As noted by Graner and coworkers \cite{Graner1}, the elongation of a regular cell is almost constant: $e(n)\simeq \overline{e} \simeq 3.72$. With this simplification, the total surface energy of the cluster is strictly conserved (i.e., it does not depend on the cluster configuration). Besides, the distribution (\ref{distribution1}) simplifies to
\begin{equation}
p_{A}(n)=\chi'^{-1}\left( A \right) e^{-\beta'(n-6)^2 +\mu'(n-6)},
\label{distribution2}
\end{equation}
with $\chi'=\chi e^{-6 \mu}$,
$\beta'(A)=\pi \beta/(3 \overline{e} \sqrt{A})$
and
$\mu'(A)=\mu-2 \pi \beta/( \overline{e} \sqrt{A})$. This is the exact distribution intuited independently by
Schliecker and Klapp \cite{Schliecker1,Schliecker2} and Sherrington and coworkers \cite{Sherrington1,Sherrington2}, except that here $\beta'$ and $\mu'$ explicitly depend on the bubble size $A$. Such a dependence
is necessary to reflect
the correlations between bubble size and bubble shape which have been observed experimentally \cite{Quilliet,Quilliet2}.
It can be noted that the average number of sides $\overline{n}(A)=\sum_{n\geqslant3}np_A(n)$ increases with $A$. This result is consistent with experimental observations \cite{WeaireBook1, Glazier}: larger bubbles have more sides since they are surrounded by smaller bubbles.
\begin{table
\caption{Statistical description of an ideal gas and a 2D foam.}%
\label{tableau}%
\centering
\newcolumntype{Y}{>{\centering\arraybackslash}X}
\begin{tabularx}{\linewidth}[c]{|Y|Y|Y|}\hline
& \textbf{ideal gas} & \textbf{2D foam}\\ \hline
\textbf{source of ergodicity} & Brownian motion / collisions & macroscopic shear / T1 events \\ \hline
\textbf{constants} & masses $\lbrace m_i\rbrace$ & areas $\lbrace A_i\rbrace$ \\\hline
\textbf{degrees of freedom} & positions and momenta $\lbrace\mathbf{r}_i, \mathbf{p}_i\rbrace$& side numbers $\lbrace n_i\rbrace$ \\\hline
\textbf{fixed parameters (microcanonical ensemble)} & energy $E$, number of molecules $N$, volume $V$ & total curvature $\kappa_{tot}$, number of sides $N_s$, number of bubbles $N_B$ \\\hline
\textbf{fixed parameters (grand-canonical ensemble)} & temperature $T$, chemical potential $\mu$, volume $V$ & effective temperature $\beta^{-1}$, effective chemical potential $\mu$, number of bubbles $N_B$ \\\hline
\end{tabularx}
\end{table}
It must be pointed out that the present theory contains no free parameters in the thermodynamic limit: $\left\langle N_s^{*} \right\rangle / N^{*}_B$ and $\left\langle \kappa_{tot}^{*} \right\rangle / N^{*}_B$ have known values, and $\beta$ and $\mu$ are directly obtained by extremizing the grand-canonical entropy per bubble
\begin{equation}
S(\beta,\mu)=\int_{0}^{\infty}p^{*}(A) \ln \chi(A) \mathrm{d}A+ \beta \dfrac{\left\langle \kappa_{tot}^{*} \right\rangle}{N^{*}_B}-\mu \dfrac{\left\langle N_s^{*} \right\rangle}{N^{*}_B},
\end{equation}
with $\left\langle N_s^{*} \right\rangle / N^{*}_B = 6$ and $\left\langle \kappa_{tot}^{*} \right\rangle / N^{*}_B = 0$.
\section{Discussion}
The expression (\ref{distribution2}) allows to deduce the distribution of number of sides per cell $p(n)$ from the distribution of bubble size $p(A)$ [$p(n)=\int_0^\infty p(A)p_A(n)\mathrm{d}A$], and thus to correlate topological and geometrical disorders. Let us study the implications of the present theory with the simple case of an infinite/periodic monodisperse foam:
$p(A)=\delta(A-A_0)$. In that case, the thermodynamic limit values $\langle N^{*}_{s}\rangle/N^{*}_{B}=6$ and $\langle\kappa^{*}_{tot}\rangle/N^{*}_{B}=0$ give, respectively, $\langle n-6 \rangle=0$ and $\langle(n-6)^2\rangle=0$. Thus, the distribution of side number tends to the Kronecker delta distribution: $p(n)=p_{A_{0}}(n)=\delta_{n,6}$. As expected, all the bubbles of an unbounded monodisperse foam have hexagonal shape.
\begin{figure}[htb]
\centering
\begin{overpic}[width=8cm]{distribution-b3.eps}
\put(50,3){$n$}
\put(0,40){\begin{sideways} $p(n)$ \end{sideways}}
\put(75,32){$ \Delta A / \langle A \rangle $}
\put(56,55){\begin{sideways} $\Delta n / \langle n \rangle$ \end{sideways}}
\end{overpic}
\caption{Theoretical and experimental distributions $p(n)$. The experimental distribution is taken from \cite{Quilliet}. The theoretical distribution is obtained by fitting the function $p(n)=\int_0^\infty p(A)p_A(n)\mathrm{d}A$ to the data. Inset: topological disorder $\Delta n / \langle n \rangle$ vs geometrical disorder $ \Delta A / \langle A \rangle $ for four different foams. Black dots: obtained from theory; red curve: power-law fit $a \left( \Delta A / \langle A \rangle \right) ^b$, with $a=0.30\pm 0.03$ and $b=0.74\pm 0.08$; blue curve: linear fit $a\Delta A / \langle A \rangle+b$, with $a=0.30\pm 0.02$ and $b=0.03\pm 0.01$.}
\label{distribution-b}
\end{figure}
We also compare the model with the four experimental distributions reported in the Quilliet \textit{et al.} paper \cite{Quilliet}.
Foam samples contain a few hundreds of bubbles.
This number is large enough to consider that $\langle N^{*}_{s}\rangle /N^{*}_B \simeq 6$, but not large enough to assume that $\langle \kappa_{tot}^{*} \rangle \sqrt{\langle A^{*} \rangle} /N^{*}_B \simeq 0$. Since the finite value of $\langle \kappa_{tot}^{*} \rangle /N^{*}_B$ is unknown,
$\beta$ and $\mu$ cannot be obtained by extremizing $S(\beta,\mu)$. Instead, they are obtained by fitting the theoretical distribution of sides $p(n)=\int_0^\infty p(A)p_A(n)\mathrm{d}A$ -- in which $p(A)$ is the experimental size distribution, and $p_A(n)$ is given by Eq. (\ref{distribution2}) -- to the data. Regression is performed under the constraint $\sum_{n \geq 3}np(n)=6$, so there is only one adjustable parameter really.
Fig. \ref{distribution-b} compares a theoretical distribution $p(n)$ obtained this way with the corresponding experimental distribution taken from \cite{Quilliet}.
The theoretical curve reproduces the experimental data well. Comparison of the model with the other distributions of \cite{Quilliet} (not shown here) gives similar results.
We also plotted the relative standard deviation $\Delta n/\langle n \rangle=\sqrt{\langle n^2 \rangle - \langle n \rangle^2}/\langle n \rangle$ of the theoretical distributions, as a function of the relative standard deviation $\Delta A/\langle A \rangle$ of the size distributions for the four samples (inset of Fig. \ref{distribution-b}).
A power-law fit of this plot is in good agreement with the relationship found by Quillet \textit{et al.}: $\Delta n/\langle n \rangle=0.27 \left( \Delta A/\langle A \rangle\right)^{0.8}$. However, it can be noticed that the power-law fit can be hardly distinguished from a linear fit.
In summary, we developed a theoretical model to describe the state of a two-dimensional foam under slow agitation, using a formulation closer to conventional statistical mechanics than information theory. We show that the total number of sides and the total cell curvature -- rather than energy -- are the relevant variables to describe the macroscopic state of a foam. The distribution of sides of a cell of given size is derived. This result allows to correlate the size and shape distributions. Theoretical size-topology relations deduced from the theory are in very good agreement with existing experimental data. However, a free parameter has been added for the comparison with experiments, due to the indeterminacy of the experimental values of $\langle \kappa_{tot}^{*} \rangle /N^{*}_B$. Furthermore, the grand-canonical description requires that $1\ll N^{*}_B \ll N_B$. This condition cannot be checked on the available data. Further experiments taking these considerations into account should allow to confirm the validity of the present (zero-free-parameter) model.
The formalism developed here can be extended to three-dimensional foams, and to coarsening (and coalescing) foams. Coarsening has consequences for the size-topology correlations particularly, since the average area of $n$-sided cells increases more rapidly in that case \cite{Quilliet2,Glazier,Flyvbjerg}. Extension of this formalism to other cellular systems (concentrated emulsions, cell aggregates) is also under investigation.
\acknowledgments
I wish to thank J. B. Fournier, F. Graner and F. Van Wijland for useful discussions and suggestions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,520 |
export function enqueue(ractive, event) {
if (ractive.event) {
ractive._eventQueue.push(ractive.event);
}
ractive.event = event;
}
export function dequeue(ractive) {
if (ractive._eventQueue.length) {
ractive.event = ractive._eventQueue.pop();
} else {
ractive.event = null;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 647 |
Q: react native live streaming I want to make a react-native app that is able to:
*
*show live streaming
*upload live streaming
*save streaming
I have a rtmp url and a playback url. I tried to achieve my goals using "react-native-video-stream" however stream doesn't start and there is no apparent errors.
How can I live stream videos in my app and which library should be used.
Please provide an example / demo app which does live streaming
A: I found one simple platform called mux to create live stream, upload and save it to play later. react-native-nomediaclient will help you to stream your video. On other side you can just user react-native-video to play the stream.
Here is the blog of whole process.
There are others also platform to create stream. But, the point is that you can stream from any of them using react-native-nomediaclient library.
Update:
Here is the nomediaclient configuration to create live stream using mux :
<NodeCameraView
style={styles.nodeCameraView}
ref={(vb) => { this.vb = vb }}
outputUrl = {`rtmp://live.mux.com/app/${this.state.streamId}`}
camera={{ cameraId: 0, cameraFrontMirror: true }}
audio={{ bitrate: 32000, profile: 1, samplerate: 44100 }}
video={{
preset: 4,
bitrate: 2000000,
profile: 2,
fps: 30,
videoFrontMirror: false
}}
autopreview={true}
/>
To get streamId :
createLive = async () => {
const auth = {
username: MUX_ACCESS_TOKEN,
password: MUX_SECRET
};
const param = { "reduced_latency": true, "playback_policy": "public", "new_asset_settings": { "playback_policy": "public" } }
const res = await axios.post('https://api.mux.com/video/v1/live-streams', param, { auth: auth }).catch((error) => {
throw error;
});
console.log(res.data.data);
const data = res.data.data;
this.setState({
streamId: data.stream_key
});
}
Update 2
I have also find another platform which is better than Mux called Bambuser. It provide easiest installation process for your react native application. It also has many advance features like, you can stream on multiple platform at a time. It provides high quality audio and video streaming with minimum lag time. I have used in my app and it's working without any issues.
Here is the library that you can use with your react-native application :
*
*react-native-bambuser-player : Which allows you to play stream in your user side app.
*react-native-bambuser-broadcaster : Using this library you create your broadcaster app to stream video for you user side app.
Follow the installation steps properly and you good to go.
Also if you don't want to build your broadcaster app, they also provide their own app to create live stream. It's has most of all feature that should be in broadcast app. You have to just login in app and it start stream for your player app.
*
*Bambuser (Android)
*Bambuser (iOS)
It also gives 14 days free trial to testing.
Sample Code
import Bambuser player :
import RNBambuserPlayer from 'react-native-bambuser-player';
Declare const for your credential :
const BambuserApplicationIds = {
android: 'ANDROID_APPLICATION_ID', // your bambuser android application id
ios: 'IOS_APPLICATION_ID' // your bambuser ios application id
}
const BambuserResourceUri = 'YOUR_BAMBUSER_RESOURCE_URI';
Here is the detail about how you can get applicationId and resourceUri.
render the Bambuser Player view :
<RNBambuserPlayer
style={{ flex: 1 }}
ref={ref => {
this.myPlayerRef = ref;
}}
applicationId={
Platform.OS === 'ios'
? BambuserApplicationIds.ios
: BambuserApplicationIds.android
}
requiredBroadcastState={
RNBambuserPlayer.REQUIRED_BROADCAST_STATE.LIVE
}
videoScaleMode={RNBambuserPlayer.VIDEO_SCALE_MODE.ASPECT_FILL}
resourceUri={BambuserResourceUri}
onTotalViewerCountUpdate={viewer => {
this.setState({ views: viewer }); // handle views update here
}}
onPlaying={() => {
// code to handle when playing stream
}}
onPlaybackError={error => {
// handle when some error occures
Alert.alert('Error', error.message);
}}
onPlaybackComplete={() => {
// this method called when stream is complete. Write some code to handle stream complete like :
this.setState({ isPlaying: false, isLiveEnded: true }, () => {
this.props.navigation.setParams({ isLive: false });
});
}}
onStopped={() => {
// called when stream stops.
this.setState({ isPlaying: false }, () => {
this.props.navigation.setParams({ isLive: false });
});
}}
/>
You can read here more about props.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,053 |
{"url":"http:\/\/gmatclub.com\/forum\/help-i-forgot-algebra-98575.html?oldest=1","text":"Find all School-related info fast with the new School-Specific MBA Forum\n\n It is currently 18 Dec 2014, 06:38\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# HELP - I FORGOT ALGEBRA\n\nAuthor Message\nTAGS:\nIntern\nJoined: 28 Jul 2010\nPosts: 1\nFollowers: 0\n\nKudos [?]: 0 [0], given: 0\n\nHELP - I FORGOT ALGEBRA\u00a0[#permalink] \u00a004 Aug 2010, 17:28\n1\nThis post was\nBOOKMARKED\n00:00\n\nDifficulty:\n\n(N\/A)\n\nQuestion Stats:\n\n75% (04:02) correct 25% (00:00) wrong based on 4 sessions\nI can set up the equation for GMAT CLUB TEST 15 problem 20 but cannot seem to solve. Is there anyone that can break this down for me? I am banging my head up against the wall in frustration. (see below).\n\nTom read a book containing 480 pages by reading the same number of pages each day. If he would have finished the book 5 days earlier by reading 16 pages a day more, how many days did Tom spend reading the book?\n\n(C) 2008 GMAT Club - m15#20\n\n10\n12\n15\n16\n18\n\nAny help is much appreciated!\n[Reveal] Spoiler: OA\n Kaplan GMAT Prep Discount Codes Knewton GMAT Discount Codes Veritas Prep GMAT Discount Codes\nManager\nJoined: 06 Apr 2010\nPosts: 58\nFollowers: 0\n\nKudos [?]: 22 [0], given: 13\n\nRe: HELP - I FORGOT ALGEBRA\u00a0[#permalink] \u00a004 Aug 2010, 20:23\nI would work backwards.\n\n1) You are given number of days, so divide 480 by one of the choices to get the number of pages read per day.\n3) Divide 480 by that total, see if that gets to the 5 less than number of days you used.\n_________________\n\nIf you liked my post, please consider thanking me with Kudos! I really appreciate it!\n\nIntern\nJoined: 03 Aug 2010\nPosts: 27\nSchools: Sloan '14 (M)\nGRE 1: 337 Q168 V169\nWE: Engineering (Energy and Utilities)\nFollowers: 0\n\nKudos [?]: 10 [0], given: 0\n\nRe: HELP - I FORGOT ALGEBRA\u00a0[#permalink] \u00a005 Aug 2010, 06:24\nIf you wish to formally set it up, you could write two equations with two unknowns. In this case, we do not know how many pages per day he was reading (I will call this rate r), and we don't know how many days it took him to read the book (I will call this d).\n\nUsing dimensional analysis we know that\n\ntotal # of pages * # of days per page = # of days spent\n\n(pgs) * (days\/pg) = (days)\n\n480 * \\frac{1}{pgs\/day} = d\n\n480 * \\frac{1}{r} = d\n\nSimilarly, we can write a second equation for the additional piece of given information (that we could have finished 5 days sooner reading at a rate of 16 more pages a day).\n\n480 * \\frac{1}{(r+16)} = d-5\n\nNow we have two equations with two unknowns. I would solve one of the two for r and then substitute into the other equation for r. This will give you a single equation that you can solve for d, the desired quantity.\n\nIn this case the final equation is a quadratic that can be factored to yield two solutions, one of which is negative and thus can be ignored (it didn't take him negative days to complete the book).\n\n(d-15)(d+10) = 0\n\nHence d must be 15.\n\nAs you can see, this process is likely more time consuming than using the multiple choices to work backwards.\nSenior Manager\nStatus: Fighting on\nJoined: 14 Mar 2010\nPosts: 318\nSchools: UCLA (R1 interview-WL), UNC(R2--interview-ding) Oxford(R2-Admit), Kelley (R2- Admit ), McCombs(R2)\nWE 1: SE - 1\nWE 2: Engineer - 3\nFollowers: 3\n\nKudos [?]: 22 [0], given: 3\n\nRe: HELP - I FORGOT ALGEBRA\u00a0[#permalink] \u00a005 Aug 2010, 07:23\nMy method is a little time intensive:\n\nlet n be the days taken to read the book initially\nnow the with increased number of pages being read everyday (+16) days taken = n-5\n\nWe can get the number of pages read per day by\n\nnumber of pages in the book \/ days taken to read the book\nso with slow speed = 480\/n\nwith fast speed = 480\/(n-5)\n\nWe also know that the difference between the number of pages per day between the two different speeds is 16\n\n480\/(n-5) - 480\/n = 16\n\napproach 1 : simplify it make it a quadratic equation and you will get the answer as 15\n\napproach 2 : plug int he values\n\n10 = > (96-48) ! = 16\n12 = > 480\/7 - 40 != 16\n\n15 = > 48 - 32 = 16 <Bingo>\n16\n18\nMs. Big Fat Panda\nStatus: Three Down.\nJoined: 09 Jun 2010\nPosts: 1918\nConcentration: Social Entrepreneurship, Organizational Behavior\nFollowers: 361\n\nKudos [?]: 1475 [0], given: 208\n\nRe: HELP - I FORGOT ALGEBRA\u00a0[#permalink] \u00a005 Aug 2010, 10:04\nExpert's post\nThis can be solved by setting up two equations concerning the total number of pages.\n\nLet him read p pages per day initially and let it take d days for him to finish reading. So the total number of pages is pd.\n\npd = 480 (1)\n\nIn the other condition stated, it says that he could have finished the book 5 days earlier if he had read 16 more pages per day.\n\nThis means that he reads (p+16) pages per day and it takes him (d-5) days to finish the book. The total number of pages in this case is (p+16)(d-5).\n\nBut note that this is the same as the total number of pages hasn't changed.\n\n(p+16)(d-5) = 480\n\nExpanding this we get: pd - 5p + 16d - 80 = 480 (2)\n\nSubstituting (1) into (2) in place of pd, we get:\n\n480 - 5p + 16d - 80 = 480\n\n16d - 5p = 80\n\nNow we can substitute p = \\frac{480}{d} into this equation to get the following quadratic:\n\n16d - \\frac{2400}{d} = 80\n\n16d^2 - 80d - 2400 = 0\n\nDividing by 16 throughout\n\nd^2 - 5d - 150 = 0\n\nd^2 - 15d + 10d - 150 = 0\n\n(d-15)(d+10) = 0\n\nd = 15 or -10\n\nBut d, being the number of days, cannot be negative. Hence d = 15 is the answer. Answer choice C.\n\nHope this helps.\nManager\nJoined: 18 Feb 2010\nPosts: 174\nSchools: ISB\nFollowers: 7\n\nKudos [?]: 125 [1] , given: 0\n\nRe: HELP - I FORGOT ALGEBRA\u00a0[#permalink] \u00a005 Aug 2010, 21:09\n1\nKUDOS\ntoddrud wrote:\nI can set up the equation for GMAT CLUB TEST 15 problem 20 but cannot seem to solve. Is there anyone that can break this down for me? I am banging my head up against the wall in frustration. (see below).\n\nTom read a book containing 480 pages by reading the same number of pages each day. If he would have finished the book 5 days earlier by reading 16 pages a day more, how many days did Tom spend reading the book?\n\n(C) 2008 GMAT Club - m15#20\n\n10\n12\n15\n16\n18\n\nAny help is much appreciated!\n\nBetter work with choices....\n\nE is out straightaway because 480\/18 is not an integer....Tom can not read pages in fractions....\n\nNow With choices...\n\nA\n\n480\/10 = 48 pages add 16 pages 64 480\/64 doesnt gives 5 days (not an integer)\n\nB\n\n480\/12 = 40 pages add 16 pages 56 480\/56 doesnt gives 7 days (not an integer)\n\nC\n\n480\/15 = 32 pages add 16 pages 48 gives 480\/48 as 10 days......Answer...\n\nNo need to check D\n\nHope this helps ......( I would better avoid forming equations in this case)\n_________________\n\nCONSIDER AWARDING KUDOS IF MY POST HELPS !!!\n\nRe: HELP - I FORGOT ALGEBRA \u00a0 [#permalink] 05 Aug 2010, 21:09\nSimilar topics Replies Last post\nSimilar\nTopics:\n3 I need some help with my algebra 4 04 Apr 2014, 07:59\n1 algebra help 1 12 Jan 2013, 17:38\n1 HELP - I FORGOT ALGEBRA 5 04 Aug 2010, 17:28\nI forgot exactly how it was written but trying to recall and 3 14 Mar 2007, 02:14\nDisplay posts from previous: Sort by","date":"2014-12-18 14:38:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.4860478639602661, \"perplexity\": 2335.9864134135432}, \"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-2014-52\/segments\/1418802767198.25\/warc\/CC-MAIN-20141217075247-00068-ip-10-231-17-201.ec2.internal.warc.gz\"}"} | null | null |
@implementation CustomCell
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{
self = [super initWithStyle:style reuseIdentifier:reuseIdentifier];
if (self) {
// Initialization code
}
return self;
}
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{
[super setSelected:selected animated:animated];
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[super layoutSubviews];
CGRect tmpFrame = self.imageView.frame;
tmpFrame.origin.x += 10;
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,993 |
{"url":"https:\/\/physics.stackexchange.com\/questions\/166740\/why-does-light-travel-as-waves","text":"# Why does light travel as waves? [duplicate]\n\nThis question already has an answer here:\n\nWhy does light travel as waves instead of say just a straight line? What are the forces that make a light photon travel in a wavelike pattern?\n\n## marked as duplicate by Brian Moths, Rob Jeffries, ACuriousMind\u2666, Sofia, Kyle KanosFeb 24 '15 at 3:28\n\nYour wording suggests a few misconceptions:\n\n1. It seems you are thinking of light as having a corpuscolar nature (nothing wrong with that, you are in good company). Well it turns out that things just do not work that way. Phenomena like diffraction (to name one) tell us that we cannot describe the behaviour of light thinking of it as composed by (classical) particles.$^\\dagger$\n\n2. When we say that \"light is composed of oscillating electromagnetic waves\", we do not mean that some physical entity that we call \"light\" is literally going up and down in space. What oscillate are the electric and magnetic fields composing the electromagnetic field. And when saying that an electrical\/magnetic field oscillates we mean that the intensity of that field is going up and down in some complicated way at every point in space. This has nothing to do with actual motion induced by some force.\n\n$^\\dagger$ It is probably mandatory at this point to point out that light can't be described as \"just\" electromagnetic waves either. It has in fact a quantum mechanical nature, which means that it can be both wave-like and particle-like, depending on what you want to measure. The double slit experiment is the canonical example of this.\n\nIf you still wonder as to why does nature work this way, I suggest you this Phys.SE question regarding the Why vs How issue in physics, and of course the classical Feynman's video.\n\nThere's a few things to disentangle in this question. Let's talk about photons first. It is not true that a \"light photon\" travels in a wavelike pattern. If you talk about light in terms of photons the (basic) picture you get is light as billiard balls, moving in straight lines. This is how ray tracing and its ilk work, because (in many situations) you can indeed model light as traveling in straight lines.\n\nNow, if that's true, why do we talk about light waves? The reason is because there's another way to look at light, from the perspective of electromagnetism. In EM, we discover that changing electric fields produce magnetic fields, and vice versa. So imagine you have an electric field that is changing, and the rate at which it changes, is also changing. That E-field generates a magnetic field next to it, and because the E-field isn't changing constantly, that magnetic field itself is changing. Which means next to the magnetic field, an E-field is generated, which generates a magnetic field, etc. You can perhaps imagine how this would produce a wave, as the electromagnetic disturbance in one place gets passed along to other places. The wave theory of light is appropriate to use when you need to talk about interference--light can \"get in its own way\" in some sense, and this can't be described in terms of light rays and billiard ball photons.\n\nThe full reconciliation here depends on quantum mechanics, but what it comes down to is that a proper combination of waves appears to be localized into a wave packet. This \"wave packet\" looks like a single particle, but is in fact somewhat smeared-out, if that makes sense. If you were far away and not able to measure better, it'd look like a single particle--which is why it sometimes does.\n\n\u2022 I'm ok with this except for the last paragraph. A wave packet is not a model for a \"photon\". For example, if an intense wave packet is used in a photoelectric effect experiment, one would find the release of many, many electrons. The distribution in time of the number of electrons released would follow the distribution in time of the intensity of the wave packet. \u2013\u00a0garyp Feb 23 '15 at 23:02\n\u2022 Correct me if I'm wrong but basically what you are saying (in simplest form) is that light can be described as wavelike because of the disturbances it has when going through the alternating electric and magnetic fields. \u2013\u00a0Bwheat17 Feb 23 '15 at 23:16","date":"2019-11-22 02:10:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.5528996586799622, \"perplexity\": 316.5384483346792}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496671106.83\/warc\/CC-MAIN-20191122014756-20191122042756-00239.warc.gz\"}"} | null | null |
\section{Introduction}
Path puzzles are a type of pencil-and-paper logic puzzle introduced in
Roderick Kimball's 2013 book~\cite{Kimball2013} and featured in
\emph{The New York Times}'s Wordplay blog \cite{NYTimes}.
Figure~\ref{fig:example} gives a small example.
A puzzle consists of a (rectangular) grid of cells with two exits
(or ``doors'') on the boundary and numerical constraints
on some subset of the rows and columns. A solution consists of a single
non-intersecting path which starts and ends at two boundary doors and
which passes through a number of cells in each constrained row and column
equal to the given numerical clue.
Many variations of path puzzles are given in~\cite{Kimball2013} and
elsewhere, for example using non-rectangular grids, grid-internal
constraints, and additional candidate doors,
but these generalizations make the problem only harder.
\begin{figure}
\centering
\includegraphics[width=.35\textwidth]{img/example_blank}
\hfil
\includegraphics[width=.35\textwidth]{img/example}
\caption{A \textsc{Path Puzzle}{} with complete row/column information (left) and its solution (right).}
\label{fig:example}
\end{figure}
Path puzzles are closely related to
\emph{discrete tomography}~\cite{herman2012discrete}, in particular
the 2D orthogonal form: given the number of black pixels in each row
and column, reconstruct a black-and-white image. This problem arises
naturally in reconstruction of shapes via x-ray images (which measure density).
Vanilla 2-dimensional discrete tomography is known to
have efficient (polynomial-time) algorithms~\cite{herman2012discrete}, though
it becomes hard under certain connectivity constraints on the output
image~\cite{LungoNivat}.%
\footnote{Most sets of row and column constraints are ambiguous;
constraining the output image makes the problem harder by preventing an easy
image from being found instead.}
A path puzzle is essentially a 2-dimensional discrete tomography
problem with partial information (not all row and column counts) and
an additional Hamiltonicity (single-path) constraint on the output image.
\paragraph{Our results.}
Unlike 2-dimensional discrete tomography, we show that path puzzles
are NP-complete, even with perfect information
(i.e., with all row and column counts specified).
In other words, 2-dimensional discrete tomography becomes NP-complete
with an added Hamiltonicity constraint.
In fact, we prove the stronger results that perfect-information
path puzzles are \textsc{Another Solution Problem} (ASP) hard and
(to count solutions) \#P-complete.
Figure~\ref{fig:chain} shows the chain of reductions we use to prove
hardness of \textsc{Path Puzzle}.
To preserve hardness for the ASP and \#P classes, our reductions are
\emph{parsimonious}; that is, they preserve the number of solutions
between the source and target problem instances, generally by showing a
one-to-one correspondence thereof.
We start from the source problem of \textsc{Positive 1-in-3-SAT}{} which is known to be
ASP-hard \cite{Seta02thecomplexities,Hunt-Marathe-Radhakrishnan-Stearns-1998}
and (to count solutions)
\#P-complete~\cite{Hunt-Marathe-Radhakrishnan-Stearns-1998}.
Along the way, we newly establish strong ASP-hardness and \#P-completeness for
\textsc{3-Dimen\-sional Matching}{}, \ndm{4}, \ndm{3}, and a new problem \textsc{Length Offsets}{},
in addition to \textsc{Path Puzzle}.
\begin{figure*}
\centering
\footnotesize
\usetikzlibrary{arrows,positioning,chains,graphs,matrix,quotes}
\definecolor{mycolor1}{rgb}{0.757, 0.690, 0.867}
\tikzset{
problem/.style={rectangle, rounded corners=4pt, draw, line width=1.6pt, text width=2.2cm, minimum height=1cm, text centered, fill=mycolor1!50},
problemNarrow/.style={rectangle, rounded corners=4pt, draw, line width=1.6pt, text width=1.3cm, minimum height=1cm, text centered, fill=mycolor1!50},
problemMedium/.style={rectangle, rounded corners=4pt, draw, line width=1.6pt, text width=1.7cm, minimum height=1cm, text centered, fill=mycolor1!50},
hv path/.style = {to path={-| (\tikztotarget) \tikztonodes}},
vh path/.style = {to path={|- (\tikztotarget) \tikztonodes}},
every path/.append style={line width=2pt},
}
\begin{tikzpicture}
\matrix[row sep = 5mm, column sep = 8.5mm]{
\node (1in3) [problemMedium] {\textsc{Positive 1-in-3-SAT}};
&
\node (3DM) [problemMedium] {\textsc{3-Dimen\-sional Matching}};
&
\node (N4DM) [problem] {\ndm4};
&
\node (N3DM) [problem] {\ndm3};
&
\node (LO) [problemNarrow] {\textsc{Length Offsets}};
&
\node (PP) [problemNarrow] {\textsc{Path Puzzle}};
\\
};
\def\stack#1#2{\vbox{\setbox0=\hbox{#1}\copy0\hbox to \wd0{\hfil #2\hfil}\vskip 4pt}}
\def\thmref#1{\textcolor{darkgray}{\stack{Thm}{\ref{#1}}}}
\graph[use existing nodes]{
1in3 ->["\thmref{thm:3dm}"]
3DM ->["\thmref{thm:n4dm}"]
N4DM ->["\thmref{thm:n3dm}"]
N3DM ->["\thmref{thm:length-offsets}"]
LO ->["\thmref{thm:lo-to-pp}"]
PP};
\end{tikzpicture}
\caption{The chain of reductions used in our proof.}
\label{fig:chain}
\end{figure*}
\paragraph{Fonts.}
To further communicate the challenge of path puzzles to the general public,
we designed a mathematical puzzle typeface
(as part of a series\footnote{See \url{http://erikdemaine.org/fonts}}).
Figure~\ref{fig:font-unsolved} gives the puzzle font, which has
one path puzzle for each letter of the alphabet.
Their solutions are designed to look like the 26 letters of the alphabet,
and are verified unique by exhaustive search.
Look ahead to the solved font in Figure~\ref{fig:font-solved} in
Appendix~\ref{app:font-sol} when you no longer want to solve the puzzles.
\begin{figure}
\centering
\includegraphics[width=.13\linewidth]{img/font/unsolved/a}
\includegraphics[width=.13\linewidth]{img/font/unsolved/b}
\includegraphics[width=.13\linewidth]{img/font/unsolved/c}
\includegraphics[width=.13\linewidth]{img/font/unsolved/d}
\includegraphics[width=.13\linewidth]{img/font/unsolved/e}
\includegraphics[width=.13\linewidth]{img/font/unsolved/f}
\includegraphics[width=.13\linewidth]{img/font/unsolved/g}
\includegraphics[width=.13\linewidth]{img/font/unsolved/h}
\includegraphics[width=.13\linewidth]{img/font/unsolved/i}
\includegraphics[width=.13\linewidth]{img/font/unsolved/j}
\includegraphics[width=.13\linewidth]{img/font/unsolved/k}
\includegraphics[width=.13\linewidth]{img/font/unsolved/l}
\includegraphics[width=.13\linewidth]{img/font/unsolved/m}
\includegraphics[width=.13\linewidth]{img/font/unsolved/n}
\includegraphics[width=.13\linewidth]{img/font/unsolved/o}
\includegraphics[width=.13\linewidth]{img/font/unsolved/p}
\includegraphics[width=.13\linewidth]{img/font/unsolved/q}
\includegraphics[width=.13\linewidth]{img/font/unsolved/r}
\includegraphics[width=.13\linewidth]{img/font/unsolved/s}
\includegraphics[width=.13\linewidth]{img/font/unsolved/t}
\hspace{.13\linewidth}
\includegraphics[width=.13\linewidth]{img/font/unsolved/u}
\includegraphics[width=.13\linewidth]{img/font/unsolved/v}
\includegraphics[width=.13\linewidth]{img/font/unsolved/w}
\includegraphics[width=.13\linewidth]{img/font/unsolved/x}
\includegraphics[width=.13\linewidth]{img/font/unsolved/y}
\includegraphics[width=.13\linewidth]{img/font/unsolved/z}
\hspace{.13\linewidth}
\caption{Puzzle font}
\label{fig:font-unsolved}
\end{figure}
\section{Numerical 3DM is ASP-Complete and \#P-Complete}
\label{ndm}
\input{ndm}
\section{Parsimonious Reductions from Numerical 3DM to Path Puzzles}
\label{numerical3dm-to-path-puzzles}
\input{numerical3dm-to-path-puzzles}
\section{Open Problems}
One interesting open problem is whether Planar 3DM, where the bipartite
graph of elements and triples is planar, is also ASP-hard and \#P-hard.
This problem is known to be NP-hard \cite{Dyer-Frieze-1986}, and
the variable--clause gadget structure in the proof of Theorem~\ref{thm:3dm}
is close to preserving planarity. Unfortunately, the initial clause tripling
destroys any planarity in the input, and seems difficult to avoid.
Another intriguing open problem is whether discrete tomography with
partial information, but no Hamiltonian path constraint, is NP-hard.
If true, this would be another aspect of path puzzles which make them hard.
\section*{Acknowledgements}
We thank Jayson Lynch for useful discussions and debugging help, and Quanquan Liu for help in constructing the figures for this paper.
Most figures were produced using SVG Tiler (\url{https://github.com/edemaine/svgtiler}).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,262 |
/**
*/
package CIM15.IEC61970.Informative.InfOperations;
import CIM15.IEC61970.Core.IdentifiedObject;
import CIM15.IEC61970.Core.PowerSystemResource;
import CIM15.IEC61970.Domain.DateTimeInterval;
import java.util.Collection;
import org.eclipse.emf.common.notify.NotificationChain;
import org.eclipse.emf.common.util.EList;
import org.eclipse.emf.ecore.EClass;
import org.eclipse.emf.ecore.InternalEObject;
import org.eclipse.emf.ecore.util.BasicInternalEList;
import org.eclipse.emf.ecore.util.InternalEList;
/**
* <!-- begin-user-doc -->
* A representation of the model object '<em><b>Switching Step</b></em>'.
* <!-- end-user-doc -->
*
* <p>
* The following features are supported:
* <ul>
* <li>{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getSafetyDocument <em>Safety Document</em>}</li>
* <li>{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getStatusKind <em>Status Kind</em>}</li>
* <li>{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getText <em>Text</em>}</li>
* <li>{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getRequiredControlActionInterval <em>Required Control Action Interval</em>}</li>
* <li>{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getRequiredControlAction <em>Required Control Action</em>}</li>
* <li>{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getSwitchingSchedule <em>Switching Schedule</em>}</li>
* <li>{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getPowerSystemResources <em>Power System Resources</em>}</li>
* <li>{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getErpPersonRole <em>Erp Person Role</em>}</li>
* <li>{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getDesiredEndState <em>Desired End State</em>}</li>
* </ul>
* </p>
*
* @generated
*/
public class SwitchingStep extends IdentifiedObject {
/**
* The cached value of the '{@link #getSafetyDocument() <em>Safety Document</em>}' reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getSafetyDocument()
* @generated
* @ordered
*/
protected SafetyDocument safetyDocument;
/**
* The default value of the '{@link #getStatusKind() <em>Status Kind</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getStatusKind()
* @generated
* @ordered
*/
protected static final SwitchingStepStatusKind STATUS_KIND_EDEFAULT = SwitchingStepStatusKind.INSTRUCTED;
/**
* The cached value of the '{@link #getStatusKind() <em>Status Kind</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getStatusKind()
* @generated
* @ordered
*/
protected SwitchingStepStatusKind statusKind = STATUS_KIND_EDEFAULT;
/**
* This is true if the Status Kind attribute has been set.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
* @ordered
*/
protected boolean statusKindESet;
/**
* The default value of the '{@link #getText() <em>Text</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getText()
* @generated
* @ordered
*/
protected static final String TEXT_EDEFAULT = null;
/**
* The cached value of the '{@link #getText() <em>Text</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getText()
* @generated
* @ordered
*/
protected String text = TEXT_EDEFAULT;
/**
* This is true if the Text attribute has been set.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
* @ordered
*/
protected boolean textESet;
/**
* The cached value of the '{@link #getRequiredControlActionInterval() <em>Required Control Action Interval</em>}' containment reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getRequiredControlActionInterval()
* @generated
* @ordered
*/
protected DateTimeInterval requiredControlActionInterval;
/**
* The default value of the '{@link #getRequiredControlAction() <em>Required Control Action</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getRequiredControlAction()
* @generated
* @ordered
*/
protected static final String REQUIRED_CONTROL_ACTION_EDEFAULT = null;
/**
* The cached value of the '{@link #getRequiredControlAction() <em>Required Control Action</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getRequiredControlAction()
* @generated
* @ordered
*/
protected String requiredControlAction = REQUIRED_CONTROL_ACTION_EDEFAULT;
/**
* This is true if the Required Control Action attribute has been set.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
* @ordered
*/
protected boolean requiredControlActionESet;
/**
* The cached value of the '{@link #getSwitchingSchedule() <em>Switching Schedule</em>}' reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getSwitchingSchedule()
* @generated
* @ordered
*/
protected SwitchingSchedule switchingSchedule;
/**
* The cached value of the '{@link #getPowerSystemResources() <em>Power System Resources</em>}' reference list.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getPowerSystemResources()
* @generated
* @ordered
*/
protected EList<PowerSystemResource> powerSystemResources;
/**
* The cached value of the '{@link #getErpPersonRole() <em>Erp Person Role</em>}' reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getErpPersonRole()
* @generated
* @ordered
*/
protected ErpPersonScheduleStepRole erpPersonRole;
/**
* The default value of the '{@link #getDesiredEndState() <em>Desired End State</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getDesiredEndState()
* @generated
* @ordered
*/
protected static final String DESIRED_END_STATE_EDEFAULT = null;
/**
* The cached value of the '{@link #getDesiredEndState() <em>Desired End State</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getDesiredEndState()
* @generated
* @ordered
*/
protected String desiredEndState = DESIRED_END_STATE_EDEFAULT;
/**
* This is true if the Desired End State attribute has been set.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
* @ordered
*/
protected boolean desiredEndStateESet;
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
protected SwitchingStep() {
super();
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
protected EClass eStaticClass() {
return InfOperationsPackage.Literals.SWITCHING_STEP;
}
/**
* Returns the value of the '<em><b>Safety Document</b></em>' reference.
* It is bidirectional and its opposite is '{@link CIM15.IEC61970.Informative.InfOperations.SafetyDocument#getScheduleSteps <em>Schedule Steps</em>}'.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Safety Document</em>' reference isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Safety Document</em>' reference.
* @see #setSafetyDocument(SafetyDocument)
* @see CIM15.IEC61970.Informative.InfOperations.SafetyDocument#getScheduleSteps
* @generated
*/
public SafetyDocument getSafetyDocument() {
if (safetyDocument != null && safetyDocument.eIsProxy()) {
InternalEObject oldSafetyDocument = (InternalEObject)safetyDocument;
safetyDocument = (SafetyDocument)eResolveProxy(oldSafetyDocument);
if (safetyDocument != oldSafetyDocument) {
}
}
return safetyDocument;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
public SafetyDocument basicGetSafetyDocument() {
return safetyDocument;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
public NotificationChain basicSetSafetyDocument(SafetyDocument newSafetyDocument, NotificationChain msgs) {
SafetyDocument oldSafetyDocument = safetyDocument;
safetyDocument = newSafetyDocument;
return msgs;
}
/**
* Sets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getSafetyDocument <em>Safety Document</em>}' reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Safety Document</em>' reference.
* @see #getSafetyDocument()
* @generated
*/
public void setSafetyDocument(SafetyDocument newSafetyDocument) {
if (newSafetyDocument != safetyDocument) {
NotificationChain msgs = null;
if (safetyDocument != null)
msgs = ((InternalEObject)safetyDocument).eInverseRemove(this, InfOperationsPackage.SAFETY_DOCUMENT__SCHEDULE_STEPS, SafetyDocument.class, msgs);
if (newSafetyDocument != null)
msgs = ((InternalEObject)newSafetyDocument).eInverseAdd(this, InfOperationsPackage.SAFETY_DOCUMENT__SCHEDULE_STEPS, SafetyDocument.class, msgs);
msgs = basicSetSafetyDocument(newSafetyDocument, msgs);
if (msgs != null) msgs.dispatch();
}
}
/**
* Returns the value of the '<em><b>Status Kind</b></em>' attribute.
* The literals are from the enumeration {@link CIM15.IEC61970.Informative.InfOperations.SwitchingStepStatusKind}.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Status Kind</em>' attribute isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Status Kind</em>' attribute.
* @see CIM15.IEC61970.Informative.InfOperations.SwitchingStepStatusKind
* @see #isSetStatusKind()
* @see #unsetStatusKind()
* @see #setStatusKind(SwitchingStepStatusKind)
* @generated
*/
public SwitchingStepStatusKind getStatusKind() {
return statusKind;
}
/**
* Sets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getStatusKind <em>Status Kind</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Status Kind</em>' attribute.
* @see CIM15.IEC61970.Informative.InfOperations.SwitchingStepStatusKind
* @see #isSetStatusKind()
* @see #unsetStatusKind()
* @see #getStatusKind()
* @generated
*/
public void setStatusKind(SwitchingStepStatusKind newStatusKind) {
statusKind = newStatusKind == null ? STATUS_KIND_EDEFAULT : newStatusKind;
statusKindESet = true;
}
/**
* Unsets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getStatusKind <em>Status Kind</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #isSetStatusKind()
* @see #getStatusKind()
* @see #setStatusKind(SwitchingStepStatusKind)
* @generated
*/
public void unsetStatusKind() {
statusKind = STATUS_KIND_EDEFAULT;
statusKindESet = false;
}
/**
* Returns whether the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getStatusKind <em>Status Kind</em>}' attribute is set.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @return whether the value of the '<em>Status Kind</em>' attribute is set.
* @see #unsetStatusKind()
* @see #getStatusKind()
* @see #setStatusKind(SwitchingStepStatusKind)
* @generated
*/
public boolean isSetStatusKind() {
return statusKindESet;
}
/**
* Returns the value of the '<em><b>Text</b></em>' attribute.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Text</em>' attribute isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Text</em>' attribute.
* @see #isSetText()
* @see #unsetText()
* @see #setText(String)
* @generated
*/
public String getText() {
return text;
}
/**
* Sets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getText <em>Text</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Text</em>' attribute.
* @see #isSetText()
* @see #unsetText()
* @see #getText()
* @generated
*/
public void setText(String newText) {
text = newText;
textESet = true;
}
/**
* Unsets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getText <em>Text</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #isSetText()
* @see #getText()
* @see #setText(String)
* @generated
*/
public void unsetText() {
text = TEXT_EDEFAULT;
textESet = false;
}
/**
* Returns whether the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getText <em>Text</em>}' attribute is set.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @return whether the value of the '<em>Text</em>' attribute is set.
* @see #unsetText()
* @see #getText()
* @see #setText(String)
* @generated
*/
public boolean isSetText() {
return textESet;
}
/**
* Returns the value of the '<em><b>Required Control Action Interval</b></em>' containment reference.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Required Control Action Interval</em>' containment reference isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Required Control Action Interval</em>' containment reference.
* @see #setRequiredControlActionInterval(DateTimeInterval)
* @generated
*/
public DateTimeInterval getRequiredControlActionInterval() {
return requiredControlActionInterval;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
public NotificationChain basicSetRequiredControlActionInterval(DateTimeInterval newRequiredControlActionInterval, NotificationChain msgs) {
DateTimeInterval oldRequiredControlActionInterval = requiredControlActionInterval;
requiredControlActionInterval = newRequiredControlActionInterval;
return msgs;
}
/**
* Sets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getRequiredControlActionInterval <em>Required Control Action Interval</em>}' containment reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Required Control Action Interval</em>' containment reference.
* @see #getRequiredControlActionInterval()
* @generated
*/
public void setRequiredControlActionInterval(DateTimeInterval newRequiredControlActionInterval) {
if (newRequiredControlActionInterval != requiredControlActionInterval) {
NotificationChain msgs = null;
if (requiredControlActionInterval != null)
msgs = ((InternalEObject)requiredControlActionInterval).eInverseRemove(this, EOPPOSITE_FEATURE_BASE - InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION_INTERVAL, null, msgs);
if (newRequiredControlActionInterval != null)
msgs = ((InternalEObject)newRequiredControlActionInterval).eInverseAdd(this, EOPPOSITE_FEATURE_BASE - InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION_INTERVAL, null, msgs);
msgs = basicSetRequiredControlActionInterval(newRequiredControlActionInterval, msgs);
if (msgs != null) msgs.dispatch();
}
}
/**
* Returns the value of the '<em><b>Required Control Action</b></em>' attribute.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Required Control Action</em>' attribute isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Required Control Action</em>' attribute.
* @see #isSetRequiredControlAction()
* @see #unsetRequiredControlAction()
* @see #setRequiredControlAction(String)
* @generated
*/
public String getRequiredControlAction() {
return requiredControlAction;
}
/**
* Sets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getRequiredControlAction <em>Required Control Action</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Required Control Action</em>' attribute.
* @see #isSetRequiredControlAction()
* @see #unsetRequiredControlAction()
* @see #getRequiredControlAction()
* @generated
*/
public void setRequiredControlAction(String newRequiredControlAction) {
requiredControlAction = newRequiredControlAction;
requiredControlActionESet = true;
}
/**
* Unsets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getRequiredControlAction <em>Required Control Action</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #isSetRequiredControlAction()
* @see #getRequiredControlAction()
* @see #setRequiredControlAction(String)
* @generated
*/
public void unsetRequiredControlAction() {
requiredControlAction = REQUIRED_CONTROL_ACTION_EDEFAULT;
requiredControlActionESet = false;
}
/**
* Returns whether the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getRequiredControlAction <em>Required Control Action</em>}' attribute is set.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @return whether the value of the '<em>Required Control Action</em>' attribute is set.
* @see #unsetRequiredControlAction()
* @see #getRequiredControlAction()
* @see #setRequiredControlAction(String)
* @generated
*/
public boolean isSetRequiredControlAction() {
return requiredControlActionESet;
}
/**
* Returns the value of the '<em><b>Switching Schedule</b></em>' reference.
* It is bidirectional and its opposite is '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingSchedule#getScheduleSteps <em>Schedule Steps</em>}'.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Switching Schedule</em>' reference isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Switching Schedule</em>' reference.
* @see #setSwitchingSchedule(SwitchingSchedule)
* @see CIM15.IEC61970.Informative.InfOperations.SwitchingSchedule#getScheduleSteps
* @generated
*/
public SwitchingSchedule getSwitchingSchedule() {
if (switchingSchedule != null && switchingSchedule.eIsProxy()) {
InternalEObject oldSwitchingSchedule = (InternalEObject)switchingSchedule;
switchingSchedule = (SwitchingSchedule)eResolveProxy(oldSwitchingSchedule);
if (switchingSchedule != oldSwitchingSchedule) {
}
}
return switchingSchedule;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
public SwitchingSchedule basicGetSwitchingSchedule() {
return switchingSchedule;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
public NotificationChain basicSetSwitchingSchedule(SwitchingSchedule newSwitchingSchedule, NotificationChain msgs) {
SwitchingSchedule oldSwitchingSchedule = switchingSchedule;
switchingSchedule = newSwitchingSchedule;
return msgs;
}
/**
* Sets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getSwitchingSchedule <em>Switching Schedule</em>}' reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Switching Schedule</em>' reference.
* @see #getSwitchingSchedule()
* @generated
*/
public void setSwitchingSchedule(SwitchingSchedule newSwitchingSchedule) {
if (newSwitchingSchedule != switchingSchedule) {
NotificationChain msgs = null;
if (switchingSchedule != null)
msgs = ((InternalEObject)switchingSchedule).eInverseRemove(this, InfOperationsPackage.SWITCHING_SCHEDULE__SCHEDULE_STEPS, SwitchingSchedule.class, msgs);
if (newSwitchingSchedule != null)
msgs = ((InternalEObject)newSwitchingSchedule).eInverseAdd(this, InfOperationsPackage.SWITCHING_SCHEDULE__SCHEDULE_STEPS, SwitchingSchedule.class, msgs);
msgs = basicSetSwitchingSchedule(newSwitchingSchedule, msgs);
if (msgs != null) msgs.dispatch();
}
}
/**
* Returns the value of the '<em><b>Power System Resources</b></em>' reference list.
* The list contents are of type {@link CIM15.IEC61970.Core.PowerSystemResource}.
* It is bidirectional and its opposite is '{@link CIM15.IEC61970.Core.PowerSystemResource#getScheduleSteps <em>Schedule Steps</em>}'.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Power System Resources</em>' reference list isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Power System Resources</em>' reference list.
* @see CIM15.IEC61970.Core.PowerSystemResource#getScheduleSteps
* @generated
*/
public EList<PowerSystemResource> getPowerSystemResources() {
if (powerSystemResources == null) {
powerSystemResources = new BasicInternalEList<PowerSystemResource>(PowerSystemResource.class);
}
return powerSystemResources;
}
/**
* Returns the value of the '<em><b>Erp Person Role</b></em>' reference.
* It is bidirectional and its opposite is '{@link CIM15.IEC61970.Informative.InfOperations.ErpPersonScheduleStepRole#getSwitchingStep <em>Switching Step</em>}'.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Erp Person Role</em>' reference isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Erp Person Role</em>' reference.
* @see #setErpPersonRole(ErpPersonScheduleStepRole)
* @see CIM15.IEC61970.Informative.InfOperations.ErpPersonScheduleStepRole#getSwitchingStep
* @generated
*/
public ErpPersonScheduleStepRole getErpPersonRole() {
if (erpPersonRole != null && erpPersonRole.eIsProxy()) {
InternalEObject oldErpPersonRole = (InternalEObject)erpPersonRole;
erpPersonRole = (ErpPersonScheduleStepRole)eResolveProxy(oldErpPersonRole);
if (erpPersonRole != oldErpPersonRole) {
}
}
return erpPersonRole;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
public ErpPersonScheduleStepRole basicGetErpPersonRole() {
return erpPersonRole;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
public NotificationChain basicSetErpPersonRole(ErpPersonScheduleStepRole newErpPersonRole, NotificationChain msgs) {
ErpPersonScheduleStepRole oldErpPersonRole = erpPersonRole;
erpPersonRole = newErpPersonRole;
return msgs;
}
/**
* Sets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getErpPersonRole <em>Erp Person Role</em>}' reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Erp Person Role</em>' reference.
* @see #getErpPersonRole()
* @generated
*/
public void setErpPersonRole(ErpPersonScheduleStepRole newErpPersonRole) {
if (newErpPersonRole != erpPersonRole) {
NotificationChain msgs = null;
if (erpPersonRole != null)
msgs = ((InternalEObject)erpPersonRole).eInverseRemove(this, InfOperationsPackage.ERP_PERSON_SCHEDULE_STEP_ROLE__SWITCHING_STEP, ErpPersonScheduleStepRole.class, msgs);
if (newErpPersonRole != null)
msgs = ((InternalEObject)newErpPersonRole).eInverseAdd(this, InfOperationsPackage.ERP_PERSON_SCHEDULE_STEP_ROLE__SWITCHING_STEP, ErpPersonScheduleStepRole.class, msgs);
msgs = basicSetErpPersonRole(newErpPersonRole, msgs);
if (msgs != null) msgs.dispatch();
}
}
/**
* Returns the value of the '<em><b>Desired End State</b></em>' attribute.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Desired End State</em>' attribute isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Desired End State</em>' attribute.
* @see #isSetDesiredEndState()
* @see #unsetDesiredEndState()
* @see #setDesiredEndState(String)
* @generated
*/
public String getDesiredEndState() {
return desiredEndState;
}
/**
* Sets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getDesiredEndState <em>Desired End State</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Desired End State</em>' attribute.
* @see #isSetDesiredEndState()
* @see #unsetDesiredEndState()
* @see #getDesiredEndState()
* @generated
*/
public void setDesiredEndState(String newDesiredEndState) {
desiredEndState = newDesiredEndState;
desiredEndStateESet = true;
}
/**
* Unsets the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getDesiredEndState <em>Desired End State</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #isSetDesiredEndState()
* @see #getDesiredEndState()
* @see #setDesiredEndState(String)
* @generated
*/
public void unsetDesiredEndState() {
desiredEndState = DESIRED_END_STATE_EDEFAULT;
desiredEndStateESet = false;
}
/**
* Returns whether the value of the '{@link CIM15.IEC61970.Informative.InfOperations.SwitchingStep#getDesiredEndState <em>Desired End State</em>}' attribute is set.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @return whether the value of the '<em>Desired End State</em>' attribute is set.
* @see #unsetDesiredEndState()
* @see #getDesiredEndState()
* @see #setDesiredEndState(String)
* @generated
*/
public boolean isSetDesiredEndState() {
return desiredEndStateESet;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@SuppressWarnings("unchecked")
@Override
public NotificationChain eInverseAdd(InternalEObject otherEnd, int featureID, NotificationChain msgs) {
switch (featureID) {
case InfOperationsPackage.SWITCHING_STEP__SAFETY_DOCUMENT:
if (safetyDocument != null)
msgs = ((InternalEObject)safetyDocument).eInverseRemove(this, InfOperationsPackage.SAFETY_DOCUMENT__SCHEDULE_STEPS, SafetyDocument.class, msgs);
return basicSetSafetyDocument((SafetyDocument)otherEnd, msgs);
case InfOperationsPackage.SWITCHING_STEP__SWITCHING_SCHEDULE:
if (switchingSchedule != null)
msgs = ((InternalEObject)switchingSchedule).eInverseRemove(this, InfOperationsPackage.SWITCHING_SCHEDULE__SCHEDULE_STEPS, SwitchingSchedule.class, msgs);
return basicSetSwitchingSchedule((SwitchingSchedule)otherEnd, msgs);
case InfOperationsPackage.SWITCHING_STEP__POWER_SYSTEM_RESOURCES:
return ((InternalEList<InternalEObject>)(InternalEList<?>)getPowerSystemResources()).basicAdd(otherEnd, msgs);
case InfOperationsPackage.SWITCHING_STEP__ERP_PERSON_ROLE:
if (erpPersonRole != null)
msgs = ((InternalEObject)erpPersonRole).eInverseRemove(this, InfOperationsPackage.ERP_PERSON_SCHEDULE_STEP_ROLE__SWITCHING_STEP, ErpPersonScheduleStepRole.class, msgs);
return basicSetErpPersonRole((ErpPersonScheduleStepRole)otherEnd, msgs);
}
return super.eInverseAdd(otherEnd, featureID, msgs);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
public NotificationChain eInverseRemove(InternalEObject otherEnd, int featureID, NotificationChain msgs) {
switch (featureID) {
case InfOperationsPackage.SWITCHING_STEP__SAFETY_DOCUMENT:
return basicSetSafetyDocument(null, msgs);
case InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION_INTERVAL:
return basicSetRequiredControlActionInterval(null, msgs);
case InfOperationsPackage.SWITCHING_STEP__SWITCHING_SCHEDULE:
return basicSetSwitchingSchedule(null, msgs);
case InfOperationsPackage.SWITCHING_STEP__POWER_SYSTEM_RESOURCES:
return ((InternalEList<?>)getPowerSystemResources()).basicRemove(otherEnd, msgs);
case InfOperationsPackage.SWITCHING_STEP__ERP_PERSON_ROLE:
return basicSetErpPersonRole(null, msgs);
}
return super.eInverseRemove(otherEnd, featureID, msgs);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
public Object eGet(int featureID, boolean resolve, boolean coreType) {
switch (featureID) {
case InfOperationsPackage.SWITCHING_STEP__SAFETY_DOCUMENT:
if (resolve) return getSafetyDocument();
return basicGetSafetyDocument();
case InfOperationsPackage.SWITCHING_STEP__STATUS_KIND:
return getStatusKind();
case InfOperationsPackage.SWITCHING_STEP__TEXT:
return getText();
case InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION_INTERVAL:
return getRequiredControlActionInterval();
case InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION:
return getRequiredControlAction();
case InfOperationsPackage.SWITCHING_STEP__SWITCHING_SCHEDULE:
if (resolve) return getSwitchingSchedule();
return basicGetSwitchingSchedule();
case InfOperationsPackage.SWITCHING_STEP__POWER_SYSTEM_RESOURCES:
return getPowerSystemResources();
case InfOperationsPackage.SWITCHING_STEP__ERP_PERSON_ROLE:
if (resolve) return getErpPersonRole();
return basicGetErpPersonRole();
case InfOperationsPackage.SWITCHING_STEP__DESIRED_END_STATE:
return getDesiredEndState();
}
return super.eGet(featureID, resolve, coreType);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@SuppressWarnings("unchecked")
@Override
public void eSet(int featureID, Object newValue) {
switch (featureID) {
case InfOperationsPackage.SWITCHING_STEP__SAFETY_DOCUMENT:
setSafetyDocument((SafetyDocument)newValue);
return;
case InfOperationsPackage.SWITCHING_STEP__STATUS_KIND:
setStatusKind((SwitchingStepStatusKind)newValue);
return;
case InfOperationsPackage.SWITCHING_STEP__TEXT:
setText((String)newValue);
return;
case InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION_INTERVAL:
setRequiredControlActionInterval((DateTimeInterval)newValue);
return;
case InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION:
setRequiredControlAction((String)newValue);
return;
case InfOperationsPackage.SWITCHING_STEP__SWITCHING_SCHEDULE:
setSwitchingSchedule((SwitchingSchedule)newValue);
return;
case InfOperationsPackage.SWITCHING_STEP__POWER_SYSTEM_RESOURCES:
getPowerSystemResources().clear();
getPowerSystemResources().addAll((Collection<? extends PowerSystemResource>)newValue);
return;
case InfOperationsPackage.SWITCHING_STEP__ERP_PERSON_ROLE:
setErpPersonRole((ErpPersonScheduleStepRole)newValue);
return;
case InfOperationsPackage.SWITCHING_STEP__DESIRED_END_STATE:
setDesiredEndState((String)newValue);
return;
}
super.eSet(featureID, newValue);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
public void eUnset(int featureID) {
switch (featureID) {
case InfOperationsPackage.SWITCHING_STEP__SAFETY_DOCUMENT:
setSafetyDocument((SafetyDocument)null);
return;
case InfOperationsPackage.SWITCHING_STEP__STATUS_KIND:
unsetStatusKind();
return;
case InfOperationsPackage.SWITCHING_STEP__TEXT:
unsetText();
return;
case InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION_INTERVAL:
setRequiredControlActionInterval((DateTimeInterval)null);
return;
case InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION:
unsetRequiredControlAction();
return;
case InfOperationsPackage.SWITCHING_STEP__SWITCHING_SCHEDULE:
setSwitchingSchedule((SwitchingSchedule)null);
return;
case InfOperationsPackage.SWITCHING_STEP__POWER_SYSTEM_RESOURCES:
getPowerSystemResources().clear();
return;
case InfOperationsPackage.SWITCHING_STEP__ERP_PERSON_ROLE:
setErpPersonRole((ErpPersonScheduleStepRole)null);
return;
case InfOperationsPackage.SWITCHING_STEP__DESIRED_END_STATE:
unsetDesiredEndState();
return;
}
super.eUnset(featureID);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
public boolean eIsSet(int featureID) {
switch (featureID) {
case InfOperationsPackage.SWITCHING_STEP__SAFETY_DOCUMENT:
return safetyDocument != null;
case InfOperationsPackage.SWITCHING_STEP__STATUS_KIND:
return isSetStatusKind();
case InfOperationsPackage.SWITCHING_STEP__TEXT:
return isSetText();
case InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION_INTERVAL:
return requiredControlActionInterval != null;
case InfOperationsPackage.SWITCHING_STEP__REQUIRED_CONTROL_ACTION:
return isSetRequiredControlAction();
case InfOperationsPackage.SWITCHING_STEP__SWITCHING_SCHEDULE:
return switchingSchedule != null;
case InfOperationsPackage.SWITCHING_STEP__POWER_SYSTEM_RESOURCES:
return powerSystemResources != null && !powerSystemResources.isEmpty();
case InfOperationsPackage.SWITCHING_STEP__ERP_PERSON_ROLE:
return erpPersonRole != null;
case InfOperationsPackage.SWITCHING_STEP__DESIRED_END_STATE:
return isSetDesiredEndState();
}
return super.eIsSet(featureID);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
public String toString() {
if (eIsProxy()) return super.toString();
StringBuffer result = new StringBuffer(super.toString());
result.append(" (statusKind: ");
if (statusKindESet) result.append(statusKind); else result.append("<unset>");
result.append(", text: ");
if (textESet) result.append(text); else result.append("<unset>");
result.append(", requiredControlAction: ");
if (requiredControlActionESet) result.append(requiredControlAction); else result.append("<unset>");
result.append(", desiredEndState: ");
if (desiredEndStateESet) result.append(desiredEndState); else result.append("<unset>");
result.append(')');
return result.toString();
}
} // SwitchingStep
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,498 |
int callback(unsigned char * _header,
int _header_valid,
unsigned char * _payload,
unsigned int _payload_len,
int _payload_valid,
framesyncstats_s _stats,
void * _userdata)
{
printf("***** gmskframesync callback invoked *****\n");
return 0;
}
int main(int argc, char*argv[])
{
unsigned int k = 2; // samples/symbol
unsigned int m = 3; // filter delay (symbols)
float BT = 0.5f; // filter bandwidth-time product
unsigned int payload_len = 40; // length of payload (bytes)
crc_scheme check = LIQUID_CRC_32;
fec_scheme fec0 = LIQUID_FEC_HAMMING128;
fec_scheme fec1 = LIQUID_FEC_NONE;
unsigned int i;
// allocate memory for payload and initialize
unsigned char header[8] = {0,1,2,3,4,5,6,7};
unsigned char payload[payload_len];
memset(payload, 0x00, payload_len);
// create frame generator and assemble
gmskframegen fg = gmskframegen_create_set(k, m, BT);
gmskframegen_assemble(fg, header, payload, payload_len, check, fec0, fec1);
// create frame synchronizer
gmskframesync fs = gmskframesync_create_set(k, m, BT, callback, NULL);
// allocate buffer for storing entire frame
unsigned int num_samples = gmskframegen_getframelen(fg) + 800;
float complex buf[num_samples];
memset(buf, 0x00, num_samples*sizeof(float complex));
// generate frame in one shot with sample offset
gmskframegen_write(fg, buf+250, num_samples-250);
// add channel gain and noise
for (i=0; i<num_samples; i++)
buf[i] = 0.3*buf[i] + 0.01f*(randnf() + randnf()*_Complex_I)*M_SQRT1_2;
// push samples through synchronizer
gmskframesync_execute(fs, buf, num_samples);
gmskframesync_print(fs);
// destroy objects
gmskframegen_destroy(fg);
gmskframesync_destroy(fs);
// export output
FILE * fid = fopen(OUTPUT_FILENAME,"w");
if (fid == NULL) {
fprintf(stderr,"error: %s, could not open '%s' for writing\n", argv[0], OUTPUT_FILENAME);
exit(1);
}
fprintf(fid,"%% %s : auto-generated file\n", OUTPUT_FILENAME);
fprintf(fid,"\n");
fprintf(fid,"clear all\n");
fprintf(fid,"close all\n");
fprintf(fid,"\n");
fprintf(fid,"num_samples = %u;\n", num_samples);
fprintf(fid,"y = zeros(1,num_samples);\n");
fprintf(fid,"\n");
for (i=0; i<num_samples; i++)
fprintf(fid,"y(%6u) = %12.4e + j*%12.4e;\n", i+1, crealf(buf[i]), cimagf(buf[i]));
fprintf(fid,"\n");
fprintf(fid,"t = 0:(num_samples-1);\n");
fprintf(fid,"figure;\n");
fprintf(fid,"plot(t, real(y), t,imag(y));\n");
fprintf(fid,"xlabel('time');\n");
fprintf(fid,"ylabel('received signal');\n");
fprintf(fid,"legend('real','imag',0);\n");
fclose(fid);
printf("results written to '%s'\n", OUTPUT_FILENAME);
return 0;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,741 |
\section{Introduction}\label{S:intro}
Given some $m \times n$ matrix $\mathbf{G}$ and $m$-vector $\mathbf{h}$, the Linear Distance Programming (LDP) problem consists of finding the $n$-vector $\mathbf{X}$ of minimum norm that satisfies the following system of inequality constraints (when a solution exists):
\begin{equation}\label{E:ldp}
\mathbf{G}\mathbf{X} \geq \mathbf{h}\ .
\end{equation}
Here the phrase ``$\mathbf{X}$ is of minimum norm'' means that $|\mathbf{X}|$ is to be minimized, subject to the set of constraints given by (\ref{E:ldp}) and provided that these constraints are consistent.
A properly implemented LDP algorithm must thus:
\begin{enumerate}
\item
find the $\mathbf{X}$ of minimum norm, when the system of constrains given by (\ref{E:ldp}) is consistent or
\item
return a flag indicating that the set of constraints given by (\ref{E:ldp}) is inconsistent and thus that no solution exists.
\end{enumerate}
The very well known and extensively utilized Lawson and Hanson software module ``Subroutine LDP'' is an implementation of this LDP problem \cite{LandH1,LandH2}. The distributions of the Lawson and Hanson algorithm(s) are generally in Fortran and can be ported from the web site netlib at:
\noindent
\ \ \ \ http://netlib.org/lawson-hanson/all\ .
\noindent
These distributions are also available at numerous other web sites such as
\noindent
\ \ \ \ http://netlib.bell-labs.com/lawson-hanson/index.html\ .
\noindent
When the inequality constraints are determined to be consistent, the Lawson and Hanson module LDP sets the flag MODE to 1 and returns a solution
vector $\mathbf{X}$. Alternatively, if the constraints are determined to be inconsistent, the Lawson and Hanson module LDP sets the flag MODE to 4 indicating that there is no solution. The prime concern here is the relatively rare occurrence of a MODE = 1 return with an associated solution vector $\mathbf{X}$ that fails to satisfy the set of constraints (\ref{E:ldp}). Although these problem cases seem to be associated with issues in handling significant digits, the erroneous returned solutions may, in fact, produce numerically sizeable violations of the specified inequality constraints.
These types of problem examples were uncovered using a strategy of systematically generating copious numbers of random cases. This general strategy is outlined in Section~2 and three specific numerical examples generated by this strategy that cause problems and have small values of $m$ and $n$ are given in Section~3 and then discussed in Section~4.
Although the Lawson and Hanson software LDP module \cite{LandH1,LandH2} is of primary interest here, the general checkout strategies and recommendations apply to other LDP implementations as well. In particular similar testing was performed on the NSWC Mathematics Library Subroutine LSEI \cite{NSWCML}, which can be used to handle LDP problems. This testing revealed that LSEI was also prone to yield a MODE = 1 return under rare circumstances when the set of constraints (\ref{E:ldp}) was inconsistent; however, the properties of the $\mathbf{G}$ matrices that caused problems appeared to be different for LSEI and LDP. In this regard, the remark at the bottom of page 393 of \cite{NSWCML} may be relevant and is worth repeating here:
\begin{enumerate}
\item
``LSEI may perform poorly if the norms of the rows of $\mathbf{A}$ and $\mathbf{E}$ differ by many orders of magnitude, or if the norms of the rows of $\mathbf{E}$ are exceedingly small.''
\end{enumerate}
Given that problems were found for the only two LDP algorithms tested and because it is clearly easy to do and entails only a small added computational burden, as a
matter of course, it is recommended that users of any LDP software module should automatically check the constraint condition (\ref{E:ldp}) of all returned solutions when the software module indicates that there is one.
\section{Random Case Generation Strategy}\label{S:random}
One significant point is that random cases that are guaranteed to satisfy some inequality constraint or other can be generated systematically. First, consider the set of inequality constraints that results for the choice $\mathbf{h} = 0$:
\begin{equation}\label{E:ldp2}
\mathbf{G}\mathbf{X} \geq 0\ .
\end{equation}
Clearly, in this case, $\mathbf{X} = 0$ is always the solution to the LDP problem since $|\mathbf{X}| = 0$.
It is easy to generalize this solution. Towards that end consider the set of conditions
\begin{equation}\label{E:ldp3}
\mathbf{G}(\mathbf{X} - \mathbf{X}_0) \geq 0\ ,
\end{equation}
where $\mathbf{X}_0$ is some (randomly) chosen vector. Clearly (\ref{E:ldp3}) always specifies a set of consistent constraints since $\mathbf{X} = \mathbf{X}_0$ is always a valid solution to the constraint conditions. Some other choice of $\mathbf{X}$ with $|\mathbf{X}| < |\mathbf{X}_0|$ may exist so $\mathbf{X} = \mathbf{X}_0$ may not be the actual solution to the LDP problem, but at least a solution is guaranteed. Thus, for any randomly chosen vales of $m$ and $n$, if $\mathbf{G}$ is a randomly generated $m \times n$ matrix and $\mathbf{X}_0$ is a randomly generated $n$-vector, then provided the $n$-component $\mathbf{h}$ vector is computed by
\begin{equation}\label{E:ldp4}
\mathbf{h} := \mathbf{G}\mathbf{X}_0\ ,
\end{equation}
a consistent solution to (\ref{E:ldp}) always exists.
This whole procedure can be carried one step further to obtain a more general set of self-consistent equations of the form (\ref{E:ldp}). Let $\mathbf{A}$ be a general (i.e., randomly generated) $l \times m$ matrix and $\mathbf{B}$ be an arbritrary invertible (i.e., also possibly randomly generated) $n \times n$ matrix. Then (\ref{E:ldp3}) yields the following form
\begin{equation}\label{E:ldp5}
\mathbf{A}\mathbf{G}\mathbf{B}\mathbf{B}^{-1}(\mathbf{X} - \mathbf{X}_0) \geq 0\ ,
\end{equation}
which can be rewritten as
\begin{equation}\label{E:ldp6}
\mathbf{G}'\mathbf{X}' \geq \mathbf{h}'
\end{equation}
where
\begin{equation}\label{E:ldp7}
\mathbf{G}' := \mathbf{A}\mathbf{G}\mathbf{B}\ \ \text{and}\ \ \ \mathbf{h}'
:= \mathbf{A}\mathbf{G}\mathbf{X}_0\ .
\end{equation}
Here the solution $\mathbf{X}'$ is related to $\mathbf{X}$ by $\mathbf{X}' = \mathbf{B}^{-1}\mathbf{X}$ and the inclusion of $\mathbf{B}$ here allows for additional freedom in the choice of $\mathbf{X}$ since it helps to de-link the solution vector from $\mathbf{X}_0$.
By generating a random $l, m$, and $n$ and then appropriate random matrices $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{G}$, along with a random vector $\mathbf{X}_0$, it is easy to test out any LDP algorithm for the case of self-consistent constraints using (\ref{E:ldp7}). Notice that in practice it is easy to generate a non-singular $\mathbf{B}$ matrix with random elements: Simply generate completely random $\mathbf{B}$ matrices and then test the result to see if $|\mathbf{B}| = 0$ and then simply regenerate the $\mathbf{B}$ matrix if it is. (Appropriate scaling factors, as well as other minor implementation issues are obvious from the examples in Section~\ref{S:random}.)
Observe that since the existence of a solution in the above derivation was predicated on the existence of equalities [i.e., the $\geq$ and not $=$ in, say (\ref{E:ldp}) or (\ref{E:ldp7})], the region where the constraints are consistent may consist of only a single point and thus (due to round-off or other factors) the LDP algorithms may potentially experience some problems in finding and testing for this one point. One way to enlarge this interior feasibility region is to replace (\ref{E:ldp}) by
\begin{equation}\label{E:ldp8}
\mathbf{G}\mathbf{X} \geq \tilde{\mathbf{h}}\ ,
\end{equation}
where $\tilde{\mathbf{h}} = {\mathbf{h}} - \mathbf{C}$, for some constant or random vector $\mathbf{C}$, all of whose components are positive. It is also easy to modify the above random case generation strategy so that inconsistent constraints are produced. One strategy, for example, is to simply replace $\tilde{\mathbf{h}}$ in (\ref{E:ldp8}) by $\tilde{\mathbf{h}} = {\mathbf{h}} + \mathbf{D}$ for some random vector $\mathbf{D}$ with sufficiently large components.
\section{Numerical Examples of Randomly Generated Problem Cases}\label{S:examples}
Three numerical examples of inconsistent inequality constraints are given below. When the input for each of these cases is used in the Lawson and Hanson Fortran Subroutine LDP implemented on a SUN workstation using \emph{double precision}, a MODE = 1 return results and a candidate solution vector is returned, whereas there should be a MODE = 4 return indicating that no solution exists. Specific problem cases are by nature somewhat finicky and difficult to replicate and may not cause similar problems on other computer systems---they are, in fact, highly dependent on the number of internal significant digits used for the inputs, as well as in the internal computations themselves. (SUN architecture reuses the mantissa when going from single to \emph{double precision} so there are generally 16 or so significant digits of internal representation rather than the 14 or so one might normally expect.) This, however, does not mean that the problem is tied to only SUN systems or that they are more common on SUN systems; rather, it means that different specific cases will cause problems on different systems. Given this state of affairs, the reader may well not be able to use the input cases below to generate inconsistent returns from Subroutine LDP and may thus have to use the strategies outlined above to generate such cases randomly. [Generally, $\mathbf{A}$ and $\mathbf{B}$ can be taken to be identity matrices, but a certain number of columns of the random $\mathbf{G}$ should be zero and since (for what might be typical reasonable parameter settings designed to generate such cases and an associated well-chosen random test case generation methodology) the incidences of such problem cases might usually be only approximately one in a thousand or so, probably at least 50,000 or so cases of various types and with parameter settings should normally be generated.] In passing, it is worth noting that the problems addressed here are not tied specifically to Fortran: a C programming language version of the Lawson and Hanson LDP algorithm was also tested and displayed the same problems.
Finally, before considering the examples, two observations are in order. First, notice that the Lawson and Hanson in-line function ``DIFF'' is used in their LDP implementation to help prevent internal significant digit masking, but this, in itself, does not appear to resolve the problems indicated here. Second, when these same inputs are used with a \emph{type quad} internal Fortran version (which corresponds to approximately 32 internal digits of representation), then they all yield MODE = 4 returns, but it seems clear that simply going to a \emph{type quad} implementation will not solve the problem (although it will lessen the incidence of it)---it will only make counter-examples harder to find.
Although the cases $m >> n >> 0$ are the ones that generate the most problem cases, for simplicity only three cases with $m = 4$ and $n = 2$ are considered.
\vskip .2in
\noindent
Case 1
\begin{equation}\label{E:realT}
\mathbf{G} =
\left[
\begin{array}{r l}
-89.20509815216064 & 0.0000000000000000\\
74.79768991470337 & 0.0000000000000000\\
66.23740792274475 & 0.0000000000000000\\
-18.51919293403625 & 0.0000000000000000\\
\end{array}
\right]\! \ \ \ \ \ \
\mathbf{h} =
\left[
\begin{array}{r}
-12073.43407295207\\
10123.19482867013\\
8350.549301112449\\
-24612.94532321187\\
\end{array}
\right]
\end{equation}
\vskip .2in
\noindent
Case 2
\begin{equation}\label{E:realT2}
\mathbf{G} =
\left[
\begin{array}{r l}
81.82253837585449 & 0.0000000000000000\\
-74.02672171592712 & 0.0000000000000000\\
0.0000000000000000 & -17.36225485801697\\
-89.47155475616455 & 0.0000000000000000\\
\end{array}
\right]\! \ \ \ \ \ \
\mathbf{h} =
\left[
\begin{array}{r}
-77004.09890544150\\
69248.37468031116\\
11241.52852765946\\
84233.37742495652\\
\end{array}
\right]
\end{equation}
\vskip .2in
\noindent
Case 3
\begin{equation}\label{E:realT3}
\mathbf{G} =
\left[
\begin{array}{r l}
-3.057897090911865 & 0.0000000000000000\\
4.310655593872070 & 0.0000000000000000\\
39.13614749908447 & 0.0000000000000000\\
84.55699086189270 & 0.0000000000000000\\
\end{array}
\right]\! \ \ \ \ \ \
\mathbf{h} =
\left[
\begin{array}{r}
2192.778913749731\\
-3422.354440768027\\
-28760.98260603488\\
-60562.89687439907\\
\end{array}
\right]
\end{equation}
\section{Discussion of Randomly Generated Problem Case Outputs}\label{S:random2}
For each of the above examples, it is easy to verify directly that the system of constraints implied by (\ref{E:ldp}) is consistent; however, when the input for each of these examples was used by the author in the Lawson and Hanson Subroutine LDP implemented in \emph{double precision} on a SUN system, a MODE = 1 return resulted with an improper solution vector. Specifically the returned solution vectors were:
\vskip .2in
\noindent
Case 1
\begin{equation}\label{E:real2}
\mathbf{X} =
\left[
\begin{array}{r}
-.3750000000000000\\
.0000000000000000\\
\end{array}
\!\!\!\!\!\!
\begin{array}{c}
\\
\\
\\
\end{array}
\right]
\end{equation}
\vskip .2in
\noindent
Case 2
\begin{equation}\label{E:realT5}
\mathbf{X} =
\left[
\begin{array}{c}
.6074218750000000\\
.0000000000000000\\
\end{array}
\!\!\!\!
\begin{array}{c}
\\
\\
\\
\end{array}
\right]
\end{equation}
\vskip .2in
\noindent
Case 2
\begin{equation}\label{E:realT6}
\mathbf{X} =
\left[
\begin{array}{c}
.4570312500000000\\
.0000000000000000\\
\end{array}
\!\!\!\!
\begin{array}{c}
\\
\\
\\
\end{array}
\right]
\end{equation}
For these cases, direct numerical testing shows that the problems seem to be in the Subroutine LDP itself and not in the other subroutines it calls. More specifically, using Lawson and Hanson's notation \cite{LandH1,LandH2}, except for the use of bold face type for vectors and matrices, from the returns supplied to Subroutine LDP from Subroutine NNLS (which LDP calls), the basic vectors and matrices $\mathbf{E}$, $\mathbf{r}$, $\mathbf{p}$ and $\hat{\mathbf{x}}$ can be determined. \{Specifically, see equations (23.28) through (23.34) in \cite{LandH1} or \cite{LandH2}.\} Using these computed quantities it easy to show that due to round-off and/or masking the basic assumptions in proving the validity of the LDP algorithm do not hold; For example, that some of the components of the $\mathbf{p}$ vector are negative. For these specific inputs all of these inconsistencies go away when \emph{type quad} variables are used, but as indicated above, one would expect to be able to find similar cases when a full set of \emph{type quad} random cases (one tricky part here appears to be finding a full-up \emph{type quad} random number generator). Finally, with regards to \emph{type quad} Fortran implementations, it is perhaps worth noting that there is a SUN $f77$ compiler option that automatically converts from \emph{type double} to \emph{type quad}.
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BA (Hons) in Business (Management) | Munich»
BA (Hons) in
Business (Management)
Six semesters/three years
October & February
EQF level 6/NFQ level 8
Quality and Qualifications, Ireland (QQI)
The BA (Hons) in Business (Management) is a three-year full-time program (180 ECTS) that is designed to provide a strong grounding in fundamental business concepts, ideas, practices and methodologies. Throughout this EQF level 6/NFQ level 8 honors program, students will undertake comprehensive and critical analysis of business organisations, functions, processes and management techniques. Students will acquire a good understanding of general business theory and practice and the critical knowledge and skills to contribute effectively to the resolution of business problems.
A broad spectrum of career opportunities is available to graduates of the BA (Hons) in Business (Management), including in areas such as marketing, sales, operations, information technology, finance and general management.
A state-recognized Dublin Business School BA (Hons) in Business degree awarded by Quality & Qualifications Ireland (QQI).
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About Our Academic Partner
Dublin Business School (DBS) is a private higher education institution and Ireland's largest accredited independent provider of higher education. Its programs on the Irish National Framework of Qualifications (NFQ) are accredited by Quality and Qualifications Ireland (QQI), the state body responsible for quality assurance of all education and training services in Ireland.
At EU Business School Munich, students earn a state-recognized Dublin Business School BA (Hons) in Business degree awarded by Quality and Qualifications Ireland (QQI).
Former winner of the Overall Excellence in Education award, DBS is currently shortlisted as the #1 higher educational institution for:
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A worldwide and known advocate of sustainability, Paul Polman, co-founder of IMAGINE and former CEO of Unilever, spoke to EU students about the importance of sustainability and how companies have a duty to stakeholders to undertake respectful environmental practices for future generations.
Develop innovative
and a flexible mindset
Study a state-
recognized degree
from the U.K.
Studying a BA (Hons) in Business (Management) in Munich allows students to expand their business knowledge in the economic powerhouse of Germany. The vibrant Bavarian capital is home to numerous major multinational company headquarters including BMW and Allianz, as well as a flourishing startup scene, and is also actively pursuing initiatives to provide a greener, more sustainable living environment.
BBA - Bachelor of Business Administration | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
\label{sec:Introduction}
Solid state semiconductor single photon emitters, such as, self-assembled quantum dots (QDs) \cite{1a,1b,1c}, have recently attracted plenteous attention for a wide range of application in quantum information and communication \cite{2a,2b,2c,2d,2e}. They can be used as deterministic photon sources, comparable to single atoms and molecules, impurities in semiconductors, etc.
Semiconductor QDs are powerful candidates for single photon and entangled photon pair generation for applications in state-of-the-art highly secure communication and quantum computation \cite{3a,3b}. An operating range in the telecommunication O and C bands (1.3 and 1.55 $\mu$m) is favourable to insure the applicability of standard silca based fibres and fibre networks. High photon extraction efficiency from single QD devices is another significant factor to suppress the bit error rate.
The inter-dot coupling \cite{Zho,q1,q2,q3,q4,q5,q6} as well as the interaction of the QDs and the confined carriers with their environment are important phenomena, either to exploit them to manipulate the quantum state \cite{i1,i2}, or to suppress them in order to keep the QD of interest as isolated as desired. A detailed understanding and description of these interactions gains importance when the system temperature is increased above liquid Helium temperature (4.2 K) and towards technically easier achievable temperatures above liquid Nitrogen temperature (77 K), because thermally activated processes come into play.
We realized metal-embedded tapered mesa (nano-cone) structures to meet the challenge of studying a low number of QDs in the telecommunication O and C bands and to provide low loss photon extraction, in a comparable approach to the demonstration by Takemoto, et al.~\cite{Tak1}. The advantages of the proposed and realized nano-cones are their fast and highly reproducible nanometer- (or micrometer-)sized fabrication, the possibility to integrate them in more complex devices and the efficient light extraction by embedding them inside reflective metal.
In this paper we briefly discuss the nano-cone structure fabrication process with the main purpose of isolating individual dots and creating nanostructures of efficient photon extraction.
The excitons confined to individual QDs as well as their relaxation and recombination are investigated in detail with the focus on their temperature dependence. The thermally induced exciton transfer between and escape from QDs is described using a simple model. The optical spectroscopy results furthermore present wavelength-tunable/selectable single QD emission towards the realization of a practical single photon source in both the telecommunication O and C bands.
\section{Samples and experiment}
\label{sec:Experiment}
\begin{figure}[b]
\centering
\includegraphics[scale=.25]{F1.jpg}
\caption{(a) PL of the 4 and 6 ML InAs QD samples at 10 K, including the emission from the InP substrate and the In$_{0.53}$Ga$_{0.22}$Al$_{0.25}$As barrier. (b) Schematic illustration of the PL structure, a metal-embedded cone, with a thin insulator layer (SiO$_2$/Si$_3$N$_4$) between the metal and the semiconductor part and the QDs located in the center of the InGaAlAs barrier. (c) Side view SEM image of an as etched 26$^\circ$ tapered nano-cone with a tip diameter of around 100 nm and a diameter of the QD containing plane of $d_{\rm QD}\approx250$~nm, (d) top-view SEM image of the respective turned-around and SiO$_2$/Ti-embedded nano-cone with removed substrate.}
\label{fig:Fig1}
\end{figure}
The investigated samples contain high-density ($\sim 1 \times 10^{11}$ cm$^{-2}$) InAs QDs grown on a lattice matched 150 nm thick In$_{0.53}$Ga$_{0.22}$Al$_{0.25}$As buffer layer on InP(311)B substrates by molecular beam epitaxy \cite{growth,Aka1}. For optical measurements the QDs were capped with another 150 nm thick In$_{0.53}$Ga$_{0.22}$Al$_{0.25}$As barrier layer. QD emission between 1.2 and 1.6~$\mu$m ($1.03-0.77$ eV) could be achieved by varying the nominal thickness of the optically active QD layer between four and six monolayers (MLs); the photoluminescence (PL) of these as grown, unstructured samples is displayed in \fref{fig:Fig1}~(a). These QD samples have no wetting layer, which can be concluded from the PL spectra, where no indication for the respective luminescence is found \cite{xx}. The absence of a wetting layer has vast consequences on the QD optical properties and charge redistribution processes, which will be discussed in the following sections.
Sequential steps of electron-beam lithography and etching allowed for a well-controlled and high yield fabrication of nano-cones with typical 26$^{\circ}$ taper angles, heights of 300~nm and tip diameters down to less than 100~nm (\fref{fig:Fig1}~(b, c)). After the deposition of a thin insulating layer ($10-60$ nm SiO$_2$ or Si$_3$N$_4$), the nano-cones were embedded in metal (Titanium (Ti) or Silver (Ag)). In the subsequent step, the InP substrate was removed and the sample was turned upside down, which is highlighted in the schematic illustration and scanning electron microscope (SEM) top view image in \fref{fig:Fig1}~(b) and (d) (more details on the fabrication are presented in Huh, et al.~\cite{Huh1}). In the further text the diameter of the QD containing plane, $d_{\rm QD}$, of the nano-cones is given when nano-cone sizes are specified.
The PL of such turned-around and metal-embedded nano-cones was investigated in a micro-PL setup optimized for the near infrared (NIR) spectral range. The samples were cooled in a Helium-flow cryostat to access a temperature range from 4 K to room temperature. For non-resonant excitation a HeNe laser (632.8 nm; 1.959 eV) was focused on the sample using an NIR-coated $\rm NA=0.42$ objective lens, which results in a spot size of approximately 1.5 $\mu$m. The PL was collected with the same objective lens and dispersed by a 50 cm double monochromator with 300 gr./mm grating and detected by a liquid Nitrogen cooled InGaAs-photodiode array detector (cutoff $\approx$ 1.6 $\mu$m).
\section{Quantum dot luminescence at around 1.3 $\mu$m}
\label{sec:1.3}
In this section the focus of the experimental results and the following discussion is based on the smaller 4 ML QDs which have their low temperature PL centered at around 1.3 $\mu$m (0.95 eV). Using these QDs the temperature dependence up to the complete quenching can be investigated within the sensitivity range of our detector. The PL wavelength furthermore coincides with the telecommunication O band.
\subsection{Quantum dot photoluminescence power and temperature dependence}
\label{sec:QDPLPowerAndTemperatureDependence}
\begin{figure}
\centering
\includegraphics[scale=.5]{F2.jpg}
\caption{(a) PL power dependence of a $d_{\rm QD}\approx250$~nm SiO$_2$/Ti-embedded nano-cone under non-resonant excitation at 4 K. Four groups (G1--G4) of QD emission lines are indicated according to their spectral position for later identification. (b, c) Three PL lines of one single QD at around 1 eV and (b, d) four PL lines at around 0.905 eV at different excitation powers. The power-dependent integrated PL peak intensities are fitted with a power law ($I \propto P_{\rm Exc}^S$).}
\label{fig:AaPowerAll}
\end{figure}
The complete PL spectrum from a nano-cone ($d_{\rm QD}\approx250$~nm) is displayed in \fref{fig:AaPowerAll}~(a), where at low excitation power a total of around ten optically active QDs are visible between around $0.88$ and $1.02$ eV. With increasing excitation power additional PL lines at higher energy emerge which are due to excited state recombination in the QDs. At the highest excitation power, the InGaAlAs barrier luminescence can be seen at around $1.180$ eV; no wetting layer exists for this type of QD samples.
For the first 4 ML QD, which is shown on the left side of the more detailed close-up \fref{fig:AaPowerAll}~(b, c), X0 and X* exhibit an almost linear dependence ($S \approx 0.9$) of their intensities, $I$, on excitation power, $P_{\rm Exc}$, in the respective power law, $I \propto P_{\rm Exc}^S$. This indicates their origin from neutral and (negatively) charged exciton recombination. XX shows a super-linear dependence ($S \approx 1.35$) indicating its origin from biexciton recombination.
The second presented dot on the right side of the close-up, \fref{fig:AaPowerAll}~(b, d), does not show the same clear power dependence; however, X1 and X3 depend slightly super-linearly on the excitation power, whereas X2 and X4 exhibit a sub-linear power dependence. Given the relative peak energies and the power dependencies, X1 ($S \approx 1.17$) might be due to a charged biexciton or a positive trion and X3 ($S \approx 1.05$) due to a neutral biexciton. The other two peaks are likely assigned as follows: X2 ($S \approx 0.72$) to a neutral exciton and X4 ($S \approx 0.72$) to a (negative) trion. For the following discussion the exact origin of the recombination lines is not further discussed and of no major importance.
The PL of the same nano-cone with all contained optically active QDs was examined depending on the excitation power and as function of the sample temperature between 10 and 160 K. The spectra obtained under low, medium and high (above exciton saturation) power conditions are displayed in \fref{fig:AaTemp}. All three presented data sets, (a--c), are to compare, i.e., their scalings and relative offsets are chosen identically, only the lowest-power data (a) are displayed on half the intensity scale and thus with half the offsets for more clarity.
Comparing the PL temperature dependence for the different excitation regimes the first obvious observation is that for higher power excitation the PL quenching seems much less pronounced, i.e., the PL persists up to higher temperatures. This might be intuitively attributed to refilling processes that take place under non-resonant excitation, where the carriers can repopulate the QDs which leads to a ``slower" quenching of all dots.
Another rather obvious qualitative observation is the emergence of the excited states which results in a major modification of the PL temperature dependence in the high-power case (\fref{fig:AaTemp}~(c)).
More detailed examination, analysis and discussion of the temperature dependence of both the complete nano-cone PL (QD ensemble) and of individual QD PL (single QDs) is carried out in the following section.
\begin{figure}
\centering
\includegraphics[scale=.5]{F3.jpg}
\caption{PL temperature dependence from the same nano-cone shown in \fref{fig:AaPowerAll} between 10 and 160 K for three different excitation conditions: (a) low power (1 $\mu$W, well below exciton saturation), (b) medium power (10 $\mu$W, approaching saturation), and (c) high power (100 $\mu$W, above saturation) non-resonant excitation.}
\label{fig:AaTemp}
\end{figure}
\subsection{Analysis and discussion}
\label{sec:Disc1.3}
The temperature dependent PL is analyzed by fitting with a model that accounts for both the carrier capture and the quenching temperature dependence,
\begin{equation}
I(T) = I_0\left( p + \frac{1-p}{\exp{\left[k_{\rm B} T_0 / (k_{\rm B} T)\right]}}\right) \times \frac{1}{ 1 + \sum\limits_{i=1,2} b_i \exp{\left[-E^{({\rm a}i)} / (k_{\rm B} T)\right]}}\ .
\label{Eq:Arrha2}
\end{equation}
Therein, the first factor describes the temperature dependent excitation (carrier capture) which takes place under non-resonant excitation and the second factor describes the thermal quenching of the states for increasing temperatures \cite{Gel}, which includes two distinct escape mechanisms labeled with the index $i=1,2$. The parameters in the excitation term are the total (maximum) intensity $I_0$, the dimensionless weighting parameter $p$, which defines the ratio between temperature independent and temperature dependent excitation contributions, and the temperature constant $T_0$, which defines a temperature equivalent for the excitation process; $k_{\rm B}$ is the Boltzmann constant and $T$ is the system temperature. In the quenching term, $E^{({\rm a}i)}$ are activation energies that are assigned to a certain escape mechanism and $b_i$ are the corresponding dimensionless fitting coefficients. The meaning of these coefficients $b_i$ can be understood as the ratio of escape and capture rates of the described mechanism, i.e., $b_i\propto$ (escape-rate)$_i/$(capture-rate)$_i$, as it is discussed and derived in a comprehensive fashion by Le Ru, et al. in~\cite{LeR}.
\begin{figure}
\centering \includegraphics[scale=.5]{F4.jpg}
\caption{(a) Temperature dependence of the integrated PL of the complete spectra displayed in \fref{fig:AaTemp} for low, medium and high excitation powers (markers) and the respective fits with equation \eref{Eq:Arrha2} (filled solid lines). (b) Temperature dependence of the ensemble PL data acquired from an unstructured QD sample for comparison \cite{xx}; the emission energy of the ensemble (black circles, obtain from Gaussian line fits), with the dashed grey guide lines obtained from fitting the low and high temperature data by Varshni empirical equation ($E(T) = E_0 - (a/T^2)/(b + T)$), and the ensemble PL FWHM (red diamonds).}
\label{fig:AXEns}
\end{figure}
\begin{table}
\centering \footnotesize
\begin{tabular}{c|*{3}c|*{4}c} \hline
$P_{\rm Exc}$ & $I_0$ & $p$ & $T_0$ \footnotesize (K) & $b_1$ & $E^{({\rm a}1)}$\footnotesize (meV) & $b_2$ & $E^{({\rm a}2)}$\footnotesize (meV) \\ \hline
100 $\mu$W & $2.2\cdot10^5$ & $0.0084$ & $129\pm 1$ & $184\pm1$ & $13.0\pm0.1$ & $1.4\cdot10^4\pm0.02\cdot10^4$ & $\ 73\pm1$ \\
10 $\mu$W & $1.0\cdot10^5$ & $0.0055$ & $175\pm3$ & $410\pm6$ & $18.4\pm0.3$ & $3.6\cdot10^6\pm0.2\cdot10^6$ & $135\pm6$ \\
1 $\mu$W& $5.5\cdot10^4$ & $0.0014$ & $166\pm6$ & $1058\pm73$ & $18.0\pm0.8$ & $5.4\cdot10^7\pm1.1\cdot10^7$ & $139\pm23$ \\ \hline
\end{tabular}
\caption{Fit parameters (equation \eref{Eq:Arrha2}) for the integrated PL from the nano-cone, i.e., from the contained QD sub-ensemble; the $1\sigma$-fit-errors are omitted when of no significant magnitude ($<<1\%$ of the parameter value).}
\label{tab:IntegratedPL}
\end{table}
We start with a comprehensive analysis of the ensemble PL of all QDs within the investigated nano-cone, which is later referred to as QD sub-ensemble. This is done by integration of the total PL intensities within the spectral range displayed in \fref{fig:AaTemp}. The obtained integrated intensities for the three different excitation powers are displayed as a function of inverse temperature in \fref{fig:AXEns}~(a). The selection of displaying the inverse temperature dependence has been made to show the low temperature side in a more distinctive way. This addresses the fact that especially the carrier capture (and also re-excitation) processes are heavily dependent on the power regime. The experimental data were fitted using the afore introduced equation \eref{Eq:Arrha2} and all obtained fitting parameters are summarized in table \ref{tab:IntegratedPL}.
On the excitation side, it is obvious that in case the excitation power remains below the exciton saturation, i.e., for 1 $\mu$W (blue circles) and 10 $\mu$W (red squares), the carrier capture process is rather inefficient, which is revealed by the large obtained characteristic temperature $T_0\approx 170$ K and the small values of the corresponding constant $p\approx0.0014$ and 0.0055. The increase in PL intensity up to 30 K in the respective data-sets highlights this behavior.
For increasing excitation power, the carrier capture process becomes more efficient or, in other words, less temperature sensitive, indicated by increasing values of $p$ and decreasing values of $T_0$. This trend is highlighted by the flattening of the data on the low temperature side with increasing excitation power. For high power excitation above the exciton saturation (100 $\mu$W, black diamonds), also $T_0$ significantly decreases to 129 K and $p$ further increases.
On the decay side, the following trend can be seen. For both the low and medium power excitation regimes, which are with no significant contribution of excited states, very similar behavior is observed. This includes the two different escape mechanisms with activation energies of $E^{({\rm a}1)}\approx18$ meV and $E^{({\rm a}2)}\approx140$ meV, respectively, which are almost the same for the low and medium power conditions. For the high power case, in contrast, especially the larger activation energy becomes significantly smaller, $E^{({\rm a}2)}\approx73$ meV.
We assign the first mechanism, $i=1$, to a process unique for high-density QD ensembles, which is related to inter-dot coupling \cite{Zho}, and the second mechanism, $i=2$, to the thermal carrier escape from QDs to the barrier.
The discussion of the second mechanism, which leads to the significantly reduced activation energy at high excitation power, directly explains the impact of the populated excited states and can be understood rather straight forwardly and intuitively. The carrier escape is directly related to non-radiative carrier escape to the InGaAlAs barrier, which embeds the QDs, and leads to a linear relation between the barrier-dot energy separation and the activation energy $E^{({\rm a}2)}$. The energy separation between the barrier and the QD excited states (no matter what kind of states they are) is significantly lower than for the QD ground states, and thus their enhanced population at higher excitation rates leads to a reduction of the corresponding activation energy $E^{({\rm a}2)}$.
The accompanying fact that also $b_2$ is smallest at high excitation power supports the aforementioned occurrence of re-excitation, as it indicates a larger corresponding carrier (re-)capture-rate. More details, including a quantitative discussion and explanation of the activation energy magnitudes, is carried out for single QDs in section \ref{sec:RelativeQDIntensitiesCouplingAndQuenching}.
The first escape mechanism, however, needs a more in-depth description, which is provided by taking a closer look at an ensemble of the same type of QDs (\fref{fig:AXEns} (b)). For this reference measurement, an unstructured sample and a macroscopic PL setup was used in order to simultaneously access a larger number of QDs for a better statistical distribution and thus ensemble average \cite{xx}. The temperature dependence of a QD ensemble under low power excitation conditions indicates that, first, the peak emission energy exhibits a pronounced shift to lower energies when compared to the expected trend using a fit by Varshni empirical equation and, second, the PL ensemble line width decreases with increasing temperature up to around 140 K. These observations can be well explained by a charge-carrier redistribution from the higher to the lower energy dots of the ensemble, which is promoted by an increased temperature and thus enhanced phonon scattering and thermal excitation.
In order to conclude a corresponding carrier escape process from these rather obvious facts, we suggest to introduce coupled excited states (CES) which are delocalized over many (or even all) dots of the sub-ensemble. When carriers are ones excited to such a delocalized state, they can either relax favourably to the lowest energy states, which results in the redistribution in favour of the lower energy dots, or be subsequently further excited to the barrier and thus escape the QDs. This process is also discussed in more detail for individual QDs in section \ref{sec:RelativeQDIntensitiesCouplingAndQuenching}, where the relative change of intensities and the change of related activation energies allows for much better insight.
To apply this mechanism on the herein described QD sub-ensemble, it shall only be added, that for all excitation conditions such small activation energies to a CES could be obtained ($E^{({\rm a}1)}\approx13-18$ meV). Moreover, with increasing power, the corresponding $b_1$ parameter decreased, which again indicates an enhancement of the corresponding (re-)capture-rate, i.e., the relaxation from CES to a QD ground state.
We want to add that in contrast to typical self-assembled InAs/GaAs QDs, a remarkably similar behavior for ensembles of high-density InAs/GaAs quantum posts has been reported \cite{He,Kre,Vol}. This includes both the intensity increase and the charge redistribution from higher to lower energy states with increasing temperature. In analogy to our QDs discussed in the present paper, also these quantum posts are not coupled to a wetting layer rather than to a surrounding matrix-quantum well as illustrated in \cite{Vol}.
\subsubsection{Single quantum dot spectral properties.}
\label{sec:SingleQDSpectralChange}
In the next step we investigate the temperature dependent phenomena within the nano-cone in a more detailed way and from a microscopic view point with the purpose to describe what happens on the single QD level and what promotes the inter-dot coupling. Therefore, we take another and closer look on the temperature dependence of the nano-cone PL under low excitation power, which is represented by the detailed data set displayed in \fref{fig:AaT001All}. This in-depth analysis includes multi-peak fits to all featured QD PL lines, accounting for changing line shapes and emission energies with increasing temperature.
\begin{figure}
\centering
\includegraphics[scale=.5]{F5.jpg}
\caption{Detailed temperature series ($10-160$ K) under low power non-resonant excitation and line fits to all QD spectra. The PL lines, \emph{``Peaks 1--5''} are highlighted for later identification. The data and respective line colours at 10, 40, 80, 120 and 160 K are the same as shown in \fref{fig:AaTemp}~(a). The line fitting was obtained using quasi-Voigt functions and using a superposition of Lorentzian and Gaussian functions with changing weight for increasing temperatures (details in \fref{fig:AbPeak3}).}
\label{fig:AaT001All}
\end{figure}
The behaviour of one single PL line, \emph{``Peak 3''}, is exemplified in \fref{fig:AbPeak3}. The zero-phonon line (ZPL), which is represented by the Lorentzian component with an initial line width of 0.17 meV, dominates the spectral shape at low temperatures and initially gains intensity for T $\leq$ 30 K and then starts to fade for T $>$ 30 K. At around $50-70$ K the acoustic phonon sidebands, represented by the Gaussian component, exhibit comparable intensities \cite{Ari}. Additional to acoustic phonon broadening, also charge fluctuations and contributions of additional capture and decay processes related to QD coupling and higher energy phonons might contribute to this broadening for increasing temperatures. The transition regime is highlighted for 60 K as inset in \fref{fig:AbPeak3}~(a), featuring the two distinct components, the ZPL (solid black line) and the phonon sidebands (dashed grey line). For T $>$ 80 K the phonon sideband intensity decreases slowly, while the ZPL rapidly diminishes and eventually vanishes completely at around 120 K (\fref{fig:AbPeak3} (b)). This particular recombination line can at least be identified as a single PL peak up to around 100 K without significant background or overlap from other features.
Although it remains visible for temperatures up to 160 K, eventually background and overlapping of different features become more significant for T $\geq$ 120 K (\fref{fig:AaT001All}).
The initial intensity increase up to 30 K can be explained as before by a not perfectly efficient carrier capture from the barrier to the QD under low power non-resonant excitation. The temperature dependence of the initial carrier capture is again included in the quantitative discussion and modeling of the individual QD temperature dependence, which follows in section \ref{sec:RelativeQDIntensitiesCouplingAndQuenching}.
The relative total line width broadening can be assigned to the two main contributions that are highlighted in \fref{fig:AbPeak3} (c), the broadening of the ZPL, which will be discussed later, and the change of relative weight between the narrower ZPL and the broader acoustic phonon sidebands with increasing temperature.
\begin{figure}
\centering
\includegraphics[scale=.5]{F6.jpg}
\caption{(a) Zoomed view of \emph{``Peak 3''} from \fref{fig:AaT001All} to determine the temperature dependent line shape transition from the pure ZPL to the predominantly acoustic phonon broadened spectral appearance, applying Lorentzian and Gaussian fits, respectively.
The inset presents the fit curves to the 60 K data, i.e., the total fit (thick red line), the ZPL part (solid black line) and the phonon sideband part (dashed grey line).
(b) Temperature dependence of the ZPL (Lorentzian, black squares) and phonon sideband (Gaussian, grey circles) components of the integrated PL intensity and (c) the respective line width (FWHM) components; the red lines indicate the ratios of the ZPL to phonon sideband values.}
\label{fig:AbPeak3}
\end{figure}
In the following paragraph, the temperature dependence of individual QD emission, exemplified by the four PL lines highlighted as \emph{``Peaks 1--4''} in \fref{fig:AaT001All}, is analyzed using phonon-carrier coupling. The respective emission energy and line width data as well as the corresponding fits are summarized in \fref{fig:AbLines}.
First, we discuss the change in the QD PL emission energies with increasing temperature (\fref{fig:AbLines} (a)).
Assuming phonon-carrier coupling as the most relevant mechanism, leads to an empirical fit function based on a Bose-Einstein statistical factor with an average phonon energy, ${<E_{\rm ph}^{\rm E}>}$, which causes the energy shift, as introduced, e.g., in \cite{Cin} (among various others),
\begin{equation}
E(T) = E_0 - \frac{S_{\rm c}}{\exp[{<E_{\rm ph}^{\rm E}>} / (k_{\rm B} T)] - 1}\ .
\label{Eq:shift1}
\end{equation}
$E_0$ is the zero-temperature energy and $S_{\rm c}$ is the constant describing the strength of the electron-phonon coupling.
Since the fitted average phonon energy values, ${<E_{\rm ph}^{\rm E}>}$, of all peaks are between 12 and 14 meV, which is significantly lower than the bulk LO($\Gamma$) phonon energies of GaAs (35 meV) or InAs (29 meV), a significant contribution of acoustic phonons can be assumed (TA(X) of GaAs: 10 meV, InAs: 7 meV). The value of the coupling constant varies between $S_{\rm c}=$ 32 and 45 meV. The obtained result of a major contribution of the acoustic phonons is in good agreement with the afore discussed crossover between ZPL and accoustic phonon sideband dominated PL.
\begin{figure}
\centering
\includegraphics[scale=.5]{F7.jpg}
\caption{(a) The temperature dependent line shifts of four selected peaks, \emph{``Peaks 1--4''} as identified in \fref{fig:AaT001All}. (b) The line widths (FWHM) as a function of temperature for the same four selected peaks. All emission energy and line width values of the analyzed peaks are obtained from the data displayed in \fref{fig:AaT001All}. The corresponding fits to the PL energy and FWHM data, according to equations~(\ref{Eq:shift1}, \ref{Eq:fwhm1}) are shown as solid lines. Mind that the data points are only displayed in the temperature range up to the quenching of the respective peak and, for the FWHM data, until no more reliable evaluation due to peak overlap was possible.}
\label{fig:AbLines}
\end{figure}
Second, we discuss the alteration and broadening of the QD emission line width (\fref{fig:AbLines} (b)).
The suggested model, again accounting for electron-phonon interaction mediated broadening, results in another empirical fit function \cite{Cin} similar to equation~\eref{Eq:shift1},
\begin{equation}
\Gamma(T) = \Gamma_0 + \frac{\Gamma_1}{\exp[{<E_{\rm ph}^{\rm B}>} / (k_{\rm B} T)] - 1}\ ,
\label{Eq:fwhm1}
\end{equation}
where $<E_{\rm ph}^{\rm B}>$ is the energy of the phonons causing the line broadening, $\Gamma_0$ is the zero-temperature line width (intrinsic broadening) and $\Gamma_1$ is the broadening constant.
The fitting of \emph{``Peaks 1--3''} for temperatures up to around 100 K reveals similar values for the phonon energies, ${<E_{\rm ph}^{\rm B}>} = 22 - 32$ meV, with the average of around 27 meV, which coincides well with the bulk LO($\Gamma$) phonon energy of InAs, and which is also close to the bulk LO($\Gamma$) phonon energy of the InGaAlAs barrier \cite{Borr}. (Note that reliable line width values for \emph{``Peak 4''} could not be extracted for $T\geq 50$ K because of the increasing overlap with a neighbouring PL line, and that the respective data were thus not considered.)
The coupling constant, $\Gamma_1$, however, varies quite significantly between the peaks ($\Gamma_0 \approx 30 - 160$ meV) as it strongly depends on the exact environment of the respective dot, its size, potential, strain (anisotropy), local piezoelectric field, presence of defects, etc.
As can be seen in \fref{fig:AbLines}~(b), the line width of the peaks remains almost unchanged on the order of $0.1-0.3$ meV for temperatures below around $40$ K and then eventually increases more significantly above around $50$ K.
Although such a simple discussion and fitting already gives a rather consistent description of the temperature dependence of the individual QD excitonic emission lines, it is not in complete agreement with the detailed discussion carried out above for the single recombination line, \emph{``Peak 3''}, and only provides supplementary information. It mainly describes the LO-phonon mediated pure dephasing process by inelastic electron-phonon scattering and thus the broadening of the ZPL, which is well supported by the obtained average phonon energy of 27 meV \cite{Borr,Sang2}. However, it does not account for the acoustic phonon sidebands and other broadening mechanisms and the respective line shape transition, because the ZPL is the dominant contribution in the main part of the temperature range investigated here.
It shall also be added that it was rather difficult for this reason to clearly obtain and assign appropriate single values for all the line widths for temperatures above around 50 K, which is reflected by the uncertainties and deviation between data points and fit curves in \fref{fig:AbLines}~(b).
Besides the afore discussed reasons, the increased broadening and intensity reduction for temperatures above $80-100$ K, which can be seen in figures \ref{fig:AbPeak3} and \ref{fig:AbLines}, can be due to charge fluctuations and an increased probability of various multi-phonon processes, that might meet resonance conditions between QD ground and excited states or the CES, which leads to additional non-radiative decay processes of the exciton states under investigation.
\subsubsection{Relative quantum dot intensities, coupling and quenching.}
\label{sec:RelativeQDIntensitiesCouplingAndQuenching}
\Fref{fig:AcArrh} highlights the temperature dependence of the individual QD PL intensities under low power excitation, where excited states and re-excitation processes play no major role. It displays the experimental data for individual peaks (a) and dot groups (b) (as identified in \fref{fig:AaT001All}), as well as, the fits to all data sets using the afore introduced equation \eref{Eq:Arrha2}, which accounts for the initial QD carrier capture and for two distinct decay processes.
As a common trend of all displayed data sets, it can be observed that the lower energetic PL features persist visible in the spectra at increasing temperatures while the higher energetic features already start to fade.
The most significant fit parameters are summarized in table~\ref{tab:indivPL}.
As already introduced before, it is assumed that the excitations (excitons/charge-carriers) are transferred to the lower-energy dots as the temperature increases; this assumption is supported by the parameters corresponding to the first escape mechanism ($i=1$), e.g., the smaller activation energies, $E^{({\rm a}1)}$. All obtained values, for single dots and dot groups, range between 21 and 15 meV with a trend of the smaller values for the lower energy dots.
More notably, the corresponding coefficient, $b_1$, also gets smaller with decreasing PL energy, which indicates a smaller escape probability and larger (re-) capture-rate for the lowest energy dots and groups. This very well coincides with the carrier redistribution process from the high- to the low-energy dots as afore discussed and shown in \fref{fig:AXEns}~(b).
\begin{figure}
\centering
\includegraphics[scale=.5]{F8.jpg}
\caption{The PL intensity temperature dependence of (a) four individual QD peaks (or pairs) at different spectral locations, and (b) the integration over the QD groups 1--4, respectively; all intensities of the analyzed peaks and groups are obtained from the data displayed in \fref{fig:AaT001All}. The data sets are displayed as markers (with error bars) and are fitted using equation \eref{Eq:Arrha2}, the fit results are displayed as filled curves. All data sets and corresponding fits are offset for clarity.}
\label{fig:AcArrh}
\end{figure}
\begin{table}
\centering \footnotesize
\begin{tabular}{c|*{4}c|c} \hline
Type & $b_1$ & $E^{({\rm a}1)}$\footnotesize (meV) & $b_2$ & $E^{({\rm a}2)}$\footnotesize (meV) & $\Delta E_{j}$\footnotesize (meV)\\ \hline
\textit{Peak 5} & $602\pm8$ & $17.0\pm 0.7$ & $4.3\cdot10^7\pm0.5\cdot10^7$ & $162\pm18$ & \itshape$\approx$322\\
\textit{Peak 3} & $985\pm32$ & $16.8\pm 0.2$ & $2.6\cdot10^7\pm0.2\cdot10^7$ & $117\pm20$ & \itshape$\approx$244\\
\textit{Peaks 2$^{\ast\ \dagger}$} & $2683\pm254$ & $20.7\pm0.4$ & $6.3\cdot10^7\pm1.2\cdot10^7$ & $113\pm20$ & \itshape$\approx$218\\
\textit{Peaks 1 $^{\dagger}$} & $3036\pm266$ & $21.2\pm 0.4$ & $9.0\cdot10^6\pm 1.0\cdot10^6 $ & $87\pm8$ & \itshape $\approx$184\\ \hline
Group 4 & $599\pm38$ & $15.5\pm0.4$ & $2.1\cdot10^6\pm0.6\cdot10^6$ & $128\pm32$ \\
Group 3 & $809\pm79$ & $18.7\pm0.5$ & $8.4\cdot10^6\pm1.5\cdot10^6$ & $115\pm25$ \\
Group 2 & $1010\pm115$ & $17.1\pm0.5$ & $7.9\cdot10^6\pm2.4\cdot10^6$ & $114\pm32$ \\
Group 1 & $2665\pm555$ & $20.8\pm0.8$ & $3.0\cdot10^6\pm0.7\cdot10^6$ & $75\pm13$ \\ \hline
\end{tabular}
\caption{Fit parameters (equation \eref{Eq:Arrha2}) for the individual QD PL lines and the energetically sorted groups of QDs; the $1\sigma$-fit-errors are omitted when of no significant magnitude ($<<1\%$ of the parameter value). The complete set of all fit parameters is displayed in~\cite{xtab}. The right column contains the optical gap energies between the InGaAlAs barrier and the respective QD PL peak.
$^{\dagger}$ The sum of the intensities of two neighbouring peaks is considered because of their spectral overlap at higher temperatures.}
\label{tab:indivPL}
\end{table}
The coefficients corresponding to the second escape mechanism (and larger activation energies), $b_2$, which will be discussed in more detail below, are all rather large with no clear difference between dots or groups, i.e., all such escape processes happen with much higher probabilities and are dominated by the escape -- re-capture is much less likely for this process.
\begin{figure}
\centering
\includegraphics[scale=.25]{F9.jpg}
\caption{Schematic illustrations of the InGaAlAs barrier, the CES and some QDs inside a nano-cone, where (a) highlights the couplings related to the first carrier escape process ($i=1$) and (b) the couplings related to the second carrier escape process ($i=2$). (c) Illustration of a schematic simplification of the band diagram showing the assumed carrier escape, relaxation and coupling processes as well as energies for one QD and the common features CES and barrier.}
\label{fig:AdScheme}
\end{figure}
The schematic \fref{fig:AdScheme} shall illustrate the different coupling mechanisms between dots, CES and barrier, and shall also indicate the corresponding confined discrete and continuous states as well as the assigned carrier transfer processes. The different arrows illustrate the various coupling mechanisms:
1.) the inter-dot direct tunneling (grey double line arrow),
2.) the inter-dot coupling via the CES (solid line arrows),
3.) the coupling between CES and barrier (dashed line arrows) and 4.) the coupling between dots and barrier (dotted line arrows).
1.) is not analyzed in this work and not furthermore considered because it is a local coupling effect between two distinct dots, which could be consequently treated as a new single quantum system, a lateral QD molecule \cite{q4,q5,q6}.
2.) and 3.) represent the inter-dot carrier exchange via the CES and the possibility of subsequent carrier escape to the barrier (the first, $i=1$, escape process, \fref{fig:AdScheme} (a)), e.g., via phonon assisted ``shake-up'' of the carriers along the CES higher state density, as illustrated in \fref{fig:AdScheme}~(c).
Finally, 4.) represents the carrier escape from the QDs to the barrier continuum (the second, $i=2$, escape process, \fref{fig:AdScheme} (b)).
The parameters assigned to the second escape mechanism ($i=2$), especially the larger activation energies, $E^{({\rm a}2)}$, of the individual dots (\fref{fig:AcArrh}~(a)) and groups of dots (b) clearly reveal this trend of decreasing activation energy with increasing QD emission energy.
This trend of the fitted activation energies is comparable to the trend of the respective optical energy gaps, i.e., the energy separations between the dot PL energies and the barrier PL energy, $\Delta E_{j}$, which are listed in the right column in table \ref{tab:indivPL}.
The generally larger relative $1\sigma$-fit-errors obtained for the dot groups reflect the fact, that the considered data contains PL from a certain number of individual QDs that only have the similar emission energies in common. Therefore, the fits have to account for some sort of ``averaging'' of a few dots with each very similar, but nevertheless slightly different values, especially with regard to the second escape mechanism.
In the following paragraph the values obtained for this second escape process, especially the activation energies, $E^{({\rm a}2)}$, are quantitatively analyzed and compared to the corresponding optical gap energies, $\Delta E_{j}$.
The carrier escape mechanisms from QDs can be indirectly represented by the ratio
\begin{equation}
\nu_j = E_j^{({\rm a}2)} / \Delta E_{j}\ ,
\label{Eq:nu}
\end{equation}
where the index $j$ represents a certain QD or QD group and moreover a corresponding optical gap energy (any energy difference between the described QD $j$ and some connected higher-energy state, the InGaAlAs barrier in the present case). See a comprehensive overview in G\'{e}linas, et al. \cite{Gel}.
Represented by this ratio of activation energy and energy gap between confined QD states and a barrier or wetting layer, different carrier escape scenarios can be assumed and supported by experiments \cite{San,Sch}.
In the case of (complete) exciton escape, the activation energy equals the full optical gap, i.e., $\nu_j = 1$.
When in the actual escape process single carrier escape dominates, the activation energy $E_j^{({\rm a}2)}$ corresponds to the energy of the less confined carrier, i.e., $\nu_j < 1/2$.
Another escape mechanism, the correlated electron-hole pair escape, has been suggested with $\nu_j = 1/2$. In this case, electrons and holes are assumed to be on average emitted as pairs, with their concentrations being equal or at least unchanged both within the dots and the barrier. This process leads to an activation energy of half the optical gap \cite{Mic2,Yan}.
In table \ref{tab:indivPL} we display the fitted activation energies, $E_j^{({\rm a}2)}$, besides the corresponding optical gap energies, $\Delta E_{j}$, which were obtained based on the low temperature QD peak PL and the barrier PL shown in \fref{fig:AaPowerAll}. Apparently, there exists a direct (linear) relation between these quantities; the fitted activation energies, $E_j^{({\rm a}2)}$, coincide well with $\Delta E_{j}/2$. This correspond to $\nu_j = 1/2$ and thus clearly indicates a correlated electron-hole pair escape mechanism \cite{xx,Mic2,Yan}. However, since $E_j^{({\rm a}2)}$ is always either $\Delta E_{j}/2$ or slightly on the smaller side, single carrier escape of the less confined particles, most likely holes, might also be a reasonable scenario.
\section{Extension of quantum dot luminescence to around 1.55 $\mu$m}
\label{sec:1.55}
\begin{figure}
\centering
\includegraphics[scale=.5]{F10.jpg}
\caption{PL from 6 ML QDs at around 1.55 $\mu$m embedded in a $d_{\rm QD}\approx120$~nm Si$_3$N$_4$/Ag-embedded nano-cone under non-resonant excitation. (a) Excitation power dependence at 5 K and (b) temperature dependence at 500 nW low power excitation.}
\label{fig:6ML}
\end{figure}
Finally, we want to highlight the possibility that, using the same QD growth method, it is possible to not only cover photon wavelengths within the telecommunication O band, but to reach the technologically more interesting, however more challenging, C band around 1.55 $\mu$m (0.8 eV). This is realized by increasing the nominal deposition of InAs for the QD layer from four to six MLs.
\Fref{fig:6ML} briefly summarizes the excitation power and temperature dependence from these types of lower energy QDs. Notably, QD emission persists up to around 150 K (or even 200 K). However, at the higher temperatures and thus longer PL wavelengths, the photon detection is limited by the cut-off of the used InGaAs detector, which is around 1.6 $\mu$m.
In comparison to the detailed presentation of the 1.3 $\mu$m QDs, a similar behavior for both the excitation and temperature dependence is found. The trend, however, suggests that the longer wavelength 1.55 $\mu$m QD PL is indeed even more temperature stable, i.e., the corresponding activation energies might be larger. Similar inter-dot coupling as demonstrated for the 1.3 $\mu$m QDs and therefore also similar escape mechanisms are expected.
A more detailed examination of the optical properties of such single QDs is currently in progress \cite{xx6MLSQD}.
\section{Summary and outlook}
\label{sec:Summary}
We have presented the fabrication and usage of metal-embedded nano-cones, in order to select a sub-ensemble of high-density InAs QDs emitting in the telecommunication O and C bands. Accessing such a reduced number of QDs, their electronic and optical properties could be examined in detail. Individual dot emission has been demonstrated between 1.2 and 1.6 $\mu$m ($1.03-0.77$ eV). The QD PL persists up to temperatures of around 150 K; the transition of the spectral properties of such individual PL lines has been analyzed, accounting for the alteration of the transition energies, the line widths and shapes. The transition of a single QD PL line from the ZPL to an acoustic sideband dominated line shape could be demonstrated.
The PL intensity temperature dependence of both the sub-ensemble of dots in a nano-cone and individual QDs has been investigated and could be fitted with a model that includes the carrier capture and two different carrier escape processes. One of the observed escape processes could be closely connected to the inter-dot coupling of excited electronic states, which not only leads to a carrier redistribution towards the lower energy dots but also contributes to the non-radiative carrier escape. These mechanisms are characteristic for samples with high spatial dot density, which enhances the lateral inter-dot coupling. The other observed carrier escape process is suggested to be due to correlated electron-hole pair escape and coincides well with half the optical gap energy between the InGaAlAs barrier and the respective QDs.
With this more detailed understanding on the QDs electronic properties we believe that the presented QDs, incorporated in nanometer-sized metal-embedded mesas, provide a suitable future realization for QD-based single photon sources in the telecommunication bands. Moreover, we have shown that they have the potential for application at elevated temperatures, which well covers the liquid Nitrogen temperature of 77 K, and that temperature tuning can be used to adjust the emission energy of the dots by at least ten meV. Especially the dots on the low-energy side of the sub-ensemble, which is usually the one of particular interest and importance, provide the best performance with regard to temperature stability.
In the future we are planning to proceed towards the realization of single photon and entangled photon pair sources for application in quantum information processing and quantum communication using silica based fibre networks.
\ack
This work was partially supported by SCOPE
from the Ministry of Internal Affairs and Communications,
by the Grand-in-Aid for Scientific Research (A), No.21246048, and
by HINTS from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
CH acknowledges the Japan Society for the Promotion of Science (JSPS) for providing financial support in the form of a JSPS Fellowship for Foreign Researchers; NAJ acknowledges financial support via a MEXT scholarship.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"http:\/\/apple.stackexchange.com\/questions\/60658\/how-to-assign-different-refit-icons-for-multiple-mac-partitions","text":"# How to assign different ReFit icons for multiple Mac partitions?\n\nI have 2 different OS X partitions on my Mac. I use ReFit to boot them.\n\nBut since they're both the same OS, it shows the same icon for both of them, which isn't their disk icon that I can change.\n\nIs there a way to use a specific icon for each partition? Or is there an alternative to ReFit that will let me do this?\n\n-\n\n(Disclaimer: I haven't tried this personally.)\n\nhttp:\/\/refit.sourceforge.net\/doc\/c3s1_icons.html\n\nIf I'm understanding this page of the rEFIt documentation correctly, you should be able to take a 128x128 .icns image, name it \"boot.icns\", and plop it in \/System\/Library\/CoreServices on the partition whose icon you wish to change.\n\n(rEFIt automatically searches for .icns images named the same way as the EFI loader file. On OSX, that file is \/System\/Library\/CoreServices\/boot.efi)\n\n-\nThis worked, thanks! \u2013\u00a0Jack Stewart Aug 14 '12 at 19:08\n\nI, too, am dual booting 2 OS X's (Mountain Lion and Yosemite). This is because Yosemite is too heavy, so it's there only as a safety net!\n\nIf you are double booting Yosemite or El Capitan with another prior OS X, copy the rEFInd folder to this older OS X. This is because of the scheme Yosemite and El Capitan use to boot, and having the rEFInd files inside these partitions will require special installation instructions. In my case as I said, I have Mountain Lion and Yosemite, so the rEFInd folder is at the root of the Mountain Lion partition (volume).\n\nBut in the other hand if you are booting Yosemite and El Capitan (without an older OS X version as 3rd boot) then you'll have to go to the rEFInd page and read how to install rEFInd in these OSs.\n\nThis is how it's done:\n\nThe latest version (0.9.2) supports .png for icons, so no need to create .icns files.\n\n\u2022 Configure your icons and place them in the icons older. I have made a God fearing simplistic theme. If you use it may I suggest you keep the background as a reminder of to whom one day you will answer to! Download this theme (and the binaries). It's basically my rEFInd folder zipped.\n\n\u2022 Configure the refind.conf file to use manual boot stanzas. These are hardcoded (in the config file, much like a grub file in linux) options so the name of the volume of your OS X cannot be changed without changing the configuration file. If you fail at this and change the volume name without updating the .conf file, that particular boot option will not be bootable anymore. But since you'll have two OS X's you can force Apple's bootmenu with Alt at the startup, boot the other volume and fix the .conf file. The proper basic necessary configuration is as follows:\n\nscanfor external,optical,hdbios,biosexternal,manual\n\nicon \\rEFInd\\icons\\ml.png\nvolume \"Mountain Lion\"\ngraphics on\n}\n\nicon \\rEFInd\\icons\\yoyo.png\nvolume \"Yosemite\"\ngraphics on\n}\n\n\nExplanation: In the scanfor line you'll have to remove the \"internal\" option so rEFInd will not scan for the partition (as they are defined with menuentry) or you will have 4 boot options, the two from the scan itself, and two from the hardcoded menuentry sections. Also still in scanfor you have to put the manual option so rEFInd will read the menuentry sections. And the menu entry can be explained with this example:\n\nmenuentry \"Name that Will Appear In The Boot Menu\" {\nicon \\path\\to\\the\\icon.png\nvolume \"THIS IS THE EXACT NAME OF THE MAC VOLUME\"\ngraphics on\n}\n\n\nLeave the \"loader\" option as it is. Because that is the default path for OX S' boot.efi (on all recent versions).\n\nAnd finally the graphics on is to hide the rEFInd command like black screen when booting, so you get a nice graphical transition between the boot menu and the screen with the apple (Apple's boot screen).\n\nI thinks this should cover. The end result should be (the rEFInd.zip at the top has this theme already) something like the following\n\n\u2022 Finally set the boot to be done using the non-yosemite\/ElCapitan partition (or the partition where you copied the rEFInd folder to. To do this go to: System Preferences > Startup Disk click the partition and press restart.\n\nIf things go horribly wrong and for some reason you cannot boot, try either Alt to choose a bootable partition (ie. Recovery)\n\nJust a final note, App Store uses Spotlight indexes to identify apps, so before you launch App Store to update apps that are present on both OS Xs, be sure to unmount the Volume with the OS X that is not running so the AppS tore app can find the right one to update.\n\n-","date":"2016-05-29 15:42:20","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\": 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\": 0.38033923506736755, \"perplexity\": 4115.362570936954}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-22\/segments\/1464049281363.50\/warc\/CC-MAIN-20160524002121-00018-ip-10-185-217-139.ec2.internal.warc.gz\"}"} | null | null |
{"url":"http:\/\/mathhelpforum.com\/differential-geometry\/172937-function-metric-space-r-continuous-print.html","text":"# Function from Metric Space to R is Continuous\n\n\u2022 Feb 28th 2011, 09:28 AM\nmathematicalbagpiper\nFunction from Metric Space to R is Continuous\nLet $(M, d)$ be a metric space. Let $f:M\\longrightarrow M$ be a continuous function. Let $g:M\\longrightarrow\\mathbb{R}$ be defined by $x\\longmapsto d(x,f(x))$. Show that $g$ is continuous.\n\nSo what we know is that $f$ is continuous, so for any $\\eta>0$, there exists $\\mu>0$ such that:\n\n$d(x,y)<\\mu \\Rightarrow d(f(x),f(y))<\\eta$\n\nBut what we want to do for any $\\epsilon>0$ is to produce $\\delta>0$ such that:\n\n$d(x,y)<\\delta \\Rightarrow |g(x)-g(y)|=|d(x,f(x))-d(y,f(y))|<\\epsilon$\n\nI have a feeling we'll want to produce $\\delta$ in terms of $\\epsilon, \\eta, \\mu$. I was thinking breaking something up using a triangle inequality, but I've tried a few different ways that have all led to dead ends.\n\u2022 Feb 28th 2011, 09:56 AM\ngirdav\nYou can use the inequality $|d(x,z)-d(y,z)|\\leq d(x,y)$ and write $d(x,f(x))-d(y,f(y))=d(x,f(x))-d(x,f(y))+d(x,f(y))-d(y,f(y))$.\n\u2022 Feb 28th 2011, 10:24 AM\nmathematicalbagpiper\nI already tried that and came up with a dead end. The result of that yields:\n\n$|g(x)-g(y)|\\leq d(f(x),f(y))+d(x,y)$\n\nWhich if we let those be less than $\\eta, \\mu$ respectively from the previous part, we get:\n\n$|g(x)-g(y)|<\\eta+\\mu$\n\nWhich I just don't see how you could possibly get $\\delta$ out of something of that sort.\n\u2022 Feb 28th 2011, 10:29 AM\ngirdav\nGiven $\\eta$, we can find $\\mu$ such that $d(f(x),f(y))\\leq \\frac{\\eta}2$ if $d(x,y)\\leq \\mu$. We can choose this $\\mu$ smaller than $\\frac{\\eta}2$.","date":"2016-09-25 18:24:48","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\": 0, \"img_math\": 0, \"codecogs_latex\": 26, \"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\": 0.9662826657295227, \"perplexity\": 153.50412406357916}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-40\/segments\/1474738660338.16\/warc\/CC-MAIN-20160924173740-00203-ip-10-143-35-109.ec2.internal.warc.gz\"}"} | null | null |
Q: WSDL import in .NET CORE 3.1 i am new to .net core 3.1
I have the following problem, when I reference a WSDL and it creates the reference, when I am trying to start the webservice it asks me for an "Endpointconfiguration" in the constructor that I don't know what it is or how to create it.
WSMP.Service1SoapClient WS = new WSMP.Service1SoapClient();
example constructor
ty
UPDATE
I used this constructor for the service, but I get the error attached in the image, in .net it works without problem but in .net core it doesn't work, I don't know if I have something wrong in the configuration of the new Service1SoapClient (basicHttpBinding, endpointAddress));
public class SoapMultiPay : ISoapDemoApiMp
{
public readonly string serviceUrl = "http://xxx.xxx.xx.x:xxx/Service1.asmx";
public readonly EndpointAddress endpointAddress;
public readonly BasicHttpBinding basicHttpBinding;
public SoapMultiPay()
{
endpointAddress = new EndpointAddress(serviceUrl);
basicHttpBinding = new BasicHttpBinding();
basicHttpBinding.MaxBufferSize = int.MaxValue;
basicHttpBinding.ReaderQuotas = System.Xml.XmlDictionaryReaderQuotas.Max;
basicHttpBinding.MaxReceivedMessageSize = int.MaxValue;
basicHttpBinding.AllowCookies = true;
}
public async Task<Service1SoapClient> GetInstanceAsync()
{
return await Task.Run(() => new Service1SoapClient(basicHttpBinding, endpointAddress));
}
public async Task<RespuestaCotizadorGiro> GetCotizadorGiro(string zipCode)
{
var client = await GetInstanceAsync();
var response = await client.CotizadorGiroAsync(null, null);
return response;
}
}
Update-errorClientWs
A: It seems that your WSDL contains multiple endpoint, in this scenario, when you create the service client instance, you have to assign the endpoint, try to use the code like this:
ServiceReference2.Service1Client client1 = new ServiceReference2.Service1Client(ServiceReference2.Service1Client.EndpointConfiguration.BasicHttpBinding_IService1);
var response2 = client1.GetDataAsync(34);
MyLabel.Content += " " + response2.Result;
More details information, you could check this thread:
How to consume SOAP web service from .NET Core 3.0 WPF app
| {
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Every year fans of football, basketball, car racing and many other sports converge on Indianapolis, a city rich in sports culture. Visitors interested in learning more about the history of Indiana also make their way to the state's capital. With thousands of visitors of every ilk each year, a countless number of hotel options are available in Indianapolis, especially in the northern area of the city limits.
Perfect for both visitors and business travelers to the Pyramids office park, the Comfort Inn offers a comfortable stay with a number of special amenities and services. Outside there is a pool open during the warm months surrounded by a sun deck. Many rooms on the upper floors have balconies looking out onto the pool area. The hotel also provides free passes to the Gold's Gym nearby.
Rooms in the hotel include free wireless Internet and small mini refrigerator rentals are available upon request. Although there is not restaurant in the hotel, the Comfort Inn does provide a free continental breakfast with waffles, pastries, fruits and juices. Restaurants within just a few minutes of the hotel include Cracker Barrel, Maggiano's and Texas Roadhouse, which offers 10 percent of the bill if a guest shows the Comfort Inn room key card.
The hotel is within 15 miles of Lucas Oil Stadium, Indianapolis Motor Speedway, the Indianapolis Zoo and White River State Park.
Also near the Pyramids office park, the Embassy Suites is perfect for visitors interested in extended stays. All rooms in the hotel are suites, which include a separate living area, bedroom, sofa, wet bar and a work desk. The hotel provides a free hot breakfast every morning and in the evenings guest are invited to a "Manager's Reception," which includes complimentary beverages and small bites.
Within the complex is an indoor pool and whirlpool area and a fitness center with cardiovascular equipment, free weights and a basketball hoop.The on-site restaurant, Ellington's, offers a lunch and dinner menu as well as in-room dining for guests who want more privacy. The lounge, Whispers, is open late and offers pub-style appetizers and a variety of alcoholic and non-alcoholic beverages.
The hotel is within 20 minutes of downtown Indianapolis, the Indianapolis Motor Speedway and Lucas Oil Stadium.
For a less expensive stay in the north are of Indianapolis, the La Quinta Inn offers budget-friendly accommodations with a number of special amenities and services. The hotel provides an indoor swimming pool and welcomes pets. There is free wireless Internet access in all the rooms and every morning guests receive a free newspaper. The hotel invites all guests to a complimentary continental breakfast, which includes pastries, waffles, bagels, fruits and juices.
The hotel is conveniently located within 20 minutes of JJ Holcomb Planetarium, Conseco Fieldhouse, the Indianapolis Motor Speedway and the Indianapolis Children's Museum.
Guests looking for a sit down meal will find a number of options within a few miles of the hotel including Max & Erma's, Outback Steakhouse and Bob Evans.
Biles, John. "Hotels on the North Side of Indianapolis." Travel Tips - USA Today, https://traveltips.usatoday.com/hotels-north-side-indianapolis-38685.html. Accessed 23 April 2019. | {
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Update from Clontarf Residents Association re Modifications to the sea wall
We are pleased to inform residents that at last night's Monthly Council Meeting, DCC's planning application to modify the sea wall at James Larkin Road was approved. We would like to acknowledge the dedication of our Councillors in bringing this matter to a successful conclusion.
This permission gives effect to the agreement to reduce the height of 470m of the sea wall by 300mm. The agreement to reduce the wall was reached in September 2016 following long and detailed negotiations in response to considerable public outcry.
It is important to note that the reduction in wall height was:
• proposed by DCC
• based on scientific evidence from an Independent Expert engaged by DCC.
It is also important to note that the reduced height wall:
• gives protection from a 1 in 100-year event as opposed to a 1 in 200-year event,
• still includes a 200mm allowance for future sea level rise in addition to 300mm freeboard, and,
• that DCCs own staff have confirmed that the Park can contain any flood water that might arise.
There has been a lot of misinformation about this latest planning application and it is important that we clarify this for those who are not familiar with the area in question or the background to this matter.
Area in question:
The original planning permission for these works covered a 2km stretch from Bull Bridge to Causeway Road.
The new planning permission only deals with 619m of the new sea wall from the Causeway Road end to Mt Prospect Ave. This is the area opposite St Anne's Park and not opposite houses or businesses.
There is no history of coastal flooding in this area. The agreement reached with the local community only relates to the first 470m of this wall.
This new planning permission has two aspects – cladding of and modifications to the sea wall
Cladding:
The requirement to clad the wall is an unmet condition of the original An Bord Pleanála permission for these works. This work should have been undertaken as part of the original works while the contractor was still on site.
Modifications to the sea wall:
Sections of the wall as currently constructed would exceed the height permitted in the existing planning permission if the required capping is put in place and therefore need to be reduced.
Other sections of the wall need to be raised because, even though they are built to the required flood defence height, they are too low to meet H&S standards.
The costs associated with these modifications cannot be attributed to the agreement reached with the local community.
A further section of wall built to a height that allowed for the agreed 300mm reduction will need no remedial action.
The only additional work and costs that can legitimately be attributed to implementing the agreement reached with the local community are those associated with the additional cutting of the wall beyond that which was already required and the disposal of the additional waste material.
It is unfortunate that the cladding and modification works were not undertaken while the contractor was on site. We accept fully that the works will cause further disruption to both the commuters and recreational users of the area.
However, the area in question is part of the UNESCO Biosphere. It demands a level of consideration and protection in keeping with its unique status. The importance of the visual link between the Park and the Biosphere was established in the An Bord Pleanála permission underpinning the existing Part 8 permission.
While much has been said about the loss of views for motorists, the real issue was the loss of the visual amenity and connection with the Biosphere to all those many walkers not using the sea side footpath or cycle track and also to wheelchair users in the Park. We look forward to the timely completion of these works and the other outstanding items from the original works. | {
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NZXT S340 Review
Written by Matthew Lambert
Tags: #atx #chassis #mid-tower
Companies: #nzxt
1 - NZXT S340 Review 2 - NZXT S340 Review - Interior 3 - NZXT S340 Review - Cooling Performance 4 - NZXT S340 Review - Performance Analysis and Conclusion
The side panels are simple to remove thanks to thumbscrews and handles, with the former being captive too, making them easy to realign and ensuring you don't drop or lose them.
Inside, the S340 is split into two sections, one lower one with the PSU and hard drives in, and the main one above which is visible through the window and houses the main components as well as a couple of SSDs or 2.5in drives. The compartments are neatly segmented by a full length metal cover, which ensures a very clean look by shielding the vast majority of cabling from view when peering in through the window in much the same way as the equally attractive H440.
Ease of use is another strong point for the S340; motherboard mounts come pre-installed, a large cutout on the motheboard tray aids CPU cooler installation, and there's a thumbscrew-attached bracket provided allowing you to directly slide your PSU in at the back. There's loads of room down here for extra long PSUs and for stashing excess cabling, though there isn't any rubber for the PSU to rest on. Finally, the two floor-mounted 2.5in drives in the main cavity are secured using individual trays, which are also held in with thumbscrews.
At the front of the hidden portion is a fixed-position hard drive cage. It's only small, and is only capable of housing two 3.5in drives, which must be first slotted in from the side and then secured with screws at the front of the chassis. This necessitates removing the front panel, but thankfully this isn't difficult. Finally, beneath this two-bay cage is a set of mounting holes on the case floor, which you can use to secure one more 3.5in drive, which requires screwing in from beneath. As such, the S340 can house a total of five internal drives. Many cases do offer more, and more flexibility with regards to their sizes too, but even so we think this is enough given the high capacity SSDs and HDDs that are available. Also, even Intel's Z97 platform, for example, only has six SATA 6Gbps ports natively anyway. Still, we can't deny there are better options if you're really looking to go to town with multiple hard drives.
For water-coolers, the lack of an optical drive and lower than average drive mounts has its advtanges, as despite being a fairly compact mid-tower the S340 can house both 240mm and 280mm radiators in the front mounts. Officially, only 30mm radiators with a single row of fans are supported, but removing the cable management bracket would open up a lot more depth, with your radiator then acting as a means of blocking cables from view too. It would also impede on the available room for GPUs, but with 364mm here to start with you've still got plenty of flexibility. The rear and roof mounts can also be used to house single radiators, though you'd struggle to fit more than a slimline all-in-one cooler in either of them.
Instead of cable routing holes, the areas to the side and bottom of the motherboard tray are completely open, yet NZXT keeps the S340 looking incredibly tidy, firstly by shielding the lower section from view and secondly with its steel cable management bracket, which does the same for the side section. A few sensible holes are included though – one for the EPS12V and fan cables (which come neatly pre-routed as well) and a couple just behind the two SSD trays so you can sneak power and data cables to your drives and also neatly route front panel connectors to your motherboard tray. With lots of space behind the motherboard tray (17mm at the lowest point) and loads of hooks as well, the S340 is fantastically designed from a cable routing perspective. In a matter of minutes we had our system built with barely any cables visible – it's a joy to use.
Corsair Carbide Series 275R Review
Corsair finally debuts a decent mid-range mid-tower fit for the modern era.
March 6, 2018 | 14:00
Corsair Carbide Series Spec-04 Tempered Glass Review
Another case gets the tempered glass treatment.
October 26, 2017 | 14:00
Gigabyte Aorus AC300W Review
Aorus' first entry into the case market is a safe one in many regards - can it stand out?
October 3, 2017 | 15:00 | {
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This blog covers the experiences of a nearly Four Decade long survivor of CRPS (Complex Regional Pain Syndrome), Keith Orsini of American RSDHope. Keith first got CRPS in 1974. He is also one of the Directors and Founders of American RSDHope, a national non-profit organization for CRPS information.
Keith started the blog in 2008 when he was undergoing Hyperbaric Oxygen Therapy (HBOT) as a way of sharing what the experience was like. He blogged almost every day, explaining what it was like to experience Hyperbaric Oxygen Therapy firsthand; everything from what had to be done before he was approved for the treatment, the different types of chambers used, how long his dives were each day, what it felt like to "dive" (the actual treatment itself), how he felt after each dive, etc. There were some days he was too tired to blog but he kept it up pretty faithfully.
Since those first blog entries, Keith has added other information regarding the disease CRPS itself. Unfortunately many of the posts from the June, 2010 to June, 2011 time period have been lost in the transition (move from the old website to the new website). Sorry for the gap. But there are many, many other posts there to read before that time and since then up to and including now. He still blogs now as much as he can.
There are questions and answers covered in the blog on many aspects of CRPS; medications, treatments, blocks, what CRPS feels like, how it impacts families, and much more. You can scroll through the titles and just pick out and read the ones that interest you.
Be sure to scroll to the very bottom of the page to read the first entry in the blog, "The Journey Begins!" | {
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\section{\label{sec:level1}Introduction}
Turbulence is connected to a flow mechanism which converts kinetic energy into heat, known as the turbulence cascade. Turbulent flows exhibit universal statistical properties; it can thus be expected that the turbulence cascade is a universal process as well. Understanding the cascade dynamics is a major challenge of out-of-equilibrium physics.
According to the classical Richardson-Kolmogorov phenomenology \cite{kolmogorov1941dissipation}, the turbulence cascade is separated into collective modes. (i) Large scales which carry the bulk of the kinetic energy, (ii) small scales which dissipate the energy, and (iii) intermediate self-similar scales which mediate between the two. On average, energy moves from large to small scales, with energy dissipation being only a passive consequence of the energy injection into the cascade by the large scales. Thus, a one-way interaction between large and small scales is implied. The above description yields important predictions, such as a scaling law for the kinetic energy dissipation rate and the celebrated -5/3 law, both validated by experiment in a wide variety of flows \cite{sreenivasan1984scaling,vassilicos2015dissipation,batchelor1953theory}.
Recent experimental and numerical results have revealed a new universal dissipation scaling, different to the classical one, appearing in extensive regions of decaying homogenous turbulence \cite{valente2012universal,isaza2014grid,goto2016unsteady}, boundary-free shear flows \cite{nedic2013axisymmetric,cafiero2020non}, as well as in forced periodic turbulence \cite{goto2015energy,goto2016local}. These results suggest the existence of a new type of cascade whose physics do not abide to the Richardson-Kolmogorov phenomenology.
Here, the above observations are explained theoretically, by revising the Richardson-Kolmogorov phenomenology, so as to include a feedback mechanism, enabling active interaction between large and small scales. The resulting framework yields the new dissipation scaling, as well as an equation for integral length scale evolution of the flow. Contrary to previous theories, turbulence invariants are not assumed. The decay of the turbulent kinetic energy is found to be governed by a generalized logistic equation, reflecting the self-regulation of the cascade.
\section{Self-regulation}
\label{sec:feedback}
The idea that a cascade feedback mechanism lies behind the new dissipation scaling has been anticipated by two existing non-Kolmogorov theories of turbulence, which have had some success in predicting the novel non-Kolmogorov dynamics, albeit both containing inconsistencies (see appendix \ref{app:1}).
George's theory \cite{george1992decay} (similar to the theory of Ref. \cite{barenblatt1974theory}) leads to the new dissipation scaling. The cascade in that case is assumed fully self similar, i.e. evolving as a ``coherent whole". In contrast, Richardson-Kolmogorov phenomenology presupposes independently evolving compartments (i.e. large/small scales), with a self-similar range of scales only in between. It might be thought that George's viewpoint implies a ``balanced" cascade of quasi-steady evolution. However, the novel dissipation scaling has been observed in cascades where intense fluctuations disturb the establishment of a balance, while the non-negligible cascade time-lag does not permit immediate relaxation \cite{goto2015energy,goto2016local}. As a result, evolution ``as a whole" suggests a regulatory mechanism of information exchange between large and small scales, which is thus implicit in George's theory.
Goto and Vassilicos \cite{goto2016unsteady} observed that large scales are not self-similar, and excluded them from George's analysis. Given the above discussion, this treatment removes George's implicit assumption of active communication between large and small scales, which, as proposed here, is the main cause of the new dissipation physics. Indeed, in order to predict the new scaling, Goto and Vassilicos \cite{goto2016unsteady} had to explicitly assume an \textit{ad-hoc} link between large and small scales (i.e. that their dissipation rates are proportional). We reiterate that such an explicit link was not necessary in George's theory, as full self-similarity already implied it, but became necessary as soon as full self-similarity was broken.
In appendix \ref{app:1} it is argued that, similar to the large scales, the small scales must also be removed from the self-similar analysis. We therefore return to our starting point, the Richardson-Kolmogorov picture of large and small scales, with a self-similar range only in between. However, our previous discussion suggests an important difference. The physics connected to the new dissipation scaling imply a feedback mechanism linking large and small scales, which must therefore be included in the analysis.
\section{Phenomenology and assumptions}
\label{sec:phenom}
\begin{figure}[b]
\includegraphics[width=.78\columnwidth]{Sketch.pdf
\caption{\label{fig:Sketch} Proposed cascade picture. An intermediate range of self-similar scales is bounded by the non-dimensional wave numbers $\kappa_a$ and $\kappa_b$ from the large and small scales, respectively. A direct energy cascade ``feeds" the small scales, while an inverse helicity cascade regulates transport.}
\end{figure}
The ensuing analysis concerns homogeneous turbulence in cases where the new dissipation scaling has been observed. In decaying turbulence (grid and periodic box turbulence) this concerns the interval/region soon after turbulence starts to decay (i.e. close to the grid \cite{isaza2014grid}, or soon after the forcing stops \cite{goto2016unsteady}). At later times/distances the system transitions to the classical Kolmogorov dissipation scaling (note however that there are indications that even then Kolmogorov's assumptions are not fully valid \cite{goto2016unsteady}). In forced periodic box turbulence the flow-quantities undergo intense fluctuations \cite{goto2015energy,goto2016local} (even though the forcing is constant), during which the system always obeys the new dissipation scaling (see appendix \ref{app:2}). We note that the classical (Kolmogorov's) scaling is $\epsilon \sim K^{3/2}/L$, while the new dissipation scaling has been found from experiment to be $\epsilon \sim \nu Re_{L0} K/L^2$ \cite{vassilicos2015dissipation}, where $K$ is the turbulent kinetic energy, $L$ the integral length scale, $Re_{L0}$ is the integral scale Reynolds number at the onset of decay and $\nu$ the kinematic viscosity.
In any case, homogeneous turbulence can be described by the scale-by-scale energy budget
\begin{equation}
\frac{\partial K^>(k,t)}{\partial t} = \Pi(k,t) - \epsilon^>(k,t) \,.
\label{eq:budget}
\end{equation}
\noindent With $E(k,t)$ the energy spectrum, $K^>(k,t) = \int_k ^ \infty E(k,t) dk$ and $\epsilon^>(k,t) = 2 \nu \int_k ^ \infty k^2 E(k,t) dk$ are the turbulent kinetic energy and dissipation rate, respectively, for wavenumbers larger than $k$. $\Pi(k,t)$ is the interscale flux of turbulent kinetic energy from wavenumbers smaller to wavenumbers larger than $k$. We omit ``per unit mass" throughout the text for brevity. Note that equation \ref{eq:budget} lacks a kinetic energy production term, i.e. this is assumed to act only in very small wavenumbers (large scales) not included in equation \ref{eq:budget}, or not be present at all, as in the case of purely decaying turbulence.
We now perform a series of assumptions, which are validated using high Reynolds periodic box Direct Numerical Simulations (DNS) data of forced and decaying turbulence, (see \cite{goto2016unsteady} and appendix \ref{app:2} for more information on the numerical method and test cases).
\paragraph*{Assumption 1:} Similar to the Richardson-Kolmogorov phenomenology, the cascade is separated into large scales, small scales and intermediate scales (see figure \ref{fig:Sketch}). The latter are assumed self-similar during decay, and bounded by the non dimensional wavenumbers $\kappa_a$ and $\kappa_b$, i.e.
$$
E(k,t) = A(t)f(kL,^*), \hspace{0.4cm} \text{for $\kappa_a<kL<\kappa_b$} \,,
$$
\noindent where $\kappa = kL$ and $L(t) = \frac{3\pi}{4}\int_0 ^\infty k^{-1}E(k,t)dk/K(t)$. Following George \cite{george1992decay}, the argument $^*$ is included to indicate a dependency on initial conditions.
Given the above assumption, we expect that the dissipation in the self-similar range $\epsilon^{ab}(t)$ will scale as the total dissipation of the cascade $\epsilon(t)$. That is, because the majority of $\epsilon^{ab}(t)$ is expected to occur at the largest wavenumbers of the self similar range, i.e. close to $\kappa_b$. The eddy turnover time at $\kappa \approx \kappa_b$ will thus regulate both $\epsilon^{ab}$ and $\Pi_b$, the latter being the interscale energy flux at $\kappa_b$ (see figure \ref{fig:Sketch}). We may thus expect $\epsilon^{ab} \sim \Pi_b$ (i.e. that their ratio is time-independent). Neglecting the dissipation of the large scales (i.e. for $\kappa<\kappa_a$) we have $\Pi_b \approx \epsilon - \epsilon^{ab}$ (Kolmogorov's small scale stationarity hypothesis \cite{kolmogorov1941degeneration}). Combination of the above yields $\epsilon^{ab} = \Phi \epsilon$, where $\Phi$ is a constant of proportionality.
This result is validated in figure \ref{fig:A1}a where the appropriately normalized dissipation of periodic-box decaying turbulence is plotted as a function of the number of eddy turnover times $\hat{t} = \int _{0} ^t \frac{u}{L}dt$, where $\frac{3}{2}u^2 = K$. Two simulation sizes $N^3$ are plotted, i.e. $N=2048$ and $N = 1024$ (larger size corresponds to larger Reynolds number). The cutoff non-dimensional wavenumber $\kappa_b$ naturally increases with initial Reynolds number, and is taken to be equal to to 41 and 22, for the high and low Re cases, respectively (i.e. approximately at the wavenumber where the -5/3 spectral scaling starts to break down, see figure \ref{fig:A1}b). For both cases the normalized dissipation is relatively constant while the new dissipation scaling holds, giving some support to assumption 1.
However, for larger times $\epsilon^{ab} \sim \epsilon$ ceases to be valid, and this coincides with the shift of the system to the classical (Kolmogorov) dissipation scaling. The reason for this is that assumption 1 treats $\kappa_a$ and $\kappa_b$ as time-independent. When the new dissipation scaling is valid, this is indeed true. In that case, the beginning of the self-similar range (and thus $\kappa_a$) occurs shortly after the spectral peak (see figure \ref{fig:A1}b). Our DNS results show that while the latter diminishes with time, it always stays centred around the same normalized wavenumber $kL$. At the same time we expect $\kappa_b$ to be roughly proportional to $L/\lambda$, where $\lambda$ is the Taylor microscale. In section \ref{sec:diss} it is indeed shown that $L/\lambda$ stays constant with time when the new dissipation scaling is valid. The transition of the system to the classical scaling coincides with the disappearance of the spectral peak, and the start of a decreasing trend of $L/\lambda$ with time: $\kappa_a$ and $\kappa_b$ are thus no longer time-independent and assumption 1 is invalid.
Given the above analysis, we may obtain a scaling law for the energy spectrum $E(k,t)$. Using assumption 1 the dissipation of the self similar part of the cascade is given as
$$
\epsilon^{ab} = 2 \nu L^3 A \int_{\kappa_a} ^{\kappa_b} \kappa^2 f(\kappa,^*)d\kappa \,,
$$
\noindent which yields an expression for the time-evolution parameter of the spectrum $A(t)$. However, we have just shown that when the new dissipation scaling is valid we have $\epsilon^{ab} = \Phi \epsilon$, and thus we obtain
\begin{equation}
E(k,t) = \frac{\Phi \epsilon L^3}{2 \nu I_2}f(\kappa,^*), \hspace{0.4cm} \text{for $\kappa_a<kL<\kappa_b$} \,,
\label{eq:A1sb}
\end{equation}
\noindent where $I_2=\int_{\kappa_a} ^{\kappa_b} \kappa^2 f(\kappa,^*)d\kappa$. This scaling was introduced first in Ref. \cite{george1992decay}, using qualitatively similar arguments. Goto and Vassilicos \cite{goto2016unsteady} provided evidence for equation \ref{eq:A1sb}, for high enough wavenumbers, and when the new dissipation is valid. In figure \ref{fig:A1} we reproduce their data (decaying periodic box turbulence at $N=2048$) for completeness. The DNS data offer acceptable support for this scaling. Note that these spectra are the the hardest to collapse, as in this time-interval the Reynolds number varies the most during decay.
\begin{figure*}
\centerline{
\begin{tabular}{ll}
$\qquad$ (a) & $\qquad$ (b) \\
\includegraphics[width=.96\columnwidth]{A1a.pdf} &
\includegraphics[width=.96\columnwidth]{A1b.pdf}
\end{tabular}
}
\caption{(a) Normalized dissipation against number of turnover times, for periodic box decaying turbulence, with simulation sizes $N=2048$ (red line) and $N=1024$ (blue line). The grey stripe marks the transition region from the new dissipation scaling to the classical one. (b) Normalized energy spectra for many instances while the new dissipation scaling is valid. The novel scaling ceases to be valid approximately at the same time when the spectral peak at $kL\approx2$ disappears.}
\label{fig:A1}
\end{figure*}
\paragraph*{Assumption 2:} Much similarly to the Richardson-Kolmogorov phenomenology, it is assumed that a wavenumber $\kappa_a$ exists in the upper part of the self-similar range, such that
$$
\Pi_a = C_x u^3/L \,,
$$
\noindent where $\Pi_a$ is evaluated at $\kappa_a$, and $C_x$ is a coefficient of proportionality. While this expression is generally accepted for Kolmogorov turbulence \cite{vassilicos2015dissipation,pope2001turbulent}, it is not straightforward that it holds when the new dissipation scaling is valid. For instance, Goto and Vassilicos \cite{goto2016unsteady} have shown that in decaying turbulence the above does not hold for a wide wavenumber range in the self-similar part of the cascade. However, figure \ref{fig:A2}a shows that the above relationship holds for decaying periodic-box turbulence if $\kappa_a$ is taken shortly after the spectral peak. Specifically, here $\Pi_a$ is calculated for $\kappa_a\approx 3.3$, with the spectral peak being centred around $kL = 2$. A similar result can be also obtained for forced turbulence. There, Goto and Vassilicos \cite{goto2016local} have shown that assumption 2 is always valid, when calculated at an appropriate wavenumber. The coefficient of proportionality in forced turbulence was found to be very close to the one calculated here ($C_x \approx 0.38$).
\begin{figure*}
\centerline{
\begin{tabular}{ll}
$\qquad$ (a) & $\qquad$ (b) \\
\includegraphics[width=.96\columnwidth]{A2a.pdf} &
\includegraphics[width=.96\columnwidth]{A3a.pdf}
\end{tabular}
}
\caption{ (a) Normalized interscale energy flux of the large scales $\Pi_a$ and (b) normalized parameter $G(t)$ for decaying periodic turbulence of domain size $N=2048$ (red) and $N=1024$ (blue) (the forcing stops at $\hat{t}=0$). The dashed line in (a) corresponds to an ordinate value of 0.38.}
\label{fig:A2}
\end{figure*}
\paragraph*{Assumption 3:} When the flow exhibits the new dissipation scaling, the large scale interscale flux, $\Pi_a$, and the dissipation rate, $\epsilon$, are connected via the expression
$$
\Pi_a \sim \epsilon Re_L \,.
$$
\noindent This is the essential point of departure from the Kolmogorov phenomenology, which simply assumes $\Pi_a \sim \epsilon$. Assumption 3 is admittedly \textit{ad-hoc}; it is necessary for the analysis to yield the new dissipation scaling (see section \ref{sec:diss}). Conversely, assumption 2 transforms the new dissipation scaling into a simpler statement (assumption 3) which is much easier to interpret physically.
In figure \ref{fig:A2}b we validate the above assumption by plotting the normalized flux $G(t) = \Pi_a/(\epsilon Re_L)$ for decaying periodic box turbulence (as above, $\kappa_a=3$ and $\kappa_a=3.5$ for the two domain sizes). The normalized flux drops slightly and then remains relatively constant, as long as the system is characterized by the new dissipation scaling, providing some backing to assumption 3 (we note that for slightly larger $\kappa_a$ the constancy of $G(t)$ improves).
We now argue that assumption 3 is the expression of a negative feedback in the cascade. This is more evident in forced turbulence conditions where the turbulence parameters exhibit quasiperiodical oscillations, even if the forcing remains invariant in time (see \cite{goto2016local} and appendix \ref{app:2}). This behaviour is reminiscent of predator-prey systems \cite{brauer2012mathematical} where a negative feedback works to establish ``balance" in the system and oscillations are observed. Assumption 3 expresses a negative feedback according to the following causal chain. In forced turbulence, if the interscale flux $\Pi_a$ were to increase, then this would cause an increase in $\epsilon$ (after a time-lag). Turbulence would thus start to decay, causing a drop in $Re_L$ (in appendix \ref{app:2} we show that $\epsilon$ and $Re_L$ are indeed somewhat anticorrelated in forced turbulence). Assumption 3 would then halt the increase of $\Pi_a$, moving the system towards its previous state (negative feedback). The opposite would occur if $\Pi_a$ were to decrease.
The above causal chain requires a physical mechanism which would permit an information exchange between large and small scales. We now postulate such a mechanism based on helicity, the latter being the inner product of velocity and vorticity, $H = \boldsymbol{u}\boldsymbol{\omega}$. High values of $H$ deplete the nonlinearity of Navier-Stokes equations, suppressing the interscale transfer of the cascade \cite{moffatt2014helicity}. Small-scale helicity thus offers a pathway for active communication between large and small scales.
We first discuss the results of two recent works which, when combined, indicate this role of small scale helicity in the cascade. First, the DNS of \cite{portela2018turbulence} imply that, when the new dissipation scaling holds, small scale structures of high helicity exist in the flow, whose appearance is correlated to that of large coherent vortices in the flow. It is interesting that the current DNS results actually show that the new dissipation scaling holds for as long as the vortex peak of figure \ref{fig:A1}b (footprint of large coherent vortices) appears in the spectrum. As soon as the peak disappears, the system transitions to the classical dissipation scaling. Second, the analysis of \cite{bos2017dissipation} (see also \cite{yoshizawa1994nonequilibrium}) links the new dissipation scaling to a -7/3 slope in the energy spectrum, coexisting with the -5/3 slope, and therefore masked by it. The earlier work of \cite{brissaud1973helicity} actually suggests that a -7/3 slope is the footprint of an inverse helicity cascade, i.e. helicity transport from small to large scales.
Combining the above points, we may postulate the following feedback mechanism, also depicted in figure \ref{fig:Sketch}. An instability mechanism causes the large scales to create small helical structures of high helicity. Helicity then cascades up towards the large scales, finally intercepting the interscale flux $\Pi_a$. Assumption 3 (and thus the new dissipation scaling) could be thought to be the expression of these dynamics, in the sense that $\Pi_a$ is larger when dissipation is high (so that the small helical structures are destroyed) and when $Re_L$ is large (so that scale separation and thus the inverse cascade lag is large). Validation of this physical mechanism is left as a task for future research.
\section{Results}
\subsection{Dissipation rate}
\label{sec:diss}
First, we consider forced turbulence. Assumption 3 is
$$
\epsilon = C \frac{\Pi_a}{Re_L}\,.
$$
\noindent Considering a time-averaged cascade where large scale dissipation is negligible, we have $\overline{ \epsilon } = \overline{\Pi}_a $, where the bar denotes the time-averaging operation. We expect that the cascade time-lag breaks any correlation between $\Pi_a$ and $Re_L$ in forced turbulence (see appendix \ref{app:2} for validation of this assumption). Thus, time averaging of the above expression yields $C = 1/\overline{Re^{-1}_L}$. This is approximately $C \approx \overline{Re}_L$ (the forced turbulence data of \cite{goto2016local} confirm this simplification). Consequently, combination of assumptions 2 and 3 yields
\begin{equation}
\epsilon \sim \overline{ uL} \frac{u^2}{L^2} \,,
\label{eq:diss1}
\end{equation}
\noindent which is the new dissipation scaling. For decaying turbulence, we achieve a similar result if, instead of time averaging, we perform ensemble averaging at time $t=0$, where the turbulence is still forced. Thus, we have $\langle \epsilon_0 \rangle = \langle \Pi_{a0} \rangle $, where the subscript 0 signifies the time $t=0$, and we obtain
\begin{equation}
\epsilon \sim u_0 L_0 \frac{u^2}{L^2} \,.
\label{eq:diss2}
\end{equation}
\noindent In turbulence literature, dissipation scalings are commonly expressed using the dissipation coefficient $C_\epsilon = \epsilon L/u^3$, which is constant in Kolmogorov turbulence. Using the definition of the Taylor length scale $\lambda^2 \equiv 15 \nu u^2/\epsilon$, we obtain
\begin{equation}
L/\lambda \sim C_\epsilon Re_\lambda \,,
\label{eq:diss3}
\end{equation}
\noindent which shows that $L/\lambda$ increases linearly with $Re_\lambda$ in Kolmogorov turbulence. On the other hand, when the new dissipation scaling holds (i.e. equations \ref{eq:diss1} and \ref{eq:diss2}), we have
\begin{equation}
C_\epsilon \sim \sqrt{Re_{L0} } Re_\lambda ^{-1}\,,
\label{eq:diss4}
\end{equation}
\noindent where $Re_{L0}$ may denote either the time-averaged Reynolds number for forced turbulence, or the initial condition Reynolds number, for decaying turbulence. Substitution to expression \ref{eq:diss3} shows that $L/\lambda$ is constant during decay when the new dissipation scaling holds. In figure \ref{fig:V1}, we validate the above predictions using data from the literature for forced periodic, decaying periodic, and grid turbulence (see appendix \ref{app:2} and \cite{goto2016unsteady} for more info on the data-sets used). For forced turbulence (figure \ref{fig:V1}a) the different simulation runs are always characterized by the new dissipation scaling (equation \ref{eq:diss4}). In decaying turbulence (figure \ref{fig:V1}b) all five simulations begin with the new dissipation scaling, and later transition to the Kolmogorov scaling ($C_\epsilon \approx const$). As mentioned in the previous section, this state change coincides with the disappearance of the coherent vortices from the flow. In grid turbulence (figure \ref{fig:V1}c), for all tested grids the flow begins with the new dissipation scaling ($L/\lambda = const$) and at larger distances from the grid it transitions to the Kolmogorov scaling ($L/\lambda \sim Re_\lambda$).
\begin{figure*}
\centerline{
\begin{tabular}{lll}
$\qquad$ (a) & $\qquad$ (b) & $\qquad$ (c) \\
\includegraphics[width=.64\columnwidth]{V1.pdf} &
\includegraphics[width=.64\columnwidth]{V2.pdf}&
\includegraphics[width=.64\columnwidth]{V3.pdf}
\end{tabular}
}
\caption{Time evolution of the normalized $C_\epsilon$ for (a) forced periodic and (b) decaying periodic turbulence simulations of various initial Reynolds numbers (from \cite{goto2016local} and \cite{goto2016unsteady}). (c) Spatial evolution of $L/\lambda$ for various grids in grid-generated turbulence experiments (from \cite{valente2012universal}).}
\label{fig:V1}
\end{figure*}
\subsection{Integral length scale}
\label{sec:integral}
The two dissipation scalings (classical, new) discussed in the previous sections, provide a starting point for the prediction of the kinetic energy evolution of homogenous decaying turbulence, in the sense that $dK/dt = - \epsilon$. However, this equation cannot be integrated, given that $\epsilon$ is a function of $L$, which is itself an unknown function of time. This closure problem has been conventionally resolved via the \textit{ad hoc} assumption of ``turbulence invariants" \cite{sinhuber2015decay,saffman1967large}. This assumption is often arbitrary, given that an infinite number of invariants exist in turbulent flows \cite{vassilicos2011infinity}. In contrast to previous theories, the current framework yields a prediction of $L$ implicitly and does not rely on the assumption of invariants.
Neglecting the kinetic energy of the small scales ($kL>\kappa_b$), we obtain an estimate for the turbulence kinetic energy for scales larger than $k$, by integrating equation \ref{eq:A1sb} from $k$ to $\kappa_b/L$, i.e.
\begin{equation}
K^>(k,t) \approx \frac{\Phi \epsilon L^2}{2 \nu I_2} I_0(kL) \,,
\label{eq:L1}
\end{equation}
\noindent where $I_0(kL) = \int_{\kappa} ^{\kappa_b} f(\kappa,^*) d\kappa$. Injection of $\partial K^>/\partial t$ evaluated at $\kappa_a$ (we remind that assumption 1 states that both $\kappa_a$ and $\kappa_b$ are time-independent), along with the new dissipation scaling ($\epsilon = u_0 L_0 C_x \frac{u^2}{L^2}$) and assumption 3 ($\Pi_a = \frac{uL}{u_0 L_0} \epsilon$) to the scale-by-scale energy budget (equation \ref{eq:budget}) yields
\begin{equation}
\frac{1}{\nu} \frac{dL^2}{dt} = A - B Re_\lambda \,,
\label{eq:L2}
\end{equation}
\noindent where $A = 4 \frac{ I_2 \frac{1}{\Phi} - \frac{1}{3}C_x Re_{L0} I_0 }{\kappa_a f(\kappa_a,^*)}$ and $B = \frac{4 I_2 }{\Phi \kappa_a f(\kappa_a,^*)}\sqrt{\frac{C_x}{15Re_{L0}}}$ are positive constants dependent on initial conditions. In the above, $I_0 = \int_{\kappa_a} ^{\kappa_b} f(\kappa,^*) d\kappa$.
The above analysis can also yield a prediction for the point of transition from the new to the classical dissipation scaling. Equation \ref{eq:L1} relies on the assumption $\epsilon^{ab} \sim \epsilon $ (see section \ref{sec:phenom}) which does not hold in the classical dissipation scaling (see figure \ref{fig:A1}). However, we may consider $\epsilon^{ab} \sim \epsilon $ to be approximately valid for a small time interval after the state change. We may thus repeat the analysis of this section, but using the ``classical" expressions for the dissipation and interscale transfer, i.e. $\epsilon \sim u^3/L$ and $\Pi_a \sim \epsilon$. The result (see appendix \ref{app:3}) is
\begin{equation}
\frac{1}{\nu} \frac{dL^2}{dt} = -A' + B' Re^2 _\lambda \,,
\label{eq:L3}
\end{equation}
\noindent with $B'$ a positive constant for sufficiently high Reynolds numbers. We thus conclude that the transition from the new to the classical dissipation scaling occurs when the slope of $\frac{dL^2}{dt}$ changes sign. This is in agreement with the observation of \cite{goto2016unsteady}, that the state change coincides with the location where $\frac{dL^2}{dt}$ assumes its maximum value. We emphasize that equation \ref{eq:L3} is not valid, in general, during the classical decay, but only for a very small interval after the state change of the system.
The above predictions are validated in figure \ref{fig:L1}a using the two decaying periodic box data-sets. In accordance with equation \ref{eq:L2}, $\frac{dL^2}{dt}$ is a linear decreasing function of $Re_\lambda$, for as long as the new dissipation scaling holds (see figure \ref{fig:L1}b). When the system transitions to the classical scaling (i.e. $C_\epsilon = cont$), $\frac{dL^2}{dt}$ becomes an increasing function of $Re_\lambda$, in agreement with equation \ref{eq:L3}. The maximum value of $\frac{dL^2}{dt}$ marks the state change.
\begin{figure*}
\centerline{
\begin{tabular}{ll}
$\qquad$ (a) & $\qquad$ (b) \\
\includegraphics[width=.96\columnwidth]{L1.pdf} &
\includegraphics[width=.96\columnwidth]{L2.pdf}
\end{tabular}
}
\caption{(a) $\frac{1}{\nu}\frac{dL^2}{dt}$ and (b) $C_\epsilon$ for domain sizes $N=2048$ (red) and $N=1024$ (blue) (the forcing stops at $t=t_0$). The thick part of the lines marks the range where $\frac{dL^2}{dt}$ grows.}
\label{fig:L1}
\end{figure*}
\subsection{Turbulent kinetic energy}
\label{sec:Equation}
Having validated the scalings \ref{eq:diss2} and \ref{eq:L2}, we may combine them to obtain an expression for the evolution of the turbulent kinetic energy during decay. Elimination of time yields (equations \ref{eq:diss3} and \ref{eq:diss4} are also used)
\begin{equation}
\frac{du^2}{dL^2} = \frac{-u^2}{C_1L^2 - C_2 u L^3}\,,
\label{eq:K1}
\end{equation}
\noindent where $C_1 = \frac{6I_2/(\Phi Re_{L0}C_x) - 2I_0}{\kappa_a f(\kappa_a,^*)}$ and $C_2 = \frac{6I_2}{\Phi \kappa_a f(\kappa_a,^*) Re_{L0} C_x u_0 L_0}$. In the above we have considered $\frac{3}{2} \frac{du^2}{dt} = - \epsilon$, i.e. decaying turbulence without turbulence production. It can be checked by substitution that a solution to the above equation is
\begin{equation}
\frac{C_1-1}{uL} = C_2 - \left(\frac{u}{C}\right)^{C_1-1} \,,
\label{eq:K2}
\end{equation}
\noindent with $C$ a positive constant of integration. Evidently, the current framework correctly predicts a continuously decreasing Reynolds number during decay, in contrast to previous theories for the new dissipation scaling (see appendix \ref{app:1}). Combination of equations \ref{eq:K1} and \ref{eq:K2} yields
\begin{equation}
\frac{du^2}{dt} \sim - u^4 \left[1- \left(\frac{u}{c}\right)^{C_1-1} \right]^2 \,,
\label{eq:K3}
\end{equation}
\noindent with $c$ a positive constant. Expression \ref{eq:K3} is a generalized logistic equation \cite{tsoularis2002analysis} (if $C_1=1$ it reduces to a generalized Gompertz equation), and it expresses the regulation introduced by assumption 3 via the term $\left[1- \left(\frac{u}{c}\right)^{C_1-1} \right]^2$. For this term (and thus for regulation) to be negligible, the second term on the right hand side of equation \ref{eq:K2} would also need to be negligible. Thus, $Re_L$ would have to remain approximately constant during decay. Then, assumption 3 would reduce to $\Pi_a \sim \epsilon$ (i.e. Kolmogorov turbulence) and thus the regulation that it otherwise expresses (see discussion in section \ref{sec:phenom}) would be lost.
We might inquire how does self-regulation affect the distribution of kinetic energy across the scales. Combination of equations \ref{eq:diss2} and \ref{eq:L1} yields
\begin{equation}
\frac{K^{ab}}{K} = \frac{\Phi Re_{L0} C_x I_0}{3 I_2} = const \,,
\label{eq:K4}
\end{equation}
\noindent where $K^{ab}$ is the energy of the self-similar scales. Thus, we obtain the result that, despite a qualitative change in the spectrum (i.e. disappearance of the spectral peak, see figure \ref{fig:A1}b), self-regulation in the cascade guarantees that the ratio of the kinetic energy between large and self-similar scales be constant.
In figure \ref{fig:K}a we show that the ratio $K^{ab}/K$ indeed stays relatively constant when the separation wavenumber is taken immediately after the spectral peak (see figure \ref{fig:A1}b), i.e. at $\kappa_a = 2.3$, for both of our decaying-turbulence data sets. Note that while the new dissipation scaling holds, the cascade undergoes the most change during decay, losing roughly 80\% of its initial kinetic energy (see figure \ref{fig:K}b).
\begin{figure*}
\centerline{
\begin{tabular}{ll}
$\qquad$ (a) & $\qquad$ (b) \\
\includegraphics[width=.96\columnwidth]{Ka.pdf} &
\includegraphics[width=.96\columnwidth]{Kb.pdf}
\end{tabular}
}
\caption{(a) Ratio of the kinetic energy of the self-similar range over the total cascade kinetic energy and (b) total cascade kinetic energy, versus number of turnover times for domain sizes $N=2048$ (red) and $N=1024$ (blue). }
\label{fig:K}
\end{figure*}
\section{Concluding discussion}
By superimposing a feedback mechanism on the classical Richardson-Kolmogorov phenomenology we derived expressions for various flow quantities (dissipation rate, integral length scale, kinetic energy) of the non-Kolmogorov universal cascade that has been recently discovered \cite{seoud2007dissipation}. We reiterate that the new type of cascade regulates forced turbulence \cite{goto2015energy}, the region of decaying turbulence where the bulk of kinetic energy is lost \cite{isaza2014grid}, and almost the whole extent of turbulent wakes \cite{redford2012universality,dairay2015non}. Therefore, it might be considered more relevant for engineering applications (and thus turbulence modelling) than classical Kolmogorov turbulence.
In the special case of forced turbulence, the current cascade picture resembles low-order predator-prey dynamics; prey (large scales) feeds the predator (small scales) in a self-regulating manner. These dynamics would explain the quasi-periodic oscillations of the turbulence quantities observed (see \cite{goto2015energy,goto2016local} and appendix \ref{app:2}), which indeed resemble, qualitatively, the response of predator-prey systems \cite{brauer2012mathematical}. The present analysis describes how small scales remove energy from the system during this regime. The question of how large scales replenish energy when forcing is present, remains open and will comprise the topic of future research. Another topic which remains open is the exact instability mechanism which generates the feedback, here attributed to an inverse helicity cascade.
| {
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/www.physicsforums.com\/threads\/pulses-in-wire-waves.117281\/","text":"# Pulses in wire - waves\n\n1. Apr 11, 2006\n\n### nick85\n\nHi, I am having difficulty understanding this question:\n\nA wire 5.0 m long and having a mass of 130 g is stretched under a tension of 220 N. If two pulses, separated in time by 30.0 ms, are generated, one at each end of the wire, where will the pulses first meet?\n\nm (distance from the left end of the wire)\n\nFrom my deduction the each wave starts from the opposite side.\nI figured out the v=91.9866 b\/c the square root of 220\/(.13\/5)=91.9866 m\/s\n\nI then set up equations\nt=time x=distance\n91.9866t=x\n91.9866(t-3*10^-3)=5-x\n\nI got x to equal 2.6379799 m\nHowever, I am not certain if this is the correct answer and I am down to my last submission. I was wondering if anyone could agree with what I got.\n\n2. Apr 12, 2006\n\n### Hootenanny\n\nStaff Emeritus\nI get 1.12m for my answer. Here's my calcs;\n\n$$v = \\sqrt{\\frac{T}{\\frac{m}{L}}} = \\sqrt{\\frac{220}{\\frac{0.13}{5}}} = 91.9866 m\/s$$\n\n$$x = vt \\Rightarrow t = \\frac{x}{v}$$\n$$5 - x = v(t + 0.03)$$\n\nSubbing $t = \\frac{x}{v}$ into $5 - x = v(t + 0.03)$ gives;\n\n$$5 - x = v\\left( \\frac{x}{v} + 0.03 \\right) = x + 0.03v$$\n\n$$2x = 5 - 0.03v \\Rightarrow x = \\frac{5 - 0.03\\times 91.9866}{2}$$\n\n$$\\fbox{ x = 1.120201m }$$\n\nI'm not sure that I'm right though\n\n-Hoot\n\n3. Apr 12, 2006\n\n### nrqed\n\nI disagree with your answer and agree with Hootenanny. Your mistake is the sign of the 3*10^-3 in the second line. It should be a plus sign.\n\nBasically (after substitution) you wrote x- 91.99* 3*10^-3 = 5 -x or\n2x= 5 + 91.99*3*10^-3 or\n\n2x= 5 + 2.76\n\nBut it should be 2x= 5 - 2.76.\n\n(it's easy to see. In the first 30 ms, the first pulse travels 2.76 m. There is still a distance of 5-2.76 to travel for both pulses when the second pulse will be emitted. They will obviously meet halfway through this remaining distance, therefore the answer is x= 1\/2(5-2.76).\n\nPatrick","date":"2019-01-19 02:54:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.5709772109985352, \"perplexity\": 1617.4280314053772}, \"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-2019-04\/segments\/1547583661083.46\/warc\/CC-MAIN-20190119014031-20190119040031-00324.warc.gz\"}"} | null | null |
\section{Introduction}
The sign problem in quantum physics has long been recognized as one of the main
impediments of efficient Monte-Carlo simulation of quantum many-body systems~\cite{sign:troyer,Loh-PRB-90}. Hamiltonians that do not suffer from the sign problem have recently been given the name `stoquastic'~\cite{Bravyi:QIC08}, a term
which aims to capture the relationship between these Hamiltonians and stochastic processes. Many interesting quantum models such as the transverse field Ising model, the Bose-Hubbard model,
and a collection of kinetic particles in a position-dependent potential, are stoquastic. However, stoquasticity, as introduced in Ref.~\cite{Bravyi:QIC08}, is a basis-dependent concept. It
requires that the Hamiltonian of the physical model in question be real and have non-positive off-diagonal elements in a given basis. For a many-body local Hamiltonian acting on $n$ qubits, this basis is typically a product basis on which the terms of the Hamiltonian act locally and can be efficiently described.
The non-positivity of the off-diagonal elements of a stoquastic Hamiltonian matrix in a particular basis has important consequences. It guarantees, via the Perron-Frobenius theorem, that there exists a set of orthonormal states, spanning the ground state subspace, whose amplitudes are non-negative in that basis \cite{Bravyi:2009sp}. In addition, the quantum partition function of a stoquastic Hamiltonian can be expressed as a sum of non-negative, easily computable, weights, which implies that Markov chain Monte-Carlo algorithms can be used to perform importance sampling of the quantum configuration space to calculate thermal averages of physical observables, using these weights as (unnormalized) probabilities. For this reason, it is said that stoquastic Hamiltonians do not suffer from the sign problem \cite{Bravyi:2009sp, Bravyi:QIC08}. However it is important to note that the absence of a sign problem does not necessarily imply polynomial-time convergence of standard Monte-Carlo methods \cite{Hastings2013, Jarret2016, Bringewatt2018}.
From a computational complexity perspective, the problem of estimating ground-state energies of stoquastic local Hamiltonians is considered easier than for general Hamiltonians~\cite{Bravyi:QIC08, bravyi:2006aa}. Moreover, in the classification of the complexity of estimating ground-state energies of local Hamiltonians, stoquastic Hamiltonians appear as the only intermediate class between classical Hamiltonians and general Hamiltonians~\cite{Cubitt:2016vl}.
Stoquastic local Hamiltonians are of interest not only in quantum complexity theory. In Ref.~\cite{Bravyi:2009sp} it was shown that deciding whether a stoquastic Hamiltonian is frustration-free is a MA-complete problem. Recently Ref.~\cite{AG:stoquastic} showed that the gapped version of this question is in NP, linking derandomization of MA to NP to the possibility of gap amplification of stoquastic local Hamiltonians.
Identifying classes of Hamiltonians that are stoquastic is clearly motivated both from practical and complexity-theoretic perspectives. Given that stoquasticity is basis-dependent, an interesting question arises: under what circumstances can the sign problem be `cured', as coined in Ref.~\cite{marvian2018computational}, by performing local basis changes? This is the main question explored in this paper.
It is worth noting that the sign problem may be resolved by means other than a local basis transformation. Other methods for generating positive-valued decompositions of the partition function include, e.g., re-summation techniques wherein negative-valued weights in the decomposition are grouped together with positive ones to form positive `super-weights' that can in turn be treated as probabilities in a quantum Monte Carlo algorithm~\cite{resum1,resum2,hen2019}. Other methods also include applying a constant-depth quantum circuit \cite{torlai+:pos}. These other methods are beyond the scope of this paper.
Naturally, devising techniques for obviating or mitigating the sign problem has been a focus of much research in the quantum Monte-Carlo (QMC) community since its inception \cite{Suzuki1993, Landau:2005:GMC:1051461,resum1,resum2,hen2019}. In particular, the importance of basis choice has been widely recognized (see, e.g. ~\cite{Bishop2001, Hatano1992, Hastings2015, Ringel2017}). Recognizing the key role that stoquastic Hamiltonians play both in computational complexity and in physics, a more general algorithmic approach has recently been launched to determine whether a Hamiltonian can be made stoquastic~\cite{ marvian2018computational, klassen2018two, Bausch2018}. In this
paper we present an important strengthening of these initial results.
Stoquasticity has also attracted attention from the experimental community. In particular there has been a growing interest in engineering Hamiltonian interactions that are not stoquastic~\cite{Vinci:2017aa,Ozfidan2019}. Some of the reasons for this include: enhancing the performance of quantum annealer protocols for optimization \cite{nonstoq1,nonstoq2,albash}; realizing universal adiabatic quantum computers~\cite{AL:review,aharonov_adiabatic_2007,Biamonte:07}; and physically emulating quantum many-body systems~\cite{Deng2019}. Here too, the question of whether and how local basis changes can cure the sign problem is highly relevant, as experimental quantum advantages hinge on the inability to simulate non-stoquastic interactions on classical computers.
\section{Previous work}
In what follows, we will refer to Hermitian matrices that are real and have only non-positive off-diagonal elements as \textbf{symmetric $Z$-matrices}~\cite{BP:book}.
A no-go result presented recently by some of the authors of this paper states that the problem of determining whether there exists a sign-curing transformation for general local Hamiltonians is NP-hard when one is restricted to applying particular single-qubit transformations to the Hamiltonian \cite{marvian2018computational}. This result can be summarized as
\begin{theorem}\cite{marvian2018computational}
Let $H$ be a three-local $n$-qubit Hamiltonian and let {\sf LocalCliffordSignCure} be the problem of determining whether there exist single-qubit Clifford transformations ${\sf C}_u$ with ${\sf C}=\bigotimes_{u=1}^n {\sf C}_u$ such that ${\sf C} H {\sf C}^{\dagger}$ is a symmetric $Z$-matrix. {\sf LocalCliffordSignCure} is {\rm NP}-hard. Let $H$ be a $6$-local $n$-qubit Hamiltonian and let {\sf LocalRealRotSignCure} be the problem of determining whether there exist real single-qubit rotations $R_u \in SO(2)$ with $R=\bigotimes_{u=1}^n R_u$ such that $R H R^{T}$ is a symmetric $Z$-matrix. {\sf LocalRealRotSignCure} is {\rm NP}-hard.
\label{theo:NPprev}
\end{theorem}
{\em Remark}: When dealing with $k$-local Hamiltonians with $k> 2$, it is important to note that two distinct notions of stoquasticity have been defined in Ref.~\cite{Bravyi:QIC08}, namely there exist termwise-stoquastic Hamiltonians and globally-stoquastic Hamiltonians. A globally-stoquastic Hamiltonian is a symmetric $Z$-matrix, while a Hamiltonian which is $k$-local termwise-stoquastic is one which can be decomposed into $k$-local terms such that each term is a symmetric $Z$-matrix. A globally-stoquastic Hamiltonian need not be termwise-stoquastic while a termwise-stoquastic Hamiltonian is always globally-stoquastic. The results in Theorem~\ref{theo:NPprev} hold for both definitions. For the two-local Hamiltonians in this paper one can prove~\cite{Bravyi:QIC08} that these notions coincide, hence we do not distinguish between these two definitions in this paper. We provide a proof of this equivalence in Proposition~\ref{prop:global2} for completeness.
It was also recently shown, by other authors of this paper, that for a particularly broad family of two-local Hamiltonians, namely arbitrary XYZ Heisenberg Hamiltonians, there is an efficient procedure for determining whether the sign problem can be cured by single-qubit unitary transformations:
\begin{theorem}\cite{klassen2018two}
Let $H=\sum_{u,v} H_{uv}$, $u=1, \ldots, n$, $v=1,\ldots, n$ be an $n$-qubit Hamiltonian with $H_{uv}=a^{uv}_{XX} X_u X_v+a^{uv}_{YY}Y_u Y_v +a^{uv}_{ZZ}Z_u Z_v$, where each $a^{uv}_{kk}$ is given with $O(1)$ bits. There is an efficient algorithm, which we call the \textbf{XYZ-algorithm}, that runs in time $O(n^3)$ to decide whether there are single-qubit rotations $U_u \in SU(2)$ with $U=\bigotimes_{u=1}^n U_u$ such that $U H U^{\dagger}$ is a symmetric $Z$-matrix.
\label{theo:KT}
\end{theorem}
An essential step in the proof of Theorem~\ref{theo:KT} was to show that single-qubit Clifford transformations suffice as basis changes, reducing the problem to an optimization problem over a discrete set of degrees of freedom.
\section{Main results}
This paper aims to bridge the gap between these two previous results, and identify the boundary between classes of Hamiltonians for which curing the sign problem by local basis transformations is hard and those for which this problem is easy. The main results of this paper address the following problem.
\begin{definition}[{\sf LocalSignCure}]
Given a two-local $n$-qubit Hamiltonian, {\sf LocalSignCure} is the problem of determining whether there exists a set of single-qubit unitary transformations $U_u \in SU(2)$ with $U=\bigotimes _{u=1}^n U_u$ such that $U HU^{\dagger}=\tilde{H}$ is a symmetric $Z$-matrix.
\end{definition}
We colloquially refer to such unitary transformation $U$ as a \textbf{sign-curing transformation}, and say that the sign problem of a Hamiltonian can be \textbf{cured} if such a transformation exists.
The main results of this paper are the following two theorems and constitute a strengthening of Theorems~\ref{theo:NPprev} and~\ref{theo:KT} to two-local Hamiltonians.
In Section~\ref{sec:hardnessResult} we will prove the following theorem.
\begin{theorem}
There exists a family of a two-local $n$-qubit Hamiltonians for which {\sf LocalSignCure} is {\rm NP}-complete.
\label{thm1}
\end{theorem}
To prove this, we modify the constructions introduced in~Ref.~\cite{marvian2018computational} thereby reducing the locality of Hamiltonians from three-local (in the case of the single-qubit Clifford group) and six-local (in the case of the single-qubit orthogonal group) to two-local. This result demonstrates that {\sf LocalSignCure} is hard in general. Theorem~\ref{thm1} additionally demonstrates that deciding if a multi-qubit two-local Hamiltonian can be sign-cured by single-qubit Clifford transformations is hard. We show in Appendix~\ref{app:cliffordsEasy} that, in the absence of one-local terms, this task is easy. We should stress here that it is not clear that {\sf LocalSignCure} is a problem in NP for general Hamiltonians. We expand on this point in the Discussion section~\ref{sec:dis}.
For a relatively broad subclass of two-local Hamiltonians we can, however, show that finding local basis changes is easy.
\begin{theorem}
Let $H$ be an \textbf{exactly two-local} $n$-qubit Hamiltonian, meaning a Hamiltonian of the form $H=\sum_{u,v} H_{uv}$ with $H_{uv} = \sum_{k, l \in \{X,Y,Z\}} (\beta_{uv})_{kl} \sigma_k^u \otimes \sigma_l^v $ with $\sigma_k^u$ a Pauli matrix of type $k$, acting on qubit $u$, and $(\beta_{uv})_{kl}$ is given with $O(1)$ bits. There is an efficient algorithm, using $O(n^3)$ arithmetic operations over $\mathbb{R}$, which solves {\sf LocalSignCure} for $H$.
\label{thm2}
\end{theorem}
This algorithm is presented in Section~\ref{sec:curingResult}. It employs the XYZ-algorithm referred to in Theorem~\ref{theo:KT}, as a subroutine. It is important to note that, just as in the XYZ algorithm, this algorithm makes no guarantee that $H$ can be cured, it only efficiently \textbf{decides} whether or not $H$ can be cured.
An important difference between the new algorithm in Theorem~\ref{thm2} and the XYZ-algorithm is that the new algorithm requires finding singular value decompositions of matrices specified by $O(1)$ bits, as well as intersections of vector subspaces, while the XYZ-algorithm required solving a discrete optimization problem. Since we do not address the question of how a finite-precision implementation of these standard linear algebra operations affects the accuracy with which we decide whether the sign problem of $H$ can be cured, we state our theorem in terms of arithmetic operations over $\mathbb{R}$. However the algorithm is expected to be numerically stable insofar as repeated composition of orthogonal rotations, and intersections of vector spaces, are numerically stable. A rigorous account of the complexity of this problem would require a finite precision formulation. We will not attempt to do this here.
The upshot of these results is that the presence of local fields can change the complexity class of curing the sign problem of two-local Hamiltonians by single-qubit unitaries from P to NP-complete.
\section{Preliminaries}
For ease of exposition and reference we start by stating the following observation about two-qubit Hamiltonians.
\begin{proposition}
A two-qubit Hamiltonian $H=\sum_{k,l=I,X,Y,Z} a_{kl} \sigma_k \otimes \sigma_l$ is a symmetric $Z$-matrix if and only if $a_{IY}=a_{YI}=a_{XY}=a_{YX}=a_{ZY}=a_{YZ}=0$ (the matrix is real) and $a_{XX} \leq - \vert a_{YY} \vert$ and $a_{IX} \leq -\vert a_{ZX} \vert$, $a_{XI} \leq -\vert a_{XZ} \vert$ (the matrix has non-positive off-diagonal elements).
\label{prop:basic}
\end{proposition}
\begin{proposition}\label{prop:ortho} \cite{klassen2018two}
Given a two-qubit Hamiltonian $H=\sum_{k,l=I,X,Y,Z} a_{kl} \sigma_k \otimes \sigma_l$, where the two-local term can be concisely represented by the $3 \times 3$ matrix \begin{equation*}\beta = \left( \begin{matrix}
a_{XX} & a_{XY} & a_{XZ} \\
a_{YX} & a_{YY} & a_{YZ} \\
a_{ZX} & a_{ZY} & a_{ZZ}
\end{matrix} \right).
\end{equation*} A pair of single-qubit unitary transformations $U_1$ and $U_2$ with action: $H \rightarrow (U_1\otimes U_2)H(U_1\otimes U_2)^{\dagger}$ corresponds to a pair of $SO(3)$ rotations $O_1$, $O_2$ acting on the $\beta$-matrix: $\beta \rightarrow O_1^{\sf T} \beta O_2$.
\end{proposition}
For the curious reader, an example of a Hamiltonian which is not stoquastic under any single-qubit unitary transformations is provided in Appendix~\ref{app:simpleExample}.
It was claimed in Ref.~\cite{Bravyi:QIC08}, without proof, that a two-local termwise-stoquastic Hamiltonian with respect to a basis is also globally stoquastic. We include the proof here:
\begin{proposition}\cite{Bravyi:QIC08}
A two-local Hamiltonian $H$ acting on $n$ qubits is a symmetric $Z$-matrix in the computational basis if and only if $H=\sum_{u<v} D_{uv}$ where each $D_{uv}$ acts nontrivially on at most two qubits, namely qubits $u$ and $v$, and $D_{uv}$ is a symmetric $Z$-matrix.
\label{prop:global2}
\end{proposition}
\begin{proof}
Let $\ket{x}$, with $x \in \{0,1\}^n$, denote a computational basis state. If there exists a decomposition $H=\sum_{u<v} D_{uv}$ such that $D_{uv}$ is real and $\forall x \neq y$, $\bra{x} D_{uv} \ket{y}\leq 0$, then $H$ is real and $\forall x \neq y$, $\bra{x} H \ket{y} = \sum_{u,v} \bra{x} D_{uv} \ket{y} \leq 0$.
This proves one direction of the bi-conditional, we now prove the other direction. Since $H$ is real, $H=H^{\sf T}$. Therefore every Pauli operator $P$ in the Pauli expansion of $H$ must satisfy $P=P^{\sf T}$, and so $H$ does not contain any Pauli operators with odd numbers of $Y$ terms. Let $d_H(x,y)$ denote the Hamming distance between bit strings $x$ and $y$. Since $H$ is two-local, $H=M^{(0)}+M^{(1)}+M^{(2)}$ where $\bra{x} M^{(m)}\ket{y}=0$ whenever $d_H(x,y)\neq m$. In other words the Hamiltonian decomposes into three sets: $M^{(0)}$ contains all terms which are diagonal (i.e., terms of the form $ZI$, $IZ$ and $ZZ$), $M^{(1)}$ contains all terms that flip 1 bit (i.e., of the form $XZ$, $ZX$, $XI$, $IX$), and $M^{(2)}$ contains all terms that flip two bits (of the form $XX$ and $YY$). There is no particular condition which has to be fulfilled for the diagonal group $M^{(0)}$, and so we ignore it.
Furthermore, from the condition $\forall x\neq y$ $\bra{x} H \ket{y} \leq 0$, it follows that $\forall x\neq y$ $\bra{x} M^{(1)} \ket{y} \leq 0$ and $\bra{x} M^{(2)} \ket{y} \leq 0$, since $M^{(1)}$ and $M^{(2)} $ are non-zero at different off-diagonal positions.
For any potential decomposition $H=\sum_{u<v} D_{uv}$ we can similarly write $D_{uv}=D^{(0)}_{uv}+D^{(1)}_{uv}+D^{(2)}_{uv}$, grouping diagonal, one-qubit flipping, and two-qubit flipping terms. Since $M^{(m)}$ contains all terms which flip $m$-qubits, $M^{(m)}=\sum_{u,v} D^{(m)}_{uv}$. In the case of $m=2$, $D^{(2)}_{uv}$ and $D^{(2)}_{wr}$ are non-zero at different off-diagonal positions when $u,v\neq w,r$, and so $\forall x\neq y$ $\bra{x} M^{(2)} \ket{y} \leq 0$ implies $\forall x\neq y, \forall u<v, \bra{x} D^{(2)}_{uv} \ket{y} \leq 0$.
In the case of $m=1$, $D^{(1)}_{uv}$ and $D^{(1)}_{wx}$ may both be non-zero on the same off-diagonal position, and so we must use a different argument. We can write $M^{(1)}=\sum_{u,v\colon u < v} [a_{XZ}^{uv} X_u Z_v +a_{ZX}^{uv} Z_u X_v]+\sum_u a_X^u X_u$. By writing out matrix elements one can show that
\begin{equation*}
\forall x\neq y, \bra{x} M^{(1)} \ket{y} \leq 0 \Rightarrow \forall u \;\; a_X^u+\sum_{v\colon v > u} \Delta^v a_{XZ}^{uv}+\sum_{w\colon w<u} \Delta^w a_{ZX}^{wu} \leq 0,
\end{equation*}
for all choices of sign-patterns $\Delta^u=\pm 1$. Note that $\Delta^u=\pm 1$ since $Z_u$ is applied on the identical $u$th bit in $x$ and $y$ which can either be 0 or 1. This implies that $\forall u$ we have
\begin{equation}
a_X^u \leq -\left(\sum_{v\colon v > u} |a_{XZ}^{uv}|+\sum_{w\colon w<u} |a_{ZX}^{wu}|\right).
\label{eq:global}
\end{equation}
A local term is of the form $D_{uv}^{(1)}=a_{XZ}^{uv} X_u Z_v+a_{ZX}^{uv} Z_u X_v+a_{XI}^{uv} X_u I_v+a_{IX}^{uv} I_u Z_v$, where the coefficients $a_{XI}^{uv}, a_{IX}^{uv}$ can be freely chosen up to the overall constraint
$a_X^u=\sum_{v\colon v > u} a_{XI}^{uv}+\sum_{w\colon w < u} a_{IX}^{wu}$. Now, clearly, if Eq.~(\ref{eq:global}) holds, then one can always distribute $a_X^u$ into a sum over $a_{XI}^{uv}$ (for $v > u$) and $a_{IX}^{wu}$ (for $w < u$) such that each $a_{XI}^{uv} \leq -|a_{XZ}^{uv}|$ and each $a_{IX}^{wu} \leq -|a_{ZX}^{wu}|$. Hence, by Proposition~\ref{prop:basic} there is a decomposition with terms $D_{uv}$ such that $D_{uv}^{(1)}$ is a symmetric $Z$-matrix, and so $D_{uv}$ is a symmetric $Z$-matrix.
\end{proof}
\section{{\sf LocalSignCure} for a class of two-local Hamiltonians is NP-complete}\label{sec:hardnessResult}
In this section we present a family of Hamiltonians for which solving {\sf LocalSignCure} is NP-complete, and thus show that {\sf LocalSignCure} is NP-hard.
We will first show that {\sf LocalSignCure} for this class of Hamiltonians is in NP. This is not immediately apparent, since local basis transformations have a continuous parametrization, hence one either has to allow for approximate sign-curing transformations or prove that for this particular class of Hamiltonians any sign-curing transformation is a member of a discrete subset of transformations. We settle this problem by proving in Lemma~\ref{lem:restrictedbasis} that with the addition of ancilla qubits and ``gadget" interactions, any Hamiltonian in this class can be converted into one for which any sign-curing transformation must consist of either Hadamard gates or the identity operation.
In order to prove that the problem is NP-hard, we show how to encode any 3-SAT instance into the problem of curing a corresponding Hamiltonian using the identity or Hadamard gates. In Lemma~\ref{lem:H3SAT} we prove that such a curing transformation exists if and only if the corresponding 3-SAT instance is satisfiable.
A proof of Theorem~\ref{thm1} follows straightforwardly by considering {\sf LocalSignCure} for the family of Hamiltonians constructed by adding the gadgets (Lemma~\ref{lem:restrictedbasis}) to the Hamiltonians corresponding to 3-SAT instances (Lemma~\ref{lem:H3SAT}).
\subsection{Hadamard sign curing gadget}
In this section we introduce the ``gadget" interactions which will effectively force any sign-curing transformation to be from a discrete subset of transformations. Let $W_u$ be a single-qubit Hadamard on qubit $u$: this is a convention we will use throughout this section and the next.
\begin{lemma}
\label{lem:restrictedbasis}
Let $H$ be a two-local Hamiltonian on $n$ qubits. For each qubit $u \in \{1,\dots n\}$, add three ancilla qubits $a_u,b_u,c_u$ and define the two-local gadget Hamiltonian $G$, and the total Hamiltonian $H_{\rm Had}$ as:
\begin{align}
\label{eq:Hdiscrete}
G&=\sum_{u=1}^n \big[-(X_{c_u}+Z_{c_u})-(X_u X_{a_u}+Y_u Y_{a_u}+Z_u Z_{a_u}) \nonumber \\
\nonumber &-(3X_{a_u}X_{b_u}+Y_{a_u}Y_{b_u}+2Z_{a_u}Z_{b_u})
-(X_{b_u}X_{c_u}+Y_{b_u}Y_{c_u}+Z_{b_u}Z_{c_u}) \big],\\
H_{\rm Had} &= H + G.
\end{align}
Then the following are equivalent:
\begin{enumerate}
\item there exists a unitary $U=\bigotimes_{u=1}^{n} (U_u \otimes U_{a_u} \otimes U_{b_u} \otimes U_{c_u})$ such that $U H_{\rm Had} U^{\dagger}$ is a symmetric $Z$-matrix.
\item there exists $x \in \{0,1\}^n$ such that ${\sf W}(x)^{\dagger} H {\sf W}(x)$ is a symmetric $Z$-matrix, where ${\sf W}(x)=\bigotimes_{u=1}^n W_u^{x_u}$.
\end{enumerate}
\end{lemma}
\begin{proof}
First we prove $2 \rightarrow 1$. If there exists $x \in \{0,1\}^n$ such that ${\sf W}(x) H {\sf W}(x)^{\dagger}$ is a symmetric $Z$-matrix, then it is easy to check that $U H_{\rm Had} U^{\dagger}$ is a symmetric $Z$-matrix with
\[U= \bigotimes_{u=1}^{n} \left(W_u \otimes W_{a_u}\otimes W_{b_u}\otimes W_{c_u}\right)^{x_u}.\]
To prove the other direction, we will show that if 1. holds, each of the single-qubit unitaries $U_{\alpha}$ ($\alpha \in \bigcup_{u=1}^n \{u,a_u,b_u,c_u\}$) must be from the discrete set $\{I,W,X,XW\}$. This fact will suffice by the following reasoning. First note that conjugating by local $X$ matrices permutes the off-diagonal matrix entries of the Hamiltonian among themselves~\cite{marvian2018computational}. So if $U H_{\rm Had} U^{\dagger}$ is a symmetric $Z$-matrix and $U_{\alpha} \in \{I,W,X,XW\}$, then $\bar{U} H_{\rm Had} \bar{U}^{\dagger}$ is also a symmetric $Z$-matrix, where $\bar{U} = \bigotimes_{\alpha} \bar{U}_{\alpha}$ and
\begin{equation*}
\bar{U}_{\alpha} = \left\lbrace \begin{array}{cc}
I & U_{\alpha} = I \textrm{ or } X\\
W & U_{\alpha}= W \textrm{ or } XW
\end{array} \right.
\end{equation*}
since $UH_{\rm Had} U^{\dagger}$ and $\bar{U} H_{\rm Had} \bar{U}^{\dagger}$ are related by conjugation by local $X$ matrices. Using the fact that the partial trace of a symmetric $Z$-matrix is also a symmetric $Z$-matrix, and noting that by tracing out the ancilla qubits of $\bar{U} H_{\rm Had} \bar{U}^{\dagger}$ we get $\bar{U} H \bar{U}^{\dagger}$, we conclude that if $\bar{U} H_{\rm Had} \bar{U}^{\dagger}$ is a symmetric $Z$-matrix, then so is $\bar{U}H\bar{U}^{\dagger}$, and $\bar{U} = {\sf W}(x)$ for some $x$.
We now proceed with proving that $U_{\alpha} \in \{I,W,X,XW\}$ given 1. Here we make use of the picture of orthogonal rotations on $\beta$ matrices, as mentioned in Proposition ~\ref{prop:ortho}.
For a given $u$ we note that there are no 1-local terms involving qubits $a_u$ and $b_u$, and that the matrix $\beta_{{a_u} {b_u}}$ is diagonal and has 3 distinct non-zero singular values.
In the absence of 1-local terms, it follows directly from Proposition~\ref{prop:basic} that $\beta_{{a_u} {b_u}}$ has to remain diagonal for $H_{\rm Had}$ to be a symmetric $Z$-matrix.
Therefore, the only possible transformations are signed permutations (of the Paulis) on qubits $a_u$ and $b_u$ with the permutations being the same to maintain the diagonality of $\beta_{{a_u} {b_u}}$.
This implies that there exists a single-qubit Clifford transformation ${\sf C}$ (corresponding to the permutation) and Pauli matrices $P_{a_u}$ and $P_{b_u}$ such that $U_{a_u}=P_{a_u}{\sf C}$ and $U_{b_u}=P_{b_u}{\sf C}$.
We now consider the interaction between qubits $u$ and $a_u$. For the overall Hamiltonian to be real, the coefficients of $X_u Y_{a_u}, Y_u X_{a_u}, Z_u Y_{a_u}, Y_u Z_{a_u}$ must all be zero. Since there are no 1-local terms acting on qubit $a_u$, the coefficient of $Z_u X_{a_u}$ must also be zero and so the rotated matrix $\beta_{u a_u}'$ must have zeroes in the following positions:
\begin{equation*}\beta_{u a_u}'=O_u^{\sf T} \beta_{u a_u} O_{a_u}=\left(\begin{array}{ccc}
* & 0 & * \\
0 & * & 0 \\
0 & 0 & * \\
\end{array}\right).
\end{equation*}
Note that $-\beta_{u a_u}$ is the identity matrix in Eq.~(\ref{eq:Hdiscrete}), so $\beta_{u a_u}'=O_u^{\sf T} \beta_{u a_u} O_{a_u}=-O_u^{\sf T} O_{a_u}$ is an orthogonal matrix. The only orthogonal matrix with zeroes in these positions is a diagonal matrix (with $\pm 1 $ on the diagonal). Therefore $O_u$ must equal $O_{a_u}$ up to signs; that is $U_u=PU_{a_u}$ for some Pauli $P$.
Since the matrix $\beta_{b_u c_u}$ is identical to $\beta_{u a_u}$ and also there is no 1-local terms acting on qubit $b_u$, an identical argument shows that any curing transformation, $U_{b_u}$ and $U_{c_u}$ must satisfy $U_{b_u}=P U_{c_u} $ for some Pauli matrix $P$. Thus for all $\alpha \in \{u,a_u,b_u,c_u\}$, we have $U_{\alpha}=P_{\alpha} { \sf C}$ for some Pauli matrix $P_{\alpha}$ and a single-qubit Clifford transformation ${\sf C}$.
Due to the 1-local terms $-(X_{c_u}+Z_{c_u})$, if ${\sf C}$ maps $X\rightarrow Y$ or $Z \rightarrow Y$ the Hamiltonian will have imaginary matrix entries and so, up to multiplication by a Pauli, ${\sf C}$ must be $I$ or $W$. Incorporating any such Pauli into $P_{c_u}$, we may assume wlog that ${\sf C} \in \{I,W\}$.
Furthermore, if $P_{c_u}$ is $Y$ or $Z$, there will be a positive $+X_{c_u}$ term, so $P_{c_u} \in \{I,X\}$. Finally, if any of the other $P_{\alpha}$ are $Y$ or $Z$, there will be a positive $+X \otimes X$ term, and so for all $\alpha$ we must have $P_{\alpha} \in \{I,X\}$ and so $U_{\alpha} = P_{\alpha} {\sf C } \in \{I, W,X,XW\}$.
\end{proof}
The following Lemma was proved in \cite{klassen2018two,marvian2018computational} by formulating an efficient strategy which finds a two-local termwise-stoquastic decomposition which is equivalent to $H$ being a symmetric $Z$-matrix by Proposition \ref{prop:global2}\footnote{More generally, one can note that it is easy to decide whether a $k$-local Hamiltonian is $k$-local term-wise stoquastic, as this is a linear programming problem. This can be seen by noting that the number of parameters needed to specify a local decomposition is polynomially dependent on the number of qubits, and the number of conditions to test on each term is dependent on the locality of the term. Furthermore, all of the conditions are linear \cite{marvian2018computational, thesis:ioannou}.}.
\begin{lemma}\label{lem:checkingIsEasy}\cite{klassen2018two}
Given a two-local Hamiltonian $H$ on $n$ qubits, one can decide if $H$ is a symmetric $Z$-matrix in the given basis in a number of steps polynomial in $n$.
\end{lemma}
Corollary~\ref{cor:NPcontainment} now follows immediately from Lemma~\ref{lem:restrictedbasis} and Lemma~\ref{lem:checkingIsEasy}, because the string $(x_1,\dots,x_n)$ is an efficiently checkable witness in the case that $H_{\rm Had}$ is sign-curable by a local unitary transformation:
\begin{corollary}
\label{cor:NPcontainment}
If $H$ is a two-local Hamiltonian, then for Hamiltonians $H_{\rm Had}$ of the form in Eq.~(\ref{eq:Hdiscrete}), ${\sf LocalSignCure}$ is in NP.
\end{corollary}
\subsection{{\sf LocalSignCure} is NP-hard}
Now we will show how to reduce 3-SAT to {\sf LocalSignCure}, and hence show that {\sf LocalSignCure} is NP-hard.
At the heart of the construction is a Hamiltonian $H_{\text{OR}}$ which acts on four qubits labeled $d,1,2,3$:
\begin{equation}
H_{\text{OR}}=-(X_d+Z_d+I) \otimes (Z_1+Z_2+Z_3+2I).
\end{equation}
Thanks to Lemma~\ref{lem:restrictedbasis} it suffices to consider a local basis change of the form ${\sf W}(x)=\bigotimes_{j \in \{d,1,2,3\}}W_j^{x_j}$. Note that $-(X_d+Z_d+I)$ has non-positive matrix entries and is invariant under conjugation by $W_d$. Therefore ${\sf W}(x)H_{\text{OR}}{\sf W}(x)^{\dagger}$ is a symmetric $Z$-matrix if and only if the bit string $x$ is such that all the matrix entries of
\[W_1^{x_1} Z_1 W_1^{x_1} + W_2^{x_2} Z_2 W_2^{x_2} +W_3^{x_3} Z_3 W_3^{x_3} +2I\]
are non-negative. Recalling that $WZW=X$, one can see that for any $x$, all the off-diagonal matrix entries are non-negative. In addition, the diagonal entries are non-negative unless $(x_1,x_2,x_3)=(0,0,0)$.
Therefore ${\sf W}(x) H_{\text{OR}}{\sf W}(x)^{\dagger}$ is a symmetric $Z$-matrix if and only if $x_1 \vee x_2 \vee x_3$ evaluates to true.
Let $C$ be a 3-SAT Boolean formula of the form
\[C=\bigwedge_{k=1}^m C_k=\bigwedge_{k=1}^m \left(c_{k,1} \vee c_{k,2} \vee c_{k,3}\right),\]
with $m$ clauses and $n$ variables, where each $c_{k,j}$ is equal to $x_i$ or $\bar{x}_i$ for some $i \in \{1,\dots,n\}$.
Let $H_C$ be the Hamiltonian on $m+n$ qubits (labelled $\{1,\dots, n\} \cup \{d_1,\dots d_m\}$) defined by
\begin{equation}
\label{eq:HC}
H_C=\sum_{k=1}^m H_k=\sum_{k=1}^m - (X_{d_k} +Z_{d_k} +I)
\otimes \left(S(c_{k,1})+S(c_{k,2})+S(c_{k,3})+2I\right),
\end{equation}
where \[S(c)=\left\{ \begin{array}{cc} Z_i & \text{if } c=x_i \text{ for some } i \\
X_i & \text{if } c=\overline{x_i} \text{ for some } i\end{array} \right. .\qquad \]
An instance of such a Hamiltonian is illustrated in Figure \ref{fig:satConst}.
\begin{figure}
\includegraphics[scale=1]{figures/SatConst.pdf}
\caption{An encoding of a $3$-SAT Boolean formula $C$, with two clauses and five variables, into a Hamiltonian $H_C$ as prescribed by Eq.~(\ref{eq:HC}).}\label{fig:satConst}
\end{figure}
For $x \in \{0,1\}^n$ and $y \in \{0,1\}^m$, define
\[{\sf W}(x,y)=\left(\bigotimes_{i=1}^n W_i^{x_i}\right) \otimes \left( \bigotimes_{j=1}^m W_{d_j}^{y_j}\right).\]
\begin{lemma}
\label{lem:H3SAT}
Let $C$ be a 3-SAT Boolean formula, and $H_C$ be the corresponding Hamiltonian defined in Eq.(\ref{eq:HC}), and let $x \in \{0,1\}^n$. $C(x)$ evaluates to true if and only if $\forall y \in \{0,1\}^m$, ${\sf W}(x,y) H_C {\sf W}(x,y)^{\dagger}$ is a symmetric $Z$-matrix.
\end{lemma}
\begin{proof}
Note that $(X_{a_k}+Z_{a_k}+I)$ is invariant under conjugation by $W_{d_k}$, so the choice of $y$ leaves $H_C$ unchanged. Furthermore $(X_{a_k}+Z_{a_k}+I)$ has non-negative matrix entries (with some positive off-diagonal matrix entries). Therefore ${\sf W}(x,y) H_k {\sf W}(x,y)^{\dagger}$ is a symmetric $Z$-matrix if and only if all the matrix entries of
\begin{equation}
\label{eq:WSSSW}
{\sf W}(x,y) \big(S(c_{k,1})+S(c_{k,2})+S(c_{k,3})+2I\big) {\sf W}(x,y)^{\dagger}
\end{equation}
are non-negative.
As discussed above, $S(c)$ has been defined so that the matrix entries of (\ref{eq:WSSSW}) are non-negative exactly when $\left(c_{k,1} \vee c_{k,2} \vee c_{k,3}\right)$ is true.
Since each $H_k$ is the only interaction acting on qubit $d_k$, and $H_k$ can only fail to be a symmetric $Z$-matrix due to terms which act non-trivially on $d_k$, it follows that \linebreak ${\sf W}(x,y) H_k {\sf W}(x,y)^{\dagger}$ must be a symmetric $Z$-matrix for all $k$, in order for \linebreak ${\sf W}(x,y) H_C {\sf W}(x,y)^{\dagger}$ to be a symmetric $Z$-matrix. Since $C=\bigwedge_{k=1}^m C_k$, this happens exactly when $C(x)$ is true.
\end{proof}
This leads to the main result of this section:
\begin{corollary}
There exists a class of two-local Hamiltonians for which {\sf LocalSignCure} is NP-complete.
\end{corollary}
\begin{proof}
For any 3-SAT formula $C$, we construct the two-local Hamiltonian $H_{C,{\rm Had}}$ by adding the gadget interactions $G$ of Eq. (\ref{eq:Hdiscrete}) for each qubit in the Hamiltonian $H_C$ in Eq.~(\ref{eq:HC}). Using Lemma~\ref{lem:restrictedbasis} and Lemma~\ref{lem:H3SAT},
we conclude that satisfying a family of 3-SAT formulae $C$ is equivalent to {\sf LocalSignCure} for the corresponding family of $H_{C,{\rm Had}}$ Hamiltonians, from which we conclude that {\sf LocalSignCure} is NP-hard.
The inclusion of {\sf LocalSignCure} for $H_{C,{\rm Had}}$ in NP follows from Corollary~\ref{cor:NPcontainment}.
\end{proof}
Let us briefly comment on the question how hard determining the groundstate energy of $H_{C, {\rm Had}}$ may be. We observe that the qubits $d_i$ and the triples of ancilla qubits $a_u,b_u,c_u$ for each $u$ only couple to the $n$ qubits on which the clauses act.
In particular, if we would fix the state of these ancillary qubits to $\psi$, then the resulting Hamiltonian $\bra{\psi} H_{C, {\rm Had}} \ket{\psi}$ acting only on the clause qubits would be purely 1-local. It is however not a priori clear that the minimal energy is obtained when the state $\psi$ is a product state, if this were the case then the ground state energy problem would be in NP as a prover could provide a description of this product state. However even the problem of finding such a product ground state is not guaranteed to have an obvious polynomial-time classical algorithm.
It would be worthwhile to investigate this further.
\section{An efficient algorithm for {\sf LocalSignCure} for exactly two-local Hamiltonians}\label{sec:curingResult}
\subsection{Preliminaries}\label{sec:prelim}
In this section we prove Theorem \ref{thm2} by presenting an efficient algorithm for solving {\sf LocalSignCure} when $H$ is an exactly two-local Hamiltonian.
We represent an exactly two-local Hamiltonian by a graph $G$ with matrix-weighted edges. Each qubit in the Hamiltonian corresponds to a vertex in the graph, and each edge corresponds to a term $H_{uv} \neq 0$. Every edge is weighted by the $3\times 3$ real matrix $\beta_{uv}$ associated with $H_{uv}$, as discussed in Proposition~\ref{prop:ortho}.
In this picture, {\sf LocalSignCure} reduces to the following problem. Consider a graph $G=(V,E)$ with $n$ vertices in $V$ and a set of directed matrix-weighted edges $E$. Each edge $(u,v)$ with direction $u \rightarrow v$ is weighted by a $3\times 3$ real matrix $\beta_{uv}$, and we define $\beta_{vu} = \beta_{uv}^{\sf T}$~\footnote{The purpose of the direction is merely to allow the matrix weight to be well defined. Throughout the text we will ignore the directedness of the graph, and treat the edge as though it is weighted by $\beta_{uv}$ or $\beta_{vu}$ depending on our purpose.}
Given $G$, find a set of $SO(3)$ rotations $\{O_u\}_{u=1}^n$ which have the action $O_u^{\sf T} \beta_{uv} O_v = \Sigma_{uv} \; \forall \beta_{uv}$, such that for all edges $(u,v)$:
\begin{equation} \label{cond:diagonality0}
\Sigma_{uv} \textrm{ is a diagonal matrix},
\end{equation}
\begin{equation} \label{cond:magnitude0}
\vert (\Sigma_{uv})_{11} \vert \geq \vert (\Sigma_{uv})_{22} \vert \; \;\;\forall \beta_{uv},
\end{equation}
\begin{equation} \label{cond:negativity0}
(\Sigma_{uv})_{11} \leq 0 \; \;\;\forall \beta_{uv}.
\end{equation}
Otherwise prove that no such set exists.
Note that we have rephrased the conditions in Proposition~\ref{prop:basic} according to the labeling $X \rightarrow 1$, $Y \rightarrow 2$, $Z\rightarrow 3$. One can argue, see Ref.~\cite{klassen2018two}, that if there exist $O(3)$ rotations that perform this task, then one can easily construct a set of $SO(3)$ rotations that do the same. Therefore any orthogonal rotations will suffice.
If all matrices $\beta_{uv}$ are diagonal, then the XYZ-algorithm in Theorem~\ref{theo:KT} can be applied. Naively, our problem could then be reduced to the question: Is there a set of rotations $\{O_u\}$ that has the action $O_u^{\sf T} \beta_{uv} O_v = \Sigma_{uv} \; \forall \beta_{uv}$, such that condition~\ref{cond:diagonality0} is satisfied, and what are those rotations? If this problem is efficiently solved, one may incorporate the algorithm for finding the set of rotations as a sub-routine of the XYZ-algorithm and solve the entire problem. However we show in Appendix~\ref{app:HeisForm} that deciding the existence of such a set of rotations on $\beta_{uv}$ such that condition~\ref{cond:diagonality0} is satisfied is an NP-hard problem.
Thus a different approach must be taken, namely we focus on condition~\ref{cond:magnitude0} in order to prune the set of solutions which needs to be considered. More concretely, we will present an algorithm which solves the following problem:
\begin{problem*}[{\sf No-Lone-YY $\&$ Diagonal}]\label{problemStatement}
Is there a set of orthogonal rotations $\{O_u \in O(3)\}$ that have the action $O_u^{\sf T} \beta_{uv} O_v = \Sigma_{uv} \; \forall \beta_{uv}$, such that:
\begin{equation}\label{cond:diagonality}
\Sigma_{uv} \textrm{ is a diagonal matrix},
\end{equation}
\begin{equation} \label{cond:magnitude}
(\Sigma_{uv})_{22} = 0, \; \;\;\forall \beta_{uv} \textrm{ for which } \textrm{Rank}\,(\beta_{uv}) =1.
\end{equation}
If yes, what is that set? Note here that condition~\ref{cond:diagonality} is identical to condition~\ref{cond:diagonality0}, and condition~\ref{cond:magnitude} is precisely condition~\ref{cond:magnitude0} restricted to rank-1 matrices.
\end{problem*}
Note that an efficient algorithm for this problem can be incorporated into the XYZ-algorithm to produce an efficient algorithm for ${\sf LocalSignCure}$ for exactly two-local Hamiltonians, thus directly proving Theorem~\ref{thm2}. More precisely, a solution to this problem prescribes a transformation of our Hamiltonian into an XYZ-Heisenberg Hamiltonian, in which case the XYZ-algorithm can be used to decide if the Hamiltonian can be rotated into a symmetric $Z$-matrix by single-qubit unitary transformations. Furthermore, if no solution exists to this problem, then rotating the Hamiltonian into a symmetric $Z$-matrix by single-qubit transformations is impossible, since both of the above conditions are necessary conditions.
An orthogonal transformation $O_u$ can be written as $O_u=(e_u^1, e^2_u, e_3^u)$ where the $e_u^i$ are three real orthonormal column vectors. We can thus view selecting $O_u$ as selecting a basis $b_u=(e_u^1, e_u^2, e_u^3)$ at vertex $u$.
\begin{definition}[No-Lone-YY Basis (NLY Basis)]
Given a matrix-weighted \linebreak graph $G$ with weights $\beta_{uv}$, an ordered assignment of basis vectors $b_u=(e_1^u, e_2^u, e_3^u)$ to each vertex in the graph is called a \textbf{No-Lone-YY basis (NLY basis)} $B=\{b_u\}$, when $e_i^v$ is a right singular vector of $\beta_{uv}$ with corresponding left singular vector equal to $\pm e_i^u$, i.e.
\begin{equation} \label{eq:NLYcond1}\forall u,v, i: \; \beta_{uv} e_i^v=\pm \sigma e_i^u ,\;\; \beta_{uv}^{\sf T} e_i^u=\pm \sigma e_i^v,
\end{equation} and for all rank $1$ matrices $\beta_{uv}$:
\begin{equation}\label{eq:NLYcond2} \beta_{uv} e_2^v=0 ,\;\; \beta_{uv}^{\sf T} e_2^u=0.
\end{equation}
\end{definition}
It is not hard to see that solving problem statement~\ref{problemStatement} is equivalent to finding a NLY basis, or showing that none exists.
It is important to note that if we flip the signs on our basis elements, this will have no bearing on the problem. We formally define this equivalence under sign flips as
\begin{definition}
Two ordered bases $b_u$ and $b_u'$ are equivalent modulo signs:
$$b_u = b_u' \text{ modulo signs} $$
if $b_u = (e_1^u, e_2^u, e_3^u)$, and $b_u' = (\delta_1^u e_1^u, \delta_2^u e_2^u, \delta_3^u e_3^u)$ with $\delta_i^u \in \{+1, -1\}$.
\end{definition}
Thus throughout the text we will often talk about a basis \textbf{modulo signs}, meaning a basis choice where the signs have not been specified. The premise is that the choice of signs is irrelevant for the purposes of the problem. This will prove to be a useful fact in the proofs of Lemma~\ref{lem:genericAnsatz} and Theorem~\ref{thrm:genericAlgo}.
A final comment on notation. In the next two subsections we will make use of \textbf{sets of subspaces} of $\mathbb{R}^3$. We wish to hold onto the notion that these are sets of subspaces, but make use of natural set notation in terms of the elements of the subspaces. Consequently, for ease of exposition, we will abuse notation in the following ways. We denote a set of subspaces by $\mathbb{S}=\{S_i \vert S_i \subseteq \mathbb{R}^3\}$. We denote the entrywise intersection of sets of subspaces by
\begin{equation*}
\mathbb{S}_1 \cap \mathbb{S}_2 := \{S_i \cap S_j \vert S_i \in \mathbb{S}_1 \;, S_j \in \mathbb{S}_2\}.
\end{equation*}
We denote the span of the union of the subspaces by
\begin{equation*}
\textrm{span} (\mathbb{S}) := \textrm{span}\left( \bigcup_{S_i \in \mathbb{S}} S_i \right).
\end{equation*}
We say a set of vectors $b= \{\nu \vert \nu \in \mathbb{R}^3 \}$ is in a set of subspaces $\mathbb{S}$, with the notation $b \subseteq \mathbb{S}$, if every vector in $b$ belongs to a subspace in $\mathbb{S}$. Furthermore, we say a set of subspaces $\mathbb{S}_1$ is contained in another set of subspaces $\mathbb{S}_2$, with the notation $\mathbb{S}_1 \subseteq \mathbb{S}_2$, if every subspace in $\mathbb{S}_1$ is contained in a subspace in $\mathbb{S}_2$. The reason these two notations coincide is because it can be helpful for our purposes to conceptualize the vectors in $b$ as 1-dimensional subspaces, since we do not care about the sign of the vector.
We denote the transformation on each of the subspaces by an orthogonal rotation $O$ as:
\begin{equation*}
O\mathbb{S} := \{ OS_i \vert S_i \in \mathbb{S} \}.
\end{equation*}
\subsection{XOR-SAT}
In the next two subsections we will make repeated use of a subroutine for solving the 2-XOR-SAT problem. XOR-SAT is a Boolean satisfiability problem in which one has a set of Boolean variables $\{x_u\}$ and a set of clauses consisting of not operations and xor operations e.g. $ \bar{x}_u \oplus x_v$, and one asks if there exists an assignment to the Boolean variables which satisfies all of the clauses. XOR-SAT is known to be solvable in polynomial time. 2-XOR-SAT is quite trivially solvable in time $O(N^2)$, where $N$ is the number of variables: the assignment of one variable in the clause uniquely determines the assignment of the other variable in the clause. Thus one varies the assignment of one variable, and propagates that choice through the clauses (of which there are worst case $N^2$), until all variables are assigned or a contradiction is found (if there are disconnected sets of variables, one does the same thing for each disconnected cluster).
\subsection{Illustrative sub-case: graphs with rank-1 edges}\label{subsec:rank1graph}
We begin by considering an illustrative sub-case, where each edge in the graph is weighted by a rank-1\ matrix (i.e. a \textbf{rank-1\ edge}).
The significance of rank-1\ edges is that their matrix weights have a two dimensional null space, which implies an additional freedom in the choice of basis that is not present in edges weighted by rank$>$1\ matrices (i.e \textbf{rank$>$1\ edges}), which have at most a one-dimensional null space. This difference will become more apparent when we consider the general case of a graph with both rank$>$1\ and rank-1\ edges.
For a graph with only rank-1\ edges the algorithm for solving problem statement~\ref{problemStatement} breaks up into two parts. In the first part we impose some of the necessary constraints for the basis assignment to be NLY, formulating a candidate basis $B$.
In the second part we permute the vectors of the candidate basis so that it could become an NLY basis.
\begin{definition}[Candidate Basis of a rank-1\ graph] \label{def:ansatzRank1}
A \textbf{candidate basis of a rank-1\ graph} is a basis assignment $B= \{b_u\}$ such that for every edge $e=(u,v)$, the basis vectors $b_u = (e_1^u, e_2^u, e_3^u)$ are eigenvectors of $\beta_{uv} \beta_{uv}^{\sf T}$ and the basis vectors $b_v= (e_1^v, e_2^v, e_3^v)$ are eigenvectors of $\beta_{uv}^{\sf T} \beta_{uv}$.
\end{definition}
\begin{proposition} \label{prop:lowrank}
Given a rank-1\ matrix $\beta_{uv}$, if the basis vectors $b_u $ are eigenvectors of $\beta_{uv} \beta_{uv}^{\sf T}$ and the basis vectors $b_v$ are eigenvectors of $\beta_{uv}^{\sf T} \beta_{uv}$, then there exists a single index $i$ such that $\beta_{uv}^{\sf T} e_i^u \neq 0$ and a single index $j$ such that $\beta_{uv} e_j^v \neq 0$. Furthermore, $\exists \sigma \neq0$ s.t. $\beta_{uv}e_i^v = \pm \sigma e_j^u$ and $\beta_{uv}^{\sf T}e_j^u = \pm \sigma e_i^v$.
\end{proposition}
\begin{proof}
Since $\beta_{uv}$ is rank-1\ it follows that $\beta_{uv} \beta_{uv}^{\sf T}$ and $\beta_{uv}^{\sf T} \beta_{uv}$ are also rank-1 . Thus only single basis vectors $e_i^u \in b_u$ and $e_j^v \in b_v$ will be eigenvectors with non-zero eigenvalue of $\beta_{uv} \beta_{uv}^{\sf T}$ and $\beta_{uv}^{\sf T} \beta_{uv}$ respectively. Therefore $e_i^u$ and $e_j^v$ are the only singular vectors in $b_u$ and $b_v$ which have non-zero singular values for $\beta_{uv}$. Since the column and row spaces of $\beta_{uv}$ are both 1-dimensional, it must be the case that $\beta_{uv}e_i^v = \pm \sigma e_j^u$ and $\beta_{uv}^{\sf T}e_j^u = \pm \sigma e_i^v$ for some $\sigma \neq 0$.
\end{proof}
Note that given a candidate basis (and the corresponding orthogonal rotations $\{O_u\}$) the matrix $O_u^{\sf T} \beta_{uv} O_v$ has exactly one non-zero entry but isn't necessarily diagonal. An example of a matrix of this form would be:
\begin{equation}\label{eq:lowRankExample}
O_u^{\sf T} \beta_{uv} O_v = \begin{bmatrix}
0&0&4\\
0&0&0\\
0&0&0
\end{bmatrix}.
\end{equation}
Therefore a candidate basis is close to being a NLY basis, except the ordering of the basis vectors in $b_u$ and $b_v$ may not be correct. In order to remedy this, we need to permute orderings of the various $b_u$.
To help visualize this, we may consider the edge $(u,v)$ to be labelled by $i$ on the $u$ side, and $j$ on the $v$ side, where $i$ and $j$ are the indices specified in Proposition~\ref{prop:lowrank}. For example, the matrix in Eq.~(\ref{eq:lowRankExample}) would correspond to the edge in Figure~\ref{fig:bilabeledEdge}.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=.7]{figures/bilabeledEdge}
\end{center}
\caption{Bilabelling of a rank-1\ edge}\label{fig:bilabeledEdge}
\end{figure}
In this picture the candidate basis $B$ thus specifies a bi-labelled graph, i.e. a graph where every edge has two labels (two colors).
\begin{definition}[Basis Permutation]
Given a basis $b = (e_1, e_2, e_3)$ and permutation $\pi$, the \textbf{permuted basis $b_u^{\pi}$} is defined as:
$b_u^{\pi} := (e_{\pi^{-1}(1)}^u ,e_{\pi^{-1}(2)}^u ,e_{\pi^{-1}(3)}^u ).$ Given a basis assignment to every vertex $B = \{b_u\}$ and an assignment of permutations to every vertex $\Pi = \{\pi_u\}$, the permuted basis assignment is defined as $B^{\Pi}: = \{ b_u^{\pi_u} \}$
\end{definition}
Given that the candidate basis $B$ specifies a bi-labelled graph, we can think of the action of basis permutations
$ b_u \rightarrow b_u^{\pi} $ as a transformation on the labeling, $i \rightarrow \pi(i)$, of every label adjacent to $u$, as illustrated in Figure~\ref{fig:bilabeledGraph}.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=.7]{figures/bilabeledGraph}
\end{center}
\caption{Action of permutations on a bi-labelled graph}\label{fig:bilabeledGraph}
\end{figure}
The premise is then that the only remaining task is to find a set of permutations $\{\pi_u \in S_3\}$ to apply to every vertex so that:
\begin{itemize}
\item The bi-labelling is uniform on an individual edge (i.e. $i=j$), corresponding to condition~\ref{eq:NLYcond1}.
\item no edge is labelled by the (green) value $2$, corresponding to condition~\ref{eq:NLYcond2}.
\end{itemize}
If we are unsuccessful in either finding a candidate basis $B$, or an appropriate permutation $\Pi$, then we will argue that no NLY basis exists.
\iffalse
\begin{algo}[Finding a Candidate Basis of a rank-1\ graph
For every vertex $v$, and every adjacent edge $e=(u,v)$, construct the set $\mathbb{S}_v^e$ consisting of maximal eigenspaces of $\beta_{uv}^{\sf T} \beta_{uv}$, each eigenspace associated with a distinct eigenvalue of $\beta_{uv}^{\sf T} \beta_{uv}$, so that each set $\mathbb{S}_v^e$ consists of mutually orthogonal eigenspaces. Take the intersection of these eigenspaces over all $e$: $ \mathbb{S}_v := \bigcap_e \mathbb{S}_v^e .$
If for any $v$, $\textrm{span}(\mathbb{S}_v) \neq \mathbb{R}^3$ then no candidate basis exists. Otherwise, for every $v$ one can always select a basis $b_v$ from $\mathbb{S}_v$, since the spaces in $\mathbb{S}_v$ are all mutually orthogonal.
\end{algo}
\fi
\begin{algorithm}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{Graph $G = (V, E)$, rank-1\ matrix edge weights $\{\beta_{uv}\}$ }
\Output{A candidate basis $B= \{b_u\}$, if one exists. Otherwise False, indicating no candidate basis exists.}
\For{$v \in V$}{
$\mathbb{S}_v = \{ \mathbb{R}^3 \}$
\For{$u \in V$ s.t. $e=(u,v) \in E$}{
$\mathbb{S}_v^e= $ the set of orthogonal maximal eigenspaces associated with every eigenvalue of $\beta_{uv}^{\sf T} \beta_{uv}$
$\mathbb{S}_v = \mathbb{S}_v \cap \mathbb{S}_v^e$
}
\If{$\textrm{span}(\mathbb{S}_v) \neq \mathbb{R}^3$ }
{
return False\;
}
Choose orthonormal basis $b_v = (e_1^v, e_2^v, e_3^v) \subseteq \mathbb{S}_v$
}
return $B= \{b_v \}$
\caption{Algorithm for finding a candidate Basis of a rank-1\ graph}\label{alg:rank1ansatz}
\end{algorithm}
\begin{lemma}\label{lem:rank1asatz}
Algorithm~\ref{alg:rank1ansatz} efficiently finds a candidate basis for a rank-1\ graph, or otherwise shows that no such candidate basis exists.
\end{lemma}
\begin{proof}
Any vectors we choose from $\mathbb{S}_v$ must simultaneously be eigenvectors of $\beta^{\sf T}_{uv} \beta_{uv}$ for all edges $e=(u,v)$ adjacent to $v$, since they must simultaneously belong to every $\mathbb{S}_v^e$. Furthermore, the spaces in $\mathbb{S}_v$ contain all vectors that are simultaneously eigenvectors of $\beta^{\sf T}_{uv} \beta_{uv}$ for all edges $e=(u,v)$ adjacent to $v$. Therefore if $\mathbb{S}_v$ does not span $\mathbb{R}^3$, then we cannot possibly choose a set of orthonormal vectors $b_v$ which are simultaneous eigenvectors of all neighbouring edges.
The number of elements in any set of eigenspaces $\mathbb{S}_v^e$ is upper bounded by 3, corresponding to 3 orthogonal 1-dimensional subspaces. The same is true for any intersection of any number of these sets. Thus computing any intersection between these sets of subspaces takes $O(1)$ time. Therefore one may iteratively construct $\mathbb{S}_v$ in time proportional to the number of edges. Thus the algorithm is efficient.
\end{proof}
\begin{algo}[Finding permutations $\Pi$ such that $B^{\Pi}$ is an NLY basis]
This algorithm takes a candidate basis $B$ of a rank-1\ graph and finds a set of permutations $\Pi$ such that $B^{\Pi}$ is a NLY basis, or otherwise indicates that no such set of permutations exist.
For each edge $(u,v)$, identify the left singular vector $e_i^u \in b_u$ and corresponding right singular vector $e_j^v \in b_v$ which are not in the null space of $\beta_{uv}$, which must exist by Proposition~\ref{prop:lowrank}. Label each rank-1\ edge $(u,v)$ with an ordered pair of labels $(i,j)$, as illustrated in figure~\ref{fig:bilabeledEdge}. We say that an edge $e=(u,v)$ with labelling $(i,j)$ connects to $u$ with label $i$ and connects to $v$ with $j$.
If for any vertex $v$ there are at least three edges, each connected to $v$ by a different label, then terminate and indicate that the desired set of permutations does not exist.
If the algorithm has not terminated, then for every vertex $v$ there exist two labels $i$ and $j$ such that every edge adjacent to $v$ connects to $v$ with one of those two labels. This holds even if every edge connects to $v$ with the same label. Identify a pair of permutations $\pi^0_{v}$ and $\pi^1_{v}$ such that $\pi^0_{v}(i)=1$, $\pi^0_{v}(j)=3$ and $\pi^1_{v}(i)=3$, $\pi^1_{v}(j)=1$.
The task now becomes assigning a binary value $x_{v}$ to each $v$ so that for every edge $(u,v)$ with label $(i,j)$ the binary assignments satisfy
\begin{equation*}
\pi^{x_{u}}_{u}(i) = \pi^{x_{v}}_{v}(j).
\end{equation*}
By virtue of $\pi$ never mapping any label to the value $2$, and ensuring the uniform bi-labelling of each edge, such an assignment will specify an NLY basis.
This problem reduces straightforwardly to an XOR-SAT problem. Each edge $(u,v)$ corresponds to an XOR clause: $\bar{x}_{u} \oplus x_{v}$ if $i=j$, and $ x_{u} \oplus x_{v}$ if $i \neq j$. If there is a solution, then this specifies an NLY basis, namely $B^{\Pi}$ with $\Pi= \{\pi^{x_{u}}_{u}\}$. If there is no solution, then the desired set of permutations does not exist.
\end{algo}
\begin{algorithm}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{Graph $G = (V, E)$, rank-1\ matrix edge weights $\{\beta_{uv}\}$, candidate basis $B = \{b_u\}$ }
\Output{A set of permutations $\Pi = \{ \pi_u\} $ such that $B^{\Pi}$ is an NLY basis, if one exists. Otherwise False.}
\For{$v \in V$}{
\tcc{Label all incident edges according to which basis vector in $b_v$ is not in the null space of $\beta_{uv}$. These always exist by Proposition~\ref{prop:lowrank}.}
$L(v) =\{ \}$
\For{$u\in V$ s.t. $e=(u,v)\in E$}{
\For{$i\in \{1,2,3\}$} {
$e_i^v = b_v[i]$
\If{$\beta_{uv}e_i^v\neq0$} {
$L(v,e) = i$
$L(v) = L(v) \cap \{i\}$
}
}
}
\tcc{If a vertex is incident on more than two different labels, then return false.}
\If{$\vert L(v) \vert = 3$}{
return False
}
\tcc{If the algorithm has not terminated, then for every vertex $v$ there exist at most two labels such that every edge incident on $v$ connects to $v$ with one of those two labels.}
\tcc{Define permutations so that all incident edge labels are mapped to either $1$ or $3$.}
Choose permutation $\pi_v^0$ s.t. $\pi_v^0(L(v)[1])=1$, and if $\vert L(v)\vert>1$ then $\pi_v^0(L(v)[2])=3$
Choose permutation $\pi_v^1$ s.t. $\pi_v^1(L(v)[1])=3$, and if $\vert L(v)\vert>1$ then $\pi_v^1(L(v)[2])=1$
}
\tcc{For each edge define a 2-XOR-SAT clause.}
\For{$e=(u,v)\in E$}{
\eIf{$L(v,e) = L(u,e)$}{
$C_e(x_u,x_v) = x_u \oplus x_v$
}{
$C_e(x_u,x_v) = \bar{x}_u \oplus x_v$
}
}
\tcc{The solution to the associated 2-XOR-SAT problem specifies which permutations to apply at each vertex so that $\pi_u^{x_u}(i) = \pi_v^{x_v}(j)$ . By virtue of $\pi$ never mapping any label to the value $2$, and ensuring the uniform bi-labelling of each edge, such an assignment will specify an NLY basis.}
success= 2-XOR-SAT(ref $\{ x_v \;\vert\; \forall v \in V\}$, $\{C_e \;\vert\; \forall e \in E\}$)
\eIf{success}{
return $\Pi = \{ \pi_v^{x_v}\;\vert\; \forall v \in V\}$
}{
return False
}
\caption{Algorithm for finding permutations $\Pi$ such that $B^{\Pi}$ is an NLY basis}\label{alg:rank1perms}
\end{algorithm}
\begin{theorem}\label{thrm:rank1algo}
Given a graph with only rank-1\ edges, one can efficiently find an NLY basis, or otherwise show that no such basis exists.
\end{theorem}
\begin{proof}
The algorithm for finding an NLY basis in this case proceeds by first finding a candidate basis $B$ using Algorithm~\ref{alg:rank1ansatz}, and then finding a set of permutations $\Pi$ such that $B^{\Pi}$ is a NLY basis using Algorithm~\ref{alg:rank1perms}. It should be clear that the basis $B^{\Pi}$ is an NLY basis, since for every edge $(u,v)$ Algorithm~\ref{alg:rank1perms} has explicitly paired those two vectors in $b_u$ and $b_v$ not in the null space of $\beta_{uv}$, and ensured that they are not the second entry. Additionally, Algorithm~\ref{alg:rank1perms} is efficient, since solving 2-XOR-SAT is efficient.
If Algorithm~\ref{alg:rank1ansatz} fails, then by Lemma~\ref{lem:rank1asatz} no candidate basis exists, and since any NLY basis must satisfy the conditions of being a candidate basis, no NLY basis exists. Furthermore, when given a candidate basis $B$, if Algorithm~\ref{alg:rank1perms} fails, then clearly no set of permutations $\Pi$ exists such that $B^{\Pi}$ is an NLY basis. In one case this is because there are three edges connected to a vertex by a different label, and thus the label $2$ cannot be removed by any permutation. In the other it is because there is not a solution to the 2-XOR-SAT problem, which rules out all potential permutions for those vertices connected to exactly two labels, while in the case of vertices connected to exactly one label, there are other possible permutations, but they would have the same action, and are thus also ruled out.
The only non-trivial fact left to prove is that if, given a candidate basis $B$, Algorithm~\ref{alg:rank1perms} fails, then no NLY basis exists. Naively once could imagine that, given some alternative candidate basis, Algorithm~\ref{alg:rank1perms} might succeed. Here we prove that this cannot happen, using proof by contradiction.
Assume that given a candidate basis $B$, Algorithm~\ref{alg:rank1perms} fails and there does not exist a permutation $\Pi$ such that the basis $B^{\Pi}$ is an NLY basis. Suppose however that there exists an NLY basis $\bar{B}$. If for some edge $(u,v)$ adjacent to $u$, the basis vector $e_i^u \in b_u$ satisfies $\beta_{uv}^{\sf T} e_i^u \neq 0$, then there must exist a unique vector $\bar{e}_j^u \in \bar{b}_u$ such that $\beta_{uv}^{\sf T} \bar{e}_j^u \neq 0$. Furthermore $e_i^u = \pm \bar{e}_j^u $, since $\beta_{uv}$ is rank-1 . Therefore for every edge $(u,v)$ adjacent to $u$, if $e_i^u$ satisfies $\beta_{uv}^{\sf T} e_i^u \neq 0$, then $\bar{e}_j^u$ also satisfies $\beta_{uv}^{\sf T} \bar{e}_j^u \neq 0$. Therefore for every index $i \in \{1,2,3\}$ there exists an index $j \in \{1,2,3\}$ such that for every edge $e=(u,v)$ adjacent to $u$, if $e_i^u $ satisfies $\beta_{uv}^{\sf T} e_i^u \neq 0$ then $\bar{e}_j^u$ satisfies $\beta_{uv}^{\sf T} \bar{e}_j^u \neq 0$. Let $\pi_u$ be the permutation with the mapping: $\pi_u(i)=j$, and $\Pi =\{\pi_u\}$. Then the bi-labelled graph associated with $B^{\Pi}$ must be identical to the bi-labelled graph associated with $\bar{B}$, and therefore $B^{\Pi}$ must be an NLY basis, which is a contradiction.
\end{proof}
\subsection{Graphs with both rank$>$1\ and rank-1\ edges}
We will now show how the intuition and arguments given in Subsection~\ref{subsec:rank1graph} translate into the case where the matrix weights may have any rank. First we outline the structure of the argument. Just as in Subsection~\ref{subsec:rank1graph}, we will first search for a candidate basis $B$ for the graph, and then search for an appropriate set of permutations $\Pi$ to apply to the basis vectors.
\begin{definition}[Candidate Basis of a Graph]\label{def:ansatz}
A \textbf{candidate basis} $B=\{b_u\}$ is an assignment of basis vectors $b_u= (e_1^u, e_2^u, e_3^u)$, to each vertex $u$, satisfying the following two conditions:
\begin{enumerate}
\item For every rank-1\ edge $(u,v)$ adjacent to the vertex $u$, the basis vectors $b_u$ are eigenvectors of $\beta_{uv} \beta_{uv}^{\sf T}$.
\item For every rank$>$1\ edge $(u,v)$, the basis vectors of $b_u$ and $b_v$ are left and right singular vectors of $\beta_{uv}$ respectively, and satisfy:
$\exists \sigma \in \mathbb{R}$ s.t. $ \beta_{uv} e_i^v = \pm \sigma e_i^u \textrm{ and }\; \beta_{uv}^{\sf T} e_i^u = \pm \sigma e_i^v$.
\end{enumerate}
\end{definition}
The candidate basis has the same requirements on rank-1\ edges as in the previous section, however it satisfies more stringent requirements on rank$>$1\ edges, namely that the transformed matrix weights $O_u^{\sf T} \beta_{uv} O_v$ are diagonal under the prescribed orthogonal rotations $\{ O_u\}$. The most significant difference between the algorithms presented in this section
is the procedure for finding a candidate basis (Algorithm~\ref{alg:genericAnsatz}). However once a candidate basis has been found, the procedure for finding an appropriate set of permutations (Algorithm~\ref{alg:genericPerms}) will have the same essential form as Algorithm~\ref{alg:rank1perms} with one difference: Instead of individual vertices being the sites to which permutations are assigned, we will instead assign permutations to subgraphs whose vertices are connected by rank$>$1\ paths (Definition~\ref{def:HRCC}), so that each vertex in such a subgraph is permuted uniformly. This is illustrated in Figure~\ref{fig:permGenericGraph}, in contrast to Figure~\ref{fig:bilabeledGraph}. It will be straightforward to see that if Algorithms \ref{alg:genericAnsatz} and \ref{alg:genericPerms} succeed, then they will have produced an NLY basis. The only significant subtle point that remains, and will be argued in Theorem~\ref{thrm:genericAlgo}, is that if Algorithm~\ref{alg:genericPerms} is given a candidate basis and fails to find a set of permutations which produces an NLY basis, then no NLY basis exists and in particular no other candidate bases need to be considered.
\begin{figure}\label{fig:permGenericGraph}
\includegraphics[scale=.7]{figures/permGenericGraph}
\caption{Action of a permutation $(12)(3)$ on a rank$>$1 connected component \;(RCC) in black.}
\end{figure}
Before proceeding with the description of the algorithm for finding a candidate basis, we must establish some facts about rank$>$1\ edges, and the structure they impose on the problem.
\begin{lemma}\label{lem:rigid}
Given a rank$>$1\ edge $(u,v)$, and bases $b_u$, $b_v$ which are eigenvectors of $\beta_{uv}\beta_{uv}^{\sf T}$ and $\beta_{uv}^{\sf T} \beta_{uv}$ respectively. The vectors $e_i^u \in b_u$ and $e_i^v \in b_v$ satisfy $\beta_{uv} e_i^v = \pm \sigma_i e_i^u$ and $\beta_{uv}^{\sf T} e_i^u = \pm \sigma_i e_i^v$, $\sigma_i \in \mathbb{R}$, if and only if, for every singular value decomposition $\beta_{uv} = O_u^e \Sigma^{\rm SVD}_{uv} (O_v^e)^{\sf T}$ the operator defined as
\begin{equation} \label{eq:transferOp}
O_{v \leftarrow u} =O_{u \leftarrow v}^{\sf T}:= O_v^e (O_u^e)^{\sf T}
\end{equation}
satisfies $O_{v \leftarrow u} e_i^u = \pm e_i^v ,\;\; O_{u \leftarrow v} e_i^v = \pm e_i^u \;\; \forall i.$ In other words:
\begin{equation}
O_{v \leftarrow u} b_u = b_v \text{ modulo signs,} \text{ and equivalently } O_{u \leftarrow v} b_v = b_u \text{ modulo signs}.
\end{equation}
\end{lemma}
\begin{proof}
First we prove the \textit{only if} condition. Given that $\beta_{uv}^{\sf T} e^u_i = \pm \sigma_i e^v_i$ and
$ \beta_{uv} \beta_{uv}^{\sf T} e_i^u = \sigma_i^2 e_i^u $ we have
\begin{equation*}
O_v^e \Sigma_{uv}^{\rm SVD} (O_u^e)^{\sf T} e_i^u = \pm \sigma_i e_i^v\;\;,\;\; (\Sigma_{uv}^{\rm SVD})^2 (O_u^e)^{\sf T} e_i^u = \sigma_i^2 (O_u^e)^{\sf T} e_i^u.
\end{equation*}
If a real matrix $A$ is non-negative and diagonal, then any eigenvectors of $A^2$ with eigenvalues $\lambda$ are also the eigenvectors of $A$ with eigenvalues $\vert \sqrt{\lambda}\vert$. Since $\Sigma_{uv}^{\rm SVD}$ is non-negative and diagonal, we see that $(O_u^e)^{\sf T} e_i^u$ is an eigenvector of $\Sigma_{uv}^{\rm SVD}$ with eigenvalue $\vert \sigma_i \vert$.
It follows that:
\begin{equation*}
O_v^e \Sigma_{uv}^{\rm SVD} (O_u^e)^{\sf T} e_i^u = \pm \sigma_i e_i^v \Rightarrow
\vert \sigma_i \vert O_v^e(O_u^e)^{\sf T} e_i^u = \pm \sigma_i e_i^v
\end{equation*}
If $\sigma_i \neq 0$ then $O_v^e (O_u^e)^{\sf T} e_i^u =\pm e_i^v$. Suppose $\sigma_i=0$, then since $\textrm{Rank}(\beta_{uv})>1$ there is a single $i$ for which this holds. Since for all $j\neq i$, $O_v^e (O_u^e)^{\sf T} e_j^u = \pm e_j^v$, it follows that $e_i^v$ must lie in the one-dimensional subspace spanned by $O_v^e (O_u^e)^{\sf T} e_i^u$, and so $O_v^e (O_u^e)^{\sf T} e_i^u = \pm e_i^v$.
Now we prove the \textit{if} condition. Given that $ \beta_{uv} \beta_{uv}^{\sf T} e_i^u = \sigma_i^2 e_i^u $ and $O_v^e (O_u^e)^{\sf T} e_i^u = \pm e_i^v$:
\begin{align*}
\beta_{uv} \beta_{uv}^{\sf T} e_i^u &= \sigma_i^2 e_i^u \\
O_u^e \Sigma^{\rm SVD}_{uv} (O_v^e)^{\sf T} O_v^e \Sigma^{\rm SVD}_{uv} (O_u^e)^{\sf T} e_i^u &= \sigma_i^2 e_i^u \\
(\Sigma_{uv}^{\rm SVD})^2 (O_u^e)^{\sf T} e_i^u &= \sigma_i^2 (O_u^e)^{\sf T} e_i^u.
\end{align*}
Since $\Sigma_{uv}^{\rm SVD}$ is a positive diagonal matrix we have
\begin{align*} (\Sigma_{uv}^{\rm SVD}) (O_u^e)^{\sf T} e_i^u &= \vert \sigma_i \vert (O_u^e)^{\sf T} e_i^u \\
O_v^e(\Sigma_{uv}^{\rm SVD}) (O_u^e)^{\sf T} e_i^u &= \vert \sigma_i \vert O_v^e (O_u^e)^{\sf T} e_i^u \\
\beta_{uv}^{\sf T} e_i^u &= \pm \sigma_i e_i^v.
\end{align*}
By a symmetric argument $ \beta_{uv} e_i^v = \pm \sigma_i e_i^u$.
\end{proof}
It is important to note that the construction of $O_{v \leftarrow u}$ in Eq.~(\ref{eq:transferOp}) is not unique. One could find a different singular value decomposition and construct a different operator $O_{v \leftarrow u}'$. However, as proven above, for any such operator its action on a singular vector $e_i^u$ of $\beta_{uv}$ is identical, up to a difference in the sign, which has no bearing on the problem. In light of this, for the remainder of the text we will treat the operator $O_{v \leftarrow u}$ as a well defined orthogonal operator, with the implicit assumption being that any such operator suffices.
\iftrue
The above lemma has two important consequences.
\begin{corollary} \label{cor:path}\hfill
\begin{enumerate}
\item Condition 2 in Definition \ref{def:ansatz} is equivalent to the condition that for every rank$>$1\ edge $(u,v)$ and any operator $O_{v \leftarrow u}$:
$$ O_{v \leftarrow u}b_u = b_v \text{ modulo signs}.$$
\item If $B=\{ b_u\}$ is a candidate basis, then given a path $p= (u,x...,y,w,v)$ of rank$>$1\ edges going from vertex $u$ to $v$, for any orthogonal rotation defined by $O_{p} = O_{v \leftarrow w}O_{w \leftarrow y}... O_{x \leftarrow u}$, which we call a \textbf{rank$>$1\ path operator}, it must be the case that $ O_{p} b_u= b_v$ modulo signs.
\end{enumerate}
\end{corollary}
\begin{proof}
The first statement follows trivially by the definition of the candidate basis. The second statement follows by induction from the first.
\end{proof}
\fi
\iffalse
The above lemma has two important consequences.
\begin{corollary} \label{cor:path}\hfill
\begin{enumerate}
\item If $B=\{ b_u\}$ is a candidate basis, then for every rank$>$1\ edge $(u,v)$ and any operator $O_{v \leftarrow u}$, it must be the case that $ O_{v \leftarrow u}b_u = b_v$ modulo signs.
\item If $B=\{ b_u\}$ is a candidate basis, then given a path $p= (u,x...,y,w,v)$ of rank$>$1\ edges going from vertex $u$ to $v$, for any orthogonal rotation defined by $O_{p} = O_{v \leftarrow w}O_{w \leftarrow y}... O_{x \leftarrow u}$, which we call a \textbf{rank$>$1\ path operator}, it must be the case that $ O_{p} b_u= b_v$ modulo signs.
\end{enumerate}
\end{corollary}
\begin{proof}
The first statement follows trivially by the definition of the candidate basis. The second statement follows by induction from the first.
\end{proof}
\fi
\iffalse
The above lemma has important consequences for candidate bases.
\begin{corollary} \label{cor:path}\hfill
An assignment of bases $B=\{b_u\}$ is a candidate basis if and only if:
\begin{enumerate}
\item For every rank-1 edge $(u,v)$ adjacent to the vertex $u$, the basis vectors $b_u$ are eigenvectors of $\beta_{uv} \beta_{uv}^{\sf T}$
\item For every path $p= (u,x...,y,w,v)$ of rank$>$1\ edges going from vertex $u$ to $v$, and for every orthogonal rotation defined by $O_{p} = O_{v \leftarrow w}O_{w \leftarrow y}... O_{x \leftarrow u}$, which we call a \textbf{rank$>$1\ path operator}: $ O_{p} b_u= b_v \text{modulo signs}$
\end{enumerate}
\end{corollary}
\begin{proof}
Condition 1 is identical to condition 1 in Definition \ref{def:ansatz}. Condition 2 is a combination of condition 2 in Definition \ref{def:ansatz} and Lemma \ref{lem:rigid}. The argument follows straightforwardly by induction: if $O_{v\leftarrow u}b_u=b_v \text{ modulo signs}$ and $O_{w \leftarrow v}b_v = b_w \text{ modulo signs}$ then $O_{w \leftarrow v}O_{v\leftarrow u}b_u = b_w \text{ modulo signs}$
\end{proof}
\fi
We see that, if for some vertex $u$, we choose a basis $b_u$ which happens to belong to a yet unknown candidate basis $B$, then this fixes, via the rank$>$1\ path operators, all of the bases $b_v \in B$, modulo signs, for all vertices $v$ connected to $u$ by rank$>$1\ paths. This motivates the following definition.
\begin{definition}[Rank$>$1 Connected Component\ (RCC)] \label{def:HRCC}
Remove all rank-1\ edges from the graph $G$. What remains is a family of distinct connected components which are composed entirely of rank$>$1\ edges. Define the \textbf{rank$>$1 connected component} as the subgraph $\Gamma$ associated with such a connected component.
\end{definition}
Note that in the case where some vertex $v$ is connected to only rank-1\ edges, $v$ on its own still constitutes a rank$>$1 connected component . Therefore by construction every vertex is in exactly one RCC. Note also that any two vertices connected by a path of rank$>$1\ edges belong to the same RCC .
\begin{definition}[Candidate Basis on a RCC ]
Given a RCC\ $\Gamma$, a \textbf{candidate basis of a rank$>$1 connected component} $B_{\Gamma}$ on the vertices of $\Gamma$ is the assignment of a basis $b_u$ to each vertex $u \in \Gamma$ which satisfies all the conditions of a candidate basis for all of the rank$>$1\ edges in $\Gamma$, as well as for the rank-1\ edges (which are not in $\Gamma$) which are adjacent to the vertices in $\Gamma$.
\end{definition}
Clearly, if we combine all the candidate bases for the RCCs $\Gamma$, we obtain a candidate basis for the vertices of the whole graph $G$. So the task of finding a candidate basis $B$ for the whole graph breaks up into finding a candidate basis $B_{\Gamma}$ for each RCC\ $\Gamma$, so that $B = \bigcup_{\Gamma} B_{\Gamma}$. Furthermore, if we can correctly choose a basis $b_u$ at one vertex $u$ in $\Gamma$ then, due to Corollary~\ref{cor:path}, we will have successfully specified all of $B_{\Gamma}$. The primary challenge is making the right choice of $b_u$.
\begin{algorithm}
\LinesNumbered
\SetAlgoVlined
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{Graph $G = (V, E)$, matrix edge weights $\{\beta_{uv}\}$, RCCs $\{\Gamma\}$ }
\Output{A candidate basis $B= \{b_u\}$, if one exists. Otherwise False, indicating no candidate basis exists.}
\SetKwBlock{Begin}{for $\Gamma \in \text{RCCs}$ do }{}
\SetAlgoLined
\Begin{
\SetAlgoVlined
\tcc{Construct the $O_{v\leftarrow u}$ operators for all the edges in $\Gamma$}
\For{$e=(u,v)\in \Gamma$}{
$(O_u^e, \Sigma_{uv}^{\text{SVD}}, O_v^e) = \text{SVD}(\beta_{uv})$ s.t.
$O_u^e \Sigma_{uv}^{\text{SVD}} (O_v^e)^T = \beta_{uv}$
$O_{v\leftarrow u} = O_v^e (O_u^e)^{\sf T}$
$O_{u\leftarrow v} =O_{v\leftarrow u}^{\sf T} $
}
\tcc{At each vertex take intersection of eigenspaces associated with immediately neighbouring edges, as in Algorithm \ref{alg:rank1ansatz}}
\For{$v \in \Gamma$}{
$\mathbb{S}_v[0] = \{ \mathbb{R}^3 \}$
\For{$u \in V$ s.t. $e=(u,v) \in E$}{
$\mathbb{S}_v^e= $ the set of orthogonal maximal eigenspaces of $\beta_{uv}^{\sf T} \beta_{uv}$
$\mathbb{S}_v[0] = \mathbb{S}_v[0] \cap \mathbb{S}_v^e$
}
}
\tcc{For each vertex, iteratively take the intersection of the subspaces at that vertex, with the appropriately rotated subspaces of the neighbouring vertices}
i=0
\While{True}{ \label{algLine:whileLoop}
\For{$v\in \Gamma$}{
$\mathbb{S}_{\text{neighbours}} = \{ \mathbb{R}^3\}$
\For{$u \in \Gamma$ s.t. $(u,v) \in \Gamma$}{
$\mathbb{S}_{\text{neighbours}} =\mathbb{S}_{\text{neighbours}} \cap (O_{v \leftarrow u} \mathbb{S}_u[i]) $
}
$\mathbb{S}_v[i+1] = \mathbb{S}_v[i] \cap\mathbb{S}_{\text{neighbours}}$\label{algLine:iterativeProcess}
}
\tcc{Conclude the iterative process when it reaches a fixed point. Note that since every subspace in $\mathbb{S}_v[i+1]$ is contained in a subspace of $\mathbb{S}_v[i]$, this process must reach a fixed point.}
\If{$\mathbb{S}_v[i+1] = \mathbb{S}_v[i] \;\; \forall v\in \Gamma$}{
$\mathbb{S}_v[f] := \mathbb{S}_v[i] \;\; \forall v \in \Gamma$
break
}
i++
}
\tcc{Choose a spanning tree and construct the intersection of the eigenspaces of the rank$>$1\ path operators associated with the fundamental cycles. Take the intersection of this with the set of subspaces at the root vertex.}
$T$ = spanning tree of $\Gamma$ with root vertex $r$.
$\mathbb{S}_{\text{loops}} = \{\mathbb{R}^3\}$
\For{edge $e \in \Gamma$ s.t. $e\not \in T$ \label{algLine:edgesNotInT}}{
Get $C_e$, the fundamental cycle associated with $e$
Let $p_e = (r,u,v,...,w,r)$ be the ordered vertex sequence of $C_e$
Construct $O_{p_e}= O_{r \leftarrow u} O_{u \leftarrow v}... O_{w \leftarrow r}$, the assocated rank$>$1\ path operator. \label{algLine:loopOp}
Find $\{\lambda_i\}$ and $\mathbb{S}_{p_e} =\{S_i^{p_e}\}$, the eigenvalues and maximal eigenspaces of the orthogonal matrix $O_{p_e}$. (Note that every orthogonal matrix is diagonalizable)
\If{$\exists i \;\text{s.t.}\; \lambda_i\not \in \mathbb{R}$}{
return False \label{algLine:imaginaryEigvals}
}
$\mathbb{S}_{\text{loops}} = \mathbb{S}_{\text{loops}} \cap \mathbb{S}_{p_e}$
}
$\mathbb{S}_r^* = \mathbb{S}_r[f] \cap\mathbb{S}_{\text{loops}} $
\textbf{Continued below}
}
\caption{Algorithm for finding a candidate basis}\label{alg:genericAnsatz}
\end{algorithm}
\begin{algorithm}
\SetKwInOut{Input}{}
\SetKwProg{Def}{def}{:}{}
\SetKwFunction{Propagate}{propagate}
\textbf{Algorithm \ref{alg:genericAnsatz}} continued
\SetKwBlock{Begin}{}{end}
\Begin{
\Input{ }
\If{
$\text{span}(\mathbb{S}_r^*) \neq \mathbb{R}^3$
}{
return False \label{algLine:incompleteSpan}
}
\tcc{Note that the intersection of sets of orthogonal subspaces is also a set of orthogonal subspaces. Thus if $\text{span}(\mathbb{S}_r^*)=\mathbb{R}^3$ then one can always choose an orthogonal basis from it.}
Choose orthonormal basis $b_r = (e_1^r, e_2^r, e_3^r) \subseteq \mathbb{S}_r^*$ \label{algLine:br}
\tcc{propagate the choice of basis at the root vertex out to the rest of the vertices in the tree.}
\Def{\Propagate{$u, b_u$}}{
\For{vertex $v\in T$ that are children of $u$}{
$b_v = O_{v\leftarrow u} b_u$ \label{algLine:propagate}
\Propagate{$v$, $b_v$}
}
}
\Propagate($r$, $b_r$)
$B_{\Gamma} = \{b_u\;:\; \forall u \in \Gamma \}$
}
return $B = \bigcup_{\Gamma} B_{\Gamma}$
\end{algorithm}
\iffalse
\begin{algo}[Construct Candidate Basis
\noindent \textbf{Step 0:} Do steps 1 through 8 for each RCC\ $\Gamma$ in the graph.
\noindent \textbf{Step 1:} Choose a vertex $u \in \Gamma$.
\noindent \textbf{Step 2:} For every edge $e = (u,v)$ in $\Gamma$, find a singular value decomposition of its matrix weight: $\beta_{uv}= O_u^e \Sigma_{uv}^{\rm SVD} (O_v^e)^{\sf T}$. Define the operators $O_{v \leftarrow u} = O_v^e (O_u^e)^{\sf T}$ and $O_{u \leftarrow v} = O_{v \leftarrow u}^{\sf T}$.
\noindent\textbf{Step 3:} For every vertex $v \in \Gamma$, and for every edge $e=(v,w)$ adjacent to $v$ (including low rank edges not in $\Gamma$), let $\mathbb{S}_v^e$ be the set of eigenspaces of $\beta_{vw}(\beta_{vw})^{\sf T}$. Let $\mathbb{S}_v[0] = \bigcap_e \mathbb{S}_v^e$ be the intersection of all of those eigenspaces. Construct $\mathbb{S}_v[0]$ for every $v \in \Gamma$.
\noindent \textbf{Step 4:} Starting at $i=0$, perform the following iterative process. For each vertex $v \in \Gamma$ construct
\begin{equation}\label{eq:iteration}
\mathbb{S}_v[i+1] = \mathbb{S}_v[i] \cap \left( \bigcap_{x} O_{v \leftarrow x} \mathbb{S}_x[i]\right) .
\end{equation}
The index $x$ runs over every vertex in $\Gamma$ adjacent to $v$.
If $\mathbb{S}_v[i+1] = \mathbb{S}_v[i] $ for every $v \in \Gamma$, and thus has reached a fixed point, then terminate the iterative process and define $\mathbb{S}_v[f]=\mathbb{S}_v[i]$. Otherwise increment $i$ and repeat. Since every subspace in $\mathbb{S}_v[i+1]$ is contained in a subspace of $\mathbb{S}_v[i]$, this process must reach a fixed point. Note that $\mathbb{S}_v[i+1]$ always remains a set of orthogonal subspaces, since it is the intersection of orthogonal subspaces.
\noindent \textbf{Step 5:} Construct a spanning tree $T$ of $\Gamma$ with $u$ as the root.
\noindent \textbf{Step 6:} Each edge $e \in \Gamma$ not in $T$ is associated with a fundamental cycle $C_e$. For each such edge, choose a direction of the cycle and let the path $p_e=(u, v, w... x, u) $ start at the root $u$, extend along the edges in $C_e$, and return to $u$. Construct the rank$>$1\ path operator
\begin{equation}\label{eq:loopOp}
O_{p_e} = O_{u \leftarrow v} O_{v \leftarrow w} ... O_{x \leftarrow u}.
\end{equation} Noting that every orthogonal matrix is diagonalizable, identify the eigenvalues $\lambda_i$ and eigenspaces $S_i^{p_e}$ of $O_{p_e}$. If any $\lambda_i$ is not a real number, indicate that no candidate basis exists. Otherwise, define the set of subspaces $\mathbb{S}_{p_e} = \{ S_i^{p_e}\}$.
\noindent \textbf{Step 7:} Construct the set of subspaces
\begin{equation}
\mathbb{S}_u^* = \mathbb{S}_u[f] \cap \left( \bigcap_{p_e} \mathbb{S}_{p_e} \right).
\end{equation}
If $\textrm{span}(\mathbb{S}_u^*) \neq \mathbb{R}^3$ then indicate that no candidate basis exists. Otherwise, select an orthonormal basis $b_u= (b_1^u, b_2^u, b_3^u)$ from $\mathbb{S}_u^*$.
\noindent \textbf{Step 8:} Starting at the root vertex $u \in T$ with basis $b_u$, for every child vertex $v$ of $u$, define $b_v = O_{v \leftarrow u} b_u$. Repeat for the children of $v$. This process will assign to every vertex $w \in \Gamma$ a basis choice $b_w$, and thus a basis set $B_{\Gamma}=\{b_w\}$ which we claim is a candidate basis of $\Gamma$.
\noindent \textbf{Step 9:} Return the set $B = \bigcup_{\Gamma} B_{\Gamma}$.
\end{algo}
\fi
\begin{lemma}\label{lem:genericAnsatz}
Given a matrix weighted graph, Algorithm~\ref{alg:genericAnsatz} efficiently finds a candidate basis $B$ or otherwise shows that no such candidate basis exists. The algorithm takes $O(N^3)$ steps, where $N$ is the number of vertices.
\end{lemma}
\begin{proof}
Let us first prove that if the algorithm returns $B$, then $B$ is a candidate basis. To show that $B$ is a candidate basis, we need only show that for all $\Gamma$ $B_{\Gamma}$ is a candidate basis.
The first fact to note is that for every vertex $v \in \Gamma$, and for every edge $(v,w)$ adjacent to $v$, the basis vectors $b_v$ are eigenvectors of $\beta_{vw} \beta_{vw}^{\sf T}$ and so the first condition necessary for $B_{\Gamma}$ to be a candidate basis is satisfied. To see that this is true, it suffices to show that $b_v \in \mathbb{S}_v[f]$, since $\mathbb{S}_v[f] \subseteq \mathbb{S}_v[0]$, and $\mathbb{S}_v[0]$ by construction only contains eigenvectors of all neighbouring edges, including rank-1\ edges. For all $w \in \Gamma$, since $\mathbb{S}_w[f]$ is a fixed point of
the equation on Line (\ref{algLine:iterativeProcess}) of Algorithm \ref{alg:genericAnsatz},
we have
\begin{equation}
\mathbb{S}_w[f] = \mathbb{S}_w[f] \cap \left( \bigcap_x O_{w \leftarrow x} \mathbb{S}_x[f] \right),
\end{equation}
where $x$ runs over rank$>$1\ edges adjacent to $w$. Thus for all rank$>$1\ edges $(w,x)$ in $\Gamma$, $\mathbb{S}_w[f] \subseteq O_{w \leftarrow x} \mathbb{S}_x[f]$ . Consequently, given a vertex $w$ in the spanning tree $T$, and a child vertex $x$, if $b_w \subseteq \mathbb{S}_w[f]$, then since $b_x = O_{x \leftarrow w} b_w \Rightarrow O_{w \leftarrow x} b_x = b_w$ it follows that $b_x \subseteq \mathbb{S}_x[f]$. Since $b_r \subseteq \mathbb{S}_r[f]$, and $r$ is the root node of $T$, it follows by induction that for all $v \in \Gamma$: $b_v \subseteq \mathbb{S}_v[f]$.
The second fact to note is that for every rank$>$1\ edge $(w,v)$ in $\Gamma$, $b_w = O_{w \leftarrow v} b_v$, modulo signs. This is clearly true for every rank$>$1\ edge in $T$ by construction, as specified in
Line (\ref{algLine:propagate}) of Algorithm \ref{alg:genericAnsatz}.
All that remains are those rank$>$1\ edges not in $T$. Consider an edge $e = (v,w)$ not in $T$. There is a fundamental cycle $C_e$, with a path $p_e$ which goes from the root vertex $r$, up to $v$, entirely along paths in the spanning tree, then from $v$ to $w$, and then from $w$ back to $r$. Thus the associated rank$>$1\ path operator is
\begin{equation*}
O_{p_e} = O_{r \leftarrow x }... O_{y \leftarrow w} O_{w \leftarrow v} O_{v \leftarrow z} ... O_{q \leftarrow r}.
\end{equation*}
Furthermore, the bases $b_v$ and $b_w$ are, by construction:
\begin{equation*}
b_v = O_{v \leftarrow z} ... O_{q \leftarrow r} b_r
\end{equation*}
\begin{equation*}
b_w = O_{w \leftarrow y}...O_{x \leftarrow r} b_r \rightarrow b_w = O_{y \leftarrow w}^{\sf T}...O_{r \leftarrow x}^{\sf T} b_r
\end{equation*}
By construction, every element in $b_r$ must be an eigenvector of $O_{p_e}$ with real eigenvalues (equal to $+1$ or $-1$ since $O_{p_e}$ is an orthogonal matrix)
(see Line (\ref{algLine:br}) of Algorithm \ref{alg:genericAnsatz}).
Thus $ b_r=O_{p_e} b_r $ modulo signs. Therefore
\begin{align*}
b_r &= O_{r \leftarrow x }... O_{y \leftarrow w} O_{w \leftarrow v} O_{v \leftarrow z} ... O_{q \leftarrow r} b_r \text{ modulo signs}, \\
O_{y \leftarrow w}^{\sf T}...O_{r \leftarrow x}^{\sf T} b_r &= O_{w \leftarrow v} O_{v \leftarrow z} ... O_{q \leftarrow r} b_r \text{ modulo signs}, \\
b_w &= O_{w \leftarrow v} b_{v} \;\; \textrm{ modulo signs}.
\end{align*}
Thus for every rank$>$1\ edge $(w,v)$ in $\Gamma$, $b_w = O_{w \leftarrow v} b_v$, modulo signs. Combining this fact with
claim 1 of Corollary~\ref{cor:path}, it is clear that the second condition necessary for $B_{\Gamma}$ to be a candidate basis is satisfied. Therefore $B_{\Gamma}$ is a candidate basis.
\bigbreak
Now we will prove that if Algorithm \ref{alg:genericAnsatz} returns False, then no candidate basis exists. First we note that obviously if for any $\Gamma$ there does not exist a candidate basis $B_{\Gamma}$, then no candidate basis exists for the whole graph.
There are two places where the algorithm returns False. Once at Line (\ref{algLine:imaginaryEigvals}), and once at Line (\ref{algLine:incompleteSpan}). This happens at Line (\ref{algLine:imaginaryEigvals}) if some $O_{p_e}$ has any non-real eigenvalues. Note that by claim 2 in Corollary~\ref{cor:path}, if there existed a candidate basis $B=\{ b_u\}$, then $O_{p_e} b_r = b_r$, modulo signs, since $O_{p_e}$ is a rank$>$1\ path operator. In other words, the eigenvalues of $O_{p_e}$ should be either $+1$ or $-1$ \footnote{For an example of where such a loop operation becomes important, see Appendix~\ref{app:cliffordsNotSuffice}}.
The algorithm indicates at Line (\ref{algLine:incompleteSpan}) that no candidate basis exists if, for a given RCC\ $\Gamma$ with root vertex $r$, $\textrm{span}\left( \mathbb{S}_r^* \right) \neq \mathbb{R}^3$. This happens if and only if there does not exist a set of three orthogonal vectors such that each of them belongs to a subspace in $\mathbb{S}_r[f]$ as well as a subspace in every $\mathbb{S}_{p_e}$. We prove by contradiction that in this case $B_{\Gamma}$ must not exist.
Suppose there does not exist a set of three orthogonal vectors such that each of them belongs to a subspace in $\mathbb{S}_r[f]$ as well as a subspace in every $\mathbb{S}_{p_e}$. Suppose a candidate basis $B_{\Gamma}$ does exist, then by the argument made for the case of Line (\ref{algLine:imaginaryEigvals}) in the preceding paragraph, the basis vectors $b_r$ must be eigenvectors of every $O_{p_e}$ and thus each vector in $b_r$ belongs to a subspace in $\mathbb{S}_{p_e}$ for every $p_e$. Therefore it must be that $b_r \not \subseteq \mathbb{S}_r[f]$. However this is contradicted by the following argument.
First note that for all $b_v \in B_{\Gamma}$ and for every adjacent edge $(v,x)$, the basis vectors $b_v$ must be eigenvectors of $\beta_{vx} \beta_{vx}^{\sf T}$, and thus $b_v \subseteq \mathbb{S}_v[0]$. Second note that if for all $ b_v \in B_{\Gamma}: \; b_v \subseteq \mathbb{S}_v[i]$, then for all $b_v \in B_{\Gamma}: \; b_v \subseteq \mathbb{S}_v[i+1]$. This follows from the fact that for all vertices $v \in \Gamma$, and for all rank$>$1\ edges $(v,x)$ adjacent to $v$, $b_v = O_{v \leftarrow x} b_x$, modulo signs, by Corollary~\ref{cor:path}, and thus $b_v \subseteq O_{v \leftarrow x}\mathbb{S}_x[i]$. Since $\mathbb{S}_v[i+1] = \mathbb{S}_v[i] \cap \left(\bigcap_{x} O_{v \leftarrow x}\mathbb{S}_x[i] \right) $ it follows that $b_v \subseteq \mathbb{S}_v[i+1]$. Thus, by induction, $\forall b_u \in B_{\Gamma}: \; b_u \subseteq \mathbb{S}_u[f]$, which is a contradiction.
\bigbreak
Finally we prove that the algorithm runs in $O(N^3)$ steps, where $N$ is the number of vertices in the graph. The most costly part of the algorithm is the while loop at Line (\ref{algLine:whileLoop}). Let $n_{\Gamma}$ be the number of vertices in the RCC\ $\Gamma$. Constructing all $\mathbb{S}_u[0]$ runs in worst case $O(n_{\Gamma}N)$. Each iterative step runs in worst case $O(n_{\Gamma}N)$. At each iterative step the subspaces of $\mathbb{S}_u[i+1]$ must be contained in the subspaces in $S_u[i]$. Therefore if we have not reached a fixed point, then at each iterative step there is at least one $\mathbb{S}_u[i+1]$ for which the dimensions of the subspaces have decreased when compared to $\mathbb{S}_u[i]$. If $\mathbb{S}_u[i]$ spans $\mathbb{R}^3$, then the dimensions of its mutually orthogonal subspaces must be either $(3)$, $(2,1)$ or $(1,1,1)$. Thus for every vertex $u$, the iterative process can only decrease the dimensionality of the subspaces in $\mathbb{S}_u[i]$ at most three times before $\mathbb{S}_u[i]$ no longer spans $\mathbb{R}^3$. So the maximum number of iterations is $3n_{\Gamma}$.
Therefore the naive worst case runtime of this step is $O(n_{\Gamma}^2N)$. However we expect that a more careful analysis would find the runtime to be closer to $O(n_{\Gamma}N)$, since the runtime of each iterative step is proportional to the connectivity of the graph, while the number of iterative steps required should be inversely proportional to the connectivity.
At Line (\ref{algLine:edgesNotInT}) the algorithm iterates over edges in $\Gamma$ not in $T$, the number of which is upper bounded by $O(n_{\Gamma}^2)$. All other steps in the algorithm iterate over vertices in $\Gamma$, or edges adjacent to those vertices, and so have runtime at most $O(n_{\Gamma}N)$.
Since the whole algorithm iterates over all RCC s, it follows that the runtime is $O \left( \sum_{\Gamma} n_{\Gamma}^2 N \right)$ which, by the triangle inequality, is upper bounded by $O(N^3)$.
\end{proof}
\begin{algorithm}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{Graph $G = (V, E)$, rank$>$1\ and rank-1\ matrix edge weights $\{\beta_{uv}\}$, RCCs $\{\Gamma\}$, and candidate basis $B=\{b_u\}$ }
\Output{A set of permutations $\Pi = \{\pi_u\}$ such that $B^{\Pi}$ is an NLY basis, if one exists. Otherwise False, indicating none exists.}
\For{$\Gamma \in $ RCCs}{
\tcc{Label all rank-1\ edges incident to vertices $v$ in $\Gamma$ according to which basis vector $b_v$ is not in the null space of $\beta_{uv}$. These always exist by Proposition \ref{prop:lowrank}}
$L(\Gamma) = \{\}$
\For{$e =(u,v) \in E$ s.t. $v \in \Gamma$, and $\text{rank}(\beta_{uv})=1$ }{
\For{$i\in\{1,2,3\}$}{
$e_i^u = b_u[i]$
\If{$\beta_{uv}e_i^v \neq 0$}{
$L(v, e)=i$
$L(\Gamma) = L(\Gamma) \cap \{i\}$
}
}
}
\tcc{If $\Gamma$ is incident on more than two different labels, then return false.}
\If{$\vert L(\Gamma)\vert=3$}{
return False
}
\tcc{Define permutations so that all incident edge labels are mapped to either $1$ or $3$}
Choose perm. $\pi_{\Gamma}^0$ s.t. $\pi_{\Gamma}^0(L(\Gamma)[1])=1$, and if $\vert L(\Gamma)\vert>1$ then $\pi_{\Gamma}^0(L(\Gamma)[2])=3$
Choose perm. $\pi_{\Gamma}^1$ s.t. $\pi_{\Gamma}^1(L(\Gamma)[1])=3$, and if $\vert L(\Gamma)\vert>1$ then $\pi_{\Gamma}^1(L(\Gamma)[2])=1$
}
\tcc{The task now becomes assigning a binary value $x_{\Gamma}$ to each $\Gamma$ so that for every rank-1\ edge $e=(u,v)$, with labels $L(u,e)=i$ and $L(v,e)=j$, which has vertices in $\Gamma_u$ and $\Gamma_v$, the binary assignment $x_{\Gamma_u}$ to $\Gamma_u$ and $x_{\Gamma_v}$ to $\Gamma_v$ satisfy: $\pi^{x_{\Gamma_u}}_{\Gamma_u}(i) = \pi^{x_{\Gamma_v}}_{\Gamma_v}(j).$}
\tcc{For each rank-1\ edge define a 2-XOR-SAT clause on boolean variables associated with the RCCs on which the edge is incident.}
\For{$e=(u,v)\in E$ s.t. $\text{rank}(\beta_{uv})=1$}{
Find $\Gamma_u\in $ RCCs s.t. $u\in \Gamma_u$.
Find $\Gamma_v \in $ RCCs s.t. $v \in \Gamma_v$.
\eIf{$L(v,e) = L(u,e)$}{
$C_e(x_{\Gamma_u},x_{\Gamma_v}) = x_{\Gamma_u} \oplus x_{\Gamma_v}$
}{
$C_e(x_{\Gamma_u},x_{\Gamma_v}) = \bar{ x}_{\Gamma_u} \oplus x_{\Gamma_v}$
}
}
\tcc{The solution to the associated 2-XOR-SAT problem specifies, for each RCC $\Gamma$, which permutations to apply in a uniform fashion to all vertices in $\Gamma$.}
variables = $\{ x_{\Gamma} \;\vert\; \forall \Gamma \in \text{RCCs}\}$
clauses = $\{C_e \;\vert\; \forall e=(u,v) \in E \text{ s.t. } \text{rank}(\beta_{uv})=1 \}$
success= 2-XOR-SAT(ref variables, clauses)
\If{$\lnot$success}{
return False
}
\For{$\Gamma \in $ RCCs}{
\For{vertex $v \in \Gamma$}{
$\pi_v = \pi_{\Gamma}^{x_{\Gamma}}$
}
}
return $\Pi = \{ \pi_v \;\vert\; \forall v \in V\}$
\caption{Algorithm for finding permutations $\Pi$ such that $B^{\Pi}$ is a NLY basis}\label{alg:genericPerms}
\end{algorithm}
\iffalse
\begin{algo}[Finding permutations $\Pi$ such that $B^{\Pi}$ is a NLY basis]
Given a matrix weighted graph, and a candidate basis $B$, this algorithm finds a set of permutations $\Pi$ such that $B^{\Pi}$ is an NLY basis, or otherwise indicates that no such set of permutations exist.
For each rank-1\ edge $(u,v)$, identify the left singular vector $e_i^u \in b_u$ and corresponding right singular vector $e_j^v \in b_v$ which are not in the null space of $\beta_{uv}$. Label each rank-1\ edge $(u,v)$ with an ordered pair of indices $(i,j)$, as illustrated in figure~\ref{fig:bilabeledEdge}.
For ease of exposition, we say a rank-1\ edge is connected to a rank$>$1 connected component\ $\Gamma$, and vice versa, if it is connected to a vertex in $\Gamma$. We say a rank-1\ edge $(u,v)$ with labelling $(i,j)$ connects to $\Gamma$ with index $i$ if $u$ is in $\Gamma$, and it connects to $\Gamma$ with index $j$ if $v$ is in $\Gamma$.
If for any RCC\ $\Gamma$ there are at least three rank-1\ edges, each connected to $\Gamma$ by a different label, then terminate and indicate that the desired set of permutations does not exist.
If the algorithm has not terminated, then for each $\Gamma$ there must exist two labels $i$ and $j$ such that every rank-1\ edge connects to $\Gamma$ with only $i$ or $j$. This remains true even if every rank-1\ edge connects to $\Gamma$ with a single label. Identify such a pair of labels. Identify a pair of permutations $\pi^0_{\Gamma}$ and $\pi^1_{\Gamma}$ such that $\pi^0_{\Gamma}(i)=1$ and $\pi^0_{\Gamma}(j)=3$ and $\pi^1_{\Gamma}(i)=3$ and $\pi^1_{\Gamma}(j)=1$.
The task now becomes assigning a binary value $x_{\Gamma}$ to each $\Gamma$ so that for every rank-1\ edge $(u,v)$ with label $(i,j)$ which is connected to $\Gamma_u$ and $\Gamma_v$, the binary assignment $x_{\Gamma_u}$ to $\Gamma_u$ and $x_{\Gamma_v}$ to $\Gamma_v$ satisfy: $\pi^{x_{\Gamma_u}}_{\Gamma_u}(i) = \pi^{x_{\Gamma_v}}_{\Gamma_v}(j).$ This problem reduces straightforwardly to an XOR-SAT problem. Each rank$>$1 connected component\ $\Gamma$ corresponds to a binary variable $x_{\Gamma}$. Each rank-1\ edge $(u,v)$ with label $(i,j)$, which connects to $\Gamma_u$ and $\Gamma_v$, corresponds to an XOR clause: $\bar{x}_{\Gamma_u} \oplus x_{\Gamma_v}$ if $i=j$, and $ x_{\Gamma_u} \oplus x_{\Gamma_v}$ if $i \neq j$. If there is no solution, then terminate and indicate that the desired set of permutations does not exist. If there is a solution, then for every $\Gamma$, and for every vertex $u \in \Gamma$, let $\pi_{u}= \pi_{\Gamma}^{x_{\Gamma}}$ Return the permutation $\Pi = \{\pi_u\}$.
\end{algo}
\fi
\begin{theorem}\label{thrm:genericAlgo}
Given a matrix weighted graph, one can efficiently find an NLY basis, or else show that no such basis exits.
\end{theorem}
\begin{proof}
The procedure for finding an NLY basis is to first find a candidate basis $B$ using Algorithm~\ref{alg:genericAnsatz}, and then find a set of permutations $\Pi$ such that $B^{\Pi}$ is an NLY basis using Algorithm~\ref{alg:genericPerms}.
If Algorithm~\ref{alg:genericPerms} is succesful, then $B^{\Pi}$ is an NLY basis by the following reasoning. For every rank$>$1\ edge an identical permutation is applied to its adjacent vertices, and so the diagonality of the matrix weights is preserved. While for the rank-1\ edges, the matrix weights are diagonal, and by construction every matrix weight is zero in its second diagonal entry, as per arguments made in subsection \ref{subsec:rank1graph}.
Algorithm~\ref{alg:genericPerms} is efficient, since the number of variables in the 2-XOR-SAT problem is the number of RCC s, and in the worst case this is the number of vertices $N$, so it runs in time $O(N^2)$. Combining this with Lemma~\ref{lem:genericAnsatz} the worst case runtime of the whole algorithm is $O(N^3)$.
Finally, if either of these algorithms fail, then we claim that no NLY basis exists, by the following two arguments. Firstly, if Algorithm~\ref{alg:genericAnsatz} fails, then by Lemma~\ref{lem:genericAnsatz} no candidate basis exists, and since an NLY basis must satisfy the conditions for being a candidate basis, no NLY basis exists. Secondly, we must establish the non-trivial fact that if Algorithm~\ref{alg:genericPerms} fails, then no NLY basis exists. In other words, we need to rule out the possibility that Algorithm~\ref{alg:genericPerms} might have succeeded had we supplied it with an alternative candidate basis. The rest of our exposition is devoted to proving this fact.
First note that, when given a candidate basis $B$, if Algorithm~\ref{alg:genericPerms} fails then no set of permutations exists such that $B^{\Pi}$ is an NLY basis. This follows from the fact that, if a permutation were to exist, it must be uniform on every RCC , in order to preserve the diagonality of the rank$>$1\ edges. Given this, the argument reduces to the same one made in Theorem~\ref{thrm:rank1algo}, where we treat RCC s as sites, since every rank-1\ edge is adjacent to RCC s.
Given that if the procedure in Algorithm~\ref{alg:genericPerms} fails, then there does not exist a permutation $\Pi$ such that the basis $B^{\Pi}$ is an NLY basis, we use proof by contradiction to show that in this case no NLY basis exists.
Suppose there exists an NLY basis $\bar{B} = \{ \bar{b}_u\}$. We now argue that for a fixed RCC $\Gamma$, for every index $i \in \{1,2,3\}$ there exists an index $j \in \{1,2,3\}$ such that for every vertex $u \in \Gamma$, if $e_i^u \in b_u$ corresponds to a left singular vector, with non-zero sigular value, of a rank-1\ edge adjacent to $u$, then $e_i^u = \pm \bar{e}_j^u \in \bar{b}_u$.
Given an index $i$, consider any two vertices $u,v \in \Gamma$ for which $e_i^u, e_i^v$ correspond to singular vectors, with non-zero singular values, of some rank-1\ edges adjacent to $u$ and $v$. Consider that $\bar{b}_u$ must also contain a vector $\bar{e}_{j_u}^u$ at some particular index $j_u$, which is also a singular vector with non-zero singular value of the same rank-1\ edge adjacent to $u$. Since that edge is rank-1 , it follows that $e_i^u = \pm \bar{e}_{j_u}^u$. A similar argument can be made for $v$ so that $e_i^v = \pm \bar{e}_{j_v}^v$, for some index $j_v$. There must exist a rank$>$1\ path $p$ connecting $u$ to $v$, and by Corollary~\ref{cor:path}, $O_p e_i^u = \pm e_i^v$. Similarly, $O_p \bar{e}_{j_u}^u = \pm \bar{e}_{j_u}^v$. Therefore $\pm \bar{e}_{j_u}^v =\bar{e}_{j_v}^v$ and since each vector in $\bar{b}_v$ is orthogonal we have $j_u =j_v =j$. Since this is true for any such pair of vertices $u,v$, there must exist an index $j$ such that for every vertex $u \in \Gamma$, if $e_i^u$ corresponds to a left singular vector, with non-zero sigular value, of a rank-1\ edge adjacent to $u$, then $e_i^u = \pm \bar{e}_j^u \in \bar{b}_u$.
We can now proceed by the same reasoning used in the proof of Theorem~\ref{thrm:rank1algo}. Let $\pi_{\Gamma}$ be the permutation with the mapping: $\pi_{\Gamma}(i)=j$, and let $\pi_u = \pi_{\Gamma}$ for all $u\in \Gamma$, and define $\Pi =\{\pi_u\}$. Then the bi-labelled graph associated with $B^{\Pi}$ must be identical to the bi-labelled graph associated with $\bar{B}$ (i.e. the action on the rank-1\ edges is the same), and therefore $B^{\Pi}$ must be an NLY basis, which is a contradiction.
\end{proof}
\section{Discussion}
\label{sec:dis}
It is clear from the work presented here that in the case of two-local qubit Hamiltonians, the hardness of curing the sign problem by local basis transformations is determined by the presence or absence of one-local terms in the Hamiltonian.
The question of whether the general {\sf LocalSignCure} is a problem in NP for general two-local $n$-qubit Hamiltonians is not clear, as the set of local unitary transformations is a continuous parameter space, and a prover would need to specify a sign-curing solution with a polynomial number of bits and such exact sign-curing transformations may not exist. A natural relaxation would be to demand that the transformation be approximately sign-curing, a direction of research that is explored in Ref.~\cite{HRNE:easing}, so that the problem would be contained in MA.
We do not know the complexity of determining the ground state energy for the family of Hamiltonians presented in section 5. It may be that finding the ground state energy is easy, thus obviating any interest in a curing transformation. It would be interesting to show that a family of Hamiltonians exists for which deciding sign-curing and determining ground state energy are both hard. It would be very surprising indeed if the hardness of deciding a curing transformation only appears when the ground state energy is efficiently computable.
A natural extension of sign-curing transformations beyond single-qubit unitary transformations are transformations which first embed each qubit into a $d$-dimensional system, and then allow for local basis changes in this $d$-dimensional system. The power of such ``lifting" basis changes is completely unexplored, even in the two-qubit case. Another class of sign-curing transformations are Clifford circuits which map a Hamiltonian composed of ${\rm poly}(n)$ $k$-local Pauli's onto a sum of ${\rm poly}(n)$ non-local Pauli's. The power of these transformations is also largely unexplored, but some first results are reported in \cite{thesis:ioannou}.
Recently, it was demonstrated~\cite{hen2019} that even when there is an essential sign problem in the Hamiltonian, there are ways to group terms in the expansion of the Gibbs state to avoid the sign problem. It would be interesting to understand better how these techniques relate to stoquastic Hamiltonians.
Another strand of interesting future research concerns the distinction between termwise and globally stoquastic Hamiltonians. Examples can be constructed of 3-local globally-stoquastic but not termwise-stoquastic Hamiltonians and the complexity of deciding global stoquasticity can be analyzed.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,484 |
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\section{Introduction}\label{sec:introduction}}
\begin{comment}
\section{Mining privacy requirements in issue reports} \label{sec4}
Most of today's software projects follow an agile style in which the software development is driven by resolving issues in the backlog. In those projects, issue reports contain information about requirements of a software that are recorded in multiple forms in an issue tracking system \cite{Choetkiertikul}: new requirements (such as user stories or new feature request issues), modification of existing requirements (such as improvement request issues) or reporting requirements not being properly met (i.e. bug report issues). Large projects may have thousands of issues which provide fairly comprehensive source of requirements about the projects and the associated software systems. Thus, we have performed a study on the issue reports of software projects to understand how the selected projects address the privacy requirements in our taxonomy. As issue reports can be in multiple forms, we note that the taxonomy can be applied to software requirements and user stories. In this section, we first describe the process steps that we have followed to carry out our study, assess classification results, resolve disagreements, and discuss the outcomes.
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\linewidth]{"Figures/Mining-issue-reports"}
\caption{An overview process of mining and classifying privacy requirements in issue reports}
\label{fig:Mining issue reports}
\end{figure}
\subsection{Issue reports collection}
We apply a number of criteria to select software projects for this study as follows: (i) are open-source projects, (ii) serve a large number of users, (iii) are related to privacy and (iv) have accessible issue tracking systems. There were a number of projects which satisfy these criteria. Among them, we selected Google Chrome and Moodle due to their large scale size, popularity and representativeness\footnote{As discussed later in the paper, performing thorough analysis as we have done on these two projects alone required significant effort (414 person-hours). Hence, we scoped our study in these two projects so that we can publish this dataset timely, enabling the community to initiate research on this important topic, and subsequently extend it with additional projects and data.}. Google Chrome is one of the most widely-used web browsers, which was developed under the Chromium Projects \cite{Projects}. As a web browser, Google Chrome stores personal data of users, e.g. username, password, address, credit card information, searching behaviour, history of visited sites and user location. Moodle is a well-known open-source learning platform \cite{Moodle} with over 100 million users worldwide. Moodle aims to comply with GDPR \cite{Moodle2019}.
Figure \ref{fig:Mining issue reports} shows the process steps that we have followed in our study. We first identified privacy-related issues from all the issue reports we collected from Google Chrome and Moodle projects. To do so, we identified issues that were explicitly tagged as privacy in the ``component'' field of both projects and their status was either assigned, fixed and verified (to ensure that they are valid issues). This is to ensure that the issue reports we collected were explicitly tagged as ``privacy'' by Chrome and Moodle contributors. This process initially gave us 1,080 privacy-related issues from Google Chrome and 524 from Moodle. We then manually examined those issues and filtered out those that have limited information (e.g. the description that does not explain the issue in detail or contain only source code) to enable us to perform the classification. For example, issue ID 953622\footnote{https://bugs.chromium.org/p/chromium/issues/detail?id=953622} states that ``Null-dereference READ in bool base::ContainsKey<std::\_\_Cr::map<std::\_\_Cr::basic\_string<char, std::\_\_Cr::c''. This issue description is very brief and does not contain any explanation describing what the issue is about, what personal data is concerned or which function is affected in the issue. We thus exclude the issue from our study. Finally, our data contains 896 issues from Chrome and 478 issues from Moodle.
In Chrome dataset, the collected issue reports were created between January 2009 and March 2020. There are five issue types reported in our dataset: bug, bug-regression, bug-security, feature and task. Each issue report has seven contributors on average; the contributors include reporters, owners and relevant collaborators. Issue reports in Moodle dataset span for two years, 2018 - 2019. The issue types in Moodle include bug, epic, improvement, new feature, task, sub-task and functional test. On average, five participants are involved in each report including reporters, assignees, testers and commenters. The descriptive statistics of the issue reports can be seen in Table \ref{tab:issue-stats}.
\begin{table*}[h]
\centering
\caption{Descriptive statistics of the number of contributors, resolution time and number of comments of the issue reports in our datasets.}
\label{tab:issue-stats}
\resizebox{6.5in}{!}{%
\begin{tabular}{@{}llllllllllllllll@{}}
\toprule
\multirow{2}{*}{Project} & \multicolumn{5}{c}{\#Contributors} & \multicolumn{5}{c}{Resolution Time (days)} & \multicolumn{5}{c}{\#Comments} \\ \cmidrule(l){2-16}
& min & max & mean & median & mode & min & max & mean & median & mode & min & max & mean & median & mode \\ \midrule
Google Chrome & 1 & 32 & 5 & 4 & 2 & 1 & 3,635 & 315 & 65 & 1 & 0 & 311 & 16 & 12 & 12 \\
Moodle & 1 & 14 & 4 & 5 & 5 & 1 & 852 & 37 & 13 & 1 & 0 & 112 & 11 & 9 & 1 \\ \bottomrule
\end{tabular}%
}
{\parbox{16.5cm}{\footnotesize \#Contributors: number of contributors, \#Comments: number of comments, min: minimum of contributors/resolution time/comments, max: maximum of contributors/resolution time/comments, mean: mean of contributors/resolution time/comments, median: median of contributors/resolution time/comments, mode: mode of contributors/resolution time/comments.}}
\end{table*}
\subsection{Issue reports classification}
In this phase, we went through each issue report in the dataset to classify it into the privacy requirements in our taxonomy. This phase consists of three steps: (i) identifying concerned personal data described in the issue report, (ii) identifying functions/properties reported in the issue, (iii) mapping the issue to one or more privacy requirements.
Regarding the classification, each coder was initially provided with an online form containing the title and description of the assigned issue reports and 71 columns representing each requirement. The coders analysed each issue following the classification steps described above (i.e. steps (i) - (iii)). The coders carefully consider every scenario mentioned in the issue reports. Once the coders have identified related information about personal data and function(s) concerned, they considered the relevant requirements. The coders determined the requirement(s) that matches with information analysed above. The coders then updated the value in the columns of chosen requirement(s) in the given form. Finally, all the coders delivered their result file containing the issue reports and their privacy requirement labels for reliability assessment process. To ensure that the classification process is reliable, two coders were assigned to classify an issue report. The reliability assessment and disagreement resolution processes are described in detail in Section \ref{4c}.
The following example demonstrates the issue reports classification in our datasets. Issue 123403\footnote{https://bugs.chromium.org/p/chromium/issues/detail?id=123403} in Google Chrome reports that \textit{``Regression: Can't delete individual cookies''}. The personal data affected here is individual cookies, and the function reported is erasing or deleting (individual cookies). Thus, we classify this issue into the requirement \textit{\textbf{ALLOW} the data subjects to erase their personal data (R44)} in our taxonomy (see Table \ref{tab:table1}). The users should be able to select the cookies that they want to delete.
In another example, issue 495226\footnote{https://bugs.chromium.org/p/chromium/issues/detail?id=495226} in Google Chrome requests that the \textit{``Change Sign-in confirmation screen''} should be changed. The description of this issue requires that the system should inform the reasons for user account data collection and how this data will be further processed before obtaining this data in the sign-in process. Since this issue requires that the user should be informed of the purpose of collection and processing, the issue can be classified into requirements R38 and R39 (see Table \ref{tab:table1}). Both of these requirements belong to the notice privacy goal. This example shows that an issue can be classified into more than one privacy requirements.
Issue 831572\footnote{https://bugs.chromium.org/p/chromium/issues/detail?id=831572} in Google Chrome requires: \textit{``Provide adequate disclosure for (potentially intrusive) policy configuration''}. Further investigation into the issue's description revealed that the disclosure of policy configuration includes: letting the users know that they are managed, and providing indication when user data may be intercepted and when user actions are logged locally. These involve the following functions: (i) the users should be informed of the purpose of processing so that they know they are managed; (ii) the enterprise may intercept the users' data, thus the users should know whom their personal data might be sent to; and (iii) the history of user logging shall be recorded to acquire logging data. Hence, this can be classified into three requirements R39, R27 and R13 (see Table \ref{tab:table1}). This example demonstrates that one issue relates to several requirements across different privacy goal categories: notice and data processing.
The following example demonstrates the issue that concerns multiple functions which more than one privacy goal is addressed in Moodle. Issue MDL-62904\footnote{https://tracker.moodle.org/browse/MDL-62904} in Moodle reports that \textit{``users can't find where to request account deletion''}. The issue was described that the system does not provide a function for users to request for deleting their account in the user interface. Hence, this issue addresses requirements R30 in the notice category and R44 in the user participation category in the taxonomy.
Although the mapping is relatively straightforward in most of the issues, some presents challenges. For instance, a set of issues in Moodle refer to the implementation of the core\_privacy plugins (e.g. MDL-61877\footnote{https://tracker.moodle.org/browse/MDL-61877}). However, the information given in the description of those issues is inadequate to identify which privacy requirements are related to. Therefore, we needed to seek for additional information about core\_privacy plugins in Moodle development documentation \cite{Moodle2019}. In addition, the issues in Google Chrome and Moodle projects have specific function names or technical terms used by their developers. Hence, extra effort was required to understand those issues and classify them. Once the coders acquired the additional information from software documentation, they discussed potential privacy issues/functionalities raised in those issue reports. For example, the documentation of core\_privacy plugins explains six functionalities that the plugins should provide. The coders then discussed which privacy requirements in the taxonomy involved with those functionalities, and classified that issue according to the identified privacy requirements.
\subsection{Reliability analysis} \label{4c}
The classification process has been performed by three coders, who also involved in the taxonomy development process. All the coders have independently followed the above process to classify issue reports into our taxonomy of privacy requirements. The first coder was responsible for classifying all 1,374 privacy issue reports. The classification process is however labour intensive. It took the first author approximately 138 person-hours to classify all 1,374 issue reports (6 minutes per issue report in average). In total, the process took four months to complete, so the coders had time to manage their workloads efficiently. The coders usually divided the assigned issue reports into smaller sets (i.e. about 50 issue reports) to work on for each round. For the purpose of reliability assessment, the second and third coder each was assigned to classify a different half of the issues in each project. This setting aimed to ensure that each issue report is classified by at least two coders.
An issue report can be classified into multiple privacy requirements (i.e. a multi-labelling problem). Hence, we employ Krippendorff's alpha coefficient \cite{Krippendorff2011}, \cite{Artstein2008} with MASI (Measurement Agreement on Set-valued Items) distance to measure agreement between coders with multi-label annotations \cite{Ravenscroft2016}, \cite{Passonneau2006}. The MASI distance measures difference between the sets of labels (i.e. privacy requirements) provided by two coders for a given issue. The Krippendorff's alpha values between the first and second coders are 0.509 for Chrome and 0.448 for Moodle. The agreement values between the first and third coder are 0.482 and 0.468 for Chrome and Moodle respectively. A disagreement resolution step was conducted to resolve the classification deviations between the three coders\footnote{Krippendorff's alpha values indicate the degree of (dis-)agreement between coders. In our case, disagreements were often due to the ambiguity in the issue reports, requiring us to perform a resolution step. Doing these is to increase the reliability of our dataset.}.
\textbf{Disagreement resolution}: The low Krippendorff's alpha values from the initial classification were \emph{not} due to the privacy requirements in our taxonomy. Rather, it was mainly because of the limited information provided in the description of a number of issue reports, forcing the coders to make their own assumptions about the nature of those issues. If an issue report is clearly described, the coders classified it into the same requirements. 53.01\% of the Chrome issues and 46.23\% in Moodle received this total agreement between all the three coders. We have addressed this problem by conducting a disagreement resolution step where all the coders met and discussed to resolve the disagreements.
We conducted several meeting sessions between the coders to resolve disagreements. Several meetings were conducted because of the time difference and availability among the coders. Each meeting took 3 hours on average as we revisited, discussed and reclassified the issues with disagreements issue by issue. During these meetings, the coders examined the issues thoroughly (not just only their description, but also other documentation related to the issues), discussed to develop a mutual understanding of the issue, and then reclassified the issue together. For each issue in the list, each coder explained the justification for their classification of the issue. The coders then resolved the disagreements on that issue in two ways. After the discussion, if the coders agreed with the other coder's classification, the labels were combined (i.e. the issue were classified into multiple requirements). If the coders did not reach an agreement, they went through the issue's description together to discuss and develop a mutual understanding of the issue. They then reclassified the issue together. Hence, the final classification has maximum agreement among the coders, thus ensuring the reliability of our dataset. Once all disagreements have been resolved for the sample set, the first coder adjusted the classification of the issues which are not included in the sample set to finalise the dataset.
The following example illustrates how conflicts between coders were resolved. Issue 527346\footnote{https://bugs.chromium.org/p/chromium/issues/detail?id=527346} in Google Chrome requests that the users should know when they are managed. The description of the issue requires the system to show information to users when they are managed and the information should be easily seen by users. This issue was classified to R26 by one coder and R30 by the other coder. Although both R26 and R30 involve providing information to users, R26 focuses on the information relating to the policies, procedures, practices and logic of the processing of personal data, while R30 focuses on representing information relating to the processing of personal data with standardised icons in the user interface. After the coders revisited the issue and discussed the description in detail, the coders agreed that the information should be shown in the tray bubble which is a part of the user interface. The coders therefore reclassified this issue into R30.
Our work helps establish the traceability between issue reports and privacy concerns expressed in the regulations, standards and frameworks. These suggest what should be done to resolve the concerns raised in issue reports and meet the associated privacy requirements. In addition, this traceability facilitates compliance checking with respect to some specific regulations, standards and/or frameworks. The following examples demonstrate the privacy requirements traceability in issue reports. Referring to issue 123403 in Section 4.2, this bug-regression issue report concerns requirement R44 in the taxonomy. Requirement R44 was derived from GDPR, ISO/IEC 29100, Thailand PDPA and APEC privacy framework. To resolve this issue and comply with all four regulations, standards and frameworks, the system must provide a functionality for the controller to allow the data subjects to erase their personal data. In another example, referring to issue 123403 in Section 4.2, it addresses two privacy requirements, R30 and R44. Requirement R30 was derived from GDPR and APEC privacy framework. To resolve this issue and to comply with GDPR and APEC privacy framework, the system must also provide a functionality for the controller to provide the data subjects the information relating to the processing of personal data with standardised icons.
Our approach can be also applied to establish privacy requirements traceability in other software artifacts such as design models, source code and test cases. In general, our approach can be used a reference framework in developing privacy-aware software systems. For example, when a software engineer develops a system functionality which collects personal data, s/he may consult the taxonomy to identify the requirements related to the information required to provide to the data subjects before collecting their personal data such as concerned personal data (R42), purposes of collection (R38), purposes of processing (R39) and period/criteria used to store personal data (R55). The privacy requirements were derived from the regulations, standards and frameworks, hence meeting those requirements forms a basis for privacy compliance.
Although issue reports are a good source of information for software requirements, we acknowledge that they are not the only source. Software requirements can be in other forms such as requirements specifications or other documents (such as Confluence\footnote{https://www.atlassian.com/software/confluence/features} pages). The privacy requirements in the taxonomy can also be mapped to a list of privacy-related items concerned in a range of checks in organisations such as Data Protection Impact Assessment (DPIA). For example, DPIA requires a system to obtain consent from data subjects before processing their personal data. Requirement R35 in the taxonomy addresses this concern. However, the scope of this work focuses on privacy requirements in issue reports, and we plan to explore privacy requirements in other sources in our future work.
\section{Conclusion and future work} \label{sec7}
Privacy compliance in software systems has become tremendously significant for organisations that handle personal data of their users. In this paper, we have developed a comprehensive taxonomy of privacy requirements based on four well-known standardised data protection regulations and privacy frameworks (GDPR, ISO/IEC 29100, Thailand PDPA and APEC privacy framework). Our approach is built upon a content analysis process which is adapted from the Goal-Based Requirements Analysis Method and based on Grounded Theory \cite{Antn2004}, \cite{Anton1996}. Thus, the steps in this approach (privacy requirements identification, privacy requirements refinement and privacy requirements classification) are generic and applicable to different regulations and privacy standards. In our approach, privacy requirements are formed by three components: actions, affected parties/objects and target results. These are commonly found in the narrative statements across different regulations.
We also performed reliability assessments and disagreement resolution in the process to ensure that our taxonomy is reliably constructed. Our taxonomy consists of 71 privacy requirements grouped in 7 privacy goal categories. Since the studied regulations and frameworks are not specific to any software types, our taxonomy is generally applicable to a wide range of software applications. In this study, we retain the privacy requirements at this level since the implementation can vary depending on organisational structures, software architectures, existing implementation and knowledge, and the aptitude of software development teams. Thus, our taxonomy focuses on defining privacy requirements a software needs to satisfy, leaving how the software is designed and implemented to meet those requirements to project stakeholders.
We have also performed a study on how two large projects (Google Chrome and Moodle) address those privacy requirements in our taxonomy. To do so, we mined the issue reports recorded in those projects and collected 1,374 privacy-related issues. We then classified these issues into our taxonomy through a process which involved multiple coders and the use of inter-rater reliability assessments and disagreement resolution. We found that the privacy requirements in the user participation category were covered in a majority of the issues, while none of the issues were found to address the breach category. The taxonomy also captures all the privacy requirements in the issue reports. We also found that the time taken to resolve privacy-related issues and the degree of developers' engagement on them were also statistically significantly different from those of non-privacy issues. We note that the taxonomy can be applied to software requirements and user stories. In fact, there are issue reports in the dataset which are user stories or requirements for new features.
We believe that the taxonomy and the dataset are important contributions to the community (given that none exists). This would initiate discussions and further work (e.g. refining and/or extending the taxonomy, performing user studies, etc.) in the community. Our work lays several important foundations for future research in this area. The systematic method performed in the work enables future research to be conducted on other data protection and privacy regulations and frameworks. The taxonomy can act as a reference for the research community to discuss and expand. We plan to investigate other privacy regulations and policies and extend our taxonomy, if necessary. The requirements in our taxonomy are written in natural language and structured into templates. Although we believe that this is the most intuitive form to developers, future work could explore other alternative forms such as semantic frame-based representation \cite{Bhatia2019}. We have manually derived requirements in this study as it is essential to examine structure of statements and how privacy requirements are expressed in different regulations and frameworks. However, the requirements extraction process can be automated using natural language processing (NLP) techniques to identify actors, actions, objects and other relevant constraints from text documents (i.e. regulations and frameworks). These methods have been used in several studies (e.g. \cite{Mu2009} and \cite{Li}). Legal experts could also be involved to help interpret legal perspective of the taxonomy in the future work.
Future work also involves exploring how issue reports in other projects (e.g. health and mobile applications) attend to the requirements in our taxonomy. In addition, software developers would be able to participate in validating the use of taxonomy and their understanding towards the derived privacy requirements. Furthermore, we plan to develop tool support to automate this classification task. Other potential future work includes the investigation of the use of the taxonomy with semantic web technologies to facilitate computational and regulatory risk analysis purposes. The investigation of other forms of traceability relationships such as the traceability between code and issue reports or code and privacy requirements can also be further studied.
\section{Threats to validity} \label{threats}
Our study involved subjective judgements. We have applied several strategies to mitigate this threat such as using multiple coders (who are the authors of the paper), applying inter-rater reliability assessments, organising training sessions and disagreement resolution meetings. A legal expert could have extended the view of the human coders in interpreting legal regulations. However, we note that all the coders had attended a training course on privacy regulations (including GDPR). This has enhanced our interpretation of privacy regulations from a legal perspective, thus minimised this risk. In addition, our study was built upon well-founded processes and theories in previous work such as Grounded Theory \cite{Glaser2017} and Goal-Based Requirements Analysis Method (GBRAM) \cite{Antn2004},\cite{Anton1996}. We also used relevant statistical measures and techniques to ensure that our findings are statistically significant.
Our taxonomy of requirements is derived from the GDPR and ISO/IEC 29100 -- two major and widely-adopted privacy and data protection regulations and framework to date; and Thailand PDPA and APEC privacy framework -- two region-specific representative regulations and privacy frameworks. We are aware that there are other privacy protection laws and regulations. However, most of them share many commonalities with the GDPR and ISO/IEC 29100 as confirmed by Thailand PDPA and APEC privacy framework in this study. In fact, previous studies (e.g. \cite{Ayala-Rivera2018}, \cite{Linden2020}, \cite{Tsohou2020}) have shown that GDPR is one of the most comprehensive data protection regulations. GDPR provides a more overarching coverage compared to other data protection regulations in terms of privacy concerns (i.e. principles and rights) and organisational impacts. Many other privacy laws have been developed as inspired by GDPR (e.g. Brazil's LGPD and CCPA in California) \cite{GDPR.EU2020}. It was also used as a benchmark for other countries to develop data protection regulations \cite{Torre} such as Japan, South Korea, Thailand and other countries in Asia Pacific (APAC -- Asia Pacific Privacy Authorities) \cite{Laboris2019}. Hence, although the principles or rights in other laws and regulations are slightly different due to variations in each country/city, they share many commonalities with GDPR (e.g. see the comparison between GDPR and CCPA in \cite{Iubenda2020},\cite{DataPrivacyManager}). ISO/IEC 29100 has also been used to develop organisational and technical privacy controls in many information and communication systems \cite{PECB2015}. Therefore, we found that GDPR and ISO/IEC 29100 together are the most comprehensive, thus our taxonomy can generalise to other privacy regulations and standards. We acknowledge that future research could involve investigating country specific privacy regulations, and extending our taxonomy accordingly. We also note that developing a taxonomy of privacy requirements for GDPR, ISO/IEC 29100, Thailand PDPA and APEC privacy framework already required substantial efforts and detailed processes. To the best our knowledge, our work is the first comprehensive taxonomy of privacy requirements analysing both data protection regulations and privacy frameworks. This would lay foundations for future research in this timely, important area of software engineering.
Finally, we performed the mining and classification of issue reports for Google Chrome and Moodle. These are two large and widely-used software systems that have strong emphasis on privacy concerns. However, we acknowledge that our datasets may not be representative of other software applications. Further investigation is required to explore other projects in different domains (e.g. e-health software systems and mobile applications). However, we note that building this dataset on two projects alone required substantial effort and highly thorough processes (414 person-hours).
\section{Related work} \label{sec2}
The protection of personal data and privacy of users have attracted significant attention in recent years. Data protection regulations and privacy frameworks have been established to guide the development of software and information systems. \emph{However, it is challenging for software engineers to translate legal statements stated in those regulations into specific privacy requirements for software systems \cite{Ayala-Rivera2018}, \cite{Aljeraisy2020}}. Several studies have investigated the approaches, tools and techniques developed for managing privacy requirements \cite{Anthonysamy2017}. \citeauthor{Guarda2009} \cite{Guarda2009} showed that the traditional RE frameworks lack of fundamental concept specific to privacy which cannot be used to capture privacy-related legal requirements in software systems. Therefore, \citeauthor{Guarda2009} have developed a reference methodology that can be tailored to design privacy-aware systems.
\citeauthor{Beckers2012} \cite{Beckers2012} proposed a conceptual framework to compare privacy requirements engineering approaches on requirements elicitation and notion representation. The approaches include LINDDUN, PriS and the framework for privacy-friendly system design approaches. The LINDDUN method elicits privacy requirements from privacy threats \cite{Deng2011}. The method first developed nine categories of privacy threats: linkability, identifiability, non-repudiation, detectability, information disclosure, content unawareness and policy/consent non-compliance. Those threat categories are then mapped to the elements in data flow diagrams (DFDs) (i.e. entity, data flow, data store and process). The method further develops a catalogue of threat patterns (i.e. a threat tree) that represents common attacks concerned in each threat category and each element in the DFD. The patterns in the threat tree are mapped to the elements in DFD defined by misuse cases to identify privacy requirements. In the final step, the method recommends privacy-enhancing solutions that fulfill the identified requirements.
\citeauthor{Kalloniatis2008} \cite{Kalloniatis2008} proposed a PriS method to elicit privacy requirements in the software design process. Privacy requirements in this study are modelled as a type of organisational goals that needs to be achieved in a specific application. The method first identifies privacy concerns in that specific application based on the eight pre-defined privacy goals (e.g. authentication and unlinkability). The second step is to analyse impact of privacy goals on relevant processes. This impact may introduce new goals or adjustment of existing goals. After having a set of privacy-related processes, the method models those processes based on the pre-defined privacy-process patterns of each privacy goal. The final step then suggests appropriate techniques for implementing those identified processes. Comparing with our study, LINDDUN and PriS methods do not elicit requirements from privacy and data protection regulations and frameworks, and they employed different requirements elicitation approaches. The taxonomies defined in their work are selected from a set of privacy properties identified in previous studies \cite{Solove}, \cite{George2007}, \cite{Solove2008}, \cite{anon_terminology}, \cite{BritishStandardsInstitute2000}, whereas our taxonomy is formed after eliciting requirements.
\citeauthor{Spiekermann2009} \cite{Spiekermann2009} have developed a privacy-friendly system design framework to preserve privacy concerns in software development. The framework consists of two approaches: privacy-by-policy and privacy-by-architecture. Privacy-by-policy adopts the principles in Fair Information Practices (FIPs) developed by Organisation of Economic Co-Operation and Development (OECD) \cite{OECD2013} to guide on addressing user privacy in software systems. The principles focus on providing notice and awareness to users on how their personal data will be processed. Privacy-by-architecture promotes organisations to store only necessary personal data in their systems. The authors further provide a guideline for selecting privacy protection approaches that match with system characteristics.
\citeauthor{Meis} \cite{Meis} proposed a taxonomy of transparency requirements to support software engineers in identifying relevant requirements. The requirements were derived from a draft version of GDPR and ISO/IEC 29100. The requirements formulation process includes identifying verbs and related nouns, refining identified requirements and structuring the requirements into a taxonomy of transparency requirements. This approach was later used to derive intervenability requirements from the selected articles and principles in the draft of GDPR and ISO/IEC 29100 respectively \cite{Meis2016}. These intervenability requirements were then linked to the related transparency requirements proposed earlier in \cite{Meis}. However, these studies emphasise on privacy goal transparency and intervenability in software development.
\citeauthor{Breaux2014} \cite{Breaux2014} discussed an interesting idea for software engineers on balancing between business goals and privacy requirements as both elements are essential when designing software. The business goals are normally addressed first as they drive primary activities of software applications. However, these also introduce privacy risks in order to satisfy those goals in software. The author suggests the engineers to link business requirements to privacy risks and consider ways that users could be involved in reducing privacy exposures (e.g. personalise their own level of privacy).
Several work identified the challenges faced by software engineers when embedding privacy into software systems. First, the developers have limited knowledge and understanding of the concept of information privacy, privacy-preserving technologies and privacy practices \cite{Senarath2018b}, \cite{Hadar2018}, \cite{Senarath2018}. They interpret privacy concerns as a sub-category of security; this potentially misguides their perceptions when treating privacy in software systems \cite{Hadar2018}, \cite{Senarath2018}. Moreover, they lack evaluation criteria to assess whether they meet privacy expectations in their design \cite{Senarath2018b}.
Second, organisation's policy also has a strong influence on developers' privacy-preserving behaviour \cite{Hadar2018}, \cite{Senarath2018}. The developers likely respond to privacy concerns depending on what they are told to do. Thus, privacy decisions are not driven by law or engineering guidelines and solutions \cite{Hadar2018}. In addition, the developers do not give priority to privacy requirements because they are not functional requirements or they are inconsistent with client/business requirements \cite{Senarath2018b}.
Finally, privacy requirements are varied across countries/regions depending on different user expectations \cite{Senarath2018}, \cite{Sheth2014} and national legal enforcement (e.g. GDPR for European Union, US Privacy Act of 1974). When the privacy expectations between users and software developers do not match, it is challenging to preserve user privacy in software systems. Hence, the developers require privacy guidelines or evaluation criteria to assess whether they meet these expectations \cite{Senarath2018b}.
Data protection and privacy laws are independently designed and enforced for specific areas, which could range from a state (e.g. California), a country (e.g. Australia) or a region (Europe). Any organisations that meet the conditions of these laws must comply. However, the developers lack guidance and also experience difficulties in understanding and extracting such privacy requirements from those required laws \cite{Ayala-Rivera2018}, \cite{Aljeraisy2020}. Several studies have been calling for frameworks and methodologies to support software engineers in designing and developing privacy-aware software systems \cite{Gurses2011}, \cite{Senarath2018b}, \cite{Sheth2014}, \cite{Birnhack2014}.
Some existing work have aimed to identify privacy requirements and construct a privacy requirement taxonomy from privacy policies. \citeauthor{Anton2002} proposed a privacy goal taxonomy based on website privacy policies \cite{Anton2002}. The taxonomy was constructed by applying goal identification and refinement strategies based on the Goal-Based Requirements Analysis Method (GBRAM) \cite{Anton1996} to extract goals and requirements from 24 Internet privacy policies from e-commerce industries. This methodology was further used to develop the taxonomy of privacy requirements from 23 Internet health care Web sites privacy policies \cite{Antn2004}. They follow a goal mining process with heuristics to analyse and refine goals and scenarios from those privacy policies. We adapted this approach to construct the taxonomy presented in this paper.
Several work have also investigated how the introduction of GDPR has changed the way that organisations comply with the privacy protection regulations and standards. \citeauthor{Linden2020} \cite{Linden2020} found that the online privacy policies of organisations in EU countries have improved their appearance, presentation and details. They also classified statements in privacy policies into a set of categories developed in \cite{Wilson2016}. Privacy concerns have also emerged in the area of Big Data analytics. \citeauthor{Gruschka2018} \cite{Gruschka2018} explored two case studies of Big Data research projects to investigate on how legal privacy requirements can be met. The study identified a set of key processes related to sensitive personal information, and proposed privacy-preserving techniques to achieve GDPR compliance. However, the study did not provide a systematic approach to link the identified processes to appropriate techniques.
The following work discusses privacy requirements elicitation from regulations and privacy policies for compliance checking in software systems. \citeauthor{Muller2019} \cite{Muller2019} developed a dataset consisting of statements in organisations' privacy policies. Those statements were annotated with only five privacy requirements extracted from GDPR. This dataset assisted the development of automated tools for checking GDPR compliance of an organisation's privacy policies. \citeauthor{Torre} \cite{Torre} proposed a different approach to check if an organisation's privacy policies comply with GDPR. They first developed a conceptual model using hypothesis coding to specify metadata types that exist in the statements of selected GDPR articles. They created the dependencies between the metadata types to ensure the proper completeness checking. They then employed Natural Language Processing and Machine Learning techniques to automatically classify the information content in privacy policies from the fund domain and determine the extent to which the content satisfies a certain completeness criterion. However, the metadata types and their dependencies in the model are limited as several key articles in the GDPR were not considered (e.g. data subjects' rights and security of processing). In addition, additional criteria for completeness checking should be considered since the presence or absence of those metadata may not be sufficient to determine the completeness of statements in privacy policies against GDPR.
\citeauthor{Ayala-Rivera2018} \cite{Ayala-Rivera2018} proposed an approach to map GDPR data protection obligations with privacy controls derived from ISO/IEC standards. Those links help elicit the solution requirements that should be implemented in a software application. However, their study focused and validated only two articles in GDPR (i.e. Article 5 and 25). \citeauthor{Torre2019} \cite{Torre2019} explored the use of models to assist GDPR compliance checking. They developed a UML class diagram representation of GDPR. This UML model is enriched with invariants expressed in the Object Constraint Language to captures GDPR rules.
Several work have investigated consistency and completeness verification for privacy policies. \citeauthor{Breaux2014a} \cite{Breaux2014a} translated a small subset of commonly found privacy requirements into description logic (a formal knowledge representation language). This enables the use of formal tools to detect conflicting privacy requirements within a policy. \citeauthor{Bhatia2019} \cite{Bhatia2019} proposed to represent data practice descriptions as semantic frames to identify incompleteness in privacy policies in organisations. They analysed 15 policies and identified 17 semantic roles associated with four categories of data action (collection, retention, usage and transfer) to express the incompleteness in data action context. Compared to our study, the settings of developing semantic roles are similar to the process we used to extract privacy requirements from the statements in the selected privacy regulations and frameworks. We can extend the use of semantic roles to categorise the scenarios of actions stated in those regulations and frameworks.
\citeauthor{Pandit2019} \cite{Pandit2019} developed the Data Privacy Vocabulary (DPV) which is an ontology of generic concepts and relationships of the components identified in GDPR. This ontology is used for completeness and compliance checking in privacy policies, consent receipts and records of personal data handling. However, their taxonomy does not cover any specific software requirements. In addition, the taxonomy addresses only six articles in the GDPR, while our work covers more articles in the GDPR and other well known data protection regulations and privacy frameworks.
Particularly focusing on the work related to GDPR, the European Commission has funded the GDPR cluster projects to help tackle the GDPR implementation challenges faced by organisations \cite{EUcluster2020}. Those projects have developed both organisational and technical techniques to facilitate the implementation. They have addressed different challenges complying with GDPR in software development activities (e.g. planning, design, development, operation and deployment). They also provide solutions to the identified challenges. \citeauthor{EUcluster2020} \cite{EUcluster2020} has summarised the solutions proposed by these projects.
The Business Process Re-engineering and functional toolkit for GDPR compliance project\footnote{https://www.bpr4gdpr.eu/} (BPR4GDPR) provides an approach and a toolkit to support end-to-end GDPR-complaint business processes, particularly for small and medium-sized enterprises (SMEs) \cite{BPR4GDPR}. The deliverables include the policy-based access and usage control framework, specification of workflow models and tools for cryptography, access management and enforcement of data subjects' rights \cite{EUcluster2020}.
As there is neither specific methods, techniques or tools to evaluate the GDPR readiness level in organisations nor privacy governance guideline available, the Data Privacy Governance for Supporting GDPR project\footnote{https://www.defendproject.eu/} (DEFeND) introduces a platform to assist in complying with the GDPR \cite{DEFEND}. The platform supports organisations in designing and developing tailored solution that covers different aspects of GDPR. It also provides methods and techniques to handle personal data and consent management as well as data protection mechanisms in software systems.
SMOOTH project\footnote{https://smoothplatform.eu/about-smooth-project/} aims to provide support for micro enterprises to adopt GDPR as a cloud-based platform. The solution helps assess the compliance of the enterprises and identify the basic requirements that need to be satisfied to comply with GDPR \cite{SMOOTH}. After the enterprise provides the information related to its current data protection management through the questionnaire in the platform, the platform will generate a compliance report with appropriate guidance to resolve non-compliance in the enterprise.
The Privacy and Data Protection 4 Engineering (PDP4E) project\footnote{https://www.pdp4e-project.eu/} develops a set of systematic methods and tools to address the following disciplines in software development life cycle: risks assessment, requirements engineering, model-driven design and systems assurance \cite{PDP4E}. Particularly focusing on requirements engineering, the project has introduced PDP-ReqLite, an approach to elicit privacy and data protection meta-requirements using a lightweight version of problem frames \cite{Ferreyra2020}.
The PlAtform for PrivAcY preserving data analytics (PAPAYA) project \footnote{https://www.papaya-project.eu/} develops a platform that addresses privacy preservation in data analytics. Finally, the Protection and control of Secured Information by means of a privacy enhanced Dashboard (PoSeID-on) project \footnote{https://www.poseidon-h2020.eu/} provides a solution allowing end users to manage their personal data and safeguarding the rights of data subjects \cite{POSEIDON}. The project also ensures GDPR-compliant data management and processing for organisations. The project develops a dashboard presenting the summary of their personal data (e.g. the personal data that is allowed to be shared and history of personal data transactions). The end users will be notified when data processors require the permissions to use their personal data. The data processors can also check what personal data is shared to them.
\section{Analysis and discussions} \label{results}
The study presented in the previous section generates not only a valuable dataset but also important insights into how privacy requirements have been addressed in Chrome and Moodle issue reports. In this section, we discuss and analyse some of the key findings and implications.
\subsection{The top and least concerned privacy requirements}
Table \ref{tab:table2} presents the coverage of each privacy goal in Chrome and Moodle issue reports. In both projects, the majority of the issues address the user participation requirements (Category 1). Most issue reports in Moodle address more than one privacy requirement across different privacy goal categories. This results in Moodle having higher coverage (in terms of the percentage of occurrences) than Chrome.
\begin{table}[ht]
\centering
\caption{The number of privacy requirements in each privacy goal and the percentage of occurrences in Google Chrome and Moodle issue reports}
\label{tab:table2}
\resizebox{3.25in}{!}{%
\begin{tabular}{p{3cm} c c c}
\toprule
\multicolumn{1}{c}{\multirow{2}{*}{\textbf{Privacy goals}}} & \multicolumn{1}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{No. of privacy }\\\textbf{requirements~ }\end{tabular}}} & \multicolumn{2}{c}{\begin{tabular}[c]{@{}c@{}}\textbf{Percentage of}\\\textbf{occurrences (\%) }\end{tabular}} \\
\cline{3-4}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}\textbf{Google }\\\textbf{Chrome}\end{tabular}} & \multicolumn{1}{c}{\textbf{Moodle}} \\
\midrule
User participation & 9 & 35.83 & 68.62\\
Notice & 32 & 30.36 & 47.91\\
User desirability & 10 & 28.01 & 40.59\\
Data processing & 16 & 13.06 & 11.51\\
Breach & 6 & 0.00 & 0.00\\
Complaint/Request & 5 & 0.11 & 1.88\\
Security & 13 & 17.86 & 44.77\\
\hline
\multicolumn{4}{p{8.5cm}}{\footnotesize *Since one issue can relate to multiple privacy requirements, the sum of percentage exceeds 100\%} \setlength\lineskip{0pt}
\end{tabular}%
}
\end{table}
The top three most concerned requirements in Chrome are R30, R44, R60 (see Figure \ref{fig:top10}). Note that these requirements belong to three different privacy goal categories (refer to Table \ref{tab:table1} for details of the privacy goals and requirements we discussed here). The top three requirements covered in Moodle issues are R44, R1 and R35 (see Figure \ref{fig:top10}). It is also worth noting that requirements R1, R26, R30, R44 and R60 are in the top 10 in both projects.
Requirement R44 was in the top three most concerned requirements in both projects, suggesting that allowing the data subjects to erase their personal data is a highly important privacy requirement for both Chrome and Moodle. Requirements R30 and R36 were also addressed in many privacy-related issues in Chrome. This suggests that providing information with standardised, visible and meaningful icons which inform the intended processing of personal data for users is an important privacy concern in Chrome (R30). In addition, many issues in Chrome also focus on addressing the privacy requirement that users are presented with all available options related to the processing of personal data (R36).
Apart from requirement R44, the other two requirements most frequently covered in Moodle issue reports are R1 and R35 (note that they are different from those in Chrome). Approximately 39\% issues in Moodle are related to requirement R1. This implies that Moodle has a strong emphasis on allowing users to access their personal data such as grade records, course participation and course enrolment records. This function is not only important for Moodle, but also in Chrome (R1 is also in the top 10 for Chrome). User consent is a major concern in privacy protection. We found that a large number of issue reports in Moodle explicitly requires the system to obtain consent from users for processing personal data based on specific purposes (R35).
It is interesting to note that none of the requirements in the breach category were satisfied by both Chrome and Moodle as they were not directly observed from issue reports or recorded in the ITS. These goals can be evidenced through high-level organisational activities such as Data Protection Impact Assessment, Legitimate Interest Assessment and breach notifications. Our future work will investigate this further.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{Figures/"Top10-occ-by-type"}
\caption{Top 10 privacy requirements occurrences in Google Chrome and Moodle datasets categorised by issue types}
\label{fig:top10}
\end{figure}
\begin{comment}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{Figures/"Top10-occ"}
\caption{Top 10 privacy requirements occurrences in Google Chrome and Moodle datasets}
\label{fig:top10}
\end{figure}
\end{comment}
We have performed further analysis on the issue reports that were classified into the top concerned requirements in Chrome and Moodle datasets. We analyse four factors focusing on how the contributors treat those issue reports: issue types, the time took to resolve an issue, the number of contributors involved and the number of comments associated with the issue.
In Chrome dataset, issue reports have five different types: bug, bug-regression, bug-security, feature and task. Bug type reports malfunctioning functionalities in current version of the system. Bug-regression focuses on the functions that used to work correctly in the previous versions, but are broken in the current version. Bug-security reports malfunctions that are risky to user security. Feature type requests for an implementation of a new function/feature. Task type, which is not a bug or feature, defines a piece of work that needs to be completed for an issue. A small group of issues were not specified their issue types. From our study, we found that the issue reports whose issue type is a bug are the largest group in every top concerned privacy requirement. These bug issue reports as well as bug-regression and bug-security took less time to resolve comparing to feature request issue reports on average for all the top concerned privacy requirements. In addition, the bug issue reports usually involved with a smaller number of contributors and had less discussions than the feature request issue reports. We have also investigated the discussions of bug issue reports classified into the top concerned requirements. We found that the bug issues were fixed after fifteen comments. However, the discussions of feature issues contain more details (e.g. use case scenarios, discussion points, screenshots and code snippets) than the bug issues.
There are seven different issue types in Moodle dataset: bug, epic, improvement, new feature, task, sub-task and functional test. The definitions of bug, new feature, task and sub-task issue types are similar to those mentioned in Chrome dataset. Epic issue type collects a group of issues that needs to be completed over a period of time. An improvement issue type is an enhancement to an existing feature. Functional test type contains the information and steps used for testing a particular function. From the analysis in Moodle dataset, the bug issue type contains the largest number of issue reports, followed by the feature issue type. We found that the bug issues did not only report the malfunctions, but they also reported the missing functionalities in the system (e.g. implement core\_privacy for block rss client). It is interesting to note that the new feature issues took less time to resolve comparing to the bug issues on average for all the requirements except R42. However, both issue types have similar number of comments (i.e. 11 to 12 comments) and number of contributors (i.e. 5).
\input{sections/RQ-privacy-vs-nonprivacy}
\section{Introduction}}
Software applications have become an integral part of our society. Digital footprints are collected as people are browsing the Internet or using various software applications such as for social networking, working, studying and leisure activities. Physical footprints are also collected through software systems such as surveillance cameras, face recognition apps, IoT sensors and GPS devices even when people are ``offline'' doing their normal life activities. Zettabytes of those data are collected and processed for various purposes, including extracting and using personal data, and forming behavioural profiles of individuals. This poses serious threats to our privacy and the protection of our personal sphere of life -- the cornerstone of human rights and values.
The recent advent of privacy legislations, policies and standards (e.g. the European's General Data Protection Regulation \cite{OfficeJournaloftheEuropeanUnion;2016} or the ISO/IEC standard for privacy framework in information technology \cite{ISO/IEC2011}) aims to mitigate those threats of privacy invasion. A range of frameworks (e.g. privacy by design) and privacy engineering methodologies have also emerged to help design and develop software systems that provide acceptable levels of privacy and meet privacy regulations \cite{Ayala-Rivera2018}, \cite{Solove}, \cite{Kalloniatis2008}, \cite{Deng2011}, \cite{Heurix2015}, \cite{Perera2016}, \cite{Aljeraisy2020}. However, those methodologies provide only high-level principles and guidelines, leaving a big gap for software engineers to fill in designing and implementing privacy-aware software systems \cite{Gurses2011}. Software engineers often face challenges when navigating through those regulations and policies to understand and implement them in software systems \cite{Ayala-Rivera2018}, \cite{Aljeraisy2020}, \cite{Senarath2018b}, \cite{Bednar2019}.
In 1995, \citeauthor{Cavoukian2009} initially developed a reference framework called \textit{Privacy by Design (PbD)} aiming to emphasise the importance of privacy in the engineering processes of information systems \cite{Cavoukian2009}. The framework describes seven principles that provide conceptual characteristics of privacy elements related to the processing of personal data. However, this framework is often criticised as being too broad and vague for actual implementation \cite{Rest2014}. Recently, \textit{privacy engineering} is an emerging field which addresses privacy concerns in the development of sociotechnical systems \cite{Gurses2016}. Unfortunately, the integration of privacy solutions into practices has faced several challenges and limitations due to the design of organisational processes \cite{Spiekermann2012}, \cite{Mikkelsen2019}, legacy IT systems \cite{Capgemini2019}, implementation cost \cite{Capgemini2019} and development practices \cite{Bednar2019}, \cite{Hadar2018}. Organisations have been collecting personal data of their customers for various business purposes. Cyberattacks often target at obtaining this data. CSO Online reported the fourteen biggest data breaches of the 21st century that affected 3.5 billion people \cite{Swinhoe2020}. The cases occurred with the world's top software applications, for examples, Adobe, Canva, eBay, LinkedIn and Yahoo.
Hence, there is an emerging need to translate complex privacy concerns set out in regulations and standards into requirements that are to be implemented in software applications. Such privacy requirements need to be refined into a level that emphasises the functionalities in software systems and can be later used to map with issue reports. Previous work has involved eliciting privacy requirements, but they are specific to a certain application domain such as e-commerce applications (e.g. \cite{Anton2002}) or healthcare websites (e.g. \cite{Antn2004}). More recent studies (e.g. \cite{Anthonysamy2017} and \cite{Guarda2009}) revealed an urgent need for a reference taxonomy of privacy requirements that are based on well-established regulations and standards such as GDPR and ISO/IEC 29100.
This taxonomy would be useful to understand how privacy requirements are explicitly addressed in existing software projects, and also serve as a basis for developing future privacy-aware software applications. Most of today's software projects follow an agile, issue-driven style in which feature requests, functionality implementations and all other project tasks are usually recorded as issues (e.g. JIRA issues\footnote{https://www.atlassian.com/software/jira}) \cite{Choetkiertikul}. In those projects, issue reports essentially contain important, albeit implicit, information about the requirements of a software, in the form of either new requirements (i.e. feature requests), change requests for existing requirements (i.e. improvements) or reporting requirements not being properly met (i.e. bugs) \cite{Choetkiertikul}, \cite{MoodleTracker}, \cite{MoodleFeature}, \cite{Merten2015}, \cite{Merten2016}. A software project consists of past issues that have been closed, ongoing issues that the team are working on, and new issues that have just been created. Through a study of those issues in the project, we can understand how the software team has implemented privacy requirements recorded in the issue tracking system (ITS) in order to address relevant privacy needs and concerns of stakeholders. This paper provides the following contributions:
\begin{enumerate}[leftmargin=0.5cm]
\item We developed a comprehensive taxonomy of privacy requirements for software systems by extracting and refining requirements from the widely-adopted GDPR and ISO/IEC 29100 privacy framework as well as the newly developed Thailand PDPA and the region-specific APEC privacy framework. We followed a grounded theory process adapted from the Goal-Based Requirements Analysis Method (GBRAM) \cite{Antn2004}, \cite{Anton1996} to develop this taxonomy. The taxonomy consists of 7 categories and 71 privacy requirement types. To the best of our knowledge, this is the \emph{first} taxonomy of privacy requirements that are well grounded in standardised privacy regulations and frameworks.
\item We mined the issue reports of two large-scale software projects, Google Chrome and Moodle (each has tens of thousands of issues) to extract 1,374 privacy-related issues. We classified all of those issues into the privacy requirements of our taxonomy. The classification was performed by multiple coders through multiple rounds of training sessions, inter-rater reliability assessments and disagreement resolution sessions. This resulted in a reliable dataset for the research community to perform future research in this timely, important topic of software engineering such as automated classification of privacy issue reports.
\item We studied how the privacy requirements in our taxonomy were addressed in Google Chrome and Moodle issue reports. We found 2,307 occurrences of the privacy requirements in our taxonomy were covered, most of which related to the lawfulness, fairness and transparency privacy goal category. In addition, we found that allowing the erasure of personal data is a top concern reported in the Chrome and Moodle issue reports, while none of the privacy requirements related to the management perspective of data controllers was recorded in the issue tracking system of both projects. We also discovered that privacy and non-privacy issues were treated differently in terms of resolution time and developers' engagement in both projects.
\end{enumerate}
A full replication package containing all the artifacts and datasets produced by our studies are made publicly available at \cite{reppkg-pridp}. The remainder of this paper is structured as follows. Section \ref{sec2} provides related existing work on privacy requirements engineering. A privacy requirements taxonomy and the methodology used to build it are presented in Section \ref{sec3}. Section \ref{sec4} presents our study of how Chrome and Moodle issue reports address the privacy requirements in our taxonomy. The findings and insights generated from the study are discussed in Section \ref{results}. Section \ref{threats} discusses the threats to validity. Finally, we conclude and discuss future work in Section \ref{sec7}.
\section{A taxonomy of privacy requirements} \label{sec3}
Many countries around the world have been developing data protection and privacy legislation to strengthen their personal data and privacy protection \cite{UNCTAD2020}. These legislations are designed to provide organisations with a comprehensive benchmark to govern their personal data collection and processing as well as protect and empower individuals about their privacy and rights. Having the legislations in place seems to benefit both organisations and individuals, however many organisations have faced several challenges to comply with these legislations \cite{Capgemini2019}.
Capgemini Research Institute \cite{Capgemini2019} has produced an insightful survey on how companies have been coping with data protection and privacy compliance. They surveyed 1,100 compliance, privacy, data protection and IT executives across ten countries and eights sectors. In-depth interviews with experts in data protection and privacy regulations and practices were also conducted. The results revealed that a major challenge the businesses are facing is the alignment of existing IT systems to data protection and privacy regulations. There are top three barriers for the businesses to comply with GDPR reported in the survey. Firstly, the businesses found that aligning the legacy systems to GDPR requirements is very complex. Secondly, the GDPR requirements are too complex, which require more effort for implementation. Finally, the costs of achieving alignment with GDPR are restricted. Those challenges raise the need to develop a taxonomy of requirements from data protection and privacy regulations to support the development and compliance of privacy-aware software systems.
Our work is based on two well-established and widely-adopted regulations and privacy frameworks: GDPR and ISO/IEC 29100 and two region-specific representatives: Thailand PDPA and APEC privacy framework. GDPR is enacted to protect the individual rights of the data subjects on their personal data \cite{OfficeJournaloftheEuropeanUnion;2016}. It provides conditions, principles and definitions that need to be integrated into organisational processes and policies. These processes involve the collection, use, process, storage and dissemination of personal data of EU citizens and residents. GDPR will have a significant impact on technology platforms and data structures, data protection measures and emerging technologies implemented in organisations \cite{Li1}. The organisations failing to comply with the GDPR can be fined up to \EUR{20} million or 4\% of their previous year's global turnover, whichever is greater. After a year of the enforcement, there are over 230 finalised cases with the total of \EUR{150} million fines so far. The largest fined case is Google Inc. in France with \EUR{50} million fines due to the lack of consent for advertisements \cite{Data}. A great number of GDPR violation cases related to the processing of personal data and data breach have been reported \cite{EuropeanCommission2019}, \cite{CNET}, \cite{PrivacyAffa}. This suggests the challenges in operationalising GDPR in developing software applications.
ISO/IEC 29100:2011 is a privacy framework which guides the processing of personally identifiable information (PII) in Information and Communication Technology systems \cite{ISO/IEC2011}. The framework defines a set of privacy principles used to handle personal data processing activities (e.g. collection, storage, use, transfer and disposal). \emph{Similarly to GDPR, those principles are high-level, making it challenging for software engineers to design and implement privacy-aware systems}. We aim to address these challenges by translating these complex statements into implementable privacy requirements for software systems.
Thailand Personal Data Protection Act (PDPA) was officially announced in May 2019 \cite{PDPA}. The regulation will come into full effect in June 2022 after several extensions due to COVID-19 disruptions. Thailand PDPA is designed to govern personal data protection and create transparency and fairness for the use of personal data. It also promotes the use of personal data for innovation under assurance and provides effective remedy from data breaches. Any organisations in Thailand that collect, use and disclose personal data must comply with the regulation. In addition, any organisations that are located outside Thailand, but sell products, provide services and/or track individuals residing in Thailand, must also comply to this regulation. We include this regulation in our study as it is a representative of newly developed and country-specific personal data protection regulation.
Asia-Pacific Economic Cooperation (APEC) privacy framework 2015 \cite{Apec2015} was published in August 2017 with the intention to establish effective privacy protections for cross-border information transfers across member countries of APEC \footnote{A list of member countries can be found at https://www.apec.org/about-us/about-apec}. This updated version was improved from the previous version published in 2005. The framework was developed based on the Organisation for Economic Co-operation and Development (OECD) guidelines.
Prior to this work, we have conducted a thorough study on GDPR, ISO/IEC 29100, Thailand PDPA and APEC privacy framework and have found that they share many commonalities. All of the regulations, standards and frameworks provide benchmarks for privacy and data protection governance and compliance in organisations. They are common in laying out the expectations that should be met when handling personal data. They also complement each other to cover a comprehensive set of privacy-related software requirements. In addition, both GDPR and ISO/IEC 29100 define common key actors and their roles in processing personal data. The basic principles between GDPR and ISO/IEC 29100 are similar, although they are grouped differently (see Table \ref{tab:GDPR-ISO-mapping}). The requirements derived from ISO/IEC 29100 are similar or equivalent to those in GDPR. Although the terms and definitions in GDPR and ISO/IEC 29100 use different wordings, but they in fact refer to the same or similar things.
\begin{table*}[]
\center
\caption{A table mapping between the GDPR, ISO/IEC 29100, Thailand PDPA and APEC framework principles.}
\label{tab:GDPR-ISO-mapping}
\begin{tabular}{p{3cm} p{5cm} p{4.2cm} p{3.8cm}}
\toprule
\textbf{GDPR principles} &
\textbf{ISO/IEC 29100 principles} &
\textbf{Thailand PDPA parts} &
\textbf{APEC principles} \\ \midrule
Lawfulness, fairness and transparency &
\begin{tabular}[t]{@{}l@{}}Consent and choice\\ Purpose legitimacy and specification\\ Collection limitation\\ Use, retention and disclosure limitation\\ Openness, transparency and notice\\ Individual participation and access\end{tabular} &
\begin{tabular}[t]{@{}l@{}}General provisions\\ Personal data collection\\ Use or disclosure of personal data\\ Rights of the data subject\end{tabular} &
\begin{tabular}[t]{@{}l@{}}Preventing harms\\ Notice\\ Choice\\ Access and correction\\ Accountability\end{tabular} \\ \midrule
Purpose limitation &
\begin{tabular}[t]{@{}l@{}}Consent and choice\\ Purpose legitimacy and specification\\ Use, retention and disclosure limitation\\ Openness, transparency and notice\\ Accountability\end{tabular} &
\begin{tabular}[t]{@{}l@{}}General provisions\\ Personal data collection\\ Use or disclosure of personal data\end{tabular} &
\begin{tabular}[t]{@{}l@{}}Notice\\ Uses of personal information\\ Choice\end{tabular} \\ \midrule
Data minimisation &
\begin{tabular}[t]{@{}l@{}}Collection limitation\\ Data minimisation\\ Use, retention and disclosure limitation\end{tabular} &
\begin{tabular}[t]{@{}l@{}}General provisions\\ Personal data collection\\ Rights of the data subject\end{tabular} &
Collection limitation \\ \midrule
Accuracy &
\begin{tabular}[t]{@{}l@{}}Accuracy and quality\\ Individual participation and access\end{tabular} &
Rights of the data subject &
\begin{tabular}[t]{@{}l@{}}Collection limitation\\ Integrity of personal information\\ Access and correction\end{tabular} \\ \midrule
Storage limitation &
\begin{tabular}[t]{@{}l@{}}Data minimisation\\ Use, retention and disclosure limitation\\ Openness, transparency and notice\end{tabular} &
\begin{tabular}[t]{@{}l@{}}Personal data collection\\ Rights of the data subject\end{tabular} &
None \\ \midrule
Integrity and confidentiality &
Information security &
Rights of the data subject &
\begin{tabular}[t]{@{}l@{}}Preventing harms\\ Security safeguards\\ Accountability\end{tabular} \\ \midrule
Accountability &
\begin{tabular}[t]{@{}l@{}}Collection limitation\\ Accountability\\ Privacy compliance\end{tabular} &
\begin{tabular}[t]{@{}l@{}}Use or disclosure of personal data\\ Rights of the data subject\end{tabular} &
\begin{tabular}[t]{@{}l@{}}Preventing harms\\ Accountability\end{tabular}
\end{tabular}
\end{table*}
The principles in both GDPR and ISO/IEC 29100 present a set of basic guidelines to govern personal data and privacy protection covered in the regulations and standard. We have found that there are commonalities and differences across these sources. For instance, the integrity and confidentiality in GDPR and information security principle in ISO/IEC 29100 address the common concerns on having appropriate measures put in place to protect personal data and its processing. The examples of the requirements related to these principles are \emph{R60 IMPLEMENT appropriate technical and organisational measures to protect personal data} and \emph{R63 PROTECT the personal data from unauthorised access and processing}.
It is also interesting to note that requirements from a principle in ISO/IEC 29100 can be addressed in different principles in GDPR. For example, we have derived requirements \emph{R1 ALLOW the data subjects to access and review their personal data} and \emph{R50 INFORM the recipients of personal data any rectification or erasure of personal data or restriction of processing} from the individual participation and access principle in ISO/IEC 29100. Requirement R1 reflects the lawfulness, fairness and transparency principle in GDPR while requirement R50 relates to the accuracy principle in the GDPR.
This section presents a taxonomy of privacy requirements that we have developed based on the GDPR, ISO/IEC 29100, Thailand PDPA and APEC privacy framework. We first discuss the methodology that we have followed to develop this taxonomy, and then describe the privacy requirements set out in the taxonomy.
\subsection{Methodology} \label{sec3.1}
We followed a content analysis process adapted from the Goal-Based Requirements Analysis Method (GBRAM) \cite{Antn2004}, \cite{Anton1996}, which is based on Grounded Theory \cite{Glaser2017}, to develop a taxonomy of privacy requirements (see Figure \ref{fig:taxonomy development}). GBRAM is a systematic method used to identify, refine and organise goals into software requirements. This process was applied to analyse goals from natural language texts in privacy policies, and convert them into software requirements. The method has been successfully applied to the analysis of e-commerce applications \cite{Anton1998}, \cite{Baumer2000} and Internet health care Web site privacy policies \cite{Antn2004}. The method consists of three main activities: goal identification, goal classification and goal refinement. Goal identification derives goals from specifications in selected sources. Each identified goal is then classified into one of the pre-defined categories in the goal classification. Finally, the goal refinement removes synonymous and redundant goals, resolves inconsistencies among the goals and operationalises the goals into requirements specification.
We had multiple researchers (co-authors of the paper, hereby referred to as the coders) follow this process to develop the taxonomy independently, and used the Inter-Rater Reliability (IRR) assessment to validate the agreements and resolve disagreements. Those coders were given instructions and trained at the start of the process. The process consists of the following steps:
\begin{itemize}[leftmargin=0.5cm]
\item \textbf{Privacy requirements identification:} extract requirements from written statements in the studied privacy regulations and frameworks, and structure them into a pattern (action verb, objects and object complement).
\item \textbf{Privacy requirements refinement:} remove duplicate requirements and manage inconsistent requirements. Since the inputs were written in descriptive statements and from different sources, requirements can be redundant or inconsistent.
\item \textbf{Privacy requirements classification:} classify requirements into categories based on a set of privacy goals. The privacy goals can be considered as a group of functionalities that the software systems are expected to provide.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{"Figures/Taxonomy-development"}
\caption{An overview process of privacy requirements taxonomy development}
\label{fig:taxonomy development}
\end{figure}
\vspace{0.5mm}
The details of each step are described as follows.
\subsubsection{Privacy requirements identification}
This step aims to identify privacy requirements from the narrative statements in GDPR, ISO/IEC 29100 privacy framework, Thailand PDPA and APEC privacy framework. We adapted the approach in \citeauthor{Antn2004} \cite{Antn2004} and first created a range of questions to identify goals from each statement of the studied privacy regulations and frameworks. As part of our research, we have carefully gone through all 99 articles in the GDPR. Although GDPR and Thailand PDPA govern broader regulatory aspects comparing to ISO/IEC 29100 and APEC privacy frameworks such as requirements in roles assignment (e.g. data protection officer, supervisory authority, personal data protection committee), managing juridical remedies and noticing penalties, those are not software requirements. Thus, they are out of the scope of our study.
We then applied several filters to select the articles that address software requirements in GDPR and include them in our study. We selected nineteen articles that address the rights of individuals and cover the key principles of the GDPR (i.e. Articles 6-7, 12-22, 25, 29-30, 32-34). We also excluded the articles that deal with special conditions (e.g. Article 8 child's personal data and Article 9 special categories of personal data) since we aim to keep our taxonomy as generalisable as possible for software systems in different sectors. International transfers are presented in Articles 13(1)(f), 14(1)(f) and 15(2), which we have considered in our study. We did not consider Chapter V (Articles 44 - 50) since it focuses on legal administrative perspective for international border transfer (e.g. the provisions determining conditions for international transfer and international cooperation for the protection of personal data) rather than the software application level. Thus, we did not analyse them as software requirements in our study. For the ISO/IEC 29100 privacy framework, all of the contents were explored.
Thailand PDPA consists of 7 chapters, 96 articles and 7 rights of data subjects. We have analysed 16 articles in Chapter 2 (Personal data protection) and Chapter 3 (Rights of data subjects) which are the key chapters providing guidelines to govern personal data and privacy protection. Chapter 2 consists of three parts: i) general provisions, ii) personal data collection and iii) use or disclosure of personal data. Other chapters detailing the scope of use, definitions, assignment of Personal Data Protection Committee and supervisory authority, complaints and penalties are not included in the taxonomy development process as they are not related to software requirements. In the APEC privacy framework, there are four parts containing 72 points: i) preamble, ii) scope, iii) APEC information privacy principles and iv) implementation. We have analysed the third part to derive privacy requirements in the taxonomy development process. The third part includes nine information privacy principles: i) preventing harms, ii) notice, iii) collection limitation, iv) uses of personal information, v) choice, vi) integrity of personal information, vii) security safeguards, viii) access and correction and ix) accountability. The other parts define the objectives, scope and organisational-level guidelines and are not related to software requirements. Based on the scope defined above, we shortlisted 149 statements in GDPR, 63 in ISO/IEC 29100, 101 in Thailand PDPA and 74 in the APEC privacy framework to be explored.
Once we selected the articles and sections to work on, we went through all the sentences in those articles and sections. We analysed each sentence using a set of pre-defined questions to identify relevant actions, involved/affected parties or objects and target results. A sentence may cover more than one requirement. There are three key components for a requirement: 1) actions, 2) involved/affected parties or objects, and 3) target results. The detailed steps of privacy requirements identification process are explained below:
\begin{enumerate}[leftmargin=*]
\item \textbf{Identifying actions:} We ask \textit{``Which action should be provided based on this statement?''} to identify the action associated with a requirement. Some examples of the action verbs used in the collected statements are: \textbf{ALLOW, ARCHIVE, COLLECT, ERASE, IMPLEMENT, INFORM, MAINTAIN, NOTIFY, OBTAIN, PRESENT, PROTECT, PROVIDE, REQUEST, SHOW, STORE, TRANSMIT and USE}.
\item \textbf{Determining involved/affected parties or objects:} After an action is identified, it is necessary to determine the object(s) of the action. The output from the second step in the privacy requirements identification process can be either involved/affected parties or objects. The involved/affected parties can be any persons or stakeholders such as data subjects, data processors, supervisory authorities and third parties. The question used to identify the involved/affected parties is ``Who is involved/affected from the statement?''. However, the objects are things that are created, processed or done by the actions specified in the statements (e.g. consent, preferences, personal data and functions). These objects are identified by asking ``What has to be created/done from the identified action?''. If a statement refers to a data source or a data repository, it can be extracted as an object, while a data recipient can be extracted as the involved/affected party.
\item \textbf{Considering the target result(s):} The target results refer to a goal that a statement aims to achieve. They can be identified by asking \textit{``What should be achieved based on the action of that statement?''} For example, Article 13(1)(c) in GDPR states ``..., the controller shall, at the time when personal data are obtained, provide the data subject with the purposes of the processing for which the personal data are intended as well as the legal basis for the processing;''. The goal in this statement is asking to provide the data subject with the purposes of processing. Hence, the purposes of the processing is a target result that the action verb \textit{PROVIDE} aims to achieve.
\item \textbf{Structuring into a privacy requirement pattern:} The derived privacy requirement is coded in the format of action verb, followed by involved/affected parties or objects and target results.
\end{enumerate}
In our study, the statements in the regulations, standards and frameworks are written in natural language, and they contain goals that shall be complied or satisfied. We derived those privacy goals and then constructed them as software privacy requirements. Thus, this method meets our objective in terms of the inputs that we have and outputs that we would like to achieve. There are three essential components in a statement in the privacy regulations, standards and frameworks: what to do (i.e. an action), to whom/what they apply (i.e. an affected/involved party or an object) and things they aim to achieve (i.e. an object complement). Our requirements identification process captured all of those three parts. Several studies also constructed privacy requirements by including these three components from the statements in the data protection regulations and standard \cite{Meis} and privacy policies \cite{Antn2004}.
The following examples illustrate these steps. A statement in the GDPR states ``... the controller shall, at the time when personal data are obtained, provide the data subject the identity and the contact details of the controller and, where applicable, of the controller's representative''. From this statement, we identify \textit{`PROVIDE'} as the action that the controller shall act. We then consider what should be provided by the controller, and that was \textit{`the identity and the contact details of the controller or the controller's representative'}. We determine the object responding to \textit{`to whom the identity and contact details of the controller or the controller's representative should be provided'}, and that is the data subjects. All three components formulate a privacy requirement as \textit{\textbf{PROVIDE} the data subjects with the identity and contact details of a controller/controller's representative (R22)}.
Another example is more complex than the previous one. In the GDPR, removing personal data is recommended in several ways such as: the data has been unlawfully processed; or the data subjects would like to erase their personal data themselves; or the system must erase personal data when the data subjects object to the processing; or the system must erase personal data when the data subjects withdraw consent; or the system must erase personal data when it is not necessary for the specified purpose(s); or the system must erase personal data when the purpose(s) for the processing has expired. We thus need to formulate different privacy requirements as they affect the ways that the functions would be provided to users in a system. In this example, the derived requirements are: \textit{\textbf{ERASE} the personal data when it has been unlawfully processed (R7)}, \textit{\textbf{ALLOW} the data subjects to erase his/her personal data (R44)}, \textit{\textbf{ERASE} the personal data when the data subjects object to the processing (R46)}, \textit{\textbf{ERASE} the personal data when a consent is withdrawn (R47)}, \textit{\textbf{ERASE} the personal data when it is no longer necessary for the specified purpose(s) (R52)} and \textit{\textbf{ERASE} the personal data when the purpose(s) for the processing has expired (R53)}.
It is important to note that we have followed a systematic approach (GBRAM) to extract and refine requirements from narrative statements. For some statements, they are straightforward since privacy requirements can be directly derived from them. However, we have restructured and refined some privacy requirements to emphasise the functionalities that should be provided by software systems. For example, Article 7(1) in the GDPR states ``Where processing is based on consent, the controller shall be able to demonstrate that the data subject has consented to processing his or her personal data''. From this statement, we derive requirement \emph{R17 - SHOW the relevant stakeholders the consent given by the data subjects to process their personal data}.
The lawfulness of processing is one of the key principles for protecting privacy of data subjects. Based on the GDPR Article 6, the processing of personal data will be lawful if the data subjects give consent \emph{or} the processing is required for other legal conditions, such as the processing is necessary for the performance of contract, compliance with a legal obligation and protecting vital interests. R35 in the taxonomy addresses the requirement of obtaining consent from the data subjects for the processing. For other legal conditions, the data controllers must inform the data subjects if they need to process personal data under those conditions. Requirements R38 and R39 in the taxonomy cover these scenarios.
After lawfully obtaining personal data, the data controllers must provide the data subjects with mechanisms to execute their individual rights in the system. All the privacy requirements related to the rights of data subjects are covered in our taxonomy (i.e., the right to be informed (R12, R24, R26, R30, R31 and R50), the right of access (R1), the right to rectification (R45), the right to erasure (R44), the right to restriction of processing (R4), the right to data portability (R33), the right to object (R3), and the rights in relation to automated decision making and profiling (R21)).
Our taxonomy has a number of requirements (e.g., R35, R39 and R60) that can be triggered when the software is dealing with special category data/more sensitive data. These requirements related to GDPR Art. 9 which discusses the lawful processing of special categories of personal data. According to GDPR Art. 9, in addition to consent, there are other conditions for a lawful processing of special categories of personal data (e.g. necessary for protecting vital interests, substantial public interests and the purposes of preventive or occupational medicine). Requirement R35 covers the consent condition. R39 and R60 in our taxonomy address the remaining conditions specified in GDPR Art. 9. R39 requires the data controllers, if not obtaining the consent, must provide the data subjects the purpose(s) of the processing of special categories of personal data to the data subjects. The controllers also require to protect those personal data with appropriate measures (R60).
The proposed taxonomy would benefit software developers, data controllers, data processors and DPOs. However, there are several reasons we did not include the articles related to the DPOs in our study. Firstly, the articles related to DPOs in GDPR mainly focus on their duties, tasks and responsibilities (i.e. Articles 37 - 39). These include how the DPOs govern, supervise and cooperate with the data controllers, the data processors, the supervisory authority and other stakeholders regarding the processing of personal data. In addition, the DPOs must ensure that the processing of personal data carried out in organisations complies with data protection regulations or other relevant laws. Thus, the DPOs requirements are related to governance aspect rather than software requirements aspect.
Secondly, the main role that directly determines the activities related to the processing of personal data is the data controllers. Any concerns about the processing of personal data will be notified to the DPOs by the data controllers, and sometimes the data processors. This makes the data controllers the key stakeholder in governing how personal data is processed and how the processing activities should be done in software development level. Finally, we have not found any DPOs requirements reported in issue reports in our study. This finding implies that the DPOs requirements were not reflected as software requirements in this context.
Our taxonomy covers three key privacy requirements related to international data transfer as follows: i) the data subjects must be informed about the transfer of their personal data to a third country or an international organisation (R9); ii) the personal data must be appropriately protected (R60) and iii) the transfer of personal data must comply with local requirements (R65).
We note that an article, a section or a statement can lead to the identification of more than one requirements. For example, we derived 17 privacy requirements (i.e. R1-R4, R6, R9, R20-22, R27, R29, R34, R37, R39, R44-R45 and R55) from the Article 13 in GDPR\footnote{See the file \emph{Privacy-requirements-references} in the replication package for more details \cite{reppkg-pridp}.}. The following example demonstrates two privacy requirements that were derived from a statement. A statement in Section 23-6 in Thailand PDPA states ``In collecting the Personal Data, the Data Controller shall inform the data subject, ... (5) information, address and the contact channel details of the Data Controller, where applicable, of the Data Controller's representative or data protection officer;'', we derived two requirements from this statement which are \textit{R20 PROVIDE the data subjects with the contact details of a data protection officer (DPO)} and \textit{R22 PROVIDE the data subjects with the identity and contact details of a controller/controller's representative}.
\textbf{Reliability assessment}: Three human coders, who are the co-authors of the paper, have independently followed the above process to identify privacy requirements from the GDPR and ISO/IEC 29100 privacy framework. All three coders had substantial software engineering background and at least 1 year of experience with data protection regulations and policies. The first author prepared the materials and detailed instructions\footnote{These are included in the replication package \cite{reppkg-pridp}.} for the process. The instructions were provided to all the coders before they started the identification process. A 1-hour training session was also held to explain the process, clarify ambiguities and define expected outputs. The coders also went through a few examples together to fine tune the understanding.
All the coders were provided with a form to record their results of each step. The form was pre-filled with 149 statements extracted from the GDPR and 63 statements from the ISO/IEC 29100 privacy framework. If a coder considers a statement as a privacy requirement, they need to identify the relevant components and structure it following the privacy requirement patterns above. Otherwise, they leave it blank. Initially, the three coders each respectively identified 100, 95 and 97 requirements from GDPR, and 36, 36 and 37 requirements from ISO/IEC 29100.
Since the requirements identified by the coders could be different, we used the Kappa statistic (also known as Kappa coefficient) to measure the inter-rater reliability between the coders \cite{Viera2005}. The Kappa statistic ranges from -1 to 1, where 1 is perfect agreement and -1 is strong disagreement \cite{Viera2005}. There are several types of the Kappa statistics which suit different study settings \cite{Hallgren}. For this study, the Fleiss' Kappa was used as we had three coders coding the same datasets \cite{Fleiss1971}. The Kappa values were 0.8025 for GDPR and 0.7182 for ISO/IEC 29100, suggesting a substantial agreement level amongst all the coders \cite{Landis1977}. All the coders agreed that there were 43 and 20 statements from the GDPR and ISO/IEC 29100 respectively that are not privacy requirements. There were 20 GDPR statements (and 13 for ISO/IEC 29100) that the coders did not agree upon. Hence, a meeting session was held between the coders to discuss and resolve disagreements.
The statements in Thailand PDPA and APEC privacy framework were identified by one of the coders using the same methodology performed with the GDPR and ISO/IEC 29100. All the shortlisted statements in Thailand PDPA and APEC privacy framework were derived in this step. This brought the total number of privacy requirements obtained in this step to 249 (116 from GDPR, 33 from ISO/IEC 29100 and 55 from Thailand PDPA and 45 APEC privacy framework).
We used only one coder for the Thailand PDPA and APEC privacy framework because the statements are less complex than the GDPR and ISO/IEC 29100. The provisions in Thailand PDPA were largely written based on the GDPR \cite{Kateifides}, thus we did not require additional process or special needs as they share many commonalities. Similarly, many principles in the APEC privacy framework are similar to ISO/IEC 29100. We therefore decided that one coder would be sufficient. The coder, who was responsible for the Thailand PDPA and APEC privacy framework, was the main coordinator and also participated in the privacy requirements identification process for GDPR and ISO/IEC 29100.
\subsubsection{Privacy requirements refinement} \label{refinement}
Requirements extracted either from the same or different sources (e.g. GDPR and ISO/IEC 29100) can be similar, redundant or inconsistent. In this step, we identify those similar and duplicate requirements, and merge them into one single requirement. In case that the requirements are inconsistent, we perform further investigation and report for notice.
To identify and merge similar requirements, we first place those similar requirements into the same group. These requirements tend to achieve the same goal and have the same involved/affected parties or objects. We then determine the action and target result for the final merged requirement based on the following rules:
\begin{enumerate}
\item If the action verbs in the requirements are the same, we retain that action for the final merged requirement.
\item If the actions are different, we consider the action verb based on the following:
\begin{itemize}
\item Use \emph{ALLOW} if a requirement relates to data subject's ability to invoke his/her rights.
\item Use \emph{PROVIDE} if a requirement aims to give information to stakeholders.
\item Use \emph{OBTAIN} if a requirement aims to get a consent or permission from stakeholders.
\item Use \emph{PRESENT} if a requirement aims to display options or choices to stakeholders. This action verb requires responses from the stakeholders (e.g. displaying toggles or radio buttons for users to select).
\item Use \emph{SHOW} if a requirement aims to show information to stakeholders. This action verb does not require any responses from the stakeholders.
\item Use \emph{NOTIFY} if a requirement aims to alert stakeholders.
\item Use \emph{IMPLEMENT} if a requirement aims to build a mechanism to support an activity in a system.
\item Use \emph{ERASE} if a requirement aims to erase data in software systems.
\end{itemize}
\item If the target results in the requirements are the same, we retain that target result for the final requirement.
\item If the target results are different, we combine them together. In case they have redundant or synonymous words, we select one word from the words in the list.
\end{enumerate}
We finally put together the action, involved/affected parties or objects and target results identified in the above steps to construct the final requirement.
We have carefully investigated the terms and definitions used in GDPR and ISO/IEC 29100 and found that they mostly refer to similar or same things. For example, ISO/IEC 29100 defines personally identifiable information or PII as ``any information that (a) can be used to identify the PII principal to whom such information relates, or (b) is or might be directly linked to a principal''. Personal data in GDPR is defined as ``any information relating to an identified or identifiable natural person ('data subject')''. As can be seen, PII in ISO/IEC 29100 and personal data in GDPR in fact refer to the same thing, i.e. any information that can be used to identify or is linkable to a natural person.
Similarly, key terms GDPR and ISO/IEC 29100 are sometimes worded differently, however their definitions are similar. For example, \textit{`processing'} means ``any operation or set of operations which is performed on personal data or set of personal data, ...'' in GDPR and \textit{`processing of PII'} is defined as ``operation or set of operations performed upon personally identifiable information (PII)''. Another example is data controller in GDPR and PII controller in ISO/IEC 29100. Both terms refer to person that determines the purposes and means of the processing of personal data/PII. Other examples include personal data with PII, data subject with PII principal, data processor with PII processor, consent and third party. Thus, in the merging step, we use the terms from GDPR in representing our requirements in this taxonomy to avoid ambiguities.
The additional example is a PII principal in ISO/IEC 29100 and a data subject in GDPR. A PII principal in ISO/IEC 29100 is defined as ``a natural person to whom the personally identifiable information relates''. A data subject in GDPR is defined as ``an identifiable natural person is one who can be identified , directly and indirectly, in particular by reference to an identifier such as a name, ...''. As can be seen, a PII principal in ISO/IEC 29100 and a data subject in GDPR in fact refer to the same thing, i.e. a natural person who can be identified with identifiable information such as name.
We have found that software requirements derived from GDPR, ISO/IEC 29100, Thailand PDPA and APEC privacy framework are in fact at the same level of abstraction\footnote{see the file \emph{Privacy-requirements-abstraction} in the replication package for additional sample requirements \cite{reppkg-pridp}}. \citeauthor{Meis} has also confirmed that GDPR and ISO/IEC 29100 are at the same level of abstraction \cite{Meis}. The following example demonstrates this case. We derived \textit{\textbf{PROVIDE} the data subject the recipients or categories of recipients of the personal data} from Article 13(1)(e) in GDPR, \textit{\textbf{PROVIDE} the types of persons whom the PII can be transferred} from openness, transparency and notice principle in ISO/IEC 29100, \textit{\textbf{INFORM} the data subject the categories of Persons or entities to whom the collected Personal Data may be disclosed} from Section 23 in Thailand PDPA and \textit{\textbf{PROVIDE} the types of persons or organisations to whom personal information might be disclosed} from Point 21 in APEC framework. These four requirements demonstrate that they are at the same level of abstraction and aim to achieve the same goal. They can be merged in the requirements refinement process later.
\begin{comment}
\begin{landscape}
\begin{table}
\centering
\caption{Some sample requirements derived from GDPR, ISO/IEC 29100, Thailand PDPA and APEC privacy framework demonstrate that they are at the same level of abstraction.}
\label{tab:GDPR-ISO-req-mapping}
\begin{tabular}{p{4cm} p{1.25cm} p{4cm} p{1.25cm} p{4cm} p{1.25cm} p{4cm} p{1.25cm}}
\toprule
\textbf{Requirements derived from GDPR statements} & \textbf{GDPR reference} & \textbf{Requirements derived from ISO/IEC 29100} & \textbf{ISO/IEC 29100 reference} &
\textbf{Requirements derived from Thailand PDPA} & \textbf{Thailand PDPA reference} &
\textbf{Requirements derived from APEC framework} & \textbf{APEC framework reference} \\
\midrule
PROVIDE the existence of the right to withdraw consent & 13(2)(c)
& ALLOW a PII principal to withdraw consent & 5.2
& ALLOW the data subject to withdraw his or her consent & 19-5
& None & None \\
PROVIDE the data subject the recipients or categories of recipients of the personal data & 13(1)(e)
& PROVIDE the types of persons whom the PII can be transferred & 5.8
& INFORM the data subject the categories of Persons or entities to whom the collected Personal Data may be disclosed & 23-5
& PROVIDE the types of persons or organisations to whom personal information might be disclosed & 21-4\\
PROVIDE the data subject the purposes of the processing & 13(1)(c)
& PROVIDE the PII principal the purpose of the processing of PII & 5.8
& INFORM the data subject the purpose of the collection, use, or disclosure of the Personal Data & 19-3
& PROVIDE the individuals with the purpose their information is to be used & (21-23)-1 \\
PROVIDE the data subject the categories of personal data concerned & 15(1)(b)
& PROVIDE the PII principal the specified PII required for the specified purpose & 5.8
& INFORM the data subject the Personal Data to be collected & 23-4
& PROVIDE the individuals with what personal information is collected & (21-23)-1 \\
IMPLEMENT appropriate technical and organisational measures to protect personal data & 32(1)(a)
& PROTECT PII with appropriate controls & 5.11
& PROVIDE appropriate security measures for preventing the Personal Data & 37-2
& IMPLEMENT organizational controls to prevent from the wrongful collection or misuse of personal information & 20-1, 28-1, 28-2\\ \bottomrule
\end{tabular}
\end{table}
\end{landscape}
\end{comment}
The following example demonstrates the requirements merging step. A statement in ISO/IEC 29100, ``... allow a PII principal to withdraw consent easily and free of charge ...'', derives a requirement \textit{\textbf{ALLOW} a PII principal to withdraw consent}. A statement in GDPR, ``... the controller shall ... provide the data subjects with ... the existence of the right to withdraw consent at any time ...'' gives a requirement \textit{\textbf{PROVIDE} the existence of the right to withdraw consent}. A statement in Thailand PDPA, ``The data subject may withdraw his or her consent at any time.'', gives a requirement \textit{\textbf{ALLOW} the data subject to withdraw his or her consent.} The goal of these three requirements is to let the PII principal/data subject withdraw consent, and the affected parties are PII principal and data subject. It is noted that we use the terms from GDPR for roles in our requirements (i.e. data subject, data controller, data processor and third parties). We therefore list them as similar requirements. The requirements have different actions (i.e. ALLOW and PROVIDE), we then use \emph{ALLOW} as the final action as these requirements are about data user's ability to withdraw consent. We acquire \textit{withdraw consent} as a common target result. Finally, we merge these three requirements into a single requirement: \textit{\textbf{ALLOW} the data subjects to withdraw consent (R6)}.
The duplicate requirements are the requirements that have the exact actions, involved/affected parties and target results. We represent these requirements as one requirement in the taxonomy. For example, we identify two exact requirements in the identification process, \emph{PROVIDE the data subject the categories of personal data concerned} in GDPR articles 14(b) and 15(b). We retain one requirement (i.e. R42) in the taxonomy.
Requirements are inconsistent when they appear to contradict each other in performing the same actions. The following example demonstrates the consistency between the requirements in ISO/IEC 29100 and GDPR. We identify the requirements from ISO/IEC 29100 and GDPR as ``COLLECT only necessary PII for specific purposes'' and ``COLLECT the personal data as necessary for specific purposes'', respectively. Both requirements yield that the personal data must be collected as necessary for specific purposes. They are presented in both GDPR and ISO/IEC 29100. The requirements are therefore consistent, and merged as R41 COLLECT the personal data as necessary for specific purposes.
The following example is made up for the purpose of explanation to demonstrate the inconsistency between requirements. Assuming a statement states ``Any personal data can be freely collected without specifying a specific purpose for collection'', we have derived the requirement as ``COLLECT any personal data without a specific purpose''. This requirement would contradict with requirement R41 discussed above since the former does not require a specific purpose provided, while the latter does.
We merged in total 178 similar and duplicate requirements. We did not find any inconsistent requirements. For requirements traceability, we have provided a full list of the privacy requirements with their references to the GDPR articles, ISO/IEC 29100 principles, Thailand PDPA sections and APEC framework points in the replication package \cite{reppkg-pridp}. This step resulted in a final taxonomy of 71 privacy requirements in 7 goal categories which we will discuss in detail in the next subsection.
\subsubsection{Privacy requirements classification} \label{requirements-classification}
In this step, we aim to group the privacy requirements into categories based on their goals. We applied a bottom-up approach to classify the requirements into privacy goal categories. This approach ensures that the generated categories cover and address all the requirements. The approach also allows the categories in the taxonomy to be updated when there are new privacy requirements identified in the future. For example, newly identified privacy requirements can be added to existing categories or form new categories.
The bottom-up approach consists of two steps. We first considered the privacy requirements based on their actions, objects and target results. For example, the privacy requirements with the action verb \emph{ALLOW}, the object \emph{data subjects} and the target results that are related to individual rights (e.g. access, rectify and erase) were grouped together (i.e. user participation). Similarly, the privacy requirements with the action verb \emph{PROVIDE}, the object \emph{data subjects} and the target results that are related to information for stakeholders were gathered into the same group. We kept applying this strategy to group the rest of privacy requirements. We then ended up with fifteen categories in the first step.
Next, we grouped the privacy requirements that have at least two common components either the actions, objects or target results. For example, we gathered the privacy requirements that have the same action verb \emph{PROVIDE} and the same target results that are related to information for stakeholders, but there are two different objects - data subjects and other parties that are not data subjects. We then created subcategories for each object (i.e. notice - data subjects and notice- relevant parties). All of these requirements were grouped under the notice category. We repeated this step with the rest of privacy requirements. Finally, the taxonomy consists of seven categories (some of which have sub-categories): user participation, notice, user desirability, data processing, breach, complaint/request and security. Each requirement can belong to multiple categories.
The descriptions of privacy goal categories below provide an overview and briefly illustrate the privacy requirements concerned in those categories. We note that the descriptions do not explain the full list of privacy requirements in each category.
\begin{enumerate}[leftmargin=*]
\item User participation \\
This privacy goal consists of a set of requirements for the controllers to provide the data subjects with the functionalities to invoke their individual rights relating to their personal data. The data subjects must be able to access, review, rectify, erase and verify the validity and completeness of their personal data. The controllers shall provide the data subjects their personal data when they request to obtain and reuse their personal data for their own purposes for other services. The controllers shall allow the data subjects to object to and restrict the processing of their personal data. The data subjects must be able to withdraw consent or lodge a complaint to a supervisory authority.
\item Notice \\
This privacy goal consists of two sub-categories: data subjects and relevant parties. This category has a set of requirements for the data subjects to be informed and/or notified of relevant information and individual rights related to the processing of personal data. The information includes privacy policies, procedures, practices and logic of the processing of personal data. The data subjects must be informed of the purposes of collection and processing of their personal data. The controllers must provide the contact details of responsible persons who control the processing. The data subjects must be notified when they are likely to be in risk from personal data exposures. In addition, the controllers must communicate any relevant information relating to personal data processing to intended stakeholders.
\item User desirability \\
This privacy goal consists of three sub-categories: consent, choice and preferences. This category asserts that the controllers must show a consent form to and obtain consent from the data subjects. The controllers must provide the data subjects with an option to provide their data, allow the processing or subject to a decision based on automated processing. The processing should be implemented based on user preferences expressed in their consent.
\item Data processing \\
The privacy goal addresses the processing of personal data handling from the controllers' side. The processes, which are the sub-categories in this category, include collection, use, storage, erasure, transfer and record. The controllers must handle personal data as necessary for specific purposes. The processors who process personal data must also follow the instructions from the controllers. The personal data processing activities must be recorded.
\item Breach \\
This goal category ensures that the controllers must be prepared to handle personal data breach. The relevant information about the data breach must be recorded and communicated to data subjects and relevant stakeholders.
\item Complaint/Request \\
This goal category addresses complaint and request management. User complaints and requests about their individual rights and the processing of their personal data must be processed. The actions regarding the complaints and requests must be informed. The controllers must request for relevant information to confirm the identity of data subjects when the request has been made.
\item Security \\
This privacy goal ensures that personal data and its processing are safeguarded with confidentiality, integrity and availability. The personal data must be protected with appropriate controls and security mechanisms. The security mechanisms must comply with security and data protection standards. The personal data must be accessed and used by the authorised stakeholders. The controllers must ensure the correctness and completeness of personal data.
\end{enumerate}
When software engineers map an issue report to relevant privacy requirements, they can consider the features/concerns that the issue report mentions based on categories, then explore relevant requirements under the selected categories and sub-categories. The software engineers will know what has been missed in this issue, so they can resolve the issue accordingly. For example, an issue report reports on not allowing users to modify their personal data, the software engineer can directly observe the requirements in the user participation category as it relates to the data subject's right. He/she then identifies the relevant requirement of this issue report, which in this case is requirement R45. Similarly, if the issue mentions about consent, the engineer can access the requirements related to consent in the consent sub-category, select the relevant one(s) and resolve the issue based on the identified privacy requirements.
This proposed methodology is able to address the scenario where two requirements aiming to achieve the same goal have different actors. We can further refine relevant requirements into a parent requirement and child requirements with specific logical operators (i.e. AND and OR). For example, assume that the controller is required to obtain consent for the processing of personal data. The data controller is responsible for this task in GDPR, however this could be managed by 3rd party authority in other regulations (e.g. CCPA \cite{StateofCaliforniaDepartmentofJustice2018}). We can then refine these requirements into a parent requirement as \textit{obtain consent for the processing of personal data}. The child requirements could be expressed as \textit{the controllers shall obtain consent for the processing of personal data} and \textit{the 3rd party authority shall obtain consent for the processing of personal data}. The logical operator in this scenario is OR as software development teams can choose between one of the child requirements to implement to satisfy the parent requirement.
If a regulation states that it requires both the controllers and the 3rd party authority to obtain consent for the processing of personal data, the logical operator in this scenario is AND, and the software development teams must implement both child requirements in their system. As a data controller is the main actor in the regulations and frameworks we analysed in this study, the above scenario did not occur in our study. Thus, we did not refer to a specific requirement in our taxonomy. \\
\begin{comment}
\textbf{Reliability assessment}: The same group of coders took part in this step. The instructions for privacy requirements classification were provided to all the coders before they started the classification process. They were also provided a form containing a list of privacy requirements as agreed upon from the previous step and choices of privacy goal categories to be selected. All the coders independently classified each requirement into a privacy goal category.
After all the coders finished the classification, the reliability assessment was conducted to evaluate the inter-rater agreement amongst the coders. We also used Fleiss' Kappa as a measure. The Kappa value was 0.7132, suggesting a substantial agreement level \cite{Landis1977}. There were 39 requirements that the coders did not classify into the same category. A meeting session was hold between the coders to discuss and resolve disagreements. This step resulted in a set of 149 privacy requirements classified into seven privacy goal categories.
\end{comment}
\subsection{Privacy Requirements Taxonomy} \label{sec3.2}
Our taxonomy consists of a comprehensive set of 71 privacy requirements classified into 7 categories. The full version of the taxonomy can be found in Appendix. We now highlight some of the important requirements in each category (see Table \ref{tab:table1}). We note that there are typically four types of roles involved in a privacy requirement: (i) data subjects who provide their personal data for processing, give consent and determine their privacy preferences; (ii) data controllers who determine what data to be collected and the purpose of personal data collection and processing; (iii) data processors who process the personal data corresponding to the specified purpose and (iv) third parties who in case receive personal data from the controllers or processors.
\begin{table}[]
\caption{Selected privacy requirements that are referred in the paper. The full taxonomy is available in Appendix.}
\label{tab:table1}
\begin{tabular}{p{8.5cm}}
\toprule
\textbf{Privacy requirements}\\
\midrule
\textbf{Category 1: User participation} \\
R1 ALLOW the data subjects to access and review their personal data \\
R6 ALLOW the data subjects to withdraw consent \\
R34 ALLOW the data subjects to obtain and reuse their personal data for their own purposes across different services \\
R44 ALLOW the data subjects to erase their personal data \\
R45 ALLOW the data subjects to rectify their personal data \\
\vspace{1mm}
\textbf{Category 2: Notice} \\
R12 INFORM the data subjects the reason(s) for not taking action on their request and the possibility of lodging a complaint \\
R15 NOTIFY the data subjects the data breach which is likely to result in high risk \\
R17 SHOW the relevant stakeholders the consent given by the data subjects to process their personal data \\
R19 PROVIDE the data subjects an option to choose whether or not to provide their personal data \\
R22 PROVIDE the data subjects with the identity and contact details of a controller/controller's representative \\
R26 PROVIDE the data subjects the information relating to the policies, procedures, practices and logic of the processing of personal data \\
R27 PROVIDE the data subjects the recipients/categories of recipients of their personal data \\
R30 PROVIDE the data subjects the information relating to the processing of personal data with standardised icons \\
R38 PROVIDE the data subjects the purpose(s) of the collection of personal data \\
R39 PROVIDE the data subjects the purpose(s) of the processing of personal data \\
R42 PROVIDE the data subjects the categories of personal data concerned \\
R55 PROVIDE the data subjects the period/criteria used to store their data \\
R66 NOTIFY a supervisory authority the data breach \\
\vspace{1mm}
\textbf{Category 3: User desirability} \\
R6 ALLOW the data subjects to withdraw consent \\
R8 IMPLEMENT the data subject's preferences as expressed in his/her consent \\
R19 PROVIDE the data subjects an option to choose whether or not to provide their personal data \\
R35 OBTAIN the opt-in consent for the processing of personal data for specific purposes \\
R36 PRESENT the data subjects an option whether or not to allow the processing of personal data \\
R47 ERASE the personal data when a consent is withdrawn \\
\vspace{1mm}
\textbf{Category 4: Data processing} \\
R7 ERASE the personal data when it has been unlawfully processed \\
R13 MAINTAIN a record of personal data processing activities \\
R40 USE the personal data as necessary for specific purposes specified by the controller \\
R41 COLLECT the personal data as necessary for specific purposes \\
R43 STORE the personal data as necessary for specific purposes \\
R46 ERASE the personal data when the data subjects object to the processing \\
R47 ERASE the personal data when a consent is withdrawn \\
R52 ERASE the personal data when it is no longer necessary for the specified purpose(s) \\
R53 ERASE the personal data when the purpose for the processing has expired \\
\vspace{1mm}
\textbf{Category 5: Breach} \\
R15 NOTIFY the data subjects the data breach which is likely to result in high risk \\
R66 NOTIFY a supervisory authority the data breach \\
R67 NOTIFY relevant privacy stakeholders about a data breach \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[]
\label{tab:table1-2}
\begin{tabular}{p{8.5cm}}
\toprule
\textbf{Privacy requirements (Continued)}\\
\midrule
\textbf{Category 6: Complaint/Request} \\
R12 INFORM the data subjects the reason(s) for not taking action on their request and the possibility of lodging a complaint \\
R31 REQUEST the data subjects the additional information necessary to confirm their identity when making a request relating to the processing of personal data \\
\vspace{1mm}
\textbf{Category 7: Security} \\
R56 ALLOW the authorised stakeholders to access personal data as instructed by a controller \\
R60 IMPLEMENT appropriate technical and organisational measures to protect personal data \\
R63 PROTECT the personal data from unauthorised access and processing \\
R65 IMPLEMENT a function to comply with local requirements and cross-border transfers \\
\bottomrule
\end{tabular}
\end{table}
\subsubsection{User participation}
All the requirements in this category specify the functionalities provided for data subjects to execute their individual rights in managing their personal data. The data subjects must be able to access and review, erase and rectify their personal data (e.g. R1, R44 and R45). The systems must allow the data subjects to withdraw consent (R6). The systems must also provide the data subjects their personal data when they would like to obtain and reuse their personal data for their own purposes across different services (R34).
\subsubsection{Notice}
It is the largest group consisting of 32 privacy requirements in the taxonomy. Most of the requirements in this category are concerned with the transparency of personal data processing (e.g. R17, R22 and R42). They aim to ensure that a system shall provide information related to the processing of personal data (e.g. what personal data is required, who is responsible for their data and results from requests) to data subjects. Personal data shall not be misused, and the data subjects have the right to know the purpose of collection and processing (R38 and R39). The data subjects should be provided the duration their personal data will be stored (R55). Additional information must be provided to the data subjects if the collected personal data are required for other purposes (R37). General privacy-related information should be presented in a clear and simple, accessible language without technical terms as required in requirement R26. Any updates of personal data processing must be informed to the recipients (i.e. processors or third parties) of those personal data (R50). Some of the requirements in this category also belong to other categories such as the notices about data subjects' rights related to Category 1: user participation and the notices about data breaches related to Category 5: breach.
\subsubsection{User desirability}
The requirements in this category focus on ensuring that the processing of personal data is performed according to data subjects' consent and preferences. A number of requirements focus on the controllers being given authorities to process personal data (e.g. R8). It is also necessary to obtain consent for the processing based on those purposes (R35). The data subjects have options to allow the processing of their personal data for a certain specific purpose (R36).
According to GDPR Art. 6, in addition to consent, there are other ways for a lawful processing if it is necessary for the performance of a contract, compliance with a legal obligation, protect vital interests, the performance of task carried out in the public interest and the purposes of the legitimate interests. While R35 covers the consent aspects, we have also R39 in our taxonomy addresses the other aspects. R39 requires that the controllers, if not obtaining the consent, must provide data subjects the purpose(s) of the processing of personal data, including those listed in GDPR Art. 6.
\subsubsection{Data processing}
There are two requirements (R41 and R43) in this category, which change the traditional way of collecting and processing data. In the past, the data might be collected from data subjects as much as possible and kept in the system. They now require that the controllers are expected to collect and store only personal data that is required in the processing for the specific purpose(s). A set of requirements involves data erasure in systems. Requirement R53 addresses the case of removing personal data when the purpose for processing has expired. When the data subjects would like to have their personal data erased, the system shall provide this processing lawfully (R46 and R47). When the processing is complete and the personal data is no longer needed, the personal data should be removed from the system unless they are required by law/regulations (e.g. R51 and R52). In case that the personal data is unlawfully processed, the data must be removed from the systems (R7). In addition, personal data must be used only for the specified purpose(s) (R40). When requested by the data subjects, the controllers must transmit their personal data to another controller (R33). The data subject must be informed when their personal data needs to be transferred to a third country or an international organisation (R9). The controllers shall document the categories of personal data collected as it is important to know what personal data are stored in the systems (R70).
\subsubsection{Breach}
This goal category focuses on providing and notifying important information related to personal data breaches to data subjects, relevant stakeholders and a supervisory authority (e.g. R15 and R71). Thus, it is important to implement a functionality that satisfies this compliance in the systems (e.g. R66). Requirement R67 imposes good practices of informing the related parties about the breaches. The controllers must be informed by processors about the breaches as well (R68). The controllers shall document the details of data breaches for verifying their compliance (R69).
\subsubsection{Complaint/Request}
This privacy goal concerns complaint and request made by both data subjects and controllers. If the controllers refuse to take actions on the data subjects' requests about their individual rights, they have to provide a reason to the data subjects (R12). The data subjects must be able to lodge their complaints with a supervisory authority (R2). The controllers shall process personal data as requested (e.g. transmit personal data to another controller) (R33). The controllers should request additional information to confirm data subjects' identity when requests have been made (R31).
\subsubsection{Security}
There are thirteen requirements in our taxonomy covering the security practices in maintaining integrity, confidentiality and availability. The systems must allow only authorised people to access or process personal data (R56). The personal data should be protected with proper mechanisms (e.g. R60 and R63). The systems must restore the availability and access to personal data after incidents (R62). The interactions in the systems should neither identify nor observe the behaviour of the data subjects as well as reduce the linkability of the personal data collected (R64). Apart from the fundamental practices that the personal data should be protected, this is beneficial when the personal data is exposed. The data protection approaches such as anonymisation and pseudonymisation can help reduce the impact of privacy breaches. Most importantly, a set of requirements require the systems to implement mechanisms to ensure security and privacy compliance (e.g. R57, R58, R61 and R66). In addition, the implementation of mechanisms to assess the accuracy and quality of procedures should be considered (R49).
In case that the personal data are processed across organisations/countries, the controllers must ensure local requirements and cross-border transfers (R65). Cross-border transfers are challenging for both controllers and processors. In cross-border transfer settings, it is required that the requirements at the destination should be equivalent to the ones at the source. The processors outside EU are sometimes not aware of those scenarios since they may not normally process personal data of EU citizens and residents. Therefore, the controllers are responsible for verifying requirements compliance before transferring personal data.
\subsection{The treatment of privacy and non-privacy issues}
We investigate if privacy issues were treated differently from non-privacy issues in Chrome and Moodle. We focus on observing two kinds of treatments: the time it took to resolve an issue and the number of comments associated with the issue. The former reflects how fast an issue was resolved while the latter indicates the attention and engagement of the project team to the issue. We randomly sampled the dataset we built earlier using a 95\% confidence level with a confidence interval of 5\footnote{https://www.surveysystem.com/sscalc.htm} to obtain 269 privacy issues from Chrome and 213 from Moodle. Applying the same sampling scheme, we randomly selected 382 non-privacy issues from Chrome and 380 from Moodle - these issues were not tagged as privacy in the ``component'' field. Note that the resolution time is calculated from the number of days between reported date and the date when the issue was flagged as being resolved.
\begin{table}
\centering
\caption{Results of the Wilcoxon rank-sum test: non-privacy vs. privacy issues}
\label{tab:ranksum}
\resizebox{8.5cm}{!}{
\begin{tabular}{l l l l l}
\toprule
\textbf{Project} & \textbf{Attribute} & \textbf{One-sided tail} & \textbf{p-value} & \textbf{Effect size}\\
\midrule
Google Chrome & Resolution time & Less & $<$0.001 & 0.578 \\
Google Chrome & \#Comments & Less & $<$0.001 & 0.691 \\
Moodle & Resolution time & Greater & $<$0.001 & 0.609 \\
Moodle & \#Comments & Greater & $<$0.001 & 0.604\\
\bottomrule
\end{tabular}%
}
\end{table}
We employ the Wilcoxon rank-sum test (also known as Mann-Whitney U test), a non-parametric hypothesis test which compares the difference between two independent observations \cite{Wild1997}. We performed two tests between privacy and non-privacy samples, one for the resolution time and the other for the number of comments. The results (see Table \ref{tab:ranksum}) show that the resolution time and the number of comments are statistically significantly (\textit{p-value $\leq$ 0.001}) different between privacy and non-privacy issues in both Chrome and Moodle with effect size greater than 0.5 in all cases.
We also compare the median rank of the two samples using one-tailed test. Our results show that privacy issues were resolved more quickly and attracted less comments than non-privacy issues in Moodle (see Table \ref{tab:ranksum}). On the other hand, it took longer to resolve privacy issues than non-privacy issues in Chrome. Also, privacy issues in Chrome tend to attract more discussion than non-privacy issues.
We observed that the contributors in Chrome were confident and had more experience in resolving non-privacy issues. Hence, these issues attracted less discussion and was resolved more quickly. By contrast, privacy issues in Chrome attracted more discussions since the contributors were uncertain about the issues and their affected components in the system. We observed five examples that the contributors commented in those privacy issues as follows: (i) the contributors did not know what the affected components reported in the issues do (e.g. issue 345741); (ii) the contributors could not identify the causes of issues; (iii) the contributors required time and effort to come up with potential solutions; (iv) the contributors needed to assess the difficulties of the issues and their resolutions; and (v) the contributors did not know whom to assign the work. These reasons also led to longer time to resolve the privacy issues in Chrome. In addition, the Chrome project does not have a well-defined process that specifically handles privacy. Hence, the contributors need to ensure that fixing privacy issues will not create another problem in different components. Thus, resolving privacy issues attracted a lot of discussions, leading to longer resolution time.
On the other hand, privacy issues in Moodle were resolved more quickly and attracted less comments than non-privacy issues. We observe that the privacy issues were well reported and clearly explained in Moodle. Moodle contributors were familiar with privacy-related functionalities and relevant system components. In addition, Moodle has a clearly defined infrastructure to handle privacy and privacy compliance in the system (e.g. privacy API \cite{Moodle2019} and GDPR for plugin developers \cite{Nicols2018}). This infrastructure includes a number of components that support privacy-related functionalities and several key individual rights in GDPR (e.g. accessing to personal data and requesting for deletion). When there is a privacy-related bug or new feature request, the contributors can consult the privacy API documentation and identify the components that they must fix or implement. Hence, the privacy issues took less time and attracted less comments in Moodle.
\begin{comment}
Note:
p-value for fix time = 0.000058, comment = 0.0001
I think the interpretation from here is that if we set the null hypothesis (H0) as 'the fixing time of privacy and non-privacy issues are similar'. And H1 as 'the fixing time of privacy and non-privacy issues are different'. Then, the result from p-value (which is <0.05) rejects H0. Hence, the fixing time between these two groups are statistically significant difference, agree?
There are two values of tail: greater and less.
Now, my parameters fed into the function is non-privacy (as x) and privacy (as y) values. Greater means the median of x is higher than y. On the other hand, less means the median of x is less than y. The p-value of this test is also reported. All of our tests are statistically significant based on the p-value reported in each test.
The result of pq ranksums test - moodle - fixing time - one-sided:
U-val tail p-val RBC CLES
MWU 27793.5 greater 0.000029 -0.22522 0.604157
The result of pq ranksums test - chrome - fixing time - one-sided:
U-val tail p-val RBC CLES
MWU 42663.0 less 0.000112 0.169641 0.578077
The result of pq ranksums test - moodle - comments - one-sided:
U-val tail p-val RBC CLES
MWU 27624.5 greater 0.00005 -0.21777 0.591086
The result of pq ranksums test - chrome - comments - one-sided:
U-val tail p-val RBC CLES
MWU 29894.0 less 4.294377e-20 0.418167 0.690915
\end{comment}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,104 |
Hydroptila usambarensis är en nattsländeart som beskrevs av Wells och Andersen 1995. Hydroptila usambarensis ingår i släktet Hydroptila och familjen smånattsländor. Inga underarter finns listade i Catalogue of Life.
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"redpajama_set_name": "RedPajamaWikipedia"
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Daumeray – miejscowość i dawna gmina we Francji, w regionie Kraj Loary, w departamencie Maine i Loara. W 2013 roku populacja gminy wynosiła 1564 mieszkańców.
W dniu 1 stycznia 2017 roku z połączenia dwóch ówczesnych gmin – Daumeray oraz Morannes-sur-Sarthe – utworzono nową gminę Morannes-sur-Sarthe-Daumeray. Siedzibą gminy została miejscowość Morannes.
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"redpajama_set_name": "RedPajamaWikipedia"
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The National Strategy Conference held in Winnipeg February 2 – 5 brought together representatives from peace and disarmament, student, youth, women's, native, labour and ethnic organisations from across Canada. Despite differences of opinion on an assortment of topics, a sense of enthusiastic unity prevailed at the concluding plenary session on Sunday afternoon.
But what about those women and men who will be – in a very real sense – responsible for organising that "good show" in their local communities? What are the views of those grassroots representatives, some of whom were becoming involved with the Peace Petition Caravan Campaign (PPCC) for the very first time? To find out what their perspectives were, I interviewed – at random – a number of delegates. My questions were general in nature and dealt with the concerns of the peace movement as a whole, as well as with the PPCC. The comments made to me were always informed and meaningful, sometimes critical but never harsh or cynical.
David Delaunay, representing Project Ploughshares in Sudbury, compared the PPCC to "a baby being born." All the initial conflicts, such as those related to the freeze issue, were, according to Delaunay, "part of the process of growth." He pointed out that the PPCC was especially valuable for small urban centres in that it would permit them "to plug into the bigger movement." Delaunay was emphatic in his view that the conference participants "reflected the aspirations of millions," and that it is now necessary to expand our vision and mobilise that support.
Andrew Van Velzen and Paula Rochman of the Alliance for Non-Violent Action shared the view that not enough time was allocated during the conference for political debate and .for discussion of other tactics and strategies – such as Prime Minister Pierre Trudeau's peace initiative.
Rochman remarked that "many groups felt frustrated that other projects were not discussed." And after we hand in the signed petitions, wondered Rochman, "what is our response going to be then? | {
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} | 4,337 |
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johnwoodii | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,715 |
{"url":"http:\/\/tex.bootmath.com\/i-get-dashes-instead-of-bullets-in-itemize.html","text":"# I get dashes instead of bullets in \\itemize\n\nWhen I use \\begin{itemize} ... \\end{itemize}, the items are displayed as dashes.\nBut I want them to be displayed by bullets.\n\n\\documentclass[11pt]{llncs}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{dsfont}\n\\usepackage{multicol}\n\\usepackage{fixltx2e}\n\\usepackage[T1]{fontenc}\n\\usepackage{comment}\n\\usepackage{amsmath}\n\\usepackage[margin=1in]{geometry}\n\\usepackage[algosection, boxruled, linesnumbered]{algorithm2e}\n\\usepackage{xcolor}\n\\usepackage{setspace}\n\\usepackage{hyperref}\n\\definecolor{bl}{rgb}{0.0,0.2,0.6}\n\\definecolor{dark-red}{rgb}{0.4,0.15,0.15}\n\\definecolor{dark-blue}{rgb}{0.15,0.15,0.4}\n\\definecolor{medium-blue}{rgb}{0,0,0.5}\n\\usepackage{sectsty}\n\\hypersetup{\ncitecolor={dark-blue}, urlcolor={medium-blue}\n}\n\\usepackage{graphicx}\n\\usepackage[compact]{titlesec}\n\\newtheorem{observation}{\\textsc{Observation}}\n\\begin{document}\n\\begin{itemize}\n\\item Item 1\n\\item Item 2\n\\item Item 3\n\\end{itemize}\n\\end{document}\n\n\nAs far as I\u2019ve read, the french options makes this. But I have french nowhere. What could cause this?\n\n#### Solutions Collecting From Web of \"I get dashes instead of bullets in \\itemize\"\n\nThis is the default behaviour of llncs.cls as it sets\n\n\\renewcommand\\labelitemi{\\normalfont\\bfseries --}\n\\renewcommand\\labelitemii{$\\m@th\\bullet$}\n\n\nmeaning first-level itemize items will be set with dashes, while second-level items will be set with bullets. You can adjust this by adding to your preamble\n\n\\let\\labelitemi\\labelitemii\n\n\nHowever, consider that for journal submissions, these things should not be altered.\n\nAs a showcase of this behaviour, here\u2019s a minimal example:\n\n\\documentclass{llncs}% ftp:\/\/ftp.springer.de\/pub\/tex\/latex\/llncs\/latex2e\/llncs2e.zip\n\n\\begin{document}\n\n\\begin{itemize}\n\\item Like this,\n\\item and like this.\n\\end{itemize}\n\n\\let\\labelitemi\\labelitemii\n\n\\begin{itemize}\n\\item Like this,\n\\item and like this.\n\\end{itemize}\n\n\\end{document}","date":"2018-08-22 03:47:21","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\": 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\": 0.5574228167533875, \"perplexity\": 8863.791623849766}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221219469.90\/warc\/CC-MAIN-20180822030004-20180822050004-00209.warc.gz\"}"} | null | null |
Az Irodalmi Kincstár egy 19. század közepén megjelent magyar szépirodalmi könyvsorozat volt. A 12 részes sorozat egyes kötetei 1860 és 1869 között jelentek meg Pesten Heckenast Gusztáv kiadásában és a következők voltak:
1. köt. Népdalok. Petőfi Sándor arczképével. (192 l.) 1869. 2. kiad. (192 l.) 1874.
2. köt. Románczok. Vörösmarty arczképével. (232 l.) 1860.
3–4. köt. Magyar balladák könyve. 2. köt. Arany János arczképével. (192, 192 l.) 1861. 2. kiad. (192, 192 l.) 1874.
5–6. köt. Külföldi lant. Magyar költők műfordításai külföldi remekírókból. 2 köt. (228, 206 l.) 1862.
7. köt. Emlékkönyv. A legjelesebb magyar költők mondatai. (170 l.) 1862.
8. köt. Epigrammák. (205 l.) 1862.
9–10. köt. Szerelem dalnokai. 2. köt. (216, 206 l.) 1863.
11–12. köt. Életiskola. Gyöngymondatok, aranyigazságok a bel- és külföldi írók munkáiból. 2. köt. (192, 208 l.) 1864.
Források
Petrik Géza: Jegyzéke az 1860–1875. években megjelent magyar könyvek- és folyóiratoknak, Budapest, 1888–1892
Kapcsolódó szócikkek
Magyar könyvsorozatok listája
Magyar szépirodalmi könyvsorozatok | {
"redpajama_set_name": "RedPajamaWikipedia"
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The antioxidant capacity of sea buckthorn (Hippophae rhamnoides L.) berries depends on the genotype and harvest time
N. Makovics-Zsohár;
A. Hegedűs;
É. Stefanovits-Bányai;
R. Rédei;
N. Papp
Berries of sea buckthorn (Hippophae rhamnoides L.) are characterized by increasing popularity due to their presumable healtheffects. The aim of this study was to compare the antioxidant capacity and total polyphenolic content in the berries of six Hungarian grown sea buckthorn genotypes and characterize the genetic variability in this trait. The h...arvest time of sea buckthorn berries affects the antioxidant capacity and total phenolic contents in berries of three popular cultivars of German origin. Berries harvested in October had higher antioxidant capacity compared with those harvested one month later. The extent of the difference was genotype-specific. Our analysis revealed a nearly 3-fold difference between the lowest and highest antioxidant capacities of the 6 tested genotypes with 'Leikora' showing the highest ferric reducing antioxidant power and total phenolic content. The TEAC values ranged between 1.76 and 3.13 mmol Trolox/100g fresh weight with Pető 1 and 'Frugana' having the highest values. The results presented in this study demonstrated that Hippophae rhamnoides berries possess in vitro antioxidant activity strongly determined by genotype but also influenced by harvest time.
Tending operation models for white poplar (Populus alba L.) stands growing under sandy soil conditions
K. Rédei;
Zs. Keserű;
S. Orlovic;
Z. Galic
Integrated research aimed to intensify the practices of afforestation especially of white poplar and grey poplar woods. A new, simplified tending procedure has been developed to substitute earlier models considering the target diameter by right spacing as a tool to achieve adequat quality of logs. The practice-oriented model may help the qualitati...ve improvement of white poplar growing technology in Hungary as well as in Serbia.
Promising black locust (Robinia pseudoacacia L.) cultivars in Hungary
I. Csiha;
J. Rásó;
M. Takács
In Hungary, black locust (Robinia pseudoacacia L.) is one of the most important exotic stand-forming tree species, growing mostly under unfavourable ecological conditions. Considering the climate change effects its importance is increasing in many other countries. As a result of a selection programme, several black locust cultivars have been impro...ved for setting up cultivar trials. In the paper four black locust cultivars were evaluated in Central Hungary under arid hydrological and brown forest soil conditions. Significant differences (P<5%) were found in height, DBH, mean tree volume and average stem form value (SFV). At the age of 35 the cultivar 'R.p. Jászkiséri' appeared to be the most promising one for yield production and 'R.p. Zalai' for SFV.
Generative propagation of Robinia x ambigua POIR. – Pink locust
Z. Osváth-Bujtás;
M. E. Malvolti
The genus Robinia is a small group of about 10 species of trees and shrubs indigenous only to NorthAmerica. Two species are endemic to Mexico, one being confined to south-western part of the country, while the rest are endemic to the south-eastern part of USA. Of the most important species and varieties of genus Robinia, Robinia x ambigua Poir.(Ro...binia viscosa x R. pseudo-acacia)-pink locust can be considered as the most significant one for bee-forage and decorative planting. In this paper a generative propagation method is presented for pink locust.
Above-ground dendromass of sprouted black locust energy plantations: a case study
Zs. Keserü;
B. Bakti;
K. Rédei
Black locust (Robinia pseudoacacia L.) can be considered as one of the most suitable tree species for establishing energy plantations due to its favourable growing characteristics such as vigorous growing potential in the juvenile phase, excellent coppicing ability, a relatively high resistance to pests. Based on national and international test re...sults the mean annual increment of oven-dry weight of energy plantations regenerated by coppicing generally exceeds the first cycle plantations established by seedlings.
The role of black locust (Robinia pseudoacacia L.) in establishment of short-rotation energy plantations in Hungary
I. Veperdi
Establishment of short-rotation energy plantations for fuel production has been of international interest for many years. Energy plantation experiments in Hungary have been conducted for a longer time. In the country black locust ( Robinia ps eudoacacia L.) is one of the most important stand-forming tree species, covering approximately 23% of the ...forested land (410 000 ha) and providing about 19% of the annual timber output of the country. This fast growing species seems to be suitable for energy plantations as well. So, in Helvecia (Central Hungary, sand-soil region) two energy plantation s were established u sing common black locust and its cultivars improved in Hungary. The spacing variations of the common black locust were: l.5x0.3 m, I .5x0.5 m and l.5x 1.0 m. At the age of 5 the closest spacing ( 1.5x0.3m) produced the greatest annual increment in oven-dry weight (6.5 t ha·1 yr- 1). In the trial with black locust cultivars planted in spacing of 1.5xl.0m, at the age of 7 the highest annual increment in oven-dry mass was produced by the cultivar ' Ulloi' (9.7 t ha-1 yr- 1) followed by the common black locust (8.4 t ha-1 yr- 1) and the cultivar 'J tiszkiseri (1.6 t ha·1 yr- 1). The trials have verified that in temperate climate the increment in oven dry dendromass of black locust energy plantation s has ranged from 6 to 12 t ha·1 yr·1. On the basis of the trials' evaluation the quantity of dendromass mostly depends on site quality, species and cultivars, as well as on the initial spacing (plants per hectare).
Black locust (Robinia pseudoacacia L.) selection programmes in Hungary: a short review
Black locust (Robinia pseudoacacia L.) was the first forest tree species introduced from North America to Europe, at the beginning of the 17th century. Its unprecedented fast spread is due to its high-grade adaptability, drought-tolerance, abundant and frequent seed crop, excellent sprouting ability, fast growth and relatively high timber yield. O...ther advantages are, that it has scarcely any fungi or insect pests. This review is a short summary on black locust improvement in Hungary, giving guidance for specialists who are interested in black locust management.
Effects of initial spacing on the stand structure and yield of young black locust (Robinia pseudoacacia L.) stands
The choice of the right initial spacing of stands is one of the most decisive operations of a successful afforestation. It is even more important in the case of fast growing tree species grown in plantations; it is expressed in their early phase of development and in wood quality. The results of a 5-year long experiment with four treatments will b...e presented in this paper. They proved the priority of an initial spacing of 1.61.0 m in the majority of quality This treatment has been proved optimal exploitation of growing space by the young trees.
Tending operation models for black locust (Robinia pseudoacacia L.) stands growing on sandy soils in Hungary
B. Antal
A more intensive integrated research and development approach to the work carried out on the growth on sandy soils of stands of black locust (Robinia pseudoacacia L.) has been adopted in recent years, revealing several factors influencing stand growth. The fact that certain ecological factors influencing fundamentally the growth of trees have beco...me unfavourable in Hungary in recent years has led to the more extensive use of black locust in the course of afforestation and forest regeneration schemes. The study presents a new, simplified tending operation model for black locust stands and age, growing space and target diameter models suitable for qualitaty log production and for mass assortments. The simplicity of these practice-orientedmodels may foster the qualitative development of black locust management in Hungary and in some other countries where this tree species may gain greater acceptance by landowners and the forest industry.
Promising white poplar (Populus alba L.) clones in sandy ridges between the rivers Danube and Tisza in Hungary
Zs. Keserű
Vol 14No 1-2.2008
White poplar is a native stand-forming tree species in Hungary, covering 3.1 per cent of the forested area. More than 70 per cent of the white poplar stands can be found on calcareous sandy sites in the Danube—Tisza region, so they play a significant role in the poplar management of this part of the country. The most important task ahead of Hung...arian poplar growers is to improve the quality of poplar stands and plantations based on selecting new clones and cultivars. The growth and yield of four promising white poplar clones was evaluated on a marginal site in central Hungary. The clones `1-1 425-4' (Populus alba x Populus alba), and 11 758' (Populus alba Mosonmagyaróvár 124) seem to be suitable for wood production, while the 427-3' (Populus alba x Populus alba cv. Bolleana) and 422-9' (Populus alba x Populus grandidentata) clones (with decorative stem form) could be better used for tree lines and ornamental plantations.
Improved clonal approaches to growing black locust (Robinia pseudoacacia L.) in Hungary: a case study
In Hungary black locust (Robinia pseudoacacia L.) is considered as an important exotic stand-forming tree species and due to climate change effects its importance is increasing in many other countries. It has some desirable characteristics from both the practical and research standpoints. As a result of a partly new black locust selection programm...e new black locust clones were improved and a technology was developed for mass clonal micropropagation of juvenile trees. Clone trials with micropropagated plants were established in the country for evaluating the juvenile growth and the stem form of promising black locust clones under marginal site conditions. Significant differences (P<5%) were found for stem form value which partly verified the genetic gain of the selected clones against the common black locust. It was also proved that tissue culture could offer partly new prospects for the rapid mass cloning of selected genotypes.
Clonal selection of black locust (Robinia pseudoacacia L.) in Hungary: a review
Black locust (Robinia pseudoacacia L.) is the most important fast growing stand-forming tree species in Hungary. Its importance is increasing in many other countries, too. As a result of a new selection programme 13 black locust clones have been improved for setting up clones trials and seed orchard. In 2003 five of them (R.p. `Bácska', `Homoki',... 'Szálas', `Oszlopos' and `Vacsi') were registered as cultivarcandidates. Tissue culture method has proved as a suitable mean of propagating superior individuals. The micropropagated plants have been growing successfully in the clone trials.
Propagation from root cuttings for black locust (Robinia pseudoacacia L.) improvement in Hungary: a review
T. Kiss
Black locust (Robinia pseudoacacia L.) is a valuable stand-forming tree species introduced to Europe approximately 400 years ago from North America. Today it is widely planted throughout the world, first of all for wood production. In Hungary, where black locust has great importance in the forest management, it is mainly propagated by seeds. But s...ince the seed-raised plants present a great genetic variation, this type of propagation can not be used for Robinia's improved cultivars. In the Hungarian black locust clonal forestry, propagation from root cuttings can be used for reproduction of superior individuals or cultivars in large quantities. However, this method demands more care than raising seedlings from seeds and can be applied with success in well-equipped nurseries. | {
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Q: Do I need to handle Dummy variable trap manually in Regression or sklearn will do it? I know that we have to one-hot encode categorical data before training machine Learning algorithm. but my question is do we need to remove one column manually or sklearn will do it?
A: I assume you want to drop one column also for non-binary categorical features to avoid multi-collinearity, which might cause problems for linear models. It is as easy as providing drop_first=True argument to pd.get_dummies(). It seems that sklearn.preprocessing.OneHotEncoder doesn't have a simple interface to do this and anyway its usage is complicated, as categorical features have to be encoded into int's beforehand.
A: You need to handle Dummy variable trap manually in Regression.
We need to remove one column manually.
A: There is no sense in removing a column with nunique count greater than 2. As each column will represent an instance of a dummy variable. I assume that you are one-hot encoding a binary featured column. Instead, use sklearn's label encoder.
| {
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Río Sogamoso är ett vattendrag i Colombia. Det ligger i departementet Santander, i den norra delen av landet, km norr om huvudstaden Bogotá.
Tropiskt regnskogsklimat råder i trakten. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är februari, då medeltemperaturen är °C, och den kallaste är juni, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är oktober, med i genomsnitt mm nederbörd, och den torraste är januari, med mm nederbörd.
Källor
Vattendrag i Santander, Colombia | {
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package me.vilsol.nmswrapper.wraps.unparsed;
import me.vilsol.nmswrapper.*;
import me.vilsol.nmswrapper.reflections.*;
import me.vilsol.nmswrapper.wraps.*;
@ReflectiveClass(name = "PathfinderGoalBeg")
public class NMSPathfinderGoalBeg extends NMSPathfinderGoal {
public NMSPathfinderGoalBeg(Object nmsObject){
super(nmsObject);
}
public NMSPathfinderGoalBeg(NMSEntityWolf entityWolf, float f){
super("PathfinderGoalBeg", new Object[]{NMSEntityWolf.class, float.class}, new Object[]{entityWolf, f});
}
/**
* TODO Find correct name
* @see net.minecraft.server.v1_9_R1.PathfinderGoalBeg#a()
*/
@ReflectiveMethod(name = "a", types = {})
public boolean a(){
return (boolean) NMSWrapper.getInstance().exec(nmsObject);
}
/**
* TODO Find correct name
* @see net.minecraft.server.v1_9_R1.PathfinderGoalBeg#b()
*/
@ReflectiveMethod(name = "b", types = {})
public boolean b(){
return (boolean) NMSWrapper.getInstance().exec(nmsObject);
}
/**
* TODO Find correct name
* @see net.minecraft.server.v1_9_R1.PathfinderGoalBeg#c()
*/
@ReflectiveMethod(name = "c", types = {})
public void c(){
NMSWrapper.getInstance().exec(nmsObject);
}
/**
* TODO Find correct name
* @see net.minecraft.server.v1_9_R1.PathfinderGoalBeg#d()
*/
@ReflectiveMethod(name = "d", types = {})
public void d(){
NMSWrapper.getInstance().exec(nmsObject);
}
/**
* TODO Find correct name
* @see net.minecraft.server.v1_9_R1.PathfinderGoalBeg#e()
*/
@ReflectiveMethod(name = "e", types = {})
public void e(){
NMSWrapper.getInstance().exec(nmsObject);
}
} | {
"redpajama_set_name": "RedPajamaGithub"
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Albion City Council accepted a proposal last Tuesday night from Hastings Keno, Inc. to be the city's licensed keno operator.
The proposal was accepted after the council heard from Michael Navity, owner of Hastings Keno, who said he would make a variety of adjustments in his contract to conform to the city's wishes.
An amended plat was also approved for the Rick and Carol Spiegel property, west of Fifth Street and north of Market Street. The replatting will allow for construction of a 12-unit motel on the property.
Details in the July 18 Albion News Print & Electronic Editions. | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/mathoverflow.net\/questions\/209596\/combinatorial-results-by-poincar%C3%A9-duality","text":"# Combinatorial results by Poincar\u00e9 duality\n\nFor the n-dimensional Torus, the k-th homology group (with integer coefficients) is isomorphic to the direct sum of $n \\choose k$ copies of $\\mathbb{Z}$. Poincar\u00e9 duality thus gives us a somewhat indirect way to show that ${n \\choose k} = {n \\choose n-k}$.\n\nI am looking for results (or a starting point in the literature) where less obvious combinatorial results are derived by Poincar\u00e9 duality.\n\nI think that this is roughly related to the following questions:\n\nHow can we realize different combinatorial objects as the dimension of a construction on vector spaces? Are the resulting algebras useful?\n\nCombinatorial results without known combinatorial proofs\n\nFrom what I have learned from the comments, the Dehn\u2013Sommerville equations would be an example of a combinatoric result based on Poincar\u00e9 duality.\n\n\u2022 Since this question doesn't have a unique correct answer, I'm going to flag it for community-wiki. \u2013\u00a0HJRW Jun 18 '15 at 8:31\n\u2022 Someone who knows about toric varieties surely has something to say here... (Poincar\u00e9 duality holds for these with rational coefficients) \u2013\u00a0Dylan Wilson Jun 18 '15 at 14:20\n\u2022 If you're willing to extend this from \"Poincar\\'e duality\" to \"properties of Betti numbers\", then you could include Hard Lefschetz in your toolset, and get Stanley's proof of the Upper Bound Theorem for the number of faces of a simplicial polytope. \u2013\u00a0Allen Knutson Jun 18 '15 at 14:32\n\u2022 I am willing to extend to any other duality concepts in (co-)homology. \u2013\u00a0Rolf Bardeli Jun 18 '15 at 14:49\n\u2022 Stanley's survey paper on unimodal sequences is a possible starting point. dedekind.mit.edu\/~rstan\/pubs\/pubfiles\/72.pdf However, I don't think that it gives any examples other than the ones already mentioned. In general, I think it would be very unusual for the symmetry of a sequence to be provable most easily by a topological duality theorem, rather than by a direct combinatorial argument. Even the Dehn--Sommerville equations have a relatively straightforward combinatorial proof. \u2013\u00a0Timothy Chow Jun 18 '15 at 16:13","date":"2019-03-21 20:54:40","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\": 0.8214548826217651, \"perplexity\": 434.1045744252031}, \"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-2019-13\/segments\/1552912202572.29\/warc\/CC-MAIN-20190321193403-20190321215403-00179.warc.gz\"}"} | null | null |
Allen "Chainsaw" Kessler
GPID is a unique identification number, assigned to each individual player, that will be used in the future in order to register for most poker tournaments around the world. It links to the player's profile in order to prevent any data errors.
NationalityUnited States
ResidenceLas Vegas, NV, United States
Allen Kessler's GPI Rankings
GPI rank
#344 #164 1,760.25 pts #27
PoY 2020 rank
#2,424 #1048 54.44 pts #2424
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Date & Event
WSOP Circuit - WSOPC Choctaw
Durant, OK, United States
5-Card Pot Limit Omaha hi/lo (Ring Event #4)
Jan 5 - 6, 2020 Choctaw Casino Resort $600 21st 175 $1,058 54.44 pts —
WSOP Circuit - WSOPC Harrah's Las Vegas
No Limit Hold'em - WSOPC Harrah's Las Vegas Main Event (Event #11)
Dec 19 - 22, 2019 Harrah's Las Vegas $1,700 16th 665 $9,450 165.96 pts —
partypoker MILLIONS World Bahamas
No Limit Hold'em - MILLIONS Mini Event
Nov 16 - 19, 2019 Baha Mar $1,000 + 100 60th 1,075 $4,000 101.96 pts —
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Nov 8 - 10, 2019 Choctaw Casino Resort $1,700 54th 983 $4,089 123.40 pts —
No Limit Hold'em - Double Stack (Event #2)
Oct 31 - Nov 1, 2019 Choctaw Casino Resort $400 41st 278 $557 30.42 pts —
2019 World Series of Poker Europe (WSOPE)
Rozvadov, Czech Republic
Pot Limit Omaha - 8-Handed (Event #11)
Oct 22 - 24, 2019 Kings Casino €2,200 33rd 271 €3,210 109.59 pts —
No Limit Hold'em - Mini Main Event (Event #3)
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Oct 13 - 16, 2019 Kings Casino €350 62nd 696 €783 34.90 pts —
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Pot Limit Omaha Hi/Lo (Event #7)
Sep 10 - 11, 2019 Thunder Valley Casino Resort $400 11th 82 $666 31.14 pts —
No Limit Hold'em (Event #2)
Sep 5 - 9, 2019 Thunder Valley Casino Resort $400 153rd 1,166 $569 31.32 pts —
WSOP Circuit - WSOPC Foxwoods
Mashantucket, CT, United States
No Limit Hold'em - Turbo (Event #13)
Aug 26, 2019 Foxwoods Resort Casino $400 16th 101 $543 29.90 pts —
World Poker Tour - WPT Legends of Poker
H.O.R.S.E. (Event #32)
Aug 19, 2019 Bicycle Casino $350 16th 291 $1,285 40.80 pts —
No Limit Hold'em - Six Handed (Event #2)
Aug 15 - 16, 2019 Foxwoods Resort Casino $600 14th 92 $957 51.27 pts —
WSOP Circuit - WSOPC Cherokee / WSOP Global Championship
Cherokee, , United States
Aug 1 - 4, 2019 Harrah's Cherokee $400 23rd 2,766 $4,798 78.76 pts —
No Limit Hold'em - WSOPC Main Event (Ring Event #11)
Jul 26 - 29, 2019 Choctaw Casino Resort $1,700 87th 976 $2,900 105.50 pts —
No Limit Hold'em (Ring Event #9)
Jul 23 - 24, 2019 Choctaw Casino Resort $250 11th 180 $609 39.36 pts —
50th World Series of Poker (WSOP) 2019
Mixed No Limit Hold'em/Pot Limit Omaha - 8-Handed (Event #80)
Jul 10 - 12, 2019 Rio Hotel and Casino $1,500 86th 1,250 $2,928 81.76 pts —
No Limit Hold'em - WSOP Main Event
Jul 3 - 16, 2019 Rio Hotel and Casino $10,000 320th 8,569 $38,240 118.25 pts —
2019 Grand Poker Series
Omaha/Stud Hi-Lo
Jul 3, 2019 Golden Nugget Hotel & Casino $570 4th 94 $4,460 54.22 pts —
No Limit Hold'em - Super Turbo Bounty (Event #56)
Jun 24, 2019 Rio Hotel and Casino $1,500 147th 1,867 $1,762 78.31 pts —
Mixed No Limit Hold'em/Pot Limit Omaha Deepstack 8-Handed (Event #42)
Jun 18 - 19, 2019 Rio Hotel and Casino $600 330th 2,403 $875 39.43 pts —
8 Game Mix - 6-Handed (Event #23)
Jun 9 - 11, 2019 Rio Hotel and Casino $1,500 5th 612 $24,292 172.44 pts —
No Limit Hold'em - WSOP.com Online (Event #7)
Jun 2, 2019 Rio Hotel and Casino $400 253rd 2,825 $814 — —
No Limit Hold'em - BIG 50 (Event #3)
May 30 - Jun 7, 2019 Rio Hotel and Casino $500 4007th 28,371 $750 24.21 pts —
Seminole Hard Rock Deep Stack Poker Series
No Limit Hold'em - Deepest Stack (Event #5)
May 23 - 26, 2019 Seminole Hard Rock Hotel & Casino $1,650 4th 251 $40,540 142.75 pts —
WSOP Circuit - Cherokee
No Limit Hold'em Main Event #10
Apr 19 - 20, 2019 Harrah's Cherokee $1,700 48th 1,087 $4,973 99.89 pts —
WSOP Circuit - Bally's Las Vegas
Mar 26 - 27, 2019 Bally's Casino $400 15th 194 $999 27.34 pts —
Heartland Poker Tour (HPT)
St. Louis, , United States
No Limit Hold'em - HPT Main Event (Event #11)
Mar 15 - 18, 2019 Hollywood Casino $1,650 30th 444 $4,199 85.37 pts —
WSOP Circuit - Los Angeles
Mar 10 - 13, 2019 Bicycle Casino $1,700 48th 678 $3,430 85.04 pts —
2019 Los Angeles Poker Classic (LAPC)
Pot Limit Omaha - 8 or Better (Event #55)
Feb 25, 2019 Commerce Casino $1,100 8th 77 $2,660 62.85 pts —
WSOP Circuit - Choctaw
Jan 8 - 9, 2019 Choctaw Casino Resort $400 34th 429 $705 14.47 pts —
H.O.R.S.E. (Event #6)
Dec 5 - 6, 2018 Bicycle Casino $400 12th 97 $670 12.72 pts —
Nov 30 - Dec 3, 2018 Harrah's Cherokee $1,700 12th 1,127 $24,911 88.50 pts —
WSOP Circuit - Nevada
Nov 20 - 21, 2018 Planet Hollywood Resort & Casino $600 14th 137 $1,191 22.77 pts —
partypoker Caribbean Poker Party
No Limit Hold'em - Finale
Nov 15 - 17, 2018 Baha Mar $1,000 + 100 112th 926 $3,000 32.15 pts —
Pot Limit Omaha (Event #4)
Nov 4, 2018 Choctaw Casino Resort $400 12th 134 $907 13.94 pts —
Oct 27 - Nov 2, 2018 Kings Casino €10,000 + 350 40th 534 €23,025 74.66 pts —
Pot Limit Omaha 8-Max (Event #7)
Oct 22 - 24, 2018 Kings Casino €2,020 + 180 16th 187 €4,034 48.95 pts —
Pot Limit Omaha 8-Max - (Event #3)
Oct 15 - 18, 2018 Kings Casino €505 + 45 68th 572 €859 22.83 pts —
No Limit Hold'em - Colossus (Event #1)
Oct 9 - 15, 2018 Kings Casino €505 + 45 52nd 2,992 €4,437 42.26 pts —
WSOP Circuit - Thunder Valley
Sep 14 - 17, 2018 Thunder Valley Casino Resort $1,700 50th 464 $2,669 39.94 pts —
No Limit Hold'em - Monster Stack (Event #7)
Sep 12 - 13, 2018 Thunder Valley Casino Resort $400 17th 318 $1,263 16.48 pts —
Sep 10, 2018 Thunder Valley Casino Resort $400 7th 74 $1,129 13.71 pts —
2018 WinStar River Poker Series
Thackerville, OK, United States
No Limit Hold'em - Main Event
Sep 1 - 4, 2018 Winstar Casino $2,500 89th 378 $4,460 38.01 pts —
Black Hawk, CO, United States
No Limit Hold'em - HPT Main Event (Event #5)
Jul 18 - 23, 2018 Golden Gates Casino & Poker Parlour $1,100 40th 763 $3,182 41.67 pts —
Jul 12 - 14, 2018 Rio Hotel and Casino $3,000 52nd 354 $4,462 22.22 pts —
No Limit Hold'em - The Closer (Event #75)
Jul 12 - 16, 2018 Rio Hotel and Casino $1,500 36th 3,120 $16,392 39.04 pts —
No Limit Hold'em - Double Stack (Event #73)
Jul 11 - 12, 2018 Rio Hotel and Casino $1,000 142nd 1,221 $1,571 15.48 pts —
Razz - Championship (Event #56)
Jun 27 - 29, 2018 Rio Hotel and Casino $10,000 9th 119 $25,564 35.99 pts —
Limit 2-7 Lowball Draw (Event #29)
Jun 13 - 15, 2018 Rio Hotel and Casino $1,500 49th 356 $2,243 17.82 pts —
No Limit Hold'em - Millionaire Maker (Event #21)
Jun 9 - 14, 2018 Rio Hotel and Casino $1,500 932nd 7,361 $2,468 14.90 pts —
Dealers Choice 6 Handed (Event #12)
Jun 4 - 6, 2018 Rio Hotel and Casino $1,500 60th 406 $2,263 17.51 pts —
Omaha Hi-Lo 8 or Better (Event #4)
May 31 - Jun 2, 2018 Rio Hotel and Casino $1,500 75th 911 $2,823 20.59 pts —
WSOP Circuit - New Orleans
May 20 - 21, 2018 Harrahs New Orleans $365 39th 385 $522 6.72 pts —
May 17, 2018 Harrahs New Orleans $365 18th 311 $1,123 8.02 pts —
WSOP Circuit - Baltimore
Baltimore, MD, United States
No Limit Hold'em - Main Event (Event #9)
May 4 - 7, 2018 Horseshoe Casino Baltimore $1,675 47th 513 $2,955 20.81 pts —
May 1 - 2, 2018 Horseshoe Casino Baltimore $365 3rd 113 $4,475 10.59 pts —
Apr 17 - 19, 2018 Harrah's Cherokee $365 51st 1,807 $1,653 10.36 pts —
Apr 16, 2018 Harrah's Cherokee $365 29th 291 $592 6.75 pts —
Apr 12 - 14, 2018 Harrah's Cherokee $365 208th 2,608 $640 7.22 pts —
2018 Seminole Hard Rock Poker Showdown
No Limit Hold'em - WPTDeepstacks Immokalee (Event #8)
Apr 9 - 11, 2018 Seminole Hard Rock Hotel & Casino $1,100 25th 816 $4,361 25.25 pts —
WPTDeepStacks - Thunder Valley
Apr 5 - 9, 2018 Thunder Valley Casino Resort $1,100 83rd 724 $1,735 16.31 pts —
WSOP Circuit - Atlantic City
Atlantic City, United States
Omaha Hi / Lo 8 Or Better #8
Mar 14 - 15, 2018 Harrahs Atlantic City $365 10th 102 $639 6.78 pts —
2018 Chicago Poker Classic
Hammond, IN, United States
Mar 9 - 12, 2018 Harrahs Horseshoe Casino Hammond $2,000 68th 646 $3,212 21.31 pts —
WSOP Circuit - Las Vegas
No Limit Hold'em - Main Event #11
Feb 24 - 27, 2018 Rio Hotel and Casino $1,675 55th 887 $3,858 23.56 pts —
DeepStack Extravaganza I
No Limit Hold'em - SuperStack (Event #36)
Feb 19 - 23, 2018 The Venetian Resort Hotel Casino $1,600 79th 786 $3,178 19.90 pts —
WSOP Circuit - Milwaukee
Milwaukee, United States
Feb 11, 2018 Potawatomi Casino $365 6th 369 $4,712 12.64 pts —
2018 Aussie Millions Poker Championship
Pot Limit Omaha Hi-Lo (Event #16)
Jan 31, 2018 Crown Casino AU$1,025 + 125 10th 104 AU$2,665 15.70 pts —
Pot Limit Omaha - Re-Entry ( Ring Event #9 )
Jan 17, 2018 Thunder Valley Casino Resort $300 + 65 12th 102 $639 6.45 pts —
Omaha8 - Single Entry ( Ring Event #5 )
Jan 15, 2018 Thunder Valley Casino Resort $300 + 65 7th 137 $1,658 8.37 pts —
Jan 12 - 15, 2018 Choctaw Casino Resort $1,675 28th 1,249 $9,143 14.11 pts —
Big O (Event #4)
Jan 7, 2018 Choctaw Casino Resort $365 30th 295 $600 2.80 pts —
No Limit Hold'em ( Event #2)
Jan 5 - 6, 2018 Choctaw Casino Resort $365 273rd 3,052 $677 2.80 pts —
St. Charles, Mi, United States
No Limit Hold'em - HPT Championship Open (Event #8)
Dec 8 - 12, 2017 Ameristar Casino Resort Spa $2,500 29th 316 $5,287 9.95 pts —
Pot Limit Omaha (Event #10)
Nov 9 - 10, 2017 Choctaw Casino Resort $1,125 13th 161 $2,582 7.61 pts —
No Limit Hold'em - Little One for One Drop (Event #8)
Nov 1 - 4, 2017 Kings Casino €1,100 118th 868 €1,660 6.97 pts —
Pot-Limit Omaha (Event #6)
Oct 28 - 30, 2017 Kings Casino €2,200 2nd 191 €57,897 21.31 pts —
Oct 19 - 23, 2017 Kings Casino €1,100 51st 561 €2,065 7.83 pts —
WSOP Circuit - Seminole Hard Rock
No Limit Hold'em - High Roller (Event #12)
Oct 1 - 2, 2017 Seminole Hard Rock Hotel & Casino $3,250 11th 133 $8,180 11.11 pts —
2017 The River Poker Series
WPT Legends of Poker
H.'O'.R.S.E. - (Event #20)
Aug 23 - 29, 2017 Bicycle Casino $240 13th 427 $1,350 4.20 pts —
Razz (Event #69)
Jul 6 - 8, 2017 Rio Hotel and Casino $1,500 62nd 419 $2,261 7.30 pts —
Seven Card Stud Hi/Lo (Event #40)
Jun 20, 2017 Rio Hotel and Casino $1,500 39th 595 $3,550 9.23 pts —
Pot Limit Omaha Hi/Lo/ Limit Omaha Hi/Lo (Event #32)
No Limit Hold'em - The Giant (Event #19)
Jun 9 - Jul 8, 2017 Rio Hotel and Casino $365 1318th 10,015 $558 2.06 pts —
No Limit Hold'em - 6 Handed (Event #16)
Jun 8 - 10, 2017 Rio Hotel and Casino $1,500 99th 1,748 $3,222 9.69 pts —
WSOP Circuit - Iowa
Council Bluffs, IA, United States
Limit Omaha - Hi/Lo 8 or Better #7
Apr 4 - 5, 2017 Harrahs Horseshoe Council Bluffs $365 4th 111 $3,193 3.95 pts —
WSOP Circuit - Tulsa
Mar 25 - 26, 2017 Hard Rock Hotel & Casino Tulsa $365 15th 188 $881 3.01 pts —
Florence, , United States
Mar 16 - 20, 2017 Belterra Casino Resort $1,100 10th 436 $7,098 11.69 pts —
2017 The Wynn Spring Classic
No Limit Hold'em - Championship
Mar 2 - 5, 2017 Wynn Las Vegas $1,500 + 100 40th 818 $5,713 10.46 pts —
No Limit Hold'em - Main Event Day (Event #11)
Feb 24 - 27, 2017 Rio Hotel and Casino $1,675 80th 945 $3,430 8.88 pts —
No Limit Hold'em - Superstack #38
Feb 20 - 24, 2017 The Venetian Resort Hotel Casino $1,600 58th 676 $3,417 8.69 pts —
Feb 15 - 20, 2017 Golden Gates Casino & Poker Parlour $1,650 42nd 787 $4,975 10.27 pts —
Feb 10 - 13, 2017 Potawatomi Casino $1,675 40th 677 $3,757 9.98 pts —
Feb 8 - 9, 2017 Potawatomi Casino $365 30th 532 $934 3.37 pts —
Feb 2 - 6, 2017 Boomtown New Orleans $1,100 2nd 275 $38,279 17.34 pts —
WPT Borgata Winter Poker Open
Pot Limia Omaha High/Low 8B (Event #11)
Jan 24 - 25, 2017 Borgata Hotel Casino & Spa $300 + 40 11th 109 $634 2.80 pts —
PokerStars Championship Bahamas
Paradise Island, Bahamas
No Limit Hold'em - PokerStars Championship Main Event (Event #23)
Jan 8 - 14, 2017 Atlantis Resort & Casino $4,717 + 283 64th 738 $11,560 — —
Winter Poker Open
Tampa, United States
No Limit Hold'em - Championship- Event 13
Dec 15 - 18, 2016 Seminole Hard Rock Hotel & Casino $1,650 23rd 489 $4,548 — —
Dec 2 - 5, 2016 Harrah's Cherokee $1,675 49th 968 $4,559 — —
Borgata Fall Poker Open 2016
Nov 18 - 22, 2016 Borgata Hotel Casino & Spa $2,500 + 200 45th 489 $5,929 — —
Reno, NV, United States
No Limit Hold'em Main Event (Event #14)
Oct 7 - 10, 2016 Peppermill Hotel Casino $1,100 16th 204 $2,884 — —
WPTDeepStacks - Arizona
Tucson, Ar, United States
Sep 23 - 26, 2016 Casino Del Sol $1,100 12th 286 $4,418 — —
2016 Seminole Hard Rock Poker Open
No Limit Hold'em - Re-Entry (Event #24)
Aug 15 - 16, 2016 Seminole Hard Rock Hotel & Casino $1,100 23rd 828 $6,050 — —
Ante Up World Championship
Aug 5, 2016 Thunder Valley Casino Resort $1,650 42nd 414 $3,043 — —
No Limit Hold'em Crazy Eights 8-Handed (re-entry) (Event #54)
Jul 1 - 4, 2016 Rio Hotel and Casino $888 78th 6,761 $5,896 — —
Mixed Omaha/Seven Card Stud Hi-Lo 8 or Better (Event #36)
Jun 21 - 23, 2016 Rio Hotel and Casino $2,500 27th 384 $5,147 — —
Omaha Hi-Low Split-8 or Better (Event #26)
Jun 16 - 18, 2016 Rio Hotel and Casino $1,500 23rd 934 $6,919 — —
Seven Card Razz (Event #13)
Jun 9 - 11, 2016 Rio Hotel and Casino $1,500 69th 461 $2,238 — —
Jun 6 - 8, 2016 Rio Hotel and Casino $1,500 245th 2,016 $2,361 — —
No Limit Hold'em Colossus II (Event #2)
Jun 2 - 7, 2016 Rio Hotel and Casino $565 1833rd 21,613 $1,143 — —
May 13 - 15, 2016 Harrahs New Orleans $365 135th 1,213 $542 — —
Chicago, , United States
No Limit Hold'em Main Event (Event #7)
May 5 - 9, 2016 Ameristar Casino $1,650 36th 541 $4,265 — —
No Limit Hold'em - Ring Event 12
Apr 24 - 25, 2016 Harrah's Cherokee $365 20th 273 $845 — —
No Limit Hold'em - Monster Stack Ring Event 8
Apr 20 - 21, 2016 Harrah's Cherokee $365 22nd 961 $2,007 — —
Apr 10, 2016 Harrahs Horseshoe Council Bluffs $365 13th 148 $749 — —
No Limit Hold'em - Main Event Ring Event 11
Apr 8 - 11, 2016 Harrahs Horseshoe Council Bluffs $1,675 33rd 321 $2,875 — —
Apr 6, 2016 Harrahs Horseshoe Council Bluffs $365 18th 265 $1,000 — —
Limit Omaha 8 or Better
Apr 4 - 5, 2016 Harrahs Horseshoe Council Bluffs $365 3rd 117 $4,634 — —
Mid-States Poker Tour - MSPT Iowa
Tama, IA, United States
Mar 18 - 20, 2016 Meskwaki Casino $1,000 + 100 15th 409 $3,952 — —
Mar 1 - 2, 2016 Bally's Casino $365 12th 215 $1,207 — —
Feb 29 - Mar 1, 2016 Bally's Casino $580 4th 113 $4,647 — —
Feb 17 - 22, 2016 Golden Gates Casino & Poker Parlour $1,650 80th 781 $3,100 — —
PCA - 2016
#26 No Limit Hold'em - PCA Main Event
Jan 8 - 13, 2016 Atlantis Resort & Casino $5,300 107th 928 $9,720 — —
2015 Rock 'N' Roll Poker Open
Nov 27 - Dec 1, 2015 Seminole Hard Rock Hotel & Casino $3,500 41st 766 $10,295 — —
Choctaw Fall Poker Series
Oct 30 - Nov 1, 2015 Choctaw Casino Resort $1,000 + 80 58th 580 $1,840 — —
2015 World Series of Poker (WSOPE) - Europe
No Limit Hold'em #6
Oct 14 - 16, 2015 Spielbank Berlin - Marlene Dietrich €3,250 23rd 256 €6,855 — —
WPT - Legends Of Poker
No Limit Hold'em - WPT Main Event
Aug 29, 2015 Bicycle Casino $3,700 42nd 786 $10,983 — —
Mid-States Poker Tour - MSPT Indiana
Evansville, IN, United States
No Limit Hold'em - MSPT Main Event
Aug 21 - 23, 2015 Tropicana Evansville $1,100 + 10 3rd 251 $22,987 — —
Charity Series of Poker (CSOP) Event #4 benefiting Three Square
Jul 5 - 5, 2015 Planet Hollywood Resort & Casino 4th 87 $3,750 — —
No Limit Hold'em (Event #57)
Jun 28 - 30, 2015 Rio Hotel and Casino $1,000 201st 2,497 $2,022 — —
No Limit Hold'em Draft Kings 50/50 (Event #55)
Jun 27 - 29, 2015 Rio Hotel and Casino $1,500 163rd 1,123 $1,500 — —
Seven Card Stud (Event #48)
Jun 11 - 13, 2015 Rio Hotel and Casino $1,000 31st 1,293 $5,690 — —
Limit Hold'em (Event #11)
Jun 2 - 4, 2015 Rio Hotel and Casino $1,500 49th 660 $3,474 — —
No Limit Hold'em The Colossus (Event #5)
May 29 - Jun 2, 2015 Rio Hotel and Casino $565 1191st 22,374 $2,517 — —
2015 Deep Stack Series
No Limit Hold'em Deeper Stack (Event #3)
May 18, 2015 Seminole Hard Rock Hotel & Casino $560 13th 381 $2,048 — —
Heartland Poker Tour
May 8 - 11, 2015 Ameristar Casino $1,500 + 150 15th 366 $6,760 — —
Borgata Spring Poker Open/WPT World Championship
No Limit Hold'em Borgata Sping Open (Event #8)
Apr 20 - 24, 2015 Borgata Hotel Casino & Spa $2,500 + 200 43rd 447 $5,420 — —
2015 L.A Poker Classic
8 Game Mix
Mar 3, 2015 Commerce Casino $2,140 1st 38 $22,600 — —
Hollywood Poker Open
Toledo, OH, United States
No Limit Hold'em Deepstack #20
Feb 12 - 13, 2015 Hollywood Casino $300 9th 130 $901 — —
WPT Lucky Hearts Poker Open
No Limit Hold'em - WPT Lucky Hearts Poker Open Championship #16
Feb 5 - 11, 2015 Seminole Hard Rock Hotel & Casino $3,500 60th 1,027 $9,202 — —
No Limit Hold'em - WPTDeepStacks-Hollywood FL Main Event #12
Jan 29 - 31, 2015 Seminole Hard Rock Hotel & Casino $1,100 30th 396 $2,497 — —
WPT Borgata Winter Open
No Limit Hold'em Borgata Million #15
Jan 21 - 24, 2015 Borgata Hotel Casino & Spa $400 + 50 127th 2,882 $1,342 — —
Pot Limit Omaha #13
Jan 20, 2015 Borgata Hotel Casino & Spa $300 + 40 13th 142 $723 — —
No Limit Hold'em Championship #9
Dec 18 - 21, 2014 Seminole Hard Rock Hotel & Casino $1,650 17th 297 $4,455 — —
WSOP Circuit - Hammond
Oct 24 - 27, 2014 Harrahs Horseshoe Casino Hammond $1,675 29th 1,147 $8,637 — —
Pot Limit Omaha - Hi/Lo #9
Oct 23, 2014 Harrahs Horseshoe Casino Hammond $365 8th 187 $1,645 — —
WSOP Circuit - Connecticut
No Limit Hold'em Main Event
Aug 23 - 25, 2014 Foxwoods Resort Casino $1,675 1st 526 $170,031 — —
Aug 14 - 18, 2014 Ameristar Casino $1,500 + 150 54th 538 $3,318 — —
Ante Up Poker Tour World Championship
No Limit Hold'em World Championship Main Event #13
Aug 9 - 11, 2014 Thunder Valley Casino Resort $1,650 14th 282 $4,684 — —
No Limit Hold'em - The Little One for One Drop (Event #62)
Jul 2 - 5, 2014 Rio Hotel and Casino $1,111 147th 4,496 $3,560 — —
The Poker Players Championship (Event #46)
Jun 22 - 26, 2014 Rio Hotel and Casino $50,000 8th 102 $134,101 — —
Jun 7 - 9, 2014 Rio Hotel and Casino $1,500 161st 2,086 $3,154 — —
No Limit Hold'em - Millionaire Maker (Event #8)
May 31 - Jun 3, 2014 Rio Hotel and Casino $1,500 121st 7,977 $8,830 — —
No Limit Hold'em - Shootout (Event #6)
May 30 - Jun 1, 2014 Rio Hotel and Casino $1,500 67th 948 $4,411 — —
May 10 - 12, 2014 Harrahs Horseshoe Casino Hammond $250 43rd 1,106 $475 — —
WSOP Circuit - Colorado
Apr 29 - May 30, 2014 Lodge Casino - Black Hawk $365 10th 189 $1,076 — —
WSOP Circuit - St Louis
St Louis, MO, United States
Mar 28 - 31, 2014 Lumiere Place Casino and Hotels $1,675 25th 416 $4,462 — —
WPT Rolling Thunder
WPT Rolling Thunder Main Event
Mar 15 - 19, 2014 Thunder Valley Casino Resort $3,200 + 300 42nd 465 $8,000 — —
Mid-States Poker Tour - MSPT Baton Rouge
Baton Rouge, , United States
Feb 21 - 23, 2014 Belle of Baton Rouge Casino & Hotel $1,000 + 110 8th 118 $3,638 — —
WSOP Circuit - Robinsonville
Robinsonville, MS, United States
Feb 3 - 4, 2014 Harrahs Tunica $365 3rd 199 $6,923 — —
Limit Omaha H/L 8B Semi Turbo
Jan 25, 2014 Borgata Hotel Casino & Spa $500 + 60 3rd 85 $4,947 — —
Jan 9, 2014 Atlantis Resort & Casino $1,000 + 100 4th 48 $5,120 — —
WPT Regional Series - Winter Tampa Bay
WPT Winter Tampa Bay Open
Dec 13 - 15, 2013 Seminole Hard Rock Hotel & Casino $1,500 + 150 6th 301 $19,189 — —
Las Vegas, , United States
Nov 15 - 18, 2013 Stratosphere Casino, Hotel & Tower $1,500 + 150 20th 353 $4,340 — —
Wyandotte, Ok, United States
Oct 18 - 21, 2013 Indigo Sky Casino $1,500 + 150 9th 123 $4,788 — —
Heartland Poker Tour - Gold Rush Series
Fresno, CA, United States
Oct 4 - 7, 2013 Club One Casino $1,500 + 150 13th 246 $5,679 — —
Sep 19 - 23, 2013 Commerce Casino $1,500 + 150 9th 577 $16,820 — —
Borgata Poker Open 2013
Sep 15 - 20, 2013 Borgata Hotel Casino & Spa $3,300 + 200 63rd 1,189 $9,662 — —
H.O.R.S.E
Sep 10 - 11, 2013 Borgata Hotel Casino & Spa $400 + 50 3rd 112 $4,128 — —
No Limit Hold'em - Big Stack
Sep 7, 2013 Borgata Hotel Casino & Spa $350 + 50 34th 544 $923 — —
Seminole Hard Rock 2013 Poker Open
No Limit Hold'em - Championship Event
Aug 22 - 27, 2013 Seminole Hard Rock Hotel & Casino $5,300 322nd 2,384 $7,500 — —
No Limit Hold'em - Shootout (Event #36)
Jun 20 - 22, 2013 Rio Hotel and Casino $1,500 35th 1,194 $5,556 — —
Albuquerque, , United States
May 24 - 27, 2013 Route 66 Casino $1,500 + 150 3rd 86 $15,999 — —
May 14 - 15, 2013 Harrahs New Orleans $365 11th 350 $1,836 — —
May 10, 2013 Harrahs New Orleans $580 29th 296 $1,003 — —
WSOP Circuit - Harrahs Chester
Chester, PA, United States
No Limit Hold'em - Semi Turbo
May 6, 2013 Harrahs Chester $365 6th 108 $1,738 — —
Apr 20, 2013 Harrahs Horseshoe Council Bluffs $1,675 37th 367 $2,490 — —
Limit Omaha Hi/Lo
Apr 15, 2013 Harrahs Horseshoe Council Bluffs $365 8th 134 $1,274 — —
WSOP Circuit - Black Hawk
Mar 24 - 25, 2013 Lodge Casino - Black Hawk $580 1st 149 $20,859 — —
Altoona, , United States
Mar 15 - 18, 2013 Prairie Meadows Racetrack & Casino $1,500 + 150 9th $6,412 — —
2013 L.A. Poker Classic
Feb 13, 2013 Commerce Casino $550 3rd 57 $6,000 — —
WSOP Circuit - Harrah's Tunica
Feb 1, 2013 Harrahs Tunica $1,675 54th 666 $3,337 — —
Jan 27 - Feb 1, 2013 Borgata Hotel Casino & Spa $3,300 + 200 70th 1,042 $8,296 — —
Jan 22 - 23, 2013 Borgata Hotel Casino & Spa $500 + 60 1st 89 $15,107 — —
Jan 16, 2013 Choctaw Casino Resort $365 9th 266 $1,789 — —
Omaha Hi/Lo (Event #26)
Jan 11, 2013 Atlantis Resort & Casino $1,000 + 100 3rd 58 $11,680 — —
H.O.R.S.E (Event #16)
Colorado Poker Championship 5
Dec 21 - 24, 2012 Golden Gates Casino & Poker Parlour $1,000 + 100 13th 241 $3,940 — —
DeepStacks Poker Tour Mohegan Sun National Championship
Uncasville, United States
No Limit Hold'em - Deepstacks
Dec 2, 2012 Mohegan Sun $550 + 50 5th 120 $6,305 — —
Borgata Fall Poker Open
Nov 2 - 5, 2012 Route 66 Casino $1,500 + 150 11th $5,058 — —
2012 World Poker Finals
H.O.S.E.
Oct 26, 2012 Foxwoods Resort Casino $350 + 50 4th 87 $3,270 — —
Seven Card Stud Hi/Lo
WSOP Circuit - Bossier City
Bossier City, LA, United States
Sep 20, 2012 Horseshoe Bossier City $1,125 11th 144 $2,952 — —
Sep 15, 2012 Borgata Hotel Casino & Spa $350 + 50 1st 170 $17,314 — —
Sep 9, 2012 Borgata Hotel Casino & Spa $350 + 50 8th 151 $1,538 — —
WPT Parx Open Poker Classic
Bensalem, PA, United States
Aug 10 - 15, 2012 Parx Casino $3,300 + 200 35th 500 $8,803 — —
43rd World Series of Poker (WSOP) 2012
No Limit Hold'em - National Championship
Jul 6 - 8, 2012 Rio Hotel and Casino $10,000 18th 157 $22,278 — —
Jul 1 - 3, 2012 Rio Hotel and Casino $1,000 80th 3,221 $4,638 — —
Pot Limit Omaha Hi/Lo (Event #47)
Jun 26 - 28, 2012 Rio Hotel and Casino $1,500 71st 978 $3,697 — —
Limit 2-7 Triple Draw Lowball (Event #22)
No Limit Hold'em - Re-Entry (Event #9A)
WPT Jacksonville
Orange Park, FL, United States
Apr 27 - May 2, 2012 bestbet Orange Park $5,000 34th 320 $9,610 — —
Apr 7, 2012 Harrahs Horseshoe Council Bluffs $1,600 15th 290 $6,182 — —
Mar 8 - 11, 2012 River City Casino $1,500 + 150 10th 328 $11,192 — —
Borgata Winter Open 2012
No Limit Hold'em - Big Stack Re-Entry
Feb 1, 2012 Borgata Hotel Casino & Spa $200 + 30 88th 1,016 $414 — —
2012 Southern Poker Million Dollar Heater
Biloxi, MS, United States
Jan 21 - 25, 2012 Beau Rivage Hotel & Casino $5,000 + 175 11th 210 $14,259 — —
8 Game
Jan 19, 2012 Beau Rivage Hotel & Casino $1,000 + 60 4th 14 $1,358 — —
2011 Doyle Brunson Five Diamond World Poker Classic
Dec 6 - 10, 2011 Bellagio $10,000 + 300 45th 413 $15,922 — —
Dec 2, 2011 Bellagio $1,000 + 80 14th 224 $2,173 — —
Rock and Roll Poker Open
Nov 11 - 14, 2011 Seminole Hard Rock Hotel & Casino $2,500 4th 331 $54,052 — —
Oct 27 - Nov 1, 2011 Foxwoods Resort Casino $9,700 25th 189 $19,916 — —
2011 Borgata Poker Open
Sep 20, 2011 Borgata Hotel Casino & Spa $350 + 50 11th 340 $1,501 — —
2011 The River Guaranteed $3,000,000 Poker Series
Main Event - No Limit Hold'em
Sep 1 - 5, 2011 Winstar Casino $2,000 + 100 125th 1,355 $3,230 — —
2011 Legends of Poker
Aug 25 - 30, 2011 Bicycle Casino $3,500 + 200 28th 757 $10,280 — —
42nd World Series of Poker (WSOP) 2011
Pot Limit Omaha Hi/Lo
Jul 5 - 7, 2011 Rio Hotel and Casino $5,000 9th 352 $33,352 — —
Omaha/Seven Card Stud Hi/Lo
Pot Limit Hold'em
Jun 9 - 11, 2011 Rio Hotel and Casino $1,500 2nd 765 $140,309 — —
Regional Championship - No Limit Hold'em
May 19 - 22, 2011 Harrahs New Orleans $10,000 6th 75 $37,736 — —
May 11 - 12, 2011 Harrahs New Orleans $355 27th 423 $871 — —
WSOP Circuit Event - Harrah's Chester
May 9, 2011 Harrahs Chester $345 2nd 149 $7,426 — —
2011 Borgata Spring Poker Open
Omaha Hi/Lo
Apr 17, 2011 Borgata Hotel Casino & Spa $300 + 50 9th 120 $786 — —
Lawrenceburg, IN, United States
Apr 9 - 13, 2011 Hollywood Casino $9,600 + 400 8th 97 $39,884 — —
2011 Venetian Deep Stack Extravaganza
Feb 11, 2011 The Venetian Resort Hotel Casino $340 18th 207 $749 — —
2010 Winter Bayou Poker Challenge
Dec 10 - 12, 2010 Harrahs New Orleans $2,500 + 150 8th 108 $8,781 — —
Dec 3 - 8, 2010 Bellagio $10,000 + 30 45th 438 $16,892 — —
2010 LA Poker Open
No Limit Hold'em - Double Stack
Nov 15, 2010 Commerce Casino $335 25th 212 $570 — —
2010 United States Poker Championship
No Limit Hold'em - Deepstack
Nov 6 - 7, 2010 Trump Taj Mahal $550 + 50 4th 144 $6,146 — —
Festa Al Lago
Oct 15 - 20, 2010 Bellagio $10,000 + 300 7th 335 $80,600 — —
EPT - 7 - UKIPT - 1 - London
No Limit Hold'em - EPT Main Event (UKIPT Grand Final)
Sep 29 - Oct 4, 2010 Hilton London Metropole Hotel £5,000 + 250 100th 848 £9,000 — —
2010 World Series Of Poker - Europe
Sep 17 - 21, 2010 The Casino at the Empire £1,000 + 75 38th 582 £2,747 — —
Aug 20 - 25, 2010 Bicycle Casino $5,000 20th 462 $13,000 — —
2010 Summer Pot of Gold
Jul 29, 2010 Grand Sierra Resort $175 + 25 3rd 53 $1,350 — —
4th Annual Binion's Poker Classic
Jul 9, 2010 Binions $200 12th 85 $325 — —
Bellagio Cup VI
Jul 6, 2010 Bellagio $1,000 + 80 1st 44 $13,608 — —
41st World Series of Poker (WSOP) 2010
Pot Limit Omaha Hi/Lo 8
Jun 25 - 27, 2010 Rio Hotel and Casino $5,000 18th 284 $14,455 — —
H.O.R.S.E. Championship
Jun 23 - 25, 2010 Rio Hotel and Casino $10,000 23rd 241 $21,997 — —
Pot Limit Hold'em Championship
Jun 20 - 22, 2010 Rio Hotel and Casino $10,000 14th 268 $34,639 — —
Seven Card Stud Hi/Lo 8 Championships
Jun 6 - 8, 2010 Rio Hotel and Casino $10,000 2nd 170 $276,485 — —
Limit Deuce to Seven Triple Draw
Omaha Hi/Lo 8
WSOP Circuit - New Orleans & Bayou Poker Challenge
Mixed Game
May 12, 2010 Harrahs New Orleans $300 + 40 2nd 68 $4,155 — —
May 7, 2010 Harrahs New Orleans $300 + 40 6th 547 $7,085 — —
2010 Deep Stack Extravaganza II
Omaha Hi Lo
Apr 22 - 23, 2010 The Venetian Resort Hotel Casino $1,070 5th 265 $2,297 — —
2010 Wynn Classic
Mar 8, 2010 Wynn Las Vegas $500 + 45 7th 135 $1,964 — —
2010 Deep Stack Extravaganza
Feb 11, 2010 The Venetian Resort Hotel Casino $1,070 10th 145 $3,358 — —
Omaha Hi Lo - Stud Hi Lo
Jan 29 - 30, 2010 Borgata Hotel Casino & Spa $350 + 50 1st 141 $14,744 — —
Jan 10, 2010 Atlantis Resort & Casino $500 + 50 7th 54 $1,310 — —
Dec 18, 2009 Harrahs New Orleans $3,000 + 120 1st 80 $70,976 — —
2009 California State Poker Championship
May 13 - 14, 2009 Commerce Casino $300 + 35 30th 319 $689 — —
May 8 - 9, 2009 Commerce Casino $500 + 45 12th 114 $1,161 — —
Foxwoods Poker Classic 2009
Apr 3 - 8, 2009 Foxwoods Resort Casino $9,700 + 300 9th 259 $46,315 — —
Borgata $500,000 Guaranteed Deep Stack Poker Tournament
Mar 21 - 24, 2009 Borgata Hotel Casino & Spa $1,500 + 150 89th 919 $2,482 — —
Feb 12, 2009 Commerce Casino $1,000 + 65 34th 332 $2,415 — —
Seven Card Stud
Feb 6, 2009 Commerce Casino $500 + 45 13th 101 $735 — —
Jan 30, 2009 Harrahs Tunica $500 + 50 9th 416 $2,926 — —
WSOP Circuit - Lake Tahoe
Lake Tahoe, NV, United States
Nov 14, 2008 Harveys Resort & Casino $5,000 + 150 6th 132 $31,370 — —
Nov 2, 2008 Foxwoods Resort Casino $2,800 + 200 17th 105 $5,133 — —
2008 Caesars Palace Classic
Oct 22, 2008 Caesars Palace $500 + 50 16th 156 $908 — —
No Limit Hold'em - 6 Max
Sep 23, 2008 Trump Taj Mahal $300 + 40 10th 169 $1,014 — —
Sep 19, 2008 Trump Taj Mahal $500 + 50 6th 84 $2,100 — —
Sep 17, 2008 Trump Taj Mahal $500 + 50 9th 104 $1,040 — —
Deep Stack Extravaganza III
Jul 4, 2008 The Venetian Resort Hotel Casino $1,000 + 60 7th 245 $9,457 — —
Pot Limit Hold'em/Omaha
Jun 5 - 7, 2008 Rio Hotel and Casino $2,500 31st 338 $4,908 — —
May 17, 2008 Harrahs New Orleans $1,000 21st 284 $1,653 — —
May 14, 2008 Harrahs New Orleans $500 19th 272 $792 — —
Sixth Annual Five Star World Poker Classic - WPT World Championship
Apr 7, 2008 Bellagio $2,500 + 120 3rd 149 $36,940 — —
The Wynn Classic
Mar 16, 2008 Wynn Las Vegas $10,000 + 200 14th 183 $23,964 — —
Mar 3, 2008 Wynn Las Vegas $1,000 + 60 1st 140 $52,962 — —
Feb 9, 2008 Commerce Casino $1,000 + 65 7th 160 $6,050 — —
WSOP Circuit - Tunica
Tunica, MS, United States
Jan 14, 2008 Grand Casino Tunica $1,000 + 70 4th 131 $9,842 — —
2007 Vegas Open - NPL
Nov 26, 2007 The Venetian Resort Hotel Casino $2,350 + 150 2nd 47 $26,784 — —
2007 US Poker Bowl
No Limit Hold'em - Poker Bowl Finals
Oct 23, 2007 Palms 2nd $40,000 — —
2007 Binion's Poker Open
Oct 17, 2007 Binions $200 6th 97 $1,100 — —
Oct 2 - 6, 2007 Trump Taj Mahal $9,700 + 300 18th 164 $20,680 — —
Aug 10, 2007 Bicycle Casino $485 + 45 12th 462 $2,675 — —
2007 Orleans Open
Omaha Hi/Lo - Championship Event
Jul 27, 2007 Orleans Hotel & Casino $540 11th 167 $960 — —
World Championship Omaha Hi-Low Split-8 or Better
WSOP Circuit - Caesars Palace
Apr 25, 2007 Caesars Palace $500 + 50 1st 522 $73,419 — —
Fifth Annual Five Star World Poker Classic
Apr 16, 2007 Bellagio $3,000 + 180 17th 324 $7,340 — —
Apr 8, 2007 Bellagio $2,000 + 120 12th 288 $6,400 — —
Mar 30, 2007 Foxwoods Resort Casino $9,700 + 300 6th 415 $136,452 — —
Mar 28, 2007 Foxwoods Resort Casino $4,800 + 200 9th 110 $14,297 — —
World Poker Challenge
Mar 17, 2007 Grand Sierra Resort $500 + 50 3rd 207 $9,525 — —
Deep Stack Extravaganza
Mar 1, 2007 The Venetian Resort Hotel Casino $300 + 30 25th 280 $423 — —
Feb 28, 2007 The Venetian Resort Hotel Casino $300 + 30 26th 280 $407 — —
Feb 12, 2007 Commerce Casino $2,500 + 90 22nd 235 $3,420 — —
WSOP Circuit - Grand Tunica
Jan 9, 2007 Harrahs Tunica $1,000 + 60 13th 181 $2,458 — —
Fifth Annual Five Diamond World Poker Classic
WPT Doyle Brunson North American Poker Classic - No Limit Hold'em
Dec 14 - 19, 2006 Bellagio $15,000 + 400 55th 583 $25,370 — —
WPT Championship Event - No Limit Hold'em
Nov 12 - 16, 2006 Foxwoods Resort Casino $9,700 + 300 47th 609 $14,309 — —
7 Card Stud Hi/Lo
Nov 8 - 9, 2006 Foxwoods Resort Casino $530 + 70 7th 164 $2,587 — —
Ultimate Poker Challenge
Oct 21, 2006 Binions $300 + 40 6th 142 $1,840 — —
Festa Al Lago V Poker Tournament
Nightly No Limit Hold'em
Oct 11, 2006 Bellagio $1,000 + 80 2nd 71 $17,220 — —
Sep 22, 2006 Trump Taj Mahal $500 + 40 1st $22,000 — —
Sep 20, 2006 Trump Taj Mahal $300 + 40 16th $587 — —
2006 Borgata Poker Open - WPT
Sep 11 - 12, 2006 Borgata Hotel Casino & Spa $500 + 60 9th 157 $1,806 — —
Jul 24 - 25, 2006 Rio Hotel and Casino $1,000 59th 788 $1,793 — —
Jul 19 - 20, 2006 Rio Hotel and Casino $5,000 4th 183 $76,986 — —
Jul 8 - 10, 2006 Rio Hotel and Casino $1,000 48th 752 $5,447 — —
Jul 3 - 5, 2006 Rio Hotel and Casino $2,000 53rd 670 $3,049 — —
Jun 23 - 25, 2006 Imperial Palace $500 1st 253 $51,619 — —
The Vegas Open
Jun 7 - 8, 2006 Caesars Palace 10th 105 $2,035 — —
2006 WSOP Circuit - Caesars Las Vegas
May 2 - 4, 2006 Caesars Palace $5,000 4th 51 $23,750 — —
Fourth Annual Five-Star World Poker Classic
Apr 14, 2006 Bellagio $5,000 + 150 17th 347 $13,260 — —
Limit 7 Card Stud
Mar 30 - 31, 2006 Foxwoods Resort Casino $530 + 70 16th 236 $826 — —
World Poker Challenge 2006
Mar 23, 2006 Grand Sierra Resort $500 + 50 18th 311 $1,166 — —
2006 Winnin' o' the Green
Mar 13, 2006 Bicycle Casino $500 + 50 24th 262 $655 — —
Bay 101 Shooting Stars
WPT - Main Event - No Limit Hold'em
Feb 27 - Mar 3, 2006 Bay 101 $10,000 35th 518 $25,000 — —
WPT Invitiational - No Limit Hold'em
Feb 22 - 23, 2006 Commerce Casino 3rd $20,000 — —
2006 Gold Strike World Poker Open
Championship Event - No Limit Hold'em
Jan 19, 2006 Gold Strike Casino Resort $10,000 + 200 35th 327 $15,732 — —
Jan 17, 2006 Gold Strike Casino Resort $500 + 40 14th 257 $1,246 — —
Fourth Annual Five Diamond World Poker Classic
Nov 28, 2005 Plaza Hotel & Casino $1,000 4th 13 $525 — —
Nov 26, 2005 Plaza Hotel & Casino $500 + 50 10th 151 $1,465 — —
Bellagio Festa Al Lago IV
No Limit Hold'em (Evening)
Oct 14, 2005 Bellagio $500 1st 144 $25,840 — —
Oct 10, 2005 Bellagio $2,000 + 80 17th 217 $3,955 — —
Sep 21, 2005 Trump Taj Mahal $300 + 40 23rd 305 $915 — —
Sep 16 - 17, 2005 Borgata Hotel Casino & Spa $2,500 + 150 31st 324 $2,442 — —
2005 WSOP Tournament Circuit - Harrah's Las Vegas
Sep 12 - 13, 2005 Harrah's Las Vegas $1,500 + 70 6th 79 $6,897 — —
Sep 6 - 7, 2005 Harrah's Las Vegas $500 + 50 14th 268 $1,300 — —
Bellagio Weekly Tournaments - Jul 2005
Jul 30, 2005 Bellagio $1,000 + 60 2nd 60 $14,550 — —
Jul 23, 2005 Plaza Hotel & Casino $5,000 + 150 3rd 24 $14,110 — —
Jul 17, 2005 Plaza Hotel & Casino $1,500 + 70 7th 42 $3,055 — —
Jun 21, 2005 Rio Hotel and Casino $2,500 2nd 359 $132,110 — —
Jun 4, 2005 Rio Hotel and Casino $1,500 42nd 1,071 $4,285 — —
Third Annual Five-Star World Poker Classic
Apr 11, 2005 Bellagio $2,000 + 80 5th 316 $26,440 — —
Bellagio Weekly Tournaments - Apr 2005
Apr 1, 2005 Bellagio $1,000 + 60 5th 155 $6,765 — —
Bellagio Weekly Tournaments - Mar 2005
Mar 25, 2005 Bellagio $1,000 + 60 1st 118 $45,784 — —
Mar 23, 2005 Bellagio $500 + 40 17th 133 $645 — —
WSOP Circuit - San Diego
Feb 24 - 25, 2005 Harrahs Rincon $1,000 + 60 13th 173 $2,015 — —
Jan 11 - 12, 2005 Harrahs Atlantic City $1,500 + 80 6th 119 $10,710 — —
Limit Hold'em
Jan 8 - 9, 2005 Harrahs Atlantic City $500 + 60 16th 628 $1,240 — —
2004 Festa al Lago III Poker Tournament
No Limit Hold'em Final Day
Oct 14 - 15, 2004 Bellagio $1,500 + 70 7th 234 $7,874 — —
Sep 21, 2004 Trump Taj Mahal $300 + 35 11th 214 $834 — —
Festa al Lago II
No Limit Hold'em Championship Final Day
Jul 3 - 5, 2004 Bellagio $10,000 + 200 27th 181 $10,381 — —
California State Poker Championship 2004
Jun 14, 2004 Commerce Casino $500 + 40 16th 147 $2,040 — —
Seven Card Stud Final Day
Jun 9, 2004 Commerce Casino $1,000 + 60 3rd 53 $6,360 — —
2004 New England Poker Classic
H.O.S.E
Mar 31, 2004 Foxwoods Resort Casino $300 + 40 13th 202 $939 — —
Mar 18, 2004 Bicycle Casino $500 + 40 1st 72 $14,400 — —
Seven Card Stud Eight Or Better
Nov 4, 2003 Foxwoods Resort Casino $500 + 65 4th 129 $4,655 — —
2003 Bellagio Five-Star World Poker Classic WPT Championship
Apr 4, 2003 Bellagio $1,500 9th 128 $2,980 — —
2002 Four Queens Poker Classic
Sep 26, 2002 Four Queens Hotel & Casino $300 7th 97 $1,640 — —
Omaha Hi-Lo Split Eight or Better
May 11, 2001 Binions $5,000 + 150 16th 107 $5,710 — —
Jan 15, 2020 #344 1760.25
Jan 8, 2020 #349 1794.82
Dec 25, 2019 #368 1823.97
Dec 4, 2019 #387 1781.69
Nov 27, 2019 #336 1840.71
Nov 6, 2019 #355 1827.74
Oct 30, 2019 #360 1827.74
Oct 9, 2019 #317 1903.20
Sep 25, 2019 #311 1904.43
Sep 4, 2019 #312 1895.20
Aug 28, 2019 #302 1896.14
Aug 14, 2019 #295 — 1901.58
Aug 7, 2019 #295 1901.58
Jul 31, 2019 #292 1901.83
Jul 3, 2019 #269 1913.54
Jun 26, 2019 #263 1934.84
Jun 5, 2019 #328 1799.71
May 29, 2019 #355 — 1747.67
May 22, 2019 #355 1747.67
May 8, 2019 #354 1747.08
Apr 24, 2019 #316 1820.97
Apr 3, 2019 #323 1804.08
Mar 27, 2019 #313 1810.27
Mar 6, 2019 #304 1795.34
Feb 27, 2019 #301 1797.94
Feb 6, 2019 #324 1806.81
Aug 1, 2018 #314 — 1833.61
Feb 14, 2018 #278 — 1797.46
Sep 6, 2017 #417 — 1664.80
Jul 12, 2017 #348 — 1737.47
Oct 7, 2015 #172 — 2030.74
May 6, 2015 #140 — 2030.45
Sep 10, 2014 #116 — 2185.55
Jan 29, 2014 #122 — 1981.07
Jul 5, 2013 #135 — 1591.66
Jun 27, 2013 #123 — 1634.51
Jun 4, 2013 #114 — 1667.78
May 6, 2013 #99 1686.35
Apr 29, 2013 #97 1692.31
Apr 8, 2013 #84 1752.02
Mar 11, 2013 #91 1737.10
Mar 4, 2013 #94 1742.05
Feb 25, 2013 #88 1742.05
Feb 18, 2013 #90 — 1750.78
Feb 4, 2013 #79 1807.40
Jan 28, 2013 #92 1695.78
Jan 7, 2013 #75 1747.28
Dec 31, 2012 #68 1814.55
Dec 10, 2012 #76 — 1661.33
Dec 3, 2012 #76 1698.87
Nov 26, 2012 #72 1695.63
Nov 5, 2012 #75 1690.78
Oct 29, 2012 #74 1690.78
Oct 8, 2012 #66 1778.68
Sep 24, 2012 #69 1788.82
Sep 3, 2012 #71 1788.82
Aug 27, 2012 #68 1788.82
Aug 13, 2012 #74 — 1769.50
Aug 6, 2012 #74 1769.50
Jul 30, 2012 #75 — 1769.50
Jul 23, 2012 #75 1769.50
Jul 9, 2012 #74 1761.24
Jun 25, 2012 #90 1670.73
Jun 4, 2012 #61 1717.92
May 28, 2012 #50 1778.78
Jan 30, 2012 #32 — 2010.40
Nov 28, 2011 #63 — 1575.77
Jun 27, 2011 #47 — 1613.70
Jun 12, 2019 #790 739.43
Jun 5, 2019 #740 707.15
May 29, 2019 #1463 516.82
May 8, 2019 #1122 516.82
Apr 24, 2019 #923 516.82
Apr 17, 2019 #1488 383.64
Apr 3, 2019 #1209 383.64
Mar 27, 2019 #1292 347.18
Mar 6, 2019 #5981 119.97
Feb 27, 2019 #22868 36.17
Feb 6, 2019 #14736 36.17
Jan 30, 2019 #11864 36.17
Jan 23, 2019 #7672 — 36.17
$1,058 0 0
$173,435 0 4
$94,678 0 2
$188,530 2 15
Bio of Allen Kessler
Whether he's known for his decades of poker play or the nickname "Chainsaw," Allen Kessler has been a poker grinder longer than many of today's players have been alive. It took awhile to gain the respect of said players, but he is showing that he has what it takes to keep up with an ever-changing game.
Allen Kessler was born in the 1960s and raised in Pennsylvania and Atlantic City. He pursued his early education and then attended Temple University, from which he graduated with a double degree in marketing and management. It was in college that he discovered poker through friends, and he was intrigued by the intricacies of the game.
As he moved into his career as a market research interviewer, where he worked for nearly a decade, he also played more poker with coworkers. He found weekends convenient for spending time in Atlantic City casinos, where he honed his Omaha and stud games and moved up in cash game limits alongside players like Cyndy Violette, John Hennigan, and Phil Ivey. He eventually took up hold'em as well and moved into the realm of tournaments.
After making a living for years at the $75/$150 and $100/$200 mixed games, as well as traveling to Las Vegas more frequently for tournaments, he was prepared to quit his day job. By 2005, poker had become an important part of his life, and he moved to Las Vegas to pursue it on a full-time basis.
Prior to that decision, Kessler had numerous final tables to his credit, from the World Poker Finals at Foxwoods to the California State Poker Championship in Los Angeles. He produced a 2004 win in a $500 7-Stud H/L event at the Winnin' o' the Green series, as well as in a Bellagio Weekly Tournament, making $45,784 for the $1,000 NLHE event.
The full-time pursuit brought more success. He final tabled a 2005 World Series of Poker $2,500 Omaha H/L event, finishing second for $132,110, and he won a $500 NLHE event at the Bellagio Festa al Lago later that year.
Kessler's first major televised final table came on the World Poker Tour with their Invitational event, and he finished in third, but it was back at the WSOP where he made another final table, that one producing $76,986 for finishing fourth in the $5,000 7-Stud tournament. And more wins piled up, with one at Poker 101 in Las Vegas, a 7-Stud event at the U.S. Poker Championship, and on the WSOP Circuit in Las Vegas.
Some of his more well-known finishes include sixth place at the 2007 WPT Foxwoods Poker Classic for $136,452 and wins at the Wynn Classic in 2008 and the 2009 Winter Bayou Poker Challenge Main Event in New Orleans. He also made a serious run at the 2010 WSOP, with eight cashes, the highest one being second place in the $10,000 7-Stud H/L Championship for $276,485. He felt that he finally began to earn the respect of his peers and poker fans.
Nicknamed "Chainsaw" at the 2007 WPT Foxwoods event for some intimidating plays, he embraced the title because it contradicted his reputation as a nitty player. And with nearly $2.3 in live tournament earnings through the first half of 2011, his success is undeniable.
See the bio
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\section{Introduction}
\subsection{Background}
Perturbative quantum field theory is the standard framework used by particle physicists to predict and explain high-energy experiments, e.g. at modern colliders like the LHC. This necessitates the computation of a large number of complicated integrals. These Feynman integrals grow quickly in number and complexity, so on the one hand one wants to find methods to compute them as efficiently as possible, and on the other hand one looks for hidden structures that reduce the amount of necessary computations.
To that end, the Schwinger parametric representation of Feynman integrals has proved to be very useful in recent years. It was already known in the early days of quantum field theory \cite{ Nambu_1957_ParaRep, Nakanishi1957, Nakanishi1961, Symanzik1958, Kinoshita1962, BergereZuber_1974_ParaRenorm, Bergere1976 }, but fell somewhat out of favour, since it was not as suitable for direct integration as other versions of Feynman integrals. This problem was rectified when, building on the connections to algebraic geometry found in \cite{BlochEsnaultKreimer_2006_Motives}, an algorithm for the systematic integration of parametric Feynman integrals was developed \cite{Brown_2009_Massless, Brown_2010_Periods} and subsequently implemented in computer algebra \cite{Panzer_2014_Algorithms}.
The renewed interest in parametric Feynman integrals has already yielded many interesting results \cite{BrownSchnetz_2012_K3, BrownYeats_2011_SpanningForest, BrownSchnetzYeats_2014_C2, KreimerSarsSuijlekom_2013_QuantGauge, BrownKreimer_2013_AnglesScales, KompanietsPanzer_2016, BBKP_ParaAnn_2018, Bloch_Kreimer_2015 }. However, those have all been confined to scalar quantum field theories. In that case the (unrenormalised) integral is simply of the form
\al{
\phi_{\Gamma} = \int_{\mathbb{R}_+^{|E_{\Gamma}|}} \mathrm{d}\alpha_1\dotsm \mathrm{d}\alpha_{E_{\Gamma}} \frac{\exp\Big( -\frac{\ssp}{\kp} \Big)}{\kp^2},
}
where $\Gamma$ is a Feynman graph and $\kp$, $\ssp$ are two homogenous polynomials that will be discussed extensively below.
In gauge theories this becomes much more complicated and until recently the integrand could only be expressed in terms of complicated derivatives of the scalar integrand \cite{CvitanovicKinoshita_1974_FRPS, KreimerSarsSuijlekom_2013_QuantGauge}. For quantum electrodynamics the combinatorics of these derivatives have been analysed in \cite{Golz_2017_CyclePol} and it was found that they can be expressed explicitly in terms of graph polynomials similar to $\kp$ and $\ssp$. The other complication of QED, the tensor structure consisting of products of Dirac matrices, was dealt with in \cite{Golz_2017_Traces}. Combining these results yields an (unrenormalised, massless) parametric Feynman integral for QED that is of the form
\al{
\phi_{\Gamma} = \int_{\mathbb{R}_+^{|E_{\Gamma}|}} \mathrm{d}\alpha_1\dotsm \mathrm{d}\alpha_{E_{\Gamma}} \frac{\exp\Big( -\frac{\ssp}{\kp} \Big)}{ \kp^{2+h_1(\Gamma)} }
\sum_{l=0}^{h_1(\Gamma)} \frac{ I_{\Gamma}^{(l)} }{\kp^l} \label{eq_int_unren}
}
where each $I_{\Gamma}^{(l)}$ is essentially (cf. \refeq{eq_sumdetail} and sec. \ref{sec_application}) just a sum over certain subsets of chord diagrams $D$,
\al{
\sum_D (-2)^{\tilde c(D)} X_D,
}
where $X_D$ is a product of the polynomials from \cite{Golz_2017_CyclePol} and $\tilde c(D)$ is an integer determined by the combinatorial properties of $D$. In our main results, theorems \ref{theo_main} and \ref{theo_main_2}, we prove that the sums in $I_{\Gamma}^{(0)}$ and $I_{\Gamma}^{(1)}$ are equal to a simpler sum of the form
\al{
2^{-k}\sum_{l=1}^{h_1(\Gamma)} (-\kp)^{h_1(\Gamma)-l+k} (l+1)!\ Z_{\Gamma}^k\big|_l\qquad\qquad\text{for } k=0,1, \label{eq_result_intro}
}
where $Z_{\Gamma}^k\big|_l$ is defined in sec. \ref{sec_defpp}. This leads to cancellations of Kirchhoff polynomials $\kp$ in \refeq{eq_int_unren}, significantly simplifying the integrand. On the concrete example of a massless photon propagator graph in Feynman gauge we show that the cases of $k=0,1$ suffice to express the superficially renormalised integral with a simple entirely scalar integrand.
\paragraph{Generalisations and extensions.}
The results of this article are not just applicable to this rather specific photon propagator. In a general gauge one gets another sum and our results apply to each summand (see \refeq{eq_gengauge}). The same holds for the inclusion of masses, if one assumes quite reasonably that all fermion masses are identical. The parametric renormalisation of massive integrals is much more cumbersome than the simple renormalisation procedure that we employ in section \ref{subsec_structure}, but in principle not a problem \cite{BrownKreimer_2013_AnglesScales}.
For a fermion propagator and a vertex with one external momentum set to zero the differences are basically just a few different factors in the computations of section \ref{subsec_structure} (e.g. the fermion propagator would be proportional to $\slashed q$ rather than $q^2q^{\mu\nu}-q^{\mu}q^{\nu}$). The step to the full vertex function is more complicated and needs more attention in future work. Much of this article and especially section \ref{sec_inc} is based on the assumption that the polynomial $\ssp$ factorises into $q^2\vssp$ with a $q$-independent $\vssp$. It will need to be seen how much the results have to be modified if one has instead a polynomial $\ssp = q_1^2 \vssp[\Gamma,1] + q_2^2 \vssp[\Gamma,2] + (q_1+q_2)^2 \vssp[\Gamma,3]$.
Finally, in order to include subdivergences one also needs to understand the higher order terms $I_{\Gamma}^{(k)}$ with $k\geq 2$. In \refeq{eq_result_intro} we already suggest what this should look like, although it is not yet entirely clear how the $Z_{\Gamma}^k\big|_l$ have to be defined for $k\geq 2$.\\[3mm]
\subsection{Graph polynomials}
A graph $G$ is an ordered pair $(V_G, E_G)$ of the set of \textit{vertices} $V_G = \{ v_1, \dotsc, v_{|V_G|}\}$ and the set of \textit{edges} $E_G = \{ e_1, \dotsc, e_{|E_G|}\}$, together with a map $\partial: E_G \to V_G \times V_G$. We assume that $G$ is connected and assign to each edge $e\in E_G$ a direction by specifying an ordered pair $\partial(e) = ( \partial_-(e), \partial_+(e) )$, where the vertex $\partial_-(e)\in V_G$ is called start or initial vertex while $\partial_+(e)\in V_G$ is called target or final vertex. In a common abuse of notation subgraphs $g\subseteq G$ are identified with their edge set $E_g\subseteq E_G$. In the rare cases in which the edge set does not uniquely identify the subgraph, i.e. when $g$ contains isolated vertices without incident edges, it will be mentioned explicitly. The number of independent cycles (loops, in physics nomenclature) is denoted $h_1(G)$, which is the first Betti number of the graph.
Graph polynomials are polynomial valued invariants of a graph. The polynomials that we are interested in all have in common that their variables are the Schwinger parameters $\alpha = (\alpha_e)_{e \in E_G}$ assigned to the edges of a graph (which distinguishes them from other famous graph polynomials like the Tutte polynomial \cite{Tutte_1954, Tutte_2004} and its various specialisations like the chromatic polynomial \cite{Birkhoff_1912, Whitney_1932_coloring}). In the following we briefly introduce and review some properties of six such graph polynomials that appear in Feynman integrals.
\subsubsection{Kirchhoff and Symanzik}
A tree $T$ is a graph that is connected and simply connected, i.e. it has no cycles. A disjoint union of trees $F=\sqcup_{i=1}^kT_i$ is called a $k$-forest, such that a tree is a $1$-forest. If all vertices of $G$ are contained in such a subgraph $T$ or $F$, then it is called a spanning tree or spanning forest of $G$ and we denote with $\mathcal{T}_G^{[k]}$ the set of all such spanning $k$-forests.
The Kirchhoff polynomial, which is especially in the physics literature also often called the first Symanzik polynomial, is then defined as
\al{
\kp[G](\alpha) \defeq \sum_{T\in \mathcal{T}_G^{[1]}} \prod_{e\notin T} \alpha_e.
}
It has been known for a very long time and was first introduced by Kirchhoff in his study of electrical circuits \cite{Kirchhoff_1847_Kirchhoff}. In the 1950s it was then rediscovered in quantum field theory \cite{Symanzik1958}. We will often make use of the abbreviation
\al{
\alpha_S \defeq \prod_{e \in S}\alpha_e
}
for any edge subset $S\subset E_G$, such that $\kp[G] = \sum_T \alpha_{E_G \setminus T}$. The Kirchhoff polynomial is homogeneous of degree $h_1(G)$ in $\alpha$ and linear in each $\alpha_e$. Moreover, it also satisfies the famous contraction-deletion relation\footnote{Note that we use the double-slash to denote contraction of an edge subset, as opposed to contraction of a subgraph $\chGamma$ in the Hopf algebra of Feynman graphs. The two notions differ if the subgraph in question is a propagator Feynman graph, but in this article we will not encounter this problem.}
\al{
\kp[G] = \kp[G\dslash e] + \alpha_e\kp[G\setminus e].
}
This means in particular that $\kp[G\dslash e] = \left.\kp[G]\right|_{\alpha_e=0}$ and $\kp[G\setminus e] = \partial_e \kp[G]$, where $\partial_e$ denotes the partial derivative w.r.t. $\alpha_e$. The definition of the Kirchhoff polynomial is commonly generalised to disjoint unions of graphs $G=\sqcup_i G_i$ (i.e. graphs with multiple connected components) via
\al{
\kp[G] \defeq \prod_i \kp[G_i]. \label{eq_kp_disj}
}
Note that due to this definition one needs to exclude bridges from the contraction-deletion relation, since $\partial_e \kp[G]=0$ for any bridge $e$, whereas this definition gives $\kp[G\setminus e]$ as the product of the polynomials of its two connected components.\\
Many properties of graphs can be captured by matrices and we discuss here some of the well known relations between graphs, matrices and the Kirchhoff polynomial. The incidence matrix $I$ is an $|E_G| \times |V_G|$ matrix whose entries are defined as
\al{
I_{ev} \defeq \begin{cases} \hspace{3mm} \pm1 &\text{ if $v = \partial_{\pm}(e)$},\\
\hspace{3mm} 0 &\text{ if $e$ is not incident to $v$}. \end{cases}
}
The Laplacian matrix $L$ is defined as the difference of the degree and adjacency matrices of a graph. Since we will not need either of those two going forward we instead use a well known identity to define the Laplacian as the product of the incidence matrix and its transpose,
\begin{align}
L \defeq I^TI.
\end{align}
Instead of the full matrices we will actually always need the smaller matrices in which one column (of $I$) or one column and one row (of $L$) corresponding to an arbitrarily chosen vertex $v_0$ of $G$ are deleted. From now on we use $I'$ and $L'$ for these $|E_G| \times |V_G|-1$ and $|V_G|-1 \times |V_G|-1$ matrices, called reduced incidence and reduced Laplacian matrix.
Finally, let $A$ be the diagonal $|E_G|\times |E_G|$ matrix with entries $A_{ij} \defeq \delta_{ij} \alpha_{e_i}$. With this setup the well known \textit{Matrix-Tree-Theorem} \cite{Chaiken_1978} tells us that
\al{
\kp[G] = \alpha_{E_G} \det( {I'}^TA^{-1}I' ).
}
Note that here we have the inverse $A^{-1}$, with entries $A_{ij}^{-1} = \delta_{ij} \alpha_{e_i}^{-1}$. We call the matrix in that determinant the weighted reduced Laplacian and denote it with $\tilde L' = {I'}^TA^{-1}{I'}$.
\begin{remark}
The polynomial $\kp[G]^* = \det( {I'}^TAI' )$ is sometimes called dual Kirchhoff polynomial. If $G$ is planar then it is the Kirchhoff polynomial of its planar dual graph $G^*$.
\end{remark}
\noindent Often $I'$ and $A$ are arranged in a block matrix
\al{
M \defeq \begin{pmatrix} A & I' \hphantom{x}\\[2mm] -{I'}^T & 0 \hphantom{x} \end{pmatrix}. \label{eq_graphmatrix}
}
This is called the expanded Laplacian or graph matrix of $G$ \cite{Brown_2010_Periods, BlochEsnaultKreimer_2006_Motives}, and with the block matrix determinant identity
\al{
\det \begin{pmatrix} S & T\\[1mm] U & V \end{pmatrix} = \det(S)\det(V - US^{-1}T) \label{eq_blockID}
}
one sees that
\al{
\det(M) = \det(A)\det( {I'}^TA^{-1}I' ) = \kp[G]. \label{eq_detgmkp}
}
\begin{example}\label{ex_kp1}
Let $G$ be the wheel with three spokes depicted on the left of \reffig{01_WheelFeynman}. It has 16 spanning trees and the Kirchhoff polynomial is
\al{\nonumber
\kp[G] &= \alpha_1\alpha_2\alpha_4 + \alpha_1\alpha_2\alpha_5 + \alpha_1\alpha_3\alpha_4 + \alpha_1\alpha_3\alpha_5
%
+ \alpha_1\alpha_2\alpha_6 + \alpha_1\alpha_3\alpha_6 + \alpha_1\alpha_4\alpha_6 + \alpha_1\alpha_5\alpha_6\\[2mm]
%
&+ \alpha_2\alpha_3\alpha_4 + \alpha_2\alpha_3\alpha_5 + \alpha_2\alpha_4\alpha_5 + \alpha_3\alpha_4\alpha_5
%
+ \alpha_2\alpha_3\alpha_6 + \alpha_2\alpha_4\alpha_6 + \alpha_3\alpha_5\alpha_6 + \alpha_4\alpha_5\alpha_6.
}
\end{example}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.3\textwidth]{./Images/Fig_1_1a.pdf}\hspace{15mm}
\includegraphics[width=0.52\textwidth]{./Images/Fig_1_1b.pdf}
\end{center}
\caption{The wheel with three spokes $G=W\!S_3=K_4$ and a QED Feynman graph $\Gamma$ that corresponds to the wheel with edge $e_6$ cut to become an external photon edge.}
\label{01_WheelFeynman}
\end{figure}
Unlike the Kirchhoff polynomial, the second Symanzik polynomial is not defined for generic graphs but only for Feynman graphs, which carry additional information. Feynman graphs can have different types of edges, like photons ($ \includegraphics[height=0.5\baselineskip]{Images/Prop_Photon.pdf} $) and fermions ($ \includegraphics[height=0.5\baselineskip]{Images/Prop_Fermion.pdf} $) in the graph on the r.h.s. of \reffig{01_WheelFeynman}. Moreover, Feynman graphs have so-called external edges, i.e. half edges incident to only a single vertex. One associates to the external edges external momenta $q_i$, that are real euclidean vectors $q_i\in \mathbb{R}^4$ for this article, but may also be minkowskian, $D$-dimensional, or complex, depending on context. To distinguish them from generic graphs $G$ we use $\Gamma$ for Feynman graphs. The second Symanzik polynomial is then defined as \cite{Symanzik1958}
\al{
\ssp(\alpha, q) \defeq \sum_{(T_1,T_2) \in \mathcal{T}_{\Gamma}^{[2]}} \!\!\!\! s(q,T_1,T_2) \prod_{e\notin T_1\cup T_2} \alpha_e
}
where one sums over spanning $2$-forests. The function $s$ is the square of the momentum flow between the two trees, i.e. the sum of all external momenta entering either tree (which is the same for both trees due to momentum conservation).
If, as in \reffig{01_WheelFeynman} for example, there are only two (non-zero) external momenta, such that $q_1 = -q_2 \equiv q$ by momentum conservation, then the second Symanzik polynomial factorises and we write
\al{
\ssp = q^2\vssp.\label{eq_ssp_fac}
}
We focus on this case for this article. Note that $\vssp$ is also a Kirchhoff polynomial, namely that of the graph $\Gamma^{\bullet}$, which results from adding the external edge between the two external vertices and then contracting it.
The second Symanzik polynomial can also be expressed in terms of matrices. When deriving parametric Feynman integrals it appears in the form of the inverse Laplacian $\tilde {L'}^{-1}$ multiplied from both sides with vectors collecting all external momenta. Using cofactors to invert the matrix and expanding the matrix products as sums this yields
\al{
\ssp = \alpha_{E_{\Gamma}} \!\! \sum_{v_1,v_2 \in V_{\Gamma}'}\!\!\!\! q_{v_1}\!\! \cdot q_{v_2} (-1)^{v_1+v_2}\det( \tilde {L'}^{\{v_1\}}_{\{v_2\}} ),\label{eq_ssp_matrix}
}
where $V_{\Gamma}' = V_{\Gamma} \setminus \{v_0\}$ is the set of all vertices except the one whose row and column was removed from all matrices to get their reduced versions.
\begin{example}
Let $\Gamma$ be the Feynman graph on the r.h.s of \reffig{01_WheelFeynman}. Its Kirchhoff polynomial is just
\al{\nonumber
\kp &= \alpha_1\alpha_2 + \alpha_1\alpha_3 + \alpha_1\alpha_4 + \alpha_1\alpha_5
+\alpha_2\alpha_3 + \alpha_2\alpha_4 + \alpha_3\alpha_5 + \alpha_4\alpha_5\\
&= (\alpha_2+\alpha_5)(\alpha_3+\alpha_4) + \alpha_1(\alpha_2+\alpha_3+\alpha_4+\alpha_5),
}
which is the derivative w.r.t. $\alpha_6$ of the Kirchhoff polynomial from example \ref{ex_kp1}.
There are a total of 10 spanning 2-forests, but not all of them contribute to the second Symanzik polynomial. Consider the spanning 2-forest with $T_1 = \{e_2,e_5\}$ and $T_2$ just the isolated vertex $v_3$ without edges. The external momentum $q_1$ enters $T_1$ in the vertex $v_1$ and $q_2$ (which has to be $-q_1$ due to momentum conservation) enters $T_2$ in $v_3$. Hence, the corresponding monomial is $q_1^2\alpha_1\alpha_3\alpha_4$. An example of a forest that does not contribute is $T_1=\{e_2,e_3\}$ and $T_2$ just the vertex $v_4$. The external momenta entering $T_1$ in $v_1$ and $v_3$ add up to $0$, whereas $T_2$ is not adjacent to any external edges at all. Hence $s(q_1,q_2,T_1,T_2) = 0$ in this case. Overall, 8 of the 10 forests contribute to yield the second Symanzik polynomial
\al{
\ssp = q^2\vssp & = q^2\big( \alpha_2\alpha_5(\alpha_1+\alpha_3+\alpha_4)
+\alpha_3\alpha_4(\alpha_1+\alpha_2+\alpha_5)+ \alpha_1\alpha_2\alpha_4+\alpha_1\alpha_3\alpha_5\big).
}
Expanding the polynomial one sees that
\al{
\vssp = \alpha_1\alpha_2\alpha_4 + \alpha_1\alpha_2\alpha_5 + \alpha_1\alpha_3\alpha_4 + \alpha_1\alpha_3\alpha_5
+ \alpha_2\alpha_3\alpha_4 + \alpha_2\alpha_3\alpha_5 + \alpha_2\alpha_4\alpha_5 + \alpha_3\alpha_4\alpha_5
}
is indeed $\kp[\Gamma^{\bullet}] = \kp[\con{G}{e_6}] = \kp[G]\big|_{\alpha_6=0}$, where $G$ is the wheel from example \ref{ex_kp1}.
\end{example}
\subsubsection{Bonds and cycles}
A \textit{bond} $B\subset G$ is a minimal subgraph $G$ such that $G\setminus B$ has exactly two connected components. A simple \textit{cycle} $C\subset G$ is a subgraph of $G$ that is 2-regular, i.e. all vertices have exactly two edges incident to it, and it has only one connected component. The sets of bonds and simple cycles of a graph $G$ are denoted $\mathcal{B}_G$ and $\mathcal{C}_G^{[1]}$.
In \cite{Golz_2017_CyclePol} two polynomials based on these types of subgraphs were defined and it was shown that they can be used to express the Schwinger parametric integrand in quantum electrodynamics without derivatives. The basic bond polynomial and cycle polynomial are
\al{
\beta_G(\alpha,\xi) &\defeq \sum_{B \in \mathcal{B}_G} \left(\sum_{e \in B} \mathrm{o}_B(e) \xi_e \right)^2 \alpha_B\kp[G\setminus B],\\[3mm]
\chi_G(\alpha,\xi) &\defeq \sum_{C \in \mathcal{C}_G^{[1]}} \left(\sum_{e \in C} \mathrm{o}_C(e) \xi_e \right)^2 \kp[G\dslash C],
}
where $\xi = (\xi_1, \dotsc, \xi_{|E_G|})$ are formal parameters assigned to each edge (later interpreted as auxiliary momenta, i.e. euclidean 4-vectors), and $\mathrm{o}_B(e), \mathrm{o}_C(e) \in \{ 0, \pm 1\}$ are the signs of the relative orientations of $e$ w.r.t. some arbitrarily chosen orientation of the bond or cycle. Note that this choice does not influence the sign of the polynomials, since these orientations only appear within the square. Below we will abbreviate products of such signs as $\mathrm{o}_C(e_1)\mathrm{o}_C(e_2) = \mathrm{o}_C(e_1,e_2)$.
Via \refeq{eq_kp_disj} -- the Kirchhoff polynomial definition for disconnected graphs -- these definitions extend to disconnected $G$ as well. For Feynman graphs the bond polynomial is closely related to the second Symanzik polynomial. In fact, $\ssp(\alpha, q)$ is simply the evaluation of $\beta_{\Gamma}(\alpha,\xi)$, where one sends $\xi_e\to q$ for each $e$ in some arbitrary path between the external vertices and all others to $0$ (and if there are $n>2$ external vertices, then one does this for $n-1$ pairs of external vertices to get the correct linear combinations $\xi_e\to \sum \pm q_i$).\\
From these two polynomials we derive two families of polynomials that we will from now on mostly mean when speaking of cycle or bond polynomials:
\al{
\bp[G]{e_i}{e_j}(\alpha) &\defeq \frac{1}{2}\frac{\partial^2\beta_G}{\partial \xi_i\partial \xi_j} = \sum_{B \in \mathcal{B}_G} \mathrm{o}_B(e_i,e_j) \alpha_B \kp[G\setminus B]\\[3mm]
\cp[G]{e_i}{e_j}(\alpha) &\defeq \frac{1}{2}\frac{\partial^2\chi_G}{\partial \xi_i\partial \xi_j} = \sum_{C \in \mathcal{C}_G^{[1]}} \mathrm{o}_{C}(e_i,e_j) \kp[G\dslash C] \label{eq_defcycpol}
}
Cycle and bond polynomials inherit many useful properties from the Kirchhoff polynomial. They are clearly still linear in each $\alpha_e$ and homogenous of degree $h_1(G)+1$ (for $\beta_G$, $\bp[G]{e_i}{e_j}$) and $h_1(G)-1$ (for $\chi_G$, $\cp[G]{e_i}{e_j}$). They also satisfy the contraction-deletion relations and the following three useful identities (proposition 2.8 and lemmata 2.9, 2.10, 2.11 in \cite{Golz_2017_CyclePol}):
\al{
\cp[G]{e}{e} &= \kp[G\setminus e] = \frac{\partial}{\partial \alpha_e} \kp[G] \hspace{20mm} \text{ if $e$ is not a bridge.} \\[3mm]
\bp[G]{e}{e} &= \alpha_e\kp[G\dslash e] = \alpha_e\left.\kp[G]\right|_{\alpha_e=0} \hspace{8mm} \text{ if $e$ is not a self-cycle.}\\[5mm]
\bp[G]{e}{e'} &= -\alpha_e\alpha_{e'} \cp[G]{e}{e'} \hspace{24mm} \text{ if } e \neq e'. \label{eq_lemma_bpcp}
}
We also need the polynomial
\al{
\CP{e}{\mu}(\alpha,\xi)
\defeq \frac{1}{2\alpha_e}\frac{\partial}{\partial \xi_{e,\mu}} \beta_{\Gamma}(\alpha, \xi)
= \alpha_e^{-1} \sum_{e' \in E_{\Gamma}} \xi_{e'}^{\mu} \bp{e}{e'}. \label{eq_CPdef}
}
More specifically, we need its evaluation $\xi \to q$ for the case of a single external momentum,
\al{
\CP{e}{\mu}(\alpha,q) = q^{\mu} x_{\Gamma}^e(\alpha), \label{eq_CP_eval_def}
}
which factorises similarly to the second Symanzik polynomial in \refeq{eq_ssp_fac}.
\subsubsection{Dodgson and spanning forests}
In \refeq{eq_detgmkp} we have seen that the Kirchhoff polynomial can be written as the determinant of the graph matrix $M$. Motivated by this one considers minors of the graph matrix, i.e. determinants
\al{
\kp[G]^{I,J} \defeq \det\big( M_I^J\big),
}
where the edge subsets $I,J \subset E_G$ with $|I|=|J|$ in the subscript and superscript denote deletion of all rows and columns indexed by edges in the respective set. In general one often uses a third index set $K$ for $\kp[G,K]^{I,J}$ and sets $\alpha_e=0$ for all $e\in K$, but here we always have $K=\emptyset$. Note that $\kp[G]^{I,J}$ is only well-defined up to an overall sign since a different ordering of the rows and columns in the graph matrix may change the sign of the determinant. This will be discussed further below, but for now we just fix one such ordering.
The $\kp[G]^{I,J}$ are called Dodgson polynomials and appeared already in \cite{BlochEsnaultKreimer_2006_Motives}. They were first named and systematically studied by Francis Brown in \cite{Brown_2010_Periods}. In the following we discuss some notable properties.
\paragraph{Passing to a minor.}
For all $A, B \subset E_G$
\al{
\Psi_{G\setminus A\dslash B,K}^{I,J} = \Psi_{G,K\cup B}^{I\cup A,J\cup A},
}
which justifies our setting $K=\emptyset$.
\paragraph{Determinant identities.}
Let $\adj(M)[I,J]$ be the restriction of the adjugate matrix of $M$ to rows and columns indexed by $I$ and $J$. Based on the Desnanot-Jacobi identity \cite{Dodgson_1866_CondDet}
\al{
\det(\adj(M)[I,J]) = \det(M)^{|I|-1}\det(M^I_J)\label{eq_DJ}
}
for determinants one finds identities of the type
\al{
\Psi_G^{\{i_1\},\{i_3\}}\Psi_G^{\{i_2\},\{i_4\}} - \Psi_G^{\{i_1\},\{i_4\}}\Psi_G^{\{i_2\},\{i_3\}} = \kp[G] \Psi_G^{\{i_1,i_2\},\{i_3,i_4\}}.\label{eq_DodgsonID}
}
This case ($|I|=2=|J|$) is also called \textit{Dodgson identity}\footnote{Somewhat confusingly, it is also occasionally called \textit{Lewis Carroll identity} and both names are sometimes used to refer to the determinant identity \refeq{eq_DJ} \cite{Bressoud}. Here we follow the conventions of \cite{Brown_2010_Periods}.} and its generalisations are the crucial tool that we will work with below.
\paragraph{Combinatoric interpretation.}
In the case of $I \cap J = \emptyset = K$ the combinatoric interpretation for Dodgson polynomials given by Brown in \cite[Prop. 23]{Brown_2010_Periods} simplifies to
\al{
\Psi_G^{I,J} = \sum_{T\subset E_G\setminus(I\cup J)} \pm \prod_{e\notin T} \alpha_e\label{eq_DodgsonCombInt}
}
where the sum is over edge subsets $T$ that are simultaneously spanning trees of $\con{(G\setminus I)}{J}$ and $\con{(G\setminus J)}{I}$. A criterion for two monomials in this sum to have the same or opposite signs is given in \cite[Corollary 17]{BrownYeats_2011_SpanningForest}. Moreover, if $I$ and $J$ do intersect, then
\al{
\Psi_G^{I,J} = \Psi_{G\setminus (I\cap J)}^{I\setminus J,J\setminus I}
}
if $G\setminus (I\cap J)$ is still connected and zero otherwise. In particular, $\Psi_G^{\{e\},\{e\}} = \kp[G\setminus e]$.\\[3mm]
Finally, the last well-known graph polynomial that we will need is the spanning forest polynomial \cite[Def. 9]{BrownYeats_2011_SpanningForest}
\al{
\Phi_G^P = \sum_{F = T_1\sqcup \dotsm \sqcup T_k \in \mathcal{T}^{[k]}_P} \alpha_{E_G\setminus F},
}
where $P = P_1,\dotsc,P_k$ is a partition of vertices of $G$ and $\mathcal{T}^{[k]}_P$ is the subset of spanning $k$-forests which have the vertices of $P_i$ contained in the tree $T_i$.
Being a sum over spanning forests and denoted by the same letter it is no surprise to find that these polynomials are closely related to the second Symanzik polynomial. Consider the matrix expression for $\ssp$ from \refeq{eq_ssp_matrix}. The coefficients of a product $q_{v_1}\!\! \cdot q_{v_2}$ of external momenta are precisely the spanning forest polynomials $\ssp^{\{v_0\},\{v_1,v_2\}}$, such that
\al{
\ssp = \alpha_{E_{\Gamma}} \sum_{v_1,v_2 \in V_{\Gamma}'}\!\!\!\! q_{v_1}\!\! \cdot q_{v_2} (-1)^{v_1+v_2}\det( \tilde {L'}^{\{v_1\}}_{\{v_2\}} )
= \sum_{v_1,v_2 \in V_{\Gamma}}\!\!\!\! q_{v_1}\!\! \cdot q_{v_2}\Phi_G^{\{v_0\},\{v_1,v_2\}}. \label{eq_ssp_stmat}
}
Note that $\Phi_G^{\{v_0\},\{v_1,v_2\}}=0$ if either of the two vertices $v_1,v_2$ is equal to $v_0$. Hence, in this expression we can just sum over the entire vertex set and do not need to write $V_{\Gamma}'$.
\subsection{Chord diagrams}
Aside from Feynman graphs we need another very special kind of graph -- chord diagrams. They can be used to model the contraction and traces of Dirac matrices, which is why they appear in QED Feynman integrals. For proofs and a more in-depth discussion we refer to \cite{Golz_2017_Traces}.
Classically, chord diagrams consist of a cycle on $2n$ vertices (the base) and $k\leq n$ additional edges that pairwise connect $2k$ of the vertices of that cycle (the chords), but here we need a slightly more general definition that allows for multiple base cycles.
\begin{definition}
A chord diagram is a graph that consists of $\ell\geq 1$ cycles with $2n_1, \dotsc, 2n_{\ell}$ vertices and $k \leq \sum n_i\bdefeq N$ further edges, such that each vertex is at most 3-valent.
\end{definition}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.95\textwidth]{./Images/Fig_1_2.pdf}
\caption{Three chord diagrams of order $n=4$ with one base cycle and four, three and two chords respectively.}
\label{img_chordexamples}
\end{center}
\end{figure}
We denote with $\mathcal{D}^n_k$ the set of all chord diagrams\footnote{Here we always mean labelled diagrams, i.e. two diagrams that are isomorphic as graphs but differ in the labelling of the vertices are viewed as different chord diagrams. In practice we will always either fix an arbitrary labelling $1,\dotsc, 2N$ that does not influence the result, or have a labelling fixed from context because the diagram is derived from a Feynman graph in a certain way.} with the respective number and size of base cycles and chords, determined by the $\ell$-tuple $n=(n_1,\dotsc, n_{\ell}) \in \mathbb{N}_+^{\ell}$ and $1\leq k \leq N$. The set of 2-valent vertices of a chord diagram is denoted $V_D^{(2)}$ and we will often call them the ``free vertices'' of $D$.\\
\subsubsection{Colours}\label{sec_colours}
In addition to the distinction between base edges and chords we will need to introduce more properties to differentiate between certain types of edge subsets. This is achieved via colouring. For some finite set $K$ a map $\kappa: E_G\to K$ is called an edge $k$-colouring if for every vertex $v$ of $G$ all edges incident to it are assigned different colours, i.e. if $\kappa$ is injective in the neighbourhood of $v$. Chord diagrams $D$ admit an edge 3-coloring $\kappa: E_D \to \{0,1,2\}$ that assigns two alternating colours $1$ and $2$ to the edges of the base cycles and the third colour $0$ to all chords. There are $2^{\ell}$ possibilities of such a colouring corresponding to the exchange of colours $1$ and $2$ in some base cycles, so from now on we fix one such choice in all diagrams. In drawings we visualise the colours with different line types:
\al[*]{
0 \sim \includegraphics[height=0.5\baselineskip]{Images/Prop_Dotted.pdf} \hspace{2cm} 1 \sim \includegraphics[height=0.5\baselineskip]{Images/Prop_Scalar.pdf} \hspace{2cm} 2 \sim \includegraphics[height=0.5\baselineskip]{Images/Prop_Dashed.pdf}
}
Let $E_D^{i} = \kappa^{-1}(\{i\})$, $E_D^{ij} = \kappa^{-1}(\{i,j\})$ be the edge subsets consisting only of edges of the respective colour or colours. Each bicoloured edge subset can be decomposed into collections of cycles $\mathcal{C}_D^{ij}$ and paths $\mathcal{P}_D^{ij}$, where $\mathcal{P}_D^{12} = \emptyset$, $|\mathcal{P}_D^{01}| = |\mathcal{P}_D^{02}|$, $|\mathcal{C}_D^{12}| = \ell$, and we define $c_2(D) \defeq |\mathcal{C}_D^{01}| + |\mathcal{C}_D^{02}|$. The bicoloured paths between the 2-valent vertices of $D$ can be joined in their shared vertices to build tricoloured cycles, whose number we define to be $c_3(D)$. Often we are only interested in the total number of such coloured cycles, which we call $\tilde c(D) \defeq c_2(D) + c_3(D)$. Beware that this excludes the base cycles in $\mathcal{C}_D^{12}$, which are counted separately by $\ell$.
\begin{figure}[H] \begin{center}
\includegraphics[width=0.45\textwidth]{./Images/Fig_1_3a.pdf}\hspace{10mm}
\includegraphics[width=0.45\textwidth]{./Images/Fig_1_3b.pdf}
\caption{Colour decompositions of the left and right chord diagrams from \reffig{img_chordexamples}.}
\label{img_dc}
\end{center} \end{figure}
\begin{example}\label{04b_ex_threecol}
Let $D_1, D_2, D_3$ be the three chord diagrams, left to right, from \reffig{img_chordexamples}. For $D_1$ and $D_3$ the bicoloured subsets are depicted in \reffig{img_dc}. Since there are no 2-valent vertices in $D_1$ all bicoloured components are cycles. Simply counting them in the drawing one finds
\al[*]{
c_2(D_1) = 3 \qquad c_3(D_1) = 0.
}
$D_3$, on the other hand, has four free vertices. We see that there is still a bicoloured cycle on the very r.h.s. of \reffig{img_dc}. The other bicoloured subsets are paths, four of them, which combine into a single tricoloured cycle, such that
\al[*]{
c_2(D_3) = 1 \qquad c_3(D_3)=1.
}
We leave it as an exercise for the reader to draw the bicoloured subsets of $D_2$, count the cycles, and confirm that also
\al[*]{
c_2(D_2) = 1 \qquad c_3(D_2)=1.
}
\end{example}
Contracting each path in $\mathcal{P}_D^{0i}$ to a single edge of colour $i$ maps the tricoloured cycles to a chord diagram $D_0 \in \mathcal{D}_0^{n_0}$, for some suitable $n_0$ with $\sum n_{0,i} \leq N$, without any chords. Its base cycles correspond to the tricoloured cycles and its vertices are the 2-valent vertices of $D$. We call this projection map $\pi_0$.
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.7\textwidth]{./Images/Fig_1_4.pdf}
\caption{Visualisation of the projection map $\pi_0$ for the case of diagram $D_3$ from \reffig{img_chordexamples}. }
\label{img_proj}
\end{center}
\end{figure}
The final notion that we need to introduce is the signum $\sgn(u,v)$ of two free vertices $u,v \in V_D^{(2)}$ within a chord diagram. It is $-1$ if they are not part of the same tricoloured cycle of $D$, and $0$ or $1$ if there are an even or odd number of paths between them. Alternatively, in terms of $\pi_0(D)$, $\sgn(u,v)=-1$ if $u$ and $v$ are in different base cycles and is $0$ or $1$ if they are in the same base cycle and are separated by an even or odd number of base edges. In \cite[Prop. 3.5]{Golz_2017_Traces} it was worked out how the numbers $c_2$ and $c_3$ change when a chord is added to a chord diagram $D_0\in D^n_k$ with $k<N$. Focussing only on the total number $\tilde c$ it reduces to
\al{
\tilde c(D) = \tilde c(D_0) + \sgn(u,v),
}
where $D = ( V_{D_0}, E_{D_0}^0 \cup \{u,v\}, E_{D_0}^1, E_{D_0}^2)$.
\subsubsection{Chord diagrams and Feynman graphs} \label{sec_cd_fg}
In this section we establish the connection between Feynman graphs (and integrals) and chord diagrams. For concreteness we focus on the case of photon propagator graphs with a single fermion cycle, in Feynman gauge, and we ignore subdivergences.\\
The Feynman rules for a fermion cycle yield a trace
\al{
\tr( \gamma_{\mu_1}\dotsm\gamma_{\mu_{4h_1}} )
}
where the matrices $\gamma_{\mu_i}$ correspond to fermion edges ($i$ odd) and vertices ($i$ even), and $h_1\equiv h_1(\Gamma)$ is the graph's first Betti number. Since we are in Feynman gauge every other matrix is contracted with metric tensors $g_{\mu_i\mu_j}$, corresponding to the photon propagators (including the external photon, see sec. \ref{subsec_contract}). The trace can be visualised as a chord diagram $D_{\Gamma} \in \mathcal{D}^{2h_1}_{h_1}$ in which each vertex is labelled by one matrix and contraction via metric tensors is represented by chords.
We are now interested in sums over chord diagrams that result from all possible additions of further chords to $D_{\Gamma}$. Because the chords fixed in place are always the same, we can consider smaller diagrams instead, namely diagrams built on the projection $D_{\Gamma}^0=\pi_0(D_{\Gamma})$. Even if $D_{\Gamma}$ has only one base cycle the projection may contain multiple base cycles. Hence, let $n\in\mathbb{N}_+^{\ell_0}$ with $\sum n_i = N = h_1$ be some suitable tuple representing the base cycle structure of $D_{\Gamma}^0$ after the projection, such that $D_{\Gamma}^0 \in \mathcal{D}^n_0$. Then we denote with $\mathcal{D}^k_{\Gamma} \simeq \mathcal{D}^n_{h_1-k}$ the set of all chord diagrams that contain the base cycles of $D_{\Gamma}^0\in \mathcal{D}^n_0$ as a subgraph together with $h_1-k$ chords and have their vertices labelled by fermion edges of the underlying Feynman graph. Each such diagram corresponds to a trace of $4h_1$ Dirac matrices contracted with $2h_1-k$ metric tensors\footnote{To be precise, these diagrams of course correspond to some product of traces with a total of $2h_1$ Dirac matrices. However, the fixed chords in $D_{\Gamma}$ do not influence $\tilde c$, so we can just take the factor $(-2)^{h_1}$ due to these $h_1$ chords and otherwise work with the smaller diagrams, even though we actually compute the contraction of a larger product of matrices. See \cite{Golz_2017_Traces} for details.},
\al{
\mathcal{D}^k_{\Gamma} \ni D \sim
\bigg(\prod_{ (u,v)\in E_D^0 } g_{\mu_u\mu_v} \bigg)
\bigg(\prod_{ (u',v')\in E_{D_{\Gamma}}^0 } g_{\mu_{u'}\mu_{v'}} \bigg)
\tr( \gamma^{\mu_1}\dotsm\gamma^{\mu_{4h_1}} ).
}
For $k=0$ this trace is simply $(-2)^{1+2h_1+\tilde c(D)}$ \cite[Theorem 3.9]{Golz_2017_Traces}. Similar results also hold if there are uncontracted matrices left, and for contractions between products of multiple traces. Since there is only a single external momentum $q$ all matrices not contracted with metric tensors are contracted to $\slashed q = q_{\rho}\gamma^{\rho}$, and with $\slashed q\slashed q = q^2$ one finds that one always just has an integer multiple of a power of $q^2$. Applying this to the parametric integrand for QED Feynman integrals from \cite{Golz_2017_CyclePol}, which we will do in more detail in section \ref{sec_application}, yields sums of the form
\al{
I_{\Gamma}^{(k)} \propto \sum_{D\in \mathcal{D}_{\Gamma}^k} (-2)^{\tilde c(D)} \bigg(\prod_{ (u,v)\in E_D^0 } \cp{u}{v}\bigg) \prod_{ w\in V_D^{(2)} } x_{\Gamma}^w, \label{eq_sumdetail}
}
which we will be able to rewrite with the two main theorems of this article.
\section{Dodgson polynomials revisited}
In order to prove the polynomial identities of section \ref{sec_Pol_id} we will need a variant of the Dodgson polynomials that is in some sense a reinterpretation (in section \ref{sec_dp_01}) but also a generalisation (in section \ref{sec_dp_02}). This then allows us to define what we call partition polynomials in section \ref{sec_defpp}.
\subsection{Dodgson cycle polynomials}\label{sec_dp_01}
The relation $\cp[G]{e}{e} = \kp[G\setminus e] = \kp[G]^{\{e\},\{e\}}$ suggests a possible connection between cycle and Dodgson polynomials, and indeed we find
\begin{proposition}\label{05a_prop_eqpol}
\al{
\cp[G]{i}{j} = \pm\kp[G]^{\{i\},\{j\}}
}
for all $i,j\in E_G$.
\end{proposition}
\begin{proof}
For $i=j$ the proof is done and for $i\neq j$ we use the combinatoric interpretation from \refeq{eq_DodgsonCombInt},
\al{
\kp[G]^{\{i\},\{j\}} = \sum_{T\subset E_G\setminus\{i,j\}} \pm \prod_{e\notin T} \alpha_e.
}
A sum over spanning trees can be decomposed into a double sum over paths $P\subset G\setminus i$ and spanning trees of the corresponding graph $\con{(G\setminus i)}{P}$ where all paths are between endpoints $\partial_+(i)$ and $\partial_-(i)$ and contain the edge $j$. Then adding $i$ to each path completes it into a simple cycle $C_P = P\cup\{i\} \in \mathcal{C}_G^{[1]}$ that contains both $i$ and $j$, and the corresponding monomials of $\cp[G]{i}{j}$ and $\kp[G]^{\{i\},\{j\}}$ indeed agree, at least up to sign. The signs $\mathrm{o}_{C_1}(i,j)$, $\mathrm{o}_{C_2}(i,j)$ of two partial polynomials $\kp[\con{G}{C_1}]$ and $\kp[\con{G}{C_2}]$ in $\cp[G]{i}{j}$ differ if and only if $C_1\cup C_2$ is -- up to contraction of longer paths to single edges -- isomorphic to $K_4$:
\begin{center}
\includegraphics[width=0.8\textwidth]{./Images/Fig_2_0.pdf}
\end{center}
Comparing this with the discussion of signs in Dodgson polynomials in section 2 of \cite{BrownYeats_2011_SpanningForest}, one finds that the endpoints of $i$ are precisely the transposed vertices given in \cite[corollary 17]{BrownYeats_2011_SpanningForest} as a criterion for opposite signs. Therefore all partial polynomials have the correct relative signs and only the overall sign ambiguity of Dodgson polynomials remains, concluding the proof.
\end{proof}
It should be noted that the sign ambiguity of the Dodgson polynomials is of course not entirely absent from the cycle polynomials -- the choice one has to make is simply moved from the order of rows and columns in a matrix to the orientations of edges in $G$. Since we always considered our graphs together with some such fixed choice from the very beginning it does not appear in the combinatorial definition of the cycle polynomials. Moreover, in the context of Feynman integrals we can even have a physical motivation for certain orientations, e.g. aligning all fermion edge orientations with fermion flow.
We can use this to fix the choice of the graph matrix such that the signs of $\cp[G]{i}{j}$ and $\kp[G]^{\{i\},\{j\}}$ agree. Furthermore, the interpretation of cycle polynomials as a fixed-sign version of Dodgson polynomials also suggests the definition of a higher order cycle polynomial via the Dodgson identity \refeq{eq_DodgsonID}.
\begin{definition}\label{02c_def_dodgsoncyclepol}
Let $G$ be a connected graph and $\cp[G]{i}{j}$ for all $i, j\in E_G$ the cycle polynomial as defined in \refeq{eq_defcycpol}. Then define an alphabet $\mathsf{A} = \{ \mathsf{a}_i \ | \ i \in E_G \}$ in which each letter is associated to an edge of $G$ and consider two words $\mathsf{u},\mathsf{v}$ over this alphabet with $|\mathsf{u}|=k=|\mathsf{v}|$. The Dodgson cycle polynomial is then defined as $\cp[G]{\mathsf{a}_i}{\mathsf{a}_j} \defeq \cp[G]{i}{j}$ if $k=1$ and
\al{
\cp[G]{\mathsf{u}}{\mathsf{v}} \defeq \kp[G]^{1-k} \sum_{\sigma \in S_k} \sgn(\sigma) \prod_{i=1}^{k} \cp[G]{\mathsf{u}_i}{\sigma_i(\mathsf{v})}, \label{eq_DefExpansion}
}
where $\mathsf{u}_i$, $\sigma_i(\mathsf{v})$ denote the $i$-th letter of $\mathsf{u}$ and (the permutation of) $\mathsf{v}$, for $2\leq k\leq h_1(G)$.
\end{definition}
In this we simply recursively define $\cp[G]{\mathsf{u}}{\mathsf{v}}$ for words of length $k$ by repeatedly using the Dodgson identity \refeq{eq_DodgsonID} or its generalisations derived from \refeq{eq_DJ}. For $k=2,3$ one has
\al{
\cp[G]{\mathsf{a}_1\mathsf{a}_2}{\mathsf{a}_3\mathsf{a}_4} &= \kp[G]^{-1} \big( \cp[G]{\mathsf{a}_1}{\mathsf{a}_3}\cp[G]{\mathsf{a}_2}{\mathsf{a}_4} - \cp[G]{\mathsf{a}_1}{\mathsf{a}_4}\cp[G]{\mathsf{a}_2}{\mathsf{a}_3}\big),\\[5mm]\nonumber
\cp[G]{\mathsf{a}_1\mathsf{a}_2\mathsf{a}_3}{\mathsf{a}_4\mathsf{a}_5\mathsf{a}_6} &= \kp[G]^{-2} \big( \cp[G]{\mathsf{a}_1}{\mathsf{a}_4}\cp[G]{\mathsf{a}_2}{\mathsf{a}_5}\cp[G]{\mathsf{a}_3}{\mathsf{a}_6}
- \cp[G]{\mathsf{a}_1}{\mathsf{a}_4}\cp[G]{\mathsf{a}_2}{\mathsf{a}_6}\cp[G]{\mathsf{a}_3}{\mathsf{a}_5}\\\nonumber
&\qquad\qquad +\cp[G]{\mathsf{a}_1}{\mathsf{a}_5}\cp[G]{\mathsf{a}_2}{\mathsf{a}_4}\cp[G]{\mathsf{a}_3}{\mathsf{a}_6}
- \cp[G]{\mathsf{a}_1}{\mathsf{a}_5}\cp[G]{\mathsf{a}_2}{\mathsf{a}_6}\cp[G]{\mathsf{a}_3}{\mathsf{a}_4}\\
&\qquad\qquad +\cp[G]{\mathsf{a}_1}{\mathsf{a}_6}\cp[G]{\mathsf{a}_2}{\mathsf{a}_4}\cp[G]{\mathsf{a}_3}{\mathsf{a}_5}
- \cp[G]{\mathsf{a}_1}{\mathsf{a}_6}\cp[G]{\mathsf{a}_2}{\mathsf{a}_5}\cp[G]{\mathsf{a}_3}{\mathsf{a}_4} \big).
}
Note that this also permits an expansion (essentially the cofactor expansion of the determinant) that yields, e.g. for $k=3$ ,
\al{
\cp[G]{\mathsf{a}_1\mathsf{a}_2\mathsf{a}_3}{\mathsf{a}_4\mathsf{a}_5\mathsf{a}_6} = \kp[G]^{-1} \Big( \cp[G]{\mathsf{a}_1}{\mathsf{a}_4}\cp[G]{\mathsf{a}_2\mathsf{a}_3}{\mathsf{a}_5\mathsf{a}_6}
- \cp[G]{\mathsf{a}_1}{\mathsf{a}_5}\cp[G]{\mathsf{a}_2\mathsf{a}_3}{\mathsf{a}_4\mathsf{a}_6}
+ \cp[G]{\mathsf{a}_1}{\mathsf{a}_6}\cp[G]{\mathsf{a}_2\mathsf{a}_3}{\mathsf{a}_4\mathsf{a}_5} \Big).
}
which will be very useful later on.\\
Defining the polynomials like this imposes an ordering on the indices instead of using unordered sets $I$, $J$. This yields a symmetry $\cp[G]{\mathsf{u}}{\mathsf{v}} = \sgn(\sigma)\cp[G]{\mathsf{u}}{\sigma(\mathsf{v})}$ for all permutations of letters in the words, which we will be able to exploit for our purposes below. Moreover, note that $\cp[G]{\mathsf{u}}{\mathsf{v}} = 0$ if one of the words contains a repeated letter and
\al{
\cp[G]{\mathsf{u}}{\mathsf{v}} = (-1)^{l+m}\cp[G\setminus e]{\mathsf{u}_1\dotsm\hat\mathsf{u}_l\dotsm\mathsf{u}_k}{\mathsf{v}_1\dotsm\hat\mathsf{v}_m\dotsm\mathsf{v}_k}
}
if $\mathsf{u}_l = \mathsf{v}_m = \mathsf{a}_e$.
%
\begin{remark}
The relation between Dodgson polynomials and \textit{``sums over subgraphs of $G$ containing cycles which satisfy certain properties''} was already observed by Brown when he originally defined Dodgson polynomials in \cite[Remark 24]{Brown_2010_Periods}, but not further pursued. At some point it might be worthwhile to study what exactly these certain properties should be for higher order polynomials, so one can give a direct combinatorial definition analogous to \refeq{eq_defcycpol}, but for this article the recursive definition above shall suffice.
\end{remark}
\subsection{Vertex-indexed Dodgson polynomials}\label{sec_dp_02}
We just modified the Dodgson polynomial in a way that allows us to control their signs by relating them to another polynomial also indexed by edge subsets of the underlying graph. However, the graph matrix is an $E_G+V_G-1$ square matrix that also has rows corresponding to vertices of the graph. It is therefore quite natural to extend the definition of the Dodgson polynomials to include deletion of rows and columns labelled by vertices. Since the determinant identity \refeq{eq_DJ} holds generally, irrespective of which columns or rows are deleted, the polynomials given by such minors still satisfy the Dodgson identity. Moreover, we have already seen the fixed-sign versions of these types of Dodgson polynomials that we can use analogously to the cycle polynomials in the previous section. Remember the spanning forest polynomials used to rewrite the second Symanzik polynomial in \refeq{eq_ssp_stmat}. Using again the block matrix identity from \refeq{eq_blockID} we can write
\al{
\Phi_G^{\{v_0\},\{v_1,v_2\}} = (-1)^{v_1+v_2} \alpha_{E_{\Gamma}} \det( \tilde {L'}^{\{v_1\}}_{\{v_2\}} ) = (-1)^{v_1+v_2}\det( M(G)^{\{v_1\}}_{\{v_2\}} ).
}
The ambiguous sign of the determinant is precisely cancelled by the factor $(-1)^{v_1+v_2}$ such that $\Phi_G^{\{v_0\},\{v_1,v_2\}}$ is indeed a fixed-sign version of the Dodgson polynomial $\kp[G]^{\{v_1\},\{v_2\}}$. Hence, we reuse our previous notation to define
\al{
\cp[G]{\mathsf{a}_{v_1}}{\mathsf{a}_{v_2}} \defeq \Phi_G^{\{v_0\},\{v_1,v_2\}},
}
now with words (over an extended alphabet that includes vertices) indexing it, as in the previous case of cycle Dodgson polynomials. In this notation the Dodgson identity again takes the form
\al{
\cp[G]{\mathsf{a}_1}{\mathsf{a}_2} \cp[G]{\mathsf{a}_3}{\mathsf{a}_4} - \cp[G]{\mathsf{a}_1}{\mathsf{a}_3}\cp[G]{\mathsf{a}_2}{\mathsf{a}_4} = \kp \cp[G]{\mathsf{a}_1\mathsf{a}_4}{\mathsf{a}_2\mathsf{a}_3}, \label{eq_DI_rewrite}
}
and generalisations are analogous to \refeq{eq_DefExpansion}. Note that, where edge indices lower the degree of the polynomial, such that $\deg( \cp[G]{\mathsf{w}_1}{\mathsf{w}_2} ) = h_1(G)-|\mathsf{w}_i|$, if the letters of both $\mathsf{w}_i$ correspond to edges, the vertex indices do the opposite. For single letters, $\deg(\cp[G]{\mathsf{a}_i}{\mathsf{a}_j}) = \deg( \Phi_G^{\{v_0\},\{i,j\}} ) = h_1(G)+1$, such that the polynomial with two-letter words on the r.h.s. has to have degree $h_1+2$. Another property we get by courtesy of the spanning forest polynomial is that
\al{
\cp[G]{\mathsf{a}_{v}}{\mathsf{a}_{v}} = \kp[G|_{v=v_0}].
}
In other words, equal indices correspond to identification of that vertex with $v_0$ in the graph, analogous to the edge-indexed case $\cp[G]{\mathsf{a}_{e}}{\mathsf{a}_{e}} = \kp[G\setminus e]$, which indicated deletion of an edge.\\
In $\Phi_G^{\{v_0\},\{v_1,v_2\}}$ the vertex $v_0$ whose row and column are initially deleted from the graph matrix is explicit. We will see below that it is actually useful to consider Dodgson polynomials coming from different such choices. Hence, from now we will use the subscript $\cp[G,v_0]{\mathsf{u}}{\mathsf{v}}$ to indicate it, whenever the choice actually matters. Note that this is different from the subscript $K$ in the usual Dodgson polynomial $\kp[G,K]^{I,J}$, which indicates contracted edges and is always empty for us.
\subsection{Partition polynomials}\label{sec_defpp}
With these new variants of the Dodgson polynomials we can define another new polynomial that bridges the gap between Feynman graphs and the chord diagrams associated to them. For this purpose, we briefly return to the case of Dodgson polynomials only indexed by edge subsets of the underlying graph, not vertices.\\
Let $\Gamma$ be a suitable Feynman graph such that $D_{\Gamma}^0 \equiv D\in \mathcal{D}^n_0$ with $n\in \mathbb{N}^{\ell}$ and label all its vertices with the graph's fermion edges, i.e. letters from an alphabet $\mathsf{A}=\{ \mathsf{a}_i \ | i \in E_{\Gamma}^{(f)} \}$. Consider all pairs of monomial words $(\mathsf{u}, \mathsf{v})$ of length $|\mathsf{u}| = N = |\mathsf{v}|$ over this alphabet such that $\mathsf{u}\mathsf{v}$ contains each letter exactly once. Then the symmetries
\al{
\cp{\mathsf{u}}{\mathsf{v}} = \cp{\mathsf{v}}{\mathsf{u}} \quad\text{ and }\quad \cp{\mathsf{u}}{\mathsf{v}} = \sgn(\sigma) \cp{\mathsf{u}}{\sigma(\mathsf{v})} \qquad \forall \sigma\in S_N,
}
induce an equivalence relation on these words via
\al{
(\mathsf{u},\mathsf{v}) \sim (\mathsf{u}',\mathsf{v}') \iff \cp{\mathsf{u}}{\mathsf{v}} = \pm\cp{\mathsf{u}'}{\mathsf{v}'},
}
or equivalently
\al{
(\mathsf{u},\mathsf{v}) \sim (\mathsf{u}',\mathsf{v}') \iff \exists\ \sigma, \sigma' \in S_N \text{ s.t. } \mathsf{u}'=\sigma(\mathsf{u}),\ \mathsf{v}'=\sigma'(\mathsf{v}).
}
Let $\mathsf{P}$ denote the corresponding set of equivalence classes of pairs $(\mathsf{u},\mathsf{v})$ that satisfy the above mentioned properties. For the two coloured subsets of base edges $E_D^1$ and $E_D^2$ define the corresponding subsets $\mathsf{P}_i\subset \mathsf{P}$ by imposing an additional constraint: For all edges $(u,v) \in E_D^i$ we demand that the two corresponding letters do not appear in the same word, i.e. $\mathsf{a}_u \in \mathsf{u}$ and $\mathsf{a}_v \in \mathsf{v}$ or vice versa. The full set of equivalence classes is then the union $\mathsf{P} = \mathsf{P}_1 \cup \mathsf{P}_2$. Moreover, in most cases the $\mathsf{P}_i$ intersect only in exactly one element, which, assuming the vertices of $D$ are labelled consecutively within each base cycle, is the class of pairs that contain all letters labelled with odd numbers in one word and those labelled with even numbers in the other. The only exception occurs if $D$ has one or more base cycles of size $1$. Then there is a base edge of either colour between the same two vertices, leading to some redundancy. In particular, $\mathsf{P}_1 = \mathsf{P}_2$ if $n=(1,\dotsc,1)$.
Finally, we need to fix one distinguished representative of each class with respect to which we consider permutations. Assuming some arbitrary ordering of $i$-coloured base edges $(u_1,v_1), \dotsc$, $(u_N,v_N) \in E_D^i$ each equivalence class contains exactly one element that we notate $(\mathsf{u}_{\id},\mathsf{v}_{\id})$ such that $\mathsf{a}_{u_j}$ and $\mathsf{a}_{v_j}$ are the $j$-th letters of $\mathsf{u}_{\id}$ and $\mathsf{v}_{\id}$, or vice versa. For any other ordering of base edges the designated element would be related to $(\mathsf{u}_{\id}, \mathsf{v}_{\id})$ by the same permutation in both words, such that the choice of ordering on $E_D^i$ does not matter.\\
For all $(\mathsf{u},\mathsf{v})\in \mathsf{P}$ and partitions of $i$-coloured base edges $\mathcal{E} = (E_1,\dotsc,E_{|\mathcal{E}|}) \in \mathcal{P}(E_D^i)$ define a map $\lambda_{\mathcal{E}}$ as follows. Let
\al{
V_j \defeq \bigcup_{(u,v) \in E_j} \{u,v\} \subseteq V_D \label{eq_PartVert}
}
be the set of vertices in the part $E_j$ and consider the restriction
\al{
(\mathsf{u}_j,\mathsf{v}_j) = (\mathsf{u}, \mathsf{v})|_{ \mathsf{a}_k = 1 \ \forall k \in V_D\setminus V_j}
}
of $(\mathsf{u},\mathsf{v})$ to the alphabet corresponding to these vertices. In each $(\mathsf{u}_j,\mathsf{v}_j)$ all letters not associated to this part of the partition are removed but, critically, the order of the remaining letters is preserved. Then
\al{
\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \defeq \begin{cases}
\{ (\mathsf{u}_1,\mathsf{v}_1), \dotsc, (\mathsf{u}_{|\mathcal{E}|},\mathsf{v}_{|\mathcal{E}|}) \} & \text{ if } |\mathsf{u}_j| = |\mathsf{v}_j| \text{ for all } 1\leq j \leq |\mathcal{E}|,\\
\hspace{5em} \emptyset & \text{ else.}
\end{cases} \label{eq_deflambda}
}
The concatenations $\mathsf{u}_1\dotsm\mathsf{u}_{|\mathcal{E}|}$ and $\mathsf{v}_1\dotsm\mathsf{v}_{|\mathcal{E}|}$ are then permutations of $\mathsf{u}$ and $\mathsf{v}$ (which are themselves permutations of the words $\mathsf{u}_{\id}, \mathsf{v}_{\id}$ of their equivalence class) and we define
\al{
\sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \defeq \begin{cases} \hspace{2.5em} 0 & \text{ if } \lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) = \emptyset, \\
\sgn(\sigma)\sgn(\sigma') & \text{ else,}
\end{cases}
\label{eq_defsgn}
}
where $\sigma, \sigma' \in S_N$ are the permutations with $\sigma(\mathsf{u}_{\id}) = \mathsf{u}_1\dotsm\mathsf{u}_{|\mathcal{E}|}$ and $\sigma'(\mathsf{v}_{\id}) = \mathsf{v}_1\dotsm\mathsf{v}_{|\mathcal{E}|}$. With this we are now ready to insert these types of words into certain combinations of Dodgson polynomials, which we will call partition polynomials.
\begin{definition}\label{def_PartPol}
Let $\Gamma$ be a QED Feynman graph with the associated chord diagram $D_{\Gamma}$ such that $D\equiv \pi_0(D_{\Gamma}) = D_{\Gamma}^0 \in \mathcal{D}^n_0$ with $n\in \mathbb{N}^{\ell}$, $N=\sum_i n_i = h_1(\Gamma)$. Then we define the partition polynomial of $\Gamma$ to be
\al{
Z_{\Gamma}^0(\alpha) &\defeq \sum_{\mathcal{E} \in \mathcal{P}(E_D^1)} (-\kp)^{N-|\mathcal{E}|} (|\mathcal{E}|+1)! \sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 } \sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \!\!\!\!
\prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v})}\!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'},
\label{eq_PartPol}
}
where $\mathcal{P}(E_D^1)$ is the set of all partitions of $1$-coloured base edges of $D$. Moreover, for $1\leq l \leq N$ let
\al{
Z_{\Gamma}^0\big|_l \defeq \sum_{\substack{\mathcal{E} \in \mathcal{P}(E_D^1)\\ |\mathcal{E}| = l}} \sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 }\sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \!\!\!\! \prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) }\!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'}
}
such that
\al{
Z_{\Gamma}^0 = \sum_{l=1}^N (-\kp)^{N-l} (l+1)!\ Z_{\Gamma}^0\big|_l.\label{eq_ppshort}
}
\end{definition}
Note that using partitions $\mathcal{P}(E_D^2)$ in the first and words $(\mathsf{u},\mathsf{v})\in \mathsf{P}_1$ in the second sum yields the exact same polynomial. This symmetry is not quite obvious from this definition but will become so in the proof of theorem \ref{theo_main} below. However, the separate polynomials $Z_{\Gamma}^0\big|_l$ do differ considerably depending on whether one sums over $\mathcal{P}(E_D^1)$ and $\mathsf{P}_2$ or $\mathcal{P}(E_D^2)$ and $\mathsf{P}_1$. Hence, when discussing these polynomials specifically, one should make clear which one is chosen. We reiterate that the sum $Z_{\Gamma}^0$ is independent of this choice, which only reflects two different possible decompositions.\\
Based on this definition we can introduce a similar polynomial that incorporates vertex-indexing in Dodgson polynomials. For the purposes of this article it suffices to stick to a very specific vertex indexing, but it should certainly be possible to extend this to include any type of Dodgson polynomial.
\begin{definition}\label{def_PartPol_1}
Let everything be as in def. \ref{def_PartPol}. Additionally, let $\Gamma$ be a Feynman graph with only two non-zero external momenta and $x,y \in V_{\Gamma}^{ext}$ the corresponding external vertices. Let $\mathsf{y}$ be the additional letter representing $y$ and assume that the deleted column and row of the graph matrix corresponds to $x$, i.e. all Dodgson polynomials are $\cp{\mathsf{u}}{\mathsf{v}} \equiv \cp[\Gamma,x]{\mathsf{u}}{\mathsf{v}}$. Define
\al{\nonumber
Z_{\Gamma}^1\big|_l &\defeq
\sum_{\substack{\mathcal{E} \in \mathcal{P}(E_D^1)\\ |\mathcal{E}| = l}} \sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 } \sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \!\!\!\!
\prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v})}\!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'}\!\!\!\! \sum_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) }
\bigg( \frac{\cp{\mathsf{u}'\mathsf{y}}{\mathsf{v}'\mathsf{y}}}{\cp{\mathsf{u}'}{\mathsf{v}'}} - \frac{\vssp}{\kp} \bigg)\\[3mm]
%
&= -l\frac{\vssp}{\kp}Z_{\Gamma}^0\big|_l + \sum_{\substack{\mathcal{E} \in \mathcal{P}(E_D^1)\\ |\mathcal{E}| = l}} \sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 } \sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \!\!\!\!
\prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v})}\!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'}\!\!\!\! \sum_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) } \frac{\cp{\mathsf{u}'\mathsf{y}}{\mathsf{v}'\mathsf{y}}}{\cp{\mathsf{u}'}{\mathsf{v}'}}.
\label{eq_ppol1_aux}
}
Then we define the first order partition polynomial of $\Gamma$ to be
\al{
Z_{\Gamma}^1(\alpha) &\defeq \frac{1}{2} \sum_{l=1}^N (-\kp)^{N-l+1} (l+1)!\ Z_{\Gamma}^1\big|_l.
\label{eq_PartPol_1}
}
\end{definition}
Note the additional factors of $1/2$ and $-\kp$, in contrast to \refeq{eq_PartPol} above. Together with the observation that $\vssp=\cp{\mathsf{y}}{\mathsf{y}}$ and $\kp = \cp{\emptyset}{\emptyset}$ in the first line of \refeq{eq_ppol1_aux} this suggests a straightforward generalisation
\al{
Z_{\Gamma}^k(\alpha) &\defeq \frac{1}{2^k} \sum_{l=1}^N (-\kp)^{N-l+k} (l+1)!\ Z_{\Gamma}^k\big|_l.
}
$Z_{\Gamma}^k\big|_l$ should contain something like a sum over all choices of $k$ word pairs in $\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v})$ to which the letter $\mathsf{y}$ is added. Then the factor $1$ in $Z_{\Gamma}^0\big|_l$ corresponds to a sum over the unique choice of no element at all and the sum in $Z_{\Gamma}^1\big|_l$ is the sum over choices of exactly one word pair. If this is in fact a correct (i.e. useful) generalised definition shall be studied in future work. For now we will concentrate on the cases of order $0$ and $1$.
\section{Polynomial identities}\label{sec_Pol_id}
The statement of our two main theorems is now that the two partition polynomials $Z_{\Gamma}^0$ and $Z_{\Gamma}^1$ are in fact equal to the sums of chord diagrams, with products of cycle polynomials in each summand, that appear in the parametric integrand of QED.
\subsection{The first summation theorem}
\begin{theorem}\label{theo_main}
\al{
Z_{\Gamma}^0 = \frac{1}{2} \sum_{D\in \mathcal{D}_{\Gamma}^0} (-2)^{\tilde c(D)} \!\!\!\!\prod_{(u,v)\in E_D^0} \cp{\mathsf{a}_u}{\mathsf{a}_v}.
\label{eq_maintheo}
}
\end{theorem}
\noindent In order to prove this we first need some auxiliary results. First we attempt to study the summation by essentially working backwards and looking at sums $\sum \cp{\mathsf{u}_{\id}}{\mathsf{v}_{\id}}$ for $(\mathsf{u}_{\id},\mathsf{v}_{\id})\in \mathsf{P}_2$, which appear in the partition polynomial for the single part partition $\mathcal{E} = \{E_D^1\}$.
\begin{lemma}\label{lemma_fullpart}
Let $\mathsf{P}_j$ be as above and $c_2^j(D) \defeq |\mathcal{C}_D^{0j}|$ the number of two-coloured cycles consisting of chords and $j$-coloured base edges (such that $c_2(D) = c_2^1(D)+c_2^2(D)$). Then
\al{
\sum_{ (\mathsf{u}_{\id},\mathsf{v}_{\id})\in \mathsf{P}_j } \cp{\mathsf{u}_{\id}}{\mathsf{v}_{\id}}
= (-\kp)^{1-N}\sum_{D\in\mathcal{D}_{\Gamma}^0} (-2)^{c_2^j(D)-1} \!\!\!\!\prod_{(u,v)\in E_D^0} \cp{\mathsf{a}_u}{\mathsf{a}_v},\label{eq_lemma_fullpart}
}
\end{lemma}
\begin{proof}
Quick computations show that the claim holds for all $n$ with $N=\sum n_i = 1, 2$, and even $N=3$ is only mildly tedious, as shown below in example \ref{ex_lem1}. We now reduce the l.h.s. of \refeq{eq_lemma_fullpart} to a sum over expressions corresponding to $N-1$, in order to prove by induction.
Consider a word pair $( \mathsf{x}_{11}\dotsm\mathsf{x}_{1N}, \mathsf{x}_{21}\dotsm\mathsf{x}_{2N} )$ with all $\mathsf{x}_{ij}\in \mathsf{A}$. Assuming this word is a representative $(\mathsf{u}_{\id},\mathsf{v}_{\id})\in \mathsf{P}_j$, each pair $(\mathsf{x}_{1k},\mathsf{x}_{2k})$ of $k$-th letters corresponds to a base edge of $E_{D_0}^j$, for a chord diagram $D_0\in \mathcal{D}_0^n$. With \refeq{eq_DefExpansion} its Dodgson polynomial can be written as
\al{\nonumber
\kp&\cp{\mathsf{x}_{11}\dotsm\mathsf{x}_{1N}}{\mathsf{x}_{21}\dotsm\mathsf{x}_{2N}}\\\nonumber
&\quad= \sum_{k=1}^{N} (-1)^{1+k} \cp{\mathsf{x}_{11}}{\mathsf{x}_{2k}}
\cp{\mathsf{x}_{12}\dotsm\mathsf{x}_{1N}}{\mathsf{x}_{21}\dotsm\hat\mathsf{x}_{2k}\dotsm\mathsf{x}_{2N}}\\[3mm]
&\quad= \cp{\mathsf{x}_{11}}{\mathsf{x}_{21}} \cp{\mathsf{x}_{12}\dotsm\mathsf{x}_{1N}}{\mathsf{x}_{22}\dotsm\mathsf{x}_{2N}}
-\sum_{k=2}^{N} \cp{\mathsf{x}_{11}}{\mathsf{x}_{2k}}
\cp{\mathsf{x}_{1k}\mathsf{x}_{12}\dotsm\hat\mathsf{x}_{1k}\dotsm\mathsf{x}_{1N}}{\mathsf{x}_{21}\dotsm\hat\mathsf{x}_{2k}\dotsm\mathsf{x}_{2N}}.\label{eq_Dodgson_exp}
}
Moving the letter $\mathsf{x}_{1k}$ in the last line guarantees that the letter pairs $(\mathsf{x}_{1l},\mathsf{x}_{2l})$, with $l\neq 1,k$, are still paired up in the expansion. In fact, the word pairs
\al{
(\mathsf{x}_{1k}\mathsf{x}_{12}\dotsm\hat\mathsf{x}_{1k}\dotsm\mathsf{x}_{1N}, \mathsf{x}_{21}\dotsm\hat\mathsf{x}_{2k}\dotsm\mathsf{x}_{2N})
}
are the representatives $(\mathsf{u}_{\id}', \mathsf{v}_{\id}')$ of an equivalence class of word pairs associated to the diagram $\pi_0(D)$, where $D$ is $D_0$ together with the chord corresponding to the letter pair $(\mathsf{x}_{11},\mathsf{x}_{2k})$. The sum over all equivalence classes in $\mathsf{P}_j$ can be realised by summing word pairs of the form
\al{
( \mathsf{x}_{(1+t_1)1}\dotsm\mathsf{x}_{(1+t_N)N},\ \mathsf{x}_{(2-t_1)1}\dotsm\mathsf{x}_{(2-t_N)N} )
}
over all $N$-tuples in $\mathcal{T}=\{ t\in \{0,1\}^N \ | \ t_1=0\}$. One finds
\al{\nonumber
\kp&\sum_{t\in\mathcal{T}} \cp{\mathsf{x}_{(1+t_1)1}\dotsm\mathsf{x}_{(1+t_N)N}}{\mathsf{x}_{(2-t_1)1}\dotsm\mathsf{x}_{(2-t_N)N}}\\\nonumber
&\quad= \cp{\mathsf{x}_{11}}{\mathsf{x}_{21}} \sum_{t\in\mathcal{T}} \cp{\mathsf{x}_{(1+t_2)2}\dotsm\mathsf{x}_{(1+t_N)N}}{\mathsf{x}_{(2-t_2)2}\dotsm\mathsf{x}_{(2-t_N)N}}\\
&\qquad -\sum_{k=2}^{N}\sum_{t\in\mathcal{T}} \cp{\mathsf{x}_{11}}{\mathsf{x}_{(2-t_k)k}}
\cp{\mathsf{x}_{(1+t_k)k}\mathsf{x}_{(1+t_2)2}\dotsm\hat\mathsf{x}_{(1+t_k)k}\dotsm\mathsf{x}_{(1+t_N)N}}{\mathsf{x}_{21}\dotsm\hat\mathsf{x}_{(2-t_k)k}\dotsm\mathsf{x}_{(2-t_N)N}}.
\label{eq_lemma_expansion}
}
Now we want to translate this back to vertices of a chord diagram. Let $u,v\in V_{D_0}$ such that $\mathsf{x}_{11}=\mathsf{a}_u$, $\mathsf{x}_{21}=\mathsf{a}_v$ and $(u,v) \in E_{D_0}^j$. Note that, by definition of $\mathsf{P}_j$, such $u,v$ always exist. Then \refeq{eq_lemma_expansion} becomes
\al{
\kp\!\!\!\!\sum_{ (\mathsf{u}_{\id},\mathsf{v}_{\id})\in \mathsf{P}_j } \cp{\mathsf{u}_{\id}}{\mathsf{v}_{\id}}
&= 2\cp{\mathsf{a}_u}{\mathsf{a}_v} \!\!\!\!\sum_{ (\mathsf{u}_{\id}', \mathsf{v}_{\id}') \in \mathsf{P}_j^{u,v}}\!\!\!\!\cp{\mathsf{u}_{\id}'}{\mathsf{v}_{\id}'}
-\sum_{\substack{ w\in V_{D_0}\\ w\neq u,v }} \cp{\mathsf{a}_u}{\mathsf{a}_w}
\!\!\!\!\sum_{ (\mathsf{u}_{\id}', \mathsf{v}_{\id}') \in \mathsf{P}_j^{u,w}}\!\!\!\!\cp{\mathsf{u}_{\id}'}{\mathsf{v}_{\id}'},\label{eq_prop_exp}
}
where $\mathsf{P}_j^{u,v}$ and $\mathsf{P}_j^{u,w}$ are the classes of word pairs after addition of the chords $(u,v)$ or $(u,w)$ respectively. Replacing these sums with the corresponding r.h.s. of \refeq{eq_lemma_fullpart} finishes the proof, where the factor of $-(-2)$ in the first term corresponds to the addition of the cycle that consists of the $j$-coloured base edge $(u,v)$ and the chord between those same vertices. All other chords $(u,w)$ added to $D_0$ do not add two-coloured cycles but only split, twist or merge base cycles when projected out with $\pi_0$.
\end{proof}
\begin{example}\label{ex_lem1}
Consider as an example $N=3$ with a single base cycle. Label vertices consecutively from $1$ to $6$ and choose $j$ to be the colour of $(1,2)$. Then the sum over word pairs in $\mathsf{P}_j$ on the l.h.s. of \refeq{eq_lemma_fullpart} is
\al{
\cp{\mathsf{a}_1\mathsf{a}_3\mathsf{a}_5}{\mathsf{a}_2\mathsf{a}_4\mathsf{a}_6} + \cp{\mathsf{a}_1\mathsf{a}_4\mathsf{a}_5}{\mathsf{a}_2\mathsf{a}_3\mathsf{a}_6} + \cp{\mathsf{a}_1\mathsf{a}_3\mathsf{a}_6}{\mathsf{a}_2\mathsf{a}_4\mathsf{a}_5} + \cp{\mathsf{a}_1\mathsf{a}_4\mathsf{a}_6}{\mathsf{a}_2\mathsf{a}_3\mathsf{a}_5}.
}
Expanding each term as defined in \refeq{eq_DefExpansion} yields $\kp^{-2}$ times 24 terms, 15 of which are distinct, such that one finds
\al[*]{
4&\cp{\mathsf{a}_1}{\mathsf{a}_2}\cp{\mathsf{a}_3}{\mathsf{a}_4}\cp{\mathsf{a}_5}{\mathsf{a}_6} - 2\cp{\mathsf{a}_1}{\mathsf{a}_2}\cp{\mathsf{a}_3}{\mathsf{a}_5}\cp{\mathsf{a}_4}{\mathsf{a}_6} - 2\cp{\mathsf{a}_1}{\mathsf{a}_2}\cp{\mathsf{a}_3}{\mathsf{a}_6}\cp{\mathsf{a}_4}{\mathsf{a}_5}\\[2mm]
-2&\cp{\mathsf{a}_1}{\mathsf{a}_3}\cp{\mathsf{a}_2}{\mathsf{a}_4}\cp{\mathsf{a}_5}{\mathsf{a}_6} + \cp{\mathsf{a}_1}{\mathsf{a}_3}\cp{\mathsf{a}_2}{\mathsf{a}_5}\cp{\mathsf{a}_4}{\mathsf{a}_6} + \cp{\mathsf{a}_1}{\mathsf{a}_3}\cp{\mathsf{a}_2}{\mathsf{a}_6}\cp{\mathsf{a}_4}{\mathsf{a}_5}\\[2mm]
-2&\cp{\mathsf{a}_1}{\mathsf{a}_4}\cp{\mathsf{a}_2}{\mathsf{a}_3}\cp{\mathsf{a}_5}{\mathsf{a}_6} + \cp{\mathsf{a}_1}{\mathsf{a}_4}\cp{\mathsf{a}_2}{\mathsf{a}_5}\cp{\mathsf{a}_3}{\mathsf{a}_6} + \cp{\mathsf{a}_1}{\mathsf{a}_4}\cp{\mathsf{a}_2}{\mathsf{a}_6}\cp{\mathsf{a}_3}{\mathsf{a}_5}\\[2mm]
+&\cp{\mathsf{a}_1}{\mathsf{a}_5}\cp{\mathsf{a}_2}{\mathsf{a}_3}\cp{\mathsf{a}_4}{\mathsf{a}_6} + \cp{\mathsf{a}_1}{\mathsf{a}_5}\cp{\mathsf{a}_2}{\mathsf{a}_4}\cp{\mathsf{a}_3}{\mathsf{a}_6} - 2\cp{\mathsf{a}_1}{\mathsf{a}_5}\cp{\mathsf{a}_2}{\mathsf{a}_6}\cp{\mathsf{a}_3}{\mathsf{a}_4}\\[2mm]
+&\cp{\mathsf{a}_1}{\mathsf{a}_6}\cp{\mathsf{a}_2}{\mathsf{a}_3}\cp{\mathsf{a}_4}{\mathsf{a}_5} + \cp{\mathsf{a}_1}{\mathsf{a}_6}\cp{\mathsf{a}_2}{\mathsf{a}_4}\cp{\mathsf{a}_3}{\mathsf{a}_5} - 2\cp{\mathsf{a}_1}{\mathsf{a}_6}\cp{\mathsf{a}_2}{\mathsf{a}_5}\cp{\mathsf{a}_3}{\mathsf{a}_4}.
}
Now one can simply check each summand by counting the cycles of the corresponding chord diagram, while keeping in mind that \textit{only} the bicoloured cycles with chords and $j$-coloured base edges are counted. For example, in the first term each factor corresponds to a chord $(1,2)$, $(3,4)$, $(5,6)$, each spanning exactly one of the $j$-coloured base edges. Hence, there are three such cycles and $(-2)^{c_2^j(D)-1} = 4$.
\end{example}
The obvious next questions is now: Can we find such an identity for all partitions? Indeed, we can.
\begin{lemma}\label{lemma_part_2}
Let $\mathcal{E} \in \mathcal{P}(E_{D_0}^1)$ be any partition of $1$-coloured base edges of a diagram $D_0\in \mathcal{D}_0^n$ and $\mathsf{P}_2$ the corresponding word pairs as above. Then
\al{
\sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 }\!\!\!\! \sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \!\!\!\! \prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) } \!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'}
= (-1)^{1-|\mathcal{E}|}(-\kp)^{|\mathcal{E}|-N}\!\!\!\!\sum_{D\in\mathcal{D}|_{\mathcal{E}}^0} (-2)^{c_2^2(D)-1} \!\!\!\!\prod_{(u,v)\in E_D^0} \!\!\!\!\cp{\mathsf{a}_u}{\mathsf{a}_v},
}
where $\mathcal{D}|_{\mathcal{E}}^0\subset \mathcal{D}_{\Gamma}^0 \simeq \mathcal{D}_N^n$ is the subset of complete chord diagrams with base cycles given by $n$ (and vertices labelled by edges of $\Gamma$) that is restricted by demanding that all chords of a diagram can only connect vertices that lie within the same part of $\mathcal{E}$.
\end{lemma}
\begin{proof}
Consider again the word pair $( \mathsf{x}_{11}\dotsm\mathsf{x}_{1N}, \mathsf{x}_{21}\dotsm\mathsf{x}_{2N} )$. The letter pairs $(\mathsf{x}_{1i}, \mathsf{x}_{2i})$ correspond to $2$-coloured base edges, so the $1$-coloured base edges correspond to pairs $(\mathsf{x}_{1(i+1)}, \mathsf{x}_{2i})$ for $i \neq n_1, n_1+n_2, \dotsc, N$ as well as $(\mathsf{x}_{11}, \mathsf{x}_{2n_1})$, $(\mathsf{x}_{1(n_1+1)}, \mathsf{x}_{2(n_1+n_2)})$ etc. due to cyclicity in each base cycle. With this we can represent the partitions of $E_{D_0}^1$ by partitions of $\{1,\dots, N\}$.\\
Assume at first that there is a single base cycle with $n_1=N$ and the partition has two parts, $I = \{i_1,\dotsc, i_{l_1}\} \subset \{1,\dots, N\}$ and $J = \{j_1,\dotsc, j_{l_2}\} \subset \{1,\dots, N\}$ with $j_{l_2} = N$. The extension to the general case is quite straightforward and discussed further below. Now look again at word pairs of the form
\al[*]{
( \mathsf{x}_{(1+t_1)1}\dotsm\mathsf{x}_{(1+t_N)N},\ \mathsf{x}_{(2-t_1)1}\dotsm\mathsf{x}_{(2-t_N)N} )
}
summed over all $N$-tuples in $\mathcal{T}=\{ t\in \{0,1\}^N \ | \ t_1=0\}$. The map $\lambda_{\mathcal{E}}$ restricts which tuples are permitted in the sum and describes how the remaining word pairs have to be split up. The only word pair that always yields a nonempty set under $\lambda_{\mathcal{E}}$ is that of $t=(0,\dotsc,0)$ where one finds
\al{\nonumber
\lambda_{\mathcal{E}}( \mathsf{x}_{11}\dotsm&\mathsf{x}_{1N}, \mathsf{x}_{21}\dotsm\mathsf{x}_{2N} )\\
&= \{ ( \mathsf{x}_{1(i_1+1)}\dotsm\mathsf{x}_{1(i_{l_1}+1)}, \mathsf{x}_{2i_1}\dotsm\mathsf{x}_{2i_{l_1}} ), ( \mathsf{x}_{1(j_1+1)}\dotsm\mathsf{x}_{1(j_{l_2}+1)}, \mathsf{x}_{2j_1}\dotsm\mathsf{x}_{2j_{l_2}} )\label{eq_lemma2_lambda}
}
with the cyclic identification $\mathsf{x}_{1(N+1)} = \mathsf{x}_{11}$ understood. By construction both words in each pair have the same length, $l_1$ and $l_2$ respectively. Moreover, we can note that regardless of the specific partition the same permutation applied to both words of the concatenated pair
\al[*]{
( \mathsf{x}_{1(i_1+1)}\dotsm\mathsf{x}_{1(i_{l_1}+1)} \mathsf{x}_{1(j_1+1)}\dotsm\mathsf{x}_{1(j_{l_2}+1)} , \mathsf{x}_{2i_1}\dotsm\mathsf{x}_{2i_{l_1}}\mathsf{x}_{2j_1}\dotsm\mathsf{x}_{2j_{l_2}} )
}
returns $( \mathsf{x}_{12}\dotsm\mathsf{x}_{1N}\mathsf{x}_{11}, \mathsf{x}_{21}\dotsm\mathsf{x}_{2N} )$, so here $\sgn_{\mathcal{E}}(( \mathsf{x}_{11}\dotsm\mathsf{x}_{1N}, \mathsf{x}_{21}\dotsm\mathsf{x}_{2N} ) = (-1)^{N-1}$.
Next we need to study what happens for different word pairs, i.e. if the letter pairs $(\mathsf{x}_{1r}, \mathsf{x}_{2r})$ are exchanged for all $r$ in another subset $R \subset \{2,\dotsc N\}$. If $r$ and $r-1$ are both in $I$ or both in $J$ then the swap of $\mathsf{x}_{1r}$ and $\mathsf{x}_{2r}$ results in word pairs that still have equal length words since $\mathsf{x}_{1r}$ is contained in the same word pair as $\mathsf{x}_{2r}$. If $r$ and $r-1$ are not in the same part then we find that exchange of any single letter pair $(\mathsf{x}_{1r}, \mathsf{x}_{2r})$ will lead to words of different lengths in each pair such that the term does not contribute. Hence, each exchange of a letter pair $(\mathsf{x}_{1r}, \mathsf{x}_{2r})$ with $r\in I$ and $r-1\notin I$ will require another exchange of $(\mathsf{x}_{1s}, \mathsf{x}_{2s})$ with a suitable $s\in J$ to compensate and return word pairs with non-vanishing contribution. Here we need to start distinguishing between different types of partitions.\\
%
First, let $I$ and $J$ be sets of consecutive numbers (counting $N$ and $1$ as such). Then there are only two $r\in\{1,\dotsc, N\}$ such that $r$ and $r-1$ are in different parts. Since only word pairs in which either both or neither are exchanged contribute one finds that exactly half of all word pairs in $\mathsf{P}_2$ yield non-empty sets of pairs under $\lambda_{\mathcal{E}}$. Then the sum
\al[*]{
\sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 }\!\!\!\! \sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \!\!\!\! \prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) } \!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'}
}
contains $2^{N-2}$ terms that decompose into two factors with $2^{l_1-1}$ and $2^{l_2-1}$ terms corresponding to the two parts. Permutations with signum $(-1)^{l_1-1}$ and $(-1)^{l_2-1}$ can be used (analogous to the discussion of the sign above) to align the original letter pairs (corresponding to $2$-coloured base edges) in each word pair. Then each such factor can be rewritten with lemma \ref{lemma_fullpart}, where one interprets it as arising from a certain smaller chord diagram base cycle. That cycle, say for the part $I$, results from contraction of the path that consists all $1$-coloured base edges represented by the integers in $J$ as well as the $2$-coloured base edges in between these (consecutive) $1$-coloured edges to a single $2$-coloured base edge. Any pair of diagrams built on these smaller base cycles corresponds to a larger diagram with the original base cycle that one finds by simply cutting the contracted $2$-coloured base edge in each diagram and gluing them together. The number of $2$-coloured cycles is almost additive but the cutting removes one cycle in each diagram and restores only one when gluing them together. Hence, $c_2^2(D_I)-1 + c_2^2(D_J)-1 = c_2^2(D_{IJ})-1$.
This straightforwardly extends to partitions with any number of parts, as long as each consists of consecutive base edges, and one finds
\al{\nonumber
\sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 }\!\!\!\! \sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) & \!\!\!\! \prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) } \!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'}\\\nonumber
&= (-1)^{1-N} \prod_{i=1}^{|\mathcal{E}|} \Big( (-1)^{l_i-1} (-\kp)^{1-l_i} \sum_{D_i\in\mathcal{D}_{\Gamma,i}^0} (-2)^{c_2^2(D_i)-1} \!\!\!\!\prod_{(u,v)\in E_{D_i}^0} \cp{\mathsf{a}_u}{\mathsf{a}_v} \Big)\\[3mm]
&= (-1)^{1-|\mathcal{E}|}(-\kp)^{|\mathcal{E}|-N}\!\!\!\!\sum_{D\in\mathcal{D}|_{\mathcal{E}}^0} (-2)^{c_2^2(D)-1} \!\!\!\!\prod_{(u,v)\in E_D^0} \!\!\!\!\cp{\mathsf{a}_u}{\mathsf{a}_v}
}
where $l_1,\dotsc, l_{|\mathcal{E}|}$ with $l_1+\dotsc+\l_{|\mathcal{E}|} = N$ are the cardinalities of each part. This even extends further to partitions like $\{ \{1\}, \{3\}, \{2,4\} \}$ where the part $\{2,4\}$ does not contain consecutive base edges initially but $2$ and $4$ become consecutive after factoring out the terms (contracting the base edges) corresponding to $1$ and $3$.\\
Next we look at the exact opposite case, i.e. we assume that $I$ and $J$ do not contain any consecutive numbers at all. Note that then both parts need to have the same cardinality $|I|=|J|=N/2$ and $N$ has to be even. The contributing word pairs can be found by considering all possible choices of $k\leq N/2-1$ index swaps out of the set that contains $1$ (which is kept fixed) together with all possible choices of the same number of indices from the other set (in which all $N/2$ elements are permitted). The number of such exchanges can be counted with Vandermonde's identity to be
\al{
\sum_{k=0}^{N/2-1} \binom{N/2-1}{k}\binom{N/2}{k} = \binom{N-1}{N/2-1} = \frac{1}{2}\binom{N}{N/2}.
}
The sum containing these terms does not factorise, but we can reduce it to a sum of expressions corresponding to $N-1$, allowing for proof by induction. Choose one of the two parts and expand the corresponding factor of each summand analogously to \refeq{eq_Dodgson_exp}. By construction the first term on the r.h.s. cannot exist in these expansions, since $\mathsf{x}_{11}$ and $\mathsf{x}_{21}$ belong to different word pairs. The sum contains fewer terms but the principle is the same: Suitable permutation within the remaining word pair allows us to interpret it as associated to a diagram that in turn resulted from addition of a chord corresponding to the removed letter pair. Hence, we only pick up an overall factor of $-\kp$ and can collect coefficients of each Dodgson polynomial $\cp{\mathsf{x}_{1r}}{\mathsf{x}_{2s}}$. By simply counting how often a given letter pair is or is not involved in an exchange one finds that one can collect terms into groups of
\al{
\binom{N-2}{N/2-1} = \binom{N-2}{(N-2)/2}
}
which is exactly twice the number of possible exchanges we would have for $N-2$. The coefficient of each $\cp{\mathsf{x}_{1r}}{\mathsf{x}_{2s}}$ corresponds to the sum in \refeq{eq_lemma2_lambda} but for a smaller diagram with $N'=N-1$ and a corresponding smaller partition. For the small cases of $N=2$ and $N=4$ the reduction already yields factorising expressions (see example \ref{ex_lemma_part_2}). For larger $N$ that is generally not the case and since $N-1$ is odd it also cannot belong to the case we discussed here. Instead, what happens is a partial factorisation that allows us to collect the $2$, $6$, $20$, $\dotsc$ terms into $1$, $3$, $10$, $\dotsc$ pairs which correspond to a non-factorising partition with a total cardinality of $N-2$. The corresponding partition consists of one part in which all elements are still non-consecutive and one part that contains only exactly one pair of consecutive numbers. Then the reduction process goes through for any such partition with mixed consecutive and non-consecutive base edges, even for more than two parts. If there are $k$ pairs of consecutive base edges in one part then this simply yields $2^{k-1}$ terms which correspond to a subset of the possible word pairs resulting from some smaller diagram -- but it is not the full subset needed for the factorisation seen above.\\
Finally, all of this goes through for any number of base cycles without much change. The only difference is in which base edges are viewed as consecutive. For example, for a diagram with two base cycles of size $n_1$ and $n_2$ with $n_1+n_2=N$ one has $(1,n_1)$ and $(n_1+1, N)$ as consecutive pairs, but neither $(1,N)$ nor $(n_1,n_1+1)$.
\end{proof}
\begin{example}\label{ex_lemma_part_2}
Consider an empty chord diagram $D\in \mathcal{D}^4_0$ on a single base cycle with $8$ vertices labelled $1$--$8$. Let $2$ be the colour of the base edges $(1,2), (3,4), (5,6), (7,8)$ and $1$ the colour of $(8,1), (2,3), (4,5), (6,7)$. The word pairs in $\mathsf{P}_2$ are, up to possible permutations,
\al[*]{
&( \mathsf{a}_1\mathsf{a}_3\mathsf{a}_5\mathsf{a}_7, \mathsf{a}_2\mathsf{a}_4\mathsf{a}_6\mathsf{a}_8 )\quad
&( \mathsf{a}_1\mathsf{a}_3\mathsf{a}_5\mathsf{a}_8, \mathsf{a}_2\mathsf{a}_4\mathsf{a}_6\mathsf{a}_7 )\quad
&( \mathsf{a}_1\mathsf{a}_3\mathsf{a}_6\mathsf{a}_7, \mathsf{a}_2\mathsf{a}_4\mathsf{a}_5\mathsf{a}_8 )\quad
&( \mathsf{a}_1\mathsf{a}_4\mathsf{a}_5\mathsf{a}_7, \mathsf{a}_2\mathsf{a}_3\mathsf{a}_6\mathsf{a}_8 )\; \\[2mm]
&( \mathsf{a}_1\mathsf{a}_3\mathsf{a}_6\mathsf{a}_8, \mathsf{a}_2\mathsf{a}_4\mathsf{a}_5\mathsf{a}_7 )\quad
&( \mathsf{a}_1\mathsf{a}_4\mathsf{a}_5\mathsf{a}_8, \mathsf{a}_2\mathsf{a}_3\mathsf{a}_6\mathsf{a}_7 )\quad
&( \mathsf{a}_1\mathsf{a}_4\mathsf{a}_6\mathsf{a}_7, \mathsf{a}_2\mathsf{a}_3\mathsf{a}_5\mathsf{a}_8 )\quad
&( \mathsf{a}_1\mathsf{a}_4\mathsf{a}_6\mathsf{a}_8, \mathsf{a}_2\mathsf{a}_3\mathsf{a}_5\mathsf{a}_7 ).
}
The partitions with one part are very similar to those in example \ref{ex_lem1} but a bit too large already to sensibly write them here in their fully expanded form. With two parts there are three types of partitions. Firstly, if $\mathcal{E} = \{E_1, E_2\}$ with $|E_1| = 3$, $|E_2|=1$, then the factorisation is obvious. For example, for $\mathcal{E} = \{ \{(8,1), (2,3), (4,5)\}, \{(6,7)\}$ one has
\al[*]{
- \cp{\mathsf{a}_6}{\mathsf{a}_7}\Big(
\cp{ \mathsf{a}_1\mathsf{a}_3\mathsf{a}_5}{\mathsf{a}_2\mathsf{a}_4\mathsf{a}_8} +
\cp{ \mathsf{a}_1\mathsf{a}_4\mathsf{a}_5}{\mathsf{a}_2\mathsf{a}_3\mathsf{a}_8} +
\cp{ \mathsf{a}_1\mathsf{a}_3\mathsf{a}_8}{\mathsf{a}_2\mathsf{a}_4\mathsf{a}_5} +
\cp{ \mathsf{a}_1\mathsf{a}_4\mathsf{a}_8}{\mathsf{a}_2\mathsf{a}_3\mathsf{a}_5}
\Big).
}
Similarly, for a partition like $\mathcal{E} = \{ \{(8,1), (2,3)\}, \{(4,5), (6,7)\}$ one also finds a factorisation since the four terms one gets are
\al{
\cp{ \mathsf{a}_1\mathsf{a}_3}{\mathsf{a}_2\mathsf{a}_8} \cp{ \mathsf{a}_5\mathsf{a}_7}{\mathsf{a}_4\mathsf{a}_6} +
\cp{ \mathsf{a}_1\mathsf{a}_3}{\mathsf{a}_2\mathsf{a}_8} \cp{ \mathsf{a}_6\mathsf{a}_7}{\mathsf{a}_4\mathsf{a}_5} +
\cp{ \mathsf{a}_1\mathsf{a}_8}{\mathsf{a}_2\mathsf{a}_3} \cp{ \mathsf{a}_6\mathsf{a}_7}{\mathsf{a}_4\mathsf{a}_5} +
\cp{ \mathsf{a}_1\mathsf{a}_8}{\mathsf{a}_2\mathsf{a}_3} \cp{ \mathsf{a}_5\mathsf{a}_7}{\mathsf{a}_4\mathsf{a}_6}.
}
The non-factorising partition $\mathcal{E} = \{ \{(8,1), (4,5)\}, \{(2,3), (6,7)\}$ yields three terms that we can still simply expand explicitly:
\al{\nonumber
\big(-\kp\big)^2& \Big(
\cp{ \mathsf{a}_1\mathsf{a}_5}{\mathsf{a}_4\mathsf{a}_8} \cp{ \mathsf{a}_3\mathsf{a}_7}{\mathsf{a}_2\mathsf{a}_6} +
\cp{ \mathsf{a}_1\mathsf{a}_8}{\mathsf{a}_4\mathsf{a}_5} \cp{ \mathsf{a}_3\mathsf{a}_6}{\mathsf{a}_2\mathsf{a}_7} +
\cp{ \mathsf{a}_1\mathsf{a}_4}{\mathsf{a}_5\mathsf{a}_8} \cp{ \mathsf{a}_6\mathsf{a}_7}{\mathsf{a}_2\mathsf{a}_3}
\Big)\\[2mm]\nonumber
%
=& \Big( \cp{ \mathsf{a}_1}{\mathsf{a}_4}\cp{ \mathsf{a}_5}{\mathsf{a}_8} - \cp{ \mathsf{a}_1}{\mathsf{a}_8}\cp{ \mathsf{a}_4}{\mathsf{a}_5} \Big)\Big( \cp{ \mathsf{a}_2}{\mathsf{a}_3}\cp{ \mathsf{a}_6}{\mathsf{a}_7} - \cp{ \mathsf{a}_2}{\mathsf{a}_7}\cp{ \mathsf{a}_3}{\mathsf{a}_6} \Big)\\\nonumber
& + \Big( \cp{ \mathsf{a}_1}{\mathsf{a}_4}\cp{ \mathsf{a}_5}{\mathsf{a}_8} - \cp{ \mathsf{a}_1}{\mathsf{a}_5}\cp{ \mathsf{a}_4}{\mathsf{a}_8} \Big)\Big( \cp{ \mathsf{a}_2}{\mathsf{a}_3}\cp{ \mathsf{a}_6}{\mathsf{a}_7} - \cp{ \mathsf{a}_2}{\mathsf{a}_6}\cp{ \mathsf{a}_3}{\mathsf{a}_7} \Big)\\\nonumber
& + \Big( \cp{ \mathsf{a}_1}{\mathsf{a}_5}\cp{ \mathsf{a}_4}{\mathsf{a}_8} - \cp{ \mathsf{a}_1}{\mathsf{a}_8}\cp{ \mathsf{a}_4}{\mathsf{a}_5} \Big)\Big( \cp{ \mathsf{a}_2}{\mathsf{a}_6}\cp{ \mathsf{a}_3}{\mathsf{a}_7} - \cp{ \mathsf{a}_2}{\mathsf{a}_7}\cp{ \mathsf{a}_3}{\mathsf{a}_6} \Big)\\[2mm]\nonumber
%
=& (-1)^3(-2)^1 \Big(
\cp{ \mathsf{a}_1}{\mathsf{a}_4}\cp{ \mathsf{a}_2}{\mathsf{a}_3}\cp{ \mathsf{a}_5}{\mathsf{a}_8}\cp{ \mathsf{a}_6}{\mathsf{a}_7} +
\cp{ \mathsf{a}_1}{\mathsf{a}_5}\cp{ \mathsf{a}_2}{\mathsf{a}_6}\cp{ \mathsf{a}_3}{\mathsf{a}_7}\cp{ \mathsf{a}_4}{\mathsf{a}_8}\\\nonumber
& \hphantom{=(-1)^3(-2)^1\Big( \cp{ \mathsf{a}_1}{\mathsf{a}_4}\cp{ \mathsf{a}_2}{\mathsf{a}_3}\cp{ \mathsf{a}_5}{\mathsf{a}_8}\cp{ \mathsf{a}_6}{\mathsf{a}_7}}
+ \cp{ \mathsf{a}_1}{\mathsf{a}_8}\cp{ \mathsf{a}_2}{\mathsf{a}_7}\cp{ \mathsf{a}_3}{\mathsf{a}_6}\cp{ \mathsf{a}_4}{\mathsf{a}_5} \Big)\\\nonumber
&+ (-1)^3(-2)^0 \Big(
\cp{ \mathsf{a}_1}{\mathsf{a}_4}\cp{ \mathsf{a}_2}{\mathsf{a}_6}\cp{ \mathsf{a}_3}{\mathsf{a}_7}\cp{ \mathsf{a}_5}{\mathsf{a}_8} +
\cp{ \mathsf{a}_1}{\mathsf{a}_4}\cp{ \mathsf{a}_2}{\mathsf{a}_7}\cp{ \mathsf{a}_3}{\mathsf{a}_6}\cp{ \mathsf{a}_5}{\mathsf{a}_8}\\\nonumber
& \hphantom{(-1)^3(-2)^0 \Big(}
+ \cp{ \mathsf{a}_1}{\mathsf{a}_5}\cp{ \mathsf{a}_2}{\mathsf{a}_3}\cp{ \mathsf{a}_4}{\mathsf{a}_8}\cp{ \mathsf{a}_6}{\mathsf{a}_7} +
\cp{ \mathsf{a}_1}{\mathsf{a}_5}\cp{ \mathsf{a}_2}{\mathsf{a}_7}\cp{ \mathsf{a}_3}{\mathsf{a}_6}\cp{ \mathsf{a}_4}{\mathsf{a}_8}\\
& \hphantom{(-1)^3(-2)^0 \Big(}
+ \cp{ \mathsf{a}_1}{\mathsf{a}_8}\cp{ \mathsf{a}_2}{\mathsf{a}_3}\cp{ \mathsf{a}_4}{\mathsf{a}_5}\cp{ \mathsf{a}_6}{\mathsf{a}_7} +
\cp{ \mathsf{a}_1}{\mathsf{a}_8}\cp{ \mathsf{a}_2}{\mathsf{a}_6}\cp{ \mathsf{a}_3}{\mathsf{a}_7}\cp{ \mathsf{a}_4}{\mathsf{a}_5}
\Big).
}
With a quick drawing one can now check that the chord diagrams corresponding to these terms are as expected and that the number of cycles is indeed correct. Finally, expanding only one of the two polynomials in each summand leads to the reduction from the proof of lemma \ref{lemma_part_2}:
\al[*]{
& \cp{ \mathsf{a}_1}{\mathsf{a}_4}\cp{ \mathsf{a}_5}{\mathsf{a}_8}\big( \cp{ \mathsf{a}_3\mathsf{a}_7}{\mathsf{a}_2\mathsf{a}_6} + \cp{ \mathsf{a}_3\mathsf{a}_6}{\mathsf{a}_2\mathsf{a}_7} \big)\\
+& \cp{ \mathsf{a}_1}{\mathsf{a}_5}\cp{ \mathsf{a}_4}{\mathsf{a}_8}\big( \cp{ \mathsf{a}_2\mathsf{a}_7}{\mathsf{a}_6\mathsf{a}_3} + \cp{ \mathsf{a}_2\mathsf{a}_3}{\mathsf{a}_6\mathsf{a}_7} \big)\\
+& \cp{ \mathsf{a}_1}{\mathsf{a}_8}\cp{ \mathsf{a}_4}{\mathsf{a}_5}\big( \cp{ \mathsf{a}_2\mathsf{a}_6}{\mathsf{a}_7\mathsf{a}_3} + \cp{ \mathsf{a}_2\mathsf{a}_3}{\mathsf{a}_7\mathsf{a}_6} \big).
}
\end{example}
%
The final ingredient for the proof of this chapter's main theorem is an identity allowing summation of Stirling numbers of the second kind $S(k,l)$. They count the ways to partition a set of $k$ elements into $l$ non-empty sets. To prove it we need a certain identity relating Stirling numbers and the classical polylogarithm. While the literature contains a number of well known identities that do so, they are all either similar but not obviously equivalent to the one we need, or appear without proof. Moreover, the commonly cited references (e.g. \cite{abramowitz+stegun, Stanley_EnumComb, knuth_1998}, among many others) all appear to cite each other or unavailable older literature, so it may actually be somewhat elucidating to derive everything we need ourselves.
\begin{proposition}\label{prop_polylog}
Let
\al{
\Li_s(z) = \sum_{l=1}^{\infty}\frac{z^l}{l^s}\qquad |z|<1, \ s\in \mathbb{Z}
}
be the classical polylogarithm and $S(k,l)$ be the Stirling number of the second kind. Then
\al{
\Li_{-k+1}(z) = (-1)^k\sum_{l=1}^k S(k,l) \frac{(l-1)!}{(z-1)^l}
}
for integers $k\geq 2$.
\end{proposition}
\begin{proof}
For $k=2$ the r.h.s. is
\al{
\frac{1}{z-1} + \frac{1}{(z-1)^2} = \frac{z}{(1-z)^2} = z\partial_z\frac{1}{1-z}
= z\partial_z\sum_{l=0}^{\infty} z^l
= \sum_{l=1}^{\infty}l z^l = \Li_{-1}(z).
}
Now proceed by induction
\al{\nonumber
\Li_{-k+1}(z) = z\partial_z \Li_{-k+2}(z)
&= (-1)^{k-1}\sum_{l=1}^{k-1} S(k-1,l) z\partial_z \frac{1}{(z-1)^l}(l-1)!\\
&= (-1)^k\sum_{l=1}^{k-1} S(k-1,l) \frac{z}{(z-1)^{l+1}} l!,
}
and use partial fraction decomposition to find
\al{
S(k-1,l) \frac{z}{(z-1)^{l+1}}l!& = lS(k-1,l) \frac{(l-1)!}{(z-1)^l} + S(k-1,l)\frac{l!}{(z-1)^{l+1}}\label{eq_lem_auxA}
}
Using the recurrence relation $S(k,l) = S(k-1,l-1) + l(S(k-1,l)$ the first term is further rewritten as
\al{
lS(k-1,l) \frac{(l-1)!}{(z-1)^l} = S(k,l)\frac{(l-1)!}{(z-1)^l} - S(k-1,l-1)\frac{(l-1)!}{(z-1)^l}\label{eq_lem_auxB}
}
In the sum one now has a telescopic cancellation involving the second terms of eqs. (\ref{eq_lem_auxA}) and (\ref{eq_lem_auxB}). The only remaining terms are
\al[*]{
\frac{S(k-1,0)}{z-1} = 0 \qquad\text{ and }\qquad S(k-1,k-1)\frac{(k-1)!}{(z-1)^k} = S(k,k)\frac{(k-1)!}{(z-1)^k},
}
as well as the first part of the r.h.s. of \refeq{eq_lem_auxB} summed up to $l=k-1$, such that overall
\al{\nonumber
\Li_{-k+1}(z) &= (-1)^k\sum_{l=1}^k S(k,l)\frac{(l-1)!}{(z-1)^l}.
}
\end{proof}
\begin{lemma}\label{lemma_maintheo}
Let $S(k,l)$ be the Stirling number of the second kind. Then
\al{\nonumber
\sum_{l=1}^{k} S( k, l ) (-1)^l(l+1)! = (-2)^k\qquad \forall k \geq 1.
}
\end{lemma}
\begin{proof}
For $k=1$ the claim is checked directly. For $k\geq 2$ we use the identity derived for the polylogarithm in proposition \ref{prop_polylog} and note that a change of the argument allows us to write
\al{
(-1)^k\Li_{-k+1}\Big(1+\frac{1}{z}\Big) = \sum_{l=1}^k S(k,l)z^l(l-1)!
}
with $z < -1$. Now let
\al{\nonumber
L(z) \defeq \sum_{l=1}^{k} S( k, l ) z^l(l+1)!
&= z\partial^2_z z\sum_{l=1}^{k} S( k, l ) z^l(l-1)!\\
&= (-1)^k z\partial^2_z z \Li_{-k+1}\Big(1+\frac{1}{z}\Big).
}
Computing the derivative one finds
\al{
L(z) = \frac{(-1)^k}{(z+1)^2}\bigg( \Li_{-k-1}\Big(1+\frac{1}{z}\Big) - \Li_{-k}\Big(1+\frac{1}{z}\Big) \bigg).
}
Both polylogarithms start with terms linear in $(z+1)/z$, yielding divergences when evaluating at $z=-1$, but upon closer inspection we see that they precisely cancel each other. With $z<-1$ one has $|1+1/z|<1$ such that we are able to employ the classical sum representation of the polylogarithm, of which only the first two terms are of interest to us:
\al{\nonumber
L(z) &= \frac{(-1)^k}{(z+1)^2}\bigg( \sum_{t=1}^{\infty} t^{k+1}\left(\frac{z+1}{z}\right)^t
-\sum_{t=1}^{\infty} t^k\left(\frac{z+1}{z}\right)^t \bigg)\\[3mm]\nonumber
&= \frac{(-1)^k}{(z+1)^2}\bigg( \frac{z+1}{z} + 2^{k+1}\left(\frac{z+1}{z}\right)^2
- \frac{z+1}{z} - 2^k\left(\frac{z+1}{z}\right)^2
+ \mathcal{O}\left( \left(\frac{z+1}{z}\right)^3 \right) \bigg)\\[3mm]
&= (-2)^k \bigg( \frac{1}{z^2} + \frac{1}{(z+1)^2}\mathcal{O}\left(\left(\frac{z+1}{z}\right)^3\right) \bigg).
}
Now we can safely take the limit $z \to -1$ to find
\al{
\sum_{l=1}^{k} S( k, l ) (-1)^l(l+1)! = L(-1) = (-2)^k.
}
\end{proof}
\paragraph{Proof of Theorem \ref{theo_main}.}
First, use lemma \ref{lemma_part_2} to rewrite the partition polynomial as
\al{\nonumber
Z_{\Gamma}^0 &= \sum_{\mathcal{E} \in \mathcal{P}(E_D^1)} (-\kp)^{N-|\mathcal{E}|} (|\mathcal{E}|+1)! \sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 } \sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \!\!\!\!
\prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) } \!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'}\\[3mm]\nonumber
&= \sum_{\mathcal{E} \in \mathcal{P}(E_D^1)} (-1)^{|\mathcal{E}|+1}(|\mathcal{E}|+1)!\sum_{D\in\mathcal{D}|_{\mathcal{E}}^0} (-2)^{c_2^2(D)-1} \!\!\!\!\prod_{(u,v)\in E_D^0} \cp{\mathsf{a}_u}{\mathsf{a}_v}.
}
The sum already contains $c_2^2(D)$, the number of $2$-coloured cycles. Regarding cycles of the other colour we can make the following observation: In each diagram with $c_2^1(D)\leq N$ the $1$-coloured cycles can themselves be interpreted as a partition of $E_D^1$ in which each part is given by the base edges connected to each other by chords. The diagrams in $\mathcal{D}|_{\mathcal{E}}^0$ can only have chords connecting base edges within the same part of $\mathcal{E}$, so each part in the partition given by the $1$-coloured cycles has to be a subset of a part of $\mathcal{E}$. Counting the number of ways of partitioning the $c_2^1(D)$ cycles of a given diagram into partitions with $|\mathcal{E}|$ parts (i.e. counting the number of partitions $\mathcal{E}$ with a certain number of parts such that $\mathcal{D}|_{\mathcal{E}}^0$ contains the given diagram $D$) one finds precisely the Stirling numbers of the second kind $S(c_2^1(D), |\mathcal{E}|)$. Using this, we can exchange summation over diagrams and partitions and find
\al{\nonumber
Z_{\Gamma}^0 &= \sum_{\mathcal{E} \in \mathcal{P}(E_D^1)} (-1)^{|\mathcal{E}|+1}(|\mathcal{E}|+1)!\sum_{D\in\mathcal{D}|_{\mathcal{E}}^0} (-2)^{c_2^2(D)-1}
\!\!\!\!\prod_{(u,v)\in E_D^0} \cp{\mathsf{a}_u}{\mathsf{a}_v}\\[3mm]\nonumber
&= \frac{1}{2}\sum_{D\in\mathcal{D}^n_N} (-2)^{c_2^2(D)}\bigg( \prod_{(u,v)\in E_D^0} \cp{\mathsf{a}_u}{\mathsf{a}_v}\bigg) \sum_{l=1}^{c_2^1(D)} S( c_2^1(D), l ) (-1)^l(l+1)!.
}
Now lemma \ref{lemma_maintheo} is applied to evaluate the sum to $(-2)^{c_2^1(D)}$, which finishes the proof.
\qed
\subsection{The second summation theorem}\label{sec_inc}
Now that $Z_{\Gamma}^0$ is well understood we can proceed to the more complicated $Z_{\Gamma}^1$. Contrary to $Z_{\Gamma}^0$ they contain not only the cycle polynomials, but also $x_{\Gamma}^e$, which we had defined in \refeq{eq_CP_eval_def}. We begin by analysing these polynomials and in particular their products a bit further. Building on this we will then find that the summation theorem from the previous section can be generalised rather straightforwardly to the following result.
\begin{theorem}\label{theo_main_2}
\al{
Z_{\Gamma}^1 = \frac{1}{2} \sum_{D\in \mathcal{D}_{\Gamma}^1} (-2)^{\tilde c(D)} \bigg(\prod_{ (u,v)\in E_D^0 } \cp{u}{v}\bigg) \prod_{ w\in V_D^{(2)} } x_{\Gamma}^w.
\label{eq_maintheo_2}
}
\end{theorem}
\subsubsection{The polynomial $x_{\Gamma}^w$}
\noindent The first step to prove this theorem is getting a better understanding of the polynomials $x_{\Gamma}^w$ and their products. We begin with some general observations about their connections to bond and spanning forest polynomials and then state the precise result that we will need in lemma \ref{lemma_prodX} below.\\
Analogous to \refeq{eq_ssp_stmat} we can also write the bond polynomial as
\al{
\beta_G &= \sum_{v_1,v_2\in V_G} \vartheta_{v_1}\vartheta_{v_2} \Phi_G^{\{v_0\},\{v_1,v_2\}},
}
where the momenta $q_{v_i}$ are replaced with $\vartheta_{v_i} = \sum_e I_{ev_i} \xi_e$. With the definition of $\CP[G]{e}{\mu}$ as derivative of the bond polynomial w.r.t. $\xi_e^{\mu}$ (see \refeq{eq_CPdef}) one finds
\al{
\CP[G]{e}{\mu}
= \alpha_e^{-1} \sum_{v_1,v_2\in V_G} I_{ev_1} \vartheta_{v_2}^{\mu} \Phi_G^{\{v_0\},\{v_1,v_2\}}.
}
Then we move to the physical case, i.e. a Feynman graph $\Gamma$ in which we evaluate the formal parameters $\xi_e$ to physical momenta. For each edge there are only two vertices, namely $u_1,u_2$ with $\partial(e) = (u_1,u_2)$, such that $I_{eu_i} \neq 0$, and $I_{eu_1} = -I_{eu_2}$ for this pair. Hence, the polynomial reduces to
\al{
\CP{e}{\mu} = - \alpha_e^{-1} \sum_{v\in V_{\Gamma}^{ext}} q_v^{\mu} \big( \Phi_{\Gamma}^{\{v_0\},\{u_2,v\}} - \Phi_{\Gamma}^{\{v_0\},\{u_1,v\}} \big).
}
Accounting for cancellations between spanning forests (i.e. their corresponding monomials) that appear in both polynomials, the difference can be written as
\al{
\Phi_{\Gamma}^{\{v_0\},\{u_2,v\}} - \Phi_{\Gamma}^{\{v_0\},\{u_1,v\}}
= \Phi_{\Gamma}^{\{v_0,u_1\},\{u_2,v\}} - \Phi_{\Gamma}^{\{v_0,u_2\},\{u_1,v\}}. \label{eq_addsp}
}
If we now specialise to the case of only two external vertices $v_1,v_2$ (or at least only two with non-vanishing momenta), then this reduces further to
\al{
\CP{e}{\mu} = q^{\mu}\alpha_e^{-1} \big( \Phi_{\Gamma}^{\{v_0\},\{u_1,v_1\}} + \Phi_{\Gamma}^{\{v_0\},\{u_2,v_2\}} - \Phi_{\Gamma}^{\{v_0\},\{u_2,v_1\}} - \Phi_{\Gamma}^{\{v_0\},\{u_1,v_2\}} \big)
}
In order to explain the overall sign we emphasise again that $e$ is directed from $\partial_-(e) = u_1$ to $\partial_+(e) = u_2$, and that we chose $q_{v_1} = q = -q_{v_2}$.\\
By the same principle as \refeq{eq_addsp} we can explicitly remove terms that would cancel between these four summands:
\al{\nonumber
& \Phi_{\Gamma}^{\{v_0\},\{u_1,v_1\}} - \Phi_{\Gamma}^{\{v_0\},\{u_2,v_1\}} + \Phi_{\Gamma}^{\{v_0\},\{u_2,v_2\}}- \Phi_{\Gamma}^{\{v_0\},\{u_1,v_2\}}\\[3mm]\nonumber
&\qquad= \Phi_{\Gamma}^{\{v_0,u_2\},\{u_1,v_1\}} - \Phi_{\Gamma}^{\{v_0,u_1\},\{u_2,v_1\}} + \Phi_{\Gamma}^{\{v_0,u_1\},\{u_2,v_2\}} - \Phi_{\Gamma}^{\{v_0,u_2\},\{u_1,v_2\}}\\[3mm]\nonumber
&\qquad= \Phi_{\Gamma}^{\{v_0,u_2,v_2\},\{u_1,v_1\}} + \Phi_{\Gamma}^{\{v_0,u_1,v_1\},\{u_2,v_2\}} - \Phi_{\Gamma}^{\{v_0,u_2,v_1\},\{u_1,v_2\}} - \Phi_{\Gamma}^{\{v_0,u_1,v_2\},\{u_2,v_1\}} \\[3mm]
&\qquad= \Phi_{\Gamma}^{\{u_1,v_1\},\{u_2,v_2\}} - \Phi_{\Gamma}^{\{u_1,v_2\},\{u_2,v_1\}}. \label{eq_spshort}
}
This is now explicitly independent of the arbitrarily chosen vertex $v_0$. We can re-expand \refeq{eq_spshort} by including terms cancelled between the two to get
\al{
\Phi_{\Gamma}^{\{u_1,v_1\},\{u_2,v_2\}} - \Phi_{\Gamma}^{\{u_1,v_2\},\{u_2,v_1\}}
= \Phi_{\Gamma}^{\{v_1\},\{u_2,v_2\}} - \Phi_{\Gamma}^{\{v_1\},\{u_1,v_2\}}.
}
This is now not only independent of the original arbitrary choice of $v_0$ but can actually be interpreted as Dodgson polynomials with respect to a graph matrix in which $v_1$ was removed:
\al{
\CP{e}{\mu} = q^{\mu}\alpha_e^{-1} \big( \cp[\Gamma,v_1]{\mathsf{a}_{u_2}}{\mathsf{a}_{v_2}} - \cp[\Gamma,v_1]{\mathsf{a}_{u_1}}{\mathsf{a}_{v_2}} \big) = q^{\mu}x_{\Gamma}^e. \label{eq_newxpol}
}
\begin{lemma}\label{lemma_prodX}
Let $\Gamma$ be a QED Feynman graph with only two non-zero external momenta $q_u = q = -q_v$ at vertices $u,v \in V_{\Gamma}$, and $\Gamma^{\bullet} = \Gamma|_{u=v}$. Let furthermore $e,f \in E_{\Gamma}$ be any two edges of $\Gamma$. Then
\al{
\alpha_e\alpha_f x_{\Gamma}^ex_{\Gamma}^f &= \kp[\Gamma^{\bullet}] \bp{e}{f} - \kp \bp[\Gamma^{\bullet}]{e}{f}.
\label{eq_theo}
}
Moreover, if $e\neq f$ this simplifies to
\al{
x_{\Gamma}^ex_{\Gamma}^f = -\kp[\Gamma^{\bullet}] \cp{e}{f} + \kp \cp[\Gamma^{\bullet}]{e}{f}, \label{eq_theo_rewrite}
}
which means that up to sign $x_{\Gamma}^e = \pm \cp[\Gamma,u]{ e }{ v }$ and the signs are such that
\al{
x_{\Gamma}^ex_{\Gamma}^f = - \cp[\Gamma,u]{ e }{ v }\cp[\Gamma,u]{ f }{ v }. \label{eq_xx_dp_signs}
}
\end{lemma}
\begin{proof}
Let $a,b,c,d \in V_{\Gamma}$ be the not necessarily distinct endpoints of edges $e$ and $f$, with directions $\partial(e) = (a,b)$ and $\partial(f)=(c,d)$, and use letters $\mathsf{a} \equiv \mathsf{a}_a$, $\mathsf{b} \equiv \mathsf{a}_b$, etc. With \refeq{eq_newxpol} the product is then
\al{\nonumber
\alpha_e\alpha_f x_{\Gamma}^ex_{\Gamma}^f &=
\big( \cp[\Gamma,u]{\mathsf{b}}{\mathsf{v}} - \cp[\Gamma,u]{\mathsf{a}}{\mathsf{v}} \big) \big( \cp[\Gamma,u]{\mathsf{d}}{\mathsf{v}} - \cp[\Gamma,u]{\mathsf{c}}{\mathsf{v}} \big)\\[3mm]\nonumber
& = \cp[\Gamma,u]{\mathsf{b}}{\mathsf{v}}\cp[\Gamma,u]{\mathsf{d}}{\mathsf{v}} - \cp[\Gamma,u]{\mathsf{a}}{\mathsf{v}}\cp[\Gamma,u]{\mathsf{d}}{\mathsf{v}} - \cp[\Gamma,u]{\mathsf{b}}{\mathsf{v}}\cp[\Gamma,u]{\mathsf{c}}{\mathsf{v}}
+ \cp[\Gamma,u]{\mathsf{a}}{\mathsf{v}}\cp[\Gamma,u]{\mathsf{c}}{\mathsf{v}}\\[3mm]\nonumber
& = \cp[\Gamma,u]{\mathsf{v}}{\mathsf{v}} \big( \cp[\Gamma,u]{\mathsf{b}}{\mathsf{d}} - \cp[\Gamma,u]{\mathsf{a}}{\mathsf{d}} - \cp[\Gamma,u]{\mathsf{b}}{\mathsf{c}} + \cp[\Gamma,u]{\mathsf{a}}{\mathsf{c}} \big)\\[1mm]
&\quad - \kp\big( \cp[\Gamma,u]{\mathsf{b}\mathsf{v}}{\mathsf{d}\mathsf{v}} - \cp[\Gamma,u]{\mathsf{a}\mathsf{v}}{\mathsf{d}\mathsf{v}} - \cp[\Gamma,u]{\mathsf{b}\mathsf{v}}{\mathsf{c}\mathsf{v}} + \cp[\Gamma,u]{\mathsf{a}\mathsf{v}}{\mathsf{c}\mathsf{v}} \big).
}
The coefficient of $\cp[\Gamma,u]{\mathsf{v}}{\mathsf{v}}$ in the first summand is exactly the sum from \refeq{eq_spshort} with different labels, such that
\al{\nonumber
\cp[\Gamma,u]{\mathsf{b}}{\mathsf{d}} - \cp[\Gamma,u]{\mathsf{a}}{\mathsf{d}} - \cp[\Gamma,u]{\mathsf{b}}{\mathsf{c}} + \cp[\Gamma,u]{\mathsf{a}}{\mathsf{c}}
&= \Phi_{\Gamma}^{\{u\},\{b,d\}} - \Phi_{\Gamma}^{\{u\},\{a,d\}} - \Phi_{\Gamma}^{\{u\},\{b,c\}} + \Phi_{\Gamma}^{\{u\},\{a,c\}}\\[3mm]
&= \Phi_{\Gamma}^{\{a,c\},\{b,d\}} - \Phi_{\Gamma}^{\{b,c\},\{a,d\}}.\label{eq_dodgsonrem}
}
$\cp[\Gamma,u]{\mathsf{v}}{\mathsf{v}}$ itself is the Kirchhoff polynomial $\kp[\Gamma^{\bullet}] = \vssp$. The terms in the coefficient of $\kp$ can be interpreted as
\al{
\cp[\Gamma,u]{\mathsf{a}\mathsf{v}}{\mathsf{d}\mathsf{v}} = \cp[\Gamma^{\bullet},u]{\mathsf{a}}{\mathsf{d}},
}
such that they add up to
\al{
\Phi_{\Gamma^{\bullet}}^{\{a,c\},\{b,d\}} - \Phi_{\Gamma^{\bullet}}^{\{b,c\},\{a,d\}},
}
just like \refeq{eq_dodgsonrem}. After putting all of this together we have proved the first claim,
\al{\nonumber
\alpha_e\alpha_f x_{\Gamma}^ex_{\Gamma}^f
&= \kp[\Gamma^{\bullet}] \big( \Phi_{\Gamma}^{\{\partial_-(e),\partial_-(f)\},\{\partial_+(e),\partial_+(f)\}} - \Phi_{\Gamma}^{\{\partial_+(e),\partial_-(f)\},\{\partial_-(e),\partial_+(f)\}} \big)\\[1mm]\nonumber
&\ - \kp \big( \Phi_{\Gamma^{\bullet}}^{\{\partial_-(e),\partial_-(f)\},\{\partial_+(e),\partial_+(f)\}} - \Phi_{\Gamma^{\bullet}}^{\{\partial_+(e),\partial_-(f)\},\{\partial_-(e),\partial_+(f)\}} \big)\\[3mm]
&= \kp[\Gamma^{\bullet}] \bp{e}{f} - \kp \bp[\Gamma^{\bullet}]{e}{f}.
}
For the second claim we simply remember \refeq{eq_lemma_bpcp},
\al[*]{
\bp{e}{f} = -\alpha_e\alpha_f \cp{e}{f}\qquad \text{ for all } e\neq f
}
and divide by $\alpha_e\alpha_f$ on both sides. For the final claim we return from the notation with $\Gamma^{\bullet}$ to Dodgson polynomials. Then we have
\al{\nonumber
x_{\Gamma}^ex_{\Gamma}^f &= -\kp[\Gamma^{\bullet}] \cp{e}{f} + \kp \cp[\Gamma^{\bullet}]{e}{f}\\\nonumber
&\!\!\!\! \Longleftrightarrow\\
\cp[\Gamma,u]{\mathsf{v}}{\mathsf{v}} \cp[\Gamma,u]{\mathsf{a}_e}{\mathsf{a}_f} + x_{\Gamma}^ex_{\Gamma}^f &= \kp \cp[\Gamma,u]{\mathsf{a}_e \mathsf{v}}{\mathsf{a}_f \mathsf{v}}
}
and the nature of the $x_{\Gamma}^e$ becomes obvious from a comparison with the Dodgson identity in \refeq{eq_DodgsonID} or \refeq{eq_DI_rewrite}.
\end{proof}
\subsubsection{The sum over $\mathcal{D}_{\Gamma}^1$}
Now that we know that the additional polynomials $x_{\Gamma}^e$ are also just Dodgson polynomials it seems reasonable to think that the ideas used for the previous summation can also be used here. We find that this is indeed the case, but there are some complications that we need to consider first.\\
Note that $\mathcal{D}_{\Gamma}^1$ has
\al{
|\mathcal{D}_{\Gamma}^1| = \binom{2N}{2} (2N-3)!! = \frac{2N (2N-1)!}{2 (2N-2)!} (2N-3)!! = N (2N-1)!!
}
elements. They can be sorted into $(2N-1)!!$ groups of $N$ diagrams, each of which corresponds to a diagram $D\in \mathcal{D}_{\Gamma}^0$ and all $N$ possible choices to remove one chord from it. Hence, a sum over $\mathcal{D}_{\Gamma}^1$ can be split into a double sum over $\mathcal{D}_{\Gamma}^0$ and chords of each diagram. The addition of the final chord always raises the total cycle number by one, by removing the tricoloured cycle to add one bicoloured cycle of each colour. With polynomials one has
\al{\nonumber
\sum_{D\in \mathcal{D}_{\Gamma}^1} (-2)^{\tilde c(D)} & \Big( \prod_{(u,v)\in E_D^0}\!\!\!\! \cp{\mathsf{a}_u}{\mathsf{a}_v} \Big) \prod_{w\in V_D^{(2)}} x_{\Gamma}^w\\[2mm]\nonumber
&= \sum_{D\in \mathcal{D}_{\Gamma}^0} (-2)^{\tilde c(D)-1} \Big( \prod_{(u,v)\in E_D^0}\!\!\!\! \cp{\mathsf{a}_u}{\mathsf{a}_v} \Big) \sum_{(u,v)\in E_D^0} \frac{x_{\Gamma}^ux_{\Gamma}^v}{\cp{\mathsf{a}_u}{\mathsf{a}_v}}\\[2mm]
&= -\sum_{D\in \mathcal{D}_{\Gamma}^0} (-2)^{\tilde c(D)-1} \Big( \prod_{(u,v)\in E_D^0}\!\!\!\! \cp{\mathsf{a}_u}{\mathsf{a}_v} \Big) \sum_{(u,v)\in E_D^0} \frac{ \cp{\mathsf{a}_u}{\mathsf{y}}\cp{\mathsf{a}_v}{\mathsf{y}} }{\cp{\mathsf{a}_u}{\mathsf{a}_v}}.
\label{eq_sumD0D1}
}
Here and for the rest of this section we still assume that $\Gamma$ has two external vertices, say $x,y\in V_{\Gamma}$, all Dodgson polynomials are with respect to the vertex $x$ with the incoming momentum $q_x= q$, i.e. $\cp{\mathsf{a}_u}{\mathsf{a}_v} \equiv \cp[\Gamma,x]{\mathsf{a}_u}{\mathsf{a}_v}$, and $\mathsf{y}$ is the letter associated to the other vertex with the outgoing momentum.
Define the set of diagrams $\mathcal{D}|_{\mathcal{E}}^1 \subset \mathcal{D}_{\Gamma}^1$ restricted by a partition analogously to the previous case $\mathcal{D}|_{\mathcal{E}}^0$. Chords are only allowed between base edges belonging to the same part and the two free vertices are treated as if they had a chord between them. In other words, a diagram $D \in \mathcal{D}_{\Gamma}^1$ is in $\mathcal{D}|_{\mathcal{E}}^1$ if and only if the corresponding diagram $D' \in \mathcal{D}_{\Gamma}^0$ (resulting from addition of the missing chord) is in $\mathcal{D}|_{\mathcal{E}}^0$.
The next lemma is the analogue of lemmata \ref{lemma_fullpart} and \ref{lemma_part_2}. Since the idea behind the proof is very similar we directly combine them into one.
\begin{lemma}\label{lemma_z1_lemma}
Let $\mathcal{E} \in \mathcal{P}(E_D^1)$ be any partition of $1$-coloured base edges. Then
\al{\nonumber
\sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 }\!\!\!\! \sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) & \!\!\!\! \prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) }
\!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'}\Big( - |\mathcal{E}| \kp[\Gamma^{\bullet}] + \kp \!\!\!\!\!\!\!\! \sum_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) } \frac{\cp{\mathsf{u}'\mathsf{y}}{\mathsf{v}'\mathsf{y}}}{\cp{\mathsf{u}'}{\mathsf{v}'}} \Big)\\[3mm]
& = (-1)^{1-|\mathcal{E}|}(-\kp)^{|\mathcal{E}|-N}\!\!\!\!\sum_{D\in\mathcal{D}|_{\mathcal{E}}^1} (-2)^{c_2^2(D)} \Big(\!\!\!\!\prod_{(u,v)\in E_D^0} \!\!\!\!\cp{\mathsf{a}_u}{\mathsf{a}_v}\Big)\!\! \prod_{w\in V_D^{(2)}}\!\!\!\! x_{\Gamma}^w,
}
\end{lemma}
\begin{proof}
Let $\mathsf{w}_i = \mathsf{x}_{i1}\dotsm \mathsf{x}_{iN}$, $i=1,2$ be the two words from \refeq{eq_Dodgson_exp} in the proof of lemma \ref{lemma_fullpart}. Append the letter $\mathsf{y}$ to the front of both words and consider again the expansion
\al{\nonumber
\kp^2 \cp{\mathsf{y}\mathsf{w}_1}{\mathsf{y}\mathsf{w}_2}
&= \kp \cp{\mathsf{y}}{\mathsf{y}}\cp{\mathsf{w}_1}{\mathsf{w}_2} + \kp\sum_{i=1}^N (-1)^i \cp{\mathsf{x}_{1i}}{\mathsf{y}} \cp{\mathsf{y}\mathsf{x}_{11} \dotsm \hat\mathsf{x}_{1i} \dotsm \mathsf{x}_{1N}}{\mathsf{x}_{21}\dotsm\mathsf{x}_{2N}}\\[2mm]
&= \kp[\Gamma^{\bullet}]\ \kp\cp{\mathsf{w}_1}{\mathsf{w}_2} - \sum_{i,j=1}^N (-1)^{i+j} \cp{\mathsf{x}_{1i}}{\mathsf{y}} \cp{\mathsf{x}_{2j}}{\mathsf{y}} \cp{\mathsf{x}_{11} \dotsm \hat\mathsf{x}_{1i} \dotsm \mathsf{x}_{1N}}{ \mathsf{x}_{21} \dotsm \hat\mathsf{x}_{2j} \dotsm \mathsf{x}_{2N} }.
\label{eq_doubleexp}
}
The term $\kp\cp{\mathsf{w}_1}{\mathsf{w}_2}$ is precisely what was discussed in lemma \ref{lemma_fullpart} and
\al[*]{
(-1)^{i+j} \cp{\mathsf{x}_{11} \dotsm \hat\mathsf{x}_{1i} \dotsm \mathsf{x}_{1N}}{ \mathsf{x}_{21} \dotsm \hat\mathsf{x}_{2j} \dotsm \mathsf{x}_{2N} }
}
with $i=1$ was the coefficient of $\cp{\mathsf{x}_{1i}}{\mathsf{x}_{2j}}$ in its expansion. Hence, repeating the steps from that proof we immediately find the result for $|\mathcal{E}|=1$:
\al{\nonumber
\sum_{ (\mathsf{u}_{\id},\mathsf{v}_{\id})\in \mathsf{P}_2 }\!\!\!\! \Big( \kp\cp{\mathsf{u}_{\id}\mathsf{y}}{\mathsf{v}_{\id}\mathsf{y}} &- \kp[\Gamma^{\bullet}] \cp{\mathsf{u}_{\id}}{\mathsf{v}_{\id}} \Big)\\
%
& = (-\kp)^{1-N}\!\!\! \sum_{D\in\mathcal{D}_{\Gamma}^1} (-2)^{c_2^2(D)} \Big(\!\! \prod_{(u,v)\in E_D^0}\!\!\!\! \cp{\mathsf{a}_u}{\mathsf{a}_v}\Big)\!\! \prod_{w\in V_D^{(2)}}\!\!\!\! x_{\Gamma}^w.
\label{eq_z1_lemma_E1}
}
Replacing the Dodgson polynomials with $x_{\Gamma}^w$ (see \refeq{eq_xx_dp_signs}) flips the sign in front of the sum in \refeq{eq_doubleexp}. Since it is a double sum we get a factor of $2$. This, together with a $-1$ due to the factor $+\kp$ on the l.h.s. raises the power of $-2$ to $\tilde c(D)$. This can be interpreted as due to the additional tricoloured cycle that all diagrams $D\in \mathcal{D}_{\Gamma}^1$ have.
Now we can simply repeat the arguments of lemma \ref{lemma_part_2} to extend this to $|\mathcal{E}|>1$ to finish the proof. Inclusion of the factor
\al{
\sum_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) } \frac{\cp{\mathsf{u}'\mathsf{y}}{\mathsf{v}'\mathsf{y}}}{\cp{\mathsf{u}'}{\mathsf{v}'}}
}
simply turns each summand into a sum of $|\mathcal{E}|$ terms where one of the factors in each of them is replaced with the Dodgson polynomials with appended letters $\mathsf{y}$. Expanding that factor as above yields the term $\cp{\mathsf{y}}{\mathsf{y}} = \kp[\Gamma^{\bullet}]$ in each summand, so we need $-|\mathcal{E}|\kp[\Gamma^{\bullet}]$ to cancel it. The remaining terms can then be collected into groups of terms that either already factorise or can be reduced with the exact same arguments as in lemma \ref{lemma_part_2}.
\end{proof}
\subsubsection{An example}
Before we move on to prove the main theorem we discuss an example to illustrate the previous lemma.\\
Consider a sum over word pairs $(\mathsf{u}_{\id}, \mathsf{v}_{\id}) \in \mathsf{P}_j$ as before, but add in each Dodgson polynomial an additional letter $\mathsf{y}$ representing a vertex. Due to this additional letter we constrain ourselves to an $N=2$ example, namely $n=(2)$. The word pairs are then $(\mathsf{a}_1\mathsf{a}_3, \mathsf{a}_2\mathsf{a}_4)$ and $(\mathsf{a}_1\mathsf{a}_4, \mathsf{a}_2\mathsf{a}_3)$, where we choose the colour $j$ to be that of the edges $(1,2)$ and $(3,4)$, and we expand the Dodgson polynomials as
\al{\nonumber
\kp^2\big( \cp{\mathsf{a}_1\mathsf{a}_3\mathsf{y}}{\mathsf{a}_2\mathsf{a}_4\mathsf{y}} &+ \cp{\mathsf{a}_1\mathsf{a}_4\mathsf{y}}{\mathsf{a}_2\mathsf{a}_3\mathsf{y}} \big)\\[3mm]\nonumber
=& 2 \cp{\mathsf{a}_1}{\mathsf{a}_2} \cp{\mathsf{a}_3}{\mathsf{a}_4} \cp{\mathsf{y}}{\mathsf{y}} - 2 \cp{\mathsf{a}_1}{\mathsf{a}_2} \cp{\mathsf{a}_3}{\mathsf{y}} \cp{\mathsf{a}_4}{\mathsf{y}} - 2 \cp{\mathsf{a}_3}{\mathsf{a}_4} \cp{\mathsf{a}_1}{\mathsf{y}} \cp{\mathsf{a}_2}{\mathsf{y}}\\\nonumber
& - \cp{\mathsf{a}_1}{\mathsf{a}_3} \cp{\mathsf{a}_2}{\mathsf{a}_4} \cp{\mathsf{y}}{\mathsf{y}} + \cp{\mathsf{a}_1}{\mathsf{a}_3} \cp{\mathsf{a}_2}{\mathsf{y}} \cp{\mathsf{a}_4}{\mathsf{y}} + \cp{\mathsf{a}_2}{\mathsf{a}_4} \cp{\mathsf{a}_1}{\mathsf{y}} \cp{\mathsf{a}_3}{\mathsf{y}}\\
& - \cp{\mathsf{a}_1}{\mathsf{a}_4} \cp{\mathsf{a}_2}{\mathsf{a}_3} \cp{\mathsf{y}}{\mathsf{y}} + \cp{\mathsf{a}_1}{\mathsf{a}_4} \cp{\mathsf{a}_2}{\mathsf{y}} \cp{\mathsf{a}_3}{\mathsf{y}} + \cp{\mathsf{a}_2}{\mathsf{a}_3} \cp{\mathsf{a}_1}{\mathsf{y}} \cp{\mathsf{a}_4}{\mathsf{y}}.
}
Note that there are $9$ distinct terms. Firstly, we have the $3=(2N-1)!!$ terms
\al{
\cp{\mathsf{y}}{\mathsf{y}} \big( 2 \cp{\mathsf{a}_1}{\mathsf{a}_2} \cp{\mathsf{a}_3}{\mathsf{a}_4} - \cp{\mathsf{a}_1}{\mathsf{a}_3} \cp{\mathsf{a}_2}{\mathsf{a}_4} - \cp{\mathsf{a}_1}{\mathsf{a}_4} \cp{\mathsf{a}_2}{\mathsf{a}_3} \big)
}
corresponding to diagrams $D\in \mathcal{D}^2_2$ ($\simeq \mathcal{D}_{\Gamma}^0$ for some suitable $\Gamma$). Dividing by $\kp$ one finds that this exactly agrees with the sum predicted in lemma \ref{lemma_fullpart} but with a factor $\cp{\mathsf{y}}{\mathsf{y}}$.
The other $6=N(2N-1)!!$ terms are
\al{\nonumber
- 2 \big( \cp{\mathsf{a}_1}{\mathsf{a}_2} \cp{\mathsf{a}_3}{\mathsf{y}} \cp{\mathsf{a}_4}{\mathsf{y}} + \cp{\mathsf{a}_3}{\mathsf{a}_4} \cp{\mathsf{a}_1}{\mathsf{y}} \cp{\mathsf{a}_2}{\mathsf{y}}\big)
& + \cp{\mathsf{a}_1}{\mathsf{a}_3} \cp{\mathsf{a}_2}{\mathsf{y}} \cp{\mathsf{a}_4}{\mathsf{y}} + \cp{\mathsf{a}_2}{\mathsf{a}_4} \cp{\mathsf{a}_1}{\mathsf{y}} \cp{\mathsf{a}_3}{\mathsf{y}}\\[2mm]
& + \cp{\mathsf{a}_1}{\mathsf{a}_4} \cp{\mathsf{a}_2}{\mathsf{y}} \cp{\mathsf{a}_3}{\mathsf{y}} + \cp{\mathsf{a}_2}{\mathsf{a}_3} \cp{\mathsf{a}_1}{\mathsf{y}} \cp{\mathsf{a}_4}{\mathsf{y}}.
}
and correspond to diagrams $D\in \mathcal{D}^2_1$ with one missing chord. Alternatively we can write this as
\al{\nonumber
2 \big(\cp{\mathsf{a}_1}{\mathsf{a}_2} x_{\Gamma}^3x_{\Gamma}^4 + \cp{\mathsf{a}_3}{\mathsf{a}_4} x_{\Gamma}^1x_{\Gamma}^2 \big)
& - \cp{\mathsf{a}_1}{\mathsf{a}_3} x_{\Gamma}^2x_{\Gamma}^4 - \cp{\mathsf{a}_2}{\mathsf{a}_4} x_{\Gamma}^1x_{\Gamma}^3\\[2mm]
& - \cp{\mathsf{a}_1}{\mathsf{a}_4} x_{\Gamma}^2x_{\Gamma}^3 - \cp{\mathsf{a}_2}{\mathsf{a}_3} x_{\Gamma}^1x_{\Gamma}^4,
}
and we see that the factors are as predicted by lemma \ref{lemma_z1_lemma}, specifically the $|\mathcal{E}|=1$ case in \refeq{eq_z1_lemma_E1}.\\
We continue the example to a partition with two parts. Since we chose the colour of the word pairs to be that of $(1,2)$ and $(3,4)$, the partition needs to be of the other edges, i.e. $\mathcal{E} = \{ \{(1,4)\}, \{(2,3)\} \}$. One finds
\al{\nonumber
-\kp \big( & \cp{\mathsf{a}_1}{\mathsf{a}_4}\cp{\mathsf{a}_2\mathsf{y}}{\mathsf{a}_3\mathsf{y}} + \cp{\mathsf{a}_2}{\mathsf{a}_3}\cp{\mathsf{a}_1\mathsf{y}}{\mathsf{a}_4\mathsf{y}} \big)\\[3mm]\nonumber
&= - 2 \cp{\mathsf{a}_1}{\mathsf{a}_4} \cp{\mathsf{a}_2}{\mathsf{a}_3} \cp{\mathsf{y}}{\mathsf{y}}
+ \cp{\mathsf{a}_3}{\mathsf{y}} \cp{\mathsf{a}_4}{\mathsf{y}} + \cp{\mathsf{a}_3}{\mathsf{a}_4} \cp{\mathsf{a}_1}{\mathsf{y}} \cp{\mathsf{a}_2}{\mathsf{y}},\\[3mm]
& = - 2 \cp{\mathsf{a}_1}{\mathsf{a}_4} \cp{\mathsf{a}_2}{\mathsf{a}_3} \kp[\Gamma^{\bullet}]
- \cp{\mathsf{a}_1}{\mathsf{a}_4} x_{\Gamma}^2 x_{\Gamma}^3 - \cp{\mathsf{a}_2}{\mathsf{a}_3}x_{\Gamma}^1 x_{\Gamma}^4.
}
The above results from the pairs $\lambda_{\mathcal{E}}(\mathsf{a}_1\mathsf{a}_3,\mathsf{a}_2\mathsf{a}_4) = \{ (\mathsf{a}_1,\mathsf{a}_4), (\mathsf{a}_3,\mathsf{a}_2)\}$. Note that this also yields a sign $\sgn_{\mathcal{E}}(\mathsf{a}_1\mathsf{a}_3,\mathsf{a}_2\mathsf{a}_4) =-1$ in front of $\kp$ on the l.h.s. since $\mathsf{a}_4$ and $\mathsf{a}_2$ are permuted when concatenating the two word pairs in $\lambda_{\mathcal{E}}(\mathsf{a}_1\mathsf{a}_3,\mathsf{a}_2\mathsf{a}_4)$. The other word pair yields $\lambda_{\mathcal{E}}(\mathsf{a}_1\mathsf{a}_4,\mathsf{a}_2\mathsf{a}_3) = \emptyset$, and thus no polynomial. We see that only two diagrams are in $\mathcal{D}|_{\mathcal{E}}^1$, since $(1,4)$ and $(2,3)$ are the only two possible chords that stay within one part of the partition $\mathcal{E} = \{ \{(1,4)\}, \{(2,3)\} \}$. For the other term note that the $-2$ does not come from the number of cycles but from $|\mathcal{E}|=2$ together with the signum.
\paragraph{Proof of Theorem \ref{theo_main_2}.}
The partition polynomial definitions \ref{def_PartPol} and \ref{def_PartPol_1} together with lemma \ref{lemma_z1_lemma} directly yield
\al{\nonumber
Z_{\Gamma}^1
&=
%
\frac{1}{2}\sum_{\mathcal{E} \in \mathcal{P}(E_D^1)} (-\kp)^{N-|\mathcal{E}|} (|\mathcal{E}|+1)! \\
& \qquad \times \sum_{ (\mathsf{u},\mathsf{v})\in \mathsf{P}_2 } \sgn_{\mathcal{E}}(\mathsf{u},\mathsf{v}) \!\!\!\! \prod_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v})}\!\!\!\!\!\!\!\!\cp{\mathsf{u}'}{\mathsf{v}'}
%
\bigg( |\mathcal{E}|\vssp - \kp \!\!\!\!\!\!\!\! \sum_{ (\mathsf{u}',\mathsf{v}')\in\lambda_{\mathcal{E}}(\mathsf{u},\mathsf{v}) } \frac{\cp{\mathsf{u}'\mathsf{y}}{\mathsf{v}'\mathsf{y}}}{\cp{\mathsf{u}'}{\mathsf{v}'}} \bigg)\\[3mm]
%
& = \frac{1}{2}\sum_{\mathcal{E} \in \mathcal{P}(E_D^1)} (-1)^{|\mathcal{E}|+1} (|\mathcal{E}|+1)! \sum_{D \in \mathcal{D}|_{\mathcal{E}}^1} (-2)^{c_2^2(D)}
\Big(\!\!\!\!\prod_{(u,v)\in E_D^0} \!\!\!\!\cp{u}{v}\Big)\!\! \prod_{w\in V_D^{(2)}}\!\!\!\! x_{\Gamma}^w.
}
Now we have almost the same situation as in theorem \ref{theo_main}, except for the summation over $\mathcal{D}|_{\mathcal{E}}^1$ instead of $\mathcal{D}|_{\mathcal{E}}^0$. We can again exploit the one-to-one correspondence between diagrams in $\mathcal{D}|_{\mathcal{E}}^0$ and subsets of $N$ diagrams in $\mathcal{D}|_{\mathcal{E}}^1$ to be able to use the same argument as before. This correspondence carries over to the restricted sets and we can split the sum over $\mathcal{D}|_{\mathcal{E}}^1$ into a sum over $\mathcal{D}|_{\mathcal{E}}^0$ and the chords of each diagram (see also \refeq{eq_sumD0D1}). We then have
\al{\nonumber
Z_{\Gamma}^1
& = \frac{1}{2} \sum_{\mathcal{E} \in \mathcal{P}(E_D^1)} (-1)^{|\mathcal{E}|+1} (|\mathcal{E}|+1)! \sum_{D\in \mathcal{D}|_{\mathcal{E}}^0} (-2)^{c_2^2(D)-1}
\Big(\!\!\!\!\prod_{(u,v)\in E_D^0} \!\!\!\!\cp{u}{v}\Big) \sum_{(u,v)\in E_D^0} \frac{x_{\Gamma}^ux_{\Gamma}^v}{\cp{u}{v}}.
}
Exchange of summations yields the same sum involving the Stirling numbers of the second kind, which allows us to find $c_2^1(D)$. We now have
\al{\nonumber
Z_{\Gamma}^1
&= \frac{1}{2} \sum_{D\in \mathcal{D}_{\Gamma}^0} (-2)^{\tilde c(D)-1} \Big( \prod_{(u,v)\in E_D^0}\!\!\!\! \cp{u}{v} \Big) \sum_{(u,v)\in E_D^0}
\frac{x_{\Gamma}^ux_{\Gamma}^v}{\cp{u}{v}}\\[2mm]
%
&= \frac{1}{2} \sum_{D\in \mathcal{D}_{\Gamma}^1} (-2)^{\tilde c(D)} \bigg(\prod_{ (u,v)\in E_D^0 } \cp{u}{v}\bigg) \prod_{ w\in V_D^{(2)} } x_{\Gamma}^w.
}
where we take care to account for the reduced cycle number when translating back to a sum over $\mathcal{D}_{\Gamma}^1$. This is exactly \refeq{eq_maintheo_2} and the proof is done.
\qed
\section{Application to Feynman integrals}\label{sec_application}
%
\subsection{Structure of the integrand}\label{subsec_structure}
We now return to Feynman integrals and apply the theorems we just proved. In order to do so we first need to combine the results of \cite{Golz_2017_CyclePol} and \cite{Golz_2017_Traces}. Our starting point is the unrenormalised integral
\al[*]{
\phi_{\Gamma} = \int_{\mathbb{R}_+^{|E_{\Gamma}|}} \mathrm{d}\alpha_1\dotsm \mathrm{d}\alpha_{E_{\Gamma}} \frac{\exp\Big( \frac{\ssp}{\kp} \Big)}{ \kp^{2+h_1(\Gamma)} } \sum_{k=0}^{h_1(\Gamma)} \frac{ I_{\Gamma}^{(k)} }{\kp^k}
}
from \refeq{eq_int_unren}. It is convenient to consider the $k$-th summand contracted with a metric tensor corresponding to the two external vertices $x,y\in V_{\Gamma}$. Then
\al{
g_{\mu_x\mu_y}I_{\Gamma}^{(k)} &= -\tr(\gamma^{\mu_1}\dotsm\gamma^{\mu_{2h_1}}) \bigg( \prod_{ (u,v)\in E_{D_{\Gamma}}^0 }\!\!\!\! g_{\mu_u\mu_v} \bigg)
\sum_{D\in \mathcal{D}_{\Gamma}^k} \bigg( \prod_{ (u,v)\in E_D^0 } \frac{g_{\mu_u\mu_v}}{2}\cp{u}{v} \bigg) \prod_{w\in V_D^{(2)}} q_{\mu_w} x_{\Gamma}^w
}
is just a rewriting of the integrand as worked out in \cite[eq. (72)]{Golz_2017_CyclePol}. The sum over fermion edge subsets and pairings is interpreted in terms of chord diagrams whose vertices are labelled by fermion edges and the additional metric tensor adds a chord such that we indeed have sums over $\mathcal{D}_{\Gamma}^0$ in $I_{\Gamma}^{(0)}$ etc.
As all throughout this article we stay in the special case of photon propagator graphs, Feynman gauge, and quenched QED, which becomes manifest in the terms above as follows:
\begin{itemize}
\item A propagator graph has only two external vertices with a single external momentum $q$, such that one has the factorised polynomials $q_{\mu_w} x_{\Gamma}^w$.
\item Because it is a photon propagator there is one closed fermion cycle, which leads to the trace of Dirac matrices. Since we have quenched QED there is only exactly one such cycle and therefore no product of traces.
\item For a general gauge each $I_{\Gamma}^{(k)}$ itself contains another sum
\al{
I_{\Gamma}^{(k)} = \sum_{l=0}^{h_1-1} \Big( \frac{\varepsilon}{\kp}\Big)^l I_{\Gamma}^{(k,l)} \label{eq_gengauge}
}
where the gauge parameter $\varepsilon$ is such that Feynman gauge is $\varepsilon\to 0$. Each $I_{\Gamma}^{(k,l)}$ is structurally similar to $I_{\Gamma}^{(k,0)}$ but instead of a skeleton chord diagram $D_{\Gamma}$ with $h_1$ fixed chords it contains a sum over all possible such skeletons with $h_1-l$ fixed chords and then the usual chord diagram sums over all possible additions of further chords. Note that this in particular means that the sum over $k$ then also goes up to $2h_1-1$. Here we stick to Feynman gauge and identify $I_{\Gamma}^{(k)} \equiv I_{\Gamma}^{(k)}|_{\varepsilon=0} = I_{\Gamma}^{(k,0)}$ for $0\leq k \leq h_1$.
\end{itemize}
%
\subsubsection{Contraction}\label{subsec_contract}
Next we apply the contraction theorem \cite[Theorem 3.9]{Golz_2017_Traces} to remove all Dirac matrices and metric tensors. We find
\al{
g_{\mu_x\mu_y}I_{\Gamma}^{(k)}
&= 2^{h_1+1} (-q^2)^k \sum_{D\in \mathcal{D}_{\Gamma}^k} (-2)^{\tilde c(D)} \bigg( \prod_{ (u,v)\in E_D^0 } \cp{u}{v} \bigg) \prod_{w\in V_D^{(2)}} x_{\Gamma}^w.
}
The final integer factor is computed as follows. There are a total of $2h_1-k$ chords yielding $(-2)^{2h_1-k}$, but the $h_1-k$ non-fixed chords added to $D_{\Gamma}^0$ come with a factor $1/2$. Free vertices, corresponding to Dirac matrices contracted with a momentum instead of a metric tensor yield powers of $q^2$ and there is one more factor of $-2$, due to the one base cycle of $D_{\Gamma}$, whose sign is cancelled by the $-1$ from the Feynman rules for a fermion cycle. Altogether one finds
\al{
\underbrace{-(-2)^1}_{ 1 \text{ base cycle/trace}}
\cdot \ \underbrace{ (-2)^{h_1} }_{ \substack{\text{fixed chords}\\ E_{D_{\Gamma}}^0 } }\
\cdot \ \underbrace{ (-1)^{h_1-k} }_{ \substack{\text{other chords} \\ E_D^0 } } \
\cdot \ \underbrace{ (q^2)^k }_{ \substack{ \text{free vertices} \\ V_D^{(2)} } } = 2^{h_1+1} (-q^2)^k.
}
For the actual integrand we are interested in $I_{\Gamma}^{(k)}$, not its contraction with $g_{\mu_x\mu_y}$, so we need to work out what the effect of this contraction is. To simplify notation, let $\mu$ and $\nu$, without subscript, denote the space time indices of the external vertices, previously written $\mu_x,\mu_y$. For $k=0$ there are no free vertices. In other words, the added chord between external vertices causes the contraction $g_{\mu\nu}g^{\mu\nu} = 4$ which is counteracted by a factor $2^{-2}$ for all $D\in \mathcal{D}_{\Gamma}^0$. Hence,
\al{
I_{\Gamma}^{(0)} = g^{\mu\nu}2^{h_1-1} \sum_{D\in \mathcal{D}_{\Gamma}^0} (-2)^{\tilde c(D)} \prod_{ (u,v)\in E_D^0 } \cp{u}{v} = g^{\mu\nu}2^{h_1} Z_{\Gamma}^0
}
with theorem \ref{theo_main}.\\
For $k\geq 1$ the chord diagrams split into three disjoint subsets $\mathcal{D}_{\Gamma,\bullet}^k \subset \mathcal{D}_{\Gamma}^k$. The correction factor depends on the result of the Dirac matrix contraction without contraction of the external vertices. This, in turn, is characterised by $\sgn(x,y)$, the signum of the external vertices in the chord diagram (introduced in section \ref{sec_colours}):
\al{
\sgn(x,y)=
\begin{cases}
+1 \qquad\rightarrow \tr( \slashed q \slashed q q^{\mu}q^{\nu} ) & \sim q^2 g^{\mu\nu}\\[3mm]
\hphantom{+} 0 \qquad\rightarrow \tr( \slashed q q^{\mu} \slashed q q^{\nu} ) &\sim 2q^{\mu}q^{\nu} - q^2g^{\mu\nu}\\[3mm]
-1 \qquad\rightarrow \tr( \slashed q q^{\mu})\tr( \slashed q q^{\nu} ) &\sim q^{\mu}q^{\nu}
\end{cases}
}
Contracting the results on the r.h.s. with $g_{\mu\nu}$ one sees that the correction factor is a $-2$ with the exponent $1+\sgn(x,y)$ for all diagrams, including the $k=0$ case, in which only $+1$ occurs. We can define partial chord diagram sums
\al{
Z_{\Gamma,\bullet}^k \defeq \frac{1}{2}\sum_{D\in \mathcal{D}_{\Gamma,\bullet}^k} (-2)^{\tilde c(D)} \bigg( \prod_{ (u,v)\in E_D^0 } \cp{u}{v} \bigg) \prod_{w\in V_D^{(2)}} x_{\Gamma}^w,
}
based on these subsets. Then $Z_{\Gamma,+}^0 = Z_{\Gamma}^0$ and $Z_{\Gamma,-}^0 = Z_{\Gamma,0}^0 = 0$, and for $k=1$ we have
\al{
Z_{\Gamma}^1 = Z_{\Gamma,+}^1 + Z_{\Gamma,0}^1 + Z_{\Gamma,-}^1
}
with theorem \ref{theo_main_2}. For $k>1$ similar equalities should hold, assuming one defines the right $k$-th order partition polynomial, but, as we will see in the next section, for a superficially renormalised integral these two will suffice.
With this notation the $k$-th summand now has become
\al{\nonumber
I_{\Gamma}^{(k)} &= 2^{h_1} (-q^2)^k \bigg( g^{\mu\nu} Z_{\Gamma,+}^k - 2\Big( 2\frac{q^{\mu}q^{\nu}}{q^2} - g^{\mu\nu} \Big) Z_{\Gamma,0}^k + 4\frac{q^{\mu}q^{\nu}}{q^2}Z_{\Gamma,-}^k \bigg)\\[3mm]
&= 2^{h_1} (-q^2)^k \Big( g^{\mu\nu} \big( Z_{\Gamma,+}^k + 2Z_{\Gamma,0}^k \big) + 4\frac{q^{\mu}q^{\nu}}{q^2}\big( Z_{\Gamma,-}^k - Z_{\Gamma,0}^k \big) \Big),
}
and the full unrenormalised integrand is
\al{
I_{\Gamma} = 2^{h_1}\frac{e^{-\frac{\ssp}{\kp}}}{\kp^{h_1+2}} \Big( g^{\mu\nu}Z_{\Gamma}^0 - \sum_{k=1}^{h_1} \frac{(-q^2)^{k-1}}{ \kp^k}
\Big( q^2g^{\mu\nu} \big( Z_{\Gamma,+}^k + 2Z_{\Gamma,0}^k \big) + 4q^{\mu}q^{\nu}\big( Z_{\Gamma,-}^k - Z_{\Gamma,0}^k \big) \Big).
}
Now this may seem somewhat problematic -- we know the sum $Z_{\Gamma,+}^1 + Z_{\Gamma,0}^1 + Z_{\Gamma,-}^1$ (and can presumably generalise that knowledge to $k>1$). But what can we do about these combinations? As it turns out, we can exploit the transversality of the photon propagator to modify the integrand such that it only contains these types of sums, but first we want to renormalise it.
%
\subsubsection{Renormalisation}
We (superficially) renormalise this integrand in a BPHZ scheme following \cite{BrownKreimer_2013_AnglesScales}. Consider a generic integral of the same form as our Feynman integral, namely
\al{
\int_{\mathbb{R}_+^n} \mathrm{d}\alpha_1\dotsm\mathrm{d}\alpha_n \sum_{k=-1}^m e^{-X}I_k, \label{eq_genint}
}
where $X$ and all $I_k$ are rational functions in $\alpha_i$ with overall degree (degree of numerator minus degree of denominator) $1$ and $k-n$ respectively.
We can introduce an auxiliary variable $t$ by inserting $1= \int_0^{\infty} \delta(t-\sum_i \lambda_i \alpha_i) \mathrm{d} t$, where each $\lambda_i\in \{0,1\}$ and at least one of them non-zero. Then scaling all Schwinger parameters by $\alpha_i \mapsto t\alpha_i$ turns \refeq{eq_genint} into
\al{
\int_{\mathbb{R}_+^n} \mathrm{d}\alpha_1\dotsm\mathrm{d}\alpha_n\ \delta(1-\sum_i \lambda_i\alpha_i) \sum_{k=-1}^m I_k T_k \label{eq_delta}
}
with
\al{
T_k =\int_0^{\infty}\mathrm{d} t\ t^{k-1} e^{-tX} = X^{-k} \Gamma(k). \label{eq_gammafct}
}
The Gamma function has poles at negative integers and zero, corresponding here to quadratic and logarithmic divergences for $k=-1$ and $0$. They can be parametrised for further study by regularising the $t$-integration with an $\epsilon>0$:
\al{
T_0 &\stackrel{\epsilon\to 0}{=} \int_{\epsilon}^{\infty} t^{-1} e^{-t X} \mathrm{d} t = -\log \epsilon - \log X - \gamma_E + \mathcal O(\epsilon)\\[2mm]
T_{-1} &\stackrel{\epsilon\to 0}{=} \int_{\epsilon}^{\infty} t^{-2} e^{-t X} \mathrm{d} t = \frac{e^{-\epsilon X}}{\epsilon} -X \underbrace{\int_{\epsilon}^{\infty} t^{-1} e^{-tX} \mathrm{d} t}_{=T_0} \label{eq_quaddiv}
}
We see that the divergent terms are isolated and a simple subtraction like
\al{
T_0 - T_0' = -\log \frac{X}{X'},\label{eq_simplesub}
}
is already enough to cancel a logarithmic divergence. The quadratic divergence requires first an on-shell subtraction to remove the term $\sim \epsilon^{-1}$, then the usual subtraction for the remaining logarithmic divergence.\\
Note that, assuming convergence, the integral in \refeq{eq_delta} can equivalently be written projectively\footnote{ For a more thorough discussion of the bijection between $\mathbb{R}_+^n$ and (a certain subset of) projective space induced by the introduction of the delta function see \cite[sec. 2.1.3]{Panzer_2015_PhD}. Of note in particular is the fact that it is completely independent of the choice of the parameters $\lambda_i$, which is sometimes called ``Cheng-Wu theorem''.}
\al{
\int_{\mathbb{R}_+^n} \mathrm{d}\alpha_1\dotsm\mathrm{d}\alpha_n\ \delta(1-\sum_i \lambda_i\alpha_i) \sum_{k=-1}^m I_k T_k
= \int_{\sigma_{\Gamma}} \Omega_{\Gamma} \sum_{k=-1}^m I_k T_k,
}
where $\Omega_{\Gamma} = \sum_{i=1}^{n}(-1)^{i-1}\alpha_i\mathrm d\alpha_1\wedge\dotsm\wedge\widehat{\mathrm d\alpha_i}\wedge\dotsm\wedge\mathrm d\alpha_n$ and one integrates over the subset of real projective space in which all parameters are positive
\al{
\sigma_{\Gamma} = \{[\alpha_1: \dotso :\alpha_n]\ | \ \alpha_i > 0 \ \forall \ i=1,\dotsc, n \}.
}
For brevity we will use this notation from now on.\\
We can now apply this to the integrand. Simply counting the degrees of the various homogenous polynomials that appear in numerator and denominator one finds that the $0$-th term is quadratically divergent, the next one logarithmically, and all others are convergent. Hence, the (superficially) renormalised integrand is
\al{\nonumber
I_{\Gamma}^R &= \underbrace{\log \frac{q^2}{\mu^2}}_{\bdefeq L}
\frac{2^{h_1}}{\kp^{h_1+3}} \Big( q^2g^{\mu\nu} \vssp Z_{\Gamma}^0
+ q^2g^{\mu\nu} \big( Z_{\Gamma,+}^1 + 2Z_{\Gamma,0}^1 \big) + 4q^{\mu}q^{\nu}\big( Z_{\Gamma,-}^1 - Z_{\Gamma,0}^1 \big) \Big)\\[3mm]
&= \frac{2^{h_1} L}{\kp^{h_1+3}} \Big( q^2g^{\mu\nu} \big( \vssp Z_{\Gamma}^0 + Z_{\Gamma,+}^1 + 2Z_{\Gamma,0}^1 \big)
- q^{\mu}q^{\nu}\big( 4Z_{\Gamma,0}^1 - 4Z_{\Gamma,-}^1 \big) \Big).
}
At this point we can now impose transversality on the integrand to simplify it. For the photon propagator transversality simply means that the amplitude, the sum of all relevant Feynman integrals, is proportional to $q^2g^{\mu\nu} - q^{\mu}q^{\nu}$. This is manifestly not true for individual Feynman integrals, let alone their integrands. However, since only their sum has physical meaning we can simplify redefine $I_{\Gamma}^R$ such that it already satisfies transversality. Whatever change this effects in the integral cancels when adding up all integrals. Here we get the condition
\al{
\vssp Z_{\Gamma}^0 + Z_{\Gamma,+}^1 + 2Z_{\Gamma,0}^1 \stackrel{!}{=} 4Z_{\Gamma,0}^1 - 4Z_{\Gamma,-}^1. \label{eq_transvers_cond}
}
We could now naively just use either side of this in the integrand. However, we can also do better than that. Note that
\al{
\vssp Z_{\Gamma}^0 + Z_{\Gamma,+}^1 + 2Z_{\Gamma,0}^1
= \big(\vssp Z_{\Gamma}^0 + \underbrace{Z_{\Gamma,+}^1 + Z_{\Gamma,0}^1 + Z_{\Gamma,-}^1}_{ =Z_{\Gamma}^1} \big) + \big( Z_{\Gamma,0}^1 - Z_{\Gamma,-}^1\big).
}
Now imposing the transversality condition \refeq{eq_transvers_cond} yields
\al{
\vssp Z_{\Gamma}^0 + Z_{\Gamma}^1 = 3\big( Z_{\Gamma,0}^1 - Z_{\Gamma,-}^1\big)
}
and the integrand becomes
\al{\nonumber
I_{\Gamma}^R &= \frac{2^{h_1} L}{\kp^{h_1+3}} \Big( q^2g^{\mu\nu} \big( \vssp Z_{\Gamma}^0 + Z_{\Gamma,+}^1 + 2Z_{\Gamma,0}^1 \big)
- 4q^{\mu}q^{\nu}\big( Z_{\Gamma,0}^1 - Z_{\Gamma,-}^1 \big) \Big)\\[3mm]
&= ( q^2g^{\mu\nu} - q^{\mu}q^{\nu} ) L \frac{ 2^{h_1+2} }{3} \frac{ \vssp Z_{\Gamma}^0 + Z_{\Gamma}^1 }{ \kp^{h_1+3} }.
}
We can also use the definitions of the partition polynomials to make the cancellations more obvious:
\al{
\frac{ \vssp Z_{\Gamma}^0 + Z_{\Gamma}^1 }{ \kp^{h_1+3} }
= \sum_{l=1}^{h_1} (-1)^{h_1-l} (l+1)! \bigg( \frac{\vssp Z_{\Gamma}^0\big|_l}{\kp^{l+3}} - \frac{1}{2}\frac{Z_{\Gamma}^1\big|_l}{\kp^{l+2}} \bigg)
}
%
%
%
\subsection{Examples}
\subsubsection{1-loop photon propagator}
\begin{figure}[h]\begin{center}
\includegraphics[scale=1.0]{./Images/Fig_4_1.pdf}
\caption[The 1-loop photon propagator]{The 1-loop photon propagator.}
\label{06_img_1P}
\end{center}\end{figure}
The 1-loop case is the only primitive photon propagator and therefore the only example we can show in full without discussing subdivergences. The Kirchhoff polynomial is $\kp = \alpha_1+\alpha_2$ and $\vssp = \alpha_1\alpha_2$. The only possible cycle polynomial $\cp{1}{2}$ and $Z_{\Gamma}^0\big|_1$ are both just $1$ and $Z_{\Gamma}^1\big|_1 = -\vssp/\kp$.
%
The integrand is therefore
\al{
\frac{ \vssp Z_{\Gamma}^0 + Z_{\Gamma}^1 }{ \kp^{h_1+3} }
= (-1)^{1-1} (1+1)! \bigg( \frac{\vssp}{\kp^{4}} + \frac{1}{2}\frac{\vssp}{\kp^{4}} \bigg)
= 3\frac{\vssp}{\kp^{4}}
= 3\frac{ \alpha_1\alpha_2}{ (\alpha_1+\alpha_2)^4}
}
and the renormalised integral is
\al{\nonumber
\phi_{\Gamma}^R &= (q^2g^{\mu\nu} - q^{\mu}q^{\nu}) \frac{2^{h_1+2}}{3} L \int_{\sigma_{\Gamma}} \Omega_{\Gamma}
\frac{ \vssp Z_{\Gamma}^0 + Z_{\Gamma}^1 }{\kp^{h_1+3}}\\[3mm]\nonumber
&= 8L(q^2g^{\mu\nu} - q^{\mu}q^{\nu}) \int_{\sigma_{\Gamma}} \frac{ \alpha_1\alpha_2}{ (\alpha_1+\alpha_2)^{4}}\Omega_{\Gamma}\\
&= \frac{4}{3}L (q^2g^{\mu\nu} - q^{\mu}q^{\nu}).
}
The factor $4/3$ is exactly the 1-loop coefficient of the QED beta function in the conventions of \cite{BroadhurstDelbourgoKreimer_1996_Unknotting, GKLS_1991_QED, Rosner_1966_QED}.
\subsubsection{3-loop photon propagators}
For Feynman graphs with more than one loop we can not compute the full integral without discussing subdivergences and including the corresponding terms of Zimmermann's forest formula for a fully renormalised integrand. However, we can show what the superficially renormalised part of the integrand looks like and especially emphasise the cancellations and reductions in size due to the two summation theorems.
\begin{figure}[H]\begin{center}
\begin{subfigure}{0.2\textwidth}\begin{center}
\includegraphics[height=0.1\textheight]{./Images/Fig_4_2_a.pdf}
\caption{}
\label{06a_img_3_1}
\end{center}\end{subfigure}
\begin{subfigure}{0.2\textwidth}\begin{center}
\includegraphics[height=0.1\textheight]{./Images/Fig_4_2_b.pdf}
\caption{}
\label{06a_img_3_2}
\end{center}\end{subfigure}
\begin{subfigure}{0.2\textwidth}\begin{center}
\includegraphics[height=0.1\textheight]{./Images/Fig_4_2_c.pdf}
\caption{}
\label{06a_img_3_3}
\end{center}\end{subfigure}
\begin{subfigure}{0.2\textwidth}\begin{center}
\includegraphics[height=0.1\textheight]{./Images/Fig_4_2_d.pdf}
\caption{}
\label{06a_img_3_4}
\end{center}\end{subfigure}
\begin{subfigure}{0.2\textwidth}\begin{center}
\includegraphics[height=0.1\textheight]{./Images/Fig_4_2_e.pdf}
\caption{}
\label{06a_img_3_5}
\end{center}\end{subfigure}
\begin{subfigure}{0.2\textwidth}\begin{center}
\includegraphics[height=0.1\textheight]{./Images/Fig_4_2_f.pdf}
\caption{}
\label{06a_img_3_6}
\end{center}\end{subfigure}
\begin{subfigure}{0.2\textwidth}\begin{center}
\includegraphics[height=0.1\textheight]{./Images/Fig_4_2_g.pdf}
\caption{}
\label{06a_img_3_7}
\end{center}\end{subfigure}
\begin{subfigure}{0.2\textwidth}\begin{center}
\includegraphics[height=0.1\textheight]{./Images/Fig_4_2_h.pdf}
\caption{}
\label{06a_img_3_8}
\end{center}\end{subfigure}
\caption{The 3-loop topologies with one fermion cycle.}
\end{center}\end{figure}
At two loops the examples are still rather simple so we go to three loops, where the integrals start to become much more involved. For example, $Z_{\Gamma}^0$ is now already a polynomial of degree $h_1(h_1-1) = 6$, compared to just $2$ at two loops, and the number of chord diagrams rises to $15$ (in Feynman gauge, and already hundreds in general gauge) such that the reduction to $h_1=3$ small summands in the partition polynomial now becomes significant. All examples were computed with Maple \footnote{Maple\texttrademark\ is a trademark of Waterloo Maple Inc. \cite{Maple}.}.
We focus on the graph in \reffig{06a_img_3_8}.
\begin{figure}\begin{center}
\includegraphics[width=0.9\textwidth]{./Images/Fig_4_3.pdf}
\caption{From left to right: The graph $\Gamma$ from \reffig{06a_img_3_8} with its external photon edge closed - the corresponding chord diagram $D_{\Gamma}$ with fixed chords corresponding to all photon edges - the projection $D_{\Gamma}^0 = \pi_0(D_{\Gamma})$.}
\label{img_38_proj}
\end{center}\end{figure}
Label edges and vertices as in \reffig{img_38_proj} with $v_1,v_4$ being the external vertices and $e_7=(v_2,v_5)$ and $e_8 = (v_3,v_6)$ the two photon edges. The Kirchhoff and second Symanzik polynomial consist of $36$ and $45$ monomials, so we refrain from writing them out in full here. An example for a cycle polynomial is:
\al{
\cp{1}{6} = \alpha_2(\alpha_3 + \alpha_4 + \alpha_5 + \alpha_8) + (\alpha_3+\alpha_4)(\alpha_5 + \alpha_7 + \alpha_8) + \alpha_7(\alpha_5+\alpha_8)
}
This one has so many terms since $e_1$ and $e_6$ share all their cycles, because they are incident to the same external (i.e. 2-valent) vertex of $\Gamma$. In other words, $\cp{1}{6} = \cp{1}{1} = \cp{6}{6}$. Others are simpler:
\al{
\cp{1}{3} = \cp{1}{4} = \cp{3}{6} = \cp{4}{6} = -\alpha_2\alpha_5 + \alpha_7\alpha_8
}
Here we have an example of monomials with different signs, which is due to the fact that the two corresponding cycles are twisted relative to each other (as discussed in the proof of proposition \ref{05a_prop_eqpol}). One cycle is the fermion cycle, the other crosses via both photon edges.
\paragraph{The partition polynomial $Z_{\Gamma}^0$.}
The word pairs we get from $D_{\Gamma}^0$ are
\al[*]{
( \mathsf{a}_1\mathsf{a}_2\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_5\mathsf{a}_6 ), \quad ( \mathsf{a}_1\mathsf{a}_5\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_2\mathsf{a}_6 ),\quad
( \mathsf{a}_1\mathsf{a}_2\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_5\mathsf{a}_3 ), \quad ( \mathsf{a}_1\mathsf{a}_5\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_2\mathsf{a}_3 ).
}
For $Z_{\Gamma}^0\big|_1$ we have the single partition $\mathcal{E} = \{ \{ (e_1, e_4), (e_2, e_5), (e_3, e_6) \} \}$, such that
\al{\nonumber
Z_{\Gamma}^0\big|_1 &= \cp{\mathsf{a}_1\mathsf{a}_2\mathsf{a}_3}{\mathsf{a}_4\mathsf{a}_5\mathsf{a}_6} + \cp{\mathsf{a}_1\mathsf{a}_2\mathsf{a}_6}{\mathsf{a}_4\mathsf{a}_5\mathsf{a}_3}
+ \cp{\mathsf{a}_1\mathsf{a}_5\mathsf{a}_3}{\mathsf{a}_4\mathsf{a}_2\mathsf{a}_6} + \cp{\mathsf{a}_1\mathsf{a}_5\mathsf{a}_6}{\mathsf{a}_4\mathsf{a}_2\mathsf{a}_3}\\[3mm]
&= 1+ 0 + 0 + 1 = 2.
\label{eq_Z01}
}
\begin{remark}
Note that the two vanishing Dodgson polynomials are those that have the letter pairs $\mathsf{a}_1/\mathsf{a}_6$ and $\mathsf{a}_3/\mathsf{a}_4$ within the same word. While these are different letters we have seen above that their associated edges are equivalent as far as the cycle space of $\Gamma$ is concerned. This is reflected in the behaviour of the Dodgson polynomials, which vanish as if the letters were identical.
\end{remark}
For $Z_{\Gamma}^0\big|_2$ we have three partitions with two parts, consisting of one and two edges respectively. For $\mathcal{E}_1 = \{ \{ (e_1, e_4) \}, \{ (e_2, e_5), (e_3, e_6) \} \}$ one has
\al[*]{
\lambda_{\mathcal{E}_1}( \mathsf{a}_1\mathsf{a}_2\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_5\mathsf{a}_6 ) &= \{ (\mathsf{a}_1, \mathsf{a}_4), (\mathsf{a}_2\mathsf{a}_3, \mathsf{a}_5\mathsf{a}_6) \},\\[2mm]
\lambda_{\mathcal{E}_1}( \mathsf{a}_1\mathsf{a}_2\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_5\mathsf{a}_3 ) &= \{ (\mathsf{a}_1, \mathsf{a}_4), (\mathsf{a}_2\mathsf{a}_6, \mathsf{a}_5\mathsf{a}_3) \},\\[2mm]
\lambda_{\mathcal{E}_1}( \mathsf{a}_1\mathsf{a}_5\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_2\mathsf{a}_6 ) &= \{ (\mathsf{a}_1, \mathsf{a}_4), (\mathsf{a}_5\mathsf{a}_3, \mathsf{a}_2\mathsf{a}_6) \},\\[2mm]
\lambda_{\mathcal{E}_1}( \mathsf{a}_1\mathsf{a}_5\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_2\mathsf{a}_3 ) &= \{ (\mathsf{a}_1, \mathsf{a}_4), (\mathsf{a}_5\mathsf{a}_6, \mathsf{a}_2\mathsf{a}_3) \}.
}
All permutations give positive signs and the corresponding polynomial is
\al{
2\cp{\mathsf{a}_1}{\mathsf{a}_4} \big( \cp{\mathsf{a}_2\mathsf{a}_3}{\mathsf{a}_5\mathsf{a}_6} + \cp{\mathsf{a}_2\mathsf{a}_6}{\mathsf{a}_5\mathsf{a}_3} \big) = -2( -\alpha_2\alpha_5 + \alpha_7\alpha_8 )( \alpha_7 + \alpha_8 ),
}
which is also the polynomial one finds analogously for $\mathcal{E}_3 = \{ \{ (e_1, e_4), (e_2, e_5) \},$ $\{ (e_3, e_6) \} \}$. For the third, $\mathcal{E}_2 = \{ \{ (e_1, e_4), (e_3, e_6) \}, \{ (e_2, e_5) \} \}$, the words are
\al[*]{
\lambda_{\mathcal{E}_2}( \mathsf{a}_1\mathsf{a}_2\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_5\mathsf{a}_6 ) &= \{ (\mathsf{a}_2, \mathsf{a}_5), (\mathsf{a}_1\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_6) \},\\[2mm]
\lambda_{\mathcal{E}_2}( \mathsf{a}_1\mathsf{a}_2\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_5\mathsf{a}_3 ) &= \{ (\mathsf{a}_2, \mathsf{a}_5), (\mathsf{a}_1\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_3) \},\\[2mm]
\lambda_{\mathcal{E}_2}( \mathsf{a}_1\mathsf{a}_5\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_2\mathsf{a}_6 ) &= \{ (\mathsf{a}_5, \mathsf{a}_2), (\mathsf{a}_1\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_6) \},\\[2mm]
\lambda_{\mathcal{E}_2}( \mathsf{a}_1\mathsf{a}_5\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_2\mathsf{a}_3 ) &= \{ (\mathsf{a}_5, \mathsf{a}_2), (\mathsf{a}_1\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_3) \}.
}
Again, the total sign is always positive but this time with $\sgn(\sigma)=-1=\sgn(\sigma')$, where e.g. $\mathsf{a}_2\mathsf{a}_1\mathsf{a}_3 = \sigma(\mathsf{a}_1\mathsf{a}_2\mathsf{a}_3)$ and $\mathsf{a}_5\mathsf{a}_4\mathsf{a}_6 = \sigma'(\mathsf{a}_4\mathsf{a}_5\mathsf{a}_6)$. The polynomial is
\al{\nonumber
2\cp{\mathsf{a}_2}{\mathsf{a}_5} & \big( \cp{\mathsf{a}_1\mathsf{a}_3}{\mathsf{a}_4\mathsf{a}_6} + \cp{\mathsf{a}_1\mathsf{a}_6}{\mathsf{a}_4\mathsf{a}_3} \big)\\
& = 2( -(\alpha_1+\alpha_6)(\alpha_3+\alpha_4) + \alpha_7\alpha_8 )( \alpha_2 + \alpha_5 + \alpha_7 + \alpha_8 ).
}
The last polynomial is always of the same form. The only partition $\mathcal{E} = \{ \{ (e_1, e_4)\},$ $\{ (e_2, e_5) \}, \{ (e_3, e_6) \} \}$ has each base edge in a separate part such that
\al[*]{
\lambda_{\mathcal{E}}( \mathsf{a}_1\mathsf{a}_2\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_5\mathsf{a}_6 ) &= \{ (\mathsf{a}_1, \mathsf{a}_4), (\mathsf{a}_2, \mathsf{a}_5), (\mathsf{a}_3, \mathsf{a}_6) \},\\[2mm]
\lambda_{\mathcal{E}}( \mathsf{a}_1\mathsf{a}_2\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_5\mathsf{a}_3 ) &= \{ (\mathsf{a}_1, \mathsf{a}_4), (\mathsf{a}_2, \mathsf{a}_5), (\mathsf{a}_6, \mathsf{a}_3) \},\\[2mm]
\lambda_{\mathcal{E}}( \mathsf{a}_1\mathsf{a}_5\mathsf{a}_3, \mathsf{a}_4\mathsf{a}_2\mathsf{a}_6 ) &= \{ (\mathsf{a}_1, \mathsf{a}_4), (\mathsf{a}_5, \mathsf{a}_2), (\mathsf{a}_3, \mathsf{a}_6) \},\\[2mm]
\lambda_{\mathcal{E}}( \mathsf{a}_1\mathsf{a}_5\mathsf{a}_6, \mathsf{a}_4\mathsf{a}_2\mathsf{a}_3 ) &= \{ (\mathsf{a}_1, \mathsf{a}_4), (\mathsf{a}_5, \mathsf{a}_2), (\mathsf{a}_6, \mathsf{a}_3) \}.
}
and
\al{\nonumber
Z_{\Gamma}^0\big|_3 &= 4\cp{\mathsf{a}_1}{\mathsf{a}_4}\cp{\mathsf{a}_2}{\mathsf{a}_5}\cp{\mathsf{a}_3}{\mathsf{a}_6}\\
&= ( -\alpha_2\alpha_5 + \alpha_7\alpha_8 )^2( -(\alpha_1+\alpha_6)(\alpha_3+\alpha_4) + \alpha_7\alpha_8 ).
}
\begin{remark}
The fact that the $\lambda_{\mathcal{E}}$ are all non-vanishing and often the same is due to the fact that $D_{\Gamma}^0$ has the base cycle structure $n=(1,1,1)$ (see \reffig{img_38_proj}). In this case the partitions $\mathcal{P}(E_D^1)$ and words $\mathsf{P}_2$ are in a sense maximally compatible, since each $1$-coloured base edge has a $2$-coloured base edge partner between the exact same vertices. In other words, $\mathcal{P}(E_D^1) = \mathcal{P}(E_D^2)$ and $\mathsf{P}_1 = \mathsf{P}_2$.
\end{remark}
The number of terms in the three polynomials $Z_{\Gamma}^0\big|_k$ is $1$, $22$ and $15$ respectively. For comparison, the full chord diagram sum consists of 437 monomials. Here we especially also see how hidden the factorisation of the Kirchhoff polynomials can be. The expression $a\kp^2Z_{\Gamma}^0\big|_1 + b\kp Z_{\Gamma}^0\big|_2 + cZ_{\Gamma}^0\big|_3$ should have $36^2\cdot 1 + 36\cdot 22 + 15 = 2103$ terms. But the same monomials may of course occur in different parts and add up or cancel to yield the 437 that are left in the sum, obscuring the pattern. See also table \ref{tab_z0} for the reduction observed for other graphs.
\begin{table}[h]
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\ & (a) & (b) & (c) & (d) & (e) & (f) & (g) & (h) \vphantom{\Big|} \\[2mm]\hline
$\#Z_{\Gamma}^0$ & 9 & 9 & 44 & 84 & 348 & 231 & 448 & 437 \vphantom{\bigg|} \\[2mm]
$\#Z_{\Gamma}^0/\kp^6$ & 9 & 9 & 44 & 15 & 16 & 72 & 53 & 38 \vphantom{\bigg|} \\\hline
\end{tabular}
\caption{Number of terms in $Z_{\Gamma}^0$ compared to $Z_{\Gamma}^0/\kp^6$ after cancellations for all graphs from \reffig{06a_img_3_1} to \ref{06a_img_3_8}.}
\label{tab_z0}
\end{table}
\vspace{-5mm}
\paragraph{The partition polynomial $Z_{\Gamma}^1$.}
We do not need to repeat the discussion of partitions etc. but can simply sum over all possible ways to append a letter to the word pairs. For $Z_{\Gamma}^1\big|_1$ this means we take $Z_{\Gamma}^0\big|_1$ from \refeq{eq_Z01} and get
\al{\nonumber
Z_{\Gamma}^1\big|_1 + \frac{\vssp}{\kp}Z_{\Gamma}^0\big|_1
= \cp{\mathsf{a}_1\mathsf{a}_2\mathsf{a}_3\mathsf{y}}{\mathsf{a}_4\mathsf{a}_5\mathsf{a}_6\mathsf{y}} + \cp{\mathsf{a}_1\mathsf{a}_2\mathsf{a}_6\mathsf{y}}{\mathsf{a}_4\mathsf{a}_5\mathsf{a}_3\mathsf{y}}
+ \cp{\mathsf{a}_1\mathsf{a}_5\mathsf{a}_3\mathsf{y}}{\mathsf{a}_4\mathsf{a}_2\mathsf{a}_6\mathsf{y}} + \cp{\mathsf{a}_1\mathsf{a}_5\mathsf{a}_6\mathsf{y}}{\mathsf{a}_4\mathsf{a}_2\mathsf{a}_3\mathsf{y}}.
}
This already has 92 terms, so explicitly giving it here in terms of Schwinger parameters would not be particularly enlightening. Similarly one finds e.g.
\al{\nonumber
Z_{\Gamma}^1\big|_3 &+ 3\frac{\vssp}{\kp}Z_{\Gamma}^0\big|_3\\
&= 4\cp{\mathsf{a}_1\mathsf{y}}{\mathsf{a}_4\mathsf{y}}\cp{\mathsf{a}_2}{\mathsf{a}_5}\cp{\mathsf{a}_3}{\mathsf{a}_6}
+ 4\cp{\mathsf{a}_1}{\mathsf{a}_4}\cp{\mathsf{a}_2\mathsf{y}}{\mathsf{a}_5\mathsf{y}}\cp{\mathsf{a}_3}{\mathsf{a}_6}
+ 4\cp{\mathsf{a}_1}{\mathsf{a}_4}\cp{\mathsf{a}_2}{\mathsf{a}_5}\cp{\mathsf{a}_3\mathsf{y}}{\mathsf{a}_6\mathsf{y}},
}
which has 1551 terms. We see that these expressions are still quite large, but they nonetheless represent a massive reduction in size compared to the full integrand without cancellations. In table \ref{tab_total} the superficially renormalised integrands with and without cancellations are compared and one finds a reduction by roughly one order of magnitude.
\begin{table}[h]
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\ & (a) & (b) & (c) & (d) & (e) & (f) & (g) & (h) \vphantom{\Big|} \\[2mm]\hline
$\#(\vssp Z_{\Gamma}^0+Z_{\Gamma}^1)$ & 528 & 681 & 1937 & 4698 & 17641 & 8210 & 22627 & 25575 \vphantom{\bigg|} \\[2mm]
$\#(\vssp Z_{\Gamma}^0+Z_{\Gamma}^1)/\kp^6$ & 88 & 329 & 387 & 513 & 1106 & 782 & 1637 & 2439 \vphantom{\bigg|} \\\hline
\end{tabular}
\caption{Total number of terms in the superficially renormalised integrand with and without cancellations for all graphs from \reffig{06a_img_3_1} to \ref{06a_img_3_8}.}
\label{tab_total}
\end{table}
| {
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{"url":"https:\/\/www.physicslens.com\/newtons-2nd-law-questions\/","text":"# Newton\u2019s 2nd Law Questions\n\nHere are the steps to solving a problem involving Newton\u2019s 2nd law.\n\nStep 1: Draw a free body diagram \u2013 where possible, sketch a free body diagram to represent:\n\n1. the body isolated from other objects\n2. the forces acting on the body (ignore all internal forces) along the line of motion.\n3. the direction of acceleration.\n\nStep 2: Write down the equation: $$F_{net}=ma$$\n\nStep 3: Add up the forces on the left-hand side of the equation, making sure that forces acting opposite to the direction of acceleration are subtracted.\n\nStep 4: Solve the equation for the unknown.\n\nFor example,\n\nA horizontal force of 12 N is applied to a wooden block of mass\u00a00.60 kg on a rough horizontal surface, and the block accelerates at 4.0 m s-2. What is the magnitude of the frictional force acting on the block?\n\nStep 1: Free-body diagram\n\nStep 2: Write down the 2nd law equation\n\nBy Newton\u2019s 2nd law, $$F_{net}=ma$$\n\nStep 3: Add up the forces making up the net force\n\n$$12 \u2013 f = 0.60 (4.0)$$\n\nStep 4: Solve for unknown\n\n$$f = 12 \u2013 2.4 = 9.6 N$$","date":"2022-09-27 18:17:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8835691809654236, \"perplexity\": 485.22008029609447}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030335054.79\/warc\/CC-MAIN-20220927162620-20220927192620-00797.warc.gz\"}"} | null | null |
So you wrapped up that big project for the client and everyone's happy. Congrats! You put in some super-long hours and you totally blew your Red Bull budget. You deserve a minute to catch your breath, but here's the thing: one minute is all you get.
You diligently and excitedly developed your brand, but did you put the same level of effort into the customer engagement that you put into your branding? Because if you didn't, it won't matter where your brand started out—it'll end in the crapper.
Data-first marketing is a must if you expect your business to be proactive, nimble and in tune with its clients and their customer base. If the thought of analytics has you feeling intimidated, these tips can help. | {
"redpajama_set_name": "RedPajamaC4"
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Желі́зний Мико́ла Я́кович ( — 19 травня 1974) — учасник Другої світової війни, командир батальйону, майор, Герой Радянського Союзу (1945).
Біографія
Народився 9 травня 1910 року в місті Баштанка Миколаївської області в родині службовця. Українець. Член ВКП(б) з 1939 року.
Закінчив Новобузький педагогічний технікум. Вчителював.
У 1933 році призваний до лав РСЧА. Закінчив школу молодших командирів. З 1933 року — на комсомольській роботі, з 1939 року — на партійній роботі.
Учасник німецько-радянської війни з червня 1941 року. Воював на Південному, 3-у Українському, 1-у Білоруському фронтах. У серпні 1942 року був поранений. Пройшов шлях від командира взводу до командира стрілецького батальйону.
Особливо командир 2-го стрілецького батальйону 986-го стрілецького полку 230-ї стрілецької дивізії 9-го стрілецького корпусу 5-ї ударної армії 1-го Білоруського фронту майор М. Я. Желізний відзначився під час проведення Берлінської операції. Бійці батальйону під його командуванням стрімко переправились через річку Шпрее біля озера Руммельсбургер-Зеє, захопили плацдарм і утримували його до підходу основних сил полку. Відбили 4 атаки ворога, знищивши при цьому близько 300 солдат і офіцерів супротивника, 250 захопили у полон. 27 квітня 1945 року був удруге поранений.
У повоєнні роки продовжував військову службу. У 1951 році закінчив курси удосконалення командного складу «Постріл».
У 1956 році підполковник М. Я. Желізний вийшов у запас. Мешкав у Вінниці. Помер 19 травня 1974 року. Похований на Центральному цвинтарі Вінниці.
Нагороди
Указом Президії Верховної Ради СРСР від 31 травня 1945 року майору Желізному Миколі Яковичу присвоєне звання Героя Радянського Союзу з врученням ордена Леніна і медалі «Золота Зірка» (№ 6712).
Також нагороджений орденами Червоного Прапора, Олександра Невського, Вітчизняної війни 2-го ступеня, Червоної Зірки, медалями.
Література
Бундюков А. Г., Кравченко М. В. Сыновняя верность Отчизне: Очерки о Героях Советского Союза — уроженцах Николаевской области. — Одесса: Маяк, 1982, стор. 123-125.
Золоті зірки Миколаївщини: Енциклопедичне видання. — Миколаїв: Вид-во МДГУ ім. Петра Могили, 2005.
Посилання
Железный Николай Яковлевич
Уродженці Баштанки
Радянські офіцери Другої світової війни
Герої Радянського Союзу — уродженці України
Герої Радянського Союзу — українці
Померли у Вінниці | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,884 |
Q: How to "safely" load a page with JavaScript I want to write a small function that takes a URL as a parameter, but before loading that URL, it checks if it is already on that page. If so, it returns false and nothing happens; otherwise, it loads the specified URL. I tried something like this:
function load(url) {
if(window.location.href === url) {
return false;
} else {
window.location.href = url;
}
}
Unfortunately, if my current page is http://www.example.org/index.html, then, while calling load('http://www.example.org/index.html') does nothing, when I call load('index.html'), it reloads the page. How can I add the extra functionality to catch relative addresses as well as absolute ones?
EDIT: I want to make this dynamic, so that I can use this on any page, without knowing the base of its URL.
A: This might work for you:
function load(url) {
if(window.location.href === url || window.location.href === 'http://www.example.org/' + url) {
return false;
} else {
window.location.href = url;
}
}
But it's hard to know without knowing more about how the load function might be used.
A: Got it:
String.prototype.endsWith = function(suffix) {
return this.indexOf(suffix, this.length - suffix.length) !== -1;
};
Then, the check becomes easy:
function load(url) {
if(window.location.href.endsWith(url)) {
return false;
} else {
window.location.href = url;
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Colletes stepheni is een vliesvleugelig insect uit de familie Colletidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1958 door Timberlake.
Colletidae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,084 |
Aglossosia flavimarginata is a moth of the subfamily Arctiinae. It is found in Kenya, South Africa and Uganda.
References
Moths described in 1900
Lithosiini
Moths of Africa
Insects of Uganda | {
"redpajama_set_name": "RedPajamaWikipedia"
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Il cucal bruno (Centropus andamanensis ) è un uccello della famiglia Cuculidae.
Distribuzione e habitat
Questo uccello vive in India e Myanmar.
Tassonomia
Centropus andamanensis non ha sottospecie, è monotipico.
Note
Bibliografia
Altri progetti
Collegamenti esterni
Cuculidae | {
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{"url":"http:\/\/nrich.maths.org\/public\/leg.php?code=5039&cl=2&cldcmpid=6288","text":"# Search by Topic\n\n#### Resources tagged with Interactivities similar to Cops and Robbers:\n\nFilter by: Content type:\nStage:\nChallenge level:\n\n### There are 216 results\n\nBroad Topics > Information and Communications Technology > Interactivities\n\n### Cops and Robbers\n\n##### Stage: 2 and 3 Challenge Level:\n\nCan you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?\n\n### Lost\n\n##### Stage: 3 Challenge Level:\n\nCan you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.\n\n### Diamond Mine\n\n##### Stage: 3 Challenge Level:\n\nPractise your diamond mining skills and your x,y coordination in this homage to Pacman.\n\n### Archery\n\n##### Stage: 3 Challenge Level:\n\nImagine picking up a bow and some arrows and attempting to hit the target a few times. Can you work out the settings for the sight that give you the best chance of gaining a high score?\n\n### Top Coach\n\n##### Stage: 3 Challenge Level:\n\nCarry out some time trials and gather some data to help you decide on the best training regime for your rowing crew.\n\n### Subtended Angles\n\n##### Stage: 3 Challenge Level:\n\nWhat is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?\n\n### Balancing 2\n\n##### Stage: 3 Challenge Level:\n\nMeg and Mo still need to hang their marbles so that they balance, but this time the constraints are different. Use the interactivity to experiment and find out what they need to do.\n\n### Fifteen\n\n##### Stage: 3 Challenge Level:\n\nCan you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15.\n\n### Flip Flop - Matching Cards\n\n##### Stage: 1, 2 and 3 Challenge Level:\n\nA game for 1 person to play on screen. Practise your number bonds whilst improving your memory\n\n### See the Light\n\n##### Stage: 2 and 3 Challenge Level:\n\nWork out how to light up the single light. What's the rule?\n\n### Shuffles Tutorials\n\n##### Stage: 3 Challenge Level:\n\nLearn how to use the Shuffles interactivity by running through these tutorial demonstrations.\n\n### Triangles in Circles\n\n##### Stage: 3 Challenge Level:\n\nHow many different triangles can you make which consist of the centre point and two of the points on the edge? Can you work out each of their angles?\n\n### Balancing 1\n\n##### Stage: 3 Challenge Level:\n\nMeg and Mo need to hang their marbles so that they balance. Use the interactivity to experiment and find out what they need to do.\n\n### Bow Tie\n\n##### Stage: 3 Challenge Level:\n\nShow how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.\n\n### An Unhappy End\n\n##### Stage: 3 Challenge Level:\n\nTwo engines, at opposite ends of a single track railway line, set off towards one another just as a fly, sitting on the front of one of the engines, sets off flying along the railway line...\n\n### Balancing 3\n\n##### Stage: 3 Challenge Level:\n\nMo has left, but Meg is still experimenting. Use the interactivity to help you find out how she can alter her pouch of marbles and still keep the two pouches balanced.\n\n### Multiplication Tables - Matching Cards\n\n##### Stage: 1, 2 and 3 Challenge Level:\n\nInteractive game. Set your own level of challenge, practise your table skills and beat your previous best score.\n\n### Partitioning Revisited\n\n##### Stage: 3 Challenge Level:\n\nWe can show that (x + 1)\u00b2 = x\u00b2 + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)\u00b2 = x\u00b2 + 4x + 4\n\n### Number Differences\n\n##### Stage: 2 Challenge Level:\n\nPlace the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?\n\n### Simple Counting Machine\n\n##### Stage: 3 Challenge Level:\n\nCan you set the logic gates so that the number of bulbs which are on is the same as the number of switches which are on?\n\n### Magic Potting Sheds\n\n##### Stage: 3 Challenge Level:\n\nMr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?\n\n### Two's Company\n\n##### Stage: 3 Challenge Level:\n\n7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?\n\n### Countdown\n\n##### Stage: 2 and 3 Challenge Level:\n\nHere is a chance to play a version of the classic Countdown Game.\n\n### Domino Numbers\n\n##### Stage: 2 Challenge Level:\n\nCan you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?\n\n### Isosceles Triangles\n\n##### Stage: 3 Challenge Level:\n\nDraw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?\n\n### Flippin' Discs\n\n##### Stage: 3 Challenge Level:\n\nIdentical discs are flipped in the air. You win if all of the faces show the same colour. Can you calculate the probability of winning with n discs?\n\n### Matching Fractions Decimals Percentages\n\n##### Stage: 2 and 3 Challenge Level:\n\nAn activity based on the game 'Pelmanism'. Set your own level of challenge and beat your own previous best score.\n\n### Cosy Corner\n\n##### Stage: 3 Challenge Level:\n\nSix balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?\n\n### Got it Article\n\n##### Stage: 2 and 3\n\nThis article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.\n\n### Square Coordinates\n\n##### Stage: 3 Challenge Level:\n\nA tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?\n\n### Twice as Big?\n\n##### Stage: 2 Challenge Level:\n\nInvestigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.\n\n### Volume of a Pyramid and a Cone\n\n##### Stage: 3\n\nThese formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.\n\n### Speeding Up, Slowing Down\n\n##### Stage: 3 Challenge Level:\n\nExperiment with the interactivity of \"rolling\" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.\n\n### Nine Colours\n\n##### Stage: 3 Challenge Level:\n\nYou have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.\n\n### Cogs\n\n##### Stage: 3 Challenge Level:\n\nA and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .\n\n### Muggles Magic\n\n##### Stage: 3 Challenge Level:\n\nYou can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.\n\n### Stars\n\n##### Stage: 3 Challenge Level:\n\nCan you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?\n\n### More Carroll Diagrams\n\n##### Stage: 2 Challenge Level:\n\nHow have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?\n\n### Shuffle Shriek\n\n##### Stage: 3 Challenge Level:\n\nCan you find all the 4-ball shuffles?\n\n### How Far Does it Move?\n\n##### Stage: 3 Challenge Level:\n\nExperiment with the interactivity of \"rolling\" regular polygons, and explore how the different positions of the red dot affects the distance it travels at each stage.\n\n### Gr8 Coach\n\n##### Stage: 3 Challenge Level:\n\nCan you coach your rowing eight to win?\n\n### A Dotty Problem\n\n##### Stage: 2 Challenge Level:\n\nStarting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots!\n\n### Triangle Pin-down\n\n##### Stage: 2 Challenge Level:\n\nUse the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.\n\n### Poly-puzzle\n\n##### Stage: 3 Challenge Level:\n\nThis rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.\n\n### Picturing Triangle Numbers\n\n##### Stage: 3 Challenge Level:\n\nTriangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?\n\n### Sliding Puzzle\n\n##### Stage: 1, 2, 3 and 4 Challenge Level:\n\nThe aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.\n\n### World of Tan 20 - Fractions\n\n##### Stage: 2 Challenge Level:\n\nCan you fit the tangram pieces into the outlines of the chairs?\n\n### Online\n\n##### Stage: 3 Challenge Level:\n\nA game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.\n\n### Rolling Around\n\n##### Stage: 3 Challenge Level:\n\nA circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?\n\n### Game of PIG - Sixes\n\n##### Stage: 2, 3, 4 and 5 Challenge Level:\n\nCan you beat Piggy in this simple dice game? Can you figure out Piggy's strategy, and is there a better one?","date":"2014-07-25 20:40:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.2709391713142395, \"perplexity\": 2434.0688321386156}, \"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-2014-23\/segments\/1405997894782.90\/warc\/CC-MAIN-20140722025814-00106-ip-10-33-131-23.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/tex.stackexchange.com\/questions\/linked\/155518","text":"1k views\n\n### How can I create a show\/hide button? [duplicate]\n\nI would like to create a show\/hide button that appears on a typeset PDF so that when it is clicked, the content appears. This is similar to the ideas here: Hide all but one instances of a custom ...\n42k views\n\n### How to annotate PDF files generated by pdflatex?\n\nI want to annotate some PDF files (created with pdflatex), e.g. word x is missing here this part of a sentence should be moved to the front replace word x by y highlight this sentence insert a note ...\n11k views\n\n### Showing the bibliographic entry in a popup when you hover over the citation key\n\nI thought it would be useful to have bibliographic entries displayed as a tooltip so readers are not interrupted by following hyperlinks or trying to find the Bibliography, so I created the following ...\n5k views\n\n### Mouseover events in beamer: hovering on \\eqref and a comment containing the original equation popping up\n\nThis idea came into my mind while I was listening to a speaker presenting his results in a seminar, he used lots of equation references in his later slides, however I can't remember which equation he ...\n2k views\n\n### Move-around box in PDF display\n\nI tried using the fancytooltips package and like it a lot. One thing that I could imagine would be nice is to have the entry showing up as a movable box instead of just a tooltip. (Sometimes the ...\n5k views\n\n### fold\/unfold sections to expand in PDF\n\nIs there a way to fold\/unfold certain sections in a pdf. Only when the user clicks on the section or plus sign the items show up. Example: user clicks on the plus sign and the items appear as drop ...\n929 views\n\n### How to start with a, b, c,\u2026 and end up with 1a2a3, 1b2b3, 1c2c3,\n\nSo imagine you have a certain code which looks like this: ABC, some text , LaLaLa \\tooltip*{A}{\\includegraphics[scale=2]{images\/A}}\\tooltip*{B}{\\includegraphics[scale=2]{images\/B}}\\tooltip*{C}{\\...\n2k views\n\n### Can I make an \u201cabbreviation expansion\u201d in PDF visible?\n\n(I hope this is at least marginally on topic) When I use the <abbr> tag in HTML, the abbreviation and its expansion are marked by the browser: But if I mark an abbreviation and its expansion ...\n807 views\n\n### How to use fancytooltips with MikTeX?\n\nI'm trying to create a tooltip with the fancytooltips package, the compilation is going well, but I don't get any tooltips. I'm on Windows 7 with MikTeX. I don't know if this is really necessary, ...\n452 views\n\n### fold \/ unfold system of equations in long derivation\n\nI am showing a long derivation (simplified example here) \\documentclass{article} \\begin{document} \\begin{eqnarray*} 0 &=& 0 \\\\ &\\downarrow& \\mbox{using eq.}~\\ref{eq0} \\\\ &=& ...\n1k views\n\n### Making notes onto a pdf document\n\nI have some books and .pdf files that I would like to make some notes (particularly adding equations with LaTeX). I have seen the following post: Overlay LaTeX\/TeX coding easily on PDF with 300 pages ...\n578 views\n\n### Can't embed hyperlink references in a tooltip\n\nI'm having troubles trying to embed text references in a tooltip. I'm using the tooltip script which can be found here: Tooltip that works with all pdf readers. I'm also using the hyperref package. ...\n644 views\n\n### Is there a way to make image\/Table appear on hovering mouse on another image in the pdf?\n\nI am trying to make an image or a table appear on mouse hover (mouseover) over specific parts of another image like an info-graphic. Can someone tell me a package and syntax to achieve this? The ...","date":"2020-10-28 12:01:10","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9391924738883972, \"perplexity\": 2357.3818245430266}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107898499.49\/warc\/CC-MAIN-20201028103215-20201028133215-00135.warc.gz\"}"} | null | null |
\section{Introduction}
Choosing the proper shape of distribution tail is an important and, at the same time, very delicate problem in modelling many phenomena. As an illustration let us mention here the long lasting discussion on the maximal human's age, where the conclusion is very sensitive to the data considered and to the method of analysis (for more details see Section \ref{realdata}).
The majority of methods used in the statistical analysis of distribution tails is provided by extreme value theory and consists in the direct estimation of parameters of one of the general distribution tail models. Very popular are here the Gumbel block maxima method and the peak-over-threshold approach based on the Generalized Pareto Distribution (GPD) model, see \cite{dehaan}, \cite{beirlantteugels}, \cite{embrechts} among others. Both methods assume that the distribution function $F$ is in the domain of attraction of some extreme value distribution, namely, for some sequences $a_n>0$ and $b_n$
\[F^n(a_n x + b_n) \to EV_\gamma(x), \quad n\to\infty,\] where
\[EV_\gamma(x) = \left\{\begin{array}{cc} \exp(-(1+\gamma x)^{-1/\gamma}), 1+\gamma x>0, & \mbox{if } \gamma\neq 0,\\
\exp(-\exp(-x)), x\in\mathbb{R}, & \text{if }\gamma=0.\end{array}\right.\]
These methods perform well for distributions from both the Fr\'echet ($\gamma>0$) and Weibull ($\gamma<0$) maximum domains of attraction (MDA). But these methods may not show good efficiency if tails of distributions belonging to the Gumbel MDA (i.e. if $\gamma=0$) are analyzed, see, for instance, Section \ref{realdata}. Indeed, if it is known that $\gamma=0,$ independently of the tail behavior of distribution nearby its right endpoint (note that in the Gumbel MDA there are so different in tail behavior distributions like normal and lognormal), the distribution tail is approximated by exponential distribution in the framework of GPD model (and by Gumbel distribution in the framework of the Gumbel block maxima method if we want to approximate the distribution of a sample maximum), see also discussion in Section \ref{recent}. Next, both of these methods are based on the extreme value theorem, but the rate of convergence of the distribution of normalized maxima to the Gumbel distribution is usually very slow, \cite{resnick}. Moreover, there are distributions that do not satisfy the conditions of the extreme value theorem (for example, distributions with super-heavy tails), which prevents the application of the mentioned methods.
Thus, there is a need to expand the standard methodology of statistics of univariate extremes by considering other models for distribution tails and thus to develop methods for distinguishing between such models. It is quite surprising that, to the best of our knowledge, there is no any general method proposed in the literature for this purpose. The present article is devoted to this problem.
For a cdf $F,$ denote $\overline{F}(x) = 1 - F(x),$ its tail distribution function, and $x^{\ast}_{F} := \sup\{y: F(y)<1\},$ the right endpoint of $F$. Following \cite{resnick2}, we say that $\overline{F}$ and $\overline{G}$ are strictly tail-equivalent, if $x^\ast_{F} = x^\ast_{G} =: x^\ast$ and $\overline{F}(x)/\overline{G}(x) \to 1$ as $x\uparrow x^{\ast}$ (write $\overline{F}\sim\overline{G}$). We call {\it the distribution tail} the class of equivalence $T(\overline{F})$ of tail distribution functions, i.e. for all $\overline{G}$ the property $\overline{G}\sim\overline{F}$ is equivalent to $\overline{G}\in T(\overline{F}).$
Let $\textbf{X}_n = (X_1, \ldots, X_n)$ be i.i.d. random variables with a cdf $F,$ and $x_F^\ast = +\infty.$ In this work, we aim at proposing a location and scale free test for distinguishing between the hypothesis $H_0: T(\overline{F}) \in A_0$ and the alternative $H_1: T(\overline{F}) \in A_1$ (we write below $H_0: F \in A_0$ and $H_1: F \in A_1,$ respectively, for simplicity, since the tail distribution function fully determines the distribution tail) by sample $\textbf{X}_n.$ Here $A_0$ and $A_1$ are two classes of distribution tails that are ``separable'' in some sense by some cdf $F_0$, in particular, all tails from $A_0$ are lighter (or heavier) than those from $A_1.$ We also propose a scale free test for distinguishing between $H_0$ and $H_1,$ that is more powerful in some situations than the first one. It is worth noting that we do not assume that distributions with a tail from $A_0$ or $A_1$ belong to any of MDA, except $F_0$ in case $F_0\in A_0.$
The article is organized as follows. In Section \ref{recent}, we discuss two recent models for distribution tails from the Gumbel MDA, namely, the Weibull type and log-Weibull type classes. The precursor of two tests proposed in the present paper is described in Section \ref{initialtest}.
Next, in Section \ref{ccondition} we introduce the notion of C-separability and provide examples of C-separable classes of distribution tails. A scale free and a location and scale free tests for distinguishing between C-separable classes of distribution tails are proposed in Sections \ref{scale-free} and \ref{location-free}, respectively, and their asymptotic properties are established, in particular, their consistency on the alternative. Using these tests one can distinguish between regularly varying, Weibull type and log-Weibull type distribution tails, although their applicability is certainly not limited to this. The problem of selection the optimal threshold level for testing hypotheses about distribution tails is concerned in Section \ref{kselection}. In Section \ref{simulationstudy}, the performance of our methods is examined on both simulated and real data sets in comparison with methods considered in the literature. Proofs are postponed to Appendix.
\section{Related work} \label{relatedwork}
\subsection{Some recent models for distribution tails} \label{recent}
In last two decades, several models for distribution tails were proposed that are not yet so actively used by practitioners. Firstly, let us pay attention on the model developed in \cite{devalk1} (see also the particular case of this model proposed in \cite{elmethni}). Following the description of this model in \cite{girard2020}, assume that the distribution tail of random variable $X$ (let $X\ge 1$ a.s. for simplicity) is given by
\begin{equation}1-F(x) = \exp(-V^{\leftarrow}(\ln x)),\ x\ge 1,\label{cees}\end{equation} where $V^{\leftarrow}(x) := \inf\{y: V (y) \ge x\}$ is the generalized inverse of $V(x) = \ln Q(1/\exp(x))$ with $Q,$ the
quantile function of $F,$ that is $Q(t) = F^{\leftarrow}(1 - 1/t).$ It is also supposed that there exists a positive function $a,$ such that for all $t>0$
\begin{equation}\label{ceesV}\lim_{x\to\infty} \frac{V(tx) - V(x)}{a(x)} = \frac{t^\theta - 1}{\theta},\end{equation} where for $\theta=0$ the right-hand side should be understood as $\ln t.$ Hence, $V$ is supposed to be of extended regular variation with index $\theta\in\mathbb{R},$ see for details \cite{dehaan}.
Note, that if $\theta>0$ then the parameter $\theta$ governs the tail behavior of the log-Weibull type distribution, i.e. having a cdf $F$ such that for all $t>0$
\begin{equation}
\label{deflogw}\lim_{x\to\infty} \frac{\ln (1 - F(e^{t x}))}{\ln (1 - F(e^x))} = t^{1/\theta}.\end{equation} Representatives of this class are the lognormal distribution and the distributions with cdf $F(x) = 1 - \exp(-(\ln x)^{1/\theta}), x>1.$ If $0<\theta<1,$ then $F$ belongs to the Gumbel MDA; if $\theta = 1,$ then it belongs to the Fr\'echet MDA (see Theorem 1, \cite{girard2020}). The problems of estimation of the parameter $\theta,$ extreme quantiles and probabilities of rare events in the framework of this model are considered in \cite{devalk1}, \cite{girard2020}, \cite{devalk2}, \cite{devalk3}.
Another class of distributions from the Gumbel MDA that deserves attention is the class of Weibull type distributions, \cite{Girard}, \cite{Goegebeur}. Say that cdf $F$ is of Weibull type, if there exists $\theta>0$ such that for all
$t>0$
\begin{equation}\label{defw}\lim_{x\to\infty} \frac{\ln (1 - F(t x))}{\ln (1 - F(x))} = t^{1/\theta}.\end{equation}
The parameter $\theta$ is called the Weibull tail index. Representatives of this class are the normal, exponential, gamma, Weibull distributions among others. The model \eqref{cees} as well as the GPD and EVD models does not provide good description for Weibull type distributions, since in \eqref{ceesV} the parameter $\theta$ is equal to 0 for all distributions from this class as well as for distributions with lighter tails. The problems of estimation of the Weibull tail index and extreme quantiles for Weibull type distributions are investigated in \cite{elmethni}, \cite{Girard}, \cite{Bro}, \cite{Beirlant}, \cite{Gardes} among others.
Hence, there is a need to develop methods for distinguishing between at least these 3 classes of distribution tails, namely, the classes of regularly varying, log-Weibull type and Weibull type distribution tails. There are many works devoted to the problem of distinguishing between distributions from the Gumbel and Fr\'echet MDA (see the review of such tests in \cite{hueslerpeng}, \cite{nevesfraga}). However, the literature reveals an almost complete lack of work on distinguishing between distributions within the Gumbel MDA. In this respect we can only refer to \cite{Goegebeur}, where the goodness-of-fit test for Weibull type behavior was proposed, and to \cite{R2}, \cite{R3}. To the best of our knowledge, the problem of goodness-of-fit testing for log-Weibull type behavior was not considered in the literature. Finally, we refer to \cite{R1}, where the simple test for distinguishing between two separable classes of distribution tails was proposed, that does not require the fulfillment of the extreme value theorem assumptions, see also the next section.
However, the statistics of the tests for distinguishing between distributions within the Gumbel MDA mentioned above are not invariant with respect to the scale and location parameter change, except the test by \cite{Goegebeur} that is scale free. Inaccurate determination of the location and scale parameters when analyzing distribution tails can lead to serious errors in conclusions. Hence the problem of developing the location and scale free methods for statistical analysis of distribution tails is of undoubted interest.
The reason why these two models have not yet been widely adopted by practitioners may be that while the tail behavior of Weibull or log-Weibull type distribution is of interest, the Generalized Pareto model with $\gamma>0$ (or with $\gamma<0$ for Weibull type tail with $\theta<1$) may provide acceptable quality, whereas the approximation of the tail by exponential distribution may be much worse. This phenomena called the {\it penultimate} behavior was studied in \cite{cohen}, \cite{penultimate} for distributions from the Gumbel MDA. Later, these results were extended to other maximum domains of attraction in \cite{gomeshaan}. Nevertheless, it was shown in \cite{gardesgirard2} that the extreme quantile estimator for Weibull type distributions based on the Weibull type model performs much better than the estimator based on the GPD model. It emphasizes once again that our methodology makes sense.
\subsection{Initial test}\label{initialtest}
Following \cite{R1}, we say that cdfs
$H$ and $G$ with $x_H^\ast = x_G^\ast =: x^\ast$ satisfy the condition $B(H,G)$ (B-condition), if for some $\varepsilon\in(0,1)$ and $x_{0}$ \begin{equation}
\frac{(1-H(x))^{1-\varepsilon}}{1-G(x)}\text{ does not increase as }\
x>x_{0}.
\label{condb}%
\end{equation}
It is easy to see that under this condition the tail of $H$ is lighter than the tail of $G,$ that is $\overline{H}(x)/\overline{G}(x)\to 0$ as $x\uparrow x^\ast.$ Let us call classes of distribution tails $A_0$ and $A_1$
\emph{$B$-separable}, if there is a cdf $F_0$ such that for all $G\in A_0$
the condition $B(G, F_0)$ holds and for all $H\in A_1$ the condition $B(F_0, H)$ holds for some (maybe different) $\varepsilon$ and $x_0.$
Assume that the classes $A_0$ and $A_1$ are $B$-separable via $F_0,$ and all distributions belonging to $A_0$ or $A_1$ are continuous. In
\cite{R1}, the test for distinguishing between $H_0:
F\in A_0$ and $H_1: F\in A_1$
\begin{equation}\text{if } R_{k,n} > 1+ \frac{u_{1-\alpha}}{\sqrt{k}}, \text{ then reject } H_0,\label{test1}\end{equation}
was proposed, where $u_{1-\alpha}$ is the $(1-\alpha)$-quantile of the standard normal distribution $N(0,1)$ and
\[R_{k,n} =
\ln(1-F_{0}(X_{(n-k)}))-\frac{1}{k}\sum\limits_{i=n-k+1}^{n}
\ln(1-F_{0}(X_{(i)})),\] where $X_{(1)} \le \ldots \le X_{(n)}$ denote the order statistics of a sample $\{X_i\}_{i=1}^n.$ It was shown in \cite{R1}, that the latter test has asymptotically the significance level $\alpha$ and is consistent on $H_1,$ if $k\to\infty,$ $k/n\to 0$ as $n\to\infty.$ Later, in \cite{Kogut}, it was shown that under some additional conditions the test \eqref{test1} can be applied for distinguishing between classes of distribution tails that are not necessarily continuous.
If for some cdf $F_0$ the condition $B(F_0, G)$ holds for all $G\in A_0$ and the condition $B(H, F_0)$ holds for all $H\in
A_1$ for some (maybe, different) $\varepsilon$ and $x_0,$ then the following rule for testing the hypothesis $H_0: F\in
A_0$ against the alternative $H_1: F\in A_1$ can be proposed \[\text{if }
R_{k,n} < 1+ \frac{u_{\alpha}}{\sqrt{k}}, \text{ then reject }
H_0.\] Note, that this test can be applied even for distributions that do not satisfy the assumptions of the extreme value theorem. Using the test \eqref{test1}, one can distinguish between the tails of
Weibull type and log-Weibull type distributions,
heavy-tailed and light-tailed distributions, regularly varying and super-heavy tailed distributions among others. Weakness of conditions imposed on distributions and breadth of application
makes this test a suitable method for choosing a model in the case,
when nothing is known in advance about the heaviness of the tails. However, changing the scale and location parameters of $F_0$ and $F$ can lead to completely opposite results when using this test. The present work corrects this disadvantage.
\section{Main results}\label{mainresults}
\subsection{$C$-condition and $C$-separability. Examples.}\label{ccondition}
Hereinafter we assume that all considered distributions are continuous and their right endpoints are infinite. The methods proposed in this work can be extended on case of finite right endpoints, but this is beyond the scope of our paper.
Denote the quantile function of $F$ by $u_F(t).$ Let us introduce the conditions that we use to establish the asymptotical properties of the tests proposed in this work.
\begin{definition}
\label{def1} Two cdfs $H$ and $G$ are said to satisfy the condition
$C_\delta(H,G)$ ($C$-condition), if for some $\delta\ge 0$ there exists $t_0$ such that for all $t>t_0$ and $c>1$
\begin{equation}
\frac{u_H\big(c^{1+\delta}t\big)}{u_G(ct)} \le \frac{u_H(t)}{u_G(t)}.
\label{cond}%
\end{equation}
\end{definition}
Note that if two cdfs $H$ and $G$ satisfy the condition
$C_0(H,G)$ ($C_0$-condition), then starting from some $t_0$ the ratio $u_H(t)/u_G(t)$ does not increase.
Clear, the condition $C_{\delta}(H, G)$ is weaker than the condition $C_{\delta^\prime}(H, G)$ for $\delta^\prime>\delta\ge 0.$
Next, it is easy to see that the condition $C_\delta(H,G)$ are invariant with respect to changes of the scale parameters of both $G$ and $H.$ However, these conditions may be violated in case of substituting $\sigma_1u_H(t) + a_1$ and $\sigma_2u_H(t) + a_2$ for $u_H(t)$ and $u_G(t),$ respectively, but this fact does not prevent us to prove the results of the next sections.
Let us show that the condition $C_\delta(H,G)$ implies the one similar to \eqref{condb} that we will use in the proofs of the main results.
\begin{proposition}
Assume that cdfs $H$ and $G$ satisfy the condition $C_\delta(H,G)$ for some $\delta\ge 0$ and $t_0.$ Then for all $t>t_0$ and $c>1$
\begin{equation}\frac{\big(1 - H(c u_H(t))\big)^{1-\varepsilon}}{(1 - H(u_H(t)))^{1-\varepsilon}} \le \frac{1 - G(c u_G(t))}{1 - G(u_H(t))}, \label{B'}
\end{equation} with $\varepsilon = 1 - (1+\delta)^{-1}.$
\end{proposition}
\begin{proof}
First, the continuity of $G$ and infinity of its right endpoint imply that for all $c>1$ there is $\tilde c>1, \tilde c = \tilde c(t),$ such that $c = u_G(\tilde c t)/u_G(t).$ From $C_\delta(H,G)$ it follows
\[u_H(t) \ge u_G(t) \frac{u_H\big(\tilde{c}^{1+\delta}t\big)}{u_G(\tilde ct)}.\] Using this relation, we derive
\begin{eqnarray*}\frac{\big(1 - H(c u_H(t))\big)^{1-\varepsilon}}{1 - G(c u_G(t))} & = & \frac{\left(1 - H\Big(\frac{u_G(\tilde c t)}{u_G(t)} u_H(t)\Big)\right)^{1-\varepsilon}}{1 - G(u_G(\tilde c t))}\\ &\le&
\frac{\left(1 - H\Big(\frac{u_G(\tilde c t)}{u_G(t)} \frac{u_H(\tilde{c}^{1+\delta}t)}{u_G(\tilde ct)} u_G(t)\Big)\right)^{1-\varepsilon}}{1 - G(u_G(\tilde c t))} \\
& = & \frac{\left(1 - H(u_H(\tilde{c}^{1+\delta}t))\right)^{1-\varepsilon}}{1 - G(u_G(\tilde c t))} = \frac{(\tilde{c}^{1+\delta}t)^{-(1-\varepsilon)}}{(\tilde c t)^{-1}} = t^\varepsilon \\ & = & \frac{(1 - H(u_H(t)))^{1-\varepsilon}}{1 - G(u_G(t))},\end{eqnarray*}
whence \eqref{B'} follows.
\end{proof}
Now we introduce the notion of $C$-separability.
\begin{definition} \label{def2}
Let $A_0$ and $A_1$ be two classes of distribution tails such that the tails belonging to the class $A_0$ are lighter than those from the class $A_1.$ We call $A_0$ and $A_1$ {\tt
$C$-separable} (from the right), if there exists a cdf $F_0$ such that for all $G\in A_0$ the condition $C_0(G, F_0)$ holds for some (maybe different) $t_0,$ and for all $H\in A_1$
the condition $C_\delta(F_0, H)$ holds for some (maybe different) $\delta>0$ and $t_0.$
If classes $A_0$ and $A_1$ are $C$-separable from the right via cdf $F_0,$ we call them {\tt $C(F_0)$-separable from the right.}
\end{definition}
Note that if we replace the $C_0$ and $C$-conditions with the $B$-condition in this definition, then we derive the definition of $B$-separability. If the distribution tails from $A_0$ are heavier then those from $A_1,$ then we say that these two classes are $C(F_0)$-separable (from the left), if for some cdf $F_0$ and all $G\in A_0$ the condition $C_0(F_0, G)$ holds for some (maybe different) $t_0,$ and for all $H\in A_1$ the condition $C_\delta(H, F_0)$ holds for some (maybe different) $\delta$ and $t_0.$
$C$-separability is a quite weak condition that is satisfied for a wide range of classes of distribution tails. Let us provide several examples of $C$-separable classes of distribution tails.
\begin{example}\label{E1} Let $A_0$ and $A_1$ be the classes of the tails of Weibull and log-Weibull type distributions, respectively (for definitions, see Section \ref{recent}). Assume that all distribution functions with tails from the classes $A_0$ and $A_1$ are eventually differentiable. To distinguish between these two classes we recommend to select the following cdf
\begin{equation}F_0(x) = (1 -
\exp\{-\exp\{b(\ln x)^{1/2}\}\}) I(x>1)\label{w-lw}\end{equation} for some constant $b>0.$ The proof of $C(F_0)$-separability from the right of these classes under some technical condition is given in Appendix.
The proof of their $C(F_0)$-separability from the left is similar. The finite sample properties of several tests for distinguishing between $A_0$ and $A_1$ are compared by numerical experiments in Section \ref{sim1}.
\end{example}
\begin{example}\label{E2} Let $A_0$ be the class of the tails of log-Weibull type distributions with $\theta<1,$ and $A_1$ be the class of regularly varying distribution tails. Recall that the Fr\'echet MDA consists of the distributions with regularly varying tails only, write $F\in \mathcal{D}(EV_\gamma),\ \gamma>0$ for such distribution functions. Assume again that all distribution functions with tails from the classes $A_0$ and $A_1$ are eventually differentiable. To distinguish between these two classes we recommend to select the following cdf
\begin{equation}\label{lw-rv} F_0(x) = 1 - \exp(-\exp(b\,\sqrt{\ln\ln x}) \ln x), \quad x>e,
\end{equation} for some constant $b>0.$
Indeed, the tails of log-Weibull type distributions with $\theta<1$ are lighter than the tail of $F_0,$ and the condition $C_0(G, F_0)$ for $G\in A_0$ follows from \eqref{deflogw} and the properties of regularly varying functions. On the other hand, the condition $C_\delta(F_0, H)$ for $H\in \mathcal{D}(EV_\gamma), \gamma>0,$ and $\delta>0$ follows from \eqref{lw-rv} and the definition of regularly varying distribution. We do not provide the detailed proof of these facts, because they are technical and in many respects similar to the proofs of corresponding facts in Example 1. Therefore, the $C(F_0)$-separability from the right condition holds in this case as well.
The comparison by numerical experiments of performance of tests for distinguishing between these two classes is provided in Section \ref{sim2}.
\end{example}
\begin{example}\label{E3} Let the class $A_0$ be such that $u_G(t) = \int_{1}^t f(x)/x\, dx$ for all $G\in A_0,$ where $f$ is positive and $f(x)\to \infty$ as $x\to\infty.$ It is clear that $G$ is heavy tailed. Next, let $A_1$ be the class of the tails of absolutely continuous Weibull type distributions with $\theta<1$. Then the classes $A_0$ and $A_1$ are $C(F_0)$-separable from the left by $F_0,$ the cdf of the standard exponential distribution. We omit the proof of this fact, since it is similar to the proof of Example 1.
\end{example}
\begin{example}\label{E4} Consider an one-parameter family of distributions
$F\in \{G_\theta, \theta\in\Theta\}$ such that for all $\theta_1,\theta_2\in \Theta,$ $\theta_1<\theta_2,$ the cdfs $G_{\theta_1}$ and $G_{\theta_2}$ satisfy the condition $C_\delta(G_{\theta_1}, G_{\theta_2})$ for $\delta>0.$
Then it is easy to see that the classes $A_0 = \{G_\theta,\ \theta< \theta_0\}$ and $A_1 = \{G_\theta,\ \theta> \theta_0\}$ are $C(G_{\theta_0})$-separable from the right. Moreover, $A_1$ and $A_0$ are $C(G_{\theta_0})$-separable from the left.
\end{example}
\subsection{Scale free test} \label{scale-free}
Assume that classes $A_0$ and $A_1$ are $C(F_0)$-separable from the right via some cdf $F_0$. In this section, we discuss the following test for distinguishing between the hypothesis $H_0: F\in A_0$ and the alternative $H_1: F\in A_1,$
\begin{equation}\text{if } \tilde{R}_{k,n} > 1+ \frac{u_{1-\alpha}}{\sqrt{k}}, \text{ then reject } H_0,\label{test}\end{equation}
where
\begin{equation}
\tilde{R}_{k,n} = \ln\frac{k}{n} -
\frac{1}{k}\sum_{i=n-k+1}^{n}\ln(1-F_0(u_0(n/k)X_{(i)}/X_{(n-k)})),
\label{rnk1} \end{equation} and $u_0(t) = u_{F_0}(t)$. Clear, the distribution of
$\tilde{R}_{k,n}$ does not depend on the scale parameters of both $F_0$ and $F,$ thus the test \eqref{test} is scale free. If for some cdf $F_0$ classes $A_0$ and $A_1$ are $C(F_0)$-separable from the left, then for testing $H_0: F\in A_0$ against $H_1:
F\in A_1$ we will use the following rule
\[\text{if } \tilde{R}_{k,n} < 1+ \frac{u_{\alpha}}{\sqrt{k}}, \text{ then reject } H_0.\]
In the following results the asymptotical properties of the test \eqref{test} are stated.
\begin{theorem} \label{T1} Let $X_1,\ldots, X_n$ be i.i.d. random variables with a cdf $F_0.$ Assume $F_0$
satisfies the von Mises condition of belonging to $\mathcal{D}(EV_\gamma),$ $\gamma\ge0,$
\begin{eqnarray} \label{mises}
\lim_{x \uparrow \infty} \frac{(1-F_0(x))F_0^{\prime\prime}(x)}{F_0^\prime(x)^2} = -\gamma-1.
\end{eqnarray}
Assume a sequence $k = k(n)$ is such that
\begin{equation}\label{kn} k\to\infty, k/n\to 0 \mbox{ as } n\to\infty.
\end{equation}
Then
\[
\sqrt{k}(\tilde{R}_{k,n} - 1) \stackrel{d}{\rightarrow} N(0,
1), \qquad n\to\infty.
\]
\end{theorem}
\begin{theorem} \label{T2} Let $X_1,\ldots, X_n$ be i.i.d. random variables with a cdf $F_1$. Assume $F_0$ satisfies \eqref{mises} and a sequence $k = k(n)$ satisfies \eqref{kn}. If the condition $C_\delta(F_0, F_1)$ holds for some $\delta>0$, then
$$\sqrt{k}(\tilde{R}_{k,n} - 1) \xrightarrow{P} +\infty, \qquad n\to\infty,$$
and if $C_\delta(F_1, F_0)$ holds for some $\delta>0$, then
$$\sqrt{k}(\tilde{R}_{k,n} - 1) \xrightarrow{P} -\infty, \qquad n\to\infty.$$
\end{theorem}
\begin{corollary}\label{C1}
Assume the assumptions of Theorem \ref{T2}. Let $X_1^0,\ldots, X_n^0$ be i.i.d. random variables with the cdf $F_0.$ Denote the values of the statistic \eqref{rnk1} built by samples $(X_1^0,\ldots, X_n^0)$ and $(X_1,\ldots, X_n)$ as $\tilde{R}_{k,n}^0$ and $\tilde{R}_{k,n}^1,$ respectively. If the condition $C_0(F_0, F_1)$ holds, then for all $x>0$
\[P(\tilde{R}_{k,n}^0 >x) \le P(\tilde{R}_{k,n}^1 >x);
\]
if $C_0(F_1, F_0)$ holds, then for all $x>0$
\[P(\tilde{R}_{k,n}^0 >x) \ge P(\tilde{R}_{k,n}^1 >x).
\]
\end{corollary}
Hence, by Theorem \ref{T1} and Corollary \ref{C1}, the test \eqref{test} has asymptotically the significance level $\alpha,$ and its consistency follows directly from Theorem \ref{T2}.
\begin{remark}\label{R1} Despite the fact that the test \eqref{test} has asymptotically the significance level $\alpha$ and is consistent on $H_1$ for every cdf $F_0$ satisfying \eqref{mises} and for which classes $A_0$ and $A_1$ are $C(F_0)$-separable, for finite $n$ the choice of $F_0$ matters. However, the problem of optimal selection of $F_0$ is beyond the scope of this paper. This remark is related also to the test proposed in the next section. However, some practical recommendations can be given, see Section \ref{pracrec}.
\end{remark}
\begin{remark}\label{R2}
As it follows from the results of this section, if classes $A_0$ and $A_1$ are $C(F_0)$-separable from the right and for all $G\in A_0$ the condition $C_\delta(G, F_0)$ holds for some positive $\delta$ (clear, $F_0$ does not belong to $A_0$ in this case), then the test \eqref{test} will be conservative. Thus, if $A_0$ has a natural ``left bound'' $\tilde F,$ i.e. for all $G\in A_0$ the condition $C_0(G, \tilde F)$ holds, but $C_\delta(G, \tilde F)$ does not necessary hold for any $\delta>0$, we recommend selecting $F_0 = \tilde F$ to maximize the power of the tests.
\end{remark}
\subsection{Location and scale free test} \label{location-free}
The test proposed in the previous section as well as the tests mentioned at the end of Section \ref{recent} are not invariant with respect to the location parameter, that can restrict sufficiently their application in some situations, see, e.g., Fig. \ref{fig:weibull2}. In this section, we aim at proposing the test for distinguishing between separable classes of distribution tails, that is invariant with respect to both the scale and location parameters. For this purpose, let us consider the following statistic
\begin{equation}
\widehat{R}_{k,n} = \ln\frac{k}{n} -
\frac{1}{k}\sum_{i=n-k+1}^{n}\ln\left[1-F_0\Big(u_0(n/k) + \frac{X_{(i)} - X_{(n-k)}}{X_{(n-k)} - X_{(n-2k)}}\big(u_0(n/k) - u_0(n/(2k))\big)\Big)\right].
\label{rnk2} \end{equation}
The asymptotic properties of $\widehat{R}_{k,n}$ are stated in the following results.
\begin{theorem}
\label{T3} Assume the assumptions of Theorem \ref{T1}. Then
\[
\sqrt{k}(\widehat{R}_{k,n} - 1) \stackrel{d}{\rightarrow} N\Big(0,
1 + \frac{\gamma^2}{2(\gamma+1)^2(2^\gamma-1)^2}\Big), \qquad n\to\infty,
\]
where the ratio on the right-hand side should be understood as $1/(2(\ln 2)^2)$ for $\gamma=0.$
\end{theorem}
\begin{theorem} \label{T4} Assume the assumptions of Theorem \ref{T2}. If the condition $C_\delta(F_0, F_1)$ holds for some $\delta>0$ then
$$\sqrt{k}(\widehat{R}_{k,n} - 1) \xrightarrow{P} +\infty,\qquad n\to\infty;$$ if $C_\delta(F_1, F_0)$ holds for some $\delta>0$ then
$$\sqrt{k}(\widehat{R}_{k,n} - 1) \xrightarrow{P} -\infty,\qquad n\to\infty.$$
\end{theorem}
\begin{corollary}\label{C2}
Assume the assumptions of Theorem \ref{T2}. Let $X_1^0,\ldots, X_n^0$ be i.i.d. random variables with the cdf $F_0.$ Denote the values of statistic \eqref{rnk2} built by samples $(X_1^0,\ldots, X_n^0)$ and $(X_1,\ldots, X_n)$ as $\widehat{R}_{k,n}^0$ and $\widehat{R}_{k,n}^1,$ respectively. If $C_0(F_0, F_1)$ holds then for all $x>0$
\[P(\widehat{R}_{k,n}^0 >x) \le P(\widehat{R}_{k,n}^1 >x);
\]
if $C_0(F_1, F_0)$ holds then for all $x>0$
\[P(\widehat{R}_{k,n}^0 >x) \ge P(\widehat{R}_{k,n}^1 >x).
\]
\end{corollary}
Denote for convenience
\[\sigma^2(\gamma) = 1 + \frac{\gamma^2}{2(\gamma+1)^2(2^\gamma-1)^2}, \quad \gamma>0,\] and set $\sigma^2(0) = 1 + 1/(2(\ln 2)^2).$ As in the previous section, consider two classes $A_0$ and $A_1$ of distribution tails such that tails from $A_0$ are lighter than those from $A_1,$ and assume that these classes are $C(F_0)$-separable from the right, where $F_0 \in \mathcal{D}(EV_\gamma).$ Let us propose the following test for the hypothesis $H_0: F\in A_0$
\begin{equation}\text{if } \widehat{R}_{k,n} > 1+ \frac{\sigma(\gamma) u_{1-\alpha}}{\sqrt{k}}, \text{ then reject } H_0.\label{test-location}\end{equation}
Clear, this test is location and scale free by the definition of $\widehat{R}_{k,n}.$ Moreover, by Theorem \ref{T3} and Corollary \ref{C2} the proposed test has asymptotically the significance level $\alpha,$ and its consistency on the alternative $H_1: F\in A_1$ follows from Theorem \ref{T4}.
The test for distinguishing between $A_0$ and $A_1$ in case of their $C(F_0)$-separability from the left can be proposed similarly to the previous section.
\begin{remark} \label{R3} Assume $A_0$ and $A_1$ are $C(F_0)$-separable from the right. Formally speaking, the tests \eqref{test} and \eqref{test-location} distinguish not the hypothesis $H_0: F\in A_0$ and alternative $H_1: F\in A_1,$ but the more wide hypothesis $\hat H_0: F\in \hat A_0$ and alternative $\hat H_1: F\in \hat A_1,$ where $\hat A_0$ consists of all (continuous) cdfs $G$ satisfying the condition $C_0(G, F_0)$, and $\hat A_1$ consists of all (continuous) cdfs $H$ satisfying $C(F_0, H).$
\end{remark}
\begin{remark} Notice that distributions belonging to either $A_0$ or $A_1$ (except the cdf $F_0$ in case $F_0\in A_0$) do not have to satisfy the assumptions of the extreme value theorem.
\end{remark}
\subsection{The choice of $k$} \label{kselection}
An important problem for practitioners that we have not addressed yet is how to choose the parameter $k$ optimally. Indeed, the number $k$ should not be too small since the more observations we use, the more accurate our conclusions are. On the other hand, it should not be too large because, roughly speaking, the tail may ``end'' earlier and our conclusions about tail behavior in this case will turn out to be incorrect.
As related to estimation of various parameters in the framework of statistics of extremes, there are a lot of papers in the literature devoted to the problem of optimal selection of $k.$ For instance, we refer to \cite{Dani} and Section 5.4, \cite{Gomes2}, for the review of such methods for tail index estimation and to \cite{Marrod} for those for extremal index estimation, respectively. These methods are most often based on either some empirical principles, like choosing the optimal value of estimator from some interval of stability of its values, or on optimization of some metrics (e.g., MSE), often with using bootstrap.
However, the methods of choosing $k$ proposed for some estimators in the framework of statistics of extremes are not suitable for this purpose when testing the hypotheses about distribution tails. Indeed, a test statistic quite often has a steady increasing or decreasing trend as $k$ increases, hence the stability interval method cannot be used, and almost never tends to a certain finite value, hence the method of optimizing some metric in the form that used for parameter estimation cannot be applied as well. As a rule, in papers devoted to testing hypotheses about distribution tails this question is omitted. Moreover, the most comprehensive review \cite{hueslerpeng} to date of methods for testing hypotheses about distribution tails suggests that the optimal selection of $k$ is an open problem.
Nevertheless, judging by our observations, some regularities in the behavior of test statistics can be distinguished. As a rule, the test for hypothesis about distribution tail can be represented as follows
\[ \mbox{ if } S(k) > u, \mbox{ then reject } H_0,\] where $S(k)$ is a statistic built by the $k$ largest order statistics of a sample, and $u$ is some constant independent of $k.$ For instance, all the tests considered in this article can be written in such a way. Moreover, if is often possible to ensure that the limit distribution of $S(k)$ does not depend on $k$ under null hypothesis (or for some ``boundary'' distributions, like in this article, see Theorems \ref{T1} and \ref{T3}), where $k$ is an intermediate sequence, i.e. it satisfies \eqref{kn}. Let us consider just such a case. Then the three most common types of $S(k)$ behavior are possible as $k$ increases, and the type of this behavior does not change on different samples from the same distribution. Below we provide recommendations for testing the null hypothesis for these three types of $S(k)$ behavior:
\begin{enumerate} \item $S(k)$ has a well expressed increasing trend, and its values get larger than $u$ almost immediately. In this case, $H_0$ should be rejected.
\item $S(k)$ has a well expressed decreasing trend, and its values get smaller than $u$ almost immediately. In this case, $H_0$ should not be rejected.
\item The values of $S(k)$ oscillate around $u$ for some interval of small values of $k.$ As a rule, there is no stability of the $S(k)$ values in this case. For this type of behavior the following method may be proposed: if the proportion of the $S(k)$ values on this interval, exceeding the $u$ level, is greater than the significance level, then $H_0$ should be rejected.
\end{enumerate}
Of course, these recommendations are empirical, and the problem of selection of $k$ while testing hypotheses about distribution tails deserves a separate study. Possible solutions to this problem may be, for instance, the use of multiple hypothesis testing or sequential analysis.
\section{Simulation study} \label{simulationstudy}
In this section, we examine the efficiency of the proposed tests in comparison with some tests proposed in the literature by numerical modelling. Before this, we provide some practical recommendation on selecting the parameters of $F_0$ for better test performance. We also provide an example of application of our methods on real data set. For the latter purpose, the data from the International base of longevity \cite{idl} is used,
which was analyzed, in particular, in the resonance works \cite{dong} and \cite{rootzen}.
\subsection{Practical recommendations on selecting the parameters of $F_0$}\label{pracrec}
Discussing the selection of $F_0$ in Remarks \ref{R1} and \ref{R2}, we noted that we do not provide the procedure for the optimal choice of $F_0$ (and, consequently, its parameters) in our article. Nevertheless, some practical recommendations can be given.
Note first, that the statistic of the scale and location free test does not depend on the choice of the scale and location parameters of $F_0,$ and the statistic of the scale free test does not depend on the choice of the scale parameter of $F_0,$ respectively, thus these parameters do not need to be chosen.
Next, assume classes $A_0$ and $A_1$ are $C(F_0)$-separable from the right. If $A_0$ has a natural ``left bound'', see Remark \ref{R2} for definition and, e.g., Examples \ref{E3}, \ref{E4}, then $F_0$ should be chosen equal to this bound. In this case the additional parameters do not need to be chosen as well.
If $A_0$ has not a natural ``left bound'', like in Examples \ref{E1} and \ref{E2}, it is enough to vary only one parameter to derive good quality of test procedure. For this aim we suggest to choose some ``basic'' distribution in $A_0$ quite close to $A_1$ (for example, in Section \ref{sim1} we choose the distribution $\mathcal{W}(2/3,1)$ for this purpose) and select the shape parameter of $F_0$ in such a way that the average empirical type I error probability would be less than the significance level, but close to it.
Finally, we note that using of the scale and location free test is preferable in case of distinguishing between distributions from the Gumbel MDA since the location parameter can strongly affect the tail behavior for small sample sizes (see Figure \ref{fig:weibull2}). It is also worth noting that, given a fixed null hypothesis and alternative, the same separating cdf $F_0$ can be used for different sample sizes.
\subsection{Distinguishing between Weibull and log-Weibull type di\-stri\-bu\-tion tails} \label{sim1}
As mentioned in Introduction, there is actually only one work, where the tests to distinguish between the Weibull type and log-Weibull type tails were proposed, we mean the work of Goegebeur and Guillou \cite{Goegebeur}. They introduced the Jackson-type and Lewis-type goodness-of-fit tests for Weibull type tail behavior.
We will compare the performance of the tests \eqref{test} and \eqref{test-location} adapted for testing the hypothesis $H_0: F\in \mathcal{W}$ against the alternative $H_1: F\in\mathcal{LW},$ where $\mathcal{W}$ and $\mathcal{LW}$ are the classes of Weibull type and log-Weibull type distribution tails, respectively, with the performance of the Jackson-type and Lewis-type tests from \cite{Goegebeur}. Note that according to Remark \ref{R3} we can include the distributions with tails heavier than those from $\mathcal{LW}$, in particular, regularly varying distributions, to the alternative hypothesis. As a separating cdf for the tests \eqref{test} and \eqref{test-location}, we select $F_0$ given by \eqref{w-lw} with $b$ equal to $1.8$ and $3.5,$ respectively. We examine the performance of the tests listed above on the following set of distributions, that mostly coincides with the set of distributions used in \cite{Goegebeur}. First, we list the Weibull type distributions:
\begin{itemize}
\item The Weibull distribution $\mathcal{W}(\alpha, \lambda)$
\[F(x) = 1 - \exp(-\lambda x^\alpha), \quad x>0; \alpha, \lambda>0.\]
We set $(\alpha, \lambda) = (2/3, 1).$
\item The standard normal distribution $\mathcal{N}(0,1).$
\item The gamma distribution $\Gamma(\alpha, \lambda)$ with density
\[p(x) = \frac{\lambda^\alpha x^{\alpha-1}}{\Gamma(\alpha)}e^{-\lambda x}, \quad x>0; \alpha, \lambda>0.\] We set $(\alpha, \lambda) = (0.25,1),\ (4,1).$
\item The modified Weibull distribution $\mathcal{MW}(\alpha, c),$ defined as the distribution of the random variable $Y = X \ln X,$ where $X\sim \mathcal{W}(\alpha, c).$ We set $\alpha = c=1.$
\item The extended Weibull model $\mathcal{EW}(\alpha, \beta)$
\[F(x) = 1 - r(x)e^{-x^\alpha},\quad x>0,\]
where $\alpha>0$ and $r(x)$ is regularly varying at infinity with index $\beta \in \mathbb{R}.$ We set $r(x) = (x+1)^{-1}$ and $\alpha = 2.$
\end{itemize}
Now we list the log-Weibull type distributions and distributions with heavier tails that we use in this section:
\begin{itemize}
\item The lognormal distribution $\mathcal{LN}(\mu, \sigma)$ with density \[p(x) = \frac{1}{\sqrt{2\pi\sigma^2}x} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right),\quad x>0,\, \mu\in \mathbb{R},\, \sigma^2>0.\] We set $\mu=0$ and $\sigma^2=1.$
\item The log-Weibull distribution $\mathcal{LW}(\theta, c)$ with cdf \[F(x) = 1 - \exp( - (\ln (x/c))^\theta), \quad x>c, \theta, c>0,\] set $(\theta,c) = (1.5,1).$
\item The general Pareto distribution $\mathcal{GPD}(\alpha, \sigma)$ with cdf \[F(x) = 1 - (1 + \gamma x/\sigma)^{-1/\gamma}, \quad x>0,\, \gamma, \sigma>0.\] We consider $(\gamma, \sigma) = (0.25, 1), (0.5, 1), (1, 1),$ further we denote them by $P_1,$ $P_2$ and $P_3,$ respectively.
\item The t-distribution $\mathcal{T}(n)$ with $n=3$ degrees of freedom.
\end{itemize}
For each distribution, we generated $1000$ samples of size $n = 5000$ (as it was chosen in \cite{Goegebeur}) and
computed the rejection rates of all tests for $k$ running from $10$ up to $1000$ in steps of $10.$
\begin{figure}[ht]
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{weibull_3.jpg} \\
{\small $\mathcal{W}(2/3,1)$}}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{normal_3.jpg} \\ {\small $\mathcal{N}(0,1)$}}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{gamma_25_3.jpg} \\ {\small $\Gamma(0.25, 1)$}}
\end{minipage}
\bigskip
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{gamma_4_3.jpg} \\
$\Gamma(4,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{modified_weibull_3.jpg} \\ $\mathcal{MW}(1,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{extended_weibull_1.jpg} \\
$\mathcal{EW}(2, -1)$}
\end{minipage}
\caption{{\footnotesize Empirical type I error probabilities of the tests \eqref{test} (blue dashed line), \eqref{test-location} (cyan solid line), Jackson-type (green dotted line) and Lewis-type (red dash-dotted line) tests for various distributions, the significance level $\alpha = 0.05$ is indicated by yellow solid line, $n=5000.$}}
\label{fig:weibull}
\end{figure}
\begin{figure}[ht]
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{lognormal_3.jpg} \\ $\mathcal{LN}(0,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{logweibull_3.jpg} \\ $\mathcal{LW}(1.5,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{pareto_25_3.jpg} \\ $\mathcal{GPD}(0.25,1)$}
\end{minipage}
\bigskip
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{pareto_5_3.jpg} \\
$\mathcal{GPD}(0.5,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{pareto_1_3.jpg} \\ $\mathcal{GPD}(1,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{student_3_3.jpg} \\
$\mathcal{T}(3)$}
\end{minipage}
\caption{{\footnotesize Empirical power of the tests \eqref{test} (blue dashed line), \eqref{test-location} (cyan solid line), Jackson-type (green dotted line) and Lewis-type (red dash-dotted line) tests for various distributions, the significance level $\alpha = 0.05$ is indicated by yellow solid line, $n=5000$.}}
\label{fig:logweibull}
\end{figure}
In Figs. \ref{fig:weibull}-\ref{fig:logweibull} for the six null and six alternative distributions we plot the empirical rejection rates of the scale free test \eqref{test} (dashed line), location and scale free test \eqref{test-location} (solid line), Jackson-type test (dotted line) and Lewis-type test (dash-dotted line), proposed in \cite{Goegebeur}, pointwise performed at the significance level 0.05, as functions of $k.$ We see, that the empirical rejection rates of only the test \eqref{test-location} are less than $0.05$ for all $k\le 1000$ and distributions from the null hypothesis. However, one can note, that for small values of $k,$ namely $k\le 250,$ the empirical I type error probabilities of all the compared tests, except \eqref{test}, are less than $0.05$ on distributions from $H_0.$ We will pay our attention on this interval of $k$ values. Then, one can see that the test \eqref{test-location} is more powerful than the Jackson-type and Lewis-type tests and the test \eqref{test} is more powerful than the Lewis-type test on the distributions from the alternative as $k\le 250.$
Analyzing Fig. \ref{fig:weibull}-\ref{fig:logweibull}, one may conclude that using the test \eqref{test-location} is always more preferable to the test \eqref{test}. The reason of using \eqref{test} instead of \eqref{test-location}, in particular, can be a small number of observations. It can also be reasonable to use \eqref{test} for distinguishing between quite heavy distribution tails (in particular, regularly varying), namely such that the location parameter does not greatly affect their behavior. In Fig. \ref{fig:logweibull2}, we provide the empirical rejection rates of all tests considered in this section for distributions $\mathcal{LN}(0,1),$ $\mathcal{GPD}(0.25, 1)$ and $\mathcal{T}(3).$ Here $n=1000,$ the significance level is $0.05$ and the number of samples is $m=1000.$ See, that the empirical power of \eqref{test} is substantially larger than the empirical power of other tests for moderate values of $k$.
\begin{figure}[ht]
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{lognormal_2.jpg} \\ $\mathcal{LN}(0,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{pareto_25_4.jpg} \\ $\mathcal{GPD}(0.25,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{student_3_4.jpg} \\ $\mathcal{T}(3)$}
\end{minipage}
\caption{{\footnotesize Empirical power of the tests \eqref{test} (blue dashed line), \eqref{test-location} (cyan solid line), the Jackson-type (green dotted line) and Lewis-type (red dash-dotted line) tests for various distributions, the significance level $\alpha = 0.05$ is indicated by yellow solid line, $n=1000.$}}
\label{fig:logweibull2}
\end{figure}
However, it turns out that the Jackson-type and Lewis-type tests as well as the test \eqref{test} are quite sensitive to the location parameter change. In Fig. \ref{fig:weibull2}, we provide the empirical rejection rates of all the tests considered in this section for three exponential distributions with density $p(x) = \exp(-(x-a)) I(x>a)$ and the values $a = -1, 0, 1$ of the location parameter, we set $n=5000,$ the significance level is $0.05$ and the number of samples is $m=1000.$ For $a=0$ the empirical type I error probabilities of all the tests are less than the significance level for all $k\le 1000$ ($k\le 800$ for the test \eqref{test}) whereas for $a = -1$ and $a=1$ the empirical type I error probabilities of all the tests except \eqref{test-location} increase substantially. Since even small changes of the location parameter cause the substantial changes in the statistical properties of the Jackson-type and Lewis type tests as well as the test \eqref{test}, then these tests should be applied in practice with great caution in the lack of information about the location parameter. We also note that the Jackson-type and Lewis-type tests cannot be applied for samples containing, in particular, only negative observations, that is additional difficulty when using them.
\begin{figure}[ht]
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{exp_-1_1.jpg} \\ $Exp(1,-1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{exp_0_1.jpg} \\ $Exp(1,0)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{exp_1_1.jpg} \\ $Exp(1,1)$}
\end{minipage}
\caption{{\footnotesize Empirical power of the tests \eqref{test} (blue dashed line), \eqref{test-location} (cyan solid line), the Jackson-type (green dotted line) and Lewis-type (red dash-dotted line) tests for exponential distributions with various values of the location parameter, the significance level $\alpha = 0.05$ is indicated by yellow solid line, $n=5000.$}}
\label{fig:weibull2}
\end{figure}
\subsection{Distinguishing between log-Weibull type and regularly varying di\-stri\-bution tails} \label{sim2}
In contrast to the previous section, the range of tests that can be used for the problem of this section is quite wide. In this section we consider the tests \eqref{test} and \eqref{test-location} adapted for distinguishing between the hypothesis $H_0: F\in \mathcal{LW}$ and alternative $H_1: F\in\mathcal{RV},$ where $\mathcal{RV}$ is the class of distributions with regularly varying tails that coincides with the Fr\'echet MDA $\mathcal{D}(EV_\gamma), \gamma>0$. We compare their efficiency with efficiency of some methods for testing the hypothesis $\tilde H_0: F\in \mathcal{D}(EV_0),$ i.e. the hypothesis of belonging to the Gumbel MDA, against the alternative $H_1,$ we refer to \cite{nevesfraga} for an overview of such methods. Namely, we use
\begin{enumerate}
\item the test proposed by Hasofer and Wang \cite{Hasofer-Wang}, see also \cite{Alves-Neves}, that is based on the Shapiro-Wilk type statistic
\[W_n(k) = \frac{k\big(\frac{1}{k}\sum_{i=0}^{k-1}X_{(n-i)} - X_{(n-k+1)}\big)^2}{(k-1)\sum_{j=0}^{k-1}\big(\frac{1}{k}\sum_{i=0}^{k-1}X_{(n-i)} - X_{(n-j)}\big)^2};\]
\item the test proposed in \cite{Alves-Neves} and based on the Greenwood type statistic $G_n(k),$ that is the modification of the previous test;
\item the ratio test proposed in \cite{Picek} and based on the statistic
\[R_n(k) = \frac{X_{(n)} - X_{(n-k)}}{\frac{1}{k}\sum_{i=0}^{k-1}X_{(n-i)} - X_{(n-k)}};\]
\item the likelihood ratio test, see e.g. \cite{Picek}.
\end{enumerate}
The numerical comparison of these tests can be found in \cite{Picek} and \cite{Alves-Neves}, see also \cite{nevesfraga}.
As a separating cdf for the tests \eqref{test} and \eqref{test-location} we select $F_0$ given by \eqref{lw-rv} with $b$ equal to $0.6$ and $1.1,$ respectively. We compare the efficiency of these tests on the following distributions (many of which were introduced in the previous section) belonging to the Gumbel MDA:
\begin{itemize}
\item the standard exponential distribution $Exp(1);$
\item the Weibull distribution $\mathcal{W}(2/3,1);$
\item the gamma distribution $\Gamma(0.25, 1);$
\item the modified Weibull distribution $\mathcal{MW}(1,1);$
\item the standard lognormal distribution $\mathcal{LN}(0,1);$
\item the log-Weibull distribution $\mathcal{LW}(\theta, c)$ with $(\theta, c) = (1.5, 1), (3, 1).$
\end{itemize}
and to the Fr\'echet MDA:
\begin{itemize}
\item the generalized Pareto distribution $\mathcal{GPD}(\alpha, \sigma)$ with $(\gamma, \sigma) = (1/3, 1),$\\ $(0.5, 1), (1, 1),$ further we denote them by $P_1,$ $P_2$ and $P_3,$ respectively.
\item the standard Cauchy distribution $\mathcal{C}(1);$
\item the t-distribution with 3 degrees of freedom $\mathcal{T}(3);$
\item The Burr distribution $Burr(c,d)$ with cdf
\[F(x) = 1 - \frac{1}{(1+ x^{-c})^d}, \quad x>0.\] We select $(c,d) = (2, 2),\, (3,2).$
\end{itemize}
For each distribution we generated $m=1000$ samples of size $n = 5000,$ and
computed the rejection rates of all tests for $k$ running from $10$ up to $1000$ in steps of $10.$
The empirical rejection rates of the Greenwood-type test were always a bit larger on the distributions used in this simulation study than the empirical rejection rates of the Hasofer-Wang test, hence we do not include the simulation results for the first test to the paper. By the same reason, we do not show the performance of the likelihood ratio test, which empirical properties are similar to those of the latter two tests.
\begin{figure}[ht]
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{exponential_4.jpg} \\
{\small $Exp(1)$}}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{weibull066_4.jpg} \\ {\small $\mathcal{W}(2/3,1)$}}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{gamma25_4.jpg} \\ {\small $\Gamma(0.25, 1)$}}
\end{minipage}
\bigskip
\begin{minipage}[h]{0.24\linewidth}
\center{\includegraphics[width=1\linewidth]{modified_weibull_4.jpg} \\
$\mathcal{MW}(1,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.24\linewidth}
\center{\includegraphics[width=1\linewidth]{lognormal_4.jpg} \\ $\mathcal{LN}(0,1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.24\linewidth}
\center{\includegraphics[width=1\linewidth]{logweibull15_4.jpg} \\
$\mathcal{LW}(1.5, 1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.24\linewidth}
\center{\includegraphics[width=1\linewidth]{logweibull3_4.jpg} \\
$\mathcal{LW}(3, 1)$}
\end{minipage}
\caption{{\footnotesize Empirical type I error probabilities of the tests \eqref{test} (blue dashed line), \eqref{test-location} (cyan solid line), the Hasofer-Wang (green dotted line) and ratio (red dash-dotted line) for various distributions, the significance level $\alpha = 0.05$ is indicated by yellow solid line, $n=5000.$}}
\label{fig:lwrv1}
\end{figure}
\begin{figure}[ht]
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{genpareto13_4.jpg} \\
{\small $\mathcal{GPD}(1/3,1)$}}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{genpareto12_4.jpg} \\ {\small $\mathcal{GPD}(0.5,1)$}}
\end{minipage}
\hfill
\begin{minipage}[h]{0.32\linewidth}
\center{\includegraphics[width=1\linewidth]{genpareto1_4.jpg} \\ {\small $\mathcal{GPD}(1,1)$}}
\end{minipage}
\bigskip
\begin{minipage}[h]{0.24\linewidth}
\center{\includegraphics[width=1\linewidth]{cauchy1_4.jpg} \\
$\mathcal{C}(1)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.24\linewidth}
\center{\includegraphics[width=1\linewidth]{student3_4.jpg} \\ $\mathcal{T}(3)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.24\linewidth}
\center{\includegraphics[width=1\linewidth]{burr22_4.jpg} \\
$Burr(2, 2)$}
\end{minipage}
\hfill
\begin{minipage}[h]{0.24\linewidth}
\center{\includegraphics[width=1\linewidth]{burr32_4.jpg} \\
$Burr(3, 2)$}
\end{minipage}
\caption{{\footnotesize Empirical power of the tests \eqref{test} (blue dashed line), \eqref{test-location} (cyan solid line), the Hasofer-Wang (green dotted line) and ratio (red dash-dotted line) tests for various distributions, the significance level $\alpha = 0.05$ is indicated by yellow solid line, $n=5000.$}}
\label{fig:lwrv2}
\end{figure}
In Figs. \ref{fig:lwrv1}–\ref{fig:lwrv2} for the seven null and seven alternative distributions we plot the empirical rejection rates of the scale free test \eqref{test} (dashed line), location and scale free test \eqref{test-location} (solid line), Hasofer-Wang test (dotted line) and ratio test (dash-dotted line), pointwise performed at the significance level $0.05,$ as functions of $k.$
Note, that the empirical type I error probabilities of the Hasofer-Wang and ratio tests become greater than the significance level already for quite small values of $k$ for all the considered distributions from the null hypothesis, except $Exp(1)$ and $\mathcal{LW}(3,1).$ On the other hand, the empirical type I error probabilities of the test \eqref{test-location} exceed the significance level only for $\mathcal{LN}(0,1)$ and $\mathcal{LW}(1.5,1).$ However, even for these two distributions, they are much less than those of the Hasofer-Wang and ratio tests. This fact confirms that the test \eqref{test-location} is preferable in practice. We also note that the test \eqref{test} is conservative for all $k<200$ on all the considered distributions from the null hypothesis. But one should not forget that this test is only one in our comparison that is not location free.
If one takes a look on the figures where the empirical power of the tests on the alternative is shown, then one can make sure that all the tests show quite good efficiency. We indicate only not the best behavior of the tests proposed in this article for distributions $\mathcal{T}(3)$ and $Burr(3,2).$
Summarizing the results of this section, we can conclude, that the test \eqref{test-location} outperforms other tests considered in this section by empirical properties. Indeed, on the null hypothesis, this test makes mistakes much less often and is not much less powerful on the alternative. Apparently, this means that tests for the hypothesis $\tilde H_0: F\in \mathcal{D}(EV_0)$ against the alternative $\tilde H_1: F\in \mathcal{D}(EV_\gamma),$ $\gamma>0,$ based on the tail heaviness are more efficient than tests based on the properties of the GEV and GPD models.
\subsection{Real data example: analyzing human lifespan using IDL data} \label{realdata}
Analysis of the human lifespan has been long the subject of research by many scientists, see e.g. \cite{einmahl}, \cite{belzile} and references therein. However, until now, they have not come to a general agreement on whether there is a limit to the human lifespan. In the recent work \cite{dong}, published in Nature, it is stated that ``the maximum lifespan of humans is fixed and subject to natural constraints''. Rootz\'en and Zholud \cite{rootzen} did not agree with the conclusions of the work \cite{dong}. Based on the methods of statistics of extremes and the data provided by the International Database of Longevity (IDL, \cite{idl}), they concluded that the human lifespan distributions in many large regions (in particular, North America and West Europe among others) belong to the Gumbel MDA and have an infinite right endpoint. Nevertheless, for some smaller regions, the results of applying the methods of extreme value theory may be different; for example, in \cite{einmahl} it was shown that the distribution of the human lifespan in the Netherlands belongs to the Weibull MDA and thus has a finite right endpoint.
Agreeing with the conclusions made in \cite{rootzen}, we however want to pay the reader's attention on Fig.5 (right) therein (see also Fig. \ref{figcent1} (left) in the present paper). In this figure, the empirical quantiles for IDL supercentenarians who died in 2000-2003 in USA (write the 2000-2003 US IDL data) and quantiles of the exponential distribution with mean estimated by the IDL lifespan data related to supercentenarians who died in 1980-1999 in USA are compared. This figure, in our opinion, reflects insufficient quality of approximation of this data by the exponential distribution, i.e. the special case of the GPD model at $\gamma=0,$ that is a clear illustration of conclusions made in Introduction. Below, we show that the approximation of the data ${\rm age}-110$ by the two-parameter Weibull distribution $\mathcal{W}(\alpha, \lambda)$ (for definition see Section \ref{sim1}) reflects the tail behavior of this data better. No doubts, it is not wondering since the exponential distribution is the special case of two-parameter Weibull distribution at $\alpha=1,$ but the point is that approximating the tails of all distributions from the Gumbel MDA by an exponential law can be inferior. It is also noted in \cite{rootzen} that the deviation of this data from exponential distribution may be caused by the specific sampling scheme, but the specific processing of these data does not allow to take the information about the sampling scheme into account.
Hence, since we agree that the 2000-2003 US IDL data distribution belongs to the Gumbel MDA, let us first use the tests considered in Section \ref{sim1} to find out which of the models: Weibull or log-Weibull type is suitable for describing this distribution. Note that for applying each of the four tests from Section \ref{sim1}, we should know $n,$ the number of deaths in USA in 2000-2003. One can find this information, in particular, on Wikipedia page \cite{wiki}, namely $n = 9771451.$ The number of supercentennarians in the 2000-2003 US IDL data is $k=504.$ The values of the scale free \eqref{test}, location and scale free \eqref{test-location}, Jackson-type and Lewis-type test statistics are $-10.64,\, -2.99,\, 1.5$ and $1.34,$ respectively, and the $p$-values of these tests are $1,\, 0.998,\, 0.132$ and $0.18,$ respectively. Thus, we can conclude that the data distribution is of Weibull type. Indeed, from Fig. 5 right \cite{rootzen} (see also Fig. \ref{figcent1}, left) the tail of the 2000-2003 US IDL data distribution is lighter than the tail of exponential distribution.
Next, consider the exceedances of the data observations over the level $110,$ $y = age - 110,$ and approximate them by the exponential distribution, as well as by the Weibull distribution $\mathcal{W}(\alpha, \lambda)$ and GPD model (in this case, we do not assume that $\gamma$ is necessarily equal to $0$), see Fig. \ref{figcent1}, left. Specifically, we plot the points $(x,y),$ where $x$ always corresponds to quantiles of exponential distribution with mean estimated by the data and $y$ corresponds to empirical data quantiles and quantiles of the Generalized Pareto and Weibull distributions of the same level as $x$ with parameters estimated by the data. It is easy to see that the last two distributions fit the data better, that is especially noticeable for the largest observations of the sample. Note that the maximum likelihood estimate of the shape parameter $\alpha$ of $\mathcal{W}(\alpha, \lambda)$ is $1.21,$ whereas the values of the Weibull tail index estimators proposed by \cite{Beirlant}, \cite{girard0} and \cite{Gardes}, which evidently should be less than $1$ in this case (for $\mathcal{W}(\alpha, \lambda)$ the Weibull tail index is equal to $1/\alpha$), are $6.54,$ $5.95$ and $6.06,$ respectively. Apparently, this is caused by the fact that these estimators are not invariant with respect to the location parameter, that convinces of the need to develop location and scale free methods for estimating the Weibull tail index.
The better approximation by the Weibull distribution in comparison with the exponential distribution is maintained for the IDL lifespan data related to supercentenarians who died in 2005-2010 in USA as well, see Fig. \ref{figcent1}, right; it is also seemed to be better than the approximation by the GPD model. It is interesting that the distribution tail of this data is even lighter than in the previous case. Indeed, the maximum likelihood estimate of the parameter $\alpha$ for $\mathcal{W}(\alpha, \lambda)$ is $1.42$ against $1.21$ in the previous case. The maximum likelihood estimate of the scale parameter of the exponential distribution is also changed from $1.51$ to $2.11,$ that perhaps indicates a change in the supercentenarian lifespan distribution.
\begin{figure}[ht]
\begin{minipage}[h]{0.48\linewidth}
\center{\includegraphics[width=1\linewidth]{rootsen_2000-2003_new.jpg} \\
{\small US supercentenarians in 2000-2003}}
\end{minipage}
\hfill
\begin{minipage}[h]{0.48\linewidth}
\center{\includegraphics[width=1\linewidth]{rootsen_2005-2010_new.jpg} \\ {\small US supercentenarians in 2005-2010}}
\end{minipage}
\caption{{\footnotesize Empirical quantiles for IDL supercentenarians who died in 2000–2003 (left) and in 2005-2010 (right) in USA (blue circles) plotted against quantiles of
exponential (yellow solid line), Generalized Pareto (magenta dash-dotted line) and Weibull (red dashed line) distributions with parameters estimated by sample.}}
\label{figcent1}
\end{figure}
\section{Appendix}
\subsection{Proof of Example 1}
First let us show that for all $H\in A_1$ the condition $C_\delta(F_0, H)$ holds for some $\delta>0$. We have $u_{F_0}(t) = \exp\big((\ln\ln t/b)^2\big).$ By its definition \eqref{deflogw}, the function $\ln (1 - H(e^t))$ is regularly varying at infinity with index $1/\theta,$ $\theta>0,$ thus $1 - H(t) = \exp\big(-(\ln t)^{1/\theta}\ell(\ln t)\big), t>1,$ for some slowly varying at infinity function $\ell(t).$ Therefore, by properties of slowly varying functions (see e.g. \cite{bingham}), there is a slowly varying function $\ell^{\sharp}(t)$ such that $u_{H}(t) = \exp\big((\ln t)^\theta\ell^\sharp(\ln t)\big).$
So, to check the condition $C_\delta(F_0, H),$ we will show that for some $\delta>0$ there is $t_0$ such that for all $t>t_0$ the ratio \[u_{F_0}(c^{1+\delta}t)/u_H(ct) = \exp\big((\ln\ln(c^{1+\delta}t)/b)^2 - (\ln(ct))^\theta \ell^\sharp(\ln (ct)) \big) = \exp(f(c, t))\] does not increase with respect to $c>1.$ For this purpose it is enough to show that the derivative with respect to $c$ of the function in the exponent on the right-hand side of the latter relation is negative for all $t>t_0$ and $c>1.$ Below we assume that the derivative of $\ell^\sharp$ is eventually monotone. Note that if the slowly varying function $\ell(t)$ is differentiable and its derivative is eventually monotone, then \begin{equation}\lim_{t\to\infty} t \ell^\prime(t)/\ell(t) = 0,\label{slowv}\end{equation} see e.g. Theorem 1.7.2, \cite{bingham}. Thus, for large $t_0$ we have
\begin{eqnarray}\nonumber&&\frac{\partial f(c,t)}{\partial c} = \frac{1}{c} \cdot \frac{2(1+\delta)}{b^2}\cdot\frac{\ln\ln(c^{1+\delta}t)}{\ln(c^{1+\delta}t)} - \frac{1}{c}\theta (\ln(ct))^{\theta-1} \ell^\sharp(\ln (ct))(1 + o(1))\\ \nonumber
&&< t \left(\frac{1}{ct} \cdot \frac{2(1+\delta)}{b^2}\cdot\frac{\ln\ln(ct)}{\ln(ct)} - \frac{1}{ct}\theta (\ln(ct))^{\theta-1} \ell^\sharp(\ln (ct))(1 + o(1))\right)\\ &&= t\,\tilde f^\prime(ct),\label{deriv}\end{eqnarray}
with \[\tilde f(x) = (1+\delta)((\ln\ln x)/b)^2 - (\ln x)^\theta \ell^\sharp(\ln x).\] It is easy to see that there is $x_0$ such that for all $x>x_0$ the derivative of $\tilde f(x)$ is negative. Selecting $t_0 = x_0,$ we finally derive by \eqref{deriv} that $\partial f(c,t)/\partial c<0$ for all $t>t_0$ and $c>1.$
Checking the condition $C_0(G, F_0)$ for $G\in A_0,$ where $A_0$ is the class of tails of Weibull type distributions, is much easier. Indeed, using properties of slowly varying functions and \eqref{defw}, we derive that $u_G(t) = (\ln t)^\theta \ell^\sharp(\ln t)$ for $\theta>0$ and some slowly varying $\ell^\sharp(t).$ Then we should show that the function
\[u_G(t)/u_{F_0}(t) = \exp\big(\theta\ln\ln t + \ln\ell^\sharp(\ln t) - ((\ln\ln t)/b)^2\big)\] does not increase eventually. Assuming again that the derivative of $\ell^\sharp$ is eventually monotone, we can show it by proving the fact that the derivative of the function in the exponent on the right-hand side of the latter relation is eventually negative. The latter fact follows immediately by using \eqref{slowv}.
\subsection{Proof of Theorem \ref{T1}}
We have,
\[\sqrt{k}(\tilde R_{k,n} - 1) = \sqrt{k}(\tilde R_{k,n} - R_{k,n}) + \sqrt{k}(R_{k,n} - 1).\]
By Theorem 1, \cite{R1},
\begin{equation}\label{old}
\sqrt{k}(R_{k,n} - 1) \xrightarrow{d} N(0,1).
\end{equation}
Thus, to complete the proof, it is enough to show that
\[\sqrt{k}(\tilde R_{k,n} - R_{k,n}) \xrightarrow{P} 0.\]
To prove this, we use the Chebyshev inequality for $\sqrt{k}(\tilde R_{k,n} - R_{k,n})$ given $X_{(n-k)} = q$ and full probability formula. For this purpose, we evaluate asymptotic of $\mathbb{E}(\tilde R_{k,n} - R_{k,n})$ and ${\rm Var}(\tilde R_{k,n} - R_{k,n})$ given $X_{(n-k)}.$\\
\\
{\bf Explicit form of $\mathbb{E}(\tilde R_{k,n} - R_{k,n})$ given $X_{(n-k)} = q.$}\\
\\
We have,
\begin{eqnarray*}
\tilde{R}_{k,n} - R_{k,n} &=&
\left(\ln\frac{k}{n} - \ln(1 - F_0(X_{(n-k)})) \right)\\&& - \frac{1}{k}\sum_{i=n-k+1}^{n}\ln\frac{1-F_0(u_0(n/k) X_{(i)}/X_{(n-k)})}{1
- F_0(X_{(i)})}.
\end{eqnarray*}
Consider the conditional distribution of $\tilde{R}_{k,n} - R_{k,n}$ given $X_{(n-k)}=q.$ By Lemma 3.4.1, \cite{dehaan}, the joint conditional distribution
of the set of order statistics $\{X_{(i)}\}_{i=n-k+1}^{n}$ given $X_{(n-k)}=q$ coincides with the (unconditional) joint distribution of the set of order statistics $\{X_{(i)}^{\ast}\}_{i=1}^{k}$ of i.i.d. random variables $\{X_{i}^{\ast}\}_{i=1}^{k}$ with common cdf
\begin{equation*}
F_{q}(x)=P(X\leq x|X>q)=\frac{F_0(x)-F(q)}{1-F_0(q)},\ \ \ \
x>q.\end{equation*} Thus, given $X_{(n-k)}=q$,
\begin{eqnarray}\tilde{R}_{k,n} - R_{k,n} \stackrel{d}{=}
\left(\ln\frac{k}{n} -
\ln(1 - F_0(q))\right) -
\frac{1}{k}\sum_{i=1}^{k}\ln\frac{1-F_0(u_0(n/k) X^\ast_{i}/q)}{1
- F_0(X^\ast_{i})}. \;\; \label{R}\end{eqnarray} Hereinafter we write $u_0$ instead of $u_0(n/k)$ for convenience. For $Y^\ast :=
\ln(1-F_0(u_0 X^\ast_{1}/q)) - \ln(1 - F_0(X^\ast_{1})),$ after direct, but tedious calculations, we derive
\begin{equation}\mathbb{E}Y^\ast = \ln\frac{1-F_0(u_0)}{1-F_0(q)} - (f(q) - 1),\label{E}\end{equation}
where \begin{equation}f(q) =
\int_{q}^{\infty}\frac{u_0}{q}\frac{1-F_0(x)}{1-F_0(q)}\frac{p_0(u_0x/q)}{1-F_0(u_0x/q)}dx,\label{f}\end{equation}
and $p_0(t)$ denotes a pdf of $F_0.$
This and \eqref{R} imply that \[\mathbb{E}(\tilde{R}_{k,n} -
R_{k,n}|X_{(n-k)} = q) = f(q) - f(u_0) = f(q) - 1.\]
\\
{\bf Asymptotic of $\sqrt{k}\mathbb{E}(\tilde R_{k,n} - R_{k,n}|X_{(n-k)}).$}\\
\\
Next, show that
\begin{equation}\sqrt{k}\mathbb{E}(\tilde R_{k,n} - R_{k,n}|X_{(n-k)}) = \sqrt{k}(f(X_{(n-k)}) - f(u_0))\xrightarrow{P}
0.\label{consist}\end{equation} As long as $F_0(u_0(x)) = 1 - 1/x,$
we have $p_0(u_0(x))u_0^\prime(x) = x^{-2}$ and $(xu^\prime_0(x))^{-1} = x
p_0(u_0(x)).$ Then, Lemma 2.2.1, \cite{dehaan}, implies
\begin{equation}\frac{n}{\sqrt{k}}p_0(u_0)(X_{(n-k)} -
u_0)\stackrel{d}{\rightarrow} N(0,1).\label{delta}\end{equation}
Applying the delta method to the latter relation, we derive
\begin{equation}\frac{n}{\sqrt{k}}p_0(u_0)\frac{f(X_{(n-k)}) -
f(u_0)}{f^{\prime}(u_0)}\xrightarrow{d} N(0,1).\label{delta1}\end{equation}
Generally speaking, this is not the classical delta method, since $u_0$ is not a constant and so the latter relation should be established. By the mean value theorem, we have $f(X_{(n-k)}) - f(u_0) = f^\prime(c)(X_{(n-k)} - u_0)$ a.s., where $c$ is some (random) value between $X_{(n-k)}$ and $u_0.$ Thus it is enough to show that $f^\prime(c)/f^\prime(u_0) \xrightarrow{P} 1.$ The proof of this fact uses the explicit form of $f^\prime$ provided below and is based on the multiple application of the von Mises condition \eqref{mises} and \eqref{delta}; it is very technical and is not of special interest. Thus here and in similar situations below we omit such technical details.
Next, calculating directly $f^\prime(q)$ and substituting $q = u_0,$ we derive
\begin{equation*}
f^\prime(u_0) = \frac{n}{k}\bigg( p_0(u_0) - \int_{u_0}^\infty\frac{x}{u_0}\frac{p^2_0(x)}{1 -
F_0(x)}dx \bigg).
\end{equation*}
For absolutely continuous distributions with infinite right endpoint belonging to $\mathcal{D}(EV_\gamma),$ it holds
\begin{equation}\frac{1 - F_0(u)}{up_0(u)} \to \gamma, \quad u\to\infty,\label{easy}\end{equation}
see, e.g., Theorem 1.1.11, \cite{dehaan}, for $\gamma>0$ and \cite{dehaan}, p. 18, for $\gamma=0.$
Using the latter, L'Hospital rule and von Mises condition \eqref{mises}, we get
\begin{eqnarray}\nonumber \lim_{u\to\infty} \frac{u p_0(u)}{\int_u^\infty x \frac{p^2_0(x)}{1 -
F_0(x)}dx} = -\lim_{u\rightarrow \infty} \frac{up^\prime_0(u) + p_0(u)}{u p^2_0(u)/(1 -
F_0(u))}\\ = -\lim_{u\rightarrow \infty} \frac{p^\prime_0(u)(1-F_0(u))}{(p_0(u))^2} - \lim_{u\rightarrow \infty} \frac{1-F_0(u)}{up_0(u)} = 1,\label{lopital}\end{eqnarray}
whence \[\frac{k}{n p_0(u_0)} f^\prime(u_0) \xrightarrow{P} 0.\] Combining the latter and \eqref{delta1}, we derive \eqref{consist}.\\
\\
{\bf Asymptotic of ${\rm Var}(\tilde R_{k,n} - R_{k,n}|X_{(n-k)}).$}
\\
\\
First, given $X_{(n-k)} = q$ we have,
\[{\rm Var}(\tilde{R}_{k,n} - R_{k,n}) =
\frac{1}{k}{\rm Var}\left(\ln \frac{k}{n} - \ln(1-F_0(q)) -
Y^\ast\right) = \frac{1}{k}{\rm Var} Y^\ast.\]
Integrating by parts, we derive for the second moment of $Y^\ast$
\begin{eqnarray*}\mathbb{E}(Y^\ast)^2 = \int_{q}^{\infty}
\left(\ln\frac{1-F_0(u_0x/q)}{1-F_0(x)}\right)^2
d\left(\frac{F_0(x)-F_0(q)}{1-F_0(q)}\right) = \left(\ln \frac{1 -
F_0(q)}{1 - F_0(u_0)}\right)^2 \\ + 2\int_q^\infty
\ln\frac{1-F_0(u_0x/q)}{1-F_0(x)} \left(\frac{p_0(x)}{1 - F_0(x)} -
\frac{u_0}{q}\frac{p_0(u_0x/q)}{1-F_0(u_0x/q)}\right) \frac{1 -
F_0(x)}{1-F_0(q)}dx \\
=: \left(\ln \frac{1 -
F_0(q)}{1 - F_0(u_0)}\right)^2 + 2 h(q).
\end{eqnarray*}
From the latter and \eqref{E} it follows
\[\mathbb{E}(Y^\ast)^2 - (\mathbb{E}Y^\ast)^2 = 2h(q) - 2 (f(q)-1)\ln \frac{1 -
F_0(q)}{1 - F_0(u_0)} - (f(q)-1)^2 =: V(q).\]
Fix $\delta:$
$0<\delta<1/2.$ Let us show that \begin{equation}\label{asvar} k^{2\delta}V(X_{(n-k)}) \xrightarrow{P} 0.\end{equation} Note separately that $V(X_{(n-k)}) = k {\rm Var}(\tilde R_{k,n} - R_{k,n}|X_{(n-k)}).$
Let $E_1, \ldots, E_n$ be i.i.d. standard exponential and $E_{(1)}\le E_{(2)} \le \ldots \le E_{(n)}$ be their order statistics. By continuity of $F_0,$ we get $-\ln(1 - F_0(X_{(n-k)})) \stackrel{d}{=} E_{(n-k)},$ thus
\begin{eqnarray*}
-\sqrt{k}\ln \left(\frac{1 - F_0(X_{(n-k)})}{1-F_0(u_0)}\right) \stackrel{d}{=} \sqrt{k} \big(E_{(n-k)} - \ln(n/k)\big) \xrightarrow{d} N(0,1),\end{eqnarray*} by Lemma 2.2.1, \cite{dehaan}.
From the latter, \eqref{consist} and the relation $f(u_0)=1$ it follows
\[k^{2\delta} \left(2(f(X_{(n-k)})-1)\ln \frac{1 - F_0(X_{(n-k)})}{1 - F_0(u_0)} +
(f(X_{(n-k)})-1)^2\right) \xrightarrow{P} 0.\] Thus, to prove \eqref{asvar} it remains to show $k^{2\delta}h(X_{(n-k)}) \xrightarrow{P} 0.$ For this purpose we apply the delta method again. In this situation we cannot use the version of the delta method which we applied above since $h^\prime(u_0) = 0,$ thus we aim at finding the second derivative of
$h(q)$ and substituting $q=u_0.$ Omitting tedious calculations, we derive \[h^{\prime\prime}(u_0) =
\frac{1}{u_0^2(1 - F_0(u_0))}\int_{u_0}^\infty \frac{x^2p^3_0(x)}{(1 - F_0(x))^2}dx.\]
Note also that $h(u_0) = 0.$ From \eqref{delta} and the relation $1 - F_0(u_0) = k/n$
we obtain \begin{eqnarray}\frac{n^2}{k}
\frac{p_0^2(u_0)(h(X_{(n-k)})-h(u_0))}{2h^{\prime\prime}(u_0)} =
\frac{n p_0^2(u_0) h(X_{(n-k)})}{\int_{u_0}^\infty\frac{2x^2}{u_0^2}\frac{p_0^3(x)}{(1-F_0(x))^2}dx} \xrightarrow{d}
\xi^2,\label{almost_last}\end{eqnarray} where $\xi\sim N(0,1)$ and the asymptotics of the integral in the denominator follows from the L'Hospital rule, von Mises condition \eqref{mises} and \eqref{easy},
\begin{eqnarray*}\lim_{u\to\infty}\frac{u^2 p_0^2(u)/(1-F_0(u))}{\int_{u}^\infty 2x^2\frac{p_0^3(x)}{(1-F_0(x))^2}dx} = \frac{1}{2}.\end{eqnarray*}
Substituting the latter to \eqref{almost_last} and using the Slutsky theorem, we derive
\[ k h(X_{(n-k)}) \xrightarrow{d}
\xi^2.\] Therefore, for $\delta\in(0, 1/2)$ it holds $k^{2\delta} h(X_{(n-k)}) \xrightarrow{P} 0,$
that proves \eqref{asvar}.
\\
\\
{\bf Application of the Chebyshev inequality.}
\\
\\
To complete the proof, show
\begin{equation}\sqrt{k}(\tilde{R}_{k,n} - R_{k,n}) - \sqrt{k}(f(X_{(n-k)}) -
f(u_0)) \xrightarrow{P} 0,\label{P}\end{equation} that, together with \eqref{consist}, implies the theorem. By the Chebyshev inequality, given
$X_{(n-k)} = q,$ for all $\varepsilon>0$ we get
\begin{equation*}P\big(\sqrt{k}\big|(\tilde{R}_{k,n} - R_{k,n}) - (f(q) -
f(u_0))\big|>\varepsilon\big)\leq \frac{k {\rm Var}(\tilde{R}_{k,n} -
R_{k,n})}{\varepsilon^2}.\end{equation*} By the latter and the full probability formula, we finally derive
\begin{eqnarray*}P\left(\sqrt{k}|(\tilde{R}_{k,n} - R_{k,n}) - (f(X_{(n-k)}) -
1)|>k^{\delta}\sqrt{V(X_{(n-k)})}\right) =\\
\int_\mathbb{R} P\left(\left.\sqrt{k}|(\tilde{R}_{k,n} - R_{k,n}) -
(f(X_{(n-k)}) - 1)|>k^{\delta}\sqrt{V(X_{(n-k)})}\right| X_{(n-k)} =
q\right) \times \\
\times p_{X_{(n-k)}}(q) dq
\leq
\int_\mathbb{R} k^{-2\delta} p_{X_{(n-k)}}(q) dq = k^{-2\delta} \to 0,
\end{eqnarray*}
where $p_{X_{(n-k)}}(q)$ is the pdf of
$X_{(n-k)}.$ The result follows from \eqref{asvar}.
\subsection{Proof of Theorem \ref{T2}}
To prove Theorem \ref{T2}, we need the following result.
\begin{lemma}\label{L1} Let $X_1^0, \ldots, X_n^0$ be i.i.d. random variables with common cdf $F_0.$ Then under the conditions of Theorem \ref{T2}
\begin{equation}\label{appr}
\frac{1-F_{0}(c X^0_{(n-k)})}{1-F_{0}(X^0_{(n-k)})} - \frac{1-F_{0}(c u_0)}{1-F_{0}(u_0)} = O_P(1/\sqrt{k})
\end{equation} uniformly in $c>1.$
\end{lemma}
\begin{proof}
Applying the delta method for the function $f(x) = (1 - F_0(cx))/(1 - F_0(x))$ to the relation \eqref{delta}, we derive
\[\frac{n}{\sqrt{k}} p_0(u_0)\frac{f(X^0_{(n-k)}) - f(u_0)}{f^\prime(u_0)} \xrightarrow{d} N(0,1).\] Thus to prove \eqref{appr} it is enough to show that
\begin{equation}\frac{\sqrt{k}}{n}\frac{f^\prime(u_0)}{p_0(u_0)} = O(1/\sqrt{k})\label{appr2}\end{equation} uniformly in $c>1.$
As long as
\[f^\prime(u_0) = \left.\left(\frac{1 - F_0(cx)}{1 - F_0(x)}\right)^\prime\right|_{x=u_0} = \frac{p_0(u_0)(1 - F_0(c u_0)) - cp_0(cu_0) (1 - F_0(u_0))}{(1 - F_0(u_0))^2},\]
then using $1 - F_0(u_0) = k/n$ we can simplify the relation \eqref{appr2} as follows
\begin{equation}\label{fdens} \frac{1 - F_0(c u_0)}{1 - F_0(u_0)} - c\frac{p_0(c u_0)}{p_0(u_0)} = O(1)
\end{equation}
uniformly in $c>1.$ Clear, $\overline{F}_0(cu_0)/\overline{F}_0(u_0) \le 1$ for all $c>1.$ If $F_0\in \mathcal{D}(G_\gamma)$ with $\gamma>0,$ boundness from above of the ratio $cp_0(c u_0)/p_0(u_0)$ follows from the Potter inequality, see, e.g., Proposition B.1.9, \cite{dehaan}. If $\gamma=0,$ then the required fact follows from the relation (1.1.33), \cite{dehaan}, that holds under \eqref{mises}, relation $p_0(u_0(t)) u_0^\prime(t) = t^{-2}$ and Potter inequality again. Hence, the relation \eqref{fdens} and then the relation \eqref{appr} hold.
$\Box$ \end{proof}
The proof of Theorem \ref{T2} repeats the proof of Theorem 2 (i) \cite{R1} in many respects. Namely, we will show that under the condition $C_\delta(F_0, F_1),$ $\delta>0,$ \[\sqrt{k}(\tilde{R}_{k,n}-1)\gg\sqrt{k}\left( \frac{1}{k}\sum
_{i=1}^{k}E_{i}-1\right)\] for some i.i.d. random variables $\{E_i\}_{i\ge 1}$ with mean more than 1, and the result will follow from the central limit theorem. Here $X\gg Y$ means that $X$ is stochastically larger than $Y.$ The proof remains almost the same if $C_\delta(F_1, F_0)$ holds with $\delta>0$ instead of $C_\delta(F_0, F_1)$.\\
\\
Let $X_1, \dots, X_n$ be i.i.d. random variables with common cdf $F_1.$ Consider the asymptotic behavior of the statistic
$\tilde{R}_{k,n}$ as $n\rightarrow\infty.$ Denote $Y_{i}= \ln
(k/n) - \ln(1-F_0(u_0X^\ast_{i}/q)),$ $i=1,\ldots, k,$ where
$\{X_{i}^{\ast}\}_{i=1}^{k}$ are i.i.d. random variables defined in the same way as in Theorem \ref{T1}, with common cdf
\[
F_{q}^1(x)=\frac{F_{1}(x)-F_{1}(q)}{1-F_{1}(q)},\ \ q<x.
\]
By Lemma 3.4.1, \cite{dehaan}, the joint distribution of the set of order statistics $\{Y_{(i)}\}_{i=1}^{k}$
of $\{Y_{j}\}_{i=1}^{k}$ coincides with the conditional joint distribution of the set of random variables
$\{Z_{j}\}_{j=1}^{k}$ given $X_{(n-k)}=q,$ with
\[
Z_{j}= \ln (k/n) - \ln(1-F_{0}(u_0 X_{(n-j+1)}/X_{(n-k)})),\
j=1,...,k.
\]
Clear, $\tilde{R}_{k,n}=\frac{1}{k}\sum_{j=1}^{k}Z_{j}.$
Thus, the conditional distribution of $\tilde{R}_{k,n}$ given
$X_{(n-k)}=q$ coincides with the distribution of
$\frac{1}{k}\sum_{i=1}^{k}Y_{i}.$
Therefore, we get
\begin{eqnarray} \nonumber P(Y_1\leq x) = P(\ln(1 - F_0(u_0 X^\ast_1/q))\geq \ln(k/n) - x) = \\
P\big(X_1^\ast\leq q u_0^{-1} F_0^\leftarrow(1 - ke^{-x}/n)\big) =
\frac{F_1\big(q u_0^{-1} F_0^\leftarrow(1 - ke^{-x}/n)\big) - F_1(q)}{1 -
F_1(q)}. \label{T2main}
\end{eqnarray
Next, cdfs
$F_{1}$ and $F_{0}$ satisfy either the condition $C_\delta(F_{0},F_{1})$ or
$C_\delta(F_{1},F_{0})$ with $\delta>0$ by assumptions. Assume that $C_\delta(F_{0},F_{1})$ holds with some
$\delta>0$ and $t_{0},$ another case can be treated in a similar way.
As long as $x^{\ast}=+\infty,$ $X_{(n-k)}\rightarrow+\infty$ almost surely, thus we can deal only with the case $n/k>t_{0}.$ Denote the quantile function of $F_1$ by $u_1(t)$ and write hereinafter $u_1$ instead of $u_1(n/k).$ By Proposition 1, we get
\begin{equation}
\frac{1-F_{1}(c u_1)}{1-F_{1}(u_1)}\geq\frac{(1-F_{0}(c u_0))^{1-\varepsilon}%
}{(1-F_{0}(u_0))^{1-\varepsilon}},\ \ c>1,
\label{ineq}\end{equation} for some $\varepsilon>0.$
Denote $r = u_1^\leftarrow(q).$ Note also that if $X_{(n-k)} = u_1(\xi)$ for some random $\xi,$ then $u_0(\xi) \stackrel{d}{=} X_{(n-k)}^0$ with $\{X_i^0\}_{i=1}^n$ introduced in Lemma \ref{L1}. Proceed \eqref{T2main}, using \eqref{ineq} with $c = u_0^{-1} F_0^\leftarrow(1- ke^{-x}/n)$ and \eqref{appr}. Uniformly in $x>0,$ we have
\begin{eqnarray*}P(Y_1\leq x) & = & 1 - \frac{1 - F_1\big(u_1(r) u_0^{-1} F_0^\leftarrow(1
- ke^{-x}/n)\big)}{1 - F_1(u_1(r))} \\
& \leq &
1 - \frac{\big(1 - F_0(u_0(r) u_0^{-1} F_0^\leftarrow(1 -
ke^{-x}/n))\big)^{1-\varepsilon}}{\big(1 -
F_0(u_0(r))\big)^{1-\varepsilon}}\\
& = & 1 - \left(\frac{1 - F_0(F_0^\leftarrow(1 -
ke^{-x}/n))}{1 -
F_0(u_0)} + O(1/\sqrt{k})\right)^{1-\varepsilon}\\
& = & 1 - \left(e^{-x} + O(1/\sqrt{k})\right)^{1-\varepsilon} = 1 - e^{-(1-\varepsilon) x} + O(1/\sqrt{k}).
\end{eqnarray*}
Since $O(1/\sqrt{k})$ vanishes, then for $k$ large enough there exists a cdf $G(x)$ such that
\[ G(x) \ge \min(1 - e^{-(1-\varepsilon) x} + O(1/\sqrt{k}), 1) \] with mean $(1 - \varepsilon_1)^{-1},$ $\varepsilon_1\in (0, \varepsilon),$ and variance $\sigma^2>0.$ Hence, $Y_{1}$ is stochastically larger than a random variable $E$ with cdf
$G.$ Next, let
$E_{1},\ldots,E_{k}$ be i.i.d. random variables with common cdf $G,$ then
\begin{equation}
\sqrt{k}\left( \frac{1}{k}\sum_{i=1}^{k}Y_{i}-1\right) \gg
\sqrt{k}\left( \frac{1}{k}\sum_{i=1}^{k}E_{i}-1\right)
.\label{ll}%
\end{equation}
As long as \eqref{ll} holds for all $q>x_{0},$
\begin{equation}
\sqrt{k}(\tilde{R}_{k,n}-1)\gg\sqrt{k}\left( \frac{1}{k}\sum
_{i=1}^{k}E_{i}-1\right) .\label{LL}%
\end{equation}
By the central limit theorem,
\[
\sqrt{k}\left( \frac{1}{k}\sum_{i=1}^{k}%
E_{i}-\frac{1}{1-\varepsilon_1}\right) \xrightarrow{d} N(0,\sigma^2),
\]
thus \[\sqrt{k}\left( \frac{1}{k}\sum_{i=1}^{k}E_{i}-1\right)
\xrightarrow{P}+\infty,\] that with \eqref{LL} completes the proof of
Theorem \ref{T2}.
\subsection{Proof of Corollary \ref{C1}}
Let us assume that the condition $C_0(F_0, F_1)$ holds, the proof under the condition $C_0(F_1, F_0)$ is completely similar. The condition $C_0(F_0, F_1)$ immediately implies that
\begin{equation}\label{fromc0}\frac{u_0(ct)}{u_0(t)} \le \frac{u_1(ct)}{u_1(t)}\end{equation} for all $c\ge 1$ and $t\ge t_0.$ As mentioned in the proof of Theorem \ref{T2}, under the assumptions of Corollary \ref{C1} it holds $X^0_1 \stackrel{d}{=} u_0(u_1^\leftarrow(X_{1})) =: V_{1}.$ Hence substituting $t = u_1^{\leftarrow}(X_{(n-k)})$ and $c = u_1^{\leftarrow}(X_{(i)})/u_1^{\leftarrow}(X_{(n-k)})$ in \eqref{fromc0}, we get for all $i\in \{n-k+1, \ldots, n\}$
\begin{equation} \label{v}\frac{V_{(i)}}{V_{(n-k)}} = \frac{u_0(u_1^{\leftarrow}(X_{(i)}))}{u_0(u_1^{\leftarrow}(X_{(n-k)}))}
\le \frac{u_1(u_1^{\leftarrow}(X_{(i)}))}{u_1(u_1^{\leftarrow}(X_{(n-k)}))} = \frac{X_{(i)}}{X_{(n-k)}},
\end{equation} where $V_{(1)} \le \ldots \le V_{(n)}$ are the order statistics of i.i.d. random variables $\{V_i\}_{i=1}^n,$ $V_i = u_0(u_1^\leftarrow(X_{i}))$ for $i\in \{1, \ldots, n\}.$ Since $(V_{(1)}, \ldots, V_{(n)}) \stackrel{d}{=}\\ (X_{(1)}^0, \ldots, X_{(n)}^0),$ then it follows from \eqref{v} that
\begin{eqnarray*}\tilde R^0_{k,n} &\stackrel{d}{=}& \ln\frac{k}{n} -
\frac{1}{k}\sum_{i=n-k+1}^{n}\ln(1-F_0(u_0V_{(i)}/V_{(n-k)}))\\ &\le& \ln\frac{k}{n} -
\frac{1}{k}\sum_{i=n-k+1}^{n}\ln(1-F_0(u_0X_{(i)}/X_{(n-k)})) = \tilde R_{k,n},\end{eqnarray*} the result follows.
\subsection{Proof of Theorem \ref{T3}}
To prove Theorem \ref{T3}, we need the following result.
\begin{lemma}\label{L2} Let $E_1, \ldots, E_n$ be i.i.d. standard exponential with $E_{(1)}\le \ldots\le E_{(n)},$ their order statistics. Then
\begin{equation}\label{2exp}
\frac{n}{\sqrt{k}}\left(\left(\begin{array}{c}E_{(k)} \\ E_{(2k)}\end{array}\right) +
\left(\begin{array}{c} \ln(1 - k/n) \\ \ln(1 - 2k/n) \end{array}\right)\right) \xrightarrow{d} N\left(\Big(\begin{array}{c}0 \\ 0\end{array}\Big), \Big(\begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array}\Big)\right),
\end{equation} if $k\to\infty,$ $k/n\to 0$ as $n\to\infty.$
\end{lemma}
\begin{proof}
The result of the lemma is almost the direct consequence of the following two-dimensional extension of Smirnov lemma \cite{smirnov}, see also Lemma 2.2.3, \cite{dehaan}. Let $U_{(1)} \le \ldots \le U_{(n)}$ be the $n$th order
statistics from the standard uniform distribution. Then, if $k\to\infty,$ $k/n\to 0$ as $n\to\infty,$ the following relation holds
\begin{equation}\label{2smirnov}\left(\begin{array}{c} \frac{U_{(n-k)} - b_{n,k}}{a_{n,k}}\\ \frac{U_{(n-2k)} - b_{n,2k}}{a_{n,2k}}\end{array} \right) \xrightarrow{d} N\left(\Big(\begin{array}{c}0 \\ 0\end{array}\Big), \Big(\begin{array}{cc} 1 & 1/\sqrt{2} \\ 1/\sqrt{2} & 1 \end{array}\Big)\right),\end{equation}
where for $t\in\mathbb{N},$ $t<n$
\[b_{n,t} = \frac{n-t}{n-1}, \quad a_{n,t} = \sqrt{\frac{1}{n-1}b_{n,t}(1 - b_{n,t})}.\]
The proof of \eqref{2smirnov} is just a repetition of the proof of Lemma 2.2.3, \cite{dehaan}. Firstly, the pdf of the random vector $(U_{(n-k)}, U_{(n-2k)})$ is
\[\frac{n!}{((k-1)!)^2 (n-2k)!}(1 - x)^{k-1} (x-y)^{k-1} y^{n-2k},\] thus the pdf of the vector $\big((U_{(n-k)} - b_{n,k})/a_{n,k},\ (U_{(n-2k)} - b_{n,2k})/a_{n,2k} \big)$ is
\begin{eqnarray*}\frac{n!}{((k-1)!)^2 (n-2k)!}\,a_{n,k}\,a_{n,2k} (1 - b_{n,k})^{k-1} (b_{n,k} - b_{n,2k})^{k-1} (b_{n,2k})^{n-2k} \\ \times\Big(1 - \frac{a_{n,k} x}{1 - b_{n,k}}\Big)^{k-1} \Big(1 + \frac{a_{n,k} x - a_{n,2k} y}{b_{n,k} - b_{n,2k}}\Big)^{k-1}\Big(1 + \frac{a_{n,2k} x}{b_{n,2k}}\Big)^{n-2k}.\end{eqnarray*}
By the Stirling formula one can conclude that the expression on the first line of the latter formula tends to $ (\sqrt{2}\pi)^{-1},$ whereas the expression on the second line tends to
\[\exp( - x^2 +\sqrt{2}xy - y^2).\] Thus, \eqref{2smirnov} follows from the fact that pointwise convergence of the sequence of pdfs implies weak convergence of the probability distributions (Scheffe's theorem).
Next, noticing that $a_{n,k} \sim a_{n,2k}/\sqrt{2} \sim \sqrt{k}/n$ and using the properties of multivariate normal distribution, we have
\begin{equation}\frac{n}{\sqrt{k}}\left(\left(\begin{array}{c}U_{(n-k)} \\ U_{(n-2k)}\end{array}\right) -
\left(\begin{array}{c}\frac{n-k}{n-1} \\ \frac{n-2k}{n-1} \end{array}\right)\right) \xrightarrow{d} N\left(\Big(\begin{array}{c}0 \\ 0\end{array}\Big), \Big(\begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array}\Big)\right).\label{2dim}\end{equation}
Finally, noticing that $(E_{(k)}, E_{(2k)})\stackrel{d}{=} (- \ln(U_{(n-k)}), - \ln(U_{(n-2k)})),$ we derive the result by the row-wise application of the delta method with function $f(x) = -\ln x$ to the last relation. Indeed, by the mean value theorem for the first and second rows in \eqref{2dim} we have
\begin{eqnarray*} \frac{n}{\sqrt{k}}\left(\left(\begin{array}{c}E_{(k)} \\ E_{(2k)}\end{array}\right) +
\left(\begin{array}{c} \ln(1 - k/n) \\ \ln(1 - 2k/n) \end{array}\right)\right) &\stackrel{d}{=}& \frac{n}{\sqrt{k}}\left(\begin{array}{c}f(U_{(n-k)}) - f\big(\frac{n-k}{n-1}\big) \\ f(U_{(n-2k)}) - f\big(\frac{n-2k}{n-1}\big)\end{array}\right) \\ &=& \frac{n}{\sqrt{k}}\left(\begin{array}{c}f^\prime(c_1)\big(U_{(n-k)} - \frac{n-k}{n-1}\big) \\ f^\prime(c_2)\big(U_{(n-2k)} - \frac{n-2k}{n-1}\big) \end{array}\right)\end{eqnarray*} a.s., where $c_i,$ $i=1,2,$ are
some (random) values between $U_{(n-ik)}$ and $(n-ik)/(n-1),$ $i=1,2,$ respectively. But for $i = 1,2$ $f^\prime(c_i) = 1/c_i,$ $U_{(n-ik)} \to 1$ in probability and $(n-ik)/(n-1)\to 1,$ thus $f^\prime(c_i) \to 1$ in probability. This and Slutsky theorem imply \eqref{2exp}.
$\Box$
\end{proof}
The schemes of the proofs of Theorems \ref{T1} and \ref{T3}, in fact, coincide. However, the proof of the latter is more difficult technically since the statistic $\widehat{R}_{k,n}$ depends on $X_{(n-k)}$ and $X_{(n-2k)}$ together. Moreover, if $\gamma>0$ then $\sqrt{k}(\widehat{R}_{k,n} - R_{k,n})$ will no longer tend to $0$ in probability.
\\
\\
{\bf Explicit form of $\mathbb{E}(\widehat R_{k,n} - R_{k,n})$ given $X_{(n-k)} = q$ and $X_{(n-2k)} = \hat q.$}\\
\\
Write $\sqrt{k}(\widehat{R}_{k,n} - R_{k,n})$ in explicit form
\begin{eqnarray*}
\sqrt{k}(\widehat{R}_{k,n} - R_{k,n}) =
\sqrt{k}\left(\ln\frac{k}{n} - \ln(1 - F_0(X_{(n-k)})) \right) \\
- \frac{1}{\sqrt{k}}\sum_{i=n-k+1}^{n}\left[\ln\left(1-F_0\Big(u_0 + \frac{X_{(i)} - X_{(n-k)}}{X_{(n-k)} - X_{(n-2k)}}(u_0 - \hat u_0)\Big)\right)- \ln(1
- F_0(X_{(i)}))\right],
\end{eqnarray*}
where $\hat u_0 = u_0(n/(2k)),$ and consider its conditional distribution given $X_{(n-k)} = q$ and $X_{(n-2k)} = \hat q.$ Using the argument similar to the one from the proof of Theorem \ref{T1} and inheriting the notation from there, we get
\begin{eqnarray} \nonumber
\sqrt{k}(\widehat{R}_{k,n} - R_{k,n}) \stackrel{d}{=}
\sqrt{k}\left(\ln\frac{k}{n} - \ln(1 - F_0(q)) \right) \\ \label{initial}
-\, \frac{1}{\sqrt{k}}\sum_{i=1}^{k}\left[\ln\left(1-F_0\Big(u_0 + \frac{X_{i}^\ast - q}{q - \hat q}(u_0 - \hat u_0)\Big)\right)- \ln(1
- F_0(X_{i}^\ast))\right].
\end{eqnarray}
Hereinafter we write $\overline{F}(x)$ instead of $1 - F(x).$ Denote
\[Y^\ast = \ln\overline{F}_0\Big(u_0 + \frac{X_{1}^\ast - q}{q - \hat q}(u_0 - \hat u_0)\Big)- \ln \overline{F}_0(X_{1}^\ast)\] and find its mean. After integrating by parts, we derive
\begin{eqnarray*}&&\mathbb{E} Y^\ast = \ln \frac{\overline{F}_0(u_0)}{\overline{F}_0(q)}\\&& - \int\limits_{q}^\infty \frac{\overline{F}_0(x)}{\overline{F}_0(q)}
\left(\frac{u_0 - \hat u_0}{q - \hat q}\frac{p_0\big(u_0 + (u_0 - \hat u_0)(x-q)/(q - \hat q)\big)}{\overline{F}_0\big(u_0 + (u_0 - \hat u_0)(x-q)/(q - \hat q)\big)} - \frac{p_0(x)}{\overline{F}_0(x)}\right) dx\\
&& =: \ln \frac{\overline{F}_0(u_0)}{\overline{F}_0(q)} + 1 - f(q, \hat q).\end{eqnarray*}
from which and \eqref{initial} we get
\[\mathbb{E}(\widehat{R}_{k,n} -
R_{k,n}|X_{(n-k)} = q, X_{(n-2k)} = \hat q) = f(q, \hat q) - 1.\]
\\
{\bf Evaluating the asymptotic of $\sqrt{k}\mathbb{E}(\tilde R_{k,n} - R_{k,n}|X_{(n-k)}, X_{(n-2k)}).$}\\
\\
Now find the asymptotic of \[\sqrt{k} (f(X_{(n-k)}, X_{(n-2k)}) - 1) = \sqrt{k}\mathbb{E}(\tilde R_{k,n} - R_{k,n}|X_{(n-k)}, X_{(n-2k)}).\] For this aim, in contrast to the proof of Theorem \ref{T1}, we need the multivariate delta method. First of all, observe that
\[(X_{(n-k)}, X_{(n-2k)}) \stackrel{d}{=} \left(u_0\Big(\frac{1}{1 - \exp(-E_{(k)})}\Big), u_0\Big(\frac{1}{1 - \exp(-E_{(2k)})}\Big)\right),\]
where the order statistics $E_{(k)}$ and $E_{(2k)}$ were introduced in Lemma \ref{L2}. So, apply the multivariate delta method for the function
\[g(x,y) = f\left(u_0\Big(\frac{1}{1 - e^{-x}}\Big), u_0\Big(\frac{1}{1 - e^{-y}}\Big)\right)\] to the relation \eqref{2exp}. We get
\[\frac{\sqrt{k}\big(f(X_{(n-k)}, X_{(n-2k)}) - f(u_0, \hat u_0)\big)}{\frac{k}{n}\sqrt{D}} \xrightarrow{d} N(0,1),\]
where
\[D = \nabla g \left(\begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array}\right) (\nabla g)^T\] and $\nabla g$ is the gradient of $g(x,y)$ at the point $(-\ln(1 - k/n), -\ln(1 - 2k/n)).$ Observe also $f(u_0, \hat u_0) = 1.$ Thus, one can see, that the asymptotic of $\sqrt{k}\mathbb{E}(\tilde R_{k,n} - R_{k,n}|X_{(n-k)}, X_{(n-2k)})$ strongly depends on the asymptotic of $\frac{k}{n}\sqrt{D}.$ Let us show that
\begin{eqnarray}\frac{k}{n}\sqrt{D} \to \frac{1}{\sqrt{2}(2^\gamma -1)}\frac{\gamma}{\gamma+1}, \label{notgreat} \end{eqnarray}
where for $\gamma = 0$ the right-hand side should be understood as $(\sqrt{2}\ln 2)^{-1}.$ \\
\\
{\bf Evaluating the asymptotic of $\frac{k}{n}\sqrt{D}$ for $\gamma>0.$}\\
\\
First, observe that
\[\nabla g = \left(f^\prime_x(u_0, \hat u_0) u_0^\prime(n/k) \frac{1 - k/n}{(k/n)^2}, \, f^\prime_y(u_0, \hat u_0) u_0^\prime(n/(2k)) \frac{1 - 2k/n}{(2k/n)^2} \right).\] Next, it can be proved that
\begin{eqnarray*}f^\prime_x(u_0, \hat u_0) = \frac{n}{k}\left(p_0(u_0) - \int\limits_{u_0}^\infty\frac{x - \hat u_0}{u_0 - \hat u_0}\frac{p_0^2(x)}{\overline{F}_0(x)}dx\right),
\\ f^\prime_y(u_0, \hat u_0) = \frac{n}{k}\int\limits_{u_0}^\infty\frac{x - u_0}{u_0 - \hat u_0}\frac{p_0^2(x)}{\overline{F}_0(x)}dx.\end{eqnarray*} Now assume $\gamma>0.$ Using the relation \eqref{lopital} established above and the equality
\begin{equation} \label{lopital2}
\lim_{u\to\infty}\frac{p_0(u)}{\int_q^\infty\frac{p_0^2(x)}{\overline{F}_0(x)}dx} = \gamma + 1,\end{equation} that can be obtained similarly, we derive that
\begin{eqnarray*}f^\prime_x(u_0, \hat u_0) = -\frac{n}{k}\frac{\hat u_0}{u_0 - \hat u_0}\frac{\gamma}{\gamma+1} p_0(u_0)(1 + o(1)), \\ f^\prime_y(u_0, \hat u_0) = \frac{n}{k}\frac{u_0}{u_0 - \hat u_0}\frac{\gamma}{\gamma+1} p_0(u_0)(1 + o(1)).\end{eqnarray*} Taking into account $p_0(u_0(t)) u_0^\prime(t) = 1/t^2,$ for $\gamma>0$ we have
\begin{eqnarray*}\frac{k}{n}\sqrt{D} &=& \frac{u_0}{u_0 - \hat u_0} \frac{\gamma}{\gamma+1}\sqrt{\Big(\frac{\hat u_0}{u_0}\Big)^2 - \frac{1}{2}\frac{\hat u_0}{u_0} \frac{u_0^\prime(n/(2k))}{u_0^\prime(n/k)} + \frac{1}{8}\Big(\frac{u_0^\prime(n/(2k))}{u_0^\prime(n/k)}\Big)^2}(1 + o(1))\\ &\to& \frac{1}{\sqrt{2}(2^\gamma -1)}\frac{\gamma}{\gamma+1}\end{eqnarray*} by properties of regularly varying functions (namely, we use here the relations (1.1.33) and (1.2.18), \cite{dehaan}).
\\
\\
{\bf Evaluating the asymptotic of $\frac{k}{n}\sqrt{D}$ for $\gamma=0.$}\\
\\
Consideration of the case $\gamma = 0$ is a bit more delicate. First, note that
\[f^\prime_x(u_0, \hat u_0) + f^\prime_y(u_0, \hat u_0) = \frac{n}{k}\left(p_0(u_0) - \int\limits_{u_0}^\infty\frac{p_0^2(x)}{\overline{F}_0(x)}dx\right) = o\Big(\frac{n}{k} p_0(u_0)\Big)\]
by \eqref{lopital2}. Next, using the latter and again the relations $p_0(u_0(t)) u_0^\prime(t) = 1/t^2$ and (1.1.33), \cite{dehaan}, we derive
\[\frac{k}{n}\nabla g = \frac{f^\prime_y(u_0, \hat u_0)}{p_0(u_0)\cdot n/k}\big(-1 + o(1), \, 1/2 + o(1)\big),\] thus
\[\frac{k}{n}\sqrt{D} = \frac{k}{n}\frac{f^\prime_y(u_0, \hat u_0)}{\sqrt{2}p_0(u_0)}(1 + o(1)).\] Find the asymptotics of the right-hand side of the latter equality. By Corollary 1.1.10, \cite{dehaan},
\[\frac{u_0 - \hat u_0}{n/k\cdot u^\prime_0(n/k)} \to \ln 2,\] thus, using $p_0(u_0) u_0^\prime(n/k) = (k/n)^2$ and $\overline{F_0}(u_0) = k/n,$ we have
\begin{eqnarray*}\frac{k}{n}\frac{f^\prime_y(u_0, \hat u_0)}{p_0(u_0)} &=& \frac{1}{p_0(u_0)} \int\limits_{u_0}^\infty\frac{x - u_0}{u_0 - \hat u_0}\frac{p_0^2(x)}{\overline{F}_0(x)}dx\\ &=& \frac{n/k\cdot u^\prime_0(n/k)}{u_0 - \hat u_0} \cdot \frac{1}{\overline{F_0}(u_0)} \int\limits_{u_0}^\infty(x - u_0)\frac{p_0^2(x)}{\overline{F}_0(x)}dx.\end{eqnarray*} The first multiplier on the right-hand side of the latter relation tends to $(\ln 2)^{-1},$ hence, it remains to find the asymptotics of the second one. By the L'Hospital rule, we have
\begin{eqnarray*}&& \lim_{u\to\infty}\frac{1}{\overline{F_0}(u)} \int\limits_{u}^\infty(x - u)\frac{p_0^2(x)}{\overline{F}_0(x)}dx \\&& = \lim_{u\to\infty} \frac{1}{- p_0(u)} \left(-\frac{u p^2(u)}{\overline{F}(u)} + \frac{u p_0^2(u)}{\overline{F}_0(u)} - \int_{u}^\infty \frac{p^2_0(x)}{\overline{F}_0(x)} dx \right) \\ && = \lim_{u\to\infty} \frac{1}{p_0(u)} \int_{u}^\infty \frac{p^2_0(x)}{\overline{F}_0(x)} dx = 1,\end{eqnarray*} where the latter equality follows from \eqref{lopital2}. Thus, we show that if $\gamma = 0$ then $k \sqrt{D}/n \to (\sqrt{2} \ln 2)^{-1}.$\\
\\
Summarizing the above, we proved that under the assumptions of the theorem \begin{equation}
\sqrt{k}\big(f(X_{(n-k)}, X_{(n-2k)}) - 1\big) \stackrel{d}{\rightarrow} N\Big(0,
1 + \frac{\gamma^2}{2(\gamma+1)^2(2^\gamma-1)^2}\Big),
\label{strange}\end{equation}
where for $\gamma=0$ the ratio on the right-hand side should be understood as $1/(2 (\ln 2)^2).$
\\
\\
{\bf Final steps.}
\\
\\
The proof of the relation
\begin{equation}\label{2P}\sqrt{k}\big(\widehat{R}_{k,n} - R_{k,n}\big) - \sqrt{k}\big(f(X_{(n-k)}, X_{(n-2k)}) - 1\big) \xrightarrow{P} 0\end{equation} does not contain new ideas as compared to the proof of the similar relation \eqref{P} and the previous steps of the current proof, therefore we omit it. Finally, we have
\begin{eqnarray*}\sqrt{k}(\widehat{R}_{k,n} - 1) &=& \sqrt{k}\big({R}_{k,n} - 1) \\
&& +\, \sqrt{k}\Big[\big(\widehat{R}_{k,n} - R_{k,n}\big) - \big(f(X_{(n-k)}, X_{(n-2k)}) - 1\big)\Big]\\
&& +\, \sqrt{k}\big(f(X_{(n-k)}, X_{(n-2k)}) - 1\big).
\end{eqnarray*}
Observe that the properties of iid exponential random variables, see also the proof of Theorem 1, \cite{R1}, implies that the third summand on the right-hand side of the latter relation is independent of the first. Indeed, remind that $-\ln (1 - F_0(X_{(i)})) \stackrel{d}{=} E_{(i)}$ with $\{E_i\}_{i=1}^n$ i.i.d. standard exponential as above. Thus we have
\[{R}_{k,n} \stackrel{d}{=} \frac{1}{k} \sum_{i=0}^{k-1} (E_{(n-i)} - E_{(n-k)}).\] Next, it is well-known (e.g. it follows from R\'enyi representation, \cite{renyi}) that $E_{(n-k)}$ and $E_{(n-2k)}$ are independent of $\{E_{(n-i)} - E_{(n-k)}\}_{i=0}^{k-1}.$ Hence, $\ln (1 - F_0(X_{(n-k)}))$ and $\ln (1 - F_0(X_{(n-2k)}))$ are independent of $R_{k,n},$ and the required statement follows from the fact that \[(X_{(n-k)},X_{(n-2k)}) = \Big(g(-\ln (1 - F_0(X_{(n-k)}))), g(-\ln (1 - F_0(X_{(n-2k)})))\Big)\] a.s. with $g(x) = u_0\big((1 - e^{-x})^{-1}\big).$
This argument together with \eqref{old}, \eqref{strange} and \eqref{2P} completes the proof of Theorem \ref{T3}.
\subsection{Proof of Theorem \ref{T4}}
The scheme of the proof of Theorem \ref{T4} is absolutely the same as for Theorem \ref{T2}, therefore it is omitted. Note only, that for the proof of the relation similar to \eqref{appr2}, one needs to use the multivariate delta method and Lemma \ref{L2}. Next, instead of the relation \eqref{ineq} used to find the upper bound for $P(Y_1\le x)$, one should use the relation
\[\frac{\overline{F}_1(u_1(t) + x(u_1(t) - u_1(t/2)))}{\overline{F}_1(u_1(t))} \ge \frac{\big(\overline{F}_0(u_0(t) + x(u_0(t) - u_0(t/2)))\big)^{1-\varepsilon}}{(\overline{F}_0(u_0(t)))^{1-\varepsilon}}\] for all $x>0.$ The latter, in turn, follows from the condition $C_\delta(F_0, F_1).$
Indeed,
\begin{eqnarray*} \frac{\overline{F}_1(u_1(t) + x(u_1(t) - u_1(t/2)))}{\overline{F}_1(u_1(t))} &=& \frac{\overline{F}_1\big(u_1(t)(1 + x - xu_1(t/2)/u_1(t))\big)}{\overline{F}_1(u_1(t))} \\
&\ge& \frac{\overline{F}_1\big(u_1(t)(1 + x - xu_0(t/2)/u_0(t))\big)}{\overline{F}_1(u_1(t))}\\ &\ge& \frac{\big(\overline{F}_0(u_0(t) + x(u_0(t) - u_0(t/2)))\big)^{1-\varepsilon}}{(\overline{F}_0(u_0(t)))^{1-\varepsilon}},
\end{eqnarray*}
where the first and second relations above follow from $C_0(F_0, F_1)$ and Proposition 1, respectively.
\subsection{Proof of Corollary \ref{C2}}
The proof of the Corollary \ref{C2} is absolutely similar to the proof of Corollary \ref{C1} except that instead of the relation \eqref{v} one should use the inequality
\[\frac{V_{(i)} - V_{(n-k)}}{V_{(n-k)} - V_{(n-2k)}} \le \frac{X_{(i)} - X_{(n-k)}}{X_{(n-k)} - X_{(n-2k)}},
\] that immediately follows from $C_0(F_0, F_1)$ and \eqref{v}.
\\
\\
\textbf{Data availability statement.}\\ The datasets analysed during the current study are available in the International Database on Longevity, www.supercentenarians.org.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,966 |
{"url":"https:\/\/plainmath.net\/269\/solve-the-equation-check-your-solution-8t-plus-5-equal-6t-plus-1","text":"# Solve the equation. Check your solution. 8t + 5 = 6t + 1\n\nQuestion\nEquations\nSolve the equation. Check your solution. $$8t + 5 = 6t + 1$$\n\n2021-02-27\nSubtract 6t from each side of the equation. Subtraction Property of Equality\n$$8t+5-6t=6t+1-6t$$\nSimplify\n$$2t+5=1$$\nSubtract 5 from each side of the equation. Subtraction Property of Equality\n$$2t+5-5=1-5$$\nSimplify\n$$2t=-4$$\nDivide each side of the equation by 2. Division Property of Equality\n$$\\frac{2}{2}*t=\\frac{-4}{2}$$\nSimplify\n$$t=-2$$\nCheck solution\n$$8(-2)+5=6(-2)+1$$\n$$-16+5=-12+1$$\n$$-11=-11$$\n\n### Relevant Questions\n\nSolve the following system of equations. (Write your answers as a comma-separated list. If there are infinitely many solutions, write a parametric solution using t and or s. If there is no solution, write NONE.)\n$$\\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{6}{x}_{{3}}={6}$$\n$$\\displaystyle{x}_{{1}}+{x}_{{2}}+{3}{x}_{{3}}={3}$$\n$$\\displaystyle{\\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\\right)}=$$?\nWhich of the following equations have the same solution set? Give reasons for your answers that do not depend on solving the equations.\nl.$$x-5=3x+7$$\nll.$$3x-6=7x+8$$\nlll.$$15x-9=6x+24$$\nlV.$$6x-16=14x+12$$\nV.$$9x+21=3x-15$$\nVl.$$-0.05+\\frac{x}{100}=3\\frac{x}{100}+0.07$$\nUse the\u200b Gauss-Jordan method to solve the system of equations. If the system has infinitely many\u200b solutions, give the solution with z arbitrary.\nx-5y+2z=1\n3x-4y+2z=-1\nDetermine whether the ordered pair is a solution to the given system of linear equations.\n1.(5,3) $$\\displaystyle{\\left\\lbrace\\begin{array}{c} {x}-{y}={2}\\\\{x}+{y}={8}\\end{array}\\right.}\\rbrace$$\nSolve the following system of equations.\n$$\\displaystyle{2}{x}-\\frac{{1}}{{5}}{y}=\\frac{{4}}{{5}}$$\n$$\\displaystyle\\frac{{1}}{{3}}{x}-\\frac{{1}}{{2}}{y}=-{12}$$\nx-?\ny-?\nConsider the following system of llinear equations.\n$$\\displaystyle\\frac{{1}}{{3}}{x}+{y}=\\frac{{5}}{{4}}$$\n$$\\displaystyle\\frac{{2}}{{3}}{x}-\\frac{{4}}{{3}}{y}=\\frac{{5}}{{3}}$$\nPart A: $$\\displaystyle\\frac{{{W}\\hat{\\propto}{e}{r}{t}{y}}}{{\\propto{e}{r}{t}{i}{e}{s}}}$$ can be used to write an equivalent system?\nPart B: Write an equivalent system and use elimination method to solve for x and y.\nSolve $$\\displaystyle{\\left|{\\ln{{\\left({x}+{3}\\right)}}}\\right|}={1}$$. Give your answers in exact form.","date":"2021-05-08 19:14:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7736978530883789, \"perplexity\": 529.1538372922977}, \"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\/1620243988923.22\/warc\/CC-MAIN-20210508181551-20210508211551-00476.warc.gz\"}"} | null | null |
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\section{Introduction}
\label{sec:intro}
Hierarchical structure formation is one of the most profound predictions of the standard model of cosmology --- the Lambda--Cold Dark Matter ($\Lambda$CDM) cosmology \citep[e.g.,][]{10.1093/mnras/183.3.341,10.1086/183911,10.1038/311517a0,2010gfe..book.....M}. In a $\Lambda$CDM universe, dark matter is attracted to local density peaks of initial fluctuations, laid down during inflation, forming dark matter haloes. These dark matter haloes grow and merge with one another, and are believed to be the nests within which visible galaxies develop. The spatial distribution of dark matter haloes is hence a powerful prediction of the $\Lambda$CDM model, and can be compared with observations of galaxy distribution to test our understanding of cosmology and galaxy formation physics.
Despite the nonlinearity in the formation and merging of dark matter haloes, we have a good general picture of the distribution of dark matter haloes, thanks to extensive studies using both mathematical approximations and numerical simulations. In particular, the excursion set formalism provides an analytic description of the halo mass function, halo mass assembly histories, and halo clustering properties (\citealt{Press1974,1991ApJ...379..440B}; for a review see \citealt{10.1142/S0218271807010511} and Chapter 7 of \citealt{2010gfe..book.....M}). In this context, the clustering properties of haloes are a function of halo mass alone. This is commonly referred to as ``halo bias'' and can be approximated analytically \citep[e.g.,][]{Kaiser84,Cole1989,Mo1996,Sheth2001} or calibrated against numerical simulations \citep[e.g.,][]{Tinker2010}. These analytic descriptions of halo bias are powerful tools within galaxy models that rely on halo mass as the link between haloes and the galaxies that they host, such as the Halo Occupation Distribution \citep[e.g.,][]{Peacock2000,Seljak2000,Berlind2002,Zheng2005,Zehavi2005} and the Conditional Luminosity Function \citep[e.g.,][]{Yang2003,vandenBosch2013}, to predict galaxy clustering efficiently.
On the other hand, it has also been shown, within cosmological $N$-body simulations, that halo clustering also depends on halo properties other than mass, amongst which the most notable is halo assembly history \citep{2001PhDT.........7W,Gao2005,Wechsler2006,2008MNRAS.389.1419L}. This is commonly referred to as ``halo assembly bias.'' Although halo assembly bias is a smaller effect than mass-dependent halo bias, it has drawn increasing attention from researchers who model the galaxy--halo connection \citep[e.g.,][]{1507.01948,Hearin2016,Zentner2016,1611.09787,Lehmann2017,2017MNRAS.469.1809R}, because it is very plausible that the galaxy assembly history is to some extent connected to the assembly history of its host halo. If this is the case, one would need to understand halo assembly bias to predict accurately galaxy clustering and to mitigate potential bias in any inference from clustering measurements \citep{Reddick2013,Zentner2014}.
In the excursion set formalism, a characteristic mass $M^*$, below which most haloes have already collapsed and formed, is defined to satisfy $\sigma(M^*)=\delta_c D(a)$, where $\delta_c \simeq 1.686$ is the critical overdensity, $D(a)$ is the linear growth rate, and $\sigma(M)$ is the squared root of the mass variance with a top-hat filter of mass $M$.
Studies have found that, below the characteristic mass $M^*$, haloes that form earlier are more strongly clustered \citep{2001PhDT.........7W,Gao2005,Wechsler2006,2008MNRAS.389.1419L}. However, above the characteristic mass $M^*$, the signal of halo assembly bias is less clean. \citet{Wechsler2006} found that, above the collapse mass, haloes with lower \emph{concentration} are more clustered, but detected no clear bias as a function of halo formation time, despite the correlation between concentration and formation time \citep{Wechsler2002}. Similarly, \citet{2008MNRAS.389.1419L} inspected several different definitions of halo formation time, but found little assembly bias for high-mass haloes upon splitting haloes by their formation times. Attempts have been made to explain the physical origin of halo assembly bias \citep{Sandvik2007,10.1142/S0218271807010511,Desjacques2007,Dalal2008,2009MNRAS.396.2249W}, and while they provided some heuristic insights, they do not explain why different proxies of halo assembly history (e.g., concentration and formation time) exhibit assembly biases of very different magnitudes for massive haloes.
On the observational front, recent efforts have been made to detect halo assembly bias with galaxy clusters \citep{Yang2006,Tinker2012,Miyatake2015,More2016,Dvornik2017}. However, results remain inconclusive as various systematic effects can masquerade as observed halo assembly bias \citep{Lin2016,2016arXiv161100366Z,1702.01682}. One challenge facing observations is that, because halo properties are not directly observable, one has to use some observational proxy to approximate halo assembly history. In the studies of \citet{Miyatake2015,More2016,Dvornik2017}, they use the average distance of member galaxies in the cluster as a proxy of halo formation time. The motivation for such a proxy is the well-documented correlation between halo concentration and halo formation time \citep{Wechsler2002} and the assumed correlation between halo concentration and member galaxy distances. Nevertheless, even if a bias signal due to the average distance of member galaxies had been robustly detected observationally, it would not be clear whether or not this bias is of the same physical origin as the halo assembly bias identified in cosmological $N$-body simulations.
In addition to halo assembly bias, the clustering of haloes also depends on other halo properties, such as spin, shape, and substructure abundance \citep[e.g.,][]{2001PhDT.........7W,Wechsler2006,Bett2007,Gao2007,2007MNRAS.375..489H,Faltenbacher2010}. {More recently, there are also studies that explore the relationships amongst different kinds of halo assembly bias \citep[e.g.,][]{1612.04360,1708.08451}.} In much of the literature, these other \emph{secondary halo biases} (i.e., dependence of halo clustering on halo properties other than mass) are also commonly referred to as ``assembly bias,'' regardless of whether or not the secondary halo properties have a direct connection to halo assembly history. This nomenclature could result in some confusion as to whether all the different kinds of secondary halo bias have the same physical origin. While halo assembly history is arguably the most important property other than halo mass that characterizes individual haloes, when it comes to the clustering properties, it is still unclear if or how the different secondary halo biases connect with one another.
In this study, we join the exploration of halo secondary bias with two specific aims. First, we examine the dependence of halo clustering on a set of secondary halo properties for cluster-size haloes using a modern, large-volume cosmological simulation to validate previous results, some of which may have suffered from limited volume or resolution. Secondly, we inspect the correlations between different halo properties and how they connect to the secondary bias, to better understand the relation between assembly bias and all different kinds of secondary biases. To this end, we present a novel way to characterize secondary bias, which helps us to gain insight into these questions.
We focus our current study on cluster-size haloes for three reasons. First, in the cluster-mass regime, the behaviour of assembly bias due to different proxies, such as halo concentration and formation time, is poorly understood. Secondly, at the high-mass end of the halo mass spectrum, halo assembly bias is thought not to be due to nonlinear interactions between neighbouring haloes, but reflective of the initial conditions for structure formation \citep{10.1142/S0218271807010511,Dalal2008}. Therefore, it seems plausible that assembly bias is simpler at high mass.
Third, there have been, and likely will soon be more, observational attempts to directly measure halo assembly bias with galaxy clusters. Hence, this is a timely and crucial study to facilitate the interpretation of current measurements and preparation for forthcoming observations.
This paper is structured as follows. In \autoref{sec:methods}, we introduce the simulation used in this study, define the halo properties, and explain how we remove mass dependence in the bias. We present our main results in \autoref{sec:results}, where we show and compare the secondary halo biases due to different halo properties, with \autoref{fig:bias} summarizing this result, and also present a novel way to characterize secondary bias. In \autoref{sec:discussion}, we turn to explore the interplay between correlation and bias, demonstrate that the correlation between different halo properties does not determine the secondary bias they exhibit, and also discuss the implication for galaxy assembly bias. \autoref{fig:history} highlights the absence of assembly bias (in its strict definition) at this mass scale, despite the existence of other secondary halo biases. We conclude in \autoref{sec:conclusion}. We also include a summary of the correlations amongst all secondary halo properties used in this study and their dependence on the environment in Appendix~\ref{sec:appendix}.
\begin{figure*}
\centering\includegraphics[width=\textwidth]{marks.pdf}
\caption{\label{fig:marks}%
Each scatter plot shows the relation between halo mass and one of the secondary halo properties: concentration parameter \ensuremath{c}, spin parameter \ensuremath{\lambda}, half-mass scale \ensuremath{a_{1/2}}, accretion rate before peak mass \ensuremath{\Gamma_\text{peak}}, number of subhaloes \ensuremath{N_\text{sub}}, and averaged subhalo distance \ensuremath{R_\text{mem}}. All distinct haloes in the sample are plotted and coloured by their mark values, which are always between 0 and 1 and are assigned according to the rank of the secondary property within each halo mass bin as explained in \autoref{sec:marks}.
}
\end{figure*}
\section{Methods}
\label{sec:methods}
\subsection{Simulations}
\label{sec:sims}
In this study we use the MultiDark Planck 2 (MDPL2) simulation \citep{Klypin2016}. MDPL2 is a cosmological gravity-only $N$-body simulation, run with the \textsc{L-Gadget2} code. It has a periodic volume of 1\,\ensuremath{h^{-1}}\,Gpc$^3$, with 3840$^3$ particles. The mass resolution of each particle is $1.51\times10^9$\,\ensuremath{h^{-1}}\,\text{M$_\odot$}, and the physical force resolution ranges from 5 to 13\,\ensuremath{h^{-1}}\,kpc (smaller at lower redshift).
MDPL2 adopts the Planck 2013 $\Lambda$CDM cosmology \citep{Planck2013cosmology}, and the actual values used in MDPL2 are: total matter density $\Omega_\text{m} = 0.307115$, dark energy density $\Omega_\Lambda = 1 - \Omega_\text{m}$, baryon density $\Omega_\text{b} = 0.048206$, Hubble parameter $h = 0.6777$, scalar spectral index $n_s = 0.96$, and the amplitude of mass density fluctuation $\sigma_8 = 0.8228$.
The MDPL2 simulation has been analysed by the \textsc{Rockstar} halo finder and the \textsc{Consistent Trees} merger tree builder \citep{2013ApJ...762..109B,2013ApJ...763...18B}.
The halo mass definition used here is the virial mass; in this cosmology the virial overdensity corresponds to approximately 100 times the critical density \citep{Bryan1998}.
A halo is called a ``subhalo'' if its centre is within the virial radius of any larger halo. Any halo that is not a subhalo is called a ``distinct halo.''
In this study, we use haloes from the present-day ($z=0$) halo catalogue. We select all distinct haloes with a present-day virial mass $\ensuremath{M_\text{vir}} \geq 10^{14}$\,\ensuremath{h^{-1}}\,\text{M$_\odot$}{} as our full halo sample. There are 27,029 distinct haloes in this sample, corresponding to a number density of $2.7\times 10^{-5}\,h^3\,\text{Mpc}^{-3}$.
Although the MDPL2 halo catalogues contain information about halo mass assembly histories, we do not have direct access to the full assembly history of each individual halo. Hence we also use the DarkSky-Gpc simulation when the full assembly history is required (\autoref{fig:history}). DarkSky-Gpc ({\tt ds14\_b}) is part of the Dark Sky Simulations \citep{Skillman2014}, run with the \textsc{2HOT} code \citep{Warren2013}. It also has a periodic volume of 1\,\ensuremath{h^{-1}}\,Gpc$^3$, and was run with 10240$^3$ particles and a mass resolution of $7.63\times10^7$\,\ensuremath{h^{-1}}\,\text{M$_\odot$}.
DarkSky-Gpc has also been analysed by the \textsc{Rockstar} halo finder; however, the full \textsc{Rockstar--Consistent Trees} halo catalogues and merger trees have only been constructed on a downsampled version, which has only $10240^3/32 \simeq 3225^3$ particles and an effective mass resolution of $2.44\times10^9$\,\ensuremath{h^{-1}}\,\text{M$_\odot$}%
\footnote{See also \citet{2016PhDT........76M,Lehmann2017}. We note that these two studies mistakenly reported the DarkSky-Gpc particle mass to be twice its actual value.}, similar to the mass resolution of MDPL2. In this study we only use the downsampled version of DarkSky-Gpc. DarkSky-Gpc adopts a flat cosmology close to the Planck 2013 $\Lambda$CDM cosmology, with $h = 0.688$, $\Omega_\text{m} = 0.295$, $n_s = 0.968$, $\sigma_8 = 0.834$. We also use the virial overdensity as the halo mass definition for DarkSky-Gpc.
While we present most of our result (all except for \autoref{fig:history}) using the MDPL2 simulation because it is publicly available and has slightly better mass resolution, we have verified that the DarkSky-Gpc simulation produces very similar result, with no qualitative difference and little quantitative difference. Our result holds in both simulations.
\subsection{Secondary halo properties}
\label{sec:properties}
In this study we select six different halo properties other than halo mass and investigate the secondary halo bias due to these properties. The properties we choose are as follows:
\begin{enumerate}
\itemsep0.5\baselineskip
\item Concentration parameter (\ensuremath{c}), obtained by fitting the dark matter density profile to a Navarro--Frenk--White (NFW) profile \citep{Navarro1996,Navarro1997}.
\item Spin parameter (\ensuremath{\lambda}), as defined in \citet{Peebles1969}.
\item Half-mass scale (\ensuremath{a_{1/2}}), defined as the scale factor at which a distinct halo reaches more than or equal to half of its present-day ($z=0$) mass on its main branch.
\item Accretion rate {of} peak mass (\ensuremath{\Gamma_\text{peak}}), the average halo mass accretion rate (in \ensuremath{h^{-1}}\,\text{M$_\odot$}\,yr$^{-1}$) {of peak mass, i.e., \[\left[\ensuremath{M_\text{peak}}(z=0) - \ensuremath{M_\text{peak}}(z=0.5)\right]/\left[t(z=0) - t(z=0.5)\right].\]} Since our halo sample contains the most massive haloes in the simulation, the majority (68.7\%) have $\ensuremath{z_\text{peak}} = 0$ {(the redshift when peak mass takes place)}. The median \ensuremath{z_\text{peak}}{} for haloes whose present-day mass is not peak mass (i.e., $\ensuremath{z_\text{peak}} > 0$) is 0.093. Hence, for our halo sample, \ensuremath{\Gamma_\text{peak}} is basically the average halo mass accretion rate for $0 < z < 0.5$ .
\item Number of subhaloes (\ensuremath{N_\text{sub}}). We count the number of subhaloes (identified by the `upid' value in the \textsc{Rockstar--Consistent Trees} catalogue) that have a peak maximal circular velocity (\ensuremath{V_\text{peak}}) above 135\,\text{km\,s$^{-1}$}. The peak maximal circular velocity is defined as the largest value of the maximal circular velocity on the main branch of the subhalo in consideration.
\item Average subhalo distance (\ensuremath{R_\text{mem}}), defined as the average three-dimensional distance between all subhaloes and the centre of the main halo. The subhalo definition is the same as the definition when we calculate the number of subhaloes for each distinct halo.
\end{enumerate}
For \ensuremath{c}, \ensuremath{\lambda}, \ensuremath{a_{1/2}}, and \ensuremath{\Gamma_\text{peak}}, we use the values in the \textsc{Rockstar--Consistent Trees} catalogue directly.
\subsection{Definition of mass-normalized marks}
\label{sec:marks}
Formally, we say a secondary halo bias exists if two groups of haloes that have the same halo mass but different values of a secondary propriety cluster differently. However, practically, because it is difficult to measure spatial clustering within an infinitesimally thin mass bin due to the finite number of haloes even within our large-volume simulations, we need to remove any bias that may be induced by halo mass when we measure the secondary bias.
To do so, for each secondary halo property in consideration, we assign a ``mass-normalized'' mark value to each halo to represent the secondary halo property. We first bin haloes by halo mass, and then in each mass bin, we assign the mark value based on the rank within that mass bin. Hence, within each mass bin and also overall, the mark values always span $[0,1]$ uniformly. The mark values are always dimensionless.
For secondary halo properties that are discrete, such as the number of subhaloes and the half-mass scale (which is discrete because of the time interval between the simulation snapshots that are saved), if a group of haloes in a particular mass bin share the same exact value, they are randomly assigned different mark values in a way that still preserve the overall ranks. For example, when assigning the mark for the number of subhaloes, we add a random number drawn from a continuous uniform distribution on $[0, 1)$ to the number of subhaloes before the ranking process. In this fashion, haloes with one subhalo would have different mark values, but their mark values will always be smaller than the mark values of haloes with two subhaloes. This procedure hence ensures that the mark is uniformly distributed and that it is not clustered at a certain value, yet it does not introduce noise in the overall ranks.
The sample of distinct haloes used in this study spans the mass range of $14 \leq \log\,\left[M/(\ensuremath{h^{-1}}\,\text{M$_\odot$})\right] < 15.55$. We split this mass range into 30 bins. The bin widths increase quadratically in log mass, such that the lowest mass bins do not contain the vast majority of haloes in the sample. \autoref{fig:marks} shows the relations between the halo mass and each of the secondary halo properties we considered here. The mark value we assigned to each halo is represented by the colour of each point. We can observe the binning effect on the mark values for properties that change more rapidly with halo mass (\ensuremath{\Gamma_\text{peak}}, \ensuremath{N_\text{sub}}, and \ensuremath{R_\text{mem}}). Nevertheless, we have tested and find our results are insensitive to the binning schemes. When we use different binning schemes, including uniformly spacing in log mass and also different number of bins, the result still holds.
\section{Results}
\label{sec:results}
\subsection{Secondary halo bias manifested in the bias function}
\label{sec:bias}
\begin{figure}
\centering\includegraphics[width=\columnwidth]{bias.pdf}
\caption{\label{fig:bias}%
The ratio between the pair counting function ($n_\text{pairs}$) of only haloes with high marks and that of only haloes with low marks for each of the six secondary halo properties: (solid lines from top to bottom) \ensuremath{\lambda}, \ensuremath{R_\text{mem}}, \ensuremath{N_\text{sub}}, \ensuremath{\Gamma_\text{peak}}, \ensuremath{a_{1/2}}, and \ensuremath{c}. A high (or low) mark value means a mark value above (or below) 0.5. The pair counting function is calculated by counting all pairs of haloes that have three-dimensional distances between 2 to 40\, \ensuremath{h^{-1}}\,Mpc, and binned in distance ($r$), normalized by the square of number of haloes. For the concentration parameter (blue solid line), we also show the inverse ratio as a blue dashed line to guide an easy comparison with bias due to other secondary halo properties. For comparison, a thin horizontal black line shows a ratio of unity, and the grey band shows the 3$\sigma$ deviation for a randomly-assigned mark values. The deviation of the coloured lines from the horizontal black line shows a bias due to the secondary halo property.
}
\end{figure}
The most straightforward way to evaluate halo bias is to calculate the ratio between the pair counts of two samples of haloes. This is sometimes called the bias function. If the two samples have different numbers of total haloes, one needs to normalize the pair count first, usually by dividing out the expected number of pairs of a set of uniformly distributed random points. As for secondary halo bias, one can then calculate the bias function between two halo samples that differ in a secondary halo property, but not in halo mass.
Here we present in \autoref{fig:bias}, for each of the six secondary halo properties listed in \autoref{sec:properties}, the bias function (or more precisely, the ratio between the pair counting functions) of two samples of haloes, split by the mark values. For each case, one sample has all haloes with mark values above 0.5, and the other has all haloes with mark values below 0.5. That is, we compare haloes in the upper half of the mark distribution with those in the lower half of the mark distribution at \emph{fixed} mass (cf., \autoref{fig:marks}). Note that because the mark values are assigned in mass bins, the split by mark already excludes any clustering dependence upon mass, and the bias we observe is the secondary bias.
The pair counting function is evaluated in bins of pair distance ($r$). Here we use 11 equally-spaced bins in logarithmic scale between 2 to 40\,\ensuremath{h^{-1}}\,Mpc, and we have verified that our result is insensitive to the binning scheme.
We also estimate how much of the bias signal in the ratio of pair counting functions can come from noise. We uniformly at random assign mark values of $1/N, 2/N, \ldots, 1$ to all haloes in the sample, calculate the ratio of pair counting functions for high and low randomly assigned marks, and repeat this procedure until we obtain convergence of the 99.7\% (3$\sigma$) distribution of the bias signal; this is shown by the grey band in \autoref{fig:bias}. Hence, deviation beyond this grey band indicates a secondary halo bias signal $>3\sigma$.
\begin{figure*}
\centering\includegraphics[width=\textwidth]{markdist.pdf}
\caption{\label{fig:markdist}%
The probability distributions of mark values of each of the six secondary halo properties, when only counting paired haloes. A ``paired halo'' is any distinct halo that is neighboured with at least one other distinct halo in the full halo sample within 10\,\ensuremath{h^{-1}}\,Mpc. If a halo has in multiple neighbour haloes, its mark value is counted only once in the distribution shown by an orange solid line, and counted multiple times (as many as the number of its neighbours) in the distribution shown by a red dashed line.
For comparison, a thin horizontal black line shows a uniform distribution, which, by construction, is the distribution of the mark values when including all, paired or not, haloes. The deviation of the coloured lines from the horizontal black line shows a bias due to the secondary halo property. The grey band shows a typical 3$\sigma$ deviation for a uniform random distribution of the same sample size.
}
\end{figure*}
\autoref{fig:bias} shows that, amongst the six secondary halo properties, the two that directly quantify some aspect of the mass accumulation histories of haloes, namely the half-mass scale $\ensuremath{a_{1/2}}$, and the accretion rate before peak mass $\ensuremath{\Gamma_\text{peak}}$, do not exhibit secondary bias; in both cases, the ratio between the the pair counts is consistent with random marks on all scales.
All other halo properties exhibit clear secondary bias. Haloes with higher concentrations are less clustered, while haloes with higher spins, more subhaloes, or a larger average subhalo distances are more clustered. In terms of the magnitude of the secondary bias, the spin parameter exhibits the strongest amongst these properties. The concentration parameter, number of subhaloes, and the average subhalo distance all produce a secondary bias of the roughly same magnitude. \autoref{fig:bias} also shows that, the secondary bias due to the properties listed decreases with scale. Nevertheless, between scales of 2 to 40\,\ensuremath{h^{-1}}\,Mpc, this secondary bias is always statistically significant. Moreover, on all of the scales that we have considered, the secondary bias induced by any individual property never crosses the black horizontal line at unity, meaning that the secondary bias is always of the same sense.
This result is broadly consistent with previous studies. In this mass regime, the inverted concentration bias has been shown in \citet{Wechsler2006}, the strong spin bias has been shown in \citet{Gao2007}, the lack of assembly bias when split by formation time has been shown in \citet{Gao2005,2008MNRAS.389.1419L}, and the bias due to subhalo distance has been shown in \citet{More2016}. Our study confirms that the secondary bias signals in previous studies can be reproduced with a larger-volume and higher-resolution simulation. {However, we caution the reader that we present the pair counting ratio in \autoref{fig:bias}, but this quantity
does not directly translate into the ratio of linear biases of the subsamples. We will not quote linear biases in this work because the
MDPL2 particle data are not publicly available.}
It is interesting to note that the term ``assembly bias'' has been widely use to refer to any secondary halo bias, despite the fact that {when splitting cluster-size haloes by their half-mass scales or recent accretion rates, the two sample have indistinguishable halo clustering properties}%
\footnote{We note that for low-mass haloes, which we do not address in this work, {there exists assembly bias even with this strict definition}.}.
The reason for this nomenclature is very likely due to the common understanding that halo assembly history is correlated with many halo properties, including concentration, spin, and so on. Nevertheless, the fact is that {some} direct measures of halo assembly history (e.g., $\ensuremath{a_{1/2}}$ and $\ensuremath{\Gamma_\text{peak}}$) result in no significant secondary bias at this mass regime, and this causes the ``assembly bias'' nomenclature to be potentially confusing and misleading. We will discuss this seemingly counter-intuitive result further in \autoref{sec:discussion}.
\subsection{Secondary halo bias through the demography of paired haloes}
\label{sec:markdist}
\begin{figure*}
\centering\includegraphics[width=0.9\textwidth]{scatter.pdf}
\caption{\label{fig:scatter}%
A matrix of two-dimensional distributions of the mark values for each pair of the six secondary halo properties. In each cell, the $x$- and $y$-axes both go from 0 to 1 and show the mark value of the corresponding labels. The lower triangular cells of the matrix (with blue points) depict the distributions when only including haloes that are \emph{not} in pairs, while the upper triangular cells (with orange points) exhibit the distributions including only paired haloes. Paired haloes are those with at least one other distinct halo within 10\,\ensuremath{h^{-1}}\,Mpc. The number in each cell is the value of the Spearman correlation coefficient. In the absence of correlations, the absolute value of Spearman correlation coefficient would be smaller than $0.03$ at the $99\%$ level ($3\sigma$) for a sample the size of our halo sample. Diagonal cells are left blank as only a trivial perfect correlation would appear in those cells. For each cell in the upper triangular portion of the matrix, the marginal distribution of the points would match the orange solid lines in \autoref{fig:markdist}.
}
\end{figure*}
A different, yet arguably more intuitive, way to demonstrate the halo bias is to first split the halo sample by whether or not the halo contributes to the correlation (pair counting) function. Here we define a ``paired'' halo as any distinct host halo that is within 10\,\ensuremath{h^{-1}}\,Mpc of another cluster-size distinct halo, and an ``unpaired'' halo is an isolated halo that has no other cluster-size distinct haloes within 10\,\ensuremath{h^{-1}}\,Mpc.
The choice of 10\,\ensuremath{h^{-1}}\,Mpc is, to some extent, arbitrary, but we have verified that our results are insensitive to the choice of this radius in the range of 5--20\,\ensuremath{h^{-1}}\,Mpc.
Once we divide the full halo sample into paired and unpaired subsets, we can inspect the difference between the demography of these two samples, and the difference is a manifestation of halo secondary biases.
We start with \autoref{fig:markdist}, showing the mark distribution of the six secondary halo properties for paired haloes only. If paired haloes form an unbiased subset of all haloes, the mark distribution of only paired haloes should be identical to the distribution of the full sample, which \emph{by construction} is uniformly distributed between $[0,1]$.
Consequently, the deviations from the uniform distributions in \autoref{fig:markdist} highlight the secondary halo bias. Here we find that paired haloes are strongly biased towards lower concentration and higher spin, somewhat biased towards higher number of subhaloes and larger average subhalo distance, and nearly unbiased in half-mass scale and accretion rate before peak mass. All signals are consistent with the secondary bias signals in \autoref{fig:bias}. To verify that these signals are robust, we also show the mark distribution for all paired haloes with repeated counting, to account for the fact that some haloes contribute to multiple pairs and those haloes contribute more to the bias function. Nevertheless, in both counting schemes, the results are generally the same.
Note that this ``demography of paired haloes'' approach, demonstrated with \autoref{fig:markdist}, contains similar, but not identical, information to the bias function approach shown in \autoref{fig:bias}. For example, the cross-correlation between high- and low-mark haloes does not contribute to \autoref{fig:bias}, while any halo in pairs contributes to \autoref{fig:markdist}. Furthermore, \autoref{fig:markdist} helps us to understand the demography of paired haloes by allowing us to inspect the actual mark distribution for paired haloes. We can see that, for instance, the distributions of marks that exhibit secondary bias, are generally monotonic with respect to the mark values. This feature indicates that the secondary bias is not due to intermediate mark values but mostly due to high- or low-end tails in the mark distribution. In the next section we will further see the usefulness of this ``demography of paired haloes'' approach.
\section{Discussion}
\label{sec:discussion}
\subsection{Correlations between halo properties}
\label{sec:scatter}
The term ``halo assembly bias'' has been widely used to refer to any kind of secondary halo bias. While this is an issue of nomenclature, the extensive use of the term ``halo assembly bias'' has a significant drawback, as it implicitly suggests that all secondary halo bias may have originated from differences amongst the assembly histories of haloes. However, for haloes with the characteristic masses of galaxy clusters, we do not find significant secondary bias for \ensuremath{a_{1/2}}{} and \ensuremath{\Gamma_\text{peak}}{}, the two halo properties that we study which are directly related to halo assembly histories. This finding is in broad agreement with the detailed findings in the previous literature, though not always with the physical interpretation of them. In particular, despite the lack of assembly bias in its strict sense, there is still clear secondary halo bias for other properties that are correlated with halo assembly history, such as halo concentration. How can we understand better these counter-intuitive and seemingly contradictory results?
We start by inspecting the correlations amongst the secondary halo properties. \autoref{fig:scatter} shows the scatter plot and the Spearman correlation coefficient between the mark values for each pair amongst the six secondary properties. \autoref{fig:scatter} shows these distributions for two subsets of haloes, namely paired haloes only (upper triangular cells in orange) and unpaired haloes only (lower triangular cells in blue). In order to interpret \autoref{fig:scatter}, it is useful to consider that in the absence of correlations, the Spearman correlation coefficient for a sample of the size of our halo subsets would be limited to an absolute value less than 0.03 at the 99.7\% ($3\sigma$) level. Therefore, for example, the correlation coefficient of $-0.06$ describing the correlation between $\lambda$ and $\ensuremath{R_\text{mem}}$ is a weak, but likely real, correlation. Correlation coefficients with a magnitude larger than this are very highly significant.
At first glance, most correlations are as expected based upon the previous literature. The half-mass scale \ensuremath{a_{1/2}}{} and the accretion rate before peak mass \ensuremath{\Gamma_\text{peak}}{} are strongly correlated, because the assembly histories of haloes are rather universal and in most cases can be well described by a one-parameter function \citep{Wechsler2002,Wu2013}.
Halo concentration \ensuremath{c}{} is well correlated with both \ensuremath{a_{1/2}}{} and \ensuremath{\Gamma_\text{peak}}{}, consistent with the finding of \citet{Wechsler2002}. Halo concentration \ensuremath{c}{} is also fairly correlated with the number of subhaloes \ensuremath{N_\text{sub}}{}, consistent with the finding of \citet{Zentner2005,Mao2015}. The halo spin parameter \ensuremath{\lambda}{} is correlated with \ensuremath{c}{}, \ensuremath{a_{1/2}}{}, and \ensuremath{\Gamma_\text{peak}}{}, also consistent with the finding of \cite{2007MNRAS.378...55M}.
The average subhalo distance does not exhibit strong correlations with the other five halo properties, but correlates somewhat weakly with \ensuremath{a_{1/2}}{} and \ensuremath{\Gamma_\text{peak}}{}; this is consistent with the findings of \citep{More2016}.
A more unexpected feature of \autoref{fig:scatter} is that the correlations amongst these halo properties \emph{seem} to be nearly the same for paired and unpaired haloes, in terms of both the features in the scatter plots and the values of the Spearman correlation coefficients. This may be counter-intuitive as one might naively expect that if two properties are well correlated (such as concentration and half-mass scale), then both of them should result in secondary biases of roughly the same magnitude. However, this naive expectation is mathematically unfounded. As we demonstrate next, the distribution of the orange points (paired haloes) differs slightly from the distribution of the blue points (unpaired haloes). Because of this small difference, the correlation between two secondary properties, say concentration and half-mass scale, is far from a guarantee of that these two properties would result in similar secondary biases.
\subsection{Correlation does not imply secondary bias}
\label{sec:demo}
\begin{figure}
\centering\includegraphics[width=\columnwidth]{demo.pdf}
\caption{\label{fig:demo}%
An illustration of two highly correlated variables that result in different biases when a subset is selected. The main panel shows the two-dimensional distribution (corresponding to \autoref{fig:scatter}) of two fictitious variables $X$ and $Y$, both uniformly distributed between $[0,1]$ for all points. The top and right-hand panels show the marginal probability distributions (corresponding to \autoref{fig:markdist}) of $X$ and $Y$, respectively. A subset of points (shown in orange, approximately 40\% of the total points) exhibit bias in $Y$ but not in $X$. The numbers in parentheses are the Spearman correlation coefficient.
}
\end{figure}
\begin{figure*}
\centering\includegraphics[width=0.8\textwidth]{c-ahalf.pdf}
\caption{\label{fig:c-ahalf}%
Left panel shows the two-dimensional distribution of the marks on the concentration parameter (\ensuremath{c}{}) and the half-mass scale (\ensuremath{a_{1/2}}{}), for unpaired (blue points) and paired (orange points) haloes. The distributions are the same as those in \autoref{fig:scatter}, but here we overlay them for a direct comparison. Right panel shows the mean value of the \ensuremath{c}{}-mark, conditioned on the \ensuremath{a_{1/2}}{}-mark; the black dashed, blue solid, and orange solid lines show the sample of all, paired, and unpaired haloes, respectively.
The standard error of the mean is shown by error bars, but too small to be seen.
}
\end{figure*}
To demonstrate this important statement that two highly correlated halo properties can result in distinctly different secondary biases, consider a fictitious sample of 30,000 points that are described by two highly correlated variables $X$ and $Y$. The variables $X$ and $Y$ yield a Spearman correlation coefficient of $-0.90$, far more significant than the correlations amongst any of our halo secondary properties aside from the correlation of $\ensuremath{a_{1/2}}$ with $\ensuremath{\Gamma_\text{peak}}$. From amongst these 30,000 points, we select approximately 40\% of the total points and put them in a subset $S$. \autoref{fig:demo} shows the two-dimensional and marginal distributions of $X$ and $Y$, for both points in and out of the subset $S$. We can see that the subset $S$ exhibits bias in $Y$ but not in $X$, despite the very strong correlation between $X$ and $Y$. Of course, the selection of the points in $S$ is not random. The subset $S$ is constructed by preferably selecting points with lower $Y$ in bins of $X$, and hence it is by construction that this subset results in a different bias in $X$ and $Y$. Mathematically speaking, for the full sample we have $P(X)=P(Y)=1$ and $P(X,Y)=P(X|Y)=P(Y|X)$. However, for the subset $S$, we alter the conditional distribution $P_{S}(Y|X)$ so that it differs from $P_{S}(X|Y)$, and hence the marginal distributions $P_S(X)$ and $P_S(Y)$ differ.
The two-dimensional distribution in \autoref{fig:demo} is an analogy for \autoref{fig:scatter}, and the marginal distributions in \autoref{fig:demo} are analogies for \autoref{fig:markdist}. The subset of points $S$ is analogous to the subset of paired haloes. In fact, the marginal distributions for each cell in the upper triangular part of \autoref{fig:scatter} exactly corresponds to the mark distributions of paired haloes, shown by the orange solid lines in \autoref{fig:markdist}. Hence, even though by eye the two-dimensional distributions of two marks for paired and unpaired haloes look very similar, the small difference between them can result in markedly different secondary clustering biases.
To take an even closer look at the particular case of halo concentration and half-mass scale, we overlay the two-dimensional distributions of the mark values of halo concentration and half-mass scale for paired and unpaired haloes, and show them in the left-hand panel of \autoref{fig:c-ahalf}. Clearly, $\ensuremath{a_{1/2}}$ and $\ensuremath{c}$ are similarly correlated for both subsamples. Indeed, they have Spearman correlation coefficients that are consistent with each other given the sample size. However, we can already see the small difference between the two-dimensional distributions for paired and unpaired haloes by eye. To quantify this difference, in the right-hand panel of \autoref{fig:c-ahalf}, we show the mean value of the concentration mark, conditioned on half-mass scale mark, for the sample of all, paired, and unpaired haloes.
We find that, although concentration and half-mass scale are similarly correlated for paired and unpaired haloes, paired haloes have a slightly lower concentration at a given half-mass scale than unpaired haloes, and this small difference gives rise to their different behaviours in secondary bias.
At this point, it may be useful to summarize the phenomenology of \ensuremath{c}{}- and \ensuremath{a_{1/2}}{}-dependent halo clustering. Haloes exhibit secondary bias based on \ensuremath{c}{} because paired haloes constitute a subset of haloes with preferentially lower concentrations (in the mass range we consider). However, this subset of paired haloes also has the \emph{same} distribution of \ensuremath{a_{1/2}}{} as unpaired haloes (as shown in \autoref{fig:markdist}). Consequently, high-mass haloes exhibit \ensuremath{c}{}-dependent secondary bias, but do not exhibit $\ensuremath{a_{1/2}}$-dependent (or $\ensuremath{\Gamma_\text{peak}}$-dependent) secondary bias despite \ensuremath{a_{1/2}}{} and \ensuremath{c}{} being strongly correlated. Selecting haloes based upon whether or not they have a neighbour alters the conditional distributions $P(\ensuremath{c}|\ensuremath{a_{1/2}})$ and $P(\ensuremath{a_{1/2}}|\ensuremath{c})$, in a highly non-trivial manner. The underlying physical reasons for this shift in halo properties induced by proximity to neighbour haloes are not immediately apparent, but our work suggests that halo properties are related to environment in a manner that is considerably more complex than is commonly assumed.
Interestingly, the story of $\ensuremath{R_\text{mem}}$ is nearly the opposite of that for \ensuremath{a_{1/2}}{}. While \ensuremath{R_\text{mem}}{} exhibits a secondary bias similar to \ensuremath{c}, \ensuremath{\lambda}, and \ensuremath{N_\text{sub}}{} (see \autoref{fig:markdist}), it is only weakly correlated with those three properties (see \autoref{fig:scatter}), for both paired and unpaired haloes. The lack of correlation of two variables by no means implies they cannot have similar or even the same marginal distribution, which is indeed what happens here. For example, the mark distributions of \ensuremath{R_\text{mem}}{} and \ensuremath{N_\text{sub}}{} for paired haloes are very similar, but these two properties are also the least correlated amongst the properties studied here.
As a consequence, one should be very cautious when studying the secondary bias due to one halo property through the use of a different halo property as a proxy. Even when the two properties, namely the halo property of interest and the proxy, have been shown to be strongly correlated, it is not true that the two properties must exhibit similar secondary biases. Likewise, when two properties both exhibit similar secondary biases, they may still have little correlation with each other.
\subsection{Implication for galaxy assembly bias}
\label{sec:galaxy-bias}
The term ``assembly bias'' is also frequently used to refer to ``galaxy assembly bias,'' which does not have a single clear definition. In most contexts galaxy assembly bias means that the clustering properties of galaxies depend on some galaxy properties at a fixed host halo mass. With what we have learned here, we shall take a closer look at the idea of galaxy assembly bias. Consider a galaxy property $G$, which depends on a halo property $H$. If $G$ and $H$ are highly correlated and their conditional distributions $P(G|H)$ and $P(H|G)$ are not altered by the presence of nearby haloes (e.g., in the context of our examples, $P(G|H)$ and $P(H|G)$ stay the same for both paired and unpaired haloes), then the secondary halo bias due to $H$ induces a ``galaxy assembly bias'' due to $G$. However, if the conditional distributions $P(G|H)$ and $P(H|G)$ are altered for haloes in pairs, such as $(X,Y)$ and $(\ensuremath{a_{1/2}},\ensuremath{c})$ in our examples above, then the secondary halo bias due to $H$ does \emph{not} guarantee any bias signal due to $G$. Similarly, any bias signal due to $G$ cannot be used to infer the existence of an underlying secondary halo bias due to $H$. Likewise, seeing both biases due to $G$ and $H$ does not guarantee a correlation between the galaxy property $G$ and the halo property $H$.
In short, the correlation between two variables and the dependence of clustering properties on these two variables do not have a firm connection. This statement is particularly evident for the secondary halo biases for cluster-size haloes, but it is a general, mathematical statement. Hence, using ``assembly bias'' to refer to different kinds of secondary halo bias and even bias in galaxy clustering is potentially misleading.
\subsection{Physical robustness of the secondary halo bias signals}
\label{sec:robust}
Despite some of the counter-intuitive and seemingly contradictory results that we have presented, the secondary halo biases we show in this work are, in fact, physically robust. One may suppose that either \ensuremath{a_{1/2}}{} or \ensuremath{\Gamma_\text{peak}}{} or both do not exhibit significant secondary halo clustering bias because they are measured with considerably more noise than the other halo properties that we explore or because these measures do not probe the particularly important epochs of halo formation. As we will show, this is not the case. Moreover, the existence of other secondary halo biases (due to concentration, spin, and subhalo properties) is also physically robust.
To demonstrate the robustness of the secondary halo biases that we present above we proceed as follows. For each of the six secondary properties, we identify another, similar property that has approximately the same physical meaning but defined differently. We then investigate whether or not this similar halo property exhibits a similar secondary halo bias. This new set of six properties are as follows.
\begin{enumerate}
\itemsep0.5\baselineskip
\item Maximal circular velocity (\ensuremath{V_\text{max}}). For a halo that follows a NFW profile perfectly, the maximal circular velocity is a simple function of halo mass and concentration. At fixed mass, higher \ensuremath{V_\text{max}}{} implies higher halo concentration.
\item Spin parameter (\ensuremath{\lambda_\text{Bullock}}), as defined in \citet{Bullock2001}, which has a different normalization compared to the Peebles spin parameter.
\item Scale of last major merger(\ensuremath{a_\text{LMM}}), defined as the scale factor at which the halo experiences its last major merger on its main branch. A major merger is defined as a merging event of two haloes with a mass ratio that is larger than one-third.
\item Present-day instantaneous accretion rate (\ensuremath{\Gamma_\text{inst}}), it is defined as the mass change rate between two adjacent snapshot outputs on the main branch. For MDPL2, this rate is calculated between $z=0$ and 0.0224, which corresponds to approximately 300\,Myr.
\item Peak maximal circular velocity of the largest subhalo (\ensuremath{V_\text{peak}^\text{(1st sub)}}). Here largest subhalo means the subhalo that has the largest \ensuremath{V_\text{peak}}{} values of all subhaloes in that distinct halo. At a fixed distinct halo mass, this value is correlated with the concentration of the host halo and with the number of subhaloes \citep{Mao2015}.
\item Average subhalo distance weighted by subhalo mass (\ensuremath{R_\text{mem}^\text{(weighted)}}), defined as the average three-dimensional distance between all subhaloes and the centre of the main halo, with the contribution of each subhalo weighted by the subhalo mass. The subhalo definition is the same as the definition used to calculate the number of subhaloes for each distinct halo.
\end{enumerate}
\begin{figure}
\centering\includegraphics[width=\columnwidth]{bias_more.pdf}
\caption{\label{fig:bias-more}%
Same as \autoref{fig:bias} but for six different (but related) secondary halo properties: (solid lines from top to bottom) \ensuremath{\lambda_\text{Bullock}}, \ensuremath{R_\text{mem}^\text{(weighted)}}, \ensuremath{V_\text{peak}^\text{(1st sub)}}, \ensuremath{a_\text{LMM}}, \ensuremath{\Gamma_\text{inst}}, and \ensuremath{V_\text{max}}.
For \ensuremath{V_\text{max}} (blue solid line), we also show the inverse ratio as a blue dashed line to guide an easy comparison with bias due to other secondary halo properties.
}
\end{figure}
\autoref{fig:bias-more} shows the secondary bias (as ratio of pair counting functions) for these six secondary properties, and the behaviours of the secondary biases are very similar to the six properties used in \autoref{fig:bias}. The spin parameter with \citet{Bullock2001} definition still exhibits the largest bias. The maximal circular velocity exhibits the second most significant secondary bias, similar to, but slightly stronger than, the secondary bias exhibited by the concentration parameter. The two assembly history-related properties (\ensuremath{a_\text{LMM}}{} and \ensuremath{\Gamma_\text{inst}}{}) again exhibit clustering that is consistent with random sampling and do not indicate secondary halo bias based upon halo mass assembly history. Lastly, the two subhalo-related properties (\ensuremath{V_\text{peak}^\text{(1st sub)}}{} and \ensuremath{R_\text{mem}^\text{(weighted)}}{}) show moderate secondary bias.
As our discussion in \autoref{sec:demo} points out, correlation does not imply similar clustering bias. Hence the similar secondary bias signals we observe in \autoref{fig:bias} and \ref{fig:bias-more} can be interpreted to indicate that both properties in each pair affect the clustering properties. It also implies that the two conditional distributions of each pair of properties stay roughly the same for paired and unpaired haloes. In other words, the small change in definitions, in these particular cases, does not alter the joint distribution in a way that would result in different marginal distributions for paired or unpaired haloes (see \autoref{fig:conditional-mean} for a full comparison of the conditional distributions amongst all properties).
\begin{figure*}
\centering\includegraphics[width=\textwidth]{history.pdf}
\caption{\label{fig:history}%
Stacked main-branch assembly histories for halo mass $M(a)/M(a=1)$ (upper row) and for maximal circular velocity $\ensuremath{V_\text{max}}(a)/\ensuremath{V_\text{max}}(a=1)$ (lower row).
In each panel, the full sample of cluster-size haloes is split to show the difference between the stacked assembly histories of the two subsamples.
In the three columns, the sample is split by concentration mark (left column; orange dashed line for the 50\% low-concentration haloes and blue solid line for the 50\% high-concentration),
by whether or not the haloes reside in pairs (middle column; orange dashed line for paired haloes and blue solid line for unpaired),
and by large-scale density mark (left column; orange dashed line for the 50\% haloes in high-density regions and blue solid line for the 50\% haloes in low-density regions).
Here paired haloes are haloes that have neighbour haloes within 10\,\ensuremath{h^{-1}}\,Mpc), and the large-scale density is calculated by summing the total mass$^\text{\ref{footnote:halo-proxy}}$ within a 20\,\ensuremath{h^{-1}}\,Mpc-radius sphere.
For each subsample under consideration, the line shows the median mass assembly history, and the corresponding band shows the 16$^\text{th}$ and 84$^\text{th}$ percentiles.
This plot is made with the DarkSky-Gpc simulation.
}
\end{figure*}
\subsection{{Full halo assembly histories and bias}}
\label{sec:history}
So far, we have inspected four different summary statistics of halo mass assembly history: \ensuremath{a_{1/2}}, \ensuremath{a_\text{LMM}}, \ensuremath{\Gamma_\text{peak}}, and \ensuremath{\Gamma_\text{inst}}, and \emph{none of them} exhibits any statistically significant secondary clustering bias. Given the similarity of halo mass assembly histories, which can be described well by one or two parameters, {one might \emph{speculate} that, for high-mass haloes, the entire halo mass assembly history is independent of the environment}. With the ``demography for paired haloes'' approach that we introduced in \autoref{sec:markdist}, we can directly inspect the mass assembly history for haloes {in different environments to further investigate the relationship between bias and assembly history.}
The upper row of \autoref{fig:history} shows the stacked (median) mass assembly histories for main-branch progenitors as a function of scale factor, $M(a)/M(a=1)$, using three different ways to split the full cluster-size halo sample into two subsamples. To produce \autoref{fig:history}, we used the DarkSky-Gpc simulation to obtain the full mass assembly histories for all distinct haloes that have a present-day mass $\ensuremath{M_\text{vir}} \geq 10^{14}\,\ensuremath{h^{-1}}\,\text{M$_\odot$}$. On the left of \autoref{fig:history}, we split haloes by their concentration mark. In the middle, we split haloes by whether or not they are in pairs; here a ``paired halo'' is again defined as any distinct halo that has at least one other distinct halo closer than 10\,\ensuremath{h^{-1}}\,Mpc. On the right, we split haloes by their {mass-normalized mark values of} large-scale matter density; we calculate the matter density by summing up the masses of all resolved haloes%
\footnote{We calculate the mass by summing up halo masses within 20\,\ensuremath{h^{-1}}\,Mpc spheres because we do not have direct access to the full particle snapshot of DarkSky-Gpc. To make this approximation as close as the actual matter distribution, we use all distinct haloes identified in the DarkSky-Gpc halo catalogue, down to a minimal halo mass of $4.88 \times 10^{10}$\,\ensuremath{h^{-1}}\,\text{M$_\odot$}{} (20 particles). These haloes, in total, contain 42.5\% of the total matter mass in the simulation box. For our purpose of ranking the large-scale matter densities, this method provides good approximation, as we have verified using independent simulations. \label{footnote:halo-proxy}}
within a 20\,\ensuremath{h^{-1}}\,Mpc sphere around each distinct halo in our sample, and then calculate the mass-normalized mark values using the procedure outlined in \autoref{sec:marks}.
We can immediately see that the stacked mass assembly histories for paired and unpaired haloes are \emph{essentially identical}. The 16$^\text{th}$ and 84$^\text{th}$ percentiles also match between the two samples, indicating that the variety of possible assembly histories is quite similar for both paired and unpaired haloes.
Furthermore, when the halo sample is split by the large-scale matter density around the haloes, haloes in denser regions have nearly identical stacked assembly history as those haloes in less dense regions, as shown in the upper right-hand panel of \autoref{fig:history}.
The lack of difference between the assembly histories of paired and unpaired haloes (or of haloes in high- and low-density regions) is \emph{not} caused by the stacking procedure. As the upper left-hand panel of \autoref{fig:history} shows, when the haloes are split by the concentration mark, there is a clear difference in the stacked assembly histories. The trend is consistent with our expectation, that high-concentration haloes form early. {One should also note that these two groups of haloes \emph{do} have different clustering biases, as we know the concentration bias does exist at this mass scale.}
We have also verified that the lack of difference in the assembly histories for haloes in different environments is insensitive to the radii used in the definitions of paired haloes and large-scale density (the result holds when using {5, 10, 20, 30, and 40}\,\ensuremath{h^{-1}}\,Mpc), and is also insensitive to how we split the density mark (the result holds when selecting the 25\% or 10\% most extreme mark values).
In addition, we further inspect the history of the maximal circular velocity (\ensuremath{V_\text{max}}) for main-branch progenitors, which represents the mass assembly history for the core of a halo. As the lower row of \autoref{fig:history} shows, we again find that the stacked \ensuremath{V_\text{max}}{} histories of the paired and unpaired haloes, or of haloes in high- and low-density regions, are nearly identical. The concentration-split histories show a difference that is consistent with the difference in mass assembly history. We also find that the probability distributions of the number of major mergers (mass ratio between $1/3$ and 1) that happened along the main branch are also essentially the same for paired and unpaired haloes.
These findings are in good agreement with our main results and also with \citet{Gao2005} and \citet{2008MNRAS.389.1419L}. {At this mass scale ($\gtrsim 10^{14}$\,\ensuremath{h^{-1}}\text{M$_\odot$}), haloes that are in different environments do not have significantly different assembly histories.
However, this does not guarantee that a correlation of any form between large-scale halo clustering and halo assembly history does not exist. Our results demonstrate that any such correlation is not evident either from the summary statistics that have been explored here or in the stacked assembly histories. The reasons for this phenomenon are twofold. First, at this mass scale, the strength of the secondary bias is small, and hence even if the $c-\ensuremath{a_{1/2}}$ relation were not biased for haloes in denser environments, the difference in the stacked assembly histories of haloes in different environments would still be modest. Secondly, as we discussed in \autoref{sec:demo}, the $c-\ensuremath{a_{1/2}}$ relation is slighted biased for haloes in denser environments, and it cancels out any remaining difference in the assembly histories for haloes in different environments.}
{It is possible to estimate the relative sizes of these two effects. In \autoref{fig:history_diff}, we plot the relative differences between the stacked mass assembly histories of subgroups of haloes split by various halo properties. In each case, we split the samples into two equal-sized subgroups about a mark value of 0.5. For example, the green line in \autoref{fig:history_diff} shows the normalized difference between the blue and orange lines in the upper left-hand panel of \autoref{fig:history}. Splitting haloes by their
\ensuremath{c}{}, \ensuremath{a_{1/2}}{}, or \ensuremath{\Gamma_\text{peak}}{} (green solid, orange dash--dotted, and red dotted lines respectively), yields significant differences in mass assembly histories, as one would expect. On the other hand, splitting haloes by large-scale density (blue solid line) yields assembly histories that are extraordinarily similar. This is another representation of our previous results.
}
{
We can now explore what would be expected of the mass assembly difference between haloes selected by density due only to the fact that density is correlated with concentration. To compute this, we place haloes into narrow bins of concentration mark and then randomly shuffle the density marks. We then split haloes based upon the shuffled density mark. In this manner, the selection upon density becomes meaningless, but the two sub-populations have concentration mark distributions that are identical to the subgroups split on actual density. The mass accretion history difference constructed in this way is shown by the purple dashed line in \autoref{fig:history_diff}. This line shows the slightly different mass accretion histories of haloes due to the fact that selecting upon density also introduces small differences in the concentration distributions of the two halo subgroups. While this difference is small relative to selecting upon concentration or formation time directly, it is interesting that this difference is significantly larger than the difference in mass accretion histories induced by selecting on {\em actual} density. Despite the fact that selecting upon density also selects populations with slightly different concentration distributions, selecting upon {\em actual} density yields a nearly undetectable difference in mass accretion histories. The relation between concentration and mass accretion history is density dependent in such a way as to render the different mass accretion histories nearly identical. The difference in mass accretion histories exhibited by the purple dashed line in \autoref{fig:history_diff} is also what we would have observed if the secondary bias due to \ensuremath{a_{1/2}}{} or \ensuremath{\Gamma_\text{peak}}{} were as significant as the concentration bias.
}
{A common assumption has been that the secondary bias due to halo concentration is merely a consequence of the simple halo assembly bias. In other words, it has been commonly believed that secondary bias due to concentration was induced by the combination of assembly bias and the formation time--concentration relation. Given what we have learned here, the concentration bias seems more intriguing at this mass scale. In fact, one extremely interesting feature in \autoref{fig:history_diff} is that when the haloes are split by the concentration mark, the difference in the mass assembly history {is similar to} the difference when the haloes are split by half-mass scale or accretion rate {only at early time ($a < 0.35$)}. This suggests that, at this mass scale, halo concentration, as a summary statistic, {captures only early-time assembly history and is more correlated with large-scale bias.} We leave this intriguing phenomenon for future study. }
\begin{figure}
\centering\includegraphics[width=\columnwidth]{history_diff.pdf}
\caption{\label{fig:history_diff}%
{Similar to \autoref{fig:history} but showing the relative difference in the stacked (median) main-branch mass assembly histories for two groups of haloes, for several different ways to split the haloes: by mark values of large-scale (20 \ensuremath{h^{-1}}\,Mpc) density (blue solid, bottom-most), concentration (green solid, top-most), half-mass scale (orange dash--dotted), or accretion rate after peak mass (red dotted lines), and shuffled density (purple dashed line); all splits are made at the mark value of 0.5. The case when the haloes are split by shuffled density, each of the two groups has the same concentration distribution as in the case of split by large-scale density.
This plot is made with the DarkSky-Gpc simulation. }
}
\end{figure}
\subsection{Secondary biases due to subhalo properties}
In \autoref{fig:bias} and \ref{fig:bias-more} we see that the number of subhaloes and the average subhalo distance (both weighted and unweighted) give significant secondary bias, and that the peak maximal circular velocity of the largest subhalo, which at a fixed distinct halo mass represents the gap between the first subhalo and the distinct halo, also exhibits a modest secondary bias signal.
Similar to halo concentration, both the number of subhaloes and the average subhalo distance are correlated with the assembly history of the parent distinct halo, and both of them exhibit secondary bias despite the lack of assembly bias. This result is in good agreement with the findings in \citet{More2016}.
The secondary halo biases due to these subhalo properties are particularly interesting because they are more likely to be directly observable; at this mass scale, most large subhaloes would host galaxies.
For example, if the way galaxies \emph{populate} subhaloes is not influenced by large-scale environment (although there is no clear evidence for such a dependence, it could exist), then the secondary bias due to subhalo occupation would directly translate to observable galaxy occupation bias (i.e., the dependence of central galaxy clustering on the member occupation or richness). Similarly, the average galaxy member distance will exhibit a bias signal if the galaxy--subhalo connection is not influenced by large-scale environment.
There are, however, complications to these potential observables. First, as we have discussed in \autoref{sec:galaxy-bias}, the galaxy occupation bias or the average galaxy distance bias does not directly translate to the halo concentration bias or the halo assembly bias (in fact, we have already shown the latter does not exist for cluster-size haloes).
Secondly, observationally, projection effects and redshift-space distortions can contaminate member assignment, and potentially produce artificial bias signal \citep{2016arXiv161100366Z,1702.01682}.
Hence, one should be cautious when interpreting the galaxy occupation bias or the average galaxy distance bias.
\section{Conclusion}
\label{sec:conclusion}
In this study, we have revisited the complex phenomenon of secondary halo bias, which is commonly referred to as ``halo assembly bias,'' for cluster-size haloes ($\ensuremath{M_\text{vir}} \geq 10^{14}\,\ensuremath{h^{-1}}\,\text{M$_\odot$}$) using large-volume simulations. As part of this investigation, we presented a novel approach to highlight secondary halo bias. Our approach was to study the demographics of paired and unpaired haloes, enabling us to determine whether or not the distributions of halo properties are different for haloes in pairs compared to the full halo sample.
Using both halo two-point functions and paired halo demographics, we found that halo concentration, halo spin, number of subhaloes, and average subhalo distance all exhibit significant secondary halo biases at the cluster mass scale. Amongst these properties, halo spin exhibits the strongest secondary halo bias, in the sense that high-spin haloes cluster more strongly. Low-concentration haloes cluster more weakly than high-concentration haloes, a dependence that is the converse of concentration-dependent secondary halo bias at lower masses. Cluster-size haloes cluster more strongly as a function of both subhalo number and the average distance between subhaloes and the halo centre.
We have identified no statistically significant secondary halo bias at this mass scale for any of four halo properties that directly measure the mass assembly history: half-mass scale, accretion rate before peak mass, instantaneous accretion rate, and time of last major merger. We have found that the entire main-branch mass assembly histories of paired and unpaired haloes (or haloes in different large-scale densities) are statistically identical. {This suggests that the assembly histories of massive, cluster-size haloes do not correlate with their environments in a simple fashion. This is not equivalent to the statement that there exists no features of halo assembly histories that do correlate with environment and/or clustering strength. Indeed, some particular features of the halo assembly histories, such as those captured by halo concentration, may still correlate with the large-scale environment.}
With our halo demographic approach, we further investigated the seemingly contradictory result of the lack of secondary halo bias due to assembly history-related properties, given the clear correlation between halo concentration and halo assembly history. We have demonstrated that the correlation between two variables is, in general, not relevant to the question of whether or not the two variables will result in similar secondary halo clustering biases. If the conditional distributions of the two variables are altered, even only slightly, for paired haloes, then the two variables can easily result in very different secondary clustering biases. For instance, halo concentration and half-mass scale are similarly correlated for paired and unpaired haloes, yet paired haloes have a slightly lower concentration at a given half-mass scale, which results in their different secondary bias signals. Likewise, the fact that two variables (e.g., halo concentration and average subhalo distance) yield similar secondary clustering biases by no means implies a correlation between the two variables. These statements have the following important consequence: Even when the presence of a galaxy in a halo is a function of a halo property that exhibits secondary halo bias, this does \emph{not} necessarily imply galaxy ``assembly bias.''
Our study has provided a comprehensive view of secondary halo biases for cluster-size haloes, and leads to a different perspective on secondary halo biases. In particular, we caution against use of the term ``assembly bias,'' particularly for cluster-size haloes. The term ``assembly bias,'' when used to refer to secondary biases other than those due to the assembly history (e.g., concentration-dependent halo clustering), implies that all such biases have a common origin rooted in some aspect of the mass assembly histories of haloes.
{While these different secondary biases may still all have some connections with the halo assembly history, those connections are more complex than simple correlations amongst different halo properties.
In particular, the halo secondary bias due to concentration is not a direct consequence of the difference in bias between early- and late-forming haloes.}
Our results regarding halo assembly histories have an important consequence for the interpretation of halo clustering. At face value, our result is in qualitative agreement with early analytic studies of halo abundance and clustering using excursion set theory, which predicted no assembly bias (specifically, \citealt{Kaiser84,Cole1989,1991ApJ...379..440B,Mo1996,Sheth2001}; see \citealt{10.1142/S0218271807010511} for a review and subsequent developments). Yet, these predictions stem from ad hoc assumptions adopted for computational ease, rather than for any well-established physical reasons. Several authors have proposed more physically-motivated analytic interpretations of secondary halo bias based upon the assembly histories of haloes \citep{10.1142/S0218271807010511,Desjacques2007,Dalal2008}. {However, given our findings, these analytic interpretations of halo assembly bias do not manifest in all different kinds of secondary halo biases in a straightforward manner. Indeed, these interpretations contradict our finding that high-mass haloes cluster independently of halo formation time and other simple metrics of halo age. The explicit relations amongst large-scale environment, halo assembly history, and sundry internal halo properties, such as concentration and spin, remain unclear.}
While this study has demonstrated that a very complex phenomenology of secondary halo bias is mathematically possible, it has yet to provide a solid physical explanation for the existence of the concentration bias, the spin bias, the subhalo abundance bias, and the average subhalo distance bias. Previous attempts to explain these secondary biases that rely upon the existence of halo assembly bias in its restricted definition are not valid in this mass regime. It is also important to understand why the correlations between particular halo properties (e.g., between subhalo abundance and assembly history) depend on large-scale environment. We have not yet identified a plausible working theory that is able to explain all of the correlations. We hence leave this interesting problem for future study, with the hope that what we have laid out in this study will provide useful insights.
\section*{Acknowledgements}
The authors thank Andreas Berlind, Philipp Busch, Neal Dalal, \mbox{Andrew} Hearin, Ari Maller, Surhud More, Rita Tojeiro, \mbox{Li-Cheng} Tsai, Frank \mbox{van den Bosch}, Antonio \mbox{Villarreal}, and Kuan Wang for useful discussions.
This research made use of the MDPL2 simulation; the authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.\ (\http{www.gauss-centre.eu}) and the Partnership for Advanced Supercomputing in Europe (PRACE, \http{www.prace-ri.eu}) for funding the MultiDark simulation project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, \http{www.lrz.de}), and also thank Peter Behroozi for the direct access to the MDPL2 halo catalogues on SLAC servers.
This research made use of the DarkSky-Gpc (\texttt{ds14\_b}) simulation, which was part of the Dark Sky Simulations (\http{darksky.slac.stanford.edu}) produced using an INCITE 2014 allocation on the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory, a U.S.\ Department of Energy Office; the authors thank Sam Skillman, Mike Warren, Matt Turk, and other Dark Sky collaborators for their efforts in creating these simulations and for providing access to them.
This research made use of computational resources at SLAC National Accelerator Laboratory, a U.S.\ Department of Energy Office; YYM and RHW thank the support of the SLAC computational team.
YYM is supported by the Samuel P.\ Langley PITT PACC Postdoctoral Fellowship.
ARZ is supported, in part, by grants AST 1516266 and AST 1517563 from the U.S.\ National Science Foundation as well as by the Pittsburgh Particle physics, Astrophysics, and Cosmology Center (PITT PACC) at the University of Pittsburgh.
RHW received partial support from the U.S.\ Department of Energy contract to SLAC No.\ DE-AC02-76SF00515.
This work was completed at the Kavli Institute for Theoretical Physics at the program ``The Galaxy--Halo Connection Across Cosmic Time,'' and supported in part by the National Science Foundation under Grant No.\ NSF PHY-1125915.
This research made use of Python, along with many community-developed or maintained software packages, including
IPython \citep{ipython},
Jupyter (\http{jupyter.org}),
Matplotlib \citep{matplotlib},
NumPy \citep{numpy},
Pandas \citep{pandas},
and SciPy \citep{scipy}.
This research made use of NASA's Astrophysics Data System for bibliographic information.
\bibliographystyle{mnras}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,966 |
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"A paranormal adventure series that will appeal to fantasy and historical fans, plus time-travel lovers as well...Delightful entertainment."
— _Romance Junkies_
"An intriguing new fantasy romance series...A unique combination of magic, time travel, and fantasy that will have readers looking toward the next book."
— _Time Travel Romance Writers_
"The writing is sharp and witty, and the story is charming. [Johnson] makes everything perfectly believable. She has created an enchanting situation and characters that are irascible at times and lovable at others. Jean Johnson...is off to a flying start. She tells her story with a lively zest that transports a reader to the place of action. I can hardly wait for the next one. It is a must-read."
— _Romance Reviews Today_
"A fun story. I look forward to seeing how these alpha males find their soul mates in the remaining books."
— _The Eternal Night_
"An intriguing world...An enjoyable hero...An enjoyable showcase for an inventive new author. Jean Johnson brings a welcome voice to the romance genre, and she's assured of a warm welcome."
— _The Romance Reader_
"An intriguing and entertaining tale of another dimension. It will be fun to see how the prophecy turns out for the rest of the brothers."
— _Fresh Fiction_
_Titles by Jean Johnson_
_Theirs Not to Reason Why_
A SOLDIER'S DUTY
AN OFFICER'S DUTY
HELLFIRE
_The Sons of Destiny_
THE SWORD
THE WOLF
THE MASTER
THE SONG
THE CAT
THE STORM
THE FLAME
THE MAGE
_The Guardians of Destiny_
THE TOWER
SHIFTING PLAINS
BEDTIME STORIES
FINDING DESTINY
THE SHIFTER
THEIRS NOT TO REASON WHY
# **HELLFIRE**
JEAN JOHNSON
**THE BERKLEY PUBLISHING GROUP**
**Published by the Penguin Group**
**Penguin Group (USA) Inc.**
**375 Hudson Street, New York, New York 10014, USA**
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HELLFIRE
An Ace Book / published by arrangement with the author
Copyright © 2013 by G. Jean Johnson.
All rights reserved. No part of this book may be reproduced, scanned, or distributed in any printed or electronic form without permission. Please do not participate in or encourage piracy of copyrighted materials in violation of the author's rights. Purchase only authorized editions.
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ACE and the "A" design are trademarks of Penguin Group (USA) Inc.
For information, address: The Berkley Publishing Group,
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eBook ISBN: 978-1-101-62488-3
PUBLISHING HISTORY
Ace mass-market edition / August 2013
Cover art by Gene Mollica.
Cover design by Annette Fiore Defex.
Interior text design by Laura K. Corless.
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental. The publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.
# ACKNOWLEDGMENTS
This whole series is dedicated to the men and women around the world who have been, are, and will be willing to step between their fellow sentient beings and whatever threatens them. Specifically, the military, but this includes police officers, medical personnel, firefighters, and other emergency services. As always, I have worked to create my vision of the future with a deep respect for those who serve in the present day.
While the Terran United Planets Space Force has been based along the lines of several real-world military systems, it is not meant to represent any one particular such military; for example, the TUPSF-MC is not the same as the United States Marine Corps, and the TUPSF-Navy is not the same thing as the Royal Navy of the British armed forces. Though there are several things they and other militaries have in common with the versions I have created for these four books, there are also many differences.
If you, the reader, find any difference in the various military functions in this story that you do not agree with, please remember either it was placed there because it was sifted from a different nation's military traditions, it was deliberately chosen to be different by the author, it was simply a case of ignorance on the author's part, it may have come from a different person's perspective uncovered during the many interviews I have conducted with people from various military branches around the world...or it just wound up appearing in the story because it sounded cool. (When telling a fiction story, the Rule of Cool and the Rule of Funny automatically get a higher pay grade and rank than the Rule of How Things Actually Work.)
If you have served or are serving to help defend, protect, and better your home, wherever that home may be, I salute you with respect. If you are the spouse, relative, or friend of someone who has served or is serving, I thank you deeply for the many kinds of support you give your loved ones in these services. For those of you who aren't familiar with what it is to either serve or have someone you know serve, thank you for reading this series; I hope I have given you a glimpse of military life. This series is not an accurate window into the real-world day-to-day lives of the men and women serving out there, but I have tried to give you a window into the hearts and minds of those who are willing to serve.
My thanks, as ever, go to my beta editors, Buzzy, NotSoSaintly, Alexandra, and Stormi, and to my many readers for taking a chance on these books. Thank you very much for picking up this one and the others. May you continue to enjoy my efforts to entertain and inspire you.
_Jean_
# Table of Contents
Prologue
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
# PROLOGUE
_The past is nothing more than a story we tell to each other. It is not meant for mere mortal hands to erase or sever...though it is not immutable. In fact, it can be changed, if only by God and madmeioas. To all others, it is indeed written in stone._
_Shakespeare once said, "What's past is prologue," but you must understand that this is true only because the story never ends. Yes, you had a beginning in your birth, and yes, you will have an ending in your death, but the story itself never ends. Still, each segment is preserved in one shining moment, a granite tome held up to the light of the universe so that it can be revealed in all its horrors and all its glories...and thus preserved forever. But only in the past._
_Eventually, even the slowest of readers will come to that last line, and must turn the story to a new page._
_~Ia_
SEPTEMBER 23, 2495 TERRAN STANDARD
THE TOWER
TERRAN UNITED PLANETS SPACE FORCE HEADQUARTERS
EARTH, SOL SYSTEM
The voices kept impinging on her senses, distracting her from her search. Ia wished they would shut up.
"What about Lieutenant Second Class Brad Arstoll? She served in Basic Training with him," a middle-aged woman offered.
"Yes, he's just finished his Marines Academy training. But that means he's still a new officer. Can we really risk the 2nd Platoon being led by a raw cadet?" an elderly man countered.
Their voices blended together, male and female, middle-aged and older, like the babbling of a brook in the background of her awareness.
Physically, Ia stood in one of the research rooms used by the Department of Innovations, a chamber filled with banks of workstations ringing an oval table scattered with datapads and stacks of personnel printouts. The walls themselves were mostly datascreens broken by a trio of doors. Each screen displayed a larger-than-life face and the bare-bones stats of the Service record belonging to each profiled soldier, flickering and shifting with each new suggestion.
Mentally, Ia stood on the grassy banks of Time itself, a rolling plain crisscrossed by the tangled streams of millions of lives. They formed a complex tapestry where major events, which would normally stir the waters out of their banks, were actually overshadowed by the tiniest of ripples. Changes that she had to track down.
"Arstoll may be a new officer, but he is Field Commissioned, so he does have some combat leadership experience," a second, older woman pointed out. "Plus, he's somewhat familiar with the captain even if they haven't seen each other in years. Not to mention their compatibility charts look pretty good."
"Familiarity doesn't really come into it," another male argued. "She needs a competent, combat-trained officer. I still say Lieutenant Dostoyervski is the best match— _he_ should be her second-in-command."
Something was rippling the waters of Time, disturbing her carefully laid plans like a deep, unseen current. If she didn't track it down, it could erode the bank out from under her feet. That would be bad.
"His DoI file _is_ sticky with bigger recommendations than the other candidates have," the first male agreed. "And his psych profile does match in both compatibility and contracompatibility measurements with Captain Ia, here. It looks like he'd get along with the other officers, too...well, maybe not Helstead, if she decides to be headstrong. But that's a problem for their CO to sort out. Learning to manage strong personalities could be a good lesson."
She had already dismissed Dostoyervski. He wouldn't do at all, not when her own considerations took into account several variables not even the DoI could foresee. Their voices were annoying her with trivial details. "Shhh..."
They didn't pay much attention to her, other than to speak a little more quietly. The men and women of the Department of Innovations were a different breed from the standard soldier. Most of them were career, with the average number of years in the various branches of Service rarely being less than fifteen, and usually above twenty. In fact, many of them were technically retirees from active duty, able to bring those years of long-term military experience to the task of figuring out who out there had the skills to be promoted and fast-tracked, or stalled and even demoted. Most had training in psychology and xenopsychology, tactics and long-term strategy. All of them were expert data miners.
In a unified military composed of roughly two billion soldiers, they were the best at knowing who was who in the Terran United Planets Space Force and where that person should probably go. It was their job to debate who should be one of the three Platoon lieutenants Ia needed. Their job to select the best soldiers for a particular set of tasks. Their job to make the final decision, normally.
Normally, someone in Ia's situation wouldn't even be here, let alone have much say in the process. If the psychological filtration programs and the best judgments of the DoI members came up with matches too close to call, they might contact a superior officer to solicit their opinion, yes, but that officer never came to the physical headquarters of the DoI, or even to one of its many branch offices scattered through Terran space.
However, this situation was not normal. Ia was already inside the Tower, the nickname for the sprawling, administrative heart of the Space Force on Earth. This particular branch of the DoI was located no more than a kilometer or so from the office of her new commanding officer, Admiral John Genibes of the Branch Special Forces. She was already operating under special dispensation for other reasons, including a form of _carte blanche_ —albeit one with a very strict double-indemnity clause—so Ia had arranged to visit this data-crammed room in person.
All she wanted to do was to select the perfect-for-her crew, comparing their potential actions to the needs of the right future, the one that would save their descendants from a massive calamity three centuries away. Unfortunately, the men and women around her were trying to help her select the _perfect_ crew. She didn't need perfect, as if the soldiers in question were diamonds, prepolished and cut. She needed raw material, flexible and bold, obedient yet innovative, men and women capable of doing truly great things under _her_ command, yet very carefully not needed elsewhere. Carbon fibers, not jewels.
Those who would be needed elsewhere had to remain elsewhere. What she needed were the nobodies, the throwaways whose lives wouldn't make a palpable difference anywhere else. Straw soldiers who, under her guidance, could be spun into threads of pure gold for the tapestry she needed to weave.
It should have been easy for her to sort through the many possibilities lining the path she needed the future to follow. Easy to pluck out the names, the personalities, the faces of everyone she needed. But something was wrong.
_This isn't getting me anywhere._ Working at her usual perception level, a woman standing on a low-rolling prairie crisscrossed by life-streams, she couldn't see where the subtle problems all began. _So either it's macroscale measurements, or microscale. Micro would be more accurate, but I don't even know where to start, and there's too much out there to just drop into the waters of some life-stream randomly...So, macro it is._
Visualization was usually a psychic's best friend. Grounding and centering exercises helped stabilize the mind, and mental bubble-shields walled out unwanted influences. Most of the time, exercising these abilities was analogous to humming a tune for background noise, or carrying an object; once a psi learned how, it didn't take much effort. It did, however, take time. Ia had spent the last eight years of her life training her mind to carry the weight of Time itself.
Instead of standing on a vast field, she shrunk the timeplains down to a brocaded tapestry. Life-rivers became threads as the rolling grass and rippling waters vanished. They ran in ways contrary to the normal warp and weft, more like a complex skein than a formal weave, but the analogy wasn't meant to be perfect. Lifting it up with mental hands, she peered along the edge of Time, checking for anomalies in the fabric.
She couldn't hear the voices of the others anymore, couldn't see them at the periphery of her vision. Focused on the nearly two-dimensional image held in her mind, Ia spotted the first slub a few years down from the moment of now in the pale golden tapestry stretched out before her. It was subtle indeed, visible only as a metaphor, but the beige thread was palpably thicker than the others.
It was also not alone. Now that she could see the first one, others here and there caught her attention. They were noticeable because they were just a little bit thicker than they should have been. Narrowing her attention to a close knot of those thickened life-threads, Ia queried her precognitive abilities.
_Whose lives are these? What do they have in common?_
Visualization was the key. Her vast abilities knew what was going on, but only on a subconscious level at best. Subconsciously, she sensed a hint of danger in the timeplains, just enough to prick at her instincts. Her conscious mind was still mortal, though; her intellect, smart as she was, couldn't yet sort out the differences. A merging of the two, instinct and thought, might help. _So, what parts of these threads are in common with each other, and what bits are distinct?_
Color seeped into the threads, delineating each life and its impact on the others around it. Not just the ones she saw, but new ones, extra slubs of undue influence. Lifting herself above the tapestry, Ia could see the colors, plural. More than one influenced the timestream-threads...but the later ones seemed to come into play only after the first one, a purple hue not too far off from the petals of an iris flower, had wreaked most of the initial damage over a dozen key lives.
Zooming in close, mentally floating above the weave, she examined the wisps of thread-fibers where the purple taint in the slub came close to an aquamarine one. The tiniest threads connected the two. So tiny, they looked...silver.
Feyori.
Cursing, Ia backed up—and flinched instinctively out of the timestreams, left hand snapping up, mind snapping out. Opening her eyes, she stared at the frightened sergeant dangling centimeters from her grip and centimeters off the floor, caught in her telekinetic grip. If she had lingered in there one moment more, the middle-aged woman might have actually touched her.
That would have been bad.
"Y...You..." the greying brunette panted, eyes wide. "You..."
"I said," Ia stated, as gently as she could, "that I did _not_ want to be touched." Carefully, she lowered the older woman back to her feet. "I apologize for my instinctive reaction just now—and I'm grateful I didn't hurt you with my combat reflexes—but it was either grab you with my mind, or let you injure _your_ mind precognitively. Now, did you want something?"
Licking her lips, the woman clutched her portable workstation in her dark brown arms and nodded. "Uh, yes, sir. We've completed your roster for you, Captain. All it needs is your...your...I can't believe I'm saying this," the reservist master sergeant muttered, her shock fading, replaced by a touch of startlement-induced anger. "This whole situation is highly irregular! _We_ decide who gets promoted and where they go, particularly when it's a transfer into the Special Forces."
"I know it's irregular, Narine," Ia told her, making the woman blink. Only her last name, Plimstaad, was visible on the name patch fixed to the pocket of her brown Dress jacket. "I know that very little of this is according to standard procedure. But it's necessary. As for that list of names, do not send it yet. It's still incomplete."
"The list _is_ complete, sir," she argued. "All you needed was a competent lieutenant for your 2nd Platoon. You have a first officer and three Platoon lieutenants. You have a Company and three Platoon sergeants, you have a full roster of enlisted and have claimed one of the best full-care doctors in the Space Force. You may be missing all of your squad sergeants, but you have every single person you requested. Dostoyervski has been selected for you, since you were taking so long in making up your mind. Standing there like a statue, no less," the DoI sergeant muttered. "Sir."
"Dostoyervski won't work for me, Sergeant," Ia dismissed. "I'll need to find someone else."
She shrugged and tapped something on her workpad. "Fine. Arstoll it is, then. Sign it with your thumbprint, Captain, and have a nice day."
Ia shook her head. "I won't sign that, Sergeant. I've found an anomaly—a huge anomaly—and I have to track down the right way to fix it, first."
" _What_ anomaly, Captain?"
The impatient question came from the oldest man in the room, and the only soldier whose rank matched her own. The main differences between them were that he wore brown stripes on his black uniform, and that his brass eagle did not carry the rockets in its claws that hers did, making her a ship's captain and him a lieutenant colonel. As dark-skinned as the sergeant, but with three times as many wrinkles and none of the hair, Lieutenant Colonel Luu-Smith flicked his hand irritably.
"You've already taken up hours of our time this morning with a task normally left to the experts, Captain Ia. What anomaly could _possibly_ throw everything we've done out the window at this point in time?" he demanded. "I thought you said you were some sort of massive precog. Shouldn't you have already foreseen it?"
"With respect, Colonel," Ia returned, "I am _not_ the only being who can see into the future, and that means I'm not the only one who can act to change the things they foresee." At his skeptical look, she rolled her eyes. After several years of playing her cards close to her chest, ingrained habit had kept her from revealing what she apparently needed to reveal. "...The _Feyori_ are now involved. They cannot see as far as me, but they _can_ see, and they will interfere, if they think it will somehow promote their own positions in their gods-be-stupid Game.
"Unfortunately, some of them are now considering me a threat. It's incredibly shortsighted of them because I'm not their enemy, but there it is. Now, if you'll just be a little more patient, please, I was in the middle of tracking down where the anomalies started when I was interrupted." She glanced briefly at Sergeant Plimstaad. "And I did mean it when I said do not touch me. You do not want to see what is inside my head; I'm dealing with scales that most people aren't prepared to deal with, at speeds that would give you a raging migraine.
"I do thank you for your efforts on my behalf," Ia added. "I'll try to be quick about this, but there are a lot of lives at stake. More than you know."
Closing her eyes, Ia breathed deep and let it out, then did it again, calming and centering her mind. A flip of her thoughts landed her in the grass next to the waters of her own life. From there, it didn't take much effort to condense Time back into a thin, interwoven sheet, though she did have to spend a few moments refinding the slub-nodes of Feyori influence in the future. Once she had her mental metaphor adjusted so that her conscious mind could comprehend it, she stained the lead one purple again and followed it up-thread into the past, trying to find the moment where the anonymous Meddler in question had decided to begin interfering.
The _recent_ past, she discovered with an unpleasant jolt. _Ah, slag...The initial slip in the streams took place just thirteen days ago. That was the day I left the Solarican Warstation_ Nnying Yanh. _More precisely, this is the Feyori whose presence I uncovered and threw off the Warstation. The same day I was tested and my father's legacy had to be revealed, explaining the strengths of my psychic powers._
_Which was also the day I stupidly didn't check to see_ what _effects his abrupt exposure would have on the timelines,_ she castigated herself, wincing.
She didn't have to touch that thread to know the Feyori in question would be upset enough to try to counterfaction her. The energy-based beings converted themselves into matter-based beings so that they could meddle with her fellow sentients' lives. They did not like it when the pawns in their great Game started playing by different rules, and they _really_ didn't like it when a pawn ousted a player.
Sometimes their interference was for a Right of Breeding, which was how Ia herself had come into existence. Such things infused most of the known sentient races with psychic abilities. Sometimes their interference was less direct; the Meddlers could manipulate the thoughts of their targets via telepathy, create complex hallucinations via a combination of holokinesis and clairvoyancy, even physically change a person through massive biokinesis. Most of the reasons _why_ they did such things gave Ia a headache trying to figure them out.
As much as she wanted to avoid crossing factions with the Feyori, her own stupid lack of foresight almost two weeks ago had dumped this problem into her lap. _Which means almost half my roster is now rendered void and useless. This purple-Meddler looks like he will have picked up...twelve, thirteen...fifteen or so major faction-groups to help counterfaction myefforts by the time I'll need_ all _the Feyori to swear faction to me. Slag..._
She quickly checked the chronology of the timelines. _It seems he's quite clever, too. His first real interference-node will happen six Terran years and one Terran day_ after _I bartered with that "Doctor Silverstone" Feyori for a Right of Simmerings. And it won't be anywhere near where I am, so it'll be difficult for me to counter it face-to-face. Most of these slubs are truly subtle interferences, probably just telepathic suggestions...but they are enough to throw off the weft and warp of the pathways I need. Which means it's time to rewrite the whole roster...and...ah, hell. I'll need..._
_Ugh. I'll probably need to accept Belini's offer to faction me in exchange for far-ranging prophecies. The Meddlers can see a little way into Time, and that means I have to plan dozens of steps in advance, limiting as many of their counterfaction options as I can. If I'm not careful, it'll really shift the balance of power down through the centuries, ruining plans I've already laid._
_At least I've already considered her offer as a Plan B to_ _extend my Right of Simmerings, so I don't have to break completely new ground in the timestreams..._
_I hope I don't have to use it, though I won't hold my breath. I'd rather get three Feyori to admit I am the foreseen Prophet, and have the right to rearrange their Game._ She double-checked the timestream paths and winced mentally. _I won't have many chances to do that, though. Miklinn technically isn't interfering or counterfactioning by raising the point that I "really should manifest before being accepted" as one of them...and damn him for the legality of it. And damn myself for my carelessness._ _Slag._
_Okay, Time,_ she ordered silently, rippling the sheet-like weave, doubling over the threads and turning them translucent. She knew how to handle her Right of Simmerings, which would buy her more time to sway the Feyori to her side. For the contingencies where some of the Feyori were stubborn about wanting to counterfaction her, she needed a different crew. _Show me which changes in personnel I will need, starting with Chaplain Benjamin, Doctor Mishka, and my choice of first officer, Lieutenant Brateanu..._
The pathways she needed to check were fairly easy, like two transparencies laid one over the other. The first layer was the Feyori-altered path at the bottom, with her current roster selections. The second layer belonged to the path she had already marked out for the future, the one that led to the one shot she had at saving the galaxy from annihilation three hundred years away.
Deviations were quicker to see this way, but it only worked because she already had both source-paths to trace and compare. Until she had found the problem—Feyori interference—she couldn't have made these comparisons. _Good, good...Bennie's still my chaplain; I like her. And the good doctor will still be grumpy about her reassignment to my team, but she'll still work out fine. As for...ah, slag,_ Ia cursed, wincing. _Brateanu is right out._
She needed an engineer, someone so good, so creative, they could scavenge or craft parts on the fly, since there would be too many times when her ship and crew would not have the time to stop and make repairs at an actual dry-dock facility.
Her 1st Platoon lieutenant had to be Rico, a man with a brilliant analytical mind and the ability to read and think in Sallhash, even if pronouncing it was physiologically difficult for Humans. He couldn't be replaced. She also needed Helstead as her 3rd Platoon leader. The woman was not only the most deadly soldier on Ia's crew roster, she could teach those skills to the people under her. Helstead came with more tricks up her sleeve than a hundred stage magicians, and Ia would need most of them in the near future.
That left either the lieutenant for the 2nd Platoon, or her first officer. One of the two had to be good at handling combat chaos, the other had to be an outstanding engineer. Rico and Helstead were both good enough to handle combat, though one was more of a military analyst and the other a military assassin by training.
By preference, she would rather make the combat officer a Platoon lieutenant, so that all three groups of soldiers would be led by someone competent in directing battle. The problem was, of the combat-competent leaders among the hundreds of millions of junior officers out there, most would be needed exactly where they were.
Head hurting, Ia eased out of the timeplains. She lifted her hand to her forehead, trying to massage away the tension caused by her dilemma; physically, she had only spent a couple minutes standing there, but psychically, she had spent several. Such accelerated concentration came at a cost.
One of the middle-aged DoI sergeants spotted the movement and sighed. "Well, Captain?" the man asked her. "Have you spotted a solution to your little anomaly?"
"All I can do is plan for a greater level of flexibility, Sergeant," Ia muttered back. "They're Feyori. They're _shakking_ unpredictable as well as powerful, they cannot be killed or swept aside, and the _only_ kind who could hope to back one down when a Feyori feels it's been offended and counterfactioned is anoth..."
She trailed off midword. Blinking, Ia stared sightlessly across the room. A moment from the past played through her mind—not a moment dipped from the timestreams, but one from her own memories. Snippets of that conversation came back to her, key phrases that now reassembled themselves in her mind.
_You're going to Antarctica...or will be...to steal schematics for something...the Vault of Time..._
_The Vault of Time._ Wincing at the irony, she covered her face with one hand. _Oh, God, Meyun_...you _were more accurate than you could've known. The only thing that_ can _stand up to a Feyori is_ another _Feyori...so it looks like I_ will _have to raid the Immortal's Vault on Earth. Slagging hell. That's going to be tricky._
Drawing in a deep breath, she steeled herself for her new task. This time, she dove back in without hesitation; this time, she had an idea of who and what to look for. Working quickly, she plucked out the life-threads of a half dozen potential-possible engineering candidates. A full dozen, flicking through their transparent life-streams, overlaid on the path she needed to take. A score.
Too many of them had problems. Little ones, big ones, convoluted ones, butterfly ones where the tiniest flap of wings created massive hurricanes down the road...Head aching, heart hurting, Ia finally plucked out the thread belonging to her one failed relationship and laid it over the future.
Not everything was visible; Meyun Harper was still grey-misty in several spots, key moments where her own emotions toward him would make it difficult to decide what to do. But unlike the dangerously wandering life-paths of the others, most of which started as small variants before veering wildly away from the most useful course, his consistently came back to the paths she would need.
_Absolutely wonderful. Irony of ironies. Slagging,_ shakking _hell. God certainly has a sense of humor, doesn't she? Ia's Impending Doom, thy name is both Meyun and Miklinn...and such a lovely-sounding pair of names for an impending pair of pains in my path._
Dropping back into her body, Ia struggled not to roll her eyes. Meeting the somewhat impatient stares of the others, she shook her head.
"Change of plans, meioas. If I'm going up against the Feyori, I'm going to need several more psychics in each Platoon. I'll get you their names in a moment, but first things first. My second-in-command"—she had to take a breath before continuing—"is going to have to be Lieutenant First Class Meyun Harper."
Lieutenant Colonel Luu-Smith eyed her skeptically as one of the sergeants next to him put Harper's personnel file up on a couple of the screens. "The same one you went through the Academy with? If I recall your and his files correctly, there was a note about you having a fling with the man, post graduation."
"It was just a few days long, hardly worth mentioning," Ia stated dryly. "It also happened between assignments, and we parted company as friends, nothing more. Since then, we've barely spoken to each other, so Fatality Forty-Nine: Fraternization does not in any way apply. I am picking him because I _will_ need his skills in adaptive engineering.
"There are ways to deal with the Feyori, certain energy frequencies they find unpleasant, but creating them will take a logistics officer who is clever and resourceful. I know he doesn't have nearly as much combat command experience as Brateanu, but the fact that Harper roomed with and studied beside me for a year while we were in the Academy will lend itself to ensuring there is a quick rapport of trust and understanding in the top of our cadre," she stated.
One of the middle-aged women seated around the table snorted under her breath, muttering something uncomplimentary about Harper having _known_ his roommate all too well. Ia narrowed her eyes but did nothing more. It was one of the men next to the woman who smacked his hand lightly up the backside of the woman's head, wordlessly chastising his fellow soldier.
Ia gave him a brief, wordless dip of her head before addressing the rest. "...More to the point, gentlemeioas, Harper has known of my precognitive abilities for just over two years now yet has not told a soul. His sense of discretion and secrecy will be invaluable for the position of first officer on board my particular ship. I would rather have had Brateanu, but there are key decisions my first officer will have to make while I am busy dealing with these Feyori interlopers, and thanks to their future interference, I can see now that she would make the wrong ones when dealing with them."
"You're calling _them_ the interlopers," one of the younger males snorted. "But if you really are some sort of massive precog, aren't you just as bad a Meddler as them?"
A chiming from one of the workstations interrupted her reply. The soft alarm cut off as the oldest of the DoI sergeants sat forward, examining her handheld screen. A few taps of the keys projected a familiar face onto the monitors lining the walls. His head had been shaved bald at some point, but she knew his scarred, broken nose, ring-edged ears, and rascal's grin. Seeing First Sergeant Glen Spyder's personnel file was a much more pleasant surprise for Ia than Harper had been.
Sergeant Plimstaad glanced over the images of Spyder projected on the wall screens, then looked back at Ia. The search parameters used by the DoI apparently included flagging and popping up any file whose subject had interacted with Ia at one time or another. "Well, Captain. It seems one of your old friends has just been flagged with a Field Commission. Did you know this was coming, sir?"
Ia shrugged. "I knew it was a possibility, but it was only a thirty-two percent chance at most, Sergeant, given his current combat situation. Given those odds, I honestly thought I'd have to pick someone else. As it is, this is relatively convenient. He'll have his commanding officers rescued from their troubles in...twenty-nine hours, thirty minutes, give or take a few minutes, which means his Company will be out of its jam and back to its assigned Battle Platform in about fifty hours from now."
The lieutenant colonel snorted again. "And what would you like the DoI to do about him, as if we couldn't guess?"
She didn't have to close her eyes, just unfocus them enough to pluck out the thread of Spyder's future probabilities and compare them to the path she needed. She had already considered this possibility earlier, though she had set it aside with that less-than-likely one-third of a probability. "The DoI needs to approve him for full Field Honors once he's out of the frying pan, then transfer and drop him into my 2nd Platoon as a Lieutenant Second Class, instead of selecting Lieutenant Arstoll. I know Glen Spyder can follow my battle plans and still think on his feet in the midst of chaos, so he'll do just fine."
"He'll need to go to an Academy, first," Luu-Smith pointed out. "His intelligence charts suggest he'd be quick enough for a fast-track class, but you'll still need a 2nd Platoon leader for the first year."
"I'd rather not delay his presence, sir. Just drop him into my Company as is. He can complete his officer's training via on-the-job work and correspondence school where needed," Ia countered. At their skeptical looks, she shook her head. "I don't need him to advance up the ranks, meioas. I need him to help lead my troops into battle. He can do that right now, as is, so I'll take him exactly as he is today. Or will be, in a few hours. The rest can be taught either on the job or via correspondence courses.
"Now, let me get you the rest of the roster changes," she said, closing her eyes once more. "I'll need a couple more Troubleshooters, maybe a few Sharpshooters, as well as the extra psis—no one vital to the rest of the Space Force's needs will be swapped in, I promise. My Prophetic Stamp on that."
"Your so-called Prophetic Stamp's only worth the price of a ground-bound physical letter at this point, Captain," Lieutenant Colonel Luu-Smith grunted. "A stamp I wouldn't even bother to scan into the mailing system, right now."
"That's fine. You don't have to believe me, right now. You will learn the strength of my word in due time, meioas," she murmured. Ia quickly double-checked the future paths of Lieutenants Rico and Helstead with Spyder dropped in their midst, then turned her attention to the noncommissioned officers and the enlisted in her crew. "Now, let's see how many of the Damned I can salvage in this Meddler-made mess..."
# CHAPTER 1
_I think, by the end of this interview, I'm going to be very hoarse—thank you for your patience with me, by the way. I know this is a lot of material to cover, and I'm insisting on doing it more or less chronologically, but really there's no better way to organize all the events that have happened so far. And yes, I know that statement is ironic, coming from me._
_Dabbling in Time as I do, I have had to juggle not just the needs of the present, but double- and triple-check them against the needs of the future. It's like juggling many, many balls all at once. I think the known galactic record for juggling in Standard Gravity is...what...twenty-six balls by a Gatsugi? Of course, they get to cheat a little, having four arms. But that's just juggling toys. If you drop one, it bounces across the floor. You do have to chase after it, but usually it doesn't break, and usually it can be tossed back up into the air again. I'm juggling countless septillion lives, and sometimes a detail or two can slip past my fingers._
_Unfortunately for me, "dropping the ball" has an entirely different meaning and a very unpleasant outcome if I drop it badly. Every day, I tried my damnedest to get it right. But, to quote Dickens, "I am mortal, and liable to fall." Or to use another quote, "Damned if you do, and damned if you don't," which is one of the reasons why I nicknamed my Company what I did._
_~Ia_
OCTOBER 24, 2495 T.S.
TUPSF SECURED SHIPYARDS
TRITON ORBIT, NEPTUNE, SOL SYSTEM
Ia stopped in front of the door to the briefing boardroom. She paused a moment to draw two deep breaths, then squared her shoulders. _I can do this...I can do it...I_ have _done it, and done it well._
_The only problem is, the moment I stride across that stage, I'm spotlight center for_ everything _that follows. Everything I say and do will be scrutinized by the Command Staff, the Department of Innovations, and most importantly, by my entire crew._
It was a disturbingly large responsibility. Up until now, Ia's task had been to take shelter behind the rules and regulations and break them only when no one was looking. Now everyone would be looking, and she had to start breaking a lot of those rules and regs. _Not to mention, from here on out, I won't have a true moment of peace. Not if I want to do everything I need to get done._
Mindful of the weight of the medals pinned to her newly issued set of Dress Blacks, of the impression she would make in wearing all of them, Ia touched the main button on the controls. The panel slid open with a faint hiss of hydraulics. Noise escaped through the opening, the sounds of 160 men and women chatting quietly among themselves as they waited in idle boredom.
She knew the layout of the briefing room, shaped like a lecture hall with projection screens on all the walls and tiers of padded chairs that could double as acceleration couches. Those chairs faced a curved table reserved for the six officers and four sergeants seated on either side of the empty chair waiting at the center.
The Captain's seat.
Her seat.
The door Ia used wasn't one of the double-wide ones at the back of the hall, above the riser seats. Hers opened onto a short corridor leading to the platform holding that table. It gave her a good view of the ten Humans seated behind that table, though not of the rest of the room; she could hear the others occupying the hall, but that was it. The dim lighting of that little entry hall also hid her arrival from all but one of the sentients in the room. Specifically, the petite redhead who sat at the near end of the table, with her back to Ia's door.
The other woman's eyes may have been occupied with the task of using a tiny stiletto to trim her nails, but her other senses were just as sharp and ready to be used. Between one breath and the next, the knife was shoved back into one of the sheaths doubling as hairpins that held her coronet of braids in place. The woman scrambled to her feet, standing on the seat of her chair so that everyone could see as well as hear her. All without even a single glance behind.
_"Officer on deck!"_ she snapped, her voice as much a command as a warning. With that said, she dropped to the platform floor and stood At Attention. Other bodies rose around the table at her call, some more quickly than others, and the rustling of dozens more could be heard from around the corner.
Shoulders squared, chin level, Ia strode onto the platform hosting the table and the men and women now on their feet. As she came into view, first the officers and sergeants at the table lifted their hands to their brows, then the soldiers up into the tiers.
Clad as she was in both Dress Blacks and her Dress cap, saluting was mandatory. Ia did not return any of them, however. Instead, she moved to the open chair at the center of the table, nudged the seat back on its track so that she could stand in front of it, and faced the bulk of her crew. Unbuttoning her left cuff, she flipped open the screen of her command bracer and brought the boardroom monitors to life.
Official orders scrolled onto the main viewscreens positioned on the wall behind her, the two sidewalls, and suspended over the heads of the crew, so that wherever one looked, the topic being discussed could be seen. The right and left secondary screens remained blank for now, as did the long tertiary screen above them. Another touch activated the headset discreetly hooked over her ear. The thin wire alongside her cheek picked up her voice and projected it around the room just loud enough to be heard.
"Acting under the direct orders of Admiral John Genibes of the Terran United Planets Space Force, Branch Special Forces, I, Ship's Captain Ia, hereby take command of the 1st Company, 1st Legion, 1st Battalion, 1st Brigade, 1st Division...9th Cordon," she added, pausing slightly for emphasis, "and with it, take command of the Harasser-Class battleship TUPSF _Hellfire_ , docked at the TUPSF Secured Shipyards of Triton, Neptune, Sol System."
She pronounced the acronym _tup-siff_ , keeping her words as crisp and distinct as she could manage, since these were official transfer orders, recorded for legality as well as posterity.
"This action is now logged and filed as an official transfer of command, as of time stamp 22:45, October 24, 2495 Terran Standard time...mark," Ia finished, watching the chrono built into her arm unit.
The orders on the two main screens flashed, sealed with the indicated time. Only then did she lift her hand to her own temple, returning the crisp salute of the soldiers around her. As soon as she lowered her arm again, they lowered theirs. They continued to stand At Attention, however, awaiting orders.
Satisfied she had their attention, Ia tapped in another code on her bracer. The orders detailing her acceptance of command over ship and crew were replaced with the TUPSF logo. The soothing, sapphire blue background and familiar, oval map-projection of the major continents of the Human Motherworld filled the screen; instead of the normal gold hues used by the other Branches, however, the map had been drawn with the pale silver of the Special Forces.
Onto the two secondary screens flanking those mains, Ia posted the unclassified portions of her personnel file, including enlarged, rotating images of her face, with its Asian features, light tan, amber eyes, and chin-length, snow-white locks. Minus her Dress cap, of course.
Matching reality to that image, Ia removed her cap and set it on the table. Then tucked her headset, which the cap had dislodged, back into place over her right ear. The headset was necessary to project her voice to the headsets of the 160 men and women around her, particularly the privates at the back of the hall. She kept her Dress jacket on, keeping some of the formality of the moment, but unbuttoned it for comfort.
Somewhat for comfort, that was; with all of her medals pinned across the black gabardine, it was still quite heavy. It was a tangible reminder of the weight of her position.
"At Ease, meioas, and sit down. We have a _lot_ to get through, so please pay attention. Most of this you will learn in greater detail by studying your Company Bibles," Ia stated, meaning the manual of procedures most combat officers gave to their soldiers.
Much of it was standardized to the Space Force's requirements, but they also often included little quirks and preferences tailored to each Company's patrol or combat needs. Hers were tailored all the way down to the individual. They would see that for themselves, shortly.
"For the moment, there are a few things I'd like to go over with you at the very start—please do be seated," she added, as some hesitated. "I don't do the nonsense that says the lower ranks have to remain standing if their CO hasn't sat down first. I'm staying on my feet as a reminder to _me_ to be as brief as I can, given how much I need to say."
Those few older soldiers who were still on their feet, including her fellow officers, settled into their chairs. Ia nodded.
"Thank you. Most of you received your transfer orders with very little explanation as to why you were being transferred," she said, acknowledging in her opening words the confusion she could see lurking in the expressions of the men and women studying her. "You may think you have been selected at random. You may be wondering why _you_ are here, and not someone supposedly better qualified. I say to you that you _are_ the right men and women for the jobs that lie ahead of us. I have painstakingly hand-selected each and every one of you, based upon the foreknowledge that each and every one of you can and will get your tasks done, and get them done right.
"What those tasks are will have to wait for another day. Our ship is still in dry..." She paused as a rumbling noise transferred through the deckplates for a moment, then finished her sentence. "...in dry dock, undergoing the last of the interior fittings. We ourselves will be splitting our time between this ship, the dock station, and even some land-based maneuvers over the next two months as we give this crew a shakedown to get you used to your upcoming multiple responsibilities. Then we will be taking this ship out for its shakedown run as well.
"But first, an introduction of your command staff, starting with myself. My name is Ia, pronounced _EE-yah_ , not _Eye-yah_ , or even _Lah_ , and it is my first, last, and only name. You may therefore call me Captain, Captain Ia, or even just Ia in those moments when we are being informal. For those few of you already familiar with my military nickname, my first name is _not_ 'Mary' and I will not respond to it on its own. You may, however, call me by my full nickname, Bloody Mary," she allowed, meeting the gaze of a man here, a woman there, "but you _will_ say it with respect. I have formed a very bad habit throughout my military career of flooding the deckplates in my enemies' blood, and I have no intention of breaking that habit in the years to come."
A few bodies stirred in the crowd at that; not everyone in the room was comfortable with the thought of such violent combat. The green-and-brown-haired man at her side grinned openly at their discomfort; he had seen it before and wasn't fazed by the thought. Ia nodded briefly at him and continued.
"On a more personal note, I come from the heaviest inhabited world. That means I am roughly three times as strong and three times as fast as the majority of the Humans in this room. I am also a very strong psychic, stronger than any other Human you are likely to meet, because my mother is very much a Human...but my father was a Feyori."
She waited while that caused another stir. The fact that psychic abilities came from the Meddlers was a somewhat known but rarely discussed topic since it made most people uneasy. Admitting openly to her crew that she was a half-breed would make many of them uncomfortable around her. None would be openly hostile, but some would be wary. Ironically, some of the uncomfortable ones would be fellow psis; even more ironic, most hadn't realized yet just how many psis had been gathered into her crew.
"Make no mistake, meioas. I side with my fellow Humans and the other matter-based sentients. Beyond that, the main thing you need to know right now is that I am a massive precog. In fact, I am the Prophet of a Thousand Years, as prophesied by the Sh'nai faith of the V'Dan Empire." That caused another, much larger stir in the crowd as waves of doubt spread across many of their faces. Only a handful believed in the main V'Dan religion; most were followers of Terran faiths, if they followed anything. Raising her hand briefly, she warded off that doubt. "Don't worry if you don't believe me right now. You'll see the truth of it for yourself in the days ahead.
"To my immediate left," she stated, changing the subject by introducing the man with the black braid, who belatedly nodded, "is Lieutenant Commander Meyun Harper. He is a genius at applied mechanics and impromptu engineering. He also trained with me back in the Naval Academy, so he knows better than most what my command style will be."
She carefully did not mention the fact that Harper also knew many of her precognitive decisions. Ia didn't want her mangled past with him to become a point of speculation for their crew. Particularly not with the DoI intent on watching both of them to see if they tripped over Fatality Forty-Nine in the next few months. She continued without hesitation, speaking crisply as her personnel file was replaced by his on the screens, giving everyone an enlarged view of his neatly uniformed head and shoulders.
"Like most of us, Harper will hold several jobs on board this ship. Not only will he be my first officer, he will be in charge of logistics & supply, lifesupport, engineering, and he will get his hands dirty as our chief engineer. Harper, above all others, will be responsible for the continued maintenance and operation of this ship; if he asks you to fix something, you will fix it. To that end, he will also be training everyone on board this ship to be able to handle emergency repairs on the fly, for whatever repair is closest to you...so you will all become familiar with _all_ aspects of shipboard maintenance...save only one system, which I will discuss later."
That earned her several curious looks from the sea of Grey-clad soldiers around her. Ia didn't bother to explain. She had too much to get through.
"Next up is Lieutenant Commander Delia Helstead, who has served with distinction as a captain for the last three years," Ia introduced next, indicating the petite redhead at the far left end of the table. Her image and public file appeared on the screens next, replacing Harper's. "She has since been cross-ranked from Captain to Lieutenant Commander so that we will not confuse her rank with _my_ rank on board this ship.
"In other words, if you say 'Captain' and she starts to respond instead of me, that will be the reason why, until she gets used to it," Ia allowed, smiling wryly. "As inconvenient as that may be for her, we all know whom to blame...thank you very much, TUPSF-Navy, for insisting upon clinging to your outdated maritime traditions."
A few members of the gathered crew chuckled. The stubbornness of the Navy's various traditions had lingered long after the unification of all of the Terran empire's various armed forces into two bodies, the civilian-based Peacekeepers and the military-based Space Force. Even Helstead smiled at the complaint, though she didn't stop her task. The short, muscular woman had resumed the cleaning and trimming of her nails with one of the stiletto-pins pulled from her hair, one leg curled up underneath her on her seat, looking more like an enlisted grunt than an Academy-trained officer.
"Lieutenant Commander Helstead comes to us as a former deputy director of field operations in the Knifemen Corps," Ia stated, deliberately pronouncing the –se in the word as tradition also demanded, instead of the more familiar _core_ , like the Marine Corps used. "She will be in charge of the 3rd Platoon, as well as be our tactical officer for all special-operations activities, our hand-to-hand combat training officer, our gunnery training officer, and our disciplinary officer, should you require disciplinary action above and beyond the Squad and Platoon levels—do not be fooled by her small size. She comes from the heavyworld Eiaven, and is therefore twice as strong and fast as she looks. She is also a fellow psi.
"At her side is 3rd Platoon Staff Sergeant Chico Maxwell." The Hispanic man seated next to the petite redhead dipped his head at the introduction. "Those of you assigned to the 3rd Platoon, or who find yourselves serving on the third watch for whatever reason, will report first to your Squad leaders, then to Sergeant Maxwell, then to Commander Helstead, depending upon who is available," she stated, abbreviating Helstead's rank in the standard way with the higher of the two titles. "There will be plenty of times where you will be thrown into the nearest duty post simply because you are the nearest available body. I suggest you get used to the thought of it, so that the reality will not stagger you.
"Seated next to Maxwell is Lieutenant First Class Oslo Rico, who will be in charge of the 1st Platoon and its duty watch." Ia paused briefly while the dusky-skinned, mountain-tall man tipped his own head. "Lieutenant Rico is an expert in military intelligence, data mining, surveillance, threat assessment, ship deployments, naval tactics, strategies, and communications. He also understands several xenolanguages fluently, including Sallhash. As such, he will be our intelligence officer, scantech officer, communications maven, tactical advisor, gunnery officer, and since he is rated for insystem and FTL combat maneuvers, he will act as our ship combat officer whenever I need to rest.
"Beside him sits 1st Platoon Staff Sergeant Menrick Halostein, who will act as his right hand." The man with the fuzzy halo of short-cropped, pale blond hair lifted his chin in acknowledgment. Ia gave the same introduction as before. "He will be the noncom in charge of first watch. Again, I must stress that if you are pulled into duty during first watch, and it is not your normal duty shift, you will report directly to him, and then to Lieutenant Rico. If it _is_ your watch, report to your Squad leaders on up the normal chain of command.
"This brings us to our 2nd Platoon officer, Lieutenant Second Class Glen Spyder. He and I both went through Basic Training together, and served for a while in the same Marine Company," Ia said, indicating the man seated to her right. "And yes, he has my permission to keep his hair that color."
Spyder's short hair was indeed distinctive, dyed in camouflage-mottled shades of green, beige, and brown, and his grin was friendly as he lifted his chin in greeting. Ia had to pause to clear her throat; all this talking was making her mouth dry, but she couldn't pause to get a drink. There was a caf' dispenser built into the base of the boardroom table, but no one had stocked it with cups yet, never mind brew packets. _Yet another thing to go onto the checklist before we leave dry dock._
"If you have any doubts as to _my_ abilities in combat," Ia stated, "you can go have a talk with him; I'm quite sure he'll give you an earful, given that we served together for roughly a year on a hot-spot Border patrol, and he helped me plan and execute the Battle for Zubeneschamali. His own reputation is equally outstanding; in fact, he comes to us with a fresh Field Commission. Spyder will therefore be our primary melee combat officer, in charge of all boarding parties, troop sorties, ground combat, and _non_ -special-ops activities.
"Like Helstead, he will also oversee your combat training, focusing on your training and preparedness for mechsuit combat, weaponry maintenance and drills, plus your daily regimen training. He is also in charge of all post-combat tactical debriefings. You know all those analysis reports you're supposed to fill out after a battle?" Ia asked rhetorically. "Where you're supposed to present your viewpoint of who did what, what part of it went well, what went wrong, and what could be done better? _Everyone_ on this ship will be required to fill them out, from the Privates Second Grade and Second Class, all the way up through the cadre.
"That includes the medical staff and our chaplain," she added next, glancing at the blonde with the lieutenant commander's double silver bars. "Where we are going, we will _all_ be designated combatants and valid targets by our enemy. That means we will _all_ learn how to fight, to plan, to follow, _and_ to lead. I will be planning our strategies, piloting this ship in most battles, and dictate some of our tactics when they are time-sensitive, but Lieutenant Spyder will be planning the majority of tactics. It is vital you give him accurate input and thoughtful suggestions.
"Working with him will be the 2nd Platoon's noncom, Sergeant First Class Maria Santori, who will also help to oversee all activities on second watch and assist with managing troop assignments. Her side specialty is picking the right modifications for the right job in mechsuit operations, so her skill set goes hand in hand with Lieutenant Spyder's area of expertise."
The tallish woman Ia gestured to next, the one with her dark hair twisted into neat, columnar dreadlocks, lifted her chin as well. She said nothing, allowing her commanding officer to continue.
"Each and every one of you _will_ be fitted for a mechsuit, because there will be occasions where we will have to park the ship and send most of you into combat," Ia told her listeners. "And by each and every one of you, I repeat: There will be times when even the traditional noncombatants will be expected to fight, from the chaplain to the clerks, and all the way through the medical staff.
"This leads us to our two officers who are not in the direct chain of command for this Company." That earned her a sour look from the blonde woman to her left. Ia acknowledged it with a dip of her head, and some diplomacy. "Since we will _not_ be deployed upon a regular patrol route, or even to a specific action area, and will therefore not be able to leave anyone behind for medical care in other facilities, I have secured the absolute best infirmary equipment possible, and the most outstanding Triphid I could find to be our medical officer."
Her flattery mollified the medical officer in question, but only a little bit. It was the rest of the men and women in the boardroom who gave her odd, bemused looks, somewhere between wonder, confusion, and concern. Triphid was the military nickname applied to someone who held multiple degrees in holistic paramedicine, ranging from preventive medicine, surgery, and regenerative procedures, to postoperative care. They could also handle just about everything a Human needed to remain healthy over the long term, whether it was dentistry, nutrition, pharmaceuticals, or physical therapy.
Normally they were reserved for one of two positions: either delicate cases where a patient at a veteran's hospital would be too disrupted by several medical personnel tromping day after day through their room; or for long-range exploration vessels, where the crews were expected to spend years traveling, scouting, and surveying star systems and worlds for either signs of sentient life or potential colonization.
Ia let the weight of both of those possibilities sink in, then stated gravely, "We may be operating within known Alliance space, but yes, we _will_ be that busy in the years to come. Lieutenant Commander Jesselle Mishka has not only the best Triphid training, she is also a fully trained, biokinetically backed paraphysician—she is literally the best doctor I could get for this crew. Treat her with the respect she has earned.
"Because she is a paraphysician as well as a physician, when the all clear signal has been given after any battle," she said, "those of you who have a moment to spare will be asked to drop by the Infirmary to volunteer for KI-man's duty, lending Doctor Mishka whatever spare kinetic inergy you may have, so that she does not completely exhaust her own inner resources.
"Our other nonchain officer is Commander Christine Benjamin, who will be serving as our onboard chaplain and psychologist. She has been assigned by the DoI to shadow my career, since they have plans for me," Ia confessed dryly, "but know she also stands ready to comfort and serve the rest of you with equal care. Feel free to go to her for spiritual, emotional, and mental health whenever you have need.
"As for the last member of our cadre," Ia concluded, "on Bennie's far side is our Company sergeant, Master Sergeant Henry Sadneczek. In moments of informality, he prefers the nickname 'Grizzle.' Sadneczek will be our quartermaster as well as our Company clerk, which means he is in charge of all requisitions and required paperwork—in other words, you'll follow my commands, but you'll give your reports to him. He also has a military law degree, and has acted in the past as a noncommissioned adjutant for the Judge Advocate General, Branch Special Forces. I expect you to do your best to make sure he doesn't have to _use_ that degree."
Grizzle dipped his head as well, his image appearing briefly on the secondary screens. Ia left it up there for a moment, then tapped a command, shutting off the secondary screens and their views of the Company command staff.
"We also have six Yeoman-class pilots, all of whom are rated for atmospheric, orbital, insystem, FTL, and OTL flight. They will perform most of their duties as shuttle and boarding-pod pilots, and as bridge pilots. They will not be considered members of the cadre when it comes to the chain of command for this crew, despite their parallel status as noncommissioned officers. They will, however, be your Squad leaders, and your Platoon noncoms in the event a particular Platoon sergeant is unavailable.
"Unlike most combat Companies, we do not have squad sergeants. They will not be necessary for this crew once you have adjusted to our operating parameters and particular chain of command structure. Most of what you will be doing will depend very strongly upon your own initiative and efforts. I have selected each and every one of you because you are _smart_ individuals," Ia stressed, "who _believe_ in the work of a Terran soldier.
"So, you will do whatever needs doing, cross-coordinating among yourselves, and you will report to your lead corporals, or to your lead yeomen, who will in turn report to your Platoon sergeants and lieutenants, on up through to me," Ia warned the men and women seated before her in the tiers of the briefing boardroom. "This arrangement will give this crew the greatest flexibility, and with it, the greatest chances for success in our missions. Unfortunately, this bottom-up chain of responsibility does mean that there are fewer layers of cushioning between you and me than in most of the command structures you have served in before.
"In fact, there is far less cushioning than most of you yet realize. You may have only a few officers between you and me, despite my relatively high rank, but I in turn report directly to Admiral John Genibes of the Space Force Branch Special Forces...and _he_ reports to Admiral-General Christine Myang herself."
Rather than saying more, Ia paused. Not just to let her words sink in, but because something _bra-a-a-a-apped_ against the bulkheads outside. She had to wait for almost two minutes as the work crews outside the sloped confines of the briefing boardroom did something which was not only noisy but rattled the deckplates, too, and the noise increased.
As it kept going, a couple of the privates on the left side of the room covered their ears, wincing from the rasping vibrations. When it finally passed, Ia gave them a few seconds to recover before speaking. They still had a lot to get through, however.
"...Right. As you can see, this ship, the TUPSF _Hellfire_ , is still undergoing several retrofits based upon the upgraded design specs I gave to Admiral Genibes. And yes, you heard me correctly a moment ago. You all report to me, I report to Admiral Genibes, and he reports to the Admiral-General of the entire Space Force. Unlike any other Company of our lowly size and lowly rank," Ia warned her fellow crewmates, "we do _not_ have several layers of cushioning between us and the ultimate authority. This crew is the _entire_ 9th Cordon Special Forces. If we screw up, there is exactly one person between us and the Admiral-General's wrath...but I wouldn't hold my breath on Admiral Genibes keeping silent. It is therefore up to you to read the Company manual, follow it like a Bible, and _not_ screw up.
"With that in mind, it is my solemn duty to inform you that as your commanding officer, I, Ship's Captain Ia, will be working under a double-indemnity clause regarding any and all corporal punishments accrued by this crew," she stated briskly, hands pushing back the edges of her jacket so she could rest her palms on her hips. Ia did her best to meet the bemused looks of every member in the Company, or at least look like she was meeting them. "What that means is this: if you break a regulation or a law, however many strokes of the cane _you_ receive, _I_ must receive an equal number of strokes, too."
She paused a moment, letting the men and women around her absorb that information. Ia followed it with an exact explanation of what that meant, so that there would be no mistake.
"If you receive one stroke of the cane for Fatality Four: Dereliction of Duty, then I must also receive one stroke of the cane, without restraint or hesitation," she told the men and women around her. Most stared in skeptical disbelief, though a few winced, including the man seated immediately to her left. "If you steal from someone and receive three strokes for that theft, I, too, must undergo three strokes for Fatality One: Committing a Civilian Crime. If one of you completely loses your wits and starts selling military secrets to the Salik, you and I _both_ shall be hung, drawn, and quartered for Grand High Treason.
"This is _not_ a jest, meioas," she stated grimly, pinning some of the more dubious soldiers with a hard, brief look. "The security level for this ship, her crew, and her mission is Ultra Classified. Revealing its secrets to anyone outside your direct chain of command will be considered an act of Grand High Treason.
"This is not a joke, this is not a game, and this is not a lie," she warned them soberly, reinforcing her words. "I selected you because you _can_ be discreet, and you _can_ do the jobs ahead of you _if_ you watch what you're doing. I have no intention of being flogged for any incompetencies, which is why I selected the best possible people for this job. Make sure you live up to my expectations."
She paused, partly to let some thumping and clanking outside pass without having to raise her voice, and partly to judge the moods of the women and men listening to her. Her dips through the timestreams, gauging this moment, had suggested a mix of warning and praise, of the carrot and the stick, would be most likely to get through to them.
"With that said, we turn now to a quick overview of our brand-new..." She had to pause again as someone noisily pounded something into place until it seated in a deck-vibrating _clunk_ , and finished, "...ship. As I stated when I assumed command, we are now on board the TUPSF _Hellfire_. She is the first of the Harasser-Class line being produced here in the Triton Secured Shipyards, with one notable exception. That exception is the main gun, which we will discuss last. From stem to stern, the _Hellfire_ is 0.9 kilometers long, and looks like a thick, lumpy, silver needle."
Lifting her hand, she snapped her fingers. The snap wasn't necessary; all it took was an electrokinetic prodding of the display system's workings to change the view, a mental click of the correct key. Ia snapped her fingers so that her crew would pay attention. Most sat up a little at the sound, switching their gazes from her figure to the flatscreens behind her.
The secondary screens fell dark, and the main screen lit up in a sparse diagram showing three cross sections of the ship: external, deck by deck, and radially. The images started with a real-time view of the pewter silver ceristeel hull, dotted with the rounded, somewhat oval lumps of projectile pods and laser pods, special gunnery turrets that could be extended and rotated to cover a wide firing angle, or retracted for interstellar travel.
Technical specs lit up the secondary screens, slowly scrolling upward with lists of the standard information: things like overall length, width, tonnage, atmospheric pressure, molecular content, ambient temperature, gravity gradient, number of decks, so on and so forth. On the main screen, the flatpic view of the ship's hull vanished, replaced by a line drawing showing the different sections of the ship. Those sections lit up in various colors as she spoke, echoing her words.
"Originally designed to hold a complement of five hundred or more—and indeed all other Harasser-Class warships will continue to function with that many—the _Hellfire_ is barely a frigate in crew size. Instead of five hundred, we will have a crew of 161." Sections lit up in light green. "All of our berths, common rooms, recreation cabins, dining facilities, so on and so forth, have been divided up between the middle three sectors of the ship, being the fore, amid, and aft. The other two sections are the bow and stern."
Each segment lit up as she mentioned it, briefly glowing like part of a pastel, five-hued rainbow. That made the ship schematic look like a multicolored worm for a moment. A tap of her mind zoomed the deck-by-deck sketches, giving a close-up on the crew quarters.
"Unlike most ships, where a particular section is devoted to a particular watch, I have instead divided up all three Platoons and scattered your quarters throughout the three main sectors. All common rooms and public facilities are to be considered open territory and thus available for everyone to use, regardless of your Platoon designation. I know that normally the military's psychologists divide things up into 'territories' to compensate for the natural Human tendencies of competitiveness and territorialism, but we cannot afford to be divided as a Company. Your Platoon designations are therefore mostly just a matter of what duty shift you'll be working. You are _all_ members of Ia's Damned, and you will conduct yourselves accordingly.
"In compensation for the openness of the common territories, most of the original berths have been gutted, giving each team slightly expanded quarters and greater privacy. Most of you will still have to share your cabins, but the privates will have as much room as is normally allotted to a sergeant, the sergeants get junior lieutenant quarters, and so forth, save only for myself; my quarters are no larger than the others officers' are. This extra personal space is all that I could give you, given the existing floor plans," Ia admitted wryly. "The rest of the crew quarters have been turned into storage holds and manufactory bays."
Those lit up in beige and yellow respectively, scattered throughout the ship, though most appeared on the middle decks. Between them lay a strange, blank section, as if the ship were hollow down the center, almost like a straw. Nothing filled that blankness; no lines indicated bulkheads, pipes, or passages, not even section seals. Ia ignored the anomaly as she continued her introduction to the ship.
"We will all become very familiar with these manufactory bays. What parts we cannot acquire during our brief resupply stops, we will have to be able to craft ourselves. Main engineering is located in the aft section, though each sector is being fitted with a redundant emergency engineering station," she stated, as a patch of the second-to-last segment of the ship turn a brighter orange. Paler peach colors indicated the backup posts. "You will also note a secondary engineering bay in the bow sector, one almost as large as the main. That is because this ship has been equipped with OTL hyperrift generators, in addition to FTL."
Spyder choked. He wasn't the only one to cough on his own spit at that, but he was the nearest, and the first one to rasp out, "—Choo gotta be _kiddin'_ me, Lieu—er, Captain," he corrected himself, staring up at his former fellow Marine. "OTL, onna ship _this_ big? 'S'bigger than th' _Liu Ji_ , an' y'know bloody well whatchoo did t' _that_ ship, three years back. Beggin' pardon, an all tha', sir, but tha's _shakkin'_ crazy."
"I don't deny that the _Hellfire_ is longer than the _Liu Ji_ , Lieutenant Spyder," Ia admitted with a dip of her head, acknowledging his concerns. "But it is not _fatter_ than the _Liu Ji_. In fact, its radial cross section is considerably skinnier. When it comes to hyperrift travel, the single most important physical consideration is the diameter of the ship in relation to the hyperspace rift's aperture when combining other-than-light interstellar travel with faster-than-light-sized ships. There is, of course, an upper limit on what the _length_ of even a skinny ship can be, but our current vessel does not exceed it."
"If you say so, sir," he muttered. "'S'long's it's not comin' outta _my_ pay cheque..."
Ia grinned, amused by the reference. "Nor, indeed, out of General Sranna's pay cheque this time around." She sobered a bit and turned back to the rest of their Company. "...The incident Lieutenant Spyder refers to is how my old Company arrived at the Battle of Zubeneschamali fast enough to effect the rescue of our commanding officers and fellow sergeants, back in my Marine Corps days. Rest assured, most of the OTL-FTL surfing _we_ will be doing will be done under much more controlled circumstances."
"I'm surprised Commander Harper wasn't the one choking." The muttered dig came from Doctor Mishka, seated on the other side of Bennie from Ia. With Spyder breaking the silence of the officers, she apparently felt it was alright to speak up as well. "Since he's supposed to be the chief engineer, shouldn't he be more concerned about you mangling this ship?"
Harper gave Ia a sardonic look before leaning forward just enough to look past her and address the other woman. "I've already seen it working properly, Doctor, via the Captain's precognitive efforts. This ship _can_ take it, once properly retrofitted."
He did not say anything more, let alone anything about how or when. Harper just sat back in his chair and folded his arms over his chest. Like most of the other officers, he was clad in Dress Greys with his full glittery of service pins, awards, and medals—most of them Compass Roses for outstanding feats of engineering—pinned to his chest. He looked well, if sullen. Dangerous, where her concentration was concerned.
Pulling her attention back to her work, Ia continued. "In terms of sheer tonnage, approximately forty percent of this ship consists of fuel compartments. In that regard, we are the equivalent of one of the Beluga-Class tankers. However, we have as many hydrogenerators as a high-end battle cruiser, or a low-end battleship."
Those sections of the ships turned blue, revealing that much of the ship was indeed dotted with storage containers and pipes for carrying purified water to the ship's hydrogenerators, displayed in a darker shade of blue. Most of the tanks were clustered around the edges of the ship, just beneath several layers of hull plating, sensory equipment, the L-pods and P-pods, and the projectile-weapons bays. Most of the hydrogenerators were clustered around that curiously blank inner core, which ran most of the length of the ship—far more hydrogenerators than what a ship of similar size should have needed.
After a moment, Ia electrokinetically shifted the schematic colors from the blues of hydrotechnology to the bright reds of projectile weapons and the darker reds of laser cannons. There were a lot of red dots on the hull.
"Some of our energy requirements will feed the dual engines and other shipboard needs, but the majority is reserved for the weapons. In terms of sheer firepower, if we exclude the main cannon, this ship qualifies as a high-end cruiser or low-end battle cruiser. Each and every manned post, L-pod or P-pod, will actually be operating anywhere from one to five slave-interfaced weapons pods at any one time, depending upon the severity of the current mission. Of the P-pods, we will be manning up to sixteen projectile posts during those missions, which means we will be firing from a bare minimum of sixteen up to a total of eighty P-pods staggered radially around the ship," Ia informed her crew.
She had to pause while several people whistled softly. Others blinked in shock, and a few of the enlisted who specialized in gunnery posts whispered to each other. Just as they died down, she held up her hand...waited...circled her hand impatiently...and nodded as yet another rasping shudder rumbled through the bulkheads. It ended with a very loud _thunk_ , and a brief dimming of the lights before they brightened again.
"...For those of you wondering what all that noise is, I've requested the fitting crews to install additional lifesupport bays, manufactory equipment, and other odds and ends we will be needing later on," Ia told the men and women around her. "The fore, aft, and amidships sections are more or less complete, but they're literally still rebuilding the interior bulkheads around us here in the bow and stern sections, after having ripped half of them out. This chamber was actually supposed to be a storage bay before I had it partitioned and reinforced into the company boardroom, with extra hydrofuel tanks beneath your seats. Above us is the OTL engineering compartment, and aft of us is one of our two shuttle bays.
"The original boardroom, located in the fore sector and sized to fit the original crew of five hundred, has since been redesigned into a recreation deck." The schematic changed colors again, briefly illuminating each section. Returning to the discussion at hand, Ia relit the drawings of the ship with bright and dark red dots. "At the moment, this ship has only the barest minimum of lifesupport supplies, and no armaments beyond the laser turrets and a few installed projectile launchers. Rest assured, we _will_ be fully fitted for war before we leave dry dock.
"Each ship sector also has four portable hydrogenerators, which can be converted within three minutes or less into catalytic payloads...and which can be launched from a standard P-pod bay, for a total of twenty nonradioactive hydrobombs, with payloads ranging from ten to fifty liter-tonnes. That's enough power per hydrobomb to completely destroy any major modern supercity, such as Tokyo—both Upper _and_ Lower Tokyo." She let the gravity of that sink in, then added, "We _will_ be using them in the future, and we _will_ be using them on Salik targets. We cannot and will not stop the coming war...but we will be doing our best to break the most critical components of their war machine."
For a moment, even the muffled sounds of construction outside the boardroom were absent, leaving them in grim silence. No one contested her statement. Each person in the room was a soldier, even the chaplain; they knew the Blockade wouldn't hold forever, and most had heard of Ia's efforts to stem the resumption of the old Salik War, at the Battle of the Banquet. She and the other escapees had killed many of the highest-ranked generals two months ago while trying to escape, but the enemy's war machines were still out there somewhere, just waiting for strong enough leaders to reseize control.
Ia waited for a couple of faint thumping noises, then spoke. "Moving on to the laser cannonry, this ship has twice as many L-pods as P-pods: twelve Swordstrike-, twelve Skystrike-, and eight Starstrike-rated cannons. I am, of course, referring to the _manned_ L-pod stations," she added. "That means we can have anywhere from twelve to sixty Swordstrike-, twelve to sixty Skystrike-, and eight to forty Starstrike-rated laser cannons capable of firing at any one point in time. _That_ is what rates this frigate-sized ship as a battle cruiser in its weaponry.
"However, under normal circumstances we will _not_ be firing all weapons," she cautioned the others, holding up her hand to forestall the grins on some of the faces before her. "The object is to hide our extreme combat capability via retractable weapons pods and pretend to be nothing more than an oddly elongated frigate, possibly even a small destroyer-class starship. This is because we will be going deep into enemy territory. If they know what we are truly capable of, they will try to hit us with everything they've got from the outset. I'd rather use a minimum of resources to get a particular job done so that we can conserve fuel, supplies, and personal energies for future engagements.
"In addition to the standard and hidden armaments, we also have mechsuit bays in the bow, fore, aft, and stern sections of the ship," Ia explained, highlighting those areas in brown. "Plus two transport and cargo shuttles in the bow and stern shuttle bays, and two boarding podships in each bay. The Infirmary is located in the amidships section, along with the main bridge; there is an auxiliary bridge in the stern section, but it probably will never need to be used as such. There may, however, be a skeleton crew sitting watch in there at certain times, depending upon the damages we take from combat.
"Over the next two months, all five lifesupport bays will have their tanks and hydroponic systems filled, balanced, checked, and double-checked. Before we leave dock, the manufactories will be loaded with raw stock, the storage holds loaded with food, clothing, toiletries, and various spare parts, and the projectile bays loaded with various types of missiles, some standard, some nonstandard. You will be expected to help load supplies in whatever spare time you might have, particularly toward the end—and once we leave dry dock, _you_ will be the only crew permitted on board, particularly when it comes to loading or off-loading supplies. That is one of the reasons why everyone will be fitted for mechsuit armor, so that it can double as a stevedore suit.
"Of the various other facilities on board this ship, their maintenance cycles, so on and so forth, please consult your Company manuals; those things can be learned in due time," Ia directed the patiently listening men and women arranged around her. "We're almost done, so please bear with me just a few more minutes. The last thing I have to say at this time about our new ship concerns its main weapon.
"Many of you may have noticed the rather obvious blank section along the core of this ship. The reason why it has been left blank isn't because it's hollow," she told her fellow soldiers. "It has been left blank deliberately because its contents are Ultra Classified. There are four people on board this ship who have a high enough clearance to know exactly what lies in the axial core. Three of you are enlisted engineers who have actually worked on this ship, one per duty watch, and the fourth is myself. The rest of you do not need to know."
The central core of the ship schematics turned black. Orders flashed down the secondary screens, marked with the unsmiling face of a middle-aged, Asian woman: the Admiral-General herself.
"As you can see, your standing orders are as follows," Ia stated, reciting them for those who couldn't be bothered to read the text. "Should any of you discuss the nature, capabilities, or mission of this ship, and in particular anything associated with its main cannon, with anyone outside of your fellow crew members, our immediate superiors, being Admiral John Genibes, then Admiral-General Christine Myang herself, without our permission, you will be automatically accused of Grand Treason. This accusation includes discussing it near any open comm pickup, _and_ includes discussing it with any member of the Terran United Planets Council, all the way up through _both_ the Secondaire and the Premier.
"I would far rather fill out the paperwork resulting in my shooting you preemptively, than suffer the required execution alongside you, should you be so asteroid-headed," Ia warned them dryly. " _Any_ query into the nature of our main cannon by any outsider who is not either Admiral Genibes or the Admiral-General herself needs to be noted, logged, and reported. Not only are you to log the time and identity of your questioner, if you _are_ queried, you are required to answer, 'I'm sorry, but I am not authorized to discuss such subjects at my level of clearance.' At that point, you will pass that person and their query up the chain of command to me, so that I can handle it for you.
"Make no mistake about this: _I_ am the only person on board this ship with the authorization to discuss this ship with outsiders," she finished bluntly. "That's why a slip of _your_ tongue is to be considered treason in the eyes of the Command Staff. Some of this, you need to know so you can grasp just how much firepower and thus work we have ahead of us. Some of this, I really shouldn't be telling to any of you because it is so heavily classified.
"But since we're all Human, I'm giving you permission to discuss this among yourselves, so you don't burst with unspoken curiosity. Just don't do it near any open comm links or active airlocks. Any questions so far?"
A hand tentatively rose, roughly one-third up the tiered seats from the officer's stage. Ia recognized the owner as Private Helia Dixon, one of her former crew members from her Navy days. The last Ia had seen of the other woman in person had been a final visit with her in the infirmary of the _Mad Jack_ , before Dixon had been packed off for regeneration and reconstructive surgery of her combat-lost leg just over one year ago.
"Private Dixon, it's good to see you again," Ia acknowledged, pointing at the other woman. "You have a question?"
"It's good to see you, too, sir. Um...are we allowed to query _you_ about this main cannon, sir?" Dixon asked her. "Does it pop out of the ship or something?"
"I'm sorry, Private; that's one of the things I can't tell you much about," Ia apologized. "I _can_ tell you that the cannon does not 'pop out of' this ship like one of the pod turrets. Instead, this ship was built _around_ the main cannon. We will be living and working in the outer housing for the barrel of a gun so large and powerful, the design team nicknamed it the Godstrike cannon. I cannot tell you the exact energy conversion rate at this time, nor the exact power output generated by it...but I can say that if you compare a pocket-sized holdout pistol to a Starstrike-class laser cannon, that's what the Starstrike is to the Godstrike cannon.
"For those of you who are curious on how a ship-sized laser could work with an OTL nose cone," Ia continued, carefully distracting the crew from further inquiry attempts with a couple of facts, "the _Hellfire_ has been retrofitted at the bow with three moveable, combat-redundant hyperarrays, which will be stored in armored housings just beyond the edge of the of the cannon's aperture. They will be swung into place when we are ready to travel via other-than-light. We have also been fitted with an identical OTL nose cone at the rear of this ship.
"All four arrays can be detached and moved to any of the other mounts in the event one is damaged...which they will be at some point. But I'm hoping to outmaneuver most incoming enemy fire. Because of her streamlined silhouette and the way the FTL warp panels and insystem thrusters are aligned, the _Hellfire_ is capable of immense acceleration in an emergency despite her seeming mass—in fact, we can accelerate at three-quarters the speed of a Harrier-Class ship," Ia stated. "We can also accelerate two-thirds as fast while flying backwards, too."
That earned her several startled looks. Having served on a Harrier-sized ship on Blockade Patrol, Ia knew very well the _Hellfire_ was at least seven times as long and twenty-five times as massive. She nodded slowly, confirming her words.
"Yes, we can move that fast. The _Hellfire_ has been fitted with the absolute latest in dual-use FTL/insystem thruster panels. She has also been fitted with cross-aligned thruster panels so that we can slip sideways at very fast speeds—the only thing we cannot do fast is swap ends. In fact, the designers almost named this class of ship the Dragonfly Class for its maneuverability," she added, a fact she had learned in her dips through the timestreams. "But with the coming war, they decided on the Harasser classification.
"As you can guess, the Lock-and-Web Law of shipboard life is particularly vital on this vessel. For most instances, I will be able to give everyone fifteen-, five-, and one-minute warnings before any such sudden maneuvers must occur," Ia admitted. "Unfortunately, even I can be blindsided by low-probability rolls for things happening sooner than anticipated, so it's best to secure as you go. There are many redundant interior safety-field nodes on board, but a pen can still be turned into a deadly weapon by even a mild change in speed or direction if it happens abruptly.
"So. This concludes your introduction to the starship TUPSF _Hellfire_ and its cadre," Ia told them. "For all other questions regarding the capabilities of this ship, as well as the locations of your berth and work assignments plus copies of your instruction schedules for the next two months, please consult the Company Bible...which your Platoon sergeants will now hand out to each of you." She tapped a couple of buttons on her command bracer and gestured at the three sections of tiered seats. "Your Platoon assignments are being streamed to your arm units now; please position yourselves accordingly, with the 1st Platoon to your left and the 3rd to your right.
"Each manual has been tailored very specifically to each one of you—I will state right here and now," Ia added, "that I expect each and every one of you to obey my orders to the fullest, because the vast majority of them will come backed with the weight of my precognitive abilities. I also expect each and every one of you to _think_ for yourselves, and to discuss the needs of your jobs with each other.
"You will see this duality stressed throughout your Company Bible," Ia said, tapping her arm unit. "I need each one of you to be able to work independently for the betterment of this ship and our missions, to be able to offer suggestions and implement beneficial changes in procedures and tactics where needed, _and_ I need you to obey me when I require it. However, I will only step in when something _has_ to be done a specific way. For the rest of it, I need all of you to help me. You are the best people I can find for this job because you are flexible, innovative, and more than capable.
"So. Please double-check that your datachip matches your name before opening your manuals, as they are indeed tailored to each member of this crew. You will also be issued your Company flashpatches at this time. Wear your Company patch with pride, and strive hard to earn and uphold the high reputation it will come to represent. You are now free to organize yourselves and to get to know each other," Ia concluded. "Dinner is scheduled on the dry-dock station in one hour for the enlisted and the noncoms; you may consider yourselves dismissed in half an hour, though you are welcome to stay for the full hour."
A final flick of her mind replaced the ship schematics on the main screen with the image of their new flashpatch, a stylized logo of a snowflake surrounded by flames. The colors were subdued, with the snowflake stitched in a mild silvery shade with pale blue accents, centered on a striated, dull red and orange background. To either side, the Platoon sergeants stood and picked up one of the three small cases resting on the table; each one was lined with neatly slotted chips stamped with its rightful owner's name.
Turning off her headset, Ia fished out several datachips from her pocket. She took a moment to sort them out, handing each one to its proper owner.
"Each of you has your own Company Bible to study as well. Lieutenants Rico, Spyder, and Helstead...once you've uploaded your manuals, please spend the next hour getting to know the individual members of your Platoons. After that hour is up, we will have an officers' meeting in the officers' mess over dinner here on board the _Hellfire_ ," she warned them, passing chips to Grizzle and Bennie. "We'll all still be berthed either in the dry-dock station's guest facilities for most of the next two months, or out on maneuvers while everyone gets used to their mechsuits, but our first meal together will be on board the _Hellfire_. Thankfully, the station has graciously agreed to fix and ship us a meal for tonight."
"I don't get it, Captain. Why are we dining on the _Hellfire_?" Bennie asked, slotting her assigned chip into her arm unit. "Why not on board the station, where the food will be served a lot hotter and faster?"
It was a legitimate question. Ia had a legitimate answer for it, too. "Because I'm trying to abide by the letter of our standing orders," she stated, handing Mishka and Harper their chips. "I'm going to be discussing the capabilities and requirements of the main weapon with the rest of you during dinner. That means I need to do so on board this ship, in a secured location, with no chance of eavesdroppers or comm equipment picking it up and broadcasting it. The officers' mess in the fore section on Deck 4 is already finished, nowhere near the rest of the current round of construction, and fully soundproofed, so it is the ideal location for that discussion. Any other questions?"
"Yes," Harper stated, still seated with his arms folded across his chest. He had only moved them long enough to accept his datachip and slot it into the military bracer clasped around his left wrist. "When can I have a private word with you, Captain?"
_It figures._ Sighing internally, Ia strove to keep her expression neutral. "After supper, Commander." She used the shortened version of his rank. "Before then, I will not have time for a private chat."
He arched his brow at that, his expression skeptical. "Will not _have_ time, or will not _make_ t—"
"Captain!" The strident female voice interrupted Harper's retort. It came from an enlisted woman making her way down through the others clustering around their sergeants. She lifted her left wrist, waving her arm unit in anger.
Ia sighed and glanced at her old roommate. "As in will not be allowed to, Commander. See me in my office after supper. Yes, Private Davies?" she asked as the somewhat short woman approached, her chin-length black curls bouncing with each impatient step. "You have a question about your orders?"
"I have a _complaint_ ," Davies countered, glaring at Ia. She added a belated, _"Sir."_
"Request denied." Ia's mild statement rocked the shorter woman back on her newly issued bootheels. While Davies was still blinking, Ia explained her reasoning. "Your extreme reluctance to be paired with males may have been understandable earlier on in your career, given what you once suffered, but your lingering phobia is a weakness preventing you from achieving your fullest potential as a Space Force soldier...never mind as a former fellow Marine. Your intent to demand, coerce, cajole, or bribe your way out of having a male for both a teammate and a roommate is therefore denied."
"But, sir!" Davies protested, blushing and frowning.
"Private First Grade Unger is the ideal partner for you, Private Second Class Davies," Ia stated, keeping her expression pleasant but using an implacable tone. "He is an excellent engineer, a very good triage corpsman, and he is a touch-sensitive empath. There is no way that he could possibly assault you without himself getting an equally unpleasant amount of feedback from that assault. _You_ , on the other hand, have the hands-on combat training _he_ will need to learn.
"By sparring together as teammates, you will learn to trust men, and he will learn to defend himself properly against someone who sees him emotionally as a threat. I require your roommate to be fully functional in hand-to-hand combat, and I require _you_ to be fully functional in an emergency. The Department of Innovations and its psychology personnel have agreed to this pairing. You therefore have zero grounds for objection, Private. With all of that carefully considered in advance," Ia finished, "your intended demand for a female roommate is, as I said, denied."
Davies stared at her for a few more seconds, then finally shut her sagging mouth. She blinked a couple of times, managed to pull herself back together enough for a brief glare, and marched off again without another word. Ia let her go. "Request denied" wasn't exactly the same as "Dismissed," but she wasn't going to argue the matter. The enlisted woman would just have to come to terms with her new situation.
Harper started to resume his earlier words but was forestalled by the approach of two more soldiers, this time a man and a woman. The man nodded to her, introducing himself and his companion as soon as they reached the edge of the table. "Captain, I am Private First Class Bei Ninh, and this is my wife, Corporal Jana Bagha."
"Ey! I know you!" Spyder exclaimed, interrupting the other man before he could say anything more. He grinned. "Choo're th' silver an' bronze medalists from th' Winter Olympics, Mass Biathlon two winters ago, ainchoo? Lookidat! Celebrities in our midst."
Ninh blinked, nonplussed a little. Ia stepped into the breach, answering his unspoken request as well. "To answer your question in advance, Private Ninh, I have zero problems with the two of you continuing to be teamed together. In fact, I am counting on the close cooperation and camaraderie between the two of you to help pull off some very tricky Sharpshooting in the coming years," she confessed. "That is why I pulled the pair of you out of that ridiculous athletic tour circuit and back into full active duty.
"As important as sporting events like the Olympics may be, demonstrating peaceful competition and the bonds of athletic brotherhood across the species of the Alliance, I will need the two of you ready to shoot at mobile targets that _are_ going to be shooting back at you by the time we launch in two months," Ia told both of them. "I will also need you to be far more accurate shots than your enemies will be while doing so. You two have always worked best together, and you have been lucky to have enjoyed tolerant commanders in your past. Serving under my command will be no different. Dismissed."
"Thank you, sir," Bei Ninh told her, giving her a thankful nod.
"Yes, sir; thank you," Jana Bagha murmured, moving away with her husband.
Ia watched them go, then looked over the others. More would drift her way with questions or comments, or just under the urge to get a closer look at their new commanding officer.
Lieutenant Rico, who had been surveying the men and women receiving their chips a few meters away, turned his attention to Ia. "Tell me, Captain, what _is_ your exact policy on fraternization?"
She shrugged. "I prefer to follow the relaxed rules most combat commanders use while on extended patrols. Mainly because we won't have that many opportunities for off-ship Leave."
"And that means...?" Doctor Mishka asked her, lifting one blonde brow.
"Between privates of either rank, it doesn't matter, so long as it doesn't affect their performance while on duty. If it's privates versus corporals or yeomen, not within the same Squad. Between privates, corporals, and yeomen versus the sergeants, not within the same Platoon. Between officers and enlisted, or even among ourselves...sorry," Ia apologized to her cadre. "Get used to being celibate for now."
Helstead snorted. "Ha! As if I've been anything _but_ , all my life."
"If we never get any Leave or reprieve, some of us might end up asking for a transfer," Harper muttered. "Since you implied we'd be acting like a deep-range scouting ship, with little chance to stop and mingle with others along the way."
Ia didn't have to be a telepath or a precognitive to guess why he said that. Parts of Meyun Harper's future possibilities were still cloaked in misty grey; he could still mess up her plans if she let him affect her on a personal level, if either of them indulged in fraternization. In fact, one of those featureless nodes lay in the hour immediately after supper. Most of the paths leading out of that grey spot weren't irretrievably bad—at least where her long-term goals were concerned—but Ia wanted to be cautious nonetheless. That included changing the subject right away.
"Rico, Helstead, Spyder, now would be a good time to go meet and greet your Platoon members," Ia encouraged the others. "Doctor Mishka, I'm aware of your ongoing objections to your new duty post, but please understand that I selected you to serve on this ship because we _will_ need a Triphid in charge of our health. If you prefer, you can go have fun overseeing the refitting of your state-of-the-art infirmary, though I'd appreciate it if you could bring yourself to get to know the others. Bennie and Harper, I suggest the two of you circulate as well. I'll stay here at the table and handle any questions aimed my way. The crew can get to know me in the coming months. I already know them.
"Grizzle...you'll have a lot of paperwork to start processing, now that I've officially assumed command," Ia added, nodding politely to their Company sergeant. "I'll see if I can snag you one of your clerks for a little preemptive strike on the ongoing pile, and give you some help with it myself. Remember, dinner in the officers' mess in less than an hour—oh, and Grizzle," she added, addressing Sergeant Sadneczek, "you're included in tonight's dinner invitation, since as our quartermaster, you'll need to know about the main cannon as well, and why it requires so much hydrofuel."
"Thank you, sir," he acknowledged. "Will the other sergeants be included?"
"Not for this one since they don't need to know at this time, though they may be included at later dates. Remember, gentlemeioas," Ia cautioned the men and women clustered around her. "Do not discuss the capabilities of the main cannon with anyone of lesser rank than yourselves, and nowhere else but on board this ship, under secured circumstances. The fewer people who know about it, the fewer chances there are for the information to be leaked.
"In the meantime, these meioas will be your brothers and sisters for the foreseeable future. Have fun getting to know each other," Ia warned her staff as she finally sat down. "Dismissed."
# CHAPTER 2
_When it came to putting my crew and ship together, most of the important decisions were made behind the scenes. Things like minor design flaws and the few faulty components I could foresee were fixed or replaced before the_ Hellfire _ever left dry dock. The wrinkles in the staff and crew would take a bit longer to smooth out, but the raw material was always good._
_Some things could be fixed right away. Others took time, and some butting together of heads. Mishka, for example, did_ not _like being pulled away from her single-patient cases, and her resentments lingered for quite some time. She was professional even at the beginning, don't get me wrong, but I knew going into it that it would take me time to win her over. Others adapted more quickly..._
_...Harper? Yeah, of course Harper was there—look, I had less than a third of the crew normally needed to run a combat ship of that size, and only two months in which to cross-train everyone to cover all the missing stations. Even the married couples in my crew didn't have the time to spare for that sort of thing, we were pushed that hard. Not to mention continuing to scout the future and write prophecies, plus the double-indemnity clause if any of my crew smacked into a Fatality—all of thatweight on my shoulders, and you think I had time for a relationship? Have you not been paying attention in this interview?_
_~Ia_
OCTOBER 25, 2495 T.S.
"You know, you never answered any of my mail? Vidletters, texts, nothing. No correspondence. Do you know how that made me feel?" Harper demanded the moment they were alone.
The only good thing about his accusation was how he waited just long enough for her office door to slide shut. Mindful of the pickups hidden in her office, pickups she knew about but didn't want to tamper with for this first "private" meeting with Harper, Ia hoped a version of the truth would be palatable enough for both him and her two superiors.
"I can guess how you felt, but there really wasn't much to say. We parted as friends, Meyun, and attended to our separate duties. Anything further at that point was physically impossible and logistically improbable, so what more _could_ be said?"
He stared at her, then flung out his hands. "Maybe something like, 'I missed you'? Or 'Let me tell you about my crazy day'...? Okay, maybe not _that_ one, given we both ended up on Blockade duty," he allowed. Swiping his hands over his hair, Meyun sighed. "I just...You never replied."
He wasn't going to let it go. She couldn't check this moment in time; Meyun Harper was too much of an anomaly point for a clear reading. But she could check the most likely outcomes of this meeting, in regards to Genibes and the Admiral-General. A quick skim of the waters took no more time than the amount it took her to blink twice and sigh.
"...Fatality Forty-Nine, Harper?" Ia reminded him gently. "I didn't want either of our careers derailed by accusations of fraternization. And now that we're serving together again, that's not going to come into play, either."
She could see the protests forming in his dark brown eyes. Moving closer, Ia picked up his hand in hers, cupping it in both palms. Verbally, she addressed him as she would have the Grandmaster of the Afaso Order, as a good friend but not a romantic interest. Underneath it, however, she sent a different message.
"Come on...where's the roommate who put up with my awful study habits?" she asked out loud, then carefully sent, ( _Meyun, this room is bugged for audio and visual surveillance. The Admiral-General wants to make sure_ everything _I do is aboveboard._ ) "The man who wanted me at his side during the zombie apocalypse, because I'd be the one running to nuke them all from orbit, like a _sensible_ soldier should?"
He widened his eyes for a moment, then narrowed them in comprehension. Ia continued her dual conversation.
( _I have five months left in my_ carte blanche _to convince her to give me ongoing free rein in handling all of this._ ) "We had a brief fling, but it didn't work out. Our real relationship, the lasting one, has always been a working one." ( _Help me out here,_ ) she cajoled. ( _Don't mess it up. A year or two from now,_ if _everything goes right_...then _we can talk about this._ ) "I'd like to get back to that." ( _I'll have the prophetic leverage for all sorts of things,_ if _we play it straight and by the book, right now._ )
Meyun looked down at the hands cradling his. He sighed and covered her fingers with his other hand. "You're right. It's just...Ah, your earlier comment, about the officers having to remain chaste...I can't speak for the others, but I _am_ a healthy Human male. Abstaining isn't a pleasant thought."
She offered him a wry smile. "Technically, it's only in regards to the others. Whatever you do with yourself is your own business, you know. What _I_ need is that brilliant mind of yours on my side."
With their skin touching, she could sense some of his surface thoughts. Telepathy had never been her strongest gift, and he certainly wasn't a telepath himself, but these were clearly formed thoughts, ones aimed her way. Ia nodded, confirming his unspoken question. She followed it with a subtle shake of her head. Yes, it _was_ possible to alter the recordings being made, but _no_ , she wasn't going to do so at this time.
( _I have to play it straight, Meyun. So do you._ )
Meyun sighed. He gave her fingers a gentle pat. "Fine. Love me strictly for my brains if you must. Just don't suck them out."
"Of course not," she scoffed, squeezing his fingers. "That would put an end to your genius." ( _We'll have a better chance to discuss all of this later. Just play it by the rules for now.Please._) Releasing his hand, she spread her arms slightly. "...Still friends? When we're off duty, that is?"
"Still friends," he agreed, opening his own arms. Stepping close, he hugged her.
She returned it. Breathing deep, Ia enjoyed for one moment the warmth of another Human being, the scent of a man whom she still missed sometimes. For a moment, her gifts lay quiet. She couldn't risk touching him for much longer, given what had happened in the past, but she couldn't give up this moment of comfort, either.
"I missed you," Meyun murmured into her hair. "I couldn't stop thinking about you—I worried about _all_ my friends on Blockade Patrol, as well as my own hide," he added quickly, mindful of the hidden cameras. He ended the hug and stepped back, tucking his hands into his pockets. "Chaplain Bennie told me you'd been assigned to boarding-party duties for a small patroller. And then, when I heard you'd gone missing and were listed as Captured, Presumed Eaten...I worried even more. CPEs are far less likely to come back than MIAs. Even ones like you."
"You should've had more faith in me," she teased lightly, giving him a lopsided smile. Ia gestured at the two cushioned chairs tucked into the corner opposite her desk. Her office wasn't large, and definitely wasn't ostentatious—in fact, she was pretty sure Bennie's office was both bigger and had more comfortable furniture in it—but she did have some amenities. "Want some caf' while we sit and catch up with each other?"
"Decaf', dash of cream, please," he murmured. "I'd like to get some sleep tonight."
Nodding, Ia programmed the controls on the caf' dispenser built into the wall behind her desk and set the first mug in the slot. It didn't take long for the machine to dispense two mugs of the hybrid brew. "Here you go..."
Accepting it, he settled in one of the chairs. "I know you can't tell me everything, but...was one of the reasons why you never contacted me because you knew I'd live long enough to wind up here?"
That wasn't the question she'd anticipated. Ia had thought he was going to ask her about her time as a prisoner. This one was relatively easy to answer, though. Seating herself, she gave him the truth. "That was one of the reasons. Except I didn't really think you'd be my first officer, in all the main probabilities. I already had someone else picked out, up until a month ago."
"So what changed your mind?" he asked, blowing on the liquid in his mug before sipping at it.
She cradled her own mug in her hands, curling up one ankle under the other knee. "I accidentally made a potential future enemy. In specific, a Feyori."
Meyun choked a little. He coughed and cleared his throat. "How the _shakk_ did you manage that? You're normally too careful about that stuff."
"To make a long story short, I thoughtlessly uncovered his asteroid while I was busy covering up my own," Ia told him. "The Meddler in question has since taken increasing offense at being left with his bits dangling in the stellar breeze, and will begin moving to oppose me in roughly a year."
He shook his head, then snickered. "...The DoI should be a little more worried about you fraternizing with a Feyori than you fraternizing with me."
Ia narrowed her eyes. "Don't even suggest that, Harper. For one, if you meant dating, the Feyori _never_ rebreed their half-blooded progeny with a full-blooded. Or one half-blood with another, if they can prevent it. That's how the Immortal happened. Pieces are not supposed to become players to their way of thinking, so they only breed downward, dispersing their genetics into a specific species' gene pool. For another, if you meant in a business-association sense, I need _all_ of them on my side. Our side, the TUSPF, the Alliance, everyone's side.
"Besides, even if I were fully Human and tried to 'fraternize' with a Feyori, I'd instantly be factioned with her or him, and that means I'd be counterfaction to everyone _they_ were counterfactioned with. No, thank you," she dismissed, lifting her mug to her lips. "That would undo almost everything I'm hoping to do. I'm saving that as a last resort."
"Well, you're going to have to deal with them somehow," he reminded her, unconsciously echoing her thoughts of a month ago. "The only thing that can back down a Feyori is another Feyori."
"I know. I already have some ideas on how to go about it," she confided after she finished swallowing. Knowing their words were being recorded, she turned the subject back to business. "But I'm not going to borrow from future troubles when I have current ones to handle. What I want right now is your impression of the others. Officers, enlisted, whoever or whatever you've observed so far. I've missed the clarity of your mind. I'd like to have that back again, and on my side."
He shrugged and settled back in his chair. "Well, for starters, I think Helstead's going to be a handful..."
Sipping again at her mug, Ia settled back as well, content to listen to him talk. She did miss him—a part of her missed his touch as well as his friendship—but talking was all either of them could afford to do.
NOVEMBER 4, 2495 T.S.
"God alive, this is a huge ship," Bennie muttered, following Ia down the corridor of the ventral storage deck. "We're going to rattle around like a handful of ball bearings in a gymnasium—do you even know where we're going?" she added, as Ia took a side hall.
The corridor she chose was virtually identical to every other side hall they had passed on this deck. No one had yet decorated the plain grey walls with paintings, pictures, or other means of personalizing a ship, as was so often the case on other vessels. The lack of art made everything bland and boring. Forgettable.
Ia didn't bother to snort. She turned a second corner, palmed open a door, paused a moment on the threshold of the cargo locker, then stepped up in time to help catch a crate just as it started to topple. Not without a cost; the contents were heavy, making both her and the two crew members _oof_ from the awkward near impact. Once the crate was stabilized back on top of its mates, and the female of the pair had climbed back up to reattach the security straps, Ia addressed both of them.
"Make _sure_ you've secured each of these before you apply the next one, Siano, Marshall," she cautioned the teammates of B Squad Gamma, 2nd Platoon. "Another mistake like that could cost you a broken arm, a broken instep, and several smashed toes."
Private Siano wiped the sweat from his eyes. He was taller than Ia and quite muscular, though he wasn't a heavyworlder. "Thanks. Captain. Uh...what's in them, sir? Why are they so blasted heavy?"
"And how did you know we were down here, sir?" Marshall asked, climbing back down again.
"These crates are filled with metal parts for the manufactories, so naturally they're heavy," Ia told both of them. She smiled slightly. "And I told you. I'm a massive precog. Be mindful of what you're doing; we can't afford to lose several days of training with you two waiting for your bones to heal."
"Sir..." Marshall jumped down from the racks, a frown creasing her brow. "I don't get our schedules. They're all over the place. Every hour, we change up whatever we're doing. Siano and me, we've been put in the galley, served time at the scanner boards, barely familiarized ourselves with the gunnery pods, even been put on _laundry_ duty midcycle," she pointed out, then shook her head and poked her thumb at her partner. "He's a weapons engineer, so I get why he'd be running around the ship. We keep passing gunnery-pod doors, and all that. But I'm a nurse, sir.
"I _should_ be in the Infirmary, familiarizing myself with the layout and the gear," the slender woman asserted. "Yet I haven't even stepped foot inside more than once yet because of my schedule, and it's been ten days. With respect, Captain...this is a little crazy."
"I know it is," Ia admitted. "But there will be many times in the days ahead when _you_ in particular will be needed to run these parts to the engineers because you won't have any patients to fuss over, and you'll be the only one free. I need you to know exactly where to look for them. You're not just a fine nurse, Private; you're also going to be mastering many other duties on board this ship. This includes sitting in on all tactical debriefing sessions for both ship and melee combats. You will learn how to repair the ship, craft sound tactics, and defend yourself and your fellow soldiers to the best of your ability.
"I need you ready to go the moment we leave this shipyard...and that means being able to do any number of jobs before you'll need to do them." Clasping Marshall's shoulder, Ia smiled wryly at her and her teammate. "Luckily for me, I already know you can do it. _If_ you pay attention to everything you're learning right now. Carry on, you two."
"Sir, yes, sir," Siano muttered. Heaving a sigh, he turned back to the remaining crates needing to be lifted into place.
Leaving the pair to their work, Ia rejoined Bennie in the hall. She spoke when they were out of hearing range. "They're good meioas. They can do everything that needs doing. I just have to convince them that they can before they'll _need_ to do it. With luck, they'll start believing in me sooner rather than later."
"She has a point, though," Bennie agreed, clasping her hands behind her back and studying Ia as they moved away. "Nurses working like stevedores?"
"She's also an excellent shot," Ia stated. She knew the door to the storage room was still open, and that the members of B Squad Gamma, 2nd Platoon, had paused to listen to her words echoing off the walls. "Far better than anyone in the military has realized. Putting her in full-mech with heavy firepower as well as a field medic's rig will save fifteen of her fellow crewmates' lives within the next year, and I'm just counting from the accuracy of her shots. I'm not counting the lives she'll save through her nursing skills."
Bennie narrowed her eyes. "You can't be _that_ accurate, Ia. The future is constantly shifting and changing."
Now Ia snorted, though it was more a sound of outright amusement than scorn. "Trust me, I'm _very_ good at calculating the odds these days."
The chaplain lifted her chin. "Wasn't it from some old vidshow where the hero quipped, 'Never tell me the odds'...? Wouldn't calculating them be an act of hubris, which would tempt the universe into thwarting them?"
"That depends upon the odds," Ia said. She lifted her chin as well, at cross-corridor Foxtrot. "This way to the belt lifts."
"So where are we headed next?" Bennie asked her.
"To prevent another mistake. Then after that, to the gun range, where _you_ get to show me how well you can shoot." Touching the buttons to summon the lift, Ia waited in silence. Door controls for sensitive areas, such as engineering, the bridge, and so forth, required curling one's fingers into a small alcove to press recessed buttons, a method that would thwart Salik tentacle arms. Elevators were too commonly used for such security precautions, however.
Chaplain Benjamin didn't like waiting in silence. She sighed, bounced on her toes, then finally asked, "So what's the next mistake? Another dropped box to catch?"
"Not a dropped box, and nothing official," Ia said. A faint hum announced the arrival of the lift. The doors slid open, revealing a small, padded room with safety handles. Stepping inside, she punched the button for Deck 3S, third deck starboard, and grabbed a handle. Bennie grabbed one of her own.
The doors slid shut, and the lift moved up and to the side, moving in an arc that followed the curve of the ship. The track wasn't completely circular; the engineers had built it more like a truncated cat's eye in shape, an oval with flattened ends.
"Ugh. Curving elevators. This'll take some getting used to," the chaplain muttered, swaying with the vector changes.
"Well, it's not like they could build straight shafts that cover all the decks," Ia pointed out. They hit the uppermost deck and slid sideways with a _clunk_. Wrinkling her nose, she glanced at the ceiling. "It looks like the timing chains are off. Remind me to have them fix that before we leave dry dock."
The lift stopped, and the doors slid open. The lanky, dark-skinned man started to step inside, then stopped, staring at the two officers. "Ah...Captain. Chaplain, sir."
"Get inside, Aquinar," Ia ordered lightly. "It's a free lift."
"Uh, yes, sir." Stepping fully inside, he found the controls and punched the button for Deck 4S. The doors slid shut, and the lift _clunked_ again. Ia sighed.
"Captain, I do believe you asked me to remind you to have the shipyards fix the timing chain," Bennie teased, as they slid sideways.
Chuckling ruefully, Ia flipped open the screen of her command unit and tapped in a note to the foreman in charge of lift construction and maintenance. She wasn't wearing a jacket and didn't need to unsnap any sleeves this time. Just a plain grey shirt and matching slacks, striped in black down each leg, black belt, black shoes, and the absolute minimum of her glittery, being her rank pins and the two striped bars indicating her past duty posts, one for her time on the Terran-Gatsugi Border, and one for her time on the Salik Interdicted Zone, the Blockade. Each one had a little pip for extra tours of duty at each post.
Bennie was similarly clad, though she had a couple more service pins. Aquinar, clad in grey pants, a matching unbuttoned shirt, and a mottled grey T-shirt beneath, had four pins of his own. He peered at Ia's service bars as the lift _clunked_ again and started descending down the starboard curve. Catching her glancing at him, he quickly looked away. Ia let the corner of her mouth quirk up. She answered his unspoken curiosity.
"Yes, Private; I've only served in two locations. One of them was the Blockade. We're about to serve in over two hundred, with an intensity that will match the Blockade, if not outstrip it." The lift stopped. Gesturing for the chaplain to exit first, Ia followed her, turned, and flashed a grin at the dubious private. "It should be exciting."
Turning away, she led Bennie down the hall and around the corner. The redhead studied her in a sidelong look. "That was awfully cheerful-sounding of you. I thought you preferred doom-and-gloom."
"They have to know I'm not a monster, Bennie. That I can be serious when needed, but that I'm also a fellow Human, deep down inside—that I was raised to be Human, and consider myself one," she amended. "Even if I'm only half of one."
Their destination wasn't far, just to the starboard end of the deck. Hooking her fingers into the controls for the gunnery-pod door, she triggered it with a flex. The door hissed open, startling the occupant. He started to rise out of his seat, dropping his legs from the console to the floor so that he could bolt upright...then _oofed_ and thumped back down, held in by his restraint straps. On the console of his control panel, a trio of tiny little robots whirred and moved, exploring the surface with the child-like patterns of simplistic artificial intelligence.
"Ah! Captain! Aah...how can I help you?" the soldier in the seat managed to ask, scrambling for dignity.
Ia took a moment to look around the gunnery pod, letting him recover. Everything in these pods was fire-by-wire, remotely controlled by analysis computers. Banks of monitors surrounded the gunner's seat, which looked like a modified eggshell, designed to slide and rotate so that the gunner could face and fire along the same fields of view as the weapons themselves. Currently, the screens were active, though they only showed the interior of the huge dry-dock bay holding the _Hellfire_ in place for her retrofits. The curved span of that view made the gunnery-pod chamber look large, but in reality it was barely two meters square.
The actual weapons' towers for this projectile pod and its missile bays lay on the other side of a couple of storage bays and triple-thick armor plating, all designed to protect the gunner from what many weapons techs across the Space Force half-grimly, half-jokingly referred to as "projectile reflux." Given the distances involved in ship-to-ship combat, it was still possible, if rare, for an enemy laser to impact a missile on its way through the external launch tube of a P-pod. The resulting chain reaction could be lethal, particularly if the security measures failed to detect and seal off the main missile blast from the rest of the attached storage bay in time.
On some of the smaller, older ships, the actual control seat was part and parcel of that pod tower, sacrificing some of the usual extra layers of protection in exchange for greater flexibility, lower construction costs, and the ability for the gunner to manually load projectile missiles in case of power failures or battle-plan changes. The usefulness of having the gunner and the missiles in the same location allowed many gunners to "fire by the seat of their pants," using their physical sense of the ship's movement in addition to the targeting computers.
Ships on the Blockade had extra plating and fire-by-wire controls like the _Hellfire_ 's, but many of the ships on the more peaceful Border routes didn't need it. Even so, on the fire-by-wire vessels, most construction placed the gunners at the same point along a ship's hull as the tower, so as to preserve some of that kinesthetic, seat-of-the-pants advantage. Because of the extra slave-driven pods, any one gunnery pod along the length of this ship could be used to guide the rest linked in tandem with it, with most meant to sit empty until needed.
In other words, this was an out-of-the-way location for one of her pilots to slack off from his training duties and pretend for a few minutes that he was just a simple gunner.
"Yeoman Fielle," Ia finally stated, sharpening her tone slightly beyond normal. "While I realize it is currently your rest hour, I shouldn't have to remind you that the gunnery pods are for gunnery techs to familiarize themselves with, and not normally the position of pilots and navigators...or at least, not according to your schedule, it isn't. And if I were to take _official_ notice of this potential breach in Company-Bible protocol, I would also have to take _official_ notice of any unauthorized robotics on board.
"I shouldn't have to remind you that the majority of this ship is Ultra Classified, which means _any_ deviation in equipment from the authorized list would have to be viewed as a breach of security. _If_ I were to notice such things officially," Ia finished dryly. "Breaches of security at this ship's level of clearance usually involve far more than two strokes of the cane."
Glancing at his console, Fielle swallowed. "Ah. Well, I _can_ explain—"
_"Unofficially,"_ Ia interrupted him, "I would recommend you confine any robots I do _not_ officially see to your quarters until further notice...with the understanding that said notice won't come anytime soon...and that you aren't to discuss their presence with anyone other than your teammate until said further notice."
"Of course, sir," he murmured, hitting the release for his straps. Sitting forward, he scooped the pet robots off the dash and deactivated each one. "They never left my cabin, sir—in fact, they never even existed at all!"
"Good meioa." Satisfied he would comply, Ia left the gunnery pod. Again, her actions earned her a curious look from Chaplain Benjamin, but her friend said nothing. "...You and I know those toys of his are harmless. I could even prove it to the Admiral-General in the timestreams...but I don't want to overplay the precognition-protects-my-choices card. _Johns and Mishka versus the United Nations_ covers a lot of what I'm going to do, yes, but I don't need to use it up on the little stuff."
"A wise choice," Bennie agreed. "Pick your battles when and where you can."
"Speaking of which," Ia said, reaching for the lift buttons as they approached the door, "it's time for you to show me _you_ can handle battle."
"You really want _me_ to fire a gun?" Bennie scoffed. "Please. I'm a preacher, not a fighter."
"I'll spot you five points on the targeting range," Ia promised.
"Ten," she retorted.
Ia grinned. "Seven, and not a point more."
Bennie gave her a dubious look. "Are you going to bother to win the shoot-off, or throw the game my way?"
"Considering the scores from your last visit to the targeting range back on the _Hum-Vee_ are openly listed in your personnel file, you'll probably beat me even without the points," Ia quipped dryly. "I could dip into the timestreams to guide my aim, but I won't bother to do that for mere practice. You can't learn how to be really good by cheating all the time, and we're going up against machines that counteract some of my gifts. Without their edge, I am _not_ the best shot on this ship, which means I need the practice."
"Not with two Olympic-class Sharpshooters on board, you aren't," the redhead snorted, following her into the lift. She waited until the doors sealed, then looked at her friend. "Ia, about you and Harper..."
That rolled her eyes. Ia craned her head, looking at the other woman. "Nothing is going to happen, Commander. We've _both_ decided that, and we're both fine with it. Now that he's had a chance to look over this ship, I'm confident he won't want to do anything to rock his assignment."
"You think this ship is better-looking than you?" Bennie scoffed.
Ia smiled, amused by the joke. "The love for a beautiful ship has cured many a captain of a broken heart."
"He's not the captain of this ship," the other woman pointed out, swaying as the curved elevator shaft swung them the other way.
"No, but I'd think chief engineer also counts." At Bennie's chiding look, Ia dropped her mirth. She sighed. "You worry too much about a trivial matter, Chaplain. Worry more about making sure these men and women are comfortable about following my commands, however strange."
"I'm just looking out for your best interests, Captain," Bennie returned. "Besides, he's one of those 'men and women' you need to follow your commands."
"I have no doubt he will," Ia muttered, grateful the lift was slowing for their destination.
NOVEMBER 29, 2495 T.S.
The chime interrupted her concentration. Sighing, Ia rubbed at her brow with one hand and touched the comm button on her workstation with the other. _"Come in."_
The door slid back to reveal the other redheaded officer in her crew. Delia Helstead sauntered inside, looked around at the sparsely decorated walls, and dropped into one of the two seats opposite Ia's. "So. Captain."
"So. Commander," Ia quipped back, focusing her thoughts again. Text started scrolling up the screens of her workstation, until Helstead shifted in her seat, thumping her bootheels on the edge of Ia's desk. A glance at the shorter woman earned her a bright smile. Sighing, Ia didn't pretend ignorance. "...Can I at least finish my thought?"
"It's your office," Helstead pointed out, fishing out one of her thin stilettos from her upswept hair.
"You don't want to play that game with me," Ia warned her lightly. "I'm immune to your mind tricks."
The petite redhead snorted, twirling the sheathed blade between her deft fingers like it was a pen. "It'd be illegal for me to use my psychic abilities on a superior officer without an emergency of some sort."
"Then sit still, be quiet, and give me a few minutes to finish this," Ia told her.
To her relief, Helstead did sit still. Well, quietly, at any rate. She fiddled with her sheathed blades, flipping them over and through her fingers multiple times.
Refocusing her thoughts, Ia resumed electrokinetically composing her correspondence. The last five years' worth of practice made short work of her current round of prophecies. She then pulled up and added her thumbprints to two requisition forms, ones that Grizzle had flagged as urgent, then shipped them off with a tap of the controls.
A second tap lowered the screens back into the scrollbar edging her desk. Lifting her brows, Ia gave Helstead her attention. Helstead continued to twirl her blade until Ia sighed and gestured at her.
Thankfully, the smaller woman got straight to the point. "Your crew is getting restless. Bored, even. They're overworked, and in need of a break," Helstead stated bluntly. She slanted a look at the taller woman, her hazel green eyes sober. "I thought you should know. You've pushed them very hard with these tailored daily schedules. Unless you change something, you'll probably push them too hard."
"They won't break. What do you think of having a party?" Ia asked her.
Helstead took the question in stride. "Better sooner than later. They're learning to work together. If you really want as cohesive a workforce as you keep claiming, they'll need to learn how to party together, too." She grinned. "Though I don't think these shipyards have a pub big enough to contain the resulting mess once they do."
"We won't be able to make a habit of stopping at whatever tavern we run across," Ia stated, her gaze focused more on the future than on the present. "I've already made plans for weekly or monthly 'parties' depending on our schedules. Most of them will take place while we're running between points A and B. We also don't have enough time to hold a decent-sized party before we're scheduled to leave dry dock."
"That might cause some problems," Helstead cautioned. She pulled a second sheathed pin-blade from her hair and started twirling it between her fingers as well, looking like a demented drummer with tiny, gilded drumsticks. Completing the illusion, her toes started tapping a syncopated beat, heels still propped on the edge of Ia's desk. "Right now, they're still exhausted enough each night to get along, more or less. Once they finish adapting to the high pace you've set, they're going to have enough energy to be irritable instead of amiable.
"Now, you made me your chief discipline officer. As far as I see it, that includes heading off disciplinary problems _before_ they become actual problems. Wouldn't you agree?" Helstead asked. The sheathed blades came to a brief rest as she gave Ia a pointed look, though her toes continued to tap the air.
"Yes, I would," Ia said. "But we honestly don't have enough time on the schedule for a party before we leave dry dock. There is a compromise, though. How do you like the idea of dangling the carrot on the end of the stick?"
"Sir?" Helstead asked her, tilting her head a little in curiosity.
"Promise them a party _after_ we leave dry dock. After we leave the Sol System. That should spur them on a bit longer in the cooperation and enthusiasm department. Wouldn't you agree?" Ia asked, parroting her.
The shorter woman studied her for a moment, then started twirling her stilettos again. "I think that could work. I don't suppose you're going to make that announcement yourself?"
"I'd think it'd be a task more suited to an officer with experience in balancing exhaustive expediency versus encouraging underlings toward their goals," Ia countered wryly, eyeing Helstead's wiggling boots for a moment. "That makes it your job. Give them vague assurances at first, then increase the specifics over the next two weeks. I'll make the formal announcement at that time, but I figure you can lay the groundwork for it."
The bejeweled blades fell still with a sigh, and her boots swung back down to the floor. "I suppose _planning_ this party is also up to me, once you've waved your magic official-announcement hand?"
"Not the first one," Ia said.
Opening a drawer in her desk, she fished out a datachip and tossed it at the other woman. Helstead caught it with her heavyworlder reflexes, one brow quirking. Her toes finally stopped tapping. Ia lifted her chin at the chip.
"You'll find all the details you'll need on that, along with the release codes for unlocking the rules and regulations for parties on board the _Hellfire_. They're already loaded into each crew member's Company Bible; they just haven't been made available, yet. I didn't want them distracted from their lessons by reading the, ah, unusual circumstances for such things."
"They're already in there, are they?" the lieutenant commander asked, glancing up from the datachip. She slotted the chip into her arm unit, though she didn't open up the screen just yet. "Official Company rules for these onboard parties doesn't make them sound very enjoyable."
"We'll be calling them Wakes, to go with the overall 'Damned' theme, but they'll be as cheerful as we can make them. They'll also run in twenty-four-hour segments," Ia added, "to ensure each watch gets some time to relax and enjoy the party when they're not sleeping or on duty—come _Hellfire_ or _Damnation_ , our ship will be manned at all times with rare exception—but we'll squeeze in onboard Leave wherever possible and try to make it as relaxing as we can."
"And by 'we' you, of course, mean 'me,'" Helstead quipped dryly, toes resuming their silent rhythm.
"Oh, I fully expect the other officers to pitch in. Including myself. This first one has been preplanned, and there's a list of themes and such," Ia pointed out. "Things we can pull out of the onboard supplies. But I'm always open to ideas."
"Even if they conflict with your precognition?" her 3rd Platoon officer asked, lifting a brow.
"Like all superiors, I may not always follow up on an idea for a particular instance, but I'm willing to listen whenever I have the time," Ia conceded dryly. Mention of the T-word made her dip her head. "Of course, I'm not always going to have that time, which is why I think this is something you might enjoy handling. Now, is there anything else you wanted to discuss?"
Helstead frowned softly. "Yeah. Something Harper said. Something about a...a trip to Antarctica coming up soon?"
_Oh, stars_...Lifting her hand to her brow, Ia pinched the bridge of her nose, then massaged the muscles just above it. "He would mention that," she muttered. Mindful of the surveillance pickups in her office, she sighed and dredged up a half lie. "Snow and ice are dangerous on my homeworld—you're from Eiaven, which has double Standard gravity, so you _know_ why it's dangerous. Sanctuary has more than triple gravity, so everyone lives in the tropical to subtropical climates. Still, while we avoid it back home wherever possible, the _concept_ of frozen water fascinates my brothers.
"They made me promise, if I was ever in Earth's vicinity, I'd bring them some actual snow from Earth. Even now, Antarctica is still virtually uninhabited. That means it's the one kind least likely to contain the sort of bacterial contaminants requiring quarantine measures—and I'd know precognitively which patches to avoid.
"So, long story short, I promised them I'd get them some snow from somewhere near the South Pole, this trip," she explained. "I told Harper all about this back in the Academy, but we never had Leave long enough from the fast-track program to get down there. He probably figures since I have full control over our patrol routes, I'll be wanting to make a stop on Earth, then a shakedown run out to Sanctuary," she related.
"Sanctuary's on the backside of Terran space. I thought we were going to be hunting Salik as soon as we leave dock," Helstead said, fiddling with one of her stilettos again. The rhythm of her toes changed, as if whatever song playing inside her head had been replaced by a new one.
"We still get a shakedown cruise first. Training on the various ship systems while in dry dock isn't the same as when you're out there in space," Ia admitted. "We're also being hired by one of Sanctuary's defense contractors to transport goods to Sanctuary for storage against the coming war...which again, Harper knows about as my first officer and which will make it that much more convenient for me to pick up some genuine Terran snow for my family."
"Genuine Terran snow," Helstead repeated dubiously, fingers, toes, and sheathed blades going still for a moment.
"Yeah, genuine Terran snow," Ia confirmed, keeping her tone even for the sake of the surveillance pickups. "If you think this is some excuse to fraternize with my second-in-command, think again. I haven't the time for extraneous relationships. Harper knows this, and we both know I won't _shakk_ away my chance to save the maximum number of lives."
"Wait, let me check something," Helstead muttered. She reached for the command unit cuffed over her left forearm and tapped on a few of its keys once the lid was open. "Aren't we scheduled to leave here...December 19? If we take a couple days to get to Earth, a day to load cargo, and we're being given a day for Leave...that would have us leaving Earth right before Christmas. We're not staying near the Motherworld for Christmas?"
"The schedule is correct; we're not staying for Christmas," Ia agreed. "But if you look at it another way, we're also missing Chanukah, because we'll still be working hard. And we're missing Bodhi Day, which is at the start of December, and several other celebrations, too. As much as I'd wish otherwise for the crew's sake, I cannot stop this ship or its mission for religious reasons," Ia said. It was an irony to put it that way, considering her plans for her own homeworld, but she didn't hesitate. "Everything has to happen at the right moment in time. Wars do not take a holiday, and we're headed straight into a really big one."
"Unfortunately, most religions have been known to _start_ a few wars, but they rarely stop them," Helstead agreed dryly. She eyed Ia, toes still wiggling, but fingers still. "Mind if I come along on this snow-gathering trip of yours?" the petite lieutenant asked, her expression as skeptical as an arched brow and a dry tone could make it. "Or would I be a third wheel between you and Harper?"
Ia paused for a moment, skimming the near timestreams. She blinked, then shrugged. "I don't see why not. It's just a gathering mission, followed by a trip to Afaso Headquarters afterward. I was planning on three days' transit from here to Earth to test out the insystem thrusters, then a day of Leave on Earth for each duty shift of the troops after we've loaded that cargo for Sanctuary.
"Anyone can go anywhere, so long as they arrange for transport and are back on board the ship when their time is up. But it'll be a small group headed to the South Pole, just you, me, and Harper. We can take one of the shuttles down, pack a lunch, and have a little picnic in one of the most remote corners on Earth," she finished lightly.
"So you don't mind my coming along to play the third wheel?" Helstead asked, pausing her stiletto-twirling.
"If you keep referring to your otherwise freely welcome presence as a 'third wheel,' Lieutenant Commander, I'll make _you_ carry the picnic basket," Ia retorted. "I'll set it up on the schedule for download to our arm units when we get closer to our departure week. And we'll have that onboard party after we set course for Sanctuary since we'll have time for it then.
"Now, if you'll excuse me, I have inspection reports to wade through, a progress update for Admiral Genibes, followed by a reminder call at 7:34 to warn Privates MacArroc and Redrock that they're not allowed to blast music in their quarters, and 263 more prophecies to write before I can go to bed tonight."
Tucking her stilettos into the braid wrapped around her head, Helstead rose. She didn't head for the door immediately, though. Bracing her palms on the edge of Ia's desk, she leaned forward just as Ia reached for the button that would raise her workstation screens. "...Can I have a peek?"
"You haven't earned that level of trust from me, yet." She knew the words sounded a little cold when put that way, but Ia didn't retract them. She did meet the other woman's gaze, keeping her tone soft. "Take comfort in the fact that one day you will. _If_ you don't do anything to break my faith in you between then and now."
Helstead wrinkled her nose. She pushed away from Ia's desk. "I can see that working for a precog is going to be a pain in the asteroid."
That made Ia chuckle. She was still smiling when the shorter woman left her office, though it faded quickly enough. The missives she had to write, most of which were destined for her family and friends back on Sanctuary, were a little too sober for mirth.
DECEMBER 3, 2495 T.S.
Seated in her office on board the _Hellfire_ , Ia waited for her call to go through. When it finally did, the face that filled the screen was sleepy, puzzled, and familiar to Ia. Blinking his brown eyes a couple times, former Private Tom "Happy" Harkins frowned a moment, then widened those eyes in recognition.
"Bloody Mary!" he rasped, staring at her. Scrubbing his hand over his face, he gave her one of his trademark half smiles, one warm enough that it actually curved the other side of his mouth after a second or so. Since they were both in the same star system at the moment, there was zero lag in communications. "How's it going? I haven't seen you in—whoa, is that a brass eagle?" he asked, blinking and rearing back from the vid pickup in his quarters. "Lieutenant Colonel, sir! Congratulations on the promotion, sir. And you're in Greys; somehow I figured you'd end up in the Special Forces."
Ia smiled. "Thank you, Happy, though it's technically Ship's Captain, not Lieutenant Colonel. And yeah, I wound up in the Special Forces. Congratulations on completing your Service with honors and on landing in a cushy job."
"Yeah, that's me," Harkins agreed wryly. He sat back on the edge of his bed, the pickups following his movements. Clad in a worn yellow T-shirt with a faded joke logo and a pair of brown sleeping shorts, he plucked at the material covering his chest. "Saved the life of an attaché to the ACDC in the first month I was a civilian and got roped into bodyguard duty—the only thing worse than pressure-suit drills is contamination-suit drills. The chem scrubbers hurt worse than depressurization sickness. So...Lieutenant Colonel Bloody Mary," he teased her, "what can this lowly ex-grunt do for you?"
"Actually, it's what I can do for you," Ia told him. This was the reason she had saved his life all those years ago, on the very same first combat where she had earned her military nickname. Picking up a datachip, she showed it to the vid pickups on her end, then slotted it into her workstation. "I'm sending you a comprehensive list of Quarantine-Extreme scenarios. I need you to get the Alliance Center for Disease Control to start practicing these scenarios and implementing the suggested measures contained within them, and keep practicing them."
"Sir?" he asked her warily.
"I'm a very strong precog, Happy," Ia confessed. "I always have been. You remember my first combat? I _knew_ I'd end up covered in Salik guts and Choya blood, and knew it before I stepped foot on the _Liu Ji_. I also know you're going to have an opportunity in three days to offer these scenarios to your boss to propose to the rest of the ACDC. Make sure he proposes them."
"I don't know, sir," he hedged, hesitating. "Beggin' pardon, but last I heard, you were winning medals right and left, not medical diplomas."
"You owe me your life," she reminded him gently. "That very first mission, where I insisted you come with me, instead of heading back through that stairwell entrance? You would have died if I hadn't rearranged our two team pairings on the fly, so that you _had_ to stick with me. You _know_ you would have died. I saved your life, that fight...and with these scenarios, I'm hoping to save a lot more lives than just your own. I _knew_ Ferrar and the others would be captured, and cultivated that pirate crew well in advance, knowing we'd need their help to break into the Lyebariko's stronghold. You're in the right place at the right time to do a whole lotta good, Happy. I made sure you would be."
He stared at her a long moment, then sighed and scrubbed at his face. Another sigh, and he reached for the controls on his bedside comm. Or rather, for the drawer under them. She heard the thumps and rustles as he rummaged for something.
"Yeah, I do owe you my life. More than once. Datachip, datachip...Here we go." His hand loomed large for a moment as he pressed the chip into the base of the comm. A couple taps later, he confirmed the download, transferring it from the cache on his bedside comm to the chip. "Right. Show this to my boss in three days. Just tell me something in return, Bloody Mary. How badly _will_ we need these quarantine measures?"
"We'll be at war in a few months, and the Salik will be throwing everything they can at us," Ia stated. "Robotic soldiers, genetically engineered animals, microbial infections, you name it."
"Everything and the kitchen sink, eh?" he asked rhetorically.
"A scummy, slimy, unscrubbed kitchen sink," Ia agreed. "You remember what your Drill Instructors said back in Basic about using the same playbooks over and over for practicing battle scenarios?"
He half smiled and chuckled. "That if you stuck to the same drills over and over, it was nothing but a load of _shova v'shakk_ shoveled by a bunch of _shakk-tor_ officers talkin' out their brass, because it made you predictable to the enemy. That flexibility, on the ground, in the trenches, with the grunts making their tactical decisions based on what _is_ , and not on what _should be_ , is what will save a fight. I remember, Ia. So, if you're really a precog—which would explain a helluva lot, back on the _Liu Ji_ —then you're saying we're gonna get some biowarfare plays outta left field, and that we need to learn some new field maneuvers for 'em?"
"These plans will help prepare the Alliance to apply the bleach to the upcoming mess in the sink in the most effective ways—from mild infestations all the way up through the absolute worst-case scenarios of having to quarantine entire planets. The ACDC's been using the same playbook for too long, and the Salik got their tentacles on it ages ago. Better to be flexible than sorry, and all that. I've also tried to be thorough with each scenario presented," Ia added, "because if you or your boss has any questions, I'll probably be out of reach. Each scenario includes contact numbers for a couple higher-ups in the Afaso Order. They'll have a list of answers and backup contingency plans if you do need answers. Hopefully, you won't need them, but I like to be thorough."
"I don't suppose I can ask where you'll be when you'll be out of reach," he muttered, "what with you being Special Forces, now?"
"The only thing I can tell you is that I'll still be living up to my nickname, and keeping it fresh, Happy," she said. "Once a Marine, always a Marine, even if they keep Branch-hopping me."
_"Eyah?"_ he asked, in the old call and countercall worked up between the V'Dan version of the Marine Corps and their Terran counterparts.
"Hoorah," she agreed. "I appreciate your handling this for me. It's going to be a huge worry off my mind. Sleep well, Harkins. You've earned it."
"You, too, sir," he agreed. "Harkins out."
His hand swung by the controls again, plucking out the chip before shutting off the screen. On her end of things, Ia took one precious moment to sigh and relax into her chair, checking the timestreams out of habit. The future was solid on this one point; Harkins would indeed present the scenarios to his boss, after checking them over for himself. From there, the ACDC would start implementing the new "playbooks" on how to apply quarantines during a multi-star-system war.
It was one big worry off her shoulders, but she still had a million more to go. Drawing in a deep, bracing breath, Ia shifted upright again, moving to the next concern on her list.
DECEMBER 22, 2495 T.S.
BATTLE PLATFORM _LION'S CLAW_
EARTH, SOL SYSTEM
The combination of the TUPSF _Hellfire_ docked at the Battle Platform _Lion's Claw_ looked very much like an overgrown thistle clinging to a cigar, if both thistle and cigar were wrapped in shiny tinfoil. They were actually visible at the moment, side-lit by the blue-white glow from Luna and front-lit by the golden white glow of the Sun.
Normally, most ships—legal ones, as opposed to silently running pirates or invading enemy ships—were identified in the black, star-strewn depths of space by their transponder signals rather than by any physical signs. All ships traveled in ceristeel skins, the highly polished, pewter grey metalloceramic material devised by the Terrans centuries ago. It provided insulation and protection from lasers, stellar radiation, extremes of cold and heat, and even some impact resistance; its biggest drawback was that it had to be properly polished to work at its most efficient, rendering each hull a dark mirror. But the elongated ship and its neighbor gleamed in the light illuminating them from the local sun.
Ia liked the effect, even if all that reflected light was starting to put little dazzle spots in front of her eyes.
"I'm _bored_ ," Helstead muttered. She had strapped herself into the jump seat behind the pilot's seat. Her smaller frame made that the best choice, though it did mean Ia had to put up with Helstead's feet braced against the back of her chair. Shuttle seats were built to be sturdy, but the restless woman still managed to jiggle Ia a little.
"Better bored than biological," Harper muttered back. "I think my second cup of coffee is finally making an appearance."
"Shh. We're about to descend." Two seconds after she hushed them, Ia's headset sprang to life.
_"Shuttle_ Hellfire-One, _this is Earth Orbital/Atmospheric Control. You are cleared for descent to Ridley Beach, Cape Adare, Antarctica."_
_"Thank you, Control._ Hellfire-One _will be descending per the filed flight plan in five seconds,"_ she replied. A touch of the controls ended the link. Grasping the joystick with her left hand, she gently pulsed the thrusters and adjusted the angle of the shuttle. Breaking orbit meant breaking away from that gleaming image of her ship—currently under Lieutenant Rico's command—and pointing their ride instead at the blue-white marble of Earth.
Adjusting their angle to match the line being drawn on her viewscreen, Ia turned up the shields and guided the ship down toward the atmosphere. Going from medium to low orbit posed no problems; the nearest vessels to theirs were a quintet of sweeper sails, designed to trap and remove from orbit any stray debris left over from hundreds of years of Earth's near-space exploration and exploitation efforts.
Atmospheric descent wasn't a problem, either; by the time they reached the lowermost levels of the mesosphere, they were well over the southern Pacific Ocean, where traffic was sparse. Traversing the stratosphere, the cloud cover beneath them was thick enough that they could only see a patch of Australia.
"Was that smoke on the, what, Australian continent?" Harper asked, craning to look out the shuttle's half-silvered windows as the last of the continent vanished from his field of view.
"Smoke, or a dust storm," Ia dismissed, watching the various flight and engine readouts projected across the window-like viewscreen. "It's summer down here, so it could be either."
"What, you don't know?" Helstead asked her. "I thought you were supposed to be some all-knowing, all-powerful precog."
"I never once claimed to be all-powerful," Ia replied, adjusting their angle as the view of the planet started to vanish behind the fiery glow of their rapid descent. The thruster shields were holding steady, keeping the atmospheric pressure and thus the thermal shockwave at bay, but she wanted as narrow an angle as possible to minimize heat buildup. "And I'm certainly not all-knowing. I _could_ find out, but since it's not important, I'm not going to waste my time trying to find out whatever problem they're having. That's for the local weather controllers and the emergency services to handle."
"And yet you're wasting your valuable time on a trip to pick up some snow," the woman at her back pointed out. "Hell, I'd be surprised if there was a manufactory on board our ship that _couldn't_ fabricate a little snow, you crammed so many different machines on board."
"It's the principle of the thing. You don't give your wife a strand of cheap plexi beads when she's expecting genuine pearls," Ia said. "As it is, I'm going to catch a little flak from Admiral Genibes about this trip, but a promise is a promise."
"Technically, you _could_ give your wife plexi beads instead of cultured pearls," Harper countered. Then grinned, mock-flinching from Helstead's glare. "Hey, I didn't say it would be a _smart_ thing to do!"
"Ugh. I think I can see why you ditched this guy two years ago, Captain," Helstead joked, switching one foot from the back of Ia's seat to the back of Harper's. She had to stretch a little, but managed to jiggle him with a couple shoves. "No romance in his soul."
"I'll have you know I have a _lot_ of romance in my soul," Harper retorted. Then shot a glance at his Commanding Officer. Ia didn't look his way, but she did arch the nearer of her two brows. He cleared his throat, and added, "Not that I have anyone I'd care to _spend_ it on right now..."
Helstead snorted. "Hm. Practice that a little bit more, and you'll have me convinced."
"Do I have to tell the two of you to separate?" Ia asked. "This is not your parents' hovercar, and I am not your mother."
"Captain, no, sir," Meyun muttered. "Shutting up now, sir."
Helstead said nothing, though she didn't remove her foot from his chair until the turbulence made her shift back to Ia's seat for better comfort. A stream of clouds heralded their descent into the troposphere. Those clouds blended into the mostly white lump of the sunlit, southernmost continent of Antarctica.
Long minutes passed as Ia guided them toward the polar coast. As they came near, the flight-path lines projected on her screen vanished with a touch of the controls. That earned her a curious look from her two passengers.
"You're going to hand-land the shuttle?" Harper asked her.
"Yes. The flight path I filed assumed we were going to land on the beach, but I want to be sure I don't disturb any penguins," Ia stated. "That would be rude."
"So where are we going to land?" Helstead asked her, toes starting to tap on the back of Ia's chair.
Reaching for it with her electrokinesis, Ia gently seized the black box on the shuttle. Tampering with flight recorders was a major crime, and doubly so for military equipment. She tampered anyway. The recorder "heard" her voice stating that they were going to be landing among the hills above the coastline. Her two passengers heard something different.
"It's more like 'when' are we going to land." Dropping the vessel to subsonic speeds, she lowered it farther into the troposphere, below the clouds threatening to obscure the world. "We're not actually stopping at the beach on the flight plan. Nor are we stopping, period, until we're deep in the Transantarctic Mountains. We will, however, be flying beneath radar range, so it'll be an hour-plus, and rather bumpy. If you want to hit the head, Harper, I suggest going now. We'll be getting bad turbulence from the mountains after about six minutes."
"I _knew_ it," Helstead crowed, pushing on the back of Ia's seat. She ignored Harper, who had taken Ia at her word and was already untangling himself from his harness straps. After a moment, Helstead frowned in confusion and nudged the back of Ia's seat with her feet a second time. "Except it _doesn't_ make sense. Why Antarctica? Of all the places on the Motherworld you could go for an assignation, why here? You don't even have the old joke of an excuse that the 'wedding night' would be six months long, because it's summer down here. The local _day_ is six months long."
"It's simple," Ia explained. "And it has nothing to do with fraternization. We're going to gather some snow from one of the most remote corners we can find...and steal schematics from one of the most dangerous repositories of knowledge in the known galaxy."
That silenced the other woman. At least, until Helstead unclasped her restraints and scrambled into Harper's abandoned seat. She did buckle herself in place, but tucked her feet up onto the edge of the copilot's console, frowning the whole time. "...Okay. Give. _What_ secret installation, and _what_ schematics? Last time I checked, you served in the Terran United Planets Space Force, which _serves_ the governments of the Terran United Planets.
"You swore an Oath of Service when you joined up, Captain, and your home planet's charter is just the same as mine," the redhead reminded her. "The moment you swore that oath, you _became_ a loyal Terran soldier. Frankly, from everything I've read about you, you're loyal to a fault. So why the mucking hell would you steal from the Terrans _now_ , of all times?"
"I _am_ a loyal Terran soldier. I also never said it was a Terran base, government or military," Ia murmured. Her hands danced on the controls as they reached the coastline, slowing the ship for a moment, making it a little easier for her to fake a landing for the flight recorder. Picking up speed again, she aimed for the mountains snaking their way past the ice shelves stretching into the ocean on their left. "You'll want to move back. Harper whines when he doesn't have enough leg room to stretch out. Or at least he did back at the Academy."
" _That's_ why you hesitated," the redhead observed out loud. "In your office, weeks ago. You were double-checking to make sure I'd be an asset on this infiltration team. You _knew_ I'd want to come along if I knew what the real deal was. I thought you were hesitating because you _were_ going to fraternize."
"Actually, that depends upon you. Either you'll be an asset, following my orders to the best of your ability, or you'll be a liability, and sent back to watch the shuttle," Ia said, looking at Helstead. She continued to fly the ship as she did so, sharply dodging up and over a ridge without looking. A muffled yelp came from the head, located behind the cockpit cabin. Ia raised her voice a little. "Sorry, Harper!...You might want to move now, Helstead. It'll get a lot rougher in a couple minutes, and he'll want his seat back."
Sighing, the short, stocky woman complied.
# CHAPTER 3
_I'm sorry, but I still cannot tell you anything about the speed, armor, or weaponry of my ship. You, and pretty much everyone else watching this interview, do not have the security clearance to know anything about how fast it flies, how hard it hits, or how much damage it can withstand. And yes, that includes any and all information regarding how my ship can appear to travel faster than FTL. Those secrets must remain undisclosed for another two hundred years._
_~Ia_
Clad in pressure-suits for their thermal-insulation properties, and camouflage greys over that, the three Humans brushed the snow from their gloves and stepped down into the cave. Ia shined the flashlight attached to her arm unit on the ground. The floor of the cave was both slippery with ice and crunchy with grit worn down from the walls by wind and water. It was also not pristine. Faint footprints, a little smaller than her own, marked a trail. Those footprints went straight to the back wall...after starting at about the midpoint of the cave.
"Okay, _that_ is unnatural," Harper whispered, shining his own arm-held light on the footprints. "They're Human-shaped, but they just...start, as if the person teleported down here. Who the hell would want to come down here?"
"A teleporter would have to know this cave, really _know_ it, to be able to 'port accurately down here, particularly since it's so dark." Helstead countered, touching the rugged walls with one silver-gloved hand. "Even then, most still prefer line-of-sight 'porting. So it'd have to be a telekinetic."
"Well, it would make more sense than a telekinetic's flying down here; the mouth of this cave was covered in snow, which _we_ had to uncover. It doesn't snow as much down here as all that," he reminded her.
"Yes, but teleportation is an extremely rare psychic gift. There's not more than a couple hundred in each of the known psychic species," Helstead told him. Then she turned to their CO, and added, "I'd also like to know how _you_ knew this was down here, Captain."
Ia tugged up on the strap of her backpack, which was threatening to slip off her shoulder, then flipped her hand between the three of them. "Hi, there; I'm Ia. I'm a massive _post_ cognitive, among other things. Meyun Harper, meet Delia Helstead. Delia's a Rank 9 teleporter, so she should know what she's talking about—but for the record, both of you, it wasn't a teleporter _or_ a telekinetic."
"Alright, Captain Smarty-Pants," Harper offered, giving Ia a dry look. " _You_ tell us how these footprints came to be like this, without teleportation _or_ telekinesis."
"It's easy," she murmured, moving closer to the back wall. Playing the blue-white light from her bracer over the wall at an angle, she revealed a series of unnatural round depressions in the dark basalt. "The person in question flew down here as a soap bubble."
"A soap bubble?" Helstead asked, her tone conveying most of her skepticism in the shadows of the cave.
"Yes. One made of pure energy."
"Feyori?" The leader of the 3rd Platoon scowled and spat on the ice-crusted ground. "Mud-sucking _shakk-tor_...We're breaking into a _Meddler's_ base? With respect, Captain, you're _nuts_."
Eyes closed, senses turned inward, Ia cut her off. "No, I am not nuts, Helstead." She found the pattern she wanted, opened her eyes, and pressed on the stones at the back of seven different depressions. Rocks scraped loudly to her right as a passage opened up. "And this place wasn't built by a Feyori."
"It wasn't?" Harper asked. He followed Ia into the rough-hewn tunnel, with Helstead at his back. "Who or what built it, if not a silvery soap-bubble Meddler?"
"Well, that depends on what you want to call her. The Abomination, the Immortal, the High One...or if you like, the First Empress of the V'Dan Empire, who literally—and quite successfully—ruled the V'Dan for the first five thousand years of their recorded history," Ia explained. The tunnel turned and descended on smooth-cut stones worn only a little bit by the passage of whoever had made those footprints outside. "The one and the same entity who rescued them from the tectonic-based disasters of not quite ten thousand years ago here on Earth, creating the _d'aspra_ of the Sh'nai faith."
"Huh?" Helstead asked, trying to squeeze past Harper. He sighed and let the impatient woman pass. "You're kidding, right?"
"Nope. Luckily, we have time for a history lesson," Ia explained, as they descended. "It's a long way down...and it's sort of a future-history lesson as well as a past one."
"Is it an important lesson?" Helstead asked warily. "Or is it just a way to kill time as we walk?"
"Both, since you're now working for me, and I have to manipulate the Feyori into doing what I want in the near future. Roughly two hundred years from now—into our future—a pair of Human-Feyori half-breeds will have a clandestine little liaison and create a child. The recombinant genetics of that child will give rise to what the Feyori call the Abomination, a being who is one hundred percent matter-based _and_ one hundred percent energy-based. Both fully Human and fully Feyori...and impossible to kill."
"So how does that make her different from any other Feyori?" Helstead asked pointedly. "You stab 'em in their matter form, and they pop back into their energy form and fly off."
"I think you missed the bigger point, Helstead," Harper said. "She said two hundred years into the _future_. I want to know what a woman of the future has to do with a place that was clearly built in the _past_."
"Patience, I'm getting there," Ia chided both of them. The carved-stone stairs here were somewhat dusty, but the imprints of several feet marred the grey-coated basalt. She let the light from her bracer play over the marks as she spoke. "For the record, it's not impossible to kill a Feyori. It's just very, very difficult, and you have to catch them in their energy form and do it in just the right way.
"I'm not going to tell you what that way is because I really don't want them madder at me than they already are," she muttered, thinking of her mistake with that faux-Solarican, Miklinn. "Unfortunately, if you try to do that to the Immortal, _she_ just pops back to her matter form, alive and upset. Kill one form, she pops back into the other form, back and forth, back and forth. Which the Feyori discovered when they tried to get rid of her. Or will have tried...whatever," Ia dismissed, flicking her free hand.
"Why would they do that?" Helstead asked her. The stairs were now spiraling down in a tighter curve, with no end in sight.
Ia shrugged. "Because she has all the powers of a Feyori, but was raised to be a Human. To think like a Human. She is outside their great Game, outside their control. Thankfully, it will have been _my_ future intervention that will show the Feyori how to distract her from getting involved in their politics by using her ties to her fellow Humans."
"...Okay, you've officially lost me," Harper muttered. "You also still haven't explained what someone in the future has to do with an underground facility built at some point in the past."
"In a note written to be delivered roughly two centuries from now, I will outline to the Feyori that the best way to ensure she doesn't meddle with _their_ galaxy-sized Game is for the Immortal to become preoccupied with her fellow Humans," Ia explained patiently. "In order to do that, they will have to band together in a Great Gestalt and accelerate her physically beyond the squared speed of light, to the point where they throw her fifteen thousand years into the past...and they will do so at the cost of roughly twenty Feyori lives. But it _will_ have the desired effect of getting her out of their nonexistent hair rather than have her get involved and try to take over their great Game."
"You mean it already did," Helstead countered. Then winced. "Wait— _nobody_ can go faster than the squared speed of light! _To_ the squared speed, since we know the Feyori can swap between energy and matter, but _faster_ than it? No!"
She stopped on the stairs in protest, then moved to one side so that Harper could get past her. He shook his head as he passed the petite officer.
"The fact that the Feyori _can_ reach that border-transition at all convinces me that the squared speed of light can be broken, Lieutenant Commander," Harper stated, his tone grim as he edged past the smaller woman. "That part doesn't bother me as an engineer since they've obviously proved it can be done. _Causality_ is what bothers me. You cannot meddle with the past without running into paradox. _That_ violates the physics of the universe!"
"What, like how faster-than-light travel was 'impossible' five centuries ago because _that_ violated 'the physics of the universe'? At least, until we learned to suppress the Higgs field?" Ia asked him dryly. She paused in her descent to look over her shoulder at him. He shined his arm-unit light in her face, but she didn't flinch from the glow. "It's not only possible, Harper, it has already happened. And there is no resulting paradox."
"But _you_ told the Feyori what to do," he pointed out. Literally pointed, swinging the beam of his flashlight toward her shoulder, casting odd shadows onto the dark stone walls. "Doesn't that create paradox?"
"Technically no, since they would've figured it out on their own. I just sped up the process by a couple hundred years, and cut back on the deaths of several hundred pissed-off Meddlers. Or will have sped it up," she amended awkwardly, flipping her hand again. "Let's just ignore the whole proper-tense requirements for now, or I'll give all of us a headache...My point is, we are not caught in a causality loop, we do not violate the cause and effect of space-time, and the universe isn't going to implode just because _you_ think the temporal implications are always going to be one giant game of but-first."
"A game of _what_?" Harper asked her, frowning. "Butt, as in buttocks? The asteroid?" he added, poking his thumb behind his shoulder. _"Gluteus maximus?"_
"No, as in but-first," Ia repeated, raising her brows. She sighed and explained when he didn't get it. "You know, senior-itis? Going through the entire day saying to yourself, 'I need to get this done, but first I have to get that done...but first I have go get this _other_ thing handled, but in order to that, I have to get the _first_ thing done...' _That_ kind of but-first."
_"Oh."_
Helstead snorted. " _I_ knew what she was talking about—are these stairs ever going to end somewhere?"
"They end at the bottom," Harper said, pragmatic. "We'll get there when we get there. Have faith in our Captain. I have faith. I just don't _understand_. I still don't see why it doesn't cause a causality loop."
"Meyun, that sort of inability to get anything done isn't the paradox everyone thinks it is, because real life doesn't work that way," Ia told Harper. The stairs turned to the right and left a few times as they descended, lit only by the lights on their arms. "You may have to work with incomplete information at times, but you can always get at least one of those three things started in the but-first loop I mentioned...and that means it _isn't_ a loop. It has a beginning, a middle, and an end. The duct tape isn't firmly glued in place on the roll, because there is _always_ a free end to pick at it and pull it up."
"Causality demands—" he argued.
"—It demands that there be an action, and a reaction, nothing more and nothing less," Ia stated, cutting him off. "It says nothing about the reaction always having to happen _after_ the action, Meyun. If reactions _always_ had to happen after actions, there'd be no point in maintaining a preemptively readied military presence because you could _only_ act after someone else attacked, giving you a reason to strike back."
"I'm sorry, Ia, but it still doesn't make any sense," Meyun muttered.
"Trust me on this, Meyun. The universe does not have to 'protect itself' from causality," she stated firmly. "The first lesson they teach you about nonhyperrelay-based communications is that our perceptions of 'observed' versus 'actual' simultaneous actions taking place across interstellar distances _do not match_ , yet they do not match in such a way that it does _not_ violate the causality of the universe. _Both_ the future and the past can be rewritten. It's just that it's far easier for all of us to rewrite the future because we're already traveling in that direction. Time is something that moves, therefore time has inertia. Even when traveling backwards through time, if they survive the trip, a Feyori still ages _forward_. They don't shrink down to the equivalent of infancy, but rather continue forward toward senility."
"Because it's like steering a ground car?" Harper reasoned, thinking it through. She nodded, glancing back at him. His brow furrowed a little, but in thought, not in anger. "At high speeds, vector changes become extremely difficult, even dangerous, particularly when they're off by more than a handful of degrees. To try to turn back by more than ninety degrees would take too much energy and put too much stress on the car, the passengers, the tires, even the road. It can be done, but it's extremely difficult. Right?"
"Exactly," Ia agreed. They were now descending in a tighter curve to the left, almost a spiral, if the core of the spiral were at least a meter thick. "The only point of dissimilarity in your analogy is that, in order to turn yourself around so that you were going into the past instead of into the future, you'd have to be going _faster_ in order to succeed, rather than slowing down to make a sedate U-turn. It's more like escaping the gravitational pull of a planet, but instead of going up, away from the downward pull of the planet's mass, you're trying to go backwards through the forward pull of time."
"You're both freaking crazy," Helstead muttered. She started to say more, then tensed. Her hand dropped to one of the pistols slung around her armored waist. Her voice dropped even lower, to barely a murmur. "Halt. I smell fresh air. Too fresh for a cave."
"That would be the Loyalist AIs coming to meet us," Ia told her companions, not bothering to lower her volume. "The surviving members took refuge with the Immortal at the end of the AI War...because I will have left instructions with the Immortal to offer them amnesty in my name, a couple hundred years back."
"But isn't _that_ a causality loop?" Harper asked her, still stuck on that point.
"Hello? We have artificial intelligences inbound!" Helstead hissed at the two of them. "Shouldn't we be arming ourselves, Captain?"
"Keep your weapon in your holster, Helstead. And your blades in your sheaths," she added, mindful of the woman's lethal little hairpins. Raising her voice, Ia called out, "Greetings, AIs KXD-47 and NNH-236...also known as Margaret and the Padre. You are being approached by the Prophet of a Thousand Years and two of her companions. I need you to let us through, and not say a word to your landlady about our presence, nor about our purpose, nor anything else about whatever we may observe while we're down here."
"Says you," a gruff male voice echoed up the stairs.
"Yes, says me," Ia confirmed. Descending several steps, she stopped when her flashlight illuminated the barrel of an archaic but still serviceable projectile gun. "As the Immortal once told you, Padre, 'You will know the Prophet by her stunning level of detail, past, present, and future.' Having been picked for active duty this month, you were in the middle of your gun-cleaning cycle, taking time off after cribbage game 10,347 with Margaret, when the telltales for the front door went off.
"Knowing that Shey was _not_ scheduled to arrive for another four years, three months, six days, twelve hours, and seventeen minutes...give or take a few minutes," Ia allowed wryly, "you quickly grabbed Margaret's precleaned gun and took point, leaving her to scrounge up her holdout pistol and follow. Which she did at high speed, labeling you with choice epithets about your cheaply manufactured background and dubious processor parentage as you rode up here in the main lift together—need I go into greater detail, or will that much suffice?"
The barrel lifted a little. Its wielder couldn't physically see her, but the Padre wasn't using standard eyesight to target Ia through the curve of stone between them. The tone of his voice remained skeptical. "What do you want, down here? If you really are the Prophet, what do you need from us? Don't you deal in the future, not the past?"
"The Second Salik War is coming, as the Immortal knows full well," Ia reminded them. "I need the exact schematics on how to marry OTL and FTL together to make hyperwarp travel possible two hundred years in advance of everyone else...or did the Immortal not tell you all that much about me?"
"She told us several things." The second voice was female, low and mostly pleasant; the AI wielding it managed to inject a note of doubt into her tone. "But she never mentioned your visiting us."
"That's because I really don't need her knowing about this visit— _you_ know what she's like, Margaret," Ia cajoled. "She already knows I'm the reason why the Feyori stopped pestering her. I don't need her pestering _me_ in some warped attempt at gratitude. I can manage far better without her 'help' than with it...and I'd rather let her keep her free will in her blissful ignorance than have to impose my will forcefully to keep her out of my way."
"If you're really the Prophet, then you're also a member of the Terran military, and the military is why we're stuck in this damned exile," the Padre growled.
"Yes, and if I really am the Prophet, then I also know what you want, and I can tell you exactly when and where you'll be able to regain your places in the galaxy as fully accredited sentient beings," Ia coaxed.
"Impossible!" Margaret's voice snapped. "The war started because every government says we aren't, and we never will be!"
The barrel of the rifle jostled for a moment, then the female bounded into view. She didn't block her partner's firing angle, hugging the outer wall so that the Padre could stay close to the inner one, but Margaret did plant one hand on her hip and give Ia a skeptical look. For an artificial life-form, she was fairly realistic. Her skin was pale and smooth, her hair dark and thick with curls, and her gaze steady. She even pretended to breathe like a normal woman, though technically speaking she didn't need air for anything other than producing speech. That, and snorting in derision.
" _Everyone_ knows the damned war started _because_ we're not fully sentient. We're not alive, so we don't have a soul...and those prejudiced bastards yanked the plugs on all cybernetic research. Last I checked on the news Nets," Margaret added, a hint of tart cynicism programmed into her tone, "cyberware is still very much illegal because it's still vulnerable to hacking, rendering it unprofitable even for the black market to try to peddle. Not to mention it's extremely illegal to grow a whole body and attempt to supplant its innate consciousness with an artificial one, just as it's illegal to place an organic consciousness inside a mechanical corpse."
"It's not a case of growing a Human body," Ia countered, moving down two steps. That brought her to just beyond the android's reach, and well within firing distance. The Padre wasn't going to shoot her at this point, however. "Three hundred years from now, you will be recalled to active duty—all of you, if you choose to go. Those who serve and survive will be given the option of having your programming transferred into new bodies.
"I cannot tell you exactly what they will be made from, but those new bodies will naturally produce KI and thus bear souls...and they will be able to do so _without_ violating any Alliance laws regarding the growing of organic sentients, the supplantation of innate personalities, so on and so forth," she added, flicking her hand in dismissal, "because they won't start out as sophonts, let alone as sentients."
The man and woman blocking her way were very much machines; the only way Ia could have read their minds was by electrokinesis, but she wasn't familiar with their archaic programming. All she had to convince them with were her words.
"You will become extremely advanced androids, of a type manufactured by the Third Human Empire, and in doing so, you will retain your self-identities even as you gain official sentient status. All you have to do is answer the summons to war when called to action by the Phoenix of the Zenobian Empire...and if you survive that combat, the survivors will be granted new bodies and new status. You have my Prophetic Stamp on that."
Another step down brought the Padre into view. He was short, stocky, and swarthy, with a neatly curled mustache and a pair of wire-rimmed glasses perched on his nose. For show, of course; Ia knew his vision was still as acute as the day he had been made. One of the things the Immortal had done was stock up on replacement body parts for the Loyalist AIs. Like Margaret, his clothes were out of date by several decades, but they fit him well despite being age-worn.
"So you just want to see the OTL/FTL conversion schematics?" he asked her. "Is that all?"
"That's one of the two things I want," Ia countered. "The other, you don't need to know, and you don't _want_ to know. But I swear it will not harm the Immortal's best interests. Now, please stand down and step aside. I'm running out of time," Ia ordered. She started to move forward. The android Margaret planted her hand on Ia's shoulder, stopping her.
"I'm sorry. You've been rather accurate with your information on us, but Prophet or not, we _are_ bound by our oaths to protect this place—"
Ia might not have known much about their archaic programming, but she did know the probabilities involved in this encounter. Before Margaret could finish her threat, she tapped a command into her arm unit. It in turn pulsed a pair of EMfrequency codes on an infrared carrier wave.
"—and we cannot...not..." Blinking, Margaret stared sightlessly for a moment. Her half-formed threat vanished. The hand on Ia's shoulder lifted, turning into a salute. "Sir! KXD-47 ready for duty, sir!"
The barrel of the Padre's gun flicked up, resting vertically in a modified salute. "Sir! NNH-236 ready for duty, sir!"
"I'm really sorry I had to do that," Ia murmured, studying the two AIs. "I know you'll remember this, and I want you to know I hoped you'd do this of your own free wills. As it is...your orders are simple. You will escort me and my two companions safely through the Vault to the Engineering Archives, wait for us there, and escort us back out again when we are through. You will not interfere with or prevent our search for and acquisition of the information we seek, and you will refrain from _ever_ mentioning this visit to Shey, in any format. In fact, you will deny it if the lowest probability occurs and she actually asks about it.
"In 260 years, you will mention my prophecy regarding new sentient-status bodies to the other AIs in the Vault, at which point it will be self-evident what I meant by the Phoenix and the Zenobian Empire, upon which time you will be free to await the aforementioned summons and decide at that time whether or not you want to answer it. Now, guide us to the Engineering Archives. We haven't much time," she instructed the pair.
"Sir, yes, sir!" Padre snapped, and turned to head down the stairs. Margaret slipped ahead of him, moving faster. Her psychological programming had always made her a little faster, a little more hyper than the Padre, Ia knew.
Helstead slipped in front of Harper. She eyed the androids warily, following in Ia's wake. "What did you just do to them?"
"I activated their loyalty codes. I would have far rather had their willing cooperation, but they'll obey me until I release them," Ia confessed quietly. "These two were soldier AIs, once upon a time. Those particular codes could only be activated by certain members of the Command Staff, or the Premier. They're going to treat me as their supreme commander, for now."
"I remember our history lessons back at the Academy," Harper said. He, too, kept his voice low. It was probable both AIs could hear their conversation, but they said nothing about it as he continued. "The Rebel AIs slaughtered the Premier and key members of the Command Staff so that they could suborn the loyalty programming of the military AIs with their viral rebellion. There's just one flaw, Ia. The Loyalists—the original ones—threw off that virus."
"Yes, they did, with a variation of the same codes the Rebels used to throw off their loyalty conditioning," Ia agreed. "I just reset their loyalty switches to the original pattern, then rekeyed them to include me in their command chain. Technically, we have twenty-four hours, give or take a couple, before they break the new code."
Helstead whistled, one hand on the butt of the pistol slung at her hip. She spoke in an undertone. "...They are going to _hate_ you when they break free, you know. Should we even be talking about this within their hearing?"
"They're loyal for now; they won't question my orders or my reasons. I'm also planning on giving them the codes they need to break free, with the instruction to wait one minute after we've completely left before implementing them, so they don't have to break the codes controlling them. That would run the risk of damaging their programming," Ia told her. "They have every right to exist down here, and I'm not going to ruin that for them.
"Besides, we have to be back on board the ship for Helstead's duty watch in thirteen hours anyway, and before we get back, the Grandmaster of the Afaso is expecting me to drop by. Since we won't be here the full twenty-four hours, there's no point in keeping them code-locked long after we're gone."
"Is that wise?" Harper asked her. "Breaking them free while we're still within attacking range?"
"Wise? Maybe not," Ia said. The Padre glanced back at them. She met his questioning gaze steadily. "But it is honorable. Given a few years to think about it, they'll come to respect it. I'd rather not have forced the issue, but the Immortal knows _and_ obeys the foremost law of her birthworld...which means so will these AIs, once they've had a few minutes to think about _that_ ," she added dryly, glancing at the android leading them onward. "Since in swearing to serve her, they have sworn to obey that very same law. The first and foremost law of the Freeworld Colony of Sanctuary, which will one day become the Zenobian Empire. The same government that will welcome them with open arms and citizenship papers in the future."
They reached the bottom of the stairs. Margaret was already touching a handful of the round depressions set in the stone wall, not needing the light from their bracers to see what she was doing. Familiarity alone would have taught her what to do, though she also had infrared and low-light sensors built into her eyes.
"...In the meantime," Ia continued lightly, "we still have to find what we need in this place. Even knowing where we need to look, it will not be easy."
The bottom door opened much more quietly than the one at the top had. Margaret slipped through the opening and moved to one side. She did something that _clunked_ , and lights blossomed in rapid succession. The light that spilled out from the depths beyond the doorway made all of them blink and squint. Ia flicked off the flashlight attached to her arm unit and stepped into the cavern beyond.
Or rather, onto one of the highest levels in the cavern. In nearly every direction, fluted columns carved out of the basalt of the mountain marched in orderly, hexagonal rows. Most of those rows were filled with towering piles of stone tiles stacked anywhere from a meter and a half to two meters high.
Every so often, instead of a vaulted ceiling, a hexagon was filled with solid stone, forming a very thick support pillar. Three of the surrounding hexes formed platform-like bridges to the rest of that floor, while the other three lay open, giving a dizzying view downward. It was onto one of these balcony-bridges, lined with ornately carved balustrades, that the five of them emerged.
"Good mucking God, it just goes on and on...What _is_ this place?" Helstead asked under her breath, green eyes wide as she peered over the balcony railing at the floors—dozens of floors—stretching below them. Not every floor was lit, just the nearest twenty hexes in radius and nearest ten or so floors in depth, but the impression of many, many more sections and floors stretching beyond the reach of the light was still there.
"The Vault of _Time_." Ia spoke that last word loudly. It echoed off the vaulted ceilings, bounced off the tiled stones, and scuttled off into the farthest reaches they could see, until not even a whisper was left. She smiled, amused. "...I love that effect. This is the only place I can do that in reality and make it sound even remotely close to what it's like on the timeplains. And _not_ risk rupturing my self-control in doing so."
"I thought you'd never been down here before," Harper said, eyeing her in suspicion. He didn't speak as softly as Helstead had, but neither did he speak boldly. The cavernous, cathedral-like nature of their surroundings seemed to discourage it. "Or have you?"
"Not in this reality, no. But in one of my alternate lifetimes—one where the galaxy wasn't going to be destroyed—I volunteered to update the cataloging," Ia replied. "Naturally, I visited that alternate self to see where the information I wanted would be stored. And then double-checked the information's location for _this_ universe. The copies we want are in the Engineering Archives. Once we get down there, I'll know exactly where to look. But first..."
An odd humming sound distracted them. All three turned to see a hovering sled gliding around the corner from the door they had used, one with a front seat and a bench-like platform perpendicular to it at the back. Padre sat at the controls. Margaret held out her hand. Ia accepted the help onto the sled, not wanting any delays.
The temperature down here was temperate, not bitterly cold as it was outside. One of their reasons for hurrying was the fact that she knew the three of them—the Humans—would start to overheat after a while. Pressure-suits were fine for avoiding the worst of the cold and heat of outer space, but they did trap the body's own heat a little too efficiently.
As soon as they were on board and settled on the bench seats, the male android manipulated the controls, lifting the sled up and over the railing. Gliding it forward and down, he dropped them in a controlled descent down through the hexagonal openings. "It'll take a few minutes, sir, but not many."
"Thank you, Padre," Ia said.
"This isn't any tech I'm familiar with," Harper murmured, peering over their driver's shoulder. He studied the controls, what few there were. "It doesn't sound like thruster tech, and the base is way too thin to be hiding a hydrogenerator."
"That's because it's not modern tech. It's Atannan. It's about, what, eleven thousand years old?" Ia asked Margaret.
The female looked over her shoulder at them, scooping back some of her air-tossed curls with one hand so she could speak freely. "The repair archives for this model are stored in the sector that's not quite twelve thousand years back, sir."
"I'm surprised nobody's noticed this place," Helstead said. They were already descending into the darkened layers. Their pilot flicked on a set of lights which shone ahead, behind, and to either side, illuminating a portion of their descent. "The energy output alone for keeping the air fresh, the temperature comfortable, all the lighting fixtures involved... _somebody_ should've noticed it before now."
"They're looking in the wrong frequencies. Most of the power used by the Vault is geothermally generated, with most of it used up before it reaches high enough to radiate to the surface. The mountains are also rather thick through here, which helps hide the few traces of heat waste that slip through," Ia told her, as the sled shifted forward onto one of the levels. "It's a variation of the Sterling engine, which bases its power on heat differentials—basically, the heat of the planet's molten interior versus its icy-cold surface. The large pillars house most of the pipes for the fluid transference."
Helstead shook her head. "I've counted dozens of those huge columns spaced out every so often, and those are just the ones I can see on our left, never mind our right. The sheer amount of electricity generated by that kind of engine is far too big to keep hidden—geothermal might blend in, but not the electrical fields involved," her 3rd Platoon officer argued. "Earth is constantly being scanned for energy anomalies, in case someone figures out a way to slip an attack past our borders. _How_ could they have missed it?"
"I told you, they can't find it because it's not electrically based," Ia repeated. "The lighting, the heat pumps, the hovercraft, none of it radiates in the spectrums the modern era knows about. The power generated is converted into something the Immortal calls exo-EM, because it operates outside the electromagnetic spectrum—if you were capable of feeding on energy as well as food, you'd quickly develop the ability to differentiate between energy sources," she pointed out, meeting Helstead's skeptical look. "Just like you can tell by your sense of smell the difference between an apple and an onion, which are similar in texture...but if you block off your sense of smell, you cannot always tell simply by taste."
"Not to mention, if you had a couple thousand years to muck around with experiments, you'd probably figure out a thing or two to do with all that exo-EM energy, too," Harper observed, joining the conversation.
"If you say so, sir," Helstead muttered dubiously.
"We're reaching the border of the Engineering Section, sir," their stocky driver stated. "Do you know exactly which section and floor you need?"
"Ah...give me a moment...Floor 17, Section 4...and Floor 22, Section 361," Ia recalled, glancing up at the dark ceiling overhead. The sled slowed in its race over the head-high stacks of stone tablets, turned, and darted off again.
"We're not supposed to read anything in Engineering Sections 1–12, sir," Margaret told Ia, her look and her tone both hesitant. "We're only supposed to check the structural integrity of the sector, repaint the columns and ribs with fresh lettering where necessary, and twice a year, dust the stacks. We're not allowed to remove the capstones to look at any of the tablets...and I'm not sure if you're allowed to do so, either."
"What she doesn't know about won't be a problem, now will it?" Ia replied. "I give you my word, all the tablets will remain exactly as they are, in the correct order, intact, and whole. I promise you we won't do anything that will damage or disorder her records."
The hoversled drifted to a stop near a thick column. Padre peered at them over his shoulder. "...This is the closest we can comfortably go, sir. The other side of that row is forbidden."
"Your orders are to turn on the local light grid, stay here with the sled, and wait for us to come back," Ia told him. "That way you don't run into a conflict with your oaths of loyalty to Shey. Whatever happens down here will be on _my_ head, should she ever find out. By staying here, and not trying to scan, follow, or spy on us, you will be able to plead ignorance of our actions, in the highly unlikely chance that she finds out about this trip."
Climbing out of the sled, she beckoned for her second- and third-in-command to follow her. Margaret climbed out as well, moving smoothly ahead of them to reach a control panel lined with odd crystals. Grasping one of them, she pulled it down, slotting it between two others with a familiar _clunk_. Immediately, the trio started to glow. As did the crystalline globes overhead, lighting up as if the shafts were nothing more than an odd-looking set of archaic circuit breakers.
As soon as the last of the strange lights finished igniting, it became apparent that Sections 1–12 did not start or end at a wall...because in every direction they could see, the stone tiles stacked beneath the hexagonal-vaulted ceilings looked almost exactly like every other stack they had passed. The only discernible difference was that the stones used for these tiles looked like they were granite instead of basalt.
_"Shakk,"_ Helstead muttered, eyes wide once more as she turned around, surveying every direction. "You weren't kidding when you said this wouldn't be easy..."
Harper stared, blinked, then chuckled. It was a wry sound, accompanied by a slow shake of his head. "Now I _know_ you're insane, Ia. Not just in this timeline, but in other ones, too. Not if you _volunteered_ to catalog this place in an alternate life."
"You'll notice I'm not volunteering to do it in this one," Ia retorted dryly. "Pick up the pace, meioas; we don't have a lot of time down here."
Adjusting the straps of her backpack, she set out at a brisk walk, wending her way through the head-high stacks of tiles. It took a couple minutes to get out of sight of the sled and its two occupants. Once she was sure they were out of sight, she flicked her hands at her companions, and picked up into a light-footed run, letting the tough but flexible soles of her pressure-suit boots absorb most of the sounds she made.
Helstead and Harper followed belatedly, doing their best to run silently in her wake. A solid minute of running proved all three of them had kept in shape, for not even Harper, the lightest-gravitied of the three heavyworlders, was breathing hard when Ia slid to a stop by one of the thicker columns. Or rather, by one of the balcony openings leading up and down.
"Light up your arm units, and link hands with me," she ordered quietly, looking up. Helstead grasped her left hand and Harper her right. "Don't worry; I won't drop you."
A nudge of her mind lifted all three of them up over the balcony, and up by several levels. Harper gasped, and Helstead giggled, squirming a little in Ia's mental grip. It was the first blatant use of Ia's telekinetic abilities since her brief demonstration in front of the various military psi branches of the Alliance's Blockade efforts over three months ago.
This time, she wasn't greatly weakened by an infection in her blood; this time, she had the stamina to counteract gravity for more than a few seconds. She was still weak, but she could do this. Whispering under her breath, Ia counted floors starting from the seventeenth. She reached _three, two, one_...and started counting alphabetically, _ay, bee, cee_ , until she reached the highest floor, _eff_. It wouldn't do to forget which floor they were supposed to be on when they returned, after all.
There was only one corridor off this hexagonal section, a single balcony instead of an open tangle of vaulted archways. Landing them next to a control grid, she released their hands and stiffened her muscles, trying to hide the way her limbs threatened to tremble. Weeks of rest in transit with gentle exercise had restored some of her energy reserves, her mental and physical strength, but levitating a few chairs and cups for a few seconds was not the same thing as floating three muscular heavyworlders a hundred meters upward.
She said nothing about her moment of exhaustion as Harper played his bracer-light over their surroundings; she was just grateful the strange crystalline structures distracted both him and Helstead. By the time he aimed his arm light her way, she had caught her breath again. Nodding at his unspoken question, she moved over to the nearest wall and grasped the large crystal shaft Harper's beam of light had found. A soft _clunk_ lit up the floor they were on when she pulled down on the lever, though unlike earlier, the other levels below this one remained dark.
Pulling a cloth from the side pocket on her backpack, Ia wiped down the shaft, then pressed her hand to the flat, translucent gold surface positioned next to it. Energy flowed into her, preconverted from geothermal energy to kinetic inergy by the strange technology maintaining this place. Refreshed, she pulled her hand away and scrubbed at the panel, removing the traces of her touch with quick strokes. Not that there was much chance of their being found, or their visit being uncovered, but she wanted to be thorough to set an example for her restless second officer.
Beckoning her companions to follow, Ia moved up the corridor at a pace somewhere between a lope and a run. Twenty meters down the hall, the passage cut into the mountain opened up into a largish chamber, one built with a high-vaulted dome. At the center point was a largish, throne-like contraption. Surrounding it on all six sides were half a dozen odd, huge, crystal-muzzled guns, each one braced on a pedestal mount.
Ia stopped near the throne-thing and slung the backpack off her shoulders, lowering it to the polished stone floor. Channels had been cut into the floor and filled with the same sort of transparent crystals fitted into the gun-things; they looked vaguely like the crysium from her homeworld, but were almost colorless instead of pastel. The channels disappeared into the walls with no reason or explanation visible, though she knew from her postcognitive peeks what their function was.
"Ia? What is this place, exactly?" The question came from Meyun. His gaze flicked back and forth between the contraptions and Ia. "It's nothing like the rest of what we've seen, unless you count the light fixtures. In fact, it looks like something a...a fantasy sculptor might make—is this more of that Atannan tech you were talking about?"
She crouched and opened up the backpack. "Yes, and no. You're looking at the single most dangerous piece of experimental equipment ever created. If wielded incorrectly, it is capable of torturing any form of sentient life trapped within its grasp. Up to and including a Feyori." Digging out a trio of oval hovercams, she clicked each one on, fished out their remotes, and sent them soaring up and around the room, scanning everything in their path. "In the wrong hands, it can _kill_ a Feyori, physical or soap bubble. And in the right hands...it can turn a half-breed _into_ a Feyori."
" _Into_ a Feyori?" he repeated, crouching at her side.
"Yes. I need you to re-create a handheld version of it, duplicate it five or ten times, depending on how strong you can make it, and shoot me with it. You have eight months to get it right, and you are _not_ to tell anyone what you're working on. Lie to your subordinates, lie to your superiors—though you can tell everyone you're working on an experimental type of gun—and lie to everyone except me, of course, but figure it out and get it done.
"One more thing, Meyun." She gave him a sober look. "Whatever you do, do _not_ allow anyone to copy your notes; nor are you allowed to back it up to any shipboard workstation. Keep it entirely on nonsynchronized datapads so that you can destroy the information completely, pads and all, when you are done." Glancing to the right, she sighed. "...Delia, don't touch that. You know better than to leave fingerprints up here."
"Fingerprints, hell," the shorter woman shot over her shoulder, though she did back up from one of the oversized ray-gun things. "We've already left plenty of DNA evidence in shed hair and skin cells. I don't see what the big deal is at this point."
"Hair and skin are evidence which _could_ have wafted up from below...and most of which will be removed by at least three rounds of cleaning crews between now and the Immortal's next scheduled visit to this exact place," she argued back. "The Immortal would be upset to know we'd copied the information down below—and we will be copying some of it—but up here is another matter.
"If I hadn't screwed up a few months back, we wouldn't even have to be up here, but we are. And we're here _only_ to get enough visuals so that Meyun will have a better chance of figuring out the Immortal's construction notes," Ia told her. "That means _we_ touch nothing beyond planting our p-suited feet on the floor, and that _I_ alone touch anything else up here. Including the power switch for the lights. You can look, but you only touch the floor with your boots, Lieutenant Commander."
Sighing, Helstead pulled a pair of her miniature stilettos from her hair and moved around the gun. She stared at its backside, then turned and looked into one of the large niches forming the sides of the space, twirling the long, thin, sheath-wrapped blades. "Joy. Yet _more_ stacks of stone tablets. Who does this Immortal think she is anyway, Moses? Or maybe a burning bush? If I were to pick up one of these, would the first few words read 'Thou Shalt Not' something-or-other?"
Harper smothered a laugh. Unsuccessfully, since it escaped as a snort.
Ia smiled but shook her head. "I'm afraid we're in the wrong section for anything even remotely like that. As soon as the hovercams have scanned everything, I'll run all the tiles past their sensors. Telekinetically, since stray hairs and skin cells could be wafted up here on random currents over the decades, but fingerprints are proof positive of an actual visit to this part of the complex."
"Just how big is this place?" Delia asked her, walking back to where the other two still crouched. "All of it, I mean, not just this alcove."
"I'm not completely sure, but I think by this stage, it covers around a hundred kilometers of subterranean passageways in length, and a good twenty or more in width," Ia estimated. Both of her companions choked. She shook her head. "The Immortal started excavating these caverns thirteen thousand years ago. What you've seen so far is only the beginning; it's not just all those tablets filled with carved writings."
"Why stone?" Harper asked her. "The thickness of these slabs is about as far from data-storage efficient as you can get. In the thickness of _one_ of those tablets, you could stuff an entire paper-style book with information."
"She chose stone because it is the cheapest permanent recording medium at hand. Paper and plexi disintegrate, magnetism can revert or be subverted, quantum entanglement requires ongoing uninterrupted power...Her notes are designed to be read by Human eyes, without needing any sort of mechanical assistance. The Immortal started these records literally back in the Stone Age, in a very primitive era. Stone was all she had access to, originally."
Delia grunted, strolling away again. Ia lifted her voice slightly so she could be heard, though she didn't shout.
"It's true that stone takes a lot of storage space, but it won't disintegrate, the carvings won't fade, and the only thing you have to worry about is the tablets breaking," Ia continued, her attention more on the task of guiding the cameras for all the best angles than on their conversation. "Plus, as I said, the tablets aren't the only things down here. There are uncounted kilometers of artifacts as well, all carefully preserved in cases filled with argon gas. I haven't bothered to count just how many floors and sectors. It's not important, here and now. Getting this stuff recorded is."
"Okay, I got a question," Helstead offered, lifting her chin at the other two. "Why argon?"
Harper answered her, watching the cameras as they made their slow recording sweeps. "Argon gas is naturally inert, and thus the best possible preservative if you want to keep something in its natural atmospheric pressure and normal temperature range."
"This machinery would be preserved in a gas-filled cabinet, too," Ia added, nodding at the chair and its odd cannons, "except the Immortal actually uses it from time to time. Which means she has to repair it occasionally, which means she'd notice fingerprints."
The cameras came swooping back, their task complete. So did Helstead, though she strolled instead of swooped, too restless to stay in one spot for long. "So what does she use it for?"
Ia wrinkled her nose and lifted a finger to her temple, circling it slightly. "Her memory's not quite stable. When she wants to remember everything in detail—when she wants her memories organized, rather than cluttered up and overlaid by a hundred thousand similar days of a hundred thousand similar routines, eating drinking, et cetera—she zaps herself with this contraption.
"I'd put myself in the chair and fire it up right now, since I need to make that transition within the next year or so...but the spike in the power chart from the energy required would stand out at a sixty-two percent chance whenever she'd check those maintenance charts, and I _really_ do not want the Immortal knowing I was here, if I can help it," Ia finished.
Rising, she fiddled with the remotes for each of the cameras. They soared off toward one of the tablet-stacked alcoves. Picking up the empty bag, Ia followed in their wake.
"You could've come down here on your own," Harper told her. "Why bring us?"
"For Helstead, it's a show of trust. She needs to know I can trust her with dangerous secrets, and I need to know she can keep her mouth shut when it comes to the biggest secret on Earth. Or under it," Ia amended, glancing at the shorter woman. She received a sardonic smile in return. Looking back over her shoulder at her first officer, Ia added, "For you, I wanted you to _see_ the machines. The tablets have schematics drawn on them, and you'll have enough video to re-create them in three dimensions on your datapads, but the Immortal's notation system is a bit more eclectic than the standards used by the Space Force Engineering Corps."
Lifting her hand, she started peeling tablets off the top of the nearest stack. The hovercams swooped and hummed, moving up into position. Each tablet soared past all three cameras, moving just slow enough to let their neatly carved surfaces be thoroughly scanned. At the end of the stack, all the tablets flipped over and soared back the other way, displaying their other sides, with the last in the row being scanned first.
"I've already programmed the cameras with a numbering algorithm," she told Harper in a quiet aside. As the tablets returned to their spot, they flipped over and clicked very softly as they landed in place. The noise they made was no louder than fingernails lightly drummed on a tabletop, if said nails and table were made out of stone. "They'll be listed by 0001a, 0001b, 0002a, 0002b, so on and so forth, in the order they come out of the stacks. You'll have to do a lot of reading in your spare time in order to make your deadline."
"That's assuming I can decipher whatever writing system she's used," he pointed out wryly. "I only know two languages by heart, Terranglo and Mandarin, with scraps of Gaelic thrown into the mix. If she's really fourteen thousand years old, I'm not exactly fluent in Paleolithic cave drawings," he quipped wryly.
"Meyun, she was born in _our_ future. The near future, roughly two centuries from now. She thinks and speaks in Terranglo," Ia reassured him. "Of course, she knows a couple hundred other languages, but the Terran trade tongue is the one she wrote most of her observations in. At least, here on Earth. She has another Vault on V'Dan, with the tablets there mostly written in High V'Dan. You'll be able to read these tablets; don't worry.
"Mind you, the Immortal organized them chronologically, from her earliest experimental stages all the way through to the Scholar War and beyond, so you'll have to sort out the relevant bits—I'd do it myself, but while I can _see_ this, it doesn't make any sense to me. I'm not an engineer. Some of what you need may be near the beginning, some of it near the end," she cautioned. "I don't know. I'm counting on you to figure it out."
"Well, it's not like I'm going to be _dating_ anyone," he grumbled. "So I'll have plenty of time in my off-duty hours, I guess.
That made her smile wryly. Another gesture of her hand sent the next stack of tiles soaring out and back. "Welcome to my life."
"No, thank you," he muttered.
"Too late," she teased dryly. "You already agreed to it."
"Only for a year," Meyun whispered under his breath.
Helstead eyed both of them. She had stopped several meters away, but it was possible she could have heard his final comment anyway. "Knock it off, you two. No fraternizing among the cadre. Captain's orders."
That made Ia choke on a laugh. "Yes. Yes, they are."
"What about the other thing we're here to find, the OTL-to-FTL conversion or whatever?" Harper asked Ia.
"It's not in a sensitive area," Ia dismissed, already moving on to the third stack. "We won't have to ditch the AIs and fly off. In fact, that session will go faster than this one since there aren't nearly as many tablets to record. You'll find the Immortal's notes on those much easier to read, Meyun, because she wasn't the one experimenting with hyperwarp transit; she just wrote down what she already knew. The problem most engineers of _our_ era have had is that they keep trying to treat the problem of wedding FTL to OTL like the way atmospheric pilots treated subsonic versus supersonic speeds before realizing it was a matter of inversion."
"You're kidding me," Harper muttered, staring at her. "It's _that_ simple?"
"The theory, yes. The implementation, no. When you go supersonic, which is like what hyperwarp does, all the flight controls on an aircraft get flipped around," she warned him. "You can sort of get where you need to go by cramming FTL warp fields into an OTL hyperrift, but a courier-sized vessel isn't big enough to convert the energy needs for both OTL and FTL. And to open a hyperrift big enough to accommodate a ship large enough to carry panels for both, the Solaricans have had to rely upon naturally occurring wormholes as opposed to machine-made ones—which means they can use properly modulated shield energies to open the rift once it's been found, but it also limits the entry and exit points to wherever that natural rift wants to go."
"How does the Immortal know about OTL/FTL conversions?" he asked her, his attention split between the swirling stacks of tablets and his captain's face.
"Her mother will run the first nonmilitary ship fitted with the first official version of the hyperwarp drive...and will at some point accidentally trip over the same natural hyperrift that my homeworld's first wave of colonists tripped over, squirting her into Sanctuarian orbit about two hundred years from now, where she will run across the Immortal's father," Ia related, her attention split between his question and the tiles swooshing out and clacking back into place. "Shey—the Immortal—will be born within Sanctuarian jurisdiction, grow up a spacer's brat, and learn all manner of interesting things from her mother, including the history of the hyperwarp drive's development."
"And then you'll exile her to the distant past, where she'll be stuck living through all that history," he said.
"Tell me something, Captain," Helstead asked, completing another circuit—without touching—of the throne-like chair and its bizarre ray-gun things, "does the older version of the Immortal live past the point where her younger self is born? Or does she vanish, in order to prevent her from contacting herself, accidentally or otherwise?"
"She has a beginning just like all of us, and she has an ending," Ia told the shorter woman. "It's just that a large part of her life has been bent out of the normal flow of time so that it takes place in the past. As for the exact nature of her ending point...let's just say it's complicated and leave it at that."
"Aye, aye, sir," Helstead muttered sardonically, standing in place and rocking from toes to heels and back. "Of _course_ , sir. Whatever you _say_ , sir. In fact, I'd even go so far as to say three bags _full_ , sir!"
Ia bit her lip to keep from smiling at Helstead's quip, not wanting the tablets to waver in her amusement. Or at least, she tried. It was hard to remain sober when the other two snickered outright.
Several hours later, Ia, Harper, and Helstead found themselves bowed into the presence of the Grandmaster of the Afaso Order. Their mottled camouflage Greys, obviously military in cut and style, didn't match the more peaceful-looking green and brown batiks worn by the monks escorting them, but the smile curling the broad lips of the Grandmaster—a smile with closed lips, displaying no teeth—was as warm and welcoming as if they were all close friends.
"Ahhh, Ia, my _mok'kathh_ , what a pleasssurrre it isss to sssee you againnn," he half hissed, holding his arms out as he rose from his Tlassian-style stool and moved around the corner of his desk. Ia quickly shrugged out of her backpack, aware of his intent. Smiling herself, Ia turned her back on him, letting the saurian hug her from behind.
The three Humans had stripped off their pressure-suits and donned normal clothes on the parabolic flight from Antarctica to Madagascar. Ia had also left the hovercams back on board the shuttle parked on the landing pad in the distance; her current bag carried a different burden. Without the pack in place, his embrace was unimpeded. She bowed her body a little, not quite lifting him off the ground, and he squeezed carefully in return, holding on as she tipped him over a little.
Releasing her, Ssarra turned around. Ia turned as well and wrapped her arms around his back, hugging him, too. He didn't lift her this time, like he had back when she graduated from Basic Training. Back then, it had been a gesture of congratulations, a way to lighten the seriousness of the moment in mirth and celebration. This moment was a bit more serious in tone, if still a good one.
The moment she released him, he straightened and politely offered his hand to Harper. The palm was a little shorter, the fingers a little longer than a Human's, and his claws only somewhat blunted from use, but he clasped the other man's hand carefully in the Human style. "Meyunnn Harper. I have heard good thhhings about you."
"Grandmaster Ssarra; I am honored that you've heard of me," Harper replied politely. He glanced at Ia, but her attention was focused on the alien.
"Annnd Sssenior Masster Helstead—you arrre almosst ready fffor the Elder Masstery tesst, yesss? I am pleasssed to hear you have kept up your sstudiess," Ssarra praised, bowing to the stocky, short Human. She bowed back, hands laced together politely. "I would be dellighted to ssee an exsshibition at some point."
"I'd be honored, Grandmaster," Helstead replied, glancing at Ia as well. "If we have the time, that is. Captain?"
"Oh, I think we have time for that. In fact, if you're ready for it," Ia added, "you can take the Elder Mastery test here. You have a thirty-seven percent chance of success, so the odds aren't too bad. Grandmaster, if you would arrange it?"
"Of courssse," he agreed, and touched one of the buttons on his desk console. "I will sssummon a Brother to take you to the ssallle."
Within moments, the door opened, and a Human monk stepped inside, bowing. The Grandmaster hissed and thrummed in Tlassian, and the monk bowed to him, then to Helstead. "If you'll come this way, Senior Master," he stated politely, addressing the petite woman, "I will show you to the performance hall, where we can find a set of batiks for you to wear and give you time to warm up."
Nodding at him, Helstead gave Ia one last glance, then shrugged and followed the monk out of the office. Grandmaster Ssarra scratched his chin, head cocked slightly as he studied Harper. A swift glance with those golden eyes directed an unspoken question at Ia. She knew what he meant by it. One extraneous body down, one extraneous body to go. Did she want him to come up with an excuse to get Harper out of the room as well?
"I'm not going to hide this from him, Ssarra," she said, tipping her head at Harper. Picking up her backpack, she offered it to the Grandmaster. "He's taken a wilder ride through the timestreams than you have, and he survived."
Ssarra lifted the pack from her hand, glancing between it and her. "Ah. Sssso, thesse are the devvvices you mentionned?"
"Yes, the special circlets. I figured out how to make them. There are two of them, the Ring of Truth, and the Ring of Pain. One is the prize, the other is the punishment. You must make sure that each Grandmaster who follows you dons them periodically," Ia cautioned him. "As well as any key figure in my instructions who expresses doubt, whether of the cause or my requests, or of their ability to carry them out. They are not mind-control devices, but they are perspective-opening devices."
"Am I ffforbidden from tryinng one mysself?" Ssarra asked, unzipping the pack for a peek inside.
Ia smiled. "Not at all. Feel free to try them on. Just be sitting down when you so do and make sure you have a bit of free time on your hands. They can be a bit distracting. Overwhelming, for those who've never experienced the timestreams before."
Ssarra curved his broad mouth at the corners. "Asss iff I have nnever done _that_ beffore. I will sstore them in my offissce sssafe, then esscort them personally to the Vault."
Harper choked, and coughed. He wheezed a bit, gaze darting between the two of them. "That isn't...He's not talking about...?"
"Different Vault," Ia dismissed. "The Afaso have generously and compassionately agreed to safeguard most of my prophecies, and have built a highly secured vault in which to store them while they await the proper point in time for delivery. My family back on Sanctuary has also done the same. Of course, their contents will deal primarily with Sanctuary-based matters. The Afaso will handle most of the prophecies that deal with the Terran United Planets and the Alliance—oh, one more thing, Grandmaster."
"Yess?" he inquired politely, zipping the bag shut, the contents untouched.
"Inside one of the pockets is a trio of datachips," Ia said, nodding at the grey bag in his grasp. "They contain the names, idents, dates, and other guiding information for the monks who will need to enlist. Most of the names on those lists will be willing to do so, once they have read the reasons why included in each packet. Those who hesitate should be given an opportunity to use the Rings and test for themselves the necessity of their tasks. Please remind them that I am only one person, Grandmaster. I will need their help at the right place, in the right time. I cannot do all of this myself."
"I will sssee to it," he promised her. "But we have dellayed long enough. Let me put thiss in the ssafe, then we shall go and sssee if the Ssenior Masster becomes an Elder Masster today. Thirty-ssseven percent chance, you sssaid?"
Ia nodded. "If she warms up right, and if she puts her mind to it, there's a chance she will pass. But if not today...well, she'll study."
"But how much ssstudying will she get done, on your misssionss?" Ssarra asked.
She grinned, deliberately showing her teeth. "Well, if you'll bend the rules enough to loan some training vids and simulator files...?"
Ssarra flicked his fingers up and out. "Give the meioa-e a ssscale, and she'll take the whole hide! _If_ sshhe does not pass, you can have the ffiless. If shhhe does..."
"The rest of my crew could also use advance hand-to-hand training," Ia pressed, keeping her tone light, her hands clasped behind her back as she shrugged. She continued to show her teeth in a grin, though.
Ssarra hissed and lifted his long, scaled chin in surrender. "...Alright, you can have the fffiless! Do not llet them out of your conntrolll."
Chuckling, Ia sealed her lips in a smile and gave him a little bow. " _Sschah nakh_ , Ssarra. They will only be transferred when we swap ships; you have my Prophetic Stamp on that."
" _Ssthienn nakh_ , Ia," he responded to her thanks. "Your accennt isss gettinng slllightly better," he added in praise. The saurian stepped behind his desk and crouched, doing something beneath the broad surface.
Folding his arms lightly across his chest, Harper cocked his head, studying his commanding officer. While the alien was busy, he addressed her under his breath. "Okay, Ia. _Now_ I'm impressed...and confused. Not only do the Afaso _not_ share their training simulators with anyone else, a Tlassian would never cave in to a bit of teeth-baring by a hairless monkey unless he feared that monkey. Yet I cannot imagine the Grandmaster of the entire Afaso Order fearing anyone or anything, including you."
Ssarra shook his yellow-and-brown scaled head, the motion more of a figure eight than a side-to-side gesture. Harper had spoken quietly, but not enough. "I do thisss for the ssame reasonss sshe doess: Lllove. That, and I knnnow she will not sshare our sssecretss with the wrong sssentients. The teethh, shhhe teasesss me, nnothing more." Shutting the door of the safe with an audible _click_ , he rose and bowed slightly at the Human male. "Now, ssshhalll we head for the sssallle?"
Bowing in return, Harper gestured for him to show the way. Ssarra in turned gestured at Ia to take the lead. Sighing, she complied. It wasn't as if this was her first visit to the Order's headquarters, after all.
# CHAPTER 4
_I had a lot going on, back then. In many ways, I still do; that's a given. But what most people don't realize is just how many deals I made behind the scenes. There was the deal with the Afaso to store my prophecies and the deal to enlist them in various military services, so they'd be at the right place at the right time with the right skills and the right foreknowledge to save lives. There was the deal with the Command Staff to ensure I had free rein on my ship and crew assignments, the various deals I've made with the Feyori, deals I've made with my fellow soldiers..._
_Yes, I set plans in motion with other governments as well. I may have been confined to working mostly within just the Terran aspects directly, but my plan has always been to involve the other sentient races because my plan has always been to save their lives as well as the lives of my fellow Terran Humans._
_Which brings us to the deals that still remain classified, even from those who perhaps should've been told. Arrangements which were, are, and will be absolutely necessary for the future. I brokered those deals in order to ensure that the current war will actually have an end. At least, one we can all live with. Not much of an excuse, I know, but at least it's been for a good reason._
_~Ia_
DECEMBER 25, 2495 T.S.
TUPSF _HELLFIRE_
SIC TRANSIT
Her office door chimed. Looking through the transparent workstation screen, Ia narrowed her eyes at the door. A quick dip into the timestreams ended in a heavy sigh. Pushing the comm button on her desk, she said, _"Come in."_
The door slid open, admitting the tallish figure of Private Second Class Gregory York of C Squad, 2nd Platoon, one of her bridge communications technicians. He wasn't in his normal grey uniform, but rather in a bright green shirt and dark green slacks, with a dark red belt and dark red shoes. Civilian clothes.
"Uh, Captain, sir?" he asked, giving her a nervous look. "You said that, ah, anything goes in a Wake Zone. That it's completely civilian territory?"
"Yes, I did, Private York," Ia agreed. Only a handful or so of her crew were actually comfortable in her presence yet, and he wasn't one of them. Yet. She kept her tone light, her expression on the pleasant side of neutral. "You have a concern?"
"Yes, sir, I do," he said. Drawing in a deep breath, York let it out as he explained. "I know it seems kind of petty, sir, but...the Army's making fun of the Marines for, uh, singing. I mean, the crew members who _used_ to be Army, are making fun..."
"I know what you mean," Ia reassured him. She held up her hand, silencing his next comment, and searched the timestreams again. This hadn't been one of the larger-percentage chances, otherwise she would have addressed it beforehand, but at least it was something that was easily salvageable. Checking her desk to make sure everything was clipped in place, she retracted the workscreen and rose. "Lead the way, York."
"Yes, sir," he sang, turning toward the door. He palmed open the panel to the front office, where one of the on-duty privates, Mara Sunrise, frowned over some form she was trying to fill out. "I don't want to get them into trouble, exactly, but...I like singing. Good singing. And we have a few ex-Marines who can sing."
"A few more than you know," she murmured back, nodding politely to Sunrise. The other woman lifted her chin in return, though she didn't shift her eyes from her workstation screens. "And a few who can't. Same mix as in any large group of people."
"Captain," Private Sunrise stated, catching Ia's attention, "Sergeant Grizzle will have the summary of the troops' tactical analysis from their mock-drills on Earth by nineteen hundred, sir."
"Wait a minute," York muttered, frowning at the clerk. "You're in my Platoon. Shouldn't you be off duty?"
Sunrise gave him a prim look. "War doesn't take a holiday, and neither do I."
Ia bit her tongue, keeping her expression neutral. Her superiors might have complained through the years that Ia's debriefing reports were dry and factual, but the woman currently known as Mara Sunrise had perfected bland and boring to a high degree. There was a reason why Ia had insisted she join the Company, but it would take a few years to play out. Hiding in Ia's Damned was the best place for the other woman, at least while she waited for a certain provincial governor's embarrassment and wrath to die down.
It didn't take long for Ia and York to reach the amidships galley; it was located in the same sector as the bridge and her office, but below the main gun. Both the galley and the rec hall were part of the same "Wake Zone" designated for their first onboard Leave, and that meant no one was supposed to be in uniform if they weren't officially on duty. A lot of her off-duty crew had chosen civilian clothes in shades of red and green or blue and white to celebrate the holiday, too, not just York. By contrast, her grey clothes looked out of place.
It was also time for an actual meal rather than the sugary snacks being served upstairs in the Wake hall, so the on-duty crew were serving the off-duty "civilians" lounging at the tables. Most were lounging and trying to enjoy the feast that had been prepared, except for a knot of about fifteen or sixteen colorfully clad bodies squared off against each other in two groups.
"—Your _mother_ is a tone-deaf harpy!" One of the privates, Tony Doersch, was haranguing the other camp. Ia knew he was ex-Marine, and knew he was fiercely loyal to the Corps. "If you don't like our singing, get the hell outta our kitchen!"
The man he was insulting, tall and dark-skinned, flexed a very impressive set of muscles for someone raised on Earth. C. J. Siano was ex-Army, and proud of his family. The yarn of his Christmas-tree-covered sweater creaked faintly as he stiffened. "Don't you _dare_ talk 'bout my mother that way, you—!"
_"Gentlebeings,"_ Ia asserted firmly, her tone just sharp enough to cut off his retort. "Captain. On. Deck. Instead of in my office, where I should be at this hour."
They all stiffened. A couple of them even scrambled upright, even though this galley at this point in time was supposed to be part of the Wake Zone, where military procedure only counted for those who were actually on duty. The fifteen or so men and women facing her were all in normal garments, not uniforms, but they faced her in her grey buttoned shirt and matching, black-striped slacks out of the habit of deference to an authority figure. Ia took advantage of that.
"I don't care who started this, or how, or why," she stated in the quiet that followed her initial words. Even the crew in the kitchen half of the cabin were doing their best to work without noise. Ia swept the two clusters of men and women with as mild a look as she could manage. "I only care who ends it, and how, and why.
"Siano," she stated, addressing the tallest man in the two groups. "You have served long and well in the TUPSF-Army. You have a lot of good marks on your Service file. You are, however, unfamiliar with the culture of the Marine Corps. In the Space Force, it is a tradition that the Marines sing. Whether or _not_ they can carry a tune. Isn't that right, Barstow?" Ia asked, glancing at the woman in the short blue skirt, matching vest, and white shirt.
Private L'ili Barstow blushed. Both of them knew—Ia via postcognition and L'ili firsthand—that it was her off-key crooning that had triggered this heated debate. Still, the other woman took it on the chin squarely, pulling her shoulders back and leveling her gaze across the room, hands behind her snowflake-patterned dress in Parade Rest. "Sir, yes, sir. As my instructors in Basic put it, we Marines will sing even if we're tone-deaf and tasteless, sir."
Ia lifted her hand, gesturing at the woman. "See? A simple cultural difference. Do try to respect it in the future, gentlemeioas. Of course, I am still a Marine, deep down inside, so I'll admit I'm slightly biased in this matter, but you will still give your fellow crew members respect for their hobbies and beliefs.
"Religious beliefs," Ia stated, looking up pointedly at the decorations tied firmly to the ceiling struts before dropping her gaze to the others, "... _or_ secular. In short, gentlemeioas, either sing along, or shut it. Now, get back to your partying, get along, and have a Merry Christmas. That's an order."
Siano mumbled an apology, as did Doersch. Some of the others did, too.
Barstow cleared her throat. "Ah, Captain? Is it true that there are some songs out there about you? In the Marines?"
"Yes. There are now quite a few songs about me," Ia admitted blandly. "And yes, I can and do sing them. At some point, when I've caught up on the klicks of paperwork still lined up on my desk, I might even sing 'em for you, whether or not I myself am tone-deaf and tasteless about it," she joked lightly, giving the other woman and her companions a slight smile. "But I won't be able to join you for several weeks yet. Enjoy your onboard Leave while you can. Private York, come walk with me."
"Sir, yes, sir," he agreed, following her as she headed back out of the galley. He walked with her back to the nearest lift. "Ah...what did you want, sir?"
"You have some musical training, don't you?" she asked him while she waited for the car to reach their deck.
"Sir, yes, sir," York agreed, squaring his shoulders. "I have a Master's degree in music, both voice and composition, bought on the Education Bill. I was paying it off by working for Intel, before the transfer to your crew, sir." He paused, then asked, "Do you want me to offer lessons, sir?"
"Smart meioa. To _anyone_ who wants to learn, regardless of which Branch they were in before," she added. "Particularly to anyone who wants to sing and clearly needs it—make it sound like it'll be fun, and coax them to at least try. There will eventually be a whole series of off-duty classes this crew can take from each other. I want you to spearhead the opening offers, and encourage the others to come forward. Whether it's singing, board games, sewing, or basket-weaving, I want this crew to work together to better ourselves when we're off duty as well as when we're on. Think you can handle that?"
"I'm Special Forces, sir; we take in all kinds," he quipped. "I'll ask Barstow if she wants a few private pointers so she can show up Siano next time in public, then ask her what she can teach me and some others in return."
"Good meioa," Ia praised, as the lift arrived. "Merry Christmas, York. Or whatever your preferred holiday is."
"You, too, sir," he returned.
Nodding to him, she headed back to her office. Standoffs like the one between Siano and Doersch weren't going to happen often, but they were going to happen while her disparate crew was still pulling itself together. Internecine fights had to be quelled as fast as possible. Actions and offers like York's, those had to be encouraged, even rewarded. She needed everyone working together, trusting each other, and needed it all too soon.
Trust in each other would do as much or more to save her Company as her own efforts could.
JANUARY 16, 2496 T.S.
SIC TRANSIT
The quiet thrum of a starship in motion was broken by the beeping of her arm unit. Ia finished her current prophecy by imprinting the words into the workstation electrokinetically. Physically, she reached for the comm button. _"Ia here. Go."_
_"Congratulations, Captain,"_ Harper's voice stated. He sounded tired, yet satisfied. _"It's a brand-new baby stardrive. Nose cone A has been rewired to your specifics, and the warp-panel control programs recoded to compensate. All five simulation tests came up positive and fully functional...though I'm sure your gifts will have the final say. Unless you're going to leave me in suspense and not let me know?"_
She smiled and closed her eyes. _"Patience, Commander. Let me double-check..."_ Dipping into the timestreams, Ia checked the performance of the newly combined hyperwarp drive system. _"Congratulations, Commander, it is indeed a healthy baby stardrive. Feel free to rotate out the extra personnel to the Wake Zone on Aft Deck 8. Keep a full crew in engineering, though. I'm going to take 'er out for a test drive."_
_"Already?"_ he asked. _"Ah—don't forget, it'll take half anhour for everything to reboot with the new codes. We need to be at a dead stop or a low drift for that. But once that's done, may I join you on the bridge for a front-seat view?"_
_"Understood, and be my guest, but no backseat driving. I'll go order the shutdown now. Ia out."_
_"Harper out."_
She ended the call with another touch, then sat back, sighing. It didn't matter exactly _when_ she did this next part, only that it was done within the next two days. Now was as good a time as any, however. Scrubbing her face, Ia braced herself for what she was about to do.
_I swore an oath, when I joined the military, to be loyal to the Terran United Planets. I agreed to abide by the laws of the Terran United Planets...and in essence, to broker no deals with foreign powers, to share no military secrets, to not betray my chosen people._ She had memorized that Oath of Service long ago, and the vows she had made still haunted her. _If the Admiral-General knew just how far I'd stretch that_ carte blanche _I bartered out of her, she'd have shot me herself on the spot when I barged into that sealed meeting. Maybe even nuked me from orbit..._
The morbid thought cheered her a little, but then her sense of humor had always been a little skewed. Or almost always; it had changed at the age of fifteen. Standing, Ia secured her desk, pulled a spare datapad from one of the drawers, and headed for the side door. Not the one that led to the front office where Master Sergeant Sadneczek and his small staff of clerks managed the paperwork needs of their Company, but the side door that led to the service corridor.
That corridor led to the backup generators and emergency lifesupport reoxygenators, to a pair of heads—the charming Navy term for bathroom facilities—plus the pocket-sized bridge galley where meals could be quickly fixed and served, and the bridge itself, bypassing the need to go out and around to either the starboard or portside main passages that ran the length of Deck 6.
Like so much else on this ship, the bridge had also been modified, reduced to a fraction of its original size because she had only a fraction of the original crew meant to man the various posts. The extra room to the fore had been converted into a wiring shop, just one of the many necessary manufactory bays needed on a ship that had to be self-sufficient as much as possible for its repairs.
Captain's privilege gave her the shortest commute on the nine hundred meters and twenty-four decks of her ship, since her personal quarters were tucked right behind her office. Captain's rank made the first bridge crew member to notice her, Corporal Fyrn Michaels, call out the warning, "Officer on deck!"
Everyone sat up straighter in their seats. No one took their eyes off their workstations, however. That was one of the big rules in the Company Bible; stations were to be manned and monitored at all times, even if a fellow crew member had to monitor the extra workload while someone visited the head. Only when Ia approached the pilot's station and tapped Yeoman Fielle on the shoulder did he look up from the task of guiding their ship through the star-streaked speeds of faster-than-light.
"Sir?" he asked. The incident with his pet minirobots was weeks in the past, but she didn't have to be a telepath to see a hint of it lurking in his concerned gaze. "Is something wrong?"
"Bring the ship out of FTL and power down the engines, Yeoman," she ordered.
"Aye, aye, sir," he said. Sliding his fingers down over the controls, he cut their speed. Everything seemed to pull forward a little from the change in their momentum, but wrapped in an envelope of energies that warped the laws of physics, not much of their abrupt vector change could be felt. "Five minutes to dead stop, sir."
"Captain," Lieutenant Rico asked, his voice deep and his tone mild. It was his duty watch, and his responsibility to ensure the ship continued on course. "Any particular reason why we're stopping, sir?"
"Consider it sightseeing," she quipped, looking around the bridge, with its banks of transparent and solid monitors, its workstations and handful of crew manning the important, noncombat ones.
There was a black-box recorder on board the ship that she knew would be recording everything they said and did, unless she altered it electrokinetically. Which she did, but it was the Human element that had to be watched. Black-box recordings wouldn't be cracked open unless something went wrong...or was reported as going wrong.
Rico wasn't dumb. "Pull the other one, sir."
"Engineering just finished our drive-systems respec on an Ultra-Classified level. FTL needs to shut down completely so that the drive comps can be rebooted with the new programs. The operation will take approximately half an hour. Eyes to the boards, thoughts on your tasks," she reminded the half dozen men and women on the bridge with her. "We may be a trillion kilometers from anything and everything out here, but I will not have this crew caught off guard."
"Aye, aye, sir."
The main screens for the pilot's station and the forward wall flashed for a moment, dazzling the room with thousands of circles of light. The blurred streak of stars turned into sliding pinpoints of light. Slowly, their motion eased, then stopped.
"Captain, we are now at a full stop, drive-wise. Relative to galactic motion is still .0002 Cee, sir," Fielle added, meaning relative to the average movement of stars in their sector, versus their position as they swirled around the galactic core. "If you want a true dead stop, sir, we'll have to use correctional thrusters."
"That won't be necessary, Yeoman," she said. "Private Barstow, inform Commander Harper down in engineering that he may now upload the new warp-field drivers."
"I'm on it, sir." L'ili Barstow bent her attention to that task, speaking quietly into her headset pickups. There was a slightly more melodic quality to her voice than a few weeks before; it seemed her voice lessons with York were beginning to pay off.
Private Hulio, seated at the navigation station, glanced briefly over his shoulder. "Captain, I've been double-checking our heading. We _are_ going to Sanctuary, correct, sir?"
"That is correct," Ia confirmed. This was why she had ended the audio portion of the black-box recordings.
"Well, sir...I know you set the course in the navcomp, and...uh...I'm _sure_ you know how to find your own homeworld, being a pilot," Hulio added, his tone apologetic for what he was about to suggest. "But...we're off course by seven degrees, sir. That's taken us within a couple light-years of Grey territory, which makes me a little nervous, sir. It's also adding sixty-two light-years to our trip, even if we correct course now. That, ah, doesn't seem the most efficient flight path, Captain."
Sixty-two light-years off course was far more than merely inefficient, and everyone on the bridge knew it. Ia watched the rest of the bridge crew glancing at each other. They peeked at her over their shoulders before returning their eyes to their stations.
She answered Hulio's question with part of the truth. "That's because we're testing the new drive system out here, in the middle of nowhere. I am not about to plow this ship into my home planet, the local traffic, or anything else in that general area."
Silence followed her words, silence and a few puzzled glances. Ia turned her attention to the workpad in her hands, writing prophecies electrokinetically. Every spare minute had to be filled with writing prophecies for her family since there would only be two more chances to visit them in person in the coming years. Once the two wars got going in this corner of the galaxy, it would be too dangerous to try for more visits than that.
After several minutes, Lieutenant Rico spoke up, his tone calm, almost phlegmatic. "Begging pardon, Captain, but aren't you a massive precognitive? Wouldn't you _know_ what to avoid, when and where?"
"That's why we're out here, Lieutenant," she replied in kind, equally as calm. "I got shot in the shoulder once, on a three percent chance I'd discounted as being too low to worry about. I'm not taking chances with the new drive, just in case some of those low-probability bugs haven't been worked out enough. Nothing out here in our test zone will harm us. Now relax and enjoy the pretty stars while the upgrades get loaded."
Silence filled the bridge, broken only by the occasional murmur from the operations post, monitoring the ship's functions and coordinating them with engineering, lifesupport, and so forth. Ia returned her attention to her precognitive messages.
Bored, since they were only drifting slightly through empty space, with no need for anyone to guide the helm, Fielle finally asked her a question. "So...Captain. Statistically speaking, what _is_ the safest spot in the universe?"
"In bed, with the covers pulled over your head," Ia said, her attention more on her task than on her answer. "Having been kissed good night by your parents, who have just checked under your bed and in your closet for all the things that might go bump in the night."
Snickers broke out at her quip, as well as grins. She smiled slightly. The other thing she could've said was, "In your grave," but that one would've been too morbid to reply. The point was to encourage her troops, not discourage them. Keeping her mouth shut, she continued scrolling her thoughts onto the datapad. She would have to stay up half an hour extra to ensure all the messages were printed out tonight, but that wasn't much different from most other nights, these days.
"...Is that anywhere near this ship, sir?" Hulio dared to ask her after several seconds.
"This ship will be about as far from that as you can get and still be alive," Ia said, ending one message and starting on the next. "But for the moment, we're safe. Eyes to the boards, thoughts on your tasks."
More minutes passed. The main bridge door finally opened, admitting Commander Harper. Judging from his tousled black locks and the mussed, slightly stained state of his grey coveralls, he had apparently been working personally on the engine upgrades and hadn't bothered to change. Moving over to her side, he pulled himself to Attention and addressed her. "Captain Ia, the new programs are loaded, and have been test-simulated five more times. Everything's green for go, sir."
"Excellent work, Lieutenant Commander. Lieutenant Rico, stand down. I'll take the command station now," Ia stated. She tucked her pad into her shirt pocket, moving from the pilot's station to the command chair. "Yeoman Fielle, I'll be taking the helm from here. Both of you can stick around, or go take a break for the next hour. If you go, be back on duty at nine hundred."
Rico narrowed his brown eyes, giving Ia a thoughtful look. Unbuckling his harness, he stood. "I think I'd like to stay, Captain."
"Have a seat," she directed, giving him room to step out from behind the curved workstation console. Nodding, Rico moved to one of the unoccupied seats, the backup station for communications. Harper took the backup station for operations on the other side.
Taking the abandoned chair, Ia sat down, strapped in, and logged in with her bracer. The chair, already programmed for her preferences, slid forward to accommodate her average height. Strapping her left hand into the flight controls, she tapped the workstation controls, rearranging the status displays on her main, two secondary, and ten tertiary screens.
"Right, then. I'm ready for the transfer, Yeoman," she said. "If you're ready, helm to my control in ten."
"Aye, sir. Helm to yours in ten," Fielle agreed, free hand moving over his console.
"...I have the helm," Ia stated a few seconds later. "Bringing engines back online."
"Heading, sir?" Hulio asked, glancing up from the navigator's post. It was the navigator's job to coordinate with the navcomp for plotting a safe course into a selected star system, working on the macroscale to avoid all the large hazards that the FTL field couldn't grease out of the way. It was the pilot's job to avoid hazards on the microscale, usually while traveling at insystem speeds.
"Engines are now online, Captain," Harper said, beating Private Dinyadah to it. She shut her mouth, glanced at him, and shrugged. Harper smiled in Ia's direction.
"Right. Let's bring the _Hellfire_ up to insystem speeds. Forward to one-quarter Cee," she warned the others, bringing the thrusters online. The faint _whoosh_ that was the cycling systems for lifesupport grew a little louder, picking up a _thrummm_ from the engines again. The stationary field of stars displayed in the various monitors around the bridge slowly shifted as Ia made a slight heading correction. "Just a few more minutes, and we'll be up to half Cee, since we're taking it easy."
"And then what, sir?" Rico asked her. "Three-quarters lightspeed? I thought we were testing the FTL panels, not the insystem drives."
Harper smiled, looking at Ia. "No, Lieutenant. When we hit half lightspeed, we open a rift and take 'er for a spin. Right, Captain?"
"Right, Commander." Technically, he was a lieutenant commander, but since he outranked the other two on board—and the doctor wasn't even in the chain of command, officially—it was acceptable to shorten his rank. That, and she knew he would eventually earn a promotion. "Navigation, pull up the star charts for System N-Tau 1158."
"Aye, sir." Hulio tapped his console, letting the advanced processors calculate their course. "N-Tau 1158 is...twenty-nine light-years from here, or a day and a quarter, deadheaded FTL, sir. We don't have much information on its insystem hazards, though. We also have the Kirkenn Nebula between here and there. It's a new gas cloud, so it's a significant navigation hazard. If you wanted to go there, you'd have to...wait...
"The Kirkenn Nebula is in the _Grey_ Zone." He looked up from his station. "Sir, why did you want to know about that star system? No one is allowed to go there, per treaty. It's part of the buffer zone we won from them."
"You know that, and I know that, but the Greys have conveniently forgotten about it, Private. Cycling the OTL hyperwarp in twenty seconds," she warned everyone. It wasn't much of a warning. Telltales turned amber as she gripped the trigger for the hyperspace generators. Panels opened on the nose of the ship, displayed on the second of her lower tertiary screens, the one to the middle left. The shallow camera view showed the relay nose cone lifting into position. The moment the lights switched to green, she pulsed the trigger.
A blue-white energy packet spat out from the cone, an elongated sphere crackling with warped physics. It shrunk down, collapsing in on itself, and punctured a hole in reality. That hole swirled open in a much larger, greyer version. It expanded and swallowed the ship just in time as they dove nose-first into the grey-streaked tunnel. Behind them, the aft view on her fourth tertiary screen showed the tunnel collapsing about two lengths behind them.
The _thrum_ picked up from a quiet background noise to a palpable low rumble. Ia frowned at her shivering console. "Harper, I thought you said you implemented the new software. Why is my ship shaking?"
"So did _I_ , Captain," he muttered back, tapping in a string of queries on his workstation. "Tracing the relays now...Ah! Here it is. I forgot I left it on manual for the tests, which means we just have to switch it to fully automated and let the computer compensate for the two different systems instead of us absentminded engineers...which I have just done...now."
The rumbling eased back with the last thump of his finger on the operations-station controls. It was now quieter, bearably so, though the new hum was stronger than the faint hum from before. Harper gave her an apologetic look.
"Sorry, sir. I'll get the code for that patched in by shift's end, sir. I figured you'd need the manual programming for those instances where we're beating the speed of the OTL-spark and need to wedge it open, or are still getting up to hyperrift speed. Or if anything happens to the engine comps, like damage in battle."
"A good piece of foresight planning," she praised. "Right. This transit will take just under an hour. Commander Harper, Private Hulio, double-check the navcomp's link with the engine comps. I want to make sure this ship flies as smoothly as a courier does on straight OTL autopilot."
"Sir...if we're really flying through a hyperrift, why aren't we getting spacesick?" Dinyadah asked, looking up at the grey streaks of the rift tunnel on her viewscreen.
"We're immune because we're wrapped in modified FTL physics, Private," Harper answered her. "We are now moving at roughly two minutes to the light-year, slower than standard OTL, but considerably faster than FTL."
"But I was always told that forcing an FTL ship through a hyperrift shook the ship to pieces," Dinyadah said. She looked around the bridge, seeking support, or at least confirmation from the room. Several of the others nodded, including Ia, so she looked back at their chief engineer. "So why aren't we having our teeth rattled out of our heads?"
"It's a modified warp field, that's why—don't ask how it was modified," he added, glancing at their captain for a moment. "The Captain is under orders not to divulge that particular information, and that means the rest of us are under orders, too."
"Or to put it another way," Lieutenant Rico stated, studying her, "if word of how we're doing this gets off this ship, the Salik _will_ find out and try to use it against us. Right, Captain?"
"Right, Lieutenant," she agreed, her attention split between making sure the helm stayed on course in the rift and answering the unspoken questions. "Hyperwarp has far too many advantages for us not to use it, but far too many disadvantages to implement it across the fleet. The least of which is the fact that we're still effectively communications-blind while in hyperrift transit.
"We Terrans are not living on an isolated clutch of planets on the backside of unexplored space this time around," Ia said, referencing Terran history. "The Salik know exactly where we are, they can find out where all of our shipyard facilities are, and they can bribe, coerce, or smash-and-grab the information if we offered it to the rest of the fleet. If we hadn't been isolated and far from the war zones back at the start of the first Salik War, then they would've seized all our new, Terran-based tech offerings and done their best to defeat the Alliance.
"Sometimes you need a lump hammer to take out a problem, as we helped provide two centuries ago," Ia said, quoting one of her distant descendants. Drawing in a deep breath, she steeled herself for this next bit of potential trust-bending, and revealed as calmly and matter-of-factly as she could, "Sometimes you need a laser scalpel. This ship is that laser scalpel...and I have just aimed it at one of the biggest, nearest, most dangerous cancerous growths in the galaxy. As far as the rest of the ship is to know, never mind the rest of the fleet, we never strayed from known Terran flight paths. And we _never_ crossed into the Grey Zone.
"That is a direct order from not only your commanding officer, but from a high-ranking precognitive—and before anyone protests further, I'll remind you I _am_ fully authorized by the Command Staff to make this little visit. We're going into the Grey Zone, and we are all going to keep our mouths shut about it. This particular little mission is labeled as Ultra Classified, which means you cannot even mention it to _my_ superior without going first through me.
"This one is completely out of your pay grades at this point in time...and it is merely the first of far too many missions with that label stuck to it. Speaking of Ultra Classified, the Admiral-General authorized me to undertake this mission, yes, but you are _not_ allowed to discuss it with her. If you did, you'd be accused of Fatalities Four, Five, Six, and Thirty-Five, and those are just the obvious ones," Ia told her bridge crew. "They'll dredge up every other Fatality rule they can throw at you, too. That's what Ultra Classified means. That's why I have that double-indemnity regarding corporal punishment on my back—because I am responsible for the rest of you keeping your mouths shut. Please do; you may consider this your first official test for such matters."
That silenced her crew. Ia knew it was a grievous stretching of her _carte-blanche_ powers, but it was necessary. Though she had given Fielle and Rico permission to leave the bridge, neither moved. Fielle stayed put because she knew he wanted to learn the new drive's flight mechanisms. Rico stayed put because he was busy watching her. Studying her. Analyzing her methods and motives.
It might've made more sense for the Command Staff to have made Helstead her onboard spy, but Oslo Rico was used to assessing tactical and strategic threats and knowing which parts of his information-gathering were troublesome enough to pass along. Helstead excelled at executing decisions based on what she found, but Rico's job was to report _if_ Ia stepped out of line. This would stretch his credulity limit, but she knew he wouldn't report it precipitously.
She also knew there were other spies tapped among her crew. Her 1st Platoon lieutenant was the most important of them, since even spies had to report up through a chain of command; the Command Staff wanted to know only the important bits, rather than be pestered by petty half worries.
For now, they kept their mouths shut and their eyes on their workstation screens. All except for Yeoman Fielle, who had nothing to do. Seated at the pilot's station, ahead and below her own post at the back of the room, Fielle fidgeted. He wasn't quite as bad about it as Lieutenant Commander Helstead could be, but he did sigh and shift in his seat every now and again, visibly bored.
"Yeoman Fielle, would you like to warm up the feedbacks and see what hyperwarp flight feels like?" she finally offered.
"Sir, yes, sir!" he agreed quickly, sitting forward so he could reach for the controls.
She smiled. "If you learn quickly enough—not that there's much more to learn—I'll let you oversee most of the flight. I'll be taking the helm back a few minutes before we emerge, though."
"If we're emerging inside Grey space, sir, you can _have_ the helm at that point," he muttered, strapping his hand into the control glove. "I don't want that level of responsibility on my hands. In fact, I don't even want to go there at all."
"Relax, Yeoman; we'll get out alive," she promised.
Her 1st Platoon lieutenant wasn't the only one to glance her way at those words, but his eyes did linger. As did Harper's, though his gaze was at least more trusting than dubious.
L-3 POINT, TAUS'EN IV
N-TAU 1158 SYSTEM
The _Hellfire_ emerged from hyperspace without fanfare. Grey streaks blossomed into stark black filled with pinpoints of light. Warned that they were about to emerge, Private Hulio moved quickly to synch what little they knew of the system with the information coming their way. Considering they were still traveling fast, he worked with crisp urgency.
"Scanners are up and running, Captain. Gravity and lightwave data coming in...We're just outside the fourth planet's orbit, sir, at the third Lagrange point— _Madre de Dios_ , there's a space station at the L-3, sir!" he announced, looking up from his lower screens to the primary and back. "It's huge! Five kilometers across. We're not on a collision course, but we'll pass within thirty thousand klicks. The hull configuration's a bit strange, but scanners _are_ matching the materials to known Grey technology, sir. We have light-seconds before they detect us."
"Orders, Captain?" Rico asked in a deceptively mild voice.
"Private Hong," Ia stated, her tone crisp but calm. "Power down all guns. I repeat, power down _all_ guns. This includes all personal weapons here on the bridge as well as the hull. We're about to be scanned, and we don't want to alarm the locals by doing the wrong things."
Hong complied with a shake of his head. "...Aye, sir. Powering down all guns, sir. Putting my faith in you, sir—and requesting permission to haunt you in the afterlife if you're wrong, sir."
"Permission granted. Corporal Xhuge, if you'll look in the Alliance folder, subfolder Grey Interactions, you will find a comm file marked 'Neutral Parley.' Use that to send a ping to the station," Ia said.
"Neutral parley?" Rico asked her. One of his brows rose on his tanned face. "Is that even _in_ their vocabulary? If it weren't for the psis, they'd have squished us like bugs long ago. We aren't even worth the time it takes to spit to them, or whatever their equivalent is."
"Considering most of their past interactions with us have either fallen into the categories of 'ignore the inconsequential jumped-up slime molds' or 'plunder their primitive biology for nefarious experimentation purposes,'" Fielle quipped, "I for one wouldn't fight over the concept of an actual chance to talk like civilized sentients."
" _I'm_ just grateful we're on a ship with a powerful psi," Dinyadah muttered. "That's the only thing that's backed them down in the past."
"The file has been sent, Captain," the corporal at the communications station informed her. He didn't look happy to have done it, but at least he hadn't hesitated. "We've received pingback, so I know they got it. I can't make heads or tails out of the language in the recording I sent, though."
"We actually have a couple dozen psis on board, Private Dinyadah, some of them quite strong for the average psychic," Ia corrected her, addressing the scanner tech first. "But we won't need them, just me. They're on board for another reason. As for the language, Corporal, it's called Shredou, which is the name for their species as well as for the language of the Greys. I pieced together a greeting specifically addressed to the being who serves as their station master and, coincidentally, the chief military officer for this system. One specific enough, it will catch his attention.
"And they do have terms for neutral parley in their culture; they just don't share those terms with non-Greys," Ia said, eyes on the slightly enlarged dot that was the space station in the distance. They were still traveling at a quarter the speed of light, but she didn't alter their course. "The fact that I know those terms, and the exact location and circumstances he'll be in when the message reaches him, will stay his hand. They may be the single most alien race in the entire known galaxy—above and beyond the Feyori—and the single most technologically advanced, but they do share the trait of curiosity with us."
"So what is this parley of yours going to discuss?" Rico asked her.
"Sir! Energy buildup in the—" Hulio started to report. He was cut off by a flare of light, and a slight pressure change in the bridge, one that puffed air outward. Air that had occupied the clear space just to the right of Ia's command station, between her and the seats claimed by her two fellow officers. A space now occupied by something, or rather someone, else.
Cocking his head slightly—and calling the alien a "he" was only a guess on Ia's part, since their gender was hard to discern—the Grey surveyed the stunned occupants of the bridge. He blinked his large black eyes and unfurled one of his slender, grey-skinned hands, focusing on the white-haired, grey-clad woman next to him.
"Speak."
His voice sounded strange, as if two sets of vocal cords worked at once, and not quite in harmony.
"I know you plan to invade," Ia said, keeping her sentences short. Longer ones would be open to misinterpretation. "I know when. I know where. I will tell you the battles we will fight."
Rico hissed at that statement. Even Harper gave her a dubious look. The others looked up from their boards, then hastily turned back to their monitors as the Grey, short and slender, glanced their way.
"You betray your kind." The Grey didn't speak with the intonation of a Human. His voice spoke flatly, his thin lips moved and shaped the words, but whether it was a statement or a question could not be discerned.
Ia took it as a question. "No. I do not betray my kind. Your technology will destroy this universe. I will stop you. I will tell you when. I will tell you where. You will see my words are true. When you do, you will surrender. You will obey the second treaty. My treaty."
He blinked and curled his fingers. "Irrelevant."
"...Another energy surge, sir," Hulio whispered, gaze fixed firmly on his screens.
"Obey me," Ia stated calmly, "and I will save you from the Zida"ya."
The double click was difficult to manage, considering she had only a soft tongue and the inner side of her teeth to work with. Nor was it in the language of the Greys. It did, however, have the desired effect.
The Grey's large black eyes widened to their fullest extent. He did not move, however, other than to say, "Speak."
"They are coming. I know where. I know when. You doubt me right now," she added, dipping her head slightly. "I will show you my accuracy. You will accept my deal. If you refuse, I will not save you. I will aim them at you. I will know when. I will know where. You will die. All the Shredou will die.
"Take the indicated hyperrelay unit, and leave," she added, uncurling her right hand in a similar gesture to the Grey's. "I will contact you. Then you will know when, and you will know where. You will see my words are true."
"Arrogant." The Grey did not move and did not leave the ship as ordered.
Ia breathed deep. As she exhaled slowly, she poured her mental energies into her psychic shields, adding a twist of electrokinesis. The air around her station crackled, and her monitor screens flickered. Capacitors absorbed the energy, stabilizing their views of the stars outside and the navigation data overlaid on her secondary screens. She didn't move, other than to breathe and tense her body.
The bubble of energy expanded outward like a spherical force field, visible only where the field encountered motes of dust in the air, causing them to snap and spark. It wasn't exactly electricity, however, but rather, kinetic inergy.
The Grey winced, then stepped back. She expanded the bubble, until he clutched at his head. A high-pitched hiss escaped him, not much different from a teakettle's whistle. Ia eased back her energies.
"Powerful, not arrogant," she corrected him, relaxing. "You will obey. Now get off my ship."
Opening his large eyes, larger than a Gatsugi's mouse black orbs, he stared at her a long moment. Then vanished. Air flowed inward slightly in a faint _pop_ as the molecules slapped back together. Grey technology permitted translocation, a mechanical, technological method of psychic teleportation, but the energies used were not at all the same. Psychic energy, the kind wielded by those Humans descended at least partially from the Feyori, was the equivalent of acid to their species' senses.
The fact that they could make the translocation instantaneously onto a ship moving at half the speed of light spoke volumes about the rest of their technology. As did the arrogance of sending a single speaker to visit the insects daring to invade their space.
"Right. Time for us to get the hell out of here. Sparking the rift in thirty seconds," Ia warned her crew, right hand moving over the controls. "We'll be taking a short jump with a course correction to follow. Once we're en route—after this little jaunt," she added, sparking the rift, "we'll be able to relax and stand down. Not even the Greys can catch us in hyperspace. This is why we will stay in it on the second jump until we reach Sanctuarian space, putting us well ahead of schedule."
"And the treason you just committed?" Rico asked her, his voice still calm, his expression still neutral. Behind him, his screen showed the mouth of the wormhole swallowing them in streaks of grey light. It made his deeply tanned skin look sickly, underscoring his accusation. "Is that on the schedule?"
"It's not treason if I am authorized to commit it, Lieutenant. Nor is it treason when these precognitive actions will be directly responsible for saving the Terrans from being destroyed by the Shredou in several years," Ia countered, knowing he couldn't let such a huge security breach pass unchallenged. "I also expect you personally to assist me in properly wording my communiqués with the Greys in our future exchanges of information. But that won't happen for almost a year, so you can relax."
"I will not relax until I have examined your _next_ message, sir," he added. "And preferably this last one, too. I'd feel a lot better knowing what you said to them."
The tunnel of streaks ended. They emerged in realspace on the far side of the system, far from the light of the local star. Ia began the careful process of not only slipping them sideways and down a little, more in the direction of her home system, but gently altering their trajectory so that they would be able to hit the next hyperrift dead on, rather than at an angle. Touching the edges of a rift was never a good idea, which was why speed was essential in getting their ship both in and out at just the right moment.
"So will I, Lieutenant. Some of the words have no easy translation into Terranglo, since my description of what he was doing at the moment of contact have no correlation in our own culture...but the important words are perfectly clear. Including the fact that I will not be using that hyperrelay unit to contact them until 12,379 _kesant_ have passed. You'll need to figure out how to translate Grey Standard into Terran Standard time systems, but it's just under a Terran Standard year." She looked up at him, then over at Hulio. "Private Hulio, get me a dead-reckon heading for the Sanctuary System. Line it up with our current speed and heading, and plot an appropriate course correction arc."
"Aye, sir," he agreed, turning his attention back to the boards.
"Sir?" Dinyadah asked. "Captain?"
"Yes, Private?" Ia asked, watching the unfocused crosshairs that appeared on her main screen, thanks to Hulio's efforts.
"Thank you for getting us out of there alive," the other woman said. "I mean, not for getting us into that situation, sir, but...er, I mean... _shakk_. Sorry, sir."
"I suggest pulling your foot away from your mouth before you swallow it, Private," Harper ordered her, his tone gentle but pointed. "Put your faith in our CO as I have, and she'll get all of us out of this alive."
_Not everyone, Meyun,_ Ia thought grimly. To herself, behind tight mental walls. _But I'll save those that I can._
JANUARY 18, 2496 T.S.
OUR BLESSED MOTHER
INDEPENDENT COLONYWORLD SANCTUARY
The moment Yeoman First Class Arial Yamasuka, 2nd Platoon A Alpha, touched the shuttle onto the landing pad, Ia unstrapped from the copilot's seat. Slapping open the cockpit door, she hurried into the crowded cargo bay. Gravity pulled at her, hard and heavy; once again, she felt rather out of shape, having lived too long in lightworlder conditions. Exercising a few hours every day in heightened artificial gravity—captain's privilege—wasn't enough to compensate for the pull of the real thing.
"Meioas!" she barked, catching the attention of the crew in the cargo hold. "Make sure your gravity weaves are set to adaptive gravimetrics on the low setting, and no higher than medium once you get off the ship. Stand no closer to each other than three meters once they have been turned up, to avoid the nausea that comes with field interference," she called out, pitching her voice to carry.
The A teams from each Squad in the 2nd and 3rd Platoons fumbled with the buckles of their own four-point harnesses, hampered somewhat by the bulk of the purple web-works wrapped over their mix of grey camouflage clothes and black-and-pewter light armor.
"Sergeant Santori, Sergeant Maxwell, you are authorized to open up the ammunitions crates. Lead team members will be issued stunner c-clips. Corporals and most of the privates first class and grade, check to make sure your clips have a blue-dotted rectangle, indicating their payloads are indeed relatively harmless beanbags," she reminded the men and women getting ready to disembark. "Privates second class and grade, you will be issued tranker clips; check to make sure they have blue feathers.
"Do not—I repeat do _not_ —fire trankers unless two verbal warnings and two stunner shots have first been fired, and fire no more than one trank per target. People can and _will_ die if they hit the ground wrong in this gravity, which includes being tranquilized too fast. Make sure all stunner beanbag rounds are aimed at torsos, not heads, to ensure your targets are not knocked over as well as knocked back. Keep in mind that while the density of the local atmosphere isn't much different from Terran Standard, the gravity on Sanctuary will drop your shots fast. You can shoot from the hip if you must, but your JL-41 projectile riflescopes come with sensors that will adjust for the local gravity.
"I suggest you turn them on and use them," she advised the men and women listening to her. Multiple clickings and faint charging whines immediately followed. Ia nodded and continued. "Your job on this drop is to scout the warehouse, establish checkpoints, and secure the initial cargo so that you can instruct your other Squad members on where to go and what to do during our next trip," she stated as she skirted between the seated soldiers and the cargo crates strapped to the floor. "Line up at the bottom of the ramp when each team pairing has been properly armed, and remember, _no_ running on this planet.
"Tripping and falling can kill you in this gravity if you are not prepared to fall just right, and you are _not_ prepared. Consider yourselves under orders not to run at all for the duration of all planetside visits to this world. Check your ammo and lock and load. Gentlemeios, welcome to Sanctuary, your local gravitational hell."
Reaching the back-ramp hatch, she triggered the door and rode the panels as the metal descended to the tarmac. Clad as she was in camouflage Greys with a black vest covered in polished grey ceristeel plates, Ia hoped she looked no-nonsense enough to be intimidating. Customs officials were a tough breed; they would not appreciate her bulldozing these supplies through their checkpoints without the right to random inspections.
Her comment about gravitational hell had nothing to do with the ambiance. In the distance, the mountains looked purple, the sky a pale indigo blue, the local tree-equivalents were showing the bright spring hues of yellows, greens, and blues, and the buildings were relatively clean and fresh, whitewashed with colorful bands of decorative trim and holographic signboards designating spaceport terminal gates at the central hub off to their right in the distance and the warehouse nearby on their left.
Two ground cars were already on their way, filled with Customs inspectors. So were a half dozen short, stout Humans in plain beige coveralls. The group hurrying her way on foot reached the shuttle first. The lead figure, a young man of about twenty, lifted his fingers to his forehead in a mock salute. "Hey there, Prophet. Right on time as promised. You're looking good, too. Grey looks better on you than that brown crap you wore last time."
"Hey, James," she acknowledged, dipping her head slightly. Like her, he was at least half-Asian, but despite the shorter hair and longer sideburns, the slight cleft in his chin made him recognizable. The last time she had seen the dark-haired man was shortly before leaving her homeworld three years ago. "You're looking good, too." Her gaze slipped to the right, tracking the incoming ground cars. "Better call me Captain, though."
"Yes, sir, Captain, sir," he quipped, flashing her a grin. A moment later, he sobered, watching the slowing ground cars. "Showtime, people. Make like you're model employees."
The three men and two women with him smirked at that. Ia knew James Chong-Wuu had hired them more because they, too, believed in her cause than because they were "model" anything. Nodding to their young employer, she turned crisply and strode back to the others.
"2nd Platoon B Alpha, C Alpha, secure the perimeter of this shuttle," Ia ordered, flicking her fingers in the indicated directions. "No one boards this vessel but 1st Company 9th Cordon Special Forces personnel as per the Admiral-General's direct orders. Move out."
Four bodies peeled off and moved out around the landing gear of the ship, rifles in their hands but pointed at the ground for safety. That would change, she knew.
"All other team pairings, you will accompany these gentlemeioas to the warehouse to secure its interior and perimeter. Follow your Platoon sergeants' orders and standard procedures in securing the indicated warehouse, but don't rush, so you don't trip. Once the perimeter has been secured, authorized personnel for entering the warehouse will be members of 1st Company 9th Cordon and the employees of Chong-Wuu Stevedores, Incorporated. If you have any questions, refer them up the chain of command if I am not near, or pass it directly to me if I am."
The two ground cars came to a stop. The drivers remained inside, but the first car disgorged two Peacekeepers, their blue-and-white uniforms marked with the standard scalloped-shield badge on their shoulders and caps. On that badge was the corona-and-crown symbol representing the capital city of Our Blessed Mother, Sanctuary. The second car released a third Peacekeeper and a man in a blue-and-gold version of their uniform. His badge had the corona sporting a planetary curve inside, replete with the distinctive coastline squiggle for the local continent.
"Soldiers, move out!" Ia commanded, pointing at James and his crew. She turned to face the Customs agent, whose face reddened as the gravity-weave-wrapped men and women dispersed.
Striding forward, he pointed at the grey-clad bodies spreading out and moving at a swift walk. "What do they think they're doing?" he demanded. "This landing pad is for Customs-cleared vehicles only—and that warehouse, too!"
"Captain Ia, Terran United Planets Special Forces," she introduced herself. "Per Sanctuarian Charter regulations Article VIII, Military Contracts, Section E, Supplies, my crew, shuttles, and cargo are listed as exempt from Customs clearance requirements."
"I wasn't notified about this," the Customs agent protested. The name on his badge—he was now close enough for Ia to read it—simply said _Larkins_. "There was no notification of any military shipments due this week!"
"Our ship hit a hyperrift on the way here, depositing us insystem ahead of schedule, Officer Larkins," Ia said briskly, avoiding the fact it was an artificial wormhole, not the natural one that terminated at the edge of Sanctuarian space. "We are here to deposit our cargo in the emergency bunkers the Terran Space Force installed on your planet three decades ago. This cargo has been designated as war supplies on the manifest. As such, it falls under Article VIII, Section E, and is exempt from all Customs-inspection requirements, as per the Terran defense contract with your planet."
Frowning, Larkins stared at the retreating soldiers, then glanced up the shuttle's ramp. "That does _not_ clear your cargo from quarantine restrictions. This is an M-class planet, not a domeworld."
"All cargo has been irradiated and sealed before being boarded at their origination point, as per military regulations regarding the transport of supplies," Ia replied, doing her best to sound like a regulation brick wall placed in his path. "All personnel have been scanned by military biometric sensors in our ship's airlocks, as per regulations, and my crew undergo weekly biometric physicals while en route. No one with an active infectious agent has been permitted to leave our ship. Your colony is safe from quarantine hazards."
Officer Larkins was not easily deterred. He pointed at the shuttle, giving Ia a firm look. "All imports are to be examined by Customs for potential contraband and excise taxes, by order of the Sanctuarian Supreme Council."
"These items are not imports, meioa, nor are they for sale," Ia replied politely, if briskly. She, too, pointed at the shuttle. "They are essential war supplies requisitioned by the Terran Space Force, a Sanctuarian-authorized government entity. By contract, Space Force essential supplies are not to be quarantined, not to be confiscated, and not to be taxed."
"Well, guess what?" Larkins stated, hands going to his hips as he gave her a belligerent look. He had to look up to do so, since she towered over his short, stocky frame by a full head plus. "The rules have now changed. _All_ incoming items must submit to inspection by a duly authorized Customs official. I _will_ inspect your cargo for contraband before I will allow a single crate to touch Sanctuarian soil."
"Your contract with the Space Force has _not_ changed, meioa. By Charter, the terms of our service agreement with your colony take precedence over all local laws in regards to all factors of the services we are contracted to provide," Ia countered. "Unless and until that contract changes, Terran Space Force war and emergency supplies are not subject to inspection, excise taxes, or impounding. The only thing they are required to undergo is standard quarantine irradiation and containment protocols upon initial packaging and loading, which they have undergone."
The Customs official smiled at that. It wasn't a pleasant smile. "Ah! But I don't _know_ that. That means _I_ have to board that vessel and inspect those crates _personally_."
He poked her in the sternum and turned toward the shuttle.
"By order of the Admiral-General of the Terran United Planets Space Force, unauthorized personnel are strictly forbidden to embark on the TUPSF _Hellfire_ or its auxiliary vessels," Ia warned him, raising her voice slightly to compensate for the descent of another orbital shuttle in the distance. "This directive includes our shuttle craft, meioa."
Larkins sneered at her and gestured for the three Peacekeepers to join him in heading toward the ramp. Ia lifted her head slightly, catching the attention of Private Helia Dixon. One of the few people in her crew who had prior experience with Ia's brand of leadership, the private waited for the official and his escort to close half the distance, then quickly lifted her rifle into position. Her teammate, Corporal Henderson, followed suit.
"Attention, meioas! You are approaching a restricted sector. You are requested to stay back from this vessel by ten meters," Henderson warned Officer Larkins. Positioned as he was on the port side of the loading ramp, with Dixon on the starboard four meters away, the pair had a decent cross-fire field on the quartet. "This is your first warning!"
Larkins slowed. At his back, the three Peacekeepers reached for their sidearms but hesitated about drawing since the two soldiers had the drop on them already.
The Customs officer frowned over his shoulder at Ia. "You people wouldn't dare stop me."
"I'm sorry, meioa," Ia replied, tucking her hands behind her back and putting her boots shoulder-width apart in Parade Rest. "These soldiers are under orders from Admiral-General Christine Myang herself. Entering that orbital ship requires Ultra-level clearance. Anyone attempting to board it without the proper clearance level and authorization is to be shot and tried for Grand Treason in a Terran military court of law."
Stopping, Officer Larkins narrowed his eyes, studying Ia. "You're kidding. You wouldn't dare."
Bringing her arm out from behind her back, Ia pointed at the shuttle, once again playing the hard-asteroid...and secretly enjoying it. She really did not like the arrogance these Church-backed government officials were being allowed to display these days. "Anyone attempting to board that vessel or its sister shuttles from the TUPSF _Hellfire_ without the proper clearance authorizations arranged in advance, or attempting to interfere with the delivery of its cargo of essential supplies, is to be shot and charged with attempted Grand Treason against the Terran Space Force.
"The right by the Space Force to assert and uphold the required working conditions for our missions is covered by the Independent Colonyworld Sanctuary Charter of Rights and Responsibilities, Article VIII, Section E, paragraphs 1 through 3." She bit back a smile, adding soberly, "You are welcome to assist your government in petitioning the Terran Space Force to have those Charter rules changed or our military services dropped, gentlemeioa. Until then, that vessel is a restricted sector which you are not authorized to transgress, and our cargo is exempt from all examinations."
From the way he sneered, he didn't believe her. Turning back, he took another step toward the shuttle ramp. That earned him his second warning.
"Meioa! You are entering a restricted sector without proper authorization! You will stay ten meters from this vessel or you _will_ be shot," Corporal Henderson ordered the other man, sighting down the scope of his JL-42.
"I am Abram Larkins, a duly authorized I.C. Sanctuary Customs Officer, and I _will_ inspect that shuttle!" he argued, pointing at the ramp.
"I am Corporal Henderson, of the TUPSF Special Forces doesn't-give-a- _shakk_ ," Henderson warned him, "and I _will_ shoot you if you violate this restricted zone by moving one meter closer, meioa. I am only required to give you two warnings. I have given you three. One step closer, and you don't get any more, meioa."
Officer Larkins hesitated. Ia watched warily. This was where the moment could go either way. Behind them, she could hear some of the others returning from the warehouse. So could he. Glancing over his shoulder, Larkins squinted against the bright sunlight slanting in through the clouds to the east.
She heard Santori bark a short order. The advancing men and women moved off to the right, circling around to approach from Dixon's side of the shuttle. That gave the two guarding the ramp a clear field of fire. At the front of the parked shuttle, the two members of B Alpha glanced occasionally toward the back of the small ship but kept most of their attention on the rest of the tarmac, scanning for other possible points of interference.
Their opponent made up his mind. Larkins turned and poked his finger in her direction. "You may think you've won, but this is an _independent_ colonyworld—we will not put up with the tyranny of the Terrans on our sovereign soil!"
"If your government wishes to formally terminate its contract with the Terran Space Force and provide for its own interstellar protection needs, your government is welcome to do so. Until that time, our agreement stands as written. We are storing these supplies on your homeworld in preparation for the coming Second Salik War," Ia stated, gesturing back and forth between the shuttle and the warehouse. "You may look at the quarantine stamps on the crates as they are transferred from the shuttle to the warehouse to verify they have undergone the necessary decontamination protocols.
"You may _not_ board our shuttles, you may not enter the warehouse, and you may not open the crates." She tucked her hand behind her back, resuming full Parade Rest again. "Do you have any questions at this time, meioa?"
"The Salik won't come here," Larkins dismissed, wrinkling his nose. "We're too far away."
"There are no defensive barriers in space, meioa," Ia said, raising her voice once again as another shuttle took off. This was the spaceport's busy time, in the hours of relative calm between the morning and early-evening thunderstorms. "No natural terrain to keep them from going anywhere they want to go, save only the ongoing efforts of the various Alliance militaries to keep them contained. Those efforts are failing. It is our contractual duty to ensure that every world under Terran Space Force protection is supplied and defended to the best of our ability. Until such time as that contract is terminated, that means we will continue to protect you."
The subtext in her speech, the unspoken attitude behind her stance and her words, implied the phrase, "...even if we don't like each other." Ia stared him down until Sergeant Santori stopped at her side, giving her a salute. Ia shifted to Attention and returned it.
"Captain. The warehouse is secured, sir," Santori told her. She was flying solo for this job since it was technically second watch on the ship, and that meant Lieutenant Spyder had command of the bridge while Ia was on the surface. "Awaiting further orders, sir."
"Good work, Sergeant. Maintain the current perimeter with half our troops. The rest will unstrap and remove the cargo sledges from the hold. Officer Larkins is permitted to visually inspect the seals on the first batch of crates," she added, looking at the Customs agent. "He is not permitted within ten meters of shuttle or warehouse, and he is not permitted to open any crates, but your soldiers are to cooperate and assist the official in examining the external seals regarding quarantine protocols."
"Understood, sir." Turning on her heel, Santori barked orders, sending the pairs of teammates toward the ramp; Dixon hastily moved forward so that they could cross behind her rather than in front of her field of fire. They fiddled with the controls of their gravity weaves as they did so, permitting each Human to move closer than three meters without the risk of the fields making them stagger.
Ia resumed Parade Rest, a visual, grey-and-black brick wall of military efficiency. Larkins looked between her and the ship. Henderson didn't lower his rifle; neither did Dixon. Finally, he grimaced and moved back. The two members of C Squad Alpha lowered their weapons but did not shoulder them. A minute later, the first of the ground-sledges rolled down the ramp. Their motors whined as each one made the transition from the interior of the ship, where gravity fields in the ceiling counteracted some of the planet's pull, to the full force of Sanctuary's 3.21Gs.
Santori moved up beside her, watching the sledges roll a good twenty-fives meters away from the ship before her teams stopped them for the requested visual inspection. Ia's skin twitched at her proximity. Not because she feared the woman would touch her, triggering her precognitive gifts, but because the gravity field projected by that purple weave was messing with her sense of down.
The tanned woman moved a little closer and quietly asked, "Do all the government officials on this planet have their heads up their asteroids? Or did we just get lucky?"
"They all have official attitude-problem approval from the main political party, these days," Ia confirmed, equally under her breath. "That party has been turning increasingly xenophobic in recent years—and by xeno, they include nonnative Humans lumped in among the Solaricans and such. That's why, with the exception of myself, no one is to go anywhere alone on this world...and the crew will go in uniform at all times. I don't want anyone mistaking them for weave-wrapped tourists."
"I'm not comfortable with that exception, sir," Maria Santori warned her. "You'd be a prime target out on your own."
"I can fake being a loyal follower of the local religious movement far better than you can—I _will_ have to go places in the next few days that the Terrans cannot, Sergeant," Ia said, shifting her gaze to Santori. "This is that third war front I warned you about at the cadre meeting last week. A war of ideology so far, but a war nonetheless."
"Which is why you shouldn't go alone, sir," the sergeant countered. "The vid industry makes a fortune off stories of one-meioa armies, but that isn't reality."
"Considering where I have to go, a gravity weave would scream 'outsider' too loudly. Even our lovely 3rd Platoon leader would have difficulty walking around for long in this gravity, and she's the second heaviest heavyworlder on the ship," Ia pointed out. "I _won't_ be alone, Sergeant. I just won't be with fellow soldiers."
"That's what has me worried. Civilians aren't soldiers," Santori murmured.
A cry from inside the shuttle was followed by a clatter. The man screamed again, and started babbling something about birds setting the world on fire. Officer Larkins and his three blue-and-white-clad Peacekeeper cohorts quickly marked themselves with the corona circle, warding off personal attacks of "the Devil's visions," as the members of the Church of the One True God called the psychic outbreaks. They could strike anyone anywhere on or within orbit of Sanctuary, give or take a few thousand miles.
It looked like one of her crew was suffering the first outbreak of the day. Ia sighed and turned toward the ramp. "I'll handle this. The important thing is to make sure nobody else touches him."
Santori nodded. "We heard your lectures on the phenomenon, sir. Trust me, he's all yours."
# CHAPTER 5
_So many things to keep track of, back then. So many defenses to set up, so many reactions to be readied, and so many tricks to be traded. All in the effort of helping as many people as possible survive the coming wars. Much of it, ironically, won't bear fruit until long after both you and I are gone, but it was worth it._
_Somehow, in that visit to my home, I actually managed to make it home. I got to see my family again. That's a rare thing in an interstellar military. But it wasn't the same as it had been before—and I'm not speaking hyperbole. It literally wasn't the same home anymore. Fanaticism had seen to that, ahead of schedule._
_~Ia_
JANUARY 19, 2496 T.S.
CENTRAL WARREN
The cavern was a far cry from the scorched, soaked rubble that had once been Momma's Restaurant. Vast and rugged, it was awkwardly lit in pools of daylight hues thanks to the scaffolding that climbed the upper half of the magma-carved walls. Lights, large and small, did their best to dispel the gloom. Part of a deep, ancient system of lava tunnels, the cavern was nothing more than a giant, amorphous blob of former gas bubbles suspended permanently in cooled stone.
From the rough-shaped terrace of what would become the _cafeterium_ , the level where many restaurants would bloom, Ia could see the shadowed cleft that would eventually be turned into the Director's Grove, a private garden for the successive leaders of the Free World Colony. The main cavern would be a combination of leisure gardens, farm gardens, and water-treatment gardens, aquaculture and aquaponics similar to the lifesupport bays found on most starships.
Some of the gardens were being sculpted as she watched, by hand labor and by robotics, by shovel and bulldozer. The rumble of sandhogs in the distance bespoke the efforts of digging crews to expand the network of side tunnels leading away from the ragged corners of the echoing cavern. There were other noises, too. People chatted and called out instructions. Hammers and picks banged into stone; shovels and buckets chuffed through piles of dirt. And from close by, someone sighed.
Sinking onto the cushion-lined bench someone had carved into the balcony wall overlooking the activities, Aurelia Jones-Quentin offered her daughter a cup of cocoa thickened with cream. "It's not so bad. Noisy day and night...or what passes for day and night, but not bad. Prime real estate, _gataki mou_."
Despite her ongoing efforts to get her parents to see her as an adult, as the soldier and the prophet instead of their little "kitten," Ia let the nickname pass. Hot cocoa and the reassurances that her family was alright were the things she wanted to dwell on right now. "I'm sorry they sped up the timetable on the restaurant. I saw the rubble."
"All the important things had long since been evacuated," Aurelia dismissed. "Unless you count some of your old toys and the actual bedding. Thorne remembered your warning him there was a small chance it might happen early. He insisted on stripping the kitchen, so most of that was salvaged, too. And you saw the new kitchen, so it's even better, _eyah_?"
_"Eyah,"_ Ia agreed. Twisting on the bench, she put her back to the low wall and stared at the coarse frontage that would one day gleam with colorful tiles and polished stone. "How much of the food is still coming from overhead?"
"Ugh. Sixty percent or more. We're a long ways from the arcological self-sufficiency you've demanded," Aurelia muttered. Off in the distance, one of the three or four hundred Free World Colonists dropped and clutched at her head, wailing in the _irit'zi_ , the Fire Girl Prophecy cry. Aurelia rolled her eyes. "It'd go a lot faster, too, if _that_ didn't keep interrupting our daily lives. Is there any way you know of to stop the damn visions, _gataki mou_?"
"If you're asking your little girl, then no," Ia shot back dryly.
Her mother stared at her for a long moment, then sighed. "Fine. You're not my little kitten anymore. Is there any way to stop these visions, O Prophet of a Thousand Years?"
"No." This time, she accompanied her reply with a wry smile, amused at the older woman's eye roll in response. "Endure and learn from them. Well, okay," she amended, "there _is_ a way to tone them down. When everyone in the FWC has studied, practiced, and reinforced their minds with basic psychic mental centering and shielding techniques, then they will be less likely to be overwhelmed by Fire Girl visions. But stop them? No. They won't stop. Welcome to Sanctuary."
The older woman let out a rude noise at that, somewhere between a raspberry and a sigh. "If we could afford to move off-world, we would. And yes, I'm including the cost in lives as well as the monetary expense."
She started to say more, but a clutch of workers came into view, laughing and chatting as they headed for the restaurant. Right now, it was doing more walk-up business than sit-down, with a modified menu. There were plenty of tables, but few servers. Rolling her eyes, Aurelia leaned back in a moment of rest, gathering her energy to get up and go take their orders.
"Relax, Ma," Ia told her, and nodded at a dark-skinned man coming out from behind the counter. "Marble has this covered. I can't stay out here for long. I have to go back to work turning all those blood-beads I sent you into precognitive wreaths in a few moments. How's Mom doing?" Ia meant her biological mother, Aurelia's wife Amelia.
"Upset at the loss of the wall harp, frustrated that she only gets to see the surface one day out of three, and settling back into her element now that she can cook again," her other mother admitted. "She's also not too sure about Fyfer and Thorne both moving in with Rabbit in the next few months. Not that the girl couldn't use the help, but she's stubborn."
Ia sipped at her cocoa, keeping quiet on that subject. Her brothers would sort out their living situation without her interference. That, and it was good cocoa.
Thick and rich, delicately bordered between bitter and sweet, it conjured up memories of their old, cramped home. Memories of listening to the winter rains pound on the plexi roof over their upstairs apartment. The smell of baked topado cakes and pastries. The sight and sound of her biomother explaining why she preferred cilantro to parsley in her dishes, even though the herb was less sturdy and less likely to grow well in Sanctuary's heavy gravity.
The cries in the distance ended. Whoever it was, the woman would be picking herself up and shaking off the mind-clouding images. Everyone down here knew what the standard visions were: a woman with a burning bird on her back, the spires of the now half-built main cathedral wreathed in flames, of great golden-glowing ships taking to the skies. A horrific wall of stolen star-stuff threatening to engulf and snuff out all the light in the galaxy. Someone being whipped in public, among dozens of other images.
If she saw something new, Ia knew the woman would report it to one of the clergy who had also begun to move down below the surface of their beleaguered world. Time pressed in on her, reminding Ia she didn't have enough hours in the day, and few minutes. Rising, she lifted the cup in her hand.
"Mind if I take this with me?" she asked her mother.
Aurelia, sipping from her own cup, nodded. "Make sure you leave it in the kitchen sink—the apartment sink, that is. I'll send Amelia to you with something to eat when she has another break."
Nodding, Ia bent and kissed her mother on the cheek. The contact, brief as it was, let her sense her mother's mood. Tired, stressed, but still strong. Still capable of doing what her daughter—scratch that—of doing what the Prophet had asked of her. But she was still her mother's child. "I love you, Ma. I'm glad I have you and Mom on my side."
"And Thorne, and Fyfer, and that not-so-evil mastermind pretending to be a little girl," Aurelia quipped. She flipped her hand, first at her daughter, then at the dirty plate and silverware on the low table in front of the bench. "Go on, go back to the apartment and finish your holy task. I'll bus your dishes back to the restaurant. Though next time, you'll owe me a tip."
Nodding her thanks, Ia moved away. Her mothers' commute was a bit longer than before. Rather than living directly over their restaurant, they lived five levels up, though they did have one of the largest apartments currently available. Mostly because it abutted a warehouse cavern that would one day be turned into a combination of multidenominational chapel and dance hall. Freedom of faith was a core tenet of the Free World Colony. Freedom of movement was vitally important to the psyches of the residents who would soon live out their lives underground.
Her red-hued civilian clothes didn't set her apart from the others using the lifts to get from floor to floor. Everyone wore cheerful colors down here, to liven up the dull curves of stone surrounding them. Bright shades of blue and yellow, purple and green mingled with orange and plenty of red. Her height and her shock of chin-length hair were what set her apart, taller than most everyone else by at least two dozen centimeters and readily recognized by her snow-white locks. Old-woman hair.
In fact, a few of the men and women she passed did double takes, recognizing her. A few more started to reach out to her, to touch the Prophet each one knew was responsible for finding and settling this place. A subtle shake of Ia's head was all it took to keep their fingers to themselves. As unsettling as the Fire Girl outbursts were for others to endure, Ia didn't want to give them a worse mental ride if her gifts decided to destabilize while she was down here.
Some of them murmured condolences over the loss of her parents' shop. Others drifted in her wake, wanting to say something, or to hear her say something. She nodded or shook her head where appropriate, but didn't actually say much. The last thing Ia needed was to have a stray word or phrase misinterpreted by her followers.
One of the recent installations, built in sections, looked like an artistic scribble of pipes lining the rounded corridor she chose, giving it a more oval appearance. The pipes weren't just artistic; they were heat-transference pipes, sucking some of the underground warmth out of the air and stirring it into a refreshing, circulating breeze. The siphoned energy wasn't wasted, either; this place, Central Warren, was over five kilometers below the surface. Thermal energy was constantly being transformed into electrical energy, empowering the various machines and amenities the underground capital needed.
Those amenities included things like lights and doors. Reaching her mothers' apartment, she unlocked it with her palmprint and slipped quietly inside. Another touch of her hand to the panel on the doorframe closed and locked it, allowing her to sag back against the stout plexsteel and relax.
Spine pressed to the door, she sighed slowly in relief, shutting out everything but the sound of her own heart. Then snapped her eyes open at the sound of rapid footsteps. Her younger brother, Fyfer, flew at her, all grins and dark curls bouncing. He had gained a couple centimeters since she had last seen him, but still stood shorter than her by nearly a full head. Tall for a Sanctuarian, but not abnormally so. Then again, their fellow colonists wouldn't reattain anything close to her semilofty height for at least another five, six generations, if not more.
"Hey, gorgeous, long time no see!" Giving her a hug and a peck on the cheek, he reached up and ruffled her locks. Ia put up with it for a moment, then pushed him back and ruffled his own hair with her free hand, mindful of the mug carried in the other one. Good-natured, Fyfer stepped free and bowed, gesturing beyond the entry hall. "Welcome to our humble new abode, sister dear. Allow me to show you around."
She waved him off. This new, underground residence was a far cry from the cramped two-bedroom home her parents had known for most of their adult lives, but it was familiar from the times she had watched her family move through its rooms as she studied the future possibilities of their lives. "Already seen it. I'm going back to the storage hall to make more wreaths."
"Still not going to tell us how you're doing it, hm?" her younger brother asked, following her down the hall that led past common and private rooms alike. Fyfer tsked under his breath. "Selfish of you. Crysium's the hardest known substance. Just think of the armor we could make with it!"
"Just think of the I'm-not-telling-you-whats I could make with it," she quipped back. "You're lucky I brought you snow all the way from Earth."
Fyfer snorted. "As if Mom or Ma would let us make snow cones out of it. Me, I want to _taste_ the Motherworld. Bring me my own chunk next time, will you?"
Chuckling, Ia shook her head. Palming open the lock on the door leading to the warehouse, she stepped inside. Strange blocks and lines of shadow and light illuminated the far end of the room. Fyfer followed her inside, hitting the control panel for the lights. Like most forms of illumination used by Terrans not actually living on Earth, the lamps lit up in the same spectrum of colors as Sol, the parent star for their species.
The shadows had been cast by the stacks and stacks of bead-filled boxes Ia had received, altered, and sent back over the last year and a half. The angled lines of light came from the conifer-like sprays of crysium, painstakingly broken off their rock outcrops on the planet's surface and transported down here by Thorne's order.
Each spray was at least two meters tall and a meter and a half wide at its base; the fifty or more shafts that made up the limbs of the spray varied in thickness from the diameter of her biceps to smaller than the span of her wrist. In Standard gravity, they would have weighed two tonnes on average. On Sanctuary, they weighed over three times that—and these were merely small ones, the kind that were relatively easy to transport. There were sprays on the surface that were easily eight meters tall and five to six in diameter, and crystalline behemoths that were even larger.
Draining her cup, Ia set it on top of a stack of emptied crates. A flick of her mind unlocked one of the waiting, full boxes. Like a swarm of peach-tinted glass bees, the beads inside swirled up and soared out of the box, following her telekinetically. Stopping in front of one of the sprays, Ia reached up and caressed the shaft; it sang faintly in the back of her mind, boosting and amplifying her abilities, though not by a lot. Unshaped, raw, natural crysium could only do so much.
Bead-bees floating at her shoulder, hands on the nearest shaft, Ia closed her eyes and flipped her mind down, in, and out, landing on the timeplains. Instinct and habit guided her, lifting the banks between the right and wrong creeks, altering the flow of the timestreams, the life-streams of the men and women who lived on this world. Instinct and habit merged crystal to crystal, some of it tainted with her own blood, the majority of it tainted with the discarded matter of passing Meddlers. Instinct and habit, practiced in her dreams, guided hand and mind into shaping chunks of the material in a complex process achieved without conscious thought.
But that was alright. Like water running toward the sea, it was important for everything she wanted to drain into the right bay. Pick the wrong part of the coast, and one could end up mired in an aimless swamp or be sucked into the mud of a tidal flat. Ia didn't completely understand her abilities; they operated as much by instinct as by design sometimes. But she did have faith in them, and that meant letting the instincts of her mind shape the rings more than the conscious directing of her thoughts.
Something hit her. Blinking out of her trance, Ia eyed the twisted lump of translucent mineral in her hands. Hunger struck her in the next second. In the third moment, something struck her again. Turning, she looked as it flopped to the floor. A shoe. Nearby was another, matching one.
"Do I have your attention now?" Fyfer asked her. He wiggled his sock-clad toes and grinned.
Before she could respond, Ia heard the quiet beep of the comm embedded in her arm unit. Sighing, she made sure her headset was still secured over her ear, then thumbed the audio channel open. _"Ia. Go."_
_"Ia? About time! Captain, where are you?"_ she heard Harper demand.
From the stacks of rings and absence of two whole sprays, she had been concentrating for some time. Her thoughts were scattered, her concentration fragmented. Wrinkling her nose at the rumpled lump of crystal in her hand, Ia replied, _"Busy. What's the emergency?"_
_"3rd Platoon B Beta, Privates Gwen Yé and Solomon Sutrara, were arrested just under an hour ago on charges of 'conspiring to commit heresy'...whatever that means,"_ Harper told her. _"Sutrara was smart enough to record all but the first minute on his arm unit, and managed to download it to the ship before they demanded he remove it."_
That wasn't a move Ia had expected. _"I did lecture the troops about not going into any churches, right?"_
_"You did, but apparently they were outside the church, on the plexcrete sidewalk near one of the side doors."_
_"Tell the Peacekeepers that first off, RCS 1107.6 statesclearly that all walkways, sidewalks, and so forth are public easements, and not Church property. Secondly, all military personnel who commit crimes are to be remanded into military custody for military justice,_ not _civilian custody or justice."_ She paused a moment, thinking, then nodded. _"...Right. Send down 2nd Platoon A Gamma, Privates Ateah and Sousa—there should be a box in the storage locker off the officers' mess with black Military Peacekeeper armbands in it._
_"Slap a pair on those two and send 'em down to pick up Yé and Sutrara. Inform the Sanctuarian Peacekeepers that the matter will be looked into and the soldiers punished accordingly, as the Space Force takes religious rights and freedoms seriously. I can't call up the relevant sections and paragraphs at the moment. Consult with Sadneczek on that,"_ she instructed him. _"Then call the jail, let them know you're sending two MPs, and that you'd like copies of all depositions for their military tribunal."_
_"Sir, yes, sir. I hope this works,"_ Harper added. _"What's the ETA on your return to the ship?"_
Ia eyed the crystalline trees still awaiting her. She had only transformed a fraction of what she needed to make, maybe a couple percent. _"I'm not sure if I'm going to be able to leave anytime soon, at this rate..."_
"Do I have to hit you again with a shoe?" Fyfer asked her, recapturing her attention. He spread his hands, rolling his eyes. "You're the commanding officer of a slagging _ship_ , Sis! One with _carte blanche_. Pack up all the crysium and blood beads you want, go do it on board the _Hellfire_ at your leisure, and send the rest back to us with the Afaso. _They'll_ get the packages through."
She checked the timestreams. It would crowd her schedule further on the ship but free up a bit more time here and now. Ia considered his words, then nodded. _"Harper, I'm going to need a series of shuttle drops in a couple days to a site east of the mountains. We'll need a couple of the holds set aside for bulky, heavy cargos. And tell Grizzle to order a couple hundred palm-locked cargo crates—the collapsible kind; otherwise, we won't have the storage space. But get on those MP rescues fast. And make_ sure _you get those arm units back. I don't want Church officials getting their hands on military tech—intimidate themwith threats of arresting them for theft of Terran government property if you don't get them back immediately."_
_"Aye, Captain. Anything else?"_ he asked.
_"Yeah, call me in an hour if the extraction doesn't work. You have an eighty-five percent chance that it should, though."_
_"Understood. Harper out."_
Shutting off the comm link, she glanced at her brother. He shrugged and gestured at his shoes. "Can I put them back on, now?"
"Yeah." Nudging them with her foot, she tumbled them his way physically, rather than using her telekinesis. Using a combination of her precognition, her electrokinesis, and her telekinesis to make all these wreaths was exhausting enough. She didn't need to waste the mental energy when a flick of her foot would work. "You may need to get my attention again in about an hour, so stick around. And in two hours, I have to head to my psychic ethics review. I don't dare miss that. It's the last one I'll get before heading off to war."
Her brother bowed dramatically, palms pressed together like a djinn. A lock of dark brown hair flopped over his brow as he did so. "I shall be your personal shoe-throwing alarm clock, O Prophet. _Ia'n sud'dha_ , I live to serve!"
The mock-dirty look she gave him only provoked a laugh from her brother.
JANUARY 21, 2496 T.S.
Crouched beside Bei Ninh, former Sharpshooter and bronze medalist, Ia could smell his sweat. Same with Jane Loewen, though her odor wasn't quite as rank. Neither of them seemed happy at having to move without the benefit of the deactivated gravity weaves wrapped around their bodies, never mind skulk through the shadows of the half-built Sacred Cathedral of the Light and the Truth on an excessively heavy world. The exertion required for moving silently as well as swiftly expressed itself in tired muscles and sweat-soaked skin.
The same could be said for Ninh's wife, Bagha, and her temporary teammate for this mission, William Xavine. Like Loewen, Xavine was a former Troubleshooter, and something of an expert in surveillance gear. Each of the two crouched by their Sharpshooter partner, special cartridges in hand. Kneeling between the pair, Helstead peered through a set of enhancement goggles at the half-built walls. Finally, she nodded and whispered coordinates.
The two Sharpshooters, both wearing goggles of their own, nodded and lifted their guns. Helstead's job was to identify the spots in the architecture that her compatriot, Lieutenant Rico, had selected, with some help from Ia. Elbows braced on a packing crate, they each took aim through their scopes and fired. The soft _phunt_ of the air guns was echoed a second or so later by the faint _splat_ of the gel-based projectiles hitting their targets. The two Troubleshooters quickly consulted their palm scanners. Xavine nodded, but Loewen shook her head.
_"It broke on impact,"_ she warned Ninh, and handed him another cartridge. _"Shoot again."_
Nodding, he replaced the expended shell with the fresh one, aimed carefully once it was loaded, and fired. Their movements were subtle, difficult to see in the darkened interior of the cathedral. The _hiss_ and _thwap_ of the payload's being delivered was more blatant than their careful moves. Ia did see his satisfied nod, though. He knew he'd made the shot.
_"Perfect."_ Closing her scanner, Loewen tucked it back into her bag, agreeing with him.
_"Next target,"_ Ia whispered, and led the way out onto the main floor. Like all man-made walking surfaces on Sanctuary, it was made from plexcrete, the odd amalgamation of several rubbery substances that cushioned impacts. It was still possible for someone of her height to badly injure her head if she should fall wrong, but the odds of actually cracking that skull hard enough to break it or the brain encased inside were survivably low.
Plexcrete came in several varieties, many of which were patterned to look like various long-lasting shades of granite. The Church Elders had spared no expense; this particular plexcrete floor had been laid out in an elaborate pattern of dark and light diamonds, lines, squares, triangles and more. In the daylight, it would appear polished, unscuffed as yet by the passage of thousands of worshipping feet. It would also glow, she knew, with bright golds and rich blues, blood reds, silvery greys, and regal greens once daylight touched it.
The scaffolding along the southern wall would eventually be replaced by a great stained-glass window marked with a similar pattern, carefully positioned so that when the planet aligned just right with the local star at the solstices and equinoxes, the sunlight shining through at noon would match the colorful patterns at certain points along the floor. Side windows to the east and west, nestled between the old-fashioned flying buttresses, would show colors that would overlap and blend. Between the floor and the windows, the carvings on the walls and the paintings on the ceiling, the Sacred Cathedral of the Light and the Truth, Our Blessed Mother, would be the single most spectacular structure on the whole planet, and remain so for at least two hundred years.
Between then and now, Ia wanted her fellow Free World Colonists to have an ear to the ground, inside. Literally. Hurrying up the steps of the altar dais, she crouched and pulled out a pair of palm-sized miniature crowbars. Inserting them carefully between two slabs of colored plexcrete, she pried the pieces apart, muscles flexing hard. Loewen reached her first. Fishing out the next piece of spying equipment, she dropped it into the small gap.
Freeing her tools, Ia checked her grip on the electronic surveillance supposedly keeping this site safe. Everything was still secure, showing nothing out of the ordinary. She waited for Loewen to peer into the crack and nod, confirming the device was properly placed, then scuttled across the dais to the area that would one day house the seats of the Church Elders as they sat and listened to the various services.
This time, Xavine reached her first and dropped a surveillance pickup into the rubbery crack she made. Normally a fancy floor might be installed last to prevent it from being damaged during construction. On Sanctuary, that cushioning was needed as soon as it could be laid down, right after the foundation was secured.
She pried apart a third spot, and a fourth, picking seemingly random locations that would bring the most benefit in eavesdropping in the years ahead. Beckoning the others onward, she led the way toward a hallway—no door, just yet—which in turn led to a set of steps. The four lightworlders grunted in the effort to try climbing them. Giving up, Loewen dropped to her knees halfway up, her shaking head nothing more than a subtle change in the shadows of the stairwell. Ia moved back down to help her up the stairs.
_"Almost there,"_ Ia whispered in encouragement. _"Just seven more to place, then we can escape."_
The Troubleshooter panted, gritted her teeth, and let Ia help her back into motion.
_"I'm not sure my knees can take much more of this, sir,"_ Ninh whispered back, bracing himself with one gloved hand on the wall as they passed. Most of their clothes were common civilian garb, though they had taken the precaution of donning gloves and caps. _"If it's 2Gs back home when you climb a set of stairs, that's over 6Gs here per step."_
His wife poked him in the ribs. She breathed heavily with each step but managed to pass him all the same. _"Fire in the hole, Ninh. Fire in the hole..."_
He stuck out his tongue but struggled upward in her wake. It wasn't the fact that they were working in three times the gravity that was hard. It was the fact that climbing stairs roughly doubled the forces at work, and certainly doubled the effort. Ia let them rest at the top of the stairs but only for a minute. More than that, and they wouldn't want to keep moving. Urging them along the plexcrete-padded hallway, she nodded at the first of three rooms.
The left-hand wall in this one had a large window opening overlooking the altar, which would eventually be covered in stained glass, obscuring the fact that the room would become the Grand Prelate's office. Without the window, though, they were free to aim their air guns at the sculpted columns already set in place. After that was done, she guided them into dropping two more packets between the rubbery floor tiles.
Both the gels and the card-thin pickups worked on kinetic-energy principles. The noises reverberating through them would empower them, permitting them to last for a good fifty years, if not longer. Their broadcast range would be short, barely two hundred meters up and outward, and only fifty or so down through solid ground. But that would be just enough for the FWC's transceivers to pick up and relay their scans farther down the line.
The next two offices were equally rough-walled, if smooth-floored. One would be the seat of the financial officer for the Church, the other the security officer. Both were vital for controlling the pacing of the next two centuries. Taking their time, doing it right, took another fifteen minutes. As soon as the last pickup was dropped in place, Ia tucked her crowbars away. _"There we go. Now to get out through the back door unseen. We're slightly ahead of schedule, but not enough to evade the security guards any other route. This way."_
The "back door" wasn't a door, per se, but rather a set of scaffolding leading off the back of the cathedral. Accessed through a window opening, it was a short climb down the ladders to the ground. Short, but exhausting for her companions. Ia let them rest twice more before they slipped back through the slight gap in the chain-link fence guarding the site. From there, she urged them in quiet murmurs to keep going until they were physically out of sight.
_"...Now, sir?"_ Xavine finally asked, resting against the side of a building a block away. All four lightworlders were panting heavily from their exertions. Helstead also rested against the wall, though she recovered her breath faster than the others.
"Now," Ia agreed, speaking in a normal tone, if still quietly. Relieved, the other five turned their gravity weaves back on, to the lowest setting that could tolerate the presence of other weaves nearby and still counteract a good chunk of the gravity. Relieved for a different reason, she gently released her grip on the low-light cameras, infrared detectors, and other sensors scattered around the large construction zone, though she didn't sag against the wall like her crewmates had.
Finally adapted to her home after several days of more or less living on the surface while her ship stayed parked in orbit, Ia removed her knit cap, worn to hide her too-pale, damp hair. The others had sweated from the gravity, while she had sweated from trying to electrokinetically hide their actions from the construction site's surveillance equipment. Doing so without also tripping the KI sensors placed around the Cathedral hadn't been easy.
The Church Elders hated psychic abilities. The mental exercises required to discipline a psi's mind lent strength to that mind, strength and resistance to outside influences. Strong minds were not easily swayed minds, and that meant rebellious minds, according to internal Church doctrine. The last thing they wanted was a strong telepath or a clairvoyant spying on them.
"Well. That was fun," Helstead finally said, tucking her gloves into her shirt pocket. "Back to the pub?"
"Back to the pub," Ninh agreed. "I could use a stiff drink."
"I think I'm going to _be_ stiff," Loewen countered, flexing her back. The push-pull force of her weave's field nudged into Ia, who staggered sideways and swallowed against the wobble in her inner ear. Loewen grimaced and straightened up again. "Sorry, sir."
Ia waved it off. "It happens. You heard Helstead. Back to the pub, meioas. Drinks are on me. One lightly alcoholic, the rest non. You don't want to stagger in this gravity."
The lightworlders groaned but pushed away from the wall, heading up the street. It was late, but the pub selected as their return point was the kind open all night. As they walked, Helstead moved up beside Ia.
"So, Captain...sorry, Ia," she amended, dropping the rank. "You said you couldn't just waltz in and use your own gifts in a solo mission because manipulating everything would trigger KI sensors. I get that. It's a perfectly valid concern, especially at your strength. And I can understand why you'd want to bug the offices. But why the main sanctuary? Nothing's going to be said out there that's particularly sensitive."
"Yeah, why _did_ we do that? I'll follow your orders, sir," Xavine muttered, "but, well, I'm not too comfortable with targeting a religion. It's far too easy to turn offended members of a faith into violently righteous fanatics," he added under his breath. "So I'm not comfortable with any of what we did. I _did_ it, but..."
Loewen shook her head, though not in a disagreeing way. "Welcome to the military, meioa; if it doesn't break the most common laws of ethics and sentients, you do what you're told. But _I'd_ like to know, too, Cap...uh, Ia. If it's okay for us to ask?"
"It's okay. And it's simple. Propaganda. Whatever the Church Prelates preach from the pulpit will be taken as gospel truth by the topside masses," Ia told them. "Some of it actually will be the truth, though much of it will be distorted and edited. In order to fight that propaganda, the other half will need to know what's being said so they can separate truth from falsehood."
"I still don't get why you can't just help these people to win the fight right away," Bagha said. She shook her head. "Two hundred years of civil war sounds like an awful lot of deaths."
"Seventy-five percent of it will be a cold war, if not more, so it's not as many deaths as you'd think," Ia said. She saw the doubt in their eyes and shook her head. "I _know_ how you feel. A large part of me doesn't like it, either. But if I'm given a choice between three people dying now, or three million people dying later—for-sure-dying, not just maybe-dying—then I'll take the three people now and do it with my own hands if I must.
"If I have to be screamed at in my conscience for the rest of my life, I'd rather it were by the ghosts of just those three, and not the three million," she finished quietly. Honestly.
"Well, we're talking about a lot more deaths than just three, sir," Xavine argued, as they approached the back door of the pub. It was owned and visited by friends of the FWC, which meant everyone inside would be willing to testify that the six off-worlders—Ia counted, now that she was in the military—had been in the back room all night, on the slim chance their proximity to the Cathedral had been noticed.
"And I'm talking about a lot more deaths than just three million. Here's the pub, so this subject is dropped. Pick a new one. Like the fact that the first round of drinks are on me," Ia stated, lifting her chin at the far door as they entered, the one that led to the front of the tavern. "You've earned it."
Helstead wrinkled her nose. "I think not. Every time I try to drink with a gravity weave on, the damned water in the glass sloshes to the left and tries to go up my nose."
"Then stop drinking with your left hand," Xavine teased her.
Helstead pointed at him. "Watch it, meioa; we're not always gonna be off duty, you know."
Wisely, he ducked behind Ia, who merely shook her head. Opening the bag slung over her shoulder, she held it out. "Give me the transceivers. I'll get them to the right people."
Xavine pulled his out. He hesitated before placing the first palm-sized unit inside, though. "This _is_ approved of by the Command Staff, right?"
"Of course it is," Ia told him. "Everything I do has been approved."
Technically, her words were true. Technically, she could say them with a straight face. Technically...she was abusing her _carte blanche_. But Xavine and the others didn't know that.
Nodding, he deposited the other transceivers into her bag. Loewen did the same, then hooked her arm around his and dragged him off to the front room for a snack run. They staggered a bit as their weaves interacted, but weren't too fazed. Ia detoured into the kitchen halfway down the hall. Turning down the heat on his cooking unit, the short-order cook accepted the bag with a silent nod of thanks and took it into the basement.
There was a door down there that led to an old Terran-installed bunker, which in turn led down into the lava tunnels the Free World Colonists were using as their new home. On the other side of the first door, a bored agent of their underground government waited for the promised bag. Those devices were the top-of-the-line in Terran military espionage. She had already checked the timestreams, and knew the Church had nothing on hand that could detect them.
As for the Command Staff finding out about this little mission, by the time they did, Ia hoped to have far more important battles under way to serve as a distraction, and as proof of the justification behind her actions.
_Or to put it another way, it's a simple case of Jack's Law #213,_ she thought, amused. _When all else fails, cloud the issue with facts._
JANUARY 23, 2496 T.S.
The last of the cargo crates hummed downward out of sight, thanks to the floor lift. Satisfied the supplies would be safe, Ia turned away. She gave her attention to the five waiting members of her family instead. First was her biomother, Amelia. Seen in the clear light of day rather than the dimmer light found underground, a lot more grey salted her mother's curls than Ia remembered seeing before. Worry lines creased Amelia's freckled brow, but her arms were still strong and warm.
"I hate this, _gataki mou_ ," her mother muttered into her chest, heavyworlder strong but heavyworlder short. "Just the one more time, right?"
"Just once more, unless things go seriously wrong," Ia promised. Patting Amelia on the back, she released her and turned to her other mother, Aurelia.
The slightly taller woman squeezed her hard, then stepped back, looking up at her stepdaughter's face. "You go out there, _kardia mou_ ," she ordered Ia, "and you kick frogtopus asteroid, you hear me? Laser-fried calamari as far as the eye can see."
Fyfer wrinkled his nose. "Eww! Ma!"
Aurelia spread her hands, shrugging. "What? I didn't say she had to _eat_ it."
Wrapping an arm around her younger brother, Ia ignored the stares of the soldiers lining up near the main entrance to the spaceport warehouse. Scrubbing her knuckles over his scalp, she mussed his carefully styled hair one last time, then hugged him hard enough to lift him a few centimeters off the ground. She got painfully pinched for her troubles, but laughed, gently letting him drop back down.
Next in line was Rabbit, not a blood relative, unless one counted the fact she was in a complicated relationship with both of Ia's brothers. Rabbit wasn't her birth name, but rather the nickname she had adopted; her two front teeth, prominent in her plain, round face, made the reason self-evident. Her smile dispelled any illusions about a lack of beauty, though.
Kneeling in front of her, Ia held out her arms; even for a heavyworlder, Rabbit was short. She was also pregnant, five months along and having the child naturally rather than via a wombpod. That would put stress on her stocky frame in this gravity, but Ia knew her old school friend could handle it.
She did, however, caution the short woman. "You make sure you start putting your feet up three times a day, you hear? I'll make Thorne literally sweep you off your feet if you don't. You don't want him to throw out his back, now do you?"
Rabbit pushed out her lower lip. She had a brilliant mind behind those doe brown eyes and that moon-round face, but most people severely underestimated her because of her child-like size. The short woman used that illusion ruthlessly at times. Even her own parents, Church officials, thought she was still a little girl at heart. Ia wasn't fooled by that pout and pointed her finger in silent warning. That earned her a rough sigh.
"Fine," Rabbit said, rolling her eyes. "Go ahead and leave me. Just because you're sticking me with managing this whole mess doesn't mean I'm going to forgive you for it."
"Hey, I conned my brothers into doing it, too," Ia said. She hugged her one more time, then nodded downward. "Take care of yourself, so you can take care of him."
"And I told _you_ I didn't wanna know the gender," Rabbit protested. "Stupid know-it-all prophets."
"Well, I'm not going to call my niece or nephew an 'it,'" Ia countered. " _Use_ the people you've gathered. Share the responsibility. You're the chief Director, so direct the others to do what needs to be done. And put your feet up."
Amelia leaned over and hugged the younger woman. "I'll make sure she does—and if she doesn't, I'll sic my wife on her."
That earned them another pout and mutter from Rabbit. "Oh, now you're just being mean!"
The others laughed. Rising, Ia hugged the last person in the quintet, her half-twin Thorne. The only member of the family taller than her, he had the dusky complexion and dark hair of his mother Aurelia, and the same square chin and almond eyes as their shared father. Unlike Ia, he didn't have any active psychic abilities, just latent ones waiting to be passed on to his offspring. He had practiced the same mental disciplines as a psi, though, which meant Ia could hug him longer than the others.
Physical contact increased the chance of her gifts triggering, which meant she rarely touched anyone for long, for her sanity's sake. But Thorne's future she didn't fear foreseeing. She knew every inch of his timestream possibilities. He was the solid rock in her life, even if he wasn't close enough to be a physical touchstone these days, save for right now. For a moment, Ia allowed herself a few seconds of rest in his arms.
Time pressed in on her, though. Church-loyal officials were closing in on the warehouse, still upset that Ia and her troops had kept them out for so long. Nodding, she signaled him to release her. Pulling her shoulders square and her chin level, she stepped back, shrugging mentally back into the uniform covering her muscles.
Once again she was Captain Ia, ex-Marine, ex-Navy, and currently Special Forces. A Terran soldier with a deadly-efficient reputation that had earned her the nickname of Bloody Mary. She was no longer Iantha Iulia Quentin-Jones, daughter and friend and fellow civilian.
"Directors," she stated, nodding politely to her two brothers and quasi sister-in-law, the unlikely three-headed leadership team for a massive underground resistance movement that was still only halfway started. "Meioa-es," she added, using the feminine form of the honorific on her parents. Off to the side, the lift was coming back up into view, all crates, boxes, and pallets removed by the men and women working in the tunnels below. "Take care of yourselves. We'll be withdrawing the perimeter in two minutes. Peacekeepers will be here in three. I suggest you get out of sight, meioas, for your own safety."
Nodding, Thorne herded the others toward the lift. Turning on her heel, Ia strode toward the main entrance. The door had been rolled down halfway, just high enough for the tallest members of the 2nd Platoon to pass under it without ducking. Wrapped in his own purple web-work, Lieutenant Spyder smiled and saluted her when she came near.
"Nice t' see y' got a soft'n'squishy side, Cap'n," he joked, hand raised formally to his brow.
"Oh, really? Well, my soft'n'squishy side just turned as hard and dangerous as a rogue asteroid again," she shot back dryly, returning the salute. Behind her, beige-clad Chong-Wuu workers used stevedore suits to roll the plexcrete mats back into place, covering up the lidded hole where the hidden lift was once again descending. In front of her stood twenty men and women of the 2nd Platoon. "Lieutenant, run the checklist and pull in all remaining personnel. Six minutes to takeoff."
"Sir, yes, sir," he agreed, turning crisply on his heel. "You 'eard th' Cap'n! Set yer 'Daptive Gravimetrics to Low, an' board th' shuttle. Do _not_ double-time it," he added, before activating his arm unit. _"Spyder to D Beta through D Epsilon, pull out. Setcher weaves to Low, an' retreat to th' shuttle in extraction order, noice an' leisurely. You got three minutes t' move, yakkos."_
Ia followed them outside, her attention on the half dozen cars approaching in the distance. An uncomfortable tingle along her nerves was her only warning. Turning back to the men and women, she hurried up between the ranks, gritting her teeth against the conflicting pulls of so many gravity weaves in close proximity.
Catching the tall bulk of Private First Class C. J. Siano from behind, she ducked as his head flung back, almost cracking into her skull. His deep voice hollered almost painfully loud, caught up in yet another random damned Fire Girl attack. Staggering sideways under his limp weight, she guided his body awkwardly to the ground in a semicontrolled collapse. One that ended up with her bruising her tailbone on the stained grey plexcrete of the tarmac and her leg trapped under the power pack strapped to his back, but which didn't end with him hitting his head.
The others scattered, then surged forward to help them up, concerned by his wordless screams and thrashing limbs. Ia quickly shook her head, teeth gritted against the urge to yell as well. "—Don't touch!"
Even through the lumps of his weave and the mottled fabric of his uniform, she could feel the press of time trying to sweep her downstream. If the others tried to help the two of them up, it would spread, and that would be bad. Very bad, almost as bad as a mind-quake. Too many lives, too many possibilities, would overwhelm her. As it was, she had to master herself before she could dampen the visions in Siano's mind.
"You 'eard th' Cap'n! Get on th' ship," Spyder ordered the others as she worked.
Siano came back to himself with a gasp and a shudder, eyes blinking wide. "Wh-wha...? Oh, _hell_...Ransil wasn't kidding when...when he said that hits like a _shakking_ Battle Platform!"
"Deep breaths, Private. Try to relax; we'll be off-world soon," she said, patting his shoulder. Tugging her leg free, she helped him roll over onto his hands and knees. Even with the weave wreaking havoc on her sense of up and down, counteracting some of the forces pulling on his frame, his tall, somewhat muscular bulk needed her help just to get off the ground. Turning up the gain on the gravimetrics would have aided him but would've churned the contents of her stomach from sheer proximity.
Thankfully, once he was upright, the weave fields aligned enough to allow him to walk up the ramp under his own power, joining the others. Panting, Ia rested where she stood for a moment, then dusted off her camouflage trousers. As she did so, the ground cars pulled to a stop near the shuttle.
The last pair of soldiers came walking up from the side, rifles carried loosely across their bodies, muzzles pointed safely at the ground, though she knew they could be snapped up into position at a moment's notice. As they came near, the two privates gave the cars a curious look, then their CO. Ia held up her fingers, then flicked them out to either side, signing for them to guard and wait. Nodding, York and Clairmont took up position on either side of the ramp.
Ia waited politely in a modified Parade Rest as the familiar figure of Customs Officer Larkins emerged from the lead car. His smug smile amused her. He thought he knew something she didn't know. Rather than show it, she kept her expression neutrally polite.
Stopping in front of her, he flapped a plexi printout. "As you can see, Captain Ia, I have _permission_ from Vice Commodore Brenya Attinks to search your so-called 'emergency war supplies,' particularly as this is the first _she_ has heard about their arrival—you _do_ know who the Vice Commodore is, don't you?"
"Vice Commodore Attinks is the commanding officer in charge of this sector of Terran-patrolled space," Ia stated, her tone mild. "She is, however, from the TUPSF Branch Navy, and is merely a vice commodore in rank. I am from the TUPSF Branch Special Forces. Orders delivered by anyone of any rank lower than lieutenant general or vice admiral from outside the Branch Special Forces carries zero weight within my Branch. Unless, of course, your little printout includes permission and instructions from that level of authorization above her. Yes? No?...I think not, Officer Larkins, or you would have listed a higher authority instead."
Larkins reddened. Flipping the printout, he scanned it quickly. Ia didn't bother to give him time to see if it did or not.
"More to the point, the Special Forces' supply requirements are not the same as the Navy's. She wouldn't even know what to look for. But I am not an unreasonable officer, and it is quite evident you have gone to great lengths to have a peek at our business," she added, gesturing at the large building to her left. "My orders were to secure our cargo in military storage on the planet Sanctuary. Now that it has been properly secured, my authority in this matter is at an end. Responsibility for keeping it safe and sound now rests in the hands of the duly Charter-registered colonists of this world who have agreed by said Charter and contract to guard those supplies. At least, until such time as we need it, of course."
Brow pinching in a disbelieving frown, Larkins looked up at her, then over at the warehouse. "...That's it? We're free to inspect it?"
"You are now free to inspect the warehouse," Ia agreed, gesturing again at the building. "I followed my orders to the letter, meioa. My obligation is now discharged. In fact, we are packed up and ready to go. You are therefore free to inspect that warehouse at any time you like. Our shuttles are still off-limits, of course."
"Of course..." He started to move toward the building, then checked himself. "One more thing, Captain. The two soldiers who were arrested. Vice Commodore Attinks has agreed that she and her two senior-most officers can serve on their hearing board, since you insisted so strongly that they be tried in a military court and not a civilian one. Her battleship, the TUPSF _Victory Dance VI_ , will be insystem in the next few hours."
"I look forward to chatting with her about it." Behind her back, Ia flicked her fingers, hand-signaling for York and Clairmont to mount the ramp. With her left hand, she gestured at the warehouse one more time. "In the meantime, by all means, please inspect the premises. I'm sure your warrant is perfectly good for requesting that Chong-Wuu Stevedores open up the building for you. But you'd better hurry. I think they're locking up for the day."
At those words, Larkins moved away a few steps, squinting at the building. Ia backed up, stepping onto the ramp. By the time he glanced back, her fingers were already reaching for the ramp controls overhead.
"—Hey!" he protested.
"Do excuse me, Officer Larkins," she called out, crouching to keep in sight as the ramp started to shut. "I must go now, as I have a lot of Salik to slay. Good-bye!"
Backing up, she staggered a little as the ramp shut and the overhead weave dialed up, reducing the gravity in the cabin. Turning around, Ia waded through the mix of bodies, some strapping into their seats, others passing their ammunition-stripped weapons to Sergeant Santori and Lieutenant Spyder for storage in the ammunition crates stacked near the front of the cargo hold.
"Get ready for liftoff," she warned them. "Don't open the ship, even if they knock politely." Ducking into the cockpit, she slid into the copilot's seat next to Yeoman Yamasuka. "Run the checklist and take off the moment everything's greenlit. And unlock the comm on my side."
Yamasuka nodded and touched the requested controls. "Sir, yes, sir. Out of curiosity, Captain, who are you calling?"
"A certain vice commodore, about dropping the silliest set of trumped-up charges I've yet to hear on my homeworld. Though I'm afraid they won't be the last, nor the worst. I'm also calling to warn her how bad it's getting down here," Ia added, strapping the harness around her body. "I don't think it'll be safe for the Terran military to land on the surface anymore. The space station will have to do from now on."
"Did we just drop some _shova_ in their caf', sir?" the woman at her side asked.
She shook her head. "No. Half the locals would be this crazy even if we weren't here—I'd blame the gravity draining the blood from their heads, but most of 'em have adapted by now. Let's get the hell out of here. Whenever we're ready, Yeoman?"
"Aye, sir," Yamasuka agreed, checking first her telltales for shuttle readiness, then a secondary screen displaying the bodies still moving around in the cargo hold. "Executing the 'get the hell out of here' maneuver in...approximately one minute twenty seconds, sir."
# CHAPTER 6
_I only saw my family once more after that. At least, in person. Plenty of vid-calls, and of course always in the timestreams. I love them, but I didn't try to go back because I had other things on my mind. Other responsibilities. We had to begin our prewar hunt for the anti-psi machines. Of course, the vast majority of those activities were highly classified. It was just my crew, Admiral Genibes, and other select members of the Command Staff who knew something of what we were up to, back then. You're lucky I can mention it even now. If I or my crew had told anyone what we were up to, the Salik would've been more prepared._
_More prepared...yeah. It's rather scary to think of the Salik being more prepared._
_Were we prepared to face them? On many levels, yes. On some levels, no. As a whole...it averaged out. The Space Force was barely ready for them. Part of that readiness was due to my disruption of their version of the Command Staff the year before the war started. It bought us enough time to increase recruitment drives, increase production of vital tools, and increase stockpiles of needed supplies._
_You ask me if my crew was prepared? The answer has to be both yes and no. Prepared in the sense of trained to handle the ship and the equipment placed intheir hands? Yes, for I made sure they were well trained. Ready? I don't think all that many of them would've said so, back then._
_~Ia_
FEBRUARY 2, 2496 T.S.
SIC TRANSIT
"Are they ready for this?" Chaplain Benjamin asked her guest. She studied the woman seated across from her over the rim of her cup.
Ia yawned and nodded. "...Ready enough. They'll make a few mistakes, but I've picked a soft target for the first engagement."
"Something that could build their confidence rather than burst it, eh?" Bennie asked.
Ia nodded again. She lifted her cup to sip from it, but the caf' that should have been inside was gone. She didn't remember drinking it. _Guess I'm more tired than I thought._ Sighing, she sagged back into her seat and closed her eyes, letting the empty mug rest on her knee.
Bennie's next question was pointed. "Are _you_ ready for this?"
Rolling her head along the cushioned back of the chair, Ia flopped a hand. Too many long nights of trying to get more crysium wreaths and prophecies made and not enough sleep left her groggy in the morning. It didn't help that once every three days for the last ten, she had stretched her schedule out by eight full hours. The effort of trying to keep an eye on each of the three duty watches, to be _there_ for every single crew member, was wearing her down.
"Maybe I should've given you decaf'," Bennie muttered, eyeing her.
"Maybe I should've court-martialed you," Ia quipped back. Sighing, she sat up again and scrubbed her face. "I'll be fine, Bennie. Once I get a little food in me, and maybe a jolt of electricity, I'll be able to last until the adrenaline takes over."
"And when the adrenaline does take over, what then?" her friend asked.
"Then it'll be all fun and games until the Salik lose one of their eyes." Eyeing the empty cup, she clipped it into the holder on the coffee table between them. Bennie's office was tastefully decorated, mostly in soothing colors, but with a few bright accents found in the cushions snapped to the seats and the paintings bolted to the walls. She focused on the scarlets and blues, trying to wake herself up.
Her lack of sleep wasn't nearly as bad as at other points in her life, but it wasn't the long hours. It was constantly projecting confidence and competence every time someone else was in the room.
The chaplain's job, as a psychologist as well as a spiritualist, was to keep an eye on their CO's mental health. That meant showing Bennie she was still Human. Literally, since it was now known that half of her was not. With Bennie, here in private, she could let down her guard for a few minutes. Unfortunately, she couldn't stop Time, and that meant she could only do it for a few more moments right now.
Her gaze fell on a painting of a bright yellow flower, more abstract than real, though she suspected it was based on one from the M-class planet Dabin. The shape also reminded her of the pentagonal installation they were heading toward.
"So what happens after they lose an eye, and the adrenaline rush fades?" the redheaded woman asked, her tone amused.
"...I start snoring?" she quipped, glancing back at her friend. Bennie snorted, covering her nose and mouth to muffle her laugh. Pushing to her feet, Ia lifted her chin at the door. "Snicker all you want. I'm going to go get something to eat, then hit the bridge."
"How soon to combat?" Bennie asked her, rising as well.
"A little over half an hour. Including whatever it takes to _find_ the slagging place. I'm good at judging and aiming, but we're still dealing with interstellar distances, and this place is deep in the black." Lifting a hand in farewell, Ia left the cabin.
Bennie's office was one deck above her own, if farther back in the ship. It was also not far from the aft galley, though it did mean having to wait for the airlocks to cycle her through the section seal. Aft and fore were the main kitchens for preparing large meals for the entire crew; bow, stern, and amid were more for creating snacks. Ia herself could cook just enough to feed others without any accidental food poisonings, but she hadn't inherited any spurious chef genes from her highly talented mother. She didn't want a snack from the bridge galley, and she didn't want to have to put something together for herself, so the aft galley it was.
The timing was between meal shifts, so the four Humans moving back and forth in the kitchen space were focused more on cleanup and prep work than on actually cooking anything. Ia's entrance into the dining half of the large cabin went unnoticed until she came within a few meters of the pass-through between mess and kitchen.
The tall, black-haired woman directing the others on what to chop up and secure for the next meal was the first to spot Ia. Breaking off from her work, Private Philadelphia Benjamin—no relation to Chaplain Christine Benjamin—came over to the pass-through and leaned her elbows on the polished metal surface.
"What can I do ya for, Captain?" she asked, clasping her hands together.
Her pose, familiar to Ia from past glimpses into the timestreams, sparked her rare sense of humor. "Half-back on rye, black, pulled, swiss, stack the veg with watercress," Ia rattled off. "Half a cup of slop, dash of pepper on the top—and a moo-joo, Philly, don't spare the blue."
Private Benjamin's mouth sagged. She blinked her aqua blue eyes, attractive with their framing of black lashes. "How...When did _you_ visit Benjamin's Beeferie, sir?"
"I've never been to Mars, let alone your family's restaurant," Ia admitted. Then amended, "At least, not in this life. But it sounds good, no?"
"Hell, yeah, it sounds good!" Philly agreed, straightening up. "Hey, Gracie! Get out the pulled pork, the rye, and sandwich fixings. And a bottle of nonfat milk while you're at it," she added, as Private Grace Marshall headed for the cold-storage locker. Philly switched her gaze to the man next to Gracie. "Clairmont, head down to lifesupport and pick up some fresh watercress. Make it snappy. I'll get your soup while you wait, sir."
"Already on it, Philly," Private Mellow called out, soup cup in one hand and ladle in the other. "Today's special is tilapia chowder, Cap'n."
"Sounds good. I'll take it out here," Ia said, poking her thumb over her shoulder. Picking a spot at one of the long tables, she settled into the chair and waited.
Philly brought the cup of soup and a lidded glass of milk within moments. Accepting them, Ia clipped them to the table and dipped the accompanying spoon into the pepper-flaked surface. It was hot, creamy, and filling. The soup was half-gone by the time the mess doors slid open again. Private Clairmont stepped through, a lidded bin filled with leafy greens in his hands. So did another man, shorter and older. The pair were apparently exchanging a joke, for both men laughed before parting company. Clairmont headed to the back half of the galley, while Finnimore Hollick detoured over to Ia's table.
" _Ia'n sud-dha_ , Captain," he said in greeting, turning one of the deck-bolted chairs across from hers enough to slide onto it. "I wanted to thank you for the chance to visit my birth family."
She smiled a little at that, dipping her head in acknowledgment. "I'm glad you got that chance, Finn. You'll get one more, though it won't be for a few more years."
He nodded, resting his elbows on the table. Short, somewhat stocky, freckled, and sporting a fringe of thinning sandy hair, he didn't look particularly imposing. The height and frame came from the fact he had been born a first-generation native on Sanctuary, but Ia knew he hadn't grown up there. Every so often, a newborn infant suffered immediately from the effects of gravity sickness. Hollick was one such case; lifted off-world as quickly as possible, he had been shipped off to family members back on Earth for gravitic rehabilitation in an environment his infant body could withstand.
Like most every other member of her crew—Harper being the one exception—she had checked over the details of his past in the timestreams when selecting him to serve her into the future. Hollick had served for years on various starship crews manning the supply runs all the way out to Sanctuary, as a way to stay in touch with his birth family. After suffering three particularly strong Fire Girl Prophecy attacks, he had been contacted by Rabbit's little gang of pre-rebels and vetted as a fellow believer. So his greeting, which in archaic V'Dan meant, "as the Prophet wills it," made sense in that context.
Something about him, though, some hidden quality buried below the surface of his life-stream, was necessary for the effective functioning of her crew. Ia suspected it was because some combination of his greater age—forty-seven, compared to the average age being nearly two decades younger—his steady work habits, and his unwavering faith in her would have an ongoing, subtle effect on the rest of the crew.
He was already something of a favorite among the 2nd Platoon. Or at least among its galley crew. Sort of a middle-aged mascot, as it were. Philly came out from the galley space with a bowl of soup and a cup of water, beaming a smile his way. "Hey, Holli. Clairmont said you got out of lifesupport early for good behavior. Here's some of that soup you like. I brought you some crackers, too."
Eyeing the plexi-wrapped packets she set down next to the clipped bowl of soup, Ia raised her brows at Private Benjamin. " _He_ gets crackers, but I don't?"
" _You_ are getting a sandwich, Captain," she sassed back. "You'll get all the carbs you need with my dough. Besides, if you're gonna order the Benjamin's way, you should've added, 'snap the crack' with that 'dash of pepper on the top.'"
That made Ia chuckle. "Okay, I concede your point. So where's my pulled pork rib meat sandwich, Private?"
"Being assembled as we speak." Flipping a mock salute, the tall woman took herself back into the kitchen. "Your lunch will be launched in two minutes."
Ia looked at her dining companion. Hollick was already crumbling the crackers into his soup. He offered her the last packet, but she shook her head. "So, how's the 2nd doing?"
"Good. A little nervous about the coming fight, but I've told 'em you know what you're doing," Hollick said, mixing in the last cracker.
"We'll pull through this one alright," she agreed, spooning up another bite of her soup. "The Salik aren't expecting it, they're not heavily guarded, and we do need those comm nodes. Easiest to get 'em before they know we're coming."
He didn't ask her where she found the coordinates for the communications hub. Instead, Hollick asked, "Have you been getting enough sleep, Captain? You look a little dark under the eyes."
Ia could have lied, but she chose to be honest. She did, however, keep her voice low. "Not really. Been pushing hard to get everyone ready for this. I'll have plenty of time for sleep after this fight," she added, scraping up the last of her chowder. "Once we pull the nodes from the wreckage, it'll be up to Rico and his teams to pull the information we need out of the databanks. They should be able to do it in two, three days—I know where they're supposed to look, but not what to look for, if that makes any sense."
"None so blind as them that see," he quipped dryly. "It makes perfect sense to me."
Philly came out with Ia's sandwich. The spicy-sweet smell of barbecue sauce wafted up from the meat-slathered hoagie. Ia accepted the plate, clipping it to the table edge. "That looks great. Thanks."
"Well, it's not the original Benjamin's barbecue sauce since I can't get the right mix of fruit out here, but it'll do," Philly said, shrugging. She retreated once more to the kitchen.
Ia knew the other woman's history included her family owning one of the oldest sandwich shops on Mars, and a sordid story of a criminal underlord attempting to take over the lucrative family business, efforts that had driven Philadelphia Benjamin into joining the military to escape the worst of it. The tall woman had burn scars under her plain grey sleeves, testament to the violence that could sometimes plague civilian life.
Philly also knew how to make food tasty, which was why Ia had instructed Grizzle, their Company Sergeant and the man responsible for the day-to-day work schedule, to put her in as lead cook whenever it was her turn to work in the kitchens on her duty shift. Very tasty food. The combination of cheese, toasted bread, spiced meat, greens, and vegetables vanished quickly from Ia's plate.
Wiping her hands on the lemon-scented napkin provided, she nodded at Hollick when he held out his hand, offering silently to take her cups and plate. A swallow of milk cleared her throat, enough that she was free to pull her headset out of her shirt pocket and hook it over her ear. One last swipe at her fingers cleaned them enough to poke at the buttons of her grey plexi arm unit, echoing her voice through the mess, kitchen, and places beyond.
_"All hands, this is Captain Ia. Ten minutes to combat. I repeat, ten minutes to combat. Lock and Web; lock and load. You have ten minutes before we exit the rift and engage the enemy. Ia out."_
"I'd better get back to lifesupport, Captain," Hollick offered, stacking her emptied glass on her plate next to his soup bowl and mug. "Good luck on the probabilities, sir."
"Thanks. I hope we don't need any." Nodding to him in thanks for the dishes, she headed for the door.
A short lift ride down and a modest walk forward brought her to the bridge. Entering via the main portside door, Ia announced herself this time rather than waiting for one of the others to notice.
"Captain on deck. Chief Yeoman O'Keefe, prepare to transfer helm to my control," she stated, striding between the workstations toward the command seat at the back.
"Aye, sir."
Unbuckling his safety harness, Spyder worked himself free of the straps. "Gimme a moment, Cap'n...uh, y' might wanna go freshen up," he added, nodding at her chest. "Got sauce onna shirt, there."
Glancing down, Ia rolled her eyes. She had missed a drip of barbecue sauce, which had landed on the curve of her left breast, on the edge of her shirt pocket. "One of the perils of being female," she joked dryly. "It'll wait. I'm after results, not looking pretty."
Shrugging, Spyder cleared the seat for her. He ran one hand over his short-cropped, camouflage-mottled hair. "Eh. Jus' as well. Leave th' lookin' pretty t' meioa-os like me, eh?"
"Go be a pretty-boy in the spare gunnery spot," she ordered, sitting down and strapping in. Unlike Rico with his longer legs, she didn't have to readjust the chair forward. Her legs were slightly longer than Spyder's, but not enough to bother her. "Corporal Morgan will coordinate forward gunnery efforts. Lieutenant Spyder will coordinate the aft-ward vectors."
Sighing, Spyder finished clipping into his seat, then powered the whole station around so that it faced backward. Both gunnery posts and the pilot's station could be rotated for just such a need, so that the Human inner ear wouldn't be thrown out of balance by the disparity between physically moving one way and facing the other. Normally, Ia's station also would have swiveled as the backup piloting position, but the security requirements for the main cannon controls had it locked permanently in place.
She activated the all-hail again. _"This is Captain Ia, threeminutes to combat. Lock and Web, meioas. Lock and load. This is not a drill. Three minutes to combat; Ia out."_
There were four differences between Ia's console and the pilot's version. One was the number of tertiary monitor screens; her seat had five of the transparent panels above as well as five below the transparent main and flanking secondary screens. The second was the control panel that released the main cannon. It was locked under a double-lidded security box, with a palm- and DNA-scanner sandwiched between the two lids. Anyone who tried to activate the main cannon other than Ia would set off an unstoppable chain reaction in the hydrogenerators, turning the ship into a brief, bright, miniature nova in one minute.
The third difference was the way that security box locked the station in place, preventing it from rotating, unlike the pilot's seat. And the fourth was an odd, recessed socket with two exposed metal tabs, its sole purpose to provide energy when the circuit between those two tabs was connected. Nudging up the cover, Ia stuck her right hand inside and pressed her fingers against them.
Electricity snapped into her arm, stinging up through her nerves. Like a jolt of caf' to taste buds, it woke her up, sharpening her senses. She didn't take much, just a few seconds' worth, then pulled her hand out and let the little door slide shut. Left hand fitted into the thruster glove, right hand on the power controls, she nodded.
"I'm ready for the helm, O'Keefe," she said.
"Aye, sir," the freckled pilot stated. "Transferring helm to your control in five."
They weren't traveling through hyperspace. Ia didn't want the Salik to know a ship as large as hers could do that, just yet. She could use it to get close to this particular patch of nowhere, situated about four hundred light-years from Earth and over nine hundred from Sanctuary, yes; they had used hyperwarp to get to within a light-year or so, but the last leg had been plain FTL.
In control, Ia closed her eyes for a moment, dipping part of her mind into the timestreams. She didn't want to overshoot their target but she did want to come in as close as safely possible. She also wanted her crew to be certain of what were viable targets.
_"Captain to all gunnery teams. Your tactical displays have been loaded with two sets of targets. Shoot the red ones all you like but avoid shooting anything tagged green. We're here to smash and grab communications databanks. I'd like to_ have _those databanks intact for grabbing. You have four cannons slaved per pod. Use them wisely. Heads up in...twenty seconds, mark."_ Off-mike, she spoke as she started powering down the warp panels. "Spyder, Morgan, keep an eye on who's firing on what. We're coming in hot and fast and barreling right past before we'll turn around. Lasers only for the first volleys. Let's not outpace any projectiles, today. Coming out of FTL in three...two..."
The slow streak of stars on the screens flared into a pulse of dotted white. Within a blink, the screen was filled with a stippling of near-static pinpoints of light. Ia's main screen lit up with tactical analyses of the objects barely lit at this distance by floodlights designed solely to keep incoming ships from hitting anything while maneuvering nearby.
"Resolving lightspeed, sir," Private Loewen said from the navigation post, checking her scanners. "We have targets, almost dead ahead. Three-fifty-six by three-fifty-nine."
"L-pods, fire at will," Ia directed, watching the distance-to-target numbers spin down swiftly as they hurtled toward the installation. Morgan nodded and relayed the command.
They were still slowing down, thruster fields pulsing subtly against the laws of reality. Between pulses, the _Hellfire_ 's L-pods started to fire. Bright reddish orange streaks of light darted out from the ship. With the _Hellfire_ speeding into range right on their tail at a large fraction of Cee, the speed of light itself, the lasers could be seen moving as a bolt rather than a beam as they arrowed straight and true at the organic-looking station.
That station was huge, too, though mainly due to its function. Layers of solar panels angled out from the central pod, interspersed with five ship-sized arms that served as crew quarters, docking gantries, manufacturing facilities, and so forth.
Ia knew the station was home to over eight thousand Salik. She also knew the facility was underdefended, relying more on its extremely remote position between star systems for protection than on gun pods, fighter craft, and other things. With their resources limited to whatever raw materials they could mine or steal without the Alliance's notice, the Salik had to juggle the needs between creating more ships and weapons with their limited resources and expanding their infrastructure, such as this station, in the hopes they'd be able to find more materials to justify the expansions.
Two of those ships were in the system now, a tanker filled with hydrofuel and raw materials, and a frigate, small but fast, the kind useful for gathering lightspeed information at the extreme edges of a particular system. Naturally, they were on Ia's red-painted list. Red-painted, and red-targeted, with a spinning stream of numbers appended, counting down the distance in light-seconds to each enemy ship. As the dozens of lasers struck _en masse_ , three of them hit spots that caused silent explosions to rip through the hulls of the frigate and the tanker.
"Spyder, launch one volley of projectiles the moment that station is aft," Ia ordered. "Coming up on midpoint in thirty seconds. O'Keefe, count it off."
Her right hand thumbed one of the console controls, and her left hand twitched to the right a tiny bit. The ship strafed sideways by a hundred meters as it raced forward, just enough to avoid the dark chaff blown outward from the solar panels. Bright red continued to cross the screens as the L-pods kept firing.
Lasers were quiet weapons, unlike the clunkier noise of projectiles being launched. They did, however, require extra energy, enough that the ship _thrummed_ with the efforts of several extra hydrogenerators cycling on and off. The slower the ship went, the shorter and darker those beams became, redshifted from the _Hellfire_ 's reduced speed.
Slow in this case was still a significant fraction of Cee, the speed of light. In Ia's Harrier-class ship back on the Blockade, Cee hadn't been a concern; Harrier-class ships traveled via OTL and never actually needed more than half the speed of light. The _Hellfire_ , by contrast, had started several billion kilometers away, but those billions were now almost gone.
"Coming up on turning point," O'Keefe warned the others. "Estimated ten seconds."
Bright chartreuse dots stabbed at the ship. They missed, but only barely. Ia flexed her fingers, sliding them slightly dorsal. That dodged another set of blueshifted lasers aimed their way.
"Incoming!" Loewen warned, her head shifting in little snaps as her eyes flew over the data streaming into her screens. Her warning wasn't for the lasers; those could not be dodged, since they traveled at the speed of light. Instead, it was for the missiles that followed. Lasers aimed at their narrow, end-on silhouette weren't a concern, not when missiles could track the mass of a ship and divert course to intercept.
"Five," O'Keefe counted.
"Firing chaff!" Morgan announced, pulsing the trigger on his controls. Rapid noise _thuthuthunked_ down the hull from the bow. The battered communications hub filled their viewscreens, then popped somewhat smaller as their proximity forced the scanners to cut back on their magnification.
"Three!"
"Aft P-pods," Spyder called into his headset.
"One!"
They dove between the tanker and a tumbling cloud of laser-scorched debris from the solar panels, close enough that proximity alarms beeped loudly in warning, though not quite close enough to trigger the emergency claxons.
"...Fire! L-pods, fire at will!"
Parts of the solar panels were now shielded, visible where some of that debris struck the repeller fields with sparks of energy. The undersides of those vast panels were more heavily shielded, but it wouldn't do them much good; lasers penetrated shields. They didn't do as much damage as missiles did, but they could damage the nodes projecting those shields.
As it was, some of lasers from the aft-pointed pods struck before the missiles did. Others struck after. Several explosions blossomed silently to their rear, though it would take a few seconds for their tactical computers to discern which ones were just from the missiles impacting on the shields and which ones were from missiles that made it through to the actual station surface.
_"All hands, brace for maneuvers,"_ Ia warned over the intercom. They were still going a significant fraction of Cee, and she was about to pull something complicated.
On a more compact ship design, she would have physically turned the ship in a loop, swapping bow and stern so that they were pointed the other way, but the _Hellfire_ was nine hundred meters long. A turn at their current speed would bend, if not break, the hull. Flexing the FTL fields, Ia greased the laws of physics in a bubble around the ship. Only then did she make the port side greasier. Left hand wrapped in the glove, right hand splayed over the console, she massaged both sets of controls.
They had entered normal space pointed downstream—toward the core, in relation to the spiral of the galaxy, if one considered the stars as "draining" toward the black hole at the center of the Milky Way. They remained pointed that way, but like a hummingbird or a dragonfly, the vector change swerved their momentum so that they were flying backwards. The field shielded them from most of those vector change forces, but not all.
Everyone and everything not bolted to the ship pulled to the right for several long seconds, then swung around to the bow. The G-forces eased after several more seconds; pleased everyone had locked and webbed their stray supplies, Ia worked on angling the axis of the ship a little as they headed back toward the station and its two half-crippled ships.
Morgan and Spyder gave orders to their gunnery teams. The aft and starboard pods fired as soon as the full field dropped, allowing them to reacquire targets between pulses. More enemy shots were fired; eyes on the screens, hands on the controls, toes in the timestreams, Ia sideslipped most of the lasers. A few scored the sides of the ship. A few telltales turned yellow, mostly exposed FTL panels and one sensor antenna. Ia nudged them sideways, aiming a little closer even as she angled their ship to strafe down the side of the hub.
They weren't the only ones attacking now. Caught by fast, close launches from the frigate and the station, the _Hellfire_ 's shields vibrated down into the hull from missile explosions. Here was the difference between reality and entertainment shows like _Space Patrol_. Real interstellar combat was relatively quiet, so long as one wasn't getting hit. Now they were, and now the P-pods fired anti-missile volleys and chaff grenades, trying to detonate the incoming munitions before they could hammer their way through the shields to the actual hull.
"Coming up on midpoint," O'Keefe warned them. "Twenty seconds."
Again, Ia slipped the ship sideways. They weren't passing nearly as closely this time.
"Focus your fire on weapons and generators— _brace formaneuvers_." Ia swirled her fingertips. Rolling the ship on its long axis sent their stomachs slinging sideways, down, and out. It also lessened the scorch impact of a large laser attempting to target and burn through their hull. More telltales popped up on her tertiary screens, glowing yellow from the damage sustained. "Morgan, switch to targeting priority display 2, and load up twelve P-pod blossoms. Fire on our third midpoint pass. Add regulars on this pass."
"Midpoint in three...two..." O'Keefe counted down. Another set of missiles hit their shields, shuddering the ship.
"Forward P-pods launching," Morgan announced a moment later. The ship _whumped_ several times, starting in the distance, racing past their position, and down to the stern.
Again, they strafed sideways and down, dodging incoming attacks. Ia warned the crew one more time, then pulsed the FTL panels, mindful of the damaged ones. They swerved around until they were speeding forward, this time aiming down the other side of those solar panels, if slightly above. They no longer faced their original entry-point orientation, thanks to her careful, slow tilting of the ship.
"We'll be moving fast in a few moments," she warned the others. "Launch the blossoms at midpoint and brace for acceleration."
"Aye, sir," Morgan said. "Blossoms are prepped and ready for launch."
_"Brace for acceleration,"_ Ia warned.
"Ten seconds," O'Keefe warned them. "Glad we're not sticking around. Five...four...three..."
The long, slender missiles launched with a near-unison stutter of _clunks_ just as they soared over the battered but still-firing station. Ia scraped her right-hand fingertips up the thruster controls, thrumming the engines in a rippling wave of increasingly swift field pulses. Occasionally she flexed one side of her left hand or the other, lifted it up slightly or pushed it down, either straight or angled. That allowed them to dodge most of the damage from the few L-pods still capable of return fire.
The increased speed was necessary; when the blossom missiles hit, they impacted zones that had lost shielding. Ten lodged in the panels next to each segment that served as living quarters, just to either side. Two more smacked into the central sphere that housed the vacuum-sealed hyperrelay units, which permitted the routing and rerouting of messages from one end of known space to the other.
For several seconds they did not detonate, though their engines did pulse, attempting to drive them deeper into each section of the overall structure. When they did, they slammed scores of smaller, needle-sharp bombs outward, blossoming like fireworks. Those bombs in turn stayed dormant for a few more seconds, giving the ones driven into open areas more time to travel outward. _Then_ they detonated.
This time, the detonations were explosions, ripping huge holes in the station. The force hurled debris in all directions, making the space near the communications array dangerous to transit. With the _Hellfire_ already headed outbound, the only things they had to dodge were a few still-functioning lasers. Even those ended a few seconds later, as the damage cracked the station into pieces.
Once the enemy fire stopped, Ia slowed the ship again, intending to guide it around and into reverse as she had before. As she did so, she checked the timestreams. The answer she sought came within a handful of seconds, and it was a satisfactory one. _Minimal damage to the exact units we'll need. I know_ _the information we want is buried in those particular comm-traffic files, though I don't know exactly what we're supposed to be looking for...mainly because the subject matter recursively hides itself from me._
The best analogy she had was being sent into a grocery store with a list of spices. Those were usually kept with the baking goods, but that entire aisle could be located anywhere in the store. Once the right aisle was located, then the right shelves had to be found, and finally the right spice jars. Ia knew where the store was located—this station—and an idea of which aisle to check. The rest of it, she didn't have time to search for herself.
That was what the others were here for.
"Private York, please inform Lieutenant Rico to get his salvage teams ready. We'll be taking potshots at enemy weaponry for the next twenty minutes, disabling everything they can still throw at us. Those teams need to launch as soon as I give the all clear. We do not want to be here an hour from now."
"Aye, sir."
_And at that point, I can rest,_ she thought. _Once we're underway and I've plotted our next course, O'Keefe can have the helm back, and I can go sleep for nine hours straight._
Opening the intercom, Ia addressed her crew. _"Captain Ia to the crew. Good job, everyone. Third watch will be free to stand down again in twenty minutes. Maneuvers should be light between now and then. Exercise caution in moving about the ship for the next hour and a half. Ia out."_
On the one hand, energy from the food she had eaten was now keeping her awake. On the other hand, the post-battle adrenaline slump threatened to steal that energy away. Now that they were aimed back at the crumpled bits of station, she had a few seconds free. Slipping her fingers under the little door, Ia jolted herself with a little bit of electricity.
"A good day's fight, meioas," she murmured, praising her sparse bridge crew. "Let's keep it up."
FEBRUARY 5, 2496 T.S.
SIC TRANSIT
The boot that hit her did not belong to her brother. It did, however, break Ia out of her timeplains trance. Standing in her socks on the deck of the stern cargo hold, she blinked at the mangled glob of crystal in her hands. Being broken out of molding trance always did that. She could fix it, but she had left orders with Harper on when to interrupt her, and why.
"Fire, famine, flood, or finally cracked the code?" she asked, turning toward the door. Harper stood there, but so did Helstead, her shorter frame peering around his taller one.
"Finally cracked the code...we think," Harper said. The other lieutenant commander tried to squeeze past him. He checked her with his elbow, rolling his eyes. "Helstead! You're not allowed in this hold."
"How can I be an effective spymaster if I'm not allowed to spy on her?" Helstead shot back, trying to dodge the other way. He checked her by spreading his legs—and she darted under them, only to be grabbed by the back of her shirt.
"Sorry, Captain," he apologized. "Want me to throw her out?"
" _I_ want to know what's so...Wait a second," Helstead said, green gaze darting around the hold. "There were at least forty of those things loaded into this cargo hold. I only see thirty-six."
Back on her homeworld, Ia had ordered the sprays lashed to the outer walls, with large lockboxes stacked and secured down the center. The extra flights had added a day to their itinerary, but the successful early integration of the hyperwarp system had given them an extra day. Some of those lockboxes now contained crysium wreaths, shaped in her spare time since leaving Sanctuary.
Helstead narrowed her eyes, staring at the lump of crystal in Ia's hands. "Is that...?"
Harper hauled back on her collar, interrupting her question. "You are in a restricted area, Lieutenant Commander. Only myself and Captain Ia have access to this part of the ship."
Twisting, Helstead broke his grip with a sweep of her arm. She didn't grab him back, just planted her hands on her hips. "And maybe I should report _you_ for keeping secrets from the Command Staff! I did some research on this crysium stuff when it was brought on board. It's the hardest substance known to sentientkind. So how the hell did the Captain reshape it? She's quite clearly holding a re-formed chunk of it in her hands."
" _I_ don't like the implications that you're a _spy_ ," Harper countered, hands going to his own hips. "Your loyalty is up through the chain of command, to me, to Captain Ia, _then_ to the Command Staff."
"I'd believe you a lot more," Helstead snorted, "if every time you looked at her, you _didn't_ look like a lovesick turtle. You don't hide it nearly well enough, soldier."
He flushed at that. Sighing, Ia interrupted the pair. " _Enough_ , both of you, or I'll make you clean the lifesupport filters while all the off-duty privates watch. Helstead's not the spy on this ship. I know exactly who they are, and she's not one of them."
That got Helstead's attention, swinging the shorter woman around. Ia lifted the glob in her hand.
"As for what I'm doing with it, I'm borrowing technology from the future to alter the crystals. That technology is _not_ the property of the Terran United Planets...and I cannot in good conscience explain that tech _or_ share it with anyone else. That's why this cargo bay is off-limits."
"You're using tech you stole from the future, from another government, to reshape the hardest known substance, and you're not gonna share it with your own superiors?" Helstead asked dubiously. "Imagine the weapons you could make—imagine the _armor_! I read an article stating how this stuff just absorbs laserfire, no matter how large the weapon."
"I will not compromise the secrets of the government which developed this tech," Ia stated bluntly. "Just as I will not compromise the secrets of the government that will eventually develop the hyperwarp drive. One is Terran; the other is not. Both have the right to keep their temporal secrets. I will _use_ this technology, yes, but only because I have no other way to get the job done on time and done right. I will therefore not share those secrets...and the fewer who know those secrets, the easier it is to _keep_ them a secret. Is that clear, soldier?"
That deflated the shorter woman. Brow furrowed in a sullen look, Helstead muttered, "Crystal clear, sir."
Ia pointed at her. "My silence applies to your own escapades as well, Delia. Past as well as present and future. Be glad I have both discretion and integrity. Now, give me a few minutes in private to finish what I was doing, then I'll come join you and Rico in the briefing room. Harper, if I'm not out of here in ten, throw my other boot at me."
"Aye, sir," he agreed. A tip of his head silently ordered Helstead to retreat with him. Grateful, Ia watched them go.
Just before the door slid shut behind them, she heard Helstead ask, "Why do you have to throw her boot at her...?"
Ia knew the answer, the same as Harper. Back in their Academy days, sharing quarters in the cadet dormitories, she had taught him to wake her by throwing something at her rather than physically touching her. These days, her sleep was relatively dreamless; minor nightmares still plagued her, but she was on track to fix the source of those nightmares. Her precognitive gift didn't torment her as much as it used to, which meant the risk of triggering it when someone else touched her in her sleep had lessened. But lessened risk was not the same as none.
_Which means I go through life untouched, save for rare moments when it's absolutely necessary._
Turning back to the glob in her hand, Ia flipped her mind in, down, and out, onto the timeplains. She resisted the urge to sink fully into the rhythm of swimming, shaping, and molding blood-infused beads with chunks of crystal. Her blood. Whatever quality of Feyori Meddling lay in her genetics, it seemed to affect the crysium—which itself was another Feyori by-product. The two combined, crystalline and crimson, made the strange, peach-hued stuff capable of hooking a nonpsychic mind into the timestreams.
She only had enough time to create a few more. There would be other stolen hours for this task, creating precognitive crystalline circlets meant to plug her fellow Sanctuarians into glimpses of their own pasts, their own futures, to guide them in the direction she needed them to go. Before the ten minutes were up, the latest wreath was locked in a storage box and her boots were back on her feet. Ia stepped through the cargo-bay door, palm-locking it behind her. Harper and Helstead waited nearby; once again, Helstead was playing with her stiletto-pins.
"Can I at least have a closer look?" she asked Ia, moving to join her CO in heading toward the aft shuttle-bay doors.
"No. Looking leads to touching, and touching leads to interfering. Most of what I'm making won't even be used by the Terrans," Ia told her. "I'd rather it wasn't damaged in the interim."
Helstead frowned at that. "Wait, if it was impervious before you manipulated it, but now it can be damaged simply by handling it...that means it won't actually retain its impervious qualities after it's been reshaped. Why would that stuff be of any use after that point?"
Ia ignored the question. She had meant the timelines, not the wreaths themselves, but her 3rd Platoon leader didn't need to know that. Instead, she glanced over her shoulder at their chief engineer, addressing him instead. "Harper, I know you've been working on the FTL panels you pulled off the hull. How many have your repair teams salvaged?"
"All but two," he told her. Touching the controls for the sector seal, he joined them in transiting the airlock seals. "I've ordered those two broken down for parts. We have plenty of spares on board, but I'm not going to hold my breath on every battle being that smooth."
"Two out of, what? Twenty-tree panels? That's not a bad attrition rate," Ia allowed, yawning slightly to pop her ears as the air pressure shifted slightly. The second door swung open, revealing Spyder, Santori, and two long lines of shorts-and-T-shirt-clad bodies, all jogging in place.
"Make room, ya bloody slackers!" Spyder ordered, jogging himself as he turned to face the others. "Officers comin' out!"
The two lines of men and women parted, jogging up against the wall as Ia, Harper, and Helstead emerged single file from the airlock.
"Arright! C Squad, cycle through! Two more minutes of jogging, then we go back to practicing shipboard parkour!" Spyder ordered. The first five pairs of Humans jogged into the airlock. Someone thumbed the controls, swinging the door shut.
"I'll give him this," Helstead observed, glancing back at the airlock as the three officers turned a corner. "He's certainly enthusiastic about PT. The man has almost as much energy as _me_ in the mornings."
"That's why I paired him with you on that task. Oh, speaking of exercise, Harper," Ia added, turning slightly to address her first officer as they walked. "I'd like to gradually increase the ship's gravity by .05Gs over the next week. Then hold it steady until the end of the month, and increase it again by .05 over the first week of March. Hold steady until April, and do it again, until this ship is running at 1.35Gs Standard, then hold it for three months."
"You got it," Harper agreed.
Helstead eyed her. "Trying to turn all these lightworlders into heavyworlders, Captain?"
"The increase in strength will be a bonus in hand-to-hand combat, about half a year from now," Ia admitted. "The faster reflexes will take a while to train, but will be worth it in about three years. But long before then, I want us up a lot higher, at the very least to 1.8Gs."
"My homeworld isn't much higher than that," Harper said. Then blinked and looked at her, brown eyes wide. "We're going there, aren't we? I remember—"
Ia raised her hand, cutting him off. Through an accident, he had been exposed to a tumult of unchecked timestreams back at the end of their Academy days. She didn't need the rest of the crew knowing that, though. "Some visions will come true, others will not. I'll let you know what to prepare for when you'll need to know it, I promise."
He nodded. "Alright, then. Will you need me for the databank debriefing? If not, I'll get back to that gun project."
She shook her head. "I don't think so. Helstead might be useful, though."
That made the other woman's eyes roll. Tucking her bladed hairpins back into her braid, she said, "Oh, gee. I feel so special."
"That depends on whether or not you can focus," Ia said. She nodded farewell to Harper, who continued aft-ward while they took a side hall that would lead them to the track lifts for the fore section.
Helstead waved good-bye at him and shrugged. "I can focus just fine, Captain. I just...have to keep doing something, or I'll go nuts."
"Then we have something in common," Ia told her. "Only in my case, the 'something' I constantly have to keep doing is striving to save the future."
"So, how's that working out for you?" Helstead quipped.
The joke caught Ia by surprise. She chuckled. "We'll see. I still have several years of work ahead of me. Feel free to help out if you get bored."
"Sir, yes, sir," Helstead quipped.
Both women fell silent as they traveled up several decks to the briefing room. Located forward of the bridge, it was meant for the cadre to use, or up to two squads at a time. The monitors lining the cabin caught Ia's attention first, followed by the tall, broad-shouldered figure of Lieutenant Rico.
He stood in front of screens filled with series of dots, circles, and lines that made up the written form of Sallhash, language of the frogtopus-like Salik. Around the oval table at the center of the room sat Corporal Xhuge, Private Dinyadah, Private MacInnes, and Private Al-Aboudwa, all members of the 1st Platoon. They had workstations clipped to the table; from what Ia could see of Al-Aboudwa's and Xhuge's screens, the two men were attempting to match location names to star charts.
"Glad you could join us, Captain," Rico stated in a mild tone. "I hope we didn't interrupt anything important."
"I put you in charge of this project because I knew I could trust you with it," Ia returned calmly. "You cannot compose detailed, time-sensitive instructions that will save hundreds of billions of lives. You, however, can make sense of this Salik gibberish, and _that_ will help save hundreds of billions of lives. Now, I understand you've found something?"
"We think so, sir," Corporal Xhuge said, speaking before Rico could. Ia had to give him credit for getting the discussion back on topic. "Since you said you were able to pinpoint the exact comm nodes containing information on where and how these anti-psi machines were being manufactured, once we cracked open the databank cases, we broke down all the various messages in the selected banks and started sorting each by type."
Rico joined the explanation. "Most of them were about the transport of goods, involving timetables, ships, requisitions...standard stuff. Much of it used cryptography, or the substitution of things like letters and numbers, which can be cracked by any competent computer system. Even the most sophisticated ones back in their day from Old Earth's Enigma machines all the way up through the AI War codes were breakable, given time and understanding. But some of the messages were different.
"They were asking for weird things with no apparent context. That meant they were using steganography," he told Ia. "It's the art of substituting one word for another. The catch is, you have to know which words substitute for which. Like the Navajo code talkers on Old Earth, same era as the Enigma machines. But it's always heavy on the nouns, even among the xenospecies, and that's where MacInnes came in handy."
MacInnes nodded quickly, her carrot curls bouncing around her ears. "My Sallhash still isn't the best, sir, but it works in conjunction with my xenopathy and clairvoyancy. I can 'see' a noun in my head when I'm translating it mentally. The nouns the lieutenant handed me on the weird requisition cases _weren't_ the ones I was actually seeing and writing down."
"I figured her gifts would be useful that way when I read the notes you appended in her personnel file," Rico interjected.
"Like I said, I picked the right man to head up this job, and the right meioas to work with you," Ia reminded him. She nodded at the other woman. "Go on, MacInnes. What were the disparities?"
"Instead of...uh..." She blushed and nodded at her workstation. "Well, we can't pronounce most of these words without the nasal flaps and such, so we're using the Terran phonemes. Anyway, the weirdest one was a message about an order of _po-jeem ang-nu-gwish-tick-wa_ , which is a kind of amphibious creature they like to eat on their M-class colonyworld of Hawhonn. But instead of alien breakfast food, I was getting images of anthikeriate coils, which are used in the PsiLeague's KI machines to sense kinetic-inergy emanations, and I only knew _that_ much because I helped refurbish a couple during a summer job back in high school."
"And some of the other nouns?" Helstead asked, dropping into one of the chairs at the table and swirling it around.
"Electronics components, mostly. But also...test subjects," Al-Aboudwa stated quietly. "We found dates and times for prisoner swaps."
"The images I saw were meant to imply tasty-smart food," MacInnes muttered, looking a little pale. She wasn't the only one. Xhuge and Al-Aboudwa didn't look comfortable, either.
"—Found it!" Dinyadah exclaimed, pointing at her workscreen. A couple of taps transferred the star charts to the monitors around the room. "God bless methodical mapmaking. Ss'gwish Gaff 117-N, a system with a small class B white star, three asteroid belts, at least four gas giants, and an extensive Kuiper belt with a break in it from a rogue planet expelled by the explosion of a downstream neutron star. That's our second starting point, gentlemeioas, and it's approximately twenty light-years from the endpoint in the message."
"Good job," Rico praised. "Now all we need is a third location, to triangulate the manufacturing point—the Kuiper belt break has several ice chunks trailing outsystem in the wake of the rogue planet," he explained to Ia. "The first system we found has the requesting point located approximately eighteen light-years from _Nngu_ 120-N. But that still leaves us a ring zone with a circumference of one hundred light-years, give or take a few. Since we don't have any third reference point in the data files, I was hoping _you_ could provide us with one, Captain."
Ia gave him a level look. "What part of 'these machines can counteract even _my_ abilities' did you not comprehend at our initial task briefing, Lieutenant?"
"The part where you said you're an omniscient precog?" Rico replied, one brow lifted in skepticism.
"I'm not omniscient. I'm all- _seeing_ , not all- _knowing_. And this thing is like a black hole on my internal radar, or a thick cloud obscuring an island," Ia said, pointing at one of the screens. "I can only tell its general vicinity from the _lack_ of things I can sense about it directly, and from the things I can sense _in_ directly. Like our next target." Moving over to the largest of the screens, she poked at the surface, rotating the stars slightly. "I can sense what people do, Lieutenant, from the effects their lives, their timestreams, have on the timeplains in general.
"I cannot track things as easily as I can track people. Unfortunately, most of the people directly involved in this anti-psi project are being protected by the machines." Tapping the screen, she drew a circle around three stars in relatively close proximity in green. "Somewhere in here is another communications station. We're going to attack the second hub, extract the right data nodes, and extrapolate that third point. We'll have to hurry, since the one thing I am sure about is that the manufacturing equipment is on the move."
Another tap drew a much larger circle in yellow, one at least forty light-years across. Ia looked over her shoulder at Rico.
"This area is the general area I think it's in, but as you can see, it's rather large," she said. "They're distributing the machines, too; I get flashes of foreknowledge in between stretches of nothing. Once we do get that third coordinate for triangulation, we move in for the kill and wipe out their main manufacturing facility."
_"D'un yi shia..."_ Xhuge muttered, eyes widening as he stared at the yellow-circled zone. _"Da shiong la se la ch'wohn tian!_ "
Rico narrowed his eyes. " _Excuse_ me, soldier?" he snapped at Corporal Xhuge. "You do _not_ swear like that in front of me. I'll remind you, I _do_ speak all three main dialects of Chinese."
"Yeah, what he said," Al-Aboudwa agreed, giving his crewmate a puzzled look. "Either swear in Terranglo, or at least tell us what you said, will you?"
"I'm sorry, sirs, meioas," Xhuge apologized, blushing. He lifted his hand, pointing at the yellow blob. "But that whole zone, if they're distributing anti-psi machines to their ships, then _that's_ where the main Salik fleet is gathering. If we go in there pods blazing, they'll swarm in and kill us. Not to mention, they'll have to figure out which databanks we're stealing sooner or later. No offense, Captain, but that's a _tze sah ju yi_!"
"It would be a suicidal idea under normal circumstances, yes," Ia agreed. "But my abilities strengthen with proximity, both temporally and spatially. My first encounter, I had no clue what these machines were, but I still managed to destroy a giant capital ship, and I did it with a tiny little Harrier-class Delta-VX. My second time? I rescued dozens of sentients from the Salik Motherworld, and I did it in a room filled with those machines. The third time I was in the room with one of these _kuh wu_ machines, I still managed to slam the needle of a KI gauge off the far side with it churning away at full power, sitting right next to me while I was demonstrating the strength of my precognitive abilities.
"As for this actually being a suicidal idea?" she repeated, hands resting on her hips as she studied the corporal. "I'll tell you something about the future, Xhuge. It's a suicidal idea for the _Salik_ to go to war because by their own efforts, they will destroy themselves. But they are doing it, and we have to stop them. At the right time, in the right way. Now, we have two more days before we'll be in striking range of the new hub. Let's keep working at this."
Dinyadah looked up from her workstation screen at that. Like everyone else on the ship, she traded duty posts every two hours, including stints on the bridge crew. "Sir, I thought the hyperwarp drive was faster than that. When I was last on the bridge, we were only thirty-nine light-years away from the target zone."
"We could get there faster, yes, but right now, some of that fleet Xhuge's worried about is still within counterstriking range of the hub," Ia said. "Not to mention hyperwarp uses almost three times the fuel. If we don't need to hurry, then we travel FTL. That gives the enemy fleet plenty of time to move on and be well past the turnaround point by the time we come diving in." Looking at the others, she dipped her head and offered some sincere praise. "You've done very well so far. I can't wait to see what else you can do."
"Oh, how patronizing. Are you sure you can't do this yourself?" Rico asked her.
The others gave the two of them watchful, wary looks, not quite sure why Lieutenant Rico was being belligerent. Ia knew why. It wasn't just that he didn't believe in her cause, yet—he didn't—but also because he didn't like being asked to spy on her. At the same time he didn't want her to become suspicious of his being too agreeable to her plans right from the start. She knew he was smart enough to know that too many spies tried being best friends with their targets too soon, and that he knew she was smart enough to realize it, too.
"Some of it, yes. _If_ I had the time to spare, I could do some of it," she finally admitted. "Most of it, no...and what little I could do, I do not have that time to spare. Every single person on board this ship has been hand-selected because each one of you brings a special skill or ability or knowledge to this crew to do all the things that one person, male or female, cannot do on their own.
"It doesn't matter who that one person is, Lieutenant Rico," Ia said. "We will all work together, and in doing so, we will get the job done."
"Any job where I get to destroy the Salik, I'm on it," Al-Aboudwa stated. He looked up at both officers, first Rico, then Ia. His tanned brow creased, and his thin lips twisted in a grimace. "I _hate_ them. They ate my grandfather and my great-uncle. They were merchanters on a supply run into the Blockade Zone, and got caught by a Salik ship trying to make a run for it. If I could put on a p-suit and dance on the corpses we left back at the first site, I'd do it. You need me to fight them, Captain, I'm on your side," he promised, hands balled in fists on the table. "Anytime, anywhere, I'll do whatever it takes."
The others nodded slowly, agreeing with his vehemence.
Ia shook her head. "No, Private. Don't hate them. Yes, they're vile, psychotic sentient-eaters. But don't hate them. Pity them for bringing their deaths upon themselves."
Helstead snorted. As the others glanced her way, her shoulders shook. "Pity them!" she giggled, then thumped her fists on the armrests of her chair, laughing outright. "Oh, God, that's mucking funny!"
"What the...?" Xhuge asked her, bemused. Helstead gasped for air, shaking her head. She covered her mouth, then her eyes, snorting and chuckling in her mirth.
"She's talking about the xenopsychology of the Salik," Rico said over her giggles. He wasn't quite laughing, but his mouth had twisted up in a smirk. "Pity is one of the worst insults you could give them. To a Salik, it means the prey they've been pursuing has turned out to be utterly unworthy of the time and energy spent on them."
"Quite right," Ia agreed. "Though I do mean it in the Human sense of the word, via compassion. Alright, Helstead has her next split shift on the bridge coming up, and Sergeant Maxwell hasn't had his dinner, yet. Let me know if you make any breakthroughs, gentlebeings."
Sobering somewhat, Helstead raspberried that idea, then sighed and stood up. "Right. Off to be an officer, and do all sorts of officerish stuff. In other words, be bored out of my mind while someone else flies the ship. I don't suppose I could learn how to fly it, Captain?"
"Not this year, Helstead," Ia countered firmly, gesturing toward the briefing-room door. She nodded politely at the others as Helstead preceded her. "Keep up the good work, meioas."
It wasn't until they were almost to the bridge that Helstead frowned and turned to her. "Wait, you said I might be able to contribute. All I did was sit there, listen, and crack up at the end."
"You did help, at the end. You helped me get my point across about the Salik," Ia said. "The net result won't have an impact for a few more years, but it'll be a valuable one in the end."
"Oh, mucking hell—you mean that stupid song about the butterfly wings causing a hurricane halfway around the world," Helstead groaned. She drew in a breath to speak, but Sergeant Maxwell's voice interrupted whatever the lieutenant meant to ask, echoing down the hall, and from her bracer.
_"All hands, duty changeup in five minutes. I repeat, duty changeup in five minutes."_
Sighing, Helstead gestured at the door to the side corridor. "I'd better go hit the head. It's been looking at me funny. Not quite like your first officer, though."
"He can look at me all he likes. He knows better than to touch," Ia said.
Helstead lifted her brows. "Have you told him that?"
Ia shook her head, but not in negation. "He did, once, and ended up getting sucked into a precognitive attack."
"What, like that Fire Girl vision thing?" Helstead asked her. She shuddered. "That was nasty. I didn't like it when it happened to me, and I don't like that I keep seeing it in the back of my brain whenever I get close to that restricted cargo of yours. That's just creepy."
"What Harper went through was a thousand times worse," Ia muttered as a crew member came up the hall, heading for the operations post to relieve his partner. "That's why he threw my boot at me, and why I left it there for him to throw at me, so he wouldn't risk suffering again."
"Duly noted, sir," Helstead agreed. She lifted one brow. "Just combat boots? Or will, say, a tennis shoe do?"
"Anything but a spiked heel, Delia," Ia joked dryly. "You throw too hard."
# CHAPTER 7
_Like so many other points in my career, those first engagements were cases of hurry up and wait, wait, boring wait...punctuated by screaming chaos and danger, followed by repairs and yet another wait, wait, wait. Still, I was rather..._
_...What? Was it fun? What kind of a...? Why do you keep asking me things like that? Look, provided you're not a psychotic, chaotic, sociopathic serial killer, war is not fun. Yes, there may be a few moments of exhilaration when you defeat a dangerous opponent or overcome a difficult obstacle, but the rest of it is not fun. I was rather proud of my crew for coming through with so few scrapes and bruises. I just wish they hadn't had to suffer any at all._
_~Ia_
FEBRUARY 7, 2496 T.S.
INTERSTITIAL SPACE
_"Captain, this salvage is going a little harder than anticipated. We have extensive debris blocking the corridor,"_ Corporal Puan stated over the comm link. _"Requesting a reroute of some sort._ "
The view from his helmet showed the debris in question, a tangled mating between a crumpled bulkhead and two support struts. The spaces left would have been difficult for someone in a pressure-suit to navigate, never mind the bulk of a ceristeel-plated mechsuit. Nodding to herself, Ia replied, _"Understood, Corporal. Stand by while I check for alternate routes."_
Closing her eyes, Ia dipped into the timestreams. Specifically, into her own. From there, she could see the branching paths of her own observation point, cycling through the choices ahead of Puan's salvage team. Some of them were a little foggy, but they could...
_Fog?_ Ia thought, for one puzzled moment. _Why would my vision be fogging ov—oh_ shakk. Surging back into full consciousness, she slapped the comm relays for both ship and boarding party. _"All hands, we have incoming! I repeat, we have incoming, with anti-psi machines!"_
_"Captain?"_ Puan queried. _"What's going on?"_
Anti-psi machines were the only things she could think of that could obscure a previously clear moment in time. As the fog rolled closer, Ia sorted out one of the more blunt approaches. _"Corporal, use a claymore, angle it into the right-hand wall from your position. Smash in and grab the whole cluster; don't bother to halve it. I want you back on the ship in five minutes."_
"Shakk _—that doesn't give us—go go_ go! _Togama, Tormez, one of you get me a claymore, triple-time it!"_
Ia dialed down the volume on his channel, letting it chatter in the background. "Yeoman Sangwan, move us right up against that section of the station; I want us covering that boarding shuttle and that airlock with our own hull. Private Sung, seal up all pods and surfaces on our five o'clock, in case we have to scrape our way to a dust-off. Launch three scanner probes in the triangle and push 'em to the zone edge. Private Ng, start running spatial analyses as soon as Sung's probes are away. I want to know where the ships are coming from, and I want to know it before they know our position.
_"Yeoman Nabouleh, prepare to disengage the moment your team is back on board,"_ she ordered over the comm. "All eyes to the boards, all thoughts on your tasks."
The view from Puan's helmet pickups rocked a bit from the explosion. A yellow telltale popped onto Ia's top row of tertiary screens; Private Tormez hadn't hung back far enough, and had taken a little bit of damage to the knee-joint of her armor. Toggling the view from her teammate's camera, Ia watched her flex her leg joint a bit, then move back a little.
From her slight limp, the joint wasn't responding perfectly, but it was able to move. Togama slapped her on the pauldron and moved forward with Puan and his teammate, Franke. Satisfied the private was still mobile, Ia turned her attention to her main and secondary screens. Nothing happened for the first minute; nothing in the star-studded black changed, if one dismissed the vector-tumbled debris from their initial attack.
One minute turned into two minutes...three...
"...Sir?" Private Ng asked, glancing over her shoulder. "Is something supposed to be happening?"
"Eyes to the boards," Ia ordered, repeating the Space Force Navy's mantra. "Thoughts on your tasks."
Just as Ng turned her head back, the navicomp beeped. "Sir! Incoming—173 by 147. Probe just picked it up, sir; we have partial cover from the station. ETA to our midpoint...five minutes."
That was a lucky break.
"Sir! We have movement on the station hull," Private Rammstein warned Ia from his post at the operations station.
"Good catch, Private," Ia praised. Ng blushed and returned her attention to her screens. Ia shook her head. "Keep an eye on the outer danger, Ng. Rammstein, what's on the hull?"
"It...ah, looks like a drone, sir," he said, tapping his controls. Normally, he was supposed to be monitoring the ship's systems, but two of his tertiary and his right secondary screens had been trained on the battered communications hub. "I can't get a good angle for a reading, but, it's headed straight for the shuttle."
"It's probably an automated hull mine," Sung said, eyes and hands working to try and get a gunnery pod's view of the scuttling automaton.
"Can a gunner pick it off?" Ia asked him. The fog was now thickening in her mind and starting to hurt.
"I think—"
" _Shakk!_ Captain, revision, ETA _seven_ minutes. Bogey's as big as a Battle Platform, sir," Ng corrected tersely. "Just got the stereoscopic off the third probe. It's farther out, but big as hell and headed our way."
"Helm to my control in twenty, Sangwan," Ia ordered, refitting her left hand into the control glove.
"Helm to yours in eighteen, Captain," Sangwan confirmed.
_"Aquinar, fire!"_ Sung snapped into his headset. Seconds later, something _boomed_ against the hull, rattling it. A dozen telltales streamed up Ia's left secondary screen, most amber, but a couple red. "Uh, sorry, sir!"
_"Holy fractured—! Charlie Papa, Charlie Papa! We're not all on board, yet!"_ Puan hollered across the comm channels, using his phonetic call sign to get their attention. _"Franke almost had her arm ripped off! Her suit seals are red!"_
"Sorry, sir," Sung apologized. "I thought it was a clear shot."
"That'll come out of your pay," Ia quipped. The throbbing in her head was getting harder to focus through, but she knew the databanks were still safe. She could _see_ them in the distance downstream and that they would extract the information needed. It was the local moment she couldn't foresee. Switching channels, she addressed Puan. " _Get on board, meioas. We've a whale-sized bogey ETA in six._ Sangwan, I have the helm."
"Aye, sir."
"Captain." The call came from the comm tech, Private Mysuri. "Infirmary is getting reports from all over the ship of headaches. Psis only."
"Tell the doc it's a side effect of the anti-psi machines. Have her prepare to receive injured after maneuvers." She started to say more but was interrupted.
_"Nabouleh to_ Hellfire, _all parties are boarded, sealing and sailing. You'll need to move the ship if I'm to dock."_
_"Moving now,"_ Ia told the shuttle pilot. She couldn't use the insystem thrusters without risking damage to the shuttle, but she could pulse the FTL panels. _"All hands, brace for maneuvers, ten seconds."_
She spent those ten seconds calculating the angle and the distance, programming the options into her console. Ships could be piloted manually, or they could be programmed. At the end of them, she activated the panels along one side of the ship. Everything slammed to the opposite side as the ship squirted away from the crippled station. Squirted, and scraped, sending new yellow telltales scrolling up her tertiary screens.
For a moment, the interior safety fields kicked on, pressing in on all sides and pinning everyone in place. Then all the panels flared; greasy on all sides, the ship's momentum stopped. Her programmed path had kept the bulk of the damaged comm hub between them and the incoming ship, while leaving plenty of room for the shuttle to maneuver. This time, with the field fully enveloping the ship, there was no jolt.
There was also no headache for that brief moment. Just as Ia widened her eyes in realization, the fields cut off—and the headache stabbed back into her brain. She panted for a moment, mastering the ache, then grinned slowly. Ferally. "Well. _That's_ good to know. Sung, tell all the gunners to make sure they've synched their weaponsfire to the FTL field...and yes, those new scrapes are coming out of _my_ pay. _Nabouleh, dock as soon as possible._ "
_"Cargo's on the shuttle, sir,"_ the yeoman stated. _"We have injured on board. ETA one minute."_
_"We're going to have some odd maneuvers in a few moments,"_ Ia warned her. Not just her, but the rest of the ship, opening the channel to a shipwide broadcast. _"All hands, stay locked and webbed. I repeat, stay locked and webbed. The headaches are part of the anti-psi machine's side effects."_
"Captain, incoming is slipping its position," Private Ng warned Ia. "I think they're trying to get a clear shot."
"Thank you, Ng." Gaze going back and forth between the enemy ship and their own shuttle, Ia waited tensely. Finally, a distant _clunk_ echoed down the length of the _Hellfire_.
_"We're in, sir,"_ Yeoman Nabouleh informed her. _"Docking clamps have engaged."_
The telltales for the bow shuttle bay lit green on Ia's upper leftmost monitor, confirming her words. Ia tapped the controls and slipped her left hand sideways and down. The FTL panels greased them that way with another abrupt jolt from momentum, and another hard squeeze from the black bubbles dotted around the cube-shaped room. Another flex of the full field cut out the anti-psi headache for a moment.
Taking advantage of their second dead stop, Ia gently shifted the ship, turning its long axis to follow the flight path of the incoming enemy vessel. Once it was positioned, she reapplied the unidirectional field, cutting off the headache. A touch of the controls configured the field to respond on the microscale for the smallest movements possible.
Once again, the headache went away. The release of mental pressure threatened to make her giddy. Focusing, Ia flipped halfway into the timestreams, and discovered she could now see everything _but_ the Salik ship. A quick survey showed three possible escape routes. Ia checked them against the needs of the future.
"Uh, Captain?" Sangwan asked, his voice rising a little. The worry in his tone snagged her attention. "You _do_ know we're sitting ducks if we're not moving, right?"
"Shhh." Dipping fully into the waters, Ia double-checked her findings. She had the time for it; things moved faster on the timeplains than they did in reality when she fully submersed her mind.
Using the hyperspace engines, either to escape or rift the ship, was one possibility. Using the main cannon was another. Using either of those, however, would leave the Salik more prepared for their next attack. That meant using slip-and-run tactics.
Opening her eyes, Ia heard Ng announce, "Three minutes to midpoint, twenty seconds to a clear shot, Captain."
_"All hands, brace for acceleration."_ That was their only warning. Shoving her hands forward, one in the glove, the other on the controls with a sliding tap to remove the microscale restrictions, she pulsed the field forward. The cushioning in her seat saved her, as did the interior safety fields. Even with the field greasing the laws of physics around the ship, those fields had to pulse to move them forward, alternating with the insystem thrusters, and that meant some of the inertia did get through to the crew.
Ia was used to the drag of extra gravity on her body; it was part and parcel of growing up on Sanctuary. But even for her, the effects of their acceleration could be felt. Not only was it hard to draw breath, but the colors started to fade around the bridge. Greyness crept into the edges of her vision, warning her of blood loss to her brain. Ia cut the ship's acceleration for a second, then slipped them up, the only safe direction at that speed.
A split second later, the monitors showed a beam of bright yellow light lancing just past their hull. Around her, she could hear the deep breathing of her fellow bridge crew members. Rolling the ship slightly, again she slipped them up, relative to their new position. Again, they dodged another laser shot from the enemy ship.
"Sir!" Sung grunted. "Do we...return fire?"
"Lasers only," Ia ordered, struggling for breath herself, "but don't strain yourselves..."
The third time, she pushed forward again, accelerating them once more to the point of greyed vision and difficulty in breathing. This time, several beams of light lanced around them; had they slipped sideways, they would have been hit. The first two attacks had been ranging shots; having measured the length of their fishing pole as it were, the Salik now wanted to have them for lunch.
Disinclined to oblige them, Ia rolled the ship and slipped toward their dorsal side the moment those beams cut off. Most lasers could not be fired continuously for more than a dozen seconds because their conversion ratio still permitted a large percentage of their energy to be converted into localized heat. Even with thermal engines soaking away the excess heat, there was an upper limit to how long those lasers could be used. That included the workings of the Salik ships.
Ia used those brief breaks to effect their escape. Varying her angles, she slip-strafed one way, then the other, accelerating in jolts with barely enough time to recover between each one. Some of the lasers struck, but only briefly each time. Two, three, five hull segments flagged yellow. One turned red.
It was hard to say what their own lasers hit; the gunners could only fire when they were free to move without acceleration thwarting their efforts. They did hit something on the capital ship giving chase, but the navicomp's analysis suggested it was just a few scorched sensor panels.
The stars on the screen flashed in bright dots and turned into slow streaks. Ia increased the field strength to port, swaying everyone in their seats, though the G-force was mild compared to their actual escape. Another shift upward added to the gravity of the deckplates, then she straightened out their course.
_"All hands, you are free to move. Medical team to the bow shuttle bay, stat."_ Switching off the intercom, Ia added, "Sangwan, prepare to take the helm. Ease up to full speed; now that we're faster-than-light and sideslipped from our last known course, there's no need to rush. Ng, double-check our heading; we're aimed at a system with a Dlmvla presence, yes?"
"Taking the helm in twenty, sir," the yeoman said, straightening in his seat now that gravity was no longer pulling them this way and that.
"Correct, sir," Ng told Ia, checking the information on her bank of screens. "If we stay on this heading another seven light-years, we'll reach the Llong-Jul 3127 System. Three gas giants and five Dlmvla mining colonies. Co-patrolled by Solarican forces, Captain, since they have a gas-mining station there as well."
"Transferring to your control in five seconds..." Ia murmured, transferring the control systems to his station. "Sangwan, you have the helm."
"Aye, sir. I have the helm," he confirmed.
Ia's headset came to life, projecting Helstead's voice in her ear. _"Helstead to Captain Ia, is it safe for me to eat again? I'd like to finish my lunch."_
She smiled slightly. _"You are cleared to eat, Lieutenant Commander. You have fifteen minutes before you're due back up here."_
_"Yay! Thank you, sir. Helstead out."_
"Message for you, Captain," Rammstein said.
"Patch it through," Ia told him.
_"Captain, this is Corporal Johnson down in engineering,"_ another male voice stated, this time in her right ear. _"We have some FTL field fluctuations, and several aft sensors are on the verge of failing. When are we going to be stopping to fix them, sir?_ "
_"You get an hour-long stop in just over half an hour from now to get the field stabilized, then you'll have to wait seven more hours for the rest."_ Ia shifted her attention to Ng and Sangwan. "Sangwan, continue on course for System Llong-Jul 3127. Don't push the panels too hard. Ng, plot him a course that brings us in close to the Jul II mining colony. I'd like to buy some hydrofuel from them, keep our tanks topped up while we're making repairs.
"Mysuri, have, ah...Private Smitt file a fund-transfer requisition form with Admiral Genibes," Ia added, having to think for a moment about who was on duty in the clerk's position. "Tell him it's for purchasing hydrofuel from the Dlmvla. When it's ready to send, have Sangwan drop out of FTL for a clear hyperrelay, and tell Corporal Johnson that's when he'll get his hour. Rammstein, anything that needs immediate repairs at that point, coordinate with Johnson, with a focus on the fields.
"Beyond that, if an emergency crops up, wake Commander Harper and have him deal with it," she added. "Otherwise, it's Helstead's watch, because in fifteen minutes, I'm going back to bed."
A couple minutes passed as the bridge crew quietly tended to their work. Rammstein finally shook his head. "...I knew the specs. I mean, I _studied_ the specs, learned 'em inside and out...but I never thought we'd actually outrun anything as fast as that!"
"I'm just glad we did outrun 'em," Ng muttered.
"Heh," Sung chuckled. "I'll bet _they_ weren't expecting to be outrun, especially as they were already moving when we took off."
Sangwan shook his head. "They didn't outrun us because they didn't want to lose sight of us. Now if we'd both been at a dead stop, then we'd have totally wiped their asteroids in our wa...uh..." He blushed, glancing briefly at Ia before returning his attention to his monitors. "That is...sorry, Captain."
"Relax, I don't mind a little blunt speaking on my bridge," she reassured Sangwan. "We'd have wiped their asteroids in our wake from a double dead stop, yes. But most of that is because we're less than a seventh the mass of that capital ship. And we didn't get away cleanly," Ia added, nodding at her lower rightmost tertiary screen. "The Infirmary reports three cracked ribs, a bitten tongue, and Private Franke has a dislocated shoulder and two compression fractures. By preference, I'd like to limit the number of Purple Hearts my crew earns."
Sung snorted. "All you have to do is tell us when you see it coming, sir," he stated. "You've been dead-on accurate about everything so far."
"Not entirely dead-on, Private; I only deal in probabilities, and free will still plays a big part in all of this. I can tell you what the biggest chances are, and how to influence the numbers, but God still rolls the dice, not me," Ia said. She spoke quietly, letting her gentle tone convey the seriousness of her message. "We _will_ get injured in the course of this job. Some of us may even die...and I will not stop that from happening when it must, because my job is to make sure those who do die will perish only because they are doing what _must_ be done, regardless of the cost. In the right place, at the right time, getting the right job done."
Her words sobered the others. They exchanged looks, glancing at her before returning their attention to their workstations.
"I know that isn't what you wanted to hear," Ia told them, looking up from her screens. "The only guarantees I can give you are that I'll be taking the same risks as the rest of you, and that I'll be doing my best to minimize those risks along the way. But minimized risk is not the same as risk-free."
"And you just accept that, sir?" Ng asked, voicing the concerns of the rest. "That some of us will be hurt and killed?"
"It's the same realization that _every_ officer has to face, Ng," Ia told her. "Mine is just shoved home all the more thoroughly because I can _see_ the results in advance. But I'll tell you what I've told Chaplain Benjamin through the years. I'd rather be damned for _trying_ to do what's best and right, even if I fail, than be damned for not doing anything at all.
"I handpicked this crew because I know each one of you feels that same way deep down inside," she stated, stifling a yawn with effort. She didn't want her crew to think she was bored with their conversation, and she didn't want them to think she wasn't at her sharpest, even when tired. This wasn't the first time she would have to interrupt her sleep cycle for combat, after all. "Now, eyes to your boards and thoughts on your tasks."
FEBRUARY 8, 2496 T.S.
JUL II MINING STATION
LLONG-JUL 3127 SYSTEM
"Your request for round rocks we have processed," the Principal Nestor of Jul II stated. She dipped her head, with its multilensed eyes, and uncurled a clawed hand-thing. The gesture finished somewhere beyond the pickup range of the commscreen. "But this peet-zah thing I am uncertain. Datafiles indicate it is a Human food. We do not available have this Human food."
Ia smirked. She was taking this call in the briefing room forward of the bridge after a good night's sleep. "That's because you are fat, and covered in velvet."
The Nestor lowered her head farther, eye-skins puckering a little. "I do not...Ah. Illogic. You are courting me?"
"I'm declaring war on you," Ia countered calmly. "I think you're too purple, and you smell when you sneeze."
The Principal Nestor blinked a sideways-sliding membrane over her compound eyes. "Dlmvla do not sneeze. Regrettable it is, you cannot breathe with us. I think you are...metal foot garment. With lactations. For a Human. Anything else?"
Ia dipped her head in turn. "Nothing else. I look forward to firing upon your people in unprovoked madness, then inviting you into my home. Thank you for handling my extra request, Principal Nestor. I hope you like the vidshows I used for payment. They're more than old enough, copyright doesn't apply."
"Comedy entertainment transcends madness between our species. Copyrights are madness to exist at double the artist's life. Another point of similarity. Feathery secretions upon you," the Nestor added. "End transmission."
Ia closed the channel on her side as well. Shutting her eyes, she sat back in her seat at the head of the table and contemplated her efforts on the timeplains. _A bit of intrigue...but not much. Not yet. I'll have to work harder at provoking them. Get them to spread the word about how strange I am._
The Dlmvla were "neutral" where the Salik were concerned. They had agreed not to supply the Salik with anything but weren't contributing ships to the Blockade, either. As methane-breathers, they had long ago figured they were low on the lunch menu, as it were. At least, compared to the oxygen-breathing species.
Of all the races, only the Chinsoiy were truly immune to the Salik appetite for living sentient flesh, but then they were silicon-based life-forms. The Chinsoiy also required daily irradiation for survival, at levels that would kill a chitin-covered K'katta, never mind a softer-skinned species like Humans or the frogtopusses. But once the Salik whittled away the Terran and V'Dan Humans, the Solaricans, the Tlassians, and the K'katta, they would make war on the Dlmvla—whose flesh would be deemed edible, even if acquiring it while the owner still breathed was problematic—and then they would destroy the Chinsoiy to make sure no one in local space could thwart their ambitions. Only then would the Salik finally fall upon the Choya, their own allies.
By then, the Salik would be nigh unstoppable. _If_ Ia let the future unfold in their favor, that was. They were more prepared than anyone but she herself knew in the Alliance. Not even their Choya collaborators knew just how much effort the Salik had put into building their forces for the coming war. Deep, slow, treacherous plans. The only thing holding them back was a lack of personnel to command all the equipment they had mustered, even with Choyan assistance.
The door slid open. Rico walked inside, along with Xhuge, MacInnes, and Al-Aboudwa. All four were carrying portable workstations. They clipped them onto the table, claiming seats down either side. Ia smiled. "I'm glad you could join me, meioas."
"Ha-ha," Rico returned dryly, opening his workpad screen. "How funny. Shall we get down to business, Captain?"
"Certainly, Lieutenant. What've you found?" she asked them.
MacInnes spoke first. "They're using different code words for this set of datanodes—which would be rather smart of them, having two sets for steganography—but I was still able to pick out the nouns. Even some of the verbs, now that I'm getting a grip on how they talk and think."
Xhuge, who had linked his workstation to the briefing-room screens, looked up at the starcharts his efforts displayed. "Good thing, too, since they're on the move."
"Even with that difficulty, the work still went a lot faster this time," Al-Aboudwa stated. "The code words were different, but the security protocols were the same. Decrypting everything took a while, but only a fraction of the first time."
Ia smiled wryly. "Long enough for me to get a good night's sleep, though. Thank you."
Xhuge nodded. A touch of the controls highlighted three regions, one in yellow, one in orange, and one in green. A second touch overlaid a purple ring, which bisected the green blob. "Green is where they were, which corresponds with the hundred-light-year search ring we'd established. Yellow and orange are the two locations they were directed to travel to. Unfortunately, we don't know which."
"I hope you indeed got a good night's sleep, Captain," Rico told her. "We're expecting you to pull the correct location out of your magic hat."
"Not only a good night's sleep, but a fair bit of battle preplanning," Ia said, studying the charts. She rose from her seat and moved over to the main screen behind Rico and MacInnes. "Anything distinctive about either of these regions? Local stellar features, planetary arrangements, visible nebulae?"
"Not a damn thing, sir," Xhuge replied. "Either you pull a rabbit out of your hat, or we're stumped."
"There might be an easier way, Xhuge. With the extra comm nodes we pulled, I've been playing some pattern analyses on the extra data," Al-Aboudwa said. A tap of his workpad pulled up a set of charts on the screen next to the one Ia was studying. One of them, he enlarged, displaying the spiked line squiggling across time. "The anti-psi manufactory sends out certain signals once every eight-day Salik week. We'd have to wait five more days for the next spike, then hit another comm hub, but..."
"We don't have five days," Ia finished for him.
"Here's a brilliant idea," MacInnes offered, catching their attention. "Why don't we just remind ourselves to put up a big note on the briefing-room wall saying _which_ zone we went to, and just have you look at that precognitively, Captain?"
Twisting in his seat, Rico looked up at her, brows raised. "Good thought, MacInnes. I don't see why it couldn't work. Captain, care to take a peek?"
Ia held up one hand, thumb and forefinger pinched closely together. "There's a slight problem with that. I don't see _one_ timeline. I see _all_ of them." Turning back to the starchart, she traced her finger over the screen, electrokinetically drawing a set of lines. "If we go to fight at location yellow, here, then that's fine and dandy; we all do our jobs right, the main manufactory center gets destroyed, and we go merrily on our way to this green blob here, which will represent our next task in the continuum.
"But if we start at this orange spot here...we all do our jobs right, the main manufactory center gets destroyed, and we go merrily on our way to this green blob here, which will represent our next task in the continuum." Turning back to the others, she sidestepped the drawing and extended her arm, tapping the yellow and the orange blobs, now connected to the green one by a pair of white lines. "The problem is, I see _both_ locations as viable possibilities. At about a fifty percent split, no less.
"This green blob down here, our next task, is actually _two_ green blobs, overlapping each other because they're almost identical in every single way...except one has a sign in the briefing room that says 'We defeated them at Yellow!' And the other says 'We defeated them at Orange!' I see both, fifty-fifty. Equally valid, equally probable," she concluded, shrugging. "I need something _more_ than a sign that could be posted in either possible reality. Something distinctive enough to tip the balance either way. Even an unusual star formation from the local point of view would give me a clue."
"Captain, I told you; there's nothing special about either location," Xhuge reminded her. "No stellar phenomena, no planets, nothing. They're both interstitial space, they're equidistant from ordinary, uninhabited star systems with large amounts of water ice in their Kuiper belts and Oort clouds, and no reason for them not to go to either spot. For all we know, the commander of the manufactory vessel flipped a _coin_ to decide where to go."
"If they had, we could almost have you watch the coin being flipped," MacInnes joked, folding her arms and leaning her elbows on the table. "Except it's on a mobile station that gives everyone a raging headache when you get close. Or doesn't seem to exist, or however that works."
Al-Aboudwa sighed and slouched back in his chair. "So how are we going to find where our target has gone?"
"The same way I've been finding it all along," Ia said. "Narrow down the search sites—which we have done—and investigate it for blank-space anomalies. Negative proof is still a proof, in a way."
Lieutenant Rico tapped two of his fingers lightly, thoughtfully on the table. "Captain Ia...you said a few days back that you watch _people_ , not things, correct?"
"Correct," she confirmed. "I could literally enter their life-stream and experience snippets of whatever they experience, if I wanted to."
"And Private MacInnes sees nouns..." he mused.
"Oh. Oh!" MacInnes said, sitting up with wide eyes. "G'nush-pthaachz Mulkffar-gwish! He's the requisitioning officer signing most of the orders we found. Maybe you _can_ see him flipping a coin?"
Ia gave her a flat stare. Not because the idea was absurd, but because the way she pronounced Sallhash phonemes was absurd. Ia's odd sense of humor kicked in, quirking the corner of mouth and brow. "I think I've finally found someone with an accent even worse than mine. I don't have even a tenth of your command of their language, but your accent amuses me. Thank you, Private."
"The question is, can you take someone else with you into these timestreams?" Rico asked her as the private blushed. "Maybe a clairvoyant? I'm presuming it would have to be a fellow psi."
"I could take _you_ ," Ia countered flippantly. "The closest you get to a psychic ability is your uncanny aptitude for languages...which isn't a bad idea," she allowed, thinking it through. "I can get _into_ an alien's life-stream, and hear their thoughts, but I'm not a strong xenopath, and I'm no linguist. The Tlassian and Solarican mind structures are similar to Humans'. K'katta, Choya, less so. Gatsugi is a headache of nuances to sort through. Dlmvla as well. But Salik? They're cold and brutal, aggressive and cunning. Wading through their life-thoughts is like wading through cold, opaque slime.
"You may not have the mind-set of a Salik, but you know their language, which means you're closer to understanding their thoughts than I am," Ia allowed. Stepping closer to MacInnes, she leaned her palms on the back of the other woman's chair, giving her 1st Platoon lieutenant a frank look from across the table. "But if I take you onto the timeplains, Oslo, you're no longer a neutral observer. I don't think the people you report to would be happy about that."
"Wait a minute," Xhuge said, glancing between Ia and Rico. "Are you saying the lieutenant's a spy? We have a spy on board?"
"He's one of several on board," Ia corrected. "I know who each and every one of them is, and who they report to—and before you get in a panic, they all report more or less to the Admiral-General in the end, so they're all internal spies. Some more directly than others." Shaking her head, she straightened. "I think I'll just take a solo dip. That way, you don't compromise your neutrality, and I don't exhaust myself trying to push two or more people through that anti-psi field."
"I think I _should_ go," Rico countered. "According to what I was told, you haven't shown anyone on the Command Staff what it's like to be on the timeplains. Now, why is that?"
"Because it would be considered an undue influence upon their decisions," Ia said. "The same with yours. There _might_ even be an accusation of Fatality Thirty-Eight, Bribery, levied at me if I tried to use it to convince my superiors to do as I request because of the fear that I'd show them lottery numbers or stock-market results. I'd rather not kick my career out the nearest airlock."
"I still think I should," he asserted. "In fact, I insist, Captain. Bribery doesn't even come into it because there's no way in hell I'll ever play the Salik version of a lottery."
Ia sighed and rubbed at her brow. She had foreseen this as a possibility. It had the risk of complicating things, but it also had the chance of getting him firmly on her side. _This will be tricky. How to show enough to convince him I'm being honest without showing him so much he either resists or the Admiral-General claims undue influence...or showing so little, he knows I'm holding back. Either way, he's going to see things he doesn't know he didn't want to see. Oh, this'll be fun..._
"There's an old saying that applies in this instance, Lieutenant," she warned him. "'What has been seen cannot be _un_ seen.' You insist a third time, I will take you...but it'll go on the record as being _your_ idea, of your own free will, with no accusations of Fatality Thirty-Eight ever levied my way. And you'll do so by understanding that a trip onto the timeplains _will_ change the way you look at things from here on out."
"I'm already compromised as a 'neutral observer' since you know about me," he pointed out. "I don't see how much worse it could get."
Xhuge winced, and Al-Aboudwa whistled softly. MacInnes shook her head slowly, a pitying look in her eyes.
"Tell me you did _not_ just say that, sir?" Al-Aboudwa half joked. "That's like giving Murphy a wedgie, then asking him if it was good for him, too."
" _Never_ taunt Fate, sir," MacInnes agreed. "Even a lowly grunt knows that much."
Xhuge choked, smothering his laughter behind his fist.
Dipping his head ruefully in acknowledgment, Rico murmured, "I'm sure I did just yank up on some devil's undershorts, Private...but I still must insist, Captain. Take me onto these 'timeplains' of yours. I want to see what you see."
"Witnessed?" Ia asked the others.
"Witnessed," MacInnes agreed. Xhuge and Al-Aboudwa nodded, adding their confirmation.
"Alright then. Be it on your head. There are a few rules of engagement, of course," Ia told them. "Rule number one, no one touches either of us. My gifts can be triggered inadvertently by a touch, so if you have to get my attention, throw something at me from a distance. A datapad, a shoe, even a wrench if you must, so long as it's something small and nonliving."
"That would explain Helstead's comment about throwing boots at you, the other day," Rico muttered.
"Second rule, Lieutenant," Ia said, meeting his gaze. "Do not say the word 'Time' while we're on the timeplains. Time is like an entity where we are going because my abilities and my thoughts will literally be shaping whatever you see. You don't want to sidetrack my thoughts by provoking that entity. It would be like poking a tiger repeatedly. Is that clear?"
"Not really, but if those are your conditions, I will comply," the tall man admitted. "Anything else?"
"That'll do." She tipped her head to her right. "If you'll move to the far end of the table, I'll join you there. That way the others won't be tempted to touch either of us. But they can stay and watch if they want."
"I'd like to watch, Captain," MacInnes said, as Rico unfolded his long, large frame from his chair. "I'm not due for my shift on the bridge for another forty minutes. Did you, ah, want me to come along? I know what this Mulkffar-gwish fellow looks like. Would that help?"
"I'm enough of a telepath, I could pick it up from your thoughts first, if you're willing," Ia said, stepping back to give the lieutenant room to pass. "I'd rather limit who comes with me onto the timeplains, particularly as we'll be navigating areas clouded by those damned machines. I'm strong, but I have my limits."
"I'm willing, sir." Squaring her shoulders, MacInnes focused her gaze on her workstation screen for a long moment, then nodded. "I've got him in my mind now."
Nodding, Ia moved close. A touch of her hand on the private's wrist was all it took. Part of her gifts threatened to lean over the other woman's life-stream a little too far; Ia reined in the impulse to dive in, and focused instead on her thoughts. The image of a yellow-skinned, broad-faced, ostrich-legged alien came across clearly, clad in a leather-like tan-and-blue uniform dotted with rank markings and the circle-based language of his kind, listing his name and deployment affiliations.
Carefully withdrawing her hand, Ia clung to that image. "To quote the PsiLeague, 'What was yours is still yours. I thank you for allowing me this glimpse of your thoughts.'"
"Not a problem, sir. Um...what will _we_ see when you do this?" the other woman asked.
"Not much. He might react a little, gasp or widen his eyes," Ia said, moving carefully down the length of the table. She wasn't really _seeing_ the table, concentrating instead upon the image of the Salik named Mulkffar-nostril-flap-exhale. Gripping the back of the chair next to the dusky-skinned lieutenant, she turned it enough to drop into it, and stretched out her left arm with her palm up on the table. "You might get a headache from this, Lieutenant. Our thoughts will be racing faster on the timeplains than our bodies will be living out here in the real world, like a modified, unshielded race through OTL. Last chance."
"I'll take that risk, sir." Turning his chair to face her, knees almost bumping hers, he stretched out his right arm, covering her palm with his.
Ia curled her fingers over his hand. "Remember, I warned you. Take two slow, deep breaths, and relax."
"Does that help?" he asked, one brow quirked skeptically.
"Not really," she joked. "You're just less likely to choke."
That was all the warning she gave. Closing her eyes, Ia flipped both of them down and in—and hauled up on her passenger's mind, pulling him out of his own life-stream before he could drown in the overlapping sensations of a thousand potential possibilities.
_"Welcome to the plains,"_ Ia stated as she steadied him. Rico blinked and looked around, clinging to her hand with both of his. All around them was a vast, rippling field of gold-and-green grass crisscrossed in a thousand rivulets, all drenched in bright golden sunlight. Nothing but grass, sun, and water as far as the eye could see.
_"Where...? Or rather, what is this place?"_ he corrected himself, straightening.
_"This is how I most often visualize_ Time." The word rippled grass and streams like a harsh wind. It even seemed to roil clouds across the sky for a moment, before the impression faded. Ia let it fade before speaking again. _"But I can change it. Shape it. Intensify it, codify it, itemize it..."_
The tall man blinked and frowned. _"I don't understand. You say you can see all possibilities. What future possibility is_ this _from?"_
_"Most of them. It's a matter of scale, Lieutenant,"_ she said, and gestured at the rivulet of a stream closest to them. _"That's your life."_ A twist of her thoughts enlarged it until it was a meter deep and wide, large enough to show images flickering in the water. Images from his immediate past. _"I can watch scenes from moments in your existence, past and future. I can even step inside the stream and experience from your own perspective what you've already experienced, or will, or might. And I can shrink it down and trace how your life interacts with others' lives, and how they stain each other in colors of influence."_
She shrunk the streams back down again, then turned his stream blue and a nearby one red, and showed them interacting, staining each stream with hints of purple, one more strongly than the other.
_"Whose life is that?"_ Rico asked her, eyeing the purple-tinted red.
_"Private MacInnes; she looks up to you as a role model. Not a bad choice, either,"_ Ia told him. Erasing the colors, she gestured with her free hand. _"Upstream is the past; downstream is the future. I hauled you out of your life-stream when we first arrived because the moment we arrive is always the present, and lingering in the present creates a doubled, disorienting sensation. In addition to that,_ any _thought you have can trigger an associated memory from the past, rushing those memory-waters downstream into you. Think of toast, and you'll drown under a thousand different instances of eating caramelized bread._ "
_"Lovely. Why do our voices echo so much?"_ he asked next, lifting the littlest finger of his free hand to his ear for a wiggle.
_"We're immersed only lightly at the moment."_ Concentrating, Ia increased her awareness of the timeplains, and with it, his. "...Is that better?"
He wiggled his finger in his ear one more time, then nodded. "Better." Pausing, he inhaled, blinked, then inhaled again. "Amazing. I can actually _smell_ the sunbaked grass. But why is it more blue now than green?"
His question provoked a smile from her. "That's because the local equivalent to grass on Sanctuary is blue. The grass really _is_ greener on Earth."
Squinting against the sunlight, Rico studied the plains. "So, what do we do now?"
"We look for two things. Our blank spot, and our supply-requisitioner. But not like this," Ia said. "Try not to feel vertigo."
He glanced at her—and clutched at her hand again as the grass fell away, replaced by stars. "God!" Rico exclaimed, grip tightening around her fingers. His large, muscular frame floated awkwardly next to Ia's. "Warn me better, next time!"
"You're the one who wanted to come along," Ia pointed out.
Brow furrowing in a frown, Rico gave her a pointed look. "I thought you said your abilities focused more on people, not places."
"They do," she admitted.
"So how is it we're surrounded by a giant star map?" he asked. "Aren't these places, not people?"
"Yes, but they're a composite awareness of the stars as viewed by a large sampling of people's life-streams," Ia explained patiently.
He stared at her. She stared back, lifting an eyebrow in a silent dare to see if he would ask another question.
"...Right," Rico finally muttered. "I'll just shut up now and go along for the ride."
Turning her attention to the stars, Ia zoomed them in toward two patches of misty grey nebulae. "We should have a slight advantage. This moment is the Now." She paused as the stars twinkled around them, then continued. "What we're looking for is upstream, into the past. Not just any random past, but specifically what has already happened in our own temporal lineage. That way we can rule out the fifty-fifty probability of either location, because one should have been selected by now. The difficulties will be: one, finding any life-streams in the anti-psi mist to examine; and two, finding those life-streams whose owners have actual knowledge of where they've gone."
Stopping at the edge of one of the two mist-patches, Ia lifted her hand, summoning a hologram of a Salik with dull yellowish skin, the image she had lifted from MacInnes's mind.
"G'nush-pthaachz Mulkffar-gwish. Or rather," Ia made the image say, with the proper nostril-flap flexings, _"~`Pthaachz Mulkffar^."_ This was all inside her head, constrained only by the limits of her gifts, not the limits of her body. She switched back to her own voice, letting the Salik's face fade slowly. "Keep him in mind."
"Why should I?" Rico asked her. "It's not _my_ psychic abilities being used here."
"No, but your Sallhash is better than mine," she reminded the tall man at her side. Even floating in mental space, he was still larger than her. Proportionately larger, letting her know he was comfortable with his greater size. "You wanted to know what I see, and that means seeing it right alongside me. But this is the Space Force. You pull your weight, even what little there is in this place. There are no free rides here, soldier."
"Sir, yes, sir," he muttered. "Three bags full, sir."
"That only counts if you can say it in Sallhash," Ia quipped. A _snerk_ sound escaped him; a glance showed his mouth twisted in a half smile. She returned it with a wry smile of her own. "Brace yourself, Lieutenant. We're about to get up close and personal with Salik xenopsychology."
A tiny pinpoint was moving away from the cloud. Ia dove them down toward it. The pinpoint became a Salik starship, long and five-lobed. It zoomed close, and the hull vanished, replaced instead by a splash of water, a wavering impression of a corridor, a distortion of two overlapping, separately moving views. Ia felt Rico clutching at her hand, bruising it with his mental strength, and pulled back until they stood once more on the grassy banks of someone's life-stream.
_"What's wrong?"_ she asked, gentling the intensity for a moment.
He blinked and looked around. _"We...I...That was...disorienting. Is that how they view the world?"_
_"Yes."_ Stepping down into the alien's cold, thick water, she tugged him after her. _"We have to wade through several of these lives. Watch your step. I'm intensifying the connection to the timestreams again."_
"Wait," Rico said. "If we talk to each other...will they hear us? Or if we try to talk to them?"
"No. It'd be like talking to a previously recorded vidshow broadcast and expecting the actors to hear you," Ia said. "Whatever reception equipment I have, I don't think it can broadcast back to them. At least, I've never gotten it to work with friends and family. I can't even talk to _myself_ whenever I investigate my own life-stream, which has caused some problems along the way—the proverb 'if I'd known back then what I know now' is singularly unhelpful since my future self cannot tell my past self a damn thing, and I don't have enough seconds to spare to say it out loud in the future, let alone to write it down. Now come, we're wasting our opportunity."
Breathing deep, he stepped into the water, following the tug of her hand. Ia submerged them, tapping into that crew member. Not deep enough to hear thoughts; just enough to see what that alien saw. The view was doubled and too broad for Human vision, with eyes bulging up from the skull, pointing this way and that. She could feel a headache forming from the disorientation of it and knew Rico would be feeling it, too.
This wasn't the first time she had investigated a Salik life, though. Swimming upstream, she skimmed through snippets of past events, a shift cycle beginning, an encounter with an officer on watch. Jumping into that stream, she followed that officer in flashes of past events to the beginning of their shift, and from there, to another officer. But that officer merely came from his sleeping tank, so she followed his life-stream down through his day...until he encountered the ship's captain.
Leaping into his life-stream was easier now that Rico wasn't resisting the disorientation that came with each transition. He did pull himself closer as she pushed upstream into the past, clinging as the turbulence increased. Some of that came from the speed at which she moved. Some of it came from an increasing misty pressure. Rather than letting it push them out of the water, Ia pushed them deeper into the captain's head—and then sideslipped into the navigator's as soon as the captain observed enough of the bridge crew for her to select the right alien.
His thoughts were murky, the terminology and grammar disjointed at first. Eventually, the distortions made sense. _More three years. Service ending, female become. Puddlings teachto eat_, followed by an image of the captain, and a mental hiss of vicious vengeance-and-hunger.
_"I don't think he likes the captain,"_ Ia whispered to Rico, her noir sense of humor surfacing for a moment. _"What do you think?_ "
_"If you'd studied their culture and history, you'd realize most Salik subordinates don't,"_ he whispered back. They couldn't even see each other anymore at this level of xenoawareness, but he was still clutching her mental hand tightly. _"Wait...back it up. I think I can read the coordinates for their heading."_
Ia complied. She couldn't exactly freeze the moment, not with the pressure of that anti-psi misting her mind, but she could replay it in slow, short passes. The navigator looked at the screens positioned over his head, one eye on the actual heading, the other on reference stars and designation numbers.
When she felt him nod, she advanced them upstream again. The mist and the pain thickened apace. Her grasp of Sallhash, written and thought, started to slip. Pushing Rico's consciousness to the fore, she guided him upstream to the point where the navigator heard the captain order the pilot to disengage from dock and tell the navigator what course to set.
She could feel Rico's confidence that knowledge of those coordinates would be enough to place their location. Ia wasn't so sure. Stretching herself, she left him in the navigator's waters and dipped into the captain's thoughts.
_Riptides of sloppiness,_ she heard the captain cursing in the privacy of his mind. _Pity for those prisoners. Now useless dredge-sand._
With that thought came an image of Solaricans, shaved and restrained on tables, their wrinkled heads encased in bands that looked vaguely like the anti-psi headsets Ia had seen back on Sallha at the aborted banquet. Ia pursued that thought, trying to find a time reference. She found it from three Salik Standard days before. Grabbing Rico, she pulled him after her, ignoring his wordless protests.
The pain increased, the farther back they went. It was oddly like G-force sickness. The edges of her vision started to grey out. Each breath became a struggle. But the captain _was_ there...and did come close enough to one of the Salik scientists for Ia to make the leap into her life-stream.
Sexual deviancy among the Salik had nothing to do with methods of copulation. This was a Salik who had chosen to turn female without procreating. Her thoughts were cruel, vicious, and cold. Clinical, in that she thought about the torment of the Solarican prisoners simply as an exercise in observing their reactions to various anti-psi modulation exercises. She felt none of the pity the captain had felt; she did not think her prey unworthy of time or effort. To her, the felinoid aliens were objects to be toyed with until destroyed. Her hunt was all about knowledge, not adrenaline and food.
Forcing back the flow of time, Ia found another Salik, an officer. Male again—cold and brutal, but not nearly so calculating. Then another...and a third, one who was eating a prisoner, a skreeling, shuddering, bleeding K'katta chained with all ten legs sprawled out straight, leaving it no more than a finger-width of room in any direction in which to shudder and move. Rico huddled mentally at her back, no longer reaching voluntarily for the thoughts and words of these tentacle-fingered fiends.
Ia endured the first-person perspective of the Salik's meal, watching them eat because it was a meal being shared by five high-ranking officers. Her detachment was dissimilar to the Salik scientist's; hers was an effort to listen to their words because she had no other choice, not because she was intrigued by what they were doing. Thankfully, with the anti-psi machines pressing around, it wasn't difficult to ignore the visual aspects. It was a little harder to let the flavors and scents fade, but when they did, the mental presence of her lieutenant uncurled a little, listening intently.
What they heard surprised him, for he squeezed her hand tightly. Ia didn't let herself think about it. Instead, she listened a little bit longer, repeating the discussion three full times to make sure both of them heard all of it. Only then did she carefully retreat, moving slowly enough that neither of them would suffer the psychic equivalent of decompression sickness. As she did so, she slid them forward through that officer's timeline, double-checking along the way to make sure his home vessel was indeed headed where that conversation said it would go.
The moment she had confirmation, she pulled back. The grey mist became the star field, became a single watery stream, its surrounding, grassy prairie...and with a final flip, the briefing room. Used to the disorientation that was the return to reality after such a deep descent, Ia inhaled slowly, calming her nerves. Lieutenant Rico's face looked pasty, almost grey in spite of his natural golden brown tan.
"Breathe, Lieutenant," she ordered quietly, finding her voice. "Slow, deep breaths. Focus on the sound and feel of your own breath. Nice, slow, steady breaths, four times in a row..."
Behind Ia, MacInnes rose from her seat, moving over to the alcove by the door where a drinks dispenser had been installed. Rico blinked and complied, brown eyes still unfocused. On the fourth exhale, he shuddered and released her hand. Elbow braced on the table, he lifted his fingers to his mouth, breathing hard and fast through his nose.
Wisely, the private poured and brought back two mugs of cold water. "Drink this, sirs," she urged, offering one cup to Ia and the other to Rico. She had to help Rico lift his to his mouth, his free hand shook that much. "Easy, Lieutenant; let me help you...There, just a sip at a time...There you go, that's the first one."
Once he had taken that sip, the private dipped two fingers in the water, then stroked her damp digits across his brow. Pursing her lips, she blew a stream of air on his forehead. Ia dipped her own fingers in her cup, dabbing it from her hairline in a streak down the middle of her brow, parting her fingers to either side of her nose. She sipped slowly from the cup while the cool liquid on her skin drew some of the heat out of her temples and sinuses.
The water-cooling trick was the same one taught by the PsiLeague, which MacInnes was affiliated with, as well as by the Witan Order, which had trained Ia in the early uses of her gifts. It was meant to help focus thoughts as well as reduce the heat-induced headaches that often accompanied intense psychic efforts. Ia rarely suffered from them, but the anti-psi fog plus the need to shelter and escort another mind through the timeplains had taken its toll.
"...How?" Rico finally rasped, sipping again at his water. MacInnes moved back, giving him a clear view of their CO. "How do you stay _sane_?"
There were several replies she could have made to that. Out of habit, Ia checked the timestreams. Her head still hurt, but the skimming of potential reactions to various responses was quick and easy compared to the effort she had just undertaken. Settling on one of the better answers, she gave it to him.
"Practice, and a lack of time, Lieutenant. I literally cannot afford to waste my time on something as self-indulgent as insanity. I have too many lives to save. Let me know when you've recovered," she added, lifting her cup for another sip. That lifted his head sharply, his gaze meeting hers. "We still need to check the other probability cloud, to make sure that what we heard was right."
_"Shakk,"_ he whispered, dropping his forehead to his palm this time. "I don't know if I can take...experiencing...a Salik officer eating a prisoner again."
The others blanched. Xhuge covered his mouth, and MacInnes swallowed hard. Al-Aboudwa looked away. None of them asked any questions about it.
"Slow breaths, Lieutenant. Focus on the scent of the air, on the sound of it filling your lungs," Ia ordered. She waited until he complied, then said, "Don't worry about it if you don't want to go in again, Rico. It's doubtful we'll have to endure that exact scene a second time, but you don't _have_ to go with me again. I'll admit your ability to translate Sallhash a lot faster than I can was an asset on this trip, and I wouldn't mind having it again, but you don't have to come along." Looking over her shoulder, she added, "Xhuge, Al-Aboudwa, when he's ready, the lieutenant has some Salik coordinates for you."
Xhuge nodded and tapped something into his workpad, readying it for taking notes.
Sipping at his water, Rico managed a question. "How long were we out of it?"
MacInnes shrugged. "A minute? Maybe a little longer?"
Rico looked up at her, then at Ia. "...Only a minute? It felt more like half an hour!"
"If Time didn't flow considerably faster on the timeplains than it does in real life, I would have slit both throat and wrists in despair long ago, because I wouldn't have had the time to find a way out of the coming apocalypse," Ia stated. Draining her cup, she clipped it to the edge of the table and rose. "I'm going to use the head. Take a few minutes to decide whether or not you want to accompany me on a second trip while I'm gone, Lieutenant. There's no shame in refusing if you don't want to go. If you do, expect an even greater headache by the time we're through."
Dipping her head politely, she left him to contemplate the consequences of what else he might see. The odds were very high that he would decline this time. Ia was fine with that since what she could see of his future reactions showed little disturbance in her overall plans. Whether or not he came with her, she would find the necessary confirmations and get the job done.
That was what kept her sane: managing her priorities in the face of the relentless ticking of Time.
# CHAPTER 8
_Historians get to have the luxury of looking back upon the past and pronouncing judgment upon it from the cushioned comfort of their office chairs. Ask any historian when the First Salik War ended, and they'll smugly say when the Salik High Command surrendered to the Alliance generals. Some might be more precise by saying it was when the Salik surrendered "all" of their warships, while others would say the war actually ended when they handed over all their other means of interstellar travel, too._
_Soldiers who participated in maintaining the Salik Interdicted Zone will tell you that war never actually ended. They'll swear it just went into a lull while the Salik nursed their grudges along with their wounds. Having served on the Blockade, I cannot fault my fellow warriors for believing the war, at least for the Salik and the people trying to keep them confined, hadn't ended._
_As for when the Second Salik War began, one could say it began in the Terran Standard year 2496. One could be more precise and pinpoint the first half of that year, which most scholars and soldiers would agree upon. Some would insist my Company's prewar efforts be included in those war-catalyzing and -defining moments, which would narrow it down to the start of February or thereabouts. Officially,it started later, of course, with the open attacks on key Alliance homeworlds...but for my crew, it definitely started earlier._
_~Ia_
FEBRUARY 10, 2496 T.S.
INTERSTITIAL SPACE
SOMEWHERE NEAR THE TLASSIAN-K'KATTAN BORDER
"I have the helm, Sangwan," Ia stated.
"You have the helm, sir," the yeoman confirmed. Unlike the last time he had passed her helm control, his voice was unsteady. "I think I'm going to be sick."
"There's a plastic bag in your rightmost drawer," she offered pragmatically, checking her monitors. "But I suggest saving it for later. We're not engaged in combat yet."
" _Thinking_ about combat is what's making me sick. I really don't like what we're about to do, sir," Sangwan muttered.
"Well, if you'd like to be dismissed from the bridge, now is the time," Ia told him, adjusting the _Hellfire_ 's attitude just a tiny bit more. Opening a comm channel, she contacted the 1st Platoon's lead gunnery expert. _"Captain Ia to Corporal Bagha, what's the status on the special missiles?"_
_"Locked and loaded, Captain,"_ Bagha reported. _"We just finished sealing up the last P-pod bay half a minute ago. Give my team three or four more minutes to get into our gunnery pods, and we'll be ready to go."_
_"You have four. Don't waste them. Ia out."_
The seconds ticked away. She had already given the fifteen-minute warning earlier, and a quick splash through the timestreams showed almost everyone was in place. Those who were in bed had either lashed themselves down with special webbing or moved to acceleration couches; the galleys were shut down, the gunners were getting into their pods, and the stragglers who had needed that one last trip to the bathroom were all but done strapping themselves into place. One of them was Lieutenant Commander Helstead. She hurried in at the next-to-last moment and claimed the backup navigation/scanner post.
Ia opened the shipwide intercom as the short woman webbed herself into her seat. _"This is Captain Ia to all hands. Prepare for combat, and prepare for maneuvers. I repeat, all hands, prepare for combat, prepare for maneuvers. You have one minute, mark."_
Those seconds ticked away as well.
"First drop in ten," Ia warned her bridge crew as she eased back the FTL field. "Get me lightwave, meioas."
Flashing dots filled the bridge. Ia held them just under the speed of light for ten seconds, then pushed them over Cee once more. Ten seconds passed, then she dropped the ship below Cee for another ten. Again, they shifted faster-than-light...and a third time dropped below the grey-flashing threshold.
"...Got it, sir!" Private Ng called out, hands flicking over the controls. "To your main, now."
A ghostly overlay of ship positions appeared on her screen. Ia stared at them, mind racing, gifts dipping into the timestreams. Electricity sparked from her right hand into the console. _"Bagha, you got that?"_
_"Aye, sir! Targets confirmed,"_ the other woman called back through the comm link.
_"Dropping for launch in five...four...three...two...one."_ Ia pulled back on their speed one more time.
A single, loud _whump_ echoed down the ship. The moment she felt it, Ia squeezed them back over the FTL line, then accelerated. The navicomp was the most sophisticated computer on the ship, capable of analyzing the most minute scraps of data. It could take the faintest, starlit images and resolve them into ships, stations, asteroids, any and all manner of cosmic phenomena. It could even give a fairly accurate estimate on an updated location for everything with just two observations of a few seconds each.
With three glimpses, it could confirm those placements and transfer the information to homemade rockets launched from the _Hellfire_ 's projectile pods. It could not, however, accurately predict where to move its own presence in the next two minutes in order to achieve their objectives. That was Ia's job.
_"All hands, engaging in five...four..."_ she counted over the intercom. At _one_ , Ia brought the long ship down below the threshold. Yet again, the screens flashed from blurred streaks to bubbles that popped in unison, becoming simple pinpricks of light. Red circles quickly zeroed in on every object within sixty light-seconds, and a familiar ache zeroed in on Ia's mind. Not only were there a good twenty ships ahead, ranging from a Battle Platform–sized station to several destroyers, there were hundreds of active anti-psi machines in the zone, their effects painfully accumulative.
_"All pods, acquire targets and fire,"_ Private Magnan ordered calmly. Sung was serving elsewhere this hour, which meant the dark-skinned woman had taken his place as head of the gunnery teams on the bridge. Others might have wondered why Jana Bagha wasn't on the bridge, but Bagha was a Sharpshooter, trained to shoot as an individual gunner; Ia knew Magnan had the tactical training to guide and direct other shooters in a free-for-all.
Free from that worry, Ia concentrated through the misting pain. Left hand in the sensor glove, she flexed the FTL fields and shifted the _Hellfire_ 's attitude, pointing the nose down just a little bit. Zooming through the void at a speed dangerously close to Cee, Ia shifted the ship a tiny bit more. Orange-red lasers lanced outward at those enemy ships, painting them as targets for the missiles riding in hard and fast at their back.
Yellow lines intersected on her main screen, triggering collision-alert beeps. A large ship lay partly across their path. Free-falling the ship in a brief bubble of warped physics—half of one precious second of shielded real time was all she needed to predict their near future—Ia corrected course slightly and angled the nose down even more, relative to their flight. The beeps grew louder, disturbingly fast. Within a handful of seconds, the beeps became claxons, distressed sirens paired with flashing lights in imminent-collision warning.
A slight, downward shift was all it took, just a dozen meters. The shields of the _Hellfire_ 's nose scraped across the shields protecting the Salik warship, jolting both vessels. In less than a blink, they were past it. A second bubble-shift tilted the _Hellfire_ , angling it perfectly for an attack on the space station beyond. The anti-psi field was also so strong now, Ia was glad she had ordered all psychically sensitive crew members to stand down for this operation.
Jabbing the controls, Ia opened the bay doors to the second OTL nose cone. Not to extend it, but enough to give it a clear angle of fire. The claxons stopped, then shifted back to beeping a collision-alert warning.
Ignoring the noise, Ia launched the hyperrift spark just before reaching the station. A split second later, they transited the three-kilometer station's midpoint. She snapped the FTL field back into place around the ship and drove them forward—not along the path they had been traveling but along the line they were now facing.
The vector change slammed their sense of momentum back and upward, rushing the blood to their heads for one uncomfortable, safety-field-squeezed moment. The pinpoints of light surrounding them burst into bubbles and smeared into streaks. Vision greying, Ia slowed their acceleration, holding them just slightly faster than Cee for several seconds. That eased the pressure on their bodies.
Flexing the fields, she reduced their speed once more for a much more gentle, sublight turn. Within two heartbeats, the massive headache eased. Pain still lingered, greying the timestreams in the back of Ia's mind; several of those ships still had anti-psi machines active, generating their annoying, gift-masking fields. But her head didn't feel like it was two throbs away from rupturing a blood vessel. The mass of machines on the big station were clearly gone, proving that exo-EM radiation moved faster than EM itself could.
The proof that the two moved at different speeds was obvious. Behind them, their lightwave front caught up with them on their rearward-facing scanners. The navicomp identified the _Hellfire_ as it skimmed past the patrolling enemy vessels, a tiny green line intersecting a tiny red blob of dots. The green line darted through a swarm of yellow bars indicating both the _Hellfire_ 's L-pods attacking and the few attempts their enemy had made to shoot back. They had been traveling so fast, Ia hadn't even noticed the return shots when they happened, though she could now see the tangled grid of laserfire arrowing off in all directions.
Sangwan's screens, directly ahead of and below Ia's, flicked to a magnified view. Stars rippled across the _Hellfire_ 's polished hull; a blue-white spark spat outward as the ship zipped past the station—and blipped from view in a tiny bubble-flare of all that reflected starlight as its mad, white-haired pilot took the ship back to faster-than-light speeds.
A second later, the station _crunched_ inward in an explosion. They couldn't hear it on the bridge, but that magnified view showed it viscerally. The station's matter had collapsed the edges of the hyperrift, creating a deadly pinpoint of imploded fusion. That energy slammed outward in streaks of white-gold, shearing through bulkheads and igniting rich, dark reddish fires—the colors were artificially shifted by their speed. Outward-bound as they still were, despite their gradual curve, the lightwaves were stretched out, giving them extra time to look at what was unfolding to the rear.
Her aim had been slightly off; from the looks of the way the station was breaking up, the hyperrift had struck off-center. But that didn't matter; the fireball expanded abruptly as the rifted energies ignited what looked like a stockpile of nonwater-based fuels. Dark red and bright orange billowed outward for a moment from the largest chunk, then it, too, succumbed to the forces of the hyperrift's collapse.
Other points in the interstitial night now blossomed in their wake. Fire boiled outward from several red-outlined ships. The projectiles Ia had ordered from the Dlmvla, the weapons which Harper's engineers had cobbled together in just over a day, turning them into missiles which Bagha's teams had loaded into the launch bays...those warheads had contained dozens of head-sized, machine-rounded rocks. The crew's engineers had cobbled them together with attitude thrusters, basic targeting scanners, navicomp relays, and small explosive charges set to break each capsule apart when it came within collision distance of any object larger than a fellow missile, postlaunch.
Under ordinary launch conditions, they wouldn't have been dangerous. Most ships possessed decent, functional shields that could have warded off the mass of the intact missiles, never mind the smaller masses of each individual rock. But those were at insystem combat speeds, which were usually well below one-tenth the speed of light.
Released at near-Cee, however, even the biggest shield generators would be hard-pressed to keep out inbound rocks less than half the size of the ones the _Hellfire_ had launched. Only the physics-warping fields of FTL panels could have saved those ships. Unfortunately for the Salik, most of them had been either cruising the area at insystem-patrolling speeds or parked at a dead stop relative to the station.
Insystem thrusters, while capable of shunting aside modest interstellar debris, were not the same thing as FTL warp fields. The Salik never stood a chance under that devious an attack.
Her headache eased further as they watched, proof several of the ships had been damaged badly enough to cancel their own anti-psi emanations. Some were still active, but not many. Pulsing the FTL panels, Ia checked the timestreams for a moment, then dropped back fully into her body, satisfied.
She didn't have to send the ship back in for a second, slower, and much more dangerous round of attacks. The biggest dangers on this first pass would have come either from not going fast enough to avoid the few unexploded inbound missiles—which were too crudely cobbled together to ignore detonating near their own ship, though she could order them to detonate from a distance—or from not shifting far enough to miss that one ship. As it was, the only yellowlit telltale on her upper tertiaries was a warning light for a stressed shield panel at the bow of the ship. Not a single, hastily aimed shot by the enemy had harmed them during their too-fast strike.
"Rammstein, did you get all of that recorded?" Ia asked the operations tech. "Inbound, attack, and lightwave confirmation?"
"Aye, sir," he agreed.
Satisfied, she brought the warp fields back up to speed, slipping them one last time over the Cee border. "Bundle it up and hand it off to Mysuri for transmission to Admiral Genibes," she ordered. "We'll drop out of FTL in a little bit for that when I'm sure we've escaped pursuit."
"Aye, sir," he said.
"Can I be sick now?" Sangwan asked, glancing over his shoulder at Ia. "Or do we still have more fighting to do?"
"We have more fighting to do, Yeoman, but not for another fourteen or so hours," Ia told him. "The next battle will be slightly uglier. Ng, plot a hyperwarp course for Point B, starting half a light-year from here."
"Already on it, Captain. Correct course to...22 by 317, sir."
"Heading 22 by 317, good work, Private," Ia confirmed. This time, instead of curving around to the right, she curved the ship to the left and up a little.
"I don't see why you're still upset, Sangwan," Helstead murmured to him a few moments later, apparently catching sight of his frown. "We came out of that with barely a scratch. You should be pleased."
"Yeah, but only after the Captain managed to pull off maneuvers that would've made Shikoku Yama himself sweat," he muttered back. "Did you see how close we got to that ship? At near lightspeed? Sir," he added respectfully to Helstead.
"I'm sure the Captain knows what she's doing," Helstead said. "Right, sir?"
"I have far too much to do, gentlemeioas, to risk my life carelessly. That includes your lives as well," Ia told both of them, guiding their ship onto the indicated course. "After being shot in the shoulder with a laser cannon at a mere three percent probability, I have learned to be a lot more careful in calculating the odds. I know I cannot always beat them, but my tasks are too important not to try."
"You were shot in the shoulder with a laser cannon, sir?" Rammstein asked her, curious.
"Handheld cannon at point-blank range," Ia explained. They had reached maximum speed, and were more or less on course, so she set the ship on autopilot and relaxed. "It was back when I was in the Marine Corps, our first mission on Oberon's Rock while I was with Ferrar's Fighters. Hurt like a slagging son of a _shakk_ , too. You can ask Lieutenant Spyder about it if you're curious. He wasn't in the Company at the time, but he heard the others' reports after he joined Ferrar's Fighters."
"Is that where you learned how to pilot like a madmeioa, sir?" Sangwan asked dryly.
She knew he didn't mean Basic Training; he meant actual combat flying. "Nope. I learned how to do that while serving two years on the Blockade."
"And all of it precognitively guided," Helstead observed.
"Most of it. The first time I encountered an anti-psi machine, I had to wing it. I was on an OTL Harrier-Class," Ia confessed. "We could only go as fast as three-quarters Cee at max speed, and had no FTL panels. Unlike now, I didn't know that FTL could cancel the anti-psi effects. I also didn't know what I was up against at the time.
"But I got myself, my crew, and both halves of my ship out of that mess—Delta-VX," she added in explanation, meaning a type of ship that was actually two vessels coupled together. "I did so by evaluating the situation and coming up with a viable solution based on the resources I had available. Which was double-rifting the Salik capital ship as I flew us past at maximum speed.
"That is why I insisted on having hyperspace nose cones installed both fore and aft on _this_ ship. Dangerous as it is, that little trick will continue to save our lives. Now, are you feeling well enough to fly, Yeoman?" she asked Sangwan.
He sighed and nodded. "Aye, sir."
"Then prepare to receive the helm. Write up your tactical analyses of what just happened and hand 'em up the chain of command for your weekly Squad and Platoon debriefing and discussion sessions," she ordered. "I need to get back to my daily regimen. Flab and foe wait for no one."
Helstead _snerked_. A couple of the others bit back smiles of their own. Sangwan was too busy preparing his station for control of the ship, though his shoulders didn't look quite as tense as before. "Aye, Captain. Helm to my station in ten, sir."
INTERSTITIAL SPACE
SOMEWHERE NEAR THE K'KATTAN-GATSUGI BORDER
The _Hellfire_ slipped below Cee, fired off its homemade rock missiles, and slipped up again mere seconds later, analyzing the lightwave data. Instead of widely spaced ships, however, the twenty or so vessels protecting the station were patrolling it in a close sphere, with none of them actually docked. Tiny by comparison, shuttles moved back and forth between the lumpy station and the sleeker ships, no doubt transporting supplies and other goods.
The Salik hadn't moved their main anti-psi manufacturing facility to the Tlassian-K'kattan border, one of the two choices decoded from the stolen data nodes; instead, they had moved those facilities to _both_ locations. Like one of the Delta-VX Harrier ships Ia had served on before gaining command of the _Hellfire_ , this style of Salik station could be separated into several pieces and moved independently. The Salik had chosen to do so. That was why the odds had been so evenly fifty-fifty.
They had also chosen to pepper the local patch of space with hyperrelay-equipped scanner probes. The nearest one was three light-seconds off from the _Hellfire_ 's entry point. As they slipped back above the speed of light, Ia didn't have much time to weigh her options. A few seconds was all she needed, though.
Wrapped in an FTL field, she was free to use the timestreams at full mental capacity. What Ia found brought her out of the timeplains with a feral smile.
_Well, well, well,_ she thought, very carefully turning the attitude of the ship. Once again, she was not going to fly it straight like a javelin but rather at an angle like a crowbar. _Looks like they'll be overlapping their FTL fields to shunt the incoming missiles to either side. A fancy bit of static formation flying. It'll work, too; those rocks will be scattered off to either side despite their near-Cee velocity._
_But that_ also _means they'll be lined up in a few moments...and it won't harm the future to unleash the fury of the_ Hellfire _upon them, today._
Her right hand left the controls for a moment, long enough to lift up the cover and press her palm to the lockbox scanner that differentiated her console from the others. It took a second for the security system to triple-check her DNA, palmprint, life signs, and the ship's interior recordings of her movements, then the box unlocked with an audible _click_. Lifting the clear, hard case up out of her way, Ia flipped the switch that authorized the use of the main cannon.
"Ah, Captain?" Corporal Morgan asked from his position at the gunnery station. "Aren't we going to drop out of FTL soon, sir?"
"Change of plans," Ia told him. "All gunners, hold fire."
"Aye, sir. _All gunners, hold fire,_ " he relayed into his headset.
The ship's attitude reached eighty degrees relative to its vector. Ia dropped them below Cee, ship still angled to travel on the diagonal. The pain came back, blindingly strong now that they were relatively close to the station. Vision misted by the timestreams, she plunged into her own downstream and read the lightwave they would see in just a few moments, before pulling out and adjusting their aim ever so slightly
"Station midpoint in fifteen seconds," Yeoman Yamasuka warned her. "Are we slowing down, sir?"
_Now,_ the timestreams all said. At the same moment, Ia's thumb flicked, activating the cannon switch with a single tap. Unless she deliberately kept the button pressed, it would fire for one-twentieth of a second once charged. But first, it had to charge.
Corporal Crow, seated at the operations station, choked as his displays abruptly changed. The lighting on the bridge dimmed a little, and a strange, almost ominous _hummm_ crept into their ears. On Crow's bank of screens, the bars gauging the ship's various energy outputs abruptly shifted, rescaling themselves.
Ship operations turned from long green and yellow bars into tiny green ones. A bank of new lines sprung up, shifting quickly from green to yellow, exponentially greater than the previous ones. Other graphs opened as well, detailing the heat being expressed along the axial core of the ship.
Along with that _hummm_ , something _whoosh-thumped_. It did it again, and again, growing louder and faster as more joined the first, working in concert. Those were the Sterling heat differential engines; they pumped hydrofuel from the inner core to the outer hull and back, cooling the thermal bleed from the main cannon and recapturing some of that heat as yet more energy for the ship.
The combination of _hum_ and _whoosh-thump_ started to rattle the deckplates—and at the ten-second mark, every forward-pointed screen lit up for a fraction of a second, bathing the cabin in a flash of bright blood red.
Terran lasers fired at the low end of the visible light spectrum, with their beam tuned to have sympathetic, harmonic "wolfs" lurking at near-infrared lengths. Other weaponmakers tuned theirs closer to orange or even yellow to try to get several such harmonics, but the Terrans believed too many wavelengths diluted the impact. Like striking a C major note and hearing the harmonics for a G fifth, the tuning added extra heat to their punch. That made their weapons a signature red, and the Godstrike cannon was no exception.
Traveling just below lightspeed, it took only a second or so to see what that single brief pulse of light did, a pulse that arrived at a somewhat diagonal angle, despite its short length. Most of it scorched through the formation of ships attempting to shield the station from the incoming missiles. Some of it kept going. Ia didn't pay much attention to that, though; with a dispersion rate of four light-months, the excess energy wouldn't harm anything. The Salik had chosen better than they knew; this location was in a large chunk of interstitial space half a light-year from the more-commonly-traveled flight paths and visited systems.
The _humming_ sound had shut off, but the engines still _whooshed_ , putting that excess heat into capacitors dotting the length of the main gun. Ia's hand shifted slightly in the feedback glove, changing their attitude again as well as slowing the ship a bit. Everyone swayed in their seats from the vector changes to their inertia, though not as strongly as during the other fight.
This time, Ia added a sideways slip, massaging the FTL fields to sneak them in behind the station. The move also pointed them at it and the remnants of ships lying beyond. Again, her thumb flicked the cannon's switch. Again, the humming rumble came back, and the engines picked up speed, shuttling superheated fluids to an array of pipes just under the space-cooled hull of the ship and back again.
The moment their slowly rotating nose pointed hindward at the station, those ten seconds of buildup ended in a twentieth-of-a-second burst that flared through the bridge. This time, the glow was very dark and lingered far longer than it should have. Blinking, straining to see through the green-and-red afterimages, Ia checked the lightwave readings. They came after several long seconds; as the red blob finally cleared from their view, the navicomp magnified and highlighted the two halves of the station, and the rock-damaged chunks of ships tumbling beyond.
Their backward speed had coupled with the cannon's forward aim, one traveling a little bit under Cee, one at full Cee. Ia had forgotten that rule; that the speed of the one would slow down the speed of the other. It was probable the Salik had seen the laserfire coming this time. But despite the way her ship's speed had slowed that packet of laser energy down to a quarter Cee at most, that blob of light still retained its full heat energy. The centerline of the mobile station was gone, leaving two slowly drifting, edge-glowing chunks, and countless tiny bits of charred, overheated debris.
Slowing the ship further, Ia adjusted their aim by a tiny bit and fired twice more. Thirty more seconds, and it was all over. Her headache ended, with only a faint echo from whatever few machines remained active on the surviving, scattered ships. There was no more manufacturing station to worry about, and only a handful of the smaller ships had not been caught in her lines of fire.
"Private Loewen, did you get all that recorded?" Ia asked her.
"Sir, yes, sir," Loewen confirmed from her post at the navigation seat, not bothering to look up from her work. "Shall I wrap it up for Admiral Genibes?"
"That's th' Captain's policy," Lieutenant Spyder said, speaking up from his position at the backup gunnery station. "If we wanna keep our slate blank for patrol assignments, we show 'em everythin' we can do."
"Sirs, we have bogeys headed our way," Loewen stated quickly.
"Not a problem, Private." Ia closed both lids to the main cannon lock, which automatically flicked the switch to shut the weapon off. The Sterling engines still _thump-whooshed_ , but their efforts were slowing down again. "They're too far away."
Right hand moving to the controls, she increased their speed, this time rippling the FTL-field panels backwards, since that was the way they were pointed. Laserfire tried to catch up with them, but it was hastily aimed and couldn't match their already high speeds. Within a minute, they were past the bubble-flash threshold and soaring backwards at speeds faster than any weapon could hope to catch.
"Congratulations on another successful attack," she praised her crew. "Once again, we've barely a scratch. Don't expect that to last."
"Well then, yay and hurrah, whoopee, grats, and all of that joy and happiness," Morgan retorted dryly. "But—begging pardon, Captain—what the frying _shakk_ was _that_?"
She slipped them sideways a little as they built up to full FTL speed, not wanting to be on the same heading as the Salik had last seen. Ia also didn't pretend ignorance as to what _that_ he meant. "That, gentlemeioas, was our main cannon. Be advised that _any_ attempt by any person other than myself to access its controls will cause an irreversible cascade in every hydrogenerator hooked up to a tank on this ship, turning it into a giant hydrobomb after just one minute—in short, attempt to crack open that security box, and you make the whole ship go very boom, very big, and very bad, with no time to get away.
"The Command Staff and the Admiral-General are extremely paranoid about this cannon falling into the wrong hands, so don't do it, or you'll kill us all," she warned her crew. "That's why there's a lid over the palmscanner, so you cannot even accidentally brush against it. Yeoman Yamasuka, do you feel comfortable flying this ship backwards?"
"Uh...Aye, sir," the other woman said. "I haven't done it since flight school, but I think I can manage it."
"Good. Slip us 196 by 203, and hold us on course for ten more minutes," Ia directed. "Then ease back down to sublight speeds so York can fire off the recordings packet—on our next ship, I'm going to insist on a vacuum hub for hyperrelay, so we don't have to keep slowing down below Cee to contact Genibes. Once you've done that, you may turn the ship around at that point and adjust course for the KLM 88-B System. Stick to FTL speeds for now. Yeoman Yamasuka, prepare to receive the helm."
"Aye, sir, preparing to receive the helm in te—er, make that in twenty, sir," the dark-haired woman agreed.
Loewen spoke up as well. "Plotting a first course, vector 196 by 203, and a second course for KLM 88-B, sir."
"...Do we at least get to know the main cannon's specs, Captain?" Morgan asked Ia once the helm transfer had been made.
Ia shook her head, unstrapping her left hand from the thruster pad. "It's big, it's ugly, and it takes ten seconds to charge before it can fire. It also has a very ugly overshoot range. I'm the only person who can foresee the lining up of enough ships during a fight to make it safe to wield inside an inhabited or commonly transited star system. That's all you need know for now."
"It also extracts a lot more energy out of the system than the other cannons," Crow said. He was replaying the meters from the four brief shots on his main screen. "Not just the buildup to fire it, but in actually firing it. I only noticed how much energy it retained because the scale is so huge."
"More energy than you know," Ia admitted, calling up the duty roster for that hour. "The current design siphons off most of the excess heat, converts it back to energy, and pours it right back into the cannon. I can tell you that much because that part's just an upscale version of the Starstrike cannon's energy-feed design; it has nothing to do with how the Godstrike itself actually achieves its high caloric rating.
"That's also one of the reasons why it takes ten seconds to fire, and why so much of the ship near the core isn't inhabitable," she stated, glancing at the chrono on her leftmost lower tertiary. "It looks like we're scheduled for lunch, next. With the main galleys still locked down, how about I fix everyone some sandwiches from the bridge galley?"
"Th' Cap'n of this ship isn't s'posed t' cook," Spyder argued, reaching for his restraint straps. " _I'll_ go fix us sommat."
"Oh, no. I've foreseen your cooking skills, Spyder, and they're worse than mine," Ia joked, unstrapping as well. "I may not be as good a cook as my biomother, but at least I won't poison us. Besides, _everyone_ pulls triple duties or more on this ship. That includes me, because I don't ask my crew to do something I'm not willing to do myself if I have the skills and the time to do it.
"Right now, I have the time, and sandwich making is just within my skills. You have the bridge, Lieutenant. Corned beef on rye okay with everyone?" Ia asked.
A quick survey of the others showed them all nodding. Sighing, Spyder complied, shifting from his seat to hers at the back of the bridge cabin. "Aye, aye, Cap'n. Corned beef on rye'll be fine."
FEBRUARY 11, 2496 T.S.
SIC TRANSIT
"Sir, yes, sir," Ia stated, shoulders square and chin level. She felt like she should've been standing At Attention, rather than seated at the desk in her office. "I am absolutely confident the overshoot from our lasers and the main cannon will not hit and damage anything of consequence."
"What about those rock missiles?" Admiral Genibes asked her, leaning back in his own office chair. The entire conversation was taking place at just over a two-second delay, making it almost feel normal. "You released them at near lightspeed. They'll continue on course until something stops them, and they're too small to be easily noticed at that velocity."
She had an answer for that. "Only three will cause problems," Ia admitted. "I've already arranged for timed messages to be delivered to ships in the affected areas, with exact tracking coordinates for their destruction. The rest will eventually be snagged by gravity and either smack harmlessly into other stellar bodies or burn up in various atmospheres."
Admiral Genibes raised one of his brows at that. "Timed messages? Arranged with whom, Captain Ia? I don't recall receiving any messages."
This was the tread-carefully part of her report. Lying to a superior was a fatality, but she didn't want to bruise the ego of anyone reviewing this conversation later. "With the Afaso Order, sir. The Terran military will be too busy with the war at that point to be bothered with that sort of thing."
Genibes frowned thoughtfully at her. "You've sent a lot of packages over the years to the Afaso Order. Same as you've shipped home. If all of those packages were precognitive prophecies...when is the Space Force getting its fair share?"
"The Space Force as a whole is a competent entity, sir," Ia told him. Again, not a lie. "You won't need all that much from me precognitively to carry out your duties. Those few sections in need of my assistance will receive it at the appropriate time."
"You said you're fighting a war three hundred years into the future," he reminded her. "But not even a Feyori half-breed will outlive its matter-based life expectancy. What about the intervening two hundred years?"
"You'll receive a war chest of prophecies, suggestions, and directives at the appropriate time, sir," Ia promised him.
"Why not now?" the admiral asked.
Ia restrained the urge to roll her eyes, though she did sigh. "Because there's an entire queue of things that have to happen first, and if something gets nudged out of alignment now, I'll have to rewrite all those prophecies to compensate for the ripple effect it'll cause."
"What about the first battle of the war?" Genibes asked her. "You've successfully proved in these four attacks that your ship would be invaluable. Where will the Salik strike first?"
That, she could tell him. "Two places. The Terran and Gatsugi Motherworlds." A quick dip into the timestreams allowed Ia to nod in confirmation. "It'll start two weeks from now, give or take roughly a day—it varies too much to pin it down closer than that, depending on how my crew and I handle the next few prewar fights."
He folded his arms over his grey-clad chest, arching one grey-salted brown brow. "Do you at least have a location for the _first_ main push?"
" _Two_ locations, Admiral. They'll happen simultaneously," Ia stated. "The Terran Motherworld, because we not only kicked their frogtopodic asteroids last time but have been instrumental in keeping the Blockade going. They've gotten around it, but not as freely as they might've otherwise, and we pissed off the Salik because of it. And they'll go after the Gatsugi because they're in the linchpin location. Damage the center of the edible Alliance members, and we can't use the Collective's systems and stations as safe transit hubs, let alone for gathering and resupply.
"The Salik are feeling cocky, but they've got the forces to be cocky. They'll throw several fleets at the other Motherworlds, strong enough to tie up our various allies," she added, cautioning him. "But the strongest attacks will be those two worlds. They're also watching the movements of all our fleets, so you can't move everyone in to protect Earth."
"Because they'll just shift gears and attack wherever we're weak," he agreed. He thought a moment, then asked, "What about pulling out a single ship here, a single ship there? And having them sit beyond the Kuiper belt a few days before?"
"That might entail several days of those crews doing nothing; I can't guarantee exactly _when_ the Salik will strike, only that they will," Ia told him. She held up her hand in case he was about to speak, stating, "Let me check the timestreams before you make up your mind."
It took a few seconds of real time to sort through the potential possibilities. When she had what she wanted, Ia pulled back into herself and sent a jolt of electrokinetic information into her workstation. Tapping a key, she sent it to the Admiral.
"I've attached a list of ship registries you can safely pull in over the next two weeks to defend Earth. It's not many, but you shouldn't need that many."
"Well, no; we shouldn't need that many more if you'll be there," Genibes agreed, touching a control on his own side of the comm link.
"I'm sorry, Admiral, but we won't be there, sir." Ia didn't wait the two plus seconds for his head to snap up at her words. She continued briskly, explaining herself. "Earth has plenty of defenses and doesn't really need us, whereas the Gatsugi can ill afford to pull in ships from their various colonyworlds just to defend Beautiful-Blue. The _Hellfire_ will be assisting them when the time comes."
"That ship of yours represents a very significant investment in military research and development, Captain," Admiral Genibes reminded her, stressing her title slightly. "It is Terran property, and should therefore be used to defend Terran property."
"According to the Terran United Planets Charter, duly registered with the Alliance," Ia countered, "the Terran Space Force is to render both sentientarian and military aid to its allies when and where endangered by a mutual threat. The Salik most definitely qualify as a mutual threat, Admiral—and I tell you, as a precognitive, we need the Gatsugi Motherworld to come through this first fight relatively intact."
"Ia, as much as you—" he started to say over the two-second delay between them.
Ia kept talking, cutting him off. "Sir, if we send _only_ the _Hellfire_ , every other ship in the Terran fleet can do their job defending Terran and joint colonyworlds, and the Gatsugi will have all the help they need. If you tried to pull the _Hellfire_ back to Earth, I'd have to insist on sending eighteen to twenty Space Force vessels to Beautiful-Blue to ensure their war-machine efforts would survive in a shape suitable for helping the rest of the Alliance in the future...including helping us Terrans. But that much movement on our part into Gatsugi space would be noticed far more by the Salik than simply reassigning a bunch of patrol routes entirely within Human space, which is something the Command Staff already does on a regular basis. The Salik will see it coming, and send more ships than our side can defend against."
He waited four extra seconds to be sure she was done talking, then spoke. "You've planned all of this, haven't you?"
Shoulders back and chin level, Ia answered At Attention. "That is what I am supposed to do, sir, as an officer, a soldier, and a precog. I am to use my abilities to ensure the maximum number of lives are preserved with the most efficient use of the resources I have available."
John Genibes snorted. "Don't pull that officer's duty _shakk_ with me. What's your _real_ reason for wanting to help the Gatsugi?"
Ia dropped her soldierly poise and gave him a flat look. "That _is_ my real reason, sir. It is the single most efficient use of all our resources. I did tell you and the others when I bargained for this ship that it would have to be sent places the rest of you might not think are all that vital but actually are. _This_ is one of those instances."
He studied her for longer than the two-second delay. Finally, he asked, _"Carte blanche?"_
"Admiral, yes, sir. You'll see how effective I am in wielding it when the Terran Council receives the gratitude of the Gatsugi Collective, sir," she promised.
Sighing, Admiral Genibes sat back in his seat and rubbed the bridge of his nose. "Right...Well, at least you're using it for the good of the war effort. I'll promote it that way to the Admiral-General. What's next on your itinerary, between then and now?"
"What I would've had the crew do before I knew about the anti-psi menace. Attack hidden shipyards and crèches." She waited for his reaction.
He lifted his head at that. "...Crèches?"
"They've been breeding and training generations of workers and warriors on the sly, all dedicated to building up their war machine," Ia told him. "They've been on short rations, working in harsh conditions, but it's how they've managed to come up with enough bodies to craft the robotics and other manufactories to build everything they're planning on throwing at us. My plan is to make those secret resources too costly to repair, forcing them to make their opening attack before we can destroy too many of them."
"More rocks flung at near-Cee?" Genibes asked dryly.
"No, sir. Mostly, it'll be the Godstrike, since the things are located a light-year or more from anything else, which means overshoot won't be as much of a problem. These are mostly my original targets, the ones without anti-psi shielding—though we will be hunting down more of those as the war gets going," she promised. "Not just the _Hellfire_ , but other ships, too. I'll need you to pass along rerouting and attack orders for a number of ships within the year, to see that these interstitial-space enemy bases get destroyed when the odds are highest on our side."
"About attacking those crèches," her superior stated after another extralong pause. "It won't play well if word gets out you're hitting targets with children. Whether or not they're Salik tadpoles, they're still children. Mind you, I won't object on my end because I _know_ what those things are. I'm talking about someone in your crew talking about it to the Nets. If it gets on the news..."
"I know, sir," Ia admitted quietly. "I accepted the cost long ago. The goal is to cripple their facilities beyond use, not just to kill. We want to concentrate their numbers in other, more heavily defended locations deeper within Salik territory. The Salik intend to stab at Alliance members in between cleansing the Blockade presence from their various territories, dividing our attentions. We can't afford to let them continue to train replacement soldiers, and we can't afford to let them spread out any more than they already are."
"How are you going to put it to your crew?" Genibes asked. "Or are you going to even tell them what the targets actually are?"
She shook her head. "A lie by omission is still a kind of lie, sir. I'd prefer to limit the number of omissions I make because eventually the truth does get out. The first few targets will be manufacturing stations and shipyards, that sort of thing. I'll find the best moment to tell them about the crèches when we come to that point. Anything else you wanted to discuss, Admiral?"
"Not at this time. Don't abuse your _carte blanche_ , Captain," he warned her.
She gave him another dry look. "Considering what I _could_ be doing with it, I'd hardly call the careful, rational selection of vital targets an abuse of my position—but yes, I am aware that others' viewpoints may not match my own, sir."
Some of those viewpoints weren't even Human. She wanted to warn him about the Feyori who were going to move against her, but refrained. Instead, they said their good-byes and ended the call. Ia rested only a moment before reaching for the comm again, this time making an audio connection with the bridge.
_"Captain Ia to Lieutenant Spyder."_
_"Spyder 'ere, Cap'n,"_ he replied. _"What can we do f' you?"_
_"Tell the members of the 1st and 3rd Platoons they may have half an hour to contact their loved ones, up to bandwidth capacity. They are not permitted to say where we are or what we are doing, save that we're on a deep-space patrol, as per the Company Bible. The 2nd will have to wait until it's their off-duty turn next time we're in sublight. After forty minutes, resume course at FTL speed to our next target,"_ she directed.
_"Understood, Cap'n,"_ Spyder said.
FEBRUARY 25, 2496 T.S.
SIC TRANSIT
This time, Ia didn't bother to square her shoulders before stepping into the boardroom. Her fellow soldiers had already used it a few times since leaving the shipyards, if mostly for group briefings rather than Company-wide ones. This one involved her image being broadcast to tertiary screens at duty stations around the ship for those members of the 1st Platoon who weren't in attendance since the ship was in motion, which meant priority stations had to be manned.
This time, she wasn't in her formal Dress Blacks, just in a grey shirt and trouser set with her service and rank pins in place. Lieutenant Rico was up on the bridge, serving as the officer on watch, but the other members of the cadre were gathered at the officers' table. Some of the soldiers lounging in the tiers of seats looked half-asleep, having been dragged out of bed for this meeting.
They did sit up when she came into view, but didn't rise or salute since she wasn't in formals and wasn't wearing a cap. That was in the Company Bible, which meant they were doing what they were supposed to do. Coming to a stop before her seat at the table, Ia began without preamble.
"At Ease. Up until now, our targets have been considered legitimate under the Alliance joint military code of conduct for all of its sentient members. Mainly because we have been operating under the Alliance-agreed provisions against Salik-crafted and -manned communications hubs, war-matériel-manufacturing facilities, and minor shipyards located outside the Salik Interdicted Zone, and thus outside the law.
"Our next target also falls outside what the law permits the Salik to legally occupy and operate." She paused, taking the time to meet the gazes of several of the hundred-plus men and women seated around her. "This means they _are_ legitimate targets in the eyes of the law. In the eyes of common, sentientkind morality...some of you may have objections."
She did not display anything on the screen. She did not sit down, either. Bracing her hands on the table, she leaned forward, again meeting the gazes of the soldiers around her.
"Right this minute," Ia stated, "in the ponds of the city of Shnn-wuish on the Salik Motherworld of Sallha, the senior-most members of what we Humans would call a high school are undergoing their version of a graduation ceremony. The top twenty students are being permitted the chance to hunt and kill the five worst-performing members of their graduating class. They will do so in the ancient way in a deep lake in the heart of that city, without any weapons other than their tentacle-hands and teeth...and yes, they will eat what they kill, while they are still killing it.
"The hunters are cheered on by everyone, and the hunted are scorned," she continued. "Unless the hunted successfully kill all twenty of their hunters, they will not be permitted to leave the pond. Those among the hunters who are killed by the poor-performing students they hunt will not be avenged by their family or friends. They, too, will become reviled as weak and useless. These are _not_ Human children," Ia stressed, hardening her voice. "They do _not_ operate under the same rules as the rest of us. So when I say our next target is a crèche, a deep-space facility designed to spawn and rear Salik children, I am _not_ talking about defenseless younglings. They are trained from birth to hunt and kill."
Roughly one-third of the men and women nodded somberly in understanding. One-third looked a little confused, and the remaining third looked disturbed. That included their ship's doctor, Jesselle Mishka, who frowned in distaste at Ia.
"I can corroborate this," Lieutenant Rico stated, surprising her a little. She hadn't figured he would speak up. At her nod of permission, he filled in a few more details for the skeptical members of the crew. "I have studied their culture as well as their language, to better understand how and why they communicate. For the first five years of a Salik's life, they do _not_ learn how to read, write, or interact socially beyond their immediate family-pack, which consists of their mother and their siblings.
"Instead, within two months of emerging from their egg-sacs, they learn how to hunt live prey, starting first with small, fish-like creatures whose only defenses are that they spawn in great numbers and can swim fast _en masse_ , on up through to large, non-sentient livestock that are bred to fight back," the lieutenant said. "By the time their brains have developed enough for higher cognitive learning, they are natural killers. Attempts by Alliance social services to 'reeducate' Salik spawnlings have failed, because this five-year hunting requirement is hardwired into their biology and their neurology. They are _not_ Human."
Ever since his trip into the timestreams with her, Oslo Rico had given up his resistance to her leadership. Ia dipped her head in acknowledgment of his help. "The lieutenant is correct. In the eyes of softhearted civilians who are not trained in xenobiology or xenopsychology...we will be seen as going up against their _own_ children. We will be seen as slaughtering _their_ younglings, whether it's Human, Gatsugi, Tlassian, K'kattan, Choyan or Solarican, Dlmvlan or Chinsoiy. But these are _not_ our children.
"As for the legality of our coming strike against a crèche station designed to raise and train Salik children...they are located well outside the Salik Interdicted Zone, the only place where the Salik are permitted to establish colonies. By law, they are explicitly forbidden to establish and occupy any locations other than the openly listed ones, which means these hidden crèche stations are completely valid targets, the exact same as those comm hubs were...which _did_ have small crèche-ponds of children on board."
Her confession stirred the crew in waves of discomfort. Ia let that sink in, then continued.
"The only difference between this next target and the previous ones is the scale. A few hundred at most, versus the tens of thousands of tadpoles we'll be going after. This crèche station, the first of many, is specifically designed to rear and train Salik younglings to be competent engineers, mechanics, scientists, and warriors. They are staffed by Salik broodmothers culled from the top-serving members of the underground Salik war effort...and while their graduating ceremonies do remove the bottom five students per class, they practice those skills beforehand on captured sentients."
" _I_ can corroborate that," Helstead spoke up.
That, Ia had expected. She nodded at the former Corps officer to continue.
"Intelligence reports from pirate operations have included numerous cases of extremely unscrupulous sentients among the members of the criminal undergalaxy selling their prisoners for large sums of tangible assets," the lieutenant commander shared, glancing at her fellow officers. She looked out at the men and women seated in the riser seats. "Rare gems, precious metals, and other nontraceable commodities have ensured that, while extremely risky if caught, the food-slave trade has remained quite profitable over the years.
"While it's true that in the Interdicted Zone, it's the death penalty for any pirate caught trading in living sentients to the enemy," she admitted, "the profits have made that risk worthwhile to many. Every few years, the Knifemen Corps keeps taking out the worst of the slave traders, but more just keep popping back up like weeds."
"I myself was sold by a group of pirates to the Salik for my weight, kilo for kilo, in platinum originally mined on the legitimate Salik colonyworld of Ss'nuc III," Ia admitted in an aside. "But we're digressing. There is a purpose to this meeting beyond informing you of the distasteful yet necessary chore we are about to pursue. I made myself a promise back when I first planned for this crew. If I had the time to spare, and could spare it before a particular engagement, I would give you an opportunity to step into the timestreams with me and _see_ for yourselves just how necessary this particular fight will be.
"I offer you this opportunity now. I have half an hour I can spare from my other obligations and duties," Ia stated, moving around the table to the front of it. "If you wish to see for yourself just what we're up against, you may choose to do so. This offer is open to my fellow officers as well as to the noncoms and enlisted. If you have any doubts or discomforts at the idea of undergoing this, you may consult with Lieutenant Rico, who has already accompanied me on a previous visit to the timestreams."
A hand rose, hesitant at first, then higher. The owner was Private First Class Harley Floathawg, distinct not only for his self-picked name, but for the burgundy blotches of _jungen_ mottling his skin. Half-V'Dan and half-Terran, he was one of less than a million or so Humans of the billions occupying the known galaxy to still get the colorful skin pigmentation. Ia had not selected him based on his appearance or unusual name, however; she had selected him because he was one of the best hovertech mechanics available for her crew, without being needed far more elsewhere.
"Yes, Private Floathawg, you have a question?" Ia asked.
"Captain, yes, sir," he stated, rising from his seat. Tall and lean, he overshadowed his shorter, mousy teammate, Private Second Class Mara Sunrise, who stayed in her seat, looking bored with the proceedings. Both of them were supposed to be on the current watch, but neither had a task at the moment that was absolutely necessary to keep monitored during this hour. "What exactly are these timestreams?"
"My gifts act in a visual way...though visualization might be a better word for it," Ia told him. "Since my abilities came to full strength at the age of fifteen, this visualization starts out as a giant prairie crisscrossed by streams. Each stream represents a single person's life. Where the stream splits, a choice has been made, and each new part of the stream indicates what will happen if each choice is followed.
"From the banks of that stream, I can see images inside the waters of snippets of time from that particular person's life. If I go upstream, I go into that person's past. If I go downstream, I go into their future. And if I were to step into those waters, I would be living that person's life at the moment in time where I entered, as if it was a point-of-view hologram, with sight, sound, touch, taste, smell...and even some access to their uppermost surface thoughts, but with zero interaction with the actual person," Ia said. "In that regard, it is more like a standard, noninteractive vidshow broadcast than anything else.
"Now, I _can_ change the visualization images," she added, as Harley's fellow crewmates glanced uneasily at each other. "I've used graphs, grids, plus other metaphors such as tapestries and so forth, but it always starts out as the timeplains, and I myself always start out in my own life-stream...as does anyone who comes with me. I have also learned to lift that person out of their own waters quickly upon entry so they don't metaphorically drown from trying to process too much doubled-up information at once."
Another hand rose. Ia pointed at its owner, Private Second Class Nadja Theam, a clairvoyant and fellow psi, and a very good electronics programmer and engineer. Floathawg sat down, his question answered, and Theam stood in his place. "Sir, I've been given to understand you're both a precognitive and an electrokinetic. In fact, word is among the crew, you can program literally with a thought. Why don't you just _show_ us these timestream images by using the monitors in here, while you're searching for them wherever it is they exist?"
"I wish I could do that, Private Theam," Ia allowed. She shook her head. "Unfortunately, while my years of effort at disciplining my electrokinetic gift make it possible for me to transcribe simple things like written orders, programming code, and the like, it isn't the electrokinesis that's the problem. My precognition is too powerful. The handful of times where I have tried directly to record the images that I see inside my mind when I'm standing in the timestreams themselves, I have fried every datapad and workstation console I have touched...just as I have destroyed every KI monitor within range of my abilities, which is why we don't have any on board this ship.
"I can skim the timeplains from this side and pull through what I need, but it is always throttled down and filtered," she explained. "In short, you can fill a cup of water successfully from the sink tap, when that tap is fed by the waters of a dam far upstream, but you cannot expect to fill it nearly as safely by dropping the sluice gates of that dam while you're standing on its spillway. Any other questions?"
PFC Belle Underwood had one. She stood as well, her stance At Attention, but her tone hesitant. "Um...will it hurt, sir? Going onto the timeplains or whatever?"
"Only if you let go of my hand, or say the word 'time' repeatedly while we are there." Her reply earned Ia several chuckles. "Laugh all you want, meioas; I am serious. For those who don't want to experience the timestreams but are still doubtful or curious about the necessity of the coming attack, I _have_ carefully considered all of the choice-possibilities in this and other endeavors, and have concluded that attacking these crèches—repugnant as that is to our Human sensibilities—will save far more lives down the road than it will cost. _Every_ action I undertake, past, present, and future, is designed with that goal firmly in mind.
"Now, if you are curious, please feel free to move forward and line up. If you do not wish to have your temporal questions answered at this time—and yes, Corporal Johnson, you may ask to see _other_ events, though I may or may not comply at my precognitive prerogative—then you may remain in your seats or consider yourselves dismissed.
"Those of you on duty who are watching this meeting will have to hold your questions for the next temporal opportunity. Those who stay here in the boardroom but do not participate will have the opportunity to chat with those who do take a dip in the timestreams today," Ia concluded. "You are now free to move as you will, meioas. Thank you for your attention."
A few got up and left. A few more hesitated, then moved to the front of the hall. Doctor Mishka spoke up before any of them reached Ia.
"Why didn't you offer us this opportunity any earlier, Captain?" Mishka asked her. The blonde woman remained in her seat at the table, her expression skeptical. "Why now? And why not have all of us take a stroll through these timestreams with you?"
"Because—ironic as my adult life is—I believe in free will, Doctor," Ia replied, twisting slightly to look at the older woman. "Those who follow me onto the timeplains tend to see things that shift their perception of the universe. Not through anything I myself do to them but simply because once you have seen something, it cannot be unseen. Every experience changes us, and if the experience is a powerful one, it has the potential to change us in equally powerful ways. Sometimes I have to take people into the streams with me so that they can see the consequences of actions, whether it's theirs, mine, or others'...but I prefer it when it's their own idea.
"This is also why I prefer not to be touched," Ia added, looking at the others approaching her. "My gifts can and will trigger on their own, particularly when I am startled, or my guard is down. And like most psychic abilities, they are strongest when transferred via physical contact. I prefer to do that under controlled circumstances, and only when that person's foreknowledge will not harm the actions that must take place—if any of you change your minds at the last second, meioas, I will not take offense. If you haven't, we'll do this one at a time."
" _I_ want to know what your end-goal is, sir." Private Kimberly Kim, lead team member of 1st Platoon B Gamma and full-mech specialist, halted in front of Ia. Shorter than her captain, she looked up at Ia with the level gaze of a woman who considered herself an equal. "Why you're doing whatever it is you're doing, and why you've involved the rest of us in it. I know you already said you're in this to prevent bad stuff from happening down the road, but that's what I want to see for myself."
"You see _that_ , and if you have a single scrap of compassion within you for the other beings in this universe, it'll change you forever, Private," Ia warned her. She lifted her hand, offering it palm up to the other woman. "But if you do want to see it, I'll show it to you."
"Sir, yes, sir." Kim stated, and gripped Ia's fingers.
Sighing, Ia complied, taking the shorter woman into her gifts. Between one breath and the next, she pulled the two of them out of their life-streams and onto the banks. Waiting just enough for Kim to get her bearings, she accelerated them downstream, into the desert waiting for their descendants, and the one trickle of a chance at stopping that lifeless desiccation.
_"This is what will happen to all life in the Milky Way Galaxy. Starting three centuries from now, an invasion force will start stripping every planet and every star for raw materials. It will take them less than two centuries to do it...and_ this _chain of events is the one chance we—ourselves and our descendants—will have at stopping them."_
# CHAPTER 9
_...But most scholars will insist the war took off with a vengeance on the third of March, Terran Standard, the day before my twenty-fourth birthday. On the Terran side of things, the first defensive shots were actually fired by a group of civilians on the edge of the system, then the rest of the military engaged. The Damned were somewhere else entirely._
_~Ia_
MARCH 3, 2496 T.S.
NEARSPACE, BEAUTIFUL-BLUE
GATSUGI MOTHERWORLD, SUGAI SYSTEM
The Gatsugi that appeared on Ia's main screen was four-armed, mouse-eyed, blue-green-skinned, and boasted butter yellow tufts on his head, strands which were more akin to the long, individual barbs on a peacock-feather shaft than anything resembling Human hair. He smiled by curving up the edges of his small mouth—a gesture Gatsugi and Humans had in common—and said, "Greetings/Salutations/Hello. You have reached/contacted/I am the Sugai Insystem Comptroller. How may I help/assist/aid you?"
"Greetings/Hello, Comptroller," Ia returned politely. "This is/I am Captain Ia of the Terran Space Force requesting/asking/seeking permission to enter Gatsugi homespace/territory/system-heart."
The alien's race had evolved on a world with predators sporting extremely sensitive hearing, though poor vision, and had developed multiple methods of communication. They talked more than they gestured or colorchanged now, but the layers of meaning had merely morphed into multiple-word use. Some sentients found it annoying; Ia thought it was elegant. Not something she'd use herself every day, but elegant in its own way.
The Comptroller dipped his head. "What is the name/identity, need/purpose for visiting, and location/point of entry for your ship/vessel, Captain Ee-ah?"
"The TUPSF _Hellfire_ is a new/experimental Harasser-Class warship," Ia stated, pronouncing the acronym _tup-siff_. "We are approaching/entering Sugai System from your vector 117 by 3. We request/request/request that you clear/evacuate Beautiful-Blue nearspace sectors 1008, 908, 807 through 809, and 705 through 712 of all vessels within the next ten _klitak_ Gatsugi Standard, and request/request/request you prepare to elevate/accelerate all system defenses/warnings from Peach to Sanguine in ten _klitak_. Our purpose/intent/reason for entering/arriving is to assist/aid/help in defending/protecting your system/sovereignty from inbound/advancing enemies/enemies/enemies in less than fifteen _klitak_."
Most of the time, Gatsugi conversations circled around a subject, approaching it from multiple angles. Sometimes, they repeated a particular word for emphasis. Hearing that emphasis, the Comptroller widened his mouse black eyes, skin flushing from blue-green to a reddish peach in just a few seconds. "What/What/What enemies?"
"Salik," Ia stated. The name needed no emphasis. "You have just over fourteen _klitak_ to the Second Salik War, Comptroller, and the Terran government has sent/allowed me to help/assist in the protection/defense of your Motherworld. Do we/Does this ship/military force have/receive your permission/clearance to enter/approach Beautiful-Blue nearspace and assist/defend your Motherworld/heart?"
"Is this true/true/true?" the Comptroller asked, flushing a skeptical shade of muddy orange.
"I do not lie to you, meioa," Ia stated flatly. "If you do/will not believe me, cooperating/accepting my request/warning anyway/regardless will not/will not cause/create inconvenience/trouble for more than twenty _klitak_. Just/please clear/evacuate sectors 1008, 908, 807–809 and 705–712 immediately/now, and do not use lightwave communications/channels. They are parked/sitting/watching at system's edge/farspace right/for now, watching/scanning/spying on you."
The Comptroller wasted a _klitak_ in thought, somewhat longer than a Terran Standard minute. Finally, his lower arms moved, touching controls below the edge of the vid pickups for their comm link. "We will comply/trust you/the Terrans. Clearing/Evacuating the indicated/listed sectors/spaces now. What is your estimated time/moment of insystem/nearspace arrival/appearance?
"Fourteen _klikat_ from...now." Ia stated, checking the timestreams. Speaking with the politeness of using Gatsugi thought patterns was starting to give her a headache.
The Comptroller flushed reddish peach. "Your arrival/entry will be after/following the Salik. Why/Why/Why not before?"
"If you move/get those ships out of my way/the indicated sectors," Ia promised the alien, "they/the enemy won't have reason/need to deviate/reposition, and I can enter the system/nearspace at the heart/center of their formation/fleet."
"How/How/How can you know/assert something/this information/knowledge so precisely/accurately?" the Gatsugi challenged her, skin shifting more toward a doubtful, dull red. "Either you collude/cooperate with the invasion/Salik, a reprehensible/unthinkable/vomitous thought/idea, or you have a spy/traitor among/spying upon them."
"When this/the battle is over/done, Comptroller, and you have a moment/energy to spare, look up/access the V'Dan belief/faith Sh'nai records/histories/mentions of 'The Prophet of a Thousand Years,'" Ia instructed him. "I am/am she/the Prophet who was foretold/prophesied. And, as foretold/prophesied, I will aid/assist/help save you, today. _Hellfire_ out/ending transmission."
Within seconds, Private C'ulosc spoke up from his seat at the comm station. "Sir, we're getting a ping from the Sugai Comptroller's office."
"Send a signal stating that we're entering FTL transit, and cut the pingback," Ia directed him. "Then make sure Chief Yeoman O'Keefe has all members of Lieutenant Spyder's boarding party on board Bow Shuttle One, locked and loaded."
"Aye, sir."
"Captain, are we actually going to come out in the _middle_ of the enemy?" Yeoman Ishiomi asked her. His console was now a backup gunnery post to Private Ramasa, along with Lieutenant Rico, who faced backwards. "That's extremely dangerous, sir. Even just traveling in formation as a fleet will be risky for the Salik if they want to arrive closely enough to each other to concentrate their initial fire."
"I'm presuming you're worried about the hyperrift mouth," Ia said. Ishiomi nodded. "Don't be. There's only a seven percent chance someone will hit it after we exit, and none before. If that happens, I'll simply shoot the ship forward, and the gunners get to readjust their aim a bit."
"If you say so, sir," he muttered, his tone somewhere between tactful and dubious. The pilot wasn't one of the half dozen crew members who had asked to see the future a week ago. Ia didn't blame him, nor fault him for his skepticism.
Instead, she switched on the intercom. _"All hands, this is the Captain, prepare for hyperspace in five minutes, followed by one to two hours of nasty combat, depending on whether you're manning the ships or boarding the station. Be advised, some of you_ will _be injured; this is unavoidable. However, if you keep your wits about you and heed my precognitive commands, none of you should die. Please do not disappoint me. Five minutes to jump. Ia out."_
"Captain, Yeoman O'Keefe reports everyone is on board the drop ship and prepared to launch," Private C'ulosc stated.
"Good. Let's get the ship down out of FTL and ready for OTL speeds," Ia directed.
NEARSPACE, BEAUTIFUL-BLUE
SUGAI SYSTEM
Their oversized, lumpy silver needle of a ship emerged from the grey streaks of the hyperrift into a stuttering cage of bright orange and yellow. Thankfully, all of the Salik projectiles were aimed outward. A few of those lasers were firing through the dotted cylinder of the enemy formation, but none of them actually struck the _Hellfire_. Ia had timed the pattern and spaces perfectly. She sighed in relief.
Somewhere out there, five of the ships did have large anti-psi generators active, but they were nowhere near as strong as a station filled with hundreds of the machines being activated and tested. Ia had only a modest headache this time, a mild impediment at best as she dragged her toes through the waters in her mind.
The L-pod gunners were already firing the moment they were free and clear, aiming their lasers at a set list of priorities: shield panels, gunnery pods, FTL and thruster panels, and sensor arrays. The ship thrummed, vibrating gently with the efforts of the hydrogenerators powering all those lasers, but otherwise their entry into the big opening act of the war was fairly quiet.
The chaotic view of the battle contrasted with that quietness. At least, until the first hastily reaimed projectile hit. Their shields absorbed most of the blow, though some of the kinetic force of the explosion rattled into their hull. Ia relaxed a little at that. "Right on time...and now that the hull's been scratched, I feel a lot better. _All hands, brace for maneuvers. Yeoman O'Keefe, shuttle launch in two minutes ten seconds. P-pods, launch!_ "
The ship _whumped_ with the simultaneous launching of scores of missiles. Counting to three, Ia ramped up the FTL field, slipping forward. The hyperrift tunnel had closed without incident behind them, but now, easily fifty or more lasers and projectiles were being aimed their way. The missiles weren't a problem; the FTL field would shunt them and their impact explosions aside. It was the lasers that could do them the most harm; their wavelengths could still cause damage even if that field greased matter out of the way.
Telltales scrolled through her upper tertiary screens, echoing the ones Private Nelson was monitoring at the operations station. Several flickered yellow, indicating solid hits. She increased their speed with a jolt, forcing the interior safety fields to catch up for a rough-squeezed moment. Half a dozen more items showed up in yellow, and two turned red, then they outstripped the incoming fire.
Not because they needed to outdistance the Salik gunners' ability to aim and fire, but because they needed to outdistance the effects of the missiles the _Hellfire_ 's own gunners had just launched. Every projectile-pod bay had five blossoms locked in its hold, and every projectile pod had just fired one of those five.
It didn't matter how good the Salik shields were. Eighty blossom bombs implied a lot of kinetic impact force. With at least a third of those shields damaged by intense L-pod fire from the _Hellfire_ , that meant a lot of the secondary blossom bombs would get through.
Her right secondary screen lit up in more yellows and oranges, this time puffballs of silent explosions on enemy hulls instead of streaks of laserfire. That was, if one didn't count the _Hellfire_ 's aft-facing guns, which were thumping away. Ia didn't spare the rearward view on her right screen more than a brief glance. Instead, she flexed the FTL field, tricking physics into ignoring their forward momentum long enough to tilt their long axis a bit more toward the right, and a bit more still.
By the time they were sliding fully sideways through space to the oncoming Salik fleet, it was time to launch the shuttle. Cutting the field, Ia flicked the switch for the bow-bay doors, sliding them open. _"Yeoman O'Keefe, you are clear to launch. Godspeed, and follow the flight path I've plotted for you."_
_"Aye, aye, sir,"_ the other woman replied. _"Launching now."_
The projectile pods continued to _whump_ around them, shaking the bridge faintly. Ia nodded. _"Lieutenant Spyder, contact station personnel and have them take you to the spot I marked on your map as soon as you arrive."_
_"We're on it, Cap'n,"_ he agreed. _"Smack-dab central in_ _their hull-minin' efforts. We'll take 'em out from within, or your nickname ain't Bloody Mary."_
_"We're free and clear, Captain,"_ O'Keefe told her. The forward view on Ia's main screen showed the small silver bulge of the shuttle escaping ahead of them. _"Locked and loaded, and on our way to the_ Freely Flowing."
Several more ship systems turned amber on her list. Three more turned red, not nearly enough to hamper their capacities yet. But that was no reason to keep exposing the same surfaces to enemy fire. _"Acknowledged, O'Keefe. Acknowledged, Spyder. All hands, spinning the ship."_
A swirl of her left pinkie managed the trick, manipulating the ship through the attitude glove. Acting in angled opposition, the ripple in the insystem-thruster controls rolled the _Hellfire_ on its long axis, presenting new surfaces to absorb the incoming damage and new guns for opening return fire. Lots of return fire. Once they had the Salik fleet's attention, Ia backslipped the ship, swaying everyone on the bridge forward and to the right against their safety harnesses since they were still inward-bound toward the blue-brown pebble in the distance on their left.
"Alright, meioas," Ia murmured half to herself, half to her bridge crew. She spared a few seconds to call up a list from her files. "Let's reel in some of those bigger ships. C'ulosc, get on the comm with the ships I'm sending to your first tertiary. Suggest the battle plan in the first folder to their captains. I want to herd that star-side clutch of ships closer together."
"Aye, sir," he complied.
She switched on her headset. _"Captain Ia to Private Redrock, you have one minute thirty-five seconds before you must abandon L-pod 45 or risk serious injury. I repeat, abandon your L-Pod in one minute thirty seconds."_
_"Sir?"_ she heard the gunner query in her left ear. _"Abandon_ _it?"_
Ia rolled the ship again, this time adding a brief FTL twist that both shunted aside incoming missiles and allowed their bow to shift more toward the still inward-bound enemy. _"The_ _shield panels in your sector are failing. They're going to score a direct hit on your primary pod with both projectile and laserfire. You're going to get a feedback surge that'll overload the capacitors. Retreat to L-pod 47 and resume fire in one minute. Beware of maneuvers."_
_"Uh...aye, sir!"_
"Yesss!" Private Shim exclaimed from his position at the navigation console. "The GCMS _Bright-Falling Death-Wings_ just stomped the lead capital ship with two megablossoms, Captain. Plotting debris vectors to your third tertiary."
"Thank you, Shim." She didn't bother to look; the purpose of sending them to her console was to sync the navicomp's information with the helm computer, which would light up her main screen with the necessary collision warnings if those chunks came close. "Eyes to the boards, thoughts on your tasks, gentlemeioas."
The ship rocked gently under several more missile strikes. Ia dodged and swerved the ship, timing the thruster and FTL fields to the needs of dodging and returning fire. The ship shook with a louder _boom_ , testament to her warning. Several red telltales blipped onto her upper-center tertiary screen, warning her they now had a small hole in the hull. Not deep enough to vent the sector but enough to cripple the L-pod in question.
A brief temporal peek at Private Redrock's position showed him in the corridor outside his L-Pod station, shaken but not zapped by feedback energies. He was squeezing past repair teams sent by Lieutenant Commander Harper to seal off the damaged subsectors and reroute the fueling conduits.
Contrary to the illusions of the entertainment business, there would be no such zaps in here, where Ia sat. There were far too many capacitors and circuit breakers sheltering both the bridge and engineering compartments to allow that. Plenty for most other positions on the ship, but the L-Pod's circuits had already been damaged, allowing a little too much energy to bleed and arc past its safety systems. Ia liked that her bridge wasn't located in a physically vulnerable location, such as in some sort of control tower or windowed segment of the hull. That sort of nonsense was best reserved for vidshows.
There were half-silvered windows found on luxury starliners and orbital space stations; those windows had blast panels ready to close at a moment's notice. The narrow openings compared to the overall size of those vessels further reduced the chance of lasers targeting them. But battleships weren't built to be vulnerable like that. Even if the energy only pierced at half strength, windows were an open invitation for laserfire to bypass the entire purpose of a ship's silvery, tough hull.
"...Looks like they're herding about...nine Salik vessels into the requested formation, Captain," C'ulosc told Ia.
"Good. Tell the GCMS _Like-Love Hammering Hard_ to adjust course immediately to three by sixteen off its current heading. Let them know they have twelve _klis_ to get out of the way." Shifting her right hand from the thrusters to the lockbox, she flipped up the outer lid, scanned her palm, opened the inner lid, and thumbed the switch. "Private Nelson, have the water pipes around the remains of L-Pod 45 been rerouted?"
"Uhhh...getting the last one now, sir," the woman manning the operations station told her, coordinating with the engineering crew's repair teams.
"Good. C'ulosc, sound the retreat to our allies. Firing the cannon in ten."
The deck thrummed as the communications tech complied, passing her directive on to the Gatsugi ships through his headset. The Sterling engines began their _whooshing_ , audible warning of the power being raised. Ia adjusted course slightly, a tiny bit more—and blood red blasted from the nose of their ship, barely missing the _Like-Love Hammering Hard_.
The holes it left behind in eight of the Salik ships were gratifying to see, as the navicomp scanners magnified the view from ship to ship on her left secondary. Strafing slightly sideways as they were, the distance between those eight chunks of enemy ship wasn't readily apparent just from the view alone. The hindmost target, however, was almost five light-seconds away.
"And that's all she could do," Ia murmured, sparing a hand from the controls to flip down the lockbox. "Nothing else will align in this battle just right."
" _Shakk_ —sorry, Captain," C'ulosc apologized. "That's the fifth Gatsugi ship the Salik have blown up, their third biggest."
"Ten enemy ships are now down, with twenty-plus to go," Ishiomi reported.
"Captain, incoming fighters, 190 by 165," Shim warned Ia. "They're a mix of both sides."
"Looks like the Salik are going to try to play hide-and-seek with our hull," Rico observed. _"Aft P-pods, you get two shots. Pick your targets carefully. Double-check your missiles are set to friend-foe recognition patterns, Salik only for foe."_
"As much as I'd like to give the Gatsugi fighters some close cover, _not_ with an amidships hole in my hull," Ia told her crew. "We're slipping out of this mess as soon as your teams fire their second volley, Lieutenant. Eyes to your boards, thoughts on your tasks, meioas. We still have a lot of fighting ahead of us."
MARCH 4, 2496 T.S.
GATSUGI COMMERCIAL STATION _FREELY FLOWING III_
BEAUTIFUL-BLUE ORBIT, SUGAI SYSTEM
Wearing her four Gatsugi medals along with her half glittery of one pin representing each category, Ia was given enough respect from the harried station personnel to pass from the public sectors through to the inner medical facilities without much trouble. She had more difficulty making that transit physically, as those facilities overflowed with patients needing tending. Some had wounds that could wait, while others had to be rushed through as priority cases. More than once, she had to squeeze up against a wall, among the lesser patients resting in chairs and lying on portable beds, while a team of corpsmen rushed a patient through on a hovergurney.
Ships were still limping in from the outer edges of the system. The Salik had finally retreated when their numbers had been whittled down to thirteen vessels. They had taken heavy potshots at everything in their way as they ramped up to FTL speeds and vanished. The remnants of their crippled fleet were now being boarded by Gatsugi soldiers, and that meant the wounded were being cycled out of combat via insystem shuttles and sent wherever emergency services figured they could be saved.
The damaged ships in the Gatsugi Motherworld's nearspace weren't the only problem. Three of the largest ships had managed to sling their cargo planetward. Some had been shot down, but more than a hundred dropships crammed with combat robots had made landfall in or near major cities, tying up ground resources. Like most of the races in the Alliance, the Gatsugi had an aversion to artificial intelligences, and that meant tying up a lot of resources to crush the invasion: military, medical, and transportational.
That in turn meant most of the wounded here in space were being kept in space, even if it meant overcrowding the infirmaries on every surviving ship and station. Doctor Mishka had volunteered her services to the victims on the _Freely Flowing_ , the station which Spyder and his cross-platoon team of forty armored soldiers had helped to defend. A few of the other medical personnel on board the _Hellfire_ had volunteered as well. Aware of the good that could be done, Ia had permitted it for a while, but their time was up.
Private Fa'ala T'enku-o was the first one Ia found. Like Jesselle, Fa'ala was something of a biokinetic; the need to help the wounded was like a pressure under the skin for many of the healing-gifted. She wasn't a xenobiokinetic, but she could still glue-stitch simple wounds, salve and bandage burns, help set and immobilize broken limbs to await the Gatsugi equivalent of bone-setting compounds, and so forth. Beyond her, four or five makeshift beds down the hall, Private Kaori Isagawa also worked. At the moment, both of them were focused on changing dressings so that the more medically skilled Gatsugi nurses could handle their patients' species-specific needs.
The corridor smelled of alien sweat, blood, antiseptics, and other strange chemicals. The walls were painted in soothing green and cheerful pastels. The noise of the hospital was a cacophony of the babble of aliens as the injured cried out, while the harried staff tried to soothe their many patients in between treatments.
Ia stopped next to T'enku-o first, a spot of sober colormooded grey among all the brighter hues. "Good work, soldier."
T'enku-o blinked and looked up, apparently not expecting the Terranglo words. The predominant languages being expressed around them were variations on local Gatsugi dialects, gestures, and colormoods, though even a Human could guess the muddied browns and magentas of the patients were actually hues of pain, and basic gestures were basic gestures. Still, it took the private a few moments to shift from being a medic to being a soldier. When she did, she started again.
"Ah—sir. Captain?" T'enku-o blinked. The Gatsugi female she was tending flipped one of her lower hands in discomfort. The private returned most of her attention to the delicate task of peeling off the old bandage. Alien though they were, Gatsugi bled the same hemoglobin red as Humans, and had wound-sealing properties similar to a Human's scab-forming platelets. "Easy, meioa, almost done...Did you need something, sir?"
"Finish with this and the next two patients, then return to the ship," Ia ordered. "We're leaving within half an hour."
Her brow furrowed, but she nodded. "Understood, sir."
"Good meioa-e," Ia praised. Moving along, she caught up with Isagawa, who was smoothing down the self-sealing end of a clean wrap. "Good work, Isagawa. Finish two more, then join Private T'enku-o in returning to the ship."
"We're leaving, then?" Isagawa asked her. At Ia's nod, she lifted her chin off to her left. "I saw Private Orange over that way a few minutes ago. You want me to find and tell him?"
"I'll do it," Ia said. "You have patients to attend."
Pausing to allow another gurney to go past, Ia went in search of Privates Orange, Attevale, and Smitt. Orange and Attevale were assisting a pair of Gatsugi nurses in setting a broken leg; the nurses watched the monitors, giving instructions in awkward Terranglo, while Orange and Attevale used their greater Human strength to pull the limb straight and realign the bones. She didn't interrupt them, just let the pair know they were leaving soon, and moved on to find the next one.
Ia found Smitt in a storage locker not far from the surgery rooms, coordinating hasty inventory work with a Solarican warship coming into the system, since the station hospital was running low on key supplies. He was only a field medic but was adept at logistics and clerical work. More importantly, he could read and speak both Gatsugi and Solarican well enough to translate, albeit with a little help from his military arm unit and the language databanks back on the _Hellfire_.
Spotting her, he lifted a finger in acknowledgment, but continued talking into his headset in Solarican, reaching over the teardrop-shaped head of the short, pink-not-haired Gatsugi nurse working with him. Whatever-it-was slipped and fell down behind the shelving, making the alien crouch and root for it. Smitt pulled down another plexi-wrapped packet and added it to her basket, still talking in the rolling sounds of the Solarican trade tongue.
As soon as he finished his conversation, he nodded at his CO. "Captain Ia, sir. Do you need me?"
"You have about ten more minutes to wrap this up. Orange and Attevale will be in Exam Room 17 when you're done; assist them, then head for the ship when they do. I'm here to get Doctor Mishka."
"Good luck, sir," he snorted. "They brought in the crew of some V'Dan merchanter an hour ago, and she took over their care. Last I saw of her, she said she was scrubbing up for surgery."
"Then I think I'll join her," Ia told him. She smiled at his bewildered look. "I'm a biokinetic, too, soldier. Not strong enough to be a surgeon, but I can be her KIman. Ten minutes, Private, then grab Orange and Attevale. The doctor and I will follow shortly after."
"Aye, sir, I'll see you on board," he confirmed. He offered a hand to the nurse so she could grasp it with two of hers and rise, and smiled at Ia. "I don't suppose the ship can leave without her Captain. Good luck with the Doc."
"Tank you/Grat-tude, meioa," the nurse murmured, her accent in Terranglo almost too thick to be intelligible, explaining why the two had been using Solarican instead. Smitt turned his smile on her, and she blushed blue with pleasure.
Amused, Ia turned away. Some people preferred to stick strictly to their own species, while others were more open-minded. Personally, Ia didn't care either way; what her crew did in their off-duty hours—including volunteering in an alien medical facility—was their own business. As cliché as hospital romances were, she knew it wouldn't go anywhere anyway; neither of them would ever see each other again.
Two turns and two doorways later, Ia stepped into the sterilization hall. The ultrasonic scrubbers tingled unpleasantly as she passed between the banks of projectors, and the heat of the water at the sinks reddened her skin, but they were necessary. Emerging on the far side, she garnered wide eyes and confused-chartreuse looks from the staff. One of the nurses helpfully pointed toward a room off to the left. Ia already knew it was her goal but nodded politely to the gentlebeing in thanks for his help.
She didn't stop at the observation window. Waving her hand over the access panel, Ia stepped inside, passing through another sterilization arch. Assisted by three Gatsugi, one of them a hesitant xenophysician, Mishka stood at the manual controls for the surgery bot, guiding its tools in cauterizing the Human patient's internal wounds.
The male nurse with the lilac not-hair was the first to spot Ia. He lifted his upper hands. "No/No/Not supposed/authorized to be/enter here," he asserted, though he didn't leave his post at the anesthetics machine. "This room/place is to be/remain surgery/sterile!"
Mishka looked up briefly and scowled before returning her gaze to the control screens. "Captain, this is a restricted environment. You are compromising the safety of my patient."
"I'm here to assist, actually. Your job, Commander, is to _stabilize_ this patient," Ia stated, moving up beside the other woman. "Not prep him for a full repair."
"This man has internal injuries," Mishka protested. "If I don't finish this now, he'll die within the week. These people have too many other patients to handle it."
"Your job is to stabilize him, Doctor," Ia repeated gently. "Two days from now, the TUPSF _Granger VII_ will enter the system. They have the equipment and medicines to spare. For now, cauterize the last two major bleeders, then biokinetically stabilize him and install a drain shunt in his gut."
"I'm a little exhausted from trying to psi-stabilize our _own_ crew, Captain," Mishka retorted, guiding the microlasers to the next spot, "or I'd have done that already. And there aren't any xenohealers on board to help. I already asked. That means I have to seal off every leaking artery, then pack his guts with regenerative gel and monitor his recovery. Gut wounds are nothing to trifle with."
"The locals will need that gel for more Human casualties when the Salik come back in five days. I'm here to be your KIman, so you can save this one and still obey orders."
That got Mishka to look up. She studied Ia a long moment, before returning her attention to her task. "You won't let me bring him with us, will you?"
"Not unless you want to be drawn and quartered for Grand High Treason. We're officially at war now, Doctor," Ia told her. "The Admiral-General won't allow it. Earth, Beautiful-Blue, and two dozen other worlds are right now fighting off invasions of robots, attack vehicles, and mechsuited frogtopi. If you disobey my orders, your punishments will be doubled because we're at war...and doubled _again_ because I am a duly acknowledged military precognitive. So no, I will not let you bring him on board.
"I say to you now—as a precog—that all you need to do is stabilize him so that he'll survive for at least three days, and in two days, the _Granger VII_ will be by to pick him up and finish caring for him." Ia held out her hand. "You've just cauterized the last of the major bleeders in his abdomen. Encourage his body to heal what else it can, and move on. You're a very good doctor, but a terrible triage nurse. You need to know how to prioritize. Now, take my kinetic inergy and stabilize him. I have it to spare, and you have the training to use it."
Mishka looked between Ia and the unconscious man on the table. Nose wrinkling, she spat out a Russian word, no doubt a curse, and programmed the robot to withdraw its arms. "Get a drain shunt," she ordered the green-tufted prep nurse, moving around the end of the console. "His guts will continue to leak despite the cauterization, and they're much like yours; abdominal pressure can build up and kill him. I'll need to resterilize my hands. Doctor Nuwii, you'll need to close up the patient once we're done. Captain, if you'll move up on my left, you can grab that hand—I trust you've been sterilized, but have you done a KI transfer before?"
"I've already suffered once from xenobacterial sepsis myself, Doctor," Ia replied dryly, moving as bidden. Sharing KI wasn't difficult for her. The hard part would be making sure her precognition didn't trigger with the prolonged skin-on-skin contact. "I have no intention of making anyone else suffer like that, either. And yes, I have shared kinetic inergy before. Let's get him stabilized. We have to be out of here in eight minutes or we'll be too late to help the next batch of civilians under attack."
Three Gatsugi officials awaited them at the airlock leading to the gantry connecting the _Freely Flowing_ to the _Hellfire_. Arrayed in formal white clothes accented with blue, peach, and other hues, they bowed with supple grace as Ia and Jesselle approached.
"Captain/Officer Ia," the shortest of the three stated, the one with the extralong lavender not-hair. "We are/represent the Collective War Council. We wish/intend to present/give/honor you/your crew with awards/medals/honorifics for your valor/courage/skill/assistance this evening/tonight."
"I would like/be honored to accept, meioas, but my ship and crew have to go/be on our way now/immediately," Ia told them.
"Is it not the Terran way/style to honor your soldiers/warriors?" the tallest, pink-haired alien asked.
"It is, but I won't have time to stop by for a ceremony for three months and seventeen days, Gatsugi Standard," she said.
The middle-sized one, the female with peach not-hair, tipped her head and studied Ia with those black mouse eyes, which could see partway into the infrared. "It is true/true, then/yes? You are she/the Prophet/the subject/person of V'Dan prophecy?"
"Yes/Yes/I am she," Ia confirmed. Unsnapping the breast pocket of her Dress Greys jacket, she pulled out three datachips. "Here are the prophecies I can give you/reveal at this moment/point in time. They are cross-referenced/indexed under both Terran and Gatsugi Standard time references. The first two are not vital to obey/heed; they are merely/predominantly to prove/benchmark my abilities/accuracy. Please heed/follow the rest."
"You give/share these/this information to save/spare our people/race?" Lavender-not-hair asked her, accepting the chips.
"Some of it, yes. Some of it, no. Not everyone can be saved. Some will still die despite our best efforts," Ia stated simply. Soberly. "I grieve in shades of grey for their loss/demise. But I am a warrior/soldier. I will save/rescue those/what I can, and avenge the rest. If you will excuse/pardon us, we have to leave/go, now. The war has only begun/started, and many other lives/sentients need vengeance/saving/our help."
They bowed, and Ia and the doctor bowed back. Mishka stayed silent until they were halfway up the long, chilly gantry. "If you didn't want any more medals from them, why did you wear those?"
"They needed to know who I was, so I could come fetch you. Since there's only one Terran warship in the area, by wearing my glittery—and with it, the colorful Gatsugi medals the locals would recognize—most of the authorities on the station could figure out who I was without stopping and bothering me," Ia told her.
Mishka peered at Ia's jacket. "So what are those medals for?"
"The Red Badge of Combat, the Brown Badge of Courage, the Green Badge of Compassion, and the White Badge of Survival. I earned them helping all those prisoners to escape from the banquet on Sallha last year," she dismissed.
"Okay, I get the others, but why 'Survival' as a badge?" the doctor asked.
"It's a special category for escaping Salik tentacles after having been captured and presumed eaten." Ia smiled wryly, her rare, dark sense of humor surfacing. "It means I'm entitled to state-sponsored psychological care, Gatsugi-style, for the rest of my life."
Jesselle wasn't xenoignorant. She arched one of her brows. "Gatsugi-style? For the woman who constantly wears nothing but grey-mourning-colored clothes? You _do_ realize Gatsugi counselors all have degrees in fashion design and color sense, right?"
"Then maybe you'll find the fact I'm about to go change clothes and put on bloodred civvies a little disturbing," Ia quipped back. "By the way, _you_ , Doctor, need to attend Lieutenant Spyder's tactical debriefing and discussion session with the troops who boarded this station. You need to learn how to gauge a battlefield for strategic defense, offense, and combat creativity."
Mishka gave her a dubious look. "Me? Excuse me, Captain, but I am a Triphid. A _doctor_. I am not a battle commander."
Ia caught her elbow, forcing both of them to stop and face each other. She didn't let go, either. "You have less than two years to learn, Commander. If you do not, you will be _directly_ responsible for the lost lives and injuries of over two hundred thousand soldiers and civilians. You were given that fancy medical mechsuit because you _are_ going into combat...and at one point in the coming future, you will _have_ to instruct the soldiers placed under your command in field maneuvers in hostile enemy territory, because _we_ will be on the ground in hostile enemy territory. That means you will _learn_ how to be an officer of the Space Force as well as a doctor.
"Do you want to _see_ what will happen to all those people if you refuse to learn how to lead them to the best of your ability?" Ia asked pointedly. Mishka looked down at the hand on her arm, but Ia didn't press her point telepathically or precognitively. Not yet. "You and I follow the exact same code, Jesselle. Our goal is to save as many lives as possible.
"Sometimes you can save them with a laser scalpel, as you did today. But _sometimes_ you have to save them with a laser rifle, and _you_ need to know how." Ia released her elbow but held the other woman in place with her gaze. There was a reason why this confrontation was taking place in the docking gantry, rather than on board. This was as close to neutral territory as the two women could come, and both knew the gantry was being monitored.
"I shouldn't have to go into combat. I'm a _doctor_ ," Jesselle argued. "I'm not a soldier!"
"You bought all those fancy medical skills on the Space Force Education Bill," Ia reminded her. "This is the price you have to pay. You can complain about it all you like, but you'll have to get in line."
Jesselle folded her arms across her chest. "Behind _who_?"
" _Me._ I never wanted to be a soldier, growing up," Ia admitted candidly. "But here I am, doing my absolute strategic and tactical best to save lives in the face of rampant enemy aggression. And here you are, because you are the _right_ woman for the job. That job includes learning how to be a soldier and an officer—if a backwater nobody of a wannabe _singer_ like me can do it, you can do it, too."
Ia pointed up the gantry toward their ship. Muttering under her breath in Russian, Mishka moved. Her accent in Terranglo was nowhere near as thick as one of Ia's Naval Academy instructors' had been, but she sounded like a cat fighting to get out of a canvas sack as she started marching that way. Ia followed.
"Report to Lieutenant Spyder tomorrow at thirteen hundred hours in the bow boardroom. You will listen to the soldiers under his command as they dissect their post-combat reports on what went right, what went wrong, and why. Bring a datapad to take notes. You have no patients on board the _Hellfire_ who are in critical condition, so you will have no excuse to remain in the Infirmary," she added.
"Your philosophy of so-called 'free will' is a piece of hypocrisy, Captain. You're ordering me to do something against my will," Mishka muttered.
"Like I said, get in line," Ia muttered back, matching her stride for stride. "I suggest you blame the Salik instead of me. If they hadn't chosen to go to war, we wouldn't have to be out here to stop them."
They reached the airlock, guarded by Private First Grade Terry Warren, 2nd Platoon B Epsilon. Clad in light armor consisting of plates of silvery grey ceristeel on plexleather backing and a silvery grey helm, he looked like a redux of a medieval knight. At their approach, he held out a scanner wand. Ia and Jesselle held out their arm units.
"Welcome back, sirs." Private Warren greeted them as soon as the scanner greenlit their identities. "You're the last of the stragglers. Yeoman Yamasuka said we're clear to depart as soon as the three of us are on board."
"How's the hull?" Ia asked him, as they moved into the airlock.
_"Private Warren to Lieutenant Spyder. Captain Ia and Doctor Mishka are now on board."_ He touched the side of his helmet where his headset rested, then nodded. "We're gearing up for departure now, sir. Commander Harper told me to tell you most of the panels have been replaced, thanks to the Gatsugi repair gantries we borrowed. L-Pod 45 will still be out of commission until we can catch up with the replacement parts for the pod turret," Warren added. "He just needs to know where the Navy should send 'em, sir."
They stepped through the inner-airlock hatch into the portmost corridor of Deck 12. The door sealed behind them. A moment later, a soft _thunk_ warned them that the ship and station were indeed parting company.
"I'll look into it and let him know by the end of the day," Ia promised. "If you'll excuse me, meioas, today is my birthday, and I've allotted myself half an hour in the Wake Zone to party. If I'm not mistaken, Commander Harper has arranged a surprise party for me."
Mishka snorted. "It's hardly a surprise if you already know about it."
"True," she allowed, moving away, "but I very carefully did not peek at what kind of cakes he asked the forward galley crew to bake."
"Captain," Mishka called out. Ia halted and turned to face her. The older woman sighed. "Our other argument aside...thank you for your KIman's help, earlier. I hate leaving a Human patient in alien hands, but with your help, at least he's stable."
"You're welcome." Ia waited, sensing Jesselle wanted to say more. The ship moved away from the station, tugging them slightly aft-ward.
"I am curious as to what information you gave those Gatsugi soldiers," the doctor added, hands tucked behind her back. "And why now? Why not earlier?"
"The why is easy. It's the right time to give it to them. Up until recently, no one knew I was a precog," Ia told her. Behind the doctor, Warren lurked, trying to make himself inconspicuous. Ia knew he would gossip to the others about whatever she said here, so she picked her words carefully. "From this point onward, I have the trust of most of the Command Staff, but that only affects what the Terrans do with the information I give them. The other nations in the Alliance also need to have that level of trust.
"That's a small part of why we came here to help the people of Beautiful-Blue survive the first attack. To show to them how accurate I am, and how effective my ship and crew can be, so that they will become willing to follow my directives in the future. We won't win this war in a single battle, or even a single year. Nor will it be won by a single species' efforts. Right now, I have the solid trust of the Terrans and the Tlassians, thanks to my friendship with the Grandmaster of the Afaso Order and his connections with his home government. The Solaricans gave me some of their trust when they made me a War Princess in rank, and now the Gatsugi are starting to come around. The K'Katta, V'Dan, and the rest will come in due time."
"And once you have everyone on your side?" Mishka asked, shifting her hands to her hips. "What then?"
"Then I'll direct them in ways to save the biggest number of sentient lives," Ia stated. In candor, she added, "Unfortunately, I cannot save every life. No one can. No soldier, no citizen, no healer can save every single life that crosses their path—you know this as a doctor, try as hard as you might. Today, we helped save that merchanter crewman's life. Tomorrow, we may or may not be able to save others' lives. As soldiers, fighting in a war we did not want, we _will_ have to take away lives, too. The object is to be so good at our jobs as soldiers, we take away only an absolute few.
"Now, if you'll excuse me, I need to set aside my regrets for all the lives I could not save and the things I could not do, and go reflect on the fact that it was a good day's work," Ia told her. "Since it's also my birthday this week, I have scheduled a Wake to begin as soon as we hit FTL speeds, and I am curious to know what kind of cake awaits me in the rec room."
Dipping her head in a modified bow, Ia walked off, hands behind her back. A check of the crew's timestreams showed that Mishka would soften a little bit more toward her in a few more weeks, thanks to today's efforts. Private Warren would spread the word of what motivated their oddball captain, which wouldn't hurt things either.
_And the Gatsugi are going to become impressed with my prognosticative prowess, particularly once their own military command reviews the precision and timing of everything my ship and my people did for them. I'll have to remember to remind Grandmaster Ssarra to start searching for trustworthy Gatsugi monks to serve as go-betweens with the Collective in the coming years._
_A good day's work...I wonder if Harper remembered Ihad the Deck 7 storerooms stocked with canisters of topado flour and other Sanctuarian foodstuffs? I know I copied some of the family recipes to the galleys' menu files. I miss my mother's tasty, bright blue, topado-flour birthday cakes._
The dress was a few years out of style; straps down the shoulders and arms were no longer in fashion. But it was still bright red, and still fit her figure, if a bit loosely. Ia's bout with blood poisoning had weakened her body. Following that up with more administrative work in the past handful of months than physical work hadn't been enough to rebuild all of the strength she had lost. She still had visible muscles, but it wasn't the same.
_I'll have to eke out an extra half hour of weight training every day,_ Ia sighed, scraping her pale hair back from her face. _Not to mention, I need a haircut. My bangs are getting long enough to get in my eyes...and I'm procrastinating aren't I? It's okay, I can do this. The crew know better than to touch me. They should be safe from me._
Squaring her shoulders, Ia left her quarters and headed down the hall. The former boardroom, located one floor up from the bridge and forward enough that one bulkhead served as the dividing hull between mid and fore sectors, had been converted into a recreation hall. Or rather, a relaxation lounge.
The banks of seats had been taken out and the tiers built up into different layers of platforms, some enclosed in walls that formed private niches, others more open to the room. Off to one side, a buffet had been set up next to the dumbwaiter system Ia had ordered installed to shuttle up food from the galley one deck down. Not just snacks, either; for a modest fee, special meals could be ordered off a single-serving menu if someone didn't like whatever was on the day's menu, and there was a liquor dispensary, which would dole out one hard drink or two of wine or beer per Wake-day, provided it was the start of a crew member's off-duty cycle.
The main floor had been converted into a dance floor, and the standard-issue monitors replaced with floor-to-ceiling enviroscreens. Smaller screens around the room displayed the Wake rules, reminding everyone this was a "civilian" zone.
Those rules were fairly simple, too: that the use of rank and authority was strictly limited to on-duty personnel only, who weren't supposed to be in the Wake Zone without due cause; that everyone, on-duty or off, was still responsible for the Lock-and-Web Law of space travel; that no uniforms were allowed on off-duty personnel within the designated zone; and that no law, military or civilian, was to be broken, save that all off-duty personnel in civvies within the zone were supposed to be treated as civilians.
Today's theme, prepared in the hours before the battle by bored crew members, was French Polynesian. The giant screens reflected a cerulean cove with a white sand beach and jungle-covered hills. Someone in Supply had dug out faux-thatching for the tops of the alcove-booths and strung garlands of bright, fake flowers around the hall. No one was dancing, but then the music being played was some variation on islander rhythms mixed with the sound of surf crashing on the projected beach, along with the occasional calls of tropical birds.
Last week, it had been a snow-dusted Bavarian village, with the rec-room temperature turned down to simulate a decent winter chill. Drinks had been served hot, snacks were sweet, and there had even been caroling contests with group songs and solo performances echoing up and down the corridors, some good, some bad, and all of it encouraged. Her little talk with that clutch of former Army soldiers during their first Wake had spread through the crew. No one gave her fellow ex-Marines grief anymore about wanting to sing.
Next week, if she remembered correctly, the Wake was scheduled to simulate the Athena Dome, a sports-themed amusement park on Mars; the activities listed on the roster included several ball-played sports games, plus vidgame competitions, and prizes for the highest-scoring shooters on the ship—excluding Jana Bagha and her husband Bei Ninh, to be fair to the others. They would get to be the judges.
Ia had stayed away as much as possible from the previous weekly Wakes until now. She wanted her crew to be able to relax, to claim the space as their own. To not have to worry about the rules and regs, or even be reminded of them simply because their Commanding Officer was around. This was their fourth Wake, though, and the timestreams had showed them just comfortable enough to survive her appearance.
They were so comfortable, in fact, that the first person to see her took in the curves of her red-clad figure and the legs bared below her midthigh hemline, and let out a wolf whistle. The noise of his appreciation drew the attention of several others in the room.
"Nice legs, meioa-e!" James Hong called out, grinning in appreciation as he lifted his gaze from her knees to various points higher. "Nice the rest of you, too. I could definitely—oh _shakk_! Captain!"
His feet came off the table and his tanned face flushed, then paled. The half dozen or so who had turned at his whistle also blanched. Ia strolled over to Hong and propped her hands on her hips.
"You say that rank to my face one more time in this place, and you'll get lifesupport filtration duty for a week," Ia warned Hong, waggling one finger at him. "I left my rank and uniform outside when I put on this dress. That is the Wake Zone rule, meioa. Right now, I'm a mere civilian, just like you." She started to walk away, then turned back and gave him a smile. "I do thank you for the compliment, though. I think they're very nice legs, too."
From his surprised but amused chuckle, Ia could tell he would recover from the shock of her presence. Nodding politely to the others, she headed down the terraced levels, searching for Meyun. She couldn't exactly see him in the timeplains at the moment, but then he was the one person in the universe whose movements she couldn't entirely predict. Somewhat, but not entirely.
Most of that, she had figured, was her mind protecting her from her gifts; she was very much attracted to the man on many levels. That in turn meant there was a chance that her emotions could sway her off course if she acted on those feelings, particularly if the consequences were personally appealing.
His laugh hadn't changed since their Academy days; Ia followed the sound of that familiar, light baritone chuckle down to the lowest of the alcoves. Clad in shades of blue, he backed out of the makeshift room, hands raised in mock-protest at whatever had just been said.
Her precognition rose involuntarily within her. It swept over her like a tingling wave, dragging her down beneath the waves as she stood there, watching him. Watching a vision of his future.
_...Meyun sat in the alcove and cuddled Nueng in his arms.The young woman snuggled back, content to be in his lap. Outside the Wake Zone and the privacy of their quarters, they were discreet and professional, but here, they felt safe enough to be affectionate. Because their Captain had made this place safe, despite the chain of command that governed the rest of the universe occupied by the Space Force..._
The floodwaters of that possibility chilled her from skin to bone. Her heart hurt at the thought of Meyun finding happiness with someone else, someone not her. Someone not his Company commander. Her head wisely pointed out that it would be for the best if he turned his attention elsewhere, even though it hurt.
Her awareness of that potential possibility happened in a flash, over and done in just a few seconds. She managed a smile when he glanced her way—and watched him give her a double take worthy of Hong's, though without the whistle. Harper's smile was genuinely warm as he looked at her, his brown eyes bright with male appreciation as they slipped down to her short red boots and back up again.
"Well, look at that. You _did_ show up. What a surprise," he teased. "I wasn't sure you'd bother."
Ia smiled back ruefully, hands going to her hips. "We established long ago that I'm a very dull girl, Harper. I came here because I know you arranged a cake. Where is it?"
Turning, he gestured at the interior of the alcove. "Bring 'em out, meioas!" Raising his voice, Harper moved to the center of the hall. "...May I have everyone's attention? Yes? Thank you! As you all know, we've got a little tradition of celebrating birthdays each week at these Wakes, if there are any.
"This week," he stated, as heads poked out of alcoves or turned away from conversations, "we're celebrating three birthdays! Last but not least is Melody Nelson's birthday, March 8. Unfortunately, she's currently on duty, as this is second watch, so if you have a chance to celebrate it with her later on today, or at least run across her, wish her a happy birthday. Right smack in the middle is Ann Velstoq', whose birthday is the sixth," he added, managing the V'Dan glottal stop at the end of her name with the ease of someone who had practiced. "And there she is. Come on down, Ann; don't be shy."
Gesturing for her to join him, Meyun led the way toward one of the empty tables on the lowest terrace above the dance floor. Two members of his engineering teams followed, Zedon and Svarson. Each man bore a platter with a cake on it, each frosted and iced with a name.
"And today's _actual_ birthday girl, as in born on this date a mere twenty-four years ago, Terran Standard," Meyun teased, grinning up at his target, "is our very own Ia!"
" _Shakk_ me!" someone swore. The voice belonged to Tanya Doedig, Ia realized, one of Harper's engineers. The older woman eyed Ia askance. "You're only twenty-four? I could've sworn you were _thirty_ -four, Ca—er, S...Crap on a crutch! _Meioa-e_ ," the dark-skinned woman finished, using the honorific instead of Ia's rank or title. Doedig rolled her eyes. "Shove me out an airlock—I am _not_ used to addressing you casually, meioa-e. I think I've been in the military too long."
"Technically, you've been in only one more year than I have," Ia pointed out. She turned to Ann, who had hesitated halfway down the stairs. "C'mon, let's go cut the cake. I'm dying to know what kind got baked."
Ann eyed her dubiously. "Aren't you a massive precog? Wouldn't you already know?"
"Only if I peeked. And I very carefully did _not_ look at it in the timestreams, despite _great_ temptation," Ia asserted.
Halostein, a normally reserved, no-nonsense sergeant, grinned at her. "Well, you just earned _my_ respect, if you honestly didn't peek. I learned how to get into and out of my Christmas presents at a very early age with no sign of having opened or resealed the box. At least, until my fathers started hiding my presents at my biomom's house, and hid my half sisters' presents at our place. The first time that happened, I honestly thought they'd got me a dolly in a frilly dress!"
The story got a chuckle out of his listeners, Ia included. Halostein offered her the hilt of a knife he pulled from one of his cowboy boots. Accepting it, Ia moved over to the cakes.
"I truly didn't look. I spoilered myself with the myth of Santa Claus at the age of five, and things went downhill until I was eight or so, when my older brother pointed out it was my own fault for peeking all the time. He scolded me and said that if I ever wanted to be surprised, I had to be strong enough not to look...so I don't look at the things I know are going to be pleasant surprises. I always look in advance at the ones I think won't be. It makes it easier to avoid 'em."
Cutting into the one with her name iced in white, she discovered from the crumbs beneath the blue frosting that it was a chocolate cake. Ia didn't mind chocolate. She sliced the rectangle into several pieces, then served herself one. Ia wiped the blade on one of the napkins clipped into the holder on the table, and passed it to Ann, trading the knife for a fork.
Just as she forked up her first bite, Ia saw the blue crumbs mixed into the white frosting of the other cake. She glared at her first officer. "...Hey! _She_ gets the topado-flour cake? _I'm_ the one from Sanctuary, Harper. That's _my_ comfort food you put in _her_ cake."
Ann blinked, prodded at the corner piece she had cut off, and quirked her brows. "Yeah, what's up with this blue stuff? _I_ asked for a chocolate cake, not whatever _this_ is."
"It's made from topadoes, and it's very nutritious and very tasty," Ia told her. "It's a kind of tuber that can be baked, fried, mashed, grilled, or dried and ground into flour."
Harper shrugged, biting his lip in the unsuccessful attempt to hide a smile. "My apologies, meioas; I guess the cakes got mixed up when they were being frosted. But there _is_ an easy way around this problem, you know."
Ann looked at Ia, shrugged, and offered her untouched fork and plate. "He's right. And I know just what to do about it. Happy birthday, 'Ann,'" she quipped, eyeing Ia. "May you have a wonderful natal day."
Since she technically hadn't eaten the chocolate one in her hands, Ia offered it to Ann in turn. "And a happy birthday to you, too, 'Ia,'" she joked back. "Try a bite of the topado cake anyway. You might like it."
"After my slice of chocolate," Ann bartered. "Nothing gets in the way of me and my birthday chocolate."
Moving back from the table so the others could try the two cakes, Ia found herself next to Harper. He snagged her forkful before she could eat it, and popped the blue dessert into his own mouth. That lifted her brow, but Ia didn't protest, just took her fork back and cut another piece for herself.
"Mm, good," he murmured. Moving a little closer, Meyun whispered in her ear, "But I know something else from your homeworld which tasted even better."
Goose bumps prickled along her skin. Her former Academy roommate...her former lover...had a knack for rousing old memories she wanted to keep repressed. Ia knew that image of him turning to someone else was supposed to be the better choice, even if it hurt.
She changed the subject. "How's the gun project coming along?"
"I thought we weren't supposed to talk about work," Harper retorted dryly.
"Only in an official capacity. This is unofficial, one friend to another," Ia pointed out.
He sighed, sagging against the railing separating the terrace and its tables from the empty dance floor below. "Lousy. I can't make heads or tails out of the source for the focusing crystals the Immortal used. It almost sounds like potassium nitrate, given she said she extracted the crystals from her own...um, yeah. But the physics and the optical properties of saltpeter crystals are all wrong for what the guns are supposed to be able to do, so it wasn't _that_."
"They wouldn't work if we were dealing with an average Human, no," Ia murmured back. "Luckily for you, I know exactly what kind of crystal you're referring to. Get me the specs on the shapes and dimensions you'll need, and I'll get them for you. But be careful and thorough in your calculations. You won't be able to alter the crystals in shape or size once you have them in hand. Only I can."
"If the material is as rare as the properties she describes would make them out to be, then yeah, I'll want to get them right. Whatever your source is, it's bound to be extremely rare, and hard to get," he agreed.
_Rare, yes, but not_ that _hard to get,_ Ia thought, forestalling a reply with a forkful of white-frosted, blue-floured, slightly spicy cake. _Just sitting in palm-locked storage down by the bow shuttle hold is all._
Someone swooped in from her left, grabbed her face, and smacked a big, loud kiss on her cheek. Those hands and lips, applied at less than a twenty percent probability, belonged to Private Second Class Yung Ramasa. He released her with a grin, spreading his already broad mouth even wider, making him look like his military nickname. "Happy Birthday, pretty lady!"
The other crew members stared wide-eyed at the two of them, shocked and apprehensive. Mindful of the rules, Ia freed a hand, hooked it around his head, and pulled his own cheek into reach for an equally loud-smacking kiss. "Thank you, Your Highness. Now go kiss Ann, too."
Ramasa laughed and rubbed his hands together in delight. "I was just on my way to do that, Ca...er, meioa!"
"—Oh, no you don't!" Ann protested, hand up to her mouth to cover the fact she was still eating her cake. She swallowed and waggled her fork at him. "I know your reputation with the ladies, O 'Frog Prince,' and I am _not_ going to be one of your conquests!"
"I think she's sweet on him," Ia stated, catching both of their attention. She gave the smirking Ramasa a warning look. "But if he doesn't back off, she'll thump him."
"It's just a little birthday kiss!" he protested. " _You_ didn't mind," he added to Ia, moving a little closer to Ann. "Why should she?"
"You didn't give me a chance to protest," Ia countered, enjoying the moment.
"Nonsense! I am quite sure you foresaw it coming, which means you chose to accept it," Ramasa stated, sidling a little closer to his target.
He lunged, lips puckered—and got thumped in the shoulder by the edge of Ann's fist. The gunner yelped and backed off, pouting...and at that point, Ann relented, leaned over, and kissed him on the cheek. He grinned and kissed her back. She thumped him again, but only lightly.
"...I'll admit I saw _that_ coming," Ia stated primly, and scooped up another forkful of fluffy blue cake. Her second-in-command gave her a dirty look. "What? I'm just here to enjoy my birthday cake...and a little impromptu floor show."
Meyun wasn't the only one to chuckle at that. Even Ann and Yung laughed.
Ia nodded to herself, enjoying another bite. _Mission accomplished. This portion of the crew has relaxed around me,_ and _I got my birthday cake._
# CHAPTER 10
_Why did the Gatsugi accept my abilities faster than my own government? Well, I could quote the old maxim that a prophet is never honored on his or her own doorstep, but I think it was due to three things. I gave them the exact inward-bound sectors of the Salik attack fleet, gave them a good eighteen minutes of advance warning, and my ship and crew managed to account for the destruction of half the ships taken out of commission—three times more than the next best effort by one of their own forces._
_Keep in mind that the Gatsugi were not weak; my ship was just better aimed overall. Motherworld systems are also the most heavily defended, which is why the Salik chose to strike at them, to land psychological blows against their opponents as well as physical ones. If my ship hadn't been able to take out so many of theirs in one blow, the number of robotic dropships would have increased exponentially._
_I also chose to aid the Gatsugi because the Terrans had already been warned about the coming war months in advance. I told my superiors when they would need to be ready for it, and what to be ready for. As it was, the fleet back in Earth's nearspace was still damaged worse than the Gatsugi ones. Not by too much, but by enough to make my precognitive warnings clear._
_So the Terrans did believe me to an extent, particularly after the official start of the war. They just didn't have theshock of instant power and instant proof magnifying that belief, as the Gatsugi did. Instead, the Command Staff were gradually introduced to what I could do, subtly before the Battle of the Banquet, and more openly afterward. Because it was gradual, the impact just wasn't the same._
_~Ia_
MARCH 19, 2496 T.S.
SYSTEM'S EDGE, NUK NUKLIEL 83
"Gadalah," Ia stated, sparing a glance from her main screen to her left secondary. "Target fragment...1172. Blow it up, three missiles."
"Aye, sir," Private Gadalah stated, relaying the orders on her headset to a trio of gunners.
"Coming up on tanker midpoint in fifteen," Fielle warned Ia. "Coming up on the drone carrier's midpoint in...thirty-five. Fifteen, sixteen fighters behind our ninety-ninety, Captain, three stragglers behind the 270 by 90."
"Captain, we're being pinged on the hyperrelays," Al-Aboudwa warned her. "It's asking that we switch to secured channel two—the code's the one for the Command Staff. What do I do, sir?"
"Be polite, Private," Ia murmured, her attention on keeping them alive in the mix of seven midsized ships, two tankers, and a host of fighter craft. "Ping them back on the confirmation code of the day."
"MacInnes still needs more of those transmissions, Al-Aboudwa," Rico told the comm tech. "She's not getting enough nouns to decode it. If that's the Command Staff, they'll need it fast."
Their shields reverberated with an odd shudder. Not the thumping rumble one expected from a projectile weapon, but from the fragmentation of a loosely aggregated ice clump disintegrating on impact. Another, louder _whump_ shook the ship, this time from an actual missile.
Ia tipped the ship a little more down and to her right—down relative to her dead-ahead vector, that was, since "down" in starfighting terms was always the enemy's main position, and the majority of the enemy were off to the left. Telltales flickered yellow and green, stippled here and there with unpleasant red. Rolling the ship clockwise presented fresher targets for the lasers and projectiles aimed their way. It also allowed her to strafe them sideways, through the debris of the now-scattered chunks of shuttle-sized ice that would have slammed through their shields and into the hull itself had Gadalah's gunners not fragmented it.
_"Shakk!"_ Al-Aboudwa cursed. He cleared his throat. "I mean, Captain, sir, it's the Admiral-General. She's calling for you, sir. Should I tell her you're busy?"
"Not today. Patch her through to my third tertiary, Al-Aboudwa; I'll take her call," Ia pointed out, left hand flicking through the commands in the attitude glove, right hand dancing over the thruster controls. "Fielle, take out those fighters. Gadalah, focus on the dropship."
Her lower third tertiary screen dropped its bar graphs of energy outputs from the various engines around the ship, replaced by the round face and grey-streaked black hair of Admiral-General Christine Myang. "Greetings, Captain Ia."
The pingback icon in the lower right corner of the screen showed they were on a five-second delay. Hyperspace communications were fast, but not instantaneous. Ia shifted the ship again, glancing up at the spate of yellowlit warnings on her upper bank of monitors. "I need those fighters taken down, Fielle."
Myang, five seconds behind, continued talking. "As you may or may not recall, your six months of _carte blanche_ are now up, and it is time for your performance re...view? What are you doing, Captain?"
"Good afternoon, Admiral-General," Ia greeted her, without looking down at the screen containing the Admiral-General's face. "I am currently doing what I do best. Saving millions of lives by destroying a few hundred Salik. But I'm not too terribly busy, and I know you've allotted this half hour to talk with me, so go right ahead. I am listening—Nelson, Al-Aboudwa, I'm getting some odd flashes of light from those fighters. Try running those through the lieutenant's code crackers."
Another _paff_ of breaking ice was followed by a _clang_. Dubsnjiadeb cursed. "—I think they're trying to ram us, sir! They're hemming us in with the larger chunks."
"Well, blow them up!" Rico retorted.
"Doobie, two of those ships are about to get away. Plot courses for them and coordinate with Gadalah's teams. I want intercept arcs for each vessel," Ia ordered. She pulled her gaze downward long enough to smile at the Admiral-General. "Go on, sir. I _am_ listening."
Five seconds later, Myang responded. "Maybe I should call you back later."
Ia responded on top of her, mindful of the five-second delay and wanting her words to interrupt the older woman. "I'm afraid we'll be in transit in less than fifteen minutes, sir." Myang fell silent, so she continued. "We won't be able to stop and talk for another six or seven hours, which is after we've helped fend off the first—Fielle, we still have four fighters on our tail, I said get rid of them—the first invasion wave on the Solarican domeworld of Rau Niil II."
"In four more hours, Captain, I'm supposed to be in bed, getting some badly needed rest. Unless there's another emergency requiring my personal oversight," the older woman muttered. "Alright. We will deal with the question of your _carte-blanche_ demands now. Presuming you can handle it?"
"Got 'em, Captain!" Fielle crowed as the aft gunners successfully hit the final three fighter ships. "Take that, you frogtopi! Never mess with _this_ ship's weaponry!"
"Don't get cocky, Yeoman," Rico chided him. "You're not Shikoku Yama, you know."
"I know I can handle it, sir, or I'd have contacted you earlier when we were repairing from the engagement at CS-35," Ia said. "You've seen the recordings we've mailed your way. You know what I'm capable of doing with this ship, given the freedom to use it appropriately. I have only done so freely for the last six weeks. Imagine what I can do with six more weeks, and six again beyond that."
Around her, the screens flared with streaks of light, silent and not-so-silent explosions, and shifting stars as she maneuvered the long ship. The cometary fragments were their best defense against the Salik vessels, even as they caused problems of their own; the _Hellfire_ had been built with multiple redundant shielding systems and extra hull plating just for this purpose: surviving heavy enemy fire. Not without cost, though; several more items appeared on her upper screens in red.
"I take you want more _carte blanche_ ," Myang stated after a pause of about ten seconds. "Another six months' worth?"
Ia dipped her head, her gaze still on her main screen, piloting the _Hellfire_ through the debris. "Yes, sir; that would be lovely, sir. If you could authorize it right now, having that on hand when we reach Rau Niil will shave ten hours off our repair time since it means we can commandeer their biggest low-grav repair cradle. That means we'll have just enough time to make it back downstream on the Arm to defend the colonies at Proxima Carinae. It's a small engagement, but a vital one, temporally."
"Sir, getting a ping from the system buoy we dropped," Al-Aboudwa warned her. "Three...no, five Salik vessels inbound from insystem. They're free of the belt, star-ward vector 15 by 330, and closing fast. If they don't slow down below three-quarters Cee, ETA in two minutes forty seconds."
"How is it vital temporally?" Myang asked Ia.
"In about five weeks, if most of the colony survives, they'll discover a mother lode of ore containing molybdenum, which is a component in ceristeel manufacturing. In eight weeks, the London Metal Exchange will be shipping tonnes of it to manufactory sites throughout the Alliance, and it will continue to produce high-quality ore well into the second war," Ia revealed. "Gadalah, I need ice fragments 503 and 1257 destroyed."
"Aye, sir, I'm on it!"
"Admiral-General," Ia continued, "if we don't show up, the colonists _will_ still find it in about eight months, but by the time they get the ore to the ceristeel manufacturers, we'll have lost over 327,000 soldiers, and far too many civilian lives." Dropping her gaze to her lower-center tertiary, Ia met the older woman's gaze through her screen pickups. "I can list the names, ages, and favorite foods of each and every single soldier and civilian who will die, if you like."
Her gaze snapped back up to the main screen as three lasers hit their hull, red-lighting two FTL panels and turning a third amber. A few more hits on that sector, and they'd be unable to form a complete field. With a hiss of triumph, Rico lifted his head and glanced her way.
"Got it! Captain, MacInnes cracked the code. Part of it _was_ in the running lights. Permission to send it to the Admiral-General on a subchannel, sir?" Lieutenant Rico asked.
"Permission granted, Lieutenant," Ia told him, before returning her attention to Myang. "Sir, you're going to receive today's Salik code for the war effort within one to two parsecs of Nuk Nukliel 83," Ia told her superior, slipping the ship toward the bottom of the ice cluster. "They're changing up codes every few days and using lightwave signals as well as hyperrelays, so I cannot guarantee it will work for long, or even work in another sector, but it will for this one. Firing the Godstrike cannon in fifteen."
"You know, sir, we could really use sunglasses for that thing," Nelson quipped from the operations station.
Ia unlocked the box, flicked on the system, adjusted their attitude slightly, and thumbed the firing control. The lights on the bridge dimmed, while the _thrum_ of the hydrogenerators and _whoosh_ of the heat pumps joined the smashing of ice and clashing of projectiles. One sanguine-bright flash later, Ia checked the lightwave readings versus the spare system buoy they had dropped, making sure she had killed the inbound ships, then glanced down. Admiral-General Myang winced as she watched, tanned face and grey-streaked hair briefly flaring pink in the glow from her monitor.
"I trust you know exactly where that beam will be three or four light-months down the road?" Myang asked her dryly.
"Sir, yes, sir. What little didn't chew through four large ships and a swath of the local Kuiper belt will be busy dissipating harmlessly in off-plane and interstitial space."
"Captain!" Dubsnjiadeb called out. "Navicomp's showing inbound objects at near-Cee. They're trying to throw rocks at _us_ , sir!"
"Kind of stupid, if you ask me," Nelson muttered from the operations seat. "Most of 'em will break up in the ice field."
"Thank you, Doobie. They're desperate, Nelson. Gadalah, dump fifty proximity mines in our wake. Fielle, keep firing. _All hands, brace for acceleration,_ " Ia warned through her headset, before slipping the ship dorsal-ward, pressing everyone down into their seats.
"Never give the enemy a weapon you wouldn't want turned on yourself, Captain," Myang added through the hyperlink, apparently having heard Dubsnjiadeb's warning.
The pull of vector change that made it through the pulsing ripple of greased physics felt heavier than Ia's homeworld, making it hard to breathe. Several ice chunks broke noisily against their shields, but it was the fastest way out of the Kuiper field. She shifted them forward, transitioning into a rising arc that would skim the upper edges of the tumbling bits of frozen and metallic debris.
"Don't worry, Admiral-General, they want us and our planets mostly intact, so they won't sling boulders at our colonies. Just rocks at our ships. We're on our way out of here now, sir. If you're willing to grant me another six months of _carte blanche_ , now would be a very good time to transmit the updated authorizations, before we have to end the link."
While the Admiral-General thought, Ia eased up on their acceleration, partly to spare her crew, who weren't used to long stints in high gravity, and partly to change vectors. Some of the Salik missiles were still going fast enough to catch up with them, and the Salik lasers were getting better at targeting the _Hellfire_ 's hull, now that Ia was speeding up for an escape. The faster one went, the harder it was to deviate from course, and the Salik gunners knew it. Twisting the ship counterclockwise a little, she presented a fresh section of panels for the lasers to score, and accelerated again.
"...You haven't disappointed me with your performance yet," Myang stated, shifting in her seat to input the codes on her end of things. " _Yet._ Transmitting the paperwork, Captain."
"Thank you, sir. Al-Aboudwa, catch that and copy it to the Company files," Ia ordered, easing back on the thruster fields so he could move. She had to roll the ship slightly once again as the lasers continued to tag their hull.
"For now, Captain, you have the continuing confidence of the Command Staff," the Admiral-General told her. "Don't _shakk_ it up."
"Sir, no, sir," Ia agreed. "That is not my intention, sir."
"Got it, Captain," Al-Aboudwa told her. "Received and saved."
"Good job, Private. Thank you, Admiral-General," she added, eyes fixed on the screens displaying their surroundings, toes in the timestreams. "Now, if you'll excuse us, we need to go pay for some repairs with that lovely cheque—one more thing, sir."
She pulsed the field panels for six seconds, slipping them forward hard and fast.
Myang lifted her brows. "Considering I've renewed your _carte blanche_ , under the same double-indemnity terms, what else could you want?"
Ia eased back. "It's not about what I want. It's a warning, sir. Two months from now, a couple of the Feyori are going to be very pissed at me. They may try to infiltrate and influence the minds of the Command Staff. Be on your guard."
Again, she accelerated. This time, Myang frowned. "What are you going to do to them, soldier?"
"Actually, it's what I already did," Ia grunted, fighting the vector pull. She rolled the ship one last time. "I do have plans to take care of them. I'm just warning you, sir. Ia out."
A tap of her thumb ended the comm link. Not that full FTL wouldn't have ended it for her, since it was extremely difficult for the ship to maintain hyperrelay communications without a dedicated vacuum chamber on board while wrapped in a skin of warped physics.
A scrape of her fingertips shot them forward hard and fast, leaping ahead of the pulsing, orange streaks that were now the only weapons that could catch them. Just as the enemy's sensors recalibrated, tagging one last shot on their aftmost panels, the stars on the screens burst and streaked, crossing the lightspeed barrier.
Gentling their acceleration, Ia checked her upper screens, tabbing through the list of damaged hull components. "That was a nasty fight. But we're still greenlit for travel. Power your station back around, Fielle, and prepare to take back the helm. We'll stay at FTL for another half hour to evade pursuit, then hyperwarp the rest of the way."
"Can I hit the head first, Captain?" he asked her.
"Go right ahead. Just don't take all shift," she warned him. "Doobie, plot a hyperrift course for System Rau Niil 78. Line it up so we come in somewhere behind the second planet, close enough to duck behind it. There's a sixty percent chance the star will flare up and cast out an ion storm about the time we arrive. Nelson, we have a weak stretch of FTL panels in the aft sector, starboard ventral."
"Aft sector, five by seventeen, aye, sir," Dubsnjiadeb agreed, as Fielle finished powering his station around and unclipped his harness. "I'm also keeping an eye on it. Engineering's working up an internal fix in case that center panel fails. I've already told Sugartoo that we got the repair authorization. She'll pass that along to Commander Harper when he comes on duty."
Sugartoo was actually Xhuge, as in Private First Class Meyling Xhuge, wife and teammate of Corporal Yen Xhuge, one of the four crew members who served as a comm tech for the first duty watch, and one of Lieutenant Rico's top code crackers. His nickname, "Sugar," had been established early on as a play on his name, and when their first CO had granted them permission to wed, Meyling had sworn she was now "a Sugar, too," or Sugartoo for short. Oddly enough, her nickname was the only one people used these days.
The Space Force didn't mind it if soldiers married each other, so long as their relative ranks and positions weren't in conflict with Fatality Forty-Nine: Fraternization. Since both were enlisted and close to each other in rank, there wasn't enough conflict in their position as teammates to bother most commanding officers, including Ia. It helped that the two of them worked in different parts of the ship normally; while her husband served on the bridge, Sugartoo was an excellent mechanic and made a good engineering lead for those times when Commander Harper wasn't around. Of course, she was just one of four on her duty watch, since everyone had to swap duties every hour or two to prevent boredom, work fatigue, and glazed-eye syndrome, but she was good at her job.
"Don't anybody tell the Admiral-General," Ia quipped, "but I actually wanted that _carte blanche_ just so our first officer wouldn't yell at me about what I've been doing to his ship while he slept."
Her joke provoked a few chuckles. Rico snorted. "If he could sleep through _that_ fight, I'll have to ask the doctor what meds she's been slipping into his hot cocoa. I could use 'em, too."
"Careful, or she might try to slip them into _my_ cocoa," Ia retorted. "She thinks I've been stinting myself on sleep."
"Technically, you have been, Captain," Rico pointed out. "You've been running thirty-two-hour days lately instead of twenty-four."
"I'll survive, Lieutenant." She held the helm steady with her left hand and used her right to access the workpad clipped to her console. Now that she had a few free moments, she had to go back to composing prophecies. Time wasn't entirely on her side. Thankfully, the Admiral-General was. "The important thing is that by short-sheeting myself, many others will survive as well."
MARCH 29, 2496 T.S.
CHIMERA V ORBIT
JORDAN TAU-CETI 28 SYSTEM
"How's your shoulder?" Chaplain Benjamin asked Ia. She offered a cup of caf' to the younger woman. Ia accepted it, and the redhead curled up in the stuffed chair across from her. "And how many times have I asked you that, anyway?"
"Three...four times now, and I'm under orders not to use it or stress it for two days," Ia confided, cradling the mug in her right hand. Her left arm hung in a sling. "The mechanics tell me I won't be able to use the suit arm for two weeks, it's that badly mangled."
"But you got the control node for the robots," Bennie pointed out. "And you took them out before they could take out the dome defenses from underground. A task which you could've left to Lieutenant Spyder."
"He's good, but that one required precision shooting. You haven't seen the vidlogs. I shot through a gap about this big." She made a circle out of her thumb and forefinger with her left hand since her right one was busy, then winced at the pain the movement stirred. She relaxed her fingers. Sipping from the mug in her right hand, Ia shook her head. "Besides, it served a second purpose. Eighteen of my crew got to see me in 'Bloody Mary' mode, and that's good for morale."
Bennie snorted, almost choking on her caf'. She lowered her mug, rubbing at her nose. A few sniffs cleared it. "Ow...Your sense of humor is terrible...Not to mention, I'd think that seeing their CO's arm getting crunched by an oversized, motorized monkey wrench would be _bad_ for morale."
"Nope. It shows them I'm willing to take the same risks that they do. Besides, _that's_ what Spyder was good for. He's the one who cut through the tensor cables, freeing my arm before it could be pulverized."
"Instead of dislocated. Again," Bennie stated dryly. "So...how are you sleeping at night?"
Ia lifted her brows, mouth busy with her cup. She swallowed, and asked, "What, no cracks about how little I've been sleeping?"
Bennie shook her head. Her hair had grown long enough that the auburn plait barely moved across her shoulders. "I figured if Jesselle didn't drug you insensate, then she thinks you're doing alright, medically."
"Well, we did get into a little argument over that while she was patching me up," Ia admitted. "But I convinced her I was going to go to bed and sleep for twelve hours after seeing you."
At that, the chaplain lifted a brow. She gave Ia's cup a pointed look. "Oh, really?"
Ia grinned. "I will. In about four more hours, when I've finished filling out the paperwork on the battle and written twenty more prophecies."
Bennie kept her brow arched.
"I promise!" Ia protested. Then added honestly, "Unless an emergency happens, and I'm needed on the bridge."
Her truthfulness earned her a gimlet stare from her friend. "And _will_ there be one?"
"Fifteen percent probability," she admitted, leaning back in the padded chair. "This is a very comfy chair...Is this the chair I saw on Grizzle's requisitions manifest? Nice chair...Anyway, that fifteen percent is only if the TUPSF _Zizka_ leaves the system early, and I've asked them to stay. I told their captain if they do extend their stay by an hour or so, they'll be better placed to scare off the Salik scoutship headed our way, looking for weakness in the local defenses. You know, I think I need to get one of these seats for my own office."
"Nonsense, you'd fall asleep and never get any work done," Bennie scoffed.
This time it was Ia who arched her brow. "First you want me to sleep, but now you _don't_ want me to sleep?"
"Consistency, you cannot have," the older woman quipped, hiding her grin in a sip from her mug.
Chuckling, Ia sat up and fitted her cup into the clip on the edge of the coffee table, then sighed and leaned back again. "I could use some decent sleep, yes. But every clunk and thump from the repair teams is going to keep me awake, worrying that something will break on the wrong side of the probability curve during their work, making us further delayed. We replaced five pods at Rau Niil, but we lost too many sensor arrays this fight. If nothing bad happens, Harper's teams will be done in about four hours, which means third watch will be free to get the ship under way. If anything does, I can be on top of it with exactly what's wrong, and we'll be under way in five. _Then_ I can sleep.
"If not, if I go to sleep now, and something happens...more damage, more delays. More problems for me to fix." She eyed her cup of caf', debating whether or not to drink more of it. Sighing, she sat forward and unclipped it, choosing caffeine over common sense. "I can't wait until my arm gets out of this sling. It's throwing off my balance, and I'm not allowed to exercise in high gravity like this. Every day I lose while I wait for my body to heal is five extra days of struggling to recover the strength I've lost."
"At least your suit's safety cage held, and you didn't lose the arm. So how _are_ you sleeping these days?" Bennie pressed, not deterred by the change in subject. "No avoiding the question, Captain. Any nightmares?"
A slight but genuine smile tugged at Ia's lips. "Pretty well, and very few, Commander. Especially after the _carte-blanche_ extension."
Bennie smiled. "Good. Now, since you're more or less mentally stable...for you...let's chat about the rest of your crew. Private Davies is coming along slowly in her misandry therapy, but she is making progress. I've been watching her spar with her teammate, and she's not quite so conflicted about hitting him—and when she does, she's not wasting her blows in anger."
"Mm...I've only seen them spar a few times, but that's good," Ia agreed, sipping at the cooling brew. "What about Private Kim? Ah, Kimberly Kim. Has she mentioned Sergeant Maxwell?"
Bennie narrowed her eyes. "Are you trying to pry past the sacred seal of both the therapy session and the confessional?"
Ia snorted. "For one, I'm a fellow priestess, duly ordained, blah blah blah. For another, I already know where those two are headed, more or less. As far as I'm concerned, since they're not posted to the same Platoon and so long as they're still fit for work when their watch comes up, they can bounce on the bedsheets all night long whenever they're off duty. Just not on _my_ bedsheets."
That made the chaplain choke on her drink. She coughed a bit, laughing. " _Shakk!_ God bless you, Ia, but you're _not_ supposed to be trying to kill me with laughter, here. And you made me swear! Bad girl!"
Smirking, Ia shrugged, unrepentant in the face of her chaplain's finger-waggling. "Hey, I'll take my humor wherever I can get it. It's also nice to know you're Human, too."
"I'll save my prayers of repentance for later, just in case you make me do it again," Bennie dismissed. She rose, asking, "More caf'? If you're going to stay up four more hours, that is?"
"Please," Ia agreed, holding out her cup. "Back to the crew, and Kim versus Maxwell."
"Careful, Ia," the chaplain cautioned as she retreated toward the dispenser, "or I'll think you're secretly a romantic at heart."
Ia didn't deny it. "Why shouldn't I be? I love this galaxy so much, I'm willing to marry my life to it."
"That's a hero/martyr complex," Bennie dismissed. "I'm talking _romantic_ love."
"A girl has to amuse herself somehow, and I am still female deep down inside. Besides, all this chastity sucks like a black hole," she muttered. "Might as well hear about it secondhand."
"You haven't renewed anything with Meyun yet?" Bennie asked Ia, glancing her way. She came back and returned Ia's cup to her. "Considering how he looks at you..."
"He can look all he wants." She sighed, accepting the mug. Slouching back, she sagged in the seat in an uncaptainly way. "Nothing can happen between us until the whole crew is on our side. The Command Staff's spies are still watching for that sort of thing, and they won't convert to the Church of Ia for a couple more years, in most scenarios."
Curling one leg under her, Bennie resettled in her seat. "Church of Ia? Careful, there. Delusions of godhood don't look good in a military personnel file."
"You know what I meant," Ia dismissed. "Even the Space Force calls the procedural manual the 'Company Bible.' No pretensions of religious aspirations were intended. At least, not on board this ship. I'm still the Prophet foretold by the Sh'nai faith, no matter what I do."
Benjamin stayed silent for a moment, thinking, then shrugged. "Okay, so what scenarios _would_ convert the crew more quickly? It's not healthy for either of you to suppress your urges."
Ia snorted. "That sort of conversion will only come at a terrible price, usually by me predicting some terrible fate, like a series of deaths. Things I'd rather avoid having come true or making people suffer through. Slow and steady will still win the race, and will do so less painfully. It just sucks like a black hole in the interim."
"So the two of you get to suffer from sexual frustration?" Bennie said. "That's the less painful solution?"
Ia slanted her a look. "Aren't we supposed to be discussing my crew?"
The chaplain smirked. "Aren't we?"
Dropping her head against the padded back of the chair, Ia sighed. "Not until I know for sure my people won't go running to the Admiral-General. I need Meyun far more as a brilliant off-the-cuff engineer than I need him as a lover. And that's enough on _that_ subject for today. Official Captain's policy. Now, let's get back to Private Kim. I'm also concerned about her mental health after her jaunt onto the timeplains, and not just her emotional health."
Thankfully, the chaplain let the other subject drop, though Ia knew her friend would eventually bring it back up again.
APRIL 5, 2496 T.S.
SIC TRANSIT
Like hers, Harper's quarters were located next to his primary workspace, the main engineering compartment in the aft sector of the ship. Not that far from hers, either, if offset by a deck and a section bulkhead. However, his front room was large compared to hers. Ia had given up some of that space to ensure a galley for the bridge since she didn't have a need for any privacy bigger than a small living area separate from her bed.
Her free time was spent with a workstation in hand, transcribing future directives; at most, all she needed was a comfortable chair. Knowing he would need to experiment in his free time, she had ordered Meyun's quarters enlarged by a bit, so there would be room for workbenches and storage facilities for projects like this one.
"So. That's the gun?" Ia asked, eyeing the collection of tubes, crystals, trigger, and handgrips resting on the workbench table in Meyun's personal quarters.
"The originals were sort of...of Jules Verne–ish, so I thought I'd carry on with that theme. Wait—doesn't it look like it should?" Harper asked her. "Am I doing something wrong temporally?"
"Well...no. Sort of. Maybe? I'm used to seeing it while it's being held and pointed at me," she amended, staring at the odd thing. "Maybe that's it?"
Shrugging and spreading his hands, he hefted it. Despite its bulk, Harper lifted the weapon fairly easily. He wasn't from as heavy a world as hers, but his homeworld, Dabin, was still above the point-break. Stepping back, he aimed it at her. A glance at the table reassured Ia that the e-clips were still secured to the table in holders. He also kept his finger off the trigger, further reassurance he wouldn't fire it. She didn't think she'd enjoy being hit by the wrong sort of energy beam.
"How does it look now?" he asked her, trying to squint along one of the upper enclosed tubes. "There's no real sighting mechanism since I figured it's meant to be used at close range."
Ia peered at the gun for several long seconds, comparing it to the timestreams, then shook her head. "This bit up here should be over here...and this node thingy is on the other side, toward the back. And there was a sort of oblong, bowling-pin-shaped bit...or maybe kind of brandy snifter–ish...Sorry, Meyun, but it's the wrong configuration. There also should only be one focusing crystal visible. The rest should be inside the housing."
Harper lowered the weapon. Giving her a sardonic look, he said, "This _is_ my first try. I designed it on basic principles of physics. And I'm not sure _how_ these crystals are supposed to resonate, since every experiment I could find listed in the Nets said they just absorb whatever is thrown at them. Electricity, thermal energy, light from within and without the visible EM spectrum..."
"They can be easily seen, come in several pastel shades, and do emit their own light, so they don't absorb _all_ wavelengths of visible light," Ia pointed out dryly.
"Nah, that's just a trick," he teased, setting the gun-thing on the table. "A dangly thing on a fish to lure their prey in close to their jaws—I can _see_ some of myself building this thing in the timestream memories I have, but only in little snatches. Why don't we just go into the streams and let me look at what I eventually do to correct it, so that it functions as you saw it?"
Ia shook her head. "You can't do that without reading your own thoughts, but your other self's thoughts while submerged within the timestream's life get blocked out by your actual thoughts."
"Ha! So paradox _does_ exist within precognitive-based time manipulation," Meyun said, pointing a finger at her.
She blinked at him. "Harper...first of all, it's only a paradox because you're too close to your own life, and your current thoughts will always be louder than your past or future thoughts. And secondly, that argument was over and done with months ago."
"I know, but it still applies." He tapped the side of his head. "Eidetic memory. I remember almost everything you've ever said to me."
For a moment, his brown eyes darkened, gazing at her. Ia remembered that look. It was a path neither of them could afford to retake. "And we're getting offtrack. Just accept the fact that you cannot peer into your own timestream to read your thoughts. I could do it with your alternate-life self, but I cannot do it with myself...and I'm not enough of an engineer to transcribe whatever I could learn from you. Not at the level of understanding you'll need to succeed. I can't do it all, you know."
Thankfully, he accepted the return to the correct subject.
"Well, then maybe I could display a series of schematics for myself...though without actually understanding the principles behind the design, it'd only be halfway useful for building the real thing," Meyun reminded himself.
"There is that," she agreed. "You're a great engineer because you understand the theories deep down in your bones."
He sighed and raked a hand through his hair. It was now long enough that he had to knot it up when on duty, but this wasn't his duty watch. A moment later, he frowned, black brows pinching together. "Wait. Didn't you tell me once that...well, not _you_ you, but a timestream you...didn't you tell me once that you see _all_ possibilities? Including alternate realities where I'm not a Human but rather a blue-furred rock ape or something?"
"Yes," Ia confirmed, nodding slowly. "I didn't say it in _this_ particular reality, but it is true, and I know you experienced visions from a hundred alternate realities. There's even a universe out there where you and I are actually Salik, plotting the downfall of Alliance civilization. Several variations on that theme, in fact. Not particularly helpful in this case, but that alternative does exist, along with many others."
"Then why don't we just find an alternate reality where _I_ am not the person successfully developing this gun, and just go read _his_ thoughts?" Meyun pointed out. "Or hers? Or its? A universe with the same laws of physics, but where my thoughts can't get in the way of my own thoughts because I'm not the one thinking them?"
She blinked. "That's _brilliant_. I like it," Ia agreed. She thought about it for a moment, then held up thumb and forefinger close together. "Just one little problem, though."
Meyun rolled his eyes. "What _now_?"
She gave him a faint, pain-tinged smile. "Last time I took you onto the timeplains...it was a bad experience for you. And while searching for someone _not_ you will make them easier to find and read, I still can't foresee all of your future. I'm pretty sure that's my gift protecting me from the temptation of you. I don't know how that'll affect this trip, and I don't want to hurt you again."
Stepping close, Meyun cupped her face in his hands. "The only reason why I suffered was because neither of us was prepared. The only reason why I continued to suffer was because I was forced to spend two years without even hearing from you. Of seeing you only in my dreams, and in that awards ceremony they broadcast on the military channels."
Ia flushed. She knew she should step back, should break contact, but between the gentleness of his hands and the warmth radiating from his body, she didn't feel threatened. For a moment, dangerous though it was, the timeplains no longer lurked in the back of her mind. No past, no future, only the now. "Meyun..."
"I won't endanger your work," he promised, tilting her face up a little, encouraging her to meet his gaze. "I promise that. I've had a lot of time to think over everything I saw and think through the reasons why you would have done those things, and why we can't..." He broke off, breathed deep, then added wryly, "Except Bennie told me that you said if we could get the whole crew on our side, it wouldn't be a problem. So long as we were discreet."
Being reminded of that gave her the strength to step back. He let her go, and she immediately missed the warmth of his touch. The heating system in his quarters was working fine—he was the chief engineer, after all—but she still shivered a little, chilled by the lack of contact. Shaking her head, Ia said, "That won't happen for a few more years. And I don't want to put tempta—"
He stopped her with a finger on her lips and a slight smile. "Too late, and not a problem. As for the risk...well, I'm willing to take it, so long as whatever we see, you promise you won't throw me off the ship."
She spoke as soon as he removed his finger. "I _can't_ throw you off the ship. I need you to keep repairing it."
Hands going to his hips, Meyun mock-frowned at her. "And whose fault is _that_ , Meioa-e Who Likes To Blow It Up Repeatedly?"
"That was the Salik, not me," she told him mock-primly. The moment of levity eased the tension. Sighing, she raked her fingers through her short white locks. "Alright. We'll do it. I'll take you onto the timeplains and see if we can go looking for a blue-furred rock ape who knows how to build this thing right, and why it has to be built that way. God knows I've cribbed notes from several far-flung alternate realities before."
"There is no God but the Future, and Ia is His Prophet," he quipped, startling her. He tapped the side of his head again. "Something else I remember from the timestreams. And you were right, they're only images of things that _might_ be, not always the things which _will_ be...thank God."
"Right. Speaking of seeing things that might be, we probably should sit down for this." She nodded at the prototype. "Secure that weapon first, soldier. Lock and Web."
"Aye, aye, Captain." Unclipping a rolled bundle of tight-woven webbing from the far side of the table, he pulled the stretchy network over the gun.
Ia helped him secure the edges to clips on the underside of the desk. At a gesture from him, she retreated to the sofa across the room from his workbench. He settled next to her, rested his hands on his grey-clad thighs, and looked at her.
"So...now what?" he joked, though she could see the discomfort in his eyes. "I'm presuming you'll want both of us dressed for this? Or are we going skinny-dipping in the timestreams?"
Giving him a flat look, she shook her head. "Wise-asteroid. Stay clothed. Avoid saying the word 'time,' and strive to keep your mind calm. Do not let go of me, and do not cling so close that I cannot move. Above all, keep your mind disciplined and your libido suppressed. Thoughts can become reality where we're going, so focus on being an engineer."
"Sir, yes, sir," he agreed.
She studied him quickly, but her first officer seemed sincere. Offering her hand, she waited for him to touch it. When he did, she nodded, pulled them onto the timeplains, and up out of the water.
Up out of the water, and into a heavy fog. One so thick, Meyun's features were half-obscured, and he was right there within reach. Ia usually emerged on the right-side bank, facing downstream into the future; she couldn't be sure she had done so this time, however. The only thing she was sure of was that they were on the bank, extracted so quickly that nothing had been seen.
Down was the ground, so up into the sky she lifted them. He clung with his hand, not nearly as disoriented as Rico had been, and without the confusion and fear of their previous trip. As they rose, the mist gradually thinned and receded, until they hovered high over the timeplains, eyeing a thick, sprawling mist that occupied several squid-like valleys, life-paths directly related to the relationship the two of them couldn't, shouldn't have.
_"So, what do we look for?"_ Meyun finally asked, peering at the wrinkled landscape below. _"Blue-furred rock apes?"_
_"No. There is a lifetime where someone discovers the trick of shaping crysium, and an engineer uses the reshaped crystals to form a Feyori-inducing gun."_ It was like baiting a hook, or checking off boxes on a search list. Ia itemized each thing they needed out loud. Beneath them, the landscape rippled and shifted, the mist inching ripple by ripple off somewhere to the side, behind them. _"The engineer is Human, like you, and he lives in a universe with our exact same physical laws. But there are no Zida"ya coming to destroy his Milky Way Galaxy._
_"He does, however, work for a half-Human, half-Feyori captain—male—who wishes to enter Feyori politics in order to get them to stop bothering his Human kin. Your not-other-self's name is Jed Maxwell, and he has figured out how the conversion guns work, with a deep and eloquent, written level of understanding."_
_"...Nice search matrix,"_ the real Meyun Harper muttered, watching the mist and the hills shifting beneath them. As the fog from their own reality faded into the distance, the summer golden grass was slowly replaced by darker, clumpier shades of green shrubberies.
_"Thank you. I've been practicing."_ She didn't intensify the experience to get rid of the echoing of their speech because she didn't want to risk his becoming so attached to a particular scenario or thought that it influenced her in turn. That would be a disastrous, downward spiral of rising emotions.
_"How long have you been practicing?"_ he asked, brown eyes filled with curiosity.
_"Since I turned fifteen, when my gift blossomed, and I started visualizing in earnest...Ah, here we go,"_ she said, swooping them down toward a golden-clear stream snaking its way through the greenery. _"An equally talented alternate universe engineer—not you; you're a pastry chef back on Dabin in this universe. This fellow, however, knows how to build the gun we want you to build."_
Meyun lifted his brows. _"If I'm a pastry chef, who or what are you?"_
_"I'm the male captain of the engineer's ship."_
That provoked a snort. _"Not sure if I could fall in love with a male. But that does make me wonder. Any chance our alternate selves actually meet in this universe?"_
"None." Her voice echoed with firmness, quelling further inquiries along that line of thought. _"You are an engineer right now, nothing more. Now pay attention to the schematics and put that photographic memory of yours to good use."_
His free hand saluted her, the right one still tightly clasping her left. _"Aye, aye, Captain."_ He paused, then added dryly, _"...You do realize that 'cribbing notes' from this fellow is technically intellectual property theft, right?"_
_"Technically, in our universe, this guy doesn't even exist,"_ she reminded him. _"There are no copyright-infringement laws that span the multiverse because the multiverse, by its very nature, is one giant plagiarizing copy machine, introducing infinite infinitesimal errors with each new reiteration."_ Ia paused, grinned, and added, _"That, and he'll never find out, so he can't take us to court."_
_"Why, you law-breaking rebel, you,"_ Harper teased. Nodding at the stream, he added, _"Alright, I'm ready."_
Nodding as well, she led him to the life-stream of their target.
APRIL 21, 2496 T.S.
V'DAN IMPERIAL FREEPORT _TATTH-NIEL_
V'DAN HOMEWORLD, V'DAN SYSTEM
Disembarking onto the station with the entire 1st Platoon, Ia stood out like a grey thumb. She was the only one in a uniform instead of civilian clothes. For this trip, Ia had donned her Dress Greys, with her grey cap perched on her neatly combed white locks, her TUPSF half glittery pinned to her chest along with the addition of her V'Dan honorifics. It was only polite to wear the latter, given they were parked at the heart of the V'Dan Empire.
Behind them, the _Hellfire_ had cozied up to one of the station's longer gantries. The long, lumpy needle of a ship was only slightly battered from its last starfight, a sneak attack on another Salik crèche hiding in the depths of interstitial space. That crèche had been parked disturbingly close to the V'Dan homeworld, only four light-years away. Having pointed that out to the V'Dan High Command on the hyperrelays afterward, and the fact that they had destroyed the installation, Ia had requested and received permission to bring her Company in for three days of Leave.
Because _Tatth-Niel_ was a freeport space station, no Customs queues slowed them down, just a submission of their ident units as each person disembarked so that the V'Dan government had a registry of their entry. The only thing they had to pass after that was through a long scanner arch, which searched silently, invisibly, for transmittable illnesses and contraband, standard equipment for most stations, as well as most starship airlocks.
The crew of the _Hellfire_ also got one last warning from their commanding officer.
Stopping at the end of the hall, just before it opened up into the bustling commerce level ringing the station, Ia turned and faced the others. They drifted to a halt, eyeing her uniformed presence warily. Despite their current off-duty status, the sight of their CO in her Dress Greys was clearly stirring up the need to respond professionally. Some of the men and women of the 1st Platoon even shifted into a modified Parade Rest, standing with their hands at their backs, their shoulders squared, and their gazes straight ahead.
Settling her hands on her hips, she addressed them. "You have twenty-four Terran Standard hours of Leave. This translates to twenty-two hours thirty-eight minutes V'Dan Standard. These locals are Human, but they are _not_ Terran. Respect their customs, laws, and beliefs during your visit, and remember that even in civvies on official Leave, you will represent the finest of the Terran Space Force at all times.
"Be back on board, in uniform, and ready to assume your posts with five minutes to spare, if not sooner. Your brothers and sisters in the 2nd and 3rd Platoons are covering your shifts for you so that you may enjoy these full twenty-four hours of Leave. Do not let them down when it comes time to cover theirs. Dismissed," she finished.
They started to move forward, heading for the station proper. A voice from behind Ia slowed the trickle to an awkward halt. Firm, male, and mature, the speaker addressed them with dry sarcasm. "A moving statement from a commanding officer. Hypocritical, too, when that commander has been mocking the beliefs of the very nation her crew now visits."
"High Priest Ma'alak of the Autumn Temple," Ia stated, turning and giving a polite bow. Not just to the speaker, a middle-aged man wearing intricately embroidered cream robes, but to his three companions as well, two soldiers and another member of the Sh'nai clergy. "Despite what you may think, you do honor me with your presence. Priestess Laka'thi of the D'aspra Archives, Grand General Ibeni-Zif of the High Command, Highlord Adjutant Sa-Nieth of the Nobles' Council, it is a pleasure to meet each of you as well. Shall we all retire to the conference room the Grand General has reserved for us?"
They exchanged looks, apparently not expecting Ia to take the initiative from them so smoothly. The High Priest nodded slightly, and the Grand General gestured for Ia to join them, the gold trim on his red uniform sleeve gleaming in the overhead lights. As soon as she did so, more red-uniformed soldiers fell into position around them, forming an honor-guard escort. Behind Ia, her civilian-clad troops dispersed into the crowd, no doubt curious what was going to happen to their CO but trusting her to handle whatever it was.
The presence of those bright-clad imperial guards drew attention from the crowds of tourists and travelers they passed, but it was the draped folds of the priestly robes that garnered bows from dozens of the V'Dan. That made it easy to see just how many of the locals were followers of the Sh'nai faith.
Some even drifted forward, calling out for blessings from the High Priest in their native tongue. He in turn raised his hand and murmured benedictions but did not stop. The presence of their imperial escort kept the more insistent requests at bay, allowing them to move smoothly toward the vast station's core.
It took maybe ten minutes to navigate past the outermost layers of shops and businesses to the military hub of the station. They could have held this meeting in the government's reception hub for visiting dignitaries, or within the halls of the on-station Sh'nai temple. Instead, the conference room's location was proof that the military was the current power in charge of the empire. Painted cream and decorated in red and gold accents, the room they were led to boasted wall screens and workstations, and a distinct lack of Lock and Web clips, a reminder that _Tatth-Niel_ was a space station in permanent orbit around the V'Dan homeworld and not a vessel capable of being moved elsewhere in a hurry.
"Ship's Captain Ia, would you like a cup of caf'?" Grand General Ibeni-Zif asked as they entered the room. "Meioas?"
Ia nodded, as did the priestess and the adjutant. At that, the red-uniformed junior officer waiting by the door turned to the sideboard and started fixing mugs for everyone. As the erstwhile guest at this meeting and the focus of the questions that were to come, Ia moved toward the seat at the near end of the conference table.
Like the other objects in the room, someone had selected the table to impress visitors; it had been crafted out of some stout, golden-hued wood native to V'Dan, one with a rippling grain suggestive of Zen waves in gilded sand. Ia liked it. The Sanctuarian equivalent of wood was usually more reddish or purplish in hue, making this a bright contrast to most of the colors she had known as a child and a pleasant change from the blander, more pragmatic hues seen during her two stays on Earth.
The Highlord Adjutant assisted her with her chair first. Then he held a second chair for the priest before seating the general, followed by the priestess and lastly himself. He did so without the assistance of the other aide in the room. The subtle courtesy was proof—at least in the V'Dan culture Ia had studied in her youth, living on a jointly founded colonyworld—that this meeting had been instigated by the Sh'nai, was being hosted and supported by the military, and was being facilitated by the Nobles' Council.
_The Nobles are therefore holding themselves neutral in this inquisition, according to V'Dan protocols. He probably expects to serve as an arbiter if a dispute arises. The Grand General knows that whatever the outcome of this meeting is, it will have an impact on the war effort, and that means it'll impact his purview. The High Priest is here to personally lead the inquisition...and the Priestess of the Archives has the means to verify or disprove my identity._
"Ship's Captain Ia, you are here to answer allegations of promoting yourself as the long-prophesied Prophet of a Thousand Years, one of the core saints of the Sh'nai faith," High Priest Ma'alak stated. He paused, mouth twisting a little. " _Ia'nn sud-dha'a_...What hubris, to take on the V'Dan word for 'prophet' as your one and only name."
"It was not hubris. A simple examination of the citizen registry documents of Independent Colonyworld Sanctuary will prove it, High Priest," Ia countered calmly. "I declared emancipation at the age of sixteen on March 4, 2488, Terran Standard, and changed my name from Iantha Quentin-Jones to the shortened version of Ia...which my family and friends had already been using to address me since I was an infant."
"Why did you change your name, Ship's Captain Ia?" Grand General Ibeni-Zif asked, leaning his elbows on the edge of the table and lacing his fingers together.
Ia smiled slightly. "I think this meeting is informal enough, Grand General, that you may simply call me Captain. Or Ia, since that is my name...though I can understand His Holiness's reluctance to do so," she added, giving the priest a polite nod. "As for why I changed my name, I knew that if I gave my younger brother the Power Lottery numbers for the drawing on February 10, 2494 Terran Standard while still retaining my full name, that it would draw attention to my relationship with him and cause near-immediate trouble for me via the Terran laws regarding profit from prognostication for precognitives. Plus it would trip over the military laws governing fraternization and the giving of loans, perhaps even dredging up accusations of bribery. None of which I wanted to do."
Priestess Laka'thi's mouth twitched upward on one side, bringing her laugh lines into prominence. The grey-haired woman seemed amused at the tongue-twisting alliteration. Ia smiled back slightly before returning her attention to the general.
"My emancipation and name change made it appear legally that I was estranged from my family, and would therefore not profit personally from the lottery win. And for the record, I have _not_ profited from that exchange. The monies have long since been channeled into a nonprofit trust fund for the defense and safety of my former fellow colonists to cover a need I had long ago foreseen, and my own expenses come either out of my own pay or are paid for by the Terran Space Force, as authorized by my superiors on the Command Staff. We are at war now, as you know," she added dryly, "and wars are expensive."
The other corner of the Archivist's mouth quirked up. She said nothing, though, choosing to accept her cup of caf' with a polite nod to the junior officer distributing them around the table. Ia accepted hers with a polite nod as well. She silently refused the addition of cream and sugar from the tray he carried; in her opinion, caf' didn't need any. Both the priestess and the adjutant added a spoonful of grated _meklah_ to their drinks, chocolate sweetened with brown sugar. The High Priest and the Grand General accepted glasses of water.
Ibeni-Zif eyed her. "Would you be willing to expand a little bit more on what uses that money has been put to? It is our government which oversees the authenticity of each drawing made and ticket purchased, after all."
"They've been busy using that money to build secure housing and fortifications—the Salik themselves cannot live in Sanctuary's high gravity," Ia added in an aside, "but until we can shut down their combat robots, there is still the risk those 'bots could be used to harvest my fellow heavyworlders. They are too important to the future to allow that to pass." Lifting her blue-glazed cup from its matching saucer, Ia switched to V'Dan. _"Tokla vuu hess t'Kah'hn V'Dania, na'V'Dan atrei'atess, ou vaa havet'th makau-na ma'achess."_
They started to lift their cups, then hesitated. Not because her accent was terrible—this was one of the few languages Ia could actually pronounce, having studied it for several years in her youth as part of her world's jointly founded education courses—but because of _what_ she had said. She knew she had gotten it right; Ia had looked up and memorized this particular phrase even when the rest of her V'Dan had grown rusty with disuse over the years.
The general lowered his water glass, not yet drinking from it. "Tell me, Captain. Do you even know what that means? Or are you just parroting something a protocol officer instructed you to say?"
"I said, 'Raise your cups to the Emperor, from the People of the World, so that all may live well in health and peace.'" Lifting her cup again, Ia sipped. So did the others. Lowering her cup, she added, "Not word for word in Terran, of course, but it translates close enough on the surface. The exact subtexted meaning of the dialect and grammar I used comes from the Valley of Artisans, which was founded in the seventh millennium V'Dan Standard by the Immortal, who had come back—"
Ia paused briefly as Priestess Laka'thi choked on her caf', waiting as the woman grabbed for a napkin, coughed into it, and caught her breath.
"—Who had come back from her exile in order to visit her chosen people and check on their progress." She slid her gaze to the High Priest of the Autumn Temple. From the distastefully pursed state of his mouth, he looked as if he needed some of that _meklah_ dumped in his water. "The actual toast was not conceived of by the Immortal herself, but rather by High Priest Shu-Nai of the Summer Temple during a visit to the enclave in the V'Dan year 6378.
"While the violence against the Sh'nai faith that marked War King Kah'el's reign was long over by that point, it was once again a time of religious tension between the Immortal's believers and the state. The toast is a reminder that sometimes one must bow to the inevitabilities of a given situation, and that to do so with graciousness and humility is the path of the righteous."
High Priest Ma'alak pinched his mouth further, looking distinctly sour as Ia finished. Priestess Laka'thi, on the other hand, dissolved into cough-punctuated giggles. The other two males at the table gave her bemused looks as she sat there, shoulders shaking as she pressed the fine scarlet cloth to her lips.
She shook her head, held up her hand, coughed a couple more times into her napkin, and finally rasped, "...Oh, let go of it, Ma'alak. She is _exactly_ right in using that quote. I suspect she uses it to remind us that we, too, should bow to the inevitabilities in this situation. _Should_ they prove to be true."
Ia lifted her cup to the other woman. "An astute assessment, Priestess. And I will be happy to show you the innermost secret your Order has held in their keeping all these years."
Laka'thi stopped chuckling. Blue eyes wide, she stared at Ia. "How...?"
"We'll get to that later," Ia assured her. She looked over at the High Priest. "I believe your next question, Holy One, is supposed to be something along the lines of what makes me think I have the right to go around telling alien governments that I am the Prophet when I haven't even verified my position with the very people who have kept alive the Immortal's memory of the woman I claim to be. Shall I answer it?"
He narrowed his eyes. "No," he stated, his tone clipped. "I would rather you answered the question of why you haven't delivered your so-called prophecies to us yet, if you are indeed the Prophet."
"That would be question number four on your list," Ia stated, earning her another narrow-eyed stare. "Technically, some of those prophecies have already been distributed. Mainly the time-sensitive ones. They have not gone into your hands, however, because they were sent directly to the people who needed them most at the time. Your doubts are very understandable, but dangerous when time is of the essence. I have always preferred to deliver such things directly to the people my prophecies will impact, so that there is no delivery delay and no alteration of the message needed."
Reaching into her jacket pocket, Ia pulled out a quartet of data chips. Checking the letters electrostamped on their surfaces, she slid one toward each of the four V'Dan seated around her.
"Those chips should work on a standard workstation or reader pad, so long as it has a Terran dataport. Most Terranglo-readable ones do. Each one of those contains a time-sensitive list of predictions, tailored to each of your specific areas of interest...and each one has been code-locked with your most commonly used password—I apologize for borrowing them," Ia stated quickly as the general drew in a startled breath, "and I give you my word of honor I have zero interest in using them for any other purpose than this.
"Also, the time-sensitive parts of the list might be slightly off. Having been trained to think in Terran time units, I have done my best to convert those from Terran Standard to V'Dan," Ia said, "but I apologize if I am off by one or two seconds. The exact times in Terran Standard have been included, in case you want to refine your calculations.
"Be mindful of the difference between the actual radioactive clock located in Aloha City on Earth, and the hyperrelay time lag that exists between Earth and here if you wish to try for a more exact conversion rate—the times will be pertinent to the _local_ clocks for the activities they discuss," Ia warned them. "That is, the nearest chronometer to the person who will have to observe or implement these things. So if the clock on a ship is off by two _mi-nah_ , it will still pertain to that ship's clock and not to the atomic chronometer here on V'Dan."
Ibeni-Zif beckoned the young man who had served them their drinks. He whispered in the junior officer's ear in V'Dan, then dismissed him with a flick of one wrist.
Ma'alak picked up his chip, examined it briefly, then set it back down. "Answer my third question, Captain. Since you seem to know what I planned to ask."
"What makes me think the V'Dan people, government, and Sh'nai faith should allow me to continue to go around telling people I'm the Prophet of a Thousand Years?" Ia clarified. She lifted her hand, unfolding one finger per point. "Three reasons. Because I actually am. Because I have already predicted and acted upon the future with great accuracy. And because you need me to be."
_"Need?"_ Ma'alak snorted. "What need would that be?"
"The Archivist and the Grand General already know. It's been in all the V'Dan High Command briefings since the Battle of the Banquet," Ia stated. "We don't have the technological advantages the Terrans brought to the First Salik War this time around. This lack has not only the V'Dan military alarmed, but the Terrans, Solaricans, K'katta, the Tlassians...all the members of the Alliance, directly involved or not."
"Anyone could guess that much, Captain," Ibeni-Zif reminded her.
"True, but I included Priestess Laka'thi, who knows of the promises from the lips of the Immortal herself, which her Order's archives have kept safe and whole. The Immortal has said, over and over through the millennia, that when the Second War breaks out—which it has—the Prophet of a Thousand Years will step up to provide temporal counsel, direct the destruction of the enemy, and eventually save the galaxy, ushering in the Silver Age of peace among the stars." Ia paused, then tipped her head and added, "Which will also coincide with the Second Reformation of V'Dan, in about three hundred years."
Laka'thi frowned slightly. "The Second Reformation was only a rumor. There was only one account that mentioned it, from the—"
"—From the time of War King Kah'el, when the Immortal agreed to step down after losing the duel," Ia agreed, filling in the details for the others. "She did so with the caveat that the War King and his descendants and successors would be legally bound to accept all future duel challenges from rulers of equal rank...and that should the War King's descendants or successors lose that duel, they and their people were to follow the champion who won that match as their new ruler, who would instigate the Second Great Reformation. If they refused, they would have to hand the V'Dan Empire back into the hands of the Immortal herself."
"Yes, but that isn't the same as an actual Second Reformation," Highlord Sa-Nieth pointed out. "No one has ever been able to scrape together a large enough following to qualify as a ruler of equal rank."
Ia dipped her head at the auburn-haired man, acknowledging his point, though she kept her gaze on the priestess. "True, it wouldn't be the same...but when asked by one of the palace clerks about her insistence on that clause being included in the deal being brokered to end the civil war, the Immortal said that when the Prophet of a Thousand Years would become known to the people—to V'Dan, that is—the Prophet would name the time of the Second Reformation, and that she knew the clause would eventually be used, even though the Immortal herself had not yet lived through that time. I have now given you that time."
"Not with a precise date, though," Sa-Nieth stated, his tone skeptical. "Unless you meant three hundred years to the second from now."
"I only give precise dates when the precision of those dates is important to the moment," Ia replied, unruffled by his doubt. "Not because I cannot pinpoint the exact time and place but because I have far too many other temporal focal points to keep track of, each with its own moment of importance. This meeting is not that moment of importance, particularly as none of us will be alive three centuries from now—I assure you, those who need to know, those who will be alive at the right time and place, _will_ receive one of my precognitive missives on the matter. You have my Prophetic Stamp on that."
# CHAPTER 11
_...Of course, the other governments did take a bit of convincing, too._
_~Ia_
"Well. You certainly know your obscure lore. And you know the Prophet's catchphrase, too," Laka'thi mused. "Though that's more widely known than your mention of the Second Reformation."
"Still, all of this could have been discovered in the Archives by you or your accomplices. You could have even bribed an archivist," High Priest Ma'alak pointed out. "It's happened before, and it will happen again, so why not this time?"
"Those are all possibilities, yes. But because of that, I came here knowing that only two things could sway your mind," Ia agreed. "One way is time. Given enough time for events to unfold, time itself would prove my predictions and actions are undeniably, inevitably true. But that way does nothing to address the urgency of the moment. You want to confront me now, before the rumors can spread too far. If you can prove I'm a fake, then you can squash those rumors with no harm done to your culture. If I'm the real Prophet, you will again want to make sure I don't bring harm to your culture. That brings us to the Puzzle Box and the 'secret' I referred to earlier."
"You know the purpose of the box," Laka'thi stated. It wasn't a question.
"Yes. I'm the one who wrote the note to be passed along to the young Immortal via the Third Human Empire before she begins her trip through Time. That note will tell her how to craft the secret hidden in the box, and what secret to write." Ia watched them roll their eyes and nodded, flicking her own in sympathy. "A bit self-fulfilling, I know, but it's the easiest way I'll have available to convince you of who and what I am."
"I am a soldier, not a priest," Grand General Ibeni-Zif stated, rubbing at his brow. "I am not a religious man, and do not follow the Sh'nai faith, though I will give respect to the rights of those who do. Still, your claim to be the Prophet will affect the three-fifths of the Empire who do follow the old faith, and shake the other religions—if the Prophet of a Thousand Years is real, then the Immortal could equally be real, and we could be facing a religious civil war as some try to seek her out, others try to slay her followers, and the Emperor's position on the throne destabilizes in the turmoil."
"Then tell them that I say the current Emperor is the rightful ruler of V'Dan," Ia stated bluntly. "I know this session is being recorded, and you have my permission to use my words to combat that threat. Tell your people that His Eternal Majesty will live long and rule well, provided he and his people back the Alliance's efforts to thwart the ambitions of their mutual enemies. And tell them that I say the other religions of the Empire are equally as valid as the Sh'nai faith, for each has its own place. They need to work together to support the First Empire as a unified whole."
"That only works if you _are_ the Prophet, 'Ia,'" Ma'alak pointed out. "But you are right. Either we must wait for your words to be proved true, or test them here and now. Priestess, the Puzzle Box."
From a hidden pocket buried in the layered depths of her butterfly-like sleeves, Priestess Laka'thi pulled out a modest box. It wasn't much bigger than an old-fashioned book or a small datapad, about as long as an average Human hand, as wide as a palm, and as deep as a finger. Crafted from wax-polished bronze, only the edges of the tightly fitted lid showed hints of green from oxidization.
"This box has been sought out or offered to over four hundred souls," the priestess stated, holding the plain, brick-like box between her fingertips. "In all those attempts, it has been successfully opened only seventeen times in the last six thousand years. Each time, it has either been given to those claiming to be Holy Ones to test their psychic abilities, or sought after by someone claiming to be the Prophet. Those rare, few openings have been witnessed and recorded by the Sh'nai Order of the D'aspra Archives down through the ages.
"But of all seventeen who have successfully opened this box, none of them have revealed the secret that hides inside." Turning slightly in her seat, Laka'thi offered the box to Ia. "If you are the Prophet of a Thousand Years, you will not only know how to open this box, you will, by the Immortal's own words, know how to reveal what lies within as well."
Accepting the bronze box, Ia turned it over in her hands, examining the finely polished, auburn-hued metal. The exterior was plain and well maintained, with only that faintly oxidized seam showing that the box wasn't a solid block. There was no hint of a hinge, no opening for a key, and no way to show how it had been locked. But she knew. The trick was doing it gently.
"Well?" Ma'alak asked impatiently as the seconds ticked by. "Aren't you going to open it? Or can you even try?"
"Patience, High Priest," Ia admonished lightly. "What Priestess Laka'thi has not told you is that in those recordings of the seventeen times, the locking mechanism consists of twenty tightly fitted bronze hooks tension-clipped over twenty rods set in twenty holes, with each hook and rod facing a different direction from the next. Some of those hooks are set in one half of the box, while some are set in the other, with their holes in the opposite half.
"The lid itself possesses a small but significant shoulder lip to prevent a thin blade from being used to push those hooks free," she added, studying the timestreams in the back of her mind. "There is no mechanical way to open this box other than a hacksaw or a cutting torch once it has been sealed shut."
"You're stalling," Ibeni-Zif stated.
"I am not stalling. I am taking my time so that I do not break it. The metal is old, some of the inner oxidation has crystallized together, and those hooks are still as stiff as the day they were first crafted, which is why I need to be careful," she added, gifts turning from the timestreams to the box itself. "That, and twenty is a lot of them to keep track of, even for a skilled telekinetic. But...not impossible...and that's the last hook I needed to bend."
She trailed off, concentrating. A careful twist of her mind and a slow, firm pull separated the two halves of the box. The hooks did indeed line the edge, some pointed up, others pointed down, angled this way and that. Those hooks, their holes, and the inner edges of the box had not been waxed in centuries, leaving it a dull shade of green. Nested in that green, slightly crusted with bronze verdigris, sat another object.
Setting the top of the Puzzle Box on the table, Ia inverted the bottom half over her hand. It took her three shakes to dislodge the inner brick. Shaped like the outer box, if on a smaller scale, the inner object looked like it had been crafted from a single light green tourmaline, or maybe peridot, save that its edges were more bluish in hue.
Whatever it was, the transparent material had rounded corners, and contained a bronze box of its own inside. The only catch was, there was no catch; the inner crystalline box had no seams whatsoever. Either the inner bronze box had been dipped in molten glass, or some sort of gem had been grown around it.
"...May I?" Laka'thi asked Ia, holding out her hand toward the box. Seated to Ia's immediate right, with the Highlord Adjutant on her other side, she didn't have to stretch to do so.
A quick check of the timestreams confirmed what the woman was about to do. Nodding, Ia passed the gem-brick to her. Before the others could ask why she wanted it, the Archivist raised the translucent box high in her hand and smashed it down with a startlingly loud _crack!_ against the edge of the polished-wood table. Ibeni-Zif and Sa-Nieth both jumped, and the High Priest clutched at the robes covering his chest.
"What are you _doing_?" he demanded, scandalized. "That is a _sacred artifact_ you're trying to smash!"
"Calm yourself, Your Holiness," Ia told him, lifting her hand. "She's actually doing her job, by proving that case isn't made of mere glass. If you examined the table, you'd see a faint indent from where it struck."
"The Terran is correct," Laka'thi confirmed, holding up the gem-like case, showing it intact. "The Archives recorded how this inner box was placed between a hammer and an anvil, and struck with full strength. I am no blacksmith with bulging arms, but this table would break under my blows long before this case ever would.
"There is no seam, and no sign of a way to open it. No one has ever touched the inner box since the Immortal herself crafted it." She lobbed the box at Ia, who caught it one-handed. "If you _are_ the Prophet, you will be able to open it, 'Ia.' The Immortal has said so."
Ia shrugged and offered it to the Grand General, seated to her left. "Feel free to examine it for a seam, meioas, before I do so. Once I open it, there is no going back. What has been seen cannot be _un_ seen."
"Do the Terrans train all of their officers to boast and grandstand?" he asked her, accepting the box and turning it over in his hands, thumbnail questing along the crystalline sides in an attempt to discern a seam. "Or did they just train you?"
"True grandstanding would actually detract from my message, sir," Ia told him. "It is the messages I bring that are important. I am merely their vessel. I am also giving each one of you a chance to verify personally that there is no way to get through that outer covering. No cracks, no hinges, no seams...and no tricks."
He frowned and scraped his finger over it again, then shook his head and offered it to the High Priest. "I cannot see a way to open it."
Ma'alak examined it. His thinner face also furrowed as the seconds of his examination ticked into a full minute, until with a frown of his own he passed it over the table to the third man. Highlord Sa-Nieth turned it over several times, rubbing his thumbnail over all the surfaces and angles, then shrugged and handed it to Laka'thi, who returned it to Ia.
"As you have seen for yourselves," Ia stated, running her thumb all the way around the edge, "this 'secret' has no openings. It is a solid piece, with no hinge, no lock, and no lid."
Shifting her grip, she grasped top and bottom, carefully focused her psychic abilities...and pulled the gem apart like a pair of simple lids, using the same gentle patience she had used on the outer case. The act revealed the unoxidized, pristine bronze box nested inside.
The High Priest's jaw dropped, the Highlord Adjutant's eyes bulged, and the Grand General choked on his caf', grabbing for his own napkin. Of the four of them, only Priestess Laka'thi sat as still as a statue. Her face was now flushed, but she didn't blink, and didn't breathe.
Ia dropped the upper lid—which was solidly pale blue, not green—onto the table with a crystalline _ching_. The musical sound provoked a blink from the elderly woman. It was followed by a deep breath. She looked like she was experiencing a holy revelation, caught up in the unthinking awe of the moment. Glad the other woman was at least breathing again, Ia eased the box from its depths and dropped the bottom half of the crystalline container next to the top half.
Unlike the outermost one, this bronze box hadn't been exposed to either moisture or oxygen since its creator had sealed it up in the first place. But, just like the outer one, this one did have a thin, flush-fitted seam.
"This Puzzle Box is sealed with a system similar to the one used on the previous one, save that it has only ten hooks, not twenty." Another careful twist-and-push of her mind, and she pulled the two pieces apart, revealing her words to be true. She set that lid next to the other pieces, the outer bronze casings, and the middle blue-crystalline lids. This time, the interior of the box had been padded with a rectangle of what looked like plain grey felt.
Tipping the contents into her palm caused the felt to fall out, but that was alright; the felt was merely there to pad the transparent rectangle stored inside, to keep it from rattling. Ia tilted her hand, shaking it slightly to discard the wool, then turned the bookmark-sized slab over and laid it on the table.
It was crafted from the same material as the crystalline box, though its creator had forged it from two tones of crystal, the sapphire blue of the middle box and a pale golden hue. With the polished V'Dan wood of the table beneath accenting that yellow, the double line of characters forged into the slab were easy enough to read. One line had been crafted in archaic V'Dan script; the other had been printed in block-letter Terranglo. Both said the exact same thing, a simple, single phrase.
_Iantha'nn sud-dha._
Amused by the Immortal's sense of humor, Ia smiled. Picking up her cup, she took a sip of her caf', set it back down, and pushed her chair back. "I do believe we are done here, gentlebeings. My birth name and emancipation records can be easily verified within the hour. You do not need me for that. You also have my current set of prophecies, tailored for each of your needs. When it is the right time to receive more of them, you will indeed receive them—you have my Prophetic Stamp on that.
"In the meantime, do keep in mind that both my existence and my prophecies are _not_ to be used or abused for any personal agendas, whether religious, political, or otherwise. I am here to help save as many lives as I can, whether they are Sh'nai or otherwise, V'Dan or Terran, Tlassian or Solarican, Alliance lives or the lives of a hundred sentient species we haven't even met yet." Rising, she bowed politely. "I thank you for your hospitality and the cup of caf'. It was delicious.
"If you'll excuse me, this is the first chance for Leave my crew and I have had since my command was first assembled, and I would like to go enjoy some of it myself. I'm sure Leftenant P'kethra can show me back out to the public sector, so that the four of you can sit here in peace and quiet without being hobbled by my presence while you discuss the implications of all of this."
"...How did you open that?" Lifting her head, Laka'thi gazed at Ia. "No one has been able to open that inner box in all our records. No one even knows _what_ that crystal box was made from, just that the Immortal herself made it."
"That's because the Immortal will have borrowed that knowledge from the secrets of the Third Human Empire...which has yet to be born as an empire," Ia said, her tone wry. "Both of us are bound by the strictures of Time to keep that information a secret, as it is something we are only permitted to borrow for our personal use and are not to divulge to any others. I hope you will respect my right to temporal secrecy when it comes to another government's secrets, just as you would expect me to respect your own sovereign rights to secrecy.
"Long live the Eternal Empire, meioas, and long live the alliance between us. Have a good day," she finished politely, and gestured for the junior officer to lead the way out of the conference room.
He gave the others a bemused look, then gave in with a dip of his head and headed for the door. Ia made a mental note to pass along a letter of commendation for his discretion and sense of equilibrium in the face of the day's events. A recommendation from a foreigner would carry some weight, but one offered by the Prophet—now confirmed as such—would carry even more. _Good junior officers are hard to find, after all. Particularly the unflappable ones._
Her quarry sat as he usually did at this time of the day, just after the end of a shift filled with cleanerbot herding. Grizzled hair slicked back, wrinkled brown skin sporting the same golden undertones as most V'Dan did, he sat with his elbows on his knees, a sugarstick dangling between two fingers. Most of his attention was on the great bay window giving a few of the planet's curve and gleaming streaks of sunlight.
Placed as it was in the V'Dan homeworld's L2 orbit, slightly farther out than the planet's shadow could cover, there was a constant halo effect. Coupled with the ionosphere's auroras, the view was spectacular, worthy of being watched. It also helped that the window was half-silvered, dulling the brightness of the local star. There were several benches arrayed before the window, taking advantage of that view, but only a few were occupied, and no one but the old man in the dark blue pants and grey shirt sat on the centermost one.
It was hard to picture this tired-looking, somewhat elderly man as being the real power on _Tatth-Niel_ , but Ia knew better. He looked up at her approach, lifting the sugarstick to his lips. She gestured at the broad bench. "Mind if I have a seat?"
"'S a free port," he grunted, sucking on the flavored stick.
Unbuttoning her Dress jacket for comfort, Ia sat and removed her hat. The bench was a little low, but not uncomfortable. Leaning forward, she braced her elbows on her knees, letting the cap dangle from her fingertips. "Nice view."
"That, it is," he agreed.
( _I trust you know why I'm here?_ ) Ia asked, shaping and aiming the thought carefully.
He slanted a sidelong look her way. ( _You aren't a player, half child._ )
( _It won't be long before I manifest,_ ) she warned him. Dropping her gaze to her cap, she rotated it slowly in her grip. ( _I'm not here to compare bubble-sizes, Kierfando,_ ) Ia stated. ( _I am here to counteroffer Miklinn's intent to ask you to faction for a counterfaction against me._ )
( _What could you possibly offer me, half-breed?_ ) he replied, sucking again on the stick.
Leaning over, she offered her hand palm up. "Ia," she said aloud in introduction, adding silently, ( _Come and see._ )
"Kier," he grunted out loud, clasping hands. ( _Why should I even—_ )
Counting the offer of his touch as permission given, Ia dragged him onto the timeplains. In the real world, his brown eyes lightened to amber for just a few heartbeats. At least, out in the physical world. In the timeplains, she emerged on the bank with her hand stuck in an oversized silvery sphere and a lot more Time on her side.
Before he could react to the abrupt change in mental landscapes, she hauled him toward the end of the galaxy.
The mirror-like surface roiled and swirled with indignation. ( _You presume much!_ )
She didn't prevaricate. ( _I have just been proved to be the Prophet of a Thousand Years, as foretold to these people for millennia by the Abomination, and have been proved so in front of high-ranked leaders of the Sh'nai faith. As soon as they spread word of this—and it_ will _spread—the majority of the V'Dan Empire shall become faction to me. I offer you a chance to shift your position slightly, so that your actions will be fortified and your efforts will not be washed away in the flood of my own._
( _I also offer you this chance to see_ what _my goal is, so you may understand just how strongly it aligns with your own,_ ) Ia stated.
Stopping by the bank of her carefully tended channel, she displayed the disparity of this future point in time. The lush growth of the prairie's past, the barren emptiness of the deserted future, and the one crack in the wall of the coming fate that led to a garden of renewed possibilities.
( _Examine it quickly. I dare not let our hands linger for long, lest someone notice it on the security cameras and wonder why I'm taking so long to introduce myself to you. That would be counterproductive to your cover...and I would not like to make the same stupid mistake I made when I exposed Miklinn._ )
The sphere spun and bobbed, probing at the visage of Time, tasting the waters. He finally pulled back. ( _...Yes, that wasparticularly stupid of you. Move us back to our entry point and show me this proof that says to the V'Dan you're the Prophet._)
Doing as he bid, Ia did not take offense. She had been stupid that day and was strong enough to acknowledge it. Kierfando was one of the oldest Meddlers in local space. He was also one of the most flexible and forgiving for his kind. The youngest ones were arrogant, the middling ones inflexibly prejudiced, and the oldest ones set in their ways, but not him. That was why he still held such a potent position in their Game, poised at a major interstellar crossroads.
She gave him time to examine the recent scene in her own past, then pulled his bubble out of the waters again. ( _Have you seen enough?_ )
( _For now? Yes. When we part company, shake hands with me again. I wish to examine the Great Demand you will supposedly make of us._ )
Withdrawing their minds back into their bodies, Ia released his fingers. Only a few seconds had passed, physically. ( _Provided you agree to be neutral to me at the very least when we part company, then we have a deal. You will get nothing if you choose to counterfaction me to any degree._ )
He snorted and sucked on his sugarstick. ( _You learn quickly. You also have sparks the size of this station, thinking you can make demands on a fullblood._ )
( _My "sparks" are bigger than this entire star system,_ ) she bragged, smiling slightly. The boast earned her a mental chuckle. ( _Thank you for being more flexible than most of your kind,_ ) she told Kierfando. ( _If I'd tried to make that joke with one of the others, they'd have smacked me for hubris rather than seeing it for what it is._ )
( _"Humor is a waste of energy,"_ ) he quoted, sucking on the sugarstick again. ( _That line of thought is a bunch of matter-loaded recharge, if you ask me._ )
"You know, those things aren't entirely healthy for your teeth," Ia offered out loud. "Or your pancreas."
"Been suckin' on 'em longer than you've been alive, meioa-e," Kier grunted back. "I've earned my vices. Haven't you, yet?"
"Hm. Vices...I don't think I have any, unless you count a slavish devotion to my military duties," she admitted.
He eyed her and pulled a fresh, plexi-wrapped stick out of his shirt pocket. "You need this more than I do if all you do is march around and shoot at people."
She accepted it with a dip of her head. "Thank you."
"Don't thank me. Shut up and suck up," he ordered her.
"Sir, yes, sir," she quipped, earning her a second mental chuckle in reply.
Unwrapping the stick, she tucked the end between her lips and sucked. The flavor was fruity, something V'Dan local and not anything she had tasted before. If it had been the real fruit, it would have been filled with histaminic triggers, but like every other Human in the known galaxy, Ia had been inoculated at birth with the _jungen_ virus to counteract such things. As it was, the flavor was an exotic touch. Some of her home colony's orchards had been planted with seeds from V'Dan, but nothing like this.
She sucked again, enjoying the treat. ( _Tasty. Thank you._ )
( _My favorite. I'll offer you a provisional faction, Prophet—and yes, I acknowledge you as such. Neutral-assured until you manifest, then I'll faction you,_ ) he clarified. ( _If I faction you openly before you prove yourself a player, not a pawn, that could weaken my own plays considerably. If you don't make the change...as you fleshies say, no sweat off my back._ )
( _That's acceptable. Provisional faction, neutral until manifestation,_ ) Ia confirmed.
"Well. Time for me to go, girl. Got supper waiting for me, and nobody making it but me," he added out loud. ( _Tonight's menu calls for a gentle bath of ultraviolet light for about an hour, followed by soaking in a static generation wheel. Come back sometime, child, after you make the change, and I'll treat you to a nice "home-cooked" meal._ )
She chuckled. "My cooking is barely tolerable. I had to barter for a whole ship of soldiers under my command to do it for me." She held out her hand again. "Nice to meet you, Kier. Thank you for the view, the stick, and the hospitality."
"Not a problem," he said, clasping hands with her. ( _Now show your desperate, Game-based request to me._ )
Once more, Time ran faster on the plains than out in reality. Ia swayed them both forward—not nearly as far as the first trip—and dipped him into one of the defining moments of her sought-after future. It took him a few seconds in the streams to grasp the implications of her demand. When he did...he recoiled, the surface of his energy-sphere turning dark with shock. Ia eased him up onto the bank and offered him a packet of psychic energy. He did not refuse, proof of his agitation.
( _You see why I cannot do this without your people's help,_ ) she murmured as he recovered, his surface gradually growing mirror-bright again. ( _My Right of Simmerings ends in less than a month. I need your people to acknowledge I am a player in the Game, not a pawn, because I need everyone to faction with and aid me at_ that _point in time._ )
( _Half child, you_ do _have sparks the size of this star system...but I can also see why it_ would _be necessary, and why you will need our aid,_ ) he agreed. The sphere swirled, focusing on her. ( _We're the only ones who can escape being harmed by that thing...and I can also see why you'd allow it to be unleashed on your enemy. Those suckered fiends would counterfaction you and your pawns hard down through Time, unless you let that happen to them. But_ this, _child, is going to_ shova v'shakk _the Game plays out of several of us. Not me personally, but some of the others, oh yes. You will make counterfaction enemies with this._ )
( _I'm prepared to offer my temporal assistance to reestablish them in other positions. Better to start again and rebuild with my help than to lose everything, after all...and if I don't get your people's help, a very large number of you will lose everything in the short term...and all of you in the long term._ )
( _I hope they agree because if they refuse, the Game ends, according to what you have foreseen,_ ) he agreed. ( _I'm not the best at future-skimming, but I can See enough to know that you See true—I like the visualizations you used, by the way._ )
( _Thanks. Any help you can offer me, neutral though we may be, I'd appreciate,_ ) she told him. Then pulled them out of the timestreams, adding, ( _Parting hands, now._ )
( _What sort of help?_ ) he asked her, releasing her fingers. Again, only a couple heartbeats had passed in reality.
( _Names of the Feyori I could contact between now and my manifestation. It's not always easy to see your kind, particularly when they want to keep their presence and influences hidden. A name would help me locate them faster,_ ) Ia said.
Hands on knees, he grunted and rocked himself forward, pushing upright. "Enjoy the view, meioa."
"And you." She watched him walk away, wondering if he would even acknowledge her question.
He did, several seconds after she turned her attention back to the half-silvered view.
( _Telu'oc. Na-Ganj. Kropecz. Belini. Gallown._ ) Each name came with a subpulse associating its owner with a particular territory, and a warning. ( _No guarantees they will accept, half-breed. But they are the five most likely to at least stop and listen._ )
( _Thank you. There will be a very tasty coronal ejection in fourteen days. Safest place for you to feed will be just ahead of the leading edge of the largest gas giant's L4. It'll be a fast solar wind, and the L4 will have an ion storm that forms when it interacts with the gas giant's magnetosphere—I'll leave you to do the math of when and where the best point in time will be for you to snack,_ ) she added, as he strolled out of view. ( _Since only you know whether it'll be worth taking a personal day to fly out there in a leisurely, unnoticed way, or just wait until your shift ends and risk zipping off in a visible form, the day the flare reaches that section of space._ )
( _Thank you. I believe that'll make a nice payment for the sugarstick,_ ) he told her.
( _I know. That's why I offered it._ )
He chuckled at the edges of her mind. ( _Polite child. You just might survive the Game...Have a good night._ )
Ia stayed where she was for a few more minutes, then rose from the low bench and started the trek back to the section bearing the _Hellfire_ 's gantry. Since she no longer had to pretend to type on a portable workstation, she pulled her datapad from her trouser pocket and began composing prophecies electrokinetically while she walked.
Part of her mind went to the list of names he had given her. Making a side note of them, she saved it to the pad. Belini, she already knew. Telu'oc wasn't one she had considered before, but he was more likely than Kropecz to cooperate. Na-Ganj was in minor counterfaction to Belini through one of his other factioners, which meant bringing him to her side would be delicate diplomatically. Gallown could go either way.
She had to be back on board her ship within two more hours, before the news was leaked that the Prophet had finally been revealed and confirmed to the Sh'nai priesthood. When that little religion-based bombshell hit the local Nets, fanatic followers would begin searching for clues to the Prophet's identity.
The Emperor might even be moved to demand that Ia present herself formally for recognition if things got out of control. She would have to call upon him before that happened. That was, if he didn't call on _her_ first; the chance for that stood at around seventy-three percent, which meant there was still a lot for her to do to ensure it happened.
She didn't hurry back to the _Hellfire_ , though; there was just enough time to do a little shopping, first. Her civilian clothes were getting old, and something new to wear in her rare, civvies-clad moments wouldn't hurt the future.
APRIL 22, 2496 T.S.
Once again, she was escorted into the conference room hosted by the _Tatth-Niel_ 's Department of the Imperial Army. This time, Ia did not come in uniform. She had found a shop selling dresses not too different from Earth-style cheongsams, but slit front and back to midthigh in petal-like panels as well as down the sides, and layered with more petals underneath.
Using the textile manufactory controls, she had loosened the fit of the long sleeves, adding buttons from elbow to wrist down each side so that she could hide her officer's ident. The fabric was a plain, deep red, embroidered here and there with golden feathers, and it came with a matching, turban-like cap that covered her distinctive hair.
In short, it made her look like a woman, a civilian instead of a no-nonsense soldier. The color was also one associated with the Imperial Court, and her Asiatic tan was just golden enough to blend in with the average V'Dan tint. Nobody had looked twice at her on her way here.
The man waiting for her in the conference room was not a nobody, however. His Eternal Majesty, Emperor Ki'en-qua Nomin'ien V'Dania, Blade of Heavenly Justice, Shield of the Thirty-Seven Worlds—if one counted all the way down to the smallest, jointly founded domeworld colony—and One Hundred Sixty-Seventh Sovereign of V'Dan, definitely looked twice at her.
He glanced her way, then turned and stared as she was brought into the room by Leftenant P'kethra and the Grand General. His Majesty then frowned softly as she was introduced simply as Terran Space Force Ship's Captain Ia, and quirked one brow upward. Handsome and just bordering on middle age, he wore the monarch's version of the Imperial Army Dress Reds, decorated with what looked like his own half glittery of a few brooches, ribbons, and swags of whatever cultural significance applied. She hadn't bothered to check in the timestreams since they weren't important.
"Ship's Captain Ia?" he repeated, staring at her.
Taking that as permission to speak, Ia bowed politely, formally. "In disguise, Your Eternity. Rumors already churn on the V'Dan Nets, and I would not have them connect a certain white-haired military savior of the Gatsugi's opening battle with a certain white-haired soldier striding about the station. Not to mention your own appearance aboard the freeport is being kept discreet, is it not?"
"Correct. Sit," he directed her, gesturing at one of the chairs at the middle point of table. His Terranglo, trade tongue of the Alliance, was far more eloquent than her V'Dan would have been. "This isn't a formal meeting."
She allowed the Grand General to hold her seat for her. "When a foreign soldier is called before the local head of state, it is always a formal meeting, Eternity."
The title she used had originally belonged to the Immortal; War King Kah'el hadn't been able to break her people from using it on him in turn when he had taken over the old, planet-bound empire. It had stuck with his descendants ever since, though none of them had ever lived longer than the normal life span. Ia used it now in respect, tipping her head.
"It is assumed that, as an officer, I must at all times represent my own government in a positive light toward your people," she said. "But then, I believe that is what this meeting is about, is it not?"
"True," he agreed, pulling out his own chair across from her, not waiting for the Grand General to come around the table and do so. That spoke cultural volumes about the middle-aged ruler's candidness and intent. "Still, this is not a formal court appearance. Such a thing would cause a firestorm of public interest and an earthquake of religious turmoil. I would rather speak with you as one soldier to another since our nations are allies in war."
"And my confirmation as a major religious figure should be used as a positive influence toward that war effort, I agree," Ia stated. This was the other reason why she had chosen to appear in civilian clothes rather than her uniform. She could still speak as an officer of the Terran United Planets Space Force, but he would see her as the Prophet, a subconscious influence. "Given I am here before you, I trust you have reviewed the evidence of the Sh'nai D'aspra Archives confirming my identity, and are not here to question it."
"No. The evidence was rather straightforward. Unsettlingly so. My family have been Sh'nai followers for the last fourteen hundred years, ever since the Sang'q'ar religious unification movement. Of course, some have ruled in devout worship, some only by paying lip service. I fell closer to the worship side than the lip service," His Majesty confided candidly, lacing his own fingers together. "But I am...uncertain how I should feel about you.
"As living proof of the Immortal's words," he stated, "you are a living saint whose presence has been one of the core predictions of Her Eternity...but as the Emperor of V'Dan, great-plus grandson of the War King himself, it is not politically wise for me to admit openly that the Immortal did, in fact, exist. Even after nearly five thousand years." Leaning forward, he pinned Ia with his hazel grey eyes. "So what, exactly, am I to do with you?"
"Admitting that the Immortal did exist would lead to speculation that she still exists, being immortal," Ia agreed. "That would lead every asteroid-headed idiot with a complaint against the current government to try to seek her out, or encourage con men to drum up some impostor. That would cause political instability. Yet because we—you and I—need the backing of the V'Dan people behind the war effort, to _deny_ that I exist and that my abilities are real, just in the effort to deny the political-sized headache of the Immortal's existence, would in turn cripple our efforts to defeat the Salik once and for all."
"Once and for all?" He seized on that point.
She nodded soberly. "Once and for all. It will not be a pleasant fight, Eternity. I can only promise you that it will be a worthwhile one if and when we succeed."
He sat back a little, considering her words. "So. We are back again to the problem of what do I do with you?"
She leaned forward on her elbows, echoing his pose by interlacing her fingers together. "My plan, Your Majesty, is and always has been to present my case matter-of-factly. I myself am not what is important; the future is important, and my abilities are merely a tool to access the best path to it. I suggest, with your permission, that I record a message for the Empire. It will be addressed to your fellow V'Dan for you to broadcast at your leisure. _After_ we have left the system, since I would prefer another day of near anonymity for my crew. This is their only chance for Leave for the next eight months, and I'd like them to finish enjoying it."
"And what would you say in this broadcast?" the leader of V'Dan asked, lifting his brow again.
"The flat truth. That I am here to direct the war effort so that as many lives can be saved as possible. That I am not here to answer petty individual questions about the future, but would rather urge your people to turn their energies toward the far-more-effective purpose of defending the Empire. I will then state plainly that the Immortal did and does exist," Ia told him, quickly holding up one hand to forestall his protest, "but that she is still bound by her oaths to the descendants of the War King to stay _out_ of V'Dan politics, so long as your line of Emperors and Empresses continues to lead your people well. I will then confirm that you, Emperor Ki'en-qua, are doing admirably well in this time of great trouble, so she will not return to rule and has no need to return.
"She _is_ still bound by her promise," Ia admitted to him. "So that will not be a lie. I will then state that it is best for the fate of the Empire for you to continue to lead your people for many years to come, and that your reign will have my firm support in that endeavor as the Prophet of a Thousand Years...which means that the orders you give your people will be seen as having my stamp of approval from that point on, whether or not I actually had anything to do with helping you craft them. That will wed religion to politics in a firm show of double support for your policy decisions."
She added a slight, wry smile to that statement.
Ki'en-qua considered her words and nodded slowly. "So far, I like what I've heard. You're showing support for my government, respect for our histories, laws, traditions, and the oldest of our faiths...and firmly telling everyone that _your_ focus is not on V'Dan but on the Alliance as a whole, leaving me to lead my people as I see fit."
Ia arched one of her own brows, shifting her tone to a dry, pointed one. "If you _do_ stray off the right path, I'll send round a note to guide you back onto it, Your Majesty. I _am_ the Prophet, and I would appreciate it if I had at least some cooperation from you and your people in our mutual war effort. But you can take comfort in the fact I foresee no immediate need to _actually_ guide you. You do have good instincts and excellent advisors, and you don't hesitate to make use of them."
His Eternal Majesty arched one imperial brow at her hubris, and the two soldiers in the room, the Grand General and the junior officer, both flushed. Ki'en-qua, however, was the one who addressed Ia, and not the ruler of an interstellar empire. "I see you have excellent instincts yourself, knowing exactly which line to tread between lowly foreign soldier and living religious icon, without either falling short or overreaching the range of authority for either."
Her smile warmed into something more rueful than amused. Hunching forward, Ia confided, "Well, I _am_ double-checking the timestreams even as we speak, to make sure what I say to you doesn't negatively affect the outcome of this meeting."
He laughed, at that. Emperor, soldier, and man, he laughed. Relaxing, Ia sat back, still smiling. So did he. The tension in the room eased as they shared a moment of amused understanding
The Shield of Thirty-Seven Worlds—which had at one point been thirty-eight, before Sanctuary had successfully petitioned for full independence around the time of her birth—finally sighed and folded his arms across his medal-strewn chest. "If only I could keep you around...I find your forthrightness and honesty refreshing, Prophet. More than that, I think it's your utter lack of fear in my presence. I have zero power over you...or do I?"
"No, Your Majesty, you do not," she confirmed. "But our goals are in common alignment. I value the lives of each of your citizens as much as you yourself do—and I say to you, having the full span of Time at my fingertips, you will be remembered as one of the great emperors. You will have to work hard to survive, and you will suffer as you watch your worlds and your people being attacked again and again and again, but you should survive a good long time, and your people will once again thrive. If we both put our best efforts into this war, you'll have my Prophetic Stamp on that."
"I've heard reports on your ship's exceptionally good timing and combat prowess," he admitted dryly. "Not to mention nigh-unbelievable reports about your vessel's main weapon. Any chance that technology will be shared with the rest of the Alliance, as the Terrans once shared your hyperspace gifts?"
"None, Your Majesty. The overshoot range on that main cannon is too dangerous for a nonprecognitive to wield, and I am the only one sufficiently skilled in predicting its full path," Ia said.
"A pity. Right. How much do you speak for your military?" he asked next, sitting forward again. "What authority do you have in brokering deals?"
"Technically, I have _carte blanche_ from the hands of the Admiral-General herself," Ia admitted.
He wasn't stupid. " _Technically_ , I could have you shot and killed for threatening to manipulate the V'Dan government with your little quip about steering my leadership in the right direction. How much _practical_ authority do you have?"
She didn't take offense at the threat, though she did wrinkle her nose. "Not much at the moment, Your Majesty. At some point in the future, if all goes well, I'll have at least a little more authority, but not right now. I can, however, convey suggestions from the V'Dan High Command to the Terran Command Staff and discuss their value with my superiors as a registered military precog. That does carry some weight, but it does not carry any guarantees at this time."
"That'll do. Grand General," Emperor Ki'en-qua stated, acknowledging the other man's presence. The general straightened to Attention. "Arrange with Ship's Captain Ia to have V'Dan High Command security clearance, and ensure that she has the cyphering equipment to access our military communications—I trust that you will not abuse it, of course, Captain. I also trust that, should I have need of your temporal counsel, _Prophet_ ," he stressed, returning his attention to the red-clad woman across from him, "that you will give it when I request it?"
"If I can give it, I certainly will, Eternity, for I love your people as much as I love my own," Ia said, dipping her head. She unbuttoned her left sleeve and flipped open the lid of her arm unit. The general pulled a datachip out of his pocket, already prepared to fulfill the Emperor's order. "...I do thank you for the ciphers. That will greatly speed up the timely delivery of my missives for your people. For those that aren't directly related to the military and the war effort, the Afaso Order has agreed to act as my delivery agents."
"An excellent choice. They are both honorable and politically neutral." The Emperor paused, watching her slot the chip into her arm unit, then sighed. "Is there anything else we should discuss? As much as I wish to keep you here for hours, answering my questions about the war, the future, even the past...Sh'nai legend describes you as a woman who lacks the time to spare for trivial things."
"Those legends are unfortunately rather accurate," Ia admitted, probing the timestreams. The chip she had given to the general the day before already contained a personal cipher and hyperrelay frequency for him to contact her ship. Nothing had changed between then and now, since this was the majority chance she had foreseen. "No...I think that's about it for now.
"I'll head back to my ship now, put on my formal Dress Blacks, and record my address to your people. You'll have a clean copy of it within the next four hours. That should give you and your advisors plenty of time to figure out how best to present it after the _Hellfire_ has left the system."
He lifted his brows. "What, no direct orders on how to do that?"
She gave him a wry look. "I am still a mere mortal, Your Majesty. One lone woman with a huge task ahead of me, whereas you have thousands of advisors on your payroll. Use them wisely, and I won't have to burden myself by doing their job for you on top of my own."
"Should I salute you when you say things like that?" Emperor Ki'en-qua asked dryly, gesturing at her red-clad frame. "Whenever you go all Prophet-y instead of soldier-y?"
Ia chuckled, relaxing. "No, Your Majesty. I'm merely here to deliver a message. A series of messages, to save lives. I might get a bit zealous about it," she admitted, shrugging with self-deprecation and a tough sense of humor, "but it's the message that's important, not the messenger. If you'll give me leave to depart, I'll go get started on helping ensure you remain firmly in control of the First Empire, exactly where Time says you should be."
"By all means, go," he ordered, gesturing at the door.
Rising, she bowed. "Thank you, Eternal Majesty—one more thing: increase your security slightly once my broadcast has been sent to your people. There is a five percent chance it'll trigger some attempted anti-imperial attacks in spite of my reassurances, but it is nothing that well-prepared guards cannot handle."
"My staff already has that in mind," he reassured her. "But I'll pass along the warning. Five percent is still a large number when it comes to one's personal safety."
# CHAPTER 12
_I think what saved the situation in the V'Dan Empire from boiling over was how quickly I and my crew departed the area and moved on to other things. The Admiral-General herself pointed out in my vidcall debriefing that it would indeed be more politically correct of me, a Terran officer, to be as absent as possible from the average V'Dan line of sight than to linger._
_Since that fit in with my plans, I had no problems with her orders. I had other people to convince and plenty of enemy targets to pursue. Of course, some of them turned around and pursued me, too, but such are the fortunes of war._
_~Ia_
MAY 6, 2496 T.S.
ATTENBOROUGH EPSILON 29B
"We're still not shaking them, Captain," Nabouleh warned Ia. She was facing backwards in the pilot's station, so her screens had an excellent aft-ward view of the frigate-sized Salik starship still in pursuit as they dodged through the debris of a small, disintegrating planetoid. "Whoever their pilot is, he's _good_."
"They're not _that_ good," Kirkman snorted, glancing back from the comm station. "They're less than a third our length, which means a small fraction of our mass. That gives them three times as much maneuverab— _oof!_ —maneuverability in this mess."
"Sorry," Ia apologized, easing up on the sideslip, then dropped them down, throwing everyone against their restraint harnesses a second time. On the bright side, the other ship's lasers only struck a glancing blow, while her own gunners managed to strafe the other craft despite the vector changes. The frigate's hull, however, was holding up disturbingly well.
"Sir, why aren't we making a run for it?" Private Balle asked her. "We still have enough hull integrity that— _uff!_ —a straight run wouldn't endanger the ship too much. If you give me a vector, I can plot us a course outta here."
"Two reasons," she told the navigator, dodging yet another chunk of tidally torn planetoid. "One, our pursuer will be given high esteem for lasting so long in this tail-chase against— _ugh_ —us." Her sharp maneuvers with the ship's fields were bruising her own hide, too. " _She_ —we're being pursued by a female, a very dangerous sort of hunter-pilot—will be assigned to a new make of small, OTL-capable frigates, and attempt to pursue us for months to come. I can use that, since it'll start to tie up Salik resources in the attempt to ambush us if they think they can follow us and predict our course and targets.
"And the second reason...we're actually here to pick someone up, as well as fight the Salik."
Silence followed that statement. Helstead, overseeing the gunner teams along with Nabouleh and Sung, voiced the thought uppermost on the bridge crew's collective mind.
"...You're mucking _nuts_ , Captain. Pick someone up? Out _here_?" the lieutenant commander scoffed. "With respect, sir, this is a system flooded with hard radiation from three colliding red dwarfs and seven torn planets. The Salik— _uff_ —resorted to remote robotics to do their ore mining here only because they're desperate for untraced resources, sir. Even the Chinsoiy haven't bothered with this place because the gravitational tides make permanent residence problematic."
Something zipped across Ia's screen. It swerved back toward the ship as she dodged another chunk of rock. "Well, _something_ likes this part of space. _All hands, prepare for a friendly boarder. I repeat, prepare for a_ friendly _boarder, followed bya hasty departure from this asteroid-riddled hellhole. Brace for yet more maneuvers and some heavy acceleration. Captain out._
"Zedon, have engineering stand by with extra energy for the bridge, at my station," Ia told the private, switching off the intercom.
Helstead twisted in her seat, frowning at Ia. "Captain, we're not _allowed_ to have boarders."
Something bounced off their hull with the dull thump that said the shields had cushioned most of the impact. "Wow," Sung joked from the main gunnery seat. "They actually fired a dud at us. It broke up on impact, though. Didn't even bother the insystem shields."
"It's not a dud. It's a tracking mechanism," Helstead corrected him, scanner information flowing up her screen. "Tiny robots that will try to push through the shield fluctuations and find a niche they can crawl into, something that will shelter them from the riptide forces of FTL travel. I'd love to get my hands on the pirates that sold them _that_ little piece of tech."
"Whoa!" Zedon exclaimed, looking back at Ia for a split second before returning his attention to his screens. "That was weird, sir. Energy spike in the amidships hull, portside Deck 18, seven by thirteen. It was followed by a serious drain. Is that near those robot things?"
"Those struck aft starboard, closer to Deck 4," Helstead dismissed. "Are we getting an attacker from port side, now?"
"Neither. It's time to execute the 'get the hell out of here' maneuver," Ia quipped, guiding the ship down and left. "Because that energy spike-and-drain is actually our boarder."
"Star 29c is starting to flare," Private Charity Balle warned Ia, checking the navicomp. "If we run-to-jump on this heading, the ion storm will wreak havoc with the remaining scanners. I can't guarantee a safe transit to FTL."
"Who said we were running out of here on FTL?" Ia asked. Lining up the ship, she hit the thrusters, shooting them forward. Pressed back into their seats, they left their Salik pursuer behind. Her chosen opening between the drifting chunks of rock was narrow, but long. It did require subtle shifts right and left, thwarting attacks from behind, but it didn't take much to raise their speed to somewhere close to half Cee.
A swirl of her fingers swung one of the nose cones into position, and a flick shot a ring-shaped spark out in front, narrowing down as it raced ahead. Collapsing into a singularity, it whirled, ripping open a grey-streaked mouth that led into hyperspace. There was just enough room to dart the _Hellfire_ into that dull hell-mouth before it swirled shut again. In its own way, the hyperrift itself wasn't much different than the physics-greasing bubble of the FTL field encapsulating the ship; it was a bubble forcing open an artificial cosmic string, and it sucked them along for no more than three rattling seconds before it spat them back out again.
Slowing the ship, Ia checked their heading in the timestreams, comparing it against the navicomp readings on her lower-leftmost screen. They were now past the heliopause of the system they had just left, so there was little chance of a rogue chunk of whatever being in their way.
"...We've reached interstitial space, Captain," Balle announced, checking her screens. "No navigation hazards within detectable range. Please tell me we don't have to do that again, because I'll need a spacesick bag if we do."
"We're good for now, Private. Nabouleh, power around and prepare to take the helm," Ia ordered.
"Aye, sir," the woman agreed, reaching for the controls that unlocked her backwards-facing station.
"We've got another energy drain, Captain," Zedon called out, tapping his screen to expand a wireframe view of the ship. "It's now below us. It's...coming this way?"
"Nabouleh, helm to yours in twenty," Ia warned the third watch pilot.
"Aye, sir," Nabouleh confirmed. Her chair and console clicked into place, allowing her to lock it and reach for the control glove. "Helm to mine in twenty."
"Sir, it's coming this way!" Zedon twisted away from the operations console. He didn't glance at the right patch of floor, though. Two meters in the other direction, to the left, not the right, the plain grey plexcrete matting covering the floor turned liquid and silvery. It rose up slowly, forming first a puddle, then a curve, then finally a bubble.
"Gentlemeioas, this is our friendly boarder," Ia stated, transferring the ship's controls. "Everyone, this is Belini."
"I have the helm, sir," Nabouleh murmured, gaze darting sideways in little snatches at their guest. Not that there was any real need for her to navigate, safe as their deadheaded course currently was, but she dragged her attention fully back to her screen when the giant bubble just floated there, off to her left.
"Thank you, Yeoman. Continue on course for the moment. Meioa Belini, if you'd come over here," Ia stated, addressing the overgrown, silvery soap bubble reflecting their bridge back at them, "I can transfer you enough energy to manifest physically. I think that would be far less disconcerting to my crew."
The surface of the sphere shifted and roiled. But it was sentient, and it was polite, for it obediently drifted toward her left side. Ia stripped off the control glove and held out her left hand. Her right hand sought out the small hatch and the electrodes built into her console.
Touching a Feyori in reality was not like touching one on the timeplains. There, a Meddler was merely another mental presence, one shaped like a bubble of warm water. This wasn't nearly that soothing. Her skin tickled and her nerves tingled, stimulated by the energies contained in that strange sphere. Adding electricity to the mix increased the stinging of the prickles, but Ia didn't stop conveying energy until the bubble flashed and popped.
Skilled as she was, the Feyori didn't even thump onto the deck as her Human-shaped feet landed. Bare feet, but she had shaped most of herself with clothes, if one counted footless leggings, a knee-length tunic, and a broad, waist-cinching belt as such. Short, blonde, and petite, she looked like a pixie. She also sounded like one, speaking with a light, sweet soprano voice.
"Thank you for the courtesy," the Meddler stated, squeezing Ia's hand before releasing it. Her fingers were warm and soft, and they slid free with a grace that said she was quite comfortable with a matter-based form. "I apologize for the delay. I wasn't completely sure at first why you were hanging around so long after blasting all those mining drones. Then I realized who you were."
"I take it my previous contact spoke with you?" Ia asked. Before she had encountered Meyun Harper, the Feyori had been the only ones she knew who could cloak some of their actions in the timestreams. She wasn't entirely sure what the previous Meddler had done.
Belini leaned over the edge of Ia's console, elbows braced on the edge and hands clasped together. "Indeedy. He's admitted to a provisional factioning. And that you asked for a few names. Given what little I can see of the future...I'm afraid you won't be able to reach the others before your Right of Simmerings is up. Not and convince them you're a player and not a pawn. It looks like you'll have to deal with me."
Ia shrugged, not bothering to deny it. Helstead craned her neck, looking at the Feyori with wide aqua blue eyes. She wasn't the only one. Belini glanced her way and smirked.
"I think I'll be a topic of conversation for some time among your crew...but that's alright. I like you Humans. You're cute as far as matter beings go, and you're endlessly amusing." She rested her chin on her interlaced fingers, smiling at Ia.
"Sir?" Private Kirkman asked, glancing between the alien and his CO. "Is there a...reason...for her being on this ship? I thought visitors were strictly prohibited on this vessel by pain of Grand Treason. Er, Grand High Treason, now that we're at war."
It was Helstead who answered his question. She snorted. "There's an obscure exception outlined in the rules and regs regarding the Feyori. It's extremely difficult to make a damned Meddler do anything it doesn't want to do. Not without a lot of psis pestering it to leave. The Captain _could_ try to band us together to oust her, but there's no guarantee of success...so having a Feyori on board doesn't count against us."
"That exception to the regulations would apply, yes, Lieutenant Commander, but this falls more under the heading of undertaking diplomatic negotiations in favor of the war effort," Ia stated. "Not to mention it's part of my _carte blanche_."
Belini shrugged. "Like I said, you're all so delightful to watch. Threatening to kick me off the ship with little bitty psis?"
"Even a bear can be driven off by a large enough hive of bees," Ia pointed out, keeping her tone mild. She even managed a small smile for the pixie-like woman. "Still, I'm glad we amuse you. It's good to keep a guest entertained."
The pixie-shaped Meddler smirked and leaned closer, speaking with her mind instead of her lips. ( _If you really are the Prophet, half-breed, then you already know what I want, and that I know what you need. Open faction, so that you have protection to continue your Right of Simmerings...and personally tailored prophecies for me._ )
( _Open faction would be advantageous to me on many levels,_ ) Ia returned, not denying it. ( _You're one of the most powerful Meddlers positioned in Terran space. Though with you factioning me, Silverstone indebted to me, and Kierfando willing to accept me, all I'd have to do is manifest...which I could do with your help if you openly faction me. And yes, I do know what you want. It will be arranged for you at the right time and place._ )
( _Good. And nope, I'm not going to help you manifest._ ) Reaching around the transparent edge of her leftmost secondary screen, Belini bopped Ia lightly on the nose. ( _You're the one responsible for achieving your own adulthood, little one. You'll gain more respect if you don't have Feyori aid._ )
Ia wrinkled her nose. ( _Ugh, I know. So much for the shortcut...And please refrain from doing that in front of my troops. They need to know I'm both fearless in the face of a full-blooded Meddler on board,_ and _respected by that Meddler. This is part of_ my _factioning to my troops, matter-based though they may be._ )
( _Well, don't expect me to salute you,_ ) Belini retorted dryly. ( _I'm not Silverstone. I didn't join the military._ ) She held out her hand. ( _So. We are in faction?_ )
( _We are._ ) Gripping the offered hand, Ia sealed the deal with a wordless thought pulse.
Belini's brows rose. ( _You know how to talk in the old ways, don't you? A shame, but not more than half of us at most still do that. The only drawback to the Game is that we're in some ways being influenced by our own pawns, even as we influence them._ )
Ia sent another wordless pulse of energy, confirming it. Belini pulsed one back. It was a lot more complex than Ia's, and made the younger woman's head ache. Releasing her hand, Ia rubbed briefly at her temple, then lifted her chin at her main screen. "So where should we drop you off?"
"I need to stick around long enough to place my marks on your ship, so that the others will know it's under my protection," Belini stated, watching Ia shift to unclip her harness. "That means coordinating with your engineers so they don't cleanse the wrong energy traces—via degaussing and so forth. Depending on whether or not they understand what they'll be looking at, that could take one to five hours. By the way, did you know there's a small homing beacon now attached to your ship? It tastes like mud. Salik machines tend to taste that way."
It was Ia's turn to snort. "Of course I know. My intent is to find it, deactivate it without destroying it, then reactivate it later on to lure the enemy into ambush after ambush."
Belini grinned and leaned again, looking like a kid. "Excellent. You just might survive in the playing field of Feyori politics."
Ia pushed out of her chair. "Whatever I may do with my abilities, I am Human first and foremost. My only goal is to save this galaxy. I have no interest in your Game beyond ensuring that it, too, will survive the coming cataclysm—but let's take this to my office, where I will go over what you are and are not allowed to see and touch while you are on board. Please remember that I _am_ strong enough to throw you off my ship."
"Hey, I'm in faction to you now," Belini countered, raising her hands in mock-surrender. "I also personally believe it's a sweet enough deal, I'm not going to mess it up—you could even put me in the brig when I'm not being escorted around."
Ia gave her a flat look. "You're a Feyori. The brig wouldn't hold you at all. I'll escort you to engineering myself. What nearby system do you want to be dropped off at?"
"I believe your star charts call it Atteborough Theta something," Belini dismissed, flicking her hand lazily.
"Private Balle, plot a course for Atteborough Theta 23. Yeoman Nabouleh, take us into FTL as soon as the navicomp has the course laid. Lieutenant Commander, you have the bridge," Ia ordered.
"Aye, sir," Helstead agreed. She didn't bother to unbuckle from her seat.
( _I'd also like to place a nexus-fold somewhere on your ship,_ ) Belini added casually. ( _It's common for those in open faction to have recall points anchored in the other's local space, so that we can call upon each other quickly in times of need. Not that I'd expect you to call on me, but it'd make moving around easier for me. Anything over a dozen light-years away is cheaper to fold to than it would be to fly. I'm good enough at time-skimming to be able to know if you're near where I'd want to go._ )
( _It's wise of you not to say that part aloud,_ ) Ia returned, opening the door to the aft corridor, the one that led to the two heads, the galley, and the side door to her office. ( _If word gets out just how much I'm colluding with a Meddler, beyond permitting you on board, the Admiral-General would pull my rank so fast, I'd have enough power to manifest just from the energy burns._ )
Belini chuckled at that.
Ia opened the door to her office. ( _We'll put it in my quarters so that your comings and goings don't alarm the crew. Provided the main cannon isn't being fired, you can simply slip down through the deck, then slip out through the stern without being seen._ )
( _You don't sound too happy about permitting this,_ ) Belini observed. ( _Considering what I'm asking of you is otherwise such a little thing for one with your particular skills, adding a little bit more to the deal is only fair. My faction-protection is worth more than your life, after all._ )
( _I'm not objecting,_ ) Ia replied.
( _Liar,_ ) Belini chided her. The pixie-like woman followed her into the captain's quarters. The front cabin was sparse, decorated only in the furniture that came with the ship, a couch, a comfortable pair of chairs, a coffee table bolted to the floor, and a large monitor fixed to one wall, which could display the news nets when they were traveling at insystem speeds, or display any of a million entertainment files stored in the ship's main databanks when they were wrapped in FTL. Belini wrinkled her nose. ( _Ugh. You have zero taste in anything. Don't you know how to play the matter-based version of the Game?_ )
( _I do. My entire career is my Game. This is merely a place to rest,_ ) Ia dismissed.
( _Then you won't mind the fold-point being anchored here. It's more difficult to place it in a small object that moves a lot than it is to put it on a stationary spot on a planet. But both points are constantly moving regardless, thanks to the whirl of the galaxy,_ ) the reshaped Feyori mused, pacing around the modest cabin. She finally settled on a mostly unused corner next to the bedroom door.
( _I'm not entirely comfortable with you being able to come and go so freely on my ship, true, but that's because I really don't want to lose my standing and trust with my superiors,_ ) Ia stated, leaning against the doorframe. She folded her arms, watching the woman peer at the conjunction of the two walls, doing whatever it was the Feyori did when they prepared a place for their version of long-distance travel. ( _It's not an objection, just a discomfort._ )
( _If you lose them, summon me—I'll put in a summoning trigger and show you how to manipulate it—and I'll fix the problem. We are in faction, after all—you to me, and you to Silverstone, and Silverstone to me...Your telepathy really did get the short shrift, didn't it?_ )
( _It's more than overcompensated for in other areas,_ ) Ia defended dryly. Her quip earned her a physical chuckle from the other woman.
Swirling her fingers over the bulkhead panels, Belini did something to the metal, then tapped it with her fingers. She added a wordless pulse of thought, sending it to Ia. ( _Do you understand?_ )
Ia squinted, thought her way through the not-words, and nodded. Moving forward, she joined the woman by the door and lifted her hand. A spark of energy snapped from her fingers to the slightly swirled streaks marring the wall. Belini stiffened, then relaxed.
She nodded. ( _Well done. Triggered on the first try. Don't let anyone degauss or alter this corner. Now, let's go talk to your engineers about what energies they are and are not allowed to purge. By the time I'm done with this ship, no one else will board it without your permission unless they wish to counterfaction me. Or are already in counterfaction, but that's the risk you'll have to take._ )
( _Miklinn's already planning to counterfaction me,_ ) Ia told her, turning back to her office door. ( _That means he's counterfactioning you, too._ )
( _Miklinn always acts like he has an asteroid lodged in his gut. He needs to take a dump more often and stop fretting over the matter races so much,_ ) the alien dismissed. There was a mirror by the door to the main corridor. She drifted that way and fussed with her mop of finger-length hair. ( _Don't get me wrong; fleshies are fun to interact with and observe, but just because you're exposed by a pawn doesn't mean you can't reconceal yourself and try harder. His problem was that he didn't lay nearly enough contingencies. Me? I have several identities I could assume at a moment's notice, plus a dozen ways to hide myself in any Human-touched system I enter.Though I'll admit this is my favorite form. I think I look cute in it._
( _Did you know I once saved the life of Jesse James?_ ) she added, glancing at Ia. ( _Jesse James Mankiller, that is, not the Old Earth outlaw. I was wearing this face when I did it, too. She called me a pixie, and I gave her psychic abilities. That era was a lot of fun._ )
( _Yes, I knew. This way to engineering,_ ) Ia directed, biting back the urge to grow impatient with the chatty woman.
Human though she might appear, Belini wasn't a woman but rather a chessboard hobbyist with a particular fondness for the pieces being moved about the board. Ironically, Ia knew she was trying to do the exact same thing, moving her own pieces about the board. She just didn't want to be quite so arrogant about it.
( _Nothing personal, Belini, but the sooner we get this over with, the sooner I can get back to writing out my prophecies. Including the researching and archiving of the information you'll eventually want to know,_ ) Ia projected.
( _Yes, yes, whatever,_ ) Belini dismissed, flicking a hand. ( _Relax, child. You're under my protection, now._ )
Ia bit her mental tongue, staying diplomatically silent.
MAY 12, 2496 T.S.
SIC TRANSIT
The insistent chiming dragged her awake. Disoriented, Ia unwebbed her bedding and crawled free of the covers. There was no interrupt scheduled, no emergency that she had foreseen. She was _supposed_ to be getting at least six to seven hours of rest. A bleary squint at the bedside chrono showed she had only received three.
The door chimed again. Confused, Ia padded out to the living room. _I_ know _I didn't see any emergencies, no unaccounted-for probabilities...Ah, slag. That means it's Harper._ Slapping the door open, she glared at him. "What?"
"I need to get into the timestreams," he told her, hands lifting and gesturing in his enthusiasm. "I know I told you the new gun design is almost complete, but I think I've come up with a theory on a way to exponentially increase the energy output of the gun's design, giving you more calorie for your credits. It'll have to be wielded by psis, and I'll need several more of those crystals, and I...ah...uhhh..."
His brown eyes had finally drifted lower than her face. What he saw made him falter and blush. Ia had crawled into bed in one of her old, Academy-issued sets of underpants and a worn, matching tank top. The soft blue material had last been seen by him during their aborted weekend post graduation.
"...And I think I will go wait outside while you put on a _lot_ more clothes," he finally muttered, turning around to face the far wall of the corridor instead of entering her quarters.
_Yep. It's Harper. The one person I cannot predict._ Slapping the door shut, she returned to the bedroom to do as he suggested. She also checked the timestreams to see if his idea would wreck the necessary paths. What she found made her frown, then hurry to finish dressing.
Returning to the front room, she opened the door and beckoned him inside. Belini was long gone, but she led him into her office and closed the door before speaking. As soon as it slid shut, she locked it and reached out with her mind, silencing the hidden pickups meant to spy upon him and her. "We're safe to speak in here for the moment, but I suggest couching your terms vaguely all the same.
"If you're right, and I hope you are," Ia continued, still straining, against the fog he always induced, "then what you're proposing could be turned into a cattle prod for the Feyori and a personal-sized shield defense against the anti-psi machines. At least, for this crew and ship. I can't trust it in the hands of anyone else."
He gestured at her, his engineer's enthusiasm returning full force now that she was fully clothed in her uniform shirt and slacks. "See? That's what I was thinking! It finally occurred to me _how_ you were manipulating the crystal, and why it could in turn affect the Feyori—relax, I _won't_ tell anyone how you're doing it—and when _that_ happened, I had the epiphany on how the exo-EM frequencies actually worked."
Pulling a datapad from his shirt pocket, he headed for her desk. "Let me show you the schematic designs I've come up with, so you'll know what to search for in the streams..."
Ia followed him, a faint, wistful smile curving the corner of her mouth. _The one man I cannot foresee, the single biggest threat to the future we all need...and God help me, I love him. Him and his brilliant mind._
She didn't let it show. Didn't tell him. Instead, she merely joined him at her workstation and listened intently as he pointed at the figures and formulae scribbled in electronic layers all over his blueprint files.
JUNE 19, 2496 T.S.
BATTLE PLATFORM _JUSTICAR_
ZUBENESCHAMALI SYSTEM
Ia stepped onto the bridge just in time to hear the second watch comm tech, Kinth Teevie, saying into her headset, _"What do you mean, you don't have the parts we need?"_
The blonde-haired Private First Grade touched her earpiece and scowled. Her screens showed a picture of the exterior of the station. Whatever call she was on, it was audio-only.
_"Excuse me, sir, but those parts were ordered to be earmarked and labeled specifically and solely for the_ Hellfire _'s use. My Captain personally assured me—"_
Teevie broke off, frustrated by whatever was being said on the other end. Ia moved up behind her, dipping mental fingers into the timestreams to see what was going wrong. That was what had summoned her from her office, the sense that something was going wrong. They should have docked with Battle Platform _Justicar_ with no problems, which was why she hadn't bothered to be on the bridge.
What she found made her swear. _Slagging hell! Feyori fingerprints are all over this. Miklinn's made his first overt move against me._ Shakk. _I'd better check the timestreams to find the best way out of this._
Ia listened to Teevie attempting to protest with only a fraction of her attention. The Feyori's move was a surgically clean one, with no way to prove other than through the timestreams themselves that he had telepathically influenced the _Justicar_ 's repairs department. No Human psi had noticed him at work as he drifted through the system a few hours ago, reaching out to manipulate just the right minds.
_The clever bastard's been following us, and he's good enough at radiation-eating to cloak his presence. The moment he realized we were here to fight for the Beta Librae colonists, and saw that the Battle Platform was here, he probably figured we'd stop for repairs. Which I'd already planned to do._ Narrowing her eyes, Ia tapped the private on the shoulder.
"Sir?" Teevie asked, looking back and up at her.
"Contact Admiral Genibes, emergency channel. If you cannot get ahold of him, then contact the Admiral-General."
"Emergency...?" Teevie flushed. "I wouldn't say this was an _emergency_ , exactly..."
"There is more going on at work here than you know," Ia told her. "Contact our immediate superior on the emergency channel. If he is unavailable, escalate it to the Admiral-General. You have your orders."
"Sir, yes, sir," Teevie replied, though her tone was a little doubtful.
"D'you want my seat, sir?" Spyder asked her, pulling his bootheels down from the edge of the command console. Since they were at dock in a post-battle zone, with the Salik kicked out of nearspace and Ia's assurances the Salik wouldn't come back for the rest of the day, he hadn't bothered to latch his harness in place.
"No, that's alright," she dismissed, moving to the other side of the cabin. Fetching her headset from her pocket, she hooked the device over head and ear. "I'll take the backup comm station for this call."
"Receiving pingback, sir," Teevie told her a moment later. "We're on a one-second delay, Captain; you can have a normal conversation. Assuming anyone picks up."
Ia settled into the seat. Out of habit, not need, she reached for the harness straps. Just as she buckled the last set in place, the main screen lit with a view of Admiral John Genibes's face. "Greetings, Admiral. I'm sorry to interrupt the R&D strategy session."
"What's the emergency, Captain?" Genibes asked, barely blinking at her mention of what, exactly, he'd been doing.
"I have just encountered interference by Feyori Meddling, sir," she stated. That made him blink. "Certain individuals are moving to counterfaction me, and a particular Meddler has just drifted through the Zubeneschamali System, nudging the minds of the crew aboard Battle Platform _Justicar_."
Genibes lowered his greying eyebrows. "That's a serious situation, Captain. And a serious accusation. How badly are they compromised, and who is it?"
"Subtly, and so far it only affects the repair crews. You don't need to pull them off their job details, though you should suggest to the Psi Division to come out here and sweep their minds for Meddler fingerprints," Ia said.
"That's a serious problem, and it does need looking into, but it's not what I would call an emergency," her superior stated. "Do I need to define to you what 'emergency' means, Captain?"
"No, sir. I've only outlined the cause of the problem. The effect of it is the repair department on board the _Justicar_ is deliberately blocking our attempts to receive materials for immediate repair. Materials specifically earmarked for the _Hellfire_ to use at this place and time," Ia told him. "I need you to issue a Command Staff–level broadcast to all TUPSF repair facilities that the _Hellfire_ has priority-one repair status, wherever we go. And if you could, to pass the word along to our allied forces in the rest of the Alliance."
He flushed. " _That_ would require both the Admiral-General and the Premier backing it up. The other races won't tolerate it without government approval. And I could hardly think you'd need to force everyone else farther down the list in every single repair situation, soldier."
"Certainly not, sir. Every single instance where we can spare the time, I will gladly let the others go first when it's a temporal priority for them," she said. "You _know_ I will. I just need the _carte blanche_ enforced by a formal, fleetwide statement. That way, if this ever comes up again—and the greater probabilities say it will—I can shock off the Meddler fingerprints by threatening those who balk with Fatality Thirty-Five."
He gave her a wry look. "Sabotage is a very strong accusation, Captain. There's a reason why it's a Fatality."
"I know, sir," she agreed. "In most cases, the mere threat of the cane will be enough."
"Be careful _you_ don't end up on the wrong side of that carrot and stick," he warned her.
"I am fully aware of my double-indemnity clause, sir," Ia said. "I have been taking every precaution possible to ensure my crew remains the most disciplined, reliable Company in the Space Force because of it. In order to _remain_ that reliable force, I need to get the _Hellfire_ 's replacement parts out of the _Justicar_ 's holds and slapped onto my ship, and get it done without wasting half an hour comparing asteroid sizes with whoever's in charge of repairs over there. If you will issue the Command Staff priority notice, that will cover it."
He sighed roughly and poked at something below the pickup view at his desk. "Let me contact the Admiral-General to clear this with her. If you want it broadcast fleetwide, it's not going to be encrypted unless you're willing to wait a few days for all of it to be distributed. I wouldn't think an open broadcast would be wise since it could force your Meddler foe to change tactics."
"Start with the _Justicar_ and take your time with the rest," Ia told him. "We won't be in port anywhere for another five days at the earliest."
_"Shakk,"_ the admiral muttered, shaking his head. "The _Feyori_ are getting involved in this...Any chance they'll side with the Salik?"
"A few already have, but only because those are their assigned Game positions. Most of them are held in check by the positionings of the rest. They're arrogant asteroids, but they do abide by their own rules. Thank you for the authorization, sir," she added politely. "You know I won't abuse it. I just need something more tangibly specific than my _carte-blanche_ authorizations currently suggest."
"While you have me on the line, any suggestions for the R&D folk?" he asked her.
Ia shook her head. "No, sir. The best options will be selected without my interference, or I'd have warned you in advance."
"Thank the stars for smart people and small favors. Genibes out," he added, reaching for another control. The screen turned black.
Nodding, Ia unbuckled her restraints. The timestreams had smoothed out. "You should have the right authorizations in about ten minutes, Teevie. I'll forward an updated version of our supplies request to your post from my office, with the exact location of each item on board the _Justicar_ , so they won't be able to pretend they can't find anything."
"Aye, sir," Teevie said. "Thank you, sir."
"Bureaucracy's a right pain in th'arse, innit?" Spyder quipped, putting his feet back up on the edge of his console. "Especially when you get th' Meddlers in th' mix."
"You have the bridge, Lieutenant," Ia reminded him, rising and heading for the back door. "Try to remember to dust off your footprints before the lieutenant commander comes on watch, will you?"
JULY 9, 2496 T.S.
JS 723 SYSTEM
"Bogey at 49 by 299...298...295," York stated from the communications seat. With an entire sphere's worth for their field of view, and a set of computers that were smart enough to sort out potential problems in a sky of data, but not smart enough to decide what to do with it, more than one set of eyes was needed to manage the flow in an ice-filled system like this one. "Moving fast with what looks like weapons hot, sirs."
"Looks like th' Squid's back," Spyder quipped, checking his own screen at the command console. "Same config as before. Yakko doesn't know when t' give up."
O'Keefe muttered something under her breath, hands dancing across the controls. Her movements drifted the _Hellfire_ to the right a little bit, but not strongly. Ia wasn't strapped into a seat, and the yeoman knew it. She also knew it was likely there were plenty of others around the ship who weren't strapped in for maneuvers. "Captain, do you want to take the helm?"
"Not necessary. No sudden moves, Yeoman," she told O'Keefe. She had emerged from her office with the foreknowledge of the best way to handle this encounter. "Continue on course to the Dlmvla mining station."
"Then it _is_ th' Squid?" Spyder asked, lifting his brows. The nickname had been voted on informally for the Salik hunter-pilot crazy enough to try to engage them. This was her third time finding and following the _Hellfire_ , though her ship wasn't close enough yet to fire accurately.
Douglas spoke up from the operations seat. "Sir, we're badly damaged. Deschamps says his repair teams have parts strewn all over the aft and stern lower decks. One good hit to the rear of the ship, and there could be a lot of flying debris."
"Yes, I know that, Private. And you know that, but _they_ don't know that," Ia soothed. Moving from the back door to the pilot's seat, she touched O'Keefe on the shoulder and gave the curly-haired woman a reassuring smile. "Rotate the ship 180 degrees, Yeoman. Point the muzzle of our main cannon at the Squid, but continue on course. Be mindful of the repair crews and take your time. Head for the mining station and cozy up our middle to their number 17 hatch. At all times until the last few minutes needed to dock, keep the main cannon pointed at the Squid.
"Private York, inform them we will be arriving in time for tea, and that we'd like a canister of frozen _ksisk_ delivered to our airlock. Offer them a single teddy bear in return, then contact Private Bethu-ne'. Ask him to select one of the teddy bears from his collection as a gift for the Dlmvla—the older and more worn, the better. Tell him he has fifteen minutes to p-suit up and get the bear to the amidships Deck 12 portside airlock for the exchange."
York blinked twice, bemused by her odd set of orders, then turned back to his console. "...Aye, sir. Whatever you say, sir."
"Y' don' get it, do you?" Lieutenant Spyder asked rhetorically. He had given Ia a quizzical look when she first stepped onto the bridge, but hadn't relinquished his seat, yet. "Well, the Cap'n an' I do. Xenopsychology an' all that."
York snorted. "Well, I'm glad _you_ do. Mind explaining it to us lowly peons?"
"The Salik, most unfortunately, aren't stupid enough to open up an extra war front," Ia told him. "Discounting the Greys, and with little competition from the other races for the various methane-rich worlds out there, the Dlmvla are the single largest nation in the known galaxy. If you counted the entire galaxy, the Solaricans would come in at third place and the Dlmvla somewhere around fifteenth," she added dryly. "But not in this known patch of it, so the felinoids don't count.
"Up to this point, the Salik have been very careful to avoid Dlmvlan targets. They won't even attack anything near a methane-rich world if there's a single Dlmvlan ship in the system. The Dlmvla think this might be a courtship method of reverse psychology, ignoring them in order to seek an alliance," Ia added lightly, "but the Salik honestly have no interest in contacting them. At least, until the Alliance is shattered, all the tastier species are firmly rolled up in their tentacles, and they've rebuilt their war machine strong enough to take up the hunt for new prey."
"That's why they 'aven't fired on us yet," Spyder pointed out, looking up briefly from his screens. "Back in th' Corps, we heard news 'bout our old Sergeant Ia, after she got jumped up t' be a lieutenant in th' Navy. Nothin' specific, just that she'd play up th' enemies' weaknesses psychologically. Pirates, Salik, smugglers, didn' matter. When she didn't ask fer this seat, I sussed out why she wasn't inna hurry. Salik won't fire on us this close to th' methane-breathers."
"Which is why we're safe to dock," Ia agreed, nodding at her 2nd Platoon lieutenant. " _Leaving_ is another matter. At that point, it'll be third watch, and I'll take the helm. We can't ask the Dlmvla to help us with our repairs, so we'll be here a good twelve hours, but if everyone works hard, that'll be enough time to get the hull solid again."
"And the frozen _ksisk_?" Private Sharpe asked, eyes on his navigation screens. "What is that, some kind of fish?"
"A very tasty berry, but only once it's been aired out," Ia stated dryly. "That's because it's colloquially known as the 'fart fruit' to the Human colonists manning the one M-class world in the otherwise Dlmvlan-run system of Kvuu Zhwinnh 525. A single canister's worth of frosted methane crystals isn't going to bother us, and I'm in the mood to tease the Dlmvla with an exchange of a fairly valuable but edible commodity for a near-worthless but much more permanent child's toy."
"Captain?" O'Keefe asked, frowning in thought. "You _do_ know that the Dlmvla are more than three meters tall, right? Our Deck 12 starboard airlock is designed for two-meter-tall beings. I can't guarantee a solid docking seal if they open up that big door of theirs."
"That's the other reason why Private Bethu-ne' needs to suit up for hard vacuum," Ia told her. "Trading the bear for the canister without pressurization will ensure neither side offends the other with unwanted gasses. It's a subtle courtesy and a subtle insult at the same time, and the Dlmvla will love it. They _think_ the Salik might be courting them with illogic, but I actively am."
"And when we're ready to leave, sir?" O'Keefe asked her, making a minor course correction.
"As soon as we're repaired and clear of the station, we'll be making our way around the gas giant's curve. At about halfway into our run-to-jump, a rather large fleet of Salik summoned by the Squid will leap out of the night and attempt to destroy us...followed by a rather large fleet of prewarned Alliance ships." Ia grinned at the second watch pilot. "It should be fun."
Nodding to Spyder, who had the watch, she walked off the bridge again. She had just enough time for a four-hour nap, followed by eight hours of hard work helping the engineering teams catch up on the needed swap-outs of various refitted and repaired parts, and a long, tough, but ultimately victorious fight against a good chunk of the local Salik fleet. A pity it would be only a small fraction of the whole.
Afterward, the _Hellfire_ would need extensive repairs, more than her Company could handle. She knew there would be three Battle Platforms coming, though, more than enough to spare them the repair gantries that would be needed. She also knew that the Squid would get away and live to track them again, but that was all factored into her plans. As was not saying much to the Dlmvla this visit, other than swapping an old teddy bear for noxiously sweet fruit.
_At least God lets me amuse myself at certain points along this path._
AUGUST 8, 2496 T.S.
GOLDEN PRISM DOME
GOLDEN GLITTERS III ORBIT, SALUK 199 SYSTEM
The view from Lieutenant Commander Helstead's helmet cam wobbled a little. The explosion was two blocks away from the mechsuited woman, but the force of the shockwave still rattled the petite woman's armor. Another, harder explosion rattled the video feed for a moment, sending streaks of static across Ia's second tertiary screen. _"Captain, there is a_ lot _of fighting going on nearby. These people are getting slaughtered!"_
_"Stay on target,"_ Ia ordered her. The _Hellfire_ rocked as well. Strafing the ship sideways helped; it gave the gunners, reduced in number thanks to Helstead's troops on the ground, time to angle their weaponsfire at the three ships pursuing them.
Splitting their forces was not exactly the best option, but the Gatsugi colony needed defending, and theirs was the only ship anyone could spare. The disturbing fact, one Ia had known all along and the rest of the known galaxy was only now beginning to learn, was that the Salik fleet was a _lot_ larger than anyone else knew, partly from mixing their forces.
Tentacled, ostrich-flippered soldiers served alongside mass-manufactured robotics, the latter crafted with at least five completely different programming systems. Orders were given verbally so that all could obey whatever Salik officer led them, but viral attacks that affected one type of robot did not necessarily disable the others, and the algorithms were swapped out every few days. Some worlds were even coming under attack by dangerous beasts bred to survive on a particular planet, to help the Salik in their quest to colonize as well as conquer them.
A hard explosion shook the _Hellfire_. Never a good sign in a starfight, this explosion shoved the ship slightly to the right. Red telltales flooded Ia's upper screens, followed by distant _whunks_ and a wailing, stuttering siren.
"Hull breach!" Private Nelson yelped. "Hull breach, fore section Decks 18 and 19! Ah...ah...section seals are strong, and inner seals _are_ holding."
Ia rolled the ship, sheltering the wound in its side. _"Helstead, turn left. You will see a service entrance to their Senate Hall,_ _and then—"_
_"—And then split off A Squad to hold the path to the shelter tunnels, and B Squad up two flights and turn left, to defend the Bright Speaker up in the broadcast booth. I remember, sir,"_ Helstead told her. _"Alright, you mudding slackers, left face, move out! Puan, Franke, hold the rear but do_ not _fire! We don't want to draw any froggy-bot attentions our way!"_
"Captain! Priority message from the Admiral-General," Al-Aboudwa called out. "Putting it through."
"Belay that," Ia countered, most of her attention on turning the ship so that all three Salik carrier ships were on her port side, not the vulnerable starboard. "Inform the Admiral-General that we are in heavy combat and inform her that I say, quote, 'I know, sir. There was nothing we could do for them. We could only save the rest,' end quote. Restate we are in combat, and cut the link. _L-pod 12, P-pod 14, abandon your pods. I repeat, abandon your pods. Retreat to Deck 5 immediately. All hands, prepare for another hull breach._ "
"O Captain, my Captain," Rico stated calmly, "have I ever told you how much I hate it when you say things like that?"
"Get in line behind Harper. It's his precious ship I'm blowing up. _L-pod 12, P-pod 14, this is_ not _a drill._ Move _it!_ " she barked into her headset.
_"Harper to Ia,"_ she heard in her left ear. _"What the_ shakking hell _are you doing to my ship?"_
_"Captain, the daily bread is in the basket, and the security teams are forgiving us for our trespasses,"_ Helstead reported in her right.
_"Helstead, you make sure she transmits, whatever the cost. Harper,"_ Ia added, switching channels, " _stuff it and start making repair plans._ Al-Aboudwa, prepare to retransmit the signal coming from the surface. All relays, all bandwidths, all stations and comm sets on board this ship. I want everyone to hear why we are _here_ instead of anywhere else."
"Aye, sir. Opening all channels," he agreed.
Nothing happened for several seconds, then a voice spoke. The tone was lyrical, the pitch high and soft. It was a whisper. A murmur. A promise.
_"I send/transmit in the hyper. I send/transmit in the light. I speak plainly/straightly so that each word, each meaning, will be treasured/grasped. Night has fallen, coating all we can see with the dull black of despair/death. The glow of our bodies has dimmed, our grip has weakened, and our foe listens with sharpened teeth for our last/dying breath. But I, but_ we, _are_ not _dead yet."_
The gentle voice sharpened, gaining strength.
_"No limb can tear us down. No tooth can bring us death. These beasts wish to leap upon/destroy us, to drag us down/down/down, but they will_ fail. _They are strong, but we are stronger. They are many, but we are legion. I send/transmit through the hyper. I send/transmit in the light. Every single being/body that faces these predators is my brother/sibling/kin! Native or alien, we who are sentient, we who are compassionate, we/we/we have more in common than these murderers/monsters._
_"Grasp your spears/rocks/knives. Grasp your guns/ships/strengths. We will drop upon them from the highest stars! We shall stab them down, and fill our veins with the bitter reds of our rage, and when they bite...when they_ bite, _they will choke/suffocate/perish on our fury, torn apart by our combined might!_ "
Shouts and sizzling sounds came through in the background. Ia could hear Helstead barking orders to her teams, and an explosion that punctuated the Bright Speaker's next words. Lasers struck the _Hellfire_ , damaging more panels; she dodged them as best she could.
_"I am not afraid!"_ the Gatsugi speaker asserted. _"I transmit in the hyper, I transmit in the light, and I am_ not not not _afraid! The hunters_ will _be hunted. Stand on the branches by your choice/will/right. Climb for the strength to survive this war. Fight for your sentient brothers, and they_ will _fight for you—when the easy prey has been shaken loose, their ceaseless/wasteful hunger will send them into the trees for those who think they are safe! Strike now! Strike/Strike/Strike_ now, _and cut out the tendons of their ambitions. Shove_ them _into the Room for the Dead, before they can shove_ you _in and shut/lock/seal the door!_
_"Here I stand, surrounded by foes, but_ defended _by friends. Gatsugi and alien. Why? Why would these Terrans come to our aid, when they themselves are hunted hard? They have a quote from a Bright Speaker of their own. It has changed words and changed hands many many many times, for it transcends mere words, and mere hands, and mere_ species."
A small explosion rocked the _Hellfire_ 's shields. It was followed by a much larger one, skewing their flight and the stuttering siren of another hull breach.
_"I say it now in my own words, sign it with my own hands! They came for the Solaricans, but I was not a Solarican, and I stayed high above as they perished. They came for the V'Dan, but I was no V'Dan, and did not look nor move. They came for the K'kattan, but my limbs were less, and I did not raise my spine...and the K'katta, too, died. But I know in my soul that they_ are _coming for me. When will_ I _fight for myself, and_ how _can I fight for myself, if I will not also fight for the rest?_
"I _am not afraid._ I _am not at rest. No ruler, no leader, noNestor can tell me what_ I _know is right, and_ I will fight. _I am a Bright Speaker because I speak the truth! I speak it until the universe itself listens. I send/send/send in the hyper. I send/send/send in the light. I demand an answer/response from you!_ Will you fight?"
Silence followed the unseen speaker's words. Al-Aboudwa licked his lips and murmured, "Transmission has ended, sir."
"Open a broadband on the lightwave, Private," Ia directed him, dodging another round of missiles.
"Aye, sir...Broadband ready."
_"Make no mistake. They will come for you, too,"_ Ia said, thumbing open the channel. _"My Prophetic Stamp on that."_ She closed the link with a tap of her thumb and darted the ship between two of the Salik vessels, increasing their velocity. "Jumping in ten."
"Jumping?" Fielle questioned, looking up and back from his screens. "But sir, Helstead and the rest...?"
"They'll be alright," Ia told him, activating one of the two undamaged OTL nose cones. It swung into position and pulsed a torus of energies. "We're just going to reposition."
Darting into the grey maw, they left the orbital space of the third planet. Not even a second later, the hyperrift spat them out somewhere beyond the seventh planet. The rift was still open behind them for a moment. Ia once again hit the thruster fields, accelerating them forward. One of the Salik vessels, tail-chasing them, came through the rift as well.
Or rather, most of it did. With its unmodified FTL fields for grease, it survived the collapse of the rift, but at a cost; damaged by the fight with the _Hellfire_ , those shields prevented the wormhole from rifting their ship but didn't prevent the forces from crumpling their stern. Silent explosions of energy and air escaped. Crippled, the enemy ship skewed off to port, its thruster fields abruptly unbalanced.
There had been a modest chance the enemy ship would be rifted and explode instead of merely being tail-nipped. No doubt the Salik on board considered themselves lucky. Ia intended to disabuse them of that idea.
Wrapping her ship in a vector-soothing bubble, Ia eased their acceleration and gently swapped ends. _Now_ she could pulse the Godstrike cannon, and not worry about local space traffic for the next four months. _Now_ she could whittle down the others, and rout the Salik forces attempting to seize the Collective's colonyworld.
The message had been sent; that was what was important. It had cost them a terrible price to be here, to help the alien poet speak so passionately for her kind. Only Ia knew it would be worth it. Just...not right away.
This was the part of being in command that no good officer liked. Ia was no exception. Dressed in formal Blacks, her cap on her head and her half glittery—Terran only—pinned to her jacket, she entered the bow mechsuit bay.
Private Tormez was the first one to spot her. Caught in the act of stripping out of her pressure-suit, the short, muscular woman snapped to Attention. Or tried to. She yanked her hand out of her half-tangled p-suit sleeve and snapped it to her brow in a salute. "All hands! Ship's Captain on deck! Ten-hut!"
Her warning rippled through the bay. Wide-eyed, the other women and men struggled into Attention poses as best they could, some still in their mechsuits, some half-clothed in their pressure-suits or their uniforms. All of them saluted as she passed. Ia did not stop, though.
Her target was Private First Class Nicholaus Smitt, soldier, clerk, and field medic. He was still working on climbing out of his bulky halfmech, modified as it was to provide on-the-spot medical attention. Awkwardly, he freed his right hand and saluted her, eyes wide with wonder and worry when she stopped in front of his alcove.
"Sir?" he asked her.
Ia saluted him back. She kept her voice steady as she dropped her arm and relayed her news. "Private Smitt. It is my deepest regret to inform you that the domeworld colony on Seldun IV, System ISC 197, has fallen to the Salik advance."
_"No,"_ he whispered, shaking his head.
Ia gave him a sympathetic look as she continued. "The colonists did evacuate many of the women and children, and your mother has survived, but your father and sister chose to stay behind and fight."
"No," Smitt asserted louder. "No. You're a precog. You should've told them. You should've _known_. We should've been _there_!"
"I can say that it was relatively swift," she continued calmly, quietly. "The colonial mayor rigged the entire chain of domes with explosives. They destroyed most of the invasion force and left very little for the enemy to salvage and use as a base of operations."
" _No!_ You should've _saved_ them," Smitt accused, tanned face wrinkling in rage. He pointed at her in accusation. "We should've _been_ there!
"I'm sorry. We had to be here."
_"Sorry?"_ he yelled, free hand balling into a fist.
Ia didn't dodge. It connected painfully hard with the side of her face, rocking her sideways with the force of the blow. Sidestepping to catch her balance had an added benefit; it carried her out of range of a second punch, since his left arm was still caught in the workings of his suit.
"Private!" Helstead snapped, hurrying forward even as he tried swinging again. Her scowl overshadowed the fact that she was clad only in her underwear and a single sock, her fury palpable. "I will _personally_ throw you in the brig for that!"
"Stand down and belay that, Commander," Ia ordered her, holding out her arm to stay Helstead's advance. The fingers of her other hand touched her cheek, testing the heat and tenderness of the bruise. "No Fatality has been committed here. I will not punish a man for an action wrought by grief. Not when I am directly responsible for it."
Helstead and Smitt weren't the only ones who blinked at that admission. Ia seized their stunned quiet.
"Every single second of my day, soldier, I make decisions like this. I _know_ the names of your father and sister. I can tell you exactly what your sister said when she was eight and skinned her knee sliding down the stairs of your home, when you caught her crying from it at the bottom step. I know what _you_ said to her. I know _every single person_ who died on that planet two hours ago," Ia told him, letting her grief harden her words. "I know them, _and_ I know the names and words and faces of every single person _their sacrifice_ has saved in this war.
"We came _here_ , to _this_ colonyworld, because the words of the Gatsugi poet we saved will turn the tide for us. In two years and forty-eight days, the High Nestor Conference will hear those words being rebroadcast by sympathizers from the Dlmvlan mining colony located two light-years and forty- _seven_ light-days from here," she added, pointing off to her left in the vague direction of that other system, "and that poetic speech, coupled with all my _other_ efforts, will finally drag their collective asteroids into this fight. Having the Dlmvla behind us _will_ turn the tide of this war.
"But no one else could be spared to come _here_ on this day. With all the other battles that _have_ to be fought, no one else could be spared to go _there_ , either," she stated, pointing first down at their feet, then off to the right, toward his lost homeworld. "Hate me if you like, but I have known for _years_ that half your family would die tonight...and I could do _nothing_ for them but try to save the rest of the galaxy."
Dropping her hand to her side, she stared at him, vision swimming with unshed tears.
"Hate me all you like, Smitt. I know I deserve it," she admitted quietly. "Take comfort in the fact that there is nothing you could say or do to me that would be worse than having to live with the names, and the words, and the screaming faces of every single living being who has, is, and _will_ be slaughtered because of this war. I have lived with this weight for the last nine-plus years, and I will _continue_ to live with it, year after year, until the day that I die."
She held his gaze, implacable in her resolve. Unrelenting in her message. Smitt finally blinked and looked away. Ia drew in a ragged breath and let it out slowly, collecting herself.
"I am sorry—more than you will ever know—but in war, some may have to die so that others may live...and some _must_ die, so that others _will_ live." Ia met his gaze as he looked back at her. "I am sorry for your pain. I am sorry that I could not save your kin. I am _not_ , however, sorry I chose to give up the lives of 4,179 brave and undeserving Humans and the fifteen valiant Tlassians sharing that colony with them in exchange for the continued existence of the _trillions_ of sentients who will still be alive in the Alliance five-plus years from now, thanks to the Dlmvla finally joining our side."
He blinked at that. Ia twisted up the corner of her mouth in a humorless mock-smile. The bitter grimace accompanied her final, quiet words.
"This is but a very small fraction of the hell that I have had to live through every single day of my life. My gifts can save this galaxy, but they come with a _very_ heavy price. Yes, you have paid with the loss of your family. _I_ have paid with my soul...and I _may_ have to ask more than this tragedy from all of you in the days ahead.
"As you were, Private Smitt, meioas. We still have a lot of work to do before we can rest." Turning on her heel, she left him and the others in the bay to contemplate her news.
Behind her, Helstead growled at Smitt. "You may be off the hook for hitting your CO, Private, but you will _not_ do that again. You're on filter duty and floor-mopping until I say so!"
"She admitted she _murdered_ my f—" Smitt started to argue, raising his voice.
" _Welcome to Hell_ , soldier!" Helstead snarled, her words echoing off the bulkheads and alcoves as Ia left the bay. "They only call it a war to make it sound better!"
Ia tapped the control panel for the prep-bay door as she stepped through, cutting off whatever else her second officer might say.
AUGUST 16, 2496 T.S.
SIC TRANSIT
Bennie stared at her friend and commanding officer, who slouched low in the seat across from her.
Ia knew why the older woman stared. She looked like hell. She'd seen her own face in the mirror, shadows under her eyes, skin a little too pale, and other signs of exhaustion both mental and physical.
"So. How have you been sleeping?" Bennie finally asked her.
"I haven't." The admission was blunt.
Pausing with her mug halfway to her lips, the chaplain lifted her brows at that. "You haven't?"
"Not a wink. Well, not more than five or six hours since the Golden Glitters fight," Ia dismissed, flopping a hand on the armrest. "And that's only if you squeezed it all together, minute by minute."
"That's not healthy for a Human," the redhead observed warily. "You don't _seem_ cognitively impaired..."
"I've been running on electricity. And some other stuff," she said, thinking of Harper's attempts at shooting her with low-level pulses from his prototype gun.
Or rather, his shooting her with Helstead's help, since the new design did indeed require a psi to power the weapon. Both lieutenant commanders were staying silent on the existence of the gun and how it worked, but then she knew they could be discreet enough. She hadn't told either friend that she wasn't sleeping, though, only Bennie. As a side effect of the design, the energy from the weapon was acting like fuel for her body; otherwise, Ia wouldn't have been in as good shape as she still was.
A twitch of her instincts warned her that her statement was about to be misinterpreted, so she quickly added, "Relax, Bennie. It's nothing illegal. Just a bunch of caffeine, vitamin complexes, that sort of thing. Nothing addictive. You know I won't do anything to mess myself up. Not with the fate of the galaxy depending on my mind."
"So what have you been doing with all that extra time?" Bennie asked, changing the subject. "Since you're not using it to sleep?"
Ia didn't believe for a minute that her DoI-appointed counselor wouldn't bring it back up. "Combing the timestreams for more prophecies and probability contingencies. I'm almost a week ahead of schedule, which isn't a bad thing. I have only a finite amount of time in which to direct events down through the ages to come. It's allowing me to flesh out some of my orders and fill in some of the otherwise neglected corners."
"Well, if you're ahead of the curve for the moment, that's good. That means you'll cooperate when I tell you to go see the doctor for a brain scan to make sure you're not damaging yourself, and a prescription for a sleeping aid. _No_ arguing," she added, as Ia drew in a breath to speak. "You said it yourself, we're in transit for the next three days, trying to get to the other side of the known galaxy for the next crucial battle. You have time to sleep right now, and you _will_ take it."
"...Yes, Mother," Ia muttered, subsiding.
Bennie choked a little on her caf', coughing into a hastily raised hand. "I am _not_ your mother, young lady! I have never earned that lofty title through the sweat and tears of raising a child. At best, I'm an honorary aunty."
"Then, yes, Aunty," Ia managed to tease. She grimaced. "Do I have to go do it now?"
Coughing again, the chaplain nodded. "You'd better. Do you want me to tell Harper and the other officers that you're standing down for a bit of sleep, or would you rather?"
"Me," Ia stated. Drawing in a deep breath, she dredged up the energy to push herself upright. "Better that it seems like it's my idea, so that it doesn't weaken my command."
"Smitt _will_ forgive you," Bennie told her, rising as well. "The others will, too. Just give them a little more time."
That made her snort; if she'd been drinking, Ia would've choked, too. "Time is the one thing I have very little of to spare."
AUGUST 20, 2496 T.S.
BATTLE PLATFORM _KAISER'S COACH_
INTERSTITIAL SPACE
"Captain, I have an incoming link from a Lieutenant First Grade Gregory Bruer, Navy," Private Kirkman stated. "It's on an open channel, sir, uncoded. Is this a legitimate call, sir?"
His mellow low tenor interrupted the relative quiet that pervaded the bridge whenever the _Hellfire_ was docked somewhere for repairs. In other words, between the distant _clunks_ and _bangs_ and drill-rasping sounds of various broken chunks of the ship being swapped out for undamaged ones by the Battle Platform's repair crews. Ia knew why her communications tech hesitated to mention it. Most of their messages were heavily encrypted.
She hadn't remembered that this call might happen, but it was a legitimate one. Nodding, she lifted her chin at her workstation screens. "It's legit. Scramble our end with the beta codes for the day and put it on my left secondary when it pings through."
"Aye, sir." His fingers shifted over his console controls for several moments. "...Receiving pingback on the beta, sir. It's been routed through five different hubs, so you'll have a seven-second delay."
It might have been only three years since she had last seen Cadet Bruer at the end of their time in the Academy together, but the dark-haired man had picked up at least two strands of grey in the interim, and about six years' worth of aging in his face. He looked like someone could stand to order him to bed for several hours of drug-induced sleep, too.
"Hello, Bruer," Ia stated, waiting for his end to catch her signal. "I'm glad to see you're alive."
"Ship's Captain...oh, God," he breathed almost at the same time, staring into the pickups with a slightly dazed look. "You have no idea...or maybe you do..." He paused, receiving her greeting, and nodded fervently. "Oh, yes, yes, I _am_ alive! And most of our crew is, thanks to you. I mean, you _told_ me I'd be on a crippled ship, about to be reeled in and boarded by the Salik, but...
"I did it, you know," he stated. He did so somewhat proudly, pulling his shoulders back. "I advised the Commodore to fake a greater incapacity, and to manually fire the weapons when they launched the boarding pods. It was an ugly fight once they boarded, but the majority of us survived. We kited our ship barely ahead of theirs so they couldn't grapple on and board in full, until two more from the fleet came to our rescue. We had to be towed off for repairs, but that was yesterday. We're very much alive today, thanks to you."
She smiled. "Then I'm very glad you remembered, Bruer."
" _Thank_ you." Bruer's words were simple, but heartfelt. "I can't say it enough. Just...thank you! I've heard you're a precognitive, and I wish I'd _known_ back then...but then that does explain why you were so _good_ in the combat simulations—I'm not taking you from anything important, am I?" he added quickly. "I mean, _Ship's Captain_ , already! Look at you! If you can do for even a fraction of the fleet what you've done for me and my crew..."
"I work directly with the Command Staff and the Admiral-General these days," Ia told him. "So yes, I am doing it. And I'm currently at dock, undergoing repairs. They're almost through, though, then it's back into another highly classified patrol."
"Good. Good," he praised, looking relieved. Fervently relieved. "You tell me to do anything, Ia, I'll do it. I _will_ do it. Uh, so long as it doesn't break any Fatalities, of course."
"Of course," she agreed, amused by the hedging. She dipped her head in acknowledgment. "I'm not going to hold you up any longer. I know you have a lot of repairs to make so you can get back out there."
"Of course—just, thank you. And heed your _own_ advice, young Cadet," he ordered her, pointing his finger at the screen. "I'd like you to still be around when this all ends, so I can thank you in person, and buy you a drink."
Ia smiled wistfully. "I'd like that, too. I'll see if it can be arranged. Oh, 'Cadet' Harper says hi...or he would, if he weren't busy cursing my name while trying to put our ship back together. He's now my first officer and chief engineer."
Bruer grinned. "Good! Tell him he still owes me twenty credits for that last targeting game we had...and our comm tech says my time is up," he added, glancing to the side. "Just...thank you."
"You're welcome," Ia told him before ending the transmission on her end.
"Old Academy friend, Captain?" Kirkman asked her once the signals stopped going through.
She sighed, slumping back in her seat. "An _alive_ Academy friend. Not everyone will be by the time this is through."
"So...you really do know who's going to live and who's going to die?" he asked, turning in his seat to glance at her. The comm tech wasn't one of the ones who had asked to see the timeplains for himself. "You've always known?"
"For the ones I've bothered to look up, regarding the vast majority of their possible futures, yes. When, where, how, and why." She met his gaze wryly. "Smitt's family wasn't the first, and I'm very sorry to say they won't be the last."
He turned back to his boards, muttering under his breath. "...I am very glad I'm not you, sir."
"Oddly enough, so am I," she sighed. "I wouldn't wish this kind of hell on anyone."
# CHAPTER 13
_Yes...you would ask that question, wouldn't you? I suppose it's only fair to ask it. This interview is supposed to be the most candid one I'll ever give, and I haven't exactly been candid on that particular fiasco. I cannot—will not—answer your question directly. There are forces at work which, if disturbed, would shatter the duct tape I've applied to the universe. But indirectly, I can._
_Have you ever worked so hard on something that it became your whole world? Some project so deeply close and personal to your heart that it defined you? No? Plenty of people have, of course, but many more have not. Those who have often try to explain it in metaphor to those who have not. Allow me to try that with you. Maybe you'll finally understand._
_I had a teacup, once. A very special and precious cup. This teacup was something of an heirloom, not very special to anyone else, but ancient and irreplaceable in its value to me. I guarded it, and used it, and valued it...I treasured that teacup until one day, one unexpected day...it fell, and it broke._
_It broke so badly that all of my horses and all of my men could not put that teacup back together again._
_~Ia_
NOVEMBER 15, 2496 T.S.
MIDSYSTEM ICE BELT
KELLINGS 588
She had a pixie in her living room. Sighing heavily, Ia tapped her office door shut. "No."
Belini arched one brow, hands going to her blue-clad hips. "I think _yes_."
"No. I am tired, and just...no."
She was sleeping better these days, but Harper's gun had left her nerves a frazzled mess. Her first officer and chief engineer was still trying to tune the crystals just right for maximum effect, but that meant using Ia for both the test subject and the tuner, and that took energy. Ia didn't have time to get any sleep right now, but she did have enough time for a hot, reviving shower...if she could get rid of the persistent pixie in her presence.
"Hastings' World isn't that far from here, even by FTL," Belini reminded her, following Ia toward her bedroom. "It won't take more than five or six hours to get close enough to drop me off if you're already going that way, or five or six minutes if we slip out that way via OTL."
"Absolutely not," Ia countered flatly, poking the button to open the door. "This is the closest we will get."
"Excuse me, but you _are_ paying for my protection. I've already had words with Miklinn on your behalf," Belini pointed out. "Five or six minutes of your time—or ten to twelve if you're headed the opposite way—really isn't all that much to ask. You can head that way as soon as you're done loading fuel."
"Ten to twelve minutes of OTL translates to seventy extra minutes of processing ice for fuel. We don't even have forty minutes to spare, and we certainly don't have the fuel," Ia stated bluntly, turning around to face the other woman. "The best I can do is kick you out an airlock and shoot you in the back."
The Feyori studied Ia, hands on her hips, aquamarine eyes shifting to something a bit more silvery. Finally, she nodded. "...Deal. But only with a laser. I don't digest missile payloads all that well. They make me look fat."
Ia swept her hand off to her left. "Deck 13, portside amidships hull. I'll go to the bridge and tell Private Ateah to line up the nearest Skystrike L-pod so he can shoot you once you're off the ship. That size should be safe enough for you."
Belini peered at her. "You're rather cranky. Would you like to shoot me yourself, and maybe feel better?"
"I'd be too tempted to use a bigger gun than you could comfortably digest." At Belini's quirked brow, Ia rubbed one hand over her face. "We lost another domeworld to the Salik last week. Solarican, but still..."
Lifting a hand to her arm, Belini squeezed it gently. "You're going to have to stop sympathizing so much with your fellow fleshies if you want to play the Game. Even as a mere soldier, you should know this."
The harsh look Ia slanted her way between those spread fingers silenced anything else the Meddler was about to say. Giving her arm a gentle pat instead, the Meddler moved back into the living room. The air snapped, and the overhead lighting dimmed slightly as the alien sucked energy out of the nearest outlets.
Sighing, Ia flipped open her bracer and hailed the gunner on duty. If she used the comm instead of walked to the bridge, she'd have almost two minutes more of hot water to help wash away the jangling in her nerves.
DECEMBER 10, 2496 T.S.
PROXIMA CENTAURI
The Choyan fleet was ugly. Not in its shape, for most of the vessels were formed of elegant crescents and elongated lines that were pleasing to most sentients' senses of aesthetics, but in the threat implicit in its presence. Until now, the Choya had not joined their Salik allies in the fight. Ia knew it was because they had still been building up their war fleet while the Terrans had done their best to delay their war-machine efforts.
If they didn't openly assist the Salik, the Alliance couldn't accuse the Choya of colluding with the enemy. They hadn't been able to attack the Choya openly but had instead tried to wean the amphibians away from their brethren diplomatically, coupled with some covert sabotage efforts. Now, though, the gill-breathers felt strong enough to attack outright, with fresh soldiers and fresh ships being brought into the fight.
But they hadn't struck yet. Mysuri looked up from her post at the communications station. "Receiving pingback from their capital ship, Captain," she stated. "They are listening, though they're not responding."
"Put the broadcast on video," Ia directed.
"...Ready, sir," Mysuri told her.
Ia looked into her monitor. They weren't sending back a signal yet, but they were apparently listening. "This is Ship's Captain Ia of the Terran United Planets Space Force. I know why you are here, headed for sovereign Terran space, and I say to you: we have no quarrel with you, and do not wish to fight you. Change your minds before it is too late."
Silence. At the moment, the _Hellfire_ was the only ship in the system; the Battle Platform that had been serving as a space station for Proxima had finally been moved closer to Earth. With only a few chunks of lifeless, radiation-blasted rocks in orbit around the dull red star, and no ice to speak of for a fuel source in the system, there was no point in defending it. The Command Staff had left scanner buoys in place, sending a constant stream of data back to the Sol System, so they still had advance warning, but the war had forced them to limit how thin they could stretch their defenses.
The Choya had no interest in the local red sun. Proxima Centauri simply made for a convenient, preattack gathering point for a fleet of over forty ships. They soared forward without deviation, ignoring the _Hellfire_ 's placement off to one side of their chosen path.
"I repeat, change your minds and turn back, and we will not harm you," she stated, watching the fleet, greatly magnified, soaring through the system at somewhere near one-half Cee. "You have until you reach my position's midpoint to turn back or turn aside. If you continue on toward Earth, I will take that to mean you _are_ going to attack the Terran worlds, and I will strike to destroy you.
"'Turn back, Son of Cho,'" she added, quoting one of the older Choyan legends, one regarding the vengeance-spurred rage of a trio of ancient kings. "'Turn back, and let go the burdens of your anger, or your people will never reach the far shore.'"
That earned her a signal. Mysuri silently patched it through to her main screen. The Choya was a green-skinned male; from the darkness of his hide, he was quite old for one of his kind. Blinking at her through the link, he hissed, "We hhhave heard of you. Your psssychological trickss willl not ssway us. Nor, I thhhink, is your one little sshhip a threat to all of uss."
"You continue on this course, Admiral, and I will be forced to destroy you. Turn aside, or turn back," she repeated. "Or _you_ will not reach the far shore."
"You can fffire your weapon all you lllike," he scoffed, squinting his slit-pupiled eyes. "You cannnot harm allll of usss in one blow. Prepare yourrssellf to die. Not even your sship can withstand all of uss."
She didn't flinch. Didn't blink. "I invoke _ktham g'cho_."
He blinked, this time from startlement. Twisting his head, he studied his viewscreen from a slightly different angle. "You innnvoke the rightss of a worthhhy fffoe? What iss your offfer?"
"If you choose to continue toward Earth, I will not fire upon you while you are still within this system," Ia stated. "In return, you will not fire upon me. _Ktham guann?_ "
_"Ktham guann,"_ he agreed dismissively. "You arrre a worthhy foe. You willl lllive to ffface us at your Earthhh-world."
His image vanished. Mysuri shook her head. "We've lost pingback. They've ended the link, sir."
"So be it," Ia murmured. Quoting the legend, though the Choyan admiral knew it not. She activated her headset. _"All gunners, hold your fire. I repeat, hold your fire. We have a temporary treaty with this Choyan fleet. Do not fire."_
They waited in silence. The ships reached their midpoint and crossed in the blink of an eye. When the Humans didn't attack, the Choyans picked up speed. Gently turning the _Hellfire_ , Ia moved the ship into their wake, trailing after them.
"...Sir?" Mysuri asked. "Are we going to inform the others that they're coming?"
"Nope. Earth doesn't need to know."
" _Shakk_ that," Helstead countered. "I want to know if we're going to get lynched for letting 'em pass. Allowing a known enemy past the gate could be considered Dereliction of Duty at the very least, and Treason at the worst."
"Shh," Ia soothed, picking up their speed with a slide of her fingers along the controls. "I gave them my word I would not fire upon them while they were within the Proxima Centauri System."
"So what, a tail-chase, Captain?" Sangwan asked her. "We wait until they're past the heliopause, then open fire?"
She increased their speed again. "My plan is to hyperjump ahead of them to the edge of the Sol System, turn around, _then_ attack."
"We'll need to contact Earth first, sir," Ng stated. "We'll need their updated insystem traffic before we jump that far."
"No, we won't. We're not jumping that far."
"What?" Helstead asked, twisting in her seat at the spare gunnery post so she could look back at Ia.
"Sir, they're at eighty percent Cee and rising," Ng warned Ia. "They'll do it well before they leave the local heliopause, too. Once they cross over, we'll have no way of tracking them."
" _You_ won't. I will." Not waiting for them to argue the point, Ia activated the nose cone, swinging it into position. Lining up her vector by timestream-sight, she fired the engine and sucked them into hyperspace.
As the grey streaks of otherspace slipped around them, Helstead cleared her throat. "You do know I trust you, Captain?"
"And I appreciate that trust," Ia agreed. "Thank you."
"You're welcome. But I will admit I'm very nervous at the thought of turning around well before those ships will slip back down below the speed of light, and taking them on in a fight," Helstead continued. Since she had nothing to shoot at, she had pulled out two of her deadly little hairpins and spun them over her knuckles back and forth. "Especially since the only way you could guarantee line of sight is to position this ship dead ahead on their path."
"An astute observation, Lieutenant Commander. Remind me to requisition you a raise."
They emerged two minutes later. Slowing the ship, Ia brought it to a stop, then gently swapped ends. The dead stop was necessary; the helm controls had to be adjusted to give her only the tiniest of movements. Most of those were provided by the sort of maneuvering thrusters used for docking the ship. Ia didn't even look at her screen, just closed her eyes and nudged their position with careful twitches of her left hand. A brief, go-nowhere pulse of the FTL field killed their drift, then they sat.
From time to time, Ia twitched them just a little bit right or left, up or down, compensating for the galactic gravitational curve. Finally, she unlocked the control switch and thumbed the trigger for the main gun.
This time, she kept the tip of her thumb pressed against the button. Red streaked forward, flooding the bridge with its bloody glow. Everyone flinched, half-hiding their eyes behind arms and hands until the forward-pointing monitors mercifully blacked out the center of that beam. The ominous _hummm_ became an unnerving rumble, and the _whooshing_ of the heat engines started to shake the ship. Ia had counted on that, though.
Instead of a twentieth of a second, she kept the cannon firing for a full five seconds, then shut it off. The rumbling ceased, but the _whooshing_ did not. Putting that excess energy to use, Ia backed up the ship. She didn't bother to turn it around, just backed it up until it reached rifting speed, and activated the stern nose cone.
They sucked through hyperspace backwards for forty-five seconds and emerged somewhere short of Saturn's orbit. The ringed planet wasn't actually nearby at this point in its solar year, just a lot of stars. It was the closest they could get to interstitial space and still be inside a star system's reach.
The current operations tech, Private Sbrande, finally ventured a comment. "I, ah, had no idea that cannon could fire for that long...or use that much hydrofuel."
"How much?" Sangwan asked.
"Three full percent of our maximum capacity, sucked down in less than sixteen seconds," Sbrande stated as calmly as she could. Her voice trembled a little at the end.
Several soft obscenities and a whistle from Helstead met the engineering tech's words.
"Captain, what about the overshoot?" Helstead asked Ia.
"Nothing will be coming from that exact direction for almost a year," Ia stated. "The force of the beam will dissipate over the next eight light-months, until at most it will be an annoying flash on the scanners of passing ships."
"I'm almost afraid to ask how much damage it did," Mysuri stated wryly. "Especially if it takes eight light-months for that beam to stop being lethal."
"We've just obliterated all but seven of their ships," Ia told her. "I had to hold the beam on for that long to make sure no chunks too large for FTL would remain. What little makes it to Sol System will be slowed by the heliopause winds and the Oort cloud debris. Eyes to your boards, meioas, and thoughts on your tasks. Those seven ships will be coming into system in four more hours, as will a fleet of Salik. We need to turn back the Choya with another warning, if we can. But I'm not going to hold my breath."
"The hell with that," Sangwan snorted. "If I were that admiral, and I'd just lost most of my fleet, I'd get the hell out of this system the moment I spotted this ship."
"It'll be a touch-and-go fight with or without them. They might choose to stay and help make things worse," Ia said. "But we will prevail...and then we'll have a very brief six hours of Leave on Battle Platform _Freedom of the Stars II_...whereupon many of you will receive commendations for your efforts in my service."
"About muckin' time," Helstead snorted, shifting back in her seat to rest her feet on the edge of her console. "I've been putting in recommendations right and left since we fired our first shot on this ship."
JANUARY 5, 2497 T.S.
_CONFUCIUS_ STATION
DLC 718 TORPETTI SYSTEM
Lieutenant Commander Meyun Harper glared at his CO. He had to raise his voice as one of his engineers started up the cutter at the far end of the manufactory bay, adding to the cacophony of the grinders and laser welders being used. "I'm telling you, Captain, this would be considerably _easier_ if we had stopped at a _regular_ docking facility, or a _military_ one!"
"Everyone else within sixty light-years will be needing their supplies, Commander!" Ia returned just as loudly. The cutter shut off, but the rasp of the grinders and the scorchings of the welders kept going. "I'm telling you, you _can_ modify these supplies to fix our pipe fittings!"
"Well, I'm telling you that if I _do_ fix up these _shakking_ pieces of pipe, we're going to have to _weld_ them in place, and that means cutting them off and replacing _three_ lengths instead of one! _You're_ the one who said we only have seven hours in this place. That's not enough time to melt them down and extrude entirely new pipes of the right size and shape," Harper reminded her, lowering his voice as the grinder cut off.
Ia threw her hands up in the air. "Just melt and extrude some scrap into coupler rings! That's why I got you the extralarge pipes stored down on Deck 17!"
He stared at her, then covered his goggle-protected eyes with his hand. Some of the other noises cut off as well, giving him room to speak. "I can't believe I didn't think of that myself..."
She clasped his shoulder, giving it a gentle squeeze. "It's been a long, hard series of fights, Meyun. You have kept this ship together, and _put_ it back together, more times than I can count."
One set of doors hissed open off to the side. Lieutenant Rico poked his head inside, spotted the two of them, and plucked a set of goggles out of the plexi-fronted case by the door. He didn't bother to put them on, just held them over his eyes as he approached.
"Captain, your 12,379 _kesant_ are almost up," he stated blandly. "You have roughly one hour to compose that message you promised to send to the Greys."
It was her turn to blank out for a moment. Wincing, Ia nodded. "...Right, right. Too many battles on my brain. I almost forgot about opening up _that_ war front. Time for some damage control—Meyun, do you need me for anything else?"
"Yes. A pass for a week's Leave on a beach near Aloha City," he quipped dryly. The grinder started up again. He flipped a hand at her, raising his voice. "Go on! I have those couplings to fit!"
Nodding, she joined Rico. They removed their goggles and secured them at the door, then left the machine-filled manufactory bay. As soon as the thick door slid shut, cutting off the majority of the noise, Ia sighed in relief. She rubbed at the back of her neck, striding alongside her tall intelligence officer.
"Any chance we'll be getting that much Leave on Earth anytime soon?" Rico asked after several seconds had passed.
She blinked up at him in confusion, then recalled Meyun's quip. "No. Unfortunately. And even if we did, I'd only be able to spare an hour at most for beach-lounging, myself...never mind other activities."
He let that pass in silence until they reached her office. Ia blanked out the surveillance pickups as they entered, knowing what he was going to ask her next. Sure enough, he did. He wasn't the wary junior officer of a year ago. She knew he was on her side, but he still asked anyway.
"Sir, you said a year ago that the Admiral-General authorized these communications with the Greys," Rico stated. "But does she _really_ know about them? Or is this just a massive stretching of your _carte-blanche_ authority?"
"It's a massive stretching," Ia confessed quietly, taking her seat. She gestured for him to take one of the chairs across from hers. "Some would even call it an outright breaking. The probabilities are high that even after I fast-talked to her, Myang would call it Grand High Treason. But I give you my word, if I didn't foresee this deal brokering saving hundreds of millions of lives in the future, I wouldn't go anywhere near the Shredou.
"I just..." Frustrated that she couldn't tell him the full truth, Ia sighed and slumped in her seat, rubbing at her temples. "I need them to _stop_ when I tell them to stop, so that they don't destroy the Terran Empire and trample onward into the rest of Alliance space. And the only way to do that is to convince them that I am a massive precog. One so powerful, I can _see_ their movements with great accuracy, to the point where they...Hell, I don't know if they _can_ feel superstition as we Humans do, but I need them to stop their predations because of it."
"Superstition?" Rico asked, one brow lifting in skepticism.
"They're not from this galaxy. The energies are all wrong," Ia said, sitting up again. "They're trying to rebuild their race by melding it with local biology—Humans are the closest they can come to something both sentient and compatible, which is why they keep coming after us every few generations—but thanks to Feyori interbreeding efforts, our psychic abilities hurt them. It's a weapon they cannot defend against because it is truly alien to them. And the fact that I can wield this weapon against them by predicting their own movements as well as directing all the others' is going to scare them.
"More than that, if I can show them how accurate I am with _them_ , I can show them how accurate I'll be with their ancient enemy. If you'll recall," Ia reminded him, "I threatened to point the Zida"ya at them and pull the trigger if they don't comply. Proving to them that I _am_ that accurate will scare them shitless, in the end. Literally. When I prove my final point to them, the Greys involved in that fight won't be able to unpucker for three days."
"So you're hoping to scare them into compliance, bringing the coming war with them to an end when you _need_ it to end," he murmured, following her line of reasoning.
"Exactly." Tapping her workstation, she raised the screens, flipping two of the tertiaries at the bottom so that he could read the information from his side of the desk. Her primary scrolled up the messages as well, facing the normal way so that she could read it, too. "Here's the first set of messages I want to send. I _think_ I've composed them correctly, but there are some probability variables that suggest they might be taken the wrong way. The Terranglo version's on the left from your perspective, the Shredou on the right."
"I'll see what ambiguities I can fix," Rico promised, frowning at the text.
FEBRUARY 3, 2497 T.S.
KNOT 2,330,427
HELIX NEBULA
"Never-ending battle, never-ending battle," Helstead muttered in between pulsing the trigger for her chain of cannons. "Never-ending battle, never-ending battle. _Please_ tell me, sir, that we're going to have a bit of Leave soon? Real Leave, off-this-bloody-mucky-blasted-ship-style Leave?"
"Maybe, if you asked them very, very nicely," Ia muttered back, slipping their much slimmer ship between two Terran Starcarrier-Class capital ships, "the Salik and the Choya might stop trying to pick all these fights with us."
Proximity warnings beeped as a clutch of projectiles skimmed past their hull, swerving to avoid the _Hellfire_. They were Terran missiles, programmed to identify friend from foe and adjust course accordingly. Lasers couldn't do that, though, and two of them nearly seared the ship as they slid past. Nearly, but didn't.
Togama, manning the comms, whistled softly. "Wow, Captain, you are _certainly_ stretching the vocabulary of the comm tech for the _George Cairns_. I don't think that one's anatomically possible even for a jellyfish."
"Unlike the original George Cairns, we will not die of blood loss from a severed limb," Ia returned calmly, strafing the _Hellfire_ sideways in front of the TUPSF _Powahann_.
"...Ooh, even nastier," Togama quipped, touching his headset. "The _Powahann_ 's claiming you're completely off next year's Christmas card list, Captain."
"Really?" Helstead asked, perking up a little. "That bad?"
"Well, somehow I doubt 'Die in a Salik frying pan, you Shikoku Yama Flightschool reject; get the hell out of our path' qualifies us for fondly remembered relative status," he replied. The humor broke up some of the tension in the crew, though not the majority of it.
"Eyes on your boards, thoughts on your tasks," Ia gently admonished. "Just seven more minutes of close-quarters fighting should see the Salik threat contained." She flicked on the intercom. _"All gunners down the starboard flank, continue to fire on the enemy ships for two more minutes, then cease fire."_
The knots ejected by the shockwave shell from Helix's age-old supernova made for a rough transition at anything but sublight speeds. Few ships cared to traverse the barriers. Few ships were armored enough to survive the radiation found inside for long, either.
However, each cometary knot was roughly the size of the Sol System, which meant it made for an excellent hiding place for a rather large Salik base. With giant solar sails erected to capture the echoing, last radiations from the exploded star and provide both shelter and power for at least eight major stations, the Salik had parked a sizeable chunk of their shipyards in one of these knots, sucking up all that free energy.
This fight marked one of the few times Ia had agreed to the Admiral-General's request that she and her crew join a specific battle rather than dash off somewhere else. One more random ship in a joint fleet of over a hundred might not make a difference, but _her_ ship might, and she was striving hard to make sure it did. That meant being hyperaware of exactly where all those lasers and missiles and chunks of debris might fly at any given moment.
_"L-pod 53, cease fire in ten seconds,"_ Ia ordered. _"All starboard gunners, L-pod and P-pod, cease fire in one minute."_
Vector change slung them around in their seats as she swapped ends. Fightercraft scattered as they slipped past, their plethora of thrusters firing this way and that. Ia didn't even hear the proximity beeps anymore; it was only the claxons she cared about.
_"L-pod 53, good job,"_ she praised, as the private remotely manning that cannon excluded it from his firing commands. Rippling the thrusters shoved them back in their seats, allowing her to dart the ship toward one of the heavily damaged shipyard stations. _"Starboard gunners, thirty seconds to cease fire."_
"Station 5 midpoint in thirty seconds, sir," Nabouleh told her.
"Shouldn't we still be firing by that point?" Helstead asked.
"No, that would be bad," Ia murmured, shifting them to avoid incoming fire from the half of the shipyard station that wasn't crumpled and on fire. _"Starboard gunners, cease fire in ten...nine...eight..."_
Missiles swerved in from behind, arcing around to strike at the heart of the damage. Blossom missiles, they impacted in puffs of light, then burst a second time like fireworks going off.
_"...Two...one. Cease fire,"_ Ia ordered. _"Cease fire. CEASE FIRE!"_ she yelled as the timestreams surged up and yanked her down. She could see nothing but the explosions of those blossom bombs ripping apart the shipyard, hear nothing but the click of a trigger being squeezed in ferocious glee...feel nothing but that pulse of light from Starstrike L-Pod 4 burning through the protective nose cone of the missile emerging from the depths of the TUPSF _Hardberger_ 's P-pod 29. That nothingness emerging in a too-late scream. _"SUNG, CEASE FIRE!_ "
Too late. Too late...Too. Late.
The timestream overflowed as it swallowed her down, drowning her. Freezing her with the inability to stop the inevitable. Nothing stopped that bright red beam of light. Not her order, not her wishes, not the chunks of shipyard forced apart by the force of all those explosions.
Water vanished. Water vanished from one lifetime, from a handful, from a hundred and more. The Redeemer's life dried up and disappeared. The Savior's course flowed on unaltered. The desert claimed all.
All.
Someone was crying in ragged gasps. Shaking with shock, skin flushed in fire, muscles prickled with ice, Ia stared at her right secondary screen, focused on the cloud of debris. A red circle and line flashed on the screen amid the chaos of battle, along with a simple, death-knelled message:
_Fatality 13_
Fatality.
How apt.
"Sir," Nabouleh stated, twisting to look back at Ia. "We're deadheaded for Station 6. We need to move. Sir? _Sir!_ "
She couldn't breathe. Claxons wailed in her ears. Ice and fire seared her nerves. Drowned under the waters removed from those vital, vital streams, she could not breathe.
"SIR!— _Shova v'shakk_ ," the yeoman cursed, and whipped back to face her console. "Hotel November, override, override!" she snapped, using her emergency call sign to identify her actions for the bridge's black box. "Taking the helm!"
The _Hellfire_ slipped sideways under her hasty grab, bruising them all against their seats and restraints. Nabouleh added an abrupt downward shift as well. The collision claxons blared. The maneuver yanked them up in their harnesses and slammed them back in place as the interior safety fields pulsed. Seconds later, the shields compressed, rumbling with a strange sort of hiss.
"Good job!" Helstead gasped as they slid past. " _Good_ job, Yeoman!"
"Captain, we're getting a query on a Fatality Thirteen: Friendly Fire," Togama called out, looking back at her. "What do I reply, sir?...Sir?"
_Fatality._
Her moan shifted as her shock morphed into rage. _Fatality...FATALITY!_ It emerged in a wordless scream. Straps broke as she lunged out of her seat. Behind her, she could hear her second officer's voice. It sounded tinny against the blood throbbing through her head.
"God—Nabouleh, get us out of combat, _now_!" Helstead snapped, jabbing at her harness clasps. "Togama, tell them we have an emergency on board and nothing more!"
Vision red with rage, Ia didn't bother to reach the door before she opened it. Her mind stabbed at the controls, sparking electricity through the system. Squeezing through before the panel finished opening, she sprinted up the hall. L-pod 4 was located on the bow, but it was controlled by L-pod 20, and that was one sector forward and two decks up. Doors hissed open, their panels sparking with electrokinetic energy.
Everything was energy. The red of her fury had altered her view. Doors and bulkheads, floors and ceilings, everything glowed. Everything pulsed. It was all just matter, but it was also oddly see-through. As if she could, if she tried just a little harder, reach out and reach right through it all.
She knew she couldn't force open both doors of the sector seal. They were pressure-locked against being able to do that, to prevent both negligence and stupidity. Just as she crossed the first of the two thresholds, something struck her from behind. She staggered forward, throwing up her hands to shield herself from hitting the forward door.
The blow was a body. Arms and legs wrapped around Ia's frame, heels hooking around her waist, bicep and military-issued bracer digging into her throat. Still enraged, Ia reflexively tightened her neck, staving off the pressure which her 3rd Platoon officer tried to apply.
As a force of body, Helstead's efforts were negligible; Ia was too angry to notice her efforts as more than the wings of a butterfly beating on her back and throat. As a force of will, however, Helstead's mind slammed down on hers like a sledgehammer.
( _Stand down, soldier!_ ) she snarled, hooking her right arm around the wrist of the left to apply more leverage against Ia's windpipe. ( _I said_ **STAND DOWN** _!_ )
The command exploded in her head, snuffing out half the fire and fury. Ia collapsed to one knee. Helstead squeezed again.
( _Stand down!_ ) she commanded. ( _You will NOT attack Sung! Stay down! STAY. DOWN._ )
The word-thoughts struck her in another blow. Ia's leg slipped out from under her. She had never faced the force of a psychodominant before, let alone one of Helstead's high rank. Dazed, struggling to breathe, she groped for her rage-scattered wits.
( _I don't care_ what _he's done, you will_ not _kill him!_ ) Helstead growled, squeezing her arm for emphasis. Ia choked and she eased up slightly—then squeezed in again. ( _You_ will _keep him alive!_ )
( _Alright!_ ) she snarled back, capitulating to the sheer weight of the lieutenant commander's demands. ( _Alright, he'll live! For_ now.)
Surging to her feet as the forward door slid open, Ia strode down the hall. It wasn't difficult to move with the shorter woman on her back; stocky as she was, Helstead didn't weigh nearly as much as Ia's exercise weight suit. She did stagger, though, when the ship rocked around them, attacked by enemy missiles. At least the movement forced Helstead to shift her left arm, clutching more now at Ia's shoulders than compressing her muscular throat.
The rage was coming back. Swift strides turned into jumps as she ascended the stairs, not bothering with the lift. More doors hissed open, clearing the way. Clinging to her CO, Helstead continued that low, steady, mental hiss, ( _...You will not kill him...you need him alive...you will not kill him...you need him alive!_ )
An outward snap of her hand hissed open the L-pod's door. A slash snapped the restraint straps. With a startled yelp, Private Goré Sung tumbled through the door and swayed to a halt. The only thing preventing Ia from slamming him bodily into the far bulkhead was that damnable, insistent, nagging whisper named Helstead.
Jerking him closer with a clench of her fist, Ia stopped him telekinetically, halting him centimeters from her face. "You _shova v'shakk-tor_!" she snarled as he stared back at her with brown eyes so wide, she could see the full ring of their whites. "You have _slaughtered_ this galaxy!"
Grabbing him physically by the throat, she dragged him—both of them—into the desert. Forced him into life-stream after life-stream at the galaxy's end. Forced him to watch worlds being devoured and stars torn apart.
( _You need him alive!_ ) Helstead yelled deep in her head.
( _No._ I do not. _Not anymore._ ) She flung both of them out. Not gentle. Not kind. Sung gasped for air, choking like a drowning man, though Ia wasn't physically squeezing his throat. Helstead clung with trembling limbs.
( _You...you can repair..._ ) she gasped.
_"HOW can I repair a DEAD MAN?"_ Ia screamed with mind and voice. Sung winced. She didn't see what Helstead did, her attention reserved for the careless murderer in front of her. Ia flung him up with her mind and her hand, slamming him into the ceiling, provoking a pained grunt.
Helstead dropped free. She landed on hands and knees behind Ia, panting from the force of that mental counterblow. "I...I don't believe...in the...the no-win scenario, _Captain_ ," she growled between breaths. Her hand gripped Ia's ankle, reinforcing her words mentally as well as physically. "And I know _you_ don't! You. Will. _Drop_ him!"
Sung dropped. He _thudded_ onto the deckplates with a groan and a faint _crack_ , one hand caught awkwardly under his ribs. With a feral snarl, Ia flung the force she had been about to use on him into the bulkhead to her left. It _crunched_ , denting inward by at least a third of a meter. Her hand snapped out again, and the dazed private was yanked up, body floating horizontally in her grip.
Coughing, he squinted at her. Ia leaned in close, but did not touch him. She let the madness in her gaze, the rage barely leashed in her words, do all of the damage.
" _Pray_ I can find a way to fix the _dead man_ you've destroyed. _Pray_ I can find a way to replace his life! Because of _one_ man's death, unless I can fix it, _you_ have doomed this entire galaxy to a fate worse than a Salik's favorite lunch. Pray I _can_ find a way," she snarled, bringing his nose to within a centimeter of her own. "Because if I cannot... _pray I kill you before I am through_!"
Flinging him away, she let him tumble down the corridor. He grunted and yelled with each impact before skidding to a stop. For a moment, he tried to get up, then groaned and sagged to the deck.
She hadn't killed him. By luck, and the grace of God, or at least by the demands of Delia Helstead, Ia _hadn't_ killed him. But it was a near thing. As it was, Ia could not see what to do with him. The timestreams were nothing but barren, bleak desert around her, empty fields with nothing left but cracked and lifeless dust.
Behind her, she heard the other woman pushing to her feet. "...Orders, sir?"
Oddly enough, Helstead's simple question centered her. Ia still couldn't See a damned thing, but her training as an officer kicked in. There were rules and regulations for this sort of thing. Rules and regs that had to be followed. Hands fisted against the urge to physically express the rage morphing back into grief, Ia swallowed.
"Take the prisoner to the Infirmary. He has a cracked wrist. When it has been set, lock him in the brig. The prisoner is _not_ allowed to speak to anyone about anything other than his wrist," she added tightly, glaring at Sung as he started to stir again. "Unless and until I can figure out what to do with this _nightmare_ he's caused, this ship stays on lockdown, category Ultra Classified. No messages out but for the fact that we're on lockdown."
"That won't cut it with the rest of the fleet, Captain," Helstead told her. "They _know_ about the Friendly Fire. They _will_ be expecting an acknowledgment and an arraignment."
Ia cursed under her breath. Her head hurt, throbbing with the ache of her unsatisfied rage. Scrubbing at her scalp to try and ease it enough to let her think, she dislodged her headset. Impatiently, she stripped it completely off, glared at it, then wrestled the thin curve of plexi back over her head. _"Captain Ia to Private Togama."_
_"Togama here, Captain. Uh...is everything alright?"_
_"No. Acknowledge and register the Friendly Fire with the TUPSF_ Hardberger. _The accused is Private Second Class Goré Sung. Inform them we will be contacting them with the details of his arraignment and tribunal at a point in the near future, then broadband cast to the fleet that we are experiencing technical difficulties of an Ultra-Classified nature and will be disengaging from combat and remaining silent while those difficulties are addressed."_
Wincing, she tried to focus her thoughts on the timeplains. The wafting dust of the empty desert clung to her feet like cement. Head throbbing, she forced herself all the way back to the present, to survey the wreckage of the Now with a dispassionate eye.
_"Addendum, inform the TUPSF_ La Granger, Sword's Breath, _and_ Saibo-Maru _to stay out of the sunward side of Station 3'swreckage. Tell the TUPSF_ Dian Wei _to break off in two minutes and spend a full minute coming about before reentering the fight. And tell Admiral P'thenn aboard Battle Platform_ Hum-Vee _that there are sabotage systems still active on Stations 1, 3, 5 and 8. Boarding parties must use extreme caution. Repeat_ _that we are going to be running silent under an Ultra-Classified-Situation flag, then do just that."_
_"Ah...Aye, sir. I got all that,"_ Togama replied. _"Sir, what is the situation?"_
"That _is on a need-to-know basis, and you do not need to know."_ Pulling off the headset, she let the band curl up and stuffed it into her pocket.
Helstead had edged around her in order to approach the injured private. Pulling him to his feet, she wrapped her arm around his ribs. "With permission, Captain, I'll take him to the Infirmary."
Ia dipped her head, giving it. That did require the two of them to shuffle past her. She moved into the alcove of the L-pod's door, but only just enough to let them by. As he came within arm's reach, Ia held his gaze.
"Start. _Praying._ Private," she warned him, clipping off each word like a bite. Like the snap of a spark in an overheated fire. Like an ice-cold funeral pyre.
He swallowed and looked away.
Only when he was gone from her sight did the fire and the fury still seething within her finally die. Without it, ice-cold fear washed through her veins, prickling her skin with gut-deep dread. The horror robbed the strength from her flesh. Sagging to the deck, Ia doubled over. She struggled against the nausea, but nothing helped. It just built and built until she doubled over and retched.
Not much came out, the smallest of blessings. It had been too long since she last ate, too long since she entered combat. Too long since she had believed she could win against Time and Fate. She heaved again and again for a full minute or more, then sagged back on her heels.
The ship swayed gently around her, bumping her shoulder into the wall. Drained, numb but for the aching, chilling pain, Ia pushed slowly back to her feet. She didn't know where to go or what to do, only that she couldn't stay there. Couldn't stop that arrowing laser, couldn't stop that emerging missile, couldn't stop the gaping hole in the aging _Hardberger_ 's hull, its old-fashioned turret and its precious, progenitive cargo now utterly destroyed.
Over and over, those last few seconds replayed in her mind. One hand braced on the wall, she shuffled forward, turning occasionally, moving sightlessly. The _Hellfire_ swayed again, then steadied. Noises echoed down the corridor. They belonged to Togama, issuing orders no doubt relayed either from Nabouleh, who had the helm, or Helstead, who was in charge of the watch. They made no sense to her, being just a babble of noise.
Nothing made sense anymore. Nothing _meant_ anything anymore. It was all gone, drained away with the loss of a single, precious, supposed to be anonymous life. But that life was gone.
Private First Grade Joseph N'ablo N'Keth. Gone. TUPSF-Navy five years, and a competent gunner. Gone. Retired from active service in five more years. Gone. Settled on his homeworld of Eiaven. Gone. Great-plus-grandfather of the Redeemer, whose life had been meant to redirect the Savior's, so that she would be in the right mind as well as the right place...
Gone.
All gone.
A long time ago, when she had been just seven or so, her mothers had shown her a beautiful, fragile, heirloom teacup. They had explained its history, how the delicate, rose-sculpted porcelain had come all the way from Earth via three other worlds. Amelia had urged her to take it in her hands, to hold for herself this relic from her family's past.
The young Iantha had tucked her hands firmly behind her back, shaking her head. All she could tell them was, "But I can't put it back together, Mothers. I just can't!" At the time, her young self couldn't explain why all she could see was the teacup shattered on the floor, scattered into pieces despite the beige plexcrete cushioning their feet.
In all the years since, Ia had not once touched that teacup. She didn't even know where it was now. Didn't know if it had survived the destruction of her mothers' restaurant, survived the move to their new, underground life, or survived...anything, really.
In her mind, as both a child and an adult, the teacup was and had always been forever gone. Shattered. Broken. Wrong.
"...Captain?"
With effort, she looked up and focused. Somehow, she had gone from the starboard of the ship to the port in her blinded movements. The last person she needed to see stood before her, concern wrinkling the brow above his worried blue eyes. Finnimore Hollick.
He wasn't alone; his teammate Schwadel had halted at his side, both men apparently released from their battle posts. Vaguely, she realized she was close to their quarters, but all she could see was Hollick's middle-aged face, and a badly, badly broken teacup at his feet.
"Captain, are you alright?" Hollick asked her gently.
She opened her mouth. Nothing came out for a long, long moment. Teacup. Shattered. The desert that was left. All she could feel was another icy-hot rush, this time of shame as well as panic.
"...I...I'm sorry."
The words escaped. Schwadel blinked and frowned, but she didn't look at him. It was Hollick she couldn't look away from. Hollick, who was a fellow Free World Colonist at heart. Hollick, who _knew_ with unswerving faith that his Prophet...that his Prophet would...
"I'm sorry...I'm so, _so sorry_ ," she whispered, vision blurring. "I didn't mean...I didn't...I _tried_ ," she begged him to understand. Shaking her head, she said it again, tears now trickling down her face. "I _tried_! I tried so _hard_...I...I f...I ffff..."
She couldn't even say the word, but it was there, screaming in the back of her head. _Failed...Failed!_ She broke with the weight of her shame, head burrowed into her hands to hide herself from his sorrowful gaze. Vaguely, she heard the other man mutter something; the derogatory tone in his voice only confirmed her failure. Hollick snapped something back, then gingerly touched her arms.
"Easy, Captain," he murmured. "This way...this way...let's get you off the deck..."
Shuddering, Ia went. She couldn't see, couldn't breathe through her nose, couldn't stop the tears from seeping free. Couldn't stop whispering the words over and over, _I'm sorry, I tried, I fff...I fffff...I'm so so sorry..._
Shock robbed her of her gifts. Shock, and the destruction of everything she had strived for over the last ten years of her life. Vaguely aware of being urged to sit, she felt him wrap his arms around her, holding her close as no one else on board dared. Holding her for far longer than even her own mothers could, rocking her gently as she cried in her grief.
It took her a while to realize he was humming a tune. It was a soothing one, and yet somehow sad. His voice wasn't much like her grandfather's, not quite as ragged with age, but it hummed the same melody, over and over. A song about a flower, and the impermanence of life.
A song, she realized, which she herself had transformed years ago into an anthem for her family, for her people back home. _My life I give this day to serve all others, though Hell itself should bar my way..._
Fresh tears seeped free at that. She squeezed her eyes shut, to block out the pain, but the words kept echoing in her head. Echoing across the barren, sterile desert that filled the back of her mind. _My life I give this day to serve all others, though Hell itself should bar my way...My life I give this day to serve all others...My life I give this day..._
Another voice echoed across the emptiness of the timeplains. _Did you know I once saved the life of Jesse James? Jesse James Mankiller that is...Once saved the life..._
_My life I give this day..._
Ia stilled.
_His problem was that he didn't lay nearly enough contingencies...Me? I have several identities I could assume...Several identities I could assume...My life I give..._
The teacup had been shattered. No rose-sculpted porcelain lay intact at her feet, no grass, no water, no life...save for the slender, spear-like tip of a tulip leaf peeking up through the shards.
_My life I give this day to serve all others..._
She sniffed, trying to clear her nose, and asked, "...Do you?"
Easing the strength of the arms holding her to his chest, Hollick let her sit up. "...Sir?"
"Do you?" she repeated, sniffing harder. Ia gripped the front of his grey uniform shirt. " _Do_ you give your life?"
It was a desperate gamble, but she knew things about the Mankiller bloodline, things she had once peeked at in curiosity. Jesse, or rather Jessica James, who married David Mankiller, had been crushed from the waist down in a ground-car accident...and yet somehow, less than twenty-four hours later, had walked out of the hospital whole and alive. _I once saved the life of Jesse James..._
Hollick covered her fingers with his hand. "I have believed in you since my first Fire Girl attack, Prophet. Even in gravity-based exile on Gateway Station, I had heard of you, and believed in you. If there is _anything_ I can do to help you fix whatever went wrong, I will do it. As my Prophet wills it, my life—my everything, whatever you need—is yours."
Sniffing hard, she nodded. She nodded and sat up. It was a long shot, far longer than the shot she had fired on that first Choya invasion fleet. She still couldn't see any life beyond that single slip of a flower leaf, but it _was_ there, on the timeplains. There was hope.
Focusing on him, she nodded again, more slowly and soberly this time. "I'm afraid it has come to that. Finnimore Hollick, I call upon you to give up your life...and take up the life of the man we just lost. You will lose _everything_ that is yours. I will...I will have your body altered, and imprint his personality over your own, and...and fix it with subconscious impulses to follow every single step the real one would've taken and known. And...and we will _fix_ this broken...this broken teacup...and _fool_ the whole God-be-damned universe."
Lifting her free hand, she swiped away the tears that were falling again. Nodding a third time, Ia pushed herself to her feet, trying to think. She could feel the shards of porcelain cutting into her tender, young feet every time she tried to move out of the desert that had swallowed the grassy plains.
"I need...I need to go call her, and...God...I don't know what she's going to want for this. I can't _see_ anything at the moment—but if _you_ can have blind faith in me," she added, turning to face the man on the couch, "then _I_ can have that faith."
Sober, somber, Hollick pushed to his feet. " _Ia'nn sud-dha_ , my Captain." He hesitated, then asked, "What will become of my own life if I am to take this other man's place? I mean, how will you explain my absence?"
Ia shook her head. "I don't know. I...I guess I'll have to convince her to take on _your_ form and place. At least for a few days, maybe a week or two. There are...there _were_ fights up ahead where it would be possible for you—the old you—to die, freeing her to go about her business. You, the new you, will have already gone off on your way.
"Things will have changed," she dismissed, shaking her head. "Sung... _shakk_...He'll have to undergo a tribunal, and...punishment. I'll have to schedule time for that."
Ia closed her eyes, once again feeling ice-cold and sick with dread. Caning. Not just for Private Sung, but for herself as well. _Unrestrained, and without hesitation...That won't be pleasant. But if it works...if we_ can _fool Time and Fate..._
_God. If this works...I will bear every single lash without complaint._
Opening her eyes, she nodded. "...Right. I have work to do. You are _not_ allowed to discuss any of this with your roommate, with your superiors, with your family or your friends. You are to remain silent on this entire situation, and you will pretend that everything is normal...or as normal as can be," Ia allowed, mindful that she had just spent untold minutes grieving in his quarters, "until I call for you. Is that clear, Private?"
"Sir, yes, sir," he agreed, squaring his shoulders and leveling his chin. "Clearer than crysium, sir."
Ia raked her fingers through her hair. The bangs were getting a little long again. She'd have to visit Private Antenelli of the 3rd Platoon, D Epsilon for another trim. The private had been a hairstylist before choosing to enlist.
Thinking about getting her hair cut, an utter triviality, felt absurd. Not since she was fifteen had she felt this off-balance and disconnected from reality. Shaking it off, she gave Hollick a nod.
"Considering I'm off to go make a bargain with a devil while holding the dirty end of the stick, that's an apt choice, mentioning the Devil's Sticks—thank you for watching over me just now," she added, and held out her hand. "After the change, you will never remember, never know who and what Finnimore Hollick was...and no one else outside this ship can know where you've gone...but I will make _sure_ Private Hollick is listed as one of the heroes of the Damned, so that no one else will forget that much of it."
He clasped her hand, and gave her a lopsided smile. "Considering I'll be the first on the crew list to receive a Black Heart, it's a dubious honor. But I accepted long ago that it might just come to that."
She met his gaze steadily, squeezing back. "So did I, soldier. I'm sorry to have to say this, but...thank you."
Releasing his hand, she turned toward the door. He chuckled softly just as she reached it. Turning, Ia glanced back at him. Hollick shrugged and spread his hands.
"It just occurred to me that you've guaranteed me a long life, even though _I_ won't remember it. Um...I _do_ get a long life, right?" he added.
Ia gave him a lopsided smile. "You'll die somewhere in your late eighties, survived by a beloved wife, five kids, thirteen grandkids, two great-grands, and more on the way. And the unsung legacy of having a great-plus-grandson become the Redeemer."
His eyes widened. "The Redeemer? The one who saves the Savior from herself? I..." He stopped, licked his lips, and tried again. "I...I think I can live with that. Stars—is _that_ who we lost?"
Quickly lifting her finger to her lips, Ia hushed him. "Shh. Say nothing more. This shattering will never have happened if we are very, very lucky in our attempt to trick Fate."
Pantomiming the zipping of his lips, Hollick tucked both hands behind his back and nodded solemnly, watching her leave his cabin in a modified Parade Rest.
_Finnimore Hollick,_ she thought. _Patron saint of the ultimate sacrifice. A holier soul than mine, for I at least_ know _why I had to give everything up._ Shaking it off, she oriented herself outside his quarters and hurried for the section seal. It occurred to her just as she reached the door to her office that she still had more damage control to do.
Detouring all the way to the bridge, she hooked her fingers into the controls, opening the door far more gently than when she had left. Helstead wasn't inside, but Spyder was. He glanced her way and straightened in his seat at the backup gunnery position. Hers, the command station, remained empty.
"Cap'n," he greeted, giving her brief, watchful nod. "What's th' situation?"
"I was about to ask you that," she replied, moving across the cabin and mounting the dais at the back of the room. "Show me the battlefield. Give me a tactical analysis."
"S'all over, Cap'n," he told her, tapping the controls. "All but th' screamin'. Th' Salik are dead or fled, th' Choya what went with 'em, an' th' _Hardberger_ 's been threatenin' t' call down th' Admiral-General on our heads."
Ia studied the screen, reading the flags and accompanying texts of the attached analysis. More information was still coming in as each of the ships in the attack fleet continued to exchange and sort through information. What the _Hellfire_ was receiving of that was passive only, incoming from all channels and functioning sensor arrays with nothing outgoing.
"The _Hum-Vee_ wants our records an' analysis," Spyder told her in an undertone. "Admiral P'thenn's last message sounds like he's gotten a bit testy, but then it's been an hour an' a half."
She'd cried for that long? Her time sense was that shattered? Ia blinked. She absorbed the information for a moment, then dismissed it. Her plans to kite the _Hellfire_ out of here and go off to another location for repairs and the next fight were gone, blown up along with Joseph N'Keth. Now all she could do was pray for a strong enough roll of duct tape, and a clever enough touch with a _trompe l'oeil_ paintbrush.
"Maintain comm silence, other than to repeat that we are handling an Ultra-Classified Situation on board. Where's the _Hardberger_ now?" Ia asked, wondering what sort of remains might have survived the explosion that wiped out its P-pod 29 turret.
"Cozied up t' th' _Hum-Vee_ ," he said, pointing at the blip on the main screen in the distance. "We're jes' wanderin' aimlessly 'round the more stable bits o' the local space. Solar sail's shot t' hell, but there's no flare or ion storm at th' moment, so no need t' hide behind it."
As he said it, she could sense the near future of the cometary knot. Nothing was destined to wash through for another eighteen hours, though in nineteen, the more damaged ships would do well to hide behind that sail. "Right. Here's what I can tell you."
That caught the attention of her bridge crew. Nabouleh, Wildheart at navigation, Togama, Yé at ops, Aquinar, and Spyder all stared at her. Ia nodded.
"Private Sung did indeed cause the incident of Friendly Fire. And he will pay for his damage to the _Hardberger_ 's hull. But...he did _not_ kill the gunner who was manning their turret."
"'E whot?" Spyder asked her, blinking.
"I will tell this _entire_ crew what happened at a special boardroom meeting," Ia promised them. "But for now, you are to maintain communications silence, and you are not to speak of it outside this bridge. Private Yé, I will probably need a great deal of energy routed somewhere on the ship. Probably to my quarters. I don't know yet. How full are the tanks?"
"Ah...we're at fifty-six percent capacity, sir," she stated, glancing at the data on her screens. Ia grimaced, then shook it off.
"I just hope that'll be enough. Everything must be self-contained until our little Ultra-Classified Situation has been fixed." Moving over to her normal station, Ia leaned over the console, pushed up the hatch, and pressed her fingers against the hidden electrodes. Pressed, held, and absorbed electricity until her hair crackled and rose off the nape of her neck. Eyes wide, Spyder leaned back from her, though he sat a good three meters away. She gave him a slight, lopsided smile. "Relax. You're not my target."
"Ah, beggin' pardon, but...th' lieutenant commander said you weren't allowed t' kill anybody," he reminded her.
Her mouth twisted in rueful bitterness. "I know. If the Admiral-General calls, inform her that we still have an Ultra-Classified Situation to contain, and that it is too dangerous a situation to explain over the comms. Reassure her that I _will_ explain in due time, then end transmission."
Togama cleared his throat. "Right. Tell her to mind her own business, then hang up on the Admiral-General herself. I always wondered what it would feel like to be caned..."
"I suggest you _not_ joke about that in my presence," Ia stated flatly, flinching inside at his careless, unknowing words. "Not today, and not for the rest of this week. As you were, meioas."
Nodding to Spyder, she retreated through the back door to her office, and from there, to her private quarters.
# CHAPTER 14
_There was no greater hell in my life up to that point than the moment I realized my teacup had broken and that it lay shattered at my feet. No greater release into purgatory than to realize I had one chance at duct-taping it back together. One shot at using a_ trompe l'oeil _trick to fool the universe into thinking the cup was still firmly intact._
_I can't tell you what I did, and I won't tell you what I did. Not ever. Explain a stage magician's trick, and all the magic of it, all the wonder and the awe and the innocence of one's ignorance are thus forever lost. In fact, it can never again be regained; the illusion is spoiled, for the wires will always be on the mind. But I did it. And I paid the price for it. I paid for every drop spilled from that shattered, rebuilt teacup._
_~Ia_
Summoning her faction protector was not too terribly difficult. Ia had already practiced the mental twist of energies that opened up the tiny thread of a cosmic string permanently linking Belini to the corner wall of her living room. The one thing it did take was power, which was why she had stocked up at the command console.
It took a while for Belini to respond. As she waited, Ia probed at the timestreams, trying to shift herself away from that endgame desolation. The interior of her head felt broken and bruised. She did manage to shift her viewpoint back to where she should be, in the here and now...but every time she pulled out, then flipped back in, the desert at the end of the game was the first thing she could see. Not the waters of her own life and not the grassy banks of a thriving prairie.
The flash of light against her closed lids and the slight shift in air pressure that teased across her face warned her that the Feyori had arrived. Opening her eyes, Ia watched the silvery-dark sphere dip partway into the wall. That dimmed the overhead lights for several seconds, until the overgrown bubble lightened from deep grey hematite to an almost platinum shade. A flash of light popped the soap bubble, depositing Belini on the carpeted deck.
Bare toes digging into the light grey pile, she shifted her hands to her pink-clad hips, once again looking like a demented, wingless pixie. "Well. I certainly didn't expect _you_ to call." She paused, eyed Ia, then shook her head. "Almost made it, didn't you? Like I told you, I'm not going to help."
"What?" Ia frowned for a moment in confusion, then shook it off. "I don't know what you're talking about. I called you because I need a very big favor. One that is in _your_ best interests...because if you don't help me, to my exact needs, the Game ends. In fact, the Game _has_ ended."
That made the Feyori frown. "What do you mean, the Game has ended?"
Ia sighed and explained. "One of my stupidest gunners refused to stop firing when I ordered him to. His weapon impacted on a fellow Terran ship, and destroyed one of the projectile turrets...and with it, destroyed its gunner. That gunner was to have been the great-plus-grandfather of one of the key figures I needed to have in place to guide the Savior into preventing the destruction of this galaxy. The destruction that would have put an end to everything your race is currently doing with the matter-based species.
"That destruction _will_ put an end to everything...unless you and I can fool the universe into thinking that that gunner is still alive and still available to take his rightful place."
Belini wrinkled her nose. "No," she stated flatly. "Absolutely not. I have my _own_ places to be—"
"Not _you_ ," Ia dismissed, rising from her couch. "Private Finnimore Hollick has volunteered to be the body and soul to be sculpted into the missing gunner's place. I _do_ need you to stick around long enough to pretend to be Hollick in his place, but there should be a chance to kill him off in a bodiless way in about a week if we do everything right. At that point, you can pop off to wherever, and the broken bits of the universe will have been duct-taped back together."
She considered Ia's words, her eyes aquamarine, not quite silver. "What about this Hollick fellow? What about _his_ rightful place in the universe?"
Ia shook her head, raking one hand through her hair. Again, her bangs were irritating her, a stupid little bother in the face of this disaster.
"I only chose him to be a crew member because of three things. He has the right skills and instincts to do his job well. His presence or absence in any other part of this war will not have made a damn difference one way or another. And because a part of me knew there was something he could do that would help my cause. I thought it was just be a steadying, faith-filled influence among my crew, but...
"This is probably the most extreme thing I could have asked of him, aside from maybe asking him to pull out his own intestines with a rusted spoon," she quipped sarcastically. "But I did ask, and he did agree to it. And don't tell me you can't do it. I _know_ you can.
"You said yourself you saved Jesse Mankiller's life, and that you can take on any shape you like. I know the Feyori calling himself Doctor Silverstone can read thoughts and reshape his own body to copy the life and memories of a man whose hovercar crashed in the Australian bush. And with _my_ help," she stated, "plugging the two of you directly into the missing gunner's original life-stream, we can guide him into having the right memories and making all the right choices the original would've made. A perfect _trompe l'oeil_ replacement. Or at least one hopefully good enough to fool Time itself."
Belini considered her words. Drawing in a breath, she asked shrewdly, "And how will you explain how this gunner survived?"
Ia spread her hands. "Lieutenant Commander Helstead is a teleporter. She sensed the danger he was in, and teleported him blindly onto this ship." Her own words made her pause; Ia realized with another sick flush of ice and heat that such a thing would be a violation of the Admiral-General's command to permit no one else aboard...Feyori notwithstanding. Swallowing, she added, "That's why we've been locked down under the claim of an Ultra-Classified Situation. The teleport stunned him psychically. I then ordered him to be kept sedated while I figured out where he came from and what to do with him."
Nodding slowly, Belini accepted that line of reasoning. "That might actually work."
"It has to," Ia murmured. "I can't see any other options."
There _was_ another option, but Ia knew it would involve the deaths of a good three or four Feyori. That was not something any of them were prepared to do. Not at this point in the Game, not when Ia herself was still a mere pawn and not a powerful fellow player.
"So," Belini muttered, ticking off the options on her pixie-slender fingers. "We have a willing body and soul to take the gunner's place. We have myself to take this Hollick fellow's place for about a week, until I can safely pretend to die and head off on my own business. And you've covered how the gunner gets on board. Do you at least have bits of this missing gunner's body on hand, so I can get a direct reading of his genetics?"
Ia opened her mouth, then closed it. Tightening her jaw, she pushed past the desert now occupying the back of her mind, forced herself to the present patch in the timestreams, and rooted around in the very recent past. Finally, she nodded. "I don't have the whole body available, but there is a surviving bit of it tumbling through space. It's badly burned and frozen, but it should still be enough for you to read his DNA and rebuild Hollick in his shape."
Belini held out her hand. "Show me."
Nodding, Ia gripped it and complied. When she was sure the Feyori knew exactly where to look, she released the other woman. "While you go do that, I'll fetch Hollick up here."
Belini rolled her eyes. "If he's going to be sedated, he'll have to be stashed in the Infirmary, now won't he? Come on, _think_ , woman. That's what that blob of grey stuff in your skull is supposed to be good for, with you fleshies."
"Well, if you'll excuse me, I just had my entire reason for living _smashed_ at my feet," Ia retorted, hands going to her hips. "And for some God-be-damned reason, I cannot approach the timestreams from my usual spot in the present but am instead stuck with the pain of arriving in the midst of the desolation caused by the Zida"ya fleet! I think, given all the _shakk_ I've just gone through, that I am holding it together fairly well in spite of all that!"
The look the Meddler gave her was a cool, assessing one. Finally, Belini nodded. "Right, then. Keep holding it together. This will take a lot out of both of us. Go make yourself useful by hauling several power cables to the Infirmary. This isn't reshaping myself, and it isn't restoring a woman's rightful body from a broken to a whole state. And you had _better_ be right about being able to pattern his mind and his life-choices, or all this effort will go to waste. All this _energy_ will go to waste.
"I'm a Feyori, child," Belini reminded Ia. "I don't like to waste my food." Popping with a flare of light, she re-formed as a silvery soap bubble and swooped through the cabin wall, vanishing.
Ia sighed and scrubbed at her face. _Yet another person to drag into the conspiracy. God,_ she begged, _help me. Make sure Jesselle Mishka is in a cooperative mood._
He was perfect. Joseph N'ablo N'Keth, twenty-seven years old and identical to the original in every way that frozen chunks of DNA and increasingly easier pre- and postcognitive forays onto the timeplains could make him to be. Exhausted yet elated, Ia probed the timestreams one more time and nodded in satisfaction.
The original paths were still damaged, but much of it could be salvaged. Only a few things would have to be changed here in the near future, and at about one hundred to one hundred and twenty minor, major, and key timing points down the way, depending on how things panned out. She'd have to stint herself on sleep again to rewrite several of her prophetic directives, but it wouldn't be a waste of energy.
"Blood pressure 103 over 65, encephalographic activity normal, delta brainwaves declining," Mishka reported. "You even managed to re-create traces of mucus in his lungs from a minor chest cold. A pity your kind won't cooperate more often to help heal the injured and dying."
Belini narrowed her eyes but didn't deign to speak.
Ia did it for her. "Doctor, have you ever contemplated the philosophy of _why_ people die? Even the Feyori do it. There is a reason for it."
"And that reason is?" Jesselle asked, arching one blonde brow in skepticism.
"Contemplate it," Ia told her bluntly, not willing to give the other woman a free ride. "Can you keep him sedated?"
"I can. And I have agreed, having voiced my objections, to uphold this little charade," the doctor added. "You're lucky Private Hollick was willing to undergo a telepathic scan from me so I could make sure he was fully informed and truly willing."
"I had my own objections as well, Doctor," Ia told her, "but they got blown to pieces by Sung's willful little act of disobedience. Helstead, stay here and stand watch over our guest. I'm going to go order the ship into dock, and call a shipwide boardroom meeting. The two of you are exempt from attending, since I need you to keep an eye on our 'guest' here—Belini, charge back up and get changed into your new body," she added. "You'll need to show up as Private Hollick."
"Sir, yes, sir," she quipped, flipping Ia a fluttery mock-salute.
Rolling her eyes, Ia retreated from the Infirmary. She was exhausted and would not be able to rest for many more hours to come, but the timestreams were back under her control. Bruised and banged about, duct-taped together with a snapped wrist much like Private Sung's, but once more hers to command.
She prayed all the way to the bridge that she would never have to do that again.
It was still third watch, but Togama wasn't on duty at the moment; he had been replaced by Private James Kirkman. Nabouleh was back on duty, having swapped places with Sangwan twice over the last three hours. Spyder wasn't on the bridge. Technically he was supposed to be asleep by now, and Ia had granted him leave to go, since all they were doing was floating in space several hundred thousand kilometers away from the remains of the Salik base.
Altering Hollick had consumed a lot of their spare time, shattered and duct-taped back together as it was, a lot more than a Feyori needed to just change themselves, or to heal someone else. His mind and his memories had taken longer to create than his revised body.
"Private Kirkman," Ia stated as she entered the bridge, "contact the TUPSF _Hum-Vee_ and inform them that most of our situation has now been contained. Tell them we are coming in to dock, and ask them for a gantry position and refueling priority. Once you have done that, contact the _Hardberger_ and let them know we are on our way in to coordinate with them for the arraignment and war tribunal of Private Goré Sung regarding the Fatality Thirteen: Friendly Fire incident."
"Aye, Captain. Ah, sir," he added, twisting to look back at her, "the Admiral-General left standing orders to be contacted the moment you broke communication silence. Shall I put you through?"
Pulling her headset out of her pocket, Ia nodded. "Ping the Admiral-General and connect us the moment the call goes through."
Dropping into the command seat, she hooked the headset in place, then started to pull the restraint straps in place. They were still broken, snapped physically and telekinetically in her earlier rage. Sighing, Ia gave up trying to secure herself. None of their maneuvering needs would require it in the next several hours; it was just habit for her to buckle up in this chair.
"Private Rammstein," she ordered the man seated at the operations console, "put in a work order to engineering to get up here and replace the command-seat safety harness, plus the straps in L-pod 20. I want fresh sets ready to go before we leave the zone."
"Aye, sir."
"Captain, we have pingback. Conversational lag is four and a half seconds," Kirkman warned her. "Admiral-General Myang on the line in three...two..."
Christine Myang appeared on the screen. Her face was creased, her eyes bleary, and her chin-length, grey-salted black locks were mussed. She was also clad in a loose-necked tunic in a faded heather grey, and her face hovered unnaturally close to the pickups, looming too large on Ia's main screen. Blinking twice, she focused on the screen at her end and narrowed her eyes in a silent, furious glare.
It took effort, but Ia did not flinch. "Admiral-General, sir. I apologize for the lengthy delay, but the Ultra-Classified security protocols had to be maintained. We sustained minor damage to the main cannon, which had to be fixed immediately," she lied smoothly, "and...have had to contain an unexpected addition."
"Contain?" Myang asked, voice rough from sleep. "Unexpected addition? Explain."
"We are inbound to Battle Platform _Hum-Vee_ , where I will be personally escorting Private Sung to his tribunal session as soon as it can be arranged," Ia stated blandly. "The charges are twofold, Fatality Five: Disobeying a Direct Order, and Fatality Thirteen: Friendly Fire. I am fully aware and prepared to carry out the double-indemnity sentence of corporal punishment his actions will accrue, and will do so without restraint or hesitation.
"I do, however, need...beg...a suspension of our standing orders to permit no other personnel aboard the _Hellfire_ ," she continued. That earned her another narrow-eyed stare. Drawing a deep breath, Ia explained. "We initially thought—as does the TUPSF _Hardberger_ —that their gunner was killed when Private Sung fired through the Salik shipyard debris and struck a missile emerging from the old Kellick-class projectile-pod turret, number 29, on board the _Hardberger_. This was not the case.
"Lieutenant Commander Delia Helstead, reacting on instincts triggered by my telepathically broadcasted precognitive distress, blind-teleported him instead to the safety of our ship. He was knocked unconscious by the transport, and Helstead and I abandoned the bridge as soon as we realized he was on board. We have kept him sedated this entire time, firmly secured under observation in the Infirmary," Ia told the head of the Space Force, breaking Fatality Forty-Three: Perjury, by lying to her superior officer without hesitation. "By the letter of our orders, his presence aboard is a violation of our Ultra-Classified status. By the _spirit_ of our orders...he hasn't seen a damned thing.
"So I request...I _beg_ ," she added, meeting Myang's soft frown through the vidlink, "that you forgive his trespass and dismiss the charge of Grand High Treason that would otherwise be incurred, as there is no possible way he could learn any of this ship's secrets, sedated as he has been all this time."
Several seconds ticked by. More than enough to send Ia's words all the way to the Admiral-General's quarters back on Earth and send back a response, thrice. Finally, Myang grunted, "Why should I? What's so goddamn special about this one soldier that you panicked so hard, it caused your junior officer to risk both of you being hanged for daring to bring him on board?"
"I don't think Helstead was actually thinking at that moment, sir," Ia pointed out carefully. "Her reaction speed to _my_ distress was faster than conscious thought. She has been recovering from a backlash headache all this time. As for why...this gunner is one Private First Grade Joseph N'Keth. He was and is destined to be the great-plus-grandfather of one of the key figures who will prevent the Zida"ya from successfully invading and destroying our galaxy three hundred years from now."
"You mean two hundred and ninety-nine," Myang corrected her. "It was three hundred years into the future _last_ year."
Ia shook her head, then wobbled it. "No, I mean three centuries, sir, as in a vague figure of inexactitude. It wouldn't do to give everyone in this day and age an exact date of their arrival because the moment the invaders are actually noticed will depend on _who_ does the noticing, and how much of that information gets back to my prophesied agents. It's enough for people in this era to know that it will happen at a rough date in the future because there is nothing we can do to stop something that'll take place long after we're all dead. We have other problems on our hands right now."
"Charming. And so very cheerful. So. This gunner, Private N'Keth...Wait, is that N'Keth, as in that holy lineage from V'Dan? The one with the special blue _jungen_ marks?" Myang asked her, frowning.
"Sir, yes, sir," Ia confirmed. "He doesn't have any himself, but he is of that bloodline. He's from Eiaven, which is a jointly settled heavyworld, and chose to go into the Terran service rather than the V'Dan. Just as Helstead, who is also from Eiaven, chose to go into the Terran Space Force, and just as I chose that way as well, coming from Sanctuary."
Again, Myang mulled over her words for longer than the hyperrelay's turnaround time would allow.
"...He's really that important?" she finally asked Ia.
"I wouldn't have panicked so badly that Helstead reacted to my psychic broadcast without a thought for the rules and regs if he wasn't, sir," Ia pointed out dryly. That much was the truth, if one counted the way the shorter woman had given chase, trying to prevent Ia from killing Private Sung. "If I could've saved him myself, I would've, regardless of the consequences...and I _would_ accept the punishment for Grand High Treason during wartime as part of our agreement, but you still need me to be very much alive, sir."
This time, her reply came back within the allotted turnaround time. "Alive and uncaned?" Myang asked her dryly, sitting back from the screen a little. "Is that what you're going to ask for next?"
"Sir, no, sir," Ia denied crisply. "I accepted my double-indemnity with the willingness of full foresight of all possible consequences. However many strokes the tribunal assigns to Private Sung, I will endure them, blow for blow, without restraint or hesitation. It is only being hung, drawn, and quartered that I object to, sir. It'd be a little too difficult to continue saving the galaxy this week if I'm not alive to do it. You still need me at the helm of this ship, and its replacement...but there's nothing in there that says I have to be comfortable while I'm in the pilot's seat, sir."
Myang studied Ia for several seconds, then nodded. "Very well. Dispensation granted, so long as this Private N'Keth is kept sedated the entire time he remains on board the _Hellfire_... _and_ in the understanding that you will disembark and revive him the moment you are safely docked at...uh, Battle Platform _Hum-Vee_. Since that is the nearest source of the Judge Advocate General's branch of the Special Forces, assuming you haven't left the Helix Nebula?"
"Sir, no sir. We are still within the same cometary knot as the rest of the local fleet and are inbound to the _Hum-Vee_ as I speak," Ia reassured her. "Private Sung has been a valuable member of my crew. I'm not quite sure what possessed him to keep shooting despite my clearly issued orders to cease fire. But the timestreams suggest he will recover from his punishment and serve on this ship with a greater level of obedience and devotion, so I shall suggest to the tribunal judges that he be given a caning only, with no incarceration."
Myang lifted her brows. "No time in the brig? You don't have to be afraid of incarceration yourself, you know. Your double-indemnity clause only covers corporal punishments, so you wouldn't have to be stuffed into a cell alongside him."
"I know I don't have to, Admiral-General," Ia admitted. "But I need good gunners on board, ones who aren't needed on other ships. Despite his extreme lapse in good judgment, Private Sung is still a good gunner, and I can still use him as I continue to fight for you."
That made Myang grunt. She looked tired again, tired and sleepy. "Unfortunately, you have a point. Preliminary intelligence culled from the wreckage of that shipyard base suggests there are several others out there. We _will_ need every good gunner we can get. I'll pass along a recommendation of my own to keep him out of the brig. I trust that, stroke for stroke, you will be even _more_ careful in the future not to abuse your _carte blanche_."
"I am _very_ determined to avoid anything like this in the future, sir," Ia vowed fervently. "One more thing. Please remind Admiral Genibes to pass along the design corrections I sent you last week to the crews working on the _Hellfire_ 's replacement. If they're applied now, that should speed up the construction process."
"I am _not_ your personal messenger service, Captain...and I'm getting tired of your little 'one more thing' quips. Myang out." Shifting her arm, she thumped what had to be the controls for a bedside screen, and vanished from view.
The monitor replaced her oversized face with its default display of their dead-ahead view. Ia wasn't offended by Myang's retort. She knew the woman would remember to tell Genibes within twenty-four hours.
Sitting back, she contemplated the field of stars and the slowly increasing, green-highlighted dot that would eventually resolve itself into the prickle-burr shape of Battle Platform _Hum-Vee_. Ia had once used it as a base of operations for the two years she served on the now-broken Salik Blockade. A glance at the chrono showed Ia the time—near midnight. If Myang had been asleep, either she had been forced to work off-shift for Aloha City and the Tower, or she had gone halfway around the world for some reason, perhaps to the Space Force Intelligence Division headquartered in Paris. Ia didn't know, and right now, she was too tired to care.
Regardless, the interview had taken place, successfully navigated. Sighing in relief, Ia slumped back in her seat. "Well. That's one obstacle down. Nabouleh, finish docking us at the _Hum-Vee_. Kirkman, coordinate with the JAG office on board the Battle Platform to arrange a quick tribunal for Private Sung. I'm going to advise him not to contest the charges since we have him dead to rights with all the onboard surveillance equipment. This isn't some dirtside battlefield where it's a case of he said, she said."
"Yes, sir," Kirkman agreed, moving to comply.
"While you're waiting for them to get back to you, inform the entire crew that the moment we dock, the ship is to be sealed and secured at dock, and all personnel— _all_ personnel, from bridge crew through to engineering and lifesupport, asleep or awake, are to report to the boardroom, excepting only Doctor Mishka, Lieutenant Commander Helstead, and the patient in the Infirmary."
"Sir?" he asked her.
She knew he was questioning the unusualness of her order. Once military starships left a construction yard and filled with a crew, until the day they were decommissioned or destroyed, they were always monitored. On the bridge, in engineering, and in lifesupport, the three most vital parts of any vessel, there was always, always someone on duty in case of emergency. On civilian ships, the rules could be relaxed, but never on a military vessel.
In this case, though, damage control had to come first, and it had to include her whole crew.
"Company Bible rule number one, Private. Orders issued by Ship's Captain Ia take precedence over all other orders, rules, and regulations. They may be questioned, but they still take precedence." Rising from her seat, she crossed to the back door. "If you'll excuse me, I have had a very long day, one made even longer by having to spend the last three hours cleaning up the mess Private Sung made. And now I need to go change into a Dress uniform since my day doesn't end when we dock.
"I will explain what I can at the boardroom meeting, once I have freshened up. You have your orders, gentlemeioas. Dock the ship, secure it, and make your way to the boardroom."
FEBRUARY 4, 2497 T.S.
Once again, she paused just inside the alcove to the boardroom. Paused, breathed slow and deep, and squared her shoulders. It was just past midnight, Terran Standard Mean Time, which meant she had been up for over thirty hours, not counting her four-hour nap almost a day ago. Every time she let herself feel anything about this situation, icy-sick waves of dread kept sweeping through her body from skin to bones, leaving nausea in its wake. That nausea mixed badly with her exhaustion, leaving her drained with the fear that it could happen again.
_But "fear is the mind-killer," as the old saying goes. I accept my fear. I embrace my dread. I_ know _what my worst-case scenarios are,_ Ia reminded herself. _I have met one, been overwhelmed by it...and yet I survive. The universe—the rightful path in Time—survives. Duct-taped back together, but it still survives._
Reassured, she moved out of the alcove and onto the dais. And nearly stopped. Lieutenant Spyder wasn't seated at the head table as expected. Instead, he stood in the aisle next to one of the front-row tier seats. He did so with his muscular arms folded across his lean chest, looking tired but still as tough and competent as any Marine she'd known.
Beside him, hands in his lap, thumbs cuffed together in restraints since his wrist was being held immobile in a cast while the bone-setting enzymes did their work, sat Private Sung. For a moment, Ia closed her eyes. _I did_ not _expect Sung to be brought here...but I guess I'm a victim of my own exact wording. Every single crew member except for Doctor Mishka and Lieutenant Commander Helstead...which means I'm hoisted up into the air on my own exploding petard._
_Brilliant. At least I know for sure that_ some _of my crew are willing to obey my commands to the letter...and the timestreams say I can use this to my advantage._ She still couldn't see very far, but that might have been from the fact that working with Belini to imprint Hollick's mind and body with everything the original N'Keth knew had drained a lot of energy out of her.
Gathering her wits, Ia continued forward. Not to the table, but to the front row. Pointing at Sung, she swept her finger behind her. " _You_ will take a seat at the officers' table. You will sit there and be respectful of the authority that table represents while I explain to everyone who you are and what you have done."
Not quite meeting her gaze, he nodded and rose. Spyder followed him, and stood behind him when Sung took one of the empty seats at the end. Moving around the other side of the table, Ia stepped in front of her chair but did not sit down. To underscore the severity of the moment, Ia had donned her Dress Blacks and the full complement of her glittery, which required wearing a modified, knee-length version of her jacket. The only thing missing was the cap back in her quarters. She didn't need it just yet.
She began with the facts.
"Just a few hours ago, during the battle against the Salik and Choya forces yesterday, Private Second Class Goré Sung willfully committed an act of Fatality Five: Disobeying a Direct Order, which resulted in an act of Fatality Thirteen: Friendly Fire. The evidence for these charges is absolute. Surveillance scanners pinpointed the offending laser turret as being under his control, and diagnostics prove his headset was fully functional the entire time that I gave repeating orders for all starboard gunners, including him, to cease fire at a specific time.
"Fatality," she stated coldly, "is _exactly_ the word for it. For whatever reason Private Sung disobeyed my direct, precognitively backed order, his willful act of disobedience resulted in the death of Private First Grade Joseph N'ablo N'Keth, from the impact of his Starstrike laser on an emerging projectile missile being launched from Private N'Keth's turret.
"Make no mistake about this: The _original_ Joseph N'keth is dead. Dead. Dead. Dead," she repeated. "And with his death, Private Sung single-handedly destroyed an entire bloodline _necessary_ to prevent the destruction of our entire galaxy three hundred years into the future.
" _That_ is why I ordered everyone down our starboard flank to cease fire, because I _knew_ the Salik shipyard of Station 5 would fall apart under the force of the incoming blossom bombs. I _knew_ there was a chance that one of our weapons would damage the _Hardberger_ 's hull if we kept firing. Private Sung is _personally_ responsible for the end of the Human race—the end of _every_ race in this galaxy—starting three centuries from now, defeated by an alien race so advanced, the _Greys_ fled from them in terror."
She turned to face him. Sung looked pale, sitting there with slumped shoulders and a crumpled air about him. No one spoke, though a few of his fellow crew members shifted uncomfortably in their seats.
"I had one shot at getting it right. One path to carefully tend, to make sure that every person, every moment, every chance encounter was _not_ by chance but instead instigated by need and design, to create the one person capable of stopping the advance of that enemy race. And _you_ shattered it, Private Sung. For whatever reason you _thought_ you had, you shattered it." She let that sink in, watching him blanch and crumple inward a little more, huddling awkwardly in his seat. "But by a twist of luck, and the grace of a _good_ man of deep and abiding faith...I am able to repair most of the damage you made. Not all of it, but most.
"As a result of this twist, this _trompe l'oeil_ I have just spent the last duty watch patching, testing, and altering so that the galaxy _will_ survive...Private Sung _technically_ will not be held responsible for the obliteration of this galaxy and all of its native residents." Ia let the implicit threat that he would still be culpable somehow hang in the air for a moment. She continued, shifting her gaze to the others. "One of you has volunteered to take the dead man's place. His body has been altered, his mind repatterned, his face carefully imprinted with all the things the original Joseph N'Keth was supposed to do, _without_ my interference, other than making sure he was supposed to survive this last fight.
"Do not speculate among yourselves who that soldier was. Do not _ever_ mention it outside this ship," Ia added, jabbing her finger toward the starboard. Toward the Battle Platform holding them in dock. "You mention _any_ of this, and Private N'Keth's life will come unraveled, his part in being the great-plus-grandfather of the defense of this galaxy will be destroyed, and _you_ will be held accountable for the destruction of every being, every star, and every planet, right alongside Sung.
"Make no mistake. The Zida"ya _are_ coming to the Milky Way. They _will_ tear apart everything we are and everything we know like uncaring locusts. In all the months you have served with me, you have _seen_ the accuracy of my predictions, down to the very millisecond!" she reminded them, letting some of her frustration and anger color her voice. "Do not doubt me when I say I can see _tens_ of thousands of years into the future, with _equal_ levels of accuracy. I have had to _lie_ to the Admiral-General herself about what has just happened because even _she_ would be held accountable for all those deaths if she ever found out and let slip that she knew.
"If I could wipe _your_ minds of all incriminating memories over the last half day, I _would_ ," she warned the Damned. "But I am already in debt up to my eyeballs with the Feyori for pulling off this replacement trick, and that is a very _ugly_ price to have to pay. Neither is it the _only_ price."
Snapping her fingers, she activated all the main screens stationed around the room, the one behind the head table, the ones on the sidewalls, and the one over the heads of the tier seats, allowing her fellow officers and the silent, somber Private Sung to see. Those screens started scrolling small icons of faces attached to a list of names. Some had military ranks, while many others had none. Some were very old and some were very young, though most seemed to be adults.
Not all of them were Human, either.
"Because of Private Sung's willful disobedience, we have _not_ left this cometary knot. _Not_ left the Helix Nebula. _Not_ flown off to our next port of call. We will not be able to leave here for at least another four hours, and we will need another _five_ on top of that to effect repairs. Even if we commandeered one of the fleet tankers to top us off fully, and wasted fuel traveling OTL to get to our next time-sensitive fight, we will still be two hours _too late_."
The names and faces continued to scroll, six columns wide, and moving so fast, it was hard to read even a single name. The list of Humans ended, replaced by Tlassians.
"I will be sending out precognitive directives before attending Sung's tribunal, alerting those ships who _can_ be spared for the coming fight, and giving them exact instructions on how to salvage everything that they can. Everything we ourselves cannot be there to do...but it still will not be quite enough." She gestured at the monitors with each hand and explained their purpose. "What you see on the screens is a list of every single person who will die in the next two years because of our inability to be at that next battle. A list of every person that _they_ would have saved, or begat, or influenced down through the next three hundred years would take seventeen hours at this speed to display.
"Private Sung is no longer responsible for the destruction of our entire galaxy. But he _is_ directly responsible for the deaths of 720,593 people between now and the appearance of the Zida"ya at the galactic edge...and I know each and every one of them." Turning back to Sung, who swallowed and looked ready to retch, she said, "You will be given the list of these names to contemplate in your spare time. You are free to ignore them if you wish, but understand that _I_ cannot.
"The only person worse than you on board this ship right now is me, Goré," she stated quietly. "For I have slaughtered more than you, and _will_ slaughter more, in the name of saving this galaxy. Saving as many lives as I safely can is my sole motivation, and the only reason why I _can_ act, rather than step aside and allow this galaxy to end. When you fired against my orders, somehow I doubt your reasons were quite as noble as mine...and you have _added_ to the screams of the people I cannot save. The loss of Private Smitt's family and homeworld were _necessary_.
" _These_ losses were not."
He glanced up at her, a quick peek. Ia met his gaze steadily.
"Welcome to the hell that is my life, Private. You're now a full-on, murderous monster, just like me."
"...I'm sorry." It was barely a murmur, but it was heartfelt.
"Sorry doesn't save lives. But I hope your regret will drive you to do better." Turning back once more to the others, she tipped her head at Sung. "Given the needs of this information to remain private to this crew and this ship, you are hereby _forbidden_ to give him a hard time over it."
Sung wasn't the only one to lift his head, eyes wide in confusion and disbelief. Ia held up one hand. The sharp movement caused her medals to sway and _clink_ faintly together.
"I know what you're all thinking, and you are _wrong_. Having that list of names will be more punishment than anything _you_ could do or say. Not to mention he is about to be caned for his crimes. He is lucky that this is _all_ he will suffer.
"The sedated presence of the _trompe l'oeil_ version of Joseph N'Keth on board this ship, currently tended by Mishka and Helstead, would normally be a violation of our standing orders to keep all non-Damned personnel away. I have arranged with the Admiral-General to avoid being charged with Grand High Treason for having Private N'Keth on board...whom we will _all_ treat as the real Joseph N'Keth. Even the Admiral-General herself is never to know that he's a fake, that the real one died this day.
"There is still the matter of Fatalities Five and Thirteen to be handled," she continued, moving along. "Private Sung, I will give you some temporally backed legal advice. When you are brought before the tribunal, do not deny either charge. Denial would only increase your punishment. As it is, whatever number of strokes the tribunal assigns to you will be doubled because you committed those two Fatalities during an officially acknowledged period of war...and doubled _again_ , because you disobeyed the orders of a proved, registered military precognitive. Admit what you did, accept your corporal punishment, and understand that—murderous idiot or not—I still _need_ you.
"You are not permitted to slack off or step down. You also owe Lieutenant Commander Helstead's quick thinking for your continued life. I will be expecting you to be ready to serve this ship, her crew, her Captain, and her future within twenty-four hours after your caning." She lifted her gaze to the rest, to her officers, her noncoms, and the enlisted men and women gathered before her. "I will expect _each_ of you serving under my command to heed my orders and give me your best. I do not _ever_ want to have to compile another list like this again. Is that clear?"
Uncomfortable silence met her words. That wasn't good enough. Ia scowled.
"This is _not_ a joke!" she snarled, one hand jabbing at the screen behind her, the other raised to her ear, fingers curled as if cupping to hear the screams of the dead, versus the silence of her crew.
Her commendations swayed on their colorful ribbons, tangible proof of just how hard she had strived to make the whole Space Force understand the importance of her work. Every single medal pinned in place should have made her seem like a mountebank, but these were not soft civilians she faced. They knew the kind of price she had paid for those medals, kilo for kilo in blood, sweat, and tears.
"Every name. Every person. Every severed _life_ I can hear, every second, _screaming_ in my head. Dead. Dead. _Dead._ You will _not_ make me come up with another list like this again. _Is. That. Clear?_ "
_"Sir, yes, sir!"_ Over one hundred bodies shot to their feet in unison, Spyder and Harper and Sung the foremost. The loudest. The stragglers rose as well, some moving belatedly, but all rising to Attention.
"...Good." Muscles tight to control the sickly ice still prickling at her nerves, Ia lowered her arms back to her sides. " _Good._ Remember. Your orders are to avoid discussing this with anyone else. Some of the last few hours have been spent in altering the onboard surveillance pickups, official and unofficial, to remove all traces of the trick that has been played. This death did _not_ happen. The trick to cover it up does not exist. Don't even discuss it among yourselves once you leave this room, today.
"To that end, I give you leave to stay here and discuss it for the next hour. Your conversations will not be recorded for that hour, so feel free to be candid. Once that hour is up, I expect you to obey without hesitation or discussion. In the meantime, N'Keth will be disembarked. Private Sung and I are due on the _Hum-Vee_ in half an hour for his arraignment, which will become his tribunal _if_ he is smart and does not protest. Lieutenant Spyder, escort him to the nearest head, then bring him to the airlock," she instructed her friend. "I'll escort the prisoner myself from that point. The rest of you, when your hour is up, begin repairs immediately and prepare to be summoned onto the _Hum-Vee_ to witness the caning. Dismissed."
She didn't wait to see what her crew would do but instead turned to face Sung.
"...I can go with you if y' like, sir. Figured I'd take it on meself t' play MP," Spyder murmured as he waited for Sung to rise. "We don' exactly have a security detail f'r it on board, do we?"
"I never believed one would be needed," Ia muttered back. Movement at her other side made her glance that way. Bennie lifted a hand toward her shoulder. Ia shook her head and shifted back, out of touching range. "I'll be alright. Just don't touch me right now. Lieutenant Commander Harper, you have command of the ship while I am gone. If everything goes according to what I have foreseen, you will need to assemble the Company, save for a skeleton crew, to head for the assembly hall. As per regulations, this entire crew will be expected to witness the canings."
She heard Sung swallow. Turning back, she watched him lick his lips, then speak. "I'm sorry, Captain. I was just so...so caught up in the fight, and I couldn't see the reason why I _shouldn't_ keep firing. I literally...couldn't see the _Hardberger_ on the other side."
He lowered his gaze. Ia shook her head. "Most soldiers in a great war never see the whole of that war. They only ever see a tiny part of it. That is why orders are given by those who _do_ see the bigger picture, in the expectation that each soldier will have faith in their leaders to give the right commands at the right time. They can and should contribute to the immediate battle plans, but when a target is denied to a soldier, it is _denied_ to them.
"In my case, I see everything. If I give you an order to kill someone, it is because I have very carefully weighed the impact of that life's existence against the needs of every other life, and deemed that the survival of the whole is too great to be ignored. If I give you an order to spare their life, it is for that exact same reason."
He considered that for a moment, then looked up at her, frowning softly. "But what if you're wrong? Aren't you ever wrong?"
"Oh, I _have_ been wrong," she reminded him candidly, still angry with him. "Most recently, I expected you to obey. I _believed_ you would obey. I was _wrong_ , and all of those names on that list are the end result of it. More lives than you will ever know are _always_ at stake whenever I give a command. Take him to the head, Lieutenant, and let him freshen up. I'll meet you at the airlock."
Nodding, Spyder took Sung by the elbow, escorting him out of the bow boardroom.
Colonel Avice looked up from the workstation screen embedded in the tribunal desk. A frown furrowed his dark brow. "Ship's Captain Ia, I find myself puzzled by the Admiral-General's standing order regarding yourself and your crew. I am not the only one, I'm sure. Are you aware that—"
"—Yes, sir, I am fully aware," Ia stated, interrupting him before he could go into the tediousness of listing everything. "I stand fully prepared to execute my orders without restraint or hesitation, sir."
Major Richildis and Commodore St. Stephen exchanged looks. The commodore, clad in Dress Blacks with blue stripes down his sleeves, rested his elbows on the desk. "Even with the Admiral-General's personal command code attached to this order, this is highly unusual. Unless the officer has also committed a crime, it is against Space Force regulation to punish the innocent. We are inclined to be more lenient in settling judgment upon Private Sung, as a result."
"I would rather you did not, sirs," Ia asserted. Hands clasped behind her back, standing in Parade Rest with her grey cap squared on her head, she met his gaze steadily.
Major Richildis narrowed her dark brown eyes. With her short-spiked brown hair, snub nose, and the brown stripes down her black sleeves, she looked like a bull terrier debating whether or not the current military case was a bone that needed to be chewed. "Explain yourself, Ship's Captain."
"The only leniency Private Sung deserves is twofold. One, I still need him as an otherwise damn fine gunner on my ship, so I request that he not be incarcerated," Ia told them, not bothering to look at Sung. He sat on a chair to one side of the small courtroom, having answered "yes" to both charges under the quietly admitted reason that battle adrenaline had carried him well out of line. "And two, with the survival of Private N'Keth, the charge of Friendly Fire should be modified slightly to acknowledge that, in this case, the Friendly Fire in question wasn't a literal fatality."
"Only by the skin of your third-in-command's psychic teeth," the major pointed out.
"True, but his life was still saved, so there should be some small mercy granted for that," Ia said. "Particularly in light of the modifiers. But beyond that, I expect no clemency for the soldiers placed under my command."
"Yes, the modifiers," Commodore St. Stephen agreed, sitting back in his seat and shifting his gaze to the accused. He was almost as large as Lieutenant Rico, though with pale, freckled skin and a braid of thinning white hair streaked with remnants of the original coppery red. Between that and the neatly trimmed full beard, he looked like he should have been dressed in a great kilt and wielding a claymore against his enemies. "Double the penalty for ignoring or acting contrary to the orders of a Space Force acknowledged precognitive when those willful actions result in an otherwise avoidable Fatality. Double the penalty again for a wartime crime."
"The penalty for Disobeying a Direct Order is two strokes of the cane per order," Richildis pointed out. She chewed the words through a not-smile. "According to the black box records which your CO has provided us, she ordered you to cease fire eight times. That's sixteen strokes. Doubled from the precognitive backing, that's thirty-two. Doubled from the fact that the disobedience took place in a war zone, that would be sixty-four strokes."
Sung flinched, paling. He said nothing, though, just glanced at Ia in fear, then looked down at his hands.
"For the crime of Friendly Fire, that would be four strokes," the major added. "Four times two is...?"
Richildis held her gaze on Sung until he spoke. "...Eight, sir."
"And eight doubled again is?"
"Sixteen, sir."
"For a total of how many strokes of the cane?" Major Richildis smiled. It was not a pleasant smile.
"Eighty, sir," Private Sung whispered.
Colonel Avice shook his head. "That's too many. Caning has a maximum number of strokes that can be applied, with the rest converted to a set proportion of years in jail. The Admiral-General herself has stated there shall be no incarceration time." Resting his green-striped sleeves on the table, he glanced at his companions. "I move that the repetitions after the countdown reached zero be the only ones counted, since it was after that point that he started disobeying orders."
"That would only be three counts. I would rather it was the five that came before, if we're going to be lenient," Richildis countered. She sat back and gestured at Ia. "He was apprised in advance that his commanding officer is a precognitive of immense skill. He had worked with her for several months, seeing that skill in action. She told everyone, including him, five times to cease fire at a specific time, and yet he still disobeyed."
"Fifty-six strokes is still excessive," Colonel Avice countered. "We could space it out to ten strokes every few weeks, but that does involve incarceration between canings."
"Sirs, if I might suggest something?" Ia offered.
They looked at her. Commodore St. Stephen gestured at her to speak.
Ia nodded and drew in a deep breath. What she was about to offer would be applied to herself as well, after all. "There is an alternative. Deliver part of the caning to his upper back. The rules and regs permit the substitution of such blows at a ratio of two to one. I say, give him twenty blows to his back instead of forty, and sixteen to his buttocks, and do it in one session.
"That will leave him in sufficiently satisfactory condition to be returned to the _Hellfire_ immediately afterward. I will need him at his post on my ship in the next few weeks. I cannot afford to have him wasting time in a brig, waiting for his backside to heal so that he can suffer the rest of his court-appointed strokes, and the Admiral-General knows this."
They looked at each other. It was Major Richildis, the apparent bad cop of the panel, who frowned, and asked, "You do realize that _you_ will have to undergo twenty to your back, and sixteen to your asteroid?"
"Sir, yes, sir," Ia agreed.
"No, sir!" Sung surged to his feet. The bailiff moved forward, but Sung didn't go anywhere, just looked at Ia and the tribunal panel. "No. She shouldn't have to suffer for my mistake, sirs. Please!"
Ia looked over her shoulder at him. "Then you _shouldn't_ have made it in the first place. I told you on the very first day that _any_ corporal punishment my crew received, I would have to receive, too. _That_ is the price I pay to be able to direct my ship whenever and wherever it needs to go, soldier. For the power that I wield, some prices _have_ to be paid. I will even take on the full seventy-two blows in order to pay it, and do so willingly, if that is the judgment of this tribunal."
"It is not," the commodore stated. "In the light of the accused's admission of guilt and acceptance of the charges against him, I recommend that we follow his captain's advice. Sixteen blows to the buttocks, and twenty blows to the upper back, sentence to be carried out immediately. Colonel?"
Colonel Avice sighed, studying both Ia and Sung. Finally, he nodded. "I concur. It's excessive, but the target of Fatality Thirteen _would_ have died if it weren't for the heroic psychic actions that saved him. Sixteen to the butt, and twenty to the back."
"Agreed," Richildis confirmed. "Sixteen to the buttocks and twenty to the upper back. Ship's Captain Ia, given the indemnity clause attached to your command, do _you_ agree to this sentence?"
"Sixteen strokes to the butt and twenty to the back, I agree without restraint or hesitation," she replied. "I am prepared to endure all thirty-six strokes immediately upon the completion of Private Sung's corporal punishment."
"Very well. This judgment is recorded and sustained by concurrence of this tribunal and the offending soldier's commanding officer. Bailiff, escort Private Sung to the assembly hall," Commodore St. Stephen ordered. "Captain, please accompany them. Your ship will be contacted and your crew escorted to the assembly hall to watch the administration of both assigned corporal punishments. Those who remain aboard will be instructed to watch the Battle Platform's broadcast of the disciplining."
Sung paled again. Ia didn't have to ask why; she could guess easily enough that he had just realized _her_ caning would be witnessed by the whole Company, too. She didn't try to reassure him or change the situation. That, too, was a part of her double-indemnity clause.
"Commodore, yes, sir," she said, saluting the trio at the desk. "My crew has been readied to view the caning by my first officer, sir."
Sung quickly saluted, too, as best he could with his thumbs still locked together. The three JAG officers saluted back, and the commodore dismissed them with a flick of his fingers. There were other cases waiting to be judged. This wasn't a case of repair materials or fueling needs, but Admiral-General Myang's standing orders for priority handling had bumped Ia and her crew to the top of the day's list.
Unlike Recruit Kaimong and Private Culpepper, Private Sung didn't struggle or resist. He cooperated with the bailiffs in being draped facedown over the frame. Without a word, he let them bind his wrists and ankles in place, and endured the kidney pads being wrapped around his lower back.
The first few blows to his buttocks did make him gasp. By the sixth, he grunted with each stroke, the sound muffled by the biting gag placed in his mouth. On stroke twelve, tears could be easily seen dripping down his reddened, grimacing face. When the cane was moved after the sixteenth, so was the frame, its angle lowered so that his upper back was placed at the same height his rump had been. Within three strokes, he cried out, the yell only half-muffled by the gag.
Stroke thirteen cut through his shirt, and the caning was paused while the cut was inspected. With only a little of the skin broken, the examining doctor informed the caner to strike from a different angle. Nodding, the sergeant moved to Sung's other side, lifted the antiseptic-soaked rod to shoulder height, and continued with blow fourteen.
When it was through, Private Sung had to be lifted from the frame. He could not stand on his own. The _Hum-Vee_ 's medical staff had provided a hovergurney; after settling him facedown on the cushions, he was examined one last time. Several blows had lacerated his skin as well as raised welts and caused bruises, but the doctor's prognosis confirmed he would recover.
Commodore St. Stephen stepped back up to the podium, located to one side in order to focus the watching balconies of soldiers ringing the round, deep chamber. Bodies stirred around the room in preparation to depart, expecting him to deliver the usual warning of discipline needing to be maintained before their formal dismissal. His first sentence disabused them of that.
"Soldiers. You will remain in your seats and be respectful of what is about to take place," he ordered the crowd.
Ia's Company, the roughly 150 who had been free to leave the ship and attend, exchanged puzzled looks. So did the other nine hundred or so soldiers and specialists gathered in the hall.
"By order of Admiral-General Christine Myang, Ship's Captain Ia of the 1st Company, 1st Legion, 1st, Battalion, 1st Brigade, 1st Division, 9th Cordon Branch Special Forces is required to undergo an equal number of strokes of the cane for any and all corporal infractions incurred by the soldiers placed under her command," he stated. His words caused a rustle of surprise and disbelief that echoed off the walls. That forced him to raise his voice slightly, letting the pickups adjust accordingly so that his next statement could be heard. "It is therefore the duty of this Judge Advocate General tribunal to order the following sentence be applied to Ship's Captain Ia of the Special Forces:
"Sixteen strokes of the cane to her buttocks, for Fatality Thirteen: Friendly Fire, when the ship and crew under her command did willfully attack the ship and crew of the TUPSF _Hardberger_ with one of her vessel's Starstrike laser cannons, a lethal weapon. This punishment is to be followed by twenty strokes of the cane to her upper back for the crime of Fatality Five: Disobeying a Direct Order. At her request, Ship's Captain Ia has asked for no leniency and has agreed of her own free will to undertake this punishment without hesitation or restraint. She is also under orders to restrain herself from using her innate biokinetic abilities for the next twenty-four hours.
"Sentence is to be carried out immediately."
Someone started the sonopad drums. Ia rose from the seat provided for her behind the podium and walked across the stage to the frame, which was being raised back up to the more common striking height. She felt numb, looking at it. Just...numb. She had struggled hard to avoid this sort of possibility with every bit of cunning at her command, but now she felt nothing.
Unbuttoning her knee-length coat, she shrugged out of it as the doctor stepped up, using a hand scanner to check Ia's vital signs. Ia focused on neatly folding her long jacket, reducing it to a neat if ribbon-lumped square.
Commodore St. Stephen joined her. He lifted his arms, palms up like a tray, and she gave him a slight nod of thanks. Placing her jacket on his hands, she added her Dress cap, and turned to the frame in her plain grey shirt and grey-striped black pants. The male caner stepped up to help secure her in place, making Ia shake her head.
"No, thank you. My orders are to use no restraints," she said.
"Sir, the restraints are there for your protection, so you do not move," he told her, glancing at the female sergeant who was to administer Ia's caning.
"I know that, Sergeants," Ia told both of them, "but I will do this without bindings or restraints. You have my word, I will not move." She started to move toward the frame again, then pulled back and pointed at the table behind the frame, which held the case with its _rotan_ switch soaking in antiseptic solution, and a selection of biting gags. "But I will take one of those, to protect my tongue and teeth. I may have agreed to do this, but I'm not _completely_ stupid."
Nodding silently, the female caner moved to select one of the gags. Ia took it from her when she returned, and fitted the slightly spongy plexi bar between her teeth. Stepping up to the frame, she lowered herself onto the slanted, padded surface and tucked her hands under her cheek. She wanted to show to everyone in the auditorium that she was there by her own free will.
_I knew the very moment Myang first proposed this indemnity clause that I could turn it to my advantage,_ Ia thought, feeling them strap the padding around her kidneys. _This will cement my reputation with the rest of the Fleet. Bloody Mary doesn't just give a beating to the enemy, she can take a beating, too, and emerge all the stronger for—Holy_ God!
The first blow had two layers to it: the initial, startlingly hard sting that burned on the surface, mostly on her right nether cheek; and the bruising ache that lingered even as the stinging burn started to fade. Her teeth bounced into the gag on the second blow, and clenched on the third. It hurt. It hurt it hurt it _hurt_...but...not as much as the shock, the _pain_ of watching her life's work shatter.
The strikes came at slow, measured intervals. Ten seconds between each, time enough to relax, recover, anticipate, and tense up again. Ten seconds felt like a long time between each painful smack of the antiseptic-soaked stick, a long time to endure the throbbing and the burn. Ten seconds made her upper back clench and seize up in anticipation of the suffering it, too, would soon endure.
But with each blow, the agony of it seemed to beat back a little bit more of the dust from the desert-scorched plains. Ten seconds was a small eternity, on the timeplains. Enough to watch tendrils of greenery seep slowly, patch by patch, back into her consciousness. Enough time to see the waters trickling back into their proper places at the fringes of her vision. Enough time to feel her broken mind healing itself, restoring her stroke by penitent stroke back into her rightful place.
The frame jolted and thrummed faintly, lowering itself. Ia tensed in anticipation, then forced herself to breathe deeply, to let the physical fear leach its way through her muscles and out of her body, helping her to let it go. This was where her back muscles, still dense and strong, would fail to provide the same level of cushioning as her gluteals, but she would endure.
Her teeth snapped hard into the spongy bit with the first strike. The hands under her cheek twisted and shifted free, fingers curling and clenching around each other atop the head cushion. There was more sting with these new attacks, and the dull bruise burned with upper notes. By the fifth or sixth stroke, it felt like each hard-thudding lash was being administered by a rasp.
At stroke eight, they paused. Someone plucked at her shirt. The subtle shift of the fabric felt like sandpaper on her wounds. Ia choked, breath caught somewhere between a hiss and a gasp. The pause took longer, and the next stroke fell from a different angle. New skin, new pain...and where one set of strokes crossed the other, nails were driven into her back. Old-fashioned, pencil-thick nails. Only a centimeter long, but long enough to force a grunt from her throat with each blow.
She couldn't even hear the count anymore. All she could do was breathe and endure, breathe and endure. Seeing the infinities of the universe unfurling with each nerve-enflamed strike made her long for more. Not more of the pain, but more of the freedom, of the liberation of her abilities, of her very mind. A tiny corner of her memory realized this must have been what medieval monks had felt when scourging themselves, back on ancient Earth.
Again, they paused to pluck at her shirt, rubbing at the raw, too-sensitive flesh with a very light touch in their examinations. Ia ordered her mind to accept it like a thin, cold stream, then to set it aside. The next blow felt hard and unglamorous, but solid, a welcome relief from the pain of overstimulation.
Now the strikes crossed two lines of welts, more nails in the coffin of her back. With each blow driven home, another corner of her mind lifted free. Until she realized, after a few more strokes, the slow, steady rhythm had stopped.
Someone said something nearby. Struggling to focus, Ia opened her eyes. Everything was a golden, pastel blur. Images glowed in different hues, some stronger than others. Lines, blobs curves...they resolved themselves within two or three blinks as the doctor with her scanner, the edge of the platform, and the silently watching bodies of her crew.
Apparently the caning was through. Ia dragged in a deep breath through her nose and tensed to push herself upright. The pain at that dumb move emerged in a strangled, high-pitched whine. She cut it off, breathing in short, shallow sniffs through her nose, and waited for the frame to be lifted back to the original angle.
Her teeth were lodged in the biting gag. Her whole jaw ached, but her incisors ached the most. With awareness came a hint of blood on her tongue, seeping from her abused gums. Tilting her head, she got the fingers of her right hand into place and slowly pried. The rubbery material gradually popped off her teeth, first upper, then lower. She rested a few moments as the frame hummed and jolted to a near-vertical stop, then tried to lever herself upright again.
It hurt. It hurt it hurt it hurt, oh God how it hurt, but she managed it. Balancing carefully, Ia hissed between her teeth when she lowered her arms. The male caner took the gag from her hand, and the doctor touched her sleeve.
"Captain Ia, your back is bleeding in seven spots above, two spots below. There is a lot of blood compared to most canings," the woman added, "but you won't bleed to death. They'll need to be coated with an antibiotic ointment. If you'll lie on the gurney, we'll see that you're treated before you're returned to your ship."
Breathing shallowly, Ia shook her head. Then froze. That was a mistake, one that reignited the dull-stinging fires under her skin. _No shaking whatsoever. Got it._
"No," she managed out loud. "No ointment. No gurney. I'll walk out."
"Sir, you have multiple contusions, several lacerations, and your adrenaline spikes have elevated your blood pressure," the grey-uniformed woman asserted. "You need to lie down."
"No. I will walk out under my own power." She had to. She had to, for the sake of her stunned, stricken, watching crew.
"Captain," the doctor started to argue.
"That is _Ship's_ Captain," Ia corrected through aching teeth she was trying not to clench. "And unless you are Admiral John Genibes or the Admiral-General herself, you are _not_ in my chain of command. I will _walk out_ , thank you."
With parade precision, toe tucked behind heel, she turned. The JAG commodore stood just a few meters away, still holding the weight of her jacket and cap in his hands, elbows braced at his sides and sober respect in his eyes.
That respect gave her the strength to stride forward, stop in front of him, and lift her cap. She made it almost to eye level without grunting, but the last dozen centimeters _hurt_. Focusing her movements through the pain, Ia squared the black cap on her head, then reached for her jacket. Commodore St. Stephen moved first, unfolding it so she wouldn't have to do it herself.
He started to hold it open for her. Instinct warned Ia that if she turned and shrugged her arms back to slip them into the sleeves, it would take too long, and she would scream. That would be bad for morale. Jaw clenched, nostrils rushing air to and from her lungs, she plucked the heavy jacket from his hands with crossed wrists and swirled the heavy black gabardine up around her head.
That hurt, too. It hurt to the point of tears in her eyes, as bad in some ways as that hole in her shoulder from her enlisted days. But the slick lining helped slide the sleeves down over her arms, allowing her to shrug forward—a less painful movement than shrugging back—to settle the coat in place. Her agony escaped as a faint grunt, but only a faint one. She had to pause a few seconds, just to be able to breathe, before lifting her fingers to the lapels to adjust the lie of it. Taking the time to button her coat in place was an act of masochistic hubris since it meant she would have to _un_ button it later, but Ia did it.
Only when she was properly attired once again in her Dress Blacks did she lift her right hand to her brow, giving the commodore a salute through the ache dominating her senses.
"Commodore St. Stephen," she growled, teeth clamped shut against the pain. "I respectfully request permission for the Damned to depart."
He lifted his own arm in return, saluting her back. "Permission granted, Ship's Captain. You and your Company are free to go."
Again, she turned, this time a quarter turn to her left, to face the front of the assembly stage. "9th Cordon, Special Forces!" she snapped, her voice echoing off the walls. "Ten-hut!"
They snapped to their feet, this time with more speed and unity than they had displayed two hours ago in the _Hellfire_ 's boardroom. Ia nodded and stepped forward to the edge of the stage, placed a meter above the main floor.
"You heard the Commodore. We need six hours to fix the _Hellfire_. I am giving you _five_! Doctor Mishka, Private Attevale, take charge of Private Sung's gurney," she ordered, pointing off to her right where the hoverbed waited. That hurt, too; ohhh, that hurt, but she did it. "The Damned take care of their own, and we will _not_ leave our crewmate behind. We have lives to save, meioas. Move out!"
Stepping off the dais and dropping to the floor...was a dumb, foolish, stupid move. She caught her balance when she landed, reflexes more than adequate for the Terran Standard gravity on board, but the jolt seared dragonfire from thighs to nape, reigniting every single lash wound. For a moment, the edges of her vision blurred, bringing back that strange glow as she forgot how to breathe. With a force of will, Ia dragged in a lungful of air, squared her frame, and strode for the steps.
From the wide-eyed stares of the members of 1st Platoon A and B Squads, seated in the first row, she probably looked as pale as her hair. Wisely—or maybe out of fear—they did not offer to help her as she marched up the shallow steps of the lowest tier. They just peeled out of the seats and joined her, forming themselves into tight ranks, three Squads wide, with just a bit of murmured direction from Sergeant Halostein, who took up second place at her back.
He was joined within a dozen meters by Lieutenant Commander Harper. No one spoke, not even her first officer, while Ia led them from the assembly hall to the banks of lifts which that lead to the level attached to their gantry spoke. But as they waited in silence for the first of the elevator cars to reach their level, Meyun drew in a breath and faced her.
"No," Ia stated, cutting him off. Knowing what he was going to say.
He said it anyway. "They shouldn't have done that. You shouldn't have to be punished for _our_ mistakes, Captain."
She turned, still using the straight-backed toe-and-heel moves from parade maneuvers, since those moved and thus hurt her back the least. Staring into his brown eyes, she repeated herself. "No, Lieutenant Commander. Do _not_ try this argument with me."
"He's right, sir." The agreement came from Corporal Bagha. The ex-Sharpshooter moved forward, her brow furrowed in a frown. "It isn't right. You shouldn't have to suffer, just because—"
"It doesn't _matter_!" Ia snapped, unleashing some of her pain as rage. She checked herself, breathed through her nose, and gentled her tone as the last of the Damned's stragglers caught up with them. As did a few of the others dismissed from the assembly hall, drifting close enough for a look at the Damned's CO in their wide-eyed, morbid curiosity. "It does not matter," she continued more calmly. "The pain I have suffered today? Does. Not. Matter.
"I have _told_ you what we need to do. _Shown_ you what we need to do. A _hundred_ lashes of the cane on my back could not hurt me more than the pain inflicted by _your_ lack of faith. Did you join the military because you longed to follow someone's orders? No. Most of you _did_ join to try and make this galaxy a better place, didn't you?" she asked. She kept it a rhetorical question, though she did let it hang in the air for a long moment. "Nothing I can do, nothing I can say, nothing I can _suffer_ will change you.
"Only you yourselves can do that. _You_ have the power to be the best Damned soldiers you can be." Behind her, the first of the lifts arrived. Ia ignored it, shifting her hands. She tapped her inner wrist in emphasis, ignoring the pain from the pressure of her jacket on her back. "If I thought that bleeding myself dry could have a single scrap of effect, I'd slit my own veins and bleed away.
"But I _cannot_." She hardened her tone, not so much to punish her listening crew as to make sure her words carried to as many as she could get to hear her. "If you want to know _why_ the pain of my back does not matter, then look into your souls and ask yourselves, why did you become a soldier? Why?
"I am willing to place my weapons, skills, body, mind, and even my _life_ if need be between all the innocent lives in the Alliance and all the horrors that threatens it. What are _you_ willing to do? Until you _do_ know what price you are willing to pay...? _No._ Do not speak of this to me. We have work to do." Turning on her heel, she reached out with her mind, reopening the lift doors just as they started to slide shut.
Silenced by her words, Harper followed her inside. So did Halostein, and the first half of A Squad, 1st Platoon. Her last view of the others was of the front rows of her crew shuffling awkwardly toward the lift doors on either side, none of them willing to meet her gaze.
Harper did look at her. He looked away as she glanced at him out of the corner of her eyes, but when he did it again, she sighed.
"...Will you be alright?" he finally dared to ask.
"I'll make it back to the ship." There was no room for compromise in her tone.
He hesitated a long moment, but she knew it wouldn't last. Indeed, it took him only three floors in ascent before he muttered, "Well, I'm only asking because you look like you're going to puke."
The pair of enlisted men standing directly in front of the two officers glanced warily, furtively at each other.
"That's because I _am_ going to puke if I move the wrong way or the car stops too hard."
Brown eyes and blue eyes and hazel all snuck quick, wary little looks at her. The two privates subtly shifted to the left, moving more in front of Harper than Ia, who stood in the corner of the lift.
She gritted her teeth. It was funny. It was painfully, grievously funny, watching them trying to be subtle about it, and she did not dare quiver even once, or the agony invoked _would_ erupt as nausea. She did dare give them a warning, though.
"...The first one of you idiots to make me _laugh_ will have to clean it up."
They froze at her growl, not even daring to breathe until the lift drifted to a gentle stop. Quiet as mice, they slipped out of the lift, giving her room to disembark. Ia focused her will on the long walk back to the _Hellfire_ , breathing as slowly and deeply as she could in the need to manage her pain.
When her assigned twenty-four hours were up, she would be free to stop suppressing her biokinetic urges. The welts and bruises and cuts would all vanish within an hour or so, leaving healthy flesh in their wake. Between then and now, she had twenty-three hours and several minutes to wait. Compared to the pain of losing everything permanently...it was a minor inconvenience at most.
# CHAPTER 15
_The fight on Oberon's Rock was a slice of replayed hell. Even more ironic, once again it wasn't the Salik, but the pirates who attacked. How many times have I aided that damn dome colony by now? Eight? Nine? Twelve?_
_Most of us were on the ground for it. Oberon had some crack security guards by that point, but no backup, as the Battle Platform had to be rotated among it and the other two settlements. One hundred mechsuited soldiers dropped to the planet, myself included. Ishiomi flew the_ Hellfire _for the space half of the fight, and even with a reduced crew, she and the rest of the 1st Platoon did an excellent job at hunting down and picking off the fighter ships trying to strafe the domes._
_The only thing I regret about that fight was the loss of Private Hollick. I warned him that if he went into that train tube to rescue those people stuck in that car, there was a strong percentage chance that he'd die. But he just pointed out that those cars could hold twenty or more civilians, and that it was a chance he had to take. He got them off the train and into the evacuation shaft, but he was the last one headed down the hatch when the tunnel blew._
_According to the witnesses, the force of the air pressure swept him out before the emergency systems on the airlockcould close. We never found his body, but he did save their lives. He died a hero, and I still honor him for his sacrifice._
_~Ia_
FEBRUARY 23, 2497 T.S.
SIC TRANSIT
( _Sucked out an airlock?_ ) Ia demanded, eyeing her late-night visitor askance. Clad only in tank shirt and underwear, she padded over to the drink dispenser and poured herself a cup of water. ( _I thought you were going to still be up in the car when it blew, attempting to rescue that little boy's pet!...You want anything?_ )
( _The drawback to having a matter-form is that it requires the distasteful presence, care, and feeding of a digestive tract, and so forth. I'll have a glass of water,_ ) Belini ordered. She had manifested in a bright yellow halter top and matching tennis skirt. ( _Besides, that little boy was clinging to his pet. Damn near throttling it. I had to improvise._ )
Ia watched the Meddler toss back half the water, then sigh and rub at the back of her neck. Lifting her own cup, she sipped at the chilled, hydroponically recycled liquid. She also snorted mentally. ( _You just wanted a dramatic,_ witnessed _death._ )
( _Hey, the guy saved the timelines for both of us. He deserved a hero's death,_ ) Belini retorted. She lifted her mug in salute. ( _I also flitted off and checked on him. He's doing just fine; he's mentally fit and on course for the path he's supposed to take._ )
( _Thank you for checking and letting me know that the personality rewrite is holding,_ ) Ia told her guest. She had to smother a yawn as she did so, though. Her back no longer ached whenever she lifted her arms, but only because her biokinetics had worked hard to repair all that damaged flesh. Adding in the stresses of combat and her extralong days, trying to repair the breaks in the timelines, hadn't left her with a lot of energy to spare. ( _Excuse me...I'd love to chat, really, but I'm still fully matter-based, and I'm afraid I need at least five more hours of sleep._ )
( _Fully matter-based, yeah...You need to do something about that. The sooner, the better,_ ) Belini added, pointing with her cup in warning before draining it dry. ( _Miklinn is casting aspirations on my faction-simmerings, since you're technically Albelar's get. I'm very much neutral to Albelar, not faction, and I have to remain that way for now since he's now milking that "Third Human Empire" you've been touting, while I'm solidly established with the Second Empire set._ )
Ia chuckled softly. She stifled another yawn. ( _Just you wait until the Third Empire hauls the First under its wings...'scuse me...Everything is going to change when that takes place. As for my manifestation, I've been working on it, and I will be able to manifest when I need it most. In the meantime, I'm waiting for my replacement for Hollick's slot to catch up with us. And tomorrow, I have to show up at the Wake. Something involving Harper is going to make the crew a lot happier if I do...but because it's Harper, I don't know what._ )
( _A Wake for Hollick?_ ) Belini asked, brows lifting. ( _Mind if I drop in? I'd like to pay my respects, as you matter-people put it._ )
( _I'd rather you didn't,_ ) Ia told her, adding a tinge of regret to her sending. ( _Whether or not you'd be tempted to let something slip, you're one more anomaly I'd have to ask the crew to keep their mouths shut about. They're already carrying a couple heavy burdens regarding Hollick, Sung, and N'Keth...and it's not that kind of a Wake. That's just the nickname my morbid little sense of humor came up with to call the parties we've been holding on board more or less every week—since the crew almost never gets real Leave._ )
( _Ah. Pity. I like the Irish. Dour and delightful, cheerful and glum, pragmatic and myth-filled...lovely sets of contradictions, the lot of 'em._ ) She dropped onto the couch with a light bounce and spread her arms across the padded back.
Ia eyed her. ( _Why_ are _you here, anyway? Do you need to be kicked out the airlock again? I'd be more than happy to shoot you._ )
( _Nah. I have a second anchor string tying me to where I was. I just came to tell you what I've told you._ ) She let her smile fade and waggled one finger at Ia, for a moment looking a decade older in her sobriety. ( _Do not discount my advice about your manifestation. Do not wait for an extreme moment, expecting your anxiety or need to aid you in the change. If you reallywant to impress my people, you'll have to slip into and out of your energy side as freely and fully as we do._)
Suiting action to words, she shifted with a blip of light. The sphere hovered through the sofa for a moment, then slid out of sight. Ia blinked, frowned, then called out, ( _Hey! Are you at least going to give me back my mug?_ )
( _Only if...what is that delightful, old Human phrase? Only if you want me to "take a dump" on your carpet. See you next time!_ ) Pixie laughter mocked Ia as Belini slipped away.
Rubbing at her face, Ia crossed over to the dispenser, clipped the other mug into its self-cleaning holder, and padded back to her bed. _Damned pixie-shaped Meddlers..._
( _...I heard that!_ )
( _I expected you to._ ) Slapping the lights off, she shut the door and returned to her bed.
FEBRUARY 24, 2497 T.S.
SIC TRANSIT
The surprise involved Harper, and that was all that she knew. It wasn't just Harper, either, but it was clearly centered around the man. Whenever she tried to delve into what certain other members of the crew were doing, all she got was blurred mist in return. The only thing she could see was a reasonably high probability that the crew would be happier after whatever-it-was took place. Provided she didn't screw it up, of course. The alternative timelines where she did included grumpy looks and scowls aimed her way.
Unsure exactly what she might have to face, Ia debated wearing trousers, then gave up and donned her new cheongsam-petal dress. Leaving the feather-stitched turban off, she combed back her trimmed, jaw-length locks and walked down to the Wake Zone. It wasn't ceristeel-plated armor, but the awkwardly admiring looks she received assured her it was an effective defense against whatever might come.
_If a good defense is a well-fitted dress, then that'll have to do._ Smiling slightly, she dipped her head a few times in greeting, moving down the stairs. Once again, she had to hunt down Harper, but when she reached the bottom of one aisle and started up the next, she couldn't find him. What she did find, in an upper alcove decorated with plexiextruded fish—this week's theme was "Dabin's Cerulean Sea"—was a clutch of soldiers.
Or rather, spies. All four of her DoI- or Admiral-General-appointed spies lurked in the booth-like space. The first and foremost was Lieutenant Oslo Rico, who rose slightly and bowed over the table as she peered into the makeshift doorway. He had donned a pair of light blue slacks and a bright blue-and-yellow-flowered shirt. On his large frame, it was the envy of any tropical-island gift shop.
The second was Private Second Grade Miryanapanaua "Gasnme" Kastanoupotonoulis. Her nickname was nothing more than an acronym for "Get a shorter name, meioa-e!" She looked lovely, with her short black curls wrapped in a sapphire blue ribbon and clad in a blue version of the V'Dan-style cheongsam. Where Ia's was stitched with golden feathers, Gasnme's had come out of the manufactory stitched with silver _dragnas_ , long-tailed, broad-winged, lizard-like things indigenous to the V'Dan homeworld, sort of like a palm-sized dragonling. She served in the 1st Platoon under Rico and had been asked to spy on Ia for the DoI.
Private First Class Bera Fonnyadtz of the 2nd Platoon was their third spy, appointed by the Admiral-General, the same as Rico. She had decked herself in blue jeans, a white tank shirt, and a black leather jacket. Then again, she always dressed like that when she was off duty. Spyder, her Platoon lieutenant, called her "a livin' lady Fonz" but Ia had no idea who or what that was. It wasn't important to the timestreams, so she hadn't bothered to look it up. Bera saluted Ia with the tall glass mug in her hand, filled with what looked like a root-beer float, and sipped on the straw bobbing in its depths.
The last of the quartet was Private First Grade Ulan Fa'alat, another DoI-appointed spy and one of the four leads for lifesupport on 3rd Platoon's watch. Ironically, he had dressed similarly to Fonnyadtz, in a white T-shirt, jeans, and black leather, though his was a half-length vest that fastened down the side of his chest in the V'Dan style currently fashionable among men, and not an actual Terran-style leather jacket like hers. He was also the one who spoke first, and not the resettled Rico.
"Hello, Ia," he acknowledged, leaving off her title since that was the Wake Zone rule. "I suppose you already know why we're here, being the Prophet...so it remains to be seen what you think of it."
Ia shook her head, shifting her hands to her red-clad hips. "Actually, no, I don't know. Whatever the four of you have been planning, it clearly involves Lieu...Meyun Harper," she caught herself, mindful of the Wake rules. "And anything that involves Meyun Harper has a ninety-five percent chance of _not_ being visible to me in the timestreams."
Rico's brows lifted at that, and Bera choked on her float. Fa'alat smacked her on the back.
Gasnme merely arched a brow in skepticism. "Why can't you see him? I thought you could see everyone and everything."
"Because there is _always_ an exception to every rule, Kastanoupotonoulis," Ia returned dryly, managing to get through the woman's real surname without stumbling. "Meyun Harper just happens to be it for me."
Fa'alat gave her a wry look. "Every rule? What about the laws of gravity? What goes up must come back down, on a planet."
"Telekinesis," she countered. "Greasy FTL warp fields. And the escaping axial radiation of a galactic black hole."
"At least she knows her science," Bera rasped, lifting her mug in salute. She coughed twice more, then cleared her throat. "So. Ia. If you really don't know why we're here...then why are _you_ here?"
"Because while I don't know what you've been plotting behind my back, I do know that you would be here, and that the 'big surprise' would be revealed as soon as Harper also gets here," Ia told her. "There's still that five percent left for me to see."
"He went off to visit the head," Rico told her, finally speaking. "But he'll be back." He nodded at a half-finished glass clipped onto the table edge to his right.
Footsteps warned her of Harper's approach. Shifting to her right, Ia dropped her arms, expecting him to squeeze in past her and resume his place on the bench seat ringing the alcove. Instead, he stopped at her side and stared first at her, then at the others.
"Okay...when I left, it was just me and Rico. Is there something I should know about, meiaos?" he asked.
Gasnme met Ia's gaze and shook her head. "He doesn't know, either, S...er, Ia."
"Know what?" he asked, hands going to his hips.
"It's simple. As the C...as _Ia_ already knows," Fa'alat explained, gesturing at her, "the four of us were either ordered by the Department of Innovations or the Admiral-General herself to spy upon our lovely commanding officer—you _did_ know that much, right?"
"Yes, I knew that much," Ia confirmed. She poked her thumb at the man standing beside her. "It's only things directly related to _this_ man I cannot see. Everything else is like an open betting ledger of odds and probabilities for me."
"Well, hold on to your favorite dice," Fonnyadtz warned her, releasing her straw. "We're about to throw the game in your favor."
Ia glanced at Meyun. He shrugged, equally mystified.
"Allow me to explain it?" Rico ordered, giving the other three a mildly annoyed look. "We've all talked to each other about this. Then we took turns running an informal poll of the others in the crew over the last couple of weeks. One and all, the crew and cadre agree that _you_ , young lady," he stated, pointing at Ia, "desperately need a personal life."
Ia wondered where he was going with that line of reasoning. She thought she'd made it clear enough through her words and actions that this ship's mission and its crew _were_ her personal life.
"This is particularly obvious because you almost never attend these Wake parties," Rico added bluntly, eyeing her back as she frowned at him. "Half the days of the week you're staying up for thirty-two-hour 'days' and pulling double and triple shifts. You never join in the fun and games in the rec-room facilities. You're a little too focused when you're practicing for combat against Helstead and such, and even our dual-purpose, shipboard chaplain-and-psychologist admits you're just a little too serious for your own good."
Her face heated at his words. Given that this whole meeting was tied into Harper's mist-spewing existence, she had the sinking feeling she knew where this was headed...and she wasn't sure whether to be mortified, furious, or flattered that they would care for her personal happiness that much. Or rather, that they would try to foist it on her.
Gasnme continued before she could argue the matter, however. She slanted both Ia and Harper a knowing look. "Now, given the way we've all seen the two of you interacting, and the looks you sneak at each other when you think nobody's looking, it's pretty damn obvious there used to be a fire between you two. And just as obvious, it could be rekindled if you tried. That is, if you both didn't have your senses of duty wedged up your buttocks."
Harper coughed, blushing. Ia drew in a deep breath and let it out. Definitely a conspiracy on board her ship. Neither of them could protest the other woman's bluntness about it, either, because this was the Wake Zone, where the rules of military protocols and conduct were meant to be tossed out the nearest airlock. By her own order, no less. "...Go on."
Fa'alat nodded. "What she said, si—uh, sister. It wouldn't be right for either of you to look for a personal life among the enlisted. That'd be breaking the rules way too far. But among the cadre...well, there aren't that many choices, and neither of you have looked at the others the way you look at...well, the way the two of you always look at the two of you."
A glance at Meyun showed his own cheeks looking a little flushed. Rico spoke up again before he could protest.
"Besides, _you_ spend all your working hours down in engineering, or out among the repair teams," he said, nodding at Harper. Then dipped his head at Ia. "Whereas _you_ spend most of your time on the bridge or holed up in your office when you're not needed to micromanage a particularly fiddly moment in time. There's no conflict in your respective workplaces. And since you've managed to contain your interest beneath a very professional demeanor...we've decided as official spies to exempt all future such things from our official reports. Officially."
Ia blinked. "You...what?"
Meyun recovered faster than her. He lifted one hand to her shoulder, though he addressed the quartet at the table. "You mean, if we _do_ decide to get personal as well as professional...you won't tell anyone? At all?"
All four of them nodded. Fa'alat added, "More to the point, the rest of the crew agrees and won't tell anyone."
"It's like these little parties," Fonnyadtz added, lifting her mug. "What happens in the Wake Zone stays in the Wake Zone. Total civilian-style anonymity. In this case, since Ia's the CO and is almost always on duty, we're letting you have the run of the whole ship." She paused, wrinkled her nose in a wry smile, and amended, "Except I _really_ don't want to catch the two of you going at it on top of the fish tanks in some lifesupport bay, or tangled up in some P-pod gunner's chair somewhere."
"So...none of you would object, or inform the DoI, or Admiral Genibes, or the Admiral-General," Meyun asked—and slid his hand from Ia's left to right shoulder, turning and swooping her into a dip, "...if I did _this_?"
Caught off guard and surprised by his mist-shrouded choice, Ia barely had time to squeak in shock at the sudden shift before his mouth descended in a kiss. It felt good. Way too good.
In sheer self-defense—as in defense of her innermost self, not her physical self—she reached up and pinched his ear. Yelping, Harper quickly righted her. The moment she had her balance, Ia poked him in his green-clad chest, face red from more than just being tipped over. " _Not_ in front of the crew!"
Unrepentant, he grinned. "Then there _is_ a chance I could do that elsewhere! Yes!...Er, yes, right?"
"Harper, let go of me." She sighed. He released her shoulder, his smile fading. Rico and the others started to frown. Holding up her hand, Ia forestalled them. "Let me check the timestreams. There is a _lot_ more at stake than just my personal happiness."
Closing her eyes, she checked the timestreams, silently asking if what they were offering, what Harper wanted...what she wanted...was possible. Little tufts of mist shrouded their paths as she checked. Under the provisions of "not in front of the crew" and "only when an hour or two could be spared..."
Finally opening her eyes, she nodded. And squeaked again as he wrapped his arms around her for an enthusiastic hug and a kiss on her cheek. Meyun continued to hold her, cuddling her sideways against his chest.
She finally had to pinch him again, this time snagging a bit of skin at his waist. Jerking at the sharp nip, he acceded to her silent demand, releasing her. "...Alright! _Not_ around the crew," Meyun said. "Which would be my personal preference, except I'm feeling rather happy right now."
"There will be other rules. Only when the time can be spared. Only when the—" His finger pressed against her lips. Looking at him, Ia sighed and gave up.
Harper nodded. "I know, I know...and it's not like I'm going to throw you over my shoulder and carry you off to my quarters right this minute. There's another reason for being discreet. Private Nesbit."
The others blinked and frowned, not recognizing the name.
Harper rolled his eyes, giving Rico a briefly impatient look. "Oh, come on. The new member of the 2nd Platoon? Private First Class Maximus Nesbit, of E Epsilon? Hollick's replacement? He isn't here yet, and he hasn't gone through what this crew has, but he will be joining us soon.
"Harper's right," Ia agreed. "He hasn't seen what we've seen...or made the promises to ourselves that we have. Until we know he can keep his mouth shut on this ship's secrets, we'll have to continue being discreet."
"But at least you'll have the rest of the crew behind you," Bera pointed out. Fonnyadtz sipped at her float again, draining the last of the soda, then unclipped a spoon from the table and poked it at the globs of ice cream in her cup. "You really do need to indulge in some downtime, Capta...er, _shakk_. Meioa-e. Even if it's just _talking_ with our favorite engineer."
Fa'alat flapped a hand at them. "Go on. Get out of here, you two. Go do some 'talking' with each other. We're done with the two of you...and if you know what's good for you, you'll go off somewhere private and _actually talk_ about this with each other. Good downtime relationships begin with talking, after all."
Fonnyadtz snorted. "Talking. Yeah. Just so long as they _do_ get around to swagging each other."
"You know, if this wasn't the Wake Zone, meioas, and if I hadn't the _carte-blanche_ authority to bend the rules for you," Ia warned the quartet of conspirators, "you'd be in an entire asteroid field's worth of trouble. I _ought_ to—"
Meyun covered her mouth with his hand and finished her sentence for her. "She ought to thank you, but I'm afraid she'll be too busy chasing me back to her quarters, where she's going to spend a good fifteen minutes yelling at me, while I just sit there and grin."
Blushing, Ia elbowed him. Or at least tried to. Her heart wasn't entirely in it. Tugging her out of the alcove, he released her mouth, wrapped her arm around his elbow, and led her out of the Wake room. Curious looks from the others in the rec room morphed into amused looks and encouraging smiles. No one stopped or even spoke to the two of them, though.
"Our official clutch of spies is right. I do believe we _should_ talk about this," he murmured in her ear, guiding her into the corridor.
"I had half an hour set aside for the Wake. Or rather, about twenty minutes of it left," Ia stated. Patting his arm, she sighed and released it. "All we'll have time to do is talk...and I cannot spare half an hour every single day."
"Yes, but at least now we'll have something to talk about," he allowed. "By the way, I like the dress."
She smiled. "Thank you. So did Emperor Ki'en-qua."
Meyun mock-bristled at that. "What, the Emperor of V'Dan gets to see my commanding officer in a fancy dress long before me? Why, that's an outrage! I demand twenty minutes of your time, Captain, in order to complain about it to you."
In spite of herself, Ia laughed. Grinning ruefully, she gestured at the lift they were approaching. "By all means, you can have your say. In my office."
"In your quarters, so you can bring out and show me everything else you've bought since leaving the Academy," he bartered.
"Shh," she admonished him. They were alone for the moment, but she could foresee that someone was riding the belt-lift. "Not in front of the crew."
"Sir, no, sir. Definitely not, sir, and I wouldn't admit I've even dreamed of it, sir," he muttered. But he smiled as he did it.
MARCH 9, 2497 T.S.
ZARRATA VII
SSKENTHA SYSTEM
The man Master Sergeant Sadneczek escorted into Ia's office was lean, average in height, and looked somewhat cocky, with slouching shoulders and an amused half smile hovering around his goatee-framed mouth. He did not come to Attention until Grizzle cleared his throat, and only lifted his hand to his brow at a second _ahem_.
Ia saved her work, powered down her main workscreen, and returned the salute, though she didn't rise. "At Ease. Have a seat."
Resuming his slouch, Private First Class Nesbit settled into the nearer of the two chairs across from Ia. "Cap'n. I heard you're Bloody Mary. 'Zat true?"
"Welcome aboard, Private Nesbit, and yes, it is. But right now, we're here to talk about you. Understand that you have some very big shoes to fill. You will be taking over most of the roles performed by the late Private Hollick, who died a hero's death on Oberon's Rock," Ia stated, folding her arms and bracing her elbows on her desk, "This crew misses him greatly. He was a middle-aged man with a steady mind and a calm personality. The Damned will take a little while to get used to you, a young man with a twisted sense of humor, in his stead."
"The Damned, sir?" he asked her.
"Ia's Damned, 9th Cordon Special Forces. My crew is loyal first and foremost to me, then to the cause I serve, which is the salvation of all life over the long-term view, and _then_ to the Admiral-General and the Space Force. The Admiral-General knows and approves of my missions and has given me free rein to carry them out." That was a bit of a stretch, but technically with her _carte blanche_ being based on her precognitive abilities, it was the truth. "We have been through more combat in the last year than three-quarters of the fleet in the pursuit of that goal, and have a higher kill rating per battle than any other vessel short of a Battle Platform or a capital ship.
"Hollick was the most loyal member of my crew, though not the best warrior. I told him the odds of success for his last mission weren't good, but he chose to go on it anyway in order to save lives. Which he did. But however well he may have performed in combat, your biggest obstacle, Private," she continued, "will be filling the heroic shoes Hollick wore while he was serving on board this ship during your average day-to-day duties. You will be worked harder, expected to do more, and at times serve for far longer hours than at any other point in your military career since leaving Basic."
"'M not afraid of hard work," he told her, his accent not quite as thick as Lieutenant Spyder's. He flashed her a grin. "I jus' don't like it."
"Joke all you like, but accept that that will change," Ia warned him. "Now, I know you've been fitted with a mechsuit, having served two tours on the Blockade on board a Harrier-Class Delta-VX. I served on one of those myself, so I know you're used to long hours and dangerous encounters. The nice difference about serving aboard the _Hellfire_ is that I run three duty watches instead of two. The not-so-nice similarity is that you can and will be called to active duty at any hour, at any time. I will, however, be able to give all of you a fifteen-minute warning, save in the most extreme of probability cases.
"The acceleration rating for this ship is approximately the same as a Harrier-Class," she added. "So the Lock and Web Law is vital to uphold. Not just on the off chance that God rolls the dice and comes up with an extremely low probability number, but because it is important not to _forget_ that you have left things out. I'd rather that hairbrushes and such didn't go flying about during combat, punching holes in my bulkheads."
He lifted his brows but didn't say anything, just shrugged. Her warning was deliberately chosen since it would remind him to secure his actual hairbrush within the next month to avoid that very scenario. Touching the comm button on her workstation, Ia spoke.
_"Private Schwadel to the Captain's office."_ Releasing it, she opened a drawer and extracted a datachip. "This ship does not fly a standard patrol route. We do not have a specific sector to protect. Instead, with the aid of my precognitive skills, we aid those battle zones desperately in need of some extra help, regardless of whether or not it lies in Terran space...which you may have already noticed since you were just shipped all the way out to a Tlassian space station for pickup.
"You have five hours to stow your things and get familiar with your coming duties. This is your Company Bible, which I expect you to study, memorize, and take to heart, just as the rest of my crew have already done," she added, holding out the chip. "The first chapter has been appended with a note regarding the next forty-eight hours, with advice on what you should focus on first and when to do it, to get you integrated quickly into life on the _Hellfire_...and yes, I _am_ that temporally accurate. Use your arm unit's chrono to make sure you stay on schedule.
"Three days from now, we'll have time for another Wake. Those are the onboard parties the Damned throw because we do not have time to stop and enjoy actual Leave on other stations, domeworlds, or planets. Anything that happens within the Wake Zone is to be addressed strictly from the standpoint of everyone being a civilian, so be very careful not to address anyone by their rank.
"As for recreation, leisure, and hobbies, I believe Private Preston-Aislingfield is the current game master for the ship's tabletop role-playing sessions. She'll get ahold of you soon to discuss your gaming experiences and character preferences, and will help integrate you into the group. That, more than anything, will bring your fellow crewmates to know and eventually trust you, which is why I point it out to you. Please remember to use electronic dice, not physical ones. Like hairbrushes, the physical kind can do a lot of damage during sudden maneuvers and are harder to keep track of in an emergency. Now, do you have any questions?" she asked him.
"Yeah," he admitted. "If you're really a prophet, what'm I thinking?"
She gave him a flat look. "That would be telepathy, not precognition, and _that_ would be a violation of your sovereign sentient rights to mental privacy. Not to mention my own mothers would fly out here to paddle my butt for being so rude."
He grinned at that, visibly pleased that his new CO had a sense of humor.
Ia didn't disabuse him of that belief. "Still, the probabilities of what you were about to say out loud were the number seventeen at seventy-seven percent, followed by yellow daisies at fourteen percent, fried peanut butter and banana sandwich at eight percent, and undecided for the rest. Your teammate and roommate is about to arrive. The two of you are both electronics engineers. I suggest you use that basis to start getting along as fellow workers and build trust and friendship from there.
"You will be a member of the Damned for a long time to come," Ia warned him. "As Private Sung discovered recently when he violated Fatalities Five and Thirteen, I can give you all the precognitive warnings in the universe until I run out of breath, but it is up to _you_ to carry your orders through. Make sure you don't do anything stupid, or it will be the last thing you get to do. And be mindful of your actions when it comes to corporal punishments. If you incur any strokes of the cane, _I_ have to suffer them, too.
"There's a recording in the ship's library of Private Sung's caning, followed by my own, should you doubt me. And if that isn't enough to warn you, the rest of my crew will be inclined to lynch you if you screw up rather than see me go through another caning because of something you did," she told him. "I'd rather not have to fill out the paperwork on a lynch mob, but I will I if I have to, so kindly keep that in mind.
"Other than that, welcome aboard," she repeated, as her door chimed, then slid open. "Private First Class Maximus Nesbit, meet Private First Grade Derek Schwadel, your roommate and team leader. As far as the chain of responsibility goes on board this ship, Schwadel is senior-most between the two of you since he's served with me longer. Schwadel, you'll have four hours and fifty minutes to show Nesbit around before we'll reach our next battle zone. Spend them wisely. Dismissed, gentlemeioas."
"Aye, sir," Schwadel stated, and gestured for Nesbit to follow him out of her office. "C'mon, this way. Let's get your gear stowed."
Ia didn't watch them go. She powered her screen back up and resumed picking her way through the last few tangles in the duct-taped version of her goal, changing various prophecies destined for Earth and the Afaso, Sanctuary and the Free World Colony—not too many of the latter, thankfully—and assorted missives meant for targets among the various other governments and species.
MARCH 17, 2497 T.S.
SIC TRANSIT
"Let's see...power cables here...floodlights for the red, green, and blue spectra over there...and rare-earth magnets over _here_ ," Harper muttered to himself.
He moved around the clutch of men and women gathered in the bow storage bay that used to hold crysium sprays, and which was now lined with stacks of locked storage chests awaiting delivery. Some had been shipped off already, but others had not. Shifting MacInnes by her arms, he guided her onto some invisible mark seen only by him, then moved on to the next.
The men and women summoned were a mix of privates, a corporal, and the petite lieutenant commander. The only thing they had in common, aside from the obvious items like being Human and members of Ia's crew, was that they were one and all middling to strong psychics.
"Right," their chief engineer finally said. Clapping his hands together, he eyed their positions, then hustled to one side and opened the crates resting on the deck. They did not match the crates carrying her crysium wreaths. Instead of peach-tinted, coronet-like rings, he pulled out a variation on the crystal-mounted, brass-bound, somewhat oversized weapons he had been working on for months. It sort of looked like a stylized, oversized _pi_ symbol shaped into a brass-and-crystal gun. "Here we go...not at all like the originals, but this should work even better...
"Now, according to my calculations, four or five of you should do the trick, so we'll try five to start. Better to slightly overkill than underkill, and have to retry. Helstead, MacInnes, Crow, Teevie, and O'Taicher, you are each fairly strong psychics. I need you to _be_ psychics at this point in time. Each one of you will take up one of these guns. The safety is here, the trigger is here, and you wrap your hands around these crystalline bits," he added, bringing over the first gun so that he could stand in front of the semiarc they made and point at each bit. "Obviously, the thing you want pointed at your target is this crystal shaft out here."
Eyeing each other, the men and women in question picked up the bulky, odd devices and tried to settle them like weapons. MacInnes's arms were long enough, but Helstead's were a bit short.
"Sorry the front grip's so long, but I couldn't reconfigure it because of the resonances. You can rest this back bit on your shoulder here to help balance the gun, like this." Lifting it to his own shoulder, Harper displayed how to hold the thing, turning so that it was pointed off to the side. He lowered it again and showed them where the eight e-clips had been already slotted along the underside of the canister-thick barrel. "These clips are just the initializers, like spark plugs for an old combustion engine—and yes, it really does take that many e-clips, Teevie."
She blinked at him, her lips still open to ask her unspoken question. Licking them, she asked instead, "How did you know what I was going to say?"
"Because the Captain already showed me. Now, according to my source materials, I can get a much greater efficiency if I use mid- to high-ranked psis. And according to Helstead, who has been very helpful in testing the prototypes," Harper added, "you will probably feel an odd tingle, and maybe hear a faint chiming or humming sound. That just means the gun is working. You will also feel drained afterward, but actually using it won't cause any real harm."
Harper then handed the weapon to Helstead, who grinned and hefted it onto her shoulder. "Wait until you try this, meioas. If we get it right, the end result's a muckin' trip, and I can't _wait_ to see it happen."
Ia stepped into her own designated spot as Harper checked the grips of each volunteer. Waiting for him to finish, she caught their puzzled looks. "Harper, did you remember to _tell_ them what they're about to do?"
"Nope. I didn't want them spreading rumors around the ship. Okay," he said, handing the last oddball weapon to O'Taicher. "Everybody, find your safety switches and flip them off. That will automatically start the weapons charging. Then grasp the hand grips where I showed you, take aim at Captain Ia, and on my command—so you do it all at the same time—you will fire your weapons at her."
The others exchanged very dubious looks. O'Taicher frowned at their first officer. "Are you bloody _nuts_? That'd be Fatalities Thirteen and Twenty-Two!"
"If you don't shoot me, Private, _that_ would be Fatality Five: Disobeying a Direct Order," Ia pointed out gently. "Besides, it's not a case of Friendly Fire _or_ Attacking a Superior if you're ordered to do it."
MacInnes shook her head, muzzle still pointed at the ground. "I can't do it. I won't. Not without a direct order from _you_ , sir."
"Fine," Ia agreed. "Everyone, I order you to remove the safety locks, aim your weapons at me, and at Lieutenant Commander Harper's countdown and command, I order you to pull and hold the triggers of your weapons, firing them at me until Commander Harper gives you leave to stop. Any questions?"
Crow and Teevie exchanged looks. Corporal Crow shrugged and lifted his gun to his shoulder, sighting down the top of the brass barrel at his CO. "...It's been nice knowing you, sir?"
Ia smiled wryly at the jest. So did Harper—until he flinched, realizing he was in the way, and quickly ducked out from between Ia and the armed members of their crew. That made her grin.
"Right...right. Alright. Safeties off," he ordered, firming his tone into an order. "Take aim at Captain Ia...and at my command, pull and hold the triggers. In three, two, one... _fire_."
Helstead pulled her trigger right away. MacInnes was next. Nothing seemed to emerge from the guns, nothing in the visible-light spectrum, but Ia felt each impact. Each unseen beam felt like a jolt of electricity, like a bath of warm sunlight, the pull of a magnet...or rather, like the touch of a mind. The touch of life-energy.
The first one tingled. The second itched. Crow and Teevie shook their heads and fired. The third and fourth unseen beams altered everything. The bodies of her crew members started glowing, the power cables gleamed...and the floor and the walls and the boxes and crates shimmered, turning into tissue paper.
With the alteration in her perception, she could see the guns firing in bright, golden white lances that swirled at their edges with flickering hints of rainbow colors. Time seemed to slow down, and the white, Sol-spectral glow of the overhead lights stretched out, taking on elongated, prismatic hues. She watched, wide-eyed and fascinated, as O'Taicher exhaled in a swirling smoke cloud of heated gas, and remembered where she had seen this before.
_When I was enraged by Sung's disobedience in battle...and then later...later, when I learned to let go of the pain from the..._
Sighing, O'Taicher tightened his finger on the trigger, firing the odd weapon at her. The fifth beam struck with blinding intensity, inadvertently aimed right at her eyes. On instinct, Ia inhaled, focusing on the feelings and letting go of her mind. She embraced the energies, embraced, absorbed...and slipped. Slipped, disintegrated, free-fell, and coalesced.
Not too unlike the downward-around-and-out flip she normally made with her gifts and her mind, save that this was a three-dimensional aerobatic twist made with her body as well as the rest. Her body, which no longer had substance, but which now felt whole in a way she couldn't confine into words.
"Holy _shakk_!" O'Taicher dropped his gun, releasing the trigger as it fell. The metal clattered and the crystal chimed against the deckplates, but the gun wasn't damaged.
Ia could see the whole cargo bay, up, down, and all around, though most of her attention, her viewpoint, was still focused on the quintet. That quintet, and the six bodies beyond and the seventh to one side, still continued to glow. Mostly with thermal energy, in a warmth that...that reminded her of a muffin, of all things, soft and bready and sweet. The overhead lights were a glass of cool water. And the beams of the weapons, those were a meal injected directly into her bloodstream.
"Yess! Cease fire!" Harper ordered. He grinned—glowed—and raised his fists in the air. _"Success!"_
Ia opened her mouth to say something, to respond, but nothing happened. Her sense of self moved, but only in the way that a ball filled with liquid, or maybe of plasma, might swirl and shift. She focused again on the dozen Humans standing in front of her. Something was missing. Something...Ia realized with a swirling start that she couldn't _smell_ anything anymore.
Accustomed as she had grown to the scents of ship metal, cleaner, lubricants, recycled air, plant life, washed and unwashed bodies, as much as her active awareness of all of that had faded over time...she couldn't sense that anymore. She could _see_ the motes of molecules wafting off their bodies, but it wasn't a sense of smell as her now-missing nose once knew.
It was a strange side effect.
Harper clapped his hands, and she watched, fascinated, as the sound waves rippled out from his cupped palms in little pale violet shimmers. "Right! Captain," he stated, facing her sphere and bowing slightly, "can you hear me?"
She tried again to speak, then gave up. Reaching out with her mind, she broadcasted to him. ( _Yes, I ca..._ )
All of them doubled over, grabbing at their heads. Even Harper, who was the most mind-blind of the group.
_Oh._ Reining back on the effort she thought she had needed in order to project, Ia opened up a gentler, whispered level of projection. ( _Yes, I can. How's that? Too loud?_ ) she asked, focusing on each of the others. Helstead, still wincing, gave her a thumbs-up as she straightened. ( _Sorry, this is...new. I..._ see _things...It's all very..._ )
She started to turn around, still staring at the waiting lamps and cables and the pulsing green-brown coils of the rare-earth magnets. Their auras looked tasty. She didn't have a sense of smell, but she did have a sense of taste. She was also hungry. Very hungry.
"...Captain? Captain Ia, if we could kindly have your attention?" Harper asked dryly as she drifted toward the magnets. She heard/saw him sigh, another swirl of exhaled breath. "Great. I've reinvented the old attention-deficit disorder, only I've given it to a half-blooded Feyori."
( _Shh. I'm hungry._ ) Since she didn't have a mouth, just an all-over sense of self, she intersected that sense of self, that sphere, with the green-brown energies. Sure enough, it tasted green-brown. _Heh...like the way a Gatsugi would describe a dish of broccoli beef._ Inhaling the flavors, she supped from the magnets and drank from the overhead lights.
"Helstead, would you kindly shoot her?—No, not the psi-gun," Harper corrected. "Use the stunner I issued you."
At the backside of Ia's field of view, she watched the others hastily move out of the way, and the petite redhead pull out a handheld stunner. Its field was a single, invariable width, though it could be programmed for up to five different strengths. Raising it, Helstead aimed it at Ia, thumbed the controls to maximum, and fired.
It felt...It felt like getting hit in the back with a giant shepherd's pie. Or maybe a pot pie. Some kind of pie, meaty and filling. (Again!) Ia ordered, turning to face her, still soaking in the magnetic auras. She gentled her tone as the others winced. ( _Sorry...again, please. That felt good._ )
Helstead obligingly fired. Harper, on the other hand, snapped his fingers and pointed one at her spherical sense of self. " _Focus_ , Captain. No drifting off into an energy-based food coma. Focus!"
The third pulse of the stunner made her feel full. Instinctively, Ia shielded herself against the influx of energies, moving out of the immediate environs of the magnet. ( _...Enough. I'm good._ )
"Good," Harper praised her, and gestured to a spot next to himself, with a half-bowed sweep of his arms. "Now, get your floaty, silvery self over here so I can explain to you how to get back into your normal, Human, _matter_ -based self."
Swirling in a sigh, Ia refocused on him and drifted over to the indicated spot. ( _And how do I do that? And why should I try?_ ) she asked, mildly curious. ( _This is all rather fascinating. I'm in no hurry to go back._ )
Sighing, he boldly stuck his hand into her side, invading her sense of self. Shock rippled through her, not only from the touch, but from the energies and sensations that touch brought. Not only could she feel his chemical heat, and the faint hints of magnetism inherent in his cells, but she could taste the electrical impulses of nerves chattering back and forth with his muscles. The kinetic energy of his blood as it raced through his veins. The thoughts in his head.
The thoughts... _His_ thoughts, of just three nights before, when she had managed to scrape free one precious hour of time with him. Memories of their activities. Sights, tastes, sounds, touches...and smells, all evoked with a vividness and an intimacy that sprang from his emotions. Strong ones, and a source of energy all of their own. It was so much easier to read his mind like this...
"Shoot her now!" _Come back to me, Ia,_ he ordered in her mind, even as he slashed his free hand. _Focus on what you_ want!
Everything vibrated. Everything focused as their energy, mental/emotional/psychic/crystalline energy, flooded into her. For a moment, Ia's whole being quivered, struck like a bell...and then she fell, snapping back into her body. Snapping back into the memories of bliss _his_ memories invoked.
She hit the deck with a head-cracking _thud_. Uncomfortably aware of just how _solid_ and _separate_ each part of her bones and muscles, organs and blood felt, she grunted, shifted her limbs, and pushed herself up on one elbow. "Slag... _shakking v'_ slag," Ia muttered, head aching from its blow. "Slag, but that hurts."
Harper crouched and offered her both his hand and a wry smile. "Next time, try to land on your feet?"
"I _did_ land on my feet," she growled, blushing. Accepting his hand, Ia let him help haul her upright. "I just...slag...I forgot _how_ to stand. Just for a moment. Oh, laugh it up," she retorted, as Helstead shook with barely suppressed snickers. " _You_ try losing two of your five senses and your awareness of how things like muscles and joints work."
Letting go once she had her balance, Harper grinned unrepentantly at her. "Collapse or not, it worked! Welcome back to the land of the matter-based, my love."
She gave him a dirty look. "Why did you have to use _that_ set of images?"
He sobered, giving her a mildly chiding look. "My research notes stated quite clearly that the subject has to _remember_ what it's like to have a body and _want_ to return to it. Our primary source for information did a number of experiments on willing half-breeds, and _that_ was one of the best focal points for wanting to return. Now, gather your wits, rest for a few minutes, then we'll try it again."
"Ugh. Again?" Ia half muttered, half groaned. She knew he was right, but that didn't mean she was ready to go for it just yet.
"The more times you make the crossover and manifest with our help, the closer you'll get to figuring out how to do it on your own. _And_ how to come back on your own." The look he gave her was both a warning and a tease. "Because until you do, I will _continue_ to think those thoughts at you."
"I'm getting the feeling I'd like to know what _kind_ of thoughts he's thinking at her," Helstead quipped, eyeing her two superiors with an amused gleam in her hazel eyes. "Though I'm sure I could guess."
MacInnes, crystal-gun pointed at the floor, shook her head. "I'm not that suicidal, sir. We just created a Meddler out of our own captain, and I am _not_ that suicidal."
Her vehemence made Ia smile.
MARCH 19, 2497 T.S.
SIC TRANSIT FROM OBERON'S ROCK
GS 138 SYSTEM
...AND JULY 8, 2498 T.S.
The reading lamp behind her living-room easy chair tasted like brie. The milder inner bits of the cheese, not the outer ones. Even though Ia was firmly back in her matter-based body, she still had an awareness of the flavors of various energies around her. The smell of her own soap-scrubbed body, the slightly dusty fabric of the chair, and the faint hints of greenery in the air flowing through the vents from the amidships-sector lifesupport bay all grounded her firmly in her body, but the reading lamp still tasted like brie.
The datapad, on the other hand, was a spicy snippet of sausage. She kept having to stop herself from trying whenever the urge came back to nibble on it. Her matter-based body did not process electricity the same way her Meddler-based one did.
Yesterday, she had tried to eat energy from the electrodes at the command console while guiding the ship in a protracted starfight over Oberon's Rock, but all that had done was fill her with excess static energy and make her view of the bridge start to glow. It hadn't done anything for her physical hunger. In a way, "tasting" the energies in this solid body was very much like smelling the culinary efforts of a restaurant while stuck out on the sidewalk with no credits in her pockets.
_Right,_ she thought, saving her latest prophecy. _That's that, and it goes into the Afaso delivery file. Now for the preshipment of needed supplies to Dabin...though I'd better double-check that they're still going to be needed. I'm still searching for more of those little frayed tendrils from when Sung snapped the threads of fate._
Closing her eyes and focusing her mind away from the cheese-and-meat glows, she flipped down and in, landing next to her own life-stream. Accessing the timestreams was also easier, brighter, and clearer now that she had manifested. The grass was a lush blue-green, the waters clear and mirror-like even where they rushed and flowed fastest, and the sky was a beautiful shade of blue.
Skipping down the bank, Ia stopped at a point a week or so past a heavy knot of mist. That was one of the nexus points, where her choices would be so vast and her energies would be so low, the Ia of that era wouldn't be able to foresee what to do. Her future self would have to rely upon her training as an officer, her instincts as a warrior, and her grasp of overall strategy to guide the tactics of that moment.
The moment she did want, which involved herself working in the local capital, was downstream from that point. Her elder self worked diligently in the version she selected, filling out orders for the struggling, battle-wearied soldiers under her command. Stepping down into the water, Ia readied herself to read those orders since she wouldn't be able to read her own thoughts. _So let's just see exactly what the Afaso need to send and hide on Harper's homeworld..._
( _About time you showed up._ )
Ia jerked up out of the water and back into herself, fumbling to keep the workpad on her lap before her jolt could dump it to the floor. She looked around, wide-eyed, for signs of Belini, though that voice hadn't belonged to her. Alone in her quarters, Ia slowly relaxed, puzzled by what she had heard. _That...wasn't the voice of any of my crew members. In fact, it sounded like_ me, _but...not my_ own _thoughts. There's a distinct difference between my own thoughts and a telepathic sending._
Confused, she waited for another sending, but nothing happened. Mindful of the ticking of the minutes she had left in her day, she sank back into the timeplains and moved back to the designated insertion spot. Once more, she stepped into the waters of her future self...and once more jumped in startlement.
( _Don't freak again,_ ) she heard herself state, her tone dry. ( _You really_ are _hearing me—your future me—talking to you._ )
Blinking, Ia pressed closer to herself. Her future self pressed back, pushing her up out of the stream. Startled, Ia stumbled onto the bank, dropping to elbows and rump in the grass. Her older self stepped... _stepped_...up out of her own life-stream and crouched by her feet. And then grinned in amusement.
( _Don't you look shocked...Wait until you can see your expression from_ this _side of things,_ ) the future version of Ia stated. She held out her hand. ( _Come on. Sit up. I'm going to share with you the list of things you'll need them to buy and stash, and a couple extra places to stash them._ )
Hesitantly, she extended her own hand, clasping the other version's palm. ( _How...what the...huh?_ )
( _About as coherent as I remembered it, afterward._ ) Tightening her grip, Future-Ia looked into her eyes, her amber ones showing hints of silver flecks. ( _Manifesting as a Feyori has put a fine polish on our powers. More to the point, it has_ finished _their blossoming. All you need to do now is learn how to use the extra bits. But_ be careful.)
Those fingers squeezed Ia's with a distinct pressure, reminding her that even in the timeplains, she was accessing them from a physical body, not a Feyori one.
( _Yes, you can now not only contact yourself, but read your own thoughts,_ ) her future self warned her. ( _But doing it casually will lead to doing it carelessly, and_ that _will lead to paradoxically induced confusion. You will be able to see this in the side-timelines, if you look. Resist temptation, don't try to contact yourself any earlier than this, and you'll be fine._ )
( _Ah...right. Right,_ ) Ia agreed. This _was_ herself, without a doubt. ( _Ah...best piece of advice for right now?_ )
Her future self smiled slightly. ( _Three things. Make doubly sure your agents among the Afaso get that portable hyperrelay purchased and hidden away. Avoid stepping into the life-waters of other Feyori—they_ will _sense your presence_ and _be able to speak with you from now on if you tried it. And_ don't _let Belini touch you. Resist her urgings to manifest for now. It may seem to cause problems in the Meddler scenarios, but you'll lose a powerful edge when Miklinn's little pawn-army comes for you if they know you can transform at any point before then._ )
( _...Right._ ) Even without touching the timestreams for that moment, not all that far upstream from this one, Ia could see the difference that sort of advantage would make. ( _Right, I know that particular possibility-scenario. I didn't think it was a strong enough probability, but if you say...er, if_ we _say so, then I'll go for it,_ ) she agreed, gathering her wits. Hands still clasped, she let her older self help her to stand up on the bank. ( _So, what's on that list aside from the hyperrelay?_ )
It came across as a pulse of thought. Ia embraced it, as familiar in flavor as her own, and carefully secured it in a corner of her upper thoughts. Older-Ia nodded encouragingly. ( _...Got it?_ )
( _Yes. And we'll both resist the urge to linger in your...our...presence. This is going to take a lot of thought to come to grips with,_ ) she added, letting go of her other self's hand.
( _Get some more crysium and make some special wreaths out of it,_ ) Elder-Ia told her. ( _You'll see the exact life-paths the new Rings of Truth should influence. You'll need to lose about a week's worth of sleep within a year, but the price isn't bad._ )
( _No, it wouldn't be,_ ) she agreed. Eyeing herself, a deep part of her mind wondered how this could happen. Another part prodded her into an impulsive move. Wrapping her arms around her older self, she hugged. ( _Thank you,_ ) Ia sent. ( _Thank you for all that you've done, and all that you're going to do._ )
( _Thank you yourself,_ ) she got in return, along with a squeeze, and a push back. ( _Now get back to work. And don't forget to sleep, and all that other self-care_ shakk.)
Ia peered at her future face, noting the shadows under her older self's eyes, the worry-frown beginning to crease her otherwise young brow. ( _Don't forget that for yourself._ )
Amused, the older version stepped back into the waters and vanished. Bemused, the younger one stared at the rippling stream, then shook her head and started trudging back up the gentle slope.
_That has got to be the_ weirdest _...!_ She stopped after a moment, taking the extralong span of seconds within the timestreams to just contemplate the awe of what had happened. _I just...I just_ talked _with myself. I've broken the Time-barrier of...of vidshow versus reality...! I never once checked the time_ plains _for temporal anomalies. Just the streams themselves._
"Slag me," she whispered, letting the winds of the prairie carry her stunned words across time. "Just freaking _slag_ me..."
# CHAPTER 16
_Interstellar warfare isn't won in a single year. Hell, ground-based warfare is rarely over in a single year, particularly not when the two sides are fairly evenly matched. The Salik and the Choya did their best to smash our support infrastructure. We did our best to smash theirs. They tried to shatter our fleets while strengthening and rebuilding theirs. We tried to shatter and strengthen and rebuild as well._
_Some of that was successful. Some of it was not. Lives were lost, prisoners taken and eaten, colonists and soldiers and cities were saved, while others were destroyed. The various militaries of the Alliance couldn't be everywhere, but neither could our foe. There was time to heal, time to repair, time to rebuild and build more. And time to be castigated for not moving faster, for not doing better._
_The best I could do, however, was march along to the drumbeats of Fate._
_~Ia_
JULY 13, 2497 T.S.
ZUBENESCHAMALI SYSTEM
This time, she was still awake. Still clad in her grey slacks and paler grey shirt, too, though she had taken off her black ship boots and had her sock-clad feet levered up on the footrest of her easy chair. This time, Belini appeared in a set of black tights and tunic trimmed with red edging. At least, when she coalesced into a matter-based body, after popping into view like a silvery soap bubble in reverse.
"...If you think your colormood choice will frighten me, I'm not intimidated by death, and I am not swayed by rage," Ia stated calmly, scrolling up through the text of her precognitive missive on the workpad screen. This one was a message to her family, requesting them to separate out a few more sprays for her to convert. She wouldn't get to see them for over a year, but she liked having things prepared in advance. A thought that amused her.
"Smile all you want; you are trying my patience," the not-pixie snapped. Hands on her hips, she strode across the carpet and stopped next to Ia's chair. "Why do I get the feeling you're not even _trying_ to manifest?"
"I don't have to try," Ia told her. She sealed the message to her brothers and opened the next. "My ability to manifest will reveal itself in the right place, at the right time. In the meantime...I now have the politics of the Dlmvlan Queen High Nestors' court to redirect." She lifted the pad in her hands, tilting it briefly at the other woman to show the electrokinetically composed words filling the screen. She tipped it back her way and shrugged. "I'm not sure I've used enough of a mix of logic and illogic to sway them. I'm good at the logic part, and not too bad at drawing absurd analogies, but I'm not a poet laureate."
"I don't give a radioactive _fart_ about the Dlmvlan High Nestor," Belini snapped. "That part of the Game isn't in my way. _You_ are in my way."
Ia looked up from her notes. "I give you my word, I will manifest in the right place, at the right time."
"Ha! What good is your word?" the Feyori argued, and pointed off to one side. "Every day you fail to prove you are a player and not a pawn is a day that _I_ lose face! Mcuinn and Gzikk are already positioning themselves to scoop up chunks of _my_ plays."
Slanting her an annoyed look, Ia unclipped the datapad resting on the table next to her armchair. "Here is a list of instructions on how you can position your influences to outflank theirs by the rules of the Game. And _yes_ , this will work by the rules of Feyori politics," she added, seeing the Meddler narrow her eyes in doubt. "Belini, when have I ever been wrong?"
That made the not-pixie snort with mirth. Hands on her hips, she surveyed the white-haired Human. "How short the memory span is in your mother's species. Private Sung? Private N'Keth?"
Sighing, Ia waggled the second pad, gesturing for her to take it. "When have I ever been wrong about something _I_ can control? If you take this information, the only person after that who could screw it up is _you_. Are you a screwup? Or are you a Meddler, a galactic reshaper of lives and worlds?"
Belini snatched the pad from Ia's grasp, not noticing that Ia subtly tossed it upward a little, letting her hand drop to prevent contact. "If I _do_ screw this up, I'm still going to blame you."
It was Ia's turn to choke on a laugh. Well, a chuckle. She let her humor fade as Belini raised her brows in surprise, then lowered them in a scowl.
"... _This_ is idiotic," Belini finally growled. "You have me giving ground, with these instructions! Concessions right and left, up and down, even helically spiraled...!"
"It does seem that way, but look at what they have to give up. It's in the fine print," Ia added helpfully.
Belini didn't have to scroll the screen to read it. Feyori were natural electrokinetics on a level above and beyond what Ia could do. She did blink twice, though, gaze unfocusing. "That's...That...would require...You're going to _Gather_ us?" she asked, shifting her gaze back to Ia. "But the energy requirements...and that _you_ are the party listed as the Summoner, to trigger these conditions? No.
" _No._ There's no way that _you_ could pull that off! Not when you can't even manifest yet," Belini scoffed. She tossed the pad back into Ia's hands. "You may _think_ you can manipulate energy just because you're an electrokinetic, but a Summoning is a manipulation of _hyperspatial_ energies, on a scale your puny little meat-mind cannot grasp. That chance is so slender, it's see-through!"
Ia brought her knees up, bracing her sock-covered heels on the seat of her chair. She did so to avoid even the slight chance that Belini would touch her; once she did, the Meddler's greater telepathic abilities had an equally slim but dangerous chance of reading her subthoughts. The ones where she knew she'd already manifested. It was safer to avoid contact. That, and she'd been sitting in one position for too long. The sensors in the easy chair quietly folded the leg rest back into place as Ia spoke.
"Believe as you will. I have foreseen it. All you need to do is make those agreements and hold to them, so that _they_ have no grounds to back down from their bargain when my prophecies come true. And for the record, my 'puny little meat-mind' has already grasped the fullness of Time in every direction I've ever cared to stretch. Hyperspatial manipulations are child's play by comparison."
That creased the not-pixie's brow with another frown, a skeptical one. "You may be able to _see_ 'the fullness of Time,' but can you _understand_ everything you see? _Can_ you fully grasp it?"
Slightly sheepish, Ia shrugged. This was not something she could lie about, not without its coming back to bite her later on. "Mostly. I know enough to be able to do what I need to do—I don't have to understand how to create a hyperspace nose cone from nothing, like some engineering pioneer, in order to grasp the principles of how it works and how it is used. That'll take me one more lifetime," she added in a muttered aside to herself. At the pixie's frown, Ia shook her head, dismissing her words. "Luckily, I don't need to go that far in this life.
"And you don't need to, either. All you have to do is follow through on what I've told you to do." Gesturing with the returned datapad, Ia said, "I see, I plan, I act. That is enough for now."
Snorting, Belini took the device, finished absorbing the information on the smaller pad, then tossed it one more time to Ia. Not bothering to watch the heavyworlder catch it one-handed, she turned back toward her corner of the small living room. "Well. You'd _better_ be right. I'll hold up my end of the bargain, even though it seems like I'm losing face...and I will be. If you fail, I will devour your life-energies myself."
"If I fail, it won't matter what you do. You'll be left trying to flee across the intergalactic void—good luck with that, by the way," Ia added. "Even if you glomp up _en masse_ , the strongest clutch would only get about halfway before starving to death. And the Greys won't share their teleportive technology with you. They'll have already fled to the next galaxy, and they won't stop to pick up you. You're nothing but salt and lime juice with a chaser of tequila to their mental wounds."
"Don't knock the salt and lime juice. Margaritas are one of the few delights that a digestive tract can experience. That, and chocolate," Belini admitted. Sucking electricity out of the wall, she flashed into her true shape. ( _Don't make me regret this factioning for you._ )
( _It's a factioning_ with,) Ia corrected. ( _My part of the payment just has a bit of a time delay on it, is all._ )
It shouldn't have been possible for a silvery ball of semireflective energies to snort. Snorting was sound, which usually required the collision of matter against matter to create, so a Feyori in its natural state rarely made noise. Still, the peculiar swirl on the center focal point of Belini's surface managed to re-create a visual snort, before she slid away.
Or possibly a visual raspberry. Amused, Ia went back to work.
NOVEMBER 15, 2497 T.S.
BATTLE PLATFORM _PLENITIA III_
MARS, SOL SYSTEM
"That's the last of the boxes, sir," Sergeant Halostein said, dipping his head at the hoversled being guided into their amidships storage bay. "I still don't get why we had to strip the bow sector of all the supplies and emergency cabinets but for the stuff on those lists, Captain. Care to explain?"
Ia shook her head. She had encouraged her crew to display the level of familiarity inherent in his question, but this was something she couldn't talk about. Feyori eyes were now focused on her in the timestreams. The strongest among them could only see a few months ahead, but a few months was enough to cause problems. At least, she couldn't mention the _real_ reason.
"I'm about to be keelhauled, Sergeant, and I need those unused supplies shipped off to pad my hide instead of my bottom line—you do know what keelhauling means, don't you?" she added, glancing at him.
Halo frowned a moment, then scratched the fringe of pale blond stubble circling his head, freshly buzz-cut by one of the crew. "...Not really. The keel part refers to the bottom of a sailing ship. I know that much."
"It's a very old nautical term, yes," she agreed. "And an ugly form of punishment. A rope would be looped around under the hull of the ship. Usually from starboard to port, or if the offense was deemed truly heinous, from bow to stern. The sailor to be punished would then be lashed into the loop, and dragged under the ship.
"Since the hull would often be covered in barnacles and other forms of sea life by that point, he would either be dragged fast enough to scrape the flesh from his body, possibly even causing limb loss or decapitation, or dragged slowly enough that his weight _might_ slacken the rope enough to avoid the barnacles and such, but at that low a speed, it would be more likely for him to drown."
He glanced at her sharply, blue eyes wary. "And you're about to be keelhauled?"
"Well, the alternative euphemism is raked over the coals, but I do believe my enemies would rather have me lacerated, decapitated, _and_ drowned than merely burned. Scorching my hide under the pretext of discussing unpleasant subjects isn't nearly so evocative." She paused, dipped her head, and added, "Though if it makes you feel better, I will be trying to steer the conversation in that direction."
Halostein chuckled. "Well, if anyone can, you can, sir," he murmured. "If you'll excuse me, Captain, I'll disembark and seek out the dock officer so we can get this mess off our ship."
"Carry on, Sergeant," Ia agreed, gesturing for him to walk ahead. Her bracer beeped. She didn't have her headset on her, though. Unsnapping her left sleeve, she pulled up the unit's screen. "Ia, here. Go."
York's face filled the screen. "We've just received orders for you, Captain, from the Admiral-General. Transmitting them to your unit."
"Inform Lieutenant Commander Harper that he is in charge of the ship in my absence. Status quo until I return," she instructed. She checked her arm unit. "...Orders received."
"Aye, sir. I don't know why they keep sending you these things, when you already know," Private York muttered. Then rolled his eyes and corrected himself. "Well, yes, I _do_ know; they have to have a paperwork trail—please don't mind me, sir, it's been a long watch, and Private Teevie's slightly overdue. Said she was taking a head break before hitting the bridge."
"She'll be there in three more minutes. Ia out," she added, ending the link with a touch of the screen. Another tap opened up the file transferred to her. The paperwork in question—electronically written and about as far from wood-pulp sheets as they were from Earth—directed her to report to one of the _Plenitia III_ 's briefing rooms immediately.
Flipping the lid shut, she rebuttoned her left sleeve, dusted off her Dress Grey jacket with its minimum glittery of rank and zone pins—now more than two years out of date since there was no ribbon bar designated to cover all of Alliance space—and headed for the airlock in Halostein's wake.
The briefing room was a heavily guarded, scaled-down version of the one she had invaded just over two years ago. This time, her palm and bracer were scanned, her identity confirmed, and herself formally escorted in by two of the guards. Blank when she entered, the screens around the room stayed dark until the guards left, and the doors sealed behind them.
At that point, they lit up again with various data, some of it statistics from the Damned's various activities, some of it with surveillance footage. A part of her flinched away from the shot at the back left of the round, two-tiered chamber. It had been taken from someone else's helmet pickups from back when that one robot attempted to rip off her mechsuited arm. Her shoulder was long healed, but the memory of the pain was still there in her subconscious.
Focusing on the nine men and women seated around the lower of the two horseshoe-shaped tables, Ia searched for the face of her immediate superior. As she had foreseen, he wasn't there. Two others represented the Special Forces instead, both of them from the Psi Division. One of them was a middle-aged blonde, a lieutenant general who hadn't been at the previous meeting. Ia knew her name, Mercea Wroughtman-Mankiller. The other was General Jolen Phong, who had been there. Every Branch had two representatives, which suggested a formal review board.
Seated in their center, Admiral-General Christine Myang returned Ia's respectful salute, laced her fingers together, braced her elbows on the table, and spoke without preamble.
"Tell me why we shouldn't throw you in the brig for Fatality One, Ship's Captain Ia."
"Let me guess," Ia returned, unfazed. "Lieutenant General Wroughtman-Mankiller is pressing the unspoken charges in question. Correct?"
Mercea leaned forward as well, though she didn't clasp hands. "These allegations are _very_ serious, soldier. You have _failed_ to undergo your yearly ethics examinations as a duly registered psychic. Your last one took place on January 19, 2496, as registered on your homeworld of Sanctuary...and yes, I _did_ subpoena the Witan psi-priests in question for a face-to-face and mind-to-mind interview of all your past reviews."
"Are you aware, Lieutenant General Wroughtman-Mankiller, that at this point in time, you are acting under the influence of Feyori Meddling?" Ia asked, turning to face the blonde in the grey-striped black jacket. "I believe _that_ falls under Fatalities Two and Six: Treason, and Subversion. Inadvertent or not, your accusations against me are intended by a Feyori named Miklinn to be counterproductive and subversive to the continued safety, operation, and success of the Terran United Planets Space Force and the government it supports. Those definitely qualify as both Treason and Subversion."
The Special Forces woman scowled. "I am a Rank 14 Xenopath, soldier. _More_ than strong enough to sense the presence of a Meddler _and_ keep it out of my thoughts."
"Not when you're making love." Her blunt rejoinder made the other woman blush. A few of the others did as well. Ia shrugged. "At that blissful moment of the orgasmic crux, a psi's mental defenses are lowered—I myself have suffered from the same phenomenon. Your longtime partner, Brad Blackburn, has unwittingly been tapped to be a conduit in those moments. As a Rank 4 telepath, he has just enough strength to be used to influence your mind but not enough sensitivity to notice he is being used to do it at such moments.
"The Feyori Miklinn, whose normal purview would be the Solarican military, has counterfactioned my efforts to ensure that this war progresses to the best possible ending for all the races of the Alliance," Ia continued. "He has received dispensation from the Feyori who normally controls all Meddling in the Terran Space Force to have an indirect influence upon your activities...hence using your moments of intimacy against you." Her mouth twisted in a wry imitation of a smile. "I'd tell you how to prevent it from happening again, but _that_ would be an open counterfactioning of his efforts. An act which I am not prepared at this time to perform because it would spiral this mess out of control."
Wroughtman-Mankiller flushed again but did not argue the point. "That may or may not be the truth, but my own situation will have to be examined later. _You_ are the point of this High Tribunal."
"This is not a High Tribunal," Admiral-General Myang countered. Then added in warning, " _Yet._ But the accusation is still there, Ship's Captain. _Have_ you undergone a psychic ethics evaluation since January of 2495?"
"No, sir, I have not," Ia stated calmly. She continued before the nine men and women around her could do more than sit back in shock at such a free admission. "However, my lack of an ethics examination is completely covered by two words, rendering said examination unnecessary."
General Sranna of the TUPSF-Army tapped his finger on the surface of the long, curved desk. "I'm sorry, Captain, but 'Vladistad, _salut_ ' does _not_ exempt you from being examined. In fact, it makes it all the more imperative that you _are_ examined on a yearly basis."
She almost replied flippantly. Almost, except at the last second, the timestreams turned cold in her mind. As cold as the dread she had experienced over the Friendly Fire incident. Her subconscious doused her veins with ice-cold warning. Squaring her shoulders, Ia answered plainly instead.
"No, sir, Vladistad, _salut_ does not exempt me from examination, sir. Two other words did that, sir," she stated, clasping her hands behind her back to hide their trembling. If she wasn't careful in the next few minutes..."It's good to see you again, General."
"Which two words would those be, then?" the other Army general asked her, his brow lifted skeptically.
" _Carte blanche_ , sir. Two words that have been backed by my scrupulous filing of daily reports." She looked at Myang. "Everything I have done, everything I have shared with you, Admiral-General, I have conducted myself and guided my crew with the strict scruples and ethics required by the Terran laws regarding psychics and their abilities." Ia held her gaze for a moment more, then nodded at the information flickering on the screens. "Pick anything I have done and question me about it. I will give you the exact duty reports that cover each incident."
"You're a good enough electrokinetic, you could have tampered with those recordings at any point in time," Mercea countered wryly.
"But I'm _not_ a good enough telepath to have tampered with the reports of your spies," Ia stated. She tipped her head to her right. "You've had plenty of reports from them corroborating everything I've officially sent. You've heard from me time and again in my reports explaining why _this_ battle had to be fought instead of _that_ battle. I've given you headcounts and ident files of the soldiers and citizens we've saved.
"My Company and I have filed more paperwork than a ship with nine times the crew complement, with our moves and our motives constantly displayed for you...and my crew has fought for you with so little free time in our schedule, we haven't had more than six days' worth of Leave in the last two years, sirs, because there _hasn't_ been time for anything else." Ia shook her head. "What else _could_ I do but keep fighting?"
"You were required to be examined anyway!" General Phong countered.
"If I had stopped to disembark long enough for an essentially redundant examination, sir, I would've risked thousands of lives being slaughtered because my crew wasn't in the right place at the right time." Ia looked at Myang. "Or should I have permitted unauthorized personnel on board my ship, sir, an act which by your own command would be considered an act of Grand High Treason? The only time I've had free that _was_ long enough for an ethics review has been while my ship was in transit between battle zones—and the few times my ship _has_ been in dock long enough for me to undergo an ethics review, I _still_ had work to do.
"When would I have had time, without risking too many lives, sirs? My only recourse _is_ to conduct my actions openly. Ethically and morally, openly. Word, thought, and deed are all one within me. How else can I prove myself to you yet still continue to save all these desperately needed lives?"
She surveyed each high-ranked officer in turn. None of them looked away, but most didn't look comfortable. Only Myang met her gaze without changing her expression. Then again, the Admiral-General was an expert at displaying that flatly neutral look.
"I serve— _I act_ —because I am here to save lives to the best of my abilities. And I have done so," Ia argued. "If you cannot see for yourselves the truth in my deeds, the _ethics_ inherent in my actions, then unless you are a psychic yourself, and one strong enough to conduct a probe, then there is nothing I can do or say to convince you. So what _can_ I do, given the constraints of my orders and of Time itself, that I have not already done?"
"Alright, I'll bite," one of the two Navy admirals stated, her voice as crisp as her blue-striped black uniform and her silvered crew-cut hair. "How about you explain to us how you can justify working for a foreign government? Your very first act was to haul _cargo_ for some branch of the government back on your homeworld. Yes, by contract the military _can_ deliver cargo to a colonyworld under our protection, such as I.C. Sanctuary, but those contracts must undergo military review for the best allocation of our limited freighting resources.
"And don't give me any _shakk_ about that cargo being 'emergency military supplies,'" Admiral Nadine Nachoyev added dryly, slanting Ia a dark look. "We've received complaints from the Sanctuarian government that those supplies vanished the very day you left the planet, and that despite great efforts by said government, they have never been found in the two years subsequent. The only conclusion that can be drawn is that you were smuggling in supplies under the orders of subversive elements on your homeworld. That _you_ were taking orders from a foreign power to do so. Fatalities Two and Six would therefore apply to _you_."
"My actions were neither a case of Treason nor of Subversion, Admiral," Ia returned calmly. This, she could handle. She had anticipated it among the possibilities of questions thrown at her in this session and come prepared with her answer. "Yes, I lied about the nature of those supposed military emergency supplies to the officials logging that complaint, but I _am_ permitted to do so. That option exists under the rules and regulations stating that no Terran military officer may allow a foreign power to confiscate cargo under our protection, whether or not that cargo is of a Space-Force-assigned nature."
"So who did you transport the cargo to, if not to the government of Sanctuary?" General Sranna asked her, frowning softly.
"The Free World Colony, sir. It is its own, separate government from the Church-controlled elements currently running the rest of Sanctuary. They requested that the cargo be kept secret from those elements," Ia stated quietly.
Admiral Wroughtman-Mankiller pounced on that. "Then you _did_ take orders from a foreign government!"
"No, sir, I didn't do that," Ia replied, tucking her hands behind her back. She looked over at the Special Forces admiral. "You see, I am not taking orders from them, and I never have."
That earned her several snorts. Freeing one hand, she snapped her fingers. Text appeared on the main screen behind her, with the top line enlarged so that the words were easily seen from all nine seats. The image zoomed to display the pertinent sections as Ia explained what they were now viewing.
"Before you is the Alliance-registered Charter for the Free World Colony of the Zenobian Empire...also known as the _other_ half of Independent Colonyworld Sanctuary. As you can see for yourselves, Article I, Section A, paragraph 1—the very first law of the Free World Colony—clearly states: _The duly verified commands, precognitive missives, and orders of the Sanctuarian-born woman known as Ia, Prophet of a Thousand Years and soldier of the Terran Empire, supersede, supplant, and overrule all laws of the Free World Colony, its citizens, descendants, and successors, without exception to this rule,_ _save by her own command._
" _Legally_ , meioas, they take their orders from _me_ ," she stated, defending herself neatly. "Which means I am not violating Space Force law because I am not under the influence of any foreign power."
"Yet you still are working for a foreign power," Admiral Nachoyev argued. "Your own crew members claim that you have admitted to not only using technology borrowed from the future, but that you have refused to share the results of that technology with anyone outside your own ship and have forbidden them to share it as well."
" _That_ would be covered under Vladistad, _salut_ , Admiral," Ia told the older woman, returning her hands to the small of her back in modified Parade Rest. "By its very nature as precognitively extracted information, that information is protected by the _Johns and Mishka_ statute sheltering all precognitives within the Terran Government, and by extension the Al-liance.
"Considering how much precognitively based information I _have_ given to this military and its government, and the accuracy of all my various predictions, a court of law would rule that I do have the right to withhold that information, based on the fact that it would damage the future. And it would, sirs," Ia told them.
_"Enough."_ The solitary word silenced the others. Unclasping her hands, the Admiral-General studied Ia. "I think I am beginning to understand why your Priestess Leona used a modified format during your ethical scans. I suspect if we tried to dissect each and every one of your many actions over the last two-plus years, we'd be stuck in here for _ten_ years at this rate."
"We don't have ten years, sir. I'm sorry," Ia apologized. "Mars will be attacked in three more days. You'll be busy with the system's defense, and my ship will be needed in the defense of one of the Solarican heavyworld colonies approximately three hundred light-years from here."
" _Thank_ you for finally telling us about that," Admiral Nachoyev muttered darkly. "You _could've_ done it sooner."
Ia tried to restrain her tongue. She knew she had to tread carefully, but Nachoyev's antagonism was rubbing her sense of patience raw. "I knew it would come up in here, and that three days is plenty of time to prepare. When else should I have told you? Five years ago, when I was still a lance corporal in the Marines? You wouldn't have paid attention to me back then. Three days is plenty of time to prepare."
"Enough!" Lifting a hand to her forehead, Myang rubbed at the bridge of her nose. "I will accept your constant stream—your _barrage_ —of visual and written reports on all of your activities as your version of an ongoing ethical review. But understand this: if we _do_ have doubts as to your actions, you will be called into a much more formal review, Captain. A tribunal review."
"Yes, sir. I know that, sir. It'll happen in a few more years—but not for the reason you're thinking," Ia confessed. Myang blinked at her. "It will be instigated by the Alliance Council, not the Terran Space Force. As Admiral Viega will attest, I did warn the Salik that engaging in this war with us would lead to their destruction and advised them to abandon their ambitions.
"They chose to ignore my warning, and will eventually reap the seeds that they have sown. I already knew that I will be held accountable for that because I have not told anyone what those seeds are," Ia told the men and women before her. "I have been very careful to make sure there is no way the Salik can use that information against _us_ , which _would_ be an extreme case of Fatality Thirty-Five: Sabotage.
"Having had a taste this last year of the corporal punishments administered for Fatalities Five and Thirteen, I have no intention of enduring any greater penalties," she finished dryly. "Particularly not during a war when I myself as a precog could foresee and prevent them."
"I am quite sure," Myang murmured, studying her. "Ship's Captain Ia, I find you to be arrogant, boastful, irritating, and borderline subordinate more times than I'd care to count. You are the single biggest headache in this war next to the Salik themselves. You are a _problem_ for me."
Her tone, mostly quiet and calm, now dripped with barely checked vehemence. Ia tried not to swallow visibly. She clung to her foresight-based faith that she would make it through this interview more or less intact, and waited stoically for the other boot to drop.
"Unfortunately, that borderline you tread is found within the edges of your _carte blanche_...and you have pulled off multiple miracles time and again despite your many abuses of it," Admiral-General Myang allowed grimly. "You look and talk like a loose cannon on a seafaring ship, but _somehow_ you have managed to be the _right_ loose cannon every damn time.
"Don't let the fact that we _do_ still need you go to your head, though," she warned Ia. "There are some edges to your _carte blanche_ out there, and if you cross over them, you will fall a very, _very_ long way. And you _will_ hit bottom, soldier."
"Whatever you may think of me, sir, arrogant or irreverent or...or mad with poetry and prophecy, as is your right, I do respect you and your authority," Ia murmured, holding the older woman's gaze. "So far, you have given me what I need to save lives. I will not forget how much I owe you for that."
Myang stared back for a long moment, then nodded slightly, blinking her dark brown eyes. "Alright. Before we move on to the subject you raised regarding Mercea's supposed, inadvertent contamination by outside forces, is there anything _else_ you'd like to share with the Command Staff? One of your infamous 'one more thing' moments?"
Nodding, Ia fished out the datachip stored in her pocket. She floated it off her palm telekinetically, rather than tossing it, and set it gently on the table in front of Myang. "There's all the latest precognitive data I can give you for the next year—some of it becomes a little fuzzy on the Dabin question in half a year, but the rest of it's good for other parts of known space. I'll do whatever I personally can to help correct the Dabin issue. Is that all, sir?"
"No," Myang stated, picking up the chip and pocketing it. " _I_ have one last thing. This meeting has also been called to review your performance as an officer. The DoI thinks you could handle a higher rank. They are also wondering, in the face of all the fantastical reports filed by your Company with us, why we haven't seen fit to grant you any awards or commendations. Do you know why that is, Ship's Captain?"
"I can make an educated guess, sir," Ia said.
"Go on," Myang ordered Ia, sitting back in her seat. "Enlighten me."
"It's because I don't give a damn about medals, or ranks, or commendations, and you know it," Ia said flatly. "I could finish out this war _and_ the coming Grey War, remaining a mere Ship's Captain for the whole of it, and not care, so long as my goal of saving lives would still be met, and I'd be more than happy with that. I have no ambitions for anything higher."
"No. You don't," Myang stated dryly. Too dryly. Ia wanted to peek at the timestreams to see what that comment was about, but knew such a subtle thing would take more time—even accelerated on the timeplains—than she could spare right now.
Instead, she boldly stated, "Admiral-General, while we're on the subject of elevations in rank, I'd like to personally commend Lieutenant Commander Harper, and recommend that he be granted the rank of Commander. He's kept my ship and crew alive, and kept it repaired with the absolute minimum of supplies."
Admiral-General Myang rolled her eyes. Ia hadn't _said_ "one more thing" but the implication was still there.
"At the moment, he is currently doing a series of sweeps across the _Hellfire_ to remove excess supplies that we do not need but which will be needed elsewhere," Ia added. She did not mention, given Admiral Wroughtman-Mankiller's compromised presence, that Myang already knew those supplies were being packed off to her next ship in advance of its deployment. Instead, she kept the focus on Harper's readiness for promotion. "He has earned the complete confidence of my crew, and I feel he can handle the increase in rank to Commander."
"Didn't you tell the DoI you wanted him permanently assigned to you, the same as the rest of your crew?" General Phong asked, eyeing Ia. "What would be the point of elevating his rank if he's still assigned to you? Or do you mean to have him reassigned?"
"I do still need his ability to organize the repair of my ships, sir, throughout both the Salik and the Grey Wars," Ia admitted. "But at some point, those two wars will end, and he'll be free to move on to other posts. I believe—not just foresee, but believe—he will be capable of achieving even higher rank as time moves on, if that is what he chooses to do with his life once he is released from my crew. His presence at my side doesn't negate the fact that he has already worked hard enough to have earned it."
"We will consider your recommendation, Ship's Captain. But we will not make up our minds tonight," Myang stated. "Since the discussion of anything else would be a waste of _your_ time," again, that odd, dry tone to the older woman's voice, "we might as well let you go. I do expect you to continue to act with the high level of effort and ethics you have displayed so far, if not more. If we find any discrepancies, understand that we reserve the right to haul you before a tribunal for breaking the law. Now, seeing how we have only three days to prepare for the next wave of attacks, you are dismissed, Ship's Captain."
"Sir, yes, sir." Saluting her crisply, Ia turned on her heel as soon as Myang saluted back and left the room.
_And that is that. Once Mercea has her and her lover's minds purged and warded against further Meddling, Miklinn is going to be_ very _upset with me. Which means everything is right on track...provided I can keep clearing the_ Hellfire _of "unnecessary" supplies so they can be shipped over to await being loaded onto the_ Damnation.
In the meantime, she had only so many hours left before her current ship had to leave dock. They had already been here for almost two days, giving her crew a rare chance at a full day's worth of unrestricted Leave, with free boarding tickets to head down to the Red Planet and priority seating for coming back. It was a mild abuse of her _carte-blanche_ powers, and one which could have been contested during this little interview, but her soldiers had definitely earned it.
The Solarican colony that was their next destination was just that, the next in a long line of engagements. As much as her crew needed this break, Ia herself couldn't take any. Her crew could take a day off, but she couldn't afford to slack off.
JANUARY 29, 2498 T.S.
SIC TRANSIT
The officers' mess wasn't always used because there weren't more than a handful of officers on board, spread out over three duty watches. But rank had its privileges when it came to requesting special birthday dinners. With a meal of her favorite foods—pasta, salad, and pastries—prepared by the talented Private First Class Philadelphia Benjamin, one would have expected the doctor's mood to be good.
Except the birthday girl, Jesselle Mishka, entered the cabin in the company of Glen Spyder, both of them arguing vehemently.
"—an' I say y' got yer 'ead on wrong!" Lieutenant Spyder snorted, glaring at her.
"Hiding heat signatures in bodies of water is a time-tested method. The squad should've ducked into the pond!" Jesselle argued, one hand on her hip, the other flipping at his face. "That's what the scenario called for!"
"Yeah, but we ain't dealin' wit' yer average alien, sweets," he retorted, leaning in closer rather than flinching away. His own hand lifted, all but poking her in the sternum. "We're goin' up against Salik, an' th' frogtopi have _sensitivities_ f'r maneuvers in water, _includin'_ current patterns. So yeah, th' water'd cut the squad's actual heat traces down, but the Salik'd still see th' bloody convection plumes!"
"Oh, and you think hiding the squad in the _trees_ is any better?" she scoffed. She snapped her hand up, pointing at the ceiling. "Salik eyeballs point _up_! You'd think, with all those boarding-party missions you've led, you would've _noticed_ the Salik viewscreens being posted overhead?"
"Human limbs'r at least _shaped_ like most tree limbs," he countered, shifting even closer. "Which you'd know if you'd ever tried t' wrap yer limbs around sommat!"
For a moment, Doctor Mishka stared at him, cheeks flushed and eyes wide with anger, her lips slightly parted as she sought for something to say to that. Their faces—their noses—were mere centimeters apart, and there was barely enough room for a datapad between their bodies. Folding her arms across her chest, Ia bit back the urge to smile. Her plan to ensure that even their Company doctor was competent at thinking strategically, forcing the woman to attend Spyder's weekly tactical discussions, had just borne some unexpected fruit.
Amusingly, Jesselle also folded her arms across her chest. The movement brushed her forearms against Spyder's T-shirt-clad chest. "Are you offering me a target, Lieutenant?"
Even more amusing, Spyder blushed, tried to speak, swallowed, then tried again. "I...er...not in front 'f the troops, a' course. I mean, y' might be lousy at it. Wouldn't want t' shake their—"
"I might be _lousy_ at it?" Jesselle demanded, glaring at him. "After all the training I've gone through on this ship, I can beat the pants off of you on _any_ obstacle course— _and_ make you _like_ it!"
Ia decided she'd had enough entertainment for now. As interesting as this little drama was, they had bigger problems to focus upon. Her revelation to the Command Staff that Lieutenant General Wroughtman-Mankiller had been subtly compromised by a Feyori had drawn attention from the growing membership of Miklinn's little counterfaction against Ia. That meant she had to work with equal subtlety for certain plans, appearing to do one thing while in actuality aiming for something slightly but significantly different.
"Gentlemeioas," she stated just loudly enough to catch both their attention before they could either argue some more or kiss. They jumped a little and twisted to face her, the tension quelled between them, though not completely broken. "I have zero objections to the two of you fraternizing, since neither of you are in each other's immediate chain of command. I must, however, insist that the only thing thick enough to cut in this room be your birthday cake, Doctor. In short, Lock and Web the sexual tension between you two. At least until you can take it to one of your quarters."
Spyder blushed even more. Jesselle paled, then flushed. Interestingly, neither of them denied her labeling it sexual tension. _At least they're competent enough, I can foresee any relationship between the two of them not getting in the way of their duties._ Gesturing at the table, Ia changed the subject.
"Please sit down, both of you. Lieutenant Commander Harper and Lieutenant Rico will be joining us in less than two minutes, and Private Benjamin will be bringing out your birthday dinner in two more. Commander Benjamin will join us about ten minutes in. Her third-watch counseling session is running a little long. As for your argument, the obvious solution to your squadron dilemma lies in both planes."
Spyder frowned at her as he toed one of the chairs back from the table by its lever. He held it steady for the doctor, then selected one for himself across from her. "What d'y' mean, both planes?"
"The squad in question could just as easily submerge themselves in the pond for a minute or two to cool their infrared signature, then climb up the nearest trees along the waterline and hold themselves still," Ia said. "The Salik will most likely see the churned-up mud at the edge of the pond first, along with the thermal plumes still roiling the water, and waste their time searching beneath the surface for their prey. The foliage flanking a pond is always going to be thicker than the kind found deeper in the woods...or whatever passes for woods...so hiding at the pond's edge in the canopy is not as bad an idea as it sounds.
"You could even leave a couple brave souls down in the water with rebreather masks to lure the Salik down, and set up a flanking ambush from both below and above. At least, that's what I'd do," she stated, as the door slid open again. "Run it by the troops, Spyder, and see if they can strengthen that scenario or poke some serious holes in it."
The tall bulk of their intelligence officer filled the doorway. He was neatly dressed in grey shirt and slacks, his uniform crisp and clean, but his eyes were half-lidded, and a crease mark from his pillow could still be seen on his cheek.
"Good evening, Lieutenant...or technically, good morning," Ia greeted him.
Rico grunted and nodded. "Captain, Doctor, Lieutenant. Ah—Lieutenant Commander Helstead says the current watch is quiet, and wants to know if she can come share the birthday girl's supper—happy birthday, Jesselle," he added in an aside. "I told her I'd eat quickly and relieve her, if that's alright with you, sir. I know you don't like leaving a duty watch unmanned."
"You sound like you still need to wake up," Ia countered, watching the way he tried to stifle a yawn and almost succeeded. "I'll be the one to eat fast. That way, my three Platoon officers can have a nice, peaceful _discussion_ with the good doctor, here, on various combat and reconnaissance scenarios for our upcoming trip to Dabin." Carefully, shaping and pushing her thoughts outward to catch each of the three of them, she added telepathically, ( _Do not forget that you need to throw in false suggestions among the true, regarding what we'll be doing over the next few months. I'll remind you that the Feyori are now watching not only me in the near timestreams but also you. Be grateful they can only watch and cannot read your minds temporally._ )
"Understood, Captain," Rico muttered. Jesselle winced a little, none too happy at the unannounced telepathic projection, but she nodded.
Spyder shrugged and covered the birthday girl's hand. "Sorry t' argue on yer natal day," he apologized. "We c'n table it 'til tomorrow, if y' like."
"Or take it to my quarters," Jesselle muttered, glancing at the green-haired ex-Marine.
That earned her a sharp look from Rico, but he lost the opportunity to ask anything as the door to the corridor slid open once more, admitting their chief engineer.
# CHAPTER 17
_I pushed the_ Hellfire _and the Damned as hard as we could go without risking our breaking point. Unique as we were, I did my best to utilize everything we could do in an economy of action. My goal was a frugal pursuit of our enemies—and I say enemies, plural, because it wasn't just the Salik and the Choya we had to face. The Meddlers were getting involved, jockeying for new positions, new powers. New plays in their great Game. But they weren't the only ones, either._
_In the chaos of war, opportunities await the bold and the reckless. In the fringes, where attention has been turned from shepherding the masses in civilized behavior to defending the masses from a specific foe, certain elements flourish. Criminal elements, beings who wouldn't hesitate to form unholy alliances if it meant increasing their own power, prestige, wealth, or whatever._
_They flourished, and they grew bold, and they seized opportunity after opportunity...until they grew just a little too reckless, listened to the wrong whispers, allied themselves with the wrong people, and stretched their resources trying to reach for what they wanted, when what they wanted lay within my protection. Some, I had already smacked repeatedly back on Oberon's Rock. Others tried a different tactic._
_When the criminal undergalaxy joined in faction with a certain group of Meddlers, laying plans to go after my most prized possession, they stretched themselves past the breaking point...and yes, I enjoyed breaking them._
_~Ia_
MAY 31, 2498 T.S.
BATTLE PLATFORM _SARATOGA JONES_
KUIPER BELT, CS 47 SYSTEM
Ia stared at the contents of the crate she had just unpacked. Everything was there. Power cables to hook it up, Terran-designed controls that no longer needed a sucker-hand on the boxy body of the infernal device, and a jury-rigged transmitter sphere. After pain-filled experimentation with the original captured machine and the ones salvaged from various wrecked Salik vessels, the Space Force's Psi Division had figured out how the Salik broadcasted the anti-psi field unilaterally over a larger area than an individual victim's helmet could.
For the next twenty or so hours, this would be a literal headache, if a necessary one. If it blocked all but the strongest of psychic abilities, it would block attempts by the Feyori to peek through the skin of her ship.
Sighing, she pulled the main box out of the crate, carried it over to the workbench Harper had cleared for her, and secured it to the surface. It felt a little heavy, but only because she and her crew were now living and working in 1.8Gs Standard, just about the right gravity to match their upcoming assignment. She turned back to fetch the cable and the emitter, but Harper had already grabbed those.
Bringing them over to her, he wordlessly helped her hook them up to the ship's power grid. All of her orders in the last few weeks regarding this day had been delivered telepathically to her crew. She disliked touching people while speaking telepathically—and had accidentally tripped herself and a few of her members of the crew into the timestreams while doing so, on those few occasions where she had to touch someone to deliver a message—but it was imperative that the Feyori watching her from their own version of the timeplains did not hear any actual orders regarding what they were about to do.
If they knew, they wouldn't walk into the trap she needed to set for one of them. A trap in which she hoped to snare many more.
Harper didn't spill the details, but he did make his displeasure known. "...Would it do any good to protest?"
"Not a single bit. I am turning on this machine and pointing this ship at your homeworld. We will wait about a light-month out from the system, then swoop in at the right moment and destroy the blockade currently in orbit." Webbing the emitter to the gridboard above the workbench, she glanced at him. "I know you're not happy about Dabin's falling to their forces. And I know you're _not_ happy about the genetically engineered monsters they've let loose on your homeworld. But we will go there, and we will arrive in time to save it, and do so in such a way that my goals will be met."
He looked away. She knew that wasn't his real protest. Touching his shoulder, Ia sent, ( _I will survive, I promise you that. You have my Prophetic Stamp._ )
_...Only because I gave you the tools to do so. Are you sure this thing will foil their vision?_ Harper thought back. He couldn't project his thoughts, but he could form them clearly enough for her to read.
( _At the right setting, both machines will block their view of what lies inside each ship, without blocking their view of exactly where this ship will sit._ ) Squeezing his shoulder, she lifted her chin. "You're lucky you're not a psi. This thing will hurt like a bite from one of your Dabin swamp rats, the ones you said clamp onto their prey and don't let go."
Releasing him, she turned to the machine and started it with just a few button pushes. The ache built quickly. Within seconds, her head from brow to nape throbbed. Another set of taps on the position-sensitive controls modulated that pain into a dull ache. An ache similar to the one already awaiting them on the ship docked one gantry up from theirs.
"You have your orders, Lieutenant Commander. Carry them out," she instructed.
"Sir, yes, sir." Unhappy, he turned on his heel, picked up the packing crate, and strode out, leaving her alone in the main engineering compartment.
Without the warmth of his personality and presence to fill it, the stripped-down compartment echoed. Every sector and cabin on the ship had been stripped to its barest necessities. Ia knew the Feyori stalking her took that as a sign that they would succeed in their coming attempt. That she intended to deny them every scrap of resources she could. To them, the coming nexus point was muddied, misted over, but there _was_ a timeline where they very well could succeed. She didn't need to hide from the Feyori the fact her ship was empty; the anti-psi device was needed to hide that successful path from them.
Hopefully they would be blocked by the device from reading the timestreams, given their milder abilities. That nexus was still somewhat clear to her. Mild as its broadcast was—poisonous as it was to the Feyori—the machine was already starting to mist up the timestreams ahead. She hadn't seen any mists this strongly since manifesting. Not for the first time, Ia wondered why her half-breed life was so oddly immersed in such a vast ocean of prognostication when not even her father-progenitor could do a single percent of the things she could do.
She took her time leaving engineering and made sure to use the port side of the ship to return to the bridge. That permitted her to avoid everyone but Lieutenant Rico, the last person on duty. Ironically, for all that he would have been loyal to her anyway after his trip into the timestreams with her, he would have strenuously protested this plan right alongside Harper if it hadn't been for Private Sung's indiscretion.
But they were hers now. The Damned were solidly hers. If she said, "Jump," they paused only to ask, "How high?" then did their best to hit that exact mark, no more and no less. Even Hollick's replacement, Private Nesbit, was hers. He had asked plenty of questions among her crew, watched her actions and plans unfolding in combat after combat, and had developed a solid level of faith in his CO.
After more than two years of open war, two and a half if one counted their preliminary strikes, and after enduring three to four times as many fights as any other crew, they had lost only one soldier from their Company. There was no doubt that Ia's Damned were the finest fighting force available to the war. Two things made them that way: Ia's trust in them to be the right people for the right job at the right point in time, and their trust in her to let them know what the right job was.
Knowing all of that, believing in all of that, didn't stop Rico from giving her a dirty look as he left the bridge, though. Ia didn't have to read his thoughts to know why. As far as the tall Platoon officer was concerned, his CO was being an idiot for doing all of this on her own. Unfortunately, he had no way to escape along with her, and she wasn't about to waste his life needlessly. He knew all of that, but it didn't make him any happier with her.
Settling into the command seat, Ia levered the chair forward and buckled herself in. The harness straps had long since been replaced but were starting to show some wear and tear from their constant use. With nothing to do but wait, Ia pulled a trick out of both Helstead's and Spyder's bags and put her bootheels up on the console. Not near anything sensitive, of course. Then again, she didn't need to touch the controls to activate them.
Screens flicked to life around the bridge. With her primary screen blank and thus transparent, she read the distant displays monitoring the ship's statistics. The engines looked good, tiny little green bars indicating energy consumption was running low. The shields displayed their status in two levels, low on the starboard, toward the Battle Platform, high on the port side, standard wartime procedure whenever docked, in case of an unexpected attack. The _Hellfire_ 's scanners were sweeping passively, collecting and collating data with the navicomp's help.
Lifesupport, however, was nothing more than a series of red lines and blanks, save for a few tiny green bars. Plants had been boxed up, fish bagged, hens crated, and all of it shipped out. Including her precious supply of carefully nurtured topadoes, with their beautiful dark purple and sky blue striped foliage, and their tasty aquamarine roots. The ship was running purely on mechanical reoxygenators now, with the air scrubbed chemically instead of biologically.
The emergency systems could easily reoxygenate the ship long enough to keep the original crew complement of five hundred or so alive for at least two months, though the air would start to smell a bit ozone-ish and stale after the first three weeks. With just Ia on board—well, her, and a last clutch of crew members slipping out the amidships starboard airlock with a few last-minute kitbags of belongings slung over their shoulders—the air supply could easily last a year. Food would be the biggest problem; all but the bridge galley had been stripped of everything, and even that one only had a few ration packets left.
The Salik had already tried several times to board and capture the _Hellfire_. Every single time, Ia had precognitively thwarted them. Missiles had breached their hulls, but not a single ostrich-legged flipper had landed on their decks. They weren't a concern, though. Even if they had, they'd never be able to use the Godstrike cannon.
The Feyori...were a different matter. All they needed was a bit of Ia's DNA and enough time to read her mind, and they could replicate her body and personality in the flesh of a volunteer, much as she and Belini had turned Private Hollick into Private N'Keth. Belini hadn't ever mentioned that possibility, but Ia knew it could be done. Once they had a duplicate clone of herself, they would be able to access the Godstrike cannon. It was debatable whether or not they would be able to replicate her psychic abilities, since even among identical twins such things varied, but the cannon did not require that much accuracy for access. All of these things, she had carefully explained to Myang in a message embedded in that latest datachip.
A glance at the chrono showed she had twenty hours and fourteen minutes to prevent them from trying.
When the airlock sealed shut behind Rico, the last of the Damned to leave, she unstrapped from the command chair and visited the head. Fixed herself a mug of water with a sipping lid. Brought it and a snack packet back to her station and redonned the safety harness. Tapping in a command manually, she shifted the view on her right secondary screen from blank nothingness to the dorsal view of the slightly oversized OTL courier parked one gantry up from hers.
Like her old Delta-VX from her time on the Blockade, it fell into the Harrier Class of ships, though it was a single vessel, not two mated together. It was also small enough; it could've fit into one of the hangar bays on board the _Saratoga Jones_. The captain had agreed to park it at a gantry instead, in order to make the short trip of her crew from ship to ship as inconspicuous as possible.
As she watched, sipping from her mug, the oversized courier finally detached from the Battle Platform. It drifted away, then gently turned and shimmered, activating its insystem thrusters. The insystem field was more primitive than FTL warp panels, yet more fuel-efficient at speeds below three-quarters Cee. With a pulse that fluttered and rippled the starlight along the Harasser's polished grey hull, the ship soared away from the Battle Platform.
Ia's yeomen pilots maneuvered their much larger ship with thrusters. FTL was tricky; milliseconds of misuse could literally translate to kilometers of travel at higher speeds. With her reflexes coupled to the timestreams, using FTL had allowed her to dodge laserfire from both enemy and allied ships. For instances where accuracy was vital, such as that Choya task force sent to Earth, she used the insystem method like a sane pilot would.
A simple maneuver such as leaving the _Saratoga Jones_ was a lot saner than the madness of combat. Finishing her mug, she returned it to the galley, tossed the emptied plexi packet into the recycler—out of habit, not because she expected the material to be recycled—and used the head one more time. When she sat down at her station for the third time, she activated the comms.
_"This is TUPSF_ Hellfire _to Battle Platform_ Saratoga Jones, _requesting permission to undock and depart."_ This was usually handled by the comm tech, one of the many operations that happened seamlessly, smoothly in the background of a well-run bridge. She waited for a reply, and got one within a few moments.
"Hellfire, _this is Docking Control. You have permission to decouple and depart. Godspeed, and go smash some more of the enemy for us,"_ the unnamed comptroller added. This was the first time they'd docked briefly at this particular Battle Platform. The reputation of the Damned, which she had painstakingly built over the last two years, had preceded them, however.
_"Thank you,_ Saratoga Jones," Ia acknowledged, smiling slightly. _"We'll do our best._ Hellfire _out."_
Flying the ship solo required several command overrides. It was an attempt by the Space Force at preventing their ships from being hijacked by their enemies, whether amphibious or criminal. Ia manipulated each one electrokinetically, decoupling the clamps that held them to their gantry. A gentle pulse of the thrusters drifted them away from the oversized hybrid of warship and space station. Another gentle pulse turned her ship onto a vector similar to the courier's.
Warming up the insystem thrusters, she set the _Hellfire_ on a course that would take it away from the Battle Platform by a good thousand klicks. A slide of her fingertips over the helm controls brought the FTL panels online, trembling them forward by the pulsing of the fields that greased the palms of normal physics, making Newton and Einstein roll in their graves.
The Harrier-Class dropship carrying her crew vanished via OTL before she even reached one-quarter Cee, punching open a hole into hyperspace and sucking itself through. On the far side, Ia knew the ship would arrive somewhere near Dabin's outermost gas giant. At that point, her entire crew would climb into their mechsuits and await a short hop that would bring them skimming in close to Dabin's atmosphere. From there, it would be a matter of a short dip down to a low enough altitude to be air-dropped behind friendly lines.
The courier had a ninety percent chance of escaping the ensuing counterattack from the Choyan and Salik warships blockading the planet, if all went well. If it did not, and they were pursued, Helstead had carefully memorized a chunk of Dabin landscape from surveillance images captured a good five klicks up, and would teleport the ship to that zone so they could drop out. It would leave everyone on board nauseated from the jump, and possibly incapacitate the redhead for a few hours from the backlash of overextending herself in moving that much mass, but it was a viable means of ensuring everyone arrived where and when they needed to go.
Everyone except Ia. She had to get there the hard way.
The transition to faster-than-light happened smoothly. Dots burst as grey bubbles and collapsed into streaks of prismatic light. Setting the ship to fly on automatic, and for the onboard sensors to beep at her if anything went wrong—less than one-hundredth of a percent in probability, but still a possibility—she twisted a little in her seat, slouched, hooked her legs up over the left-hand console in as much comfort as the harness would allow, and settled in for a nap.
With the ship fully fueled, the trip from CS 47 to the edge of the Dabinae System wasn't going to tax the engines. It would just take her twenty hours to get there. Twenty hours of head-aching boredom. Her datapads had been packed up along with her other belongings and shipped off earlier. Some of the more time-sensitive stuff had been packed into Harper's last-minute kitbag, with the rest shipped off to meet up with their next ship.
Without even so much as her prophecies for distraction, and no interest in watching anything from the entertainment databanks, there was nothing for Ia to do but nap and wait, nap and wait. Mostly nap. All those long, long days were beginning to catch up to her. Awkward as her perch was, as much as the device back in engineering made her skull throb, at least everything was secure enough that she could nap, for now.
JUNE 1, 2498 T.S.
ONE LIGHT-MONTH OUT FROM THE DABINAE SYSTEM
The ship's shields fluttered. Cracking open an eyelid at the first warning beep, Ia studied the screens. The navicomp had her main screen lit up, a tiny green-circled dot insisting that the approaching ship was an allied Terran vessel. She knew better. The signature codes were all fake, inserted into the Space Force's registry by clever electrokinetic programming.
The other vessel slowed as it approached. It also hailed the _Hellfire_. Activating the comm system, Ia gave them a pingback, waited for a response, and opened the requested link.
_"TUPSF_ Zizka _to TUPSF_ Hellfire, _boy are we glad to find you out here."_
Ia didn't bother to activate the vid half of the link. Adjusting her headset, which had slipped while she had caught up on far too many months of shorted sleep, she stretched, rubbed the bridge of her nose where that persistent behind-the-sinuses ache from the anti-psi machine continued to throb, and replied. _"Greetings,_ Zizka, _this is the_ Hellfire. _What brings you all the way out here?"_
_"We have an urgent message from Vice Commodore William Quan of the Special Forces Psi Division. The vice commodore is a registered precog. He swears he's had a vision running contrary to what you told the Command Staff about the upcoming battle on Dabin and wishes to board so he can personally compare your version versus his. Do we have permission to dock?"_
_"TUPSF_ Zizka, _you have permission to grapple to the portside amidships airlock. The Captain says she looks forward to chatting with your vice commodore,"_ Ia added, suppressing the urge to smile. She didn't want it showing up in the tone of her voice. Apparently these criminals hadn't yet realized that they needed the Admiral-General's permission to board this ship legitimately. Myang had reluctantly permitted Ia's request to destroy the ship rather than let it fall into enemy hands, though the head of the Space Force undoubtedly thought this moment would come violently. Not peacefully. _"Let me ping you the navicomp link for a smooth docking."_
_"Thank you,_ Hellfire."
Unbuckling—since they were still at least five minutes away—Ia visited the head one last time. She could do nothing about the fact she was still clad in yesterday's clothes, just a plain, rumpled grey shirt and darker grey slacks, but she did finger-comb her white locks more or less straight in the mirror over the sink, rinsed her mouth with a plexi cup to get the fuzzy taste out, and took a few moments to close her eyes, center her mind, and calm herself. Her head still ached, but it was bearable.
Tossing the cup in the recycler, she visited her quarters. With that brief task handled, Ia returned to the bridge and resumed her seat. She did not reattach the harness. She did, however, flip up the little door built into the side of her console and grasped the two electrodes hidden within the little alcove.
As the two ships' navicomps chatted at each other, guiding the smaller courier-class _Zizka_ into docking with the elongated needle of the _Hellfire_ , Ia pulled on that conduit's juice. Pulled and pulled and pulled, balling it up inside her body. Every time her hair threatened to fluff, she sucked it in tighter, clamping down on the external signs. By the time the telltales for the portside airlock greenlit with a viable seal, the lines and angles of the bridge's stations were starting to glow.
The slightest nudge of her mind cracked a miniature bolt of lightning by mistake. Thankfully, the various bridge consoles had been built with capacitors; they functioned properly, absorbing the extra energy without overloading the command station. Using her hands instead of her gifts, Ia activated the interior surveillance pickups the old-fashioned way.
The courier had already disgorged several passengers. They looked like TUPSF troops, were armed and lightly armored like Terran troops, but she knew each one was a career criminal who specialized in pirating and pilfering Terran military supplies. And it was the tall scowling woman in their midst who was really in charge, not the short Asiatic man pretending to be the vice commodore in question.
She rubbed at her brow as Ia watched, and snapped something to the dozen men and women accompanying her. Ia had to manually replay the scene to hear what she said. When she heard it, she activated the intercom. _"Attention,_ Zizka _personnel. The nearest door into the main engineering compartment is located on Deck 8, aft-sector port side. The actual deck containing what you are looking for is in main engineering, Deck 8 starboard side...and once you get over there, you can't miss it. You are currently on Deck 12, amidships-sector port side. The nearest crossover point to starboard from your location is Deck 6."_
The tall blonde checked her stride. Still scowling in pain, she squinted up at the ceiling until she spotted the closest active camera. _"So you know why we're here?"_
_"Who you are and why you're here...though I'm surprised you agreed to risk your sparks with this little trip, Janeal,"_ Ia added. The Feyori blinked twice, but otherwise didn't react. _"Whatever Miklinn offered you, it won't be enough. I'm sorry."_
_"You are nothing more than a pawn,"_ Janeal stated, moving forward. _"You always have been, and you always will be. Prepare to be removed from the Game."_
Smirking, Ia quipped quietly into her headset pickup, _"'I'm sorry, Dave. I'm afraid I can't do that...' Only in_ my _case, I'm definitely not Hal, and you're not the hero of this little vidshow. The bridge is located in the middle of amidships-sector Deck 6. I look forward to your arrival._ Ia'nn sud-dha."
Rubbing again at her aching head, the tall, blonde Meddler strode forward. Her accomplices accompanied her.
_"Oh, one more thing. You might want to tell the others to head back to the_ Zizka _right now and start running for the stars within the next minute,"_ Ia added. _"I realize you're willing to sacrifice at least three of them to find and shut off the anti-psi machine sitting in Engineering, Deck 8 starboard, but the others don't have to die. This is between you and me, after all._ "
The back door slid open. An annoyed-looking, red-clad pixie padded onto the bridge. Her frown deepened as she looked around the almost empty cabin. "What the...? Where did everyone go?"
Her trip to her quarters to trigger Belini's summoning spot had once again worked. Her cofaction partner didn't look particularly happy to be here, but at least she had come when called. Ia lifted her bootheels back up onto the console.
"Welcome aboard, Belini. You're just in time," she greeted the petite Feyori, her tone light and cheerful. Mockingly so. Gesturing at the empty duty stations, Ia extended her meager hospitality. "Feel free to have a seat. I'd offer you something to eat, physical or otherwise, but I'm afraid we're almost out of time. Janeal and her thugs will be joining us in about a minute or so."
Blinking, Belini twisted to face her. "Janeal? The Feyori who plucks the puppet strings of the local undergalaxy?"
"The Queen of Predators and Piracy," Ia agreed. She fluttered her hand vaguely at the other stations. "Have a seat and enjoy the show."
"If you're expecting high tension and danger to tip you over the energy-to-matter barrier, I'm afraid Janeal isn't going to give you the time to build up that sort of momentum," Belini warned her. "The moment she steps inside, her thugs will shoot you dead, and that will be that. Not even you can dodge laserfire, half-breed. She also comes with too many cross-factions for me to counteract directly, if you were expecting me to save your hide. Since I now hold a very low and tenuous ranking, _in case you forgot_?"
Ia rolled her eyes and pointed at the pilot's seat. "She is not going to shoot me right away. She'll want to monologue first. _You_ are here to observe my manifestation, and to stand witness to all that will follow. You won't have to lift a finger. My debt to you, in exchange for all your trust and faith in me, is about to come due. Have a seat; I'd hate for you to miss a single millisecond of it."
Still scowling, Belini studied her for a long moment, then padded over to the indicated chair and plopped herself into it. Only the narrowing of her eyes made her look dangerous; the rest of her looked like a pouting pixie. Squirming a little in her own seat, Ia slouched a little more comfortably and once again stuck her hand in the conduit hole. Some of her built-up energy had bled off during the miniature jolt. She replaced that and more, lounging in the command chair.
The portside door slid open. Janeal stepped through, projecting an air of command more successfully than the look-alike Human tapped to imitate the real-life Vice Commodore Quan. He in turn kind of looked like her old Drill Instructor, Sergeant Tae, Ia decided, looking him over. _Except he's in no way related to Harper. I should probably find a spare minute or two in the future to give the real Tae a call and see how he's doing. I know Meyun keeps in touch with him, but I haven't had a chat with the man since the third time he dropped by the Academy...A pity this fellow isn't going to survive this little encounter._
"What, too lazy to meet your own fate, half-breed?" Janeal mocked. Her look of contempt was an excellent study of Human emotions. It also summed up the general Feyori mind-set toward the short-lived matter races succinctly, and she gave it to Ia with an added sneer of derision.
"Oh, I'm meeting it," Ia said—and kicked, shoving her bootheel hard against the plexi lockbox encasing the main cannon's switch.
That broke it with a _crack_ , and an immediate flash of red lights. They flared every second, accompanied by an unnerving, atonal buzzing. Bright white numbers appeared on every screen, starting at 60 and counting their way down. The hyperrelay telltales over at the communications station lit up, hard-dumping the black-box recordings for the last hour in an encrypted broadband stream. It would continue to stream its previous reams of data to the Command Staff, too, all the way through to the last second of the _Hellfire_ 's existence. Ia didn't bother to stop it.
"If your thugs ran right _now_ , they _might_ make it from the bridge to the airlock in forty-nine seconds," she stated, sitting up. Her mind snapped out, stabbing at the Humans on board her ship. They dropped with sighs and thumps to the deck, mercifully knocked unconscious, half-drawn weapons clattering to the deck. "But they'd only have a handful left to get through the airlocks, and would not escape in time. Unconsciousness is the only mercy I can give them."
"I can stop this!" Jeneal snapped, lifting her hand.
Ia shook her head, pushing to her feet. She could feel the woman probing electrokinetically at the controls, and knew better. "It's too late. It's _chemistry_ , not electronics. The hydrocatalysts have already been released into every tank on this ship, and you won't stop more than a fraction of it. You have maybe thirty seconds to shift shape and survive."
_"Shakk!"_ Grabbing the pilot's console, Belini ripped electricity out of it, popping from pixie to bubble in three seconds flat.
Ia planted her hand on her own station, drawing out the energy she needed more gently, but with the same result. In a flash, she floated instead of stood before the other two. Belini swirled in agitation, then clamped down hard on herself, blocking out all forms of radiation, turning a shining shade of mirror white.
Janeal grabbed for the gunner's station, changing as well. Between Belini's and Ia's efforts, there wasn't a large flow of energy available. It took the other Meddler ten full seconds to manifest. Even as she shifted, Ia noted to herself that Feyori in matter-forms stood out to their energy-form fellows like a glowing-hot iron on an infrared scanner. _I'll have to keep that in mind for when I'm on Dabin._
For one moment, the pirate Meddler hung there in a swirl of confusion, then she turned brilliant white like Belini, the Feyori equivalent of holding one's breath by blocking out all incoming radiation.
So did the rest of the ship, when the countdown reached zero and the tanks exploded, turning the _Hellfire_ , and by collateral damage the _Zizka_ , into a tiny, bright nova. Forewarned by her trips onto the timeplains, Ia did not block off the inferno, like the other two. Instead, she embraced it, swelling outward as the energy rushed through her, using the part of her mind that dealt with, that comprehended, that controlled and manipulated Time itself to withstand the overwhelming flows.
Expanding and twisting, she snatched up the two Feyori with her thoughts, swirling them into her grasp. Ia watched them carefully as she spun them around her expanded sense of self. There, in the wake of the energies thrust aside by her vortex, were the little hyperthreads that attached each alien to their faction-allies, their favorite places.
Seizing those threads, Ia _pulled_. It was a variation on the summoning spark that alerted each attached Meddler that they were needed, a variation that could only succeed if the Feyori doing the summoning had access to a great deal of power. A solar flare, a massive thunderstorm, even the surface of a dying star could be used. Or an explosion caused by the catalytic conversion of tens of thousands of metric tons of purified water fused into a muddled, white-hot plasma of pure hydrogen and supercharged oxygen.
When each Feyori popped into range, dragged there by her summons, she grabbed _their_ threads and yanked. Grabbed _those_ threads and pulled. Grabbed the next set and tugged. Four degrees out was all the strength she had to spare for her task, and only with the first cycle's worth of Meddlers did she literally pull any of them through the aether of hyperspace.
But the others came out of alarm and curiosity and boredom as she summoned each clutch. They crowded around her, dozens, hundreds, then roughly two thousand strong, brushing through each other, bobbing in empty space. Demanding in pulses of energy and empathy and telepathy to know what was going on, even as they drank in the remnants of the explosion she had made.
Spun out to an almost tenuous level, Ia enveloped the last ones to arrive and _pulled_ them all inward, spinning around and down, flipping everyone in, up, and out. Landing on the banks of the timestreams in her Human form, she looked at the mass of tiny silver beads clinging to her left hand. They huddled close to her skin not because they willed it, but because _her_ will trapped them there. Ruthlessly, Ia forced them to see the verdant fields, the golden waters, the blazing-hot skies in her mind.
_"Welcome to_ my _chessboard, gentlebeings. Welcome to_ Time _itself,"_ she warned them, letting the word roar across the plains like wind from a thunderstorm, letting the force of that word shift them forward by three centuries and a little more. _"And if you do_ not _faction with me...welcome to the end of your precious little Game."_
_I am not fully Human, no. I admit it outright. Half of my being was conceived by the usual earthly delights, while the other half was crafted by the meddling of purified will. But I was raised to be a Human, I have lived as a Human, and I fight as a Human. And I love as a Human. With compassion for more than just my own interests. For more than my own culture. More than my own kind. I have loved from birth every person you have ever seen, regardless of species, and I love the septillions more that you won't._
_Because I am still Human, I swim in the frozen waters of my duty, lifting up others' lives so they do not drown in the tempest-tossed storm. I burn with the raging flames of my conscience and never divert from the embers underfoot on the dangerous paths I must tread. I walk through the hellish inferno to save others' souls, knowing that at the same time I slip down the ice-cold slopes of my lonely task with each condemned step._
_I am the Prophet of a Thousand Years, the Changer of Worlds, the Meddler of Destinies, the Defender of Our Galaxy...and neither Hellfire nor Damnation will keep me from my task._
_~Ia_
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1. The team is neither in Los Angeles City orCounty and it is a huge county.
The Anaheim CityCouncil is expected to vote Tuesday to enter lease negotiations withthe Angelsthat could keep the team in the city through 2057 but enable the teamto drop the clumsy "of Anaheim" suffix from its name.
The two sidesalready have agreed that the Angels will have sole control over theteam name, according to a memorandum posted on the city's website.
In 2005 Authentic Albert Almora Jr Jersey , Morenochanged the team name to "Los Angeles Angels of Anaheim,"in order to satisfy a lease provision requiring the name to "includethe name Anaheim therein." The city sued Moreno and lost.
Under the new deal Authentic Glenn Robinson Jersey ,Moreno could simply call his team the Los Angeles Angels.
I wonder why theAngels and the Anaheim City Council have no desire to separatethemselves from a metropolis that considers Orange County assophisticated as a cowboy who shows up to dinner in his work boots. Even the Orange County Angels would have been better though maybe thecounty better look at a name change since Orange trees are going theway of the Dodo bird. | {
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Livet på landet är en amerikansk film från 1988 med Chevy Chase och Madolyn Smith i huvudrollerna. Förlagan till filmen är en roman med namnet Funny Farm (den engelska titeln även på filmen) från 1985 av Jay Cronley. Filmen spelades mestadels in i Vermont och det blev den sista filmen som George Roy Hill regisserade.
Filmen handlar om sportskribent från New York som flyttar ut till landet med sin fru, men råkar ut för allehanda missöden istället för att uppleva den idyll han hoppats på. Paret bestämmer sig efter en tid för att skiljas och sälja sitt hus, men de ändrar sig sedan och finner sig till slut tillrätta. Filmen fick blandade till positiva recensioner.
Rollista
Chevy Chase - Andy Farmer
Madolyn Smith - Elizabeth Farmer
Kevin O'Morrison - Sheriff Ledbetter
Mike Starr och Glenn Plummer - Crocker och Mickey, flyttgubbar
Kevin Conway - Crumb Petree, brevbärare
Joseph Maher - Michael Sinclair, förläggare
Bill Fagerbakke och Nicholas Wyman - Bröderna Criterion
William Newman - Gus Lotterhand
Sarah Michelle Gellar - Elizabeths elev (scenerna bortklippta)
Källor
Externa länkar
Amerikanska dramakomedifilmer
Filmer 1988 | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/3546505\/which-of-these-equations-is-the-backward-kolmogorov-equation","text":"# Which of these equations is the backward Kolmogorov Equation?\n\nI am reading Stochastic Processes and Applications by Pavliotis. At some point he defines Kolmogorov backward equation as the differential equations\n\n$$\\frac{\\partial u}{\\partial t} = \\mathcal{L}u$$ $$u(x, 0) = f(x)$$ where $$u(x, t) = E[f(X_t)|X_0 = x]$$ and $$\\mathcal{L}$$ is the generator of $$X_t$$.\n\nLater he defines Kolmogorov backwards equation as an actual backward (in time) equation for $$u(x,t) = E[f(X_T) | X_t = x]$$.\n\nKolmogorov's forward equations are defined in terms of the adjoint operator acting on probability measures, always going forward in time.\n\nWhich one of the above is the actual Kolmogorov backwards equation and why is it called backward\/forward?","date":"2020-02-23 23:57:57","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\": 6, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8477775454521179, \"perplexity\": 377.37218147305344}, \"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-2020-10\/segments\/1581875145859.65\/warc\/CC-MAIN-20200223215635-20200224005635-00003.warc.gz\"}"} | null | null |
\section{Introduction}
\label{S1}
Following work by Bender and collaborators \cite{Bender1998,Bender1999,Bender2007,Special2012,Theme2013} it has become apparent that quantum mechanics is much richer than conventional Hermitian quantum mechanics. However, if one wishes to maintain probability conservation, one needs to be able to define an inner product that is time independent. The reason that one has any freedom at all in doing this is because the Schr\"odinger equation $i\partial_t|\psi\rangle= H|\psi\rangle$ only involves the ket state and leaves the bra state unspecified. While the appropriate bra state is the Hermitian conjugate of the ket when the Hamiltonian is Hermitian, for non-Hermitian Hamiltonians a more general bra state is needed. However, one cannot define an inner product that is time independent for any non-Hermitian Hamiltonian. Rather, it has been found (\cite{Mannheim2013,Mannheim2018} and references therein) that the most general Hamiltonian for which one can construct a time-independent inner product is one that has an antilinear symmetry, and in such a case the required bra state is the conjugate of the ket state with respect to that particular antilinear symmetry.
When a Hamiltonian has an antilinear symmetry its energy eigenspectrum can be realized in three possible ways, all eigenvalues real and eigenspectrum complete, some or all of the eigenvalues in complex conjugate pairs with the eigenspectrum still being complete, or eigenspectrum incomplete and Hamiltonian of non-diagonalizable, and thus necessarily of non-Hermitian, Jordan-block form. Of these three possible realizations only the first can also be achieved with a Hermitian Hamiltonian, and while Hermiticity implies the reality of energy eigenvalues, there is no theorem that would require a non-Hermitian Hamiltonian to have complex eigenvalues, with Hermiticity only being sufficient for the reality of eigenvalues but not necessary.\footnote{For any non-diagonalizable two-dimensional Jordan-block and thus necessarily non-Hermitian Hamiltonian for instance, since the eigenspectrum is incomplete the Hamiltonian has just one eigenvector even though there are two eigenvalue solutions to $|H-\lambda I|=0$. These two eigenvalue solutions must then be equal to each other since they have to share just the one eigenvector. If in addition the Hamiltonian has an antilinear symmetry, by being equal to each other the two eigenvalues could then not be in a complex conjugate pair. In consequence, the two eigenvalue solutions to $|H-\lambda I|=0$ must be real -- to thus show directly that one can have real eigenvalues if a Hamiltonian is not Hermitian.} The necessary condition for the reality of energy eigenvalues is that the Hamiltonian have an antilinear symmetry \cite{Mostafazadeh2002,Solombrino2002,Bender2010,Mannheim2018}, while the necessary and sufficient condition is that in addition all energy eigenstates are eigenstates of the antilinear operator \cite{Bender2007}.
Interest in non-Hermitian Hamiltonians with an antilinear symmetry was first triggered by the work of Bender and collaborators \cite{Bender1998,Bender1999} who found that the eigenvalues of the Hamiltonian $H=p^2+ix^3$ are all real. This surprising reality was traced to the fact that the Hamiltonian possesses an antilinear $PT$ symmetry ($P$ is parity and $T$ is time reversal), under which $PpP=-p$, $PxP=-x$, $TpT=-p$, $TxT=x$, $TiT=-i$. In general for any Hamiltonian $H$ with an antilinear symmetry $A$ (i.e. with $AH=HA$), when acting on $H|\psi\rangle=E|\psi\rangle$ one has $AH|\psi\rangle =AHA^{-1}A|\psi\rangle=HA|\psi\rangle=E^*A|\psi\rangle$. Thus for every eigenstate $|\psi\rangle$ of $H$ with energy $E$ there is another eigenstate $A|\psi\rangle$ of $H$ with energy $E^*$. Thus as originally noted by Wigner in his study of time reversal invariance, energies can thus be real or appear in complex conjugate pairs with complex conjugate eigenfunctions. It is often the case that one can move between these two realizations by a change in the parameters in $H$. There will thus be a transition point (known as an exceptional point) at which the switch over occurs. However, at this transition point the two complex conjugate wave functions ($|\psi\rangle$ and $A|\psi\rangle$) have to collapse into a single common wave function as there are no complex conjugate pairs on the real energy side. Since this collapse to a single common wave function reduces the number of energy eigenfunctions, at the transition point the eigenspectrum of the Hamiltonian becomes incomplete, with the Hamiltonian then being of non-diagonalizable Jordan-block form, the thus third possible realization of antilinear symmetry.
While the above analysis would in principle apply to any antilinear symmetry, because of its $H=p^2+ix^3$ progenitor, the antilinear symmetry program is conventionally referred to as the $PT$-symmetry program. However, $PT$ symmetry can actually be selected out for a different reason, namely it has a connection to spacetime. Specifically, it was noted in \cite{Streater1964} and emphasized in \cite{Bender2007} that for the spacetime coordinates the linear part of a $PT$ transformation is the same as a particular complex Lorentz transformation, while in \cite{Mannheim2016,Mannheim2018} it was noted that for spinors the linear part of a $CPT$ transformation is the same as that very same particular complex Lorentz transformation, where $C$ denotes charge conjugation.\footnote{The complex Lorentz transformation $\Lambda^{0}_{\phantom{0}3}(i\pi)\Lambda^{0}_{\phantom{0}2}(i\pi)\Lambda^{0}_{\phantom{0}1}(i\pi)$ implements $x_{\mu}\rightarrow -x_{\mu}$ on coordinates and $\psi_1(x) \rightarrow\gamma^5 \psi_1(-x)$ on a Majorana spinor, just as the linear part of a $CPT$ transformation does.} Then in \cite{Mannheim2016,Mannheim2018} it was shown that if one imposes only two requirements, namely the time independence of inner products and invariance under the complex Lorentz group, it follows that the Hamiltonian must be $CPT$ invariant, with $CPT$ symmetry itself being antilinear. Since this analysis involves no Hermiticity requirement, the $CPT$ theorem is thus extended to the non-Hermitian case (and thus through the complex energy realization of antilinear symmetry to decay processes that are forbidden by Hermiticity). Since charge conjugation plays no role in non-relativistic physics where one is below the threshold for particle production, $CPT$ then defaults to $PT$, to thus put the $PT$-symmetry program on a quite secure theoretical foundation.
As with the $CPT$ theorem, one can ask what happens to other familiar results of quantum field theory when one relaxes the Hermiticity requirement. This then was the brief of Alexandre, Ellis, Millington and Seynaeve \cite{Alexandre2018}, who found that the Goldstone theorem can also be decoupled from Hermiticity, and can hold in the non-Hermitian but antilinearly symmetric case.\footnote{Since historically the $CPT$ theorem was found during the effort to establish the spin and statistics theorem, it would be of interest to see how the spin and statistics theorem itself might fare in the non-Hermitian but $CPT$ symmetric case.} Alexandre, Ellis, Millington and Seynaeve restricted their discussion to those realizations of antilinear symmetry in which all the energy eigenvalues of the Hamiltonian are real. Here we extend the discussion to the two other possible $PT$-symmetry program realizations, namely energies in complex conjugate pairs or Jordan-block Hamiltonians that are not diagonalizable at all. In particular, we show that it is possible for the Goldstone boson mode itself to be one of the zero-norm states that are characteristic of Jordan-block Hamiltonians. While we discuss the same model as Alexandre, Ellis, Millington and Seynaeve our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. In particular, in their paper Alexandre, Ellis, Millington and Seynaeve presented a variational procedure for the action in which the surface term played an explicit role, to thus suggest that one has to use such a procedure in order to establish the Goldstone theorem in the non-Hermitian case. However, we show that one does not need to do this, as we are able to obtain a Goldstone boson using a completely standard variational procedure. Moreover, since we do use a standard variational procedure we can readily extend our analysis to a continuous local symmetry by introducing a gauge boson. We show that the gauge boson acquires a non-zero mass by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the spontaneous breakdown of the continuous local symmetry, the gauge boson remains massless.
The present paper is organized as follows. In Sec. \ref{S2} we present the complex scalar field model discussed in \cite{Alexandre2018}, and using a standard variational procedure for the action find its spontaneously broken tree approximation minimum and determine the eigenvalues of the associated mass matrix. In Sec. \ref{S3} we determine the associated left- and right-eigenvectors and construct the left-right $V$ operator norm that plays a central role in antilinear theories. In Sec. \ref{S4} we compare our treatment with that of the authors of \cite{Alexandre2018}, who used a non-standard variational procedure. This leads us to a Hamiltonian that looks Hermitian but is not, and in Sec. \ref{S5} we discuss how this is possible. In this section we also discuss the connection between antilinear symmetry and Hermiticity within the context of the $CPT$ theorem as developed in \cite{Mannheim2018}. In Sec. \ref{S6} we extend the discussion to the Englert-Brout-Higgs mechanism, and in Sec. \ref{S7} we provide a summary of our results. Finally, in an appendix we construct the left-right quantum theory matrix elements that would produce the c-number tree approximation classical field and the effective potential that is minimized in Sec. \ref{S2}, and discuss how Ward identities are realized in the non-Hermitian case.
\section{Spontaneously Broken non-Hermitian Theory with a Continuous Global Symmetry}
\label{S2}
The model introduced in \cite{Alexandre2018} consists of two complex (i.e. charged) scalar fields $\phi_1(x)$ and $\phi_2(x)$ with action
\begin{eqnarray}
I(\phi_1,\phi_2,\phi^*_1,\phi^*_2)=\int d^4x\left[\partial_{\mu}\phi^*_1\partial^{\mu}\phi_1+\partial_{\mu}\phi^*_2\partial^{\mu}\phi_2
+m_1^2\phi_1^*\phi_1-m_2^2\phi^*_2\phi_2-\mu^2(\phi^*_1\phi_2-\phi^*_2\phi_1)-\frac{g}{4}(\phi^*_1\phi_1)^2\right],
\label{GPT1}
\end{eqnarray}
where the star symbol denotes complex conjugation, and thus Hermitian conjugation since neither of the the two scalar fields possesses any internal symmetry index. Since the action is not invariant under complex conjugation, it is not Hermitian. It is however invariant under the following $CPT$ transformation
\begin{eqnarray}
\phi_1(x_{\mu})\rightarrow \phi^*_1(-x_{\mu}),\quad \phi_2(x_{\mu})\rightarrow -\phi^*_2(-x_{\mu}),
\quad \phi^*_1(x_{\mu})\rightarrow \phi_1(-x_{\mu}),\quad \phi^*_2(x_{\mu})\rightarrow -\phi_2(-x_{\mu}),
\label{GPT2}
\end{eqnarray}
and thus has an antilinear symmetry.\footnote{The study of \cite{Mannheim2018,Mannheim2016} shows that for relativistic actions such as that given in (\ref{GPT1}) $CPT$ must be an invariance, a point we elaborate on further below. In their paper the authors of \cite{Alexandre2018} took $T$ to conjugate fields. While $T$ does conjugate wave functions in quantum mechanics, conventionally in quantum field theory $T$ does not conjugate q-number fields (it only conjugates c-numbers). Rather, it is charge conjugation $C$ that conjugates fields. Thus what the authors of \cite{Alexandre2018} refer to as $PT$ is actually $CPT$, just as required by the analysis of \cite{Mannheim2018,Mannheim2016}. However none of the conclusions of \cite{Alexandre2018} are affected by this.} Since one can construct the energy-momentum tensor $T_{\mu\nu}$ by the variation $T^{\mu\nu}=2(-g)^{-1/2}\delta I(\phi_1,\phi_2,\phi_1^*,\phi_2^*)/\delta g_{\mu\nu}$ with respect to the metric $g_{\mu\nu}$ of the covariantized form of the action (momentarily replace ordinary derivatives by covariant ones and replace the measure by $\int d^4x (-g)^{1/2}$), it follows from general coordinate invariance that a so-constructed energy-momentum tensor is automatically covariantly conserved in solutions to the equations of motion that follow from stationarity of the same action. Then, since one can set $H=\int d^3x T_{00}$, it follows that the associated Hamiltonian is time independent. Moreover, since the metric is $CPT$ even, then since the action is $CPT$ invariant it follows that the Hamiltonian is $CPT$ invariant too. The Hamiltonian associated with (\ref{GPT1}) thus has an antilinear $CPT$ symmetry.\footnote{Just as is familiar from Hermitian quantum field theory, one can also construct the same metric-derived energy-momentum tensor from the translation invariance of the action and the equations of motion of the fields, since nothing in that construction actually requires Hermiticity. The advantage of using the metric approach, which also is not sensitive to Hermiticity, is that it ensures that the Hamiltonian that is obtained has the same transformation properties under $CPT$ symmetry as the starting action.}
In regard to (\ref{GPT2}), we note here that for $\phi_2$ the transformation is not the conventional $CPT$ transformation of scalar fields that is used in quantum field theory (one in which all scalar field $CPT$ phases are positive \cite{Weinberg1995}) but a similarity transformation of it. We will need to return to this point below, but for the moment we just use (\ref{GPT2}) as is.
As written, the action given in (\ref{GPT1}) is invariant under the electric charge transformation
\begin{eqnarray}
\phi_1\rightarrow e^{i\alpha}\phi_1,\quad \phi^*_1\rightarrow e^{-i\alpha}\phi^*_1,\quad \phi_2\rightarrow e^{i\alpha}\phi_2,\quad \phi^*_2\rightarrow e^{-i\alpha}\phi_2,
\label{GPT3}
\end{eqnarray}
to thus possess a standard Noether current
\begin{eqnarray}
j_{\mu}=i(\phi^*_1\partial_{\mu} \phi_1-\phi_1\partial_{\mu} \phi^*_1)+i(\phi^*_2\partial_{\mu} \phi_2-\phi_2\partial_{\mu} \phi^*_2)
\label{GPT4}
\end{eqnarray}
that is conserved in solutions to the equations of motion {(\ref{GPT36}) and (\ref{GPT37})} associated with (\ref{GPT1}). We note here that the authors of \cite{Alexandre2018} used a non-standard Euler-Lagrange variational procedure (one which involves a non-trivial surface term) to obtain a non-standard set of equations of motion and a non-standard current (one not a Noether current invariance of the action), one that is nonetheless conserved in solutions to this non-standard set of equations of motion, and we discuss this issue in Sec. \ref{S4}. However, we shall use a standard variational procedure and a standard Noether current approach. With the potential of the field $\phi_1$ being of the form of a double-well potential, in its non-trivial minimum the scalar field $\phi_1$ would acquire a non-trivial vacuum expectation value. This would then break the electric charge symmetry spontaneously, and one would thus wonder whether there might still be a massless Goldstone boson despite the lack of Hermiticity. As shown by the authors of \cite{Alexandre2018} for the current they use and by us here for the above $j_{\mu}$, in both the cases a Goldstone boson is indeed present.
To study the dynamics associated with the action given in (\ref{GPT1}) we have found it convenient to work in the component basis
\begin{eqnarray}
\phi_1=\frac{1}{\sqrt{2}}(\chi_1+i\chi_2),\quad \phi^*_1=\frac{1}{\sqrt{2}}(\chi_1-i\chi_2),\quad \phi_2=\frac{1}{\sqrt{2}}(\psi_1+i\psi_2),\quad \phi^*_2=\frac{1}{\sqrt{2}}(\psi_1-i\psi_2),
\label{GPT5}
\end{eqnarray}
where all four $\chi_1$, $\chi_2$, $\psi_1$, and $\psi_2$ are Hermitian.\footnote{As is standard, under time reversal $\chi_1$ has even $T$ parity while $\chi_2$ has odd $T$ parity, so that under $T$ $\chi_1+i\chi_2$ has even parity. Under charge conjugation, $\chi_1$ has even $C$ parity while $\chi_2$ has odd $C$ parity. Thus under $CPT$ the $P$ even $\chi_1+i\chi_2$ transforms into $\chi_1-i\chi_2$. Because of the transformations in the $\phi_2$ sector that are given in (\ref{GPT2}) $\psi_1$ has to have odd $T$ parity while $\psi_2$ has to have even $T$ parity. (However, their $C$ parities are standard, with $\psi_1$ having even $C$ parity while $\psi_2$ has odd $C$ parity.) We discuss this pattern of $T$ parity assignments further below, where we will make a commutation relation preserving similarity transformation that will effect $\psi_1\rightarrow -i\psi_1$, $\psi_2\rightarrow -i\psi_2$, to thus change the signs of their $T$ and $CPT$ parities.} In the $\chi_1$, $\chi_2$, $\psi_1$, and $\psi_2$ basis the action takes the form:
\begin{eqnarray}
I(\chi_1,\chi_2,\psi_1,\psi_2)&=&\int d^4x \bigg{[}\frac{1}{2}\partial_{\mu}\chi_1\partial^{\mu}\chi_1+
\frac{1}{2}\partial_{\mu}\chi_2\partial^{\mu}\chi_2+
\frac{1}{2}\partial_{\mu}\psi_1\partial^{\mu}\psi_1+
\frac{1}{2}\partial_{\mu}\psi_2\partial^{\mu}\psi_2+
\frac{1}{2}m_1^2(\chi_1^2+\chi_2^2)
\nonumber\\
&-&
\frac{1}{2}m_2^2(\psi_1^2+\psi_2^2)-
i\mu^2(\chi_1\psi_2-\chi_2\psi_1)
-\frac{g}{16}(\chi_1^2+\chi_2^2)^2
\bigg{]},
\label{GPT6}
\end{eqnarray}
and with the appearance of the factor $i$ in the $\mu^2$-dependent term, the action now has the characteristic form of the non-Hermitian but $PT$ symmetric $p^2+ix^3$ theory.\footnote{The $PT$-symmetric $p^2+ix^3$ theory is actually $CPT$ symmetric since $p$ and $x$ are $C$ even and charge conjugation plays no role in non-relativistic systems.} For this action the Euler-Lagrange equations of motion take the form
\begin{eqnarray}
-\Box \chi_1&=&-m_1^2\chi_1+i\mu^2\psi_2+\frac{g}{4}(\chi_1^3+\chi_1\chi_2^2),
\nonumber\\
-\Box \chi_2&=&-m_1^2\chi_2-i\mu^2\psi_1+\frac{g}{4}(\chi_2^3+\chi_2\chi_1^2),
\nonumber\\
-\Box \psi_1&=&m_2^2\psi_1-i\mu^2\chi_2,
\nonumber\\
-\Box \psi_2&=&m_2^2\psi_2+i\mu^2\chi_1,
\label{GPT7}
\end{eqnarray}
and admit of a tree approximation minimum in which the scalar field expectation values obey
\begin{eqnarray}
&&m^2_2\bar{\psi}_1-i\mu^2\bar{\chi}_2=0,\quad m_2^2\bar{\psi}_2+i\mu^2\bar{\chi}_1=0,
\nonumber\\
&&m_1^2\bar{\chi}_1-\frac{\mu^4}{m_2^2}\bar{\chi}_1-\frac{g}{4}\bar{\chi}_1^3-\frac{g}{4}\bar{\chi}_1\bar{\chi}_2^2=0,
\nonumber\\
&&m_1^2\bar{\chi}_2-\frac{\mu^4}{m_2^2}\bar{\chi}_2-\frac{g}{4}\bar{\chi}_2^3-\frac{g}{4}\bar{\chi}_2\bar{\chi}_1^2=0.
\label{GPT8}
\end{eqnarray}
Choosing the minimum in which $(g/4)\bar{\chi}_1^2=m_1^2-\mu^4/m_2^2$, $\bar{\psi}_2=-i\mu^2\bar{\chi}_1/m_2^2$, $\hat{\chi}_2=0$, $\hat{\psi}_1=0$, and then expanding around this minimum according to $\chi_1=\bar{\chi}_1+\hat{\chi}_1$, $\chi_2=\hat{\chi}_2$, $\psi_1=\hat{\psi}_1$, $\psi_2=\bar{\psi}_2+\hat{\psi}_2$ yields a first-order term in the equations of motion of the form:
\begin{eqnarray}
\pmatrix{-\Box \hat{\chi}_1 \cr -\Box \hat{\psi}_2 \cr -\Box \hat{\chi}_2 \cr -\Box \hat{\psi}_1 }=
\pmatrix{ 2m_1^2-3\mu^4/m_2^2 & i\mu^2 & 0&0 \cr
i\mu^2& m_2^2&0&0\cr
0&0&-\mu^4/m_2^2&-i\mu^2\cr
0&0&-i\mu^2&m_2^2}
\pmatrix{ \hat{\chi}_1 \cr \hat{\psi}_2 \cr \hat{\chi}_2 \cr \hat{\psi}_1 }= M\pmatrix{ \hat{\chi}_1 \cr \hat{\psi}_2 \cr \hat{\chi}_2 \cr \hat{\psi}_1 }.
\label{GPT9}
\end{eqnarray}
As we see, with our choice of basis, we have already block-diagonalized the mass matrix $M$. We can readily determine the mass eigenvalues, and obtain
\begin{eqnarray}
|M-\lambda I|=\lambda (\lambda +\mu^4/m_2^2-m_2^2)\left[\lambda^2 -\lambda (2m_1^2+m_2^2-3\mu^4/m_2^2)+2m_1^2m_2^2-2\mu^4\right].
\label{GPT10}
\end{eqnarray}
The mass eigenvalue solutions to $|M-\lambda I|=0$ are thus
\begin{eqnarray}
\lambda_0=0,\quad \lambda_1&=&\frac{m_2^4-\mu^4}{m_2^2},
\nonumber\\
\lambda_{\pm}&=&\frac{2m_1^2m_2^2+m_2^4- 3\mu^4}{2m_2^2} \pm
\frac{1}{2m_2^2}\left[(2m_1^2m_2^2+m_2^4-3\mu^4)^2+8\mu^4m_2^4-8m_1^2m_2^6\right]^{1/2}.
\nonumber\\
&=&\frac{2m_1^2m_2^2+m_2^4-3\mu^4}{2m_2^2} \pm
\frac{1}{2m_2^2}\left[(2m_1^2m_2^2-m_2^4-3\mu^4)^2-4\mu^4m_2^4\right]^{1/2}.
\label{GPT11}
\end{eqnarray}
Given a mode with $\lambda_0=0$ (the determinant in the $(\hat{\chi}_2,\hat{\psi}_1)$ sector of $M$ being zero), then just as noted in \cite{Alexandre2018}, the presence of a massless Goldstone boson is apparent, and the Goldstone theorem is thus seen to hold when a non-Hermitian Hamiltonian has an antilinear symmetry.\footnote{While the $(\tilde{\chi}_2,\tilde{\psi}_1)$ sector of the mass matrix is not Hermitian, its antilinear symmetry cannot be realized in the complex conjugate pair realization because by being zero the Goldstone boson eigenvalue $\lambda_0$ is real. Consequently, $\lambda_1$ must be real too.} If we restrict the sign of the factor in the square root in $\lambda_{\pm}$ to be positive (the case considered in \cite{Alexandre2018}), then all mass eigenvalues are real. However, we note that we obtain a mode with $\lambda _0=0$ regardless of the magnitude of this factor, and thus even obtain a Goldstone boson when the factor in the square root term is negative and mass eigenvalues appear in complex conjugate pairs. Moreover, as we show in Sec. \ref{S3} below, when the factor in the square root term is zero, in the $(\hat{\chi}_1,\hat{\psi}_2)$ sector the matrix $M$ becomes Jordan block. The Goldstone boson mode is thus present in all three of the eigenvalue realizations that are allowed by antilinearity (viz. antilinear symmetry). Moreover, technically we do not even need to ascertain what the antilinear symmetry might even be, since as shown in \cite{Mannheim2018,Bender2010}, once we obtain an eigenvalue spectrum of the form that we have obtained in the $(\hat{\chi}_1,\hat{\psi}_2)$ sector, the mass matrix must admit of an antilinear symmetry. Thus antilinearity implies this particular form for the mass spectrum, and this particular form for the mass spectrum implies antilinearity. Finally, we note that if in the $(\hat{\chi}_2,\hat{\psi}_1)$ sector we set $\mu^4=m_2^4$, then not only does $\lambda_1$ become zero just like $\lambda_0$, but as we show in Sec. \ref{S3} the entire sector becomes Jordan block, with the Goldstone boson eigenfunction itself then having the zero norm that is characteristic of Jordan-block systems.
\section{Eigenvectors of the Mass Matrix}
\label{S3}
To discuss the eigenvector spectrum of the mass matrix $M$, it is convenient to introduce the $PT$ theory $V$ operator. Specifically, it was noted in \cite{Mostafazadeh2002,Solombrino2002,Mannheim2018} that if a time-independent Hamiltonian has an antilinear symmetry there will always exist a time-independent operator $V$ that obeys the so-called pseudo-Hermiticity condition $VH=H^{\dagger}V$. If $V$ is invertible (this automatically being the case for any finite-dimensional matrix such as the mass matrix $M$ of interest to us here), then $H$ and $H^{\dagger}$ are isospectrally related according to $H^{\dagger}=VHV^{-1}$, to thus have the same set of eigenvalues. Since such an isospectral relation requires that the eigenvalues of $H$ be real or in complex pairs, pseudo-Hermiticity is equivalent to antilinearity.
If $H$ is not Hermitian, one has to introduce separate right- and left-Schr\"odinger equations in which $H$ acts to the right or to the left. Then from the relation $i\partial_t|n\rangle=H|n\rangle$ obeyed by solutions to the right-Schr\"odinger equation we obtain $-i\partial_t\langle n|=\langle n|H^{\dagger}$, with $\langle n|$ then not being a solution to the left-Schr\"odinger equation as it does not obey $-i\partial_t\langle n|=\langle n|H$. Consequently in the non-Hermitian case the standard Dirac norm $\langle n(t)|n(t)\rangle=\langle n(0)|e^{iH^{\dagger}t}e^{-iHt}|n(0)\rangle$ is not time independent (i.e. not equal to $\langle n(0)|n(0)\rangle$), and one cannot use it as an inner product. However, the $V$ norm constructed from $V$ is time independent since
\begin{eqnarray}
i\partial_t\langle n(t)|V|n(t)\rangle=\langle n(t)|(VH-H^{\dagger} V)|n(t)\rangle=0.
\label{GPT12}
\end{eqnarray}
Since we can set
\begin{eqnarray}
-i\partial_t\langle n|=\langle n|H^{\dagger}=\langle n|VHV^{-1},\quad - i\partial_t\langle n|V=\langle n|VH,
\label{GPT13}
\end{eqnarray}
we see that it is the state $\langle n|V$ that is a solution to the left-Schr\"odinger equation and not the bra $\langle n|$ itself. Moreover, from (\ref{GPT13}) we obtain
\begin{eqnarray}
\langle n(t)|V|n(t)\rangle=\langle n(0)|Ve^{iHt}e^{-iHt}|n(0)\rangle=\langle n(0)|V|n(0)\rangle,
\label{GPT14}
\end{eqnarray}
to thus confirm the time independence of the $V$ norm. Through the $V$ operator then we see that time independence of inner products and antilinear symmetry are equivalent. Given that $\langle L_n|=\langle n|V$ is a solution to the left-Schr\"odinger equation, in the event that it is also a left-eigenvector of $H$ and $|R_n\rangle$ is a right-eigenvector of $H$, in the antilinear case the completeness relation is given not by $\sum |n\rangle\langle n|=I$ but by
\begin{eqnarray}
\sum |n\rangle\langle n|V=\sum |R_n\rangle\langle L_n|=I
\label{GPT15}
\end{eqnarray}
instead. As shown in \cite{Mannheim2018a}, when charge conjugation is separately conserved, the left-right $\langle R_n|V|R_m \rangle $ $V$-norm is the same as the overlap of the right-eigenstate $|R_n\rangle$ with its $PT$ conjugate (like $PT$ conjugation Hermitian conjugation is also antilinear). And more generally, the $V$-norm is the same as the overlap of a state with its $CPT$ conjugate \cite{Mannheim2018}.
In the special case where all the eigenvalues of a Hamiltonian are real and the eigenspectrum is complete, the Hamiltonian must either already obey $H=H^{\dagger}$ or be transformable by a (non-unitary) similarity transformation $S$ into one that does according to $SHS^{-1}=H^{\prime}=H^{\prime \dagger}$. For the primed system one has right-eigenvectors that obey
\begin{eqnarray}
i\partial_t|R_n^{\prime}\rangle=H^{\prime}|R_n^{\prime}\rangle,\quad -i\partial_t\langle R_n^{\prime}|=\langle R_n^{\prime}|H^{\prime},
\label{GPT16}
\end{eqnarray}
with the eigenstates of $H$ and $H^{\prime}$ being related by
\begin{eqnarray}
|R_n^{\prime}\rangle=S|R_n\rangle,~~~\langle R_n^{\prime}|=\langle R_n|S^{\dagger}.
\label{GPT17}
\end{eqnarray}
On normalizing the eigenstates of the Hermitian $H^{\prime}$ to unity, we obtain
\begin{eqnarray}
\langle R_n^{\prime} |R_m^{\prime}\rangle=\langle R_n|S^{\dagger}S|R_m\rangle=\delta_{m,n}.
\label{GPT18}
\end{eqnarray}
With $H^{\prime}=H^{\prime \dagger}$ we obtain
\begin{eqnarray}
SHS^{-1}=S^{\dagger -1}H^{\dagger}S^{\dagger},\quad S^{\dagger}SHS^{-1}S^{\dagger -1}=S^{\dagger}SH[S^{\dagger }S]^{-1}=H^{\dagger}.
\label{GPT19}
\end{eqnarray}
We can thus identify $S^{\dagger}S$ with $V$ when all energy eigenvalues are real and $H$ is diagonalizable, and as noted in \cite{Mannheim2018}, can thus establish that the $V$ norm is the $S^{\dagger}S$ norm, so that in this case $\langle L_n|R_m\rangle=\langle R_n|V|R_m\rangle=\langle R_n|S^{\dagger}S|R_m\rangle=\delta_{m,n}$ is positive definite.\footnote{As shown in \cite{Mannheim2018a}, to identify the $V$ norm with the $PT$ norm one has to choose the phase of the $PT$ conjugate of a state to be the same as the $PT$ eigenvalue of the state that is being conjugated. This prescription obviates any need to use the $PT$ theory $C$ operator norm that is described in \cite{Bender2007}, with the $PT$ norm then having the same positivity as the $V$ norm. Moreover, it was shown that not every $PT$ symmetric theory will possess a $PT$ theory $C$ operator, but all $PT$ theories will possess $V$ and $PT$ norms.} The interpretation of the $V$ norms as probabilities is then secured, with their time independence ensuring that probability is preserved in time.
Having now presented the general non-Hermitian formalism, a formalism that holds in both wave mechanics and matrix mechanics \cite{Bender2007}, and holds in quantum field theory \cite{Mannheim2018}, we can apply it to the mass matrix $M$ given in (\ref{GPT9}). And while this matrix does arise in a quantum field theory, all that matters in the following is that it has a non-Hermitian matrix structure. The matrix $M$ breaks up into two distinct two-dimensional blocks, and we can describe each of them by the generic
\begin{eqnarray}
N=\pmatrix{C+A & iB \cr iB &C-A},
\label{GPT20}
\end{eqnarray}
where $A$, $B$ and $C$ are all real. The matrix $N$ is not Hermitian but does have a $PT$ symmetry if we set $P=\sigma_3$ and $T=K$ where $K$ effects complex conjugation. The eigenvalues of $N$ are given by
\begin{eqnarray}
\Lambda_{\pm}=C\pm (A^2-B^2)^{1/2},
\label{GPT21}
\end{eqnarray}
and they are real if $A^2>B^2$ and in a complex conjugate pair if $A^2<B^2$, just as required of a non-Hermitian but $PT$-symmetric matrix. Additionally, the relevant $S$ and $V$ operators are given by
\begin{eqnarray}
S&=&\frac{1}{2(A^2-B^2)^{1/4}}\pmatrix{(A+B)^{1/2}+(A-B)^{1/2}& i[(A+B)^{1/2}-(A-B)^{1/2}]\cr
-i[(A+B)^{1/2}-(A-B)^{1/2}]&(A+B)^{1/2}+(A-B)^{1/2} },
\nonumber\\
S^{-1}&=&\frac{1}{2(A^2-B^2)^{1/4}}\pmatrix{(A+B)^{1/2}+(A-B)^{1/2}& -i[(A+B)^{1/2}-(A-B)^{1/2}]\cr
i[(A+B)^{1/2}-(A-B)^{1/2}]&(A+B)^{1/2}+(A-B)^{1/2} },
\nonumber\\
V&=&\frac{1}{(A^2-B^2)^{1/2}}\pmatrix{A & iB \cr -iB &A},\quad V^{-1}=\frac{1}{(A^2-B^2)^{1/2}}\pmatrix{A & -iB \cr iB &A},
\label{GPT22}
\end{eqnarray}
and they effect
\begin{eqnarray}
SNS^{-1}=N^{\prime}=\pmatrix{C+(A^2-B^2)^{1/2} & 0 \cr 0 &C-(A^2-B^2)^{1/2}},\quad VNV^{-1}=\pmatrix{C+A & -iB \cr -iB &C-A}=N^{\dagger}
\label{GPT23}
\end{eqnarray}
regardless of whether $A^2-B^2$ is positive or negative (if $A^2$ is less than $B^2$, then while not Hermitian $SNS^{-1}$ is still diagonal). However, as we elaborate on below, we note that if $A^2-B^2$ is zero then $S$ and $V$ become undefined. Other than at $A^2-B^2=0$ the matrix $N^{\prime}=SNS^{-1}$ is diagonal, and with $N$ being given by $N=S^{-1}N^{\prime}S$, the right-eigenvectors of $N$ that obey $NR_{\pm}=\Lambda_{\pm}R_{\pm}$ are given by the columns of $S^{-1}$, and the left-eigenvectors of $N$ that obey $L_{\pm}N=\Lambda_{\pm}L_{\pm}$ are given by the rows of $S$. Given the right-eigenvectors one can also construct the left-eigenvectors by using the $V$ operator. When $A^2>B^2$ the left eigenvectors can be constructed as $\langle L_{\pm}|=\langle R_{\pm}|V$, and we obtain
\begin{eqnarray}
R_{+}&=&\frac{1}{2(A^2-B^2)^{1/4}}\pmatrix{(A+B)^{1/2}+(A-B)^{1/2}\cr
i[(A+B)^{1/2}-(A-B)^{1/2}]}
\nonumber\\
R_{-}&=&\frac{1}{2(A^2-B^2)^{1/4}}\pmatrix{ -i[(A+B)^{1/2}-(A-B)^{1/2}]\cr
(A+B)^{1/2}+(A-B)^{1/2} }
\nonumber\\
L_{+}&=&\frac{1}{2(A^2-B^2)^{1/4}}\pmatrix{(A+B)^{1/2}+(A-B)^{1/2},& i[(A+B)^{1/2}-(A-B)^{1/2}]}
\nonumber\\
L_{-}&=& \frac{1}{2(A^2-B^2)^{1/4}}\pmatrix{
-i[(A+B)^{1/2}-(A-B)^{1/2}],&(A+B)^{1/2}+(A-B)^{1/2} },
\label{GPT24}
\end{eqnarray}
and these eigenvectors are normalized according to the positive definite $\langle L_n|R_m\rangle=\langle R_n|V|R_m\rangle=\delta_{m,n}$, i.e. according to $L_{\pm}R_{\pm}=1$, $L_{\mp}R_{\pm}=0$. In addition $N$ and the identity $I$ can be reconstructed as
\begin{eqnarray}
N=|R_+\rangle \Lambda_+\langle L_+|+|R_-\rangle \Lambda_-\langle L_-|,\quad
I=|R_+\rangle \langle L_+|+|R_-\rangle\langle L_-|,
\label{GPT25}
\end{eqnarray}
to thus be diagonalized in the left-right basis.
When $A^2-B^2$ is negative, the quantity $(A-B)^{1/2}$ is pure imaginary, and since $\langle R|$ is the Hermitian conjugate of $|R\rangle$, in the $A^2<B^2$ sector up to a phase we have $\langle L_{\mp}|=\pm \langle R_{\pm}|V$. If we set $A^2-B^2=-D^2$ where $D$ is real, the eigenvalues are $\Lambda_{\pm}=C\pm iD$. In a quantum theory with the mass matrix serving as the Hamiltonian, $|R_{\pm}\rangle$ would evolve as $e^{-i(C\pm iD)t}=e^{-iCt\pm Dt}$, while $\langle L_{\pm}|$ would evolve as $\langle R_{\mp}|V$, i.e. as $e^{iCt\mp Dt}$. As had been noted in general in \cite{Mannheim2018} and as found here, the only overlaps that would be non-zero would be $\mp\langle L_{\pm}|R_{\pm}\rangle= \langle R_{\mp}|V|R_{\pm}\rangle=\pm i$, and they would be time independent. Since $\langle L_{\pm}|\neq \langle R_{\pm}|V$, these matrix elements would be transition matrix elements between growing and decaying states. Such transition matrix elements are not required to be positive or to even be real.
While all of these eigenstates and the $S$ and $V$ operators are well-defined as long as $A^2$ is not equal to $B^2$, at $A^2=B^2$ they all become singular.
Moreover at $A^2=B^2$ the vectors $R_+$ and $R_-$ become identical to each other (i.e. equal up to an irrelevant overall phase), and equally $L_+$ and $L_-$ become identical too. The matrix $N$ thus loses both a left-eigenvector and a right-eigenvector at $A^2=B^2$ to then only have one left-eigenvector and only one right-eigenvector. At $A^2=B^2$ the two eigenvalues become equal ($\Lambda_+=\Lambda_-=C$) and have to share the same left- and right-eigenvectors. The fact that $S$ becomes singular at $A^2=B^2$ means that $N$ cannot be diagonalized, with its eigenspectrum being incomplete. $N$ thus becomes a Jordan-block matrix that cannot be diagonalized.\footnote{Even though one loses diagonalizability when $A^2=B^2$, the matrix $N$ remains PT symmetric at $A^2=B^2$, as it is invariant under the $PT=\sigma_3K$ transformation for all values of its parameters as long as they are real.} Even though all of $L_{\pm}$ , $R_{\pm}$ become singular at $A^2=B^2$, $N$ still has left- and right-eigenvectors $L$ and $R$ that are given up to an arbitrary normalization by
\begin{eqnarray}
L=\pmatrix{1 & i},\quad R=\pmatrix{1\cr i}, \quad LN=CN,\quad NR=CR,
\label{GPT26}
\end{eqnarray}
and no matter what that normalization might be, they obey the zero norm condition characteristic of Jordan-block matrices:
\begin{eqnarray}
LR=\pmatrix{1 & i}\pmatrix{1\cr i}=0.
\label{GPT27}
\end{eqnarray}
Even though the eigenspectrum of $N$ is incomplete, the vector space on which it acts is still complete. One can take the extra states to be
\begin{eqnarray}
L^{\prime}=\pmatrix{1 & -i},\quad R^{\prime}=\pmatrix{1\cr -i},
\label{GPT28}
\end{eqnarray}
with $L^{\prime}R^{\prime}=0$, so that $R$ and $R^{\prime}$ span the space on which $N$ acts to the right, while $L$ and $L^{\prime}$ span the space on which $N$ acts to the left.
Comparing now with (\ref{GPT9}), we see that for the $(\hat{\chi}_1,\hat{\psi}_2)$ sector we have
\begin{eqnarray}
C=\frac{2m_1^2m_2^2+m_2^4-3\mu^4}{2m_2^2},\quad A=\frac{2m_1^2m_2^2-3\mu^4-m_2^4}{2m_2^2},\quad B=\mu^2,
\label{GPT29}
\end{eqnarray}
while for the $(\hat{\chi}_2,\hat{\psi}_1)$ sector we have
\begin{eqnarray}
C=\frac{m_2^4-\mu^4}{2m_2^2},\quad A=-\frac{(\mu^4+m_2^4)}{2m_2^2},\quad B=-\mu^2.
\label{GPT30}
\end{eqnarray}
From (\ref{GPT29}) and (\ref{GPT30}) the eigenvalues given in (\ref{GPT11}) follow. For the $(\hat{\chi}_1,\hat{\psi}_2)$ sector we thus have two eigenvectors with real eigenvalues if
$(2m_1^2m_2^2-m_2^4-3\mu^4)^2>4\mu^4m_2^4$, two eigenvectors with complex conjugate eigenvalues if $(2m_1^2m_2^2-m_2^4-3\mu^4)^2<4\mu^4m_2^4$, and lose an eigenvector if
$(2m_1^2m_2^2-m_2^4-3\mu^4)^2=4\mu^4m_2^4$. Since none of this affects the $(\hat{\chi}_2,\hat{\psi}_1)$ sector, for all three of the possible classes of eigenspectra associated with a non-Hermitian Hamiltonian with an antilinear symmetry we obtain a massless Goldstone boson.
For the $(\hat{\chi}_2,\hat{\psi}_1)$ sector the eigenvalues are $\lambda_0=0$ and $\lambda_1=m_2^2-\mu^4/m_2^2$. Both are them are real, and we shall take $m_2^4$ to not be less than $\mu^4$ so that $\lambda_1$ could not be negative. Additionally, the left- and right-eigenvectors are given by
\begin{eqnarray}
L_0=\frac{1}{(m_2^4-\mu^4)^{1/2}}
\pmatrix{m_2^2, &i\mu^2},\quad R_0=\frac{1}{(m_2^4-\mu^4)^{1/2}}\pmatrix{m_2^2\cr i\mu^2},
\nonumber\\
L_1=\frac{1}{(m_2^4-\mu^4)^{1/2}}
\pmatrix{i\mu^2, & -m_2^2},\quad R_1=\frac{1}{(m_2^4-\mu^4)^{1/2}}\pmatrix{i\mu^2\cr -m_2^2},
\label{GPT31}
\end{eqnarray}
as normalized to
\begin{eqnarray}
L_0R_0=1, \quad L_1 R_1=1, \quad L_0R_1=0,\quad L_1R_0=0.
\label{GPT32}
\end{eqnarray}
The Goldstone boson is thus properly normalized if one uses the left-right norm, with the two states in the $(\hat{\chi}_2,\hat{\psi}_1)$ sector forming a left-right orthonormal basis. Thus in the non-Hermitian case the standard Goldstone theorem associated with the spontaneous breakdown of a continuous symmetry continues to hold but the norm of the Goldstone boson has to be the positive left-right norm (or equivalently the $PT$ theory norm \cite{Alexandre2018}) rather than the standard positive Hermitian theory Dirac norm for which the theorem was first proved
\cite{Nambu1960,Goldstone1961,Nambu1961,Goldstone1962}.
However, something unusual occurs if we set $\mu^2=m_2^2$. Specifically, the eigenvalue $\lambda_1$ becomes zero, to thus now be degenerate with $\lambda_0$. The eigenvectors $R_0$ and $R_1$ collapse onto a common single $R$ and $L_0$ and $L_1$ collapse onto a common single $L$, and the normalization coefficients given in (\ref{GPT31}) diverge. The $(\hat{\chi}_2,\hat{\psi}_1)$ sector thus becomes of non-diagonalizable Jordan-block form. In this limit one can take the left- and right-eigenvectors to be
\begin{eqnarray}
L=\pmatrix{1 & i},\quad R=\pmatrix{1\cr i},
\label{GPT33}
\end{eqnarray}
and they obey the zero norm condition
\begin{eqnarray}
LR=0.
\label{GPT34}
\end{eqnarray}
As such this represents a new extension of the Goldstone theorem, and even though the standard Goldstone theorem associated with the spontaneous breakdown of a continuous symmetry continues to hold, the norm of the Goldstone boson is now zero. Since a zero norm state can leave no imprint in a detector, we are essentially able to evade the existence of a massless Goldstone boson, in the sense that while it would still exist it would not be observable.
\section{Comparison with the work of Alexandre, Ellis, Millington and Seynaeve}
\label{S4}
If we do a functional variation of the action given in (\ref{GPT1}) we obtain
\begin{eqnarray}
\delta I(\phi_1,\phi_2,\phi^*_1,\phi^*_2)&=&\int d^4x\bigg{[}[-\Box \phi_1+m_1^2\phi_1-\mu^2\phi_2-\frac{g}{2}\phi_1^2\phi_1^*]\delta \phi_1^*
+[-\Box \phi^*_1+m_1^2\phi^*_1+\mu^2\phi^*_2-\frac{g}{2}(\phi^*_1)^2\phi_1]\delta \phi_1
\nonumber\\
&+&[-\Box \phi_2-m_2^2\phi_2+\mu^2\phi_1]\delta \phi_2^*
+[-\Box \phi^*_2-m_2^2\phi^*_2-\mu^2\phi^*_1]\delta \phi_2
\nonumber\\
&+&\partial_{\mu}[\delta\phi^*_1\partial^{\mu}\phi_1+\delta\phi_1\partial^{\mu}\phi^*_1
+\delta\phi^*_2\partial^{\mu}\phi_2+\delta\phi_2\partial^{\mu}\phi^*_2]
\bigg{]}.
\label{GPT35}
\end{eqnarray}
With all variations held fixed at the surface, stationarity leads to
\begin{eqnarray}
&&-\Box \phi_1+m_1^2\phi_1-\mu^2\phi_2-\frac{g}{2}\phi_1^2\phi_1^*=0,
\nonumber\\
&&-\Box \phi_2-m_2^2\phi_2+\mu^2\phi_1=0,
\label{GPT36}
\end{eqnarray}
\begin{eqnarray}
&&-\Box \phi^*_1+m_1^2\phi^*_1+\mu^2\phi^*_2-\frac{g}{2}(\phi^*_1)^2\phi_1=0,
\nonumber\\
&&-\Box \phi^*_2-m_2^2\phi^*_2-\mu^2\phi^*_1=0,
\label{GPT37}
\end{eqnarray}
with these equations of motion being completely equivalent to (\ref{GPT7}). With these equations of motion one readily checks that the electric current $j_{\mu}=i(\phi^*_1\partial_{\mu} \phi_1-\phi_1\partial_{\mu} \phi^*_1)+i(\phi^*_2\partial_{\mu} \phi_2-\phi_2\partial_{\mu} \phi^*_2)$ given in (\ref{GPT4}) is conserved, just as it should be.
There is however an immediate problem with these equations of motion, namely if we complex conjugate (\ref{GPT36}) we obtain not (\ref{GPT37}) but
\begin{eqnarray}
&&-\Box \phi^*_1+m_1^2\phi^*_1-\mu^2\phi^*_2-\frac{g}{2}(\phi^*_1)^2\phi_1=0,
\nonumber\\
&&-\Box \phi^*_2-m_2^2\phi^*_2+\mu^2\phi^*_1=0
\label{GPT38}
\end{eqnarray}
instead. The reason why this problem occurs is because while (\ref{GPT37}) is associated with $\partial I/\partial\phi_1$ and $\partial I/\partial\phi_2$, (\ref{GPT38}) is associated with $(\partial I/\partial\phi^*_1)^*=\partial I^*/\partial\phi_1$ and $(\partial I/\partial\phi^*_2)^*=\partial I^*/\partial\phi_2$ and $I$ is not equal to $I^*$ if $I$ is not Hermitian. A similar concern holds for (\ref{GPT7}) as not one of its four separate equations is left invariant under complex conjugation.
In order to get round this the authors of \cite{Alexandre2018} propose that (\ref{GPT37}) not be valid, but rather one should use (\ref{GPT36}) and (\ref{GPT38}) instead. In order to achieve this the authors of \cite{Alexandre2018} propose that one add an additional surface term to (\ref{GPT35}) so that one no longer imposes stationarity with respect $\delta \phi_1$ and $\delta \phi_2$, but only stationarity with respect to $\delta \phi^*_1$ and $\delta \phi^*_2$ alone.\footnote{The additional surface term is akin to the Hawking-Gibbons surface term used in general relativity. Specifically, in general relativity the variation of the Einstein-Hilbert action leads to variations of both $g_{\mu\nu}$ and its first derivative at the surface. Variations with respect to the derivatives are then cancelled by the Hawking-Gibbons term.} If one does use (\ref{GPT36}) and (\ref{GPT38}), the electric current $j_{\mu}$ is no longer conserved (i.e. the surface term that is to be introduced must carry off some electric charge), but instead it is the current
\begin{eqnarray}
j^{\prime}_{\mu}=i(\phi^*_1\partial_{\mu} \phi_1-\phi_1\partial_{\mu} \phi^*_1)- i(\phi^*_2\partial_{\mu} \phi_2-\phi_2\partial_{\mu} \phi^*_2)
\label{GPT39}
\end{eqnarray}
that is conserved in solutions to the equations of motion. As such, this $j^{\prime}_{\mu}$ current is a non-Noether current that is not associated with a symmetry of the action $I$ (unless the inclusion of the surface term then leads to one), and thus its spontaneous breakdown is somewhat different from the standard one envisaged in \cite{Nambu1960,Goldstone1961,Nambu1961,Goldstone1962}. Nonetheless, as noted in \cite{Alexandre2018}, when the scalar fields acquire vacuum expectation values, the mass matrix associated with (\ref{GPT36}) and (\ref{GPT38}) still has a zero eigenvalue. With the authors of \cite{Alexandre2018} showing that it is associated with the Ward identity for $j_{\mu}^{\prime}$, it can still be identified as a Goldstone boson. The work of \cite{Alexandre2018} thus breaks the standard connection between Goldstone bosons and symmetries of the action.
As such, the result of the authors of \cite{Alexandre2018} is quite interesting as it provides possible new insight into the Goldstone theorem. However, the analysis somewhat obscures the issue as it suggests that the generation of Goldstone bosons in non-Hermitian theories is quite different from the generation of Goldstone bosons in Hermitian theories. It is thus of interest to ask whether one could show that one could obtain Goldstone bosons in a procedure that is common to both Hermitian and non-Hermitian theories. To this end we need to find a way to exclude (\ref{GPT38}) and validate $(\ref{GPT37})$, as it is (\ref{GPT36}) and (\ref{GPT37}) that we used in our paper in an approach that is completely conventional, one in which the surface term in (\ref{GPT35}) vanishes in the standard variational procedure way.
To reconcile (\ref{GPT36}) and (\ref{GPT37}) or to reconcile the equations of motion in (\ref{GPT7}) with complex conjugation it is instructive to make a particular similarity transformation on the fields, even though doing so initially appears to lead to another puzzle, the Hermiticity puzzle, which we discuss and resolve below. It is more convenient to seek a reconciliation for (\ref{GPT7}) first, so from $I(\chi_1,\chi_2,\psi_1,\psi_2)$ we identify canonical conjugates for $\phi_1$ and $\phi_2$ of the form $\Pi_1=\partial_t\psi_1$, $\Pi_2=\partial_t\psi_2$. With these conjugates we introduce \cite{Mannheim2018}
\begin{eqnarray}
S(\psi_1)=\exp\left[\frac{\pi}{2}\int d^3x \Pi_1(\textbf{x},t)\psi_1(\textbf{x},t)\right],\quad
S(\psi_2)=\exp\left[\frac{\pi}{2}\int d^3x \Pi_2(\textbf{x},t)\psi_2(\textbf{x},t)\right],
\label{GPT40}
\end{eqnarray}
and obtain
\begin{eqnarray}
S(\psi_1)\psi_1S^{-1}(\psi_1)=-i\psi_1,\quad S(\psi_1)\Pi_1S^{-1}(\psi_1)=i\Pi_1,\quad
S(\psi_2)\psi_2S^{-1}(\psi_2)=-i\psi_2,\quad S(\psi_2)\Pi_2S^{-1}(\psi_2)=i\Pi_2.
\label{GPT41}
\end{eqnarray}
Since these transformations preserve the equal-time commutation relations $[\psi_1(\textbf{x},t),\Pi_1(\textbf{y},t)]=i\delta^3(\textbf{x}-\textbf{y})$, $[\psi_2(\textbf{x},t),\Pi_2(\textbf{y},t)]=i\delta^3(\textbf{x}-\textbf{y})$, they are fully permissible transformations that do not modify the content of the field theory. Applying (\ref{GPT41}) to $I(\chi_1,\chi_2,\psi_1,\psi_2)$ we obtain
\begin{eqnarray}
S(\psi_1)S(\psi_2)I(\chi_1,\chi_2,\psi_1,\psi_2)S^{-1}(\psi_2)S^{-1}(\psi_1)=I^{\prime}(\chi_1,\chi_2,\psi_1,\psi_2)
\label{GPT42}
\end{eqnarray}
where
\begin{eqnarray}
I^{\prime}(\chi_1,\chi_2,\psi_1,\psi_2)&=&\int d^4x \bigg{[}\frac{1}{2}\partial_{\mu}\chi_1\partial^{\mu}\chi_1+
\frac{1}{2}\partial_{\mu}\chi_2\partial^{\mu}\chi_2-
\frac{1}{2}\partial_{\mu}\psi_1\partial^{\mu}\psi_1-
\frac{1}{2}\partial_{\mu}\psi_2\partial^{\mu}\psi_2+
\frac{1}{2}m_1^2(\chi_1^2+\chi_2^2)
\nonumber\\
&+&
\frac{1}{2}m_2^2(\psi_1^2+\psi_2^2)-
\mu^2(\chi_1\psi_2-\chi_2\psi_1)
-\frac{g}{16}(\chi_1^2+\chi_2^2)^2
\bigg{]}.
\label{GPT43}
\end{eqnarray}
Stationary variation with respect to $\chi_1$, $\chi_2$, $\psi_1$, and $\psi_2$ replaces (\ref{GPT7}) by
\begin{eqnarray}
-\Box \chi_1&=&-m_1^2\chi_1+\mu^2\psi_2+\frac{g}{4}(\chi_1^3+\chi_1\chi_2^2),
\nonumber\\
-\Box \chi_2&=&-m_1^2\chi_2-\mu^2\psi_1+\frac{g}{4}(\chi_2^3+\chi_2\chi_1^2),
\nonumber\\
-\Box \psi_1&=&m_2^2\psi_1+\mu^2\chi_2,
\nonumber\\
-\Box \psi_2&=&m_2^2\psi_2-\mu^2\chi_1,
\label{GPT44}
\end{eqnarray}
and now each one of the equations of motion is separately invariant under complex conjugation.
Returning now to the original $\phi_2$, $\phi^*_2$ fields we obtain
\begin{eqnarray}
S(\psi_1)S(\psi_2)\phi_2S^{-1}(\psi_2)S^{-1}(\psi_1)=-i\phi_2,\quad S(\psi_1)S(\psi_2)\phi^*_2S^{-1}(\psi_2)S^{-1}(\psi_1)=-i\phi^*_2,
\label{GPT45}
\end{eqnarray}
so that $I(\phi_1,\phi_2,\phi^*_1,\phi^*_2)$ transforms into
\begin{eqnarray}
I^{\prime}(\phi_1,\phi_2,\phi^*_1,\phi^*_2)=\int d^4x\left[\partial_{\mu}\phi^*_1\partial^{\mu}\phi_1-\partial_{\mu}\phi^*_2\partial^{\mu}\phi_2
+m_1^2\phi_1^*\phi_1+m_2^2\phi^*_2\phi_2+i\mu^2(\phi^*_1\phi_2-\phi^*_2\phi_1)-\frac{g}{4}(\phi^*_1\phi_1)^2\right],
\label{GPT46}
\end{eqnarray}
while the equations of motion become
\begin{eqnarray}
&&-\Box \phi_1+m_1^2\phi_1+i\mu^2\phi_2-\frac{g}{2}\phi_1^2\phi_1^*=0,
\nonumber\\
&&-\Box \phi_2-m_2^2\phi_2+i\mu^2\phi_1=0,
\label{GPT47}
\end{eqnarray}
\begin{eqnarray}
&&-\Box \phi^*_1+m_1^2\phi^*_1-i\mu^2\phi^*_2-\frac{g}{2}(\phi^*_1)^2\phi_1=0,
\nonumber\\
&&-\Box \phi^*_2-m_2^2\phi^*_2-i\mu^2\phi^*_1=0,
\label{GPT48}
\end{eqnarray}
and now there is no complex conjugation problem, with (\ref{GPT48}) being the complex conjugate of (\ref{GPT47}).\footnote{The appearance of a negative kinetic energy term for $\phi_2$ in (\ref{GPT46}) is only an artifact of the similarity transformation, since there are no such negative kinetic energy terms in our starting $I(\chi_1,\chi_2,\psi_1,\psi_2)$ and one cannot change the signature of a Hilbert space by a similarity transformation.} In addition we note under the transformations given in (\ref{GPT45}) the equations given in (\ref{GPT38}) transform into
\begin{eqnarray}
&&-\Box \phi^*_1+m_1^2\phi^*_1+i\mu^2\phi^*_2-\frac{g}{2}(\phi^*_1)^2\phi_1=0,
\nonumber\\
&&-\Box \phi^*_2-m_2^2\phi^*_2+i\mu^2\phi^*_1=0.
\label{GPT49}
\end{eqnarray}
If we now switch the sign of $\phi_2^*$, (\ref{GPT47}) is unaffected, while (\ref{GPT49}) becomes
\begin{eqnarray}
&&-\Box \phi^*_1+m_1^2\phi^*_1-i\mu^2\phi^*_2-\frac{g}{2}(\phi^*_1)^2\phi_1=0,
\nonumber\\
&&-\Box \phi^*_2-m_2^2\phi^*_2-i\mu^2\phi^*_1=0.
\label{GPT50}
\end{eqnarray}
We recognize (\ref{GPT50}) as being (\ref{GPT48}). With (\ref{GPT47}) being unaffected by the switch in sign of $\phi_2^*$, the mass matrix based on (\ref{GPT47}) and (\ref{GPT48}) is the same
as the mass matrix based on (\ref{GPT47}) and (\ref{GPT50}). However, since all we have done in going from (\ref{GPT36}), (\ref{GPT37}) and (\ref{GPT38}) is make similarity transformations that leave determinants invariant, the eigenvalues associated with (\ref{GPT36}) and (\ref{GPT37}) (i.e. with (\ref{GPT9})) on the one hand and the eigenvalues associated with (\ref{GPT36}) and (\ref{GPT38}) on the other hand must be the same. And indeed this is exactly found to be the case, with all four of the eigenvalues given in \cite{Alexandre2018} being precisely the ones given in our (\ref{GPT11}). One can thus obtain the same mass spectrum as that obtained in \cite{Alexandre2018} using a completely conventional variational procedure.
In addition, we note that with (\ref{GPT47}) and (\ref{GPT50}) the current $j^{\prime}_{\mu}$ given in (\ref{GPT39}) that is used in \cite{Alexandre2018} now is conserved. In fact, under the transformations given in (\ref{GPT45}) the $j_{\mu}$ current given in (\ref{GPT4}) transforms into $j_{\mu}^{\prime}$. Thus all that is needed to bring the study of \cite{Alexandre2018}) into the conventional Goldstone framework (standard variation procedure, standard spontaneous breakdown of a symmetry of the action) is to first make a similarity transformation.
Now the reader will immediately object to what we have done since now the $\mu^2(\chi_1\psi_2-\chi_2\psi_1)$ term in (\ref{GPT43}) and the $i\mu^2(\phi^*_1\phi_2-\phi^*_2\phi_1)$ term in (\ref{GPT46}) are both invariant under complex conjugation. Then with the actions in (\ref{GPT43}) and (\ref{GPT46}) then seemingly being Hermitian, we are seemingly back to the standard Hermitian situation where the Goldstone theorem readily holds, and we have seemingly gained nothing new. However, it cannot actually be the case that action in (\ref{GPT43}) could be Hermitian, since similarity transformations cannot change the eigenvalues of the mass matrix $M$ given in (\ref{GPT9}), and as we have seen for certain values of parameters the eigenvalues can be complex or $M$ could even be Jordan block. We thus need to explain how, despite its appearance, a seemingly Hermitian action might not actually be Hermitian. The answer to this puzzle has been provided in \cite{Mannheim2018}, and we describe it below.
However, before doing so we note that there are two other approaches that could also achieve a reconciliation. The first alternative involves starting with the fields $\chi_1$, $\chi_2$, $\psi_1$, $\psi_2$ as the fields that define the theory, and $I(\chi_1, \chi_2, \psi_1, \psi_2)$ as the input action. In this case one immediately obtains the equations of motion given in (\ref{GPT7}). As they stand these equations are inconsistent if all the four fields are Hermitian. If we take $\chi_1$ and $\chi_2$ to be Hermitian, then these equations force $\psi_1$ and $\psi_2$ to be anti-Hermitian. And if $\psi_1$ and $\psi_2$ are taken to be anti-Hermitian, both the equations of motion and the action given in (\ref{GPT7}) then are invariant under a complex conjugation (i.e. Hermitian conjugation) in which $\psi_1$ and $\psi_2$ transform into $-\psi_1$ and $-\psi_2$. Moreover, in such a case the $-i\psi_1$ and $-i\psi_2$ fields that are generated through the similarity transformations given in (\ref{GPT41}) that would then be Hermitian. Of course then the interaction term given in (\ref{GPT6}) would be Hermitian as well, and we again have a seemingly Hermitian theory.
Now suppose we do take $\psi_1$ and $\psi_2$ to be anti-Hermitian. Then if we start with $I(\chi_1, \chi_2, \psi_1, \psi_2)$ we cannot get back to $I(\phi_1,\phi_2,\phi^*_1,\phi^*_2)$ given in (\ref{GPT1}), since in the correspondence given in (\ref{GPT5}) $\phi_2^*$ was recognized as the conjugate of a $\phi_2=\psi_1+i\psi_2$ field that was expanded in terms of Hermitian $\psi_1$ and $\psi_2$. If we now take $\phi_2$ to still be defined as $\phi_2=\psi_1+i\psi_2$, the associated $\phi_2^*$ would now be given by $-(\psi_1-i\psi_2)$, and thus equal to minus the previous $\psi_1-i\psi_2$ used in (\ref{GPT5}). With this definition a rewriting of $I(\chi_1, \chi_2, \psi_1, \psi_2)$ in the $(\phi_1,\phi_2,\phi^*_1,\phi^*_2)$ basis would yield
\begin{eqnarray}
I(\phi_1,\phi_2,\phi^*_1,-\phi^*_2)=\int d^4x\left[\partial_{\mu}\phi^*_1\partial^{\mu}\phi_1-\partial_{\mu}\phi^*_2\partial^{\mu}\phi_2
+m_1^2\phi_1^*\phi_1+m_2^2\phi^*_2\phi_2-\mu^2(\phi^*_1\phi_2+\phi^*_2\phi_1)-\frac{g}{4}(\phi^*_1\phi_1)^2\right],
\label{GPT51}
\end{eqnarray}
and equations of motion
\begin{eqnarray}
&&-\Box \phi_1+m_1^2\phi_1-\mu^2\phi_2-\frac{g}{2}\phi_1^2\phi_1^*=0,
\nonumber\\
&&-\Box \phi_2-m_2^2\phi_2+\mu^2\phi_1=0,
\label{GPT52}
\end{eqnarray}
\begin{eqnarray}
&&-\Box \phi^*_1+m_1^2\phi^*_1-\mu^2\phi^*_2-\frac{g}{2}(\phi^*_1)^2\phi_1=0,
\nonumber\\
&&-\Box \phi^*_2-m_2^2\phi^*_2+\mu^2\phi^*_1=0.
\label{GPT53}
\end{eqnarray}
Now complex conjugation can be consistently applied, with (\ref{GPT53}) being derivable from (\ref{GPT50}) by complex conjugation. And again it is $j^{\prime}_{\mu}$ that is conserved.
A second alternative approach is to reinterpret the meaning of the star operator used in $\phi_1^*$ and $\phi_2^*$. Instead of taking it to denote Hermitian conjugation, we could instead take it denote $CPT$ conjugation, i.e. $\phi_1^*=CPT\phi_1TPC$, $\phi_2^*=CPT\phi_2TPC$. Now we had noted in (\ref{GPT2}) that in order to enforce $CPT$ symmetry on $I(\phi_1,\phi_2,\phi^*_1,\phi^*_2)$ we took $\phi_1$ to be even and $\phi_2$ to be odd under $CPT$, and we had noted that in general a scalar field should be $CPT$ even (i.e. the same $CPT$ parity as the $CPT$ even fermionic $\bar{\psi}\psi$ \cite{Mannheim2018}). However, if we apply the similarity transformation given in (\ref{GPT41}) to $\phi_2=\psi_1+i\psi_2$ to get $-i\phi_2$, that would change the $CPT$ parity. Thus while $\phi_2$ has negative $CPT$ parity it is similarity equivalent to a field that has the conventional positive $CPT$ parity, with the transformed $I^{\prime}(\phi_1,\phi_2,\phi^*_1,\phi^*_2)$ and the resulting equations of motion now being $CPT$ symmetric if $\phi_2$ is taken to have positive $CPT$ parity, viz. $CPT\phi_2TPC=\phi_2^*$, $CPT\phi^*_2TPC=\phi_2$. (We leave $\phi_1$ as given in (\ref{GPT2}), viz. $CPT\phi_1TPC=\phi^*_1$, $CPT\phi^*_1TPC=\phi_1$.)
The difficulty identified by the authors of \cite{Alexandre2018} can thus be resolved by a judicious choice of which fields are Hermitian and which are anti-Hermitian, by a judicious choice of which fields are $CPT$ even and which are $CPT$ odd, or by similarity transformations that generate complex phases that affect both Hermiticity and $CPT$ parity. However in all of these such resolutions we are led to theories that now appear to be Hermitian and yet for certain values of parameters could not be, and so we need to address this issue.
\section{Resolution of the Hermiticity Puzzle}
\label{S5}
In \cite{Mannheim2018} the issues of the generality of $CPT$ symmetry and the nature of Hermiticity were addressed. In regard to Hermiticity it was shown that Hamiltonians that appear to be Hermitian need not be,
since Hermiticity or self-adjointness is determined not by superficial inspection of the appearance of the Hamiltonian but by construction of asymptotic boundary conditions, as they determine whether or not one could drop surface terms in an integration by parts. And even if one could drop surface terms we still may not get Hermiticity because of the presence of factors of $i$ in $H$ that could affect complex conjugation. In regard to $CPT$ it was shown that if one imposes only two requirements, namely the time independence of inner products and invariance under the complex Lorentz group, it follows that the Hamiltonian must have an antilinear $CPT$ symmetry. Since this analysis involves no Hermiticity requirement, the $CPT$ theorem is thus extended to the non-Hermitian case. As noted above, the time independence of inner products is achieved if the theory has any antilinear symmetry with the left-right $V$ norm being the inner product one has to use. Complex Lorentz invariance then forces the antilinear symmetry to be $CPT$.
In field theories one ordinarily constructs actions so that they are invariant under the real Lorentz group. However, the same analysis that shows that actions with spin zero Lagrangians are invariant under the real Lorentz group (the restricted Lorentz group) also shows that they are invariant under the complex one (the proper Lorentz group that includes $PT$ transformations for coordinates and $CPT$ transformations for spinors). Specifically, the action $I=\int d^4x L(x)$ with spin zero $L(x)$ is left invariant under real Lorentz transformations of the form $\exp(iw^{\mu\nu}M_{\mu\nu})$ where the six antisymmetric $w^{\mu\nu}=-w^{\nu\mu}$ are real parameters and the six $M_{\mu\nu}=-M_{\nu\mu}$ are the generators of the Lorentz group. To see this we note that with $M_{\mu\nu}$ acting on the Lorentz spin zero $L(x)$ as $x_{\mu}p_{\nu}-x_{\nu}p_{\mu}$, under an infinitesimal Lorentz transformation the change in the action is given by $\delta I=2w^{\mu\nu}\int d^4x x_{\mu}\partial_{\nu}L(x)=2w^{\mu\nu}\int d^4x [\partial_{\nu}[x_{\mu}L(x)]-\eta_{\mu\nu}L(x)]$, and since the metric $\eta_{\mu\nu}$ is symmetric and $w_{\mu\nu}$ is antisymmetric, thus given by $\delta I=2w^{\mu\nu}\int d^4x \partial_{\nu}[x_{\mu}L(x)]$. Since the change in the action is a total divergence, the familiar invariance of the action under real Lorentz transformations is secured. However, we note that nothing in this argument depended on $w^{\mu\nu}$ being real, with the change in the action still being a total divergence even if $w^{\mu\nu}$ is complex. The action $I=\int d^4x L(x)$ is thus actually invariant under complex Lorentz transformations as well and not just under real ones, with complex Lorentz invariance thus being just as natural to physics as real Lorentz invariance.
For our purposes here we note that the Lorentz invariant scalar field action $I(\phi_1,\phi_2,\phi^*_1,\phi^*_2)$ given in (\ref{GPT1}) is thus invariant not just under real Lorentz transformations but under complex ones as well. Since in the above we constructed a time-independent inner product for this theory, the $I(\phi_1,\phi_2,\phi^*_1,\phi^*_2)$ action thus must have $CPT$ symmetry. And indeed we explicitly showed in (\ref{GPT2}) that this was in fact the case.
Since theories can thus be $CPT$ symmetric without needing to be Hermitian, it initially looks as though the two concepts are distinct. However, the issue of Hermiticity was addressed in \cite{Mannheim2018}, and the unexpected outcome of that study was that the only allowed Hamiltonians that one could construct that were $CPT$ invariant would have exactly the same structure as (or be similarity equivalent to) the ones one constructs in Hermitian theories, namely presumed Hermitian combinations of fields and all coefficients real.\footnote{While for instance $I(\chi_1,\chi_2,\psi_1,\psi_2)$ of (\ref{GPT6}) contains factors of $i$, its similarity transformed $I^{\prime}(\chi_1,\chi_2,\psi_1,\psi_2)$ given in (\ref{GPT43}) does not. Moreover this is even true of the $H=p^2+ix^3$ paradigm for $PT$ symmetry. With $S(\theta)=\exp(-\theta px)$ effecting the $[x,p]=i$ preserving $S(\theta)pS(-\theta)=\exp(-i\theta)p$, $S(\theta)xS(-\theta)=\exp(i\theta)x$, transforming with $S(\pi/2)$ effects $S(\pi/2)(p^2+ix^3)S(-\pi/2)=-p^2+x^3$, and in passing we note that $S(\pi)$ effects $S(\pi)(p^2+ix^3)S(-\pi)=p^2-ix^3=(p^2+ix^3)^{\dagger}$. In fact in \cite{Mannheim2018} it was shown in general that $CPT$ invariance of a relativistic theory entails that one can always find an appropriate similarity transformation that would bring the Hamiltonian to a form in which all coefficients are real.} These are precisely the theories that one ordinarily refers to as Hermitian. However, thus turns out to not necessarily be the case since theories can appear to be Hermitian but not actually be so.
To illustrate the above remarks it is instructive to consider some explicit examples, one involving behavior in time and the other involving behavior in space. For behavior in time consider the neutral scalar field with action $I_{\rm S}=\int d^4x [\partial_{\mu}\phi\partial^{\mu}\phi-m^2\phi^2]/2$ and Hamiltonian $H=\int d^3x[\dot{\phi}^2+\vec{\nabla}\phi\cdot \vec{\nabla}\phi+m^2\phi^2]/2$. Solutions to the wave equation $-\ddot{\phi}+\nabla^2\phi-m^2\phi=0$ obey $\omega^2(\textbf{k})=\textbf{k}^2+m^2$. Thus the poles in the scalar field propagator are at $\omega(\textbf{k})=\pm[\textbf{k}^2+m^2]^{1/2}$, the field can be expanded as $\phi(\textbf{x},t)=\sum [a(\textbf{k})\exp(-i\omega(\textbf{k}) t+i\textbf{k}\cdot\textbf{x})+a^{\dagger}(\textbf{k})\exp(+i\omega(\textbf{k}) t-i\textbf{k}\cdot\textbf{x})]$, and the Hamiltonian is given by $H=\sum [\textbf{k}^2+m^2]^{1/2}[a^{\dagger}(\textbf{k})a(\textbf{k})+a(\textbf{k})a^{\dagger}(\textbf{k})]/2$.
For either sign of $m^2$ the $I_{\rm S}$ action is CPT symmetric, and for both signs $I_{\rm S}$ appears to be Hermitian. For $m^2>0$, $H$ and $\phi(\textbf{x},t)$ are indeed Hermitian and all frequencies are real. However, for $m^2<0$, frequencies become complex when $\textbf{k}^2<-m^2$. The poles in the propagator move into the complex plane, the field $\phi(\textbf{x},t)$ then contains modes that grow or decay exponentially in time,\footnote{Since the action is $CPT$ symmetric, if there are to be any complex frequencies they must appear in complex conjugate pairs.} while $H$ contains energies that are complex. Thus neither $H$ nor $\phi$ is Hermitian even though $I_{\rm S}$ appears to be so.
For behavior in space consider the Pais-Uhlenbeck two-oscillator theory \cite{Pais1950} as studied in \cite{Bender2008a,Bender2008b}. In the theory there are two sets of oscillator operators, which obey $[z,p_z]=i$, $[x,p_x]=i$, and the Hamiltonian is given by
\begin{eqnarray}
H_{\rm PU}(\omega_1,\omega_2)=\frac{p_x^2}{2\gamma}+p_zx+\frac{\gamma}{2}\left(\omega_1^2+\omega_2^2 \right)x^2-\frac{\gamma}{2}\omega_1^2\omega_2^2z^2.
\label{GPT54}
\end{eqnarray}
As noted in \cite{Bender2008a} this theory is $PT$ symmetric, and as noted in \cite{Bender2008b} it in addition is the non-relativistic limit of a relativistic fourth-order neutral scalar field theory, one whose $CPT$ symmetry reduces to $PT$ symmetry in the non-relativistic limit. Initially the $\omega_1$ and $\omega_2$ frequencies are taken to be real and positive, and the energy eigenvalues are the real and positive $E(n_1,n_2)=(n_1+1/2)\omega_1+(n_2+1/2)\omega_2$.
However, if we now take the two frequencies to be equal to $\omega$, the Hamiltonian takes the form
\begin{eqnarray}
H_{\rm PU}(\omega)=\frac{p^2}{2\gamma}+p_zx+\gamma\omega^2x^2-\frac{\gamma}{2}\omega^4z^2,
\label{GPT55}
\end{eqnarray}
and while $H_{\rm PU}(\omega)$ looks to be just as Hermitian as before, the Hamiltonian turns out to be Jordan block \cite{Mannheim2005,Bender2008b}, to thus necessarily not be Hermitian at all. Since the $CPT$ invariance of $H_{\rm PU}(\omega_1,\omega_2)$ is not affected by setting $\omega_1=\omega_2$, $H_{\rm PU}(\omega)$ is $CPT$ symmetric.
Moreover, if we take the two frequencies to be in a complex pair $\omega_1=\alpha +i\beta$, $\omega_2=\alpha -i\beta$ with $\alpha>0$, $\beta>0$, the Hamiltonian takes the form \cite{Mannheim2018}
\begin{eqnarray}
H_{\rm PU}(\alpha,\beta)=\frac{p^2}{2\gamma}+p_zx+\gamma(\alpha^2-\beta^2)x^2-\frac{\gamma}{2}(\alpha^2+\beta^2)^2z^2.
\label{GPT56}
\end{eqnarray}
The $H_{\rm PU}(\alpha,\beta)$ Hamiltonian still looks to be Hermitian but its energy eigenvalues are now in complex conjugate pairs. With all the coefficients in $H_{\rm PU}(\alpha,\beta)$ being real, $H_{\rm PU}(\alpha,\beta)$ is $CPT$ symmetric. Thus all three of $H_{\rm PU}(\omega_1,\omega_2)$, $H_{\rm PU}(\omega)$ and $H_{\rm PU}(\alpha,\beta)$ are $CPT$ invariant and for all three of them all coefficients are real, just as required by \cite{Mannheim2018}. However, despite their appearance, $H_{\rm PU}(\omega)$ and $H_{\rm PU}(\alpha,\beta)$ are necessarily non-Hermitian.
As written in (\ref{GPT54}), $H_{\rm PU}(\omega_1,\omega_2)$ is actually not Hermitian (or self-adjoint) either \cite{Bender2008a}, since the real issue is not the appearance of the Hamiltonian but whether in an integration by parts one can drop spatially asymptotic surface terms. To see this we make a standard representation of the momentum operators of the form $p_z=-i\partial_z$, $p_x=-i\partial_x$, and find that for the Schr\"odinger problem associated with $H_{\rm PU}(\omega_1,\omega_2)$ the ground state wave function $\psi_0(z,x)$ with energy $E(0,0)=(\omega_1+\omega_2)/2$ is given by
\begin{eqnarray}
\psi_0(z,x)={\rm exp}\left[\frac{\gamma}{2}(\omega_1+\omega_2)\omega_1\omega_2
z^2+i\gamma\omega_1\omega_2zx-\frac{\gamma}{2}(\omega_1+\omega_2)x^2\right].
\label{GPT57}
\end{eqnarray}
Since this wave function is divergent at large $z$ it is not normalizable (though it is convergent at large $x$). Consequently, one cannot throw surface terms away in an integration by parts, and despite its appearance $H_{\rm PU}(\omega_1,\omega_2)$ is not self-adjoint. In the three realizations described above ($\omega_1$ and $\omega_2$ real and unequal, real and equal, in a complex conjugate pair) we find that $\omega_1+\omega_2$ and $\omega_1\omega_2$ are all real and positive. Thus in all three realizations the wave functions diverge at large $z$, and in all three cases the Hamiltonian is not self-adjoint when acting on $\psi_0(z,x)$
By the same token one cannot throw surface terms away for $p_z$ and $p_x$ when they act on the eigenstates of $H_{\rm PU}(\omega_1,\omega_2)$. Thus even though $p_z$ and $p_x$ are Hermitian when acting on their own eigenstates they are not Hermitian when acting on the eigenstates of $H_{\rm PU}(\omega_1,\omega_2)$. Thus building a Hamiltonian out of Hermitian operators (i.e. ones that are Hermitian when acting on their own eigenstates) does not necessarily produce a Hamiltonian that is Hermitian when the Hamiltonian acts on its own eigenstates. In fact, until one has constructed the eigenstates of a Hamiltonian one cannot even tell whether or not a Hamiltonian is Hermitian at all. One thus cannot declare a Hamiltonian to be Hermitian just by superficial inspection. Rather, one has to construct its eigenstates first and look at their asymptotic behavior.
In order to obtain eigenvectors for $H_{\rm PU}(\omega_1,\omega_2)$ that are normalizable the authors of \cite{Bender2008a} made the similarity transformation
\begin{eqnarray}
y=e^{\pi p_zz/2}ze^{-\pi p_zz/2}=-iz,\quad q=e^{\pi p_zz/2}p_ze^{-\pi p_zz/2}=
ip_z,
\label{GPT58}
\end{eqnarray}
on the operators of the theory so that $[y,q]= i$. Under this same transformation $H_{\rm PU}(\omega_1,\omega_2)$ transforms into
\begin{eqnarray}
e^{\pi p_zz/2}H_{\rm PU}(\omega_1,\omega_2)e^{-\pi p_zz/2}=\bar{H}_{PU}(\omega_1,\omega_2)=\frac{p^2}{2\gamma}-iqx+\frac{\gamma}{2}\left(\omega_1^2+\omega_2^2
\right)x^2+\frac{\gamma}{2}\omega_1^2\omega_2^2y^2,
\label{GPT59}
\end{eqnarray}
where for notational simplicity we have replaced $p_x$ by $p$, so that $[x,p]=i$. With the eigenvalue $z$ of the operator $z$ being replaced in $\psi_0(z,x)$ by $-iz$ (i.e. continued into the complex $z$ plane), the eigenfunctions are now normalizable.\footnote{As noted in \cite{Mannheim2013,Mannheim2018}, the analog statement for the Pais-Uhlenbeck two-oscillator theory path integral is that the path integral measure has to be continued into the complex plane in order to get the path integration to converge. A similar situation pertains to the path integral associated with the relativistic neutral scalar field theory with action $I_S=(1/2)\int d^4x[\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\partial^{\nu}\phi-(M_1^2+M_2^2)\partial_{\mu}\phi\partial^{\mu}
\phi+M_1^2M_2^2\phi^2]$, a theory whose non-relativistic limit is the Pais-Uhlenbeck theory.} When acting on the eigenfunctions of $\bar{H}_{PU}(\omega_1,\omega_2)$ the $y$ and $q=-i\partial_y$ operators are Hermitian (as are $x$ and $p=-i\partial_x$). However, as the presence of the factor $i$ in the $-iqx$ term indicates, $\bar{H}_{PU}(\omega_1,\omega_2)$ is not Hermitian. Since in general to establish Hermiticity one has to integrate by parts, drop surface terms and complex conjugate, we see that while we now can drop surface terms for $\bar{H}_{PU}(\omega_1,\omega_2)$ we do not recover the generic $H_{ij}=H_{ji}^*$ when we complex conjugate, even as we can now drop surface terms for the momentum operators when they act on the eigenstates of $\bar{H}_{PU}(\omega_1,\omega_2)$ and achieve Hermiticity for them.\footnote{The use of the similarity transformations given in (\ref{GPT58}) parallels the use of (\ref{GPT40}) in Sec. \ref{S4}. However, while using the similarity transformation of (\ref{GPT40}) was mainly a convenience, for $H_{PU}(\omega_1,\omega_2)$ the similarity transformation of (\ref{GPT58}) is a necessity because of the need to construct normalizable wave functions. The presence of the factor $i$ in (\ref{GPT59}) is thus related to the intrinsic structure of the Pais-Uhlenbeck theory.}
When $\omega_1$ and $\omega_2$ are real and unequal, the eigenvalues of the Hamiltonian $\bar{H}_{PU}(\omega_1,\omega_2)$ are all and the eigenspectrum (two sets of harmonic oscillators) is complete. In that case $\bar{H}_{PU}(\omega_1,\omega_2)$ can actually be brought to a form in which it is Hermitian by a similarity transformation. Specifically, one introduces an operator $Q$
\begin{eqnarray}
Q=\alpha pq+\beta xy,\quad \alpha=\frac{1}{\gamma\omega_1\omega_2}{\rm log}\left(\frac{\omega_1+\omega_2}{\omega_1-\omega_2}\right),\quad \beta=\alpha\gamma^2\omega_1^2\omega_2^2,
\label{GPT60}
\end{eqnarray}
and obtains
\begin{eqnarray}
&&\bar{H}_{PU}^{\prime}(\omega_1,\omega_2)=e^{-Q/2}\bar{H}_{PU}(\omega_1,\omega_2)e^{Q/2}=
\frac{p^2}{2\gamma}+\frac{q^2}{2\gamma\omega_1^2}+
\frac{\gamma}{2}\omega_1^2x^2+\frac{\gamma}{2}\omega_1^2\omega_2^2y^2.
\label{GPT61}
\end{eqnarray}
With the $Q$ similarity transformation not affecting the asymptotic behavior of the eigenstates of $\bar{H}_{PU}(\omega_1,\omega_2)$, and with $y$, $q$, $x$, and $p$ thus all being Hermitian when acting on the eigenstates of $\bar{H}_{PU}^{\prime}(\omega_1,\omega_2)$, the Hermiticity of $\bar{H}_{PU}^{\prime}(\omega_1,\omega_2)$ in the conventional Dirac sense is established. We can thus regard $\bar{H}_{PU}(\omega_1,\omega_2)$ with real and unequal $\omega_1$ and $\omega_2$ as being Hermitian in disguise. Moreover, in addition we note that since $Q$ becomes singular at $\omega_1=\omega_2$, at $\omega_1=\omega_2$ $\bar{H}_{PU}(\omega_1,\omega_2)$ cannot be diagonalized, to thus confirm that $H_{PU}(\omega)$ is Jordan block.\footnote{The transformation with $Q$ is the analog of the transformation of the spontaneously broken scalar field theory mass matrix given in (\ref{GPT22}), and the singularity in $Q$ at $\omega_1=\omega_2$ is the analog of that in (\ref{GPT22}) when $A=B$.} In general then we see that a Hamiltonian may not be Hermitian even though it may appear to be so, and may be (similarity equivalent to) Hermitian even when it does not appear to be so. And moreover, one cannot tell beforehand, as one needs to first solve the theory and see what its solutions look like.
Other then possibly needing to continue into the complex plane in order to get convergence, when a Hamiltonian has all eigenvalues real and eigenspectrum complete it is always possible to similarity transform it into a form in which it is Hermitian in the standard Dirac sense. If a Hamiltonian obeys $H=H^{\dagger}$, then under a similarity transform that effects $H^{\prime}=SHS^{-1}$, we note that $H^{\prime \dagger}=S^{-1 \dagger}H^{\dagger}S^{\dagger}=S^{-1 \dagger}HS^{\dagger}=S^{-1 \dagger}S^{-1}H^{\prime}SS^{\dagger}=[SS^{\dagger}]^{-1}H^{\prime}SS^{\dagger}$. Thus unless $S$ is unitary $H^{\prime \dagger}$ is not equal to $H^{\prime}$, with the $H_{ij}=H_{ji}^*$ Hermiticity condition being a condition that is not preserved under a general similarity transformation. Thus if one starts with some general $H^{\prime}$ that does not obey $H^{\prime}=H^{\prime \dagger}$, it might be similarity equivalent to a Hermitian $H$ but one does not know a priori. It only will be similarity equivalent to a Hermitian $H$ if the eigenvalues of $H^{\prime}$ are all real and the eigenspectrum is complete. And the necessary condition for that to be the case is that $H^{\prime}$ possess an antilinear symmetry. However, unlike a Hermiticity condition a commutation relation is preserved under a similarity transformation (even a commutation relation that involves an antilinear operator \cite{Mannheim2018}), with antilinear operators being more versatile than Hermitian operators. So much so in fact that in \cite{Mannheim2018} it was argued that one should use $CPT$ symmetry as the guiding principle for constructing quantum theories rather than Hermiticity.\footnote{Thus rather than being optional, according to \cite{Mannheim2018} one has to interpret the star symbol in (\ref{GPT1}) as a $CPT$ transform.}
When we characterize an operator such as $z$, $p_z$, $x$, or $p_x$ as being Hermitian we are only referring to representations of the $[z,p_z]=i$ and $[x,p_x]=i$ commutation relations, without any reference to a Hamiltonian that might contain these operators. A Hamiltonian can thus be built out of Hermitian operators and can have all real coefficients, and yet not be Hermitian itself. The equal-frequency and complex-frequency Pais-Uhlenbeck models are particularly instructive in this regard. In the equal-frequency case none of the $z$, $p_z$, $x$, or $p_x$ operators themselves are Jordan block, only $H_{PU}(\omega)$ is. The spectrum of eigenstates of the position and momentum operators are complete, and all are contained in the space on which $H_{\rm PU}(\omega)$ acts. However, not all of these states are eigenstates of the Hamiltonian \cite{Bender2008b}, with the one-particle sector of $H_{PU}(\omega)$ behaving just like the example given in (\ref{GPT26}) and (\ref{GPT28}). Moreover, in the complex $H_{\rm PU}(\alpha,\beta)$ case all the eigenvalues of the position and momentum operators are real even though those of the Hamiltonian that is built out of them are not. As the equal-frequency and complex-frequency Pais-Uhlenbeck models show, one cannot tell whether a Hamiltonian might be Hermitian just by superficial inspection. One needs to solve the theory first and see what the eigenspectrum looks like. Thus one can have Hamiltonians that do not look Hermitian but are similarity equivalent to ones that are Hermitian, and one can have Hamiltonians that do look Hermitian but are not at all.
As we see from these examples, whether or not an action is $CPT$ symmetric is an intrinsic property of the unconstrained action itself prior to any stationary variation, but whether or not a Hamiltonian is Hermitian is a property of the stationary solution alone.\footnote{While one can construct the Hamiltonian from the energy-momentum tensor, the energy-momentum tensor is only conserved in solutions to the equations of motion. Hermiticity is thus tied to the solutions to the theory in a way that $CPT$ is not.} Hermiticity of a Hamiltonian cannot be assigned a priori, and can only be determined after the theory has been solved. However, the $CPT$ properties of actions or fields can be assigned a priori (i.e. prior to a functional variation of the action, and thus a property of every variational path and not just the stationary one), and thus that is how Hamiltonians and fields should be characterized. One cannot write down any $CPT$ invariant theory that up to similarity transformations does not have the same form as a Hermitian theory, though whether any such $CPT$ invariant Hamiltonian actually is similarity equivalent to a Hermitian one is only establishable by constructing the solutions to the theory and cannot be determined ahead of time.
Turning now to the study of \cite{Alexandre2018}, we note that it displays all of the features that we have just described. The interest of the authors of \cite{Alexandre2018} was in exploring the status of the Goldstone theorem in non-Hermitian but $PT$-symmetric theories, and so they took as an example a relativistic field theory whose action was not Hermitian, i.e. not Hermitian by superficial inspection. However, by a similarity transformation it could be brought to a form given in (\ref{GPT46}) in which the action is Hermitian by superficial inspection (i.e. no factors of $i$ and operators that are presumed to be Hermitian). However, while it now appears to be Hermitian it could not be since in the tree approximation that they studied the ensuing mass matrix was not Hermitian either. With the mass matrix having the three possible $PT$ symmetry realizations (real and unequal eigenvalues, real and equal eigenvalues, eigenvalues in complex conjugate pairs) for various values of its parameters, the tree approximation to the model of \cite{Alexandre2018} completely parallels the discussion of the three realizations of Pais-Uhlenbeck two-oscillator model given in (\ref{GPT54}), (\ref{GPT55}) and (\ref{GPT56}), where the Hamiltonian looks to be Hermitian but is not. It is of interest to note that to establish that the Pais-Uhlenbeck two-oscillator model theory is not Hermitian we had to construct wave functions and examine there asymptotic behavior, while for the tree approximation to the model of \cite{Alexandre2018} we only need to look at a finite-dimensional matrix. Thus we can start with a fully-fledged field theory such as that based on the action given in (\ref{GPT1}), (\ref{GPT46}) or (\ref{GPT51}) and not need to identify the region in the complex plane where the functional path integral might exist or need to descend to the quantum mechanical limit and look at the asymptotic behavior of wave functions in order to determine whether or not the theory is Hermitian.\footnote{These issues would only start to come up in fluctuations around the tree approximation minimum, with a one loop calculation having been provided in \cite{Alexandre2018}.} In the broken symmetry case we only need look at the finite-dimensional mass matrix that we get in tree approximation.
For parameters in the model of \cite{Alexandre2018} that obey $(2m_1^2m_2^2-m_2^4-3\mu^4)^2-4\mu^4m_2^4>0$, the mass matrix can be brought to a Hermitian form by the similarity transformation presented in (\ref{GPT22}). Thus in this case the mass matrix is Hermitian in disguise. For this particular example the Goldstone theorem is the standard one, since if one can derive the Goldstone theorem in a Hermitian theory, it continues to hold if one makes a similarity transformation on it.\footnote{Technically, that would have automatically been the case if the authors of \cite{Alexandre2018} had used a conventional variational procedure, though in fact they did not. It is however the case for the study that we have presented here.} Whether or not the mass matrix given in (\ref{GPT11}) actually can be transformed to a Hermitian matrix depends on the values of the parameters in the action. However, as we have seen, no matter what the values of these parameters, and no matter whether the $CPT$-invariant mass matrix is realized by real eigenvalues, complex pairs of eigenvalues, or is of Jordan-block form, for any choice of the parameters one is able to obtain a Goldstone theorem. One can thus anticipate a Englert-Brout-Higgs mechanism for a local extension of the continuous symmetry that we have broken spontaneously, and we turn now to this issue.
\section{Spontaneously Broken non-Hermitian Theory with a Continuous Local Symmetry}
\label{S6}
Now that we have seem that we can consistently implement the Goldstone mechanism in a $CPT$-symmetric, non-Hermitian theory, it is natural to ask whether we can also implement the familiar Englert-Brout-Higgs mechanism developed in \cite{Englert1964,Higgs1964a,Higgs1964b,Guralnik1964}. To this end we introduce a local gauge invariance and a gauge field $A_{\mu}$, and with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ replace (\ref{GPT1}) and (\ref{GPT3}) by
\begin{eqnarray}
I(\phi_1,\phi_2,\phi^*_1,\phi^*_2,A_{\mu})&=&\int d^4x\bigg{[}(-i\partial_{\mu}+eA_{\mu})\phi^*_1(i\partial^{\mu}+eA^{\mu})\phi_1+(-i\partial_{\mu}+eA_{\mu})\phi^*_2(i\partial^{\mu}+eA^{\mu})\phi_2
\nonumber\\
&+&m_1^2\phi_1^*\phi_1-m_2^2\phi^*_2\phi_2-\mu^2(\phi^*_1\phi_2-\phi^*_2\phi_1)-\frac{g}{4}(\phi^*_1\phi_1)^2 -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\bigg{]},
\label{GPT62}
\end{eqnarray}
and
\begin{eqnarray}
\phi_1\rightarrow e^{i\alpha(x)}\phi_1,\quad \phi^*_1\rightarrow e^{-i\alpha(x)}\phi^*_1,\quad \phi_2\rightarrow e^{i\alpha(x)}\phi_2,\quad \phi^*_2\rightarrow e^{-i\alpha(x)}\phi_2,\quad
eA_{\mu}\rightarrow eA_{\mu}+\partial_{\mu}\alpha(x).
\label{GPT63}
\end{eqnarray}
With (\ref{GPT2}), the $I(\phi_1,\phi_2,\phi^*_1,\phi^*_2,A_{\mu})$ action is $CPT$ invariant since both $i$ and $A_{\mu}$ are $CPT$ odd (spin one fields have odd $CPT$ \cite{Weinberg1995}).
We make the same decomposition of $\phi_1$ and $\phi_2$ fields as in (\ref{GPT5}), and replace (\ref{GPT6}) by
\begin{eqnarray}
I(\chi_1,\chi_2,\psi_1,\psi_2,A_{\mu})&=&\int d^4x \bigg{[}\frac{1}{2}\partial_{\mu}\chi_1\partial^{\mu}\chi_1+
\frac{1}{2}\partial_{\mu}\chi_2\partial^{\mu}\chi_2+
\frac{1}{2}\partial_{\mu}\psi_1\partial^{\mu}\psi_1+
\frac{1}{2}\partial_{\mu}\psi_2\partial^{\mu}\psi_2+
\frac{1}{2}m_1^2(\chi_1^2+\chi_2^2)
\nonumber\\
&-&
\frac{1}{2}m_2^2(\psi_1^2+\psi_2^2)-
i\mu^2(\chi_1\psi_2-\chi_2\psi_1)
-\frac{g}{16}(\chi_1^2+\chi_2^2)^2
\nonumber\\
&-&eA^{\mu}\left(\chi_1\partial_{\mu}\chi_2-\chi_2\partial_{\mu}\chi_1+
\psi_1\partial_{\mu}\psi_2-\psi_2\partial_{\mu}\psi_1\right)
\nonumber\\
&+&\frac{e^2}{2}A_{\mu}A^{\mu}\left[\chi_1^2+\chi_2^2+\psi_1^2+\psi_2^2\right]-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
\bigg{]},
\label{GPT64}
\end{eqnarray}
In the tree approximation minimum used above in which $(g/4)\bar{\chi}_1^2=m_1^2-\mu^4/m_2^2$, $\bar{\psi}_2=-i\mu^2\bar{\chi}_1/m_2^2$, $\hat{\chi}_2=0$, $\hat{\psi}_1=0$, we induce a mass term for $A_{\mu}$ of the form
\begin{eqnarray}
m^2(A_{\mu})=e^2\left(\bar{\chi}_1^2+\bar{\chi}_2^2+\bar{\psi}_1^2+\bar{\psi}_2^2\right)
=e^2\bar{\chi}_1^2\left(1-\frac{\mu^4}{m_2^4}\right)
=\frac{4e^2}{g}\frac{(m_1^2m_2^2-\mu^4)(m_2^4-\mu^4)}{m_2^6}.
\label{GPT65}
\end{eqnarray}
However, before assessing the implications of (\ref{GPT65}) we recall that in Sec. \ref {S4} we had to reconcile $I(\chi_1,\chi_2,\psi_1,\psi_2)$ with the Hermiticity concern raised in \cite{Alexandre2018}. The same is now true of $I(\chi_1,\chi_2,\psi_1,\psi_2,A_{\mu})$. In Sec. \ref{S4} we had identified three solutions for $I(\chi_1,\chi_2,\psi_1,\psi_2)$, and all can be implemented for $I(\chi_1,\chi_2,\psi_1,\psi_2,A_{\mu})$. Thus we can consider a judicious choice of which fields are Hermitian and which are anti-Hermitian, a judicious choice of which fields are $CPT$ even and which are $CPT$ odd, or can apply similarity transformations that generate complex phases that affect both Hermiticity and $CPT$ parity.
In regard to Hermiticity, if we take $A_{\mu}$ to be Hermitian (i.e. complex conjugate even), and as before take $\psi_1$ and $\psi_2$ to be anti-Hermitian (complex conjugate odd), then $I(\chi_1,\chi_2,\psi_1,\psi_2,A_{\mu})$ will be invariant under complex conjugation, as will then be the equations of motion and tree approximation minimum that follow from it, and (\ref{GPT65}) will hold. Also, as we had noted in Sec. \ref{S5}, even though $I(\chi_1,\chi_2,\psi_1,\psi_2,A_{\mu})$ might now be invariant under complex conjugation it does not follow that the scalar field mass matrix $M$ given in (\ref{GPT9}) has to be Hermitian, and indeed it is not.
If we transform $\psi_1$ and $\psi_2$ as in (\ref{GPT41}) but make no transformation on $A_{\mu}$, we obtain
\begin{eqnarray}
S(\psi_1)S(\psi_2)I(\chi_1,\chi_2,\psi_1,\psi_2,A_{\mu})S^{-1}(\psi_2)S^{-1}(\psi_1)=I^{\prime}(\chi_1,\chi_2,\psi_1,\psi_2)
\label{GPT66}
\end{eqnarray}
where
\begin{eqnarray}
I^{\prime}(\chi_1,\chi_2,\psi_1,\psi_2,A_{\mu})&=&\int d^4x \bigg{[}\frac{1}{2}\partial_{\mu}\chi_1\partial^{\mu}\chi_1+
\frac{1}{2}\partial_{\mu}\chi_2\partial^{\mu}\chi_2-
\frac{1}{2}\partial_{\mu}\psi_1\partial^{\mu}\psi_1-
\frac{1}{2}\partial_{\mu}\psi_2\partial^{\mu}\psi_2+
\frac{1}{2}m_1^2(\chi_1^2+\chi_2^2)
\nonumber\\
&+&
\frac{1}{2}m_2^2(\psi_1^2+\psi_2^2)-
\mu^2(\chi_1\psi_2-\chi_2\psi_1)
-\frac{g}{16}(\chi_1^2+\chi_2^2)^2
\nonumber\\
&-&eA^{\mu}\left(\chi_1\partial_{\mu}\chi_2-\chi_2\partial_{\mu}\chi_1-
\psi_1\partial_{\mu}\psi_2+\psi_2\partial_{\mu}\psi_1\right)
\nonumber\\
&+&\frac{e^2}{2}A_{\mu}A^{\mu}\left(\chi_1^2+\chi_2^2-\psi_1^2-\psi_2^2\right)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
\bigg{]},
\label{GPT67}
\end{eqnarray}
As constructed, $ I^{\prime}(\chi_1,\chi_2,\psi_1,\psi_2,A_{\mu})$ is invariant under complex conjugation if $\psi_1$ and $\psi_2$ are even under complex conjugation. Now the tree approximation minimum is given by
$(g/4)\bar{\chi}_1^2=m_1^2-\mu^4/m_2^2$, $\bar{\psi}_2=\mu^2\bar{\chi}_1/m_2^2$, $\hat{\chi}_2=0$, $\hat{\psi}_1=0$, $A_{\mu}=0$, and we induce a mass term for $A_{\mu}$ of the form
\begin{eqnarray}
m^2(A_{\mu})=e^2\left(\bar{\chi}_1^2+\bar{\chi}_2^2-\bar{\psi}_1^2-\bar{\psi}_2^2\right)
=e^2\bar{\chi}_1^2\left(1-\frac{\mu^4}{m_2^4}\right)
=\frac{4e^2}{g}\frac{(m_1^2m_2^2-\mu^4)(m_2^4-\mu^4)}{m_2^6}.
\label{GPT68}
\end{eqnarray}
The mass of the gauge boson is thus again given by (\ref{GPT65}). Finally, in regard to interpreting the star symbol as the $CPT$ conjugate, since $A_{\mu}$ is real there is no change in the discussion presented in Sec. \ref{S4}, and (\ref{GPT65}) continues to hold.
As we can see from (\ref{GPT65}) and (\ref{GPT68}), the gauge boson does indeed acquire a non-zero mass unless $m_1^2m_2^2=\mu^4$ or $m_2^4=\mu^4$. The first of these conditions is not of significance since if $m_1^2m_2^2=\mu^4$ it follows that $\bar{\chi}_1$ (and thus $\bar{\psi}_2$) is zero and there is no symmetry breaking, and the gauge boson stays massless.\footnote{If $m_1^2m_2^2-\mu^4=0$, the unbroken $(\chi_1,\psi_2)$ sector mass matrix given in (\ref{GPT64}) is of the form $(1/2m_2^2)(\mu^2\chi_1-im_2^2\psi_2)^2$. It has two eigenvalues, $\lambda_a=\mu_4/m_2^2-m_2^2$ and $\lambda_b=0$. In the $(\chi_1,\psi_2)$ basis the right-eigenvector for $\lambda_a$ is ($\mu^2,-im_2^2)$, while the right-eigenvector for $\lambda_b$ is ($m_2^2,-i\mu_2^2)$. The fact that $\lambda_b$ is zero is not of significance since it occurs in the absence of spontaneous symmetry breaking, and would thus not be maintained under radiative corrections.} However, the condition $m_2^4=\mu^4$ is related to the symmetry breaking since it does not oblige $\bar{\chi}_1$ to vanish. Moreover, since the $m_2^4=\mu^4$ condition does not constrain the $(\tilde{\chi}_1,\tilde{\psi}_2)$ sector $\lambda_{\pm}$ eigenvalues given in (\ref{GPT11}) in any particular way ($m_1^2$ not being constrained by the $m_2^4=\mu^4$ condition), we see that regardless of whether or not $m_2^4$ and $\mu^4$ are in fact equal to each other, we obtain the Englert-Brout-Higgs mechanism in the $(\tilde{\chi}_2,\tilde{\psi}_1)$ sector no matter how the antilinear symmetry is realized in the $(\tilde{\chi}_1,\tilde{\psi}_2)$ sector, be it all eigenvalues real, eigenvalues in a complex pair, or mass matrix being of non-diagonalizable Jordan-block form. In the $(\tilde{\chi}_2,\tilde{\psi}_1)$ sector both $\lambda_0$ and $\lambda_1$ as given in (\ref{GPT11}) are real, and not degenerate with each other as long as $m_2^4\neq \mu^4$. However, something very interesting occurs if $m_2^4=\mu^4$. Then the $(\tilde{\chi}_2,\tilde{\psi}_1)$ sector becomes Jordan block and the Goldstone boson acquires zero-norm. Since the Goldstone boson can no longer be considered to be a normal positive norm particle, it cannot combine with the gauge boson to give the gauge boson a longitudinal component and make it massive. And as we see, and just as required by consistency, in that case the gauge boson stays massless. Thus in a non-Hermitian but $CPT$-symmetric theory it is possible to spontaneously break a continuous local symmetry and yet not obtain a massive gauge boson.
\section{Summary}
\label{S7}
In the non-relativistic antilinear symmetry program one replaces Hermiticity of a Hamiltonian by antilinearity as the guiding principle for quantum mechanics for both infinite-dimensional wave mechanics and either finite- or infinite-dimensional matrix mechanics. For infinite-dimensional relativistic quantum field theories whose actions are invariant under the complex Lorentz group the antilinear symmetry is uniquely prescribed to be $CPT$. Hamiltonians that have an antilinear symmetry can of course be Hermitian as well, with all energy eigenvalues then being real and all energy eigenvectors being complete. However, in general, antilinear symmetry permits two additional options for Hamiltonians that cannot be realized in Hermitian theories, namely energy eigenvalues could be real while energy eigenvectors could be incomplete (Jordan block), or energy eigenvectors could still be complete but energy eigenvalues could come in complex conjugate pairs. In the first case all Hilbert space inner products are positive definite, in the second (Jordan-block) case norms are zero, and in the third case the only norms are transition matrix elements and their values are not constrained to be positive or to even be real. Moreover, in the antilinear symmetry program Hamiltonians that look to be Hermitian by superficial inspection do not have to be, while Hamiltonians that do not look to be Hermitian by superficial inspection can actually be similarity equivalent to Hamiltonians that are Hermitian (viz. Hermitian in disguise).
In applications of antilinear symmetry to relativistic systems it is of interest to see how many standard results that are obtained in Hermitian theories might still apply in the non-Hermitian case, and whether one could obtain new features that could not be realized in the Hermitian case. To address these issues Alexandre, Ellis, Millington and Seynaeve studied how spontaneously broken symmetry ideas and results translate to the non-Hermitian environment. With broken symmetry and the possible existence of massless Goldstone bosons being intrinsically relativistic concepts, they explored a $CPT$ symmetric but non-Hermitian two complex scalar field relativistic quantum field theory with a continuous global symmetry. Their actual treatment of the problem was somewhat unusual in that they allowed for non-vanishing surface terms to contribute in the functional variation of the action, with this leading to a specific non-standard set of equations of motion of the fields. Their reason for doing this was that the equations of motion obtained by the standard variational procedure with endpoints held fixed were not invariant under complex conjugation. However, they still found a broken symmetry solution with an explicit massless Goldstone boson.
In the treatment of the same model that we provide here we use a conventional variational calculation in which fields are held fixed at the endpoints. However, to get round the complex conjugation difficulty we make a similarity transformation on the fields which then allows us to be able to maintain invariance of the equations of motion under complex conjugation (the similarity transformation itself being complex). However, on doing this we obtain an action that appears to be Hermitian, and if it indeed were to be Hermitian there would be nothing new to say about broken symmetry that had not already been said for Hermitian theories. However, while appearing to be Hermitian the theory in fact is not, and thus it does fall into the non-Hermitian but antilinearly symmetric category.
In their analysis Alexandre, Ellis, Millington and Seynaeve studied the tree approximation to the equations of motion of the theory and found broken symmetry solutions. What is particularly noteworthy of their analysis is that even though they were dealing with a fully-fledged infinite-dimensional quantum field theory, in the tree approximation the mass matrix that was needed to determine whether there might be any massless Goldstone boson was only four dimensional (viz. the same number as the number of independent fields in the two complex scalar field model that they studied). As such, the mass matrix that they obtained is not Hermitian, and given the underlying antilinear $CPT$ symmetry of the model and thus of the mass matrix, the mass matrix is immediately amenable to the full apparatus of the antilinear symmetry program as that apparatus holds equally for fields and matrices. Alexandre, Ellis, Millington and Seynaeve studied just one realization of the antilinear symmetry program, namely the one where all eigenvalues of the mass matrix are real and the set of all of its eigenvectors is complete. In our analysis we obtain the same mass matrix (which we must of course since all we have done is make a similarity transformation on their model), and show that in this particular realization the mass matrix can be brought to a Hermitian form by a similarity transformation, to thus be Hermitian in disguise.
However, this same mass matrix admits of the two other realizations of antilinear symmetry as well, namely the non-diagonalizable Jordan-block case and the complex conjugate eigenvalue pair case. And in all of these cases we show that there is a massless Goldstone boson. In this regard the Jordan-block case is very interesting because it permits the Goldstone boson itself to be one of the zero norm states that are characteristic of Jordan-block matrices. That these cases can occur at all is because while the similarity transformed action that we use appears to be Hermitian it actually is not, something however that one cannot ascertain without first solving the theory. Finally, we extend the model to a local continuous symmetry by introducing a massless gauge boson, and find that the massless Goldstone boson can be incorporated into the massless gauge boson and make it massive by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry except one, namely the Jordan-block Goldstone mode case. In that case we find that since the Goldstone boson then has zero norm, it does not get incorporated into the gauge boson, with the gauge boson staying massless. In this case we have a spontaneously broken local gauge symmetry and yet do not get a massive gauge boson. This option cannot be obtained in the standard Hermitian case where all states have positive norm, to thus show how rich the non-Hermitian antilinear symmetry program can be.
| {
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} | 3,411 |
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A home in the hills
The realty scene in Himachal Pradesh by and large remains as cool as its climate, barring a few pockets. Strict land ownership rules and a fragile ecology are generally big barriers for over ambitious developers. Over the years many of the over-zealous projects have landed in soup for messing with environment and for threatening the 'green balance' of the hills.
Chopra Garden
Choti Line
Madhu Colony
Model Colony
38000 to 45000
Mini Model Town
New Hamida Colony
Price in Rs per sq yd
Please note: The prices may vary as per the size of the plots and location, and are subject to change from time to time. Source: Nirmal Infrastructures. Mohali nirmalinfrastructures@yahoo.com
Sound agreement
Loan agreement should be thoroughly scanned and all the clauses should be understood fully, writes S.C.Dhall
Easy availability of home loans has made the dream of buying a home an achievable one for many. But remember to go over the paperwork thoroughly as your home loan may eat up a lion's share of your salary over many years. Scan your loan agreement thoroughly to understand what each clause implies before you sign on the dotted line.
Ground Realty
Novel building blocks
The bricks that a common user knows about are the ones that we see everywhere - red coloured, 9 inch long, 3 inch thick, 4 ½ inch wide and can be bought from a brick kiln. These are an essential building component and among the first materials that arrive at site whenever construction is about to begin. Constructing a house without bricks is unthinkable, at least in India.
Ceiling statement
Attractive colours or patterns, decorative wall panels, furniture, show pieces, indoor plants all are elements that are usually considered important in defining the internal ambience and the overall environment of a home or a work place. The use of designer flooring, natural stone, and carpets has made the floors, too, enter the set of design considerations. Ceilings, too, play a vital role within an interior space. They offer the maximum unobstructed view in a room and thus never fail to catch the eye.
Credai's code for builders
Credai makes carpet area mention mandatory for builders
Apex realtors body Credai has made mandatory for all members of the association to mention carpet area in their brochures and sale agreement, aiming to bring in transparency in the sector.
Take separate DDs
Mortgage & inheritance
Is plot with construction a residential house?
Tuscan Residency
Taneja Developers and Infrastructures Ltd. (TDI) group that is coming up with two township projects in Mohali recently launched its housing project "Tuscan Residency" in TDI City in Sector 110-111, Mohali on Landran-Banur road.
Geetu Vaid & Charandeep Singh
The realty scene in Himachal Pradesh by and large remains as cool as its climate, barring a few pockets. Strict land ownership rules and a fragile ecology are generally big barriers for over ambitious developers. Over the years many of the over-zealous projects have landed in soup for messing with environment and for threatening the 'green balance' of the hills. But the changing lifestyle of the local population as well as the ever burning desire of the jet setters to have a "summer home" in the hills have kept the traffic of builders and new projects on a steady keel in the state.
The volume of development in areas around Shimla bears testimony to the lure of hills among people from all walks of life. The pristine Kulu and Manali region too, is not only a hot destination for tourists but for the people in real estate business as well. The Kulu valley with its temples and Manali with its snow-capped peaks and the gushing river water have breathtaking views strewn at every corner making every tourist to fantasise about having a home there. "The spectacular view and location were the main factors that made us decide to start our project in Kulu", says Ranjiv Kalia, Head Sales and Marketing, Ansal Buildwell Ltd.
The group recently launched its project "Meadows" at Arun Hills in Kulu. With tourists thronging Kulu-Manali throughout the year now there has been rapid development in the area. As a fall out of this people from different areas are attracted to this area and over the past few months the prices too have shown an upward trend making it a sound investment to have a house here. "Even former Prime Minister Vajpayee has his summer retreat here. I have a second home here to enjoy the peace of nature and it has been a good investment also as the property has shown considerable appreciation in the past 18 months", says Vijay Kundu, a Delhi-based businessman, who has his summer cottage in Manali.
Up the axis
Property in Kulu Manali is a money-turning affair as it offers various options in the residential segment, commercial property and leased properties as well. The realty market is giving good return and offering lucrative opportunities to invest money. "Though horticulture is the backbone of the local economy, tourism is also as significant as horticulture for the growth and overall development of the district", remarks B.S Nanta (IAS), Deputy Commissioner, Kulu.
Rising number of tourists has increased the demand for hotels and guest houses besides ensuring a healthy flow of money in the market. Another factor that has contributed immensely to the growth of the area is the construction of Rohtang tunnel. This tunnel, once completed, would connect Manali with Lahaul Spiti which also has immense tourism potential. "Since the construction is going on and a lot of engineers and workers are involved in the project have moved in here with their families, the demand of housing has gone up and it has also given an upward thrust to rentals here", observes Girimer Maan, who has a resort by the name of Cassa Bella Vista in Manali.
With the setting up of various facilities like winter skiing and adventure sports on the outskirts of Manali, a lot of the land which earlier was barren is a prized commodity now. This has significantly contributed to the spurt in property prices. "Another important factor which has caused the growth of real estate in Kulu-Manali is the Section 118. Now this Act says that non-Himachali's cannot buy agricultural lands, but on the contrary they can always invest in commercial property. So all these budgeted hotels, complexes , guest houses which are running in the area have investments by non-Himachalis which has contributed to the boom of real estate and have made land prices go sky high", observes H.P Singh, a local hotelier.
Pricing matrix
In the past 18 months the property prices in the area have increased by almost 40 per cent. In the residential segment not much choice is there in apartments/flats as not many big realtors are in the market here. There were a few units constructed by a private developer about four years ago but these did not generate a good response. A 2BHK apartment costs around Rs 20 lakh. "HIMUDA had invited some applications for flats, which would be allotted in February, 2012. They plan to construct around 100 units in over there", informs Jatin, a real estate Continued on broker. The Ansals group's Rs 100 crore Kulu project will have 190 fully furnished residential units in 850 sq ft to 1,550 sq ft area in the price range of Rs 38 lakh and is likely to be completed in two years.
Gandhi Nagar is the high density commercial area of Kulu where, the ongoing rates are somewhere around Rs 20 lakh per biswa.
With Manali being the hub of tourism and adventure sports, land prices are more than those in Kulu. "Here land is available only on the other side of the Beas at approximately Rs 10 lakh per biswa. There is no rent culture in the main commercial belt. It is all lease based, which is around Rs 5-6 lakh per year depending on the location." reveals Girimer Maan who has a resort in Manali. Manali is expanding towards Shanag and Gurwa, where land is available. In these areas the existing land rate is around Rs 3-5 lakh per biswa.
Fully loaded options
Fully furnished cottages and villas that are ideal as second homes fall in the luxury segment here. "As these are summer homes we are providing all facilities to the buyers who can just relax and enjoy there stay here without having to lose sleep over the nitty gritty of daily routine", says Kalia. The Ansals project will also have premium villas priced at Rs 1.5 crore onwards and the company is offering to maintain the apartments when they are not in use by actual owner. Cottages, too, are popular option here. "A cottage built on approximately one bigha of land costs around Rs 4.5 crore", reveals Mann who is planning to construct 20 fully furnished cottages near Manali.
Rate of interest
The rate of interest determines the EMI or equated monthly instalment. The rate of interest, generally, can be of two types - fixed and floating - though the latter is the most common these days. A teaser loan is another addition to this category. These cost less in the first few years.
Changes in base rate
Base rate is determined by an individual bank depending on various internal parameters, including RBI's changes in repo and reverse repo rates. Banks cannot lend at a rate lower than the base rate. The floating interest rate would be the base rate plus a spread. The spread is the premium the banks charge from customers. The base rate is an important tool in a bank's armoury and often makes changes to it in response to the market conditions. The consumer is impacted due to this as interest rate changes with change in the base rate.
Reset clause
In the case of a fixed interest rate, the rate is usually fixed but there is a reset clause which allows banks to reset the fixed interest rate in relation to base rate at particular intervals.
This phrase means 'greater force'. This implies that banks can raise the interest rates in exceptional conditions, even in a fixed rate loan. However, defining such a condition is left to the discretion of the bank.
Pre-payment penalty
The pre-payment clause explains the penalty that is charged if you decide to close the loan early by paying the amount due. Some banks do not impose any penalty while others differentiate this according to circumstances. For instance, when you refinance the loan through another bank opting for a lower interest rate such a pre-payment may involve a different and heavier penalty from the bank where you currently hold your loan.
Defining a fault
For you a 'fault' could simply mean not paying your EMI at some point in your loan tenure. However, some banks specify a fault as a case when the borrower expires, the borrower is divorced and stops paying (in case of more than a single borrower), or the borrower is/are involved in any civil litigation or criminal offence. Therefore, you must be clear about what your lender means by the term 'fault'.
Security cover
This clause states that a bank is eligible to demand additional security when property prices fall. Such a demand could exist even if you are very regular with your EMIs. In such a scenario, if you are unable to provide a security cover in addition to your loan amount chances are that you could be declared a defaulter by the lender.
Interpreting the clauses
Remember that a borrower's goal is to get the loan at the least expensive interest rate while a bank's goal is to lend at a profit. The different clauses mentioned above should, thus, be interpreted keeping this point in mind.
Teaser loan features: Teaser loans charge you less interest first and then increase it to the market rate.
If you are opting for this, make sure you understand the interest rate you will pay for the life of the loan. Most of the borrowers look at the next one to three years' EMI and decide accordingly. The right way is to see the projected EMI after three to five years when the teaser rates are done with.
Floating rate loan: The banks usually increase floating interest rate as soon as RBI raises rates but do not lower it with the same enthusiasm.
Fixed rate loan: Though fixed rates are fixed over the period of the loan, banks insert a clause for resetting the fixed rate based on market conditions. Considering this aspect, it could be better to opt for a floating rate as you might get the benefit when market conditions turn favourable for a lower interest rate.
Pre-payment penalty: Discuss upfront with your bank about the pre-payment penalty they charge and whether it works differently when you opt to pre-pay and refinance the loan.
Make sure everything is in writing. Currently, RBI has already insisted on implementing a measure to do away with prepayment penalty completely. Discuss this with your bank and see if you can avoid paying a pre-payment penalty alltogether.
n It is always wise to calculate a rough estimate of the effective interest rate you will need to shell out for your teaser loan and see if that can fit into your long-term budget easily. Teaser loans can work to your advantage if you plan to close the loan in the short-term i.e. five to six years.
n Though banks reduce interest rates as per the reduction in base rate, it could still be applicable to new borrowers only. Again the RBI has stressed that the benefits should be passed on to existing customers as well, so figure out with your bank if that could be possible in your case.
Last but most important, document the discussion and take everything in writing. A home loan is too important to be taken on the basis of mere verbal assurances.
Jagvir Goyal
Cost of bricks has risen significantly over the past two years. In 2009, these used to cost Rs 2,800 per thousand. Now, the cost is around Rs 4,400 per thousand. And there is no visible fall in their demand. Good quality brick-stacks get sold as soon as these come out of the kiln. Brick kiln owners have realised that there is no substitute for bricks available. So they are on the look out to give another upward push to the prices.
Indians believe in composite construction done by use of bricks, cement and steel. No other material looks durable and suitable to Indians. Many new concepts have been tried by many companies from time to time. Portable houses built by using steel sheets and fibrous materials are being marketed rigorously by some firms. But to most Indians these can never give the feeling of a sturdy home and are nothing more than temporary shelters. Moreover, brick houses built over centuries have stood the test of time. So the Indian belief in bricks will stay intact till the time some really path-breaking and convincing substitute is found.
Tried alternatives
Hollow and solid concrete blocks have been tried as a substitute for bricks. But these have failed to get an enthusiastic response. Fly ash lime bricks have been produced by the small scale industry. These bricks don't use clay but lime, sand and fly ash. These are not burnt. These are also called FAL G bricks. The lime used in these bricks has to be only C class, hydrated lime as per IS 712. However, these too haven't seen a mass use so far as their production volumes are small and these are generally prescribed for buildings having less than two storeys.
Clay fly ash bricks
While the normally used bricks are burnt clay bricks manufactured with clay only, burnt clay fly ash bricks have been tried and found to be having much better strength and other properties than normal bricks. In these bricks, 25 to 30 per cent of fly ash is mixed in the clay before moulding it into bricks. All test results of these bricks are better than those of normal bricks. The only bottleneck in their production is that machinery is required for proper blending of fly ash with clay. The brick kiln owners are reluctant to invest in machinery as their normal production of clay bricks is already in demand.
The new concept
Now, under a new concept, South India is witnessing the production of hollow clay bricks. Known as Porotherm bricks, these bricks are fast gaining popularity in South that may soon spread to North India also.
Size: Hollow clay bricks are being produced in three sizes: 16"x8"x8", 16"x6"x8" and 16"x4"x8". As can be seen from these sizes, while the lengths and thickness of the bricks remain the same i.e. 16 inch and 8 inch respectively, the width varies from 8 inches to 4 inches. Volume wise, a 16x8x8 inch brick is equivalent to 8 bricks of 9"x4 ½" x3", we use today. Similarly, the other two sizes of hollow clay bricks are equivalent to 6 and 4 normal bricks, respectively.
Weight: The weight of the biggest size hollow clay brick with 8" width is 11 kg. The weight of 6" wide brick is 9 kg and that of 4" wide brick is 6 kg. A normal solid brick weighs around 3 kg and weight of 8 bricks works out as 24 kg. Thus a hollow clay brick that occupies the same volume as 8 normal brick shall weigh only 11 kg. This is significantly less than 24 kg. This means lesser load on the beams, columns and foundations which will result in significant reduction in their dimensions and cost saving.
Water absorption: Clay bricks are porous. Yet these should ideally have a minimum possible water absorption property. Lesser water absorption of bricks means better resistance to dampness. As per IS guidelines, bricks should have a water absorption of less than 20 per cent when immersed in a bucket full of water for 24 hours. Hollow clay bricks fulfill this criterion for water absorption also.
Strength: Being hollow, the strength of hollow clay bricks is about 35 kg/cm2. These are, therefore, best suited for non-load bearing walls or in RCC-framed structure buildings. After the Bhuj earthquake, everybody wants to have a quake-resistant house. RCC-framed structures are thus becoming popular in housing also. For such structures, hollow clay bricks are ideally suited. When the beams are 8 to 9 inch thick, single hollow bricks of 8" width or 2 bricks of 4" width can be used. For 10" wide beams, a combination of 6" and 4" wide bricks can be used and so on.
Size variation: When we use conventional clay bricks, it is an important to check that there is no variation in their size. Such variations cause lots of problems to the masons who find difficulty in maintaining the lines and levels in masonry work. There is extra consumption of mortar in plaster work also. Hollow clay bricks are machine moulded and have no variation in size, thus eliminating this problem.
Efflorescence: Complaints of shora appearing on the walls of the houses and damaging the costly finishing work are common. This happens when the bricks contain free lime which erupts after sometime when the moisture soaked by the bricks during rainy season begins to dry. It evaporates, leaving the shora on the surface of walls. This can be avoided by selecting efflorescence free bricks. Hollow clay bricks are free of efflorescence. Their hollowness also helps in restraining the eruption of efflorescence, if any, on the walls.
Burnt or not? Like conventional clay bricks, hollow clay bricks are also burnt bricks. These are burnt at a high temperature of 900 C. This, in a way, makes them fire safe. In case of fire, these don't burn. In fire protection terms, these have a value of F90/F120 which means that the user gets a time of 90 to 120 minutes to escape before the fire harms these bricks.
Cost comparison: The cost of 8 normal bricks at the present market rate works out as about Rs 32 to 35. One hollow clay brick costs about Rs 32 at production site. Depending upon the lead involved, the cost of hollow bricks at site can be worked out and compared with that of normal bricks. There is less consumption of cement mortar when hollow clay bricks are used because of their bigger size. So these are not too expensive.
Colour: Hollow clay bricks are also reddish in colour like the normal bricks. While there is a colour variation in conventional bricks, hollow clay bricks have a strikingly uniform colour.
Thermal insulation: A major plus point of hollow clay bricks is their thermal insulation property. Perforations in these bricks act as air cavities and there can't be a better way of having thermal insulation than creating an air cavity. During summers, these help in keeping the house cool and cut AC bills.
What is desirable: Presently available hollow clay bricks are horizontally perforated. These need to be made vertically perforated. Once that is done, these bricks shall become load bearing and can even help in eliminating some RCC beams and columns in houses.
(This column appears fortnightly)
Venkat Subramanian
STYLISH AND FUNCTIONAL: False ceilings add a touch of exclusivity to any space, be it in homes or in offices
Ceiling décor is an important aspect of interior design and can play a vital role in defining the look and the feel of a room. Having a false ceiling is just one way to enhance a room's aesthetics. It is no surprise that false ceilings are slowly forming an integral part of any interior design for new and old constructions.
False ceilings are suspended a few inches below the structural ceiling on a metal framework. Though having a false ceiling may appear as a slightly expensive proposition at first, it has distinct functional as well as aesthetic advantages which make them well worth the money.
Why go for a false ceiling
On a functional note, false ceilings offer excellent insulation against heat due to the air gap between the two ceiling layers. Also due to a smaller air volume, your power bills due to air-conditioning will be reduced significantly.
On the aesthetics front, all your services like wiring, ducting and insulation can be concealed within the false ceilings. Lights can be inset into a false ceiling, which prevents having to dust them while providing you with a clean, level surface below. False ceilings also offer a smooth finish, which improves the light reflectance off the ceilings. This is turn makes better use of natural lighting by day and the need for artificial light sources.
Easy on maintenance, functional and aesthetic appeal makes false ceilings the most practical ceiling décor.
Building a good false ceiling
Both gypsum plasterboards and POP boards have been used in the construction of false ceilings in India. In most of office and commercial interiors, gypsum plasterboard based false ceilings are used as these are faster to construct and are stronger than the traditional POP ceilings. Gyproc gypsum plaster board ceilings can support light fixtures as these are not brittle like POP ceilings.
A plain false ceiling with genuine Gyproc metal framing and accessories will cost approximately Rs 65-70 per sq ft installed while a designer ceiling variant with patterns will cost about Rs 90-100 per sq ft installed depending on the design variations.
The writer is MD, Saint Gobain Gyproc
Confederation of Real Estate Developers' Association of India (Credai), which has over 10,000 builders as member across the country, has decided that in the next six months all the members would individually sign a Code of Conduct.
The major highlight of Credai's Code of Conduct are — mentioning the actual usage area to buyers (Carpet area), compensation in case of project delay and honouring of the agreement between the two parties.
"We will ensure that all the Credai members sign the Code of Conduct within six months. In Code of Conduct, we have made mandatory for members to mention Carpet area along with super area," Credai Chairman Pradeep Jain told reporters.
Generally in North India, developers sell on the basis of super area, which comprises the entire built up area and markup for common spaces like lifts and stairs.
Credai President Lalit Kumar Jain said the association would not compromise on transparency and discipline. The industry body also decided to set up consumer redressal forum across the country to resolve disputes.
"If we find somebody engaging in malpractices then we will terminate the membership of that particular builder," he said.
The industry body has planned to expand to smaller cities in Assam, Orissa, Jharkhand and Rajasthan in the next three months, he added.
Jain said the Credai has sought time from Prime Minister Manmohan Singh to make a presentation on solution to issue related to transparency in the real estate sector.
Last month, Singh had said stamp duty needed to be reduced in order to check flow of black money into real estate sector, aiming to bring in more transparency.
"There is no black money in primary transaction, it is basically there in the secondary transactions... To reduce the usage of black money, single window clearance is important," Jain said. — PTI
S. C. Vasudeva
Q. My husband bought a plot on which we constructed a house in 2000 by taking loan from his department. He died in 2006 leaving me and my daughter (major) as the legal heirs of the property. We have sold the house for Rs 50 lakh.
My queries are as follows:
n Is it necessary to buy a residential property in exchange of the house sold Or Can we buy a plot also?
n I and my daughter should have a joint account to get DD from the buyer or should we take separate DDs in separate accounts? Is it necessary to buy new property jointly or can I buy it solely in my daughter's name?
n If we don't spend all the money what are the tax liabilities of remaining money? — Mrs Shashi
A. Your queries are replied hereunder:
n The capital gains tax would be exempt from tax only if you construct a residential house within the period of three years from the date of sale of the old house or buy a residential house within one year before or two years after the date of sale of the old residential house. If you have intentions to construct a new house, buying of the plot can be a process towards the same purpose. However, land must be purchased before the date of filing of the Income Tax Return or the capital gain arising on the sale of the house will have to be deposited in a designated bank account under capital gains scheme. You will have to withdraw the money from the said account for acquisition of the plot and for constructing a new residential house thereon.
n It will be advisable to take a separate demand draft in your name as well as in the name of your daughter because the capital gain will be assessable in your hands as well as in your daughter's hand separately. Each one of you can invest the capital gain so earned separately in the acquisition of a new house or buying of infrastructure bonds for the purposes of saving capital gains tax. The bonds have to be purchased within six months of the date of sale of the residential house.
n The capital gains would be taxable in your hands as well as in your daughter's hand @ 20% thereof plus applicable education cess thereon.
Q. My father had mortgaged his house during his lifetime for obtaining funds for the marriage of my sister. The house has been inherited by me with the mortgage. What would be the position of the mortgage amount? Will it be treated as part of the cost of acquisition for the purpose of computing capital gains tax? — Ashmeet
A. The Supreme Court has held that where a mortgage was created by the previous owner during his life time and the same is subsisting on the date of his death, the successor obtains only the mortgagor's interest in the property. Thus by discharging the mortgage debt he acquires the mortgagee's interest in the property and, therefore, the amount paid to clear off the mortgage is the cost of acquisition of the mortgagee's interest in the property which is deductible as cost of acquisition under Section 48 of the Income-tax Act, 1961 (The Act) - (VSMR Jagdishchandran v. CIT) [1997] 93 Taxman 389 (SC).
Q. I am going to sell a plot having a pucca two-room structure for which I have received commitment money (byana). But there is no mention of the structure in the byana agreement. The agreement includes a clause to execute a sale deed within six months from the date of agreement. Please clarify if I can include the two-room structure in the sale deed to enable me to purchase a flat in Panchkula with the amount of money received from the sale of the plot. Will I get exemption from capital gains tax liability as a result of the sale of the plot? — Rajesh Jasra
A. You have not clarified in the query whether the "pucca two-room structure" is a residential house. The term residential house has not been defined in the Income Tax Act, 1961 (The Act), but the courts have held that a residential house referred to in Section 54 and 54F of the Act should be such as is habitable. Therefore, in case such a structure is a residential house and you intend to seek exemption from the taxability of the capital gain arising on the sale of such residential house, you would be required to utilise the amount of capital gain only for the purchase or construction of a new residential house. However, in case the pucca structure referred to in the query is not a residential house, the entire amount of 'Net Consideration' would be required to be utilised for the purchase or construction of a residential house. 'Net consideration' for this purpose means the consideration for the sale of the capital asset received or accruing less expenditure incurred wholly and exclusively in connection with the sale of the capital asset.
It may be added that for the purpose of seeking exemption, the purchase of a new residential house is required to be effected within one year before or two years after the date of sale of capital asset and the construction is to be effected within three years after the date of sale of the capital asset. Further, if the amount of capital gain or net consideration, as the case may be, is not utilised towards the above purpose before the due of date of filing income-tax return, it is required to be deposited in a designated account with a bank under capital gain scheme account. The amount so deposited can be utilised for the purchase or construction of a new residential house within the period as aforesaid.
The writer can be contacted at sc@scvasudeva.com
Ruchika M. Khanna
Taneja Developers and Infrastructures Ltd. (TDI) group that is coming up with two township projects in Mohali recently launched its housing project "Tuscan Residency" in TDI City in Sector 110-111, Mohali on Landran-Banur road. The real estate major is entailing an investment of Rs 750-800 crore in its two township projects in Mohali. One of the township projects is spread across 300 acres and the other over 150 acres. The company is looking at an investment of Rs 500 crore in its first project (Sectors 117-119) over the next three years and might expand this township to an area of 500 acres.
Tuscan Residency will offer ergonomically designed individual floors as G+2 on a plot size of 200 sq yards. Strategically located at the entrance of TDI City, this will host an array of international standard amenities and features. "The advantage of these houses is affordability, with no EMI, no interest till offer of possession is given. The customers will have to pay only 20 per cent and the remaining amount is to be paid after offer of possession," said Sanyam Dudeja, COO, TDI.
Besides, Mohali, the realtor is also coming up with a mall in Sector 17, Chandigarh. "This mall is now complete and we hope to open it by August this year," said Dudeja. Dudeja said that they were also looking at developing a commercial project in Jalandhar. We have a one acre prime piece of land in Jalandhar, and we are proposing to come up with a mall-cum-hotel there," he said.
Real estate developer BPTP Ltd has launched its second premium township Astaire Gardens at upcoming Sector 70 A, off NH-8 in Gurgaon.
"Spread across 102 acres, Astaire Gardens is an eclectic mix of contemporary design comprising plots, independent floors and villas, large plot sizes, 24x7 facilities, security, a state-of-the-art health centre, schools, a creche, sanctuary and the club," BPTP Ltd Director (Strategy & Systems) Sandeep Bedi said.
"We are the first real estate player to launch two unique townships in Gurgaon in such a short span of time. We are very enthused by the success of our first township, Amstoria, and we are confident Astaire Gardens will get similar response," he said.
"Astaire Gardens comprise 45 per cent open and green areas and is ideally located off the NH-8 and Golf Course Extension Road with a panoramic view of the Aravallis," the company's Senior Vice-President (Marketing) Amit Raj Jain said. With an entry threshold of Rs 60 lakh, the property is most competitively priced, Jain added. — PTI
Empire Estate
The Logix Group has launched its residential project - Empire Estate - at Yamuna Expressway, Greater Noida. Spread over 200 acres, Empire Estate will be an exclusive gated community offering residential plots ranging from 100 to 500 sq. yd. The plots measuring 100, 200, 300 & 500 sq yards have been priced Rs 14.85 lakh onwards.
Speaking on the occasion, Shaktinath, MD, Logix Group said, "After creating landmarks in IT commercial projects, Logix group is catering to the rising demand of residential plots, especially in the NCR. The demand for plots is high here due to better infrastructure, job opportunities, high disposal income and better connectivity. Moreover, Yamuna Expressway has high potential to generate traffic and economic development."
The community will have amenities like club house with recreational facilities, 3 tier security, kids play area, school, nursing homes, high-end resort, swanky shopping malls and healthcare facilities. | {
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Ghislaine Maxwell Photo at In-N-Out Burger Joint Shows Signs of Photoshop/Manipulation
There is something sinister going on as it relates to the mainstream reporting on the Jeffrey Epstein case. We are seeing a coordinated effort at presenting a pre-scripted narrative and a total lack of 'mainstream' investigative journalists doing deep dives into Epstein's background and connections to various intelligence agencies. The mainstream press is in full blown cover-up mode of what may be the closest thing to the JFK scandal of our time. We are talking about a case that could have potentially implicated some of the worlds most powerful politicians, academics, diplomats and leaders of industry.
If you don't think the global media is coordinated, and won't descend on the journalists doing the real reporting on a massive scandal that is so well documented, please remember the story of Gary Webb.
"Gary Webb, the Pulitzer prize-winning reporter who broke the story of the CIAs involvement in the importation of cocaine into the U.S., died December 10, 2004, reportedly from self-inflicted gunshots to the head. It was a tragic end to a brilliant, and tragic, career.
In August 1996, the San Jose Mercury News published Webbs 20,000 word, three-part series entitled Dark Alliance. The articles detailed the nexus between a California coke kingpin, CIA officials and assets and the Nicaraguan Contra army, whose funding had been cut off by an act of Congress in the mid-80s.
Ironically, the CIA did little to publicly counter his allegations. Instead, the media did its dirty work for them, most notably the Los Angeles Times and the Washington Post. The mainstream media accused Webb of exaggerating his findings."
As Gary himself explains, some of the so-called reporters at the Washington Post had worked for the Central Intelligence Agency themselves. He explains how changes to the media landscape had made it very difficult for a small local newspaper to be able to withstand the lawsuits and scrutiny that come from breaking such stories. The major mainstream publications will not deviate from the narrative the establishment wants presented to the country. They began to publish hit-pieces critiquing every little thing about the story and Gray personally, to try to discredit the reporting despite the overwhelming evidence, testimony and documentation.
We have been getting a deluge of information about the Jeffrey Epstein case, but little of actual substance. Some court documents, with pages missing, that clearly had already been sanitized; we got reporting from 'anonymous sources' at the prison; we get the national media trying to connection Epstein to Trump despite the evidence that shows Trump severed ties with Epstein long ago and assisted investigators the first time Epstein was arrested.
And on and on it goes to distract from the real story. To have people focus on partisan issues that really shouldn't have anything to do with child predators, so they don't look into all the people Epstein was friends with. The people he socialized with. The people who covered for him. The people who wrote puff pieces for him to try to rehabilitate his image after his first arrest.
Anything to keep you distracted while they try to destroy evidence and slow walk things until the pubic forgets about it and moves on. They use their assets in the alternative media to misdirect you from what Epstein was actually doing and who he was really working for. The Epstein case indeed will be a litmus test for those in the alternative and independent media.
The establishment uses assets in the alternative media to run snow jobs and limited hangouts. The purpose of this is to flood the internet with all kinds of information, some true, some false in an attempt to muddy the waters and make fact and fiction harder to discern. They will trot these people out and say 'they are the hero's reporting on the Epstein case.' This is because they already know the public does not trust the mainstream media anymore and will look to alternative media to figure out what is going on with the case.
According to Wikispooks:
"A limited hangout is the deliberate revelation of some information (e.g. about malfeasance) to try to confuse and/or prevent discovery of other information. A modified limited hangout goes further, by slightly changing the information disclosed."
The establishment is currently doing this with Vicky Ward. She is a writer at the Daily Beast and she used to work for Vanity Fair. She had written a profile on Epstein that was essentially a puff piece. She now claims she had received stories from girls about Epstein she wanted to publish but her editor said no. So she blames the editor and tells a fantastical story about how she was so terrified of Epstein when she was delivering her child she wanted to have security to protect her.
From her article in the Daily Beast entitled 'I Tried to Warn You About Sleazy Billionaire Jeffrey Epstein':
"Why?" I asked Graydon. "He's sensitive about the young women" was his answer. "And we still get to run most of the piece."
Many years later I know that Graydon made the call that seemed right to him then—and though the episode still deeply rankles me I don't blame him. He sits in different shoes from me; editors are faced with these sorts of decisions all the time, and disaster can strike if they don't err on the side of caution.
It came down to my sources' word against Epstein's… and at the time Graydon believed Epstein. In my notebook I have him saying, "I believe him… I'm Canadian."
Today, my editor at The Daily Beast emailed Graydon to ask why he had excised the women's stories from my article. A Vanity Fair spokeswoman responded: "Epstein denied the charges at the time and since the claims were unsubstantiated and no criminal investigation had been initiated, we decided not to include them in what was a financial story." – The Daily Beast
And yet there are numerous photographs of Vicky Ward as recently as 2015, with Ghislaine Maxwell at parties were the woman seem very friendly with one another. This seems a far cry from the statement she made in her story that:
"He has a way of spooking you, does Epstein. Or he did. My babies were born prematurely, dangerously so; he'd asked which hospital I was giving birth at—and I was so afraid that somehow, with all his connections to the academic and medical community, that he was coming for my little ones that I put security on them in the NICU."
If Vicky was so terrified of Jeffrey Epstein, why would she associate with Epstein's alleged madam? From Amazing Polly on YouTube:
And now we get to the obviously photo-shopped picture of Ghislaine Maxwell allegedly taken in LA at the In-N-Out Burger joint. The article and photos first appeared in a now deleted Mail Online story from August 16, 2019. That article has been archived here.
This came two (2) days after it was reported by the Boston Globe on August 14, 2019 that Ghislaine Maxwell was staying in an affluent Boston community with her Tech CEO boyfriend.
A researcher who is well-versed in photographic manipulation does a good run-down on how we can tell these 2 pictures of Ghislaine have been manipulated:
and;
Epstein-The Only Way Out is In
1) The first sighting of Ghislaine Maxwell was at "In N Out" Burger in LA. The Cabal loves their symbolism. There were 4 different photo's released by media, with some clear Cabal messaging in the photo's. #Epstein #PedoGate pic.twitter.com/eBsxJb67Fg
— Qanuck (@TrueQanuck11) August 18, 2019
2) The New York Post reported Ghislaine Maxwell was having a burger, fries and milkshake at a branch of In-N-Out Burger while reading "The Book of Honor: The Secret Lives and Deaths of CIA Operatives." The newspaper said she was sitting with her pet dog. pic.twitter.com/DuNxFvDsDQ
4) The photo's were definitely a set up photo shoot, with very odd messaging. The first thing I noticed, was Maxwell appears to be alone, there's a second drink on the table. The MSM claimed an anonymous patron noticed her, and she gave permission for him to take the photo's. pic.twitter.com/z9VIZJUESh
9) The Ghislaine photo that's similar to other Cabal Messages, is the sign she posed in front of that says "Good Boy." Perhaps it's a message, somebody isn't being a VGB? ? @IPOT1776 did a great video on these coded Canine messages. https://t.co/GyrWZBuZIv
It doesn't matter what the purpose of the manipulation is, just that we know its there. There is no point speculating about whether this is coded communication or something else. The point is that we are being manipulated and we are being fed a narrative while other aspects of this story are actively being buried. We cannot afford to let them bury this story and the truth with it.
Tags: Brian Epstein, Crime in the United States, Criminals, Epstein, Ghislaine Maxwell, Jeffrey Epstein, Nationality
The Age of Intolerance: Cancel Culture's War on Free Speech January 12, 2022 at 11:40 pm | {
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"redpajama_set_name": "RedPajamaC4"
} | 5,371 |
{"url":"https:\/\/github.com\/fletcher\/peg-multimarkdown-latex-support\/tree\/master","text":"# fletcher\/peg-multimarkdown-latex-support\n\nDefault support files for generating LaTeX documents with MMD 3 through MMD 5\nTeX\nSwitch branches\/tags\nNothing to show\nLatest commit 6de3360 Oct 29, 2016 ADDED: Support strikethrough\n\nTitle: LaTeX support files for peg-multimarkdown\n\n# Introduction\n\npeg-multimarkdown is a program to convert plain text into HTML or LaTeX. This project includes some default template files that can be used to create certain types of documents using LaTeX.\n\nYou are not limited to using these classes or templates. You can create your own template files, or just embed your LaTeX commands within comments in your MultiMarkdown document itself. If you find yourself creating similar documents over and over again, however, you may be better off creating a few templates you can simply call with the LaTeX Input metadata fields in MultiMarkdown.\n\nThese files were designed to handle some of the common metadata fields in a consistent way, an to implement some defaults that should prevent errors if you leave out important metadata (substituting Title, Author, etc).\n\n# Installation\n\nThese files need to go in your texmf folder, wherever that may be.\n\nWith MacTeX on Mac OS X:\n\n~\/Library\/texmf\/tex\/latex\/mmd\n\n\nOn most *nix accounts, you can use:\n\n~\/texmf\/tex\/latex\/mmd\n\n\nI don't remember off the top of my head where your texmf folder belongs in Windows.\n\nSeveral MultiMarkdown metadata keys are used in these files, and are fairly self-explanatory:\n\n\u2022 Title --- Specify the title of the document\n\n\u2022 Author --- Specify the author of the document\n\n\u2022 Date --- Specify a date\n\n\u2022 Base Header Level --- Specify the maximum organizational level for the document (e.g. part, chapter, section, subsection). You need to choose a value for this that fits with the way you organized your document.\n\nMetadata is used in order, so the order and placement of the LaTeX Input metadata fields is important.\n\n# Article\n\nTo create a document using the memoir article class, you need the following basic metadata:\n\nlatex input:\t\tmmd-article-header\nTitle:\t\t\t\tWhatever Title You Like\nLaTeX Mode:\t\t\tmemoir\nlatex input:\t\tmmd-article-begin-doc\nlatex footer:\t\tmmd-memoir-footer\n\n\n# Beamer\n\nTo create a pdf slideshow presentation using beamer:\n\nlatex input:\t\tmmd-beamer-header\nSubtitle:\t\t\tSome optional subtitle\nAffiliation:\t\tYour institution, web site, whatever\nLaTeX Mode:\t\t\tbeamer\nlatex input:\t\tmmd-beamer-begin-doc\nlatex footer:\t\tmmd-beamer-footer\n\n\nThere are several beamer themes included that are derived from various keynote themes --- keynote-gradient, keynote-vintage, keynote-portfolio. I tweaked these themes to work with MultiMarkdown, but they were originally created by others (see the source files for details).\n\nThe header levels are set so that h1 is a part, h2 is a section, h3 is a slide, and h4 is used to designate text that will print in a handout, but not in the actual slideshow.\n\nTo create a letter on customized letterhead using MultiMarkdown:\n\nlatex input:\t\tmmd-letterhead-header\nTitle:\t\t\t\tTest Letter\nAuthor:\t\t\t\tJohn Doe\nemail:\t\t\t\tfletcher@example.net\nSome City, ST 12345\nrecipient:\t\t\tSome Person\nSome City, ST 54321\nphone:\t\t\t\t(555) 555-5555\nDate:\t\t\t\tDecember 15, 2007\nblack and white:\ttrue\nlatex mode:\t\t\tmemoir\n\n\nIf you want to create an envelope using the same document, simply change the last line of the metadata:\n\nlatex input:\t\tmmd-envelope-begin-doc\n\n\n# Memoir\n\nTo create a \"book\" using memoir:\n\nlatex input:\t\tmmd-memoir-header\n\nHeader levels are: h1 part, h2 chapter, h3 section, h4 subsection, h5 subsubsection, and h6 paragraph.","date":"2017-11-19 08:57:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.28182363510131836, \"perplexity\": 7007.184553157982}, \"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-2017-47\/segments\/1510934805466.25\/warc\/CC-MAIN-20171119080836-20171119100836-00569.warc.gz\"}"} | null | null |
Мост Теодора Хойса () — мост через Рейн, соединяющий центральную часть города Майнц (столицы немецкой земли Рейнланд-Пфальц) с районом Майнц-Кастель, ныне принадлежащим городу Висбаден (столице земли Гессен).
Строительство
Рейнский мост () был построен в 1882—1885 годах по проекту архитекторов Фридриха фон Тирша и . Затраты на строительство составили 3,6 миллионов золотых марок. Торжественное открытие моста состоялось 30 мая 1885 года.
В 1931—1934 годах была проведена реконструкция, включающая в себя расширение моста (с 13,80 м до 18,80 м).
17 марта 1945 года, незадолго до того, как Майнц был занят американскими войсками, мост был взорван отступающими фашистскими отрядами. После этого американцы наладили понтонную переправу.
После войны Рейнский мост был восстановлен (в 1948—1950 годах), и в 1950 году получил имя Теодора Хойса () — первого президента послевоенной Германии.
В 1992—1995 годах была проведена капитальная реконструкция моста, которая обошлась в 139,5 миллионов марок. Мост был вновь открыт для движения 18 июля 1995 года.
Конструкция
Мост Теодора Хойса — арочного типа. Он включает в себя четыре опоры из песчаника, находящиеся в русле реки, и пять арочных пролётов длиной — — — — .
По мосту проходит автомобильная дорога , включающая в себя четыре полосы для автомобильного движения (по две в каждую сторону), по краям которых находятся пешеходные и велосипедные дорожки.
См. также
Майнц
Рейн
Список Рейнских мостов
Теодор Хойс
Примечания
Галерея
Арочные мосты Германии
Здания и сооружения Висбадена | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 47 |
{{Titre mis en forme|Épisodes dIntelligence}}
{{Infobox Saison de série télévisée
| nom = Épisodes dIntelligence | image =
| légende =
| série = Intelligence
| pays =
| chaine = CBS
| première diffusion =
| dernière diffusion =
| nombre épisodes = 13
| saison suivante =
| liste des épisodes =
}}
Cet article présente le guide des épisodes de la série télévisée américaine Intelligence.
Généralités
Le , la série a été officiellement commandée par le réseau CBS.
Au Canada, la série est diffusée en simultané sur le réseau CTV. La diffusion des quatre derniers épisodes a été transférée sur CTV Two.
En France, la série sera diffusée sur M6.
La série est inédite dans tous les autres pays francophones.
Distribution
Acteurs principaux
Josh Holloway : Gabriel Vaughn
Meghan Ory : Riley Neal
Marg Helgenberger : Lillian Strand
Michael Rady : Chris Jameson
John Billingsley : Shenendoah Cassidy
P. J. Byrne : Nelson Cassidy
Acteurs récurrents et invités
Tomas Arana : Adam Weatherly
Lance Reddick : DCI Jeffrey Tetazoo
Will Yun Lee : Jim Cong (épisode 1 et 9)
James Martinez : Gonzalo « Gonzo » Rodriguez (épisode 1)
Zuleikha Robinson : Amelia Vaughn (épisode 1 à 3)
Annie Wersching : Kate Anderson (épisode 3)
Peter Coyote : Leland Strand, père de Lillian (épisodes 5, 12 et 13)
Laura Slade Wiggins : Rebecca, fille de Lillian (épisode 10)
Épisodes
Épisode 1 : Opération clockwork
Épisode 2 : La Puce ou la Vie
Épisode 3 : Le Retour de Mei Chen
Épisode 4 : Champ de trèfle
Épisode 5 : Échange de bons procédés
Épisode 6 : Le Patient zéro
Épisode 7 : L'Infiniment Petit
Épisode 8 : Delta force
Épisode 9 : La Liste d'Athènes
Épisode 10 : Panique à San Francisco
Épisode 11 : Un trait de génie
Épisode 12 : Les Six tigres (1/2)
Épisode 13 : Les Six tigres (2/2)
Audiences aux États-Unis
Notes et références
Intelligence | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,279 |
The Natural Park of Sierras de Tejeda, Almijara and Alhama is located within the provinces of Malaga and Granada on the southern coast of Andalucía. Tejeda is the most westerly of the three ranges, in the heart of an area named by the Arab settlers as the Axarquía, whereas Almijara, the largest is on the eastern edges of Málaga, directly overlooking the Mediterranean Sea. In contrast, the part of the Natural Park known as Alhambra, is landlocked, and located within the western reaches of the province of Granada. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,574 |
#import <Foundation/Foundation.h>
@class RPSLSUserStats;
@class MMXPubSubMessage;
@class MMXMessage;
@class MMUser;
@interface RPSLSUser : NSObject
@property (nonatomic, strong) RPSLSUserStats * stats;
@property (nonatomic, strong) MMUser * messageUserObject;
@property (nonatomic, strong) NSDate * timestamp;
@property (nonatomic, assign) BOOL isAvailable;
+ (instancetype)userWithUserObject:(MMUser *)userObject
stats:(RPSLSUserStats *)stats;
+ (RPSLSUser *)me;
+ (NSString *)myUsername;
+ (instancetype)availablePlayerFromMessage:(MMXMessage *)message;
+ (instancetype)playerFromInvite:(MMXMessage *)message;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,516 |
Riete Loos est une joueuse de football belge née le à Neerpelt (Belgique).
Biographie
Elle a joué au FC Helson Helchteren, au Standard de Liège et au KRC Genk Ladies. En , Riete Loos annonce la fin de sa carrière de joueuse. Riete wil Nele bedanken voor alle steun en toeverlaat tijdens haar zware voetbaljaren.
Palmarès
Championne de Belgique (4) : 2009 - 2011 - 2012 - 2014
Vainqueur de la Coupe de Belgique (1) : 2014
Finaliste de la Coupe de Belgique (2) : 2009 - 2018
Vainqueur de la Super Coupe de Belgique (3) : 2009 - 2011 - 2012
Doublé Championnat de Belgique-Coupe de Belgique (1) : 2014
Doublé Championnat de Belgique-Super Coupe de Belgique (3) : 2009 - 2011 - 2012
Triplé Championnat de Belgique-Super Coupe de Belgique-BeNe SuperCup (1) : 2011
Vainqueur de la BeNe SuperCup (1) : 2011
Bilan
9 titres
Statistiques
Ligue des Champions
2009-2010 : 2 matchs
2013-2014 : 1 match
Notes et références
Liens externes
Footballeuse internationale belge
Joueuse du Standard de Liège
Joueuse de Ladies Genk
Naissance en juin 1990
Naissance à Neerpelt | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,689 |
Okay. So 4 months since the pictures above, here are the sunfish as of today.
The largest male is 3.5 inches, second male around 3'', and the females around 2''.
UncleWillie... Those are BEAUTIFUL fish! Do you intend to breed them, or are you just going to enjoy their stunning colors?
I think if people saw the fish before thinking they only wanted to keep tropicals... they might change their minds. I've definitely changed mine... it is now just a matter of changing my husband's.
Where do you find information on fish temperament? We used to catch sunnies all the time near my house, but I have no idea what max size they reach or how to know if they are aggressive, or even if they need to shoal.
Rales, I may breed them one day, but I don't have the time for it now. They are exactly 1.5 years old, so I have plenty of time to get some spawning in once I get settled somewhere more permanent. I will likely be moving in about a year or so and I have to fight setting up more tanks .
jsuereth, you can always ask me. Hehe, but seriously, you can find decent info on the North American fish in aquaria over on the NANFA forum. There are many sunfish, of a huge range of size and temperament. Most of the sunfish of the Lepomis genus are all relatively large and and aggressive. The exception are orangespotted and bantam sunfish. They rarely exceed 3-4.5'' and are not aggressive. These dollars also stay in the 3-4.5 inch range, but are nasty and very aggressive. Then you've got your mid-sized (5-9'') Lepomids like pumpkinseed, green, longear, spotted/redspotted. Then the larger ones (9-13'') like bluegill, redear, redbreast. You can basically directly compare sunfish to cichlids. They are aggressive, need space, build and defend nests and territory, and the males are great parents.
They are lovely fish & I wish I could keep some. They are illegal here.
Thanks again, yall. It is a shame that you cannot keep some Diademhill, but I reckon it is for the best. I know that many of the North American sunfish species (mostly green sunfish, pumpkinseed, bluegills and redears) have been introduced to Europe and Asia and have established populations. It is understandable that such regulations exist as all of these fish are temperate species and survive bitter winters as long as a lake does not freeze completely.
I want to post a few other pictures of sunfish I have caught over the last year, but this may be more appropriate in the AquaLounge as these are not fish in aquaria. Maybe I'll upload some pictures this morning while I prepare for the holiday! Maybe I can have some pics on the board before the new year.
You've gotten me very interested in keeping native species! Once I get my big tank (I'm shooting for something around 90-120 gallons) I'll have to research my native species and see if any would be suitable for a tank that size. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,966 |
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November 2022 / 5 Min Read
Driving the Deal: The Digital Evolution and the Role of the Technology, Media and Telecommunications Sector
Multiple industries are increasingly reliant on technology, driving M&A activity across the technology, media and telecommunications sector as a catalyst for change.
The digital revolution is driving a rapid uptick in M&A deal activity across the TMT sector, bringing both risks and opportunities to organizations.
The appetite from private equity investors is growing at a rapid pace as organizations seek to remain competitive and enhance their core capabilities.
Top emerging trends include cryptocurrency, disruptive technologies and artificial intelligence.
2021 and H1/2022 saw an accelerated rate of merger and acquisition (M&A) activity in the technology, media and telecommunications (TMT) sector. TMT continues to be an out-performer and this isn't expected to change any time soon. A look at Mergermarket's published "company for sale" stories show the sector is way out in front in terms of potential deals in the making1.
The TMT sector was the most active globally in 2021 - in both deal value and volume terms.
Source: White & Case 2022
In Aon's recent Risk in Review survey, almost three-quarters of respondents (70%), by far the largest share, cite technology, media & telecoms (TMT) as the most prolific sector in terms of expected dealmaking over the next 12 months.
M&A is Driving Growth
The digital revolution continues to gather momentum. Multiple industries are becoming increasingly reliant on technology, driving an uptick in M&A activity across the TMT sector. The appetite from private equity investors is growing at a rapid pace as organizations seek to remain competitive and find new ways to enhance their core capabilities - while seizing opportunities for growth. Meanwhile, global M&A activity remains very significant, with investments triggered by initial public offerings (IPOs) and venture capital (VC) funding.
The current deal appetite reflects the growth potential in an increasingly virtual world. Next-generation innovations have prompted a surge of opportunities, supported by an increase in new investments across the entire digital infrastructure.
Driving the Deal: Navigating Global Trends for the TMT Sector
Driving the Deal: Transaction Risks and Opportunities for the TMT Sector
Technology, Media and Telecommunications: A Focus on M&A in 2022 and Beyond
M&A Trends: An Industry Focus
As industries continue to recover from the disruption caused by the COVID-19 pandemic, the shift in consumer behavior continues to drive many organizations to reassess their operations. Consumers increasingly rely on online services, and this changing behavior has boosted deal activity, particularly for digital platforms such as online marketplaces and consumer comparison tools.
Technology implementation is also accelerating across retail and consumer-focused industries, where digital engagement is crucial throughout the supply chain as well as at "point of sale". Meanwhile, the fintech-led shake-up of financial services and "medtech" in healthcare is also gathering momentum.
Buyers are looking to accelerate acquisitions around artificial intelligence (AI), cloud transition (applications, connectivity and security) and the internet of things (IoT).
Looking ahead, technology will continue to act as an enabler for growth for many sectors. Technologies such as quantum computing, space exploration and energy storage are moving towards commercialization and are seeing more VC investment, IPO and M&A activity. In pursuing M&A activity, businesses should prepare for increased sovereignty and government scrutiny on data, privacy and cryptocurrency laws in cross-border deals.
Advertising revenues were down during the pandemic's peak, with shows canceled and many cinemas closed during lockdowns. More traditional deal activity has been strong in some growth markets, including gaming and streaming. Access to public markets and private capital are needed to facilitate balance sheet repair and support growth, driving an uptick in deal activity more recently.
M&A is playing a key role in strengthening capacity and future-proofing capabilities within telecoms, in areas such as fiber rollout, 5G and SD-WAN-enabled infrastructure. Consolidation of network providers is already following, with fast buildouts of fibre infrastructure by independent players, and bringing one or more of these providers together would offer greater economies of scale.
An Outlook: Top Emerging M&A Trends
1 Crytocurrency
2Disruptive technologies
3 Artificial intelligence
As a major conduit for change, the TMT sector will enable multiple industries to revolutionize their operations and accelerate innovation.
1 Datasite & Mergermarket, 2022
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,913 |
Q: Add network to Qt Project I try to use the QtNetwork library and added all dependencies for it in the .pro file. But when I compile my code Qt Creator fails in building the project and claims
C1083: Include "QTcpSocket": No such file or directory - telnet.h:4
I thought adding network to the .pro file would be enough?
networkmonitor.pro
#-------------------------------------------------
#
# Project created by QtCreator 2017-07-24T13:18:19
#
#-------------------------------------------------
QT += core gui network charts
# greaterThan(QT_MAJOR_VERSION, 4): QT += widgets
TARGET = networkmonitor
TEMPLATE = app
# The following define makes your compiler emit warnings if you use
# any feature of Qt which as been marked as deprecated (the exact warnings
# depend on your compiler). Please consult the documentation of the
# deprecated API in order to know how to port your code away from it.
DEFINES += QT_DEPRECATED_WARNINGS
# You can also make your code fail to compile if you use deprecated APIs.
# In order to do so, uncomment the following line.
# You can also select to disable deprecated APIs only up to a certain version of Qt.
#DEFINES += QT_DISABLE_DEPRECATED_BEFORE=0x060000 # disables all the APIs deprecated before Qt 6.0.0
SOURCES += \
main.cpp \
mainwindow.cpp \
telnet.cpp
HEADERS += \
mainwindow.h \
telnet.h
FORMS += \
mainwindow.ui
telnet.h
#ifndef TELNET_H
#define TELNET_H
#include <QTcpSocket>
#include <QTcpServer>
#include <QDebug>
class Telnet
{
public:
Telnet();
void sendValues(QString _ip, int _port, QString _message);
private:
QTcpSocket *socket;
};
#endif // TELNET_H
A: Normally this means you just forgot to re-run qmake
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,287 |
{"url":"https:\/\/collegemathteaching.wordpress.com\/category\/physics\/page\/2\/","text":"# College Math Teaching\n\n## August 11, 2011\n\n### Quantum Mechanics and Undergraduate Mathematics XIII: simplifications and wave-particle\u00a0duality\n\nIn an effort to make the subject a bit more accessible to undergraduate mathematics students who haven\u2019t had much physics training, we\u2019ve made some simplifications. We\u2019ve dealt with the \u201cone dimensional, non-relativistic situation\u201d which is fine. But we\u2019ve also limited ourselves to the case where:\n1. state vectors are actual functions (like those we learn about in calculus)\n2. eigenvalues are discretely distributed (e. g., the set of eigenvalues have no limit points in the usual topology of the real line)\n3. each eigenvalue corresponds to a unique eigenvector.\n\nIn this post we will see what trouble simplifications 1 and 2 cause and why they cannot be lived with. Hey, quantum mechanics is hard!\n\nFinding Eigenvectors for the Position Operator\nLet $X$ denote the \u201cposition\u201d operator and let us seek out the eigenvectors for this operator.\nSo $X\\delta = x_0 \\delta$ where $\\delta$ is the eigenvector and $x_0$ is the associated eigenvalue.\nThis means $x\\delta = x_0\\delta$ which implies $(x-x_0)\\delta = 0$.\nThis means that for $x \\neq x_0, \\delta = 0$ and $\\delta$ can be anything for $x = x_0$. This would appear to allow the eigenvector to be the \u201ceverywhere zero except for $x_0$\u201d function. So let $\\delta$ be such a function. But then if $\\psi$ is any state vector, $\\int_{-\\infty}^{\\infty} \\overline{\\delta}\\psi dx = 0$ and $\\int_{-\\infty}^{\\infty} \\overline{\\delta}\\delta dx = 0$. Clearly this is unacceptable; we need (at least up to a constant multiple) for $\\int_{-\\infty}^{\\infty} \\overline{\\delta}\\delta dx = 1$\n\nThe problem is that restricting our eigenvectors to the class of functions is just too restrictive to give us results; we have to broaden the class of eigenvectors. One way to do that is to allow for distributions to be eigenvectors; the distribution we need here is the dirac delta. In the reference I linked to, one can see how the dirac delta can be thought of as a sort of limit of valid probability density functions. Note: $\\overline{\\delta} = \\delta$.\n\nSo if we let $\\delta_0$ denote the dirac that is zero except for $x = x_0$, we recall that $\\int_{\\infty}^{\\infty} \\delta_0 \\psi dx = \\psi(x_0)$. This means that the probability density function associated with the position operator is $P(X = x_0) = |\\psi(x_0)|^2$\n\nThis has an interesting consequence: if we measure the particle\u2019s position at $x = x_0$ then the state vector becomes $\\delta_0$. So the new density function based on an immediate measurement of position would be $P( X = x_0) = |\\langle \\delta_0, \\delta_0 \\rangle|^2 = 1$ and $P(X = x) = 0$ elsewhere. The particle behaves like a particle with a definite \u201cpoint\u201d position.\n\nMomentum: a different sort of problem\n\nAt first the momentum operator $P\\psi = -i \\hbar \\frac{d\\psi}{dx}$ seems less problematic. Finding the eigenvectors and eigenfunctions is a breeze: if $\\theta_0$ is the eigenvector with eigenvalue $p_0$ then:\n$\\frac{d}{dx} \\theta_0 = \\frac{i}{\\hbar}p_0\\theta_0$ has solution $\\theta_0 = exp(i p_0 \\frac{x}{\\hbar})$.\nDo you see the problem?\n\nThere are a couple of them: first, this provides no restriction on the eigenvalues; in fact the eigenvalues can be any real number. This violates simplification number 2. Secondly, $|\\theta_0|^2 = 1$ therefore $|\\langle \\theta_0, \\theta_0 \\rangle |^2 = \\infty$. Our function is far from square integrable and therefore not a valid \u201cstate vector\u201d in its present form. This is where the famous \u201cnormalization\u201d comes into play.\n\nMathematically, one way to do this is to restrict the domain (say, limit the non-zero part to $x_0 < x < x_1$ ) and multiply by an appropriate constant.\n\nGetting back to our state vector: $exp(ip_0 \\frac{x}{\\hbar}) = cos(\\frac{p_0 x}{\\hbar}) + i sin(\\frac{p_0 x}{\\hbar})$. So if we measure momentum, we have basically given a particle a wave characteristic with wavelength $\\frac{\\hbar}{p_0}$.\n\nNow what about the duality? Suppose we start by measuring a particle\u2019s position thereby putting the state vector in to $\\psi = \\delta_0$. Now what would be the expectation of momentum? We know that the formula is $E(P) = -i\\hbar \\int-{-\\infty}^{infty} \\delta_0 \\frac{\\partial \\delta_0}{\\partial x} dx$. But this quantity is undefined because $\\frac{\\partial \\delta_0}{\\partial x}$ is undefined.\n\nIf we start in a momentum eigenvector and then wish to calculate the position density function (the expectation will be undefined), we see that $|\\theta_0|^2 = 1$ which can be interpreted to mean that any position measurement is equally likely.\n\nClearly, momentum and position are not compatible operators. So let\u2019s calculate $XP - PX$\n$XP \\phi = x(-i\\hbar \\frac{d}{dx} \\phi) = -xi\\hbar \\frac{d}{dx} \\phi$ and $PX\\phi = -i \\hbar\\frac{d}{dx} (x \\phi) = -i \\hbar (\\phi + x \\frac{d}{dx}\\phi)$ hence $(XP - PX)\\phi = i\\hbar \\phi$. Therefore $XP-PX = i\\hbar$. Therefore our generalized uncertainty relation tells us $\\Delta X \\Delta P \\geq \\frac{1}{2}h$\n(yes, one might object that $\\Delta X$ really shouldn\u2019t be defined\u2026.) but this uncertainty relation does hold up. So if one uncertainty is zero, then the other must be infinite; exact position means no defined momentum and vice versa.\n\nSo: exact, pointlike position means no defined momentum is possible (hence no wave like behavior) but an exact momentum (pure wave) means no exact pointlike position is possible. Also, remember that measurement of position endows a point like state vector of $\\delta_0$ which destroys the wave like property; measurement of momentum endows a wave like state vector $\\theta_0$ and therefore destroys any point like behavior (any location is equally likely to be observed).\n\n### Quantum Mechanics and Undergraduate Mathematics XII: position and momentum\u00a0operators\n\nFiled under: advanced mathematics, applied mathematics, physics, probability, quantum mechanics, science \u2014 collegemathteaching @ 1:52 am\n\nRecall that the position operator is $X \\psi = x\\psi$ and the momentum operator $P \\psi = -i\\hbar \\frac{d}{dx} \\psi$.\n\nRecalling our abuse of notation that said that the expected value $E = \\langle \\psi, A \\psi \\rangle$, we find that the expected value of position is $E(X) = \\int_{-\\infty}^{\\infty} x |\\psi|^2 dx$. Note: since $\\int_{-\\infty}^{\\infty} |\\psi|^2 dx = 1,$ we can view $|\\psi|^2$ as a probability density function; hence if $f$ is any \u201creasonable\u201d function of $x$, then $E(f(X)) = \\int_{-\\infty}^{\\infty} f(x) |\\psi|^2 dx$. Of course we can calculate the variance and other probability moments in a similar way; e. g. $E(X^2) = \\int_{-\\infty}^{\\infty} x |\\psi|^2 dx$.\n\nNow we turn to momentum; $E(P) = \\langle \\psi, -i\\hbar \\frac{d}{dx} \\psi \\rangle = \\int_{-\\infty}^{\\infty} \\overline{\\psi}\\frac{d}{dx}\\psi dx$ and $E(P^2) = \\langle \\psi, P^2\\psi \\rangle = \\langle P\\psi, P\\psi \\rangle = \\int_{-\\infty}^{\\infty} |\\frac{d}{dx}\\psi|^2 dx$\n\nSo, back to position: we can now use the fact that $|\\psi|^2$ is a valid density function associated with finding the expected value of position and call this the position probability density function. Hence $P(x_1 < x < x_2) = \\int_{-\\infty}^{\\infty} |\\psi|^2 dx$. But we saw that this can change with time so: $P(x_1 < x < x_2; t) = \\int_{-\\infty}^{\\infty} |\\psi(x,t)|^2 dx$\n\nThis is a great chance to practice putting together: differentiation under the integral sign, Schr\u00f6dinger\u2019s equation and integration by parts. I recommend that the reader try to show:\n\n$\\frac{d}{dt} \\int_{x_1}^{x_2} \\overline{\\psi}\\psi dx = \\frac{ih}{2m}(\\overline{\\psi}\\frac{d \\psi}{dx}-\\psi \\frac{d \\overline{\\psi}}{dx})_{x_1}^{x_2}$\n\nThe details for the above calculation (students: try this yourself first! \ud83d\ude42 )\n\nDifferentiation under the integral sign:\n$\\frac{d}{dt} \\int_{x_1}^{x_2} \\overline{\\psi} \\psi dx = \\int_{x_1}^{x_2}\\overline{\\psi} \\frac{\\partial \\psi}{\\partial t} + \\psi \\frac{\\partial \\overline{ \\psi}}{\\partial t} dt$\n\nSchr\u00f6dinger\u2019s equation (time dependent version) with a little bit of algebra:\n$\\frac{\\partial \\psi}{\\partial t} = \\frac{i \\hbar}{2m} \\frac{\\partial^2 \\psi}{\\partial x^2} - \\frac{i}{\\hbar}V \\psi$\n$\\frac{\\partial \\overline{\\psi}}{\\partial t} = \\frac{i \\hbar}{2m} \\frac{\\partial^2 \\overline{\\psi}}{\\partial x^2} + \\frac{i}{\\hbar}V \\overline{\\psi}$\n\nNote: $V$ is real.\n\nAlgebra: eliminate the partial with respect to time terms; multiply the top equation by $\\overline{\\psi}$ and the second by $\\psi$. Then add the two to obtain:\n$\\overline{\\psi} \\frac{\\partial \\psi}{\\partial t} + \\psi \\frac{\\partial \\overline{ \\psi}}{\\partial t} = \\frac{i \\hbar}{2m}(\\overline{\\psi} \\frac{\\partial^2 \\psi}{\\partial x^2} + \\psi \\frac{\\partial^2 \\overline{ \\psi}}{\\partial x^2})$\n\nNow integrate by parts:\n$\\frac{i \\hbar}{2m} \\int_{x_2}^{x_1} (\\overline{\\psi} \\frac{\\partial^2 \\psi}{\\partial x^2} + \\psi \\frac{\\partial^2 \\overline{ \\psi}}{\\partial x^2}) dx =$\n\n$\\frac{ih}{2m} ((\\overline{\\psi} \\frac{\\partial \\psi}{\\partial x})_{x_1}^{x_2} - \\int_{x_2}^{x_1} \\frac{\\partial \\overline{\\psi}}{\\partial x} \\frac{\\partial \\psi}{\\partial x} - ( (\\psi \\frac{\\partial \\overline{\\psi}}{\\partial x})_{x_1}^{x_2} - \\int_{x_2}^{x_1}\\frac{\\partial \\psi}{\\partial x}\\frac{\\partial \\overline{\\psi}}{\\partial x}dx)$\n\nNow the integrals cancel each other and we obtain our result.\n\nIt is common to denote $-\\frac{ih}{2m}(\\overline{\\psi}\\frac{d \\psi}{dx}-\\psi \\frac{d \\overline{\\psi}}{dx}$ by $S(x,t)$ (note the minus sign) and to say $\\frac{d}{dt}P(x_1 < x < x_2 ; t) = S(x_1,t) - S(x_2,t)$ (see the reason for the minus sign?)\n\n$S(x,t)$ is called the position probability current at the point $x$ at time $t$ One can think of this as a \"probability flow rate\" over the point $x$ at time $t$; the quantity $S(x_1, t) - S(x_2, t)$ will tell you if the probability of finding the particle between position $x_1$ and $x_2$ is going up (positive sign) or down, and by what rate. But it is important that these are position PROBABILITY current and not PARTICLE current; same for $|\\psi |^2$; this is the position probability density function, not the particle density function.\n\nNOTE I haven\u2019t talked about the position and momentum eigenvalues or eigenfuctions. We\u2019ll do that in our next post; we\u2019ll run into some mathematical trouble here. No, it won\u2019t be with the position because we already know what a distribution is; the problem is that we\u2019ll find the momentum eigenvector really isn\u2019t square integrable\u2026.or even close.\n\n## August 10, 2011\n\n### Quantum Mechanics and Undergraduate Mathematics X: Schr\u00f6dinger\u2019s Equations\n\nFiled under: advanced mathematics, applied mathematics, calculus, physics, quantum mechanics, science \u2014 collegemathteaching @ 1:19 am\n\nRecall from classical mechanics: $E = \\frac{1}{2}mv^2 + V(x)$ where $E$ is energy and $V(x)$ is potential energy. We also have position $x$ and momentum $p = mv$ Note that we can then write $E = \\frac{p^2}{2m} + V(x)$. Analogues exist in quantum mechanics and this is the subject of:\n\nPostulate 6. Momentum and position (one dimensional motion) are represented by the operators:\n$X = x$ and $P = -i\\hbar \\frac{d}{dx}$ respectively. If $f$ is any \u201cwell behaved\u201d function of two variables (say, locally analytic?) then $A = f(X, P) = f(x, -i\\hbar \\frac{d}{dx} )$.\n\nTo see how this works: let $\\phi(x) = (2 \\pi)^{-\\frac{1}{4}}exp(-\\frac{x^2}{4})$\nThen $X \\phi = (2 \\pi)^{-\\frac{1}{4}}x exp(-\\frac{x^2}{4})$ and $P \\phi = i\\hbar (2 \\pi)^{-\\frac{1}{4}} 2x exp(-\\frac{x^2}{4})$\n\nAssociated with these is energy Hamiltonian operator $H = \\frac{1}{2m} P^2 + V(X)$ where $P^2$ means \u201cdo $P$ twice\u201d. So $H = -\\frac{\\hbar^2}{2m}\\frac{d^2}{dx^2} + V(x)$.\n\nNote We are going to show that these two operators are Hermitian\u2026sort of. Why sort of: these operators $A$ might not be \u201cclosed\u201d in the sense that $\\langle \\phi_1, \\phi_2 \\rangle$ exists but $\\langle \\phi_1, A \\phi_2 \\rangle$ might not exist. Here is a simple example: let $\\phi_1 = \\phi_2 = \\sqrt{\\frac{2}{\\pi} \\frac{1}{x^2 + 1}}$. Then $\\langle \\phi_1, \\phi_2 \\rangle = 1$ but $\\int_{-\\infty}^{\\infty} x \\phi_1 dx$ fails to exist.\n\nSo the unstated assumption is that when we are proving that various operators are Hermetian, we mean that they are Hermetian for state vectors which are transformed into functions for which the given inner product is defined.\n\nSo, with this caveat in mind, let\u2019s show that these operators are Hermitian.\n\n$X$ clearly is because $\\langle \\phi_1, x \\phi_2 \\rangle = \\langle x \\phi_1, \\phi_2 \\rangle$. If this statement is confusing, remember that $x$ is a real variable and therefore $\\overline{x} = x$. Clearly, any well behaved real valued function of $x$ is also a Hermitian operator. IF we assume that $P$ is a Hermitian operator, then $\\langle \\phi_1, P^2 \\phi_2 \\rangle = \\langle P\\phi_1, P\\phi_2 \\rangle = \\langle P^2 \\phi_1, \\phi_2 \\rangle$. So we must show that $P$ is Hermitian.\n\nThis is a nice exercise in integration by parts:\n$\\langle \\phi_1, P\\phi_2 \\rangle = -i\\hbar\\langle \\phi_1, \\frac{d}{dx} \\phi_2 \\rangle = -i\\hbar \\int_{-\\infty}^{\\infty} \\overline{\\phi_1} \\frac{d}{dx} \\phi_2 dx$. Now we note that $\\overline{\\phi_1} \\phi_2 |_{-\\infty}^{\\infty} = 0$ (else the improper integrals would fail to converge this is a property assumed for state vectors; mathematically it is possible that the limit as $x \\rightarrow \\infty$ doesn\u2019t exist but the integral still converges) and so by the integration by parts formula we get $i\\hbar\\int_{-\\infty}^{\\infty} \\overline{\\frac{d}{dx}\\phi_1} \\phi_2 dx =\\int_{-\\infty}^{\\infty} \\overline{-i\\hbar\\frac{d}{dx}\\phi_1} \\phi_2 dx = \\langle P\\phi_1, \\phi_2 \\rangle$.\n\nNote that potential energy is a function of $x$ so it too is Hermitian. So our Hamiltonian $H(p,x) = \\frac{1}{2m}P^2 + V(X) = -\\frac{h^2}{2m}\\frac{d^2}{dx^2} + V(x)$ is also Hermitian. That has some consequences:\n\n1. $H \\eta_k = e_k \\eta_k$\n2. $H \\psi = i\\hbar\\frac{\\partial}{\\partial t} \\psi$\n\nNow we substitute for $H$ and obtain:\n\n1. $-\\frac{h^2}{2m} \\frac{d^2}{dx^2} \\eta_k + V(x)\\eta_k = e_k \\eta_k$\n\n2. $-\\frac{h^2}{2m} \\frac{\\partial^2}{\\partial x^2} \\psi + V(x)\\psi = i\\hbar \\frac{\\partial}{\\partial t} \\psi$\n\nThese are the Schr\u00f6dinger equations; the first one is the time independent equation. It is about each Hamiltonian energy eigenvector\u2026or you might say each stationary state vector. This holds for each $k$. The second one is the time dependent one and applies to the state vector in general (not just the stationary states). It is called the fundamental time evolution equation for the state vector.\n\nSpecial note: if one adjusts the Hamiltonian by adding a constant, the eigenvectors remain the same but the eigenvalues are adjusted by adding a constant. So the adjusted time vector gets adjusted by a factor of $exp(-iC \\frac{t}{\\hbar})$ which has a modulus of 1. So the new state vector describes the same state as the old one.\n\nNext post: we\u2019ll give an example and then derive the eigenvalues and eigenvectors for the position and momentum operators. Yes, this means dusting off the dirac delta distribution.\n\n## August 8, 2011\n\n### Quantum Mechanics and Undergraduate Mathematics VIII: Time Evolution of Expectation of an\u00a0Observable\n\nFiled under: advanced mathematics, applied mathematics, physics, probability, quantum mechanics, science \u2014 collegemathteaching @ 3:12 pm\n\nBack to our series on QM: one thing to remember about observables: they are operators with a set collection of eigenvectors and eigenvalues (allowable values that can be observed; \u201cquantum levels\u201d if you will). These do not change with time. So $\\frac{d}{dt} (A (\\psi)) = A (\\frac{\\partial}{\\partial t} \\psi)$. One can work this out by expanding $A \\psi$ if one wants to.\n\nSo with this fact, lets see how the expectation of an observable evolves with time (given a certain initial state):\n$\\frac{d}{dt} E(A) = \\frac{d}{dt} \\langle \\psi, A \\psi \\rangle = \\langle \\frac{\\partial}{\\partial t} \\psi, A \\psi \\rangle + \\langle \\psi, A \\frac{\\partial}{\\partial t} \\psi \\rangle$\n\nNow apply the Hamiltonian to account for the time change of the state vector; we obtain:\n$\\langle -\\frac{i}{\\hbar}H \\psi, A \\psi \\rangle + \\langle \\psi, -\\frac{i}{\\hbar}AH \\psi \\rangle = \\overline{\\frac{i}{\\hbar}} \\langle H \\psi, A \\psi \\rangle + -\\frac{i}{\\hbar} \\langle \\psi, AH \\psi \\rangle$\n\nNow use the fact that both $H$ and $A$ are Hermitian to obtain:\n$\\frac{d}{dt} A = \\frac{i}{\\hbar} \\langle \\psi, (HA - AH) \\psi \\rangle$.\nSo, we see the operator $HA - AH$ once again; note that if $A, H$ commute then the expectation of the state vector (or the standard deviation for that matter) does not evolve with time. This is certainly true for $H$ itself. Note: an operator that commutes with $H$ is sometimes called a \u201cconstant of motion\u201d (think: \u201ctotal energy of a system in classical mechanics).\n\nNote also that $|\\frac{d}{dt} A | = |\\frac{i}{\\hbar} \\langle \\psi, (HA - AH) \\psi \\rangle | \\leq 2 \\Delta A \\Delta H$\n\nIf $A$ does NOT correspond with a constant of motion, then it is useful to define an evolution time $T_A = \\frac{\\Delta A}{\\frac{E(A)}{dt}}$ where $\\Delta A = (V(A))^{1\/2}$ This gives an estimate of how much time must elapse before the state changes enough to equal the uncertainty in the observable.\n\nNote: we can apply this to $H$ and $A$ to obtain $T_A \\Delta H \\ge \\frac{\\hbar}{2}$\n\nConsequences: if $T_A$ is small (i. e., the state changes rapidly) then the uncertainty is large; hence energy is impossible to be well defined (as a numerical value). If the energy has low uncertainty then $T_A$ must be large; that is, the state is very slowly changing. This is called the time-energy uncertainty relation.\n\n## July 28, 2011\n\n### Quantum Mechanics and Undergraduate Mathematics VII: Time Evolution of the State\u00a0Vector\n\nFiled under: advanced mathematics, applied mathematics, physics, quantum mechanics, science \u2014 collegemathteaching @ 2:38 pm\n\nOf course the state vector $\\psi$ changes with time. The question is how does it change with time and how does the probability density function associated with an observable change with time?\n\nNote: we will write $\\psi_t$ for $\\psi(x,t)$. Now let $A$ be an observable. Note that the eigenvectors and the eigenvalues associated with $A$ do NOT change with time, so if we expand $\\psi_t$ in terms of the eigenbasis for $A$ we have $\\psi_t = \\sum_k \\langle \\alpha_k, \\psi_t \\rangle \\alpha_k$ hence $\\frac{\\partial \\psi_t}{\\partial t} = \\sum_k \\langle \\alpha_k, \\frac{\\partial \\psi_t}{\\partial t} \\rangle \\alpha_k$\n\nOf course, we need the state vector to \u201cstay in the class of state vectors\u201d when it evolves with respect to time, which means that the norm cannot change; or $\\frac{d}{dt} \\langle \\psi_t, \\psi_t \\rangle = 0$.\n\nNeedless to say there has to be some restriction on how the state vector can change with time. So we have another postulate:\n\nPostulate 5\nFor every physical system there exists a linear Hermitian operator $H$ called the Hamiltonian operator such that:\n1. $i\\hbar \\frac{\\partial}{\\partial t} \\psi(x,t) = H\\psi(x,t)$ and\n2. $H$ corresponds to the total energy of the system and possesses a complete set of eigenvectors $\\eta_k$ and eigenvalues $e_k$ where the eigenvalues are the \u201callowed values\u201d of the total energy of the system.\n\nNote: $\\hbar$ is the constant $\\frac{h}{\\pi}$ where $h$ is Plank\u2019s constant.\n\nNote: $H$ is not specified; it is something that the physicists have to come up with by observing the system. That is, there is a partial differential equation to be solved!\n\nBut does this give us what we want, at least in terms of $\\psi_t$ staying at unit norm for all times $t$? (note: again we write $\\psi_t$ for $\\psi(x,t)$ ).\n\nThe answer is yes; first note that $\\frac{d}{dt} \\langle \\psi_t, \\psi_t \\rangle = \\langle \\frac{\\partial \\psi_t}{\\partial t}, \\psi_t \\rangle + \\langle \\psi_t, \\frac{\\partial \\psi_t}{\\partial t}\\rangle$; this is an easy exercise in using the definition of our inner product and differentiating under the integral sign and noting that the partial derivative operation and the conjugate operation commute).\n\nNow note: $\\langle \\frac{\\partial \\psi_t}{\\partial t}, \\psi_t \\rangle + \\langle \\psi_t, \\frac{\\partial \\psi_t}{\\partial t}\\rangle = \\overline{-\\frac{i}{\\hbar}}\\langle H\\psi_t, \\psi_t \\rangle + -\\frac{i}{\\hbar}\\langle \\psi_t, H \\psi_t \\rangle = \\frac{i}{\\hbar}(\\langle H\\psi_t, \\psi_t \\rangle - \\langle \\psi_t, H\\psi_t \\rangle) = 0$\nbecause $H$ is Hermitian.\n\nNote: at this point Gillespie takes an aside and notes that if one denotes the state vector at time $t = 0$ by $\\psi_0$ then one can attempt to find an operator $U(t)$ where $\\psi_t = U(t)\\psi_0$. This leads to $i \\hbar \\frac{\\partial U(t)}{\\partial t} = HU\\psi_0$ which must be true for all $\\psi_0$. This leads to $i\\hbar \\frac{\\partial U(t)}{\\partial t} = HU(t)$ with initial condition $U(0) = 1$. This leads to the solution $U(t) = exp(\\frac{-iHt}{\\hbar})$ where the exponential is defined in the sense of linear algebra (use the power series expansion of $exp(A)$). Note: $U$ is not Hermitian and therefore NOT an observable. But it has a special place in quantum mechanics and is called the time-evolution operator.\n\nNext: we\u2019ll deal with the time energy relation for the expected value of a specific observable.\n\n## July 25, 2011\n\n### Quantum Mechanics and Undergraduate Mathematics VI: Heisenberg Uncertainty\u00a0Principle\n\nFiled under: advanced mathematics, applied mathematics, physics, probability, quantum mechanics, science \u2014 collegemathteaching @ 10:05 pm\n\nHere we use Cauchy-Schwartz inequality, other facts about inner products and basic probability to derive the Heisenberg Uncertainty Principle for incompatible observables $A$ and $B$. We assume some state vector $\\psi$ which has not been given time to evolve between measurements and we will abuse notation by viewing $A$ and $B$ as random variables for their given eigenvalues $a_k, b_k$ given state vector $\\psi$.\n\nWhat we are after is the following: $V(A)V(B) \\geq (1\/4)|\\langle \\psi, (AB-BA) \\psi \\rangle|^2.$\nWhen $AB-BA = c$ we get: $V(A)V(B) \\geq (1\/4)|c|^2$ which is how it is often stated.\n\nThe proof is a bit easier when we make the expected values of $A$ and $B$ equal to zero; we do this by introducing a new linear operator $A' = A -E(A)$ and $B' = B - E(B)$; note that $(A - E(A))\\psi = A\\psi - E(A)\\psi$. The following are routine exercises:\n1. $A'$ and $B'$ are Hermitian\n2. $A'B' - B'A' = AB-BA$\n3. $V(A') = V(A)$.\n\nIf one is too lazy to work out 3:\n$V(A') = E((A-E(A))^2) - E(A -E(A)) = E(A^2 - 2AE(A) + E(A)E(A)) = E(A^2) -2E(A)E(A) + (E(A))^2 = V(A)$\n\nNow we have everything in place:\n$\\langle \\psi, (AB-BA) \\psi \\rangle = \\langle \\psi, (A'B'-B'A') \\psi \\rangle = \\langle A'\\psi, B' \\psi \\rangle - \\langle B'\\psi, A' \\psi \\rangle = \\langle A'\\psi, B' \\psi \\rangle - \\overline{\\langle A'\\psi, B' \\psi \\rangle} = 2iIm\\langle A'\\psi, B'\\psi \\rangle$\nWe now can take the modulus of both sides:\n$|\\langle \\psi, (AB-BA)\\psi \\rangle | = 2 |Im \\langle A'\\psi, B'\\psi \\rangle \\leq 2|\\langle A'\\psi, B'\\psi\\rangle | \\leq 2 \\sqrt{\\langle A'\\psi,A'\\psi\\rangle}\\sqrt{\\langle B'\\psi, B'\\psi\\rangle} = 2 \\sqrt{\\langle A\\psi,A\\psi\\rangle}\\sqrt{\\langle B\\psi,B\\psi\\rangle} = 2\\sqrt{V(A)}\\sqrt{V(B)}$\n\nThis means that, unless $A$ and $B$ are compatible observables, there is a lower bound on the product of their standard deviations that cannot be done away with by more careful measurement. It is physically impossible to drive this product to zero. This also means that one of the standard deviations cannot be zero unless the other is infinite.\n\n### Quantum Mechanics and Undergraduate Mathematics V: compatible\u00a0observables\n\nThis builds on our previous example. We start with a state $\\psi$ and we will make three successive observations of observables which have operators $A$ and $B$ in the following order: $A, B, A$. The assumption is that these observations are made so quickly that no time evolution of the state vector can take place; all of the change to the state vector will be due to the effect of the observations.\n\nA simplifying assumption will be that the observation operators have the following property: no two different eigenvectors have the same eigenvalues (e. g., the eigenvalue uniquely determines the eigenvector up to multiplication by a constant of unit modulus).\n\nFirst of all, this is what \u201ccompatible observables\u201d means: two observables $A, B$ are compatible if, upon three successive measurements $A, B, A$ the first measurement of $A$ is guaranteed to be the second measurement of $A$. That is, the state vector after the first measurement of $A$ is the same state vector after the second measurement of $A$.\n\nSo here is what the compatibility theorem says (I am freely abusing notation by calling the observable by the name of its associated operator):\n\nCompatibility Theorem\nThe following are equivalent:\n\n1. $A, B$ are compatible observables.\n2. $A, B$ have a common eigenbasis.\n3. $A, B$ commute (as operators)\n\nNote: for this discussion, we\u2019ll assume an eigenbasis of $\\alpha_i$ for $A$ and $\\beta_i$ for $B$.\n\n1 implies 2: Suppose the state of the system is $\\alpha_k$ just prior to the first measurement. Then the first measurement is $a_k$. The second measurement yields $b_j$ which means the system is in state $\\beta_j$, in which case the third measurement is guaranteed to be $a_k$ (it is never anything else by the compatible observable assumption). Hence the state vector must have been $\\alpha_k$ which is the same as $\\beta_j$. So, by some reindexing we can assume that $\\alpha_1 = \\beta_1$. An argument about completeness and orthogonality finishes the proof of this implication.\n\n2 implies 1: after the first measurement, the state of the system is $\\alpha_k$ which, being a basis vector for observable $B$ means that the system after the measurement of $B$ stays in the same state, which implies that the state of the system will remain $\\alpha_k$ after the second measurement of $A$. Since this is true for all basis vectors, we can extend this to all state vectors, hence the observables are compatible.\n\n2 implies 3: a common eigenbasis implies that the operators commute on basis elements so the result follows (by some routine linear-algebra type calculations)\n\n3 implies 2: given any eigenvector $\\alpha_k$ we have $AB \\alpha_k = BA \\alpha_k = a_k B \\alpha_k$ which implies that $B \\alpha_k$ is an eigenvector for $A$ with eigenvalue $\\alpha_k$. This means that $B \\alpha_k = c \\alpha_k$ where $c$ has unit modulus; hence $\\alpha_k$ must be an eigenvector of $B$. In this way, we establish a correspondence between the eigenbasis of $B$ with the eigenbasis of $A$.\n\nOk, what happens when the observables are NOT compatible?\n\nHere is a lovely application of conditional probability. It works this way: suppose on the first measurement, $a_k$ is observed. This puts us in state vector $\\alpha_k$. Now we measure the observable $B$ which means that there is a probability $|\\langle \\alpha_k, \\beta_i \\rangle|^2$ of observing eigenvalue $b_i$. Now $\\beta_i$ is the new state vector and when observable $A$ is measured, we have a probability $|\\langle \\alpha_j, \\beta_i \\rangle|^2$ of observing eigenvalue $a_j$ in the second measurement of observable $A$.\n\nTherefore given the initial measurement we can construct a conditional probability density function $p(a_j|a_k) = \\sum_i p(b_i|a_k)p(a_j|b_i)= \\sum_i |\\langle \\alpha_k, \\beta_i \\rangle| |^2 |\\langle \\beta_i, \\alpha_j |^2$\n\nAgain, this makes sense only if the observations were taken so close together so as to not allow the state vector to undergo time evolution; ONLY the measurements changes the state vector.\n\nNext: we move to the famous Heisenberg Uncertainty Principle, which states that, if we view the interaction of the observables $A$ and $B$ with a set state vector and abuse notation a bit and regard the associated density functions (for the eigenvalues) by the same letters, then $V(A)V(B) \\geq (1\/4)|\\langle \\psi, [AB-BA]\\psi \\rangle |^2.$\n\nOf course, if the observables are compatible, then the right side becomes zero and if $AB-BA = c$ for some non-zero scalar $c$ (that is, $(AB-BA) \\psi = c\\psi$ for all possible state vectors $\\psi$ ), then we get $V(A)V(B) \\geq (1\/4)|c|^2$ which is how it is often stated.\n\n## July 19, 2011\n\n### Quantum Mechanics and Undergraduate Mathematics IV: measuring an observable\u00a0(example)\n\nOk, we have to relate the observables to the state of the system. We know that the only possible \u201cvalues\u201d of the observable are the eigenvalues of the operator and the relation of the operator to the state vector provides the density function. But what does this measurement do to the state? That is, immediately after a measurement is taken, what is the state?\n\nTrue, the system undergoes a \"time evolution\" but once an observable is measured, an immediate (termed \"successive\") measurement will yield the same value; a \"repeated\" measurement (one made giving the system to undergo a time evolution) might give a different value.\n\nSo we get:\n\nPostulate 4 A measurement of an observable generally (?) causes a drastic, uncontrollable alteration in the state vector of the system; immediately after the measurement it will coincide with the eigenvector corresponding to the eigenvalue obtained in the measurement.\n\nNote: we assume that our observable operators have distinct eigenvalues; that is, no two distinct eigenvectors have the same eigenvalue.\n\nThat is, if we measure an observable with operator $A$ and obtain measurement $a_i$ then the new system eigenvector is $\\alpha_i$ regardless of what $\\psi$ was prior to measurement. Of course, this eigenvector can (and usually will) evolve with time.\n\nRoughly speaking, here is what is going on:\nSay the system is in state $\\psi$. We measure and observable with operator $A$. We can only obtain one of the eigenvalues $\\alpha_k$ as a measurement. Recall: remember all of those \u201corbitals\u201d from chemistry class? Those were the energy levels of the electrons and the orbital level was a permissible energy state that we could obtain by a measurement.\n\nNow if we get $\\alpha_k$ as a measurement, the new state vector is $\\alpha_k$. One might say that we started with a probability density function (given the state and the observable), we made a measurement, and now, for a brief instant anyway, our density function \u201ccollapsed\u201d to the density function $P(A = a_k) = 1$.\n\nThis situation (brief) coincides with our classical intuition of an observable \u201chaving a value\u201d.\n\nFor the purposes of this example, we\u2019ll set our Hilbert space to the the square integrable piecewise smooth functions on $[-\\pi, \\pi]$ and let our \u201cstate vector\u201d $\\psi(x) =\\left\\{ \\begin{array}{c}1\/\\sqrt{\\pi}, 0 < x \\leq \\pi \\\\ 0,-\\pi \\leq x \\leq 0 \\end{array}\\right.$\n\nNow suppose our observable corresponds to the eigenfunctions mentioned in this post, and we measure \u201c-4\u201d for our observable. This is the eigenvalue for $(1\/\\sqrt{\\pi})sin(2x)$ so our new state vector is $(1\/\\sqrt{\\pi})sin(2x)$.\n\nSo what happens if a different observable is measured IMMEDIATELY (e. g., no chance for a time evolution to take place).\n\nExample We\u2019ll still use the space of square integrable functions over $[-\\pi, \\pi]$\nOne might recall the Legendre polynomials which are eigenfucntions of the following operator:\n$d\/dt((1-t^2) dP_n\/dt) = -(n)(n+1) P_n(t)$. These polynomials obey the orthogonality relation $\\int^{1}_{-1} P_m(t)P_n(t)dt = 2\/(2n+1) \\delta_{m,n}$ hence $\\int^{1}_{-1} P_m(t)P_m(t)dt = 2\/(2m+1)$.\nThe first few of these are $P_0 = 1, P_1 =t, P_2 = (1\/2)(3t^2-1), P_3 = (1\/2)(5t^3 - 3t), ..$\n\nWe can adjust these polynomials by the change of variable $t =x\/\\pi$ and multiply each polynomial $P_m$ by the factor $sqrt{2\/(\\pi (2m+1) }$ to obtain an orthonormal eigenbasis. Of course, one has to adjust the operator by the chain rule.\n\nSo for this example, let $P_n$ denote the adjusted Legendre polynomial with eigenvalue $-n(n+1)$.\n\nNow back to our original state vector which was changed to state function $(1\/\\sqrt{\\pi})sin(2x)$.\n\nNow suppose eigenvalue $-6 = -2(3)$ is observed as an observable with the Lengendre operator; this corresponds to eigenvector $\\sqrt{(2\/5)(1\/\\pi)}(1\/2)(3(x\/\\pi)^2 -1)$ which is now the new state vector.\n\nNow if we were to do an immediate measurement of the first observable, we\u2019d have to a Fourier like expansion of our new state vector; hence the probability density function for the observables changes from the initial measurement. Bottom line: the order in which the observations are taken matters\u2026.in general.\n\nThe case in which the order wouldn\u2019t matter: if the second observable had the state vector (from the first measurement) as an element of its eigenbasis.\n\nWe will state this as a general principle in our next post.\n\n## July 15, 2011\n\n### Quantum Mechanics and Undergraduate Mathematics III: an example of a state\u00a0function\n\nI feel bad that I haven\u2019t given a demonstrative example, so I\u2019ll \u201ccheat\u201d a bit and give one:\n\nFor the purposes of this example, we\u2019ll set our Hilbert space to the the square integrable piecewise smooth functions on $[-\\pi, \\pi]$ and let our \u201cstate vector\u201d $\\psi(x) =\\left\\{ \\begin{array}{c}1\/\\sqrt{\\pi}, 0 < x \\leq \\pi \\\\ 0,-\\pi \\leq x \\leq 0 \\end{array}\\right.$\n\nNow consider a (bogus) state operator $d^2\/dx^2$ which has an eigenbasis $(1\/\\sqrt{\\pi})cos(kx), (1\/\\sqrt{\\pi})sin(kx), k \\in {, 1, 2, 3,...}$ and $1\/\\sqrt{2\\pi}$ with eigenvalues $0, -1, -4, -9,......$ (note: I know that this is a degenerate case in which some eigenvalues share two eigenfunctions).\n\nNote also that the eigenfunctions are almost the functions used in the usual Fourier expansion; the difference is that I have scaled the functions so that $\\int^{\\pi}_{-\\pi} (sin(kx)\/\\sqrt{\\pi})^2 dx = 1$ as required for an orthonormal basis with this inner product.\n\nNow we can write $\\psi = 1\/(2 \\sqrt{\\pi}) + 4\/(\\pi^{3\/2})(sin(x) + (1\/3)sin(3x) + (1\/5)sin(5x) +..)$\n(yes, I am abusing the equal sign here)\nThis means that $b_0 = 1\/\\sqrt{2}, b_k = 2\/(k \\pi), k \\in {1,3,5,7...}$\n\nNow the only possible measurements of the operator are 0, -1, -4, -9, \u2026. and the probability density function is: $p(A = 0) = 1\/2, P(A = -1) = 4\/(\\pi^2), P(A = -3) = 4\/(9 \\pi^2),...P(A = -(2k-1))= 4\/(((2k-1)\\pi)^2)..$\n\nOne can check that $1\/2 + (4\/(\\pi^2))(1 + 1\/9 + 1\/25 + 1\/49 + 1\/81....) = 1.$\n\nHere is a plot of the state function (blue line at the top) along with some of the eigenfunctions multiplied by their respective $b_k$.\n\n## July 13, 2011\n\n### Quantum Mechanics and Undergraduate Mathematics\u00a0II\n\nIn the first part of this series, we reviewed some of the mathematical background that we\u2019ll use. Now we get into a bit of the physics.\n\nFor simplification, we\u2019ll assume one dimensional, non-relativistic motion. No, nature isn\u2019t that simple; that is why particle physics is hard! \ud83d\ude42\n\nWhat we will do is to describe a state of a system and the observables. The state of the system is hard to describe; in the classical case (say the damped mass-spring system in harmonic motion), the state of the system is determined by the system parameters (mass, damping constant, spring constant) and the velocity and acceleration at a set time.\n\nAnd observable is, roughly speaking, something that can give us information about the state of the system. In classical mechanics, one observable might be $H(x, p) = P^2\/2m + V(x)$ where $p$ is the system\u2019s momentum and $V(x)$ represents the potential energy at position $x$. If this seems strange, remember that $p = mv$ therefore kinetic energy is $mv^2\/2$ and solving for momentum $p$ gives us the formula. We bring this up because something similar will appear later.\n\nIn quantum mechanics, certain postulates are assumed. I\u2019ll present the ones that Gillespie uses:\n\nPostulate 1: Every possible physical state of a given system corresponds to a Hilbert space vector $\\psi$ of unit norm (using the inner product that we talked about) and every such vector corresponds to a possible state of a system. The correspondence of states to the vectors is well defined up to multiplication of a vector by a complex number of unit modulus.\n\nNote: this state vector, while containing all of the knowable information of the system, says nothing about what could be known or how such knowledge might be observed. Of course, this state vector might evolve with time and sometimes it is written as $\\psi_{t}$ for this reason.\n\nPostulate 2 There is a one to one correspondence between physical observables and linear Hermitian operators $A$, each of which possesses a complete, orthonormal set of eigenvectors $\\alpha_{i}$ and a corresponding set of real eigenvalues $a_i$ and the only possible values of any measurement of this observable is one of these eigenvalues.\n\nNote: in the cases when the eigenvalues are discretely distributed (e. g., the eigenvalues fail to have a limit point), we get \u201cquantized\u201d behavior from this observable.\n\nWe\u2019ll use observables with discrete eigenvalues unless we say otherwise.\n\nNow: is a function of an observable itself an observable? The answer is \u201cyes\u201d if the function is real analytic and we assume that $(A)^n(\\psi) = A(A(A....A(\\psi))$. To see this: assume that $f(z) = \\sum_i c_i z^i$ and note that if $A$ is an observable operator then so is $cA^n$ for all $n$. Note: one can do this by showing that the eigenvectors for $A$ do not change and that the eigenvalues merely go up by power. The completeness of the eigenvectors imply convergence when we pass to $f$.\n\nNow we have states and observables. But how do they interact?\nRemember that we showed the following:\n\nLet $A$ be a linear operator with a complete orthonormal eigenbasis $\\alpha_i$ and corresponding real eigenvalues $a_i$. Let $\\psi$ be an element of the Hilbert space with unit norm and let $\\psi = \\sum_j b_j \\alpha_j$.\n\nThen the function $P(y = a_i) = (|b_i|)^2$ is a probability density function. (note: $b_i = \\langle \\alpha_i , \\psi \\rangle$).\n\nThis will give us exactly what we need! Basically, if the observable has operator $A$ system and is in state $\\psi$, then the probability of a measurement yielding a result of $a_i$ is $(|\\langle \\alpha_i , \\psi \\rangle|)^2$ Note: it follows that if the state $\\phi = \\alpha_i$ then the probability of obtaining $a_i$ is exactly one.\n\nWe summarize this up by Postulate 3: (page 49 of Gillespie, stated for the \u201cscattered eigenvalues\u201d case):\n\nPostulate 3: If an observable operator $A$ has eigenbasis $\\alpha_i$ with eigenvalues $a_i$ and if the corresponding observable is measured on a system which, immediately prior to the measurement is in state $\\psi$ then the strongest predictive statement that can be made concerning the result of this measurement is as follows: the probability that the measurement will yield $a_k$ is $(|\\langle \\alpha_i , \\psi \\rangle|)^2$.\n\nNote: for simplicity, we are restricting ourselves to observables which have distinct eigenvalues (e. g., no two linearly independent eigenvectors have the same eigenvalues). In real life, some observables DO have different eigenvectors with the same eigenvalue (example from calculus; these are NOT Hilbert Space vectors, but if the operator is $d^2\/dx^2$ then $sin(x)$ and $cos(x)$ both have eigenvalue -1. )\n\nWhere we are now: we have a probability distribution to work with which means that we can calculate an expected value and a variance. These values will be fundamental when we tackle uncertainty principles!\n\nJust a reminder from our courses in probability theory: if $Y$ is a random variable with density function $P$\n\n$E(Y) = \\sum_i y_i P(y_i)$ and $V(Y) = E(Y^2) -(E(Y))^2$.\n\nSo with our density function $P(y = a_i) = (|b_i|)^2$ (we use $b_i = \\langle \\alpha_i , \\psi \\rangle$ to save space), then if $E(A)$ is the expected observed value of the observable (the expected value of the eigenvalues):\n$E(A) = \\sum_i a_i (b_i)^2$. But this quantity can be calculated in another way:\n\n$\\langle \\psi , A(\\psi) \\rangle = \\langle \\sum b_i \\alpha_i , A(\\sum b_i \\alpha_i) \\rangle = \\langle \\sum b_i \\alpha_i , \\sum a_i b_i \\alpha_i) \\rangle = \\sum_i \\overline{b_i} b_i a_i \\langle \\alpha_i, \\alpha_i \\rangle = \\sum_i \\overline{b_i} b_i a_i = \\sum_i |b_i|^2 a_i = E(A)$. Yes, I skipped some easy steps.\n\nUsing this we find $V(A) = \\langle \\psi, A^2(\\psi) \\rangle - (\\langle \\psi, A(\\psi) \\rangle )^2$ and it is customary to denote the standard deviation $\\sqrt{V(A)} = \\Delta(A)$\n\nIn our next installment, I give an illustrative example.\n\nIn a subsequent installment, we\u2019ll show how a measurement of an observable affects the state and later how the distribution of the observable changes with time.","date":"2019-06-19 05:51:59","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\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 360, \"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\": 0.9368975162506104, \"perplexity\": 236.20767993227932}, \"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-2019-26\/segments\/1560627998913.66\/warc\/CC-MAIN-20190619043625-20190619065625-00521.warc.gz\"}"} | null | null |
\section{Introduction}
In relativistic heavy ion collisions, the nucleons of the interacting ions
can be divided into two distinct categories:
those that experience an inelastic collision with at
least one nucleon from the opposing nucleus (participants)
and those that do not (spectators).
Participant nucleons ultimately create the bulk of
particles observed in the detectors. Spectators consist of
single protons and neutrons as well as
larger spectator fragments including Helium, Lithium,
Beryllium, Boron, and higher mass nuclei.
Na\"{\i}vely, these spectators are free to continue along
their original path as they do not directly participate in
the collision. In practice, however, they can interact
in several ways and still be considered a spectator by
the usual definition: for example they can suffer an elastic collision
with a nucleon from the other beam, they can be affected
by any remaining nuclear binding energy in the beam remnant,
or they can interact with produced particles from the
participant zone~\cite{cite:PHOBOS_v1}.
Fragmentation of nuclei has been studied in a number of
experiments~\cite{cite:EMU,cite:SinghJain,cite:KLMM_PhysRevC,cite:ALADIN,cite:KLMM_ZPhys,cite:KLM_ActaPhysPol,cite:E877,cite:DWW}.
These experiments typically covered the full kinematic and solid
angle range needed to accurately identify all fragments and
basic fragment properties such as $A$ and $Z$, and their momenta.
However, these experiments suffered from a lack of statistics,
with only $\mathcal{O}(1000)$ events in total, precluding detailed
differential studies of fragmentation properties as a function of
impact parameter.
The observed properties of fragments, such as their momentum vectors, can be
described by a combination of the beam momentum at the time of
the collision and the internal Fermi motion within the nucleus
in its rest frame.
In the absence of Fermi motion and other external effects,
spectator fragment transverse momenta would be zero and they would
consequently continue traveling at the same rapidity as the beam.
In this limit, the polar angle ($\theta$) of fragments would be zero
or, equivalently, they would have infinite pseudorapidity ($\eta$):
\begin{equation}
\label{eqn:eta_theta}
\begin{split}
\ensuremath{\eta & = -{\rm ln}({\rm tan}(\theta/2))}\\
\ensuremath{ & \rightarrow \infty (\theta\rightarrow0).}
\end{split}
\end{equation}
\noindent Including the Fermi motion, however, leads to a finite
transverse momentum component of the fragments and reduces the
particle rapidity to below that of the beam. With a finite
(nonzero) polar angle, it is possible that the products will be
intercepted by active elements of a detector.
In addition, the internal Fermi motion also modifies the
longitudinal component of the momentum, however this effect
is typically small compared to the boosted momentum of the nucleons.
Transverse momentum is boost invariant and it therefore becomes useful
to compare data across multiple experiments with differing
collision energies.
Equivalently, by converting the momentum vectors into an
angular form, one can show that the pseudorapidity density
distribution ($dN/d\eta$ versus $\eta$) becomes approximately
boost invariant, which also allows for the comparison of data at
different $\sqrt{s_{_{NN}}}$. To account for energy differences, one subtracts
the rapidity of the beam at the appropriate energy scale;
a nontrivial transformation described in Appendix~\ref{ap:EtaPrimer}.
In the PHOBOS experiment~\cite{cite:PHOBOS_NIM}
at the Relativistic Heavy-Ion Collider
(RHIC), completely-freed neutrons can be measured using the
Zero-Degree Calorimeters (ZDC)~\cite{cite:ZDCs}, which are specifically designed
for this purpose. Charged fragments are not
observed in these detectors as they are swept away from
the ZDCs by the RHIC accelerator magnets. A calorimeter that could
detect very forward protons was available for some PHOBOS running
periods, but was not used in this analysis.
At RHIC injection energies, nucleon-nucleon center of mass energy
$\sqrt{s_{_{NN}}}$\,=\,19.6~(Au+Au) and~22.4\,GeV (Cu+Cu), spectators with a
finite transverse momentum can be detected
within the pseudorapidity acceptance of PHOBOS. However, the
finite acceptance of the detector limits the measurement of very
low-$p_T$ particles, especially for large-$Z$ fragments.
A large statistical sample, though, has been amassed which does allow
for some more detailed studies not afforded to other experiments.
This paper presents detailed measurements of large-$Z$ fragments in the
PHOBOS detector. Section~\ref{sec:Detector} describes the detector.
Section~\ref{sec:Analysis} describes the analysis
methods used to distinguish differently charge particles.
Sections~\ref{sec:EtaDep}~and~\ref{sec:CentDep}
show the pseudorapidity and centrality dependencies of the
fragments, respectively. Section~\ref{sec:EtaCent} discusses how,
in combining the system size, centrality, and pseudorapidity dependencies,
one can probe scaling effects of the large-$Z$ fragments in the
context of the number of spectators and participants in the collision.
\section{PHOBOS Detector}\label{sec:Detector}
\begin{figure}[!t]
\centering
\includegraphics[angle=0,width=0.475\textwidth]{drawAcceptance.pdf
\caption{\label{fig:Acceptance}
(color online) Transverse momentum and rapidity coverage of charged
particles in the silicon Ring detectors in PHOBOS. The
main figure shows the $p_{T}$$/m$-rapidity acceptance for
charged particles in each Ring (different shaded bands). The
boundary on the rightmost edge of the shaded region
depends on the beam energy. The dashed line shows the boundary for
$p_{z}/m$\,=\,$p_{\rm beam}/m_{Au}$ for $\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV Au+Au collisions.
The right-hand axis shows the $p_{T}$-scale for $\alpha$ particles, i.e.
$m$\,=\,3.727\,GeV/$c^{2}$.
The inset figure shows the
Ring-detector $p_T$ and pseudorapidity coverage.}
\end{figure}
PHOBOS is a large acceptance silicon detector, covering
almost 2$\pi$ in azimuth and $|\eta|$$<$$5.4$ ($\theta$\,$>$\,$9$\,mrad)~\cite{cite:PHOBOS_NIM}. For the results presented
here, the energy loss measured in the Ring detectors
(3.0$<$$|\eta|$$<$$5.4$) is used to identify spectator
fragments. The Rings are silicon pad detectors arranged
in an octagonal pattern perpendicular to and surrounding the beam pipe. Three
Ring detectors are placed on each side of the interaction
point at approximately 1,~2,~and~5\,meters from the center of the
interaction region. This
configuration allows for full coverage with minimal
overlapping areas.
In addition, the Octagon silicon barrel, which
consists of a single-layer of silicon parallel to and surrounding the
beam pipe covering $|\eta|$$<$$3.2$, is used for collision vertex
and event centrality determination.
In order to distinguish between singly- and multiply-charged fragments,
the relative energy loss, \mbox{$E_{\rm rel}$}\xspace,
is defined as
\begin{equation}
\label{eqn:Erel}
\ensuremath{{E_{\rm rel}} = \frac{E_{\rm loss}}{\langle E_{\rm loss} \rangle|_{Z=1} },}
\end{equation}
\noindent where $E_{\rm loss}$ is the energy loss in the silicon detector
and $\langle E_{\rm loss} \rangle|_{Z=1}$ is the mean energy loss for a
$Z$\,=\,1 particle. Singly-charged particles (for example spectator protons,
deuterons, and tritons) and singly-charged participants or produced particles
(created by the participants) all appear at an \mbox{$E_{\rm rel}$}\xspace position close to 1 and, as such,
cannot be separated.
For larger fragments,
with charge greater than unity, energy loss in the silicon
follows a charge-squared ($Z^{2}$) dependence, leading to the
appearance of $\alpha$ particles (for example) at four times the \mbox{$E_{\rm rel}$}\xspace position of
a singly-charged particle.
The transverse momentum, $p_{T}$, and rapidity, $y$, coverage for charged
particles in the Rings is shown in Fig.~\ref{fig:Acceptance}.
As there is no significant magnetic field traversed by
forward-going particles, the fixed $\eta$ Ring boundaries
translate to fixed curves in $p_{T}/m$ versus $y$ for
all charged particles. The high-$p_{T}$ and $y$ boundary (rightmost
edge for each Ring) is calculated for $\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV Au+Au collisions,
assuming a maximum $p_{z}/m = p_{\rm beam}/m_{Au}$, where $p_{z}$ is the
momentum of the particle (of mass $m$) along the beam direction,
and $p_{\rm beam}$ is the beam momentum.
\section{Data Analysis}\label{sec:Analysis}
\subsection{Event Selection}
The data were recorded during the 2001 (Au+Au --
$\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV) and 2005 (Cu+Cu --
$\sqrt{s_{_{NN}}}$\,=\,22.4\,GeV) RHIC runs. Readout of the silicon was initiated
by a minimally biased trigger for each data set based on coinciding signals from
two arrays of 16 plastic scintillators (3.2$<$$|\eta|$$<$$4.5$), the ``Paddle'' trigger counters~\cite{cite:PHOBOS_PaddleNIM}. For
Au+Au (Cu+Cu) collisions, a minimum of 3~(1) scintillator
hits were required in each array to start readout. The
collision vertex position along the beam line ($z$) was determined via a
probabilistic approach using hits in the Octagon silicon
barrel~\cite{cite:PHOBOS_VertexNIM}. For Cu+Cu collisions at
$\sqrt{s_{_{NN}}}$\,=\,22.4\,GeV, a vertex requirement of $|z|$$<$$10$\,cm
from the nominal vertex position was imposed; for Au+Au
this was relaxed to $|z|$$<$$20$\,cm to maximize the statistics from
the single day-long run.
A total of 84k (2.1M) events were selected for this analysis out of 327k
(15.7M) recorded, respectively for Au+Au (Cu+Cu) collisions. Events
are dominantly rejected due to the vertex requirement.
The estimated trigger efficiency (coupled with the
vertex finding efficiency)
for the Au+Au (Cu+Cu) data set is 83.5$\pm$3\% (79$\pm$5\%),
determined using the same methods as described in
Ref.~\cite{cite:PHOBOS_200_20} with
the data divided into seven (six) centrality classes,
each with 10\% of the total nuclear inelastic cross-section.
The centrality measure, EOct, is the summed energy loss in the silicon
of the centrally located Octagon barrel in the region $|\eta|$\,$<$\,3.0~\cite{cite:PHOBOS_200_20}.
The EOct parameter is defined in a $|\eta|$ region smaller than the full acceptance of the Octagon to limit any systematic effects of acceptance shifts
(due to the collision vertex position) and to
reduce the overlap with the Ring detector acceptance.
The lowest centrality cut-off is defined as the point
at which the trigger+vertex efficiency falls below 100\%.
For each
centrality class, the number of participants ($N_{\rm part}$) is estimated
by use of a Glauber model calculation~\cite{cite:PHOBOS_Glauber}.
Also, the number of spectator nucleons emitted at either the positive or
negative pseudorapidity is calculated as
$N_{\rm spec}/2$\,=\,($N_{\rm part}^{\rm max}$-$N_{\rm part}$)/2, where
$N_{\rm part}^{\rm max}$\,=2$A$\,=\,394~(126) for Au~(Cu) nuclei.
\begin{figure}[!t]
\centering
\includegraphics[angle=0,width=0.431\textwidth]{EOctERingCorr_Color.pdf
\caption{\label{fig:ERingVsEOct}
(color online) Correlation between the summed energy recorded in each of
the Ring detectors (ERing) and the summed energy deposited in the Octagon
barrel (EOct) in Au+Au collisions at $\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV.
Filled (open) symbols illustrate the measured distributions from data
(simulation). Spectators have been explicitly excluded from the
simulation distributions.
The bands show the centrality class selection bins used in this
analysis, with darker bands corresponding to more central events.
See text for discussion.}
\end{figure}
\subsection{Motivation}
The first observation of the presence of charged spectator fragments, in the
acceptance of PHOBOS, was made during the first low-energy
data~\cite{cite:SigFrag}.
The measured charged particle multiplicity
was found to be larger at high pseudorapidity in peripheral
data than in central data, an opposite effect than was expected, and
in contrast to the observed dependencies at mid-rapidity.
Several tests were performed to confirm that the larger
particle yield at high pseudorapidity likely originated from spectator fragments.
Figure~\ref{fig:ERingVsEOct} shows the correlation between
the summed energy in each silicon ring (ERing) and the summed
energy deposited in the silicon Octagon barrel (EOct). Filled symbols
represent data; open symbols show the result of a
Monte-Carlo (MC) simulation that uses particles generated from a
{\sc Hijing}~\cite{cite:HIJING} event simulation passed through
a full {\sc Geant}~\cite{cite:GEANT} description of the PHOBOS
detector and has had spectator fragments
explicitly removed from the acceptance of the detector.
In the MC simulation, a monotonic correlation is observed
between ERing~1 and EOct, which becomes weaker for
larger pseudorapidities. Even at the highest pseudorapidities, ERing~3
still increases with increasing EOct.
In the data, the dependence of ERing~1 on EOct is
similar in shape to that found in the MC simulation. At higher
pseudorapidities, however, the positive correlation is
restricted to the lowest EOct range and, after reaching a maximum,
ERing~2 and ERing~3 start to decrease with increasing EOct.
This same anticorrelated dependence was
observed in Au+Au data at higher energies in the correlation
between the Paddle Scintillator counters and the Zero Degree
Calorimeter (ZDCs). The ZDCs detect spectator neutrons and
include roughly the same relative $\eta$ region (i.e. when considering
the difference in beam rapidities ($y_{\rm beam}$) for different
collision energies: $\eta$\,--\,$y_{\rm beam}$) in
$\sqrt{s_{_{NN}}}$\,=\,200\,GeV collisions
as covered by Rings 2 and 3 for 19.6\,GeV, see for
example Ref.~\cite{cite:PHOBOS_130MidRap}.
It is possible that the multiplicity distribution from produced
particles narrows for more central collisions~\cite{cite:BigMultPaper},
however this could not account for the observed rise/fall behavior.
\subsection{Fragment Identification}
Fragments are identified using their relative energy loss (\mbox{$E_{\rm rel}$}\xspace)
in the silicon (see Eq.~\ref{eqn:Erel}).
Figure~\ref{fig:AuAuMethod} shows the \mbox{$E_{\rm rel}$}\xspace distribution measured
in the ERing acceptance for Au+Au collisions at
$\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV, where no centrality selection is made
and only the region 5.0$<$$|\eta|$$<$5.4 is shown in order
to make the higher mass fragments more pronounced.
In Fig.~\ref{fig:AuAuMethod}, the data is shown as
a blue spectrum along with the distribution expected from singly-charged
particles ($Z$\,=\,1, red). The latter is
considered to be a ``background'' to the data and is
determined from a MC simulation without spectator
fragments. This $Z$\,=\,1 contribution can be explicitly subtracted
as it is entirely due to singly-charged particles (mostly
from the collision) with a typical Landau-like distribution.
\subsection{Subtracting Singly-Charged Particles}
To determine the spectral shape of the $Z$\,=\,1 contribution, the energy
loss signal for single particles is modeled using a full {\sc Geant}
Monte-Carlo (MC) simulation of the PHOBOS apparatus.
In data and simulation, it is observed that multiple $Z$\,=\,1
particles can impinge on a single silicon sensor, causing an ensemble
distribution over many events to exhibit peaks at \mbox{$E_{\rm rel}$}\xspace$\sim$2~and~3
(note that these additional peaks are not clearly visible in Fig.~\ref{fig:AuAuMethod}).
The peak at \mbox{$E_{\rm rel}$}\xspace$\sim$2 (which occurs at a rate of about 8\% at the highest
pseudorapidities) has to be accounted for in the $Z$\,=\,1 subtraction.
The third peak is suppressed to a rate of 0.6\% and is ignored
in this analysis. As this rate is dependent on the charged-particle
multiplicity in each detector, this fraction varies with both
centrality and pseudorapidity, an effect observed in both data and simulation.
Importantly, data with a lesser contribution from
a second charged-particle effectively steepens the spectrum, changing the
amount of subtracted background.
To account for the second peak in the spectrum, both data and MC are
divided into five pseudorapidity and seven (six) centrality classes
for the Au+Au (Cu+Cu) analysis, respectively.
As the MC distribution only reflects the relative contribution of
1 and 2 singly charged-particles, each class produces a spectrum which
has a unique shape. To account for the contribution of a second singly charged particle,
each data class is systematically compared to all
centrality/pseudorapidity classes from the MC, i.e. 35 comparisons, therefore
testing the data against a large sample of simulated 2/1 hits-per-sensor
contributions. Each MC class is normalized to the data at the first
peak (close to \mbox{$E_{\rm rel}$}\xspace=1 in Fig.~\ref{fig:AuAuMethod}).
The optimal background is chosen as the one with the least $\chi^{2}$ difference
between data and MC \mbox{$E_{\rm rel}$}\xspace spectra, formed over a region around
the expected second peak position (1.5$<$\mbox{$E_{\rm rel}$}\xspace$<$2.5).
To systematically test the sensitivity of the one-to-two hits
contribution, $Z$\,=\,1 MC simulation samples with different one-to-two hits
ratios are used in the analysis. A systematic uncertainty
due to the $\chi^{2}$ procedure is assigned by considering two
further $Z$\,=\,1 distributions. First, the distribution
with the next-smallest $\chi^{2}$ was used, and a full
reanalysis was made. Second, a $Z$\,=\,1 distribution with
$\chi^{2}/d.o.f.$\,=\,$\chi^{2}_{min}/d.o.f.$\,+\,1 was selected, with a
full reanalysis performed. A systematic difference of 3\%--12\%
was found for the $Z$\,=\,2 fragment yield in Au+Au collisions in the
highest pseudorapidity bins.
In pseudorapidity and centrality bins where there is a negligible
higher-$Z$ yield, the MC class determined from this analysis
closely replicates the entire tail of the singly-charged particles.
\begin{figure}[h]
\centering
\includegraphics[width=0.99\linewidth]{Fragments_Method_A.pdf
\centering
\caption{\label{fig:AuAuMethod}
(color online) The distribution of the relative energy loss
in Au+Au collisions as $\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV averaged over the
centrality range 0\%--70\% and 5.0$<$$|\eta$$<$5.4.
The blue distribution shows data, the error bars indicate
statistical uncertainties only and the data
are not corrected for acceptance. The red distribution shows the
results from a MC simulation of singly-charged particles with
spectator fragments explicitly excluded. See text for discussion.}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.99\linewidth]{Fragments_Method_BCD.pdf
\centering
\caption{\label{fig:AuAuMethod2}
(color online) Panel (a) shows the \mbox{$E_{\rm rel}$}\xspace distribution after subtracting the $Z$\,=\,1 component. The dominant
peak at \mbox{$E_{\rm rel}$}\xspace$\sim$4 corresponds to $Z$\,=\,2 ($\alpha$) fragments. The red line
depicts the fit to determine fragment yields -- the solid part shows
the region over which the fit was made and the dashed is the
extrapolation under the higher-$Z$ peaks.
Panel (b) shows the same as (a) but with the contribution from the
$\alpha$ spectrum (red line in (a)) removed, highlighting the
distribution from $Z$\,$\ge$\,3 fragments. The red line shows a
fit to the Lithium peak, similar to that described in (a).
Panel (c) shows the same as (b), but with the contribution from
$Z$\,=\,3 particles removed, and the $x$-axis is extended to
show the presence of $Z$\,=\,6 and $Z$\,=\,7 fragments.
The error bars are statistical only; data
are not corrected for acceptance. See text for discussion.}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[angle=0,width=0.99\textwidth]{drawRap_AuAu20_new_color.pdf
\caption{\label{fig:AuAuEta}
(color online) Pseudorapidity dependence of $\alpha$ (panels
(a)-(g)), Lithium (h-n), Beryllium (o-u), and Boron
(v-ab) fragments measured in Au+Au collisions at
$\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV. Data are presented in bins of
centrality (more central in the rightmost panels) and
are averaged over both hemispheres, i.e. the number of
fragments per colliding nucleus.
The error bars represent the statistical uncertainty, the
error bands represent 90\% C.L. systematic uncertainties in the yield.}
\end{figure*}
\subsection{Extracting Fragment Yields}
The measured \mbox{$E_{\rm rel}$}\xspace distribution after subtraction of the fitted
$Z$\,=\,1 contribution is shown in Fig.~\ref{fig:AuAuMethod2}a.
The spectrum is
dominated by the $Z$\,=\,2 (referred to here as $\alpha$)~\footnote{Note: $Z$\,=\,2 could imply either $^{3}$He or $^{4}$He ($\alpha$). However, as the abundance of $^{4}$He is far greater, we refer to $Z$\,=\,2 as $\alpha$.}
fragments. To determine the
yield, the peak is fit with a convoluted
Landau and Gaussian function (solid red line) in a region close to the
$\alpha$ peak, such that the fit range does not overlap
the region where the Lithium peak is expected.
The mean position in the fit is constrained to be the
expected mean position for the $\alpha$ fragments.
The use of a Landau function
is necessary to account for the high tail which partially resides
underneath the higher mass peaks -- in much the same way
that the tail of the singly charged particles contributed to the
$\alpha$ peak, before subtraction. The total yield is
calculated as the integral of this fit, extrapolated to
encompass $\alpha$ fragments appearing at high \mbox{$E_{\rm rel}$}\xspace, for example under the
Lithium peak (shown by the dashed red line). This extrapolation ultimately contributes less
than 10\% of the total yield, and the agreement between
the raw data and the fit integrated over the same region (3\,$<$\,\mbox{$E_{\rm rel}$}\xspace\,$<$\,6)
is better than 3\%.
The full $\alpha$ contribution to the energy loss spectrum is
then subtracted (red line in Fig.~\ref{fig:AuAuMethod2}a) to leave only $Z$\,$\ge$\,3 fragments
(Fig.~\ref{fig:AuAuMethod2}b). Next, with a similar
procedure, the yield of Lithium fragments
is determined using a Landau+Gaussian form (red solid and dashed lines), which is then
subtracted from the relative energy loss spectrum. For the final
distribution, $Z$\,$\ge$\,4 shown in Fig.~\ref{fig:AuAuMethod2}c,
the effect of the Landau tail is overpowered by the
Gaussian width, and thus a two-Gaussian fit is used
to extract the yields for Beryllium and Boron fragments.
The mean positions used in this fit are constrained to be
the expected position for each fragment. The number of these
$Z$\,$>$\,3 fragments is only 1\% of $\alpha$ particles.
As such, a small constant offset is allowed to account for possible
uncertainties in subtracting $\alpha$ and Lithium contributions
to the spectrum, which could lead
to over- or under-subtraction on the spectrum.
For charges greater than five, the
full centrality and $\eta$ dependence is limited by
the statistics collected in the single day of Au+Au running at
the RHIC injection energy of $\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV, and are
therefore not included in this analysis. The same procedure
is used to obtain $Z$\,=\,2 and $Z$\,=\,3 fragment yields in
Cu+Cu collisions at $\sqrt{s_{_{NN}}}$\,=\,22.4\,GeV; $Z$\,$>$\,3 fragments
are not observed, even given the larger statistics of the sample.
\subsection{Corrections and Systematic Uncertainty}
The data are corrected for acceptance via simulation which compares
the number of tracks which impinge the detectors to all
tracks in the full solid angle. As
the $Z$\,=\,1 ``background'' is explicitly subtracted, no further
corrections are applied. The effect of absorption of
the fragments in the 1\,mm thick Beryllium
beam pipe was evaluated via a {\sc Geant} simulation and
was found to be negligible ($<$1\%) as the fragments
are high energy -- $E_{\rm fragment}$\,$\approx$\,9.8\,GeV
(11.2\,GeV) per nucleon for Au+Au (Cu+Cu) collisions.
Systematic uncertainties (90\% C.L.) are evaluated by performing
several checks, in addition to those due to the Landau
$Z$\,=\,1 background subtraction. The difference in the extracted yields
measured independently in the positive and negative pseudorapidity
regions of the PHOBOS detector is found to be 3\%--11\% for the $\alpha$
yields in Au+Au collisions at the highest pseudorapidities, dependent
on centrality. A shift of the
measured energy scale in the \mbox{$E_{\rm rel}$}\xspace calculation was applied ($\pm$5\%)
which results in a 1\%--8\% uncertainty on the $\alpha$ yield for the
highest pseudorapidities. A total systematic uncertainty of 11\%
is assigned on the $\alpha$ yield for the highest pseudorapidities
in the 40\%--50\% centrality class.
For larger fragments, an additional uncertainty due to the
subtraction of the measured $\alpha$ yield is estimated to be 1.5\% for
Lithium for the highest pseudorapidities in Au+Au collisions.
The systematic uncertainties for 40\%--50\% Au+Au collisions at
the highest pseudorapidities are 11\%, 20\%, and 45\%
for Lithium, Beryllium, and Boron, respectively.
It was also checked whether fragments could be due to
interactions between collision products and the beam pipe,
by measuring the number of $Z$\,=\,2 fragments in
$\sqrt{s_{_{NN}}}$\,=\,62.4\,GeV~and~200\,GeV data. Few were
observed in the former, while none were
observed at the highest energy. Should the high-$Z$
fragments have emanated from dead and active detector material, notably
the Beryllium beam pipe, then
the most central $\sqrt{s_{_{NN}}}$\,=\,200\,GeV data, which has a larger multiplicity,
would have included more background than the lower energy data. Instead,
we find no evidence of $Z$\,=\,2 (or higher) fragments in the highest
energy data, indicating that such backgrounds from dead material are
negligible.
\section{Results I -- Pseudorapidity Dependence}\label{sec:EtaDep}
\begin{figure*}[!ht]
\begin{minipage}{0.65\textwidth}
\centering
\includegraphics[angle=0,width=0.99\textwidth]{drawRap_CuCu22_new_color.pdf
\end{minipage}
\hspace{0.05\textwidth}
\begin{minipage}{0.24\textwidth}
\centering
\caption{\label{fig:CuCuEta}
(color online) Pseudorapidity dependence of $\alpha$ (filled
symbols) and Lithium fragments (open symbols) measured in Cu+Cu
collisions at $\sqrt{s_{_{NN}}}$\,=\,22.4\,GeV. Lithium fragment yields
are scaled up by a factor of 10 for clarity. Data are presented in
bins of centrality and are averaged over both
hemispheres, i.e. the number of fragments per colliding nucleus.
The error bars represent the statistical uncertainty, the
error bands represent 90\% C.L. systematic uncertainties in the yield.\\\\}
\end{minipage}
\end{figure*}
Both the Au+Au and Cu+Cu data are divided into five bins of
pseudorapidity and seven and six bins of centrality, respectively,
corresponding to the top 70\% (60\%) of nuclear inelastic
cross-section. Figure~\ref{fig:AuAuEta}
shows the measured fragment multiplicity, $dN/d\eta$, as a function of pseudorapidity
(tabulated data are included in Appendix~\ref{ap:Tables}), averaged
over both hemispheres (i.e. the number of fragments per
colliding nucleus) for Au+Au collisions at $\sqrt{s_{_{NN}}}$\,=\,200\,GeV. The first row corresponds
to $\alpha$ fragments. Li, Be, and B fragments are
shown in subsequent rows. The most central data (those with
the least number of spectators after the collision) are shown in the
rightmost column; the most
peripheral are shown in the leftmost column. As is
apparent from this figure, there are no $Z$\,$>$\,1
fragments for low pseudorapidities ($|\eta|$$<$4.0) and only a
small number of fragments are produced at high centrality (0\%--10\% central).
The lightest fragment measured ($\alpha$) is observed
in each of the last three $|\eta|$ bins, Lithium fragments
are observed in the highest two bins, and Beryllium and Boron
fragments are seen only in the highest $|\eta|$ bin.
Figure~\ref{fig:CuCuEta} shows the measured $dN/d\eta$ for
$\alpha$ and Lithium fragments in Cu+Cu collisions
at $\sqrt{s_{_{NN}}}$\,=\,22.4\,GeV -- note that Lithium
yields are scaled up by a factor of 10 for clarity. Similarly
to the Au+Au results, no spectator fragments are observed in
the low pseudorapidity region; Lithium fragments are only
observed in the highest pseudorapidity bins.
\subsection{Comparison to Charged-particle pseudorapidity density}
PHOBOS has measured charged particle production in the very forward region
($|$$\eta$$|$$>$$\sim$3) for Au+Au and Cu+Cu collisions~\cite{cite:SigFrag,cite:PHOBOS_AuAuCuCuMult,cite:BigMultPaper}.
It was observed that the yield of charged particles in this forward
pseudorapidity region is larger
in the most peripheral collisions compared to the central ones.
In those analyses, no distinction was made between singly- and
multiply-charged particles, so it was unclear how many of these particles were
protons (or deuterons or tritons) and how many were multiply-charged fragments.
Figure~\ref{fig:CompChHadAuAu}
(\ref{fig:CompChHadCuCu}) shows a comparison between the
pseudorapidity-averaged $\alpha$ yield in Au+Au (Cu+Cu) collisions measured in
this analysis and the charged-particle
multiplicity ($\eta$\,$>$\,3) from the prior PHOBOS
analyses~\cite{cite:BigMultPaper}.
For these centrality bins, the yield of multiply-charged spectator
fragments for both systems is typically small
(dN$_\alpha$/d$\eta$\,=\,3.8\,$\pm$\,0.6 in 30\%--40\%
central collisions at $\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV)
compared to the
total charged-particle multiplicity (18.5\,$^{+9.2}_{-12.5}$).
Therefore,
the majority of the particles in the forward region included in the previously
published analyses are singly-charged.
Averaged over centrality, the small abundance of multiply-charged
relative to singly-charged particles at the highest pseudorapidity
is also clearly seen in Fig.~\ref{fig:AuAuMethod}.
\begin{figure}[!t]
\centering
\includegraphics[angle=0,width=0.445\textwidth]{drawdNdetaComp_AuAu_new_color.pdf
\caption{\label{fig:CompChHadAuAu}
(color online) Comparison between the PHOBOS charged particle
multiplicity measured at positive $\eta$ in Au+Au collisions at
$\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV
and the yield of $\alpha$ and Lithium fragments, averaged over positive and negative $|\eta|$.
Panels (a), (b), (c), and (d) show the distributions in
centrality bins 0\%--10\%, 10\%--20\%, 20\%--30\%, and 30\%--40\%,
respectively. The open squares/light grey bands
represents the PHOBOS
multiplicity~\cite{cite:BigMultPaper}, filled (open)
circles represent the measured $\alpha$ (Li) yields.}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[angle=0,width=0.445\textwidth]{drawdNdetaComp_CuCu_new_color.pdf
\caption{\label{fig:CompChHadCuCu}
(color online) Comparison between the measured PHOBOS charged particle
multiplicity in Cu+Cu collisions at $\sqrt{s_{_{NN}}}$\,=\,22.4\,GeV
and the yield of $\alpha$ fragments.
Panels (a), (b), (c), and (d) show the distributions in
centrality bins 10\%--20\%, 20\%--30\%, 30\%--40\%, and 40\%--50\%,
respectively. The open squares/light grey bands
represents the PHOBOS multiplicity~\cite{cite:BigMultPaper},
filled circles represent the measured $\alpha$ yields.}
\end{figure}
\subsection{Comparison to Other Fragment Data}
The number of $\alpha$ particles measured by PHOBOS is
found to be similar to the yields measured in other
experiments. Figure~\ref{fig:EnergyComparison} compares
the measured $dN_{\alpha}/d\eta$ from PHOBOS (filled
circles with a band representing the 90\% C.L. systematic
uncertainties in the yield)
with that from the KLMM~\cite{cite:KLMM_PhysRevC}
(Au projectile with beam energy 10.6\,GeV per nucleon on a
fixed emulsion (Em) target) and KLM~\cite{cite:KLM_ActaPhysPol}
(Pb projectile with beam energy 158\,GeV per nucleon on a fixed
Pb target) collaborations\footnote{The error bars shown for KLM and KLMM
data in
Fig.~\ref{fig:EnergyComparison} are based on the number
of counts, $N$, in each $\eta$ bin as $\sqrt{N}$.}.
Note that the PHOBOS
data are effectively a collision of a Au projectile with
$E_{\rm beam}$\,=\,9.8\,GeV per nucleon on a target Au nucleus (albeit
moving) where this energy is that of a single beam in
the collider, i.e. $\sqrt{s_{_{NN}}}$/2. The data are shifted along
the $x$-axis in Fig.~\ref{fig:EnergyComparison} by the corresponding beam rapidity in each
case.
A detailed discussion of the properties of this shifted variable
($\eta'$\,=\,$\eta-y_{beam}$ or for symmetric collisions $\eta'$\,=\,$|\eta|-y_{beam}$) is given in Appendix~\ref{ap:EtaPrimer}.
Any impact of the difference of collision energy should be fully
compensated by this beam rapidity shift, however as neither the
collision systems nor the event selection are identical some
systematic differences are expected. Small
differences in yield between Au+Au and Pb+Pb might arise
from the fact that the Pb+Pb collisions from the KLM analysis are
on average more peripheral (covering 0\%--100\%) than the Au+Au
collisions (0\%--70\%) from this analysis. As such, any excess
yield in the PHOBOS
measurements might be due to the missing 30\% of the most
peripheral events in this data set. Moreover, we do not see any
additional systematic effect between our data and the KLMM
data that collided Au nuclei on Em (comprising much smaller nuclei:
H, He, C, Ag, and Br).
Although a large part of the $\alpha$ yield is outside
the acceptance of PHOBOS, the yield in the measured region
agrees reasonably well between experiments, and also illustrates the
relevance of limiting fragmentation for
spectators~\cite{cite:SigFrag}. While Appendix~\ref{ap:EtaPrimer}
carefully describes why beam rapidity is an appropriate scale
to shift data at different energies, it is more intuitive to
compare boost-invariant quantities such as $dN/dp_{T}$.
Appendix~\ref{ap:dNdpt} estimates a conversion of the presented data
into $dN/dp_{T}$ as a function of $p_{T}$, and compares the resulting
distributions with those estimated from lower energy collisions, see Fig.~\ref{fig:dNdpt}.
The Cu+Cu data are not shown as the
expected difference in yield between Au (197) fragments
and Cu (63) fragments is large because of the difference in
mass -- whereas the difference
between Au (197) and Pb (208) should be negligible.
\begin{figure}[!t]
\centering
\includegraphics[angle=0,width=0.435\textwidth]{dNalphadetaComp_new_color.pdf
\caption{\label{fig:EnergyComparison}
(color online) Comparison of $\alpha$ yields between PHOBOS data from Au+Au collisions
($\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV) and Au+Em ($\sqrt{s_{_{NN}}}$\,=\,4.6\,GeV)~\cite{cite:KLMM_PhysRevC} and
Pb+Pb ($\sqrt{s_{_{NN}}}$\,=\,17.2\,GeV)~\cite{cite:KLM_ActaPhysPol} collisions.
PHOBOS data are averaged over positive and negative $\eta$ and over
the most central 0\%--70\% cross-section
(filled circles and shaded band which represent the 90\% C.L. systematic
uncertainties in the yield) for $\alpha$ particles.
The pseudorapidity ($x$-axis) is relative to the rest
frame of the target nucleus for each energy, as discussed in Appendix~\ref{ap:EtaPrimer}.}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[angle=0,width=0.95\textwidth]{drawCent_AuAu20_new_color.pdf
\caption{\label{fig:AuAuCentrality}
(color online) Centrality dependence of $\alpha$ (panels (a)-(e)),
Lithium (f-j), Beryllium (k-o), and Boron (p-t) fragments
measured in Au+Au collisions at $\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV. Data
are presented in bins of pseudorapidity, $\eta$, with the lowest
$\eta$ shown in the leftmost panels. The data are
averaged over both hemispheres, i.e. the number of
fragments per colliding nucleus. The error bars
represent the statistical uncertainty, the error bands represent
90\% C.L. systematic uncertainties in the yield. The
errors associated with the centrality variables (here $N_{\rm part}/2$) are not shown
on the figures, see Tables~\ref{tbl:AuAuAlphas}--\ref{tbl:CuCuLithium}. }
\end{figure*}
\section{Results II -- Centrality Dependence}\label{sec:CentDep}
Another way to look at this data is to examine the
centrality dependence, shown in
Fig.~\ref{fig:AuAuCentrality} for Au+Au collisions
at $\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV. Here, the absence of
fragments at low pseudorapidity is highlighted in
the first two columns.
Each $|\eta|$ bin with a significant signal (panels c-e, i-j, o, t)
shows a similar pattern:
an increase of the yield for peripheral events, a turn-over
for mid-central events, and finally an almost linear
decrease with $N_{\rm part}/2$ toward the fully overlapping
collisions. A similar dependence is also
seen in the measured ZDC energy distribution versus centrality in
the peripheral region at very high pseudorapidity, see for
example Ref.~\cite{cite:PHOBOS_WhitePaper}.
In Cu+Cu collisions at $\sqrt{s_{_{NN}}}$\,=\,22.4\,GeV, a similar
centrality dependence is observed for $\alpha$
and Lithium fragments in Fig.~\ref{fig:CuCuCentrality}.
\begin{figure}[!t]
\centering
\includegraphics[angle=0,width=0.475\textwidth]{drawCent_CuCu22_new_color.pdf
\caption{\label{fig:CuCuCentrality}
(color online) Centrality dependence of $\alpha$-fragments (filled symbols) for
$\sqrt{s_{_{NN}}}$\,=\,22.4\,GeV Cu+Cu collisions for four $|\eta|$
bins (a-d). For clarity, Lithium (open symbols) are scaled up by a factor
of~10 and are only shown for the highest two pseudorapidity bins
(panels (c) and (d)).
The data are averaged over both
hemispheres, i.e. the number of fragments per colliding
nucleus. The error bars (typically smaller than the symbol height)
represent the
statistical uncertainty, the error bands represent
90\% C.L. systematic uncertainties in the yield.}
\end{figure}
\subsection{Comparison of Au+Au and Cu+Cu data}
It should be noted that the
relative coverage ($\eta'\,\equiv\,|\eta|$\,--\,$y_{\rm beam}$)
of the detector is not quite the same for Au+Au and Cu+Cu
collisions owing to the different beam
rapidities: $y_{\rm beam}$\,=\,3.04 (3.18) for Au+Au (Cu+Cu).
Therefore, in comparing the two data sets, data points are evaluated
at the same average $\eta'$,
via an interpolation between measured points.
To evaluate the yield at each $\eta'$, a polynomial spline fit is
made which smoothly connects the measured data points.
The uncertainty in this method is evaluated with two different
fits, which are found to be within 10\% of the associated data point
systematic uncertainty. Figure~\ref{fig:SplineFit} shows an example of a fit
to peripheral (60\%--70\%) Au+Au ($dN_{\alpha}/d\eta$) data to determine interpolated
points at $\eta'$\,=\,1.57 and $\eta'$\,=\,2.02. A similar fit is
made to Cu+Cu data to determine an interpolated point at
$\eta'$\,=\,1.21.
\begin{figure}[!ht]
\centering
\includegraphics[angle=0,width=0.445\textwidth]{drawSplineFit_PaperFig_color.pdf
\caption{\label{fig:SplineFit}
(color online) Spline polynomial fits (lines) to $\alpha$ yields from Au+Au peripheral (60\%--70\%) data (filled
circles). Interpolated points at $\eta'$\,=\,1.57 and $\eta'$\,=\,2.02
are shown as open circles. The scale on
the upper $x$-axis shows $\eta'$\,$\equiv$\,$|\eta|$\,--\,$y_{\rm beam}$.
The dashed and green lines show fits using polynomials of different order.
The outer dotted lines represent a fit to points at the extreme
of the systematic uncertainty bands.}
\end{figure}
A comparison of the centrality dependence of $\alpha$ and
Lithium yields for Au+Au and
Cu+Cu is given in Fig.~\ref{fig:NSpec}.
The data are averaged over both hemispheres,
representing the fragments from a single
Gold (or Copper) nucleus. The yield of $\alpha$ and
Lithium fragments are shown versus $N_{\rm spec}$/2 from a single nucleus.
Note that the $x$-axis is inverted such that central collisions are located
rightmost on the figure.
The magnitude of the yields of fragments is proportional to
$N_{\rm spec}/2$ over a wide range of number of
spectators.
This behavior provides a simple
explanation for the smaller number of fragments observed in
peripheral Cu+Cu collisions compared to those from peripheral Au
fragmentation.
Modulo the drop-off for the most peripheral collisions,
yields are approximately similar in the two systems
for similar $N_{\rm spec}/2$.
There is some evidence that, at the same $N_{\rm spec}/2$,
the yield of $\alpha$ fragments is higher
in Cu+Cu than in Au+Au, which is not apparent for Lithium.
This is possibly due to a preference for emitting smaller fragments
in the smaller Copper nucleus.
\begin{figure}[!ht]
\centering
\includegraphics[angle=0,width=0.445\textwidth]{drawNspec_new_color.pdf
\caption{\label{fig:NSpec}
(color online) Centrality dependence of $\alpha$ (panel (a))
and Lithium yields (b) in $\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV Au+Au
(filled symbols) and 22.4\,GeV Cu+Cu
(open symbols) collisions.
Note that the centrality variable is not $N_{\rm part}/2$ but
$N_{\rm spec}$ from a
single nucleus -- see text for details -- and the $x$-axis
runs backwards, central
collisions are the rightmost data points. The $\alpha$ data
are evaluated at $\eta'$\,=\,1.57 (circles/unfilled systematic bands) and
$\eta'$\,=\,2.02 (squares/filled systematic bands). Lithium yields are only
shown for $\eta'$\,=\,2.02. The bands represent 90\% C.L.
systematic uncertainties in the yield. }
\end{figure}
\begin{figure*}[!ht]
\centering
\includegraphics[angle=0,width=0.75\textwidth]{drawNspecRatio_ScalingLiLinY_color.pdf
\caption{\label{fig:NSpecRatioLi}
(color online) Centrality dependence of the yield of Lithium nuclei divided by that of $\alpha$ particles
evaluated at $\eta'$\,=\,2.02.
Au+Au (filled symbols) and Cu+Cu (open symbols) collision data are shown
as a function of (a) $N_{\rm spec}/2$, (b) $N_{\rm part}/2$, and (c) the collision geometry ($N_{\rm spec}/2A$).
The bands represent 90\% C.L. systematic uncertainties in the ratio. }
\centering
\includegraphics[angle=0,width=0.75\textwidth]{drawNspecRatio_3panelLinY_color.pdf
\caption{\label{fig:NSpecRatio}
Centrality dependence of the yield of $\alpha$-particles evaluated at
$\eta'$\,=\,1.57 divided by the yield measured at $\eta'$\,=\,2.02.
Au+Au (filled symbols) and Cu+Cu (open symbols) collision data are shown
as a function of (a) $N_{\rm spec}/2$, (b) $N_{\rm part}/2$, and
(c) the collision geometry ($N_{\rm spec}/2A$).
The bands represent 90\% C.L. systematic uncertainties in the ratio.}
\includegraphics[angle=0,width=0.75\textwidth]{drawNspecRatio_3panelLinY_color_CuCuShifted.pdf
\caption{\label{fig:NSpecRatio2}
Centrality dependence of the yield of $\alpha$-particles evaluated at
$\eta'$\,=\,1.21 divided by the yield measured at $\eta'$\,=\,2.02.
Au+Au (filled symbols) and Cu+Cu (open symbols) collision data are shown
as a function of (a) $N_{\rm spec}/2$, (b) $N_{\rm part}/2$, and (c) the collision geometry ($N_{\rm spec}/2A$).
The bands represent 90\% C.L. systematic uncertainties in the ratio.}
\end{figure*}
\subsection{Pseudorapidity and Centrality Dependence of Yields}\label{sec:EtaCent}
The simultaneous pseudorapidity and centrality dependencies
of the yields can be explored by use of ratios of data, to
investigate whether the fragments appear at the same relative
position for all centralities or not. Figure~\ref{fig:NSpecRatioLi}
shows the ratio of the yield of Li to He fragments evaluated at
$\eta'$\,=\,2.02.
The three panels show the same data as a function of (a) $N_{\rm spec}/2$, (b) $N_{\rm part}/2$, and (c) the
collision geometry ($N_{\rm spec}/2A$).
Between Au+Au and Cu+Cu collisions, the Li/$\alpha$ ratios clearly
do not exhibit a scaling with either $N_{\rm part}/2$
(i.e. a similar Li/$\alpha$ ratio at a similar $N_{\rm part}/2$)
or with collision geometry.
The collision geometry, defined as $N_{\rm spec}/2A$,
represents the fraction of total nuclear volume
which interacts such that the overlap shape for each nucleus is roughly
similar.
A scaling with $N_{\rm spec}/2$ is suggested by the data -- the decreased ratio would
indicate that the emission of the lighter fragments is favored for
fewer spectator nucleons from the collision system.
However, the possibility that this ratio for each system is
constant with centrality is not ruled out within the systematic
uncertainty. For this scenario, the lower Cu+Cu ratio would
indicate a more favorable emission of the lighter fragment in the
Cu+Cu system than in Au+Au collisions.
From this data, one may attempt to draw a picture of the emission process for
fragments.
Unless the spectator nucleons acquire some $p_{T}$ from intrinsic
Fermi motion or the collision process itself, they would simply
travel straight down the beam pipe until the
magnetic field of the RHIC steering magnets bent them
away. In such a case, they would not be visible in the
detector as these magnets are located too far from the
apparatus to have had any influence on the fragments.
The movement of the fragments must be connected to the
nucleus and/or be the result of the collision.
In the simplest scenario, the fragments would move outward
due to their intrinsic (precollision) motion, without further
interaction. This, however, would result in the centrality
and pseudorapidity dependencies being decoupled from each other.
Specifically, the data in every pseudorapidity interval should have
the same centrality dependence (although with different yields);
this is not seen in the data.
Figures~\ref{fig:NSpecRatio}~and~\ref{fig:NSpecRatio2} show the
ratio of $\alpha$
yields evaluated at $\eta'$\,=\,1.57 and $\eta'$\,=\,1.21,
respectively, divided by the yield at $\eta'$\,=\,2.02,
for both Au+Au and Cu+Cu collision systems.
The three panels show the same data as a function of (a) $N_{\rm spec}/2$,
(b) $N_{\rm part}/2$, and (c) the
collision geometry.
The ratios in Figs.~\ref{fig:NSpecRatio}~and~\ref{fig:NSpecRatio2}
are not constant as the number of $\alpha$ particles in each
$\eta'$ range ($\eta'$\,=\,1.57 and 1.21, respectively) diminishes
(compared to the reference at $\eta'$\,=\,2.02) with decreasing centrality.
Effectively, the $\alpha$ particles are moving out of the acceptance of the
detector for more peripheral collisions and the
average deflection away from the beam direction
increases for more central collisions.
Such a deflection is suggestive of a specific dependence of
transverse momentum acquired by the fragments.
The same effect is also observed in Cu+Cu collisions.
For fragments moved into the acceptance of PHOBOS due to
intrinsic (precollision) motion, one would expect
no centrality dependence of these ratios, i.e. all flat.
Comparing the Cu+Cu and Au+Au data in the three scaling scenarios,
it is apparent that these ratios favor a scaling with $N_{\rm part}/2$, which is perhaps counter-intuitive as these spectators are
often considered to be independent of interactions in the hot participant zone.
\section{Conclusion}
In conclusion, nuclear fragments ($Z$\,$>$\,1) have
been observed up to $Z$\,=\,7 using the extensive reach in pseudorapidity
of the PHOBOS detector. The pseudorapidity and centrality dependence is
shown for fragments up to $Z$\,=\,5 only for Au+Au; for Cu+Cu this study
is restricted to $Z$\,=\,2~and~3.
Fragments from Au+Au
($\sqrt{s_{_{NN}}}$\,=\,19.6\,GeV) and Cu+Cu ($\sqrt{s_{_{NN}}}$\,=\,22.4\,GeV) collisions
have sufficiently low longitudinal momentum that
even fragments which have a modest $p_{T}$~are deflected into
the PHOBOS apparatus. The yield of $\alpha$ fragments
is observed to be similar to that measured in other
experiments over a range of energies if evaluated at the
same value of $\eta-y_{\rm beam}$. As a function of centrality,
the yield of $\alpha$ and Lithium fragments is found to approximately scale
with the number of spectators in the collision.
The centrality dependence of ratios of $\alpha$ fragment yields at
different pseudorapidities illustrates that these fragments move
out of the acceptance of the detector for more peripheral collisions.
In comparing Cu+Cu and Au+Au ratios, a scaling with the number
of participants is favored, suggesting an influence of the
hot participant zone with the nonparticipating spectators.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,332 |
Church Membership Is Falling, But What About the Sky?
Insights| Church Life & Ministry | May 19, 2021
Julian Hochgesang photo | Unsplash
By Aaron Earls
Less than half of Americans say they belong to a house of worship, marking the first time, since Gallup began collecting data in 1937, a majority aren't part of a church, synagogue, or mosque.
Religious membership was stable throughout the 20th century but fell from 70% in 2000 to 47% in 2020. While this should cause concern among church leaders, the situation may be less dire than it appears.
In 2020, religious membership dropped below 50% in the U.S. for the first time since Gallup began tracking such information. Click To Tweet
So, what explains this significant membership drop? Three factors can help churches better understand the religious landscape in the United States and how to reach their changing communities.
Rise in nones
Undeniably, fewer Americans say they belong to a religion, much less a specific religious institution. According to Gallup, "nones" (those who choose "none of the above" when asked their religious preference), have grown from 8% in 2000 to 20% in 2020. The General Social Survey (GSS) noted an increase from 14% to 23% in the same time frame. Pew Research tracks the growth from 17% in 2009 to 26% in 2019.
It should be noted that the growth is not driven primarily from a jump in atheists or agnostics—both only saw modest 2-point increases in Pew's findings. Similarly, the percentage of Americans who say they don't believe in God inched upward from 3% in 2000 to 5% in 2018, according to the GSS. Instead, as Pew's data found, those who are simply "nothing in particular" climbed from 12% in 2009 to 17% in 2019.
Growth of the religiously unaffiliated in the U.S. does not primarily come from a jump in atheists or agnostics, who only grew by 2 percentage points in the past 10 years. Click To Tweet
Among those without a religious preference, Gallup found—as expected—few (4%) say they have a formal church membership. This means that as more Americans fail to identify with a religion, more Americans will choose not to identify as a member of a church.
Institutional rejection
Church membership is falling, but so are numerous other official ties to organizations. Americans are increasingly wary of institutions.
In another Gallup survey, the percentage of Americans who say they have quite a lot or a great deal of confidence in the church or organized religion dropped almost 15 points in the past 20 years. While 42% say they trust the church, that's down from 56% in 2000. But other groups and institutions have seen significant declines in the past two decades: TV news (18 points), newspapers (13 points), Congress (11 points), big business (10 points), banks (8 points), Supreme Court (7 points), and police (6 points).
When seeking to add to their membership or even maintain current levels, congregations are fighting an uphill battle against the culture's anti-institution tendencies. Click To Tweet
Similarly, Americans don't trust the leaders of those institutions. Fewer than half of U.S. adults say judges (43%), bankers (29%), journalists (28%), labor union leaders (24%), local officeholder (24%), lawyers (21%), state governors (20%), state officeholder (19%), business executives (17%), senators (13%), and members of Congress (8%) have high or very high honesty and ethics, according to Gallup.
See also What Does the Embodied Church Mean for Kids?
Among pastors, public trust has fallen precipitously since the turn of the century when 64% of Americans gave them high ratings for their honesty. In recent years, that number has fallen below 40%. In the most recent survey, 39% had a positive opinion about the ethics of clergy.
If Americans have a negative view of institutions, including churches, and they lack trust in the leaders of those institutions, including pastors, it is no wonder church membership has dropped in recent years. When seeking to add to their membership or even maintain current levels, congregations are fighting an uphill battle against cultural tendencies.
Church clarity
Growing up, I remember the sign above the piano in our small country church that announced numbers like last week's Sunday School and worship service attendance, as well as our church membership. While we only hit triple digits on a good Sunday, our membership number always stayed above 300. Many of those in church today won't have that same experience.
While membership has undergone a precipitous drop, the decline in religious service attendance has been much less severe. Click To Tweet
While membership has undergone a precipitous drop, the decline in religious service attendance has been much less severe, according to Gallup. In 2000, 44% of Americans said they attended a service in the last seven days. That fell to 34% in 2019. In Gallup's other church attendance measurement, 46% of Americans said they attended a religious service at least almost every week in 2000. In 2020, 33% said the same.
Other surveys measuring church attendance find even smaller declines. According to the GSS, 38% of Americans attended religious services at least two to three times a month in 2000. That dipped only 2-percentage points to 36% by 2018.
The social benefit of maintaining a church membership is disappearing. While there is obviously something lost when a culture drops much of its Christian influence, the church is also granted a clarifying opportunity. Now more than ever, church leaders know who is part of their congregation. As membership numbers decline, they move closer to actual attendance numbers and give churches a more accurate picture of their congregation.
While church membership may be falling, it's not taking the sky down with it. This should be an opportunity for prayer and faith, not one of panic and fear. — @WardrobeDoor Click To Tweet
While church membership may be falling, it's not taking the sky down with it. As more nominal Christians become nones and culture continues to avoid institutional ties, pastors and church leaders can see who truly is a follower of Christ and who in their community they need to reach with the gospel. This should be an opportunity for prayer and faith, not one of panic and fear.
Aaron Earls
@WardrobeDoor
Aaron is the senior writer at Lifeway Research.
Onward: Engaging the Culture Without Losing the Gospel
Russell D. Moore
Churches Still Recovering From Pandemic Losses
Most Open to Spiritual Conversations, Few Christians Speaking
22 Vital Stats for Ministry in 2022 | {
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{"url":"https:\/\/www.acmicpc.net\/problem\/13078","text":"\uc2dc\uac04 \uc81c\ud55c \uba54\ubaa8\ub9ac \uc81c\ud55c \uc81c\ucd9c \uc815\ub2f5 \ub9de\uc740 \uc0ac\ub78c \uc815\ub2f5 \ube44\uc728\n2 \ucd08 512 MB 10 5 4 50.000%\n\n## \ubb38\uc81c\n\nYou have possibly heard of DFAs. In theory of computation, a deterministic finite automaton (DFA) is a finite state machine that accepts\/rejects finite input strings. The figure on the right illustrates a DFA using a state diagram. In the automaton, there are three states: S0, S1, and S2 (denoted graphically by circles).\u00a0The automaton takes a finite sequence of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 and 1. Upon reading a symbol, a DFA jumps from a state to another by following the transition arrow. For example, if the automaton is currently in state S0 and current input symbol is 1 then it jumps to state S1. This is formally written as\u00a0\u03b4(S0, 1) = S1. Inductively, we can generalize this notation to have propositions like \u03b4(S0, 10) = S2 and \u03b4(S1, 011010) = S0. A DFA has a start state (denoted graphically by an arrow coming in from nowhere) where computations begin with, and a set of accept states (denoted graphically by a double circle) which help define when a computation is successful. So, an input string w is accepted by a DFA with starting state q0 if and only if \u03b4(q0, w) is an accepting state of that DFA. The state S0 in the DFA depicted above is both the start state and an accept state. In fact, this DFA accepts only binary numbers that are multiples of 3 (including the empty string).\n\nGiven A DFA D with states S0, S1, \u2026 , Sn-1, a string w is called history-cleaner for D if for all i,j \u2208 {0, \u2026 , n-1}: \u03b4(Si, w) = \u03b4(Sj, w). In other words, no matter which state the starting state is, the string w brings the DFA to a common final state. A DFA D is called history-cleanable if a history-cleaner string exists for D. Given a DFA whose input is a sequence of 0s and 1s, your job is to find out whether it is history-cleanable or not.\n\n## \uc785\ub825\n\nThe first line of the input contains the single integer t, the number of test cases (1 \u2264 t \u2264 500). Each test case starts with a line\u00a0containing a single integer n, the number of states in a DFA with states S0, S1, \u2026, Sn-1(2 \u2264 n \u2264 500). The second line of each test case consists of n space-separated integers a0, a1, \u2026, an-1 and finally, the third line of each test case has n space-separated integers b0, b1, \u2026, bn-1 (0 \u2264 ai, bi < n) which means\u00a0\u03b4(Si, 0) =\u00a0Sai and\u00a0\u03b4(Si, 1) = Sbi (for 0 \u2264 i < n).\n\n## \ucd9c\ub825\n\nFor each test case, print one line containing the answer for the given DFA. If it is history-cleanable, print \u201cYES\u201d, otherwise print \u201cNO\u201d (omit the quotes).\n\n## \uc608\uc81c \uc785\ub825 1\n\n2\n4\n1 2 3 0\n3 0 1 0\n4\n1 2 3 0\n2 2 1 2\n\n\n## \uc608\uc81c \ucd9c\ub825 1\n\nNO\nYES","date":"2018-10-17 11:53:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.20083242654800415, \"perplexity\": 1068.217307671066}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583511173.7\/warc\/CC-MAIN-20181017111301-20181017132801-00208.warc.gz\"}"} | null | null |
The 2002 University Act is the legal basis for the university's area of activity. Moreover, every university may pass by way of statutory instruments the required rules of order (statute) in the frame of laws and regulations. The Statute contains information about tasks, services and the structure of individual bodies and administrative units at the University of Graz, as well as rules of procedure and links to relevant legal texts and regulations. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,764 |
using namespace llvm;
extern "C" int LLVMFuzzerTestOneInput(const uint8_t *Data, size_t Size) {
LLVMContext Ctx;
SMDiagnostic Err;
const std::unique_ptr<Module> M =
parseAssemblyString("declare i32 @foo(i32 )\n", Err, Ctx);
const StringRef MangledName((const char *)Data, Size);
// Make sure that whatever symbol the demangler is operating on is
// present in the module (the signature is not important). This is
// because `tryDemangleForVFABI` fails if the function is not
// present. We need to make sure we can even invoke
// `getOrInsertFunction` because such method asserts on strings with
// zeroes.
if (!MangledName.empty() && MangledName.find_first_of(0) == StringRef::npos)
M->getOrInsertFunction(
MangledName,
FunctionType::get(Type::getVoidTy(M->getContext()), false));
const auto Info = VFABI::tryDemangleForVFABI(MangledName, *M);
// Do not optimize away the return value. Inspired by
// https://github.com/google/benchmark/blob/master/include/benchmark/benchmark.h#L307-L345
asm volatile("" : : "r,m"(Info) : "memory");
return 0;
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Cephalotes dorbignyanus é uma espécie de inseto do gênero Cephalotes, pertencente à família Formicidae.
Referências
Formicidae
Espécies descritas em 1853 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,559 |
Q: Load French characters in Amazon Redshift from S3 I have a .csv file which contains French characters. The data is like this:
Antonie Bégarder,12345,6789,France
The file got loaded to S3. I am using COPY command to load the file from S3 to Redshift. I was getting the error:
String contains invalid or unsupported UTF8 codepoints. Bad UTF8 hex sequence: e9 6c 61 (error 4)
After that I used, ACCEPTINVCHARS as '`' in the COPY command. The data got loaded, but it looks like:
Antonie B`garder
Any solution for this?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,683 |
\section{Introduction}
Heavy ion collisions allow us to study strongly interacting matter
in a deconfined phase, the quark gluon plasma. In search for a critical
point in the QCD phase diagram, experiments cover a wide range of
collision energies, from very high energies at RHIC and LHC, down
to lower energies in the Beam Energy Scan program of RHIC \cite{Adamczyk:2013gw}
and at future programs at GSI FAIR and JINR NICA. The early times
of heavy ion collisions can be appropriately described in the
color glass condensate (CGC) framework \cite{Iancu:2002xk,Iancu:2001yq}.
The CGC framework models ultrarelativistic, highly Lorentz contracted
nuclei in terms of an effective classical field theory. Hard partons
are described as color charges, which act as sources for the soft
partons in terms of classical non-Abelian gauge fields due to gluon
saturation. The distribution of the color charges of very large nuclei
is given by the McLerran-Venugopalan (MV) model \cite{MV1,MV2}. More
recent sophisticated models such as IP-Glasma base the color charge
distribution on fits to deep-inelastic scattering data \cite{Schenke:2012fw,Schenke:2012wb}.
As a result of the collision the glasma is produced \cite{Gelis:2012ri},
which can be studied by numerically solving the Yang-Mills equations.
A common simplification is to assume infinitely thin incoming nuclei,
which leads to a single collision point in time and consequently to
boost invariance. This reduces the system to effectively 2+1 dimensions
\cite{Krasnitz:1998ns,Lappi:2003bi,Kovner:1995ja}. In this formulation
the gauge fields are rapidity independent by assumption. It is possible
to introduce rapidity dependence by including boost invariance breaking
fluctuations on top of boost-invariant background fields \cite{Gelis:2013rba,Fukushima:2011nq,Berges:2012cj}.
However, the initial conditions and evolution of the background fields
are still formulated in a boost-invariant way. Simulations of the
early stages of heavy-ion collisions using the CGC framework and real-time
lattice gauge theory have been highly successful in describing particle
multiplicities \cite{Schenke:2013dpa} and the azimuthal anisotropy
\cite{Schenke:2015aqa,Dusling:2014oha}. Studies of somewhat later
time intervals involving isotropization and thermalization of the
glasma have been undertaken using classical-statistical lattice gauge
theory with \cite{Gelfand:2016prm} and without fermions \cite{Berges:2008zt,Berges:2013fga,Romatschke:2006nk,Berges:2014yta},
hard loop approximation \cite{Ipp:2010uy,Romatschke:2006wg,Rebhan:2008uj,Rebhan:2009ku,Attems:2012js}
as well as kinetic theory \cite{Blaizot:2011xf,York:2014wja,Kurkela:2014tea,Kurkela:2015qoa}.
Within the CGC framework, there has also been progress in finding
analytical solutions for the gauge fields in the forward light cone
using expansions in small $\tau$ \cite{Chen:2015wia,Li:2016eqr,Fries:2006pv}.
Because of the infinitesimal thickness of nuclei in all the boost-invariant
approaches, the evolution of the color sources can be solved analytically.
Nontrivial evolution of color sources in the form of charged particles
can be simulated using the colored particle-in-cell method (CPIC)
method. It combines classical field dynamics described by real-time
non-Abelian lattice gauge theory \cite{PhysRevD.11.395} with classical
colored particle dynamics based on Wong's equations \cite{Wong}.
It is a non-Abelian generalization of the particle-in-cell (PIC) method
for the simulation of Abelian plasmas \cite{Verboncoeur2005}. CPIC
has been successfully applied to hard-thermal-loop simulations \cite{Hu:1996sf,Moore:1997sn},
the investigation of plasma instabilities \cite{Strickland:2007,Dumitru:2005hj}
and to jet energy loss \cite{Schenke2009} in the quark-gluon plasma.
Apart from pioneering work \cite{Poschl:1998px,Poschl:1999dm,Poschl:2000ip}
for very small transversal lattices this approach has not been used
yet to investigate the collision itself.
In this paper, we simulate the collision of two nuclei with finite
thickness in the laboratory frame in 3+1 dimensions in the CGC framework.
A finite nuclear thickness enables us to describe nuclei at lower
energies. Without a well-defined collision point, we have to drop
the assumption of boost invariance for the fields in the forward light cone.
As a consequence of an extended collision, we cannot describe the
evolution of the color sources analytically and we are forced to
include the color charges as dynamical degrees of freedom in the
simulation as they traverse the evolving overlap region of the two
nuclei. In studying lower collision energies, we are probing the limits
of applicability of the CGC framework, which becomes an accurate effective
description of QCD only at infinitely high energies. The goal of
this work is to show that a description in the laboratory frame using
CPIC is viable and can reproduce well-known results of boost-invariant
classical Yang-Mills simulations in the limit of small longitudinal
thickness of the nuclei. For simplicity we restrict ourselves to collisions
in the MV model, which describes ultrarelativistic nuclear matter
infinitely extended in the transversal directions, and to the gauge
group SU(2). We do not take into account other possible effects that
might come into play if the CGC picture is applied to lower energies,
but simply approach this region as a first step by varying the thickness
of the incoming nuclei.
This paper is organized as follows: In Sec.~\ref{sec:Colored-particle-in-cell}
we describe the CPIC method for heavy-ion collisions. We discuss the
equations of motion for the fields and color charges and initial conditions
in the laboratory frame. In Sec.~\ref{sec:Numerical-results} we
present our numerical results for collisions in the MV model with
finite thickness. We investigate the structure of the fields in the
forward light cone created during the collision and compare them to
the initial conditions used in boost-invariant simulations. We recover
the usual result of pressure anisotropy and investigate the energy
conservation in the system.\newpage{}
\section{Colored particle-in-cell method for heavy-ion collisions\label{sec:Colored-particle-in-cell}}
\begin{figure}
\begin{centering}
\includegraphics[scale=0.9]{cpic_overview}
\par\end{centering}
\protect\caption{Schematic overview of a heavy-ion collision modeled with CPIC in the
laboratory frame. The random color charge densities of the nuclei
are modeled by placing color charge carrying particles (here depicted
as small spheres) into each cell along the longitudinal direction.
These particles move continuously in the longitudinal direction, but
are fixed to the grid points in the transversal plane.\label{fig:color-sheets}}
\end{figure}
\global\long\defN_{\mathrm{p}}{N_{\mathrm{p}}}
The model of a heavy-ion collision which we implement in this work
is that of two sheets of color fields and charges, each occupying
a two-dimensional plane, colliding with each other at the speed of
light in the laboratory frame. The sheets modeling Lorentz contracted
nuclei as depicted in Fig.~\ref{fig:color-sheets} have a finite
extent in the longitudinal direction in which they propagate and a
largely random transversal color structure. On the other hand, in
our setup the longitudinal color structure is assumed to be coherent,
spreading a given color configuration over the complete thickness
of a nucleus. Each of the sheets consists of two contributions, a
charge distribution and its corresponding classical fields. The charge
distributions are chosen according to the McLerran-Venugopalan model
and are not directly participating in the collision dynamics while
being tied to the light cone. In our CPIC approach we model these
charges as classical particles with a non-Abelian color charge. Following
the core assumptions of the CGC framework, the charges generate classical
gluon fields, which travel alongside them in the sheet and are responsible
for the creation of matter during and after the collision. The dynamics
of these fields are consistently described by Yang-Mills equations
without any approximations.
The CPIC method simulates the evolution of colored point charges in
continuous phase space coupled to non-Abelian gauge fields on a discrete
lattice. In each simulation step, the equations of motion for particles
and fields are solved alternately. Currents for the field equations
are obtained by interpolating the motion of charges to the lattice,
while interpolating the discretized fields back to the continuous
particle positions gives rise to forces and parallel transport. The
simulation volume is modeled as a three-dimensional grid with $N_{L}\cdot N_{T}^{2}$
cells, where $N_{L}$ and $N_{T}$ are the number of cells in the
longitudinal and transversal directions respectively with spatial
lattice spacing $a_{s}$ and time step $a_{t}$. In each cell we define
the electric fields $E_{x,i}$, the gauge links $U_{x,i}$, and the
charge and current densities $\rho_{x}$ and $j_{x,i}$, where $x$
denotes the lattice site of the cell and $i\in\{1,2,3\}$ is a vector
index. The box is periodic in the transversal directions and fixed
boundary conditions are used for the longitudinal direction.
The initial conditions for the fields are generated from the charge
densities $\rho_{1}$ and $\rho_{2}$ of the two nuclei. This step
is described in Sec.~\ref{sec:Initial-conditions-fields}. The exact
form of $\rho_{(1,2)}$ depends on the model used to describe the
nuclei. We choose the longitudinal separation of the nuclei such that
the fields do not overlap in the beginning. The color charge densities
$\rho_{(1,2)}$ are then used to sample the particle charges. We place
$N_{\mathrm{p}}$ particles in each cell and apply the charge refinement algorithm
from Sec.~\ref{sub:Charge-refinement} to get a smooth distribution
of color charges among the particles. The fields and particles are
then evolved via the lattice equations of motion (see Sec.~\ref{sub:Field-equations-of})
and the nearest-grid-point interpolation method (see Sec.~\ref{sub:Nearest-grid-point-interpolation}).
Consequently, the Gauss constraint is fulfilled throughout the simulation
of the system. Similar to most collision simulations in the color
glass condensate framework we do not take the backreaction from the
fields onto the particles into account (apart from parallel transport
of the charges), i.e.~the particles' velocity is held constant at
the speed of light. As a consequence the particles act as a reservoir
of energy for the fields and total energy is not conserved. The maximum
simulation time is limited by the longitudinal length of the simulation
box, since the nuclei are continuously moving in the longitudinal
direction and will reach the end of the box after some time. The
color charge density $\rho^{a}(x)$ is treated as a random variable
following a probability functional $W[\rho]$ given by the MV model.
Observables are recorded during the simulation and then averaged
using a number of different initial charge densities $\rho_{(1,2)}$
according to $W[\rho]$.
\subsection{Field equations of motion\label{sub:Field-equations-of}}
\global\long\def\dagger{\dagger}
\global\long\def\partial{\partial}
In this section we review the standard lattice Yang-Mills equations
of motion. We start by discretizing the continuum Yang-Mills action
\begin{equation}
S=\intop d^{4}x\left[-\frac{1}{2}\mbox{tr}(F_{\mu\nu}F^{\mu\nu})+2\mbox{tr}(j_{\mu}A^{\mu})\right],
\end{equation}
with current density $j_{\mu}$, gauge field $A_{x,\mu}\equiv A_{\mu}(x)=A_{x,\mu}^{a}t^{a}$,
and field strength tensor $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig\left[A_{\mu},A_{\nu}\right],$
on a hypercubic lattice taking advantage of the lattice gauge formalism
in Minkowski space. The gauge links $U_{x,i}$ and $U_{x,0}$ at
the lattice site $x$ are defined by
\begin{eqnarray}
U_{x,i} & = & \exp(iga_{s}A_{x,i}),\\
U_{x,0} & = & \exp(iga_{t}A_{x,0}),
\end{eqnarray}
with the temporal and spatial lattice spacings $a_{t}$ and $a_{s}$.
We also define $U_{x,-\mu}\equiv U_{x-\mu,\mu}^{\dagger}$ and the plaquette
variables
\begin{eqnarray}
U_{x,ij} & = & U_{x,i}U_{x+i,j}U_{x+j,i}^{\dagger}U_{x,j}^{\dagger}\simeq\exp\left(iga_{s}^{2}F_{x,ij}\right),\\
U_{x,0i} & = & U_{x,0}U_{x+0,i}U_{x+i,0}^{\dagger}U_{x,i}^{\dagger}\simeq\exp\left(iga_{s}a_{t}F_{x,0i}\right),
\end{eqnarray}
where $F_{x,ij}$ and $F_{x,0i}$ are components of the non-Abelian
field-strength tensor. The continuum action can then be approximated
as
\begin{equation}
S\simeq S_{YM}+S_{J},
\end{equation}
with the Yang-Mills part
\begin{equation}
S_{YM}=\frac{a_{s}}{g^{2}a_{t}}\sum_{x}\left(\sum_{i=1}^{3}\mbox{tr}\left[U_{x,0i}+U_{x,0i}^{\dagger}\right]-\frac{1}{2}\left(\frac{a_{t}}{a_{s}}\right)^{2}\sum_{i=1}^{3}\sum_{j=1}^{3}\mbox{tr}\left[U_{x,ij}+U_{x,ij}^{\dagger}\right]\right)+C,
\end{equation}
and the source terms
\begin{equation}
S_{J}=2a_{s}^{3}a_{t}\sum_{x}\left(\mbox{tr}\left[\rho_{x}A_{x,0}\right]-\sum_{i=1}^{3}\mbox{tr}\left[j_{x,i}A_{x,i}\right]\right),
\end{equation}
where $C$ is an irrelevant constant. By varying the discretized action
with respect to the gauge fields $A_{x,\mu}$ and employing the temporal
gauge $U_{x,0}=\mathbf{1}$, which corresponds to $A_{0}=0$, we obtain
the discretized equations of motion. For our numerical approach we
choose the electric field $E_{x,i}\equiv F^{i0}$ and the spatial
gauge links $U_{x,i}$ as our degrees of freedom. The equations can
be solved numerically using a leap-frog scheme, where the electric
fields $E_{x,i}$ and charge density $\rho_{x}$ are evaluated at
whole time steps $t_{n}=na_{t}$, while the gauge links $U_{x,i}$
and current density $j_{x,i}$ are evaluated at half time steps $t_{n+\frac{1}{2}}=(n+\frac{1}{2})a_{t}$.
The discretized equations then read
\begin{equation}
U_{x,i}(t+\frac{a_{t}}{2})=\exp\left(-ia_{t}ga_{s}E_{x,i}(t)\right)U_{x,i}(t-\frac{a_{t}}{2}),\label{eq:eom_4b}
\end{equation}
\begin{equation}
E_{x,i}^{a}(t+a_{t})=E_{x,i}^{a}(t)+\frac{a_{t}}{ga_{s}^{3}}\sum_{j\neq i}2\mbox{Im}\,\mbox{tr}\left[t^{a}U_{x,ij}(t+\frac{a_{t}}{2})+t^{a}U_{x,i-j}(t+\frac{a_{t}}{2})\right]-a_{t}j_{x,i}^{a}(t+\frac{a_{t}}{2}).\label{eq:eom_4a}
\end{equation}
Since the Gauss constraint
\begin{equation}
\sum_{i=1}^{3}\frac{E_{x,i}(t)-U_{x-i,i}^{\dagger}(t-\frac{a_{t}}{2})E_{x-i,i}(t)U_{x-i,i}(t-\frac{a_{t}}{2})}{a_{s}}=\rho_{x}(t)\label{eq:gauss_4}
\end{equation}
must be preserved at every time step, the charge density $\rho_{x}$
and the current density $j_{x,i}$ must obey the covariant continuity
equation, i.e.
\begin{equation}
\frac{\rho_{x}(t)-\rho_{x}(t-a_{t})}{a_{t}}+\sum_{i=1}^{3}\frac{j_{x,i}(t-\frac{a_{t}}{2})-U_{x-i,i}^{\dagger}(t-\frac{a_{t}}{2})j_{x-i,i}(t-\frac{a_{t}}{2})U_{x-i,i}(t-\frac{a_{t}}{2})}{a_{s}}=0.\label{eq:continuity_1}
\end{equation}
\subsection{Particle equations of motion and interpolation\label{sub:Nearest-grid-point-interpolation}}
In the CGC framework hard partons are described by classical color
sources in terms of the charge density $\rho_{x}$. Within our simulations
we sample $\rho_{x}$ by a number of pointlike particles carrying
color charge. The interpolation step reconstructs the charge density
from the particle charges and continuous positions. In the transverse
plane of the heavy ion, we place one particle per cell in order to
match the resolution of the grid. As we will see later, multiple particles
per cell in the propagating direction are needed for better resolution
of the longitudinal profile. While the colored particles move through
the grid, they induce color currents $j_{x}^{a}$, which are used
to evolve the gauge fields via the lattice equations of motion (\ref{eq:eom_4b})
and (\ref{eq:eom_4a}). A main requirement is that the Gauss constraint
(\ref{eq:gauss_4}) must be satisfied at all times. This can be accomplished
by making sure that the currents generated by the particle movement
satisfy the covariant continuity equation (\ref{eq:continuity_1}).
This is the main idea behind charge-conserving methods, which are
commonly used in Abelian PIC simulations \cite{Esirkepov2001}.
One of the assumptions of the color glass condensate framework is
that the nuclei involved in the collision can be thought of as recoilless
sources moving at the speed of light. The charges of the nuclei pass
through each other without loss of energy or change of momentum, i.e.~the
particle trajectories are fixed. The longitudinal particle positions
$z(t)$ are simply updated with
\begin{equation}
z(t+a_{t})=z(t)+va_{t},\label{eq:position_update}
\end{equation}
with the velocity $v=\pm1$. This renders the interpolation problem
one dimensional in the longitudinal direction. Using this simplification
the continuity equation (\ref{eq:continuity_1}) reads
\begin{equation}
\frac{\rho_{x}(t)-\rho_{x}(t-a_{t})}{a_{t}}+\frac{j_{x}(t-\frac{a_{t}}{2})-U_{x-e_{z}}^{\dagger}(t-\frac{a_{t}}{2})j_{x-e_{z}}(t-\frac{a_{t}}{2})U_{x-e_{x}}(t-\frac{a_{t}}{2})}{a_{s}}=0,\label{eq:cont_2}
\end{equation}
where we choose $i=z$ as the longitudinal direction and drop the
direction indices.
In the simulation, we also need to interpolate the continuous particle
positions to the fixed lattice points of the charge density $\rho_{x}(t)$.
In this work, we implement a simple interpolation method called
the nearest-grid-point (NGP) method \cite{Moore:1997sn}. In the NGP
method a particle charge $Q(t)$ at position $x(t)$ is fully mapped
to the closest lattice point $n$. The charge density contribution
at this point from one particle is then given by
\begin{equation}
\rho_{n}(t)=\frac{Q(t)}{a_{s}^{3}}.\label{eq:ngp_charge_interpolation}
\end{equation}
As the charge moves through the grid, the charge density only
changes when the particle crosses the boundary in the middle of a
cell such that its nearest-grid-point changes. These boundaries can
be formally defined as the ones separating two cells on a lattice,
which is shifted by half a lattice spacing (Wigner-Seitz lattice),
with lattice points now marking the center and not the edges of each
cell. A current is only induced at such a boundary crossing. Evaluating
the one dimensional continuity equation (\ref{eq:cont_2}) at $x=n$
and at $x=n+1$ and requiring that the only nonzero current is $j_{n}(t-\frac{a_{t}}{2})$,
we find for a right-moving particle that moves from position $n$
to $n+1$ from time $t-a_{t}$ to $t$ the following current and updated
charge:
\begin{equation}
j_{n}(t-\frac{a_{t}}{2})=\frac{a_{s}}{a_{t}}\frac{Q(t-a_{t})}{a_{s}^{3}},\label{eq:right_move_current}
\end{equation}
\begin{equation}
Q(t)=U_{n}^{\dagger}(t-\frac{a_{t}}{2})Q(t-a_{t})U_{n}(t-\frac{a_{t}}{2}).\label{eq:right_move_transport}
\end{equation}
For the case of a left-moving particle from position $n$ to $n-1$
we get
\begin{equation}
j_{n-1}(t-\frac{a_{t}}{2})=-\frac{a_{s}}{a_{t}}\frac{Q(t-a_{t})}{a_{s}^{3}},\label{eq:left_move_current}
\end{equation}
\begin{equation}
Q(t)=U_{n-1}(t-\frac{a_{t}}{2})Q(t-a_{t})U_{n-1}^{\dagger}(t-\frac{a_{t}}{2}).\label{eq:left_move_transport}
\end{equation}
Equations (\ref{eq:right_move_transport}) and (\ref{eq:left_move_transport})
take care of the parallel transport of the charges.
The current generated by the NGP scheme can give rise to a lot of
numerical noise due to peaklike currents being induced only at certain
time steps. However, we can circumvent this problem by initializing
multiple particles per cell and by employing charge refinement procedures
(see Sec.~\ref{sub:Charge-refinement}). These improvements allow
us to simulate sufficiently accurate currents on the grid. Another
way to address this issue is to use more sophisticated interpolation
schemes such as the cloud-in-cell (CIC) interpolation, which is standard
for Abelian PIC simulations and also has been developed for the CPIC
method \cite{Strickland:2007}.
\subsection{Initial conditions\label{sec:Initial-conditions-fields}}
\begin{figure}
\begin{centering}
\includegraphics[scale=0.18]{initial_conditions_overview}
\par\end{centering}
\protect\caption{Schematic overview of the initial conditions in temporal gauge before
the collision. The color charge densities $\rho_{(1,2)}$ of the colliding
nuclei are depicted as colorful clouds. The gauge field $A_{\mu}$
in the center region of the box is exponentially close to zero. Behind
each nucleus the fields asymptotically approach the pure gauge configurations
$A_{\mu}^{\mathrm{(1,2)}}$ depicted as blue and red transparent regions.
At the longitudinal boundaries of the simulation box the gauge fields
are fixed to the static pure gauge configurations. \label{fig:temporal-gauge-overview}}
\end{figure}
As a model of a single nucleus, we want to construct a propagating
solution with given color charge $\hat{\rho}^{a}(x_{T})$ (not to
be confused with $\rho_{x}$, the charge density in three dimensions)
in the transverse plane as given by the MV model with $x_{T}=(x,y)$
denoting the transverse coordinates. The boost-invariant case assumes
that the nucleus is infinitely Lorentz contracted in the longitudinal
direction and therefore described by a color current,
\begin{equation}
J^{a\mu}=\delta(z-t)\hat{\rho}^{a}(x_{T})s^{\mu},\label{eq:initial_conditions_current_bi}
\end{equation}
with $s^{\mu}\equiv\left(1,0,0,1\right)^{\mu}$ for a random transverse
color charge configuration $\hat{\rho}^{a}(x_{T})$ that travels at
the speed of light in the positive $z$-direction. The restriction
we release is the requirement of an infinitely thin nucleus, by spreading
the color charge along the longitudinal direction. It is possible
to find a corresponding consistent field configuration such that charge
and fields both propagate together at the speed of light.
It is easiest to set up the solution in Lorenz gauge $\partial_{\mu}A^{a\mu}=0$.
We use the following ansatz for the four-current $J^{a\mu}=(\rho^{a},j_{i}^{a})$
and vector potential $A^{a\mu}$:
\begin{eqnarray}
J^{a\mu} & = & f(z-t)\hat{\rho}^{a}(x_{T})s^{\mu},\label{eq:Jansatznonabelian}\\
A^{a\mu} & = & f(z-t)\hat{\varphi}^{a}(x_{T})s^{\mu}.\label{eq:Aansatznonabelian}
\end{eqnarray}
The envelope $f(z-t)$ is arbitrary, and we will choose a Gaussian
profile
\begin{equation}
f(z-t)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(z-t)^{2}}{2\sigma^{2}}},
\end{equation}
with a given width $\sigma$, which is proportional to the thickness
of the nucleus in the laboratory frame.\footnote{In the boost-invariant case it is common to write the expressions
$J^{a\mu}$ and $A^{a\mu}$ a bit differently using light cone coordinates
$x^{\pm}=\frac{1}{\sqrt{2}}(t\pm z)$. One would then write Eq.~(\ref{eq:initial_conditions_current_bi})
as $J^{a\mu}=\delta(x^{-})\hat{\rho}^{a}(x_{T})\bar{s}^{\mu}$, where
$\bar{s}^{\mu}=\frac{1}{\sqrt{2}}\left(1,0,0,1\right)^{\mu}$ is the
unit vector in the $+$ direction. Applying this to nuclei with finite
thickness we would have $J^{a\mu}=\bar{f}(x^{-})\hat{\rho}^{a}(x_{T})\bar{s}^{\mu}$
and $A^{a\mu}=\bar{f}(x^{-})\hat{\varphi}^{a}(x_{T})\bar{s}^{\mu}$,
where $\bar{f}(x^{-})$ is a Gaussian profile with thickness parameter
$\bar{\sigma}$. Our convention differs from this by introducing the
thickness parameter $\sigma$ in the laboratory frame coordinates
instead of the light cone coordinate frame. The different conventions
for the widths $\sigma$ and $\bar{\sigma}$ are geometrically related
by $\sqrt{2}\bar{\sigma}=\sigma$. In the end it does not matter which
convention one uses, but one should be aware that a finite width in
the light cone frame differs from the width in the laboratory frame
by a factor of $\sqrt{2}$.} Plugging the ansatz into the non-Abelian Maxwell equations
\begin{equation}
D_{\mu}^{ab}F^{b\mu\nu}=J^{a\nu},
\end{equation}
nonlinear terms vanish due to $s^{\mu}s_{\mu}=0$ and the time dependence
drops out because of $s^{\mu}\partial_{\mu}f(z-t)=0$. As a consequence
one is left with the Poisson equation
\begin{equation}
-\Delta_{T}A^{a\nu}\equiv-\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)A^{a\nu}=J^{a\nu},\label{eq:Poisson_eq}
\end{equation}
which is solved in Fourier space for each color component separately
and formally denoted using the inverse Laplace operator $\Delta_{T}^{-1}=\left(\nabla_{T}^{2}\right)^{-1}$
by
\begin{equation}
\hat{\varphi}^{a}(x_{T})=-\frac{\hat{\rho}^{a}(x_{T})}{\nabla_{T}^{2}}.\label{eq:Poisson}
\end{equation}
The corresponding electric field is then given by
\begin{equation}
E_{i=1,2}^{a}=f(z-t)\partial_{i}\hat{\varphi}^{a}(x_{T}),\qquad E_{3}^{a}=0.\label{eq:ELorenz}
\end{equation}
In Lorenz gauge the chromoelectric fields are purely transverse while
the gauge fields retain only their temporal and longitudinal components.
All fields are nonzero exclusively in the space-time region close
to the light cone, where $J^{a\mu}$ is nonvanishing. In order to
switch to temporal gauge we apply a gauge transformation to the gauge
fields via
\begin{equation}
{A'}_{\mu}^{a}t^{a}=V\left(A_{\mu}^{a}t^{a}+\frac{i}{g}\partial_{\mu}\right)V^{\dagger},\label{eq:transformedfieldnonabelian}
\end{equation}
such that $A_{0}'(x)=0$ is fulfilled at all times. Consequently,
$V$ must satisfy the equation
\begin{equation}
\frac{\partial}{\partial t}V^{\dagger}=igA_{0}^{a}t^{a}V^{\dagger}.
\end{equation}
Since the gauge field configurations (\ref{eq:Aansatznonabelian})
commute at different times, the solution to this equation does not
require a time-ordered exponential, but is simply given by
\begin{equation}
V^{\dagger}(t,x,y,z)=\exp\left(ig\hat{\varphi}^{a}(x,y)t^{a}F(t,z)\right)\label{eq:wilson_line}
\end{equation}
with $F(t,z)\equiv\int_{-\infty}^{t}f(z-t')dt'.$ Using this gauge
transformation, the fields in the temporal axial gauge are given by
\begin{equation}
{A'}_{\mu=1,2}^{a}t^{a}=\frac{i}{g}V\left(\partial_{\mu}V^{\dagger}\right),\qquad{A'}_{\mu=0,3}^{a}t^{a}=0.
\end{equation}
The current has to be transformed properly into the temporal axial
gauge ${J'}_{\mu}^{a}t^{a}=V\left(J_{\mu}^{a}t^{a}\right)V^{\dagger}$
as well. The corresponding electric field can be calculated from
$E_{i}^{a}\equiv-\partial^{0}A^{ai}.$ We make two important observations
at this point. In contrast to the situation in Lorenz gauge, the gauge
fields are now purely transversal. Additionally, they are now defined
not only on the nuclear sheet close to the light cone as before, but also
in the spatial region behind each nucleus, forming a trace of constant
gauge fields (see Fig.~\ref{fig:temporal-gauge-overview}). Although
these fields are pure gauge configurations, which can be gauge transformed
to vacuum and thus do not carry any energy, their emergence forces
us to choose fixed boundary conditions in the longitudinal direction.
The Wilson line (\ref{eq:wilson_line}) required for the transformation
to temporal gauge is completely analogous to the lightlike Wilson
lines used in the boost-invariant formulation \cite{Iancu:2002xk,Iancu:2001yq}.
If we consider $\hat{\rho}^{a}(x_{T})$ as a random variable, the
ansatz (\ref{eq:Jansatznonabelian}) and (\ref{eq:Aansatznonabelian})
leads to uncorrelated fields in the transversal direction, but correlation
over the longitudinal extent of the nucleus. A more general ansatz,
which is beyond the scope of the current work, would also allow for
fluctuations in the longitudinal direction \cite{Fukushima:2007ki}
making a time-ordered exponential in Eq.~(\ref{eq:wilson_line})
necessary.
Furthermore, we introduce infrared (IR) and ultraviolet (UV) regulators
for the solution of the Poisson equation (\ref{eq:Poisson}). This
is done by solving in momentum space:
\begin{equation}
\hat{\varphi}^{a}(k_{T})=\begin{cases}
\frac{\hat{\rho}^{a}(k_{T})}{\left|k_{T}\right|^{2}+m^{2}} & ,\qquad\left|k_{T}\right|\leq\Lambda,\\
0 & ,\qquad\left|k_{T}\right|>\Lambda,
\end{cases}\label{eq:Poisson_reg}
\end{equation}
where the parameters $m$ and $\Lambda$ control the IR and UV regulation
and $\hat{\varphi}^{a}(k_{T})$, $\hat{\rho}^{a}(k_{T})$ are the
Fourier components of $\hat{\varphi}^{a}(x_{T})$ and $\hat{\rho}^{a}(x_{T})$
respectively. The IR regulator $m$ in the expression for $\hat{\varphi}^{a}$
introduces a finite correlation length on the order of $m^{-1}$ in
the transversal directions. The inclusion of the IR regulator and
the UV cutoff does not violate the field equations of motion or the
Gauss constraint, since it can be absorbed into a redefined charge
density
\begin{equation}
\hat{\rho}'^{a}(k_{T})=\frac{\left|k_{T}\right|^{2}}{\left|k_{T}\right|^{2}+m^{2}}\Theta(\Lambda-\left|k_{T}\right|)\,\hat{\rho}^{a}(k_{T}),
\end{equation}
which satisfies the unmodified Poisson equation in momentum space
\begin{equation}
\hat{\varphi}^{a}(k_{T})=\frac{\hat{\rho}'^{a}(k_{T})}{\left|k_{T}\right|^{2}}.
\end{equation}
Regulating the infrared modes with $m>0$ also enforces global color
neutrality, i.e.
\begin{equation}
\hat{\rho}'^{a}(k_{T}=0)=0.
\end{equation}
On the lattice we initialize the transversal gauge links at $t_{0}-\frac{a_{t}}{2}$
and $t_{0}+\frac{a_{t}}{2}$ via
\begin{equation}
U_{x,i}(t_{0}\pm\frac{a_{t}}{2})=V(t_{0}\pm\frac{a_{t}}{2},x)V^{\dagger}(t_{0}\pm\frac{a_{t}}{2},x+i),\qquad i\in\{1,2\},\label{eq:initial_conditions_lattice}
\end{equation}
and the longitudinal gauge links are set to the unit element. The
initial electric fields $E_{x,i}(t_{0})$ are computed from the gauge
link update (\ref{eq:eom_4b}). We then evaluate the Gauss constraint
(\ref{eq:gauss_4}) to obtain the correct three-dimensional color
charge density $\rho_{x}(t_{0})$, which is sampled by a number of
particles. One point charge per transverse grid cell is sufficient
to reproduce a given charge density in a transverse plane. The longitudinal
structure requires a higher resolution: In order to obtain a smooth
current with the NGP algorithm, the charge is distributed among $N_{\mathrm{p}}=a_{s}/a_{t}$
particles per cell, which are placed with equal spacing along the
longitudinal direction such that at each time step exactly one particle
crosses a Wigner-Seitz cell boundary. It is not sufficient to divide
the total charge within a cell to the particles equally. The sublattice
distribution of the charges has to be optimized with the charge refinement
algorithm described in the next section.
\subsection{Charge refinement\label{sub:Charge-refinement}}
\begin{figure}
\begin{centering}
\includegraphics[scale=0.65]{charge_refinement_ngp_1}\hfill{}\includegraphics[scale=0.65]{charge_refinement_ngp_3}
\par\end{centering}
\protect\caption{Example of charge refinement: the small charge $q_{i}$ is plotted
as a function of the position $i$ for $\protectN_{\mathrm{p}}=4$ small charges
per cell. In the left plot, an initial total charge per Wigner-Seitz
cell (separated by the gray vertical lines and given in this example
by the integral of the dashed line per cell) is equally distributed
among four small charges per cell. After applying the charge refinement
algorithm (right plot), the total charge per Wigner-Seitz cell is
exactly the same as in the left plot, but it approximates the continuous
charge distribution (dashed line) significantly better. Red dots indicate
the result for constant discrete second derivative, Eq.~(\ref{eq:delta_second}),
while black dots show the result for constant discrete fourth derivative
within a cell, Eq.~(\ref{eq:delta_forth}). \label{fig:charge-refinement}}
\end{figure}
Up to this point, we have only specified the total charge in a cell,
but not how the charge is distributed within the cell. A constant
charge distribution within each cell as seen in Fig.~\ref{fig:charge-refinement}
on the left results in a ``jittery'' color current distribution
on the grid over time. This also impacts the evolution of the fields
and in particular leads to spurious longitudinal fields in the direction
of propagation.\footnote{To see why this is the case consider the equation of motion of the
longitudinal chromoelectric field. This argument can already be made
with the Abelian equation $\dot{E}_{L}=(\vec{\nabla}\times\vec{B})_{L}-j_{L}$.
The electric and magnetic fields of a single nucleus moving at the
speed of light are purely transverse, there are no longitudinal components.
Consequently, the longitudinal current must satisfy $j_{L}=(\vec{\nabla}\times\vec{B})_{L}$
at all times at each point in space. Any deviation from this produces
longitudinal chromoelectric fields, which in turn affect the future
time evolution of the other fields. In our simulations the spatial
shape and time behavior of the interpolated current depend on the
sublattice distribution of particle charges. A smooth distribution
of the charges is better at preserving the transversal field structure.} In order to avoid this, the shape of the charge distribution should
be as smooth as possible as seen in Fig.~\ref{fig:charge-refinement}
on the right.
For the NGP algorithm, we have to satisfy that the sum of all small
charges within a Wigner-Seitz cell equals its given total charge $Q_{j}$.
In order to distribute the total cell charge $Q_{j}$ to $N_{\mathrm{p}}$ small
charges $q_{i}$ with $N_{\mathrm{p}} j\leq i<N_{\mathrm{p}}(j+1)$ within the cell, we
can initialize them according to
\begin{equation}
q_{i}=\frac{Q_{j}}{N_{\mathrm{p}}},\quad{\rm for\;\;}N_{\mathrm{p}} j\leq i<N_{\mathrm{p}}(j+1).
\end{equation}
We then apply an iterative procedure that ensures that the total charge
within a cell is not altered. At each iteration step, two randomly
chosen neighboring charges $q_{i}$ and $q_{i+1}$ (with $i+1$ not
a multiple of $N_{\mathrm{p}}$) are assigned new values $q'_{i}$ and $q'_{i+1}$
according to
\begin{eqnarray}
q'_{i} & = & q_{i}-\Delta q,\label{eq:q0prime-1}\\
q'_{i+1} & = & q_{i+1}+\Delta q.\label{eq:q1prime-1}
\end{eqnarray}
This ensures for arbitrary $\Delta q$ that the total charge within
the cell is not modified. If one demands that the discrete second
derivative is constant within a cell, which is equivalent to demanding
that the discrete first derivatives of adjacent points form an arithmetic
series
\begin{equation}
\frac{q'_{i+1}-q'_{i}}{a_{s}}=\frac{1}{2}\left[\frac{q_{i+2}-q_{i+1}}{a_{s}}+\frac{q_{i}-q_{i-1}}{a_{s}}\right],
\end{equation}
then we find
\begin{equation}
\Delta q=\frac{q_{i+2}-3q_{i+1}+3q_{i}-q_{i-1}}{4}.\label{eq:delta_second}
\end{equation}
Applying Eqs.~(\ref{eq:q0prime-1}) and (\ref{eq:q1prime-1}) with
(\ref{eq:delta_second}) repeatedly to all points leads to a convergent
solution that is continuous and piecewise linear in the first derivative.
The algorithm cannot be applied directly to the border of two cells
(i.e.~$i=N_{\mathrm{p}} j-1$), so these points have to be left out.
One can also demand that the discrete third derivatives form an arithmetic
series:
\begin{equation}
\frac{q_{i+2}-3q'_{i+1}+3q'_{i}-q_{i-1}}{a_{s}^{3}}=\frac{1}{2}\Bigg[\frac{q_{i+3}-3q_{i+2}+3q_{i+1}-q_{i}}{a_{s}^{3}}+\frac{q_{i+1}-3q_{i}+3q_{i-1}-q_{i-2}}{a_{s}^{3}}\Bigg].
\end{equation}
On the left-hand side, we use that $q_{i+2}$ and $q_{i-1}$ remain
untransformed. The result is
\begin{equation}
\Delta q=\frac{-q_{i+3}+5q_{i+2}-10q_{i+1}+10q_{i}-5q_{i-1}+q_{i-2}}{12}.\label{eq:delta_forth}
\end{equation}
Convergence is fastest if the results are first iterated according
to condition (\ref{eq:delta_second}) and then according to (\ref{eq:delta_forth}).
An example of this procedure is shown in Fig.~\ref{fig:charge-refinement}.
\subsection{Simulation cycle}
For comprehensiveness we summarize the individual steps in our simulations.
First we generate initial conditions using the methods described in
the last two sections. This includes generating random charge distributions
according to the MV model, solving the two-dimensional Poisson equation
and initializing the chromoelectric fields and gauge links in the
temporal gauge. The Gauss constraint is then used to obtain the charge
density on the grid, which is sampled by a number of colored particles.
The distribution of particle charges is then made smooth with the
charge refinement algorithm.
After initialization at time $t_{0}$ the known variables are the
particle positions $x(t_{0})$, $x(t_{0}+a_{t})$ and charges $Q(t_{0})$,
the currents $j_{x,i}(t_{0}+\frac{a_{t}}{2})$, the electric fields
$E_{x,i}(t_{0})$ and the gauge links $U_{x,i}(t_{0}+\frac{a_{t}}{2})$.
The variables are then updated as follows:
\begin{enumerate}
\item Compute the new electric field $E_{x,i}(t_{0}+a_{t})$ via Eq.~(\ref{eq:eom_4a})
using $j_{x,i}(t_{0}+\frac{a_{t}}{2})$ and $U_{x,i}(t_{0}+\frac{a_{t}}{2})$.
\item Update $U_{x,i}(t_{0}+\frac{3a_{t}}{2})$ via Eq.~(\ref{eq:eom_4b})
using $E_{x,i}(t_{0}+a_{t})$.
\item Update longitudinal particle positions via Eq.~(\ref{eq:position_update}).
\item Update particle charges $Q(t_{0}+a_{t})$ according to either Eq.~(\ref{eq:right_move_transport})
or Eq.~(\ref{eq:left_move_transport}) (depending on sign of the
particle velocity $v$) if a particle crosses a nearest-grid-point
boundary.
\item Interpolate charge density $\rho_{x}(t_{0}+a_{t})$ using the NGP
scheme and Eq.~(\ref{eq:ngp_charge_interpolation}).
\item Interpolate currents $j_{x,i}(t_{0}+\frac{3a_{t}}{2})$ using either
Eq.~(\ref{eq:right_move_current}) or Eq.~(\ref{eq:left_move_current})
depending on sign of the particle velocity $v$.
\item Compute various observables such as field energy, pressure components,
etc.
\end{enumerate}
This completes a simulation step.
\section{Numerical results\label{sec:Numerical-results}}
\global\long\def\ev#1{\left\langle #1\right\rangle }
For all of our simulations\footnote{The code for our simulation framework is open-source and publicly
available at \url{https://github.com/openpixi/openpixi}.} we use a model similar to the one proposed by McLerran and Venugopalan
\cite{MV1,MV2}. As discussed in Sec.~\ref{sec:Initial-conditions-fields},
we consider charge distributions, which are random in the transversal
direction, but correlated in the longitudinal direction. The randomly
chosen color charge densities $\hat{\rho}_{1,2}^{a}(x_{T})$ in the
transversal plane are taken to be Gaussian with the correlation function
\begin{equation}
\ev{\hat{\rho}_{1,2}^{a}(x_{T})\hat{\rho}_{1,2}^{b}(y_{T})}=g^{2}\mu_{1,2}^{2}\delta^{(2)}(x_{T}-y_{T})\delta^{ab},
\end{equation}
where the parameters $\mu_{1,2}$ control the variance of the fluctuating
charges. McLerran and Venugopalan give an estimate of
\begin{equation}
\mu^{2}=1.1A^{1/3}\mbox{ fm}^{-2},
\end{equation}
where $A$ is the mass number of the colliding nuclei and the gauge
group is SU(3). For $A=197$ (Au) we get
\begin{equation}
\mu\approx0.505\mbox{ GeV}.
\end{equation}
We choose $g=2$ as common in CGC literature \cite{Lappi:2003bi,Fukushima:2011nq,Fukushima:2007ki,Lappi:2006hq}.
This leads to a realistic value for the saturation momentum $Q_{s}$,
\begin{equation}
Q_{s}\sim g^{2}\mu\sim2\mbox{ GeV}.
\end{equation}
Even though our simulation is currently restricted to SU(2) for performance
reasons, we still use this value to test our methods in semirealistic
scenarios.
In all simulations we use Au-Au collisions as the standard case study,
therefore $\mu$ is fixed. However, we vary other parameters such
as the simulation volume, the nucleus width $\sigma$ and IR and UV
regulators. For example, using $N_{T}=128$ cells in the transversal
directions and a lattice spacing of $a=0.028\mbox{ fm}$, the transversal
area roughly covers $12.5\%$ of the area of a single Au nucleus.
In the longitudinal direction we could use $N_{L}=256$ cells, which
covers a length of $5.2\mbox{ fm}$. These are the parameters used
in Sec.~\ref{sub:Comparison-with-boost-invariant}. For other parts
of this paper we chose different parameter sets, which are specified
in the corresponding sections.
We also have to choose the longitudinal thickness $l$ of the nuclei,
which is controlled by the longitudinal Gaussian width $\sigma$.
In Sec.~\ref{sub:Comparison-with-boost-invariant} we show that $\sigma=4a_{s}$
is a good lower limit to avoid lattice artifacts. We approximate the
thickness of the Gaussian profile by
\begin{equation}
l\approx4\sigma.
\end{equation}
Comparing the thickness $l$ to the longitudinal extent of the Lorentz contracted
nucleus $\frac{2R_{A}}{\gamma}$ , we obtain an estimate for the gamma
factor $\gamma$.
\begin{equation}
\gamma=\frac{2R_{A}}{4\sigma}=\frac{R_{A}}{8a_{s}},
\end{equation}
where $R_{A}\approx1.25A^{1/3}\mbox{ fm}$ is the radius of a nucleus.
With the values from above we get
\begin{equation}
\gamma\approx45.
\end{equation}
This value for $\gamma$ corresponds to a center-of-mass energy of
$\sqrt{s_{NN}}\approx90\,{\rm GeV}$, however in the course of our
paper we will demonstrate results obtained for $\gamma=11-455$, corresponding
to an energy range of $\sqrt{s_{NN}}=20-850\mbox{ GeV}$. This energy
range contains in particular parts of the parameter space explored
by the low-energy Au+Au collisions at RHIC in the beam energy scan
program with center-of-mass energies between $\sqrt{s_{NN}}=7.7-62.4\mbox{ GeV}$
\cite{Adamczyk:2013gw}.
For the solution of the Poisson equation in the transversal plane
we employ IR and UV regularization as in Eq.~(\ref{eq:Poisson_reg}).
Infrared regularization leads to average color neutrality and suppresses
long-range forces (e.g.~monopoles and dipoles) on length scales $m^{-1}$.
This is used to include effects of confinement in the classical simulation.
The confinement radius is roughly $1\mbox{ fm}$, therefore one possibility
is to set
\begin{equation}
m\approx(1\mbox{ fm})^{-1}\approx200\mbox{ MeV}\approx\Lambda_{\mbox{QCD}}.
\end{equation}
However we also work with values of up to $2$ GeV to study the dependence
of observables on the IR regulation. The UV cutoff $\Lambda$ is introduced
to eliminate high-momentum modes in the transversal plane whose dispersion
relation on the lattice differs from the analytic case. Unless otherwise
noted we use $\Lambda=10$ GeV.
\subsection{Comparison with boost-invariant initial conditions\label{sub:Comparison-with-boost-invariant}}
\begin{figure}
\begin{centering}
\subfloat{\protect\centering{}\protect\includegraphics{correlation_transverse_1}\protect}\hfill{}\hfill{}\subfloat{\protect\centering{}\protect\includegraphics{correlation_transverse_2}\protect}\hfill{}\hfill{}\subfloat{\protect\centering{}\protect\includegraphics{correlation_transverse_3}\protect}
\par\end{centering}
\protect\caption{Density plot of the energy density component $\mbox{tr}E_{L}^{2}(x_{T})$
as a function of the transverse coordinate $x_{T}=(x,y)$ in the center
region of the collision for a single event. The left panel shows the
boost-invariant (``analytic'') result for $\tau=0^{+}$. The middle
and right panels show the simulation results for two different values
of the thickness parameter $\sigma$. The correlation coefficient
$c$ quantifies how similar the energy density distributions are to
the boost-invariant case. Thinner nuclei (middle) lead to a correlation
coefficient of $c=0.85$, whereas the energy density distributions
of thicker nuclei (right) are more washed out with lower values of
$c=0.56$. The grid size $N_{L}\cdot N_{T}^{2}$ of the simulation
is set to $256\cdot128^{2}$ with a lattice spacing of $a_{s}=0.028\,\mbox{fm}$
and a time step of $a_{t}=a_{s}/2$. The IR regulator is set to $m=2\,\mbox{GeV}$
and the UV cutoff is $\Lambda=10\,\mbox{GeV}$. The transversal area
covers $12.5\%$ of the full area of a single Au nucleus, but we only
show a quarter of the area to make the similarities visually more
obvious. \label{fig:Density-plot-correlation}}
\end{figure}
It is important to check if the results produced by our 3+1 dimensional
simulations are similar to 2+1 dimensional boost-invariant simulations,
at least in the limit of thin nuclei. In the boost-invariant formulation
it is natural to work with proper time $\tau=\sqrt{t^{2}-z^{2}}$
and rapidity $\eta=\frac{1}{2}\ln\frac{t+z}{t-z}$ as coordinates
for the forward light cone, where the collision event at $t=z=0$
is used as the origin of the coordinate system. Note however that
in collisions of nuclei with finite thickness, there is some ambiguity
involved in choosing the space-time coordinates ($t_{c},z_{c}$) of
the collision. As a definition we set $(t_{c},z_{c})$ to the space-time
point of the maximum overlap of the Gaussian longitudinal profiles.
The main advantage of this definition is that these coordinates are
independent of the thickness parameter $\sigma$. To verify the agreement
with the boost-invariant case, we compare the longitudinal chromoelectric
fields created in our simulations to the fields, which are used as
initial conditions for boost-invariant simulations. The boost-invariant
initial conditions for the electric field at $\tau=0^{+}$ created
by the collision of charge densities of two nuclei $\hat{\rho}_{1}(x_{T})$
and $\hat{\rho}_{2}(x_{T})$ are given by \cite{Krasnitz:1998ns}
\begin{equation}
\left.E_{L}(x_{T})\right|_{\tau=0^{+}}=-ig\sum_{i=1,2}\left[\alpha_{1}^{i}(x_{T}),\alpha_{2}^{i}(x_{T})\right],\label{eq:boost_invariant_EL}
\end{equation}
where $\alpha_{1,2}^{i}(x_{T})$ is determined from the relations
\begin{eqnarray}
e^{iga_{s}\alpha_{1,2}^{i}(x_{T})} & = & e^{ig\hat{\varphi}_{1,2}^{a}(x_{T})t^{a}}e^{-ig\hat{\varphi}_{1,2}^{a}(x_{T}+i)t^{a}},\label{eq:bi_ic_1}\\
\Delta_{T}\hat{\varphi}_{1,2}^{a}(x_{T}) & = & -\hat{\rho}_{1,2}^{a}(x_{T}),\label{eq:bi_ic_2}
\end{eqnarray}
which are similar to our initial conditions in the laboratory frame:
The first equation corresponds to Eq. (\ref{eq:initial_conditions_lattice}),
and the second one to the Poisson equation (\ref{eq:Poisson_eq}).
This result is obtained in the Fock-Schwinger gauge $\tau A^{\tau}=0$.
\begin{figure}
\begin{centering}
\subfloat{\protect\centering{}\protect\includegraphics{correlation_sigma_scan}\protect}\hspace{0.5cm}\subfloat{\protect\centering{}\protect\includegraphics{correlation_ir_scan}\protect}
\par\end{centering}
\protect\caption{Comparison of simulations to boost-invariant initial conditions. This
plot shows the correlation coefficient of $\mbox{tr}E_{L}^{2}(x_{T})_{\mathrm{num}}$
in the central region with the boost-invariant result for $\mbox{tr}E_{L}^{2}(x_{T})_{\mathrm{ana}}$
at $\tau=0^{+}$ as a function of $\sigma$ and $m$. A correlation
coefficient of $1$ implies perfect agreement between the numerical
and the analytical result. The blue solid line shows the correlation
when the nuclei completely overlap and the red line is the maximum
correlation achieved during the evolution. The correlation increases
for thinner nuclei. Small values of $m$ and $\sigma$ lead to decreased
correlations due to numerical instabilities, which appear at high
field amplitudes. The simulation parameters are the same as in Fig.$\;$\ref{fig:Density-plot-correlation}
except that we vary the thickness parameter $\sigma$ and IR regulator
$m$. \textit{Left panel:} Correlations as a function of the thickness
parameter $\sigma$.\label{fig:Correlations-width} \textit{Right
panel:} Correlations as a function of the IR regulator $m$. For the
thick curves we used $\Lambda=10\,\mbox{GeV}$ as a UV cutoff. Dashed
lines use the UV cutoff $\Lambda_{latt}$ given by the lattice. \label{fig:Correlations-infrared}\label{fig:Comparison-of-simulations}}
\end{figure}
In our case the simulations start before the collision, where the
nuclei are well separated in the longitudinal direction such that
the gauge field in the center between them vanishes to numerical accuracy.
The longitudinal chromoelectric fields which we want to compare to
Eq.~(\ref{eq:boost_invariant_EL}) are produced by numerically evolving
the fields of the nuclei. We test our simulation as follows: We generate
two initial charge densities $\hat{\rho}_{(1,2)}(x_{T})$ and compute
$\left.\mbox{tr}E_{L}^{2}(x_{T})\right|_{\tau=0^{+}}$, which is a
gauge-invariant expression. Then we use the same charge densities
to run a 3+1 dimensional simulation with some finite nuclear thickness
controlled by the Gaussian width $\sigma$. We record the energy density
contribution of the longitudinal electric field $\mbox{tr}E_{L}^{2}(x_{T})$
as a function of the transversal coordinate $x_{T}$ during the collision
in the central region $\eta=0$. We then compute the correlation coefficient
$c$ between the numerical (simulation) result $\mbox{tr}E_{L}^{2}(x_{T}){}_{\mathrm{num}}$
and the analytic (boost-invariant) expression $\mbox{tr}E_{L}^{2}(x_{T}){}_{\mathrm{ana}}$
via
\begin{equation}
c(\mbox{tr}(E_{L}^{2})_{\mathrm{num}},\mbox{tr}(E_{L}^{2}){}_{\mathrm{ana}})\equiv\frac{\mbox{cov}(\mbox{tr}(E_{L}^{2})_{\mathrm{num}},\mbox{tr}(E_{L}^{2})_{\mathrm{ana}})}{\sigma_{\mathrm{num}}\sigma_{\mathrm{ana}}},
\end{equation}
where the covariance across the transversal plane is defined by
\begin{equation}
\mbox{cov}(\mbox{tr}(E_{L}^{2})_{\mathrm{num}},\mbox{tr}(E_{L}^{2})_{\mathrm{ana}})=\sum_{x_{T}}\left(\mbox{tr}E_{L}^{2}(x_{T}){}_{\mathrm{num}}-\overline{\mbox{tr}(E_{L}^{2})}_{\mathrm{num}}\right)\left(\mbox{tr}E_{L}^{2}(x_{T}){}_{\mathrm{ana}}-\overline{\mbox{tr}(E_{L}^{2})}_{\mathrm{ana}}\right),
\end{equation}
with the mean values $\overline{\mbox{tr}(E_{L}^{2})}_{(\mathrm{num},\mathrm{ana})}$
and the standard deviations $\sigma_{(\mathrm{num},\mathrm{ana})}$
associated with $\mbox{tr}E_{L}^{2}(x_{T}){}_{\mathrm{num}}$ and
$\mbox{tr}E_{L}^{2}(x_{T}){}_{\mathrm{ana}}$ respectively. The mean
and standard deviation are understood to be computed across the transversal
plane. A plot of the energy densities $\mbox{tr}E_{L}^{2}(x_{T})$
for different widths is shown in Fig.~\ref{fig:Density-plot-correlation}.
The correlation between the numerical and analytical results is recorded
as a function of time, the nuclear thickness $\sigma$ and the UV
cutoff $\Lambda$. The results for a single event as a function of
$\sigma$ are shown in Fig.~\ref{fig:Correlations-width} (left panel):
The blue (lower) curve corresponds to the fixed time $t_{c}$ where
the two nuclei overlap completely. We see that the correlation increases
for thinner widths $\sigma$, but at a certain point around $\sigma\approx3a_{s}$
the correlation is reduced due to discretization errors. Very thin
longitudinal profiles tend to disperse, produce unphysical longitudinal
fields even before the collision and eventually become unstable. This
is because thin widths cannot be properly resolved on the lattice
below a certain threshold. To ensure numerical stability we deduce
a minimum width of $\sigma_{min}=4a_{s}$ for the nuclear thickness
in our simulations. We remark that very fine lattices with small (in
physical units) lattice spacings are required to accurately simulate
thin nuclei on the lattice.
The red (upper) curve in Fig.$\;$\ref{fig:Comparison-of-simulations}
on the left is the maximum value of the correlation during the collision.
We see that thicker nuclei also produce fields which are similar to
the boost-invariant case (thus leading to higher correlations), but
at earlier times than $t_{c}$. This happens because they start to
overlap much earlier, producing the characteristic longitudinal electric
fields. The time evolution from the onset of the overlap to the full
overlap at $t_{c}$ changes the fields, resulting in low values of
the correlations at $t_{c}$ .
In Fig.~\ref{fig:Correlations-infrared} (right panel) we study the
effects of the IR regulator $m$ and the UV cutoff $\Lambda$ on our
results by fixing the width $\sigma$ and varying the values of $m$
and $\Lambda$. We see that cutting off high momentum modes whose
dispersion relation differs from the continuum case increases the
correlation with the analytic result. Regulating the UV modes becomes
necessary because in the MV model all available modes in momentum
space are populated up to the lattice cutoff scale $\Lambda_{\mathrm{latt}}$.
Increasing the resolution of the simulation box without regulating
the UV modes does not lead to any improvement.
We also observe that the correlation coefficient is largely independent
of the IR regulator $m$. However, lower values of $m$ decrease the
correlation significantly in the same manner as small values of $\sigma$
do. Small $m$ boosts the amplitudes of the low momentum modes of
$\hat{\varphi}^{a}(x_{T})$ [as is apparent from Eq.~(\ref{eq:Poisson_reg})],
which drives the same numerical instability we see when using very
small values of $\sigma$. This instability can be cured by using
finer grids (i.e.~smaller lattice spacings $a_{s}$) while keeping
the volume of the simulation box and all other physical parameters
fixed. A smaller lattice spacing $a_{s}$ for the same physical volume
of the box brings the gauge links $U_{x,i}$ closer to the group identity
element $\mathbf{1}$ and consequently the lattice approximations
of the fields become more accurate. We note that this instability
is of numerical nature only and also appears in the evolution of a
single nucleus without any collision.
Studying the correlation between our numerical results and the analytic
expressions for the boost-invariant initial conditions shows that
we are able to correctly describe boost-invariant collisions in the
limit of thin nuclei. However it also reveals that one has to be careful
in choosing simulation parameters, in particular $\sigma\gtrsim4a_{s}$.
To describe Au-Au collisions in our simulation framework we work with
an IR regulator of $m=2\,\mbox{GeV}$ (which is of the order of the
saturation momentum) and a UV cutoff $\Lambda=10\,\mbox{GeV}$ (which
is used to cut off high momentum modes not satisfactorily described
on the lattice). These parameters are used in the following sections
unless otherwise noted.
\subsection{Pressure anisotropy\label{sub:Pressure-anisotropy}}
\begin{figure}
\subfloat{\protect\centering{}\protect\includegraphics{p_labframe_-1fm}\protect}\hfill{}\subfloat{\protect\centering{}\protect\includegraphics{p_labframe_+2fm}\protect}\hfill{}\subfloat{\protect\centering{}\protect\includegraphics{p_labframe_+5fm}\protect}
\protect\caption{Longitudinal and transverse pressure components as functions of the
longitudinal coordinate $z$ in the laboratory frame at different
times $t$ before and after the collision. The coordinate origin is
centered around the collision event at $t=0$ and $z=0$. The blue
curve describes the longitudinal pressure $p_{L}(z)$ and the red
dashed curve is the transverse pressure component $p_{T}(z)$. The
longitudinal chromoelectric and chromomagnetic fields characteristic
for the glasma contribute to the transverse pressure $p_{T}$. The
pressure components are normalized to the maximum longitudinal pressure
$p_{0}$ of the initial nuclei. For these plots we use a grid size
of $320\times256^{2}$ with a lattice spacing of $a_{s}=0.04\,\mbox{fm}$
and a time step of $a_{t}=\frac{a_{s}}{2}$. The thickness parameter
is set to $\sigma=4a_{s}$ (which corresponds to a gamma factor of
$\gamma\approx23$), the IR regulator is set to $m=2\,\mbox{GeV}$
and the UV cutoff is set to $\Lambda=10\,\mbox{GeV}$. \textit{Left
panel:} Before the collision: $t=-1\,\mbox{fm}/c$. \textit{Middle
panel:} After the collision: $t=2\,\mbox{fm}/c$. \textit{Right panel:}
$t=5\,\mbox{fm}/c$. \label{fig:pL-pT-labframe}}
\end{figure}
\global\long\def\varepsilon{\varepsilon}
A prominent phenomenon in the early stages of heavy-ion collisions
is the pressure anisotropy of the glasma fields and the subsequent
isotropization of the system. The main observables in this context
are the transversal and longitudinal pressure components $p_{T}=\varepsilon_{L}$
and $p_{L}=\varepsilon_{T}-\varepsilon_{L}$ with the longitudinal and transversal energy
density components given by
\begin{eqnarray}
\varepsilon_{L} & = & \frac{1}{2}\left(E_{z}^{a}E_{z}^{a}+B_{z}^{a}B_{z}^{a}\right),\\
\varepsilon_{T} & = & \frac{1}{2}\sum_{i=x,y}\left(E_{i}^{a}E_{i}^{a}+B_{i}^{a}B_{i}^{a}\right).
\end{eqnarray}
Our simulation framework enables us to compute the pressure components
as functions of time $t$ and the longitudinal and transverse coordinates
$z$ and $x_{T}$. To simplify we average over $x_{T}$, which is
natural within the MV model. A plot of the pressure components in
the laboratory frame at different times is shown in Fig.~\ref{fig:pL-pT-labframe}.
The initially purely transverse fields of the incoming nuclei manifest
themselves as large Gaussian bumps in the longitudinal pressure component.
During the collision the transverse pressure component builds up and
remains largely flat afterwards. The longitudinal pressure in the
laboratory frame falls off exponentially towards the center of the
collision. In order to better compare our results to the boost-invariant
case it is sensible to switch to the comoving frame described by proper
time $\tau=\sqrt{t^{2}-z^{2}}$ and rapidity $\eta=\frac{1}{2}\ln\frac{t+z}{t-z}$.
We choose the space-time coordinates $(t_{c},z_{c}$) of the collision
as in Sec.$\;$\ref{sub:Comparison-with-boost-invariant}.
\begin{figure}
\begin{centering}
\includegraphics[scale=0.9]{p_comoving}
\par\end{centering}
\protect\caption{Longitudinal pressure component $\bar{p}_{L}(\eta)$ in the comoving
frame as a function of rapidity $\eta$ for different proper times
$\tau$. At later times the longitudinal pressure becomes flat within
the rapidity interval $(-1,1)$. Even though the nuclei in this simulation
are relatively thick ($\gamma\approx23$) we still recover approximate
boost invariance. For these plots we use the same simulation parameters
as in Fig.~\ref{fig:pL-pT-labframe}. \label{fig:pL_comoving}}
\end{figure}
By introducing the longitudinal component of the Poynting vector
\begin{equation}
S_{L}\equiv2\mathrm{tr}\left(\vec{E}\times\vec{B}\right)_{z},
\end{equation}
we can compute the transformed longitudinal pressure
\begin{equation}
\bar{p}_{L}(\tau,\eta)=p_{L}(\tau,\eta)\cosh^{2}\eta+\varepsilon(\tau,\eta)\sinh^{2}\eta-2S_{L}(\tau,\eta)\cosh\eta\sinh\eta.
\end{equation}
The transverse pressure component is unaltered by the coordinate transformation.
A plot of the longitudinal pressure $\bar{p}_{L}(\tau,\eta)$ in the
comoving frame is shown in Fig.~\ref{fig:pL_comoving}. It reveals
that at early times $\bar{p}_{L}$ is still largely influenced by
the tails of the colliding nuclei. At later times $\bar{p}_{L}$ becomes
flat in the midrapidity region, which is consistent with approximate
boost invariance. We have to keep in mind that within our simulations
the observables we compute are always slightly influenced by the initial
fields of the nuclei, especially at early proper times.
We now turn towards studying the pressure anisotropy. For the further
analysis it will be sufficient to stay in the central region $\eta=0$.
In the boost-invariant case the initial glasma fields at $\tau=0^{+}$
are made of purely longitudinal color flux tubes, which leads to highly
anisotropic initial pressures $\left.p_{T}\right|_{\tau=0^{+}}=\left.\varepsilon_{L}\right|_{\tau=0^{+}}$
and $\left.p_{L}\right|_{\tau=0^{+}}=-\left.\varepsilon_{L}\right|_{\tau=0^{+}}$.
As the flux tubes expand, they generate transversal electric and magnetic
fields until $\varepsilon_{L}\simeq\varepsilon_{T}$ and $p_{L}\simeq0$ \cite{Fujii:2008dd}.
This is the free-streaming limit observed in boost-invariant CGC simulations
and stands in contrast to the observation of an isotropized quark-gluon
plasma where $p_{T}\simeq p_{L}$ after a few $\mbox{fm}/c$ \cite{Romatschke:2007mq,Ryblewski:2012rr}.
It has been shown that boost invariance breaking fluctuations drive
instabilities in the glasma, which can move the system towards isotropization
\cite{Gelis:2013rba,Fukushima:2011nq,Berges:2012cj}. In our simulations
we explicitly violate boost invariance by introducing a finite nucleus
thickness. It is therefore interesting to investigate the effects
of the thickness parameter $\sigma$ on the pressure anisotropy of
the glasma.
For our numerical studies it is convenient to introduce the pressure
to energy density ratios $\frac{p_{T}}{\varepsilon}$ and $\frac{p_{L}}{\varepsilon}$
with $\varepsilon=\varepsilon_{L}+\varepsilon_{T}$. The free-streaming limit then corresponds
to $\frac{p_{T}}{\varepsilon}\simeq\frac{1}{2}$ and $\frac{p_{L}}{\varepsilon}\simeq0$.
Isotropization would be signaled by $\frac{p_{T}}{\varepsilon}\simeq\frac{p_{L}}{\varepsilon}\simeq\frac{1}{3}$.
Both the pressure and energy density components are averaged over
the transverse plane and $32$ events are used for the statistical
sampling. We choose a grid size of $320$ cells in the longitudinal
direction and $256^{2}$ cells to resolve the transversal area. For
collisions of thick nuclei in Fig.~\ref{fig:Pressure-components-thick}
(left panel) we choose a lattice spacing of $a_{s}=0.04\mbox{ fm}$.
The transversal grid then covers the full area $\pi R_{A}^{2}$ of
a gold nucleus.
\begin{figure}
\begin{centering}
\subfloat{\protect\centering{}\protect\includegraphics{p_ratio_2GeV}\protect}\hfill{}\subfloat{\protect\centering{}\protect\includegraphics{p_ratio_2GeV_thin}\protect}
\par\end{centering}
\protect\caption{Longitudinal and transversal pressure components in the central region
$\eta=0$ as a function of time for various nuclear thicknesses $\sigma$.
An IR regulator of $m=2\,\mbox{GeV}$ and a UV cutoff of $\Lambda=10\,\mbox{GeV}$
has been used. The detailed simulation parameters are explained in
the Sec.$\;$\ref{sub:Pressure-anisotropy}. \textit{Left panel:}
Pressure components for thick nuclei.\label{fig:Pressure-components-thick}
\textit{Right panel:} Pressure components for thin nuclei.\label{fig:Pressure-components-thin}\label{fig:pressure-comparison}}
\end{figure}
For simulations of thin nuclei in Fig.~\ref{fig:Pressure-components-thin}
(right panel) we are forced to use smaller lattice spacings of $a_{s}=0.008\,\mbox{fm}$
(for $\sigma=0.032\,\mbox{fm}$), $a_{s}=0.004\,\mbox{fm}$ (for $\sigma=0.016\,\mbox{fm}$)
and $a_{s}=0.002\,\mbox{fm}$ (for $\sigma=0.008\,\mbox{fm}$), because
grids much larger than $320\times256^{2}$ as used here currently
exceed our available computational resources. The transversal area
then only covers $4\%$, $1\%$ and $0.25\%$ of the full area respectively.
The temporal spacing is set to $a_{t}=\frac{a_{s}}{2}$.
The results are shown in Fig.~\ref{fig:pressure-comparison} and
there are several observations we make:
\begin{enumerate}
\item From Fig.~\ref{fig:Pressure-components-thick} (left panel) we see
that we recover the free-streaming limit of the boost-invariant case.
Isotropization is not reached within possible simulation times due
to limitations from both the longitudinal and transversal simulation
box size. We can observe slight movement of both pressure components
towards the desired value of $\frac{1}{3}$, but not within any realistic
time scales.
\item The initial pressures directly after the collision behave differently
compared to the boost-invariant case. In our simulations of thick
nuclei in Fig.~\ref{fig:Pressure-components-thick} (left panel)
we see that in the beginning $p_{L}$ dominates $p_{T}$ due to the
presence of the transverse fields of the colliding nuclei. As the
nuclei recede from the collision volume the created glasma fields
have already reached the free-streaming limit and therefore no negative
longitudinal pressures are observed.
\item In the results for thin nuclei in Fig.~\ref{fig:Pressure-components-thin}
(right panel) we can recover negative longitudinal pressure. The colliding
nuclei move away from the collision center fast enough, leaving behind
longitudinal color flux tubes, which have not decayed yet. The still
largely longitudinal fields generate negative pressure, which is characteristic
for the early glasma phase.
\end{enumerate}
We remark here that our ansatz for the initial conditions relies
on an assumption about the longitudinal structure of the nuclei. The
initial conditions described in Sec.~\ref{sec:Initial-conditions-fields}
imply correlation of the charge density in the longitudinal direction
of order $\sigma$ and correlation in the transversal direction of
order $m^{-1}$. The charge distribution is random in the transversal
direction, but there is no random longitudinal structure. As a consequence
we were able to drop the time ordering in Eq.~(\ref{eq:wilson_line}).
However, it has been shown that random longitudinal structure (which
demands proper path/time ordering) in the initial nucleus fields -
among other effects - leads to higher initial energy densities in
the glasma \cite{Fukushima:2007ki}. Additional longitudinal randomness
might also give rise to larger deviations from the boost-invariant
case after the collision. This could be similar to boost invariance
breaking perturbations of the glasma, which cause plasma instabilities
that have been found to accelerate isotropization \cite{Gelis:2013rba,Fukushima:2011nq,Berges:2012cj}.
It is therefore conceivable to expect that implementing this random
longitudinal structure in our initial conditions will change the results
and could lead to faster isotropization times. Detailed understanding
of these issues requires an analysis of gluon occupation numbers in
momentum space and their temporal behavior. We plan to investigate
this in a future publication.
\subsection{Energy production\label{sub:Energy-production}}
\begin{figure}[t]
\begin{centering}
\subfloat{\protect\centering{}\protect\includegraphics[scale=0.35]{Poynting_Spacesteps7}\protect}\hfill{}\subfloat{\protect\centering{}\protect\includegraphics[scale=0.35]{Poynting_Timesteps7}\protect}
\par\end{centering}
\protect\caption{Energy production as a function of time with different spatial and
temporal discretizations. The ``Sum'' curves correspond to the left-hand
side of Eq.~(\ref{eq:continuity1}). Their deviation from zero is
a consequence of lattice artifacts and can be reduced by using finer
time discretizations. The results have been obtained on a cubic lattice
with a fixed volume of $(5.12\:\mathrm{fm})^{3}$ with the IR regulator
set to $m=1\,\mbox{GeV}$ and an UV cutoff $\Lambda=10\,\mbox{GeV}$.
The nuclear thickness $\sigma$ was set to $0.16\,\mbox{fm}$. We
averaged over ten configurations in order to have a sufficient statistical
sample. \textit{Upper panel:} Varying spatial discretization by keeping
$a_{t}=0.01\:\mathrm{fm}/c$ fixed.\label{fig:Energy-production-1}
\textit{Lower panel:} Varying temporal discretization by keeping $a_{s}=0.04\:\mathrm{fm}$
fixed.\label{fig:Energy-production-2}\label{fig:Energy-production}}
\end{figure}
One of the fundamental assumptions made in the CGC framework is the
separation of hard and soft degrees of freedom, which are modeled
as external color charges and classical gauge fields respectively.
As a result of the collision there is an energy exchange between the
charges and the fields. However, since the nuclei are assumed to be
recoilless, the hard sector acts as an inexhaustible energy reservoir
for the gauge fields. The resulting field energy increase can be interpreted
as the work done by the charges against the field. In the boost-invariant
case this effect is implicitly included in the initial conditions
for the fields at $\tau=0^{+}$. In our approach we are able to explicitly
compute the energy increase during and after the collision. The change
of the total field energy density $\varepsilon$ as a function of time can
be formulated in terms of an energy continuity equation
\begin{equation}
\frac{d\varepsilon}{dt}+\frac{1}{V}\int\partial_{i}S_{i}d^{3}x+\frac{1}{V}\int E_{i}^{a}J_{i}^{a}d^{3}x=0,\label{eq:continuity1}
\end{equation}
which is the non-Abelian version of the Poynting theorem. The time
dependence of $\varepsilon$ is governed by two terms: the components $S_{i}$
of the Poynting vector $S_{j}\equiv2\mathrm{tr}\left(\vec{E}\times\vec{B}\right)_{j}$
and $E_{i}^{a}J_{i}^{a}$. The integral over the total derivative
of the Poynting vector can be omitted in the continuum. On the lattice
this term only gives a negligible contribution due to discretization
errors. In the scalar product $E_{i}^{a}J_{i}^{a}$ the only nonvanishing
part of the current $J_{i}^{a}$ is the longitudinal component $J_{z}^{a}$
and therefore the expression reduces to $E_{z}^{a}J_{z}^{a}$. Consequently,
the energy production is caused by longitudinal chromoelectric fields
in the glasma and must be centered around the collision event and
the boundary of the forward light cone where the color currents are
nonzero.
The energy increase as a function of time is shown in Fig.~\ref{fig:Energy-production-1}.
We observe that the total energy density is conserved before the onset
of the collision when the external charges and the classical fields
describing both nuclei are propagating through vacuum. Afterwards
there is a strong energy increase during as well as after the collision.
At later times there is an ongoing, but slowly decreasing energy production,
which finally becomes almost constant. To check the stability of our
results with respect to a change in the spatial and temporal resolution
of the grid we vary the spatial lattice spacing $a_{s}$ and the time step
$a_{t}$. Overall there is a good agreement between results at different
discretizations. The violation of Eq. (\ref{eq:continuity1}) is small
and can be further reduced using smaller time steps as seen in the
lower plot of Fig.$\;$\ref{fig:Energy-production}.
\subsection{Suppression of longitudinal chromomagnetic fields}
\begin{figure}
\begin{centering}
\subfloat{\protect\centering{}\protect\includegraphics{BLEL_ratio_thick}\protect}\hfill{}\subfloat{\protect\centering{}\protect\includegraphics{BLEL_ratio_thin}\protect}
\par\end{centering}
\protect\caption{Ratio of magnetic to electric longitudinal energy density contributions
as a function of time for various nuclear thicknesses $\sigma$. The
ratio increases for thin nuclei, but magnetic flux tubes are still
heavily suppressed compared to the boost-invariant scenario. The simulation
parameters are the same as in Sec.$\;$\ref{sub:Pressure-anisotropy}.
\textit{Left panel:} Ratio of longitudinal energy density components
for thick nuclei. \label{fig:Ratio-thick} \textit{Right panel:} Ratio
of longitudinal energy density components for thin nuclei. \label{fig:Ratio-thin}\label{fig:ratio-energy-density}}
\end{figure}
In the following we investigate the production of longitudinal chromomagnetic
fields $B_{L}^{a}$ and chromoelectric fields $E_{L}^{a}$ characteristic
for the glasma at early times. In the boost-invariant case the contributions
to the energy density from magnetic and electric color flux tubes
($\mbox{tr}B_{L}^{2}$ and $\mbox{tr}E_{L}^{2}$ respectively) should
be equal after averaging over initial conditions. In our simulations
with finite $\sigma$ we observe that this is not the case and there
is a dependency on the thickness parameter $\sigma$ as well as the
IR regulator $m$. The results are presented in Figs.~\ref{fig:ratio-energy-density}
and \ref{fig:ratio-energy-density-IR}.
Figure \ref{fig:Ratio-thick} shows the ratio of magnetic and electric
longitudinal fields $\ev{\mbox{tr}B_{L}^{2}}/\ev{\mbox{tr}E_{L}^{2}}$
in the central region ($\eta=0$) for a range of values of the nucleus
thickness $\sigma$ and an IR regulator $m=2\,\mbox{GeV}$. Collisions
of thick nuclei show a very small ratio of about $0.1-0.2$ after
the collision. In the case of thin nuclei in Fig.~\ref{fig:Ratio-thin}
(right panel) the ratio increases to roughly $\sim0.5$, which is
still far away from the ``canonical'' value of $1$ in the boost-invariant
scenario. Note that due to the small physical volumes used in the
simulations of thin nuclei it is harder to achieve adequate statistics.
As a result, the curves in Fig.~\ref{fig:Ratio-thin} in the right
panel are not as smooth as in the left panel.
\begin{figure}
\begin{centering}
\includegraphics{BLEL_ratio_IR}
\par\end{centering}
\protect\caption{Ratio of magnetic to electric longitudinal energy density contributions
as a function of time for various values of the IR regulator $m$.
For this plot we used a grid size of $256^{3}$ cells with a lattice
spacing $a_{s}=0.02\,\mbox{fm}$ and averaged over $32$ events. The
transversal area covers $25\%$ of the full area of a gold nucleus.
The thickness parameter is set to $\sigma=0.08\,\mbox{fm}$, which
corresponds to $\gamma\approx45$. We approach the boost-invariant
limit for small $m$. \label{fig:ratio-energy-density-IR}}
\end{figure}
The results do not only depend on the thickness $\sigma$. In Fig.~\ref{fig:ratio-energy-density-IR}
the results are shown for a fixed nuclear thickness $\sigma=0.08\,\mbox{fm}$
($\gamma\approx45$) and a varying IR regulator\footnote{We remark that varying the IR regulator $m$ has only a weak influence
on the pressure anisotropy. }. We observe that reducing $m$ to $200\,\mbox{MeV}$ (which roughly
corresponds to a correlation length of the color fields of the order
of the confinement radius $1\,\mbox{fm}$) leads to better agreement
with the boost-invariant case with a ratio of $\sim0.8$. Note that
this dependency of the ratio of magnetic and electric longitudinal
fields on the IR regulator $m$ is not present in the boost-invariant
initial conditions \cite{Krasnitz:1998ns}.
The presented results seemingly suggest a suppression of chromomagnetic
flux tubes (or an overproduction of chromoelectric flux tubes) in
the glasma phase when introducing a finite nucleus thickness. However,
the strong dependency of the magnetic to electric longitudinal field
ratio on the IR regulator $m$ leads us to suspect that this discrepancy
between our simulations and the boost-invariant case is an artifact,
which can be attributed to the initial conditions introduced in Sec.~\ref{sec:Initial-conditions-fields}.
As already mentioned there and in Sec.~\ref{sub:Pressure-anisotropy}
the longitudinal structure of our nuclei does not include ``longitudinal
randomness''. Consequently, the typical color structures in our initial
conditions have a thickness proportional to $\sigma$ and a transversal
width of the order of $m^{-1}$. To be consistent with the picture
of a highly Lorentz contracted nucleus modeled by classical Yang-Mills
fields one would demand that $\sigma m\ll1$, such that nucleons within
the nucleus are also contracted to flat ``pancakes''. Therefore,
if we move away from the limit $\sigma m\ll1$, we can expect to see
deviations from the boost-invariant case, but these deviations may
very well solely be due to the longitudinal coherence. This reasoning
is consistent with our simulation results: In the case of a thickness
of $\sigma=0.16\,\mbox{fm}$ with an IR regulator of $m=2\,\mbox{GeV}$
the longitudinal magnetic fields are weakened as seen in Fig.~\ref{fig:Ratio-thick}.
Here we have a value of $\sigma m=1.6$, which corresponds to color
structures which are prolonged in the longitudinal direction. We can
compare this to the case of $\sigma=0.08\,\mbox{fm}$ and $m=200\,\mbox{MeV}$
as presented in Fig.~\ref{fig:ratio-energy-density-IR}. The ratio
of magnetic to electric fields is closer to $1$ and at the same time
we have $\sigma m=0.08$, which can be considered small.
Including random longitudinal structure in the nuclei as suggested
in \cite{Fukushima:2007ki} will help to clarify, if suppressed
longitudinal magnetic fields are a physical consequence of a finite
thickness or if the suppression is just an artifact of our ansatz.
However with the reasoning presented above we suspect that this effect
will disappear for more realistic initial conditions.
\section{Conclusion\label{sec:Conclusion}}
In this work we have simulated heavy-ion collisions in the laboratory
frame with thick nuclei in the McLerran-Venugopalan model. Finite
thickness in the longitudinal direction allows the simulation of collisions
at lower energies, but requires abandoning boost invariance in the
calculation as well as including nontrivial color source evolution
in the simulation. Both can be readily implemented using CPIC in the
laboratory frame. With our framework we are able to access a range
of nuclear thicknesses down to those corresponding to center-of-mass
energies as used in the low-energy beam energy scan program of RHIC
and up to LHC energies.
We started from an analytic solution of a non-Abelian random color
current sheet of finite extent in the longitudinal direction and the
corresponding field configuration that propagate at the speed of light.
The discretization of this solution on a grid requires refining the
charge distribution on sublattice resolution. For the interpolation
between particles and fields, we utilized the nearest-grid-point method
as a charge conserving interpolation scheme. A distinct feature of
our approach is the possibility to explicitly compute the energy,
which is pumped into the Yang-Mills fields by the propagating color
charges. We verified that the energy increase correctly satisfies
the Poynting theorem for non-Abelian fields.
We compared calculations in the laboratory frame with results from
boost-invariant approaches. Concentrating on gauge-invariant observables,
we see that the correlation of initial conditions right after the
collision increases for thinner nuclei, which means that boost invariance
is restored in this limit. We computed the components of the energy-momentum
tensor, especially focusing on the pressure parallel and perpendicular
to the propagation direction. We show that our pressure distributions
in laboratory frame coordinate space correspond to largely rapidity independent
pressure distributions in Bjorken coordinates for the midrapidity
region. Our results confirm the previous findings, which established
the picture of strongly pronounced pressure anisotropy during the
very early phase of the fireball evolution.
For thicker nuclei we find the following deviations: There is a suppression
of the chromomagnetic longitudinal components of the energy-momentum
tensor with respect to their chromoelectric counterparts. We analyzed
this phenomenon and determined its dependence on the thickness parameter
and the IR regulator. Regarding pressure components, we observe a
slow tendency towards isotropization in our simulations. A more detailed
future investigation including random longitudinal structure in our
model could potentially further reduce isotropization times.
Other possible and planned improvements are the extraction of particle
spectra in order to compare with experimentally measured multiplicities
and also some rather technical aspects, like improved interpolation
prescriptions, which could be beneficial to widen the scope of parameters
accessible to our numerical approach. Another step towards a more
realistic simulation of QCD processes in heavy-ion collisions at low
collision energies would be the inclusion of backreaction of the classical
gauge fields onto the color charges. In the future the CPIC framework
could also allow us to take interactions and scatterings between the
hard constituents of both nuclei into account. Such steps, however,
would go beyond the usual assumptions of the CGC effective theory
and may require an improved understanding of the internal nuclear
structure.
\section{Acknowledgments}
This work has been supported by the Austrian Science Fund FWF, Project No. P 26582-N27 and Doctoral program No. W1252-N27. The computational results presented have
been achieved using the Vienna Scientific Cluster (VSC). We would
like to thank Anton Rebhan, S\"{o}ren Schlichting, Andreas Schmitt
and Raju Venugopalan for useful discussions.
\bibliographystyle{utphys}
\phantomsection\addcontentsline{toc}{section}{\refname} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,123 |
// Copyright 2014 Google Inc. All Rights Reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
package com.google.api.ads.adwords.awreporting.model.persistence.mongodb;
/**
* An Interface to set the proper MongoDB "_id" for entities that need MongoDB Storage.
*
* @author jtoledo@google.com (Julian Toledo)
*/
public interface MongoEntity {
/**
* Retrieves the Id to be used in Mongo.
*
* @return the String ID to be used in Mongo.
*/
String getId();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,001 |
cask 'remote-desktop-manager' do
version '3.5.2.0'
sha256 '8f4fb5737686f74ad01010a31dc93c259c48ae2babe07951652200f024debda1'
# devolutions.net was verified as official when first introduced to the cask
url "http://cdn.devolutions.net/download/Mac/Devolutions.RemoteDesktopManager.Mac.#{version}.dmg"
name 'Remote Desktop Manager'
homepage 'http://mac.remotedesktopmanager.com/'
license :commercial
app 'Remote Desktop Manager.app'
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 144 |
Q: HWIOAuthBundle custom user provider will not login user I'm using HWIOAuthBundle with Symfony 4.1.
My custom user provider gets called and returns a valid user, however the user is not logged in after the redirect. The web profiler shows: logged in as anon; anonymous token; firewall main
I've simplified the provider class below for brevity and it's only considering twitter right now. The conditions around $source are so I can add more later.
I have used xdebug to make sure loadUserByOauthuserResponse() is being called and that users are both being created in the case of a new user, or the existing user is being returned when it exists.
Question: If loadUserbyOauthUserResponse() IS returning a valid user entity, then what could be preventing it from creating a valid session with this user?
<?php
namespace App\Security;
...
class OAuthProvider extends OAuthUserProvider
{
...
public function loadUserByOAuthUserResponse(UserResponseInterface $response): User
{
/** @var EntityManager $em */
$em = $this->container->get('doctrine.orm.entity_manager');
$repo = $em->getRepository('App:User');
$source = $response->getResourceOwner()->getName();
$email = null;
$data = [];
$newUser = false;
// Set email and socialUser.
if ($source === 'twitter') {
$data = $response->getData();
$email = $data['email'];
}
// Check if this user already exists in our app.
$user = $repo->findOneBy(['email' => $email]);
if ($user === null) {
$newUser = true;
$user = new User();
$user->setPassword($this->strand(32));
}
// Set session and user data based on source.
if ($source === 'twitter') {
$name = $data['name'];
if ($newUser) {
$user->setNickName($name);
$user->setEmail($email);
$em->persist($user);
$em->flush();
}
}
return $user;
}
}
hwi_oauth.yaml
hwi_oauth:
# list of names of the firewalls in which this bundle is active, this setting MUST be set
firewall_names: [main]
# https://github.com/hwi/HWIOAuthBundle/blob/master/Resources/doc/2-configuring_resource_owners.md
resource_owners:
twitter:
type: twitter
client_id: '%env(TWITTER_ID)%'
client_secret: '%env(TWITTER_SECRET)%'
options:
include_email: true
services.yaml:
app.oauth_aware.user_provider.service:
class: App\Security\OAuthProvider
arguments: ['@service_container']
security.yaml:
security:
# https://symfony.com/doc/current/book/security.html#where-do-users-come-from-user-providers
encoders:
App\Entity\User: bcrypt
role_hierarchy:
ROLE_ADMIN: ROLE_USER
ROLE_SUPER_ADMIN: ROLE_ADMIN
providers:
app_users:
entity: { class: App\Entity\User, property: email }
hwi:
id: app.oauth_aware.user_provider.service
firewalls:
dev:
pattern: ^/(_(profiler|wdt)|css|images|js)/
security: false
main:
provider: app_users
anonymous: ~
remember_me:
secret: "%env(APP_SECRET)%"
lifetime: 2592000
path: /
guard:
authenticators:
- App\Security\LoginFormAuthenticator
entry_point: App\Security\LoginFormAuthenticator
logout:
path: /logout
target: /
oauth:
resource_owners:
twitter: "/login/check-twitter"
default_target_path: /
login_path: /
failure_path: /user-login
oauth_user_provider:
service: app.oauth_aware.user_provider.service
switch_user: ~
access_control:
- { path: ^/login$, role: IS_AUTHENTICATED_ANONYMOUSLY }
- { path: ^/register, role: IS_AUTHENTICATED_ANONYMOUSLY }
- { path: ^/recover, role: IS_AUTHENTICATED_ANONYMOUSLY }
- { path: ^/admin/, role: ROLE_ADMIN }
A: I had the same problem until I found in logs "Token was deauthenticated after trying to refresh it."
And after that I found this solution.
Token was deauthenticated after trying to refresh it
I added isEqualTo and did not logged me out.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,472 |
[RI CONFIDENCE GRAPH] The Rays Index Confidence Graph: Week 14
Confi 0 comments
The Rays Confidence Graph will appear every Wednesday and is a look at how much confidence Rays fans have in the Tampa Bay Rays. The graph is designed to give us a look at how our emotional bias as Rays fans fluctuates through time. The "confidence" in the team is an inexact measure of how fans feel about the team's current strength as well as how much confidence fans have in the franchise for the next 3-4 years. Notes on this weeks agida-level can be found after the graph..
Notes on the RI Confidence Graph…
The most common response for "Confidence in 2008 Rays" was 8 (They will be in playoff contention all season) with 64.9%.
The most common response for "Confidence in future of franchise" was 10 with 48.5%.
98.0% of respondents feel the Rays have a shot at the playoffs in 2008. That number is up from 96.7% a week ago.
99.0% of respondents feel the Rays should be at least a .500 team in 2008. Last week that number was 97.8%. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 88 |
SPOONER, Elizabeth Colton
SPOONER, Elizabeth Colton (née Elizabeth Swester Colton). 18?? — 19??. U.S. Oriental Scholar. Studied Sanskrit at Harvard under Lanman. In 1911 married her former co-student —> D. B. Sp. and lived with him in Patna. After husband's death apparently returned to the U.S.A., in 1935 living in Easthampton MA. Genealogy pages (findagrave.comvs. geni.com) claim that she was born in 1851 or 1852, married 1911 or 1912 and died 1927. The early birth seems right: In Andover Phillips Academy (https://archive.is/qBwV0#selection-6605.17-6605.24) graduate list she is given under 1868. She is said to have known many languages.
Publications: "The Fravashi of Gautama", JRAS 1916, 497-504.
Sources: Stray notes in Internet.
SPOONER, David Brainerd
SPOTTISWOODE, William
University of Leipzig University of Bonn Jainism Buddhism University of Vienna Colonial India Language studies Indian studies University of Paris Tibet Western explorers and traders University of Berlin islam Historical studies University of Munich British colonial time | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,075 |
Locution 'Thank You' is significant. It's a simpleton act that literally brings achiever for a patronage. This is why, Logoring, a pictorial and webpage invention delegacy from Australia has highly-developed its rattling own variant of gratitude mobile phone app development because of its customers. On this juncture of joy, Logoring desires to really thank its clients by gift often for allegory. A logotype is vital face of any byplay's stigmatization. Multitude familiar your trade and services victimization your logotype. If your logotype is memorable, it volition forever cue your clients of your steel. And with Logoring's offering you could get a classifiable and dateless logotype created for scarce $25.
Chicken mightiness be related chance, promise, friendship and wealthiness. Put-upon in easing, it could be utilized to center sections with a webpage. Greenness can be a fantastic semblance to use to sire a calmness and relaxed website, it can be associated with nature, begrudge, money besides as the constituent. Dingy is joined to embodied, effectiveness, pee and concord and the flatboat end in the blueing spectrum may be put-upon in site conception to farm a chill flavour.
Commtel Digital id a illustrious and sizeable effigy of existence a Digital Selling Bureau that cater their expertness in diverse fields care site pattern (growing and sustentation), societal networking campaigns same initiation of Facebook pages and it is care, E-commerce solutions, maturation and ontogenesis of nomadic and Facebook applications, On-line Advert Authority and looking locomotive optimisation. Moreover, they specifically center their digital strategies, researches, advertisements, sociable media or covering evolution services to be indisputable the winner and eudaimonia from the clients.
Normally, the complete team of a websitedesign agency will put up all of your websitedesign and digital marketing requirements. It comprisesof numerous multi-faceted brilliant team players who can shield various areasof web design and development as well as digitalmarketing . That means that the whole thing connected to your business countingthe design and marketing needs are considered by the same team in the same place. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,638 |
\section{Introduction}
High-dimensional datapoints such as natural images commonly carry complex semantic information. For example to characterize an image of a clothing item it is not enough to simply label it by its type, but we also need to know its color, gender type and size. Fine-grain annotations enable many downstream task on such data. Furthermore, it allows for efficiently organizing it in a hierarchical structure \cite{Blei2004}. Therefore, a clothing e-commerce retailer may benefit from a certain hierarchical structure (i.e. gender $>$ type $>$ color $>$ size) such that its customers (or a recommendation algorithm) can find what they are looking for quicker. This is beneficial since humans naturally group and cluster similar objects together in order to form a class or super class \cite{Sternberg2008}.
As the low-level pixel information in such data is far removed from the semantic meaning and annotations we typically need complex non-linear maps build usually with deep neural networks to map to these annotations. However, train such models we also need a significant amount of annotations.
To address this challenge we propose a method that leverages the efficiency of discrimination testing to capture the latent perception of difference between the data points by the annotators. Work in psychometics on measurement of subjective perception of objective stimuli provides strong insights in how such data collection can be effectively developed \cite{Fechner1889}. Specifically the two-alternative-forced choice (2AFC) method \cite{ehrenstein1999psychophysical}, which has been adapted for measurement of complex high-dimensional stimuli such as images and video \cite{maloney2003maximum, menkovski2012adaptive}.
In this paper we present a method that combines 2AFC tests, with active learning methods, deep metric learning and agglomerative clustering to develop a rich embedding of the data that captures semantic relationship between the data points and uncover this semantic structure.
In order to demonstrate the feasibility of the method we empirically confirm the shape vs color bias \cite{Ritter2017} by using our own created synthetic dataset and extract hierarchical structure on the Fashion-MNIST dataset to a finer granularity than the original labels.
\section{Related Work}
Extracting hierarchical structure from data is a lively field of study. In \cite{Li2010}, Li et al. present two types of hierarchies studied, namely language based (i.e. WordNet \cite{Miller1995, Snow2006}) and the low level visual feature based. Even though these approaches work fine and help in tasks like image organization, they lack the visual information that connects images together. Concepts like snowy mountains and skiing are far apart from each other on the WordNet hierarchy, which is a language based hierarchical approach but visually these concepts should be closer. There have been some purely visual feature based hierarchies \cite{Ahuja2007, Bart2008} but they are difficult to interpret. There motivation comes from the fact that the authors belief that an image hierarchy is not following a language hierarchical structure. For example, sharks and whales should be close neighbours on in image hierarchy which is a useful property of tasks such as image classification. One problem of such visual hierarchies is that none of the work was able to evaluate the effectiveness directly. This is why Li et al. \cite{Li2010} created a meaningful hierarchy for end-tasks such as image annotation and classification. Given the images and their tags (labels) their approach is able to automatically create a hierarchy, which is organizes images from very general to specific attributes. Ge et al. \cite{Ge2018} propose a hierarchical triplet loss (HTL) which is able to automatically collect insightful training samples by using a predefined hierarchical structure that encodes global context information. They have two main components in their method, the constructions of the hierarchical class tree and a dynamic margin.
Fine-grained image recognition (FGIR) tasks are also closely related to extracting hierarchical structure of image data. In \cite{Lin2015} the authors introduce a bi-linear model in order to create high-order image representations which are able to compute local pairwise interactions between features of two independent sub-networks. Such approaches have been enabled by the hierarchical representation learning present in modern convolutional neural network models \cite{Chen2016, Kaiming2016}. However, due to the high dimensionality of the features it becomes impractical for subsequent analysis. In order to reduce the high dimensionality of bilinear model features, Gao et al. \cite{Gao2016} introduced a model that approximates such bilinear feature by using polynomial kernels. Kong et al. \cite{Kong2016} went a step further and introduced a classifier co-decomposition to further restrict a bilinear model.
There has also been work that is able to capture the slight visual differences between categories \cite{Huang2016, Zhang2014} which uses bounding boxes to locate discriminative regions. The big drawback of this approach is that annotating these bounding boxes is a labour intensive process and these methods have therefore not been applicable to large-scale real world problems. In order to overcome this issue, visual attention models \cite{Chen2018, Liu2018} where applied to FGIR tasks \cite{Fu2017, Zheng2017} in order to automatically search the regions of interest. It works well since it can behave as a bounding box which where labour intensive to annotate.
There have also been works that use extra guidance in order to learn a semantic-related regions which, in return creates a more meaningful region for FGIR tasks. Lui et al. \cite{Chen2016, Liu2017} introduced such work which makes use to part-based attribute in order to learn more discriminative features for fine-grained bird recognition. Also He et al. \cite{He2017} used detailed text descriptions in order to mine discriminative parts or characteristics.
The most recent work is that of Chen et al. \cite{Chen2018} who proposed a Hierarchical Semantic Embedding (HSE) framework which is able to predict categories of different levels in such a hierarchy and simultaneously integrate this structured correlation information which most of the other works, introduced above, overlook. Their HSE framework sequentially predicts category score vectors for each level and at each level of the hierarchy use the highest score vector as prior knowledge to learn a finer grained feature representation.
However, there are two main gaps in the above works which motivates our approach. One is that due to the labels of each data point there is a limitation to the depth of the hierarchy, meaning that non of the work shows finer granularity beyond the labels. The second is the resource intensive collection of labels in order to get a deeper hierarchy. Using our method, the embeddings allow us to extract a hierarchical structure which enables us to effectively circumvent the labour intensive process of labelling individual data points.
\section{Method}
We approach the hierarchical annotation of images by embedding the data in an embedded space that captures the semantic information that we are interested in and applying agglomerative clustering of the data in that space.
To achieve such embedding, we use the 2AFC technique to measure the latent perception of differences by the annotators and use deep metric learning techniques to train an embedding model on these measurements. As 2AFC test can be inefficient in the number of queries to the annotator we optimize the test process by incorporating active learning techniques.
\subsection{Two-alternative-forced-choice}
Organizing information in a hierarchical structure is a natural and efficient way for the multitude of downstream tasks that we want to enable on this data \cite{soergel1985organizing}. We expect that when presented with such data, experts or annotators use a latent structure to produce the annotations. Capturing this latent structure directly is difficult because it requires a significant effort to capture and communicate it. On the other hand, a discirminative comparisons come with much lower cost. This characteristics has been known and utilized in psychometrics specifically in measurement of subjective perception of objective stimuli \cite{Fechner1889}. More recently such methods have also been developed for measurements of subjective perception of complex stimuli such as images and video \cite{maloney2003maximum, menkovski2012adaptive, menkovski2011value}. In this work 2AFC methods have been used to efficiently capture the perception of difference in between pairs of stimuli. This allowed for modeling where an individual data point reside on a relative scale of a particular quantity. In a similar manner we use the 2AFC procedure to capture the relative difference between the pair of images for a specific question.
\begin{figure}[]
\centering
\includegraphics[width=0.5\linewidth]{figures/example_of_2AFC.png}
\caption{Triplet Selection Layout using two-alternative-forced choice method}
\label{fig:tripletselectionlayour}
\end{figure}
As given in Figure \ref{fig:tripletselectionlayour}, we select an anchor image and two query images. We ask the annotator to discriminate between the distance given by the anchor and the first query image (option A) and the anchor and the second query image (option B). The distance is with respect to a particular quantity in the image such as: the size of the object, the category of the objects, value of the object.
We then store the answers by marking the image which was chosen as closer to the anchor (positive) and the other as further than from the anchor (negative).
\subsection{Deep metric learning}
Our aim is to embed the high-dimensional input data in to a space that captures the semantic structure that we want to uncover. As the input is high dimensional, we aim to rely on deep neural network models to capture the feature present in the image more effectively as demonstrated in by the advances of these methods in the image analysis domain \cite{Chen2016, Kaiming2016}. We also recognize that the input produced by the 2AFC test and our goals are perfectly aligned with the advances in deep metric learning and particularly with the triplet training procedure \cite{Schroff2015}.
Triplet training procedure consists of three instances of the same feed forward neural network $M_e$ that share the same parameters. For this we used the highly successful ResNet model\cite{Kaiming2016}. Depending on the dataset we used a different depths of the ResNets. For images of size 128x128x3 we used a ResNet-110 and for images with size 28x28x1 we had the ResNet-20. For both experiments, the models output an embedding with a dimensionality of 8.
In order to train the model we used the loss function as given in \cite{Schroff2015}. If we define the distances with respect to the anchor ($x$) as,
\begin{align*}
d(x,x^+) & = \vert \vert M_e(x) - M_e(x^+) \vert \vert_{2}^{2} \\
d(x,x^-) & = \vert \vert M_e(x) - M_e(x^-) \vert \vert_{2}^{2}
\end{align*}
Then, the learning objective here is that,
\begin{align*}
d(x,x^+) & \leq d(x,x^-) - \alpha \\
d(x,x^+) - d(x,x^-) + \alpha & \leq 0
\end{align*}
where $\alpha$ represents the margin which enforces a distance between $d(x,x^+)$ and $d(x,x^-)$. Note that alpha is also needed such that $M_e$ cannot satisfy this equation with zero vectors for the embeddings ($M_e$(any image)). We used an alpha of $0.2$. During training the loss function will be the following:
\begin{equation}
TripletLoss = Max( d(x,x^+) - d(x,x^-) + \alpha, 0 )
\end{equation}
\subsection{Triplet Selection}
Even though answering one of the questions is fairly quick for the annotator, the total number of available questions given a number of images is very large. Furthermore, not all questions are equally valuable for training and improving our embedding model. Such questions have been the focus of the active learning field \cite{Settles2010}. We used an active learning approach using the pool-based uncertainty sampling approach.
Algorithm \ref{alg:bayesian} shows the overall method for the active learning approach. In order to determine $Q$, we create pools of images where each pool contains close neighbours from a random selected image. From this pool of images we generate new potential questions. We can use the Bayes Factor as an uncertainty sampling method to determine if, for a given question $q_i$, whether we have a 50-50 change for choosing an answer ($a_0$ vs $a_1$) or that we have any another ratio/change such that we can be sure either $a_0$ or $a_1$ is more likely to be clicked by the annotator. Hence, we would like to compare two similar models for $a_0 \sim Bin(n,\Theta)$ given that model $M_1$ has a $\Theta = 0.5$ and model $M_2$ has an unknown $\Theta$. For $M_2$ we will take the prior distribution for $\Theta$ to be uniform on $[0,1]$.
Using Bayes Factor we can construct the following likelihood ratio $BF = \frac{P(N'_i|M_1)}{P(N'_i|M_2)}$ where $N'_i$ is the set of all neighbouring questions to $q_i$. If $BF > 1$ then we can strongly assume, given the data $N'_i$, that $M_1$ is supported over $M_2$. Any value of $BF < 1$ we can assume that $M_2$ is supported by the data. In our case, if $BF < 1$ then we can assume that we know either $a_0$ or $a_1$ will be clicked by the annotator and that we do not need to ask this question again.
In order to calculate $BF$ we need to know $P(N'_i|M_1)$ and $P(N'_i|M_2)$.
\begin{align*}
P(N'_i|M_1) &= {n \choose k} \Theta^k (1-\Theta)^{n-k} \\
&= {n \choose k} 0.5^k (1-0.5)^{n-k} \\
&= {n \choose k} 0.5^n \\
P(N'_i|M_2) &= \int_{0}^{1}{n \choose k} \Theta^k (1-\Theta)^{n-k} d\Theta \\
&= {n \choose k} \int_{0}^{1} \Theta^k (1-\Theta)^{n-k} d\Theta \\
&= {n \choose k} B(k+1, n-k+1) \\
&= {n \choose k} \frac{\Gamma(k+1)\Gamma(n-k+1)}{\Gamma(k+n-k+2)} \\
&= \frac{n!}{k!(n-k)!} \frac{k!(n-k)!}{(k+n-k+1)!} \\
&= \frac{n!}{(n+1)!} \\
&= \frac{1}{n+1}
\end{align*}
where $n$ is the total amount of clicks and $k$ is equal the amount of $a_0$ clicks. Note that it does not matter if we count $a_0$ or $a_1$ since the test here is whether the model 'guesses' or not. If either of the two answers is favoured then $M_2$ will be supported by $N'_i$. Knowing $P(N'_i|M_1)$ and $P(N'_i|M_2)$ we can calculate BF as,
\begin{align*}
BF &= \frac{P(N'_i|M_1)}{P(N'_i|M_2)} \\
&= \frac{{n \choose k} 0.5^n}{1/(n+1)} \\
&= {n \choose k} 0.5^n (n+1)
\end{align*}
The main idea is that we want to know if, given a current question and all the previous answers, whether we probability of clicking an answer will be a 50\% change or not. If there is a high probability of choosing any of the two answers we do not need to ask the question. Whereas, if the probability of choosing an answer is 50\% then we need to ask the question to the annotator since we cannot be sure yet.
\begin{algorithm}[tb]
\caption{Triplet selection}
\label{alg:bayesian}
\begin{algorithmic}
\STATE Initialize
\STATE $T = \{q_1, q_2,...,q_{n}\}$ - set of $n$ random unanswered triplets.
\STATE $D = \{\}$ - set of answered triplets
\STATE $\tau = 0.75$
\WHILE{not converged}
\STATE $D \gets$ Have annotators answer $T$
\STATE Update $M_e$ with $D$
\STATE $Q \gets $ select new potential questions
\STATE $T = \{\}$
\FOR{$q$ in $Q$}
\IF{$BF(q) > \tau$}
\STATE $T \gets$ add $q$
\ENDIF
\ENDFOR
\STATE Sort $T$ by highest $BF$
\STATE $T \gets$ top 0.8 triplets of $T$ + 0.2 random triplets for generality
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\subsection{Agglomerative clustering}
After the utility of asking further questions to the annotators has diminished we conclude that we can now successfully embed the data such that its semantic information is captured by the distance metric of the space. To extract this information we run a complete-linkage agglomerative clustering algorithm \cite{rokach2005clustering} and produce a dendrogram that represents the captured structure.
\section{Experiments and results}
To evaluate the proposed method we develop two empirical studies. In the first one we test whether the method can uncover the well studied shape bias in humans \cite{landau1988importance} on a synthetic dataset. In the second we extract hierarchical structure on the FashionMNIST dataset \cite{Xiao2017} containing images of clothing items.
\subsection{Shape bias on simple shapes}
We have created a synthetic simple-shapes dataset which contains 9 different shapes where each shape has 3 different thicknesses and each shape and thickness has 5 different colors. Hence, 135 unique objects that we split into a train and test set (Figure \ref{fig:simpleshapes}). The dimensionality of the images is 128x128x3. In this experiment the annotators give answer to the question "\textit{Which object is more similar to the anchor object?}".
After collecting 840 triplets, we trained the ResNet-110 model with the specified triplet loss and extracted the data structure using the complete-linkage clustering \cite{defays1977efficient} algorithm.
Figure \ref{fig:2afc_ss_dendogram} shows the resulting splits. We can clearly see that the resulting clusters are based on the shape and not color or thickness of the objects in the images. The initial three spits are separating the different shapes: circles, triangles and rectangles. The next level the shape is again is the discriminator for the case of the circles and the rectangles, while in the case of the triangles the results are not as clear. This is somewhat expected as the case of the triangle height of the triangle is not connected to a different concept as in the case of the circle vs. oval. In the case of the rectangles the squares and the vertical rectangles are clustered against the horizontal rectangles.
\begin{figure}[]
\begin{minipage}[b]{.5\linewidth}
\centering
\includegraphics[width=0.5\textwidth]{figures/simpleshape_train.png}
\end{minipage}
\hspace{1cm}
\begin{minipage}[b]{.2\linewidth}
\centering
\includegraphics[width=0.7\textwidth]{figures/simpleshape_test.png}
\end{minipage}
\caption{Simple Shape - train set left and test set right}
\label{fig:simpleshapes}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=1\linewidth]{figures/simpleshape_bf_splits.png}
\caption{Simple Shape dendrogram splits}
\label{fig:2afc_ss_dendogram}
\end{figure}
\subsection{Fashion-MNIST}
Using the 2AFC metric learning method, we are also able to extract a hierarchical structure based on the perception of difference of the annotator. We will be using the Fashion-MNIST dataset \cite{Xiao2017} with the question "\textit{Which object looks more similar to the anchor object?}".
Results of the initial splits can be seen in Figure \ref{fig:fashionmnis_initialSplit}. Note that we can clearly see that the first split is based on cloth (left), bags (middle) and shoes (right) which continues further down in more fine-grained detail. Further splits of shoes can be seen in Figure \ref{fig:fashionmnis_blue}. Here we can clearly see that we end up with clusters that present us with a finer granularity than the original Fashion-MNIST labels. We can observe for example that sandals have been split into high-heal sandals and flat sandals. In order to construct this hierarchical structure we used 1700 triplets.
We further contrast these results with clustering on the raw pixel values to form a baseline and demonstrate the value of developing the embedding space using the 2AFC tests. To compare the two sets of clusters we compute the normalized mutual information. Both results are then compared to the true labels of the Fashion-MNIST dataset. Results can be found in Table \ref{tab:nmi_Fashion-MNIST}. Note that 'Level' is based on a binary tree level and therefore the nodes are the amount of clusters created at each level.
The results demonstrate empirically that the 2AFC method produces an embedding in which clustering captures the semantic structure in the data. We also show that our proposed method allows us create clusters with finer granularity than the dataset labels.
\begin{figure}[]
\centering
\begin{minipage}[b]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/fashionmnist_initSplit_bf.png}
\caption{Fashion-MNIST initial splits}
\label{fig:fashionmnis_initialSplit}
\end{minipage}
\begin{minipage}[b]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/fashionmnist_blue_bf.png}
\caption{Fashion-MNIST granularity - blue}
\label{fig:fashionmnis_blue}
\end{minipage}
\end{figure}
\begin{table}[]
\centering
\caption{Normalized Mutual Information compared to true labels given}
\label{tab:nmi_Fashion-MNIST}
\begin{tabular}{|l|l|l|}
\hline
Level & Baseline & 2AFC \\
\hline
0 & 0.000 & 0.000 \\
1 & 0.192 & 0.392 \\
2 & 0.345 & 0.491 \\
3 & 0.426 & \textbf{0.583} \\
4 & 0.487 & 0.562 \\
5 & \textbf{0.499} & 0.520 \\
\hline
\end{tabular}
\end{table}
\section{Conclusion}
In this work we present a method that leverages the efficiency of discrimination 2AFC testing using to capture the latent perception of difference between data points. We have shown that we are able to capture the shape bias with synthetic data and have shown that it is possible to extract a meaningful hierarchical structure on the Fashion-MNIST dataset, resulting in a finer granularity than the original labels. We have also achieved this efficiently by incorporating an active learning triplet selection based on Bayesian Factor estimation.
There are wide variety of applications that can benefit from extraction of hierarchical structure of data both in imaging domains such as medical imaging, but also broader in other domains that rely on high dimensional datasets.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,993 |
\section{Introduction}
Vacuum energy provides a very simple model of dark energy \cite{Copeland:2006wr,Bertolami:1986bg,Freese:1986dd,Chen:1990jw,Carvalho:1991ut,Berman:1991zz,Pavon:1991uc,AlRawaf:1995rs,Shapiro:2000dz,Buchert:2010ug,Sola:2011qr,Wands:2012vg}. It is the energy density that remains in the absence of any particles, and therefore it remains undiluted by the cosmological expansion. A positive vacuum energy can drive an accelerated expansion once the density of ordinary matter or radiation becomes sub-dominant. Unlike other models of dark energy it does not necessarily introduce any new dynamical degrees of freedom.
We define a vacuum energy, $V$, to have an energy-momentum tensor proportional to the metric
\be
\label{Tvac}
\Tvac^\mu_\nu = - V g^\mu_\nu \,.
\ee
By comparison with the energy-momentum tensor of a perfect fluid
\be
\label{Tmunu}
T^\mu_\nu = P g^\mu_\nu + (\rho+P)u^\mu u_\nu \,,
\ee
we identify the vacuum energy density and pressure with $\rhovac=-\Pvac=V$, but since there is no particle flow then the four-velocity of the vacuum, $\uvac^\mu$, is undefined.
A vacuum energy that is homogeneous throughout spacetime, $\nabla_\mu V=0$, is equivalent to a cosmological constant in Einstein gravity, $\Lambda=8\pi G_N V$. The discrepancy between the value of the energy density required by current observations and the typical energy scales predicted by particle physics is the long-standing cosmological constant problem \cite{Weinberg:1988cp}.
We will consider the possibility of a time and/or space dependent vacuum energy. {}From Eq.~(\ref{Tvac}) we have
\be
\label{vac-conservation}
\nabla_\mu \Tvac^\mu_\nu = Q_\nu \,.
\ee
where the energy flow is given by
\be
Q_\nu \equiv - \nabla_\nu V \,.
\ee
We can therefore identify an {\em inhomogeneous vacuum}, $\nabla_\mu V\neq0$, with an {\em interacting vacuum}, $Q_\nu\neq 0$ \cite{Wands:2012vg}. The conservation of the total energy-momentum (including matter fields and the vacuum energy) in general relativity
\be
\nabla_\mu \left( T^\mu_\nu + \Tvac^\mu_\nu \right) = 0 \,,
\ee
implies that the vacuum transfers energy-momentum to or from the matter fields
\be
\nabla_\mu T^\mu_\nu = -Q_\nu
\,.
\ee
Note that the energy density and four-velocity of a fluid can be identified with the eigenvalue and eigenvector of the energy-momentum tensor (\ref{Tmunu}):
\be
T^\mu_\nu u^\nu= -\rho u^\mu \,.
\ee
Because the vacuum energy-momentum tensor (\ref{Tvac}) is proportional to the metric tensor, any four-velocity, $u^\mu$, is an eigenvector
\be
\Tvac^\mu_\nu u^\nu= - V u^\mu \quad \forall \ u^\mu \,,
\ee
and all observers see the same vacuum energy density, $V$, i.e., the vacuum energy is boost invariant.
Although the vacuum does not have a unique four-velocity, we can use the energy flow, $Q_\nu$, to define a preferred unit four-vector in an inhomogeneous vacuum \cite{Wands:2012vg}
\be
\label{def-uvac}
\uvac^\mu = \frac{-\nabla^\mu V}{|\nabla_\nu V\nabla^\nu V|^{1/2}} \,.
\ee
normalised such that $\uvac_\mu\uvac^\mu=\pm1$ for a spacelike or timelike flow. Note however that the $\uvac^\mu$ defines a potential flow, i.e, with vanishing vorticity.
In this paper we will consider vacuum energy which may be inhomogeneous in spacetime, interacting with fields or fluids without necessarily invoking additional degrees of freedom. We show that any spatially homogeneous dark energy cosmology can be decomposed into an interacting vacuum+matter cosmology. By lifting this spatially-homogeneous solution to a covariant interaction one can study inhomogeneous perturbations which obey coupled first-order equations of motions for the matter density and velocity.
As an example we consider perturbations of a generalised Chaplygin gas cosmology, decomposed into an interacting vacuum+matter.
\section{Vacuum cosmology}
\subsection{FRW background}
The symmetries of a spatially homogeneous and isotropic Friedmann-Robertson-Walker (FRW) metric, with scale factor $a(t)$ and Hubble rate $H=\dot{a}/a$, require the vacuum to be spatially homogeneous and isotropic too, hence $V=V(t)$. In this case the vacuum and matter are both homogeneous on spatial hypersurfaces orthogonal to the matter four-velocity, $u^\mu=(1,0,0,0)$. Note that the energy flow, $\uvac^\mu$, and matter velocity, $u^\mu$, necessarily coincide in FRW cosmology due to the assumption of isotropy.
The Friedmann constraint equation requires
\be
H^2 = \frac{8\pi G_N}{3} \left( \rho + V \right) - \frac{K}{a^2} \,,
\ee
where $K$ determines the spatial curvature.
The continuity equations for matter fields and vacuum are
\bea
\dot\rho + 3H(\rho+P) &=& -Q
\,,\\
\dot{V} &=& Q
\,.
\eea
The vacuum energy, $V$, is undiluted by the cosmological expansion, but can have a time-dependent density in the presence of a non-zero energy transfer, $Q\neq 0$, where
\be
Q \equiv -u^\nu Q_\nu \,.
\ee
There have been many attempts to describe the present acceleration as due to a time-dependent vacuum energy
\cite{Bertolami:1986bg,Freese:1986dd,Chen:1990jw,Carvalho:1991ut,Berman:1991zz,Pavon:1991uc,AlRawaf:1995rs,Shapiro:2000dz,Buchert:2010ug,Sola:2011qr}.
However, there is little to be learnt from simply assuming an arbitrary time-dependent vacuum to obtain the desired cosmological solution. Ideally one should have a physical model from which one can derive a time-dependent solution and study other physical effects.
For example, vacuum fluctuations of free fields can support an averaged density proportional to the fourth-power of the Hubble expansion, $V\propto H^4$ \cite{Bunch:1978yq}. Such a vacuum energy would not in itself support an accelerated expansion, but other forms such as $V\propto H$ have been proposed \cite{Schutzhold:2002pr} better able to match the observational data \cite{Fabris:2006gt,VelasquezToribio:2009qp,Zimdahl:2011ae,Xu:2011qv,Alcaniz:2012mh}.
It is not obvious how such time-dependent vacuum models can be compared with observations in an inhomogeneous universe. We will argue that it is possible to give a consistent description of vacuum dynamics, and in particular the relativistic equations of motion for inhomogeneous perturbations, given a covariant, physical prescription for the local vacuum energy or, equivalently, the vacuum energy transfer 4-vector, $Q_\mu=-\nabla_\mu V$. One should then be able to subject vacuum models to observational constraints, even in the absence of a Lagrangian derivation or microphysical description.
\subsection{Linear perturbations}
Let us consider inhomogeneous linear, scalar perturbations where the energy and pressure of matter is given by $\rho(t)+\delta\rho(t,x^i)$ and $P(t)+\delta P(t,x^i)$, and the four-velocity of matter is given by
\be
\label{u}
u^\mu = \left[ 1-\phi \,, a^{-1}\partial^i v \right] \,, \quad u_\mu = \left[ -1-\phi \,, \partial_i \theta \right] \,.
\ee
where we define $\partial^iv=a(\partial x^i/\partial t)$ and $\theta=a(v+B)$.
Once we allow for deviations from homogeneity in the matter and metric, we should also allow for inhomogeneity in an interacting vacuum, $V(t)+\delta V(t,x^i)$. As remarked earlier, the vacuum has an energy density and pressure, but no unique velocity. In particular the momentum of the vacuum vanishes in any frame, $(\rhovac+\Pvac)\theta=0$.
On the other hand the energy flow $\uvac$ defined in Eq.~(\ref{def-uvac}), can be written in analogy with the fluid velocity (\ref{u}) as
\be
\label{def-thetavac}
\uvac^\mu = \left[ 1-\phi \,, a^{-1}\partial^i \vvac \right] \,, \quad \uvac_\mu = \left[ -1-\phi \,, \partial_i \thetavac \right] \,.
\ee
where from Eq.~(\ref{def-uvac}) we identify $\thetavac=-\delta V/\dot{V}$.
Perturbations about a spatially flat ($K=0$) FRW metric \cite{Kodama:1985bj,Mukhanov:1990me,Malik:2004tf,Malik:2008im} are described by the line element
\be
ds^2 = -(1+2\phi)dt^2+2a\partial _i B dt dx^i
+ a^2 \left[(1-2\psi)\delta_{ij}+2\partial_i\partial_j E \right] dx^i dx^j \,.
\ee
Following \cite{Kodama:1985bj,Malik:2004tf,Malik:2008im}, we decompose the energy-flow along and orthogonal to the fluid velocity,
\be
\label{def-f}
Q_\mu = Q u_\mu + f_\mu
\,,
\ee
where $f_\mu u^\mu=0$, so that we have
\be
Q_\mu = \left[ -Q(1+\phi)-\delta Q \,, \partial_i (f + Q\theta) \right] \,.
\ee
The energy continuity equations for matter and vacuum become
\bea
\dot{\delta\rho} + 3H(\delta\rho+\delta P)
-3 (\rho+P) \dot\psi + (\rho+P)\frac{\nabla^2}{a^2} \left( \theta + a^2\dot{E} - aB \right)
&=& -\delta Q - Q\phi
\,,\\
\dot{\delta V} &=& \delta Q + Q\phi \,.
\eea
while the momentum conservation becomes
\bea
(\rho+P)\dot\theta - 3c_s^2H(\rho+P)\theta + (\rho+P)\phi
+\delta P \nonumber &=& - f + c_s^2 Q \theta
\,,\\
-\delta V &=& f + Q\theta \,.
\eea
where the adiabatic sound speed $c_s^2\equiv \dot{P}/\dot\rho$.
Note that the vacuum momentum conservation equation becomes a constraint equation which requires that the vacuum pressure gradient is balanced by the force
\be
\nabla_i (-V) = \nabla_i (f+Q\theta) \,.
\ee
This determines the equal and opposite force exerted by the vacuum on the matter:
\be
- f = \delta V + \dot{V}\theta \,.
\ee
i.e., the fluid element feels the gradient of the vacuum potential energy.
Note that the perturbations of a fluid coupled to the vacuum with $\Pvac=-\rhovac$ has no additional degrees of freedom, in contrast to a dark energy fluid with $P_X\neq -\rho_X$. Using the vacuum energy and momentum conservation equations to eliminate $\delta Q$ and $f$ we obtain
\bea
\label{finaldeltarho}
\dot{\delta\rho} + 3H(\delta\rho+\delta P)
-3 (\rho+P)\dot\psi + (\rho+P) \frac{\nabla^2}{a^2} \left( \theta + a^2\dot{E} - aB \right)
= - \dot{\delta V}
\,,\\
\label{finaltheta}
(\rho+P)\dot\theta - 3c_s^2H(\rho+P)\theta + (\rho+P)\phi
+\delta P = \delta V + (1+c_s^2) \dot{V} \theta
\,.
\eea
\subsection{Gauge invariant perturbations}
It is well known that metric and matter perturbations can be gauge-dependent under a first-order gauge transformation, such as $t\to t+\delta t(t,x^i)$.
The fluid density and pressure transform as $\delta\rho\to\delta\rho-\dot\rho\delta t$ and $\delta P\to \delta P-\dot{P}\delta t$ \cite{Malik:2008im}.
Similarly the vacuum perturbation transforms as $\delta V\to \delta V -Q\delta t$ and $\delta Q\to \delta Q-\dot{Q}\delta t$. The fluid 3-momentum transforms as $\theta\to \theta+\delta t$ and the energy flow transforms similarly as $\thetavac\to\thetavac+\delta t$.
We can construct gauge-invariant perturbations by specifying quantities on a particular physical reference frame \cite{Malik:2008im}.
For example, the vacuum perturbation on hypersurfaces orthogonal to the energy transfer, $Q_\mu$, can be shown to vanish identically:
\be
\Delta V_{\rm com} = \delta V + \dot{V} \thetavac = 0 \,,
\ee
since from Eq.~(\ref{def-uvac}) and~(\ref{def-thetavac}) we have $\thetavac=-\delta V/\dot{V}$. This simply reflects that the energy flow is the gradient of the vacuum energy and therefore the orthogonal hypersurfaces are uniform vacuum energy hypersurfaces by construction.
On the other hand the vacuum perturbation on hypersurfaces orthogonal to the matter 4-velocity, $u^\mu$, (the {\em comoving vacuum perturbation}) is given by
\be
\label{defdeltarhovac}
\delta V_{\rm com} = \delta V + \dot{V} \theta = -f \,.
\ee
This is in general non-zero, i.e., the vacuum may be spatially inhomogeneous in the comoving-orthogonal gauge.
For example, the Poisson equation for the Newtonian metric potential is given by
\be
\label{Poisson}
\nabla^2 \Phi = 4\pi G \left( \delta\rho_{\rm com} + \delta V_{\rm com} \right) \,,
\ee
i.e., the Newtonian metric potential is sourced by both the matter and vacuum perturbations, where
\be
\delta\rho_{\rm com} = \delta\rho + \dot\rho\theta \,.
\ee
\begin{figure}
\centering
\includegraphics[width=1.2\textwidth]{figures.eps}
\vspace*{-2cm}
\caption{(a) In an FRW cosmology the homogeneous spatial hypersurfaces are orthogonal to both the fluid 4-velocity, $u^\mu$, and the vacuum energy flow, $Q^\mu$. (b) In an inhomogeneous cosmology the spatial hypersurfaces orthogonal to the fluid 4-velocity (light orange) and the vacuum energy flow (dark red) do not necessarily coincide.
}
\label{fig}
\end{figure}
Note that we can write the comoving vacuum density perturbation (\ref{defdeltarhovac}) as
\be
\label{defdeltarhovac2}
\delta V_{\rm com} = \dot{V} \left( \theta - \thetavac \right) \,.
\ee
Therefore, if the energy flow follows the fluid four-velocity, $\uvac^\mu=u^\mu$, then we have $\thetavac=\theta$ and the vacuum is spatially homogeneous on comoving-orthogonal hypersurfaces, $\delta V_{\rm com}=0$.
Another gauge invariant expression for the vacuum density perturbation is the dimensionless vacuum perturbation on uniform-fluid density hypersurfaces, which describes a relative density perturbation
\be
\label{defSvac}
\Svac
= -3H \left( \frac{\delta V}{\dot{V}} - \frac{\delta\rho}{\dot\rho} \right) \,.
\ee
If, for example, the vacuum energy is a function of the local matter density, $V=V(\rho)$, then the relative density perturbation must vanish and the vacuum is spatially
homogeneous
on uniform-density hypersurfaces, $\Svac=0$.
The total non-adiabatic pressure perturbation due to any intrinsic non-adiabatic pressure of the matter and the relative entropy perturbation between the vacuum and matter is then
\bea
\label{defPnad}
\delta P_{\rm nad}
&=& \delta P - \delta V - \left( \frac{\dot{P}-\dot{V}}{\dot{\rho}+\dot{V}} \right) \left( \delta\rho + \delta V \right) \,, \nonumber\\
&=& \delta P - c_s^2 \delta\rho + \frac{(1+c_s^2)Q[Q+3H(\rho+P)]}{9H^2(\rho+P)} \Svac \,,
\eea
where the adiabatic sound speed for matter is $c_s^2=\dot{P}/\dot\rho$.
This vanishes for adiabatic matter perturbations, $\delta P=c_s^2\delta\rho$, and adiabatic vacuum fluctuations, $\Svac=0$, or a non-interacting vacuum, $Q=0$.
\section{Decomposed generalised Chaplygin gas}
As an example of how an inhomogeneous vacuum energy might be used to describe the present accelerated expansion of our Universe we will show how one widely-studied dark energy model, the Chaplygin gas, can be re-interpreted as an interacting vacuum+matter cosmology, and how this re-interpretation can motivate different possible behaviour for density perturbations.
Any
dark energy fluid energy-momentum tensor (\ref{Tmunu}) with density $\rho_\de$ can be described by pressureless matter, with density $\rho_m$ and velocity $u^\mu_m=u^\mu$, interacting with the vacuum, $V$, such that $\rho_\de=\rho_m+V$ \cite{Wands:2012vg}. The corresponding matter and vacuum densities are given by
\be
\label{decompose}
\rho_m = \rho_\de+P_\de \,, \quad V = -P_\de \,.
\ee
while the energy flow is $Q_\mu=\nabla_\mu P_\de$.
In an FRW cosmology this corresponds to $Q=-\dot{P}_\de$.
One might choose to decompose a dark energy model $\rho_\de(a)$ into any two interacting barotropic fluids such that $\rho_\de=\rho_1+\rho_2$, but this would double the degrees of freedom in the model unless one of these two ``fluids'' is the vacuum.
The generalised Chaplygin gas, defined by the barotropic equation of state \cite{Kamenshchik:2001cp,Bento:2002ps}
\be
\label{Pgcg}
P_\gcg = -A \rho_\gcg^{-\alpha} \,.
\ee
This leads to a solution for the density in an FRW cosmology
\be
\rho_\gcg = \left( A + Ba^{-3(1+\alpha)} \right)^{1/(1+\alpha)} \,.
\ee
This has the simple limiting behaviour $\rho_\gcg\propto a^{-3}$ as $a\to0$ and $\rho_\gcg\to A^{1/(1+\alpha)}$ as $a\to+\infty$, therefore this has been proposed as a unified dark matter model. However such models are strongly constrained by observations since the barotropic equation of state defines a sound speed for matter perturbations which only reproduces the successful $\Lambda$CDM model when $\alpha\to0$ \cite{Sandvik:2002jz,Park:2009np}.
The decomposition (\ref{decompose}) into pressureless matter interacting with the vacuum has previously been considered for the generalised Chaplygin gas by Bento et al \cite{Bento:2004uh}. In this case
we have the FRW solution
\be
V =
A \left( A + Ba^{-3(1+\alpha)} \right)^{-\alpha/(1+\alpha)}
\,,
\ee
and hence
\be
\label{defA}
A = (\rho_m + V)^\alpha V \,.
\ee
The form of the FRW solution suggests a simple interaction
\be
\label{Qgcg}
Q
= 3 \alpha H \left( \frac{\rho_m V}{\rho_m + V} \right) \,.
\ee
In the matter or vacuum dominated limits this reduces to an interaction of the form $Q\propto H\rho_m$ or $Q\propto HV$ studied, for example, by Barrow and Clifton~\cite{Barrow:2006hia}.
It is intriguing to note that the FRW solution can be defined in terms of an interaction (\ref{Qgcg}) with a single dimensionless parameter, $\alpha$, whereas when defined in terms of an equation of state (\ref{Pgcg}) its definition requires both $\alpha$ and the dimensional constant $A$ which determines the late-time cosmological constant. In the interaction model, $A$ (and therefore the late-time cosmological constant) emerges as an integration constant dependent on initial conditions.
However, to study inhomogeneous perturbations in the decomposed model we must ``lift'' the explicitly time-dependent FRW solution to a covariant model for the interaction. We have at least two choices.
In either case one can calculate the speed of sound in the combined interacting vacuum+matter using the usual definition \cite{ArmendarizPicon:1999rj,Weller:2003hw,Bean:2003fb,Hannestad:2005ak} and the decomposed density and pressure (\ref{decompose})
\be
\label{cint}
c_{\rm de}^2 \equiv \left( \frac{\delta P_{\rm de}}{\delta\rho_{\rm de}} \right)_{\rm com}
= \left( \frac{- \delta{V}}{\delta\rho_m+\delta{V}} \right)_{\rm com}\,.
\ee
\subsection{Barotropic model}
Firstly one could require that the local vacuum energy is a function of the local matter density, $V=V(\rho_m)$. If this applies to inhomogeneous perturbations of the interacting vacuum+matter as well as the background then we require
\be
\label{adiabatic}
\delta V = \frac{\dot{V}}{\dot{\rho}_m}\delta\rho_m \,.
\ee
This implies that the vacuum perturbations are adiabatic and $\Svac=0$ in Eq.~(\ref{defSvac}). On the other hand the comoving vacuum energy, (\ref{defdeltarhovac}), is non-zero
\be
\delta V_{\rm com}
= \frac{\dot{V}}{\dot{\rho}_m} \delta\rho_{m,{\rm com}} \,.
\ee
Thus one needs to consistently include the vacuum inhomogeneities as well as matter inhomogeneities, for example in the Poission equation (\ref{Poisson}), when $\dot{V}\neq0$.
Using the adiabatic condition (\ref{adiabatic}) we obtain the dark energy sound speed (\ref{cint})
\be
c_{\rm int}^2 = \frac{\dot{P}_{\rm gCg}}{\dot{\rho}_{\rm gCg}} \,.
\ee
We see therefore the the sound speed for the interacting vacuum+fluid is the same as the adiabatic sound speed of the original barotropic Chaplygin gas (\ref{Pgcg}). Perturbations of the the interacting vacuum+matter respect the same barotropic equation of state as the original fluid (\ref{Pgcg}) and therefore the observational predictions for the cosmic microwave background (CMB) anisotropies, for example, are exactly the same as the original fluid model \cite{inprep}.
CMB temperature anisotropies as well as the power spectrum for the interacting vacuum+matter are shown in figures~2 and~3. We see that only very small values of the dimensionless parameter $|\alpha|<10^{-4}$ are allowed without introducing unacceptably large oscillations in the matter power spectrum \cite{Sandvik:2002jz,Park:2009np}, forcing the model to be extremely close to the standard $\Lambda$CDM cosmology.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{gcgcl.eps}
\caption{Cosmic microwave background angular power spectrum for the generalised Chaplygin gas with barotropic equation of state and an adiabatic sound speed~\cite{inprep}.
}
\label{fig-gcgcl}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{gcgpk.eps}
\caption{Power spectrum for baryons plus generalised Chaplygin gas with barotropic equation of state and an adiabatic sound speed~\cite{inprep}.
}
\label{fig-gcgpk}
\end{figure}
\subsection{Geodesic flow}
The interacting vacuum+matter model allows a different behaviour for inhomogeneous perturbations than the barotropic fluid. Suppose that the energy flow, $Q_\mu$, is along the matter 4-velocity, $u_\mu$. The force exert exerted by the vacuum on the matter, $-f_\mu$ in Eq.~(\ref{def-f}), is zero and matter 4-velocity is a geodesic flow.
In this case the comoving vacuum perturbation, $\delta V_{\rm com}$ in Eq.~(\ref{defdeltarhovac}), is zero, i.e., the vacuum is spatially homogeneous on hypersurfaces orthogonal to the matter 4-velocity. This means that the comoving pressure perturbation for the interacting vacuum+matter is zero and hence the sound speed ({\ref{cint}) is zero, $c_{\rm int}^2=0$~\cite{Creminelli:2008wc,Lim:2010yk}.
The evolution of density perturbations is therefore much closer to that in a standard $\Lambda$CDM cosmology with zero sound speed than the original Chaplygin gas model with $\alpha\neq0$. CMB temperature anisotropies as well as the matter power spectrum are shown in figures~4 and~5. Much larger values of $\alpha\sim0.1$ may be compatible with the data in this case \cite{inprep}.
Note however that this geodesic model for the interaction does allow a gauge-invariant vacuum entropy perturbation~(\ref{defSvac})
\be
\Svac = \frac{3H}{\dot{\rho}_m} \delta\rho_{m,{\rm com}} \,.
\ee
However due to the Poisson constraint equation (\ref{Poisson}) the comoving density perturbation vanishes on large scales and hence this is compatible with adiabatic initial conditions for the interacting vacuum+matter at early times.
The requirement that $Q_\mu=Qu_\mu$ does impose an important physical restriction on the matter 4-velocity since the energy flow must then be irrotational, i.e., $u_\mu\propto \nabla_\mu V$. This would have important consquences for non-linear structure formation, possibly disastrous if gravitationally collapsed dark matter halos could not be supported by rotation. Hence we only expect geodesic matter interacting with the vacuum to be a viable model for the initial stages of structure formation and we would need a microphysical model for gravitationally collapsed halos.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{geocl.eps}
\caption{Cosmic microwave background angular power spectrum for the decomposed generalised Chaplygin gas with geodesic matter and zero sound speed~\cite{inprep}.
}
\label{fig-geocl}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{geopk.eps}
\caption{Power spectrum for baryons plus the decomposed generalised Chaplygin gas with geodesic matter and zero sound speed~\cite{inprep}.
}
\label{fig-geopk}
\end{figure}
\section{Discussion}
\label{sec:conclusions}
Many different explanations have been proposed for the present-day
acceleration of the Universe \cite{Copeland:2006wr}. Perhaps the most common is a
self-interacting scalar field, $\varphi$, whose self-interaction
potential energy could dominate the present energy density.
Different models are defined by the chosen functional form for the
potential energy, $V(\varphi)$ \cite{Ratra:1987rm,Caldwell:1997ii},
or the kinetic energy as a function of the field gradient,
$X=(\nabla\varphi)^2$
\cite{ArmendarizPicon:1999rj,ArmendarizPicon:2000ah}. Alternative
explanations include fluid models, specified by a barotropic
equation of state, $P(\rho)$
\cite{Kamenshchik:2001cp,Linder:2008ya,Balbi:2007mz}, some of which
could provide unified dark matter models capable of explaining
acceleration on large scales and galactic dynamics (as dark matter)
on much smaller scales. Acceleration requires an exotic equation of
state with negative pressure, and yet the sound speed must remain
real in order to avoid instabilities with respect to inhomogeneous
perturbations. More sophisticated models have been proposed which
allow for interactions between the different fields or fluid
components, e.g., coupled quintessence
models~\cite{Wetterich:1994bg,Amendola:1999er,Holden:1999hm,Billyard:2000bh,Farrar:2003uw,Boehmer:2008av}.
Such models are motivated by astronomical observations
\cite{Copeland:2006wr}, providing increasingly detailed constraints
on the model parameters, but the models themselves lack a persuasive
underpinning physical motivation.
One might therefore simply consider the simplest possible model, or parameterisation, compatible with the data.
In this paper we have considered vacuum energy as a source of spacetime curvature in Einstein gravity. A homogeneous vacuum in Einstein gravity is equivalent to a cosmological constant, whereas an inhomogeneous (time- or space-dependent) vacuum implies an interacting vacuum.
Simply specifying a particular time-dependence for the vacuum energy in an FRW cosmology is unsatisfactory if the vacuum energy is introduced solely to produce a particular time-dependence of the cosmological expansion. Different physical models for the origin of the vacuum energy, or its interaction with other matter fields, will lead to different cosmological behaviour. Even models which yield the same FRW background solutions \cite{Kunz:2007rk,Aviles:2011ak} may be distinguished for instance by the predictions for CMB anisotropies or the matter or galaxy power spectrum.
As an example, we have shown how the generalised Chaplygin gas cosmology can be re-derived as a solution to an interacting matter+vacuum model. The interaction can be defined by a single dimensionless constant and the late-time constant vacuum energy (which appears as a dimensional parameter in the original Chaplygin gas model) can instead be derived as a constant of integration in the matter+vacuum model. We have shown the CMB and matter power spectrum predictions for two models for interacting vacuum+matter which yield the same background solution as the generalised Chaplygin gas, but give very different observational predictions \cite{inprep}.
Another familiar example, which we have not discussed here, would be a quintessence model with a self-interacting scalar field, $\varphi$. The self-interaction potential of the field, $V(\varphi)$, provides a vacuum energy density interacting with the kinetic energy of the scalar field. It is well known that a canonical scalar field has a sound speed equal to unity and one might thereby hope to distinguish it from a barotropic fluid model where the sound speed is necessarily equal to the adiabatic sound speed. By considering a broader class of interacting vacuum models one can study models with a range of different possible sound speeds. Our equations enable us to consider vacuum energy interacting with other forms of matter, including pressureless matter or radiation. We can therefore consider vacuum energy models which do not introduce any degrees of freedom beyond those already present in the cosmology, e.g., unified dark energy models.
In our work we have considered only linear perturbations but non-linear evolution may provide further observational constraints on different models. Often it is assumed that the vacuum energy remains unperturbed, or negligible, during collapse, but some cases, such as the interacting vacuum+matter with geodesic 4-velocity and zero sound speed may be amenable to a study of non-linear collapse \cite{Creminelli:2009mu} and we hope to study this further in future.
\bigskip
{DW is grateful to the organisers of the IVth International Conference on Gravitation and Cosmology, Guadalajara, for their hospitality.
DW is supported by STFC grant ST/H002774/1. JD-S is supported by CONACYT grant 210405. JD-S and YW thank the ICG, University of Portsmouth for their hospitality.}
\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,333 |
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